LMFDB - Embedded newform 45.5.g.e.37.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [45,5,Mod(28,45)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(45, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 3]))

N = Newforms(chi, 5, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("45.28");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 45 = 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	45.g (of order $4$, degree $2$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$4.65164833877$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(i)$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 60x^{5} + 1973x^{4} - 3300x^{3} + 1800x^{2} + 31560x + 276676 $ x^8 - 60x^5 + 1973x^4 - 3300x^3 + 1800x^2 + 31560*x + 276676
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 15)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			37.4
Root			$-5.02811 + 5.02811i$ of defining polynomial
Character	$\chi$	$=$	45.37
Dual form			45.5.g.e.28.4

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(5.02811 + 5.02811i) q^{2} +34.5637i q^{4} +(23.8949 + 7.35070i) q^{5} +(-38.5593 - 38.5593i) q^{7} +(-93.3405 + 93.3405i) q^{8} +O(q^{10})$	\(q+(5.02811 + 5.02811i) q^{2} +34.5637i q^{4} +(23.8949 + 7.35070i) q^{5} +(-38.5593 - 38.5593i) q^{7} +(-93.3405 + 93.3405i) q^{8} +(83.1861 + 157.106i) q^{10} +40.4333 q^{11} +(20.8688 - 20.8688i) q^{13} -387.760i q^{14} -385.633 q^{16} +(-15.8713 - 15.8713i) q^{17} -314.926i q^{19} +(-254.068 + 825.898i) q^{20} +(203.303 + 203.303i) q^{22} +(572.869 - 572.869i) q^{23} +(516.934 + 351.289i) q^{25} +209.862 q^{26} +(1332.75 - 1332.75i) q^{28} +824.433i q^{29} -1347.19 q^{31} +(-445.555 - 445.555i) q^{32} -159.606i q^{34} +(-637.933 - 1204.81i) q^{35} +(-589.843 - 589.843i) q^{37} +(1583.48 - 1583.48i) q^{38} +(-2916.48 + 1544.25i) q^{40} -1856.55 q^{41} +(-671.078 + 671.078i) q^{43} +1397.53i q^{44} +5760.90 q^{46} +(-504.865 - 504.865i) q^{47} +572.636i q^{49} +(832.883 + 4365.52i) q^{50} +(721.305 + 721.305i) q^{52} +(-2251.32 + 2251.32i) q^{53} +(966.151 + 297.213i) q^{55} +7198.29 q^{56} +(-4145.34 + 4145.34i) q^{58} +2585.03i q^{59} -3276.74 q^{61} +(-6773.81 - 6773.81i) q^{62} +1689.53i q^{64} +(652.060 - 345.259i) q^{65} +(-3428.94 - 3428.94i) q^{67} +(548.573 - 548.573i) q^{68} +(2850.31 - 9265.50i) q^{70} +5679.51 q^{71} +(4450.06 - 4450.06i) q^{73} -5931.59i q^{74} +10885.0 q^{76} +(-1559.08 - 1559.08i) q^{77} +6465.77i q^{79} +(-9214.66 - 2834.67i) q^{80} +(-9334.92 - 9334.92i) q^{82} +(621.380 - 621.380i) q^{83} +(-262.579 - 495.910i) q^{85} -6748.50 q^{86} +(-3774.07 + 3774.07i) q^{88} +1856.76i q^{89} -1609.37 q^{91} +(19800.5 + 19800.5i) q^{92} -5077.03i q^{94} +(2314.93 - 7525.13i) q^{95} +(12383.5 + 12383.5i) q^{97} +(-2879.27 + 2879.27i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 84 q^{5} + 20 q^{7} - 180 q^{8}+O(q^{10}) $ 8 * q + 84 * q^5 + 20 * q^7 - 180 * q^8	$ 8 q + 84 q^{5} + 20 q^{7} - 180 q^{8} + 104 q^{10} + 288 q^{11} - 340 q^{13} + 620 q^{16} - 900 q^{17} - 564 q^{20} - 1100 q^{22} + 1560 q^{23} - 1204 q^{25} + 3024 q^{26} + 3580 q^{28} - 512 q^{31} - 4980 q^{32} - 6600 q^{35} - 3820 q^{37} + 7680 q^{38} - 2952 q^{40} + 2712 q^{41} - 1240 q^{43} + 13528 q^{46} - 4800 q^{47} - 3744 q^{50} - 1240 q^{52} - 1020 q^{53} - 3644 q^{55} + 30720 q^{56} + 2340 q^{58} - 4760 q^{61} - 28680 q^{62} + 1212 q^{65} - 8920 q^{67} + 1920 q^{68} + 7380 q^{70} - 7536 q^{71} + 11600 q^{73} + 4344 q^{76} + 360 q^{77} - 10644 q^{80} - 27200 q^{82} + 32400 q^{83} - 15628 q^{85} - 14592 q^{86} - 14340 q^{88} + 16528 q^{91} + 31800 q^{92} - 18864 q^{95} + 58640 q^{97} - 46440 q^{98}+O(q^{100}) $ 8 * q + 84 * q^5 + 20 * q^7 - 180 * q^8 + 104 * q^10 + 288 * q^11 - 340 * q^13 + 620 * q^16 - 900 * q^17 - 564 * q^20 - 1100 * q^22 + 1560 * q^23 - 1204 * q^25 + 3024 * q^26 + 3580 * q^28 - 512 * q^31 - 4980 * q^32 - 6600 * q^35 - 3820 * q^37 + 7680 * q^38 - 2952 * q^40 + 2712 * q^41 - 1240 * q^43 + 13528 * q^46 - 4800 * q^47 - 3744 * q^50 - 1240 * q^52 - 1020 * q^53 - 3644 * q^55 + 30720 * q^56 + 2340 * q^58 - 4760 * q^61 - 28680 * q^62 + 1212 * q^65 - 8920 * q^67 + 1920 * q^68 + 7380 * q^70 - 7536 * q^71 + 11600 * q^73 + 4344 * q^76 + 360 * q^77 - 10644 * q^80 - 27200 * q^82 + 32400 * q^83 - 15628 * q^85 - 14592 * q^86 - 14340 * q^88 + 16528 * q^91 + 31800 * q^92 - 18864 * q^95 + 58640 * q^97 - 46440 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/45\mathbb{Z}\right)^\times$.

$n$	$11$	$37$
$\chi(n)$	$1$	$e\left(\frac{1}{4}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	5.02811	+	5.02811i	1.25703	+	1.25703i	0.952505	+	0.304522i	$0.0984966\pi$
$2$	5.02811	+	5.02811i	1.25703	+	1.25703i	0.304522	+	0.952505i	$0.401503\pi$
$3$		0			0
$3$		0			0
$4$			34.5637i			2.16023i
$5$	23.8949	+	7.35070i	0.955797	+	0.294028i
$5$	23.8949	+	7.35070i	0.955797	+	0.294028i
$6$		0			0
$7$	−38.5593	−	38.5593i	−0.786924	−	0.786924i	0.194065	−	0.980989i	$-0.437833\pi$
$7$	−38.5593	−	38.5593i	−0.786924	−	0.786924i	−0.980989	+	0.194065i	$0.937833\pi$
$8$	−93.3405	+	93.3405i	−1.45845	+	1.45845i
$9$		0			0
$10$	83.1861	+	157.106i	0.831861	+	1.57106i
$11$	40.4333			0.334160			0.167080	−	0.985943i	$-0.446566\pi$
$11$	40.4333			0.334160			0.167080	+	0.985943i	$0.446566\pi$
$12$		0			0
$13$	20.8688	−	20.8688i	0.123484	−	0.123484i	−0.642664	−	0.766148i	$-0.722171\pi$
$13$	20.8688	−	20.8688i	0.123484	−	0.123484i	0.766148	+	0.642664i	$0.222171\pi$
$14$		−	387.760i		−	1.97837i
$15$		0			0
$16$	−385.633			−1.50638
$17$	−15.8713	−	15.8713i	−0.0549181	−	0.0549181i	0.679114	−	0.734032i	$-0.262364\pi$
$17$	−15.8713	−	15.8713i	−0.0549181	−	0.0549181i	−0.734032	+	0.679114i	$0.762364\pi$
$18$		0			0
$19$		−	314.926i		−	0.872371i	−0.899857	−	0.436185i	$-0.856329\pi$
$19$		−	314.926i		−	0.872371i	0.899857	−	0.436185i	$-0.143671\pi$
$20$	−254.068	+	825.898i	−0.635170	+	2.06474i
$21$		0			0
$22$	203.303	+	203.303i	0.420048	+	0.420048i
$23$	572.869	−	572.869i	1.08293	−	1.08293i	0.0866935	−	0.996235i	$-0.472370\pi$
$23$	572.869	−	572.869i	1.08293	−	1.08293i	0.996235	−	0.0866935i	$-0.0276301\pi$
$24$		0			0
$25$	516.934	+	351.289i	0.827095	+	0.562062i
$26$	209.862			0.310446
$27$		0			0
$28$	1332.75	−	1332.75i	1.69994	−	1.69994i
$29$			824.433i			0.980301i	0.871638	+	0.490151i	$0.163058\pi$
$29$			824.433i			0.980301i	−0.871638	+	0.490151i	$0.836942\pi$
$30$		0			0
$31$	−1347.19			−1.40186			−0.700931	−	0.713229i	$-0.747232\pi$
$31$	−1347.19			−1.40186			−0.700931	+	0.713229i	$0.747232\pi$
$32$	−445.555	−	445.555i	−0.435112	−	0.435112i
$33$		0			0
$34$		−	159.606i		−	0.138067i
$35$	−637.933	−	1204.81i	−0.520762	−	0.983517i
$36$		0			0
$37$	−589.843	−	589.843i	−0.430857	−	0.430857i	0.458063	−	0.888920i	$-0.348543\pi$
$37$	−589.843	−	589.843i	−0.430857	−	0.430857i	−0.888920	+	0.458063i	$0.848543\pi$
$38$	1583.48	−	1583.48i	1.09659	−	1.09659i
$39$		0			0
$40$	−2916.48	+	1544.25i	−1.82280	+	0.965154i
$41$	−1856.55			−1.10443			−0.552215	−	0.833702i	$-0.686217\pi$
$41$	−1856.55			−1.10443			−0.552215	+	0.833702i	$0.686217\pi$
$42$		0			0
$43$	−671.078	+	671.078i	−0.362941	+	0.362941i	−0.864894	−	0.501954i	$-0.832615\pi$
$43$	−671.078	+	671.078i	−0.362941	+	0.362941i	0.501954	+	0.864894i	$0.332615\pi$
$44$			1397.53i			0.721863i
$45$		0			0
$46$	5760.90			2.72254
$47$	−504.865	−	504.865i	−0.228549	−	0.228549i	0.583537	−	0.812086i	$-0.301668\pi$
$47$	−504.865	−	504.865i	−0.228549	−	0.228549i	−0.812086	+	0.583537i	$0.801668\pi$
$48$		0			0
$49$			572.636i			0.238499i
$50$	832.883	+	4365.52i	0.333153	+	1.74621i
$51$		0			0
$52$	721.305	+	721.305i	0.266755	+	0.266755i
$53$	−2251.32	+	2251.32i	−0.801468	+	0.801468i	−0.983325	−	0.181857i	$-0.941789\pi$
$53$	−2251.32	+	2251.32i	−0.801468	+	0.801468i	0.181857	+	0.983325i	$0.441789\pi$
$54$		0			0
$55$	966.151	+	297.213i	0.319389	+	0.0982523i
$56$	7198.29			2.29537
$57$		0			0
$58$	−4145.34	+	4145.34i	−1.23227	+	1.23227i
$59$			2585.03i			0.742610i	0.928511	+	0.371305i	$0.121090\pi$
$59$			2585.03i			0.742610i	−0.928511	+	0.371305i	$0.878910\pi$
$60$		0			0
$61$	−3276.74			−0.880607			−0.440304	−	0.897849i	$-0.645129\pi$
$61$	−3276.74			−0.880607			−0.440304	+	0.897849i	$0.645129\pi$
$62$	−6773.81	−	6773.81i	−1.76218	−	1.76218i
$63$		0			0
$64$			1689.53i			0.412483i
$65$	652.060	−	345.259i	0.154334	−	0.0817180i
$66$		0			0
$67$	−3428.94	−	3428.94i	−0.763854	−	0.763854i	0.213163	−	0.977017i	$-0.431624\pi$
$67$	−3428.94	−	3428.94i	−0.763854	−	0.763854i	−0.977017	+	0.213163i	$0.931624\pi$
$68$	548.573	−	548.573i	0.118636	−	0.118636i
$69$		0			0
$70$	2850.31	−	9265.50i	0.581696	−	1.89092i
$71$	5679.51			1.12666			0.563331	−	0.826231i	$-0.309519\pi$
$71$	5679.51			1.12666			0.563331	+	0.826231i	$0.309519\pi$
$72$		0			0
$73$	4450.06	−	4450.06i	0.835065	−	0.835065i	−0.153140	−	0.988205i	$-0.548939\pi$
$73$	4450.06	−	4450.06i	0.835065	−	0.835065i	0.988205	+	0.153140i	$0.0489385\pi$
$74$		−	5931.59i		−	1.08320i
$75$		0			0
$76$	10885.0			1.88453
$77$	−1559.08	−	1559.08i	−0.262958	−	0.262958i
$78$		0			0
$79$			6465.77i			1.03602i	0.855376	+	0.518008i	$0.173326\pi$
$79$			6465.77i			1.03602i	−0.855376	+	0.518008i	$0.826674\pi$
$80$	−9214.66	−	2834.67i	−1.43979	−	0.442918i
$81$		0			0
$82$	−9334.92	−	9334.92i	−1.38830	−	1.38830i
$83$	621.380	−	621.380i	0.0901988	−	0.0901988i	−0.660568	−	0.750766i	$-0.729684\pi$
$83$	621.380	−	621.380i	0.0901988	−	0.0901988i	0.750766	+	0.660568i	$0.229684\pi$
$84$		0			0
$85$	−262.579	−	495.910i	−0.0363431	−	0.0686380i
$86$	−6748.50			−0.912453
$87$		0			0
$88$	−3774.07	+	3774.07i	−0.487354	+	0.487354i
$89$			1856.76i			0.234410i	0.993108	+	0.117205i	$0.0373935\pi$
$89$			1856.76i			0.234410i	−0.993108	+	0.117205i	$0.962606\pi$
$90$		0			0
$91$	−1609.37			−0.194345
$92$	19800.5	+	19800.5i	2.33938	+	2.33938i
$93$		0			0
$94$		−	5077.03i		−	0.574585i
$95$	2314.93	−	7525.13i	0.256502	−	0.833809i
$96$		0			0
$97$	12383.5	+	12383.5i	1.31614	+	1.31614i	0.916806	+	0.399332i	$0.130758\pi$
$97$	12383.5	+	12383.5i	1.31614	+	1.31614i	0.399332	+	0.916806i	$0.369242\pi$
$98$	−2879.27	+	2879.27i	−0.299800	+	0.299800i
$99$		0			0
$100$	−12141.9	+	17867.2i	−1.21419	+	1.78672i
$101$	189.344			0.0185613			0.00928067	−	0.999957i	$-0.497046\pi$
$101$	189.344			0.0185613			0.00928067	+	0.999957i	$0.497046\pi$
$102$		0			0
$103$	14330.9	−	14330.9i	1.35083	−	1.35083i	0.466092	−	0.884736i	$-0.345662\pi$
$103$	14330.9	−	14330.9i	1.35083	−	1.35083i	0.884736	−	0.466092i	$-0.154338\pi$
$104$			3895.82i			0.360190i
$105$		0			0
$106$	−22639.8			−2.01493
$107$	384.065	+	384.065i	0.0335457	+	0.0335457i	0.723681	−	0.690135i	$-0.242449\pi$
$107$	384.065	+	384.065i	0.0335457	+	0.0335457i	−0.690135	+	0.723681i	$0.742449\pi$
$108$		0			0
$109$			14104.7i			1.18717i	0.804773	+	0.593583i	$0.202287\pi$
$109$			14104.7i			1.18717i	−0.804773	+	0.593583i	$0.797713\pi$
$110$	3363.49	+	6352.33i	0.277974	+	0.524986i
$111$		0			0
$112$	14869.7	+	14869.7i	1.18540	+	1.18540i
$113$	12599.1	−	12599.1i	0.986693	−	0.986693i	−0.0132199	−	0.999913i	$-0.504208\pi$
$113$	12599.1	−	12599.1i	0.986693	−	0.986693i	0.999913	+	0.0132199i	$0.00420816\pi$
$114$		0			0
$115$	17899.7	−	9477.67i	1.35347	−	0.716648i
$116$	−28495.5			−2.11768
$117$		0			0
$118$	−12997.8	+	12997.8i	−0.933481	+	0.933481i
$119$			1223.97i			0.0864328i
$120$		0			0
$121$	−13006.1			−0.888337
$122$	−16475.8	−	16475.8i	−1.10695	−	1.10695i
$123$		0			0
$124$		−	46563.9i		−	3.02835i
$125$	9769.88	+	12193.9i	0.625272	+	0.780407i
$126$		0			0
$127$	−10957.5	−	10957.5i	−0.679367	−	0.679367i	0.280490	−	0.959857i	$-0.409503\pi$
$127$	−10957.5	−	10957.5i	−0.679367	−	0.679367i	−0.959857	+	0.280490i	$0.909503\pi$
$128$	−15624.0	+	15624.0i	−0.953614	+	0.953614i
$129$		0			0
$130$	5014.63	+	1542.63i	0.296723	+	0.0912799i
$131$	−10872.8			−0.633579			−0.316789	−	0.948496i	$-0.602605\pi$
$131$	−10872.8			−0.633579			−0.316789	+	0.948496i	$0.602605\pi$
$132$		0			0
$133$	−12143.3	+	12143.3i	−0.686489	+	0.686489i
$134$		−	34482.2i		−	1.92037i
$135$		0			0
$136$	2962.88			0.160190
$137$	3387.27	+	3387.27i	0.180471	+	0.180471i	0.791561	−	0.611090i	$-0.209269\pi$
$137$	3387.27	+	3387.27i	0.180471	+	0.180471i	−0.611090	+	0.791561i	$0.709269\pi$
$138$		0			0
$139$		−	6096.09i		−	0.315516i	−0.987478	−	0.157758i	$-0.949573\pi$
$139$		−	6096.09i		−	0.315516i	0.987478	−	0.157758i	$-0.0504266\pi$
$140$	41642.7	−	22049.4i	2.12463	−	1.12497i
$141$		0			0
$142$	28557.2	+	28557.2i	1.41625	+	1.41625i
$143$	843.796	−	843.796i	0.0412634	−	0.0412634i
$144$		0			0
$145$	−6060.17	+	19699.8i	−0.288236	+	0.936969i
$146$	44750.8			2.09940
$147$		0			0
$148$	20387.2	−	20387.2i	0.930752	−	0.930752i
$149$			41060.3i			1.84948i	0.380600	+	0.924740i	$0.375718\pi$
$149$			41060.3i			1.84948i	−0.380600	+	0.924740i	$0.624282\pi$
$150$		0			0
$151$	12096.5			0.530523			0.265262	−	0.964176i	$-0.414542\pi$
$151$	12096.5			0.530523			0.265262	+	0.964176i	$0.414542\pi$
$152$	29395.3	+	29395.3i	1.27231	+	1.27231i
$153$		0			0
$154$		−	15678.4i		−	0.661091i
$155$	−32191.0	−	9902.79i	−1.33989	−	0.412187i
$156$		0			0
$157$	−21184.3	−	21184.3i	−0.859440	−	0.859440i	0.131832	−	0.991272i	$-0.457914\pi$
$157$	−21184.3	−	21184.3i	−0.859440	−	0.859440i	−0.991272	+	0.131832i	$0.957914\pi$
$158$	−32510.6	+	32510.6i	−1.30230	+	1.30230i
$159$		0			0
$160$	−7371.35	−	13921.6i	−0.287943	−	0.543814i
$161$	−44178.8			−1.70436
$162$		0			0
$163$	3953.29	−	3953.29i	0.148793	−	0.148793i	−0.628786	−	0.777579i	$-0.716448\pi$
$163$	3953.29	−	3953.29i	0.148793	−	0.148793i	0.777579	+	0.628786i	$0.216448\pi$
$164$		−	64169.2i		−	2.38583i
$165$		0			0
$166$	6248.73			0.226765
$167$	7863.94	+	7863.94i	0.281973	+	0.281973i	0.833895	−	0.551922i	$-0.186106\pi$
$167$	7863.94	+	7863.94i	0.281973	+	0.281973i	−0.551922	+	0.833895i	$0.686106\pi$
$168$		0			0
$169$			27690.0i			0.969503i
$170$	1173.21	−	3813.76i	0.0405956	−	0.131964i
$171$		0			0
$172$	−23195.0	−	23195.0i	−0.784037	−	0.784037i
$173$	20563.0	−	20563.0i	0.687059	−	0.687059i	−0.274522	−	0.961581i	$-0.588520\pi$
$173$	20563.0	−	20563.0i	0.687059	−	0.687059i	0.961581	+	0.274522i	$0.0885195\pi$
$174$		0			0
$175$	−6387.16	−	33478.1i	−0.208560	−	1.09316i
$176$	−15592.4			−0.503371
$177$		0			0
$178$	−9336.01	+	9336.01i	−0.294660	+	0.294660i
$179$		−	11578.3i		−	0.361360i	−0.983542	−	0.180680i	$-0.942170\pi$
$179$		−	11578.3i		−	0.361360i	0.983542	−	0.180680i	$-0.0578298\pi$
$180$		0			0
$181$	15410.3			0.470385			0.235193	−	0.971949i	$-0.424428\pi$
$181$	15410.3			0.470385			0.235193	+	0.971949i	$0.424428\pi$
$182$	−8092.11	−	8092.11i	−0.244298	−	0.244298i
$183$		0			0
$184$			106944.i			3.15879i
$185$	−9758.49	−	18430.0i	−0.285128	−	0.538496i
$186$		0			0
$187$	−641.731	−	641.731i	−0.0183514	−	0.0183514i
$188$	17450.0	−	17450.0i	0.493720	−	0.493720i
$189$		0			0
$190$	49476.9	−	26197.5i	1.37055	−	0.725691i
$191$	8129.00			0.222828			0.111414	−	0.993774i	$-0.464462\pi$
$191$	8129.00			0.222828			0.111414	+	0.993774i	$0.464462\pi$
$192$		0			0
$193$	−28495.5	+	28495.5i	−0.765001	+	0.765001i	−0.977222	−	0.212220i	$-0.931931\pi$
$193$	−28495.5	+	28495.5i	−0.765001	+	0.765001i	0.212220	+	0.977222i	$0.431931\pi$
$194$			124532.i			3.30884i
$195$		0			0
$196$	−19792.4			−0.515213
$197$	14908.5	+	14908.5i	0.384150	+	0.384150i	0.872595	−	0.488445i	$-0.162436\pi$
$197$	14908.5	+	14908.5i	0.384150	+	0.384150i	−0.488445	+	0.872595i	$0.662436\pi$
$198$		0			0
$199$		−	3870.42i		−	0.0977353i	−0.998805	−	0.0488677i	$-0.984439\pi$
$199$		−	3870.42i		−	0.0977353i	0.998805	−	0.0488677i	$-0.0155613\pi$
$200$	−81040.4	+	15461.4i	−2.02601	+	0.386536i
$201$		0			0
$202$	952.043	+	952.043i	0.0233321	+	0.0233321i
$203$	31789.6	−	31789.6i	0.771423	−	0.771423i
$204$		0			0
$205$	−44362.1	−	13646.9i	−1.05561	−	0.324734i
$206$	144115.			3.39606
$207$		0			0
$208$	−8047.71	+	8047.71i	−0.186014	+	0.186014i
$209$		−	12733.5i		−	0.291511i
$210$		0			0
$211$	−17666.9			−0.396822			−0.198411	−	0.980119i	$-0.563578\pi$
$211$	−17666.9			−0.396822			−0.198411	+	0.980119i	$0.563578\pi$
$212$	−77814.2	−	77814.2i	−1.73136	−	1.73136i
$213$		0			0
$214$			3862.24i			0.0843358i
$215$	−20968.2	+	11102.4i	−0.453612	+	0.240183i
$216$		0			0
$217$	51946.6	+	51946.6i	1.10316	+	1.10316i
$218$	−70920.0	+	70920.0i	−1.49230	+	1.49230i
$219$		0			0
$220$	−10272.8	+	33393.8i	−0.212248	+	0.689954i
$221$	−662.433			−0.0135630
$222$		0			0
$223$	10730.0	−	10730.0i	0.215768	−	0.215768i	−0.590944	−	0.806713i	$-0.701245\pi$
$223$	10730.0	−	10730.0i	0.215768	−	0.215768i	0.806713	+	0.590944i	$0.201245\pi$
$224$			34360.5i			0.684800i
$225$		0			0
$226$	126699.			2.48060
$227$	34199.9	+	34199.9i	0.663703	+	0.663703i	0.956251	−	0.292548i	$-0.0945032\pi$
$227$	34199.9	+	34199.9i	0.663703	+	0.663703i	−0.292548	+	0.956251i	$0.594503\pi$
$228$		0			0
$229$		−	73430.0i		−	1.40024i	−0.714026	−	0.700120i	$-0.753130\pi$
$229$		−	73430.0i		−	1.40024i	0.714026	−	0.700120i	$-0.246870\pi$
$230$	137656.	+	42346.7i	2.60220	+	0.800504i
$231$		0			0
$232$	−76953.1	−	76953.1i	−1.42972	−	1.42972i
$233$	−18203.1	+	18203.1i	−0.335300	+	0.335300i	−0.854595	−	0.519295i	$-0.826195\pi$
$233$	−18203.1	+	18203.1i	−0.335300	+	0.335300i	0.519295	+	0.854595i	$0.326195\pi$
$234$		0			0
$235$	−8352.59	−	15774.8i	−0.151247	−	0.285646i
$236$	−89348.2			−1.60421
$237$		0			0
$238$	−6154.28	+	6154.28i	−0.108648	+	0.108648i
$239$		−	102775.i		−	1.79925i	−0.436661	−	0.899626i	$-0.643839\pi$
$239$		−	102775.i		−	1.79925i	0.436661	−	0.899626i	$-0.356161\pi$
$240$		0			0
$241$	69403.6			1.19495			0.597473	−	0.801889i	$-0.296172\pi$
$241$	69403.6			1.19495			0.597473	+	0.801889i	$0.296172\pi$
$242$	−65396.3	−	65396.3i	−1.11666	−	1.11666i
$243$		0			0
$244$		−	113256.i		−	1.90232i
$245$	−4209.28	+	13683.1i	−0.0701254	+	0.227956i
$246$		0			0
$247$	−6572.14	−	6572.14i	−0.107724	−	0.107724i
$248$	125747.	−	125747.i	2.04454	−	2.04454i
$249$		0			0
$250$	−12188.0	+	110436.i	−0.195008	+	1.76698i
$251$	40858.0			0.648530			0.324265	−	0.945966i	$-0.394883\pi$
$251$	40858.0			0.648530			0.324265	+	0.945966i	$0.394883\pi$
$252$		0			0
$253$	23163.0	−	23163.0i	0.361871	−	0.361871i
$254$		−	110191.i		−	1.70796i
$255$		0			0
$256$	−130086.			−1.98495
$257$	−53271.7	−	53271.7i	−0.806548	−	0.806548i	0.177562	−	0.984110i	$-0.443179\pi$
$257$	−53271.7	−	53271.7i	−0.806548	−	0.806548i	−0.984110	+	0.177562i	$0.943179\pi$
$258$		0			0
$259$			45487.9i			0.678103i
$260$	11933.4	+	22537.6i	0.176530	+	0.333397i
$261$		0			0
$262$	−54669.8	−	54669.8i	−0.796425	−	0.796425i
$263$	−47788.6	+	47788.6i	−0.690896	+	0.690896i	−0.962429	−	0.271533i	$-0.912469\pi$
$263$	−47788.6	+	47788.6i	−0.690896	+	0.690896i	0.271533	+	0.962429i	$0.412469\pi$
$264$		0			0
$265$	−70344.0	+	37246.4i	−1.00170	+	0.530387i
$266$	−122116.			−1.72587
$267$		0			0
$268$	118517.	−	118517.i	1.65010	−	1.65010i
$269$			130832.i			1.80805i	0.427483	+	0.904023i	$0.359400\pi$
$269$			130832.i			1.80805i	−0.427483	+	0.904023i	$0.640600\pi$
$270$		0			0
$271$	−116029.			−1.57989			−0.789945	−	0.613178i	$-0.789891\pi$
$271$	−116029.			−1.57989			−0.789945	+	0.613178i	$0.789891\pi$
$272$	6120.51	+	6120.51i	0.0827274	+	0.0827274i
$273$		0			0
$274$			34063.1i			0.453715i
$275$	20901.4	+	14203.8i	0.276382	+	0.187819i
$276$		0			0
$277$	41111.1	+	41111.1i	0.535796	+	0.535796i	0.922291	−	0.386495i	$-0.126314\pi$
$277$	41111.1	+	41111.1i	0.535796	+	0.535796i	−0.386495	+	0.922291i	$0.626314\pi$
$278$	30651.8	−	30651.8i	0.396613	−	0.396613i
$279$		0			0
$280$	172003.	+	52912.5i	2.19391	+	0.674904i
$281$	61086.4			0.773627			0.386814	−	0.922158i	$-0.373576\pi$
$281$	61086.4			0.773627			0.386814	+	0.922158i	$0.373576\pi$
$282$		0			0
$283$	54739.9	−	54739.9i	0.683489	−	0.683489i	−0.277296	−	0.960785i	$-0.589438\pi$
$283$	54739.9	−	54739.9i	0.683489	−	0.683489i	0.960785	+	0.277296i	$0.0894382\pi$
$284$			196305.i			2.43386i
$285$		0			0
$286$	8485.40			0.103739
$287$	71587.1	+	71587.1i	0.869103	+	0.869103i
$288$		0			0
$289$		−	83017.2i		−	0.993968i
$290$	−129524.	+	68581.4i	−1.54012	+	0.815474i
$291$		0			0
$292$	153811.	+	153811.i	1.80394	+	1.80394i
$293$	33601.9	−	33601.9i	0.391406	−	0.391406i	−0.483782	−	0.875189i	$-0.660737\pi$
$293$	33601.9	−	33601.9i	0.391406	−	0.391406i	0.875189	+	0.483782i	$0.160737\pi$
$294$		0			0
$295$	−19001.8	+	61769.0i	−0.218348	+	0.709785i
$296$	110113.			1.25676
$297$		0			0
$298$	−206456.	+	206456.i	−2.32485	+	2.32485i
$299$		−	23910.2i		−	0.267449i
$300$		0			0
$301$	51752.5			0.571214
$302$	60822.3	+	60822.3i	0.666882	+	0.666882i
$303$		0			0
$304$			121446.i			1.31412i
$305$	−78297.4	−	24086.3i	−0.841681	−	0.258923i
$306$		0			0
$307$	84437.4	+	84437.4i	0.895897	+	0.895897i	0.995070	−	0.0991732i	$-0.0316198\pi$
$307$	84437.4	+	84437.4i	0.895897	+	0.895897i	−0.0991732	+	0.995070i	$0.531620\pi$
$308$	53887.6	−	53887.6i	0.568051	−	0.568051i
$309$		0			0
$310$	−112067.	−	211652.i	−1.16615	−	2.20241i
$311$	−93477.0			−0.966460			−0.483230	−	0.875493i	$-0.660537\pi$
$311$	−93477.0			−0.966460			−0.483230	+	0.875493i	$0.660537\pi$
$312$		0			0
$313$	−6979.59	+	6979.59i	−0.0712429	+	0.0712429i	−0.741830	−	0.670588i	$-0.766042\pi$
$313$	−6979.59	+	6979.59i	−0.0712429	+	0.0712429i	0.670588	+	0.741830i	$0.266042\pi$
$314$		−	213034.i		−	2.16068i
$315$		0			0
$316$	−223481.			−2.23804
$317$	−41396.6	−	41396.6i	−0.411951	−	0.411951i	0.470467	−	0.882418i	$-0.344086\pi$
$317$	−41396.6	−	41396.6i	−0.411951	−	0.411951i	−0.882418	+	0.470467i	$0.844086\pi$
$318$		0			0
$319$			33334.6i			0.327577i
$320$	−12419.2	+	40371.2i	−0.121282	+	0.394250i
$321$		0			0
$322$	−222136.	−	222136.i	−2.14243	−	2.14243i
$323$	−4998.29	+	4998.29i	−0.0479090	+	0.0479090i
$324$		0			0
$325$	18118.8	−	3456.82i	0.171539	−	0.0327273i
$326$	39755.1			0.374074
$327$		0			0
$328$	173291.	−	173291.i	1.61075	−	1.61075i
$329$			38934.5i			0.359702i
$330$		0			0
$331$	65879.3			0.601303			0.300651	−	0.953734i	$-0.402796\pi$
$331$	65879.3			0.601303			0.300651	+	0.953734i	$0.402796\pi$
$332$	21477.2	+	21477.2i	0.194851	+	0.194851i
$333$		0			0
$334$			79081.5i			0.708895i
$335$	−56729.1	−	107139.i	−0.505495	−	0.954684i
$336$		0			0
$337$	18022.9	+	18022.9i	0.158696	+	0.158696i	0.781989	−	0.623293i	$-0.214205\pi$
$337$	18022.9	+	18022.9i	0.158696	+	0.158696i	−0.623293	+	0.781989i	$0.714205\pi$
$338$	−139228.	+	139228.i	−1.21869	+	1.21869i
$339$		0			0
$340$	17140.5	−	9075.71i	0.148274	−	0.0785096i
$341$	−54471.3			−0.468445
$342$		0			0
$343$	−70500.4	+	70500.4i	−0.599244	+	0.599244i
$344$		−	125277.i		−	1.05866i
$345$		0			0
$346$	206786.			1.72730
$347$	32583.8	+	32583.8i	0.270610	+	0.270610i	0.829346	−	0.558736i	$-0.188713\pi$
$347$	32583.8	+	32583.8i	0.270610	+	0.270610i	−0.558736	+	0.829346i	$0.688713\pi$
$348$		0			0
$349$		−	54340.9i		−	0.446145i	−0.974802	−	0.223072i	$-0.928391\pi$
$349$		−	54340.9i		−	0.446145i	0.974802	−	0.223072i	$-0.0716086\pi$
$350$	136216.	−	200447.i	1.11197	−	1.63630i
$351$		0			0
$352$	−18015.2	−	18015.2i	−0.145397	−	0.145397i
$353$	11345.9	−	11345.9i	0.0910524	−	0.0910524i	−0.660114	−	0.751166i	$-0.729492\pi$
$353$	11345.9	−	11345.9i	0.0910524	−	0.0910524i	0.751166	+	0.660114i	$0.229492\pi$
$354$		0			0
$355$	135711.	+	41748.4i	1.07686	+	0.331271i
$356$	−64176.7			−0.506381
$357$		0			0
$358$	58217.1	−	58217.1i	0.454239	−	0.454239i
$359$			26830.8i			0.208183i	0.994568	+	0.104091i	$0.0331934\pi$
$359$			26830.8i			0.208183i	−0.994568	+	0.104091i	$0.966807\pi$
$360$		0			0
$361$	31142.7			0.238969
$362$	77484.6	+	77484.6i	0.591287	+	0.591287i
$363$		0			0
$364$		−	55626.0i		−	0.419832i
$365$	139045.	−	73622.7i	1.04368	−	0.552620i
$366$		0			0
$367$	−112367.	−	112367.i	−0.834267	−	0.834267i	0.153830	−	0.988097i	$-0.450839\pi$
$367$	−112367.	−	112367.i	−0.834267	−	0.834267i	−0.988097	+	0.153830i	$0.950839\pi$
$368$	−220917.	+	220917.i	−1.63130	+	1.63130i
$369$		0			0
$370$	43601.4	−	141735.i	0.318491	−	1.03532i
$371$	173619.			1.26139
$372$		0			0
$373$	−16877.0	+	16877.0i	−0.121305	+	0.121305i	−0.765153	−	0.643848i	$-0.777337\pi$
$373$	−16877.0	+	16877.0i	−0.121305	+	0.121305i	0.643848	+	0.765153i	$0.277337\pi$
$374$		−	6453.38i		−	0.0461364i
$375$		0			0
$376$	94248.7			0.666653
$377$	17205.0	+	17205.0i	0.121052	+	0.121052i
$378$		0			0
$379$		−	50523.2i		−	0.351733i	−0.984414	−	0.175866i	$-0.943727\pi$
$379$		−	50523.2i		−	0.351733i	0.984414	−	0.175866i	$-0.0562726\pi$
$380$	260097.	+	80012.5i	1.80122	+	0.554104i
$381$		0			0
$382$	40873.5	+	40873.5i	0.280101	+	0.280101i
$383$	−17592.0	+	17592.0i	−0.119927	+	0.119927i	−0.764523	−	0.644596i	$-0.777026\pi$
$383$	−17592.0	+	17592.0i	−0.119927	+	0.119927i	0.644596	+	0.764523i	$0.277026\pi$
$384$		0			0
$385$	−25793.7	−	48714.4i	−0.174017	−	0.328652i
$386$	−286557.			−1.92325
$387$		0			0
$388$	−428022.	+	428022.i	−2.84317	+	2.84317i
$389$			6343.53i			0.0419210i	0.999780	+	0.0209605i	$0.00667242\pi$
$389$			6343.53i			0.0419210i	−0.999780	+	0.0209605i	$0.993328\pi$
$390$		0			0
$391$	−18184.4			−0.118945
$392$	−53450.1	−	53450.1i	−0.347838	−	0.347838i
$393$		0			0
$394$			149923.i			0.965773i
$395$	−47528.0	+	154499.i	−0.304618	+	0.990221i
$396$		0			0
$397$	−145003.	−	145003.i	−0.920018	−	0.920018i	0.0770121	−	0.997030i	$-0.475462\pi$
$397$	−145003.	−	145003.i	−0.920018	−	0.920018i	−0.997030	+	0.0770121i	$0.975462\pi$
$398$	19460.9	−	19460.9i	0.122856	−	0.122856i
$399$		0			0
$400$	−199347.	−	135469.i	−1.24592	−	0.846678i
$401$	299021.			1.85957			0.929786	−	0.368100i	$-0.119992\pi$
$401$	299021.			1.85957			0.929786	+	0.368100i	$0.119992\pi$
$402$		0			0
$403$	−28114.3	+	28114.3i	−0.173108	+	0.173108i
$404$			6544.44i			0.0400968i
$405$		0			0
$406$	319683.			1.93940
$407$	−23849.3	−	23849.3i	−0.143975	−	0.143975i
$408$		0			0
$409$		−	17363.2i		−	0.103796i	−0.998652	−	0.0518982i	$-0.983473\pi$
$409$		−	17363.2i		−	0.103796i	0.998652	−	0.0518982i	$-0.0165271\pi$
$410$	−154439.	−	291675.i	−0.918733	−	1.73513i
$411$		0			0
$412$	495331.	+	495331.i	2.91811	+	2.91811i
$413$	99676.8	−	99676.8i	0.584378	−	0.584378i
$414$		0			0
$415$	19415.4	−	10280.2i	0.112733	−	0.0596907i
$416$	−18596.4			−0.107459
$417$		0			0
$418$	64025.4	−	64025.4i	0.366437	−	0.366437i
$419$		−	36236.3i		−	0.206403i	−0.994660	−	0.103201i	$-0.967091\pi$
$419$		−	36236.3i		−	0.206403i	0.994660	−	0.103201i	$-0.0329086\pi$
$420$		0			0
$421$	−121899.			−0.687761			−0.343881	−	0.939013i	$-0.611742\pi$
$421$	−121899.			−0.687761			−0.343881	+	0.939013i	$0.611742\pi$
$422$	−88831.2	−	88831.2i	−0.498817	−	0.498817i
$423$		0			0
$424$		−	420280.i		−	2.33780i
$425$	−2629.01	−	13779.9i	−0.0145551	−	0.0762899i
$426$		0			0
$427$	126349.	+	126349.i	0.692971	+	0.692971i
$428$	−13274.7	+	13274.7i	−0.0724666	+	0.0724666i
$429$		0			0
$430$	−161255.	−	49606.2i	−0.872119	−	0.268287i
$431$	−330169.			−1.77739			−0.888693	−	0.458504i	$-0.848386\pi$
$431$	−330169.			−1.77739			−0.888693	+	0.458504i	$0.848386\pi$
$432$		0			0
$433$	140458.	−	140458.i	0.749151	−	0.749151i	−0.225169	−	0.974320i	$-0.572293\pi$
$433$	140458.	−	140458.i	0.749151	−	0.749151i	0.974320	+	0.225169i	$0.0722934\pi$
$434$			522387.i			2.77340i
$435$		0			0
$436$	−487512.			−2.56455
$437$	−180411.	−	180411.i	−0.944715	−	0.944715i
$438$		0			0
$439$			25797.9i			0.133861i	0.997758	+	0.0669306i	$0.0213206\pi$
$439$			25797.9i			0.133861i	−0.997758	+	0.0669306i	$0.978679\pi$
$440$	−117923.	+	62439.0i	−0.609107	+	0.322515i
$441$		0			0
$442$	−3330.78	−	3330.78i	−0.0170491	−	0.0170491i
$443$	−131776.	+	131776.i	−0.671473	+	0.671473i	−0.958056	−	0.286583i	$-0.907481\pi$
$443$	−131776.	+	131776.i	−0.671473	+	0.671473i	0.286583	+	0.958056i	$0.407481\pi$
$444$		0			0
$445$	−13648.5	+	44367.2i	−0.0689232	+	0.224049i
$446$	107903.			0.542454
$447$		0			0
$448$	65147.1	−	65147.1i	0.324593	−	0.324593i
$449$		−	133600.i		−	0.662697i	−0.943508	−	0.331349i	$-0.892496\pi$
$449$		−	133600.i		−	0.662697i	0.943508	−	0.331349i	$-0.107504\pi$
$450$		0			0
$451$	−75066.4			−0.369056
$452$	435471.	+	435471.i	2.13149	+	2.13149i
$453$		0			0
$454$			343922.i			1.66858i
$455$	−38455.9	−	11830.0i	−0.185755	−	0.0571430i
$456$		0			0
$457$	−147483.	−	147483.i	−0.706170	−	0.706170i	0.259558	−	0.965728i	$-0.416423\pi$
$457$	−147483.	−	147483.i	−0.706170	−	0.706170i	−0.965728	+	0.259558i	$0.916423\pi$
$458$	369214.	−	369214.i	1.76014	−	1.76014i
$459$		0			0
$460$	327584.	+	618679.i	1.54813	+	2.92381i
$461$	−206578.			−0.972034			−0.486017	−	0.873949i	$-0.661551\pi$
$461$	−206578.			−0.972034			−0.486017	+	0.873949i	$0.661551\pi$
$462$		0			0
$463$	−156702.	+	156702.i	−0.730990	+	0.730990i	−0.970816	−	0.239826i	$-0.922910\pi$
$463$	−156702.	+	156702.i	−0.730990	+	0.730990i	0.239826	+	0.970816i	$0.422910\pi$
$464$		−	317928.i		−	1.47670i
$465$		0			0
$466$	−183055.			−0.842963
$467$	100008.	+	100008.i	0.458566	+	0.458566i	0.898184	−	0.439619i	$-0.144886\pi$
$467$	100008.	+	100008.i	0.458566	+	0.458566i	−0.439619	+	0.898184i	$0.644886\pi$
$468$		0			0
$469$			264435.i			1.20219i
$470$	37319.8	−	121315.i	0.168944	−	0.549186i
$471$		0			0
$472$	−241288.	−	241288.i	−1.08306	−	1.08306i
$473$	−27133.9	+	27133.9i	−0.121280	+	0.121280i
$474$		0			0
$475$	110630.	−	162796.i	0.490327	−	0.721533i
$476$	−42305.1			−0.186715
$477$		0			0
$478$	516764.	−	516764.i	2.26171	−	2.26171i
$479$			124818.i			0.544010i	0.962296	+	0.272005i	$0.0876867\pi$
$479$			124818.i			0.544010i	−0.962296	+	0.272005i	$0.912313\pi$
$480$		0			0
$481$	−24618.7			−0.106408
$482$	348969.	+	348969.i	1.50208	+	1.50208i
$483$		0			0
$484$		−	449541.i		−	1.91902i
$485$	204876.	+	386932.i	0.870979	+	1.64494i
$486$		0			0
$487$	−106165.	−	106165.i	−0.447635	−	0.447635i	0.446932	−	0.894568i	$-0.352516\pi$
$487$	−106165.	−	106165.i	−0.447635	−	0.447635i	−0.894568	+	0.446932i	$0.852516\pi$
$488$	305853.	−	305853.i	1.28432	−	1.28432i
$489$		0			0
$490$	−89964.7	+	47635.3i	−0.374697	+	0.198398i
$491$	−185092.			−0.767758			−0.383879	−	0.923383i	$-0.625412\pi$
$491$	−185092.			−0.767758			−0.383879	+	0.923383i	$0.625412\pi$
$492$		0			0
$493$	13084.9	−	13084.9i	0.0538363	−	0.0538363i
$494$		−	66090.8i		−	0.270824i
$495$		0			0
$496$	519520.			2.11173
$497$	−218998.	−	218998.i	−0.886598	−	0.886598i
$498$		0			0
$499$			482480.i			1.93766i	0.247723	+	0.968831i	$0.420318\pi$
$499$			482480.i			1.93766i	−0.247723	+	0.968831i	$0.579682\pi$
$500$	−421465.	+	337684.i	−1.68586	+	1.35073i
$501$		0			0
$502$	205439.	+	205439.i	0.815220	+	0.815220i
$503$	−224311.	+	224311.i	−0.886572	+	0.886572i	−0.994192	−	0.107620i	$-0.965677\pi$
$503$	−224311.	+	224311.i	−0.886572	+	0.886572i	0.107620	+	0.994192i	$0.465677\pi$
$504$		0			0
$505$	4524.36	+	1391.81i	0.0177409	+	0.00545756i
$506$	232932.			0.909763
$507$		0			0
$508$	378732.	−	378732.i	1.46759	−	1.46759i
$509$		−	390482.i		−	1.50718i	−0.657344	−	0.753591i	$-0.728320\pi$
$509$		−	390482.i		−	1.50718i	0.657344	−	0.753591i	$-0.271680\pi$
$510$		0			0
$511$	−343182.			−1.31426
$512$	−404102.	−	404102.i	−1.54153	−	1.54153i
$513$		0			0
$514$		−	535711.i		−	2.02770i
$515$	447779.	−	237094.i	1.68830	−	0.893936i
$516$		0			0
$517$	−20413.4	−	20413.4i	−0.0763719	−	0.0763719i
$518$	−228718.	+	228718.i	−0.852394	+	0.852394i
$519$		0			0
$520$	−28637.0	+	93090.2i	−0.105906	+	0.344269i
$521$	71234.7			0.262432			0.131216	−	0.991354i	$-0.458112\pi$
$521$	71234.7			0.262432			0.131216	+	0.991354i	$0.458112\pi$
$522$		0			0
$523$	−26895.3	+	26895.3i	−0.0983272	+	0.0983272i	−0.754559	−	0.656232i	$-0.772149\pi$
$523$	−26895.3	+	26895.3i	−0.0983272	+	0.0983272i	0.656232	+	0.754559i	$0.272149\pi$
$524$		−	375806.i		−	1.36868i
$525$		0			0
$526$	−480573.			−1.73695
$527$	21381.7	+	21381.7i	0.0769876	+	0.0769876i
$528$		0			0
$529$		−	376517.i		−	1.34547i
$530$	−540976.	−	166419.i	−1.92587	−	0.592448i
$531$		0			0
$532$	−419718.	−	419718.i	−1.48298	−	1.48298i
$533$	−38744.0	+	38744.0i	−0.136380	+	0.136380i
$534$		0			0
$535$	6354.05	+	12000.3i	0.0221995	+	0.0419263i
$536$	640119.			2.22808
$537$		0			0
$538$	−657838.	+	657838.i	−2.27276	+	2.27276i
$539$			23153.6i			0.0796967i
$540$		0			0
$541$	293490.			1.00276			0.501382	−	0.865226i	$-0.332825\pi$
$541$	293490.			1.00276			0.501382	+	0.865226i	$0.332825\pi$
$542$	−583405.	−	583405.i	−1.98596	−	1.98596i
$543$		0			0
$544$			14143.1i			0.0477911i
$545$	−103680.	+	337031.i	−0.349060	+	1.13469i
$546$		0			0
$547$	178128.	+	178128.i	0.595330	+	0.595330i	0.939066	−	0.343736i	$-0.111693\pi$
$547$	178128.	+	178128.i	0.595330	+	0.595330i	−0.343736	+	0.939066i	$0.611693\pi$
$548$	−117077.	+	117077.i	−0.389860	+	0.389860i
$549$		0			0
$550$	33676.2	+	176512.i	0.111326	+	0.583512i
$551$	259635.			0.855186
$552$		0			0
$553$	249316.	−	249316.i	0.815266	−	0.815266i
$554$			413422.i			1.34702i
$555$		0			0
$556$	210704.			0.681589
$557$	−122805.	−	122805.i	−0.395827	−	0.395827i	0.480931	−	0.876758i	$-0.340299\pi$
$557$	−122805.	−	122805.i	−0.395827	−	0.395827i	−0.876758	+	0.480931i	$0.840299\pi$
$558$		0			0
$559$			28009.2i			0.0896349i
$560$	246008.	+	464614.i	0.784464	+	1.48155i
$561$		0			0
$562$	307149.	+	307149.i	0.972470	+	0.972470i
$563$	86662.0	−	86662.0i	0.273408	−	0.273408i	−0.557062	−	0.830471i	$-0.688072\pi$
$563$	86662.0	−	86662.0i	0.273408	−	0.273408i	0.830471	+	0.557062i	$0.188072\pi$
$564$		0			0
$565$	393666.	−	208442.i	1.23319	−	0.652962i
$566$	550477.			1.71833
$567$		0			0
$568$	−530128.	+	530128.i	−1.64318	+	1.64318i
$569$			358179.i			1.10631i	0.833080	+	0.553153i	$0.186575\pi$
$569$			358179.i			1.10631i	−0.833080	+	0.553153i	$0.813425\pi$
$570$		0			0
$571$	420094.			1.28847			0.644234	−	0.764828i	$-0.277176\pi$
$571$	420094.			1.28847			0.644234	+	0.764828i	$0.277176\pi$
$572$	29164.8	+	29164.8i	0.0891387	+	0.0891387i
$573$		0			0
$574$			719896.i			2.18497i
$575$	497378.	−	94893.1i	1.50436	−	0.287011i
$576$		0			0
$577$	−167153.	−	167153.i	−0.502068	−	0.502068i	0.410012	−	0.912080i	$-0.365524\pi$
$577$	−167153.	−	167153.i	−0.502068	−	0.502068i	−0.912080	+	0.410012i	$0.865524\pi$
$578$	417419.	−	417419.i	1.24944	−	1.24944i
$579$		0			0
$580$	−680898.	−	209462.i	−2.02407	−	0.622658i
$581$	−47919.9			−0.141959
$582$		0			0
$583$	−91028.5	+	91028.5i	−0.267818	+	0.267818i
$584$			830742.i			2.43579i
$585$		0			0
$586$	337908.			0.984017
$587$	43094.6	+	43094.6i	0.125068	+	0.125068i	0.766870	−	0.641802i	$-0.221813\pi$
$587$	43094.6	+	43094.6i	0.125068	+	0.125068i	−0.641802	+	0.766870i	$0.721813\pi$
$588$		0			0
$589$			424265.i			1.22294i
$590$	−406124.	+	215038.i	−1.16669	+	0.617749i
$591$		0			0
$592$	227463.	+	227463.i	0.649033	+	0.649033i
$593$	114499.	−	114499.i	0.325605	−	0.325605i	−0.525307	−	0.850912i	$-0.676050\pi$
$593$	114499.	−	114499.i	0.325605	−	0.325605i	0.850912	+	0.525307i	$0.176050\pi$
$594$		0			0
$595$	−8997.08	+	29246.8i	−0.0254137	+	0.0826122i
$596$	−1.41920e6			−3.99531
$597$		0			0
$598$	120223.	−	120223.i	0.336191	−	0.336191i
$599$		−	282109.i		−	0.786254i	−0.919484	−	0.393127i	$-0.871393\pi$
$599$		−	282109.i		−	0.786254i	0.919484	−	0.393127i	$-0.128607\pi$
$600$		0			0
$601$	−155225.			−0.429747			−0.214874	−	0.976642i	$-0.568934\pi$
$601$	−155225.			−0.429747			−0.214874	+	0.976642i	$0.568934\pi$
$602$	260217.	+	260217.i	0.718031	+	0.718031i
$603$		0			0
$604$			418099.i			1.14605i
$605$	−310781.	−	95604.4i	−0.849070	−	0.261196i
$606$		0			0
$607$	−262539.	−	262539.i	−0.712553	−	0.712553i	0.254516	−	0.967069i	$-0.418084\pi$
$607$	−262539.	−	262539.i	−0.712553	−	0.712553i	−0.967069	+	0.254516i	$0.918084\pi$
$608$	−140317.	+	140317.i	−0.379579	+	0.379579i
$609$		0			0
$610$	−272579.	−	514797.i	−0.732543	−	1.38349i
$611$	−21071.9			−0.0564444
$612$		0			0
$613$	−98919.2	+	98919.2i	−0.263245	+	0.263245i	−0.826371	−	0.563126i	$-0.809598\pi$
$613$	−98919.2	+	98919.2i	−0.263245	+	0.263245i	0.563126	+	0.826371i	$0.309598\pi$
$614$			849121.i			2.25233i
$615$		0			0
$616$	291051.			0.767021
$617$	440524.	+	440524.i	1.15717	+	1.15717i	0.985080	+	0.172095i	$0.0550535\pi$
$617$	440524.	+	440524.i	1.15717	+	1.15717i	0.172095	+	0.985080i	$0.444947\pi$
$618$		0			0
$619$		−	468975.i		−	1.22396i	−0.790871	−	0.611982i	$-0.790372\pi$
$619$		−	468975.i		−	1.22396i	0.790871	−	0.611982i	$-0.209628\pi$
$620$	342277.	−	1.11264e6i	0.890420	−	2.89449i
$621$		0			0
$622$	−470012.	−	470012.i	−1.21487	−	1.21487i
$623$	71595.5	−	71595.5i	0.184463	−	0.184463i
$624$		0			0
$625$	143817.	+	363187.i	0.368172	+	0.929758i
$626$	−70188.3			−0.179108
$627$		0			0
$628$	732210.	−	732210.i	1.85659	−	1.85659i
$629$			18723.2i			0.0473237i
$630$		0			0
$631$	327190.			0.821752			0.410876	−	0.911691i	$-0.365223\pi$
$631$	327190.			0.821752			0.410876	+	0.911691i	$0.365223\pi$
$632$	−603519.	−	603519.i	−1.51097	−	1.51097i
$633$		0			0
$634$		−	416293.i		−	1.03567i
$635$	−181283.	−	342374.i	−0.449583	−	0.849089i
$636$		0			0
$637$	11950.2	+	11950.2i	0.0294509	+	0.0294509i
$638$	−167610.	+	167610.i	−0.411773	+	0.411773i
$639$		0			0
$640$	−488182.	+	258487.i	−1.19185	+	0.631072i
$641$	584738.			1.42313			0.711567	−	0.702619i	$-0.247986\pi$
$641$	584738.			1.42313			0.711567	+	0.702619i	$0.247986\pi$
$642$		0			0
$643$	−202062.	+	202062.i	−0.488723	+	0.488723i	−0.907903	−	0.419180i	$-0.862318\pi$
$643$	−202062.	+	202062.i	−0.488723	+	0.488723i	0.419180	+	0.907903i	$0.362318\pi$
$644$		−	1.52699e6i		−	3.68183i
$645$		0			0
$646$	−50263.9			−0.120446
$647$	−119543.	−	119543.i	−0.285573	−	0.285573i	0.549754	−	0.835327i	$-0.314722\pi$
$647$	−119543.	−	119543.i	−0.285573	−	0.285573i	−0.835327	+	0.549754i	$0.814722\pi$
$648$		0			0
$649$			104521.i			0.248150i
$650$	108485.	+	73722.1i	0.256768	+	0.174490i
$651$		0			0
$652$	136640.	+	136640.i	0.321428	+	0.321428i
$653$	279985.	−	279985.i	0.656611	−	0.656611i	−0.297965	−	0.954577i	$-0.596308\pi$
$653$	279985.	−	279985.i	0.656611	−	0.656611i	0.954577	+	0.297965i	$0.0963080\pi$
$654$		0			0
$655$	−259806.	−	79923.0i	−0.605572	−	0.186290i
$656$	715946.			1.66369
$657$		0			0
$658$	−195767.	+	195767.i	−0.452155	+	0.452155i
$659$			171782.i			0.395555i	0.980247	+	0.197778i	$0.0633724\pi$
$659$			171782.i			0.395555i	−0.980247	+	0.197778i	$0.936628\pi$
$660$		0			0
$661$	566188.			1.29586			0.647929	−	0.761701i	$-0.275635\pi$
$661$	566188.			1.29586			0.647929	+	0.761701i	$0.275635\pi$
$662$	331248.	+	331248.i	0.755854	+	0.755854i
$663$		0			0
$664$			116000.i			0.263100i
$665$	−379425.	+	200902.i	−0.857992	+	0.454297i
$666$		0			0
$667$	472292.	+	472292.i	1.06160	+	1.06160i
$668$	−271807.	+	271807.i	−0.609128	+	0.609128i
$669$		0			0
$670$	253468.	−	823949.i	0.564643	−	1.83548i
$671$	−132489.			−0.294263
$672$		0			0
$673$	−115466.	+	115466.i	−0.254932	+	0.254932i	−0.822989	−	0.568057i	$-0.807695\pi$
$673$	−115466.	+	115466.i	−0.254932	+	0.254932i	0.568057	+	0.822989i	$0.307695\pi$
$674$			181243.i			0.398970i
$675$		0			0
$676$	−957070.			−2.09435
$677$	−255557.	−	255557.i	−0.557584	−	0.557584i	0.371035	−	0.928619i	$-0.379003\pi$
$677$	−255557.	−	255557.i	−0.557584	−	0.557584i	−0.928619	+	0.371035i	$0.879003\pi$
$678$		0			0
$679$		−	955001.i		−	2.07140i
$680$	70797.7	+	21779.2i	0.153109	+	0.0471004i
$681$		0			0
$682$	−273888.	−	273888.i	−0.588849	−	0.588849i
$683$	−97131.8	+	97131.8i	−0.208219	+	0.208219i	−0.803510	−	0.595291i	$-0.797037\pi$
$683$	−97131.8	+	97131.8i	−0.208219	+	0.208219i	0.595291	+	0.803510i	$0.297037\pi$
$684$		0			0
$685$	56039.6	+	105837.i	0.119430	+	0.225558i
$686$	−708967.			−1.50653
$687$		0			0
$688$	258789.	−	258789.i	0.546726	−	0.546726i
$689$			93965.1i			0.197937i
$690$		0			0
$691$	−40145.6			−0.0840778			−0.0420389	−	0.999116i	$-0.513385\pi$
$691$	−40145.6			−0.0840778			−0.0420389	+	0.999116i	$0.513385\pi$
$692$	710734.	+	710734.i	1.48421	+	1.48421i
$693$		0			0
$694$			327670.i			0.680327i
$695$	44810.6	−	145666.i	0.0927707	−	0.301569i
$696$		0			0
$697$	29465.9	+	29465.9i	0.0606532	+	0.0606532i
$698$	273232.	−	273232.i	0.560816	−	0.560816i
$699$		0			0
$700$	1.15713e6	−	220764.i	2.36148	−	0.450539i
$701$	−225909.			−0.459724			−0.229862	−	0.973223i	$-0.573828\pi$
$701$	−225909.			−0.459724			−0.229862	+	0.973223i	$0.573828\pi$
$702$		0			0
$703$	−185757.	+	185757.i	−0.375867	+	0.375867i
$704$			68313.3i			0.137835i
$705$		0			0
$706$	114097.			0.228911
$707$	−7300.97	−	7300.97i	−0.0146064	−	0.0146064i
$708$		0			0
$709$			459779.i			0.914654i	0.889299	+	0.457327i	$0.151193\pi$
$709$			459779.i			0.914654i	−0.889299	+	0.457327i	$0.848807\pi$
$710$	472456.	+	892287.i	0.937227	+	1.77006i
$711$		0			0
$712$	−173311.	−	173311.i	−0.341875	−	0.341875i
$713$	−771763.	+	771763.i	−1.51812	+	1.51812i
$714$		0			0
$715$	26364.9	−	13959.9i	0.0515721	−	0.0273068i
$716$	400191.			0.780622
$717$		0			0
$718$	−134908.	+	134908.i	−0.261691	+	0.261691i
$719$		−	633365.i		−	1.22517i	−0.790405	−	0.612585i	$-0.790130\pi$
$719$		−	633365.i		−	1.22517i	0.790405	−	0.612585i	$-0.209870\pi$
$720$		0			0
$721$	−1.10518e6			−2.12600
$722$	156589.	+	156589.i	0.300391	+	0.300391i
$723$		0			0
$724$			532637.i			1.01614i
$725$	−289614.	+	426178.i	−0.550990	+	0.810802i
$726$		0			0
$727$	483916.	+	483916.i	0.915590	+	0.915590i	0.996705	−	0.0811152i	$-0.0258482\pi$
$727$	483916.	+	483916.i	0.915590	+	0.915590i	−0.0811152	+	0.996705i	$0.525848\pi$
$728$	150220.	−	150220.i	0.283442	−	0.283442i
$729$		0			0
$730$	1.06932e6	+	328950.i	2.00660	+	0.617282i
$731$	21301.8			0.0398641
$732$		0			0
$733$	−79446.4	+	79446.4i	−0.147865	+	0.147865i	−0.777164	−	0.629298i	$-0.783342\pi$
$733$	−79446.4	+	79446.4i	−0.147865	+	0.147865i	0.629298	+	0.777164i	$0.283342\pi$
$734$		−	1.12998e6i		−	2.09739i
$735$		0			0
$736$	−510489.			−0.942390
$737$	−138643.	−	138643.i	−0.255249	−	0.255249i
$738$		0			0
$739$		−	885328.i		−	1.62112i	−0.585655	−	0.810561i	$-0.699163\pi$
$739$		−	885328.i		−	1.62112i	0.585655	−	0.810561i	$-0.300837\pi$
$740$	637010.	−	337290.i	1.16328	−	0.615942i
$741$		0			0
$742$	872975.	+	872975.i	1.58560	+	1.58560i
$743$	−693877.	+	693877.i	−1.25691	+	1.25691i	−0.304352	+	0.952560i	$0.598440\pi$
$743$	−693877.	+	693877.i	−1.25691	+	1.25691i	−0.952560	+	0.304352i	$0.901560\pi$
$744$		0			0
$745$	−301822.	+	981133.i	−0.543799	+	1.76773i
$746$	−169719.			−0.304967
$747$		0			0
$748$	22180.6	−	22180.6i	0.0396434	−	0.0396434i
$749$		−	29618.5i		−	0.0527959i
$750$		0			0
$751$	−1.03267e6			−1.83097			−0.915487	−	0.402347i	$-0.868194\pi$
$751$	−1.03267e6			−1.83097			−0.915487	+	0.402347i	$0.868194\pi$
$752$	194692.	+	194692.i	0.344281	+	0.344281i
$753$		0			0
$754$			173017.i			0.304331i
$755$	289044.	+	88917.5i	0.507073	+	0.155989i
$756$		0			0
$757$	−366937.	−	366937.i	−0.640323	−	0.640323i	0.310312	−	0.950635i	$-0.399567\pi$
$757$	−366937.	−	366937.i	−0.640323	−	0.640323i	−0.950635	+	0.310312i	$0.899567\pi$
$758$	254036.	−	254036.i	0.442137	−	0.442137i
$759$		0			0
$760$	486323.	+	918476.i	0.841972	+	1.59016i
$761$	721432.			1.24574			0.622868	−	0.782327i	$-0.285967\pi$
$761$	721432.			1.24574			0.622868	+	0.782327i	$0.285967\pi$
$762$		0			0
$763$	543867.	−	543867.i	0.934209	−	0.934209i
$764$			280969.i			0.481361i
$765$		0			0
$766$	−176909.			−0.301504
$767$	53946.5	+	53946.5i	0.0917007	+	0.0917007i
$768$		0			0
$769$		−	543034.i		−	0.918279i	−0.888364	−	0.459139i	$-0.848158\pi$
$769$		−	543034.i		−	0.918279i	0.888364	−	0.459139i	$-0.151842\pi$
$770$	115248.	−	374635.i	0.194379	−	0.631869i
$771$		0			0
$772$	−984913.	−	984913.i	−1.65258	−	1.65258i
$773$	501793.	−	501793.i	0.839780	−	0.839780i	−0.149050	−	0.988830i	$-0.547622\pi$
$773$	501793.	−	501793.i	0.839780	−	0.839780i	0.988830	+	0.149050i	$0.0476215\pi$
$774$		0			0
$775$	−696408.	−	473253.i	−1.15947	−	0.787934i
$776$	−2.31177e6			−3.83903
$777$		0			0
$778$	−31895.9	+	31895.9i	−0.0526958	+	0.0526958i
$779$			584675.i			0.963473i
$780$		0			0
$781$	229641.			0.376485
$782$	−91433.1	−	91433.1i	−0.149517	−	0.149517i
$783$		0			0
$784$		−	220827.i		−	0.359269i
$785$	−350478.	−	661918.i	−0.568750	−	1.07415i
$786$		0			0
$787$	−219605.	−	219605.i	−0.354562	−	0.354562i	0.507242	−	0.861804i	$-0.330665\pi$
$787$	−219605.	−	219605.i	−0.354562	−	0.354562i	−0.861804	+	0.507242i	$0.830665\pi$
$788$	−515293.	+	515293.i	−0.829854	+	0.829854i
$789$		0			0
$790$	−1.01581e6	+	537863.i	−1.62765	+	0.861821i
$791$	−971623.			−1.55290
$792$		0			0
$793$	−68381.7	+	68381.7i	−0.108741	+	0.108741i
$794$		−	1.45818e6i		−	2.31298i
$795$		0			0
$796$	133776.			0.211131
$797$	311150.	+	311150.i	0.489839	+	0.489839i	0.908255	−	0.418416i	$-0.137415\pi$
$797$	311150.	+	311150.i	0.489839	+	0.489839i	−0.418416	+	0.908255i	$0.637415\pi$
$798$		0			0
$799$			16025.8i			0.0251030i
$800$	−73804.0	−	386841.i	−0.115319	−	0.604439i
$801$		0			0
$802$	1.50351e6	+	1.50351e6i	2.33753	+	2.33753i
$803$	179931.	−	179931.i	0.279045	−	0.279045i
$804$		0			0
$805$	−1.05565e6	−	324746.i	−1.62903	−	0.501131i
$806$	−282723.			−0.435202
$807$		0			0
$808$	−17673.5	+	17673.5i	−0.0270707	+	0.0270707i
$809$		−	827176.i		−	1.26387i	−0.775023	−	0.631933i	$-0.782262\pi$
$809$		−	827176.i		−	1.26387i	0.775023	−	0.631933i	$-0.217738\pi$
$810$		0			0
$811$	1.22035e6			1.85542			0.927711	−	0.373300i	$-0.121774\pi$
$811$	1.22035e6			1.85542			0.927711	+	0.373300i	$0.121774\pi$
$812$	1.09877e6	+	1.09877e6i	1.66645	+	1.66645i
$813$		0			0
$814$		−	239834.i		−	0.361961i
$815$	123523.	−	65404.0i	0.185966	−	0.0984667i
$816$		0			0
$817$	211340.	+	211340.i	0.316619	+	0.316619i
$818$	87303.8	−	87303.8i	0.130475	−	0.130475i
$819$		0			0
$820$	471689.	−	1.53332e6i	0.701501	−	2.28037i
$821$	1.13223e6			1.67977			0.839884	−	0.542766i	$-0.182623\pi$
$821$	1.13223e6			1.67977			0.839884	+	0.542766i	$0.182623\pi$
$822$		0			0
$823$	504583.	−	504583.i	0.744961	−	0.744961i	−0.228567	−	0.973528i	$-0.573404\pi$
$823$	504583.	−	504583.i	0.744961	−	0.744961i	0.973528	+	0.228567i	$0.0734041\pi$
$824$			2.67531e6i			3.94022i
$825$		0			0
$826$	1.00237e6			1.46916
$827$	707633.	+	707633.i	1.03466	+	1.03466i	0.999377	+	0.0352807i	$0.0112325\pi$
$827$	707633.	+	707633.i	1.03466	+	1.03466i	0.0352807	+	0.999377i	$0.488767\pi$
$828$		0			0
$829$			846141.i			1.23121i	0.788053	+	0.615607i	$0.211089\pi$
$829$			846141.i			1.23121i	−0.788053	+	0.615607i	$0.788911\pi$
$830$	149313.	+	45932.6i	0.216741	+	0.0666752i
$831$		0			0
$832$	35258.5	+	35258.5i	0.0509352	+	0.0509352i
$833$	9088.50	−	9088.50i	0.0130979	−	0.0130979i
$834$		0			0
$835$	130103.	+	245714.i	0.186601	+	0.352417i
$836$	440117.			0.629732
$837$		0			0
$838$	182200.	−	182200.i	0.259454	−	0.259454i
$839$			499671.i			0.709839i	0.934897	+	0.354920i	$0.115492\pi$
$839$			499671.i			0.709839i	−0.934897	+	0.354920i	$0.884508\pi$
$840$		0			0
$841$	27590.7			0.0390096
$842$	−612924.	−	612924.i	−0.864534	−	0.864534i
$843$		0			0
$844$		−	610635.i		−	0.857229i
$845$	−203541.	+	661650.i	−0.285061	+	0.926648i
$846$		0			0
$847$	501508.	+	501508.i	0.699054	+	0.699054i
$848$	868184.	−	868184.i	1.20731	−	1.20731i
$849$		0			0
$850$	56067.7	−	82505.6i	0.0776023	−	0.114195i
$851$	−675806.			−0.933175
$852$		0			0
$853$	−175495.	+	175495.i	−0.241194	+	0.241194i	−0.817344	−	0.576150i	$-0.804554\pi$
$853$	−175495.	+	175495.i	−0.241194	+	0.241194i	0.576150	+	0.817344i	$0.304554\pi$
$854$			1.27059e6i			1.74217i
$855$		0			0
$856$	−71697.7			−0.0978492
$857$	−932907.	−	932907.i	−1.27021	−	1.27021i	−0.945976	−	0.324237i	$-0.894892\pi$
$857$	−932907.	−	932907.i	−1.27021	−	1.27021i	−0.324237	−	0.945976i	$-0.605108\pi$
$858$		0			0
$859$			318689.i			0.431897i	0.976405	+	0.215948i	$0.0692843\pi$
$859$			318689.i			0.431897i	−0.976405	+	0.215948i	$0.930716\pi$
$860$	−383742.	−	724741.i	−0.518851	−	0.979909i
$861$		0			0
$862$	−1.66012e6	−	1.66012e6i	−2.23422	−	2.23422i
$863$	834366.	−	834366.i	1.12030	−	1.12030i	0.128606	−	0.991696i	$-0.458950\pi$
$863$	834366.	−	834366.i	1.12030	−	1.12030i	0.991696	−	0.128606i	$-0.0410502\pi$
$864$		0			0
$865$	642504.	−	340199.i	0.858704	−	0.454674i
$866$	1.41247e6			1.88341
$867$		0			0
$868$	−1.79547e6	+	1.79547e6i	−2.38308	+	2.38308i
$869$			261433.i			0.346195i
$870$		0			0
$871$	−143116.			−0.188648
$872$	−1.31654e6	−	1.31654e6i	−1.73142	−	1.73142i
$873$		0			0
$874$		−	1.81426e6i		−	2.37507i
$875$	93466.6	−	846906.i	0.122079	−	1.10616i
$876$		0			0
$877$	−218162.	−	218162.i	−0.283648	−	0.283648i	0.550914	−	0.834562i	$-0.314279\pi$
$877$	−218162.	−	218162.i	−0.283648	−	0.283648i	−0.834562	+	0.550914i	$0.814279\pi$
$878$	−129714.	+	129714.i	−0.168267	+	0.168267i
$879$		0			0
$880$	−372579.	−	114615.i	−0.481120	−	0.148005i
$881$	279844.			0.360549			0.180274	−	0.983616i	$-0.442301\pi$
$881$	279844.			0.360549			0.180274	+	0.983616i	$0.442301\pi$
$882$		0			0
$883$	269641.	−	269641.i	0.345831	−	0.345831i	−0.512723	−	0.858554i	$-0.671363\pi$
$883$	269641.	−	269641.i	0.345831	−	0.345831i	0.858554	+	0.512723i	$0.171363\pi$
$884$		−	22896.2i		−	0.0292994i
$885$		0			0
$886$	−1.32517e6			−1.68812
$887$	−566987.	−	566987.i	−0.720653	−	0.720653i	0.248085	−	0.968738i	$-0.420199\pi$
$887$	−566987.	−	566987.i	−0.720653	−	0.720653i	−0.968738	+	0.248085i	$0.920199\pi$
$888$		0			0
$889$			845027.i			1.06922i
$890$	−291710.	+	154457.i	−0.368274	+	0.194997i
$891$		0			0
$892$	370867.	+	370867.i	0.466110	+	0.466110i
$893$	−158995.	+	158995.i	−0.199380	+	0.199380i
$894$		0			0
$895$	85108.9	−	276663.i	0.106250	−	0.345387i
$896$	1.20490e6			1.50084
$897$		0			0
$898$	671758.	−	671758.i	0.833029	−	0.833029i
$899$		−	1.11067e6i		−	1.37425i
$900$		0			0
$901$	71463.1			0.0880303
$902$	−377442.	−	377442.i	−0.463913	−	0.463913i
$903$		0			0
$904$			2.35201e6i			2.87808i
$905$	368228.	+	113276.i	0.449593	+	0.138306i
$906$		0			0
$907$	256644.	+	256644.i	0.311973	+	0.311973i	0.845674	−	0.533700i	$-0.179199\pi$
$907$	256644.	+	256644.i	0.311973	+	0.311973i	−0.533700	+	0.845674i	$0.679199\pi$
$908$	−1.18208e6	+	1.18208e6i	−1.43375	+	1.43375i
$909$		0			0
$910$	−133878.	−	252843.i	−0.161668	−	0.305329i
$911$	143826.			0.173301			0.0866506	−	0.996239i	$-0.472384\pi$
$911$	143826.			0.173301			0.0866506	+	0.996239i	$0.472384\pi$
$912$		0			0
$913$	25124.4	−	25124.4i	0.0301408	−	0.0301408i
$914$		−	1.48312e6i		−	1.77535i
$915$		0			0
$916$	2.53801e6			3.02485
$917$	419249.	+	419249.i	0.498578	+	0.498578i
$918$		0			0
$919$		−	993411.i		−	1.17625i	−0.808772	−	0.588123i	$-0.799867\pi$
$919$		−	993411.i		−	1.17625i	0.808772	−	0.588123i	$-0.200133\pi$
$920$	−786113.	+	2.55541e6i	−0.928772	+	3.01916i
$921$		0			0
$922$	−1.03869e6	−	1.03869e6i	−1.22187	−	1.22187i
$923$	118525.	−	118525.i	0.139125	−	0.139125i
$924$		0			0
$925$	−97704.7	−	512116.i	−0.114191	−	0.598528i
$926$	−1.57582e6			−1.83775
$927$		0			0
$928$	367330.	−	367330.i	0.426541	−	0.426541i
$929$			1.39909e6i			1.62112i	0.585658	+	0.810558i	$0.300836\pi$
$929$			1.39909e6i			1.62112i	−0.585658	+	0.810558i	$0.699164\pi$
$930$		0			0
$931$	180338.			0.208059
$932$	−629168.	−	629168.i	−0.724327	−	0.724327i
$933$		0			0
$934$			1.00570e6i			1.15286i
$935$	−10616.9	−	20051.3i	−0.0121444	−	0.0229361i
$936$		0			0
$937$	482443.	+	482443.i	0.549499	+	0.549499i	0.926296	−	0.376797i	$-0.122975\pi$
$937$	482443.	+	482443.i	0.549499	+	0.549499i	−0.376797	+	0.926296i	$0.622975\pi$
$938$	−1.32961e6	+	1.32961e6i	−1.51119	+	1.51119i
$939$		0			0
$940$	545237.	−	288697.i	0.617063	−	0.326728i
$941$	−494897.			−0.558902			−0.279451	−	0.960160i	$-0.590152\pi$
$941$	−494897.			−0.558902			−0.279451	+	0.960160i	$0.590152\pi$
$942$		0			0
$943$	−1.06356e6	+	1.06356e6i	−1.19602	+	1.19602i
$944$		−	996871.i		−	1.11865i
$945$		0			0
$946$	−272864.			−0.304905
$947$	1.05460e6	+	1.05460e6i	1.17595	+	1.17595i	0.980767	+	0.195183i	$0.0625300\pi$
$947$	1.05460e6	+	1.05460e6i	1.17595	+	1.17595i	0.195183	+	0.980767i	$0.437470\pi$
$948$		0			0
$949$		−	185735.i		−	0.206235i
$950$	1.37482e6	−	262296.i	1.52334	−	0.290633i
$951$		0			0
$952$	−114246.	−	114246.i	−0.126058	−	0.126058i
$953$	319054.	−	319054.i	0.351300	−	0.351300i	−0.509293	−	0.860593i	$-0.670093\pi$
$953$	319054.	−	319054.i	0.351300	−	0.351300i	0.860593	+	0.509293i	$0.170093\pi$
$954$		0			0
$955$	194242.	+	59753.9i	0.212979	+	0.0655178i
$956$	3.55229e6			3.88681
$957$		0			0
$958$	−627599.	+	627599.i	−0.683835	+	0.683835i
$959$		−	261221.i		−	0.284034i
$960$		0			0
$961$	891397.			0.965216
$962$	−123785.	−	123785.i	−0.133758	−	0.133758i
$963$		0			0
$964$			2.39885e6i			2.58136i
$965$	−890361.	+	471436.i	−0.956118	+	0.506254i
$966$		0			0
$967$	−1.01810e6	−	1.01810e6i	−1.08877	−	1.08877i	−0.995655	−	0.0931182i	$-0.970317\pi$
$967$	−1.01810e6	−	1.01810e6i	−1.08877	−	1.08877i	−0.0931182	−	0.995655i	$-0.529683\pi$
$968$	1.21400e6	−	1.21400e6i	1.29559	−	1.29559i
$969$		0			0
$970$	−915395.	+	2.97567e6i	−0.972893	+	3.16258i
$971$	−610675.			−0.647696			−0.323848	−	0.946109i	$-0.604977\pi$
$971$	−610675.			−0.647696			−0.323848	+	0.946109i	$0.604977\pi$
$972$		0			0
$973$	−235061.	+	235061.i	−0.248287	+	0.248287i
$974$		−	1.06762e6i		−	1.12538i
$975$		0			0
$976$	1.26362e6			1.32653
$977$	90031.7	+	90031.7i	0.0943206	+	0.0943206i	0.752693	−	0.658372i	$-0.228755\pi$
$977$	90031.7	+	90031.7i	0.0943206	+	0.0943206i	−0.658372	+	0.752693i	$0.728755\pi$
$978$		0			0
$979$			75075.1i			0.0783305i
$980$	−472939.	−	145488.i	−0.492439	−	0.151487i
$981$		0			0
$982$	−930662.	−	930662.i	−0.965093	−	0.965093i
$983$	−406452.	+	406452.i	−0.420632	+	0.420632i	−0.885421	−	0.464790i	$-0.846130\pi$
$983$	−406452.	+	406452.i	−0.420632	+	0.420632i	0.464790	+	0.885421i	$0.346130\pi$
$984$		0			0
$985$	246649.	+	465824.i	0.254218	+	0.480120i
$986$	131584.			0.135347
$987$		0			0
$988$	227158.	−	227158.i	0.232709	−	0.232709i
$989$			768879.i			0.786078i
$990$		0			0
$991$	−563990.			−0.574281			−0.287140	−	0.957889i	$-0.592705\pi$
$991$	−563990.			−0.574281			−0.287140	+	0.957889i	$0.592705\pi$
$992$	600246.	+	600246.i	0.609967	+	0.609967i
$993$		0			0
$994$		−	2.20229e6i		−	2.22896i
$995$	28450.3	−	92483.3i	0.0287369	−	0.0934151i
$996$		0			0
$997$	49988.1	+	49988.1i	0.0502894	+	0.0502894i	0.731804	−	0.681515i	$-0.238679\pi$
$997$	49988.1	+	49988.1i	0.0502894	+	0.0502894i	−0.681515	+	0.731804i	$0.738679\pi$
$998$	−2.42596e6	+	2.42596e6i	−2.43569	+	2.43569i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	45.5.g.e.37.4		8
3.2	odd	2		15.5.f.a.7.1	✓	8
5.2	odd	4		225.5.g.m.118.1		8
5.3	odd	4	inner	45.5.g.e.28.4		8
5.4	even	2		225.5.g.m.82.1		8
12.11	even	2		240.5.bg.c.97.3		8
15.2	even	4		75.5.f.e.43.4		8
15.8	even	4		15.5.f.a.13.1	yes	8
15.14	odd	2		75.5.f.e.7.4		8
60.23	odd	4		240.5.bg.c.193.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
15.5.f.a.7.1	✓	8	3.2	odd	2
15.5.f.a.13.1	yes	8	15.8	even	4
45.5.g.e.28.4		8	5.3	odd	4	inner
45.5.g.e.37.4		8	1.1	even	1	trivial
75.5.f.e.7.4		8	15.14	odd	2
75.5.f.e.43.4		8	15.2	even	4
225.5.g.m.82.1		8	5.4	even	2
225.5.g.m.118.1		8	5.2	odd	4
240.5.bg.c.97.3		8	12.11	even	2
240.5.bg.c.193.3		8	60.23	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 45 = 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	45.g (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(5.02811 + 5.02811i) q^{2} +34.5637i q^{4} +(23.8949 + 7.35070i) q^{5} +(-38.5593 - 38.5593i) q^{7} +(-93.3405 + 93.3405i) q^{8} +O(q^{10})\)	\(q+(5.02811 + 5.02811i) q^{2} +34.5637i q^{4} +(23.8949 + 7.35070i) q^{5} +(-38.5593 - 38.5593i) q^{7} +(-93.3405 + 93.3405i) q^{8} +(83.1861 + 157.106i) q^{10} +40.4333 q^{11} +(20.8688 - 20.8688i) q^{13} -387.760i q^{14} -385.633 q^{16} +(-15.8713 - 15.8713i) q^{17} -314.926i q^{19} +(-254.068 + 825.898i) q^{20} +(203.303 + 203.303i) q^{22} +(572.869 - 572.869i) q^{23} +(516.934 + 351.289i) q^{25} +209.862 q^{26} +(1332.75 - 1332.75i) q^{28} +824.433i q^{29} -1347.19 q^{31} +(-445.555 - 445.555i) q^{32} -159.606i q^{34} +(-637.933 - 1204.81i) q^{35} +(-589.843 - 589.843i) q^{37} +(1583.48 - 1583.48i) q^{38} +(-2916.48 + 1544.25i) q^{40} -1856.55 q^{41} +(-671.078 + 671.078i) q^{43} +1397.53i q^{44} +5760.90 q^{46} +(-504.865 - 504.865i) q^{47} +572.636i q^{49} +(832.883 + 4365.52i) q^{50} +(721.305 + 721.305i) q^{52} +(-2251.32 + 2251.32i) q^{53} +(966.151 + 297.213i) q^{55} +7198.29 q^{56} +(-4145.34 + 4145.34i) q^{58} +2585.03i q^{59} -3276.74 q^{61} +(-6773.81 - 6773.81i) q^{62} +1689.53i q^{64} +(652.060 - 345.259i) q^{65} +(-3428.94 - 3428.94i) q^{67} +(548.573 - 548.573i) q^{68} +(2850.31 - 9265.50i) q^{70} +5679.51 q^{71} +(4450.06 - 4450.06i) q^{73} -5931.59i q^{74} +10885.0 q^{76} +(-1559.08 - 1559.08i) q^{77} +6465.77i q^{79} +(-9214.66 - 2834.67i) q^{80} +(-9334.92 - 9334.92i) q^{82} +(621.380 - 621.380i) q^{83} +(-262.579 - 495.910i) q^{85} -6748.50 q^{86} +(-3774.07 + 3774.07i) q^{88} +1856.76i q^{89} -1609.37 q^{91} +(19800.5 + 19800.5i) q^{92} -5077.03i q^{94} +(2314.93 - 7525.13i) q^{95} +(12383.5 + 12383.5i) q^{97} +(-2879.27 + 2879.27i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 84 q^{5} + 20 q^{7} - 180 q^{8}+O(q^{10}) \) 8 * q + 84 * q^5 + 20 * q^7 - 180 * q^8	\( 8 q + 84 q^{5} + 20 q^{7} - 180 q^{8} + 104 q^{10} + 288 q^{11} - 340 q^{13} + 620 q^{16} - 900 q^{17} - 564 q^{20} - 1100 q^{22} + 1560 q^{23} - 1204 q^{25} + 3024 q^{26} + 3580 q^{28} - 512 q^{31} - 4980 q^{32} - 6600 q^{35} - 3820 q^{37} + 7680 q^{38} - 2952 q^{40} + 2712 q^{41} - 1240 q^{43} + 13528 q^{46} - 4800 q^{47} - 3744 q^{50} - 1240 q^{52} - 1020 q^{53} - 3644 q^{55} + 30720 q^{56} + 2340 q^{58} - 4760 q^{61} - 28680 q^{62} + 1212 q^{65} - 8920 q^{67} + 1920 q^{68} + 7380 q^{70} - 7536 q^{71} + 11600 q^{73} + 4344 q^{76} + 360 q^{77} - 10644 q^{80} - 27200 q^{82} + 32400 q^{83} - 15628 q^{85} - 14592 q^{86} - 14340 q^{88} + 16528 q^{91} + 31800 q^{92} - 18864 q^{95} + 58640 q^{97} - 46440 q^{98}+O(q^{100}) \) 8 * q + 84 * q^5 + 20 * q^7 - 180 * q^8 + 104 * q^10 + 288 * q^11 - 340 * q^13 + 620 * q^16 - 900 * q^17 - 564 * q^20 - 1100 * q^22 + 1560 * q^23 - 1204 * q^25 + 3024 * q^26 + 3580 * q^28 - 512 * q^31 - 4980 * q^32 - 6600 * q^35 - 3820 * q^37 + 7680 * q^38 - 2952 * q^40 + 2712 * q^41 - 1240 * q^43 + 13528 * q^46 - 4800 * q^47 - 3744 * q^50 - 1240 * q^52 - 1020 * q^53 - 3644 * q^55 + 30720 * q^56 + 2340 * q^58 - 4760 * q^61 - 28680 * q^62 + 1212 * q^65 - 8920 * q^67 + 1920 * q^68 + 7380 * q^70 - 7536 * q^71 + 11600 * q^73 + 4344 * q^76 + 360 * q^77 - 10644 * q^80 - 27200 * q^82 + 32400 * q^83 - 15628 * q^85 - 14592 * q^86 - 14340 * q^88 + 16528 * q^91 + 31800 * q^92 - 18864 * q^95 + 58640 * q^97 - 46440 * q^98

Introduction

L-functions

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Database

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