LMFDB - Embedded newform 45.7.d.a.44.9

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [45,7,Mod(44,45)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(45, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1])) N = Newforms(chi, 7, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("45.44"); S:= CuspForms(chi, 7); N := Newforms(S);

Level:	$ N $	$=$	$ 45 = 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 7 $
Character orbit:	$[\chi]$	$=$	45.d (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)

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

Self dual:	no
Analytic conductor:	$10.3524337629$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 630x^{10} + 143853x^{8} - 14514820x^{6} + 700911828x^{4} - 15238290240x^{2} + 141093389376 $ x^12 - 630x^10 + 143853x^8 - 14514820x^6 + 700911828x^4 - 15238290240*x^2 + 141093389376
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4}\cdot 3^{20}\cdot 5^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			44.9
Root			$8.37795 + 1.41421i$ of defining polynomial
Character	$\chi$	$=$	45.44
Dual form			45.7.d.a.44.10

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+8.37795 q^{2} +6.18997 q^{4} +(-60.3283 - 109.478i) q^{5} -270.176i q^{7} -484.329 q^{8} +(-505.427 - 917.203i) q^{10} -1187.10i q^{11} -526.105i q^{13} -2263.52i q^{14} -4453.84 q^{16} +6589.09 q^{17} -9088.18 q^{19} +(-373.430 - 677.668i) q^{20} -9945.48i q^{22} +8922.54 q^{23} +(-8346.00 + 13209.3i) q^{25} -4407.68i q^{26} -1672.38i q^{28} -7731.97i q^{29} +3114.29 q^{31} -6316.98 q^{32} +55203.1 q^{34} +(-29578.4 + 16299.2i) q^{35} -75996.2i q^{37} -76140.3 q^{38} +(29218.7 + 53023.5i) q^{40} +56742.3i q^{41} +126636. i q^{43} -7348.13i q^{44} +74752.5 q^{46} +182063. q^{47} +44654.2 q^{49} +(-69922.3 + 110667. i) q^{50} -3256.57i q^{52} +31815.2 q^{53} +(-129962. + 71615.8i) q^{55} +130854. i q^{56} -64778.1i q^{58} -319642. i q^{59} -359529. q^{61} +26091.3 q^{62} +232123. q^{64} +(-57597.1 + 31739.0i) q^{65} -487383. i q^{67} +40786.3 q^{68} +(-247806. + 136554. i) q^{70} -370686. i q^{71} -323083. i q^{73} -636692. i q^{74} -56255.6 q^{76} -320726. q^{77} -66000.9 q^{79} +(268693. + 487599. i) q^{80} +475384. i q^{82} +7403.28 q^{83} +(-397509. - 721363. i) q^{85} +1.06095e6i q^{86} +574948. i q^{88} -991440. i q^{89} -142141. q^{91} +55230.3 q^{92} +1.52532e6 q^{94} +(548274. + 994959. i) q^{95} +1.52751e6i q^{97} +374110. q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 516 q^{4} - 1368 q^{10} + 36372 q^{16} + 4320 q^{19} - 63384 q^{25} + 60192 q^{31} + 106296 q^{34} - 221772 q^{40} - 1078968 q^{46} + 711516 q^{49} - 104112 q^{55} - 449784 q^{61} + 3964572 q^{64}+ \cdots + 5793408 q^{94}+O(q^{100}) $ 12 * q + 516 * q^4 - 1368 * q^10 + 36372 * q^16 + 4320 * q^19 - 63384 * q^25 + 60192 * q^31 + 106296 * q^34 - 221772 * q^40 - 1078968 * q^46 + 711516 * q^49 - 104112 * q^55 - 449784 * q^61 + 3964572 * q^64 - 3326616 * q^70 - 584400 * q^76 + 4324608 * q^79 - 3305772 * q^85 + 631152 * q^91 + 5793408 * q^94

Character values

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

$n$	$11$	$37$
$\chi(n)$	$-1$	$-1$

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 $

$2$	8.37795			1.04724			0.523622	−	0.851951i	$-0.324581\pi$
$2$	8.37795			1.04724			0.523622	+	0.851951i	$0.324581\pi$
$3$		0			0
$3$		0			0
$4$	6.18997			0.0967183
$5$	−60.3283	−	109.478i	−0.482626	−	0.875826i
$5$	−60.3283	−	109.478i	−0.482626	−	0.875826i
$6$		0			0
$7$		−	270.176i		−	0.787684i	−0.919178	−	0.393842i	$-0.871146\pi$
$7$		−	270.176i		−	0.787684i	0.919178	−	0.393842i	$-0.128854\pi$
$8$	−484.329			−0.945956
$9$		0			0
$10$	−505.427	−	917.203i	−0.505427	−	0.917203i
$11$		−	1187.10i		−	0.891888i	−0.895061	−	0.445944i	$-0.852868\pi$
$11$		−	1187.10i		−	0.891888i	0.895061	−	0.445944i	$-0.147132\pi$
$12$		0			0
$13$		−	526.105i		−	0.239465i	−0.992806	−	0.119733i	$-0.961796\pi$
$13$		−	526.105i		−	0.239465i	0.992806	−	0.119733i	$-0.0382037\pi$
$14$		−	2263.52i		−	0.824896i
$15$		0			0
$16$	−4453.84			−1.08736
$17$	6589.09			1.34115			0.670577	−	0.741840i	$-0.266046\pi$
$17$	6589.09			1.34115			0.670577	+	0.741840i	$0.266046\pi$
$18$		0			0
$19$	−9088.18			−1.32500			−0.662500	−	0.749062i	$-0.730505\pi$
$19$	−9088.18			−1.32500			−0.662500	+	0.749062i	$0.730505\pi$
$20$	−373.430	−	677.668i	−0.0466788	−	0.0847085i
$21$		0			0
$22$		−	9945.48i		−	0.934023i
$23$	8922.54			0.733339			0.366670	−	0.930351i	$-0.380498\pi$
$23$	8922.54			0.733339			0.366670	+	0.930351i	$0.380498\pi$
$24$		0			0
$25$	−8346.00	+	13209.3i	−0.534144	+	0.845394i
$26$		−	4407.68i		−	0.250778i
$27$		0			0
$28$		−	1672.38i		−	0.0761834i
$29$		−	7731.97i		−	0.317027i	−0.987357	−	0.158514i	$-0.949330\pi$
$29$		−	7731.97i		−	0.317027i	0.987357	−	0.158514i	$-0.0506702\pi$
$30$		0			0
$31$	3114.29			0.104538			0.0522689	−	0.998633i	$-0.483355\pi$
$31$	3114.29			0.104538			0.0522689	+	0.998633i	$0.483355\pi$
$32$	−6316.98			−0.192779
$33$		0			0
$34$	55203.1			1.40451
$35$	−29578.4	+	16299.2i	−0.689874	+	0.380157i
$36$		0			0
$37$		−	75996.2i		−	1.50033i	−0.661251	−	0.750165i	$-0.729974\pi$
$37$		−	75996.2i		−	1.50033i	0.661251	−	0.750165i	$-0.270026\pi$
$38$	−76140.3			−1.38760
$39$		0			0
$40$	29218.7	+	53023.5i	0.456543	+	0.828493i
$41$			56742.3i			0.823295i	0.911343	+	0.411648i	$0.135047\pi$
$41$			56742.3i			0.823295i	−0.911343	+	0.411648i	$0.864953\pi$
$42$		0			0
$43$			126636.i			1.59277i	0.604790	+	0.796385i	$0.293257\pi$
$43$			126636.i			1.59277i	−0.604790	+	0.796385i	$0.706743\pi$
$44$		−	7348.13i		−	0.0862619i
$45$		0			0
$46$	74752.5			0.767984
$47$	182063.			1.75359			0.876797	−	0.480861i	$-0.159676\pi$
$47$	182063.			1.75359			0.876797	+	0.480861i	$0.159676\pi$
$48$		0			0
$49$	44654.2			0.379554
$50$	−69922.3	+	110667.i	−0.559379	+	0.885333i
$51$		0			0
$52$		−	3256.57i		−	0.0231607i
$53$	31815.2			0.213701			0.106851	−	0.994275i	$-0.465923\pi$
$53$	31815.2			0.213701			0.106851	+	0.994275i	$0.465923\pi$
$54$		0			0
$55$	−129962.	+	71615.8i	−0.781139	+	0.430448i
$56$			130854.i			0.745114i
$57$		0			0
$58$		−	64778.1i		−	0.332005i
$59$		−	319642.i		−	1.55635i	−0.628048	−	0.778175i	$-0.716146\pi$
$59$		−	319642.i		−	1.55635i	0.628048	−	0.778175i	$-0.283854\pi$
$60$		0			0
$61$	−359529.			−1.58396			−0.791981	−	0.610546i	$-0.790950\pi$
$61$	−359529.			−1.58396			−0.791981	+	0.610546i	$0.790950\pi$
$62$	26091.3			0.109476
$63$		0			0
$64$	232123.			0.885478
$65$	−57597.1	+	31739.0i	−0.209730	+	0.115572i
$66$		0			0
$67$		−	487383.i		−	1.62049i	−0.586093	−	0.810244i	$-0.699335\pi$
$67$		−	487383.i		−	1.62049i	0.586093	−	0.810244i	$-0.300665\pi$
$68$	40786.3			0.129714
$69$		0			0
$70$	−247806.	+	136554.i	−0.722466	+	0.398117i
$71$		−	370686.i		−	1.03569i	−0.855473	−	0.517847i	$-0.826734\pi$
$71$		−	370686.i		−	1.03569i	0.855473	−	0.517847i	$-0.173266\pi$
$72$		0			0
$73$		−	323083.i		−	0.830511i	−0.909705	−	0.415255i	$-0.863692\pi$
$73$		−	323083.i		−	0.830511i	0.909705	−	0.415255i	$-0.136308\pi$
$74$		−	636692.i		−	1.57121i
$75$		0			0
$76$	−56255.6			−0.128152
$77$	−320726.			−0.702525
$78$		0			0
$79$	−66000.9			−0.133866			−0.0669328	−	0.997757i	$-0.521321\pi$
$79$	−66000.9			−0.133866			−0.0669328	+	0.997757i	$0.521321\pi$
$80$	268693.	+	487599.i	0.524790	+	0.952342i
$81$		0			0
$82$			475384.i			0.862190i
$83$	7403.28			0.0129476			0.00647381	−	0.999979i	$-0.497939\pi$
$83$	7403.28			0.0129476			0.00647381	+	0.999979i	$0.497939\pi$
$84$		0			0
$85$	−397509.	−	721363.i	−0.647276	−	1.17462i
$86$			1.06095e6i			1.66802i
$87$		0			0
$88$			574948.i			0.843686i
$89$		−	991440.i		−	1.40636i	−0.711012	−	0.703180i	$-0.751763\pi$
$89$		−	991440.i		−	1.40636i	0.711012	−	0.703180i	$-0.248237\pi$
$90$		0			0
$91$	−142141.			−0.188623
$92$	55230.3			0.0709273
$93$		0			0
$94$	1.52532e6			1.83644
$95$	548274.	+	994959.i	0.639480	+	1.16047i
$96$		0			0
$97$			1.52751e6i			1.67367i	0.547458	+	0.836833i	$0.315596\pi$
$97$			1.52751e6i			1.67367i	−0.547458	+	0.836833i	$0.684404\pi$
$98$	374110.			0.397486
$99$		0			0
$100$	−51661.5	+	81765.0i	−0.0516615	+	0.0817650i
$101$			452341.i			0.439037i	0.975608	+	0.219519i	$0.0704487\pi$
$101$			452341.i			0.439037i	−0.975608	+	0.219519i	$0.929551\pi$
$102$		0			0
$103$		−	104961.i		−	0.0960542i	−0.998846	−	0.0480271i	$-0.984707\pi$
$103$		−	104961.i		−	0.0960542i	0.998846	−	0.0480271i	$-0.0152934\pi$
$104$			254808.i			0.226523i
$105$		0			0
$106$	266546.			0.223797
$107$	−657099.			−0.536388			−0.268194	−	0.963365i	$-0.586427\pi$
$107$	−657099.			−0.536388			−0.268194	+	0.963365i	$0.586427\pi$
$108$		0			0
$109$	1.65657e6			1.27918			0.639589	−	0.768717i	$-0.279105\pi$
$109$	1.65657e6			1.27918			0.639589	+	0.768717i	$0.279105\pi$
$110$	−1.08881e6	+	599994.i	−0.818042	+	0.450784i
$111$		0			0
$112$			1.20332e6i			0.856499i
$113$	−1.02621e6			−0.711216			−0.355608	−	0.934635i	$-0.615726\pi$
$113$	−1.02621e6			−0.711216			−0.355608	+	0.934635i	$0.615726\pi$
$114$		0			0
$115$	−538281.	−	976824.i	−0.353929	−	0.642278i
$116$		−	47860.7i		−	0.0306623i
$117$		0			0
$118$		−	2.67794e6i		−	1.62988i
$119$		−	1.78021e6i		−	1.05641i
$120$		0			0
$121$	362349.			0.204536
$122$	−3.01212e6			−1.65879
$123$		0			0
$124$	19277.3			0.0101107
$125$	1.94963e6	+	116813.i	0.998210	+	0.0598083i
$126$		0			0
$127$		−	375925.i		−	0.183523i	−0.995781	−	0.0917615i	$-0.970750\pi$
$127$		−	375925.i		−	0.183523i	0.995781	−	0.0917615i	$-0.0292497\pi$
$128$	2.34900e6			1.12009
$129$		0			0
$130$	−482545.	+	265908.i	−0.219638	+	0.121032i
$131$			1.42739e6i			0.634934i	0.948269	+	0.317467i	$0.102832\pi$
$131$			1.42739e6i			0.634934i	−0.948269	+	0.317467i	$0.897168\pi$
$132$		0			0
$133$			2.45540e6i			1.04368i
$134$		−	4.08327e6i		−	1.69705i
$135$		0			0
$136$	−3.19129e6			−1.26867
$137$	2.76280e6			1.07445			0.537226	−	0.843438i	$-0.319472\pi$
$137$	2.76280e6			1.07445			0.537226	+	0.843438i	$0.319472\pi$
$138$		0			0
$139$	−2.33540e6			−0.869596			−0.434798	−	0.900528i	$-0.643180\pi$
$139$	−2.33540e6			−0.869596			−0.434798	+	0.900528i	$0.643180\pi$
$140$	−183089.	+	100892.i	−0.0667235	+	0.0367681i
$141$		0			0
$142$		−	3.10559e6i		−	1.08462i
$143$	−624540.			−0.213576
$144$		0			0
$145$	−846484.	+	466457.i	−0.277661	+	0.153006i
$146$		−	2.70677e6i		−	0.869747i
$147$		0			0
$148$		−	470414.i		−	0.145109i
$149$			5.57895e6i			1.68653i	0.537499	+	0.843265i	$0.319369\pi$
$149$			5.57895e6i			1.68653i	−0.537499	+	0.843265i	$0.680631\pi$
$150$		0			0
$151$	−187475.			−0.0544520			−0.0272260	−	0.999629i	$-0.508667\pi$
$151$	−187475.			−0.0544520			−0.0272260	+	0.999629i	$0.508667\pi$
$152$	4.40167e6			1.25339
$153$		0			0
$154$	−2.68703e6			−0.735715
$155$	−187879.	−	340947.i	−0.0504527	−	0.0915570i
$156$		0			0
$157$		−	2.11041e6i		−	0.545340i	−0.962108	−	0.272670i	$-0.912093\pi$
$157$		−	2.11041e6i		−	0.545340i	0.962108	−	0.272670i	$-0.0879066\pi$
$158$	−552952.			−0.140190
$159$		0			0
$160$	381092.	+	691572.i	0.0930401	+	0.168841i
$161$		−	2.41065e6i		−	0.577639i
$162$		0			0
$163$			4.41451e6i			1.01934i	0.860370	+	0.509671i	$0.170233\pi$
$163$			4.41451e6i			1.01934i	−0.860370	+	0.509671i	$0.829767\pi$
$164$			351233.i			0.0796277i
$165$		0			0
$166$	62024.3			0.0135593
$167$	279043.			0.0599131			0.0299565	−	0.999551i	$-0.490463\pi$
$167$	279043.			0.0599131			0.0299565	+	0.999551i	$0.490463\pi$
$168$		0			0
$169$	4.55002e6			0.942656
$170$	−3.33031e6	−	6.04354e6i	−0.677856	−	1.23011i
$171$		0			0
$172$			783876.i			0.154050i
$173$	−4.97675e6			−0.961187			−0.480594	−	0.876943i	$-0.659579\pi$
$173$	−4.97675e6			−0.961187			−0.480594	+	0.876943i	$0.659579\pi$
$174$		0			0
$175$	3.56882e6	+	2.25488e6i	0.665903	+	0.420736i
$176$			5.28717e6i			0.969806i
$177$		0			0
$178$		−	8.30623e6i		−	1.47280i
$179$			3.25828e6i			0.568106i	0.958809	+	0.284053i	$0.0916791\pi$
$179$			3.25828e6i			0.568106i	−0.958809	+	0.284053i	$0.908321\pi$
$180$		0			0
$181$	1.15053e7			1.94027			0.970137	−	0.242559i	$-0.0779869\pi$
$181$	1.15053e7			1.94027			0.970137	+	0.242559i	$0.0779869\pi$
$182$	−1.19085e6			−0.197534
$183$		0			0
$184$	−4.32145e6			−0.693706
$185$	−8.31994e6	+	4.58472e6i	−1.31403	+	0.724098i
$186$		0			0
$187$		−	7.82193e6i		−	1.19616i
$188$	1.12697e6			0.169605
$189$		0			0
$190$	4.59341e6	+	8.33571e6i	0.669691	+	1.21530i
$191$		−	6.07903e6i		−	0.872437i	−0.899841	−	0.436218i	$-0.856317\pi$
$191$		−	6.07903e6i		−	0.872437i	0.899841	−	0.436218i	$-0.143683\pi$
$192$		0			0
$193$			1.12931e7i			1.57088i	0.618941	+	0.785438i	$0.287562\pi$
$193$			1.12931e7i			1.57088i	−0.618941	+	0.785438i	$0.712438\pi$
$194$			1.27974e7i			1.75274i
$195$		0			0
$196$	276408.			0.0367099
$197$	3.46877e6			0.453708			0.226854	−	0.973929i	$-0.427156\pi$
$197$	3.46877e6			0.453708			0.226854	+	0.973929i	$0.427156\pi$
$198$		0			0
$199$	−5.80303e6			−0.736369			−0.368185	−	0.929753i	$-0.620021\pi$
$199$	−5.80303e6			−0.736369			−0.368185	+	0.929753i	$0.620021\pi$
$200$	4.04221e6	−	6.39764e6i	0.505276	−	0.799705i
$201$		0			0
$202$			3.78969e6i			0.459779i
$203$	−2.08899e6			−0.249717
$204$		0			0
$205$	6.21205e6	−	3.42317e6i	0.721064	−	0.397344i
$206$		−	879358.i		−	0.100592i
$207$		0			0
$208$			2.34319e6i			0.260386i
$209$			1.07886e7i			1.18175i
$210$		0			0
$211$	−1.31986e6			−0.140502			−0.0702509	−	0.997529i	$-0.522380\pi$
$211$	−1.31986e6			−0.140502			−0.0702509	+	0.997529i	$0.522380\pi$
$212$	196935.			0.0206688
$213$		0			0
$214$	−5.50514e6			−0.561729
$215$	1.38639e7	−	7.63975e6i	1.39499	−	0.768712i
$216$		0			0
$217$		−	841404.i		−	0.0823427i
$218$	1.38787e7			1.33961
$219$		0			0
$220$	−804461.	+	443300.i	−0.0755504	+	0.0416322i
$221$		−	3.46655e6i		−	0.321160i
$222$		0			0
$223$		−	1.33313e6i		−	0.120215i	−0.998192	−	0.0601076i	$-0.980856\pi$
$223$		−	1.33313e6i		−	0.120215i	0.998192	−	0.0601076i	$-0.0191444\pi$
$224$			1.70669e6i			0.151849i
$225$		0			0
$226$	−8.59754e6			−0.744816
$227$	−7.97225e6			−0.681559			−0.340779	−	0.940143i	$-0.610691\pi$
$227$	−7.97225e6			−0.681559			−0.340779	+	0.940143i	$0.610691\pi$
$228$		0			0
$229$	−1.11916e7			−0.931938			−0.465969	−	0.884801i	$-0.654294\pi$
$229$	−1.11916e7			−0.931938			−0.465969	+	0.884801i	$0.654294\pi$
$230$	−4.50969e6	−	8.18378e6i	−0.370649	−	0.672621i
$231$		0			0
$232$			3.74482e6i			0.299894i
$233$	−1.33843e7			−1.05810			−0.529052	−	0.848589i	$-0.677452\pi$
$233$	−1.33843e7			−1.05810			−0.529052	+	0.848589i	$0.677452\pi$
$234$		0			0
$235$	−1.09836e7	−	1.99320e7i	−0.846330	−	1.53584i
$236$		−	1.97857e6i		−	0.150528i
$237$		0			0
$238$		−	1.49145e7i		−	1.10631i
$239$		−	3.17061e6i		−	0.232247i	−0.993235	−	0.116123i	$-0.962953\pi$
$239$		−	3.17061e6i		−	0.232247i	0.993235	−	0.116123i	$-0.0370468\pi$
$240$		0			0
$241$	3.36319e6			0.240270			0.120135	−	0.992758i	$-0.461667\pi$
$241$	3.36319e6			0.240270			0.120135	+	0.992758i	$0.461667\pi$
$242$	3.03574e6			0.214199
$243$		0			0
$244$	−2.22548e6			−0.153198
$245$	−2.69391e6	−	4.88867e6i	−0.183183	−	0.332424i
$246$		0			0
$247$			4.78134e6i			0.317291i
$248$	−1.50834e6			−0.0988881
$249$		0			0
$250$	1.63339e7	+	978654.i	1.04537	+	0.0626339i
$251$		−	2.64563e7i		−	1.67305i	−0.547932	−	0.836523i	$-0.684585\pi$
$251$		−	2.64563e7i		−	1.67305i	0.547932	−	0.836523i	$-0.315415\pi$
$252$		0			0
$253$		−	1.05920e7i		−	0.654056i
$254$		−	3.14948e6i		−	0.192193i
$255$		0			0
$256$	4.82392e6			0.287528
$257$	−4.28701e6			−0.252554			−0.126277	−	0.991995i	$-0.540303\pi$
$257$	−4.28701e6			−0.252554			−0.126277	+	0.991995i	$0.540303\pi$
$258$		0			0
$259$	−2.05323e7			−1.18179
$260$	−356524.	+	196463.i	−0.0202847	+	0.0111779i
$261$		0			0
$262$			1.19586e7i			0.664930i
$263$	2.94305e7			1.61782			0.808910	−	0.587933i	$-0.200058\pi$
$263$	2.94305e7			1.61782			0.808910	+	0.587933i	$0.200058\pi$
$264$		0			0
$265$	−1.91936e6	−	3.48308e6i	−0.103138	−	0.187165i
$266$			2.05712e7i			1.09299i
$267$		0			0
$268$		−	3.01689e6i		−	0.156731i
$269$			1.40558e7i			0.722100i	0.932547	+	0.361050i	$0.117582\pi$
$269$			1.40558e7i			0.722100i	−0.932547	+	0.361050i	$0.882418\pi$
$270$		0			0
$271$	−2.69073e6			−0.135196			−0.0675978	−	0.997713i	$-0.521533\pi$
$271$	−2.69073e6			−0.135196			−0.0675978	+	0.997713i	$0.521533\pi$
$272$	−2.93468e7			−1.45832
$273$		0			0
$274$	2.31465e7			1.12521
$275$	1.56808e7	+	9.90756e6i	0.753996	+	0.476396i
$276$		0			0
$277$			4.16627e7i			1.96024i	0.198414	+	0.980118i	$0.436421\pi$
$277$			4.16627e7i			1.96024i	−0.198414	+	0.980118i	$0.563579\pi$
$278$	−1.95659e7			−0.910679
$279$		0			0
$280$	1.43257e7	−	7.89419e6i	0.652590	−	0.359611i
$281$			1.07292e7i			0.483556i	0.970332	+	0.241778i	$0.0777305\pi$
$281$			1.07292e7i			0.483556i	−0.970332	+	0.241778i	$0.922269\pi$
$282$		0			0
$283$		−	5.50312e6i		−	0.242800i	−0.992604	−	0.121400i	$-0.961262\pi$
$283$		−	5.50312e6i		−	0.242800i	0.992604	−	0.121400i	$-0.0387385\pi$
$284$		−	2.29454e6i		−	0.100171i
$285$		0			0
$286$	−5.23236e6			−0.223666
$287$	1.53304e7			0.648496
$288$		0			0
$289$	1.92786e7			0.798696
$290$	−7.09179e6	+	3.90795e6i	−0.290778	+	0.160234i
$291$		0			0
$292$		−	1.99987e6i		−	0.0803256i
$293$	1.23547e7			0.491169			0.245584	−	0.969375i	$-0.421020\pi$
$293$	1.23547e7			0.491169			0.245584	+	0.969375i	$0.421020\pi$
$294$		0			0
$295$	−3.49938e7	+	1.92834e7i	−1.36309	+	0.751135i
$296$			3.68072e7i			1.41925i
$297$		0			0
$298$			4.67402e7i			1.76621i
$299$		−	4.69419e6i		−	0.175609i
$300$		0			0
$301$	3.42140e7			1.25460
$302$	−1.57066e6			−0.0570245
$303$		0			0
$304$	4.04773e7			1.44076
$305$	2.16898e7	+	3.93606e7i	0.764461	+	1.38728i
$306$		0			0
$307$		−	4.10198e7i		−	1.41768i	−0.705370	−	0.708840i	$-0.749219\pi$
$307$		−	4.10198e7i		−	1.41768i	0.705370	−	0.708840i	$-0.250781\pi$
$308$	−1.98529e6			−0.0679471
$309$		0			0
$310$	−1.57404e6	−	2.85643e6i	−0.0528362	−	0.0958824i
$311$		−	4.55195e7i		−	1.51327i	−0.653837	−	0.756635i	$-0.726842\pi$
$311$		−	4.55195e7i		−	1.51327i	0.653837	−	0.756635i	$-0.273158\pi$
$312$		0			0
$313$			1.47679e7i			0.481598i	0.970575	+	0.240799i	$0.0774095\pi$
$313$			1.47679e7i			0.481598i	−0.970575	+	0.240799i	$0.922591\pi$
$314$		−	1.76809e7i		−	0.571103i
$315$		0			0
$316$	−408544.			−0.0129472
$317$	−7.38900e6			−0.231957			−0.115979	−	0.993252i	$-0.537000\pi$
$317$	−7.38900e6			−0.231957			−0.115979	+	0.993252i	$0.537000\pi$
$318$		0			0
$319$	−9.17865e6			−0.282753
$320$	−1.40036e7	−	2.54124e7i	−0.427355	−	0.775525i
$321$		0			0
$322$		−	2.01963e7i		−	0.604929i
$323$	−5.98829e7			−1.77703
$324$		0			0
$325$	6.94946e6	+	4.39087e6i	0.202442	+	0.127909i
$326$			3.69845e7i			1.06750i
$327$		0			0
$328$		−	2.74820e7i		−	0.778801i
$329$		−	4.91891e7i		−	1.38128i
$330$		0			0
$331$	−4.89533e7			−1.34989			−0.674944	−	0.737869i	$-0.735832\pi$
$331$	−4.89533e7			−1.34989			−0.674944	+	0.737869i	$0.735832\pi$
$332$	45826.1			0.00125227
$333$		0			0
$334$	2.33781e6			0.0627436
$335$	−5.33579e7	+	2.94030e7i	−1.41927	+	0.782090i
$336$		0			0
$337$		−	2.52313e7i		−	0.659250i	−0.944112	−	0.329625i	$-0.893078\pi$
$337$		−	2.52313e7i		−	0.659250i	0.944112	−	0.329625i	$-0.106922\pi$
$338$	3.81198e7			0.987191
$339$		0			0
$340$	−2.46057e6	−	4.46521e6i	−0.0626035	−	0.113607i
$341$		−	3.69698e6i		−	0.0932360i
$342$		0			0
$343$		−	4.38503e7i		−	1.08665i
$344$		−	6.13337e7i		−	1.50669i
$345$		0			0
$346$	−4.16950e7			−1.00660
$347$	4.18476e7			1.00157			0.500787	−	0.865571i	$-0.333044\pi$
$347$	4.18476e7			1.00157			0.500787	+	0.865571i	$0.333044\pi$
$348$		0			0
$349$	1.11598e7			0.262532			0.131266	−	0.991347i	$-0.458096\pi$
$349$	1.11598e7			0.262532			0.131266	+	0.991347i	$0.458096\pi$
$350$	2.98994e7	+	1.88913e7i	0.697362	+	0.440613i
$351$		0			0
$352$			7.49890e6i			0.171937i
$353$	2.13705e7			0.485837			0.242918	−	0.970047i	$-0.421895\pi$
$353$	2.13705e7			0.485837			0.242918	+	0.970047i	$0.421895\pi$
$354$		0			0
$355$	−4.05821e7	+	2.23629e7i	−0.907088	+	0.499853i
$356$		−	6.13699e6i		−	0.136021i
$357$		0			0
$358$			2.72977e7i			0.594945i
$359$		−	2.27851e7i		−	0.492455i	−0.969212	−	0.246228i	$-0.920809\pi$
$359$		−	2.27851e7i		−	0.492455i	0.969212	−	0.246228i	$-0.0791911\pi$
$360$		0			0
$361$	3.55491e7			0.755627
$362$	9.63909e7			2.03194
$363$		0			0
$364$	−879847.			−0.0182433
$365$	−3.53706e7	+	1.94910e7i	−0.727383	+	0.400826i
$366$		0			0
$367$		−	3.26916e7i		−	0.661360i	−0.943743	−	0.330680i	$-0.892722\pi$
$367$		−	3.26916e7i		−	0.661360i	0.943743	−	0.330680i	$-0.107278\pi$
$368$	−3.97396e7			−0.797407
$369$		0			0
$370$	−6.97040e7	+	3.84105e7i	−1.37611	+	0.758307i
$371$		−	8.59569e6i		−	0.168329i
$372$		0			0
$373$			2.20835e7i			0.425542i	0.977102	+	0.212771i	$0.0682488\pi$
$373$			2.20835e7i			0.425542i	−0.977102	+	0.212771i	$0.931751\pi$
$374$		−	6.55317e7i		−	1.25267i
$375$		0			0
$376$	−8.81786e7			−1.65882
$377$	−4.06783e6			−0.0759169
$378$		0			0
$379$	2.12751e7			0.390800			0.195400	−	0.980724i	$-0.437399\pi$
$379$	2.12751e7			0.390800			0.195400	+	0.980724i	$0.437399\pi$
$380$	3.39380e6	+	6.15877e6i	0.0618494	+	0.112239i
$381$		0			0
$382$		−	5.09298e7i		−	0.913654i
$383$	7.76311e7			1.38178			0.690891	−	0.722959i	$-0.257219\pi$
$383$	7.76311e7			1.38178			0.690891	+	0.722959i	$0.257219\pi$
$384$		0			0
$385$	1.93488e7	+	3.51125e7i	0.339057	+	0.615290i
$386$			9.46131e7i			1.64509i
$387$		0			0
$388$			9.45525e6i			0.161874i
$389$			7.43035e7i			1.26229i	0.775663	+	0.631147i	$0.217415\pi$
$389$			7.43035e7i			1.26229i	−0.775663	+	0.631147i	$0.782585\pi$
$390$		0			0
$391$	5.87914e7			0.983521
$392$	−2.16273e7			−0.359042
$393$		0			0
$394$	2.90611e7			0.475143
$395$	3.98172e6	+	7.22567e6i	0.0646070	+	0.117243i
$396$		0			0
$397$			2.25395e7i			0.360224i	0.983646	+	0.180112i	$0.0576459\pi$
$397$			2.25395e7i			0.360224i	−0.983646	+	0.180112i	$0.942354\pi$
$398$	−4.86175e7			−0.771157
$399$		0			0
$400$	3.71718e7	−	5.88320e7i	0.580809	−	0.919250i
$401$		−	4.85810e7i		−	0.753414i	−0.926332	−	0.376707i	$-0.877056\pi$
$401$		−	4.85810e7i		−	0.753414i	0.926332	−	0.376707i	$-0.122944\pi$
$402$		0			0
$403$		−	1.63844e6i		−	0.0250332i
$404$			2.79998e6i			0.0424630i
$405$		0			0
$406$	−1.75014e7			−0.261515
$407$	−9.02153e7			−1.33813
$408$		0			0
$409$	6.71469e7			0.981423			0.490711	−	0.871322i	$-0.336737\pi$
$409$	6.71469e7			0.981423			0.490711	+	0.871322i	$0.336737\pi$
$410$	5.20442e7	−	2.86791e7i	0.755129	−	0.416116i
$411$		0			0
$412$		−	649706.i		−	0.00929020i
$413$	−8.63593e7			−1.22591
$414$		0			0
$415$	−446627.	−	810499.i	−0.00624886	−	0.0113399i
$416$			3.32339e6i			0.0461638i
$417$		0			0
$418$			9.03863e7i			1.23758i
$419$		−	4.06857e7i		−	0.553095i	−0.961000	−	0.276548i	$-0.910810\pi$
$419$		−	4.06857e7i		−	0.553095i	0.961000	−	0.276548i	$-0.0891904\pi$
$420$		0			0
$421$	9.91589e7			1.32888			0.664440	−	0.747342i	$-0.268670\pi$
$421$	9.91589e7			1.32888			0.664440	+	0.747342i	$0.268670\pi$
$422$	−1.10578e7			−0.147140
$423$		0			0
$424$	−1.54090e7			−0.202152
$425$	−5.49926e7	+	8.70371e7i	−0.716370	+	1.13380i
$426$		0			0
$427$			9.71360e7i			1.24766i
$428$	−4.06742e6			−0.0518786
$429$		0			0
$430$	1.16151e8	−	6.40054e7i	1.46089	−	0.805029i
$431$			2.37118e7i			0.296164i	0.988975	+	0.148082i	$0.0473099\pi$
$431$			2.37118e7i			0.296164i	−0.988975	+	0.148082i	$0.952690\pi$
$432$		0			0
$433$		−	6.09661e7i		−	0.750974i	−0.926828	−	0.375487i	$-0.877476\pi$
$433$		−	6.09661e7i		−	0.750974i	0.926828	−	0.375487i	$-0.122524\pi$
$434$		−	7.04923e6i		−	0.0862328i
$435$		0			0
$436$	1.02541e7			0.123720
$437$	−8.10896e7			−0.971675
$438$		0			0
$439$	1.53212e7			0.181092			0.0905458	−	0.995892i	$-0.471139\pi$
$439$	1.53212e7			0.181092			0.0905458	+	0.995892i	$0.471139\pi$
$440$	6.29444e7	−	3.46856e7i	0.738923	−	0.407185i
$441$		0			0
$442$		−	2.90426e7i		−	0.336332i
$443$	1.12121e8			1.28966			0.644828	−	0.764327i	$-0.276929\pi$
$443$	1.12121e8			1.28966			0.644828	+	0.764327i	$0.276929\pi$
$444$		0			0
$445$	−1.08541e8	+	5.98119e7i	−1.23173	+	0.678746i
$446$		−	1.11689e7i		−	0.125895i
$447$		0			0
$448$		−	6.27138e7i		−	0.697476i
$449$			1.27219e8i			1.40544i	0.711468	+	0.702719i	$0.248031\pi$
$449$			1.27219e8i			1.40544i	−0.711468	+	0.702719i	$0.751969\pi$
$450$		0			0
$451$	6.73589e7			0.734287
$452$	−6.35222e6			−0.0687876
$453$		0			0
$454$	−6.67911e7			−0.713758
$455$	8.57510e6	+	1.55613e7i	0.0910343	+	0.165201i
$456$		0			0
$457$			7.74410e7i			0.811377i	0.914011	+	0.405688i	$0.132968\pi$
$457$			7.74410e7i			0.811377i	−0.914011	+	0.405688i	$0.867032\pi$
$458$	−9.37629e7			−0.975965
$459$		0			0
$460$	−3.33195e6	−	6.04652e6i	−0.0342314	−	0.0621200i
$461$		−	2.35645e7i		−	0.240523i	−0.992742	−	0.120261i	$-0.961627\pi$
$461$		−	2.35645e7i		−	0.240523i	0.992742	−	0.120261i	$-0.0383733\pi$
$462$		0			0
$463$		−	4.67479e7i		−	0.470998i	−0.971875	−	0.235499i	$-0.924328\pi$
$463$		−	4.67479e7i		−	0.470998i	0.971875	−	0.235499i	$-0.0756725\pi$
$464$			3.44370e7i			0.344724i
$465$		0			0
$466$	−1.12133e8			−1.10809
$467$	1.04192e8			1.02302			0.511511	−	0.859277i	$-0.329086\pi$
$467$	1.04192e8			1.02302			0.511511	+	0.859277i	$0.329086\pi$
$468$		0			0
$469$	−1.31679e8			−1.27643
$470$	−9.20197e7	−	1.66989e8i	−0.886313	−	1.60840i
$471$		0			0
$472$			1.54812e8i			1.47224i
$473$	1.50330e8			1.42057
$474$		0			0
$475$	7.58499e7	−	1.20048e8i	0.707741	−	1.12015i
$476$		−	1.10195e7i		−	0.102174i
$477$		0			0
$478$		−	2.65632e7i		−	0.243219i
$479$			9.95625e7i			0.905919i	0.891531	+	0.452959i	$0.149632\pi$
$479$			9.95625e7i			0.905919i	−0.891531	+	0.452959i	$0.850368\pi$
$480$		0			0
$481$	−3.99820e7			−0.359277
$482$	2.81766e7			0.251622
$483$		0			0
$484$	2.24293e6			0.0197824
$485$	1.67229e8	−	9.21521e7i	1.46584	−	0.807755i
$486$		0			0
$487$			1.90543e8i			1.64971i	0.565347	+	0.824853i	$0.308742\pi$
$487$			1.90543e8i			1.64971i	−0.565347	+	0.824853i	$0.691258\pi$
$488$	1.74130e8			1.49836
$489$		0			0
$490$	−2.25694e7	−	4.09570e7i	−0.191837	−	0.348129i
$491$		−	1.08462e8i		−	0.916287i	−0.888878	−	0.458143i	$-0.848515\pi$
$491$		−	1.08462e8i		−	0.916287i	0.888878	−	0.458143i	$-0.151485\pi$
$492$		0			0
$493$		−	5.09467e7i		−	0.425182i
$494$			4.00578e7i			0.332281i
$495$		0			0
$496$	−1.38705e7			−0.113671
$497$	−1.00150e8			−0.815799
$498$		0			0
$499$	−1.85918e8			−1.49631			−0.748153	−	0.663527i	$-0.769059\pi$
$499$	−1.85918e8			−1.49631			−0.748153	+	0.663527i	$0.769059\pi$
$500$	1.20681e7	+	723070.i	0.0965452	+	0.00578456i
$501$		0			0
$502$		−	2.21649e8i		−	1.75209i
$503$	−1.39688e8			−1.09763			−0.548815	−	0.835944i	$-0.684921\pi$
$503$	−1.39688e8			−1.09763			−0.548815	+	0.835944i	$0.684921\pi$
$504$		0			0
$505$	4.95215e7	−	2.72889e7i	0.384521	−	0.211891i
$506$		−	8.87389e7i		−	0.684956i
$507$		0			0
$508$		−	2.32697e6i		−	0.0177500i
$509$			3.55184e7i			0.269340i	0.990891	+	0.134670i	$0.0429974\pi$
$509$			3.55184e7i			0.269340i	−0.990891	+	0.134670i	$0.957003\pi$
$510$		0			0
$511$	−8.72890e7			−0.654180
$512$	−1.09921e8			−0.818977
$513$		0			0
$514$	−3.59163e7			−0.264486
$515$	−1.14910e7	+	6.33212e6i	−0.0841268	+	0.0463583i
$516$		0			0
$517$		−	2.16128e8i		−	1.56401i
$518$	−1.72019e8			−1.23762
$519$		0			0
$520$	2.78959e7	−	1.53721e7i	0.198395	−	0.109326i
$521$		−	1.93385e8i		−	1.36745i	−0.729742	−	0.683723i	$-0.760360\pi$
$521$		−	1.93385e8i		−	1.36745i	0.729742	−	0.683723i	$-0.239640\pi$
$522$		0			0
$523$			9.39198e6i			0.0656527i	0.999461	+	0.0328263i	$0.0104508\pi$
$523$			9.39198e6i			0.0656527i	−0.999461	+	0.0328263i	$0.989549\pi$
$524$			8.83550e6i			0.0614098i
$525$		0			0
$526$	2.46567e8			1.69425
$527$	2.05203e7			0.140201
$528$		0			0
$529$	−6.84242e7			−0.462214
$530$	−1.60803e7	−	2.91810e7i	−0.108010	−	0.196008i
$531$		0			0
$532$			1.51989e7i			0.100943i
$533$	2.98524e7			0.197150
$534$		0			0
$535$	3.96416e7	+	7.19381e7i	0.258875	+	0.469783i
$536$			2.36054e8i			1.53291i
$537$		0			0
$538$			1.17758e8i			0.756214i
$539$		−	5.30091e7i		−	0.338520i
$540$		0			0
$541$	2.07632e7			0.131130			0.0655651	−	0.997848i	$-0.479115\pi$
$541$	2.07632e7			0.131130			0.0655651	+	0.997848i	$0.479115\pi$
$542$	−2.25428e7			−0.141583
$543$		0			0
$544$	−4.16231e7			−0.258546
$545$	−9.99382e7	−	1.81359e8i	−0.617365	−	1.12034i
$546$		0			0
$547$		−	3.73970e7i		−	0.228494i	−0.993452	−	0.114247i	$-0.963554\pi$
$547$		−	3.73970e7i		−	0.228494i	0.993452	−	0.114247i	$-0.0364456\pi$
$548$	1.71016e7			0.103919
$549$		0			0
$550$	1.31373e8	+	8.30050e7i	0.789617	+	0.498903i
$551$			7.02696e7i			0.420061i
$552$		0			0
$553$			1.78318e7i			0.105444i
$554$			3.49048e8i			2.05284i
$555$		0			0
$556$	−1.44561e7			−0.0841059
$557$	2.13408e8			1.23494			0.617468	−	0.786596i	$-0.288159\pi$
$557$	2.13408e8			1.23494			0.617468	+	0.786596i	$0.288159\pi$
$558$		0			0
$559$	6.66240e7			0.381413
$560$	1.31737e8	−	7.25942e7i	0.750144	−	0.413369i
$561$		0			0
$562$			8.98882e7i			0.506400i
$563$	−1.01902e8			−0.571029			−0.285515	−	0.958374i	$-0.592165\pi$
$563$	−1.01902e8			−0.571029			−0.285515	+	0.958374i	$0.592165\pi$
$564$		0			0
$565$	6.19095e7	+	1.12348e8i	0.343251	+	0.622902i
$566$		−	4.61048e7i		−	0.254271i
$567$		0			0
$568$			1.79534e8i			0.979720i
$569$			1.12478e8i			0.610566i	0.952262	+	0.305283i	$0.0987510\pi$
$569$			1.12478e8i			0.610566i	−0.952262	+	0.305283i	$0.901249\pi$
$570$		0			0
$571$	1.75933e8			0.945016			0.472508	−	0.881326i	$-0.343349\pi$
$571$	1.75933e8			0.945016			0.472508	+	0.881326i	$0.343349\pi$
$572$	−3.86589e6			−0.0206567
$573$		0			0
$574$	1.28437e8			0.679133
$575$	−7.44675e7	+	1.17860e8i	−0.391709	+	0.619960i
$576$		0			0
$577$		−	1.61356e8i		−	0.839957i	−0.907534	−	0.419978i	$-0.862038\pi$
$577$		−	1.61356e8i		−	0.839957i	0.907534	−	0.419978i	$-0.137962\pi$
$578$	1.61515e8			0.836428
$579$		0			0
$580$	−5.23971e6	+	2.88735e6i	−0.0268549	+	0.0147984i
$581$		−	2.00019e6i		−	0.0101986i
$582$		0			0
$583$		−	3.77679e7i		−	0.190598i
$584$			1.56478e8i			0.785626i
$585$		0			0
$586$	1.03507e8			0.514373
$587$	−4.39079e7			−0.217084			−0.108542	−	0.994092i	$-0.534618\pi$
$587$	−4.39079e7			−0.217084			−0.108542	+	0.994092i	$0.534618\pi$
$588$		0			0
$589$	−2.83032e7			−0.138513
$590$	−2.93176e8	+	1.61555e8i	−1.42749	+	0.786621i
$591$		0			0
$592$			3.38475e8i			1.63140i
$593$	−3.94038e7			−0.188962			−0.0944808	−	0.995527i	$-0.530119\pi$
$593$	−3.94038e7			−0.188962			−0.0944808	+	0.995527i	$0.530119\pi$
$594$		0			0
$595$	−1.94895e8	+	1.07397e8i	−0.925228	+	0.509849i
$596$			3.45336e7i			0.163118i
$597$		0			0
$598$		−	3.93277e7i		−	0.183905i
$599$			1.11994e8i			0.521092i	0.965461	+	0.260546i	$0.0839026\pi$
$599$			1.11994e8i			0.521092i	−0.965461	+	0.260546i	$0.916097\pi$
$600$		0			0
$601$	−3.96028e8			−1.82432			−0.912162	−	0.409830i	$-0.865588\pi$
$601$	−3.96028e8			−1.82432			−0.912162	+	0.409830i	$0.865588\pi$
$602$	2.86643e8			1.31387
$603$		0			0
$604$	−1.16047e6			−0.00526650
$605$	−2.18599e7	−	3.96693e7i	−0.0987146	−	0.179138i
$606$		0			0
$607$		−	3.06455e8i		−	1.37025i	−0.728425	−	0.685125i	$-0.759747\pi$
$607$		−	3.06455e8i		−	1.37025i	0.728425	−	0.685125i	$-0.240253\pi$
$608$	5.74098e7			0.255432
$609$		0			0
$610$	1.81716e8	+	3.29761e8i	0.800577	+	1.45281i
$611$		−	9.57844e7i		−	0.419924i
$612$		0			0
$613$		−	1.80074e7i		−	0.0781751i	−0.999236	−	0.0390875i	$-0.987555\pi$
$613$		−	1.80074e7i		−	0.0781751i	0.999236	−	0.0390875i	$-0.0124451\pi$
$614$		−	3.43661e8i		−	1.48465i
$615$		0			0
$616$	1.55337e8			0.664558
$617$	1.66217e8			0.707651			0.353825	−	0.935311i	$-0.384881\pi$
$617$	1.66217e8			0.707651			0.353825	+	0.935311i	$0.384881\pi$
$618$		0			0
$619$	1.89841e8			0.800422			0.400211	−	0.916423i	$-0.368937\pi$
$619$	1.89841e8			0.800422			0.400211	+	0.916423i	$0.368937\pi$
$620$	−1.16297e6	−	2.11045e6i	−0.00487970	−	0.00885524i
$621$		0			0
$622$		−	3.81360e8i		−	1.58476i
$623$	−2.67863e8			−1.10777
$624$		0			0
$625$	−1.04829e8	−	2.20489e8i	−0.429380	−	0.903124i
$626$			1.23724e8i			0.504350i
$627$		0			0
$628$		−	1.30634e7i		−	0.0527443i
$629$		−	5.00746e8i		−	2.01217i
$630$		0			0
$631$	2.56171e8			1.01963			0.509814	−	0.860285i	$-0.329714\pi$
$631$	2.56171e8			1.01963			0.509814	+	0.860285i	$0.329714\pi$
$632$	3.19662e7			0.126631
$633$		0			0
$634$	−6.19046e7			−0.242916
$635$	−4.11557e7	+	2.26789e7i	−0.160734	+	0.0885730i
$636$		0			0
$637$		−	2.34928e7i		−	0.0908900i
$638$	−7.68982e7			−0.296111
$639$		0			0
$640$	−1.41711e8	−	2.57164e8i	−0.540584	−	0.981004i
$641$		−	4.41284e7i		−	0.167550i	−0.996485	−	0.0837749i	$-0.973302\pi$
$641$		−	4.41284e7i		−	0.167550i	0.996485	−	0.0837749i	$-0.0266977\pi$
$642$		0			0
$643$			1.96340e8i			0.738545i	0.929321	+	0.369272i	$0.120393\pi$
$643$			1.96340e8i			0.738545i	−0.929321	+	0.369272i	$0.879607\pi$
$644$		−	1.49219e7i		−	0.0558683i
$645$		0			0
$646$	−5.01695e8			−1.86098
$647$	−9.99293e7			−0.368961			−0.184480	−	0.982836i	$-0.559060\pi$
$647$	−9.99293e7			−0.368961			−0.184480	+	0.982836i	$0.559060\pi$
$648$		0			0
$649$	−3.79447e8			−1.38809
$650$	5.82222e7	+	3.67865e7i	0.212006	+	0.133952i
$651$		0			0
$652$			2.73257e7i			0.0985890i
$653$	−2.49531e7			−0.0896160			−0.0448080	−	0.998996i	$-0.514268\pi$
$653$	−2.49531e7			−0.0896160			−0.0448080	+	0.998996i	$0.514268\pi$
$654$		0			0
$655$	1.56268e8	−	8.61120e7i	0.556092	−	0.306436i
$656$		−	2.52721e8i		−	0.895221i
$657$		0			0
$658$		−	4.12103e8i		−	1.44653i
$659$		−	1.49890e8i		−	0.523739i	−0.965103	−	0.261870i	$-0.915661\pi$
$659$		−	1.49890e8i		−	0.523739i	0.965103	−	0.261870i	$-0.0843390\pi$
$660$		0			0
$661$	−4.68646e6			−0.0162271			−0.00811355	−	0.999967i	$-0.502583\pi$
$661$	−4.68646e6			−0.0162271			−0.00811355	+	0.999967i	$0.502583\pi$
$662$	−4.10128e8			−1.41366
$663$		0			0
$664$	−3.58563e6			−0.0122479
$665$	2.68813e8	−	1.48130e8i	0.914084	−	0.503708i
$666$		0			0
$667$		−	6.89888e7i		−	0.232488i
$668$	1.72727e6			0.00579469
$669$		0			0
$670$	−4.47029e8	+	2.46336e8i	−1.48632	+	0.819038i
$671$			4.26798e8i			1.41272i
$672$		0			0
$673$			3.58399e8i			1.17577i	0.808945	+	0.587884i	$0.200039\pi$
$673$			3.58399e8i			1.17577i	−0.808945	+	0.587884i	$0.799961\pi$
$674$		−	2.11387e8i		−	0.690395i
$675$		0			0
$676$	2.81645e7			0.0911721
$677$	2.26026e7			0.0728437			0.0364218	−	0.999337i	$-0.488404\pi$
$677$	2.26026e7			0.0728437			0.0364218	+	0.999337i	$0.488404\pi$
$678$		0			0
$679$	4.12696e8			1.31832
$680$	1.92525e8	+	3.49377e8i	0.612295	+	1.11114i
$681$		0			0
$682$		−	3.09731e7i		−	0.0976407i
$683$	5.32455e8			1.67117			0.835585	−	0.549361i	$-0.185129\pi$
$683$	5.32455e8			1.67117			0.835585	+	0.549361i	$0.185129\pi$
$684$		0			0
$685$	−1.66675e8	−	3.02466e8i	−0.518559	−	0.941033i
$686$		−	3.67376e8i		−	1.13799i
$687$		0			0
$688$		−	5.64018e8i		−	1.73192i
$689$		−	1.67381e7i		−	0.0511740i
$690$		0			0
$691$	−5.20725e8			−1.57825			−0.789123	−	0.614236i	$-0.789464\pi$
$691$	−5.20725e8			−1.57825			−0.789123	+	0.614236i	$0.789464\pi$
$692$	−3.08060e7			−0.0929644
$693$		0			0
$694$	3.50597e8			1.04889
$695$	1.40891e8	+	2.55676e8i	0.419690	+	0.761615i
$696$		0			0
$697$			3.73880e8i			1.10417i
$698$	9.34965e7			0.274934
$699$		0			0
$700$	2.20909e7	+	1.39577e7i	0.0644050	+	0.0406929i
$701$			2.97763e8i			0.864404i	0.901777	+	0.432202i	$0.142263\pi$
$701$			2.97763e8i			0.864404i	−0.901777	+	0.432202i	$0.857737\pi$
$702$		0			0
$703$			6.90667e8i			1.98794i
$704$		−	2.75553e8i		−	0.789746i
$705$		0			0
$706$	1.79041e8			0.508789
$707$	1.22211e8			0.345823
$708$		0			0
$709$	1.89039e8			0.530411			0.265205	−	0.964192i	$-0.414560\pi$
$709$	1.89039e8			0.530411			0.265205	+	0.964192i	$0.414560\pi$
$710$	−3.39995e8	+	1.87355e8i	−0.949942	+	0.523468i
$711$		0			0
$712$			4.80183e8i			1.33035i
$713$	2.77873e7			0.0766617
$714$		0			0
$715$	3.76774e7	+	6.83736e7i	0.103077	+	0.187055i
$716$			2.01687e7i			0.0549462i
$717$		0			0
$718$		−	1.90892e8i		−	0.515721i
$719$		−	1.30902e8i		−	0.352176i	−0.984374	−	0.176088i	$-0.943656\pi$
$719$		−	1.30902e8i		−	0.352176i	0.984374	−	0.176088i	$-0.0563443\pi$
$720$		0			0
$721$	−2.83579e7			−0.0756603
$722$	2.97829e8			0.791325
$723$		0			0
$724$	7.12176e7			0.187660
$725$	1.02134e8	+	6.45311e7i	0.268013	+	0.169338i
$726$		0			0
$727$			4.44143e8i			1.15590i	0.816073	+	0.577949i	$0.196147\pi$
$727$			4.44143e8i			1.15590i	−0.816073	+	0.577949i	$0.803853\pi$
$728$	6.88429e7			0.178429
$729$		0			0
$730$	−2.96333e8	+	1.63295e8i	−0.761747	+	0.419762i
$731$			8.34419e8i			2.13615i
$732$		0			0
$733$		−	4.71631e8i		−	1.19754i	−0.800921	−	0.598770i	$-0.795656\pi$
$733$		−	4.71631e8i		−	1.19754i	0.800921	−	0.598770i	$-0.204344\pi$
$734$		−	2.73888e8i		−	0.692605i
$735$		0			0
$736$	−5.63635e7			−0.141372
$737$	−5.78573e8			−1.44529
$738$		0			0
$739$	6.40610e8			1.58731			0.793653	−	0.608371i	$-0.208177\pi$
$739$	6.40610e8			1.58731			0.793653	+	0.608371i	$0.208177\pi$
$740$	−5.15002e7	+	2.83793e7i	−0.127091	+	0.0700336i
$741$		0			0
$742$		−	7.20142e7i		−	0.176281i
$743$	−5.44941e8			−1.32857			−0.664283	−	0.747481i	$-0.731263\pi$
$743$	−5.44941e8			−1.32857			−0.664283	+	0.747481i	$0.731263\pi$
$744$		0			0
$745$	6.10774e8	−	3.36569e8i	1.47711	−	0.813963i
$746$			1.85015e8i			0.445646i
$747$		0			0
$748$		−	4.84175e7i		−	0.115691i
$749$			1.77532e8i			0.422504i
$750$		0			0
$751$	5.56224e8			1.31320			0.656598	−	0.754241i	$-0.271995\pi$
$751$	5.56224e8			1.31320			0.656598	+	0.754241i	$0.271995\pi$
$752$	−8.10881e8			−1.90679
$753$		0			0
$754$	−3.40801e7			−0.0795035
$755$	1.13101e7	+	2.05245e7i	0.0262799	+	0.0476905i
$756$		0			0
$757$			4.02524e8i			0.927906i	0.885860	+	0.463953i	$0.153569\pi$
$757$			4.02524e8i			0.927906i	−0.885860	+	0.463953i	$0.846431\pi$
$758$	1.78242e8			0.409263
$759$		0			0
$760$	−2.65545e8	−	4.81888e8i	−0.604920	−	1.09775i
$761$		−	1.44544e8i		−	0.327979i	−0.986462	−	0.163989i	$-0.947564\pi$
$761$		−	1.44544e8i		−	0.327979i	0.986462	−	0.163989i	$-0.0524363\pi$
$762$		0			0
$763$		−	4.47565e8i		−	1.00759i
$764$		−	3.76290e7i		−	0.0843806i
$765$		0			0
$766$	6.50389e8			1.44706
$767$	−1.68165e8			−0.372691
$768$		0			0
$769$	−2.64281e8			−0.581148			−0.290574	−	0.956853i	$-0.593846\pi$
$769$	−2.64281e8			−0.581148			−0.290574	+	0.956853i	$0.593846\pi$
$770$	1.62104e8	+	2.94171e8i	0.355075	+	0.644359i
$771$		0			0
$772$			6.99040e7i			0.151932i
$773$	−1.66034e8			−0.359466			−0.179733	−	0.983715i	$-0.557523\pi$
$773$	−1.66034e8			−0.359466			−0.179733	+	0.983715i	$0.557523\pi$
$774$		0			0
$775$	−2.59918e7	+	4.11374e7i	−0.0558382	+	0.0883756i
$776$		−	7.39818e8i		−	1.58321i
$777$		0			0
$778$			6.22511e8i			1.32193i
$779$		−	5.15684e8i		−	1.09087i
$780$		0			0
$781$	−4.40043e8			−0.923723
$782$	4.92551e8			1.02999
$783$		0			0
$784$	−1.98883e8			−0.412714
$785$	−2.31044e8	+	1.27317e8i	−0.477623	+	0.263195i
$786$		0			0
$787$			2.25184e8i			0.461970i	0.972957	+	0.230985i	$0.0741948\pi$
$787$			2.25184e8i			0.461970i	−0.972957	+	0.230985i	$0.925805\pi$
$788$	2.14716e7			0.0438819
$789$		0			0
$790$	3.33586e7	+	6.05363e7i	0.0676592	+	0.122782i
$791$			2.77257e8i			0.560213i
$792$		0			0
$793$			1.89150e8i			0.379303i
$794$			1.88834e8i			0.377242i
$795$		0			0
$796$	−3.59206e7			−0.0712204
$797$	5.51423e8			1.08921			0.544603	−	0.838694i	$-0.316680\pi$
$797$	5.51423e8			1.08921			0.544603	+	0.838694i	$0.316680\pi$
$798$		0			0
$799$	1.19963e9			2.35184
$800$	5.27215e7	−	8.34427e7i	0.102972	−	0.162974i
$801$		0			0
$802$		−	4.07009e8i		−	0.789008i
$803$	−3.83532e8			−0.740722
$804$		0			0
$805$	−2.63914e8	+	1.45430e8i	−0.505912	+	0.278784i
$806$		−	1.37268e7i		−	0.0262158i
$807$		0			0
$808$		−	2.19082e8i		−	0.415310i
$809$			3.95120e7i			0.0746249i	0.999304	+	0.0373125i	$0.0118797\pi$
$809$			3.95120e7i			0.0746249i	−0.999304	+	0.0373125i	$0.988120\pi$
$810$		0			0
$811$	−8.11405e8			−1.52116			−0.760580	−	0.649244i	$-0.775085\pi$
$811$	−8.11405e8			−1.52116			−0.760580	+	0.649244i	$0.775085\pi$
$812$	−1.29308e7			−0.0241522
$813$		0			0
$814$	−7.55819e8			−1.40134
$815$	4.83293e8	−	2.66320e8i	0.892766	−	0.491961i
$816$		0			0
$817$		−	1.15089e9i		−	2.11042i
$818$	5.62553e8			1.02779
$819$		0			0
$820$	3.84524e7	−	2.11893e7i	0.0697401	−	0.0384304i
$821$			1.22832e8i			0.221963i	0.993822	+	0.110982i	$0.0353994\pi$
$821$			1.22832e8i			0.221963i	−0.993822	+	0.110982i	$0.964601\pi$
$822$		0			0
$823$			8.46296e8i			1.51818i	0.650987	+	0.759089i	$0.274355\pi$
$823$			8.46296e8i			1.51818i	−0.650987	+	0.759089i	$0.725645\pi$
$824$			5.08357e7i			0.0908630i
$825$		0			0
$826$	−7.23514e8			−1.28383
$827$	−5.80388e8			−1.02613			−0.513065	−	0.858350i	$-0.671490\pi$
$827$	−5.80388e8			−1.02613			−0.513065	+	0.858350i	$0.671490\pi$
$828$		0			0
$829$	−6.30126e8			−1.10602			−0.553011	−	0.833174i	$-0.686521\pi$
$829$	−6.30126e8			−1.10602			−0.553011	+	0.833174i	$0.686521\pi$
$830$	−3.74182e6	−	6.79032e6i	−0.00654408	−	0.0118756i
$831$		0			0
$832$		−	1.22121e8i		−	0.212041i
$833$	2.94231e8			0.509041
$834$		0			0
$835$	−1.68342e7	−	3.05491e7i	−0.0289156	−	0.0524735i
$836$			6.67811e7i			0.114297i
$837$		0			0
$838$		−	3.40863e8i		−	0.579225i
$839$			2.51051e7i			0.0425085i	0.999774	+	0.0212542i	$0.00676595\pi$
$839$			2.51051e7i			0.0425085i	−0.999774	+	0.0212542i	$0.993234\pi$
$840$		0			0
$841$	5.35040e8			0.899494
$842$	8.30748e8			1.39166
$843$		0			0
$844$	−8.16992e6			−0.0135891
$845$	−2.74495e8	−	4.98129e8i	−0.454951	−	0.825603i
$846$		0			0
$847$		−	9.78978e7i		−	0.161110i
$848$	−1.41700e8			−0.232371
$849$		0			0
$850$	−4.60725e8	+	7.29192e8i	−0.750213	+	1.18737i
$851$		−	6.78079e8i		−	1.10025i
$852$		0			0
$853$		−	3.53908e8i		−	0.570221i	−0.958495	−	0.285111i	$-0.907970\pi$
$853$		−	3.53908e8i		−	0.570221i	0.958495	−	0.285111i	$-0.0920303\pi$
$854$			8.13800e8i			1.30660i
$855$		0			0
$856$	3.18252e8			0.507400
$857$	8.06395e8			1.28117			0.640583	−	0.767889i	$-0.278693\pi$
$857$	8.06395e8			1.28117			0.640583	+	0.767889i	$0.278693\pi$
$858$		0			0
$859$	−9.26709e8			−1.46206			−0.731028	−	0.682348i	$-0.760959\pi$
$859$	−9.26709e8			−1.46206			−0.731028	+	0.682348i	$0.760959\pi$
$860$	8.58174e7	−	4.72899e7i	0.134921	−	0.0743486i
$861$		0			0
$862$			1.98656e8i			0.310155i
$863$	6.89660e8			1.07301			0.536504	−	0.843898i	$-0.319745\pi$
$863$	6.89660e8			1.07301			0.536504	+	0.843898i	$0.319745\pi$
$864$		0			0
$865$	3.00239e8	+	5.44847e8i	0.463894	+	0.841833i
$866$		−	5.10771e8i		−	0.786452i
$867$		0			0
$868$		−	5.20827e6i		−	0.00796405i
$869$			7.83499e7i			0.119393i
$870$		0			0
$871$	−2.56414e8			−0.388050
$872$	−8.02327e8			−1.21005
$873$		0			0
$874$	−6.79365e8			−1.01758
$875$	3.15601e7	−	5.26742e8i	0.0471101	−	0.786274i
$876$		0			0
$877$		−	2.94275e8i		−	0.436269i	−0.975919	−	0.218134i	$-0.930003\pi$
$877$		−	2.94275e8i		−	0.436269i	0.975919	−	0.218134i	$-0.0699971\pi$
$878$	1.28360e8			0.189647
$879$		0			0
$880$	5.78830e8	−	3.18966e8i	0.849382	−	0.468054i
$881$			6.79591e7i			0.0993847i	0.998765	+	0.0496924i	$0.0158241\pi$
$881$			6.79591e7i			0.0993847i	−0.998765	+	0.0496924i	$0.984176\pi$
$882$		0			0
$883$		−	9.49523e8i		−	1.37919i	−0.724196	−	0.689594i	$-0.757789\pi$
$883$		−	9.49523e8i		−	1.37919i	0.724196	−	0.689594i	$-0.242211\pi$
$884$		−	2.14579e7i		−	0.0310620i
$885$		0			0
$886$	9.39340e8			1.35058
$887$	1.87598e8			0.268817			0.134409	−	0.990926i	$-0.457086\pi$
$887$	1.87598e8			0.268817			0.134409	+	0.990926i	$0.457086\pi$
$888$		0			0
$889$	−1.01566e8			−0.144558
$890$	−9.09352e8	+	5.01101e8i	−1.28992	+	0.710812i
$891$		0			0
$892$		−	8.25207e6i		−	0.0116270i
$893$	−1.65462e9			−2.32351
$894$		0			0
$895$	3.56711e8	−	1.96566e8i	0.497562	−	0.274183i
$896$		−	6.34642e8i		−	0.882276i
$897$		0			0
$898$			1.06583e9i			1.47183i
$899$		−	2.40796e7i		−	0.0331413i
$900$		0			0
$901$	2.09633e8			0.286607
$902$	5.64330e8			0.768977
$903$		0			0
$904$	4.97024e8			0.672778
$905$	−6.94096e8	−	1.25958e9i	−0.936427	−	1.69934i
$906$		0			0
$907$		−	1.30184e9i		−	1.74476i	−0.488832	−	0.872378i	$-0.662577\pi$
$907$		−	1.30184e9i		−	1.74476i	0.488832	−	0.872378i	$-0.337423\pi$
$908$	−4.93480e7			−0.0659192
$909$		0			0
$910$	7.18417e7	+	1.30372e8i	0.0953350	+	0.173005i
$911$			1.25808e9i			1.66400i	0.554775	+	0.832000i	$0.312804\pi$
$911$			1.25808e9i			1.66400i	−0.554775	+	0.832000i	$0.687196\pi$
$912$		0			0
$913$		−	8.78846e6i		−	0.0115478i
$914$			6.48797e8i			0.849709i
$915$		0			0
$916$	−6.92759e7			−0.0901354
$917$	3.85646e8			0.500127
$918$		0			0
$919$	−2.32755e8			−0.299883			−0.149942	−	0.988695i	$-0.547909\pi$
$919$	−2.32755e8			−0.299883			−0.149942	+	0.988695i	$0.547909\pi$
$920$	2.60705e8	+	4.73105e8i	0.334801	+	0.607566i
$921$		0			0
$922$		−	1.97422e8i		−	0.251886i
$923$	−1.95020e8			−0.248013
$924$		0			0
$925$	1.00385e9	+	6.34264e8i	1.26837	+	0.801392i
$926$		−	3.91651e8i		−	0.493250i
$927$		0			0
$928$			4.88427e7i			0.0611161i
$929$			1.02421e9i			1.27745i	0.769436	+	0.638724i	$0.220538\pi$
$929$			1.02421e9i			1.27745i	−0.769436	+	0.638724i	$0.779462\pi$
$930$		0			0
$931$	−4.05825e8			−0.502910
$932$	−8.28485e7			−0.102338
$933$		0			0
$934$	8.72917e8			1.07135
$935$	−8.56331e8	+	4.71883e8i	−1.04763	+	0.577298i
$936$		0			0
$937$			5.87353e8i			0.713971i	0.934110	+	0.356985i	$0.116195\pi$
$937$			5.87353e8i			0.713971i	−0.934110	+	0.356985i	$0.883805\pi$
$938$	−1.10320e9			−1.33673
$939$		0			0
$940$	−6.79880e7	−	1.23378e8i	−0.0818556	−	0.148544i
$941$		−	1.44532e8i		−	0.173458i	−0.996232	−	0.0867292i	$-0.972359\pi$
$941$		−	1.44532e8i		−	0.173458i	0.996232	−	0.0867292i	$-0.0276415\pi$
$942$		0			0
$943$			5.06285e8i			0.603755i
$944$			1.42363e9i			1.69232i
$945$		0			0
$946$	1.25946e9			1.48768
$947$	−2.40595e8			−0.283294			−0.141647	−	0.989917i	$-0.545240\pi$
$947$	−2.40595e8			−0.283294			−0.141647	+	0.989917i	$0.545240\pi$
$948$		0			0
$949$	−1.69975e8			−0.198878
$950$	6.35467e8	−	1.00576e9i	0.741177	−	1.17307i
$951$		0			0
$952$			8.62208e8i			0.999313i
$953$	1.59931e9			1.84779			0.923896	−	0.382644i	$-0.124986\pi$
$953$	1.59931e9			1.84779			0.923896	+	0.382644i	$0.124986\pi$
$954$		0			0
$955$	−6.65522e8	+	3.66737e8i	−0.764103	+	0.421061i
$956$		−	1.96260e7i		−	0.0224625i
$957$		0			0
$958$			8.34129e8i			0.948717i
$959$		−	7.46440e8i		−	0.846328i
$960$		0			0
$961$	−8.77805e8			−0.989072
$962$	−3.34967e8			−0.376250
$963$		0			0
$964$	2.08181e7			0.0232386
$965$	1.23635e9	−	6.81294e8i	1.37581	−	0.758145i
$966$		0			0
$967$			3.53519e8i			0.390960i	0.980708	+	0.195480i	$0.0626265\pi$
$967$			3.53519e8i			0.390960i	−0.980708	+	0.195480i	$0.937373\pi$
$968$	−1.75496e8			−0.193482
$969$		0			0
$970$	1.40104e9	−	7.72045e8i	1.53509	−	0.845916i
$971$			1.33953e8i			0.146317i	0.997320	+	0.0731583i	$0.0233078\pi$
$971$			1.33953e8i			0.146317i	−0.997320	+	0.0731583i	$0.976692\pi$
$972$		0			0
$973$			6.30969e8i			0.684967i
$974$			1.59636e9i			1.72764i
$975$		0			0
$976$	1.60129e9			1.72234
$977$	2.99781e8			0.321455			0.160728	−	0.986999i	$-0.448616\pi$
$977$	2.99781e8			0.321455			0.160728	+	0.986999i	$0.448616\pi$
$978$		0			0
$979$	−1.17694e9			−1.25431
$980$	−1.66752e7	−	3.02607e7i	−0.0177171	−	0.0321515i
$981$		0			0
$982$		−	9.08685e8i		−	0.959575i
$983$	1.04294e9			1.09799			0.548996	−	0.835825i	$-0.315010\pi$
$983$	1.04294e9			1.09799			0.548996	+	0.835825i	$0.315010\pi$
$984$		0			0
$985$	−2.09265e8	−	3.79755e8i	−0.218971	−	0.397369i
$986$		−	4.26829e8i		−	0.445269i
$987$		0			0
$988$			2.95963e7i			0.0306879i
$989$			1.12992e9i			1.16804i
$990$		0			0
$991$	3.39676e8			0.349014			0.174507	−	0.984656i	$-0.444167\pi$
$991$	3.39676e8			0.349014			0.174507	+	0.984656i	$0.444167\pi$
$992$	−1.96729e7			−0.0201527
$993$		0			0
$994$	−8.39054e8			−0.854340
$995$	3.50087e8	+	6.35306e8i	0.355391	+	0.644931i
$996$		0			0
$997$			1.79345e9i			1.80969i	0.425739	+	0.904846i	$0.360014\pi$
$997$			1.79345e9i			1.80969i	−0.425739	+	0.904846i	$0.639986\pi$
$998$	−1.55761e9			−1.56700
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	45.7.d.a.44.9	yes	12
3.2	odd	2	inner	45.7.d.a.44.4	yes	12
4.3	odd	2		720.7.c.a.449.3		12
5.2	odd	4		225.7.c.e.26.10		12
5.3	odd	4		225.7.c.e.26.3		12
5.4	even	2	inner	45.7.d.a.44.3	✓	12
12.11	even	2		720.7.c.a.449.10		12
15.2	even	4		225.7.c.e.26.4		12
15.8	even	4		225.7.c.e.26.9		12
15.14	odd	2	inner	45.7.d.a.44.10	yes	12
20.19	odd	2		720.7.c.a.449.9		12
60.59	even	2		720.7.c.a.449.4		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.7.d.a.44.3	✓	12	5.4	even	2	inner
45.7.d.a.44.4	yes	12	3.2	odd	2	inner
45.7.d.a.44.9	yes	12	1.1	even	1	trivial
45.7.d.a.44.10	yes	12	15.14	odd	2	inner
225.7.c.e.26.3		12	5.3	odd	4
225.7.c.e.26.4		12	15.2	even	4
225.7.c.e.26.9		12	15.8	even	4
225.7.c.e.26.10		12	5.2	odd	4
720.7.c.a.449.3		12	4.3	odd	2
720.7.c.a.449.4		12	60.59	even	2
720.7.c.a.449.9		12	20.19	odd	2
720.7.c.a.449.10		12	12.11	even	2

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 45 = 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	45.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+8.37795 q^{2} +6.18997 q^{4} +(-60.3283 - 109.478i) q^{5} -270.176i q^{7} -484.329 q^{8} +(-505.427 - 917.203i) q^{10} -1187.10i q^{11} -526.105i q^{13} -2263.52i q^{14} -4453.84 q^{16} +6589.09 q^{17} -9088.18 q^{19} +(-373.430 - 677.668i) q^{20} -9945.48i q^{22} +8922.54 q^{23} +(-8346.00 + 13209.3i) q^{25} -4407.68i q^{26} -1672.38i q^{28} -7731.97i q^{29} +3114.29 q^{31} -6316.98 q^{32} +55203.1 q^{34} +(-29578.4 + 16299.2i) q^{35} -75996.2i q^{37} -76140.3 q^{38} +(29218.7 + 53023.5i) q^{40} +56742.3i q^{41} +126636. i q^{43} -7348.13i q^{44} +74752.5 q^{46} +182063. q^{47} +44654.2 q^{49} +(-69922.3 + 110667. i) q^{50} -3256.57i q^{52} +31815.2 q^{53} +(-129962. + 71615.8i) q^{55} +130854. i q^{56} -64778.1i q^{58} -319642. i q^{59} -359529. q^{61} +26091.3 q^{62} +232123. q^{64} +(-57597.1 + 31739.0i) q^{65} -487383. i q^{67} +40786.3 q^{68} +(-247806. + 136554. i) q^{70} -370686. i q^{71} -323083. i q^{73} -636692. i q^{74} -56255.6 q^{76} -320726. q^{77} -66000.9 q^{79} +(268693. + 487599. i) q^{80} +475384. i q^{82} +7403.28 q^{83} +(-397509. - 721363. i) q^{85} +1.06095e6i q^{86} +574948. i q^{88} -991440. i q^{89} -142141. q^{91} +55230.3 q^{92} +1.52532e6 q^{94} +(548274. + 994959. i) q^{95} +1.52751e6i q^{97} +374110. q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 516 q^{4} - 1368 q^{10} + 36372 q^{16} + 4320 q^{19} - 63384 q^{25} + 60192 q^{31} + 106296 q^{34} - 221772 q^{40} - 1078968 q^{46} + 711516 q^{49} - 104112 q^{55} - 449784 q^{61} + 3964572 q^{64}+ \cdots + 5793408 q^{94}+O(q^{100}) \) 12 * q + 516 * q^4 - 1368 * q^10 + 36372 * q^16 + 4320 * q^19 - 63384 * q^25 + 60192 * q^31 + 106296 * q^34 - 221772 * q^40 - 1078968 * q^46 + 711516 * q^49 - 104112 * q^55 - 449784 * q^61 + 3964572 * q^64 - 3326616 * q^70 - 584400 * q^76 + 4324608 * q^79 - 3305772 * q^85 + 631152 * q^91 + 5793408 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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