LMFDB - Embedded newform 450.5.g.a.307.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [450,5,Mod(307,450)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("450.307"); S:= CuspForms(chi, 5); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(450, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 5, names="a")

Level:	$ N $	$=$	$ 450 = 2 \cdot 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	450.g (of order $4$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [2,-4,0,0,0,0,-58,32,0,0,236,0,-138,0,0,-128,-542] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	no
Analytic conductor:	$46.5164833877$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 10)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			307.1
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	450.307
Dual form			450.5.g.a.343.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-2.00000 - 2.00000i) q^{2} +8.00000i q^{4} +(-29.0000 - 29.0000i) q^{7} +(16.0000 - 16.0000i) q^{8} +118.000 q^{11} +(-69.0000 + 69.0000i) q^{13} +116.000i q^{14} -64.0000 q^{16} +(-271.000 - 271.000i) q^{17} +280.000i q^{19} +(-236.000 - 236.000i) q^{22} +(269.000 - 269.000i) q^{23} +276.000 q^{26} +(232.000 - 232.000i) q^{28} -680.000i q^{29} +202.000 q^{31} +(128.000 + 128.000i) q^{32} +1084.00i q^{34} +(651.000 + 651.000i) q^{37} +(560.000 - 560.000i) q^{38} -1682.00 q^{41} +(-1089.00 + 1089.00i) q^{43} +944.000i q^{44} -1076.00 q^{46} +(1269.00 + 1269.00i) q^{47} -719.000i q^{49} +(-552.000 - 552.000i) q^{52} +(-611.000 + 611.000i) q^{53} -928.000 q^{56} +(-1360.00 + 1360.00i) q^{58} -1160.00i q^{59} -5598.00 q^{61} +(-404.000 - 404.000i) q^{62} -512.000i q^{64} +(751.000 + 751.000i) q^{67} +(2168.00 - 2168.00i) q^{68} -6442.00 q^{71} +(2951.00 - 2951.00i) q^{73} -2604.00i q^{74} -2240.00 q^{76} +(-3422.00 - 3422.00i) q^{77} +10560.0i q^{79} +(3364.00 + 3364.00i) q^{82} +(-6231.00 + 6231.00i) q^{83} +4356.00 q^{86} +(1888.00 - 1888.00i) q^{88} +14480.0i q^{89} +4002.00 q^{91} +(2152.00 + 2152.00i) q^{92} -5076.00i q^{94} +(7311.00 + 7311.00i) q^{97} +(-1438.00 + 1438.00i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{2} - 58 q^{7} + 32 q^{8} + 236 q^{11} - 138 q^{13} - 128 q^{16} - 542 q^{17} - 472 q^{22} + 538 q^{23} + 552 q^{26} + 464 q^{28} + 404 q^{31} + 256 q^{32} + 1302 q^{37} + 1120 q^{38} - 3364 q^{41}+ \cdots - 2876 q^{98}+O(q^{100}) $ 2 * q - 4 * q^2 - 58 * q^7 + 32 * q^8 + 236 * q^11 - 138 * q^13 - 128 * q^16 - 542 * q^17 - 472 * q^22 + 538 * q^23 + 552 * q^26 + 464 * q^28 + 404 * q^31 + 256 * q^32 + 1302 * q^37 + 1120 * q^38 - 3364 * q^41 - 2178 * q^43 - 2152 * q^46 + 2538 * q^47 - 1104 * q^52 - 1222 * q^53 - 1856 * q^56 - 2720 * q^58 - 11196 * q^61 - 808 * q^62 + 1502 * q^67 + 4336 * q^68 - 12884 * q^71 + 5902 * q^73 - 4480 * q^76 - 6844 * q^77 + 6728 * q^82 - 12462 * q^83 + 8712 * q^86 + 3776 * q^88 + 8004 * q^91 + 4304 * q^92 + 14622 * q^97 - 2876 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/450\mathbb{Z}\right)^\times$.

$n$	$101$	$127$
$\chi(n)$	$1$	$e\left(\frac{1}{4}\right)$

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 $

$2$	−2.00000	−	2.00000i	−0.500000	−	0.500000i
$2$	−2.00000	−	2.00000i	−0.500000	−	0.500000i
$3$		0			0
$3$		0			0
$4$			8.00000i			0.500000i
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−29.0000	−	29.0000i	−0.591837	−	0.591837i	0.346291	−	0.938127i	$-0.387441\pi$
$7$	−29.0000	−	29.0000i	−0.591837	−	0.591837i	−0.938127	+	0.346291i	$0.887441\pi$
$8$	16.0000	−	16.0000i	0.250000	−	0.250000i
$9$		0			0
$10$		0			0
$11$	118.000			0.975207			0.487603	−	0.873065i	$-0.337871\pi$
$11$	118.000			0.975207			0.487603	+	0.873065i	$0.337871\pi$
$12$		0			0
$13$	−69.0000	+	69.0000i	−0.408284	+	0.408284i	−0.881140	−	0.472856i	$-0.843223\pi$
$13$	−69.0000	+	69.0000i	−0.408284	+	0.408284i	0.472856	+	0.881140i	$0.343223\pi$
$14$			116.000i			0.591837i
$15$		0			0
$16$	−64.0000			−0.250000
$17$	−271.000	−	271.000i	−0.937716	−	0.937716i	0.0604547	−	0.998171i	$-0.480745\pi$
$17$	−271.000	−	271.000i	−0.937716	−	0.937716i	−0.998171	+	0.0604547i	$0.980745\pi$
$18$		0			0
$19$			280.000i			0.775623i	0.921739	+	0.387812i	$0.126769\pi$
$19$			280.000i			0.775623i	−0.921739	+	0.387812i	$0.873231\pi$
$20$		0			0
$21$		0			0
$22$	−236.000	−	236.000i	−0.487603	−	0.487603i
$23$	269.000	−	269.000i	0.508507	−	0.508507i	−0.405561	−	0.914068i	$-0.632924\pi$
$23$	269.000	−	269.000i	0.508507	−	0.508507i	0.914068	+	0.405561i	$0.132924\pi$
$24$		0			0
$25$		0			0
$26$	276.000			0.408284
$27$		0			0
$28$	232.000	−	232.000i	0.295918	−	0.295918i
$29$		−	680.000i		−	0.808561i	−0.914635	−	0.404281i	$-0.867522\pi$
$29$		−	680.000i		−	0.808561i	0.914635	−	0.404281i	$-0.132478\pi$
$30$		0			0
$31$	202.000			0.210198			0.105099	−	0.994462i	$-0.466484\pi$
$31$	202.000			0.210198			0.105099	+	0.994462i	$0.466484\pi$
$32$	128.000	+	128.000i	0.125000	+	0.125000i
$33$		0			0
$34$			1084.00i			0.937716i
$35$		0			0
$36$		0			0
$37$	651.000	+	651.000i	0.475530	+	0.475530i	0.903699	−	0.428169i	$-0.140841\pi$
$37$	651.000	+	651.000i	0.475530	+	0.475530i	−0.428169	+	0.903699i	$0.640841\pi$
$38$	560.000	−	560.000i	0.387812	−	0.387812i
$39$		0			0
$40$		0			0
$41$	−1682.00			−1.00059			−0.500297	−	0.865854i	$-0.666776\pi$
$41$	−1682.00			−1.00059			−0.500297	+	0.865854i	$0.666776\pi$
$42$		0			0
$43$	−1089.00	+	1089.00i	−0.588967	+	0.588967i	−0.937352	−	0.348385i	$-0.886730\pi$
$43$	−1089.00	+	1089.00i	−0.588967	+	0.588967i	0.348385	+	0.937352i	$0.386730\pi$
$44$			944.000i			0.487603i
$45$		0			0
$46$	−1076.00			−0.508507
$47$	1269.00	+	1269.00i	0.574468	+	0.574468i	0.933374	−	0.358906i	$-0.116850\pi$
$47$	1269.00	+	1269.00i	0.574468	+	0.574468i	−0.358906	+	0.933374i	$0.616850\pi$
$48$		0			0
$49$		−	719.000i		−	0.299459i
$50$		0			0
$51$		0			0
$52$	−552.000	−	552.000i	−0.204142	−	0.204142i
$53$	−611.000	+	611.000i	−0.217515	+	0.217515i	−0.807450	−	0.589935i	$-0.799153\pi$
$53$	−611.000	+	611.000i	−0.217515	+	0.217515i	0.589935	+	0.807450i	$0.299153\pi$
$54$		0			0
$55$		0			0
$56$	−928.000			−0.295918
$57$		0			0
$58$	−1360.00	+	1360.00i	−0.404281	+	0.404281i
$59$		−	1160.00i		−	0.333238i	−0.986021	−	0.166619i	$-0.946715\pi$
$59$		−	1160.00i		−	0.333238i	0.986021	−	0.166619i	$-0.0532849\pi$
$60$		0			0
$61$	−5598.00			−1.50443			−0.752217	−	0.658915i	$-0.771016\pi$
$61$	−5598.00			−1.50443			−0.752217	+	0.658915i	$0.771016\pi$
$62$	−404.000	−	404.000i	−0.105099	−	0.105099i
$63$		0			0
$64$		−	512.000i		−	0.125000i
$65$		0			0
$66$		0			0
$67$	751.000	+	751.000i	0.167298	+	0.167298i	0.785791	−	0.618493i	$-0.212256\pi$
$67$	751.000	+	751.000i	0.167298	+	0.167298i	−0.618493	+	0.785791i	$0.712256\pi$
$68$	2168.00	−	2168.00i	0.468858	−	0.468858i
$69$		0			0
$70$		0			0
$71$	−6442.00			−1.27792			−0.638961	−	0.769240i	$-0.720635\pi$
$71$	−6442.00			−1.27792			−0.638961	+	0.769240i	$0.720635\pi$
$72$		0			0
$73$	2951.00	−	2951.00i	0.553762	−	0.553762i	−0.373762	−	0.927525i	$-0.621932\pi$
$73$	2951.00	−	2951.00i	0.553762	−	0.553762i	0.927525	+	0.373762i	$0.121932\pi$
$74$		−	2604.00i		−	0.475530i
$75$		0			0
$76$	−2240.00			−0.387812
$77$	−3422.00	−	3422.00i	−0.577163	−	0.577163i
$78$		0			0
$79$			10560.0i			1.69204i	0.533154	+	0.846018i	$0.321007\pi$
$79$			10560.0i			1.69204i	−0.533154	+	0.846018i	$0.678993\pi$
$80$		0			0
$81$		0			0
$82$	3364.00	+	3364.00i	0.500297	+	0.500297i
$83$	−6231.00	+	6231.00i	−0.904485	+	0.904485i	−0.995820	−	0.0913348i	$-0.970887\pi$
$83$	−6231.00	+	6231.00i	−0.904485	+	0.904485i	0.0913348	+	0.995820i	$0.470887\pi$
$84$		0			0
$85$		0			0
$86$	4356.00			0.588967
$87$		0			0
$88$	1888.00	−	1888.00i	0.243802	−	0.243802i
$89$			14480.0i			1.82805i	0.405656	+	0.914026i	$0.367043\pi$
$89$			14480.0i			1.82805i	−0.405656	+	0.914026i	$0.632957\pi$
$90$		0			0
$91$	4002.00			0.483275
$92$	2152.00	+	2152.00i	0.254253	+	0.254253i
$93$		0			0
$94$		−	5076.00i		−	0.574468i
$95$		0			0
$96$		0			0
$97$	7311.00	+	7311.00i	0.777022	+	0.777022i	0.979323	−	0.202301i	$-0.0648420\pi$
$97$	7311.00	+	7311.00i	0.777022	+	0.777022i	−0.202301	+	0.979323i	$0.564842\pi$
$98$	−1438.00	+	1438.00i	−0.149729	+	0.149729i
$99$		0			0
$100$		0			0
$101$	878.000			0.0860700			0.0430350	−	0.999074i	$-0.486297\pi$
$101$	878.000			0.0860700			0.0430350	+	0.999074i	$0.486297\pi$
$102$		0			0
$103$	−10429.0	+	10429.0i	−0.983033	+	0.983033i	−0.999858	−	0.0168252i	$-0.994644\pi$
$103$	−10429.0	+	10429.0i	−0.983033	+	0.983033i	0.0168252	+	0.999858i	$0.494644\pi$
$104$			2208.00i			0.204142i
$105$		0			0
$106$	2444.00			0.217515
$107$	−4711.00	−	4711.00i	−0.411477	−	0.411477i	0.470776	−	0.882253i	$-0.343974\pi$
$107$	−4711.00	−	4711.00i	−0.411477	−	0.411477i	−0.882253	+	0.470776i	$0.843974\pi$
$108$		0			0
$109$			22040.0i			1.85506i	0.373745	+	0.927531i	$0.378073\pi$
$109$			22040.0i			1.85506i	−0.373745	+	0.927531i	$0.621927\pi$
$110$		0			0
$111$		0			0
$112$	1856.00	+	1856.00i	0.147959	+	0.147959i
$113$	−2111.00	+	2111.00i	−0.165322	+	0.165322i	−0.784920	−	0.619597i	$-0.787296\pi$
$113$	−2111.00	+	2111.00i	−0.165322	+	0.165322i	0.619597	+	0.784920i	$0.287296\pi$
$114$		0			0
$115$		0			0
$116$	5440.00			0.404281
$117$		0			0
$118$	−2320.00	+	2320.00i	−0.166619	+	0.166619i
$119$			15718.0i			1.10995i
$120$		0			0
$121$	−717.000			−0.0489721
$122$	11196.0	+	11196.0i	0.752217	+	0.752217i
$123$		0			0
$124$			1616.00i			0.105099i
$125$		0			0
$126$		0			0
$127$	−5909.00	−	5909.00i	−0.366359	−	0.366359i	0.499789	−	0.866147i	$-0.333411\pi$
$127$	−5909.00	−	5909.00i	−0.366359	−	0.366359i	−0.866147	+	0.499789i	$0.833411\pi$
$128$	−1024.00	+	1024.00i	−0.0625000	+	0.0625000i
$129$		0			0
$130$		0			0
$131$	6358.00			0.370491			0.185246	−	0.982692i	$-0.440692\pi$
$131$	6358.00			0.370491			0.185246	+	0.982692i	$0.440692\pi$
$132$		0			0
$133$	8120.00	−	8120.00i	0.459042	−	0.459042i
$134$		−	3004.00i		−	0.167298i
$135$		0			0
$136$	−8672.00			−0.468858
$137$	20409.0	+	20409.0i	1.08738	+	1.08738i	0.995798	+	0.0915804i	$0.0291919\pi$
$137$	20409.0	+	20409.0i	1.08738	+	1.08738i	0.0915804	+	0.995798i	$0.470808\pi$
$138$		0			0
$139$		−	9400.00i		−	0.486517i	−0.969961	−	0.243259i	$-0.921784\pi$
$139$		−	9400.00i		−	0.486517i	0.969961	−	0.243259i	$-0.0782164\pi$
$140$		0			0
$141$		0			0
$142$	12884.0	+	12884.0i	0.638961	+	0.638961i
$143$	−8142.00	+	8142.00i	−0.398161	+	0.398161i
$144$		0			0
$145$		0			0
$146$	−11804.0			−0.553762
$147$		0			0
$148$	−5208.00	+	5208.00i	−0.237765	+	0.237765i
$149$		−	13800.0i		−	0.621594i	−0.950476	−	0.310797i	$-0.899404\pi$
$149$		−	13800.0i		−	0.621594i	0.950476	−	0.310797i	$-0.100596\pi$
$150$		0			0
$151$	−18998.0			−0.833209			−0.416605	−	0.909088i	$-0.636780\pi$
$151$	−18998.0			−0.833209			−0.416605	+	0.909088i	$0.636780\pi$
$152$	4480.00	+	4480.00i	0.193906	+	0.193906i
$153$		0			0
$154$			13688.0i			0.577163i
$155$		0			0
$156$		0			0
$157$	16371.0	+	16371.0i	0.664165	+	0.664165i	0.956359	−	0.292194i	$-0.0943853\pi$
$157$	16371.0	+	16371.0i	0.664165	+	0.664165i	−0.292194	+	0.956359i	$0.594385\pi$
$158$	21120.0	−	21120.0i	0.846018	−	0.846018i
$159$		0			0
$160$		0			0
$161$	−15602.0			−0.601906
$162$		0			0
$163$	−20009.0	+	20009.0i	−0.753096	+	0.753096i	−0.975056	−	0.221960i	$-0.928755\pi$
$163$	−20009.0	+	20009.0i	−0.753096	+	0.753096i	0.221960	+	0.975056i	$0.428755\pi$
$164$		−	13456.0i		−	0.500297i
$165$		0			0
$166$	24924.0			0.904485
$167$	1549.00	+	1549.00i	0.0555416	+	0.0555416i	0.734332	−	0.678790i	$-0.237496\pi$
$167$	1549.00	+	1549.00i	0.0555416	+	0.0555416i	−0.678790	+	0.734332i	$0.737496\pi$
$168$		0			0
$169$			19039.0i			0.666608i
$170$		0			0
$171$		0			0
$172$	−8712.00	−	8712.00i	−0.294484	−	0.294484i
$173$	2789.00	−	2789.00i	0.0931872	−	0.0931872i	−0.658976	−	0.752164i	$-0.729010\pi$
$173$	2789.00	−	2789.00i	0.0931872	−	0.0931872i	0.752164	+	0.658976i	$0.229010\pi$
$174$		0			0
$175$		0			0
$176$	−7552.00			−0.243802
$177$		0			0
$178$	28960.0	−	28960.0i	0.914026	−	0.914026i
$179$			2600.00i			0.0811460i	0.999177	+	0.0405730i	$0.0129183\pi$
$179$			2600.00i			0.0811460i	−0.999177	+	0.0405730i	$0.987082\pi$
$180$		0			0
$181$	−44398.0			−1.35521			−0.677604	−	0.735427i	$-0.736982\pi$
$181$	−44398.0			−1.35521			−0.677604	+	0.735427i	$0.736982\pi$
$182$	−8004.00	−	8004.00i	−0.241637	−	0.241637i
$183$		0			0
$184$		−	8608.00i		−	0.254253i
$185$		0			0
$186$		0			0
$187$	−31978.0	−	31978.0i	−0.914467	−	0.914467i
$188$	−10152.0	+	10152.0i	−0.287234	+	0.287234i
$189$		0			0
$190$		0			0
$191$	14678.0			0.402346			0.201173	−	0.979556i	$-0.435525\pi$
$191$	14678.0			0.402346			0.201173	+	0.979556i	$0.435525\pi$
$192$		0			0
$193$	−42849.0	+	42849.0i	−1.15034	+	1.15034i	−0.163855	+	0.986484i	$0.552393\pi$
$193$	−42849.0	+	42849.0i	−1.15034	+	1.15034i	−0.986484	+	0.163855i	$0.947607\pi$
$194$		−	29244.0i		−	0.777022i
$195$		0			0
$196$	5752.00			0.149729
$197$	−10971.0	−	10971.0i	−0.282692	−	0.282692i	0.551490	−	0.834182i	$-0.314060\pi$
$197$	−10971.0	−	10971.0i	−0.282692	−	0.282692i	−0.834182	+	0.551490i	$0.814060\pi$
$198$		0			0
$199$		−	38160.0i		−	0.963612i	−0.876278	−	0.481806i	$-0.839981\pi$
$199$		−	38160.0i		−	0.963612i	0.876278	−	0.481806i	$-0.160019\pi$
$200$		0			0
$201$		0			0
$202$	−1756.00	−	1756.00i	−0.0430350	−	0.0430350i
$203$	−19720.0	+	19720.0i	−0.478536	+	0.478536i
$204$		0			0
$205$		0			0
$206$	41716.0			0.983033
$207$		0			0
$208$	4416.00	−	4416.00i	0.102071	−	0.102071i
$209$			33040.0i			0.756393i
$210$		0			0
$211$	72842.0			1.63613			0.818063	−	0.575128i	$-0.195048\pi$
$211$	72842.0			1.63613			0.818063	+	0.575128i	$0.195048\pi$
$212$	−4888.00	−	4888.00i	−0.108758	−	0.108758i
$213$		0			0
$214$			18844.0i			0.411477i
$215$		0			0
$216$		0			0
$217$	−5858.00	−	5858.00i	−0.124403	−	0.124403i
$218$	44080.0	−	44080.0i	0.927531	−	0.927531i
$219$		0			0
$220$		0			0
$221$	37398.0			0.765709
$222$		0			0
$223$	30891.0	−	30891.0i	0.621187	−	0.621187i	−0.324648	−	0.945835i	$-0.605246\pi$
$223$	30891.0	−	30891.0i	0.621187	−	0.621187i	0.945835	+	0.324648i	$0.105246\pi$
$224$		−	7424.00i		−	0.147959i
$225$		0			0
$226$	8444.00			0.165322
$227$	−54911.0	−	54911.0i	−1.06563	−	1.06563i	−0.997689	−	0.0679438i	$-0.978356\pi$
$227$	−54911.0	−	54911.0i	−1.06563	−	1.06563i	−0.0679438	−	0.997689i	$-0.521644\pi$
$228$		0			0
$229$		−	50280.0i		−	0.958792i	−0.877599	−	0.479396i	$-0.840856\pi$
$229$		−	50280.0i		−	0.958792i	0.877599	−	0.479396i	$-0.159144\pi$
$230$		0			0
$231$		0			0
$232$	−10880.0	−	10880.0i	−0.202140	−	0.202140i
$233$	−2391.00	+	2391.00i	−0.0440421	+	0.0440421i	−0.728785	−	0.684743i	$-0.759915\pi$
$233$	−2391.00	+	2391.00i	−0.0440421	+	0.0440421i	0.684743	+	0.728785i	$0.259915\pi$
$234$		0			0
$235$		0			0
$236$	9280.00			0.166619
$237$		0			0
$238$	31436.0	−	31436.0i	0.554975	−	0.554975i
$239$			17760.0i			0.310919i	0.987842	+	0.155459i	$0.0496858\pi$
$239$			17760.0i			0.310919i	−0.987842	+	0.155459i	$0.950314\pi$
$240$		0			0
$241$	−28238.0			−0.486183			−0.243092	−	0.970003i	$-0.578162\pi$
$241$	−28238.0			−0.486183			−0.243092	+	0.970003i	$0.578162\pi$
$242$	1434.00	+	1434.00i	0.0244860	+	0.0244860i
$243$		0			0
$244$		−	44784.0i		−	0.752217i
$245$		0			0
$246$		0			0
$247$	−19320.0	−	19320.0i	−0.316675	−	0.316675i
$248$	3232.00	−	3232.00i	0.0525494	−	0.0525494i
$249$		0			0
$250$		0			0
$251$	−121002.			−1.92064			−0.960318	−	0.278907i	$-0.910028\pi$
$251$	−121002.			−1.92064			−0.960318	+	0.278907i	$0.910028\pi$
$252$		0			0
$253$	31742.0	−	31742.0i	0.495899	−	0.495899i
$254$			23636.0i			0.366359i
$255$		0			0
$256$	4096.00			0.0625000
$257$	−72431.0	−	72431.0i	−1.09663	−	1.09663i	−0.994803	−	0.101823i	$-0.967533\pi$
$257$	−72431.0	−	72431.0i	−1.09663	−	1.09663i	−0.101823	−	0.994803i	$-0.532467\pi$
$258$		0			0
$259$		−	37758.0i		−	0.562872i
$260$		0			0
$261$		0			0
$262$	−12716.0	−	12716.0i	−0.185246	−	0.185246i
$263$	−14771.0	+	14771.0i	−0.213549	+	0.213549i	−0.805773	−	0.592224i	$-0.798250\pi$
$263$	−14771.0	+	14771.0i	−0.213549	+	0.213549i	0.592224	+	0.805773i	$0.298250\pi$
$264$		0			0
$265$		0			0
$266$	−32480.0			−0.459042
$267$		0			0
$268$	−6008.00	+	6008.00i	−0.0836489	+	0.0836489i
$269$		−	89720.0i		−	1.23989i	−0.784644	−	0.619947i	$-0.787154\pi$
$269$		−	89720.0i		−	1.23989i	0.784644	−	0.619947i	$-0.212846\pi$
$270$		0			0
$271$	68202.0			0.928664			0.464332	−	0.885661i	$-0.346294\pi$
$271$	68202.0			0.928664			0.464332	+	0.885661i	$0.346294\pi$
$272$	17344.0	+	17344.0i	0.234429	+	0.234429i
$273$		0			0
$274$		−	81636.0i		−	1.08738i
$275$		0			0
$276$		0			0
$277$	−18549.0	−	18549.0i	−0.241747	−	0.241747i	0.575826	−	0.817573i	$-0.304681\pi$
$277$	−18549.0	−	18549.0i	−0.241747	−	0.241747i	−0.817573	+	0.575826i	$0.804681\pi$
$278$	−18800.0	+	18800.0i	−0.243259	+	0.243259i
$279$		0			0
$280$		0			0
$281$	−2322.00			−0.0294069			−0.0147035	−	0.999892i	$-0.504680\pi$
$281$	−2322.00			−0.0294069			−0.0147035	+	0.999892i	$0.504680\pi$
$282$		0			0
$283$	91711.0	−	91711.0i	1.14511	−	1.14511i	0.157613	−	0.987501i	$-0.449620\pi$
$283$	91711.0	−	91711.0i	1.14511	−	1.14511i	0.987501	−	0.157613i	$-0.0503797\pi$
$284$		−	51536.0i		−	0.638961i
$285$		0			0
$286$	32568.0			0.398161
$287$	48778.0	+	48778.0i	0.592189	+	0.592189i
$288$		0			0
$289$			63361.0i			0.758624i
$290$		0			0
$291$		0			0
$292$	23608.0	+	23608.0i	0.276881	+	0.276881i
$293$	−4851.00	+	4851.00i	−0.0565062	+	0.0565062i	−0.734795	−	0.678289i	$-0.762722\pi$
$293$	−4851.00	+	4851.00i	−0.0565062	+	0.0565062i	0.678289	+	0.734795i	$0.262722\pi$
$294$		0			0
$295$		0			0
$296$	20832.0			0.237765
$297$		0			0
$298$	−27600.0	+	27600.0i	−0.310797	+	0.310797i
$299$			37122.0i			0.415230i
$300$		0			0
$301$	63162.0			0.697145
$302$	37996.0	+	37996.0i	0.416605	+	0.416605i
$303$		0			0
$304$		−	17920.0i		−	0.193906i
$305$		0			0
$306$		0			0
$307$	−42849.0	−	42849.0i	−0.454636	−	0.454636i	0.442254	−	0.896890i	$-0.354179\pi$
$307$	−42849.0	−	42849.0i	−0.454636	−	0.454636i	−0.896890	+	0.442254i	$0.854179\pi$
$308$	27376.0	−	27376.0i	0.288582	−	0.288582i
$309$		0			0
$310$		0			0
$311$	72278.0			0.747283			0.373642	−	0.927573i	$-0.378109\pi$
$311$	72278.0			0.747283			0.373642	+	0.927573i	$0.378109\pi$
$312$		0			0
$313$	−18249.0	+	18249.0i	−0.186273	+	0.186273i	−0.794083	−	0.607810i	$-0.792048\pi$
$313$	−18249.0	+	18249.0i	−0.186273	+	0.186273i	0.607810	+	0.794083i	$0.292048\pi$
$314$		−	65484.0i		−	0.664165i
$315$		0			0
$316$	−84480.0			−0.846018
$317$	25149.0	+	25149.0i	0.250266	+	0.250266i	0.821080	−	0.570814i	$-0.193372\pi$
$317$	25149.0	+	25149.0i	0.250266	+	0.250266i	−0.570814	+	0.821080i	$0.693372\pi$
$318$		0			0
$319$		−	80240.0i		−	0.788514i
$320$		0			0
$321$		0			0
$322$	31204.0	+	31204.0i	0.300953	+	0.300953i
$323$	75880.0	−	75880.0i	0.727315	−	0.727315i
$324$		0			0
$325$		0			0
$326$	80036.0			0.753096
$327$		0			0
$328$	−26912.0	+	26912.0i	−0.250149	+	0.250149i
$329$		−	73602.0i		−	0.679983i
$330$		0			0
$331$	−54038.0			−0.493223			−0.246611	−	0.969114i	$-0.579317\pi$
$331$	−54038.0			−0.493223			−0.246611	+	0.969114i	$0.579317\pi$
$332$	−49848.0	−	49848.0i	−0.452243	−	0.452243i
$333$		0			0
$334$		−	6196.00i		−	0.0555416i
$335$		0			0
$336$		0			0
$337$	−8529.00	−	8529.00i	−0.0750997	−	0.0750997i	0.668559	−	0.743659i	$-0.266911\pi$
$337$	−8529.00	−	8529.00i	−0.0750997	−	0.0750997i	−0.743659	+	0.668559i	$0.766911\pi$
$338$	38078.0	−	38078.0i	0.333304	−	0.333304i
$339$		0			0
$340$		0			0
$341$	23836.0			0.204986
$342$		0			0
$343$	−90480.0	+	90480.0i	−0.769067	+	0.769067i
$344$			34848.0i			0.294484i
$345$		0			0
$346$	−11156.0			−0.0931872
$347$	−56551.0	−	56551.0i	−0.469658	−	0.469658i	0.432146	−	0.901804i	$-0.357756\pi$
$347$	−56551.0	−	56551.0i	−0.469658	−	0.469658i	−0.901804	+	0.432146i	$0.857756\pi$
$348$		0			0
$349$		−	22520.0i		−	0.184892i	−0.995718	−	0.0924459i	$-0.970531\pi$
$349$		−	22520.0i		−	0.184892i	0.995718	−	0.0924459i	$-0.0294685\pi$
$350$		0			0
$351$		0			0
$352$	15104.0	+	15104.0i	0.121901	+	0.121901i
$353$	−44511.0	+	44511.0i	−0.357205	+	0.357205i	−0.862782	−	0.505576i	$-0.831280\pi$
$353$	−44511.0	+	44511.0i	−0.357205	+	0.357205i	0.505576	+	0.862782i	$0.331280\pi$
$354$		0			0
$355$		0			0
$356$	−115840.			−0.914026
$357$		0			0
$358$	5200.00	−	5200.00i	0.0405730	−	0.0405730i
$359$			9680.00i			0.0751080i	0.999295	+	0.0375540i	$0.0119566\pi$
$359$			9680.00i			0.0751080i	−0.999295	+	0.0375540i	$0.988043\pi$
$360$		0			0
$361$	51921.0			0.398409
$362$	88796.0	+	88796.0i	0.677604	+	0.677604i
$363$		0			0
$364$			32016.0i			0.241637i
$365$		0			0
$366$		0			0
$367$	14971.0	+	14971.0i	0.111152	+	0.111152i	0.760496	−	0.649343i	$-0.224956\pi$
$367$	14971.0	+	14971.0i	0.111152	+	0.111152i	−0.649343	+	0.760496i	$0.724956\pi$
$368$	−17216.0	+	17216.0i	−0.127127	+	0.127127i
$369$		0			0
$370$		0			0
$371$	35438.0			0.257467
$372$		0			0
$373$	13811.0	−	13811.0i	0.0992676	−	0.0992676i	−0.655729	−	0.754996i	$-0.727639\pi$
$373$	13811.0	−	13811.0i	0.0992676	−	0.0992676i	0.754996	+	0.655729i	$0.227639\pi$
$374$			127912.i			0.914467i
$375$		0			0
$376$	40608.0			0.287234
$377$	46920.0	+	46920.0i	0.330123	+	0.330123i
$378$		0			0
$379$			251080.i			1.74797i	0.485954	+	0.873984i	$0.338472\pi$
$379$			251080.i			1.74797i	−0.485954	+	0.873984i	$0.661528\pi$
$380$		0			0
$381$		0			0
$382$	−29356.0	−	29356.0i	−0.201173	−	0.201173i
$383$	−86091.0	+	86091.0i	−0.586895	+	0.586895i	−0.936789	−	0.349894i	$-0.886217\pi$
$383$	−86091.0	+	86091.0i	−0.586895	+	0.586895i	0.349894	+	0.936789i	$0.386217\pi$
$384$		0			0
$385$		0			0
$386$	171396.			1.15034
$387$		0			0
$388$	−58488.0	+	58488.0i	−0.388511	+	0.388511i
$389$			75000.0i			0.495635i	0.968807	+	0.247818i	$0.0797134\pi$
$389$			75000.0i			0.495635i	−0.968807	+	0.247818i	$0.920287\pi$
$390$		0			0
$391$	−145798.			−0.953670
$392$	−11504.0	−	11504.0i	−0.0748646	−	0.0748646i
$393$		0			0
$394$			43884.0i			0.282692i
$395$		0			0
$396$		0			0
$397$	−29149.0	−	29149.0i	−0.184945	−	0.184945i	0.608562	−	0.793507i	$-0.291747\pi$
$397$	−29149.0	−	29149.0i	−0.184945	−	0.184945i	−0.793507	+	0.608562i	$0.791747\pi$
$398$	−76320.0	+	76320.0i	−0.481806	+	0.481806i
$399$		0			0
$400$		0			0
$401$	45918.0			0.285558			0.142779	−	0.989755i	$-0.454396\pi$
$401$	45918.0			0.285558			0.142779	+	0.989755i	$0.454396\pi$
$402$		0			0
$403$	−13938.0	+	13938.0i	−0.0858204	+	0.0858204i
$404$			7024.00i			0.0430350i
$405$		0			0
$406$	78880.0			0.478536
$407$	76818.0	+	76818.0i	0.463740	+	0.463740i
$408$		0			0
$409$			78720.0i			0.470585i	0.971925	+	0.235293i	$0.0756049\pi$
$409$			78720.0i			0.470585i	−0.971925	+	0.235293i	$0.924395\pi$
$410$		0			0
$411$		0			0
$412$	−83432.0	−	83432.0i	−0.491517	−	0.491517i
$413$	−33640.0	+	33640.0i	−0.197222	+	0.197222i
$414$		0			0
$415$		0			0
$416$	−17664.0			−0.102071
$417$		0			0
$418$	66080.0	−	66080.0i	0.378196	−	0.378196i
$419$			14760.0i			0.0840733i	0.999116	+	0.0420367i	$0.0133846\pi$
$419$			14760.0i			0.0840733i	−0.999116	+	0.0420367i	$0.986615\pi$
$420$		0			0
$421$	221282.			1.24848			0.624240	−	0.781232i	$-0.285409\pi$
$421$	221282.			1.24848			0.624240	+	0.781232i	$0.285409\pi$
$422$	−145684.	−	145684.i	−0.818063	−	0.818063i
$423$		0			0
$424$			19552.0i			0.108758i
$425$		0			0
$426$		0			0
$427$	162342.	+	162342.i	0.890379	+	0.890379i
$428$	37688.0	−	37688.0i	0.205738	−	0.205738i
$429$		0			0
$430$		0			0
$431$	−212522.			−1.14406			−0.572031	−	0.820232i	$-0.693844\pi$
$431$	−212522.			−1.14406			−0.572031	+	0.820232i	$0.693844\pi$
$432$		0			0
$433$	−145409.	+	145409.i	−0.775560	+	0.775560i	−0.979072	−	0.203512i	$-0.934764\pi$
$433$	−145409.	+	145409.i	−0.775560	+	0.775560i	0.203512	+	0.979072i	$0.434764\pi$
$434$			23432.0i			0.124403i
$435$		0			0
$436$	−176320.			−0.927531
$437$	75320.0	+	75320.0i	0.394410	+	0.394410i
$438$		0			0
$439$		−	299440.i		−	1.55375i	−0.629656	−	0.776874i	$-0.716804\pi$
$439$		−	299440.i		−	1.55375i	0.629656	−	0.776874i	$-0.283196\pi$
$440$		0			0
$441$		0			0
$442$	−74796.0	−	74796.0i	−0.382855	−	0.382855i
$443$	240609.	−	240609.i	1.22604	−	1.22604i	0.260590	−	0.965450i	$-0.416083\pi$
$443$	240609.	−	240609.i	1.22604	−	1.22604i	0.965450	−	0.260590i	$-0.0839170\pi$
$444$		0			0
$445$		0			0
$446$	−123564.			−0.621187
$447$		0			0
$448$	−14848.0	+	14848.0i	−0.0739796	+	0.0739796i
$449$			82480.0i			0.409125i	0.978854	+	0.204562i	$0.0655772\pi$
$449$			82480.0i			0.409125i	−0.978854	+	0.204562i	$0.934423\pi$
$450$		0			0
$451$	−198476.			−0.975787
$452$	−16888.0	−	16888.0i	−0.0826611	−	0.0826611i
$453$		0			0
$454$			219644.i			1.06563i
$455$		0			0
$456$		0			0
$457$	188151.	+	188151.i	0.900895	+	0.900895i	0.995514	−	0.0946187i	$-0.0301632\pi$
$457$	188151.	+	188151.i	0.900895	+	0.900895i	−0.0946187	+	0.995514i	$0.530163\pi$
$458$	−100560.	+	100560.i	−0.479396	+	0.479396i
$459$		0			0
$460$		0			0
$461$	326158.			1.53471			0.767355	−	0.641223i	$-0.221573\pi$
$461$	326158.			1.53471			0.767355	+	0.641223i	$0.221573\pi$
$462$		0			0
$463$	218731.	−	218731.i	1.02035	−	1.02035i	0.0205595	−	0.999789i	$-0.493455\pi$
$463$	218731.	−	218731.i	1.02035	−	1.02035i	0.999789	−	0.0205595i	$-0.00654474\pi$
$464$			43520.0i			0.202140i
$465$		0			0
$466$	9564.00			0.0440421
$467$	59249.0	+	59249.0i	0.271673	+	0.271673i	0.829774	−	0.558100i	$-0.188470\pi$
$467$	59249.0	+	59249.0i	0.271673	+	0.271673i	−0.558100	+	0.829774i	$0.688470\pi$
$468$		0			0
$469$		−	43558.0i		−	0.198026i
$470$		0			0
$471$		0			0
$472$	−18560.0	−	18560.0i	−0.0833094	−	0.0833094i
$473$	−128502.	+	128502.i	−0.574365	+	0.574365i
$474$		0			0
$475$		0			0
$476$	−125744.			−0.554975
$477$		0			0
$478$	35520.0	−	35520.0i	0.155459	−	0.155459i
$479$		−	273440.i		−	1.19177i	−0.803071	−	0.595883i	$-0.796802\pi$
$479$		−	273440.i		−	1.19177i	0.803071	−	0.595883i	$-0.203198\pi$
$480$		0			0
$481$	−89838.0			−0.388302
$482$	56476.0	+	56476.0i	0.243092	+	0.243092i
$483$		0			0
$484$		−	5736.00i		−	0.0244860i
$485$		0			0
$486$		0			0
$487$	123651.	+	123651.i	0.521362	+	0.521362i	0.917983	−	0.396620i	$-0.129817\pi$
$487$	123651.	+	123651.i	0.521362	+	0.521362i	−0.396620	+	0.917983i	$0.629817\pi$
$488$	−89568.0	+	89568.0i	−0.376109	+	0.376109i
$489$		0			0
$490$		0			0
$491$	−198442.			−0.823134			−0.411567	−	0.911379i	$-0.635018\pi$
$491$	−198442.			−0.823134			−0.411567	+	0.911379i	$0.635018\pi$
$492$		0			0
$493$	−184280.	+	184280.i	−0.758201	+	0.758201i
$494$			77280.0i			0.316675i
$495$		0			0
$496$	−12928.0			−0.0525494
$497$	186818.	+	186818.i	0.756321	+	0.756321i
$498$		0			0
$499$			269240.i			1.08128i	0.841254	+	0.540640i	$0.181818\pi$
$499$			269240.i			1.08128i	−0.841254	+	0.540640i	$0.818182\pi$
$500$		0			0
$501$		0			0
$502$	242004.	+	242004.i	0.960318	+	0.960318i
$503$	109869.	−	109869.i	0.434249	−	0.434249i	−0.455822	−	0.890071i	$-0.650655\pi$
$503$	109869.	−	109869.i	0.434249	−	0.434249i	0.890071	+	0.455822i	$0.150655\pi$
$504$		0			0
$505$		0			0
$506$	−126968.			−0.495899
$507$		0			0
$508$	47272.0	−	47272.0i	0.183179	−	0.183179i
$509$			211000.i			0.814417i	0.913335	+	0.407209i	$0.133498\pi$
$509$			211000.i			0.814417i	−0.913335	+	0.407209i	$0.866502\pi$
$510$		0			0
$511$	−171158.			−0.655474
$512$	−8192.00	−	8192.00i	−0.0312500	−	0.0312500i
$513$		0			0
$514$			289724.i			1.09663i
$515$		0			0
$516$		0			0
$517$	149742.	+	149742.i	0.560225	+	0.560225i
$518$	−75516.0	+	75516.0i	−0.281436	+	0.281436i
$519$		0			0
$520$		0			0
$521$	−297282.			−1.09520			−0.547600	−	0.836740i	$-0.684458\pi$
$521$	−297282.			−1.09520			−0.547600	+	0.836740i	$0.684458\pi$
$522$		0			0
$523$	25071.0	−	25071.0i	0.0916576	−	0.0916576i	−0.659791	−	0.751449i	$-0.729355\pi$
$523$	25071.0	−	25071.0i	0.0916576	−	0.0916576i	0.751449	+	0.659791i	$0.229355\pi$
$524$			50864.0i			0.185246i
$525$		0			0
$526$	59084.0			0.213549
$527$	−54742.0	−	54742.0i	−0.197106	−	0.197106i
$528$		0			0
$529$			135119.i			0.482842i
$530$		0			0
$531$		0			0
$532$	64960.0	+	64960.0i	0.229521	+	0.229521i
$533$	116058.	−	116058.i	0.408527	−	0.408527i
$534$		0			0
$535$		0			0
$536$	24032.0			0.0836489
$537$		0			0
$538$	−179440.	+	179440.i	−0.619947	+	0.619947i
$539$		−	84842.0i		−	0.292034i
$540$		0			0
$541$	−142478.			−0.486803			−0.243402	−	0.969926i	$-0.578263\pi$
$541$	−142478.			−0.486803			−0.243402	+	0.969926i	$0.578263\pi$
$542$	−136404.	−	136404.i	−0.464332	−	0.464332i
$543$		0			0
$544$		−	69376.0i		−	0.234429i
$545$		0			0
$546$		0			0
$547$	−291009.	−	291009.i	−0.972594	−	0.972594i	0.0270399	−	0.999634i	$-0.491392\pi$
$547$	−291009.	−	291009.i	−0.972594	−	0.972594i	−0.999634	+	0.0270399i	$0.991392\pi$
$548$	−163272.	+	163272.i	−0.543689	+	0.543689i
$549$		0			0
$550$		0			0
$551$	190400.			0.627139
$552$		0			0
$553$	306240.	−	306240.i	1.00141	−	1.00141i
$554$			74196.0i			0.241747i
$555$		0			0
$556$	75200.0			0.243259
$557$	−83091.0	−	83091.0i	−0.267820	−	0.267820i	0.560401	−	0.828221i	$-0.310647\pi$
$557$	−83091.0	−	83091.0i	−0.267820	−	0.267820i	−0.828221	+	0.560401i	$0.810647\pi$
$558$		0			0
$559$		−	150282.i		−	0.480932i
$560$		0			0
$561$		0			0
$562$	4644.00	+	4644.00i	0.0147035	+	0.0147035i
$563$	43449.0	−	43449.0i	0.137076	−	0.137076i	−0.635239	−	0.772316i	$-0.719098\pi$
$563$	43449.0	−	43449.0i	0.137076	−	0.137076i	0.772316	+	0.635239i	$0.219098\pi$
$564$		0			0
$565$		0			0
$566$	−366844.			−1.14511
$567$		0			0
$568$	−103072.	+	103072.i	−0.319480	+	0.319480i
$569$			270560.i			0.835678i	0.908521	+	0.417839i	$0.137212\pi$
$569$			270560.i			0.835678i	−0.908521	+	0.417839i	$0.862788\pi$
$570$		0			0
$571$	57482.0			0.176303			0.0881515	−	0.996107i	$-0.471904\pi$
$571$	57482.0			0.176303			0.0881515	+	0.996107i	$0.471904\pi$
$572$	−65136.0	−	65136.0i	−0.199081	−	0.199081i
$573$		0			0
$574$		−	195112.i		−	0.592189i
$575$		0			0
$576$		0			0
$577$	−195889.	−	195889.i	−0.588381	−	0.588381i	0.348812	−	0.937193i	$-0.386585\pi$
$577$	−195889.	−	195889.i	−0.588381	−	0.588381i	−0.937193	+	0.348812i	$0.886585\pi$
$578$	126722.	−	126722.i	0.379312	−	0.379312i
$579$		0			0
$580$		0			0
$581$	361398.			1.07062
$582$		0			0
$583$	−72098.0	+	72098.0i	−0.212122	+	0.212122i
$584$		−	94432.0i		−	0.276881i
$585$		0			0
$586$	19404.0			0.0565062
$587$	−404631.	−	404631.i	−1.17431	−	1.17431i	−0.981171	−	0.193139i	$-0.938133\pi$
$587$	−404631.	−	404631.i	−1.17431	−	1.17431i	−0.193139	−	0.981171i	$-0.561867\pi$
$588$		0			0
$589$			56560.0i			0.163034i
$590$		0			0
$591$		0			0
$592$	−41664.0	−	41664.0i	−0.118882	−	0.118882i
$593$	−210991.	+	210991.i	−0.600005	+	0.600005i	−0.940314	−	0.340309i	$-0.889468\pi$
$593$	−210991.	+	210991.i	−0.600005	+	0.600005i	0.340309	+	0.940314i	$0.389468\pi$
$594$		0			0
$595$		0			0
$596$	110400.			0.310797
$597$		0			0
$598$	74244.0	−	74244.0i	0.207615	−	0.207615i
$599$		−	300560.i		−	0.837679i	−0.908060	−	0.418839i	$-0.862437\pi$
$599$		−	300560.i		−	0.837679i	0.908060	−	0.418839i	$-0.137563\pi$
$600$		0			0
$601$	367442.			1.01728			0.508639	−	0.860980i	$-0.330149\pi$
$601$	367442.			1.01728			0.508639	+	0.860980i	$0.330149\pi$
$602$	−126324.	−	126324.i	−0.348572	−	0.348572i
$603$		0			0
$604$		−	151984.i		−	0.416605i
$605$		0			0
$606$		0			0
$607$	−146469.	−	146469.i	−0.397529	−	0.397529i	0.479832	−	0.877360i	$-0.340698\pi$
$607$	−146469.	−	146469.i	−0.397529	−	0.397529i	−0.877360	+	0.479832i	$0.840698\pi$
$608$	−35840.0	+	35840.0i	−0.0969529	+	0.0969529i
$609$		0			0
$610$		0			0
$611$	−175122.			−0.469092
$612$		0			0
$613$	−160989.	+	160989.i	−0.428425	+	0.428425i	−0.888092	−	0.459666i	$-0.847969\pi$
$613$	−160989.	+	160989.i	−0.428425	+	0.428425i	0.459666	+	0.888092i	$0.347969\pi$
$614$			171396.i			0.454636i
$615$		0			0
$616$	−109504.			−0.288582
$617$	320409.	+	320409.i	0.841656	+	0.841656i	0.989074	−	0.147419i	$-0.0470965\pi$
$617$	320409.	+	320409.i	0.841656	+	0.841656i	−0.147419	+	0.989074i	$0.547096\pi$
$618$		0			0
$619$			341160.i			0.890383i	0.895435	+	0.445191i	$0.146864\pi$
$619$			341160.i			0.890383i	−0.895435	+	0.445191i	$0.853136\pi$
$620$		0			0
$621$		0			0
$622$	−144556.	−	144556.i	−0.373642	−	0.373642i
$623$	419920.	−	419920.i	1.08191	−	1.08191i
$624$		0			0
$625$		0			0
$626$	72996.0			0.186273
$627$		0			0
$628$	−130968.	+	130968.i	−0.332082	+	0.332082i
$629$		−	352842.i		−	0.891824i
$630$		0			0
$631$	−390998.			−0.982010			−0.491005	−	0.871157i	$-0.663370\pi$
$631$	−390998.			−0.982010			−0.491005	+	0.871157i	$0.663370\pi$
$632$	168960.	+	168960.i	0.423009	+	0.423009i
$633$		0			0
$634$		−	100596.i		−	0.250266i
$635$		0			0
$636$		0			0
$637$	49611.0	+	49611.0i	0.122264	+	0.122264i
$638$	−160480.	+	160480.i	−0.394257	+	0.394257i
$639$		0			0
$640$		0			0
$641$	585038.			1.42386			0.711931	−	0.702249i	$-0.247821\pi$
$641$	585038.			1.42386			0.711931	+	0.702249i	$0.247821\pi$
$642$		0			0
$643$	31911.0	−	31911.0i	0.0771824	−	0.0771824i	−0.667462	−	0.744644i	$-0.732619\pi$
$643$	31911.0	−	31911.0i	0.0771824	−	0.0771824i	0.744644	+	0.667462i	$0.232619\pi$
$644$		−	124816.i		−	0.300953i
$645$		0			0
$646$	−303520.			−0.727315
$647$	−280931.	−	280931.i	−0.671106	−	0.671106i	0.286865	−	0.957971i	$-0.407387\pi$
$647$	−280931.	−	280931.i	−0.671106	−	0.671106i	−0.957971	+	0.286865i	$0.907387\pi$
$648$		0			0
$649$		−	136880.i		−	0.324975i
$650$		0			0
$651$		0			0
$652$	−160072.	−	160072.i	−0.376548	−	0.376548i
$653$	523989.	−	523989.i	1.22884	−	1.22884i	0.264439	−	0.964402i	$-0.414813\pi$
$653$	523989.	−	523989.i	1.22884	−	1.22884i	0.964402	−	0.264439i	$-0.0851867\pi$
$654$		0			0
$655$		0			0
$656$	107648.			0.250149
$657$		0			0
$658$	−147204.	+	147204.i	−0.339991	+	0.339991i
$659$			404360.i			0.931102i	0.885021	+	0.465551i	$0.154144\pi$
$659$			404360.i			0.931102i	−0.885021	+	0.465551i	$0.845856\pi$
$660$		0			0
$661$	−5278.00			−0.0120800			−0.00603999	−	0.999982i	$-0.501923\pi$
$661$	−5278.00			−0.0120800			−0.00603999	+	0.999982i	$0.501923\pi$
$662$	108076.	+	108076.i	0.246611	+	0.246611i
$663$		0			0
$664$			199392.i			0.452243i
$665$		0			0
$666$		0			0
$667$	−182920.	−	182920.i	−0.411159	−	0.411159i
$668$	−12392.0	+	12392.0i	−0.0277708	+	0.0277708i
$669$		0			0
$670$		0			0
$671$	−660564.			−1.46713
$672$		0			0
$673$	332111.	−	332111.i	0.733252	−	0.733252i	−0.238011	−	0.971263i	$-0.576495\pi$
$673$	332111.	−	332111.i	0.733252	−	0.733252i	0.971263	+	0.238011i	$0.0764953\pi$
$674$			34116.0i			0.0750997i
$675$		0			0
$676$	−152312.			−0.333304
$677$	578309.	+	578309.i	1.26178	+	1.26178i	0.950231	+	0.311546i	$0.100847\pi$
$677$	578309.	+	578309.i	1.26178	+	1.26178i	0.311546	+	0.950231i	$0.399153\pi$
$678$		0			0
$679$		−	424038.i		−	0.919740i
$680$		0			0
$681$		0			0
$682$	−47672.0	−	47672.0i	−0.102493	−	0.102493i
$683$	−349311.	+	349311.i	−0.748809	+	0.748809i	−0.974255	−	0.225447i	$-0.927616\pi$
$683$	−349311.	+	349311.i	−0.748809	+	0.748809i	0.225447	+	0.974255i	$0.427616\pi$
$684$		0			0
$685$		0			0
$686$	361920.			0.769067
$687$		0			0
$688$	69696.0	−	69696.0i	0.147242	−	0.147242i
$689$		−	84318.0i		−	0.177616i
$690$		0			0
$691$	282762.			0.592195			0.296098	−	0.955158i	$-0.404315\pi$
$691$	282762.			0.592195			0.296098	+	0.955158i	$0.404315\pi$
$692$	22312.0	+	22312.0i	0.0465936	+	0.0465936i
$693$		0			0
$694$			226204.i			0.469658i
$695$		0			0
$696$		0			0
$697$	455822.	+	455822.i	0.938274	+	0.938274i
$698$	−45040.0	+	45040.0i	−0.0924459	+	0.0924459i
$699$		0			0
$700$		0			0
$701$	−270242.			−0.549942			−0.274971	−	0.961453i	$-0.588668\pi$
$701$	−270242.			−0.549942			−0.274971	+	0.961453i	$0.588668\pi$
$702$		0			0
$703$	−182280.	+	182280.i	−0.368832	+	0.368832i
$704$		−	60416.0i		−	0.121901i
$705$		0			0
$706$	178044.			0.357205
$707$	−25462.0	−	25462.0i	−0.0509394	−	0.0509394i
$708$		0			0
$709$		−	297800.i		−	0.592423i	−0.955122	−	0.296212i	$-0.904277\pi$
$709$		−	297800.i		−	0.592423i	0.955122	−	0.296212i	$-0.0957234\pi$
$710$		0			0
$711$		0			0
$712$	231680.	+	231680.i	0.457013	+	0.457013i
$713$	54338.0	−	54338.0i	0.106887	−	0.106887i
$714$		0			0
$715$		0			0
$716$	−20800.0			−0.0405730
$717$		0			0
$718$	19360.0	−	19360.0i	0.0375540	−	0.0375540i
$719$		−	913760.i		−	1.76756i	−0.467902	−	0.883780i	$-0.654990\pi$
$719$		−	913760.i		−	1.76756i	0.467902	−	0.883780i	$-0.345010\pi$
$720$		0			0
$721$	604882.			1.16359
$722$	−103842.	−	103842.i	−0.199204	−	0.199204i
$723$		0			0
$724$		−	355184.i		−	0.677604i
$725$		0			0
$726$		0			0
$727$	417651.	+	417651.i	0.790214	+	0.790214i	0.981529	−	0.191315i	$-0.0612751\pi$
$727$	417651.	+	417651.i	0.790214	+	0.790214i	−0.191315	+	0.981529i	$0.561275\pi$
$728$	64032.0	−	64032.0i	0.120819	−	0.120819i
$729$		0			0
$730$		0			0
$731$	590238.			1.10457
$732$		0			0
$733$	−394549.	+	394549.i	−0.734333	+	0.734333i	−0.971475	−	0.237142i	$-0.923789\pi$
$733$	−394549.	+	394549.i	−0.734333	+	0.734333i	0.237142	+	0.971475i	$0.423789\pi$
$734$		−	59884.0i		−	0.111152i
$735$		0			0
$736$	68864.0			0.127127
$737$	88618.0	+	88618.0i	0.163150	+	0.163150i
$738$		0			0
$739$			109880.i			0.201201i	0.994927	+	0.100600i	$0.0320764\pi$
$739$			109880.i			0.201201i	−0.994927	+	0.100600i	$0.967924\pi$
$740$		0			0
$741$		0			0
$742$	−70876.0	−	70876.0i	−0.128733	−	0.128733i
$743$	−466451.	+	466451.i	−0.844945	+	0.844945i	−0.989497	−	0.144552i	$-0.953826\pi$
$743$	−466451.	+	466451.i	−0.844945	+	0.844945i	0.144552	+	0.989497i	$0.453826\pi$
$744$		0			0
$745$		0			0
$746$	−55244.0			−0.0992676
$747$		0			0
$748$	255824.	−	255824.i	0.457234	−	0.457234i
$749$			273238.i			0.487054i
$750$		0			0
$751$	1.01092e6			1.79241			0.896206	−	0.443638i	$-0.146313\pi$
$751$	1.01092e6			1.79241			0.896206	+	0.443638i	$0.146313\pi$
$752$	−81216.0	−	81216.0i	−0.143617	−	0.143617i
$753$		0			0
$754$		−	187680.i		−	0.330123i
$755$		0			0
$756$		0			0
$757$	−313269.	−	313269.i	−0.546671	−	0.546671i	0.378806	−	0.925476i	$-0.376335\pi$
$757$	−313269.	−	313269.i	−0.546671	−	0.546671i	−0.925476	+	0.378806i	$0.876335\pi$
$758$	502160.	−	502160.i	0.873984	−	0.873984i
$759$		0			0
$760$		0			0
$761$	−142082.			−0.245341			−0.122670	−	0.992447i	$-0.539146\pi$
$761$	−142082.			−0.245341			−0.122670	+	0.992447i	$0.539146\pi$
$762$		0			0
$763$	639160.	−	639160.i	1.09789	−	1.09789i
$764$			117424.i			0.201173i
$765$		0			0
$766$	344364.			0.586895
$767$	80040.0	+	80040.0i	0.136056	+	0.136056i
$768$		0			0
$769$			13280.0i			0.0224567i	0.999937	+	0.0112283i	$0.00357417\pi$
$769$			13280.0i			0.0224567i	−0.999937	+	0.0112283i	$0.996426\pi$
$770$		0			0
$771$		0			0
$772$	−342792.	−	342792.i	−0.575170	−	0.575170i
$773$	−782211.	+	782211.i	−1.30908	+	1.30908i	−0.386994	+	0.922082i	$0.626487\pi$
$773$	−782211.	+	782211.i	−1.30908	+	1.30908i	−0.922082	+	0.386994i	$0.873513\pi$
$774$		0			0
$775$		0			0
$776$	233952.			0.388511
$777$		0			0
$778$	150000.	−	150000.i	0.247818	−	0.247818i
$779$		−	470960.i		−	0.776085i
$780$		0			0
$781$	−760156.			−1.24624
$782$	291596.	+	291596.i	0.476835	+	0.476835i
$783$		0			0
$784$			46016.0i			0.0748646i
$785$		0			0
$786$		0			0
$787$	−201409.	−	201409.i	−0.325184	−	0.325184i	0.525568	−	0.850752i	$-0.323853\pi$
$787$	−201409.	−	201409.i	−0.325184	−	0.325184i	−0.850752	+	0.525568i	$0.823853\pi$
$788$	87768.0	−	87768.0i	0.141346	−	0.141346i
$789$		0			0
$790$		0			0
$791$	122438.			0.195688
$792$		0			0
$793$	386262.	−	386262.i	0.614236	−	0.614236i
$794$			116596.i			0.184945i
$795$		0			0
$796$	305280.			0.481806
$797$	−36291.0	−	36291.0i	−0.0571324	−	0.0571324i	0.677963	−	0.735096i	$-0.262863\pi$
$797$	−36291.0	−	36291.0i	−0.0571324	−	0.0571324i	−0.735096	+	0.677963i	$0.762863\pi$
$798$		0			0
$799$		−	687798.i		−	1.07738i
$800$		0			0
$801$		0			0
$802$	−91836.0	−	91836.0i	−0.142779	−	0.142779i
$803$	348218.	−	348218.i	0.540033	−	0.540033i
$804$		0			0
$805$		0			0
$806$	55752.0			0.0858204
$807$		0			0
$808$	14048.0	−	14048.0i	0.0215175	−	0.0215175i
$809$			71600.0i			0.109400i	0.998503	+	0.0546998i	$0.0174202\pi$
$809$			71600.0i			0.109400i	−0.998503	+	0.0546998i	$0.982580\pi$
$810$		0			0
$811$	−103318.			−0.157085			−0.0785424	−	0.996911i	$-0.525027\pi$
$811$	−103318.			−0.157085			−0.0785424	+	0.996911i	$0.525027\pi$
$812$	−157760.	−	157760.i	−0.239268	−	0.239268i
$813$		0			0
$814$		−	307272.i		−	0.463740i
$815$		0			0
$816$		0			0
$817$	−304920.	−	304920.i	−0.456817	−	0.456817i
$818$	157440.	−	157440.i	0.235293	−	0.235293i
$819$		0			0
$820$		0			0
$821$	157438.			0.233573			0.116787	−	0.993157i	$-0.462741\pi$
$821$	157438.			0.233573			0.116787	+	0.993157i	$0.462741\pi$
$822$		0			0
$823$	−791309.	+	791309.i	−1.16828	+	1.16828i	−0.185666	+	0.982613i	$0.559444\pi$
$823$	−791309.	+	791309.i	−1.16828	+	1.16828i	−0.982613	+	0.185666i	$0.940556\pi$
$824$			333728.i			0.491517i
$825$		0			0
$826$	134560.			0.197222
$827$	−889671.	−	889671.i	−1.30082	−	1.30082i	−0.927837	−	0.372987i	$-0.878334\pi$
$827$	−889671.	−	889671.i	−1.30082	−	1.30082i	−0.372987	−	0.927837i	$-0.621666\pi$
$828$		0			0
$829$		−	618280.i		−	0.899655i	−0.893115	−	0.449828i	$-0.851485\pi$
$829$		−	618280.i		−	0.899655i	0.893115	−	0.449828i	$-0.148515\pi$
$830$		0			0
$831$		0			0
$832$	35328.0	+	35328.0i	0.0510355	+	0.0510355i
$833$	−194849.	+	194849.i	−0.280807	+	0.280807i
$834$		0			0
$835$		0			0
$836$	−264320.			−0.378196
$837$		0			0
$838$	29520.0	−	29520.0i	0.0420367	−	0.0420367i
$839$			821360.i			1.16684i	0.812172	+	0.583418i	$0.198285\pi$
$839$			821360.i			1.16684i	−0.812172	+	0.583418i	$0.801715\pi$
$840$		0			0
$841$	244881.			0.346229
$842$	−442564.	−	442564.i	−0.624240	−	0.624240i
$843$		0			0
$844$			582736.i			0.818063i
$845$		0			0
$846$		0			0
$847$	20793.0	+	20793.0i	0.0289835	+	0.0289835i
$848$	39104.0	−	39104.0i	0.0543788	−	0.0543788i
$849$		0			0
$850$		0			0
$851$	350238.			0.483620
$852$		0			0
$853$	698291.	−	698291.i	0.959706	−	0.959706i	−0.0395127	−	0.999219i	$-0.512581\pi$
$853$	698291.	−	698291.i	0.959706	−	0.959706i	0.999219	+	0.0395127i	$0.0125806\pi$
$854$		−	649368.i		−	0.890379i
$855$		0			0
$856$	−150752.			−0.205738
$857$	144489.	+	144489.i	0.196731	+	0.196731i	0.798597	−	0.601866i	$-0.205576\pi$
$857$	144489.	+	144489.i	0.196731	+	0.196731i	−0.601866	+	0.798597i	$0.705576\pi$
$858$		0			0
$859$		−	943480.i		−	1.27863i	−0.768943	−	0.639317i	$-0.779217\pi$
$859$		−	943480.i		−	1.27863i	0.768943	−	0.639317i	$-0.220783\pi$
$860$		0			0
$861$		0			0
$862$	425044.	+	425044.i	0.572031	+	0.572031i
$863$	438149.	−	438149.i	0.588302	−	0.588302i	−0.348869	−	0.937171i	$-0.613434\pi$
$863$	438149.	−	438149.i	0.588302	−	0.588302i	0.937171	+	0.348869i	$0.113434\pi$
$864$		0			0
$865$		0			0
$866$	581636.			0.775560
$867$		0			0
$868$	46864.0	−	46864.0i	0.0622014	−	0.0622014i
$869$			1.24608e6i			1.65009i
$870$		0			0
$871$	−103638.			−0.136610
$872$	352640.	+	352640.i	0.463766	+	0.463766i
$873$		0			0
$874$		−	301280.i		−	0.394410i
$875$		0			0
$876$		0			0
$877$	−281469.	−	281469.i	−0.365958	−	0.365958i	0.500043	−	0.866001i	$-0.333318\pi$
$877$	−281469.	−	281469.i	−0.365958	−	0.365958i	−0.866001	+	0.500043i	$0.833318\pi$
$878$	−598880.	+	598880.i	−0.776874	+	0.776874i
$879$		0			0
$880$		0			0
$881$	−876722.			−1.12956			−0.564781	−	0.825241i	$-0.691039\pi$
$881$	−876722.			−1.12956			−0.564781	+	0.825241i	$0.691039\pi$
$882$		0			0
$883$	327431.	−	327431.i	0.419951	−	0.419951i	−0.465236	−	0.885187i	$-0.654031\pi$
$883$	327431.	−	327431.i	0.419951	−	0.419951i	0.885187	+	0.465236i	$0.154031\pi$
$884$			299184.i			0.382855i
$885$		0			0
$886$	−962436.			−1.22604
$887$	−477171.	−	477171.i	−0.606494	−	0.606494i	0.335534	−	0.942028i	$-0.391083\pi$
$887$	−477171.	−	477171.i	−0.606494	−	0.606494i	−0.942028	+	0.335534i	$0.891083\pi$
$888$		0			0
$889$			342722.i			0.433649i
$890$		0			0
$891$		0			0
$892$	247128.	+	247128.i	0.310593	+	0.310593i
$893$	−355320.	+	355320.i	−0.445571	+	0.445571i
$894$		0			0
$895$		0			0
$896$	59392.0			0.0739796
$897$		0			0
$898$	164960.	−	164960.i	0.204562	−	0.204562i
$899$		−	137360.i		−	0.169958i
$900$		0			0
$901$	331162.			0.407935
$902$	396952.	+	396952.i	0.487893	+	0.487893i
$903$		0			0
$904$			67552.0i			0.0826611i
$905$		0			0
$906$		0			0
$907$	−1.11209e6	−	1.11209e6i	−1.35184	−	1.35184i	−0.883604	−	0.468235i	$-0.844890\pi$
$907$	−1.11209e6	−	1.11209e6i	−1.35184	−	1.35184i	−0.468235	−	0.883604i	$-0.655110\pi$
$908$	439288.	−	439288.i	0.532816	−	0.532816i
$909$		0			0
$910$		0			0
$911$	883958.			1.06511			0.532556	−	0.846395i	$-0.321232\pi$
$911$	883958.			1.06511			0.532556	+	0.846395i	$0.321232\pi$
$912$		0			0
$913$	−735258.	+	735258.i	−0.882060	+	0.882060i
$914$		−	752604.i		−	0.900895i
$915$		0			0
$916$	402240.			0.479396
$917$	−184382.	−	184382.i	−0.219270	−	0.219270i
$918$		0			0
$919$			1.24040e6i			1.46869i	0.678775	+	0.734346i	$0.262511\pi$
$919$			1.24040e6i			1.46869i	−0.678775	+	0.734346i	$0.737489\pi$
$920$		0			0
$921$		0			0
$922$	−652316.	−	652316.i	−0.767355	−	0.767355i
$923$	444498.	−	444498.i	0.521755	−	0.521755i
$924$		0			0
$925$		0			0
$926$	−874924.			−1.02035
$927$		0			0
$928$	87040.0	−	87040.0i	0.101070	−	0.101070i
$929$		−	1.22744e6i		−	1.42223i	−0.703077	−	0.711113i	$-0.748191\pi$
$929$		−	1.22744e6i		−	1.42223i	0.703077	−	0.711113i	$-0.251809\pi$
$930$		0			0
$931$	201320.			0.232267
$932$	−19128.0	−	19128.0i	−0.0220210	−	0.0220210i
$933$		0			0
$934$		−	236996.i		−	0.271673i
$935$		0			0
$936$		0			0
$937$	1.07047e6	+	1.07047e6i	1.21926	+	1.21926i	0.967892	+	0.251366i	$0.0808799\pi$
$937$	1.07047e6	+	1.07047e6i	1.21926	+	1.21926i	0.251366	+	0.967892i	$0.419120\pi$
$938$	−87116.0	+	87116.0i	−0.0990130	+	0.0990130i
$939$		0			0
$940$		0			0
$941$	−558642.			−0.630891			−0.315446	−	0.948944i	$-0.602154\pi$
$941$	−558642.			−0.630891			−0.315446	+	0.948944i	$0.602154\pi$
$942$		0			0
$943$	−452458.	+	452458.i	−0.508809	+	0.508809i
$944$			74240.0i			0.0833094i
$945$		0			0
$946$	514008.			0.574365
$947$	−191711.	−	191711.i	−0.213770	−	0.213770i	0.592097	−	0.805867i	$-0.298300\pi$
$947$	−191711.	−	191711.i	−0.213770	−	0.213770i	−0.805867	+	0.592097i	$0.798300\pi$
$948$		0			0
$949$			407238.i			0.452185i
$950$		0			0
$951$		0			0
$952$	251488.	+	251488.i	0.277487	+	0.277487i
$953$	−630231.	+	630231.i	−0.693927	+	0.693927i	−0.963094	−	0.269166i	$-0.913252\pi$
$953$	−630231.	+	630231.i	−0.693927	+	0.693927i	0.269166	+	0.963094i	$0.413252\pi$
$954$		0			0
$955$		0			0
$956$	−142080.			−0.155459
$957$		0			0
$958$	−546880.	+	546880.i	−0.595883	+	0.595883i
$959$		−	1.18372e6i		−	1.28710i
$960$		0			0
$961$	−882717.			−0.955817
$962$	179676.	+	179676.i	0.194151	+	0.194151i
$963$		0			0
$964$		−	225904.i		−	0.243092i
$965$		0			0
$966$		0			0
$967$	345491.	+	345491.i	0.369474	+	0.369474i	0.867285	−	0.497811i	$-0.165863\pi$
$967$	345491.	+	345491.i	0.369474	+	0.369474i	−0.497811	+	0.867285i	$0.665863\pi$
$968$	−11472.0	+	11472.0i	−0.0122430	+	0.0122430i
$969$		0			0
$970$		0			0
$971$	−1.08308e6			−1.14874			−0.574372	−	0.818595i	$-0.694753\pi$
$971$	−1.08308e6			−1.14874			−0.574372	+	0.818595i	$0.694753\pi$
$972$		0			0
$973$	−272600.	+	272600.i	−0.287939	+	0.287939i
$974$		−	494604.i		−	0.521362i
$975$		0			0
$976$	358272.			0.376109
$977$	−146751.	−	146751.i	−0.153742	−	0.153742i	0.626045	−	0.779787i	$-0.284673\pi$
$977$	−146751.	−	146751.i	−0.153742	−	0.153742i	−0.779787	+	0.626045i	$0.784673\pi$
$978$		0			0
$979$			1.70864e6i			1.78273i
$980$		0			0
$981$		0			0
$982$	396884.	+	396884.i	0.411567	+	0.411567i
$983$	466909.	−	466909.i	0.483198	−	0.483198i	−0.422953	−	0.906151i	$-0.639007\pi$
$983$	466909.	−	466909.i	0.483198	−	0.483198i	0.906151	+	0.422953i	$0.139007\pi$
$984$		0			0
$985$		0			0
$986$	737120.			0.758201
$987$		0			0
$988$	154560.	−	154560.i	0.158337	−	0.158337i
$989$			585882.i			0.598987i
$990$		0			0
$991$	−901238.			−0.917682			−0.458841	−	0.888518i	$-0.651735\pi$
$991$	−901238.			−0.917682			−0.458841	+	0.888518i	$0.651735\pi$
$992$	25856.0	+	25856.0i	0.0262747	+	0.0262747i
$993$		0			0
$994$		−	747272.i		−	0.756321i
$995$		0			0
$996$		0			0
$997$	−152149.	−	152149.i	−0.153066	−	0.153066i	0.626420	−	0.779486i	$-0.284520\pi$
$997$	−152149.	−	152149.i	−0.153066	−	0.153066i	−0.779486	+	0.626420i	$0.784520\pi$
$998$	538480.	−	538480.i	0.540640	−	0.540640i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	450.5.g.a.307.1		2
3.2	odd	2		50.5.c.b.7.1		2
5.2	odd	4		90.5.g.b.73.1		2
5.3	odd	4	inner	450.5.g.a.343.1		2
5.4	even	2		90.5.g.b.37.1		2
12.11	even	2		400.5.p.c.257.1		2
15.2	even	4		10.5.c.a.3.1	✓	2
15.8	even	4		50.5.c.b.43.1		2
15.14	odd	2		10.5.c.a.7.1	yes	2
60.23	odd	4		400.5.p.c.193.1		2
60.47	odd	4		80.5.p.b.33.1		2
60.59	even	2		80.5.p.b.17.1		2
120.29	odd	2		320.5.p.b.257.1		2
120.59	even	2		320.5.p.i.257.1		2
120.77	even	4		320.5.p.b.193.1		2
120.107	odd	4		320.5.p.i.193.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
10.5.c.a.3.1	✓	2	15.2	even	4
10.5.c.a.7.1	yes	2	15.14	odd	2
50.5.c.b.7.1		2	3.2	odd	2
50.5.c.b.43.1		2	15.8	even	4
80.5.p.b.17.1		2	60.59	even	2
80.5.p.b.33.1		2	60.47	odd	4
90.5.g.b.37.1		2	5.4	even	2
90.5.g.b.73.1		2	5.2	odd	4
320.5.p.b.193.1		2	120.77	even	4
320.5.p.b.257.1		2	120.29	odd	2
320.5.p.i.193.1		2	120.107	odd	4
320.5.p.i.257.1		2	120.59	even	2
400.5.p.c.193.1		2	60.23	odd	4
400.5.p.c.257.1		2	12.11	even	2
450.5.g.a.307.1		2	1.1	even	1	trivial
450.5.g.a.343.1		2	5.3	odd	4	inner

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:	\( N \)	\(=\)	\( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	450.g (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-2.00000 - 2.00000i) q^{2} +8.00000i q^{4} +(-29.0000 - 29.0000i) q^{7} +(16.0000 - 16.0000i) q^{8} +118.000 q^{11} +(-69.0000 + 69.0000i) q^{13} +116.000i q^{14} -64.0000 q^{16} +(-271.000 - 271.000i) q^{17} +280.000i q^{19} +(-236.000 - 236.000i) q^{22} +(269.000 - 269.000i) q^{23} +276.000 q^{26} +(232.000 - 232.000i) q^{28} -680.000i q^{29} +202.000 q^{31} +(128.000 + 128.000i) q^{32} +1084.00i q^{34} +(651.000 + 651.000i) q^{37} +(560.000 - 560.000i) q^{38} -1682.00 q^{41} +(-1089.00 + 1089.00i) q^{43} +944.000i q^{44} -1076.00 q^{46} +(1269.00 + 1269.00i) q^{47} -719.000i q^{49} +(-552.000 - 552.000i) q^{52} +(-611.000 + 611.000i) q^{53} -928.000 q^{56} +(-1360.00 + 1360.00i) q^{58} -1160.00i q^{59} -5598.00 q^{61} +(-404.000 - 404.000i) q^{62} -512.000i q^{64} +(751.000 + 751.000i) q^{67} +(2168.00 - 2168.00i) q^{68} -6442.00 q^{71} +(2951.00 - 2951.00i) q^{73} -2604.00i q^{74} -2240.00 q^{76} +(-3422.00 - 3422.00i) q^{77} +10560.0i q^{79} +(3364.00 + 3364.00i) q^{82} +(-6231.00 + 6231.00i) q^{83} +4356.00 q^{86} +(1888.00 - 1888.00i) q^{88} +14480.0i q^{89} +4002.00 q^{91} +(2152.00 + 2152.00i) q^{92} -5076.00i q^{94} +(7311.00 + 7311.00i) q^{97} +(-1438.00 + 1438.00i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{2} - 58 q^{7} + 32 q^{8} + 236 q^{11} - 138 q^{13} - 128 q^{16} - 542 q^{17} - 472 q^{22} + 538 q^{23} + 552 q^{26} + 464 q^{28} + 404 q^{31} + 256 q^{32} + 1302 q^{37} + 1120 q^{38} - 3364 q^{41}+ \cdots - 2876 q^{98}+O(q^{100}) \) 2 * q - 4 * q^2 - 58 * q^7 + 32 * q^8 + 236 * q^11 - 138 * q^13 - 128 * q^16 - 542 * q^17 - 472 * q^22 + 538 * q^23 + 552 * q^26 + 464 * q^28 + 404 * q^31 + 256 * q^32 + 1302 * q^37 + 1120 * q^38 - 3364 * q^41 - 2178 * q^43 - 2152 * q^46 + 2538 * q^47 - 1104 * q^52 - 1222 * q^53 - 1856 * q^56 - 2720 * q^58 - 11196 * q^61 - 808 * q^62 + 1502 * q^67 + 4336 * q^68 - 12884 * q^71 + 5902 * q^73 - 4480 * q^76 - 6844 * q^77 + 6728 * q^82 - 12462 * q^83 + 8712 * q^86 + 3776 * q^88 + 8004 * q^91 + 4304 * q^92 + 14622 * q^97 - 2876 * q^98

Introduction

L-functions

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