LMFDB - Embedded newform 4536.2.a.z.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4536,2,Mod(1,4536)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4536, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("4536.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4536 = 2^{3} \cdot 3^{4} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4536.a (trivial)

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$36.2201423569$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.45729.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 11x^{2} + 12x - 3 $ x^4 - x^3 - 11x^2 + 12x - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 504)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$3.31050$ of defining polynomial
Character	$\chi$	$=$	4536.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.31050 q^{5} -1.00000 q^{7} +O(q^{10})$	$q-2.31050 q^{5} -1.00000 q^{7} +1.31050 q^{11} -3.86561 q^{13} -0.326936 q^{17} +3.08232 q^{19} +3.63744 q^{23} +0.338407 q^{25} +9.50305 q^{29} +6.49808 q^{31} +2.31050 q^{35} -2.31050 q^{37} -9.49808 q^{41} +0.0987572 q^{43} +0.216707 q^{47} +1.00000 q^{49} +13.7921 q^{53} -3.02791 q^{55} -2.44985 q^{59} -15.3687 q^{61} +8.93150 q^{65} -5.87058 q^{67} -1.77182 q^{71} -5.99504 q^{73} -1.31050 q^{77} -14.5919 q^{79} -6.08232 q^{83} +0.755385 q^{85} +5.52224 q^{89} +3.86561 q^{91} -7.12170 q^{95} -5.98356 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 3 q^{5} - 4 q^{7}+O(q^{10}) $ 4 * q + 3 * q^5 - 4 * q^7	$ 4 q + 3 q^{5} - 4 q^{7} - 7 q^{11} - 3 q^{13} + 3 q^{17} - 4 q^{19} - 2 q^{23} + 5 q^{25} + 9 q^{29} - 3 q^{31} - 3 q^{35} + 3 q^{37} - 9 q^{41} - 8 q^{43} - 3 q^{47} + 4 q^{49} + 6 q^{53} - 28 q^{55} - 10 q^{59} - 20 q^{61} - q^{65} - 11 q^{67} - 3 q^{71} - 24 q^{73} + 7 q^{77} - 21 q^{79} - 8 q^{83} - 9 q^{85} + 6 q^{89} + 3 q^{91} - 36 q^{95} - 16 q^{97}+O(q^{100}) $ 4 * q + 3 * q^5 - 4 * q^7 - 7 * q^11 - 3 * q^13 + 3 * q^17 - 4 * q^19 - 2 * q^23 + 5 * q^25 + 9 * q^29 - 3 * q^31 - 3 * q^35 + 3 * q^37 - 9 * q^41 - 8 * q^43 - 3 * q^47 + 4 * q^49 + 6 * q^53 - 28 * q^55 - 10 * q^59 - 20 * q^61 - q^65 - 11 * q^67 - 3 * q^71 - 24 * q^73 + 7 * q^77 - 21 * q^79 - 8 * q^83 - 9 * q^85 + 6 * q^89 + 3 * q^91 - 36 * q^95 - 16 * q^97

Display 100 coefficients 10 coefficients

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 $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−2.31050	−1.03329	−0.516643	−	0.856201i	$-0.672819\pi$
$5$	−2.31050	−1.03329	−0.516643	+	0.856201i	$0.672819\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	1.31050	0.395130	0.197565	−	0.980290i	$-0.436697\pi$
$11$	1.31050	0.395130	0.197565	+	0.980290i	$0.436697\pi$
$12$	0	0
$13$	−3.86561	−1.07213	−0.536064	−	0.844177i	$-0.680089\pi$
$13$	−3.86561	−1.07213	−0.536064	+	0.844177i	$0.680089\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−0.326936	−0.0792936	−0.0396468	−	0.999214i	$-0.512623\pi$
$17$	−0.326936	−0.0792936	−0.0396468	+	0.999214i	$0.512623\pi$
$18$	0	0
$19$	3.08232	0.707133	0.353566	−	0.935409i	$-0.384969\pi$
$19$	3.08232	0.707133	0.353566	+	0.935409i	$0.384969\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	3.63744	0.758458	0.379229	−	0.925303i	$-0.376189\pi$
$23$	3.63744	0.758458	0.379229	+	0.925303i	$0.376189\pi$
$24$	0	0
$25$	0.338407	0.0676815
$26$	0	0
$27$	0	0
$28$	0	0
$29$	9.50305	1.76467	0.882336	−	0.470620i	$-0.155970\pi$
$29$	9.50305	1.76467	0.882336	+	0.470620i	$0.155970\pi$
$30$	0	0
$31$	6.49808	1.16709	0.583545	−	0.812081i	$-0.301665\pi$
$31$	6.49808	1.16709	0.583545	+	0.812081i	$0.301665\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	2.31050	0.390546
$36$	0	0
$37$	−2.31050	−0.379844	−0.189922	−	0.981799i	$-0.560823\pi$
$37$	−2.31050	−0.379844	−0.189922	+	0.981799i	$0.560823\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−9.49808	−1.48335	−0.741676	−	0.670759i	$-0.765969\pi$
$41$	−9.49808	−1.48335	−0.741676	+	0.670759i	$0.765969\pi$
$42$	0	0
$43$	0.0987572	0.0150603	0.00753017	−	0.999972i	$-0.497603\pi$
$43$	0.0987572	0.0150603	0.00753017	+	0.999972i	$0.497603\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	0.216707	0.0316100	0.0158050	−	0.999875i	$-0.494969\pi$
$47$	0.216707	0.0316100	0.0158050	+	0.999875i	$0.494969\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	13.7921	1.89450	0.947249	−	0.320500i	$-0.103851\pi$
$53$	13.7921	1.89450	0.947249	+	0.320500i	$0.103851\pi$
$54$	0	0
$55$	−3.02791	−0.408283
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−2.44985	−0.318943	−0.159472	−	0.987203i	$-0.550979\pi$
$59$	−2.44985	−0.318943	−0.159472	+	0.987203i	$0.550979\pi$
$60$	0	0
$61$	−15.3687	−1.96776	−0.983878	−	0.178842i	$-0.942765\pi$
$61$	−15.3687	−1.96776	−0.983878	+	0.178842i	$0.942765\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	8.93150	1.10782
$66$	0	0
$67$	−5.87058	−0.717205	−0.358603	−	0.933490i	$-0.616747\pi$
$67$	−5.87058	−0.717205	−0.358603	+	0.933490i	$0.616747\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−1.77182	−0.210277	−0.105138	−	0.994458i	$-0.533529\pi$
$71$	−1.77182	−0.210277	−0.105138	+	0.994458i	$0.533529\pi$
$72$	0	0
$73$	−5.99504	−0.701666	−0.350833	−	0.936438i	$-0.614101\pi$
$73$	−5.99504	−0.701666	−0.350833	+	0.936438i	$0.614101\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−1.31050	−0.149345
$78$	0	0
$79$	−14.5919	−1.64171	−0.820857	−	0.571134i	$-0.806504\pi$
$79$	−14.5919	−1.64171	−0.820857	+	0.571134i	$0.806504\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−6.08232	−0.667621	−0.333811	−	0.942640i	$-0.608335\pi$
$83$	−6.08232	−0.667621	−0.333811	+	0.942640i	$0.608335\pi$
$84$	0	0
$85$	0.755385	0.0819330
$86$	0	0
$87$	0	0
$88$	0	0
$89$	5.52224	0.585356	0.292678	−	0.956211i	$-0.405454\pi$
$89$	5.52224	0.585356	0.292678	+	0.956211i	$0.405454\pi$
$90$	0	0
$91$	3.86561	0.405226
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−7.12170	−0.730671
$96$	0	0
$97$	−5.98356	−0.607539	−0.303769	−	0.952746i	$-0.598245\pi$
$97$	−5.98356	−0.607539	−0.303769	+	0.952746i	$0.598245\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−7.70211	−0.766388	−0.383194	−	0.923668i	$-0.625176\pi$
$101$	−7.70211	−0.766388	−0.383194	+	0.923668i	$0.625176\pi$
$102$	0	0
$103$	4.06092	0.400134	0.200067	−	0.979782i	$-0.435884\pi$
$103$	4.06092	0.400134	0.200067	+	0.979782i	$0.435884\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−14.0774	−1.36091	−0.680455	−	0.732790i	$-0.738218\pi$
$107$	−14.0774	−1.36091	−0.680455	+	0.732790i	$0.738218\pi$
$108$	0	0
$109$	−6.67306	−0.639164	−0.319582	−	0.947559i	$-0.603542\pi$
$109$	−6.67306	−0.639164	−0.319582	+	0.947559i	$0.603542\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−7.69835	−0.724200	−0.362100	−	0.932139i	$-0.617940\pi$
$113$	−7.69835	−0.724200	−0.362100	+	0.932139i	$0.617940\pi$
$114$	0	0
$115$	−8.40429	−0.783704
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0.326936	0.0299702
$120$	0	0
$121$	−9.28259	−0.843872
$122$	0	0
$123$	0	0
$124$	0	0
$125$	10.7706	0.963352
$126$	0	0
$127$	14.9847	1.32968	0.664838	−	0.746987i	$-0.268500\pi$
$127$	14.9847	1.32968	0.664838	+	0.746987i	$0.268500\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	10.4158	0.910029	0.455015	−	0.890484i	$-0.349634\pi$
$131$	10.4158	0.910029	0.455015	+	0.890484i	$0.349634\pi$
$132$	0	0
$133$	−3.08232	−0.267271
$134$	0	0
$135$	0	0
$136$	0	0
$137$	6.26340	0.535118	0.267559	−	0.963541i	$-0.413783\pi$
$137$	6.26340	0.535118	0.267559	+	0.963541i	$0.413783\pi$
$138$	0	0
$139$	−20.1597	−1.70992	−0.854961	−	0.518693i	$-0.826419\pi$
$139$	−20.1597	−1.70992	−0.854961	+	0.518693i	$0.826419\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−5.06588	−0.423631
$144$	0	0
$145$	−21.9568	−1.82341
$146$	0	0
$147$	0	0
$148$	0	0
$149$	17.0176	1.39413	0.697067	−	0.717006i	$-0.254488\pi$
$149$	17.0176	1.39413	0.697067	+	0.717006i	$0.254488\pi$
$150$	0	0
$151$	−16.3928	−1.33403	−0.667014	−	0.745045i	$-0.732428\pi$
$151$	−16.3928	−1.33403	−0.667014	+	0.745045i	$0.732428\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−15.0138	−1.20594
$156$	0	0
$157$	20.4549	1.63248	0.816238	−	0.577715i	$-0.196056\pi$
$157$	20.4549	1.63248	0.816238	+	0.577715i	$0.196056\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−3.63744	−0.286670
$162$	0	0
$163$	16.8810	1.32222	0.661110	−	0.750289i	$-0.270086\pi$
$163$	16.8810	1.32222	0.661110	+	0.750289i	$0.270086\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−3.21174	−0.248532	−0.124266	−	0.992249i	$-0.539658\pi$
$167$	−3.21174	−0.248532	−0.124266	+	0.992249i	$0.539658\pi$
$168$	0	0
$169$	1.94297	0.149459
$170$	0	0
$171$	0	0
$172$	0	0
$173$	0.612145	0.0465405	0.0232703	−	0.999729i	$-0.492592\pi$
$173$	0.612145	0.0465405	0.0232703	+	0.999729i	$0.492592\pi$
$174$	0	0
$175$	−0.338407	−0.0255812
$176$	0	0
$177$	0	0
$178$	0	0
$179$	4.38135	0.327477	0.163739	−	0.986504i	$-0.447645\pi$
$179$	4.38135	0.327477	0.163739	+	0.986504i	$0.447645\pi$
$180$	0	0
$181$	−3.50192	−0.260295	−0.130148	−	0.991495i	$-0.541545\pi$
$181$	−3.50192	−0.260295	−0.130148	+	0.991495i	$0.541545\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	5.33841	0.392488
$186$	0	0
$187$	−0.428449	−0.0313313
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−5.43066	−0.392949	−0.196474	−	0.980509i	$-0.562949\pi$
$191$	−5.43066	−0.392949	−0.196474	+	0.980509i	$0.562949\pi$
$192$	0	0
$193$	−9.08386	−0.653871	−0.326935	−	0.945047i	$-0.606016\pi$
$193$	−9.08386	−0.653871	−0.326935	+	0.945047i	$0.606016\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−8.58040	−0.611329	−0.305664	−	0.952139i	$-0.598879\pi$
$197$	−8.58040	−0.611329	−0.305664	+	0.952139i	$0.598879\pi$
$198$	0	0
$199$	−25.4905	−1.80697	−0.903487	−	0.428616i	$-0.859001\pi$
$199$	−25.4905	−1.80697	−0.903487	+	0.428616i	$0.859001\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−9.50305	−0.666983
$204$	0	0
$205$	21.9453	1.53273
$206$	0	0
$207$	0	0
$208$	0	0
$209$	4.03938	0.279410
$210$	0	0
$211$	−16.6413	−1.14564	−0.572818	−	0.819683i	$-0.694150\pi$
$211$	−16.6413	−1.14564	−0.572818	+	0.819683i	$0.694150\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−0.228179	−0.0155616
$216$	0	0
$217$	−6.49808	−0.441119
$218$	0	0
$219$	0	0
$220$	0	0
$221$	1.26381	0.0850129
$222$	0	0
$223$	18.5701	1.24354	0.621772	−	0.783198i	$-0.286413\pi$
$223$	18.5701	1.24354	0.621772	+	0.783198i	$0.286413\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−15.6819	−1.04085	−0.520423	−	0.853909i	$-0.674226\pi$
$227$	−15.6819	−1.04085	−0.520423	+	0.853909i	$0.674226\pi$
$228$	0	0
$229$	11.1405	0.736184	0.368092	−	0.929789i	$-0.380011\pi$
$229$	11.1405	0.736184	0.368092	+	0.929789i	$0.380011\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−21.6831	−1.42050	−0.710252	−	0.703948i	$-0.751419\pi$
$233$	−21.6831	−1.42050	−0.710252	+	0.703948i	$0.751419\pi$
$234$	0	0
$235$	−0.500702	−0.0326622
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−12.4054	−0.802440	−0.401220	−	0.915982i	$-0.631414\pi$
$239$	−12.4054	−0.802440	−0.401220	+	0.915982i	$0.631414\pi$
$240$	0	0
$241$	−16.7006	−1.07578	−0.537889	−	0.843016i	$-0.680778\pi$
$241$	−16.7006	−1.07578	−0.537889	+	0.843016i	$0.680778\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−2.31050	−0.147612
$246$	0	0
$247$	−11.9151	−0.758137
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−27.0215	−1.70558	−0.852790	−	0.522255i	$-0.825091\pi$
$251$	−27.0215	−1.70558	−0.852790	+	0.522255i	$0.825091\pi$
$252$	0	0
$253$	4.76686	0.299690
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−19.9529	−1.24463	−0.622314	−	0.782768i	$-0.713807\pi$
$257$	−19.9529	−1.24463	−0.622314	+	0.782768i	$0.713807\pi$
$258$	0	0
$259$	2.31050	0.143567
$260$	0	0
$261$	0	0
$262$	0	0
$263$	15.8568	0.977773	0.488887	−	0.872347i	$-0.337403\pi$
$263$	15.8568	0.977773	0.488887	+	0.872347i	$0.337403\pi$
$264$	0	0
$265$	−31.8667	−1.95756
$266$	0	0
$267$	0	0
$268$	0	0
$269$	0.694466	0.0423423	0.0211712	−	0.999776i	$-0.493261\pi$
$269$	0.694466	0.0423423	0.0211712	+	0.999776i	$0.493261\pi$
$270$	0	0
$271$	−9.08883	−0.552107	−0.276053	−	0.961142i	$-0.589027\pi$
$271$	−9.08883	−0.552107	−0.276053	+	0.961142i	$0.589027\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0.443483	0.0267430
$276$	0	0
$277$	14.4017	0.865313	0.432656	−	0.901559i	$-0.357576\pi$
$277$	14.4017	0.865313	0.432656	+	0.901559i	$0.357576\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−9.36712	−0.558796	−0.279398	−	0.960175i	$-0.590135\pi$
$281$	−9.36712	−0.558796	−0.279398	+	0.960175i	$0.590135\pi$
$282$	0	0
$283$	−19.0862	−1.13456	−0.567279	−	0.823526i	$-0.692004\pi$
$283$	−19.0862	−1.13456	−0.567279	+	0.823526i	$0.692004\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	9.49808	0.560654
$288$	0	0
$289$	−16.8931	−0.993713
$290$	0	0
$291$	0	0
$292$	0	0
$293$	18.2052	1.06356	0.531781	−	0.846882i	$-0.321523\pi$
$293$	18.2052	1.06356	0.531781	+	0.846882i	$0.321523\pi$
$294$	0	0
$295$	5.66038	0.329560
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−14.0609	−0.813164
$300$	0	0
$301$	−0.0987572	−0.00569227
$302$	0	0
$303$	0	0
$304$	0	0
$305$	35.5093	2.03326
$306$	0	0
$307$	−26.8537	−1.53262	−0.766312	−	0.642469i	$-0.777910\pi$
$307$	−26.8537	−1.53262	−0.766312	+	0.642469i	$0.777910\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−1.58049	−0.0896212	−0.0448106	−	0.998996i	$-0.514268\pi$
$311$	−1.58049	−0.0896212	−0.0448106	+	0.998996i	$0.514268\pi$
$312$	0	0
$313$	20.5234	1.16005	0.580025	−	0.814599i	$-0.303043\pi$
$313$	20.5234	1.16005	0.580025	+	0.814599i	$0.303043\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	31.8645	1.78969	0.894845	−	0.446376i	$-0.147286\pi$
$317$	31.8645	1.78969	0.894845	+	0.446376i	$0.147286\pi$
$318$	0	0
$319$	12.4537	0.697276
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−1.00772	−0.0560711
$324$	0	0
$325$	−1.30815	−0.0725632
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−0.216707	−0.0119474
$330$	0	0
$331$	34.1428	1.87666	0.938330	−	0.345741i	$-0.112373\pi$
$331$	34.1428	1.87666	0.938330	+	0.345741i	$0.112373\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	13.5640	0.741079
$336$	0	0
$337$	0.505666	0.0275454	0.0137727	−	0.999905i	$-0.495616\pi$
$337$	0.505666	0.0275454	0.0137727	+	0.999905i	$0.495616\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	8.51573	0.461153
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	28.4307	1.52624	0.763120	−	0.646256i	$-0.223666\pi$
$347$	28.4307	1.52624	0.763120	+	0.646256i	$0.223666\pi$
$348$	0	0
$349$	−35.1608	−1.88211	−0.941057	−	0.338247i	$-0.890166\pi$
$349$	−35.1608	−1.88211	−0.941057	+	0.338247i	$0.890166\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	0.992279	0.0528137	0.0264068	−	0.999651i	$-0.491593\pi$
$353$	0.992279	0.0528137	0.0264068	+	0.999651i	$0.491593\pi$
$354$	0	0
$355$	4.09379	0.217276
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−23.2914	−1.22928	−0.614638	−	0.788810i	$-0.710698\pi$
$359$	−23.2914	−1.22928	−0.614638	+	0.788810i	$0.710698\pi$
$360$	0	0
$361$	−9.49930	−0.499963
$362$	0	0
$363$	0	0
$364$	0	0
$365$	13.8515	0.725022
$366$	0	0
$367$	−11.7829	−0.615062	−0.307531	−	0.951538i	$-0.599503\pi$
$367$	−11.7829	−0.615062	−0.307531	+	0.951538i	$0.599503\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−13.7921	−0.716053
$372$	0	0
$373$	−13.8343	−0.716312	−0.358156	−	0.933662i	$-0.616594\pi$
$373$	−13.8343	−0.716312	−0.358156	+	0.933662i	$0.616594\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−36.7351	−1.89195
$378$	0	0
$379$	−21.5877	−1.10889	−0.554443	−	0.832222i	$-0.687069\pi$
$379$	−21.5877	−1.10889	−0.554443	+	0.832222i	$0.687069\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	16.6398	0.850253	0.425127	−	0.905134i	$-0.360230\pi$
$383$	16.6398	0.850253	0.425127	+	0.905134i	$0.360230\pi$
$384$	0	0
$385$	3.02791	0.154316
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−18.9342	−0.960001	−0.480000	−	0.877268i	$-0.659363\pi$
$389$	−18.9342	−0.960001	−0.480000	+	0.877268i	$0.659363\pi$
$390$	0	0
$391$	−1.18921	−0.0601408
$392$	0	0
$393$	0	0
$394$	0	0
$395$	33.7145	1.69636
$396$	0	0
$397$	16.0746	0.806761	0.403381	−	0.915032i	$-0.367835\pi$
$397$	16.0746	0.806761	0.403381	+	0.915032i	$0.367835\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	38.8818	1.94166	0.970832	−	0.239762i	$-0.0770694\pi$
$401$	38.8818	1.94166	0.970832	+	0.239762i	$0.0770694\pi$
$402$	0	0
$403$	−25.1191	−1.25127
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−3.02791	−0.150088
$408$	0	0
$409$	12.7999	0.632913	0.316456	−	0.948607i	$-0.397507\pi$
$409$	12.7999	0.632913	0.316456	+	0.948607i	$0.397507\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	2.44985	0.120549
$414$	0	0
$415$	14.0532	0.689844
$416$	0	0
$417$	0	0
$418$	0	0
$419$	1.19409	0.0583352	0.0291676	−	0.999575i	$-0.490714\pi$
$419$	1.19409	0.0583352	0.0291676	+	0.999575i	$0.490714\pi$
$420$	0	0
$421$	16.3813	0.798378	0.399189	−	0.916869i	$-0.369292\pi$
$421$	16.3813	0.798378	0.399189	+	0.916869i	$0.369292\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−0.110637	−0.00536671
$426$	0	0
$427$	15.3687	0.743742
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−26.3928	−1.27130	−0.635649	−	0.771978i	$-0.719267\pi$
$431$	−26.3928	−1.27130	−0.635649	+	0.771978i	$0.719267\pi$
$432$	0	0
$433$	23.7503	1.14137	0.570684	−	0.821170i	$-0.306678\pi$
$433$	23.7503	1.14137	0.570684	+	0.821170i	$0.306678\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	11.2117	0.536330
$438$	0	0
$439$	−20.4862	−0.977753	−0.488877	−	0.872353i	$-0.662593\pi$
$439$	−20.4862	−0.977753	−0.488877	+	0.872353i	$0.662593\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−7.79819	−0.370503	−0.185252	−	0.982691i	$-0.559310\pi$
$443$	−7.79819	−0.370503	−0.185252	+	0.982691i	$0.559310\pi$
$444$	0	0
$445$	−12.7591	−0.604841
$446$	0	0
$447$	0	0
$448$	0	0
$449$	2.35981	0.111366	0.0556831	−	0.998448i	$-0.482266\pi$
$449$	2.35981	0.111366	0.0556831	+	0.998448i	$0.482266\pi$
$450$	0	0
$451$	−12.4472	−0.586117
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−8.93150	−0.418715
$456$	0	0
$457$	20.2025	0.945032	0.472516	−	0.881322i	$-0.343346\pi$
$457$	20.2025	0.945032	0.472516	+	0.881322i	$0.343346\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−12.0439	−0.560942	−0.280471	−	0.959862i	$-0.590491\pi$
$461$	−12.0439	−0.560942	−0.280471	+	0.959862i	$0.590491\pi$
$462$	0	0
$463$	−23.2875	−1.08226	−0.541130	−	0.840939i	$-0.682003\pi$
$463$	−23.2875	−1.08226	−0.541130	+	0.840939i	$0.682003\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	34.0682	1.57649	0.788243	−	0.615364i	$-0.210991\pi$
$467$	34.0682	1.57649	0.788243	+	0.615364i	$0.210991\pi$
$468$	0	0
$469$	5.87058	0.271078
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0.129421	0.00595080
$474$	0	0
$475$	1.04308	0.0478598
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−18.7695	−0.857602	−0.428801	−	0.903399i	$-0.641064\pi$
$479$	−18.7695	−0.857602	−0.428801	+	0.903399i	$0.641064\pi$
$480$	0	0
$481$	8.93150	0.407241
$482$	0	0
$483$	0	0
$484$	0	0
$485$	13.8250	0.627762
$486$	0	0
$487$	−13.8837	−0.629132	−0.314566	−	0.949236i	$-0.601859\pi$
$487$	−13.8837	−0.629132	−0.314566	+	0.949236i	$0.601859\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	9.08500	0.410000	0.205000	−	0.978762i	$-0.434281\pi$
$491$	9.08500	0.410000	0.205000	+	0.978762i	$0.434281\pi$
$492$	0	0
$493$	−3.10689	−0.139927
$494$	0	0
$495$	0	0
$496$	0	0
$497$	1.77182	0.0794771
$498$	0	0
$499$	−23.7577	−1.06354	−0.531769	−	0.846889i	$-0.678473\pi$
$499$	−23.7577	−1.06354	−0.531769	+	0.846889i	$0.678473\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−22.5612	−1.00595	−0.502977	−	0.864300i	$-0.667762\pi$
$503$	−22.5612	−1.00595	−0.502977	+	0.864300i	$0.667762\pi$
$504$	0	0
$505$	17.7957	0.791899
$506$	0	0
$507$	0	0
$508$	0	0
$509$	7.89628	0.349996	0.174998	−	0.984569i	$-0.444008\pi$
$509$	7.89628	0.349996	0.174998	+	0.984569i	$0.444008\pi$
$510$	0	0
$511$	5.99504	0.265205
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−9.38275	−0.413453
$516$	0	0
$517$	0.283994	0.0124901
$518$	0	0
$519$	0	0
$520$	0	0
$521$	9.86440	0.432167	0.216084	−	0.976375i	$-0.430672\pi$
$521$	9.86440	0.432167	0.216084	+	0.976375i	$0.430672\pi$
$522$	0	0
$523$	22.9097	1.00177	0.500885	−	0.865514i	$-0.333008\pi$
$523$	22.9097	1.00177	0.500885	+	0.865514i	$0.333008\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−2.12446	−0.0925428
$528$	0	0
$529$	−9.76907	−0.424742
$530$	0	0
$531$	0	0
$532$	0	0
$533$	36.7159	1.59034
$534$	0	0
$535$	32.5257	1.40621
$536$	0	0
$537$	0	0
$538$	0	0
$539$	1.31050	0.0564472
$540$	0	0
$541$	29.7187	1.27771	0.638853	−	0.769329i	$-0.279409\pi$
$541$	29.7187	1.27771	0.638853	+	0.769329i	$0.279409\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	15.4181	0.660439
$546$	0	0
$547$	−2.60945	−0.111572	−0.0557859	−	0.998443i	$-0.517766\pi$
$547$	−2.60945	−0.111572	−0.0557859	+	0.998443i	$0.517766\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	29.2914	1.24786
$552$	0	0
$553$	14.5919	0.620510
$554$	0	0
$555$	0	0
$556$	0	0
$557$	26.0406	1.10338	0.551688	−	0.834051i	$-0.313984\pi$
$557$	26.0406	1.10338	0.551688	+	0.834051i	$0.313984\pi$
$558$	0	0
$559$	−0.381757	−0.0161466
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−44.7752	−1.88705	−0.943525	−	0.331301i	$-0.892513\pi$
$563$	−44.7752	−1.88705	−0.943525	+	0.331301i	$0.892513\pi$
$564$	0	0
$565$	17.7870	0.748307
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−29.2580	−1.22656	−0.613280	−	0.789865i	$-0.710150\pi$
$569$	−29.2580	−1.22656	−0.613280	+	0.789865i	$0.710150\pi$
$570$	0	0
$571$	36.5256	1.52855	0.764275	−	0.644890i	$-0.223097\pi$
$571$	36.5256	1.52855	0.764275	+	0.644890i	$0.223097\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	1.23093	0.0513335
$576$	0	0
$577$	9.88205	0.411395	0.205698	−	0.978616i	$-0.434054\pi$
$577$	9.88205	0.411395	0.205698	+	0.978616i	$0.434054\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	6.08232	0.252337
$582$	0	0
$583$	18.0746	0.748573
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−8.11908	−0.335110	−0.167555	−	0.985863i	$-0.553587\pi$
$587$	−8.11908	−0.335110	−0.167555	+	0.985863i	$0.553587\pi$
$588$	0	0
$589$	20.0292	0.825288
$590$	0	0
$591$	0	0
$592$	0	0
$593$	29.6969	1.21950	0.609752	−	0.792592i	$-0.291269\pi$
$593$	29.6969	1.21950	0.609752	+	0.792592i	$0.291269\pi$
$594$	0	0
$595$	−0.755385	−0.0309678
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−16.3906	−0.669702	−0.334851	−	0.942271i	$-0.608686\pi$
$599$	−16.3906	−0.669702	−0.334851	+	0.942271i	$0.608686\pi$
$600$	0	0
$601$	−48.8849	−1.99406	−0.997028	−	0.0770454i	$-0.975451\pi$
$601$	−48.8849	−1.99406	−0.997028	+	0.0770454i	$0.975451\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	21.4474	0.871962
$606$	0	0
$607$	47.2065	1.91605	0.958027	−	0.286680i	$-0.0925515\pi$
$607$	47.2065	1.91605	0.958027	+	0.286680i	$0.0925515\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−0.837706	−0.0338899
$612$	0	0
$613$	−15.6396	−0.631679	−0.315840	−	0.948813i	$-0.602286\pi$
$613$	−15.6396	−0.631679	−0.315840	+	0.948813i	$0.602286\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	31.5586	1.27050	0.635251	−	0.772306i	$-0.280897\pi$
$617$	31.5586	1.27050	0.635251	+	0.772306i	$0.280897\pi$
$618$	0	0
$619$	14.6324	0.588125	0.294063	−	0.955786i	$-0.404993\pi$
$619$	14.6324	0.588125	0.294063	+	0.955786i	$0.404993\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−5.52224	−0.221244
$624$	0	0
$625$	−26.5775	−1.06310
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0.755385	0.0301192
$630$	0	0
$631$	−36.3101	−1.44548	−0.722742	−	0.691118i	$-0.757119\pi$
$631$	−36.3101	−1.44548	−0.722742	+	0.691118i	$0.757119\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−34.6221	−1.37394
$636$	0	0
$637$	−3.86561	−0.153161
$638$	0	0
$639$	0	0
$640$	0	0
$641$	10.8146	0.427152	0.213576	−	0.976926i	$-0.431489\pi$
$641$	10.8146	0.427152	0.213576	+	0.976926i	$0.431489\pi$
$642$	0	0
$643$	−13.9507	−0.550162	−0.275081	−	0.961421i	$-0.588705\pi$
$643$	−13.9507	−0.550162	−0.275081	+	0.961421i	$0.588705\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−34.5634	−1.35883	−0.679414	−	0.733755i	$-0.737766\pi$
$647$	−34.5634	−1.35883	−0.679414	+	0.733755i	$0.737766\pi$
$648$	0	0
$649$	−3.21053	−0.126024
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−27.5854	−1.07950	−0.539751	−	0.841825i	$-0.681481\pi$
$653$	−27.5854	−1.07950	−0.539751	+	0.841825i	$0.681481\pi$
$654$	0	0
$655$	−24.0656	−0.940321
$656$	0	0
$657$	0	0
$658$	0	0
$659$	20.6117	0.802919	0.401460	−	0.915877i	$-0.368503\pi$
$659$	20.6117	0.802919	0.401460	+	0.915877i	$0.368503\pi$
$660$	0	0
$661$	−27.1630	−1.05652	−0.528259	−	0.849083i	$-0.677155\pi$
$661$	−27.1630	−1.05652	−0.528259	+	0.849083i	$0.677155\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	7.12170	0.276168
$666$	0	0
$667$	34.5667	1.33843
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−20.1406	−0.777520
$672$	0	0
$673$	19.0529	0.734434	0.367217	−	0.930135i	$-0.380311\pi$
$673$	19.0529	0.734434	0.367217	+	0.930135i	$0.380311\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−21.7389	−0.835496	−0.417748	−	0.908563i	$-0.637180\pi$
$677$	−21.7389	−0.835496	−0.417748	+	0.908563i	$0.637180\pi$
$678$	0	0
$679$	5.98356	0.229628
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−41.0808	−1.57191	−0.785957	−	0.618281i	$-0.787829\pi$
$683$	−41.0808	−1.57191	−0.785957	+	0.618281i	$0.787829\pi$
$684$	0	0
$685$	−14.4716	−0.552931
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−53.3151	−2.03114
$690$	0	0
$691$	18.0720	0.687491	0.343745	−	0.939063i	$-0.388304\pi$
$691$	18.0720	0.687491	0.343745	+	0.939063i	$0.388304\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	46.5789	1.76684
$696$	0	0
$697$	3.10526	0.117620
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−41.5130	−1.56793	−0.783963	−	0.620808i	$-0.786805\pi$
$701$	−41.5130	−1.56793	−0.783963	+	0.620808i	$0.786805\pi$
$702$	0	0
$703$	−7.12170	−0.268600
$704$	0	0
$705$	0	0
$706$	0	0
$707$	7.70211	0.289667
$708$	0	0
$709$	1.36907	0.0514166	0.0257083	−	0.999669i	$-0.491816\pi$
$709$	1.36907	0.0514166	0.0257083	+	0.999669i	$0.491816\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	23.6364	0.885189
$714$	0	0
$715$	11.7047	0.437732
$716$	0	0
$717$	0	0
$718$	0	0
$719$	20.1854	0.752787	0.376394	−	0.926460i	$-0.377164\pi$
$719$	20.1854	0.752787	0.376394	+	0.926460i	$0.377164\pi$
$720$	0	0
$721$	−4.06092	−0.151237
$722$	0	0
$723$	0	0
$724$	0	0
$725$	3.21590	0.119436
$726$	0	0
$727$	3.38786	0.125649	0.0628243	−	0.998025i	$-0.479989\pi$
$727$	3.38786	0.125649	0.0628243	+	0.998025i	$0.479989\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−0.0322873	−0.00119419
$732$	0	0
$733$	−17.8121	−0.657904	−0.328952	−	0.944347i	$-0.606695\pi$
$733$	−17.8121	−0.657904	−0.328952	+	0.944347i	$0.606695\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−7.69339	−0.283390
$738$	0	0
$739$	−1.15512	−0.0424918	−0.0212459	−	0.999774i	$-0.506763\pi$
$739$	−1.15512	−0.0424918	−0.0212459	+	0.999774i	$0.506763\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−27.1727	−0.996869	−0.498435	−	0.866927i	$-0.666092\pi$
$743$	−27.1727	−0.996869	−0.498435	+	0.866927i	$0.666092\pi$
$744$	0	0
$745$	−39.3191	−1.44054
$746$	0	0
$747$	0	0
$748$	0	0
$749$	14.0774	0.514375
$750$	0	0
$751$	22.5919	0.824389	0.412195	−	0.911096i	$-0.364762\pi$
$751$	22.5919	0.824389	0.412195	+	0.911096i	$0.364762\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	37.8756	1.37843
$756$	0	0
$757$	13.9989	0.508797	0.254399	−	0.967099i	$-0.418122\pi$
$757$	13.9989	0.508797	0.254399	+	0.967099i	$0.418122\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−0.504672	−0.0182944	−0.00914718	−	0.999958i	$-0.502912\pi$
$761$	−0.504672	−0.0182944	−0.00914718	+	0.999958i	$0.502912\pi$
$762$	0	0
$763$	6.67306	0.241581
$764$	0	0
$765$	0	0
$766$	0	0
$767$	9.47018	0.341948
$768$	0	0
$769$	−9.78437	−0.352833	−0.176417	−	0.984316i	$-0.556451\pi$
$769$	−9.78437	−0.352833	−0.176417	+	0.984316i	$0.556451\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	26.5907	0.956403	0.478201	−	0.878250i	$-0.341289\pi$
$773$	26.5907	0.956403	0.478201	+	0.878250i	$0.341289\pi$
$774$	0	0
$775$	2.19900	0.0789904
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−29.2761	−1.04893
$780$	0	0
$781$	−2.32197	−0.0830867
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−47.2610	−1.68682
$786$	0	0
$787$	−38.2216	−1.36245	−0.681227	−	0.732072i	$-0.738553\pi$
$787$	−38.2216	−1.36245	−0.681227	+	0.732072i	$0.738553\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	7.69835	0.273722
$792$	0	0
$793$	59.4093	2.10969
$794$	0	0
$795$	0	0
$796$	0	0
$797$	22.3732	0.792500	0.396250	−	0.918143i	$-0.370311\pi$
$797$	22.3732	0.792500	0.396250	+	0.918143i	$0.370311\pi$
$798$	0	0
$799$	−0.0708493	−0.00250647
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−7.85649	−0.277249
$804$	0	0
$805$	8.40429	0.296212
$806$	0	0
$807$	0	0
$808$	0	0
$809$	5.50319	0.193482	0.0967409	−	0.995310i	$-0.469158\pi$
$809$	5.50319	0.193482	0.0967409	+	0.995310i	$0.469158\pi$
$810$	0	0
$811$	−12.3713	−0.434414	−0.217207	−	0.976126i	$-0.569695\pi$
$811$	−12.3713	−0.434414	−0.217207	+	0.976126i	$0.569695\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−39.0035	−1.36623
$816$	0	0
$817$	0.304402	0.0106497
$818$	0	0
$819$	0	0
$820$	0	0
$821$	42.5535	1.48513	0.742564	−	0.669776i	$-0.233610\pi$
$821$	42.5535	1.48513	0.742564	+	0.669776i	$0.233610\pi$
$822$	0	0
$823$	−16.2351	−0.565919	−0.282960	−	0.959132i	$-0.591316\pi$
$823$	−16.2351	−0.565919	−0.282960	+	0.959132i	$0.591316\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	34.0199	1.18299	0.591494	−	0.806309i	$-0.298538\pi$
$827$	34.0199	1.18299	0.591494	+	0.806309i	$0.298538\pi$
$828$	0	0
$829$	38.6394	1.34200	0.671000	−	0.741457i	$-0.265865\pi$
$829$	38.6394	1.34200	0.671000	+	0.741457i	$0.265865\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−0.326936	−0.0113277
$834$	0	0
$835$	7.42073	0.256805
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−7.77223	−0.268327	−0.134164	−	0.990959i	$-0.542835\pi$
$839$	−7.77223	−0.268327	−0.134164	+	0.990959i	$0.542835\pi$
$840$	0	0
$841$	61.3079	2.11407
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−4.48923	−0.154434
$846$	0	0
$847$	9.28259	0.318954
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−8.40429	−0.288095
$852$	0	0
$853$	20.0543	0.686647	0.343324	−	0.939217i	$-0.388447\pi$
$853$	20.0543	0.686647	0.343324	+	0.939217i	$0.388447\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	1.67582	0.0572449	0.0286225	−	0.999590i	$-0.490888\pi$
$857$	1.67582	0.0572449	0.0286225	+	0.999590i	$0.490888\pi$
$858$	0	0
$859$	37.4511	1.27782	0.638908	−	0.769283i	$-0.279386\pi$
$859$	37.4511	1.27782	0.638908	+	0.769283i	$0.279386\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−30.0265	−1.02211	−0.511057	−	0.859547i	$-0.670746\pi$
$863$	−30.0265	−1.02211	−0.511057	+	0.859547i	$0.670746\pi$
$864$	0	0
$865$	−1.41436	−0.0480897
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−19.1226	−0.648691
$870$	0	0
$871$	22.6934	0.768936
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−10.7706	−0.364113
$876$	0	0
$877$	22.7033	0.766637	0.383318	−	0.923616i	$-0.374781\pi$
$877$	22.7033	0.766637	0.383318	+	0.923616i	$0.374781\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−36.0266	−1.21377	−0.606883	−	0.794791i	$-0.707580\pi$
$881$	−36.0266	−1.21377	−0.606883	+	0.794791i	$0.707580\pi$
$882$	0	0
$883$	47.7159	1.60577	0.802884	−	0.596135i	$-0.203298\pi$
$883$	47.7159	1.60577	0.802884	+	0.596135i	$0.203298\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−11.4192	−0.383421	−0.191710	−	0.981452i	$-0.561403\pi$
$887$	−11.4192	−0.383421	−0.191710	+	0.981452i	$0.561403\pi$
$888$	0	0
$889$	−14.9847	−0.502571
$890$	0	0
$891$	0	0
$892$	0	0
$893$	0.667961	0.0223525
$894$	0	0
$895$	−10.1231	−0.338378
$896$	0	0
$897$	0	0
$898$	0	0
$899$	61.7516	2.05953
$900$	0	0
$901$	−4.50915	−0.150221
$902$	0	0
$903$	0	0
$904$	0	0
$905$	8.09117	0.268960
$906$	0	0
$907$	3.63093	0.120563	0.0602815	−	0.998181i	$-0.480800\pi$
$907$	3.63093	0.120563	0.0602815	+	0.998181i	$0.480800\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	2.49695	0.0827277	0.0413638	−	0.999144i	$-0.486830\pi$
$911$	2.49695	0.0827277	0.0413638	+	0.999144i	$0.486830\pi$
$912$	0	0
$913$	−7.97088	−0.263798
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−10.4158	−0.343959
$918$	0	0
$919$	0.454816	0.0150030	0.00750149	−	0.999972i	$-0.497612\pi$
$919$	0.454816	0.0150030	0.00750149	+	0.999972i	$0.497612\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	6.84918	0.225443
$924$	0	0
$925$	−0.781890	−0.0257084
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−33.3691	−1.09481	−0.547403	−	0.836869i	$-0.684383\pi$
$929$	−33.3691	−1.09481	−0.547403	+	0.836869i	$0.684383\pi$
$930$	0	0
$931$	3.08232	0.101019
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0.989932	0.0323742
$936$	0	0
$937$	−5.28675	−0.172711	−0.0863553	−	0.996264i	$-0.527522\pi$
$937$	−5.28675	−0.172711	−0.0863553	+	0.996264i	$0.527522\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	44.6168	1.45447	0.727234	−	0.686390i	$-0.240806\pi$
$941$	44.6168	1.45447	0.727234	+	0.686390i	$0.240806\pi$
$942$	0	0
$943$	−34.5487	−1.12506
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−28.7389	−0.933890	−0.466945	−	0.884286i	$-0.654645\pi$
$947$	−28.7389	−0.933890	−0.466945	+	0.884286i	$0.654645\pi$
$948$	0	0
$949$	23.1745	0.752276
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−52.6957	−1.70698	−0.853491	−	0.521107i	$-0.825519\pi$
$953$	−52.6957	−1.70698	−0.853491	+	0.521107i	$0.825519\pi$
$954$	0	0
$955$	12.5475	0.406029
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−6.26340	−0.202256
$960$	0	0
$961$	11.2251	0.362100
$962$	0	0
$963$	0	0
$964$	0	0
$965$	20.9883	0.675636
$966$	0	0
$967$	44.3127	1.42500	0.712500	−	0.701672i	$-0.247563\pi$
$967$	44.3127	1.42500	0.712500	+	0.701672i	$0.247563\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−49.8993	−1.60134	−0.800672	−	0.599103i	$-0.795524\pi$
$971$	−49.8993	−1.60134	−0.800672	+	0.599103i	$0.795524\pi$
$972$	0	0
$973$	20.1597	0.646290
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−32.1961	−1.03004	−0.515022	−	0.857177i	$-0.672216\pi$
$977$	−32.1961	−1.03004	−0.515022	+	0.857177i	$0.672216\pi$
$978$	0	0
$979$	7.23689	0.231292
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−12.3471	−0.393812	−0.196906	−	0.980422i	$-0.563089\pi$
$983$	−12.3471	−0.393812	−0.196906	+	0.980422i	$0.563089\pi$
$984$	0	0
$985$	19.8250	0.631678
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0.359223	0.0114226
$990$	0	0
$991$	6.80483	0.216163	0.108081	−	0.994142i	$-0.465529\pi$
$991$	6.80483	0.216163	0.108081	+	0.994142i	$0.465529\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	58.8958	1.86712
$996$	0	0
$997$	1.62483	0.0514589	0.0257295	−	0.999669i	$-0.491809\pi$
$997$	1.62483	0.0514589	0.0257295	+	0.999669i	$0.491809\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4536.2.a.z.1.1		4
3.2	odd	2		4536.2.a.y.1.4		4
4.3	odd	2		9072.2.a.cj.1.1		4
9.2	odd	6		1512.2.r.e.1009.1		8
9.4	even	3		504.2.r.e.169.3	✓	8
9.5	odd	6		1512.2.r.e.505.1		8
9.7	even	3		504.2.r.e.337.3	yes	8
12.11	even	2		9072.2.a.cg.1.4		4
36.7	odd	6		1008.2.r.l.337.2		8
36.11	even	6		3024.2.r.m.1009.1		8
36.23	even	6		3024.2.r.m.2017.1		8
36.31	odd	6		1008.2.r.l.673.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
504.2.r.e.169.3	✓	8	9.4	even	3
504.2.r.e.337.3	yes	8	9.7	even	3
1008.2.r.l.337.2		8	36.7	odd	6
1008.2.r.l.673.2		8	36.31	odd	6
1512.2.r.e.505.1		8	9.5	odd	6
1512.2.r.e.1009.1		8	9.2	odd	6
3024.2.r.m.1009.1		8	36.11	even	6
3024.2.r.m.2017.1		8	36.23	even	6
4536.2.a.y.1.4		4	3.2	odd	2
4536.2.a.z.1.1		4	1.1	even	1	trivial
9072.2.a.cg.1.4		4	12.11	even	2
9072.2.a.cj.1.1		4	4.3	odd	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 4536 = 2^{3} \cdot 3^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4536.a (trivial)

\(f(q)\)	\(=\)	\(q-2.31050 q^{5} -1.00000 q^{7} +O(q^{10})\)	\(q-2.31050 q^{5} -1.00000 q^{7} +1.31050 q^{11} -3.86561 q^{13} -0.326936 q^{17} +3.08232 q^{19} +3.63744 q^{23} +0.338407 q^{25} +9.50305 q^{29} +6.49808 q^{31} +2.31050 q^{35} -2.31050 q^{37} -9.49808 q^{41} +0.0987572 q^{43} +0.216707 q^{47} +1.00000 q^{49} +13.7921 q^{53} -3.02791 q^{55} -2.44985 q^{59} -15.3687 q^{61} +8.93150 q^{65} -5.87058 q^{67} -1.77182 q^{71} -5.99504 q^{73} -1.31050 q^{77} -14.5919 q^{79} -6.08232 q^{83} +0.755385 q^{85} +5.52224 q^{89} +3.86561 q^{91} -7.12170 q^{95} -5.98356 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 3 q^{5} - 4 q^{7}+O(q^{10}) \) 4 * q + 3 * q^5 - 4 * q^7	\( 4 q + 3 q^{5} - 4 q^{7} - 7 q^{11} - 3 q^{13} + 3 q^{17} - 4 q^{19} - 2 q^{23} + 5 q^{25} + 9 q^{29} - 3 q^{31} - 3 q^{35} + 3 q^{37} - 9 q^{41} - 8 q^{43} - 3 q^{47} + 4 q^{49} + 6 q^{53} - 28 q^{55} - 10 q^{59} - 20 q^{61} - q^{65} - 11 q^{67} - 3 q^{71} - 24 q^{73} + 7 q^{77} - 21 q^{79} - 8 q^{83} - 9 q^{85} + 6 q^{89} + 3 q^{91} - 36 q^{95} - 16 q^{97}+O(q^{100}) \) 4 * q + 3 * q^5 - 4 * q^7 - 7 * q^11 - 3 * q^13 + 3 * q^17 - 4 * q^19 - 2 * q^23 + 5 * q^25 + 9 * q^29 - 3 * q^31 - 3 * q^35 + 3 * q^37 - 9 * q^41 - 8 * q^43 - 3 * q^47 + 4 * q^49 + 6 * q^53 - 28 * q^55 - 10 * q^59 - 20 * q^61 - q^65 - 11 * q^67 - 3 * q^71 - 24 * q^73 + 7 * q^77 - 21 * q^79 - 8 * q^83 - 9 * q^85 + 6 * q^89 + 3 * q^91 - 36 * q^95 - 16 * q^97

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