LMFDB - Embedded newform 3600.3.c.l.449.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3600,3,Mod(449,3600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3600, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1, 1]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("3600.449");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 3600 = 2^{4} \cdot 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	3600.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$98.0928951697$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.40960000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 7x^{4} + 1 $ x^8 + 7*x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{37}]$
Coefficient ring index:	$ 2^{11} $
Twist minimal:	no (minimal twist has level 360)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			449.6
Root			$1.14412 + 1.14412i$ of defining polynomial
Character	$\chi$	$=$	3600.449
Dual form			3600.3.c.l.449.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.83772i q^{7} +O(q^{10})$	$q+4.83772i q^{7} +14.8306i q^{11} +20.1359i q^{13} +6.57484 q^{17} +17.6754 q^{19} +25.1891 q^{23} +38.4133i q^{29} +32.9737 q^{31} -44.7851i q^{37} +21.2132i q^{41} +16.2719i q^{43} -24.0044 q^{47} +25.5964 q^{49} +42.4264 q^{53} -86.3848i q^{59} -38.2719 q^{61} -37.2982i q^{67} +19.3400i q^{71} +108.974i q^{73} -71.7464 q^{77} +119.570 q^{79} -123.458 q^{83} -63.0317i q^{89} -97.4121 q^{91} +109.623i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q + 192 q^{19} + 112 q^{31} - 200 q^{49} + 48 q^{61} + 400 q^{79} + 688 q^{91}+O(q^{100}) $ 8 * q + 192 * q^19 + 112 * q^31 - 200 * q^49 + 48 * q^61 + 400 * q^79 + 688 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3600\mathbb{Z}\right)^\times$.

$n$	$577$	$901$	$2801$	$3151$
$\chi(n)$	$-1$	$1$	$-1$	$1$

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 $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	4.83772i	0.691103i	0.938400	+	0.345552i	$0.112308\pi$
$7$	4.83772i	0.691103i	−0.938400	+	0.345552i	$0.887692\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	14.8306i	1.34824i	0.738623	+	0.674119i	$0.235477\pi$
$11$	14.8306i	1.34824i	−0.738623	+	0.674119i	$0.764523\pi$
$12$	0	0
$13$	20.1359i	1.54892i	0.632624	+	0.774459i	$0.281978\pi$
$13$	20.1359i	1.54892i	−0.632624	+	0.774459i	$0.718022\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	6.57484	0.386755	0.193378	−	0.981124i	$-0.438056\pi$
$17$	6.57484	0.386755	0.193378	+	0.981124i	$0.438056\pi$
$18$	0	0
$19$	17.6754	0.930287	0.465143	−	0.885235i	$-0.346003\pi$
$19$	17.6754	0.930287	0.465143	+	0.885235i	$0.346003\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	25.1891	1.09518	0.547589	−	0.836747i	$-0.315546\pi$
$23$	25.1891	1.09518	0.547589	+	0.836747i	$0.315546\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0	0
$28$	0	0
$29$	38.4133i	1.32460i	0.749241	+	0.662298i	$0.230419\pi$
$29$	38.4133i	1.32460i	−0.749241	+	0.662298i	$0.769581\pi$
$30$	0	0
$31$	32.9737	1.06367	0.531833	−	0.846849i	$-0.321503\pi$
$31$	32.9737	1.06367	0.531833	+	0.846849i	$0.321503\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 44.7851i	− 1.21041i	−0.796071	−	0.605203i	$-0.793092\pi$
$37$	− 44.7851i	− 1.21041i	0.796071	−	0.605203i	$-0.206908\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	21.2132i	0.517395i	0.965958	+	0.258698i	$0.0832933\pi$
$41$	21.2132i	0.517395i	−0.965958	+	0.258698i	$0.916707\pi$
$42$	0	0
$43$	16.2719i	0.378416i	0.981937	+	0.189208i	$0.0605920\pi$
$43$	16.2719i	0.378416i	−0.981937	+	0.189208i	$0.939408\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−24.0044	−0.510732	−0.255366	−	0.966845i	$-0.582196\pi$
$47$	−24.0044	−0.510732	−0.255366	+	0.966845i	$0.582196\pi$
$48$	0	0
$49$	25.5964	0.522376
$50$	0	0
$51$	0	0
$52$	0	0
$53$	42.4264	0.800498	0.400249	−	0.916406i	$-0.368924\pi$
$53$	42.4264	0.800498	0.400249	+	0.916406i	$0.368924\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 86.3848i	− 1.46415i	−0.681225	−	0.732075i	$-0.738552\pi$
$59$	− 86.3848i	− 1.46415i	0.681225	−	0.732075i	$-0.261448\pi$
$60$	0	0
$61$	−38.2719	−0.627408	−0.313704	−	0.949521i	$-0.601570\pi$
$61$	−38.2719	−0.627408	−0.313704	+	0.949521i	$0.601570\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 37.2982i	− 0.556690i	−0.960481	−	0.278345i	$-0.910214\pi$
$67$	− 37.2982i	− 0.556690i	0.960481	−	0.278345i	$-0.0897858\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	19.3400i	0.272394i	0.990682	+	0.136197i	$0.0434881\pi$
$71$	19.3400i	0.272394i	−0.990682	+	0.136197i	$0.956512\pi$
$72$	0	0
$73$	108.974i	1.49279i	0.665503	+	0.746395i	$0.268217\pi$
$73$	108.974i	1.49279i	−0.665503	+	0.746395i	$0.731783\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−71.7464	−0.931772
$78$	0	0
$79$	119.570	1.51355	0.756773	−	0.653678i	$-0.226775\pi$
$79$	119.570	1.51355	0.756773	+	0.653678i	$0.226775\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−123.458	−1.48745	−0.743725	−	0.668486i	$-0.766943\pi$
$83$	−123.458	−1.48745	−0.743725	+	0.668486i	$0.766943\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 63.0317i	− 0.708221i	−0.935204	−	0.354110i	$-0.884784\pi$
$89$	− 63.0317i	− 0.708221i	0.935204	−	0.354110i	$-0.115216\pi$
$90$	0	0
$91$	−97.4121	−1.07046
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	109.623i	1.13013i	0.825046	+	0.565066i	$0.191149\pi$
$97$	109.623i	1.13013i	−0.825046	+	0.565066i	$0.808851\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	140.696i	1.39303i	0.717544	+	0.696513i	$0.245266\pi$
$101$	140.696i	1.39303i	−0.717544	+	0.696513i	$0.754734\pi$
$102$	0	0
$103$	− 143.057i	− 1.38890i	−0.719540	−	0.694451i	$-0.755647\pi$
$103$	− 143.057i	− 1.38890i	0.719540	−	0.694451i	$-0.244353\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−88.5992	−0.828030	−0.414015	−	0.910270i	$-0.635874\pi$
$107$	−88.5992	−0.828030	−0.414015	+	0.910270i	$0.635874\pi$
$108$	0	0
$109$	−213.517	−1.95888	−0.979438	−	0.201746i	$-0.935338\pi$
$109$	−213.517	−1.95888	−0.979438	+	0.201746i	$0.935338\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	11.2392	0.0994622	0.0497311	−	0.998763i	$-0.484164\pi$
$113$	11.2392	0.0994622	0.0497311	+	0.998763i	$0.484164\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	31.8072i	0.267288i
$120$	0	0
$121$	−98.9473	−0.817747
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	36.8377i	0.290061i	0.989427	+	0.145030i	$0.0463280\pi$
$127$	36.8377i	0.290061i	−0.989427	+	0.145030i	$0.953672\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 11.5432i	− 0.0881161i	−0.999029	−	0.0440580i	$-0.985971\pi$
$131$	− 11.5432i	− 0.0881161i	0.999029	−	0.0440580i	$-0.0140286\pi$
$132$	0	0
$133$	85.5089i	0.642924i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−145.701	−1.06351	−0.531756	−	0.846897i	$-0.678468\pi$
$137$	−145.701	−1.06351	−0.531756	+	0.846897i	$0.678468\pi$
$138$	0	0
$139$	182.649	1.31402	0.657011	−	0.753881i	$-0.271820\pi$
$139$	182.649	1.31402	0.657011	+	0.753881i	$0.271820\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−298.629	−2.08831
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	142.532i	0.956588i	0.878200	+	0.478294i	$0.158745\pi$
$149$	142.532i	0.956588i	−0.878200	+	0.478294i	$0.841255\pi$
$150$	0	0
$151$	230.438	1.52608	0.763041	−	0.646350i	$-0.223705\pi$
$151$	230.438	1.52608	0.763041	+	0.646350i	$0.223705\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 164.136i	− 1.04545i	−0.852501	−	0.522726i	$-0.824915\pi$
$157$	− 164.136i	− 1.04545i	0.852501	−	0.522726i	$-0.175085\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	121.858i	0.756881i
$162$	0	0
$163$	222.763i	1.36664i	0.730117	+	0.683322i	$0.239465\pi$
$163$	222.763i	1.36664i	−0.730117	+	0.683322i	$0.760535\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	236.180	1.41425	0.707125	−	0.707089i	$-0.249992\pi$
$167$	236.180	1.41425	0.707125	+	0.707089i	$0.249992\pi$
$168$	0	0
$169$	−236.456	−1.39915
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−204.106	−1.17980	−0.589901	−	0.807476i	$-0.700833\pi$
$173$	−204.106	−1.17980	−0.589901	+	0.807476i	$0.700833\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 314.110i	− 1.75481i	−0.479753	−	0.877403i	$-0.659274\pi$
$179$	− 314.110i	− 1.75481i	0.479753	−	0.877403i	$-0.340726\pi$
$180$	0	0
$181$	304.763	1.68377	0.841887	−	0.539654i	$-0.181445\pi$
$181$	304.763	1.68377	0.841887	+	0.539654i	$0.181445\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	97.5089i	0.521438i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	12.3062i	0.0644302i	0.999481	+	0.0322151i	$0.0102562\pi$
$191$	12.3062i	0.0644302i	−0.999481	+	0.0322151i	$0.989744\pi$
$192$	0	0
$193$	293.895i	1.52277i	0.648300	+	0.761385i	$0.275480\pi$
$193$	293.895i	1.52277i	−0.648300	+	0.761385i	$0.724520\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	160.346	0.813937	0.406969	−	0.913442i	$-0.366586\pi$
$197$	160.346	0.813937	0.406969	+	0.913442i	$0.366586\pi$
$198$	0	0
$199$	−146.982	−0.738604	−0.369302	−	0.929309i	$-0.620403\pi$
$199$	−146.982	−0.738604	−0.369302	+	0.929309i	$0.620403\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−185.833	−0.915432
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	262.138i	1.25425i
$210$	0	0
$211$	398.982	1.89091	0.945455	−	0.325751i	$-0.105617\pi$
$211$	398.982	1.89091	0.945455	+	0.325751i	$0.105617\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	159.517i	0.735103i
$218$	0	0
$219$	0	0
$220$	0	0
$221$	132.391i	0.599052i
$222$	0	0
$223$	− 226.092i	− 1.01386i	−0.861986	−	0.506932i	$-0.830779\pi$
$223$	− 226.092i	− 1.01386i	0.861986	−	0.506932i	$-0.169221\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−153.504	−0.676229	−0.338115	−	0.941105i	$-0.609789\pi$
$227$	−153.504	−0.676229	−0.338115	+	0.941105i	$0.609789\pi$
$228$	0	0
$229$	−72.8071	−0.317935	−0.158967	−	0.987284i	$-0.550817\pi$
$229$	−72.8071	−0.317935	−0.158967	+	0.987284i	$0.550817\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−14.9856	−0.0643160	−0.0321580	−	0.999483i	$-0.510238\pi$
$233$	−14.9856	−0.0643160	−0.0321580	+	0.999483i	$0.510238\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 185.299i	− 0.775311i	−0.921804	−	0.387655i	$-0.873285\pi$
$239$	− 185.299i	− 0.775311i	0.921804	−	0.387655i	$-0.126715\pi$
$240$	0	0
$241$	134.930	0.559874	0.279937	−	0.960018i	$-0.409686\pi$
$241$	134.930	0.559874	0.279937	+	0.960018i	$0.409686\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	355.912i	1.44094i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	388.419i	1.54748i	0.633501	+	0.773742i	$0.281617\pi$
$251$	388.419i	1.54748i	−0.633501	+	0.773742i	$0.718383\pi$
$252$	0	0
$253$	373.570i	1.47656i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	260.227	1.01256	0.506279	−	0.862370i	$-0.331021\pi$
$257$	260.227	1.01256	0.506279	+	0.862370i	$0.331021\pi$
$258$	0	0
$259$	216.658	0.836516
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−243.673	−0.926512	−0.463256	−	0.886225i	$-0.653319\pi$
$263$	−243.673	−0.926512	−0.463256	+	0.886225i	$0.653319\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	4.62110i	0.0171788i	0.999963	+	0.00858941i	$0.00273413\pi$
$269$	4.62110i	0.0171788i	−0.999963	+	0.00858941i	$0.997266\pi$
$270$	0	0
$271$	307.947	1.13634	0.568169	−	0.822912i	$-0.307652\pi$
$271$	307.947	1.13634	0.568169	+	0.822912i	$0.307652\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	332.680i	1.20101i	0.799621	+	0.600505i	$0.205034\pi$
$277$	332.680i	1.20101i	−0.799621	+	0.600505i	$0.794966\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 323.408i	− 1.15092i	−0.817831	−	0.575459i	$-0.804823\pi$
$281$	− 323.408i	− 1.15092i	0.817831	−	0.575459i	$-0.195177\pi$
$282$	0	0
$283$	− 181.737i	− 0.642179i	−0.947049	−	0.321090i	$-0.895951\pi$
$283$	− 181.737i	− 0.642179i	0.947049	−	0.321090i	$-0.104049\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−102.624	−0.357573
$288$	0	0
$289$	−245.772	−0.850421
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−378.978	−1.29344	−0.646720	−	0.762727i	$-0.723860\pi$
$293$	−378.978	−1.29344	−0.646720	+	0.762727i	$0.723860\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	507.206i	1.69634i
$300$	0	0
$301$	−78.7189	−0.261524
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 16.5438i	− 0.0538885i	−0.999637	−	0.0269443i	$-0.991422\pi$
$307$	− 16.5438i	− 0.0538885i	0.999637	−	0.0269443i	$-0.00857766\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	− 178.662i	− 0.574476i	−0.957859	−	0.287238i	$-0.907263\pi$
$311$	− 178.662i	− 0.574476i	0.957859	−	0.287238i	$-0.0927370\pi$
$312$	0	0
$313$	− 135.088i	− 0.431590i	−0.976439	−	0.215795i	$-0.930766\pi$
$313$	− 135.088i	− 0.431590i	0.976439	−	0.215795i	$-0.0692342\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−378.445	−1.19383	−0.596916	−	0.802304i	$-0.703607\pi$
$317$	−378.445	−1.19383	−0.596916	+	0.802304i	$0.703607\pi$
$318$	0	0
$319$	−569.693	−1.78587
$320$	0	0
$321$	0	0
$322$	0	0
$323$	116.213	0.359793
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 116.127i	− 0.352968i
$330$	0	0
$331$	−371.956	−1.12373	−0.561867	−	0.827228i	$-0.689917\pi$
$331$	−371.956	−1.12373	−0.561867	+	0.827228i	$0.689917\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 193.956i	− 0.575537i	−0.957700	−	0.287768i	$-0.907087\pi$
$337$	− 193.956i	− 0.575537i	0.957700	−	0.287768i	$-0.0929133\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	489.020i	1.43408i
$342$	0	0
$343$	360.877i	1.05212i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−649.856	−1.87278	−0.936392	−	0.350957i	$-0.885856\pi$
$347$	−649.856	−1.87278	−0.936392	+	0.350957i	$0.885856\pi$
$348$	0	0
$349$	343.675	0.984743	0.492372	−	0.870385i	$-0.336130\pi$
$349$	343.675	0.984743	0.492372	+	0.870385i	$0.336130\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−461.195	−1.30650	−0.653250	−	0.757142i	$-0.726595\pi$
$353$	−461.195	−1.30650	−0.653250	+	0.757142i	$0.726595\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 248.380i	− 0.691867i	−0.938259	−	0.345933i	$-0.887562\pi$
$359$	− 248.380i	− 0.691867i	0.938259	−	0.345933i	$-0.112438\pi$
$360$	0	0
$361$	−48.5787	−0.134567
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 59.6007i	− 0.162400i	−0.996698	−	0.0811999i	$-0.974125\pi$
$367$	− 59.6007i	− 0.162400i	0.996698	−	0.0811999i	$-0.0258752\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	205.247i	0.553227i
$372$	0	0
$373$	− 110.460i	− 0.296141i	−0.988977	−	0.148070i	$-0.952694\pi$
$373$	− 110.460i	− 0.296141i	0.988977	−	0.148070i	$-0.0473062\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−773.487	−2.05169
$378$	0	0
$379$	637.579	1.68227	0.841133	−	0.540829i	$-0.181889\pi$
$379$	637.579	1.68227	0.841133	+	0.540829i	$0.181889\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−574.728	−1.50060	−0.750298	−	0.661100i	$-0.770090\pi$
$383$	−574.728	−1.50060	−0.750298	+	0.661100i	$0.770090\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 71.4364i	− 0.183641i	−0.995776	−	0.0918206i	$-0.970731\pi$
$389$	− 71.4364i	− 0.183641i	0.995776	−	0.0918206i	$-0.0292686\pi$
$390$	0	0
$391$	165.614	0.423566
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	365.767i	0.921328i	0.887575	+	0.460664i	$0.152389\pi$
$397$	365.767i	0.921328i	−0.887575	+	0.460664i	$0.847611\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	728.791i	1.81743i	0.417413	+	0.908717i	$0.362937\pi$
$401$	728.791i	1.81743i	−0.417413	+	0.908717i	$0.637063\pi$
$402$	0	0
$403$	663.956i	1.64753i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	664.190	1.63192
$408$	0	0
$409$	401.035	0.980525	0.490263	−	0.871575i	$-0.336901\pi$
$409$	401.035	0.980525	0.490263	+	0.871575i	$0.336901\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	417.906	1.01188
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	356.623i	0.851130i	0.904928	+	0.425565i	$0.139925\pi$
$419$	356.623i	0.851130i	−0.904928	+	0.425565i	$0.860075\pi$
$420$	0	0
$421$	272.719	0.647788	0.323894	−	0.946093i	$-0.395008\pi$
$421$	272.719	0.647788	0.323894	+	0.946093i	$0.395008\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 185.149i	− 0.433604i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	469.680i	1.08974i	0.838519	+	0.544872i	$0.183422\pi$
$431$	469.680i	1.08974i	−0.838519	+	0.544872i	$0.816578\pi$
$432$	0	0
$433$	− 361.412i	− 0.834670i	−0.908753	−	0.417335i	$-0.862964\pi$
$433$	− 361.412i	− 0.834670i	0.908753	−	0.417335i	$-0.137036\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	445.229	1.01883
$438$	0	0
$439$	−250.105	−0.569716	−0.284858	−	0.958570i	$-0.591946\pi$
$439$	−250.105	−0.569716	−0.284858	+	0.958570i	$0.591946\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−356.319	−0.804333	−0.402166	−	0.915567i	$-0.631743\pi$
$443$	−356.319	−0.804333	−0.402166	+	0.915567i	$0.631743\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 282.657i	− 0.629525i	−0.949171	−	0.314762i	$-0.898075\pi$
$449$	− 282.657i	− 0.629525i	0.949171	−	0.314762i	$-0.101925\pi$
$450$	0	0
$451$	−314.605	−0.697572
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 413.851i	− 0.905581i	−0.891617	−	0.452791i	$-0.850429\pi$
$457$	− 413.851i	− 0.905581i	0.891617	−	0.452791i	$-0.149571\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 177.453i	− 0.384931i	−0.981304	−	0.192465i	$-0.938352\pi$
$461$	− 177.453i	− 0.384931i	0.981304	−	0.192465i	$-0.0616483\pi$
$462$	0	0
$463$	267.101i	0.576892i	0.957496	+	0.288446i	$0.0931386\pi$
$463$	267.101i	0.576892i	−0.957496	+	0.288446i	$0.906861\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−260.525	−0.557870	−0.278935	−	0.960310i	$-0.589981\pi$
$467$	−260.525	−0.557870	−0.278935	+	0.960310i	$0.589981\pi$
$468$	0	0
$469$	180.438	0.384730
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−241.322	−0.510195
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 149.746i	− 0.312621i	−0.987708	−	0.156311i	$-0.950040\pi$
$479$	− 149.746i	− 0.312621i	0.987708	−	0.156311i	$-0.0499601\pi$
$480$	0	0
$481$	901.789	1.87482
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 628.749i	− 1.29107i	−0.763732	−	0.645533i	$-0.776635\pi$
$487$	− 628.749i	− 1.29107i	0.763732	−	0.645533i	$-0.223365\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	390.341i	0.794992i	0.917604	+	0.397496i	$0.130121\pi$
$491$	390.341i	0.794992i	−0.917604	+	0.397496i	$0.869879\pi$
$492$	0	0
$493$	252.561i	0.512294i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−93.5615	−0.188253
$498$	0	0
$499$	−254.097	−0.509212	−0.254606	−	0.967045i	$-0.581946\pi$
$499$	−254.097	−0.509212	−0.254606	+	0.967045i	$0.581946\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	830.503	1.65110	0.825549	−	0.564330i	$-0.190865\pi$
$503$	830.503	1.65110	0.825549	+	0.564330i	$0.190865\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 808.291i	− 1.58800i	−0.607919	−	0.793999i	$-0.707995\pi$
$509$	− 808.291i	− 1.58800i	0.607919	−	0.793999i	$-0.292005\pi$
$510$	0	0
$511$	−527.184	−1.03167
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 356.000i	− 0.688588i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	591.948i	1.13618i	0.822968	+	0.568088i	$0.192317\pi$
$521$	591.948i	1.13618i	−0.822968	+	0.568088i	$0.807683\pi$
$522$	0	0
$523$	529.781i	1.01297i	0.862250	+	0.506483i	$0.169055\pi$
$523$	529.781i	1.01297i	−0.862250	+	0.506483i	$0.830945\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	216.796	0.411378
$528$	0	0
$529$	105.491	0.199416
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−427.148	−0.801403
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	379.611i	0.704288i
$540$	0	0
$541$	−413.307	−0.763968	−0.381984	−	0.924169i	$-0.624759\pi$
$541$	−413.307	−0.763968	−0.381984	+	0.924169i	$0.624759\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	691.851i	1.26481i	0.774638	+	0.632405i	$0.217932\pi$
$547$	691.851i	1.26481i	−0.774638	+	0.632405i	$0.782068\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	678.971i	1.23225i
$552$	0	0
$553$	578.447i	1.04602i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−111.996	−0.201069	−0.100535	−	0.994934i	$-0.532055\pi$
$557$	−111.996	−0.201069	−0.100535	+	0.994934i	$0.532055\pi$
$558$	0	0
$559$	−327.650	−0.586136
$560$	0	0
$561$	0	0
$562$	0	0
$563$	397.455	0.705960	0.352980	−	0.935631i	$-0.385168\pi$
$563$	397.455	0.705960	0.352980	+	0.935631i	$0.385168\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 340.056i	− 0.597639i	−0.954310	−	0.298819i	$-0.903407\pi$
$569$	− 340.056i	− 0.597639i	0.954310	−	0.298819i	$-0.0965928\pi$
$570$	0	0
$571$	547.412	0.958690	0.479345	−	0.877627i	$-0.340874\pi$
$571$	547.412	0.958690	0.479345	+	0.877627i	$0.340874\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	140.974i	0.244322i	0.992510	+	0.122161i	$0.0389824\pi$
$577$	140.974i	0.244322i	−0.992510	+	0.122161i	$0.961018\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 597.257i	− 1.02798i
$582$	0	0
$583$	629.210i	1.07926i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−222.057	−0.378291	−0.189145	−	0.981949i	$-0.560572\pi$
$587$	−222.057	−0.378291	−0.189145	+	0.981949i	$0.560572\pi$
$588$	0	0
$589$	582.824	0.989515
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−948.975	−1.60029	−0.800147	−	0.599804i	$-0.795245\pi$
$593$	−948.975	−1.60029	−0.800147	+	0.599804i	$0.795245\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	31.7951i	0.0530804i	0.999648	+	0.0265402i	$0.00844900\pi$
$599$	31.7951i	0.0530804i	−0.999648	+	0.0265402i	$0.991551\pi$
$600$	0	0
$601$	−520.561	−0.866158	−0.433079	−	0.901356i	$-0.642573\pi$
$601$	−520.561	−0.866158	−0.433079	+	0.901356i	$0.642573\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 676.505i	− 1.11451i	−0.830343	−	0.557253i	$-0.811856\pi$
$607$	− 676.505i	− 1.11451i	0.830343	−	0.557253i	$-0.188144\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 483.351i	− 0.791082i
$612$	0	0
$613$	− 479.820i	− 0.782741i	−0.920233	−	0.391370i	$-0.872001\pi$
$613$	− 479.820i	− 0.782741i	0.920233	−	0.391370i	$-0.127999\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	514.687	0.834177	0.417088	−	0.908866i	$-0.363051\pi$
$617$	514.687	0.834177	0.417088	+	0.908866i	$0.363051\pi$
$618$	0	0
$619$	−830.894	−1.34232	−0.671158	−	0.741314i	$-0.734203\pi$
$619$	−830.894	−1.34232	−0.671158	+	0.741314i	$0.734203\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	304.930	0.489454
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 294.454i	− 0.468131i
$630$	0	0
$631$	481.324	0.762795	0.381398	−	0.924411i	$-0.375443\pi$
$631$	481.324	0.762795	0.381398	+	0.924411i	$0.375443\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	515.409i	0.809119i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 1076.40i	− 1.67925i	−0.543163	−	0.839627i	$-0.682774\pi$
$641$	− 1076.40i	− 1.67925i	0.543163	−	0.839627i	$-0.317226\pi$
$642$	0	0
$643$	1021.36i	1.58843i	0.607638	+	0.794214i	$0.292117\pi$
$643$	1021.36i	1.58843i	−0.607638	+	0.794214i	$0.707883\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−972.866	−1.50366	−0.751829	−	0.659359i	$-0.770828\pi$
$647$	−972.866	−1.50366	−0.751829	+	0.659359i	$0.770828\pi$
$648$	0	0
$649$	1281.14	1.97402
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−309.260	−0.473599	−0.236799	−	0.971559i	$-0.576098\pi$
$653$	−309.260	−0.473599	−0.236799	+	0.971559i	$0.576098\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	294.758i	0.447281i	0.974672	+	0.223641i	$0.0717942\pi$
$659$	294.758i	0.447281i	−0.974672	+	0.223641i	$0.928206\pi$
$660$	0	0
$661$	232.552	0.351819	0.175909	−	0.984406i	$-0.443713\pi$
$661$	232.552	0.351819	0.175909	+	0.984406i	$0.443713\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	967.596i	1.45067i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	− 567.596i	− 0.845895i
$672$	0	0
$673$	− 37.5174i	− 0.0557466i	−0.999611	−	0.0278733i	$-0.991127\pi$
$673$	− 37.5174i	− 0.0557466i	0.999611	−	0.0278733i	$-0.00887349\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−111.630	−0.164890	−0.0824448	−	0.996596i	$-0.526273\pi$
$677$	−111.630	−0.164890	−0.0824448	+	0.996596i	$0.526273\pi$
$678$	0	0
$679$	−530.325	−0.781038
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−80.5971	−0.118005	−0.0590023	−	0.998258i	$-0.518792\pi$
$683$	−80.5971	−0.118005	−0.0590023	+	0.998258i	$0.518792\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	854.296i	1.23991i
$690$	0	0
$691$	−280.377	−0.405756	−0.202878	−	0.979204i	$-0.565029\pi$
$691$	−280.377	−0.405756	−0.202878	+	0.979204i	$0.565029\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	139.473i	0.200105i
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 182.365i	− 0.260150i	−0.991504	−	0.130075i	$-0.958478\pi$
$701$	− 182.365i	− 0.260150i	0.991504	−	0.130075i	$-0.0415218\pi$
$702$	0	0
$703$	− 791.596i	− 1.12603i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−680.646	−0.962725
$708$	0	0
$709$	42.7374	0.0602784	0.0301392	−	0.999546i	$-0.490405\pi$
$709$	42.7374	0.0602784	0.0301392	+	0.999546i	$0.490405\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	830.577	1.16490
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	− 765.586i	− 1.06479i	−0.846495	−	0.532396i	$-0.821292\pi$
$719$	− 765.586i	− 1.06479i	0.846495	−	0.532396i	$-0.178708\pi$
$720$	0	0
$721$	692.070	0.959875
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 201.653i	− 0.277377i	−0.990336	−	0.138689i	$-0.955711\pi$
$727$	− 201.653i	− 0.277377i	0.990336	−	0.138689i	$-0.0442887\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	106.985i	0.146354i
$732$	0	0
$733$	− 489.057i	− 0.667199i	−0.942715	−	0.333600i	$-0.891737\pi$
$733$	− 489.057i	− 0.667199i	0.942715	−	0.333600i	$-0.108263\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	553.156	0.750551
$738$	0	0
$739$	−760.306	−1.02883	−0.514415	−	0.857541i	$-0.671991\pi$
$739$	−760.306	−1.02883	−0.514415	+	0.857541i	$0.671991\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	771.776	1.03873	0.519365	−	0.854553i	$-0.326169\pi$
$743$	771.776	1.03873	0.519365	+	0.854553i	$0.326169\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 428.618i	− 0.572254i
$750$	0	0
$751$	−1366.53	−1.81962	−0.909809	−	0.415026i	$-0.863772\pi$
$751$	−1366.53	−1.81962	−0.909809	+	0.415026i	$0.863772\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	10.6271i	0.0140384i	0.999975	+	0.00701919i	$0.00223430\pi$
$757$	10.6271i	0.0140384i	−0.999975	+	0.00701919i	$0.997766\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 20.6052i	− 0.0270765i	−0.999908	−	0.0135383i	$-0.995691\pi$
$761$	− 20.6052i	− 0.0270765i	0.999908	−	0.0135383i	$-0.00430950\pi$
$762$	0	0
$763$	− 1032.94i	− 1.35379i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	1739.44	2.26785
$768$	0	0
$769$	−1235.93	−1.60719	−0.803595	−	0.595177i	$-0.797082\pi$
$769$	−1235.93	−1.60719	−0.803595	+	0.595177i	$0.797082\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	54.0189	0.0698822	0.0349411	−	0.999389i	$-0.488876\pi$
$773$	54.0189	0.0698822	0.0349411	+	0.999389i	$0.488876\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	374.953i	0.481326i
$780$	0	0
$781$	−286.824	−0.367253
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 28.0612i	− 0.0356559i	−0.999841	−	0.0178280i	$-0.994325\pi$
$787$	− 28.0612i	− 0.0356559i	0.999841	−	0.0178280i	$-0.00567512\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	54.3722i	0.0687386i
$792$	0	0
$793$	− 770.641i	− 0.971804i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−835.564	−1.04839	−0.524193	−	0.851599i	$-0.675633\pi$
$797$	−835.564	−1.04839	−0.524193	+	0.851599i	$0.675633\pi$
$798$	0	0
$799$	−157.825	−0.197528
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−1616.15	−2.01264
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 793.187i	− 0.980453i	−0.871595	−	0.490227i	$-0.836914\pi$
$809$	− 793.187i	− 0.980453i	0.871595	−	0.490227i	$-0.163086\pi$
$810$	0	0
$811$	−298.105	−0.367577	−0.183789	−	0.982966i	$-0.558836\pi$
$811$	−298.105	−0.367577	−0.183789	+	0.982966i	$0.558836\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	287.613i	0.352035i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 1143.48i	− 1.39280i	−0.717656	−	0.696398i	$-0.754785\pi$
$821$	− 1143.48i	− 1.39280i	0.717656	−	0.696398i	$-0.245215\pi$
$822$	0	0
$823$	− 460.399i	− 0.559416i	−0.960085	−	0.279708i	$-0.909762\pi$
$823$	− 460.399i	− 0.559416i	0.960085	−	0.279708i	$-0.0902376\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−250.204	−0.302544	−0.151272	−	0.988492i	$-0.548337\pi$
$827$	−250.204	−0.302544	−0.151272	+	0.988492i	$0.548337\pi$
$828$	0	0
$829$	171.815	0.207256	0.103628	−	0.994616i	$-0.466955\pi$
$829$	171.815	0.207256	0.103628	+	0.994616i	$0.466955\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	168.292	0.202032
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	915.442i	1.09111i	0.838075	+	0.545555i	$0.183681\pi$
$839$	915.442i	1.09111i	−0.838075	+	0.545555i	$0.816319\pi$
$840$	0	0
$841$	−634.579	−0.754553
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 478.680i	− 0.565147i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 1128.10i	− 1.32561i
$852$	0	0
$853$	− 1242.33i	− 1.45642i	−0.685353	−	0.728211i	$-0.740352\pi$
$853$	− 1242.33i	− 1.45642i	0.685353	−	0.728211i	$-0.259648\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	716.993	0.836632	0.418316	−	0.908302i	$-0.362621\pi$
$857$	716.993	0.836632	0.418316	+	0.908302i	$0.362621\pi$
$858$	0	0
$859$	551.509	0.642036	0.321018	−	0.947073i	$-0.395975\pi$
$859$	551.509	0.642036	0.321018	+	0.947073i	$0.395975\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	1513.33	1.75356	0.876782	−	0.480887i	$-0.159685\pi$
$863$	1513.33	1.75356	0.876782	+	0.480887i	$0.159685\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	1773.30i	2.04062i
$870$	0	0
$871$	751.035	0.862267
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 946.355i	− 1.07908i	−0.841959	−	0.539541i	$-0.818598\pi$
$877$	− 946.355i	− 1.07908i	0.841959	−	0.539541i	$-0.181402\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	1239.83i	1.40730i	0.710547	+	0.703650i	$0.248447\pi$
$881$	1239.83i	1.40730i	−0.710547	+	0.703650i	$0.751553\pi$
$882$	0	0
$883$	− 1156.41i	− 1.30964i	−0.755785	−	0.654820i	$-0.772745\pi$
$883$	− 1156.41i	− 1.30964i	0.755785	−	0.654820i	$-0.227255\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	1100.19	1.24035	0.620174	−	0.784464i	$-0.287062\pi$
$887$	1100.19	1.24035	0.620174	+	0.784464i	$0.287062\pi$
$888$	0	0
$889$	−178.211	−0.200462
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−424.288	−0.475127
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	1266.63i	1.40893i
$900$	0	0
$901$	278.947	0.309597
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	570.430i	0.628919i	0.949271	+	0.314460i	$0.101823\pi$
$907$	570.430i	0.628919i	−0.949271	+	0.314460i	$0.898177\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	1321.74i	1.45087i	0.688290	+	0.725435i	$0.258362\pi$
$911$	1321.74i	1.45087i	−0.688290	+	0.725435i	$0.741638\pi$
$912$	0	0
$913$	− 1830.96i	− 2.00544i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	55.8428	0.0608973
$918$	0	0
$919$	1748.03	1.90210	0.951048	−	0.309044i	$-0.100009\pi$
$919$	1748.03	1.90210	0.951048	+	0.309044i	$0.100009\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−389.429	−0.421917
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 556.666i	− 0.599210i	−0.954063	−	0.299605i	$-0.903145\pi$
$929$	− 556.666i	− 0.599210i	0.954063	−	0.299605i	$-0.0968548\pi$
$930$	0	0
$931$	452.429	0.485960
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 117.701i	− 0.125615i	−0.998026	−	0.0628074i	$-0.979995\pi$
$937$	− 117.701i	− 0.125615i	0.998026	−	0.0628074i	$-0.0200054\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	1714.17i	1.82165i	0.412797	+	0.910823i	$0.364552\pi$
$941$	1714.17i	1.82165i	−0.412797	+	0.910823i	$0.635448\pi$
$942$	0	0
$943$	534.342i	0.566640i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	1151.68	1.21613	0.608066	−	0.793886i	$-0.291946\pi$
$947$	1151.68	1.21613	0.608066	+	0.793886i	$0.291946\pi$
$948$	0	0
$949$	−2194.29	−2.31221
$950$	0	0
$951$	0	0
$952$	0	0
$953$	1272.18	1.33493	0.667463	−	0.744643i	$-0.267380\pi$
$953$	1272.18	1.33493	0.667463	+	0.744643i	$0.267380\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	− 704.862i	− 0.734997i
$960$	0	0
$961$	126.263	0.131387
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 1358.70i	− 1.40506i	−0.711652	−	0.702532i	$-0.752053\pi$
$967$	− 1358.70i	− 1.40506i	0.711652	−	0.702532i	$-0.247947\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 1818.71i	− 1.87303i	−0.350632	−	0.936513i	$-0.614033\pi$
$971$	− 1818.71i	− 1.87303i	0.350632	−	0.936513i	$-0.385967\pi$
$972$	0	0
$973$	883.606i	0.908125i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	594.974	0.608981	0.304490	−	0.952515i	$-0.401514\pi$
$977$	594.974	0.608981	0.304490	+	0.952515i	$0.401514\pi$
$978$	0	0
$979$	934.799	0.954850
$980$	0	0
$981$	0	0
$982$	0	0
$983$	1209.81	1.23073	0.615366	−	0.788241i	$-0.289008\pi$
$983$	1209.81	1.23073	0.615366	+	0.788241i	$0.289008\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	409.874i	0.414433i
$990$	0	0
$991$	236.596	0.238745	0.119373	−	0.992850i	$-0.461912\pi$
$991$	236.596	0.238745	0.119373	+	0.992850i	$0.461912\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	614.355i	0.616204i	0.951353	+	0.308102i	$0.0996938\pi$
$997$	614.355i	0.616204i	−0.951353	+	0.308102i	$0.900306\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3600.3.c.l.449.6		8
3.2	odd	2	inner	3600.3.c.l.449.5		8
4.3	odd	2		1800.3.c.b.449.3		8
5.2	odd	4		3600.3.l.m.1601.4		4
5.3	odd	4		720.3.l.d.161.3		4
5.4	even	2	inner	3600.3.c.l.449.4		8
12.11	even	2		1800.3.c.b.449.4		8
15.2	even	4		3600.3.l.m.1601.3		4
15.8	even	4		720.3.l.d.161.1		4
15.14	odd	2	inner	3600.3.c.l.449.3		8
20.3	even	4		360.3.l.a.161.4	yes	4
20.7	even	4		1800.3.l.g.1601.1		4
20.19	odd	2		1800.3.c.b.449.5		8
40.3	even	4		2880.3.l.a.1601.2		4
40.13	odd	4		2880.3.l.h.1601.1		4
60.23	odd	4		360.3.l.a.161.2	✓	4
60.47	odd	4		1800.3.l.g.1601.2		4
60.59	even	2		1800.3.c.b.449.6		8
120.53	even	4		2880.3.l.h.1601.3		4
120.83	odd	4		2880.3.l.a.1601.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
360.3.l.a.161.2	✓	4	60.23	odd	4
360.3.l.a.161.4	yes	4	20.3	even	4
720.3.l.d.161.1		4	15.8	even	4
720.3.l.d.161.3		4	5.3	odd	4
1800.3.c.b.449.3		8	4.3	odd	2
1800.3.c.b.449.4		8	12.11	even	2
1800.3.c.b.449.5		8	20.19	odd	2
1800.3.c.b.449.6		8	60.59	even	2
1800.3.l.g.1601.1		4	20.7	even	4
1800.3.l.g.1601.2		4	60.47	odd	4
2880.3.l.a.1601.2		4	40.3	even	4
2880.3.l.a.1601.4		4	120.83	odd	4
2880.3.l.h.1601.1		4	40.13	odd	4
2880.3.l.h.1601.3		4	120.53	even	4
3600.3.c.l.449.3		8	15.14	odd	2	inner
3600.3.c.l.449.4		8	5.4	even	2	inner
3600.3.c.l.449.5		8	3.2	odd	2	inner
3600.3.c.l.449.6		8	1.1	even	1	trivial
3600.3.l.m.1601.3		4	15.2	even	4
3600.3.l.m.1601.4		4	5.2	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3600 = 2^{4} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	3600.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+4.83772i q^{7} +O(q^{10})\)	\(q+4.83772i q^{7} +14.8306i q^{11} +20.1359i q^{13} +6.57484 q^{17} +17.6754 q^{19} +25.1891 q^{23} +38.4133i q^{29} +32.9737 q^{31} -44.7851i q^{37} +21.2132i q^{41} +16.2719i q^{43} -24.0044 q^{47} +25.5964 q^{49} +42.4264 q^{53} -86.3848i q^{59} -38.2719 q^{61} -37.2982i q^{67} +19.3400i q^{71} +108.974i q^{73} -71.7464 q^{77} +119.570 q^{79} -123.458 q^{83} -63.0317i q^{89} -97.4121 q^{91} +109.623i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q + 192 q^{19} + 112 q^{31} - 200 q^{49} + 48 q^{61} + 400 q^{79} + 688 q^{91}+O(q^{100}) \) 8 * q + 192 * q^19 + 112 * q^31 - 200 * q^49 + 48 * q^61 + 400 * q^79 + 688 * q^91

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