LMFDB - Embedded newform 3600.3.c.l.449.7

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3600,3,Mod(449,3600)] 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("3600.449"); S:= CuspForms(chi, 3); N := Newforms(S);

sage: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")

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:traces = [8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,192,0,0,0,0,0,0,0,0,0,0, 0,112] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(31)] == traces)

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			449.7
Root			$-0.437016 - 0.437016i$ of defining polynomial
Character	$\chi$	$=$	3600.449
Dual form			3600.3.c.l.449.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+11.1623i q^{7} -12.0022i q^{11} -24.1359i q^{13} -29.2023 q^{17} +30.3246 q^{19} -19.5323 q^{23} +29.4690i q^{29} -4.97367 q^{31} +24.7851i q^{37} +21.2132i q^{41} -72.2719i q^{43} +29.6612 q^{47} -75.5964 q^{49} +42.4264 q^{53} +83.5564i q^{59} +50.2719 q^{61} +13.2982i q^{67} +37.2285i q^{71} +71.0263i q^{73} +133.972 q^{77} -19.5701 q^{79} -51.9042 q^{83} +133.742i q^{89} +269.412 q^{91} +46.3772i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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

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
$7$	11.1623i	1.59461i	0.603576	+	0.797306i	$0.293742\pi$
$7$	11.1623i	1.59461i	−0.603576	+	0.797306i	$0.706258\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 12.0022i	− 1.09111i	−0.838075	−	0.545554i	$-0.816319\pi$
$11$	− 12.0022i	− 1.09111i	0.838075	−	0.545554i	$-0.183681\pi$
$12$	0	0
$13$	− 24.1359i	− 1.85661i	−0.371818	−	0.928306i	$-0.621266\pi$
$13$	− 24.1359i	− 1.85661i	0.371818	−	0.928306i	$-0.378734\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−29.2023	−1.71778	−0.858890	−	0.512160i	$-0.828845\pi$
$17$	−29.2023	−1.71778	−0.858890	+	0.512160i	$0.828845\pi$
$18$	0	0
$19$	30.3246	1.59603	0.798015	−	0.602638i	$-0.205884\pi$
$19$	30.3246	1.59603	0.798015	+	0.602638i	$0.205884\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−19.5323	−0.849228	−0.424614	−	0.905374i	$-0.639590\pi$
$23$	−19.5323	−0.849228	−0.424614	+	0.905374i	$0.639590\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0	0
$28$	0	0
$29$	29.4690i	1.01617i	0.861306	+	0.508086i	$0.169647\pi$
$29$	29.4690i	1.01617i	−0.861306	+	0.508086i	$0.830353\pi$
$30$	0	0
$31$	−4.97367	−0.160441	−0.0802204	−	0.996777i	$-0.525562\pi$
$31$	−4.97367	−0.160441	−0.0802204	+	0.996777i	$0.525562\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	24.7851i	0.669866i	0.942242	+	0.334933i	$0.108714\pi$
$37$	24.7851i	0.669866i	−0.942242	+	0.334933i	$0.891286\pi$
$38$	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
$43$	− 72.2719i	− 1.68074i	−0.542012	−	0.840371i	$-0.682337\pi$
$43$	− 72.2719i	− 1.68074i	0.542012	−	0.840371i	$-0.317663\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	29.6612	0.631090	0.315545	−	0.948911i	$-0.397813\pi$
$47$	29.6612	0.631090	0.315545	+	0.948911i	$0.397813\pi$
$48$	0	0
$49$	−75.5964	−1.54278
$50$	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
$59$	83.5564i	1.41621i	0.706107	+	0.708105i	$0.250450\pi$
$59$	83.5564i	1.41621i	−0.706107	+	0.708105i	$0.749550\pi$
$60$	0	0
$61$	50.2719	0.824129	0.412065	−	0.911155i	$-0.364808\pi$
$61$	50.2719	0.824129	0.412065	+	0.911155i	$0.364808\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	13.2982i	0.198481i	0.995063	+	0.0992405i	$0.0316413\pi$
$67$	13.2982i	0.198481i	−0.995063	+	0.0992405i	$0.968359\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	37.2285i	0.524346i	0.965021	+	0.262173i	$0.0844391\pi$
$71$	37.2285i	0.524346i	−0.965021	+	0.262173i	$0.915561\pi$
$72$	0	0
$73$	71.0263i	0.972963i	0.873691	+	0.486482i	$0.161720\pi$
$73$	71.0263i	0.972963i	−0.873691	+	0.486482i	$0.838280\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	133.972	1.73989
$78$	0	0
$79$	−19.5701	−0.247723	−0.123861	−	0.992300i	$-0.539528\pi$
$79$	−19.5701	−0.247723	−0.123861	+	0.992300i	$0.539528\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−51.9042	−0.625351	−0.312676	−	0.949860i	$-0.601225\pi$
$83$	−51.9042	−0.625351	−0.312676	+	0.949860i	$0.601225\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	133.742i	1.50272i	0.659891	+	0.751361i	$0.270602\pi$
$89$	133.742i	1.50272i	−0.659891	+	0.751361i	$0.729398\pi$
$90$	0	0
$91$	269.412	2.96057
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	46.3772i	0.478116i	0.971005	+	0.239058i	$0.0768386\pi$
$97$	46.3772i	0.478116i	−0.971005	+	0.239058i	$0.923161\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	113.863i	1.12735i	0.825995	+	0.563677i	$0.190614\pi$
$101$	113.863i	1.12735i	−0.825995	+	0.563677i	$0.809386\pi$
$102$	0	0
$103$	15.0569i	0.146184i	0.997325	+	0.0730919i	$0.0232867\pi$
$103$	15.0569i	0.146184i	−0.997325	+	0.0730919i	$0.976713\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−52.8221	−0.493665	−0.246832	−	0.969058i	$-0.579390\pi$
$107$	−52.8221	−0.493665	−0.246832	+	0.969058i	$0.579390\pi$
$108$	0	0
$109$	1.51744	0.0139215	0.00696074	−	0.999976i	$-0.497784\pi$
$109$	1.51744	0.0139215	0.00696074	+	0.999976i	$0.497784\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−96.0920	−0.850372	−0.425186	−	0.905106i	$-0.639791\pi$
$113$	−96.0920	−0.850372	−0.425186	+	0.905106i	$0.639791\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 325.964i	− 2.73919i
$120$	0	0
$121$	−23.0527	−0.190518
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	43.1623i	0.339860i	0.985456	+	0.169930i	$0.0543542\pi$
$127$	43.1623i	0.339860i	−0.985456	+	0.169930i	$0.945646\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 2.59893i	− 0.0198392i	−0.999951	−	0.00991959i	$-0.996842\pi$
$131$	− 2.59893i	− 0.0198392i	0.999951	−	0.00991959i	$-0.00315755\pi$
$132$	0	0
$133$	338.491i	2.54505i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−199.367	−1.45523	−0.727616	−	0.685984i	$-0.759372\pi$
$137$	−199.367	−1.45523	−0.727616	+	0.685984i	$0.759372\pi$
$138$	0	0
$139$	157.351	1.13202	0.566010	−	0.824398i	$-0.308486\pi$
$139$	157.351	1.13202	0.566010	+	0.824398i	$0.308486\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−289.684	−2.02576
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	44.1446i	0.296272i	0.988967	+	0.148136i	$0.0473274\pi$
$149$	44.1446i	0.296272i	−0.988967	+	0.148136i	$0.952673\pi$
$150$	0	0
$151$	−98.4384	−0.651910	−0.325955	−	0.945385i	$-0.605686\pi$
$151$	−98.4384	−0.651910	−0.325955	+	0.945385i	$0.605686\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 119.864i	− 0.763465i	−0.924273	−	0.381733i	$-0.875328\pi$
$157$	− 119.864i	− 0.763465i	0.924273	−	0.381733i	$-0.124672\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 218.024i	− 1.35419i
$162$	0	0
$163$	− 118.763i	− 0.728607i	−0.931280	−	0.364304i	$-0.881307\pi$
$163$	− 118.763i	− 0.728607i	0.931280	−	0.364304i	$-0.118693\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−94.7584	−0.567415	−0.283708	−	0.958911i	$-0.591565\pi$
$167$	−94.7584	−0.567415	−0.283708	+	0.958911i	$0.591565\pi$
$168$	0	0
$169$	−413.544	−2.44700
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−186.217	−1.07640	−0.538200	−	0.842817i	$-0.680895\pi$
$173$	−186.217	−1.07640	−0.538200	+	0.842817i	$0.680895\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 197.835i	− 1.10522i	−0.833439	−	0.552611i	$-0.813631\pi$
$179$	− 197.835i	− 1.10522i	0.833439	−	0.552611i	$-0.186369\pi$
$180$	0	0
$181$	−36.7630	−0.203110	−0.101555	−	0.994830i	$-0.532382\pi$
$181$	−36.7630	−0.203110	−0.101555	+	0.994830i	$0.532382\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	350.491i	1.87428i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	83.8603i	0.439059i	0.975606	+	0.219530i	$0.0704523\pi$
$191$	83.8603i	0.439059i	−0.975606	+	0.219530i	$0.929548\pi$
$192$	0	0
$193$	142.105i	0.736297i	0.929767	+	0.368149i	$0.120008\pi$
$193$	142.105i	0.736297i	−0.929767	+	0.368149i	$0.879992\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−81.1497	−0.411927	−0.205964	−	0.978560i	$-0.566033\pi$
$197$	−81.1497	−0.411927	−0.205964	+	0.978560i	$0.566033\pi$
$198$	0	0
$199$	358.982	1.80393	0.901965	−	0.431808i	$-0.142124\pi$
$199$	358.982	1.80393	0.901965	+	0.431808i	$0.142124\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−328.941	−1.62040
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 363.961i	− 1.74144i
$210$	0	0
$211$	−106.982	−0.507025	−0.253512	−	0.967332i	$-0.581586\pi$
$211$	−106.982	−0.507025	−0.253512	+	0.967332i	$0.581586\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 55.5174i	− 0.255841i
$218$	0	0
$219$	0	0
$220$	0	0
$221$	704.824i	3.18925i
$222$	0	0
$223$	362.092i	1.62373i	0.583845	+	0.811865i	$0.301548\pi$
$223$	362.092i	1.62373i	−0.583845	+	0.811865i	$0.698452\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	96.9355	0.427029	0.213514	−	0.976940i	$-0.431509\pi$
$227$	96.9355	0.427029	0.213514	+	0.976940i	$0.431509\pi$
$228$	0	0
$229$	−275.193	−1.20172	−0.600858	−	0.799356i	$-0.705174\pi$
$229$	−275.193	−1.20172	−0.600858	+	0.799356i	$0.705174\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	128.123	0.549883	0.274941	−	0.961461i	$-0.411342\pi$
$233$	128.123	0.549883	0.274941	+	0.961461i	$0.411342\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 238.965i	− 0.999853i	−0.866068	−	0.499926i	$-0.833360\pi$
$239$	− 238.965i	− 0.999853i	0.866068	−	0.499926i	$-0.166640\pi$
$240$	0	0
$241$	−446.930	−1.85448	−0.927240	−	0.374468i	$-0.877825\pi$
$241$	−446.930	−1.85448	−0.927240	+	0.374468i	$0.877825\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 731.912i	− 2.96321i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	21.7034i	0.0864677i	0.999065	+	0.0432338i	$0.0137661\pi$
$251$	21.7034i	0.0864677i	−0.999065	+	0.0432338i	$0.986234\pi$
$252$	0	0
$253$	234.430i	0.926600i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−401.649	−1.56284	−0.781418	−	0.624008i	$-0.785503\pi$
$257$	−401.649	−1.56284	−0.781418	+	0.624008i	$0.785503\pi$
$258$	0	0
$259$	−276.658	−1.06818
$260$	0	0
$261$	0	0
$262$	0	0
$263$	158.820	0.603877	0.301939	−	0.953327i	$-0.402366\pi$
$263$	158.820	0.603877	0.301939	+	0.953327i	$0.402366\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	210.339i	0.781931i	0.920405	+	0.390965i	$0.127859\pi$
$269$	210.339i	0.781931i	−0.920405	+	0.390965i	$0.872141\pi$
$270$	0	0
$271$	232.053	0.856283	0.428141	−	0.903712i	$-0.359169\pi$
$271$	232.053	0.856283	0.428141	+	0.903712i	$0.359169\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	111.320i	0.401878i	0.979604	+	0.200939i	$0.0643993\pi$
$277$	111.320i	0.401878i	−0.979604	+	0.200939i	$0.935601\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	320.580i	1.14085i	0.821349	+	0.570426i	$0.193222\pi$
$281$	320.580i	1.14085i	−0.821349	+	0.570426i	$0.806778\pi$
$282$	0	0
$283$	197.737i	0.698716i	0.936989	+	0.349358i	$0.113600\pi$
$283$	197.737i	0.698716i	−0.936989	+	0.349358i	$0.886400\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−236.788	−0.825044
$288$	0	0
$289$	563.772	1.95077
$290$	0	0
$291$	0	0
$292$	0	0
$293$	5.62566	0.0192002	0.00960010	−	0.999954i	$-0.496944\pi$
$293$	5.62566	0.0192002	0.00960010	+	0.999954i	$0.496944\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	471.429i	1.57669i
$300$	0	0
$301$	806.719	2.68013
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	160.544i	0.522944i	0.965211	+	0.261472i	$0.0842079\pi$
$307$	160.544i	0.522944i	−0.965211	+	0.261472i	$0.915792\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	501.103i	1.61126i	0.592417	+	0.805631i	$0.298174\pi$
$311$	501.103i	1.61126i	−0.592417	+	0.805631i	$0.701826\pi$
$312$	0	0
$313$	219.088i	0.699960i	0.936757	+	0.349980i	$0.113812\pi$
$313$	219.088i	0.699960i	−0.936757	+	0.349980i	$0.886188\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	95.6018	0.301583	0.150792	−	0.988566i	$-0.451818\pi$
$317$	95.6018	0.301583	0.150792	+	0.988566i	$0.451818\pi$
$318$	0	0
$319$	353.693	1.10875
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−885.545	−2.74163
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	331.087i	1.00634i
$330$	0	0
$331$	171.956	0.519504	0.259752	−	0.965675i	$-0.416359\pi$
$331$	171.956	0.519504	0.259752	+	0.965675i	$0.416359\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	349.956i	1.03844i	0.854639	+	0.519222i	$0.173778\pi$
$337$	349.956i	1.03844i	−0.854639	+	0.519222i	$0.826222\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	59.6949i	0.175058i
$342$	0	0
$343$	− 296.877i	− 0.865530i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−345.751	−0.996399	−0.498200	−	0.867062i	$-0.666005\pi$
$347$	−345.751	−0.996399	−0.498200	+	0.867062i	$0.666005\pi$
$348$	0	0
$349$	356.325	1.02099	0.510494	−	0.859882i	$-0.329463\pi$
$349$	356.325	1.02099	0.510494	+	0.859882i	$0.329463\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−13.9811	−0.0396065	−0.0198032	−	0.999804i	$-0.506304\pi$
$353$	−13.9811	−0.0396065	−0.0198032	+	0.999804i	$0.506304\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	502.939i	1.40094i	0.713680	+	0.700472i	$0.247027\pi$
$359$	502.939i	1.40094i	−0.713680	+	0.700472i	$0.752973\pi$
$360$	0	0
$361$	558.579	1.54731
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	275.601i	0.750956i	0.926831	+	0.375478i	$0.122521\pi$
$367$	275.601i	0.750956i	−0.926831	+	0.375478i	$0.877479\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	473.575i	1.27648i
$372$	0	0
$373$	− 53.5395i	− 0.143538i	−0.997421	−	0.0717688i	$-0.977136\pi$
$373$	− 53.5395i	− 0.143538i	0.997421	−	0.0717688i	$-0.0228644\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	711.262	1.88664
$378$	0	0
$379$	30.4213	0.0802674	0.0401337	−	0.999194i	$-0.487222\pi$
$379$	30.4213	0.0802674	0.0401337	+	0.999194i	$0.487222\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	659.581	1.72214	0.861072	−	0.508483i	$-0.169794\pi$
$383$	659.581	1.72214	0.861072	+	0.508483i	$0.169794\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 98.2692i	− 0.252620i	−0.991991	−	0.126310i	$-0.959687\pi$
$389$	− 98.2692i	− 0.252620i	0.991991	−	0.126310i	$-0.0403134\pi$
$390$	0	0
$391$	570.386	1.45879
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 209.767i	− 0.528381i	−0.964471	−	0.264191i	$-0.914895\pi$
$397$	− 209.767i	− 0.528381i	0.964471	−	0.264191i	$-0.0851048\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	49.0264i	0.122260i	0.998130	+	0.0611302i	$0.0194705\pi$
$401$	49.0264i	0.122260i	−0.998130	+	0.0611302i	$0.980530\pi$
$402$	0	0
$403$	120.044i	0.297876i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	297.475	0.730897
$408$	0	0
$409$	−29.0349	−0.0709899	−0.0354950	−	0.999370i	$-0.511301\pi$
$409$	−29.0349	−0.0709899	−0.0354950	+	0.999370i	$0.511301\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−932.679	−2.25830
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 314.197i	− 0.749873i	−0.927050	−	0.374937i	$-0.877664\pi$
$419$	− 314.197i	− 0.749873i	0.927050	−	0.374937i	$-0.122336\pi$
$420$	0	0
$421$	−612.719	−1.45539	−0.727695	−	0.685901i	$-0.759408\pi$
$421$	−612.719	−1.45539	−0.727695	+	0.685901i	$0.759408\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	561.149i	1.31417i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	22.4664i	0.0521261i	0.999660	+	0.0260631i	$0.00829707\pi$
$431$	22.4664i	0.0521261i	−0.999660	+	0.0260631i	$0.991703\pi$
$432$	0	0
$433$	5.41210i	0.0124991i	0.999980	+	0.00624954i	$0.00198930\pi$
$433$	5.41210i	0.0124991i	−0.999980	+	0.00624954i	$0.998011\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−592.307	−1.35539
$438$	0	0
$439$	−401.895	−0.915478	−0.457739	−	0.889087i	$-0.651341\pi$
$439$	−401.895	−0.915478	−0.457739	+	0.889087i	$0.651341\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	412.888	0.932027	0.466014	−	0.884778i	$-0.345690\pi$
$443$	412.888	0.932027	0.466014	+	0.884778i	$0.345690\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 14.3283i	− 0.0319117i	−0.999873	−	0.0159558i	$-0.994921\pi$
$449$	− 14.3283i	− 0.0319117i	0.999873	−	0.0159558i	$-0.00507912\pi$
$450$	0	0
$451$	254.605	0.564534
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	281.851i	0.616741i	0.951266	+	0.308370i	$0.0997836\pi$
$457$	281.851i	0.616741i	−0.951266	+	0.308370i	$0.900216\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 812.496i	− 1.76247i	−0.472683	−	0.881233i	$-0.656714\pi$
$461$	− 812.496i	− 1.76247i	0.472683	−	0.881233i	$-0.343286\pi$
$462$	0	0
$463$	652.899i	1.41015i	0.709133	+	0.705074i	$0.249086\pi$
$463$	652.899i	1.41015i	−0.709133	+	0.705074i	$0.750914\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−27.9742	−0.0599020	−0.0299510	−	0.999551i	$-0.509535\pi$
$467$	−27.9742	−0.0599020	−0.0299510	+	0.999551i	$0.509535\pi$
$468$	0	0
$469$	−148.438	−0.316500
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−867.421	−1.83387
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 596.959i	− 1.24626i	−0.782118	−	0.623131i	$-0.785860\pi$
$479$	− 596.959i	− 1.24626i	0.782118	−	0.623131i	$-0.214140\pi$
$480$	0	0
$481$	598.211	1.24368
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	452.749i	0.929670i	0.885397	+	0.464835i	$0.153886\pi$
$487$	452.749i	0.929670i	−0.885397	+	0.464835i	$0.846114\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 602.473i	− 1.22703i	−0.789682	−	0.613516i	$-0.789754\pi$
$491$	− 602.473i	− 1.22703i	0.789682	−	0.613516i	$-0.210246\pi$
$492$	0	0
$493$	− 860.561i	− 1.74556i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−415.555	−0.836127
$498$	0	0
$499$	−873.903	−1.75131	−0.875655	−	0.482938i	$-0.839570\pi$
$499$	−873.903	−1.75131	−0.875655	+	0.482938i	$0.839570\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−10.2588	−0.0203953	−0.0101977	−	0.999948i	$-0.503246\pi$
$503$	−10.2588	−0.0203953	−0.0101977	+	0.999948i	$0.503246\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 226.913i	− 0.445802i	−0.974841	−	0.222901i	$-0.928447\pi$
$509$	− 226.913i	− 0.445802i	0.974841	−	0.222901i	$-0.0715527\pi$
$510$	0	0
$511$	−792.816	−1.55150
$512$	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
$521$	395.174i	0.758490i	0.925296	+	0.379245i	$0.123816\pi$
$521$	395.174i	0.758490i	−0.925296	+	0.379245i	$0.876184\pi$
$522$	0	0
$523$	694.219i	1.32738i	0.748008	+	0.663690i	$0.231010\pi$
$523$	694.219i	1.32738i	−0.748008	+	0.663690i	$0.768990\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	145.242	0.275602
$528$	0	0
$529$	−147.491	−0.278811
$530$	0	0
$531$	0	0
$532$	0	0
$533$	512.001	0.960602
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	907.323i	1.68335i
$540$	0	0
$541$	105.307	0.194652	0.0973260	−	0.995253i	$-0.468971\pi$
$541$	105.307	0.194652	0.0973260	+	0.995253i	$0.468971\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 3.85054i	− 0.00703938i	−0.999994	−	0.00351969i	$-0.998880\pi$
$547$	− 3.85054i	− 0.00703938i	0.999994	−	0.00351969i	$-0.00112035\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	893.634i	1.62184i
$552$	0	0
$553$	− 218.447i	− 0.395022i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	174.221	0.312785	0.156392	−	0.987695i	$-0.450014\pi$
$557$	174.221	0.312785	0.156392	+	0.987695i	$0.450014\pi$
$558$	0	0
$559$	−1744.35	−3.12048
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−872.631	−1.54997	−0.774983	−	0.631982i	$-0.782242\pi$
$563$	−872.631	−1.54997	−0.774983	+	0.631982i	$0.782242\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 590.496i	− 1.03778i	−0.854841	−	0.518889i	$-0.826346\pi$
$569$	− 590.496i	− 1.03778i	0.854841	−	0.518889i	$-0.173654\pi$
$570$	0	0
$571$	180.588	0.316266	0.158133	−	0.987418i	$-0.449453\pi$
$571$	180.588	0.316266	0.158133	+	0.987418i	$0.449453\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	103.026i	0.178555i	0.996007	+	0.0892776i	$0.0284558\pi$
$577$	103.026i	0.178555i	−0.996007	+	0.0892776i	$0.971544\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 579.369i	− 0.997192i
$582$	0	0
$583$	− 509.210i	− 0.873430i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−937.598	−1.59727	−0.798636	−	0.601815i	$-0.794445\pi$
$587$	−937.598	−1.59727	−0.798636	+	0.601815i	$0.794445\pi$
$588$	0	0
$589$	−150.824	−0.256068
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−1002.64	−1.69079	−0.845396	−	0.534139i	$-0.820636\pi$
$593$	−1002.64	−1.69079	−0.845396	+	0.534139i	$0.820636\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	335.900i	0.560769i	0.959888	+	0.280384i	$0.0904619\pi$
$599$	335.900i	0.560769i	−0.959888	+	0.280384i	$0.909538\pi$
$600$	0	0
$601$	592.561	0.985958	0.492979	−	0.870041i	$-0.335908\pi$
$601$	592.561	0.985958	0.492979	+	0.870041i	$0.335908\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 1163.50i	− 1.91680i	−0.285435	−	0.958398i	$-0.592138\pi$
$607$	− 1163.50i	− 1.91680i	0.285435	−	0.958398i	$-0.407862\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 715.902i	− 1.17169i
$612$	0	0
$613$	19.8199i	0.0323327i	0.999869	+	0.0161663i	$0.00514613\pi$
$613$	19.8199i	0.0323327i	−0.999869	+	0.0161663i	$0.994854\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	1069.23	1.73295	0.866477	−	0.499218i	$-0.166379\pi$
$617$	1069.23	1.73295	0.866477	+	0.499218i	$0.166379\pi$
$618$	0	0
$619$	762.894	1.23246	0.616231	−	0.787565i	$-0.288659\pi$
$619$	762.894	1.23246	0.616231	+	0.787565i	$0.288659\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−1492.87	−2.39626
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 723.779i	− 1.15068i
$630$	0	0
$631$	−973.324	−1.54251	−0.771255	−	0.636526i	$-0.780371\pi$
$631$	−973.324	−1.54251	−0.771255	+	0.636526i	$0.780371\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	1824.59i	2.86435i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	694.564i	1.08356i	0.840519	+	0.541782i	$0.182250\pi$
$641$	694.564i	1.08356i	−0.840519	+	0.541782i	$0.817750\pi$
$642$	0	0
$643$	578.641i	0.899908i	0.893052	+	0.449954i	$0.148560\pi$
$643$	578.641i	0.899908i	−0.893052	+	0.449954i	$0.851440\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	1227.42	1.89710	0.948551	−	0.316625i	$-0.102550\pi$
$647$	1227.42	1.89710	0.948551	+	0.316625i	$0.102550\pi$
$648$	0	0
$649$	1002.86	1.54524
$650$	0	0
$651$	0	0
$652$	0	0
$653$	3.78969	0.00580351	0.00290176	−	0.999996i	$-0.499076\pi$
$653$	3.78969	0.00580351	0.00290176	+	0.999996i	$0.499076\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	822.470i	1.24806i	0.781401	+	0.624029i	$0.214505\pi$
$659$	822.470i	1.24806i	−0.781401	+	0.624029i	$0.785495\pi$
$660$	0	0
$661$	−412.552	−0.624134	−0.312067	−	0.950060i	$-0.601021\pi$
$661$	−412.552	−0.624134	−0.312067	+	0.950060i	$0.601021\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 575.596i	− 0.862962i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	− 603.373i	− 0.899215i
$672$	0	0
$673$	177.517i	0.263770i	0.991265	+	0.131885i	$0.0421030\pi$
$673$	177.517i	0.263770i	−0.991265	+	0.131885i	$0.957897\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−997.113	−1.47284	−0.736420	−	0.676524i	$-0.763485\pi$
$677$	−997.113	−1.47284	−0.736420	+	0.676524i	$0.763485\pi$
$678$	0	0
$679$	−517.675	−0.762409
$680$	0	0
$681$	0	0
$682$	0	0
$683$	1296.82	1.89871	0.949356	−	0.314201i	$-0.101737\pi$
$683$	1296.82	1.89871	0.949356	+	0.314201i	$0.101737\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 1024.00i	− 1.48621i
$690$	0	0
$691$	−343.623	−0.497283	−0.248642	−	0.968596i	$-0.579984\pi$
$691$	−343.623	−0.497283	−0.248642	+	0.968596i	$0.579984\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 619.473i	− 0.888771i
$698$	0	0
$699$	0	0
$700$	0	0
$701$	255.904i	0.365056i	0.983201	+	0.182528i	$0.0584280\pi$
$701$	255.904i	0.365056i	−0.983201	+	0.182528i	$0.941572\pi$
$702$	0	0
$703$	751.596i	1.06913i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−1270.97	−1.79769
$708$	0	0
$709$	1105.26	1.55890	0.779452	−	0.626462i	$-0.215498\pi$
$709$	1105.26	1.55890	0.779452	+	0.626462i	$0.215498\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	97.1469	0.136251
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	− 801.363i	− 1.11455i	−0.830327	−	0.557276i	$-0.811847\pi$
$719$	− 801.363i	− 1.11455i	0.830327	−	0.557276i	$-0.188153\pi$
$720$	0	0
$721$	−168.070	−0.233106
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	57.6534i	0.0793031i	0.999214	+	0.0396516i	$0.0126248\pi$
$727$	57.6534i	0.0793031i	−0.999214	+	0.0396516i	$0.987375\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	2110.50i	2.88714i
$732$	0	0
$733$	− 330.943i	− 0.451491i	−0.974186	−	0.225746i	$-0.927518\pi$
$733$	− 330.943i	− 0.451491i	0.974186	−	0.225746i	$-0.0724818\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	159.608	0.216564
$738$	0	0
$739$	1200.31	1.62423	0.812115	−	0.583497i	$-0.198316\pi$
$739$	1200.31	1.62423	0.812115	+	0.583497i	$0.198316\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	896.996	1.20726	0.603631	−	0.797264i	$-0.293720\pi$
$743$	896.996	1.20726	0.603631	+	0.797264i	$0.293720\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 589.615i	− 0.787203i
$750$	0	0
$751$	1226.53	1.63320	0.816600	−	0.577203i	$-0.195856\pi$
$751$	1226.53	1.63320	0.816600	+	0.577203i	$0.195856\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 286.627i	− 0.378635i	−0.981916	−	0.189318i	$-0.939372\pi$
$757$	− 286.627i	− 0.378635i	0.981916	−	0.189318i	$-0.0606276\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	176.169i	0.231496i	0.993279	+	0.115748i	$0.0369266\pi$
$761$	176.169i	0.231496i	−0.993279	+	0.115748i	$0.963073\pi$
$762$	0	0
$763$	16.9381i	0.0221993i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	2016.71	2.62935
$768$	0	0
$769$	787.929	1.02461	0.512307	−	0.858802i	$-0.328791\pi$
$769$	787.929	1.02461	0.512307	+	0.858802i	$0.328791\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−563.136	−0.728507	−0.364253	−	0.931300i	$-0.618676\pi$
$773$	−563.136	−0.728507	−0.364253	+	0.931300i	$0.618676\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	643.281i	0.825778i
$780$	0	0
$781$	446.824	0.572118
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	364.061i	0.462594i	0.972883	+	0.231297i	$0.0742969\pi$
$787$	364.061i	0.462594i	−0.972883	+	0.231297i	$0.925703\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 1072.61i	− 1.35601i
$792$	0	0
$793$	− 1213.36i	− 1.53009i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	863.848	1.08387	0.541937	−	0.840419i	$-0.317691\pi$
$797$	863.848	1.08387	0.541937	+	0.840419i	$0.317691\pi$
$798$	0	0
$799$	−866.175	−1.08407
$800$	0	0
$801$	0	0
$802$	0	0
$803$	852.472	1.06161
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	1514.44i	1.87198i	0.352020	+	0.935992i	$0.385495\pi$
$809$	1514.44i	1.87198i	−0.352020	+	0.935992i	$0.614505\pi$
$810$	0	0
$811$	−449.895	−0.554741	−0.277370	−	0.960763i	$-0.589463\pi$
$811$	−449.895	−0.554741	−0.277370	+	0.960763i	$0.589463\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 2191.61i	− 2.68251i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 1277.65i	− 1.55621i	−0.628134	−	0.778105i	$-0.716181\pi$
$821$	− 1277.65i	− 1.55621i	0.628134	−	0.778105i	$-0.283819\pi$
$822$	0	0
$823$	− 795.601i	− 0.966708i	−0.875425	−	0.483354i	$-0.839418\pi$
$823$	− 795.601i	− 0.966708i	0.875425	−	0.483354i	$-0.160582\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−89.2072	−0.107868	−0.0539342	−	0.998544i	$-0.517176\pi$
$827$	−89.2072	−0.107868	−0.0539342	+	0.998544i	$0.517176\pi$
$828$	0	0
$829$	−1535.81	−1.85261	−0.926306	−	0.376773i	$-0.877034\pi$
$829$	−1535.81	−1.85261	−0.926306	+	0.376773i	$0.877034\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	2207.59	2.65016
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	− 479.864i	− 0.571948i	−0.958237	−	0.285974i	$-0.907683\pi$
$839$	− 479.864i	− 0.571948i	0.958237	−	0.285974i	$-0.0923171\pi$
$840$	0	0
$841$	−27.4213	−0.0326056
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 257.320i	− 0.303802i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 484.108i	− 0.568869i
$852$	0	0
$853$	446.328i	0.523245i	0.965170	+	0.261623i	$0.0842576\pi$
$853$	446.328i	0.523245i	−0.965170	+	0.261623i	$0.915742\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−660.425	−0.770624	−0.385312	−	0.922786i	$-0.625906\pi$
$857$	−660.425	−0.770624	−0.385312	+	0.922786i	$0.625906\pi$
$858$	0	0
$859$	804.491	0.936544	0.468272	−	0.883584i	$-0.344877\pi$
$859$	804.491	0.936544	0.468272	+	0.883584i	$0.344877\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	1343.39	1.55665	0.778323	−	0.627864i	$-0.216071\pi$
$863$	1343.39	1.55665	0.778323	+	0.627864i	$0.216071\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	234.884i	0.270293i
$870$	0	0
$871$	320.965	0.368502
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 737.645i	− 0.841100i	−0.907269	−	0.420550i	$-0.861837\pi$
$877$	− 737.645i	− 0.841100i	0.907269	−	0.420550i	$-0.138163\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 66.0332i	− 0.0749526i	−0.999298	−	0.0374763i	$-0.988068\pi$
$881$	− 66.0332i	− 0.0749526i	0.999298	−	0.0374763i	$-0.0119319\pi$
$882$	0	0
$883$	652.411i	0.738858i	0.929259	+	0.369429i	$0.120447\pi$
$883$	652.411i	0.738858i	−0.929259	+	0.369429i	$0.879553\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−1377.37	−1.55285	−0.776423	−	0.630212i	$-0.782968\pi$
$887$	−1377.37	−1.55285	−0.776423	+	0.630212i	$0.782968\pi$
$888$	0	0
$889$	−481.789	−0.541945
$890$	0	0
$891$	0	0
$892$	0	0
$893$	899.464	1.00724
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 146.569i	− 0.163035i
$900$	0	0
$901$	−1238.95	−1.37508
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	709.570i	0.782326i	0.920321	+	0.391163i	$0.127927\pi$
$907$	709.570i	0.782326i	−0.920321	+	0.391163i	$0.872073\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	− 145.117i	− 0.159295i	−0.996823	−	0.0796473i	$-0.974621\pi$
$911$	− 145.117i	− 0.159295i	0.996823	−	0.0796473i	$-0.0253794\pi$
$912$	0	0
$913$	622.964i	0.682326i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	29.0100	0.0316358
$918$	0	0
$919$	343.974	0.374292	0.187146	−	0.982332i	$-0.440076\pi$
$919$	343.974	0.374292	0.187146	+	0.982332i	$0.440076\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	898.546	0.973506
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	1572.07i	1.69222i	0.533009	+	0.846109i	$0.321061\pi$
$929$	1572.07i	1.69222i	−0.533009	+	0.846109i	$0.678939\pi$
$930$	0	0
$931$	−2292.43	−2.46233
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	1273.70i	1.35934i	0.733518	+	0.679670i	$0.237877\pi$
$937$	1273.70i	1.35934i	−0.733518	+	0.679670i	$0.762123\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	220.475i	0.234299i	0.993114	+	0.117150i	$0.0373757\pi$
$941$	220.475i	0.234299i	−0.993114	+	0.117150i	$0.962624\pi$
$942$	0	0
$943$	− 414.342i	− 0.439387i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−1513.72	−1.59843	−0.799216	−	0.601043i	$-0.794752\pi$
$947$	−1513.72	−1.59843	−0.799216	+	0.601043i	$0.794752\pi$
$948$	0	0
$949$	1714.29	1.80641
$950$	0	0
$951$	0	0
$952$	0	0
$953$	1075.41	1.12845	0.564224	−	0.825622i	$-0.309176\pi$
$953$	1075.41	1.12845	0.564224	+	0.825622i	$0.309176\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	− 2225.39i	− 2.32053i
$960$	0	0
$961$	−936.263	−0.974259
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 201.303i	− 0.208173i	−0.994568	−	0.104086i	$-0.966808\pi$
$967$	− 201.303i	− 0.208173i	0.994568	−	0.104086i	$-0.0331919\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 164.019i	− 0.168917i	−0.996427	−	0.0844586i	$-0.973084\pi$
$971$	− 164.019i	− 0.168917i	0.996427	−	0.0844586i	$-0.0269161\pi$
$972$	0	0
$973$	1756.39i	1.80513i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	4.65230	0.00476182	0.00238091	−	0.999997i	$-0.499242\pi$
$977$	4.65230	0.00476182	0.00238091	+	0.999997i	$0.499242\pi$
$978$	0	0
$979$	1605.20	1.63963
$980$	0	0
$981$	0	0
$982$	0	0
$983$	798.373	0.812180	0.406090	−	0.913833i	$-0.366892\pi$
$983$	798.373	0.812180	0.406090	+	0.913833i	$0.366892\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	1411.63i	1.42733i
$990$	0	0
$991$	135.404	0.136633	0.0683166	−	0.997664i	$-0.478237\pi$
$991$	135.404	0.136633	0.0683166	+	0.997664i	$0.478237\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	405.645i	0.406865i	0.979089	+	0.203433i	$0.0652098\pi$
$997$	405.645i	0.406865i	−0.979089	+	0.203433i	$0.934790\pi$
$998$	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.449.7		8
3.2	odd	2	inner	3600.3.c.l.449.8		8
4.3	odd	2		1800.3.c.b.449.2		8
5.2	odd	4		3600.3.l.m.1601.1		4
5.3	odd	4		720.3.l.d.161.2		4
5.4	even	2	inner	3600.3.c.l.449.1		8
12.11	even	2		1800.3.c.b.449.1		8
15.2	even	4		3600.3.l.m.1601.2		4
15.8	even	4		720.3.l.d.161.4		4
15.14	odd	2	inner	3600.3.c.l.449.2		8
20.3	even	4		360.3.l.a.161.1	✓	4
20.7	even	4		1800.3.l.g.1601.4		4
20.19	odd	2		1800.3.c.b.449.8		8
40.3	even	4		2880.3.l.a.1601.3		4
40.13	odd	4		2880.3.l.h.1601.4		4
60.23	odd	4		360.3.l.a.161.3	yes	4
60.47	odd	4		1800.3.l.g.1601.3		4
60.59	even	2		1800.3.c.b.449.7		8
120.53	even	4		2880.3.l.h.1601.2		4
120.83	odd	4		2880.3.l.a.1601.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
360.3.l.a.161.1	✓	4	20.3	even	4
360.3.l.a.161.3	yes	4	60.23	odd	4
720.3.l.d.161.2		4	5.3	odd	4
720.3.l.d.161.4		4	15.8	even	4
1800.3.c.b.449.1		8	12.11	even	2
1800.3.c.b.449.2		8	4.3	odd	2
1800.3.c.b.449.7		8	60.59	even	2
1800.3.c.b.449.8		8	20.19	odd	2
1800.3.l.g.1601.3		4	60.47	odd	4
1800.3.l.g.1601.4		4	20.7	even	4
2880.3.l.a.1601.1		4	120.83	odd	4
2880.3.l.a.1601.3		4	40.3	even	4
2880.3.l.h.1601.2		4	120.53	even	4
2880.3.l.h.1601.4		4	40.13	odd	4
3600.3.c.l.449.1		8	5.4	even	2	inner
3600.3.c.l.449.2		8	15.14	odd	2	inner
3600.3.c.l.449.7		8	1.1	even	1	trivial
3600.3.c.l.449.8		8	3.2	odd	2	inner
3600.3.l.m.1601.1		4	5.2	odd	4
3600.3.l.m.1601.2		4	15.2	even	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 $\F_{q}$
Belyi maps

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)

\(f(q)\)	\(=\)	\(q+11.1623i q^{7} -12.0022i q^{11} -24.1359i q^{13} -29.2023 q^{17} +30.3246 q^{19} -19.5323 q^{23} +29.4690i q^{29} -4.97367 q^{31} +24.7851i q^{37} +21.2132i q^{41} -72.2719i q^{43} +29.6612 q^{47} -75.5964 q^{49} +42.4264 q^{53} +83.5564i q^{59} +50.2719 q^{61} +13.2982i q^{67} +37.2285i q^{71} +71.0263i q^{73} +133.972 q^{77} -19.5701 q^{79} -51.9042 q^{83} +133.742i q^{89} +269.412 q^{91} +46.3772i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 192 q^{19} + 112 q^{31} - 200 q^{49} + 48 q^{61} + 400 q^{79} + 688 q^{91}+O(q^{100}) \) 8 * q + 192 * q^19 + 112 * q^31 - 200 * q^49 + 48 * q^61 + 400 * q^79 + 688 * q^91

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