LMFDB - Embedded newform 432.4.c.g.431.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [432,4,Mod(431,432)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(432, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 1])) N = Newforms(chi, 4, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("432.431"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$ N $	$=$	$ 432 = 2^{4} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	432.c (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,0,0,0,0,0,0,0,0,28] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	no
Analytic conductor:	$25.4888251225$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{12})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{6}\cdot 3^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			431.2
Root			$-0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	432.431
Dual form			432.4.c.g.431.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-12.0000i q^{5} +5.19615i q^{7} -62.3538 q^{11} +7.00000 q^{13} +84.0000i q^{17} +67.5500i q^{19} +62.3538 q^{23} -19.0000 q^{25} -168.000i q^{29} +259.808i q^{31} +62.3538 q^{35} -97.0000 q^{37} -72.0000i q^{41} +363.731i q^{43} +436.477 q^{47} +316.000 q^{49} +504.000i q^{53} +748.246i q^{55} -436.477 q^{59} -133.000 q^{61} -84.0000i q^{65} +545.596i q^{67} -498.831 q^{71} -497.000 q^{73} -324.000i q^{77} +1127.57i q^{79} -872.954 q^{83} +1008.00 q^{85} -1164.00i q^{89} +36.3731i q^{91} +810.600 q^{95} -749.000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 28 q^{13} - 76 q^{25} - 388 q^{37} + 1264 q^{49} - 532 q^{61} - 1988 q^{73} + 4032 q^{85} - 2996 q^{97}+O(q^{100}) $ 4 * q + 28 * q^13 - 76 * q^25 - 388 * q^37 + 1264 * q^49 - 532 * q^61 - 1988 * q^73 + 4032 * q^85 - 2996 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/432\mathbb{Z}\right)^\times$.

$n$	$271$	$325$	$353$
$\chi(n)$	$-1$	$1$	$-1$

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
$5$	− 12.0000i	− 1.07331i	−0.843801	−	0.536656i	$-0.819687\pi$
$5$	− 12.0000i	− 1.07331i	0.843801	−	0.536656i	$-0.180313\pi$
$6$	0	0
$7$	5.19615i	0.280566i	0.990111	+	0.140283i	$0.0448012\pi$
$7$	5.19615i	0.280566i	−0.990111	+	0.140283i	$0.955199\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−62.3538	−1.70913	−0.854563	−	0.519348i	$-0.826175\pi$
$11$	−62.3538	−1.70913	−0.854563	+	0.519348i	$0.826175\pi$
$12$	0	0
$13$	7.00000	0.149342	0.0746712	−	0.997208i	$-0.476209\pi$
$13$	7.00000	0.149342	0.0746712	+	0.997208i	$0.476209\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	84.0000i	1.19841i	0.800595	+	0.599206i	$0.204517\pi$
$17$	84.0000i	1.19841i	−0.800595	+	0.599206i	$0.795483\pi$
$18$	0	0
$19$	67.5500i	0.815633i	0.913064	+	0.407817i	$0.133710\pi$
$19$	67.5500i	0.815633i	−0.913064	+	0.407817i	$0.866290\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	62.3538	0.565290	0.282645	−	0.959225i	$-0.408788\pi$
$23$	62.3538	0.565290	0.282645	+	0.959225i	$0.408788\pi$
$24$	0	0
$25$	−19.0000	−0.152000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 168.000i	− 1.07575i	−0.843024	−	0.537876i	$-0.819227\pi$
$29$	− 168.000i	− 1.07575i	0.843024	−	0.537876i	$-0.180773\pi$
$30$	0	0
$31$	259.808i	1.50525i	0.658448	+	0.752626i	$0.271213\pi$
$31$	259.808i	1.50525i	−0.658448	+	0.752626i	$0.728787\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	62.3538	0.301135
$36$	0	0
$37$	−97.0000	−0.430992	−0.215496	−	0.976505i	$-0.569137\pi$
$37$	−97.0000	−0.430992	−0.215496	+	0.976505i	$0.569137\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 72.0000i	− 0.274256i	−0.990553	−	0.137128i	$-0.956213\pi$
$41$	− 72.0000i	− 0.274256i	0.990553	−	0.137128i	$-0.0437872\pi$
$42$	0	0
$43$	363.731i	1.28996i	0.764198	+	0.644981i	$0.223135\pi$
$43$	363.731i	1.28996i	−0.764198	+	0.644981i	$0.776865\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	436.477	1.35461	0.677305	−	0.735702i	$-0.263148\pi$
$47$	436.477	1.35461	0.677305	+	0.735702i	$0.263148\pi$
$48$	0	0
$49$	316.000	0.921283
$50$	0	0
$51$	0	0
$52$	0	0
$53$	504.000i	1.30622i	0.757263	+	0.653111i	$0.226536\pi$
$53$	504.000i	1.30622i	−0.757263	+	0.653111i	$0.773464\pi$
$54$	0	0
$55$	748.246i	1.83443i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−436.477	−0.963126	−0.481563	−	0.876411i	$-0.659931\pi$
$59$	−436.477	−0.963126	−0.481563	+	0.876411i	$0.659931\pi$
$60$	0	0
$61$	−133.000	−0.279162	−0.139581	−	0.990211i	$-0.544576\pi$
$61$	−133.000	−0.279162	−0.139581	+	0.990211i	$0.544576\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 84.0000i	− 0.160291i
$66$	0	0
$67$	545.596i	0.994853i	0.867506	+	0.497427i	$0.165722\pi$
$67$	545.596i	0.994853i	−0.867506	+	0.497427i	$0.834278\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−498.831	−0.833807	−0.416904	−	0.908951i	$-0.636885\pi$
$71$	−498.831	−0.833807	−0.416904	+	0.908951i	$0.636885\pi$
$72$	0	0
$73$	−497.000	−0.796842	−0.398421	−	0.917203i	$-0.630442\pi$
$73$	−497.000	−0.796842	−0.398421	+	0.917203i	$0.630442\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 324.000i	− 0.479522i
$78$	0	0
$79$	1127.57i	1.60584i	0.596090	+	0.802918i	$0.296720\pi$
$79$	1127.57i	1.60584i	−0.596090	+	0.802918i	$0.703280\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−872.954	−1.15445	−0.577224	−	0.816586i	$-0.695864\pi$
$83$	−872.954	−1.15445	−0.577224	+	0.816586i	$0.695864\pi$
$84$	0	0
$85$	1008.00	1.28627
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 1164.00i	− 1.38633i	−0.720777	−	0.693167i	$-0.756215\pi$
$89$	− 1164.00i	− 1.38633i	0.720777	−	0.693167i	$-0.243785\pi$
$90$	0	0
$91$	36.3731i	0.0419004i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	810.600	0.875429
$96$	0	0
$97$	−749.000	−0.784015	−0.392007	−	0.919962i	$-0.628219\pi$
$97$	−749.000	−0.784015	−0.392007	+	0.919962i	$0.628219\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	1008.00i	0.993067i	0.868018	+	0.496533i	$0.165394\pi$
$101$	1008.00i	0.993067i	−0.868018	+	0.496533i	$0.834606\pi$
$102$	0	0
$103$	732.657i	0.700882i	0.936585	+	0.350441i	$0.113968\pi$
$103$	732.657i	0.700882i	−0.936585	+	0.350441i	$0.886032\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1309.43	1.18306	0.591530	−	0.806283i	$-0.298524\pi$
$107$	1309.43	1.18306	0.591530	+	0.806283i	$0.298524\pi$
$108$	0	0
$109$	−1226.00	−1.07733	−0.538667	−	0.842518i	$-0.681072\pi$
$109$	−1226.00	−1.07733	−0.538667	+	0.842518i	$0.681072\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	2100.00i	1.74824i	0.485708	+	0.874121i	$0.338562\pi$
$113$	2100.00i	1.74824i	−0.485708	+	0.874121i	$0.661438\pi$
$114$	0	0
$115$	− 748.246i	− 0.606733i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−436.477	−0.336233
$120$	0	0
$121$	2557.00	1.92111
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 1272.00i	− 0.910169i
$126$	0	0
$127$	509.223i	0.355797i	0.984049	+	0.177899i	$0.0569299\pi$
$127$	509.223i	0.355797i	−0.984049	+	0.177899i	$0.943070\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	872.954	0.582216	0.291108	−	0.956690i	$-0.405976\pi$
$131$	872.954	0.582216	0.291108	+	0.956690i	$0.405976\pi$
$132$	0	0
$133$	−351.000	−0.228839
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 84.0000i	− 0.0523840i	−0.999657	−	0.0261920i	$-0.991662\pi$
$137$	− 84.0000i	− 0.0523840i	0.999657	−	0.0261920i	$-0.00833812\pi$
$138$	0	0
$139$	− 2301.90i	− 1.40463i	−0.711864	−	0.702317i	$-0.752149\pi$
$139$	− 2301.90i	− 1.40463i	0.711864	−	0.702317i	$-0.247851\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−436.477	−0.255245
$144$	0	0
$145$	−2016.00	−1.15462
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 2688.00i	− 1.47792i	−0.673751	−	0.738958i	$-0.735318\pi$
$149$	− 2688.00i	− 1.47792i	0.673751	−	0.738958i	$-0.264682\pi$
$150$	0	0
$151$	− 763.834i	− 0.411655i	−0.978588	−	0.205828i	$-0.934011\pi$
$151$	− 763.834i	− 0.411655i	0.978588	−	0.205828i	$-0.0659886\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	3117.69	1.61561
$156$	0	0
$157$	3206.00	1.62972	0.814862	−	0.579655i	$-0.196813\pi$
$157$	3206.00	1.62972	0.814862	+	0.579655i	$0.196813\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	324.000i	0.158601i
$162$	0	0
$163$	− 3673.68i	− 1.76531i	−0.470026	−	0.882653i	$-0.655755\pi$
$163$	− 3673.68i	− 1.76531i	0.470026	−	0.882653i	$-0.344245\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−3928.29	−1.82024	−0.910121	−	0.414343i	$-0.864011\pi$
$167$	−3928.29	−1.82024	−0.910121	+	0.414343i	$0.864011\pi$
$168$	0	0
$169$	−2148.00	−0.977697
$170$	0	0
$171$	0	0
$172$	0	0
$173$	1272.00i	0.559008i	0.960145	+	0.279504i	$0.0901700\pi$
$173$	1272.00i	0.559008i	−0.960145	+	0.279504i	$0.909830\pi$
$174$	0	0
$175$	− 98.7269i	− 0.0426460i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−3491.81	−1.45805	−0.729024	−	0.684489i	$-0.760026\pi$
$179$	−3491.81	−1.45805	−0.729024	+	0.684489i	$0.760026\pi$
$180$	0	0
$181$	−3269.00	−1.34245	−0.671223	−	0.741255i	$-0.734231\pi$
$181$	−3269.00	−1.34245	−0.671223	+	0.741255i	$0.734231\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	1164.00i	0.462589i
$186$	0	0
$187$	− 5237.72i	− 2.04824i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−1683.55	−0.637789	−0.318894	−	0.947790i	$-0.603312\pi$
$191$	−1683.55	−0.637789	−0.318894	+	0.947790i	$0.603312\pi$
$192$	0	0
$193$	4583.00	1.70928	0.854641	−	0.519219i	$-0.173777\pi$
$193$	4583.00	1.70928	0.854641	+	0.519219i	$0.173777\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	4116.00i	1.48859i	0.667849	+	0.744297i	$0.267215\pi$
$197$	4116.00i	1.48859i	−0.667849	+	0.744297i	$0.732785\pi$
$198$	0	0
$199$	− 223.435i	− 0.0795922i	−0.999208	−	0.0397961i	$-0.987329\pi$
$199$	− 223.435i	− 0.0795922i	0.999208	−	0.0397961i	$-0.0126708\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	872.954	0.301819
$204$	0	0
$205$	−864.000	−0.294363
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 4212.00i	− 1.39402i
$210$	0	0
$211$	− 4401.14i	− 1.43596i	−0.696065	−	0.717979i	$-0.745067\pi$
$211$	− 4401.14i	− 1.43596i	0.696065	−	0.717979i	$-0.254933\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	4364.77	1.38453
$216$	0	0
$217$	−1350.00	−0.422322
$218$	0	0
$219$	0	0
$220$	0	0
$221$	588.000i	0.178974i
$222$	0	0
$223$	758.638i	0.227812i	0.993492	+	0.113906i	$0.0363363\pi$
$223$	758.638i	0.227812i	−0.993492	+	0.113906i	$0.963664\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−1745.91	−0.510484	−0.255242	−	0.966877i	$-0.582155\pi$
$227$	−1745.91	−0.510484	−0.255242	+	0.966877i	$0.582155\pi$
$228$	0	0
$229$	3598.00	1.03826	0.519132	−	0.854694i	$-0.326255\pi$
$229$	3598.00	1.03826	0.519132	+	0.854694i	$0.326255\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 1344.00i	− 0.377890i	−0.981988	−	0.188945i	$-0.939493\pi$
$233$	− 1344.00i	− 0.377890i	0.981988	−	0.188945i	$-0.0605068\pi$
$234$	0	0
$235$	− 5237.72i	− 1.45392i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	5237.72	1.41757	0.708787	−	0.705423i	$-0.249243\pi$
$239$	5237.72	1.41757	0.708787	+	0.705423i	$0.249243\pi$
$240$	0	0
$241$	−2513.00	−0.671687	−0.335843	−	0.941918i	$-0.609021\pi$
$241$	−2513.00	−0.671687	−0.335843	+	0.941918i	$0.609021\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 3792.00i	− 0.988824i
$246$	0	0
$247$	472.850i	0.121809i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	872.954	0.219523	0.109762	−	0.993958i	$-0.464991\pi$
$251$	872.954	0.219523	0.109762	+	0.993958i	$0.464991\pi$
$252$	0	0
$253$	−3888.00	−0.966152
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 3288.00i	− 0.798054i	−0.916939	−	0.399027i	$-0.869348\pi$
$257$	− 3288.00i	− 0.798054i	0.916939	−	0.399027i	$-0.130652\pi$
$258$	0	0
$259$	− 504.027i	− 0.120922i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−1247.08	−0.292388	−0.146194	−	0.989256i	$-0.546702\pi$
$263$	−1247.08	−0.292388	−0.146194	+	0.989256i	$0.546702\pi$
$264$	0	0
$265$	6048.00	1.40198
$266$	0	0
$267$	0	0
$268$	0	0
$269$	4884.00i	1.10700i	0.832850	+	0.553500i	$0.186708\pi$
$269$	4884.00i	1.10700i	−0.832850	+	0.553500i	$0.813292\pi$
$270$	0	0
$271$	4099.76i	0.918978i	0.888184	+	0.459489i	$0.151967\pi$
$271$	4099.76i	0.918978i	−0.888184	+	0.459489i	$0.848033\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	1184.72	0.259787
$276$	0	0
$277$	1870.00	0.405622	0.202811	−	0.979218i	$-0.434992\pi$
$277$	1870.00	0.405622	0.202811	+	0.979218i	$0.434992\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	7896.00i	1.67628i	0.545452	+	0.838142i	$0.316358\pi$
$281$	7896.00i	1.67628i	−0.545452	+	0.838142i	$0.683642\pi$
$282$	0	0
$283$	− 3626.91i	− 0.761829i	−0.924610	−	0.380915i	$-0.875609\pi$
$283$	− 3626.91i	− 0.761829i	0.924610	−	0.380915i	$-0.124391\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	374.123	0.0769470
$288$	0	0
$289$	−2143.00	−0.436190
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 588.000i	− 0.117240i	−0.998280	−	0.0586200i	$-0.981330\pi$
$293$	− 588.000i	− 0.117240i	0.998280	−	0.0586200i	$-0.0186700\pi$
$294$	0	0
$295$	5237.72i	1.03374i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	436.477	0.0844218
$300$	0	0
$301$	−1890.00	−0.361920
$302$	0	0
$303$	0	0
$304$	0	0
$305$	1596.00i	0.299629i
$306$	0	0
$307$	2650.04i	0.492657i	0.969186	+	0.246328i	$0.0792242\pi$
$307$	2650.04i	0.492657i	−0.969186	+	0.246328i	$0.920776\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−4801.24	−0.875414	−0.437707	−	0.899118i	$-0.644209\pi$
$311$	−4801.24	−0.875414	−0.437707	+	0.899118i	$0.644209\pi$
$312$	0	0
$313$	−6181.00	−1.11620	−0.558100	−	0.829774i	$-0.688470\pi$
$313$	−6181.00	−1.11620	−0.558100	+	0.829774i	$0.688470\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 2856.00i	− 0.506022i	−0.967463	−	0.253011i	$-0.918579\pi$
$317$	− 2856.00i	− 0.506022i	0.967463	−	0.253011i	$-0.0814209\pi$
$318$	0	0
$319$	10475.4i	1.83860i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−5674.20	−0.977464
$324$	0	0
$325$	−133.000	−0.0227000
$326$	0	0
$327$	0	0
$328$	0	0
$329$	2268.00i	0.380057i
$330$	0	0
$331$	− 1345.80i	− 0.223481i	−0.993737	−	0.111740i	$-0.964358\pi$
$331$	− 1345.80i	− 0.223481i	0.993737	−	0.111740i	$-0.0356425\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	6547.15	1.06779
$336$	0	0
$337$	3499.00	0.565587	0.282793	−	0.959181i	$-0.408739\pi$
$337$	3499.00	0.565587	0.282793	+	0.959181i	$0.408739\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 16200.0i	− 2.57267i
$342$	0	0
$343$	3424.26i	0.539046i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−11722.5	−1.81354	−0.906769	−	0.421627i	$-0.861459\pi$
$347$	−11722.5	−1.81354	−0.906769	+	0.421627i	$0.861459\pi$
$348$	0	0
$349$	7175.00	1.10048	0.550242	−	0.835005i	$-0.314535\pi$
$349$	7175.00	1.10048	0.550242	+	0.835005i	$0.314535\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 1584.00i	− 0.238832i	−0.992844	−	0.119416i	$-0.961898\pi$
$353$	− 1584.00i	− 0.238832i	0.992844	−	0.119416i	$-0.0381023\pi$
$354$	0	0
$355$	5985.97i	0.894936i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−4801.24	−0.705850	−0.352925	−	0.935652i	$-0.614813\pi$
$359$	−4801.24	−0.705850	−0.352925	+	0.935652i	$0.614813\pi$
$360$	0	0
$361$	2296.00	0.334743
$362$	0	0
$363$	0	0
$364$	0	0
$365$	5964.00i	0.855260i
$366$	0	0
$367$	10834.0i	1.54095i	0.637470	+	0.770476i	$0.279981\pi$
$367$	10834.0i	1.54095i	−0.637470	+	0.770476i	$0.720019\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−2618.86	−0.366481
$372$	0	0
$373$	−4201.00	−0.583162	−0.291581	−	0.956546i	$-0.594181\pi$
$373$	−4201.00	−0.583162	−0.291581	+	0.956546i	$0.594181\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 1176.00i	− 0.160655i
$378$	0	0
$379$	5201.35i	0.704948i	0.935822	+	0.352474i	$0.114660\pi$
$379$	5201.35i	0.704948i	−0.935822	+	0.352474i	$0.885340\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	8729.54	1.16464	0.582322	−	0.812958i	$-0.302144\pi$
$383$	8729.54	1.16464	0.582322	+	0.812958i	$0.302144\pi$
$384$	0	0
$385$	−3888.00	−0.514677
$386$	0	0
$387$	0	0
$388$	0	0
$389$	14364.0i	1.87219i	0.351741	+	0.936097i	$0.385590\pi$
$389$	14364.0i	1.87219i	−0.351741	+	0.936097i	$0.614410\pi$
$390$	0	0
$391$	5237.72i	0.677450i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	13530.8	1.72356
$396$	0	0
$397$	2198.00	0.277870	0.138935	−	0.990301i	$-0.455632\pi$
$397$	2198.00	0.277870	0.138935	+	0.990301i	$0.455632\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	672.000i	0.0836860i	0.999124	+	0.0418430i	$0.0133229\pi$
$401$	672.000i	0.0836860i	−0.999124	+	0.0418430i	$0.986677\pi$
$402$	0	0
$403$	1818.65i	0.224798i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	6048.32	0.736620
$408$	0	0
$409$	−6041.00	−0.730338	−0.365169	−	0.930941i	$-0.618989\pi$
$409$	−6041.00	−0.730338	−0.365169	+	0.930941i	$0.618989\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 2268.00i	− 0.270220i
$414$	0	0
$415$	10475.4i	1.23908i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	10039.0	1.17049	0.585245	−	0.810856i	$-0.300998\pi$
$419$	10039.0	1.17049	0.585245	+	0.810856i	$0.300998\pi$
$420$	0	0
$421$	6343.00	0.734297	0.367148	−	0.930162i	$-0.380334\pi$
$421$	6343.00	0.734297	0.367148	+	0.930162i	$0.380334\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 1596.00i	− 0.182159i
$426$	0	0
$427$	− 691.088i	− 0.0783234i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−7794.23	−0.871078	−0.435539	−	0.900170i	$-0.643442\pi$
$431$	−7794.23	−0.871078	−0.435539	+	0.900170i	$0.643442\pi$
$432$	0	0
$433$	10514.0	1.16691	0.583453	−	0.812147i	$-0.301701\pi$
$433$	10514.0	1.16691	0.583453	+	0.812147i	$0.301701\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	4212.00i	0.461069i
$438$	0	0
$439$	1465.31i	0.159307i	0.996823	+	0.0796534i	$0.0253813\pi$
$439$	1465.31i	0.159307i	−0.996823	+	0.0796534i	$0.974619\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−8355.41	−0.896112	−0.448056	−	0.894005i	$-0.647884\pi$
$443$	−8355.41	−0.896112	−0.448056	+	0.894005i	$0.647884\pi$
$444$	0	0
$445$	−13968.0	−1.48797
$446$	0	0
$447$	0	0
$448$	0	0
$449$	3780.00i	0.397303i	0.980070	+	0.198652i	$0.0636563\pi$
$449$	3780.00i	0.397303i	−0.980070	+	0.198652i	$0.936344\pi$
$450$	0	0
$451$	4489.48i	0.468739i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	436.477	0.0449722
$456$	0	0
$457$	−2486.00	−0.254464	−0.127232	−	0.991873i	$-0.540609\pi$
$457$	−2486.00	−0.254464	−0.127232	+	0.991873i	$0.540609\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 10404.0i	− 1.05111i	−0.850759	−	0.525556i	$-0.823857\pi$
$461$	− 10404.0i	− 1.05111i	0.850759	−	0.525556i	$-0.176143\pi$
$462$	0	0
$463$	− 909.327i	− 0.0912743i	−0.998958	−	0.0456371i	$-0.985468\pi$
$463$	− 909.327i	− 0.0912743i	0.998958	−	0.0456371i	$-0.0145318\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	11784.9	1.16775	0.583875	−	0.811844i	$-0.301536\pi$
$467$	11784.9	1.16775	0.583875	+	0.811844i	$0.301536\pi$
$468$	0	0
$469$	−2835.00	−0.279122
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 22680.0i	− 2.20471i
$474$	0	0
$475$	− 1283.45i	− 0.123976i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−18332.0	−1.74867	−0.874334	−	0.485325i	$-0.838701\pi$
$479$	−18332.0	−1.74867	−0.874334	+	0.485325i	$0.838701\pi$
$480$	0	0
$481$	−679.000	−0.0643654
$482$	0	0
$483$	0	0
$484$	0	0
$485$	8988.00i	0.841493i
$486$	0	0
$487$	− 12257.7i	− 1.14056i	−0.821452	−	0.570278i	$-0.806836\pi$
$487$	− 12257.7i	− 1.14056i	0.821452	−	0.570278i	$-0.193164\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	6173.03	0.567383	0.283691	−	0.958916i	$-0.408441\pi$
$491$	6173.03	0.567383	0.283691	+	0.958916i	$0.408441\pi$
$492$	0	0
$493$	14112.0	1.28919
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 2592.00i	− 0.233938i
$498$	0	0
$499$	9384.25i	0.841877i	0.907089	+	0.420939i	$0.138299\pi$
$499$	9384.25i	0.841877i	−0.907089	+	0.420939i	$0.861701\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−1309.43	−0.116073	−0.0580364	−	0.998314i	$-0.518484\pi$
$503$	−1309.43	−0.116073	−0.0580364	+	0.998314i	$0.518484\pi$
$504$	0	0
$505$	12096.0	1.06587
$506$	0	0
$507$	0	0
$508$	0	0
$509$	5292.00i	0.460833i	0.973092	+	0.230416i	$0.0740088\pi$
$509$	5292.00i	0.460833i	−0.973092	+	0.230416i	$0.925991\pi$
$510$	0	0
$511$	− 2582.49i	− 0.223567i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	8791.89	0.752266
$516$	0	0
$517$	−27216.0	−2.31520
$518$	0	0
$519$	0	0
$520$	0	0
$521$	9900.00i	0.832489i	0.909253	+	0.416245i	$0.136654\pi$
$521$	9900.00i	0.832489i	−0.909253	+	0.416245i	$0.863346\pi$
$522$	0	0
$523$	− 16289.9i	− 1.36197i	−0.732298	−	0.680984i	$-0.761552\pi$
$523$	− 16289.9i	− 1.36197i	0.732298	−	0.680984i	$-0.238448\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−21823.8	−1.80391
$528$	0	0
$529$	−8279.00	−0.680447
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 504.000i	− 0.0409581i
$534$	0	0
$535$	− 15713.2i	− 1.26979i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−19703.8	−1.57459
$540$	0	0
$541$	15919.0	1.26509	0.632543	−	0.774525i	$-0.282011\pi$
$541$	15919.0	1.26509	0.632543	+	0.774525i	$0.282011\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	14712.0i	1.15632i
$546$	0	0
$547$	3600.93i	0.281471i	0.990047	+	0.140736i	$0.0449468\pi$
$547$	3600.93i	0.281471i	−0.990047	+	0.140736i	$0.955053\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	11348.4	0.877419
$552$	0	0
$553$	−5859.00	−0.450543
$554$	0	0
$555$	0	0
$556$	0	0
$557$	2100.00i	0.159748i	0.996805	+	0.0798742i	$0.0254519\pi$
$557$	2100.00i	0.159748i	−0.996805	+	0.0798742i	$0.974548\pi$
$558$	0	0
$559$	2546.11i	0.192646i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−9602.49	−0.718822	−0.359411	−	0.933179i	$-0.617022\pi$
$563$	−9602.49	−0.718822	−0.359411	+	0.933179i	$0.617022\pi$
$564$	0	0
$565$	25200.0	1.87641
$566$	0	0
$567$	0	0
$568$	0	0
$569$	2604.00i	0.191855i	0.995388	+	0.0959274i	$0.0305817\pi$
$569$	2604.00i	0.191855i	−0.995388	+	0.0959274i	$0.969418\pi$
$570$	0	0
$571$	− 20550.8i	− 1.50617i	−0.657923	−	0.753085i	$-0.728565\pi$
$571$	− 20550.8i	− 1.50617i	0.657923	−	0.753085i	$-0.271435\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−1184.72	−0.0859241
$576$	0	0
$577$	9947.00	0.717676	0.358838	−	0.933400i	$-0.383173\pi$
$577$	9947.00	0.717676	0.358838	+	0.933400i	$0.383173\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 4536.00i	− 0.323898i
$582$	0	0
$583$	− 31426.3i	− 2.23250i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	14403.7	1.01279	0.506393	−	0.862303i	$-0.330978\pi$
$587$	14403.7	1.01279	0.506393	+	0.862303i	$0.330978\pi$
$588$	0	0
$589$	−17550.0	−1.22773
$590$	0	0
$591$	0	0
$592$	0	0
$593$	6792.00i	0.470344i	0.971954	+	0.235172i	$0.0755653\pi$
$593$	6792.00i	0.470344i	−0.971954	+	0.235172i	$0.924435\pi$
$594$	0	0
$595$	5237.72i	0.360883i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	11847.2	0.808121	0.404061	−	0.914732i	$-0.367598\pi$
$599$	11847.2	0.808121	0.404061	+	0.914732i	$0.367598\pi$
$600$	0	0
$601$	−10150.0	−0.688897	−0.344449	−	0.938805i	$-0.611934\pi$
$601$	−10150.0	−0.688897	−0.344449	+	0.938805i	$0.611934\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 30684.0i	− 2.06195i
$606$	0	0
$607$	− 13338.5i	− 0.891918i	−0.895053	−	0.445959i	$-0.852863\pi$
$607$	− 13338.5i	− 0.891918i	0.895053	−	0.445959i	$-0.147137\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	3055.34	0.202301
$612$	0	0
$613$	−2941.00	−0.193778	−0.0968890	−	0.995295i	$-0.530889\pi$
$613$	−2941.00	−0.193778	−0.0968890	+	0.995295i	$0.530889\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 14700.0i	− 0.959157i	−0.877499	−	0.479578i	$-0.840790\pi$
$617$	− 14700.0i	− 0.959157i	0.877499	−	0.479578i	$-0.159210\pi$
$618$	0	0
$619$	2478.56i	0.160940i	0.996757	+	0.0804701i	$0.0256421\pi$
$619$	2478.56i	0.160940i	−0.996757	+	0.0804701i	$0.974358\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	6048.32	0.388958
$624$	0	0
$625$	−17639.0	−1.12890
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 8148.00i	− 0.516506i
$630$	0	0
$631$	− 30153.3i	− 1.90235i	−0.308652	−	0.951175i	$-0.599878\pi$
$631$	− 30153.3i	− 1.90235i	0.308652	−	0.951175i	$-0.400122\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	6110.68	0.381881
$636$	0	0
$637$	2212.00	0.137587
$638$	0	0
$639$	0	0
$640$	0	0
$641$	8736.00i	0.538301i	0.963098	+	0.269151i	$0.0867430\pi$
$641$	8736.00i	0.538301i	−0.963098	+	0.269151i	$0.913257\pi$
$642$	0	0
$643$	2650.04i	0.162531i	0.996692	+	0.0812654i	$0.0258961\pi$
$643$	2650.04i	0.162531i	−0.996692	+	0.0812654i	$0.974104\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	27061.6	1.64436	0.822179	−	0.569229i	$-0.192758\pi$
$647$	27061.6	1.64436	0.822179	+	0.569229i	$0.192758\pi$
$648$	0	0
$649$	27216.0	1.64610
$650$	0	0
$651$	0	0
$652$	0	0
$653$	2520.00i	0.151019i	0.997145	+	0.0755094i	$0.0240583\pi$
$653$	2520.00i	0.151019i	−0.997145	+	0.0755094i	$0.975942\pi$
$654$	0	0
$655$	− 10475.4i	− 0.624900i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−1371.78	−0.0810882	−0.0405441	−	0.999178i	$-0.512909\pi$
$659$	−1371.78	−0.0810882	−0.0405441	+	0.999178i	$0.512909\pi$
$660$	0	0
$661$	−17773.0	−1.04582	−0.522912	−	0.852387i	$-0.675154\pi$
$661$	−17773.0	−1.04582	−0.522912	+	0.852387i	$0.675154\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	4212.00i	0.245616i
$666$	0	0
$667$	− 10475.4i	− 0.608112i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	8293.06	0.477124
$672$	0	0
$673$	3323.00	0.190330	0.0951651	−	0.995462i	$-0.469662\pi$
$673$	3323.00	0.190330	0.0951651	+	0.995462i	$0.469662\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	29052.0i	1.64927i	0.565662	+	0.824637i	$0.308621\pi$
$677$	29052.0i	1.64927i	−0.565662	+	0.824637i	$0.691379\pi$
$678$	0	0
$679$	− 3891.92i	− 0.219968i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−14466.1	−0.810438	−0.405219	−	0.914220i	$-0.632805\pi$
$683$	−14466.1	−0.810438	−0.405219	+	0.914220i	$0.632805\pi$
$684$	0	0
$685$	−1008.00	−0.0562244
$686$	0	0
$687$	0	0
$688$	0	0
$689$	3528.00i	0.195074i
$690$	0	0
$691$	4063.39i	0.223703i	0.993725	+	0.111851i	$0.0356781\pi$
$691$	4063.39i	0.223703i	−0.993725	+	0.111851i	$0.964322\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−27622.7	−1.50761
$696$	0	0
$697$	6048.00	0.328672
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 588.000i	− 0.0316811i	−0.999875	−	0.0158406i	$-0.994958\pi$
$701$	− 588.000i	− 0.0316811i	0.999875	−	0.0158406i	$-0.00504242\pi$
$702$	0	0
$703$	− 6552.35i	− 0.351531i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−5237.72	−0.278621
$708$	0	0
$709$	29455.0	1.56023	0.780117	−	0.625634i	$-0.215160\pi$
$709$	29455.0	1.56023	0.780117	+	0.625634i	$0.215160\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	16200.0i	0.850904i
$714$	0	0
$715$	5237.72i	0.273958i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	9602.49	0.498070	0.249035	−	0.968494i	$-0.419887\pi$
$719$	9602.49	0.498070	0.249035	+	0.968494i	$0.419887\pi$
$720$	0	0
$721$	−3807.00	−0.196644
$722$	0	0
$723$	0	0
$724$	0	0
$725$	3192.00i	0.163514i
$726$	0	0
$727$	− 25471.5i	− 1.29943i	−0.760177	−	0.649716i	$-0.774888\pi$
$727$	− 25471.5i	− 1.29943i	0.760177	−	0.649716i	$-0.225112\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−30553.4	−1.54591
$732$	0	0
$733$	−30170.0	−1.52027	−0.760133	−	0.649768i	$-0.774866\pi$
$733$	−30170.0	−1.52027	−0.760133	+	0.649768i	$0.774866\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 34020.0i	− 1.70033i
$738$	0	0
$739$	31499.1i	1.56795i	0.620795	+	0.783973i	$0.286810\pi$
$739$	31499.1i	1.56795i	−0.620795	+	0.783973i	$0.713190\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	13904.9	0.686570	0.343285	−	0.939231i	$-0.388460\pi$
$743$	13904.9	0.686570	0.343285	+	0.939231i	$0.388460\pi$
$744$	0	0
$745$	−32256.0	−1.58627
$746$	0	0
$747$	0	0
$748$	0	0
$749$	6804.00i	0.331926i
$750$	0	0
$751$	− 6147.05i	− 0.298680i	−0.988786	−	0.149340i	$-0.952285\pi$
$751$	− 6147.05i	− 0.298680i	0.988786	−	0.149340i	$-0.0477150\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−9166.01	−0.441835
$756$	0	0
$757$	13687.0	0.657150	0.328575	−	0.944478i	$-0.393432\pi$
$757$	13687.0	0.657150	0.328575	+	0.944478i	$0.393432\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	5460.00i	0.260085i	0.991508	+	0.130043i	$0.0415114\pi$
$761$	5460.00i	0.260085i	−0.991508	+	0.130043i	$0.958489\pi$
$762$	0	0
$763$	− 6370.48i	− 0.302263i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−3055.34	−0.143836
$768$	0	0
$769$	20783.0	0.974583	0.487291	−	0.873239i	$-0.337985\pi$
$769$	20783.0	0.974583	0.487291	+	0.873239i	$0.337985\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 10680.0i	− 0.496938i	−0.968640	−	0.248469i	$-0.920073\pi$
$773$	− 10680.0i	− 0.496938i	0.968640	−	0.248469i	$-0.0799274\pi$
$774$	0	0
$775$	− 4936.34i	− 0.228798i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	4863.60	0.223693
$780$	0	0
$781$	31104.0	1.42508
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 38472.0i	− 1.74920i
$786$	0	0
$787$	42509.7i	1.92542i	0.270530	+	0.962712i	$0.412801\pi$
$787$	42509.7i	1.92542i	−0.270530	+	0.962712i	$0.587199\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−10911.9	−0.490497
$792$	0	0
$793$	−931.000	−0.0416908
$794$	0	0
$795$	0	0
$796$	0	0
$797$	22080.0i	0.981322i	0.871351	+	0.490661i	$0.163245\pi$
$797$	22080.0i	0.981322i	−0.871351	+	0.490661i	$0.836755\pi$
$798$	0	0
$799$	36664.1i	1.62338i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	30989.9	1.36190
$804$	0	0
$805$	3888.00	0.170229
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 37632.0i	− 1.63544i	−0.575616	−	0.817720i	$-0.695238\pi$
$809$	− 37632.0i	− 1.63544i	0.575616	−	0.817720i	$-0.304762\pi$
$810$	0	0
$811$	8843.85i	0.382922i	0.981500	+	0.191461i	$0.0613225\pi$
$811$	8843.85i	0.382922i	−0.981500	+	0.191461i	$0.938677\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−44084.2	−1.89472
$816$	0	0
$817$	−24570.0	−1.05214
$818$	0	0
$819$	0	0
$820$	0	0
$821$	10500.0i	0.446349i	0.974779	+	0.223174i	$0.0716420\pi$
$821$	10500.0i	0.446349i	−0.974779	+	0.223174i	$0.928358\pi$
$822$	0	0
$823$	− 42956.6i	− 1.81941i	−0.415257	−	0.909704i	$-0.636309\pi$
$823$	− 42956.6i	− 1.81941i	0.415257	−	0.909704i	$-0.363691\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−22634.4	−0.951725	−0.475863	−	0.879520i	$-0.657864\pi$
$827$	−22634.4	−0.951725	−0.475863	+	0.879520i	$0.657864\pi$
$828$	0	0
$829$	−16877.0	−0.707072	−0.353536	−	0.935421i	$-0.615021\pi$
$829$	−16877.0	−0.707072	−0.353536	+	0.935421i	$0.615021\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	26544.0i	1.10408i
$834$	0	0
$835$	47139.5i	1.95369i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	21823.8	0.898025	0.449012	−	0.893526i	$-0.351776\pi$
$839$	21823.8	0.898025	0.449012	+	0.893526i	$0.351776\pi$
$840$	0	0
$841$	−3835.00	−0.157243
$842$	0	0
$843$	0	0
$844$	0	0
$845$	25776.0i	1.04937i
$846$	0	0
$847$	13286.6i	0.538998i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−6048.32	−0.243635
$852$	0	0
$853$	−1897.00	−0.0761454	−0.0380727	−	0.999275i	$-0.512122\pi$
$853$	−1897.00	−0.0761454	−0.0380727	+	0.999275i	$0.512122\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	21264.0i	0.847566i	0.905764	+	0.423783i	$0.139298\pi$
$857$	21264.0i	0.847566i	−0.905764	+	0.423783i	$0.860702\pi$
$858$	0	0
$859$	27565.6i	1.09491i	0.836836	+	0.547454i	$0.184403\pi$
$859$	27565.6i	1.09491i	−0.836836	+	0.547454i	$0.815597\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	26625.1	1.05021	0.525103	−	0.851038i	$-0.324027\pi$
$863$	26625.1	1.05021	0.525103	+	0.851038i	$0.324027\pi$
$864$	0	0
$865$	15264.0	0.599990
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 70308.0i	− 2.74457i
$870$	0	0
$871$	3819.17i	0.148574i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	6609.51	0.255362
$876$	0	0
$877$	−2329.00	−0.0896747	−0.0448374	−	0.998994i	$-0.514277\pi$
$877$	−2329.00	−0.0896747	−0.0448374	+	0.998994i	$0.514277\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	41004.0i	1.56806i	0.620724	+	0.784029i	$0.286839\pi$
$881$	41004.0i	1.56806i	−0.620724	+	0.784029i	$0.713161\pi$
$882$	0	0
$883$	25715.8i	0.980073i	0.871702	+	0.490036i	$0.163016\pi$
$883$	25715.8i	0.980073i	−0.871702	+	0.490036i	$0.836984\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	30553.4	1.15657	0.578287	−	0.815833i	$-0.303721\pi$
$887$	30553.4	1.15657	0.578287	+	0.815833i	$0.303721\pi$
$888$	0	0
$889$	−2646.00	−0.0998245
$890$	0	0
$891$	0	0
$892$	0	0
$893$	29484.0i	1.10486i
$894$	0	0
$895$	41901.8i	1.56494i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	43647.7	1.61928
$900$	0	0
$901$	−42336.0	−1.56539
$902$	0	0
$903$	0	0
$904$	0	0
$905$	39228.0i	1.44087i
$906$	0	0
$907$	− 5856.06i	− 0.214385i	−0.994238	−	0.107193i	$-0.965814\pi$
$907$	− 5856.06i	− 0.214385i	0.994238	−	0.107193i	$-0.0341862\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	51504.3	1.87312	0.936560	−	0.350507i	$-0.113991\pi$
$911$	51504.3	1.87312	0.936560	+	0.350507i	$0.113991\pi$
$912$	0	0
$913$	54432.0	1.97310
$914$	0	0
$915$	0	0
$916$	0	0
$917$	4536.00i	0.163350i
$918$	0	0
$919$	8947.77i	0.321175i	0.987022	+	0.160588i	$0.0513389\pi$
$919$	8947.77i	0.321175i	−0.987022	+	0.160588i	$0.948661\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−3491.81	−0.124523
$924$	0	0
$925$	1843.00	0.0655108
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 34944.0i	− 1.23410i	−0.786925	−	0.617048i	$-0.788328\pi$
$929$	− 34944.0i	− 1.23410i	0.786925	−	0.617048i	$-0.211672\pi$
$930$	0	0
$931$	21345.8i	0.751429i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−62852.7	−2.19840
$936$	0	0
$937$	875.000	0.0305069	0.0152535	−	0.999884i	$-0.495144\pi$
$937$	875.000	0.0305069	0.0152535	+	0.999884i	$0.495144\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 45204.0i	− 1.56600i	−0.622020	−	0.783001i	$-0.713688\pi$
$941$	− 45204.0i	− 1.56600i	0.622020	−	0.783001i	$-0.286312\pi$
$942$	0	0
$943$	− 4489.48i	− 0.155034i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−31364.0	−1.07623	−0.538116	−	0.842871i	$-0.680864\pi$
$947$	−31364.0	−1.07623	−0.538116	+	0.842871i	$0.680864\pi$
$948$	0	0
$949$	−3479.00	−0.119002
$950$	0	0
$951$	0	0
$952$	0	0
$953$	9492.00i	0.322640i	0.986902	+	0.161320i	$0.0515751\pi$
$953$	9492.00i	0.322640i	−0.986902	+	0.161320i	$0.948425\pi$
$954$	0	0
$955$	20202.6i	0.684547i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	436.477	0.0146972
$960$	0	0
$961$	−37709.0	−1.26578
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 54996.0i	− 1.83459i
$966$	0	0
$967$	2291.50i	0.0762045i	0.999274	+	0.0381023i	$0.0121313\pi$
$967$	2291.50i	0.0762045i	−0.999274	+	0.0381023i	$0.987869\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	40592.3	1.34158	0.670788	−	0.741649i	$-0.265956\pi$
$971$	40592.3	1.34158	0.670788	+	0.741649i	$0.265956\pi$
$972$	0	0
$973$	11961.0	0.394093
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 25704.0i	− 0.841703i	−0.907130	−	0.420851i	$-0.861731\pi$
$977$	− 25704.0i	− 0.841703i	0.907130	−	0.420851i	$-0.138269\pi$
$978$	0	0
$979$	72579.9i	2.36942i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−42338.2	−1.37373	−0.686867	−	0.726783i	$-0.741015\pi$
$983$	−42338.2	−1.37373	−0.686867	+	0.726783i	$0.741015\pi$
$984$	0	0
$985$	49392.0	1.59773
$986$	0	0
$987$	0	0
$988$	0	0
$989$	22680.0i	0.729203i
$990$	0	0
$991$	− 3382.70i	− 0.108431i	−0.998529	−	0.0542154i	$-0.982734\pi$
$991$	− 3382.70i	− 0.108431i	0.998529	−	0.0542154i	$-0.0172658\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−2681.21	−0.0854274
$996$	0	0
$997$	35966.0	1.14248	0.571241	−	0.820782i	$-0.306462\pi$
$997$	35966.0	1.14248	0.571241	+	0.820782i	$0.306462\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	432.4.c.g.431.2	yes	4
3.2	odd	2	inner	432.4.c.g.431.4	yes	4
4.3	odd	2	inner	432.4.c.g.431.1	✓	4
8.3	odd	2		1728.4.c.f.1727.3		4
8.5	even	2		1728.4.c.f.1727.4		4
12.11	even	2	inner	432.4.c.g.431.3	yes	4
24.5	odd	2		1728.4.c.f.1727.2		4
24.11	even	2		1728.4.c.f.1727.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
432.4.c.g.431.1	✓	4	4.3	odd	2	inner
432.4.c.g.431.2	yes	4	1.1	even	1	trivial
432.4.c.g.431.3	yes	4	12.11	even	2	inner
432.4.c.g.431.4	yes	4	3.2	odd	2	inner
1728.4.c.f.1727.1		4	24.11	even	2
1728.4.c.f.1727.2		4	24.5	odd	2
1728.4.c.f.1727.3		4	8.3	odd	2
1728.4.c.f.1727.4		4	8.5	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 432 = 2^{4} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	432.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-12.0000i q^{5} +5.19615i q^{7} -62.3538 q^{11} +7.00000 q^{13} +84.0000i q^{17} +67.5500i q^{19} +62.3538 q^{23} -19.0000 q^{25} -168.000i q^{29} +259.808i q^{31} +62.3538 q^{35} -97.0000 q^{37} -72.0000i q^{41} +363.731i q^{43} +436.477 q^{47} +316.000 q^{49} +504.000i q^{53} +748.246i q^{55} -436.477 q^{59} -133.000 q^{61} -84.0000i q^{65} +545.596i q^{67} -498.831 q^{71} -497.000 q^{73} -324.000i q^{77} +1127.57i q^{79} -872.954 q^{83} +1008.00 q^{85} -1164.00i q^{89} +36.3731i q^{91} +810.600 q^{95} -749.000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 28 q^{13} - 76 q^{25} - 388 q^{37} + 1264 q^{49} - 532 q^{61} - 1988 q^{73} + 4032 q^{85} - 2996 q^{97}+O(q^{100}) \) 4 * q + 28 * q^13 - 76 * q^25 - 388 * q^37 + 1264 * q^49 - 532 * q^61 - 1988 * q^73 + 4032 * q^85 - 2996 * q^97

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