LMFDB - Embedded newform 1152.3.j.b.161.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1152,3,Mod(161,1152)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1152, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 3, 2]))

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

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

chi := DirichletCharacter("1152.161");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1152 = 2^{7} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1152.j (of order $4$, degree $2$, 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:	$31.3897264543$
Analytic rank:	$0$
Dimension:	$32$
Relative dimension:	$16$ over $\Q(i)$
Twist minimal:	no (minimal twist has level 144)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			161.6
Character	$\chi$	$=$	1152.161
Dual form			1152.3.j.b.737.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.84570 - 1.84570i) q^{5} +0.226665i q^{7} +O(q^{10})$	$q+(-1.84570 - 1.84570i) q^{5} +0.226665i q^{7} +(11.5821 + 11.5821i) q^{11} +(8.11369 + 8.11369i) q^{13} -6.20082i q^{17} +(-4.43355 - 4.43355i) q^{19} -28.6638 q^{23} -18.1868i q^{25} +(-15.7561 + 15.7561i) q^{29} +33.5751 q^{31} +(0.418355 - 0.418355i) q^{35} +(-43.8384 + 43.8384i) q^{37} -62.2969 q^{41} +(18.3048 - 18.3048i) q^{43} +13.8029i q^{47} +48.9486 q^{49} +(37.9167 + 37.9167i) q^{53} -42.7539i q^{55} +(70.4759 + 70.4759i) q^{59} +(60.3861 + 60.3861i) q^{61} -29.9508i q^{65} +(61.9736 + 61.9736i) q^{67} +32.5259 q^{71} +130.954i q^{73} +(-2.62525 + 2.62525i) q^{77} +132.387 q^{79} +(-19.2783 + 19.2783i) q^{83} +(-11.4448 + 11.4448i) q^{85} -91.2401 q^{89} +(-1.83909 + 1.83909i) q^{91} +16.3660i q^{95} -20.0808 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q+O(q^{10}) $ 32 * q	$ 32 q + 64 q^{19} + 128 q^{43} - 224 q^{49} - 64 q^{61} - 64 q^{67} + 512 q^{79} - 320 q^{85} - 192 q^{91}+O(q^{100}) $ 32 * q + 64 * q^19 + 128 * q^43 - 224 * q^49 - 64 * q^61 - 64 * q^67 + 512 * q^79 - 320 * q^85 - 192 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1152\mathbb{Z}\right)^\times$.

$n$	$127$	$641$	$901$
$\chi(n)$	$1$	$-1$	$e\left(\frac{3}{4}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−1.84570	−	1.84570i	−0.369140	−	0.369140i	0.498024	−	0.867163i	$-0.334059\pi$
$5$	−1.84570	−	1.84570i	−0.369140	−	0.369140i	−0.867163	+	0.498024i	$0.834059\pi$
$6$		0			0
$7$			0.226665i			0.0323807i	0.999869	+	0.0161904i	$0.00515378\pi$
$7$			0.226665i			0.0323807i	−0.999869	+	0.0161904i	$0.994846\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	11.5821	+	11.5821i	1.05291	+	1.05291i	0.998520	+	0.0543946i	$0.0173229\pi$
$11$	11.5821	+	11.5821i	1.05291	+	1.05291i	0.0543946	+	0.998520i	$0.482677\pi$
$12$		0			0
$13$	8.11369	+	8.11369i	0.624130	+	0.624130i	0.946585	−	0.322455i	$-0.104508\pi$
$13$	8.11369	+	8.11369i	0.624130	+	0.624130i	−0.322455	+	0.946585i	$0.604508\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	6.20082i		−	0.364754i	−0.983229	−	0.182377i	$-0.941621\pi$
$17$		−	6.20082i		−	0.364754i	0.983229	−	0.182377i	$-0.0583791\pi$
$18$		0			0
$19$	−4.43355	−	4.43355i	−0.233345	−	0.233345i	0.580743	−	0.814087i	$-0.302762\pi$
$19$	−4.43355	−	4.43355i	−0.233345	−	0.233345i	−0.814087	+	0.580743i	$0.802762\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−28.6638			−1.24625			−0.623127	−	0.782121i	$-0.714138\pi$
$23$	−28.6638			−1.24625			−0.623127	+	0.782121i	$0.714138\pi$
$24$		0			0
$25$		−	18.1868i		−	0.727472i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−15.7561	+	15.7561i	−0.543312	+	0.543312i	−0.924498	−	0.381186i	$-0.875516\pi$
$29$	−15.7561	+	15.7561i	−0.543312	+	0.543312i	0.381186	+	0.924498i	$0.375516\pi$
$30$		0			0
$31$	33.5751			1.08307			0.541534	−	0.840679i	$-0.317844\pi$
$31$	33.5751			1.08307			0.541534	+	0.840679i	$0.317844\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	0.418355	−	0.418355i	0.0119530	−	0.0119530i
$36$		0			0
$37$	−43.8384	+	43.8384i	−1.18482	+	1.18482i	−0.206342	+	0.978480i	$0.566156\pi$
$37$	−43.8384	+	43.8384i	−1.18482	+	1.18482i	−0.978480	+	0.206342i	$0.933844\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−62.2969			−1.51944			−0.759719	−	0.650252i	$-0.774663\pi$
$41$	−62.2969			−1.51944			−0.759719	+	0.650252i	$0.774663\pi$
$42$		0			0
$43$	18.3048	−	18.3048i	0.425692	−	0.425692i	−0.461466	−	0.887158i	$-0.652676\pi$
$43$	18.3048	−	18.3048i	0.425692	−	0.425692i	0.887158	+	0.461466i	$0.152676\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			13.8029i			0.293678i	0.989160	+	0.146839i	$0.0469099\pi$
$47$			13.8029i			0.293678i	−0.989160	+	0.146839i	$0.953090\pi$
$48$		0			0
$49$	48.9486			0.998951
$50$		0			0
$51$		0			0
$52$		0			0
$53$	37.9167	+	37.9167i	0.715409	+	0.715409i	0.967661	−	0.252252i	$-0.0811713\pi$
$53$	37.9167	+	37.9167i	0.715409	+	0.715409i	−0.252252	+	0.967661i	$0.581171\pi$
$54$		0			0
$55$		−	42.7539i		−	0.777344i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	70.4759	+	70.4759i	1.19451	+	1.19451i	0.975788	+	0.218719i	$0.0701879\pi$
$59$	70.4759	+	70.4759i	1.19451	+	1.19451i	0.218719	+	0.975788i	$0.429812\pi$
$60$		0			0
$61$	60.3861	+	60.3861i	0.989936	+	0.989936i	0.999950	−	0.0100137i	$-0.00318750\pi$
$61$	60.3861	+	60.3861i	0.989936	+	0.989936i	−0.0100137	+	0.999950i	$0.503187\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	29.9508i		−	0.460782i
$66$		0			0
$67$	61.9736	+	61.9736i	0.924979	+	0.924979i	0.997376	−	0.0723967i	$-0.0230648\pi$
$67$	61.9736	+	61.9736i	0.924979	+	0.924979i	−0.0723967	+	0.997376i	$0.523065\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	32.5259			0.458111			0.229055	−	0.973413i	$-0.426436\pi$
$71$	32.5259			0.458111			0.229055	+	0.973413i	$0.426436\pi$
$72$		0			0
$73$			130.954i			1.79389i	0.442145	+	0.896944i	$0.354218\pi$
$73$			130.954i			1.79389i	−0.442145	+	0.896944i	$0.645782\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−2.62525	+	2.62525i	−0.0340941	+	0.0340941i
$78$		0			0
$79$	132.387			1.67579			0.837894	−	0.545833i	$-0.183787\pi$
$79$	132.387			1.67579			0.837894	+	0.545833i	$0.183787\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−19.2783	+	19.2783i	−0.232268	+	0.232268i	−0.813639	−	0.581371i	$-0.802517\pi$
$83$	−19.2783	+	19.2783i	−0.232268	+	0.232268i	0.581371	+	0.813639i	$0.302517\pi$
$84$		0			0
$85$	−11.4448	+	11.4448i	−0.134645	+	0.134645i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−91.2401			−1.02517			−0.512585	−	0.858637i	$-0.671312\pi$
$89$	−91.2401			−1.02517			−0.512585	+	0.858637i	$0.671312\pi$
$90$		0			0
$91$	−1.83909	+	1.83909i	−0.0202098	+	0.0202098i
$92$		0			0
$93$		0			0
$94$		0			0
$95$			16.3660i			0.172274i
$96$		0			0
$97$	−20.0808			−0.207019			−0.103509	−	0.994628i	$-0.533007\pi$
$97$	−20.0808			−0.207019			−0.103509	+	0.994628i	$0.533007\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−29.7693	−	29.7693i	−0.294745	−	0.294745i	0.544206	−	0.838951i	$-0.316831\pi$
$101$	−29.7693	−	29.7693i	−0.294745	−	0.294745i	−0.838951	+	0.544206i	$0.816831\pi$
$102$		0			0
$103$		−	111.006i		−	1.07773i	−0.842392	−	0.538865i	$-0.818853\pi$
$103$		−	111.006i		−	1.07773i	0.842392	−	0.538865i	$-0.181147\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	13.1409	+	13.1409i	0.122812	+	0.122812i	0.765842	−	0.643029i	$-0.222323\pi$
$107$	13.1409	+	13.1409i	0.122812	+	0.122812i	−0.643029	+	0.765842i	$0.722323\pi$
$108$		0			0
$109$	−3.89888	−	3.89888i	−0.0357696	−	0.0357696i	0.688996	−	0.724765i	$-0.258052\pi$
$109$	−3.89888	−	3.89888i	−0.0357696	−	0.0357696i	−0.724765	+	0.688996i	$0.758052\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			117.686i			1.04147i	0.853718	+	0.520735i	$0.174342\pi$
$113$			117.686i			1.04147i	−0.853718	+	0.520735i	$0.825658\pi$
$114$		0			0
$115$	52.9048	+	52.9048i	0.460041	+	0.460041i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	1.40551			0.0118110
$120$		0			0
$121$			147.288i			1.21726i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−79.7098	+	79.7098i	−0.637678	+	0.637678i
$126$		0			0
$127$	43.1157			0.339494			0.169747	−	0.985488i	$-0.445705\pi$
$127$	43.1157			0.339494			0.169747	+	0.985488i	$0.445705\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−97.5874	+	97.5874i	−0.744942	+	0.744942i	−0.973524	−	0.228583i	$-0.926591\pi$
$131$	−97.5874	+	97.5874i	−0.744942	+	0.744942i	0.228583	+	0.973524i	$0.426591\pi$
$132$		0			0
$133$	1.00493	−	1.00493i	0.00755588	−	0.00755588i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−49.2731			−0.359658			−0.179829	−	0.983698i	$-0.557554\pi$
$137$	−49.2731			−0.359658			−0.179829	+	0.983698i	$0.557554\pi$
$138$		0			0
$139$	172.904	−	172.904i	1.24392	−	1.24392i	0.285552	−	0.958363i	$-0.407823\pi$
$139$	172.904	−	172.904i	1.24392	−	1.24392i	0.958363	−	0.285552i	$-0.0921769\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			187.946i			1.31431i
$144$		0			0
$145$	58.1618			0.401116
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−128.537	−	128.537i	−0.862663	−	0.862663i	0.128984	−	0.991647i	$-0.458828\pi$
$149$	−128.537	−	128.537i	−0.862663	−	0.862663i	−0.991647	+	0.128984i	$0.958828\pi$
$150$		0			0
$151$			89.9523i			0.595710i	0.954611	+	0.297855i	$0.0962713\pi$
$151$			89.9523i			0.595710i	−0.954611	+	0.297855i	$0.903729\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−61.9695	−	61.9695i	−0.399803	−	0.399803i
$156$		0			0
$157$	95.2413	+	95.2413i	0.606633	+	0.606633i	0.942065	−	0.335432i	$-0.108882\pi$
$157$	95.2413	+	95.2413i	0.606633	+	0.606633i	−0.335432	+	0.942065i	$0.608882\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	6.49709i		−	0.0403546i
$162$		0			0
$163$	−84.2958	−	84.2958i	−0.517152	−	0.517152i	0.399556	−	0.916709i	$-0.369164\pi$
$163$	−84.2958	−	84.2958i	−0.517152	−	0.517152i	−0.916709	+	0.399556i	$0.869164\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	193.153			1.15661			0.578304	−	0.815821i	$-0.303715\pi$
$167$	193.153			1.15661			0.578304	+	0.815821i	$0.303715\pi$
$168$		0			0
$169$		−	37.3362i		−	0.220924i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−60.1097	+	60.1097i	−0.347455	+	0.347455i	−0.859161	−	0.511706i	$-0.829014\pi$
$173$	−60.1097	+	60.1097i	−0.347455	+	0.347455i	0.511706	+	0.859161i	$0.329014\pi$
$174$		0			0
$175$	4.12231			0.0235561
$176$		0			0
$177$		0			0
$178$		0			0
$179$	103.680	−	103.680i	0.579217	−	0.579217i	−0.355471	−	0.934687i	$-0.615679\pi$
$179$	103.680	−	103.680i	0.579217	−	0.579217i	0.934687	+	0.355471i	$0.115679\pi$
$180$		0			0
$181$	3.08029	−	3.08029i	0.0170182	−	0.0170182i	−0.698547	−	0.715565i	$-0.746169\pi$
$181$	3.08029	−	3.08029i	0.0170182	−	0.0170182i	0.715565	+	0.698547i	$0.246169\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	161.825			0.874729
$186$		0			0
$187$	71.8182	−	71.8182i	0.384054	−	0.384054i
$188$		0			0
$189$		0			0
$190$		0			0
$191$			235.549i			1.23324i	0.787261	+	0.616620i	$0.211499\pi$
$191$			235.549i			1.23324i	−0.787261	+	0.616620i	$0.788501\pi$
$192$		0			0
$193$	−9.38546			−0.0486293			−0.0243147	−	0.999704i	$-0.507740\pi$
$193$	−9.38546			−0.0486293			−0.0243147	+	0.999704i	$0.507740\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−75.8630	−	75.8630i	−0.385091	−	0.385091i	0.487841	−	0.872932i	$-0.337785\pi$
$197$	−75.8630	−	75.8630i	−0.385091	−	0.385091i	−0.872932	+	0.487841i	$0.837785\pi$
$198$		0			0
$199$			315.653i			1.58620i	0.609093	+	0.793099i	$0.291534\pi$
$199$			315.653i			1.58620i	−0.609093	+	0.793099i	$0.708466\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−3.57135	−	3.57135i	−0.0175928	−	0.0175928i
$204$		0			0
$205$	114.981	+	114.981i	0.560884	+	0.560884i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		−	102.699i		−	0.491384i
$210$		0			0
$211$	−20.9669	−	20.9669i	−0.0993692	−	0.0993692i	0.655675	−	0.755044i	$-0.272384\pi$
$211$	−20.9669	−	20.9669i	−0.0993692	−	0.0993692i	−0.755044	+	0.655675i	$0.772384\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−67.5701			−0.314280
$216$		0			0
$217$			7.61030i			0.0350705i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	50.3115	−	50.3115i	0.227654	−	0.227654i
$222$		0			0
$223$	−317.091			−1.42193			−0.710966	−	0.703226i	$-0.751742\pi$
$223$	−317.091			−1.42193			−0.710966	+	0.703226i	$0.751742\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−101.358	+	101.358i	−0.446513	+	0.446513i	−0.894193	−	0.447681i	$-0.852250\pi$
$227$	−101.358	+	101.358i	−0.446513	+	0.446513i	0.447681	+	0.894193i	$0.352250\pi$
$228$		0			0
$229$	83.7775	−	83.7775i	0.365841	−	0.365841i	−0.500117	−	0.865958i	$-0.666710\pi$
$229$	83.7775	−	83.7775i	0.365841	−	0.365841i	0.865958	+	0.500117i	$0.166710\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	289.948			1.24441			0.622207	−	0.782853i	$-0.286236\pi$
$233$	289.948			1.24441			0.622207	+	0.782853i	$0.286236\pi$
$234$		0			0
$235$	25.4759	−	25.4759i	0.108408	−	0.108408i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	441.865i		−	1.84881i	−0.381413	−	0.924405i	$-0.624562\pi$
$239$		−	441.865i		−	1.84881i	0.381413	−	0.924405i	$-0.375438\pi$
$240$		0			0
$241$	−336.119			−1.39469			−0.697343	−	0.716738i	$-0.745634\pi$
$241$	−336.119			−1.39469			−0.697343	+	0.716738i	$0.745634\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−90.3444	−	90.3444i	−0.368752	−	0.368752i
$246$		0			0
$247$		−	71.9449i		−	0.291275i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−73.9741	−	73.9741i	−0.294717	−	0.294717i	0.544223	−	0.838941i	$-0.316824\pi$
$251$	−73.9741	−	73.9741i	−0.294717	−	0.294717i	−0.838941	+	0.544223i	$0.816824\pi$
$252$		0			0
$253$	−331.986	−	331.986i	−1.31220	−	1.31220i
$254$		0			0
$255$		0			0
$256$		0			0
$257$			441.535i			1.71803i	0.511947	+	0.859017i	$0.328924\pi$
$257$			441.535i			1.71803i	−0.511947	+	0.859017i	$0.671076\pi$
$258$		0			0
$259$	−9.93664	−	9.93664i	−0.0383654	−	0.0383654i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−116.115			−0.441503			−0.220752	−	0.975330i	$-0.570851\pi$
$263$	−116.115			−0.441503			−0.220752	+	0.975330i	$0.570851\pi$
$264$		0			0
$265$		−	139.965i		−	0.528171i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−46.5610	+	46.5610i	−0.173089	+	0.173089i	−0.788335	−	0.615246i	$-0.789057\pi$
$269$	−46.5610	+	46.5610i	−0.173089	+	0.173089i	0.615246	+	0.788335i	$0.289057\pi$
$270$		0			0
$271$	343.577			1.26781			0.633907	−	0.773410i	$-0.281450\pi$
$271$	343.577			1.26781			0.633907	+	0.773410i	$0.281450\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	210.641	−	210.641i	0.765966	−	0.765966i
$276$		0			0
$277$	−10.4668	+	10.4668i	−0.0377863	+	0.0377863i	−0.725747	−	0.687961i	$-0.758506\pi$
$277$	−10.4668	+	10.4668i	−0.0377863	+	0.0377863i	0.687961	+	0.725747i	$0.258506\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−56.0486			−0.199461			−0.0997306	−	0.995014i	$-0.531798\pi$
$281$	−56.0486			−0.199461			−0.0997306	+	0.995014i	$0.531798\pi$
$282$		0			0
$283$	−248.129	+	248.129i	−0.876782	+	0.876782i	−0.993200	−	0.116418i	$-0.962859\pi$
$283$	−248.129	+	248.129i	−0.876782	+	0.876782i	0.116418	+	0.993200i	$0.462859\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	14.1205i		−	0.0492005i
$288$		0			0
$289$	250.550			0.866955
$290$		0			0
$291$		0			0
$292$		0			0
$293$	243.305	+	243.305i	0.830392	+	0.830392i	0.987570	−	0.157178i	$-0.0502397\pi$
$293$	243.305	+	243.305i	0.830392	+	0.830392i	−0.157178	+	0.987570i	$0.550240\pi$
$294$		0			0
$295$		−	260.154i		−	0.881880i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−232.569	−	232.569i	−0.777824	−	0.777824i
$300$		0			0
$301$	4.14905	+	4.14905i	0.0137842	+	0.0137842i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	222.909i		−	0.730849i
$306$		0			0
$307$	−29.1965	−	29.1965i	−0.0951028	−	0.0951028i	0.657955	−	0.753057i	$-0.271422\pi$
$307$	−29.1965	−	29.1965i	−0.0951028	−	0.0951028i	−0.753057	+	0.657955i	$0.771422\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	5.31281			0.0170830			0.00854150	−	0.999964i	$-0.497281\pi$
$311$	5.31281			0.0170830			0.00854150	+	0.999964i	$0.497281\pi$
$312$		0			0
$313$		−	71.0570i		−	0.227019i	−0.993537	−	0.113510i	$-0.963791\pi$
$313$		−	71.0570i		−	0.227019i	0.993537	−	0.113510i	$-0.0362093\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	340.594	−	340.594i	1.07443	−	1.07443i	0.0774326	−	0.996998i	$-0.475328\pi$
$317$	340.594	−	340.594i	1.07443	−	1.07443i	0.996998	−	0.0774326i	$-0.0246723\pi$
$318$		0			0
$319$	−364.975			−1.14412
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−27.4916	+	27.4916i	−0.0851134	+	0.0851134i
$324$		0			0
$325$	147.562	−	147.562i	0.454037	−	0.454037i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−3.12863			−0.00950951
$330$		0			0
$331$	−283.647	+	283.647i	−0.856939	+	0.856939i	−0.990976	−	0.134038i	$-0.957206\pi$
$331$	−283.647	+	283.647i	−0.856939	+	0.856939i	0.134038	+	0.990976i	$0.457206\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		−	228.769i		−	0.682893i
$336$		0			0
$337$	−44.9447			−0.133367			−0.0666835	−	0.997774i	$-0.521242\pi$
$337$	−44.9447			−0.133367			−0.0666835	+	0.997774i	$0.521242\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	388.869	+	388.869i	1.14038	+	1.14038i
$342$		0			0
$343$			22.2015i			0.0647275i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	272.333	+	272.333i	0.784821	+	0.784821i	0.980640	−	0.195819i	$-0.0627365\pi$
$347$	272.333	+	272.333i	0.784821	+	0.784821i	−0.195819	+	0.980640i	$0.562737\pi$
$348$		0			0
$349$	−178.956	−	178.956i	−0.512768	−	0.512768i	0.402605	−	0.915374i	$-0.368105\pi$
$349$	−178.956	−	178.956i	−0.512768	−	0.512768i	−0.915374	+	0.402605i	$0.868105\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	698.328i		−	1.97827i	−0.147025	−	0.989133i	$-0.546970\pi$
$353$		−	698.328i		−	1.97827i	0.147025	−	0.989133i	$-0.453030\pi$
$354$		0			0
$355$	−60.0329	−	60.0329i	−0.169107	−	0.169107i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	364.766			1.01606			0.508030	−	0.861339i	$-0.330374\pi$
$359$	364.766			1.01606			0.508030	+	0.861339i	$0.330374\pi$
$360$		0			0
$361$		−	321.687i		−	0.891100i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	241.701	−	241.701i	0.662195	−	0.662195i
$366$		0			0
$367$	−632.894			−1.72451			−0.862254	−	0.506476i	$-0.830948\pi$
$367$	−632.894			−1.72451			−0.862254	+	0.506476i	$0.830948\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−8.59439	+	8.59439i	−0.0231655	+	0.0231655i
$372$		0			0
$373$	−33.3679	+	33.3679i	−0.0894582	+	0.0894582i	−0.750420	−	0.660962i	$-0.770149\pi$
$373$	−33.3679	+	33.3679i	−0.0894582	+	0.0894582i	0.660962	+	0.750420i	$0.270149\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−255.679			−0.678194
$378$		0			0
$379$	−439.149	+	439.149i	−1.15870	+	1.15870i	−0.173950	+	0.984755i	$0.555653\pi$
$379$	−439.149	+	439.149i	−1.15870	+	1.15870i	−0.984755	+	0.173950i	$0.944347\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	168.628i		−	0.440283i	−0.975468	−	0.220142i	$-0.929348\pi$
$383$		−	168.628i		−	0.440283i	0.975468	−	0.220142i	$-0.0706519\pi$
$384$		0			0
$385$	9.69083			0.0251710
$386$		0			0
$387$		0			0
$388$		0			0
$389$	54.4610	+	54.4610i	0.140003	+	0.140003i	0.773635	−	0.633632i	$-0.218437\pi$
$389$	54.4610	+	54.4610i	0.140003	+	0.140003i	−0.633632	+	0.773635i	$0.718437\pi$
$390$		0			0
$391$			177.739i			0.454576i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−244.347	−	244.347i	−0.618600	−	0.618600i
$396$		0			0
$397$	−446.859	−	446.859i	−1.12559	−	1.12559i	−0.990886	−	0.134704i	$-0.956992\pi$
$397$	−446.859	−	446.859i	−1.12559	−	1.12559i	−0.134704	−	0.990886i	$-0.543008\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			7.95350i			0.0198342i	0.999951	+	0.00991709i	$0.00315676\pi$
$401$			7.95350i			0.0198342i	−0.999951	+	0.00991709i	$0.996843\pi$
$402$		0			0
$403$	272.418	+	272.418i	0.675975	+	0.675975i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−1015.48			−2.49503
$408$		0			0
$409$		−	624.006i		−	1.52569i	−0.646583	−	0.762843i	$-0.723803\pi$
$409$		−	624.006i		−	1.52569i	0.646583	−	0.762843i	$-0.276197\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−15.9744	+	15.9744i	−0.0386790	+	0.0386790i
$414$		0			0
$415$	71.1637			0.171479
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−57.4026	+	57.4026i	−0.136999	+	0.136999i	−0.772281	−	0.635282i	$-0.780884\pi$
$419$	−57.4026	+	57.4026i	−0.136999	+	0.136999i	0.635282	+	0.772281i	$0.280884\pi$
$420$		0			0
$421$	4.45855	−	4.45855i	0.0105904	−	0.0105904i	−0.701792	−	0.712382i	$-0.747616\pi$
$421$	4.45855	−	4.45855i	0.0105904	−	0.0105904i	0.712382	+	0.701792i	$0.247616\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−112.773			−0.265348
$426$		0			0
$427$	−13.6874	+	13.6874i	−0.0320549	+	0.0320549i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	515.590i		−	1.19626i	−0.801397	−	0.598132i	$-0.795910\pi$
$431$		−	515.590i		−	1.19626i	0.801397	−	0.598132i	$-0.204090\pi$
$432$		0			0
$433$	301.416			0.696111			0.348056	−	0.937474i	$-0.386842\pi$
$433$	301.416			0.696111			0.348056	+	0.937474i	$0.386842\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	127.083	+	127.083i	0.290807	+	0.290807i
$438$		0			0
$439$		−	656.561i		−	1.49558i	−0.663933	−	0.747792i	$-0.731114\pi$
$439$		−	656.561i		−	1.49558i	0.663933	−	0.747792i	$-0.268886\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	229.799	+	229.799i	0.518733	+	0.518733i	0.917188	−	0.398455i	$-0.130454\pi$
$443$	229.799	+	229.799i	0.518733	+	0.518733i	−0.398455	+	0.917188i	$0.630454\pi$
$444$		0			0
$445$	168.402	+	168.402i	0.378431	+	0.378431i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			184.489i			0.410888i	0.978669	+	0.205444i	$0.0658639\pi$
$449$			184.489i			0.410888i	−0.978669	+	0.205444i	$0.934136\pi$
$450$		0			0
$451$	−721.526	−	721.526i	−1.59984	−	1.59984i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	6.78881			0.0149205
$456$		0			0
$457$		−	376.876i		−	0.824673i	−0.911032	−	0.412336i	$-0.864713\pi$
$457$		−	376.876i		−	0.824673i	0.911032	−	0.412336i	$-0.135287\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	28.5551	−	28.5551i	0.0619418	−	0.0619418i	−0.675457	−	0.737399i	$-0.736054\pi$
$461$	28.5551	−	28.5551i	0.0619418	−	0.0619418i	0.737399	+	0.675457i	$0.236054\pi$
$462$		0			0
$463$	144.828			0.312803			0.156401	−	0.987694i	$-0.450011\pi$
$463$	144.828			0.312803			0.156401	+	0.987694i	$0.450011\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	366.519	−	366.519i	0.784837	−	0.784837i	−0.195806	−	0.980643i	$-0.562732\pi$
$467$	366.519	−	366.519i	0.784837	−	0.784837i	0.980643	+	0.195806i	$0.0627323\pi$
$468$		0			0
$469$	−14.0473	+	14.0473i	−0.0299515	+	0.0299515i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	424.014			0.896435
$474$		0			0
$475$	−80.6321	+	80.6321i	−0.169752	+	0.169752i
$476$		0			0
$477$		0			0
$478$		0			0
$479$			119.680i			0.249853i	0.992166	+	0.124927i	$0.0398696\pi$
$479$			119.680i			0.249853i	−0.992166	+	0.124927i	$0.960130\pi$
$480$		0			0
$481$	−711.382			−1.47896
$482$		0			0
$483$		0			0
$484$		0			0
$485$	37.0631	+	37.0631i	0.0764188	+	0.0764188i
$486$		0			0
$487$			637.829i			1.30971i	0.755754	+	0.654855i	$0.227270\pi$
$487$			637.829i			1.30971i	−0.755754	+	0.654855i	$0.772730\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−130.249	−	130.249i	−0.265273	−	0.265273i	0.561919	−	0.827192i	$-0.310063\pi$
$491$	−130.249	−	130.249i	−0.265273	−	0.265273i	−0.827192	+	0.561919i	$0.810063\pi$
$492$		0			0
$493$	97.7004	+	97.7004i	0.198175	+	0.198175i
$494$		0			0
$495$		0			0
$496$		0			0
$497$			7.37248i			0.0148340i
$498$		0			0
$499$	7.28612	+	7.28612i	0.0146014	+	0.0146014i	0.714370	−	0.699768i	$-0.246713\pi$
$499$	7.28612	+	7.28612i	0.0146014	+	0.0146014i	−0.699768	+	0.714370i	$0.746713\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−311.009			−0.618307			−0.309154	−	0.951012i	$-0.600046\pi$
$503$	−311.009			−0.618307			−0.309154	+	0.951012i	$0.600046\pi$
$504$		0			0
$505$			109.890i			0.217604i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−431.114	+	431.114i	−0.846983	+	0.846983i	−0.989756	−	0.142773i	$-0.954398\pi$
$509$	−431.114	+	431.114i	−0.846983	+	0.846983i	0.142773	+	0.989756i	$0.454398\pi$
$510$		0			0
$511$	−29.6827			−0.0580874
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−204.884	+	204.884i	−0.397832	+	0.397832i
$516$		0			0
$517$	−159.866	+	159.866i	−0.309218	+	0.309218i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	241.849			0.464201			0.232101	−	0.972692i	$-0.425440\pi$
$521$	241.849			0.464201			0.232101	+	0.972692i	$0.425440\pi$
$522$		0			0
$523$	183.752	−	183.752i	0.351342	−	0.351342i	−0.509267	−	0.860609i	$-0.670083\pi$
$523$	183.752	−	183.752i	0.351342	−	0.351342i	0.860609	+	0.509267i	$0.170083\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	208.193i		−	0.395053i
$528$		0			0
$529$	292.615			0.553147
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−505.458	−	505.458i	−0.948326	−	0.948326i
$534$		0			0
$535$		−	48.5083i		−	0.0906698i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	566.926	+	566.926i	1.05181	+	1.05181i
$540$		0			0
$541$	186.010	+	186.010i	0.343826	+	0.343826i	0.857803	−	0.513978i	$-0.171829\pi$
$541$	186.010	+	186.010i	0.343826	+	0.343826i	−0.513978	+	0.857803i	$0.671829\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$			14.3923i			0.0264079i
$546$		0			0
$547$	244.192	+	244.192i	0.446421	+	0.446421i	0.894163	−	0.447742i	$-0.147772\pi$
$547$	244.192	+	244.192i	0.446421	+	0.446421i	−0.447742	+	0.894163i	$0.647772\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	139.711			0.253558
$552$		0			0
$553$			30.0076i			0.0542632i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	367.416	−	367.416i	0.659634	−	0.659634i	−0.295659	−	0.955293i	$-0.595539\pi$
$557$	367.416	−	367.416i	0.659634	−	0.659634i	0.955293	+	0.295659i	$0.0955393\pi$
$558$		0			0
$559$	297.038			0.531374
$560$		0			0
$561$		0			0
$562$		0			0
$563$	418.290	−	418.290i	0.742967	−	0.742967i	−0.230181	−	0.973148i	$-0.573932\pi$
$563$	418.290	−	418.290i	0.742967	−	0.742967i	0.973148	+	0.230181i	$0.0739318\pi$
$564$		0			0
$565$	217.213	−	217.213i	0.384448	−	0.384448i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	439.363			0.772166			0.386083	−	0.922464i	$-0.373828\pi$
$569$	439.363			0.772166			0.386083	+	0.922464i	$0.373828\pi$
$570$		0			0
$571$	−81.8476	+	81.8476i	−0.143341	+	0.143341i	−0.775136	−	0.631795i	$-0.782318\pi$
$571$	−81.8476	+	81.8476i	−0.143341	+	0.143341i	0.631795	+	0.775136i	$0.282318\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			521.303i			0.906614i
$576$		0			0
$577$	303.497			0.525992			0.262996	−	0.964797i	$-0.415289\pi$
$577$	303.497			0.525992			0.262996	+	0.964797i	$0.415289\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−4.36971	−	4.36971i	−0.00752101	−	0.00752101i
$582$		0			0
$583$			878.306i			1.50653i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−670.665	−	670.665i	−1.14253	−	1.14253i	−0.987986	−	0.154543i	$-0.950609\pi$
$587$	−670.665	−	670.665i	−1.14253	−	1.14253i	−0.154543	−	0.987986i	$-0.549391\pi$
$588$		0			0
$589$	−148.857	−	148.857i	−0.252728	−	0.252728i
$590$		0			0
$591$		0			0
$592$		0			0
$593$			390.656i			0.658779i	0.944194	+	0.329390i	$0.106843\pi$
$593$			390.656i			0.658779i	−0.944194	+	0.329390i	$0.893157\pi$
$594$		0			0
$595$	−2.59414	−	2.59414i	−0.00435991	−	0.00435991i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	788.713			1.31672			0.658358	−	0.752705i	$-0.271251\pi$
$599$	788.713			1.31672			0.658358	+	0.752705i	$0.271251\pi$
$600$		0			0
$601$		−	30.8170i		−	0.0512762i	−0.999671	−	0.0256381i	$-0.991838\pi$
$601$		−	30.8170i		−	0.0512762i	0.999671	−	0.0256381i	$-0.00816176\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	271.849	−	271.849i	0.449337	−	0.449337i
$606$		0			0
$607$	−607.774			−1.00128			−0.500638	−	0.865657i	$-0.666901\pi$
$607$	−607.774			−1.00128			−0.500638	+	0.865657i	$0.666901\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−111.992	+	111.992i	−0.183293	+	0.183293i
$612$		0			0
$613$	−507.089	+	507.089i	−0.827225	+	0.827225i	−0.987132	−	0.159907i	$-0.948881\pi$
$613$	−507.089	+	507.089i	−0.827225	+	0.827225i	0.159907	+	0.987132i	$0.448881\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−202.840			−0.328752			−0.164376	−	0.986398i	$-0.552561\pi$
$617$	−202.840			−0.328752			−0.164376	+	0.986398i	$0.552561\pi$
$618$		0			0
$619$	699.181	−	699.181i	1.12953	−	1.12953i	0.139280	−	0.990253i	$-0.455521\pi$
$619$	699.181	−	699.181i	1.12953	−	1.12953i	0.990253	−	0.139280i	$-0.0444787\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		−	20.6809i		−	0.0331957i
$624$		0			0
$625$	−160.430			−0.256687
$626$		0			0
$627$		0			0
$628$		0			0
$629$	271.834	+	271.834i	0.432168	+	0.432168i
$630$		0			0
$631$		−	269.488i		−	0.427081i	−0.976934	−	0.213540i	$-0.931501\pi$
$631$		−	269.488i		−	0.427081i	0.976934	−	0.213540i	$-0.0684995\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−79.5786	−	79.5786i	−0.125321	−	0.125321i
$636$		0			0
$637$	397.154	+	397.154i	0.623475	+	0.623475i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	638.869i		−	0.996676i	−0.866983	−	0.498338i	$-0.833944\pi$
$641$		−	638.869i		−	0.996676i	0.866983	−	0.498338i	$-0.166056\pi$
$642$		0			0
$643$	161.282	+	161.282i	0.250827	+	0.250827i	0.821310	−	0.570483i	$-0.193244\pi$
$643$	161.282	+	161.282i	0.250827	+	0.250827i	−0.570483	+	0.821310i	$0.693244\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	921.712			1.42459			0.712297	−	0.701878i	$-0.247655\pi$
$647$	921.712			1.42459			0.712297	+	0.701878i	$0.247655\pi$
$648$		0			0
$649$			1632.51i			2.51543i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−661.262	+	661.262i	−1.01265	+	1.01265i	−0.0127331	+	0.999919i	$0.504053\pi$
$653$	−661.262	+	661.262i	−1.01265	+	1.01265i	−0.999919	+	0.0127331i	$0.995947\pi$
$654$		0			0
$655$	360.234			0.549975
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−702.485	+	702.485i	−1.06599	+	1.06599i	−0.0683229	+	0.997663i	$0.521765\pi$
$659$	−702.485	+	702.485i	−1.06599	+	1.06599i	−0.997663	+	0.0683229i	$0.978235\pi$
$660$		0			0
$661$	451.292	−	451.292i	0.682741	−	0.682741i	−0.277876	−	0.960617i	$-0.589630\pi$
$661$	451.292	−	451.292i	0.682741	−	0.682741i	0.960617	+	0.277876i	$0.0896304\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−3.70960			−0.00557835
$666$		0			0
$667$	451.629	−	451.629i	0.677104	−	0.677104i
$668$		0			0
$669$		0			0
$670$		0			0
$671$			1398.79i			2.08464i
$672$		0			0
$673$	−626.462			−0.930849			−0.465425	−	0.885088i	$-0.654098\pi$
$673$	−626.462			−0.930849			−0.465425	+	0.885088i	$0.654098\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−652.167	−	652.167i	−0.963318	−	0.963318i	0.0360323	−	0.999351i	$-0.488528\pi$
$677$	−652.167	−	652.167i	−0.963318	−	0.963318i	−0.999351	+	0.0360323i	$0.988528\pi$
$678$		0			0
$679$		−	4.55162i		−	0.00670342i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−331.471	−	331.471i	−0.485316	−	0.485316i	0.421509	−	0.906824i	$-0.361501\pi$
$683$	−331.471	−	331.471i	−0.485316	−	0.485316i	−0.906824	+	0.421509i	$0.861501\pi$
$684$		0			0
$685$	90.9433	+	90.9433i	0.132764	+	0.132764i
$686$		0			0
$687$		0			0
$688$		0			0
$689$			615.288i			0.893016i
$690$		0			0
$691$	648.386	+	648.386i	0.938331	+	0.938331i	0.998206	−	0.0598752i	$-0.0190703\pi$
$691$	648.386	+	648.386i	0.938331	+	0.938331i	−0.0598752	+	0.998206i	$0.519070\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−638.258			−0.918357
$696$		0			0
$697$			386.292i			0.554221i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−557.872	+	557.872i	−0.795823	+	0.795823i	−0.982434	−	0.186611i	$-0.940250\pi$
$701$	−557.872	+	557.872i	−0.795823	+	0.795823i	0.186611	+	0.982434i	$0.440250\pi$
$702$		0			0
$703$	388.720			0.552944
$704$		0			0
$705$		0			0
$706$		0			0
$707$	6.74765	−	6.74765i	0.00954407	−	0.00954407i
$708$		0			0
$709$	409.754	−	409.754i	0.577932	−	0.577932i	−0.356401	−	0.934333i	$-0.615996\pi$
$709$	409.754	−	409.754i	0.577932	−	0.577932i	0.934333	+	0.356401i	$0.115996\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−962.391			−1.34978
$714$		0			0
$715$	346.892	−	346.892i	0.485164	−	0.485164i
$716$		0			0
$717$		0			0
$718$		0			0
$719$		−	700.847i		−	0.974752i	−0.873192	−	0.487376i	$-0.837954\pi$
$719$		−	700.847i		−	0.974752i	0.873192	−	0.487376i	$-0.162046\pi$
$720$		0			0
$721$	25.1612			0.0348977
$722$		0			0
$723$		0			0
$724$		0			0
$725$	286.552	+	286.552i	0.395244	+	0.395244i
$726$		0			0
$727$		−	1106.88i		−	1.52253i	−0.648439	−	0.761267i	$-0.724578\pi$
$727$		−	1106.88i		−	1.52253i	0.648439	−	0.761267i	$-0.275422\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−113.504	−	113.504i	−0.155273	−	0.155273i
$732$		0			0
$733$	−185.291	−	185.291i	−0.252784	−	0.252784i	0.569327	−	0.822111i	$-0.307204\pi$
$733$	−185.291	−	185.291i	−0.252784	−	0.252784i	−0.822111	+	0.569327i	$0.807204\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			1435.56i			1.94785i
$738$		0			0
$739$	−764.710	−	764.710i	−1.03479	−	1.03479i	−0.999373	−	0.0354181i	$-0.988724\pi$
$739$	−764.710	−	764.710i	−1.03479	−	1.03479i	−0.0354181	−	0.999373i	$-0.511276\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	860.679			1.15838			0.579192	−	0.815191i	$-0.303368\pi$
$743$	860.679			1.15838			0.579192	+	0.815191i	$0.303368\pi$
$744$		0			0
$745$			474.480i			0.636886i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−2.97859	+	2.97859i	−0.00397675	+	0.00397675i
$750$		0			0
$751$	1060.33			1.41189			0.705943	−	0.708268i	$-0.250523\pi$
$751$	1060.33			1.41189			0.705943	+	0.708268i	$0.250523\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	166.025	−	166.025i	0.219900	−	0.219900i
$756$		0			0
$757$	−85.8730	+	85.8730i	−0.113439	+	0.113439i	−0.761548	−	0.648109i	$-0.775560\pi$
$757$	−85.8730	+	85.8730i	−0.113439	+	0.113439i	0.648109	+	0.761548i	$0.275560\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−587.026			−0.771388			−0.385694	−	0.922627i	$-0.626038\pi$
$761$	−587.026			−0.771388			−0.385694	+	0.922627i	$0.626038\pi$
$762$		0			0
$763$	0.883741	−	0.883741i	0.00115825	−	0.00115825i
$764$		0			0
$765$		0			0
$766$		0			0
$767$			1143.64i			1.49105i
$768$		0			0
$769$	1048.36			1.36328			0.681642	−	0.731686i	$-0.261266\pi$
$769$	1048.36			1.36328			0.681642	+	0.731686i	$0.261266\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−1027.31	−	1027.31i	−1.32899	−	1.32899i	−0.906249	−	0.422744i	$-0.861067\pi$
$773$	−1027.31	−	1027.31i	−1.32899	−	1.32899i	−0.422744	−	0.906249i	$-0.638933\pi$
$774$		0			0
$775$		−	610.623i		−	0.787901i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	276.197	+	276.197i	0.354553	+	0.354553i
$780$		0			0
$781$	376.716	+	376.716i	0.482351	+	0.482351i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		−	351.573i		−	0.447864i
$786$		0			0
$787$	998.571	+	998.571i	1.26883	+	1.26883i	0.946692	+	0.322141i	$0.104402\pi$
$787$	998.571	+	998.571i	1.26883	+	1.26883i	0.322141	+	0.946692i	$0.395598\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−26.6754			−0.0337236
$792$		0			0
$793$			979.908i			1.23570i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	261.834	−	261.834i	0.328524	−	0.328524i	−0.523501	−	0.852025i	$-0.675374\pi$
$797$	261.834	−	261.834i	0.328524	−	0.328524i	0.852025	+	0.523501i	$0.175374\pi$
$798$		0			0
$799$	85.5890			0.107120
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−1516.71	+	1516.71i	−1.88881	+	1.88881i
$804$		0			0
$805$	−11.9917	+	11.9917i	−0.0148965	+	0.0148965i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	652.165			0.806137			0.403069	−	0.915170i	$-0.367944\pi$
$809$	652.165			0.806137			0.403069	+	0.915170i	$0.367944\pi$
$810$		0			0
$811$	−150.173	+	150.173i	−0.185171	+	0.185171i	−0.793605	−	0.608434i	$-0.791798\pi$
$811$	−150.173	+	150.173i	−0.185171	+	0.185171i	0.608434	+	0.793605i	$0.291798\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$			311.169i			0.381803i
$816$		0			0
$817$	−162.310			−0.198666
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−979.062	−	979.062i	−1.19252	−	1.19252i	−0.976356	−	0.216167i	$-0.930644\pi$
$821$	−979.062	−	979.062i	−1.19252	−	1.19252i	−0.216167	−	0.976356i	$-0.569356\pi$
$822$		0			0
$823$			453.787i			0.551381i	0.961246	+	0.275691i	$0.0889065\pi$
$823$			453.787i			0.551381i	−0.961246	+	0.275691i	$0.911093\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	1162.48	+	1162.48i	1.40566	+	1.40566i	0.780488	+	0.625171i	$0.214971\pi$
$827$	1162.48	+	1162.48i	1.40566	+	1.40566i	0.625171	+	0.780488i	$0.285029\pi$
$828$		0			0
$829$	−126.113	−	126.113i	−0.152127	−	0.152127i	0.626940	−	0.779067i	$-0.284307\pi$
$829$	−126.113	−	126.113i	−0.152127	−	0.152127i	−0.779067	+	0.626940i	$0.784307\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		−	303.521i		−	0.364371i
$834$		0			0
$835$	−356.503	−	356.503i	−0.426950	−	0.426950i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	82.9161			0.0988273			0.0494137	−	0.998778i	$-0.484265\pi$
$839$	82.9161			0.0988273			0.0494137	+	0.998778i	$0.484265\pi$
$840$		0			0
$841$			344.494i			0.409624i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−68.9113	+	68.9113i	−0.0815519	+	0.0815519i
$846$		0			0
$847$	−33.3851			−0.0394157
$848$		0			0
$849$		0			0
$850$		0			0
$851$	1256.58	−	1256.58i	1.47659	−	1.47659i
$852$		0			0
$853$	−558.406	+	558.406i	−0.654638	+	0.654638i	−0.954106	−	0.299468i	$-0.903191\pi$
$853$	−558.406	+	558.406i	−0.654638	+	0.654638i	0.299468	+	0.954106i	$0.403191\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−429.366			−0.501011			−0.250505	−	0.968115i	$-0.580597\pi$
$857$	−429.366			−0.501011			−0.250505	+	0.968115i	$0.580597\pi$
$858$		0			0
$859$	−63.4781	+	63.4781i	−0.0738977	+	0.0738977i	−0.743090	−	0.669192i	$-0.766640\pi$
$859$	−63.4781	+	63.4781i	−0.0738977	+	0.0738977i	0.669192	+	0.743090i	$0.266640\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			176.985i			0.205081i	0.994729	+	0.102540i	$0.0326971\pi$
$863$			176.985i			0.205081i	−0.994729	+	0.102540i	$0.967303\pi$
$864$		0			0
$865$	221.889			0.256519
$866$		0			0
$867$		0			0
$868$		0			0
$869$	1533.32	+	1533.32i	1.76446	+	1.76446i
$870$		0			0
$871$			1005.67i			1.15461i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−18.0674	−	18.0674i	−0.0206485	−	0.0206485i
$876$		0			0
$877$	852.573	+	852.573i	0.972147	+	0.972147i	0.999622	−	0.0274758i	$-0.00874693\pi$
$877$	852.573	+	852.573i	0.972147	+	0.972147i	−0.0274758	+	0.999622i	$0.508747\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	756.244i		−	0.858393i	−0.903211	−	0.429196i	$-0.858797\pi$
$881$		−	756.244i		−	0.858393i	0.903211	−	0.429196i	$-0.141203\pi$
$882$		0			0
$883$	617.889	+	617.889i	0.699761	+	0.699761i	0.964359	−	0.264598i	$-0.0852393\pi$
$883$	617.889	+	617.889i	0.699761	+	0.699761i	−0.264598	+	0.964359i	$0.585239\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−311.194			−0.350838			−0.175419	−	0.984494i	$-0.556128\pi$
$887$	−311.194			−0.350838			−0.175419	+	0.984494i	$0.556128\pi$
$888$		0			0
$889$			9.77283i			0.0109931i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	61.1957	−	61.1957i	0.0685282	−	0.0685282i
$894$		0			0
$895$	−382.723			−0.427624
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−529.011	+	529.011i	−0.588444	+	0.588444i
$900$		0			0
$901$	235.114	−	235.114i	0.260948	−	0.260948i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−11.3706			−0.0125641
$906$		0			0
$907$	593.896	−	593.896i	0.654792	−	0.654792i	−0.299351	−	0.954143i	$-0.596770\pi$
$907$	593.896	−	593.896i	0.654792	−	0.654792i	0.954143	+	0.299351i	$0.0967702\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	499.219i		−	0.547991i	−0.961731	−	0.273995i	$-0.911655\pi$
$911$		−	499.219i		−	0.547991i	0.961731	−	0.273995i	$-0.0883453\pi$
$912$		0			0
$913$	−446.564			−0.489117
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−22.1197	−	22.1197i	−0.0241218	−	0.0241218i
$918$		0			0
$919$		−	405.667i		−	0.441422i	−0.975339	−	0.220711i	$-0.929162\pi$
$919$		−	405.667i		−	0.441422i	0.975339	−	0.220711i	$-0.0708378\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	263.905	+	263.905i	0.285921	+	0.285921i
$924$		0			0
$925$	797.280	+	797.280i	0.861925	+	0.861925i
$926$		0			0
$927$		0			0
$928$		0			0
$929$		−	640.107i		−	0.689028i	−0.938781	−	0.344514i	$-0.888044\pi$
$929$		−	640.107i		−	0.689028i	0.938781	−	0.344514i	$-0.111956\pi$
$930$		0			0
$931$	−217.016	−	217.016i	−0.233100	−	0.233100i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−265.109			−0.283539
$936$		0			0
$937$		−	1648.87i		−	1.75973i	−0.475224	−	0.879865i	$-0.657633\pi$
$937$		−	1648.87i		−	1.75973i	0.475224	−	0.879865i	$-0.342367\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−783.146	+	783.146i	−0.832248	+	0.832248i	−0.987824	−	0.155576i	$-0.950277\pi$
$941$	−783.146	+	783.146i	−0.832248	+	0.832248i	0.155576	+	0.987824i	$0.450277\pi$
$942$		0			0
$943$	1785.67			1.89360
$944$		0			0
$945$		0			0
$946$		0			0
$947$	510.366	−	510.366i	0.538929	−	0.538929i	−0.384286	−	0.923214i	$-0.625552\pi$
$947$	510.366	−	510.366i	0.538929	−	0.538929i	0.923214	+	0.384286i	$0.125552\pi$
$948$		0			0
$949$	−1062.52	+	1062.52i	−1.11962	+	1.11962i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−1381.65			−1.44979			−0.724897	−	0.688857i	$-0.758113\pi$
$953$	−1381.65			−1.44979			−0.724897	+	0.688857i	$0.758113\pi$
$954$		0			0
$955$	434.752	−	434.752i	0.455238	−	0.455238i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		−	11.1685i		−	0.0116460i
$960$		0			0
$961$	166.287			0.173035
$962$		0			0
$963$		0			0
$964$		0			0
$965$	17.3227	+	17.3227i	0.0179510	+	0.0179510i
$966$		0			0
$967$		−	783.542i		−	0.810281i	−0.914254	−	0.405141i	$-0.867223\pi$
$967$		−	783.542i		−	0.810281i	0.914254	−	0.405141i	$-0.132777\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−73.9020	−	73.9020i	−0.0761091	−	0.0761091i	0.668027	−	0.744137i	$-0.267139\pi$
$971$	−73.9020	−	73.9020i	−0.0761091	−	0.0761091i	−0.744137	+	0.668027i	$0.767139\pi$
$972$		0			0
$973$	39.1914	+	39.1914i	0.0402789	+	0.0402789i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			584.085i			0.597835i	0.954279	+	0.298918i	$0.0966256\pi$
$977$			584.085i			0.597835i	−0.954279	+	0.298918i	$0.903374\pi$
$978$		0			0
$979$	−1056.75	−	1056.75i	−1.07942	−	1.07942i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−1608.50			−1.63632			−0.818158	−	0.574993i	$-0.805005\pi$
$983$	−1608.50			−1.63632			−0.818158	+	0.574993i	$0.805005\pi$
$984$		0			0
$985$			280.040i			0.284305i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−524.685	+	524.685i	−0.530520	+	0.530520i
$990$		0			0
$991$	477.041			0.481374			0.240687	−	0.970603i	$-0.422627\pi$
$991$	477.041			0.481374			0.240687	+	0.970603i	$0.422627\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	582.600	−	582.600i	0.585528	−	0.585528i
$996$		0			0
$997$	−783.852	+	783.852i	−0.786210	+	0.786210i	−0.980871	−	0.194660i	$-0.937640\pi$
$997$	−783.852	+	783.852i	−0.786210	+	0.786210i	0.194660	+	0.980871i	$0.437640\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1152.3.j.b.161.6		32
3.2	odd	2	inner	1152.3.j.b.161.11		32
4.3	odd	2		1152.3.j.a.161.6		32
8.3	odd	2		144.3.j.a.125.13	yes	32
8.5	even	2		576.3.j.a.17.11		32
12.11	even	2		1152.3.j.a.161.11		32
16.3	odd	4		144.3.j.a.53.4	✓	32
16.5	even	4	inner	1152.3.j.b.737.11		32
16.11	odd	4		1152.3.j.a.737.11		32
16.13	even	4		576.3.j.a.305.6		32
24.5	odd	2		576.3.j.a.17.6		32
24.11	even	2		144.3.j.a.125.4	yes	32
48.5	odd	4	inner	1152.3.j.b.737.6		32
48.11	even	4		1152.3.j.a.737.6		32
48.29	odd	4		576.3.j.a.305.11		32
48.35	even	4		144.3.j.a.53.13	yes	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
144.3.j.a.53.4	✓	32	16.3	odd	4
144.3.j.a.53.13	yes	32	48.35	even	4
144.3.j.a.125.4	yes	32	24.11	even	2
144.3.j.a.125.13	yes	32	8.3	odd	2
576.3.j.a.17.6		32	24.5	odd	2
576.3.j.a.17.11		32	8.5	even	2
576.3.j.a.305.6		32	16.13	even	4
576.3.j.a.305.11		32	48.29	odd	4
1152.3.j.a.161.6		32	4.3	odd	2
1152.3.j.a.161.11		32	12.11	even	2
1152.3.j.a.737.6		32	48.11	even	4
1152.3.j.a.737.11		32	16.11	odd	4
1152.3.j.b.161.6		32	1.1	even	1	trivial
1152.3.j.b.161.11		32	3.2	odd	2	inner
1152.3.j.b.737.6		32	48.5	odd	4	inner
1152.3.j.b.737.11		32	16.5	even	4	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1152 = 2^{7} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1152.j (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.84570 - 1.84570i) q^{5} +0.226665i q^{7} +O(q^{10})\)	\(q+(-1.84570 - 1.84570i) q^{5} +0.226665i q^{7} +(11.5821 + 11.5821i) q^{11} +(8.11369 + 8.11369i) q^{13} -6.20082i q^{17} +(-4.43355 - 4.43355i) q^{19} -28.6638 q^{23} -18.1868i q^{25} +(-15.7561 + 15.7561i) q^{29} +33.5751 q^{31} +(0.418355 - 0.418355i) q^{35} +(-43.8384 + 43.8384i) q^{37} -62.2969 q^{41} +(18.3048 - 18.3048i) q^{43} +13.8029i q^{47} +48.9486 q^{49} +(37.9167 + 37.9167i) q^{53} -42.7539i q^{55} +(70.4759 + 70.4759i) q^{59} +(60.3861 + 60.3861i) q^{61} -29.9508i q^{65} +(61.9736 + 61.9736i) q^{67} +32.5259 q^{71} +130.954i q^{73} +(-2.62525 + 2.62525i) q^{77} +132.387 q^{79} +(-19.2783 + 19.2783i) q^{83} +(-11.4448 + 11.4448i) q^{85} -91.2401 q^{89} +(-1.83909 + 1.83909i) q^{91} +16.3660i q^{95} -20.0808 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q+O(q^{10}) \) 32 * q	\( 32 q + 64 q^{19} + 128 q^{43} - 224 q^{49} - 64 q^{61} - 64 q^{67} + 512 q^{79} - 320 q^{85} - 192 q^{91}+O(q^{100}) \) 32 * q + 64 * q^19 + 128 * q^43 - 224 * q^49 - 64 * q^61 - 64 * q^67 + 512 * q^79 - 320 * q^85 - 192 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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