LMFDB - Embedded newform 756.3.bk.f.485.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [756,3,Mod(53,756)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(756, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 3, 4])) N = Newforms(chi, 3, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("756.53"); S:= CuspForms(chi, 3); N := Newforms(S);

Level:	$ N $	$=$	$ 756 = 2^{2} \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	756.bk (of order $6$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [12,0,0,0,0,0,-2,0,0,0,0,0,-32] 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:	$20.5995079856$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$6$ over $\Q(\zeta_{6})$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 52x^{10} + 846x^{8} + 5348x^{6} + 11241x^{4} + 2656x^{2} + 64 $ x^12 + 52x^10 + 846x^8 + 5348x^6 + 11241x^4 + 2656*x^2 + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{25}]$
Coefficient ring index:	$ 2^{6}\cdot 3^{7} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			485.2
Root			$-1.88606i$ of defining polynomial
Character	$\chi$	$=$	756.485
Dual form			756.3.bk.f.53.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-5.08547 - 2.93610i) q^{5} +(6.70780 + 2.00135i) q^{7} +(-7.89428 + 4.55776i) q^{11} +8.17424 q^{13} +(-14.0290 + 8.09965i) q^{17} +(2.46644 - 4.27201i) q^{19} +(32.0942 + 18.5296i) q^{23} +(4.74136 + 8.21227i) q^{25} -35.8476i q^{29} +(13.0363 + 22.5795i) q^{31} +(-28.2362 - 29.8726i) q^{35} +(3.60441 - 6.24302i) q^{37} +49.0093i q^{41} +32.3139 q^{43} +(26.4916 + 15.2950i) q^{47} +(40.9892 + 26.8493i) q^{49} +(67.1604 - 38.7751i) q^{53} +53.5282 q^{55} +(61.9118 - 35.7448i) q^{59} +(20.9021 - 36.2034i) q^{61} +(-41.5699 - 24.0004i) q^{65} +(40.1061 + 69.4658i) q^{67} +42.9314i q^{71} +(48.1288 + 83.3616i) q^{73} +(-62.0749 + 14.7733i) q^{77} +(54.1597 - 93.8073i) q^{79} +96.3783i q^{83} +95.1255 q^{85} +(-103.469 - 59.7379i) q^{89} +(54.8312 + 16.3595i) q^{91} +(-25.0861 + 14.4834i) q^{95} -1.06915 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 2 q^{7} - 32 q^{13} - 18 q^{19} + 22 q^{25} - 26 q^{31} + 48 q^{37} - 12 q^{43} + 54 q^{49} - 208 q^{55} - 122 q^{61} + 164 q^{67} - 130 q^{73} + 220 q^{79} - 280 q^{85} + 328 q^{91} - 292 q^{97}+O(q^{100}) $ 12 * q - 2 * q^7 - 32 * q^13 - 18 * q^19 + 22 * q^25 - 26 * q^31 + 48 * q^37 - 12 * q^43 + 54 * q^49 - 208 * q^55 - 122 * q^61 + 164 * q^67 - 130 * q^73 + 220 * q^79 - 280 * q^85 + 328 * q^91 - 292 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/756\mathbb{Z}\right)^\times$.

$n$	$29$	$325$	$379$
$\chi(n)$	$-1$	$e\left(\frac{1}{3}\right)$	$1$

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
$5$	−5.08547	−	2.93610i	−1.01709	−	0.587220i	−0.103834	−	0.994595i	$-0.533111\pi$
$5$	−5.08547	−	2.93610i	−1.01709	−	0.587220i	−0.913261	+	0.407375i	$0.866444\pi$
$6$		0			0
$7$	6.70780	+	2.00135i	0.958257	+	0.285907i
$7$	6.70780	+	2.00135i	0.958257	+	0.285907i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−7.89428	+	4.55776i	−0.717662	+	0.414342i	−0.813891	−	0.581017i	$-0.802655\pi$
$11$	−7.89428	+	4.55776i	−0.717662	+	0.414342i	0.0962298	+	0.995359i	$0.469322\pi$
$12$		0			0
$13$	8.17424			0.628788			0.314394	−	0.949293i	$-0.398199\pi$
$13$	8.17424			0.628788			0.314394	+	0.949293i	$0.398199\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−14.0290	+	8.09965i	−0.825236	+	0.476450i	−0.852219	−	0.523186i	$-0.824743\pi$
$17$	−14.0290	+	8.09965i	−0.825236	+	0.476450i	0.0269829	+	0.999636i	$0.491410\pi$
$18$		0			0
$19$	2.46644	−	4.27201i	0.129813	−	0.224842i	−0.793791	−	0.608190i	$-0.791896\pi$
$19$	2.46644	−	4.27201i	0.129813	−	0.224842i	0.923604	+	0.383348i	$0.125229\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	32.0942	+	18.5296i	1.39540	+	0.805635i	0.993907	−	0.110226i	$-0.0351575\pi$
$23$	32.0942	+	18.5296i	1.39540	+	0.805635i	0.401495	+	0.915861i	$0.368491\pi$
$24$		0			0
$25$	4.74136	+	8.21227i	0.189654	+	0.328491i
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	35.8476i		−	1.23613i	−0.786129	−	0.618063i	$-0.787918\pi$
$29$		−	35.8476i		−	1.23613i	0.786129	−	0.618063i	$-0.212082\pi$
$30$		0			0
$31$	13.0363	+	22.5795i	0.420525	+	0.728371i	0.995991	−	0.0894550i	$-0.0285125\pi$
$31$	13.0363	+	22.5795i	0.420525	+	0.728371i	−0.575466	+	0.817826i	$0.695179\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−28.2362	−	29.8726i	−0.806748	−	0.853503i
$36$		0			0
$37$	3.60441	−	6.24302i	0.0974165	−	0.168730i	−0.813198	−	0.581987i	$-0.802275\pi$
$37$	3.60441	−	6.24302i	0.0974165	−	0.168730i	0.910615	+	0.413257i	$0.135609\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			49.0093i			1.19535i	0.801739	+	0.597675i	$0.203909\pi$
$41$			49.0093i			1.19535i	−0.801739	+	0.597675i	$0.796091\pi$
$42$		0			0
$43$	32.3139			0.751486			0.375743	−	0.926724i	$-0.377387\pi$
$43$	32.3139			0.751486			0.375743	+	0.926724i	$0.377387\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	26.4916	+	15.2950i	0.563652	+	0.325425i	0.754610	−	0.656174i	$-0.227826\pi$
$47$	26.4916	+	15.2950i	0.563652	+	0.325425i	−0.190958	+	0.981598i	$0.561159\pi$
$48$		0			0
$49$	40.9892	+	26.8493i	0.836514	+	0.547946i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	67.1604	−	38.7751i	1.26718	−	0.731605i	0.292724	−	0.956197i	$-0.405438\pi$
$53$	67.1604	−	38.7751i	1.26718	−	0.731605i	0.974453	+	0.224592i	$0.0721048\pi$
$54$		0			0
$55$	53.5282			0.973240
$56$		0			0
$57$		0			0
$58$		0			0
$59$	61.9118	−	35.7448i	1.04935	−	0.605844i	0.126884	−	0.991918i	$-0.459502\pi$
$59$	61.9118	−	35.7448i	1.04935	−	0.605844i	0.922468	+	0.386074i	$0.126169\pi$
$60$		0			0
$61$	20.9021	−	36.2034i	0.342657	−	0.593499i	−0.642269	−	0.766480i	$-0.722007\pi$
$61$	20.9021	−	36.2034i	0.342657	−	0.593499i	0.984925	+	0.172981i	$0.0553399\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−41.5699	−	24.0004i	−0.639537	−	0.369237i
$66$		0			0
$67$	40.1061	+	69.4658i	0.598599	+	1.03680i	0.993028	+	0.117877i	$0.0376089\pi$
$67$	40.1061	+	69.4658i	0.598599	+	1.03680i	−0.394429	+	0.918926i	$0.629058\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			42.9314i			0.604668i	0.953202	+	0.302334i	$0.0977658\pi$
$71$			42.9314i			0.604668i	−0.953202	+	0.302334i	$0.902234\pi$
$72$		0			0
$73$	48.1288	+	83.3616i	0.659299	+	1.14194i	0.980797	+	0.195030i	$0.0624803\pi$
$73$	48.1288	+	83.3616i	0.659299	+	1.14194i	−0.321498	+	0.946910i	$0.604186\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−62.0749	+	14.7733i	−0.806168	+	0.191862i
$78$		0			0
$79$	54.1597	−	93.8073i	0.685566	−	1.18743i	−0.287693	−	0.957723i	$-0.592888\pi$
$79$	54.1597	−	93.8073i	0.685566	−	1.18743i	0.973259	−	0.229712i	$-0.0737784\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			96.3783i			1.16118i	0.814194	+	0.580592i	$0.197179\pi$
$83$			96.3783i			1.16118i	−0.814194	+	0.580592i	$0.802821\pi$
$84$		0			0
$85$	95.1255			1.11912
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−103.469	−	59.7379i	−1.16257	−	0.671212i	−0.210654	−	0.977561i	$-0.567559\pi$
$89$	−103.469	−	59.7379i	−1.16257	−	0.671212i	−0.951919	+	0.306349i	$0.900893\pi$
$90$		0			0
$91$	54.8312	+	16.3595i	0.602541	+	0.179775i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−25.0861	+	14.4834i	−0.264064	+	0.152457i
$96$		0			0
$97$	−1.06915			−0.0110221			−0.00551107	−	0.999985i	$-0.501754\pi$
$97$	−1.06915			−0.0110221			−0.00551107	+	0.999985i	$0.501754\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	30.4978	−	17.6079i	0.301959	−	0.174336i	−0.341364	−	0.939931i	$-0.610889\pi$
$101$	30.4978	−	17.6079i	0.301959	−	0.174336i	0.643322	+	0.765595i	$0.277555\pi$
$102$		0			0
$103$	−22.9854	+	39.8119i	−0.223160	+	0.386524i	−0.955766	−	0.294129i	$-0.904970\pi$
$103$	−22.9854	+	39.8119i	−0.223160	+	0.386524i	0.732606	+	0.680653i	$0.238304\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	54.3738	+	31.3927i	0.508167	+	0.293390i	0.732080	−	0.681219i	$-0.238550\pi$
$107$	54.3738	+	31.3927i	0.508167	+	0.293390i	−0.223913	+	0.974609i	$0.571883\pi$
$108$		0			0
$109$	88.3231	+	152.980i	0.810304	+	1.40349i	0.912652	+	0.408739i	$0.134031\pi$
$109$	88.3231	+	152.980i	0.810304	+	1.40349i	−0.102348	+	0.994749i	$0.532635\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	69.0317i		−	0.610900i	−0.952208	−	0.305450i	$-0.901193\pi$
$113$		−	69.0317i		−	0.610900i	0.952208	−	0.305450i	$-0.0988070\pi$
$114$		0			0
$115$	−108.810	−	188.464i	−0.946170	−	1.63881i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−110.314	+	26.2539i	−0.927009	+	0.220621i
$120$		0			0
$121$	−18.9536	+	32.8286i	−0.156641	+	0.271311i
$122$		0			0
$123$		0			0
$124$		0			0
$125$			91.1206i			0.728965i
$126$		0			0
$127$	−18.6244			−0.146649			−0.0733245	−	0.997308i	$-0.523361\pi$
$127$	−18.6244			−0.146649			−0.0733245	+	0.997308i	$0.523361\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−139.095	−	80.3065i	−1.06179	−	0.613027i	−0.135866	−	0.990727i	$-0.543382\pi$
$131$	−139.095	−	80.3065i	−1.06179	−	0.613027i	−0.925928	+	0.377700i	$0.876715\pi$
$132$		0			0
$133$	25.0942	−	23.7195i	0.188678	−	0.178342i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	96.3127	−	55.6062i	0.703013	−	0.405885i	−0.105456	−	0.994424i	$-0.533630\pi$
$137$	96.3127	−	55.6062i	0.703013	−	0.405885i	0.808468	+	0.588539i	$0.200297\pi$
$138$		0			0
$139$	−93.8528			−0.675200			−0.337600	−	0.941290i	$-0.609615\pi$
$139$	−93.8528			−0.675200			−0.337600	+	0.941290i	$0.609615\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−64.5298	+	37.2563i	−0.451257	+	0.260533i
$144$		0			0
$145$	−105.252	+	182.302i	−0.725878	+	1.25726i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	27.5259	+	15.8921i	0.184737	+	0.106658i	0.589517	−	0.807756i	$-0.299318\pi$
$149$	27.5259	+	15.8921i	0.184737	+	0.106658i	−0.404779	+	0.914414i	$0.632652\pi$
$150$		0			0
$151$	−119.642	−	207.227i	−0.792334	−	1.37236i	−0.924518	−	0.381138i	$-0.875532\pi$
$151$	−119.642	−	207.227i	−0.792334	−	1.37236i	0.132184	−	0.991225i	$-0.457801\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		−	153.103i		−	0.987763i
$156$		0			0
$157$	51.0455	+	88.4133i	0.325130	+	0.563142i	0.981539	−	0.191263i	$-0.0612585\pi$
$157$	51.0455	+	88.4133i	0.325130	+	0.563142i	−0.656408	+	0.754406i	$0.727925\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	178.197	+	188.525i	1.10682	+	1.17096i
$162$		0			0
$163$	100.747	−	174.499i	0.618079	−	1.07054i	−0.371757	−	0.928330i	$-0.621245\pi$
$163$	100.747	−	174.499i	0.618079	−	1.07054i	0.989836	−	0.142214i	$-0.0454220\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	93.5784i		−	0.560350i	−0.959949	−	0.280175i	$-0.909608\pi$
$167$		−	93.5784i		−	0.560350i	0.959949	−	0.280175i	$-0.0903925\pi$
$168$		0			0
$169$	−102.182			−0.604626
$170$		0			0
$171$		0			0
$172$		0			0
$173$	124.697	+	71.9939i	0.720792	+	0.416149i	0.815044	−	0.579399i	$-0.196713\pi$
$173$	124.697	+	71.9939i	0.720792	+	0.416149i	−0.0942521	+	0.995548i	$0.530046\pi$
$174$		0			0
$175$	15.3684	+	64.5754i	0.0878196	+	0.369002i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	249.025	−	143.775i	1.39120	−	0.803210i	0.397753	−	0.917493i	$-0.369790\pi$
$179$	249.025	−	143.775i	1.39120	−	0.803210i	0.993448	+	0.114282i	$0.0364568\pi$
$180$		0			0
$181$	−180.940			−0.999667			−0.499833	−	0.866122i	$-0.666605\pi$
$181$	−180.940			−0.999667			−0.499833	+	0.866122i	$0.666605\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−36.6602	+	21.1658i	−0.198164	+	0.114410i
$186$		0			0
$187$	73.8326	−	127.882i	0.394827	−	0.683860i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	196.587	+	113.499i	1.02925	+	0.594237i	0.916769	−	0.399417i	$-0.130787\pi$
$191$	196.587	+	113.499i	1.02925	+	0.594237i	0.112480	+	0.993654i	$0.464121\pi$
$192$		0			0
$193$	161.945	+	280.497i	0.839092	+	1.45335i	0.890655	+	0.454681i	$0.150247\pi$
$193$	161.945	+	280.497i	0.839092	+	1.45335i	−0.0515623	+	0.998670i	$0.516420\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			193.203i			0.980725i	0.871519	+	0.490362i	$0.163136\pi$
$197$			193.203i			0.980725i	−0.871519	+	0.490362i	$0.836864\pi$
$198$		0			0
$199$	−112.833	−	195.432i	−0.567000	−	0.982072i	−0.996861	−	0.0791770i	$-0.974771\pi$
$199$	−112.833	−	195.432i	−0.567000	−	0.982072i	0.429861	−	0.902895i	$-0.358563\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	71.7438	−	240.459i	0.353418	−	1.18453i
$204$		0			0
$205$	143.896	−	249.236i	0.701933	−	1.21578i
$206$		0			0
$207$		0			0
$208$		0			0
$209$			44.9659i			0.215148i
$210$		0			0
$211$	−161.585			−0.765804			−0.382902	−	0.923789i	$-0.625075\pi$
$211$	−161.585			−0.765804			−0.382902	+	0.923789i	$0.625075\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−164.332	−	94.8769i	−0.764333	−	0.441288i
$216$		0			0
$217$	42.2552	+	177.549i	0.194725	+	0.818198i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−114.677	+	66.2085i	−0.518898	+	0.299586i
$222$		0			0
$223$	−427.779			−1.91829			−0.959146	−	0.282911i	$-0.908700\pi$
$223$	−427.779			−1.91829			−0.959146	+	0.282911i	$0.908700\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−198.700	+	114.720i	−0.875331	+	0.505373i	−0.869116	−	0.494608i	$-0.835312\pi$
$227$	−198.700	+	114.720i	−0.875331	+	0.505373i	−0.00621479	+	0.999981i	$0.501978\pi$
$228$		0			0
$229$	−120.254	+	208.286i	−0.525126	+	0.909546i	0.474445	+	0.880285i	$0.342649\pi$
$229$	−120.254	+	208.286i	−0.525126	+	0.909546i	−0.999572	+	0.0292607i	$0.990685\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−324.983	−	187.629i	−1.39478	−	0.805275i	−0.400938	−	0.916105i	$-0.631316\pi$
$233$	−324.983	−	187.629i	−1.39478	−	0.805275i	−0.993839	+	0.110830i	$0.964649\pi$
$234$		0			0
$235$	−89.8150	−	155.564i	−0.382192	−	0.661975i
$236$		0			0
$237$		0			0
$238$		0			0
$239$			260.626i			1.09049i	0.838278	+	0.545243i	$0.183563\pi$
$239$			260.626i			1.09049i	−0.838278	+	0.545243i	$0.816437\pi$
$240$		0			0
$241$	−168.240	−	291.401i	−0.698093	−	1.20913i	−0.969127	−	0.246563i	$-0.920699\pi$
$241$	−168.240	−	291.401i	−0.698093	−	1.20913i	0.271034	−	0.962570i	$-0.412634\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−129.617	−	256.890i	−0.529049	−	1.04853i
$246$		0			0
$247$	20.1613	−	34.9204i	0.0816248	−	0.141378i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	408.816i		−	1.62875i	−0.580339	−	0.814375i	$-0.697080\pi$
$251$		−	408.816i		−	1.62875i	0.580339	−	0.814375i	$-0.302920\pi$
$252$		0			0
$253$	−337.814			−1.33523
$254$		0			0
$255$		0			0
$256$		0			0
$257$	338.121	+	195.214i	1.31565	+	0.759589i	0.983025	−	0.183471i	$-0.0587332\pi$
$257$	338.121	+	195.214i	1.31565	+	0.759589i	0.332622	+	0.943060i	$0.392067\pi$
$258$		0			0
$259$	36.6721	−	34.6632i	0.141591	−	0.133835i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−396.356	+	228.836i	−1.50706	+	0.870099i	−0.507089	+	0.861894i	$0.669279\pi$
$263$	−396.356	+	228.836i	−1.50706	+	0.870099i	−0.999966	+	0.00820543i	$0.997388\pi$
$264$		0			0
$265$	−455.390			−1.71845
$266$		0			0
$267$		0			0
$268$		0			0
$269$	254.655	−	147.025i	0.946674	−	0.546563i	0.0546279	−	0.998507i	$-0.482603\pi$
$269$	254.655	−	147.025i	0.946674	−	0.546563i	0.892046	+	0.451944i	$0.149269\pi$
$270$		0			0
$271$	−22.6498	+	39.2306i	−0.0835786	+	0.144762i	−0.904785	−	0.425869i	$-0.859968\pi$
$271$	−22.6498	+	39.2306i	−0.0835786	+	0.144762i	0.821206	+	0.570632i	$0.193302\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−74.8592	−	43.2200i	−0.272215	−	0.157164i
$276$		0			0
$277$	−48.0872	−	83.2895i	−0.173600	−	0.300684i	0.766076	−	0.642750i	$-0.222207\pi$
$277$	−48.0872	−	83.2895i	−0.173600	−	0.300684i	−0.939676	+	0.342066i	$0.888873\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	38.6849i		−	0.137669i	−0.997628	−	0.0688344i	$-0.978072\pi$
$281$		−	38.6849i		−	0.137669i	0.997628	−	0.0688344i	$-0.0219280\pi$
$282$		0			0
$283$	−102.398	−	177.359i	−0.361831	−	0.626711i	0.626431	−	0.779477i	$-0.284515\pi$
$283$	−102.398	−	177.359i	−0.361831	−	0.626711i	−0.988262	+	0.152766i	$0.951182\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−98.0850	+	328.745i	−0.341759	+	1.14545i
$288$		0			0
$289$	−13.2913	+	23.0212i	−0.0459907	+	0.0796583i
$290$		0			0
$291$		0			0
$292$		0			0
$293$			208.611i			0.711984i	0.934489	+	0.355992i	$0.115857\pi$
$293$			208.611i			0.711984i	−0.934489	+	0.355992i	$0.884143\pi$
$294$		0			0
$295$	−419.801			−1.42305
$296$		0			0
$297$		0			0
$298$		0			0
$299$	262.346	+	151.466i	0.877412	+	0.506574i
$300$		0			0
$301$	216.755	+	64.6715i	0.720117	+	0.214856i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−212.594	+	122.741i	−0.697028	+	0.402430i
$306$		0			0
$307$	−190.413			−0.620238			−0.310119	−	0.950698i	$-0.600369\pi$
$307$	−190.413			−0.620238			−0.310119	+	0.950698i	$0.600369\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−83.0994	+	47.9775i	−0.267201	+	0.154268i	−0.627615	−	0.778524i	$-0.715969\pi$
$311$	−83.0994	+	47.9775i	−0.267201	+	0.154268i	0.360414	+	0.932792i	$0.382635\pi$
$312$		0			0
$313$	−34.9383	+	60.5148i	−0.111624	+	0.193338i	−0.916425	−	0.400206i	$-0.868939\pi$
$313$	−34.9383	+	60.5148i	−0.111624	+	0.193338i	0.804801	+	0.593544i	$0.202272\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−150.094	−	86.6570i	−0.473484	−	0.273366i	0.244213	−	0.969722i	$-0.421470\pi$
$317$	−150.094	−	86.6570i	−0.473484	−	0.273366i	−0.717697	+	0.696356i	$0.754804\pi$
$318$		0			0
$319$	163.385	+	282.991i	0.512179	+	0.887120i
$320$		0			0
$321$		0			0
$322$		0			0
$323$			79.9093i			0.247397i
$324$		0			0
$325$	38.7570	+	67.1291i	0.119252	+	0.206551i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	147.090	+	155.615i	0.447082	+	0.472993i
$330$		0			0
$331$	279.207	−	483.600i	0.843525	−	1.46103i	−0.0433706	−	0.999059i	$-0.513810\pi$
$331$	279.207	−	483.600i	0.843525	−	1.46103i	0.886896	−	0.461970i	$-0.152857\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		−	471.022i		−	1.40604i
$336$		0			0
$337$	−30.1017			−0.0893225			−0.0446613	−	0.999002i	$-0.514221\pi$
$337$	−30.1017			−0.0893225			−0.0446613	+	0.999002i	$0.514221\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−205.824	−	118.833i	−0.603590	−	0.348483i
$342$		0			0
$343$	221.212	+	262.134i	0.644934	+	0.764239i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−1.06894	+	0.617154i	−0.00308053	+	0.00177854i	−0.501539	−	0.865135i	$-0.667233\pi$
$347$	−1.06894	+	0.617154i	−0.00308053	+	0.00177854i	0.498459	+	0.866913i	$0.333899\pi$
$348$		0			0
$349$	−74.1959			−0.212596			−0.106298	−	0.994334i	$-0.533900\pi$
$349$	−74.1959			−0.212596			−0.106298	+	0.994334i	$0.533900\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	464.043	−	267.916i	1.31457	−	0.758968i	0.331721	−	0.943377i	$-0.392371\pi$
$353$	464.043	−	267.916i	1.31457	−	0.758968i	0.982849	+	0.184410i	$0.0590373\pi$
$354$		0			0
$355$	126.051	−	218.327i	0.355073	−	0.615004i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	250.458	+	144.602i	0.697655	+	0.402792i	0.806474	−	0.591270i	$-0.201373\pi$
$359$	250.458	+	144.602i	0.697655	+	0.402792i	−0.108818	+	0.994062i	$0.534707\pi$
$360$		0			0
$361$	168.333	+	291.562i	0.466297	+	0.807651i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		−	565.244i		−	1.54861i
$366$		0			0
$367$	−180.020	−	311.803i	−0.490517	−	0.849600i	0.509424	−	0.860516i	$-0.329859\pi$
$367$	−180.020	−	311.803i	−0.490517	−	0.849600i	−0.999940	+	0.0109158i	$0.996525\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	528.101	−	125.684i	1.42345	−	0.338770i
$372$		0			0
$373$	121.777	−	210.924i	0.326480	−	0.565480i	−0.655331	−	0.755342i	$-0.727471\pi$
$373$	121.777	−	210.924i	0.326480	−	0.565480i	0.981811	+	0.189862i	$0.0608042\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	293.027i		−	0.777261i
$378$		0			0
$379$	487.602			1.28655			0.643275	−	0.765635i	$-0.277575\pi$
$379$	487.602			1.28655			0.643275	+	0.765635i	$0.277575\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	561.942	+	324.438i	1.46721	+	0.847095i	0.999327	−	0.0366936i	$-0.0116826\pi$
$383$	561.942	+	324.438i	1.46721	+	0.847095i	0.467886	+	0.883789i	$0.345016\pi$
$384$		0			0
$385$	359.056	+	107.129i	0.932614	+	0.278256i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−626.974	+	361.984i	−1.61176	+	0.930549i	−0.622796	+	0.782384i	$0.714003\pi$
$389$	−626.974	+	361.984i	−1.61176	+	0.930549i	−0.988963	+	0.148165i	$0.952663\pi$
$390$		0			0
$391$	−600.334			−1.53538
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−550.855	+	318.036i	−1.39457	+	0.805155i
$396$		0			0
$397$	−241.457	+	418.216i	−0.608205	+	1.05344i	0.383331	+	0.923611i	$0.374777\pi$
$397$	−241.457	+	418.216i	−0.608205	+	1.05344i	−0.991536	+	0.129831i	$0.958557\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−98.0122	−	56.5874i	−0.244419	−	0.141116i	0.372787	−	0.927917i	$-0.378402\pi$
$401$	−98.0122	−	56.5874i	−0.244419	−	0.141116i	−0.617206	+	0.786801i	$0.711736\pi$
$402$		0			0
$403$	106.562	+	184.570i	0.264421	+	0.457991i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			65.7122i			0.161455i
$408$		0			0
$409$	144.356	+	250.032i	0.352949	+	0.611326i	0.986765	−	0.162158i	$-0.0518454\pi$
$409$	144.356	+	250.032i	0.352949	+	0.611326i	−0.633815	+	0.773484i	$0.718512\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	486.830	−	115.862i	1.17876	−	0.280537i
$414$		0			0
$415$	282.976	−	490.129i	0.681871	−	1.18103i
$416$		0			0
$417$		0			0
$418$		0			0
$419$			200.898i			0.479471i	0.970838	+	0.239735i	$0.0770607\pi$
$419$			200.898i			0.479471i	−0.970838	+	0.239735i	$0.922939\pi$
$420$		0			0
$421$	65.5283			0.155649			0.0778246	−	0.996967i	$-0.475203\pi$
$421$	65.5283			0.155649			0.0778246	+	0.996967i	$0.475203\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−133.033	−	76.8067i	−0.313019	−	0.180722i
$426$		0			0
$427$	212.663	−	201.013i	0.498039	−	0.470756i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	422.697	−	244.044i	0.980735	−	0.566228i	0.0782431	−	0.996934i	$-0.475069\pi$
$431$	422.697	−	244.044i	0.980735	−	0.566228i	0.902492	+	0.430707i	$0.141736\pi$
$432$		0			0
$433$	205.749			0.475170			0.237585	−	0.971367i	$-0.423644\pi$
$433$	205.749			0.475170			0.237585	+	0.971367i	$0.423644\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	158.317	−	91.4045i	0.362282	−	0.209164i
$438$		0			0
$439$	−293.613	+	508.553i	−0.668822	+	1.15843i	0.309411	+	0.950928i	$0.399868\pi$
$439$	−293.613	+	508.553i	−0.668822	+	1.15843i	−0.978234	+	0.207506i	$0.933465\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−127.890	−	73.8372i	−0.288690	−	0.166675i	0.348661	−	0.937249i	$-0.386637\pi$
$443$	−127.890	−	73.8372i	−0.288690	−	0.166675i	−0.637351	+	0.770574i	$0.719970\pi$
$444$		0			0
$445$	350.793	+	607.591i	0.788298	+	1.36537i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			112.984i			0.251634i	0.992053	+	0.125817i	$0.0401552\pi$
$449$			112.984i			0.251634i	−0.992053	+	0.125817i	$0.959845\pi$
$450$		0			0
$451$	−223.373	−	386.893i	−0.495284	−	0.857857i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−230.809	−	244.186i	−0.507273	−	0.536672i
$456$		0			0
$457$	358.491	−	620.925i	0.784445	−	1.35870i	−0.144885	−	0.989448i	$-0.546281\pi$
$457$	358.491	−	620.925i	0.784445	−	1.35870i	0.929330	−	0.369250i	$-0.120385\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			522.058i			1.13245i	0.824252	+	0.566223i	$0.191596\pi$
$461$			522.058i			1.13245i	−0.824252	+	0.566223i	$0.808404\pi$
$462$		0			0
$463$	340.342			0.735079			0.367540	−	0.930008i	$-0.380200\pi$
$463$	340.342			0.735079			0.367540	+	0.930008i	$0.380200\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	134.846	+	77.8533i	0.288749	+	0.166710i	0.637378	−	0.770552i	$-0.280019\pi$
$467$	134.846	+	77.8533i	0.288749	+	0.166710i	−0.348628	+	0.937261i	$0.613352\pi$
$468$		0			0
$469$	129.998	+	546.229i	0.277182	+	1.16467i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−255.095	+	147.279i	−0.539313	+	0.311372i
$474$		0			0
$475$	46.7772			0.0984782
$476$		0			0
$477$		0			0
$478$		0			0
$479$	80.9255	−	46.7224i	0.168947	−	0.0975415i	−0.413142	−	0.910666i	$-0.635569\pi$
$479$	80.9255	−	46.7224i	0.168947	−	0.0975415i	0.582089	+	0.813125i	$0.302235\pi$
$480$		0			0
$481$	29.4633	−	51.0320i	0.0612543	−	0.106096i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	5.43712	+	3.13912i	0.0112106	+	0.00647241i
$486$		0			0
$487$	12.4895	+	21.6324i	0.0256457	+	0.0444197i	0.878563	−	0.477626i	$-0.158502\pi$
$487$	12.4895	+	21.6324i	0.0256457	+	0.0444197i	−0.852918	+	0.522045i	$0.825169\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			168.856i			0.343903i	0.985105	+	0.171952i	$0.0550073\pi$
$491$			168.856i			0.343903i	−0.985105	+	0.171952i	$0.944993\pi$
$492$		0			0
$493$	290.353	+	502.907i	0.588952	+	1.02010i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−85.9209	+	287.975i	−0.172879	+	0.579427i
$498$		0			0
$499$	206.746	−	358.094i	0.414320	−	0.717623i	−0.581037	−	0.813877i	$-0.697353\pi$
$499$	206.746	−	358.094i	0.414320	−	0.717623i	0.995357	+	0.0962542i	$0.0306862\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			8.38205i			0.0166641i	0.999965	+	0.00833206i	$0.00265221\pi$
$503$			8.38205i			0.0166641i	−0.999965	+	0.00833206i	$0.997348\pi$
$504$		0			0
$505$	−206.794			−0.409494
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−838.295	−	483.990i	−1.64695	−	0.950865i	−0.978276	−	0.207305i	$-0.933531\pi$
$509$	−838.295	−	483.990i	−1.64695	−	0.950865i	−0.668669	−	0.743560i	$-0.733136\pi$
$510$		0			0
$511$	156.003	+	655.496i	0.305289	+	1.28277i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	233.784	−	134.975i	0.453949	−	0.262087i
$516$		0			0
$517$	−278.843			−0.539348
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−653.506	+	377.302i	−1.25433	+	0.724188i	−0.971967	−	0.235119i	$-0.924452\pi$
$521$	−653.506	+	377.302i	−1.25433	+	0.724188i	−0.282364	+	0.959307i	$0.591119\pi$
$522$		0			0
$523$	170.253	−	294.888i	0.325532	−	0.563839i	−0.656088	−	0.754685i	$-0.727790\pi$
$523$	170.253	−	294.888i	0.325532	−	0.563839i	0.981620	+	0.190846i	$0.0611232\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−365.772	−	211.179i	−0.694065	−	0.400718i
$528$		0			0
$529$	422.193	+	731.260i	0.798097	+	1.38234i
$530$		0			0
$531$		0			0
$532$		0			0
$533$			400.614i			0.751622i
$534$		0			0
$535$	−184.344	−	319.294i	−0.344569	−	0.596811i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−445.953	−	25.1372i	−0.827371	−	0.0466368i
$540$		0			0
$541$	297.677	−	515.592i	0.550235	−	0.953035i	−0.448022	−	0.894023i	$-0.647871\pi$
$541$	297.677	−	515.592i	0.550235	−	0.953035i	0.998257	−	0.0590129i	$-0.0187953\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		−	1037.30i		−	1.90331i
$546$		0			0
$547$	636.823			1.16421			0.582105	−	0.813114i	$-0.302229\pi$
$547$	636.823			1.16421			0.582105	+	0.813114i	$0.302229\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−153.141	−	88.4162i	−0.277934	−	0.160465i
$552$		0			0
$553$	551.034	−	520.848i	0.996445	−	0.941859i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	667.634	−	385.458i	1.19862	−	0.692026i	0.238376	−	0.971173i	$-0.423385\pi$
$557$	667.634	−	385.458i	1.19862	−	0.692026i	0.960248	+	0.279147i	$0.0900517\pi$
$558$		0			0
$559$	264.142			0.472526
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−256.653	+	148.179i	−0.455867	+	0.263195i	−0.710305	−	0.703894i	$-0.751443\pi$
$563$	−256.653	+	148.179i	−0.455867	+	0.263195i	0.254438	+	0.967089i	$0.418110\pi$
$564$		0			0
$565$	−202.684	+	351.059i	−0.358733	+	0.621343i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	208.296	+	120.260i	0.366074	+	0.211353i	0.671742	−	0.740785i	$-0.265546\pi$
$569$	208.296	+	120.260i	0.366074	+	0.211353i	−0.305668	+	0.952138i	$0.598880\pi$
$570$		0			0
$571$	379.395	+	657.132i	0.664440	+	1.15084i	0.979437	+	0.201751i	$0.0646632\pi$
$571$	379.395	+	657.132i	0.664440	+	1.15084i	−0.314997	+	0.949093i	$0.602003\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			351.422i			0.611169i
$576$		0			0
$577$	−257.346	−	445.736i	−0.446006	−	0.772505i	0.552116	−	0.833768i	$-0.313821\pi$
$577$	−257.346	−	445.736i	−0.446006	−	0.772505i	−0.998122	+	0.0612622i	$0.980487\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−192.887	+	646.487i	−0.331991	+	1.11271i
$582$		0			0
$583$	−353.455	+	612.202i	−0.606270	+	1.05009i
$584$		0			0
$585$		0			0
$586$		0			0
$587$			104.491i			0.178008i	0.996031	+	0.0890040i	$0.0283684\pi$
$587$			104.491i			0.178008i	−0.996031	+	0.0890040i	$0.971632\pi$
$588$		0			0
$589$	128.613			0.218358
$590$		0			0
$591$		0			0
$592$		0			0
$593$	537.873	+	310.541i	0.907037	+	0.523678i	0.879477	−	0.475942i	$-0.157893\pi$
$593$	537.873	+	310.541i	0.907037	+	0.523678i	0.0275606	+	0.999620i	$0.491226\pi$
$594$		0			0
$595$	638.083	+	190.380i	1.07241	+	0.319966i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	232.385	−	134.168i	0.387955	−	0.223986i	−0.293319	−	0.956015i	$-0.594760\pi$
$599$	232.385	−	134.168i	0.387955	−	0.223986i	0.681274	+	0.732029i	$0.261426\pi$
$600$		0			0
$601$	950.264			1.58114			0.790569	−	0.612373i	$-0.209785\pi$
$601$	950.264			1.58114			0.790569	+	0.612373i	$0.209785\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	192.776	−	111.299i	0.318638	−	0.183966i
$606$		0			0
$607$	238.717	−	413.469i	0.393273	−	0.681169i	−0.599606	−	0.800295i	$-0.704676\pi$
$607$	238.717	−	413.469i	0.393273	−	0.681169i	0.992879	+	0.119127i	$0.0380094\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	216.549	+	125.025i	0.354418	+	0.204623i
$612$		0			0
$613$	−66.9196	−	115.908i	−0.109167	−	0.189084i	0.806266	−	0.591553i	$-0.201485\pi$
$613$	−66.9196	−	115.908i	−0.109167	−	0.189084i	−0.915433	+	0.402470i	$0.868152\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			42.7777i			0.0693318i	0.999399	+	0.0346659i	$0.0110367\pi$
$617$			42.7777i			0.0693318i	−0.999399	+	0.0346659i	$0.988963\pi$
$618$		0			0
$619$	263.304	+	456.056i	0.425370	+	0.736763i	0.996455	−	0.0841280i	$-0.0268105\pi$
$619$	263.304	+	456.056i	0.425370	+	0.736763i	−0.571084	+	0.820891i	$0.693477\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−574.493	−	607.788i	−0.922140	−	0.975582i
$624$		0			0
$625$	386.073	−	668.698i	0.617717	−	1.06992i
$626$		0			0
$627$		0			0
$628$		0			0
$629$			116.778i			0.185656i
$630$		0			0
$631$	−153.010			−0.242488			−0.121244	−	0.992623i	$-0.538688\pi$
$631$	−153.010			−0.242488			−0.121244	+	0.992623i	$0.538688\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	94.7140	+	54.6831i	0.149156	+	0.0861152i
$636$		0			0
$637$	335.056	+	219.473i	0.525990	+	0.344542i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	826.691	−	477.290i	1.28969	−	0.744603i	0.311091	−	0.950380i	$-0.399306\pi$
$641$	826.691	−	477.290i	1.28969	−	0.744603i	0.978599	+	0.205777i	$0.0659723\pi$
$642$		0			0
$643$	−1183.88			−1.84119			−0.920593	−	0.390523i	$-0.872294\pi$
$643$	−1183.88			−1.84119			−0.920593	+	0.390523i	$0.872294\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	963.931	−	556.526i	1.48985	−	0.860163i	0.489914	−	0.871771i	$-0.337028\pi$
$647$	963.931	−	556.526i	1.48985	−	0.860163i	0.999933	+	0.0116078i	$0.00369497\pi$
$648$		0			0
$649$	−325.833	+	564.359i	−0.502053	+	0.869582i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	787.576	+	454.707i	1.20609	+	0.696336i	0.961903	−	0.273393i	$-0.0881458\pi$
$653$	787.576	+	454.707i	1.20609	+	0.696336i	0.244186	+	0.969728i	$0.421479\pi$
$654$		0			0
$655$	471.576	+	816.794i	0.719963	+	1.24701i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	236.989i		−	0.359619i	−0.983701	−	0.179810i	$-0.942452\pi$
$659$		−	236.989i		−	0.359619i	0.983701	−	0.179810i	$-0.0575482\pi$
$660$		0			0
$661$	−18.7316	−	32.4441i	−0.0283383	−	0.0490834i	0.851508	−	0.524341i	$-0.175688\pi$
$661$	−18.7316	−	32.4441i	−0.0283383	−	0.0490834i	−0.879847	+	0.475257i	$0.842355\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−197.259	+	46.9460i	−0.296630	+	0.0705955i
$666$		0			0
$667$	664.243	−	1150.50i	0.995867	−	1.72489i
$668$		0			0
$669$		0			0
$670$		0			0
$671$			381.066i			0.567908i
$672$		0			0
$673$	−487.829			−0.724858			−0.362429	−	0.932011i	$-0.618052\pi$
$673$	−487.829			−0.724858			−0.362429	+	0.932011i	$0.618052\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−458.084	−	264.475i	−0.676637	−	0.390657i	0.121949	−	0.992536i	$-0.461085\pi$
$677$	−458.084	−	264.475i	−0.676637	−	0.390657i	−0.798587	+	0.601879i	$0.794419\pi$
$678$		0			0
$679$	−7.17162	−	2.13974i	−0.0105620	−	0.00315131i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−674.584	+	389.471i	−0.987678	+	0.570236i	−0.904579	−	0.426305i	$-0.859815\pi$
$683$	−674.584	+	389.471i	−0.987678	+	0.570236i	−0.0830985	+	0.996541i	$0.526482\pi$
$684$		0			0
$685$	−653.061			−0.953374
$686$		0			0
$687$		0			0
$688$		0			0
$689$	548.986	−	316.957i	0.796786	−	0.460025i
$690$		0			0
$691$	−300.966	+	521.288i	−0.435551	+	0.754397i	−0.997340	−	0.0728836i	$-0.976780\pi$
$691$	−300.966	+	521.288i	−0.435551	+	0.754397i	0.561789	+	0.827280i	$0.310113\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	477.286	+	275.561i	0.686743	+	0.396491i
$696$		0			0
$697$	−396.959	−	687.552i	−0.569524	−	0.986445i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	356.605i		−	0.508709i	−0.967111	−	0.254354i	$-0.918137\pi$
$701$		−	356.605i		−	0.508709i	0.967111	−	0.254354i	$-0.0818630\pi$
$702$		0			0
$703$	−17.7801	−	30.7961i	−0.0252918	−	0.0438067i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	239.813	−	57.0736i	0.339198	−	0.0807264i
$708$		0			0
$709$	136.696	−	236.764i	0.192801	−	0.333941i	−0.753376	−	0.657590i	$-0.771576\pi$
$709$	136.696	−	236.764i	0.192801	−	0.333941i	0.946177	+	0.323648i	$0.104909\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			966.229i			1.35516i
$714$		0			0
$715$	437.552			0.611961
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−844.958	−	487.836i	−1.17518	−	0.678493i	−0.220289	−	0.975435i	$-0.570700\pi$
$719$	−844.958	−	487.836i	−1.17518	−	0.678493i	−0.954896	+	0.296942i	$0.904033\pi$
$720$		0			0
$721$	−233.859	+	221.049i	−0.324354	+	0.306586i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	294.391	−	169.966i	0.406056	−	0.234437i
$726$		0			0
$727$	−1156.41			−1.59065			−0.795327	−	0.606181i	$-0.792701\pi$
$727$	−1156.41			−1.59065			−0.795327	+	0.606181i	$0.792701\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−453.332	+	261.731i	−0.620153	+	0.358046i
$732$		0			0
$733$	507.481	−	878.983i	0.692335	−	1.19916i	−0.278736	−	0.960368i	$-0.589916\pi$
$733$	507.481	−	878.983i	0.692335	−	1.19916i	0.971071	−	0.238791i	$-0.0767511\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−633.218	−	365.588i	−0.859183	−	0.496049i
$738$		0			0
$739$	−304.939	−	528.171i	−0.412638	−	0.714710i	0.582539	−	0.812802i	$-0.302059\pi$
$739$	−304.939	−	528.171i	−0.412638	−	0.714710i	−0.995177	+	0.0980926i	$0.968726\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	709.331i		−	0.954685i	−0.878717	−	0.477343i	$-0.841600\pi$
$743$		−	709.331i		−	0.954685i	0.878717	−	0.477343i	$-0.158400\pi$
$744$		0			0
$745$	−93.3214	−	161.637i	−0.125264	−	0.216963i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	301.901	+	319.397i	0.403072	+	0.426432i
$750$		0			0
$751$	−660.043	+	1143.23i	−0.878886	+	1.52227i	−0.0263204	+	0.999654i	$0.508379\pi$
$751$	−660.043	+	1143.23i	−0.878886	+	1.52227i	−0.852565	+	0.522621i	$0.824954\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$			1405.13i			1.86110i
$756$		0			0
$757$	−1462.13			−1.93148			−0.965739	−	0.259515i	$-0.916437\pi$
$757$	−1462.13			−1.93148			−0.965739	+	0.259515i	$0.916437\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−32.2297	−	18.6078i	−0.0423517	−	0.0244518i	0.478675	−	0.877992i	$-0.341117\pi$
$761$	−32.2297	−	18.6078i	−0.0423517	−	0.0244518i	−0.521026	+	0.853541i	$0.674451\pi$
$762$		0			0
$763$	286.287	+	1202.93i	0.375212	+	1.57657i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	506.082	−	292.187i	0.659820	−	0.380947i
$768$		0			0
$769$	−1292.41			−1.68064			−0.840319	−	0.542092i	$-0.817633\pi$
$769$	−1292.41			−1.68064			−0.840319	+	0.542092i	$0.817633\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	427.371	−	246.743i	0.552873	−	0.319201i	−0.197407	−	0.980322i	$-0.563252\pi$
$773$	427.371	−	246.743i	0.552873	−	0.319201i	0.750280	+	0.661120i	$0.229919\pi$
$774$		0			0
$775$	−123.619	+	214.115i	−0.159509	+	0.276277i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	209.368	+	120.879i	0.268765	+	0.155172i
$780$		0			0
$781$	−195.671	−	338.913i	−0.250539	−	0.433947i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		−	599.498i		−	0.763692i
$786$		0			0
$787$	−533.606	−	924.233i	−0.678025	−	1.17437i	−0.975575	−	0.219668i	$-0.929503\pi$
$787$	−533.606	−	924.233i	−0.678025	−	1.17437i	0.297549	−	0.954706i	$-0.403831\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	138.157	−	463.051i	0.174661	−	0.585400i
$792$		0			0
$793$	170.859	−	295.936i	0.215458	−	0.373185i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			915.049i			1.14812i	0.818815	+	0.574058i	$0.194632\pi$
$797$			915.049i			1.14812i	−0.818815	+	0.574058i	$0.805368\pi$
$798$		0			0
$799$	−495.535			−0.620194
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−759.885	−	438.720i	−0.946308	−	0.546351i
$804$		0			0
$805$	−352.691	−	1481.94i	−0.438125	−	1.84092i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−709.030	+	409.358i	−0.876427	+	0.506006i	−0.869479	−	0.493970i	$-0.835545\pi$
$809$	−709.030	+	409.358i	−0.876427	+	0.506006i	−0.00694855	+	0.999976i	$0.502212\pi$
$810$		0			0
$811$	1355.39			1.67126			0.835631	−	0.549291i	$-0.185102\pi$
$811$	1355.39			1.67126			0.835631	+	0.549291i	$0.185102\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−1024.69	+	591.605i	−1.25729	+	0.725896i
$816$		0			0
$817$	79.7005	−	138.045i	0.0975526	−	0.168966i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−673.145	−	388.640i	−0.819908	−	0.473374i	0.0304765	−	0.999535i	$-0.490298\pi$
$821$	−673.145	−	388.640i	−0.819908	−	0.473374i	−0.850385	+	0.526161i	$0.823631\pi$
$822$		0			0
$823$	483.838	+	838.032i	0.587895	+	1.01826i	0.994508	+	0.104664i	$0.0333766\pi$
$823$	483.838	+	838.032i	0.587895	+	1.01826i	−0.406612	+	0.913601i	$0.633290\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	1320.39i		−	1.59660i	−0.602257	−	0.798302i	$-0.705732\pi$
$827$		−	1320.39i		−	1.59660i	0.602257	−	0.798302i	$-0.294268\pi$
$828$		0			0
$829$	41.7309	+	72.2800i	0.0503388	+	0.0871894i	0.890097	−	0.455771i	$-0.150637\pi$
$829$	41.7309	+	72.2800i	0.0503388	+	0.0871894i	−0.839758	+	0.542961i	$0.817303\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−792.508	−	44.6716i	−0.951390	−	0.0536274i
$834$		0			0
$835$	−274.756	+	475.891i	−0.329049	+	0.569929i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	650.213i		−	0.774986i	−0.921873	−	0.387493i	$-0.873341\pi$
$839$		−	650.213i		−	0.774986i	0.921873	−	0.387493i	$-0.126659\pi$
$840$		0			0
$841$	−444.054			−0.528007
$842$		0			0
$843$		0			0
$844$		0			0
$845$	519.642	+	300.016i	0.614961	+	0.355048i
$846$		0			0
$847$	−192.838	+	182.275i	−0.227672	+	0.215200i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	231.361	−	133.577i	0.271870	−	0.156964i
$852$		0			0
$853$	−96.7015			−0.113366			−0.0566832	−	0.998392i	$-0.518052\pi$
$853$	−96.7015			−0.113366			−0.0566832	+	0.998392i	$0.518052\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−1153.36	+	665.891i	−1.34581	+	0.777002i	−0.987653	−	0.156660i	$-0.949927\pi$
$857$	−1153.36	+	665.891i	−1.34581	+	0.777002i	−0.358154	+	0.933662i	$0.616594\pi$
$858$		0			0
$859$	293.029	−	507.542i	0.341129	−	0.590852i	−0.643514	−	0.765434i	$-0.722524\pi$
$859$	293.029	−	507.542i	0.341129	−	0.590852i	0.984643	+	0.174582i	$0.0558575\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−492.769	−	284.500i	−0.570995	−	0.329664i	0.186552	−	0.982445i	$-0.440269\pi$
$863$	−492.769	−	284.500i	−0.570995	−	0.329664i	−0.757547	+	0.652781i	$0.773602\pi$
$864$		0			0
$865$	−422.762	−	732.246i	−0.488742	−	0.846527i
$866$		0			0
$867$		0			0
$868$		0			0
$869$			987.388i			1.13623i
$870$		0			0
$871$	327.837	+	567.831i	0.376392	+	0.651930i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−182.364	+	611.219i	−0.208416	+	0.698536i
$876$		0			0
$877$	627.891	−	1087.54i	0.715953	−	1.24007i	−0.246638	−	0.969108i	$-0.579326\pi$
$877$	627.891	−	1087.54i	0.715953	−	1.24007i	0.962591	−	0.270959i	$-0.0873409\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			715.846i			0.812538i	0.913754	+	0.406269i	$0.133170\pi$
$881$			715.846i			0.812538i	−0.913754	+	0.406269i	$0.866830\pi$
$882$		0			0
$883$	462.652			0.523955			0.261977	−	0.965074i	$-0.415625\pi$
$883$	462.652			0.523955			0.261977	+	0.965074i	$0.415625\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	329.401	+	190.180i	0.371365	+	0.214408i	0.674055	−	0.738681i	$-0.264551\pi$
$887$	329.401	+	190.180i	0.371365	+	0.214408i	−0.302689	+	0.953089i	$0.597884\pi$
$888$		0			0
$889$	−124.929	−	37.2740i	−0.140527	−	0.0419280i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	130.680	−	75.4483i	0.146338	−	0.0844886i
$894$		0			0
$895$	−1688.55			−1.88664
$896$		0			0
$897$		0			0
$898$		0			0
$899$	809.422	−	467.320i	0.900358	−	0.519822i
$900$		0			0
$901$	−628.129	+	1087.95i	−0.697147	+	1.20749i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	920.164	+	531.257i	1.01676	+	0.587024i
$906$		0			0
$907$	−480.774	−	832.725i	−0.530071	−	0.918109i	−0.999385	−	0.0350779i	$-0.988832\pi$
$907$	−480.774	−	832.725i	−0.530071	−	0.918109i	0.469314	−	0.883031i	$-0.344501\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			666.377i			0.731478i	0.930717	+	0.365739i	$0.119184\pi$
$911$			666.377i			0.731478i	−0.930717	+	0.365739i	$0.880816\pi$
$912$		0			0
$913$	−439.270	−	760.837i	−0.481128	−	0.833338i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−772.300	−	817.058i	−0.842203	−	0.891012i
$918$		0			0
$919$	−272.734	+	472.389i	−0.296773	+	0.514025i	−0.975396	−	0.220462i	$-0.929244\pi$
$919$	−272.734	+	472.389i	−0.296773	+	0.514025i	0.678623	+	0.734487i	$0.262577\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$			350.932i			0.380208i
$924$		0			0
$925$	68.3592			0.0739018
$926$		0			0
$927$		0			0
$928$		0			0
$929$	1086.77	+	627.445i	1.16982	+	0.675399i	0.953639	−	0.300954i	$-0.0973049\pi$
$929$	1086.77	+	627.445i	1.16982	+	0.675399i	0.216186	+	0.976352i	$0.430638\pi$
$930$		0			0
$931$	215.798	−	108.884i	0.231792	−	0.116953i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−750.947	+	433.560i	−0.803152	+	0.463700i
$936$		0			0
$937$	250.164			0.266984			0.133492	−	0.991050i	$-0.457381\pi$
$937$	250.164			0.266984			0.133492	+	0.991050i	$0.457381\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	1290.84	−	745.265i	1.37177	−	0.791992i	0.380620	−	0.924732i	$-0.375711\pi$
$941$	1290.84	−	745.265i	1.37177	−	0.791992i	0.991151	+	0.132739i	$0.0423773\pi$
$942$		0			0
$943$	−908.124	+	1572.92i	−0.963016	+	1.66799i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	285.236	+	164.681i	0.301200	+	0.173898i	0.642982	−	0.765881i	$-0.277697\pi$
$947$	285.236	+	164.681i	0.301200	+	0.173898i	−0.341782	+	0.939779i	$0.611030\pi$
$948$		0			0
$949$	393.417	+	681.418i	0.414560	+	0.718038i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	1230.89i		−	1.29159i	−0.763509	−	0.645797i	$-0.776525\pi$
$953$		−	1230.89i		−	1.29159i	0.763509	−	0.645797i	$-0.223475\pi$
$954$		0			0
$955$	−666.490	−	1154.40i	−0.697896	−	1.20879i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	757.334	−	180.239i	0.789712	−	0.187945i
$960$		0			0
$961$	140.611	−	243.545i	0.146317	−	0.253429i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		−	1901.94i		−	1.97093i
$966$		0			0
$967$	−671.972			−0.694904			−0.347452	−	0.937698i	$-0.612953\pi$
$967$	−671.972			−0.694904			−0.347452	+	0.937698i	$0.612953\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	476.719	+	275.234i	0.490957	+	0.283454i	0.724971	−	0.688779i	$-0.241853\pi$
$971$	476.719	+	275.234i	0.490957	+	0.283454i	−0.234014	+	0.972233i	$0.575186\pi$
$972$		0			0
$973$	−629.546	−	187.833i	−0.647016	−	0.193045i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−253.982	+	146.637i	−0.259961	+	0.150089i	−0.624317	−	0.781171i	$-0.714623\pi$
$977$	−253.982	+	146.637i	−0.259961	+	0.150089i	0.364356	+	0.931260i	$0.381289\pi$
$978$		0			0
$979$	1089.08			1.11245
$980$		0			0
$981$		0			0
$982$		0			0
$983$	1128.68	−	651.646i	1.14820	−	0.662916i	0.199755	−	0.979846i	$-0.435985\pi$
$983$	1128.68	−	651.646i	1.14820	−	0.662916i	0.948449	+	0.316930i	$0.102652\pi$
$984$		0			0
$985$	567.262	−	982.527i	0.575901	−	0.997490i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	1037.09	+	598.764i	1.04863	+	0.605424i
$990$		0			0
$991$	−395.105	−	684.343i	−0.398694	−	0.690558i	0.594871	−	0.803821i	$-0.297203\pi$
$991$	−395.105	−	684.343i	−0.398694	−	0.690558i	−0.993565	+	0.113263i	$0.963870\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$			1325.15i			1.33181i
$996$		0			0
$997$	−762.395	−	1320.51i	−0.764690	−	1.32448i	−0.940411	−	0.340041i	$-0.889559\pi$
$997$	−762.395	−	1320.51i	−0.764690	−	1.32448i	0.175721	−	0.984440i	$-0.443774\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	756.3.bk.f.485.2	yes	12
3.2	odd	2	inner	756.3.bk.f.485.5	yes	12
7.4	even	3	inner	756.3.bk.f.53.5	yes	12
21.11	odd	6	inner	756.3.bk.f.53.2	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
756.3.bk.f.53.2	✓	12	21.11	odd	6	inner
756.3.bk.f.53.5	yes	12	7.4	even	3	inner
756.3.bk.f.485.2	yes	12	1.1	even	1	trivial
756.3.bk.f.485.5	yes	12	3.2	odd	2	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 756 = 2^{2} \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	756.bk (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-5.08547 - 2.93610i) q^{5} +(6.70780 + 2.00135i) q^{7} +(-7.89428 + 4.55776i) q^{11} +8.17424 q^{13} +(-14.0290 + 8.09965i) q^{17} +(2.46644 - 4.27201i) q^{19} +(32.0942 + 18.5296i) q^{23} +(4.74136 + 8.21227i) q^{25} -35.8476i q^{29} +(13.0363 + 22.5795i) q^{31} +(-28.2362 - 29.8726i) q^{35} +(3.60441 - 6.24302i) q^{37} +49.0093i q^{41} +32.3139 q^{43} +(26.4916 + 15.2950i) q^{47} +(40.9892 + 26.8493i) q^{49} +(67.1604 - 38.7751i) q^{53} +53.5282 q^{55} +(61.9118 - 35.7448i) q^{59} +(20.9021 - 36.2034i) q^{61} +(-41.5699 - 24.0004i) q^{65} +(40.1061 + 69.4658i) q^{67} +42.9314i q^{71} +(48.1288 + 83.3616i) q^{73} +(-62.0749 + 14.7733i) q^{77} +(54.1597 - 93.8073i) q^{79} +96.3783i q^{83} +95.1255 q^{85} +(-103.469 - 59.7379i) q^{89} +(54.8312 + 16.3595i) q^{91} +(-25.0861 + 14.4834i) q^{95} -1.06915 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 2 q^{7} - 32 q^{13} - 18 q^{19} + 22 q^{25} - 26 q^{31} + 48 q^{37} - 12 q^{43} + 54 q^{49} - 208 q^{55} - 122 q^{61} + 164 q^{67} - 130 q^{73} + 220 q^{79} - 280 q^{85} + 328 q^{91} - 292 q^{97}+O(q^{100}) \) 12 * q - 2 * q^7 - 32 * q^13 - 18 * q^19 + 22 * q^25 - 26 * q^31 + 48 * q^37 - 12 * q^43 + 54 * q^49 - 208 * q^55 - 122 * q^61 + 164 * q^67 - 130 * q^73 + 220 * q^79 - 280 * q^85 + 328 * q^91 - 292 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

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