LMFDB - Embedded newform 3150.2.b.c.251.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3150,2,Mod(251,3150)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3150, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("3150.251");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3150 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3150.b (of order $2$, degree $1$, 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:	$25.1528766367$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.40960000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 7x^{4} + 1 $ x^8 + 7*x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 630)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			251.6
Root			$-1.14412 - 1.14412i$ of defining polynomial
Character	$\chi$	$=$	3150.251
Dual form			3150.2.b.c.251.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -1.00000 q^{4} +(-2.12132 + 1.58114i) q^{7} -1.00000i q^{8} +O(q^{10})$	$q+1.00000i q^{2} -1.00000 q^{4} +(-2.12132 + 1.58114i) q^{7} -1.00000i q^{8} +1.41421i q^{11} -3.16228i q^{13} +(-1.58114 - 2.12132i) q^{14} +1.00000 q^{16} +4.47214 q^{17} -1.41421 q^{22} -6.00000i q^{23} +3.16228 q^{26} +(2.12132 - 1.58114i) q^{28} +2.82843i q^{29} +1.00000i q^{32} +4.47214i q^{34} +4.24264 q^{37} -9.48683 q^{41} +8.48528 q^{43} -1.41421i q^{44} +6.00000 q^{46} +4.47214 q^{47} +(2.00000 - 6.70820i) q^{49} +3.16228i q^{52} -6.00000i q^{53} +(1.58114 + 2.12132i) q^{56} -2.82843 q^{58} +9.48683 q^{59} +13.4164i q^{61} -1.00000 q^{64} -4.47214 q^{68} +5.65685i q^{71} +6.32456i q^{73} +4.24264i q^{74} +(-2.23607 - 3.00000i) q^{77} +4.00000 q^{79} -9.48683i q^{82} +8.94427 q^{83} +8.48528i q^{86} +1.41421 q^{88} -9.48683 q^{89} +(5.00000 + 6.70820i) q^{91} +6.00000i q^{92} +4.47214i q^{94} +12.6491i q^{97} +(6.70820 + 2.00000i) q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{4}+O(q^{10}) $ 8 * q - 8 * q^4	$ 8 q - 8 q^{4} + 8 q^{16} + 48 q^{46} + 16 q^{49} - 8 q^{64} + 32 q^{79} + 40 q^{91}+O(q^{100}) $ 8 * q - 8 * q^4 + 8 * q^16 + 48 * q^46 + 16 * q^49 - 8 * q^64 + 32 * q^79 + 40 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3150\mathbb{Z}\right)^\times$.

$n$	$127$	$451$	$2801$
$\chi(n)$	$1$	$-1$	$-1$

Coefficient data

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

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

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

$2$			1.00000i			0.707107i
$2$			1.00000i			0.707107i
$3$		0			0
$3$		0			0
$4$	−1.00000			−0.500000
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−2.12132	+	1.58114i	−0.801784	+	0.597614i
$7$	−2.12132	+	1.58114i	−0.801784	+	0.597614i
$8$		−	1.00000i		−	0.353553i
$9$		0			0
$10$		0			0
$11$			1.41421i			0.426401i	0.977008	+	0.213201i	$0.0683888\pi$
$11$			1.41421i			0.426401i	−0.977008	+	0.213201i	$0.931611\pi$
$12$		0			0
$13$		−	3.16228i		−	0.877058i	−0.898717	−	0.438529i	$-0.855500\pi$
$13$		−	3.16228i		−	0.877058i	0.898717	−	0.438529i	$-0.144500\pi$
$14$	−1.58114	−	2.12132i	−0.422577	−	0.566947i
$15$		0			0
$16$	1.00000			0.250000
$17$	4.47214			1.08465			0.542326	−	0.840168i	$-0.317544\pi$
$17$	4.47214			1.08465			0.542326	+	0.840168i	$0.317544\pi$
$18$		0			0
$19$		0			0		1.00000			$0$
$19$		0			0		−1.00000			$\pi$
$20$		0			0
$21$		0			0
$22$	−1.41421			−0.301511
$23$		−	6.00000i		−	1.25109i	−0.780189	−	0.625543i	$-0.784877\pi$
$23$		−	6.00000i		−	1.25109i	0.780189	−	0.625543i	$-0.215123\pi$
$24$		0			0
$25$		0			0
$26$	3.16228			0.620174
$27$		0			0
$28$	2.12132	−	1.58114i	0.400892	−	0.298807i
$29$			2.82843i			0.525226i	0.964901	+	0.262613i	$0.0845842\pi$
$29$			2.82843i			0.525226i	−0.964901	+	0.262613i	$0.915416\pi$
$30$		0			0
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$			1.00000i			0.176777i
$33$		0			0
$34$			4.47214i			0.766965i
$35$		0			0
$36$		0			0
$37$	4.24264			0.697486			0.348743	−	0.937218i	$-0.386609\pi$
$37$	4.24264			0.697486			0.348743	+	0.937218i	$0.386609\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−9.48683			−1.48159			−0.740797	−	0.671729i	$-0.765552\pi$
$41$	−9.48683			−1.48159			−0.740797	+	0.671729i	$0.765552\pi$
$42$		0			0
$43$	8.48528			1.29399			0.646997	−	0.762493i	$-0.276025\pi$
$43$	8.48528			1.29399			0.646997	+	0.762493i	$0.276025\pi$
$44$		−	1.41421i		−	0.213201i
$45$		0			0
$46$	6.00000			0.884652
$47$	4.47214			0.652328			0.326164	−	0.945313i	$-0.394244\pi$
$47$	4.47214			0.652328			0.326164	+	0.945313i	$0.394244\pi$
$48$		0			0
$49$	2.00000	−	6.70820i	0.285714	−	0.958315i
$50$		0			0
$51$		0			0
$52$			3.16228i			0.438529i
$53$		−	6.00000i		−	0.824163i	−0.911147	−	0.412082i	$-0.864802\pi$
$53$		−	6.00000i		−	0.824163i	0.911147	−	0.412082i	$-0.135198\pi$
$54$		0			0
$55$		0			0
$56$	1.58114	+	2.12132i	0.211289	+	0.283473i
$57$		0			0
$58$	−2.82843			−0.371391
$59$	9.48683			1.23508			0.617540	−	0.786539i	$-0.288129\pi$
$59$	9.48683			1.23508			0.617540	+	0.786539i	$0.288129\pi$
$60$		0			0
$61$			13.4164i			1.71780i	0.512148	+	0.858898i	$0.328850\pi$
$61$			13.4164i			1.71780i	−0.512148	+	0.858898i	$0.671150\pi$
$62$		0			0
$63$		0			0
$64$	−1.00000			−0.125000
$65$		0			0
$66$		0			0
$67$		0			0			−	1.00000i	$-0.5\pi$
$67$		0			0				1.00000i	$0.5\pi$
$68$	−4.47214			−0.542326
$69$		0			0
$70$		0			0
$71$			5.65685i			0.671345i	0.941979	+	0.335673i	$0.108964\pi$
$71$			5.65685i			0.671345i	−0.941979	+	0.335673i	$0.891036\pi$
$72$		0			0
$73$			6.32456i			0.740233i	0.928985	+	0.370117i	$0.120682\pi$
$73$			6.32456i			0.740233i	−0.928985	+	0.370117i	$0.879318\pi$
$74$			4.24264i			0.493197i
$75$		0			0
$76$		0			0
$77$	−2.23607	−	3.00000i	−0.254824	−	0.341882i
$78$		0			0
$79$	4.00000			0.450035			0.225018	−	0.974355i	$-0.427756\pi$
$79$	4.00000			0.450035			0.225018	+	0.974355i	$0.427756\pi$
$80$		0			0
$81$		0			0
$82$		−	9.48683i		−	1.04765i
$83$	8.94427			0.981761			0.490881	−	0.871227i	$-0.336675\pi$
$83$	8.94427			0.981761			0.490881	+	0.871227i	$0.336675\pi$
$84$		0			0
$85$		0			0
$86$			8.48528i			0.914991i
$87$		0			0
$88$	1.41421			0.150756
$89$	−9.48683			−1.00560			−0.502801	−	0.864402i	$-0.667697\pi$
$89$	−9.48683			−1.00560			−0.502801	+	0.864402i	$0.667697\pi$
$90$		0			0
$91$	5.00000	+	6.70820i	0.524142	+	0.703211i
$92$			6.00000i			0.625543i
$93$		0			0
$94$			4.47214i			0.461266i
$95$		0			0
$96$		0			0
$97$			12.6491i			1.28432i	0.766570	+	0.642161i	$0.221962\pi$
$97$			12.6491i			1.28432i	−0.766570	+	0.642161i	$0.778038\pi$
$98$	6.70820	+	2.00000i	0.677631	+	0.202031i
$99$		0			0
$100$		0			0
$101$	−18.9737			−1.88795			−0.943975	−	0.330017i	$-0.892946\pi$
$101$	−18.9737			−1.88795			−0.943975	+	0.330017i	$0.892946\pi$
$102$		0			0
$103$			15.8114i			1.55794i	0.627060	+	0.778971i	$0.284258\pi$
$103$			15.8114i			1.55794i	−0.627060	+	0.778971i	$0.715742\pi$
$104$	−3.16228			−0.310087
$105$		0			0
$106$	6.00000			0.582772
$107$		−	12.0000i		−	1.16008i	−0.814587	−	0.580042i	$-0.803036\pi$
$107$		−	12.0000i		−	1.16008i	0.814587	−	0.580042i	$-0.196964\pi$
$108$		0			0
$109$	10.0000			0.957826			0.478913	−	0.877862i	$-0.341031\pi$
$109$	10.0000			0.957826			0.478913	+	0.877862i	$0.341031\pi$
$110$		0			0
$111$		0			0
$112$	−2.12132	+	1.58114i	−0.200446	+	0.149404i
$113$			6.00000i			0.564433i	0.959351	+	0.282216i	$0.0910696\pi$
$113$			6.00000i			0.564433i	−0.959351	+	0.282216i	$0.908930\pi$
$114$		0			0
$115$		0			0
$116$		−	2.82843i		−	0.262613i
$117$		0			0
$118$			9.48683i			0.873334i
$119$	−9.48683	+	7.07107i	−0.869657	+	0.648204i
$120$		0			0
$121$	9.00000			0.818182
$122$	−13.4164			−1.21466
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	21.2132			1.88237			0.941184	−	0.337895i	$-0.109715\pi$
$127$	21.2132			1.88237			0.941184	+	0.337895i	$0.109715\pi$
$128$		−	1.00000i		−	0.0883883i
$129$		0			0
$130$		0			0
$131$	9.48683			0.828868			0.414434	−	0.910079i	$-0.363979\pi$
$131$	9.48683			0.828868			0.414434	+	0.910079i	$0.363979\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		−	4.47214i		−	0.383482i
$137$		−	18.0000i		−	1.53784i	−0.639343	−	0.768922i	$-0.720793\pi$
$137$		−	18.0000i		−	1.53784i	0.639343	−	0.768922i	$-0.279207\pi$
$138$		0			0
$139$			13.4164i			1.13796i	0.822350	+	0.568982i	$0.192663\pi$
$139$			13.4164i			1.13796i	−0.822350	+	0.568982i	$0.807337\pi$
$140$		0			0
$141$		0			0
$142$	−5.65685			−0.474713
$143$	4.47214			0.373979
$144$		0			0
$145$		0			0
$146$	−6.32456			−0.523424
$147$		0			0
$148$	−4.24264			−0.348743
$149$			11.3137i			0.926855i	0.886135	+	0.463428i	$0.153381\pi$
$149$			11.3137i			0.926855i	−0.886135	+	0.463428i	$0.846619\pi$
$150$		0			0
$151$	20.0000			1.62758			0.813788	−	0.581161i	$-0.197401\pi$
$151$	20.0000			1.62758			0.813788	+	0.581161i	$0.197401\pi$
$152$		0			0
$153$		0			0
$154$	3.00000	−	2.23607i	0.241747	−	0.180187i
$155$		0			0
$156$		0			0
$157$		−	15.8114i		−	1.26189i	−0.775829	−	0.630943i	$-0.782668\pi$
$157$		−	15.8114i		−	1.26189i	0.775829	−	0.630943i	$-0.217332\pi$
$158$			4.00000i			0.318223i
$159$		0			0
$160$		0			0
$161$	9.48683	+	12.7279i	0.747667	+	1.00310i
$162$		0			0
$163$		0			0			−	1.00000i	$-0.5\pi$
$163$		0			0				1.00000i	$0.5\pi$
$164$	9.48683			0.740797
$165$		0			0
$166$			8.94427i			0.694210i
$167$	−8.94427			−0.692129			−0.346064	−	0.938211i	$-0.612482\pi$
$167$	−8.94427			−0.692129			−0.346064	+	0.938211i	$0.612482\pi$
$168$		0			0
$169$	3.00000			0.230769
$170$		0			0
$171$		0			0
$172$	−8.48528			−0.646997
$173$	8.94427			0.680020			0.340010	−	0.940422i	$-0.389569\pi$
$173$	8.94427			0.680020			0.340010	+	0.940422i	$0.389569\pi$
$174$		0			0
$175$		0			0
$176$			1.41421i			0.106600i
$177$		0			0
$178$		−	9.48683i		−	0.711068i
$179$		−	18.3848i		−	1.37414i	−0.726590	−	0.687071i	$-0.758896\pi$
$179$		−	18.3848i		−	1.37414i	0.726590	−	0.687071i	$-0.241104\pi$
$180$		0			0
$181$			13.4164i			0.997234i	0.866822	+	0.498617i	$0.166159\pi$
$181$			13.4164i			0.997234i	−0.866822	+	0.498617i	$0.833841\pi$
$182$	−6.70820	+	5.00000i	−0.497245	+	0.370625i
$183$		0			0
$184$	−6.00000			−0.442326
$185$		0			0
$186$		0			0
$187$			6.32456i			0.462497i
$188$	−4.47214			−0.326164
$189$		0			0
$190$		0			0
$191$			22.6274i			1.63726i	0.574320	+	0.818631i	$0.305267\pi$
$191$			22.6274i			1.63726i	−0.574320	+	0.818631i	$0.694733\pi$
$192$		0			0
$193$	−8.48528			−0.610784			−0.305392	−	0.952227i	$-0.598787\pi$
$193$	−8.48528			−0.610784			−0.305392	+	0.952227i	$0.598787\pi$
$194$	−12.6491			−0.908153
$195$		0			0
$196$	−2.00000	+	6.70820i	−0.142857	+	0.479157i
$197$			12.0000i			0.854965i	0.904024	+	0.427482i	$0.140599\pi$
$197$			12.0000i			0.854965i	−0.904024	+	0.427482i	$0.859401\pi$
$198$		0			0
$199$			26.8328i			1.90213i	0.308994	+	0.951064i	$0.400008\pi$
$199$			26.8328i			1.90213i	−0.308994	+	0.951064i	$0.599992\pi$
$200$		0			0
$201$		0			0
$202$		−	18.9737i		−	1.33498i
$203$	−4.47214	−	6.00000i	−0.313882	−	0.421117i
$204$		0			0
$205$		0			0
$206$	−15.8114			−1.10163
$207$		0			0
$208$		−	3.16228i		−	0.219265i
$209$		0			0
$210$		0			0
$211$	20.0000			1.37686			0.688428	−	0.725304i	$-0.258301\pi$
$211$	20.0000			1.37686			0.688428	+	0.725304i	$0.258301\pi$
$212$			6.00000i			0.412082i
$213$		0			0
$214$	12.0000			0.820303
$215$		0			0
$216$		0			0
$217$		0			0
$218$			10.0000i			0.677285i
$219$		0			0
$220$		0			0
$221$		−	14.1421i		−	0.951303i
$222$		0			0
$223$		−	22.1359i		−	1.48233i	−0.671322	−	0.741166i	$-0.734273\pi$
$223$		−	22.1359i		−	1.48233i	0.671322	−	0.741166i	$-0.265727\pi$
$224$	−1.58114	−	2.12132i	−0.105644	−	0.141737i
$225$		0			0
$226$	−6.00000			−0.399114
$227$	17.8885			1.18730			0.593652	−	0.804722i	$-0.297686\pi$
$227$	17.8885			1.18730			0.593652	+	0.804722i	$0.297686\pi$
$228$		0			0
$229$		−	13.4164i		−	0.886581i	−0.896378	−	0.443291i	$-0.853811\pi$
$229$		−	13.4164i		−	0.886581i	0.896378	−	0.443291i	$-0.146189\pi$
$230$		0			0
$231$		0			0
$232$	2.82843			0.185695
$233$			6.00000i			0.393073i	0.980497	+	0.196537i	$0.0629694\pi$
$233$			6.00000i			0.393073i	−0.980497	+	0.196537i	$0.937031\pi$
$234$		0			0
$235$		0			0
$236$	−9.48683			−0.617540
$237$		0			0
$238$	−7.07107	−	9.48683i	−0.458349	−	0.614940i
$239$			11.3137i			0.731823i	0.930650	+	0.365911i	$0.119243\pi$
$239$			11.3137i			0.731823i	−0.930650	+	0.365911i	$0.880757\pi$
$240$		0			0
$241$		−	13.4164i		−	0.864227i	−0.901819	−	0.432113i	$-0.857768\pi$
$241$		−	13.4164i		−	0.864227i	0.901819	−	0.432113i	$-0.142232\pi$
$242$			9.00000i			0.578542i
$243$		0			0
$244$		−	13.4164i		−	0.858898i
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$	9.48683			0.598804			0.299402	−	0.954127i	$-0.403213\pi$
$251$	9.48683			0.598804			0.299402	+	0.954127i	$0.403213\pi$
$252$		0			0
$253$	8.48528			0.533465
$254$			21.2132i			1.33103i
$255$		0			0
$256$	1.00000			0.0625000
$257$	4.47214			0.278964			0.139482	−	0.990225i	$-0.455456\pi$
$257$	4.47214			0.278964			0.139482	+	0.990225i	$0.455456\pi$
$258$		0			0
$259$	−9.00000	+	6.70820i	−0.559233	+	0.416828i
$260$		0			0
$261$		0			0
$262$			9.48683i			0.586098i
$263$		−	6.00000i		−	0.369976i	−0.982741	−	0.184988i	$-0.940775\pi$
$263$		−	6.00000i		−	0.369976i	0.982741	−	0.184988i	$-0.0592246\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$	18.9737			1.15684			0.578422	−	0.815737i	$-0.303669\pi$
$269$	18.9737			1.15684			0.578422	+	0.815737i	$0.303669\pi$
$270$		0			0
$271$		0			0		1.00000			$0$
$271$		0			0		−1.00000			$\pi$
$272$	4.47214			0.271163
$273$		0			0
$274$	18.0000			1.08742
$275$		0			0
$276$		0			0
$277$	−21.2132			−1.27458			−0.637289	−	0.770625i	$-0.719944\pi$
$277$	−21.2132			−1.27458			−0.637289	+	0.770625i	$0.719944\pi$
$278$	−13.4164			−0.804663
$279$		0			0
$280$		0			0
$281$			1.41421i			0.0843649i	0.999110	+	0.0421825i	$0.0134311\pi$
$281$			1.41421i			0.0843649i	−0.999110	+	0.0421825i	$0.986569\pi$
$282$		0			0
$283$			6.32456i			0.375956i	0.982173	+	0.187978i	$0.0601933\pi$
$283$			6.32456i			0.375956i	−0.982173	+	0.187978i	$0.939807\pi$
$284$		−	5.65685i		−	0.335673i
$285$		0			0
$286$			4.47214i			0.264443i
$287$	20.1246	−	15.0000i	1.18792	−	0.885422i
$288$		0			0
$289$	3.00000			0.176471
$290$		0			0
$291$		0			0
$292$		−	6.32456i		−	0.370117i
$293$	−4.47214			−0.261265			−0.130632	−	0.991431i	$-0.541701\pi$
$293$	−4.47214			−0.261265			−0.130632	+	0.991431i	$0.541701\pi$
$294$		0			0
$295$		0			0
$296$		−	4.24264i		−	0.246598i
$297$		0			0
$298$	−11.3137			−0.655386
$299$	−18.9737			−1.09728
$300$		0			0
$301$	−18.0000	+	13.4164i	−1.03750	+	0.773309i
$302$			20.0000i			1.15087i
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$			12.6491i			0.721923i	0.932581	+	0.360961i	$0.117551\pi$
$307$			12.6491i			0.721923i	−0.932581	+	0.360961i	$0.882449\pi$
$308$	2.23607	+	3.00000i	0.127412	+	0.170941i
$309$		0			0
$310$		0			0
$311$	−18.9737			−1.07590			−0.537949	−	0.842977i	$-0.680801\pi$
$311$	−18.9737			−1.07590			−0.537949	+	0.842977i	$0.680801\pi$
$312$		0			0
$313$		−	31.6228i		−	1.78743i	−0.448640	−	0.893713i	$-0.648091\pi$
$313$		−	31.6228i		−	1.78743i	0.448640	−	0.893713i	$-0.351909\pi$
$314$	15.8114			0.892288
$315$		0			0
$316$	−4.00000			−0.225018
$317$			12.0000i			0.673987i	0.941507	+	0.336994i	$0.109410\pi$
$317$			12.0000i			0.673987i	−0.941507	+	0.336994i	$0.890590\pi$
$318$		0			0
$319$	−4.00000			−0.223957
$320$		0			0
$321$		0			0
$322$	−12.7279	+	9.48683i	−0.709299	+	0.528681i
$323$		0			0
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$			9.48683i			0.523823i
$329$	−9.48683	+	7.07107i	−0.523026	+	0.389841i
$330$		0			0
$331$	−10.0000			−0.549650			−0.274825	−	0.961494i	$-0.588620\pi$
$331$	−10.0000			−0.549650			−0.274825	+	0.961494i	$0.588620\pi$
$332$	−8.94427			−0.490881
$333$		0			0
$334$		−	8.94427i		−	0.489409i
$335$		0			0
$336$		0			0
$337$	25.4558			1.38667			0.693334	−	0.720616i	$-0.256141\pi$
$337$	25.4558			1.38667			0.693334	+	0.720616i	$0.256141\pi$
$338$			3.00000i			0.163178i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	6.36396	+	17.3925i	0.343622	+	0.939108i
$344$		−	8.48528i		−	0.457496i
$345$		0			0
$346$			8.94427i			0.480847i
$347$		−	12.0000i		−	0.644194i	−0.946707	−	0.322097i	$-0.895612\pi$
$347$		−	12.0000i		−	0.644194i	0.946707	−	0.322097i	$-0.104388\pi$
$348$		0			0
$349$		−	13.4164i		−	0.718164i	−0.933306	−	0.359082i	$-0.883090\pi$
$349$		−	13.4164i		−	0.718164i	0.933306	−	0.359082i	$-0.116910\pi$
$350$		0			0
$351$		0			0
$352$	−1.41421			−0.0753778
$353$	−31.3050			−1.66619			−0.833097	−	0.553127i	$-0.813435\pi$
$353$	−31.3050			−1.66619			−0.833097	+	0.553127i	$0.813435\pi$
$354$		0			0
$355$		0			0
$356$	9.48683			0.502801
$357$		0			0
$358$	18.3848			0.971666
$359$		−	31.1127i		−	1.64207i	−0.570881	−	0.821033i	$-0.693398\pi$
$359$		−	31.1127i		−	1.64207i	0.570881	−	0.821033i	$-0.306602\pi$
$360$		0			0
$361$	19.0000			1.00000
$362$	−13.4164			−0.705151
$363$		0			0
$364$	−5.00000	−	6.70820i	−0.262071	−	0.351605i
$365$		0			0
$366$		0			0
$367$			22.1359i			1.15549i	0.816218	+	0.577743i	$0.196067\pi$
$367$			22.1359i			1.15549i	−0.816218	+	0.577743i	$0.803933\pi$
$368$		−	6.00000i		−	0.312772i
$369$		0			0
$370$		0			0
$371$	9.48683	+	12.7279i	0.492532	+	0.660801i
$372$		0			0
$373$	−21.2132			−1.09838			−0.549189	−	0.835698i	$-0.685063\pi$
$373$	−21.2132			−1.09838			−0.549189	+	0.835698i	$0.685063\pi$
$374$	−6.32456			−0.327035
$375$		0			0
$376$		−	4.47214i		−	0.230633i
$377$	8.94427			0.460653
$378$		0			0
$379$	−20.0000			−1.02733			−0.513665	−	0.857991i	$-0.671713\pi$
$379$	−20.0000			−1.02733			−0.513665	+	0.857991i	$0.671713\pi$
$380$		0			0
$381$		0			0
$382$	−22.6274			−1.15772
$383$	−4.47214			−0.228515			−0.114258	−	0.993451i	$-0.536449\pi$
$383$	−4.47214			−0.228515			−0.114258	+	0.993451i	$0.536449\pi$
$384$		0			0
$385$		0			0
$386$		−	8.48528i		−	0.431889i
$387$		0			0
$388$		−	12.6491i		−	0.642161i
$389$			11.3137i			0.573628i	0.957986	+	0.286814i	$0.0925961\pi$
$389$			11.3137i			0.573628i	−0.957986	+	0.286814i	$0.907404\pi$
$390$		0			0
$391$		−	26.8328i		−	1.35699i
$392$	−6.70820	−	2.00000i	−0.338815	−	0.101015i
$393$		0			0
$394$	−12.0000			−0.604551
$395$		0			0
$396$		0			0
$397$			22.1359i			1.11097i	0.831526	+	0.555486i	$0.187468\pi$
$397$			22.1359i			1.11097i	−0.831526	+	0.555486i	$0.812532\pi$
$398$	−26.8328			−1.34501
$399$		0			0
$400$		0			0
$401$			1.41421i			0.0706225i	0.999376	+	0.0353112i	$0.0112422\pi$
$401$			1.41421i			0.0706225i	−0.999376	+	0.0353112i	$0.988758\pi$
$402$		0			0
$403$		0			0
$404$	18.9737			0.943975
$405$		0			0
$406$	6.00000	−	4.47214i	0.297775	−	0.221948i
$407$			6.00000i			0.297409i
$408$		0			0
$409$		0			0		1.00000			$0$
$409$		0			0		−1.00000			$\pi$
$410$		0			0
$411$		0			0
$412$		−	15.8114i		−	0.778971i
$413$	−20.1246	+	15.0000i	−0.990267	+	0.738102i
$414$		0			0
$415$		0			0
$416$	3.16228			0.155043
$417$		0			0
$418$		0			0
$419$	9.48683			0.463462			0.231731	−	0.972780i	$-0.425561\pi$
$419$	9.48683			0.463462			0.231731	+	0.972780i	$0.425561\pi$
$420$		0			0
$421$	−22.0000			−1.07221			−0.536107	−	0.844150i	$-0.680106\pi$
$421$	−22.0000			−1.07221			−0.536107	+	0.844150i	$0.680106\pi$
$422$			20.0000i			0.973585i
$423$		0			0
$424$	−6.00000			−0.291386
$425$		0			0
$426$		0			0
$427$	−21.2132	−	28.4605i	−1.02658	−	1.37730i
$428$			12.0000i			0.580042i
$429$		0			0
$430$		0			0
$431$		−	36.7696i		−	1.77113i	−0.464518	−	0.885564i	$-0.653773\pi$
$431$		−	36.7696i		−	1.77113i	0.464518	−	0.885564i	$-0.346227\pi$
$432$		0			0
$433$			6.32456i			0.303939i	0.988385	+	0.151969i	$0.0485615\pi$
$433$			6.32456i			0.303939i	−0.988385	+	0.151969i	$0.951438\pi$
$434$		0			0
$435$		0			0
$436$	−10.0000			−0.478913
$437$		0			0
$438$		0			0
$439$		0			0		1.00000			$0$
$439$		0			0		−1.00000			$\pi$
$440$		0			0
$441$		0			0
$442$	14.1421			0.672673
$443$		−	24.0000i		−	1.14027i	−0.821549	−	0.570137i	$-0.806890\pi$
$443$		−	24.0000i		−	1.14027i	0.821549	−	0.570137i	$-0.193110\pi$
$444$		0			0
$445$		0			0
$446$	22.1359			1.04817
$447$		0			0
$448$	2.12132	−	1.58114i	0.100223	−	0.0747018i
$449$		−	18.3848i		−	0.867631i	−0.901002	−	0.433816i	$-0.857167\pi$
$449$		−	18.3848i		−	0.867631i	0.901002	−	0.433816i	$-0.142833\pi$
$450$		0			0
$451$		−	13.4164i		−	0.631754i
$452$		−	6.00000i		−	0.282216i
$453$		0			0
$454$			17.8885i			0.839551i
$455$		0			0
$456$		0			0
$457$	16.9706			0.793849			0.396925	−	0.917851i	$-0.370077\pi$
$457$	16.9706			0.793849			0.396925	+	0.917851i	$0.370077\pi$
$458$	13.4164			0.626908
$459$		0			0
$460$		0			0
$461$		0			0			−	1.00000i	$-0.5\pi$
$461$		0			0				1.00000i	$0.5\pi$
$462$		0			0
$463$	21.2132			0.985861			0.492931	−	0.870069i	$-0.335926\pi$
$463$	21.2132			0.985861			0.492931	+	0.870069i	$0.335926\pi$
$464$			2.82843i			0.131306i
$465$		0			0
$466$	−6.00000			−0.277945
$467$	−8.94427			−0.413892			−0.206946	−	0.978352i	$-0.566352\pi$
$467$	−8.94427			−0.413892			−0.206946	+	0.978352i	$0.566352\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		−	9.48683i		−	0.436667i
$473$			12.0000i			0.551761i
$474$		0			0
$475$		0			0
$476$	9.48683	−	7.07107i	0.434828	−	0.324102i
$477$		0			0
$478$	−11.3137			−0.517477
$479$	−18.9737			−0.866929			−0.433464	−	0.901171i	$-0.642709\pi$
$479$	−18.9737			−0.866929			−0.433464	+	0.901171i	$0.642709\pi$
$480$		0			0
$481$		−	13.4164i		−	0.611736i
$482$	13.4164			0.611101
$483$		0			0
$484$	−9.00000			−0.409091
$485$		0			0
$486$		0			0
$487$	−4.24264			−0.192252			−0.0961262	−	0.995369i	$-0.530645\pi$
$487$	−4.24264			−0.192252			−0.0961262	+	0.995369i	$0.530645\pi$
$488$	13.4164			0.607332
$489$		0			0
$490$		0			0
$491$		−	15.5563i		−	0.702048i	−0.936366	−	0.351024i	$-0.885834\pi$
$491$		−	15.5563i		−	0.702048i	0.936366	−	0.351024i	$-0.114166\pi$
$492$		0			0
$493$			12.6491i			0.569687i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−8.94427	−	12.0000i	−0.401205	−	0.538274i
$498$		0			0
$499$	10.0000			0.447661			0.223831	−	0.974628i	$-0.428144\pi$
$499$	10.0000			0.447661			0.223831	+	0.974628i	$0.428144\pi$
$500$		0			0
$501$		0			0
$502$			9.48683i			0.423418i
$503$	22.3607			0.997013			0.498507	−	0.866886i	$-0.333882\pi$
$503$	22.3607			0.997013			0.498507	+	0.866886i	$0.333882\pi$
$504$		0			0
$505$		0			0
$506$			8.48528i			0.377217i
$507$		0			0
$508$	−21.2132			−0.941184
$509$	18.9737			0.840993			0.420496	−	0.907294i	$-0.361856\pi$
$509$	18.9737			0.840993			0.420496	+	0.907294i	$0.361856\pi$
$510$		0			0
$511$	−10.0000	−	13.4164i	−0.442374	−	0.593507i
$512$			1.00000i			0.0441942i
$513$		0			0
$514$			4.47214i			0.197257i
$515$		0			0
$516$		0			0
$517$			6.32456i			0.278154i
$518$	−6.70820	−	9.00000i	−0.294742	−	0.395437i
$519$		0			0
$520$		0			0
$521$	28.4605			1.24688			0.623439	−	0.781872i	$-0.285735\pi$
$521$	28.4605			1.24688			0.623439	+	0.781872i	$0.285735\pi$
$522$		0			0
$523$			6.32456i			0.276553i	0.990394	+	0.138277i	$0.0441563\pi$
$523$			6.32456i			0.276553i	−0.990394	+	0.138277i	$0.955844\pi$
$524$	−9.48683			−0.414434
$525$		0			0
$526$	6.00000			0.261612
$527$		0			0
$528$		0			0
$529$	−13.0000			−0.565217
$530$		0			0
$531$		0			0
$532$		0			0
$533$			30.0000i			1.29944i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$			18.9737i			0.818013i
$539$	9.48683	+	2.82843i	0.408627	+	0.121829i
$540$		0			0
$541$	−22.0000			−0.945854			−0.472927	−	0.881102i	$-0.656803\pi$
$541$	−22.0000			−0.945854			−0.472927	+	0.881102i	$0.656803\pi$
$542$		0			0
$543$		0			0
$544$			4.47214i			0.191741i
$545$		0			0
$546$		0			0
$547$	42.4264			1.81402			0.907011	−	0.421107i	$-0.138358\pi$
$547$	42.4264			1.81402			0.907011	+	0.421107i	$0.138358\pi$
$548$			18.0000i			0.768922i
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−8.48528	+	6.32456i	−0.360831	+	0.268947i
$554$		−	21.2132i		−	0.901263i
$555$		0			0
$556$		−	13.4164i		−	0.568982i
$557$		−	18.0000i		−	0.762684i	−0.924434	−	0.381342i	$-0.875462\pi$
$557$		−	18.0000i		−	0.762684i	0.924434	−	0.381342i	$-0.124538\pi$
$558$		0			0
$559$		−	26.8328i		−	1.13491i
$560$		0			0
$561$		0			0
$562$	−1.41421			−0.0596550
$563$	35.7771			1.50782			0.753912	−	0.656975i	$-0.228164\pi$
$563$	35.7771			1.50782			0.753912	+	0.656975i	$0.228164\pi$
$564$		0			0
$565$		0			0
$566$	−6.32456			−0.265841
$567$		0			0
$568$	5.65685			0.237356
$569$		−	9.89949i		−	0.415008i	−0.978234	−	0.207504i	$-0.933466\pi$
$569$		−	9.89949i		−	0.415008i	0.978234	−	0.207504i	$-0.0665341\pi$
$570$		0			0
$571$	−22.0000			−0.920671			−0.460336	−	0.887745i	$-0.652271\pi$
$571$	−22.0000			−0.920671			−0.460336	+	0.887745i	$0.652271\pi$
$572$	−4.47214			−0.186989
$573$		0			0
$574$	15.0000	+	20.1246i	0.626088	+	0.839985i
$575$		0			0
$576$		0			0
$577$		−	25.2982i		−	1.05318i	−0.850120	−	0.526589i	$-0.823471\pi$
$577$		−	25.2982i		−	1.05318i	0.850120	−	0.526589i	$-0.176529\pi$
$578$			3.00000i			0.124784i
$579$		0			0
$580$		0			0
$581$	−18.9737	+	14.1421i	−0.787160	+	0.586715i
$582$		0			0
$583$	8.48528			0.351424
$584$	6.32456			0.261712
$585$		0			0
$586$		−	4.47214i		−	0.184742i
$587$	−8.94427			−0.369170			−0.184585	−	0.982817i	$-0.559094\pi$
$587$	−8.94427			−0.369170			−0.184585	+	0.982817i	$0.559094\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$	4.24264			0.174371
$593$	22.3607			0.918243			0.459122	−	0.888373i	$-0.348164\pi$
$593$	22.3607			0.918243			0.459122	+	0.888373i	$0.348164\pi$
$594$		0			0
$595$		0			0
$596$		−	11.3137i		−	0.463428i
$597$		0			0
$598$		−	18.9737i		−	0.775891i
$599$			2.82843i			0.115566i	0.998329	+	0.0577832i	$0.0184032\pi$
$599$			2.82843i			0.115566i	−0.998329	+	0.0577832i	$0.981597\pi$
$600$		0			0
$601$		−	13.4164i		−	0.547267i	−0.961834	−	0.273633i	$-0.911775\pi$
$601$		−	13.4164i		−	0.547267i	0.961834	−	0.273633i	$-0.0882255\pi$
$602$	−13.4164	−	18.0000i	−0.546812	−	0.733625i
$603$		0			0
$604$	−20.0000			−0.813788
$605$		0			0
$606$		0			0
$607$		−	15.8114i		−	0.641764i	−0.947119	−	0.320882i	$-0.896021\pi$
$607$		−	15.8114i		−	0.641764i	0.947119	−	0.320882i	$-0.103979\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	14.1421i		−	0.572130i
$612$		0			0
$613$	−21.2132			−0.856793			−0.428397	−	0.903591i	$-0.640921\pi$
$613$	−21.2132			−0.856793			−0.428397	+	0.903591i	$0.640921\pi$
$614$	−12.6491			−0.510477
$615$		0			0
$616$	−3.00000	+	2.23607i	−0.120873	+	0.0900937i
$617$			18.0000i			0.724653i	0.932051	+	0.362326i	$0.118017\pi$
$617$			18.0000i			0.724653i	−0.932051	+	0.362326i	$0.881983\pi$
$618$		0			0
$619$		−	40.2492i		−	1.61775i	−0.587979	−	0.808876i	$-0.700076\pi$
$619$		−	40.2492i		−	1.61775i	0.587979	−	0.808876i	$-0.299924\pi$
$620$		0			0
$621$		0			0
$622$		−	18.9737i		−	0.760775i
$623$	20.1246	−	15.0000i	0.806276	−	0.600962i
$624$		0			0
$625$		0			0
$626$	31.6228			1.26390
$627$		0			0
$628$			15.8114i			0.630943i
$629$	18.9737			0.756530
$630$		0			0
$631$	−28.0000			−1.11466			−0.557331	−	0.830290i	$-0.688175\pi$
$631$	−28.0000			−1.11466			−0.557331	+	0.830290i	$0.688175\pi$
$632$		−	4.00000i		−	0.159111i
$633$		0			0
$634$	−12.0000			−0.476581
$635$		0			0
$636$		0			0
$637$	−21.2132	−	6.32456i	−0.840498	−	0.250588i
$638$		−	4.00000i		−	0.158362i
$639$		0			0
$640$		0			0
$641$		−	15.5563i		−	0.614439i	−0.951639	−	0.307219i	$-0.900601\pi$
$641$		−	15.5563i		−	0.614439i	0.951639	−	0.307219i	$-0.0993986\pi$
$642$		0			0
$643$			6.32456i			0.249416i	0.992193	+	0.124708i	$0.0397994\pi$
$643$			6.32456i			0.249416i	−0.992193	+	0.124708i	$0.960201\pi$
$644$	−9.48683	−	12.7279i	−0.373834	−	0.501550i
$645$		0			0
$646$		0			0
$647$	−22.3607			−0.879089			−0.439545	−	0.898221i	$-0.644860\pi$
$647$	−22.3607			−0.879089			−0.439545	+	0.898221i	$0.644860\pi$
$648$		0			0
$649$			13.4164i			0.526640i
$650$		0			0
$651$		0			0
$652$		0			0
$653$			24.0000i			0.939193i	0.882881	+	0.469596i	$0.155601\pi$
$653$			24.0000i			0.939193i	−0.882881	+	0.469596i	$0.844399\pi$
$654$		0			0
$655$		0			0
$656$	−9.48683			−0.370399
$657$		0			0
$658$	−7.07107	−	9.48683i	−0.275659	−	0.369835i
$659$			32.5269i			1.26707i	0.773715	+	0.633534i	$0.218396\pi$
$659$			32.5269i			1.26707i	−0.773715	+	0.633534i	$0.781604\pi$
$660$		0			0
$661$			40.2492i			1.56551i	0.622328	+	0.782757i	$0.286187\pi$
$661$			40.2492i			1.56551i	−0.622328	+	0.782757i	$0.713813\pi$
$662$		−	10.0000i		−	0.388661i
$663$		0			0
$664$		−	8.94427i		−	0.347105i
$665$		0			0
$666$		0			0
$667$	16.9706			0.657103
$668$	8.94427			0.346064
$669$		0			0
$670$		0			0
$671$	−18.9737			−0.732470
$672$		0			0
$673$	−8.48528			−0.327084			−0.163542	−	0.986536i	$-0.552292\pi$
$673$	−8.48528			−0.327084			−0.163542	+	0.986536i	$0.552292\pi$
$674$			25.4558i			0.980522i
$675$		0			0
$676$	−3.00000			−0.115385
$677$	−22.3607			−0.859391			−0.429695	−	0.902974i	$-0.641379\pi$
$677$	−22.3607			−0.859391			−0.429695	+	0.902974i	$0.641379\pi$
$678$		0			0
$679$	−20.0000	−	26.8328i	−0.767530	−	1.02975i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	36.0000i		−	1.37750i	−0.724998	−	0.688751i	$-0.758159\pi$
$683$		−	36.0000i		−	1.37750i	0.724998	−	0.688751i	$-0.241841\pi$
$684$		0			0
$685$		0			0
$686$	−17.3925	+	6.36396i	−0.664050	+	0.242977i
$687$		0			0
$688$	8.48528			0.323498
$689$	−18.9737			−0.722839
$690$		0			0
$691$		0			0		1.00000			$0$
$691$		0			0		−1.00000			$\pi$
$692$	−8.94427			−0.340010
$693$		0			0
$694$	12.0000			0.455514
$695$		0			0
$696$		0			0
$697$	−42.4264			−1.60701
$698$	13.4164			0.507819
$699$		0			0
$700$		0			0
$701$		−	36.7696i		−	1.38877i	−0.719605	−	0.694383i	$-0.755677\pi$
$701$		−	36.7696i		−	1.38877i	0.719605	−	0.694383i	$-0.244323\pi$
$702$		0			0
$703$		0			0
$704$		−	1.41421i		−	0.0533002i
$705$		0			0
$706$		−	31.3050i		−	1.17818i
$707$	40.2492	−	30.0000i	1.51373	−	1.12827i
$708$		0			0
$709$	10.0000			0.375558			0.187779	−	0.982211i	$-0.439871\pi$
$709$	10.0000			0.375558			0.187779	+	0.982211i	$0.439871\pi$
$710$		0			0
$711$		0			0
$712$			9.48683i			0.355534i
$713$		0			0
$714$		0			0
$715$		0			0
$716$			18.3848i			0.687071i
$717$		0			0
$718$	31.1127			1.16112
$719$	37.9473			1.41520			0.707598	−	0.706615i	$-0.249779\pi$
$719$	37.9473			1.41520			0.707598	+	0.706615i	$0.249779\pi$
$720$		0			0
$721$	−25.0000	−	33.5410i	−0.931049	−	1.24913i
$722$			19.0000i			0.707107i
$723$		0			0
$724$		−	13.4164i		−	0.498617i
$725$		0			0
$726$		0			0
$727$			3.16228i			0.117282i	0.998279	+	0.0586412i	$0.0186768\pi$
$727$			3.16228i			0.117282i	−0.998279	+	0.0586412i	$0.981323\pi$
$728$	6.70820	−	5.00000i	0.248623	−	0.185312i
$729$		0			0
$730$		0			0
$731$	37.9473			1.40353
$732$		0			0
$733$		−	41.1096i		−	1.51842i	−0.650847	−	0.759209i	$-0.725586\pi$
$733$		−	41.1096i		−	1.51842i	0.650847	−	0.759209i	$-0.274414\pi$
$734$	−22.1359			−0.817053
$735$		0			0
$736$	6.00000			0.221163
$737$		0			0
$738$		0			0
$739$	34.0000			1.25071			0.625355	−	0.780340i	$-0.284954\pi$
$739$	34.0000			1.25071			0.625355	+	0.780340i	$0.284954\pi$
$740$		0			0
$741$		0			0
$742$	−12.7279	+	9.48683i	−0.467257	+	0.348273i
$743$		−	24.0000i		−	0.880475i	−0.897881	−	0.440237i	$-0.854894\pi$
$743$		−	24.0000i		−	0.880475i	0.897881	−	0.440237i	$-0.145106\pi$
$744$		0			0
$745$		0			0
$746$		−	21.2132i		−	0.776671i
$747$		0			0
$748$		−	6.32456i		−	0.231249i
$749$	18.9737	+	25.4558i	0.693283	+	0.930136i
$750$		0			0
$751$	−40.0000			−1.45962			−0.729810	−	0.683650i	$-0.760392\pi$
$751$	−40.0000			−1.45962			−0.729810	+	0.683650i	$0.760392\pi$
$752$	4.47214			0.163082
$753$		0			0
$754$			8.94427i			0.325731i
$755$		0			0
$756$		0			0
$757$	46.6690			1.69622			0.848108	−	0.529824i	$-0.177742\pi$
$757$	46.6690			1.69622			0.848108	+	0.529824i	$0.177742\pi$
$758$		−	20.0000i		−	0.726433i
$759$		0			0
$760$		0			0
$761$	9.48683			0.343897			0.171949	−	0.985106i	$-0.444994\pi$
$761$	9.48683			0.343897			0.171949	+	0.985106i	$0.444994\pi$
$762$		0			0
$763$	−21.2132	+	15.8114i	−0.767970	+	0.572411i
$764$		−	22.6274i		−	0.818631i
$765$		0			0
$766$		−	4.47214i		−	0.161585i
$767$		−	30.0000i		−	1.08324i
$768$		0			0
$769$			40.2492i			1.45142i	0.687999	+	0.725712i	$0.258490\pi$
$769$			40.2492i			1.45142i	−0.687999	+	0.725712i	$0.741510\pi$
$770$		0			0
$771$		0			0
$772$	8.48528			0.305392
$773$	−31.3050			−1.12596			−0.562980	−	0.826470i	$-0.690345\pi$
$773$	−31.3050			−1.12596			−0.562980	+	0.826470i	$0.690345\pi$
$774$		0			0
$775$		0			0
$776$	12.6491			0.454077
$777$		0			0
$778$	−11.3137			−0.405616
$779$		0			0
$780$		0			0
$781$	−8.00000			−0.286263
$782$	26.8328			0.959540
$783$		0			0
$784$	2.00000	−	6.70820i	0.0714286	−	0.239579i
$785$		0			0
$786$		0			0
$787$			31.6228i			1.12723i	0.826038	+	0.563615i	$0.190590\pi$
$787$			31.6228i			1.12723i	−0.826038	+	0.563615i	$0.809410\pi$
$788$		−	12.0000i		−	0.427482i
$789$		0			0
$790$		0			0
$791$	−9.48683	−	12.7279i	−0.337313	−	0.452553i
$792$		0			0
$793$	42.4264			1.50661
$794$	−22.1359			−0.785575
$795$		0			0
$796$		−	26.8328i		−	0.951064i
$797$	−8.94427			−0.316822			−0.158411	−	0.987373i	$-0.550637\pi$
$797$	−8.94427			−0.316822			−0.158411	+	0.987373i	$0.550637\pi$
$798$		0			0
$799$	20.0000			0.707549
$800$		0			0
$801$		0			0
$802$	−1.41421			−0.0499376
$803$	−8.94427			−0.315637
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$			18.9737i			0.667491i
$809$		−	52.3259i		−	1.83968i	−0.392293	−	0.919840i	$-0.628318\pi$
$809$		−	52.3259i		−	1.83968i	0.392293	−	0.919840i	$-0.371682\pi$
$810$		0			0
$811$			40.2492i			1.41334i	0.707543	+	0.706671i	$0.249804\pi$
$811$			40.2492i			1.41334i	−0.707543	+	0.706671i	$0.750196\pi$
$812$	4.47214	+	6.00000i	0.156941	+	0.210559i
$813$		0			0
$814$	−6.00000			−0.210300
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	19.7990i		−	0.690990i	−0.938421	−	0.345495i	$-0.887711\pi$
$821$		−	19.7990i		−	0.690990i	0.938421	−	0.345495i	$-0.112289\pi$
$822$		0			0
$823$	21.2132			0.739446			0.369723	−	0.929142i	$-0.379453\pi$
$823$	21.2132			0.739446			0.369723	+	0.929142i	$0.379453\pi$
$824$	15.8114			0.550816
$825$		0			0
$826$	−15.0000	−	20.1246i	−0.521917	−	0.700225i
$827$			48.0000i			1.66912i	0.550914	+	0.834562i	$0.314279\pi$
$827$			48.0000i			1.66912i	−0.550914	+	0.834562i	$0.685721\pi$
$828$		0			0
$829$		−	13.4164i		−	0.465971i	−0.972480	−	0.232986i	$-0.925151\pi$
$829$		−	13.4164i		−	0.465971i	0.972480	−	0.232986i	$-0.0748495\pi$
$830$		0			0
$831$		0			0
$832$			3.16228i			0.109632i
$833$	8.94427	−	30.0000i	0.309901	−	1.03944i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$			9.48683i			0.327717i
$839$	−37.9473			−1.31009			−0.655044	−	0.755591i	$-0.727350\pi$
$839$	−37.9473			−1.31009			−0.655044	+	0.755591i	$0.727350\pi$
$840$		0			0
$841$	21.0000			0.724138
$842$		−	22.0000i		−	0.758170i
$843$		0			0
$844$	−20.0000			−0.688428
$845$		0			0
$846$		0			0
$847$	−19.0919	+	14.2302i	−0.656005	+	0.488957i
$848$		−	6.00000i		−	0.206041i
$849$		0			0
$850$		0			0
$851$		−	25.4558i		−	0.872615i
$852$		0			0
$853$		−	3.16228i		−	0.108274i	−0.998534	−	0.0541372i	$-0.982759\pi$
$853$		−	3.16228i		−	0.108274i	0.998534	−	0.0541372i	$-0.0172408\pi$
$854$	28.4605	−	21.2132i	0.973898	−	0.725901i
$855$		0			0
$856$	−12.0000			−0.410152
$857$	31.3050			1.06936			0.534678	−	0.845056i	$-0.320433\pi$
$857$	31.3050			1.06936			0.534678	+	0.845056i	$0.320433\pi$
$858$		0			0
$859$			13.4164i			0.457762i	0.973454	+	0.228881i	$0.0735067\pi$
$859$			13.4164i			0.457762i	−0.973454	+	0.228881i	$0.926493\pi$
$860$		0			0
$861$		0			0
$862$	36.7696			1.25238
$863$			6.00000i			0.204242i	0.994772	+	0.102121i	$0.0325630\pi$
$863$			6.00000i			0.204242i	−0.994772	+	0.102121i	$0.967437\pi$
$864$		0			0
$865$		0			0
$866$	−6.32456			−0.214917
$867$		0			0
$868$		0			0
$869$			5.65685i			0.191896i
$870$		0			0
$871$		0			0
$872$		−	10.0000i		−	0.338643i
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−21.2132			−0.716319			−0.358159	−	0.933660i	$-0.616596\pi$
$877$	−21.2132			−0.716319			−0.358159	+	0.933660i	$0.616596\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	47.4342			1.59810			0.799049	−	0.601266i	$-0.205337\pi$
$881$	47.4342			1.59810			0.799049	+	0.601266i	$0.205337\pi$
$882$		0			0
$883$	33.9411			1.14221			0.571105	−	0.820877i	$-0.306515\pi$
$883$	33.9411			1.14221			0.571105	+	0.820877i	$0.306515\pi$
$884$			14.1421i			0.475651i
$885$		0			0
$886$	24.0000			0.806296
$887$	44.7214			1.50160			0.750798	−	0.660532i	$-0.229669\pi$
$887$	44.7214			1.50160			0.750798	+	0.660532i	$0.229669\pi$
$888$		0			0
$889$	−45.0000	+	33.5410i	−1.50925	+	1.12493i
$890$		0			0
$891$		0			0
$892$			22.1359i			0.741166i
$893$		0			0
$894$		0			0
$895$		0			0
$896$	1.58114	+	2.12132i	0.0528221	+	0.0708683i
$897$		0			0
$898$	18.3848			0.613508
$899$		0			0
$900$		0			0
$901$		−	26.8328i		−	0.893931i
$902$	13.4164			0.446718
$903$		0			0
$904$	6.00000			0.199557
$905$		0			0
$906$		0			0
$907$	−25.4558			−0.845247			−0.422624	−	0.906305i	$-0.638891\pi$
$907$	−25.4558			−0.845247			−0.422624	+	0.906305i	$0.638891\pi$
$908$	−17.8885			−0.593652
$909$		0			0
$910$		0			0
$911$			48.0833i			1.59307i	0.604593	+	0.796535i	$0.293336\pi$
$911$			48.0833i			1.59307i	−0.604593	+	0.796535i	$0.706664\pi$
$912$		0			0
$913$			12.6491i			0.418624i
$914$			16.9706i			0.561336i
$915$		0			0
$916$			13.4164i			0.443291i
$917$	−20.1246	+	15.0000i	−0.664573	+	0.495344i
$918$		0			0
$919$	−20.0000			−0.659739			−0.329870	−	0.944027i	$-0.607005\pi$
$919$	−20.0000			−0.659739			−0.329870	+	0.944027i	$0.607005\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	17.8885			0.588809
$924$		0			0
$925$		0			0
$926$			21.2132i			0.697109i
$927$		0			0
$928$	−2.82843			−0.0928477
$929$	9.48683			0.311253			0.155626	−	0.987816i	$-0.450260\pi$
$929$	9.48683			0.311253			0.155626	+	0.987816i	$0.450260\pi$
$930$		0			0
$931$		0			0
$932$		−	6.00000i		−	0.196537i
$933$		0			0
$934$		−	8.94427i		−	0.292666i
$935$		0			0
$936$		0			0
$937$			12.6491i			0.413228i	0.978422	+	0.206614i	$0.0662445\pi$
$937$			12.6491i			0.413228i	−0.978422	+	0.206614i	$0.933755\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	37.9473			1.23705			0.618524	−	0.785766i	$-0.287731\pi$
$941$	37.9473			1.23705			0.618524	+	0.785766i	$0.287731\pi$
$942$		0			0
$943$			56.9210i			1.85360i
$944$	9.48683			0.308770
$945$		0			0
$946$	−12.0000			−0.390154
$947$			12.0000i			0.389948i	0.980808	+	0.194974i	$0.0624622\pi$
$947$			12.0000i			0.389948i	−0.980808	+	0.194974i	$0.937538\pi$
$948$		0			0
$949$	20.0000			0.649227
$950$		0			0
$951$		0			0
$952$	7.07107	+	9.48683i	0.229175	+	0.307470i
$953$			6.00000i			0.194359i	0.995267	+	0.0971795i	$0.0309821\pi$
$953$			6.00000i			0.194359i	−0.995267	+	0.0971795i	$0.969018\pi$
$954$		0			0
$955$		0			0
$956$		−	11.3137i		−	0.365911i
$957$		0			0
$958$		−	18.9737i		−	0.613011i
$959$	28.4605	+	38.1838i	0.919037	+	1.23302i
$960$		0			0
$961$	31.0000			1.00000
$962$	13.4164			0.432562
$963$		0			0
$964$			13.4164i			0.432113i
$965$		0			0
$966$		0			0
$967$	21.2132			0.682171			0.341085	−	0.940032i	$-0.389205\pi$
$967$	21.2132			0.682171			0.341085	+	0.940032i	$0.389205\pi$
$968$		−	9.00000i		−	0.289271i
$969$		0			0
$970$		0			0
$971$	−28.4605			−0.913341			−0.456670	−	0.889636i	$-0.650958\pi$
$971$	−28.4605			−0.913341			−0.456670	+	0.889636i	$0.650958\pi$
$972$		0			0
$973$	−21.2132	−	28.4605i	−0.680064	−	0.912402i
$974$		−	4.24264i		−	0.135943i
$975$		0			0
$976$			13.4164i			0.429449i
$977$			18.0000i			0.575871i	0.957650	+	0.287936i	$0.0929689\pi$
$977$			18.0000i			0.575871i	−0.957650	+	0.287936i	$0.907031\pi$
$978$		0			0
$979$		−	13.4164i		−	0.428790i
$980$		0			0
$981$		0			0
$982$	15.5563			0.496423
$983$	−4.47214			−0.142639			−0.0713195	−	0.997454i	$-0.522721\pi$
$983$	−4.47214			−0.142639			−0.0713195	+	0.997454i	$0.522721\pi$
$984$		0			0
$985$		0			0
$986$	−12.6491			−0.402830
$987$		0			0
$988$		0			0
$989$		−	50.9117i		−	1.61890i
$990$		0			0
$991$	20.0000			0.635321			0.317660	−	0.948205i	$-0.397103\pi$
$991$	20.0000			0.635321			0.317660	+	0.948205i	$0.397103\pi$
$992$		0			0
$993$		0			0
$994$	12.0000	−	8.94427i	0.380617	−	0.283695i
$995$		0			0
$996$		0			0
$997$		−	53.7587i		−	1.70256i	−0.524715	−	0.851278i	$-0.675828\pi$
$997$		−	53.7587i		−	1.70256i	0.524715	−	0.851278i	$-0.324172\pi$
$998$			10.0000i			0.316544i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3150.2.b.c.251.6		8
3.2	odd	2	inner	3150.2.b.c.251.2		8
5.2	odd	4		630.2.d.a.629.3	yes	4
5.3	odd	4		630.2.d.d.629.2	yes	4
5.4	even	2	inner	3150.2.b.c.251.3		8
7.6	odd	2	inner	3150.2.b.c.251.5		8
15.2	even	4		630.2.d.d.629.1	yes	4
15.8	even	4		630.2.d.a.629.4	yes	4
15.14	odd	2	inner	3150.2.b.c.251.7		8
20.3	even	4		5040.2.k.a.1889.1		4
20.7	even	4		5040.2.k.d.1889.4		4
21.20	even	2	inner	3150.2.b.c.251.1		8
35.13	even	4		630.2.d.d.629.3	yes	4
35.27	even	4		630.2.d.a.629.2	yes	4
35.34	odd	2	inner	3150.2.b.c.251.4		8
60.23	odd	4		5040.2.k.d.1889.3		4
60.47	odd	4		5040.2.k.a.1889.2		4
105.62	odd	4		630.2.d.d.629.4	yes	4
105.83	odd	4		630.2.d.a.629.1	✓	4
105.104	even	2	inner	3150.2.b.c.251.8		8
140.27	odd	4		5040.2.k.d.1889.1		4
140.83	odd	4		5040.2.k.a.1889.4		4
420.83	even	4		5040.2.k.d.1889.2		4
420.167	even	4		5040.2.k.a.1889.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
630.2.d.a.629.1	✓	4	105.83	odd	4
630.2.d.a.629.2	yes	4	35.27	even	4
630.2.d.a.629.3	yes	4	5.2	odd	4
630.2.d.a.629.4	yes	4	15.8	even	4
630.2.d.d.629.1	yes	4	15.2	even	4
630.2.d.d.629.2	yes	4	5.3	odd	4
630.2.d.d.629.3	yes	4	35.13	even	4
630.2.d.d.629.4	yes	4	105.62	odd	4
3150.2.b.c.251.1		8	21.20	even	2	inner
3150.2.b.c.251.2		8	3.2	odd	2	inner
3150.2.b.c.251.3		8	5.4	even	2	inner
3150.2.b.c.251.4		8	35.34	odd	2	inner
3150.2.b.c.251.5		8	7.6	odd	2	inner
3150.2.b.c.251.6		8	1.1	even	1	trivial
3150.2.b.c.251.7		8	15.14	odd	2	inner
3150.2.b.c.251.8		8	105.104	even	2	inner
5040.2.k.a.1889.1		4	20.3	even	4
5040.2.k.a.1889.2		4	60.47	odd	4
5040.2.k.a.1889.3		4	420.167	even	4
5040.2.k.a.1889.4		4	140.83	odd	4
5040.2.k.d.1889.1		4	140.27	odd	4
5040.2.k.d.1889.2		4	420.83	even	4
5040.2.k.d.1889.3		4	60.23	odd	4
5040.2.k.d.1889.4		4	20.7	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3150 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3150.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000 q^{4} +(-2.12132 + 1.58114i) q^{7} -1.00000i q^{8} +O(q^{10})\)	\(q+1.00000i q^{2} -1.00000 q^{4} +(-2.12132 + 1.58114i) q^{7} -1.00000i q^{8} +1.41421i q^{11} -3.16228i q^{13} +(-1.58114 - 2.12132i) q^{14} +1.00000 q^{16} +4.47214 q^{17} -1.41421 q^{22} -6.00000i q^{23} +3.16228 q^{26} +(2.12132 - 1.58114i) q^{28} +2.82843i q^{29} +1.00000i q^{32} +4.47214i q^{34} +4.24264 q^{37} -9.48683 q^{41} +8.48528 q^{43} -1.41421i q^{44} +6.00000 q^{46} +4.47214 q^{47} +(2.00000 - 6.70820i) q^{49} +3.16228i q^{52} -6.00000i q^{53} +(1.58114 + 2.12132i) q^{56} -2.82843 q^{58} +9.48683 q^{59} +13.4164i q^{61} -1.00000 q^{64} -4.47214 q^{68} +5.65685i q^{71} +6.32456i q^{73} +4.24264i q^{74} +(-2.23607 - 3.00000i) q^{77} +4.00000 q^{79} -9.48683i q^{82} +8.94427 q^{83} +8.48528i q^{86} +1.41421 q^{88} -9.48683 q^{89} +(5.00000 + 6.70820i) q^{91} +6.00000i q^{92} +4.47214i q^{94} +12.6491i q^{97} +(6.70820 + 2.00000i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{4}+O(q^{10}) \) 8 * q - 8 * q^4	\( 8 q - 8 q^{4} + 8 q^{16} + 48 q^{46} + 16 q^{49} - 8 q^{64} + 32 q^{79} + 40 q^{91}+O(q^{100}) \) 8 * q - 8 * q^4 + 8 * q^16 + 48 * q^46 + 16 * q^49 - 8 * q^64 + 32 * q^79 + 40 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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