LMFDB - Embedded newform 630.2.d.d.629.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [630,2,Mod(629,630)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(630, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("630.629");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 630 = 2 \cdot 3^{2} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	630.d (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:	$5.03057532734$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{-5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 4x^{2} + 9 $ x^4 - 4*x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			629.4
Root			$-1.58114 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	630.629
Dual form			630.2.d.d.629.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +2.23607i q^{5} +(1.58114 - 2.12132i) q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{4} +2.23607i q^{5} +(1.58114 - 2.12132i) q^{7} +1.00000 q^{8} +2.23607i q^{10} -1.41421i q^{11} +3.16228 q^{13} +(1.58114 - 2.12132i) q^{14} +1.00000 q^{16} +4.47214i q^{17} +2.23607i q^{20} -1.41421i q^{22} +6.00000 q^{23} -5.00000 q^{25} +3.16228 q^{26} +(1.58114 - 2.12132i) q^{28} +2.82843i q^{29} +1.00000 q^{32} +4.47214i q^{34} +(4.74342 + 3.53553i) q^{35} +4.24264i q^{37} +2.23607i q^{40} -9.48683 q^{41} -8.48528i q^{43} -1.41421i q^{44} +6.00000 q^{46} +4.47214i q^{47} +(-2.00000 - 6.70820i) q^{49} -5.00000 q^{50} +3.16228 q^{52} +6.00000 q^{53} +3.16228 q^{55} +(1.58114 - 2.12132i) q^{56} +2.82843i q^{58} -9.48683 q^{59} -13.4164i q^{61} +1.00000 q^{64} +7.07107i q^{65} +4.47214i q^{68} +(4.74342 + 3.53553i) q^{70} -5.65685i q^{71} -6.32456 q^{73} +4.24264i q^{74} +(-3.00000 - 2.23607i) q^{77} -4.00000 q^{79} +2.23607i q^{80} -9.48683 q^{82} -8.94427i q^{83} -10.0000 q^{85} -8.48528i q^{86} -1.41421i q^{88} +9.48683 q^{89} +(5.00000 - 6.70820i) q^{91} +6.00000 q^{92} +4.47214i q^{94} +12.6491 q^{97} +(-2.00000 - 6.70820i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{2} + 4 q^{4} + 4 q^{8}+O(q^{10}) $ 4 * q + 4 * q^2 + 4 * q^4 + 4 * q^8	$ 4 q + 4 q^{2} + 4 q^{4} + 4 q^{8} + 4 q^{16} + 24 q^{23} - 20 q^{25} + 4 q^{32} + 24 q^{46} - 8 q^{49} - 20 q^{50} + 24 q^{53} + 4 q^{64} - 12 q^{77} - 16 q^{79} - 40 q^{85} + 20 q^{91} + 24 q^{92} - 8 q^{98}+O(q^{100}) $ 4 * q + 4 * q^2 + 4 * q^4 + 4 * q^8 + 4 * q^16 + 24 * q^23 - 20 * q^25 + 4 * q^32 + 24 * q^46 - 8 * q^49 - 20 * q^50 + 24 * q^53 + 4 * q^64 - 12 * q^77 - 16 * q^79 - 40 * q^85 + 20 * q^91 + 24 * q^92 - 8 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/630\mathbb{Z}\right)^\times$.

$n$	$127$	$281$	$451$
$\chi(n)$	$-1$	$-1$	$-1$

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 $

$2$	1.00000			0.707107
$2$	1.00000			0.707107
$3$		0			0
$3$		0			0
$4$	1.00000			0.500000
$5$			2.23607i			1.00000i
$5$			2.23607i			1.00000i
$6$		0			0
$7$	1.58114	−	2.12132i	0.597614	−	0.801784i
$7$	1.58114	−	2.12132i	0.597614	−	0.801784i
$8$	1.00000			0.353553
$9$		0			0
$10$			2.23607i			0.707107i
$11$		−	1.41421i		−	0.426401i	−0.977008	−	0.213201i	$-0.931611\pi$
$11$		−	1.41421i		−	0.426401i	0.977008	−	0.213201i	$-0.0683888\pi$
$12$		0			0
$13$	3.16228			0.877058			0.438529	−	0.898717i	$-0.355500\pi$
$13$	3.16228			0.877058			0.438529	+	0.898717i	$0.355500\pi$
$14$	1.58114	−	2.12132i	0.422577	−	0.566947i
$15$		0			0
$16$	1.00000			0.250000
$17$			4.47214i			1.08465i	0.840168	+	0.542326i	$0.182456\pi$
$17$			4.47214i			1.08465i	−0.840168	+	0.542326i	$0.817544\pi$
$18$		0			0
$19$		0			0		1.00000			$0$
$19$		0			0		−1.00000			$\pi$
$20$			2.23607i			0.500000i
$21$		0			0
$22$		−	1.41421i		−	0.301511i
$23$	6.00000			1.25109			0.625543	−	0.780189i	$-0.284877\pi$
$23$	6.00000			1.25109			0.625543	+	0.780189i	$0.284877\pi$
$24$		0			0
$25$	−5.00000			−1.00000
$26$	3.16228			0.620174
$27$		0			0
$28$	1.58114	−	2.12132i	0.298807	−	0.400892i
$29$			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			$\pi$
$32$	1.00000			0.176777
$33$		0			0
$34$			4.47214i			0.766965i
$35$	4.74342	+	3.53553i	0.801784	+	0.597614i
$36$		0			0
$37$			4.24264i			0.697486i	0.937218	+	0.348743i	$0.113391\pi$
$37$			4.24264i			0.697486i	−0.937218	+	0.348743i	$0.886609\pi$
$38$		0			0
$39$		0			0
$40$			2.23607i			0.353553i
$41$	−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
$43$		−	8.48528i		−	1.29399i	−0.762493	−	0.646997i	$-0.776025\pi$
$43$		−	8.48528i		−	1.29399i	0.762493	−	0.646997i	$-0.223975\pi$
$44$		−	1.41421i		−	0.213201i
$45$		0			0
$46$	6.00000			0.884652
$47$			4.47214i			0.652328i	0.945313	+	0.326164i	$0.105756\pi$
$47$			4.47214i			0.652328i	−0.945313	+	0.326164i	$0.894244\pi$
$48$		0			0
$49$	−2.00000	−	6.70820i	−0.285714	−	0.958315i
$50$	−5.00000			−0.707107
$51$		0			0
$52$	3.16228			0.438529
$53$	6.00000			0.824163			0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000			0.824163			0.412082	+	0.911147i	$0.364802\pi$
$54$		0			0
$55$	3.16228			0.426401
$56$	1.58114	−	2.12132i	0.211289	−	0.283473i
$57$		0			0
$58$			2.82843i			0.371391i
$59$	−9.48683			−1.23508			−0.617540	−	0.786539i	$-0.711871\pi$
$59$	−9.48683			−1.23508			−0.617540	+	0.786539i	$0.711871\pi$
$60$		0			0
$61$		−	13.4164i		−	1.71780i	−0.512148	−	0.858898i	$-0.671150\pi$
$61$		−	13.4164i		−	1.71780i	0.512148	−	0.858898i	$-0.328850\pi$
$62$		0			0
$63$		0			0
$64$	1.00000			0.125000
$65$			7.07107i			0.877058i
$66$		0			0
$67$		0			0		1.00000			$0$
$67$		0			0		−1.00000			$\pi$
$68$			4.47214i			0.542326i
$69$		0			0
$70$	4.74342	+	3.53553i	0.566947	+	0.422577i
$71$		−	5.65685i		−	0.671345i	−0.941979	−	0.335673i	$-0.891036\pi$
$71$		−	5.65685i		−	0.671345i	0.941979	−	0.335673i	$-0.108964\pi$
$72$		0			0
$73$	−6.32456			−0.740233			−0.370117	−	0.928985i	$-0.620682\pi$
$73$	−6.32456			−0.740233			−0.370117	+	0.928985i	$0.620682\pi$
$74$			4.24264i			0.493197i
$75$		0			0
$76$		0			0
$77$	−3.00000	−	2.23607i	−0.341882	−	0.254824i
$78$		0			0
$79$	−4.00000			−0.450035			−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000			−0.450035			−0.225018	+	0.974355i	$0.572244\pi$
$80$			2.23607i			0.250000i
$81$		0			0
$82$	−9.48683			−1.04765
$83$		−	8.94427i		−	0.981761i	−0.871227	−	0.490881i	$-0.836675\pi$
$83$		−	8.94427i		−	0.981761i	0.871227	−	0.490881i	$-0.163325\pi$
$84$		0			0
$85$	−10.0000			−1.08465
$86$		−	8.48528i		−	0.914991i
$87$		0			0
$88$		−	1.41421i		−	0.150756i
$89$	9.48683			1.00560			0.502801	−	0.864402i	$-0.332303\pi$
$89$	9.48683			1.00560			0.502801	+	0.864402i	$0.332303\pi$
$90$		0			0
$91$	5.00000	−	6.70820i	0.524142	−	0.703211i
$92$	6.00000			0.625543
$93$		0			0
$94$			4.47214i			0.461266i
$95$		0			0
$96$		0			0
$97$	12.6491			1.28432			0.642161	−	0.766570i	$-0.278038\pi$
$97$	12.6491			1.28432			0.642161	+	0.766570i	$0.278038\pi$
$98$	−2.00000	−	6.70820i	−0.202031	−	0.677631i
$99$		0			0
$100$	−5.00000			−0.500000
$101$	−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
$103$	−15.8114			−1.55794			−0.778971	−	0.627060i	$-0.784258\pi$
$103$	−15.8114			−1.55794			−0.778971	+	0.627060i	$0.784258\pi$
$104$	3.16228			0.310087
$105$		0			0
$106$	6.00000			0.582772
$107$	−12.0000			−1.16008			−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−12.0000			−1.16008			−0.580042	+	0.814587i	$0.696964\pi$
$108$		0			0
$109$	−10.0000			−0.957826			−0.478913	−	0.877862i	$-0.658969\pi$
$109$	−10.0000			−0.957826			−0.478913	+	0.877862i	$0.658969\pi$
$110$	3.16228			0.301511
$111$		0			0
$112$	1.58114	−	2.12132i	0.149404	−	0.200446i
$113$	−6.00000			−0.564433			−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000			−0.564433			−0.282216	+	0.959351i	$0.591070\pi$
$114$		0			0
$115$			13.4164i			1.25109i
$116$			2.82843i			0.262613i
$117$		0			0
$118$	−9.48683			−0.873334
$119$	9.48683	+	7.07107i	0.869657	+	0.648204i
$120$		0			0
$121$	9.00000			0.818182
$122$		−	13.4164i		−	1.21466i
$123$		0			0
$124$		0			0
$125$		−	11.1803i		−	1.00000i
$126$		0			0
$127$			21.2132i			1.88237i	0.337895	+	0.941184i	$0.390285\pi$
$127$			21.2132i			1.88237i	−0.337895	+	0.941184i	$0.609715\pi$
$128$	1.00000			0.0883883
$129$		0			0
$130$			7.07107i			0.620174i
$131$	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
$137$	−18.0000			−1.53784			−0.768922	−	0.639343i	$-0.779207\pi$
$137$	−18.0000			−1.53784			−0.768922	+	0.639343i	$0.779207\pi$
$138$		0			0
$139$			13.4164i			1.13796i	0.822350	+	0.568982i	$0.192663\pi$
$139$			13.4164i			1.13796i	−0.822350	+	0.568982i	$0.807337\pi$
$140$	4.74342	+	3.53553i	0.400892	+	0.298807i
$141$		0			0
$142$		−	5.65685i		−	0.474713i
$143$		−	4.47214i		−	0.373979i
$144$		0			0
$145$	−6.32456			−0.525226
$146$	−6.32456			−0.523424
$147$		0			0
$148$			4.24264i			0.348743i
$149$			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
$157$	−15.8114			−1.26189			−0.630943	−	0.775829i	$-0.717332\pi$
$157$	−15.8114			−1.26189			−0.630943	+	0.775829i	$0.717332\pi$
$158$	−4.00000			−0.318223
$159$		0			0
$160$			2.23607i			0.176777i
$161$	9.48683	−	12.7279i	0.747667	−	1.00310i
$162$		0			0
$163$		0			0		1.00000			$0$
$163$		0			0		−1.00000			$\pi$
$164$	−9.48683			−0.740797
$165$		0			0
$166$		−	8.94427i		−	0.694210i
$167$		−	8.94427i		−	0.692129i	−0.938211	−	0.346064i	$-0.887518\pi$
$167$		−	8.94427i		−	0.692129i	0.938211	−	0.346064i	$-0.112482\pi$
$168$		0			0
$169$	−3.00000			−0.230769
$170$	−10.0000			−0.766965
$171$		0			0
$172$		−	8.48528i		−	0.646997i
$173$		−	8.94427i		−	0.680020i	−0.940422	−	0.340010i	$-0.889569\pi$
$173$		−	8.94427i		−	0.680020i	0.940422	−	0.340010i	$-0.110431\pi$
$174$		0			0
$175$	−7.90569	+	10.6066i	−0.597614	+	0.801784i
$176$		−	1.41421i		−	0.106600i
$177$		0			0
$178$	9.48683			0.711068
$179$		−	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	$-0.833841\pi$
$181$		−	13.4164i		−	0.997234i	0.866822	−	0.498617i	$-0.166159\pi$
$182$	5.00000	−	6.70820i	0.370625	−	0.497245i
$183$		0			0
$184$	6.00000			0.442326
$185$	−9.48683			−0.697486
$186$		0			0
$187$	6.32456			0.462497
$188$			4.47214i			0.326164i
$189$		0			0
$190$		0			0
$191$		−	22.6274i		−	1.63726i	−0.574320	−	0.818631i	$-0.694733\pi$
$191$		−	22.6274i		−	1.63726i	0.574320	−	0.818631i	$-0.305267\pi$
$192$		0			0
$193$			8.48528i			0.610784i	0.952227	+	0.305392i	$0.0987875\pi$
$193$			8.48528i			0.610784i	−0.952227	+	0.305392i	$0.901213\pi$
$194$	12.6491			0.908153
$195$		0			0
$196$	−2.00000	−	6.70820i	−0.142857	−	0.479157i
$197$	12.0000			0.854965			0.427482	−	0.904024i	$-0.359401\pi$
$197$	12.0000			0.854965			0.427482	+	0.904024i	$0.359401\pi$
$198$		0			0
$199$			26.8328i			1.90213i	0.308994	+	0.951064i	$0.400008\pi$
$199$			26.8328i			1.90213i	−0.308994	+	0.951064i	$0.599992\pi$
$200$	−5.00000			−0.353553
$201$		0			0
$202$	−18.9737			−1.33498
$203$	6.00000	+	4.47214i	0.421117	+	0.313882i
$204$		0			0
$205$		−	21.2132i		−	1.48159i
$206$	−15.8114			−1.10163
$207$		0			0
$208$	3.16228			0.219265
$209$		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	$0.258301\pi$
$212$	6.00000			0.412082
$213$		0			0
$214$	−12.0000			−0.820303
$215$	18.9737			1.29399
$216$		0			0
$217$		0			0
$218$	−10.0000			−0.677285
$219$		0			0
$220$	3.16228			0.213201
$221$			14.1421i			0.951303i
$222$		0			0
$223$	22.1359			1.48233			0.741166	−	0.671322i	$-0.234273\pi$
$223$	22.1359			1.48233			0.741166	+	0.671322i	$0.234273\pi$
$224$	1.58114	−	2.12132i	0.105644	−	0.141737i
$225$		0			0
$226$	−6.00000			−0.399114
$227$			17.8885i			1.18730i	0.804722	+	0.593652i	$0.202314\pi$
$227$			17.8885i			1.18730i	−0.804722	+	0.593652i	$0.797686\pi$
$228$		0			0
$229$		−	13.4164i		−	0.886581i	−0.896378	−	0.443291i	$-0.853811\pi$
$229$		−	13.4164i		−	0.886581i	0.896378	−	0.443291i	$-0.146189\pi$
$230$			13.4164i			0.884652i
$231$		0			0
$232$			2.82843i			0.185695i
$233$	−6.00000			−0.393073			−0.196537	−	0.980497i	$-0.562969\pi$
$233$	−6.00000			−0.393073			−0.196537	+	0.980497i	$0.562969\pi$
$234$		0			0
$235$	−10.0000			−0.652328
$236$	−9.48683			−0.617540
$237$		0			0
$238$	9.48683	+	7.07107i	0.614940	+	0.458349i
$239$			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	$0.142232\pi$
$241$			13.4164i			0.864227i	−0.901819	+	0.432113i	$0.857768\pi$
$242$	9.00000			0.578542
$243$		0			0
$244$		−	13.4164i		−	0.858898i
$245$	15.0000	−	4.47214i	0.958315	−	0.285714i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		−	11.1803i		−	0.707107i
$251$	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
$253$		−	8.48528i		−	0.533465i
$254$			21.2132i			1.33103i
$255$		0			0
$256$	1.00000			0.0625000
$257$			4.47214i			0.278964i	0.990225	+	0.139482i	$0.0445438\pi$
$257$			4.47214i			0.278964i	−0.990225	+	0.139482i	$0.955456\pi$
$258$		0			0
$259$	9.00000	+	6.70820i	0.559233	+	0.416828i
$260$			7.07107i			0.438529i
$261$		0			0
$262$	9.48683			0.586098
$263$	6.00000			0.369976			0.184988	−	0.982741i	$-0.440775\pi$
$263$	6.00000			0.369976			0.184988	+	0.982741i	$0.440775\pi$
$264$		0			0
$265$			13.4164i			0.824163i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−18.9737			−1.15684			−0.578422	−	0.815737i	$-0.696331\pi$
$269$	−18.9737			−1.15684			−0.578422	+	0.815737i	$0.696331\pi$
$270$		0			0
$271$		0			0		1.00000			$0$
$271$		0			0		−1.00000			$\pi$
$272$			4.47214i			0.271163i
$273$		0			0
$274$	−18.0000			−1.08742
$275$			7.07107i			0.426401i
$276$		0			0
$277$		−	21.2132i		−	1.27458i	−0.770625	−	0.637289i	$-0.780056\pi$
$277$		−	21.2132i		−	1.27458i	0.770625	−	0.637289i	$-0.219944\pi$
$278$			13.4164i			0.804663i
$279$		0			0
$280$	4.74342	+	3.53553i	0.283473	+	0.211289i
$281$		−	1.41421i		−	0.0843649i	−0.999110	−	0.0421825i	$-0.986569\pi$
$281$		−	1.41421i		−	0.0843649i	0.999110	−	0.0421825i	$-0.0134311\pi$
$282$		0			0
$283$	−6.32456			−0.375956			−0.187978	−	0.982173i	$-0.560193\pi$
$283$	−6.32456			−0.375956			−0.187978	+	0.982173i	$0.560193\pi$
$284$		−	5.65685i		−	0.335673i
$285$		0			0
$286$		−	4.47214i		−	0.264443i
$287$	−15.0000	+	20.1246i	−0.885422	+	1.18792i
$288$		0			0
$289$	−3.00000			−0.176471
$290$	−6.32456			−0.371391
$291$		0			0
$292$	−6.32456			−0.370117
$293$			4.47214i			0.261265i	0.991431	+	0.130632i	$0.0417008\pi$
$293$			4.47214i			0.261265i	−0.991431	+	0.130632i	$0.958299\pi$
$294$		0			0
$295$		−	21.2132i		−	1.23508i
$296$			4.24264i			0.246598i
$297$		0			0
$298$			11.3137i			0.655386i
$299$	18.9737			1.09728
$300$		0			0
$301$	−18.0000	−	13.4164i	−1.03750	−	0.773309i
$302$	20.0000			1.15087
$303$		0			0
$304$		0			0
$305$	30.0000			1.71780
$306$		0			0
$307$	12.6491			0.721923			0.360961	−	0.932581i	$-0.382449\pi$
$307$	12.6491			0.721923			0.360961	+	0.932581i	$0.382449\pi$
$308$	−3.00000	−	2.23607i	−0.170941	−	0.127412i
$309$		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
$313$	31.6228			1.78743			0.893713	−	0.448640i	$-0.148091\pi$
$313$	31.6228			1.78743			0.893713	+	0.448640i	$0.148091\pi$
$314$	−15.8114			−0.892288
$315$		0			0
$316$	−4.00000			−0.225018
$317$	12.0000			0.673987			0.336994	−	0.941507i	$-0.390590\pi$
$317$	12.0000			0.673987			0.336994	+	0.941507i	$0.390590\pi$
$318$		0			0
$319$	4.00000			0.223957
$320$			2.23607i			0.125000i
$321$		0			0
$322$	9.48683	−	12.7279i	0.528681	−	0.709299i
$323$		0			0
$324$		0			0
$325$	−15.8114			−0.877058
$326$		0			0
$327$		0			0
$328$	−9.48683			−0.523823
$329$	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	$0.588620\pi$
$332$		−	8.94427i		−	0.490881i
$333$		0			0
$334$		−	8.94427i		−	0.489409i
$335$		0			0
$336$		0			0
$337$			25.4558i			1.38667i	0.720616	+	0.693334i	$0.243859\pi$
$337$			25.4558i			1.38667i	−0.720616	+	0.693334i	$0.756141\pi$
$338$	−3.00000			−0.163178
$339$		0			0
$340$	−10.0000			−0.542326
$341$		0			0
$342$		0			0
$343$	−17.3925	−	6.36396i	−0.939108	−	0.343622i
$344$		−	8.48528i		−	0.457496i
$345$		0			0
$346$		−	8.94427i		−	0.480847i
$347$	−12.0000			−0.644194			−0.322097	−	0.946707i	$-0.604388\pi$
$347$	−12.0000			−0.644194			−0.322097	+	0.946707i	$0.604388\pi$
$348$		0			0
$349$		−	13.4164i		−	0.718164i	−0.933306	−	0.359082i	$-0.883090\pi$
$349$		−	13.4164i		−	0.718164i	0.933306	−	0.359082i	$-0.116910\pi$
$350$	−7.90569	+	10.6066i	−0.422577	+	0.566947i
$351$		0			0
$352$		−	1.41421i		−	0.0753778i
$353$			31.3050i			1.66619i	0.553127	+	0.833097i	$0.313435\pi$
$353$			31.3050i			1.66619i	−0.553127	+	0.833097i	$0.686565\pi$
$354$		0			0
$355$	12.6491			0.671345
$356$	9.48683			0.502801
$357$		0			0
$358$		−	18.3848i		−	0.971666i
$359$		−	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
$362$		−	13.4164i		−	0.705151i
$363$		0			0
$364$	5.00000	−	6.70820i	0.262071	−	0.351605i
$365$		−	14.1421i		−	0.740233i
$366$		0			0
$367$	22.1359			1.15549			0.577743	−	0.816218i	$-0.303933\pi$
$367$	22.1359			1.15549			0.577743	+	0.816218i	$0.303933\pi$
$368$	6.00000			0.312772
$369$		0			0
$370$	−9.48683			−0.493197
$371$	9.48683	−	12.7279i	0.492532	−	0.660801i
$372$		0			0
$373$			21.2132i			1.09838i	0.835698	+	0.549189i	$0.185063\pi$
$373$			21.2132i			1.09838i	−0.835698	+	0.549189i	$0.814937\pi$
$374$	6.32456			0.327035
$375$		0			0
$376$			4.47214i			0.230633i
$377$			8.94427i			0.460653i
$378$		0			0
$379$	20.0000			1.02733			0.513665	−	0.857991i	$-0.328287\pi$
$379$	20.0000			1.02733			0.513665	+	0.857991i	$0.328287\pi$
$380$		0			0
$381$		0			0
$382$		−	22.6274i		−	1.15772i
$383$			4.47214i			0.228515i	0.993451	+	0.114258i	$0.0364490\pi$
$383$			4.47214i			0.228515i	−0.993451	+	0.114258i	$0.963551\pi$
$384$		0			0
$385$	5.00000	−	6.70820i	0.254824	−	0.341882i
$386$			8.48528i			0.431889i
$387$		0			0
$388$	12.6491			0.642161
$389$			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
$392$	−2.00000	−	6.70820i	−0.101015	−	0.338815i
$393$		0			0
$394$	12.0000			0.604551
$395$		−	8.94427i		−	0.450035i
$396$		0			0
$397$	22.1359			1.11097			0.555486	−	0.831526i	$-0.312532\pi$
$397$	22.1359			1.11097			0.555486	+	0.831526i	$0.312532\pi$
$398$			26.8328i			1.34501i
$399$		0			0
$400$	−5.00000			−0.250000
$401$		−	1.41421i		−	0.0706225i	−0.999376	−	0.0353112i	$-0.988758\pi$
$401$		−	1.41421i		−	0.0706225i	0.999376	−	0.0353112i	$-0.0112422\pi$
$402$		0			0
$403$		0			0
$404$	−18.9737			−0.943975
$405$		0			0
$406$	6.00000	+	4.47214i	0.297775	+	0.221948i
$407$	6.00000			0.297409
$408$		0			0
$409$		0			0		1.00000			$0$
$409$		0			0		−1.00000			$\pi$
$410$		−	21.2132i		−	1.04765i
$411$		0			0
$412$	−15.8114			−0.778971
$413$	−15.0000	+	20.1246i	−0.738102	+	0.990267i
$414$		0			0
$415$	20.0000			0.981761
$416$	3.16228			0.155043
$417$		0			0
$418$		0			0
$419$	−9.48683			−0.463462			−0.231731	−	0.972780i	$-0.574439\pi$
$419$	−9.48683			−0.463462			−0.231731	+	0.972780i	$0.574439\pi$
$420$		0			0
$421$	−22.0000			−1.07221			−0.536107	−	0.844150i	$-0.680106\pi$
$421$	−22.0000			−1.07221			−0.536107	+	0.844150i	$0.680106\pi$
$422$	20.0000			0.973585
$423$		0			0
$424$	6.00000			0.291386
$425$		−	22.3607i		−	1.08465i
$426$		0			0
$427$	−28.4605	−	21.2132i	−1.37730	−	1.02658i
$428$	−12.0000			−0.580042
$429$		0			0
$430$	18.9737			0.914991
$431$			36.7696i			1.77113i	0.464518	+	0.885564i	$0.346227\pi$
$431$			36.7696i			1.77113i	−0.464518	+	0.885564i	$0.653773\pi$
$432$		0			0
$433$	−6.32456			−0.303939			−0.151969	−	0.988385i	$-0.548562\pi$
$433$	−6.32456			−0.303939			−0.151969	+	0.988385i	$0.548562\pi$
$434$		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			$\pi$
$440$	3.16228			0.150756
$441$		0			0
$442$			14.1421i			0.672673i
$443$	24.0000			1.14027			0.570137	−	0.821549i	$-0.306890\pi$
$443$	24.0000			1.14027			0.570137	+	0.821549i	$0.306890\pi$
$444$		0			0
$445$			21.2132i			1.00560i
$446$	22.1359			1.04817
$447$		0			0
$448$	1.58114	−	2.12132i	0.0747018	−	0.100223i
$449$		−	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
$452$	−6.00000			−0.282216
$453$		0			0
$454$			17.8885i			0.839551i
$455$	15.0000	+	11.1803i	0.703211	+	0.524142i
$456$		0			0
$457$			16.9706i			0.793849i	0.917851	+	0.396925i	$0.129923\pi$
$457$			16.9706i			0.793849i	−0.917851	+	0.396925i	$0.870077\pi$
$458$		−	13.4164i		−	0.626908i
$459$		0			0
$460$			13.4164i			0.625543i
$461$		0			0			−	1.00000i	$-0.5\pi$
$461$		0			0				1.00000i	$0.5\pi$
$462$		0			0
$463$		−	21.2132i		−	0.985861i	−0.870069	−	0.492931i	$-0.835926\pi$
$463$		−	21.2132i		−	0.985861i	0.870069	−	0.492931i	$-0.164074\pi$
$464$			2.82843i			0.131306i
$465$		0			0
$466$	−6.00000			−0.277945
$467$		−	8.94427i		−	0.413892i	−0.978352	−	0.206946i	$-0.933648\pi$
$467$		−	8.94427i		−	0.413892i	0.978352	−	0.206946i	$-0.0663524\pi$
$468$		0			0
$469$		0			0
$470$	−10.0000			−0.461266
$471$		0			0
$472$	−9.48683			−0.436667
$473$	−12.0000			−0.551761
$474$		0			0
$475$		0			0
$476$	9.48683	+	7.07107i	0.434828	+	0.324102i
$477$		0			0
$478$			11.3137i			0.517477i
$479$	18.9737			0.866929			0.433464	−	0.901171i	$-0.357291\pi$
$479$	18.9737			0.866929			0.433464	+	0.901171i	$0.357291\pi$
$480$		0			0
$481$			13.4164i			0.611736i
$482$			13.4164i			0.611101i
$483$		0			0
$484$	9.00000			0.409091
$485$			28.2843i			1.28432i
$486$		0			0
$487$		−	4.24264i		−	0.192252i	−0.995369	−	0.0961262i	$-0.969355\pi$
$487$		−	4.24264i		−	0.192252i	0.995369	−	0.0961262i	$-0.0306452\pi$
$488$		−	13.4164i		−	0.607332i
$489$		0			0
$490$	15.0000	−	4.47214i	0.677631	−	0.202031i
$491$			15.5563i			0.702048i	0.936366	+	0.351024i	$0.114166\pi$
$491$			15.5563i			0.702048i	−0.936366	+	0.351024i	$0.885834\pi$
$492$		0			0
$493$	−12.6491			−0.569687
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−12.0000	−	8.94427i	−0.538274	−	0.401205i
$498$		0			0
$499$	−10.0000			−0.447661			−0.223831	−	0.974628i	$-0.571856\pi$
$499$	−10.0000			−0.447661			−0.223831	+	0.974628i	$0.571856\pi$
$500$		−	11.1803i		−	0.500000i
$501$		0			0
$502$	9.48683			0.423418
$503$		−	22.3607i		−	0.997013i	−0.866886	−	0.498507i	$-0.833882\pi$
$503$		−	22.3607i		−	0.997013i	0.866886	−	0.498507i	$-0.166118\pi$
$504$		0			0
$505$		−	42.4264i		−	1.88795i
$506$		−	8.48528i		−	0.377217i
$507$		0			0
$508$			21.2132i			0.941184i
$509$	−18.9737			−0.840993			−0.420496	−	0.907294i	$-0.638144\pi$
$509$	−18.9737			−0.840993			−0.420496	+	0.907294i	$0.638144\pi$
$510$		0			0
$511$	−10.0000	+	13.4164i	−0.442374	+	0.593507i
$512$	1.00000			0.0441942
$513$		0			0
$514$			4.47214i			0.197257i
$515$		−	35.3553i		−	1.55794i
$516$		0			0
$517$	6.32456			0.278154
$518$	9.00000	+	6.70820i	0.395437	+	0.294742i
$519$		0			0
$520$			7.07107i			0.310087i
$521$	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
$523$	−6.32456			−0.276553			−0.138277	−	0.990394i	$-0.544156\pi$
$523$	−6.32456			−0.276553			−0.138277	+	0.990394i	$0.544156\pi$
$524$	9.48683			0.414434
$525$		0			0
$526$	6.00000			0.261612
$527$		0			0
$528$		0			0
$529$	13.0000			0.565217
$530$			13.4164i			0.582772i
$531$		0			0
$532$		0			0
$533$	−30.0000			−1.29944
$534$		0			0
$535$		−	26.8328i		−	1.16008i
$536$		0			0
$537$		0			0
$538$	−18.9737			−0.818013
$539$	−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
$545$		−	22.3607i		−	0.957826i
$546$		0			0
$547$			42.4264i			1.81402i	0.421107	+	0.907011i	$0.361642\pi$
$547$			42.4264i			1.81402i	−0.421107	+	0.907011i	$0.638358\pi$
$548$	−18.0000			−0.768922
$549$		0			0
$550$			7.07107i			0.301511i
$551$		0			0
$552$		0			0
$553$	−6.32456	+	8.48528i	−0.268947	+	0.360831i
$554$		−	21.2132i		−	0.901263i
$555$		0			0
$556$			13.4164i			0.568982i
$557$	−18.0000			−0.762684			−0.381342	−	0.924434i	$-0.624538\pi$
$557$	−18.0000			−0.762684			−0.381342	+	0.924434i	$0.624538\pi$
$558$		0			0
$559$		−	26.8328i		−	1.13491i
$560$	4.74342	+	3.53553i	0.200446	+	0.149404i
$561$		0			0
$562$		−	1.41421i		−	0.0596550i
$563$		−	35.7771i		−	1.50782i	−0.656975	−	0.753912i	$-0.728164\pi$
$563$		−	35.7771i		−	1.50782i	0.656975	−	0.753912i	$-0.271836\pi$
$564$		0			0
$565$		−	13.4164i		−	0.564433i
$566$	−6.32456			−0.265841
$567$		0			0
$568$		−	5.65685i		−	0.237356i
$569$		−	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	$0.652271\pi$
$572$		−	4.47214i		−	0.186989i
$573$		0			0
$574$	−15.0000	+	20.1246i	−0.626088	+	0.839985i
$575$	−30.0000			−1.25109
$576$		0			0
$577$	−25.2982			−1.05318			−0.526589	−	0.850120i	$-0.676529\pi$
$577$	−25.2982			−1.05318			−0.526589	+	0.850120i	$0.676529\pi$
$578$	−3.00000			−0.124784
$579$		0			0
$580$	−6.32456			−0.262613
$581$	−18.9737	−	14.1421i	−0.787160	−	0.586715i
$582$		0			0
$583$		−	8.48528i		−	0.351424i
$584$	−6.32456			−0.261712
$585$		0			0
$586$			4.47214i			0.184742i
$587$		−	8.94427i		−	0.369170i	−0.982817	−	0.184585i	$-0.940906\pi$
$587$		−	8.94427i		−	0.369170i	0.982817	−	0.184585i	$-0.0590940\pi$
$588$		0			0
$589$		0			0
$590$		−	21.2132i		−	0.873334i
$591$		0			0
$592$			4.24264i			0.174371i
$593$		−	22.3607i		−	0.918243i	−0.888373	−	0.459122i	$-0.848164\pi$
$593$		−	22.3607i		−	0.918243i	0.888373	−	0.459122i	$-0.151836\pi$
$594$		0			0
$595$	−15.8114	+	21.2132i	−0.648204	+	0.869657i
$596$			11.3137i			0.463428i
$597$		0			0
$598$	18.9737			0.775891
$599$			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	$0.0882255\pi$
$601$			13.4164i			0.547267i	−0.961834	+	0.273633i	$0.911775\pi$
$602$	−18.0000	−	13.4164i	−0.733625	−	0.546812i
$603$		0			0
$604$	20.0000			0.813788
$605$			20.1246i			0.818182i
$606$		0			0
$607$	−15.8114			−0.641764			−0.320882	−	0.947119i	$-0.603979\pi$
$607$	−15.8114			−0.641764			−0.320882	+	0.947119i	$0.603979\pi$
$608$		0			0
$609$		0			0
$610$	30.0000			1.21466
$611$			14.1421i			0.572130i
$612$		0			0
$613$			21.2132i			0.856793i	0.903591	+	0.428397i	$0.140921\pi$
$613$			21.2132i			0.856793i	−0.903591	+	0.428397i	$0.859079\pi$
$614$	12.6491			0.510477
$615$		0			0
$616$	−3.00000	−	2.23607i	−0.120873	−	0.0900937i
$617$	18.0000			0.724653			0.362326	−	0.932051i	$-0.381983\pi$
$617$	18.0000			0.724653			0.362326	+	0.932051i	$0.381983\pi$
$618$		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
$622$	−18.9737			−0.760775
$623$	15.0000	−	20.1246i	0.600962	−	0.806276i
$624$		0			0
$625$	25.0000			1.00000
$626$	31.6228			1.26390
$627$		0			0
$628$	−15.8114			−0.630943
$629$	−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	$0.688175\pi$
$632$	−4.00000			−0.159111
$633$		0			0
$634$	12.0000			0.476581
$635$	−47.4342			−1.88237
$636$		0			0
$637$	−6.32456	−	21.2132i	−0.250588	−	0.840498i
$638$	4.00000			0.158362
$639$		0			0
$640$			2.23607i			0.0883883i
$641$			15.5563i			0.614439i	0.951639	+	0.307219i	$0.0993986\pi$
$641$			15.5563i			0.614439i	−0.951639	+	0.307219i	$0.900601\pi$
$642$		0			0
$643$	−6.32456			−0.249416			−0.124708	−	0.992193i	$-0.539799\pi$
$643$	−6.32456			−0.249416			−0.124708	+	0.992193i	$0.539799\pi$
$644$	9.48683	−	12.7279i	0.373834	−	0.501550i
$645$		0			0
$646$		0			0
$647$		−	22.3607i		−	0.879089i	−0.898221	−	0.439545i	$-0.855140\pi$
$647$		−	22.3607i		−	0.879089i	0.898221	−	0.439545i	$-0.144860\pi$
$648$		0			0
$649$			13.4164i			0.526640i
$650$	−15.8114			−0.620174
$651$		0			0
$652$		0			0
$653$	−24.0000			−0.939193			−0.469596	−	0.882881i	$-0.655601\pi$
$653$	−24.0000			−0.939193			−0.469596	+	0.882881i	$0.655601\pi$
$654$		0			0
$655$			21.2132i			0.828868i
$656$	−9.48683			−0.370399
$657$		0			0
$658$	9.48683	+	7.07107i	0.369835	+	0.275659i
$659$			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	$-0.713813\pi$
$661$		−	40.2492i		−	1.56551i	0.622328	−	0.782757i	$-0.286187\pi$
$662$	−10.0000			−0.388661
$663$		0			0
$664$		−	8.94427i		−	0.347105i
$665$		0			0
$666$		0			0
$667$			16.9706i			0.657103i
$668$		−	8.94427i		−	0.346064i
$669$		0			0
$670$		0			0
$671$	−18.9737			−0.732470
$672$		0			0
$673$			8.48528i			0.327084i	0.986536	+	0.163542i	$0.0522919\pi$
$673$			8.48528i			0.327084i	−0.986536	+	0.163542i	$0.947708\pi$
$674$			25.4558i			0.980522i
$675$		0			0
$676$	−3.00000			−0.115385
$677$		−	22.3607i		−	0.859391i	−0.902974	−	0.429695i	$-0.858621\pi$
$677$		−	22.3607i		−	0.859391i	0.902974	−	0.429695i	$-0.141379\pi$
$678$		0			0
$679$	20.0000	−	26.8328i	0.767530	−	1.02975i
$680$	−10.0000			−0.383482
$681$		0			0
$682$		0			0
$683$	36.0000			1.37750			0.688751	−	0.724998i	$-0.258159\pi$
$683$	36.0000			1.37750			0.688751	+	0.724998i	$0.258159\pi$
$684$		0			0
$685$		−	40.2492i		−	1.53784i
$686$	−17.3925	−	6.36396i	−0.664050	−	0.242977i
$687$		0			0
$688$		−	8.48528i		−	0.323498i
$689$	18.9737			0.722839
$690$		0			0
$691$		0			0		1.00000			$0$
$691$		0			0		−1.00000			$\pi$
$692$		−	8.94427i		−	0.340010i
$693$		0			0
$694$	−12.0000			−0.455514
$695$	−30.0000			−1.13796
$696$		0			0
$697$		−	42.4264i		−	1.60701i
$698$		−	13.4164i		−	0.507819i
$699$		0			0
$700$	−7.90569	+	10.6066i	−0.298807	+	0.400892i
$701$			36.7696i			1.38877i	0.719605	+	0.694383i	$0.244323\pi$
$701$			36.7696i			1.38877i	−0.719605	+	0.694383i	$0.755677\pi$
$702$		0			0
$703$		0			0
$704$		−	1.41421i		−	0.0533002i
$705$		0			0
$706$			31.3050i			1.17818i
$707$	−30.0000	+	40.2492i	−1.12827	+	1.51373i
$708$		0			0
$709$	−10.0000			−0.375558			−0.187779	−	0.982211i	$-0.560129\pi$
$709$	−10.0000			−0.375558			−0.187779	+	0.982211i	$0.560129\pi$
$710$	12.6491			0.474713
$711$		0			0
$712$	9.48683			0.355534
$713$		0			0
$714$		0			0
$715$	10.0000			0.373979
$716$		−	18.3848i		−	0.687071i
$717$		0			0
$718$		−	31.1127i		−	1.16112i
$719$	−37.9473			−1.41520			−0.707598	−	0.706615i	$-0.750221\pi$
$719$	−37.9473			−1.41520			−0.707598	+	0.706615i	$0.750221\pi$
$720$		0			0
$721$	−25.0000	+	33.5410i	−0.931049	+	1.24913i
$722$	19.0000			0.707107
$723$		0			0
$724$		−	13.4164i		−	0.498617i
$725$		−	14.1421i		−	0.525226i
$726$		0			0
$727$	3.16228			0.117282			0.0586412	−	0.998279i	$-0.481323\pi$
$727$	3.16228			0.117282			0.0586412	+	0.998279i	$0.481323\pi$
$728$	5.00000	−	6.70820i	0.185312	−	0.248623i
$729$		0			0
$730$		−	14.1421i		−	0.523424i
$731$	37.9473			1.40353
$732$		0			0
$733$	41.1096			1.51842			0.759209	−	0.650847i	$-0.225586\pi$
$733$	41.1096			1.51842			0.759209	+	0.650847i	$0.225586\pi$
$734$	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	$-0.715046\pi$
$739$	−34.0000			−1.25071			−0.625355	+	0.780340i	$0.715046\pi$
$740$	−9.48683			−0.348743
$741$		0			0
$742$	9.48683	−	12.7279i	0.348273	−	0.467257i
$743$	24.0000			0.880475			0.440237	−	0.897881i	$-0.354894\pi$
$743$	24.0000			0.880475			0.440237	+	0.897881i	$0.354894\pi$
$744$		0			0
$745$	−25.2982			−0.926855
$746$			21.2132i			0.776671i
$747$		0			0
$748$	6.32456			0.231249
$749$	−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	$0.760392\pi$
$752$			4.47214i			0.163082i
$753$		0			0
$754$			8.94427i			0.325731i
$755$			44.7214i			1.62758i
$756$		0			0
$757$			46.6690i			1.69622i	0.529824	+	0.848108i	$0.322258\pi$
$757$			46.6690i			1.69622i	−0.529824	+	0.848108i	$0.677742\pi$
$758$	20.0000			0.726433
$759$		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
$763$	−15.8114	+	21.2132i	−0.572411	+	0.767970i
$764$		−	22.6274i		−	0.818631i
$765$		0			0
$766$			4.47214i			0.161585i
$767$	−30.0000			−1.08324
$768$		0			0
$769$			40.2492i			1.45142i	0.687999	+	0.725712i	$0.258490\pi$
$769$			40.2492i			1.45142i	−0.687999	+	0.725712i	$0.741510\pi$
$770$	5.00000	−	6.70820i	0.180187	−	0.241747i
$771$		0			0
$772$			8.48528i			0.305392i
$773$			31.3050i			1.12596i	0.826470	+	0.562980i	$0.190345\pi$
$773$			31.3050i			1.12596i	−0.826470	+	0.562980i	$0.809655\pi$
$774$		0			0
$775$		0			0
$776$	12.6491			0.454077
$777$		0			0
$778$			11.3137i			0.405616i
$779$		0			0
$780$		0			0
$781$	−8.00000			−0.286263
$782$			26.8328i			0.959540i
$783$		0			0
$784$	−2.00000	−	6.70820i	−0.0714286	−	0.239579i
$785$		−	35.3553i		−	1.26189i
$786$		0			0
$787$	31.6228			1.12723			0.563615	−	0.826038i	$-0.309410\pi$
$787$	31.6228			1.12723			0.563615	+	0.826038i	$0.309410\pi$
$788$	12.0000			0.427482
$789$		0			0
$790$		−	8.94427i		−	0.318223i
$791$	−9.48683	+	12.7279i	−0.337313	+	0.452553i
$792$		0			0
$793$		−	42.4264i		−	1.50661i
$794$	22.1359			0.785575
$795$		0			0
$796$			26.8328i			0.951064i
$797$		−	8.94427i		−	0.316822i	−0.987373	−	0.158411i	$-0.949363\pi$
$797$		−	8.94427i		−	0.316822i	0.987373	−	0.158411i	$-0.0506372\pi$
$798$		0			0
$799$	−20.0000			−0.707549
$800$	−5.00000			−0.176777
$801$		0			0
$802$		−	1.41421i		−	0.0499376i
$803$			8.94427i			0.315637i
$804$		0			0
$805$	28.4605	+	21.2132i	1.00310	+	0.747667i
$806$		0			0
$807$		0			0
$808$	−18.9737			−0.667491
$809$		−	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	$-0.750196\pi$
$811$		−	40.2492i		−	1.41334i	0.707543	−	0.706671i	$-0.249804\pi$
$812$	6.00000	+	4.47214i	0.210559	+	0.156941i
$813$		0			0
$814$	6.00000			0.210300
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		−	21.2132i		−	0.740797i
$821$			19.7990i			0.690990i	0.938421	+	0.345495i	$0.112289\pi$
$821$			19.7990i			0.690990i	−0.938421	+	0.345495i	$0.887711\pi$
$822$		0			0
$823$		−	21.2132i		−	0.739446i	−0.929142	−	0.369723i	$-0.879453\pi$
$823$		−	21.2132i		−	0.739446i	0.929142	−	0.369723i	$-0.120547\pi$
$824$	−15.8114			−0.550816
$825$		0			0
$826$	−15.0000	+	20.1246i	−0.521917	+	0.700225i
$827$	48.0000			1.66912			0.834562	−	0.550914i	$-0.185721\pi$
$827$	48.0000			1.66912			0.834562	+	0.550914i	$0.185721\pi$
$828$		0			0
$829$		−	13.4164i		−	0.465971i	−0.972480	−	0.232986i	$-0.925151\pi$
$829$		−	13.4164i		−	0.465971i	0.972480	−	0.232986i	$-0.0748495\pi$
$830$	20.0000			0.694210
$831$		0			0
$832$	3.16228			0.109632
$833$	30.0000	−	8.94427i	1.03944	−	0.309901i
$834$		0			0
$835$	20.0000			0.692129
$836$		0			0
$837$		0			0
$838$	−9.48683			−0.327717
$839$	37.9473			1.31009			0.655044	−	0.755591i	$-0.272650\pi$
$839$	37.9473			1.31009			0.655044	+	0.755591i	$0.272650\pi$
$840$		0			0
$841$	21.0000			0.724138
$842$	−22.0000			−0.758170
$843$		0			0
$844$	20.0000			0.688428
$845$		−	6.70820i		−	0.230769i
$846$		0			0
$847$	14.2302	−	19.0919i	0.488957	−	0.656005i
$848$	6.00000			0.206041
$849$		0			0
$850$		−	22.3607i		−	0.766965i
$851$			25.4558i			0.872615i
$852$		0			0
$853$	3.16228			0.108274			0.0541372	−	0.998534i	$-0.482759\pi$
$853$	3.16228			0.108274			0.0541372	+	0.998534i	$0.482759\pi$
$854$	−28.4605	−	21.2132i	−0.973898	−	0.725901i
$855$		0			0
$856$	−12.0000			−0.410152
$857$			31.3050i			1.06936i	0.845056	+	0.534678i	$0.179567\pi$
$857$			31.3050i			1.06936i	−0.845056	+	0.534678i	$0.820433\pi$
$858$		0			0
$859$			13.4164i			0.457762i	0.973454	+	0.228881i	$0.0735067\pi$
$859$			13.4164i			0.457762i	−0.973454	+	0.228881i	$0.926493\pi$
$860$	18.9737			0.646997
$861$		0			0
$862$			36.7696i			1.25238i
$863$	−6.00000			−0.204242			−0.102121	−	0.994772i	$-0.532563\pi$
$863$	−6.00000			−0.204242			−0.102121	+	0.994772i	$0.532563\pi$
$864$		0			0
$865$	20.0000			0.680020
$866$	−6.32456			−0.214917
$867$		0			0
$868$		0			0
$869$			5.65685i			0.191896i
$870$		0			0
$871$		0			0
$872$	−10.0000			−0.338643
$873$		0			0
$874$		0			0
$875$	−23.7171	−	17.6777i	−0.801784	−	0.597614i
$876$		0			0
$877$		−	21.2132i		−	0.716319i	−0.933660	−	0.358159i	$-0.883404\pi$
$877$		−	21.2132i		−	0.716319i	0.933660	−	0.358159i	$-0.116596\pi$
$878$		0			0
$879$		0			0
$880$	3.16228			0.106600
$881$	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
$883$		−	33.9411i		−	1.14221i	−0.820877	−	0.571105i	$-0.806515\pi$
$883$		−	33.9411i		−	1.14221i	0.820877	−	0.571105i	$-0.193485\pi$
$884$			14.1421i			0.475651i
$885$		0			0
$886$	24.0000			0.806296
$887$			44.7214i			1.50160i	0.660532	+	0.750798i	$0.270331\pi$
$887$			44.7214i			1.50160i	−0.660532	+	0.750798i	$0.729669\pi$
$888$		0			0
$889$	45.0000	+	33.5410i	1.50925	+	1.12493i
$890$			21.2132i			0.711068i
$891$		0			0
$892$	22.1359			0.741166
$893$		0			0
$894$		0			0
$895$	41.1096			1.37414
$896$	1.58114	−	2.12132i	0.0528221	−	0.0708683i
$897$		0			0
$898$		−	18.3848i		−	0.613508i
$899$		0			0
$900$		0			0
$901$			26.8328i			0.893931i
$902$			13.4164i			0.446718i
$903$		0			0
$904$	−6.00000			−0.199557
$905$	30.0000			0.997234
$906$		0			0
$907$		−	25.4558i		−	0.845247i	−0.906305	−	0.422624i	$-0.861109\pi$
$907$		−	25.4558i		−	0.845247i	0.906305	−	0.422624i	$-0.138891\pi$
$908$			17.8885i			0.593652i
$909$		0			0
$910$	15.0000	+	11.1803i	0.497245	+	0.370625i
$911$		−	48.0833i		−	1.59307i	−0.604593	−	0.796535i	$-0.706664\pi$
$911$		−	48.0833i		−	1.59307i	0.604593	−	0.796535i	$-0.293336\pi$
$912$		0			0
$913$	−12.6491			−0.418624
$914$			16.9706i			0.561336i
$915$		0			0
$916$		−	13.4164i		−	0.443291i
$917$	15.0000	−	20.1246i	0.495344	−	0.664573i
$918$		0			0
$919$	20.0000			0.659739			0.329870	−	0.944027i	$-0.392995\pi$
$919$	20.0000			0.659739			0.329870	+	0.944027i	$0.392995\pi$
$920$			13.4164i			0.442326i
$921$		0			0
$922$		0			0
$923$		−	17.8885i		−	0.588809i
$924$		0			0
$925$		−	21.2132i		−	0.697486i
$926$		−	21.2132i		−	0.697109i
$927$		0			0
$928$			2.82843i			0.0928477i
$929$	−9.48683			−0.311253			−0.155626	−	0.987816i	$-0.549740\pi$
$929$	−9.48683			−0.311253			−0.155626	+	0.987816i	$0.549740\pi$
$930$		0			0
$931$		0			0
$932$	−6.00000			−0.196537
$933$		0			0
$934$		−	8.94427i		−	0.292666i
$935$			14.1421i			0.462497i
$936$		0			0
$937$	12.6491			0.413228			0.206614	−	0.978422i	$-0.433755\pi$
$937$	12.6491			0.413228			0.206614	+	0.978422i	$0.433755\pi$
$938$		0			0
$939$		0			0
$940$	−10.0000			−0.326164
$941$	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
$943$	−56.9210			−1.85360
$944$	−9.48683			−0.308770
$945$		0			0
$946$	−12.0000			−0.390154
$947$	12.0000			0.389948			0.194974	−	0.980808i	$-0.437538\pi$
$947$	12.0000			0.389948			0.194974	+	0.980808i	$0.437538\pi$
$948$		0			0
$949$	−20.0000			−0.649227
$950$		0			0
$951$		0			0
$952$	9.48683	+	7.07107i	0.307470	+	0.229175i
$953$	−6.00000			−0.194359			−0.0971795	−	0.995267i	$-0.530982\pi$
$953$	−6.00000			−0.194359			−0.0971795	+	0.995267i	$0.530982\pi$
$954$		0			0
$955$	50.5964			1.63726
$956$			11.3137i			0.365911i
$957$		0			0
$958$	18.9737			0.613011
$959$	−28.4605	+	38.1838i	−0.919037	+	1.23302i
$960$		0			0
$961$	31.0000			1.00000
$962$			13.4164i			0.432562i
$963$		0			0
$964$			13.4164i			0.432113i
$965$	−18.9737			−0.610784
$966$		0			0
$967$			21.2132i			0.682171i	0.940032	+	0.341085i	$0.110795\pi$
$967$			21.2132i			0.682171i	−0.940032	+	0.341085i	$0.889205\pi$
$968$	9.00000			0.289271
$969$		0			0
$970$			28.2843i			0.908153i
$971$	−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
$973$	28.4605	+	21.2132i	0.912402	+	0.680064i
$974$		−	4.24264i		−	0.135943i
$975$		0			0
$976$		−	13.4164i		−	0.429449i
$977$	18.0000			0.575871			0.287936	−	0.957650i	$-0.407031\pi$
$977$	18.0000			0.575871			0.287936	+	0.957650i	$0.407031\pi$
$978$		0			0
$979$		−	13.4164i		−	0.428790i
$980$	15.0000	−	4.47214i	0.479157	−	0.142857i
$981$		0			0
$982$			15.5563i			0.496423i
$983$			4.47214i			0.142639i	0.997454	+	0.0713195i	$0.0227210\pi$
$983$			4.47214i			0.142639i	−0.997454	+	0.0713195i	$0.977279\pi$
$984$		0			0
$985$			26.8328i			0.854965i
$986$	−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
$995$	−60.0000			−1.90213
$996$		0			0
$997$	−53.7587			−1.70256			−0.851278	−	0.524715i	$-0.824172\pi$
$997$	−53.7587			−1.70256			−0.851278	+	0.524715i	$0.824172\pi$
$998$	−10.0000			−0.316544
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	630.2.d.d.629.4	yes	4
3.2	odd	2		630.2.d.a.629.2	yes	4
4.3	odd	2		5040.2.k.a.1889.3		4
5.2	odd	4		3150.2.b.c.251.8		8
5.3	odd	4		3150.2.b.c.251.1		8
5.4	even	2		630.2.d.a.629.1	✓	4
7.6	odd	2	inner	630.2.d.d.629.1	yes	4
12.11	even	2		5040.2.k.d.1889.1		4
15.2	even	4		3150.2.b.c.251.4		8
15.8	even	4		3150.2.b.c.251.5		8
15.14	odd	2	inner	630.2.d.d.629.3	yes	4
20.19	odd	2		5040.2.k.d.1889.2		4
21.20	even	2		630.2.d.a.629.3	yes	4
28.27	even	2		5040.2.k.a.1889.2		4
35.13	even	4		3150.2.b.c.251.2		8
35.27	even	4		3150.2.b.c.251.7		8
35.34	odd	2		630.2.d.a.629.4	yes	4
60.59	even	2		5040.2.k.a.1889.4		4
84.83	odd	2		5040.2.k.d.1889.4		4
105.62	odd	4		3150.2.b.c.251.3		8
105.83	odd	4		3150.2.b.c.251.6		8
105.104	even	2	inner	630.2.d.d.629.2	yes	4
140.139	even	2		5040.2.k.d.1889.3		4
420.419	odd	2		5040.2.k.a.1889.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
630.2.d.a.629.1	✓	4	5.4	even	2
630.2.d.a.629.2	yes	4	3.2	odd	2
630.2.d.a.629.3	yes	4	21.20	even	2
630.2.d.a.629.4	yes	4	35.34	odd	2
630.2.d.d.629.1	yes	4	7.6	odd	2	inner
630.2.d.d.629.2	yes	4	105.104	even	2	inner
630.2.d.d.629.3	yes	4	15.14	odd	2	inner
630.2.d.d.629.4	yes	4	1.1	even	1	trivial
3150.2.b.c.251.1		8	5.3	odd	4
3150.2.b.c.251.2		8	35.13	even	4
3150.2.b.c.251.3		8	105.62	odd	4
3150.2.b.c.251.4		8	15.2	even	4
3150.2.b.c.251.5		8	15.8	even	4
3150.2.b.c.251.6		8	105.83	odd	4
3150.2.b.c.251.7		8	35.27	even	4
3150.2.b.c.251.8		8	5.2	odd	4
5040.2.k.a.1889.1		4	420.419	odd	2
5040.2.k.a.1889.2		4	28.27	even	2
5040.2.k.a.1889.3		4	4.3	odd	2
5040.2.k.a.1889.4		4	60.59	even	2
5040.2.k.d.1889.1		4	12.11	even	2
5040.2.k.d.1889.2		4	20.19	odd	2
5040.2.k.d.1889.3		4	140.139	even	2
5040.2.k.d.1889.4		4	84.83	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 630 = 2 \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	630.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +2.23607i q^{5} +(1.58114 - 2.12132i) q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} +2.23607i q^{5} +(1.58114 - 2.12132i) q^{7} +1.00000 q^{8} +2.23607i q^{10} -1.41421i q^{11} +3.16228 q^{13} +(1.58114 - 2.12132i) q^{14} +1.00000 q^{16} +4.47214i q^{17} +2.23607i q^{20} -1.41421i q^{22} +6.00000 q^{23} -5.00000 q^{25} +3.16228 q^{26} +(1.58114 - 2.12132i) q^{28} +2.82843i q^{29} +1.00000 q^{32} +4.47214i q^{34} +(4.74342 + 3.53553i) q^{35} +4.24264i q^{37} +2.23607i q^{40} -9.48683 q^{41} -8.48528i q^{43} -1.41421i q^{44} +6.00000 q^{46} +4.47214i q^{47} +(-2.00000 - 6.70820i) q^{49} -5.00000 q^{50} +3.16228 q^{52} +6.00000 q^{53} +3.16228 q^{55} +(1.58114 - 2.12132i) q^{56} +2.82843i q^{58} -9.48683 q^{59} -13.4164i q^{61} +1.00000 q^{64} +7.07107i q^{65} +4.47214i q^{68} +(4.74342 + 3.53553i) q^{70} -5.65685i q^{71} -6.32456 q^{73} +4.24264i q^{74} +(-3.00000 - 2.23607i) q^{77} -4.00000 q^{79} +2.23607i q^{80} -9.48683 q^{82} -8.94427i q^{83} -10.0000 q^{85} -8.48528i q^{86} -1.41421i q^{88} +9.48683 q^{89} +(5.00000 - 6.70820i) q^{91} +6.00000 q^{92} +4.47214i q^{94} +12.6491 q^{97} +(-2.00000 - 6.70820i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{2} + 4 q^{4} + 4 q^{8}+O(q^{10}) \) 4 * q + 4 * q^2 + 4 * q^4 + 4 * q^8	\( 4 q + 4 q^{2} + 4 q^{4} + 4 q^{8} + 4 q^{16} + 24 q^{23} - 20 q^{25} + 4 q^{32} + 24 q^{46} - 8 q^{49} - 20 q^{50} + 24 q^{53} + 4 q^{64} - 12 q^{77} - 16 q^{79} - 40 q^{85} + 20 q^{91} + 24 q^{92} - 8 q^{98}+O(q^{100}) \) 4 * q + 4 * q^2 + 4 * q^4 + 4 * q^8 + 4 * q^16 + 24 * q^23 - 20 * q^25 + 4 * q^32 + 24 * q^46 - 8 * q^49 - 20 * q^50 + 24 * q^53 + 4 * q^64 - 12 * q^77 - 16 * q^79 - 40 * q^85 + 20 * q^91 + 24 * q^92 - 8 * q^98

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