LMFDB - Embedded newform 3150.2.d.b.3149.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3150,2,Mod(3149,3150)]

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, 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("3150.3149");

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.d (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$25.1528766367$
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:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{6}\cdot 5^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3149.7
Root			$-0.437016 - 0.437016i$ of defining polynomial
Character	$\chi$	$=$	3150.3149
Dual form			3150.2.d.b.3149.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} +(2.23607 + 1.41421i) q^{7} -1.00000 q^{8} +O(q^{10})$	$q-1.00000 q^{2} +1.00000 q^{4} +(2.23607 + 1.41421i) q^{7} -1.00000 q^{8} +1.41421i q^{11} -0.926210 q^{13} +(-2.23607 - 1.41421i) q^{14} +1.00000 q^{16} -2.23607i q^{17} +7.63441i q^{19} -1.41421i q^{22} -1.00000 q^{23} +0.926210 q^{26} +(2.23607 + 1.41421i) q^{28} -0.757359i q^{29} +4.08849i q^{31} -1.00000 q^{32} +2.23607i q^{34} -2.82843i q^{37} -7.63441i q^{38} -8.56062 q^{41} +3.58579i q^{43} +1.41421i q^{44} +1.00000 q^{46} +1.30986i q^{47} +(3.00000 + 6.32456i) q^{49} -0.926210 q^{52} -8.07107 q^{53} +(-2.23607 - 1.41421i) q^{56} +0.757359i q^{58} +7.25077 q^{59} +0.926210i q^{61} -4.08849i q^{62} +1.00000 q^{64} -2.23607i q^{68} +15.6569i q^{71} -13.9590 q^{73} +2.82843i q^{74} +7.63441i q^{76} +(-2.00000 + 3.16228i) q^{77} +13.0711 q^{79} +8.56062 q^{82} -14.3426i q^{83} -3.58579i q^{86} -1.41421i q^{88} -2.61972 q^{89} +(-2.07107 - 1.30986i) q^{91} -1.00000 q^{92} -1.30986i q^{94} +0.542561 q^{97} +(-3.00000 - 6.32456i) q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{2} + 8 q^{4} - 8 q^{8}+O(q^{10}) $ 8 * q - 8 * q^2 + 8 * q^4 - 8 * q^8	$ 8 q - 8 q^{2} + 8 q^{4} - 8 q^{8} + 8 q^{16} - 8 q^{23} - 8 q^{32} + 8 q^{46} + 24 q^{49} - 8 q^{53} + 8 q^{64} - 16 q^{77} + 48 q^{79} + 40 q^{91} - 8 q^{92} - 24 q^{98}+O(q^{100}) $ 8 * q - 8 * q^2 + 8 * q^4 - 8 * q^8 + 8 * q^16 - 8 * q^23 - 8 * q^32 + 8 * q^46 + 24 * q^49 - 8 * q^53 + 8 * q^64 - 16 * q^77 + 48 * q^79 + 40 * q^91 - 8 * q^92 - 24 * q^98

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 $

$2$	−1.00000			−0.707107
$2$	−1.00000			−0.707107
$3$		0			0
$3$		0			0
$4$	1.00000			0.500000
$5$		0			0
$5$		0			0
$6$		0			0
$7$	2.23607	+	1.41421i	0.845154	+	0.534522i
$7$	2.23607	+	1.41421i	0.845154	+	0.534522i
$8$	−1.00000			−0.353553
$9$		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
$13$	−0.926210			−0.256884			−0.128442	−	0.991717i	$-0.540998\pi$
$13$	−0.926210			−0.256884			−0.128442	+	0.991717i	$0.540998\pi$
$14$	−2.23607	−	1.41421i	−0.597614	−	0.377964i
$15$		0			0
$16$	1.00000			0.250000
$17$		−	2.23607i		−	0.542326i	−0.962533	−	0.271163i	$-0.912592\pi$
$17$		−	2.23607i		−	0.542326i	0.962533	−	0.271163i	$-0.0874083\pi$
$18$		0			0
$19$			7.63441i			1.75145i	0.482806	+	0.875727i	$0.339618\pi$
$19$			7.63441i			1.75145i	−0.482806	+	0.875727i	$0.660382\pi$
$20$		0			0
$21$		0			0
$22$		−	1.41421i		−	0.301511i
$23$	−1.00000			−0.208514			−0.104257	−	0.994550i	$-0.533247\pi$
$23$	−1.00000			−0.208514			−0.104257	+	0.994550i	$0.533247\pi$
$24$		0			0
$25$		0			0
$26$	0.926210			0.181645
$27$		0			0
$28$	2.23607	+	1.41421i	0.422577	+	0.267261i
$29$		−	0.757359i		−	0.140638i	−0.997525	−	0.0703190i	$-0.977598\pi$
$29$		−	0.757359i		−	0.140638i	0.997525	−	0.0703190i	$-0.0224017\pi$
$30$		0			0
$31$			4.08849i			0.734314i	0.930159	+	0.367157i	$0.119669\pi$
$31$			4.08849i			0.734314i	−0.930159	+	0.367157i	$0.880331\pi$
$32$	−1.00000			−0.176777
$33$		0			0
$34$			2.23607i			0.383482i
$35$		0			0
$36$		0			0
$37$		−	2.82843i		−	0.464991i	−0.972598	−	0.232495i	$-0.925311\pi$
$37$		−	2.82843i		−	0.464991i	0.972598	−	0.232495i	$-0.0746890\pi$
$38$		−	7.63441i		−	1.23847i
$39$		0			0
$40$		0			0
$41$	−8.56062			−1.33694			−0.668472	−	0.743737i	$-0.733052\pi$
$41$	−8.56062			−1.33694			−0.668472	+	0.743737i	$0.733052\pi$
$42$		0			0
$43$			3.58579i			0.546827i	0.961897	+	0.273414i	$0.0881528\pi$
$43$			3.58579i			0.546827i	−0.961897	+	0.273414i	$0.911847\pi$
$44$			1.41421i			0.213201i
$45$		0			0
$46$	1.00000			0.147442
$47$			1.30986i			0.191062i	0.995426	+	0.0955312i	$0.0304550\pi$
$47$			1.30986i			0.191062i	−0.995426	+	0.0955312i	$0.969545\pi$
$48$		0			0
$49$	3.00000	+	6.32456i	0.428571	+	0.903508i
$50$		0			0
$51$		0			0
$52$	−0.926210			−0.128442
$53$	−8.07107			−1.10865			−0.554323	−	0.832301i	$-0.687023\pi$
$53$	−8.07107			−1.10865			−0.554323	+	0.832301i	$0.687023\pi$
$54$		0			0
$55$		0			0
$56$	−2.23607	−	1.41421i	−0.298807	−	0.188982i
$57$		0			0
$58$			0.757359i			0.0994461i
$59$	7.25077			0.943969			0.471985	−	0.881607i	$-0.343538\pi$
$59$	7.25077			0.943969			0.471985	+	0.881607i	$0.343538\pi$
$60$		0			0
$61$			0.926210i			0.118589i	0.998241	+	0.0592945i	$0.0188851\pi$
$61$			0.926210i			0.118589i	−0.998241	+	0.0592945i	$0.981115\pi$
$62$		−	4.08849i		−	0.519238i
$63$		0			0
$64$	1.00000			0.125000
$65$		0			0
$66$		0			0
$67$		0			0		1.00000			$0$
$67$		0			0		−1.00000			$\pi$
$68$		−	2.23607i		−	0.271163i
$69$		0			0
$70$		0			0
$71$			15.6569i			1.85813i	0.369921	+	0.929063i	$0.379385\pi$
$71$			15.6569i			1.85813i	−0.369921	+	0.929063i	$0.620615\pi$
$72$		0			0
$73$	−13.9590			−1.63377			−0.816887	−	0.576798i	$-0.804302\pi$
$73$	−13.9590			−1.63377			−0.816887	+	0.576798i	$0.804302\pi$
$74$			2.82843i			0.328798i
$75$		0			0
$76$			7.63441i			0.875727i
$77$	−2.00000	+	3.16228i	−0.227921	+	0.360375i
$78$		0			0
$79$	13.0711			1.47061			0.735305	−	0.677736i	$-0.237039\pi$
$79$	13.0711			1.47061			0.735305	+	0.677736i	$0.237039\pi$
$80$		0			0
$81$		0			0
$82$	8.56062			0.945363
$83$		−	14.3426i		−	1.57431i	−0.616757	−	0.787153i	$-0.711554\pi$
$83$		−	14.3426i		−	1.57431i	0.616757	−	0.787153i	$-0.288446\pi$
$84$		0			0
$85$		0			0
$86$		−	3.58579i		−	0.386665i
$87$		0			0
$88$		−	1.41421i		−	0.150756i
$89$	−2.61972			−0.277689			−0.138845	−	0.990314i	$-0.544339\pi$
$89$	−2.61972			−0.277689			−0.138845	+	0.990314i	$0.544339\pi$
$90$		0			0
$91$	−2.07107	−	1.30986i	−0.217107	−	0.137310i
$92$	−1.00000			−0.104257
$93$		0			0
$94$		−	1.30986i		−	0.135102i
$95$		0			0
$96$		0			0
$97$	0.542561			0.0550887			0.0275444	−	0.999621i	$-0.491231\pi$
$97$	0.542561			0.0550887			0.0275444	+	0.999621i	$0.491231\pi$
$98$	−3.00000	−	6.32456i	−0.303046	−	0.638877i
$99$		0			0
$100$		0			0
$101$	−12.1065			−1.20465			−0.602323	−	0.798252i	$-0.705758\pi$
$101$	−12.1065			−1.20465			−0.602323	+	0.798252i	$0.705758\pi$
$102$		0			0
$103$	−10.4130			−1.02603			−0.513014	−	0.858380i	$-0.671471\pi$
$103$	−10.4130			−1.02603			−0.513014	+	0.858380i	$0.671471\pi$
$104$	0.926210			0.0908223
$105$		0			0
$106$	8.07107			0.783931
$107$	2.00000			0.193347			0.0966736	−	0.995316i	$-0.469180\pi$
$107$	2.00000			0.193347			0.0966736	+	0.995316i	$0.469180\pi$
$108$		0			0
$109$	7.07107			0.677285			0.338643	−	0.940915i	$-0.390032\pi$
$109$	7.07107			0.677285			0.338643	+	0.940915i	$0.390032\pi$
$110$		0			0
$111$		0			0
$112$	2.23607	+	1.41421i	0.211289	+	0.133631i
$113$	−1.07107			−0.100758			−0.0503788	−	0.998730i	$-0.516043\pi$
$113$	−1.07107			−0.100758			−0.0503788	+	0.998730i	$0.516043\pi$
$114$		0			0
$115$		0			0
$116$		−	0.757359i		−	0.0703190i
$117$		0			0
$118$	−7.25077			−0.667487
$119$	3.16228	−	5.00000i	0.289886	−	0.458349i
$120$		0			0
$121$	9.00000			0.818182
$122$		−	0.926210i		−	0.0838551i
$123$		0			0
$124$			4.08849i			0.367157i
$125$		0			0
$126$		0			0
$127$		−	10.0000i		−	0.887357i	−0.896186	−	0.443678i	$-0.853673\pi$
$127$		−	10.0000i		−	0.887357i	0.896186	−	0.443678i	$-0.146327\pi$
$128$	−1.00000			−0.0883883
$129$		0			0
$130$		0			0
$131$	−12.6491			−1.10516			−0.552579	−	0.833461i	$-0.686356\pi$
$131$	−12.6491			−1.10516			−0.552579	+	0.833461i	$0.686356\pi$
$132$		0			0
$133$	−10.7967	+	17.0711i	−0.936192	+	1.48025i
$134$		0			0
$135$		0			0
$136$			2.23607i			0.191741i
$137$	12.1421			1.03737			0.518686	−	0.854965i	$-0.326421\pi$
$137$	12.1421			1.03737			0.518686	+	0.854965i	$0.326421\pi$
$138$		0			0
$139$			10.7967i			0.915763i	0.889013	+	0.457882i	$0.151392\pi$
$139$			10.7967i			0.915763i	−0.889013	+	0.457882i	$0.848608\pi$
$140$		0			0
$141$		0			0
$142$		−	15.6569i		−	1.31389i
$143$		−	1.30986i		−	0.109536i
$144$		0			0
$145$		0			0
$146$	13.9590			1.15525
$147$		0			0
$148$		−	2.82843i		−	0.232495i
$149$			0.757359i			0.0620453i	0.999519	+	0.0310226i	$0.00987640\pi$
$149$			0.757359i			0.0620453i	−0.999519	+	0.0310226i	$0.990124\pi$
$150$		0			0
$151$	−14.1421			−1.15087			−0.575435	−	0.817847i	$-0.695167\pi$
$151$	−14.1421			−1.15087			−0.575435	+	0.817847i	$0.695167\pi$
$152$		−	7.63441i		−	0.619233i
$153$		0			0
$154$	2.00000	−	3.16228i	0.161165	−	0.254824i
$155$		0			0
$156$		0			0
$157$	10.7967			0.861670			0.430835	−	0.902431i	$-0.358219\pi$
$157$	10.7967			0.861670			0.430835	+	0.902431i	$0.358219\pi$
$158$	−13.0711			−1.03988
$159$		0			0
$160$		0			0
$161$	−2.23607	−	1.41421i	−0.176227	−	0.111456i
$162$		0			0
$163$			7.92893i			0.621042i	0.950567	+	0.310521i	$0.100503\pi$
$163$			7.92893i			0.621042i	−0.950567	+	0.310521i	$0.899497\pi$
$164$	−8.56062			−0.668472
$165$		0			0
$166$			14.3426i			1.11320i
$167$			12.1065i			0.936833i	0.883508	+	0.468416i	$0.155175\pi$
$167$			12.1065i			0.936833i	−0.883508	+	0.468416i	$0.844825\pi$
$168$		0			0
$169$	−12.1421			−0.934010
$170$		0			0
$171$		0			0
$172$			3.58579i			0.273414i
$173$		−	10.2541i		−	0.779607i	−0.920898	−	0.389804i	$-0.872543\pi$
$173$		−	10.2541i		−	0.779607i	0.920898	−	0.389804i	$-0.127457\pi$
$174$		0			0
$175$		0			0
$176$			1.41421i			0.106600i
$177$		0			0
$178$	2.61972			0.196356
$179$			11.3137i			0.845626i	0.906217	+	0.422813i	$0.138957\pi$
$179$			11.3137i			0.845626i	−0.906217	+	0.422813i	$0.861043\pi$
$180$		0			0
$181$		−	2.61972i		−	0.194722i	−0.995249	−	0.0973610i	$-0.968960\pi$
$181$		−	2.61972i		−	0.194722i	0.995249	−	0.0973610i	$-0.0310401\pi$
$182$	2.07107	+	1.30986i	0.153518	+	0.0970932i
$183$		0			0
$184$	1.00000			0.0737210
$185$		0			0
$186$		0			0
$187$	3.16228			0.231249
$188$			1.30986i			0.0955312i
$189$		0			0
$190$		0			0
$191$		−	0.656854i		−	0.0475283i	−0.999718	−	0.0237642i	$-0.992435\pi$
$191$		−	0.656854i		−	0.0475283i	0.999718	−	0.0237642i	$-0.00756508\pi$
$192$		0			0
$193$			8.48528i			0.610784i	0.952227	+	0.305392i	$0.0987875\pi$
$193$			8.48528i			0.610784i	−0.952227	+	0.305392i	$0.901213\pi$
$194$	−0.542561			−0.0389536
$195$		0			0
$196$	3.00000	+	6.32456i	0.214286	+	0.451754i
$197$	5.92893			0.422419			0.211209	−	0.977441i	$-0.432260\pi$
$197$	5.92893			0.422419			0.211209	+	0.977441i	$0.432260\pi$
$198$		0			0
$199$			21.5934i			1.53071i	0.643606	+	0.765357i	$0.277438\pi$
$199$			21.5934i			1.53071i	−0.643606	+	0.765357i	$0.722562\pi$
$200$		0			0
$201$		0			0
$202$	12.1065			0.851814
$203$	1.07107	−	1.69351i	0.0751742	−	0.118861i
$204$		0			0
$205$		0			0
$206$	10.4130			0.725511
$207$		0			0
$208$	−0.926210			−0.0642211
$209$	−10.7967			−0.746823
$210$		0			0
$211$	16.2132			1.11616			0.558081	−	0.829786i	$-0.311538\pi$
$211$	16.2132			1.11616			0.558081	+	0.829786i	$0.311538\pi$
$212$	−8.07107			−0.554323
$213$		0			0
$214$	−2.00000			−0.136717
$215$		0			0
$216$		0			0
$217$	−5.78199	+	9.14214i	−0.392507	+	0.620609i
$218$	−7.07107			−0.478913
$219$		0			0
$220$		0			0
$221$			2.07107i			0.139315i
$222$		0			0
$223$	−13.0328			−0.872738			−0.436369	−	0.899768i	$-0.643736\pi$
$223$	−13.0328			−0.872738			−0.436369	+	0.899768i	$0.643736\pi$
$224$	−2.23607	−	1.41421i	−0.149404	−	0.0944911i
$225$		0			0
$226$	1.07107			0.0712464
$227$			25.1393i			1.66855i	0.551345	+	0.834277i	$0.314115\pi$
$227$			25.1393i			1.66855i	−0.551345	+	0.834277i	$0.685885\pi$
$228$		0			0
$229$		−	10.7967i		−	0.713465i	−0.934206	−	0.356733i	$-0.883891\pi$
$229$		−	10.7967i		−	0.713465i	0.934206	−	0.356733i	$-0.116109\pi$
$230$		0			0
$231$		0			0
$232$			0.757359i			0.0497231i
$233$	23.0711			1.51144			0.755718	−	0.654897i	$-0.227288\pi$
$233$	23.0711			1.51144			0.755718	+	0.654897i	$0.227288\pi$
$234$		0			0
$235$		0			0
$236$	7.25077			0.471985
$237$		0			0
$238$	−3.16228	+	5.00000i	−0.204980	+	0.324102i
$239$		−	7.17157i		−	0.463890i	−0.972729	−	0.231945i	$-0.925491\pi$
$239$		−	7.17157i		−	0.463890i	0.972729	−	0.231945i	$-0.0745090\pi$
$240$		0			0
$241$		−	4.47214i		−	0.288076i	−0.989572	−	0.144038i	$-0.953991\pi$
$241$		−	4.47214i		−	0.288076i	0.989572	−	0.144038i	$-0.0460087\pi$
$242$	−9.00000			−0.578542
$243$		0			0
$244$			0.926210i			0.0592945i
$245$		0			0
$246$		0			0
$247$		−	7.07107i		−	0.449921i
$248$		−	4.08849i		−	0.259619i
$249$		0			0
$250$		0			0
$251$	−18.8148			−1.18758			−0.593788	−	0.804621i	$-0.702368\pi$
$251$	−18.8148			−1.18758			−0.593788	+	0.804621i	$0.702368\pi$
$252$		0			0
$253$		−	1.41421i		−	0.0889108i
$254$			10.0000i			0.627456i
$255$		0			0
$256$	1.00000			0.0625000
$257$			21.2097i			1.32303i	0.749933	+	0.661513i	$0.230086\pi$
$257$			21.2097i			1.32303i	−0.749933	+	0.661513i	$0.769914\pi$
$258$		0			0
$259$	4.00000	−	6.32456i	0.248548	−	0.392989i
$260$		0			0
$261$		0			0
$262$	12.6491			0.781465
$263$	−29.2843			−1.80575			−0.902873	−	0.429908i	$-0.858546\pi$
$263$	−29.2843			−1.80575			−0.902873	+	0.429908i	$0.858546\pi$
$264$		0			0
$265$		0			0
$266$	10.7967	−	17.0711i	0.661988	−	1.04669i
$267$		0			0
$268$		0			0
$269$	−27.9179			−1.70219			−0.851093	−	0.525014i	$-0.824060\pi$
$269$	−27.9179			−1.70219			−0.851093	+	0.525014i	$0.824060\pi$
$270$		0			0
$271$		−	7.09185i		−	0.430799i	−0.976526	−	0.215400i	$-0.930895\pi$
$271$		−	7.09185i		−	0.430799i	0.976526	−	0.215400i	$-0.0691054\pi$
$272$		−	2.23607i		−	0.135582i
$273$		0			0
$274$	−12.1421			−0.733533
$275$		0			0
$276$		0			0
$277$			21.2132i			1.27458i	0.770625	+	0.637289i	$0.219944\pi$
$277$			21.2132i			1.27458i	−0.770625	+	0.637289i	$0.780056\pi$
$278$		−	10.7967i		−	0.647543i
$279$		0			0
$280$		0			0
$281$			28.4853i			1.69929i	0.527356	+	0.849645i	$0.323184\pi$
$281$			28.4853i			1.69929i	−0.527356	+	0.849645i	$0.676816\pi$
$282$		0			0
$283$	7.63441			0.453819			0.226909	−	0.973916i	$-0.427138\pi$
$283$	7.63441			0.453819			0.226909	+	0.973916i	$0.427138\pi$
$284$			15.6569i			0.929063i
$285$		0			0
$286$			1.30986i			0.0774535i
$287$	−19.1421	−	12.1065i	−1.12992	−	0.714627i
$288$		0			0
$289$	12.0000			0.705882
$290$		0			0
$291$		0			0
$292$	−13.9590			−0.816887
$293$			25.2982i			1.47794i	0.673740	+	0.738969i	$0.264687\pi$
$293$			25.2982i			1.47794i	−0.673740	+	0.738969i	$0.735313\pi$
$294$		0			0
$295$		0			0
$296$			2.82843i			0.164399i
$297$		0			0
$298$		−	0.757359i		−	0.0438726i
$299$	0.926210			0.0535641
$300$		0			0
$301$	−5.07107	+	8.01806i	−0.292291	+	0.462153i
$302$	14.1421			0.813788
$303$		0			0
$304$			7.63441i			0.437864i
$305$		0			0
$306$		0			0
$307$	−3.70484			−0.211446			−0.105723	−	0.994396i	$-0.533716\pi$
$307$	−3.70484			−0.211446			−0.105723	+	0.994396i	$0.533716\pi$
$308$	−2.00000	+	3.16228i	−0.113961	+	0.180187i
$309$		0			0
$310$		0			0
$311$	5.23943			0.297101			0.148550	−	0.988905i	$-0.452539\pi$
$311$	5.23943			0.297101			0.148550	+	0.988905i	$0.452539\pi$
$312$		0			0
$313$	−16.5787			−0.937083			−0.468541	−	0.883442i	$-0.655220\pi$
$313$	−16.5787			−0.937083			−0.468541	+	0.883442i	$0.655220\pi$
$314$	−10.7967			−0.609293
$315$		0			0
$316$	13.0711			0.735305
$317$	30.0711			1.68896			0.844480	−	0.535588i	$-0.179910\pi$
$317$	30.0711			1.68896			0.844480	+	0.535588i	$0.179910\pi$
$318$		0			0
$319$	1.07107			0.0599683
$320$		0			0
$321$		0			0
$322$	2.23607	+	1.41421i	0.124611	+	0.0788110i
$323$	17.0711			0.949860
$324$		0			0
$325$		0			0
$326$		−	7.92893i		−	0.439143i
$327$		0			0
$328$	8.56062			0.472681
$329$	−1.85242	+	2.92893i	−0.102127	+	0.161477i
$330$		0			0
$331$	7.92893			0.435814			0.217907	−	0.975970i	$-0.430077\pi$
$331$	7.92893			0.435814			0.217907	+	0.975970i	$0.430077\pi$
$332$		−	14.3426i		−	0.787153i
$333$		0			0
$334$		−	12.1065i		−	0.662441i
$335$		0			0
$336$		0			0
$337$		−	11.9706i		−	0.652078i	−0.945356	−	0.326039i	$-0.894286\pi$
$337$		−	11.9706i		−	0.652078i	0.945356	−	0.326039i	$-0.105714\pi$
$338$	12.1421			0.660445
$339$		0			0
$340$		0			0
$341$	−5.78199			−0.313113
$342$		0			0
$343$	−2.23607	+	18.3848i	−0.120736	+	0.992685i
$344$		−	3.58579i		−	0.193333i
$345$		0			0
$346$			10.2541i			0.551265i
$347$	9.07107			0.486960			0.243480	−	0.969906i	$-0.421711\pi$
$347$	9.07107			0.486960			0.243480	+	0.969906i	$0.421711\pi$
$348$		0			0
$349$		−	6.16564i		−	0.330039i	−0.986290	−	0.165020i	$-0.947231\pi$
$349$		−	6.16564i		−	0.330039i	0.986290	−	0.165020i	$-0.0527688\pi$
$350$		0			0
$351$		0			0
$352$		−	1.41421i		−	0.0753778i
$353$			18.9737i			1.00987i	0.863158	+	0.504933i	$0.168483\pi$
$353$			18.9737i			1.00987i	−0.863158	+	0.504933i	$0.831517\pi$
$354$		0			0
$355$		0			0
$356$	−2.61972			−0.138845
$357$		0			0
$358$		−	11.3137i		−	0.597948i
$359$		−	16.3137i		−	0.861005i	−0.902589	−	0.430502i	$-0.858336\pi$
$359$		−	16.3137i		−	0.861005i	0.902589	−	0.430502i	$-0.141664\pi$
$360$		0			0
$361$	−39.2843			−2.06759
$362$			2.61972i			0.137689i
$363$		0			0
$364$	−2.07107	−	1.30986i	−0.108553	−	0.0686552i
$365$		0			0
$366$		0			0
$367$	−24.5967			−1.28394			−0.641970	−	0.766730i	$-0.721883\pi$
$367$	−24.5967			−1.28394			−0.641970	+	0.766730i	$0.721883\pi$
$368$	−1.00000			−0.0521286
$369$		0			0
$370$		0			0
$371$	−18.0475	−	11.4142i	−0.936977	−	0.592596i
$372$		0			0
$373$		−	7.07107i		−	0.366126i	−0.983101	−	0.183063i	$-0.941399\pi$
$373$		−	7.07107i		−	0.366126i	0.983101	−	0.183063i	$-0.0586012\pi$
$374$	−3.16228			−0.163517
$375$		0			0
$376$		−	1.30986i		−	0.0675508i
$377$			0.701474i			0.0361277i
$378$		0			0
$379$	−22.0711			−1.13371			−0.566857	−	0.823816i	$-0.691841\pi$
$379$	−22.0711			−1.13371			−0.566857	+	0.823816i	$0.691841\pi$
$380$		0			0
$381$		0			0
$382$			0.656854i			0.0336076i
$383$			16.0361i			0.819408i	0.912219	+	0.409704i	$0.134368\pi$
$383$			16.0361i			0.819408i	−0.912219	+	0.409704i	$0.865632\pi$
$384$		0			0
$385$		0			0
$386$		−	8.48528i		−	0.431889i
$387$		0			0
$388$	0.542561			0.0275444
$389$			22.8284i			1.15745i	0.815524	+	0.578724i	$0.196449\pi$
$389$			22.8284i			1.15745i	−0.815524	+	0.578724i	$0.803551\pi$
$390$		0			0
$391$			2.23607i			0.113083i
$392$	−3.00000	−	6.32456i	−0.151523	−	0.319438i
$393$		0			0
$394$	−5.92893			−0.298695
$395$		0			0
$396$		0			0
$397$	−29.6114			−1.48616			−0.743078	−	0.669205i	$-0.766635\pi$
$397$	−29.6114			−1.48616			−0.743078	+	0.669205i	$0.766635\pi$
$398$		−	21.5934i		−	1.08238i
$399$		0			0
$400$		0			0
$401$		−	9.79899i		−	0.489338i	−0.969607	−	0.244669i	$-0.921321\pi$
$401$		−	9.79899i		−	0.489338i	0.969607	−	0.244669i	$-0.0786793\pi$
$402$		0			0
$403$		−	3.78680i		−	0.188634i
$404$	−12.1065			−0.602323
$405$		0			0
$406$	−1.07107	+	1.69351i	−0.0531562	+	0.0840473i
$407$	4.00000			0.198273
$408$		0			0
$409$		−	22.9032i		−	1.13249i	−0.824236	−	0.566246i	$-0.808395\pi$
$409$		−	22.9032i		−	1.13249i	0.824236	−	0.566246i	$-0.191605\pi$
$410$		0			0
$411$		0			0
$412$	−10.4130			−0.513014
$413$	16.2132	+	10.2541i	0.797800	+	0.504573i
$414$		0			0
$415$		0			0
$416$	0.926210			0.0454112
$417$		0			0
$418$	10.7967			0.528083
$419$	28.8441			1.40913			0.704564	−	0.709640i	$-0.251142\pi$
$419$	28.8441			1.40913			0.704564	+	0.709640i	$0.251142\pi$
$420$		0			0
$421$	33.3553			1.62564			0.812820	−	0.582515i	$-0.197931\pi$
$421$	33.3553			1.62564			0.812820	+	0.582515i	$0.197931\pi$
$422$	−16.2132			−0.789246
$423$		0			0
$424$	8.07107			0.391966
$425$		0			0
$426$		0			0
$427$	−1.30986	+	2.07107i	−0.0633885	+	0.100226i
$428$	2.00000			0.0966736
$429$		0			0
$430$		0			0
$431$			4.79899i			0.231159i	0.993298	+	0.115580i	$0.0368725\pi$
$431$			4.79899i			0.231159i	−0.993298	+	0.115580i	$0.963127\pi$
$432$		0			0
$433$	−9.71157			−0.466708			−0.233354	−	0.972392i	$-0.574970\pi$
$433$	−9.71157			−0.466708			−0.233354	+	0.972392i	$0.574970\pi$
$434$	5.78199	−	9.14214i	0.277545	−	0.438837i
$435$		0			0
$436$	7.07107			0.338643
$437$		−	7.63441i		−	0.365204i
$438$		0			0
$439$			4.08849i			0.195133i	0.995229	+	0.0975664i	$0.0311058\pi$
$439$			4.08849i			0.195133i	−0.995229	+	0.0975664i	$0.968894\pi$
$440$		0			0
$441$		0			0
$442$		−	2.07107i		−	0.0985106i
$443$	23.0711			1.09614			0.548070	−	0.836433i	$-0.315363\pi$
$443$	23.0711			1.09614			0.548070	+	0.836433i	$0.315363\pi$
$444$		0			0
$445$		0			0
$446$	13.0328			0.617119
$447$		0			0
$448$	2.23607	+	1.41421i	0.105644	+	0.0668153i
$449$			18.3848i			0.867631i	0.901002	+	0.433816i	$0.142833\pi$
$449$			18.3848i			0.867631i	−0.901002	+	0.433816i	$0.857167\pi$
$450$		0			0
$451$		−	12.1065i		−	0.570075i
$452$	−1.07107			−0.0503788
$453$		0			0
$454$		−	25.1393i		−	1.17985i
$455$		0			0
$456$		0			0
$457$			17.8284i			0.833979i	0.908911	+	0.416989i	$0.136915\pi$
$457$			17.8284i			0.833979i	−0.908911	+	0.416989i	$0.863085\pi$
$458$			10.7967i			0.504496i
$459$		0			0
$460$		0			0
$461$	38.9394			1.81359			0.906794	−	0.421575i	$-0.138523\pi$
$461$	38.9394			1.81359			0.906794	+	0.421575i	$0.138523\pi$
$462$		0			0
$463$		−	38.2843i		−	1.77922i	−0.456720	−	0.889610i	$-0.650976\pi$
$463$		−	38.2843i		−	1.77922i	0.456720	−	0.889610i	$-0.349024\pi$
$464$		−	0.757359i		−	0.0351595i
$465$		0			0
$466$	−23.0711			−1.06875
$467$		−	29.6114i		−	1.37025i	−0.728424	−	0.685127i	$-0.759747\pi$
$467$		−	29.6114i		−	1.37025i	0.728424	−	0.685127i	$-0.240253\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$	−7.25077			−0.333744
$473$	−5.07107			−0.233168
$474$		0			0
$475$		0			0
$476$	3.16228	−	5.00000i	0.144943	−	0.229175i
$477$		0			0
$478$			7.17157i			0.328020i
$479$	33.6999			1.53979			0.769895	−	0.638171i	$-0.220309\pi$
$479$	33.6999			1.53979			0.769895	+	0.638171i	$0.220309\pi$
$480$		0			0
$481$			2.61972i			0.119449i
$482$			4.47214i			0.203700i
$483$		0			0
$484$	9.00000			0.409091
$485$		0			0
$486$		0			0
$487$			29.8995i			1.35488i	0.735580	+	0.677438i	$0.236910\pi$
$487$			29.8995i			1.35488i	−0.735580	+	0.677438i	$0.763090\pi$
$488$		−	0.926210i		−	0.0419275i
$489$		0			0
$490$		0			0
$491$			21.5147i			0.970946i	0.874252	+	0.485473i	$0.161353\pi$
$491$			21.5147i			0.970946i	−0.874252	+	0.485473i	$0.838647\pi$
$492$		0			0
$493$	−1.69351			−0.0762717
$494$			7.07107i			0.318142i
$495$		0			0
$496$			4.08849i			0.183579i
$497$	−22.1421	+	35.0098i	−0.993211	+	1.57040i
$498$		0			0
$499$	22.0711			0.988037			0.494018	−	0.869451i	$-0.335528\pi$
$499$	22.0711			0.988037			0.494018	+	0.869451i	$0.335528\pi$
$500$		0			0
$501$		0			0
$502$	18.8148			0.839744
$503$			21.8181i			0.972822i	0.873730	+	0.486411i	$0.161694\pi$
$503$			21.8181i			0.972822i	−0.873730	+	0.486411i	$0.838306\pi$
$504$		0			0
$505$		0			0
$506$			1.41421i			0.0628695i
$507$		0			0
$508$		−	10.0000i		−	0.443678i
$509$	0.224736			0.00996125			0.00498063	−	0.999988i	$-0.498415\pi$
$509$	0.224736			0.00996125			0.00498063	+	0.999988i	$0.498415\pi$
$510$		0			0
$511$	−31.2132	−	19.7410i	−1.38079	−	0.873289i
$512$	−1.00000			−0.0441942
$513$		0			0
$514$		−	21.2097i		−	0.935521i
$515$		0			0
$516$		0			0
$517$	−1.85242			−0.0814693
$518$	−4.00000	+	6.32456i	−0.175750	+	0.277885i
$519$		0			0
$520$		0			0
$521$	−17.5049			−0.766903			−0.383452	−	0.923561i	$-0.625265\pi$
$521$	−17.5049			−0.766903			−0.383452	+	0.923561i	$0.625265\pi$
$522$		0			0
$523$	28.4605			1.24449			0.622245	−	0.782822i	$-0.286221\pi$
$523$	28.4605			1.24449			0.622245	+	0.782822i	$0.286221\pi$
$524$	−12.6491			−0.552579
$525$		0			0
$526$	29.2843			1.27685
$527$	9.14214			0.398238
$528$		0			0
$529$	−22.0000			−0.956522
$530$		0			0
$531$		0			0
$532$	−10.7967	+	17.0711i	−0.468096	+	0.740125i
$533$	7.92893			0.343440
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$	27.9179			1.20363
$539$	−8.94427	+	4.24264i	−0.385257	+	0.182743i
$540$		0			0
$541$	5.07107			0.218022			0.109011	−	0.994041i	$-0.465232\pi$
$541$	5.07107			0.218022			0.109011	+	0.994041i	$0.465232\pi$
$542$			7.09185i			0.304621i
$543$		0			0
$544$			2.23607i			0.0958706i
$545$		0			0
$546$		0			0
$547$		−	16.2132i		−	0.693227i	−0.938008	−	0.346613i	$-0.887332\pi$
$547$		−	16.2132i		−	0.693227i	0.938008	−	0.346613i	$-0.112668\pi$
$548$	12.1421			0.518686
$549$		0			0
$550$		0			0
$551$	5.78199			0.246321
$552$		0			0
$553$	29.2278	+	18.4853i	1.24289	+	0.786074i
$554$		−	21.2132i		−	0.901263i
$555$		0			0
$556$			10.7967i			0.457882i
$557$	−26.1421			−1.10768			−0.553839	−	0.832624i	$-0.686838\pi$
$557$	−26.1421			−1.10768			−0.553839	+	0.832624i	$0.686838\pi$
$558$		0			0
$559$		−	3.32119i		−	0.140471i
$560$		0			0
$561$		0			0
$562$		−	28.4853i		−	1.20158i
$563$		−	24.3720i		−	1.02716i	−0.858042	−	0.513579i	$-0.828319\pi$
$563$		−	24.3720i		−	1.02716i	0.858042	−	0.513579i	$-0.171681\pi$
$564$		0			0
$565$		0			0
$566$	−7.63441			−0.320898
$567$		0			0
$568$		−	15.6569i		−	0.656947i
$569$		−	22.5269i		−	0.944377i	−0.881498	−	0.472189i	$-0.843464\pi$
$569$		−	22.5269i		−	0.944377i	0.881498	−	0.472189i	$-0.156536\pi$
$570$		0			0
$571$	−38.2132			−1.59917			−0.799586	−	0.600551i	$-0.794948\pi$
$571$	−38.2132			−1.59917			−0.799586	+	0.600551i	$0.794948\pi$
$572$		−	1.30986i		−	0.0547679i
$573$		0			0
$574$	19.1421	+	12.1065i	0.798977	+	0.505318i
$575$		0			0
$576$		0			0
$577$	−23.4458			−0.976062			−0.488031	−	0.872826i	$-0.662285\pi$
$577$	−23.4458			−0.976062			−0.488031	+	0.872826i	$0.662285\pi$
$578$	−12.0000			−0.499134
$579$		0			0
$580$		0			0
$581$	20.2835	−	32.0711i	0.841502	−	1.33053i
$582$		0			0
$583$		−	11.4142i		−	0.472728i
$584$	13.9590			0.577626
$585$		0			0
$586$		−	25.2982i		−	1.04506i
$587$		−	15.4277i		−	0.636771i	−0.947961	−	0.318385i	$-0.896859\pi$
$587$		−	15.4277i		−	0.636771i	0.947961	−	0.318385i	$-0.103141\pi$
$588$		0			0
$589$	−31.2132			−1.28612
$590$		0			0
$591$		0			0
$592$		−	2.82843i		−	0.116248i
$593$			46.8916i			1.92561i	0.270202	+	0.962804i	$0.412910\pi$
$593$			46.8916i			1.92561i	−0.270202	+	0.962804i	$0.587090\pi$
$594$		0			0
$595$		0			0
$596$			0.757359i			0.0310226i
$597$		0			0
$598$	−0.926210			−0.0378755
$599$		−	3.68629i		−	0.150618i	−0.997160	−	0.0753089i	$-0.976006\pi$
$599$		−	3.68629i		−	0.150618i	0.997160	−	0.0753089i	$-0.0239943\pi$
$600$		0			0
$601$		−	12.1065i		−	0.493836i	−0.969036	−	0.246918i	$-0.920582\pi$
$601$		−	12.1065i		−	0.493836i	0.969036	−	0.246918i	$-0.0794179\pi$
$602$	5.07107	−	8.01806i	0.206681	−	0.326792i
$603$		0			0
$604$	−14.1421			−0.575435
$605$		0			0
$606$		0			0
$607$	43.9541			1.78404			0.892020	−	0.451996i	$-0.149288\pi$
$607$	43.9541			1.78404			0.892020	+	0.451996i	$0.149288\pi$
$608$		−	7.63441i		−	0.309616i
$609$		0			0
$610$		0			0
$611$		−	1.21320i		−	0.0490810i
$612$		0			0
$613$		−	15.8579i		−	0.640493i	−0.947334	−	0.320247i	$-0.896234\pi$
$613$		−	15.8579i		−	0.640493i	0.947334	−	0.320247i	$-0.103766\pi$
$614$	3.70484			0.149515
$615$		0			0
$616$	2.00000	−	3.16228i	0.0805823	−	0.127412i
$617$	32.0000			1.28827			0.644136	−	0.764911i	$-0.277217\pi$
$617$	32.0000			1.28827			0.644136	+	0.764911i	$0.277217\pi$
$618$		0			0
$619$		−	18.2064i		−	0.731776i	−0.930659	−	0.365888i	$-0.880765\pi$
$619$		−	18.2064i		−	0.731776i	0.930659	−	0.365888i	$-0.119235\pi$
$620$		0			0
$621$		0			0
$622$	−5.23943			−0.210082
$623$	−5.85786	−	3.70484i	−0.234690	−	0.148431i
$624$		0			0
$625$		0			0
$626$	16.5787			0.662618
$627$		0			0
$628$	10.7967			0.430835
$629$	−6.32456			−0.252177
$630$		0			0
$631$	10.2843			0.409410			0.204705	−	0.978824i	$-0.434376\pi$
$631$	10.2843			0.409410			0.204705	+	0.978824i	$0.434376\pi$
$632$	−13.0711			−0.519939
$633$		0			0
$634$	−30.0711			−1.19427
$635$		0			0
$636$		0			0
$637$	−2.77863	−	5.85786i	−0.110093	−	0.232097i
$638$	−1.07107			−0.0424040
$639$		0			0
$640$		0			0
$641$		−	28.4853i		−	1.12510i	−0.826763	−	0.562550i	$-0.809820\pi$
$641$		−	28.4853i		−	1.12510i	0.826763	−	0.562550i	$-0.190180\pi$
$642$		0			0
$643$	29.9951			1.18289			0.591446	−	0.806345i	$-0.298557\pi$
$643$	29.9951			1.18289			0.591446	+	0.806345i	$0.298557\pi$
$644$	−2.23607	−	1.41421i	−0.0881134	−	0.0557278i
$645$		0			0
$646$	−17.0711			−0.671652
$647$		−	37.0869i		−	1.45804i	−0.684493	−	0.729019i	$-0.739977\pi$
$647$		−	37.0869i		−	1.45804i	0.684493	−	0.729019i	$-0.260023\pi$
$648$		0			0
$649$			10.2541i			0.402510i
$650$		0			0
$651$		0			0
$652$			7.92893i			0.310521i
$653$	8.14214			0.318626			0.159313	−	0.987228i	$-0.449072\pi$
$653$	8.14214			0.318626			0.159313	+	0.987228i	$0.449072\pi$
$654$		0			0
$655$		0			0
$656$	−8.56062			−0.334236
$657$		0			0
$658$	1.85242	−	2.92893i	0.0722148	−	0.114182i
$659$		−	48.3848i		−	1.88480i	−0.334483	−	0.942402i	$-0.608562\pi$
$659$		−	48.3848i		−	1.88480i	0.334483	−	0.942402i	$-0.391438\pi$
$660$		0			0
$661$		−	18.2064i		−	0.708146i	−0.935218	−	0.354073i	$-0.884796\pi$
$661$		−	18.2064i		−	0.708146i	0.935218	−	0.354073i	$-0.115204\pi$
$662$	−7.92893			−0.308167
$663$		0			0
$664$			14.3426i			0.556602i
$665$		0			0
$666$		0			0
$667$			0.757359i			0.0293251i
$668$			12.1065i			0.468416i
$669$		0			0
$670$		0			0
$671$	−1.30986			−0.0505665
$672$		0			0
$673$		−	34.7990i		−	1.34140i	−0.741728	−	0.670701i	$-0.765993\pi$
$673$		−	34.7990i		−	1.34140i	0.741728	−	0.670701i	$-0.234007\pi$
$674$			11.9706i			0.461089i
$675$		0			0
$676$	−12.1421			−0.467005
$677$			22.9032i			0.880243i	0.897938	+	0.440122i	$0.145065\pi$
$677$			22.9032i			0.880243i	−0.897938	+	0.440122i	$0.854935\pi$
$678$		0			0
$679$	1.21320	+	0.767297i	0.0465585	+	0.0294462i
$680$		0			0
$681$		0			0
$682$	5.78199			0.221404
$683$	−36.0000			−1.37750			−0.688751	−	0.724998i	$-0.741841\pi$
$683$	−36.0000			−1.37750			−0.688751	+	0.724998i	$0.741841\pi$
$684$		0			0
$685$		0			0
$686$	2.23607	−	18.3848i	0.0853735	−	0.701934i
$687$		0			0
$688$			3.58579i			0.136707i
$689$	7.47550			0.284794
$690$		0			0
$691$		−	23.9884i		−	0.912560i	−0.889836	−	0.456280i	$-0.849181\pi$
$691$		−	23.9884i		−	0.912560i	0.889836	−	0.456280i	$-0.150819\pi$
$692$		−	10.2541i		−	0.389804i
$693$		0			0
$694$	−9.07107			−0.344333
$695$		0			0
$696$		0			0
$697$			19.1421i			0.725060i
$698$			6.16564i			0.233373i
$699$		0			0
$700$		0			0
$701$			41.8701i			1.58141i	0.612197	+	0.790705i	$0.290286\pi$
$701$			41.8701i			1.58141i	−0.612197	+	0.790705i	$0.709714\pi$
$702$		0			0
$703$	21.5934			0.814410
$704$			1.41421i			0.0533002i
$705$		0			0
$706$		−	18.9737i		−	0.714083i
$707$	−27.0711	−	17.1212i	−1.01811	−	0.643911i
$708$		0			0
$709$	−24.1421			−0.906677			−0.453338	−	0.891338i	$-0.649767\pi$
$709$	−24.1421			−0.906677			−0.453338	+	0.891338i	$0.649767\pi$
$710$		0			0
$711$		0			0
$712$	2.61972			0.0981780
$713$		−	4.08849i		−	0.153115i
$714$		0			0
$715$		0			0
$716$			11.3137i			0.422813i
$717$		0			0
$718$			16.3137i			0.608822i
$719$	23.9884			0.894615			0.447307	−	0.894380i	$-0.352383\pi$
$719$	23.9884			0.894615			0.447307	+	0.894380i	$0.352383\pi$
$720$		0			0
$721$	−23.2843	−	14.7263i	−0.867152	−	0.548435i
$722$	39.2843			1.46201
$723$		0			0
$724$		−	2.61972i		−	0.0973610i
$725$		0			0
$726$		0			0
$727$	−6.70820			−0.248794			−0.124397	−	0.992233i	$-0.539700\pi$
$727$	−6.70820			−0.248794			−0.124397	+	0.992233i	$0.539700\pi$
$728$	2.07107	+	1.30986i	0.0767589	+	0.0485466i
$729$		0			0
$730$		0			0
$731$	8.01806			0.296559
$732$		0			0
$733$	39.6408			1.46417			0.732084	−	0.681214i	$-0.238548\pi$
$733$	39.6408			1.46417			0.732084	+	0.681214i	$0.238548\pi$
$734$	24.5967			0.907883
$735$		0			0
$736$	1.00000			0.0368605
$737$		0			0
$738$		0			0
$739$	26.3553			0.969497			0.484748	−	0.874654i	$-0.338911\pi$
$739$	26.3553			0.969497			0.484748	+	0.874654i	$0.338911\pi$
$740$		0			0
$741$		0			0
$742$	18.0475	+	11.4142i	0.662543	+	0.419029i
$743$	35.1421			1.28924			0.644620	−	0.764503i	$-0.277016\pi$
$743$	35.1421			1.28924			0.644620	+	0.764503i	$0.277016\pi$
$744$		0			0
$745$		0			0
$746$			7.07107i			0.258890i
$747$		0			0
$748$	3.16228			0.115624
$749$	4.47214	+	2.82843i	0.163408	+	0.103348i
$750$		0			0
$751$	31.2132			1.13899			0.569493	−	0.821996i	$-0.307140\pi$
$751$	31.2132			1.13899			0.569493	+	0.821996i	$0.307140\pi$
$752$			1.30986i			0.0477656i
$753$		0			0
$754$		−	0.701474i		−	0.0255462i
$755$		0			0
$756$		0			0
$757$		−	44.0416i		−	1.60072i	−0.599520	−	0.800360i	$-0.704642\pi$
$757$		−	44.0416i		−	1.60072i	0.599520	−	0.800360i	$-0.295358\pi$
$758$	22.0711			0.801657
$759$		0			0
$760$		0			0
$761$	31.3050			1.13480			0.567402	−	0.823441i	$-0.307949\pi$
$761$	31.3050			1.13480			0.567402	+	0.823441i	$0.307949\pi$
$762$		0			0
$763$	15.8114	+	10.0000i	0.572411	+	0.362024i
$764$		−	0.656854i		−	0.0237642i
$765$		0			0
$766$		−	16.0361i		−	0.579409i
$767$	−6.71573			−0.242491
$768$		0			0
$769$		−	15.0441i		−	0.542504i	−0.962508	−	0.271252i	$-0.912562\pi$
$769$		−	15.0441i		−	0.542504i	0.962508	−	0.271252i	$-0.0874376\pi$
$770$		0			0
$771$		0			0
$772$			8.48528i			0.305392i
$773$			17.3460i			0.623892i	0.950100	+	0.311946i	$0.100981\pi$
$773$			17.3460i			0.623892i	−0.950100	+	0.311946i	$0.899019\pi$
$774$		0			0
$775$		0			0
$776$	−0.542561			−0.0194768
$777$		0			0
$778$		−	22.8284i		−	0.818439i
$779$		−	65.3553i		−	2.34160i
$780$		0			0
$781$	−22.1421			−0.792308
$782$		−	2.23607i		−	0.0799616i
$783$		0			0
$784$	3.00000	+	6.32456i	0.107143	+	0.225877i
$785$		0			0
$786$		0			0
$787$	33.9247			1.20928			0.604642	−	0.796497i	$-0.293316\pi$
$787$	33.9247			1.20928			0.604642	+	0.796497i	$0.293316\pi$
$788$	5.92893			0.211209
$789$		0			0
$790$		0			0
$791$	−2.39498	−	1.51472i	−0.0851557	−	0.0538572i
$792$		0			0
$793$		−	0.857864i		−	0.0304637i
$794$	29.6114			1.05087
$795$		0			0
$796$			21.5934i			0.765357i
$797$			11.0214i			0.390399i	0.980764	+	0.195199i	$0.0625354\pi$
$797$			11.0214i			0.390399i	−0.980764	+	0.195199i	$0.937465\pi$
$798$		0			0
$799$	2.92893			0.103618
$800$		0			0
$801$		0			0
$802$			9.79899i			0.346014i
$803$		−	19.7410i		−	0.696643i
$804$		0			0
$805$		0			0
$806$			3.78680i			0.133384i
$807$		0			0
$808$	12.1065			0.425907
$809$			19.8995i			0.699629i	0.936819	+	0.349814i	$0.113755\pi$
$809$			19.8995i			0.699629i	−0.936819	+	0.349814i	$0.886245\pi$
$810$		0			0
$811$		−	41.1096i		−	1.44355i	−0.692126	−	0.721777i	$-0.743326\pi$
$811$		−	41.1096i		−	1.44355i	0.692126	−	0.721777i	$-0.256674\pi$
$812$	1.07107	−	1.69351i	0.0375871	−	0.0594304i
$813$		0			0
$814$	−4.00000			−0.140200
$815$		0			0
$816$		0			0
$817$	−27.3754			−0.957743
$818$			22.9032i			0.800793i
$819$		0			0
$820$		0			0
$821$		−	43.9411i		−	1.53356i	−0.641912	−	0.766778i	$-0.721859\pi$
$821$		−	43.9411i		−	1.53356i	0.641912	−	0.766778i	$-0.278141\pi$
$822$		0			0
$823$		−	17.0711i		−	0.595060i	−0.954712	−	0.297530i	$-0.903837\pi$
$823$		−	17.0711i		−	0.595060i	0.954712	−	0.297530i	$-0.0961628\pi$
$824$	10.4130			0.362756
$825$		0			0
$826$	−16.2132	−	10.2541i	−0.564129	−	0.356787i
$827$	−30.4264			−1.05803			−0.529015	−	0.848613i	$-0.677438\pi$
$827$	−30.4264			−1.05803			−0.529015	+	0.848613i	$0.677438\pi$
$828$		0			0
$829$		−	21.4345i		−	0.744450i	−0.928143	−	0.372225i	$-0.878595\pi$
$829$		−	21.4345i		−	0.744450i	0.928143	−	0.372225i	$-0.121405\pi$
$830$		0			0
$831$		0			0
$832$	−0.926210			−0.0321105
$833$	14.1421	−	6.70820i	0.489996	−	0.232425i
$834$		0			0
$835$		0			0
$836$	−10.7967			−0.373411
$837$		0			0
$838$	−28.8441			−0.996405
$839$	35.0098			1.20867			0.604336	−	0.796729i	$-0.293438\pi$
$839$	35.0098			1.20867			0.604336	+	0.796729i	$0.293438\pi$
$840$		0			0
$841$	28.4264			0.980221
$842$	−33.3553			−1.14950
$843$		0			0
$844$	16.2132			0.558081
$845$		0			0
$846$		0			0
$847$	20.1246	+	12.7279i	0.691490	+	0.437337i
$848$	−8.07107			−0.277162
$849$		0			0
$850$		0			0
$851$			2.82843i			0.0969572i
$852$		0			0
$853$	9.10318			0.311687			0.155844	−	0.987782i	$-0.450190\pi$
$853$	9.10318			0.311687			0.155844	+	0.987782i	$0.450190\pi$
$854$	1.30986	−	2.07107i	0.0448224	−	0.0708705i
$855$		0			0
$856$	−2.00000			−0.0683586
$857$			17.8885i			0.611061i	0.952182	+	0.305531i	$0.0988338\pi$
$857$			17.8885i			0.611061i	−0.952182	+	0.305531i	$0.901166\pi$
$858$		0			0
$859$			2.61972i			0.0893836i	0.999001	+	0.0446918i	$0.0142306\pi$
$859$			2.61972i			0.0893836i	−0.999001	+	0.0446918i	$0.985769\pi$
$860$		0			0
$861$		0			0
$862$		−	4.79899i		−	0.163454i
$863$	−19.8579			−0.675970			−0.337985	−	0.941152i	$-0.609745\pi$
$863$	−19.8579			−0.675970			−0.337985	+	0.941152i	$0.609745\pi$
$864$		0			0
$865$		0			0
$866$	9.71157			0.330013
$867$		0			0
$868$	−5.78199	+	9.14214i	−0.196254	+	0.310304i
$869$			18.4853i			0.627070i
$870$		0			0
$871$		0			0
$872$	−7.07107			−0.239457
$873$		0			0
$874$			7.63441i			0.258238i
$875$		0			0
$876$		0			0
$877$			5.85786i			0.197806i	0.995097	+	0.0989030i	$0.0315334\pi$
$877$			5.85786i			0.197806i	−0.995097	+	0.0989030i	$0.968467\pi$
$878$		−	4.08849i		−	0.137980i
$879$		0			0
$880$		0			0
$881$	−32.0064			−1.07832			−0.539162	−	0.842202i	$-0.681259\pi$
$881$	−32.0064			−1.07832			−0.539162	+	0.842202i	$0.681259\pi$
$882$		0			0
$883$		−	36.0122i		−	1.21191i	−0.795500	−	0.605953i	$-0.792792\pi$
$883$		−	36.0122i		−	1.21191i	0.795500	−	0.605953i	$-0.207208\pi$
$884$			2.07107i			0.0696575i
$885$		0			0
$886$	−23.0711			−0.775088
$887$		−	16.3539i		−	0.549112i	−0.961571	−	0.274556i	$-0.911469\pi$
$887$		−	16.3539i		−	0.549112i	0.961571	−	0.274556i	$-0.0885308\pi$
$888$		0			0
$889$	14.1421	−	22.3607i	0.474312	−	0.749953i
$890$		0			0
$891$		0			0
$892$	−13.0328			−0.436369
$893$	−10.0000			−0.334637
$894$		0			0
$895$		0			0
$896$	−2.23607	−	1.41421i	−0.0747018	−	0.0472456i
$897$		0			0
$898$		−	18.3848i		−	0.613508i
$899$	3.09645			0.103273
$900$		0			0
$901$			18.0475i			0.601248i
$902$			12.1065i			0.403104i
$903$		0			0
$904$	1.07107			0.0356232
$905$		0			0
$906$		0			0
$907$			14.8995i			0.494730i	0.968922	+	0.247365i	$0.0795646\pi$
$907$			14.8995i			0.494730i	−0.968922	+	0.247365i	$0.920435\pi$
$908$			25.1393i			0.834277i
$909$		0			0
$910$		0			0
$911$			12.3726i			0.409922i	0.978770	+	0.204961i	$0.0657067\pi$
$911$			12.3726i			0.409922i	−0.978770	+	0.204961i	$0.934293\pi$
$912$		0			0
$913$	20.2835			0.671287
$914$		−	17.8284i		−	0.589712i
$915$		0			0
$916$		−	10.7967i		−	0.356733i
$917$	−28.2843	−	17.8885i	−0.934029	−	0.590732i
$918$		0			0
$919$	−7.07107			−0.233253			−0.116627	−	0.993176i	$-0.537208\pi$
$919$	−7.07107			−0.233253			−0.116627	+	0.993176i	$0.537208\pi$
$920$		0			0
$921$		0			0
$922$	−38.9394			−1.28240
$923$		−	14.5015i		−	0.477324i
$924$		0			0
$925$		0			0
$926$			38.2843i			1.25810i
$927$		0			0
$928$			0.757359i			0.0248615i
$929$	25.3640			0.832167			0.416084	−	0.909326i	$-0.363402\pi$
$929$	25.3640			0.832167			0.416084	+	0.909326i	$0.363402\pi$
$930$		0			0
$931$	−48.2843	+	22.9032i	−1.58245	+	0.750623i
$932$	23.0711			0.755718
$933$		0			0
$934$			29.6114i			0.968916i
$935$		0			0
$936$		0			0
$937$	−5.23943			−0.171165			−0.0855824	−	0.996331i	$-0.527275\pi$
$937$	−5.23943			−0.171165			−0.0855824	+	0.996331i	$0.527275\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	44.2719			1.44322			0.721611	−	0.692299i	$-0.243402\pi$
$941$	44.2719			1.44322			0.721611	+	0.692299i	$0.243402\pi$
$942$		0			0
$943$	8.56062			0.278772
$944$	7.25077			0.235992
$945$		0			0
$946$	5.07107			0.164875
$947$	33.3553			1.08390			0.541951	−	0.840410i	$-0.317686\pi$
$947$	33.3553			1.08390			0.541951	+	0.840410i	$0.317686\pi$
$948$		0			0
$949$	12.9289			0.419691
$950$		0			0
$951$		0			0
$952$	−3.16228	+	5.00000i	−0.102490	+	0.162051i
$953$	36.0000			1.16615			0.583077	−	0.812417i	$-0.301849\pi$
$953$	36.0000			1.16615			0.583077	+	0.812417i	$0.301849\pi$
$954$		0			0
$955$		0			0
$956$		−	7.17157i		−	0.231945i
$957$		0			0
$958$	−33.6999			−1.08880
$959$	27.1506	+	17.1716i	0.876740	+	0.554499i
$960$		0			0
$961$	14.2843			0.460783
$962$		−	2.61972i		−	0.0844631i
$963$		0			0
$964$		−	4.47214i		−	0.144038i
$965$		0			0
$966$		0			0
$967$		−	49.4975i		−	1.59173i	−0.605473	−	0.795866i	$-0.707016\pi$
$967$		−	49.4975i		−	1.59173i	0.605473	−	0.795866i	$-0.292984\pi$
$968$	−9.00000			−0.289271
$969$		0			0
$970$		0			0
$971$	−27.6001			−0.885730			−0.442865	−	0.896588i	$-0.646038\pi$
$971$	−27.6001			−0.885730			−0.442865	+	0.896588i	$0.646038\pi$
$972$		0			0
$973$	−15.2688	+	24.1421i	−0.489496	+	0.773961i
$974$		−	29.8995i		−	0.958042i
$975$		0			0
$976$			0.926210i			0.0296472i
$977$	12.0000			0.383914			0.191957	−	0.981403i	$-0.438517\pi$
$977$	12.0000			0.383914			0.191957	+	0.981403i	$0.438517\pi$
$978$		0			0
$979$		−	3.70484i		−	0.118407i
$980$		0			0
$981$		0			0
$982$		−	21.5147i		−	0.686562i
$983$		−	6.32456i		−	0.201722i	−0.994901	−	0.100861i	$-0.967840\pi$
$983$		−	6.32456i		−	0.201722i	0.994901	−	0.100861i	$-0.0321597\pi$
$984$		0			0
$985$		0			0
$986$	1.69351			0.0539322
$987$		0			0
$988$		−	7.07107i		−	0.224961i
$989$		−	3.58579i		−	0.114021i
$990$		0			0
$991$	−25.3553			−0.805439			−0.402719	−	0.915323i	$-0.631935\pi$
$991$	−25.3553			−0.805439			−0.402719	+	0.915323i	$0.631935\pi$
$992$		−	4.08849i		−	0.129810i
$993$		0			0
$994$	22.1421	−	35.0098i	0.702306	−	1.11044i
$995$		0			0
$996$		0			0
$997$	1.85242			0.0586667			0.0293334	−	0.999570i	$-0.490662\pi$
$997$	1.85242			0.0586667			0.0293334	+	0.999570i	$0.490662\pi$
$998$	−22.0711			−0.698647
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3150.2.d.b.3149.7		8
3.2	odd	2		3150.2.d.e.3149.7		8
5.2	odd	4		3150.2.b.d.251.2	yes	8
5.3	odd	4		3150.2.b.a.251.7	yes	8
5.4	even	2		3150.2.d.e.3149.2		8
7.6	odd	2	inner	3150.2.d.b.3149.4		8
15.2	even	4		3150.2.b.d.251.6	yes	8
15.8	even	4		3150.2.b.a.251.3	✓	8
15.14	odd	2	inner	3150.2.d.b.3149.2		8
21.20	even	2		3150.2.d.e.3149.4		8
35.13	even	4		3150.2.b.a.251.8	yes	8
35.27	even	4		3150.2.b.d.251.1	yes	8
35.34	odd	2		3150.2.d.e.3149.5		8
105.62	odd	4		3150.2.b.d.251.5	yes	8
105.83	odd	4		3150.2.b.a.251.4	yes	8
105.104	even	2	inner	3150.2.d.b.3149.5		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3150.2.b.a.251.3	✓	8	15.8	even	4
3150.2.b.a.251.4	yes	8	105.83	odd	4
3150.2.b.a.251.7	yes	8	5.3	odd	4
3150.2.b.a.251.8	yes	8	35.13	even	4
3150.2.b.d.251.1	yes	8	35.27	even	4
3150.2.b.d.251.2	yes	8	5.2	odd	4
3150.2.b.d.251.5	yes	8	105.62	odd	4
3150.2.b.d.251.6	yes	8	15.2	even	4
3150.2.d.b.3149.2		8	15.14	odd	2	inner
3150.2.d.b.3149.4		8	7.6	odd	2	inner
3150.2.d.b.3149.5		8	105.104	even	2	inner
3150.2.d.b.3149.7		8	1.1	even	1	trivial
3150.2.d.e.3149.2		8	5.4	even	2
3150.2.d.e.3149.4		8	21.20	even	2
3150.2.d.e.3149.5		8	35.34	odd	2
3150.2.d.e.3149.7		8	3.2	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:	\( N \)	\(=\)	\( 3150 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3150.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} +(2.23607 + 1.41421i) q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} +(2.23607 + 1.41421i) q^{7} -1.00000 q^{8} +1.41421i q^{11} -0.926210 q^{13} +(-2.23607 - 1.41421i) q^{14} +1.00000 q^{16} -2.23607i q^{17} +7.63441i q^{19} -1.41421i q^{22} -1.00000 q^{23} +0.926210 q^{26} +(2.23607 + 1.41421i) q^{28} -0.757359i q^{29} +4.08849i q^{31} -1.00000 q^{32} +2.23607i q^{34} -2.82843i q^{37} -7.63441i q^{38} -8.56062 q^{41} +3.58579i q^{43} +1.41421i q^{44} +1.00000 q^{46} +1.30986i q^{47} +(3.00000 + 6.32456i) q^{49} -0.926210 q^{52} -8.07107 q^{53} +(-2.23607 - 1.41421i) q^{56} +0.757359i q^{58} +7.25077 q^{59} +0.926210i q^{61} -4.08849i q^{62} +1.00000 q^{64} -2.23607i q^{68} +15.6569i q^{71} -13.9590 q^{73} +2.82843i q^{74} +7.63441i q^{76} +(-2.00000 + 3.16228i) q^{77} +13.0711 q^{79} +8.56062 q^{82} -14.3426i q^{83} -3.58579i q^{86} -1.41421i q^{88} -2.61972 q^{89} +(-2.07107 - 1.30986i) q^{91} -1.00000 q^{92} -1.30986i q^{94} +0.542561 q^{97} +(-3.00000 - 6.32456i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{2} + 8 q^{4} - 8 q^{8}+O(q^{10}) \) 8 * q - 8 * q^2 + 8 * q^4 - 8 * q^8	\( 8 q - 8 q^{2} + 8 q^{4} - 8 q^{8} + 8 q^{16} - 8 q^{23} - 8 q^{32} + 8 q^{46} + 24 q^{49} - 8 q^{53} + 8 q^{64} - 16 q^{77} + 48 q^{79} + 40 q^{91} - 8 q^{92} - 24 q^{98}+O(q^{100}) \) 8 * q - 8 * q^2 + 8 * q^4 - 8 * q^8 + 8 * q^16 - 8 * q^23 - 8 * q^32 + 8 * q^46 + 24 * q^49 - 8 * q^53 + 8 * q^64 - 16 * q^77 + 48 * q^79 + 40 * q^91 - 8 * q^92 - 24 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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