LMFDB - Embedded newform 2156.2.c.c.1077.16

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2156,2,Mod(1077,2156)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2156, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("2156.1077");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2156 = 2^{2} \cdot 7^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2156.c (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:	$17.2157466758$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 2x^{14} - 17x^{12} - 343x^{10} + 490x^{8} - 16807x^{6} - 40817x^{4} + 235298x^{2} + 5764801 $ x^16 + 2x^14 - 17x^12 - 343x^10 + 490x^8 - 16807x^6 - 40817x^4 + 235298*x^2 + 5764801
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{8}\cdot 7^{2} $
Twist minimal:	no (minimal twist has level 308)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1077.16
Root			$2.64407 - 0.0942693i$ of defining polynomial
Character	$\chi$	$=$	2156.1077
Dual form			2156.2.c.c.1077.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.73846i q^{3} -0.365169i q^{5} -4.49914 q^{9} +O(q^{10})$	$q+2.73846i q^{3} -0.365169i q^{5} -4.49914 q^{9} +(-0.305755 + 3.30250i) q^{11} +4.51413 q^{13} +1.00000 q^{15} +3.32366 q^{17} +3.32366 q^{19} +5.11065 q^{23} +4.86665 q^{25} -4.10534i q^{27} +5.32202i q^{29} -5.05429i q^{31} +(-9.04376 - 0.837298i) q^{33} +0.611511 q^{37} +12.3617i q^{39} -4.51413 q^{41} +7.03973i q^{43} +1.64295i q^{45} -0.583781i q^{47} +9.10169i q^{51} -13.7414 q^{53} +(1.20597 + 0.111652i) q^{55} +9.10169i q^{57} -3.41134i q^{59} -11.4402 q^{61} -1.64842i q^{65} +10.8748 q^{67} +13.9953i q^{69} -9.37564 q^{71} +3.13391 q^{73} +13.3271i q^{75} +11.9270i q^{79} -2.25514 q^{81} +14.7639 q^{83} -1.21370i q^{85} -14.5741 q^{87} -8.52778i q^{89} +13.8409 q^{93} -1.21370i q^{95} -9.15963i q^{97} +(1.37564 - 14.8584i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 8 q^{9}+O(q^{10}) $ 16 * q - 8 * q^9	$ 16 q - 8 q^{9} + 2 q^{11} + 16 q^{15} + 4 q^{23} + 12 q^{25} - 4 q^{37} - 16 q^{53} + 36 q^{67} - 76 q^{71} + 16 q^{81} - 20 q^{93} - 52 q^{99}+O(q^{100}) $ 16 * q - 8 * q^9 + 2 * q^11 + 16 * q^15 + 4 * q^23 + 12 * q^25 - 4 * q^37 - 16 * q^53 + 36 * q^67 - 76 * q^71 + 16 * q^81 - 20 * q^93 - 52 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2156\mathbb{Z}\right)^\times$.

$n$	$981$	$1079$	$1277$
$\chi(n)$	$-1$	$1$	$-1$

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 $

$2$		0			0
$2$		0			0
$3$			2.73846i			1.58105i	0.612431	+	0.790524i	$0.290192\pi$
$3$			2.73846i			1.58105i	−0.612431	+	0.790524i	$0.709808\pi$
$4$		0			0
$5$		−	0.365169i		−	0.163309i	−0.996661	−	0.0816543i	$-0.973980\pi$
$5$		−	0.365169i		−	0.163309i	0.996661	−	0.0816543i	$-0.0260203\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$	−4.49914			−1.49971
$10$		0			0
$11$	−0.305755	+	3.30250i	−0.0921887	+	0.995742i
$11$	−0.305755	+	3.30250i	−0.0921887	+	0.995742i
$12$		0			0
$13$	4.51413			1.25199			0.625997	−	0.779826i	$-0.284692\pi$
$13$	4.51413			1.25199			0.625997	+	0.779826i	$0.284692\pi$
$14$		0			0
$15$	1.00000			0.258199
$16$		0			0
$17$	3.32366			0.806105			0.403052	−	0.915177i	$-0.367949\pi$
$17$	3.32366			0.806105			0.403052	+	0.915177i	$0.367949\pi$
$18$		0			0
$19$	3.32366			0.762499			0.381249	−	0.924472i	$-0.375494\pi$
$19$	3.32366			0.762499			0.381249	+	0.924472i	$0.375494\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	5.11065			1.06564			0.532822	−	0.846227i	$-0.321131\pi$
$23$	5.11065			1.06564			0.532822	+	0.846227i	$0.321131\pi$
$24$		0			0
$25$	4.86665			0.973330
$26$		0			0
$27$		−	4.10534i		−	0.790073i
$28$		0			0
$29$			5.32202i			0.988274i	0.869384	+	0.494137i	$0.164516\pi$
$29$			5.32202i			0.988274i	−0.869384	+	0.494137i	$0.835484\pi$
$30$		0			0
$31$		−	5.05429i		−	0.907777i	−0.891059	−	0.453888i	$-0.850036\pi$
$31$		−	5.05429i		−	0.907777i	0.891059	−	0.453888i	$-0.149964\pi$
$32$		0			0
$33$	−9.04376	−	0.837298i	−1.57432	−	0.145755i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	0.611511			0.100532			0.0502659	−	0.998736i	$-0.483993\pi$
$37$	0.611511			0.100532			0.0502659	+	0.998736i	$0.483993\pi$
$38$		0			0
$39$			12.3617i			1.97946i
$40$		0			0
$41$	−4.51413			−0.704988			−0.352494	−	0.935814i	$-0.614666\pi$
$41$	−4.51413			−0.704988			−0.352494	+	0.935814i	$0.614666\pi$
$42$		0			0
$43$			7.03973i			1.07355i	0.843726	+	0.536774i	$0.180357\pi$
$43$			7.03973i			1.07355i	−0.843726	+	0.536774i	$0.819643\pi$
$44$		0			0
$45$			1.64295i			0.244916i
$46$		0			0
$47$		−	0.583781i		−	0.0851532i	−0.999093	−	0.0425766i	$-0.986443\pi$
$47$		−	0.583781i		−	0.0851532i	0.999093	−	0.0425766i	$-0.0135567\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$			9.10169i			1.27449i
$52$		0			0
$53$	−13.7414			−1.88753			−0.943765	−	0.330616i	$-0.892744\pi$
$53$	−13.7414			−1.88753			−0.943765	+	0.330616i	$0.892744\pi$
$54$		0			0
$55$	1.20597	+	0.111652i	0.162613	+	0.0150552i
$56$		0			0
$57$			9.10169i			1.20555i
$58$		0			0
$59$		−	3.41134i		−	0.444119i	−0.975033	−	0.222059i	$-0.928722\pi$
$59$		−	3.41134i		−	0.444119i	0.975033	−	0.222059i	$-0.0712779\pi$
$60$		0			0
$61$	−11.4402			−1.46477			−0.732384	−	0.680892i	$-0.761592\pi$
$61$	−11.4402			−1.46477			−0.732384	+	0.680892i	$0.761592\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	1.64842i		−	0.204461i
$66$		0			0
$67$	10.8748			1.32857			0.664283	−	0.747481i	$-0.268737\pi$
$67$	10.8748			1.32857			0.664283	+	0.747481i	$0.268737\pi$
$68$		0			0
$69$			13.9953i			1.68484i
$70$		0			0
$71$	−9.37564			−1.11268			−0.556342	−	0.830954i	$-0.687795\pi$
$71$	−9.37564			−1.11268			−0.556342	+	0.830954i	$0.687795\pi$
$72$		0			0
$73$	3.13391			0.366796			0.183398	−	0.983039i	$-0.441290\pi$
$73$	3.13391			0.366796			0.183398	+	0.983039i	$0.441290\pi$
$74$		0			0
$75$			13.3271i			1.53888i
$76$		0			0
$77$		0			0
$78$		0			0
$79$			11.9270i			1.34189i	0.741505	+	0.670947i	$0.234112\pi$
$79$			11.9270i			1.34189i	−0.741505	+	0.670947i	$0.765888\pi$
$80$		0			0
$81$	−2.25514			−0.250571
$82$		0			0
$83$	14.7639			1.62054			0.810272	−	0.586054i	$-0.199319\pi$
$83$	14.7639			1.62054			0.810272	+	0.586054i	$0.199319\pi$
$84$		0			0
$85$		−	1.21370i		−	0.131644i
$86$		0			0
$87$	−14.5741			−1.56251
$88$		0			0
$89$		−	8.52778i		−	0.903942i	−0.892033	−	0.451971i	$-0.850721\pi$
$89$		−	8.52778i		−	0.903942i	0.892033	−	0.451971i	$-0.149279\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	13.8409			1.43524
$94$		0			0
$95$		−	1.21370i		−	0.124523i
$96$		0			0
$97$		−	9.15963i		−	0.930019i	−0.885306	−	0.465010i	$-0.846051\pi$
$97$		−	9.15963i		−	0.930019i	0.885306	−	0.465010i	$-0.153949\pi$
$98$		0			0
$99$	1.37564	−	14.8584i	0.138257	−	1.49333i
$100$		0			0
$101$	−13.3836			−1.33172			−0.665861	−	0.746076i	$-0.731936\pi$
$101$	−13.3836			−1.33172			−0.665861	+	0.746076i	$0.731936\pi$
$102$		0			0
$103$			6.83910i			0.673877i	0.941527	+	0.336938i	$0.109391\pi$
$103$			6.83910i			0.673877i	−0.941527	+	0.336938i	$0.890609\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			11.1480i			1.07772i	0.842395	+	0.538861i	$0.181145\pi$
$107$			11.1480i			1.07772i	−0.842395	+	0.538861i	$0.818855\pi$
$108$		0			0
$109$		−	11.1480i		−	1.06779i	−0.845551	−	0.533895i	$-0.820728\pi$
$109$		−	11.1480i		−	1.06779i	0.845551	−	0.533895i	$-0.179272\pi$
$110$		0			0
$111$			1.67460i			0.158946i
$112$		0			0
$113$	8.84567			0.832131			0.416065	−	0.909335i	$-0.363409\pi$
$113$	8.84567			0.832131			0.416065	+	0.909335i	$0.363409\pi$
$114$		0			0
$115$		−	1.86625i		−	0.174029i
$116$		0			0
$117$	−20.3097			−1.87763
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−10.8130	−	2.01951i	−0.983002	−	0.183592i
$122$		0			0
$123$		−	12.3617i		−	1.11462i
$124$		0			0
$125$		−	3.60300i		−	0.322262i
$126$		0			0
$127$		−	7.03973i		−	0.624675i	−0.949971	−	0.312337i	$-0.898888\pi$
$127$		−	7.03973i		−	0.624675i	0.949971	−	0.312337i	$-0.101112\pi$
$128$		0			0
$129$	−19.2780			−1.69733
$130$		0			0
$131$	8.86951			0.774933			0.387466	−	0.921884i	$-0.373350\pi$
$131$	8.86951			0.774933			0.387466	+	0.921884i	$0.373350\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	−1.49914			−0.129026
$136$		0			0
$137$	8.57596			0.732694			0.366347	−	0.930478i	$-0.380608\pi$
$137$	8.57596			0.732694			0.366347	+	0.930478i	$0.380608\pi$
$138$		0			0
$139$	12.8204			1.08741			0.543707	−	0.839275i	$-0.317020\pi$
$139$	12.8204			1.08741			0.543707	+	0.839275i	$0.317020\pi$
$140$		0			0
$141$	1.59866			0.134631
$142$		0			0
$143$	−1.38022	+	14.9079i	−0.115420	+	1.24666i
$144$		0			0
$145$	1.94344			0.161394
$146$		0			0
$147$		0			0
$148$		0			0
$149$			21.4634i			1.75835i	0.476497	+	0.879176i	$0.341906\pi$
$149$			21.4634i			1.75835i	−0.476497	+	0.879176i	$0.658094\pi$
$150$		0			0
$151$			1.21370i			0.0987692i	0.998780	+	0.0493846i	$0.0157260\pi$
$151$			1.21370i			0.0987692i	−0.998780	+	0.0493846i	$0.984274\pi$
$152$		0			0
$153$	−14.9536			−1.20893
$154$		0			0
$155$	−1.84567			−0.148248
$156$		0			0
$157$			16.0445i			1.28049i	0.768172	+	0.640243i	$0.221167\pi$
$157$			16.0445i			1.28049i	−0.768172	+	0.640243i	$0.778833\pi$
$158$		0			0
$159$		−	37.6303i		−	2.98428i
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−19.2376			−1.50680			−0.753401	−	0.657561i	$-0.771588\pi$
$163$	−19.2376			−1.50680			−0.753401	+	0.657561i	$0.771588\pi$
$164$		0			0
$165$	−0.305755	+	3.30250i	−0.0238030	+	0.257099i
$166$		0			0
$167$	−0.189746			−0.0146830			−0.00734151	−	0.999973i	$-0.502337\pi$
$167$	−0.189746			−0.0146830			−0.00734151	+	0.999973i	$0.502337\pi$
$168$		0			0
$169$	7.37735			0.567489
$170$		0			0
$171$	−14.9536			−1.14353
$172$		0			0
$173$	−22.6016			−1.71837			−0.859185	−	0.511664i	$-0.829029\pi$
$173$	−22.6016			−1.71837			−0.859185	+	0.511664i	$0.829029\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	9.34180			0.702173
$178$		0			0
$179$	−6.88462			−0.514581			−0.257290	−	0.966334i	$-0.582830\pi$
$179$	−6.88462			−0.514581			−0.257290	+	0.966334i	$0.582830\pi$
$180$		0			0
$181$		−	20.3397i		−	1.51184i	−0.654664	−	0.755920i	$-0.727190\pi$
$181$		−	20.3397i		−	1.51184i	0.654664	−	0.755920i	$-0.272810\pi$
$182$		0			0
$183$		−	31.3285i		−	2.31587i
$184$		0			0
$185$		−	0.223305i		−	0.0164177i
$186$		0			0
$187$	−1.01623	+	10.9764i	−0.0743138	+	0.802672i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	3.00000			0.217072			0.108536	−	0.994092i	$-0.465384\pi$
$191$	3.00000			0.217072			0.108536	+	0.994092i	$0.465384\pi$
$192$		0			0
$193$		−	25.9372i		−	1.86700i	−0.358579	−	0.933500i	$-0.616738\pi$
$193$		−	25.9372i		−	1.86700i	0.358579	−	0.933500i	$-0.383262\pi$
$194$		0			0
$195$	4.51413			0.323263
$196$		0			0
$197$			15.7067i			1.11906i	0.828812	+	0.559528i	$0.189017\pi$
$197$			15.7067i			1.11906i	−0.828812	+	0.559528i	$0.810983\pi$
$198$		0			0
$199$		−	1.88330i		−	0.133504i	−0.997770	−	0.0667519i	$-0.978736\pi$
$199$		−	1.88330i		−	0.133504i	0.997770	−	0.0667519i	$-0.0212636\pi$
$200$		0			0
$201$			29.7801i			2.10053i
$202$		0			0
$203$		0			0
$204$		0			0
$205$			1.64842i			0.115131i
$206$		0			0
$207$	−22.9936			−1.59816
$208$		0			0
$209$	−1.01623	+	10.9764i	−0.0702938	+	0.759252i
$210$		0			0
$211$			11.5135i			0.792621i	0.918117	+	0.396310i	$0.129710\pi$
$211$			11.5135i			0.792621i	−0.918117	+	0.396310i	$0.870290\pi$
$212$		0			0
$213$		−	25.6748i		−	1.75921i
$214$		0			0
$215$	2.57069			0.175320
$216$		0			0
$217$		0			0
$218$		0			0
$219$			8.58207i			0.579923i
$220$		0			0
$221$	15.0034			1.00924
$222$		0			0
$223$			0.120749i			0.00808595i	0.999992	+	0.00404298i	$0.00128692\pi$
$223$			0.120749i			0.00808595i	−0.999992	+	0.00404298i	$0.998713\pi$
$224$		0			0
$225$	−21.8958			−1.45972
$226$		0			0
$227$	−1.19047			−0.0790144			−0.0395072	−	0.999219i	$-0.512579\pi$
$227$	−1.19047			−0.0790144			−0.0395072	+	0.999219i	$0.512579\pi$
$228$		0			0
$229$			15.9812i			1.05606i	0.849224	+	0.528032i	$0.177070\pi$
$229$			15.9812i			1.05606i	−0.849224	+	0.528032i	$0.822930\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	8.25342i		−	0.540700i	−0.962762	−	0.270350i	$-0.912861\pi$
$233$		−	8.25342i		−	0.540700i	0.962762	−	0.270350i	$-0.0871394\pi$
$234$		0			0
$235$	−0.213179			−0.0139063
$236$		0			0
$237$	−32.6616			−2.12160
$238$		0			0
$239$		−	2.89462i		−	0.187238i	−0.995608	−	0.0936188i	$-0.970157\pi$
$239$		−	2.89462i		−	0.187238i	0.995608	−	0.0936188i	$-0.0298435\pi$
$240$		0			0
$241$	−0.752964			−0.0485027			−0.0242513	−	0.999706i	$-0.507720\pi$
$241$	−0.752964			−0.0485027			−0.0242513	+	0.999706i	$0.507720\pi$
$242$		0			0
$243$		−	18.4916i		−	1.18624i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	15.0034			0.954644
$248$		0			0
$249$			40.4302i			2.56216i
$250$		0			0
$251$			3.66515i			0.231342i	0.993288	+	0.115671i	$0.0369018\pi$
$251$			3.66515i			0.231342i	−0.993288	+	0.115671i	$0.963098\pi$
$252$		0			0
$253$	−1.56261	+	16.8779i	−0.0982404	+	1.06111i
$254$		0			0
$255$	3.32366			0.208135
$256$		0			0
$257$			4.05554i			0.252978i	0.991968	+	0.126489i	$0.0403708\pi$
$257$			4.05554i			0.252978i	−0.991968	+	0.126489i	$0.959629\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		−	23.9445i		−	1.48213i
$262$		0			0
$263$		−	9.10169i		−	0.561234i	−0.959820	−	0.280617i	$-0.909461\pi$
$263$		−	9.10169i		−	0.561234i	0.959820	−	0.280617i	$-0.0905391\pi$
$264$		0			0
$265$			5.01795i			0.308250i
$266$		0			0
$267$	23.3529			1.42918
$268$		0			0
$269$			6.51497i			0.397224i	0.980078	+	0.198612i	$0.0636434\pi$
$269$			6.51497i			0.397224i	−0.980078	+	0.198612i	$0.936357\pi$
$270$		0			0
$271$	−8.02753			−0.487638			−0.243819	−	0.969821i	$-0.578400\pi$
$271$	−8.02753			−0.487638			−0.243819	+	0.969821i	$0.578400\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−1.48800	+	16.0721i	−0.0897301	+	0.969185i
$276$		0			0
$277$		−	2.13125i		−	0.128054i	−0.997948	−	0.0640272i	$-0.979606\pi$
$277$		−	2.13125i		−	0.128054i	0.997948	−	0.0640272i	$-0.0203944\pi$
$278$		0			0
$279$			22.7400i			1.36141i
$280$		0			0
$281$			4.54304i			0.271015i	0.990776	+	0.135508i	$0.0432665\pi$
$281$			4.54304i			0.271015i	−0.990776	+	0.135508i	$0.956733\pi$
$282$		0			0
$283$	31.8196			1.89148			0.945741	−	0.324923i	$-0.105338\pi$
$283$	31.8196			1.89148			0.945741	+	0.324923i	$0.105338\pi$
$284$		0			0
$285$	3.32366			0.196876
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−5.95331			−0.350195
$290$		0			0
$291$	25.0832			1.47041
$292$		0			0
$293$	14.3263			0.836954			0.418477	−	0.908227i	$-0.362564\pi$
$293$	14.3263			0.836954			0.418477	+	0.908227i	$0.362564\pi$
$294$		0			0
$295$	−1.24572			−0.0725284
$296$		0			0
$297$	13.5579	+	1.25523i	0.786708	+	0.0728358i
$298$		0			0
$299$	23.0701			1.33418
$300$		0			0
$301$		0			0
$302$		0			0
$303$		−	36.6505i		−	2.10552i
$304$		0			0
$305$			4.17761i			0.239209i
$306$		0			0
$307$	25.9253			1.47963			0.739817	−	0.672808i	$-0.234912\pi$
$307$	25.9253			1.47963			0.739817	+	0.672808i	$0.234912\pi$
$308$		0			0
$309$	−18.7286			−1.06543
$310$		0			0
$311$		−	24.1519i		−	1.36953i	−0.728763	−	0.684766i	$-0.759905\pi$
$311$		−	24.1519i		−	1.36953i	0.728763	−	0.684766i	$-0.240095\pi$
$312$		0			0
$313$		−	27.5217i		−	1.55562i	−0.628499	−	0.777810i	$-0.716330\pi$
$313$		−	27.5217i		−	1.55562i	0.628499	−	0.777810i	$-0.283670\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−14.5760			−0.818668			−0.409334	−	0.912385i	$-0.634239\pi$
$317$	−14.5760			−0.818668			−0.409334	+	0.912385i	$0.634239\pi$
$318$		0			0
$319$	−17.5760	−	1.62723i	−0.984065	−	0.0911077i
$320$		0			0
$321$	−30.5284			−1.70393
$322$		0			0
$323$	11.0467			0.614654
$324$		0			0
$325$	21.9687			1.21860
$326$		0			0
$327$	30.5284			1.68823
$328$		0			0
$329$		0			0
$330$		0			0
$331$	24.4829			1.34570			0.672850	−	0.739779i	$-0.265070\pi$
$331$	24.4829			1.34570			0.672850	+	0.739779i	$0.265070\pi$
$332$		0			0
$333$	−2.75127			−0.150769
$334$		0			0
$335$		−	3.97114i		−	0.216966i
$336$		0			0
$337$		−	4.14510i		−	0.225798i	−0.993606	−	0.112899i	$-0.963986\pi$
$337$		−	4.14510i		−	0.225798i	0.993606	−	0.112899i	$-0.0360137\pi$
$338$		0			0
$339$			24.2235i			1.31564i
$340$		0			0
$341$	16.6918	+	1.54538i	0.903911	+	0.0836868i
$342$		0			0
$343$		0			0
$344$		0			0
$345$	5.11065			0.275148
$346$		0			0
$347$			16.9204i			0.908334i	0.890917	+	0.454167i	$0.150063\pi$
$347$			16.9204i			0.908334i	−0.890917	+	0.454167i	$0.849937\pi$
$348$		0			0
$349$	19.0882			1.02177			0.510885	−	0.859649i	$-0.329318\pi$
$349$	19.0882			1.02177			0.510885	+	0.859649i	$0.329318\pi$
$350$		0			0
$351$		−	18.5320i		−	0.989166i
$352$		0			0
$353$		−	22.5184i		−	1.19853i	−0.800550	−	0.599266i	$-0.795459\pi$
$353$		−	22.5184i		−	1.19853i	0.800550	−	0.599266i	$-0.204541\pi$
$354$		0			0
$355$			3.42369i			0.181711i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	24.7391i		−	1.30568i	−0.757496	−	0.652839i	$-0.773578\pi$
$359$		−	24.7391i		−	1.30568i	0.757496	−	0.652839i	$-0.226422\pi$
$360$		0			0
$361$	−7.95331			−0.418596
$362$		0			0
$363$	5.53035	−	29.6110i	0.290268	−	1.55417i
$364$		0			0
$365$		−	1.14441i		−	0.0599010i
$366$		0			0
$367$			7.51371i			0.392212i	0.980583	+	0.196106i	$0.0628297\pi$
$367$			7.51371i			0.392212i	−0.980583	+	0.196106i	$0.937170\pi$
$368$		0			0
$369$	20.3097			1.05728
$370$		0			0
$371$		0			0
$372$		0			0
$373$		−	4.95658i		−	0.256642i	−0.991733	−	0.128321i	$-0.959041\pi$
$373$		−	4.95658i		−	0.256642i	0.991733	−	0.128321i	$-0.0409588\pi$
$374$		0			0
$375$	9.86665			0.509512
$376$		0			0
$377$			24.0243i			1.23731i
$378$		0			0
$379$	−31.3080			−1.60818			−0.804091	−	0.594506i	$-0.797348\pi$
$379$	−31.3080			−1.60818			−0.804091	+	0.594506i	$0.797348\pi$
$380$		0			0
$381$	19.2780			0.987641
$382$		0			0
$383$			16.3299i			0.834419i	0.908810	+	0.417210i	$0.136992\pi$
$383$			16.3299i			0.834419i	−0.908810	+	0.417210i	$0.863008\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		−	31.6727i		−	1.61002i
$388$		0			0
$389$	17.4893			0.886743			0.443371	−	0.896338i	$-0.353782\pi$
$389$	17.4893			0.886743			0.443371	+	0.896338i	$0.353782\pi$
$390$		0			0
$391$	16.9861			0.859022
$392$		0			0
$393$			24.2888i			1.22521i
$394$		0			0
$395$	4.35538			0.219143
$396$		0			0
$397$		−	22.3092i		−	1.11966i	−0.828606	−	0.559832i	$-0.810866\pi$
$397$		−	22.3092i		−	1.11966i	0.828606	−	0.559832i	$-0.189134\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	30.8358			1.53987			0.769933	−	0.638124i	$-0.220289\pi$
$401$	30.8358			1.53987			0.769933	+	0.638124i	$0.220289\pi$
$402$		0			0
$403$		−	22.8157i		−	1.13653i
$404$		0			0
$405$			0.823508i			0.0409204i
$406$		0			0
$407$	−0.186973	+	2.01951i	−0.00926789	+	0.100104i
$408$		0			0
$409$	29.3690			1.45220			0.726101	−	0.687588i	$-0.241331\pi$
$409$	29.3690			1.45220			0.726101	+	0.687588i	$0.241331\pi$
$410$		0			0
$411$			23.4849i			1.15842i
$412$		0			0
$413$		0			0
$414$		0			0
$415$		−	5.39131i		−	0.264649i
$416$		0			0
$417$			35.1082i			1.71925i
$418$		0			0
$419$		−	35.2447i		−	1.72181i	−0.508762	−	0.860907i	$-0.669897\pi$
$419$		−	35.2447i		−	1.72181i	0.508762	−	0.860907i	$-0.330103\pi$
$420$		0			0
$421$	−9.47173			−0.461624			−0.230812	−	0.972998i	$-0.574138\pi$
$421$	−9.47173			−0.461624			−0.230812	+	0.972998i	$0.574138\pi$
$422$		0			0
$423$			2.62651i			0.127705i
$424$		0			0
$425$	16.1751			0.784606
$426$		0			0
$427$		0			0
$428$		0			0
$429$	−40.8247	−	3.77967i	−1.97103	−	0.182484i
$430$		0			0
$431$			26.4568i			1.27438i	0.770707	+	0.637189i	$0.219903\pi$
$431$			26.4568i			1.27438i	−0.770707	+	0.637189i	$0.780097\pi$
$432$		0			0
$433$			23.5184i			1.13022i	0.825015	+	0.565110i	$0.191166\pi$
$433$			23.5184i			1.13022i	−0.825015	+	0.565110i	$0.808834\pi$
$434$		0			0
$435$			5.32202i			0.255171i
$436$		0			0
$437$	16.9861			0.812553
$438$		0			0
$439$	25.1723			1.20141			0.600705	−	0.799471i	$-0.294887\pi$
$439$	25.1723			1.20141			0.600705	+	0.799471i	$0.294887\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−24.5871			−1.16817			−0.584084	−	0.811693i	$-0.698546\pi$
$443$	−24.5871			−1.16817			−0.584084	+	0.811693i	$0.698546\pi$
$444$		0			0
$445$	−3.11408			−0.147622
$446$		0			0
$447$	−58.7767			−2.78004
$448$		0			0
$449$	18.8474			0.889462			0.444731	−	0.895664i	$-0.353299\pi$
$449$	18.8474			0.889462			0.444731	+	0.895664i	$0.353299\pi$
$450$		0			0
$451$	1.38022	−	14.9079i	0.0649920	−	0.701986i
$452$		0			0
$453$	−3.32366			−0.156159
$454$		0			0
$455$		0			0
$456$		0			0
$457$			0.763374i			0.0357091i	0.999841	+	0.0178546i	$0.00568359\pi$
$457$			0.763374i			0.0357091i	−0.999841	+	0.0178546i	$0.994316\pi$
$458$		0			0
$459$		−	13.6447i		−	0.636881i
$460$		0			0
$461$	−12.0034			−0.559055			−0.279527	−	0.960138i	$-0.590178\pi$
$461$	−12.0034			−0.559055			−0.279527	+	0.960138i	$0.590178\pi$
$462$		0			0
$463$	22.8940			1.06398			0.531988	−	0.846752i	$-0.321445\pi$
$463$	22.8940			1.06398			0.531988	+	0.846752i	$0.321445\pi$
$464$		0			0
$465$		−	5.05429i		−	0.234387i
$466$		0			0
$467$		−	36.6890i		−	1.69777i	−0.528581	−	0.848883i	$-0.677276\pi$
$467$		−	36.6890i		−	1.69777i	0.528581	−	0.848883i	$-0.322724\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	−43.9370			−2.02451
$472$		0			0
$473$	−23.2487	−	2.15243i	−1.06898	−	0.0989690i
$474$		0			0
$475$	16.1751			0.742163
$476$		0			0
$477$	61.8247			2.83076
$478$		0			0
$479$	13.4534			0.614700			0.307350	−	0.951597i	$-0.400558\pi$
$479$	13.4534			0.614700			0.307350	+	0.951597i	$0.400558\pi$
$480$		0			0
$481$	2.76044			0.125865
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−3.34481			−0.151880
$486$		0			0
$487$	−30.5725			−1.38537			−0.692687	−	0.721238i	$-0.743573\pi$
$487$	−30.5725			−1.38537			−0.692687	+	0.721238i	$0.743573\pi$
$488$		0			0
$489$		−	52.6812i		−	2.38233i
$490$		0			0
$491$			40.4514i			1.82554i	0.408469	+	0.912772i	$0.366063\pi$
$491$			40.4514i			1.82554i	−0.408469	+	0.912772i	$0.633937\pi$
$492$		0			0
$493$			17.6885i			0.796652i
$494$		0			0
$495$	−5.42584	−	0.502340i	−0.243873	−	0.0225785i
$496$		0			0
$497$		0			0
$498$		0			0
$499$	16.0612			0.718999			0.359500	−	0.933145i	$-0.382947\pi$
$499$	16.0612			0.718999			0.359500	+	0.933145i	$0.382947\pi$
$500$		0			0
$501$		−	0.519612i		−	0.0232146i
$502$		0			0
$503$	−22.2531			−0.992219			−0.496109	−	0.868260i	$-0.665238\pi$
$503$	−22.2531			−0.992219			−0.496109	+	0.868260i	$0.665238\pi$
$504$		0			0
$505$			4.88729i			0.217482i
$506$		0			0
$507$			20.2026i			0.897227i
$508$		0			0
$509$			25.8981i			1.14791i	0.818886	+	0.573956i	$0.194592\pi$
$509$			25.8981i			1.14791i	−0.818886	+	0.573956i	$0.805408\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		−	13.6447i		−	0.602429i
$514$		0			0
$515$	2.49743			0.110050
$516$		0			0
$517$	1.92794	+	0.178494i	0.0847906	+	0.00785016i
$518$		0			0
$519$		−	61.8936i		−	2.71683i
$520$		0			0
$521$			22.9861i			1.00704i	0.863984	+	0.503520i	$0.167962\pi$
$521$			22.9861i			1.00704i	−0.863984	+	0.503520i	$0.832038\pi$
$522$		0			0
$523$	−18.9992			−0.830778			−0.415389	−	0.909644i	$-0.636355\pi$
$523$	−18.9992			−0.830778			−0.415389	+	0.909644i	$0.636355\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	16.7987i		−	0.731763i
$528$		0			0
$529$	3.11878			0.135599
$530$		0			0
$531$			15.3481i			0.666051i
$532$		0			0
$533$	−20.3774			−0.882641
$534$		0			0
$535$	4.07092			0.176001
$536$		0			0
$537$		−	18.8532i		−	0.813577i
$538$		0			0
$539$		0			0
$540$		0			0
$541$		−	0.413538i		−	0.0177794i	−0.999960	−	0.00888970i	$-0.997170\pi$
$541$		−	0.413538i		−	0.0177794i	0.999960	−	0.00888970i	$-0.00282972\pi$
$542$		0			0
$543$	55.6994			2.39029
$544$		0			0
$545$	−4.07092			−0.174379
$546$		0			0
$547$			1.64842i			0.0704814i	0.999379	+	0.0352407i	$0.0112198\pi$
$547$			1.64842i			0.0704814i	−0.999379	+	0.0352407i	$0.988780\pi$
$548$		0			0
$549$	51.4711			2.19673
$550$		0			0
$551$			17.6885i			0.753557i
$552$		0			0
$553$		0			0
$554$		0			0
$555$	0.611511			0.0259572
$556$		0			0
$557$		−	32.4361i		−	1.37436i	−0.726486	−	0.687181i	$-0.758848\pi$
$557$		−	32.4361i		−	1.37436i	0.726486	−	0.687181i	$-0.241152\pi$
$558$		0			0
$559$			31.7782i			1.34408i
$560$		0			0
$561$	−30.0583	−	2.78289i	−1.26906	−	0.117494i
$562$		0			0
$563$	−25.9563			−1.09393			−0.546964	−	0.837156i	$-0.684216\pi$
$563$	−25.9563			−1.09393			−0.546964	+	0.837156i	$0.684216\pi$
$564$		0			0
$565$		−	3.23017i		−	0.135894i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			34.6946i			1.45447i	0.686386	+	0.727237i	$0.259196\pi$
$569$			34.6946i			1.45447i	−0.686386	+	0.727237i	$0.740804\pi$
$570$		0			0
$571$		−	34.6041i		−	1.44814i	−0.689727	−	0.724069i	$-0.742270\pi$
$571$		−	34.6041i		−	1.44814i	0.689727	−	0.724069i	$-0.257730\pi$
$572$		0			0
$573$			8.21537i			0.343202i
$574$		0			0
$575$	24.8718			1.03722
$576$		0			0
$577$			0.468870i			0.0195193i	0.999952	+	0.00975966i	$0.00310665\pi$
$577$			0.468870i			0.0195193i	−0.999952	+	0.00975966i	$0.996893\pi$
$578$		0			0
$579$	71.0278			2.95182
$580$		0			0
$581$		0			0
$582$		0			0
$583$	4.20152	−	45.3811i	0.174009	−	1.87949i
$584$		0			0
$585$			7.41648i			0.306634i
$586$		0			0
$587$			44.9718i			1.85618i	0.372352	+	0.928091i	$0.378551\pi$
$587$			44.9718i			1.85618i	−0.372352	+	0.928091i	$0.621449\pi$
$588$		0			0
$589$		−	16.7987i		−	0.692179i
$590$		0			0
$591$	−43.0121			−1.76928
$592$		0			0
$593$	−15.5168			−0.637199			−0.318600	−	0.947889i	$-0.603213\pi$
$593$	−15.5168			−0.637199			−0.318600	+	0.947889i	$0.603213\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	5.15734			0.211076
$598$		0			0
$599$	8.88462			0.363016			0.181508	−	0.983389i	$-0.441902\pi$
$599$	8.88462			0.363016			0.181508	+	0.983389i	$0.441902\pi$
$600$		0			0
$601$	−10.2497			−0.418095			−0.209048	−	0.977905i	$-0.567036\pi$
$601$	−10.2497			−0.418095			−0.209048	+	0.977905i	$0.567036\pi$
$602$		0			0
$603$	−48.9272			−1.99247
$604$		0			0
$605$	−0.737465	+	3.94858i	−0.0299822	+	0.160533i
$606$		0			0
$607$	−43.1011			−1.74942			−0.874710	−	0.484647i	$-0.838948\pi$
$607$	−43.1011			−1.74942			−0.874710	+	0.484647i	$0.838948\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	2.63526i		−	0.106611i
$612$		0			0
$613$		−	13.9522i		−	0.563524i	−0.959484	−	0.281762i	$-0.909081\pi$
$613$		−	13.9522i		−	0.563524i	0.959484	−	0.281762i	$-0.0909188\pi$
$614$		0			0
$615$	−4.51413			−0.182027
$616$		0			0
$617$	−13.7915			−0.555226			−0.277613	−	0.960693i	$-0.589543\pi$
$617$	−13.7915			−0.555226			−0.277613	+	0.960693i	$0.589543\pi$
$618$		0			0
$619$		−	34.7049i		−	1.39491i	−0.716630	−	0.697454i	$-0.754316\pi$
$619$		−	34.7049i		−	1.39491i	0.716630	−	0.697454i	$-0.245684\pi$
$620$		0			0
$621$		−	20.9810i		−	0.841937i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	23.0176			0.920702
$626$		0			0
$627$	−30.0583	−	2.78289i	−1.20041	−	0.111138i
$628$		0			0
$629$	2.03245			0.0810391
$630$		0			0
$631$	−10.9790			−0.437067			−0.218533	−	0.975829i	$-0.570127\pi$
$631$	−10.9790			−0.437067			−0.218533	+	0.975829i	$0.570127\pi$
$632$		0			0
$633$	−31.5292			−1.25317
$634$		0			0
$635$	−2.57069			−0.102015
$636$		0			0
$637$		0			0
$638$		0			0
$639$	42.1823			1.66871
$640$		0			0
$641$	13.7080			0.541434			0.270717	−	0.962659i	$-0.412739\pi$
$641$	13.7080			0.541434			0.270717	+	0.962659i	$0.412739\pi$
$642$		0			0
$643$		−	10.7909i		−	0.425553i	−0.977101	−	0.212777i	$-0.931749\pi$
$643$		−	10.7909i		−	0.425553i	0.977101	−	0.212777i	$-0.0682507\pi$
$644$		0			0
$645$			7.03973i			0.277189i
$646$		0			0
$647$			0.223305i			0.00877902i	0.999990	+	0.00438951i	$0.00139723\pi$
$647$			0.223305i			0.00877902i	−0.999990	+	0.00438951i	$0.998603\pi$
$648$		0			0
$649$	11.2660	+	1.04304i	0.442227	+	0.0409427i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	23.6995			0.927432			0.463716	−	0.885984i	$-0.346516\pi$
$653$	23.6995			0.927432			0.463716	+	0.885984i	$0.346516\pi$
$654$		0			0
$655$		−	3.23887i		−	0.126553i
$656$		0			0
$657$	−14.0999			−0.550090
$658$		0			0
$659$		−	32.3521i		−	1.26026i	−0.776490	−	0.630130i	$-0.783002\pi$
$659$		−	32.3521i		−	1.26026i	0.776490	−	0.630130i	$-0.216998\pi$
$660$		0			0
$661$		−	41.5335i		−	1.61546i	−0.589550	−	0.807732i	$-0.700695\pi$
$661$		−	41.5335i		−	1.61546i	0.589550	−	0.807732i	$-0.299305\pi$
$662$		0			0
$663$			41.0862i			1.59565i
$664$		0			0
$665$		0			0
$666$		0			0
$667$			27.1990i			1.05315i
$668$		0			0
$669$	−0.330666			−0.0127843
$670$		0			0
$671$	3.49790	−	37.7813i	0.135035	−	1.45853i
$672$		0			0
$673$		−	24.8621i		−	0.958362i	−0.877716	−	0.479181i	$-0.840934\pi$
$673$		−	24.8621i		−	0.958362i	0.877716	−	0.479181i	$-0.159066\pi$
$674$		0			0
$675$		−	19.9792i		−	0.769002i
$676$		0			0
$677$	25.2670			0.971091			0.485546	−	0.874211i	$-0.338621\pi$
$677$	25.2670			0.971091			0.485546	+	0.874211i	$0.338621\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		−	3.26006i		−	0.124926i
$682$		0			0
$683$	−9.29539			−0.355678			−0.177839	−	0.984060i	$-0.556911\pi$
$683$	−9.29539			−0.355678			−0.177839	+	0.984060i	$0.556911\pi$
$684$		0			0
$685$		−	3.13168i		−	0.119655i
$686$		0			0
$687$	−43.7637			−1.66969
$688$		0			0
$689$	−62.0306			−2.36318
$690$		0			0
$691$		−	16.3515i		−	0.622042i	−0.950403	−	0.311021i	$-0.899329\pi$
$691$		−	16.3515i		−	0.622042i	0.950403	−	0.311021i	$-0.100671\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	4.68162i		−	0.177584i
$696$		0			0
$697$	−15.0034			−0.568295
$698$		0			0
$699$	22.6016			0.854872
$700$		0			0
$701$		−	42.9480i		−	1.62213i	−0.584959	−	0.811063i	$-0.698890\pi$
$701$		−	42.9480i		−	1.62213i	0.584959	−	0.811063i	$-0.301110\pi$
$702$		0			0
$703$	2.03245			0.0766553
$704$		0			0
$705$		−	0.583781i		−	0.0219865i
$706$		0			0
$707$		0			0
$708$		0			0
$709$	4.79625			0.180127			0.0900634	−	0.995936i	$-0.471293\pi$
$709$	4.79625			0.180127			0.0900634	+	0.995936i	$0.471293\pi$
$710$		0			0
$711$		−	53.6614i		−	2.01246i
$712$		0			0
$713$		−	25.8307i		−	0.967368i
$714$		0			0
$715$	5.44391	+	0.504013i	0.203591	+	0.0188490i
$716$		0			0
$717$	7.92680			0.296032
$718$		0			0
$719$		−	2.64639i		−	0.0986935i	−0.998782	−	0.0493468i	$-0.984286\pi$
$719$		−	2.64639i		−	0.0986935i	0.998782	−	0.0493468i	$-0.0157139\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		−	2.06196i		−	0.0766851i
$724$		0			0
$725$			25.9004i			0.961917i
$726$		0			0
$727$			17.1118i			0.634642i	0.948318	+	0.317321i	$0.102783\pi$
$727$			17.1118i			0.634642i	−0.948318	+	0.317321i	$0.897217\pi$
$728$		0			0
$729$	43.8731			1.62493
$730$		0			0
$731$			23.3976i			0.865392i
$732$		0			0
$733$	12.3519			0.456228			0.228114	−	0.973634i	$-0.426744\pi$
$733$	12.3519			0.456228			0.228114	+	0.973634i	$0.426744\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−3.32502	+	35.9140i	−0.122479	+	1.32291i
$738$		0			0
$739$		−	20.1960i		−	0.742923i	−0.928448	−	0.371462i	$-0.878857\pi$
$739$		−	20.1960i		−	0.742923i	0.928448	−	0.371462i	$-0.121143\pi$
$740$		0			0
$741$			41.0862i			1.50934i
$742$		0			0
$743$		−	9.44593i		−	0.346538i	−0.984875	−	0.173269i	$-0.944567\pi$
$743$		−	9.44593i		−	0.346538i	0.984875	−	0.173269i	$-0.0554330\pi$
$744$		0			0
$745$	7.83778			0.287154
$746$		0			0
$747$	−66.4247			−2.43035
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−1.18473			−0.0432316			−0.0216158	−	0.999766i	$-0.506881\pi$
$751$	−1.18473			−0.0432316			−0.0216158	+	0.999766i	$0.506881\pi$
$752$		0			0
$753$	−10.0368			−0.365763
$754$		0			0
$755$	0.443205			0.0161299
$756$		0			0
$757$	39.1275			1.42211			0.711057	−	0.703134i	$-0.248217\pi$
$757$	39.1275			1.42211			0.711057	+	0.703134i	$0.248217\pi$
$758$		0			0
$759$	−46.2195	−	4.27914i	−1.67766	−	0.155323i
$760$		0			0
$761$	−4.35538			−0.157882			−0.0789412	−	0.996879i	$-0.525154\pi$
$761$	−4.35538			−0.157882			−0.0789412	+	0.996879i	$0.525154\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$			5.46060i			0.197428i
$766$		0			0
$767$		−	15.3992i		−	0.556034i
$768$		0			0
$769$	21.5312			0.776434			0.388217	−	0.921568i	$-0.373091\pi$
$769$	21.5312			0.776434			0.388217	+	0.921568i	$0.373091\pi$
$770$		0			0
$771$	−11.1059			−0.399970
$772$		0			0
$773$			8.44743i			0.303833i	0.988393	+	0.151917i	$0.0485445\pi$
$773$			8.44743i			0.303833i	−0.988393	+	0.151917i	$0.951456\pi$
$774$		0			0
$775$		−	24.5975i		−	0.883567i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−15.0034			−0.537553
$780$		0			0
$781$	2.86665	−	30.9631i	0.102577	−	1.10795i
$782$		0			0
$783$	21.8487			0.780808
$784$		0			0
$785$	5.85894			0.209115
$786$		0			0
$787$	9.30702			0.331759			0.165880	−	0.986146i	$-0.446954\pi$
$787$	9.30702			0.331759			0.165880	+	0.986146i	$0.446954\pi$
$788$		0			0
$789$	24.9246			0.887338
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−51.6425			−1.83388
$794$		0			0
$795$	−13.7414			−0.487358
$796$		0			0
$797$		−	13.1607i		−	0.466175i	−0.972456	−	0.233088i	$-0.925117\pi$
$797$		−	13.1607i		−	0.466175i	0.972456	−	0.233088i	$-0.0748829\pi$
$798$		0			0
$799$		−	1.94029i		−	0.0686424i
$800$		0			0
$801$			38.3677i			1.35566i
$802$		0			0
$803$	−0.958210	+	10.3497i	−0.0338145	+	0.365234i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−17.8409			−0.628031
$808$		0			0
$809$			34.5349i			1.21418i	0.794633	+	0.607090i	$0.207663\pi$
$809$			34.5349i			1.21418i	−0.794633	+	0.607090i	$0.792337\pi$
$810$		0			0
$811$	14.6941			0.515980			0.257990	−	0.966148i	$-0.416940\pi$
$811$	14.6941			0.515980			0.257990	+	0.966148i	$0.416940\pi$
$812$		0			0
$813$		−	21.9830i		−	0.770979i
$814$		0			0
$815$			7.02497i			0.246074i
$816$		0			0
$817$			23.3976i			0.818579i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	32.6483i		−	1.13943i	−0.821842	−	0.569716i	$-0.807053\pi$
$821$		−	32.6483i		−	1.13943i	0.821842	−	0.569716i	$-0.192947\pi$
$822$		0			0
$823$	17.9901			0.627097			0.313548	−	0.949572i	$-0.398482\pi$
$823$	17.9901			0.627097			0.313548	+	0.949572i	$0.398482\pi$
$824$		0			0
$825$	−44.0128	−	4.07484i	−1.53233	−	0.141868i
$826$		0			0
$827$			46.5793i			1.61972i	0.586623	+	0.809860i	$0.300457\pi$
$827$			46.5793i			1.61972i	−0.586623	+	0.809860i	$0.699543\pi$
$828$		0			0
$829$		−	25.9579i		−	0.901554i	−0.892637	−	0.450777i	$-0.851147\pi$
$829$		−	25.9579i		−	0.901554i	0.892637	−	0.450777i	$-0.148853\pi$
$830$		0			0
$831$	5.83633			0.202460
$832$		0			0
$833$		0			0
$834$		0			0
$835$			0.0692895i			0.00239786i
$836$		0			0
$837$	−20.7496			−0.717210
$838$		0			0
$839$		−	16.9653i		−	0.585707i	−0.956157	−	0.292854i	$-0.905395\pi$
$839$		−	16.9653i		−	0.585707i	0.956157	−	0.292854i	$-0.0946049\pi$
$840$		0			0
$841$	0.676145			0.0233154
$842$		0			0
$843$	−12.4409			−0.428488
$844$		0			0
$845$		−	2.69398i		−	0.0926758i
$846$		0			0
$847$		0			0
$848$		0			0
$849$			87.1367i			2.99052i
$850$		0			0
$851$	3.12522			0.107131
$852$		0			0
$853$	−10.5342			−0.360684			−0.180342	−	0.983604i	$-0.557720\pi$
$853$	−10.5342			−0.360684			−0.180342	+	0.983604i	$0.557720\pi$
$854$		0			0
$855$			5.46060i			0.186748i
$856$		0			0
$857$	−39.0245			−1.33305			−0.666525	−	0.745482i	$-0.732219\pi$
$857$	−39.0245			−1.33305			−0.666525	+	0.745482i	$0.732219\pi$
$858$		0			0
$859$			9.06062i			0.309144i	0.987982	+	0.154572i	$0.0493999\pi$
$859$			9.06062i			0.309144i	−0.987982	+	0.154572i	$0.950600\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−26.9180			−0.916301			−0.458150	−	0.888875i	$-0.651488\pi$
$863$	−26.9180			−0.916301			−0.458150	+	0.888875i	$0.651488\pi$
$864$		0			0
$865$			8.25342i			0.280625i
$866$		0			0
$867$		−	16.3029i		−	0.553675i
$868$		0			0
$869$	−39.3890	−	3.64675i	−1.33618	−	0.123708i
$870$		0			0
$871$	49.0901			1.66336
$872$		0			0
$873$			41.2105i			1.39476i
$874$		0			0
$875$		0			0
$876$		0			0
$877$		−	18.6381i		−	0.629364i	−0.949197	−	0.314682i	$-0.898102\pi$
$877$		−	18.6381i		−	0.629364i	0.949197	−	0.314682i	$-0.101898\pi$
$878$		0			0
$879$			39.2321i			1.32327i
$880$		0			0
$881$			17.4946i			0.589408i	0.955589	+	0.294704i	$0.0952211\pi$
$881$			17.4946i			0.589408i	−0.955589	+	0.294704i	$0.904779\pi$
$882$		0			0
$883$	27.8846			0.938392			0.469196	−	0.883094i	$-0.344544\pi$
$883$	27.8846			0.938392			0.469196	+	0.883094i	$0.344544\pi$
$884$		0			0
$885$		−	3.41134i		−	0.114671i
$886$		0			0
$887$	−32.7894			−1.10096			−0.550480	−	0.834849i	$-0.685555\pi$
$887$	−32.7894			−1.10096			−0.550480	+	0.834849i	$0.685555\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	0.689521	−	7.44760i	0.0230998	−	0.249504i
$892$		0			0
$893$		−	1.94029i		−	0.0649292i
$894$		0			0
$895$			2.51405i			0.0840355i
$896$		0			0
$897$			63.1766i			2.10940i
$898$		0			0
$899$	26.8990			0.897132
$900$		0			0
$901$	−45.6718			−1.52155
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−7.42744			−0.246897
$906$		0			0
$907$	6.55396			0.217621			0.108810	−	0.994063i	$-0.465296\pi$
$907$	6.55396			0.217621			0.108810	+	0.994063i	$0.465296\pi$
$908$		0			0
$909$	60.2149			1.99720
$910$		0			0
$911$	−32.4216			−1.07418			−0.537088	−	0.843526i	$-0.680476\pi$
$911$	−32.4216			−1.07418			−0.537088	+	0.843526i	$0.680476\pi$
$912$		0			0
$913$	−4.51413	+	48.7576i	−0.149396	+	1.61364i
$914$		0			0
$915$	−11.4402			−0.378201
$916$		0			0
$917$		0			0
$918$		0			0
$919$			5.82603i			0.192183i	0.995373	+	0.0960915i	$0.0306341\pi$
$919$			5.82603i			0.192183i	−0.995373	+	0.0960915i	$0.969366\pi$
$920$		0			0
$921$			70.9953i			2.33937i
$922$		0			0
$923$	−42.3228			−1.39307
$924$		0			0
$925$	2.97601			0.0978506
$926$		0			0
$927$		−	30.7701i		−	1.01062i
$928$		0			0
$929$		−	16.9366i		−	0.555671i	−0.960629	−	0.277835i	$-0.910383\pi$
$929$		−	16.9366i		−	0.555671i	0.960629	−	0.277835i	$-0.0896170\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$	66.1391			2.16530
$934$		0			0
$935$	4.00823	+	0.371094i	0.131083	+	0.0121361i
$936$		0			0
$937$	−8.52101			−0.278369			−0.139185	−	0.990266i	$-0.544448\pi$
$937$	−8.52101			−0.278369			−0.139185	+	0.990266i	$0.544448\pi$
$938$		0			0
$939$	75.3671			2.45951
$940$		0			0
$941$	−5.89435			−0.192150			−0.0960751	−	0.995374i	$-0.530629\pi$
$941$	−5.89435			−0.192150			−0.0960751	+	0.995374i	$0.530629\pi$
$942$		0			0
$943$	−23.0701			−0.751267
$944$		0			0
$945$		0			0
$946$		0			0
$947$	4.32726			0.140617			0.0703085	−	0.997525i	$-0.477602\pi$
$947$	4.32726			0.140617			0.0703085	+	0.997525i	$0.477602\pi$
$948$		0			0
$949$	14.1469			0.459227
$950$		0			0
$951$		−	39.9156i		−	1.29435i
$952$		0			0
$953$			42.1267i			1.36462i	0.731064	+	0.682309i	$0.239024\pi$
$953$			42.1267i			1.36462i	−0.731064	+	0.682309i	$0.760976\pi$
$954$		0			0
$955$		−	1.09551i		−	0.0354498i
$956$		0			0
$957$	4.45611	−	48.1310i	0.144046	−	1.55585i
$958$		0			0
$959$		0			0
$960$		0			0
$961$	5.45417			0.175941
$962$		0			0
$963$		−	50.1567i		−	1.61628i
$964$		0			0
$965$	−9.47146			−0.304897
$966$		0			0
$967$			3.74288i			0.120363i	0.998187	+	0.0601815i	$0.0191680\pi$
$967$			3.74288i			0.120363i	−0.998187	+	0.0601815i	$0.980832\pi$
$968$		0			0
$969$			30.2509i			0.971798i
$970$		0			0
$971$		−	38.9280i		−	1.24926i	−0.780921	−	0.624630i	$-0.785250\pi$
$971$		−	38.9280i		−	1.24926i	0.780921	−	0.624630i	$-0.214750\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$			60.1603i			1.92667i
$976$		0			0
$977$	28.5327			0.912842			0.456421	−	0.889764i	$-0.349131\pi$
$977$	28.5327			0.912842			0.456421	+	0.889764i	$0.349131\pi$
$978$		0			0
$979$	28.1630	+	2.60741i	0.900093	+	0.0833333i
$980$		0			0
$981$			50.1567i			1.60138i
$982$		0			0
$983$			21.9698i			0.700728i	0.936614	+	0.350364i	$0.113942\pi$
$983$			21.9698i			0.700728i	−0.936614	+	0.350364i	$0.886058\pi$
$984$		0			0
$985$	5.73560			0.182751
$986$		0			0
$987$		0			0
$988$		0			0
$989$			35.9776i			1.14402i
$990$		0			0
$991$	27.9478			0.887790			0.443895	−	0.896079i	$-0.353596\pi$
$991$	27.9478			0.887790			0.443895	+	0.896079i	$0.353596\pi$
$992$		0			0
$993$			67.0453i			2.12762i
$994$		0			0
$995$	−0.687724			−0.0218023
$996$		0			0
$997$	45.4820			1.44043			0.720215	−	0.693751i	$-0.244043\pi$
$997$	45.4820			1.44043			0.720215	+	0.693751i	$0.244043\pi$
$998$		0			0
$999$		−	2.51046i		−	0.0794274i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2156.2.c.c.1077.16		16
7.2	even	3		2156.2.q.c.2089.1		16
7.3	odd	6		2156.2.q.c.901.2		16
7.4	even	3		308.2.q.a.285.7	yes	16
7.5	odd	6		308.2.q.a.241.8	yes	16
7.6	odd	2	inner	2156.2.c.c.1077.2		16
11.10	odd	2	inner	2156.2.c.c.1077.15		16
21.5	even	6		2772.2.cs.a.2089.4		16
21.11	odd	6		2772.2.cs.a.901.3		16
28.11	odd	6		1232.2.bn.c.593.2		16
28.19	even	6		1232.2.bn.c.241.1		16
77.10	even	6		2156.2.q.c.901.1		16
77.32	odd	6		308.2.q.a.285.8	yes	16
77.54	even	6		308.2.q.a.241.7	✓	16
77.65	odd	6		2156.2.q.c.2089.2		16
77.76	even	2	inner	2156.2.c.c.1077.1		16
231.32	even	6		2772.2.cs.a.901.4		16
231.131	odd	6		2772.2.cs.a.2089.3		16
308.131	odd	6		1232.2.bn.c.241.2		16
308.263	even	6		1232.2.bn.c.593.1		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
308.2.q.a.241.7	✓	16	77.54	even	6
308.2.q.a.241.8	yes	16	7.5	odd	6
308.2.q.a.285.7	yes	16	7.4	even	3
308.2.q.a.285.8	yes	16	77.32	odd	6
1232.2.bn.c.241.1		16	28.19	even	6
1232.2.bn.c.241.2		16	308.131	odd	6
1232.2.bn.c.593.1		16	308.263	even	6
1232.2.bn.c.593.2		16	28.11	odd	6
2156.2.c.c.1077.1		16	77.76	even	2	inner
2156.2.c.c.1077.2		16	7.6	odd	2	inner
2156.2.c.c.1077.15		16	11.10	odd	2	inner
2156.2.c.c.1077.16		16	1.1	even	1	trivial
2156.2.q.c.901.1		16	77.10	even	6
2156.2.q.c.901.2		16	7.3	odd	6
2156.2.q.c.2089.1		16	7.2	even	3
2156.2.q.c.2089.2		16	77.65	odd	6
2772.2.cs.a.901.3		16	21.11	odd	6
2772.2.cs.a.901.4		16	231.32	even	6
2772.2.cs.a.2089.3		16	231.131	odd	6
2772.2.cs.a.2089.4		16	21.5	even	6

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:	\( N \)	\(=\)	\( 2156 = 2^{2} \cdot 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2156.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.73846i q^{3} -0.365169i q^{5} -4.49914 q^{9} +O(q^{10})\)	\(q+2.73846i q^{3} -0.365169i q^{5} -4.49914 q^{9} +(-0.305755 + 3.30250i) q^{11} +4.51413 q^{13} +1.00000 q^{15} +3.32366 q^{17} +3.32366 q^{19} +5.11065 q^{23} +4.86665 q^{25} -4.10534i q^{27} +5.32202i q^{29} -5.05429i q^{31} +(-9.04376 - 0.837298i) q^{33} +0.611511 q^{37} +12.3617i q^{39} -4.51413 q^{41} +7.03973i q^{43} +1.64295i q^{45} -0.583781i q^{47} +9.10169i q^{51} -13.7414 q^{53} +(1.20597 + 0.111652i) q^{55} +9.10169i q^{57} -3.41134i q^{59} -11.4402 q^{61} -1.64842i q^{65} +10.8748 q^{67} +13.9953i q^{69} -9.37564 q^{71} +3.13391 q^{73} +13.3271i q^{75} +11.9270i q^{79} -2.25514 q^{81} +14.7639 q^{83} -1.21370i q^{85} -14.5741 q^{87} -8.52778i q^{89} +13.8409 q^{93} -1.21370i q^{95} -9.15963i q^{97} +(1.37564 - 14.8584i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 8 q^{9}+O(q^{10}) \) 16 * q - 8 * q^9	\( 16 q - 8 q^{9} + 2 q^{11} + 16 q^{15} + 4 q^{23} + 12 q^{25} - 4 q^{37} - 16 q^{53} + 36 q^{67} - 76 q^{71} + 16 q^{81} - 20 q^{93} - 52 q^{99}+O(q^{100}) \) 16 * q - 8 * q^9 + 2 * q^11 + 16 * q^15 + 4 * q^23 + 12 * q^25 - 4 * q^37 - 16 * q^53 + 36 * q^67 - 76 * q^71 + 16 * q^81 - 20 * q^93 - 52 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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