LMFDB - Embedded newform 352.2.c.a.177.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [352,2,Mod(177,352)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(352, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("352.177"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$N$	$=$	$352 = 2^{5} \cdot 11$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	352.c (of order $2$ , degree $1$ , not minimal)

Newform invariants

comment:select newform

sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)

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

Self dual:	no
Analytic conductor:	$2.81073415115$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	10.0.578281160704.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{10} + 2x^{8} - 2x^{7} - 3x^{6} - 6x^{5} - 6x^{4} - 8x^{3} + 16x^{2} + 32$ x^10 + 2x^8 - 2x^7 - 3x^6 - 6x^5 - 6x^4 - 8x^3 + 16*x^2 + 32
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$2^{6}$
Twist minimal:	no (minimal twist has level 88)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			177.6
Root			$1.41363 + 0.0406696i$ of defining polynomial
Character	$\chi$	$=$	352.177
Dual form			352.2.c.a.177.5

$q$ -expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+0.229967i q^{3} -2.51595i q^{5} -1.47743 q^{7} +2.94712 q^{9} -1.00000i q^{11} -3.47743i q^{13} +0.578585 q^{15} -3.31475 q^{17} -7.13195i q^{19} -0.339760i q^{21} +6.45332 q^{23} -1.33001 q^{25} +1.36764i q^{27} +1.41480i q^{29} -0.636125 q^{31} +0.229967 q^{33} +3.71715i q^{35} +6.97588i q^{37} +0.799694 q^{39} +6.72955 q^{41} -3.21471i q^{43} -7.41480i q^{45} -0.862328 q^{47} -4.81719 q^{49} -0.762283i q^{51} +13.2515i q^{53} -2.51595 q^{55} +1.64011 q^{57} +2.63236i q^{59} -6.45993i q^{61} -4.35416 q^{63} -8.74905 q^{65} +7.66426i q^{67} +1.48405i q^{69} -12.2900 q^{71} -13.2440 q^{73} -0.305858i q^{75} +1.47743i q^{77} +16.3409 q^{79} +8.52683 q^{81} +13.8040i q^{83} +8.33976i q^{85} -0.325357 q^{87} -1.04979 q^{89} +5.13767i q^{91} -0.146288i q^{93} -17.9436 q^{95} +16.2110 q^{97} -2.94712i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$10 q - 10 q^{9} - 8 q^{15} - 4 q^{17} + 12 q^{23} - 6 q^{25} + 4 q^{31} - 24 q^{39} + 4 q^{41} + 4 q^{47} - 6 q^{49} + 8 q^{55} + 16 q^{57} + 40 q^{63} + 16 q^{65} + 12 q^{71} - 4 q^{73} - 16 q^{79} - 6 q^{81}+ \cdots - 20 q^{97}+O(q^{100})$ 10 * q - 10 * q^9 - 8 * q^15 - 4 * q^17 + 12 * q^23 - 6 * q^25 + 4 * q^31 - 24 * q^39 + 4 * q^41 + 4 * q^47 - 6 * q^49 + 8 * q^55 + 16 * q^57 + 40 * q^63 + 16 * q^65 + 12 * q^71 - 4 * q^73 - 16 * q^79 - 6 * q^81 - 32 * q^87 - 4 * q^89 - 24 * q^95 - 20 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/352\mathbb{Z}\right)^\times$ .

$n$	$133$	$287$	$321$
$\chi(n)$	$-1$	$1$	$1$

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
$3$	0.229967i	0.132771i	0.997794	+	0.0663857i	$0.0211468\pi$
$3$	0.229967i	0.132771i	−0.997794	+	0.0663857i	$0.978853\pi$
$4$	0	0
$5$	− 2.51595i	− 1.12517i	−0.826740	−	0.562584i	$-0.809807\pi$
$5$	− 2.51595i	− 1.12517i	0.826740	−	0.562584i	$-0.190193\pi$
$6$	0	0
$7$	−1.47743	−0.558417	−0.279209	−	0.960230i	$-0.590072\pi$
$7$	−1.47743	−0.558417	−0.279209	+	0.960230i	$0.590072\pi$
$8$	0	0
$9$	2.94712	0.982372
$10$	0	0
$11$	− 1.00000i	− 0.301511i
$11$	− 1.00000i	− 0.301511i
$12$	0	0
$13$	− 3.47743i	− 0.964466i	−0.876043	−	0.482233i	$-0.839826\pi$
$13$	− 3.47743i	− 0.964466i	0.876043	−	0.482233i	$-0.160174\pi$
$14$	0	0
$15$	0.578585	0.149390
$16$	0	0
$17$	−3.31475	−0.803946	−0.401973	−	0.915652i	$-0.631675\pi$
$17$	−3.31475	−0.803946	−0.401973	+	0.915652i	$0.631675\pi$
$18$	0	0
$19$	− 7.13195i	− 1.63618i	−0.575090	−	0.818090i	$-0.695033\pi$
$19$	− 7.13195i	− 1.63618i	0.575090	−	0.818090i	$-0.304967\pi$
$20$	0	0
$21$	− 0.339760i	− 0.0741418i
$22$	0	0
$23$	6.45332	1.34561	0.672805	−	0.739820i	$-0.265089\pi$
$23$	6.45332	1.34561	0.672805	+	0.739820i	$0.265089\pi$
$24$	0	0
$25$	−1.33001	−0.266002
$26$	0	0
$27$	1.36764i	0.263202i
$28$	0	0
$29$	1.41480i	0.262722i	0.991335	+	0.131361i	$0.0419346\pi$
$29$	1.41480i	0.262722i	−0.991335	+	0.131361i	$0.958065\pi$
$30$	0	0
$31$	−0.636125	−0.114251	−0.0571257	−	0.998367i	$-0.518194\pi$
$31$	−0.636125	−0.114251	−0.0571257	+	0.998367i	$0.518194\pi$
$32$	0	0
$33$	0.229967	0.0400321
$34$	0	0
$35$	3.71715i	0.628313i
$36$	0	0
$37$	6.97588i	1.14683i	0.819266	+	0.573414i	$0.194381\pi$
$37$	6.97588i	1.14683i	−0.819266	+	0.573414i	$0.805619\pi$
$38$	0	0
$39$	0.799694	0.128054
$40$	0	0
$41$	6.72955	1.05098	0.525490	−	0.850800i	$-0.323882\pi$
$41$	6.72955	1.05098	0.525490	+	0.850800i	$0.323882\pi$
$42$	0	0
$43$	− 3.21471i	− 0.490239i	−0.969493	−	0.245119i	$-0.921173\pi$
$43$	− 3.21471i	− 0.490239i	0.969493	−	0.245119i	$-0.0788271\pi$
$44$	0	0
$45$	− 7.41480i	− 1.10533i
$46$	0	0
$47$	−0.862328	−0.125783	−0.0628917	−	0.998020i	$-0.520032\pi$
$47$	−0.862328	−0.125783	−0.0628917	+	0.998020i	$0.520032\pi$
$48$	0	0
$49$	−4.81719	−0.688170
$50$	0	0
$51$	− 0.762283i	− 0.106741i
$52$	0	0
$53$	13.2515i	1.82023i	0.414353	+	0.910116i	$0.364008\pi$
$53$	13.2515i	1.82023i	−0.414353	+	0.910116i	$0.635992\pi$
$54$	0	0
$55$	−2.51595	−0.339251
$56$	0	0
$57$	1.64011	0.217238
$58$	0	0
$59$	2.63236i	0.342704i	0.985210	+	0.171352i	$0.0548136\pi$
$59$	2.63236i	0.342704i	−0.985210	+	0.171352i	$0.945186\pi$
$60$	0	0
$61$	− 6.45993i	− 0.827110i	−0.910479	−	0.413555i	$-0.864287\pi$
$61$	− 6.45993i	− 0.827110i	0.910479	−	0.413555i	$-0.135713\pi$
$62$	0	0
$63$	−4.35416	−0.548573
$64$	0	0
$65$	−8.74905	−1.08519
$66$	0	0
$67$	7.66426i	0.936339i	0.883639	+	0.468169i	$0.155086\pi$
$67$	7.66426i	0.936339i	−0.883639	+	0.468169i	$0.844914\pi$
$68$	0	0
$69$	1.48405i	0.178658i
$70$	0	0
$71$	−12.2900	−1.45856	−0.729278	−	0.684218i	$-0.760144\pi$
$71$	−12.2900	−1.45856	−0.729278	+	0.684218i	$0.760144\pi$
$72$	0	0
$73$	−13.2440	−1.55009	−0.775045	−	0.631905i	$-0.782273\pi$
$73$	−13.2440	−1.55009	−0.775045	+	0.631905i	$0.782273\pi$
$74$	0	0
$75$	− 0.305858i	− 0.0353175i
$76$	0	0
$77$	1.47743i	0.168369i
$78$	0	0
$79$	16.3409	1.83850	0.919249	−	0.393676i	$-0.128797\pi$
$79$	16.3409	1.83850	0.919249	+	0.393676i	$0.128797\pi$
$80$	0	0
$81$	8.52683	0.947426
$82$	0	0
$83$	13.8040i	1.51518i	0.652730	+	0.757591i	$0.273624\pi$
$83$	13.8040i	1.51518i	−0.652730	+	0.757591i	$0.726376\pi$
$84$	0	0
$85$	8.33976i	0.904574i
$86$	0	0
$87$	−0.325357	−0.0348819
$88$	0	0
$89$	−1.04979	−0.111277	−0.0556387	−	0.998451i	$-0.517720\pi$
$89$	−1.04979	−0.111277	−0.0556387	+	0.998451i	$0.517720\pi$
$90$	0	0
$91$	5.13767i	0.538574i
$92$	0	0
$93$	− 0.146288i	− 0.0151693i
$94$	0	0
$95$	−17.9436	−1.84098
$96$	0	0
$97$	16.2110	1.64598	0.822989	−	0.568057i	$-0.192305\pi$
$97$	16.2110	1.64598	0.822989	+	0.568057i	$0.192305\pi$
$98$	0	0
$99$	− 2.94712i	− 0.296196i
$100$	0	0
$101$	8.63460i	0.859175i	0.903025	+	0.429588i	$0.141341\pi$
$101$	8.63460i	0.859175i	−0.903025	+	0.429588i	$0.858659\pi$
$102$	0	0
$103$	4.95487	0.488217	0.244109	−	0.969748i	$-0.421505\pi$
$103$	4.95487	0.488217	0.244109	+	0.969748i	$0.421505\pi$
$104$	0	0
$105$	−0.854821	−0.0834220
$106$	0	0
$107$	6.34666i	0.613554i	0.951781	+	0.306777i	$0.0992507\pi$
$107$	6.34666i	0.613554i	−0.951781	+	0.306777i	$0.900749\pi$
$108$	0	0
$109$	− 7.94246i	− 0.760750i	−0.924832	−	0.380375i	$-0.875795\pi$
$109$	− 7.94246i	− 0.760750i	0.924832	−	0.380375i	$-0.124205\pi$
$110$	0	0
$111$	−1.60422	−0.152266
$112$	0	0
$113$	4.88248	0.459305	0.229653	−	0.973273i	$-0.426241\pi$
$113$	4.88248	0.459305	0.229653	+	0.973273i	$0.426241\pi$
$114$	0	0
$115$	− 16.2362i	− 1.51404i
$116$	0	0
$117$	− 10.2484i	− 0.947464i
$118$	0	0
$119$	4.89733	0.448937
$120$	0	0
$121$	−1.00000	−0.0909091
$122$	0	0
$123$	1.54757i	0.139540i
$124$	0	0
$125$	− 9.23351i	− 0.825871i
$126$	0	0
$127$	4.12017	0.365606	0.182803	−	0.983150i	$-0.441483\pi$
$127$	4.12017	0.365606	0.182803	+	0.983150i	$0.441483\pi$
$128$	0	0
$129$	0.739276	0.0650897
$130$	0	0
$131$	13.3015i	1.16216i	0.813847	+	0.581080i	$0.197370\pi$
$131$	13.3015i	1.16216i	−0.813847	+	0.581080i	$0.802630\pi$
$132$	0	0
$133$	10.5370i	0.913671i
$134$	0	0
$135$	3.44091	0.296147
$136$	0	0
$137$	−6.84444	−0.584760	−0.292380	−	0.956302i	$-0.594447\pi$
$137$	−6.84444	−0.584760	−0.292380	+	0.956302i	$0.594447\pi$
$138$	0	0
$139$	− 12.1966i	− 1.03450i	−0.855834	−	0.517250i	$-0.826956\pi$
$139$	− 12.1966i	− 1.03450i	0.855834	−	0.517250i	$-0.173044\pi$
$140$	0	0
$141$	− 0.198307i	− 0.0167004i
$142$	0	0
$143$	−3.47743	−0.290798
$144$	0	0
$145$	3.55956	0.295606
$146$	0	0
$147$	− 1.10779i	− 0.0913693i
$148$	0	0
$149$	3.04004i	0.249050i	0.992216	+	0.124525i	$0.0397407\pi$
$149$	3.04004i	0.249050i	−0.992216	+	0.124525i	$0.960259\pi$
$150$	0	0
$151$	1.83669	0.149468	0.0747339	−	0.997204i	$-0.476189\pi$
$151$	1.83669	0.149468	0.0747339	+	0.997204i	$0.476189\pi$
$152$	0	0
$153$	−9.76896	−0.789774
$154$	0	0
$155$	1.60046i	0.128552i
$156$	0	0
$157$	− 10.0560i	− 0.802558i	−0.915956	−	0.401279i	$-0.868566\pi$
$157$	− 10.0560i	− 0.802558i	0.915956	−	0.401279i	$-0.131434\pi$
$158$	0	0
$159$	−3.04740	−0.241675
$160$	0	0
$161$	−9.53434	−0.751411
$162$	0	0
$163$	− 12.8934i	− 1.00989i	−0.863152	−	0.504945i	$-0.831513\pi$
$163$	− 12.8934i	− 1.00989i	0.863152	−	0.504945i	$-0.168487\pi$
$164$	0	0
$165$	− 0.578585i	− 0.0450428i
$166$	0	0
$167$	−4.47434	−0.346235	−0.173117	−	0.984901i	$-0.555384\pi$
$167$	−4.47434	−0.346235	−0.173117	+	0.984901i	$0.555384\pi$
$168$	0	0
$169$	0.907463	0.0698048
$170$	0	0
$171$	− 21.0187i	− 1.60734i
$172$	0	0
$173$	− 0.182807i	− 0.0138986i	−0.999976	−	0.00694928i	$-0.997788\pi$
$173$	− 0.182807i	− 0.0138986i	0.999976	−	0.00694928i	$-0.00221204\pi$
$174$	0	0
$175$	1.96500	0.148540
$176$	0	0
$177$	−0.605356	−0.0455013
$178$	0	0
$179$	− 20.0915i	− 1.50171i	−0.660469	−	0.750853i	$-0.729642\pi$
$179$	− 20.0915i	− 1.50171i	0.660469	−	0.750853i	$-0.270358\pi$
$180$	0	0
$181$	7.63413i	0.567440i	0.958907	+	0.283720i	$0.0915686\pi$
$181$	7.63413i	0.567440i	−0.958907	+	0.283720i	$0.908431\pi$
$182$	0	0
$183$	1.48557	0.109817
$184$	0	0
$185$	17.5510	1.29037
$186$	0	0
$187$	3.31475i	0.242399i
$188$	0	0
$189$	− 2.02059i	− 0.146977i
$190$	0	0
$191$	22.9943	1.66381	0.831906	−	0.554917i	$-0.187250\pi$
$191$	22.9943	1.66381	0.831906	+	0.554917i	$0.187250\pi$
$192$	0	0
$193$	0.735278	0.0529265	0.0264632	−	0.999650i	$-0.491576\pi$
$193$	0.735278	0.0529265	0.0264632	+	0.999650i	$0.491576\pi$
$194$	0	0
$195$	− 2.01199i	− 0.144082i
$196$	0	0
$197$	− 11.1665i	− 0.795579i	−0.917477	−	0.397789i	$-0.869777\pi$
$197$	− 11.1665i	− 0.795579i	0.917477	−	0.397789i	$-0.130223\pi$
$198$	0	0
$199$	18.1315	1.28531	0.642655	−	0.766156i	$-0.277833\pi$
$199$	18.1315	1.28531	0.642655	+	0.766156i	$0.277833\pi$
$200$	0	0
$201$	−1.76253	−0.124319
$202$	0	0
$203$	− 2.09027i	− 0.146708i
$204$	0	0
$205$	− 16.9312i	− 1.18253i
$206$	0	0
$207$	19.0187	1.32189
$208$	0	0
$209$	−7.13195	−0.493327
$210$	0	0
$211$	18.4180i	1.26795i	0.773355	+	0.633973i	$0.218577\pi$
$211$	18.4180i	1.26795i	−0.773355	+	0.633973i	$0.781423\pi$
$212$	0	0
$213$	− 2.82629i	− 0.193654i
$214$	0	0
$215$	−8.08805	−0.551601
$216$	0	0
$217$	0.939831	0.0637999
$218$	0	0
$219$	− 3.04568i	− 0.205808i
$220$	0	0
$221$	11.5268i	0.775379i
$222$	0	0
$223$	−14.5654	−0.975368	−0.487684	−	0.873020i	$-0.662158\pi$
$223$	−14.5654	−0.975368	−0.487684	+	0.873020i	$0.662158\pi$
$224$	0	0
$225$	−3.91970	−0.261313
$226$	0	0
$227$	− 7.15145i	− 0.474658i	−0.971429	−	0.237329i	$-0.923728\pi$
$227$	− 7.15145i	− 0.474658i	0.971429	−	0.237329i	$-0.0762720\pi$
$228$	0	0
$229$	− 6.48500i	− 0.428541i	−0.976774	−	0.214271i	$-0.931263\pi$
$229$	− 6.48500i	− 0.428541i	0.976774	−	0.214271i	$-0.0687374\pi$
$230$	0	0
$231$	−0.339760	−0.0223546
$232$	0	0
$233$	11.4786	0.751988	0.375994	−	0.926622i	$-0.377301\pi$
$233$	11.4786	0.751988	0.375994	+	0.926622i	$0.377301\pi$
$234$	0	0
$235$	2.16957i	0.141527i
$236$	0	0
$237$	3.75787i	0.244100i
$238$	0	0
$239$	−0.112036	−0.00724698	−0.00362349	−	0.999993i	$-0.501153\pi$
$239$	−0.112036	−0.00724698	−0.00362349	+	0.999993i	$0.501153\pi$
$240$	0	0
$241$	2.93937	0.189341	0.0946706	−	0.995509i	$-0.469820\pi$
$241$	2.93937	0.189341	0.0946706	+	0.995509i	$0.469820\pi$
$242$	0	0
$243$	6.06381i	0.388993i
$244$	0	0
$245$	12.1198i	0.774307i
$246$	0	0
$247$	−24.8009	−1.57804
$248$	0	0
$249$	−3.17445	−0.201173
$250$	0	0
$251$	31.2199i	1.97058i	0.170882	+	0.985291i	$0.445338\pi$
$251$	31.2199i	1.97058i	−0.170882	+	0.985291i	$0.554662\pi$
$252$	0	0
$253$	− 6.45332i	− 0.405717i
$254$	0	0
$255$	−1.91787	−0.120102
$256$	0	0
$257$	7.09741	0.442725	0.221362	−	0.975192i	$-0.428950\pi$
$257$	7.09741	0.442725	0.221362	+	0.975192i	$0.428950\pi$
$258$	0	0
$259$	− 10.3064i	− 0.640409i
$260$	0	0
$261$	4.16957i	0.258090i
$262$	0	0
$263$	−16.3863	−1.01042	−0.505211	−	0.862996i	$-0.668585\pi$
$263$	−16.3863	−1.01042	−0.505211	+	0.862996i	$0.668585\pi$
$264$	0	0
$265$	33.3401	2.04807
$266$	0	0
$267$	− 0.241417i	− 0.0147745i
$268$	0	0
$269$	− 17.1262i	− 1.04420i	−0.852883	−	0.522102i	$-0.825148\pi$
$269$	− 17.1262i	− 1.04420i	0.852883	−	0.522102i	$-0.174852\pi$
$270$	0	0
$271$	−19.5142	−1.18540	−0.592702	−	0.805422i	$-0.701939\pi$
$271$	−19.5142	−1.18540	−0.592702	+	0.805422i	$0.701939\pi$
$272$	0	0
$273$	−1.18149	−0.0715073
$274$	0	0
$275$	1.33001i	0.0802027i
$276$	0	0
$277$	− 10.5618i	− 0.634596i	−0.948326	−	0.317298i	$-0.897224\pi$
$277$	− 10.5618i	− 0.634596i	0.948326	−	0.317298i	$-0.102776\pi$
$278$	0	0
$279$	−1.87473	−0.112237
$280$	0	0
$281$	−11.5994	−0.691961	−0.345981	−	0.938242i	$-0.612454\pi$
$281$	−11.5994	−0.691961	−0.345981	+	0.938242i	$0.612454\pi$
$282$	0	0
$283$	21.6234i	1.28538i	0.766128	+	0.642688i	$0.222181\pi$
$283$	21.6234i	1.28538i	−0.766128	+	0.642688i	$0.777819\pi$
$284$	0	0
$285$	− 4.12644i	− 0.244429i
$286$	0	0
$287$	−9.94246	−0.586885
$288$	0	0
$289$	−6.01240	−0.353671
$290$	0	0
$291$	3.72799i	0.218539i
$292$	0	0
$293$	20.1305i	1.17604i	0.808848	+	0.588018i	$0.200092\pi$
$293$	20.1305i	1.17604i	−0.808848	+	0.588018i	$0.799908\pi$
$294$	0	0
$295$	6.62289	0.385600
$296$	0	0
$297$	1.36764	0.0793585
$298$	0	0
$299$	− 22.4410i	− 1.29780i
$300$	0	0
$301$	4.74952i	0.273758i
$302$	0	0
$303$	−1.98567	−0.114074
$304$	0	0
$305$	−16.2529	−0.930637
$306$	0	0
$307$	2.24089i	0.127894i	0.997953	+	0.0639471i	$0.0203689\pi$
$307$	2.24089i	0.127894i	−0.997953	+	0.0639471i	$0.979631\pi$
$308$	0	0
$309$	1.13945i	0.0648213i
$310$	0	0
$311$	15.6919	0.889807	0.444904	−	0.895578i	$-0.353238\pi$
$311$	15.6919	0.889807	0.444904	+	0.895578i	$0.353238\pi$
$312$	0	0
$313$	−12.1167	−0.684876	−0.342438	−	0.939540i	$-0.611253\pi$
$313$	−12.1167	−0.684876	−0.342438	+	0.939540i	$0.611253\pi$
$314$	0	0
$315$	10.9549i	0.617237i
$316$	0	0
$317$	11.8958i	0.668132i	0.942550	+	0.334066i	$0.108421\pi$
$317$	11.8958i	0.668132i	−0.942550	+	0.334066i	$0.891579\pi$
$318$	0	0
$319$	1.41480	0.0792135
$320$	0	0
$321$	−1.45952	−0.0814625
$322$	0	0
$323$	23.6407i	1.31540i
$324$	0	0
$325$	4.62502i	0.256550i
$326$	0	0
$327$	1.82650	0.101006
$328$	0	0
$329$	1.27403	0.0702396
$330$	0	0
$331$	− 24.7692i	− 1.36144i	−0.732544	−	0.680719i	$-0.761667\pi$
$331$	− 24.7692i	− 1.36144i	0.732544	−	0.680719i	$-0.238333\pi$
$332$	0	0
$333$	20.5587i	1.12661i
$334$	0	0
$335$	19.2829	1.05354
$336$	0	0
$337$	−11.2285	−0.611654	−0.305827	−	0.952087i	$-0.598933\pi$
$337$	−11.2285	−0.611654	−0.305827	+	0.952087i	$0.598933\pi$
$338$	0	0
$339$	1.12281i	0.0609826i
$340$	0	0
$341$	0.636125i	0.0344481i
$342$	0	0
$343$	17.4591	0.942703
$344$	0	0
$345$	3.73379	0.201021
$346$	0	0
$347$	16.0230i	0.860160i	0.902791	+	0.430080i	$0.141515\pi$
$347$	16.0230i	0.860160i	−0.902791	+	0.430080i	$0.858485\pi$
$348$	0	0
$349$	− 9.27513i	− 0.496486i	−0.968698	−	0.248243i	$-0.920147\pi$
$349$	− 9.27513i	− 0.496486i	0.968698	−	0.248243i	$-0.0798532\pi$
$350$	0	0
$351$	4.75587	0.253850
$352$	0	0
$353$	−9.58150	−0.509972	−0.254986	−	0.966945i	$-0.582071\pi$
$353$	−9.58150	−0.509972	−0.254986	+	0.966945i	$0.582071\pi$
$354$	0	0
$355$	30.9211i	1.64112i
$356$	0	0
$357$	1.12622i	0.0596060i
$358$	0	0
$359$	7.80501	0.411932	0.205966	−	0.978559i	$-0.433966\pi$
$359$	7.80501	0.411932	0.205966	+	0.978559i	$0.433966\pi$
$360$	0	0
$361$	−31.8647	−1.67709
$362$	0	0
$363$	− 0.229967i	− 0.0120701i
$364$	0	0
$365$	33.3212i	1.74411i
$366$	0	0
$367$	−31.4393	−1.64112	−0.820558	−	0.571563i	$-0.806337\pi$
$367$	−31.4393	−1.64112	−0.820558	+	0.571563i	$0.806337\pi$
$368$	0	0
$369$	19.8328	1.03245
$370$	0	0
$371$	− 19.5782i	− 1.01645i
$372$	0	0
$373$	32.0424i	1.65909i	0.558437	+	0.829547i	$0.311401\pi$
$373$	32.0424i	1.65909i	−0.558437	+	0.829547i	$0.688599\pi$
$374$	0	0
$375$	2.12340	0.109652
$376$	0	0
$377$	4.91987	0.253386
$378$	0	0
$379$	11.4465i	0.587965i	0.955811	+	0.293983i	$0.0949808\pi$
$379$	11.4465i	0.587965i	−0.955811	+	0.293983i	$0.905019\pi$
$380$	0	0
$381$	0.947503i	0.0485420i
$382$	0	0
$383$	6.82520	0.348751	0.174376	−	0.984679i	$-0.444209\pi$
$383$	6.82520	0.348751	0.174376	+	0.984679i	$0.444209\pi$
$384$	0	0
$385$	3.71715	0.189443
$386$	0	0
$387$	− 9.47412i	− 0.481597i
$388$	0	0
$389$	6.65984i	0.337667i	0.985645	+	0.168834i	$0.0540000\pi$
$389$	6.65984i	0.337667i	−0.985645	+	0.168834i	$0.946000\pi$
$390$	0	0
$391$	−21.3912	−1.08180
$392$	0	0
$393$	−3.05891	−0.154302
$394$	0	0
$395$	− 41.1130i	− 2.06862i
$396$	0	0
$397$	2.42189i	0.121551i	0.998151	+	0.0607757i	$0.0193574\pi$
$397$	2.42189i	0.121551i	−0.998151	+	0.0607757i	$0.980643\pi$
$398$	0	0
$399$	−2.42315	−0.121309
$400$	0	0
$401$	−9.09122	−0.453994	−0.226997	−	0.973895i	$-0.572891\pi$
$401$	−9.09122	−0.453994	−0.226997	+	0.973895i	$0.572891\pi$
$402$	0	0
$403$	2.21208i	0.110192i
$404$	0	0
$405$	− 21.4531i	− 1.06601i
$406$	0	0
$407$	6.97588	0.345782
$408$	0	0
$409$	9.08013	0.448984	0.224492	−	0.974476i	$-0.427928\pi$
$409$	9.08013	0.448984	0.224492	+	0.974476i	$0.427928\pi$
$410$	0	0
$411$	− 1.57399i	− 0.0776394i
$412$	0	0
$413$	− 3.88914i	− 0.191372i
$414$	0	0
$415$	34.7301	1.70483
$416$	0	0
$417$	2.80481	0.137352
$418$	0	0
$419$	2.99512i	0.146321i	0.997320	+	0.0731607i	$0.0233086\pi$
$419$	2.99512i	0.146321i	−0.997320	+	0.0731607i	$0.976691\pi$
$420$	0	0
$421$	− 26.6646i	− 1.29955i	−0.760126	−	0.649776i	$-0.774863\pi$
$421$	− 26.6646i	− 1.29955i	0.760126	−	0.649776i	$-0.225137\pi$
$422$	0	0
$423$	−2.54138	−0.123566
$424$	0	0
$425$	4.40866	0.213851
$426$	0	0
$427$	9.54412i	0.461872i
$428$	0	0
$429$	− 0.799694i	− 0.0386096i
$430$	0	0
$431$	−19.5700	−0.942656	−0.471328	−	0.881958i	$-0.656225\pi$
$431$	−19.5700	−0.942656	−0.471328	+	0.881958i	$0.656225\pi$
$432$	0	0
$433$	−6.49028	−0.311903	−0.155951	−	0.987765i	$-0.549844\pi$
$433$	−6.49028	−0.311903	−0.155951	+	0.987765i	$0.549844\pi$
$434$	0	0
$435$	0.818582i	0.0392480i
$436$	0	0
$437$	− 46.0247i	− 2.20166i
$438$	0	0
$439$	−25.3338	−1.20912	−0.604559	−	0.796560i	$-0.706651\pi$
$439$	−25.3338	−1.20912	−0.604559	+	0.796560i	$0.706651\pi$
$440$	0	0
$441$	−14.1968	−0.676039
$442$	0	0
$443$	− 1.46196i	− 0.0694597i	−0.999397	−	0.0347299i	$-0.988943\pi$
$443$	− 1.46196i	− 0.0694597i	0.999397	−	0.0347299i	$-0.0110571\pi$
$444$	0	0
$445$	2.64122i	0.125206i
$446$	0	0
$447$	−0.699108	−0.0330667
$448$	0	0
$449$	−6.00066	−0.283188	−0.141594	−	0.989925i	$-0.545223\pi$
$449$	−6.00066	−0.283188	−0.141594	+	0.989925i	$0.545223\pi$
$450$	0	0
$451$	− 6.72955i	− 0.316882i
$452$	0	0
$453$	0.422378i	0.0198450i
$454$	0	0
$455$	12.9261	0.605986
$456$	0	0
$457$	17.5689	0.821837	0.410919	−	0.911672i	$-0.365208\pi$
$457$	17.5689	0.821837	0.410919	+	0.911672i	$0.365208\pi$
$458$	0	0
$459$	− 4.53339i	− 0.211600i
$460$	0	0
$461$	26.1231i	1.21668i	0.793678	+	0.608338i	$0.208163\pi$
$461$	26.1231i	1.21668i	−0.793678	+	0.608338i	$0.791837\pi$
$462$	0	0
$463$	−20.3258	−0.944619	−0.472310	−	0.881433i	$-0.656580\pi$
$463$	−20.3258	−0.944619	−0.472310	+	0.881433i	$0.656580\pi$
$464$	0	0
$465$	−0.368052	−0.0170680
$466$	0	0
$467$	1.29764i	0.0600478i	0.999549	+	0.0300239i	$0.00955833\pi$
$467$	1.29764i	0.0600478i	−0.999549	+	0.0300239i	$0.990442\pi$
$468$	0	0
$469$	− 11.3234i	− 0.522868i
$470$	0	0
$471$	2.31255	0.106557
$472$	0	0
$473$	−3.21471	−0.147813
$474$	0	0
$475$	9.48557i	0.435228i
$476$	0	0
$477$	39.0537i	1.78814i
$478$	0	0
$479$	28.5133	1.30281	0.651404	−	0.758731i	$-0.274180\pi$
$479$	28.5133	1.30281	0.651404	+	0.758731i	$0.274180\pi$
$480$	0	0
$481$	24.2582	1.10608
$482$	0	0
$483$	− 2.19258i	− 0.0997659i
$484$	0	0
$485$	− 40.7861i	− 1.85200i
$486$	0	0
$487$	6.03364	0.273410	0.136705	−	0.990612i	$-0.456349\pi$
$487$	6.03364	0.273410	0.136705	+	0.990612i	$0.456349\pi$
$488$	0	0
$489$	2.96505	0.134084
$490$	0	0
$491$	4.15408i	0.187471i	0.995597	+	0.0937354i	$0.0298808\pi$
$491$	4.15408i	0.187471i	−0.995597	+	0.0937354i	$0.970119\pi$
$492$	0	0
$493$	− 4.68971i	− 0.211214i
$494$	0	0
$495$	−7.41480	−0.333270
$496$	0	0
$497$	18.1577	0.814482
$498$	0	0
$499$	− 30.1028i	− 1.34759i	−0.738920	−	0.673793i	$-0.764664\pi$
$499$	− 30.1028i	− 1.34759i	0.738920	−	0.673793i	$-0.235336\pi$
$500$	0	0
$501$	− 1.02895i	− 0.0459700i
$502$	0	0
$503$	−14.0368	−0.625869	−0.312935	−	0.949775i	$-0.601312\pi$
$503$	−14.0368	−0.625869	−0.312935	+	0.949775i	$0.601312\pi$
$504$	0	0
$505$	21.7242	0.966716
$506$	0	0
$507$	0.208686i	0.00926808i
$508$	0	0
$509$	− 16.9844i	− 0.752821i	−0.926453	−	0.376411i	$-0.877158\pi$
$509$	− 16.9844i	− 0.752821i	0.926453	−	0.376411i	$-0.122842\pi$
$510$	0	0
$511$	19.5671	0.865597
$512$	0	0
$513$	9.75393	0.430646
$514$	0	0
$515$	− 12.4662i	− 0.549326i
$516$	0	0
$517$	0.862328i	0.0379251i
$518$	0	0
$519$	0.0420395	0.00184533
$520$	0	0
$521$	−3.66607	−0.160613	−0.0803066	−	0.996770i	$-0.525590\pi$
$521$	−3.66607	−0.160613	−0.0803066	+	0.996770i	$0.525590\pi$
$522$	0	0
$523$	− 2.91807i	− 0.127598i	−0.997963	−	0.0637990i	$-0.979678\pi$
$523$	− 2.91807i	− 0.127598i	0.997963	−	0.0637990i	$-0.0203217\pi$
$524$	0	0
$525$	0.451885i	0.0197219i
$526$	0	0
$527$	2.10860	0.0918519
$528$	0	0
$529$	18.6453	0.810666
$530$	0	0
$531$	7.75787i	0.336663i
$532$	0	0
$533$	− 23.4016i	− 1.01363i
$534$	0	0
$535$	15.9679	0.690352
$536$	0	0
$537$	4.62037	0.199384
$538$	0	0
$539$	4.81719i	0.207491i
$540$	0	0
$541$	− 16.4673i	− 0.707984i	−0.935248	−	0.353992i	$-0.884824\pi$
$541$	− 16.4673i	− 0.707984i	0.935248	−	0.353992i	$-0.115176\pi$
$542$	0	0
$543$	−1.75560	−0.0753398
$544$	0	0
$545$	−19.9828	−0.855971
$546$	0	0
$547$	25.9536i	1.10969i	0.831952	+	0.554847i	$0.187223\pi$
$547$	25.9536i	1.10969i	−0.831952	+	0.554847i	$0.812777\pi$
$548$	0	0
$549$	− 19.0382i	− 0.812529i
$550$	0	0
$551$	10.0903	0.429860
$552$	0	0
$553$	−24.1426	−1.02665
$554$	0	0
$555$	4.03614i	0.171325i
$556$	0	0
$557$	13.0152i	0.551473i	0.961233	+	0.275737i	$0.0889217\pi$
$557$	13.0152i	0.551473i	−0.961233	+	0.275737i	$0.911078\pi$
$558$	0	0
$559$	−11.1789	−0.472819
$560$	0	0
$561$	−0.762283	−0.0321836
$562$	0	0
$563$	− 27.9837i	− 1.17937i	−0.807632	−	0.589687i	$-0.799251\pi$
$563$	− 27.9837i	− 1.17937i	0.807632	−	0.589687i	$-0.200749\pi$
$564$	0	0
$565$	− 12.2841i	− 0.516795i
$566$	0	0
$567$	−12.5978	−0.529059
$568$	0	0
$569$	−10.6048	−0.444574	−0.222287	−	0.974981i	$-0.571352\pi$
$569$	−10.6048	−0.444574	−0.222287	+	0.974981i	$0.571352\pi$
$570$	0	0
$571$	− 1.82472i	− 0.0763622i	−0.999271	−	0.0381811i	$-0.987844\pi$
$571$	− 1.82472i	− 0.0763622i	0.999271	−	0.0381811i	$-0.0121564\pi$
$572$	0	0
$573$	5.28793i	0.220907i
$574$	0	0
$575$	−8.58298	−0.357935
$576$	0	0
$577$	23.3377	0.971560	0.485780	−	0.874081i	$-0.338536\pi$
$577$	23.3377	0.971560	0.485780	+	0.874081i	$0.338536\pi$
$578$	0	0
$579$	0.169089i	0.00702712i
$580$	0	0
$581$	− 20.3944i	− 0.846103i
$582$	0	0
$583$	13.2515	0.548821
$584$	0	0
$585$	−25.7845	−1.06606
$586$	0	0
$587$	− 30.3967i	− 1.25461i	−0.778775	−	0.627303i	$-0.784159\pi$
$587$	− 30.3967i	− 1.25461i	0.778775	−	0.627303i	$-0.215841\pi$
$588$	0	0
$589$	4.53681i	0.186936i
$590$	0	0
$591$	2.56792	0.105630
$592$	0	0
$593$	39.3809	1.61718	0.808590	−	0.588373i	$-0.200231\pi$
$593$	39.3809	1.61718	0.808590	+	0.588373i	$0.200231\pi$
$594$	0	0
$595$	− 12.3214i	− 0.505130i
$596$	0	0
$597$	4.16965i	0.170652i
$598$	0	0
$599$	35.9183	1.46758	0.733792	−	0.679374i	$-0.237749\pi$
$599$	35.9183	1.46758	0.733792	+	0.679374i	$0.237749\pi$
$600$	0	0
$601$	13.8292	0.564104	0.282052	−	0.959399i	$-0.408985\pi$
$601$	13.8292	0.564104	0.282052	+	0.959399i	$0.408985\pi$
$602$	0	0
$603$	22.5875i	0.919833i
$604$	0	0
$605$	2.51595i	0.102288i
$606$	0	0
$607$	−19.7004	−0.799615	−0.399807	−	0.916599i	$-0.630923\pi$
$607$	−19.7004	−0.799615	−0.399807	+	0.916599i	$0.630923\pi$
$608$	0	0
$609$	0.480693	0.0194786
$610$	0	0
$611$	2.99869i	0.121314i
$612$	0	0
$613$	31.7465i	1.28223i	0.767445	+	0.641114i	$0.221528\pi$
$613$	31.7465i	1.28223i	−0.767445	+	0.641114i	$0.778472\pi$
$614$	0	0
$615$	3.89362	0.157006
$616$	0	0
$617$	−27.5402	−1.10873	−0.554363	−	0.832275i	$-0.687038\pi$
$617$	−27.5402	−1.10873	−0.554363	+	0.832275i	$0.687038\pi$
$618$	0	0
$619$	− 19.6108i	− 0.788224i	−0.919063	−	0.394112i	$-0.871052\pi$
$619$	− 19.6108i	− 0.788224i	0.919063	−	0.394112i	$-0.128948\pi$
$620$	0	0
$621$	8.82581i	0.354167i
$622$	0	0
$623$	1.55099	0.0621392
$624$	0	0
$625$	−29.8811	−1.19525
$626$	0	0
$627$	− 1.64011i	− 0.0654997i
$628$	0	0
$629$	− 23.1233i	− 0.921988i
$630$	0	0
$631$	−17.6518	−0.702705	−0.351353	−	0.936243i	$-0.614278\pi$
$631$	−17.6518	−0.702705	−0.351353	+	0.936243i	$0.614278\pi$
$632$	0	0
$633$	−4.23552	−0.168347
$634$	0	0
$635$	− 10.3662i	− 0.411368i
$636$	0	0
$637$	16.7515i	0.663717i
$638$	0	0
$639$	−36.2201	−1.43284
$640$	0	0
$641$	16.7617	0.662047	0.331024	−	0.943622i	$-0.392606\pi$
$641$	16.7617	0.662047	0.331024	+	0.943622i	$0.392606\pi$
$642$	0	0
$643$	22.1367i	0.872984i	0.899708	+	0.436492i	$0.143779\pi$
$643$	22.1367i	0.872984i	−0.899708	+	0.436492i	$0.856221\pi$
$644$	0	0
$645$	− 1.85998i	− 0.0732368i
$646$	0	0
$647$	−22.7792	−0.895544	−0.447772	−	0.894148i	$-0.647782\pi$
$647$	−22.7792	−0.895544	−0.447772	+	0.894148i	$0.647782\pi$
$648$	0	0
$649$	2.63236	0.103329
$650$	0	0
$651$	0.216130i	0.00847080i
$652$	0	0
$653$	− 35.9327i	− 1.40615i	−0.711114	−	0.703077i	$-0.751809\pi$
$653$	− 35.9327i	− 1.40615i	0.711114	−	0.703077i	$-0.248191\pi$
$654$	0	0
$655$	33.4660	1.30762
$656$	0	0
$657$	−39.0315	−1.52277
$658$	0	0
$659$	− 1.49362i	− 0.0581831i	−0.999577	−	0.0290916i	$-0.990739\pi$
$659$	− 1.49362i	− 0.0581831i	0.999577	−	0.0290916i	$-0.00926144\pi$
$660$	0	0
$661$	44.3160i	1.72369i	0.507170	+	0.861846i	$0.330692\pi$
$661$	44.3160i	1.72369i	−0.507170	+	0.861846i	$0.669308\pi$
$662$	0	0
$663$	−2.65079	−0.102948
$664$	0	0
$665$	26.5105	1.02803
$666$	0	0
$667$	9.13015i	0.353521i
$668$	0	0
$669$	− 3.34955i	− 0.129501i
$670$	0	0
$671$	−6.45993	−0.249383
$672$	0	0
$673$	−33.8921	−1.30644	−0.653222	−	0.757166i	$-0.726583\pi$
$673$	−33.8921	−1.30644	−0.653222	+	0.757166i	$0.726583\pi$
$674$	0	0
$675$	− 1.81898i	− 0.0700124i
$676$	0	0
$677$	− 7.57568i	− 0.291157i	−0.989347	−	0.145578i	$-0.953496\pi$
$677$	− 7.57568i	− 0.291157i	0.989347	−	0.145578i	$-0.0465043\pi$
$678$	0	0
$679$	−23.9507	−0.919143
$680$	0	0
$681$	1.64459	0.0630210
$682$	0	0
$683$	16.1346i	0.617374i	0.951164	+	0.308687i	$0.0998896\pi$
$683$	16.1346i	0.617374i	−0.951164	+	0.308687i	$0.900110\pi$
$684$	0	0
$685$	17.2203i	0.657953i
$686$	0	0
$687$	1.49134	0.0568980
$688$	0	0
$689$	46.0812	1.75555
$690$	0	0
$691$	− 10.5531i	− 0.401457i	−0.979647	−	0.200729i	$-0.935669\pi$
$691$	− 10.5531i	− 0.401457i	0.979647	−	0.200729i	$-0.0643309\pi$
$692$	0	0
$693$	4.35416i	0.165401i
$694$	0	0
$695$	−30.6860	−1.16399
$696$	0	0
$697$	−22.3068	−0.844931
$698$	0	0
$699$	2.63970i	0.0998425i
$700$	0	0
$701$	43.8782i	1.65726i	0.559800	+	0.828628i	$0.310878\pi$
$701$	43.8782i	1.65726i	−0.559800	+	0.828628i	$0.689122\pi$
$702$	0	0
$703$	49.7516	1.87642
$704$	0	0
$705$	−0.498930	−0.0187908
$706$	0	0
$707$	− 12.7570i	− 0.479778i
$708$	0	0
$709$	4.31359i	0.162000i	0.996714	+	0.0810002i	$0.0258115\pi$
$709$	4.31359i	0.162000i	−0.996714	+	0.0810002i	$0.974189\pi$
$710$	0	0
$711$	48.1586	1.80609
$712$	0	0
$713$	−4.10511	−0.153738
$714$	0	0
$715$	8.74905i	0.327196i
$716$	0	0
$717$	− 0.0257645i	0 0.000962192i
$718$	0	0
$719$	0.793243	0.0295830	0.0147915	−	0.999891i	$-0.495292\pi$
$719$	0.793243	0.0295830	0.0147915	+	0.999891i	$0.495292\pi$
$720$	0	0
$721$	−7.32048	−0.272629
$722$	0	0
$723$	0.675956i	0.0251391i
$724$	0	0
$725$	− 1.88170i	− 0.0698845i
$726$	0	0
$727$	38.5654	1.43031	0.715155	−	0.698966i	$-0.246356\pi$
$727$	38.5654	1.43031	0.715155	+	0.698966i	$0.246356\pi$
$728$	0	0
$729$	24.1860	0.895779
$730$	0	0
$731$	10.6560i	0.394125i
$732$	0	0
$733$	35.3659i	1.30627i	0.757242	+	0.653134i	$0.226546\pi$
$733$	35.3659i	1.30627i	−0.757242	+	0.653134i	$0.773454\pi$
$734$	0	0
$735$	−2.78716	−0.102806
$736$	0	0
$737$	7.66426	0.282317
$738$	0	0
$739$	41.6624i	1.53258i	0.642497	+	0.766288i	$0.277899\pi$
$739$	41.6624i	1.53258i	−0.642497	+	0.766288i	$0.722101\pi$
$740$	0	0
$741$	− 5.70338i	− 0.209519i
$742$	0	0
$743$	−33.2692	−1.22053	−0.610264	−	0.792198i	$-0.708937\pi$
$743$	−33.2692	−1.22053	−0.610264	+	0.792198i	$0.708937\pi$
$744$	0	0
$745$	7.64859	0.280223
$746$	0	0
$747$	40.6819i	1.48847i
$748$	0	0
$749$	− 9.37676i	− 0.342619i
$750$	0	0
$751$	3.69284	0.134754	0.0673768	−	0.997728i	$-0.478537\pi$
$751$	3.69284	0.134754	0.0673768	+	0.997728i	$0.478537\pi$
$752$	0	0
$753$	−7.17954	−0.261637
$754$	0	0
$755$	− 4.62103i	− 0.168176i
$756$	0	0
$757$	19.2210i	0.698600i	0.937011	+	0.349300i	$0.113581\pi$
$757$	19.2210i	0.698600i	−0.937011	+	0.349300i	$0.886419\pi$
$758$	0	0
$759$	1.48405	0.0538676
$760$	0	0
$761$	−9.79341	−0.355011	−0.177505	−	0.984120i	$-0.556803\pi$
$761$	−9.79341	−0.355011	−0.177505	+	0.984120i	$0.556803\pi$
$762$	0	0
$763$	11.7345i	0.424816i
$764$	0	0
$765$	24.5782i	0.888628i
$766$	0	0
$767$	9.15386	0.330527
$768$	0	0
$769$	−48.2853	−1.74121	−0.870606	−	0.491981i	$-0.836273\pi$
$769$	−48.2853	−1.74121	−0.870606	+	0.491981i	$0.836273\pi$
$770$	0	0
$771$	1.63217i	0.0587812i
$772$	0	0
$773$	− 21.0475i	− 0.757025i	−0.925596	−	0.378513i	$-0.876436\pi$
$773$	− 21.0475i	− 0.757025i	0.925596	−	0.378513i	$-0.123564\pi$
$774$	0	0
$775$	0.846053	0.0303911
$776$	0	0
$777$	2.37013	0.0850279
$778$	0	0
$779$	− 47.9948i	− 1.71959i
$780$	0	0
$781$	12.2900i	0.439771i
$782$	0	0
$783$	−1.93493	−0.0691489
$784$	0	0
$785$	−25.3005	−0.903012
$786$	0	0
$787$	− 18.2077i	− 0.649034i	−0.945880	−	0.324517i	$-0.894798\pi$
$787$	− 18.2077i	− 0.649034i	0.945880	−	0.324517i	$-0.105202\pi$
$788$	0	0
$789$	− 3.76830i	− 0.134155i
$790$	0	0
$791$	−7.21354	−0.256484
$792$	0	0
$793$	−22.4640	−0.797719
$794$	0	0
$795$	7.66712i	0.271925i
$796$	0	0
$797$	40.2274i	1.42493i	0.701709	+	0.712463i	$0.252421\pi$
$797$	40.2274i	1.42493i	−0.701709	+	0.712463i	$0.747579\pi$
$798$	0	0
$799$	2.85840	0.101123
$800$	0	0
$801$	−3.09385	−0.109316
$802$	0	0
$803$	13.2440i	0.467370i
$804$	0	0
$805$	23.9879i	0.845464i
$806$	0	0
$807$	3.93846	0.138640
$808$	0	0
$809$	27.8044	0.977552	0.488776	−	0.872409i	$-0.337444\pi$
$809$	27.8044	0.977552	0.488776	+	0.872409i	$0.337444\pi$
$810$	0	0
$811$	− 7.59987i	− 0.266868i	−0.991058	−	0.133434i	$-0.957400\pi$
$811$	− 7.59987i	− 0.266868i	0.991058	−	0.133434i	$-0.0426004\pi$
$812$	0	0
$813$	− 4.48762i	− 0.157388i
$814$	0	0
$815$	−32.4392	−1.13629
$816$	0	0
$817$	−22.9271	−0.802119
$818$	0	0
$819$	15.1413i	0.529080i
$820$	0	0
$821$	− 26.6456i	− 0.929937i	−0.885327	−	0.464968i	$-0.846066\pi$
$821$	− 26.6456i	− 0.929937i	0.885327	−	0.464968i	$-0.153934\pi$
$822$	0	0
$823$	−35.5588	−1.23950	−0.619751	−	0.784799i	$-0.712766\pi$
$823$	−35.5588	−1.23950	−0.619751	+	0.784799i	$0.712766\pi$
$824$	0	0
$825$	−0.305858	−0.0106486
$826$	0	0
$827$	− 3.03188i	− 0.105429i	−0.998610	−	0.0527145i	$-0.983213\pi$
$827$	− 3.03188i	− 0.105429i	0.998610	−	0.0527145i	$-0.0167873\pi$
$828$	0	0
$829$	− 35.0569i	− 1.21758i	−0.793332	−	0.608789i	$-0.791656\pi$
$829$	− 35.0569i	− 1.21758i	0.793332	−	0.608789i	$-0.208344\pi$
$830$	0	0
$831$	2.42886	0.0842562
$832$	0	0
$833$	15.9678	0.553252
$834$	0	0
$835$	11.2572i	0.389572i
$836$	0	0
$837$	− 0.869989i	− 0.0300712i
$838$	0	0
$839$	6.41520	0.221477	0.110739	−	0.993850i	$-0.464678\pi$
$839$	6.41520	0.221477	0.110739	+	0.993850i	$0.464678\pi$
$840$	0	0
$841$	26.9983	0.930977
$842$	0	0
$843$	− 2.66747i	− 0.0918727i
$844$	0	0
$845$	− 2.28313i	− 0.0785421i
$846$	0	0
$847$	1.47743	0.0507652
$848$	0	0
$849$	−4.97266	−0.170661
$850$	0	0
$851$	45.0176i	1.54318i
$852$	0	0
$853$	− 10.5605i	− 0.361585i	−0.983521	−	0.180793i	$-0.942134\pi$
$853$	− 10.5605i	− 0.361585i	0.983521	−	0.180793i	$-0.0578663\pi$
$854$	0	0
$855$	−52.8820	−1.80852
$856$	0	0
$857$	−5.03900	−0.172129	−0.0860644	−	0.996290i	$-0.527429\pi$
$857$	−5.03900	−0.172129	−0.0860644	+	0.996290i	$0.527429\pi$
$858$	0	0
$859$	24.5174i	0.836521i	0.908327	+	0.418261i	$0.137360\pi$
$859$	24.5174i	0.836521i	−0.908327	+	0.418261i	$0.862640\pi$
$860$	0	0
$861$	− 2.28644i	− 0.0779215i
$862$	0	0
$863$	−17.8155	−0.606448	−0.303224	−	0.952919i	$-0.598063\pi$
$863$	−17.8155	−0.606448	−0.303224	+	0.952919i	$0.598063\pi$
$864$	0	0
$865$	−0.459934	−0.0156382
$866$	0	0
$867$	− 1.38265i	− 0.0469574i
$868$	0	0
$869$	− 16.3409i	− 0.554328i
$870$	0	0
$871$	26.6520	0.903067
$872$	0	0
$873$	47.7757	1.61696
$874$	0	0
$875$	13.6419i	0.461180i
$876$	0	0
$877$	− 22.4071i	− 0.756634i	−0.925676	−	0.378317i	$-0.876503\pi$
$877$	− 22.4071i	− 0.756634i	0.925676	−	0.378317i	$-0.123497\pi$
$878$	0	0
$879$	−4.62934	−0.156144
$880$	0	0
$881$	−12.0016	−0.404345	−0.202172	−	0.979350i	$-0.564800\pi$
$881$	−12.0016	−0.404345	−0.202172	+	0.979350i	$0.564800\pi$
$882$	0	0
$883$	− 27.9363i	− 0.940130i	−0.882632	−	0.470065i	$-0.844230\pi$
$883$	− 27.9363i	− 0.940130i	0.882632	−	0.470065i	$-0.155770\pi$
$884$	0	0
$885$	1.52305i	0.0511966i
$886$	0	0
$887$	48.6903	1.63486	0.817430	−	0.576028i	$-0.195398\pi$
$887$	48.6903	1.63486	0.817430	+	0.576028i	$0.195398\pi$
$888$	0	0
$889$	−6.08728	−0.204161
$890$	0	0
$891$	− 8.52683i	− 0.285660i
$892$	0	0
$893$	6.15008i	0.205804i
$894$	0	0
$895$	−50.5492	−1.68967
$896$	0	0
$897$	5.16068	0.172310
$898$	0	0
$899$	− 0.899988i	− 0.0300163i
$900$	0	0
$901$	− 43.9254i	− 1.46337i
$902$	0	0
$903$	−1.09223	−0.0363472
$904$	0	0
$905$	19.2071	0.638465
$906$	0	0
$907$	− 48.2773i	− 1.60302i	−0.597981	−	0.801510i	$-0.704030\pi$
$907$	− 48.2773i	− 1.60302i	0.597981	−	0.801510i	$-0.295970\pi$
$908$	0	0
$909$	25.4472i	0.844029i
$910$	0	0
$911$	−43.1688	−1.43025	−0.715123	−	0.698998i	$-0.753630\pi$
$911$	−43.1688	−1.43025	−0.715123	+	0.698998i	$0.753630\pi$
$912$	0	0
$913$	13.8040	0.456844
$914$	0	0
$915$	− 3.73762i	− 0.123562i
$916$	0	0
$917$	− 19.6521i	− 0.648970i
$918$	0	0
$919$	−36.1457	−1.19234	−0.596168	−	0.802860i	$-0.703311\pi$
$919$	−36.1457	−1.19234	−0.596168	+	0.802860i	$0.703311\pi$
$920$	0	0
$921$	−0.515330	−0.0169807
$922$	0	0
$923$	42.7377i	1.40673i
$924$	0	0
$925$	− 9.27801i	− 0.305059i
$926$	0	0
$927$	14.6026	0.479611
$928$	0	0
$929$	26.1901	0.859269	0.429634	−	0.903003i	$-0.358642\pi$
$929$	26.1901	0.859269	0.429634	+	0.903003i	$0.358642\pi$
$930$	0	0
$931$	34.3560i	1.12597i
$932$	0	0
$933$	3.60862i	0.118141i
$934$	0	0
$935$	8.33976	0.272739
$936$	0	0
$937$	−14.6547	−0.478749	−0.239375	−	0.970927i	$-0.576942\pi$
$937$	−14.6547	−0.478749	−0.239375	+	0.970927i	$0.576942\pi$
$938$	0	0
$939$	− 2.78644i	− 0.0909319i
$940$	0	0
$941$	− 5.50159i	− 0.179347i	−0.995971	−	0.0896734i	$-0.971418\pi$
$941$	− 5.50159i	− 0.179347i	0.995971	−	0.0896734i	$-0.0285823\pi$
$942$	0	0
$943$	43.4279	1.41421
$944$	0	0
$945$	−5.08372	−0.165373
$946$	0	0
$947$	− 1.00019i	− 0.0325019i	−0.999868	−	0.0162509i	$-0.994827\pi$
$947$	− 1.00019i	− 0.0325019i	0.999868	−	0.0162509i	$-0.00517306\pi$
$948$	0	0
$949$	46.0551i	1.49501i
$950$	0	0
$951$	−2.73563	−0.0887088
$952$	0	0
$953$	55.6770	1.80356	0.901778	−	0.432200i	$-0.142263\pi$
$953$	55.6770	1.80356	0.901778	+	0.432200i	$0.142263\pi$
$954$	0	0
$955$	− 57.8526i	− 1.87207i
$956$	0	0
$957$	0.325357i	0.0105173i
$958$	0	0
$959$	10.1122	0.326540
$960$	0	0
$961$	−30.5953	−0.986947
$962$	0	0
$963$	18.7043i	0.602739i
$964$	0	0
$965$	− 1.84992i	− 0.0595511i
$966$	0	0
$967$	25.9869	0.835682	0.417841	−	0.908520i	$-0.362787\pi$
$967$	25.9869	0.835682	0.417841	+	0.908520i	$0.362787\pi$
$968$	0	0
$969$	−5.43656	−0.174648
$970$	0	0
$971$	16.0587i	0.515349i	0.966232	+	0.257675i	$0.0829563\pi$
$971$	16.0587i	0.515349i	−0.966232	+	0.257675i	$0.917044\pi$
$972$	0	0
$973$	18.0196i	0.577683i
$974$	0	0
$975$	−1.06360	−0.0340625
$976$	0	0
$977$	−5.40488	−0.172918	−0.0864588	−	0.996255i	$-0.527555\pi$
$977$	−5.40488	−0.172918	−0.0864588	+	0.996255i	$0.527555\pi$
$978$	0	0
$979$	1.04979i	0.0335514i
$980$	0	0
$981$	− 23.4073i	− 0.747339i
$982$	0	0
$983$	36.4792	1.16350	0.581752	−	0.813366i	$-0.302367\pi$
$983$	36.4792	1.16350	0.581752	+	0.813366i	$0.302367\pi$
$984$	0	0
$985$	−28.0943	−0.895159
$986$	0	0
$987$	0.292985i	0.00932581i
$988$	0	0
$989$	− 20.7455i	− 0.659670i
$990$	0	0
$991$	−48.2714	−1.53339	−0.766696	−	0.642010i	$-0.778101\pi$
$991$	−48.2714	−1.53339	−0.766696	+	0.642010i	$0.778101\pi$
$992$	0	0
$993$	5.69609	0.180760
$994$	0	0
$995$	− 45.6181i	− 1.44619i
$996$	0	0
$997$	8.78994i	0.278380i	0.990266	+	0.139190i	$0.0444499\pi$
$997$	8.78994i	0.278380i	−0.990266	+	0.139190i	$0.955550\pi$
$998$	0	0
$999$	−9.54049	−0.301848

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	352.2.c.a.177.6		10
3.2	odd	2		3168.2.f.g.1585.9		10
4.3	odd	2		88.2.c.a.45.2	yes	10
8.3	odd	2		88.2.c.a.45.1	✓	10
8.5	even	2	inner	352.2.c.a.177.5		10
11.10	odd	2		3872.2.c.f.1937.6		10
12.11	even	2		792.2.f.g.397.9		10
16.3	odd	4		2816.2.a.r.1.3		5
16.5	even	4		2816.2.a.p.1.3		5
16.11	odd	4		2816.2.a.o.1.3		5
16.13	even	4		2816.2.a.q.1.3		5
24.5	odd	2		3168.2.f.g.1585.2		10
24.11	even	2		792.2.f.g.397.10		10
44.3	odd	10		968.2.o.g.493.6		40
44.7	even	10		968.2.o.h.269.8		40
44.15	odd	10		968.2.o.g.269.3		40
44.19	even	10		968.2.o.h.493.5		40
44.27	odd	10		968.2.o.g.245.7		40
44.31	odd	10		968.2.o.g.565.10		40
44.35	even	10		968.2.o.h.565.1		40
44.39	even	10		968.2.o.h.245.4		40
44.43	even	2		968.2.c.d.485.9		10
88.3	odd	10		968.2.o.g.493.3		40
88.19	even	10		968.2.o.h.493.8		40
88.21	odd	2		3872.2.c.f.1937.5		10
88.27	odd	10		968.2.o.g.245.10		40
88.35	even	10		968.2.o.h.565.4		40
88.43	even	2		968.2.c.d.485.10		10
88.51	even	10		968.2.o.h.269.5		40
88.59	odd	10		968.2.o.g.269.6		40
88.75	odd	10		968.2.o.g.565.7		40
88.83	even	10		968.2.o.h.245.1		40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
88.2.c.a.45.1	✓	10	8.3	odd	2
88.2.c.a.45.2	yes	10	4.3	odd	2
352.2.c.a.177.5		10	8.5	even	2	inner
352.2.c.a.177.6		10	1.1	even	1	trivial
792.2.f.g.397.9		10	12.11	even	2
792.2.f.g.397.10		10	24.11	even	2
968.2.c.d.485.9		10	44.43	even	2
968.2.c.d.485.10		10	88.43	even	2
968.2.o.g.245.7		40	44.27	odd	10
968.2.o.g.245.10		40	88.27	odd	10
968.2.o.g.269.3		40	44.15	odd	10
968.2.o.g.269.6		40	88.59	odd	10
968.2.o.g.493.3		40	88.3	odd	10
968.2.o.g.493.6		40	44.3	odd	10
968.2.o.g.565.7		40	88.75	odd	10
968.2.o.g.565.10		40	44.31	odd	10
968.2.o.h.245.1		40	88.83	even	10
968.2.o.h.245.4		40	44.39	even	10
968.2.o.h.269.5		40	88.51	even	10
968.2.o.h.269.8		40	44.7	even	10
968.2.o.h.493.5		40	44.19	even	10
968.2.o.h.493.8		40	88.19	even	10
968.2.o.h.565.1		40	44.35	even	10
968.2.o.h.565.4		40	88.35	even	10
2816.2.a.o.1.3		5	16.11	odd	4
2816.2.a.p.1.3		5	16.5	even	4
2816.2.a.q.1.3		5	16.13	even	4
2816.2.a.r.1.3		5	16.3	odd	4
3168.2.f.g.1585.2		10	24.5	odd	2
3168.2.f.g.1585.9		10	3.2	odd	2
3872.2.c.f.1937.5		10	88.21	odd	2
3872.2.c.f.1937.6		10	11.10	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 $\F_{q}$
Belyi maps

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Character values

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	352.2.c.a.177.6

Level	$352$
Weight	$2$
Character	352.177
Analytic conductor	$2.811$
Analytic rank	$0$
Dimension	$10$
Inner twists	$2$