LMFDB - Embedded newform 352.2.c.a.177.4

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)

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);

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")

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			177.4
Root			$-0.239536 + 1.39378i$ of defining polynomial
Character	$\chi$	$=$	352.177
Dual form			352.2.c.a.177.7

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.33544i q^{3} +1.93119i q^{5} +1.83930 q^{7} +1.21660 q^{9} -1.00000i q^{11} -0.160700i q^{13} +2.57899 q^{15} +5.41442 q^{17} +2.79744i q^{19} -2.45628i q^{21} -4.44137 q^{23} +1.27050 q^{25} -5.63102i q^{27} -8.34948i q^{29} +9.05835 q^{31} -1.33544 q^{33} +3.55204i q^{35} -0.602073i q^{37} -0.214606 q^{39} -11.7639 q^{41} +4.47936i q^{43} +2.34948i q^{45} -6.29558 q^{47} -3.61698 q^{49} -7.23064i q^{51} +11.7211i q^{53} +1.93119 q^{55} +3.73582 q^{57} +9.63102i q^{59} -3.32912i q^{61} +2.23769 q^{63} +0.310343 q^{65} +5.76864i q^{67} +5.93119i q^{69} -9.62916 q^{71} -0.681940 q^{73} -1.69668i q^{75} -1.83930i q^{77} -5.77867 q^{79} -3.87010 q^{81} -2.92400i q^{83} +10.4563i q^{85} -11.1502 q^{87} +5.74214 q^{89} -0.295576i q^{91} -12.0969i q^{93} -5.40240 q^{95} -5.37829 q^{97} -1.21660i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	− 1.33544i	− 0.771017i	−0.922704	−	0.385509i	$-0.874026\pi$
$3$	− 1.33544i	− 0.771017i	0.922704	−	0.385509i	$-0.125974\pi$
$4$	0	0
$5$	1.93119i	0.863655i	0.901956	+	0.431827i	$0.142131\pi$
$5$	1.93119i	0.863655i	−0.901956	+	0.431827i	$0.857869\pi$
$6$	0	0
$7$	1.83930	0.695190	0.347595	−	0.937645i	$-0.386998\pi$
$7$	1.83930	0.695190	0.347595	+	0.937645i	$0.386998\pi$
$8$	0	0
$9$	1.21660	0.405532
$10$	0	0
$11$	− 1.00000i	− 0.301511i
$11$	− 1.00000i	− 0.301511i
$12$	0	0
$13$	− 0.160700i	− 0.0445702i	−0.999752	−	0.0222851i	$-0.992906\pi$
$13$	− 0.160700i	− 0.0445702i	0.999752	−	0.0222851i	$-0.00709416\pi$
$14$	0	0
$15$	2.57899	0.665893
$16$	0	0
$17$	5.41442	1.31319	0.656595	−	0.754243i	$-0.271996\pi$
$17$	5.41442	1.31319	0.656595	+	0.754243i	$0.271996\pi$
$18$	0	0
$19$	2.79744i	0.641777i	0.947117	+	0.320889i	$0.103982\pi$
$19$	2.79744i	0.641777i	−0.947117	+	0.320889i	$0.896018\pi$
$20$	0	0
$21$	− 2.45628i	− 0.536003i
$22$	0	0
$23$	−4.44137	−0.926090	−0.463045	−	0.886335i	$-0.653243\pi$
$23$	−4.44137	−0.926090	−0.463045	+	0.886335i	$0.653243\pi$
$24$	0	0
$25$	1.27050	0.254101
$26$	0	0
$27$	− 5.63102i	− 1.08369i
$28$	0	0
$29$	− 8.34948i	− 1.55046i	−0.631679	−	0.775230i	$-0.717634\pi$
$29$	− 8.34948i	− 1.55046i	0.631679	−	0.775230i	$-0.282366\pi$
$30$	0	0
$31$	9.05835	1.62693	0.813464	−	0.581616i	$-0.197579\pi$
$31$	9.05835	1.62693	0.813464	+	0.581616i	$0.197579\pi$
$32$	0	0
$33$	−1.33544	−0.232470
$34$	0	0
$35$	3.55204i	0.600404i
$36$	0	0
$37$	− 0.602073i	− 0.0989801i	−0.998775	−	0.0494901i	$-0.984240\pi$
$37$	− 0.602073i	− 0.0989801i	0.998775	−	0.0494901i	$-0.0157596\pi$
$38$	0	0
$39$	−0.214606	−0.0343644
$40$	0	0
$41$	−11.7639	−1.83721	−0.918606	−	0.395174i	$-0.870684\pi$
$41$	−11.7639	−1.83721	−0.918606	+	0.395174i	$0.870684\pi$
$42$	0	0
$43$	4.47936i	0.683096i	0.939864	+	0.341548i	$0.110951\pi$
$43$	4.47936i	0.683096i	−0.939864	+	0.341548i	$0.889049\pi$
$44$	0	0
$45$	2.34948i	0.350240i
$46$	0	0
$47$	−6.29558	−0.918304	−0.459152	−	0.888358i	$-0.651847\pi$
$47$	−6.29558	−0.918304	−0.459152	+	0.888358i	$0.651847\pi$
$48$	0	0
$49$	−3.61698	−0.516711
$50$	0	0
$51$	− 7.23064i	− 1.01249i
$52$	0	0
$53$	11.7211i	1.61001i	0.593268	+	0.805005i	$0.297838\pi$
$53$	11.7211i	1.61001i	−0.593268	+	0.805005i	$0.702162\pi$
$54$	0	0
$55$	1.93119	0.260402
$56$	0	0
$57$	3.73582	0.494821
$58$	0	0
$59$	9.63102i	1.25385i	0.779079	+	0.626926i	$0.215687\pi$
$59$	9.63102i	1.25385i	−0.779079	+	0.626926i	$0.784313\pi$
$60$	0	0
$61$	− 3.32912i	− 0.426250i	−0.977025	−	0.213125i	$-0.931636\pi$
$61$	− 3.32912i	− 0.426250i	0.977025	−	0.213125i	$-0.0683641\pi$
$62$	0	0
$63$	2.23769	0.281922
$64$	0	0
$65$	0.310343	0.0384933
$66$	0	0
$67$	5.76864i	0.704751i	0.935859	+	0.352376i	$0.114626\pi$
$67$	5.76864i	0.704751i	−0.935859	+	0.352376i	$0.885374\pi$
$68$	0	0
$69$	5.93119i	0.714031i
$70$	0	0
$71$	−9.62916	−1.14277	−0.571386	−	0.820682i	$-0.693594\pi$
$71$	−9.62916	−1.14277	−0.571386	+	0.820682i	$0.693594\pi$
$72$	0	0
$73$	−0.681940	−0.0798150	−0.0399075	−	0.999203i	$-0.512706\pi$
$73$	−0.681940	−0.0798150	−0.0399075	+	0.999203i	$0.512706\pi$
$74$	0	0
$75$	− 1.69668i	− 0.195916i
$76$	0	0
$77$	− 1.83930i	− 0.209608i
$78$	0	0
$79$	−5.77867	−0.650151	−0.325075	−	0.945688i	$-0.605390\pi$
$79$	−5.77867	−0.650151	−0.325075	+	0.945688i	$0.605390\pi$
$80$	0	0
$81$	−3.87010	−0.430011
$82$	0	0
$83$	− 2.92400i	− 0.320951i	−0.987040	−	0.160476i	$-0.948697\pi$
$83$	− 2.92400i	− 0.320951i	0.987040	−	0.160476i	$-0.0513028\pi$
$84$	0	0
$85$	10.4563i	1.13414i
$86$	0	0
$87$	−11.1502	−1.19543
$88$	0	0
$89$	5.74214	0.608666	0.304333	−	0.952566i	$-0.401566\pi$
$89$	5.74214	0.608666	0.304333	+	0.952566i	$0.401566\pi$
$90$	0	0
$91$	− 0.295576i	− 0.0309848i
$92$	0	0
$93$	− 12.0969i	− 1.25439i
$94$	0	0
$95$	−5.40240	−0.554274
$96$	0	0
$97$	−5.37829	−0.546082	−0.273041	−	0.962002i	$-0.588030\pi$
$97$	−5.37829	−0.546082	−0.273041	+	0.962002i	$0.588030\pi$
$98$	0	0
$99$	− 1.21660i	− 0.122273i
$100$	0	0
$101$	9.31868i	0.927244i	0.886033	+	0.463622i	$0.153450\pi$
$101$	9.31868i	0.927244i	−0.886033	+	0.463622i	$0.846550\pi$
$102$	0	0
$103$	−1.67860	−0.165397	−0.0826987	−	0.996575i	$-0.526354\pi$
$103$	−1.67860	−0.165397	−0.0826987	+	0.996575i	$0.526354\pi$
$104$	0	0
$105$	4.74354	0.462922
$106$	0	0
$107$	− 11.2768i	− 1.09017i	−0.838381	−	0.545085i	$-0.816497\pi$
$107$	− 11.2768i	− 1.09017i	0.838381	−	0.545085i	$-0.183503\pi$
$108$	0	0
$109$	− 19.6373i	− 1.88092i	−0.339910	−	0.940458i	$-0.610397\pi$
$109$	− 19.6373i	− 1.88092i	0.339910	−	0.940458i	$-0.389603\pi$
$110$	0	0
$111$	−0.804032	−0.0763154
$112$	0	0
$113$	−12.9156	−1.21499	−0.607497	−	0.794322i	$-0.707826\pi$
$113$	−12.9156	−1.21499	−0.607497	+	0.794322i	$0.707826\pi$
$114$	0	0
$115$	− 8.57714i	− 0.799822i
$116$	0	0
$117$	− 0.195508i	− 0.0180747i
$118$	0	0
$119$	9.95874	0.912916
$120$	0	0
$121$	−1.00000	−0.0909091
$122$	0	0
$123$	15.7100i	1.41652i
$124$	0	0
$125$	12.1095i	1.08311i
$126$	0	0
$127$	−1.12716	−0.100019	−0.0500096	−	0.998749i	$-0.515925\pi$
$127$	−1.12716	−0.100019	−0.0500096	+	0.998749i	$0.515925\pi$
$128$	0	0
$129$	5.98192	0.526679
$130$	0	0
$131$	− 10.9554i	− 0.957178i	−0.878039	−	0.478589i	$-0.841148\pi$
$131$	− 10.9554i	− 0.957178i	0.878039	−	0.478589i	$-0.158852\pi$
$132$	0	0
$133$	5.14534i	0.446157i
$134$	0	0
$135$	10.8746	0.935934
$136$	0	0
$137$	−10.1753	−0.869338	−0.434669	−	0.900590i	$-0.643135\pi$
$137$	−10.1753	−0.869338	−0.434669	+	0.900590i	$0.643135\pi$
$138$	0	0
$139$	− 18.3347i	− 1.55513i	−0.628802	−	0.777565i	$-0.716455\pi$
$139$	− 18.3347i	− 1.55513i	0.628802	−	0.777565i	$-0.283545\pi$
$140$	0	0
$141$	8.40737i	0.708028i
$142$	0	0
$143$	−0.160700	−0.0134384
$144$	0	0
$145$	16.1244	1.33906
$146$	0	0
$147$	4.83026i	0.398393i
$148$	0	0
$149$	− 8.46892i	− 0.693801i	−0.937902	−	0.346901i	$-0.887234\pi$
$149$	− 8.46892i	− 0.693801i	0.937902	−	0.346901i	$-0.112766\pi$
$150$	0	0
$151$	10.0705	0.819528	0.409764	−	0.912191i	$-0.365611\pi$
$151$	10.0705	0.819528	0.409764	+	0.912191i	$0.365611\pi$
$152$	0	0
$153$	6.58717	0.532541
$154$	0	0
$155$	17.4934i	1.40510i
$156$	0	0
$157$	− 8.73969i	− 0.697503i	−0.937215	−	0.348752i	$-0.886606\pi$
$157$	− 8.73969i	− 0.697503i	0.937215	−	0.348752i	$-0.113394\pi$
$158$	0	0
$159$	15.6528	1.24135
$160$	0	0
$161$	−8.16901	−0.643809
$162$	0	0
$163$	24.4237i	1.91301i	0.291710	+	0.956507i	$0.405776\pi$
$163$	24.4237i	1.91301i	−0.291710	+	0.956507i	$0.594224\pi$
$164$	0	0
$165$	− 2.57899i	− 0.200774i
$166$	0	0
$167$	7.36485	0.569909	0.284954	−	0.958541i	$-0.408022\pi$
$167$	7.36485	0.569909	0.284954	+	0.958541i	$0.408022\pi$
$168$	0	0
$169$	12.9742	0.998013
$170$	0	0
$171$	3.40336i	0.260262i
$172$	0	0
$173$	− 1.38302i	− 0.105149i	−0.998617	−	0.0525747i	$-0.983257\pi$
$173$	− 1.38302i	− 0.105149i	0.998617	−	0.0525747i	$-0.0167428\pi$
$174$	0	0
$175$	2.33684	0.176648
$176$	0	0
$177$	12.8617	0.966741
$178$	0	0
$179$	9.89679i	0.739720i	0.929087	+	0.369860i	$0.120594\pi$
$179$	9.89679i	0.739720i	−0.929087	+	0.369860i	$0.879406\pi$
$180$	0	0
$181$	− 11.6803i	− 0.868192i	−0.900867	−	0.434096i	$-0.857068\pi$
$181$	− 11.6803i	− 0.868192i	0.900867	−	0.434096i	$-0.142932\pi$
$182$	0	0
$183$	−4.44584	−0.328646
$184$	0	0
$185$	1.16272	0.0854846
$186$	0	0
$187$	− 5.41442i	− 0.395942i
$188$	0	0
$189$	− 10.3571i	− 0.753370i
$190$	0	0
$191$	−12.0902	−0.874814	−0.437407	−	0.899264i	$-0.644103\pi$
$191$	−12.0902	−0.874814	−0.437407	+	0.899264i	$0.644103\pi$
$192$	0	0
$193$	−13.2620	−0.954622	−0.477311	−	0.878734i	$-0.658388\pi$
$193$	−13.2620	−0.954622	−0.477311	+	0.878734i	$0.658388\pi$
$194$	0	0
$195$	− 0.414445i	− 0.0296790i
$196$	0	0
$197$	11.6835i	0.832415i	0.909270	+	0.416208i	$0.136641\pi$
$197$	11.6835i	0.832415i	−0.909270	+	0.416208i	$0.863359\pi$
$198$	0	0
$199$	24.9329	1.76745	0.883725	−	0.468007i	$-0.155028\pi$
$199$	24.9329	1.76745	0.883725	+	0.468007i	$0.155028\pi$
$200$	0	0
$201$	7.70367	0.543375
$202$	0	0
$203$	− 15.3572i	− 1.07786i
$204$	0	0
$205$	− 22.7183i	− 1.58672i
$206$	0	0
$207$	−5.40336	−0.375560
$208$	0	0
$209$	2.79744	0.193503
$210$	0	0
$211$	− 5.96245i	− 0.410472i	−0.978713	−	0.205236i	$-0.934204\pi$
$211$	− 5.96245i	− 0.410472i	0.978713	−	0.205236i	$-0.0657962\pi$
$212$	0	0
$213$	12.8592i	0.881096i
$214$	0	0
$215$	−8.65049	−0.589959
$216$	0	0
$217$	16.6610	1.13102
$218$	0	0
$219$	0.910690i	0.0615387i
$220$	0	0
$221$	− 0.870099i	− 0.0585292i
$222$	0	0
$223$	−1.03801	−0.0695103	−0.0347552	−	0.999396i	$-0.511065\pi$
$223$	−1.03801	−0.0695103	−0.0347552	+	0.999396i	$0.511065\pi$
$224$	0	0
$225$	1.54569	0.103046
$226$	0	0
$227$	− 6.65612i	− 0.441782i	−0.975299	−	0.220891i	$-0.929104\pi$
$227$	− 6.65612i	− 0.441782i	0.975299	−	0.220891i	$-0.0708965\pi$
$228$	0	0
$229$	16.3885i	1.08298i	0.840707	+	0.541491i	$0.182140\pi$
$229$	16.3885i	1.08298i	−0.840707	+	0.541491i	$0.817860\pi$
$230$	0	0
$231$	−2.45628	−0.161611
$232$	0	0
$233$	−16.0742	−1.05306	−0.526529	−	0.850157i	$-0.676507\pi$
$233$	−16.0742	−1.05306	−0.526529	+	0.850157i	$0.676507\pi$
$234$	0	0
$235$	− 12.1580i	− 0.793098i
$236$	0	0
$237$	7.71707i	0.501278i
$238$	0	0
$239$	2.52062	0.163045	0.0815226	−	0.996671i	$-0.474022\pi$
$239$	2.52062	0.163045	0.0815226	+	0.996671i	$0.474022\pi$
$240$	0	0
$241$	6.11179	0.393695	0.196848	−	0.980434i	$-0.436930\pi$
$241$	6.11179	0.393695	0.196848	+	0.980434i	$0.436930\pi$
$242$	0	0
$243$	− 11.7248i	− 0.752144i
$244$	0	0
$245$	− 6.98507i	− 0.446260i
$246$	0	0
$247$	0.449550	0.0286042
$248$	0	0
$249$	−3.90483	−0.247459
$250$	0	0
$251$	− 0.489390i	− 0.0308900i	−0.999881	−	0.0154450i	$-0.995084\pi$
$251$	− 0.489390i	− 0.0308900i	0.999881	−	0.0154450i	$-0.00491649\pi$
$252$	0	0
$253$	4.44137i	0.279227i
$254$	0	0
$255$	13.9637	0.874443
$256$	0	0
$257$	10.0886	0.629311	0.314655	−	0.949206i	$-0.398111\pi$
$257$	10.0886	0.629311	0.314655	+	0.949206i	$0.398111\pi$
$258$	0	0
$259$	− 1.10739i	− 0.0688100i
$260$	0	0
$261$	− 10.1580i	− 0.628762i
$262$	0	0
$263$	−3.98466	−0.245705	−0.122852	−	0.992425i	$-0.539204\pi$
$263$	−3.98466	−0.245705	−0.122852	+	0.992425i	$0.539204\pi$
$264$	0	0
$265$	−22.6356	−1.39049
$266$	0	0
$267$	− 7.66829i	− 0.469292i
$268$	0	0
$269$	− 2.70069i	− 0.164664i	−0.996605	−	0.0823320i	$-0.973763\pi$
$269$	− 2.70069i	− 0.164664i	0.996605	−	0.0823320i	$-0.0262368\pi$
$270$	0	0
$271$	−22.3611	−1.35834	−0.679170	−	0.733981i	$-0.737660\pi$
$271$	−22.3611	−1.35834	−0.679170	+	0.733981i	$0.737660\pi$
$272$	0	0
$273$	−0.394724	−0.0238898
$274$	0	0
$275$	− 1.27050i	− 0.0766142i
$276$	0	0
$277$	31.4865i	1.89184i	0.324396	+	0.945921i	$0.394839\pi$
$277$	31.4865i	1.89184i	−0.324396	+	0.945921i	$0.605161\pi$
$278$	0	0
$279$	11.0204	0.659772
$280$	0	0
$281$	−9.57079	−0.570945	−0.285473	−	0.958387i	$-0.592151\pi$
$281$	−9.57079	−0.570945	−0.285473	+	0.958387i	$0.592151\pi$
$282$	0	0
$283$	16.3997i	0.974860i	0.873162	+	0.487430i	$0.162066\pi$
$283$	16.3997i	0.974860i	−0.873162	+	0.487430i	$0.837934\pi$
$284$	0	0
$285$	7.21458i	0.427355i
$286$	0	0
$287$	−21.6373	−1.27721
$288$	0	0
$289$	12.3159	0.724467
$290$	0	0
$291$	7.18239i	0.421039i
$292$	0	0
$293$	− 10.0087i	− 0.584715i	−0.956309	−	0.292357i	$-0.905560\pi$
$293$	− 10.0087i	− 0.584715i	0.956309	−	0.292357i	$-0.0944397\pi$
$294$	0	0
$295$	−18.5993	−1.08289
$296$	0	0
$297$	−5.63102	−0.326745
$298$	0	0
$299$	0.713730i	0.0412761i
$300$	0	0
$301$	8.23888i	0.474881i
$302$	0	0
$303$	12.4446	0.714921
$304$	0	0
$305$	6.42916	0.368133
$306$	0	0
$307$	− 18.8436i	− 1.07546i	−0.843117	−	0.537731i	$-0.819282\pi$
$307$	− 18.8436i	− 1.07546i	0.843117	−	0.537731i	$-0.180718\pi$
$308$	0	0
$309$	2.24167i	0.127524i
$310$	0	0
$311$	1.59661	0.0905356	0.0452678	−	0.998975i	$-0.485586\pi$
$311$	1.59661	0.0905356	0.0452678	+	0.998975i	$0.485586\pi$
$312$	0	0
$313$	3.94136	0.222779	0.111389	−	0.993777i	$-0.464470\pi$
$313$	3.94136	0.222779	0.111389	+	0.993777i	$0.464470\pi$
$314$	0	0
$315$	4.32140i	0.243483i
$316$	0	0
$317$	− 1.94384i	− 0.109177i	−0.998509	−	0.0545884i	$-0.982615\pi$
$317$	− 1.94384i	− 0.109177i	0.998509	−	0.0545884i	$-0.0173847\pi$
$318$	0	0
$319$	−8.34948	−0.467481
$320$	0	0
$321$	−15.0595	−0.840539
$322$	0	0
$323$	15.1465i	0.842776i
$324$	0	0
$325$	− 0.204170i	− 0.0113253i
$326$	0	0
$327$	−26.2245	−1.45022
$328$	0	0
$329$	−11.5795	−0.638396
$330$	0	0
$331$	7.52148i	0.413418i	0.978402	+	0.206709i	$0.0662753\pi$
$331$	7.52148i	0.413418i	−0.978402	+	0.206709i	$0.933725\pi$
$332$	0	0
$333$	− 0.732480i	− 0.0401397i
$334$	0	0
$335$	−11.1403	−0.608662
$336$	0	0
$337$	−8.47233	−0.461517	−0.230759	−	0.973011i	$-0.574121\pi$
$337$	−8.47233	−0.461517	−0.230759	+	0.973011i	$0.574121\pi$
$338$	0	0
$339$	17.2480i	0.936781i
$340$	0	0
$341$	− 9.05835i	− 0.490537i
$342$	0	0
$343$	−19.5278	−1.05440
$344$	0	0
$345$	−11.4543	−0.616677
$346$	0	0
$347$	17.2487i	0.925960i	0.886369	+	0.462980i	$0.153220\pi$
$347$	17.2487i	0.925960i	−0.886369	+	0.462980i	$0.846780\pi$
$348$	0	0
$349$	4.67588i	0.250294i	0.992138	+	0.125147i	$0.0399403\pi$
$349$	4.67588i	0.250294i	−0.992138	+	0.125147i	$0.960060\pi$
$350$	0	0
$351$	−0.904906	−0.0483003
$352$	0	0
$353$	−5.45055	−0.290104	−0.145052	−	0.989424i	$-0.546335\pi$
$353$	−5.45055	−0.290104	−0.145052	+	0.989424i	$0.546335\pi$
$354$	0	0
$355$	− 18.5957i	− 0.986960i
$356$	0	0
$357$	− 13.2993i	− 0.703874i
$358$	0	0
$359$	28.0176	1.47871	0.739357	−	0.673313i	$-0.235129\pi$
$359$	28.0176	1.47871	0.739357	+	0.673313i	$0.235129\pi$
$360$	0	0
$361$	11.1743	0.588122
$362$	0	0
$363$	1.33544i	0.0700925i
$364$	0	0
$365$	− 1.31696i	− 0.0689326i
$366$	0	0
$367$	28.8393	1.50540	0.752699	−	0.658365i	$-0.228752\pi$
$367$	28.8393	1.50540	0.752699	+	0.658365i	$0.228752\pi$
$368$	0	0
$369$	−14.3119	−0.745049
$370$	0	0
$371$	21.5585i	1.11926i
$372$	0	0
$373$	1.34080i	0.0694239i	0.999397	+	0.0347119i	$0.0110514\pi$
$373$	1.34080i	0.0694239i	−0.999397	+	0.0347119i	$0.988949\pi$
$374$	0	0
$375$	16.1716	0.835096
$376$	0	0
$377$	−1.34176	−0.0691044
$378$	0	0
$379$	8.72245i	0.448042i	0.974584	+	0.224021i	$0.0719184\pi$
$379$	8.72245i	0.448042i	−0.974584	+	0.224021i	$0.928082\pi$
$380$	0	0
$381$	1.50525i	0.0771165i
$382$	0	0
$383$	−7.76275	−0.396658	−0.198329	−	0.980136i	$-0.563551\pi$
$383$	−7.76275	−0.396658	−0.198329	+	0.980136i	$0.563551\pi$
$384$	0	0
$385$	3.55204	0.181029
$386$	0	0
$387$	5.44957i	0.277017i
$388$	0	0
$389$	− 38.5938i	− 1.95678i	−0.206760	−	0.978392i	$-0.566292\pi$
$389$	− 38.5938i	− 1.95678i	0.206760	−	0.978392i	$-0.433708\pi$
$390$	0	0
$391$	−24.0475	−1.21613
$392$	0	0
$393$	−14.6303	−0.738001
$394$	0	0
$395$	− 11.1597i	− 0.561506i
$396$	0	0
$397$	20.4200i	1.02485i	0.858732	+	0.512426i	$0.171253\pi$
$397$	20.4200i	1.02485i	−0.858732	+	0.512426i	$0.828747\pi$
$398$	0	0
$399$	6.87129	0.343995
$400$	0	0
$401$	4.96247	0.247814	0.123907	−	0.992294i	$-0.460458\pi$
$401$	4.96247	0.247814	0.123907	+	0.992294i	$0.460458\pi$
$402$	0	0
$403$	− 1.45568i	− 0.0725125i
$404$	0	0
$405$	− 7.47390i	− 0.371381i
$406$	0	0
$407$	−0.602073	−0.0298436
$408$	0	0
$409$	15.3418	0.758601	0.379301	−	0.925273i	$-0.376165\pi$
$409$	15.3418	0.758601	0.379301	+	0.925273i	$0.376165\pi$
$410$	0	0
$411$	13.5886i	0.670274i
$412$	0	0
$413$	17.7143i	0.871665i
$414$	0	0
$415$	5.64681	0.277191
$416$	0	0
$417$	−24.4849	−1.19903
$418$	0	0
$419$	− 12.0628i	− 0.589306i	−0.955604	−	0.294653i	$-0.904796\pi$
$419$	− 12.0628i	− 0.589306i	0.955604	−	0.294653i	$-0.0952041\pi$
$420$	0	0
$421$	14.3042i	0.697143i	0.937282	+	0.348572i	$0.113333\pi$
$421$	14.3042i	0.697143i	−0.937282	+	0.348572i	$0.886667\pi$
$422$	0	0
$423$	−7.65918	−0.372402
$424$	0	0
$425$	6.87904	0.333682
$426$	0	0
$427$	− 6.12325i	− 0.296325i
$428$	0	0
$429$	0.214606i	0.0103613i
$430$	0	0
$431$	18.9520	0.912885	0.456443	−	0.889753i	$-0.349123\pi$
$431$	18.9520	0.912885	0.456443	+	0.889753i	$0.349123\pi$
$432$	0	0
$433$	−16.4130	−0.788760	−0.394380	−	0.918948i	$-0.629041\pi$
$433$	−16.4130	−0.788760	−0.394380	+	0.918948i	$0.629041\pi$
$434$	0	0
$435$	− 21.5332i	− 1.03244i
$436$	0	0
$437$	− 12.4245i	− 0.594344i
$438$	0	0
$439$	24.5482	1.17162	0.585810	−	0.810448i	$-0.300776\pi$
$439$	24.5482	1.17162	0.585810	+	0.810448i	$0.300776\pi$
$440$	0	0
$441$	−4.40040	−0.209543
$442$	0	0
$443$	11.0679i	0.525854i	0.964816	+	0.262927i	$0.0846878\pi$
$443$	11.0679i	0.525854i	−0.964816	+	0.262927i	$0.915312\pi$
$444$	0	0
$445$	11.0892i	0.525677i
$446$	0	0
$447$	−11.3097	−0.534933
$448$	0	0
$449$	26.6647	1.25838	0.629192	−	0.777250i	$-0.283386\pi$
$449$	26.6647	1.25838	0.629192	+	0.777250i	$0.283386\pi$
$450$	0	0
$451$	11.7639i	0.553940i
$452$	0	0
$453$	− 13.4486i	− 0.631871i
$454$	0	0
$455$	0.570814	0.0267602
$456$	0	0
$457$	3.28296	0.153570	0.0767851	−	0.997048i	$-0.475534\pi$
$457$	3.28296	0.153570	0.0767851	+	0.997048i	$0.475534\pi$
$458$	0	0
$459$	− 30.4887i	− 1.42309i
$460$	0	0
$461$	3.17514i	0.147881i	0.997263	+	0.0739406i	$0.0235575\pi$
$461$	3.17514i	0.147881i	−0.997263	+	0.0739406i	$0.976442\pi$
$462$	0	0
$463$	−21.8696	−1.01637	−0.508184	−	0.861248i	$-0.669683\pi$
$463$	−21.8696	−1.01637	−0.508184	+	0.861248i	$0.669683\pi$
$464$	0	0
$465$	23.3614	1.08336
$466$	0	0
$467$	− 4.95735i	− 0.229399i	−0.993400	−	0.114699i	$-0.963410\pi$
$467$	− 4.95735i	− 0.229399i	0.993400	−	0.114699i	$-0.0365905\pi$
$468$	0	0
$469$	10.6103i	0.489936i
$470$	0	0
$471$	−11.6713	−0.537787
$472$	0	0
$473$	4.47936	0.205961
$474$	0	0
$475$	3.55416i	0.163076i
$476$	0	0
$477$	14.2598i	0.652912i
$478$	0	0
$479$	35.5423	1.62397	0.811984	−	0.583680i	$-0.198388\pi$
$479$	35.5423	1.62397	0.811984	+	0.583680i	$0.198388\pi$
$480$	0	0
$481$	−0.0967532	−0.00441157
$482$	0	0
$483$	10.9092i	0.496387i
$484$	0	0
$485$	− 10.3865i	− 0.471627i
$486$	0	0
$487$	−10.1547	−0.460153	−0.230076	−	0.973173i	$-0.573898\pi$
$487$	−10.1547	−0.460153	−0.230076	+	0.973173i	$0.573898\pi$
$488$	0	0
$489$	32.6164	1.47497
$490$	0	0
$491$	− 0.367564i	− 0.0165879i	−0.999966	−	0.00829395i	$-0.997360\pi$
$491$	− 0.367564i	− 0.0165879i	0.999966	−	0.00829395i	$-0.00264008\pi$
$492$	0	0
$493$	− 45.2076i	− 2.03605i
$494$	0	0
$495$	2.34948	0.105601
$496$	0	0
$497$	−17.7109	−0.794443
$498$	0	0
$499$	− 31.1824i	− 1.39591i	−0.716140	−	0.697957i	$-0.754093\pi$
$499$	− 31.1824i	− 1.39591i	0.716140	−	0.697957i	$-0.245907\pi$
$500$	0	0
$501$	− 9.83532i	− 0.439410i
$502$	0	0
$503$	−20.2004	−0.900692	−0.450346	−	0.892854i	$-0.648699\pi$
$503$	−20.2004	−0.900692	−0.450346	+	0.892854i	$0.648699\pi$
$504$	0	0
$505$	−17.9962	−0.800818
$506$	0	0
$507$	− 17.3262i	− 0.769486i
$508$	0	0
$509$	22.0209i	0.976058i	0.872827	+	0.488029i	$0.162284\pi$
$509$	22.0209i	0.976058i	−0.872827	+	0.488029i	$0.837716\pi$
$510$	0	0
$511$	−1.25429	−0.0554866
$512$	0	0
$513$	15.7524	0.695488
$514$	0	0
$515$	− 3.24170i	− 0.142846i
$516$	0	0
$517$	6.29558i	0.276879i
$518$	0	0
$519$	−1.84695	−0.0810719
$520$	0	0
$521$	36.1120	1.58209	0.791047	−	0.611755i	$-0.209536\pi$
$521$	36.1120	1.58209	0.791047	+	0.611755i	$0.209536\pi$
$522$	0	0
$523$	− 34.5388i	− 1.51028i	−0.655565	−	0.755139i	$-0.727569\pi$
$523$	− 34.5388i	− 1.51028i	0.655565	−	0.755139i	$-0.272431\pi$
$524$	0	0
$525$	− 3.12071i	− 0.136199i
$526$	0	0
$527$	49.0457	2.13646
$528$	0	0
$529$	−3.27421	−0.142357
$530$	0	0
$531$	11.7171i	0.508478i
$532$	0	0
$533$	1.89046i	0.0818850i
$534$	0	0
$535$	21.7777	0.941530
$536$	0	0
$537$	13.2166	0.570337
$538$	0	0
$539$	3.61698i	0.155794i
$540$	0	0
$541$	− 6.14527i	− 0.264206i	−0.991236	−	0.132103i	$-0.957827\pi$
$541$	− 6.14527i	− 0.264206i	0.991236	−	0.132103i	$-0.0421729\pi$
$542$	0	0
$543$	−15.5984	−0.669391
$544$	0	0
$545$	37.9234	1.62446
$546$	0	0
$547$	17.3331i	0.741110i	0.928811	+	0.370555i	$0.120832\pi$
$547$	17.3331i	0.741110i	−0.928811	+	0.370555i	$0.879168\pi$
$548$	0	0
$549$	− 4.05020i	− 0.172858i
$550$	0	0
$551$	23.3572	0.995050
$552$	0	0
$553$	−10.6287	−0.451978
$554$	0	0
$555$	− 1.55274i	− 0.0659101i
$556$	0	0
$557$	38.1630i	1.61702i	0.588485	+	0.808508i	$0.299725\pi$
$557$	38.1630i	1.61702i	−0.588485	+	0.808508i	$0.700275\pi$
$558$	0	0
$559$	0.719834	0.0304457
$560$	0	0
$561$	−7.23064	−0.305278
$562$	0	0
$563$	− 33.2131i	− 1.39976i	−0.714258	−	0.699882i	$-0.753236\pi$
$563$	− 33.2131i	− 1.39976i	0.714258	−	0.699882i	$-0.246764\pi$
$564$	0	0
$565$	− 24.9424i	− 1.04934i
$566$	0	0
$567$	−7.11827	−0.298939
$568$	0	0
$569$	8.23504	0.345231	0.172615	−	0.984989i	$-0.444778\pi$
$569$	8.23504	0.345231	0.172615	+	0.984989i	$0.444778\pi$
$570$	0	0
$571$	32.7618i	1.37104i	0.728055	+	0.685518i	$0.240424\pi$
$571$	32.7618i	1.37104i	−0.728055	+	0.685518i	$0.759576\pi$
$572$	0	0
$573$	16.1457i	0.674496i
$574$	0	0
$575$	−5.64278	−0.235320
$576$	0	0
$577$	−6.50814	−0.270937	−0.135469	−	0.990782i	$-0.543254\pi$
$577$	−6.50814	−0.270937	−0.135469	+	0.990782i	$0.543254\pi$
$578$	0	0
$579$	17.7107i	0.736030i
$580$	0	0
$581$	− 5.37812i	− 0.223122i
$582$	0	0
$583$	11.7211	0.485436
$584$	0	0
$585$	0.377562	0.0156103
$586$	0	0
$587$	9.95325i	0.410815i	0.978677	+	0.205407i	$0.0658519\pi$
$587$	9.95325i	0.410815i	−0.978677	+	0.205407i	$0.934148\pi$
$588$	0	0
$589$	25.3402i	1.04413i
$590$	0	0
$591$	15.6026	0.641806
$592$	0	0
$593$	8.80877	0.361733	0.180867	−	0.983508i	$-0.442110\pi$
$593$	8.80877	0.361733	0.180867	+	0.983508i	$0.442110\pi$
$594$	0	0
$595$	19.2322i	0.788444i
$596$	0	0
$597$	− 33.2965i	− 1.36273i
$598$	0	0
$599$	−31.9145	−1.30399	−0.651996	−	0.758223i	$-0.726068\pi$
$599$	−31.9145	−1.30399	−0.651996	+	0.758223i	$0.726068\pi$
$600$	0	0
$601$	11.0314	0.449981	0.224991	−	0.974361i	$-0.427765\pi$
$601$	11.0314	0.449981	0.224991	+	0.974361i	$0.427765\pi$
$602$	0	0
$603$	7.01811i	0.285799i
$604$	0	0
$605$	− 1.93119i	− 0.0785141i
$606$	0	0
$607$	−12.2159	−0.495828	−0.247914	−	0.968782i	$-0.579745\pi$
$607$	−12.2159	−0.495828	−0.247914	+	0.968782i	$0.579745\pi$
$608$	0	0
$609$	−20.5086	−0.831052
$610$	0	0
$611$	1.01170i	0.0409290i
$612$	0	0
$613$	3.57482i	0.144386i	0.997391	+	0.0721928i	$0.0229997\pi$
$613$	3.57482i	0.144386i	−0.997391	+	0.0721928i	$0.977000\pi$
$614$	0	0
$615$	−30.3390	−1.22339
$616$	0	0
$617$	30.5057	1.22811	0.614057	−	0.789262i	$-0.289537\pi$
$617$	30.5057	1.22811	0.614057	+	0.789262i	$0.289537\pi$
$618$	0	0
$619$	− 10.6118i	− 0.426526i	−0.976995	−	0.213263i	$-0.931591\pi$
$619$	− 10.6118i	− 0.426526i	0.976995	−	0.213263i	$-0.0684091\pi$
$620$	0	0
$621$	25.0094i	1.00359i
$622$	0	0
$623$	10.5615	0.423138
$624$	0	0
$625$	−17.0333	−0.681332
$626$	0	0
$627$	− 3.73582i	− 0.149194i
$628$	0	0
$629$	− 3.25987i	− 0.129980i
$630$	0	0
$631$	−4.29235	−0.170876	−0.0854379	−	0.996343i	$-0.527229\pi$
$631$	−4.29235	−0.170876	−0.0854379	+	0.996343i	$0.527229\pi$
$632$	0	0
$633$	−7.96250	−0.316481
$634$	0	0
$635$	− 2.17676i	− 0.0863820i
$636$	0	0
$637$	0.581249i	0.0230299i
$638$	0	0
$639$	−11.7148	−0.463431
$640$	0	0
$641$	−26.5606	−1.04908	−0.524540	−	0.851386i	$-0.675763\pi$
$641$	−26.5606	−1.04908	−0.524540	+	0.851386i	$0.675763\pi$
$642$	0	0
$643$	28.0614i	1.10663i	0.832971	+	0.553317i	$0.186638\pi$
$643$	28.0614i	1.10663i	−0.832971	+	0.553317i	$0.813362\pi$
$644$	0	0
$645$	11.5522i	0.454868i
$646$	0	0
$647$	32.2983	1.26978	0.634888	−	0.772604i	$-0.281046\pi$
$647$	32.2983	1.26978	0.634888	+	0.772604i	$0.281046\pi$
$648$	0	0
$649$	9.63102	0.378050
$650$	0	0
$651$	− 22.2498i	− 0.872039i
$652$	0	0
$653$	32.9200i	1.28826i	0.764916	+	0.644130i	$0.222780\pi$
$653$	32.9200i	1.28826i	−0.764916	+	0.644130i	$0.777220\pi$
$654$	0	0
$655$	21.1570	0.826671
$656$	0	0
$657$	−0.829646	−0.0323676
$658$	0	0
$659$	3.99602i	0.155663i	0.996967	+	0.0778314i	$0.0247996\pi$
$659$	3.99602i	0.155663i	−0.996967	+	0.0778314i	$0.975200\pi$
$660$	0	0
$661$	− 19.2377i	− 0.748258i	−0.927377	−	0.374129i	$-0.877942\pi$
$661$	− 19.2377i	− 0.748258i	0.927377	−	0.374129i	$-0.122058\pi$
$662$	0	0
$663$	−1.16197	−0.0451270
$664$	0	0
$665$	−9.93663	−0.385326
$666$	0	0
$667$	37.0832i	1.43587i
$668$	0	0
$669$	1.38620i	0.0535937i
$670$	0	0
$671$	−3.32912	−0.128519
$672$	0	0
$673$	5.41202	0.208618	0.104309	−	0.994545i	$-0.466737\pi$
$673$	5.41202	0.208618	0.104309	+	0.994545i	$0.466737\pi$
$674$	0	0
$675$	− 7.15422i	− 0.275366i
$676$	0	0
$677$	− 41.1062i	− 1.57984i	−0.613209	−	0.789921i	$-0.710122\pi$
$677$	− 41.1062i	− 1.57984i	0.613209	−	0.789921i	$-0.289878\pi$
$678$	0	0
$679$	−9.89229	−0.379631
$680$	0	0
$681$	−8.88885	−0.340621
$682$	0	0
$683$	− 35.8592i	− 1.37211i	−0.727548	−	0.686057i	$-0.759340\pi$
$683$	− 35.8592i	− 1.37211i	0.727548	−	0.686057i	$-0.240660\pi$
$684$	0	0
$685$	− 19.6505i	− 0.750808i
$686$	0	0
$687$	21.8859	0.834997
$688$	0	0
$689$	1.88358	0.0717585
$690$	0	0
$691$	− 45.1462i	− 1.71744i	−0.512444	−	0.858721i	$-0.671260\pi$
$691$	− 45.1462i	− 1.71744i	0.512444	−	0.858721i	$-0.328740\pi$
$692$	0	0
$693$	− 2.23769i	− 0.0850027i
$694$	0	0
$695$	35.4078	1.34310
$696$	0	0
$697$	−63.6947	−2.41261
$698$	0	0
$699$	21.4662i	0.811926i
$700$	0	0
$701$	− 18.5867i	− 0.702009i	−0.936374	−	0.351005i	$-0.885840\pi$
$701$	− 18.5867i	− 0.702009i	0.936374	−	0.351005i	$-0.114160\pi$
$702$	0	0
$703$	1.68426	0.0635232
$704$	0	0
$705$	−16.2362	−0.611492
$706$	0	0
$707$	17.1399i	0.644610i
$708$	0	0
$709$	27.2703i	1.02416i	0.858938	+	0.512079i	$0.171124\pi$
$709$	27.2703i	1.02416i	−0.858938	+	0.512079i	$0.828876\pi$
$710$	0	0
$711$	−7.03031	−0.263657
$712$	0	0
$713$	−40.2315	−1.50668
$714$	0	0
$715$	− 0.310343i	− 0.0116062i
$716$	0	0
$717$	− 3.36614i	− 0.125711i
$718$	0	0
$719$	−34.1800	−1.27470	−0.637349	−	0.770575i	$-0.719969\pi$
$719$	−34.1800	−1.27470	−0.637349	+	0.770575i	$0.719969\pi$
$720$	0	0
$721$	−3.08745	−0.114983
$722$	0	0
$723$	− 8.16194i	− 0.303546i
$724$	0	0
$725$	− 10.6080i	− 0.393973i
$726$	0	0
$727$	25.0380	0.928608	0.464304	−	0.885676i	$-0.346304\pi$
$727$	25.0380	0.928608	0.464304	+	0.885676i	$0.346304\pi$
$728$	0	0
$729$	−27.2680	−1.00993
$730$	0	0
$731$	24.2531i	0.897034i
$732$	0	0
$733$	− 17.2695i	− 0.637862i	−0.947778	−	0.318931i	$-0.896676\pi$
$733$	− 17.2695i	− 0.637862i	0.947778	−	0.318931i	$-0.103324\pi$
$734$	0	0
$735$	−9.32815	−0.344074
$736$	0	0
$737$	5.76864	0.212490
$738$	0	0
$739$	17.2687i	0.635240i	0.948218	+	0.317620i	$0.102884\pi$
$739$	17.2687i	0.635240i	−0.948218	+	0.317620i	$0.897116\pi$
$740$	0	0
$741$	− 0.600347i	− 0.0220543i
$742$	0	0
$743$	−34.6374	−1.27072	−0.635361	−	0.772215i	$-0.719149\pi$
$743$	−34.6374	−1.27072	−0.635361	+	0.772215i	$0.719149\pi$
$744$	0	0
$745$	16.3551	0.599205
$746$	0	0
$747$	− 3.55734i	− 0.130156i
$748$	0	0
$749$	− 20.7414i	− 0.757875i
$750$	0	0
$751$	−51.5526	−1.88118	−0.940591	−	0.339543i	$-0.889728\pi$
$751$	−51.5526	−1.88118	−0.940591	+	0.339543i	$0.889728\pi$
$752$	0	0
$753$	−0.653551	−0.0238167
$754$	0	0
$755$	19.4481i	0.707790i
$756$	0	0
$757$	− 32.6049i	− 1.18504i	−0.805554	−	0.592522i	$-0.798132\pi$
$757$	− 32.6049i	− 1.18504i	0.805554	−	0.592522i	$-0.201868\pi$
$758$	0	0
$759$	5.93119	0.215289
$760$	0	0
$761$	−2.79094	−0.101172	−0.0505858	−	0.998720i	$-0.516109\pi$
$761$	−2.79094	−0.101172	−0.0505858	+	0.998720i	$0.516109\pi$
$762$	0	0
$763$	− 36.1190i	− 1.30759i
$764$	0	0
$765$	12.7211i	0.459932i
$766$	0	0
$767$	1.54771	0.0558845
$768$	0	0
$769$	23.1604	0.835187	0.417593	−	0.908634i	$-0.362874\pi$
$769$	23.1604	0.835187	0.417593	+	0.908634i	$0.362874\pi$
$770$	0	0
$771$	− 13.4728i	− 0.485209i
$772$	0	0
$773$	20.7913i	0.747811i	0.927467	+	0.373905i	$0.121982\pi$
$773$	20.7913i	0.747811i	−0.927467	+	0.373905i	$0.878018\pi$
$774$	0	0
$775$	11.5087	0.413403
$776$	0	0
$777$	−1.47886	−0.0530537
$778$	0	0
$779$	− 32.9088i	− 1.17908i
$780$	0	0
$781$	9.62916i	0.344559i
$782$	0	0
$783$	−47.0161	−1.68022
$784$	0	0
$785$	16.8780	0.602402
$786$	0	0
$787$	40.3874i	1.43965i	0.694153	+	0.719827i	$0.255779\pi$
$787$	40.3874i	1.43965i	−0.694153	+	0.719827i	$0.744221\pi$
$788$	0	0
$789$	5.32128i	0.189443i
$790$	0	0
$791$	−23.7556	−0.844651
$792$	0	0
$793$	−0.534990	−0.0189981
$794$	0	0
$795$	30.2285i	1.07209i
$796$	0	0
$797$	31.1190i	1.10229i	0.834409	+	0.551145i	$0.185809\pi$
$797$	31.1190i	1.10229i	−0.834409	+	0.551145i	$0.814191\pi$
$798$	0	0
$799$	−34.0869	−1.20591
$800$	0	0
$801$	6.98588	0.246834
$802$	0	0
$803$	0.681940i	0.0240651i
$804$	0	0
$805$	− 15.7759i	− 0.556028i
$806$	0	0
$807$	−3.60661	−0.126959
$808$	0	0
$809$	23.6252	0.830619	0.415309	−	0.909680i	$-0.363673\pi$
$809$	23.6252	0.830619	0.415309	+	0.909680i	$0.363673\pi$
$810$	0	0
$811$	26.2978i	0.923442i	0.887025	+	0.461721i	$0.152768\pi$
$811$	26.2978i	0.923442i	−0.887025	+	0.461721i	$0.847232\pi$
$812$	0	0
$813$	29.8619i	1.04730i
$814$	0	0
$815$	−47.1669	−1.65218
$816$	0	0
$817$	−12.5307	−0.438395
$818$	0	0
$819$	− 0.359597i	− 0.0125653i
$820$	0	0
$821$	55.6264i	1.94137i	0.240348	+	0.970687i	$0.422738\pi$
$821$	55.6264i	1.94137i	−0.240348	+	0.970687i	$0.577262\pi$
$822$	0	0
$823$	−28.0971	−0.979402	−0.489701	−	0.871890i	$-0.662894\pi$
$823$	−28.0971	−0.979402	−0.489701	+	0.871890i	$0.662894\pi$
$824$	0	0
$825$	−1.69668	−0.0590709
$826$	0	0
$827$	− 38.5555i	− 1.34070i	−0.742043	−	0.670352i	$-0.766143\pi$
$827$	− 38.5555i	− 1.34070i	0.742043	−	0.670352i	$-0.233857\pi$
$828$	0	0
$829$	− 44.4581i	− 1.54409i	−0.635566	−	0.772047i	$-0.719233\pi$
$829$	− 44.4581i	− 1.54409i	0.635566	−	0.772047i	$-0.280767\pi$
$830$	0	0
$831$	42.0484	1.45864
$832$	0	0
$833$	−19.5838	−0.678539
$834$	0	0
$835$	14.2229i	0.492204i
$836$	0	0
$837$	− 51.0077i	− 1.76308i
$838$	0	0
$839$	39.7880	1.37364	0.686818	−	0.726830i	$-0.259007\pi$
$839$	39.7880	1.37364	0.686818	+	0.726830i	$0.259007\pi$
$840$	0	0
$841$	−40.7138	−1.40393
$842$	0	0
$843$	12.7812i	0.440209i
$844$	0	0
$845$	25.0556i	0.861939i
$846$	0	0
$847$	−1.83930	−0.0631991
$848$	0	0
$849$	21.9008	0.751634
$850$	0	0
$851$	2.67403i	0.0916645i
$852$	0	0
$853$	4.19523i	0.143642i	0.997418	+	0.0718210i	$0.0228810\pi$
$853$	4.19523i	0.143642i	−0.997418	+	0.0718210i	$0.977119\pi$
$854$	0	0
$855$	−6.57254	−0.224776
$856$	0	0
$857$	43.4106	1.48288	0.741438	−	0.671021i	$-0.234144\pi$
$857$	43.4106	1.48288	0.741438	+	0.671021i	$0.234144\pi$
$858$	0	0
$859$	31.7672i	1.08388i	0.840416	+	0.541941i	$0.182310\pi$
$859$	31.7672i	1.08388i	−0.840416	+	0.541941i	$0.817690\pi$
$860$	0	0
$861$	28.8954i	0.984752i
$862$	0	0
$863$	51.0969	1.73936	0.869679	−	0.493618i	$-0.164326\pi$
$863$	51.0969	1.73936	0.869679	+	0.493618i	$0.164326\pi$
$864$	0	0
$865$	2.67088	0.0908127
$866$	0	0
$867$	− 16.4472i	− 0.558577i
$868$	0	0
$869$	5.77867i	0.196028i
$870$	0	0
$871$	0.927021	0.0314109
$872$	0	0
$873$	−6.54321	−0.221454
$874$	0	0
$875$	22.2731i	0.752967i
$876$	0	0
$877$	45.8912i	1.54963i	0.632185	+	0.774817i	$0.282158\pi$
$877$	45.8912i	1.54963i	−0.632185	+	0.774817i	$0.717842\pi$
$878$	0	0
$879$	−13.3660	−0.450825
$880$	0	0
$881$	−19.3333	−0.651356	−0.325678	−	0.945481i	$-0.605593\pi$
$881$	−19.3333	−0.651356	−0.325678	+	0.945481i	$0.605593\pi$
$882$	0	0
$883$	− 22.5862i	− 0.760088i	−0.924968	−	0.380044i	$-0.875909\pi$
$883$	− 22.5862i	− 0.760088i	0.924968	−	0.380044i	$-0.124091\pi$
$884$	0	0
$885$	24.8383i	0.834931i
$886$	0	0
$887$	34.2005	1.14834	0.574170	−	0.818736i	$-0.305325\pi$
$887$	34.2005	1.14834	0.574170	+	0.818736i	$0.305325\pi$
$888$	0	0
$889$	−2.07318	−0.0695323
$890$	0	0
$891$	3.87010i	0.129653i
$892$	0	0
$893$	− 17.6115i	− 0.589347i
$894$	0	0
$895$	−19.1126	−0.638863
$896$	0	0
$897$	0.953144	0.0318246
$898$	0	0
$899$	− 75.6325i	− 2.52249i
$900$	0	0
$901$	63.4627i	2.11425i
$902$	0	0
$903$	11.0025	0.366142
$904$	0	0
$905$	22.5569	0.749818
$906$	0	0
$907$	17.2305i	0.572129i	0.958210	+	0.286065i	$0.0923472\pi$
$907$	17.2305i	0.572129i	−0.958210	+	0.286065i	$0.907653\pi$
$908$	0	0
$909$	11.3371i	0.376027i
$910$	0	0
$911$	28.1534	0.932763	0.466382	−	0.884584i	$-0.345557\pi$
$911$	28.1534	0.932763	0.466382	+	0.884584i	$0.345557\pi$
$912$	0	0
$913$	−2.92400	−0.0967704
$914$	0	0
$915$	− 8.58577i	− 0.283837i
$916$	0	0
$917$	− 20.1503i	− 0.665420i
$918$	0	0
$919$	−1.87464	−0.0618387	−0.0309193	−	0.999522i	$-0.509843\pi$
$919$	−1.87464	−0.0618387	−0.0309193	+	0.999522i	$0.509843\pi$
$920$	0	0
$921$	−25.1645	−0.829199
$922$	0	0
$923$	1.54741i	0.0509336i
$924$	0	0
$925$	− 0.764935i	− 0.0251509i
$926$	0	0
$927$	−2.04218	−0.0670740
$928$	0	0
$929$	23.2555	0.762989	0.381495	−	0.924371i	$-0.375409\pi$
$929$	23.2555	0.762989	0.381495	+	0.924371i	$0.375409\pi$
$930$	0	0
$931$	− 10.1183i	− 0.331613i
$932$	0	0
$933$	− 2.13218i	− 0.0698045i
$934$	0	0
$935$	10.4563	0.341957
$936$	0	0
$937$	−11.1266	−0.363490	−0.181745	−	0.983346i	$-0.558174\pi$
$937$	−11.1266	−0.363490	−0.181745	+	0.983346i	$0.558174\pi$
$938$	0	0
$939$	− 5.26345i	− 0.171766i
$940$	0	0
$941$	− 23.5923i	− 0.769088i	−0.923107	−	0.384544i	$-0.874359\pi$
$941$	− 23.5923i	− 0.769088i	0.923107	−	0.384544i	$-0.125641\pi$
$942$	0	0
$943$	52.2479	1.70142
$944$	0	0
$945$	20.0016	0.650652
$946$	0	0
$947$	− 23.1038i	− 0.750772i	−0.926869	−	0.375386i	$-0.877510\pi$
$947$	− 23.1038i	− 0.750772i	0.926869	−	0.375386i	$-0.122490\pi$
$948$	0	0
$949$	0.109588i	0.00355737i
$950$	0	0
$951$	−2.59588	−0.0841771
$952$	0	0
$953$	2.75964	0.0893934	0.0446967	−	0.999001i	$-0.485768\pi$
$953$	2.75964	0.0893934	0.0446967	+	0.999001i	$0.485768\pi$
$954$	0	0
$955$	− 23.3484i	− 0.755537i
$956$	0	0
$957$	11.1502i	0.360436i
$958$	0	0
$959$	−18.7155	−0.604355
$960$	0	0
$961$	51.0537	1.64689
$962$	0	0
$963$	− 13.7193i	− 0.442099i
$964$	0	0
$965$	− 25.6115i	− 0.824464i
$966$	0	0
$967$	−50.7176	−1.63097	−0.815484	−	0.578780i	$-0.803529\pi$
$967$	−50.7176	−1.63097	−0.815484	+	0.578780i	$0.803529\pi$
$968$	0	0
$969$	20.2273	0.649794
$970$	0	0
$971$	− 38.8913i	− 1.24808i	−0.781392	−	0.624041i	$-0.785490\pi$
$971$	− 38.8913i	− 1.24808i	0.781392	−	0.624041i	$-0.214510\pi$
$972$	0	0
$973$	− 33.7230i	− 1.08111i
$974$	0	0
$975$	−0.272657	−0.00873202
$976$	0	0
$977$	−17.1186	−0.547673	−0.273837	−	0.961776i	$-0.588293\pi$
$977$	−17.1186	−0.547673	−0.273837	+	0.961776i	$0.588293\pi$
$978$	0	0
$979$	− 5.74214i	− 0.183520i
$980$	0	0
$981$	− 23.8907i	− 0.762773i
$982$	0	0
$983$	5.78625	0.184553	0.0922763	−	0.995733i	$-0.470586\pi$
$983$	5.78625	0.184553	0.0922763	+	0.995733i	$0.470586\pi$
$984$	0	0
$985$	−22.5631	−0.718919
$986$	0	0
$987$	15.4637i	0.492214i
$988$	0	0
$989$	− 19.8945i	− 0.632608i
$990$	0	0
$991$	4.97362	0.157992	0.0789961	−	0.996875i	$-0.474829\pi$
$991$	4.97362	0.157992	0.0789961	+	0.996875i	$0.474829\pi$
$992$	0	0
$993$	10.0445	0.318752
$994$	0	0
$995$	48.1503i	1.52647i
$996$	0	0
$997$	− 22.3402i	− 0.707521i	−0.935336	−	0.353760i	$-0.884903\pi$
$997$	− 22.3402i	− 0.707521i	0.935336	−	0.353760i	$-0.115097\pi$
$998$	0	0
$999$	−3.39028	−0.107264

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	352.2.c.a.177.4		10
3.2	odd	2		3168.2.f.g.1585.3		10
4.3	odd	2		88.2.c.a.45.4	yes	10
8.3	odd	2		88.2.c.a.45.3	✓	10
8.5	even	2	inner	352.2.c.a.177.7		10
11.10	odd	2		3872.2.c.f.1937.4		10
12.11	even	2		792.2.f.g.397.7		10
16.3	odd	4		2816.2.a.r.1.2		5
16.5	even	4		2816.2.a.p.1.2		5
16.11	odd	4		2816.2.a.o.1.4		5
16.13	even	4		2816.2.a.q.1.4		5
24.5	odd	2		3168.2.f.g.1585.8		10
24.11	even	2		792.2.f.g.397.8		10
44.3	odd	10		968.2.o.g.493.8		40
44.7	even	10		968.2.o.h.269.10		40
44.15	odd	10		968.2.o.g.269.1		40
44.19	even	10		968.2.o.h.493.3		40
44.27	odd	10		968.2.o.g.245.5		40
44.31	odd	10		968.2.o.g.565.9		40
44.35	even	10		968.2.o.h.565.2		40
44.39	even	10		968.2.o.h.245.6		40
44.43	even	2		968.2.c.d.485.7		10
88.3	odd	10		968.2.o.g.493.1		40
88.19	even	10		968.2.o.h.493.10		40
88.21	odd	2		3872.2.c.f.1937.7		10
88.27	odd	10		968.2.o.g.245.9		40
88.35	even	10		968.2.o.h.565.6		40
88.43	even	2		968.2.c.d.485.8		10
88.51	even	10		968.2.o.h.269.3		40
88.59	odd	10		968.2.o.g.269.8		40
88.75	odd	10		968.2.o.g.565.5		40
88.83	even	10		968.2.o.h.245.2		40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
88.2.c.a.45.3	✓	10	8.3	odd	2
88.2.c.a.45.4	yes	10	4.3	odd	2
352.2.c.a.177.4		10	1.1	even	1	trivial
352.2.c.a.177.7		10	8.5	even	2	inner
792.2.f.g.397.7		10	12.11	even	2
792.2.f.g.397.8		10	24.11	even	2
968.2.c.d.485.7		10	44.43	even	2
968.2.c.d.485.8		10	88.43	even	2
968.2.o.g.245.5		40	44.27	odd	10
968.2.o.g.245.9		40	88.27	odd	10
968.2.o.g.269.1		40	44.15	odd	10
968.2.o.g.269.8		40	88.59	odd	10
968.2.o.g.493.1		40	88.3	odd	10
968.2.o.g.493.8		40	44.3	odd	10
968.2.o.g.565.5		40	88.75	odd	10
968.2.o.g.565.9		40	44.31	odd	10
968.2.o.h.245.2		40	88.83	even	10
968.2.o.h.245.6		40	44.39	even	10
968.2.o.h.269.3		40	88.51	even	10
968.2.o.h.269.10		40	44.7	even	10
968.2.o.h.493.3		40	44.19	even	10
968.2.o.h.493.10		40	88.19	even	10
968.2.o.h.565.2		40	44.35	even	10
968.2.o.h.565.6		40	88.35	even	10
2816.2.a.o.1.4		5	16.11	odd	4
2816.2.a.p.1.2		5	16.5	even	4
2816.2.a.q.1.4		5	16.13	even	4
2816.2.a.r.1.2		5	16.3	odd	4
3168.2.f.g.1585.3		10	3.2	odd	2
3168.2.f.g.1585.8		10	24.5	odd	2
3872.2.c.f.1937.4		10	11.10	odd	2
3872.2.c.f.1937.7		10	88.21	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

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

\(f(q)\)	\(=\)	\(q-1.33544i q^{3} +1.93119i q^{5} +1.83930 q^{7} +1.21660 q^{9} -1.00000i q^{11} -0.160700i q^{13} +2.57899 q^{15} +5.41442 q^{17} +2.79744i q^{19} -2.45628i q^{21} -4.44137 q^{23} +1.27050 q^{25} -5.63102i q^{27} -8.34948i q^{29} +9.05835 q^{31} -1.33544 q^{33} +3.55204i q^{35} -0.602073i q^{37} -0.214606 q^{39} -11.7639 q^{41} +4.47936i q^{43} +2.34948i q^{45} -6.29558 q^{47} -3.61698 q^{49} -7.23064i q^{51} +11.7211i q^{53} +1.93119 q^{55} +3.73582 q^{57} +9.63102i q^{59} -3.32912i q^{61} +2.23769 q^{63} +0.310343 q^{65} +5.76864i q^{67} +5.93119i q^{69} -9.62916 q^{71} -0.681940 q^{73} -1.69668i q^{75} -1.83930i q^{77} -5.77867 q^{79} -3.87010 q^{81} -2.92400i q^{83} +10.4563i q^{85} -11.1502 q^{87} +5.74214 q^{89} -0.295576i q^{91} -12.0969i q^{93} -5.40240 q^{95} -5.37829 q^{97} -1.21660i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 10 q^{9} - 8 q^{15} - 4 q^{17} + 12 q^{23} - 6 q^{25} + 4 q^{31} - 24 q^{39} + 4 q^{41} + 4 q^{47} - 6 q^{49} + 8 q^{55} + 16 q^{57} + 40 q^{63} + 16 q^{65} + 12 q^{71} - 4 q^{73} - 16 q^{79} - 6 q^{81}+ \cdots - 20 q^{97}+O(q^{100}) \) 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

Introduction

L-functions

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