LMFDB - Embedded newform 2368.2.a.ba.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2368,2,Mod(1,2368)] 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("2368.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2368, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2368 = 2^{6} \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2368.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,3,0,1,0,-2,0,5,0,-1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$18.9085751986$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{13}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 3 $ x^2 - x - 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 74)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.30278$ of defining polynomial
Character	$\chi$	$=$	2368.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-0.302776 q^{3} -1.30278 q^{5} -4.60555 q^{7} -2.90833 q^{9} +1.30278 q^{11} +2.30278 q^{13} +0.394449 q^{15} -6.00000 q^{17} +2.00000 q^{19} +1.39445 q^{21} +6.90833 q^{23} -3.30278 q^{25} +1.78890 q^{27} -6.90833 q^{29} -3.30278 q^{31} -0.394449 q^{33} +6.00000 q^{35} -1.00000 q^{37} -0.697224 q^{39} -0.908327 q^{41} -6.60555 q^{43} +3.78890 q^{45} +2.60555 q^{47} +14.2111 q^{49} +1.81665 q^{51} +6.00000 q^{53} -1.69722 q^{55} -0.605551 q^{57} +3.39445 q^{59} +10.5139 q^{61} +13.3944 q^{63} -3.00000 q^{65} +14.5139 q^{67} -2.09167 q^{69} -6.00000 q^{71} -8.69722 q^{73} +1.00000 q^{75} -6.00000 q^{77} +16.1194 q^{79} +8.18335 q^{81} +17.2111 q^{83} +7.81665 q^{85} +2.09167 q^{87} +5.21110 q^{89} -10.6056 q^{91} +1.00000 q^{93} -2.60555 q^{95} +12.4222 q^{97} -3.78890 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 3 q^{3} + q^{5} - 2 q^{7} + 5 q^{9} - q^{11} + q^{13} + 8 q^{15} - 12 q^{17} + 4 q^{19} + 10 q^{21} + 3 q^{23} - 3 q^{25} + 18 q^{27} - 3 q^{29} - 3 q^{31} - 8 q^{33} + 12 q^{35} - 2 q^{37} - 5 q^{39}+ \cdots - 22 q^{99}+O(q^{100}) $ 2 * q + 3 * q^3 + q^5 - 2 * q^7 + 5 * q^9 - q^11 + q^13 + 8 * q^15 - 12 * q^17 + 4 * q^19 + 10 * q^21 + 3 * q^23 - 3 * q^25 + 18 * q^27 - 3 * q^29 - 3 * q^31 - 8 * q^33 + 12 * q^35 - 2 * q^37 - 5 * q^39 + 9 * q^41 - 6 * q^43 + 22 * q^45 - 2 * q^47 + 14 * q^49 - 18 * q^51 + 12 * q^53 - 7 * q^55 + 6 * q^57 + 14 * q^59 + 3 * q^61 + 34 * q^63 - 6 * q^65 + 11 * q^67 - 15 * q^69 - 12 * q^71 - 21 * q^73 + 2 * q^75 - 12 * q^77 + 7 * q^79 + 38 * q^81 + 20 * q^83 - 6 * q^85 + 15 * q^87 - 4 * q^89 - 14 * q^91 + 2 * q^93 + 2 * q^95 - 4 * q^97 - 22 * q^99

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
$3$	−0.302776	−0.174808	−0.0874038	−	0.996173i	$-0.527857\pi$
$3$	−0.302776	−0.174808	−0.0874038	+	0.996173i	$0.527857\pi$
$4$	0	0
$5$	−1.30278	−0.582619	−0.291309	−	0.956629i	$-0.594091\pi$
$5$	−1.30278	−0.582619	−0.291309	+	0.956629i	$0.594091\pi$
$6$	0	0
$7$	−4.60555	−1.74073	−0.870367	−	0.492403i	$-0.836119\pi$
$7$	−4.60555	−1.74073	−0.870367	+	0.492403i	$0.836119\pi$
$8$	0	0
$9$	−2.90833	−0.969442
$10$	0	0
$11$	1.30278	0.392802	0.196401	−	0.980524i	$-0.437075\pi$
$11$	1.30278	0.392802	0.196401	+	0.980524i	$0.437075\pi$
$12$	0	0
$13$	2.30278	0.638675	0.319338	−	0.947641i	$-0.396540\pi$
$13$	2.30278	0.638675	0.319338	+	0.947641i	$0.396540\pi$
$14$	0	0
$15$	0.394449	0.101846
$16$	0	0
$17$	−6.00000	−1.45521	−0.727607	−	0.685994i	$-0.759367\pi$
$17$	−6.00000	−1.45521	−0.727607	+	0.685994i	$0.759367\pi$
$18$	0	0
$19$	2.00000	0.458831	0.229416	−	0.973329i	$-0.426318\pi$
$19$	2.00000	0.458831	0.229416	+	0.973329i	$0.426318\pi$
$20$	0	0
$21$	1.39445	0.304294
$22$	0	0
$23$	6.90833	1.44049	0.720243	−	0.693722i	$-0.244030\pi$
$23$	6.90833	1.44049	0.720243	+	0.693722i	$0.244030\pi$
$24$	0	0
$25$	−3.30278	−0.660555
$26$	0	0
$27$	1.78890	0.344273
$28$	0	0
$29$	−6.90833	−1.28284	−0.641422	−	0.767188i	$-0.721655\pi$
$29$	−6.90833	−1.28284	−0.641422	+	0.767188i	$0.721655\pi$
$30$	0	0
$31$	−3.30278	−0.593196	−0.296598	−	0.955002i	$-0.595852\pi$
$31$	−3.30278	−0.593196	−0.296598	+	0.955002i	$0.595852\pi$
$32$	0	0
$33$	−0.394449	−0.0686647
$34$	0	0
$35$	6.00000	1.01419
$36$	0	0
$37$	−1.00000	−0.164399
$37$	−1.00000	−0.164399
$38$	0	0
$39$	−0.697224	−0.111645
$40$	0	0
$41$	−0.908327	−0.141857	−0.0709284	−	0.997481i	$-0.522596\pi$
$41$	−0.908327	−0.141857	−0.0709284	+	0.997481i	$0.522596\pi$
$42$	0	0
$43$	−6.60555	−1.00734	−0.503669	−	0.863897i	$-0.668017\pi$
$43$	−6.60555	−1.00734	−0.503669	+	0.863897i	$0.668017\pi$
$44$	0	0
$45$	3.78890	0.564815
$46$	0	0
$47$	2.60555	0.380059	0.190029	−	0.981778i	$-0.439142\pi$
$47$	2.60555	0.380059	0.190029	+	0.981778i	$0.439142\pi$
$48$	0	0
$49$	14.2111	2.03016
$50$	0	0
$51$	1.81665	0.254382
$52$	0	0
$53$	6.00000	0.824163	0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000	0.824163	0.412082	+	0.911147i	$0.364802\pi$
$54$	0	0
$55$	−1.69722	−0.228854
$56$	0	0
$57$	−0.605551	−0.0802072
$58$	0	0
$59$	3.39445	0.441920	0.220960	−	0.975283i	$-0.429081\pi$
$59$	3.39445	0.441920	0.220960	+	0.975283i	$0.429081\pi$
$60$	0	0
$61$	10.5139	1.34616	0.673082	−	0.739568i	$-0.264970\pi$
$61$	10.5139	1.34616	0.673082	+	0.739568i	$0.264970\pi$
$62$	0	0
$63$	13.3944	1.68754
$64$	0	0
$65$	−3.00000	−0.372104
$66$	0	0
$67$	14.5139	1.77315	0.886576	−	0.462583i	$-0.153077\pi$
$67$	14.5139	1.77315	0.886576	+	0.462583i	$0.153077\pi$
$68$	0	0
$69$	−2.09167	−0.251808
$70$	0	0
$71$	−6.00000	−0.712069	−0.356034	−	0.934473i	$-0.615871\pi$
$71$	−6.00000	−0.712069	−0.356034	+	0.934473i	$0.615871\pi$
$72$	0	0
$73$	−8.69722	−1.01793	−0.508967	−	0.860786i	$-0.669972\pi$
$73$	−8.69722	−1.01793	−0.508967	+	0.860786i	$0.669972\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	−6.00000	−0.683763
$78$	0	0
$79$	16.1194	1.81358	0.906789	−	0.421585i	$-0.138526\pi$
$79$	16.1194	1.81358	0.906789	+	0.421585i	$0.138526\pi$
$80$	0	0
$81$	8.18335	0.909261
$82$	0	0
$83$	17.2111	1.88916	0.944582	−	0.328276i	$-0.106467\pi$
$83$	17.2111	1.88916	0.944582	+	0.328276i	$0.106467\pi$
$84$	0	0
$85$	7.81665	0.847835
$86$	0	0
$87$	2.09167	0.224251
$88$	0	0
$89$	5.21110	0.552376	0.276188	−	0.961104i	$-0.410929\pi$
$89$	5.21110	0.552376	0.276188	+	0.961104i	$0.410929\pi$
$90$	0	0
$91$	−10.6056	−1.11176
$92$	0	0
$93$	1.00000	0.103695
$94$	0	0
$95$	−2.60555	−0.267324
$96$	0	0
$97$	12.4222	1.26128	0.630642	−	0.776074i	$-0.282792\pi$
$97$	12.4222	1.26128	0.630642	+	0.776074i	$0.282792\pi$
$98$	0	0
$99$	−3.78890	−0.380799
$100$	0	0
$101$	−16.4222	−1.63407	−0.817035	−	0.576588i	$-0.804384\pi$
$101$	−16.4222	−1.63407	−0.817035	+	0.576588i	$0.804384\pi$
$102$	0	0
$103$	−3.30278	−0.325432	−0.162716	−	0.986673i	$-0.552025\pi$
$103$	−3.30278	−0.325432	−0.162716	+	0.986673i	$0.552025\pi$
$104$	0	0
$105$	−1.81665	−0.177287
$106$	0	0
$107$	4.30278	0.415965	0.207983	−	0.978133i	$-0.433310\pi$
$107$	4.30278	0.415965	0.207983	+	0.978133i	$0.433310\pi$
$108$	0	0
$109$	−2.00000	−0.191565	−0.0957826	−	0.995402i	$-0.530535\pi$
$109$	−2.00000	−0.191565	−0.0957826	+	0.995402i	$0.530535\pi$
$110$	0	0
$111$	0.302776	0.0287382
$112$	0	0
$113$	11.2111	1.05465	0.527326	−	0.849663i	$-0.323195\pi$
$113$	11.2111	1.05465	0.527326	+	0.849663i	$0.323195\pi$
$114$	0	0
$115$	−9.00000	−0.839254
$116$	0	0
$117$	−6.69722	−0.619159
$118$	0	0
$119$	27.6333	2.53314
$120$	0	0
$121$	−9.30278	−0.845707
$122$	0	0
$123$	0.275019	0.0247977
$124$	0	0
$125$	10.8167	0.967471
$126$	0	0
$127$	4.78890	0.424946	0.212473	−	0.977167i	$-0.431848\pi$
$127$	4.78890	0.424946	0.212473	+	0.977167i	$0.431848\pi$
$128$	0	0
$129$	2.00000	0.176090
$130$	0	0
$131$	3.39445	0.296574	0.148287	−	0.988944i	$-0.452624\pi$
$131$	3.39445	0.296574	0.148287	+	0.988944i	$0.452624\pi$
$132$	0	0
$133$	−9.21110	−0.798704
$134$	0	0
$135$	−2.33053	−0.200580
$136$	0	0
$137$	−9.90833	−0.846525	−0.423263	−	0.906007i	$-0.639115\pi$
$137$	−9.90833	−0.846525	−0.423263	+	0.906007i	$0.639115\pi$
$138$	0	0
$139$	8.90833	0.755594	0.377797	−	0.925888i	$-0.376682\pi$
$139$	8.90833	0.755594	0.377797	+	0.925888i	$0.376682\pi$
$140$	0	0
$141$	−0.788897	−0.0664372
$142$	0	0
$143$	3.00000	0.250873
$144$	0	0
$145$	9.00000	0.747409
$146$	0	0
$147$	−4.30278	−0.354887
$148$	0	0
$149$	1.81665	0.148826	0.0744130	−	0.997228i	$-0.476292\pi$
$149$	1.81665	0.148826	0.0744130	+	0.997228i	$0.476292\pi$
$150$	0	0
$151$	13.3944	1.09002	0.545012	−	0.838428i	$-0.316525\pi$
$151$	13.3944	1.09002	0.545012	+	0.838428i	$0.316525\pi$
$152$	0	0
$153$	17.4500	1.41075
$154$	0	0
$155$	4.30278	0.345607
$156$	0	0
$157$	−7.21110	−0.575509	−0.287754	−	0.957704i	$-0.592909\pi$
$157$	−7.21110	−0.575509	−0.287754	+	0.957704i	$0.592909\pi$
$158$	0	0
$159$	−1.81665	−0.144070
$160$	0	0
$161$	−31.8167	−2.50750
$162$	0	0
$163$	−20.4222	−1.59959	−0.799795	−	0.600273i	$-0.795059\pi$
$163$	−20.4222	−1.59959	−0.799795	+	0.600273i	$0.795059\pi$
$164$	0	0
$165$	0.513878	0.0400054
$166$	0	0
$167$	−12.5139	−0.968353	−0.484176	−	0.874970i	$-0.660881\pi$
$167$	−12.5139	−0.968353	−0.484176	+	0.874970i	$0.660881\pi$
$168$	0	0
$169$	−7.69722	−0.592094
$170$	0	0
$171$	−5.81665	−0.444811
$172$	0	0
$173$	23.2111	1.76471	0.882354	−	0.470587i	$-0.155958\pi$
$173$	23.2111	1.76471	0.882354	+	0.470587i	$0.155958\pi$
$174$	0	0
$175$	15.2111	1.14985
$176$	0	0
$177$	−1.02776	−0.0772509
$178$	0	0
$179$	7.81665	0.584244	0.292122	−	0.956381i	$-0.405639\pi$
$179$	7.81665	0.584244	0.292122	+	0.956381i	$0.405639\pi$
$180$	0	0
$181$	−20.0000	−1.48659	−0.743294	−	0.668965i	$-0.766738\pi$
$181$	−20.0000	−1.48659	−0.743294	+	0.668965i	$0.766738\pi$
$182$	0	0
$183$	−3.18335	−0.235320
$184$	0	0
$185$	1.30278	0.0957820
$186$	0	0
$187$	−7.81665	−0.571610
$188$	0	0
$189$	−8.23886	−0.599289
$190$	0	0
$191$	−12.5139	−0.905472	−0.452736	−	0.891644i	$-0.649552\pi$
$191$	−12.5139	−0.905472	−0.452736	+	0.891644i	$0.649552\pi$
$192$	0	0
$193$	−4.00000	−0.287926	−0.143963	−	0.989583i	$-0.545985\pi$
$193$	−4.00000	−0.287926	−0.143963	+	0.989583i	$0.545985\pi$
$194$	0	0
$195$	0.908327	0.0650466
$196$	0	0
$197$	6.00000	0.427482	0.213741	−	0.976890i	$-0.431435\pi$
$197$	6.00000	0.427482	0.213741	+	0.976890i	$0.431435\pi$
$198$	0	0
$199$	2.42221	0.171706	0.0858528	−	0.996308i	$-0.472639\pi$
$199$	2.42221	0.171706	0.0858528	+	0.996308i	$0.472639\pi$
$200$	0	0
$201$	−4.39445	−0.309961
$202$	0	0
$203$	31.8167	2.23309
$204$	0	0
$205$	1.18335	0.0826485
$206$	0	0
$207$	−20.0917	−1.39647
$208$	0	0
$209$	2.60555	0.180230
$210$	0	0
$211$	6.69722	0.461056	0.230528	−	0.973066i	$-0.425955\pi$
$211$	6.69722	0.461056	0.230528	+	0.973066i	$0.425955\pi$
$212$	0	0
$213$	1.81665	0.124475
$214$	0	0
$215$	8.60555	0.586894
$216$	0	0
$217$	15.2111	1.03260
$218$	0	0
$219$	2.63331	0.177942
$220$	0	0
$221$	−13.8167	−0.929409
$222$	0	0
$223$	−15.8167	−1.05916	−0.529581	−	0.848260i	$-0.677651\pi$
$223$	−15.8167	−1.05916	−0.529581	+	0.848260i	$0.677651\pi$
$224$	0	0
$225$	9.60555	0.640370
$226$	0	0
$227$	−7.81665	−0.518810	−0.259405	−	0.965769i	$-0.583526\pi$
$227$	−7.81665	−0.518810	−0.259405	+	0.965769i	$0.583526\pi$
$228$	0	0
$229$	−17.3944	−1.14946	−0.574729	−	0.818344i	$-0.694892\pi$
$229$	−17.3944	−1.14946	−0.574729	+	0.818344i	$0.694892\pi$
$230$	0	0
$231$	1.81665	0.119527
$232$	0	0
$233$	−9.51388	−0.623275	−0.311637	−	0.950201i	$-0.600877\pi$
$233$	−9.51388	−0.623275	−0.311637	+	0.950201i	$0.600877\pi$
$234$	0	0
$235$	−3.39445	−0.221429
$236$	0	0
$237$	−4.88057	−0.317027
$238$	0	0
$239$	−0.513878	−0.0332400	−0.0166200	−	0.999862i	$-0.505291\pi$
$239$	−0.513878	−0.0332400	−0.0166200	+	0.999862i	$0.505291\pi$
$240$	0	0
$241$	8.00000	0.515325	0.257663	−	0.966235i	$-0.417048\pi$
$241$	8.00000	0.515325	0.257663	+	0.966235i	$0.417048\pi$
$242$	0	0
$243$	−7.84441	−0.503219
$244$	0	0
$245$	−18.5139	−1.18281
$246$	0	0
$247$	4.60555	0.293044
$248$	0	0
$249$	−5.21110	−0.330240
$250$	0	0
$251$	−6.78890	−0.428511	−0.214256	−	0.976778i	$-0.568733\pi$
$251$	−6.78890	−0.428511	−0.214256	+	0.976778i	$0.568733\pi$
$252$	0	0
$253$	9.00000	0.565825
$254$	0	0
$255$	−2.36669	−0.148208
$256$	0	0
$257$	−11.2111	−0.699329	−0.349665	−	0.936875i	$-0.613704\pi$
$257$	−11.2111	−0.699329	−0.349665	+	0.936875i	$0.613704\pi$
$258$	0	0
$259$	4.60555	0.286175
$260$	0	0
$261$	20.0917	1.24364
$262$	0	0
$263$	7.81665	0.481996	0.240998	−	0.970526i	$-0.422525\pi$
$263$	7.81665	0.481996	0.240998	+	0.970526i	$0.422525\pi$
$264$	0	0
$265$	−7.81665	−0.480173
$266$	0	0
$267$	−1.57779	−0.0965595
$268$	0	0
$269$	6.78890	0.413926	0.206963	−	0.978349i	$-0.433642\pi$
$269$	6.78890	0.413926	0.206963	+	0.978349i	$0.433642\pi$
$270$	0	0
$271$	−6.42221	−0.390121	−0.195061	−	0.980791i	$-0.562490\pi$
$271$	−6.42221	−0.390121	−0.195061	+	0.980791i	$0.562490\pi$
$272$	0	0
$273$	3.21110	0.194345
$274$	0	0
$275$	−4.30278	−0.259467
$276$	0	0
$277$	25.1194	1.50928	0.754640	−	0.656139i	$-0.227811\pi$
$277$	25.1194	1.50928	0.754640	+	0.656139i	$0.227811\pi$
$278$	0	0
$279$	9.60555	0.575069
$280$	0	0
$281$	−12.0000	−0.715860	−0.357930	−	0.933748i	$-0.616517\pi$
$281$	−12.0000	−0.715860	−0.357930	+	0.933748i	$0.616517\pi$
$282$	0	0
$283$	17.3944	1.03399	0.516996	−	0.855988i	$-0.327050\pi$
$283$	17.3944	1.03399	0.516996	+	0.855988i	$0.327050\pi$
$284$	0	0
$285$	0.788897	0.0467303
$286$	0	0
$287$	4.18335	0.246935
$288$	0	0
$289$	19.0000	1.11765
$290$	0	0
$291$	−3.76114	−0.220482
$292$	0	0
$293$	25.0278	1.46214	0.731069	−	0.682304i	$-0.239022\pi$
$293$	25.0278	1.46214	0.731069	+	0.682304i	$0.239022\pi$
$294$	0	0
$295$	−4.42221	−0.257471
$296$	0	0
$297$	2.33053	0.135231
$298$	0	0
$299$	15.9083	0.920002
$300$	0	0
$301$	30.4222	1.75351
$302$	0	0
$303$	4.97224	0.285648
$304$	0	0
$305$	−13.6972	−0.784301
$306$	0	0
$307$	7.09167	0.404743	0.202372	−	0.979309i	$-0.435135\pi$
$307$	7.09167	0.404743	0.202372	+	0.979309i	$0.435135\pi$
$308$	0	0
$309$	1.00000	0.0568880
$310$	0	0
$311$	−5.09167	−0.288722	−0.144361	−	0.989525i	$-0.546113\pi$
$311$	−5.09167	−0.288722	−0.144361	+	0.989525i	$0.546113\pi$
$312$	0	0
$313$	27.0278	1.52770	0.763850	−	0.645394i	$-0.223307\pi$
$313$	27.0278	1.52770	0.763850	+	0.645394i	$0.223307\pi$
$314$	0	0
$315$	−17.4500	−0.983194
$316$	0	0
$317$	5.21110	0.292685	0.146342	−	0.989234i	$-0.453250\pi$
$317$	5.21110	0.292685	0.146342	+	0.989234i	$0.453250\pi$
$318$	0	0
$319$	−9.00000	−0.503903
$320$	0	0
$321$	−1.30278	−0.0727138
$322$	0	0
$323$	−12.0000	−0.667698
$324$	0	0
$325$	−7.60555	−0.421880
$326$	0	0
$327$	0.605551	0.0334871
$328$	0	0
$329$	−12.0000	−0.661581
$330$	0	0
$331$	1.21110	0.0665682	0.0332841	−	0.999446i	$-0.489403\pi$
$331$	1.21110	0.0665682	0.0332841	+	0.999446i	$0.489403\pi$
$332$	0	0
$333$	2.90833	0.159375
$334$	0	0
$335$	−18.9083	−1.03307
$336$	0	0
$337$	−19.1194	−1.04150	−0.520751	−	0.853709i	$-0.674348\pi$
$337$	−19.1194	−1.04150	−0.520751	+	0.853709i	$0.674348\pi$
$338$	0	0
$339$	−3.39445	−0.184361
$340$	0	0
$341$	−4.30278	−0.233008
$342$	0	0
$343$	−33.2111	−1.79323
$344$	0	0
$345$	2.72498	0.146708
$346$	0	0
$347$	31.8167	1.70801	0.854004	−	0.520267i	$-0.174168\pi$
$347$	31.8167	1.70801	0.854004	+	0.520267i	$0.174168\pi$
$348$	0	0
$349$	22.2389	1.19042	0.595209	−	0.803571i	$-0.297069\pi$
$349$	22.2389	1.19042	0.595209	+	0.803571i	$0.297069\pi$
$350$	0	0
$351$	4.11943	0.219879
$352$	0	0
$353$	31.8167	1.69343	0.846715	−	0.532047i	$-0.178577\pi$
$353$	31.8167	1.69343	0.846715	+	0.532047i	$0.178577\pi$
$354$	0	0
$355$	7.81665	0.414865
$356$	0	0
$357$	−8.36669	−0.442812
$358$	0	0
$359$	11.2111	0.591699	0.295850	−	0.955235i	$-0.404397\pi$
$359$	11.2111	0.591699	0.295850	+	0.955235i	$0.404397\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	0	0
$363$	2.81665	0.147836
$364$	0	0
$365$	11.3305	0.593067
$366$	0	0
$367$	17.8167	0.930022	0.465011	−	0.885305i	$-0.346050\pi$
$367$	17.8167	0.930022	0.465011	+	0.885305i	$0.346050\pi$
$368$	0	0
$369$	2.64171	0.137522
$370$	0	0
$371$	−27.6333	−1.43465
$372$	0	0
$373$	−3.81665	−0.197619	−0.0988094	−	0.995106i	$-0.531503\pi$
$373$	−3.81665	−0.197619	−0.0988094	+	0.995106i	$0.531503\pi$
$374$	0	0
$375$	−3.27502	−0.169121
$376$	0	0
$377$	−15.9083	−0.819321
$378$	0	0
$379$	−15.3305	−0.787477	−0.393738	−	0.919223i	$-0.628818\pi$
$379$	−15.3305	−0.787477	−0.393738	+	0.919223i	$0.628818\pi$
$380$	0	0
$381$	−1.44996	−0.0742838
$382$	0	0
$383$	−20.8444	−1.06510	−0.532550	−	0.846399i	$-0.678766\pi$
$383$	−20.8444	−1.06510	−0.532550	+	0.846399i	$0.678766\pi$
$384$	0	0
$385$	7.81665	0.398374
$386$	0	0
$387$	19.2111	0.976555
$388$	0	0
$389$	11.8806	0.602369	0.301184	−	0.953566i	$-0.402618\pi$
$389$	11.8806	0.602369	0.301184	+	0.953566i	$0.402618\pi$
$390$	0	0
$391$	−41.4500	−2.09621
$392$	0	0
$393$	−1.02776	−0.0518435
$394$	0	0
$395$	−21.0000	−1.05662
$396$	0	0
$397$	−27.8167	−1.39608	−0.698039	−	0.716060i	$-0.745944\pi$
$397$	−27.8167	−1.39608	−0.698039	+	0.716060i	$0.745944\pi$
$398$	0	0
$399$	2.78890	0.139620
$400$	0	0
$401$	13.8167	0.689971	0.344985	−	0.938608i	$-0.387884\pi$
$401$	13.8167	0.689971	0.344985	+	0.938608i	$0.387884\pi$
$402$	0	0
$403$	−7.60555	−0.378859
$404$	0	0
$405$	−10.6611	−0.529753
$406$	0	0
$407$	−1.30278	−0.0645762
$408$	0	0
$409$	−5.02776	−0.248607	−0.124303	−	0.992244i	$-0.539670\pi$
$409$	−5.02776	−0.248607	−0.124303	+	0.992244i	$0.539670\pi$
$410$	0	0
$411$	3.00000	0.147979
$412$	0	0
$413$	−15.6333	−0.769265
$414$	0	0
$415$	−22.4222	−1.10066
$416$	0	0
$417$	−2.69722	−0.132084
$418$	0	0
$419$	−25.1472	−1.22852	−0.614260	−	0.789104i	$-0.710545\pi$
$419$	−25.1472	−1.22852	−0.614260	+	0.789104i	$0.710545\pi$
$420$	0	0
$421$	−28.7250	−1.39997	−0.699985	−	0.714158i	$-0.746810\pi$
$421$	−28.7250	−1.39997	−0.699985	+	0.714158i	$0.746810\pi$
$422$	0	0
$423$	−7.57779	−0.368445
$424$	0	0
$425$	19.8167	0.961249
$426$	0	0
$427$	−48.4222	−2.34331
$428$	0	0
$429$	−0.908327	−0.0438544
$430$	0	0
$431$	5.21110	0.251010	0.125505	−	0.992093i	$-0.459945\pi$
$431$	5.21110	0.251010	0.125505	+	0.992093i	$0.459945\pi$
$432$	0	0
$433$	−11.9361	−0.573612	−0.286806	−	0.957989i	$-0.592593\pi$
$433$	−11.9361	−0.573612	−0.286806	+	0.957989i	$0.592593\pi$
$434$	0	0
$435$	−2.72498	−0.130653
$436$	0	0
$437$	13.8167	0.660940
$438$	0	0
$439$	9.33053	0.445322	0.222661	−	0.974896i	$-0.428526\pi$
$439$	9.33053	0.445322	0.222661	+	0.974896i	$0.428526\pi$
$440$	0	0
$441$	−41.3305	−1.96812
$442$	0	0
$443$	0.275019	0.0130666	0.00653328	−	0.999979i	$-0.497920\pi$
$443$	0.275019	0.0130666	0.00653328	+	0.999979i	$0.497920\pi$
$444$	0	0
$445$	−6.78890	−0.321825
$446$	0	0
$447$	−0.550039	−0.0260159
$448$	0	0
$449$	−0.788897	−0.0372304	−0.0186152	−	0.999827i	$-0.505926\pi$
$449$	−0.788897	−0.0372304	−0.0186152	+	0.999827i	$0.505926\pi$
$450$	0	0
$451$	−1.18335	−0.0557216
$452$	0	0
$453$	−4.05551	−0.190545
$454$	0	0
$455$	13.8167	0.647735
$456$	0	0
$457$	4.60555	0.215439	0.107719	−	0.994181i	$-0.465645\pi$
$457$	4.60555	0.215439	0.107719	+	0.994181i	$0.465645\pi$
$458$	0	0
$459$	−10.7334	−0.500991
$460$	0	0
$461$	16.4222	0.764858	0.382429	−	0.923985i	$-0.375088\pi$
$461$	16.4222	0.764858	0.382429	+	0.923985i	$0.375088\pi$
$462$	0	0
$463$	−30.3028	−1.40829	−0.704145	−	0.710056i	$-0.748669\pi$
$463$	−30.3028	−1.40829	−0.704145	+	0.710056i	$0.748669\pi$
$464$	0	0
$465$	−1.30278	−0.0604148
$466$	0	0
$467$	0	0		−	1.00000i	$-0.5\pi$
$467$	0	0			1.00000i	$0.5\pi$
$468$	0	0
$469$	−66.8444	−3.08659
$470$	0	0
$471$	2.18335	0.100603
$472$	0	0
$473$	−8.60555	−0.395684
$474$	0	0
$475$	−6.60555	−0.303083
$476$	0	0
$477$	−17.4500	−0.798979
$478$	0	0
$479$	−12.1194	−0.553751	−0.276875	−	0.960906i	$-0.589299\pi$
$479$	−12.1194	−0.553751	−0.276875	+	0.960906i	$0.589299\pi$
$480$	0	0
$481$	−2.30278	−0.104998
$482$	0	0
$483$	9.63331	0.438331
$484$	0	0
$485$	−16.1833	−0.734848
$486$	0	0
$487$	22.7889	1.03266	0.516332	−	0.856389i	$-0.327297\pi$
$487$	22.7889	1.03266	0.516332	+	0.856389i	$0.327297\pi$
$488$	0	0
$489$	6.18335	0.279621
$490$	0	0
$491$	−14.7250	−0.664529	−0.332265	−	0.943186i	$-0.607813\pi$
$491$	−14.7250	−0.664529	−0.332265	+	0.943186i	$0.607813\pi$
$492$	0	0
$493$	41.4500	1.86681
$494$	0	0
$495$	4.93608	0.221860
$496$	0	0
$497$	27.6333	1.23952
$498$	0	0
$499$	8.23886	0.368822	0.184411	−	0.982849i	$-0.440962\pi$
$499$	8.23886	0.368822	0.184411	+	0.982849i	$0.440962\pi$
$500$	0	0
$501$	3.78890	0.169275
$502$	0	0
$503$	24.5139	1.09302	0.546510	−	0.837453i	$-0.315956\pi$
$503$	24.5139	1.09302	0.546510	+	0.837453i	$0.315956\pi$
$504$	0	0
$505$	21.3944	0.952040
$506$	0	0
$507$	2.33053	0.103503
$508$	0	0
$509$	25.8167	1.14430	0.572152	−	0.820148i	$-0.306109\pi$
$509$	25.8167	1.14430	0.572152	+	0.820148i	$0.306109\pi$
$510$	0	0
$511$	40.0555	1.77195
$512$	0	0
$513$	3.57779	0.157964
$514$	0	0
$515$	4.30278	0.189603
$516$	0	0
$517$	3.39445	0.149288
$518$	0	0
$519$	−7.02776	−0.308484
$520$	0	0
$521$	−9.63331	−0.422043	−0.211021	−	0.977481i	$-0.567679\pi$
$521$	−9.63331	−0.422043	−0.211021	+	0.977481i	$0.567679\pi$
$522$	0	0
$523$	32.2389	1.40971	0.704853	−	0.709353i	$-0.251013\pi$
$523$	32.2389	1.40971	0.704853	+	0.709353i	$0.251013\pi$
$524$	0	0
$525$	−4.60555	−0.201003
$526$	0	0
$527$	19.8167	0.863227
$528$	0	0
$529$	24.7250	1.07500
$530$	0	0
$531$	−9.87217	−0.428416
$532$	0	0
$533$	−2.09167	−0.0906004
$534$	0	0
$535$	−5.60555	−0.242349
$536$	0	0
$537$	−2.36669	−0.102130
$538$	0	0
$539$	18.5139	0.797449
$540$	0	0
$541$	20.9361	0.900113	0.450056	−	0.893000i	$-0.351404\pi$
$541$	20.9361	0.900113	0.450056	+	0.893000i	$0.351404\pi$
$542$	0	0
$543$	6.05551	0.259867
$544$	0	0
$545$	2.60555	0.111610
$546$	0	0
$547$	−13.3944	−0.572705	−0.286353	−	0.958124i	$-0.592443\pi$
$547$	−13.3944	−0.572705	−0.286353	+	0.958124i	$0.592443\pi$
$548$	0	0
$549$	−30.5778	−1.30503
$550$	0	0
$551$	−13.8167	−0.588609
$552$	0	0
$553$	−74.2389	−3.15696
$554$	0	0
$555$	−0.394449	−0.0167434
$556$	0	0
$557$	6.51388	0.276002	0.138001	−	0.990432i	$-0.455932\pi$
$557$	6.51388	0.276002	0.138001	+	0.990432i	$0.455932\pi$
$558$	0	0
$559$	−15.2111	−0.643361
$560$	0	0
$561$	2.36669	0.0999218
$562$	0	0
$563$	44.0555	1.85672	0.928359	−	0.371684i	$-0.121220\pi$
$563$	44.0555	1.85672	0.928359	+	0.371684i	$0.121220\pi$
$564$	0	0
$565$	−14.6056	−0.614460
$566$	0	0
$567$	−37.6888	−1.58278
$568$	0	0
$569$	−10.4222	−0.436922	−0.218461	−	0.975846i	$-0.570104\pi$
$569$	−10.4222	−0.436922	−0.218461	+	0.975846i	$0.570104\pi$
$570$	0	0
$571$	−20.3028	−0.849645	−0.424822	−	0.905277i	$-0.639663\pi$
$571$	−20.3028	−0.849645	−0.424822	+	0.905277i	$0.639663\pi$
$572$	0	0
$573$	3.78890	0.158283
$574$	0	0
$575$	−22.8167	−0.951520
$576$	0	0
$577$	−28.2389	−1.17560	−0.587800	−	0.809007i	$-0.700006\pi$
$577$	−28.2389	−1.17560	−0.587800	+	0.809007i	$0.700006\pi$
$578$	0	0
$579$	1.21110	0.0503317
$580$	0	0
$581$	−79.2666	−3.28853
$582$	0	0
$583$	7.81665	0.323733
$584$	0	0
$585$	8.72498	0.360734
$586$	0	0
$587$	−2.36669	−0.0976838	−0.0488419	−	0.998807i	$-0.515553\pi$
$587$	−2.36669	−0.0976838	−0.0488419	+	0.998807i	$0.515553\pi$
$588$	0	0
$589$	−6.60555	−0.272177
$590$	0	0
$591$	−1.81665	−0.0747272
$592$	0	0
$593$	36.5139	1.49945	0.749723	−	0.661752i	$-0.230187\pi$
$593$	36.5139	1.49945	0.749723	+	0.661752i	$0.230187\pi$
$594$	0	0
$595$	−36.0000	−1.47586
$596$	0	0
$597$	−0.733385	−0.0300154
$598$	0	0
$599$	35.2111	1.43869	0.719343	−	0.694655i	$-0.244443\pi$
$599$	35.2111	1.43869	0.719343	+	0.694655i	$0.244443\pi$
$600$	0	0
$601$	−20.6972	−0.844257	−0.422129	−	0.906536i	$-0.638717\pi$
$601$	−20.6972	−0.844257	−0.422129	+	0.906536i	$0.638717\pi$
$602$	0	0
$603$	−42.2111	−1.71897
$604$	0	0
$605$	12.1194	0.492725
$606$	0	0
$607$	31.5139	1.27911	0.639554	−	0.768746i	$-0.279119\pi$
$607$	31.5139	1.27911	0.639554	+	0.768746i	$0.279119\pi$
$608$	0	0
$609$	−9.63331	−0.390361
$610$	0	0
$611$	6.00000	0.242734
$612$	0	0
$613$	8.18335	0.330522	0.165261	−	0.986250i	$-0.447153\pi$
$613$	8.18335	0.330522	0.165261	+	0.986250i	$0.447153\pi$
$614$	0	0
$615$	−0.358288	−0.0144476
$616$	0	0
$617$	47.5694	1.91507	0.957536	−	0.288314i	$-0.0930948\pi$
$617$	47.5694	1.91507	0.957536	+	0.288314i	$0.0930948\pi$
$618$	0	0
$619$	−2.69722	−0.108411	−0.0542053	−	0.998530i	$-0.517263\pi$
$619$	−2.69722	−0.108411	−0.0542053	+	0.998530i	$0.517263\pi$
$620$	0	0
$621$	12.3583	0.495921
$622$	0	0
$623$	−24.0000	−0.961540
$624$	0	0
$625$	2.42221	0.0968882
$626$	0	0
$627$	−0.788897	−0.0315055
$628$	0	0
$629$	6.00000	0.239236
$630$	0	0
$631$	−18.3028	−0.728622	−0.364311	−	0.931277i	$-0.618695\pi$
$631$	−18.3028	−0.728622	−0.364311	+	0.931277i	$0.618695\pi$
$632$	0	0
$633$	−2.02776	−0.0805961
$634$	0	0
$635$	−6.23886	−0.247582
$636$	0	0
$637$	32.7250	1.29661
$638$	0	0
$639$	17.4500	0.690310
$640$	0	0
$641$	−2.48612	−0.0981959	−0.0490980	−	0.998794i	$-0.515635\pi$
$641$	−2.48612	−0.0981959	−0.0490980	+	0.998794i	$0.515635\pi$
$642$	0	0
$643$	−29.8167	−1.17585	−0.587927	−	0.808914i	$-0.700056\pi$
$643$	−29.8167	−1.17585	−0.587927	+	0.808914i	$0.700056\pi$
$644$	0	0
$645$	−2.60555	−0.102593
$646$	0	0
$647$	25.9361	1.01965	0.509826	−	0.860277i	$-0.329710\pi$
$647$	25.9361	1.01965	0.509826	+	0.860277i	$0.329710\pi$
$648$	0	0
$649$	4.42221	0.173587
$650$	0	0
$651$	−4.60555	−0.180506
$652$	0	0
$653$	−6.90833	−0.270344	−0.135172	−	0.990822i	$-0.543159\pi$
$653$	−6.90833	−0.270344	−0.135172	+	0.990822i	$0.543159\pi$
$654$	0	0
$655$	−4.42221	−0.172790
$656$	0	0
$657$	25.2944	0.986827
$658$	0	0
$659$	−42.1194	−1.64074	−0.820370	−	0.571833i	$-0.806233\pi$
$659$	−42.1194	−1.64074	−0.820370	+	0.571833i	$0.806233\pi$
$660$	0	0
$661$	12.4861	0.485654	0.242827	−	0.970070i	$-0.421925\pi$
$661$	12.4861	0.485654	0.242827	+	0.970070i	$0.421925\pi$
$662$	0	0
$663$	4.18335	0.162468
$664$	0	0
$665$	12.0000	0.465340
$666$	0	0
$667$	−47.7250	−1.84792
$668$	0	0
$669$	4.78890	0.185149
$670$	0	0
$671$	13.6972	0.528775
$672$	0	0
$673$	24.3028	0.936803	0.468402	−	0.883516i	$-0.344830\pi$
$673$	24.3028	0.936803	0.468402	+	0.883516i	$0.344830\pi$
$674$	0	0
$675$	−5.90833	−0.227412
$676$	0	0
$677$	36.2389	1.39277	0.696386	−	0.717667i	$-0.254790\pi$
$677$	36.2389	1.39277	0.696386	+	0.717667i	$0.254790\pi$
$678$	0	0
$679$	−57.2111	−2.19556
$680$	0	0
$681$	2.36669	0.0906918
$682$	0	0
$683$	12.0000	0.459167	0.229584	−	0.973289i	$-0.426264\pi$
$683$	12.0000	0.459167	0.229584	+	0.973289i	$0.426264\pi$
$684$	0	0
$685$	12.9083	0.493202
$686$	0	0
$687$	5.26662	0.200934
$688$	0	0
$689$	13.8167	0.526373
$690$	0	0
$691$	8.00000	0.304334	0.152167	−	0.988355i	$-0.451375\pi$
$691$	8.00000	0.304334	0.152167	+	0.988355i	$0.451375\pi$
$692$	0	0
$693$	17.4500	0.662869
$694$	0	0
$695$	−11.6056	−0.440224
$696$	0	0
$697$	5.44996	0.206432
$698$	0	0
$699$	2.88057	0.108953
$700$	0	0
$701$	−14.8806	−0.562031	−0.281016	−	0.959703i	$-0.590671\pi$
$701$	−14.8806	−0.562031	−0.281016	+	0.959703i	$0.590671\pi$
$702$	0	0
$703$	−2.00000	−0.0754314
$704$	0	0
$705$	1.02776	0.0387075
$706$	0	0
$707$	75.6333	2.84448
$708$	0	0
$709$	1.66947	0.0626982	0.0313491	−	0.999508i	$-0.490020\pi$
$709$	1.66947	0.0626982	0.0313491	+	0.999508i	$0.490020\pi$
$710$	0	0
$711$	−46.8806	−1.75816
$712$	0	0
$713$	−22.8167	−0.854490
$714$	0	0
$715$	−3.90833	−0.146163
$716$	0	0
$717$	0.155590	0.00581061
$718$	0	0
$719$	8.36669	0.312025	0.156012	−	0.987755i	$-0.450136\pi$
$719$	8.36669	0.312025	0.156012	+	0.987755i	$0.450136\pi$
$720$	0	0
$721$	15.2111	0.566491
$722$	0	0
$723$	−2.42221	−0.0900828
$724$	0	0
$725$	22.8167	0.847389
$726$	0	0
$727$	−29.9083	−1.10924	−0.554619	−	0.832104i	$-0.687136\pi$
$727$	−29.9083	−1.10924	−0.554619	+	0.832104i	$0.687136\pi$
$728$	0	0
$729$	−22.1749	−0.821294
$730$	0	0
$731$	39.6333	1.46589
$732$	0	0
$733$	−29.6333	−1.09453	−0.547266	−	0.836959i	$-0.684331\pi$
$733$	−29.6333	−1.09453	−0.547266	+	0.836959i	$0.684331\pi$
$734$	0	0
$735$	5.60555	0.206764
$736$	0	0
$737$	18.9083	0.696497
$738$	0	0
$739$	−42.3305	−1.55715	−0.778577	−	0.627549i	$-0.784058\pi$
$739$	−42.3305	−1.55715	−0.778577	+	0.627549i	$0.784058\pi$
$740$	0	0
$741$	−1.39445	−0.0512264
$742$	0	0
$743$	−35.4500	−1.30053	−0.650266	−	0.759706i	$-0.725343\pi$
$743$	−35.4500	−1.30053	−0.650266	+	0.759706i	$0.725343\pi$
$744$	0	0
$745$	−2.36669	−0.0867089
$746$	0	0
$747$	−50.0555	−1.83144
$748$	0	0
$749$	−19.8167	−0.724085
$750$	0	0
$751$	−14.0000	−0.510867	−0.255434	−	0.966827i	$-0.582218\pi$
$751$	−14.0000	−0.510867	−0.255434	+	0.966827i	$0.582218\pi$
$752$	0	0
$753$	2.05551	0.0749070
$754$	0	0
$755$	−17.4500	−0.635069
$756$	0	0
$757$	−9.30278	−0.338115	−0.169058	−	0.985606i	$-0.554072\pi$
$757$	−9.30278	−0.338115	−0.169058	+	0.985606i	$0.554072\pi$
$758$	0	0
$759$	−2.72498	−0.0989105
$760$	0	0
$761$	42.1194	1.52683	0.763414	−	0.645909i	$-0.223522\pi$
$761$	42.1194	1.52683	0.763414	+	0.645909i	$0.223522\pi$
$762$	0	0
$763$	9.21110	0.333464
$764$	0	0
$765$	−22.7334	−0.821927
$766$	0	0
$767$	7.81665	0.282243
$768$	0	0
$769$	−22.0000	−0.793340	−0.396670	−	0.917961i	$-0.629834\pi$
$769$	−22.0000	−0.793340	−0.396670	+	0.917961i	$0.629834\pi$
$770$	0	0
$771$	3.39445	0.122248
$772$	0	0
$773$	50.0555	1.80037	0.900186	−	0.435506i	$-0.143431\pi$
$773$	50.0555	1.80037	0.900186	+	0.435506i	$0.143431\pi$
$774$	0	0
$775$	10.9083	0.391839
$776$	0	0
$777$	−1.39445	−0.0500256
$778$	0	0
$779$	−1.81665	−0.0650884
$780$	0	0
$781$	−7.81665	−0.279702
$782$	0	0
$783$	−12.3583	−0.441649
$784$	0	0
$785$	9.39445	0.335302
$786$	0	0
$787$	25.2111	0.898679	0.449339	−	0.893361i	$-0.351659\pi$
$787$	25.2111	0.898679	0.449339	+	0.893361i	$0.351659\pi$
$788$	0	0
$789$	−2.36669	−0.0842565
$790$	0	0
$791$	−51.6333	−1.83587
$792$	0	0
$793$	24.2111	0.859761
$794$	0	0
$795$	2.36669	0.0839379
$796$	0	0
$797$	17.3305	0.613879	0.306939	−	0.951729i	$-0.400695\pi$
$797$	17.3305	0.613879	0.306939	+	0.951729i	$0.400695\pi$
$798$	0	0
$799$	−15.6333	−0.553067
$800$	0	0
$801$	−15.1556	−0.535496
$802$	0	0
$803$	−11.3305	−0.399846
$804$	0	0
$805$	41.4500	1.46092
$806$	0	0
$807$	−2.05551	−0.0723575
$808$	0	0
$809$	29.4500	1.03541	0.517703	−	0.855561i	$-0.326787\pi$
$809$	29.4500	1.03541	0.517703	+	0.855561i	$0.326787\pi$
$810$	0	0
$811$	54.1472	1.90136	0.950682	−	0.310166i	$-0.100385\pi$
$811$	54.1472	1.90136	0.950682	+	0.310166i	$0.100385\pi$
$812$	0	0
$813$	1.94449	0.0681961
$814$	0	0
$815$	26.6056	0.931952
$816$	0	0
$817$	−13.2111	−0.462198
$818$	0	0
$819$	30.8444	1.07779
$820$	0	0
$821$	−11.2111	−0.391270	−0.195635	−	0.980677i	$-0.562677\pi$
$821$	−11.2111	−0.391270	−0.195635	+	0.980677i	$0.562677\pi$
$822$	0	0
$823$	12.8444	0.447728	0.223864	−	0.974620i	$-0.428133\pi$
$823$	12.8444	0.447728	0.223864	+	0.974620i	$0.428133\pi$
$824$	0	0
$825$	1.30278	0.0453568
$826$	0	0
$827$	27.3944	0.952598	0.476299	−	0.879283i	$-0.341978\pi$
$827$	27.3944	0.952598	0.476299	+	0.879283i	$0.341978\pi$
$828$	0	0
$829$	−4.72498	−0.164105	−0.0820527	−	0.996628i	$-0.526148\pi$
$829$	−4.72498	−0.164105	−0.0820527	+	0.996628i	$0.526148\pi$
$830$	0	0
$831$	−7.60555	−0.263834
$832$	0	0
$833$	−85.2666	−2.95431
$834$	0	0
$835$	16.3028	0.564181
$836$	0	0
$837$	−5.90833	−0.204222
$838$	0	0
$839$	49.0278	1.69263	0.846313	−	0.532686i	$-0.178817\pi$
$839$	49.0278	1.69263	0.846313	+	0.532686i	$0.178817\pi$
$840$	0	0
$841$	18.7250	0.645689
$842$	0	0
$843$	3.63331	0.125138
$844$	0	0
$845$	10.0278	0.344965
$846$	0	0
$847$	42.8444	1.47215
$848$	0	0
$849$	−5.26662	−0.180750
$850$	0	0
$851$	−6.90833	−0.236814
$852$	0	0
$853$	11.5416	0.395178	0.197589	−	0.980285i	$-0.436689\pi$
$853$	11.5416	0.395178	0.197589	+	0.980285i	$0.436689\pi$
$854$	0	0
$855$	7.57779	0.259155
$856$	0	0
$857$	−14.8444	−0.507075	−0.253538	−	0.967326i	$-0.581594\pi$
$857$	−14.8444	−0.507075	−0.253538	+	0.967326i	$0.581594\pi$
$858$	0	0
$859$	−24.0555	−0.820764	−0.410382	−	0.911914i	$-0.634605\pi$
$859$	−24.0555	−0.820764	−0.410382	+	0.911914i	$0.634605\pi$
$860$	0	0
$861$	−1.26662	−0.0431661
$862$	0	0
$863$	−12.0000	−0.408485	−0.204242	−	0.978920i	$-0.565473\pi$
$863$	−12.0000	−0.408485	−0.204242	+	0.978920i	$0.565473\pi$
$864$	0	0
$865$	−30.2389	−1.02815
$866$	0	0
$867$	−5.75274	−0.195373
$868$	0	0
$869$	21.0000	0.712376
$870$	0	0
$871$	33.4222	1.13247
$872$	0	0
$873$	−36.1278	−1.22274
$874$	0	0
$875$	−49.8167	−1.68411
$876$	0	0
$877$	−7.21110	−0.243502	−0.121751	−	0.992561i	$-0.538851\pi$
$877$	−7.21110	−0.243502	−0.121751	+	0.992561i	$0.538851\pi$
$878$	0	0
$879$	−7.57779	−0.255593
$880$	0	0
$881$	25.5416	0.860520	0.430260	−	0.902705i	$-0.358422\pi$
$881$	25.5416	0.860520	0.430260	+	0.902705i	$0.358422\pi$
$882$	0	0
$883$	−2.42221	−0.0815137	−0.0407568	−	0.999169i	$-0.512977\pi$
$883$	−2.42221	−0.0815137	−0.0407568	+	0.999169i	$0.512977\pi$
$884$	0	0
$885$	1.33894	0.0450078
$886$	0	0
$887$	−28.4222	−0.954324	−0.477162	−	0.878815i	$-0.658335\pi$
$887$	−28.4222	−0.954324	−0.477162	+	0.878815i	$0.658335\pi$
$888$	0	0
$889$	−22.0555	−0.739718
$890$	0	0
$891$	10.6611	0.357159
$892$	0	0
$893$	5.21110	0.174383
$894$	0	0
$895$	−10.1833	−0.340392
$896$	0	0
$897$	−4.81665	−0.160823
$898$	0	0
$899$	22.8167	0.760978
$900$	0	0
$901$	−36.0000	−1.19933
$902$	0	0
$903$	−9.21110	−0.306526
$904$	0	0
$905$	26.0555	0.866115
$906$	0	0
$907$	26.0000	0.863316	0.431658	−	0.902037i	$-0.357929\pi$
$907$	26.0000	0.863316	0.431658	+	0.902037i	$0.357929\pi$
$908$	0	0
$909$	47.7611	1.58414
$910$	0	0
$911$	46.4222	1.53804	0.769018	−	0.639227i	$-0.220746\pi$
$911$	46.4222	1.53804	0.769018	+	0.639227i	$0.220746\pi$
$912$	0	0
$913$	22.4222	0.742067
$914$	0	0
$915$	4.14719	0.137102
$916$	0	0
$917$	−15.6333	−0.516257
$918$	0	0
$919$	38.4222	1.26743	0.633716	−	0.773566i	$-0.281529\pi$
$919$	38.4222	1.26743	0.633716	+	0.773566i	$0.281529\pi$
$920$	0	0
$921$	−2.14719	−0.0707522
$922$	0	0
$923$	−13.8167	−0.454781
$924$	0	0
$925$	3.30278	0.108595
$926$	0	0
$927$	9.60555	0.315488
$928$	0	0
$929$	−36.5139	−1.19798	−0.598991	−	0.800756i	$-0.704431\pi$
$929$	−36.5139	−1.19798	−0.598991	+	0.800756i	$0.704431\pi$
$930$	0	0
$931$	28.4222	0.931500
$932$	0	0
$933$	1.54163	0.0504708
$934$	0	0
$935$	10.1833	0.333031
$936$	0	0
$937$	−28.9083	−0.944394	−0.472197	−	0.881493i	$-0.656539\pi$
$937$	−28.9083	−0.944394	−0.472197	+	0.881493i	$0.656539\pi$
$938$	0	0
$939$	−8.18335	−0.267053
$940$	0	0
$941$	7.81665	0.254816	0.127408	−	0.991850i	$-0.459334\pi$
$941$	7.81665	0.254816	0.127408	+	0.991850i	$0.459334\pi$
$942$	0	0
$943$	−6.27502	−0.204343
$944$	0	0
$945$	10.7334	0.349157
$946$	0	0
$947$	39.6333	1.28791	0.643955	−	0.765064i	$-0.277293\pi$
$947$	39.6333	1.28791	0.643955	+	0.765064i	$0.277293\pi$
$948$	0	0
$949$	−20.0278	−0.650128
$950$	0	0
$951$	−1.57779	−0.0511635
$952$	0	0
$953$	−18.7527	−0.607461	−0.303730	−	0.952758i	$-0.598232\pi$
$953$	−18.7527	−0.607461	−0.303730	+	0.952758i	$0.598232\pi$
$954$	0	0
$955$	16.3028	0.527545
$956$	0	0
$957$	2.72498	0.0880861
$958$	0	0
$959$	45.6333	1.47358
$960$	0	0
$961$	−20.0917	−0.648118
$962$	0	0
$963$	−12.5139	−0.403254
$964$	0	0
$965$	5.21110	0.167751
$966$	0	0
$967$	−25.7250	−0.827260	−0.413630	−	0.910445i	$-0.635739\pi$
$967$	−25.7250	−0.827260	−0.413630	+	0.910445i	$0.635739\pi$
$968$	0	0
$969$	3.63331	0.116719
$970$	0	0
$971$	31.5416	1.01222	0.506110	−	0.862469i	$-0.331083\pi$
$971$	31.5416	1.01222	0.506110	+	0.862469i	$0.331083\pi$
$972$	0	0
$973$	−41.0278	−1.31529
$974$	0	0
$975$	2.30278	0.0737478
$976$	0	0
$977$	18.0000	0.575871	0.287936	−	0.957650i	$-0.407031\pi$
$977$	18.0000	0.575871	0.287936	+	0.957650i	$0.407031\pi$
$978$	0	0
$979$	6.78890	0.216974
$980$	0	0
$981$	5.81665	0.185711
$982$	0	0
$983$	12.0000	0.382741	0.191370	−	0.981518i	$-0.438707\pi$
$983$	12.0000	0.382741	0.191370	+	0.981518i	$0.438707\pi$
$984$	0	0
$985$	−7.81665	−0.249059
$986$	0	0
$987$	3.63331	0.115649
$988$	0	0
$989$	−45.6333	−1.45105
$990$	0	0
$991$	−54.3028	−1.72498	−0.862492	−	0.506070i	$-0.831098\pi$
$991$	−54.3028	−1.72498	−0.862492	+	0.506070i	$0.831098\pi$
$992$	0	0
$993$	−0.366692	−0.0116366
$994$	0	0
$995$	−3.15559	−0.100039
$996$	0	0
$997$	23.5778	0.746716	0.373358	−	0.927687i	$-0.378206\pi$
$997$	23.5778	0.746716	0.373358	+	0.927687i	$0.378206\pi$
$998$	0	0
$999$	−1.78890	−0.0565982

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2368.2.a.ba.1.1		2
4.3	odd	2		2368.2.a.s.1.2		2
8.3	odd	2		74.2.a.a.1.1	✓	2
8.5	even	2		592.2.a.f.1.2		2
24.5	odd	2		5328.2.a.bf.1.1		2
24.11	even	2		666.2.a.j.1.1		2
40.3	even	4		1850.2.b.i.149.3		4
40.19	odd	2		1850.2.a.u.1.2		2
40.27	even	4		1850.2.b.i.149.2		4
56.27	even	2		3626.2.a.a.1.2		2
88.43	even	2		8954.2.a.p.1.1		2
296.147	odd	2		2738.2.a.l.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
74.2.a.a.1.1	✓	2	8.3	odd	2
592.2.a.f.1.2		2	8.5	even	2
666.2.a.j.1.1		2	24.11	even	2
1850.2.a.u.1.2		2	40.19	odd	2
1850.2.b.i.149.2		4	40.27	even	4
1850.2.b.i.149.3		4	40.3	even	4
2368.2.a.s.1.2		2	4.3	odd	2
2368.2.a.ba.1.1		2	1.1	even	1	trivial
2738.2.a.l.1.1		2	296.147	odd	2
3626.2.a.a.1.2		2	56.27	even	2
5328.2.a.bf.1.1		2	24.5	odd	2
8954.2.a.p.1.1		2	88.43	even	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:	\( N \)	\(=\)	\( 2368 = 2^{6} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2368.a (trivial)

\(f(q)\)	\(=\)	\(q-0.302776 q^{3} -1.30278 q^{5} -4.60555 q^{7} -2.90833 q^{9} +1.30278 q^{11} +2.30278 q^{13} +0.394449 q^{15} -6.00000 q^{17} +2.00000 q^{19} +1.39445 q^{21} +6.90833 q^{23} -3.30278 q^{25} +1.78890 q^{27} -6.90833 q^{29} -3.30278 q^{31} -0.394449 q^{33} +6.00000 q^{35} -1.00000 q^{37} -0.697224 q^{39} -0.908327 q^{41} -6.60555 q^{43} +3.78890 q^{45} +2.60555 q^{47} +14.2111 q^{49} +1.81665 q^{51} +6.00000 q^{53} -1.69722 q^{55} -0.605551 q^{57} +3.39445 q^{59} +10.5139 q^{61} +13.3944 q^{63} -3.00000 q^{65} +14.5139 q^{67} -2.09167 q^{69} -6.00000 q^{71} -8.69722 q^{73} +1.00000 q^{75} -6.00000 q^{77} +16.1194 q^{79} +8.18335 q^{81} +17.2111 q^{83} +7.81665 q^{85} +2.09167 q^{87} +5.21110 q^{89} -10.6056 q^{91} +1.00000 q^{93} -2.60555 q^{95} +12.4222 q^{97} -3.78890 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 3 q^{3} + q^{5} - 2 q^{7} + 5 q^{9} - q^{11} + q^{13} + 8 q^{15} - 12 q^{17} + 4 q^{19} + 10 q^{21} + 3 q^{23} - 3 q^{25} + 18 q^{27} - 3 q^{29} - 3 q^{31} - 8 q^{33} + 12 q^{35} - 2 q^{37} - 5 q^{39}+ \cdots - 22 q^{99}+O(q^{100}) \) 2 * q + 3 * q^3 + q^5 - 2 * q^7 + 5 * q^9 - q^11 + q^13 + 8 * q^15 - 12 * q^17 + 4 * q^19 + 10 * q^21 + 3 * q^23 - 3 * q^25 + 18 * q^27 - 3 * q^29 - 3 * q^31 - 8 * q^33 + 12 * q^35 - 2 * q^37 - 5 * q^39 + 9 * q^41 - 6 * q^43 + 22 * q^45 - 2 * q^47 + 14 * q^49 - 18 * q^51 + 12 * q^53 - 7 * q^55 + 6 * q^57 + 14 * q^59 + 3 * q^61 + 34 * q^63 - 6 * q^65 + 11 * q^67 - 15 * q^69 - 12 * q^71 - 21 * q^73 + 2 * q^75 - 12 * q^77 + 7 * q^79 + 38 * q^81 + 20 * q^83 - 6 * q^85 + 15 * q^87 - 4 * q^89 - 14 * q^91 + 2 * q^93 + 2 * q^95 - 4 * q^97 - 22 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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