LMFDB - Embedded newform 2376.2.a.h.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2376,2,Mod(1,2376)] 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("2376.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2376, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2376 = 2^{3} \cdot 3^{3} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2376.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,0,0,-2,0,0,0,2,0,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	yes
Analytic conductor:	$18.9724555203$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	2376.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.41421 q^{5} +0.414214 q^{7} +1.00000 q^{11} -2.41421 q^{13} -6.24264 q^{17} -5.24264 q^{19} -5.65685 q^{23} -3.00000 q^{25} -0.585786 q^{29} +3.65685 q^{31} +0.585786 q^{35} -3.82843 q^{37} -2.82843 q^{41} -11.6569 q^{43} +8.24264 q^{47} -6.82843 q^{49} +5.89949 q^{53} +1.41421 q^{55} +2.58579 q^{59} -7.58579 q^{61} -3.41421 q^{65} +8.31371 q^{67} +4.00000 q^{71} +10.8995 q^{73} +0.414214 q^{77} +0.0710678 q^{79} +9.65685 q^{83} -8.82843 q^{85} +0.242641 q^{89} -1.00000 q^{91} -7.41421 q^{95} -14.6569 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{7} + 2 q^{11} - 2 q^{13} - 4 q^{17} - 2 q^{19} - 6 q^{25} - 4 q^{29} - 4 q^{31} + 4 q^{35} - 2 q^{37} - 12 q^{43} + 8 q^{47} - 8 q^{49} - 8 q^{53} + 8 q^{59} - 18 q^{61} - 4 q^{65} - 6 q^{67}+ \cdots - 18 q^{97}+O(q^{100}) $ 2 * q - 2 * q^7 + 2 * q^11 - 2 * q^13 - 4 * q^17 - 2 * q^19 - 6 * q^25 - 4 * q^29 - 4 * q^31 + 4 * q^35 - 2 * q^37 - 12 * q^43 + 8 * q^47 - 8 * q^49 - 8 * q^53 + 8 * q^59 - 18 * q^61 - 4 * q^65 - 6 * q^67 + 8 * q^71 + 2 * q^73 - 2 * q^77 - 14 * q^79 + 8 * q^83 - 12 * q^85 - 8 * q^89 - 2 * q^91 - 12 * q^95 - 18 * q^97

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	0
$3$	0	0
$4$	0	0
$5$	1.41421	0.632456	0.316228	−	0.948683i	$-0.397584\pi$
$5$	1.41421	0.632456	0.316228	+	0.948683i	$0.397584\pi$
$6$	0	0
$7$	0.414214	0.156558	0.0782790	−	0.996931i	$-0.475058\pi$
$7$	0.414214	0.156558	0.0782790	+	0.996931i	$0.475058\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−2.41421	−0.669582	−0.334791	−	0.942292i	$-0.608666\pi$
$13$	−2.41421	−0.669582	−0.334791	+	0.942292i	$0.608666\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−6.24264	−1.51406	−0.757031	−	0.653379i	$-0.773351\pi$
$17$	−6.24264	−1.51406	−0.757031	+	0.653379i	$0.773351\pi$
$18$	0	0
$19$	−5.24264	−1.20274	−0.601372	−	0.798969i	$-0.705379\pi$
$19$	−5.24264	−1.20274	−0.601372	+	0.798969i	$0.705379\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−5.65685	−1.17954	−0.589768	−	0.807573i	$-0.700781\pi$
$23$	−5.65685	−1.17954	−0.589768	+	0.807573i	$0.700781\pi$
$24$	0	0
$25$	−3.00000	−0.600000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−0.585786	−0.108778	−0.0543889	−	0.998520i	$-0.517321\pi$
$29$	−0.585786	−0.108778	−0.0543889	+	0.998520i	$0.517321\pi$
$30$	0	0
$31$	3.65685	0.656790	0.328395	−	0.944540i	$-0.393492\pi$
$31$	3.65685	0.656790	0.328395	+	0.944540i	$0.393492\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0.585786	0.0990160
$36$	0	0
$37$	−3.82843	−0.629390	−0.314695	−	0.949193i	$-0.601902\pi$
$37$	−3.82843	−0.629390	−0.314695	+	0.949193i	$0.601902\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−2.82843	−0.441726	−0.220863	−	0.975305i	$-0.570887\pi$
$41$	−2.82843	−0.441726	−0.220863	+	0.975305i	$0.570887\pi$
$42$	0	0
$43$	−11.6569	−1.77765	−0.888827	−	0.458243i	$-0.848479\pi$
$43$	−11.6569	−1.77765	−0.888827	+	0.458243i	$0.848479\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	8.24264	1.20231	0.601156	−	0.799131i	$-0.294707\pi$
$47$	8.24264	1.20231	0.601156	+	0.799131i	$0.294707\pi$
$48$	0	0
$49$	−6.82843	−0.975490
$50$	0	0
$51$	0	0
$52$	0	0
$53$	5.89949	0.810358	0.405179	−	0.914237i	$-0.367209\pi$
$53$	5.89949	0.810358	0.405179	+	0.914237i	$0.367209\pi$
$54$	0	0
$55$	1.41421	0.190693
$56$	0	0
$57$	0	0
$58$	0	0
$59$	2.58579	0.336641	0.168320	−	0.985732i	$-0.446166\pi$
$59$	2.58579	0.336641	0.168320	+	0.985732i	$0.446166\pi$
$60$	0	0
$61$	−7.58579	−0.971260	−0.485630	−	0.874164i	$-0.661410\pi$
$61$	−7.58579	−0.971260	−0.485630	+	0.874164i	$0.661410\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−3.41421	−0.423481
$66$	0	0
$67$	8.31371	1.01568	0.507841	−	0.861451i	$-0.330444\pi$
$67$	8.31371	1.01568	0.507841	+	0.861451i	$0.330444\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	4.00000	0.474713	0.237356	−	0.971423i	$-0.423719\pi$
$71$	4.00000	0.474713	0.237356	+	0.971423i	$0.423719\pi$
$72$	0	0
$73$	10.8995	1.27569	0.637845	−	0.770165i	$-0.279826\pi$
$73$	10.8995	1.27569	0.637845	+	0.770165i	$0.279826\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0.414214	0.0472040
$78$	0	0
$79$	0.0710678	0.00799575	0.00399788	−	0.999992i	$-0.498727\pi$
$79$	0.0710678	0.00799575	0.00399788	+	0.999992i	$0.498727\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	9.65685	1.05998	0.529989	−	0.848005i	$-0.322196\pi$
$83$	9.65685	1.05998	0.529989	+	0.848005i	$0.322196\pi$
$84$	0	0
$85$	−8.82843	−0.957577
$86$	0	0
$87$	0	0
$88$	0	0
$89$	0.242641	0.0257199	0.0128599	−	0.999917i	$-0.495906\pi$
$89$	0.242641	0.0257199	0.0128599	+	0.999917i	$0.495906\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−7.41421	−0.760682
$96$	0	0
$97$	−14.6569	−1.48818	−0.744089	−	0.668080i	$-0.767116\pi$
$97$	−14.6569	−1.48818	−0.744089	+	0.668080i	$0.767116\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−14.2426	−1.41720	−0.708598	−	0.705613i	$-0.750672\pi$
$101$	−14.2426	−1.41720	−0.708598	+	0.705613i	$0.750672\pi$
$102$	0	0
$103$	−11.4853	−1.13168	−0.565839	−	0.824516i	$-0.691448\pi$
$103$	−11.4853	−1.13168	−0.565839	+	0.824516i	$0.691448\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	14.2426	1.37689	0.688444	−	0.725289i	$-0.258294\pi$
$107$	14.2426	1.37689	0.688444	+	0.725289i	$0.258294\pi$
$108$	0	0
$109$	−7.65685	−0.733394	−0.366697	−	0.930341i	$-0.619511\pi$
$109$	−7.65685	−0.733394	−0.366697	+	0.930341i	$0.619511\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	5.41421	0.509326	0.254663	−	0.967030i	$-0.418035\pi$
$113$	5.41421	0.509326	0.254663	+	0.967030i	$0.418035\pi$
$114$	0	0
$115$	−8.00000	−0.746004
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−2.58579	−0.237039
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−11.3137	−1.01193
$126$	0	0
$127$	6.82843	0.605925	0.302962	−	0.953002i	$-0.402024\pi$
$127$	6.82843	0.605925	0.302962	+	0.953002i	$0.402024\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	6.24264	0.545422	0.272711	−	0.962096i	$-0.412080\pi$
$131$	6.24264	0.545422	0.272711	+	0.962096i	$0.412080\pi$
$132$	0	0
$133$	−2.17157	−0.188299
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−2.82843	−0.241649	−0.120824	−	0.992674i	$-0.538554\pi$
$137$	−2.82843	−0.241649	−0.120824	+	0.992674i	$0.538554\pi$
$138$	0	0
$139$	−22.5563	−1.91320	−0.956602	−	0.291397i	$-0.905880\pi$
$139$	−22.5563	−1.91320	−0.956602	+	0.291397i	$0.905880\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−2.41421	−0.201887
$144$	0	0
$145$	−0.828427	−0.0687971
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−19.3137	−1.58224	−0.791120	−	0.611661i	$-0.790502\pi$
$149$	−19.3137	−1.58224	−0.791120	+	0.611661i	$0.790502\pi$
$150$	0	0
$151$	−4.75736	−0.387148	−0.193574	−	0.981086i	$-0.562008\pi$
$151$	−4.75736	−0.387148	−0.193574	+	0.981086i	$0.562008\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	5.17157	0.415391
$156$	0	0
$157$	15.3137	1.22217	0.611083	−	0.791566i	$-0.290734\pi$
$157$	15.3137	1.22217	0.611083	+	0.791566i	$0.290734\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−2.34315	−0.184666
$162$	0	0
$163$	−3.48528	−0.272988	−0.136494	−	0.990641i	$-0.543583\pi$
$163$	−3.48528	−0.272988	−0.136494	+	0.990641i	$0.543583\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	19.2132	1.48676	0.743381	−	0.668868i	$-0.233221\pi$
$167$	19.2132	1.48676	0.743381	+	0.668868i	$0.233221\pi$
$168$	0	0
$169$	−7.17157	−0.551659
$170$	0	0
$171$	0	0
$172$	0	0
$173$	10.1421	0.771092	0.385546	−	0.922689i	$-0.374013\pi$
$173$	10.1421	0.771092	0.385546	+	0.922689i	$0.374013\pi$
$174$	0	0
$175$	−1.24264	−0.0939348
$176$	0	0
$177$	0	0
$178$	0	0
$179$	17.4142	1.30160	0.650800	−	0.759249i	$-0.274434\pi$
$179$	17.4142	1.30160	0.650800	+	0.759249i	$0.274434\pi$
$180$	0	0
$181$	19.9706	1.48440	0.742200	−	0.670178i	$-0.233782\pi$
$181$	19.9706	1.48440	0.742200	+	0.670178i	$0.233782\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−5.41421	−0.398061
$186$	0	0
$187$	−6.24264	−0.456507
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−3.31371	−0.239772	−0.119886	−	0.992788i	$-0.538253\pi$
$191$	−3.31371	−0.239772	−0.119886	+	0.992788i	$0.538253\pi$
$192$	0	0
$193$	−6.41421	−0.461705	−0.230853	−	0.972989i	$-0.574152\pi$
$193$	−6.41421	−0.461705	−0.230853	+	0.972989i	$0.574152\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−0.585786	−0.0417356	−0.0208678	−	0.999782i	$-0.506643\pi$
$197$	−0.585786	−0.0417356	−0.0208678	+	0.999782i	$0.506643\pi$
$198$	0	0
$199$	−10.6569	−0.755444	−0.377722	−	0.925919i	$-0.623293\pi$
$199$	−10.6569	−0.755444	−0.377722	+	0.925919i	$0.623293\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−0.242641	−0.0170300
$204$	0	0
$205$	−4.00000	−0.279372
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−5.24264	−0.362641
$210$	0	0
$211$	2.41421	0.166201	0.0831007	−	0.996541i	$-0.473518\pi$
$211$	2.41421	0.166201	0.0831007	+	0.996541i	$0.473518\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−16.4853	−1.12429
$216$	0	0
$217$	1.51472	0.102826
$218$	0	0
$219$	0	0
$220$	0	0
$221$	15.0711	1.01379
$222$	0	0
$223$	−17.6569	−1.18239	−0.591195	−	0.806529i	$-0.701344\pi$
$223$	−17.6569	−1.18239	−0.591195	+	0.806529i	$0.701344\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	4.10051	0.272160	0.136080	−	0.990698i	$-0.456550\pi$
$227$	4.10051	0.272160	0.136080	+	0.990698i	$0.456550\pi$
$228$	0	0
$229$	4.00000	0.264327	0.132164	−	0.991228i	$-0.457808\pi$
$229$	4.00000	0.264327	0.132164	+	0.991228i	$0.457808\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−18.2426	−1.19512	−0.597558	−	0.801826i	$-0.703862\pi$
$233$	−18.2426	−1.19512	−0.597558	+	0.801826i	$0.703862\pi$
$234$	0	0
$235$	11.6569	0.760409
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−14.1421	−0.914779	−0.457389	−	0.889267i	$-0.651215\pi$
$239$	−14.1421	−0.914779	−0.457389	+	0.889267i	$0.651215\pi$
$240$	0	0
$241$	−18.0711	−1.16406	−0.582030	−	0.813167i	$-0.697741\pi$
$241$	−18.0711	−1.16406	−0.582030	+	0.813167i	$0.697741\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−9.65685	−0.616954
$246$	0	0
$247$	12.6569	0.805336
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−19.5563	−1.23439	−0.617193	−	0.786812i	$-0.711730\pi$
$251$	−19.5563	−1.23439	−0.617193	+	0.786812i	$0.711730\pi$
$252$	0	0
$253$	−5.65685	−0.355643
$254$	0	0
$255$	0	0
$256$	0	0
$257$	23.0711	1.43913	0.719567	−	0.694423i	$-0.244340\pi$
$257$	23.0711	1.43913	0.719567	+	0.694423i	$0.244340\pi$
$258$	0	0
$259$	−1.58579	−0.0985360
$260$	0	0
$261$	0	0
$262$	0	0
$263$	21.5563	1.32922	0.664611	−	0.747190i	$-0.268597\pi$
$263$	21.5563	1.32922	0.664611	+	0.747190i	$0.268597\pi$
$264$	0	0
$265$	8.34315	0.512515
$266$	0	0
$267$	0	0
$268$	0	0
$269$	8.00000	0.487769	0.243884	−	0.969804i	$-0.421578\pi$
$269$	8.00000	0.487769	0.243884	+	0.969804i	$0.421578\pi$
$270$	0	0
$271$	10.0711	0.611774	0.305887	−	0.952068i	$-0.401047\pi$
$271$	10.0711	0.611774	0.305887	+	0.952068i	$0.401047\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−3.00000	−0.180907
$276$	0	0
$277$	−8.48528	−0.509831	−0.254916	−	0.966963i	$-0.582048\pi$
$277$	−8.48528	−0.509831	−0.254916	+	0.966963i	$0.582048\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−14.3431	−0.855640	−0.427820	−	0.903864i	$-0.640718\pi$
$281$	−14.3431	−0.855640	−0.427820	+	0.903864i	$0.640718\pi$
$282$	0	0
$283$	16.4853	0.979948	0.489974	−	0.871737i	$-0.337006\pi$
$283$	16.4853	0.979948	0.489974	+	0.871737i	$0.337006\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−1.17157	−0.0691558
$288$	0	0
$289$	21.9706	1.29239
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−22.2426	−1.29943	−0.649714	−	0.760178i	$-0.725112\pi$
$293$	−22.2426	−1.29943	−0.649714	+	0.760178i	$0.725112\pi$
$294$	0	0
$295$	3.65685	0.212910
$296$	0	0
$297$	0	0
$298$	0	0
$299$	13.6569	0.789796
$300$	0	0
$301$	−4.82843	−0.278306
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−10.7279	−0.614279
$306$	0	0
$307$	22.8284	1.30289	0.651444	−	0.758697i	$-0.274164\pi$
$307$	22.8284	1.30289	0.651444	+	0.758697i	$0.274164\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−9.89949	−0.561349	−0.280674	−	0.959803i	$-0.590558\pi$
$311$	−9.89949	−0.561349	−0.280674	+	0.959803i	$0.590558\pi$
$312$	0	0
$313$	12.7990	0.723442	0.361721	−	0.932286i	$-0.382189\pi$
$313$	12.7990	0.723442	0.361721	+	0.932286i	$0.382189\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−29.4142	−1.65207	−0.826033	−	0.563621i	$-0.809408\pi$
$317$	−29.4142	−1.65207	−0.826033	+	0.563621i	$0.809408\pi$
$318$	0	0
$319$	−0.585786	−0.0327977
$320$	0	0
$321$	0	0
$322$	0	0
$323$	32.7279	1.82103
$324$	0	0
$325$	7.24264	0.401749
$326$	0	0
$327$	0	0
$328$	0	0
$329$	3.41421	0.188232
$330$	0	0
$331$	−14.1716	−0.778940	−0.389470	−	0.921039i	$-0.627342\pi$
$331$	−14.1716	−0.778940	−0.389470	+	0.921039i	$0.627342\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	11.7574	0.642373
$336$	0	0
$337$	−16.2132	−0.883189	−0.441595	−	0.897215i	$-0.645587\pi$
$337$	−16.2132	−0.883189	−0.441595	+	0.897215i	$0.645587\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	3.65685	0.198030
$342$	0	0
$343$	−5.72792	−0.309279
$344$	0	0
$345$	0	0
$346$	0	0
$347$	2.72792	0.146443	0.0732213	−	0.997316i	$-0.476672\pi$
$347$	2.72792	0.146443	0.0732213	+	0.997316i	$0.476672\pi$
$348$	0	0
$349$	−1.58579	−0.0848852	−0.0424426	−	0.999099i	$-0.513514\pi$
$349$	−1.58579	−0.0848852	−0.0424426	+	0.999099i	$0.513514\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−29.2132	−1.55486	−0.777431	−	0.628968i	$-0.783478\pi$
$353$	−29.2132	−1.55486	−0.777431	+	0.628968i	$0.783478\pi$
$354$	0	0
$355$	5.65685	0.300235
$356$	0	0
$357$	0	0
$358$	0	0
$359$	7.79899	0.411615	0.205807	−	0.978593i	$-0.434018\pi$
$359$	7.79899	0.411615	0.205807	+	0.978593i	$0.434018\pi$
$360$	0	0
$361$	8.48528	0.446594
$362$	0	0
$363$	0	0
$364$	0	0
$365$	15.4142	0.806817
$366$	0	0
$367$	−13.1421	−0.686014	−0.343007	−	0.939333i	$-0.611445\pi$
$367$	−13.1421	−0.686014	−0.343007	+	0.939333i	$0.611445\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	2.44365	0.126868
$372$	0	0
$373$	18.8995	0.978579	0.489289	−	0.872121i	$-0.337256\pi$
$373$	18.8995	0.978579	0.489289	+	0.872121i	$0.337256\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	1.41421	0.0728357
$378$	0	0
$379$	17.8284	0.915785	0.457892	−	0.889008i	$-0.348604\pi$
$379$	17.8284	0.915785	0.457892	+	0.889008i	$0.348604\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−16.2426	−0.829960	−0.414980	−	0.909830i	$-0.636211\pi$
$383$	−16.2426	−0.829960	−0.414980	+	0.909830i	$0.636211\pi$
$384$	0	0
$385$	0.585786	0.0298544
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−14.8284	−0.751831	−0.375916	−	0.926654i	$-0.622672\pi$
$389$	−14.8284	−0.751831	−0.375916	+	0.926654i	$0.622672\pi$
$390$	0	0
$391$	35.3137	1.78589
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0.100505	0.00505696
$396$	0	0
$397$	12.6863	0.636707	0.318353	−	0.947972i	$-0.396870\pi$
$397$	12.6863	0.636707	0.318353	+	0.947972i	$0.396870\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	23.7990	1.18846	0.594232	−	0.804293i	$-0.297456\pi$
$401$	23.7990	1.18846	0.594232	+	0.804293i	$0.297456\pi$
$402$	0	0
$403$	−8.82843	−0.439775
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−3.82843	−0.189768
$408$	0	0
$409$	9.38478	0.464047	0.232024	−	0.972710i	$-0.425465\pi$
$409$	9.38478	0.464047	0.232024	+	0.972710i	$0.425465\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	1.07107	0.0527038
$414$	0	0
$415$	13.6569	0.670389
$416$	0	0
$417$	0	0
$418$	0	0
$419$	26.6274	1.30083	0.650417	−	0.759577i	$-0.274594\pi$
$419$	26.6274	1.30083	0.650417	+	0.759577i	$0.274594\pi$
$420$	0	0
$421$	−24.7990	−1.20863	−0.604314	−	0.796746i	$-0.706553\pi$
$421$	−24.7990	−1.20863	−0.604314	+	0.796746i	$0.706553\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	18.7279	0.908438
$426$	0	0
$427$	−3.14214	−0.152059
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−5.17157	−0.249106	−0.124553	−	0.992213i	$-0.539750\pi$
$431$	−5.17157	−0.249106	−0.124553	+	0.992213i	$0.539750\pi$
$432$	0	0
$433$	24.0000	1.15337	0.576683	−	0.816968i	$-0.304347\pi$
$433$	24.0000	1.15337	0.576683	+	0.816968i	$0.304347\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	29.6569	1.41868
$438$	0	0
$439$	−2.82843	−0.134993	−0.0674967	−	0.997719i	$-0.521501\pi$
$439$	−2.82843	−0.134993	−0.0674967	+	0.997719i	$0.521501\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	27.7574	1.31879	0.659396	−	0.751796i	$-0.270812\pi$
$443$	27.7574	1.31879	0.659396	+	0.751796i	$0.270812\pi$
$444$	0	0
$445$	0.343146	0.0162667
$446$	0	0
$447$	0	0
$448$	0	0
$449$	2.14214	0.101094	0.0505468	−	0.998722i	$-0.483904\pi$
$449$	2.14214	0.101094	0.0505468	+	0.998722i	$0.483904\pi$
$450$	0	0
$451$	−2.82843	−0.133185
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−1.41421	−0.0662994
$456$	0	0
$457$	15.6569	0.732397	0.366198	−	0.930537i	$-0.380659\pi$
$457$	15.6569	0.732397	0.366198	+	0.930537i	$0.380659\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−7.41421	−0.345314	−0.172657	−	0.984982i	$-0.555235\pi$
$461$	−7.41421	−0.345314	−0.172657	+	0.984982i	$0.555235\pi$
$462$	0	0
$463$	−10.5147	−0.488660	−0.244330	−	0.969692i	$-0.578568\pi$
$463$	−10.5147	−0.488660	−0.244330	+	0.969692i	$0.578568\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	12.6863	0.587052	0.293526	−	0.955951i	$-0.405171\pi$
$467$	12.6863	0.587052	0.293526	+	0.955951i	$0.405171\pi$
$468$	0	0
$469$	3.44365	0.159013
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−11.6569	−0.535983
$474$	0	0
$475$	15.7279	0.721647
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−30.2426	−1.38182	−0.690911	−	0.722940i	$-0.742790\pi$
$479$	−30.2426	−1.38182	−0.690911	+	0.722940i	$0.742790\pi$
$480$	0	0
$481$	9.24264	0.421428
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−20.7279	−0.941206
$486$	0	0
$487$	17.3431	0.785893	0.392946	−	0.919561i	$-0.371456\pi$
$487$	17.3431	0.785893	0.392946	+	0.919561i	$0.371456\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−6.44365	−0.290798	−0.145399	−	0.989373i	$-0.546447\pi$
$491$	−6.44365	−0.290798	−0.145399	+	0.989373i	$0.546447\pi$
$492$	0	0
$493$	3.65685	0.164696
$494$	0	0
$495$	0	0
$496$	0	0
$497$	1.65685	0.0743201
$498$	0	0
$499$	−18.2843	−0.818516	−0.409258	−	0.912419i	$-0.634212\pi$
$499$	−18.2843	−0.818516	−0.409258	+	0.912419i	$0.634212\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	19.3137	0.861156	0.430578	−	0.902553i	$-0.358310\pi$
$503$	19.3137	0.861156	0.430578	+	0.902553i	$0.358310\pi$
$504$	0	0
$505$	−20.1421	−0.896313
$506$	0	0
$507$	0	0
$508$	0	0
$509$	39.0711	1.73179	0.865897	−	0.500222i	$-0.166748\pi$
$509$	39.0711	1.73179	0.865897	+	0.500222i	$0.166748\pi$
$510$	0	0
$511$	4.51472	0.199719
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−16.2426	−0.715736
$516$	0	0
$517$	8.24264	0.362511
$518$	0	0
$519$	0	0
$520$	0	0
$521$	16.9706	0.743494	0.371747	−	0.928334i	$-0.378759\pi$
$521$	16.9706	0.743494	0.371747	+	0.928334i	$0.378759\pi$
$522$	0	0
$523$	19.2426	0.841422	0.420711	−	0.907195i	$-0.361781\pi$
$523$	19.2426	0.841422	0.420711	+	0.907195i	$0.361781\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−22.8284	−0.994422
$528$	0	0
$529$	9.00000	0.391304
$530$	0	0
$531$	0	0
$532$	0	0
$533$	6.82843	0.295772
$534$	0	0
$535$	20.1421	0.870820
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−6.82843	−0.294121
$540$	0	0
$541$	30.0711	1.29286	0.646428	−	0.762975i	$-0.276262\pi$
$541$	30.0711	1.29286	0.646428	+	0.762975i	$0.276262\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−10.8284	−0.463839
$546$	0	0
$547$	12.4142	0.530793	0.265397	−	0.964139i	$-0.414497\pi$
$547$	12.4142	0.530793	0.265397	+	0.964139i	$0.414497\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	3.07107	0.130832
$552$	0	0
$553$	0.0294373	0.00125180
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−18.0416	−0.764448	−0.382224	−	0.924070i	$-0.624842\pi$
$557$	−18.0416	−0.764448	−0.382224	+	0.924070i	$0.624842\pi$
$558$	0	0
$559$	28.1421	1.19029
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−35.5980	−1.50028	−0.750138	−	0.661281i	$-0.770013\pi$
$563$	−35.5980	−1.50028	−0.750138	+	0.661281i	$0.770013\pi$
$564$	0	0
$565$	7.65685	0.322126
$566$	0	0
$567$	0	0
$568$	0	0
$569$	23.7990	0.997706	0.498853	−	0.866687i	$-0.333755\pi$
$569$	23.7990	0.997706	0.498853	+	0.866687i	$0.333755\pi$
$570$	0	0
$571$	−1.58579	−0.0663631	−0.0331815	−	0.999449i	$-0.510564\pi$
$571$	−1.58579	−0.0663631	−0.0331815	+	0.999449i	$0.510564\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	16.9706	0.707721
$576$	0	0
$577$	−34.1127	−1.42013	−0.710065	−	0.704136i	$-0.751335\pi$
$577$	−34.1127	−1.42013	−0.710065	+	0.704136i	$0.751335\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	4.00000	0.165948
$582$	0	0
$583$	5.89949	0.244332
$584$	0	0
$585$	0	0
$586$	0	0
$587$	26.3848	1.08902	0.544508	−	0.838756i	$-0.316716\pi$
$587$	26.3848	1.08902	0.544508	+	0.838756i	$0.316716\pi$
$588$	0	0
$589$	−19.1716	−0.789951
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−33.9411	−1.39379	−0.696897	−	0.717171i	$-0.745437\pi$
$593$	−33.9411	−1.39379	−0.696897	+	0.717171i	$0.745437\pi$
$594$	0	0
$595$	−3.65685	−0.149916
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−40.4853	−1.65418	−0.827092	−	0.562067i	$-0.810006\pi$
$599$	−40.4853	−1.65418	−0.827092	+	0.562067i	$0.810006\pi$
$600$	0	0
$601$	28.6274	1.16774	0.583868	−	0.811848i	$-0.301538\pi$
$601$	28.6274	1.16774	0.583868	+	0.811848i	$0.301538\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	1.41421	0.0574960
$606$	0	0
$607$	9.10051	0.369378	0.184689	−	0.982797i	$-0.440872\pi$
$607$	9.10051	0.369378	0.184689	+	0.982797i	$0.440872\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−19.8995	−0.805047
$612$	0	0
$613$	−31.5269	−1.27336	−0.636680	−	0.771128i	$-0.719693\pi$
$613$	−31.5269	−1.27336	−0.636680	+	0.771128i	$0.719693\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−24.2843	−0.977648	−0.488824	−	0.872382i	$-0.662574\pi$
$617$	−24.2843	−0.977648	−0.488824	+	0.872382i	$0.662574\pi$
$618$	0	0
$619$	16.7990	0.675208	0.337604	−	0.941288i	$-0.390383\pi$
$619$	16.7990	0.675208	0.337604	+	0.941288i	$0.390383\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0.100505	0.00402665
$624$	0	0
$625$	−1.00000	−0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	23.8995	0.952935
$630$	0	0
$631$	3.97056	0.158066	0.0790328	−	0.996872i	$-0.474817\pi$
$631$	3.97056	0.158066	0.0790328	+	0.996872i	$0.474817\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	9.65685	0.383221
$636$	0	0
$637$	16.4853	0.653171
$638$	0	0
$639$	0	0
$640$	0	0
$641$	34.5858	1.36606	0.683028	−	0.730392i	$-0.260663\pi$
$641$	34.5858	1.36606	0.683028	+	0.730392i	$0.260663\pi$
$642$	0	0
$643$	34.2843	1.35204	0.676020	−	0.736883i	$-0.263703\pi$
$643$	34.2843	1.35204	0.676020	+	0.736883i	$0.263703\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−30.8284	−1.21199	−0.605995	−	0.795468i	$-0.707225\pi$
$647$	−30.8284	−1.21199	−0.605995	+	0.795468i	$0.707225\pi$
$648$	0	0
$649$	2.58579	0.101501
$650$	0	0
$651$	0	0
$652$	0	0
$653$	13.8579	0.542300	0.271150	−	0.962537i	$-0.412596\pi$
$653$	13.8579	0.542300	0.271150	+	0.962537i	$0.412596\pi$
$654$	0	0
$655$	8.82843	0.344955
$656$	0	0
$657$	0	0
$658$	0	0
$659$	25.0711	0.976630	0.488315	−	0.872667i	$-0.337612\pi$
$659$	25.0711	0.976630	0.488315	+	0.872667i	$0.337612\pi$
$660$	0	0
$661$	41.1421	1.60024	0.800122	−	0.599838i	$-0.204768\pi$
$661$	41.1421	1.60024	0.800122	+	0.599838i	$0.204768\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−3.07107	−0.119091
$666$	0	0
$667$	3.31371	0.128307
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−7.58579	−0.292846
$672$	0	0
$673$	−12.0711	−0.465305	−0.232653	−	0.972560i	$-0.574741\pi$
$673$	−12.0711	−0.465305	−0.232653	+	0.972560i	$0.574741\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−31.5147	−1.21121	−0.605605	−	0.795766i	$-0.707069\pi$
$677$	−31.5147	−1.21121	−0.605605	+	0.795766i	$0.707069\pi$
$678$	0	0
$679$	−6.07107	−0.232986
$680$	0	0
$681$	0	0
$682$	0	0
$683$	21.2132	0.811701	0.405850	−	0.913940i	$-0.366975\pi$
$683$	21.2132	0.811701	0.405850	+	0.913940i	$0.366975\pi$
$684$	0	0
$685$	−4.00000	−0.152832
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−14.2426	−0.542601
$690$	0	0
$691$	−16.9706	−0.645591	−0.322795	−	0.946469i	$-0.604623\pi$
$691$	−16.9706	−0.645591	−0.322795	+	0.946469i	$0.604623\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−31.8995	−1.21002
$696$	0	0
$697$	17.6569	0.668801
$698$	0	0
$699$	0	0
$700$	0	0
$701$	33.0711	1.24908	0.624538	−	0.780994i	$-0.285287\pi$
$701$	33.0711	1.24908	0.624538	+	0.780994i	$0.285287\pi$
$702$	0	0
$703$	20.0711	0.756995
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−5.89949	−0.221873
$708$	0	0
$709$	−50.1127	−1.88202	−0.941011	−	0.338376i	$-0.890122\pi$
$709$	−50.1127	−1.88202	−0.941011	+	0.338376i	$0.890122\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−20.6863	−0.774708
$714$	0	0
$715$	−3.41421	−0.127684
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−24.7279	−0.922196	−0.461098	−	0.887349i	$-0.652544\pi$
$719$	−24.7279	−0.922196	−0.461098	+	0.887349i	$0.652544\pi$
$720$	0	0
$721$	−4.75736	−0.177173
$722$	0	0
$723$	0	0
$724$	0	0
$725$	1.75736	0.0652667
$726$	0	0
$727$	14.0000	0.519231	0.259616	−	0.965712i	$-0.416404\pi$
$727$	14.0000	0.519231	0.259616	+	0.965712i	$0.416404\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	72.7696	2.69148
$732$	0	0
$733$	45.5980	1.68420	0.842100	−	0.539322i	$-0.181319\pi$
$733$	45.5980	1.68420	0.842100	+	0.539322i	$0.181319\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	8.31371	0.306239
$738$	0	0
$739$	−15.5147	−0.570718	−0.285359	−	0.958421i	$-0.592113\pi$
$739$	−15.5147	−0.570718	−0.285359	+	0.958421i	$0.592113\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−41.7574	−1.53193	−0.765964	−	0.642883i	$-0.777738\pi$
$743$	−41.7574	−1.53193	−0.765964	+	0.642883i	$0.777738\pi$
$744$	0	0
$745$	−27.3137	−1.00070
$746$	0	0
$747$	0	0
$748$	0	0
$749$	5.89949	0.215563
$750$	0	0
$751$	−52.5980	−1.91933	−0.959664	−	0.281150i	$-0.909284\pi$
$751$	−52.5980	−1.91933	−0.959664	+	0.281150i	$0.909284\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−6.72792	−0.244854
$756$	0	0
$757$	40.1127	1.45792	0.728960	−	0.684556i	$-0.240004\pi$
$757$	40.1127	1.45792	0.728960	+	0.684556i	$0.240004\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	5.85786	0.212347	0.106174	−	0.994348i	$-0.466140\pi$
$761$	5.85786	0.212347	0.106174	+	0.994348i	$0.466140\pi$
$762$	0	0
$763$	−3.17157	−0.114819
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−6.24264	−0.225409
$768$	0	0
$769$	−16.8995	−0.609411	−0.304706	−	0.952447i	$-0.598558\pi$
$769$	−16.8995	−0.609411	−0.304706	+	0.952447i	$0.598558\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	15.7574	0.566753	0.283376	−	0.959009i	$-0.408545\pi$
$773$	15.7574	0.566753	0.283376	+	0.959009i	$0.408545\pi$
$774$	0	0
$775$	−10.9706	−0.394074
$776$	0	0
$777$	0	0
$778$	0	0
$779$	14.8284	0.531284
$780$	0	0
$781$	4.00000	0.143131
$782$	0	0
$783$	0	0
$784$	0	0
$785$	21.6569	0.772966
$786$	0	0
$787$	−25.8701	−0.922168	−0.461084	−	0.887357i	$-0.652539\pi$
$787$	−25.8701	−0.922168	−0.461084	+	0.887357i	$0.652539\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	2.24264	0.0797391
$792$	0	0
$793$	18.3137	0.650339
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−2.62742	−0.0930679	−0.0465339	−	0.998917i	$-0.514818\pi$
$797$	−2.62742	−0.0930679	−0.0465339	+	0.998917i	$0.514818\pi$
$798$	0	0
$799$	−51.4558	−1.82038
$800$	0	0
$801$	0	0
$802$	0	0
$803$	10.8995	0.384635
$804$	0	0
$805$	−3.31371	−0.116793
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−27.4142	−0.963832	−0.481916	−	0.876217i	$-0.660059\pi$
$809$	−27.4142	−0.963832	−0.481916	+	0.876217i	$0.660059\pi$
$810$	0	0
$811$	−14.1421	−0.496598	−0.248299	−	0.968683i	$-0.579871\pi$
$811$	−14.1421	−0.496598	−0.248299	+	0.968683i	$0.579871\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−4.92893	−0.172653
$816$	0	0
$817$	61.1127	2.13806
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−39.1127	−1.36504	−0.682521	−	0.730866i	$-0.739117\pi$
$821$	−39.1127	−1.36504	−0.682521	+	0.730866i	$0.739117\pi$
$822$	0	0
$823$	−23.1421	−0.806684	−0.403342	−	0.915049i	$-0.632152\pi$
$823$	−23.1421	−0.806684	−0.403342	+	0.915049i	$0.632152\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	2.24264	0.0779843	0.0389921	−	0.999240i	$-0.487585\pi$
$827$	2.24264	0.0779843	0.0389921	+	0.999240i	$0.487585\pi$
$828$	0	0
$829$	22.5980	0.784860	0.392430	−	0.919782i	$-0.371634\pi$
$829$	22.5980	0.784860	0.392430	+	0.919782i	$0.371634\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	42.6274	1.47695
$834$	0	0
$835$	27.1716	0.940311
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−28.7696	−0.993235	−0.496618	−	0.867969i	$-0.665425\pi$
$839$	−28.7696	−0.993235	−0.496618	+	0.867969i	$0.665425\pi$
$840$	0	0
$841$	−28.6569	−0.988167
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−10.1421	−0.348900
$846$	0	0
$847$	0.414214	0.0142325
$848$	0	0
$849$	0	0
$850$	0	0
$851$	21.6569	0.742387
$852$	0	0
$853$	−53.8701	−1.84448	−0.922238	−	0.386623i	$-0.873641\pi$
$853$	−53.8701	−1.84448	−0.922238	+	0.386623i	$0.873641\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	0.786797	0.0268765	0.0134382	−	0.999910i	$-0.495722\pi$
$857$	0.786797	0.0268765	0.0134382	+	0.999910i	$0.495722\pi$
$858$	0	0
$859$	−14.5147	−0.495236	−0.247618	−	0.968858i	$-0.579648\pi$
$859$	−14.5147	−0.495236	−0.247618	+	0.968858i	$0.579648\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	9.41421	0.320464	0.160232	−	0.987079i	$-0.448776\pi$
$863$	9.41421	0.320464	0.160232	+	0.987079i	$0.448776\pi$
$864$	0	0
$865$	14.3431	0.487682
$866$	0	0
$867$	0	0
$868$	0	0
$869$	0.0710678	0.00241081
$870$	0	0
$871$	−20.0711	−0.680082
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−4.68629	−0.158426
$876$	0	0
$877$	41.6690	1.40706	0.703532	−	0.710664i	$-0.251605\pi$
$877$	41.6690	1.40706	0.703532	+	0.710664i	$0.251605\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	2.14214	0.0721704	0.0360852	−	0.999349i	$-0.488511\pi$
$881$	2.14214	0.0721704	0.0360852	+	0.999349i	$0.488511\pi$
$882$	0	0
$883$	11.6274	0.391294	0.195647	−	0.980674i	$-0.437319\pi$
$883$	11.6274	0.391294	0.195647	+	0.980674i	$0.437319\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−57.0711	−1.91626	−0.958129	−	0.286335i	$-0.907563\pi$
$887$	−57.0711	−1.91626	−0.958129	+	0.286335i	$0.907563\pi$
$888$	0	0
$889$	2.82843	0.0948624
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−43.2132	−1.44607
$894$	0	0
$895$	24.6274	0.823204
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−2.14214	−0.0714442
$900$	0	0
$901$	−36.8284	−1.22693
$902$	0	0
$903$	0	0
$904$	0	0
$905$	28.2426	0.938817
$906$	0	0
$907$	35.1421	1.16688	0.583438	−	0.812158i	$-0.301707\pi$
$907$	35.1421	1.16688	0.583438	+	0.812158i	$0.301707\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−31.5147	−1.04413	−0.522065	−	0.852906i	$-0.674838\pi$
$911$	−31.5147	−1.04413	−0.522065	+	0.852906i	$0.674838\pi$
$912$	0	0
$913$	9.65685	0.319595
$914$	0	0
$915$	0	0
$916$	0	0
$917$	2.58579	0.0853902
$918$	0	0
$919$	−21.4558	−0.707763	−0.353881	−	0.935290i	$-0.615138\pi$
$919$	−21.4558	−0.707763	−0.353881	+	0.935290i	$0.615138\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−9.65685	−0.317859
$924$	0	0
$925$	11.4853	0.377634
$926$	0	0
$927$	0	0
$928$	0	0
$929$	55.0711	1.80682	0.903412	−	0.428774i	$-0.141055\pi$
$929$	55.0711	1.80682	0.903412	+	0.428774i	$0.141055\pi$
$930$	0	0
$931$	35.7990	1.17326
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−8.82843	−0.288720
$936$	0	0
$937$	−11.7279	−0.383135	−0.191567	−	0.981479i	$-0.561357\pi$
$937$	−11.7279	−0.383135	−0.191567	+	0.981479i	$0.561357\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−9.17157	−0.298985	−0.149492	−	0.988763i	$-0.547764\pi$
$941$	−9.17157	−0.298985	−0.149492	+	0.988763i	$0.547764\pi$
$942$	0	0
$943$	16.0000	0.521032
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−33.6985	−1.09505	−0.547527	−	0.836788i	$-0.684431\pi$
$947$	−33.6985	−1.09505	−0.547527	+	0.836788i	$0.684431\pi$
$948$	0	0
$949$	−26.3137	−0.854179
$950$	0	0
$951$	0	0
$952$	0	0
$953$	23.5980	0.764414	0.382207	−	0.924077i	$-0.375164\pi$
$953$	23.5980	0.764414	0.382207	+	0.924077i	$0.375164\pi$
$954$	0	0
$955$	−4.68629	−0.151645
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−1.17157	−0.0378321
$960$	0	0
$961$	−17.6274	−0.568626
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−9.07107	−0.292008
$966$	0	0
$967$	8.21320	0.264119	0.132059	−	0.991242i	$-0.457841\pi$
$967$	8.21320	0.264119	0.132059	+	0.991242i	$0.457841\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	17.4142	0.558849	0.279424	−	0.960168i	$-0.409856\pi$
$971$	17.4142	0.558849	0.279424	+	0.960168i	$0.409856\pi$
$972$	0	0
$973$	−9.34315	−0.299528
$974$	0	0
$975$	0	0
$976$	0	0
$977$	31.0711	0.994052	0.497026	−	0.867736i	$-0.334425\pi$
$977$	31.0711	0.994052	0.497026	+	0.867736i	$0.334425\pi$
$978$	0	0
$979$	0.242641	0.00775483
$980$	0	0
$981$	0	0
$982$	0	0
$983$	6.38478	0.203643	0.101821	−	0.994803i	$-0.467533\pi$
$983$	6.38478	0.203643	0.101821	+	0.994803i	$0.467533\pi$
$984$	0	0
$985$	−0.828427	−0.0263959
$986$	0	0
$987$	0	0
$988$	0	0
$989$	65.9411	2.09681
$990$	0	0
$991$	−49.6274	−1.57647	−0.788233	−	0.615376i	$-0.789004\pi$
$991$	−49.6274	−1.57647	−0.788233	+	0.615376i	$0.789004\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−15.0711	−0.477785
$996$	0	0
$997$	15.3726	0.486855	0.243427	−	0.969919i	$-0.421728\pi$
$997$	15.3726	0.486855	0.243427	+	0.969919i	$0.421728\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2376.2.a.h.1.2	yes	2
3.2	odd	2		2376.2.a.f.1.1	✓	2
4.3	odd	2		4752.2.a.ba.1.2		2
12.11	even	2		4752.2.a.bc.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2376.2.a.f.1.1	✓	2	3.2	odd	2
2376.2.a.h.1.2	yes	2	1.1	even	1	trivial
4752.2.a.ba.1.2		2	4.3	odd	2
4752.2.a.bc.1.1		2	12.11	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:	\( N \)	\(=\)	\( 2376 = 2^{3} \cdot 3^{3} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2376.a (trivial)

\(f(q)\)	\(=\)	\(q+1.41421 q^{5} +0.414214 q^{7} +1.00000 q^{11} -2.41421 q^{13} -6.24264 q^{17} -5.24264 q^{19} -5.65685 q^{23} -3.00000 q^{25} -0.585786 q^{29} +3.65685 q^{31} +0.585786 q^{35} -3.82843 q^{37} -2.82843 q^{41} -11.6569 q^{43} +8.24264 q^{47} -6.82843 q^{49} +5.89949 q^{53} +1.41421 q^{55} +2.58579 q^{59} -7.58579 q^{61} -3.41421 q^{65} +8.31371 q^{67} +4.00000 q^{71} +10.8995 q^{73} +0.414214 q^{77} +0.0710678 q^{79} +9.65685 q^{83} -8.82843 q^{85} +0.242641 q^{89} -1.00000 q^{91} -7.41421 q^{95} -14.6569 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{7} + 2 q^{11} - 2 q^{13} - 4 q^{17} - 2 q^{19} - 6 q^{25} - 4 q^{29} - 4 q^{31} + 4 q^{35} - 2 q^{37} - 12 q^{43} + 8 q^{47} - 8 q^{49} - 8 q^{53} + 8 q^{59} - 18 q^{61} - 4 q^{65} - 6 q^{67}+ \cdots - 18 q^{97}+O(q^{100}) \) 2 * q - 2 * q^7 + 2 * q^11 - 2 * q^13 - 4 * q^17 - 2 * q^19 - 6 * q^25 - 4 * q^29 - 4 * q^31 + 4 * q^35 - 2 * q^37 - 12 * q^43 + 8 * q^47 - 8 * q^49 - 8 * q^53 + 8 * q^59 - 18 * q^61 - 4 * q^65 - 6 * q^67 + 8 * q^71 + 2 * q^73 - 2 * q^77 - 14 * q^79 + 8 * q^83 - 12 * q^85 - 8 * q^89 - 2 * q^91 - 12 * q^95 - 18 * q^97

Introduction

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