LMFDB - Embedded newform 32.12.a.c.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [32,12,Mod(1,32)] 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("32.1"); S:= CuspForms(chi, 12); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(32, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 12, names="a")

Level:	$ N $	$=$	$ 32 = 2^{5} $
Weight:	$ k $	$=$	$ 12 $
Character orbit:	$[\chi]$	$=$	32.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,14060] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$24.5869817779$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{273}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 68 $ x^2 - x - 68
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-7.76136$ of defining polynomial
Character	$\chi$	$=$	32.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+528.727 q^{3} +7030.00 q^{5} +34896.0 q^{7} +102405. q^{9} +724884. q^{11} -1.18207e6 q^{13} +3.71695e6 q^{15} +5.44387e6 q^{17} -2.09772e7 q^{19} +1.84504e7 q^{21} +6.60168e6 q^{23} +592775. q^{25} -3.95181e7 q^{27} +1.64046e8 q^{29} +1.97841e8 q^{31} +3.83266e8 q^{33} +2.45319e8 q^{35} +5.40227e8 q^{37} -6.24990e8 q^{39} +2.79999e8 q^{41} -3.02692e8 q^{43} +7.19907e8 q^{45} -1.87326e9 q^{47} -7.59598e8 q^{49} +2.87832e9 q^{51} +4.77652e9 q^{53} +5.09594e9 q^{55} -1.10912e10 q^{57} -4.44316e9 q^{59} -7.47969e9 q^{61} +3.57352e9 q^{63} -8.30992e9 q^{65} -1.09155e10 q^{67} +3.49049e9 q^{69} -2.70127e9 q^{71} -2.18032e10 q^{73} +3.13416e8 q^{75} +2.52955e10 q^{77} -3.37233e10 q^{79} -3.90350e10 q^{81} +6.18211e10 q^{83} +3.82704e10 q^{85} +8.67358e10 q^{87} +2.21958e10 q^{89} -4.12493e10 q^{91} +1.04604e11 q^{93} -1.47470e11 q^{95} +1.00703e11 q^{97} +7.42318e10 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 14060 q^{5} + 204810 q^{9} - 2364132 q^{13} + 10887748 q^{17} + 36900864 q^{21} + 1185550 q^{25} + 328092988 q^{29} + 766531584 q^{33} + 1080453740 q^{37} + 559998644 q^{41} + 1439814300 q^{45} - 1519196462 q^{49}+ \cdots + 201406465188 q^{97}+O(q^{100}) $ 2 * q + 14060 * q^5 + 204810 * q^9 - 2364132 * q^13 + 10887748 * q^17 + 36900864 * q^21 + 1185550 * q^25 + 328092988 * q^29 + 766531584 * q^33 + 1080453740 * q^37 + 559998644 * q^41 + 1439814300 * q^45 - 1519196462 * q^49 + 9553038796 * q^53 - 22182451200 * q^57 - 14959370564 * q^61 - 16619847960 * q^65 + 6980972544 * q^69 - 43606336236 * q^73 + 50591084544 * q^77 - 78070028238 * q^81 + 76540868440 * q^85 + 44391542900 * q^89 + 209207771136 * q^93 + 201406465188 * 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
$3$	528.727	1.25622	0.628108	−	0.778126i	$-0.283830\pi$
$3$	528.727	1.25622	0.628108	+	0.778126i	$0.283830\pi$
$4$	0	0
$5$	7030.00	1.00605	0.503026	−	0.864271i	$-0.332220\pi$
$5$	7030.00	1.00605	0.503026	+	0.864271i	$0.332220\pi$
$6$	0	0
$7$	34896.0	0.784758	0.392379	−	0.919804i	$-0.371652\pi$
$7$	34896.0	0.784758	0.392379	+	0.919804i	$0.371652\pi$
$8$	0	0
$9$	102405.	0.578079
$10$	0	0
$11$	724884.	1.35709	0.678546	−	0.734558i	$-0.262611\pi$
$11$	724884.	1.35709	0.678546	+	0.734558i	$0.262611\pi$
$12$	0	0
$13$	−1.18207e6	−0.882985	−0.441492	−	0.897265i	$-0.645551\pi$
$13$	−1.18207e6	−0.882985	−0.441492	+	0.897265i	$0.645551\pi$
$14$	0	0
$15$	3.71695e6	1.26382
$16$	0	0
$17$	5.44387e6	0.929906	0.464953	−	0.885335i	$-0.346071\pi$
$17$	5.44387e6	0.929906	0.464953	+	0.885335i	$0.346071\pi$
$18$	0	0
$19$	−2.09772e7	−1.94358	−0.971792	−	0.235839i	$-0.924216\pi$
$19$	−2.09772e7	−1.94358	−0.971792	+	0.235839i	$0.924216\pi$
$20$	0	0
$21$	1.84504e7	0.985826
$22$	0	0
$23$	6.60168e6	0.213871	0.106935	−	0.994266i	$-0.465896\pi$
$23$	6.60168e6	0.213871	0.106935	+	0.994266i	$0.465896\pi$
$24$	0	0
$25$	592775.	0.0121400
$26$	0	0
$27$	−3.95181e7	−0.530024
$28$	0	0
$29$	1.64046e8	1.48518	0.742588	−	0.669748i	$-0.233598\pi$
$29$	1.64046e8	1.48518	0.742588	+	0.669748i	$0.233598\pi$
$30$	0	0
$31$	1.97841e8	1.24116	0.620579	−	0.784144i	$-0.286898\pi$
$31$	1.97841e8	1.24116	0.620579	+	0.784144i	$0.286898\pi$
$32$	0	0
$33$	3.83266e8	1.70480
$34$	0	0
$35$	2.45319e8	0.789508
$36$	0	0
$37$	5.40227e8	1.28076	0.640378	−	0.768060i	$-0.278778\pi$
$37$	5.40227e8	1.28076	0.640378	+	0.768060i	$0.278778\pi$
$38$	0	0
$39$	−6.24990e8	−1.10922
$40$	0	0
$41$	2.79999e8	0.377438	0.188719	−	0.982031i	$-0.439566\pi$
$41$	2.79999e8	0.377438	0.188719	+	0.982031i	$0.439566\pi$
$42$	0	0
$43$	−3.02692e8	−0.313997	−0.156998	−	0.987599i	$-0.550182\pi$
$43$	−3.02692e8	−0.313997	−0.156998	+	0.987599i	$0.550182\pi$
$44$	0	0
$45$	7.19907e8	0.581578
$46$	0	0
$47$	−1.87326e9	−1.19141	−0.595703	−	0.803205i	$-0.703126\pi$
$47$	−1.87326e9	−1.19141	−0.595703	+	0.803205i	$0.703126\pi$
$48$	0	0
$49$	−7.59598e8	−0.384154
$50$	0	0
$51$	2.87832e9	1.16816
$52$	0	0
$53$	4.77652e9	1.56890	0.784448	−	0.620194i	$-0.212946\pi$
$53$	4.77652e9	1.56890	0.784448	+	0.620194i	$0.212946\pi$
$54$	0	0
$55$	5.09594e9	1.36530
$56$	0	0
$57$	−1.10912e10	−2.44156
$58$	0	0
$59$	−4.44316e9	−0.809107	−0.404554	−	0.914514i	$-0.632573\pi$
$59$	−4.44316e9	−0.809107	−0.404554	+	0.914514i	$0.632573\pi$
$60$	0	0
$61$	−7.47969e9	−1.13389	−0.566943	−	0.823757i	$-0.691874\pi$
$61$	−7.47969e9	−1.13389	−0.566943	+	0.823757i	$0.691874\pi$
$62$	0	0
$63$	3.57352e9	0.453653
$64$	0	0
$65$	−8.30992e9	−0.888328
$66$	0	0
$67$	−1.09155e10	−0.987720	−0.493860	−	0.869541i	$-0.664415\pi$
$67$	−1.09155e10	−0.987720	−0.493860	+	0.869541i	$0.664415\pi$
$68$	0	0
$69$	3.49049e9	0.268668
$70$	0	0
$71$	−2.70127e9	−0.177683	−0.0888417	−	0.996046i	$-0.528317\pi$
$71$	−2.70127e9	−0.177683	−0.0888417	+	0.996046i	$0.528317\pi$
$72$	0	0
$73$	−2.18032e10	−1.23096	−0.615480	−	0.788153i	$-0.711038\pi$
$73$	−2.18032e10	−1.23096	−0.615480	+	0.788153i	$0.711038\pi$
$74$	0	0
$75$	3.13416e8	0.0152505
$76$	0	0
$77$	2.52955e10	1.06499
$78$	0	0
$79$	−3.37233e10	−1.23305	−0.616525	−	0.787335i	$-0.711460\pi$
$79$	−3.37233e10	−1.23305	−0.616525	+	0.787335i	$0.711460\pi$
$80$	0	0
$81$	−3.90350e10	−1.24390
$82$	0	0
$83$	6.18211e10	1.72269	0.861345	−	0.508021i	$-0.169623\pi$
$83$	6.18211e10	1.72269	0.861345	+	0.508021i	$0.169623\pi$
$84$	0	0
$85$	3.82704e10	0.935533
$86$	0	0
$87$	8.67358e10	1.86570
$88$	0	0
$89$	2.21958e10	0.421333	0.210666	−	0.977558i	$-0.432437\pi$
$89$	2.21958e10	0.421333	0.210666	+	0.977558i	$0.432437\pi$
$90$	0	0
$91$	−4.12493e10	−0.692930
$92$	0	0
$93$	1.04604e11	1.55916
$94$	0	0
$95$	−1.47470e11	−1.95535
$96$	0	0
$97$	1.00703e11	1.19069	0.595345	−	0.803470i	$-0.297015\pi$
$97$	1.00703e11	1.19069	0.595345	+	0.803470i	$0.297015\pi$
$98$	0	0
$99$	7.42318e10	0.784506
$100$	0	0
$101$	−1.68408e10	−0.159439	−0.0797194	−	0.996817i	$-0.525402\pi$
$101$	−1.68408e10	−0.159439	−0.0797194	+	0.996817i	$0.525402\pi$
$102$	0	0
$103$	−2.37472e10	−0.201840	−0.100920	−	0.994895i	$-0.532179\pi$
$103$	−2.37472e10	−0.201840	−0.100920	+	0.994895i	$0.532179\pi$
$104$	0	0
$105$	1.29707e11	0.991792
$106$	0	0
$107$	−2.58781e11	−1.78370	−0.891850	−	0.452331i	$-0.850593\pi$
$107$	−2.58781e11	−1.78370	−0.891850	+	0.452331i	$0.850593\pi$
$108$	0	0
$109$	−9.93062e10	−0.618202	−0.309101	−	0.951029i	$-0.600028\pi$
$109$	−9.93062e10	−0.618202	−0.309101	+	0.951029i	$0.600028\pi$
$110$	0	0
$111$	2.85632e11	1.60891
$112$	0	0
$113$	−1.60348e11	−0.818716	−0.409358	−	0.912374i	$-0.634247\pi$
$113$	−1.60348e11	−0.818716	−0.409358	+	0.912374i	$0.634247\pi$
$114$	0	0
$115$	4.64098e10	0.215165
$116$	0	0
$117$	−1.21049e11	−0.510435
$118$	0	0
$119$	1.89969e11	0.729752
$120$	0	0
$121$	2.40146e11	0.841696
$122$	0	0
$123$	1.48043e11	0.474144
$124$	0	0
$125$	−3.39095e11	−0.993838
$126$	0	0
$127$	−2.75925e11	−0.741088	−0.370544	−	0.928815i	$-0.620829\pi$
$127$	−2.75925e11	−0.741088	−0.370544	+	0.928815i	$0.620829\pi$
$128$	0	0
$129$	−1.60042e11	−0.394448
$130$	0	0
$131$	5.67091e10	0.128428	0.0642141	−	0.997936i	$-0.479546\pi$
$131$	5.67091e10	0.128428	0.0642141	+	0.997936i	$0.479546\pi$
$132$	0	0
$133$	−7.32021e11	−1.52524
$134$	0	0
$135$	−2.77812e11	−0.533231
$136$	0	0
$137$	−3.27687e11	−0.580091	−0.290046	−	0.957013i	$-0.593670\pi$
$137$	−3.27687e11	−0.580091	−0.290046	+	0.957013i	$0.593670\pi$
$138$	0	0
$139$	1.52515e11	0.249305	0.124652	−	0.992200i	$-0.460218\pi$
$139$	1.52515e11	0.249305	0.124652	+	0.992200i	$0.460218\pi$
$140$	0	0
$141$	−9.90443e11	−1.49666
$142$	0	0
$143$	−8.56861e11	−1.19829
$144$	0	0
$145$	1.15325e12	1.49416
$146$	0	0
$147$	−4.01620e11	−0.482581
$148$	0	0
$149$	5.68879e11	0.634593	0.317296	−	0.948326i	$-0.397225\pi$
$149$	5.68879e11	0.634593	0.317296	+	0.948326i	$0.397225\pi$
$150$	0	0
$151$	−4.64770e10	−0.0481798	−0.0240899	−	0.999710i	$-0.507669\pi$
$151$	−4.64770e10	−0.0481798	−0.0240899	+	0.999710i	$0.507669\pi$
$152$	0	0
$153$	5.57480e11	0.537559
$154$	0	0
$155$	1.39082e12	1.24867
$156$	0	0
$157$	1.71773e11	0.143717	0.0718585	−	0.997415i	$-0.477107\pi$
$157$	1.71773e11	0.143717	0.0718585	+	0.997415i	$0.477107\pi$
$158$	0	0
$159$	2.52547e12	1.97087
$160$	0	0
$161$	2.30372e11	0.167837
$162$	0	0
$163$	−7.11815e11	−0.484546	−0.242273	−	0.970208i	$-0.577893\pi$
$163$	−7.11815e11	−0.484546	−0.242273	+	0.970208i	$0.577893\pi$
$164$	0	0
$165$	2.69436e12	1.71512
$166$	0	0
$167$	1.30459e11	0.0777202	0.0388601	−	0.999245i	$-0.487627\pi$
$167$	1.30459e11	0.0777202	0.0388601	+	0.999245i	$0.487627\pi$
$168$	0	0
$169$	−3.94880e11	−0.220338
$170$	0	0
$171$	−2.14817e12	−1.12355
$172$	0	0
$173$	1.40825e12	0.690919	0.345460	−	0.938434i	$-0.387723\pi$
$173$	1.40825e12	0.690919	0.345460	+	0.938434i	$0.387723\pi$
$174$	0	0
$175$	2.06855e10	0.00952699
$176$	0	0
$177$	−2.34922e12	−1.01641
$178$	0	0
$179$	−1.44332e12	−0.587042	−0.293521	−	0.955953i	$-0.594827\pi$
$179$	−1.44332e12	−0.587042	−0.293521	+	0.955953i	$0.594827\pi$
$180$	0	0
$181$	3.12844e12	1.19700	0.598502	−	0.801122i	$-0.295763\pi$
$181$	3.12844e12	1.19700	0.598502	+	0.801122i	$0.295763\pi$
$182$	0	0
$183$	−3.95471e12	−1.42441
$184$	0	0
$185$	3.79779e12	1.28851
$186$	0	0
$187$	3.94618e12	1.26197
$188$	0	0
$189$	−1.37902e12	−0.415941
$190$	0	0
$191$	3.15325e12	0.897583	0.448792	−	0.893636i	$-0.351854\pi$
$191$	3.15325e12	0.897583	0.448792	+	0.893636i	$0.351854\pi$
$192$	0	0
$193$	−3.47912e12	−0.935199	−0.467599	−	0.883940i	$-0.654881\pi$
$193$	−3.47912e12	−0.935199	−0.467599	+	0.883940i	$0.654881\pi$
$194$	0	0
$195$	−4.39368e12	−1.11593
$196$	0	0
$197$	−1.71468e12	−0.411736	−0.205868	−	0.978580i	$-0.566002\pi$
$197$	−1.71468e12	−0.411736	−0.205868	+	0.978580i	$0.566002\pi$
$198$	0	0
$199$	−1.04789e12	−0.238026	−0.119013	−	0.992893i	$-0.537973\pi$
$199$	−1.04789e12	−0.238026	−0.119013	+	0.992893i	$0.537973\pi$
$200$	0	0
$201$	−5.77134e12	−1.24079
$202$	0	0
$203$	5.72456e12	1.16550
$204$	0	0
$205$	1.96840e12	0.379722
$206$	0	0
$207$	6.76045e11	0.123634
$208$	0	0
$209$	−1.52061e13	−2.63762
$210$	0	0
$211$	1.14266e13	1.88089	0.940445	−	0.339946i	$-0.110409\pi$
$211$	1.14266e13	1.88089	0.940445	+	0.339946i	$0.110409\pi$
$212$	0	0
$213$	−1.42823e12	−0.223209
$214$	0	0
$215$	−2.12793e12	−0.315897
$216$	0	0
$217$	6.90386e12	0.974010
$218$	0	0
$219$	−1.15279e13	−1.54635
$220$	0	0
$221$	−6.43502e12	−0.821093
$222$	0	0
$223$	−9.17386e12	−1.11397	−0.556987	−	0.830521i	$-0.688043\pi$
$223$	−9.17386e12	−1.11397	−0.556987	+	0.830521i	$0.688043\pi$
$224$	0	0
$225$	6.07031e10	0.00701790
$226$	0	0
$227$	3.88316e11	0.0427605	0.0213803	−	0.999771i	$-0.493194\pi$
$227$	3.88316e11	0.0427605	0.0213803	+	0.999771i	$0.493194\pi$
$228$	0	0
$229$	1.37176e13	1.43941	0.719705	−	0.694280i	$-0.244277\pi$
$229$	1.37176e13	1.43941	0.719705	+	0.694280i	$0.244277\pi$
$230$	0	0
$231$	1.33744e13	1.33786
$232$	0	0
$233$	1.27243e12	0.121388	0.0606942	−	0.998156i	$-0.480669\pi$
$233$	1.27243e12	0.121388	0.0606942	+	0.998156i	$0.480669\pi$
$234$	0	0
$235$	−1.31690e13	−1.19862
$236$	0	0
$237$	−1.78304e13	−1.54898
$238$	0	0
$239$	−9.28197e12	−0.769930	−0.384965	−	0.922931i	$-0.625787\pi$
$239$	−9.28197e12	−0.769930	−0.384965	+	0.922931i	$0.625787\pi$
$240$	0	0
$241$	7.46994e12	0.591866	0.295933	−	0.955209i	$-0.404370\pi$
$241$	7.46994e12	0.591866	0.295933	+	0.955209i	$0.404370\pi$
$242$	0	0
$243$	−1.36383e13	−1.03259
$244$	0	0
$245$	−5.33998e12	−0.386479
$246$	0	0
$247$	2.47965e13	1.71616
$248$	0	0
$249$	3.26865e13	2.16407
$250$	0	0
$251$	1.53661e13	0.973548	0.486774	−	0.873528i	$-0.338173\pi$
$251$	1.53661e13	0.973548	0.486774	+	0.873528i	$0.338173\pi$
$252$	0	0
$253$	4.78546e12	0.290242
$254$	0	0
$255$	2.02346e13	1.17523
$256$	0	0
$257$	1.14664e13	0.637962	0.318981	−	0.947761i	$-0.396659\pi$
$257$	1.14664e13	0.637962	0.318981	+	0.947761i	$0.396659\pi$
$258$	0	0
$259$	1.88517e13	1.00508
$260$	0	0
$261$	1.67992e13	0.858549
$262$	0	0
$263$	−3.22103e13	−1.57848	−0.789238	−	0.614088i	$-0.789524\pi$
$263$	−3.22103e13	−1.57848	−0.789238	+	0.614088i	$0.789524\pi$
$264$	0	0
$265$	3.35789e13	1.57839
$266$	0	0
$267$	1.17355e13	0.529285
$268$	0	0
$269$	−2.79224e13	−1.20869	−0.604345	−	0.796723i	$-0.706565\pi$
$269$	−2.79224e13	−1.20869	−0.604345	+	0.796723i	$0.706565\pi$
$270$	0	0
$271$	−4.40610e13	−1.83115	−0.915573	−	0.402152i	$-0.868262\pi$
$271$	−4.40610e13	−1.83115	−0.915573	+	0.402152i	$0.868262\pi$
$272$	0	0
$273$	−2.18096e13	−0.870470
$274$	0	0
$275$	4.29693e11	0.0164751
$276$	0	0
$277$	4.52311e13	1.66647	0.833237	−	0.552916i	$-0.186485\pi$
$277$	4.52311e13	1.66647	0.833237	+	0.552916i	$0.186485\pi$
$278$	0	0
$279$	2.02599e13	0.717488
$280$	0	0
$281$	−1.58668e12	−0.0540261	−0.0270131	−	0.999635i	$-0.508600\pi$
$281$	−1.58668e12	−0.0540261	−0.0270131	+	0.999635i	$0.508600\pi$
$282$	0	0
$283$	−5.49142e13	−1.79829	−0.899144	−	0.437653i	$-0.855810\pi$
$283$	−5.49142e13	−1.79829	−0.899144	+	0.437653i	$0.855810\pi$
$284$	0	0
$285$	−7.79713e13	−2.45634
$286$	0	0
$287$	9.77085e12	0.296198
$288$	0	0
$289$	−4.63613e12	−0.135275
$290$	0	0
$291$	5.32445e13	1.49576
$292$	0	0
$293$	4.14937e12	0.112256	0.0561281	−	0.998424i	$-0.482124\pi$
$293$	4.14937e12	0.112256	0.0561281	+	0.998424i	$0.482124\pi$
$294$	0	0
$295$	−3.12354e13	−0.814004
$296$	0	0
$297$	−2.86461e13	−0.719290
$298$	0	0
$299$	−7.80362e12	−0.188845
$300$	0	0
$301$	−1.05627e13	−0.246412
$302$	0	0
$303$	−8.90416e12	−0.200290
$304$	0	0
$305$	−5.25822e13	−1.14075
$306$	0	0
$307$	8.76112e11	0.0183357	0.00916787	−	0.999958i	$-0.497082\pi$
$307$	8.76112e11	0.0183357	0.00916787	+	0.999958i	$0.497082\pi$
$308$	0	0
$309$	−1.25558e13	−0.253555
$310$	0	0
$311$	6.63328e13	1.29284	0.646422	−	0.762980i	$-0.276265\pi$
$311$	6.63328e13	1.29284	0.646422	+	0.762980i	$0.276265\pi$
$312$	0	0
$313$	7.98746e13	1.50285	0.751424	−	0.659820i	$-0.229367\pi$
$313$	7.98746e13	1.50285	0.751424	+	0.659820i	$0.229367\pi$
$314$	0	0
$315$	2.51219e13	0.456398
$316$	0	0
$317$	−6.12431e11	−0.0107456	−0.00537280	−	0.999986i	$-0.501710\pi$
$317$	−6.12431e11	−0.0107456	−0.00537280	+	0.999986i	$0.501710\pi$
$318$	0	0
$319$	1.18915e14	2.01552
$320$	0	0
$321$	−1.36825e14	−2.24071
$322$	0	0
$323$	−1.14197e14	−1.80735
$324$	0	0
$325$	−7.00699e11	−0.0107195
$326$	0	0
$327$	−5.25058e13	−0.776596
$328$	0	0
$329$	−6.53692e13	−0.934966
$330$	0	0
$331$	7.69109e13	1.06398	0.531991	−	0.846750i	$-0.321444\pi$
$331$	7.69109e13	1.06398	0.531991	+	0.846750i	$0.321444\pi$
$332$	0	0
$333$	5.53219e13	0.740379
$334$	0	0
$335$	−7.67363e13	−0.993698
$336$	0	0
$337$	−6.66260e12	−0.0834986	−0.0417493	−	0.999128i	$-0.513293\pi$
$337$	−6.66260e12	−0.0834986	−0.0417493	+	0.999128i	$0.513293\pi$
$338$	0	0
$339$	−8.47805e13	−1.02848
$340$	0	0
$341$	1.43412e14	1.68436
$342$	0	0
$343$	−9.55076e13	−1.08623
$344$	0	0
$345$	2.45381e13	0.270294
$346$	0	0
$347$	1.59901e13	0.170623	0.0853116	−	0.996354i	$-0.472811\pi$
$347$	1.59901e13	0.170623	0.0853116	+	0.996354i	$0.472811\pi$
$348$	0	0
$349$	1.50287e14	1.55375	0.776876	−	0.629654i	$-0.216803\pi$
$349$	1.50287e14	1.55375	0.776876	+	0.629654i	$0.216803\pi$
$350$	0	0
$351$	4.67130e13	0.468003
$352$	0	0
$353$	−9.89478e13	−0.960827	−0.480414	−	0.877042i	$-0.659513\pi$
$353$	−9.89478e13	−0.960827	−0.480414	+	0.877042i	$0.659513\pi$
$354$	0	0
$355$	−1.89899e13	−0.178759
$356$	0	0
$357$	1.00442e14	0.916726
$358$	0	0
$359$	4.58069e13	0.405426	0.202713	−	0.979238i	$-0.435024\pi$
$359$	4.58069e13	0.405426	0.202713	+	0.979238i	$0.435024\pi$
$360$	0	0
$361$	3.23554e14	2.77752
$362$	0	0
$363$	1.26971e14	1.05735
$364$	0	0
$365$	−1.53276e14	−1.23841
$366$	0	0
$367$	6.94482e12	0.0544500	0.0272250	−	0.999629i	$-0.491333\pi$
$367$	6.94482e12	0.0544500	0.0272250	+	0.999629i	$0.491333\pi$
$368$	0	0
$369$	2.86733e13	0.218189
$370$	0	0
$371$	1.66681e14	1.23120
$372$	0	0
$373$	6.39415e13	0.458548	0.229274	−	0.973362i	$-0.426365\pi$
$373$	6.39415e13	0.458548	0.229274	+	0.973362i	$0.426365\pi$
$374$	0	0
$375$	−1.79288e14	−1.24848
$376$	0	0
$377$	−1.93914e14	−1.31139
$378$	0	0
$379$	6.60185e13	0.433661	0.216830	−	0.976209i	$-0.430428\pi$
$379$	6.60185e13	0.433661	0.216830	+	0.976209i	$0.430428\pi$
$380$	0	0
$381$	−1.45889e14	−0.930967
$382$	0	0
$383$	1.81624e14	1.12611	0.563055	−	0.826419i	$-0.309626\pi$
$383$	1.81624e14	1.12611	0.563055	+	0.826419i	$0.309626\pi$
$384$	0	0
$385$	1.77828e14	1.07143
$386$	0	0
$387$	−3.09972e13	−0.181515
$388$	0	0
$389$	2.19065e13	0.124695	0.0623476	−	0.998054i	$-0.480141\pi$
$389$	2.19065e13	0.124695	0.0623476	+	0.998054i	$0.480141\pi$
$390$	0	0
$391$	3.59387e13	0.198880
$392$	0	0
$393$	2.99836e13	0.161334
$394$	0	0
$395$	−2.37075e14	−1.24051
$396$	0	0
$397$	3.00980e13	0.153176	0.0765879	−	0.997063i	$-0.475597\pi$
$397$	3.00980e13	0.153176	0.0765879	+	0.997063i	$0.475597\pi$
$398$	0	0
$399$	−3.87039e14	−1.91604
$400$	0	0
$401$	−3.17010e13	−0.152679	−0.0763395	−	0.997082i	$-0.524323\pi$
$401$	−3.17010e13	−0.152679	−0.0763395	+	0.997082i	$0.524323\pi$
$402$	0	0
$403$	−2.33861e14	−1.09592
$404$	0	0
$405$	−2.74416e14	−1.25143
$406$	0	0
$407$	3.91602e14	1.73810
$408$	0	0
$409$	−2.07090e14	−0.894706	−0.447353	−	0.894357i	$-0.647633\pi$
$409$	−2.07090e14	−0.894706	−0.447353	+	0.894357i	$0.647633\pi$
$410$	0	0
$411$	−1.73257e14	−0.728720
$412$	0	0
$413$	−1.55048e14	−0.634954
$414$	0	0
$415$	4.34602e14	1.73311
$416$	0	0
$417$	8.06387e13	0.313181
$418$	0	0
$419$	1.03612e14	0.391952	0.195976	−	0.980609i	$-0.437213\pi$
$419$	1.03612e14	0.391952	0.195976	+	0.980609i	$0.437213\pi$
$420$	0	0
$421$	−2.80328e14	−1.03303	−0.516517	−	0.856277i	$-0.672772\pi$
$421$	−2.80328e14	−1.03303	−0.516517	+	0.856277i	$0.672772\pi$
$422$	0	0
$423$	−1.91831e14	−0.688727
$424$	0	0
$425$	3.22699e12	0.0112891
$426$	0	0
$427$	−2.61011e14	−0.889826
$428$	0	0
$429$	−4.53045e14	−1.50531
$430$	0	0
$431$	−5.16298e14	−1.67215	−0.836076	−	0.548614i	$-0.815156\pi$
$431$	−5.16298e14	−1.67215	−0.836076	+	0.548614i	$0.815156\pi$
$432$	0	0
$433$	1.27910e14	0.403852	0.201926	−	0.979401i	$-0.435280\pi$
$433$	1.27910e14	0.403852	0.201926	+	0.979401i	$0.435280\pi$
$434$	0	0
$435$	6.09752e14	1.87699
$436$	0	0
$437$	−1.38485e14	−0.415676
$438$	0	0
$439$	2.90652e14	0.850781	0.425391	−	0.905010i	$-0.360137\pi$
$439$	2.90652e14	0.850781	0.425391	+	0.905010i	$0.360137\pi$
$440$	0	0
$441$	−7.77867e13	−0.222072
$442$	0	0
$443$	4.18452e14	1.16527	0.582633	−	0.812735i	$-0.302023\pi$
$443$	4.18452e14	1.16527	0.582633	+	0.812735i	$0.302023\pi$
$444$	0	0
$445$	1.56036e14	0.423883
$446$	0	0
$447$	3.00781e14	0.797186
$448$	0	0
$449$	2.33717e14	0.604416	0.302208	−	0.953242i	$-0.402276\pi$
$449$	2.33717e14	0.604416	0.302208	+	0.953242i	$0.402276\pi$
$450$	0	0
$451$	2.02967e14	0.512218
$452$	0	0
$453$	−2.45736e13	−0.0605242
$454$	0	0
$455$	−2.89983e14	−0.697123
$456$	0	0
$457$	2.93493e14	0.688745	0.344373	−	0.938833i	$-0.388092\pi$
$457$	2.93493e14	0.688745	0.344373	+	0.938833i	$0.388092\pi$
$458$	0	0
$459$	−2.15132e14	−0.492872
$460$	0	0
$461$	7.48297e14	1.67386	0.836930	−	0.547310i	$-0.184348\pi$
$461$	7.48297e14	1.67386	0.836930	+	0.547310i	$0.184348\pi$
$462$	0	0
$463$	−4.98429e14	−1.08870	−0.544349	−	0.838859i	$-0.683223\pi$
$463$	−4.98429e14	−1.08870	−0.544349	+	0.838859i	$0.683223\pi$
$464$	0	0
$465$	7.35365e14	1.56860
$466$	0	0
$467$	−6.24887e14	−1.30184	−0.650921	−	0.759145i	$-0.725617\pi$
$467$	−6.24887e14	−1.30184	−0.650921	+	0.759145i	$0.725617\pi$
$468$	0	0
$469$	−3.80908e14	−0.775122
$470$	0	0
$471$	9.08212e13	0.180540
$472$	0	0
$473$	−2.19417e14	−0.426122
$474$	0	0
$475$	−1.24348e13	−0.0235952
$476$	0	0
$477$	4.89139e14	0.906946
$478$	0	0
$479$	−5.09440e14	−0.923098	−0.461549	−	0.887115i	$-0.652706\pi$
$479$	−5.09440e14	−0.923098	−0.461549	+	0.887115i	$0.652706\pi$
$480$	0	0
$481$	−6.38584e14	−1.13089
$482$	0	0
$483$	1.21804e14	0.210839
$484$	0	0
$485$	7.07944e14	1.19790
$486$	0	0
$487$	−6.48709e12	−0.0107310	−0.00536551	−	0.999986i	$-0.501708\pi$
$487$	−6.48709e12	−0.0107310	−0.00536551	+	0.999986i	$0.501708\pi$
$488$	0	0
$489$	−3.76356e14	−0.608695
$490$	0	0
$491$	−2.25931e13	−0.0357296	−0.0178648	−	0.999840i	$-0.505687\pi$
$491$	−2.25931e13	−0.0357296	−0.0178648	+	0.999840i	$0.505687\pi$
$492$	0	0
$493$	8.93048e14	1.38107
$494$	0	0
$495$	5.21849e14	0.789254
$496$	0	0
$497$	−9.42634e13	−0.139439
$498$	0	0
$499$	3.84953e14	0.556999	0.278500	−	0.960436i	$-0.410163\pi$
$499$	3.84953e14	0.556999	0.278500	+	0.960436i	$0.410163\pi$
$500$	0	0
$501$	6.89772e13	0.0976333
$502$	0	0
$503$	1.27749e15	1.76903	0.884515	−	0.466512i	$-0.154490\pi$
$503$	1.27749e15	1.76903	0.884515	+	0.466512i	$0.154490\pi$
$504$	0	0
$505$	−1.18390e14	−0.160404
$506$	0	0
$507$	−2.08784e14	−0.276792
$508$	0	0
$509$	1.06672e15	1.38389	0.691945	−	0.721951i	$-0.256754\pi$
$509$	1.06672e15	1.38389	0.691945	+	0.721951i	$0.256754\pi$
$510$	0	0
$511$	−7.60843e14	−0.966006
$512$	0	0
$513$	8.28980e14	1.03015
$514$	0	0
$515$	−1.66943e14	−0.203062
$516$	0	0
$517$	−1.35790e15	−1.61685
$518$	0	0
$519$	7.44581e14	0.867944
$520$	0	0
$521$	−1.15548e15	−1.31873	−0.659365	−	0.751823i	$-0.729175\pi$
$521$	−1.15548e15	−1.31873	−0.659365	+	0.751823i	$0.729175\pi$
$522$	0	0
$523$	2.80759e11	0.000313744 0	0.000156872	−	1.00000i	$-0.499950\pi$
$523$	2.80759e11	0.000313744 0	0.000156872		1.00000i	$0.499950\pi$
$524$	0	0
$525$	1.09370e13	0.0119680
$526$	0	0
$527$	1.07702e15	1.15416
$528$	0	0
$529$	−9.09228e14	−0.954259
$530$	0	0
$531$	−4.55002e14	−0.467728
$532$	0	0
$533$	−3.30978e14	−0.333272
$534$	0	0
$535$	−1.81923e15	−1.79449
$536$	0	0
$537$	−7.63119e14	−0.737452
$538$	0	0
$539$	−5.50621e14	−0.521332
$540$	0	0
$541$	−6.20845e14	−0.575968	−0.287984	−	0.957635i	$-0.592985\pi$
$541$	−6.20845e14	−0.575968	−0.287984	+	0.957635i	$0.592985\pi$
$542$	0	0
$543$	1.65409e15	1.50369
$544$	0	0
$545$	−6.98123e14	−0.621944
$546$	0	0
$547$	−8.54098e14	−0.745723	−0.372862	−	0.927887i	$-0.621623\pi$
$547$	−8.54098e14	−0.745723	−0.372862	+	0.927887i	$0.621623\pi$
$548$	0	0
$549$	−7.65957e14	−0.655476
$550$	0	0
$551$	−3.44124e15	−2.88657
$552$	0	0
$553$	−1.17681e15	−0.967646
$554$	0	0
$555$	2.00800e15	1.61864
$556$	0	0
$557$	4.36204e14	0.344736	0.172368	−	0.985033i	$-0.444858\pi$
$557$	4.36204e14	0.344736	0.172368	+	0.985033i	$0.444858\pi$
$558$	0	0
$559$	3.57802e14	0.277254
$560$	0	0
$561$	2.08645e15	1.58530
$562$	0	0
$563$	7.00249e14	0.521743	0.260871	−	0.965374i	$-0.415990\pi$
$563$	7.00249e14	0.521743	0.260871	+	0.965374i	$0.415990\pi$
$564$	0	0
$565$	−1.12725e15	−0.823671
$566$	0	0
$567$	−1.36216e15	−0.976164
$568$	0	0
$569$	−1.90009e15	−1.33554	−0.667768	−	0.744369i	$-0.732750\pi$
$569$	−1.90009e15	−1.33554	−0.667768	+	0.744369i	$0.732750\pi$
$570$	0	0
$571$	1.19112e15	0.821215	0.410608	−	0.911812i	$-0.365317\pi$
$571$	1.19112e15	0.821215	0.410608	+	0.911812i	$0.365317\pi$
$572$	0	0
$573$	1.66721e15	1.12756
$574$	0	0
$575$	3.91331e12	0.00259640
$576$	0	0
$577$	1.82145e15	1.18563	0.592817	−	0.805337i	$-0.298016\pi$
$577$	1.82145e15	1.18563	0.592817	+	0.805337i	$0.298016\pi$
$578$	0	0
$579$	−1.83950e15	−1.17481
$580$	0	0
$581$	2.15731e15	1.35190
$582$	0	0
$583$	3.46242e15	2.12914
$584$	0	0
$585$	−8.50978e14	−0.513524
$586$	0	0
$587$	1.09273e15	0.647149	0.323574	−	0.946203i	$-0.395115\pi$
$587$	1.09273e15	0.647149	0.323574	+	0.946203i	$0.395115\pi$
$588$	0	0
$589$	−4.15016e15	−2.41230
$590$	0	0
$591$	−9.06597e14	−0.517229
$592$	0	0
$593$	1.47734e15	0.827330	0.413665	−	0.910429i	$-0.364249\pi$
$593$	1.47734e15	0.827330	0.413665	+	0.910429i	$0.364249\pi$
$594$	0	0
$595$	1.33548e15	0.734168
$596$	0	0
$597$	−5.54048e14	−0.299012
$598$	0	0
$599$	8.43489e14	0.446922	0.223461	−	0.974713i	$-0.428264\pi$
$599$	8.43489e14	0.446922	0.223461	+	0.974713i	$0.428264\pi$
$600$	0	0
$601$	3.76961e15	1.96104	0.980520	−	0.196417i	$-0.0629306\pi$
$601$	3.76961e15	1.96104	0.980520	+	0.196417i	$0.0629306\pi$
$602$	0	0
$603$	−1.11781e15	−0.570980
$604$	0	0
$605$	1.68822e15	0.846790
$606$	0	0
$607$	−9.80174e14	−0.482798	−0.241399	−	0.970426i	$-0.577606\pi$
$607$	−9.80174e14	−0.482798	−0.241399	+	0.970426i	$0.577606\pi$
$608$	0	0
$609$	3.02673e15	1.46413
$610$	0	0
$611$	2.21432e15	1.05199
$612$	0	0
$613$	−2.87519e15	−1.34163	−0.670817	−	0.741623i	$-0.734056\pi$
$613$	−2.87519e15	−1.34163	−0.670817	+	0.741623i	$0.734056\pi$
$614$	0	0
$615$	1.04074e15	0.477013
$616$	0	0
$617$	1.72435e15	0.776348	0.388174	−	0.921586i	$-0.373106\pi$
$617$	1.72435e15	0.776348	0.388174	+	0.921586i	$0.373106\pi$
$618$	0	0
$619$	−2.85002e15	−1.26052	−0.630260	−	0.776384i	$-0.717052\pi$
$619$	−2.85002e15	−1.26052	−0.630260	+	0.776384i	$0.717052\pi$
$620$	0	0
$621$	−2.60886e14	−0.113357
$622$	0	0
$623$	7.74543e14	0.330644
$624$	0	0
$625$	−2.41278e15	−1.01199
$626$	0	0
$627$	−8.03986e15	−3.31342
$628$	0	0
$629$	2.94093e15	1.19098
$630$	0	0
$631$	−1.03945e15	−0.413659	−0.206829	−	0.978377i	$-0.566314\pi$
$631$	−1.03945e15	−0.413659	−0.206829	+	0.978377i	$0.566314\pi$
$632$	0	0
$633$	6.04155e15	2.36280
$634$	0	0
$635$	−1.93975e15	−0.745573
$636$	0	0
$637$	8.97895e14	0.339202
$638$	0	0
$639$	−2.76623e14	−0.102715
$640$	0	0
$641$	−2.10449e12	−0.000768117 0	−0.000384058	−	1.00000i	$-0.500122\pi$
$641$	−2.10449e12	−0.000768117 0	−0.000384058		1.00000i	$0.500122\pi$
$642$	0	0
$643$	4.20911e15	1.51018	0.755092	−	0.655619i	$-0.227592\pi$
$643$	4.20911e15	1.51018	0.755092	+	0.655619i	$0.227592\pi$
$644$	0	0
$645$	−1.12509e15	−0.396835
$646$	0	0
$647$	2.25242e15	0.781044	0.390522	−	0.920594i	$-0.372294\pi$
$647$	2.25242e15	0.781044	0.390522	+	0.920594i	$0.372294\pi$
$648$	0	0
$649$	−3.22078e15	−1.09803
$650$	0	0
$651$	3.65025e15	1.22357
$652$	0	0
$653$	3.41913e15	1.12692	0.563460	−	0.826144i	$-0.309470\pi$
$653$	3.41913e15	1.12692	0.563460	+	0.826144i	$0.309470\pi$
$654$	0	0
$655$	3.98665e14	0.129205
$656$	0	0
$657$	−2.23275e15	−0.711592
$658$	0	0
$659$	5.08319e15	1.59318	0.796592	−	0.604517i	$-0.206634\pi$
$659$	5.08319e15	1.59318	0.796592	+	0.604517i	$0.206634\pi$
$660$	0	0
$661$	−5.21637e15	−1.60791	−0.803953	−	0.594693i	$-0.797274\pi$
$661$	−5.21637e15	−1.60791	−0.803953	+	0.594693i	$0.797274\pi$
$662$	0	0
$663$	−3.40237e15	−1.03147
$664$	0	0
$665$	−5.14611e15	−1.53447
$666$	0	0
$667$	1.08298e15	0.317636
$668$	0	0
$669$	−4.85046e15	−1.39939
$670$	0	0
$671$	−5.42191e15	−1.53879
$672$	0	0
$673$	5.16121e14	0.144102	0.0720508	−	0.997401i	$-0.477046\pi$
$673$	5.16121e14	0.144102	0.0720508	+	0.997401i	$0.477046\pi$
$674$	0	0
$675$	−2.34253e13	−0.00643451
$676$	0	0
$677$	3.68680e15	0.996349	0.498175	−	0.867077i	$-0.334004\pi$
$677$	3.68680e15	0.996349	0.498175	+	0.867077i	$0.334004\pi$
$678$	0	0
$679$	3.51414e15	0.934404
$680$	0	0
$681$	2.05313e14	0.0537164
$682$	0	0
$683$	−4.15977e15	−1.07092	−0.535458	−	0.844562i	$-0.679861\pi$
$683$	−4.15977e15	−1.07092	−0.535458	+	0.844562i	$0.679861\pi$
$684$	0	0
$685$	−2.30364e15	−0.583602
$686$	0	0
$687$	7.25289e15	1.80821
$688$	0	0
$689$	−5.64616e15	−1.38531
$690$	0	0
$691$	2.43277e15	0.587451	0.293725	−	0.955890i	$-0.405105\pi$
$691$	2.43277e15	0.587451	0.293725	+	0.955890i	$0.405105\pi$
$692$	0	0
$693$	2.59039e15	0.615648
$694$	0	0
$695$	1.07218e15	0.250813
$696$	0	0
$697$	1.52428e15	0.350982
$698$	0	0
$699$	6.72769e14	0.152490
$700$	0	0
$701$	2.95533e15	0.659413	0.329707	−	0.944083i	$-0.393050\pi$
$701$	2.95533e15	0.659413	0.329707	+	0.944083i	$0.393050\pi$
$702$	0	0
$703$	−1.13325e16	−2.48926
$704$	0	0
$705$	−6.96281e15	−1.50572
$706$	0	0
$707$	−5.87674e14	−0.125121
$708$	0	0
$709$	1.09522e14	0.0229587	0.0114794	−	0.999934i	$-0.496346\pi$
$709$	1.09522e14	0.0229587	0.0114794	+	0.999934i	$0.496346\pi$
$710$	0	0
$711$	−3.45343e15	−0.712800
$712$	0	0
$713$	1.30608e15	0.265448
$714$	0	0
$715$	−6.02373e15	−1.20554
$716$	0	0
$717$	−4.90762e15	−0.967199
$718$	0	0
$719$	2.37930e15	0.461786	0.230893	−	0.972979i	$-0.425835\pi$
$719$	2.37930e15	0.461786	0.230893	+	0.972979i	$0.425835\pi$
$720$	0	0
$721$	−8.28683e14	−0.158396
$722$	0	0
$723$	3.94956e15	0.743511
$724$	0	0
$725$	9.72427e13	0.0180301
$726$	0	0
$727$	3.97241e15	0.725463	0.362731	−	0.931894i	$-0.381844\pi$
$727$	3.97241e15	0.725463	0.362731	+	0.931894i	$0.381844\pi$
$728$	0	0
$729$	−2.96022e14	−0.0532504
$730$	0	0
$731$	−1.64782e15	−0.291987
$732$	0	0
$733$	5.77077e15	1.00731	0.503654	−	0.863906i	$-0.331989\pi$
$733$	5.77077e15	1.00731	0.503654	+	0.863906i	$0.331989\pi$
$734$	0	0
$735$	−2.82339e15	−0.485501
$736$	0	0
$737$	−7.91251e15	−1.34043
$738$	0	0
$739$	−7.09611e15	−1.18434	−0.592170	−	0.805813i	$-0.701729\pi$
$739$	−7.09611e15	−1.18434	−0.592170	+	0.805813i	$0.701729\pi$
$740$	0	0
$741$	1.31106e16	2.15586
$742$	0	0
$743$	1.40436e15	0.227531	0.113766	−	0.993508i	$-0.463709\pi$
$743$	1.40436e15	0.227531	0.113766	+	0.993508i	$0.463709\pi$
$744$	0	0
$745$	3.99922e15	0.638433
$746$	0	0
$747$	6.33079e15	0.995851
$748$	0	0
$749$	−9.03042e15	−1.39977
$750$	0	0
$751$	1.24303e16	1.89873	0.949366	−	0.314173i	$-0.101727\pi$
$751$	1.24303e16	1.89873	0.949366	+	0.314173i	$0.101727\pi$
$752$	0	0
$753$	8.12446e15	1.22299
$754$	0	0
$755$	−3.26733e14	−0.0484714
$756$	0	0
$757$	1.32729e16	1.94061	0.970305	−	0.241883i	$-0.0777649\pi$
$757$	1.32729e16	1.94061	0.970305	+	0.241883i	$0.0777649\pi$
$758$	0	0
$759$	2.53020e15	0.364607
$760$	0	0
$761$	−1.95021e15	−0.276991	−0.138495	−	0.990363i	$-0.544227\pi$
$761$	−1.95021e15	−0.276991	−0.138495	+	0.990363i	$0.544227\pi$
$762$	0	0
$763$	−3.46539e15	−0.485140
$764$	0	0
$765$	3.91908e15	0.540812
$766$	0	0
$767$	5.25211e15	0.714430
$768$	0	0
$769$	−2.20177e15	−0.295242	−0.147621	−	0.989044i	$-0.547162\pi$
$769$	−2.20177e15	−0.295242	−0.147621	+	0.989044i	$0.547162\pi$
$770$	0	0
$771$	6.06259e15	0.801418
$772$	0	0
$773$	1.21701e16	1.58602	0.793010	−	0.609209i	$-0.208513\pi$
$773$	1.21701e16	1.58602	0.793010	+	0.609209i	$0.208513\pi$
$774$	0	0
$775$	1.17275e14	0.0150677
$776$	0	0
$777$	9.96742e15	1.26260
$778$	0	0
$779$	−5.87361e15	−0.733583
$780$	0	0
$781$	−1.95811e15	−0.241133
$782$	0	0
$783$	−6.48281e15	−0.787179
$784$	0	0
$785$	1.20757e15	0.144587
$786$	0	0
$787$	−3.49830e15	−0.413043	−0.206522	−	0.978442i	$-0.566214\pi$
$787$	−3.49830e15	−0.413043	−0.206522	+	0.978442i	$0.566214\pi$
$788$	0	0
$789$	−1.70304e16	−1.98291
$790$	0	0
$791$	−5.59551e15	−0.642494
$792$	0	0
$793$	8.84148e15	1.00120
$794$	0	0
$795$	1.77541e16	1.98280
$796$	0	0
$797$	−5.11204e15	−0.563085	−0.281542	−	0.959549i	$-0.590846\pi$
$797$	−5.11204e15	−0.563085	−0.281542	+	0.959549i	$0.590846\pi$
$798$	0	0
$799$	−1.01978e16	−1.10790
$800$	0	0
$801$	2.27296e15	0.243564
$802$	0	0
$803$	−1.58048e16	−1.67052
$804$	0	0
$805$	1.61952e15	0.168853
$806$	0	0
$807$	−1.47633e16	−1.51838
$808$	0	0
$809$	−2.92781e15	−0.297048	−0.148524	−	0.988909i	$-0.547452\pi$
$809$	−2.92781e15	−0.297048	−0.148524	+	0.988909i	$0.547452\pi$
$810$	0	0
$811$	1.13555e16	1.13655	0.568277	−	0.822838i	$-0.307610\pi$
$811$	1.13555e16	1.13655	0.568277	+	0.822838i	$0.307610\pi$
$812$	0	0
$813$	−2.32962e16	−2.30031
$814$	0	0
$815$	−5.00406e15	−0.487478
$816$	0	0
$817$	6.34965e15	0.610279
$818$	0	0
$819$	−4.22414e15	−0.400568
$820$	0	0
$821$	−3.50553e14	−0.0327994	−0.0163997	−	0.999866i	$-0.505220\pi$
$821$	−3.50553e14	−0.0327994	−0.0163997	+	0.999866i	$0.505220\pi$
$822$	0	0
$823$	−3.72462e15	−0.343861	−0.171930	−	0.985109i	$-0.555000\pi$
$823$	−3.72462e15	−0.343861	−0.171930	+	0.985109i	$0.555000\pi$
$824$	0	0
$825$	2.27190e14	0.0206963
$826$	0	0
$827$	9.10441e15	0.818411	0.409205	−	0.912442i	$-0.365806\pi$
$827$	9.10441e15	0.818411	0.409205	+	0.912442i	$0.365806\pi$
$828$	0	0
$829$	−3.32943e15	−0.295338	−0.147669	−	0.989037i	$-0.547177\pi$
$829$	−3.32943e15	−0.295338	−0.147669	+	0.989037i	$0.547177\pi$
$830$	0	0
$831$	2.39149e16	2.09345
$832$	0	0
$833$	−4.13516e15	−0.357227
$834$	0	0
$835$	9.17127e14	0.0781905
$836$	0	0
$837$	−7.81830e15	−0.657843
$838$	0	0
$839$	7.78240e15	0.646283	0.323141	−	0.946351i	$-0.395261\pi$
$839$	7.78240e15	0.646283	0.323141	+	0.946351i	$0.395261\pi$
$840$	0	0
$841$	1.47107e16	1.20575
$842$	0	0
$843$	−8.38919e14	−0.0678685
$844$	0	0
$845$	−2.77601e15	−0.221671
$846$	0	0
$847$	8.38012e15	0.660528
$848$	0	0
$849$	−2.90346e16	−2.25904
$850$	0	0
$851$	3.56641e15	0.273917
$852$	0	0
$853$	2.40598e15	0.182420	0.0912099	−	0.995832i	$-0.470927\pi$
$853$	2.40598e15	0.182420	0.0912099	+	0.995832i	$0.470927\pi$
$854$	0	0
$855$	−1.51017e16	−1.13035
$856$	0	0
$857$	−7.74432e15	−0.572254	−0.286127	−	0.958192i	$-0.592368\pi$
$857$	−7.74432e15	−0.572254	−0.286127	+	0.958192i	$0.592368\pi$
$858$	0	0
$859$	−6.81239e15	−0.496978	−0.248489	−	0.968635i	$-0.579934\pi$
$859$	−6.81239e15	−0.496978	−0.248489	+	0.968635i	$0.579934\pi$
$860$	0	0
$861$	5.16611e15	0.372088
$862$	0	0
$863$	6.57977e15	0.467898	0.233949	−	0.972249i	$-0.424835\pi$
$863$	6.57977e15	0.467898	0.233949	+	0.972249i	$0.424835\pi$
$864$	0	0
$865$	9.90002e15	0.695101
$866$	0	0
$867$	−2.45125e15	−0.169935
$868$	0	0
$869$	−2.44455e16	−1.67336
$870$	0	0
$871$	1.29029e16	0.872142
$872$	0	0
$873$	1.03125e16	0.688313
$874$	0	0
$875$	−1.18330e16	−0.779923
$876$	0	0
$877$	−1.74871e16	−1.13820	−0.569101	−	0.822267i	$-0.692709\pi$
$877$	−1.74871e16	−1.13820	−0.569101	+	0.822267i	$0.692709\pi$
$878$	0	0
$879$	2.19389e15	0.141018
$880$	0	0
$881$	−1.04152e16	−0.661149	−0.330575	−	0.943780i	$-0.607243\pi$
$881$	−1.04152e16	−0.661149	−0.330575	+	0.943780i	$0.607243\pi$
$882$	0	0
$883$	1.62388e15	0.101805	0.0509025	−	0.998704i	$-0.483790\pi$
$883$	1.62388e15	0.101805	0.0509025	+	0.998704i	$0.483790\pi$
$884$	0	0
$885$	−1.65150e16	−1.02256
$886$	0	0
$887$	1.02417e16	0.626313	0.313157	−	0.949701i	$-0.398614\pi$
$887$	1.02417e16	0.626313	0.313157	+	0.949701i	$0.398614\pi$
$888$	0	0
$889$	−9.62865e15	−0.581575
$890$	0	0
$891$	−2.82959e16	−1.68809
$892$	0	0
$893$	3.92958e16	2.31560
$894$	0	0
$895$	−1.01465e16	−0.590595
$896$	0	0
$897$	−4.12599e15	−0.237230
$898$	0	0
$899$	3.24551e16	1.84334
$900$	0	0
$901$	2.60028e16	1.45893
$902$	0	0
$903$	−5.58481e15	−0.309546
$904$	0	0
$905$	2.19929e16	1.20425
$906$	0	0
$907$	−9.93380e15	−0.537372	−0.268686	−	0.963228i	$-0.586589\pi$
$907$	−9.93380e15	−0.537372	−0.268686	+	0.963228i	$0.586589\pi$
$908$	0	0
$909$	−1.72458e15	−0.0921682
$910$	0	0
$911$	−2.83862e16	−1.49884	−0.749420	−	0.662094i	$-0.769668\pi$
$911$	−2.83862e16	−1.49884	−0.749420	+	0.662094i	$0.769668\pi$
$912$	0	0
$913$	4.48131e16	2.33785
$914$	0	0
$915$	−2.78016e16	−1.43303
$916$	0	0
$917$	1.97892e15	0.100785
$918$	0	0
$919$	−3.04547e15	−0.153257	−0.0766284	−	0.997060i	$-0.524416\pi$
$919$	−3.04547e15	−0.153257	−0.0766284	+	0.997060i	$0.524416\pi$
$920$	0	0
$921$	4.63224e14	0.0230337
$922$	0	0
$923$	3.19308e15	0.156892
$924$	0	0
$925$	3.20233e14	0.0155484
$926$	0	0
$927$	−2.43183e15	−0.116680
$928$	0	0
$929$	−2.24387e16	−1.06392	−0.531962	−	0.846768i	$-0.678545\pi$
$929$	−2.24387e16	−1.06392	−0.531962	+	0.846768i	$0.678545\pi$
$930$	0	0
$931$	1.59343e16	0.746636
$932$	0	0
$933$	3.50719e16	1.62409
$934$	0	0
$935$	2.77416e16	1.26960
$936$	0	0
$937$	−1.14631e16	−0.518482	−0.259241	−	0.965813i	$-0.583472\pi$
$937$	−1.14631e16	−0.518482	−0.259241	+	0.965813i	$0.583472\pi$
$938$	0	0
$939$	4.22318e16	1.88790
$940$	0	0
$941$	−6.08259e15	−0.268748	−0.134374	−	0.990931i	$-0.542902\pi$
$941$	−6.08259e15	−0.268748	−0.134374	+	0.990931i	$0.542902\pi$
$942$	0	0
$943$	1.84847e15	0.0807230
$944$	0	0
$945$	−9.69453e15	−0.418458
$946$	0	0
$947$	1.95888e16	0.835762	0.417881	−	0.908502i	$-0.362773\pi$
$947$	1.95888e16	0.835762	0.417881	+	0.908502i	$0.362773\pi$
$948$	0	0
$949$	2.57728e16	1.08692
$950$	0	0
$951$	−3.23808e14	−0.0134988
$952$	0	0
$953$	−1.57650e16	−0.649654	−0.324827	−	0.945773i	$-0.605306\pi$
$953$	−1.57650e16	−0.649654	−0.324827	+	0.945773i	$0.605306\pi$
$954$	0	0
$955$	2.21673e16	0.903015
$956$	0	0
$957$	6.28734e16	2.53193
$958$	0	0
$959$	−1.14350e16	−0.455231
$960$	0	0
$961$	1.37326e16	0.540474
$962$	0	0
$963$	−2.65005e16	−1.03112
$964$	0	0
$965$	−2.44582e16	−0.940858
$966$	0	0
$967$	−1.75917e16	−0.669055	−0.334527	−	0.942386i	$-0.608577\pi$
$967$	−1.75917e16	−0.669055	−0.334527	+	0.942386i	$0.608577\pi$
$968$	0	0
$969$	−6.03792e16	−2.27042
$970$	0	0
$971$	−1.46904e16	−0.546169	−0.273084	−	0.961990i	$-0.588044\pi$
$971$	−1.46904e16	−0.546169	−0.273084	+	0.961990i	$0.588044\pi$
$972$	0	0
$973$	5.32215e15	0.195644
$974$	0	0
$975$	−3.70478e14	−0.0134660
$976$	0	0
$977$	2.10824e16	0.757705	0.378853	−	0.925457i	$-0.376319\pi$
$977$	2.10824e16	0.757705	0.378853	+	0.925457i	$0.376319\pi$
$978$	0	0
$979$	1.60894e16	0.571787
$980$	0	0
$981$	−1.01695e16	−0.357370
$982$	0	0
$983$	2.57442e16	0.894613	0.447306	−	0.894381i	$-0.352383\pi$
$983$	2.57442e16	0.894613	0.447306	+	0.894381i	$0.352383\pi$
$984$	0	0
$985$	−1.20542e16	−0.414228
$986$	0	0
$987$	−3.45625e16	−1.17452
$988$	0	0
$989$	−1.99828e15	−0.0671547
$990$	0	0
$991$	2.31512e16	0.769428	0.384714	−	0.923036i	$-0.374300\pi$
$991$	2.31512e16	0.769428	0.384714	+	0.923036i	$0.374300\pi$
$992$	0	0
$993$	4.06649e16	1.33659
$994$	0	0
$995$	−7.36667e15	−0.239466
$996$	0	0
$997$	1.95960e16	0.630006	0.315003	−	0.949091i	$-0.397995\pi$
$997$	1.95960e16	0.630006	0.315003	+	0.949091i	$0.397995\pi$
$998$	0	0
$999$	−2.13487e16	−0.678832

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	32.12.a.c.1.2	yes	2
4.3	odd	2	inner	32.12.a.c.1.1	✓	2
8.3	odd	2		64.12.a.i.1.2		2
8.5	even	2		64.12.a.i.1.1		2
16.3	odd	4		256.12.b.i.129.1		4
16.5	even	4		256.12.b.i.129.2		4
16.11	odd	4		256.12.b.i.129.4		4
16.13	even	4		256.12.b.i.129.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
32.12.a.c.1.1	✓	2	4.3	odd	2	inner
32.12.a.c.1.2	yes	2	1.1	even	1	trivial
64.12.a.i.1.1		2	8.5	even	2
64.12.a.i.1.2		2	8.3	odd	2
256.12.b.i.129.1		4	16.3	odd	4
256.12.b.i.129.2		4	16.5	even	4
256.12.b.i.129.3		4	16.13	even	4
256.12.b.i.129.4		4	16.11	odd	4

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:	\( N \)	\(=\)	\( 32 = 2^{5} \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	32.a (trivial)

\(f(q)\)	\(=\)	\(q+528.727 q^{3} +7030.00 q^{5} +34896.0 q^{7} +102405. q^{9} +724884. q^{11} -1.18207e6 q^{13} +3.71695e6 q^{15} +5.44387e6 q^{17} -2.09772e7 q^{19} +1.84504e7 q^{21} +6.60168e6 q^{23} +592775. q^{25} -3.95181e7 q^{27} +1.64046e8 q^{29} +1.97841e8 q^{31} +3.83266e8 q^{33} +2.45319e8 q^{35} +5.40227e8 q^{37} -6.24990e8 q^{39} +2.79999e8 q^{41} -3.02692e8 q^{43} +7.19907e8 q^{45} -1.87326e9 q^{47} -7.59598e8 q^{49} +2.87832e9 q^{51} +4.77652e9 q^{53} +5.09594e9 q^{55} -1.10912e10 q^{57} -4.44316e9 q^{59} -7.47969e9 q^{61} +3.57352e9 q^{63} -8.30992e9 q^{65} -1.09155e10 q^{67} +3.49049e9 q^{69} -2.70127e9 q^{71} -2.18032e10 q^{73} +3.13416e8 q^{75} +2.52955e10 q^{77} -3.37233e10 q^{79} -3.90350e10 q^{81} +6.18211e10 q^{83} +3.82704e10 q^{85} +8.67358e10 q^{87} +2.21958e10 q^{89} -4.12493e10 q^{91} +1.04604e11 q^{93} -1.47470e11 q^{95} +1.00703e11 q^{97} +7.42318e10 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 14060 q^{5} + 204810 q^{9} - 2364132 q^{13} + 10887748 q^{17} + 36900864 q^{21} + 1185550 q^{25} + 328092988 q^{29} + 766531584 q^{33} + 1080453740 q^{37} + 559998644 q^{41} + 1439814300 q^{45} - 1519196462 q^{49}+ \cdots + 201406465188 q^{97}+O(q^{100}) \) 2 * q + 14060 * q^5 + 204810 * q^9 - 2364132 * q^13 + 10887748 * q^17 + 36900864 * q^21 + 1185550 * q^25 + 328092988 * q^29 + 766531584 * q^33 + 1080453740 * q^37 + 559998644 * q^41 + 1439814300 * q^45 - 1519196462 * q^49 + 9553038796 * q^53 - 22182451200 * q^57 - 14959370564 * q^61 - 16619847960 * q^65 + 6980972544 * q^69 - 43606336236 * q^73 + 50591084544 * q^77 - 78070028238 * q^81 + 76540868440 * q^85 + 44391542900 * q^89 + 209207771136 * q^93 + 201406465188 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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