LMFDB - Embedded newform 336.4.h.b.239.13

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [336,4,Mod(239,336)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(336, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 1, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("336.239");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 336 = 2^{4} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	336.h (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$19.8246417619$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			239.13
Character	$\chi$	$=$	336.239
Dual form			336.4.h.b.239.14

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.13969 - 5.06963i) q^{3} +14.6453i q^{5} -7.00000i q^{7} +(-24.4022 - 11.5556i) q^{9} +O(q^{10})$	\(q+(1.13969 - 5.06963i) q^{3} +14.6453i q^{5} -7.00000i q^{7} +(-24.4022 - 11.5556i) q^{9} +13.9027 q^{11} -45.4743 q^{13} +(74.2459 + 16.6910i) q^{15} +120.873i q^{17} +34.9253i q^{19} +(-35.4874 - 7.97783i) q^{21} -151.440 q^{23} -89.4834 q^{25} +(-86.3935 + 110.540i) q^{27} -224.601i q^{29} +44.2927i q^{31} +(15.8448 - 70.4815i) q^{33} +102.517 q^{35} -224.904 q^{37} +(-51.8266 + 230.538i) q^{39} +459.995i q^{41} +497.722i q^{43} +(169.235 - 357.377i) q^{45} +134.391 q^{47} -49.0000 q^{49} +(612.781 + 137.758i) q^{51} +282.447i q^{53} +203.608i q^{55} +(177.058 + 39.8040i) q^{57} +48.3707 q^{59} -343.594 q^{61} +(-80.8892 + 170.816i) q^{63} -665.982i q^{65} +678.793i q^{67} +(-172.594 + 767.743i) q^{69} +820.561 q^{71} +370.525 q^{73} +(-101.983 + 453.647i) q^{75} -97.3188i q^{77} -986.876i q^{79} +(461.936 + 563.965i) q^{81} -484.889 q^{83} -1770.22 q^{85} +(-1138.64 - 255.976i) q^{87} -980.557i q^{89} +318.320i q^{91} +(224.548 + 50.4800i) q^{93} -511.490 q^{95} -488.599 q^{97} +(-339.256 - 160.654i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 136 q^{9}+O(q^{10}) $ 24 * q - 136 * q^9	$ 24 q - 136 q^{9} + 96 q^{13} - 112 q^{21} + 168 q^{25} - 320 q^{33} - 864 q^{37} + 592 q^{45} - 1176 q^{49} - 1120 q^{57} - 2304 q^{61} + 1168 q^{69} + 3312 q^{73} - 968 q^{81} - 5808 q^{85} + 1776 q^{93} + 6480 q^{97}+O(q^{100}) $ 24 * q - 136 * q^9 + 96 * q^13 - 112 * q^21 + 168 * q^25 - 320 * q^33 - 864 * q^37 + 592 * q^45 - 1176 * q^49 - 1120 * q^57 - 2304 * q^61 + 1168 * q^69 + 3312 * q^73 - 968 * q^81 - 5808 * q^85 + 1776 * q^93 + 6480 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/336\mathbb{Z}\right)^\times$.

$n$	$85$	$113$	$127$	$241$
$\chi(n)$	$1$	$-1$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$	1.13969	−	5.06963i	0.219333	−	0.975650i
$3$	1.13969	−	5.06963i	0.219333	−	0.975650i
$4$		0			0
$5$			14.6453i			1.30991i	0.755667	+	0.654956i	$0.227313\pi$
$5$			14.6453i			1.30991i	−0.755667	+	0.654956i	$0.772687\pi$
$6$		0			0
$7$		−	7.00000i		−	0.377964i
$7$		−	7.00000i		−	0.377964i
$8$		0			0
$9$	−24.4022	−	11.5556i	−0.903786	−	0.427985i
$10$		0			0
$11$	13.9027			0.381075			0.190537	−	0.981680i	$-0.438977\pi$
$11$	13.9027			0.381075			0.190537	+	0.981680i	$0.438977\pi$
$12$		0			0
$13$	−45.4743			−0.970176			−0.485088	−	0.874465i	$-0.661213\pi$
$13$	−45.4743			−0.970176			−0.485088	+	0.874465i	$0.661213\pi$
$14$		0			0
$15$	74.2459	+	16.6910i	1.27801	+	0.287307i
$16$		0			0
$17$			120.873i			1.72447i	0.506506	+	0.862236i	$0.330937\pi$
$17$			120.873i			1.72447i	−0.506506	+	0.862236i	$0.669063\pi$
$18$		0			0
$19$			34.9253i			0.421706i	0.977518	+	0.210853i	$0.0676242\pi$
$19$			34.9253i			0.421706i	−0.977518	+	0.210853i	$0.932376\pi$
$20$		0			0
$21$	−35.4874	−	7.97783i	−0.368761	−	0.0829002i
$22$		0			0
$23$	−151.440			−1.37293			−0.686465	−	0.727163i	$-0.740838\pi$
$23$	−151.440			−1.37293			−0.686465	+	0.727163i	$0.740838\pi$
$24$		0			0
$25$	−89.4834			−0.715867
$26$		0			0
$27$	−86.3935	+	110.540i	−0.615794	+	0.787907i
$28$		0			0
$29$		−	224.601i		−	1.43819i	−0.694913	−	0.719093i	$-0.744557\pi$
$29$		−	224.601i		−	1.43819i	0.694913	−	0.719093i	$-0.255443\pi$
$30$		0			0
$31$			44.2927i			0.256620i	0.991734	+	0.128310i	$0.0409552\pi$
$31$			44.2927i			0.256620i	−0.991734	+	0.128310i	$0.959045\pi$
$32$		0			0
$33$	15.8448	−	70.4815i	0.0835824	−	0.371795i
$34$		0			0
$35$	102.517			0.495100
$36$		0			0
$37$	−224.904			−0.999299			−0.499649	−	0.866228i	$-0.666538\pi$
$37$	−224.904			−0.999299			−0.499649	+	0.866228i	$0.666538\pi$
$38$		0			0
$39$	−51.8266	+	230.538i	−0.212792	+	0.946553i
$40$		0			0
$41$			459.995i			1.75217i	0.482153	+	0.876087i	$0.339855\pi$
$41$			459.995i			1.75217i	−0.482153	+	0.876087i	$0.660145\pi$
$42$		0			0
$43$			497.722i			1.76516i	0.470161	+	0.882581i	$0.344196\pi$
$43$			497.722i			1.76516i	−0.470161	+	0.882581i	$0.655804\pi$
$44$		0			0
$45$	169.235	−	357.377i	0.560623	−	1.18388i
$46$		0			0
$47$	134.391			0.417085			0.208543	−	0.978013i	$-0.433128\pi$
$47$	134.391			0.417085			0.208543	+	0.978013i	$0.433128\pi$
$48$		0			0
$49$	−49.0000			−0.142857
$50$		0			0
$51$	612.781	+	137.758i	1.68248	+	0.378234i
$52$		0			0
$53$			282.447i			0.732020i	0.930611	+	0.366010i	$0.119276\pi$
$53$			282.447i			0.732020i	−0.930611	+	0.366010i	$0.880724\pi$
$54$		0			0
$55$			203.608i			0.499174i
$56$		0			0
$57$	177.058	+	39.8040i	0.411438	+	0.0924942i
$58$		0			0
$59$	48.3707			0.106734			0.0533672	−	0.998575i	$-0.483005\pi$
$59$	48.3707			0.106734			0.0533672	+	0.998575i	$0.483005\pi$
$60$		0			0
$61$	−343.594			−0.721192			−0.360596	−	0.932722i	$-0.617427\pi$
$61$	−343.594			−0.721192			−0.360596	+	0.932722i	$0.617427\pi$
$62$		0			0
$63$	−80.8892	+	170.816i	−0.161763	+	0.341599i
$64$		0			0
$65$		−	665.982i		−	1.27084i
$66$		0			0
$67$			678.793i			1.23773i	0.785498	+	0.618864i	$0.212407\pi$
$67$			678.793i			1.23773i	−0.785498	+	0.618864i	$0.787593\pi$
$68$		0			0
$69$	−172.594	+	767.743i	−0.301129	+	1.33950i
$70$		0			0
$71$	820.561			1.37159			0.685794	−	0.727796i	$-0.259455\pi$
$71$	820.561			1.37159			0.685794	+	0.727796i	$0.259455\pi$
$72$		0			0
$73$	370.525			0.594064			0.297032	−	0.954868i	$-0.404003\pi$
$73$	370.525			0.594064			0.297032	+	0.954868i	$0.404003\pi$
$74$		0			0
$75$	−101.983	+	453.647i	−0.157014	+	0.698436i
$76$		0			0
$77$		−	97.3188i		−	0.144033i
$78$		0			0
$79$		−	986.876i		−	1.40547i	−0.711451	−	0.702736i	$-0.751962\pi$
$79$		−	986.876i		−	1.40547i	0.711451	−	0.702736i	$-0.248038\pi$
$80$		0			0
$81$	461.936	+	563.965i	0.633657	+	0.773614i
$82$		0			0
$83$	−484.889			−0.641247			−0.320624	−	0.947207i	$-0.603892\pi$
$83$	−484.889			−0.641247			−0.320624	+	0.947207i	$0.603892\pi$
$84$		0			0
$85$	−1770.22			−2.25891
$86$		0			0
$87$	−1138.64	−	255.976i	−1.40317	−	0.315442i
$88$		0			0
$89$		−	980.557i		−	1.16785i	−0.811807	−	0.583926i	$-0.801516\pi$
$89$		−	980.557i		−	1.16785i	0.811807	−	0.583926i	$-0.198484\pi$
$90$		0			0
$91$			318.320i			0.366692i
$92$		0			0
$93$	224.548	+	50.4800i	0.250371	+	0.0562852i
$94$		0			0
$95$	−511.490			−0.552398
$96$		0			0
$97$	−488.599			−0.511440			−0.255720	−	0.966751i	$-0.582313\pi$
$97$	−488.599			−0.511440			−0.255720	+	0.966751i	$0.582313\pi$
$98$		0			0
$99$	−339.256	−	160.654i	−0.344410	−	0.163094i
$100$		0			0
$101$			171.807i			0.169262i	0.996412	+	0.0846309i	$0.0269711\pi$
$101$			171.807i			0.169262i	−0.996412	+	0.0846309i	$0.973029\pi$
$102$		0			0
$103$		−	1178.61i		−	1.12750i	−0.825946	−	0.563749i	$-0.809358\pi$
$103$		−	1178.61i		−	1.12750i	0.825946	−	0.563749i	$-0.190642\pi$
$104$		0			0
$105$	116.837	−	519.722i	0.108592	−	0.483044i
$106$		0			0
$107$	−401.039			−0.362336			−0.181168	−	0.983452i	$-0.557988\pi$
$107$	−401.039			−0.362336			−0.181168	+	0.983452i	$0.557988\pi$
$108$		0			0
$109$	−866.058			−0.761039			−0.380520	−	0.924773i	$-0.624255\pi$
$109$	−866.058			−0.761039			−0.380520	+	0.924773i	$0.624255\pi$
$110$		0			0
$111$	−256.321	+	1140.18i	−0.219180	+	0.974966i
$112$		0			0
$113$			419.806i			0.349487i	0.984614	+	0.174744i	$0.0559096\pi$
$113$			419.806i			0.349487i	−0.984614	+	0.174744i	$0.944090\pi$
$114$		0			0
$115$		−	2217.87i		−	1.79842i
$116$		0			0
$117$	1109.67	+	525.483i	0.876832	+	0.415221i
$118$		0			0
$119$	846.112			0.651789
$120$		0			0
$121$	−1137.72			−0.854782
$122$		0			0
$123$	2332.00	+	524.251i	1.70951	+	0.384310i
$124$		0			0
$125$			520.150i			0.372189i
$126$		0			0
$127$		−	204.993i		−	0.143230i	−0.997432	−	0.0716151i	$-0.977185\pi$
$127$		−	204.993i		−	0.143230i	0.997432	−	0.0716151i	$-0.0228153\pi$
$128$		0			0
$129$	2523.27	+	567.249i	1.72218	+	0.387159i
$130$		0			0
$131$	−44.9236			−0.0299618			−0.0149809	−	0.999888i	$-0.504769\pi$
$131$	−44.9236			−0.0299618			−0.0149809	+	0.999888i	$0.504769\pi$
$132$		0			0
$133$	244.477			0.159390
$134$		0			0
$135$	−1618.89	−	1265.25i	−1.03209	−	0.806636i
$136$		0			0
$137$		−	729.133i		−	0.454701i	−0.973813	−	0.227350i	$-0.926994\pi$
$137$		−	729.133i		−	0.454701i	0.973813	−	0.227350i	$-0.0730063\pi$
$138$		0			0
$139$		−	1890.32i		−	1.15349i	−0.816926	−	0.576743i	$-0.804323\pi$
$139$		−	1890.32i		−	1.15349i	0.816926	−	0.576743i	$-0.195677\pi$
$140$		0			0
$141$	153.165	−	681.314i	0.0914807	−	0.406929i
$142$		0			0
$143$	−632.215			−0.369709
$144$		0			0
$145$	3289.34			1.88390
$146$		0			0
$147$	−55.8448	+	248.412i	−0.0313333	+	0.139379i
$148$		0			0
$149$			734.918i			0.404073i	0.979378	+	0.202036i	$0.0647559\pi$
$149$			734.918i			0.404073i	−0.979378	+	0.202036i	$0.935244\pi$
$150$		0			0
$151$		−	990.698i		−	0.533920i	−0.963708	−	0.266960i	$-0.913981\pi$
$151$		−	990.698i		−	0.533920i	0.963708	−	0.266960i	$-0.0860191\pi$
$152$		0			0
$153$	1396.76	−	2949.57i	0.738049	−	1.55855i
$154$		0			0
$155$	−648.678			−0.336149
$156$		0			0
$157$	1520.81			0.773080			0.386540	−	0.922273i	$-0.373670\pi$
$157$	1520.81			0.773080			0.386540	+	0.922273i	$0.373670\pi$
$158$		0			0
$159$	1431.90	+	321.902i	0.714195	+	0.160556i
$160$		0			0
$161$			1060.08i			0.518919i
$162$		0			0
$163$		−	3135.82i		−	1.50685i	−0.657535	−	0.753424i	$-0.728401\pi$
$163$		−	3135.82i		−	1.50685i	0.657535	−	0.753424i	$-0.271599\pi$
$164$		0			0
$165$	1032.22	+	232.050i	0.487019	+	0.109485i
$166$		0			0
$167$	4075.35			1.88839			0.944193	−	0.329394i	$-0.106844\pi$
$167$	4075.35			1.88839			0.944193	+	0.329394i	$0.106844\pi$
$168$		0			0
$169$	−129.091			−0.0587578
$170$		0			0
$171$	403.583	−	852.255i	0.180484	−	0.381132i
$172$		0			0
$173$		−	721.361i		−	0.317018i	−0.987358	−	0.158509i	$-0.949331\pi$
$173$		−	721.361i		−	0.317018i	0.987358	−	0.158509i	$-0.0506687\pi$
$174$		0			0
$175$			626.384i			0.270572i
$176$		0			0
$177$	55.1276	−	245.222i	0.0234104	−	0.104135i
$178$		0			0
$179$	−4087.07			−1.70660			−0.853301	−	0.521418i	$-0.825403\pi$
$179$	−4087.07			−1.70660			−0.853301	+	0.521418i	$0.825403\pi$
$180$		0			0
$181$	−3549.84			−1.45778			−0.728889	−	0.684632i	$-0.759963\pi$
$181$	−3549.84			−1.45778			−0.728889	+	0.684632i	$0.759963\pi$
$182$		0			0
$183$	−391.591	+	1741.89i	−0.158182	+	0.703631i
$184$		0			0
$185$		−	3293.78i		−	1.30899i
$186$		0			0
$187$			1680.46i			0.657152i
$188$		0			0
$189$	773.782	+	604.755i	0.297801	+	0.232748i
$190$		0			0
$191$	−1269.54			−0.480945			−0.240473	−	0.970656i	$-0.577302\pi$
$191$	−1269.54			−0.480945			−0.240473	+	0.970656i	$0.577302\pi$
$192$		0			0
$193$	1979.22			0.738172			0.369086	−	0.929395i	$-0.379671\pi$
$193$	1979.22			0.738172			0.369086	+	0.929395i	$0.379671\pi$
$194$		0			0
$195$	−3376.28	−	759.013i	−1.23990	−	0.278739i
$196$		0			0
$197$			4655.85i			1.68384i	0.539606	+	0.841918i	$0.318573\pi$
$197$			4655.85i			1.68384i	−0.539606	+	0.841918i	$0.681427\pi$
$198$		0			0
$199$		−	166.378i		−	0.0592676i	−0.999561	−	0.0296338i	$-0.990566\pi$
$199$		−	166.378i		−	0.0592676i	0.999561	−	0.0296338i	$-0.00943411\pi$
$200$		0			0
$201$	3441.23	+	773.613i	1.20759	+	0.271475i
$202$		0			0
$203$	−1572.21			−0.543584
$204$		0			0
$205$	−6736.74			−2.29519
$206$		0			0
$207$	3695.47	+	1749.98i	1.24083	+	0.587594i
$208$		0			0
$209$			485.556i			0.160701i
$210$		0			0
$211$		−	1923.89i		−	0.627707i	−0.949471	−	0.313853i	$-0.898380\pi$
$211$		−	1923.89i		−	0.627707i	0.949471	−	0.313853i	$-0.101620\pi$
$212$		0			0
$213$	935.185	−	4159.94i	0.300835	−	1.33819i
$214$		0			0
$215$	−7289.27			−2.31220
$216$		0			0
$217$	310.049			0.0969931
$218$		0			0
$219$	422.283	−	1878.42i	0.130298	−	0.579598i
$220$		0			0
$221$		−	5496.62i		−	1.67304i
$222$		0			0
$223$		−	3286.15i		−	0.986803i	−0.869802	−	0.493402i	$-0.835753\pi$
$223$		−	3286.15i		−	0.986803i	0.869802	−	0.493402i	$-0.164247\pi$
$224$		0			0
$225$	2183.59	+	1034.03i	0.646990	+	0.306380i
$226$		0			0
$227$	4610.89			1.34817			0.674087	−	0.738652i	$-0.264537\pi$
$227$	4610.89			1.34817			0.674087	+	0.738652i	$0.264537\pi$
$228$		0			0
$229$	−5100.93			−1.47196			−0.735980	−	0.677003i	$-0.763278\pi$
$229$	−5100.93			−1.47196			−0.735980	+	0.677003i	$0.763278\pi$
$230$		0			0
$231$	−493.370	−	110.913i	−0.140525	−	0.0315912i
$232$		0			0
$233$			6943.92i			1.95241i	0.216853	+	0.976204i	$0.430421\pi$
$233$			6943.92i			1.95241i	−0.216853	+	0.976204i	$0.569579\pi$
$234$		0			0
$235$			1968.20i			0.546345i
$236$		0			0
$237$	−5003.09	−	1124.73i	−1.37125	−	0.308267i
$238$		0			0
$239$	3416.16			0.924572			0.462286	−	0.886731i	$-0.347029\pi$
$239$	3416.16			0.924572			0.462286	+	0.886731i	$0.347029\pi$
$240$		0			0
$241$	3724.80			0.995584			0.497792	−	0.867296i	$-0.334144\pi$
$241$	3724.80			0.995584			0.497792	+	0.867296i	$0.334144\pi$
$242$		0			0
$243$	3385.55	−	1699.10i	0.893759	−	0.448548i
$244$		0			0
$245$		−	717.617i		−	0.187130i
$246$		0			0
$247$		−	1588.20i		−	0.409129i
$248$		0			0
$249$	−552.624	+	2458.21i	−0.140647	+	0.625633i
$250$		0			0
$251$	4455.29			1.12038			0.560189	−	0.828365i	$-0.310728\pi$
$251$	4455.29			1.12038			0.560189	+	0.828365i	$0.310728\pi$
$252$		0			0
$253$	−2105.42			−0.523188
$254$		0			0
$255$	−2017.50	+	8974.34i	−0.495453	+	2.20390i
$256$		0			0
$257$			1067.17i			0.259020i	0.991578	+	0.129510i	$0.0413405\pi$
$257$			1067.17i			0.259020i	−0.991578	+	0.129510i	$0.958660\pi$
$258$		0			0
$259$			1574.33i			0.377699i
$260$		0			0
$261$	−2595.40	+	5480.77i	−0.615523	+	1.29981i
$262$		0			0
$263$	2881.25			0.675533			0.337767	−	0.941230i	$-0.390329\pi$
$263$	2881.25			0.675533			0.337767	+	0.941230i	$0.390329\pi$
$264$		0			0
$265$	−4136.51			−0.958881
$266$		0			0
$267$	−4971.05	−	1117.53i	−1.13941	−	0.256149i
$268$		0			0
$269$			5561.17i			1.26049i	0.776398	+	0.630243i	$0.217045\pi$
$269$			5561.17i			1.26049i	−0.776398	+	0.630243i	$0.782955\pi$
$270$		0			0
$271$			5418.65i			1.21461i	0.794468	+	0.607306i	$0.207750\pi$
$271$			5418.65i			1.21461i	−0.794468	+	0.607306i	$0.792250\pi$
$272$		0			0
$273$	1613.76	+	362.786i	0.357763	+	0.0804278i
$274$		0			0
$275$	−1244.06			−0.272799
$276$		0			0
$277$	3291.26			0.713908			0.356954	−	0.934122i	$-0.383815\pi$
$277$	3291.26			0.713908			0.356954	+	0.934122i	$0.383815\pi$
$278$		0			0
$279$	511.829	−	1080.84i	0.109829	−	0.231929i
$280$		0			0
$281$		−	6394.15i		−	1.35745i	−0.734393	−	0.678724i	$-0.762533\pi$
$281$		−	6394.15i		−	1.35745i	0.734393	−	0.678724i	$-0.237467\pi$
$282$		0			0
$283$			9183.66i			1.92902i	0.264050	+	0.964509i	$0.414942\pi$
$283$			9183.66i			1.92902i	−0.264050	+	0.964509i	$0.585058\pi$
$284$		0			0
$285$	−582.940	+	2593.06i	−0.121159	+	0.538947i
$286$		0			0
$287$	3219.96			0.662260
$288$		0			0
$289$	−9697.30			−1.97381
$290$		0			0
$291$	−556.851	+	2477.01i	−0.112176	+	0.498987i
$292$		0			0
$293$		−	3914.97i		−	0.780598i	−0.920688	−	0.390299i	$-0.872372\pi$
$293$		−	3914.97i		−	0.780598i	0.920688	−	0.390299i	$-0.127628\pi$
$294$		0			0
$295$			708.402i			0.139813i
$296$		0			0
$297$	−1201.10	+	1536.81i	−0.234663	+	0.300251i
$298$		0			0
$299$	6886.61			1.33198
$300$		0			0
$301$	3484.06			0.667168
$302$		0			0
$303$	870.998	+	195.807i	0.165140	+	0.0371248i
$304$		0			0
$305$		−	5032.02i		−	0.944698i
$306$		0			0
$307$			1579.23i			0.293588i	0.989167	+	0.146794i	$0.0468955\pi$
$307$			1579.23i			0.293588i	−0.989167	+	0.146794i	$0.953104\pi$
$308$		0			0
$309$	−5975.14	−	1343.26i	−1.10004	−	0.247298i
$310$		0			0
$311$	515.431			0.0939788			0.0469894	−	0.998895i	$-0.485037\pi$
$311$	515.431			0.0939788			0.0469894	+	0.998895i	$0.485037\pi$
$312$		0			0
$313$	10135.1			1.83026			0.915129	−	0.403161i	$-0.132089\pi$
$313$	10135.1			1.83026			0.915129	+	0.403161i	$0.132089\pi$
$314$		0			0
$315$	−2501.64	−	1184.64i	−0.447464	−	0.211895i
$316$		0			0
$317$		−	7471.40i		−	1.32377i	−0.749604	−	0.661886i	$-0.769756\pi$
$317$		−	7471.40i		−	1.32377i	0.749604	−	0.661886i	$-0.230244\pi$
$318$		0			0
$319$		−	3122.56i		−	0.548056i
$320$		0			0
$321$	−457.060	+	2033.12i	−0.0794723	+	0.353513i
$322$		0			0
$323$	−4221.53			−0.727221
$324$		0			0
$325$	4069.19			0.694517
$326$		0			0
$327$	−987.037	+	4390.59i	−0.166921	+	0.742508i
$328$		0			0
$329$		−	940.740i		−	0.157643i
$330$		0			0
$331$			6725.92i			1.11689i	0.829542	+	0.558444i	$0.188601\pi$
$331$			6725.92i			1.11689i	−0.829542	+	0.558444i	$0.811399\pi$
$332$		0			0
$333$	5488.16	+	2598.90i	0.903152	+	0.427685i
$334$		0			0
$335$	−9941.09			−1.62131
$336$		0			0
$337$	−75.0546			−0.0121320			−0.00606600	−	0.999982i	$-0.501931\pi$
$337$	−75.0546			−0.0121320			−0.00606600	+	0.999982i	$0.501931\pi$
$338$		0			0
$339$	2128.26	+	478.449i	0.340977	+	0.0766542i
$340$		0			0
$341$			615.788i			0.0977912i
$342$		0			0
$343$			343.000i			0.0539949i
$344$		0			0
$345$	−11243.8	−	2527.69i	−1.75462	−	0.394453i
$346$		0			0
$347$	−4435.86			−0.686252			−0.343126	−	0.939289i	$-0.611486\pi$
$347$	−4435.86			−0.686252			−0.343126	+	0.939289i	$0.611486\pi$
$348$		0			0
$349$	2611.13			0.400489			0.200245	−	0.979746i	$-0.435826\pi$
$349$	2611.13			0.400489			0.200245	+	0.979746i	$0.435826\pi$
$350$		0			0
$351$	3928.68	−	5026.74i	0.597429	−	0.764409i
$352$		0			0
$353$			916.862i			0.138243i	0.997608	+	0.0691213i	$0.0220196\pi$
$353$			916.862i			0.138243i	−0.997608	+	0.0691213i	$0.977980\pi$
$354$		0			0
$355$			12017.3i			1.79666i
$356$		0			0
$357$	964.305	−	4289.47i	0.142959	−	0.635918i
$358$		0			0
$359$	−3916.49			−0.575778			−0.287889	−	0.957664i	$-0.592953\pi$
$359$	−3916.49			−0.575778			−0.287889	+	0.957664i	$0.592953\pi$
$360$		0			0
$361$	5639.22			0.822164
$362$		0			0
$363$	−1296.64	+	5767.79i	−0.187482	+	0.833968i
$364$		0			0
$365$			5426.43i			0.778171i
$366$		0			0
$367$			9856.78i			1.40196i	0.713180	+	0.700980i	$0.247254\pi$
$367$			9856.78i			1.40196i	−0.713180	+	0.700980i	$0.752746\pi$
$368$		0			0
$369$	5315.52	−	11224.9i	0.749905	−	1.58359i
$370$		0			0
$371$	1977.13			0.276678
$372$		0			0
$373$	−11058.4			−1.53508			−0.767538	−	0.641003i	$-0.778519\pi$
$373$	−11058.4			−1.53508			−0.767538	+	0.641003i	$0.778519\pi$
$374$		0			0
$375$	2636.97	+	592.809i	0.363126	+	0.0816335i
$376$		0			0
$377$			10213.6i			1.39529i
$378$		0			0
$379$		−	3764.44i		−	0.510202i	−0.966914	−	0.255101i	$-0.917891\pi$
$379$		−	3764.44i		−	0.510202i	0.966914	−	0.255101i	$-0.0821087\pi$
$380$		0			0
$381$	−1039.24	−	233.629i	−0.139742	−	0.0314152i
$382$		0			0
$383$	10258.1			1.36858			0.684290	−	0.729210i	$-0.260112\pi$
$383$	10258.1			1.36858			0.684290	+	0.729210i	$0.260112\pi$
$384$		0			0
$385$	1425.26			0.188670
$386$		0			0
$387$	5751.48	−	12145.5i	0.755463	−	1.59533i
$388$		0			0
$389$			11610.0i			1.51324i	0.653857	+	0.756618i	$0.273150\pi$
$389$			11610.0i			1.51324i	−0.653857	+	0.756618i	$0.726850\pi$
$390$		0			0
$391$		−	18305.0i		−	2.36758i
$392$		0			0
$393$	−51.1989	+	227.746i	−0.00657162	+	0.0292322i
$394$		0			0
$395$	14453.1			1.84104
$396$		0			0
$397$	7004.43			0.885497			0.442749	−	0.896646i	$-0.354003\pi$
$397$	7004.43			0.885497			0.442749	+	0.896646i	$0.354003\pi$
$398$		0			0
$399$	278.628	−	1239.41i	0.0349595	−	0.155509i
$400$		0			0
$401$			2223.31i			0.276875i	0.990371	+	0.138438i	$0.0442080\pi$
$401$			2223.31i			0.276875i	−0.990371	+	0.138438i	$0.955792\pi$
$402$		0			0
$403$		−	2014.18i		−	0.248966i
$404$		0			0
$405$	−8259.40	+	6765.17i	−1.01337	+	0.830035i
$406$		0			0
$407$	−3126.78			−0.380807
$408$		0			0
$409$	−2310.73			−0.279359			−0.139680	−	0.990197i	$-0.544607\pi$
$409$	−2310.73			−0.279359			−0.139680	+	0.990197i	$0.544607\pi$
$410$		0			0
$411$	−3696.43	−	830.985i	−0.443629	−	0.0997311i
$412$		0			0
$413$		−	338.595i		−	0.0403418i
$414$		0			0
$415$		−	7101.33i		−	0.839977i
$416$		0			0
$417$	−9583.20	−	2154.37i	−1.12540	−	0.252998i
$418$		0			0
$419$	−9808.83			−1.14366			−0.571829	−	0.820373i	$-0.693766\pi$
$419$	−9808.83			−1.14366			−0.571829	+	0.820373i	$0.693766\pi$
$420$		0			0
$421$	−2381.84			−0.275733			−0.137867	−	0.990451i	$-0.544024\pi$
$421$	−2381.84			−0.275733			−0.137867	+	0.990451i	$0.544024\pi$
$422$		0			0
$423$	−3279.45	−	1552.97i	−0.376956	−	0.178506i
$424$		0			0
$425$		−	10816.1i		−	1.23449i
$426$		0			0
$427$			2405.16i			0.272585i
$428$		0			0
$429$	−720.529	+	3205.09i	−0.0810896	+	0.360707i
$430$		0			0
$431$	−8717.18			−0.974227			−0.487114	−	0.873339i	$-0.661950\pi$
$431$	−8717.18			−0.974227			−0.487114	+	0.873339i	$0.661950\pi$
$432$		0			0
$433$	14091.6			1.56397			0.781983	−	0.623300i	$-0.214209\pi$
$433$	14091.6			1.56397			0.781983	+	0.623300i	$0.214209\pi$
$434$		0			0
$435$	3748.83	−	16675.7i	0.413201	−	1.83802i
$436$		0			0
$437$		−	5289.08i		−	0.578973i
$438$		0			0
$439$		−	6819.29i		−	0.741383i	−0.928756	−	0.370691i	$-0.879121\pi$
$439$		−	6819.29i		−	0.741383i	0.928756	−	0.370691i	$-0.120879\pi$
$440$		0			0
$441$	1195.71	+	566.224i	0.129112	+	0.0611407i
$442$		0			0
$443$	9565.27			1.02587			0.512934	−	0.858428i	$-0.328558\pi$
$443$	9565.27			1.02587			0.512934	+	0.858428i	$0.328558\pi$
$444$		0			0
$445$	14360.5			1.52978
$446$		0			0
$447$	3725.76	+	837.578i	0.394233	+	0.0886266i
$448$		0			0
$449$		−	752.859i		−	0.0791306i	−0.999217	−	0.0395653i	$-0.987403\pi$
$449$		−	752.859i		−	0.0791306i	0.999217	−	0.0395653i	$-0.0125973\pi$
$450$		0			0
$451$			6395.17i			0.667709i
$452$		0			0
$453$	−5022.47	−	1129.09i	−0.520919	−	0.117106i
$454$		0			0
$455$	−4661.87			−0.480334
$456$		0			0
$457$	−14222.2			−1.45577			−0.727884	−	0.685700i	$-0.759496\pi$
$457$	−14222.2			−1.45577			−0.727884	+	0.685700i	$0.759496\pi$
$458$		0			0
$459$	−13361.3	−	10442.7i	−1.35872	−	1.06192i
$460$		0			0
$461$			12808.3i			1.29401i	0.762484	+	0.647007i	$0.223980\pi$
$461$			12808.3i			1.29401i	−0.762484	+	0.647007i	$0.776020\pi$
$462$		0			0
$463$		−	1416.85i		−	0.142217i	−0.997469	−	0.0711086i	$-0.977346\pi$
$463$		−	1416.85i		−	0.142217i	0.997469	−	0.0711086i	$-0.0226537\pi$
$464$		0			0
$465$	−739.292	+	3288.56i	−0.0737287	+	0.327964i
$466$		0			0
$467$	−12720.2			−1.26043			−0.630215	−	0.776421i	$-0.717033\pi$
$467$	−12720.2			−1.26043			−0.630215	+	0.776421i	$0.717033\pi$
$468$		0			0
$469$	4751.55			0.467817
$470$		0			0
$471$	1733.25	−	7709.92i	0.169562	−	0.754256i
$472$		0			0
$473$			6919.68i			0.672658i
$474$		0			0
$475$		−	3125.24i		−	0.301886i
$476$		0			0
$477$	3263.84	−	6892.33i	0.313294	−	0.661589i
$478$		0			0
$479$	−16314.1			−1.55618			−0.778092	−	0.628150i	$-0.783812\pi$
$479$	−16314.1			−1.55618			−0.778092	+	0.628150i	$0.783812\pi$
$480$		0			0
$481$	10227.4			0.969496
$482$		0			0
$483$	5374.20	+	1208.16i	0.506283	+	0.113816i
$484$		0			0
$485$		−	7155.65i		−	0.669941i
$486$		0			0
$487$			4903.33i			0.456244i	0.973632	+	0.228122i	$0.0732586\pi$
$487$			4903.33i			0.456244i	−0.973632	+	0.228122i	$0.926741\pi$
$488$		0			0
$489$	−15897.4	−	3573.86i	−1.47016	−	0.330502i
$490$		0			0
$491$	−1213.14			−0.111503			−0.0557517	−	0.998445i	$-0.517756\pi$
$491$	−1213.14			−0.111503			−0.0557517	+	0.998445i	$0.517756\pi$
$492$		0			0
$493$	27148.3			2.48011
$494$		0			0
$495$	2352.82	−	4968.50i	0.213639	−	0.451146i
$496$		0			0
$497$		−	5743.93i		−	0.518411i
$498$		0			0
$499$		−	8093.93i		−	0.726121i	−0.931766	−	0.363060i	$-0.881732\pi$
$499$		−	8093.93i		−	0.726121i	0.931766	−	0.363060i	$-0.118268\pi$
$500$		0			0
$501$	4644.64	−	20660.5i	0.414186	−	1.84240i
$502$		0			0
$503$	3991.77			0.353846			0.176923	−	0.984225i	$-0.443386\pi$
$503$	3991.77			0.353846			0.176923	+	0.984225i	$0.443386\pi$
$504$		0			0
$505$	−2516.16			−0.221718
$506$		0			0
$507$	−147.123	+	654.442i	−0.0128875	+	0.0573270i
$508$		0			0
$509$			7309.19i			0.636491i	0.948008	+	0.318246i	$0.103094\pi$
$509$			7309.19i			0.636491i	−0.948008	+	0.318246i	$0.896906\pi$
$510$		0			0
$511$		−	2593.67i		−	0.224535i
$512$		0			0
$513$	−3860.66	−	3017.32i	−0.332265	−	0.259684i
$514$		0			0
$515$	17261.1			1.47692
$516$		0			0
$517$	1868.40			0.158941
$518$		0			0
$519$	−3657.03	−	822.128i	−0.309299	−	0.0695326i
$520$		0			0
$521$			1657.13i			0.139348i	0.997570	+	0.0696739i	$0.0221959\pi$
$521$			1657.13i			0.139348i	−0.997570	+	0.0696739i	$0.977804\pi$
$522$		0			0
$523$			5621.88i			0.470034i	0.971991	+	0.235017i	$0.0755145\pi$
$523$			5621.88i			0.470034i	−0.971991	+	0.235017i	$0.924485\pi$
$524$		0			0
$525$	3175.53	+	713.883i	0.263984	+	0.0593455i
$526$		0			0
$527$	−5353.80			−0.442533
$528$		0			0
$529$	10767.0			0.884936
$530$		0			0
$531$	−1180.35	−	558.953i	−0.0964651	−	0.0456808i
$532$		0			0
$533$		−	20917.9i		−	1.69992i
$534$		0			0
$535$		−	5873.32i		−	0.474628i
$536$		0			0
$537$	−4657.99	+	20719.9i	−0.374315	+	1.66505i
$538$		0			0
$539$	−681.232			−0.0544392
$540$		0			0
$541$	16776.8			1.33326			0.666629	−	0.745390i	$-0.267737\pi$
$541$	16776.8			1.33326			0.666629	+	0.745390i	$0.267737\pi$
$542$		0			0
$543$	−4045.72	+	17996.4i	−0.319739	+	1.42228i
$544$		0			0
$545$		−	12683.6i		−	0.996894i
$546$		0			0
$547$			19586.5i			1.53100i	0.643437	+	0.765499i	$0.277508\pi$
$547$			19586.5i			1.53100i	−0.643437	+	0.765499i	$0.722492\pi$
$548$		0			0
$549$	8384.46	+	3970.44i	0.651803	+	0.308660i
$550$		0			0
$551$	7844.27			0.606492
$552$		0			0
$553$	−6908.13			−0.531218
$554$		0			0
$555$	−16698.2	−	3753.89i	−1.27712	−	0.287106i
$556$		0			0
$557$		−	13365.8i		−	1.01674i	−0.861138	−	0.508372i	$-0.830248\pi$
$557$		−	13365.8i		−	1.01674i	0.861138	−	0.508372i	$-0.169752\pi$
$558$		0			0
$559$		−	22633.6i		−	1.71252i
$560$		0			0
$561$	8519.31	+	1915.20i	0.641151	+	0.144135i
$562$		0			0
$563$	−10979.2			−0.821877			−0.410939	−	0.911663i	$-0.634799\pi$
$563$	−10979.2			−0.821877			−0.410939	+	0.911663i	$0.634799\pi$
$564$		0			0
$565$	−6148.17			−0.457797
$566$		0			0
$567$	3947.75	−	3233.55i	0.292399	−	0.239500i
$568$		0			0
$569$			24903.3i			1.83480i	0.397969	+	0.917399i	$0.369715\pi$
$569$			24903.3i			1.83480i	−0.397969	+	0.917399i	$0.630285\pi$
$570$		0			0
$571$		−	3896.72i		−	0.285591i	−0.989752	−	0.142796i	$-0.954391\pi$
$571$		−	3896.72i		−	0.285591i	0.989752	−	0.142796i	$-0.0456091\pi$
$572$		0			0
$573$	−1446.88	+	6436.09i	−0.105487	+	0.469234i
$574$		0			0
$575$	13551.3			0.982835
$576$		0			0
$577$	11824.8			0.853161			0.426581	−	0.904450i	$-0.359718\pi$
$577$	11824.8			0.853161			0.426581	+	0.904450i	$0.359718\pi$
$578$		0			0
$579$	2255.69	−	10033.9i	0.161906	−	0.720198i
$580$		0			0
$581$			3394.23i			0.242369i
$582$		0			0
$583$			3926.77i			0.278954i
$584$		0			0
$585$	−7695.82	+	16251.4i	−0.543903	+	1.14857i
$586$		0			0
$587$	5957.49			0.418896			0.209448	−	0.977820i	$-0.432833\pi$
$587$	5957.49			0.418896			0.209448	+	0.977820i	$0.432833\pi$
$588$		0			0
$589$	−1546.94			−0.108218
$590$		0			0
$591$	23603.4	+	5306.22i	1.64283	+	0.369321i
$592$		0			0
$593$		−	11547.6i		−	0.799671i	−0.916587	−	0.399836i	$-0.869067\pi$
$593$		−	11547.6i		−	0.799671i	0.916587	−	0.399836i	$-0.130933\pi$
$594$		0			0
$595$			12391.5i			0.853786i
$596$		0			0
$597$	−843.476	−	189.620i	−0.0578244	−	0.0129994i
$598$		0			0
$599$	27338.0			1.86477			0.932386	−	0.361464i	$-0.117723\pi$
$599$	27338.0			1.86477			0.932386	+	0.361464i	$0.117723\pi$
$600$		0			0
$601$	−15844.1			−1.07536			−0.537682	−	0.843148i	$-0.680700\pi$
$601$	−15844.1			−1.07536			−0.537682	+	0.843148i	$0.680700\pi$
$602$		0			0
$603$	7843.86	−	16564.1i	0.529729	−	1.11864i
$604$		0			0
$605$		−	16662.1i		−	1.11969i
$606$		0			0
$607$		−	575.804i		−	0.0385027i	−0.999815	−	0.0192514i	$-0.993872\pi$
$607$		−	575.804i		−	0.0385027i	0.999815	−	0.0192514i	$-0.00612828\pi$
$608$		0			0
$609$	−1791.83	+	7970.51i	−0.119226	+	0.530347i
$610$		0			0
$611$	−6111.35			−0.404646
$612$		0			0
$613$	9646.06			0.635564			0.317782	−	0.948164i	$-0.397062\pi$
$613$	9646.06			0.635564			0.317782	+	0.948164i	$0.397062\pi$
$614$		0			0
$615$	−7677.79	+	34152.8i	−0.503412	+	2.23930i
$616$		0			0
$617$			18960.4i			1.23714i	0.785728	+	0.618572i	$0.212288\pi$
$617$			18960.4i			1.23714i	−0.785728	+	0.618572i	$0.787712\pi$
$618$		0			0
$619$		−	26238.0i		−	1.70370i	−0.523783	−	0.851852i	$-0.675480\pi$
$619$		−	26238.0i		−	1.70370i	0.523783	−	0.851852i	$-0.324520\pi$
$620$		0			0
$621$	13083.4	−	16740.2i	0.845442	−	1.08174i
$622$		0			0
$623$	−6863.90			−0.441406
$624$		0			0
$625$	−18803.1			−1.20340
$626$		0			0
$627$	2461.59	+	553.383i	0.156788	+	0.0352472i
$628$		0			0
$629$		−	27184.9i		−	1.72326i
$630$		0			0
$631$			31385.8i			1.98011i	0.140680	+	0.990055i	$0.455071\pi$
$631$			31385.8i			1.98011i	−0.140680	+	0.990055i	$0.544929\pi$
$632$		0			0
$633$	−9753.41	−	2192.64i	−0.612422	−	0.137677i
$634$		0			0
$635$	3002.18			0.187619
$636$		0			0
$637$	2228.24			0.138597
$638$		0			0
$639$	−20023.5	−	9482.08i	−1.23962	−	0.587019i
$640$		0			0
$641$			6203.66i			0.382262i	0.981565	+	0.191131i	$0.0612155\pi$
$641$			6203.66i			0.382262i	−0.981565	+	0.191131i	$0.938785\pi$
$642$		0			0
$643$			1900.46i			0.116558i	0.998300	+	0.0582791i	$0.0185613\pi$
$643$			1900.46i			0.116558i	−0.998300	+	0.0582791i	$0.981439\pi$
$644$		0			0
$645$	−8307.51	+	36953.9i	−0.507144	+	2.25590i
$646$		0			0
$647$	5220.82			0.317236			0.158618	−	0.987340i	$-0.449296\pi$
$647$	5220.82			0.317236			0.158618	+	0.987340i	$0.449296\pi$
$648$		0			0
$649$	672.484			0.0406738
$650$		0			0
$651$	353.360	−	1571.83i	0.0212738	−	0.0946313i
$652$		0			0
$653$			5266.00i			0.315581i	0.987473	+	0.157790i	$0.0504370\pi$
$653$			5266.00i			0.315581i	−0.987473	+	0.157790i	$0.949563\pi$
$654$		0			0
$655$		−	657.917i		−	0.0392473i
$656$		0			0
$657$	−9041.63	−	4281.64i	−0.536906	−	0.254251i
$658$		0			0
$659$	−19994.5			−1.18191			−0.590954	−	0.806706i	$-0.701248\pi$
$659$	−19994.5			−1.18191			−0.590954	+	0.806706i	$0.701248\pi$
$660$		0			0
$661$	−16431.9			−0.966911			−0.483456	−	0.875369i	$-0.660619\pi$
$661$	−16431.9			−0.966911			−0.483456	+	0.875369i	$0.660619\pi$
$662$		0			0
$663$	−27865.8	−	6264.44i	−1.63230	−	0.366954i
$664$		0			0
$665$			3580.43i			0.208787i
$666$		0			0
$667$			34013.6i			1.97453i
$668$		0			0
$669$	−16659.6	−	3745.20i	−0.962775	−	0.216439i
$670$		0			0
$671$	−4776.88			−0.274828
$672$		0			0
$673$	−8167.51			−0.467807			−0.233904	−	0.972260i	$-0.575150\pi$
$673$	−8167.51			−0.467807			−0.233904	+	0.972260i	$0.575150\pi$
$674$		0			0
$675$	7730.78	−	9891.52i	0.440827	−	0.564037i
$676$		0			0
$677$			34191.6i			1.94105i	0.241000	+	0.970525i	$0.422524\pi$
$677$			34191.6i			1.94105i	−0.241000	+	0.970525i	$0.577476\pi$
$678$		0			0
$679$			3420.19i			0.193306i
$680$		0			0
$681$	5254.99	−	23375.5i	0.295700	−	1.31535i
$682$		0			0
$683$	−22.6262			−0.00126760			−0.000633799	−	1.00000i	$-0.500202\pi$
$683$	−22.6262			−0.00126760			−0.000633799		1.00000i	$0.500202\pi$
$684$		0			0
$685$	10678.3			0.595618
$686$		0			0
$687$	−5813.47	+	25859.8i	−0.322850	+	1.43612i
$688$		0			0
$689$		−	12844.1i		−	0.710188i
$690$		0			0
$691$			27943.6i			1.53839i	0.639016	+	0.769193i	$0.279342\pi$
$691$			27943.6i			1.53839i	−0.639016	+	0.769193i	$0.720658\pi$
$692$		0			0
$693$	−1124.58	+	2374.80i	−0.0616438	+	0.130175i
$694$		0			0
$695$	27684.2			1.51096
$696$		0			0
$697$	−55601.0			−3.02158
$698$		0			0
$699$	35203.1	+	7913.91i	1.90487	+	0.428228i
$700$		0			0
$701$			4454.40i			0.240001i	0.992774	+	0.120000i	$0.0382896\pi$
$701$			4454.40i			0.240001i	−0.992774	+	0.120000i	$0.961710\pi$
$702$		0			0
$703$		−	7854.86i		−	0.421410i
$704$		0			0
$705$	9978.02	+	2243.13i	0.533041	+	0.119832i
$706$		0			0
$707$	1202.65			0.0639750
$708$		0			0
$709$	1733.18			0.0918067			0.0459034	−	0.998946i	$-0.485383\pi$
$709$	1733.18			0.0918067			0.0459034	+	0.998946i	$0.485383\pi$
$710$		0			0
$711$	−11403.9	+	24082.0i	−0.601521	+	1.27025i
$712$		0			0
$713$		−	6707.68i		−	0.352321i
$714$		0			0
$715$		−	9258.94i		−	0.484287i
$716$		0			0
$717$	3893.36	−	17318.6i	0.202789	−	0.902059i
$718$		0			0
$719$	5157.90			0.267534			0.133767	−	0.991013i	$-0.457293\pi$
$719$	5157.90			0.267534			0.133767	+	0.991013i	$0.457293\pi$
$720$		0			0
$721$	−8250.30			−0.426154
$722$		0			0
$723$	4245.12	−	18883.4i	0.218365	−	0.971341i
$724$		0			0
$725$			20098.1i			1.02955i
$726$		0			0
$727$			6924.59i			0.353259i	0.984277	+	0.176629i	$0.0565194\pi$
$727$			6924.59i			0.353259i	−0.984277	+	0.176629i	$0.943481\pi$
$728$		0			0
$729$	−4755.32	−	19099.9i	−0.241595	−	0.970377i
$730$		0			0
$731$	−60161.2			−3.04397
$732$		0			0
$733$	−17794.3			−0.896655			−0.448327	−	0.893869i	$-0.647980\pi$
$733$	−17794.3			−0.896655			−0.448327	+	0.893869i	$0.647980\pi$
$734$		0			0
$735$	−3638.05	−	817.861i	−0.182574	−	0.0410439i
$736$		0			0
$737$			9437.05i			0.471667i
$738$		0			0
$739$		−	619.992i		−	0.0308617i	−0.999881	−	0.0154308i	$-0.995088\pi$
$739$		−	619.992i		−	0.0308617i	0.999881	−	0.0154308i	$-0.00491198\pi$
$740$		0			0
$741$	−8051.60	−	1810.06i	−0.399167	−	0.0897357i
$742$		0			0
$743$	−9464.45			−0.467318			−0.233659	−	0.972319i	$-0.575070\pi$
$743$	−9464.45			−0.467318			−0.233659	+	0.972319i	$0.575070\pi$
$744$		0			0
$745$	−10763.1			−0.529299
$746$		0			0
$747$	11832.4	+	5603.19i	0.579550	+	0.274444i
$748$		0			0
$749$			2807.27i			0.136950i
$750$		0			0
$751$		−	19159.6i		−	0.930953i	−0.885060	−	0.465476i	$-0.845883\pi$
$751$		−	19159.6i		−	0.930953i	0.885060	−	0.465476i	$-0.154117\pi$
$752$		0			0
$753$	5077.64	−	22586.6i	0.245736	−	1.09310i
$754$		0			0
$755$	14509.0			0.699387
$756$		0			0
$757$	−14913.4			−0.716031			−0.358016	−	0.933716i	$-0.616547\pi$
$757$	−14913.4			−0.716031			−0.358016	+	0.933716i	$0.616547\pi$
$758$		0			0
$759$	−2399.53	+	10673.7i	−0.114753	+	0.510449i
$760$		0			0
$761$		−	12228.8i		−	0.582515i	−0.956645	−	0.291258i	$-0.905926\pi$
$761$		−	12228.8i		−	0.582515i	0.956645	−	0.291258i	$-0.0940737\pi$
$762$		0			0
$763$			6062.40i			0.287646i
$764$		0			0
$765$	43197.2	+	20455.9i	2.04157	+	0.966778i
$766$		0			0
$767$	−2199.62			−0.103551
$768$		0			0
$769$	16296.5			0.764197			0.382099	−	0.924122i	$-0.375202\pi$
$769$	16296.5			0.764197			0.382099	+	0.924122i	$0.375202\pi$
$770$		0			0
$771$	5410.15	+	1216.24i	0.252713	+	0.0568118i
$772$		0			0
$773$			19766.2i			0.919714i	0.887993	+	0.459857i	$0.152099\pi$
$773$			19766.2i			0.919714i	−0.887993	+	0.459857i	$0.847901\pi$
$774$		0			0
$775$		−	3963.46i		−	0.183706i
$776$		0			0
$777$	7981.27	+	1794.25i	0.368502	+	0.0828421i
$778$		0			0
$779$	−16065.5			−0.738903
$780$		0			0
$781$	11408.0			0.522677
$782$		0			0
$783$	24827.5	+	19404.1i	1.13316	+	0.885627i
$784$		0			0
$785$			22272.6i			1.01267i
$786$		0			0
$787$		−	16693.4i		−	0.756108i	−0.925784	−	0.378054i	$-0.876593\pi$
$787$		−	16693.4i		−	0.756108i	0.925784	−	0.378054i	$-0.123407\pi$
$788$		0			0
$789$	3283.73	−	14606.8i	0.148167	−	0.659084i
$790$		0			0
$791$	2938.64			0.132094
$792$		0			0
$793$	15624.7			0.699684
$794$		0			0
$795$	−4714.33	+	20970.5i	−0.210315	+	0.935532i
$796$		0			0
$797$		−	6666.68i		−	0.296293i	−0.988965	−	0.148147i	$-0.952669\pi$
$797$		−	6666.68i		−	0.296293i	0.988965	−	0.148147i	$-0.0473308\pi$
$798$		0			0
$799$			16244.3i			0.719252i
$800$		0			0
$801$	−11330.9	+	23927.8i	−0.499823	+	1.05549i
$802$		0			0
$803$	5151.29			0.226383
$804$		0			0
$805$	−15525.1			−0.679737
$806$		0			0
$807$	28193.1	+	6338.01i	1.22979	+	0.276467i
$808$		0			0
$809$		−	9909.58i		−	0.430658i	−0.976542	−	0.215329i	$-0.930918\pi$
$809$		−	9909.58i		−	0.430658i	0.976542	−	0.215329i	$-0.0690824\pi$
$810$		0			0
$811$		−	6564.87i		−	0.284246i	−0.989849	−	0.142123i	$-0.954607\pi$
$811$		−	6564.87i		−	0.284246i	0.989849	−	0.142123i	$-0.0453929\pi$
$812$		0			0
$813$	27470.5	+	6175.58i	1.18504	+	0.266405i
$814$		0			0
$815$	45924.8			1.97384
$816$		0			0
$817$	−17383.1			−0.744380
$818$		0			0
$819$	3678.38	−	7767.71i	0.156939	−	0.331411i
$820$		0			0
$821$		−	11978.5i		−	0.509199i	−0.967047	−	0.254599i	$-0.918056\pi$
$821$		−	11978.5i		−	0.509199i	0.967047	−	0.254599i	$-0.0819436\pi$
$822$		0			0
$823$		−	18871.4i		−	0.799290i	−0.916670	−	0.399645i	$-0.869133\pi$
$823$		−	18871.4i		−	0.799290i	0.916670	−	0.399645i	$-0.130867\pi$
$824$		0			0
$825$	−1417.84	+	6306.92i	−0.0598339	+	0.266156i
$826$		0			0
$827$	39287.6			1.65195			0.825976	−	0.563705i	$-0.190624\pi$
$827$	39287.6			1.65195			0.825976	+	0.563705i	$0.190624\pi$
$828$		0			0
$829$	112.278			0.00470397			0.00235198	−	0.999997i	$-0.499251\pi$
$829$	112.278			0.00470397			0.00235198	+	0.999997i	$0.499251\pi$
$830$		0			0
$831$	3751.01	−	16685.4i	0.156584	−	0.696524i
$832$		0			0
$833$		−	5922.78i		−	0.246353i
$834$		0			0
$835$			59684.6i			2.47362i
$836$		0			0
$837$	−4896.13	−	3826.60i	−0.202192	−	0.158025i
$838$		0			0
$839$	−11797.5			−0.485454			−0.242727	−	0.970095i	$-0.578042\pi$
$839$	−11797.5			−0.485454			−0.242727	+	0.970095i	$0.578042\pi$
$840$		0			0
$841$	−26056.8			−1.06838
$842$		0			0
$843$	−32416.0	−	7287.35i	−1.32439	−	0.297734i
$844$		0			0
$845$		−	1890.57i		−	0.0769675i
$846$		0			0
$847$			7964.01i			0.323077i
$848$		0			0
$849$	46557.7	+	10466.5i	1.88205	+	0.423098i
$850$		0			0
$851$	34059.5			1.37197
$852$		0			0
$853$	26383.7			1.05904			0.529520	−	0.848297i	$-0.322372\pi$
$853$	26383.7			1.05904			0.529520	+	0.848297i	$0.322372\pi$
$854$		0			0
$855$	12481.5	+	5910.58i	0.499249	+	0.236418i
$856$		0			0
$857$		−	2222.10i		−	0.0885712i	−0.999019	−	0.0442856i	$-0.985899\pi$
$857$		−	2222.10i		−	0.0885712i	0.999019	−	0.0442856i	$-0.0141012\pi$
$858$		0			0
$859$		−	6113.29i		−	0.242820i	−0.992602	−	0.121410i	$-0.961258\pi$
$859$		−	6113.29i		−	0.242820i	0.992602	−	0.121410i	$-0.0387416\pi$
$860$		0			0
$861$	3669.76	−	16324.0i	0.145256	−	0.646133i
$862$		0			0
$863$	−23982.7			−0.945978			−0.472989	−	0.881068i	$-0.656825\pi$
$863$	−23982.7			−0.945978			−0.472989	+	0.881068i	$0.656825\pi$
$864$		0			0
$865$	10564.5			0.415265
$866$		0			0
$867$	−11051.9	+	49161.7i	−0.432921	+	1.92574i
$868$		0			0
$869$		−	13720.2i		−	0.535589i
$870$		0			0
$871$		−	30867.6i		−	1.20081i
$872$		0			0
$873$	11922.9	+	5646.05i	0.462232	+	0.218889i
$874$		0			0
$875$	3641.05			0.140674
$876$		0			0
$877$	4314.94			0.166140			0.0830702	−	0.996544i	$-0.473527\pi$
$877$	4314.94			0.166140			0.0830702	+	0.996544i	$0.473527\pi$
$878$		0			0
$879$	−19847.4	−	4461.85i	−0.761590	−	0.171211i
$880$		0			0
$881$			13908.4i			0.531880i	0.963990	+	0.265940i	$0.0856823\pi$
$881$			13908.4i			0.531880i	−0.963990	+	0.265940i	$0.914318\pi$
$882$		0			0
$883$		−	31012.0i		−	1.18192i	−0.806700	−	0.590961i	$-0.798749\pi$
$883$		−	31012.0i		−	1.18192i	0.806700	−	0.590961i	$-0.201251\pi$
$884$		0			0
$885$	3591.33	+	807.358i	0.136408	+	0.0306656i
$886$		0			0
$887$	−38618.3			−1.46187			−0.730934	−	0.682449i	$-0.760915\pi$
$887$	−38618.3			−1.46187			−0.730934	+	0.682449i	$0.760915\pi$
$888$		0			0
$889$	−1434.95			−0.0541359
$890$		0			0
$891$	6422.16	+	7840.63i	0.241471	+	0.294805i
$892$		0			0
$893$			4693.67i			0.175887i
$894$		0			0
$895$		−	59856.1i		−	2.23550i
$896$		0			0
$897$	7848.60	−	34912.6i	0.292149	−	1.29955i
$898$		0			0
$899$	9948.20			0.369067
$900$		0			0
$901$	−34140.2			−1.26235
$902$		0			0
$903$	3970.74	−	17662.9i	0.146332	−	0.650923i
$904$		0			0
$905$		−	51988.4i		−	1.90956i
$906$		0			0
$907$		−	27884.1i		−	1.02081i	−0.859933	−	0.510407i	$-0.829495\pi$
$907$		−	27884.1i		−	1.02081i	0.859933	−	0.510407i	$-0.170505\pi$
$908$		0			0
$909$	1985.33	−	4192.47i	0.0724416	−	0.152976i
$910$		0			0
$911$	−39542.4			−1.43809			−0.719043	−	0.694965i	$-0.755420\pi$
$911$	−39542.4			−1.43809			−0.719043	+	0.694965i	$0.755420\pi$
$912$		0			0
$913$	−6741.27			−0.244363
$914$		0			0
$915$	−25510.5	−	5734.94i	−0.921694	−	0.207204i
$916$		0			0
$917$			314.465i			0.0113245i
$918$		0			0
$919$			13478.3i			0.483796i	0.970302	+	0.241898i	$0.0777699\pi$
$919$			13478.3i			0.483796i	−0.970302	+	0.241898i	$0.922230\pi$
$920$		0			0
$921$	8006.12	+	1799.84i	0.286439	+	0.0643937i
$922$		0			0
$923$	−37314.4			−1.33068
$924$		0			0
$925$	20125.2			0.715365
$926$		0			0
$927$	−13619.6	+	28760.8i	−0.482553	+	1.01902i
$928$		0			0
$929$			21231.5i			0.749819i	0.927061	+	0.374910i	$0.122326\pi$
$929$			21231.5i			0.749819i	−0.927061	+	0.374910i	$0.877674\pi$
$930$		0			0
$931$		−	1711.34i		−	0.0602437i
$932$		0			0
$933$	587.431	−	2613.04i	0.0206127	−	0.0916904i
$934$		0			0
$935$	−24610.8			−0.860811
$936$		0			0
$937$	30386.4			1.05942			0.529712	−	0.848177i	$-0.322300\pi$
$937$	30386.4			1.05942			0.529712	+	0.848177i	$0.322300\pi$
$938$		0			0
$939$	11550.9	−	51381.3i	0.401437	−	1.78569i
$940$		0			0
$941$			28605.3i			0.990973i	0.868616	+	0.495487i	$0.165010\pi$
$941$			28605.3i			0.990973i	−0.868616	+	0.495487i	$0.834990\pi$
$942$		0			0
$943$		−	69661.5i		−	2.40561i
$944$		0			0
$945$	−8856.78	+	11332.2i	−0.304880	+	0.390093i
$946$		0			0
$947$	−4517.34			−0.155009			−0.0775047	−	0.996992i	$-0.524695\pi$
$947$	−4517.34			−0.155009			−0.0775047	+	0.996992i	$0.524695\pi$
$948$		0			0
$949$	−16849.3			−0.576347
$950$		0			0
$951$	−37877.2	−	8515.08i	−1.29154	−	0.290347i
$952$		0			0
$953$		−	32311.2i		−	1.09828i	−0.835729	−	0.549141i	$-0.814955\pi$
$953$		−	32311.2i		−	1.09828i	0.835729	−	0.549141i	$-0.185045\pi$
$954$		0			0
$955$		−	18592.7i		−	0.629996i
$956$		0			0
$957$	−15830.2	−	3558.75i	−0.534711	−	0.120207i
$958$		0			0
$959$	−5103.93			−0.171861
$960$		0			0
$961$	27829.2			0.934146
$962$		0			0
$963$	9786.24	+	4634.25i	0.327474	+	0.155074i
$964$		0			0
$965$			28986.1i			0.966940i
$966$		0			0
$967$			50510.3i			1.67973i	0.542793	+	0.839867i	$0.317367\pi$
$967$			50510.3i			1.67973i	−0.542793	+	0.839867i	$0.682633\pi$
$968$		0			0
$969$	−4811.24	+	21401.6i	−0.159504	+	0.709513i
$970$		0			0
$971$	46193.8			1.52671			0.763353	−	0.645982i	$-0.223552\pi$
$971$	46193.8			1.52671			0.763353	+	0.645982i	$0.223552\pi$
$972$		0			0
$973$	−13232.2			−0.435977
$974$		0			0
$975$	4637.62	−	20629.3i	0.152331	−	0.677606i
$976$		0			0
$977$			34716.0i			1.13681i	0.822749	+	0.568405i	$0.192439\pi$
$977$			34716.0i			1.13681i	−0.822749	+	0.568405i	$0.807561\pi$
$978$		0			0
$979$		−	13632.4i		−	0.445038i
$980$		0			0
$981$	21133.7	+	10007.8i	0.687816	+	0.325714i
$982$		0			0
$983$	33172.7			1.07634			0.538171	−	0.842836i	$-0.319115\pi$
$983$	33172.7			1.07634			0.538171	+	0.842836i	$0.319115\pi$
$984$		0			0
$985$	−68186.1			−2.20567
$986$		0			0
$987$	−4769.20	−	1072.15i	−0.153805	−	0.0345765i
$988$		0			0
$989$		−	75375.0i		−	2.42344i
$990$		0			0
$991$		−	46668.6i		−	1.49594i	−0.663733	−	0.747969i	$-0.731029\pi$
$991$		−	46668.6i		−	1.49594i	0.663733	−	0.747969i	$-0.268971\pi$
$992$		0			0
$993$	34097.9	+	7665.46i	1.08969	+	0.244971i
$994$		0			0
$995$	2436.65			0.0776353
$996$		0			0
$997$	−47034.8			−1.49409			−0.747045	−	0.664774i	$-0.768528\pi$
$997$	−47034.8			−1.49409			−0.747045	+	0.664774i	$0.768528\pi$
$998$		0			0
$999$	19430.3	−	24861.0i	0.615362	−	0.787354i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	336.4.h.b.239.13	yes	24
3.2	odd	2	inner	336.4.h.b.239.11	✓	24
4.3	odd	2	inner	336.4.h.b.239.12	yes	24
12.11	even	2	inner	336.4.h.b.239.14	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
336.4.h.b.239.11	✓	24	3.2	odd	2	inner
336.4.h.b.239.12	yes	24	4.3	odd	2	inner
336.4.h.b.239.13	yes	24	1.1	even	1	trivial
336.4.h.b.239.14	yes	24	12.11	even	2	inner

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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	336.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(1.13969 - 5.06963i) q^{3} +14.6453i q^{5} -7.00000i q^{7} +(-24.4022 - 11.5556i) q^{9} +O(q^{10})\)	\(q+(1.13969 - 5.06963i) q^{3} +14.6453i q^{5} -7.00000i q^{7} +(-24.4022 - 11.5556i) q^{9} +13.9027 q^{11} -45.4743 q^{13} +(74.2459 + 16.6910i) q^{15} +120.873i q^{17} +34.9253i q^{19} +(-35.4874 - 7.97783i) q^{21} -151.440 q^{23} -89.4834 q^{25} +(-86.3935 + 110.540i) q^{27} -224.601i q^{29} +44.2927i q^{31} +(15.8448 - 70.4815i) q^{33} +102.517 q^{35} -224.904 q^{37} +(-51.8266 + 230.538i) q^{39} +459.995i q^{41} +497.722i q^{43} +(169.235 - 357.377i) q^{45} +134.391 q^{47} -49.0000 q^{49} +(612.781 + 137.758i) q^{51} +282.447i q^{53} +203.608i q^{55} +(177.058 + 39.8040i) q^{57} +48.3707 q^{59} -343.594 q^{61} +(-80.8892 + 170.816i) q^{63} -665.982i q^{65} +678.793i q^{67} +(-172.594 + 767.743i) q^{69} +820.561 q^{71} +370.525 q^{73} +(-101.983 + 453.647i) q^{75} -97.3188i q^{77} -986.876i q^{79} +(461.936 + 563.965i) q^{81} -484.889 q^{83} -1770.22 q^{85} +(-1138.64 - 255.976i) q^{87} -980.557i q^{89} +318.320i q^{91} +(224.548 + 50.4800i) q^{93} -511.490 q^{95} -488.599 q^{97} +(-339.256 - 160.654i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 136 q^{9}+O(q^{10}) \) 24 * q - 136 * q^9	\( 24 q - 136 q^{9} + 96 q^{13} - 112 q^{21} + 168 q^{25} - 320 q^{33} - 864 q^{37} + 592 q^{45} - 1176 q^{49} - 1120 q^{57} - 2304 q^{61} + 1168 q^{69} + 3312 q^{73} - 968 q^{81} - 5808 q^{85} + 1776 q^{93} + 6480 q^{97}+O(q^{100}) \) 24 * q - 136 * q^9 + 96 * q^13 - 112 * q^21 + 168 * q^25 - 320 * q^33 - 864 * q^37 + 592 * q^45 - 1176 * q^49 - 1120 * q^57 - 2304 * q^61 + 1168 * q^69 + 3312 * q^73 - 968 * q^81 - 5808 * q^85 + 1776 * q^93 + 6480 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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