LMFDB - Embedded newform 1296.5.e.h.161.9

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1296,5,Mod(161,1296)] 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("1296.161"); S:= CuspForms(chi, 5); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1296, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 5, names="a")

Level:	$ N $	$=$	$ 1296 = 2^{4} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	1296.e (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [12,0,0,0,0,0,-96,0,0,0,0,0,-72] 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:	no
Analytic conductor:	$133.967472157$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 160x^{10} + 9733x^{8} + 278004x^{6} + 3678300x^{4} + 18632592x^{2} + 25765776 $ x^12 + 160x^10 + 9733x^8 + 278004x^6 + 3678300x^4 + 18632592*x^2 + 25765776
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{10}\cdot 3^{12} $
Twist minimal:	no (minimal twist has level 648)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			161.9
Root			$-6.45812i$ of defining polynomial
Character	$\chi$	$=$	1296.161
Dual form			1296.5.e.h.161.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+21.6069i q^{5} -69.7156 q^{7} -101.749i q^{11} -132.364 q^{13} +18.2471i q^{17} -715.640 q^{19} +9.97161i q^{23} +158.144 q^{25} +1470.12i q^{29} -10.6290 q^{31} -1506.33i q^{35} -1588.06 q^{37} -1302.47i q^{41} -1416.55 q^{43} +1230.20i q^{47} +2459.26 q^{49} -4136.29i q^{53} +2198.49 q^{55} -2208.66i q^{59} +744.695 q^{61} -2859.98i q^{65} +548.306 q^{67} -4094.86i q^{71} -7716.83 q^{73} +7093.52i q^{77} -3867.59 q^{79} -6856.18i q^{83} -394.261 q^{85} +14349.6i q^{89} +9227.87 q^{91} -15462.7i q^{95} +9932.84 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 96 q^{7} - 72 q^{13} - 336 q^{19} + 84 q^{25} + 1536 q^{31} - 492 q^{37} - 10128 q^{43} - 6828 q^{49} + 13968 q^{55} + 26268 q^{61} + 20784 q^{67} - 9984 q^{73} - 44592 q^{79} - 45348 q^{85} + 11808 q^{91}+ \cdots + 76608 q^{97}+O(q^{100}) $ 12 * q - 96 * q^7 - 72 * q^13 - 336 * q^19 + 84 * q^25 + 1536 * q^31 - 492 * q^37 - 10128 * q^43 - 6828 * q^49 + 13968 * q^55 + 26268 * q^61 + 20784 * q^67 - 9984 * q^73 - 44592 * q^79 - 45348 * q^85 + 11808 * q^91 + 76608 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1296\mathbb{Z}\right)^\times$.

$n$	$325$	$1135$	$1217$
$\chi(n)$	$1$	$1$	$-1$

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	21.6069i	0.864274i	0.901808	+	0.432137i	$0.142240\pi$
$5$	21.6069i	0.864274i	−0.901808	+	0.432137i	$0.857760\pi$
$6$	0	0
$7$	−69.7156	−1.42277	−0.711384	−	0.702804i	$-0.751931\pi$
$7$	−69.7156	−1.42277	−0.711384	+	0.702804i	$0.751931\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 101.749i	− 0.840905i	−0.907315	−	0.420452i	$-0.861871\pi$
$11$	− 101.749i	− 0.840905i	0.907315	−	0.420452i	$-0.138129\pi$
$12$	0	0
$13$	−132.364	−0.783222	−0.391611	−	0.920131i	$-0.628082\pi$
$13$	−132.364	−0.783222	−0.391611	+	0.920131i	$0.628082\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	18.2471i	0.0631386i	0.999502	+	0.0315693i	$0.0100505\pi$
$17$	18.2471i	0.0631386i	−0.999502	+	0.0315693i	$0.989950\pi$
$18$	0	0
$19$	−715.640	−1.98238	−0.991191	−	0.132437i	$-0.957720\pi$
$19$	−715.640	−1.98238	−0.991191	+	0.132437i	$0.957720\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	9.97161i	0.0188499i	0.999956	+	0.00942496i	$0.00300010\pi$
$23$	9.97161i	0.0188499i	−0.999956	+	0.00942496i	$0.997000\pi$
$24$	0	0
$25$	158.144	0.253030
$26$	0	0
$27$	0	0
$28$	0	0
$29$	1470.12i	1.74806i	0.485872	+	0.874030i	$0.338502\pi$
$29$	1470.12i	1.74806i	−0.485872	+	0.874030i	$0.661498\pi$
$30$	0	0
$31$	−10.6290	−0.0110603	−0.00553017	−	0.999985i	$-0.501760\pi$
$31$	−10.6290	−0.0110603	−0.00553017	+	0.999985i	$0.501760\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 1506.33i	− 1.22966i
$36$	0	0
$37$	−1588.06	−1.16002	−0.580009	−	0.814610i	$-0.696951\pi$
$37$	−1588.06	−1.16002	−0.580009	+	0.814610i	$0.696951\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 1302.47i	− 0.774819i	−0.921908	−	0.387410i	$-0.873370\pi$
$41$	− 1302.47i	− 0.774819i	0.921908	−	0.387410i	$-0.126630\pi$
$42$	0	0
$43$	−1416.55	−0.766115	−0.383058	−	0.923724i	$-0.625129\pi$
$43$	−1416.55	−0.766115	−0.383058	+	0.923724i	$0.625129\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	1230.20i	0.556901i	0.960451	+	0.278451i	$0.0898209\pi$
$47$	1230.20i	0.556901i	−0.960451	+	0.278451i	$0.910179\pi$
$48$	0	0
$49$	2459.26	1.02427
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 4136.29i	− 1.47251i	−0.676703	−	0.736256i	$-0.736592\pi$
$53$	− 4136.29i	− 1.47251i	0.676703	−	0.736256i	$-0.263408\pi$
$54$	0	0
$55$	2198.49	0.726772
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 2208.66i	− 0.634490i	−0.948344	−	0.317245i	$-0.897242\pi$
$59$	− 2208.66i	− 0.634490i	0.948344	−	0.317245i	$-0.102758\pi$
$60$	0	0
$61$	744.695	0.200133	0.100067	−	0.994981i	$-0.468094\pi$
$61$	744.695	0.200133	0.100067	+	0.994981i	$0.468094\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 2859.98i	− 0.676918i
$66$	0	0
$67$	548.306	0.122144	0.0610722	−	0.998133i	$-0.480548\pi$
$67$	548.306	0.122144	0.0610722	+	0.998133i	$0.480548\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 4094.86i	− 0.812311i	−0.913804	−	0.406155i	$-0.866869\pi$
$71$	− 4094.86i	− 0.812311i	0.913804	−	0.406155i	$-0.133131\pi$
$72$	0	0
$73$	−7716.83	−1.44808	−0.724042	−	0.689756i	$-0.757718\pi$
$73$	−7716.83	−1.44808	−0.724042	+	0.689756i	$0.757718\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	7093.52i	1.19641i
$78$	0	0
$79$	−3867.59	−0.619707	−0.309853	−	0.950784i	$-0.600280\pi$
$79$	−3867.59	−0.619707	−0.309853	+	0.950784i	$0.600280\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 6856.18i	− 0.995236i	−0.867396	−	0.497618i	$-0.834208\pi$
$83$	− 6856.18i	− 0.995236i	0.867396	−	0.497618i	$-0.165792\pi$
$84$	0	0
$85$	−394.261	−0.0545691
$86$	0	0
$87$	0	0
$88$	0	0
$89$	14349.6i	1.81159i	0.423712	+	0.905797i	$0.360727\pi$
$89$	14349.6i	1.81159i	−0.423712	+	0.905797i	$0.639273\pi$
$90$	0	0
$91$	9227.87	1.11434
$92$	0	0
$93$	0	0
$94$	0	0
$95$	− 15462.7i	− 1.71332i
$96$	0	0
$97$	9932.84	1.05567	0.527837	−	0.849346i	$-0.323003\pi$
$97$	9932.84	1.05567	0.527837	+	0.849346i	$0.323003\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	10010.8i	0.981351i	0.871342	+	0.490675i	$0.163250\pi$
$101$	10010.8i	0.981351i	−0.871342	+	0.490675i	$0.836750\pi$
$102$	0	0
$103$	15903.5	1.49906	0.749529	−	0.661971i	$-0.230280\pi$
$103$	15903.5	1.49906	0.749529	+	0.661971i	$0.230280\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 480.920i	− 0.0420054i	−0.999779	−	0.0210027i	$-0.993314\pi$
$107$	− 480.920i	− 0.0420054i	0.999779	−	0.0210027i	$-0.00668586\pi$
$108$	0	0
$109$	8398.92	0.706920	0.353460	−	0.935450i	$-0.385005\pi$
$109$	8398.92	0.706920	0.353460	+	0.935450i	$0.385005\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 13429.9i	− 1.05176i	−0.850560	−	0.525878i	$-0.823737\pi$
$113$	− 13429.9i	− 1.05176i	0.850560	−	0.525878i	$-0.176263\pi$
$114$	0	0
$115$	−215.455	−0.0162915
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 1272.10i	− 0.0898315i
$120$	0	0
$121$	4288.04	0.292879
$122$	0	0
$123$	0	0
$124$	0	0
$125$	16921.3i	1.08296i
$126$	0	0
$127$	25615.2	1.58815	0.794073	−	0.607823i	$-0.207957\pi$
$127$	25615.2	1.58815	0.794073	+	0.607823i	$0.207957\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 8338.79i	− 0.485915i	−0.970037	−	0.242958i	$-0.921882\pi$
$131$	− 8338.79i	− 0.485915i	0.970037	−	0.242958i	$-0.0781176\pi$
$132$	0	0
$133$	49891.3	2.82047
$134$	0	0
$135$	0	0
$136$	0	0
$137$	34114.4i	1.81759i	0.417238	+	0.908797i	$0.362998\pi$
$137$	34114.4i	1.81759i	−0.417238	+	0.908797i	$0.637002\pi$
$138$	0	0
$139$	−26994.7	−1.39717	−0.698584	−	0.715528i	$-0.746186\pi$
$139$	−26994.7	−1.39717	−0.698584	+	0.715528i	$0.746186\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	13468.0i	0.658615i
$144$	0	0
$145$	−31764.6	−1.51080
$146$	0	0
$147$	0	0
$148$	0	0
$149$	27790.7i	1.25178i	0.779913	+	0.625888i	$0.215263\pi$
$149$	27790.7i	1.25178i	−0.779913	+	0.625888i	$0.784737\pi$
$150$	0	0
$151$	−24058.4	−1.05515	−0.527574	−	0.849509i	$-0.676898\pi$
$151$	−24058.4	−1.05515	−0.527574	+	0.849509i	$0.676898\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 229.659i	− 0.00955916i
$156$	0	0
$157$	31626.1	1.28306	0.641528	−	0.767099i	$-0.278301\pi$
$157$	31626.1	1.28306	0.641528	+	0.767099i	$0.278301\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 695.177i	− 0.0268191i
$162$	0	0
$163$	39116.7	1.47227	0.736133	−	0.676836i	$-0.236650\pi$
$163$	39116.7	1.47227	0.736133	+	0.676836i	$0.236650\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	15445.7i	0.553829i	0.960895	+	0.276915i	$0.0893119\pi$
$167$	15445.7i	0.553829i	−0.960895	+	0.276915i	$0.910688\pi$
$168$	0	0
$169$	−11040.6	−0.386564
$170$	0	0
$171$	0	0
$172$	0	0
$173$	19177.5i	0.640765i	0.947288	+	0.320382i	$0.103811\pi$
$173$	19177.5i	0.640765i	−0.947288	+	0.320382i	$0.896189\pi$
$174$	0	0
$175$	−11025.1	−0.360003
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 61163.7i	− 1.90892i	−0.298341	−	0.954459i	$-0.596433\pi$
$179$	− 61163.7i	− 1.90892i	0.298341	−	0.954459i	$-0.403567\pi$
$180$	0	0
$181$	12496.9	0.381455	0.190728	−	0.981643i	$-0.438915\pi$
$181$	12496.9	0.381455	0.190728	+	0.981643i	$0.438915\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 34313.1i	− 1.00257i
$186$	0	0
$187$	1856.63	0.0530935
$188$	0	0
$189$	0	0
$190$	0	0
$191$	43580.8i	1.19462i	0.802012	+	0.597308i	$0.203763\pi$
$191$	43580.8i	1.19462i	−0.802012	+	0.597308i	$0.796237\pi$
$192$	0	0
$193$	−50329.8	−1.35117	−0.675585	−	0.737282i	$-0.736109\pi$
$193$	−50329.8	−1.35117	−0.675585	+	0.737282i	$0.736109\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 33260.4i	− 0.857028i	−0.903535	−	0.428514i	$-0.859037\pi$
$197$	− 33260.4i	− 0.857028i	0.903535	−	0.428514i	$-0.140963\pi$
$198$	0	0
$199$	52566.2	1.32740	0.663698	−	0.748000i	$-0.268986\pi$
$199$	52566.2	1.32740	0.663698	+	0.748000i	$0.268986\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 102490.i	− 2.48708i
$204$	0	0
$205$	28142.3	0.669657
$206$	0	0
$207$	0	0
$208$	0	0
$209$	72816.0i	1.66700i
$210$	0	0
$211$	51732.5	1.16198	0.580989	−	0.813911i	$-0.302666\pi$
$211$	51732.5	1.16198	0.580989	+	0.813911i	$0.302666\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	− 30607.1i	− 0.662134i
$216$	0	0
$217$	741.006	0.0157363
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 2415.26i	− 0.0494515i
$222$	0	0
$223$	−28926.1	−0.581675	−0.290837	−	0.956773i	$-0.593934\pi$
$223$	−28926.1	−0.581675	−0.290837	+	0.956773i	$0.593934\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	67779.5i	1.31537i	0.753295	+	0.657683i	$0.228463\pi$
$227$	67779.5i	1.31537i	−0.753295	+	0.657683i	$0.771537\pi$
$228$	0	0
$229$	22431.8	0.427753	0.213877	−	0.976861i	$-0.431391\pi$
$229$	22431.8	0.427753	0.213877	+	0.976861i	$0.431391\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	58686.1i	1.08099i	0.841346	+	0.540497i	$0.181764\pi$
$233$	58686.1i	1.08099i	−0.841346	+	0.540497i	$0.818236\pi$
$234$	0	0
$235$	−26580.7	−0.481316
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 75957.1i	− 1.32976i	−0.746951	−	0.664879i	$-0.768483\pi$
$239$	− 75957.1i	− 1.32976i	0.746951	−	0.664879i	$-0.231517\pi$
$240$	0	0
$241$	36766.5	0.633021	0.316510	−	0.948589i	$-0.397489\pi$
$241$	36766.5	0.633021	0.316510	+	0.948589i	$0.397489\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	53136.9i	0.885247i
$246$	0	0
$247$	94725.3	1.55265
$248$	0	0
$249$	0	0
$250$	0	0
$251$	12072.2i	0.191620i	0.995400	+	0.0958098i	$0.0305441\pi$
$251$	12072.2i	0.191620i	−0.995400	+	0.0958098i	$0.969456\pi$
$252$	0	0
$253$	1014.61	0.0158510
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 43388.7i	− 0.656917i	−0.944518	−	0.328459i	$-0.893471\pi$
$257$	− 43388.7i	− 0.656917i	0.944518	−	0.328459i	$-0.106529\pi$
$258$	0	0
$259$	110713.	1.65043
$260$	0	0
$261$	0	0
$262$	0	0
$263$	21944.0i	0.317253i	0.987339	+	0.158626i	$0.0507065\pi$
$263$	21944.0i	0.317253i	−0.987339	+	0.158626i	$0.949294\pi$
$264$	0	0
$265$	89372.2	1.27265
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 82701.3i	− 1.14290i	−0.820637	−	0.571450i	$-0.806381\pi$
$269$	− 82701.3i	− 1.14290i	0.820637	−	0.571450i	$-0.193619\pi$
$270$	0	0
$271$	−60271.4	−0.820678	−0.410339	−	0.911933i	$-0.634590\pi$
$271$	−60271.4	−0.820678	−0.410339	+	0.911933i	$0.634590\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 16091.0i	− 0.212774i
$276$	0	0
$277$	−38453.7	−0.501163	−0.250581	−	0.968096i	$-0.580622\pi$
$277$	−38453.7	−0.501163	−0.250581	+	0.968096i	$0.580622\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 61364.6i	− 0.777150i	−0.921417	−	0.388575i	$-0.872967\pi$
$281$	− 61364.6i	− 0.777150i	0.921417	−	0.388575i	$-0.127033\pi$
$282$	0	0
$283$	−86332.7	−1.07796	−0.538980	−	0.842319i	$-0.681190\pi$
$283$	−86332.7	−1.07796	−0.538980	+	0.842319i	$0.681190\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	90802.6i	1.10239i
$288$	0	0
$289$	83188.0	0.996014
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 163937.i	− 1.90960i	−0.297249	−	0.954800i	$-0.596069\pi$
$293$	− 163937.i	− 1.90960i	0.297249	−	0.954800i	$-0.403931\pi$
$294$	0	0
$295$	47722.2	0.548373
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 1319.89i	− 0.0147637i
$300$	0	0
$301$	98755.4	1.09000
$302$	0	0
$303$	0	0
$304$	0	0
$305$	16090.5i	0.172970i
$306$	0	0
$307$	76361.8	0.810214	0.405107	−	0.914269i	$-0.367234\pi$
$307$	76361.8	0.810214	0.405107	+	0.914269i	$0.367234\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	117976.i	1.21976i	0.792495	+	0.609878i	$0.208781\pi$
$311$	117976.i	1.21976i	−0.792495	+	0.609878i	$0.791219\pi$
$312$	0	0
$313$	91794.9	0.936979	0.468490	−	0.883469i	$-0.344798\pi$
$313$	91794.9	0.936979	0.468490	+	0.883469i	$0.344798\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 2094.20i	− 0.0208401i	−0.999946	−	0.0104201i	$-0.996683\pi$
$317$	− 2094.20i	− 0.0208401i	0.999946	−	0.0104201i	$-0.00331687\pi$
$318$	0	0
$319$	149584.	1.46995
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 13058.3i	− 0.125165i
$324$	0	0
$325$	−20932.6	−0.198178
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 85763.8i	− 0.792341i
$330$	0	0
$331$	101827.	0.929405	0.464702	−	0.885467i	$-0.346161\pi$
$331$	101827.	0.929405	0.464702	+	0.885467i	$0.346161\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	11847.2i	0.105566i
$336$	0	0
$337$	111745.	0.983940	0.491970	−	0.870612i	$-0.336277\pi$
$337$	111745.	0.983940	0.491970	+	0.870612i	$0.336277\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	1081.49i	0.00930069i
$342$	0	0
$343$	−4061.82	−0.0345249
$344$	0	0
$345$	0	0
$346$	0	0
$347$	2705.37i	0.0224681i	0.999937	+	0.0112341i	$0.00357599\pi$
$347$	2705.37i	0.0224681i	−0.999937	+	0.0112341i	$0.996424\pi$
$348$	0	0
$349$	1045.42	0.00858301	0.00429151	−	0.999991i	$-0.498634\pi$
$349$	1045.42	0.00858301	0.00429151	+	0.999991i	$0.498634\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	42334.9i	0.339742i	0.985466	+	0.169871i	$0.0543351\pi$
$353$	42334.9i	0.339742i	−0.985466	+	0.169871i	$0.945665\pi$
$354$	0	0
$355$	88477.0	0.702059
$356$	0	0
$357$	0	0
$358$	0	0
$359$	156787.i	1.21652i	0.793737	+	0.608261i	$0.208133\pi$
$359$	156787.i	1.21652i	−0.793737	+	0.608261i	$0.791867\pi$
$360$	0	0
$361$	381820.	2.92984
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 166737.i	− 1.25154i
$366$	0	0
$367$	226135.	1.67894	0.839470	−	0.543406i	$-0.182865\pi$
$367$	226135.	1.67894	0.839470	+	0.543406i	$0.182865\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	288364.i	2.09504i
$372$	0	0
$373$	−49361.2	−0.354788	−0.177394	−	0.984140i	$-0.556767\pi$
$373$	−49361.2	−0.354788	−0.177394	+	0.984140i	$0.556767\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 194591.i	− 1.36912i
$378$	0	0
$379$	−241069.	−1.67828	−0.839138	−	0.543918i	$-0.816940\pi$
$379$	−241069.	−1.67828	−0.839138	+	0.543918i	$0.816940\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 267148.i	− 1.82119i	−0.413300	−	0.910595i	$-0.635624\pi$
$383$	− 267148.i	− 1.82119i	0.413300	−	0.910595i	$-0.364376\pi$
$384$	0	0
$385$	−153269.	−1.03403
$386$	0	0
$387$	0	0
$388$	0	0
$389$	122781.i	0.811394i	0.914008	+	0.405697i	$0.132971\pi$
$389$	122781.i	0.811394i	−0.914008	+	0.405697i	$0.867029\pi$
$390$	0	0
$391$	−181.953	−0.00119016
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 83566.5i	− 0.535597i
$396$	0	0
$397$	261605.	1.65984	0.829919	−	0.557885i	$-0.188387\pi$
$397$	261605.	1.65984	0.829919	+	0.557885i	$0.188387\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	53729.9i	0.334139i	0.985945	+	0.167069i	$0.0534304\pi$
$401$	53729.9i	0.334139i	−0.985945	+	0.167069i	$0.946570\pi$
$402$	0	0
$403$	1406.90	0.00866269
$404$	0	0
$405$	0	0
$406$	0	0
$407$	161585.i	0.975464i
$408$	0	0
$409$	−189509.	−1.13288	−0.566438	−	0.824104i	$-0.691679\pi$
$409$	−189509.	−1.13288	−0.566438	+	0.824104i	$0.691679\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	153978.i	0.902731i
$414$	0	0
$415$	148140.	0.860157
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 3031.26i	− 0.0172662i	−0.999963	−	0.00863308i	$-0.997252\pi$
$419$	− 3031.26i	− 0.0172662i	0.999963	−	0.00863308i	$-0.00274803\pi$
$420$	0	0
$421$	−230447.	−1.30019	−0.650094	−	0.759854i	$-0.725271\pi$
$421$	−230447.	−1.30019	−0.650094	+	0.759854i	$0.725271\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	2885.66i	0.0159759i
$426$	0	0
$427$	−51916.9	−0.284743
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 72632.4i	− 0.390999i	−0.980704	−	0.195499i	$-0.937367\pi$
$431$	− 72632.4i	− 0.390999i	0.980704	−	0.195499i	$-0.0626328\pi$
$432$	0	0
$433$	−135194.	−0.721076	−0.360538	−	0.932745i	$-0.617407\pi$
$433$	−135194.	−0.721076	−0.360538	+	0.932745i	$0.617407\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 7136.09i	− 0.0373678i
$438$	0	0
$439$	−707.513	−0.00367118	−0.00183559	−	0.999998i	$-0.500584\pi$
$439$	−707.513	−0.00367118	−0.00183559	+	0.999998i	$0.500584\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 61072.1i	− 0.311197i	−0.987820	−	0.155599i	$-0.950269\pi$
$443$	− 61072.1i	− 0.311197i	0.987820	−	0.155599i	$-0.0497306\pi$
$444$	0	0
$445$	−310051.	−1.56571
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 312118.i	− 1.54820i	−0.633064	−	0.774099i	$-0.718203\pi$
$449$	− 312118.i	− 1.54820i	0.633064	−	0.774099i	$-0.281797\pi$
$450$	0	0
$451$	−132526.	−0.651549
$452$	0	0
$453$	0	0
$454$	0	0
$455$	199385.i	0.963097i
$456$	0	0
$457$	−69590.7	−0.333211	−0.166605	−	0.986024i	$-0.553281\pi$
$457$	−69590.7	−0.333211	−0.166605	+	0.986024i	$0.553281\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 258360.i	− 1.21569i	−0.794056	−	0.607845i	$-0.792034\pi$
$461$	− 258360.i	− 1.21569i	0.794056	−	0.607845i	$-0.207966\pi$
$462$	0	0
$463$	−180479.	−0.841908	−0.420954	−	0.907082i	$-0.638305\pi$
$463$	−180479.	−0.841908	−0.420954	+	0.907082i	$0.638305\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	109307.i	0.501203i	0.968090	+	0.250602i	$0.0806284\pi$
$467$	109307.i	0.501203i	−0.968090	+	0.250602i	$0.919372\pi$
$468$	0	0
$469$	−38225.5	−0.173783
$470$	0	0
$471$	0	0
$472$	0	0
$473$	144133.i	0.644230i
$474$	0	0
$475$	−113174.	−0.501602
$476$	0	0
$477$	0	0
$478$	0	0
$479$	369464.i	1.61028i	0.593086	+	0.805139i	$0.297909\pi$
$479$	369464.i	1.61028i	−0.593086	+	0.805139i	$0.702091\pi$
$480$	0	0
$481$	210203.	0.908551
$482$	0	0
$483$	0	0
$484$	0	0
$485$	214617.i	0.912392i
$486$	0	0
$487$	−88956.5	−0.375076	−0.187538	−	0.982257i	$-0.560051\pi$
$487$	−88956.5	−0.375076	−0.187538	+	0.982257i	$0.560051\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	342110.i	1.41907i	0.704671	+	0.709534i	$0.251095\pi$
$491$	342110.i	1.41907i	−0.704671	+	0.709534i	$0.748905\pi$
$492$	0	0
$493$	−26825.3	−0.110370
$494$	0	0
$495$	0	0
$496$	0	0
$497$	285475.i	1.15573i
$498$	0	0
$499$	−45872.0	−0.184224	−0.0921120	−	0.995749i	$-0.529362\pi$
$499$	−45872.0	−0.184224	−0.0921120	+	0.995749i	$0.529362\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	224886.i	0.888845i	0.895817	+	0.444422i	$0.146591\pi$
$503$	224886.i	0.888845i	−0.895817	+	0.444422i	$0.853409\pi$
$504$	0	0
$505$	−216301.	−0.848156
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 387111.i	− 1.49417i	−0.664729	−	0.747085i	$-0.731453\pi$
$509$	− 387111.i	− 1.49417i	0.664729	−	0.747085i	$-0.268547\pi$
$510$	0	0
$511$	537984.	2.06028
$512$	0	0
$513$	0	0
$514$	0	0
$515$	343625.i	1.29560i
$516$	0	0
$517$	125172.	0.468301
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 303979.i	− 1.11987i	−0.828536	−	0.559936i	$-0.810826\pi$
$521$	− 303979.i	− 1.11987i	0.828536	−	0.559936i	$-0.189174\pi$
$522$	0	0
$523$	52206.3	0.190862	0.0954310	−	0.995436i	$-0.469577\pi$
$523$	52206.3	0.190862	0.0954310	+	0.995436i	$0.469577\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 193.948i	0 0.000698334i
$528$	0	0
$529$	279742.	0.999645
$530$	0	0
$531$	0	0
$532$	0	0
$533$	172401.i	0.606855i
$534$	0	0
$535$	10391.2	0.0363042
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 250229.i	− 0.861310i
$540$	0	0
$541$	166809.	0.569935	0.284968	−	0.958537i	$-0.408017\pi$
$541$	166809.	0.569935	0.284968	+	0.958537i	$0.408017\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	181474.i	0.610973i
$546$	0	0
$547$	−326231.	−1.09031	−0.545156	−	0.838335i	$-0.683530\pi$
$547$	−326231.	−1.09031	−0.545156	+	0.838335i	$0.683530\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	− 1.05208e6i	− 3.46532i
$552$	0	0
$553$	269631.	0.881698
$554$	0	0
$555$	0	0
$556$	0	0
$557$	129945.i	0.418840i	0.977826	+	0.209420i	$0.0671576\pi$
$557$	129945.i	0.418840i	−0.977826	+	0.209420i	$0.932842\pi$
$558$	0	0
$559$	187500.	0.600038
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 259438.i	− 0.818496i	−0.912423	−	0.409248i	$-0.865791\pi$
$563$	− 259438.i	− 0.818496i	0.912423	−	0.409248i	$-0.134209\pi$
$564$	0	0
$565$	290178.	0.909006
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 363906.i	− 1.12400i	−0.827139	−	0.561998i	$-0.810033\pi$
$569$	− 363906.i	− 1.12400i	0.827139	−	0.561998i	$-0.189967\pi$
$570$	0	0
$571$	−197294.	−0.605122	−0.302561	−	0.953130i	$-0.597842\pi$
$571$	−197294.	−0.605122	−0.302561	+	0.953130i	$0.597842\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	1576.95i	0.00476959i
$576$	0	0
$577$	186408.	0.559904	0.279952	−	0.960014i	$-0.409682\pi$
$577$	186408.	0.559904	0.279952	+	0.960014i	$0.409682\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	477983.i	1.41599i
$582$	0	0
$583$	−420865.	−1.23824
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 101765.i	− 0.295339i	−0.989037	−	0.147670i	$-0.952823\pi$
$587$	− 101765.i	− 0.295339i	0.989037	−	0.147670i	$-0.0471773\pi$
$588$	0	0
$589$	7606.53	0.0219258
$590$	0	0
$591$	0	0
$592$	0	0
$593$	416289.i	1.18382i	0.806004	+	0.591910i	$0.201626\pi$
$593$	416289.i	1.18382i	−0.806004	+	0.591910i	$0.798374\pi$
$594$	0	0
$595$	27486.2	0.0776391
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 290224.i	− 0.808872i	−0.914566	−	0.404436i	$-0.867468\pi$
$599$	− 290224.i	− 0.808872i	0.914566	−	0.404436i	$-0.132532\pi$
$600$	0	0
$601$	−35413.7	−0.0980444	−0.0490222	−	0.998798i	$-0.515611\pi$
$601$	−35413.7	−0.0980444	−0.0490222	+	0.998798i	$0.515611\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	92651.1i	0.253128i
$606$	0	0
$607$	581459.	1.57813	0.789063	−	0.614313i	$-0.210567\pi$
$607$	581459.	1.57813	0.789063	+	0.614313i	$0.210567\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 162834.i	− 0.436177i
$612$	0	0
$613$	−560291.	−1.49105	−0.745526	−	0.666476i	$-0.767802\pi$
$613$	−560291.	−1.49105	−0.745526	+	0.666476i	$0.767802\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 328808.i	− 0.863718i	−0.901941	−	0.431859i	$-0.857858\pi$
$617$	− 328808.i	− 0.863718i	0.901941	−	0.431859i	$-0.142142\pi$
$618$	0	0
$619$	−467306.	−1.21961	−0.609804	−	0.792553i	$-0.708752\pi$
$619$	−467306.	−1.21961	−0.609804	+	0.792553i	$0.708752\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 1.00039e6i	− 2.57748i
$624$	0	0
$625$	−266776.	−0.682946
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 28977.5i	− 0.0732418i
$630$	0	0
$631$	−355301.	−0.892356	−0.446178	−	0.894944i	$-0.647215\pi$
$631$	−355301.	−0.892356	−0.446178	+	0.894944i	$0.647215\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	553464.i	1.37259i
$636$	0	0
$637$	−325519.	−0.802227
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 684806.i	− 1.66668i	−0.552763	−	0.833339i	$-0.686426\pi$
$641$	− 684806.i	− 1.66668i	0.552763	−	0.833339i	$-0.313574\pi$
$642$	0	0
$643$	356625.	0.862560	0.431280	−	0.902218i	$-0.358062\pi$
$643$	356625.	0.862560	0.431280	+	0.902218i	$0.358062\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	192531.i	0.459930i	0.973199	+	0.229965i	$0.0738612\pi$
$647$	192531.i	0.459930i	−0.973199	+	0.229965i	$0.926139\pi$
$648$	0	0
$649$	−224730.	−0.533545
$650$	0	0
$651$	0	0
$652$	0	0
$653$	608250.i	1.42645i	0.700936	+	0.713224i	$0.252766\pi$
$653$	608250.i	1.42645i	−0.700936	+	0.713224i	$0.747234\pi$
$654$	0	0
$655$	180175.	0.419964
$656$	0	0
$657$	0	0
$658$	0	0
$659$	104953.i	0.241671i	0.992673	+	0.120835i	$0.0385573\pi$
$659$	104953.i	0.241671i	−0.992673	+	0.120835i	$0.961443\pi$
$660$	0	0
$661$	−406962.	−0.931432	−0.465716	−	0.884934i	$-0.654203\pi$
$661$	−406962.	−0.931432	−0.465716	+	0.884934i	$0.654203\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	1.07799e6i	2.43766i
$666$	0	0
$667$	−14659.4	−0.0329508
$668$	0	0
$669$	0	0
$670$	0	0
$671$	− 75772.3i	− 0.168293i
$672$	0	0
$673$	163990.	0.362065	0.181033	−	0.983477i	$-0.442056\pi$
$673$	163990.	0.362065	0.181033	+	0.983477i	$0.442056\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 354958.i	− 0.774461i	−0.921983	−	0.387230i	$-0.873432\pi$
$677$	− 354958.i	− 0.774461i	0.921983	−	0.387230i	$-0.126568\pi$
$678$	0	0
$679$	−692474.	−1.50198
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 278372.i	− 0.596738i	−0.954451	−	0.298369i	$-0.903557\pi$
$683$	− 278372.i	− 0.596738i	0.954451	−	0.298369i	$-0.0964427\pi$
$684$	0	0
$685$	−737106.	−1.57090
$686$	0	0
$687$	0	0
$688$	0	0
$689$	547498.i	1.15330i
$690$	0	0
$691$	−720308.	−1.50856	−0.754279	−	0.656554i	$-0.772013\pi$
$691$	−720308.	−1.50856	−0.754279	+	0.656554i	$0.772013\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 583270.i	− 1.20754i
$696$	0	0
$697$	23766.3	0.0489210
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 742057.i	− 1.51008i	−0.655677	−	0.755042i	$-0.727617\pi$
$701$	− 742057.i	− 1.51008i	0.655677	−	0.755042i	$-0.272383\pi$
$702$	0	0
$703$	1.13648e6	2.29960
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 697906.i	− 1.39623i
$708$	0	0
$709$	634340.	1.26191	0.630957	−	0.775818i	$-0.282663\pi$
$709$	634340.	1.26191	0.630957	+	0.775818i	$0.282663\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 105.988i	0 0.000208486i
$714$	0	0
$715$	−291002.	−0.569224
$716$	0	0
$717$	0	0
$718$	0	0
$719$	− 256604.i	− 0.496369i	−0.968713	−	0.248185i	$-0.920166\pi$
$719$	− 256604.i	− 0.496369i	0.968713	−	0.248185i	$-0.0798340\pi$
$720$	0	0
$721$	−1.10872e6	−2.13281
$722$	0	0
$723$	0	0
$724$	0	0
$725$	232490.i	0.442311i
$726$	0	0
$727$	−151785.	−0.287184	−0.143592	−	0.989637i	$-0.545865\pi$
$727$	−151785.	−0.287184	−0.143592	+	0.989637i	$0.545865\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 25847.8i	− 0.0483714i
$732$	0	0
$733$	615126.	1.14487	0.572435	−	0.819950i	$-0.305999\pi$
$733$	615126.	1.14487	0.572435	+	0.819950i	$0.305999\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 55789.9i	− 0.102712i
$738$	0	0
$739$	−58065.2	−0.106323	−0.0531614	−	0.998586i	$-0.516930\pi$
$739$	−58065.2	−0.106323	−0.0531614	+	0.998586i	$0.516930\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	71109.7i	0.128810i	0.997924	+	0.0644052i	$0.0205150\pi$
$743$	71109.7i	0.128810i	−0.997924	+	0.0644052i	$0.979485\pi$
$744$	0	0
$745$	−600469.	−1.08188
$746$	0	0
$747$	0	0
$748$	0	0
$749$	33527.6i	0.0597639i
$750$	0	0
$751$	−518490.	−0.919306	−0.459653	−	0.888099i	$-0.652026\pi$
$751$	−518490.	−0.919306	−0.459653	+	0.888099i	$0.652026\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 519827.i	− 0.911937i
$756$	0	0
$757$	−185939.	−0.324473	−0.162237	−	0.986752i	$-0.551871\pi$
$757$	−185939.	−0.324473	−0.162237	+	0.986752i	$0.551871\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	191342.i	0.330400i	0.986260	+	0.165200i	$0.0528270\pi$
$761$	191342.i	0.330400i	−0.986260	+	0.165200i	$0.947173\pi$
$762$	0	0
$763$	−585536.	−1.00578
$764$	0	0
$765$	0	0
$766$	0	0
$767$	292348.i	0.496946i
$768$	0	0
$769$	−475470.	−0.804027	−0.402013	−	0.915634i	$-0.631689\pi$
$769$	−475470.	−0.804027	−0.402013	+	0.915634i	$0.631689\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	13977.8i	0.0233926i	0.999932	+	0.0116963i	$0.00372313\pi$
$773$	13977.8i	0.0233926i	−0.999932	+	0.0116963i	$0.996277\pi$
$774$	0	0
$775$	−1680.91	−0.00279859
$776$	0	0
$777$	0	0
$778$	0	0
$779$	932101.i	1.53599i
$780$	0	0
$781$	−416650.	−0.683076
$782$	0	0
$783$	0	0
$784$	0	0
$785$	683340.i	1.10891i
$786$	0	0
$787$	−240778.	−0.388747	−0.194374	−	0.980928i	$-0.562267\pi$
$787$	−240778.	−0.388747	−0.194374	+	0.980928i	$0.562267\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	936272.i	1.49640i
$792$	0	0
$793$	−98571.2	−0.156749
$794$	0	0
$795$	0	0
$796$	0	0
$797$	996311.i	1.56848i	0.620459	+	0.784239i	$0.286946\pi$
$797$	996311.i	1.56848i	−0.620459	+	0.784239i	$0.713054\pi$
$798$	0	0
$799$	−22447.4	−0.0351620
$800$	0	0
$801$	0	0
$802$	0	0
$803$	785184.i	1.21770i
$804$	0	0
$805$	15020.6	0.0231790
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 204246.i	− 0.312073i	−0.987751	−	0.156036i	$-0.950128\pi$
$809$	− 204246.i	− 0.312073i	0.987751	−	0.156036i	$-0.0498717\pi$
$810$	0	0
$811$	−489147.	−0.743700	−0.371850	−	0.928293i	$-0.621276\pi$
$811$	−489147.	−0.743700	−0.371850	+	0.928293i	$0.621276\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	845188.i	1.27244i
$816$	0	0
$817$	1.01374e6	1.51873
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 106409.i	− 0.157867i	−0.996880	−	0.0789335i	$-0.974849\pi$
$821$	− 106409.i	− 0.157867i	0.996880	−	0.0789335i	$-0.0251515\pi$
$822$	0	0
$823$	−150725.	−0.222529	−0.111265	−	0.993791i	$-0.535490\pi$
$823$	−150725.	−0.222529	−0.111265	+	0.993791i	$0.535490\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 1.33944e6i	− 1.95846i	−0.202763	−	0.979228i	$-0.564992\pi$
$827$	− 1.33944e6i	− 1.95846i	0.202763	−	0.979228i	$-0.435008\pi$
$828$	0	0
$829$	345624.	0.502915	0.251457	−	0.967868i	$-0.419090\pi$
$829$	345624.	0.502915	0.251457	+	0.967868i	$0.419090\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	44874.3i	0.0646707i
$834$	0	0
$835$	−333734.	−0.478660
$836$	0	0
$837$	0	0
$838$	0	0
$839$	− 1.37155e6i	− 1.94844i	−0.225599	−	0.974220i	$-0.572434\pi$
$839$	− 1.37155e6i	− 1.94844i	0.225599	−	0.974220i	$-0.427566\pi$
$840$	0	0
$841$	−1.45397e6	−2.05571
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 238554.i	− 0.334097i
$846$	0	0
$847$	−298943.	−0.416699
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 15835.5i	− 0.0218662i
$852$	0	0
$853$	626580.	0.861150	0.430575	−	0.902555i	$-0.358311\pi$
$853$	626580.	0.861150	0.430575	+	0.902555i	$0.358311\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	393713.i	0.536066i	0.963410	+	0.268033i	$0.0863737\pi$
$857$	393713.i	0.536066i	−0.963410	+	0.268033i	$0.913626\pi$
$858$	0	0
$859$	−960937.	−1.30229	−0.651146	−	0.758952i	$-0.725712\pi$
$859$	−960937.	−1.30229	−0.651146	+	0.758952i	$0.725712\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	560804.i	0.752991i	0.926418	+	0.376495i	$0.122871\pi$
$863$	560804.i	0.752991i	−0.926418	+	0.376495i	$0.877129\pi$
$864$	0	0
$865$	−414365.	−0.553797
$866$	0	0
$867$	0	0
$868$	0	0
$869$	393525.i	0.521114i
$870$	0	0
$871$	−72576.2	−0.0956661
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 1.17968e6i	− 1.54080i
$876$	0	0
$877$	−391716.	−0.509298	−0.254649	−	0.967033i	$-0.581960\pi$
$877$	−391716.	−0.509298	−0.254649	+	0.967033i	$0.581960\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	1.48558e6i	1.91401i	0.290079	+	0.957003i	$0.406318\pi$
$881$	1.48558e6i	1.91401i	−0.290079	+	0.957003i	$0.593682\pi$
$882$	0	0
$883$	−682700.	−0.875605	−0.437803	−	0.899071i	$-0.644243\pi$
$883$	−682700.	−0.875605	−0.437803	+	0.899071i	$0.644243\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	981548.i	1.24757i	0.781597	+	0.623784i	$0.214406\pi$
$887$	981548.i	1.24757i	−0.781597	+	0.623784i	$0.785594\pi$
$888$	0	0
$889$	−1.78578e6	−2.25956
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 880377.i	− 1.10399i
$894$	0	0
$895$	1.32155e6	1.64983
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 15625.9i	− 0.0193341i
$900$	0	0
$901$	75475.1	0.0929724
$902$	0	0
$903$	0	0
$904$	0	0
$905$	270018.i	0.329682i
$906$	0	0
$907$	−940567.	−1.14334	−0.571670	−	0.820484i	$-0.693704\pi$
$907$	−940567.	−1.14334	−0.571670	+	0.820484i	$0.693704\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	− 985946.i	− 1.18800i	−0.804465	−	0.594000i	$-0.797548\pi$
$911$	− 985946.i	− 1.18800i	0.804465	−	0.594000i	$-0.202452\pi$
$912$	0	0
$913$	−697613.	−0.836899
$914$	0	0
$915$	0	0
$916$	0	0
$917$	581343.i	0.691344i
$918$	0	0
$919$	768111.	0.909480	0.454740	−	0.890624i	$-0.349732\pi$
$919$	768111.	0.909480	0.454740	+	0.890624i	$0.349732\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	542014.i	0.636219i
$924$	0	0
$925$	−251142.	−0.293519
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 130876.i	− 0.151645i	−0.997121	−	0.0758223i	$-0.975842\pi$
$929$	− 130876.i	− 0.151645i	0.997121	−	0.0758223i	$-0.0241582\pi$
$930$	0	0
$931$	−1.75995e6	−2.03049
$932$	0	0
$933$	0	0
$934$	0	0
$935$	40115.9i	0.0458874i
$936$	0	0
$937$	1.16253e6	1.32411	0.662055	−	0.749455i	$-0.269684\pi$
$937$	1.16253e6	1.32411	0.662055	+	0.749455i	$0.269684\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 361510.i	− 0.408264i	−0.978943	−	0.204132i	$-0.934563\pi$
$941$	− 361510.i	− 0.408264i	0.978943	−	0.204132i	$-0.0654371\pi$
$942$	0	0
$943$	12987.7	0.0146053
$944$	0	0
$945$	0	0
$946$	0	0
$947$	1.29315e6i	1.44194i	0.692965	+	0.720971i	$0.256304\pi$
$947$	1.29315e6i	1.44194i	−0.692965	+	0.720971i	$0.743696\pi$
$948$	0	0
$949$	1.02143e6	1.13417
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 1.60264e6i	− 1.76462i	−0.470672	−	0.882308i	$-0.655989\pi$
$953$	− 1.60264e6i	− 1.76462i	0.470672	−	0.882308i	$-0.344011\pi$
$954$	0	0
$955$	−941643.	−1.03248
$956$	0	0
$957$	0	0
$958$	0	0
$959$	− 2.37831e6i	− 2.58601i
$960$	0	0
$961$	−923408.	−0.999878
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 1.08747e6i	− 1.16778i
$966$	0	0
$967$	850224.	0.909244	0.454622	−	0.890685i	$-0.349774\pi$
$967$	850224.	0.909244	0.454622	+	0.890685i	$0.349774\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	819691.i	0.869384i	0.900579	+	0.434692i	$0.143143\pi$
$971$	819691.i	0.869384i	−0.900579	+	0.434692i	$0.856857\pi$
$972$	0	0
$973$	1.88195e6	1.98784
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 816791.i	− 0.855700i	−0.903850	−	0.427850i	$-0.859271\pi$
$977$	− 816791.i	− 0.855700i	0.903850	−	0.427850i	$-0.140729\pi$
$978$	0	0
$979$	1.46007e6	1.52338
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 321055.i	− 0.332255i	−0.986104	−	0.166128i	$-0.946874\pi$
$983$	− 321055.i	− 0.332255i	0.986104	−	0.166128i	$-0.0531264\pi$
$984$	0	0
$985$	718652.	0.740707
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 14125.3i	− 0.0144412i
$990$	0	0
$991$	251405.	0.255992	0.127996	−	0.991775i	$-0.459146\pi$
$991$	251405.	0.255992	0.127996	+	0.991775i	$0.459146\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	1.13579e6i	1.14723i
$996$	0	0
$997$	107149.	0.107795	0.0538974	−	0.998546i	$-0.482836\pi$
$997$	107149.	0.107795	0.0538974	+	0.998546i	$0.482836\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1296.5.e.h.161.9		12
3.2	odd	2	inner	1296.5.e.h.161.4		12
4.3	odd	2		648.5.e.b.161.9	yes	12
12.11	even	2		648.5.e.b.161.4	✓	12
36.7	odd	6		648.5.m.g.377.4		24
36.11	even	6		648.5.m.g.377.9		24
36.23	even	6		648.5.m.g.593.4		24
36.31	odd	6		648.5.m.g.593.9		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
648.5.e.b.161.4	✓	12	12.11	even	2
648.5.e.b.161.9	yes	12	4.3	odd	2
648.5.m.g.377.4		24	36.7	odd	6
648.5.m.g.377.9		24	36.11	even	6
648.5.m.g.593.4		24	36.23	even	6
648.5.m.g.593.9		24	36.31	odd	6
1296.5.e.h.161.4		12	3.2	odd	2	inner
1296.5.e.h.161.9		12	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 1296 = 2^{4} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	1296.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+21.6069i q^{5} -69.7156 q^{7} -101.749i q^{11} -132.364 q^{13} +18.2471i q^{17} -715.640 q^{19} +9.97161i q^{23} +158.144 q^{25} +1470.12i q^{29} -10.6290 q^{31} -1506.33i q^{35} -1588.06 q^{37} -1302.47i q^{41} -1416.55 q^{43} +1230.20i q^{47} +2459.26 q^{49} -4136.29i q^{53} +2198.49 q^{55} -2208.66i q^{59} +744.695 q^{61} -2859.98i q^{65} +548.306 q^{67} -4094.86i q^{71} -7716.83 q^{73} +7093.52i q^{77} -3867.59 q^{79} -6856.18i q^{83} -394.261 q^{85} +14349.6i q^{89} +9227.87 q^{91} -15462.7i q^{95} +9932.84 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 96 q^{7} - 72 q^{13} - 336 q^{19} + 84 q^{25} + 1536 q^{31} - 492 q^{37} - 10128 q^{43} - 6828 q^{49} + 13968 q^{55} + 26268 q^{61} + 20784 q^{67} - 9984 q^{73} - 44592 q^{79} - 45348 q^{85} + 11808 q^{91}+ \cdots + 76608 q^{97}+O(q^{100}) \) 12 * q - 96 * q^7 - 72 * q^13 - 336 * q^19 + 84 * q^25 + 1536 * q^31 - 492 * q^37 - 10128 * q^43 - 6828 * q^49 + 13968 * q^55 + 26268 * q^61 + 20784 * q^67 - 9984 * q^73 - 44592 * q^79 - 45348 * q^85 + 11808 * q^91 + 76608 * q^97

Introduction

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