LMFDB - Embedded newform 1728.4.c.j.1727.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1728,4,Mod(1727,1728)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1728, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1728.1727");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1728 = 2^{6} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1728.c (of order $2$, degree $1$, not 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:	$101.955300490$
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} + 3 x^{10} - 12 x^{9} + 73 x^{8} - 12 x^{7} + 589 x^{6} + 84 x^{5} + 2452 x^{4} + 852 x^{3} + \cdots + 9496 $ x^12 + 3x^10 - 12x^9 + 73x^8 - 12x^7 + 589x^6 + 84x^5 + 2452x^4 + 852x^3 + 6854x^2 - 888x + 9496
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{32}\cdot 3^{12} $
Twist minimal:	no (minimal twist has level 108)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1727.2
Root			$2.61836 - 1.60260i$ of defining polynomial
Character	$\chi$	$=$	1728.1727
Dual form			1728.4.c.j.1727.11

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-20.8488i q^{5} +13.9048i q^{7} +O(q^{10})$	$q-20.8488i q^{5} +13.9048i q^{7} -34.5116 q^{11} +31.3361 q^{13} -34.4719i q^{17} +120.723i q^{19} +137.155 q^{23} -309.672 q^{25} -93.1005i q^{29} +111.365i q^{31} +289.899 q^{35} +146.680 q^{37} +8.44531i q^{41} +427.523i q^{43} -318.826 q^{47} +149.655 q^{49} +291.451i q^{53} +719.525i q^{55} -364.665 q^{59} +289.983 q^{61} -653.320i q^{65} +305.907i q^{67} -102.802 q^{71} +442.688 q^{73} -479.878i q^{77} -245.350i q^{79} +478.981 q^{83} -718.697 q^{85} +1417.36i q^{89} +435.723i q^{91} +2516.92 q^{95} +1153.69 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q+O(q^{10}) $ 12 * q	$ 12 q + 72 q^{13} - 384 q^{25} + 240 q^{37} + 288 q^{49} - 144 q^{61} + 156 q^{73} + 168 q^{85} + 516 q^{97}+O(q^{100}) $ 12 * q + 72 * q^13 - 384 * q^25 + 240 * q^37 + 288 * q^49 - 144 * q^61 + 156 * q^73 + 168 * q^85 + 516 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1728\mathbb{Z}\right)^\times$.

$n$	$325$	$703$	$1217$
$\chi(n)$	$1$	$-1$	$-1$

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
$5$	− 20.8488i	− 1.86477i	−0.361464	−	0.932386i	$-0.617723\pi$
$5$	− 20.8488i	− 1.86477i	0.361464	−	0.932386i	$-0.382277\pi$
$6$	0	0
$7$	13.9048i	0.750791i	0.926865	+	0.375395i	$0.122493\pi$
$7$	13.9048i	0.750791i	−0.926865	+	0.375395i	$0.877507\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−34.5116	−0.945967	−0.472984	−	0.881071i	$-0.656823\pi$
$11$	−34.5116	−0.945967	−0.472984	+	0.881071i	$0.656823\pi$
$12$	0	0
$13$	31.3361	0.668544	0.334272	−	0.942477i	$-0.391510\pi$
$13$	31.3361	0.668544	0.334272	+	0.942477i	$0.391510\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 34.4719i	− 0.491803i	−0.969295	−	0.245902i	$-0.920916\pi$
$17$	− 34.4719i	− 0.491803i	0.969295	−	0.245902i	$-0.0790840\pi$
$18$	0	0
$19$	120.723i	1.45767i	0.684691	+	0.728833i	$0.259937\pi$
$19$	120.723i	1.45767i	−0.684691	+	0.728833i	$0.740063\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	137.155	1.24343	0.621714	−	0.783244i	$-0.286436\pi$
$23$	137.155	1.24343	0.621714	+	0.783244i	$0.286436\pi$
$24$	0	0
$25$	−309.672	−2.47738
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 93.1005i	− 0.596149i	−0.954543	−	0.298075i	$-0.903656\pi$
$29$	− 93.1005i	− 0.596149i	0.954543	−	0.298075i	$-0.0963444\pi$
$30$	0	0
$31$	111.365i	0.645217i	0.946532	+	0.322609i	$0.104560\pi$
$31$	111.365i	0.645217i	−0.946532	+	0.322609i	$0.895440\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	289.899	1.40005
$36$	0	0
$37$	146.680	0.651733	0.325867	−	0.945416i	$-0.394344\pi$
$37$	146.680	0.651733	0.325867	+	0.945416i	$0.394344\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	8.44531i	0.0321692i	0.999871	+	0.0160846i	$0.00512011\pi$
$41$	8.44531i	0.0321692i	−0.999871	+	0.0160846i	$0.994880\pi$
$42$	0	0
$43$	427.523i	1.51620i	0.652137	+	0.758101i	$0.273873\pi$
$43$	427.523i	1.51620i	−0.652137	+	0.758101i	$0.726127\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−318.826	−0.989481	−0.494740	−	0.869041i	$-0.664737\pi$
$47$	−318.826	−0.989481	−0.494740	+	0.869041i	$0.664737\pi$
$48$	0	0
$49$	149.655	0.436313
$50$	0	0
$51$	0	0
$52$	0	0
$53$	291.451i	0.755357i	0.925937	+	0.377679i	$0.123278\pi$
$53$	291.451i	0.755357i	−0.925937	+	0.377679i	$0.876722\pi$
$54$	0	0
$55$	719.525i	1.76401i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−364.665	−0.804666	−0.402333	−	0.915493i	$-0.631801\pi$
$59$	−364.665	−0.804666	−0.402333	+	0.915493i	$0.631801\pi$
$60$	0	0
$61$	289.983	0.608664	0.304332	−	0.952566i	$-0.401567\pi$
$61$	289.983	0.608664	0.304332	+	0.952566i	$0.401567\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 653.320i	− 1.24668i
$66$	0	0
$67$	305.907i	0.557797i	0.960321	+	0.278899i	$0.0899694\pi$
$67$	305.907i	0.557797i	−0.960321	+	0.278899i	$0.910031\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−102.802	−0.171836	−0.0859178	−	0.996302i	$-0.527382\pi$
$71$	−102.802	−0.171836	−0.0859178	+	0.996302i	$0.527382\pi$
$72$	0	0
$73$	442.688	0.709764	0.354882	−	0.934911i	$-0.384521\pi$
$73$	442.688	0.709764	0.354882	+	0.934911i	$0.384521\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 479.878i	− 0.710224i
$78$	0	0
$79$	− 245.350i	− 0.349419i	−0.984620	−	0.174709i	$-0.944101\pi$
$79$	− 245.350i	− 0.349419i	0.984620	−	0.174709i	$-0.0558986\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	478.981	0.633433	0.316717	−	0.948520i	$-0.397420\pi$
$83$	478.981	0.633433	0.316717	+	0.948520i	$0.397420\pi$
$84$	0	0
$85$	−718.697	−0.917101
$86$	0	0
$87$	0	0
$88$	0	0
$89$	1417.36i	1.68809i	0.536273	+	0.844045i	$0.319832\pi$
$89$	1417.36i	1.68809i	−0.536273	+	0.844045i	$0.680168\pi$
$90$	0	0
$91$	435.723i	0.501937i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	2516.92	2.71822
$96$	0	0
$97$	1153.69	1.20762	0.603811	−	0.797128i	$-0.293648\pi$
$97$	1153.69	1.20762	0.603811	+	0.797128i	$0.293648\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 767.096i	− 0.755732i	−0.925860	−	0.377866i	$-0.876658\pi$
$101$	− 767.096i	− 0.755732i	0.925860	−	0.377866i	$-0.123342\pi$
$102$	0	0
$103$	1202.04i	1.14991i	0.818187	+	0.574953i	$0.194979\pi$
$103$	1202.04i	1.14991i	−0.818187	+	0.574953i	$0.805021\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1309.28	1.18292	0.591462	−	0.806333i	$-0.298551\pi$
$107$	1309.28	1.18292	0.591462	+	0.806333i	$0.298551\pi$
$108$	0	0
$109$	−1102.04	−0.968407	−0.484204	−	0.874955i	$-0.660891\pi$
$109$	−1102.04	−0.968407	−0.484204	+	0.874955i	$0.660891\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	69.9098i	0.0581997i	0.999577	+	0.0290998i	$0.00926407\pi$
$113$	69.9098i	0.0581997i	−0.999577	+	0.0290998i	$0.990736\pi$
$114$	0	0
$115$	− 2859.52i	− 2.31871i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	479.326	0.369241
$120$	0	0
$121$	−139.949	−0.105146
$122$	0	0
$123$	0	0
$124$	0	0
$125$	3850.19i	2.75497i
$126$	0	0
$127$	− 2558.29i	− 1.78749i	−0.448574	−	0.893746i	$-0.648068\pi$
$127$	− 2558.29i	− 1.78749i	0.448574	−	0.893746i	$-0.351932\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	86.6249	0.0577744	0.0288872	−	0.999583i	$-0.490804\pi$
$131$	86.6249	0.0577744	0.0288872	+	0.999583i	$0.490804\pi$
$132$	0	0
$133$	−1678.63	−1.09440
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 699.734i	− 0.436367i	−0.975908	−	0.218184i	$-0.929987\pi$
$137$	− 699.734i	− 0.436367i	0.975908	−	0.218184i	$-0.0700131\pi$
$138$	0	0
$139$	789.049i	0.481484i	0.970589	+	0.240742i	$0.0773908\pi$
$139$	789.049i	0.481484i	−0.970589	+	0.240742i	$0.922609\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−1081.46	−0.632421
$144$	0	0
$145$	−1941.03	−1.11168
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 1593.12i	− 0.875930i	−0.898992	−	0.437965i	$-0.855699\pi$
$149$	− 1593.12i	− 0.875930i	0.898992	−	0.437965i	$-0.144301\pi$
$150$	0	0
$151$	− 1502.12i	− 0.809544i	−0.914418	−	0.404772i	$-0.867351\pi$
$151$	− 1502.12i	− 0.809544i	0.914418	−	0.404772i	$-0.132649\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	2321.82	1.20318
$156$	0	0
$157$	3596.08	1.82802	0.914008	−	0.405696i	$-0.132971\pi$
$157$	3596.08	1.82802	0.914008	+	0.405696i	$0.132971\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	1907.12i	0.933555i
$162$	0	0
$163$	− 3313.82i	− 1.59238i	−0.605044	−	0.796192i	$-0.706844\pi$
$163$	− 3313.82i	− 1.59238i	0.605044	−	0.796192i	$-0.293156\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	487.046	0.225681	0.112841	−	0.993613i	$-0.464005\pi$
$167$	487.046	0.225681	0.112841	+	0.993613i	$0.464005\pi$
$168$	0	0
$169$	−1215.05	−0.553049
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 1752.30i	− 0.770086i	−0.922899	−	0.385043i	$-0.874187\pi$
$173$	− 1752.30i	− 0.770086i	0.922899	−	0.385043i	$-0.125813\pi$
$174$	0	0
$175$	− 4305.94i	− 1.85999i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	2807.62	1.17236	0.586178	−	0.810182i	$-0.300632\pi$
$179$	2807.62	1.17236	0.586178	+	0.810182i	$0.300632\pi$
$180$	0	0
$181$	2307.37	0.947546	0.473773	−	0.880647i	$-0.342892\pi$
$181$	2307.37	0.947546	0.473773	+	0.880647i	$0.342892\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 3058.11i	− 1.21533i
$186$	0	0
$187$	1189.68i	0.465230i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	3375.74	1.27885	0.639425	−	0.768854i	$-0.279173\pi$
$191$	3375.74	1.27885	0.639425	+	0.768854i	$0.279173\pi$
$192$	0	0
$193$	561.917	0.209573	0.104787	−	0.994495i	$-0.466584\pi$
$193$	561.917	0.209573	0.104787	+	0.994495i	$0.466584\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 16.7420i	− 0.00605491i	−0.999995	−	0.00302746i	$-0.999036\pi$
$197$	− 16.7420i	− 0.00605491i	0.999995	−	0.00302746i	$-0.000963671\pi$
$198$	0	0
$199$	2760.01i	0.983176i	0.870828	+	0.491588i	$0.163583\pi$
$199$	2760.01i	0.983176i	−0.870828	+	0.491588i	$0.836417\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	1294.55	0.447583
$204$	0	0
$205$	176.075	0.0599882
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 4166.33i	− 1.37890i
$210$	0	0
$211$	− 2766.47i	− 0.902615i	−0.892368	−	0.451308i	$-0.850958\pi$
$211$	− 2766.47i	− 0.902615i	0.892368	−	0.451308i	$-0.149042\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	8913.34	2.82737
$216$	0	0
$217$	−1548.51	−0.484423
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 1080.21i	− 0.328792i
$222$	0	0
$223$	− 110.636i	− 0.0332231i	−0.999862	−	0.0166115i	$-0.994712\pi$
$223$	− 110.636i	− 0.0332231i	0.999862	−	0.0166115i	$-0.00528786\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	1800.57	0.526468	0.263234	−	0.964732i	$-0.415211\pi$
$227$	1800.57	0.526468	0.263234	+	0.964732i	$0.415211\pi$
$228$	0	0
$229$	1491.95	0.430528	0.215264	−	0.976556i	$-0.430939\pi$
$229$	1491.95	0.430528	0.215264	+	0.976556i	$0.430939\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	4545.75i	1.27812i	0.769157	+	0.639060i	$0.220676\pi$
$233$	4545.75i	1.27812i	−0.769157	+	0.639060i	$0.779324\pi$
$234$	0	0
$235$	6647.14i	1.84516i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−3305.97	−0.894751	−0.447376	−	0.894346i	$-0.647641\pi$
$239$	−3305.97	−0.894751	−0.447376	+	0.894346i	$0.647641\pi$
$240$	0	0
$241$	−2337.95	−0.624898	−0.312449	−	0.949934i	$-0.601149\pi$
$241$	−2337.95	−0.624898	−0.312449	+	0.949934i	$0.601149\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 3120.13i	− 0.813624i
$246$	0	0
$247$	3782.97i	0.974514i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−3625.36	−0.911675	−0.455838	−	0.890063i	$-0.650660\pi$
$251$	−3625.36	−0.911675	−0.455838	+	0.890063i	$0.650660\pi$
$252$	0	0
$253$	−4733.45	−1.17624
$254$	0	0
$255$	0	0
$256$	0	0
$257$	2124.02i	0.515537i	0.966207	+	0.257768i	$0.0829871\pi$
$257$	2124.02i	0.515537i	−0.966207	+	0.257768i	$0.917013\pi$
$258$	0	0
$259$	2039.57i	0.489315i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	4785.87	1.12209	0.561044	−	0.827786i	$-0.310400\pi$
$263$	4785.87	1.12209	0.561044	+	0.827786i	$0.310400\pi$
$264$	0	0
$265$	6076.41	1.40857
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 241.884i	− 0.0548249i	−0.999624	−	0.0274125i	$-0.991273\pi$
$269$	− 241.884i	− 0.0548249i	0.999624	−	0.0274125i	$-0.00872675\pi$
$270$	0	0
$271$	− 828.799i	− 0.185778i	−0.995676	−	0.0928892i	$-0.970390\pi$
$271$	− 828.799i	− 0.185778i	0.995676	−	0.0928892i	$-0.0296102\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	10687.3	2.34352
$276$	0	0
$277$	5897.19	1.27916	0.639581	−	0.768724i	$-0.279108\pi$
$277$	5897.19	1.27916	0.639581	+	0.768724i	$0.279108\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	3055.95i	0.648763i	0.945926	+	0.324382i	$0.105156\pi$
$281$	3055.95i	0.648763i	−0.945926	+	0.324382i	$0.894844\pi$
$282$	0	0
$283$	196.045i	0.0411790i	0.999788	+	0.0205895i	$0.00655430\pi$
$283$	196.045i	0.0411790i	−0.999788	+	0.0205895i	$0.993446\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−117.431	−0.0241523
$288$	0	0
$289$	3724.69	0.758130
$290$	0	0
$291$	0	0
$292$	0	0
$293$	8748.98i	1.74444i	0.489114	+	0.872220i	$0.337320\pi$
$293$	8748.98i	1.74444i	−0.489114	+	0.872220i	$0.662680\pi$
$294$	0	0
$295$	7602.82i	1.50052i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	4297.91	0.831287
$300$	0	0
$301$	−5944.65	−1.13835
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 6045.79i	− 1.13502i
$306$	0	0
$307$	− 2095.99i	− 0.389656i	−0.980837	−	0.194828i	$-0.937585\pi$
$307$	− 2095.99i	− 0.389656i	0.980837	−	0.194828i	$-0.0624149\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	3393.74	0.618783	0.309391	−	0.950935i	$-0.399875\pi$
$311$	3393.74	0.618783	0.309391	+	0.950935i	$0.399875\pi$
$312$	0	0
$313$	3579.49	0.646405	0.323203	−	0.946330i	$-0.395240\pi$
$313$	3579.49	0.646405	0.323203	+	0.946330i	$0.395240\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	4356.56i	0.771889i	0.922522	+	0.385945i	$0.126124\pi$
$317$	4356.56i	0.771889i	−0.922522	+	0.385945i	$0.873876\pi$
$318$	0	0
$319$	3213.05i	0.563937i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	4161.53	0.716885
$324$	0	0
$325$	−9703.91	−1.65623
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 4433.23i	− 0.742893i
$330$	0	0
$331$	9191.99i	1.52640i	0.646164	+	0.763198i	$0.276372\pi$
$331$	9191.99i	1.52640i	−0.646164	+	0.763198i	$0.723628\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	6377.78	1.04017
$336$	0	0
$337$	1663.33	0.268865	0.134432	−	0.990923i	$-0.457079\pi$
$337$	1663.33	0.268865	0.134432	+	0.990923i	$0.457079\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 3843.38i	− 0.610354i
$342$	0	0
$343$	6850.30i	1.07837i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	9668.26	1.49573	0.747866	−	0.663850i	$-0.231078\pi$
$347$	9668.26	1.49573	0.747866	+	0.663850i	$0.231078\pi$
$348$	0	0
$349$	−9928.24	−1.52277	−0.761384	−	0.648301i	$-0.775480\pi$
$349$	−9928.24	−1.52277	−0.761384	+	0.648301i	$0.775480\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	4460.01i	0.672471i	0.941778	+	0.336236i	$0.109154\pi$
$353$	4460.01i	0.672471i	−0.941778	+	0.336236i	$0.890846\pi$
$354$	0	0
$355$	2143.29i	0.320434i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	2158.32	0.317303	0.158652	−	0.987335i	$-0.449285\pi$
$359$	2158.32	0.317303	0.158652	+	0.987335i	$0.449285\pi$
$360$	0	0
$361$	−7714.95	−1.12479
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 9229.52i	− 1.32355i
$366$	0	0
$367$	1156.43i	0.164482i	0.996612	+	0.0822411i	$0.0262078\pi$
$367$	1156.43i	0.164482i	−0.996612	+	0.0822411i	$0.973792\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−4052.59	−0.567115
$372$	0	0
$373$	6286.39	0.872647	0.436323	−	0.899790i	$-0.356280\pi$
$373$	6286.39	0.872647	0.436323	+	0.899790i	$0.356280\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 2917.41i	− 0.398552i
$378$	0	0
$379$	− 6921.76i	− 0.938119i	−0.883167	−	0.469059i	$-0.844593\pi$
$379$	− 6921.76i	− 0.938119i	0.883167	−	0.469059i	$-0.155407\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−1802.67	−0.240502	−0.120251	−	0.992744i	$-0.538370\pi$
$383$	−1802.67	−0.240502	−0.120251	+	0.992744i	$0.538370\pi$
$384$	0	0
$385$	−10004.9	−1.32441
$386$	0	0
$387$	0	0
$388$	0	0
$389$	4421.02i	0.576233i	0.957595	+	0.288117i	$0.0930291\pi$
$389$	4421.02i	0.576233i	−0.957595	+	0.288117i	$0.906971\pi$
$390$	0	0
$391$	− 4728.00i	− 0.611522i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−5115.26	−0.651586
$396$	0	0
$397$	13769.8	1.74077	0.870383	−	0.492375i	$-0.163871\pi$
$397$	13769.8	1.74077	0.870383	+	0.492375i	$0.163871\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	13534.1i	1.68544i	0.538350	+	0.842722i	$0.319048\pi$
$401$	13534.1i	1.68544i	−0.538350	+	0.842722i	$0.680952\pi$
$402$	0	0
$403$	3489.74i	0.431356i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−5062.18	−0.616518
$408$	0	0
$409$	7230.24	0.874114	0.437057	−	0.899434i	$-0.356021\pi$
$409$	7230.24	0.874114	0.437057	+	0.899434i	$0.356021\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 5070.61i	− 0.604136i
$414$	0	0
$415$	− 9986.16i	− 1.18121i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−9066.85	−1.05715	−0.528574	−	0.848887i	$-0.677273\pi$
$419$	−9066.85	−1.05715	−0.528574	+	0.848887i	$0.677273\pi$
$420$	0	0
$421$	−6017.63	−0.696630	−0.348315	−	0.937378i	$-0.613246\pi$
$421$	−6017.63	−0.696630	−0.348315	+	0.937378i	$0.613246\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	10675.0i	1.21838i
$426$	0	0
$427$	4032.17i	0.456979i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	10533.7	1.17724	0.588619	−	0.808410i	$-0.299672\pi$
$431$	10533.7	1.17724	0.588619	+	0.808410i	$0.299672\pi$
$432$	0	0
$433$	−79.2056	−0.00879071	−0.00439536	−	0.999990i	$-0.501399\pi$
$433$	−79.2056	−0.00879071	−0.00439536	+	0.999990i	$0.501399\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	16557.7i	1.81250i
$438$	0	0
$439$	− 9484.31i	− 1.03112i	−0.856854	−	0.515560i	$-0.827584\pi$
$439$	− 9484.31i	− 1.03112i	0.856854	−	0.515560i	$-0.172416\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−9734.58	−1.04403	−0.522013	−	0.852937i	$-0.674819\pi$
$443$	−9734.58	−1.04403	−0.522013	+	0.852937i	$0.674819\pi$
$444$	0	0
$445$	29550.3	3.14790
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 9270.01i	− 0.974340i	−0.873307	−	0.487170i	$-0.838029\pi$
$449$	− 9270.01i	− 0.974340i	0.873307	−	0.487170i	$-0.161971\pi$
$450$	0	0
$451$	− 291.461i	− 0.0304310i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	9084.31	0.935997
$456$	0	0
$457$	−2542.86	−0.260285	−0.130142	−	0.991495i	$-0.541543\pi$
$457$	−2542.86	−0.260285	−0.130142	+	0.991495i	$0.541543\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	10664.8i	1.07746i	0.842479	+	0.538729i	$0.181095\pi$
$461$	10664.8i	1.07746i	−0.842479	+	0.538729i	$0.818905\pi$
$462$	0	0
$463$	1963.48i	0.197086i	0.995133	+	0.0985428i	$0.0314181\pi$
$463$	1963.48i	0.197086i	−0.995133	+	0.0985428i	$0.968582\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	19778.8	1.95986	0.979928	−	0.199350i	$-0.0638829\pi$
$467$	19778.8	1.95986	0.979928	+	0.199350i	$0.0638829\pi$
$468$	0	0
$469$	−4253.58	−0.418789
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 14754.5i	− 1.43428i
$474$	0	0
$475$	− 37384.4i	− 3.61119i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−16062.6	−1.53219	−0.766095	−	0.642728i	$-0.777803\pi$
$479$	−16062.6	−1.53219	−0.766095	+	0.642728i	$0.777803\pi$
$480$	0	0
$481$	4596.39	0.435712
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 24053.0i	− 2.25194i
$486$	0	0
$487$	− 15907.5i	− 1.48016i	−0.672519	−	0.740080i	$-0.734788\pi$
$487$	− 15907.5i	− 1.48016i	0.672519	−	0.740080i	$-0.265212\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−5924.49	−0.544538	−0.272269	−	0.962221i	$-0.587774\pi$
$491$	−5924.49	−0.544538	−0.272269	+	0.962221i	$0.587774\pi$
$492$	0	0
$493$	−3209.35	−0.293188
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 1429.44i	− 0.129013i
$498$	0	0
$499$	− 5331.33i	− 0.478283i	−0.970985	−	0.239142i	$-0.923134\pi$
$499$	− 5331.33i	− 0.478283i	0.970985	−	0.239142i	$-0.0768660\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−2144.13	−0.190064	−0.0950319	−	0.995474i	$-0.530295\pi$
$503$	−2144.13	−0.190064	−0.0950319	+	0.995474i	$0.530295\pi$
$504$	0	0
$505$	−15993.0	−1.40927
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 14329.9i	− 1.24786i	−0.781481	−	0.623929i	$-0.785535\pi$
$509$	− 14329.9i	− 1.24786i	0.781481	−	0.623929i	$-0.214465\pi$
$510$	0	0
$511$	6155.51i	0.532884i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	25061.0	2.14431
$516$	0	0
$517$	11003.2	0.936016
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 10822.8i	− 0.910087i	−0.890469	−	0.455043i	$-0.849624\pi$
$521$	− 10822.8i	− 0.910087i	0.890469	−	0.455043i	$-0.150376\pi$
$522$	0	0
$523$	12111.8i	1.01265i	0.862344	+	0.506323i	$0.168995\pi$
$523$	12111.8i	1.01265i	−0.862344	+	0.506323i	$0.831005\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	3838.96	0.317320
$528$	0	0
$529$	6644.58	0.546115
$530$	0	0
$531$	0	0
$532$	0	0
$533$	264.643i	0.0215065i
$534$	0	0
$535$	− 27296.9i	− 2.20588i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−5164.85	−0.412738
$540$	0	0
$541$	−22063.1	−1.75336	−0.876681	−	0.481073i	$-0.840247\pi$
$541$	−22063.1	−1.75336	−0.876681	+	0.481073i	$0.840247\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	22976.2i	1.80586i
$546$	0	0
$547$	17842.7i	1.39470i	0.716731	+	0.697350i	$0.245637\pi$
$547$	17842.7i	1.39470i	−0.716731	+	0.697350i	$0.754363\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	11239.3	0.868987
$552$	0	0
$553$	3411.56	0.262340
$554$	0	0
$555$	0	0
$556$	0	0
$557$	10942.0i	0.832365i	0.909281	+	0.416183i	$0.136632\pi$
$557$	10942.0i	0.832365i	−0.909281	+	0.416183i	$0.863368\pi$
$558$	0	0
$559$	13396.9i	1.01365i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−1991.56	−0.149084	−0.0745418	−	0.997218i	$-0.523749\pi$
$563$	−1991.56	−0.149084	−0.0745418	+	0.997218i	$0.523749\pi$
$564$	0	0
$565$	1457.53	0.108529
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 12048.3i	− 0.887684i	−0.896105	−	0.443842i	$-0.853615\pi$
$569$	− 12048.3i	− 0.887684i	0.896105	−	0.443842i	$-0.146385\pi$
$570$	0	0
$571$	− 24916.7i	− 1.82615i	−0.407792	−	0.913075i	$-0.633701\pi$
$571$	− 24916.7i	− 1.82615i	0.407792	−	0.913075i	$-0.366299\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−42473.2	−3.08044
$576$	0	0
$577$	−17705.3	−1.27743	−0.638717	−	0.769441i	$-0.720535\pi$
$577$	−17705.3	−1.27743	−0.638717	+	0.769441i	$0.720535\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	6660.15i	0.475576i
$582$	0	0
$583$	− 10058.5i	− 0.714543i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−22453.6	−1.57880	−0.789402	−	0.613876i	$-0.789609\pi$
$587$	−22453.6	−1.57880	−0.789402	+	0.613876i	$0.789609\pi$
$588$	0	0
$589$	−13444.3	−0.940511
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 26839.1i	− 1.85860i	−0.369328	−	0.929299i	$-0.620412\pi$
$593$	− 26839.1i	− 1.85860i	0.369328	−	0.929299i	$-0.379588\pi$
$594$	0	0
$595$	− 9993.36i	− 0.688551i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	9254.44	0.631262	0.315631	−	0.948882i	$-0.397784\pi$
$599$	9254.44	0.631262	0.315631	+	0.948882i	$0.397784\pi$
$600$	0	0
$601$	1102.45	0.0748250	0.0374125	−	0.999300i	$-0.488088\pi$
$601$	1102.45	0.0748250	0.0374125	+	0.999300i	$0.488088\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	2917.77i	0.196073i
$606$	0	0
$607$	21383.2i	1.42985i	0.699202	+	0.714924i	$0.253539\pi$
$607$	21383.2i	1.42985i	−0.699202	+	0.714924i	$0.746461\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−9990.77	−0.661511
$612$	0	0
$613$	−19769.0	−1.30255	−0.651274	−	0.758843i	$-0.725765\pi$
$613$	−19769.0	−1.30255	−0.651274	+	0.758843i	$0.725765\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 2702.48i	− 0.176333i	−0.996106	−	0.0881667i	$-0.971899\pi$
$617$	− 2702.48i	− 0.176333i	0.996106	−	0.0881667i	$-0.0281008\pi$
$618$	0	0
$619$	6967.79i	0.452438i	0.974076	+	0.226219i	$0.0726365\pi$
$619$	6967.79i	0.452438i	−0.974076	+	0.226219i	$0.927364\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−19708.2	−1.26740
$624$	0	0
$625$	41562.7	2.66001
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 5056.35i	− 0.320524i
$630$	0	0
$631$	19202.4i	1.21147i	0.795667	+	0.605734i	$0.207120\pi$
$631$	19202.4i	1.21147i	−0.795667	+	0.605734i	$0.792880\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−53337.2	−3.33326
$636$	0	0
$637$	4689.61	0.291694
$638$	0	0
$639$	0	0
$640$	0	0
$641$	23580.8i	1.45302i	0.687155	+	0.726511i	$0.258860\pi$
$641$	23580.8i	1.45302i	−0.687155	+	0.726511i	$0.741140\pi$
$642$	0	0
$643$	25167.2i	1.54354i	0.635899	+	0.771772i	$0.280629\pi$
$643$	25167.2i	1.54354i	−0.635899	+	0.771772i	$0.719371\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−2247.67	−0.136577	−0.0682883	−	0.997666i	$-0.521754\pi$
$647$	−2247.67	−0.136577	−0.0682883	+	0.997666i	$0.521754\pi$
$648$	0	0
$649$	12585.2	0.761188
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 11278.6i	− 0.675906i	−0.941163	−	0.337953i	$-0.890266\pi$
$653$	− 11278.6i	− 0.675906i	0.941163	−	0.337953i	$-0.109734\pi$
$654$	0	0
$655$	− 1806.02i	− 0.107736i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	658.636	0.0389329	0.0194665	−	0.999811i	$-0.493803\pi$
$659$	658.636	0.0389329	0.0194665	+	0.999811i	$0.493803\pi$
$660$	0	0
$661$	−1740.41	−0.102412	−0.0512060	−	0.998688i	$-0.516306\pi$
$661$	−1740.41	−0.102412	−0.0512060	+	0.998688i	$0.516306\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	34997.4i	2.04081i
$666$	0	0
$667$	− 12769.2i	− 0.741269i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−10007.8	−0.575776
$672$	0	0
$673$	−273.146	−0.0156449	−0.00782243	−	0.999969i	$-0.502490\pi$
$673$	−273.146	−0.0156449	−0.00782243	+	0.999969i	$0.502490\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 11373.1i	− 0.645649i	−0.946459	−	0.322824i	$-0.895368\pi$
$677$	− 11373.1i	− 0.645649i	0.946459	−	0.322824i	$-0.104632\pi$
$678$	0	0
$679$	16041.8i	0.906671i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	1223.56	0.0685480	0.0342740	−	0.999412i	$-0.489088\pi$
$683$	1223.56	0.0685480	0.0342740	+	0.999412i	$0.489088\pi$
$684$	0	0
$685$	−14588.6	−0.813725
$686$	0	0
$687$	0	0
$688$	0	0
$689$	9132.95i	0.504989i
$690$	0	0
$691$	9247.55i	0.509108i	0.967059	+	0.254554i	$0.0819286\pi$
$691$	9247.55i	0.509108i	−0.967059	+	0.254554i	$0.918071\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	16450.7	0.897858
$696$	0	0
$697$	291.126	0.0158209
$698$	0	0
$699$	0	0
$700$	0	0
$701$	14448.0i	0.778447i	0.921143	+	0.389224i	$0.127257\pi$
$701$	14448.0i	0.778447i	−0.921143	+	0.389224i	$0.872743\pi$
$702$	0	0
$703$	17707.6i	0.950009i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	10666.4	0.567397
$708$	0	0
$709$	−7819.28	−0.414188	−0.207094	−	0.978321i	$-0.566401\pi$
$709$	−7819.28	−0.414188	−0.207094	+	0.978321i	$0.566401\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	15274.3i	0.802281i
$714$	0	0
$715$	22547.1i	1.17932i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−7251.47	−0.376126	−0.188063	−	0.982157i	$-0.560221\pi$
$719$	−7251.47	−0.376126	−0.188063	+	0.982157i	$0.560221\pi$
$720$	0	0
$721$	−16714.1	−0.863338
$722$	0	0
$723$	0	0
$724$	0	0
$725$	28830.6i	1.47689i
$726$	0	0
$727$	− 6605.59i	− 0.336985i	−0.985703	−	0.168492i	$-0.946110\pi$
$727$	− 6605.59i	− 0.336985i	0.985703	−	0.168492i	$-0.0538899\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	14737.5	0.745673
$732$	0	0
$733$	37147.9	1.87188	0.935942	−	0.352155i	$-0.114551\pi$
$733$	37147.9	1.87188	0.935942	+	0.352155i	$0.114551\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 10557.3i	− 0.527658i
$738$	0	0
$739$	34209.1i	1.70285i	0.524479	+	0.851423i	$0.324260\pi$
$739$	34209.1i	1.70285i	−0.524479	+	0.851423i	$0.675740\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−15746.4	−0.777494	−0.388747	−	0.921345i	$-0.627092\pi$
$743$	−15746.4	−0.777494	−0.388747	+	0.921345i	$0.627092\pi$
$744$	0	0
$745$	−33214.6	−1.63341
$746$	0	0
$747$	0	0
$748$	0	0
$749$	18205.3i	0.888128i
$750$	0	0
$751$	27773.5i	1.34949i	0.738049	+	0.674747i	$0.235747\pi$
$751$	27773.5i	1.34949i	−0.738049	+	0.674747i	$0.764253\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−31317.5	−1.50961
$756$	0	0
$757$	12694.0	0.609471	0.304736	−	0.952437i	$-0.401432\pi$
$757$	12694.0	0.609471	0.304736	+	0.952437i	$0.401432\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 18061.4i	− 0.860349i	−0.902746	−	0.430175i	$-0.858452\pi$
$761$	− 18061.4i	− 0.860349i	0.902746	−	0.430175i	$-0.141548\pi$
$762$	0	0
$763$	− 15323.7i	− 0.727071i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−11427.2	−0.537955
$768$	0	0
$769$	32615.8	1.52946	0.764731	−	0.644350i	$-0.222872\pi$
$769$	32615.8	1.52946	0.764731	+	0.644350i	$0.222872\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	8518.40i	0.396359i	0.980166	+	0.198179i	$0.0635029\pi$
$773$	8518.40i	0.396359i	−0.980166	+	0.198179i	$0.936497\pi$
$774$	0	0
$775$	− 34486.6i	− 1.59844i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−1019.54	−0.0468919
$780$	0	0
$781$	3547.86	0.162551
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 74973.9i	− 3.40883i
$786$	0	0
$787$	12371.2i	0.560338i	0.959951	+	0.280169i	$0.0903906\pi$
$787$	12371.2i	0.560338i	−0.959951	+	0.280169i	$0.909609\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−972.085	−0.0436958
$792$	0	0
$793$	9086.93	0.406919
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 37675.0i	− 1.67443i	−0.546877	−	0.837213i	$-0.684184\pi$
$797$	− 37675.0i	− 1.67443i	0.546877	−	0.837213i	$-0.315816\pi$
$798$	0	0
$799$	10990.5i	0.486630i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−15277.9	−0.671413
$804$	0	0
$805$	39761.2	1.74087
$806$	0	0
$807$	0	0
$808$	0	0
$809$	41809.2i	1.81697i	0.417914	+	0.908487i	$0.362761\pi$
$809$	41809.2i	1.81697i	−0.417914	+	0.908487i	$0.637239\pi$
$810$	0	0
$811$	19357.2i	0.838129i	0.907956	+	0.419065i	$0.137642\pi$
$811$	19357.2i	0.838129i	−0.907956	+	0.419065i	$0.862358\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−69089.2	−2.96943
$816$	0	0
$817$	−51611.7	−2.21012
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 24774.0i	− 1.05313i	−0.850135	−	0.526565i	$-0.823480\pi$
$821$	− 24774.0i	− 1.05313i	0.850135	−	0.526565i	$-0.176520\pi$
$822$	0	0
$823$	10955.1i	0.463997i	0.972716	+	0.231999i	$0.0745265\pi$
$823$	10955.1i	0.463997i	−0.972716	+	0.231999i	$0.925474\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−15446.2	−0.649475	−0.324738	−	0.945804i	$-0.605276\pi$
$827$	−15446.2	−0.649475	−0.324738	+	0.945804i	$0.605276\pi$
$828$	0	0
$829$	−4599.69	−0.192707	−0.0963533	−	0.995347i	$-0.530718\pi$
$829$	−4599.69	−0.192707	−0.0963533	+	0.995347i	$0.530718\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 5158.90i	− 0.214580i
$834$	0	0
$835$	− 10154.3i	− 0.420844i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−26299.3	−1.08219	−0.541093	−	0.840963i	$-0.681989\pi$
$839$	−26299.3	−1.08219	−0.541093	+	0.840963i	$0.681989\pi$
$840$	0	0
$841$	15721.3	0.644606
$842$	0	0
$843$	0	0
$844$	0	0
$845$	25332.3i	1.03131i
$846$	0	0
$847$	− 1945.97i	− 0.0789426i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	20118.0	0.810384
$852$	0	0
$853$	−26223.5	−1.05261	−0.526305	−	0.850296i	$-0.676423\pi$
$853$	−26223.5	−1.05261	−0.526305	+	0.850296i	$0.676423\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	13156.2i	0.524396i	0.965014	+	0.262198i	$0.0844473\pi$
$857$	13156.2i	0.524396i	−0.965014	+	0.262198i	$0.915553\pi$
$858$	0	0
$859$	− 33557.9i	− 1.33292i	−0.745539	−	0.666462i	$-0.767808\pi$
$859$	− 33557.9i	− 1.33292i	0.745539	−	0.666462i	$-0.232192\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	16298.7	0.642889	0.321445	−	0.946928i	$-0.395832\pi$
$863$	16298.7	0.642889	0.321445	+	0.946928i	$0.395832\pi$
$864$	0	0
$865$	−36533.3	−1.43603
$866$	0	0
$867$	0	0
$868$	0	0
$869$	8467.44i	0.330539i
$870$	0	0
$871$	9585.92i	0.372912i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−53536.2	−2.06841
$876$	0	0
$877$	28884.4	1.11215	0.556075	−	0.831132i	$-0.312307\pi$
$877$	28884.4	1.11215	0.556075	+	0.831132i	$0.312307\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	47233.0i	1.80626i	0.429362	+	0.903132i	$0.358738\pi$
$881$	47233.0i	1.80626i	−0.429362	+	0.903132i	$0.641262\pi$
$882$	0	0
$883$	− 12998.5i	− 0.495394i	−0.968838	−	0.247697i	$-0.920326\pi$
$883$	− 12998.5i	− 0.495394i	0.968838	−	0.247697i	$-0.0796738\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	16177.5	0.612386	0.306193	−	0.951970i	$-0.400945\pi$
$887$	16177.5	0.612386	0.306193	+	0.951970i	$0.400945\pi$
$888$	0	0
$889$	35572.6	1.34203
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 38489.5i	− 1.44233i
$894$	0	0
$895$	− 58535.6i	− 2.18618i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	10368.1	0.384646
$900$	0	0
$901$	10046.9	0.371487
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 48106.0i	− 1.76696i
$906$	0	0
$907$	41803.5i	1.53039i	0.643800	+	0.765194i	$0.277357\pi$
$907$	41803.5i	1.53039i	−0.643800	+	0.765194i	$0.722643\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−25834.2	−0.939543	−0.469772	−	0.882788i	$-0.655664\pi$
$911$	−25834.2	−0.939543	−0.469772	+	0.882788i	$0.655664\pi$
$912$	0	0
$913$	−16530.4	−0.599207
$914$	0	0
$915$	0	0
$916$	0	0
$917$	1204.51i	0.0433765i
$918$	0	0
$919$	18742.0i	0.672734i	0.941731	+	0.336367i	$0.109198\pi$
$919$	18742.0i	0.672734i	−0.941731	+	0.336367i	$0.890802\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−3221.41	−0.114880
$924$	0	0
$925$	−45422.8	−1.61459
$926$	0	0
$927$	0	0
$928$	0	0
$929$	9052.31i	0.319695i	0.987142	+	0.159847i	$0.0511002\pi$
$929$	9052.31i	0.319695i	−0.987142	+	0.159847i	$0.948900\pi$
$930$	0	0
$931$	18066.8i	0.635999i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	24803.4	0.867547
$936$	0	0
$937$	−28189.2	−0.982819	−0.491409	−	0.870929i	$-0.663518\pi$
$937$	−28189.2	−0.982819	−0.491409	+	0.870929i	$0.663518\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	11490.8i	0.398074i	0.979992	+	0.199037i	$0.0637814\pi$
$941$	11490.8i	0.398074i	−0.979992	+	0.199037i	$0.936219\pi$
$942$	0	0
$943$	1158.32i	0.0400001i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−15309.8	−0.525344	−0.262672	−	0.964885i	$-0.584604\pi$
$947$	−15309.8	−0.525344	−0.262672	+	0.964885i	$0.584604\pi$
$948$	0	0
$949$	13872.1	0.474508
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 9896.90i	− 0.336403i	−0.985753	−	0.168201i	$-0.946204\pi$
$953$	− 9896.90i	− 0.336403i	0.985753	−	0.168201i	$-0.0537959\pi$
$954$	0	0
$955$	− 70380.1i	− 2.38476i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	9729.69	0.327620
$960$	0	0
$961$	17388.9	0.583695
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 11715.3i	− 0.390807i
$966$	0	0
$967$	− 16779.4i	− 0.558003i	−0.960291	−	0.279001i	$-0.909997\pi$
$967$	− 16779.4i	− 0.558003i	0.960291	−	0.279001i	$-0.0900034\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−53400.1	−1.76487	−0.882437	−	0.470431i	$-0.844098\pi$
$971$	−53400.1	−1.76487	−0.882437	+	0.470431i	$0.844098\pi$
$972$	0	0
$973$	−10971.6	−0.361494
$974$	0	0
$975$	0	0
$976$	0	0
$977$	9474.13i	0.310240i	0.987896	+	0.155120i	$0.0495764\pi$
$977$	9474.13i	0.310240i	−0.987896	+	0.155120i	$0.950424\pi$
$978$	0	0
$979$	− 48915.4i	− 1.59688i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	56693.6	1.83952	0.919758	−	0.392485i	$-0.128385\pi$
$983$	56693.6	1.83952	0.919758	+	0.392485i	$0.128385\pi$
$984$	0	0
$985$	−349.050	−0.0112910
$986$	0	0
$987$	0	0
$988$	0	0
$989$	58637.1i	1.88529i
$990$	0	0
$991$	− 38944.6i	− 1.24835i	−0.781285	−	0.624175i	$-0.785435\pi$
$991$	− 38944.6i	− 1.24835i	0.781285	−	0.624175i	$-0.214565\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	57542.9	1.83340
$996$	0	0
$997$	33995.1	1.07987	0.539937	−	0.841705i	$-0.318448\pi$
$997$	33995.1	1.07987	0.539937	+	0.841705i	$0.318448\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1728.4.c.j.1727.2		12
3.2	odd	2	inner	1728.4.c.j.1727.12		12
4.3	odd	2	inner	1728.4.c.j.1727.1		12
8.3	odd	2		108.4.b.b.107.1	✓	12
8.5	even	2		108.4.b.b.107.11	yes	12
12.11	even	2	inner	1728.4.c.j.1727.11		12
24.5	odd	2		108.4.b.b.107.2	yes	12
24.11	even	2		108.4.b.b.107.12	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
108.4.b.b.107.1	✓	12	8.3	odd	2
108.4.b.b.107.2	yes	12	24.5	odd	2
108.4.b.b.107.11	yes	12	8.5	even	2
108.4.b.b.107.12	yes	12	24.11	even	2
1728.4.c.j.1727.1		12	4.3	odd	2	inner
1728.4.c.j.1727.2		12	1.1	even	1	trivial
1728.4.c.j.1727.11		12	12.11	even	2	inner
1728.4.c.j.1727.12		12	3.2	odd	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:	\( N \)	\(=\)	\( 1728 = 2^{6} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1728.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-20.8488i q^{5} +13.9048i q^{7} +O(q^{10})\)	\(q-20.8488i q^{5} +13.9048i q^{7} -34.5116 q^{11} +31.3361 q^{13} -34.4719i q^{17} +120.723i q^{19} +137.155 q^{23} -309.672 q^{25} -93.1005i q^{29} +111.365i q^{31} +289.899 q^{35} +146.680 q^{37} +8.44531i q^{41} +427.523i q^{43} -318.826 q^{47} +149.655 q^{49} +291.451i q^{53} +719.525i q^{55} -364.665 q^{59} +289.983 q^{61} -653.320i q^{65} +305.907i q^{67} -102.802 q^{71} +442.688 q^{73} -479.878i q^{77} -245.350i q^{79} +478.981 q^{83} -718.697 q^{85} +1417.36i q^{89} +435.723i q^{91} +2516.92 q^{95} +1153.69 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q+O(q^{10}) \) 12 * q	\( 12 q + 72 q^{13} - 384 q^{25} + 240 q^{37} + 288 q^{49} - 144 q^{61} + 156 q^{73} + 168 q^{85} + 516 q^{97}+O(q^{100}) \) 12 * q + 72 * q^13 - 384 * q^25 + 240 * q^37 + 288 * q^49 - 144 * q^61 + 156 * q^73 + 168 * q^85 + 516 * q^97

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