LMFDB - Embedded newform 1728.3.e.u.1025.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1728,3,Mod(1025,1728)]

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([0, 0, 1]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("1728.1025");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1728 = 2^{6} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1728.e (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:	$47.0845896815$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\Q(\zeta_{24})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{4} + 1 $ x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{10}\cdot 3^{8} $
Twist minimal:	no (minimal twist has level 864)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1025.6
Root			$0.965926 + 0.258819i$ of defining polynomial
Character	$\chi$	$=$	1728.1025
Dual form			1728.3.e.u.1025.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.78194i q^{5} +0.953512 q^{7} +O(q^{10})$	$q+3.78194i q^{5} +0.953512 q^{7} +11.6969i q^{11} +8.69694 q^{13} +4.73545i q^{17} +29.2699 q^{19} -2.69694i q^{23} +10.6969 q^{25} -21.7060i q^{29} -22.5953 q^{31} +3.60612i q^{35} -24.0908 q^{37} -19.7990i q^{41} +49.0047 q^{43} +59.3939i q^{47} -48.0908 q^{49} +48.1796i q^{53} -44.2371 q^{55} +12.6061i q^{59} +4.00000 q^{61} +32.8913i q^{65} +111.358 q^{67} +56.6969i q^{71} +35.6969 q^{73} +11.1532i q^{77} -144.442 q^{79} +101.697i q^{83} -17.9092 q^{85} +19.0060i q^{89} +8.29263 q^{91} +110.697i q^{95} +27.0000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q - 48 q^{13} - 32 q^{25} + 160 q^{37} - 32 q^{49} + 32 q^{61} + 168 q^{73} - 496 q^{85} + 216 q^{97}+O(q^{100}) $ 8 * q - 48 * q^13 - 32 * q^25 + 160 * q^37 - 32 * q^49 + 32 * q^61 + 168 * q^73 - 496 * q^85 + 216 * 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
$5$	3.78194i	0.756388i	0.925726	+	0.378194i	$0.123455\pi$
$5$	3.78194i	0.756388i	−0.925726	+	0.378194i	$0.876545\pi$
$6$	0	0
$7$	0.953512	0.136216	0.0681080	−	0.997678i	$-0.478304\pi$
$7$	0.953512	0.136216	0.0681080	+	0.997678i	$0.478304\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	11.6969i	1.06336i	0.846946	+	0.531679i	$0.178439\pi$
$11$	11.6969i	1.06336i	−0.846946	+	0.531679i	$0.821561\pi$
$12$	0	0
$13$	8.69694	0.668995	0.334498	−	0.942397i	$-0.391433\pi$
$13$	8.69694	0.668995	0.334498	+	0.942397i	$0.391433\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	4.73545i	0.278556i	0.990253	+	0.139278i	$0.0444781\pi$
$17$	4.73545i	0.278556i	−0.990253	+	0.139278i	$0.955522\pi$
$18$	0	0
$19$	29.2699	1.54052	0.770260	−	0.637730i	$-0.220126\pi$
$19$	29.2699	1.54052	0.770260	+	0.637730i	$0.220126\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 2.69694i	− 0.117258i	−0.998280	−	0.0586291i	$-0.981327\pi$
$23$	− 2.69694i	− 0.117258i	0.998280	−	0.0586291i	$-0.0186729\pi$
$24$	0	0
$25$	10.6969	0.427878
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 21.7060i	− 0.748483i	−0.927331	−	0.374242i	$-0.877903\pi$
$29$	− 21.7060i	− 0.748483i	0.927331	−	0.374242i	$-0.122097\pi$
$30$	0	0
$31$	−22.5953	−0.728881	−0.364440	−	0.931227i	$-0.618740\pi$
$31$	−22.5953	−0.728881	−0.364440	+	0.931227i	$0.618740\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	3.60612i	0.103032i
$36$	0	0
$37$	−24.0908	−0.651103	−0.325552	−	0.945524i	$-0.605550\pi$
$37$	−24.0908	−0.651103	−0.325552	+	0.945524i	$0.605550\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 19.7990i	− 0.482902i	−0.970413	−	0.241451i	$-0.922377\pi$
$41$	− 19.7990i	− 0.482902i	0.970413	−	0.241451i	$-0.0776233\pi$
$42$	0	0
$43$	49.0047	1.13964	0.569822	−	0.821768i	$-0.307012\pi$
$43$	49.0047	1.13964	0.569822	+	0.821768i	$0.307012\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	59.3939i	1.26370i	0.775091	+	0.631850i	$0.217704\pi$
$47$	59.3939i	1.26370i	−0.775091	+	0.631850i	$0.782296\pi$
$48$	0	0
$49$	−48.0908	−0.981445
$50$	0	0
$51$	0	0
$52$	0	0
$53$	48.1796i	0.909049i	0.890734	+	0.454524i	$0.150191\pi$
$53$	48.1796i	0.909049i	−0.890734	+	0.454524i	$0.849809\pi$
$54$	0	0
$55$	−44.2371	−0.804311
$56$	0	0
$57$	0	0
$58$	0	0
$59$	12.6061i	0.213663i	0.994277	+	0.106832i	$0.0340706\pi$
$59$	12.6061i	0.213663i	−0.994277	+	0.106832i	$0.965929\pi$
$60$	0	0
$61$	4.00000	0.0655738	0.0327869	−	0.999462i	$-0.489562\pi$
$61$	4.00000	0.0655738	0.0327869	+	0.999462i	$0.489562\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	32.8913i	0.506020i
$66$	0	0
$67$	111.358	1.66207	0.831034	−	0.556222i	$-0.187750\pi$
$67$	111.358	1.66207	0.831034	+	0.556222i	$0.187750\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	56.6969i	0.798548i	0.916832	+	0.399274i	$0.130738\pi$
$71$	56.6969i	0.798548i	−0.916832	+	0.399274i	$0.869262\pi$
$72$	0	0
$73$	35.6969	0.488999	0.244500	−	0.969649i	$-0.421376\pi$
$73$	35.6969	0.488999	0.244500	+	0.969649i	$0.421376\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	11.1532i	0.144846i
$78$	0	0
$79$	−144.442	−1.82839	−0.914193	−	0.405280i	$-0.867174\pi$
$79$	−144.442	−1.82839	−0.914193	+	0.405280i	$0.867174\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	101.697i	1.22526i	0.790368	+	0.612632i	$0.209889\pi$
$83$	101.697i	1.22526i	−0.790368	+	0.612632i	$0.790111\pi$
$84$	0	0
$85$	−17.9092	−0.210696
$86$	0	0
$87$	0	0
$88$	0	0
$89$	19.0060i	0.213551i	0.994283	+	0.106775i	$0.0340526\pi$
$89$	19.0060i	0.213551i	−0.994283	+	0.106775i	$0.965947\pi$
$90$	0	0
$91$	8.29263	0.0911278
$92$	0	0
$93$	0	0
$94$	0	0
$95$	110.697i	1.16523i
$96$	0	0
$97$	27.0000	0.278351	0.139175	−	0.990268i	$-0.455555\pi$
$97$	27.0000	0.278351	0.139175	+	0.990268i	$0.455555\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 189.665i	− 1.87787i	−0.344091	−	0.938936i	$-0.611813\pi$
$101$	− 189.665i	− 1.87787i	0.344091	−	0.938936i	$-0.388187\pi$
$102$	0	0
$103$	−74.4605	−0.722918	−0.361459	−	0.932388i	$-0.617721\pi$
$103$	−74.4605	−0.722918	−0.361459	+	0.932388i	$0.617721\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	138.576i	1.29510i	0.762024	+	0.647549i	$0.224206\pi$
$107$	138.576i	1.29510i	−0.762024	+	0.647549i	$0.775794\pi$
$108$	0	0
$109$	114.879	1.05393	0.526966	−	0.849886i	$-0.323330\pi$
$109$	114.879	1.05393	0.526966	+	0.849886i	$0.323330\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 157.663i	− 1.39525i	−0.716463	−	0.697625i	$-0.754240\pi$
$113$	− 157.663i	− 1.39525i	0.716463	−	0.697625i	$-0.245760\pi$
$114$	0	0
$115$	10.1997	0.0886927
$116$	0	0
$117$	0	0
$118$	0	0
$119$	4.51531i	0.0379438i
$120$	0	0
$121$	−15.8184	−0.130730
$122$	0	0
$123$	0	0
$124$	0	0
$125$	135.004i	1.08003i
$126$	0	0
$127$	−125.661	−0.989458	−0.494729	−	0.869047i	$-0.664733\pi$
$127$	−125.661	−0.989458	−0.494729	+	0.869047i	$0.664733\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	223.182i	1.70368i	0.523805	+	0.851838i	$0.324512\pi$
$131$	223.182i	1.70368i	−0.523805	+	0.851838i	$0.675488\pi$
$132$	0	0
$133$	27.9092	0.209843
$134$	0	0
$135$	0	0
$136$	0	0
$137$	162.142i	1.18352i	0.806116	+	0.591758i	$0.201566\pi$
$137$	162.142i	1.18352i	−0.806116	+	0.591758i	$0.798434\pi$
$138$	0	0
$139$	−69.9819	−0.503467	−0.251734	−	0.967797i	$-0.581001\pi$
$139$	−69.9819	−0.503467	−0.251734	+	0.967797i	$0.581001\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	101.728i	0.711381i
$144$	0	0
$145$	82.0908	0.566144
$146$	0	0
$147$	0	0
$148$	0	0
$149$	216.171i	1.45081i	0.688322	+	0.725405i	$0.258348\pi$
$149$	216.171i	1.45081i	−0.688322	+	0.725405i	$0.741652\pi$
$150$	0	0
$151$	92.5772	0.613094	0.306547	−	0.951855i	$-0.400826\pi$
$151$	92.5772	0.613094	0.306547	+	0.951855i	$0.400826\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 85.4541i	− 0.551317i
$156$	0	0
$157$	−292.545	−1.86334	−0.931672	−	0.363301i	$-0.881649\pi$
$157$	−292.545	−1.86334	−0.931672	+	0.363301i	$0.881649\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 2.57156i	− 0.0159724i
$162$	0	0
$163$	112.601	0.690804	0.345402	−	0.938455i	$-0.387743\pi$
$163$	112.601	0.690804	0.345402	+	0.938455i	$0.387743\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	106.182i	0.635818i	0.948121	+	0.317909i	$0.102981\pi$
$167$	106.182i	0.635818i	−0.948121	+	0.317909i	$0.897019\pi$
$168$	0	0
$169$	−93.3633	−0.552445
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 224.271i	− 1.29636i	−0.761486	−	0.648182i	$-0.775530\pi$
$173$	− 224.271i	− 1.29636i	0.761486	−	0.648182i	$-0.224470\pi$
$174$	0	0
$175$	10.1997	0.0582838
$176$	0	0
$177$	0	0
$178$	0	0
$179$	113.424i	0.633656i	0.948483	+	0.316828i	$0.102618\pi$
$179$	113.424i	0.633656i	−0.948483	+	0.316828i	$0.897382\pi$
$180$	0	0
$181$	160.182	0.884981	0.442491	−	0.896773i	$-0.354095\pi$
$181$	160.182	0.884981	0.442491	+	0.896773i	$0.354095\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 91.1100i	− 0.492486i
$186$	0	0
$187$	−55.3903	−0.296205
$188$	0	0
$189$	0	0
$190$	0	0
$191$	200.697i	1.05077i	0.850865	+	0.525385i	$0.176079\pi$
$191$	200.697i	1.05077i	−0.850865	+	0.525385i	$0.823921\pi$
$192$	0	0
$193$	−152.909	−0.792276	−0.396138	−	0.918191i	$-0.629650\pi$
$193$	−152.909	−0.792276	−0.396138	+	0.918191i	$0.629650\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 15.2883i	− 0.0776056i	−0.999247	−	0.0388028i	$-0.987646\pi$
$197$	− 15.2883i	− 0.0776056i	0.999247	−	0.0388028i	$-0.0123544\pi$
$198$	0	0
$199$	231.299	1.16230	0.581152	−	0.813795i	$-0.302602\pi$
$199$	231.299	1.16230	0.581152	+	0.813795i	$0.302602\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 20.6969i	− 0.101955i
$204$	0	0
$205$	74.8786	0.365261
$206$	0	0
$207$	0	0
$208$	0	0
$209$	342.368i	1.63812i
$210$	0	0
$211$	−185.819	−0.880659	−0.440329	−	0.897836i	$-0.645138\pi$
$211$	−185.819	−0.880659	−0.440329	+	0.897836i	$0.645138\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	185.333i	0.862012i
$216$	0	0
$217$	−21.5449	−0.0992852
$218$	0	0
$219$	0	0
$220$	0	0
$221$	41.1839i	0.186353i
$222$	0	0
$223$	272.964	1.22405	0.612027	−	0.790837i	$-0.290354\pi$
$223$	272.964	1.22405	0.612027	+	0.790837i	$0.290354\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	307.757i	1.35576i	0.735173	+	0.677879i	$0.237101\pi$
$227$	307.757i	1.35576i	−0.735173	+	0.677879i	$0.762899\pi$
$228$	0	0
$229$	−200.424	−0.875216	−0.437608	−	0.899166i	$-0.644174\pi$
$229$	−200.424	−0.875216	−0.437608	+	0.899166i	$0.644174\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 381.687i	− 1.63814i	−0.573693	−	0.819070i	$-0.694490\pi$
$233$	− 381.687i	− 1.63814i	0.573693	−	0.819070i	$-0.305510\pi$
$234$	0	0
$235$	−224.624	−0.955847
$236$	0	0
$237$	0	0
$238$	0	0
$239$	254.697i	1.06568i	0.846217	+	0.532839i	$0.178875\pi$
$239$	254.697i	1.06568i	−0.846217	+	0.532839i	$0.821125\pi$
$240$	0	0
$241$	283.757	1.17742	0.588708	−	0.808346i	$-0.299637\pi$
$241$	283.757	1.17742	0.588708	+	0.808346i	$0.299637\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 181.877i	− 0.742353i
$246$	0	0
$247$	254.558	1.03060
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 54.0000i	− 0.215139i	−0.994198	−	0.107570i	$-0.965693\pi$
$251$	− 54.0000i	− 0.215139i	0.994198	−	0.107570i	$-0.0343069\pi$
$252$	0	0
$253$	31.5459	0.124687
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 202.725i	− 0.788815i	−0.918936	−	0.394407i	$-0.870950\pi$
$257$	− 202.725i	− 0.788815i	0.918936	−	0.394407i	$-0.129050\pi$
$258$	0	0
$259$	−22.9709	−0.0886906
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 162.879i	− 0.619310i	−0.950849	−	0.309655i	$-0.899786\pi$
$263$	− 162.879i	− 0.619310i	0.950849	−	0.309655i	$-0.100214\pi$
$264$	0	0
$265$	−182.212	−0.687593
$266$	0	0
$267$	0	0
$268$	0	0
$269$	119.715i	0.445038i	0.974928	+	0.222519i	$0.0714280\pi$
$269$	119.715i	0.445038i	−0.974928	+	0.222519i	$0.928572\pi$
$270$	0	0
$271$	337.601	1.24576	0.622879	−	0.782318i	$-0.285963\pi$
$271$	337.601	1.24576	0.622879	+	0.782318i	$0.285963\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	125.121i	0.454987i
$276$	0	0
$277$	268.182	0.968165	0.484082	−	0.875022i	$-0.339154\pi$
$277$	268.182	0.968165	0.484082	+	0.875022i	$0.339154\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	272.322i	0.969117i	0.874759	+	0.484559i	$0.161020\pi$
$281$	272.322i	0.969117i	−0.874759	+	0.484559i	$0.838980\pi$
$282$	0	0
$283$	67.4104	0.238199	0.119100	−	0.992882i	$-0.461999\pi$
$283$	67.4104	0.238199	0.119100	+	0.992882i	$0.461999\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 18.8786i	− 0.0657790i
$288$	0	0
$289$	266.576	0.922407
$290$	0	0
$291$	0	0
$292$	0	0
$293$	378.345i	1.29128i	0.763642	+	0.645639i	$0.223409\pi$
$293$	378.345i	1.29128i	−0.763642	+	0.645639i	$0.776591\pi$
$294$	0	0
$295$	−47.6756	−0.161612
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 23.4551i	− 0.0784452i
$300$	0	0
$301$	46.7265	0.155238
$302$	0	0
$303$	0	0
$304$	0	0
$305$	15.1278i	0.0495992i
$306$	0	0
$307$	−425.699	−1.38664	−0.693321	−	0.720629i	$-0.743853\pi$
$307$	−425.699	−1.38664	−0.693321	+	0.720629i	$0.743853\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	− 556.999i	− 1.79099i	−0.445068	−	0.895497i	$-0.646821\pi$
$311$	− 556.999i	− 1.79099i	0.445068	−	0.895497i	$-0.353179\pi$
$312$	0	0
$313$	169.636	0.541967	0.270984	−	0.962584i	$-0.412651\pi$
$313$	169.636	0.541967	0.270984	+	0.962584i	$0.412651\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 225.128i	− 0.710183i	−0.934832	−	0.355091i	$-0.884450\pi$
$317$	− 225.128i	− 0.710183i	0.934832	−	0.355091i	$-0.115550\pi$
$318$	0	0
$319$	253.894	0.795906
$320$	0	0
$321$	0	0
$322$	0	0
$323$	138.606i	0.429121i
$324$	0	0
$325$	93.0306	0.286248
$326$	0	0
$327$	0	0
$328$	0	0
$329$	56.6328i	0.172136i
$330$	0	0
$331$	−315.005	−0.951677	−0.475839	−	0.879533i	$-0.657855\pi$
$331$	−315.005	−0.951677	−0.475839	+	0.879533i	$0.657855\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	421.151i	1.25717i
$336$	0	0
$337$	−642.545	−1.90666	−0.953331	−	0.301928i	$-0.902370\pi$
$337$	−642.545	−1.90666	−0.953331	+	0.301928i	$0.902370\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 264.296i	− 0.775061i
$342$	0	0
$343$	−92.5772	−0.269904
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 301.515i	− 0.868920i	−0.900691	−	0.434460i	$-0.856939\pi$
$347$	− 301.515i	− 0.868920i	0.900691	−	0.434460i	$-0.143061\pi$
$348$	0	0
$349$	−180.454	−0.517060	−0.258530	−	0.966003i	$-0.583238\pi$
$349$	−180.454	−0.517060	−0.258530	+	0.966003i	$0.583238\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	331.825i	0.940014i	0.882663	+	0.470007i	$0.155749\pi$
$353$	331.825i	0.940014i	−0.882663	+	0.470007i	$0.844251\pi$
$354$	0	0
$355$	−214.424	−0.604012
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 566.908i	− 1.57913i	−0.613666	−	0.789566i	$-0.710306\pi$
$359$	− 566.908i	− 1.57913i	0.613666	−	0.789566i	$-0.289694\pi$
$360$	0	0
$361$	495.727	1.37320
$362$	0	0
$363$	0	0
$364$	0	0
$365$	135.004i	0.369873i
$366$	0	0
$367$	219.192	0.597253	0.298627	−	0.954370i	$-0.403471\pi$
$367$	219.192	0.597253	0.298627	+	0.954370i	$0.403471\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	45.9398i	0.123827i
$372$	0	0
$373$	−606.454	−1.62588	−0.812941	−	0.582346i	$-0.802135\pi$
$373$	−606.454	−1.62588	−0.812941	+	0.582346i	$0.802135\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 188.776i	− 0.500732i
$378$	0	0
$379$	124.708	0.329044	0.164522	−	0.986373i	$-0.447392\pi$
$379$	124.708	0.329044	0.164522	+	0.986373i	$0.447392\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	58.4541i	0.152622i	0.997084	+	0.0763108i	$0.0243141\pi$
$383$	58.4541i	0.152622i	−0.997084	+	0.0763108i	$0.975686\pi$
$384$	0	0
$385$	−42.1806	−0.109560
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 121.526i	− 0.312406i	−0.987725	−	0.156203i	$-0.950075\pi$
$389$	− 121.526i	− 0.312406i	0.987725	−	0.156203i	$-0.0499255\pi$
$390$	0	0
$391$	12.7712	0.0326630
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 546.272i	− 1.38297i
$396$	0	0
$397$	330.091	0.831463	0.415732	−	0.909487i	$-0.363526\pi$
$397$	330.091	0.831463	0.415732	+	0.909487i	$0.363526\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 506.202i	− 1.26235i	−0.775641	−	0.631174i	$-0.782573\pi$
$401$	− 506.202i	− 1.26235i	0.775641	−	0.631174i	$-0.217427\pi$
$402$	0	0
$403$	−196.510	−0.487618
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 281.789i	− 0.692356i
$408$	0	0
$409$	−521.302	−1.27458	−0.637289	−	0.770625i	$-0.719944\pi$
$409$	−521.302	−1.27458	−0.637289	+	0.770625i	$0.719944\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	12.0201i	0.0291043i
$414$	0	0
$415$	−384.612	−0.926775
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 133.151i	− 0.317783i	−0.987296	−	0.158891i	$-0.949208\pi$
$419$	− 133.151i	− 0.317783i	0.987296	−	0.158891i	$-0.0507920\pi$
$420$	0	0
$421$	129.031	0.306486	0.153243	−	0.988189i	$-0.451028\pi$
$421$	129.031	0.306486	0.153243	+	0.988189i	$0.451028\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	50.6548i	0.119188i
$426$	0	0
$427$	3.81405	0.00893219
$428$	0	0
$429$	0	0
$430$	0	0
$431$	509.333i	1.18175i	0.806764	+	0.590873i	$0.201217\pi$
$431$	509.333i	1.18175i	−0.806764	+	0.590873i	$0.798783\pi$
$432$	0	0
$433$	−5.54489	−0.0128058	−0.00640288	−	0.999980i	$-0.502038\pi$
$433$	−5.54489	−0.0128058	−0.00640288	+	0.999980i	$0.502038\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 78.9391i	− 0.180639i
$438$	0	0
$439$	−337.514	−0.768825	−0.384412	−	0.923162i	$-0.625596\pi$
$439$	−337.514	−0.768825	−0.384412	+	0.923162i	$0.625596\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 475.212i	− 1.07271i	−0.843991	−	0.536357i	$-0.819800\pi$
$443$	− 475.212i	− 1.07271i	0.843991	−	0.536357i	$-0.180200\pi$
$444$	0	0
$445$	−71.8796	−0.161527
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 251.858i	− 0.560932i	−0.959864	−	0.280466i	$-0.909511\pi$
$449$	− 251.858i	− 0.560932i	0.959864	−	0.280466i	$-0.0904890\pi$
$450$	0	0
$451$	231.588	0.513498
$452$	0	0
$453$	0	0
$454$	0	0
$455$	31.3622i	0.0689280i
$456$	0	0
$457$	42.5755	0.0931630	0.0465815	−	0.998914i	$-0.485167\pi$
$457$	42.5755	0.0931630	0.0465815	+	0.998914i	$0.485167\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	552.346i	1.19815i	0.800694	+	0.599074i	$0.204464\pi$
$461$	552.346i	1.19815i	−0.800694	+	0.599074i	$0.795536\pi$
$462$	0	0
$463$	−524.084	−1.13193	−0.565966	−	0.824429i	$-0.691496\pi$
$463$	−524.084	−1.13193	−0.565966	+	0.824429i	$0.691496\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	130.423i	0.279279i	0.990202	+	0.139640i	$0.0445944\pi$
$467$	130.423i	0.279279i	−0.990202	+	0.139640i	$0.955406\pi$
$468$	0	0
$469$	106.182	0.226400
$470$	0	0
$471$	0	0
$472$	0	0
$473$	573.205i	1.21185i
$474$	0	0
$475$	313.098	0.659154
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 572.302i	− 1.19479i	−0.801949	−	0.597393i	$-0.796203\pi$
$479$	− 572.302i	− 1.19479i	0.801949	−	0.597393i	$-0.203797\pi$
$480$	0	0
$481$	−209.516	−0.435585
$482$	0	0
$483$	0	0
$484$	0	0
$485$	102.112i	0.210541i
$486$	0	0
$487$	−141.957	−0.291494	−0.145747	−	0.989322i	$-0.546559\pi$
$487$	−141.957	−0.291494	−0.145747	+	0.989322i	$0.546559\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 464.394i	− 0.945812i	−0.881113	−	0.472906i	$-0.843205\pi$
$491$	− 464.394i	− 0.945812i	0.881113	−	0.472906i	$-0.156795\pi$
$492$	0	0
$493$	102.788	0.208494
$494$	0	0
$495$	0	0
$496$	0	0
$497$	54.0612i	0.108775i
$498$	0	0
$499$	−144.356	−0.289290	−0.144645	−	0.989484i	$-0.546204\pi$
$499$	−144.356	−0.289290	−0.144645	+	0.989484i	$0.546204\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 733.485i	− 1.45822i	−0.684396	−	0.729110i	$-0.739934\pi$
$503$	− 733.485i	− 1.45822i	0.684396	−	0.729110i	$-0.260066\pi$
$504$	0	0
$505$	717.302	1.42040
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 300.680i	− 0.590727i	−0.955385	−	0.295364i	$-0.904559\pi$
$509$	− 300.680i	− 0.590727i	0.955385	−	0.295364i	$-0.0954408\pi$
$510$	0	0
$511$	34.0374	0.0666095
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 281.605i	− 0.546806i
$516$	0	0
$517$	−694.727	−1.34377
$518$	0	0
$519$	0	0
$520$	0	0
$521$	375.988i	0.721666i	0.932630	+	0.360833i	$0.117508\pi$
$521$	375.988i	0.721666i	−0.932630	+	0.360833i	$0.882492\pi$
$522$	0	0
$523$	665.001	1.27151	0.635757	−	0.771890i	$-0.280688\pi$
$523$	665.001	1.27151	0.635757	+	0.771890i	$0.280688\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 106.999i	− 0.203034i
$528$	0	0
$529$	521.727	0.986251
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 172.191i	− 0.323059i
$534$	0	0
$535$	−524.084	−0.979596
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 562.515i	− 1.04363i
$540$	0	0
$541$	818.302	1.51257	0.756287	−	0.654241i	$-0.227012\pi$
$541$	818.302	1.51257	0.756287	+	0.654241i	$0.227012\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	434.464i	0.797181i
$546$	0	0
$547$	625.532	1.14357	0.571784	−	0.820404i	$-0.306251\pi$
$547$	625.532	1.14357	0.571784	+	0.820404i	$0.306251\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	− 635.333i	− 1.15305i
$552$	0	0
$553$	−137.728	−0.249055
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 27.9087i	− 0.0501054i	−0.999686	−	0.0250527i	$-0.992025\pi$
$557$	− 27.9087i	− 0.0501054i	0.999686	−	0.0250527i	$-0.00797536\pi$
$558$	0	0
$559$	426.191	0.762416
$560$	0	0
$561$	0	0
$562$	0	0
$563$	461.574i	0.819848i	0.912120	+	0.409924i	$0.134445\pi$
$563$	461.574i	0.819848i	−0.912120	+	0.409924i	$0.865555\pi$
$564$	0	0
$565$	596.272	1.05535
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 983.092i	− 1.72775i	−0.503703	−	0.863877i	$-0.668029\pi$
$569$	− 983.092i	− 1.72775i	0.503703	−	0.863877i	$-0.331971\pi$
$570$	0	0
$571$	692.856	1.21341	0.606704	−	0.794928i	$-0.292491\pi$
$571$	692.856	1.21341	0.606704	+	0.794928i	$0.292491\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	− 28.8490i	− 0.0501721i
$576$	0	0
$577$	−154.545	−0.267842	−0.133921	−	0.990992i	$-0.542757\pi$
$577$	−154.545	−0.267842	−0.133921	+	0.990992i	$0.542757\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	96.9692i	0.166901i
$582$	0	0
$583$	−563.554	−0.966644
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 867.545i	− 1.47793i	−0.673744	−	0.738965i	$-0.735315\pi$
$587$	− 867.545i	− 1.47793i	0.673744	−	0.738965i	$-0.264685\pi$
$588$	0	0
$589$	−661.362	−1.12286
$590$	0	0
$591$	0	0
$592$	0	0
$593$	343.675i	0.579553i	0.957094	+	0.289776i	$0.0935809\pi$
$593$	343.675i	0.579553i	−0.957094	+	0.289776i	$0.906419\pi$
$594$	0	0
$595$	−17.0766	−0.0287002
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 908.969i	− 1.51748i	−0.651395	−	0.758739i	$-0.725816\pi$
$599$	− 908.969i	− 1.51748i	0.651395	−	0.758739i	$-0.274184\pi$
$600$	0	0
$601$	−419.817	−0.698531	−0.349266	−	0.937024i	$-0.613569\pi$
$601$	−419.817	−0.698531	−0.349266	+	0.937024i	$0.613569\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 59.8241i	− 0.0988828i
$606$	0	0
$607$	131.671	0.216921	0.108461	−	0.994101i	$-0.465408\pi$
$607$	131.671	0.216921	0.108461	+	0.994101i	$0.465408\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	516.545i	0.845409i
$612$	0	0
$613$	494.727	0.807058	0.403529	−	0.914967i	$-0.367783\pi$
$613$	494.727	0.807058	0.403529	+	0.914967i	$0.367783\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 500.330i	− 0.810908i	−0.914116	−	0.405454i	$-0.867114\pi$
$617$	− 500.330i	− 0.810908i	0.914116	−	0.405454i	$-0.132886\pi$
$618$	0	0
$619$	636.974	1.02904	0.514519	−	0.857479i	$-0.327971\pi$
$619$	636.974	1.02904	0.514519	+	0.857479i	$0.327971\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	18.1225i	0.0290890i
$624$	0	0
$625$	−243.152	−0.389043
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 114.081i	− 0.181369i
$630$	0	0
$631$	−365.541	−0.579305	−0.289652	−	0.957132i	$-0.593540\pi$
$631$	−365.541	−0.579305	−0.289652	+	0.957132i	$0.593540\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 475.243i	− 0.748414i
$636$	0	0
$637$	−418.243	−0.656582
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 843.812i	− 1.31640i	−0.752843	−	0.658200i	$-0.771318\pi$
$641$	− 843.812i	− 1.31640i	0.752843	−	0.658200i	$-0.228682\pi$
$642$	0	0
$643$	1252.22	1.94746	0.973732	−	0.227696i	$-0.0731192\pi$
$643$	1252.22	1.94746	0.973732	+	0.227696i	$0.0731192\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 799.151i	− 1.23516i	−0.786507	−	0.617582i	$-0.788112\pi$
$647$	− 799.151i	− 1.23516i	0.786507	−	0.617582i	$-0.211888\pi$
$648$	0	0
$649$	−147.453	−0.227200
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 798.227i	− 1.22240i	−0.791477	−	0.611199i	$-0.790687\pi$
$653$	− 798.227i	− 1.22240i	0.791477	−	0.611199i	$-0.209313\pi$
$654$	0	0
$655$	−844.059	−1.28864
$656$	0	0
$657$	0	0
$658$	0	0
$659$	106.151i	0.161079i	0.996751	+	0.0805395i	$0.0256643\pi$
$659$	106.151i	0.161079i	−0.996751	+	0.0805395i	$0.974336\pi$
$660$	0	0
$661$	−772.272	−1.16834	−0.584170	−	0.811631i	$-0.698580\pi$
$661$	−772.272	−1.16834	−0.584170	+	0.811631i	$0.698580\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	105.551i	0.158723i
$666$	0	0
$667$	−58.5398	−0.0877658
$668$	0	0
$669$	0	0
$670$	0	0
$671$	46.7878i	0.0697284i
$672$	0	0
$673$	494.637	0.734973	0.367486	−	0.930029i	$-0.380218\pi$
$673$	494.637	0.734973	0.367486	+	0.930029i	$0.380218\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	1089.14i	1.60877i	0.594109	+	0.804385i	$0.297505\pi$
$677$	1089.14i	1.60877i	−0.594109	+	0.804385i	$0.702495\pi$
$678$	0	0
$679$	25.7448	0.0379158
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 1126.60i	− 1.64949i	−0.565502	−	0.824747i	$-0.691318\pi$
$683$	− 1126.60i	− 1.64949i	0.565502	−	0.824747i	$-0.308682\pi$
$684$	0	0
$685$	−613.210	−0.895197
$686$	0	0
$687$	0	0
$688$	0	0
$689$	419.015i	0.608149i
$690$	0	0
$691$	−462.019	−0.668624	−0.334312	−	0.942462i	$-0.608504\pi$
$691$	−462.019	−0.668624	−0.334312	+	0.942462i	$0.608504\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 264.667i	− 0.380816i
$696$	0	0
$697$	93.7571	0.134515
$698$	0	0
$699$	0	0
$700$	0	0
$701$	189.193i	0.269891i	0.990853	+	0.134945i	$0.0430859\pi$
$701$	189.193i	0.269891i	−0.990853	+	0.134945i	$0.956914\pi$
$702$	0	0
$703$	−705.136	−1.00304
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 180.848i	− 0.255796i
$708$	0	0
$709$	926.302	1.30649	0.653245	−	0.757146i	$-0.273407\pi$
$709$	926.302	1.30649	0.653245	+	0.757146i	$0.273407\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	60.9382i	0.0854673i
$714$	0	0
$715$	−384.727	−0.538080
$716$	0	0
$717$	0	0
$718$	0	0
$719$	957.453i	1.33165i	0.746110	+	0.665823i	$0.231919\pi$
$719$	957.453i	1.33165i	−0.746110	+	0.665823i	$0.768081\pi$
$720$	0	0
$721$	−70.9990	−0.0984729
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 232.188i	− 0.320259i
$726$	0	0
$727$	979.545	1.34738	0.673690	−	0.739015i	$-0.264709\pi$
$727$	979.545	1.34738	0.673690	+	0.739015i	$0.264709\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	232.059i	0.317454i
$732$	0	0
$733$	452.636	0.617511	0.308756	−	0.951141i	$-0.400087\pi$
$733$	452.636	0.617511	0.308756	+	0.951141i	$0.400087\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1302.55i	1.76737i
$738$	0	0
$739$	460.026	0.622497	0.311249	−	0.950328i	$-0.399253\pi$
$739$	460.026	0.622497	0.311249	+	0.950328i	$0.399253\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	393.119i	0.529097i	0.964372	+	0.264549i	$0.0852230\pi$
$743$	393.119i	0.529097i	−0.964372	+	0.264549i	$0.914777\pi$
$744$	0	0
$745$	−817.545	−1.09738
$746$	0	0
$747$	0	0
$748$	0	0
$749$	132.133i	0.176413i
$750$	0	0
$751$	611.229	0.813887	0.406944	−	0.913453i	$-0.366595\pi$
$751$	611.229	0.813887	0.406944	+	0.913453i	$0.366595\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	350.121i	0.463737i
$756$	0	0
$757$	−106.182	−0.140266	−0.0701332	−	0.997538i	$-0.522342\pi$
$757$	−106.182	−0.140266	−0.0701332	+	0.997538i	$0.522342\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	844.843i	1.11017i	0.831792	+	0.555087i	$0.187315\pi$
$761$	844.843i	1.11017i	−0.831792	+	0.555087i	$0.812685\pi$
$762$	0	0
$763$	109.538	0.143562
$764$	0	0
$765$	0	0
$766$	0	0
$767$	109.635i	0.142940i
$768$	0	0
$769$	−588.728	−0.765575	−0.382788	−	0.923836i	$-0.625036\pi$
$769$	−588.728	−0.765575	−0.382788	+	0.923836i	$0.625036\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 9.87858i	− 0.0127795i	−0.999980	−	0.00638976i	$-0.997966\pi$
$773$	− 9.87858i	− 0.0127795i	0.999980	−	0.00638976i	$-0.00203394\pi$
$774$	0	0
$775$	−241.701	−0.311872
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 579.514i	− 0.743921i
$780$	0	0
$781$	−663.181	−0.849143
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 1106.39i	− 1.40941i
$786$	0	0
$787$	−118.409	−0.150456	−0.0752279	−	0.997166i	$-0.523968\pi$
$787$	−118.409	−0.150456	−0.0752279	+	0.997166i	$0.523968\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 150.334i	− 0.190055i
$792$	0	0
$793$	34.7878	0.0438685
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 371.477i	− 0.466095i	−0.972465	−	0.233047i	$-0.925130\pi$
$797$	− 371.477i	− 0.466095i	0.972465	−	0.233047i	$-0.0748697\pi$
$798$	0	0
$799$	−281.257	−0.352011
$800$	0	0
$801$	0	0
$802$	0	0
$803$	417.545i	0.519981i
$804$	0	0
$805$	9.72549	0.0120814
$806$	0	0
$807$	0	0
$808$	0	0
$809$	64.7746i	0.0800675i	0.999198	+	0.0400337i	$0.0127465\pi$
$809$	64.7746i	0.0800675i	−0.999198	+	0.0400337i	$0.987253\pi$
$810$	0	0
$811$	−782.832	−0.965268	−0.482634	−	0.875822i	$-0.660320\pi$
$811$	−782.832	−0.965268	−0.482634	+	0.875822i	$0.660320\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	425.850i	0.522515i
$816$	0	0
$817$	1434.36	1.75564
$818$	0	0
$819$	0	0
$820$	0	0
$821$	589.748i	0.718329i	0.933274	+	0.359164i	$0.116938\pi$
$821$	589.748i	0.718329i	−0.933274	+	0.359164i	$0.883062\pi$
$822$	0	0
$823$	−492.821	−0.598810	−0.299405	−	0.954126i	$-0.596788\pi$
$823$	−492.821	−0.598810	−0.299405	+	0.954126i	$0.596788\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 475.212i	− 0.574622i	−0.957837	−	0.287311i	$-0.907239\pi$
$827$	− 475.212i	− 0.574622i	0.957837	−	0.287311i	$-0.0927613\pi$
$828$	0	0
$829$	597.637	0.720913	0.360456	−	0.932776i	$-0.382621\pi$
$829$	597.637	0.720913	0.360456	+	0.932776i	$0.382621\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 227.732i	− 0.273387i
$834$	0	0
$835$	−401.572	−0.480925
$836$	0	0
$837$	0	0
$838$	0	0
$839$	290.636i	0.346407i	0.984886	+	0.173204i	$0.0554119\pi$
$839$	290.636i	0.346407i	−0.984886	+	0.173204i	$0.944588\pi$
$840$	0	0
$841$	369.849	0.439773
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 353.094i	− 0.417863i
$846$	0	0
$847$	−15.0830	−0.0178076
$848$	0	0
$849$	0	0
$850$	0	0
$851$	64.9714i	0.0763472i
$852$	0	0
$853$	201.273	0.235960	0.117980	−	0.993016i	$-0.462358\pi$
$853$	201.273	0.235960	0.117980	+	0.993016i	$0.462358\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 1009.64i	− 1.17811i	−0.808093	−	0.589055i	$-0.799500\pi$
$857$	− 1009.64i	− 1.17811i	0.808093	−	0.589055i	$-0.200500\pi$
$858$	0	0
$859$	−1172.12	−1.36452	−0.682261	−	0.731109i	$-0.739003\pi$
$859$	−1172.12	−1.36452	−0.682261	+	0.731109i	$0.739003\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 755.755i	− 0.875730i	−0.899041	−	0.437865i	$-0.855735\pi$
$863$	− 755.755i	− 0.875730i	0.899041	−	0.437865i	$-0.144265\pi$
$864$	0	0
$865$	848.179	0.980553
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 1689.53i	− 1.94423i
$870$	0	0
$871$	968.478	1.11192
$872$	0	0
$873$	0	0
$874$	0	0
$875$	128.728i	0.147117i
$876$	0	0
$877$	317.818	0.362393	0.181196	−	0.983447i	$-0.442003\pi$
$877$	317.818	0.362393	0.181196	+	0.983447i	$0.442003\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 705.755i	− 0.801084i	−0.916278	−	0.400542i	$-0.868822\pi$
$881$	− 705.755i	− 0.801084i	0.916278	−	0.400542i	$-0.131178\pi$
$882$	0	0
$883$	505.389	0.572355	0.286177	−	0.958177i	$-0.407615\pi$
$883$	505.389	0.572355	0.286177	+	0.958177i	$0.407615\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 1581.00i	− 1.78241i	−0.453602	−	0.891205i	$-0.649861\pi$
$887$	− 1581.00i	− 1.78241i	0.453602	−	0.891205i	$-0.350139\pi$
$888$	0	0
$889$	−119.819	−0.134780
$890$	0	0
$891$	0	0
$892$	0	0
$893$	1738.45i	1.94676i
$894$	0	0
$895$	−428.964	−0.479290
$896$	0	0
$897$	0	0
$898$	0	0
$899$	490.454i	0.545555i
$900$	0	0
$901$	−228.152	−0.253221
$902$	0	0
$903$	0	0
$904$	0	0
$905$	605.797i	0.669389i
$906$	0	0
$907$	−773.875	−0.853225	−0.426612	−	0.904435i	$-0.640293\pi$
$907$	−773.875	−0.853225	−0.426612	+	0.904435i	$0.640293\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	1785.51i	1.95995i	0.199121	+	0.979975i	$0.436191\pi$
$911$	1785.51i	1.95995i	−0.199121	+	0.979975i	$0.563809\pi$
$912$	0	0
$913$	−1189.54	−1.30289
$914$	0	0
$915$	0	0
$916$	0	0
$917$	212.806i	0.232068i
$918$	0	0
$919$	992.402	1.07987	0.539936	−	0.841706i	$-0.318448\pi$
$919$	992.402	1.07987	0.539936	+	0.841706i	$0.318448\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	493.090i	0.534225i
$924$	0	0
$925$	−257.698	−0.278592
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 887.568i	− 0.955401i	−0.878523	−	0.477701i	$-0.841470\pi$
$929$	− 887.568i	− 0.955401i	0.878523	−	0.477701i	$-0.158530\pi$
$930$	0	0
$931$	−1407.61	−1.51194
$932$	0	0
$933$	0	0
$934$	0	0
$935$	− 209.483i	− 0.224046i
$936$	0	0
$937$	493.000	0.526147	0.263074	−	0.964776i	$-0.415264\pi$
$937$	493.000	0.526147	0.263074	+	0.964776i	$0.415264\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	379.770i	0.403581i	0.979429	+	0.201791i	$0.0646761\pi$
$941$	379.770i	0.403581i	−0.979429	+	0.201791i	$0.935324\pi$
$942$	0	0
$943$	−53.3967	−0.0566242
$944$	0	0
$945$	0	0
$946$	0	0
$947$	414.092i	0.437267i	0.975807	+	0.218633i	$0.0701599\pi$
$947$	414.092i	0.437267i	−0.975807	+	0.218633i	$0.929840\pi$
$948$	0	0
$949$	310.454	0.327138
$950$	0	0
$951$	0	0
$952$	0	0
$953$	1254.36i	1.31623i	0.752919	+	0.658113i	$0.228645\pi$
$953$	1254.36i	1.31623i	−0.752919	+	0.658113i	$0.771355\pi$
$954$	0	0
$955$	−759.024	−0.794789
$956$	0	0
$957$	0	0
$958$	0	0
$959$	154.604i	0.161214i
$960$	0	0
$961$	−450.452	−0.468733
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 578.293i	− 0.599268i
$966$	0	0
$967$	522.755	0.540595	0.270297	−	0.962777i	$-0.412878\pi$
$967$	522.755	0.540595	0.270297	+	0.962777i	$0.412878\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	1044.66i	1.07586i	0.842988	+	0.537932i	$0.180794\pi$
$971$	1044.66i	1.07586i	−0.842988	+	0.537932i	$0.819206\pi$
$972$	0	0
$973$	−66.7286	−0.0685803
$974$	0	0
$975$	0	0
$976$	0	0
$977$	1375.71i	1.40809i	0.710154	+	0.704047i	$0.248625\pi$
$977$	1375.71i	1.40809i	−0.710154	+	0.704047i	$0.751375\pi$
$978$	0	0
$979$	−222.312	−0.227081
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 451.757i	− 0.459570i	−0.973241	−	0.229785i	$-0.926198\pi$
$983$	− 451.757i	− 0.459570i	0.973241	−	0.229785i	$-0.0738023\pi$
$984$	0	0
$985$	57.8194	0.0586999
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 132.163i	− 0.133633i
$990$	0	0
$991$	1811.29	1.82774	0.913872	−	0.406002i	$-0.133077\pi$
$991$	1811.29	1.82774	0.913872	+	0.406002i	$0.133077\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	874.757i	0.879153i
$996$	0	0
$997$	377.546	0.378682	0.189341	−	0.981911i	$-0.439365\pi$
$997$	377.546	0.378682	0.189341	+	0.981911i	$0.439365\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1728.3.e.u.1025.6		8
3.2	odd	2	inner	1728.3.e.u.1025.4		8
4.3	odd	2	inner	1728.3.e.u.1025.5		8
8.3	odd	2		864.3.e.f.161.3	✓	8
8.5	even	2		864.3.e.f.161.4	yes	8
12.11	even	2	inner	1728.3.e.u.1025.3		8
24.5	odd	2		864.3.e.f.161.6	yes	8
24.11	even	2		864.3.e.f.161.5	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
864.3.e.f.161.3	✓	8	8.3	odd	2
864.3.e.f.161.4	yes	8	8.5	even	2
864.3.e.f.161.5	yes	8	24.11	even	2
864.3.e.f.161.6	yes	8	24.5	odd	2
1728.3.e.u.1025.3		8	12.11	even	2	inner
1728.3.e.u.1025.4		8	3.2	odd	2	inner
1728.3.e.u.1025.5		8	4.3	odd	2	inner
1728.3.e.u.1025.6		8	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 1728 = 2^{6} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1728.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+3.78194i q^{5} +0.953512 q^{7} +O(q^{10})\)	\(q+3.78194i q^{5} +0.953512 q^{7} +11.6969i q^{11} +8.69694 q^{13} +4.73545i q^{17} +29.2699 q^{19} -2.69694i q^{23} +10.6969 q^{25} -21.7060i q^{29} -22.5953 q^{31} +3.60612i q^{35} -24.0908 q^{37} -19.7990i q^{41} +49.0047 q^{43} +59.3939i q^{47} -48.0908 q^{49} +48.1796i q^{53} -44.2371 q^{55} +12.6061i q^{59} +4.00000 q^{61} +32.8913i q^{65} +111.358 q^{67} +56.6969i q^{71} +35.6969 q^{73} +11.1532i q^{77} -144.442 q^{79} +101.697i q^{83} -17.9092 q^{85} +19.0060i q^{89} +8.29263 q^{91} +110.697i q^{95} +27.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q - 48 q^{13} - 32 q^{25} + 160 q^{37} - 32 q^{49} + 32 q^{61} + 168 q^{73} - 496 q^{85} + 216 q^{97}+O(q^{100}) \) 8 * q - 48 * q^13 - 32 * q^25 + 160 * q^37 - 32 * q^49 + 32 * q^61 + 168 * q^73 - 496 * q^85 + 216 * q^97

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