LMFDB - Embedded newform 1728.3.e.s.1025.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1728,3,Mod(1025,1728)] 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("1728.1025"); S:= CuspForms(chi, 3); N := Newforms(S);

sage: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")

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:traces = [4,0,0,0,0,0,12,0,0,0,0,0,8] 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:	$47.0845896815$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{8})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 216)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1025.2
Root			$0.707107 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	1728.1025
Dual form			1728.3.e.s.1025.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-0.171573i q^{5} -5.48528 q^{7} -3.34315i q^{11} -14.9706 q^{13} +17.3137i q^{17} +24.9706 q^{19} +40.2843i q^{23} +24.9706 q^{25} -42.0000i q^{29} -7.48528 q^{31} +0.941125i q^{35} +19.0294 q^{37} -38.9117i q^{41} -30.9706 q^{43} +8.05887i q^{47} -18.9117 q^{49} -99.0833i q^{53} -0.573593 q^{55} -97.1960i q^{59} +15.8823 q^{61} +2.56854i q^{65} -74.9706 q^{67} -46.2843i q^{71} -121.912 q^{73} +18.3381i q^{77} +107.941 q^{79} +29.4020i q^{83} +2.97056 q^{85} -73.0294i q^{89} +82.1177 q^{91} -4.28427i q^{95} +78.8823 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 12 q^{7} + 8 q^{13} + 32 q^{19} + 32 q^{25} + 4 q^{31} + 144 q^{37} - 56 q^{43} + 128 q^{49} - 172 q^{55} - 208 q^{61} - 232 q^{67} - 284 q^{73} + 296 q^{79} - 56 q^{85} + 600 q^{91} + 44 q^{97}+O(q^{100}) $ 4 * q + 12 * q^7 + 8 * q^13 + 32 * q^19 + 32 * q^25 + 4 * q^31 + 144 * q^37 - 56 * q^43 + 128 * q^49 - 172 * q^55 - 208 * q^61 - 232 * q^67 - 284 * q^73 + 296 * q^79 - 56 * q^85 + 600 * q^91 + 44 * q^97

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
$5$	− 0.171573i	− 0.0343146i	−0.999853	−	0.0171573i	$-0.994538\pi$
$5$	− 0.171573i	− 0.0343146i	0.999853	−	0.0171573i	$-0.00546160\pi$
$6$	0	0
$7$	−5.48528	−0.783612	−0.391806	−	0.920048i	$-0.628149\pi$
$7$	−5.48528	−0.783612	−0.391806	+	0.920048i	$0.628149\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 3.34315i	− 0.303922i	−0.988386	−	0.151961i	$-0.951441\pi$
$11$	− 3.34315i	− 0.303922i	0.988386	−	0.151961i	$-0.0485589\pi$
$12$	0	0
$13$	−14.9706	−1.15158	−0.575791	−	0.817597i	$-0.695306\pi$
$13$	−14.9706	−1.15158	−0.575791	+	0.817597i	$0.695306\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	17.3137i	1.01845i	0.860632	+	0.509227i	$0.170069\pi$
$17$	17.3137i	1.01845i	−0.860632	+	0.509227i	$0.829931\pi$
$18$	0	0
$19$	24.9706	1.31424	0.657120	−	0.753786i	$-0.271774\pi$
$19$	24.9706	1.31424	0.657120	+	0.753786i	$0.271774\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	40.2843i	1.75149i	0.482774	+	0.875745i	$0.339629\pi$
$23$	40.2843i	1.75149i	−0.482774	+	0.875745i	$0.660371\pi$
$24$	0	0
$25$	24.9706	0.998823
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 42.0000i	− 1.44828i	−0.689655	−	0.724138i	$-0.742238\pi$
$29$	− 42.0000i	− 1.44828i	0.689655	−	0.724138i	$-0.257762\pi$
$30$	0	0
$31$	−7.48528	−0.241461	−0.120730	−	0.992685i	$-0.538524\pi$
$31$	−7.48528	−0.241461	−0.120730	+	0.992685i	$0.538524\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0.941125i	0.0268893i
$36$	0	0
$37$	19.0294	0.514309	0.257155	−	0.966370i	$-0.417215\pi$
$37$	19.0294	0.514309	0.257155	+	0.966370i	$0.417215\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 38.9117i	− 0.949066i	−0.880238	−	0.474533i	$-0.842617\pi$
$41$	− 38.9117i	− 0.949066i	0.880238	−	0.474533i	$-0.157383\pi$
$42$	0	0
$43$	−30.9706	−0.720246	−0.360123	−	0.932905i	$-0.617265\pi$
$43$	−30.9706	−0.720246	−0.360123	+	0.932905i	$0.617265\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	8.05887i	0.171465i	0.996318	+	0.0857327i	$0.0273231\pi$
$47$	8.05887i	0.171465i	−0.996318	+	0.0857327i	$0.972677\pi$
$48$	0	0
$49$	−18.9117	−0.385953
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 99.0833i	− 1.86950i	−0.355312	−	0.934748i	$-0.615625\pi$
$53$	− 99.0833i	− 1.86950i	0.355312	−	0.934748i	$-0.384375\pi$
$54$	0	0
$55$	−0.573593	−0.0104290
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 97.1960i	− 1.64739i	−0.567034	−	0.823695i	$-0.691909\pi$
$59$	− 97.1960i	− 1.64739i	0.567034	−	0.823695i	$-0.308091\pi$
$60$	0	0
$61$	15.8823	0.260365	0.130182	−	0.991490i	$-0.458444\pi$
$61$	15.8823	0.260365	0.130182	+	0.991490i	$0.458444\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	2.56854i	0.0395160i
$66$	0	0
$67$	−74.9706	−1.11896	−0.559482	−	0.828843i	$-0.689000\pi$
$67$	−74.9706	−1.11896	−0.559482	+	0.828843i	$0.689000\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 46.2843i	− 0.651891i	−0.945389	−	0.325946i	$-0.894317\pi$
$71$	− 46.2843i	− 0.651891i	0.945389	−	0.325946i	$-0.105683\pi$
$72$	0	0
$73$	−121.912	−1.67002	−0.835012	−	0.550232i	$-0.814539\pi$
$73$	−121.912	−1.67002	−0.835012	+	0.550232i	$0.814539\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	18.3381i	0.238157i
$78$	0	0
$79$	107.941	1.36634	0.683172	−	0.730258i	$-0.260600\pi$
$79$	107.941	1.36634	0.683172	+	0.730258i	$0.260600\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	29.4020i	0.354241i	0.984189	+	0.177121i	$0.0566783\pi$
$83$	29.4020i	0.354241i	−0.984189	+	0.177121i	$0.943322\pi$
$84$	0	0
$85$	2.97056	0.0349478
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 73.0294i	− 0.820555i	−0.911961	−	0.410278i	$-0.865432\pi$
$89$	− 73.0294i	− 0.820555i	0.911961	−	0.410278i	$-0.134568\pi$
$90$	0	0
$91$	82.1177	0.902393
$92$	0	0
$93$	0	0
$94$	0	0
$95$	− 4.28427i	− 0.0450976i
$96$	0	0
$97$	78.8823	0.813219	0.406610	−	0.913602i	$-0.366711\pi$
$97$	78.8823	0.813219	0.406610	+	0.913602i	$0.366711\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 52.4558i	− 0.519365i	−0.965694	−	0.259682i	$-0.916382\pi$
$101$	− 52.4558i	− 0.519365i	0.965694	−	0.259682i	$-0.0836179\pi$
$102$	0	0
$103$	−0.0588745	−0.000571597 0	−0.000285799	−	1.00000i	$-0.500091\pi$
$103$	−0.0588745	−0.000571597 0	−0.000285799		1.00000i	$0.500091\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	70.8823i	0.662451i	0.943552	+	0.331225i	$0.107462\pi$
$107$	70.8823i	0.662451i	−0.943552	+	0.331225i	$0.892538\pi$
$108$	0	0
$109$	84.9706	0.779546	0.389773	−	0.920911i	$-0.372553\pi$
$109$	84.9706	0.779546	0.389773	+	0.920911i	$0.372553\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 175.029i	− 1.54893i	−0.632615	−	0.774467i	$-0.718018\pi$
$113$	− 175.029i	− 1.54893i	0.632615	−	0.774467i	$-0.281982\pi$
$114$	0	0
$115$	6.91169	0.0601016
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 94.9706i	− 0.798072i
$120$	0	0
$121$	109.823	0.907631
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 8.57359i	− 0.0685887i
$126$	0	0
$127$	−190.161	−1.49733	−0.748667	−	0.662946i	$-0.769306\pi$
$127$	−190.161	−1.49733	−0.748667	+	0.662946i	$0.769306\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	140.314i	1.07110i	0.844504	+	0.535549i	$0.179895\pi$
$131$	140.314i	1.07110i	−0.844504	+	0.535549i	$0.820105\pi$
$132$	0	0
$133$	−136.971	−1.02985
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 256.794i	− 1.87441i	−0.348781	−	0.937204i	$-0.613404\pi$
$137$	− 256.794i	− 1.87441i	0.348781	−	0.937204i	$-0.386596\pi$
$138$	0	0
$139$	−40.1177	−0.288617	−0.144308	−	0.989533i	$-0.546096\pi$
$139$	−40.1177	−0.288617	−0.144308	+	0.989533i	$0.546096\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	50.0488i	0.349991i
$144$	0	0
$145$	−7.20606	−0.0496970
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 223.534i	− 1.50023i	−0.661308	−	0.750114i	$-0.729998\pi$
$149$	− 223.534i	− 1.50023i	0.661308	−	0.750114i	$-0.270002\pi$
$150$	0	0
$151$	7.60303	0.0503512	0.0251756	−	0.999683i	$-0.491986\pi$
$151$	7.60303	0.0503512	0.0251756	+	0.999683i	$0.491986\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	1.28427i	0.00828562i
$156$	0	0
$157$	−177.823	−1.13263	−0.566316	−	0.824188i	$-0.691632\pi$
$157$	−177.823	−1.13263	−0.566316	+	0.824188i	$0.691632\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 220.971i	− 1.37249i
$162$	0	0
$163$	203.823	1.25045	0.625225	−	0.780444i	$-0.285007\pi$
$163$	203.823	1.25045	0.625225	+	0.780444i	$0.285007\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 195.941i	− 1.17330i	−0.809841	−	0.586650i	$-0.800446\pi$
$167$	− 195.941i	− 1.17330i	0.809841	−	0.586650i	$-0.199554\pi$
$168$	0	0
$169$	55.1177	0.326141
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 58.4558i	− 0.337895i	−0.985625	−	0.168948i	$-0.945963\pi$
$173$	− 58.4558i	− 0.337895i	0.985625	−	0.168948i	$-0.0540368\pi$
$174$	0	0
$175$	−136.971	−0.782689
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 34.8823i	− 0.194873i	−0.995242	−	0.0974365i	$-0.968936\pi$
$179$	− 34.8823i	− 0.194873i	0.995242	−	0.0974365i	$-0.0310643\pi$
$180$	0	0
$181$	62.0589	0.342867	0.171433	−	0.985196i	$-0.445160\pi$
$181$	62.0589	0.342867	0.171433	+	0.985196i	$0.445160\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 3.26494i	− 0.0176483i
$186$	0	0
$187$	57.8823	0.309531
$188$	0	0
$189$	0	0
$190$	0	0
$191$	186.853i	0.978287i	0.872203	+	0.489143i	$0.162691\pi$
$191$	186.853i	0.978287i	−0.872203	+	0.489143i	$0.837309\pi$
$192$	0	0
$193$	−85.8528	−0.444833	−0.222417	−	0.974952i	$-0.571395\pi$
$193$	−85.8528	−0.444833	−0.222417	+	0.974952i	$0.571395\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	84.1615i	0.427216i	0.976920	+	0.213608i	$0.0685215\pi$
$197$	84.1615i	0.427216i	−0.976920	+	0.213608i	$0.931479\pi$
$198$	0	0
$199$	204.103	1.02564	0.512821	−	0.858496i	$-0.328601\pi$
$199$	204.103	1.02564	0.512821	+	0.858496i	$0.328601\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	230.382i	1.13489i
$204$	0	0
$205$	−6.67619	−0.0325668
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 83.4802i	− 0.399427i
$210$	0	0
$211$	−276.971	−1.31266	−0.656328	−	0.754475i	$-0.727891\pi$
$211$	−276.971	−1.31266	−0.656328	+	0.754475i	$0.727891\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	5.31371i	0.0247149i
$216$	0	0
$217$	41.0589	0.189211
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 259.196i	− 1.17283i
$222$	0	0
$223$	209.765	0.940648	0.470324	−	0.882494i	$-0.344137\pi$
$223$	209.765	0.940648	0.470324	+	0.882494i	$0.344137\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	56.0589i	0.246955i	0.992347	+	0.123478i	$0.0394047\pi$
$227$	56.0589i	0.246955i	−0.992347	+	0.123478i	$0.960595\pi$
$228$	0	0
$229$	273.529	1.19445	0.597225	−	0.802074i	$-0.296270\pi$
$229$	273.529	1.19445	0.597225	+	0.802074i	$0.296270\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 138.167i	− 0.592989i	−0.955035	−	0.296495i	$-0.904182\pi$
$233$	− 138.167i	− 0.592989i	0.955035	−	0.296495i	$-0.0958177\pi$
$234$	0	0
$235$	1.38268	0.00588376
$236$	0	0
$237$	0	0
$238$	0	0
$239$	143.833i	0.601814i	0.953654	+	0.300907i	$0.0972893\pi$
$239$	143.833i	0.601814i	−0.953654	+	0.300907i	$0.902711\pi$
$240$	0	0
$241$	−207.588	−0.861360	−0.430680	−	0.902505i	$-0.641726\pi$
$241$	−207.588	−0.861360	−0.430680	+	0.902505i	$0.641726\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	3.24473i	0.0132438i
$246$	0	0
$247$	−373.823	−1.51345
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 452.039i	− 1.80095i	−0.434907	−	0.900475i	$-0.643219\pi$
$251$	− 452.039i	− 1.80095i	0.434907	−	0.900475i	$-0.356781\pi$
$252$	0	0
$253$	134.676	0.532317
$254$	0	0
$255$	0	0
$256$	0	0
$257$	314.902i	1.22530i	0.790355	+	0.612649i	$0.209896\pi$
$257$	314.902i	1.22530i	−0.790355	+	0.612649i	$0.790104\pi$
$258$	0	0
$259$	−104.382	−0.403019
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 45.7746i	− 0.174048i	−0.996206	−	0.0870240i	$-0.972264\pi$
$263$	− 45.7746i	− 0.174048i	0.996206	−	0.0870240i	$-0.0277357\pi$
$264$	0	0
$265$	−17.0000	−0.0641509
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 352.960i	− 1.31212i	−0.754708	−	0.656060i	$-0.772222\pi$
$269$	− 352.960i	− 1.31212i	0.754708	−	0.656060i	$-0.227778\pi$
$270$	0	0
$271$	480.397	1.77268	0.886341	−	0.463033i	$-0.153239\pi$
$271$	480.397	1.77268	0.886341	+	0.463033i	$0.153239\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 83.4802i	− 0.303564i
$276$	0	0
$277$	−33.8234	−0.122106	−0.0610530	−	0.998135i	$-0.519446\pi$
$277$	−33.8234	−0.122106	−0.0610530	+	0.998135i	$0.519446\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	236.725i	0.842438i	0.906959	+	0.421219i	$0.138398\pi$
$281$	236.725i	0.842438i	−0.906959	+	0.421219i	$0.861602\pi$
$282$	0	0
$283$	−442.676	−1.56423	−0.782113	−	0.623136i	$-0.785858\pi$
$283$	−442.676	−1.56423	−0.782113	+	0.623136i	$0.785858\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	213.442i	0.743699i
$288$	0	0
$289$	−10.7645	−0.0372474
$290$	0	0
$291$	0	0
$292$	0	0
$293$	27.9411i	0.0953622i	0.998863	+	0.0476811i	$0.0151831\pi$
$293$	27.9411i	0.0953622i	−0.998863	+	0.0476811i	$0.984817\pi$
$294$	0	0
$295$	−16.6762	−0.0565295
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 603.078i	− 2.01698i
$300$	0	0
$301$	169.882	0.564393
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 2.72496i	− 0.00893431i
$306$	0	0
$307$	387.235	1.26135	0.630675	−	0.776047i	$-0.282778\pi$
$307$	387.235	1.26135	0.630675	+	0.776047i	$0.282778\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	− 292.617i	− 0.940892i	−0.882429	−	0.470446i	$-0.844093\pi$
$311$	− 292.617i	− 0.940892i	0.882429	−	0.470446i	$-0.155907\pi$
$312$	0	0
$313$	−269.558	−0.861209	−0.430605	−	0.902541i	$-0.641700\pi$
$313$	−269.558	−0.861209	−0.430605	+	0.902541i	$0.641700\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 129.250i	− 0.407728i	−0.978999	−	0.203864i	$-0.934650\pi$
$317$	− 129.250i	− 0.407728i	0.978999	−	0.203864i	$-0.0653500\pi$
$318$	0	0
$319$	−140.412	−0.440163
$320$	0	0
$321$	0	0
$322$	0	0
$323$	432.333i	1.33849i
$324$	0	0
$325$	−373.823	−1.15023
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 44.2052i	− 0.134362i
$330$	0	0
$331$	482.500	1.45770	0.728851	−	0.684672i	$-0.240055\pi$
$331$	482.500	1.45770	0.728851	+	0.684672i	$0.240055\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	12.8629i	0.0383968i
$336$	0	0
$337$	447.235	1.32711	0.663553	−	0.748129i	$-0.269048\pi$
$337$	447.235	1.32711	0.663553	+	0.748129i	$0.269048\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	25.0244i	0.0733853i
$342$	0	0
$343$	372.515	1.08605
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 573.500i	− 1.65274i	−0.563130	−	0.826368i	$-0.690403\pi$
$347$	− 573.500i	− 1.65274i	0.563130	−	0.826368i	$-0.309597\pi$
$348$	0	0
$349$	74.9117	0.214647	0.107323	−	0.994224i	$-0.465772\pi$
$349$	74.9117	0.214647	0.107323	+	0.994224i	$0.465772\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 195.245i	− 0.553101i	−0.960999	−	0.276551i	$-0.910809\pi$
$353$	− 195.245i	− 0.553101i	0.960999	−	0.276551i	$-0.0891913\pi$
$354$	0	0
$355$	−7.94113	−0.0223694
$356$	0	0
$357$	0	0
$358$	0	0
$359$	15.9411i	0.0444042i	0.999754	+	0.0222021i	$0.00706774\pi$
$359$	15.9411i	0.0444042i	−0.999754	+	0.0222021i	$0.992932\pi$
$360$	0	0
$361$	262.529	0.727227
$362$	0	0
$363$	0	0
$364$	0	0
$365$	20.9167i	0.0573061i
$366$	0	0
$367$	−511.250	−1.39305	−0.696526	−	0.717532i	$-0.745272\pi$
$367$	−511.250	−1.39305	−0.696526	+	0.717532i	$0.745272\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	543.500i	1.46496i
$372$	0	0
$373$	−214.382	−0.574750	−0.287375	−	0.957818i	$-0.592783\pi$
$373$	−214.382	−0.574750	−0.287375	+	0.957818i	$0.592783\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	628.764i	1.66781i
$378$	0	0
$379$	−160.000	−0.422164	−0.211082	−	0.977468i	$-0.567699\pi$
$379$	−160.000	−0.422164	−0.211082	+	0.977468i	$0.567699\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	620.205i	1.61933i	0.586889	+	0.809667i	$0.300352\pi$
$383$	620.205i	1.61933i	−0.586889	+	0.809667i	$0.699648\pi$
$384$	0	0
$385$	3.14632	0.00817226
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 258.848i	− 0.665418i	−0.943029	−	0.332709i	$-0.892037\pi$
$389$	− 258.848i	− 0.665418i	0.943029	−	0.332709i	$-0.107963\pi$
$390$	0	0
$391$	−697.470	−1.78381
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 18.5198i	− 0.0468855i
$396$	0	0
$397$	−592.558	−1.49259	−0.746295	−	0.665615i	$-0.768169\pi$
$397$	−592.558	−1.49259	−0.746295	+	0.665615i	$0.768169\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 343.529i	− 0.856681i	−0.903617	−	0.428340i	$-0.859098\pi$
$401$	− 343.529i	− 0.856681i	0.903617	−	0.428340i	$-0.140902\pi$
$402$	0	0
$403$	112.059	0.278062
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 63.6182i	− 0.156310i
$408$	0	0
$409$	−570.765	−1.39551	−0.697756	−	0.716335i	$-0.745818\pi$
$409$	−570.765	−1.39551	−0.697756	+	0.716335i	$0.745818\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	533.147i	1.29091i
$414$	0	0
$415$	5.04459	0.0121556
$416$	0	0
$417$	0	0
$418$	0	0
$419$	249.765i	0.596097i	0.954551	+	0.298048i	$0.0963357\pi$
$419$	249.765i	0.596097i	−0.954551	+	0.298048i	$0.903664\pi$
$420$	0	0
$421$	177.823	0.422383	0.211192	−	0.977445i	$-0.432266\pi$
$421$	177.823	0.422383	0.211192	+	0.977445i	$0.432266\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	432.333i	1.01725i
$426$	0	0
$427$	−87.1186	−0.204025
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 252.843i	− 0.586642i	−0.956014	−	0.293321i	$-0.905240\pi$
$431$	− 252.843i	− 0.586642i	0.956014	−	0.293321i	$-0.0947605\pi$
$432$	0	0
$433$	469.177	1.08355	0.541774	−	0.840524i	$-0.317753\pi$
$433$	469.177	1.08355	0.541774	+	0.840524i	$0.317753\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	1005.92i	2.30188i
$438$	0	0
$439$	400.103	0.911395	0.455698	−	0.890135i	$-0.349390\pi$
$439$	400.103	0.911395	0.455698	+	0.890135i	$0.349390\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	406.431i	0.917451i	0.888578	+	0.458725i	$0.151694\pi$
$443$	406.431i	0.917451i	−0.888578	+	0.458725i	$0.848306\pi$
$444$	0	0
$445$	−12.5299	−0.0281570
$446$	0	0
$447$	0	0
$448$	0	0
$449$	531.245i	1.18317i	0.806241	+	0.591587i	$0.201498\pi$
$449$	531.245i	1.18317i	−0.806241	+	0.591587i	$0.798502\pi$
$450$	0	0
$451$	−130.087	−0.288442
$452$	0	0
$453$	0	0
$454$	0	0
$455$	− 14.0892i	− 0.0309652i
$456$	0	0
$457$	727.705	1.59235	0.796176	−	0.605065i	$-0.206853\pi$
$457$	727.705	1.59235	0.796176	+	0.605065i	$0.206853\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	735.416i	1.59526i	0.603145	+	0.797632i	$0.293914\pi$
$461$	735.416i	1.59526i	−0.603145	+	0.797632i	$0.706086\pi$
$462$	0	0
$463$	−567.721	−1.22618	−0.613089	−	0.790014i	$-0.710073\pi$
$463$	−567.721	−1.22618	−0.613089	+	0.790014i	$0.710073\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	472.029i	1.01077i	0.862894	+	0.505385i	$0.168649\pi$
$467$	472.029i	1.01077i	−0.862894	+	0.505385i	$0.831351\pi$
$468$	0	0
$469$	411.235	0.876833
$470$	0	0
$471$	0	0
$472$	0	0
$473$	103.539i	0.218899i
$474$	0	0
$475$	623.529	1.31269
$476$	0	0
$477$	0	0
$478$	0	0
$479$	213.098i	0.444882i	0.974946	+	0.222441i	$0.0714025\pi$
$479$	213.098i	0.444882i	−0.974946	+	0.222441i	$0.928598\pi$
$480$	0	0
$481$	−284.881	−0.592269
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 13.5341i	− 0.0279053i
$486$	0	0
$487$	726.118	1.49100	0.745501	−	0.666505i	$-0.232210\pi$
$487$	726.118	1.49100	0.745501	+	0.666505i	$0.232210\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 366.235i	− 0.745895i	−0.927852	−	0.372948i	$-0.878347\pi$
$491$	− 366.235i	− 0.745895i	0.927852	−	0.372948i	$-0.121653\pi$
$492$	0	0
$493$	727.176	1.47500
$494$	0	0
$495$	0	0
$496$	0	0
$497$	253.882i	0.510829i
$498$	0	0
$499$	−634.617	−1.27178	−0.635889	−	0.771780i	$-0.719366\pi$
$499$	−634.617	−1.27178	−0.635889	+	0.771780i	$0.719366\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 932.735i	− 1.85434i	−0.374636	−	0.927172i	$-0.622232\pi$
$503$	− 932.735i	− 1.85434i	0.374636	−	0.927172i	$-0.377768\pi$
$504$	0	0
$505$	−9.00000	−0.0178218
$506$	0	0
$507$	0	0
$508$	0	0
$509$	578.543i	1.13663i	0.822812	+	0.568314i	$0.192404\pi$
$509$	578.543i	1.13663i	−0.822812	+	0.568314i	$0.807596\pi$
$510$	0	0
$511$	668.720	1.30865
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0.0101013i	0 1.96141e-5i
$516$	0	0
$517$	26.9420	0.0521122
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 585.744i	− 1.12427i	−0.827046	−	0.562135i	$-0.809980\pi$
$521$	− 585.744i	− 1.12427i	0.827046	−	0.562135i	$-0.190020\pi$
$522$	0	0
$523$	−197.411	−0.377459	−0.188730	−	0.982029i	$-0.560437\pi$
$523$	−197.411	−0.377459	−0.188730	+	0.982029i	$0.560437\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 129.598i	− 0.245916i
$528$	0	0
$529$	−1093.82	−2.06772
$530$	0	0
$531$	0	0
$532$	0	0
$533$	582.530i	1.09293i
$534$	0	0
$535$	12.1615	0.0227317
$536$	0	0
$537$	0	0
$538$	0	0
$539$	63.2245i	0.117300i
$540$	0	0
$541$	−587.294	−1.08557	−0.542785	−	0.839872i	$-0.682630\pi$
$541$	−587.294	−1.08557	−0.542785	+	0.839872i	$0.682630\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 14.5786i	− 0.0267498i
$546$	0	0
$547$	394.235	0.720723	0.360362	−	0.932813i	$-0.382653\pi$
$547$	394.235	0.720723	0.360362	+	0.932813i	$0.382653\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	− 1048.76i	− 1.90338i
$552$	0	0
$553$	−592.087	−1.07068
$554$	0	0
$555$	0	0
$556$	0	0
$557$	18.5349i	0.0332763i	0.999862	+	0.0166382i	$0.00529634\pi$
$557$	18.5349i	0.0332763i	−0.999862	+	0.0166382i	$0.994704\pi$
$558$	0	0
$559$	463.647	0.829422
$560$	0	0
$561$	0	0
$562$	0	0
$563$	622.892i	1.10638i	0.833055	+	0.553190i	$0.186590\pi$
$563$	622.892i	1.10638i	−0.833055	+	0.553190i	$0.813410\pi$
$564$	0	0
$565$	−30.0303	−0.0531510
$566$	0	0
$567$	0	0
$568$	0	0
$569$	157.393i	0.276613i	0.990389	+	0.138306i	$0.0441659\pi$
$569$	157.393i	0.276613i	−0.990389	+	0.138306i	$0.955834\pi$
$570$	0	0
$571$	−500.705	−0.876891	−0.438445	−	0.898758i	$-0.644471\pi$
$571$	−500.705	−0.876891	−0.438445	+	0.898758i	$0.644471\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	1005.92i	1.74943i
$576$	0	0
$577$	−223.588	−0.387501	−0.193750	−	0.981051i	$-0.562065\pi$
$577$	−223.588	−0.387501	−0.193750	+	0.981051i	$0.562065\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 161.278i	− 0.277588i
$582$	0	0
$583$	−331.250	−0.568181
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 451.254i	− 0.768746i	−0.923178	−	0.384373i	$-0.874418\pi$
$587$	− 451.254i	− 0.768746i	0.923178	−	0.384373i	$-0.125582\pi$
$588$	0	0
$589$	−186.912	−0.317337
$590$	0	0
$591$	0	0
$592$	0	0
$593$	369.245i	0.622672i	0.950300	+	0.311336i	$0.100777\pi$
$593$	369.245i	0.622672i	−0.950300	+	0.311336i	$0.899223\pi$
$594$	0	0
$595$	−16.2944	−0.0273855
$596$	0	0
$597$	0	0
$598$	0	0
$599$	264.530i	0.441619i	0.975317	+	0.220810i	$0.0708699\pi$
$599$	264.530i	0.441619i	−0.975317	+	0.220810i	$0.929130\pi$
$600$	0	0
$601$	461.971	0.768670	0.384335	−	0.923194i	$-0.374431\pi$
$601$	461.971	0.768670	0.384335	+	0.923194i	$0.374431\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 18.8427i	− 0.0311450i
$606$	0	0
$607$	−314.000	−0.517298	−0.258649	−	0.965971i	$-0.583277\pi$
$607$	−314.000	−0.517298	−0.258649	+	0.965971i	$0.583277\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 120.646i	− 0.197456i
$612$	0	0
$613$	617.882	1.00796	0.503982	−	0.863714i	$-0.331868\pi$
$613$	617.882	1.00796	0.503982	+	0.863714i	$0.331868\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	171.618i	0.278149i	0.990282	+	0.139075i	$0.0444128\pi$
$617$	171.618i	0.278149i	−0.990282	+	0.139075i	$0.955587\pi$
$618$	0	0
$619$	−847.117	−1.36852	−0.684262	−	0.729236i	$-0.739876\pi$
$619$	−847.117	−1.36852	−0.684262	+	0.729236i	$0.739876\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	400.587i	0.642997i
$624$	0	0
$625$	622.793	0.996469
$626$	0	0
$627$	0	0
$628$	0	0
$629$	329.470i	0.523800i
$630$	0	0
$631$	−281.368	−0.445907	−0.222954	−	0.974829i	$-0.571570\pi$
$631$	−281.368	−0.445907	−0.222954	+	0.974829i	$0.571570\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	32.6266i	0.0513804i
$636$	0	0
$637$	283.119	0.444456
$638$	0	0
$639$	0	0
$640$	0	0
$641$	635.127i	0.990838i	0.868654	+	0.495419i	$0.164985\pi$
$641$	635.127i	0.990838i	−0.868654	+	0.495419i	$0.835015\pi$
$642$	0	0
$643$	−498.177	−0.774769	−0.387385	−	0.921918i	$-0.626621\pi$
$643$	−498.177	−0.774769	−0.387385	+	0.921918i	$0.626621\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 904.784i	− 1.39843i	−0.714912	−	0.699215i	$-0.753533\pi$
$647$	− 904.784i	− 1.39843i	0.714912	−	0.699215i	$-0.246467\pi$
$648$	0	0
$649$	−324.940	−0.500678
$650$	0	0
$651$	0	0
$652$	0	0
$653$	302.897i	0.463854i	0.972733	+	0.231927i	$0.0745030\pi$
$653$	302.897i	0.463854i	−0.972733	+	0.231927i	$0.925497\pi$
$654$	0	0
$655$	24.0740	0.0367542
$656$	0	0
$657$	0	0
$658$	0	0
$659$	17.2355i	0.0261540i	0.999914	+	0.0130770i	$0.00416266\pi$
$659$	17.2355i	0.0261540i	−0.999914	+	0.0130770i	$0.995837\pi$
$660$	0	0
$661$	152.971	0.231423	0.115711	−	0.993283i	$-0.463085\pi$
$661$	152.971	0.231423	0.115711	+	0.993283i	$0.463085\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	23.5004i	0.0353390i
$666$	0	0
$667$	1691.94	2.53664
$668$	0	0
$669$	0	0
$670$	0	0
$671$	− 53.0967i	− 0.0791307i
$672$	0	0
$673$	578.176	0.859102	0.429551	−	0.903043i	$-0.358672\pi$
$673$	578.176	0.859102	0.429551	+	0.903043i	$0.358672\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	658.431i	0.972571i	0.873800	+	0.486286i	$0.161649\pi$
$677$	658.431i	0.972571i	−0.873800	+	0.486286i	$0.838351\pi$
$678$	0	0
$679$	−432.691	−0.637248
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 1011.25i	− 1.48061i	−0.672273	−	0.740304i	$-0.734682\pi$
$683$	− 1011.25i	− 1.48061i	0.672273	−	0.740304i	$-0.265318\pi$
$684$	0	0
$685$	−44.0589	−0.0643195
$686$	0	0
$687$	0	0
$688$	0	0
$689$	1483.33i	2.15288i
$690$	0	0
$691$	414.940	0.600492	0.300246	−	0.953862i	$-0.402931\pi$
$691$	414.940	0.600492	0.300246	+	0.953862i	$0.402931\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	6.88312i	0.00990377i
$696$	0	0
$697$	673.706	0.966579
$698$	0	0
$699$	0	0
$700$	0	0
$701$	641.955i	0.915770i	0.889012	+	0.457885i	$0.151393\pi$
$701$	641.955i	0.915770i	−0.889012	+	0.457885i	$0.848607\pi$
$702$	0	0
$703$	475.176	0.675926
$704$	0	0
$705$	0	0
$706$	0	0
$707$	287.735i	0.406980i
$708$	0	0
$709$	−1275.18	−1.79856	−0.899278	−	0.437378i	$-0.855907\pi$
$709$	−1275.18	−1.79856	−0.899278	+	0.437378i	$0.855907\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 301.539i	− 0.422916i
$714$	0	0
$715$	8.58701	0.0120098
$716$	0	0
$717$	0	0
$718$	0	0
$719$	1020.67i	1.41956i	0.704422	+	0.709782i	$0.251206\pi$
$719$	1020.67i	1.41956i	−0.704422	+	0.709782i	$0.748794\pi$
$720$	0	0
$721$	0.322943	0.000447910 0
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 1048.76i	− 1.44657i
$726$	0	0
$727$	−210.484	−0.289525	−0.144762	−	0.989466i	$-0.546242\pi$
$727$	−210.484	−0.289525	−0.144762	+	0.989466i	$0.546242\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 536.215i	− 0.733537i
$732$	0	0
$733$	230.794	0.314862	0.157431	−	0.987530i	$-0.449679\pi$
$733$	230.794	0.314862	0.157431	+	0.987530i	$0.449679\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	250.638i	0.340078i
$738$	0	0
$739$	638.353	0.863807	0.431903	−	0.901920i	$-0.357842\pi$
$739$	638.353	0.863807	0.431903	+	0.901920i	$0.357842\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 1393.01i	− 1.87484i	−0.348195	−	0.937422i	$-0.613205\pi$
$743$	− 1393.01i	− 1.87484i	0.348195	−	0.937422i	$-0.386795\pi$
$744$	0	0
$745$	−38.3524	−0.0514797
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 388.809i	− 0.519104i
$750$	0	0
$751$	−574.072	−0.764411	−0.382205	−	0.924077i	$-0.624835\pi$
$751$	−574.072	−0.764411	−0.382205	+	0.924077i	$0.624835\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 1.30447i	− 0.00172778i
$756$	0	0
$757$	−770.999	−1.01849	−0.509246	−	0.860621i	$-0.670076\pi$
$757$	−770.999	−1.01849	−0.509246	+	0.860621i	$0.670076\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	600.666i	0.789312i	0.918829	+	0.394656i	$0.129136\pi$
$761$	600.666i	0.789312i	−0.918829	+	0.394656i	$0.870864\pi$
$762$	0	0
$763$	−466.087	−0.610862
$764$	0	0
$765$	0	0
$766$	0	0
$767$	1455.08i	1.89710i
$768$	0	0
$769$	691.029	0.898607	0.449303	−	0.893379i	$-0.351672\pi$
$769$	691.029	0.898607	0.449303	+	0.893379i	$0.351672\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	394.118i	0.509855i	0.966960	+	0.254927i	$0.0820516\pi$
$773$	394.118i	0.509855i	−0.966960	+	0.254927i	$0.917948\pi$
$774$	0	0
$775$	−186.912	−0.241176
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 971.647i	− 1.24730i
$780$	0	0
$781$	−154.735	−0.198124
$782$	0	0
$783$	0	0
$784$	0	0
$785$	30.5097i	0.0388658i
$786$	0	0
$787$	343.411	0.436355	0.218177	−	0.975909i	$-0.429989\pi$
$787$	343.411	0.436355	0.218177	+	0.975909i	$0.429989\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	960.086i	1.21376i
$792$	0	0
$793$	−237.766	−0.299831
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 537.729i	− 0.674692i	−0.941381	−	0.337346i	$-0.890471\pi$
$797$	− 537.729i	− 0.674692i	0.941381	−	0.337346i	$-0.109529\pi$
$798$	0	0
$799$	−139.529	−0.174630
$800$	0	0
$801$	0	0
$802$	0	0
$803$	407.569i	0.507557i
$804$	0	0
$805$	−37.9126	−0.0470963
$806$	0	0
$807$	0	0
$808$	0	0
$809$	1172.23i	1.44898i	0.689285	+	0.724490i	$0.257925\pi$
$809$	1172.23i	1.44898i	−0.689285	+	0.724490i	$0.742075\pi$
$810$	0	0
$811$	31.2649	0.0385511	0.0192755	−	0.999814i	$-0.493864\pi$
$811$	31.2649	0.0385511	0.0192755	+	0.999814i	$0.493864\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	− 34.9706i	− 0.0429087i
$816$	0	0
$817$	−773.352	−0.946576
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 969.765i	− 1.18120i	−0.806965	−	0.590600i	$-0.798891\pi$
$821$	− 969.765i	− 1.18120i	0.806965	−	0.590600i	$-0.201109\pi$
$822$	0	0
$823$	275.014	0.334161	0.167080	−	0.985943i	$-0.446566\pi$
$823$	275.014	0.334161	0.167080	+	0.985943i	$0.446566\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	38.9016i	0.0470394i	0.999723	+	0.0235197i	$0.00748724\pi$
$827$	38.9016i	0.0470394i	−0.999723	+	0.0235197i	$0.992513\pi$
$828$	0	0
$829$	1197.53	1.44455	0.722273	−	0.691608i	$-0.243097\pi$
$829$	1197.53	1.44455	0.722273	+	0.691608i	$0.243097\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 327.431i	− 0.393075i
$834$	0	0
$835$	−33.6182	−0.0402613
$836$	0	0
$837$	0	0
$838$	0	0
$839$	753.558i	0.898162i	0.893491	+	0.449081i	$0.148248\pi$
$839$	753.558i	0.898162i	−0.893491	+	0.449081i	$0.851752\pi$
$840$	0	0
$841$	−923.000	−1.09750
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 9.45671i	− 0.0111914i
$846$	0	0
$847$	−602.412	−0.711230
$848$	0	0
$849$	0	0
$850$	0	0
$851$	766.587i	0.900807i
$852$	0	0
$853$	−912.940	−1.07027	−0.535135	−	0.844767i	$-0.679739\pi$
$853$	−912.940	−1.07027	−0.535135	+	0.844767i	$0.679739\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 756.469i	− 0.882695i	−0.897336	−	0.441347i	$-0.854501\pi$
$857$	− 756.469i	− 0.882695i	0.897336	−	0.441347i	$-0.145499\pi$
$858$	0	0
$859$	−1177.65	−1.37095	−0.685474	−	0.728097i	$-0.740405\pi$
$859$	−1177.65	−1.37095	−0.685474	+	0.728097i	$0.740405\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	397.373i	0.460455i	0.973137	+	0.230227i	$0.0739470\pi$
$863$	397.373i	0.460455i	−0.973137	+	0.230227i	$0.926053\pi$
$864$	0	0
$865$	−10.0294	−0.0115947
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 360.863i	− 0.415262i
$870$	0	0
$871$	1122.35	1.28858
$872$	0	0
$873$	0	0
$874$	0	0
$875$	47.0286i	0.0537469i
$876$	0	0
$877$	−1052.41	−1.20001	−0.600007	−	0.799995i	$-0.704836\pi$
$877$	−1052.41	−1.20001	−0.600007	+	0.799995i	$0.704836\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 1467.92i	− 1.66620i	−0.553123	−	0.833099i	$-0.686564\pi$
$881$	− 1467.92i	− 1.66620i	0.553123	−	0.833099i	$-0.313436\pi$
$882$	0	0
$883$	207.736	0.235262	0.117631	−	0.993057i	$-0.462470\pi$
$883$	207.736	0.235262	0.117631	+	0.993057i	$0.462470\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	297.951i	0.335909i	0.985795	+	0.167954i	$0.0537162\pi$
$887$	297.951i	0.335909i	−0.985795	+	0.167954i	$0.946284\pi$
$888$	0	0
$889$	1043.09	1.17333
$890$	0	0
$891$	0	0
$892$	0	0
$893$	201.235i	0.225347i
$894$	0	0
$895$	−5.98485	−0.00668698
$896$	0	0
$897$	0	0
$898$	0	0
$899$	314.382i	0.349702i
$900$	0	0
$901$	1715.50	1.90399
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 10.6476i	− 0.0117653i
$906$	0	0
$907$	−279.675	−0.308352	−0.154176	−	0.988043i	$-0.549272\pi$
$907$	−279.675	−0.308352	−0.154176	+	0.988043i	$0.549272\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	− 1395.21i	− 1.53152i	−0.643127	−	0.765760i	$-0.722363\pi$
$911$	− 1395.21i	− 1.53152i	0.643127	−	0.765760i	$-0.277637\pi$
$912$	0	0
$913$	98.2952	0.107662
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 769.660i	− 0.839324i
$918$	0	0
$919$	−619.219	−0.673797	−0.336899	−	0.941541i	$-0.609378\pi$
$919$	−619.219	−0.673797	−0.336899	+	0.941541i	$0.609378\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	692.902i	0.750706i
$924$	0	0
$925$	475.176	0.513704
$926$	0	0
$927$	0	0
$928$	0	0
$929$	339.431i	0.365373i	0.983171	+	0.182686i	$0.0584793\pi$
$929$	339.431i	0.365373i	−0.983171	+	0.182686i	$0.941521\pi$
$930$	0	0
$931$	−472.235	−0.507235
$932$	0	0
$933$	0	0
$934$	0	0
$935$	− 9.93102i	− 0.0106214i
$936$	0	0
$937$	−782.647	−0.835269	−0.417634	−	0.908615i	$-0.637141\pi$
$937$	−782.647	−0.835269	−0.417634	+	0.908615i	$0.637141\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	135.103i	0.143574i	0.997420	+	0.0717872i	$0.0228702\pi$
$941$	135.103i	0.143574i	−0.997420	+	0.0717872i	$0.977130\pi$
$942$	0	0
$943$	1567.53	1.66228
$944$	0	0
$945$	0	0
$946$	0	0
$947$	86.8234i	0.0916826i	0.998949	+	0.0458413i	$0.0145968\pi$
$947$	86.8234i	0.0916826i	−0.998949	+	0.0458413i	$0.985403\pi$
$948$	0	0
$949$	1825.09	1.92317
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 965.823i	− 1.01346i	−0.862106	−	0.506728i	$-0.830855\pi$
$953$	− 965.823i	− 1.01346i	0.862106	−	0.506728i	$-0.169145\pi$
$954$	0	0
$955$	32.0589	0.0335695
$956$	0	0
$957$	0	0
$958$	0	0
$959$	1408.59i	1.46881i
$960$	0	0
$961$	−904.971	−0.941697
$962$	0	0
$963$	0	0
$964$	0	0
$965$	14.7300i	0.0152643i
$966$	0	0
$967$	1274.07	1.31755	0.658776	−	0.752339i	$-0.271075\pi$
$967$	1274.07	1.31755	0.658776	+	0.752339i	$0.271075\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 764.087i	− 0.786907i	−0.919345	−	0.393453i	$-0.871280\pi$
$971$	− 764.087i	− 0.786907i	0.919345	−	0.393453i	$-0.128720\pi$
$972$	0	0
$973$	220.057	0.226164
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 713.087i	− 0.729874i	−0.931032	−	0.364937i	$-0.881091\pi$
$977$	− 713.087i	− 0.729874i	0.931032	−	0.364937i	$-0.118909\pi$
$978$	0	0
$979$	−244.148	−0.249385
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 788.412i	− 0.802047i	−0.916068	−	0.401023i	$-0.868655\pi$
$983$	− 788.412i	− 0.802047i	0.916068	−	0.401023i	$-0.131345\pi$
$984$	0	0
$985$	14.4398	0.0146597
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 1247.63i	− 1.26150i
$990$	0	0
$991$	−118.868	−0.119947	−0.0599737	−	0.998200i	$-0.519102\pi$
$991$	−118.868	−0.119947	−0.0599737	+	0.998200i	$0.519102\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 35.0185i	− 0.0351944i
$996$	0	0
$997$	−319.795	−0.320757	−0.160379	−	0.987056i	$-0.551271\pi$
$997$	−319.795	−0.320757	−0.160379	+	0.987056i	$0.551271\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1728.3.e.s.1025.2		4
3.2	odd	2	inner	1728.3.e.s.1025.3		4
4.3	odd	2		1728.3.e.p.1025.2		4
8.3	odd	2		216.3.e.c.161.3	yes	4
8.5	even	2		432.3.e.h.161.3		4
12.11	even	2		1728.3.e.p.1025.3		4
24.5	odd	2		432.3.e.h.161.2		4
24.11	even	2		216.3.e.c.161.2	✓	4
72.5	odd	6		1296.3.q.l.593.2		8
72.11	even	6		648.3.m.f.377.3		8
72.13	even	6		1296.3.q.l.593.3		8
72.29	odd	6		1296.3.q.l.1025.3		8
72.43	odd	6		648.3.m.f.377.2		8
72.59	even	6		648.3.m.f.593.2		8
72.61	even	6		1296.3.q.l.1025.2		8
72.67	odd	6		648.3.m.f.593.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
216.3.e.c.161.2	✓	4	24.11	even	2
216.3.e.c.161.3	yes	4	8.3	odd	2
432.3.e.h.161.2		4	24.5	odd	2
432.3.e.h.161.3		4	8.5	even	2
648.3.m.f.377.2		8	72.43	odd	6
648.3.m.f.377.3		8	72.11	even	6
648.3.m.f.593.2		8	72.59	even	6
648.3.m.f.593.3		8	72.67	odd	6
1296.3.q.l.593.2		8	72.5	odd	6
1296.3.q.l.593.3		8	72.13	even	6
1296.3.q.l.1025.2		8	72.61	even	6
1296.3.q.l.1025.3		8	72.29	odd	6
1728.3.e.p.1025.2		4	4.3	odd	2
1728.3.e.p.1025.3		4	12.11	even	2
1728.3.e.s.1025.2		4	1.1	even	1	trivial
1728.3.e.s.1025.3		4	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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q-0.171573i q^{5} -5.48528 q^{7} -3.34315i q^{11} -14.9706 q^{13} +17.3137i q^{17} +24.9706 q^{19} +40.2843i q^{23} +24.9706 q^{25} -42.0000i q^{29} -7.48528 q^{31} +0.941125i q^{35} +19.0294 q^{37} -38.9117i q^{41} -30.9706 q^{43} +8.05887i q^{47} -18.9117 q^{49} -99.0833i q^{53} -0.573593 q^{55} -97.1960i q^{59} +15.8823 q^{61} +2.56854i q^{65} -74.9706 q^{67} -46.2843i q^{71} -121.912 q^{73} +18.3381i q^{77} +107.941 q^{79} +29.4020i q^{83} +2.97056 q^{85} -73.0294i q^{89} +82.1177 q^{91} -4.28427i q^{95} +78.8823 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 12 q^{7} + 8 q^{13} + 32 q^{19} + 32 q^{25} + 4 q^{31} + 144 q^{37} - 56 q^{43} + 128 q^{49} - 172 q^{55} - 208 q^{61} - 232 q^{67} - 284 q^{73} + 296 q^{79} - 56 q^{85} + 600 q^{91} + 44 q^{97}+O(q^{100}) \) 4 * q + 12 * q^7 + 8 * q^13 + 32 * q^19 + 32 * q^25 + 4 * q^31 + 144 * q^37 - 56 * q^43 + 128 * q^49 - 172 * q^55 - 208 * q^61 - 232 * q^67 - 284 * q^73 + 296 * q^79 - 56 * q^85 + 600 * q^91 + 44 * q^97

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