LMFDB - Embedded newform 2736.3.m.f.1711.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2736,3,Mod(1711,2736)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2736, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2736.1711");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 2736 = 2^{4} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2736.m (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$74.5506003290$
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} - 2 x^{11} + 50 x^{10} - 136 x^{9} + 2215 x^{8} - 5020 x^{7} + 18282 x^{6} - 12094 x^{5} + \cdots + 1024 $ x^12 - 2x^11 + 50x^10 - 136x^9 + 2215x^8 - 5020x^7 + 18282x^6 - 12094x^5 + 48457x^4 - 30372x^3 + 89392x^2 + 9344*x + 1024
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{16} $
Twist minimal:	no (minimal twist has level 912)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1711.3
Root			$3.06079 + 5.30145i$ of defining polynomial
Character	$\chi$	$=$	2736.1711
Dual form			2736.3.m.f.1711.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.38490 q^{5} -7.59081i q^{7} +O(q^{10})$	$q-1.38490 q^{5} -7.59081i q^{7} +7.52404i q^{11} +14.7359 q^{13} +30.7401 q^{17} -4.35890i q^{19} -4.11943i q^{23} -23.0821 q^{25} +18.9459 q^{29} -3.84118i q^{31} +10.5125i q^{35} -42.0326 q^{37} +49.6292 q^{41} +5.08009i q^{43} +32.6549i q^{47} -8.62045 q^{49} -18.1252 q^{53} -10.4200i q^{55} +75.5116i q^{59} -75.0766 q^{61} -20.4078 q^{65} -50.7368i q^{67} +45.1030i q^{71} +115.575 q^{73} +57.1136 q^{77} +67.4150i q^{79} -22.6202i q^{83} -42.5720 q^{85} -95.6879 q^{89} -111.858i q^{91} +6.03664i q^{95} +43.1069 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 10 q^{5}+O(q^{10}) $ 12 * q + 10 * q^5	$ 12 q + 10 q^{5} + 36 q^{13} + 10 q^{17} + 58 q^{25} - 12 q^{29} + 32 q^{37} - 136 q^{41} - 22 q^{49} + 236 q^{53} - 210 q^{61} + 52 q^{65} - 158 q^{73} + 70 q^{77} + 242 q^{85} - 444 q^{89} + 320 q^{97}+O(q^{100}) $ 12 * q + 10 * q^5 + 36 * q^13 + 10 * q^17 + 58 * q^25 - 12 * q^29 + 32 * q^37 - 136 * q^41 - 22 * q^49 + 236 * q^53 - 210 * q^61 + 52 * q^65 - 158 * q^73 + 70 * q^77 + 242 * q^85 - 444 * q^89 + 320 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2736\mathbb{Z}\right)^\times$.

$n$	$1009$	$1217$	$1711$	$2053$
$\chi(n)$	$1$	$1$	$-1$	$1$

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−1.38490	−0.276980	−0.138490	−	0.990364i	$-0.544225\pi$
$5$	−1.38490	−0.276980	−0.138490	+	0.990364i	$0.544225\pi$
$6$	0	0
$7$	− 7.59081i	− 1.08440i	−0.840249	−	0.542201i	$-0.817591\pi$
$7$	− 7.59081i	− 1.08440i	0.840249	−	0.542201i	$-0.182409\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	7.52404i	0.684003i	0.939699	+	0.342002i	$0.111105\pi$
$11$	7.52404i	0.684003i	−0.939699	+	0.342002i	$0.888895\pi$
$12$	0	0
$13$	14.7359	1.13353	0.566767	−	0.823878i	$-0.308194\pi$
$13$	14.7359	1.13353	0.566767	+	0.823878i	$0.308194\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	30.7401	1.80824	0.904121	−	0.427277i	$-0.140527\pi$
$17$	30.7401	1.80824	0.904121	+	0.427277i	$0.140527\pi$
$18$	0	0
$19$	− 4.35890i	− 0.229416i
$19$	− 4.35890i	− 0.229416i
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 4.11943i	− 0.179106i	−0.995982	−	0.0895528i	$-0.971456\pi$
$23$	− 4.11943i	− 0.179106i	0.995982	−	0.0895528i	$-0.0285438\pi$
$24$	0	0
$25$	−23.0821	−0.923282
$26$	0	0
$27$	0	0
$28$	0	0
$29$	18.9459	0.653308	0.326654	−	0.945144i	$-0.394079\pi$
$29$	18.9459	0.653308	0.326654	+	0.945144i	$0.394079\pi$
$30$	0	0
$31$	− 3.84118i	− 0.123909i	−0.998079	−	0.0619546i	$-0.980267\pi$
$31$	− 3.84118i	− 0.123909i	0.998079	−	0.0619546i	$-0.0197334\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	10.5125i	0.300357i
$36$	0	0
$37$	−42.0326	−1.13602	−0.568008	−	0.823023i	$-0.692286\pi$
$37$	−42.0326	−1.13602	−0.568008	+	0.823023i	$0.692286\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	49.6292	1.21047	0.605234	−	0.796047i	$-0.293079\pi$
$41$	49.6292	1.21047	0.605234	+	0.796047i	$0.293079\pi$
$42$	0	0
$43$	5.08009i	0.118142i	0.998254	+	0.0590708i	$0.0188138\pi$
$43$	5.08009i	0.118142i	−0.998254	+	0.0590708i	$0.981186\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	32.6549i	0.694786i	0.937720	+	0.347393i	$0.112933\pi$
$47$	32.6549i	0.694786i	−0.937720	+	0.347393i	$0.887067\pi$
$48$	0	0
$49$	−8.62045	−0.175928
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−18.1252	−0.341984	−0.170992	−	0.985272i	$-0.554697\pi$
$53$	−18.1252	−0.341984	−0.170992	+	0.985272i	$0.554697\pi$
$54$	0	0
$55$	− 10.4200i	− 0.189455i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	75.5116i	1.27986i	0.768434	+	0.639929i	$0.221036\pi$
$59$	75.5116i	1.27986i	−0.768434	+	0.639929i	$0.778964\pi$
$60$	0	0
$61$	−75.0766	−1.23076	−0.615382	−	0.788229i	$-0.710998\pi$
$61$	−75.0766	−1.23076	−0.615382	+	0.788229i	$0.710998\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−20.4078	−0.313966
$66$	0	0
$67$	− 50.7368i	− 0.757265i	−0.925547	−	0.378633i	$-0.876394\pi$
$67$	− 50.7368i	− 0.757265i	0.925547	−	0.378633i	$-0.123606\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	45.1030i	0.635253i	0.948216	+	0.317627i	$0.102886\pi$
$71$	45.1030i	0.635253i	−0.948216	+	0.317627i	$0.897114\pi$
$72$	0	0
$73$	115.575	1.58322	0.791609	−	0.611028i	$-0.209244\pi$
$73$	115.575	1.58322	0.791609	+	0.611028i	$0.209244\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	57.1136	0.741734
$78$	0	0
$79$	67.4150i	0.853355i	0.904404	+	0.426677i	$0.140316\pi$
$79$	67.4150i	0.853355i	−0.904404	+	0.426677i	$0.859684\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 22.6202i	− 0.272533i	−0.990672	−	0.136266i	$-0.956490\pi$
$83$	− 22.6202i	− 0.272533i	0.990672	−	0.136266i	$-0.0435103\pi$
$84$	0	0
$85$	−42.5720	−0.500847
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−95.6879	−1.07514	−0.537572	−	0.843218i	$-0.680658\pi$
$89$	−95.6879	−1.07514	−0.537572	+	0.843218i	$0.680658\pi$
$90$	0	0
$91$	− 111.858i	− 1.22921i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	6.03664i	0.0635435i
$96$	0	0
$97$	43.1069	0.444402	0.222201	−	0.975001i	$-0.428676\pi$
$97$	43.1069	0.444402	0.222201	+	0.975001i	$0.428676\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	62.0963	0.614815	0.307408	−	0.951578i	$-0.400538\pi$
$101$	62.0963	0.614815	0.307408	+	0.951578i	$0.400538\pi$
$102$	0	0
$103$	− 122.744i	− 1.19169i	−0.803098	−	0.595847i	$-0.796816\pi$
$103$	− 122.744i	− 1.19169i	0.803098	−	0.595847i	$-0.203184\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	77.8830i	0.727878i	0.931423	+	0.363939i	$0.118568\pi$
$107$	77.8830i	0.727878i	−0.931423	+	0.363939i	$0.881432\pi$
$108$	0	0
$109$	113.683	1.04297	0.521483	−	0.853262i	$-0.325379\pi$
$109$	113.683	1.04297	0.521483	+	0.853262i	$0.325379\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	180.531	1.59762	0.798810	−	0.601583i	$-0.205463\pi$
$113$	180.531	1.59762	0.798810	+	0.601583i	$0.205463\pi$
$114$	0	0
$115$	5.70499i	0.0496086i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 233.342i	− 1.96086i
$120$	0	0
$121$	64.3889	0.532140
$122$	0	0
$123$	0	0
$124$	0	0
$125$	66.5888	0.532710
$126$	0	0
$127$	− 197.750i	− 1.55709i	−0.627589	−	0.778545i	$-0.715958\pi$
$127$	− 197.750i	− 1.55709i	0.627589	−	0.778545i	$-0.284042\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 61.4446i	− 0.469043i	−0.972111	−	0.234521i	$-0.924648\pi$
$131$	− 61.4446i	− 0.469043i	0.972111	−	0.234521i	$-0.0753523\pi$
$132$	0	0
$133$	−33.0876	−0.248779
$134$	0	0
$135$	0	0
$136$	0	0
$137$	252.417	1.84246	0.921230	−	0.389018i	$-0.127186\pi$
$137$	252.417	1.84246	0.921230	+	0.389018i	$0.127186\pi$
$138$	0	0
$139$	− 222.442i	− 1.60030i	−0.599799	−	0.800151i	$-0.704753\pi$
$139$	− 222.442i	− 1.60030i	0.599799	−	0.800151i	$-0.295247\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	110.874i	0.775341i
$144$	0	0
$145$	−26.2382	−0.180953
$146$	0	0
$147$	0	0
$148$	0	0
$149$	6.53232	0.0438411	0.0219205	−	0.999760i	$-0.493022\pi$
$149$	6.53232	0.0438411	0.0219205	+	0.999760i	$0.493022\pi$
$150$	0	0
$151$	− 119.980i	− 0.794568i	−0.917696	−	0.397284i	$-0.869953\pi$
$151$	− 119.980i	− 0.794568i	0.917696	−	0.397284i	$-0.130047\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	5.31965i	0.0343203i
$156$	0	0
$157$	−115.415	−0.735126	−0.367563	−	0.929999i	$-0.619808\pi$
$157$	−115.415	−0.735126	−0.367563	+	0.929999i	$0.619808\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−31.2698	−0.194222
$162$	0	0
$163$	216.195i	1.32635i	0.748464	+	0.663176i	$0.230792\pi$
$163$	216.195i	1.32635i	−0.748464	+	0.663176i	$0.769208\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 218.936i	− 1.31099i	−0.755199	−	0.655496i	$-0.772460\pi$
$167$	− 218.936i	− 1.31099i	0.755199	−	0.655496i	$-0.227540\pi$
$168$	0	0
$169$	48.1480	0.284900
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−22.5865	−0.130558	−0.0652788	−	0.997867i	$-0.520794\pi$
$173$	−22.5865	−0.130558	−0.0652788	+	0.997867i	$0.520794\pi$
$174$	0	0
$175$	175.212i	1.00121i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 268.892i	− 1.50219i	−0.660195	−	0.751094i	$-0.729526\pi$
$179$	− 268.892i	− 1.50219i	0.660195	−	0.751094i	$-0.270474\pi$
$180$	0	0
$181$	−95.7614	−0.529068	−0.264534	−	0.964376i	$-0.585218\pi$
$181$	−95.7614	−0.529068	−0.264534	+	0.964376i	$0.585218\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	58.2109	0.314653
$186$	0	0
$187$	231.290i	1.23684i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 219.722i	− 1.15038i	−0.818020	−	0.575189i	$-0.804928\pi$
$191$	− 219.722i	− 1.15038i	0.818020	−	0.575189i	$-0.195072\pi$
$192$	0	0
$193$	313.329	1.62347	0.811733	−	0.584028i	$-0.198524\pi$
$193$	313.329	1.62347	0.811733	+	0.584028i	$0.198524\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	39.2210	0.199092	0.0995458	−	0.995033i	$-0.468261\pi$
$197$	39.2210	0.199092	0.0995458	+	0.995033i	$0.468261\pi$
$198$	0	0
$199$	− 53.7814i	− 0.270258i	−0.990828	−	0.135129i	$-0.956855\pi$
$199$	− 53.7814i	− 0.270258i	0.990828	−	0.135129i	$-0.0431449\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 143.815i	− 0.708448i
$204$	0	0
$205$	−68.7315	−0.335275
$206$	0	0
$207$	0	0
$208$	0	0
$209$	32.7965	0.156921
$210$	0	0
$211$	− 130.638i	− 0.619136i	−0.950877	−	0.309568i	$-0.899816\pi$
$211$	− 130.638i	− 0.619136i	0.950877	−	0.309568i	$-0.100184\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	− 7.03542i	− 0.0327229i
$216$	0	0
$217$	−29.1577	−0.134367
$218$	0	0
$219$	0	0
$220$	0	0
$221$	452.985	2.04970
$222$	0	0
$223$	101.971i	0.457270i	0.973512	+	0.228635i	$0.0734262\pi$
$223$	101.971i	0.457270i	−0.973512	+	0.228635i	$0.926574\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	237.887i	1.04796i	0.851731	+	0.523980i	$0.175553\pi$
$227$	237.887i	1.04796i	−0.851731	+	0.523980i	$0.824447\pi$
$228$	0	0
$229$	58.9496	0.257422	0.128711	−	0.991682i	$-0.458916\pi$
$229$	58.9496	0.257422	0.128711	+	0.991682i	$0.458916\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−238.100	−1.02189	−0.510945	−	0.859613i	$-0.670705\pi$
$233$	−238.100	−1.02189	−0.510945	+	0.859613i	$0.670705\pi$
$234$	0	0
$235$	− 45.2238i	− 0.192442i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	441.268i	1.84631i	0.384430	+	0.923154i	$0.374398\pi$
$239$	441.268i	1.84631i	−0.384430	+	0.923154i	$0.625602\pi$
$240$	0	0
$241$	61.5679	0.255468	0.127734	−	0.991808i	$-0.459230\pi$
$241$	61.5679	0.255468	0.127734	+	0.991808i	$0.459230\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	11.9385	0.0487284
$246$	0	0
$247$	− 64.2325i	− 0.260051i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	147.646i	0.588232i	0.955770	+	0.294116i	$0.0950252\pi$
$251$	147.646i	0.588232i	−0.955770	+	0.294116i	$0.904975\pi$
$252$	0	0
$253$	30.9947	0.122509
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−108.191	−0.420978	−0.210489	−	0.977596i	$-0.567506\pi$
$257$	−108.191	−0.420978	−0.210489	+	0.977596i	$0.567506\pi$
$258$	0	0
$259$	319.061i	1.23190i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 445.063i	− 1.69226i	−0.532980	−	0.846128i	$-0.678928\pi$
$263$	− 445.063i	− 1.69226i	0.532980	−	0.846128i	$-0.321072\pi$
$264$	0	0
$265$	25.1015	0.0947228
$266$	0	0
$267$	0	0
$268$	0	0
$269$	108.597	0.403707	0.201854	−	0.979416i	$-0.435303\pi$
$269$	108.597	0.403707	0.201854	+	0.979416i	$0.435303\pi$
$270$	0	0
$271$	389.889i	1.43871i	0.694645	+	0.719353i	$0.255562\pi$
$271$	389.889i	1.43871i	−0.694645	+	0.719353i	$0.744438\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 173.670i	− 0.631528i
$276$	0	0
$277$	501.009	1.80870	0.904349	−	0.426794i	$-0.140357\pi$
$277$	501.009	1.80870	0.904349	+	0.426794i	$0.140357\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	41.6793	0.148325	0.0741625	−	0.997246i	$-0.476372\pi$
$281$	41.6793	0.148325	0.0741625	+	0.997246i	$0.476372\pi$
$282$	0	0
$283$	− 408.676i	− 1.44408i	−0.691849	−	0.722042i	$-0.743204\pi$
$283$	− 408.676i	− 1.44408i	0.691849	−	0.722042i	$-0.256796\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 376.726i	− 1.31263i
$288$	0	0
$289$	655.955	2.26974
$290$	0	0
$291$	0	0
$292$	0	0
$293$	88.5997	0.302388	0.151194	−	0.988504i	$-0.451688\pi$
$293$	88.5997	0.302388	0.151194	+	0.988504i	$0.451688\pi$
$294$	0	0
$295$	− 104.576i	− 0.354495i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 60.7036i	− 0.203022i
$300$	0	0
$301$	38.5620	0.128113
$302$	0	0
$303$	0	0
$304$	0	0
$305$	103.974	0.340897
$306$	0	0
$307$	545.946i	1.77832i	0.457592	+	0.889162i	$0.348712\pi$
$307$	545.946i	1.77832i	−0.457592	+	0.889162i	$0.651288\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	− 237.088i	− 0.762341i	−0.924505	−	0.381171i	$-0.875521\pi$
$311$	− 237.088i	− 0.762341i	0.924505	−	0.381171i	$-0.124479\pi$
$312$	0	0
$313$	227.920	0.728179	0.364089	−	0.931364i	$-0.381380\pi$
$313$	227.920	0.728179	0.364089	+	0.931364i	$0.381380\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	556.139	1.75438	0.877191	−	0.480141i	$-0.159415\pi$
$317$	556.139	1.75438	0.877191	+	0.480141i	$0.159415\pi$
$318$	0	0
$319$	142.550i	0.446865i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 133.993i	− 0.414839i
$324$	0	0
$325$	−340.136	−1.04657
$326$	0	0
$327$	0	0
$328$	0	0
$329$	247.877	0.753427
$330$	0	0
$331$	293.735i	0.887417i	0.896171	+	0.443709i	$0.146337\pi$
$331$	293.735i	0.887417i	−0.896171	+	0.443709i	$0.853663\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	70.2653i	0.209747i
$336$	0	0
$337$	174.257	0.517082	0.258541	−	0.966000i	$-0.416758\pi$
$337$	174.257	0.517082	0.258541	+	0.966000i	$0.416758\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	28.9012	0.0847543
$342$	0	0
$343$	− 306.514i	− 0.893626i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 28.3145i	− 0.0815981i	−0.999167	−	0.0407990i	$-0.987010\pi$
$347$	− 28.3145i	− 0.0815981i	0.999167	−	0.0407990i	$-0.0129903\pi$
$348$	0	0
$349$	−66.2115	−0.189718	−0.0948589	−	0.995491i	$-0.530240\pi$
$349$	−66.2115	−0.189718	−0.0948589	+	0.995491i	$0.530240\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−375.616	−1.06407	−0.532034	−	0.846723i	$-0.678572\pi$
$353$	−375.616	−1.06407	−0.532034	+	0.846723i	$0.678572\pi$
$354$	0	0
$355$	− 62.4631i	− 0.175952i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 156.040i	− 0.434653i	−0.976099	−	0.217326i	$-0.930266\pi$
$359$	− 156.040i	− 0.434653i	0.976099	−	0.217326i	$-0.0697336\pi$
$360$	0	0
$361$	−19.0000	−0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−160.060	−0.438519
$366$	0	0
$367$	102.555i	0.279441i	0.990191	+	0.139721i	$0.0446205\pi$
$367$	102.555i	0.279441i	−0.990191	+	0.139721i	$0.955380\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	137.585i	0.370848i
$372$	0	0
$373$	−268.383	−0.719526	−0.359763	−	0.933044i	$-0.617142\pi$
$373$	−268.383	−0.719526	−0.359763	+	0.933044i	$0.617142\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	279.186	0.740547
$378$	0	0
$379$	− 448.588i	− 1.18361i	−0.806081	−	0.591805i	$-0.798415\pi$
$379$	− 448.588i	− 1.18361i	0.806081	−	0.591805i	$-0.201585\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	1.54436i	0.00403227i	0.999998	+	0.00201614i	$0.000641757\pi$
$383$	1.54436i	0.00403227i	−0.999998	+	0.00201614i	$0.999358\pi$
$384$	0	0
$385$	−79.0965	−0.205445
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−44.6311	−0.114733	−0.0573665	−	0.998353i	$-0.518270\pi$
$389$	−44.6311	−0.114733	−0.0573665	+	0.998353i	$0.518270\pi$
$390$	0	0
$391$	− 126.632i	− 0.323866i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 93.3630i	− 0.236362i
$396$	0	0
$397$	319.804	0.805552	0.402776	−	0.915299i	$-0.368045\pi$
$397$	319.804	0.805552	0.402776	+	0.915299i	$0.368045\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−707.466	−1.76425	−0.882127	−	0.471011i	$-0.843889\pi$
$401$	−707.466	−1.76425	−0.882127	+	0.471011i	$0.843889\pi$
$402$	0	0
$403$	− 56.6035i	− 0.140455i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 316.254i	− 0.777038i
$408$	0	0
$409$	−455.563	−1.11385	−0.556923	−	0.830564i	$-0.688018\pi$
$409$	−455.563	−1.11385	−0.556923	+	0.830564i	$0.688018\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	573.195	1.38788
$414$	0	0
$415$	31.3267i	0.0754860i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	119.062i	0.284157i	0.989855	+	0.142079i	$0.0453786\pi$
$419$	119.062i	0.284157i	−0.989855	+	0.142079i	$0.954621\pi$
$420$	0	0
$421$	202.144	0.480153	0.240076	−	0.970754i	$-0.422827\pi$
$421$	202.144	0.480153	0.240076	+	0.970754i	$0.422827\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−709.545	−1.66952
$426$	0	0
$427$	569.893i	1.33464i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 622.305i	− 1.44386i	−0.691964	−	0.721932i	$-0.743255\pi$
$431$	− 622.305i	− 1.44386i	0.691964	−	0.721932i	$-0.256745\pi$
$432$	0	0
$433$	−68.8999	−0.159122	−0.0795611	−	0.996830i	$-0.525352\pi$
$433$	−68.8999	−0.159122	−0.0795611	+	0.996830i	$0.525352\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−17.9562	−0.0410896
$438$	0	0
$439$	− 63.3650i	− 0.144339i	−0.997392	−	0.0721697i	$-0.977008\pi$
$439$	− 63.3650i	− 0.144339i	0.997392	−	0.0721697i	$-0.0229923\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	329.910i	0.744717i	0.928089	+	0.372359i	$0.121451\pi$
$443$	329.910i	0.744717i	−0.928089	+	0.372359i	$0.878549\pi$
$444$	0	0
$445$	132.518	0.297793
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−192.344	−0.428384	−0.214192	−	0.976792i	$-0.568712\pi$
$449$	−192.344	−0.428384	−0.214192	+	0.976792i	$0.568712\pi$
$450$	0	0
$451$	373.412i	0.827965i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	154.912i	0.340465i
$456$	0	0
$457$	148.258	0.324416	0.162208	−	0.986757i	$-0.448138\pi$
$457$	148.258	0.324416	0.162208	+	0.986757i	$0.448138\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−10.8198	−0.0234702	−0.0117351	−	0.999931i	$-0.503735\pi$
$461$	−10.8198	−0.0234702	−0.0117351	+	0.999931i	$0.503735\pi$
$462$	0	0
$463$	− 790.701i	− 1.70778i	−0.520455	−	0.853889i	$-0.674238\pi$
$463$	− 790.701i	− 1.70778i	0.520455	−	0.853889i	$-0.325762\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	42.0061i	0.0899487i	0.998988	+	0.0449744i	$0.0143206\pi$
$467$	42.0061i	0.0899487i	−0.998988	+	0.0449744i	$0.985679\pi$
$468$	0	0
$469$	−385.133	−0.821180
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−38.2228	−0.0808093
$474$	0	0
$475$	100.612i	0.211815i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 653.635i	− 1.36458i	−0.731080	−	0.682292i	$-0.760983\pi$
$479$	− 653.635i	− 1.36458i	0.731080	−	0.682292i	$-0.239017\pi$
$480$	0	0
$481$	−619.389	−1.28771
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−59.6988	−0.123090
$486$	0	0
$487$	− 294.437i	− 0.604593i	−0.953214	−	0.302296i	$-0.902247\pi$
$487$	− 294.437i	− 0.604593i	0.953214	−	0.302296i	$-0.0977532\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	135.380i	0.275722i	0.990452	+	0.137861i	$0.0440228\pi$
$491$	135.380i	0.275722i	−0.990452	+	0.137861i	$0.955977\pi$
$492$	0	0
$493$	582.400	1.18134
$494$	0	0
$495$	0	0
$496$	0	0
$497$	342.368	0.688870
$498$	0	0
$499$	− 385.063i	− 0.771670i	−0.922568	−	0.385835i	$-0.873913\pi$
$499$	− 385.063i	− 0.771670i	0.922568	−	0.385835i	$-0.126087\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	467.243i	0.928912i	0.885596	+	0.464456i	$0.153750\pi$
$503$	467.243i	0.928912i	−0.885596	+	0.464456i	$0.846250\pi$
$504$	0	0
$505$	−85.9972	−0.170291
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−255.444	−0.501855	−0.250928	−	0.968006i	$-0.580736\pi$
$509$	−255.444	−0.501855	−0.250928	+	0.968006i	$0.580736\pi$
$510$	0	0
$511$	− 877.307i	− 1.71684i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	169.989i	0.330075i
$516$	0	0
$517$	−245.697	−0.475236
$518$	0	0
$519$	0	0
$520$	0	0
$521$	376.843	0.723307	0.361653	−	0.932313i	$-0.382212\pi$
$521$	376.843	0.723307	0.361653	+	0.932313i	$0.382212\pi$
$522$	0	0
$523$	314.604i	0.601537i	0.953697	+	0.300769i	$0.0972432\pi$
$523$	314.604i	0.601537i	−0.953697	+	0.300769i	$0.902757\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 118.078i	− 0.224058i
$528$	0	0
$529$	512.030	0.967921
$530$	0	0
$531$	0	0
$532$	0	0
$533$	731.333	1.37211
$534$	0	0
$535$	− 107.860i	− 0.201608i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 64.8606i	− 0.120335i
$540$	0	0
$541$	−805.075	−1.48812	−0.744062	−	0.668111i	$-0.767103\pi$
$541$	−805.075	−1.48812	−0.744062	+	0.668111i	$0.767103\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−157.440	−0.288881
$546$	0	0
$547$	− 154.290i	− 0.282065i	−0.990005	−	0.141033i	$-0.954958\pi$
$547$	− 154.290i	− 0.282065i	0.990005	−	0.141033i	$-0.0450423\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	− 82.5834i	− 0.149879i
$552$	0	0
$553$	511.735	0.925380
$554$	0	0
$555$	0	0
$556$	0	0
$557$	819.397	1.47109	0.735545	−	0.677476i	$-0.236926\pi$
$557$	819.397	1.47109	0.735545	+	0.677476i	$0.236926\pi$
$558$	0	0
$559$	74.8599i	0.133918i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	44.1064i	0.0783417i	0.999233	+	0.0391708i	$0.0124717\pi$
$563$	44.1064i	0.0783417i	−0.999233	+	0.0391708i	$0.987528\pi$
$564$	0	0
$565$	−250.017	−0.442509
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−124.301	−0.218455	−0.109228	−	0.994017i	$-0.534838\pi$
$569$	−124.301	−0.218455	−0.109228	+	0.994017i	$0.534838\pi$
$570$	0	0
$571$	210.662i	0.368935i	0.982839	+	0.184467i	$0.0590560\pi$
$571$	210.662i	0.368935i	−0.982839	+	0.184467i	$0.940944\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	95.0848i	0.165365i
$576$	0	0
$577$	−169.784	−0.294254	−0.147127	−	0.989118i	$-0.547003\pi$
$577$	−169.784	−0.294254	−0.147127	+	0.989118i	$0.547003\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−171.706	−0.295535
$582$	0	0
$583$	− 136.374i	− 0.233918i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	272.890i	0.464889i	0.972610	+	0.232444i	$0.0746724\pi$
$587$	272.890i	0.464889i	−0.972610	+	0.232444i	$0.925328\pi$
$588$	0	0
$589$	−16.7433	−0.0284267
$590$	0	0
$591$	0	0
$592$	0	0
$593$	662.738	1.11760	0.558801	−	0.829302i	$-0.311262\pi$
$593$	662.738	1.11760	0.558801	+	0.829302i	$0.311262\pi$
$594$	0	0
$595$	323.156i	0.543119i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	792.064i	1.32231i	0.750249	+	0.661155i	$0.229933\pi$
$599$	792.064i	1.32231i	−0.750249	+	0.661155i	$0.770067\pi$
$600$	0	0
$601$	−898.697	−1.49534	−0.747668	−	0.664073i	$-0.768827\pi$
$601$	−898.697	−1.49534	−0.747668	+	0.664073i	$0.768827\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−89.1721	−0.147392
$606$	0	0
$607$	305.256i	0.502894i	0.967871	+	0.251447i	$0.0809064\pi$
$607$	305.256i	0.502894i	−0.967871	+	0.251447i	$0.919094\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	481.201i	0.787563i
$612$	0	0
$613$	−654.587	−1.06784	−0.533921	−	0.845534i	$-0.679282\pi$
$613$	−654.587	−1.06784	−0.533921	+	0.845534i	$0.679282\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−998.320	−1.61802	−0.809011	−	0.587793i	$-0.799997\pi$
$617$	−998.320	−1.61802	−0.809011	+	0.587793i	$0.799997\pi$
$618$	0	0
$619$	538.922i	0.870634i	0.900277	+	0.435317i	$0.143364\pi$
$619$	538.922i	0.870634i	−0.900277	+	0.435317i	$0.856636\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	726.349i	1.16589i
$624$	0	0
$625$	484.833	0.775732
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−1292.09	−2.05419
$630$	0	0
$631$	403.068i	0.638777i	0.947624	+	0.319389i	$0.103478\pi$
$631$	403.068i	0.638777i	−0.947624	+	0.319389i	$0.896522\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	273.864i	0.431283i
$636$	0	0
$637$	−127.030	−0.199420
$638$	0	0
$639$	0	0
$640$	0	0
$641$	204.064	0.318352	0.159176	−	0.987250i	$-0.449116\pi$
$641$	204.064	0.318352	0.159176	+	0.987250i	$0.449116\pi$
$642$	0	0
$643$	− 1210.26i	− 1.88221i	−0.338110	−	0.941107i	$-0.609788\pi$
$643$	− 1210.26i	− 1.88221i	0.338110	−	0.941107i	$-0.390212\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	522.563i	0.807671i	0.914832	+	0.403835i	$0.132323\pi$
$647$	522.563i	0.807671i	−0.914832	+	0.403835i	$0.867677\pi$
$648$	0	0
$649$	−568.152	−0.875427
$650$	0	0
$651$	0	0
$652$	0	0
$653$	793.199	1.21470	0.607350	−	0.794434i	$-0.292232\pi$
$653$	793.199	1.21470	0.607350	+	0.794434i	$0.292232\pi$
$654$	0	0
$655$	85.0946i	0.129915i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	603.726i	0.916124i	0.888920	+	0.458062i	$0.151456\pi$
$659$	603.726i	0.916124i	−0.888920	+	0.458062i	$0.848544\pi$
$660$	0	0
$661$	268.569	0.406308	0.203154	−	0.979147i	$-0.434881\pi$
$661$	268.569	0.406308	0.203154	+	0.979147i	$0.434881\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	45.8230	0.0689067
$666$	0	0
$667$	− 78.0464i	− 0.117011i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	− 564.879i	− 0.841847i
$672$	0	0
$673$	−313.116	−0.465255	−0.232627	−	0.972566i	$-0.574732\pi$
$673$	−313.116	−0.465255	−0.232627	+	0.972566i	$0.574732\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	684.838	1.01158	0.505789	−	0.862657i	$-0.331201\pi$
$677$	684.838	1.01158	0.505789	+	0.862657i	$0.331201\pi$
$678$	0	0
$679$	− 327.217i	− 0.481910i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	567.375i	0.830711i	0.909659	+	0.415355i	$0.136343\pi$
$683$	567.375i	0.830711i	−0.909659	+	0.415355i	$0.863657\pi$
$684$	0	0
$685$	−349.572	−0.510324
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−267.091	−0.387651
$690$	0	0
$691$	− 898.570i	− 1.30039i	−0.759767	−	0.650195i	$-0.774687\pi$
$691$	− 898.570i	− 1.30039i	0.759767	−	0.650195i	$-0.225313\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	308.060i	0.443251i
$696$	0	0
$697$	1525.61	2.18882
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−772.661	−1.10223	−0.551113	−	0.834430i	$-0.685797\pi$
$701$	−772.661	−1.10223	−0.551113	+	0.834430i	$0.685797\pi$
$702$	0	0
$703$	183.216i	0.260620i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 471.362i	− 0.666707i
$708$	0	0
$709$	371.779	0.524370	0.262185	−	0.965018i	$-0.415557\pi$
$709$	371.779	0.524370	0.262185	+	0.965018i	$0.415557\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−15.8235	−0.0221928
$714$	0	0
$715$	− 153.549i	− 0.214754i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	940.181i	1.30762i	0.756658	+	0.653811i	$0.226831\pi$
$719$	940.181i	1.30762i	−0.756658	+	0.653811i	$0.773169\pi$
$720$	0	0
$721$	−931.730	−1.29228
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−437.311	−0.603187
$726$	0	0
$727$	290.090i	0.399024i	0.979895	+	0.199512i	$0.0639356\pi$
$727$	290.090i	0.399024i	−0.979895	+	0.199512i	$0.936064\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	156.163i	0.213629i
$732$	0	0
$733$	77.1358	0.105233	0.0526165	−	0.998615i	$-0.483244\pi$
$733$	77.1358	0.105233	0.0526165	+	0.998615i	$0.483244\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	381.745	0.517972
$738$	0	0
$739$	484.447i	0.655544i	0.944757	+	0.327772i	$0.106298\pi$
$739$	484.447i	0.655544i	−0.944757	+	0.327772i	$0.893702\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	506.529i	0.681734i	0.940111	+	0.340867i	$0.110721\pi$
$743$	506.529i	0.681734i	−0.940111	+	0.340867i	$0.889279\pi$
$744$	0	0
$745$	−9.04660	−0.0121431
$746$	0	0
$747$	0	0
$748$	0	0
$749$	591.195	0.789313
$750$	0	0
$751$	439.108i	0.584698i	0.956312	+	0.292349i	$0.0944368\pi$
$751$	439.108i	0.584698i	−0.956312	+	0.292349i	$0.905563\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	166.160i	0.220079i
$756$	0	0
$757$	722.165	0.953983	0.476991	−	0.878908i	$-0.341727\pi$
$757$	722.165	0.953983	0.476991	+	0.878908i	$0.341727\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	731.962	0.961842	0.480921	−	0.876764i	$-0.340302\pi$
$761$	731.962	0.961842	0.480921	+	0.876764i	$0.340302\pi$
$762$	0	0
$763$	− 862.949i	− 1.13099i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	1112.73i	1.45076i
$768$	0	0
$769$	753.718	0.980127	0.490063	−	0.871687i	$-0.336974\pi$
$769$	753.718	0.980127	0.490063	+	0.871687i	$0.336974\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−1261.03	−1.63135	−0.815675	−	0.578510i	$-0.803634\pi$
$773$	−1261.03	−1.63135	−0.815675	+	0.578510i	$0.803634\pi$
$774$	0	0
$775$	88.6624i	0.114403i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 216.329i	− 0.277701i
$780$	0	0
$781$	−339.357	−0.434515
$782$	0	0
$783$	0	0
$784$	0	0
$785$	159.838	0.203615
$786$	0	0
$787$	− 232.259i	− 0.295120i	−0.989053	−	0.147560i	$-0.952858\pi$
$787$	− 232.259i	− 0.295120i	0.989053	−	0.147560i	$-0.0471420\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 1370.38i	− 1.73246i
$792$	0	0
$793$	−1106.33	−1.39511
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−143.202	−0.179676	−0.0898381	−	0.995956i	$-0.528635\pi$
$797$	−143.202	−0.179676	−0.0898381	+	0.995956i	$0.528635\pi$
$798$	0	0
$799$	1003.82i	1.25634i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	869.590i	1.08293i
$804$	0	0
$805$	43.3055	0.0537957
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−720.927	−0.891134	−0.445567	−	0.895249i	$-0.646998\pi$
$809$	−720.927	−0.891134	−0.445567	+	0.895249i	$0.646998\pi$
$810$	0	0
$811$	89.8749i	0.110820i	0.998464	+	0.0554099i	$0.0176466\pi$
$811$	89.8749i	0.110820i	−0.998464	+	0.0554099i	$0.982353\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	− 299.409i	− 0.367373i
$816$	0	0
$817$	22.1436	0.0271036
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−1227.03	−1.49456	−0.747279	−	0.664510i	$-0.768640\pi$
$821$	−1227.03	−1.49456	−0.747279	+	0.664510i	$0.768640\pi$
$822$	0	0
$823$	− 163.609i	− 0.198796i	−0.995048	−	0.0993982i	$-0.968308\pi$
$823$	− 163.609i	− 0.198796i	0.995048	−	0.0993982i	$-0.0316918\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	1444.29i	1.74642i	0.487347	+	0.873208i	$0.337965\pi$
$827$	1444.29i	1.74642i	−0.487347	+	0.873208i	$0.662035\pi$
$828$	0	0
$829$	−733.691	−0.885031	−0.442515	−	0.896761i	$-0.645914\pi$
$829$	−733.691	−0.885031	−0.442515	+	0.896761i	$0.645914\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−264.994	−0.318120
$834$	0	0
$835$	303.204i	0.363118i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	− 1349.11i	− 1.60800i	−0.594631	−	0.803999i	$-0.702702\pi$
$839$	− 1349.11i	− 1.60800i	0.594631	−	0.803999i	$-0.297298\pi$
$840$	0	0
$841$	−482.052	−0.573189
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−66.6802	−0.0789114
$846$	0	0
$847$	− 488.764i	− 0.577053i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	173.150i	0.203467i
$852$	0	0
$853$	1160.72	1.36075	0.680376	−	0.732863i	$-0.261817\pi$
$853$	1160.72	1.36075	0.680376	+	0.732863i	$0.261817\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−957.128	−1.11684	−0.558418	−	0.829560i	$-0.688591\pi$
$857$	−957.128	−1.11684	−0.558418	+	0.829560i	$0.688591\pi$
$858$	0	0
$859$	128.616i	0.149728i	0.997194	+	0.0748640i	$0.0238523\pi$
$859$	128.616i	0.149728i	−0.997194	+	0.0748640i	$0.976148\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 1172.59i	− 1.35873i	−0.733799	−	0.679366i	$-0.762255\pi$
$863$	− 1172.59i	− 1.35873i	0.733799	−	0.679366i	$-0.237745\pi$
$864$	0	0
$865$	31.2800	0.0361618
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−507.233	−0.583697
$870$	0	0
$871$	− 747.654i	− 0.858386i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 505.463i	− 0.577672i
$876$	0	0
$877$	−1282.16	−1.46198	−0.730991	−	0.682387i	$-0.760942\pi$
$877$	−1282.16	−1.46198	−0.730991	+	0.682387i	$0.760942\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−635.867	−0.721756	−0.360878	−	0.932613i	$-0.617523\pi$
$881$	−635.867	−0.721756	−0.360878	+	0.932613i	$0.617523\pi$
$882$	0	0
$883$	479.337i	0.542850i	0.962460	+	0.271425i	$0.0874949\pi$
$883$	479.337i	0.542850i	−0.962460	+	0.271425i	$0.912505\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	1242.57i	1.40087i	0.713716	+	0.700435i	$0.247011\pi$
$887$	1242.57i	1.40087i	−0.713716	+	0.700435i	$0.752989\pi$
$888$	0	0
$889$	−1501.09	−1.68851
$890$	0	0
$891$	0	0
$892$	0	0
$893$	142.340	0.159395
$894$	0	0
$895$	372.388i	0.416076i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 72.7748i	− 0.0809508i
$900$	0	0
$901$	−557.170	−0.618390
$902$	0	0
$903$	0	0
$904$	0	0
$905$	132.620	0.146541
$906$	0	0
$907$	− 97.9894i	− 0.108037i	−0.998540	−	0.0540184i	$-0.982797\pi$
$907$	− 97.9894i	− 0.108037i	0.998540	−	0.0540184i	$-0.0172030\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	− 1494.02i	− 1.63998i	−0.572376	−	0.819991i	$-0.693978\pi$
$911$	− 1494.02i	− 1.63998i	0.572376	−	0.819991i	$-0.306022\pi$
$912$	0	0
$913$	170.195	0.186413
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−466.414	−0.508631
$918$	0	0
$919$	1373.35i	1.49440i	0.664601	+	0.747198i	$0.268602\pi$
$919$	1373.35i	1.49440i	−0.664601	+	0.747198i	$0.731398\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	664.635i	0.720081i
$924$	0	0
$925$	970.198	1.04886
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−604.998	−0.651235	−0.325618	−	0.945502i	$-0.605572\pi$
$929$	−604.998	−0.651235	−0.325618	+	0.945502i	$0.605572\pi$
$930$	0	0
$931$	37.5757i	0.0403605i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	− 320.313i	− 0.342581i
$936$	0	0
$937$	−752.109	−0.802677	−0.401339	−	0.915930i	$-0.631455\pi$
$937$	−752.109	−0.802677	−0.401339	+	0.915930i	$0.631455\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−864.351	−0.918545	−0.459273	−	0.888295i	$-0.651890\pi$
$941$	−864.351	−0.918545	−0.459273	+	0.888295i	$0.651890\pi$
$942$	0	0
$943$	− 204.444i	− 0.216802i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	1437.86i	1.51834i	0.650895	+	0.759168i	$0.274394\pi$
$947$	1437.86i	1.51834i	−0.650895	+	0.759168i	$0.725606\pi$
$948$	0	0
$949$	1703.10	1.79463
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−632.945	−0.664161	−0.332080	−	0.943251i	$-0.607751\pi$
$953$	−632.945	−0.664161	−0.332080	+	0.943251i	$0.607751\pi$
$954$	0	0
$955$	304.293i	0.318632i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	− 1916.05i	− 1.99797i
$960$	0	0
$961$	946.245	0.984647
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−433.929	−0.449668
$966$	0	0
$967$	309.101i	0.319650i	0.987145	+	0.159825i	$0.0510930\pi$
$967$	309.101i	0.319650i	−0.987145	+	0.159825i	$0.948907\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	1362.86i	1.40357i	0.712390	+	0.701783i	$0.247613\pi$
$971$	1362.86i	1.40357i	−0.712390	+	0.701783i	$0.752387\pi$
$972$	0	0
$973$	−1688.51	−1.73537
$974$	0	0
$975$	0	0
$976$	0	0
$977$	1233.20	1.26223	0.631113	−	0.775691i	$-0.282598\pi$
$977$	1233.20	1.26223	0.631113	+	0.775691i	$0.282598\pi$
$978$	0	0
$979$	− 719.959i	− 0.735402i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 1015.07i	− 1.03263i	−0.856399	−	0.516314i	$-0.827304\pi$
$983$	− 1015.07i	− 1.03263i	0.856399	−	0.516314i	$-0.172696\pi$
$984$	0	0
$985$	−54.3172	−0.0551444
$986$	0	0
$987$	0	0
$988$	0	0
$989$	20.9271	0.0211598
$990$	0	0
$991$	− 1192.33i	− 1.20316i	−0.798812	−	0.601581i	$-0.794538\pi$
$991$	− 1192.33i	− 1.20316i	0.798812	−	0.601581i	$-0.205462\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	74.4818i	0.0748561i
$996$	0	0
$997$	−1335.83	−1.33985	−0.669925	−	0.742428i	$-0.733674\pi$
$997$	−1335.83	−1.33985	−0.669925	+	0.742428i	$0.733674\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2736.3.m.f.1711.3		12
3.2	odd	2		912.3.m.a.799.11	yes	12
4.3	odd	2	inner	2736.3.m.f.1711.4		12
12.11	even	2		912.3.m.a.799.5	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
912.3.m.a.799.5	✓	12	12.11	even	2
912.3.m.a.799.11	yes	12	3.2	odd	2
2736.3.m.f.1711.3		12	1.1	even	1	trivial
2736.3.m.f.1711.4		12	4.3	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:	\( N \)	\(=\)	\( 2736 = 2^{4} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	2736.m (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.38490 q^{5} -7.59081i q^{7} +O(q^{10})\)	\(q-1.38490 q^{5} -7.59081i q^{7} +7.52404i q^{11} +14.7359 q^{13} +30.7401 q^{17} -4.35890i q^{19} -4.11943i q^{23} -23.0821 q^{25} +18.9459 q^{29} -3.84118i q^{31} +10.5125i q^{35} -42.0326 q^{37} +49.6292 q^{41} +5.08009i q^{43} +32.6549i q^{47} -8.62045 q^{49} -18.1252 q^{53} -10.4200i q^{55} +75.5116i q^{59} -75.0766 q^{61} -20.4078 q^{65} -50.7368i q^{67} +45.1030i q^{71} +115.575 q^{73} +57.1136 q^{77} +67.4150i q^{79} -22.6202i q^{83} -42.5720 q^{85} -95.6879 q^{89} -111.858i q^{91} +6.03664i q^{95} +43.1069 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 10 q^{5}+O(q^{10}) \) 12 * q + 10 * q^5	\( 12 q + 10 q^{5} + 36 q^{13} + 10 q^{17} + 58 q^{25} - 12 q^{29} + 32 q^{37} - 136 q^{41} - 22 q^{49} + 236 q^{53} - 210 q^{61} + 52 q^{65} - 158 q^{73} + 70 q^{77} + 242 q^{85} - 444 q^{89} + 320 q^{97}+O(q^{100}) \) 12 * q + 10 * q^5 + 36 * q^13 + 10 * q^17 + 58 * q^25 - 12 * q^29 + 32 * q^37 - 136 * q^41 - 22 * q^49 + 236 * q^53 - 210 * q^61 + 52 * q^65 - 158 * q^73 + 70 * q^77 + 242 * q^85 - 444 * q^89 + 320 * q^97

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