LMFDB - Embedded newform 1386.2.c.b.197.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1386,2,Mod(197,1386)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1386, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("1386.197");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1386 = 2 \cdot 3^{2} \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1386.c (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:	$11.0672657201$
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} + 4x^{10} - 8x^{9} + 17x^{8} - 16x^{7} + 88x^{6} - 48x^{5} + 153x^{4} - 216x^{3} + 324x^{2} + 729 $ x^12 + 4x^10 - 8x^9 + 17x^8 - 16x^7 + 88x^6 - 48x^5 + 153x^4 - 216x^3 + 324*x^2 + 729
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{5}\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			197.4
Root			$-0.748321 - 1.56205i$ of defining polynomial
Character	$\chi$	$=$	1386.197
Dual form			1386.2.c.b.197.9

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} -1.15079i q^{5} -1.00000i q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{4} -1.15079i q^{5} -1.00000i q^{7} +1.00000 q^{8} -1.15079i q^{10} +(1.65877 + 2.87202i) q^{11} +0.143255i q^{13} -1.00000i q^{14} +1.00000 q^{16} +5.46996 q^{17} +3.61321i q^{19} -1.15079i q^{20} +(1.65877 + 2.87202i) q^{22} -9.38476i q^{23} +3.67567 q^{25} +0.143255i q^{26} -1.00000i q^{28} -8.59638 q^{29} +5.05969 q^{31} +1.00000 q^{32} +5.46996 q^{34} -1.15079 q^{35} +1.83514 q^{37} +3.61321i q^{38} -1.15079i q^{40} +10.3772 q^{41} -0.569763i q^{43} +(1.65877 + 2.87202i) q^{44} -9.38476i q^{46} -5.48829i q^{47} -1.00000 q^{49} +3.67567 q^{50} +0.143255i q^{52} -13.0766i q^{53} +(3.30510 - 1.90890i) q^{55} -1.00000i q^{56} -8.59638 q^{58} +1.86327i q^{59} -2.44484i q^{61} +5.05969 q^{62} +1.00000 q^{64} +0.164857 q^{65} -8.26424 q^{67} +5.46996 q^{68} -1.15079 q^{70} -7.08317i q^{71} +12.6683i q^{73} +1.83514 q^{74} +3.61321i q^{76} +(2.87202 - 1.65877i) q^{77} +10.8523i q^{79} -1.15079i q^{80} +10.3772 q^{82} -16.5772 q^{83} -6.29479i q^{85} -0.569763i q^{86} +(1.65877 + 2.87202i) q^{88} +17.0377i q^{89} +0.143255 q^{91} -9.38476i q^{92} -5.48829i q^{94} +4.15806 q^{95} -8.25745 q^{97} -1.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 12 q^{2} + 12 q^{4} + 12 q^{8}+O(q^{10}) $ 12 * q + 12 * q^2 + 12 * q^4 + 12 * q^8	$ 12 q + 12 q^{2} + 12 q^{4} + 12 q^{8} - 4 q^{11} + 12 q^{16} + 16 q^{17} - 4 q^{22} - 4 q^{25} + 16 q^{29} + 12 q^{32} + 16 q^{34} - 8 q^{35} + 24 q^{37} + 16 q^{41} - 4 q^{44} - 12 q^{49} - 4 q^{50} - 8 q^{55} + 16 q^{58} + 12 q^{64} - 48 q^{67} + 16 q^{68} - 8 q^{70} + 24 q^{74} + 8 q^{77} + 16 q^{82} + 16 q^{83} - 4 q^{88} - 48 q^{95} + 48 q^{97} - 12 q^{98}+O(q^{100}) $ 12 * q + 12 * q^2 + 12 * q^4 + 12 * q^8 - 4 * q^11 + 12 * q^16 + 16 * q^17 - 4 * q^22 - 4 * q^25 + 16 * q^29 + 12 * q^32 + 16 * q^34 - 8 * q^35 + 24 * q^37 + 16 * q^41 - 4 * q^44 - 12 * q^49 - 4 * q^50 - 8 * q^55 + 16 * q^58 + 12 * q^64 - 48 * q^67 + 16 * q^68 - 8 * q^70 + 24 * q^74 + 8 * q^77 + 16 * q^82 + 16 * q^83 - 4 * q^88 - 48 * q^95 + 48 * q^97 - 12 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1386\mathbb{Z}\right)^\times$.

$n$	$155$	$199$	$1135$
$\chi(n)$	$-1$	$1$	$-1$

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 $

$2$	1.00000			0.707107
$2$	1.00000			0.707107
$3$		0			0
$3$		0			0
$4$	1.00000			0.500000
$5$		−	1.15079i		−	0.514650i	−0.966325	−	0.257325i	$-0.917159\pi$
$5$		−	1.15079i		−	0.514650i	0.966325	−	0.257325i	$-0.0828412\pi$
$6$		0			0
$7$		−	1.00000i		−	0.377964i
$7$		−	1.00000i		−	0.377964i
$8$	1.00000			0.353553
$9$		0			0
$10$		−	1.15079i		−	0.363913i
$11$	1.65877	+	2.87202i	0.500137	+	0.865947i
$11$	1.65877	+	2.87202i	0.500137	+	0.865947i
$12$		0			0
$13$			0.143255i			0.0397319i	0.999803	+	0.0198659i	$0.00632394\pi$
$13$			0.143255i			0.0397319i	−0.999803	+	0.0198659i	$0.993676\pi$
$14$		−	1.00000i		−	0.267261i
$15$		0			0
$16$	1.00000			0.250000
$17$	5.46996			1.32666			0.663330	−	0.748327i	$-0.269143\pi$
$17$	5.46996			1.32666			0.663330	+	0.748327i	$0.269143\pi$
$18$		0			0
$19$			3.61321i			0.828928i	0.910066	+	0.414464i	$0.136031\pi$
$19$			3.61321i			0.828928i	−0.910066	+	0.414464i	$0.863969\pi$
$20$		−	1.15079i		−	0.257325i
$21$		0			0
$22$	1.65877	+	2.87202i	0.353650	+	0.612317i
$23$		−	9.38476i		−	1.95686i	−0.206584	−	0.978429i	$-0.566235\pi$
$23$		−	9.38476i		−	1.95686i	0.206584	−	0.978429i	$-0.433765\pi$
$24$		0			0
$25$	3.67567			0.735135
$26$			0.143255i			0.0280947i
$27$		0			0
$28$		−	1.00000i		−	0.188982i
$29$	−8.59638			−1.59631			−0.798154	−	0.602454i	$-0.794190\pi$
$29$	−8.59638			−1.59631			−0.798154	+	0.602454i	$0.794190\pi$
$30$		0			0
$31$	5.05969			0.908747			0.454374	−	0.890811i	$-0.349863\pi$
$31$	5.05969			0.908747			0.454374	+	0.890811i	$0.349863\pi$
$32$	1.00000			0.176777
$33$		0			0
$34$	5.46996			0.938090
$35$	−1.15079			−0.194520
$36$		0			0
$37$	1.83514			0.301696			0.150848	−	0.988557i	$-0.451800\pi$
$37$	1.83514			0.301696			0.150848	+	0.988557i	$0.451800\pi$
$38$			3.61321i			0.586141i
$39$		0			0
$40$		−	1.15079i		−	0.181956i
$41$	10.3772			1.62065			0.810325	−	0.585981i	$-0.199291\pi$
$41$	10.3772			1.62065			0.810325	+	0.585981i	$0.199291\pi$
$42$		0			0
$43$		−	0.569763i		−	0.0868881i	−0.999056	−	0.0434441i	$-0.986167\pi$
$43$		−	0.569763i		−	0.0868881i	0.999056	−	0.0434441i	$-0.0138330\pi$
$44$	1.65877	+	2.87202i	0.250068	+	0.432973i
$45$		0			0
$46$		−	9.38476i		−	1.38371i
$47$		−	5.48829i		−	0.800550i	−0.916395	−	0.400275i	$-0.868915\pi$
$47$		−	5.48829i		−	0.800550i	0.916395	−	0.400275i	$-0.131085\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$	3.67567			0.519819
$51$		0			0
$52$			0.143255i			0.0198659i
$53$		−	13.0766i		−	1.79622i	−0.439775	−	0.898108i	$-0.644942\pi$
$53$		−	13.0766i		−	1.79622i	0.439775	−	0.898108i	$-0.355058\pi$
$54$		0			0
$55$	3.30510	−	1.90890i	0.445660	−	0.257396i
$56$		−	1.00000i		−	0.133631i
$57$		0			0
$58$	−8.59638			−1.12876
$59$			1.86327i			0.242577i	0.992617	+	0.121289i	$0.0387026\pi$
$59$			1.86327i			0.242577i	−0.992617	+	0.121289i	$0.961297\pi$
$60$		0			0
$61$		−	2.44484i		−	0.313030i	−0.987676	−	0.156515i	$-0.949974\pi$
$61$		−	2.44484i		−	0.313030i	0.987676	−	0.156515i	$-0.0500259\pi$
$62$	5.05969			0.642581
$63$		0			0
$64$	1.00000			0.125000
$65$	0.164857			0.0204480
$66$		0			0
$67$	−8.26424			−1.00964			−0.504819	−	0.863225i	$-0.668441\pi$
$67$	−8.26424			−1.00964			−0.504819	+	0.863225i	$0.668441\pi$
$68$	5.46996			0.663330
$69$		0			0
$70$	−1.15079			−0.137546
$71$		−	7.08317i		−	0.840618i	−0.907381	−	0.420309i	$-0.861922\pi$
$71$		−	7.08317i		−	0.840618i	0.907381	−	0.420309i	$-0.138078\pi$
$72$		0			0
$73$			12.6683i			1.48271i	0.671114	+	0.741354i	$0.265816\pi$
$73$			12.6683i			1.48271i	−0.671114	+	0.741354i	$0.734184\pi$
$74$	1.83514			0.213331
$75$		0			0
$76$			3.61321i			0.414464i
$77$	2.87202	−	1.65877i	0.327297	−	0.189034i
$78$		0			0
$79$			10.8523i			1.22098i	0.792023	+	0.610492i	$0.209028\pi$
$79$			10.8523i			1.22098i	−0.792023	+	0.610492i	$0.790972\pi$
$80$		−	1.15079i		−	0.128663i
$81$		0			0
$82$	10.3772			1.14597
$83$	−16.5772			−1.81958			−0.909790	−	0.415070i	$-0.863757\pi$
$83$	−16.5772			−1.81958			−0.909790	+	0.415070i	$0.863757\pi$
$84$		0			0
$85$		−	6.29479i		−	0.682766i
$86$		−	0.569763i		−	0.0614392i
$87$		0			0
$88$	1.65877	+	2.87202i	0.176825	+	0.306158i
$89$			17.0377i			1.80599i	0.429649	+	0.902996i	$0.358637\pi$
$89$			17.0377i			1.80599i	−0.429649	+	0.902996i	$0.641363\pi$
$90$		0			0
$91$	0.143255			0.0150172
$92$		−	9.38476i		−	0.978429i
$93$		0			0
$94$		−	5.48829i		−	0.566074i
$95$	4.15806			0.426608
$96$		0			0
$97$	−8.25745			−0.838417			−0.419208	−	0.907890i	$-0.637692\pi$
$97$	−8.25745			−0.838417			−0.419208	+	0.907890i	$0.637692\pi$
$98$	−1.00000			−0.101015
$99$		0			0
$100$	3.67567			0.367567
$101$	14.6619			1.45892			0.729459	−	0.684024i	$-0.239772\pi$
$101$	14.6619			1.45892			0.729459	+	0.684024i	$0.239772\pi$
$102$		0			0
$103$	18.7689			1.84935			0.924677	−	0.380752i	$-0.124335\pi$
$103$	18.7689			1.84935			0.924677	+	0.380752i	$0.124335\pi$
$104$			0.143255i			0.0140473i
$105$		0			0
$106$		−	13.0766i		−	1.27012i
$107$	−9.78967			−0.946403			−0.473201	−	0.880954i	$-0.656902\pi$
$107$	−9.78967			−0.946403			−0.473201	+	0.880954i	$0.656902\pi$
$108$		0			0
$109$			5.77066i			0.552729i	0.961053	+	0.276364i	$0.0891296\pi$
$109$			5.77066i			0.552729i	−0.961053	+	0.276364i	$0.910870\pi$
$110$	3.30510	−	1.90890i	0.315129	−	0.182006i
$111$		0			0
$112$		−	1.00000i		−	0.0944911i
$113$			5.93292i			0.558122i	0.960273	+	0.279061i	$0.0900232\pi$
$113$			5.93292i			0.558122i	−0.960273	+	0.279061i	$0.909977\pi$
$114$		0			0
$115$	−10.7999			−1.00710
$116$	−8.59638			−0.798154
$117$		0			0
$118$			1.86327i			0.171528i
$119$		−	5.46996i		−	0.501430i
$120$		0			0
$121$	−5.49699	+	9.52801i	−0.499727	+	0.866183i
$122$		−	2.44484i		−	0.221346i
$123$		0			0
$124$	5.05969			0.454374
$125$		−	9.98391i		−	0.892988i
$126$		0			0
$127$			11.9538i			1.06073i	0.847769	+	0.530366i	$0.177946\pi$
$127$			11.9538i			1.06073i	−0.847769	+	0.530366i	$0.822054\pi$
$128$	1.00000			0.0883883
$129$		0			0
$130$	0.164857			0.0144589
$131$	−3.16510			−0.276536			−0.138268	−	0.990395i	$-0.544154\pi$
$131$	−3.16510			−0.276536			−0.138268	+	0.990395i	$0.544154\pi$
$132$		0			0
$133$	3.61321			0.313305
$134$	−8.26424			−0.713922
$135$		0			0
$136$	5.46996			0.469045
$137$		−	7.76237i		−	0.663184i	−0.943423	−	0.331592i	$-0.892414\pi$
$137$		−	7.76237i		−	0.663184i	0.943423	−	0.331592i	$-0.107586\pi$
$138$		0			0
$139$			12.5506i			1.06453i	0.846578	+	0.532265i	$0.178659\pi$
$139$			12.5506i			1.06453i	−0.846578	+	0.532265i	$0.821341\pi$
$140$	−1.15079			−0.0972598
$141$		0			0
$142$		−	7.08317i		−	0.594407i
$143$	−0.411432	+	0.237627i	−0.0344057	+	0.0198714i
$144$		0			0
$145$			9.89266i			0.821540i
$146$			12.6683i			1.04843i
$147$		0			0
$148$	1.83514			0.150848
$149$	6.49699			0.532254			0.266127	−	0.963938i	$-0.414256\pi$
$149$	6.49699			0.532254			0.266127	+	0.963938i	$0.414256\pi$
$150$		0			0
$151$			2.34721i			0.191013i	0.995429	+	0.0955067i	$0.0304471\pi$
$151$			2.34721i			0.191013i	−0.995429	+	0.0955067i	$0.969553\pi$
$152$			3.61321i			0.293070i
$153$		0			0
$154$	2.87202	−	1.65877i	0.231434	−	0.133667i
$155$		−	5.82266i		−	0.467687i
$156$		0			0
$157$	−18.5760			−1.48253			−0.741263	−	0.671215i	$-0.765773\pi$
$157$	−18.5760			−1.48253			−0.741263	+	0.671215i	$0.765773\pi$
$158$			10.8523i			0.863366i
$159$		0			0
$160$		−	1.15079i		−	0.0909782i
$161$	−9.38476			−0.739623
$162$		0			0
$163$	−9.24284			−0.723955			−0.361978	−	0.932187i	$-0.617898\pi$
$163$	−9.24284			−0.723955			−0.361978	+	0.932187i	$0.617898\pi$
$164$	10.3772			0.810325
$165$		0			0
$166$	−16.5772			−1.28664
$167$	−3.34432			−0.258791			−0.129396	−	0.991593i	$-0.541304\pi$
$167$	−3.34432			−0.258791			−0.129396	+	0.991593i	$0.541304\pi$
$168$		0			0
$169$	12.9795			0.998421
$170$		−	6.29479i		−	0.482789i
$171$		0			0
$172$		−	0.569763i		−	0.0434441i
$173$	−11.2721			−0.856998			−0.428499	−	0.903542i	$-0.640958\pi$
$173$	−11.2721			−0.856998			−0.428499	+	0.903542i	$0.640958\pi$
$174$		0			0
$175$		−	3.67567i		−	0.277855i
$176$	1.65877	+	2.87202i	0.125034	+	0.216487i
$177$		0			0
$178$			17.0377i			1.27703i
$179$		−	17.5972i		−	1.31528i	−0.753332	−	0.657640i	$-0.771555\pi$
$179$		−	17.5972i		−	1.31528i	0.753332	−	0.657640i	$-0.228445\pi$
$180$		0			0
$181$	10.2124			0.759079			0.379540	−	0.925176i	$-0.376082\pi$
$181$	10.2124			0.759079			0.379540	+	0.925176i	$0.376082\pi$
$182$	0.143255			0.0106188
$183$		0			0
$184$		−	9.38476i		−	0.691854i
$185$		−	2.11187i		−	0.155268i
$186$		0			0
$187$	9.07338	+	15.7098i	0.663511	+	1.14882i
$188$		−	5.48829i		−	0.400275i
$189$		0			0
$190$	4.15806			0.301658
$191$			16.6717i			1.20632i	0.797618	+	0.603162i	$0.206093\pi$
$191$			16.6717i			1.20632i	−0.797618	+	0.603162i	$0.793907\pi$
$192$		0			0
$193$		−	21.8366i		−	1.57184i	−0.618331	−	0.785918i	$-0.712191\pi$
$193$		−	21.8366i		−	1.57184i	0.618331	−	0.785918i	$-0.287809\pi$
$194$	−8.25745			−0.592850
$195$		0			0
$196$	−1.00000			−0.0714286
$197$	−24.2527			−1.72793			−0.863966	−	0.503550i	$-0.832027\pi$
$197$	−24.2527			−1.72793			−0.863966	+	0.503550i	$0.832027\pi$
$198$		0			0
$199$	−16.4360			−1.16512			−0.582558	−	0.812789i	$-0.697948\pi$
$199$	−16.4360			−1.16512			−0.582558	+	0.812789i	$0.697948\pi$
$200$	3.67567			0.259909
$201$		0			0
$202$	14.6619			1.03161
$203$			8.59638i			0.603348i
$204$		0			0
$205$		−	11.9420i		−	0.834068i
$206$	18.7689			1.30769
$207$		0			0
$208$			0.143255i			0.00993296i
$209$	−10.3772	+	5.99347i	−0.717807	+	0.414577i
$210$		0			0
$211$			7.57023i			0.521156i	0.965453	+	0.260578i	$0.0839132\pi$
$211$			7.57023i			0.521156i	−0.965453	+	0.260578i	$0.916087\pi$
$212$		−	13.0766i		−	0.898108i
$213$		0			0
$214$	−9.78967			−0.669208
$215$	−0.655680			−0.0447170
$216$		0			0
$217$		−	5.05969i		−	0.343474i
$218$			5.77066i			0.390838i
$219$		0			0
$220$	3.30510	−	1.90890i	0.222830	−	0.128698i
$221$			0.783600i			0.0527107i
$222$		0			0
$223$	−27.5906			−1.84760			−0.923802	−	0.382872i	$-0.874935\pi$
$223$	−27.5906			−1.84760			−0.923802	+	0.382872i	$0.874935\pi$
$224$		−	1.00000i		−	0.0668153i
$225$		0			0
$226$			5.93292i			0.394652i
$227$	13.2040			0.876382			0.438191	−	0.898882i	$-0.355619\pi$
$227$	13.2040			0.876382			0.438191	+	0.898882i	$0.355619\pi$
$228$		0			0
$229$	−14.7899			−0.977345			−0.488672	−	0.872467i	$-0.662519\pi$
$229$	−14.7899			−0.977345			−0.488672	+	0.872467i	$0.662519\pi$
$230$	−10.7999			−0.712126
$231$		0			0
$232$	−8.59638			−0.564380
$233$	27.3915			1.79448			0.897239	−	0.441544i	$-0.145569\pi$
$233$	27.3915			1.79448			0.897239	+	0.441544i	$0.145569\pi$
$234$		0			0
$235$	−6.31589			−0.412003
$236$			1.86327i			0.121289i
$237$		0			0
$238$		−	5.46996i		−	0.354565i
$239$	−6.93096			−0.448326			−0.224163	−	0.974552i	$-0.571965\pi$
$239$	−6.93096			−0.448326			−0.224163	+	0.974552i	$0.571965\pi$
$240$		0			0
$241$		−	9.11277i		−	0.587005i	−0.955958	−	0.293503i	$-0.905179\pi$
$241$		−	9.11277i		−	0.587005i	0.955958	−	0.293503i	$-0.0948210\pi$
$242$	−5.49699	+	9.52801i	−0.353360	+	0.612484i
$243$		0			0
$244$		−	2.44484i		−	0.156515i
$245$			1.15079i			0.0735215i
$246$		0			0
$247$	−0.517612			−0.0329348
$248$	5.05969			0.321291
$249$		0			0
$250$		−	9.98391i		−	0.631438i
$251$		−	10.2935i		−	0.649722i	−0.945762	−	0.324861i	$-0.894682\pi$
$251$		−	10.2935i		−	0.649722i	0.945762	−	0.324861i	$-0.105318\pi$
$252$		0			0
$253$	26.9532	−	15.5671i	1.69453	−	0.978696i
$254$			11.9538i			0.750051i
$255$		0			0
$256$	1.00000			0.0625000
$257$			13.8535i			0.864156i	0.901836	+	0.432078i	$0.142220\pi$
$257$			13.8535i			0.864156i	−0.901836	+	0.432078i	$0.857780\pi$
$258$		0			0
$259$		−	1.83514i		−	0.114030i
$260$	0.164857			0.0102240
$261$		0			0
$262$	−3.16510			−0.195541
$263$	−23.6425			−1.45786			−0.728929	−	0.684590i	$-0.759981\pi$
$263$	−23.6425			−1.45786			−0.728929	+	0.684590i	$0.759981\pi$
$264$		0			0
$265$	−15.0485			−0.924423
$266$	3.61321			0.221540
$267$		0			0
$268$	−8.26424			−0.504819
$269$			25.9952i			1.58495i	0.609902	+	0.792477i	$0.291209\pi$
$269$			25.9952i			1.58495i	−0.609902	+	0.792477i	$0.708791\pi$
$270$		0			0
$271$		−	2.78687i		−	0.169290i	−0.996411	−	0.0846451i	$-0.973024\pi$
$271$		−	2.78687i		−	0.169290i	0.996411	−	0.0846451i	$-0.0269756\pi$
$272$	5.46996			0.331665
$273$		0			0
$274$		−	7.76237i		−	0.468942i
$275$	6.09708	+	10.5566i	0.367668	+	0.636587i
$276$		0			0
$277$		−	6.05717i		−	0.363940i	−0.983304	−	0.181970i	$-0.941753\pi$
$277$		−	6.05717i		−	0.363940i	0.983304	−	0.181970i	$-0.0582473\pi$
$278$			12.5506i			0.752737i
$279$		0			0
$280$	−1.15079			−0.0687731
$281$	−21.0235			−1.25416			−0.627080	−	0.778955i	$-0.715750\pi$
$281$	−21.0235			−1.25416			−0.627080	+	0.778955i	$0.715750\pi$
$282$		0			0
$283$			26.3403i			1.56577i	0.622168	+	0.782884i	$0.286252\pi$
$283$			26.3403i			1.56577i	−0.622168	+	0.782884i	$0.713748\pi$
$284$		−	7.08317i		−	0.420309i
$285$		0			0
$286$	−0.411432	+	0.237627i	−0.0243285	+	0.0140512i
$287$		−	10.3772i		−	0.612548i
$288$		0			0
$289$	12.9204			0.760026
$290$			9.89266i			0.580917i
$291$		0			0
$292$			12.6683i			0.741354i
$293$	13.3538			0.780136			0.390068	−	0.920786i	$-0.372452\pi$
$293$	13.3538			0.780136			0.390068	+	0.920786i	$0.372452\pi$
$294$		0			0
$295$	2.14424			0.124842
$296$	1.83514			0.106666
$297$		0			0
$298$	6.49699			0.376361
$299$	1.34442			0.0777496
$300$		0			0
$301$	−0.569763			−0.0328406
$302$			2.34721i			0.135067i
$303$		0			0
$304$			3.61321i			0.207232i
$305$	−2.81351			−0.161101
$306$		0			0
$307$		−	11.6014i		−	0.662127i	−0.943608	−	0.331064i	$-0.892593\pi$
$307$		−	11.6014i		−	0.662127i	0.943608	−	0.331064i	$-0.107407\pi$
$308$	2.87202	−	1.65877i	0.163649	−	0.0945169i
$309$		0			0
$310$		−	5.82266i		−	0.330705i
$311$			25.0959i			1.42306i	0.702656	+	0.711530i	$0.251997\pi$
$311$			25.0959i			1.42306i	−0.702656	+	0.711530i	$0.748003\pi$
$312$		0			0
$313$	8.49467			0.480147			0.240073	−	0.970755i	$-0.422828\pi$
$313$	8.49467			0.480147			0.240073	+	0.970755i	$0.422828\pi$
$314$	−18.5760			−1.04830
$315$		0			0
$316$			10.8523i			0.610492i
$317$		−	10.8746i		−	0.610779i	−0.952228	−	0.305389i	$-0.901213\pi$
$317$		−	10.8746i		−	0.610779i	0.952228	−	0.305389i	$-0.0987866\pi$
$318$		0			0
$319$	−14.2594	−	24.6890i	−0.798372	−	1.38232i
$320$		−	1.15079i		−	0.0643313i
$321$		0			0
$322$	−9.38476			−0.522992
$323$			19.7641i			1.09971i
$324$		0			0
$325$			0.526560i			0.0292083i
$326$	−9.24284			−0.511914
$327$		0			0
$328$	10.3772			0.572986
$329$	−5.48829			−0.302579
$330$		0			0
$331$	−24.1944			−1.32984			−0.664922	−	0.746913i	$-0.731535\pi$
$331$	−24.1944			−1.32984			−0.664922	+	0.746913i	$0.731535\pi$
$332$	−16.5772			−0.909790
$333$		0			0
$334$	−3.34432			−0.182993
$335$			9.51044i			0.519611i
$336$		0			0
$337$		−	8.92360i		−	0.486099i	−0.970014	−	0.243050i	$-0.921852\pi$
$337$		−	8.92360i		−	0.486099i	0.970014	−	0.243050i	$-0.0781478\pi$
$338$	12.9795			0.705991
$339$		0			0
$340$		−	6.29479i		−	0.341383i
$341$	8.39284	+	14.5315i	0.454498	+	0.786926i
$342$		0			0
$343$			1.00000i			0.0539949i
$344$		−	0.569763i		−	0.0307196i
$345$		0			0
$346$	−11.2721			−0.605989
$347$	−14.4562			−0.776051			−0.388025	−	0.921649i	$-0.626843\pi$
$347$	−14.4562			−0.776051			−0.388025	+	0.921649i	$0.626843\pi$
$348$		0			0
$349$		−	15.0950i		−	0.808016i	−0.914755	−	0.404008i	$-0.867617\pi$
$349$		−	15.0950i		−	0.808016i	0.914755	−	0.404008i	$-0.132383\pi$
$350$		−	3.67567i		−	0.196473i
$351$		0			0
$352$	1.65877	+	2.87202i	0.0884125	+	0.153079i
$353$		−	20.0091i		−	1.06498i	−0.846437	−	0.532488i	$-0.821257\pi$
$353$		−	20.0091i		−	1.06498i	0.846437	−	0.532488i	$-0.178743\pi$
$354$		0			0
$355$	−8.15127			−0.432624
$356$			17.0377i			0.902996i
$357$		0			0
$358$		−	17.5972i		−	0.930043i
$359$	−9.27682			−0.489612			−0.244806	−	0.969572i	$-0.578724\pi$
$359$	−9.27682			−0.489612			−0.244806	+	0.969572i	$0.578724\pi$
$360$		0			0
$361$	5.94469			0.312878
$362$	10.2124			0.536750
$363$		0			0
$364$	0.143255			0.00750861
$365$	14.5785			0.763076
$366$		0			0
$367$	−10.8754			−0.567690			−0.283845	−	0.958870i	$-0.591610\pi$
$367$	−10.8754			−0.567690			−0.283845	+	0.958870i	$0.591610\pi$
$368$		−	9.38476i		−	0.489214i
$369$		0			0
$370$		−	2.11187i		−	0.109791i
$371$	−13.0766			−0.678906
$372$		0			0
$373$			2.51248i			0.130091i	0.997882	+	0.0650456i	$0.0207193\pi$
$373$			2.51248i			0.130091i	−0.997882	+	0.0650456i	$0.979281\pi$
$374$	9.07338	+	15.7098i	0.469173	+	0.812336i
$375$		0			0
$376$		−	5.48829i		−	0.283037i
$377$		−	1.23148i		−	0.0634243i
$378$		0			0
$379$	0.239642			0.0123096			0.00615478	−	0.999981i	$-0.498041\pi$
$379$	0.239642			0.0123096			0.00615478	+	0.999981i	$0.498041\pi$
$380$	4.15806			0.213304
$381$		0			0
$382$			16.6717i			0.853000i
$383$			14.0879i			0.719860i	0.932979	+	0.359930i	$0.117199\pi$
$383$			14.0879i			0.719860i	−0.932979	+	0.359930i	$0.882801\pi$
$384$		0			0
$385$	−1.90890	−	3.30510i	−0.0972864	−	0.168444i
$386$		−	21.8366i		−	1.11146i
$387$		0			0
$388$	−8.25745			−0.419208
$389$			10.2059i			0.517458i	0.965950	+	0.258729i	$0.0833037\pi$
$389$			10.2059i			0.517458i	−0.965950	+	0.258729i	$0.916696\pi$
$390$		0			0
$391$		−	51.3342i		−	2.59608i
$392$	−1.00000			−0.0505076
$393$		0			0
$394$	−24.2527			−1.22183
$395$	12.4888			0.628380
$396$		0			0
$397$	25.1802			1.26376			0.631880	−	0.775066i	$-0.282284\pi$
$397$	25.1802			1.26376			0.631880	+	0.775066i	$0.282284\pi$
$398$	−16.4360			−0.823862
$399$		0			0
$400$	3.67567			0.183784
$401$			27.3876i			1.36767i	0.729636	+	0.683835i	$0.239689\pi$
$401$			27.3876i			1.36767i	−0.729636	+	0.683835i	$0.760311\pi$
$402$		0			0
$403$			0.724827i			0.0361062i
$404$	14.6619			0.729459
$405$		0			0
$406$			8.59638i			0.426631i
$407$	3.04407	+	5.27057i	0.150889	+	0.261252i
$408$		0			0
$409$		−	15.5159i		−	0.767212i	−0.923497	−	0.383606i	$-0.874682\pi$
$409$		−	15.5159i		−	0.767212i	0.923497	−	0.383606i	$-0.125318\pi$
$410$		−	11.9420i		−	0.589775i
$411$		0			0
$412$	18.7689			0.924677
$413$	1.86327			0.0916855
$414$		0			0
$415$			19.0769i			0.936447i
$416$			0.143255i			0.00702367i
$417$		0			0
$418$	−10.3772	+	5.99347i	−0.507566	+	0.293150i
$419$		−	12.8834i		−	0.629396i	−0.949192	−	0.314698i	$-0.898097\pi$
$419$		−	12.8834i		−	0.629396i	0.949192	−	0.314698i	$-0.101903\pi$
$420$		0			0
$421$	−2.34110			−0.114098			−0.0570490	−	0.998371i	$-0.518169\pi$
$421$	−2.34110			−0.114098			−0.0570490	+	0.998371i	$0.518169\pi$
$422$			7.57023i			0.368513i
$423$		0			0
$424$		−	13.0766i		−	0.635058i
$425$	20.1058			0.975274
$426$		0			0
$427$	−2.44484			−0.118314
$428$	−9.78967			−0.473201
$429$		0			0
$430$	−0.655680			−0.0316197
$431$	−1.13836			−0.0548328			−0.0274164	−	0.999624i	$-0.508728\pi$
$431$	−1.13836			−0.0548328			−0.0274164	+	0.999624i	$0.508728\pi$
$432$		0			0
$433$	31.9390			1.53489			0.767445	−	0.641114i	$-0.221528\pi$
$433$	31.9390			1.53489			0.767445	+	0.641114i	$0.221528\pi$
$434$		−	5.05969i		−	0.242873i
$435$		0			0
$436$			5.77066i			0.276364i
$437$	33.9091			1.62209
$438$		0			0
$439$			7.64917i			0.365075i	0.983199	+	0.182538i	$0.0584311\pi$
$439$			7.64917i			0.365075i	−0.983199	+	0.182538i	$0.941569\pi$
$440$	3.30510	−	1.90890i	0.157565	−	0.0910031i
$441$		0			0
$442$			0.783600i			0.0372721i
$443$			4.94335i			0.234866i	0.993081	+	0.117433i	$0.0374665\pi$
$443$			4.94335i			0.234866i	−0.993081	+	0.117433i	$0.962534\pi$
$444$		0			0
$445$	19.6069			0.929455
$446$	−27.5906			−1.30645
$447$		0			0
$448$		−	1.00000i		−	0.0472456i
$449$			12.2684i			0.578983i	0.957181	+	0.289491i	$0.0934861\pi$
$449$			12.2684i			0.578983i	−0.957181	+	0.289491i	$0.906514\pi$
$450$		0			0
$451$	17.2134	+	29.8036i	0.810546	+	1.40340i
$452$			5.93292i			0.279061i
$453$		0			0
$454$	13.2040			0.619696
$455$		−	0.164857i		−	0.00772862i
$456$		0			0
$457$		−	10.5999i		−	0.495843i	−0.968780	−	0.247921i	$-0.920253\pi$
$457$		−	10.5999i		−	0.495843i	0.968780	−	0.247921i	$-0.0797475\pi$
$458$	−14.7899			−0.691087
$459$		0			0
$460$	−10.7999			−0.503549
$461$	3.96146			0.184503			0.0922517	−	0.995736i	$-0.470594\pi$
$461$	3.96146			0.184503			0.0922517	+	0.995736i	$0.470594\pi$
$462$		0			0
$463$	−0.411528			−0.0191253			−0.00956266	−	0.999954i	$-0.503044\pi$
$463$	−0.411528			−0.0191253			−0.00956266	+	0.999954i	$0.503044\pi$
$464$	−8.59638			−0.399077
$465$		0			0
$466$	27.3915			1.26889
$467$		−	2.07385i		−	0.0959663i	−0.998848	−	0.0479832i	$-0.984721\pi$
$467$		−	2.07385i		−	0.0959663i	0.998848	−	0.0479832i	$-0.0152794\pi$
$468$		0			0
$469$			8.26424i			0.381607i
$470$	−6.31589			−0.291330
$471$		0			0
$472$			1.86327i			0.0857640i
$473$	1.63637	−	0.945104i	0.0752405	−	0.0434559i
$474$		0			0
$475$			13.2810i			0.609374i
$476$		−	5.46996i		−	0.250715i
$477$		0			0
$478$	−6.93096			−0.317015
$479$	1.37120			0.0626519			0.0313260	−	0.999509i	$-0.490027\pi$
$479$	1.37120			0.0626519			0.0313260	+	0.999509i	$0.490027\pi$
$480$		0			0
$481$			0.262894i			0.0119869i
$482$		−	9.11277i		−	0.415075i
$483$		0			0
$484$	−5.49699	+	9.52801i	−0.249863	+	0.433092i
$485$			9.50262i			0.431492i
$486$		0			0
$487$	19.1410			0.867362			0.433681	−	0.901066i	$-0.357214\pi$
$487$	19.1410			0.867362			0.433681	+	0.901066i	$0.357214\pi$
$488$		−	2.44484i		−	0.110673i
$489$		0			0
$490$			1.15079i			0.0519875i
$491$	−16.4375			−0.741815			−0.370908	−	0.928670i	$-0.620953\pi$
$491$	−16.4375			−0.741815			−0.370908	+	0.928670i	$0.620953\pi$
$492$		0			0
$493$	−47.0218			−2.11776
$494$	−0.517612			−0.0232885
$495$		0			0
$496$	5.05969			0.227187
$497$	−7.08317			−0.317724
$498$		0			0
$499$	12.3791			0.554165			0.277083	−	0.960846i	$-0.410632\pi$
$499$	12.3791			0.554165			0.277083	+	0.960846i	$0.410632\pi$
$500$		−	9.98391i		−	0.446494i
$501$		0			0
$502$		−	10.2935i		−	0.459423i
$503$	−33.8364			−1.50869			−0.754345	−	0.656478i	$-0.772045\pi$
$503$	−33.8364			−1.50869			−0.754345	+	0.656478i	$0.772045\pi$
$504$		0			0
$505$		−	16.8729i		−	0.750833i
$506$	26.9532	−	15.5671i	1.19822	−	0.692043i
$507$		0			0
$508$			11.9538i			0.530366i
$509$		−	12.2742i		−	0.544046i	−0.962291	−	0.272023i	$-0.912307\pi$
$509$		−	12.2742i		−	0.544046i	0.962291	−	0.272023i	$-0.0876926\pi$
$510$		0			0
$511$	12.6683			0.560411
$512$	1.00000			0.0441942
$513$		0			0
$514$			13.8535i			0.611051i
$515$		−	21.5991i		−	0.951771i
$516$		0			0
$517$	15.7625	−	9.10379i	0.693233	−	0.400384i
$518$		−	1.83514i		−	0.0806315i
$519$		0			0
$520$	0.164857			0.00722947
$521$			4.58836i			0.201020i	0.994936	+	0.100510i	$0.0320474\pi$
$521$			4.58836i			0.201020i	−0.994936	+	0.100510i	$0.967953\pi$
$522$		0			0
$523$			19.4618i			0.851006i	0.904957	+	0.425503i	$0.139903\pi$
$523$			19.4618i			0.851006i	−0.904957	+	0.425503i	$0.860097\pi$
$524$	−3.16510			−0.138268
$525$		0			0
$526$	−23.6425			−1.03086
$527$	27.6763			1.20560
$528$		0			0
$529$	−65.0737			−2.82929
$530$	−15.0485			−0.653666
$531$		0			0
$532$	3.61321			0.156653
$533$			1.48659i			0.0643914i
$534$		0			0
$535$			11.2659i			0.487067i
$536$	−8.26424			−0.356961
$537$		0			0
$538$			25.9952i			1.12073i
$539$	−1.65877	−	2.87202i	−0.0714481	−	0.123707i
$540$		0			0
$541$		−	1.60893i		−	0.0691732i	−0.999402	−	0.0345866i	$-0.988989\pi$
$541$		−	1.60893i		−	0.0691732i	0.999402	−	0.0345866i	$-0.0110115\pi$
$542$		−	2.78687i		−	0.119706i
$543$		0			0
$544$	5.46996			0.234523
$545$	6.64083			0.284462
$546$		0			0
$547$		−	17.3649i		−	0.742470i	−0.928539	−	0.371235i	$-0.878934\pi$
$547$		−	17.3649i		−	0.742470i	0.928539	−	0.371235i	$-0.121066\pi$
$548$		−	7.76237i		−	0.331592i
$549$		0			0
$550$	6.09708	+	10.5566i	0.259980	+	0.450135i
$551$		−	31.0606i		−	1.32322i
$552$		0			0
$553$	10.8523			0.461488
$554$		−	6.05717i		−	0.257344i
$555$		0			0
$556$			12.5506i			0.532265i
$557$	−5.60396			−0.237447			−0.118724	−	0.992927i	$-0.537880\pi$
$557$	−5.60396			−0.237447			−0.118724	+	0.992927i	$0.537880\pi$
$558$		0			0
$559$	0.0816216			0.00345223
$560$	−1.15079			−0.0486299
$561$		0			0
$562$	−21.0235			−0.886825
$563$	46.3765			1.95454			0.977269	−	0.212004i	$-0.0679989\pi$
$563$	46.3765			1.95454			0.977269	+	0.212004i	$0.0679989\pi$
$564$		0			0
$565$	6.82757			0.287238
$566$			26.3403i			1.10717i
$567$		0			0
$568$		−	7.08317i		−	0.297203i
$569$	1.56795			0.0657319			0.0328659	−	0.999460i	$-0.489537\pi$
$569$	1.56795			0.0657319			0.0328659	+	0.999460i	$0.489537\pi$
$570$		0			0
$571$		−	29.5685i		−	1.23740i	−0.785626	−	0.618702i	$-0.787659\pi$
$571$		−	29.5685i		−	1.23740i	0.785626	−	0.618702i	$-0.212341\pi$
$572$	−0.411432	+	0.237627i	−0.0172028	+	0.00993568i
$573$		0			0
$574$		−	10.3772i		−	0.433137i
$575$		−	34.4953i		−	1.43855i
$576$		0			0
$577$	24.8720			1.03544			0.517718	−	0.855552i	$-0.326782\pi$
$577$	24.8720			1.03544			0.517718	+	0.855552i	$0.326782\pi$
$578$	12.9204			0.537420
$579$		0			0
$580$			9.89266i			0.410770i
$581$			16.5772i			0.687736i
$582$		0			0
$583$	37.5564	−	21.6911i	1.55543	−	0.898353i
$584$			12.6683i			0.524216i
$585$		0			0
$586$	13.3538			0.551639
$587$		−	34.3530i		−	1.41790i	−0.705258	−	0.708951i	$-0.749169\pi$
$587$		−	34.3530i		−	1.41790i	0.705258	−	0.708951i	$-0.250831\pi$
$588$		0			0
$589$			18.2817i			0.753286i
$590$	2.14424			0.0882769
$591$		0			0
$592$	1.83514			0.0754239
$593$	13.9639			0.573428			0.286714	−	0.958016i	$-0.407437\pi$
$593$	13.9639			0.573428			0.286714	+	0.958016i	$0.407437\pi$
$594$		0			0
$595$	−6.29479			−0.258061
$596$	6.49699			0.266127
$597$		0			0
$598$	1.34442			0.0549773
$599$		−	20.4290i		−	0.834705i	−0.908745	−	0.417353i	$-0.862958\pi$
$599$		−	20.4290i		−	0.834705i	0.908745	−	0.417353i	$-0.137042\pi$
$600$		0			0
$601$		−	32.9645i		−	1.34465i	−0.740256	−	0.672325i	$-0.765296\pi$
$601$		−	32.9645i		−	1.34465i	0.740256	−	0.672325i	$-0.234704\pi$
$602$	−0.569763			−0.0232218
$603$		0			0
$604$			2.34721i			0.0955067i
$605$	10.9648	+	6.32590i	0.445782	+	0.257185i
$606$		0			0
$607$			33.6230i			1.36472i	0.731018	+	0.682358i	$0.239046\pi$
$607$			33.6230i			1.36472i	−0.731018	+	0.682358i	$0.760954\pi$
$608$			3.61321i			0.146535i
$609$		0			0
$610$	−2.81351			−0.113916
$611$	0.786227			0.0318073
$612$		0			0
$613$		−	23.6780i		−	0.956344i	−0.878266	−	0.478172i	$-0.841300\pi$
$613$		−	23.6780i		−	0.956344i	0.878266	−	0.478172i	$-0.158700\pi$
$614$		−	11.6014i		−	0.468195i
$615$		0			0
$616$	2.87202	−	1.65877i	0.115717	−	0.0668336i
$617$		−	31.0257i		−	1.24905i	−0.781005	−	0.624525i	$-0.785293\pi$
$617$		−	31.0257i		−	1.24905i	0.781005	−	0.624525i	$-0.214707\pi$
$618$		0			0
$619$	13.7959			0.554505			0.277253	−	0.960797i	$-0.410576\pi$
$619$	13.7959			0.554505			0.277253	+	0.960797i	$0.410576\pi$
$620$		−	5.82266i		−	0.233844i
$621$		0			0
$622$			25.0959i			1.00625i
$623$	17.0377			0.682601
$624$		0			0
$625$	6.88896			0.275558
$626$	8.49467			0.339515
$627$		0			0
$628$	−18.5760			−0.741263
$629$	10.0382			0.400247
$630$		0			0
$631$	11.9596			0.476105			0.238053	−	0.971252i	$-0.423491\pi$
$631$	11.9596			0.476105			0.238053	+	0.971252i	$0.423491\pi$
$632$			10.8523i			0.431683i
$633$		0			0
$634$		−	10.8746i		−	0.431886i
$635$	13.7564			0.545906
$636$		0			0
$637$		−	0.143255i		−	0.00567598i
$638$	−14.2594	−	24.6890i	−0.564534	−	0.977446i
$639$		0			0
$640$		−	1.15079i		−	0.0454891i
$641$			10.9956i			0.434300i	0.976138	+	0.217150i	$0.0696761\pi$
$641$			10.9956i			0.434300i	−0.976138	+	0.217150i	$0.930324\pi$
$642$		0			0
$643$	−26.4305			−1.04232			−0.521159	−	0.853460i	$-0.674500\pi$
$643$	−26.4305			−1.04232			−0.521159	+	0.853460i	$0.674500\pi$
$644$	−9.38476			−0.369811
$645$		0			0
$646$			19.7641i			0.777609i
$647$			3.71578i			0.146082i	0.997329	+	0.0730411i	$0.0232704\pi$
$647$			3.71578i			0.146082i	−0.997329	+	0.0730411i	$0.976730\pi$
$648$		0			0
$649$	−5.35135	+	3.09073i	−0.210059	+	0.121322i
$650$			0.526560i			0.0206534i
$651$		0			0
$652$	−9.24284			−0.361978
$653$			14.1287i			0.552898i	0.961029	+	0.276449i	$0.0891577\pi$
$653$			14.1287i			0.552898i	−0.961029	+	0.276449i	$0.910842\pi$
$654$		0			0
$655$			3.64238i			0.142320i
$656$	10.3772			0.405162
$657$		0			0
$658$	−5.48829			−0.213956
$659$	−30.8720			−1.20260			−0.601301	−	0.799022i	$-0.705351\pi$
$659$	−30.8720			−1.20260			−0.601301	+	0.799022i	$0.705351\pi$
$660$		0			0
$661$	−19.2123			−0.747272			−0.373636	−	0.927575i	$-0.621889\pi$
$661$	−19.2123			−0.747272			−0.373636	+	0.927575i	$0.621889\pi$
$662$	−24.1944			−0.940342
$663$		0			0
$664$	−16.5772			−0.643318
$665$		−	4.15806i		−	0.161243i
$666$		0			0
$667$			80.6749i			3.12375i
$668$	−3.34432			−0.129396
$669$		0			0
$670$			9.51044i			0.367420i
$671$	7.02163	−	4.05542i	0.271067	−	0.156558i
$672$		0			0
$673$			10.8853i			0.419599i	0.977744	+	0.209799i	$0.0672811\pi$
$673$			10.8853i			0.419599i	−0.977744	+	0.209799i	$0.932719\pi$
$674$		−	8.92360i		−	0.343724i
$675$		0			0
$676$	12.9795			0.499211
$677$	−16.0044			−0.615097			−0.307549	−	0.951532i	$-0.599509\pi$
$677$	−16.0044			−0.615097			−0.307549	+	0.951532i	$0.599509\pi$
$678$		0			0
$679$			8.25745i			0.316892i
$680$		−	6.29479i		−	0.241394i
$681$		0			0
$682$	8.39284	+	14.5315i	0.321378	+	0.556441i
$683$		−	16.4957i		−	0.631192i	−0.948894	−	0.315596i	$-0.897796\pi$
$683$		−	16.4957i		−	0.631192i	0.948894	−	0.315596i	$-0.102204\pi$
$684$		0			0
$685$	−8.93289			−0.341308
$686$			1.00000i			0.0381802i
$687$		0			0
$688$		−	0.569763i		−	0.0217220i
$689$	1.87330			0.0713670
$690$		0			0
$691$	−31.2134			−1.18741			−0.593706	−	0.804682i	$-0.702336\pi$
$691$	−31.2134			−1.18741			−0.593706	+	0.804682i	$0.702336\pi$
$692$	−11.2721			−0.428499
$693$		0			0
$694$	−14.4562			−0.548751
$695$	14.4432			0.547861
$696$		0			0
$697$	56.7630			2.15005
$698$		−	15.0950i		−	0.571353i
$699$		0			0
$700$		−	3.67567i		−	0.138927i
$701$	35.7531			1.35038			0.675189	−	0.737645i	$-0.264062\pi$
$701$	35.7531			1.35038			0.675189	+	0.737645i	$0.264062\pi$
$702$		0			0
$703$			6.63076i			0.250084i
$704$	1.65877	+	2.87202i	0.0625171	+	0.108243i
$705$		0			0
$706$		−	20.0091i		−	0.753052i
$707$		−	14.6619i		−	0.551419i
$708$		0			0
$709$	2.15599			0.0809701			0.0404850	−	0.999180i	$-0.487110\pi$
$709$	2.15599			0.0809701			0.0404850	+	0.999180i	$0.487110\pi$
$710$	−8.15127			−0.305912
$711$		0			0
$712$			17.0377i			0.638515i
$713$		−	47.4840i		−	1.77829i
$714$		0			0
$715$	0.273459	+	0.473473i	0.0102268	+	0.0177069i
$716$		−	17.5972i		−	0.657640i
$717$		0			0
$718$	−9.27682			−0.346208
$719$			50.3699i			1.87848i	0.343260	+	0.939240i	$0.388469\pi$
$719$			50.3699i			1.87848i	−0.343260	+	0.939240i	$0.611531\pi$
$720$		0			0
$721$		−	18.7689i		−	0.698990i
$722$	5.94469			0.221238
$723$		0			0
$724$	10.2124			0.379540
$725$	−31.5975			−1.17350
$726$		0			0
$727$	−19.2834			−0.715182			−0.357591	−	0.933878i	$-0.616402\pi$
$727$	−19.2834			−0.715182			−0.357591	+	0.933878i	$0.616402\pi$
$728$	0.143255			0.00530939
$729$		0			0
$730$	14.5785			0.539576
$731$		−	3.11658i		−	0.115271i
$732$		0			0
$733$		−	13.3862i		−	0.494430i	−0.968961	−	0.247215i	$-0.920485\pi$
$733$		−	13.3862i		−	0.494430i	0.968961	−	0.247215i	$-0.0795155\pi$
$734$	−10.8754			−0.401418
$735$		0			0
$736$		−	9.38476i		−	0.345927i
$737$	−13.7084	−	23.7351i	−0.504957	−	0.874292i
$738$		0			0
$739$			26.5893i			0.978104i	0.872255	+	0.489052i	$0.162657\pi$
$739$			26.5893i			0.978104i	−0.872255	+	0.489052i	$0.837343\pi$
$740$		−	2.11187i		−	0.0776339i
$741$		0			0
$742$	−13.0766			−0.480059
$743$	0.341095			0.0125136			0.00625678	−	0.999980i	$-0.498008\pi$
$743$	0.341095			0.0125136			0.00625678	+	0.999980i	$0.498008\pi$
$744$		0			0
$745$		−	7.47670i		−	0.273925i
$746$			2.51248i			0.0919883i
$747$		0			0
$748$	9.07338	+	15.7098i	0.331756	+	0.574408i
$749$			9.78967i			0.357707i
$750$		0			0
$751$	26.2524			0.957965			0.478983	−	0.877824i	$-0.341006\pi$
$751$	26.2524			0.957965			0.478983	+	0.877824i	$0.341006\pi$
$752$		−	5.48829i		−	0.200137i
$753$		0			0
$754$		−	1.23148i		−	0.0448477i
$755$	2.70116			0.0983052
$756$		0			0
$757$	17.2810			0.628089			0.314045	−	0.949408i	$-0.398316\pi$
$757$	17.2810			0.628089			0.314045	+	0.949408i	$0.398316\pi$
$758$	0.239642			0.00870417
$759$		0			0
$760$	4.15806			0.150829
$761$	14.8366			0.537828			0.268914	−	0.963164i	$-0.413335\pi$
$761$	14.8366			0.537828			0.268914	+	0.963164i	$0.413335\pi$
$762$		0			0
$763$	5.77066			0.208912
$764$			16.6717i			0.603162i
$765$		0			0
$766$			14.0879i			0.509018i
$767$	−0.266923			−0.00963804
$768$		0			0
$769$		−	12.3646i		−	0.445881i	−0.974832	−	0.222940i	$-0.928435\pi$
$769$		−	12.3646i		−	0.445881i	0.974832	−	0.222940i	$-0.0715655\pi$
$770$	−1.90890	−	3.30510i	−0.0687919	−	0.119108i
$771$		0			0
$772$		−	21.8366i		−	0.785918i
$773$			8.45911i			0.304253i	0.988361	+	0.152127i	$0.0486121\pi$
$773$			8.45911i			0.304253i	−0.988361	+	0.152127i	$0.951388\pi$
$774$		0			0
$775$	18.5978			0.668052
$776$	−8.25745			−0.296425
$777$		0			0
$778$			10.2059i			0.365898i
$779$			37.4951i			1.34340i
$780$		0			0
$781$	20.3430	−	11.7493i	0.727930	−	0.420424i
$782$		−	51.3342i		−	1.83571i
$783$		0			0
$784$	−1.00000			−0.0357143
$785$			21.3771i			0.762982i
$786$		0			0
$787$			37.1771i			1.32522i	0.748964	+	0.662610i	$0.230551\pi$
$787$			37.1771i			1.32522i	−0.748964	+	0.662610i	$0.769449\pi$
$788$	−24.2527			−0.863966
$789$		0			0
$790$	12.4888			0.444332
$791$	5.93292			0.210950
$792$		0			0
$793$	0.350236			0.0124373
$794$	25.1802			0.893613
$795$		0			0
$796$	−16.4360			−0.582558
$797$		−	41.0382i		−	1.45365i	−0.686825	−	0.726823i	$-0.740996\pi$
$797$		−	41.0382i		−	1.45365i	0.686825	−	0.726823i	$-0.259004\pi$
$798$		0			0
$799$		−	30.0207i		−	1.06206i
$800$	3.67567			0.129955
$801$		0			0
$802$			27.3876i			0.967089i
$803$	−36.3835	+	21.0137i	−1.28395	+	0.741556i
$804$		0			0
$805$			10.7999i			0.380647i
$806$			0.724827i			0.0255309i
$807$		0			0
$808$	14.6619			0.515805
$809$	27.7384			0.975231			0.487615	−	0.873059i	$-0.337867\pi$
$809$	27.7384			0.975231			0.487615	+	0.873059i	$0.337867\pi$
$810$		0			0
$811$		−	27.2973i		−	0.958538i	−0.877668	−	0.479269i	$-0.840902\pi$
$811$		−	27.2973i		−	0.958538i	0.877668	−	0.479269i	$-0.159098\pi$
$812$			8.59638i			0.301674i
$813$		0			0
$814$	3.04407	+	5.27057i	0.106695	+	0.184733i
$815$			10.6366i			0.372584i
$816$		0			0
$817$	2.05868			0.0720240
$818$		−	15.5159i		−	0.542501i
$819$		0			0
$820$		−	11.9420i		−	0.417034i
$821$	12.5441			0.437792			0.218896	−	0.975748i	$-0.429754\pi$
$821$	12.5441			0.437792			0.218896	+	0.975748i	$0.429754\pi$
$822$		0			0
$823$	−46.5594			−1.62296			−0.811480	−	0.584380i	$-0.801338\pi$
$823$	−46.5594			−1.62296			−0.811480	+	0.584380i	$0.801338\pi$
$824$	18.7689			0.653846
$825$		0			0
$826$	1.86327			0.0648315
$827$	−14.7117			−0.511575			−0.255788	−	0.966733i	$-0.582335\pi$
$827$	−14.7117			−0.511575			−0.255788	+	0.966733i	$0.582335\pi$
$828$		0			0
$829$	36.1533			1.25566			0.627828	−	0.778352i	$-0.283944\pi$
$829$	36.1533			1.25566			0.627828	+	0.778352i	$0.283944\pi$
$830$			19.0769i			0.662168i
$831$		0			0
$832$			0.143255i			0.00496648i
$833$	−5.46996			−0.189523
$834$		0			0
$835$			3.84862i			0.133187i
$836$	−10.3772	+	5.99347i	−0.358904	+	0.207289i
$837$		0			0
$838$		−	12.8834i		−	0.445050i
$839$			2.72097i			0.0939385i	0.998896	+	0.0469692i	$0.0149563\pi$
$839$			2.72097i			0.0939385i	−0.998896	+	0.0469692i	$0.985044\pi$
$840$		0			0
$841$	44.8977			1.54820
$842$	−2.34110			−0.0806795
$843$		0			0
$844$			7.57023i			0.260578i
$845$		−	14.9367i		−	0.513838i
$846$		0			0
$847$	9.52801	+	5.49699i	0.327386	+	0.188879i
$848$		−	13.0766i		−	0.449054i
$849$		0			0
$850$	20.1058			0.689623
$851$		−	17.2224i		−	0.590375i
$852$		0			0
$853$			35.9890i			1.23224i	0.787652	+	0.616120i	$0.211296\pi$
$853$			35.9890i			1.23224i	−0.787652	+	0.616120i	$0.788704\pi$
$854$	−2.44484			−0.0836608
$855$		0			0
$856$	−9.78967			−0.334604
$857$	−10.6877			−0.365084			−0.182542	−	0.983198i	$-0.558433\pi$
$857$	−10.6877			−0.365084			−0.182542	+	0.983198i	$0.558433\pi$
$858$		0			0
$859$	−22.2894			−0.760506			−0.380253	−	0.924883i	$-0.624163\pi$
$859$	−22.2894			−0.760506			−0.380253	+	0.924883i	$0.624163\pi$
$860$	−0.655680			−0.0223585
$861$		0			0
$862$	−1.13836			−0.0387726
$863$			21.9854i			0.748393i	0.927350	+	0.374196i	$0.122081\pi$
$863$			21.9854i			0.748393i	−0.927350	+	0.374196i	$0.877919\pi$
$864$		0			0
$865$			12.9718i			0.441055i
$866$	31.9390			1.08533
$867$		0			0
$868$		−	5.05969i		−	0.171737i
$869$	−31.1681	+	18.0015i	−1.05731	+	0.610659i
$870$		0			0
$871$		−	1.18390i		−	0.0401148i
$872$			5.77066i			0.195419i
$873$		0			0
$874$	33.9091			1.14699
$875$	−9.98391			−0.337518
$876$		0			0
$877$			6.53893i			0.220804i	0.993887	+	0.110402i	$0.0352138\pi$
$877$			6.53893i			0.220804i	−0.993887	+	0.110402i	$0.964786\pi$
$878$			7.64917i			0.258147i
$879$		0			0
$880$	3.30510	−	1.90890i	0.111415	−	0.0643489i
$881$		−	31.5248i		−	1.06210i	−0.847341	−	0.531049i	$-0.821798\pi$
$881$		−	31.5248i		−	1.06210i	0.847341	−	0.531049i	$-0.178202\pi$
$882$		0			0
$883$	29.2542			0.984483			0.492242	−	0.870459i	$-0.336178\pi$
$883$	29.2542			0.984483			0.492242	+	0.870459i	$0.336178\pi$
$884$			0.783600i			0.0263553i
$885$		0			0
$886$			4.94335i			0.166075i
$887$	50.7954			1.70554			0.852771	−	0.522285i	$-0.174920\pi$
$887$	50.7954			1.70554			0.852771	+	0.522285i	$0.174920\pi$
$888$		0			0
$889$	11.9538			0.400919
$890$	19.6069			0.657224
$891$		0			0
$892$	−27.5906			−0.923802
$893$	19.8304			0.663598
$894$		0			0
$895$	−20.2508			−0.676909
$896$		−	1.00000i		−	0.0334077i
$897$		0			0
$898$			12.2684i			0.409403i
$899$	−43.4950			−1.45064
$900$		0			0
$901$		−	71.5287i		−	2.38297i
$902$	17.2134	+	29.8036i	0.573143	+	0.992351i
$903$		0			0
$904$			5.93292i			0.197326i
$905$		−	11.7523i		−	0.390660i
$906$		0			0
$907$	0.942105			0.0312821			0.0156410	−	0.999878i	$-0.495021\pi$
$907$	0.942105			0.0312821			0.0156410	+	0.999878i	$0.495021\pi$
$908$	13.2040			0.438191
$909$		0			0
$910$		−	0.164857i		−	0.00546496i
$911$		−	29.3335i		−	0.971862i	−0.873997	−	0.485931i	$-0.838481\pi$
$911$		−	29.3335i		−	0.971862i	0.873997	−	0.485931i	$-0.161519\pi$
$912$		0			0
$913$	−27.4976	−	47.6099i	−0.910038	−	1.57566i
$914$		−	10.5999i		−	0.350614i
$915$		0			0
$916$	−14.7899			−0.488672
$917$			3.16510i			0.104521i
$918$		0			0
$919$		−	28.6857i		−	0.946254i	−0.880994	−	0.473127i	$-0.843125\pi$
$919$		−	28.6857i		−	0.946254i	0.880994	−	0.473127i	$-0.156875\pi$
$920$	−10.7999			−0.356063
$921$		0			0
$922$	3.96146			0.130464
$923$	1.01470			0.0333993
$924$		0			0
$925$	6.74539			0.221787
$926$	−0.411528			−0.0135236
$927$		0			0
$928$	−8.59638			−0.282190
$929$			16.3383i			0.536041i	0.963413	+	0.268021i	$0.0863695\pi$
$929$			16.3383i			0.536041i	−0.963413	+	0.268021i	$0.913630\pi$
$930$		0			0
$931$		−	3.61321i		−	0.118418i
$932$	27.3915			0.897239
$933$		0			0
$934$		−	2.07385i		−	0.0678584i
$935$	18.0788	−	10.4416i	0.591239	−	0.341476i
$936$		0			0
$937$			18.1887i			0.594197i	0.954847	+	0.297099i	$0.0960190\pi$
$937$			18.1887i			0.594197i	−0.954847	+	0.297099i	$0.903981\pi$
$938$			8.26424i			0.269837i
$939$		0			0
$940$	−6.31589			−0.206002
$941$	38.8223			1.26557			0.632785	−	0.774328i	$-0.281912\pi$
$941$	38.8223			1.26557			0.632785	+	0.774328i	$0.281912\pi$
$942$		0			0
$943$		−	97.3877i		−	3.17138i
$944$			1.86327i			0.0606443i
$945$		0			0
$946$	1.63637	−	0.945104i	0.0532030	−	0.0307280i
$947$		−	25.3915i		−	0.825114i	−0.910932	−	0.412557i	$-0.864636\pi$
$947$		−	25.3915i		−	0.825114i	0.910932	−	0.412557i	$-0.135364\pi$
$948$		0			0
$949$	−1.81479			−0.0589107
$950$			13.2810i			0.430892i
$951$		0			0
$952$		−	5.46996i		−	0.177282i
$953$	57.2709			1.85519			0.927593	−	0.373592i	$-0.121874\pi$
$953$	57.2709			1.85519			0.927593	+	0.373592i	$0.121874\pi$
$954$		0			0
$955$	19.1857			0.620836
$956$	−6.93096			−0.224163
$957$		0			0
$958$	1.37120			0.0443016
$959$	−7.76237			−0.250660
$960$		0			0
$961$	−5.39954			−0.174179
$962$			0.262894i			0.00847604i
$963$		0			0
$964$		−	9.11277i		−	0.293503i
$965$	−25.1295			−0.808946
$966$		0			0
$967$			36.5328i			1.17482i	0.809291	+	0.587408i	$0.199851\pi$
$967$			36.5328i			1.17482i	−0.809291	+	0.587408i	$0.800149\pi$
$968$	−5.49699	+	9.52801i	−0.176680	+	0.306242i
$969$		0			0
$970$			9.50262i			0.305111i
$971$		−	7.28698i		−	0.233850i	−0.993141	−	0.116925i	$-0.962696\pi$
$971$		−	7.28698i		−	0.233850i	0.993141	−	0.116925i	$-0.0373038\pi$
$972$		0			0
$973$	12.5506			0.402355
$974$	19.1410			0.613318
$975$		0			0
$976$		−	2.44484i		−	0.0782575i
$977$		−	4.29587i		−	0.137437i	−0.997636	−	0.0687186i	$-0.978109\pi$
$977$		−	4.29587i		−	0.137437i	0.997636	−	0.0687186i	$-0.0218911\pi$
$978$		0			0
$979$	−48.9326	+	28.2615i	−1.56389	+	0.903243i
$980$			1.15079i			0.0367607i
$981$		0			0
$982$	−16.4375			−0.524543
$983$			0.940131i			0.0299855i	0.999888	+	0.0149928i	$0.00477252\pi$
$983$			0.940131i			0.0299855i	−0.999888	+	0.0149928i	$0.995227\pi$
$984$		0			0
$985$			27.9098i			0.889281i
$986$	−47.0218			−1.49748
$987$		0			0
$988$	−0.517612			−0.0164674
$989$	−5.34709			−0.170028
$990$		0			0
$991$	31.4839			1.00012			0.500060	−	0.865991i	$-0.333311\pi$
$991$	31.4839			1.00012			0.500060	+	0.865991i	$0.333311\pi$
$992$	5.05969			0.160645
$993$		0			0
$994$	−7.08317			−0.224665
$995$			18.9144i			0.599628i
$996$		0			0
$997$			41.0243i			1.29925i	0.760254	+	0.649626i	$0.225075\pi$
$997$			41.0243i			1.29925i	−0.760254	+	0.649626i	$0.774925\pi$
$998$	12.3791			0.391854
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1386.2.c.b.197.4	yes	12
3.2	odd	2		1386.2.c.a.197.9	yes	12
11.10	odd	2		1386.2.c.a.197.4	✓	12
33.32	even	2	inner	1386.2.c.b.197.9	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1386.2.c.a.197.4	✓	12	11.10	odd	2
1386.2.c.a.197.9	yes	12	3.2	odd	2
1386.2.c.b.197.4	yes	12	1.1	even	1	trivial
1386.2.c.b.197.9	yes	12	33.32	even	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:	\( N \)	\(=\)	\( 1386 = 2 \cdot 3^{2} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1386.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} -1.15079i q^{5} -1.00000i q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} -1.15079i q^{5} -1.00000i q^{7} +1.00000 q^{8} -1.15079i q^{10} +(1.65877 + 2.87202i) q^{11} +0.143255i q^{13} -1.00000i q^{14} +1.00000 q^{16} +5.46996 q^{17} +3.61321i q^{19} -1.15079i q^{20} +(1.65877 + 2.87202i) q^{22} -9.38476i q^{23} +3.67567 q^{25} +0.143255i q^{26} -1.00000i q^{28} -8.59638 q^{29} +5.05969 q^{31} +1.00000 q^{32} +5.46996 q^{34} -1.15079 q^{35} +1.83514 q^{37} +3.61321i q^{38} -1.15079i q^{40} +10.3772 q^{41} -0.569763i q^{43} +(1.65877 + 2.87202i) q^{44} -9.38476i q^{46} -5.48829i q^{47} -1.00000 q^{49} +3.67567 q^{50} +0.143255i q^{52} -13.0766i q^{53} +(3.30510 - 1.90890i) q^{55} -1.00000i q^{56} -8.59638 q^{58} +1.86327i q^{59} -2.44484i q^{61} +5.05969 q^{62} +1.00000 q^{64} +0.164857 q^{65} -8.26424 q^{67} +5.46996 q^{68} -1.15079 q^{70} -7.08317i q^{71} +12.6683i q^{73} +1.83514 q^{74} +3.61321i q^{76} +(2.87202 - 1.65877i) q^{77} +10.8523i q^{79} -1.15079i q^{80} +10.3772 q^{82} -16.5772 q^{83} -6.29479i q^{85} -0.569763i q^{86} +(1.65877 + 2.87202i) q^{88} +17.0377i q^{89} +0.143255 q^{91} -9.38476i q^{92} -5.48829i q^{94} +4.15806 q^{95} -8.25745 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 12 q^{2} + 12 q^{4} + 12 q^{8}+O(q^{10}) \) 12 * q + 12 * q^2 + 12 * q^4 + 12 * q^8	\( 12 q + 12 q^{2} + 12 q^{4} + 12 q^{8} - 4 q^{11} + 12 q^{16} + 16 q^{17} - 4 q^{22} - 4 q^{25} + 16 q^{29} + 12 q^{32} + 16 q^{34} - 8 q^{35} + 24 q^{37} + 16 q^{41} - 4 q^{44} - 12 q^{49} - 4 q^{50} - 8 q^{55} + 16 q^{58} + 12 q^{64} - 48 q^{67} + 16 q^{68} - 8 q^{70} + 24 q^{74} + 8 q^{77} + 16 q^{82} + 16 q^{83} - 4 q^{88} - 48 q^{95} + 48 q^{97} - 12 q^{98}+O(q^{100}) \) 12 * q + 12 * q^2 + 12 * q^4 + 12 * q^8 - 4 * q^11 + 12 * q^16 + 16 * q^17 - 4 * q^22 - 4 * q^25 + 16 * q^29 + 12 * q^32 + 16 * q^34 - 8 * q^35 + 24 * q^37 + 16 * q^41 - 4 * q^44 - 12 * q^49 - 4 * q^50 - 8 * q^55 + 16 * q^58 + 12 * q^64 - 48 * q^67 + 16 * q^68 - 8 * q^70 + 24 * q^74 + 8 * q^77 + 16 * q^82 + 16 * q^83 - 4 * q^88 - 48 * q^95 + 48 * q^97 - 12 * q^98

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