LMFDB - Embedded newform 1386.2.a.l.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1386,2,Mod(1,1386)]

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([0, 0, 0]))

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

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

chi := DirichletCharacter("1386.1");

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.a (trivial)

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:	yes
Analytic conductor:	$11.0672657201$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 154)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	1386.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{2} +1.00000 q^{4} +4.00000 q^{5} -1.00000 q^{7} +1.00000 q^{8} +O(q^{10})$

\(q+1.00000 q^{2} +1.00000 q^{4} +4.00000 q^{5} -1.00000 q^{7} +1.00000 q^{8} +4.00000 q^{10} +1.00000 q^{11} +2.00000 q^{13} -1.00000 q^{14} +1.00000 q^{16} +4.00000 q^{17} -6.00000 q^{19} +4.00000 q^{20} +1.00000 q^{22} -4.00000 q^{23} +11.0000 q^{25} +2.00000 q^{26} -1.00000 q^{28} +2.00000 q^{29} -2.00000 q^{31} +1.00000 q^{32} +4.00000 q^{34} -4.00000 q^{35} +10.0000 q^{37} -6.00000 q^{38} +4.00000 q^{40} -4.00000 q^{41} -8.00000 q^{43} +1.00000 q^{44} -4.00000 q^{46} -2.00000 q^{47} +1.00000 q^{49} +11.0000 q^{50} +2.00000 q^{52} -6.00000 q^{53} +4.00000 q^{55} -1.00000 q^{56} +2.00000 q^{58} +12.0000 q^{59} -14.0000 q^{61} -2.00000 q^{62} +1.00000 q^{64} +8.00000 q^{65} -12.0000 q^{67} +4.00000 q^{68} -4.00000 q^{70} +8.00000 q^{71} +4.00000 q^{73} +10.0000 q^{74} -6.00000 q^{76} -1.00000 q^{77} +4.00000 q^{80} -4.00000 q^{82} +6.00000 q^{83} +16.0000 q^{85} -8.00000 q^{86} +1.00000 q^{88} +6.00000 q^{89} -2.00000 q^{91} -4.00000 q^{92} -2.00000 q^{94} -24.0000 q^{95} -14.0000 q^{97} +1.00000 q^{98} +O(q^{100})\)

Display 100 coefficients 10 coefficients

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
$5$	4.00000	1.78885	0.894427	−	0.447214i	$-0.147584\pi$
$5$	4.00000	1.78885	0.894427	+	0.447214i	$0.147584\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	1.00000	0.353553
$9$	0	0
$10$	4.00000	1.26491
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	−1.00000	−0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	4.00000	0.970143	0.485071	−	0.874475i	$-0.338794\pi$
$17$	4.00000	0.970143	0.485071	+	0.874475i	$0.338794\pi$
$18$	0	0
$19$	−6.00000	−1.37649	−0.688247	−	0.725476i	$-0.741620\pi$
$19$	−6.00000	−1.37649	−0.688247	+	0.725476i	$0.741620\pi$
$20$	4.00000	0.894427
$21$	0	0
$22$	1.00000	0.213201
$23$	−4.00000	−0.834058	−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000	−0.834058	−0.417029	+	0.908893i	$0.636929\pi$
$24$	0	0
$25$	11.0000	2.20000
$26$	2.00000	0.392232
$27$	0	0
$28$	−1.00000	−0.188982
$29$	2.00000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	0	0
$31$	−2.00000	−0.359211	−0.179605	−	0.983739i	$-0.557482\pi$
$31$	−2.00000	−0.359211	−0.179605	+	0.983739i	$0.557482\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	4.00000	0.685994
$35$	−4.00000	−0.676123
$36$	0	0
$37$	10.0000	1.64399	0.821995	−	0.569495i	$-0.192861\pi$
$37$	10.0000	1.64399	0.821995	+	0.569495i	$0.192861\pi$
$38$	−6.00000	−0.973329
$39$	0	0
$40$	4.00000	0.632456
$41$	−4.00000	−0.624695	−0.312348	−	0.949968i	$-0.601115\pi$
$41$	−4.00000	−0.624695	−0.312348	+	0.949968i	$0.601115\pi$
$42$	0	0
$43$	−8.00000	−1.21999	−0.609994	−	0.792406i	$-0.708828\pi$
$43$	−8.00000	−1.21999	−0.609994	+	0.792406i	$0.708828\pi$
$44$	1.00000	0.150756
$45$	0	0
$46$	−4.00000	−0.589768
$47$	−2.00000	−0.291730	−0.145865	−	0.989305i	$-0.546597\pi$
$47$	−2.00000	−0.291730	−0.145865	+	0.989305i	$0.546597\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	11.0000	1.55563
$51$	0	0
$52$	2.00000	0.277350
$53$	−6.00000	−0.824163	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	0	0
$55$	4.00000	0.539360
$56$	−1.00000	−0.133631
$57$	0	0
$58$	2.00000	0.262613
$59$	12.0000	1.56227	0.781133	−	0.624364i	$-0.214642\pi$
$59$	12.0000	1.56227	0.781133	+	0.624364i	$0.214642\pi$
$60$	0	0
$61$	−14.0000	−1.79252	−0.896258	−	0.443533i	$-0.853725\pi$
$61$	−14.0000	−1.79252	−0.896258	+	0.443533i	$0.853725\pi$
$62$	−2.00000	−0.254000
$63$	0	0
$64$	1.00000	0.125000
$65$	8.00000	0.992278
$66$	0	0
$67$	−12.0000	−1.46603	−0.733017	−	0.680211i	$-0.761888\pi$
$67$	−12.0000	−1.46603	−0.733017	+	0.680211i	$0.761888\pi$
$68$	4.00000	0.485071
$69$	0	0
$70$	−4.00000	−0.478091
$71$	8.00000	0.949425	0.474713	−	0.880141i	$-0.342552\pi$
$71$	8.00000	0.949425	0.474713	+	0.880141i	$0.342552\pi$
$72$	0	0
$73$	4.00000	0.468165	0.234082	−	0.972217i	$-0.424791\pi$
$73$	4.00000	0.468165	0.234082	+	0.972217i	$0.424791\pi$
$74$	10.0000	1.16248
$75$	0	0
$76$	−6.00000	−0.688247
$77$	−1.00000	−0.113961
$78$	0	0
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	4.00000	0.447214
$81$	0	0
$82$	−4.00000	−0.441726
$83$	6.00000	0.658586	0.329293	−	0.944228i	$-0.393190\pi$
$83$	6.00000	0.658586	0.329293	+	0.944228i	$0.393190\pi$
$84$	0	0
$85$	16.0000	1.73544
$86$	−8.00000	−0.862662
$87$	0	0
$88$	1.00000	0.106600
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	0	0
$91$	−2.00000	−0.209657
$92$	−4.00000	−0.417029
$93$	0	0
$94$	−2.00000	−0.206284
$95$	−24.0000	−2.46235
$96$	0	0
$97$	−14.0000	−1.42148	−0.710742	−	0.703452i	$-0.751641\pi$
$97$	−14.0000	−1.42148	−0.710742	+	0.703452i	$0.751641\pi$
$98$	1.00000	0.101015
$99$	0	0
$100$	11.0000	1.10000
$101$	−6.00000	−0.597022	−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000	−0.597022	−0.298511	+	0.954406i	$0.596490\pi$
$102$	0	0
$103$	18.0000	1.77359	0.886796	−	0.462160i	$-0.152926\pi$
$103$	18.0000	1.77359	0.886796	+	0.462160i	$0.152926\pi$
$104$	2.00000	0.196116
$105$	0	0
$106$	−6.00000	−0.582772
$107$	16.0000	1.54678	0.773389	−	0.633932i	$-0.218560\pi$
$107$	16.0000	1.54678	0.773389	+	0.633932i	$0.218560\pi$
$108$	0	0
$109$	−14.0000	−1.34096	−0.670478	−	0.741929i	$-0.733911\pi$
$109$	−14.0000	−1.34096	−0.670478	+	0.741929i	$0.733911\pi$
$110$	4.00000	0.381385
$111$	0	0
$112$	−1.00000	−0.0944911
$113$	−14.0000	−1.31701	−0.658505	−	0.752577i	$-0.728811\pi$
$113$	−14.0000	−1.31701	−0.658505	+	0.752577i	$0.728811\pi$
$114$	0	0
$115$	−16.0000	−1.49201
$116$	2.00000	0.185695
$117$	0	0
$118$	12.0000	1.10469
$119$	−4.00000	−0.366679
$120$	0	0
$121$	1.00000	0.0909091
$122$	−14.0000	−1.26750
$123$	0	0
$124$	−2.00000	−0.179605
$125$	24.0000	2.14663
$126$	0	0
$127$	8.00000	0.709885	0.354943	−	0.934888i	$-0.384500\pi$
$127$	8.00000	0.709885	0.354943	+	0.934888i	$0.384500\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	8.00000	0.701646
$131$	−6.00000	−0.524222	−0.262111	−	0.965038i	$-0.584419\pi$
$131$	−6.00000	−0.524222	−0.262111	+	0.965038i	$0.584419\pi$
$132$	0	0
$133$	6.00000	0.520266
$134$	−12.0000	−1.03664
$135$	0	0
$136$	4.00000	0.342997
$137$	−6.00000	−0.512615	−0.256307	−	0.966595i	$-0.582506\pi$
$137$	−6.00000	−0.512615	−0.256307	+	0.966595i	$0.582506\pi$
$138$	0	0
$139$	14.0000	1.18746	0.593732	−	0.804663i	$-0.297654\pi$
$139$	14.0000	1.18746	0.593732	+	0.804663i	$0.297654\pi$
$140$	−4.00000	−0.338062
$141$	0	0
$142$	8.00000	0.671345
$143$	2.00000	0.167248
$144$	0	0
$145$	8.00000	0.664364
$146$	4.00000	0.331042
$147$	0	0
$148$	10.0000	0.821995
$149$	−2.00000	−0.163846	−0.0819232	−	0.996639i	$-0.526106\pi$
$149$	−2.00000	−0.163846	−0.0819232	+	0.996639i	$0.526106\pi$
$150$	0	0
$151$	−24.0000	−1.95309	−0.976546	−	0.215308i	$-0.930924\pi$
$151$	−24.0000	−1.95309	−0.976546	+	0.215308i	$0.930924\pi$
$152$	−6.00000	−0.486664
$153$	0	0
$154$	−1.00000	−0.0805823
$155$	−8.00000	−0.642575
$156$	0	0
$157$	−8.00000	−0.638470	−0.319235	−	0.947676i	$-0.603426\pi$
$157$	−8.00000	−0.638470	−0.319235	+	0.947676i	$0.603426\pi$
$158$	0	0
$159$	0	0
$160$	4.00000	0.316228
$161$	4.00000	0.315244
$162$	0	0
$163$	4.00000	0.313304	0.156652	−	0.987654i	$-0.449930\pi$
$163$	4.00000	0.313304	0.156652	+	0.987654i	$0.449930\pi$
$164$	−4.00000	−0.312348
$165$	0	0
$166$	6.00000	0.465690
$167$	−4.00000	−0.309529	−0.154765	−	0.987951i	$-0.549462\pi$
$167$	−4.00000	−0.309529	−0.154765	+	0.987951i	$0.549462\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	16.0000	1.22714
$171$	0	0
$172$	−8.00000	−0.609994
$173$	14.0000	1.06440	0.532200	−	0.846619i	$-0.321365\pi$
$173$	14.0000	1.06440	0.532200	+	0.846619i	$0.321365\pi$
$174$	0	0
$175$	−11.0000	−0.831522
$176$	1.00000	0.0753778
$177$	0	0
$178$	6.00000	0.449719
$179$	−4.00000	−0.298974	−0.149487	−	0.988764i	$-0.547762\pi$
$179$	−4.00000	−0.298974	−0.149487	+	0.988764i	$0.547762\pi$
$180$	0	0
$181$	20.0000	1.48659	0.743294	−	0.668965i	$-0.233262\pi$
$181$	20.0000	1.48659	0.743294	+	0.668965i	$0.233262\pi$
$182$	−2.00000	−0.148250
$183$	0	0
$184$	−4.00000	−0.294884
$185$	40.0000	2.94086
$186$	0	0
$187$	4.00000	0.292509
$188$	−2.00000	−0.145865
$189$	0	0
$190$	−24.0000	−1.74114
$191$	4.00000	0.289430	0.144715	−	0.989473i	$-0.453773\pi$
$191$	4.00000	0.289430	0.144715	+	0.989473i	$0.453773\pi$
$192$	0	0
$193$	2.00000	0.143963	0.0719816	−	0.997406i	$-0.477068\pi$
$193$	2.00000	0.143963	0.0719816	+	0.997406i	$0.477068\pi$
$194$	−14.0000	−1.00514
$195$	0	0
$196$	1.00000	0.0714286
$197$	−6.00000	−0.427482	−0.213741	−	0.976890i	$-0.568565\pi$
$197$	−6.00000	−0.427482	−0.213741	+	0.976890i	$0.568565\pi$
$198$	0	0
$199$	−14.0000	−0.992434	−0.496217	−	0.868199i	$-0.665278\pi$
$199$	−14.0000	−0.992434	−0.496217	+	0.868199i	$0.665278\pi$
$200$	11.0000	0.777817
$201$	0	0
$202$	−6.00000	−0.422159
$203$	−2.00000	−0.140372
$204$	0	0
$205$	−16.0000	−1.11749
$206$	18.0000	1.25412
$207$	0	0
$208$	2.00000	0.138675
$209$	−6.00000	−0.415029
$210$	0	0
$211$	−8.00000	−0.550743	−0.275371	−	0.961338i	$-0.588801\pi$
$211$	−8.00000	−0.550743	−0.275371	+	0.961338i	$0.588801\pi$
$212$	−6.00000	−0.412082
$213$	0	0
$214$	16.0000	1.09374
$215$	−32.0000	−2.18238
$216$	0	0
$217$	2.00000	0.135769
$218$	−14.0000	−0.948200
$219$	0	0
$220$	4.00000	0.269680
$221$	8.00000	0.538138
$222$	0	0
$223$	−2.00000	−0.133930	−0.0669650	−	0.997755i	$-0.521332\pi$
$223$	−2.00000	−0.133930	−0.0669650	+	0.997755i	$0.521332\pi$
$224$	−1.00000	−0.0668153
$225$	0	0
$226$	−14.0000	−0.931266
$227$	2.00000	0.132745	0.0663723	−	0.997795i	$-0.478857\pi$
$227$	2.00000	0.132745	0.0663723	+	0.997795i	$0.478857\pi$
$228$	0	0
$229$	20.0000	1.32164	0.660819	−	0.750546i	$-0.270209\pi$
$229$	20.0000	1.32164	0.660819	+	0.750546i	$0.270209\pi$
$230$	−16.0000	−1.05501
$231$	0	0
$232$	2.00000	0.131306
$233$	−30.0000	−1.96537	−0.982683	−	0.185296i	$-0.940675\pi$
$233$	−30.0000	−1.96537	−0.982683	+	0.185296i	$0.940675\pi$
$234$	0	0
$235$	−8.00000	−0.521862
$236$	12.0000	0.781133
$237$	0	0
$238$	−4.00000	−0.259281
$239$	−16.0000	−1.03495	−0.517477	−	0.855697i	$-0.673129\pi$
$239$	−16.0000	−1.03495	−0.517477	+	0.855697i	$0.673129\pi$
$240$	0	0
$241$	12.0000	0.772988	0.386494	−	0.922292i	$-0.373686\pi$
$241$	12.0000	0.772988	0.386494	+	0.922292i	$0.373686\pi$
$242$	1.00000	0.0642824
$243$	0	0
$244$	−14.0000	−0.896258
$245$	4.00000	0.255551
$246$	0	0
$247$	−12.0000	−0.763542
$248$	−2.00000	−0.127000
$249$	0	0
$250$	24.0000	1.51789
$251$	−12.0000	−0.757433	−0.378717	−	0.925513i	$-0.623635\pi$
$251$	−12.0000	−0.757433	−0.378717	+	0.925513i	$0.623635\pi$
$252$	0	0
$253$	−4.00000	−0.251478
$254$	8.00000	0.501965
$255$	0	0
$256$	1.00000	0.0625000
$257$	6.00000	0.374270	0.187135	−	0.982334i	$-0.440080\pi$
$257$	6.00000	0.374270	0.187135	+	0.982334i	$0.440080\pi$
$258$	0	0
$259$	−10.0000	−0.621370
$260$	8.00000	0.496139
$261$	0	0
$262$	−6.00000	−0.370681
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	0	0
$265$	−24.0000	−1.47431
$266$	6.00000	0.367884
$267$	0	0
$268$	−12.0000	−0.733017
$269$	−12.0000	−0.731653	−0.365826	−	0.930683i	$-0.619214\pi$
$269$	−12.0000	−0.731653	−0.365826	+	0.930683i	$0.619214\pi$
$270$	0	0
$271$	−20.0000	−1.21491	−0.607457	−	0.794353i	$-0.707810\pi$
$271$	−20.0000	−1.21491	−0.607457	+	0.794353i	$0.707810\pi$
$272$	4.00000	0.242536
$273$	0	0
$274$	−6.00000	−0.362473
$275$	11.0000	0.663325
$276$	0	0
$277$	−30.0000	−1.80253	−0.901263	−	0.433273i	$-0.857359\pi$
$277$	−30.0000	−1.80253	−0.901263	+	0.433273i	$0.857359\pi$
$278$	14.0000	0.839664
$279$	0	0
$280$	−4.00000	−0.239046
$281$	10.0000	0.596550	0.298275	−	0.954480i	$-0.403589\pi$
$281$	10.0000	0.596550	0.298275	+	0.954480i	$0.403589\pi$
$282$	0	0
$283$	6.00000	0.356663	0.178331	−	0.983970i	$-0.442930\pi$
$283$	6.00000	0.356663	0.178331	+	0.983970i	$0.442930\pi$
$284$	8.00000	0.474713
$285$	0	0
$286$	2.00000	0.118262
$287$	4.00000	0.236113
$288$	0	0
$289$	−1.00000	−0.0588235
$290$	8.00000	0.469776
$291$	0	0
$292$	4.00000	0.234082
$293$	−18.0000	−1.05157	−0.525786	−	0.850617i	$-0.676229\pi$
$293$	−18.0000	−1.05157	−0.525786	+	0.850617i	$0.676229\pi$
$294$	0	0
$295$	48.0000	2.79467
$296$	10.0000	0.581238
$297$	0	0
$298$	−2.00000	−0.115857
$299$	−8.00000	−0.462652
$300$	0	0
$301$	8.00000	0.461112
$302$	−24.0000	−1.38104
$303$	0	0
$304$	−6.00000	−0.344124
$305$	−56.0000	−3.20655
$306$	0	0
$307$	−10.0000	−0.570730	−0.285365	−	0.958419i	$-0.592115\pi$
$307$	−10.0000	−0.570730	−0.285365	+	0.958419i	$0.592115\pi$
$308$	−1.00000	−0.0569803
$309$	0	0
$310$	−8.00000	−0.454369
$311$	−14.0000	−0.793867	−0.396934	−	0.917847i	$-0.629926\pi$
$311$	−14.0000	−0.793867	−0.396934	+	0.917847i	$0.629926\pi$
$312$	0	0
$313$	−2.00000	−0.113047	−0.0565233	−	0.998401i	$-0.518002\pi$
$313$	−2.00000	−0.113047	−0.0565233	+	0.998401i	$0.518002\pi$
$314$	−8.00000	−0.451466
$315$	0	0
$316$	0	0
$317$	−6.00000	−0.336994	−0.168497	−	0.985702i	$-0.553891\pi$
$317$	−6.00000	−0.336994	−0.168497	+	0.985702i	$0.553891\pi$
$318$	0	0
$319$	2.00000	0.111979
$320$	4.00000	0.223607
$321$	0	0
$322$	4.00000	0.222911
$323$	−24.0000	−1.33540
$324$	0	0
$325$	22.0000	1.22034
$326$	4.00000	0.221540
$327$	0	0
$328$	−4.00000	−0.220863
$329$	2.00000	0.110264
$330$	0	0
$331$	−20.0000	−1.09930	−0.549650	−	0.835395i	$-0.685239\pi$
$331$	−20.0000	−1.09930	−0.549650	+	0.835395i	$0.685239\pi$
$332$	6.00000	0.329293
$333$	0	0
$334$	−4.00000	−0.218870
$335$	−48.0000	−2.62252
$336$	0	0
$337$	−18.0000	−0.980522	−0.490261	−	0.871576i	$-0.663099\pi$
$337$	−18.0000	−0.980522	−0.490261	+	0.871576i	$0.663099\pi$
$338$	−9.00000	−0.489535
$339$	0	0
$340$	16.0000	0.867722
$341$	−2.00000	−0.108306
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	−8.00000	−0.431331
$345$	0	0
$346$	14.0000	0.752645
$347$	−8.00000	−0.429463	−0.214731	−	0.976673i	$-0.568888\pi$
$347$	−8.00000	−0.429463	−0.214731	+	0.976673i	$0.568888\pi$
$348$	0	0
$349$	−10.0000	−0.535288	−0.267644	−	0.963518i	$-0.586245\pi$
$349$	−10.0000	−0.535288	−0.267644	+	0.963518i	$0.586245\pi$
$350$	−11.0000	−0.587975
$351$	0	0
$352$	1.00000	0.0533002
$353$	6.00000	0.319348	0.159674	−	0.987170i	$-0.448956\pi$
$353$	6.00000	0.319348	0.159674	+	0.987170i	$0.448956\pi$
$354$	0	0
$355$	32.0000	1.69838
$356$	6.00000	0.317999
$357$	0	0
$358$	−4.00000	−0.211407
$359$	16.0000	0.844448	0.422224	−	0.906492i	$-0.361250\pi$
$359$	16.0000	0.844448	0.422224	+	0.906492i	$0.361250\pi$
$360$	0	0
$361$	17.0000	0.894737
$362$	20.0000	1.05118
$363$	0	0
$364$	−2.00000	−0.104828
$365$	16.0000	0.837478
$366$	0	0
$367$	22.0000	1.14839	0.574195	−	0.818718i	$-0.305315\pi$
$367$	22.0000	1.14839	0.574195	+	0.818718i	$0.305315\pi$
$368$	−4.00000	−0.208514
$369$	0	0
$370$	40.0000	2.07950
$371$	6.00000	0.311504
$372$	0	0
$373$	−10.0000	−0.517780	−0.258890	−	0.965907i	$-0.583357\pi$
$373$	−10.0000	−0.517780	−0.258890	+	0.965907i	$0.583357\pi$
$374$	4.00000	0.206835
$375$	0	0
$376$	−2.00000	−0.103142
$377$	4.00000	0.206010
$378$	0	0
$379$	4.00000	0.205466	0.102733	−	0.994709i	$-0.467241\pi$
$379$	4.00000	0.205466	0.102733	+	0.994709i	$0.467241\pi$
$380$	−24.0000	−1.23117
$381$	0	0
$382$	4.00000	0.204658
$383$	10.0000	0.510976	0.255488	−	0.966812i	$-0.417764\pi$
$383$	10.0000	0.510976	0.255488	+	0.966812i	$0.417764\pi$
$384$	0	0
$385$	−4.00000	−0.203859
$386$	2.00000	0.101797
$387$	0	0
$388$	−14.0000	−0.710742
$389$	30.0000	1.52106	0.760530	−	0.649303i	$-0.224939\pi$
$389$	30.0000	1.52106	0.760530	+	0.649303i	$0.224939\pi$
$390$	0	0
$391$	−16.0000	−0.809155
$392$	1.00000	0.0505076
$393$	0	0
$394$	−6.00000	−0.302276
$395$	0	0
$396$	0	0
$397$	24.0000	1.20453	0.602263	−	0.798298i	$-0.294266\pi$
$397$	24.0000	1.20453	0.602263	+	0.798298i	$0.294266\pi$
$398$	−14.0000	−0.701757
$399$	0	0
$400$	11.0000	0.550000
$401$	18.0000	0.898877	0.449439	−	0.893311i	$-0.351624\pi$
$401$	18.0000	0.898877	0.449439	+	0.893311i	$0.351624\pi$
$402$	0	0
$403$	−4.00000	−0.199254
$404$	−6.00000	−0.298511
$405$	0	0
$406$	−2.00000	−0.0992583
$407$	10.0000	0.495682
$408$	0	0
$409$	16.0000	0.791149	0.395575	−	0.918434i	$-0.370545\pi$
$409$	16.0000	0.791149	0.395575	+	0.918434i	$0.370545\pi$
$410$	−16.0000	−0.790184
$411$	0	0
$412$	18.0000	0.886796
$413$	−12.0000	−0.590481
$414$	0	0
$415$	24.0000	1.17811
$416$	2.00000	0.0980581
$417$	0	0
$418$	−6.00000	−0.293470
$419$	32.0000	1.56330	0.781651	−	0.623716i	$-0.214378\pi$
$419$	32.0000	1.56330	0.781651	+	0.623716i	$0.214378\pi$
$420$	0	0
$421$	−2.00000	−0.0974740	−0.0487370	−	0.998812i	$-0.515520\pi$
$421$	−2.00000	−0.0974740	−0.0487370	+	0.998812i	$0.515520\pi$
$422$	−8.00000	−0.389434
$423$	0	0
$424$	−6.00000	−0.291386
$425$	44.0000	2.13431
$426$	0	0
$427$	14.0000	0.677507
$428$	16.0000	0.773389
$429$	0	0
$430$	−32.0000	−1.54318
$431$	−16.0000	−0.770693	−0.385346	−	0.922772i	$-0.625918\pi$
$431$	−16.0000	−0.770693	−0.385346	+	0.922772i	$0.625918\pi$
$432$	0	0
$433$	2.00000	0.0961139	0.0480569	−	0.998845i	$-0.484697\pi$
$433$	2.00000	0.0961139	0.0480569	+	0.998845i	$0.484697\pi$
$434$	2.00000	0.0960031
$435$	0	0
$436$	−14.0000	−0.670478
$437$	24.0000	1.14808
$438$	0	0
$439$	28.0000	1.33637	0.668184	−	0.743996i	$-0.267072\pi$
$439$	28.0000	1.33637	0.668184	+	0.743996i	$0.267072\pi$
$440$	4.00000	0.190693
$441$	0	0
$442$	8.00000	0.380521
$443$	36.0000	1.71041	0.855206	−	0.518289i	$-0.173431\pi$
$443$	36.0000	1.71041	0.855206	+	0.518289i	$0.173431\pi$
$444$	0	0
$445$	24.0000	1.13771
$446$	−2.00000	−0.0947027
$447$	0	0
$448$	−1.00000	−0.0472456
$449$	18.0000	0.849473	0.424736	−	0.905317i	$-0.360367\pi$
$449$	18.0000	0.849473	0.424736	+	0.905317i	$0.360367\pi$
$450$	0	0
$451$	−4.00000	−0.188353
$452$	−14.0000	−0.658505
$453$	0	0
$454$	2.00000	0.0938647
$455$	−8.00000	−0.375046
$456$	0	0
$457$	2.00000	0.0935561	0.0467780	−	0.998905i	$-0.485105\pi$
$457$	2.00000	0.0935561	0.0467780	+	0.998905i	$0.485105\pi$
$458$	20.0000	0.934539
$459$	0	0
$460$	−16.0000	−0.746004
$461$	−30.0000	−1.39724	−0.698620	−	0.715493i	$-0.746202\pi$
$461$	−30.0000	−1.39724	−0.698620	+	0.715493i	$0.746202\pi$
$462$	0	0
$463$	−32.0000	−1.48717	−0.743583	−	0.668644i	$-0.766875\pi$
$463$	−32.0000	−1.48717	−0.743583	+	0.668644i	$0.766875\pi$
$464$	2.00000	0.0928477
$465$	0	0
$466$	−30.0000	−1.38972
$467$	0	0		−	1.00000i	$-0.5\pi$
$467$	0	0			1.00000i	$0.5\pi$
$468$	0	0
$469$	12.0000	0.554109
$470$	−8.00000	−0.369012
$471$	0	0
$472$	12.0000	0.552345
$473$	−8.00000	−0.367840
$474$	0	0
$475$	−66.0000	−3.02829
$476$	−4.00000	−0.183340
$477$	0	0
$478$	−16.0000	−0.731823
$479$	16.0000	0.731059	0.365529	−	0.930800i	$-0.380888\pi$
$479$	16.0000	0.731059	0.365529	+	0.930800i	$0.380888\pi$
$480$	0	0
$481$	20.0000	0.911922
$482$	12.0000	0.546585
$483$	0	0
$484$	1.00000	0.0454545
$485$	−56.0000	−2.54283
$486$	0	0
$487$	−28.0000	−1.26880	−0.634401	−	0.773004i	$-0.718753\pi$
$487$	−28.0000	−1.26880	−0.634401	+	0.773004i	$0.718753\pi$
$488$	−14.0000	−0.633750
$489$	0	0
$490$	4.00000	0.180702
$491$	36.0000	1.62466	0.812329	−	0.583200i	$-0.198200\pi$
$491$	36.0000	1.62466	0.812329	+	0.583200i	$0.198200\pi$
$492$	0	0
$493$	8.00000	0.360302
$494$	−12.0000	−0.539906
$495$	0	0
$496$	−2.00000	−0.0898027
$497$	−8.00000	−0.358849
$498$	0	0
$499$	44.0000	1.96971	0.984855	−	0.173379i	$-0.0554684\pi$
$499$	44.0000	1.96971	0.984855	+	0.173379i	$0.0554684\pi$
$500$	24.0000	1.07331
$501$	0	0
$502$	−12.0000	−0.535586
$503$	36.0000	1.60516	0.802580	−	0.596544i	$-0.203460\pi$
$503$	36.0000	1.60516	0.802580	+	0.596544i	$0.203460\pi$
$504$	0	0
$505$	−24.0000	−1.06799
$506$	−4.00000	−0.177822
$507$	0	0
$508$	8.00000	0.354943
$509$	28.0000	1.24108	0.620539	−	0.784176i	$-0.286914\pi$
$509$	28.0000	1.24108	0.620539	+	0.784176i	$0.286914\pi$
$510$	0	0
$511$	−4.00000	−0.176950
$512$	1.00000	0.0441942
$513$	0	0
$514$	6.00000	0.264649
$515$	72.0000	3.17270
$516$	0	0
$517$	−2.00000	−0.0879599
$518$	−10.0000	−0.439375
$519$	0	0
$520$	8.00000	0.350823
$521$	−10.0000	−0.438108	−0.219054	−	0.975713i	$-0.570297\pi$
$521$	−10.0000	−0.438108	−0.219054	+	0.975713i	$0.570297\pi$
$522$	0	0
$523$	−34.0000	−1.48672	−0.743358	−	0.668894i	$-0.766768\pi$
$523$	−34.0000	−1.48672	−0.743358	+	0.668894i	$0.766768\pi$
$524$	−6.00000	−0.262111
$525$	0	0
$526$	0	0
$527$	−8.00000	−0.348485
$528$	0	0
$529$	−7.00000	−0.304348
$530$	−24.0000	−1.04249
$531$	0	0
$532$	6.00000	0.260133
$533$	−8.00000	−0.346518
$534$	0	0
$535$	64.0000	2.76696
$536$	−12.0000	−0.518321
$537$	0	0
$538$	−12.0000	−0.517357
$539$	1.00000	0.0430730
$540$	0	0
$541$	14.0000	0.601907	0.300954	−	0.953639i	$-0.402695\pi$
$541$	14.0000	0.601907	0.300954	+	0.953639i	$0.402695\pi$
$542$	−20.0000	−0.859074
$543$	0	0
$544$	4.00000	0.171499
$545$	−56.0000	−2.39878
$546$	0	0
$547$	−12.0000	−0.513083	−0.256541	−	0.966533i	$-0.582583\pi$
$547$	−12.0000	−0.513083	−0.256541	+	0.966533i	$0.582583\pi$
$548$	−6.00000	−0.256307
$549$	0	0
$550$	11.0000	0.469042
$551$	−12.0000	−0.511217
$552$	0	0
$553$	0	0
$554$	−30.0000	−1.27458
$555$	0	0
$556$	14.0000	0.593732
$557$	−14.0000	−0.593199	−0.296600	−	0.955002i	$-0.595853\pi$
$557$	−14.0000	−0.593199	−0.296600	+	0.955002i	$0.595853\pi$
$558$	0	0
$559$	−16.0000	−0.676728
$560$	−4.00000	−0.169031
$561$	0	0
$562$	10.0000	0.421825
$563$	34.0000	1.43293	0.716465	−	0.697623i	$-0.245759\pi$
$563$	34.0000	1.43293	0.716465	+	0.697623i	$0.245759\pi$
$564$	0	0
$565$	−56.0000	−2.35594
$566$	6.00000	0.252199
$567$	0	0
$568$	8.00000	0.335673
$569$	−6.00000	−0.251533	−0.125767	−	0.992060i	$-0.540139\pi$
$569$	−6.00000	−0.251533	−0.125767	+	0.992060i	$0.540139\pi$
$570$	0	0
$571$	28.0000	1.17176	0.585882	−	0.810397i	$-0.300748\pi$
$571$	28.0000	1.17176	0.585882	+	0.810397i	$0.300748\pi$
$572$	2.00000	0.0836242
$573$	0	0
$574$	4.00000	0.166957
$575$	−44.0000	−1.83493
$576$	0	0
$577$	14.0000	0.582828	0.291414	−	0.956597i	$-0.405874\pi$
$577$	14.0000	0.582828	0.291414	+	0.956597i	$0.405874\pi$
$578$	−1.00000	−0.0415945
$579$	0	0
$580$	8.00000	0.332182
$581$	−6.00000	−0.248922
$582$	0	0
$583$	−6.00000	−0.248495
$584$	4.00000	0.165521
$585$	0	0
$586$	−18.0000	−0.743573
$587$	12.0000	0.495293	0.247647	−	0.968850i	$-0.420343\pi$
$587$	12.0000	0.495293	0.247647	+	0.968850i	$0.420343\pi$
$588$	0	0
$589$	12.0000	0.494451
$590$	48.0000	1.97613
$591$	0	0
$592$	10.0000	0.410997
$593$	12.0000	0.492781	0.246390	−	0.969171i	$-0.420755\pi$
$593$	12.0000	0.492781	0.246390	+	0.969171i	$0.420755\pi$
$594$	0	0
$595$	−16.0000	−0.655936
$596$	−2.00000	−0.0819232
$597$	0	0
$598$	−8.00000	−0.327144
$599$	−24.0000	−0.980613	−0.490307	−	0.871550i	$-0.663115\pi$
$599$	−24.0000	−0.980613	−0.490307	+	0.871550i	$0.663115\pi$
$600$	0	0
$601$	8.00000	0.326327	0.163163	−	0.986599i	$-0.447830\pi$
$601$	8.00000	0.326327	0.163163	+	0.986599i	$0.447830\pi$
$602$	8.00000	0.326056
$603$	0	0
$604$	−24.0000	−0.976546
$605$	4.00000	0.162623
$606$	0	0
$607$	8.00000	0.324710	0.162355	−	0.986732i	$-0.448091\pi$
$607$	8.00000	0.324710	0.162355	+	0.986732i	$0.448091\pi$
$608$	−6.00000	−0.243332
$609$	0	0
$610$	−56.0000	−2.26737
$611$	−4.00000	−0.161823
$612$	0	0
$613$	46.0000	1.85792	0.928961	−	0.370177i	$-0.120703\pi$
$613$	46.0000	1.85792	0.928961	+	0.370177i	$0.120703\pi$
$614$	−10.0000	−0.403567
$615$	0	0
$616$	−1.00000	−0.0402911
$617$	−18.0000	−0.724653	−0.362326	−	0.932051i	$-0.618017\pi$
$617$	−18.0000	−0.724653	−0.362326	+	0.932051i	$0.618017\pi$
$618$	0	0
$619$	8.00000	0.321547	0.160774	−	0.986991i	$-0.448601\pi$
$619$	8.00000	0.321547	0.160774	+	0.986991i	$0.448601\pi$
$620$	−8.00000	−0.321288
$621$	0	0
$622$	−14.0000	−0.561349
$623$	−6.00000	−0.240385
$624$	0	0
$625$	41.0000	1.64000
$626$	−2.00000	−0.0799361
$627$	0	0
$628$	−8.00000	−0.319235
$629$	40.0000	1.59490
$630$	0	0
$631$	−12.0000	−0.477712	−0.238856	−	0.971055i	$-0.576772\pi$
$631$	−12.0000	−0.477712	−0.238856	+	0.971055i	$0.576772\pi$
$632$	0	0
$633$	0	0
$634$	−6.00000	−0.238290
$635$	32.0000	1.26988
$636$	0	0
$637$	2.00000	0.0792429
$638$	2.00000	0.0791808
$639$	0	0
$640$	4.00000	0.158114
$641$	6.00000	0.236986	0.118493	−	0.992955i	$-0.462194\pi$
$641$	6.00000	0.236986	0.118493	+	0.992955i	$0.462194\pi$
$642$	0	0
$643$	4.00000	0.157745	0.0788723	−	0.996885i	$-0.474868\pi$
$643$	4.00000	0.157745	0.0788723	+	0.996885i	$0.474868\pi$
$644$	4.00000	0.157622
$645$	0	0
$646$	−24.0000	−0.944267
$647$	−6.00000	−0.235884	−0.117942	−	0.993020i	$-0.537630\pi$
$647$	−6.00000	−0.235884	−0.117942	+	0.993020i	$0.537630\pi$
$648$	0	0
$649$	12.0000	0.471041
$650$	22.0000	0.862911
$651$	0	0
$652$	4.00000	0.156652
$653$	−10.0000	−0.391330	−0.195665	−	0.980671i	$-0.562687\pi$
$653$	−10.0000	−0.391330	−0.195665	+	0.980671i	$0.562687\pi$
$654$	0	0
$655$	−24.0000	−0.937758
$656$	−4.00000	−0.156174
$657$	0	0
$658$	2.00000	0.0779681
$659$	8.00000	0.311636	0.155818	−	0.987786i	$-0.450199\pi$
$659$	8.00000	0.311636	0.155818	+	0.987786i	$0.450199\pi$
$660$	0	0
$661$	20.0000	0.777910	0.388955	−	0.921257i	$-0.372836\pi$
$661$	20.0000	0.777910	0.388955	+	0.921257i	$0.372836\pi$
$662$	−20.0000	−0.777322
$663$	0	0
$664$	6.00000	0.232845
$665$	24.0000	0.930680
$666$	0	0
$667$	−8.00000	−0.309761
$668$	−4.00000	−0.154765
$669$	0	0
$670$	−48.0000	−1.85440
$671$	−14.0000	−0.540464
$672$	0	0
$673$	22.0000	0.848038	0.424019	−	0.905653i	$-0.360619\pi$
$673$	22.0000	0.848038	0.424019	+	0.905653i	$0.360619\pi$
$674$	−18.0000	−0.693334
$675$	0	0
$676$	−9.00000	−0.346154
$677$	−26.0000	−0.999261	−0.499631	−	0.866239i	$-0.666531\pi$
$677$	−26.0000	−0.999261	−0.499631	+	0.866239i	$0.666531\pi$
$678$	0	0
$679$	14.0000	0.537271
$680$	16.0000	0.613572
$681$	0	0
$682$	−2.00000	−0.0765840
$683$	−36.0000	−1.37750	−0.688751	−	0.724998i	$-0.741841\pi$
$683$	−36.0000	−1.37750	−0.688751	+	0.724998i	$0.741841\pi$
$684$	0	0
$685$	−24.0000	−0.916993
$686$	−1.00000	−0.0381802
$687$	0	0
$688$	−8.00000	−0.304997
$689$	−12.0000	−0.457164
$690$	0	0
$691$	36.0000	1.36950	0.684752	−	0.728776i	$-0.259910\pi$
$691$	36.0000	1.36950	0.684752	+	0.728776i	$0.259910\pi$
$692$	14.0000	0.532200
$693$	0	0
$694$	−8.00000	−0.303676
$695$	56.0000	2.12420
$696$	0	0
$697$	−16.0000	−0.606043
$698$	−10.0000	−0.378506
$699$	0	0
$700$	−11.0000	−0.415761
$701$	30.0000	1.13308	0.566542	−	0.824033i	$-0.308281\pi$
$701$	30.0000	1.13308	0.566542	+	0.824033i	$0.308281\pi$
$702$	0	0
$703$	−60.0000	−2.26294
$704$	1.00000	0.0376889
$705$	0	0
$706$	6.00000	0.225813
$707$	6.00000	0.225653
$708$	0	0
$709$	18.0000	0.676004	0.338002	−	0.941145i	$-0.390249\pi$
$709$	18.0000	0.676004	0.338002	+	0.941145i	$0.390249\pi$
$710$	32.0000	1.20094
$711$	0	0
$712$	6.00000	0.224860
$713$	8.00000	0.299602
$714$	0	0
$715$	8.00000	0.299183
$716$	−4.00000	−0.149487
$717$	0	0
$718$	16.0000	0.597115
$719$	26.0000	0.969636	0.484818	−	0.874615i	$-0.338886\pi$
$719$	26.0000	0.969636	0.484818	+	0.874615i	$0.338886\pi$
$720$	0	0
$721$	−18.0000	−0.670355
$722$	17.0000	0.632674
$723$	0	0
$724$	20.0000	0.743294
$725$	22.0000	0.817059
$726$	0	0
$727$	−10.0000	−0.370879	−0.185440	−	0.982656i	$-0.559371\pi$
$727$	−10.0000	−0.370879	−0.185440	+	0.982656i	$0.559371\pi$
$728$	−2.00000	−0.0741249
$729$	0	0
$730$	16.0000	0.592187
$731$	−32.0000	−1.18356
$732$	0	0
$733$	−22.0000	−0.812589	−0.406294	−	0.913742i	$-0.633179\pi$
$733$	−22.0000	−0.812589	−0.406294	+	0.913742i	$0.633179\pi$
$734$	22.0000	0.812035
$735$	0	0
$736$	−4.00000	−0.147442
$737$	−12.0000	−0.442026
$738$	0	0
$739$	−12.0000	−0.441427	−0.220714	−	0.975339i	$-0.570839\pi$
$739$	−12.0000	−0.441427	−0.220714	+	0.975339i	$0.570839\pi$
$740$	40.0000	1.47043
$741$	0	0
$742$	6.00000	0.220267
$743$	24.0000	0.880475	0.440237	−	0.897881i	$-0.354894\pi$
$743$	24.0000	0.880475	0.440237	+	0.897881i	$0.354894\pi$
$744$	0	0
$745$	−8.00000	−0.293097
$746$	−10.0000	−0.366126
$747$	0	0
$748$	4.00000	0.146254
$749$	−16.0000	−0.584627
$750$	0	0
$751$	28.0000	1.02173	0.510867	−	0.859660i	$-0.329324\pi$
$751$	28.0000	1.02173	0.510867	+	0.859660i	$0.329324\pi$
$752$	−2.00000	−0.0729325
$753$	0	0
$754$	4.00000	0.145671
$755$	−96.0000	−3.49380
$756$	0	0
$757$	26.0000	0.944986	0.472493	−	0.881334i	$-0.343354\pi$
$757$	26.0000	0.944986	0.472493	+	0.881334i	$0.343354\pi$
$758$	4.00000	0.145287
$759$	0	0
$760$	−24.0000	−0.870572
$761$	48.0000	1.74000	0.869999	−	0.493053i	$-0.164119\pi$
$761$	48.0000	1.74000	0.869999	+	0.493053i	$0.164119\pi$
$762$	0	0
$763$	14.0000	0.506834
$764$	4.00000	0.144715
$765$	0	0
$766$	10.0000	0.361315
$767$	24.0000	0.866590
$768$	0	0
$769$	16.0000	0.576975	0.288487	−	0.957484i	$-0.406848\pi$
$769$	16.0000	0.576975	0.288487	+	0.957484i	$0.406848\pi$
$770$	−4.00000	−0.144150
$771$	0	0
$772$	2.00000	0.0719816
$773$	48.0000	1.72644	0.863220	−	0.504828i	$-0.168444\pi$
$773$	48.0000	1.72644	0.863220	+	0.504828i	$0.168444\pi$
$774$	0	0
$775$	−22.0000	−0.790263
$776$	−14.0000	−0.502571
$777$	0	0
$778$	30.0000	1.07555
$779$	24.0000	0.859889
$780$	0	0
$781$	8.00000	0.286263
$782$	−16.0000	−0.572159
$783$	0	0
$784$	1.00000	0.0357143
$785$	−32.0000	−1.14213
$786$	0	0
$787$	−22.0000	−0.784215	−0.392108	−	0.919919i	$-0.628254\pi$
$787$	−22.0000	−0.784215	−0.392108	+	0.919919i	$0.628254\pi$
$788$	−6.00000	−0.213741
$789$	0	0
$790$	0	0
$791$	14.0000	0.497783
$792$	0	0
$793$	−28.0000	−0.994309
$794$	24.0000	0.851728
$795$	0	0
$796$	−14.0000	−0.496217
$797$	16.0000	0.566749	0.283375	−	0.959009i	$-0.408546\pi$
$797$	16.0000	0.566749	0.283375	+	0.959009i	$0.408546\pi$
$798$	0	0
$799$	−8.00000	−0.283020
$800$	11.0000	0.388909
$801$	0	0
$802$	18.0000	0.635602
$803$	4.00000	0.141157
$804$	0	0
$805$	16.0000	0.563926
$806$	−4.00000	−0.140894
$807$	0	0
$808$	−6.00000	−0.211079
$809$	30.0000	1.05474	0.527372	−	0.849635i	$-0.323177\pi$
$809$	30.0000	1.05474	0.527372	+	0.849635i	$0.323177\pi$
$810$	0	0
$811$	38.0000	1.33436	0.667180	−	0.744896i	$-0.267501\pi$
$811$	38.0000	1.33436	0.667180	+	0.744896i	$0.267501\pi$
$812$	−2.00000	−0.0701862
$813$	0	0
$814$	10.0000	0.350500
$815$	16.0000	0.560456
$816$	0	0
$817$	48.0000	1.67931
$818$	16.0000	0.559427
$819$	0	0
$820$	−16.0000	−0.558744
$821$	−18.0000	−0.628204	−0.314102	−	0.949389i	$-0.601703\pi$
$821$	−18.0000	−0.628204	−0.314102	+	0.949389i	$0.601703\pi$
$822$	0	0
$823$	−4.00000	−0.139431	−0.0697156	−	0.997567i	$-0.522209\pi$
$823$	−4.00000	−0.139431	−0.0697156	+	0.997567i	$0.522209\pi$
$824$	18.0000	0.627060
$825$	0	0
$826$	−12.0000	−0.417533
$827$	−20.0000	−0.695468	−0.347734	−	0.937593i	$-0.613049\pi$
$827$	−20.0000	−0.695468	−0.347734	+	0.937593i	$0.613049\pi$
$828$	0	0
$829$	−20.0000	−0.694629	−0.347314	−	0.937749i	$-0.612906\pi$
$829$	−20.0000	−0.694629	−0.347314	+	0.937749i	$0.612906\pi$
$830$	24.0000	0.833052
$831$	0	0
$832$	2.00000	0.0693375
$833$	4.00000	0.138592
$834$	0	0
$835$	−16.0000	−0.553703
$836$	−6.00000	−0.207514
$837$	0	0
$838$	32.0000	1.10542
$839$	−30.0000	−1.03572	−0.517858	−	0.855467i	$-0.673270\pi$
$839$	−30.0000	−1.03572	−0.517858	+	0.855467i	$0.673270\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	−2.00000	−0.0689246
$843$	0	0
$844$	−8.00000	−0.275371
$845$	−36.0000	−1.23844
$846$	0	0
$847$	−1.00000	−0.0343604
$848$	−6.00000	−0.206041
$849$	0	0
$850$	44.0000	1.50919
$851$	−40.0000	−1.37118
$852$	0	0
$853$	2.00000	0.0684787	0.0342393	−	0.999414i	$-0.489099\pi$
$853$	2.00000	0.0684787	0.0342393	+	0.999414i	$0.489099\pi$
$854$	14.0000	0.479070
$855$	0	0
$856$	16.0000	0.546869
$857$	−32.0000	−1.09310	−0.546550	−	0.837427i	$-0.684059\pi$
$857$	−32.0000	−1.09310	−0.546550	+	0.837427i	$0.684059\pi$
$858$	0	0
$859$	28.0000	0.955348	0.477674	−	0.878537i	$-0.341480\pi$
$859$	28.0000	0.955348	0.477674	+	0.878537i	$0.341480\pi$
$860$	−32.0000	−1.09119
$861$	0	0
$862$	−16.0000	−0.544962
$863$	44.0000	1.49778	0.748889	−	0.662696i	$-0.230588\pi$
$863$	44.0000	1.49778	0.748889	+	0.662696i	$0.230588\pi$
$864$	0	0
$865$	56.0000	1.90406
$866$	2.00000	0.0679628
$867$	0	0
$868$	2.00000	0.0678844
$869$	0	0
$870$	0	0
$871$	−24.0000	−0.813209
$872$	−14.0000	−0.474100
$873$	0	0
$874$	24.0000	0.811812
$875$	−24.0000	−0.811348
$876$	0	0
$877$	34.0000	1.14810	0.574049	−	0.818821i	$-0.305372\pi$
$877$	34.0000	1.14810	0.574049	+	0.818821i	$0.305372\pi$
$878$	28.0000	0.944954
$879$	0	0
$880$	4.00000	0.134840
$881$	−30.0000	−1.01073	−0.505363	−	0.862907i	$-0.668641\pi$
$881$	−30.0000	−1.01073	−0.505363	+	0.862907i	$0.668641\pi$
$882$	0	0
$883$	−44.0000	−1.48072	−0.740359	−	0.672212i	$-0.765344\pi$
$883$	−44.0000	−1.48072	−0.740359	+	0.672212i	$0.765344\pi$
$884$	8.00000	0.269069
$885$	0	0
$886$	36.0000	1.20944
$887$	−16.0000	−0.537227	−0.268614	−	0.963248i	$-0.586566\pi$
$887$	−16.0000	−0.537227	−0.268614	+	0.963248i	$0.586566\pi$
$888$	0	0
$889$	−8.00000	−0.268311
$890$	24.0000	0.804482
$891$	0	0
$892$	−2.00000	−0.0669650
$893$	12.0000	0.401565
$894$	0	0
$895$	−16.0000	−0.534821
$896$	−1.00000	−0.0334077
$897$	0	0
$898$	18.0000	0.600668
$899$	−4.00000	−0.133407
$900$	0	0
$901$	−24.0000	−0.799556
$902$	−4.00000	−0.133185
$903$	0	0
$904$	−14.0000	−0.465633
$905$	80.0000	2.65929
$906$	0	0
$907$	−52.0000	−1.72663	−0.863316	−	0.504664i	$-0.831616\pi$
$907$	−52.0000	−1.72663	−0.863316	+	0.504664i	$0.831616\pi$
$908$	2.00000	0.0663723
$909$	0	0
$910$	−8.00000	−0.265197
$911$	36.0000	1.19273	0.596367	−	0.802712i	$-0.296610\pi$
$911$	36.0000	1.19273	0.596367	+	0.802712i	$0.296610\pi$
$912$	0	0
$913$	6.00000	0.198571
$914$	2.00000	0.0661541
$915$	0	0
$916$	20.0000	0.660819
$917$	6.00000	0.198137
$918$	0	0
$919$	24.0000	0.791687	0.395843	−	0.918318i	$-0.370452\pi$
$919$	24.0000	0.791687	0.395843	+	0.918318i	$0.370452\pi$
$920$	−16.0000	−0.527504
$921$	0	0
$922$	−30.0000	−0.987997
$923$	16.0000	0.526646
$924$	0	0
$925$	110.000	3.61678
$926$	−32.0000	−1.05159
$927$	0	0
$928$	2.00000	0.0656532
$929$	30.0000	0.984268	0.492134	−	0.870519i	$-0.336217\pi$
$929$	30.0000	0.984268	0.492134	+	0.870519i	$0.336217\pi$
$930$	0	0
$931$	−6.00000	−0.196642
$932$	−30.0000	−0.982683
$933$	0	0
$934$	0	0
$935$	16.0000	0.523256
$936$	0	0
$937$	12.0000	0.392023	0.196011	−	0.980602i	$-0.437201\pi$
$937$	12.0000	0.392023	0.196011	+	0.980602i	$0.437201\pi$
$938$	12.0000	0.391814
$939$	0	0
$940$	−8.00000	−0.260931
$941$	14.0000	0.456387	0.228193	−	0.973616i	$-0.426718\pi$
$941$	14.0000	0.456387	0.228193	+	0.973616i	$0.426718\pi$
$942$	0	0
$943$	16.0000	0.521032
$944$	12.0000	0.390567
$945$	0	0
$946$	−8.00000	−0.260102
$947$	4.00000	0.129983	0.0649913	−	0.997886i	$-0.479298\pi$
$947$	4.00000	0.129983	0.0649913	+	0.997886i	$0.479298\pi$
$948$	0	0
$949$	8.00000	0.259691
$950$	−66.0000	−2.14132
$951$	0	0
$952$	−4.00000	−0.129641
$953$	22.0000	0.712650	0.356325	−	0.934362i	$-0.384030\pi$
$953$	22.0000	0.712650	0.356325	+	0.934362i	$0.384030\pi$
$954$	0	0
$955$	16.0000	0.517748
$956$	−16.0000	−0.517477
$957$	0	0
$958$	16.0000	0.516937
$959$	6.00000	0.193750
$960$	0	0
$961$	−27.0000	−0.870968
$962$	20.0000	0.644826
$963$	0	0
$964$	12.0000	0.386494
$965$	8.00000	0.257529
$966$	0	0
$967$	32.0000	1.02905	0.514525	−	0.857475i	$-0.327968\pi$
$967$	32.0000	1.02905	0.514525	+	0.857475i	$0.327968\pi$
$968$	1.00000	0.0321412
$969$	0	0
$970$	−56.0000	−1.79805
$971$	−56.0000	−1.79713	−0.898563	−	0.438845i	$-0.855388\pi$
$971$	−56.0000	−1.79713	−0.898563	+	0.438845i	$0.855388\pi$
$972$	0	0
$973$	−14.0000	−0.448819
$974$	−28.0000	−0.897178
$975$	0	0
$976$	−14.0000	−0.448129
$977$	−2.00000	−0.0639857	−0.0319928	−	0.999488i	$-0.510185\pi$
$977$	−2.00000	−0.0639857	−0.0319928	+	0.999488i	$0.510185\pi$
$978$	0	0
$979$	6.00000	0.191761
$980$	4.00000	0.127775
$981$	0	0
$982$	36.0000	1.14881
$983$	−18.0000	−0.574111	−0.287055	−	0.957914i	$-0.592676\pi$
$983$	−18.0000	−0.574111	−0.287055	+	0.957914i	$0.592676\pi$
$984$	0	0
$985$	−24.0000	−0.764704
$986$	8.00000	0.254772
$987$	0	0
$988$	−12.0000	−0.381771
$989$	32.0000	1.01754
$990$	0	0
$991$	16.0000	0.508257	0.254128	−	0.967170i	$-0.418211\pi$
$991$	16.0000	0.508257	0.254128	+	0.967170i	$0.418211\pi$
$992$	−2.00000	−0.0635001
$993$	0	0
$994$	−8.00000	−0.253745
$995$	−56.0000	−1.77532
$996$	0	0
$997$	−42.0000	−1.33015	−0.665077	−	0.746775i	$-0.731601\pi$
$997$	−42.0000	−1.33015	−0.665077	+	0.746775i	$0.731601\pi$
$998$	44.0000	1.39280
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1386.2.a.l.1.1		1
3.2	odd	2		154.2.a.a.1.1	✓	1
7.6	odd	2		9702.2.a.ba.1.1		1
12.11	even	2		1232.2.a.e.1.1		1
15.2	even	4		3850.2.c.j.1849.1		2
15.8	even	4		3850.2.c.j.1849.2		2
15.14	odd	2		3850.2.a.u.1.1		1
21.2	odd	6		1078.2.e.j.67.1		2
21.5	even	6		1078.2.e.i.67.1		2
21.11	odd	6		1078.2.e.j.177.1		2
21.17	even	6		1078.2.e.i.177.1		2
21.20	even	2		1078.2.a.d.1.1		1
24.5	odd	2		4928.2.a.v.1.1		1
24.11	even	2		4928.2.a.w.1.1		1
33.32	even	2		1694.2.a.g.1.1		1
84.83	odd	2		8624.2.a.r.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
154.2.a.a.1.1	✓	1	3.2	odd	2
1078.2.a.d.1.1		1	21.20	even	2
1078.2.e.i.67.1		2	21.5	even	6
1078.2.e.i.177.1		2	21.17	even	6
1078.2.e.j.67.1		2	21.2	odd	6
1078.2.e.j.177.1		2	21.11	odd	6
1232.2.a.e.1.1		1	12.11	even	2
1386.2.a.l.1.1		1	1.1	even	1	trivial
1694.2.a.g.1.1		1	33.32	even	2
3850.2.a.u.1.1		1	15.14	odd	2
3850.2.c.j.1849.1		2	15.2	even	4
3850.2.c.j.1849.2		2	15.8	even	4
4928.2.a.v.1.1		1	24.5	odd	2
4928.2.a.w.1.1		1	24.11	even	2
8624.2.a.r.1.1		1	84.83	odd	2
9702.2.a.ba.1.1		1	7.6	odd	2

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.a (trivial)

Introduction

L-functions

Modular forms

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Representations

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