LMFDB - Embedded newform 1110.2.a.g.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1110,2,Mod(1,1110)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1110, 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("1110.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1110 = 2 \cdot 3 \cdot 5 \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1110.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:	$8.86339462436$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	1110.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} +4.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$

\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} +4.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} +4.00000 q^{11} +1.00000 q^{12} +2.00000 q^{13} -4.00000 q^{14} -1.00000 q^{15} +1.00000 q^{16} -2.00000 q^{17} -1.00000 q^{18} -1.00000 q^{20} +4.00000 q^{21} -4.00000 q^{22} -1.00000 q^{24} +1.00000 q^{25} -2.00000 q^{26} +1.00000 q^{27} +4.00000 q^{28} -6.00000 q^{29} +1.00000 q^{30} -1.00000 q^{32} +4.00000 q^{33} +2.00000 q^{34} -4.00000 q^{35} +1.00000 q^{36} -1.00000 q^{37} +2.00000 q^{39} +1.00000 q^{40} -6.00000 q^{41} -4.00000 q^{42} +4.00000 q^{43} +4.00000 q^{44} -1.00000 q^{45} +12.0000 q^{47} +1.00000 q^{48} +9.00000 q^{49} -1.00000 q^{50} -2.00000 q^{51} +2.00000 q^{52} -6.00000 q^{53} -1.00000 q^{54} -4.00000 q^{55} -4.00000 q^{56} +6.00000 q^{58} +8.00000 q^{59} -1.00000 q^{60} +10.0000 q^{61} +4.00000 q^{63} +1.00000 q^{64} -2.00000 q^{65} -4.00000 q^{66} +4.00000 q^{67} -2.00000 q^{68} +4.00000 q^{70} -1.00000 q^{72} -6.00000 q^{73} +1.00000 q^{74} +1.00000 q^{75} +16.0000 q^{77} -2.00000 q^{78} -1.00000 q^{80} +1.00000 q^{81} +6.00000 q^{82} -4.00000 q^{83} +4.00000 q^{84} +2.00000 q^{85} -4.00000 q^{86} -6.00000 q^{87} -4.00000 q^{88} -14.0000 q^{89} +1.00000 q^{90} +8.00000 q^{91} -12.0000 q^{94} -1.00000 q^{96} -10.0000 q^{97} -9.00000 q^{98} +4.00000 q^{99} +O(q^{100})\)

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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	−1.00000	−0.408248
$7$	4.00000	1.51186	0.755929	−	0.654654i	$-0.227186\pi$
$7$	4.00000	1.51186	0.755929	+	0.654654i	$0.227186\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	1.00000	0.316228
$11$	4.00000	1.20605	0.603023	−	0.797724i	$-0.293963\pi$
$11$	4.00000	1.20605	0.603023	+	0.797724i	$0.293963\pi$
$12$	1.00000	0.288675
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	−4.00000	−1.06904
$15$	−1.00000	−0.258199
$16$	1.00000	0.250000
$17$	−2.00000	−0.485071	−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000	−0.485071	−0.242536	+	0.970143i	$0.577979\pi$
$18$	−1.00000	−0.235702
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	−1.00000	−0.223607
$21$	4.00000	0.872872
$22$	−4.00000	−0.852803
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	−1.00000	−0.204124
$25$	1.00000	0.200000
$26$	−2.00000	−0.392232
$27$	1.00000	0.192450
$28$	4.00000	0.755929
$29$	−6.00000	−1.11417	−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000	−1.11417	−0.557086	+	0.830455i	$0.688081\pi$
$30$	1.00000	0.182574
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	−1.00000	−0.176777
$33$	4.00000	0.696311
$34$	2.00000	0.342997
$35$	−4.00000	−0.676123
$36$	1.00000	0.166667
$37$	−1.00000	−0.164399
$37$	−1.00000	−0.164399
$38$	0	0
$39$	2.00000	0.320256
$40$	1.00000	0.158114
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	−4.00000	−0.617213
$43$	4.00000	0.609994	0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000	0.609994	0.304997	+	0.952353i	$0.401344\pi$
$44$	4.00000	0.603023
$45$	−1.00000	−0.149071
$46$	0	0
$47$	12.0000	1.75038	0.875190	−	0.483779i	$-0.160736\pi$
$47$	12.0000	1.75038	0.875190	+	0.483779i	$0.160736\pi$
$48$	1.00000	0.144338
$49$	9.00000	1.28571
$50$	−1.00000	−0.141421
$51$	−2.00000	−0.280056
$52$	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	$0.635198\pi$
$54$	−1.00000	−0.136083
$55$	−4.00000	−0.539360
$56$	−4.00000	−0.534522
$57$	0	0
$58$	6.00000	0.787839
$59$	8.00000	1.04151	0.520756	−	0.853706i	$-0.325650\pi$
$59$	8.00000	1.04151	0.520756	+	0.853706i	$0.325650\pi$
$60$	−1.00000	−0.129099
$61$	10.0000	1.28037	0.640184	−	0.768221i	$-0.278858\pi$
$61$	10.0000	1.28037	0.640184	+	0.768221i	$0.278858\pi$
$62$	0	0
$63$	4.00000	0.503953
$64$	1.00000	0.125000
$65$	−2.00000	−0.248069
$66$	−4.00000	−0.492366
$67$	4.00000	0.488678	0.244339	−	0.969690i	$-0.421429\pi$
$67$	4.00000	0.488678	0.244339	+	0.969690i	$0.421429\pi$
$68$	−2.00000	−0.242536
$69$	0	0
$70$	4.00000	0.478091
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	−1.00000	−0.117851
$73$	−6.00000	−0.702247	−0.351123	−	0.936329i	$-0.614200\pi$
$73$	−6.00000	−0.702247	−0.351123	+	0.936329i	$0.614200\pi$
$74$	1.00000	0.116248
$75$	1.00000	0.115470
$76$	0	0
$77$	16.0000	1.82337
$78$	−2.00000	−0.226455
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	−1.00000	−0.111803
$81$	1.00000	0.111111
$82$	6.00000	0.662589
$83$	−4.00000	−0.439057	−0.219529	−	0.975606i	$-0.570452\pi$
$83$	−4.00000	−0.439057	−0.219529	+	0.975606i	$0.570452\pi$
$84$	4.00000	0.436436
$85$	2.00000	0.216930
$86$	−4.00000	−0.431331
$87$	−6.00000	−0.643268
$88$	−4.00000	−0.426401
$89$	−14.0000	−1.48400	−0.741999	−	0.670402i	$-0.766122\pi$
$89$	−14.0000	−1.48400	−0.741999	+	0.670402i	$0.766122\pi$
$90$	1.00000	0.105409
$91$	8.00000	0.838628
$92$	0	0
$93$	0	0
$94$	−12.0000	−1.23771
$95$	0	0
$96$	−1.00000	−0.102062
$97$	−10.0000	−1.01535	−0.507673	−	0.861550i	$-0.669494\pi$
$97$	−10.0000	−1.01535	−0.507673	+	0.861550i	$0.669494\pi$
$98$	−9.00000	−0.909137
$99$	4.00000	0.402015
$100$	1.00000	0.100000
$101$	6.00000	0.597022	0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000	0.597022	0.298511	+	0.954406i	$0.403510\pi$
$102$	2.00000	0.198030
$103$	8.00000	0.788263	0.394132	−	0.919054i	$-0.371045\pi$
$103$	8.00000	0.788263	0.394132	+	0.919054i	$0.371045\pi$
$104$	−2.00000	−0.196116
$105$	−4.00000	−0.390360
$106$	6.00000	0.582772
$107$	12.0000	1.16008	0.580042	−	0.814587i	$-0.303036\pi$
$107$	12.0000	1.16008	0.580042	+	0.814587i	$0.303036\pi$
$108$	1.00000	0.0962250
$109$	−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
$111$	−1.00000	−0.0949158
$112$	4.00000	0.377964
$113$	14.0000	1.31701	0.658505	−	0.752577i	$-0.271189\pi$
$113$	14.0000	1.31701	0.658505	+	0.752577i	$0.271189\pi$
$114$	0	0
$115$	0	0
$116$	−6.00000	−0.557086
$117$	2.00000	0.184900
$118$	−8.00000	−0.736460
$119$	−8.00000	−0.733359
$120$	1.00000	0.0912871
$121$	5.00000	0.454545
$122$	−10.0000	−0.905357
$123$	−6.00000	−0.541002
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	−4.00000	−0.356348
$127$	12.0000	1.06483	0.532414	−	0.846484i	$-0.321285\pi$
$127$	12.0000	1.06483	0.532414	+	0.846484i	$0.321285\pi$
$128$	−1.00000	−0.0883883
$129$	4.00000	0.352180
$130$	2.00000	0.175412
$131$	−8.00000	−0.698963	−0.349482	−	0.936943i	$-0.613642\pi$
$131$	−8.00000	−0.698963	−0.349482	+	0.936943i	$0.613642\pi$
$132$	4.00000	0.348155
$133$	0	0
$134$	−4.00000	−0.345547
$135$	−1.00000	−0.0860663
$136$	2.00000	0.171499
$137$	10.0000	0.854358	0.427179	−	0.904167i	$-0.359507\pi$
$137$	10.0000	0.854358	0.427179	+	0.904167i	$0.359507\pi$
$138$	0	0
$139$	−4.00000	−0.339276	−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000	−0.339276	−0.169638	+	0.985506i	$0.554260\pi$
$140$	−4.00000	−0.338062
$141$	12.0000	1.01058
$142$	0	0
$143$	8.00000	0.668994
$144$	1.00000	0.0833333
$145$	6.00000	0.498273
$146$	6.00000	0.496564
$147$	9.00000	0.742307
$148$	−1.00000	−0.0821995
$149$	−10.0000	−0.819232	−0.409616	−	0.912258i	$-0.634337\pi$
$149$	−10.0000	−0.819232	−0.409616	+	0.912258i	$0.634337\pi$
$150$	−1.00000	−0.0816497
$151$	0	0		−	1.00000i	$-0.5\pi$
$151$	0	0			1.00000i	$0.5\pi$
$152$	0	0
$153$	−2.00000	−0.161690
$154$	−16.0000	−1.28932
$155$	0	0
$156$	2.00000	0.160128
$157$	10.0000	0.798087	0.399043	−	0.916932i	$-0.369342\pi$
$157$	10.0000	0.798087	0.399043	+	0.916932i	$0.369342\pi$
$158$	0	0
$159$	−6.00000	−0.475831
$160$	1.00000	0.0790569
$161$	0	0
$162$	−1.00000	−0.0785674
$163$	4.00000	0.313304	0.156652	−	0.987654i	$-0.449930\pi$
$163$	4.00000	0.313304	0.156652	+	0.987654i	$0.449930\pi$
$164$	−6.00000	−0.468521
$165$	−4.00000	−0.311400
$166$	4.00000	0.310460
$167$	8.00000	0.619059	0.309529	−	0.950890i	$-0.399829\pi$
$167$	8.00000	0.619059	0.309529	+	0.950890i	$0.399829\pi$
$168$	−4.00000	−0.308607
$169$	−9.00000	−0.692308
$170$	−2.00000	−0.153393
$171$	0	0
$172$	4.00000	0.304997
$173$	2.00000	0.152057	0.0760286	−	0.997106i	$-0.475776\pi$
$173$	2.00000	0.152057	0.0760286	+	0.997106i	$0.475776\pi$
$174$	6.00000	0.454859
$175$	4.00000	0.302372
$176$	4.00000	0.301511
$177$	8.00000	0.601317
$178$	14.0000	1.04934
$179$	−16.0000	−1.19590	−0.597948	−	0.801535i	$-0.704017\pi$
$179$	−16.0000	−1.19590	−0.597948	+	0.801535i	$0.704017\pi$
$180$	−1.00000	−0.0745356
$181$	14.0000	1.04061	0.520306	−	0.853980i	$-0.325818\pi$
$181$	14.0000	1.04061	0.520306	+	0.853980i	$0.325818\pi$
$182$	−8.00000	−0.592999
$183$	10.0000	0.739221
$184$	0	0
$185$	1.00000	0.0735215
$186$	0	0
$187$	−8.00000	−0.585018
$188$	12.0000	0.875190
$189$	4.00000	0.290957
$190$	0	0
$191$	−16.0000	−1.15772	−0.578860	−	0.815427i	$-0.696502\pi$
$191$	−16.0000	−1.15772	−0.578860	+	0.815427i	$0.696502\pi$
$192$	1.00000	0.0721688
$193$	−2.00000	−0.143963	−0.0719816	−	0.997406i	$-0.522932\pi$
$193$	−2.00000	−0.143963	−0.0719816	+	0.997406i	$0.522932\pi$
$194$	10.0000	0.717958
$195$	−2.00000	−0.143223
$196$	9.00000	0.642857
$197$	2.00000	0.142494	0.0712470	−	0.997459i	$-0.477302\pi$
$197$	2.00000	0.142494	0.0712470	+	0.997459i	$0.477302\pi$
$198$	−4.00000	−0.284268
$199$	−24.0000	−1.70131	−0.850657	−	0.525720i	$-0.823796\pi$
$199$	−24.0000	−1.70131	−0.850657	+	0.525720i	$0.823796\pi$
$200$	−1.00000	−0.0707107
$201$	4.00000	0.282138
$202$	−6.00000	−0.422159
$203$	−24.0000	−1.68447
$204$	−2.00000	−0.140028
$205$	6.00000	0.419058
$206$	−8.00000	−0.557386
$207$	0	0
$208$	2.00000	0.138675
$209$	0	0
$210$	4.00000	0.276026
$211$	−20.0000	−1.37686	−0.688428	−	0.725304i	$-0.741699\pi$
$211$	−20.0000	−1.37686	−0.688428	+	0.725304i	$0.741699\pi$
$212$	−6.00000	−0.412082
$213$	0	0
$214$	−12.0000	−0.820303
$215$	−4.00000	−0.272798
$216$	−1.00000	−0.0680414
$217$	0	0
$218$	14.0000	0.948200
$219$	−6.00000	−0.405442
$220$	−4.00000	−0.269680
$221$	−4.00000	−0.269069
$222$	1.00000	0.0671156
$223$	−20.0000	−1.33930	−0.669650	−	0.742677i	$-0.733556\pi$
$223$	−20.0000	−1.33930	−0.669650	+	0.742677i	$0.733556\pi$
$224$	−4.00000	−0.267261
$225$	1.00000	0.0666667
$226$	−14.0000	−0.931266
$227$	−4.00000	−0.265489	−0.132745	−	0.991150i	$-0.542379\pi$
$227$	−4.00000	−0.265489	−0.132745	+	0.991150i	$0.542379\pi$
$228$	0	0
$229$	−18.0000	−1.18947	−0.594737	−	0.803921i	$-0.702744\pi$
$229$	−18.0000	−1.18947	−0.594737	+	0.803921i	$0.702744\pi$
$230$	0	0
$231$	16.0000	1.05272
$232$	6.00000	0.393919
$233$	10.0000	0.655122	0.327561	−	0.944830i	$-0.393773\pi$
$233$	10.0000	0.655122	0.327561	+	0.944830i	$0.393773\pi$
$234$	−2.00000	−0.130744
$235$	−12.0000	−0.782794
$236$	8.00000	0.520756
$237$	0	0
$238$	8.00000	0.518563
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	−1.00000	−0.0645497
$241$	18.0000	1.15948	0.579741	−	0.814801i	$-0.303154\pi$
$241$	18.0000	1.15948	0.579741	+	0.814801i	$0.303154\pi$
$242$	−5.00000	−0.321412
$243$	1.00000	0.0641500
$244$	10.0000	0.640184
$245$	−9.00000	−0.574989
$246$	6.00000	0.382546
$247$	0	0
$248$	0	0
$249$	−4.00000	−0.253490
$250$	1.00000	0.0632456
$251$	24.0000	1.51487	0.757433	−	0.652913i	$-0.226453\pi$
$251$	24.0000	1.51487	0.757433	+	0.652913i	$0.226453\pi$
$252$	4.00000	0.251976
$253$	0	0
$254$	−12.0000	−0.752947
$255$	2.00000	0.125245
$256$	1.00000	0.0625000
$257$	−18.0000	−1.12281	−0.561405	−	0.827541i	$-0.689739\pi$
$257$	−18.0000	−1.12281	−0.561405	+	0.827541i	$0.689739\pi$
$258$	−4.00000	−0.249029
$259$	−4.00000	−0.248548
$260$	−2.00000	−0.124035
$261$	−6.00000	−0.371391
$262$	8.00000	0.494242
$263$	−12.0000	−0.739952	−0.369976	−	0.929041i	$-0.620634\pi$
$263$	−12.0000	−0.739952	−0.369976	+	0.929041i	$0.620634\pi$
$264$	−4.00000	−0.246183
$265$	6.00000	0.368577
$266$	0	0
$267$	−14.0000	−0.856786
$268$	4.00000	0.244339
$269$	−18.0000	−1.09748	−0.548740	−	0.835993i	$-0.684892\pi$
$269$	−18.0000	−1.09748	−0.548740	+	0.835993i	$0.684892\pi$
$270$	1.00000	0.0608581
$271$	8.00000	0.485965	0.242983	−	0.970031i	$-0.421874\pi$
$271$	8.00000	0.485965	0.242983	+	0.970031i	$0.421874\pi$
$272$	−2.00000	−0.121268
$273$	8.00000	0.484182
$274$	−10.0000	−0.604122
$275$	4.00000	0.241209
$276$	0	0
$277$	10.0000	0.600842	0.300421	−	0.953807i	$-0.402873\pi$
$277$	10.0000	0.600842	0.300421	+	0.953807i	$0.402873\pi$
$278$	4.00000	0.239904
$279$	0	0
$280$	4.00000	0.239046
$281$	−14.0000	−0.835170	−0.417585	−	0.908638i	$-0.637123\pi$
$281$	−14.0000	−0.835170	−0.417585	+	0.908638i	$0.637123\pi$
$282$	−12.0000	−0.714590
$283$	−4.00000	−0.237775	−0.118888	−	0.992908i	$-0.537933\pi$
$283$	−4.00000	−0.237775	−0.118888	+	0.992908i	$0.537933\pi$
$284$	0	0
$285$	0	0
$286$	−8.00000	−0.473050
$287$	−24.0000	−1.41668
$288$	−1.00000	−0.0589256
$289$	−13.0000	−0.764706
$290$	−6.00000	−0.352332
$291$	−10.0000	−0.586210
$292$	−6.00000	−0.351123
$293$	−14.0000	−0.817889	−0.408944	−	0.912559i	$-0.634103\pi$
$293$	−14.0000	−0.817889	−0.408944	+	0.912559i	$0.634103\pi$
$294$	−9.00000	−0.524891
$295$	−8.00000	−0.465778
$296$	1.00000	0.0581238
$297$	4.00000	0.232104
$298$	10.0000	0.579284
$299$	0	0
$300$	1.00000	0.0577350
$301$	16.0000	0.922225
$302$	0	0
$303$	6.00000	0.344691
$304$	0	0
$305$	−10.0000	−0.572598
$306$	2.00000	0.114332
$307$	12.0000	0.684876	0.342438	−	0.939540i	$-0.388747\pi$
$307$	12.0000	0.684876	0.342438	+	0.939540i	$0.388747\pi$
$308$	16.0000	0.911685
$309$	8.00000	0.455104
$310$	0	0
$311$	−16.0000	−0.907277	−0.453638	−	0.891186i	$-0.649874\pi$
$311$	−16.0000	−0.907277	−0.453638	+	0.891186i	$0.649874\pi$
$312$	−2.00000	−0.113228
$313$	6.00000	0.339140	0.169570	−	0.985518i	$-0.445762\pi$
$313$	6.00000	0.339140	0.169570	+	0.985518i	$0.445762\pi$
$314$	−10.0000	−0.564333
$315$	−4.00000	−0.225374
$316$	0	0
$317$	2.00000	0.112331	0.0561656	−	0.998421i	$-0.482113\pi$
$317$	2.00000	0.112331	0.0561656	+	0.998421i	$0.482113\pi$
$318$	6.00000	0.336463
$319$	−24.0000	−1.34374
$320$	−1.00000	−0.0559017
$321$	12.0000	0.669775
$322$	0	0
$323$	0	0
$324$	1.00000	0.0555556
$325$	2.00000	0.110940
$326$	−4.00000	−0.221540
$327$	−14.0000	−0.774202
$328$	6.00000	0.331295
$329$	48.0000	2.64633
$330$	4.00000	0.220193
$331$	−32.0000	−1.75888	−0.879440	−	0.476011i	$-0.842082\pi$
$331$	−32.0000	−1.75888	−0.879440	+	0.476011i	$0.842082\pi$
$332$	−4.00000	−0.219529
$333$	−1.00000	−0.0547997
$334$	−8.00000	−0.437741
$335$	−4.00000	−0.218543
$336$	4.00000	0.218218
$337$	−30.0000	−1.63420	−0.817102	−	0.576493i	$-0.804421\pi$
$337$	−30.0000	−1.63420	−0.817102	+	0.576493i	$0.804421\pi$
$338$	9.00000	0.489535
$339$	14.0000	0.760376
$340$	2.00000	0.108465
$341$	0	0
$342$	0	0
$343$	8.00000	0.431959
$344$	−4.00000	−0.215666
$345$	0	0
$346$	−2.00000	−0.107521
$347$	−12.0000	−0.644194	−0.322097	−	0.946707i	$-0.604388\pi$
$347$	−12.0000	−0.644194	−0.322097	+	0.946707i	$0.604388\pi$
$348$	−6.00000	−0.321634
$349$	30.0000	1.60586	0.802932	−	0.596071i	$-0.203272\pi$
$349$	30.0000	1.60586	0.802932	+	0.596071i	$0.203272\pi$
$350$	−4.00000	−0.213809
$351$	2.00000	0.106752
$352$	−4.00000	−0.213201
$353$	−26.0000	−1.38384	−0.691920	−	0.721974i	$-0.743235\pi$
$353$	−26.0000	−1.38384	−0.691920	+	0.721974i	$0.743235\pi$
$354$	−8.00000	−0.425195
$355$	0	0
$356$	−14.0000	−0.741999
$357$	−8.00000	−0.423405
$358$	16.0000	0.845626
$359$	−24.0000	−1.26667	−0.633336	−	0.773877i	$-0.718315\pi$
$359$	−24.0000	−1.26667	−0.633336	+	0.773877i	$0.718315\pi$
$360$	1.00000	0.0527046
$361$	−19.0000	−1.00000
$362$	−14.0000	−0.735824
$363$	5.00000	0.262432
$364$	8.00000	0.419314
$365$	6.00000	0.314054
$366$	−10.0000	−0.522708
$367$	−20.0000	−1.04399	−0.521996	−	0.852948i	$-0.674812\pi$
$367$	−20.0000	−1.04399	−0.521996	+	0.852948i	$0.674812\pi$
$368$	0	0
$369$	−6.00000	−0.312348
$370$	−1.00000	−0.0519875
$371$	−24.0000	−1.24602
$372$	0	0
$373$	2.00000	0.103556	0.0517780	−	0.998659i	$-0.483511\pi$
$373$	2.00000	0.103556	0.0517780	+	0.998659i	$0.483511\pi$
$374$	8.00000	0.413670
$375$	−1.00000	−0.0516398
$376$	−12.0000	−0.618853
$377$	−12.0000	−0.618031
$378$	−4.00000	−0.205738
$379$	−20.0000	−1.02733	−0.513665	−	0.857991i	$-0.671713\pi$
$379$	−20.0000	−1.02733	−0.513665	+	0.857991i	$0.671713\pi$
$380$	0	0
$381$	12.0000	0.614779
$382$	16.0000	0.818631
$383$	24.0000	1.22634	0.613171	−	0.789950i	$-0.289894\pi$
$383$	24.0000	1.22634	0.613171	+	0.789950i	$0.289894\pi$
$384$	−1.00000	−0.0510310
$385$	−16.0000	−0.815436
$386$	2.00000	0.101797
$387$	4.00000	0.203331
$388$	−10.0000	−0.507673
$389$	34.0000	1.72387	0.861934	−	0.507020i	$-0.169253\pi$
$389$	34.0000	1.72387	0.861934	+	0.507020i	$0.169253\pi$
$390$	2.00000	0.101274
$391$	0	0
$392$	−9.00000	−0.454569
$393$	−8.00000	−0.403547
$394$	−2.00000	−0.100759
$395$	0	0
$396$	4.00000	0.201008
$397$	18.0000	0.903394	0.451697	−	0.892171i	$-0.350819\pi$
$397$	18.0000	0.903394	0.451697	+	0.892171i	$0.350819\pi$
$398$	24.0000	1.20301
$399$	0	0
$400$	1.00000	0.0500000
$401$	−6.00000	−0.299626	−0.149813	−	0.988714i	$-0.547867\pi$
$401$	−6.00000	−0.299626	−0.149813	+	0.988714i	$0.547867\pi$
$402$	−4.00000	−0.199502
$403$	0	0
$404$	6.00000	0.298511
$405$	−1.00000	−0.0496904
$406$	24.0000	1.19110
$407$	−4.00000	−0.198273
$408$	2.00000	0.0990148
$409$	−30.0000	−1.48340	−0.741702	−	0.670729i	$-0.765981\pi$
$409$	−30.0000	−1.48340	−0.741702	+	0.670729i	$0.765981\pi$
$410$	−6.00000	−0.296319
$411$	10.0000	0.493264
$412$	8.00000	0.394132
$413$	32.0000	1.57462
$414$	0	0
$415$	4.00000	0.196352
$416$	−2.00000	−0.0980581
$417$	−4.00000	−0.195881
$418$	0	0
$419$	20.0000	0.977064	0.488532	−	0.872546i	$-0.337533\pi$
$419$	20.0000	0.977064	0.488532	+	0.872546i	$0.337533\pi$
$420$	−4.00000	−0.195180
$421$	−6.00000	−0.292422	−0.146211	−	0.989253i	$-0.546708\pi$
$421$	−6.00000	−0.292422	−0.146211	+	0.989253i	$0.546708\pi$
$422$	20.0000	0.973585
$423$	12.0000	0.583460
$424$	6.00000	0.291386
$425$	−2.00000	−0.0970143
$426$	0	0
$427$	40.0000	1.93574
$428$	12.0000	0.580042
$429$	8.00000	0.386244
$430$	4.00000	0.192897
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	1.00000	0.0481125
$433$	−30.0000	−1.44171	−0.720854	−	0.693087i	$-0.756250\pi$
$433$	−30.0000	−1.44171	−0.720854	+	0.693087i	$0.756250\pi$
$434$	0	0
$435$	6.00000	0.287678
$436$	−14.0000	−0.670478
$437$	0	0
$438$	6.00000	0.286691
$439$	−24.0000	−1.14546	−0.572729	−	0.819745i	$-0.694115\pi$
$439$	−24.0000	−1.14546	−0.572729	+	0.819745i	$0.694115\pi$
$440$	4.00000	0.190693
$441$	9.00000	0.428571
$442$	4.00000	0.190261
$443$	−4.00000	−0.190046	−0.0950229	−	0.995475i	$-0.530292\pi$
$443$	−4.00000	−0.190046	−0.0950229	+	0.995475i	$0.530292\pi$
$444$	−1.00000	−0.0474579
$445$	14.0000	0.663664
$446$	20.0000	0.947027
$447$	−10.0000	−0.472984
$448$	4.00000	0.188982
$449$	42.0000	1.98210	0.991051	−	0.133482i	$-0.0426157\pi$
$449$	42.0000	1.98210	0.991051	+	0.133482i	$0.0426157\pi$
$450$	−1.00000	−0.0471405
$451$	−24.0000	−1.13012
$452$	14.0000	0.658505
$453$	0	0
$454$	4.00000	0.187729
$455$	−8.00000	−0.375046
$456$	0	0
$457$	−2.00000	−0.0935561	−0.0467780	−	0.998905i	$-0.514895\pi$
$457$	−2.00000	−0.0935561	−0.0467780	+	0.998905i	$0.514895\pi$
$458$	18.0000	0.841085
$459$	−2.00000	−0.0933520
$460$	0	0
$461$	26.0000	1.21094	0.605470	−	0.795868i	$-0.292985\pi$
$461$	26.0000	1.21094	0.605470	+	0.795868i	$0.292985\pi$
$462$	−16.0000	−0.744387
$463$	8.00000	0.371792	0.185896	−	0.982569i	$-0.440481\pi$
$463$	8.00000	0.371792	0.185896	+	0.982569i	$0.440481\pi$
$464$	−6.00000	−0.278543
$465$	0	0
$466$	−10.0000	−0.463241
$467$	36.0000	1.66588	0.832941	−	0.553362i	$-0.186655\pi$
$467$	36.0000	1.66588	0.832941	+	0.553362i	$0.186655\pi$
$468$	2.00000	0.0924500
$469$	16.0000	0.738811
$470$	12.0000	0.553519
$471$	10.0000	0.460776
$472$	−8.00000	−0.368230
$473$	16.0000	0.735681
$474$	0	0
$475$	0	0
$476$	−8.00000	−0.366679
$477$	−6.00000	−0.274721
$478$	0	0
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	1.00000	0.0456435
$481$	−2.00000	−0.0911922
$482$	−18.0000	−0.819878
$483$	0	0
$484$	5.00000	0.227273
$485$	10.0000	0.454077
$486$	−1.00000	−0.0453609
$487$	8.00000	0.362515	0.181257	−	0.983436i	$-0.441983\pi$
$487$	8.00000	0.362515	0.181257	+	0.983436i	$0.441983\pi$
$488$	−10.0000	−0.452679
$489$	4.00000	0.180886
$490$	9.00000	0.406579
$491$	36.0000	1.62466	0.812329	−	0.583200i	$-0.198200\pi$
$491$	36.0000	1.62466	0.812329	+	0.583200i	$0.198200\pi$
$492$	−6.00000	−0.270501
$493$	12.0000	0.540453
$494$	0	0
$495$	−4.00000	−0.179787
$496$	0	0
$497$	0	0
$498$	4.00000	0.179244
$499$	40.0000	1.79065	0.895323	−	0.445418i	$-0.146945\pi$
$499$	40.0000	1.79065	0.895323	+	0.445418i	$0.146945\pi$
$500$	−1.00000	−0.0447214
$501$	8.00000	0.357414
$502$	−24.0000	−1.07117
$503$	40.0000	1.78351	0.891756	−	0.452517i	$-0.149474\pi$
$503$	40.0000	1.78351	0.891756	+	0.452517i	$0.149474\pi$
$504$	−4.00000	−0.178174
$505$	−6.00000	−0.266996
$506$	0	0
$507$	−9.00000	−0.399704
$508$	12.0000	0.532414
$509$	30.0000	1.32973	0.664863	−	0.746965i	$-0.268490\pi$
$509$	30.0000	1.32973	0.664863	+	0.746965i	$0.268490\pi$
$510$	−2.00000	−0.0885615
$511$	−24.0000	−1.06170
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	18.0000	0.793946
$515$	−8.00000	−0.352522
$516$	4.00000	0.176090
$517$	48.0000	2.11104
$518$	4.00000	0.175750
$519$	2.00000	0.0877903
$520$	2.00000	0.0877058
$521$	42.0000	1.84005	0.920027	−	0.391856i	$-0.128167\pi$
$521$	42.0000	1.84005	0.920027	+	0.391856i	$0.128167\pi$
$522$	6.00000	0.262613
$523$	−36.0000	−1.57417	−0.787085	−	0.616844i	$-0.788411\pi$
$523$	−36.0000	−1.57417	−0.787085	+	0.616844i	$0.788411\pi$
$524$	−8.00000	−0.349482
$525$	4.00000	0.174574
$526$	12.0000	0.523225
$527$	0	0
$528$	4.00000	0.174078
$529$	−23.0000	−1.00000
$530$	−6.00000	−0.260623
$531$	8.00000	0.347170
$532$	0	0
$533$	−12.0000	−0.519778
$534$	14.0000	0.605839
$535$	−12.0000	−0.518805
$536$	−4.00000	−0.172774
$537$	−16.0000	−0.690451
$538$	18.0000	0.776035
$539$	36.0000	1.55063
$540$	−1.00000	−0.0430331
$541$	26.0000	1.11783	0.558914	−	0.829226i	$-0.311218\pi$
$541$	26.0000	1.11783	0.558914	+	0.829226i	$0.311218\pi$
$542$	−8.00000	−0.343629
$543$	14.0000	0.600798
$544$	2.00000	0.0857493
$545$	14.0000	0.599694
$546$	−8.00000	−0.342368
$547$	−28.0000	−1.19719	−0.598597	−	0.801050i	$-0.704275\pi$
$547$	−28.0000	−1.19719	−0.598597	+	0.801050i	$0.704275\pi$
$548$	10.0000	0.427179
$549$	10.0000	0.426790
$550$	−4.00000	−0.170561
$551$	0	0
$552$	0	0
$553$	0	0
$554$	−10.0000	−0.424859
$555$	1.00000	0.0424476
$556$	−4.00000	−0.169638
$557$	−30.0000	−1.27114	−0.635570	−	0.772043i	$-0.719235\pi$
$557$	−30.0000	−1.27114	−0.635570	+	0.772043i	$0.719235\pi$
$558$	0	0
$559$	8.00000	0.338364
$560$	−4.00000	−0.169031
$561$	−8.00000	−0.337760
$562$	14.0000	0.590554
$563$	−44.0000	−1.85438	−0.927189	−	0.374593i	$-0.877783\pi$
$563$	−44.0000	−1.85438	−0.927189	+	0.374593i	$0.877783\pi$
$564$	12.0000	0.505291
$565$	−14.0000	−0.588984
$566$	4.00000	0.168133
$567$	4.00000	0.167984
$568$	0	0
$569$	2.00000	0.0838444	0.0419222	−	0.999121i	$-0.486652\pi$
$569$	2.00000	0.0838444	0.0419222	+	0.999121i	$0.486652\pi$
$570$	0	0
$571$	−4.00000	−0.167395	−0.0836974	−	0.996491i	$-0.526673\pi$
$571$	−4.00000	−0.167395	−0.0836974	+	0.996491i	$0.526673\pi$
$572$	8.00000	0.334497
$573$	−16.0000	−0.668410
$574$	24.0000	1.00174
$575$	0	0
$576$	1.00000	0.0416667
$577$	−34.0000	−1.41544	−0.707719	−	0.706494i	$-0.750276\pi$
$577$	−34.0000	−1.41544	−0.707719	+	0.706494i	$0.750276\pi$
$578$	13.0000	0.540729
$579$	−2.00000	−0.0831172
$580$	6.00000	0.249136
$581$	−16.0000	−0.663792
$582$	10.0000	0.414513
$583$	−24.0000	−0.993978
$584$	6.00000	0.248282
$585$	−2.00000	−0.0826898
$586$	14.0000	0.578335
$587$	−12.0000	−0.495293	−0.247647	−	0.968850i	$-0.579657\pi$
$587$	−12.0000	−0.495293	−0.247647	+	0.968850i	$0.579657\pi$
$588$	9.00000	0.371154
$589$	0	0
$590$	8.00000	0.329355
$591$	2.00000	0.0822690
$592$	−1.00000	−0.0410997
$593$	18.0000	0.739171	0.369586	−	0.929197i	$-0.379500\pi$
$593$	18.0000	0.739171	0.369586	+	0.929197i	$0.379500\pi$
$594$	−4.00000	−0.164122
$595$	8.00000	0.327968
$596$	−10.0000	−0.409616
$597$	−24.0000	−0.982255
$598$	0	0
$599$	32.0000	1.30748	0.653742	−	0.756717i	$-0.273198\pi$
$599$	32.0000	1.30748	0.653742	+	0.756717i	$0.273198\pi$
$600$	−1.00000	−0.0408248
$601$	26.0000	1.06056	0.530281	−	0.847822i	$-0.322086\pi$
$601$	26.0000	1.06056	0.530281	+	0.847822i	$0.322086\pi$
$602$	−16.0000	−0.652111
$603$	4.00000	0.162893
$604$	0	0
$605$	−5.00000	−0.203279
$606$	−6.00000	−0.243733
$607$	24.0000	0.974130	0.487065	−	0.873366i	$-0.338067\pi$
$607$	24.0000	0.974130	0.487065	+	0.873366i	$0.338067\pi$
$608$	0	0
$609$	−24.0000	−0.972529
$610$	10.0000	0.404888
$611$	24.0000	0.970936
$612$	−2.00000	−0.0808452
$613$	10.0000	0.403896	0.201948	−	0.979396i	$-0.435273\pi$
$613$	10.0000	0.403896	0.201948	+	0.979396i	$0.435273\pi$
$614$	−12.0000	−0.484281
$615$	6.00000	0.241943
$616$	−16.0000	−0.644658
$617$	2.00000	0.0805170	0.0402585	−	0.999189i	$-0.487182\pi$
$617$	2.00000	0.0805170	0.0402585	+	0.999189i	$0.487182\pi$
$618$	−8.00000	−0.321807
$619$	−44.0000	−1.76851	−0.884255	−	0.467005i	$-0.845333\pi$
$619$	−44.0000	−1.76851	−0.884255	+	0.467005i	$0.845333\pi$
$620$	0	0
$621$	0	0
$622$	16.0000	0.641542
$623$	−56.0000	−2.24359
$624$	2.00000	0.0800641
$625$	1.00000	0.0400000
$626$	−6.00000	−0.239808
$627$	0	0
$628$	10.0000	0.399043
$629$	2.00000	0.0797452
$630$	4.00000	0.159364
$631$	−16.0000	−0.636950	−0.318475	−	0.947931i	$-0.603171\pi$
$631$	−16.0000	−0.636950	−0.318475	+	0.947931i	$0.603171\pi$
$632$	0	0
$633$	−20.0000	−0.794929
$634$	−2.00000	−0.0794301
$635$	−12.0000	−0.476205
$636$	−6.00000	−0.237915
$637$	18.0000	0.713186
$638$	24.0000	0.950169
$639$	0	0
$640$	1.00000	0.0395285
$641$	18.0000	0.710957	0.355479	−	0.934684i	$-0.384318\pi$
$641$	18.0000	0.710957	0.355479	+	0.934684i	$0.384318\pi$
$642$	−12.0000	−0.473602
$643$	44.0000	1.73519	0.867595	−	0.497271i	$-0.165665\pi$
$643$	44.0000	1.73519	0.867595	+	0.497271i	$0.165665\pi$
$644$	0	0
$645$	−4.00000	−0.157500
$646$	0	0
$647$	24.0000	0.943537	0.471769	−	0.881722i	$-0.343616\pi$
$647$	24.0000	0.943537	0.471769	+	0.881722i	$0.343616\pi$
$648$	−1.00000	−0.0392837
$649$	32.0000	1.25611
$650$	−2.00000	−0.0784465
$651$	0	0
$652$	4.00000	0.156652
$653$	−30.0000	−1.17399	−0.586995	−	0.809590i	$-0.699689\pi$
$653$	−30.0000	−1.17399	−0.586995	+	0.809590i	$0.699689\pi$
$654$	14.0000	0.547443
$655$	8.00000	0.312586
$656$	−6.00000	−0.234261
$657$	−6.00000	−0.234082
$658$	−48.0000	−1.87123
$659$	−44.0000	−1.71400	−0.856998	−	0.515319i	$-0.827673\pi$
$659$	−44.0000	−1.71400	−0.856998	+	0.515319i	$0.827673\pi$
$660$	−4.00000	−0.155700
$661$	10.0000	0.388955	0.194477	−	0.980907i	$-0.437699\pi$
$661$	10.0000	0.388955	0.194477	+	0.980907i	$0.437699\pi$
$662$	32.0000	1.24372
$663$	−4.00000	−0.155347
$664$	4.00000	0.155230
$665$	0	0
$666$	1.00000	0.0387492
$667$	0	0
$668$	8.00000	0.309529
$669$	−20.0000	−0.773245
$670$	4.00000	0.154533
$671$	40.0000	1.54418
$672$	−4.00000	−0.154303
$673$	−38.0000	−1.46479	−0.732396	−	0.680879i	$-0.761598\pi$
$673$	−38.0000	−1.46479	−0.732396	+	0.680879i	$0.761598\pi$
$674$	30.0000	1.15556
$675$	1.00000	0.0384900
$676$	−9.00000	−0.346154
$677$	−30.0000	−1.15299	−0.576497	−	0.817099i	$-0.695581\pi$
$677$	−30.0000	−1.15299	−0.576497	+	0.817099i	$0.695581\pi$
$678$	−14.0000	−0.537667
$679$	−40.0000	−1.53506
$680$	−2.00000	−0.0766965
$681$	−4.00000	−0.153280
$682$	0	0
$683$	−12.0000	−0.459167	−0.229584	−	0.973289i	$-0.573736\pi$
$683$	−12.0000	−0.459167	−0.229584	+	0.973289i	$0.573736\pi$
$684$	0	0
$685$	−10.0000	−0.382080
$686$	−8.00000	−0.305441
$687$	−18.0000	−0.686743
$688$	4.00000	0.152499
$689$	−12.0000	−0.457164
$690$	0	0
$691$	20.0000	0.760836	0.380418	−	0.924815i	$-0.375780\pi$
$691$	20.0000	0.760836	0.380418	+	0.924815i	$0.375780\pi$
$692$	2.00000	0.0760286
$693$	16.0000	0.607790
$694$	12.0000	0.455514
$695$	4.00000	0.151729
$696$	6.00000	0.227429
$697$	12.0000	0.454532
$698$	−30.0000	−1.13552
$699$	10.0000	0.378235
$700$	4.00000	0.151186
$701$	−30.0000	−1.13308	−0.566542	−	0.824033i	$-0.691719\pi$
$701$	−30.0000	−1.13308	−0.566542	+	0.824033i	$0.691719\pi$
$702$	−2.00000	−0.0754851
$703$	0	0
$704$	4.00000	0.150756
$705$	−12.0000	−0.451946
$706$	26.0000	0.978523
$707$	24.0000	0.902613
$708$	8.00000	0.300658
$709$	−14.0000	−0.525781	−0.262891	−	0.964826i	$-0.584676\pi$
$709$	−14.0000	−0.525781	−0.262891	+	0.964826i	$0.584676\pi$
$710$	0	0
$711$	0	0
$712$	14.0000	0.524672
$713$	0	0
$714$	8.00000	0.299392
$715$	−8.00000	−0.299183
$716$	−16.0000	−0.597948
$717$	0	0
$718$	24.0000	0.895672
$719$	−8.00000	−0.298350	−0.149175	−	0.988811i	$-0.547662\pi$
$719$	−8.00000	−0.298350	−0.149175	+	0.988811i	$0.547662\pi$
$720$	−1.00000	−0.0372678
$721$	32.0000	1.19174
$722$	19.0000	0.707107
$723$	18.0000	0.669427
$724$	14.0000	0.520306
$725$	−6.00000	−0.222834
$726$	−5.00000	−0.185567
$727$	0	0		−	1.00000i	$-0.5\pi$
$727$	0	0			1.00000i	$0.5\pi$
$728$	−8.00000	−0.296500
$729$	1.00000	0.0370370
$730$	−6.00000	−0.222070
$731$	−8.00000	−0.295891
$732$	10.0000	0.369611
$733$	34.0000	1.25582	0.627909	−	0.778287i	$-0.283911\pi$
$733$	34.0000	1.25582	0.627909	+	0.778287i	$0.283911\pi$
$734$	20.0000	0.738213
$735$	−9.00000	−0.331970
$736$	0	0
$737$	16.0000	0.589368
$738$	6.00000	0.220863
$739$	−28.0000	−1.03000	−0.514998	−	0.857191i	$-0.672207\pi$
$739$	−28.0000	−1.03000	−0.514998	+	0.857191i	$0.672207\pi$
$740$	1.00000	0.0367607
$741$	0	0
$742$	24.0000	0.881068
$743$	28.0000	1.02722	0.513610	−	0.858024i	$-0.328308\pi$
$743$	28.0000	1.02722	0.513610	+	0.858024i	$0.328308\pi$
$744$	0	0
$745$	10.0000	0.366372
$746$	−2.00000	−0.0732252
$747$	−4.00000	−0.146352
$748$	−8.00000	−0.292509
$749$	48.0000	1.75388
$750$	1.00000	0.0365148
$751$	−32.0000	−1.16770	−0.583848	−	0.811863i	$-0.698454\pi$
$751$	−32.0000	−1.16770	−0.583848	+	0.811863i	$0.698454\pi$
$752$	12.0000	0.437595
$753$	24.0000	0.874609
$754$	12.0000	0.437014
$755$	0	0
$756$	4.00000	0.145479
$757$	10.0000	0.363456	0.181728	−	0.983349i	$-0.441831\pi$
$757$	10.0000	0.363456	0.181728	+	0.983349i	$0.441831\pi$
$758$	20.0000	0.726433
$759$	0	0
$760$	0	0
$761$	−54.0000	−1.95750	−0.978749	−	0.205061i	$-0.934261\pi$
$761$	−54.0000	−1.95750	−0.978749	+	0.205061i	$0.934261\pi$
$762$	−12.0000	−0.434714
$763$	−56.0000	−2.02734
$764$	−16.0000	−0.578860
$765$	2.00000	0.0723102
$766$	−24.0000	−0.867155
$767$	16.0000	0.577727
$768$	1.00000	0.0360844
$769$	−14.0000	−0.504853	−0.252426	−	0.967616i	$-0.581229\pi$
$769$	−14.0000	−0.504853	−0.252426	+	0.967616i	$0.581229\pi$
$770$	16.0000	0.576600
$771$	−18.0000	−0.648254
$772$	−2.00000	−0.0719816
$773$	42.0000	1.51064	0.755318	−	0.655359i	$-0.227483\pi$
$773$	42.0000	1.51064	0.755318	+	0.655359i	$0.227483\pi$
$774$	−4.00000	−0.143777
$775$	0	0
$776$	10.0000	0.358979
$777$	−4.00000	−0.143499
$778$	−34.0000	−1.21896
$779$	0	0
$780$	−2.00000	−0.0716115
$781$	0	0
$782$	0	0
$783$	−6.00000	−0.214423
$784$	9.00000	0.321429
$785$	−10.0000	−0.356915
$786$	8.00000	0.285351
$787$	−12.0000	−0.427754	−0.213877	−	0.976861i	$-0.568609\pi$
$787$	−12.0000	−0.427754	−0.213877	+	0.976861i	$0.568609\pi$
$788$	2.00000	0.0712470
$789$	−12.0000	−0.427211
$790$	0	0
$791$	56.0000	1.99113
$792$	−4.00000	−0.142134
$793$	20.0000	0.710221
$794$	−18.0000	−0.638796
$795$	6.00000	0.212798
$796$	−24.0000	−0.850657
$797$	34.0000	1.20434	0.602171	−	0.798367i	$-0.294303\pi$
$797$	34.0000	1.20434	0.602171	+	0.798367i	$0.294303\pi$
$798$	0	0
$799$	−24.0000	−0.849059
$800$	−1.00000	−0.0353553
$801$	−14.0000	−0.494666
$802$	6.00000	0.211867
$803$	−24.0000	−0.846942
$804$	4.00000	0.141069
$805$	0	0
$806$	0	0
$807$	−18.0000	−0.633630
$808$	−6.00000	−0.211079
$809$	−6.00000	−0.210949	−0.105474	−	0.994422i	$-0.533636\pi$
$809$	−6.00000	−0.210949	−0.105474	+	0.994422i	$0.533636\pi$
$810$	1.00000	0.0351364
$811$	28.0000	0.983213	0.491606	−	0.870817i	$-0.336410\pi$
$811$	28.0000	0.983213	0.491606	+	0.870817i	$0.336410\pi$
$812$	−24.0000	−0.842235
$813$	8.00000	0.280572
$814$	4.00000	0.140200
$815$	−4.00000	−0.140114
$816$	−2.00000	−0.0700140
$817$	0	0
$818$	30.0000	1.04893
$819$	8.00000	0.279543
$820$	6.00000	0.209529
$821$	−42.0000	−1.46581	−0.732905	−	0.680331i	$-0.761836\pi$
$821$	−42.0000	−1.46581	−0.732905	+	0.680331i	$0.761836\pi$
$822$	−10.0000	−0.348790
$823$	4.00000	0.139431	0.0697156	−	0.997567i	$-0.477791\pi$
$823$	4.00000	0.139431	0.0697156	+	0.997567i	$0.477791\pi$
$824$	−8.00000	−0.278693
$825$	4.00000	0.139262
$826$	−32.0000	−1.11342
$827$	36.0000	1.25184	0.625921	−	0.779886i	$-0.284723\pi$
$827$	36.0000	1.25184	0.625921	+	0.779886i	$0.284723\pi$
$828$	0	0
$829$	34.0000	1.18087	0.590434	−	0.807086i	$-0.298956\pi$
$829$	34.0000	1.18087	0.590434	+	0.807086i	$0.298956\pi$
$830$	−4.00000	−0.138842
$831$	10.0000	0.346896
$832$	2.00000	0.0693375
$833$	−18.0000	−0.623663
$834$	4.00000	0.138509
$835$	−8.00000	−0.276851
$836$	0	0
$837$	0	0
$838$	−20.0000	−0.690889
$839$	−24.0000	−0.828572	−0.414286	−	0.910147i	$-0.635969\pi$
$839$	−24.0000	−0.828572	−0.414286	+	0.910147i	$0.635969\pi$
$840$	4.00000	0.138013
$841$	7.00000	0.241379
$842$	6.00000	0.206774
$843$	−14.0000	−0.482186
$844$	−20.0000	−0.688428
$845$	9.00000	0.309609
$846$	−12.0000	−0.412568
$847$	20.0000	0.687208
$848$	−6.00000	−0.206041
$849$	−4.00000	−0.137280
$850$	2.00000	0.0685994
$851$	0	0
$852$	0	0
$853$	10.0000	0.342393	0.171197	−	0.985237i	$-0.445237\pi$
$853$	10.0000	0.342393	0.171197	+	0.985237i	$0.445237\pi$
$854$	−40.0000	−1.36877
$855$	0	0
$856$	−12.0000	−0.410152
$857$	−18.0000	−0.614868	−0.307434	−	0.951569i	$-0.599470\pi$
$857$	−18.0000	−0.614868	−0.307434	+	0.951569i	$0.599470\pi$
$858$	−8.00000	−0.273115
$859$	−8.00000	−0.272956	−0.136478	−	0.990643i	$-0.543578\pi$
$859$	−8.00000	−0.272956	−0.136478	+	0.990643i	$0.543578\pi$
$860$	−4.00000	−0.136399
$861$	−24.0000	−0.817918
$862$	0	0
$863$	−12.0000	−0.408485	−0.204242	−	0.978920i	$-0.565473\pi$
$863$	−12.0000	−0.408485	−0.204242	+	0.978920i	$0.565473\pi$
$864$	−1.00000	−0.0340207
$865$	−2.00000	−0.0680020
$866$	30.0000	1.01944
$867$	−13.0000	−0.441503
$868$	0	0
$869$	0	0
$870$	−6.00000	−0.203419
$871$	8.00000	0.271070
$872$	14.0000	0.474100
$873$	−10.0000	−0.338449
$874$	0	0
$875$	−4.00000	−0.135225
$876$	−6.00000	−0.202721
$877$	26.0000	0.877958	0.438979	−	0.898497i	$-0.355340\pi$
$877$	26.0000	0.877958	0.438979	+	0.898497i	$0.355340\pi$
$878$	24.0000	0.809961
$879$	−14.0000	−0.472208
$880$	−4.00000	−0.134840
$881$	−46.0000	−1.54978	−0.774890	−	0.632096i	$-0.782195\pi$
$881$	−46.0000	−1.54978	−0.774890	+	0.632096i	$0.782195\pi$
$882$	−9.00000	−0.303046
$883$	−36.0000	−1.21150	−0.605748	−	0.795656i	$-0.707126\pi$
$883$	−36.0000	−1.21150	−0.605748	+	0.795656i	$0.707126\pi$
$884$	−4.00000	−0.134535
$885$	−8.00000	−0.268917
$886$	4.00000	0.134383
$887$	−44.0000	−1.47738	−0.738688	−	0.674048i	$-0.764554\pi$
$887$	−44.0000	−1.47738	−0.738688	+	0.674048i	$0.764554\pi$
$888$	1.00000	0.0335578
$889$	48.0000	1.60987
$890$	−14.0000	−0.469281
$891$	4.00000	0.134005
$892$	−20.0000	−0.669650
$893$	0	0
$894$	10.0000	0.334450
$895$	16.0000	0.534821
$896$	−4.00000	−0.133631
$897$	0	0
$898$	−42.0000	−1.40156
$899$	0	0
$900$	1.00000	0.0333333
$901$	12.0000	0.399778
$902$	24.0000	0.799113
$903$	16.0000	0.532447
$904$	−14.0000	−0.465633
$905$	−14.0000	−0.465376
$906$	0	0
$907$	4.00000	0.132818	0.0664089	−	0.997792i	$-0.478846\pi$
$907$	4.00000	0.132818	0.0664089	+	0.997792i	$0.478846\pi$
$908$	−4.00000	−0.132745
$909$	6.00000	0.199007
$910$	8.00000	0.265197
$911$	40.0000	1.32526	0.662630	−	0.748947i	$-0.269440\pi$
$911$	40.0000	1.32526	0.662630	+	0.748947i	$0.269440\pi$
$912$	0	0
$913$	−16.0000	−0.529523
$914$	2.00000	0.0661541
$915$	−10.0000	−0.330590
$916$	−18.0000	−0.594737
$917$	−32.0000	−1.05673
$918$	2.00000	0.0660098
$919$	16.0000	0.527791	0.263896	−	0.964551i	$-0.414993\pi$
$919$	16.0000	0.527791	0.263896	+	0.964551i	$0.414993\pi$
$920$	0	0
$921$	12.0000	0.395413
$922$	−26.0000	−0.856264
$923$	0	0
$924$	16.0000	0.526361
$925$	−1.00000	−0.0328798
$926$	−8.00000	−0.262896
$927$	8.00000	0.262754
$928$	6.00000	0.196960
$929$	50.0000	1.64045	0.820223	−	0.572043i	$-0.193849\pi$
$929$	50.0000	1.64045	0.820223	+	0.572043i	$0.193849\pi$
$930$	0	0
$931$	0	0
$932$	10.0000	0.327561
$933$	−16.0000	−0.523816
$934$	−36.0000	−1.17796
$935$	8.00000	0.261628
$936$	−2.00000	−0.0653720
$937$	2.00000	0.0653372	0.0326686	−	0.999466i	$-0.489599\pi$
$937$	2.00000	0.0653372	0.0326686	+	0.999466i	$0.489599\pi$
$938$	−16.0000	−0.522419
$939$	6.00000	0.195803
$940$	−12.0000	−0.391397
$941$	−10.0000	−0.325991	−0.162995	−	0.986627i	$-0.552116\pi$
$941$	−10.0000	−0.325991	−0.162995	+	0.986627i	$0.552116\pi$
$942$	−10.0000	−0.325818
$943$	0	0
$944$	8.00000	0.260378
$945$	−4.00000	−0.130120
$946$	−16.0000	−0.520205
$947$	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
$949$	−12.0000	−0.389536
$950$	0	0
$951$	2.00000	0.0648544
$952$	8.00000	0.259281
$953$	−22.0000	−0.712650	−0.356325	−	0.934362i	$-0.615970\pi$
$953$	−22.0000	−0.712650	−0.356325	+	0.934362i	$0.615970\pi$
$954$	6.00000	0.194257
$955$	16.0000	0.517748
$956$	0	0
$957$	−24.0000	−0.775810
$958$	0	0
$959$	40.0000	1.29167
$960$	−1.00000	−0.0322749
$961$	−31.0000	−1.00000
$962$	2.00000	0.0644826
$963$	12.0000	0.386695
$964$	18.0000	0.579741
$965$	2.00000	0.0643823
$966$	0	0
$967$	48.0000	1.54358	0.771788	−	0.635880i	$-0.219363\pi$
$967$	48.0000	1.54358	0.771788	+	0.635880i	$0.219363\pi$
$968$	−5.00000	−0.160706
$969$	0	0
$970$	−10.0000	−0.321081
$971$	20.0000	0.641831	0.320915	−	0.947108i	$-0.396010\pi$
$971$	20.0000	0.641831	0.320915	+	0.947108i	$0.396010\pi$
$972$	1.00000	0.0320750
$973$	−16.0000	−0.512936
$974$	−8.00000	−0.256337
$975$	2.00000	0.0640513
$976$	10.0000	0.320092
$977$	−18.0000	−0.575871	−0.287936	−	0.957650i	$-0.592969\pi$
$977$	−18.0000	−0.575871	−0.287936	+	0.957650i	$0.592969\pi$
$978$	−4.00000	−0.127906
$979$	−56.0000	−1.78977
$980$	−9.00000	−0.287494
$981$	−14.0000	−0.446986
$982$	−36.0000	−1.14881
$983$	−4.00000	−0.127580	−0.0637901	−	0.997963i	$-0.520319\pi$
$983$	−4.00000	−0.127580	−0.0637901	+	0.997963i	$0.520319\pi$
$984$	6.00000	0.191273
$985$	−2.00000	−0.0637253
$986$	−12.0000	−0.382158
$987$	48.0000	1.52786
$988$	0	0
$989$	0	0
$990$	4.00000	0.127128
$991$	−40.0000	−1.27064	−0.635321	−	0.772248i	$-0.719132\pi$
$991$	−40.0000	−1.27064	−0.635321	+	0.772248i	$0.719132\pi$
$992$	0	0
$993$	−32.0000	−1.01549
$994$	0	0
$995$	24.0000	0.760851
$996$	−4.00000	−0.126745
$997$	−54.0000	−1.71020	−0.855099	−	0.518465i	$-0.826503\pi$
$997$	−54.0000	−1.71020	−0.855099	+	0.518465i	$0.826503\pi$
$998$	−40.0000	−1.26618
$999$	−1.00000	−0.0316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1110.2.a.g.1.1	✓	1
3.2	odd	2		3330.2.a.ba.1.1		1
4.3	odd	2		8880.2.a.a.1.1		1
5.4	even	2		5550.2.a.w.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1110.2.a.g.1.1	✓	1	1.1	even	1	trivial
3330.2.a.ba.1.1		1	3.2	odd	2
5550.2.a.w.1.1		1	5.4	even	2
8880.2.a.a.1.1		1	4.3	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:	\( N \)	\(=\)	\( 1110 = 2 \cdot 3 \cdot 5 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1110.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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