LMFDB - Embedded newform 113.2.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [113,2,Mod(1,113)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(113, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("113.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 113 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	113.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:	$0.902309542840$
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	$\chi$	$=$	113.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{2} +2.00000 q^{3} -1.00000 q^{4} +2.00000 q^{5} -2.00000 q^{6} +3.00000 q^{8} +1.00000 q^{9} +O(q^{10})$

\(q-1.00000 q^{2} +2.00000 q^{3} -1.00000 q^{4} +2.00000 q^{5} -2.00000 q^{6} +3.00000 q^{8} +1.00000 q^{9} -2.00000 q^{10} -2.00000 q^{12} +2.00000 q^{13} +4.00000 q^{15} -1.00000 q^{16} -6.00000 q^{17} -1.00000 q^{18} +6.00000 q^{19} -2.00000 q^{20} -6.00000 q^{23} +6.00000 q^{24} -1.00000 q^{25} -2.00000 q^{26} -4.00000 q^{27} -6.00000 q^{29} -4.00000 q^{30} -4.00000 q^{31} -5.00000 q^{32} +6.00000 q^{34} -1.00000 q^{36} +2.00000 q^{37} -6.00000 q^{38} +4.00000 q^{39} +6.00000 q^{40} -2.00000 q^{41} +6.00000 q^{43} +2.00000 q^{45} +6.00000 q^{46} +6.00000 q^{47} -2.00000 q^{48} -7.00000 q^{49} +1.00000 q^{50} -12.0000 q^{51} -2.00000 q^{52} +10.0000 q^{53} +4.00000 q^{54} +12.0000 q^{57} +6.00000 q^{58} +6.00000 q^{59} -4.00000 q^{60} +6.00000 q^{61} +4.00000 q^{62} +7.00000 q^{64} +4.00000 q^{65} +2.00000 q^{67} +6.00000 q^{68} -12.0000 q^{69} -6.00000 q^{71} +3.00000 q^{72} +2.00000 q^{73} -2.00000 q^{74} -2.00000 q^{75} -6.00000 q^{76} -4.00000 q^{78} +10.0000 q^{79} -2.00000 q^{80} -11.0000 q^{81} +2.00000 q^{82} -4.00000 q^{83} -12.0000 q^{85} -6.00000 q^{86} -12.0000 q^{87} -14.0000 q^{89} -2.00000 q^{90} +6.00000 q^{92} -8.00000 q^{93} -6.00000 q^{94} +12.0000 q^{95} -10.0000 q^{96} -14.0000 q^{97} +7.00000 q^{98} +O(q^{100})\)

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	−0.353553	−	0.935414i	$-0.615027\pi$
$2$	−1.00000	−0.707107	−0.353553	+	0.935414i	$0.615027\pi$
$3$	2.00000	1.15470	0.577350	−	0.816497i	$-0.304087\pi$
$3$	2.00000	1.15470	0.577350	+	0.816497i	$0.304087\pi$
$4$	−1.00000	−0.500000
$5$	2.00000	0.894427	0.447214	−	0.894427i	$-0.352416\pi$
$5$	2.00000	0.894427	0.447214	+	0.894427i	$0.352416\pi$
$6$	−2.00000	−0.816497
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	3.00000	1.06066
$9$	1.00000	0.333333
$10$	−2.00000	−0.632456
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	−2.00000	−0.577350
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	0	0
$15$	4.00000	1.03280
$16$	−1.00000	−0.250000
$17$	−6.00000	−1.45521	−0.727607	−	0.685994i	$-0.759367\pi$
$17$	−6.00000	−1.45521	−0.727607	+	0.685994i	$0.759367\pi$
$18$	−1.00000	−0.235702
$19$	6.00000	1.37649	0.688247	−	0.725476i	$-0.258380\pi$
$19$	6.00000	1.37649	0.688247	+	0.725476i	$0.258380\pi$
$20$	−2.00000	−0.447214
$21$	0	0
$22$	0	0
$23$	−6.00000	−1.25109	−0.625543	−	0.780189i	$-0.715123\pi$
$23$	−6.00000	−1.25109	−0.625543	+	0.780189i	$0.715123\pi$
$24$	6.00000	1.22474
$25$	−1.00000	−0.200000
$26$	−2.00000	−0.392232
$27$	−4.00000	−0.769800
$28$	0	0
$29$	−6.00000	−1.11417	−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000	−1.11417	−0.557086	+	0.830455i	$0.688081\pi$
$30$	−4.00000	−0.730297
$31$	−4.00000	−0.718421	−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000	−0.718421	−0.359211	+	0.933257i	$0.616954\pi$
$32$	−5.00000	−0.883883
$33$	0	0
$34$	6.00000	1.02899
$35$	0	0
$36$	−1.00000	−0.166667
$37$	2.00000	0.328798	0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000	0.328798	0.164399	+	0.986394i	$0.447432\pi$
$38$	−6.00000	−0.973329
$39$	4.00000	0.640513
$40$	6.00000	0.948683
$41$	−2.00000	−0.312348	−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000	−0.312348	−0.156174	+	0.987730i	$0.549916\pi$
$42$	0	0
$43$	6.00000	0.914991	0.457496	−	0.889212i	$-0.348747\pi$
$43$	6.00000	0.914991	0.457496	+	0.889212i	$0.348747\pi$
$44$	0	0
$45$	2.00000	0.298142
$46$	6.00000	0.884652
$47$	6.00000	0.875190	0.437595	−	0.899172i	$-0.355830\pi$
$47$	6.00000	0.875190	0.437595	+	0.899172i	$0.355830\pi$
$48$	−2.00000	−0.288675
$49$	−7.00000	−1.00000
$50$	1.00000	0.141421
$51$	−12.0000	−1.68034
$52$	−2.00000	−0.277350
$53$	10.0000	1.37361	0.686803	−	0.726844i	$-0.259014\pi$
$53$	10.0000	1.37361	0.686803	+	0.726844i	$0.259014\pi$
$54$	4.00000	0.544331
$55$	0	0
$56$	0	0
$57$	12.0000	1.58944
$58$	6.00000	0.787839
$59$	6.00000	0.781133	0.390567	−	0.920575i	$-0.372279\pi$
$59$	6.00000	0.781133	0.390567	+	0.920575i	$0.372279\pi$
$60$	−4.00000	−0.516398
$61$	6.00000	0.768221	0.384111	−	0.923287i	$-0.374508\pi$
$61$	6.00000	0.768221	0.384111	+	0.923287i	$0.374508\pi$
$62$	4.00000	0.508001
$63$	0	0
$64$	7.00000	0.875000
$65$	4.00000	0.496139
$66$	0	0
$67$	2.00000	0.244339	0.122169	−	0.992509i	$-0.461015\pi$
$67$	2.00000	0.244339	0.122169	+	0.992509i	$0.461015\pi$
$68$	6.00000	0.727607
$69$	−12.0000	−1.44463
$70$	0	0
$71$	−6.00000	−0.712069	−0.356034	−	0.934473i	$-0.615871\pi$
$71$	−6.00000	−0.712069	−0.356034	+	0.934473i	$0.615871\pi$
$72$	3.00000	0.353553
$73$	2.00000	0.234082	0.117041	−	0.993127i	$-0.462659\pi$
$73$	2.00000	0.234082	0.117041	+	0.993127i	$0.462659\pi$
$74$	−2.00000	−0.232495
$75$	−2.00000	−0.230940
$76$	−6.00000	−0.688247
$77$	0	0
$78$	−4.00000	−0.452911
$79$	10.0000	1.12509	0.562544	−	0.826767i	$-0.309823\pi$
$79$	10.0000	1.12509	0.562544	+	0.826767i	$0.309823\pi$
$80$	−2.00000	−0.223607
$81$	−11.0000	−1.22222
$82$	2.00000	0.220863
$83$	−4.00000	−0.439057	−0.219529	−	0.975606i	$-0.570452\pi$
$83$	−4.00000	−0.439057	−0.219529	+	0.975606i	$0.570452\pi$
$84$	0	0
$85$	−12.0000	−1.30158
$86$	−6.00000	−0.646997
$87$	−12.0000	−1.28654
$88$	0	0
$89$	−14.0000	−1.48400	−0.741999	−	0.670402i	$-0.766122\pi$
$89$	−14.0000	−1.48400	−0.741999	+	0.670402i	$0.766122\pi$
$90$	−2.00000	−0.210819
$91$	0	0
$92$	6.00000	0.625543
$93$	−8.00000	−0.829561
$94$	−6.00000	−0.618853
$95$	12.0000	1.23117
$96$	−10.0000	−1.02062
$97$	−14.0000	−1.42148	−0.710742	−	0.703452i	$-0.751641\pi$
$97$	−14.0000	−1.42148	−0.710742	+	0.703452i	$0.751641\pi$
$98$	7.00000	0.707107
$99$	0	0
$100$	1.00000	0.100000
$101$	−6.00000	−0.597022	−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000	−0.597022	−0.298511	+	0.954406i	$0.596490\pi$
$102$	12.0000	1.18818
$103$	14.0000	1.37946	0.689730	−	0.724066i	$-0.257729\pi$
$103$	14.0000	1.37946	0.689730	+	0.724066i	$0.257729\pi$
$104$	6.00000	0.588348
$105$	0	0
$106$	−10.0000	−0.971286
$107$	18.0000	1.74013	0.870063	−	0.492941i	$-0.164078\pi$
$107$	18.0000	1.74013	0.870063	+	0.492941i	$0.164078\pi$
$108$	4.00000	0.384900
$109$	18.0000	1.72409	0.862044	−	0.506834i	$-0.169184\pi$
$109$	18.0000	1.72409	0.862044	+	0.506834i	$0.169184\pi$
$110$	0	0
$111$	4.00000	0.379663
$112$	0	0
$113$	1.00000	0.0940721
$113$	1.00000	0.0940721
$114$	−12.0000	−1.12390
$115$	−12.0000	−1.11901
$116$	6.00000	0.557086
$117$	2.00000	0.184900
$118$	−6.00000	−0.552345
$119$	0	0
$120$	12.0000	1.09545
$121$	−11.0000	−1.00000
$122$	−6.00000	−0.543214
$123$	−4.00000	−0.360668
$124$	4.00000	0.359211
$125$	−12.0000	−1.07331
$126$	0	0
$127$	16.0000	1.41977	0.709885	−	0.704317i	$-0.248747\pi$
$127$	16.0000	1.41977	0.709885	+	0.704317i	$0.248747\pi$
$128$	3.00000	0.265165
$129$	12.0000	1.05654
$130$	−4.00000	−0.350823
$131$	8.00000	0.698963	0.349482	−	0.936943i	$-0.386358\pi$
$131$	8.00000	0.698963	0.349482	+	0.936943i	$0.386358\pi$
$132$	0	0
$133$	0	0
$134$	−2.00000	−0.172774
$135$	−8.00000	−0.688530
$136$	−18.0000	−1.54349
$137$	−14.0000	−1.19610	−0.598050	−	0.801459i	$-0.704058\pi$
$137$	−14.0000	−1.19610	−0.598050	+	0.801459i	$0.704058\pi$
$138$	12.0000	1.02151
$139$	−16.0000	−1.35710	−0.678551	−	0.734553i	$-0.737392\pi$
$139$	−16.0000	−1.35710	−0.678551	+	0.734553i	$0.737392\pi$
$140$	0	0
$141$	12.0000	1.01058
$142$	6.00000	0.503509
$143$	0	0
$144$	−1.00000	−0.0833333
$145$	−12.0000	−0.996546
$146$	−2.00000	−0.165521
$147$	−14.0000	−1.15470
$148$	−2.00000	−0.164399
$149$	14.0000	1.14692	0.573462	−	0.819232i	$-0.305600\pi$
$149$	14.0000	1.14692	0.573462	+	0.819232i	$0.305600\pi$
$150$	2.00000	0.163299
$151$	−2.00000	−0.162758	−0.0813788	−	0.996683i	$-0.525932\pi$
$151$	−2.00000	−0.162758	−0.0813788	+	0.996683i	$0.525932\pi$
$152$	18.0000	1.45999
$153$	−6.00000	−0.485071
$154$	0	0
$155$	−8.00000	−0.642575
$156$	−4.00000	−0.320256
$157$	2.00000	0.159617	0.0798087	−	0.996810i	$-0.474569\pi$
$157$	2.00000	0.159617	0.0798087	+	0.996810i	$0.474569\pi$
$158$	−10.0000	−0.795557
$159$	20.0000	1.58610
$160$	−10.0000	−0.790569
$161$	0	0
$162$	11.0000	0.864242
$163$	16.0000	1.25322	0.626608	−	0.779334i	$-0.284443\pi$
$163$	16.0000	1.25322	0.626608	+	0.779334i	$0.284443\pi$
$164$	2.00000	0.156174
$165$	0	0
$166$	4.00000	0.310460
$167$	14.0000	1.08335	0.541676	−	0.840587i	$-0.317790\pi$
$167$	14.0000	1.08335	0.541676	+	0.840587i	$0.317790\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	12.0000	0.920358
$171$	6.00000	0.458831
$172$	−6.00000	−0.457496
$173$	−14.0000	−1.06440	−0.532200	−	0.846619i	$-0.678635\pi$
$173$	−14.0000	−1.06440	−0.532200	+	0.846619i	$0.678635\pi$
$174$	12.0000	0.909718
$175$	0	0
$176$	0	0
$177$	12.0000	0.901975
$178$	14.0000	1.04934
$179$	−18.0000	−1.34538	−0.672692	−	0.739923i	$-0.734862\pi$
$179$	−18.0000	−1.34538	−0.672692	+	0.739923i	$0.734862\pi$
$180$	−2.00000	−0.149071
$181$	18.0000	1.33793	0.668965	−	0.743294i	$-0.266738\pi$
$181$	18.0000	1.33793	0.668965	+	0.743294i	$0.266738\pi$
$182$	0	0
$183$	12.0000	0.887066
$184$	−18.0000	−1.32698
$185$	4.00000	0.294086
$186$	8.00000	0.586588
$187$	0	0
$188$	−6.00000	−0.437595
$189$	0	0
$190$	−12.0000	−0.870572
$191$	−10.0000	−0.723575	−0.361787	−	0.932261i	$-0.617833\pi$
$191$	−10.0000	−0.723575	−0.361787	+	0.932261i	$0.617833\pi$
$192$	14.0000	1.01036
$193$	−14.0000	−1.00774	−0.503871	−	0.863779i	$-0.668091\pi$
$193$	−14.0000	−1.00774	−0.503871	+	0.863779i	$0.668091\pi$
$194$	14.0000	1.00514
$195$	8.00000	0.572892
$196$	7.00000	0.500000
$197$	18.0000	1.28245	0.641223	−	0.767354i	$-0.278427\pi$
$197$	18.0000	1.28245	0.641223	+	0.767354i	$0.278427\pi$
$198$	0	0
$199$	−26.0000	−1.84309	−0.921546	−	0.388270i	$-0.873073\pi$
$199$	−26.0000	−1.84309	−0.921546	+	0.388270i	$0.873073\pi$
$200$	−3.00000	−0.212132
$201$	4.00000	0.282138
$202$	6.00000	0.422159
$203$	0	0
$204$	12.0000	0.840168
$205$	−4.00000	−0.279372
$206$	−14.0000	−0.975426
$207$	−6.00000	−0.417029
$208$	−2.00000	−0.138675
$209$	0	0
$210$	0	0
$211$	12.0000	0.826114	0.413057	−	0.910705i	$-0.364461\pi$
$211$	12.0000	0.826114	0.413057	+	0.910705i	$0.364461\pi$
$212$	−10.0000	−0.686803
$213$	−12.0000	−0.822226
$214$	−18.0000	−1.23045
$215$	12.0000	0.818393
$216$	−12.0000	−0.816497
$217$	0	0
$218$	−18.0000	−1.21911
$219$	4.00000	0.270295
$220$	0	0
$221$	−12.0000	−0.807207
$222$	−4.00000	−0.268462
$223$	26.0000	1.74109	0.870544	−	0.492090i	$-0.163767\pi$
$223$	26.0000	1.74109	0.870544	+	0.492090i	$0.163767\pi$
$224$	0	0
$225$	−1.00000	−0.0666667
$226$	−1.00000	−0.0665190
$227$	4.00000	0.265489	0.132745	−	0.991150i	$-0.457621\pi$
$227$	4.00000	0.265489	0.132745	+	0.991150i	$0.457621\pi$
$228$	−12.0000	−0.794719
$229$	−14.0000	−0.925146	−0.462573	−	0.886581i	$-0.653074\pi$
$229$	−14.0000	−0.925146	−0.462573	+	0.886581i	$0.653074\pi$
$230$	12.0000	0.791257
$231$	0	0
$232$	−18.0000	−1.18176
$233$	22.0000	1.44127	0.720634	−	0.693316i	$-0.243851\pi$
$233$	22.0000	1.44127	0.720634	+	0.693316i	$0.243851\pi$
$234$	−2.00000	−0.130744
$235$	12.0000	0.782794
$236$	−6.00000	−0.390567
$237$	20.0000	1.29914
$238$	0	0
$239$	−12.0000	−0.776215	−0.388108	−	0.921614i	$-0.626871\pi$
$239$	−12.0000	−0.776215	−0.388108	+	0.921614i	$0.626871\pi$
$240$	−4.00000	−0.258199
$241$	14.0000	0.901819	0.450910	−	0.892570i	$-0.351100\pi$
$241$	14.0000	0.901819	0.450910	+	0.892570i	$0.351100\pi$
$242$	11.0000	0.707107
$243$	−10.0000	−0.641500
$244$	−6.00000	−0.384111
$245$	−14.0000	−0.894427
$246$	4.00000	0.255031
$247$	12.0000	0.763542
$248$	−12.0000	−0.762001
$249$	−8.00000	−0.506979
$250$	12.0000	0.758947
$251$	16.0000	1.00991	0.504956	−	0.863145i	$-0.331509\pi$
$251$	16.0000	1.00991	0.504956	+	0.863145i	$0.331509\pi$
$252$	0	0
$253$	0	0
$254$	−16.0000	−1.00393
$255$	−24.0000	−1.50294
$256$	−17.0000	−1.06250
$257$	−14.0000	−0.873296	−0.436648	−	0.899632i	$-0.643834\pi$
$257$	−14.0000	−0.873296	−0.436648	+	0.899632i	$0.643834\pi$
$258$	−12.0000	−0.747087
$259$	0	0
$260$	−4.00000	−0.248069
$261$	−6.00000	−0.371391
$262$	−8.00000	−0.494242
$263$	−10.0000	−0.616626	−0.308313	−	0.951285i	$-0.599764\pi$
$263$	−10.0000	−0.616626	−0.308313	+	0.951285i	$0.599764\pi$
$264$	0	0
$265$	20.0000	1.22859
$266$	0	0
$267$	−28.0000	−1.71357
$268$	−2.00000	−0.122169
$269$	18.0000	1.09748	0.548740	−	0.835993i	$-0.315108\pi$
$269$	18.0000	1.09748	0.548740	+	0.835993i	$0.315108\pi$
$270$	8.00000	0.486864
$271$	10.0000	0.607457	0.303728	−	0.952759i	$-0.401768\pi$
$271$	10.0000	0.607457	0.303728	+	0.952759i	$0.401768\pi$
$272$	6.00000	0.363803
$273$	0	0
$274$	14.0000	0.845771
$275$	0	0
$276$	12.0000	0.722315
$277$	−18.0000	−1.08152	−0.540758	−	0.841178i	$-0.681862\pi$
$277$	−18.0000	−1.08152	−0.540758	+	0.841178i	$0.681862\pi$
$278$	16.0000	0.959616
$279$	−4.00000	−0.239474
$280$	0	0
$281$	−6.00000	−0.357930	−0.178965	−	0.983855i	$-0.557275\pi$
$281$	−6.00000	−0.357930	−0.178965	+	0.983855i	$0.557275\pi$
$282$	−12.0000	−0.714590
$283$	−20.0000	−1.18888	−0.594438	−	0.804141i	$-0.702626\pi$
$283$	−20.0000	−1.18888	−0.594438	+	0.804141i	$0.702626\pi$
$284$	6.00000	0.356034
$285$	24.0000	1.42164
$286$	0	0
$287$	0	0
$288$	−5.00000	−0.294628
$289$	19.0000	1.11765
$290$	12.0000	0.704664
$291$	−28.0000	−1.64139
$292$	−2.00000	−0.117041
$293$	−6.00000	−0.350524	−0.175262	−	0.984522i	$-0.556077\pi$
$293$	−6.00000	−0.350524	−0.175262	+	0.984522i	$0.556077\pi$
$294$	14.0000	0.816497
$295$	12.0000	0.698667
$296$	6.00000	0.348743
$297$	0	0
$298$	−14.0000	−0.810998
$299$	−12.0000	−0.693978
$300$	2.00000	0.115470
$301$	0	0
$302$	2.00000	0.115087
$303$	−12.0000	−0.689382
$304$	−6.00000	−0.344124
$305$	12.0000	0.687118
$306$	6.00000	0.342997
$307$	−20.0000	−1.14146	−0.570730	−	0.821138i	$-0.693340\pi$
$307$	−20.0000	−1.14146	−0.570730	+	0.821138i	$0.693340\pi$
$308$	0	0
$309$	28.0000	1.59286
$310$	8.00000	0.454369
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	12.0000	0.679366
$313$	30.0000	1.69570	0.847850	−	0.530236i	$-0.177897\pi$
$313$	30.0000	1.69570	0.847850	+	0.530236i	$0.177897\pi$
$314$	−2.00000	−0.112867
$315$	0	0
$316$	−10.0000	−0.562544
$317$	18.0000	1.01098	0.505490	−	0.862832i	$-0.331312\pi$
$317$	18.0000	1.01098	0.505490	+	0.862832i	$0.331312\pi$
$318$	−20.0000	−1.12154
$319$	0	0
$320$	14.0000	0.782624
$321$	36.0000	2.00932
$322$	0	0
$323$	−36.0000	−2.00309
$324$	11.0000	0.611111
$325$	−2.00000	−0.110940
$326$	−16.0000	−0.886158
$327$	36.0000	1.99080
$328$	−6.00000	−0.331295
$329$	0	0
$330$	0	0
$331$	−20.0000	−1.09930	−0.549650	−	0.835395i	$-0.685239\pi$
$331$	−20.0000	−1.09930	−0.549650	+	0.835395i	$0.685239\pi$
$332$	4.00000	0.219529
$333$	2.00000	0.109599
$334$	−14.0000	−0.766046
$335$	4.00000	0.218543
$336$	0	0
$337$	−2.00000	−0.108947	−0.0544735	−	0.998515i	$-0.517348\pi$
$337$	−2.00000	−0.108947	−0.0544735	+	0.998515i	$0.517348\pi$
$338$	9.00000	0.489535
$339$	2.00000	0.108625
$340$	12.0000	0.650791
$341$	0	0
$342$	−6.00000	−0.324443
$343$	0	0
$344$	18.0000	0.970495
$345$	−24.0000	−1.29212
$346$	14.0000	0.752645
$347$	20.0000	1.07366	0.536828	−	0.843692i	$-0.319622\pi$
$347$	20.0000	1.07366	0.536828	+	0.843692i	$0.319622\pi$
$348$	12.0000	0.643268
$349$	−22.0000	−1.17763	−0.588817	−	0.808267i	$-0.700406\pi$
$349$	−22.0000	−1.17763	−0.588817	+	0.808267i	$0.700406\pi$
$350$	0	0
$351$	−8.00000	−0.427008
$352$	0	0
$353$	14.0000	0.745145	0.372572	−	0.928003i	$-0.378476\pi$
$353$	14.0000	0.745145	0.372572	+	0.928003i	$0.378476\pi$
$354$	−12.0000	−0.637793
$355$	−12.0000	−0.636894
$356$	14.0000	0.741999
$357$	0	0
$358$	18.0000	0.951330
$359$	6.00000	0.316668	0.158334	−	0.987386i	$-0.449388\pi$
$359$	6.00000	0.316668	0.158334	+	0.987386i	$0.449388\pi$
$360$	6.00000	0.316228
$361$	17.0000	0.894737
$362$	−18.0000	−0.946059
$363$	−22.0000	−1.15470
$364$	0	0
$365$	4.00000	0.209370
$366$	−12.0000	−0.627250
$367$	16.0000	0.835193	0.417597	−	0.908633i	$-0.362873\pi$
$367$	16.0000	0.835193	0.417597	+	0.908633i	$0.362873\pi$
$368$	6.00000	0.312772
$369$	−2.00000	−0.104116
$370$	−4.00000	−0.207950
$371$	0	0
$372$	8.00000	0.414781
$373$	−22.0000	−1.13912	−0.569558	−	0.821951i	$-0.692886\pi$
$373$	−22.0000	−1.13912	−0.569558	+	0.821951i	$0.692886\pi$
$374$	0	0
$375$	−24.0000	−1.23935
$376$	18.0000	0.928279
$377$	−12.0000	−0.618031
$378$	0	0
$379$	−10.0000	−0.513665	−0.256833	−	0.966456i	$-0.582679\pi$
$379$	−10.0000	−0.513665	−0.256833	+	0.966456i	$0.582679\pi$
$380$	−12.0000	−0.615587
$381$	32.0000	1.63941
$382$	10.0000	0.511645
$383$	−4.00000	−0.204390	−0.102195	−	0.994764i	$-0.532587\pi$
$383$	−4.00000	−0.204390	−0.102195	+	0.994764i	$0.532587\pi$
$384$	6.00000	0.306186
$385$	0	0
$386$	14.0000	0.712581
$387$	6.00000	0.304997
$388$	14.0000	0.710742
$389$	−6.00000	−0.304212	−0.152106	−	0.988364i	$-0.548606\pi$
$389$	−6.00000	−0.304212	−0.152106	+	0.988364i	$0.548606\pi$
$390$	−8.00000	−0.405096
$391$	36.0000	1.82060
$392$	−21.0000	−1.06066
$393$	16.0000	0.807093
$394$	−18.0000	−0.906827
$395$	20.0000	1.00631
$396$	0	0
$397$	18.0000	0.903394	0.451697	−	0.892171i	$-0.350819\pi$
$397$	18.0000	0.903394	0.451697	+	0.892171i	$0.350819\pi$
$398$	26.0000	1.30326
$399$	0	0
$400$	1.00000	0.0500000
$401$	6.00000	0.299626	0.149813	−	0.988714i	$-0.452133\pi$
$401$	6.00000	0.299626	0.149813	+	0.988714i	$0.452133\pi$
$402$	−4.00000	−0.199502
$403$	−8.00000	−0.398508
$404$	6.00000	0.298511
$405$	−22.0000	−1.09319
$406$	0	0
$407$	0	0
$408$	−36.0000	−1.78227
$409$	10.0000	0.494468	0.247234	−	0.968956i	$-0.420478\pi$
$409$	10.0000	0.494468	0.247234	+	0.968956i	$0.420478\pi$
$410$	4.00000	0.197546
$411$	−28.0000	−1.38114
$412$	−14.0000	−0.689730
$413$	0	0
$414$	6.00000	0.294884
$415$	−8.00000	−0.392705
$416$	−10.0000	−0.490290
$417$	−32.0000	−1.56705
$418$	0	0
$419$	−26.0000	−1.27018	−0.635092	−	0.772437i	$-0.719038\pi$
$419$	−26.0000	−1.27018	−0.635092	+	0.772437i	$0.719038\pi$
$420$	0	0
$421$	−18.0000	−0.877266	−0.438633	−	0.898666i	$-0.644537\pi$
$421$	−18.0000	−0.877266	−0.438633	+	0.898666i	$0.644537\pi$
$422$	−12.0000	−0.584151
$423$	6.00000	0.291730
$424$	30.0000	1.45693
$425$	6.00000	0.291043
$426$	12.0000	0.581402
$427$	0	0
$428$	−18.0000	−0.870063
$429$	0	0
$430$	−12.0000	−0.578691
$431$	−30.0000	−1.44505	−0.722525	−	0.691345i	$-0.757018\pi$
$431$	−30.0000	−1.44505	−0.722525	+	0.691345i	$0.757018\pi$
$432$	4.00000	0.192450
$433$	10.0000	0.480569	0.240285	−	0.970702i	$-0.422759\pi$
$433$	10.0000	0.480569	0.240285	+	0.970702i	$0.422759\pi$
$434$	0	0
$435$	−24.0000	−1.15071
$436$	−18.0000	−0.862044
$437$	−36.0000	−1.72211
$438$	−4.00000	−0.191127
$439$	4.00000	0.190910	0.0954548	−	0.995434i	$-0.469569\pi$
$439$	4.00000	0.190910	0.0954548	+	0.995434i	$0.469569\pi$
$440$	0	0
$441$	−7.00000	−0.333333
$442$	12.0000	0.570782
$443$	−16.0000	−0.760183	−0.380091	−	0.924949i	$-0.624107\pi$
$443$	−16.0000	−0.760183	−0.380091	+	0.924949i	$0.624107\pi$
$444$	−4.00000	−0.189832
$445$	−28.0000	−1.32733
$446$	−26.0000	−1.23114
$447$	28.0000	1.32435
$448$	0	0
$449$	34.0000	1.60456	0.802280	−	0.596948i	$-0.203620\pi$
$449$	34.0000	1.60456	0.802280	+	0.596948i	$0.203620\pi$
$450$	1.00000	0.0471405
$451$	0	0
$452$	−1.00000	−0.0470360
$453$	−4.00000	−0.187936
$454$	−4.00000	−0.187729
$455$	0	0
$456$	36.0000	1.68585
$457$	−30.0000	−1.40334	−0.701670	−	0.712502i	$-0.747562\pi$
$457$	−30.0000	−1.40334	−0.701670	+	0.712502i	$0.747562\pi$
$458$	14.0000	0.654177
$459$	24.0000	1.12022
$460$	12.0000	0.559503
$461$	22.0000	1.02464	0.512321	−	0.858794i	$-0.328786\pi$
$461$	22.0000	1.02464	0.512321	+	0.858794i	$0.328786\pi$
$462$	0	0
$463$	20.0000	0.929479	0.464739	−	0.885448i	$-0.346148\pi$
$463$	20.0000	0.929479	0.464739	+	0.885448i	$0.346148\pi$
$464$	6.00000	0.278543
$465$	−16.0000	−0.741982
$466$	−22.0000	−1.01913
$467$	−28.0000	−1.29569	−0.647843	−	0.761774i	$-0.724329\pi$
$467$	−28.0000	−1.29569	−0.647843	+	0.761774i	$0.724329\pi$
$468$	−2.00000	−0.0924500
$469$	0	0
$470$	−12.0000	−0.553519
$471$	4.00000	0.184310
$472$	18.0000	0.828517
$473$	0	0
$474$	−20.0000	−0.918630
$475$	−6.00000	−0.275299
$476$	0	0
$477$	10.0000	0.457869
$478$	12.0000	0.548867
$479$	−18.0000	−0.822441	−0.411220	−	0.911536i	$-0.634897\pi$
$479$	−18.0000	−0.822441	−0.411220	+	0.911536i	$0.634897\pi$
$480$	−20.0000	−0.912871
$481$	4.00000	0.182384
$482$	−14.0000	−0.637683
$483$	0	0
$484$	11.0000	0.500000
$485$	−28.0000	−1.27141
$486$	10.0000	0.453609
$487$	−34.0000	−1.54069	−0.770344	−	0.637629i	$-0.779915\pi$
$487$	−34.0000	−1.54069	−0.770344	+	0.637629i	$0.779915\pi$
$488$	18.0000	0.814822
$489$	32.0000	1.44709
$490$	14.0000	0.632456
$491$	30.0000	1.35388	0.676941	−	0.736038i	$-0.263305\pi$
$491$	30.0000	1.35388	0.676941	+	0.736038i	$0.263305\pi$
$492$	4.00000	0.180334
$493$	36.0000	1.62136
$494$	−12.0000	−0.539906
$495$	0	0
$496$	4.00000	0.179605
$497$	0	0
$498$	8.00000	0.358489
$499$	−2.00000	−0.0895323	−0.0447661	−	0.998997i	$-0.514254\pi$
$499$	−2.00000	−0.0895323	−0.0447661	+	0.998997i	$0.514254\pi$
$500$	12.0000	0.536656
$501$	28.0000	1.25095
$502$	−16.0000	−0.714115
$503$	4.00000	0.178351	0.0891756	−	0.996016i	$-0.471577\pi$
$503$	4.00000	0.178351	0.0891756	+	0.996016i	$0.471577\pi$
$504$	0	0
$505$	−12.0000	−0.533993
$506$	0	0
$507$	−18.0000	−0.799408
$508$	−16.0000	−0.709885
$509$	−2.00000	−0.0886484	−0.0443242	−	0.999017i	$-0.514113\pi$
$509$	−2.00000	−0.0886484	−0.0443242	+	0.999017i	$0.514113\pi$
$510$	24.0000	1.06274
$511$	0	0
$512$	11.0000	0.486136
$513$	−24.0000	−1.05963
$514$	14.0000	0.617514
$515$	28.0000	1.23383
$516$	−12.0000	−0.528271
$517$	0	0
$518$	0	0
$519$	−28.0000	−1.22906
$520$	12.0000	0.526235
$521$	−6.00000	−0.262865	−0.131432	−	0.991325i	$-0.541958\pi$
$521$	−6.00000	−0.262865	−0.131432	+	0.991325i	$0.541958\pi$
$522$	6.00000	0.262613
$523$	34.0000	1.48672	0.743358	−	0.668894i	$-0.233232\pi$
$523$	34.0000	1.48672	0.743358	+	0.668894i	$0.233232\pi$
$524$	−8.00000	−0.349482
$525$	0	0
$526$	10.0000	0.436021
$527$	24.0000	1.04546
$528$	0	0
$529$	13.0000	0.565217
$530$	−20.0000	−0.868744
$531$	6.00000	0.260378
$532$	0	0
$533$	−4.00000	−0.173259
$534$	28.0000	1.21168
$535$	36.0000	1.55642
$536$	6.00000	0.259161
$537$	−36.0000	−1.55351
$538$	−18.0000	−0.776035
$539$	0	0
$540$	8.00000	0.344265
$541$	2.00000	0.0859867	0.0429934	−	0.999075i	$-0.486311\pi$
$541$	2.00000	0.0859867	0.0429934	+	0.999075i	$0.486311\pi$
$542$	−10.0000	−0.429537
$543$	36.0000	1.54491
$544$	30.0000	1.28624
$545$	36.0000	1.54207
$546$	0	0
$547$	−32.0000	−1.36822	−0.684111	−	0.729378i	$-0.739809\pi$
$547$	−32.0000	−1.36822	−0.684111	+	0.729378i	$0.739809\pi$
$548$	14.0000	0.598050
$549$	6.00000	0.256074
$550$	0	0
$551$	−36.0000	−1.53365
$552$	−36.0000	−1.53226
$553$	0	0
$554$	18.0000	0.764747
$555$	8.00000	0.339581
$556$	16.0000	0.678551
$557$	18.0000	0.762684	0.381342	−	0.924434i	$-0.375462\pi$
$557$	18.0000	0.762684	0.381342	+	0.924434i	$0.375462\pi$
$558$	4.00000	0.169334
$559$	12.0000	0.507546
$560$	0	0
$561$	0	0
$562$	6.00000	0.253095
$563$	−4.00000	−0.168580	−0.0842900	−	0.996441i	$-0.526862\pi$
$563$	−4.00000	−0.168580	−0.0842900	+	0.996441i	$0.526862\pi$
$564$	−12.0000	−0.505291
$565$	2.00000	0.0841406
$566$	20.0000	0.840663
$567$	0	0
$568$	−18.0000	−0.755263
$569$	−22.0000	−0.922288	−0.461144	−	0.887325i	$-0.652561\pi$
$569$	−22.0000	−0.922288	−0.461144	+	0.887325i	$0.652561\pi$
$570$	−24.0000	−1.00525
$571$	26.0000	1.08807	0.544033	−	0.839064i	$-0.316897\pi$
$571$	26.0000	1.08807	0.544033	+	0.839064i	$0.316897\pi$
$572$	0	0
$573$	−20.0000	−0.835512
$574$	0	0
$575$	6.00000	0.250217
$576$	7.00000	0.291667
$577$	−22.0000	−0.915872	−0.457936	−	0.888985i	$-0.651411\pi$
$577$	−22.0000	−0.915872	−0.457936	+	0.888985i	$0.651411\pi$
$578$	−19.0000	−0.790296
$579$	−28.0000	−1.16364
$580$	12.0000	0.498273
$581$	0	0
$582$	28.0000	1.16064
$583$	0	0
$584$	6.00000	0.248282
$585$	4.00000	0.165380
$586$	6.00000	0.247858
$587$	0	0		−	1.00000i	$-0.5\pi$
$587$	0	0			1.00000i	$0.5\pi$
$588$	14.0000	0.577350
$589$	−24.0000	−0.988903
$590$	−12.0000	−0.494032
$591$	36.0000	1.48084
$592$	−2.00000	−0.0821995
$593$	18.0000	0.739171	0.369586	−	0.929197i	$-0.379500\pi$
$593$	18.0000	0.739171	0.369586	+	0.929197i	$0.379500\pi$
$594$	0	0
$595$	0	0
$596$	−14.0000	−0.573462
$597$	−52.0000	−2.12822
$598$	12.0000	0.490716
$599$	−38.0000	−1.55264	−0.776319	−	0.630340i	$-0.782915\pi$
$599$	−38.0000	−1.55264	−0.776319	+	0.630340i	$0.782915\pi$
$600$	−6.00000	−0.244949
$601$	−22.0000	−0.897399	−0.448699	−	0.893683i	$-0.648113\pi$
$601$	−22.0000	−0.897399	−0.448699	+	0.893683i	$0.648113\pi$
$602$	0	0
$603$	2.00000	0.0814463
$604$	2.00000	0.0813788
$605$	−22.0000	−0.894427
$606$	12.0000	0.487467
$607$	26.0000	1.05531	0.527654	−	0.849460i	$-0.323072\pi$
$607$	26.0000	1.05531	0.527654	+	0.849460i	$0.323072\pi$
$608$	−30.0000	−1.21666
$609$	0	0
$610$	−12.0000	−0.485866
$611$	12.0000	0.485468
$612$	6.00000	0.242536
$613$	26.0000	1.05013	0.525065	−	0.851062i	$-0.324041\pi$
$613$	26.0000	1.05013	0.525065	+	0.851062i	$0.324041\pi$
$614$	20.0000	0.807134
$615$	−8.00000	−0.322591
$616$	0	0
$617$	−18.0000	−0.724653	−0.362326	−	0.932051i	$-0.618017\pi$
$617$	−18.0000	−0.724653	−0.362326	+	0.932051i	$0.618017\pi$
$618$	−28.0000	−1.12633
$619$	−10.0000	−0.401934	−0.200967	−	0.979598i	$-0.564408\pi$
$619$	−10.0000	−0.401934	−0.200967	+	0.979598i	$0.564408\pi$
$620$	8.00000	0.321288
$621$	24.0000	0.963087
$622$	0	0
$623$	0	0
$624$	−4.00000	−0.160128
$625$	−19.0000	−0.760000
$626$	−30.0000	−1.19904
$627$	0	0
$628$	−2.00000	−0.0798087
$629$	−12.0000	−0.478471
$630$	0	0
$631$	−10.0000	−0.398094	−0.199047	−	0.979990i	$-0.563785\pi$
$631$	−10.0000	−0.398094	−0.199047	+	0.979990i	$0.563785\pi$
$632$	30.0000	1.19334
$633$	24.0000	0.953914
$634$	−18.0000	−0.714871
$635$	32.0000	1.26988
$636$	−20.0000	−0.793052
$637$	−14.0000	−0.554700
$638$	0	0
$639$	−6.00000	−0.237356
$640$	6.00000	0.237171
$641$	34.0000	1.34292	0.671460	−	0.741041i	$-0.265668\pi$
$641$	34.0000	1.34292	0.671460	+	0.741041i	$0.265668\pi$
$642$	−36.0000	−1.42081
$643$	46.0000	1.81406	0.907031	−	0.421063i	$-0.138343\pi$
$643$	46.0000	1.81406	0.907031	+	0.421063i	$0.138343\pi$
$644$	0	0
$645$	24.0000	0.944999
$646$	36.0000	1.41640
$647$	12.0000	0.471769	0.235884	−	0.971781i	$-0.424201\pi$
$647$	12.0000	0.471769	0.235884	+	0.971781i	$0.424201\pi$
$648$	−33.0000	−1.29636
$649$	0	0
$650$	2.00000	0.0784465
$651$	0	0
$652$	−16.0000	−0.626608
$653$	22.0000	0.860927	0.430463	−	0.902608i	$-0.358350\pi$
$653$	22.0000	0.860927	0.430463	+	0.902608i	$0.358350\pi$
$654$	−36.0000	−1.40771
$655$	16.0000	0.625172
$656$	2.00000	0.0780869
$657$	2.00000	0.0780274
$658$	0	0
$659$	14.0000	0.545363	0.272681	−	0.962104i	$-0.412090\pi$
$659$	14.0000	0.545363	0.272681	+	0.962104i	$0.412090\pi$
$660$	0	0
$661$	−30.0000	−1.16686	−0.583432	−	0.812162i	$-0.698291\pi$
$661$	−30.0000	−1.16686	−0.583432	+	0.812162i	$0.698291\pi$
$662$	20.0000	0.777322
$663$	−24.0000	−0.932083
$664$	−12.0000	−0.465690
$665$	0	0
$666$	−2.00000	−0.0774984
$667$	36.0000	1.39393
$668$	−14.0000	−0.541676
$669$	52.0000	2.01044
$670$	−4.00000	−0.154533
$671$	0	0
$672$	0	0
$673$	10.0000	0.385472	0.192736	−	0.981251i	$-0.438264\pi$
$673$	10.0000	0.385472	0.192736	+	0.981251i	$0.438264\pi$
$674$	2.00000	0.0770371
$675$	4.00000	0.153960
$676$	9.00000	0.346154
$677$	−42.0000	−1.61419	−0.807096	−	0.590421i	$-0.798962\pi$
$677$	−42.0000	−1.61419	−0.807096	+	0.590421i	$0.798962\pi$
$678$	−2.00000	−0.0768095
$679$	0	0
$680$	−36.0000	−1.38054
$681$	8.00000	0.306561
$682$	0	0
$683$	−18.0000	−0.688751	−0.344375	−	0.938832i	$-0.611909\pi$
$683$	−18.0000	−0.688751	−0.344375	+	0.938832i	$0.611909\pi$
$684$	−6.00000	−0.229416
$685$	−28.0000	−1.06983
$686$	0	0
$687$	−28.0000	−1.06827
$688$	−6.00000	−0.228748
$689$	20.0000	0.761939
$690$	24.0000	0.913664
$691$	8.00000	0.304334	0.152167	−	0.988355i	$-0.451375\pi$
$691$	8.00000	0.304334	0.152167	+	0.988355i	$0.451375\pi$
$692$	14.0000	0.532200
$693$	0	0
$694$	−20.0000	−0.759190
$695$	−32.0000	−1.21383
$696$	−36.0000	−1.36458
$697$	12.0000	0.454532
$698$	22.0000	0.832712
$699$	44.0000	1.66423
$700$	0	0
$701$	2.00000	0.0755390	0.0377695	−	0.999286i	$-0.487975\pi$
$701$	2.00000	0.0755390	0.0377695	+	0.999286i	$0.487975\pi$
$702$	8.00000	0.301941
$703$	12.0000	0.452589
$704$	0	0
$705$	24.0000	0.903892
$706$	−14.0000	−0.526897
$707$	0	0
$708$	−12.0000	−0.450988
$709$	−22.0000	−0.826227	−0.413114	−	0.910679i	$-0.635559\pi$
$709$	−22.0000	−0.826227	−0.413114	+	0.910679i	$0.635559\pi$
$710$	12.0000	0.450352
$711$	10.0000	0.375029
$712$	−42.0000	−1.57402
$713$	24.0000	0.898807
$714$	0	0
$715$	0	0
$716$	18.0000	0.672692
$717$	−24.0000	−0.896296
$718$	−6.00000	−0.223918
$719$	−12.0000	−0.447524	−0.223762	−	0.974644i	$-0.571834\pi$
$719$	−12.0000	−0.447524	−0.223762	+	0.974644i	$0.571834\pi$
$720$	−2.00000	−0.0745356
$721$	0	0
$722$	−17.0000	−0.632674
$723$	28.0000	1.04133
$724$	−18.0000	−0.668965
$725$	6.00000	0.222834
$726$	22.0000	0.816497
$727$	32.0000	1.18681	0.593407	−	0.804902i	$-0.297782\pi$
$727$	32.0000	1.18681	0.593407	+	0.804902i	$0.297782\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	−4.00000	−0.148047
$731$	−36.0000	−1.33151
$732$	−12.0000	−0.443533
$733$	−46.0000	−1.69905	−0.849524	−	0.527549i	$-0.823111\pi$
$733$	−46.0000	−1.69905	−0.849524	+	0.527549i	$0.823111\pi$
$734$	−16.0000	−0.590571
$735$	−28.0000	−1.03280
$736$	30.0000	1.10581
$737$	0	0
$738$	2.00000	0.0736210
$739$	−32.0000	−1.17714	−0.588570	−	0.808447i	$-0.700309\pi$
$739$	−32.0000	−1.17714	−0.588570	+	0.808447i	$0.700309\pi$
$740$	−4.00000	−0.147043
$741$	24.0000	0.881662
$742$	0	0
$743$	−30.0000	−1.10059	−0.550297	−	0.834969i	$-0.685485\pi$
$743$	−30.0000	−1.10059	−0.550297	+	0.834969i	$0.685485\pi$
$744$	−24.0000	−0.879883
$745$	28.0000	1.02584
$746$	22.0000	0.805477
$747$	−4.00000	−0.146352
$748$	0	0
$749$	0	0
$750$	24.0000	0.876356
$751$	22.0000	0.802791	0.401396	−	0.915905i	$-0.368525\pi$
$751$	22.0000	0.802791	0.401396	+	0.915905i	$0.368525\pi$
$752$	−6.00000	−0.218797
$753$	32.0000	1.16614
$754$	12.0000	0.437014
$755$	−4.00000	−0.145575
$756$	0	0
$757$	−14.0000	−0.508839	−0.254419	−	0.967094i	$-0.581884\pi$
$757$	−14.0000	−0.508839	−0.254419	+	0.967094i	$0.581884\pi$
$758$	10.0000	0.363216
$759$	0	0
$760$	36.0000	1.30586
$761$	−42.0000	−1.52250	−0.761249	−	0.648459i	$-0.775414\pi$
$761$	−42.0000	−1.52250	−0.761249	+	0.648459i	$0.775414\pi$
$762$	−32.0000	−1.15924
$763$	0	0
$764$	10.0000	0.361787
$765$	−12.0000	−0.433861
$766$	4.00000	0.144526
$767$	12.0000	0.433295
$768$	−34.0000	−1.22687
$769$	−42.0000	−1.51456	−0.757279	−	0.653091i	$-0.773472\pi$
$769$	−42.0000	−1.51456	−0.757279	+	0.653091i	$0.773472\pi$
$770$	0	0
$771$	−28.0000	−1.00840
$772$	14.0000	0.503871
$773$	30.0000	1.07903	0.539513	−	0.841978i	$-0.318609\pi$
$773$	30.0000	1.07903	0.539513	+	0.841978i	$0.318609\pi$
$774$	−6.00000	−0.215666
$775$	4.00000	0.143684
$776$	−42.0000	−1.50771
$777$	0	0
$778$	6.00000	0.215110
$779$	−12.0000	−0.429945
$780$	−8.00000	−0.286446
$781$	0	0
$782$	−36.0000	−1.28736
$783$	24.0000	0.857690
$784$	7.00000	0.250000
$785$	4.00000	0.142766
$786$	−16.0000	−0.570701
$787$	28.0000	0.998092	0.499046	−	0.866575i	$-0.333684\pi$
$787$	28.0000	0.998092	0.499046	+	0.866575i	$0.333684\pi$
$788$	−18.0000	−0.641223
$789$	−20.0000	−0.712019
$790$	−20.0000	−0.711568
$791$	0	0
$792$	0	0
$793$	12.0000	0.426132
$794$	−18.0000	−0.638796
$795$	40.0000	1.41865
$796$	26.0000	0.921546
$797$	26.0000	0.920967	0.460484	−	0.887668i	$-0.347676\pi$
$797$	26.0000	0.920967	0.460484	+	0.887668i	$0.347676\pi$
$798$	0	0
$799$	−36.0000	−1.27359
$800$	5.00000	0.176777
$801$	−14.0000	−0.494666
$802$	−6.00000	−0.211867
$803$	0	0
$804$	−4.00000	−0.141069
$805$	0	0
$806$	8.00000	0.281788
$807$	36.0000	1.26726
$808$	−18.0000	−0.633238
$809$	−18.0000	−0.632846	−0.316423	−	0.948618i	$-0.602482\pi$
$809$	−18.0000	−0.632846	−0.316423	+	0.948618i	$0.602482\pi$
$810$	22.0000	0.773001
$811$	−2.00000	−0.0702295	−0.0351147	−	0.999383i	$-0.511180\pi$
$811$	−2.00000	−0.0702295	−0.0351147	+	0.999383i	$0.511180\pi$
$812$	0	0
$813$	20.0000	0.701431
$814$	0	0
$815$	32.0000	1.12091
$816$	12.0000	0.420084
$817$	36.0000	1.25948
$818$	−10.0000	−0.349642
$819$	0	0
$820$	4.00000	0.139686
$821$	22.0000	0.767805	0.383903	−	0.923374i	$-0.374580\pi$
$821$	22.0000	0.767805	0.383903	+	0.923374i	$0.374580\pi$
$822$	28.0000	0.976612
$823$	−16.0000	−0.557725	−0.278862	−	0.960331i	$-0.589957\pi$
$823$	−16.0000	−0.557725	−0.278862	+	0.960331i	$0.589957\pi$
$824$	42.0000	1.46314
$825$	0	0
$826$	0	0
$827$	−16.0000	−0.556375	−0.278187	−	0.960527i	$-0.589734\pi$
$827$	−16.0000	−0.556375	−0.278187	+	0.960527i	$0.589734\pi$
$828$	6.00000	0.208514
$829$	−22.0000	−0.764092	−0.382046	−	0.924143i	$-0.624780\pi$
$829$	−22.0000	−0.764092	−0.382046	+	0.924143i	$0.624780\pi$
$830$	8.00000	0.277684
$831$	−36.0000	−1.24883
$832$	14.0000	0.485363
$833$	42.0000	1.45521
$834$	32.0000	1.10807
$835$	28.0000	0.968980
$836$	0	0
$837$	16.0000	0.553041
$838$	26.0000	0.898155
$839$	−22.0000	−0.759524	−0.379762	−	0.925084i	$-0.623994\pi$
$839$	−22.0000	−0.759524	−0.379762	+	0.925084i	$0.623994\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	18.0000	0.620321
$843$	−12.0000	−0.413302
$844$	−12.0000	−0.413057
$845$	−18.0000	−0.619219
$846$	−6.00000	−0.206284
$847$	0	0
$848$	−10.0000	−0.343401
$849$	−40.0000	−1.37280
$850$	−6.00000	−0.205798
$851$	−12.0000	−0.411355
$852$	12.0000	0.411113
$853$	26.0000	0.890223	0.445112	−	0.895475i	$-0.353164\pi$
$853$	26.0000	0.890223	0.445112	+	0.895475i	$0.353164\pi$
$854$	0	0
$855$	12.0000	0.410391
$856$	54.0000	1.84568
$857$	10.0000	0.341593	0.170797	−	0.985306i	$-0.445366\pi$
$857$	10.0000	0.341593	0.170797	+	0.985306i	$0.445366\pi$
$858$	0	0
$859$	−14.0000	−0.477674	−0.238837	−	0.971060i	$-0.576766\pi$
$859$	−14.0000	−0.477674	−0.238837	+	0.971060i	$0.576766\pi$
$860$	−12.0000	−0.409197
$861$	0	0
$862$	30.0000	1.02180
$863$	−20.0000	−0.680808	−0.340404	−	0.940279i	$-0.610564\pi$
$863$	−20.0000	−0.680808	−0.340404	+	0.940279i	$0.610564\pi$
$864$	20.0000	0.680414
$865$	−28.0000	−0.952029
$866$	−10.0000	−0.339814
$867$	38.0000	1.29055
$868$	0	0
$869$	0	0
$870$	24.0000	0.813676
$871$	4.00000	0.135535
$872$	54.0000	1.82867
$873$	−14.0000	−0.473828
$874$	36.0000	1.21772
$875$	0	0
$876$	−4.00000	−0.135147
$877$	−14.0000	−0.472746	−0.236373	−	0.971662i	$-0.575959\pi$
$877$	−14.0000	−0.472746	−0.236373	+	0.971662i	$0.575959\pi$
$878$	−4.00000	−0.134993
$879$	−12.0000	−0.404750
$880$	0	0
$881$	42.0000	1.41502	0.707508	−	0.706705i	$-0.249819\pi$
$881$	42.0000	1.41502	0.707508	+	0.706705i	$0.249819\pi$
$882$	7.00000	0.235702
$883$	−14.0000	−0.471138	−0.235569	−	0.971858i	$-0.575695\pi$
$883$	−14.0000	−0.471138	−0.235569	+	0.971858i	$0.575695\pi$
$884$	12.0000	0.403604
$885$	24.0000	0.806751
$886$	16.0000	0.537531
$887$	−6.00000	−0.201460	−0.100730	−	0.994914i	$-0.532118\pi$
$887$	−6.00000	−0.201460	−0.100730	+	0.994914i	$0.532118\pi$
$888$	12.0000	0.402694
$889$	0	0
$890$	28.0000	0.938562
$891$	0	0
$892$	−26.0000	−0.870544
$893$	36.0000	1.20469
$894$	−28.0000	−0.936460
$895$	−36.0000	−1.20335
$896$	0	0
$897$	−24.0000	−0.801337
$898$	−34.0000	−1.13459
$899$	24.0000	0.800445
$900$	1.00000	0.0333333
$901$	−60.0000	−1.99889
$902$	0	0
$903$	0	0
$904$	3.00000	0.0997785
$905$	36.0000	1.19668
$906$	4.00000	0.132891
$907$	−38.0000	−1.26177	−0.630885	−	0.775877i	$-0.717308\pi$
$907$	−38.0000	−1.26177	−0.630885	+	0.775877i	$0.717308\pi$
$908$	−4.00000	−0.132745
$909$	−6.00000	−0.199007
$910$	0	0
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	−12.0000	−0.397360
$913$	0	0
$914$	30.0000	0.992312
$915$	24.0000	0.793416
$916$	14.0000	0.462573
$917$	0	0
$918$	−24.0000	−0.792118
$919$	−16.0000	−0.527791	−0.263896	−	0.964551i	$-0.585007\pi$
$919$	−16.0000	−0.527791	−0.263896	+	0.964551i	$0.585007\pi$
$920$	−36.0000	−1.18688
$921$	−40.0000	−1.31804
$922$	−22.0000	−0.724531
$923$	−12.0000	−0.394985
$924$	0	0
$925$	−2.00000	−0.0657596
$926$	−20.0000	−0.657241
$927$	14.0000	0.459820
$928$	30.0000	0.984798
$929$	−46.0000	−1.50921	−0.754606	−	0.656179i	$-0.772172\pi$
$929$	−46.0000	−1.50921	−0.754606	+	0.656179i	$0.772172\pi$
$930$	16.0000	0.524661
$931$	−42.0000	−1.37649
$932$	−22.0000	−0.720634
$933$	0	0
$934$	28.0000	0.916188
$935$	0	0
$936$	6.00000	0.196116
$937$	−46.0000	−1.50275	−0.751377	−	0.659873i	$-0.770610\pi$
$937$	−46.0000	−1.50275	−0.751377	+	0.659873i	$0.770610\pi$
$938$	0	0
$939$	60.0000	1.95803
$940$	−12.0000	−0.391397
$941$	10.0000	0.325991	0.162995	−	0.986627i	$-0.447884\pi$
$941$	10.0000	0.325991	0.162995	+	0.986627i	$0.447884\pi$
$942$	−4.00000	−0.130327
$943$	12.0000	0.390774
$944$	−6.00000	−0.195283
$945$	0	0
$946$	0	0
$947$	38.0000	1.23483	0.617417	−	0.786636i	$-0.288179\pi$
$947$	38.0000	1.23483	0.617417	+	0.786636i	$0.288179\pi$
$948$	−20.0000	−0.649570
$949$	4.00000	0.129845
$950$	6.00000	0.194666
$951$	36.0000	1.16738
$952$	0	0
$953$	−42.0000	−1.36051	−0.680257	−	0.732974i	$-0.738132\pi$
$953$	−42.0000	−1.36051	−0.680257	+	0.732974i	$0.738132\pi$
$954$	−10.0000	−0.323762
$955$	−20.0000	−0.647185
$956$	12.0000	0.388108
$957$	0	0
$958$	18.0000	0.581554
$959$	0	0
$960$	28.0000	0.903696
$961$	−15.0000	−0.483871
$962$	−4.00000	−0.128965
$963$	18.0000	0.580042
$964$	−14.0000	−0.450910
$965$	−28.0000	−0.901352
$966$	0	0
$967$	−52.0000	−1.67221	−0.836104	−	0.548572i	$-0.815172\pi$
$967$	−52.0000	−1.67221	−0.836104	+	0.548572i	$0.815172\pi$
$968$	−33.0000	−1.06066
$969$	−72.0000	−2.31297
$970$	28.0000	0.899026
$971$	−14.0000	−0.449281	−0.224641	−	0.974442i	$-0.572121\pi$
$971$	−14.0000	−0.449281	−0.224641	+	0.974442i	$0.572121\pi$
$972$	10.0000	0.320750
$973$	0	0
$974$	34.0000	1.08943
$975$	−4.00000	−0.128103
$976$	−6.00000	−0.192055
$977$	42.0000	1.34370	0.671850	−	0.740688i	$-0.265500\pi$
$977$	42.0000	1.34370	0.671850	+	0.740688i	$0.265500\pi$
$978$	−32.0000	−1.02325
$979$	0	0
$980$	14.0000	0.447214
$981$	18.0000	0.574696
$982$	−30.0000	−0.957338
$983$	−6.00000	−0.191370	−0.0956851	−	0.995412i	$-0.530504\pi$
$983$	−6.00000	−0.191370	−0.0956851	+	0.995412i	$0.530504\pi$
$984$	−12.0000	−0.382546
$985$	36.0000	1.14706
$986$	−36.0000	−1.14647
$987$	0	0
$988$	−12.0000	−0.381771
$989$	−36.0000	−1.14473
$990$	0	0
$991$	−60.0000	−1.90596	−0.952981	−	0.303029i	$-0.902002\pi$
$991$	−60.0000	−1.90596	−0.952981	+	0.303029i	$0.902002\pi$
$992$	20.0000	0.635001
$993$	−40.0000	−1.26936
$994$	0	0
$995$	−52.0000	−1.64851
$996$	8.00000	0.253490
$997$	42.0000	1.33015	0.665077	−	0.746775i	$-0.268399\pi$
$997$	42.0000	1.33015	0.665077	+	0.746775i	$0.268399\pi$
$998$	2.00000	0.0633089
$999$	−8.00000	−0.253109

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	113.2.a.a.1.1	✓	1
3.2	odd	2		1017.2.a.h.1.1		1
4.3	odd	2		1808.2.a.a.1.1		1
5.4	even	2		2825.2.a.b.1.1		1
7.6	odd	2		5537.2.a.a.1.1		1
8.3	odd	2		7232.2.a.f.1.1		1
8.5	even	2		7232.2.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
113.2.a.a.1.1	✓	1	1.1	even	1	trivial
1017.2.a.h.1.1		1	3.2	odd	2
1808.2.a.a.1.1		1	4.3	odd	2
2825.2.a.b.1.1		1	5.4	even	2
5537.2.a.a.1.1		1	7.6	odd	2
7232.2.a.a.1.1		1	8.5	even	2
7232.2.a.f.1.1		1	8.3	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:	\( N \)	\(=\)	\( 113 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	113.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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