LMFDB - Embedded newform 122.2.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [122,2,Mod(1,122)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(122, 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("122.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 122 = 2 \cdot 61 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	122.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:	$0.974174904660$
Analytic rank:	$1$
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	$\chi$	$=$	122.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{2} -2.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +2.00000 q^{6} -5.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$

\(q-1.00000 q^{2} -2.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +2.00000 q^{6} -5.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} -3.00000 q^{11} -2.00000 q^{12} -3.00000 q^{13} +5.00000 q^{14} -2.00000 q^{15} +1.00000 q^{16} -1.00000 q^{18} +1.00000 q^{20} +10.0000 q^{21} +3.00000 q^{22} +5.00000 q^{23} +2.00000 q^{24} -4.00000 q^{25} +3.00000 q^{26} +4.00000 q^{27} -5.00000 q^{28} +6.00000 q^{29} +2.00000 q^{30} -1.00000 q^{32} +6.00000 q^{33} -5.00000 q^{35} +1.00000 q^{36} -12.0000 q^{37} +6.00000 q^{39} -1.00000 q^{40} -3.00000 q^{41} -10.0000 q^{42} -8.00000 q^{43} -3.00000 q^{44} +1.00000 q^{45} -5.00000 q^{46} +12.0000 q^{47} -2.00000 q^{48} +18.0000 q^{49} +4.00000 q^{50} -3.00000 q^{52} -2.00000 q^{53} -4.00000 q^{54} -3.00000 q^{55} +5.00000 q^{56} -6.00000 q^{58} -9.00000 q^{59} -2.00000 q^{60} -1.00000 q^{61} -5.00000 q^{63} +1.00000 q^{64} -3.00000 q^{65} -6.00000 q^{66} +7.00000 q^{67} -10.0000 q^{69} +5.00000 q^{70} -16.0000 q^{71} -1.00000 q^{72} -3.00000 q^{73} +12.0000 q^{74} +8.00000 q^{75} +15.0000 q^{77} -6.00000 q^{78} +1.00000 q^{79} +1.00000 q^{80} -11.0000 q^{81} +3.00000 q^{82} -12.0000 q^{83} +10.0000 q^{84} +8.00000 q^{86} -12.0000 q^{87} +3.00000 q^{88} +12.0000 q^{89} -1.00000 q^{90} +15.0000 q^{91} +5.00000 q^{92} -12.0000 q^{94} +2.00000 q^{96} +2.00000 q^{97} -18.0000 q^{98} -3.00000 q^{99} +O(q^{100})\)

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
$3$	−2.00000	−1.15470	−0.577350	−	0.816497i	$-0.695913\pi$
$3$	−2.00000	−1.15470	−0.577350	+	0.816497i	$0.695913\pi$
$4$	1.00000	0.500000
$5$	1.00000	0.447214	0.223607	−	0.974679i	$-0.428217\pi$
$5$	1.00000	0.447214	0.223607	+	0.974679i	$0.428217\pi$
$6$	2.00000	0.816497
$7$	−5.00000	−1.88982	−0.944911	−	0.327327i	$-0.893852\pi$
$7$	−5.00000	−1.88982	−0.944911	+	0.327327i	$0.893852\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	−1.00000	−0.316228
$11$	−3.00000	−0.904534	−0.452267	−	0.891883i	$-0.649385\pi$
$11$	−3.00000	−0.904534	−0.452267	+	0.891883i	$0.649385\pi$
$12$	−2.00000	−0.577350
$13$	−3.00000	−0.832050	−0.416025	−	0.909353i	$-0.636577\pi$
$13$	−3.00000	−0.832050	−0.416025	+	0.909353i	$0.636577\pi$
$14$	5.00000	1.33631
$15$	−2.00000	−0.516398
$16$	1.00000	0.250000
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	−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
$21$	10.0000	2.18218
$22$	3.00000	0.639602
$23$	5.00000	1.04257	0.521286	−	0.853382i	$-0.325452\pi$
$23$	5.00000	1.04257	0.521286	+	0.853382i	$0.325452\pi$
$24$	2.00000	0.408248
$25$	−4.00000	−0.800000
$26$	3.00000	0.588348
$27$	4.00000	0.769800
$28$	−5.00000	−0.944911
$29$	6.00000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	2.00000	0.365148
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	−1.00000	−0.176777
$33$	6.00000	1.04447
$34$	0	0
$35$	−5.00000	−0.845154
$36$	1.00000	0.166667
$37$	−12.0000	−1.97279	−0.986394	−	0.164399i	$-0.947432\pi$
$37$	−12.0000	−1.97279	−0.986394	+	0.164399i	$0.947432\pi$
$38$	0	0
$39$	6.00000	0.960769
$40$	−1.00000	−0.158114
$41$	−3.00000	−0.468521	−0.234261	−	0.972174i	$-0.575267\pi$
$41$	−3.00000	−0.468521	−0.234261	+	0.972174i	$0.575267\pi$
$42$	−10.0000	−1.54303
$43$	−8.00000	−1.21999	−0.609994	−	0.792406i	$-0.708828\pi$
$43$	−8.00000	−1.21999	−0.609994	+	0.792406i	$0.708828\pi$
$44$	−3.00000	−0.452267
$45$	1.00000	0.149071
$46$	−5.00000	−0.737210
$47$	12.0000	1.75038	0.875190	−	0.483779i	$-0.160736\pi$
$47$	12.0000	1.75038	0.875190	+	0.483779i	$0.160736\pi$
$48$	−2.00000	−0.288675
$49$	18.0000	2.57143
$50$	4.00000	0.565685
$51$	0	0
$52$	−3.00000	−0.416025
$53$	−2.00000	−0.274721	−0.137361	−	0.990521i	$-0.543862\pi$
$53$	−2.00000	−0.274721	−0.137361	+	0.990521i	$0.543862\pi$
$54$	−4.00000	−0.544331
$55$	−3.00000	−0.404520
$56$	5.00000	0.668153
$57$	0	0
$58$	−6.00000	−0.787839
$59$	−9.00000	−1.17170	−0.585850	−	0.810419i	$-0.699239\pi$
$59$	−9.00000	−1.17170	−0.585850	+	0.810419i	$0.699239\pi$
$60$	−2.00000	−0.258199
$61$	−1.00000	−0.128037
$61$	−1.00000	−0.128037
$62$	0	0
$63$	−5.00000	−0.629941
$64$	1.00000	0.125000
$65$	−3.00000	−0.372104
$66$	−6.00000	−0.738549
$67$	7.00000	0.855186	0.427593	−	0.903971i	$-0.359362\pi$
$67$	7.00000	0.855186	0.427593	+	0.903971i	$0.359362\pi$
$68$	0	0
$69$	−10.0000	−1.20386
$70$	5.00000	0.597614
$71$	−16.0000	−1.89885	−0.949425	−	0.313993i	$-0.898333\pi$
$71$	−16.0000	−1.89885	−0.949425	+	0.313993i	$0.898333\pi$
$72$	−1.00000	−0.117851
$73$	−3.00000	−0.351123	−0.175562	−	0.984468i	$-0.556174\pi$
$73$	−3.00000	−0.351123	−0.175562	+	0.984468i	$0.556174\pi$
$74$	12.0000	1.39497
$75$	8.00000	0.923760
$76$	0	0
$77$	15.0000	1.70941
$78$	−6.00000	−0.679366
$79$	1.00000	0.112509	0.0562544	−	0.998416i	$-0.482084\pi$
$79$	1.00000	0.112509	0.0562544	+	0.998416i	$0.482084\pi$
$80$	1.00000	0.111803
$81$	−11.0000	−1.22222
$82$	3.00000	0.331295
$83$	−12.0000	−1.31717	−0.658586	−	0.752506i	$-0.728845\pi$
$83$	−12.0000	−1.31717	−0.658586	+	0.752506i	$0.728845\pi$
$84$	10.0000	1.09109
$85$	0	0
$86$	8.00000	0.862662
$87$	−12.0000	−1.28654
$88$	3.00000	0.319801
$89$	12.0000	1.27200	0.635999	−	0.771690i	$-0.280588\pi$
$89$	12.0000	1.27200	0.635999	+	0.771690i	$0.280588\pi$
$90$	−1.00000	−0.105409
$91$	15.0000	1.57243
$92$	5.00000	0.521286
$93$	0	0
$94$	−12.0000	−1.23771
$95$	0	0
$96$	2.00000	0.204124
$97$	2.00000	0.203069	0.101535	−	0.994832i	$-0.467625\pi$
$97$	2.00000	0.203069	0.101535	+	0.994832i	$0.467625\pi$
$98$	−18.0000	−1.81827
$99$	−3.00000	−0.301511
$100$	−4.00000	−0.400000
$101$	−12.0000	−1.19404	−0.597022	−	0.802225i	$-0.703650\pi$
$101$	−12.0000	−1.19404	−0.597022	+	0.802225i	$0.703650\pi$
$102$	0	0
$103$	16.0000	1.57653	0.788263	−	0.615338i	$-0.210980\pi$
$103$	16.0000	1.57653	0.788263	+	0.615338i	$0.210980\pi$
$104$	3.00000	0.294174
$105$	10.0000	0.975900
$106$	2.00000	0.194257
$107$	−6.00000	−0.580042	−0.290021	−	0.957020i	$-0.593662\pi$
$107$	−6.00000	−0.580042	−0.290021	+	0.957020i	$0.593662\pi$
$108$	4.00000	0.384900
$109$	−5.00000	−0.478913	−0.239457	−	0.970907i	$-0.576969\pi$
$109$	−5.00000	−0.478913	−0.239457	+	0.970907i	$0.576969\pi$
$110$	3.00000	0.286039
$111$	24.0000	2.27798
$112$	−5.00000	−0.472456
$113$	1.00000	0.0940721	0.0470360	−	0.998893i	$-0.485022\pi$
$113$	1.00000	0.0940721	0.0470360	+	0.998893i	$0.485022\pi$
$114$	0	0
$115$	5.00000	0.466252
$116$	6.00000	0.557086
$117$	−3.00000	−0.277350
$118$	9.00000	0.828517
$119$	0	0
$120$	2.00000	0.182574
$121$	−2.00000	−0.181818
$122$	1.00000	0.0905357
$123$	6.00000	0.541002
$124$	0	0
$125$	−9.00000	−0.804984
$126$	5.00000	0.445435
$127$	6.00000	0.532414	0.266207	−	0.963916i	$-0.414230\pi$
$127$	6.00000	0.532414	0.266207	+	0.963916i	$0.414230\pi$
$128$	−1.00000	−0.0883883
$129$	16.0000	1.40872
$130$	3.00000	0.263117
$131$	−4.00000	−0.349482	−0.174741	−	0.984614i	$-0.555909\pi$
$131$	−4.00000	−0.349482	−0.174741	+	0.984614i	$0.555909\pi$
$132$	6.00000	0.522233
$133$	0	0
$134$	−7.00000	−0.604708
$135$	4.00000	0.344265
$136$	0	0
$137$	−7.00000	−0.598050	−0.299025	−	0.954245i	$-0.596661\pi$
$137$	−7.00000	−0.598050	−0.299025	+	0.954245i	$0.596661\pi$
$138$	10.0000	0.851257
$139$	−13.0000	−1.10265	−0.551323	−	0.834292i	$-0.685877\pi$
$139$	−13.0000	−1.10265	−0.551323	+	0.834292i	$0.685877\pi$
$140$	−5.00000	−0.422577
$141$	−24.0000	−2.02116
$142$	16.0000	1.34269
$143$	9.00000	0.752618
$144$	1.00000	0.0833333
$145$	6.00000	0.498273
$146$	3.00000	0.248282
$147$	−36.0000	−2.96923
$148$	−12.0000	−0.986394
$149$	15.0000	1.22885	0.614424	−	0.788976i	$-0.289388\pi$
$149$	15.0000	1.22885	0.614424	+	0.788976i	$0.289388\pi$
$150$	−8.00000	−0.653197
$151$	−15.0000	−1.22068	−0.610341	−	0.792139i	$-0.708968\pi$
$151$	−15.0000	−1.22068	−0.610341	+	0.792139i	$0.708968\pi$
$152$	0	0
$153$	0	0
$154$	−15.0000	−1.20873
$155$	0	0
$156$	6.00000	0.480384
$157$	−8.00000	−0.638470	−0.319235	−	0.947676i	$-0.603426\pi$
$157$	−8.00000	−0.638470	−0.319235	+	0.947676i	$0.603426\pi$
$158$	−1.00000	−0.0795557
$159$	4.00000	0.317221
$160$	−1.00000	−0.0790569
$161$	−25.0000	−1.97028
$162$	11.0000	0.864242
$163$	−22.0000	−1.72317	−0.861586	−	0.507611i	$-0.830529\pi$
$163$	−22.0000	−1.72317	−0.861586	+	0.507611i	$0.830529\pi$
$164$	−3.00000	−0.234261
$165$	6.00000	0.467099
$166$	12.0000	0.931381
$167$	8.00000	0.619059	0.309529	−	0.950890i	$-0.399829\pi$
$167$	8.00000	0.619059	0.309529	+	0.950890i	$0.399829\pi$
$168$	−10.0000	−0.771517
$169$	−4.00000	−0.307692
$170$	0	0
$171$	0	0
$172$	−8.00000	−0.609994
$173$	18.0000	1.36851	0.684257	−	0.729241i	$-0.260127\pi$
$173$	18.0000	1.36851	0.684257	+	0.729241i	$0.260127\pi$
$174$	12.0000	0.909718
$175$	20.0000	1.51186
$176$	−3.00000	−0.226134
$177$	18.0000	1.35296
$178$	−12.0000	−0.899438
$179$	18.0000	1.34538	0.672692	−	0.739923i	$-0.265138\pi$
$179$	18.0000	1.34538	0.672692	+	0.739923i	$0.265138\pi$
$180$	1.00000	0.0745356
$181$	0	0		−	1.00000i	$-0.5\pi$
$181$	0	0			1.00000i	$0.5\pi$
$182$	−15.0000	−1.11187
$183$	2.00000	0.147844
$184$	−5.00000	−0.368605
$185$	−12.0000	−0.882258
$186$	0	0
$187$	0	0
$188$	12.0000	0.875190
$189$	−20.0000	−1.45479
$190$	0	0
$191$	1.00000	0.0723575	0.0361787	−	0.999345i	$-0.488481\pi$
$191$	1.00000	0.0723575	0.0361787	+	0.999345i	$0.488481\pi$
$192$	−2.00000	−0.144338
$193$	0	0		−	1.00000i	$-0.5\pi$
$193$	0	0			1.00000i	$0.5\pi$
$194$	−2.00000	−0.143592
$195$	6.00000	0.429669
$196$	18.0000	1.28571
$197$	−7.00000	−0.498729	−0.249365	−	0.968410i	$-0.580222\pi$
$197$	−7.00000	−0.498729	−0.249365	+	0.968410i	$0.580222\pi$
$198$	3.00000	0.213201
$199$	26.0000	1.84309	0.921546	−	0.388270i	$-0.126927\pi$
$199$	26.0000	1.84309	0.921546	+	0.388270i	$0.126927\pi$
$200$	4.00000	0.282843
$201$	−14.0000	−0.987484
$202$	12.0000	0.844317
$203$	−30.0000	−2.10559
$204$	0	0
$205$	−3.00000	−0.209529
$206$	−16.0000	−1.11477
$207$	5.00000	0.347524
$208$	−3.00000	−0.208013
$209$	0	0
$210$	−10.0000	−0.690066
$211$	−12.0000	−0.826114	−0.413057	−	0.910705i	$-0.635539\pi$
$211$	−12.0000	−0.826114	−0.413057	+	0.910705i	$0.635539\pi$
$212$	−2.00000	−0.137361
$213$	32.0000	2.19260
$214$	6.00000	0.410152
$215$	−8.00000	−0.545595
$216$	−4.00000	−0.272166
$217$	0	0
$218$	5.00000	0.338643
$219$	6.00000	0.405442
$220$	−3.00000	−0.202260
$221$	0	0
$222$	−24.0000	−1.61077
$223$	13.0000	0.870544	0.435272	−	0.900299i	$-0.356652\pi$
$223$	13.0000	0.870544	0.435272	+	0.900299i	$0.356652\pi$
$224$	5.00000	0.334077
$225$	−4.00000	−0.266667
$226$	−1.00000	−0.0665190
$227$	11.0000	0.730096	0.365048	−	0.930989i	$-0.381053\pi$
$227$	11.0000	0.730096	0.365048	+	0.930989i	$0.381053\pi$
$228$	0	0
$229$	21.0000	1.38772	0.693860	−	0.720110i	$-0.255909\pi$
$229$	21.0000	1.38772	0.693860	+	0.720110i	$0.255909\pi$
$230$	−5.00000	−0.329690
$231$	−30.0000	−1.97386
$232$	−6.00000	−0.393919
$233$	−10.0000	−0.655122	−0.327561	−	0.944830i	$-0.606227\pi$
$233$	−10.0000	−0.655122	−0.327561	+	0.944830i	$0.606227\pi$
$234$	3.00000	0.196116
$235$	12.0000	0.782794
$236$	−9.00000	−0.585850
$237$	−2.00000	−0.129914
$238$	0	0
$239$	−6.00000	−0.388108	−0.194054	−	0.980991i	$-0.562164\pi$
$239$	−6.00000	−0.388108	−0.194054	+	0.980991i	$0.562164\pi$
$240$	−2.00000	−0.129099
$241$	−11.0000	−0.708572	−0.354286	−	0.935137i	$-0.615276\pi$
$241$	−11.0000	−0.708572	−0.354286	+	0.935137i	$0.615276\pi$
$242$	2.00000	0.128565
$243$	10.0000	0.641500
$244$	−1.00000	−0.0640184
$245$	18.0000	1.14998
$246$	−6.00000	−0.382546
$247$	0	0
$248$	0	0
$249$	24.0000	1.52094
$250$	9.00000	0.569210
$251$	−4.00000	−0.252478	−0.126239	−	0.992000i	$-0.540291\pi$
$251$	−4.00000	−0.252478	−0.126239	+	0.992000i	$0.540291\pi$
$252$	−5.00000	−0.314970
$253$	−15.0000	−0.943042
$254$	−6.00000	−0.376473
$255$	0	0
$256$	1.00000	0.0625000
$257$	2.00000	0.124757	0.0623783	−	0.998053i	$-0.480131\pi$
$257$	2.00000	0.124757	0.0623783	+	0.998053i	$0.480131\pi$
$258$	−16.0000	−0.996116
$259$	60.0000	3.72822
$260$	−3.00000	−0.186052
$261$	6.00000	0.371391
$262$	4.00000	0.247121
$263$	12.0000	0.739952	0.369976	−	0.929041i	$-0.379366\pi$
$263$	12.0000	0.739952	0.369976	+	0.929041i	$0.379366\pi$
$264$	−6.00000	−0.369274
$265$	−2.00000	−0.122859
$266$	0	0
$267$	−24.0000	−1.46878
$268$	7.00000	0.427593
$269$	−10.0000	−0.609711	−0.304855	−	0.952399i	$-0.598608\pi$
$269$	−10.0000	−0.609711	−0.304855	+	0.952399i	$0.598608\pi$
$270$	−4.00000	−0.243432
$271$	22.0000	1.33640	0.668202	−	0.743980i	$-0.267064\pi$
$271$	22.0000	1.33640	0.668202	+	0.743980i	$0.267064\pi$
$272$	0	0
$273$	−30.0000	−1.81568
$274$	7.00000	0.422885
$275$	12.0000	0.723627
$276$	−10.0000	−0.601929
$277$	−18.0000	−1.08152	−0.540758	−	0.841178i	$-0.681862\pi$
$277$	−18.0000	−1.08152	−0.540758	+	0.841178i	$0.681862\pi$
$278$	13.0000	0.779688
$279$	0	0
$280$	5.00000	0.298807
$281$	8.00000	0.477240	0.238620	−	0.971113i	$-0.423305\pi$
$281$	8.00000	0.477240	0.238620	+	0.971113i	$0.423305\pi$
$282$	24.0000	1.42918
$283$	6.00000	0.356663	0.178331	−	0.983970i	$-0.442930\pi$
$283$	6.00000	0.356663	0.178331	+	0.983970i	$0.442930\pi$
$284$	−16.0000	−0.949425
$285$	0	0
$286$	−9.00000	−0.532181
$287$	15.0000	0.885422
$288$	−1.00000	−0.0589256
$289$	−17.0000	−1.00000
$290$	−6.00000	−0.352332
$291$	−4.00000	−0.234484
$292$	−3.00000	−0.175562
$293$	26.0000	1.51894	0.759468	−	0.650545i	$-0.225459\pi$
$293$	26.0000	1.51894	0.759468	+	0.650545i	$0.225459\pi$
$294$	36.0000	2.09956
$295$	−9.00000	−0.524000
$296$	12.0000	0.697486
$297$	−12.0000	−0.696311
$298$	−15.0000	−0.868927
$299$	−15.0000	−0.867472
$300$	8.00000	0.461880
$301$	40.0000	2.30556
$302$	15.0000	0.863153
$303$	24.0000	1.37876
$304$	0	0
$305$	−1.00000	−0.0572598
$306$	0	0
$307$	19.0000	1.08439	0.542194	−	0.840254i	$-0.317594\pi$
$307$	19.0000	1.08439	0.542194	+	0.840254i	$0.317594\pi$
$308$	15.0000	0.854704
$309$	−32.0000	−1.82042
$310$	0	0
$311$	3.00000	0.170114	0.0850572	−	0.996376i	$-0.472893\pi$
$311$	3.00000	0.170114	0.0850572	+	0.996376i	$0.472893\pi$
$312$	−6.00000	−0.339683
$313$	−14.0000	−0.791327	−0.395663	−	0.918396i	$-0.629485\pi$
$313$	−14.0000	−0.791327	−0.395663	+	0.918396i	$0.629485\pi$
$314$	8.00000	0.451466
$315$	−5.00000	−0.281718
$316$	1.00000	0.0562544
$317$	18.0000	1.01098	0.505490	−	0.862832i	$-0.331312\pi$
$317$	18.0000	1.01098	0.505490	+	0.862832i	$0.331312\pi$
$318$	−4.00000	−0.224309
$319$	−18.0000	−1.00781
$320$	1.00000	0.0559017
$321$	12.0000	0.669775
$322$	25.0000	1.39320
$323$	0	0
$324$	−11.0000	−0.611111
$325$	12.0000	0.665640
$326$	22.0000	1.21847
$327$	10.0000	0.553001
$328$	3.00000	0.165647
$329$	−60.0000	−3.30791
$330$	−6.00000	−0.330289
$331$	−7.00000	−0.384755	−0.192377	−	0.981321i	$-0.561620\pi$
$331$	−7.00000	−0.384755	−0.192377	+	0.981321i	$0.561620\pi$
$332$	−12.0000	−0.658586
$333$	−12.0000	−0.657596
$334$	−8.00000	−0.437741
$335$	7.00000	0.382451
$336$	10.0000	0.545545
$337$	6.00000	0.326841	0.163420	−	0.986557i	$-0.447747\pi$
$337$	6.00000	0.326841	0.163420	+	0.986557i	$0.447747\pi$
$338$	4.00000	0.217571
$339$	−2.00000	−0.108625
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−55.0000	−2.96972
$344$	8.00000	0.431331
$345$	−10.0000	−0.538382
$346$	−18.0000	−0.967686
$347$	−28.0000	−1.50312	−0.751559	−	0.659665i	$-0.770698\pi$
$347$	−28.0000	−1.50312	−0.751559	+	0.659665i	$0.770698\pi$
$348$	−12.0000	−0.643268
$349$	−20.0000	−1.07058	−0.535288	−	0.844670i	$-0.679797\pi$
$349$	−20.0000	−1.07058	−0.535288	+	0.844670i	$0.679797\pi$
$350$	−20.0000	−1.06904
$351$	−12.0000	−0.640513
$352$	3.00000	0.159901
$353$	3.00000	0.159674	0.0798369	−	0.996808i	$-0.474560\pi$
$353$	3.00000	0.159674	0.0798369	+	0.996808i	$0.474560\pi$
$354$	−18.0000	−0.956689
$355$	−16.0000	−0.849192
$356$	12.0000	0.635999
$357$	0	0
$358$	−18.0000	−0.951330
$359$	28.0000	1.47778	0.738892	−	0.673824i	$-0.235349\pi$
$359$	28.0000	1.47778	0.738892	+	0.673824i	$0.235349\pi$
$360$	−1.00000	−0.0527046
$361$	−19.0000	−1.00000
$362$	0	0
$363$	4.00000	0.209946
$364$	15.0000	0.786214
$365$	−3.00000	−0.157027
$366$	−2.00000	−0.104542
$367$	−6.00000	−0.313197	−0.156599	−	0.987662i	$-0.550053\pi$
$367$	−6.00000	−0.313197	−0.156599	+	0.987662i	$0.550053\pi$
$368$	5.00000	0.260643
$369$	−3.00000	−0.156174
$370$	12.0000	0.623850
$371$	10.0000	0.519174
$372$	0	0
$373$	26.0000	1.34623	0.673114	−	0.739538i	$-0.264956\pi$
$373$	26.0000	1.34623	0.673114	+	0.739538i	$0.264956\pi$
$374$	0	0
$375$	18.0000	0.929516
$376$	−12.0000	−0.618853
$377$	−18.0000	−0.927047
$378$	20.0000	1.02869
$379$	−22.0000	−1.13006	−0.565032	−	0.825069i	$-0.691136\pi$
$379$	−22.0000	−1.13006	−0.565032	+	0.825069i	$0.691136\pi$
$380$	0	0
$381$	−12.0000	−0.614779
$382$	−1.00000	−0.0511645
$383$	−33.0000	−1.68622	−0.843111	−	0.537740i	$-0.819278\pi$
$383$	−33.0000	−1.68622	−0.843111	+	0.537740i	$0.819278\pi$
$384$	2.00000	0.102062
$385$	15.0000	0.764471
$386$	0	0
$387$	−8.00000	−0.406663
$388$	2.00000	0.101535
$389$	2.00000	0.101404	0.0507020	−	0.998714i	$-0.483854\pi$
$389$	2.00000	0.101404	0.0507020	+	0.998714i	$0.483854\pi$
$390$	−6.00000	−0.303822
$391$	0	0
$392$	−18.0000	−0.909137
$393$	8.00000	0.403547
$394$	7.00000	0.352655
$395$	1.00000	0.0503155
$396$	−3.00000	−0.150756
$397$	4.00000	0.200754	0.100377	−	0.994949i	$-0.467995\pi$
$397$	4.00000	0.200754	0.100377	+	0.994949i	$0.467995\pi$
$398$	−26.0000	−1.30326
$399$	0	0
$400$	−4.00000	−0.200000
$401$	0	0		−	1.00000i	$-0.5\pi$
$401$	0	0			1.00000i	$0.5\pi$
$402$	14.0000	0.698257
$403$	0	0
$404$	−12.0000	−0.597022
$405$	−11.0000	−0.546594
$406$	30.0000	1.48888
$407$	36.0000	1.78445
$408$	0	0
$409$	−10.0000	−0.494468	−0.247234	−	0.968956i	$-0.579522\pi$
$409$	−10.0000	−0.494468	−0.247234	+	0.968956i	$0.579522\pi$
$410$	3.00000	0.148159
$411$	14.0000	0.690569
$412$	16.0000	0.788263
$413$	45.0000	2.21431
$414$	−5.00000	−0.245737
$415$	−12.0000	−0.589057
$416$	3.00000	0.147087
$417$	26.0000	1.27323
$418$	0	0
$419$	−12.0000	−0.586238	−0.293119	−	0.956076i	$-0.594693\pi$
$419$	−12.0000	−0.586238	−0.293119	+	0.956076i	$0.594693\pi$
$420$	10.0000	0.487950
$421$	20.0000	0.974740	0.487370	−	0.873195i	$-0.337956\pi$
$421$	20.0000	0.974740	0.487370	+	0.873195i	$0.337956\pi$
$422$	12.0000	0.584151
$423$	12.0000	0.583460
$424$	2.00000	0.0971286
$425$	0	0
$426$	−32.0000	−1.55041
$427$	5.00000	0.241967
$428$	−6.00000	−0.290021
$429$	−18.0000	−0.869048
$430$	8.00000	0.385794
$431$	−22.0000	−1.05970	−0.529851	−	0.848091i	$-0.677752\pi$
$431$	−22.0000	−1.05970	−0.529851	+	0.848091i	$0.677752\pi$
$432$	4.00000	0.192450
$433$	16.0000	0.768911	0.384455	−	0.923144i	$-0.374389\pi$
$433$	16.0000	0.768911	0.384455	+	0.923144i	$0.374389\pi$
$434$	0	0
$435$	−12.0000	−0.575356
$436$	−5.00000	−0.239457
$437$	0	0
$438$	−6.00000	−0.286691
$439$	−36.0000	−1.71819	−0.859093	−	0.511819i	$-0.828972\pi$
$439$	−36.0000	−1.71819	−0.859093	+	0.511819i	$0.828972\pi$
$440$	3.00000	0.143019
$441$	18.0000	0.857143
$442$	0	0
$443$	−12.0000	−0.570137	−0.285069	−	0.958507i	$-0.592016\pi$
$443$	−12.0000	−0.570137	−0.285069	+	0.958507i	$0.592016\pi$
$444$	24.0000	1.13899
$445$	12.0000	0.568855
$446$	−13.0000	−0.615568
$447$	−30.0000	−1.41895
$448$	−5.00000	−0.236228
$449$	21.0000	0.991051	0.495526	−	0.868593i	$-0.334975\pi$
$449$	21.0000	0.991051	0.495526	+	0.868593i	$0.334975\pi$
$450$	4.00000	0.188562
$451$	9.00000	0.423793
$452$	1.00000	0.0470360
$453$	30.0000	1.40952
$454$	−11.0000	−0.516256
$455$	15.0000	0.703211
$456$	0	0
$457$	−16.0000	−0.748448	−0.374224	−	0.927338i	$-0.622091\pi$
$457$	−16.0000	−0.748448	−0.374224	+	0.927338i	$0.622091\pi$
$458$	−21.0000	−0.981266
$459$	0	0
$460$	5.00000	0.233126
$461$	14.0000	0.652045	0.326023	−	0.945362i	$-0.394291\pi$
$461$	14.0000	0.652045	0.326023	+	0.945362i	$0.394291\pi$
$462$	30.0000	1.39573
$463$	4.00000	0.185896	0.0929479	−	0.995671i	$-0.470371\pi$
$463$	4.00000	0.185896	0.0929479	+	0.995671i	$0.470371\pi$
$464$	6.00000	0.278543
$465$	0	0
$466$	10.0000	0.463241
$467$	13.0000	0.601568	0.300784	−	0.953692i	$-0.402752\pi$
$467$	13.0000	0.601568	0.300784	+	0.953692i	$0.402752\pi$
$468$	−3.00000	−0.138675
$469$	−35.0000	−1.61615
$470$	−12.0000	−0.553519
$471$	16.0000	0.737241
$472$	9.00000	0.414259
$473$	24.0000	1.10352
$474$	2.00000	0.0918630
$475$	0	0
$476$	0	0
$477$	−2.00000	−0.0915737
$478$	6.00000	0.274434
$479$	−14.0000	−0.639676	−0.319838	−	0.947472i	$-0.603629\pi$
$479$	−14.0000	−0.639676	−0.319838	+	0.947472i	$0.603629\pi$
$480$	2.00000	0.0912871
$481$	36.0000	1.64146
$482$	11.0000	0.501036
$483$	50.0000	2.27508
$484$	−2.00000	−0.0909091
$485$	2.00000	0.0908153
$486$	−10.0000	−0.453609
$487$	−8.00000	−0.362515	−0.181257	−	0.983436i	$-0.558017\pi$
$487$	−8.00000	−0.362515	−0.181257	+	0.983436i	$0.558017\pi$
$488$	1.00000	0.0452679
$489$	44.0000	1.98975
$490$	−18.0000	−0.813157
$491$	−8.00000	−0.361035	−0.180517	−	0.983572i	$-0.557777\pi$
$491$	−8.00000	−0.361035	−0.180517	+	0.983572i	$0.557777\pi$
$492$	6.00000	0.270501
$493$	0	0
$494$	0	0
$495$	−3.00000	−0.134840
$496$	0	0
$497$	80.0000	3.58849
$498$	−24.0000	−1.07547
$499$	−37.0000	−1.65635	−0.828174	−	0.560471i	$-0.810620\pi$
$499$	−37.0000	−1.65635	−0.828174	+	0.560471i	$0.810620\pi$
$500$	−9.00000	−0.402492
$501$	−16.0000	−0.714827
$502$	4.00000	0.178529
$503$	−38.0000	−1.69434	−0.847168	−	0.531325i	$-0.821694\pi$
$503$	−38.0000	−1.69434	−0.847168	+	0.531325i	$0.821694\pi$
$504$	5.00000	0.222718
$505$	−12.0000	−0.533993
$506$	15.0000	0.666831
$507$	8.00000	0.355292
$508$	6.00000	0.266207
$509$	−16.0000	−0.709188	−0.354594	−	0.935020i	$-0.615381\pi$
$509$	−16.0000	−0.709188	−0.354594	+	0.935020i	$0.615381\pi$
$510$	0	0
$511$	15.0000	0.663561
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−2.00000	−0.0882162
$515$	16.0000	0.705044
$516$	16.0000	0.704361
$517$	−36.0000	−1.58328
$518$	−60.0000	−2.63625
$519$	−36.0000	−1.58022
$520$	3.00000	0.131559
$521$	40.0000	1.75243	0.876216	−	0.481919i	$-0.160060\pi$
$521$	40.0000	1.75243	0.876216	+	0.481919i	$0.160060\pi$
$522$	−6.00000	−0.262613
$523$	1.00000	0.0437269	0.0218635	−	0.999761i	$-0.493040\pi$
$523$	1.00000	0.0437269	0.0218635	+	0.999761i	$0.493040\pi$
$524$	−4.00000	−0.174741
$525$	−40.0000	−1.74574
$526$	−12.0000	−0.523225
$527$	0	0
$528$	6.00000	0.261116
$529$	2.00000	0.0869565
$530$	2.00000	0.0868744
$531$	−9.00000	−0.390567
$532$	0	0
$533$	9.00000	0.389833
$534$	24.0000	1.03858
$535$	−6.00000	−0.259403
$536$	−7.00000	−0.302354
$537$	−36.0000	−1.55351
$538$	10.0000	0.431131
$539$	−54.0000	−2.32594
$540$	4.00000	0.172133
$541$	−12.0000	−0.515920	−0.257960	−	0.966156i	$-0.583050\pi$
$541$	−12.0000	−0.515920	−0.257960	+	0.966156i	$0.583050\pi$
$542$	−22.0000	−0.944981
$543$	0	0
$544$	0	0
$545$	−5.00000	−0.214176
$546$	30.0000	1.28388
$547$	37.0000	1.58201	0.791003	−	0.611812i	$-0.209559\pi$
$547$	37.0000	1.58201	0.791003	+	0.611812i	$0.209559\pi$
$548$	−7.00000	−0.299025
$549$	−1.00000	−0.0426790
$550$	−12.0000	−0.511682
$551$	0	0
$552$	10.0000	0.425628
$553$	−5.00000	−0.212622
$554$	18.0000	0.764747
$555$	24.0000	1.01874
$556$	−13.0000	−0.551323
$557$	−28.0000	−1.18640	−0.593199	−	0.805056i	$-0.702135\pi$
$557$	−28.0000	−1.18640	−0.593199	+	0.805056i	$0.702135\pi$
$558$	0	0
$559$	24.0000	1.01509
$560$	−5.00000	−0.211289
$561$	0	0
$562$	−8.00000	−0.337460
$563$	−2.00000	−0.0842900	−0.0421450	−	0.999112i	$-0.513419\pi$
$563$	−2.00000	−0.0842900	−0.0421450	+	0.999112i	$0.513419\pi$
$564$	−24.0000	−1.01058
$565$	1.00000	0.0420703
$566$	−6.00000	−0.252199
$567$	55.0000	2.30978
$568$	16.0000	0.671345
$569$	9.00000	0.377300	0.188650	−	0.982044i	$-0.439589\pi$
$569$	9.00000	0.377300	0.188650	+	0.982044i	$0.439589\pi$
$570$	0	0
$571$	18.0000	0.753277	0.376638	−	0.926360i	$-0.377080\pi$
$571$	18.0000	0.753277	0.376638	+	0.926360i	$0.377080\pi$
$572$	9.00000	0.376309
$573$	−2.00000	−0.0835512
$574$	−15.0000	−0.626088
$575$	−20.0000	−0.834058
$576$	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	$0.750276\pi$
$578$	17.0000	0.707107
$579$	0	0
$580$	6.00000	0.249136
$581$	60.0000	2.48922
$582$	4.00000	0.165805
$583$	6.00000	0.248495
$584$	3.00000	0.124141
$585$	−3.00000	−0.124035
$586$	−26.0000	−1.07405
$587$	0	0		−	1.00000i	$-0.5\pi$
$587$	0	0			1.00000i	$0.5\pi$
$588$	−36.0000	−1.48461
$589$	0	0
$590$	9.00000	0.370524
$591$	14.0000	0.575883
$592$	−12.0000	−0.493197
$593$	−18.0000	−0.739171	−0.369586	−	0.929197i	$-0.620500\pi$
$593$	−18.0000	−0.739171	−0.369586	+	0.929197i	$0.620500\pi$
$594$	12.0000	0.492366
$595$	0	0
$596$	15.0000	0.614424
$597$	−52.0000	−2.12822
$598$	15.0000	0.613396
$599$	39.0000	1.59350	0.796748	−	0.604311i	$-0.206552\pi$
$599$	39.0000	1.59350	0.796748	+	0.604311i	$0.206552\pi$
$600$	−8.00000	−0.326599
$601$	−11.0000	−0.448699	−0.224350	−	0.974509i	$-0.572026\pi$
$601$	−11.0000	−0.448699	−0.224350	+	0.974509i	$0.572026\pi$
$602$	−40.0000	−1.63028
$603$	7.00000	0.285062
$604$	−15.0000	−0.610341
$605$	−2.00000	−0.0813116
$606$	−24.0000	−0.974933
$607$	−30.0000	−1.21766	−0.608831	−	0.793300i	$-0.708361\pi$
$607$	−30.0000	−1.21766	−0.608831	+	0.793300i	$0.708361\pi$
$608$	0	0
$609$	60.0000	2.43132
$610$	1.00000	0.0404888
$611$	−36.0000	−1.45640
$612$	0	0
$613$	26.0000	1.05013	0.525065	−	0.851062i	$-0.324041\pi$
$613$	26.0000	1.05013	0.525065	+	0.851062i	$0.324041\pi$
$614$	−19.0000	−0.766778
$615$	6.00000	0.241943
$616$	−15.0000	−0.604367
$617$	34.0000	1.36879	0.684394	−	0.729112i	$-0.260067\pi$
$617$	34.0000	1.36879	0.684394	+	0.729112i	$0.260067\pi$
$618$	32.0000	1.28723
$619$	−4.00000	−0.160774	−0.0803868	−	0.996764i	$-0.525616\pi$
$619$	−4.00000	−0.160774	−0.0803868	+	0.996764i	$0.525616\pi$
$620$	0	0
$621$	20.0000	0.802572
$622$	−3.00000	−0.120289
$623$	−60.0000	−2.40385
$624$	6.00000	0.240192
$625$	11.0000	0.440000
$626$	14.0000	0.559553
$627$	0	0
$628$	−8.00000	−0.319235
$629$	0	0
$630$	5.00000	0.199205
$631$	−29.0000	−1.15447	−0.577236	−	0.816577i	$-0.695869\pi$
$631$	−29.0000	−1.15447	−0.577236	+	0.816577i	$0.695869\pi$
$632$	−1.00000	−0.0397779
$633$	24.0000	0.953914
$634$	−18.0000	−0.714871
$635$	6.00000	0.238103
$636$	4.00000	0.158610
$637$	−54.0000	−2.13956
$638$	18.0000	0.712627
$639$	−16.0000	−0.632950
$640$	−1.00000	−0.0395285
$641$	6.00000	0.236986	0.118493	−	0.992955i	$-0.462194\pi$
$641$	6.00000	0.236986	0.118493	+	0.992955i	$0.462194\pi$
$642$	−12.0000	−0.473602
$643$	4.00000	0.157745	0.0788723	−	0.996885i	$-0.474868\pi$
$643$	4.00000	0.157745	0.0788723	+	0.996885i	$0.474868\pi$
$644$	−25.0000	−0.985138
$645$	16.0000	0.629999
$646$	0	0
$647$	−27.0000	−1.06148	−0.530740	−	0.847535i	$-0.678086\pi$
$647$	−27.0000	−1.06148	−0.530740	+	0.847535i	$0.678086\pi$
$648$	11.0000	0.432121
$649$	27.0000	1.05984
$650$	−12.0000	−0.470679
$651$	0	0
$652$	−22.0000	−0.861586
$653$	−34.0000	−1.33052	−0.665261	−	0.746611i	$-0.731680\pi$
$653$	−34.0000	−1.33052	−0.665261	+	0.746611i	$0.731680\pi$
$654$	−10.0000	−0.391031
$655$	−4.00000	−0.156293
$656$	−3.00000	−0.117130
$657$	−3.00000	−0.117041
$658$	60.0000	2.33904
$659$	−10.0000	−0.389545	−0.194772	−	0.980848i	$-0.562397\pi$
$659$	−10.0000	−0.389545	−0.194772	+	0.980848i	$0.562397\pi$
$660$	6.00000	0.233550
$661$	14.0000	0.544537	0.272268	−	0.962221i	$-0.412226\pi$
$661$	14.0000	0.544537	0.272268	+	0.962221i	$0.412226\pi$
$662$	7.00000	0.272063
$663$	0	0
$664$	12.0000	0.465690
$665$	0	0
$666$	12.0000	0.464991
$667$	30.0000	1.16160
$668$	8.00000	0.309529
$669$	−26.0000	−1.00522
$670$	−7.00000	−0.270434
$671$	3.00000	0.115814
$672$	−10.0000	−0.385758
$673$	−36.0000	−1.38770	−0.693849	−	0.720121i	$-0.744086\pi$
$673$	−36.0000	−1.38770	−0.693849	+	0.720121i	$0.744086\pi$
$674$	−6.00000	−0.231111
$675$	−16.0000	−0.615840
$676$	−4.00000	−0.153846
$677$	24.0000	0.922395	0.461197	−	0.887298i	$-0.347420\pi$
$677$	24.0000	0.922395	0.461197	+	0.887298i	$0.347420\pi$
$678$	2.00000	0.0768095
$679$	−10.0000	−0.383765
$680$	0	0
$681$	−22.0000	−0.843042
$682$	0	0
$683$	2.00000	0.0765279	0.0382639	−	0.999268i	$-0.487817\pi$
$683$	2.00000	0.0765279	0.0382639	+	0.999268i	$0.487817\pi$
$684$	0	0
$685$	−7.00000	−0.267456
$686$	55.0000	2.09991
$687$	−42.0000	−1.60240
$688$	−8.00000	−0.304997
$689$	6.00000	0.228582
$690$	10.0000	0.380693
$691$	−8.00000	−0.304334	−0.152167	−	0.988355i	$-0.548625\pi$
$691$	−8.00000	−0.304334	−0.152167	+	0.988355i	$0.548625\pi$
$692$	18.0000	0.684257
$693$	15.0000	0.569803
$694$	28.0000	1.06287
$695$	−13.0000	−0.493118
$696$	12.0000	0.454859
$697$	0	0
$698$	20.0000	0.757011
$699$	20.0000	0.756469
$700$	20.0000	0.755929
$701$	−6.00000	−0.226617	−0.113308	−	0.993560i	$-0.536145\pi$
$701$	−6.00000	−0.226617	−0.113308	+	0.993560i	$0.536145\pi$
$702$	12.0000	0.452911
$703$	0	0
$704$	−3.00000	−0.113067
$705$	−24.0000	−0.903892
$706$	−3.00000	−0.112906
$707$	60.0000	2.25653
$708$	18.0000	0.676481
$709$	2.00000	0.0751116	0.0375558	−	0.999295i	$-0.488043\pi$
$709$	2.00000	0.0751116	0.0375558	+	0.999295i	$0.488043\pi$
$710$	16.0000	0.600469
$711$	1.00000	0.0375029
$712$	−12.0000	−0.449719
$713$	0	0
$714$	0	0
$715$	9.00000	0.336581
$716$	18.0000	0.672692
$717$	12.0000	0.448148
$718$	−28.0000	−1.04495
$719$	42.0000	1.56634	0.783168	−	0.621810i	$-0.213603\pi$
$719$	42.0000	1.56634	0.783168	+	0.621810i	$0.213603\pi$
$720$	1.00000	0.0372678
$721$	−80.0000	−2.97936
$722$	19.0000	0.707107
$723$	22.0000	0.818189
$724$	0	0
$725$	−24.0000	−0.891338
$726$	−4.00000	−0.148454
$727$	8.00000	0.296704	0.148352	−	0.988935i	$-0.452603\pi$
$727$	8.00000	0.296704	0.148352	+	0.988935i	$0.452603\pi$
$728$	−15.0000	−0.555937
$729$	13.0000	0.481481
$730$	3.00000	0.111035
$731$	0	0
$732$	2.00000	0.0739221
$733$	−11.0000	−0.406294	−0.203147	−	0.979148i	$-0.565117\pi$
$733$	−11.0000	−0.406294	−0.203147	+	0.979148i	$0.565117\pi$
$734$	6.00000	0.221464
$735$	−36.0000	−1.32788
$736$	−5.00000	−0.184302
$737$	−21.0000	−0.773545
$738$	3.00000	0.110432
$739$	3.00000	0.110357	0.0551784	−	0.998477i	$-0.482427\pi$
$739$	3.00000	0.110357	0.0551784	+	0.998477i	$0.482427\pi$
$740$	−12.0000	−0.441129
$741$	0	0
$742$	−10.0000	−0.367112
$743$	25.0000	0.917161	0.458581	−	0.888653i	$-0.348358\pi$
$743$	25.0000	0.917161	0.458581	+	0.888653i	$0.348358\pi$
$744$	0	0
$745$	15.0000	0.549557
$746$	−26.0000	−0.951928
$747$	−12.0000	−0.439057
$748$	0	0
$749$	30.0000	1.09618
$750$	−18.0000	−0.657267
$751$	10.0000	0.364905	0.182453	−	0.983215i	$-0.441596\pi$
$751$	10.0000	0.364905	0.182453	+	0.983215i	$0.441596\pi$
$752$	12.0000	0.437595
$753$	8.00000	0.291536
$754$	18.0000	0.655521
$755$	−15.0000	−0.545906
$756$	−20.0000	−0.727393
$757$	47.0000	1.70824	0.854122	−	0.520073i	$-0.174095\pi$
$757$	47.0000	1.70824	0.854122	+	0.520073i	$0.174095\pi$
$758$	22.0000	0.799076
$759$	30.0000	1.08893
$760$	0	0
$761$	18.0000	0.652499	0.326250	−	0.945284i	$-0.394215\pi$
$761$	18.0000	0.652499	0.326250	+	0.945284i	$0.394215\pi$
$762$	12.0000	0.434714
$763$	25.0000	0.905061
$764$	1.00000	0.0361787
$765$	0	0
$766$	33.0000	1.19234
$767$	27.0000	0.974913
$768$	−2.00000	−0.0721688
$769$	24.0000	0.865462	0.432731	−	0.901523i	$-0.357550\pi$
$769$	24.0000	0.865462	0.432731	+	0.901523i	$0.357550\pi$
$770$	−15.0000	−0.540562
$771$	−4.00000	−0.144056
$772$	0	0
$773$	−6.00000	−0.215805	−0.107903	−	0.994161i	$-0.534413\pi$
$773$	−6.00000	−0.215805	−0.107903	+	0.994161i	$0.534413\pi$
$774$	8.00000	0.287554
$775$	0	0
$776$	−2.00000	−0.0717958
$777$	−120.000	−4.30498
$778$	−2.00000	−0.0717035
$779$	0	0
$780$	6.00000	0.214834
$781$	48.0000	1.71758
$782$	0	0
$783$	24.0000	0.857690
$784$	18.0000	0.642857
$785$	−8.00000	−0.285532
$786$	−8.00000	−0.285351
$787$	−32.0000	−1.14068	−0.570338	−	0.821410i	$-0.693188\pi$
$787$	−32.0000	−1.14068	−0.570338	+	0.821410i	$0.693188\pi$
$788$	−7.00000	−0.249365
$789$	−24.0000	−0.854423
$790$	−1.00000	−0.0355784
$791$	−5.00000	−0.177780
$792$	3.00000	0.106600
$793$	3.00000	0.106533
$794$	−4.00000	−0.141955
$795$	4.00000	0.141865
$796$	26.0000	0.921546
$797$	−3.00000	−0.106265	−0.0531327	−	0.998587i	$-0.516921\pi$
$797$	−3.00000	−0.106265	−0.0531327	+	0.998587i	$0.516921\pi$
$798$	0	0
$799$	0	0
$800$	4.00000	0.141421
$801$	12.0000	0.423999
$802$	0	0
$803$	9.00000	0.317603
$804$	−14.0000	−0.493742
$805$	−25.0000	−0.881134
$806$	0	0
$807$	20.0000	0.704033
$808$	12.0000	0.422159
$809$	−31.0000	−1.08990	−0.544951	−	0.838468i	$-0.683452\pi$
$809$	−31.0000	−1.08990	−0.544951	+	0.838468i	$0.683452\pi$
$810$	11.0000	0.386501
$811$	−7.00000	−0.245803	−0.122902	−	0.992419i	$-0.539220\pi$
$811$	−7.00000	−0.245803	−0.122902	+	0.992419i	$0.539220\pi$
$812$	−30.0000	−1.05279
$813$	−44.0000	−1.54315
$814$	−36.0000	−1.26180
$815$	−22.0000	−0.770626
$816$	0	0
$817$	0	0
$818$	10.0000	0.349642
$819$	15.0000	0.524142
$820$	−3.00000	−0.104765
$821$	4.00000	0.139601	0.0698005	−	0.997561i	$-0.477764\pi$
$821$	4.00000	0.139601	0.0698005	+	0.997561i	$0.477764\pi$
$822$	−14.0000	−0.488306
$823$	52.0000	1.81261	0.906303	−	0.422628i	$-0.138892\pi$
$823$	52.0000	1.81261	0.906303	+	0.422628i	$0.138892\pi$
$824$	−16.0000	−0.557386
$825$	−24.0000	−0.835573
$826$	−45.0000	−1.56575
$827$	28.0000	0.973655	0.486828	−	0.873498i	$-0.338154\pi$
$827$	28.0000	0.973655	0.486828	+	0.873498i	$0.338154\pi$
$828$	5.00000	0.173762
$829$	−1.00000	−0.0347314	−0.0173657	−	0.999849i	$-0.505528\pi$
$829$	−1.00000	−0.0347314	−0.0173657	+	0.999849i	$0.505528\pi$
$830$	12.0000	0.416526
$831$	36.0000	1.24883
$832$	−3.00000	−0.104006
$833$	0	0
$834$	−26.0000	−0.900306
$835$	8.00000	0.276851
$836$	0	0
$837$	0	0
$838$	12.0000	0.414533
$839$	6.00000	0.207143	0.103572	−	0.994622i	$-0.466973\pi$
$839$	6.00000	0.207143	0.103572	+	0.994622i	$0.466973\pi$
$840$	−10.0000	−0.345033
$841$	7.00000	0.241379
$842$	−20.0000	−0.689246
$843$	−16.0000	−0.551069
$844$	−12.0000	−0.413057
$845$	−4.00000	−0.137604
$846$	−12.0000	−0.412568
$847$	10.0000	0.343604
$848$	−2.00000	−0.0686803
$849$	−12.0000	−0.411839
$850$	0	0
$851$	−60.0000	−2.05677
$852$	32.0000	1.09630
$853$	33.0000	1.12990	0.564949	−	0.825126i	$-0.308896\pi$
$853$	33.0000	1.12990	0.564949	+	0.825126i	$0.308896\pi$
$854$	−5.00000	−0.171096
$855$	0	0
$856$	6.00000	0.205076
$857$	37.0000	1.26390	0.631948	−	0.775011i	$-0.282256\pi$
$857$	37.0000	1.26390	0.631948	+	0.775011i	$0.282256\pi$
$858$	18.0000	0.614510
$859$	−10.0000	−0.341196	−0.170598	−	0.985341i	$-0.554570\pi$
$859$	−10.0000	−0.341196	−0.170598	+	0.985341i	$0.554570\pi$
$860$	−8.00000	−0.272798
$861$	−30.0000	−1.02240
$862$	22.0000	0.749323
$863$	−32.0000	−1.08929	−0.544646	−	0.838666i	$-0.683336\pi$
$863$	−32.0000	−1.08929	−0.544646	+	0.838666i	$0.683336\pi$
$864$	−4.00000	−0.136083
$865$	18.0000	0.612018
$866$	−16.0000	−0.543702
$867$	34.0000	1.15470
$868$	0	0
$869$	−3.00000	−0.101768
$870$	12.0000	0.406838
$871$	−21.0000	−0.711558
$872$	5.00000	0.169321
$873$	2.00000	0.0676897
$874$	0	0
$875$	45.0000	1.52128
$876$	6.00000	0.202721
$877$	−52.0000	−1.75592	−0.877958	−	0.478738i	$-0.841094\pi$
$877$	−52.0000	−1.75592	−0.877958	+	0.478738i	$0.841094\pi$
$878$	36.0000	1.21494
$879$	−52.0000	−1.75392
$880$	−3.00000	−0.101130
$881$	35.0000	1.17918	0.589590	−	0.807703i	$-0.299289\pi$
$881$	35.0000	1.17918	0.589590	+	0.807703i	$0.299289\pi$
$882$	−18.0000	−0.606092
$883$	31.0000	1.04323	0.521617	−	0.853180i	$-0.325329\pi$
$883$	31.0000	1.04323	0.521617	+	0.853180i	$0.325329\pi$
$884$	0	0
$885$	18.0000	0.605063
$886$	12.0000	0.403148
$887$	8.00000	0.268614	0.134307	−	0.990940i	$-0.457119\pi$
$887$	8.00000	0.268614	0.134307	+	0.990940i	$0.457119\pi$
$888$	−24.0000	−0.805387
$889$	−30.0000	−1.00617
$890$	−12.0000	−0.402241
$891$	33.0000	1.10554
$892$	13.0000	0.435272
$893$	0	0
$894$	30.0000	1.00335
$895$	18.0000	0.601674
$896$	5.00000	0.167038
$897$	30.0000	1.00167
$898$	−21.0000	−0.700779
$899$	0	0
$900$	−4.00000	−0.133333
$901$	0	0
$902$	−9.00000	−0.299667
$903$	−80.0000	−2.66223
$904$	−1.00000	−0.0332595
$905$	0	0
$906$	−30.0000	−0.996683
$907$	−4.00000	−0.132818	−0.0664089	−	0.997792i	$-0.521154\pi$
$907$	−4.00000	−0.132818	−0.0664089	+	0.997792i	$0.521154\pi$
$908$	11.0000	0.365048
$909$	−12.0000	−0.398015
$910$	−15.0000	−0.497245
$911$	−14.0000	−0.463841	−0.231920	−	0.972735i	$-0.574501\pi$
$911$	−14.0000	−0.463841	−0.231920	+	0.972735i	$0.574501\pi$
$912$	0	0
$913$	36.0000	1.19143
$914$	16.0000	0.529233
$915$	2.00000	0.0661180
$916$	21.0000	0.693860
$917$	20.0000	0.660458
$918$	0	0
$919$	−16.0000	−0.527791	−0.263896	−	0.964551i	$-0.585007\pi$
$919$	−16.0000	−0.527791	−0.263896	+	0.964551i	$0.585007\pi$
$920$	−5.00000	−0.164845
$921$	−38.0000	−1.25214
$922$	−14.0000	−0.461065
$923$	48.0000	1.57994
$924$	−30.0000	−0.986928
$925$	48.0000	1.57823
$926$	−4.00000	−0.131448
$927$	16.0000	0.525509
$928$	−6.00000	−0.196960
$929$	−11.0000	−0.360898	−0.180449	−	0.983584i	$-0.557755\pi$
$929$	−11.0000	−0.360898	−0.180449	+	0.983584i	$0.557755\pi$
$930$	0	0
$931$	0	0
$932$	−10.0000	−0.327561
$933$	−6.00000	−0.196431
$934$	−13.0000	−0.425373
$935$	0	0
$936$	3.00000	0.0980581
$937$	−37.0000	−1.20874	−0.604369	−	0.796705i	$-0.706575\pi$
$937$	−37.0000	−1.20874	−0.604369	+	0.796705i	$0.706575\pi$
$938$	35.0000	1.14279
$939$	28.0000	0.913745
$940$	12.0000	0.391397
$941$	−34.0000	−1.10837	−0.554184	−	0.832394i	$-0.686970\pi$
$941$	−34.0000	−1.10837	−0.554184	+	0.832394i	$0.686970\pi$
$942$	−16.0000	−0.521308
$943$	−15.0000	−0.488467
$944$	−9.00000	−0.292925
$945$	−20.0000	−0.650600
$946$	−24.0000	−0.780307
$947$	−23.0000	−0.747400	−0.373700	−	0.927550i	$-0.621911\pi$
$947$	−23.0000	−0.747400	−0.373700	+	0.927550i	$0.621911\pi$
$948$	−2.00000	−0.0649570
$949$	9.00000	0.292152
$950$	0	0
$951$	−36.0000	−1.16738
$952$	0	0
$953$	58.0000	1.87880	0.939402	−	0.342817i	$-0.111381\pi$
$953$	58.0000	1.87880	0.939402	+	0.342817i	$0.111381\pi$
$954$	2.00000	0.0647524
$955$	1.00000	0.0323592
$956$	−6.00000	−0.194054
$957$	36.0000	1.16371
$958$	14.0000	0.452319
$959$	35.0000	1.13021
$960$	−2.00000	−0.0645497
$961$	−31.0000	−1.00000
$962$	−36.0000	−1.16069
$963$	−6.00000	−0.193347
$964$	−11.0000	−0.354286
$965$	0	0
$966$	−50.0000	−1.60872
$967$	−18.0000	−0.578841	−0.289420	−	0.957202i	$-0.593463\pi$
$967$	−18.0000	−0.578841	−0.289420	+	0.957202i	$0.593463\pi$
$968$	2.00000	0.0642824
$969$	0	0
$970$	−2.00000	−0.0642161
$971$	−34.0000	−1.09111	−0.545556	−	0.838074i	$-0.683681\pi$
$971$	−34.0000	−1.09111	−0.545556	+	0.838074i	$0.683681\pi$
$972$	10.0000	0.320750
$973$	65.0000	2.08380
$974$	8.00000	0.256337
$975$	−24.0000	−0.768615
$976$	−1.00000	−0.0320092
$977$	−42.0000	−1.34370	−0.671850	−	0.740688i	$-0.734500\pi$
$977$	−42.0000	−1.34370	−0.671850	+	0.740688i	$0.734500\pi$
$978$	−44.0000	−1.40696
$979$	−36.0000	−1.15056
$980$	18.0000	0.574989
$981$	−5.00000	−0.159638
$982$	8.00000	0.255290
$983$	−36.0000	−1.14822	−0.574111	−	0.818778i	$-0.694652\pi$
$983$	−36.0000	−1.14822	−0.574111	+	0.818778i	$0.694652\pi$
$984$	−6.00000	−0.191273
$985$	−7.00000	−0.223039
$986$	0	0
$987$	120.000	3.81964
$988$	0	0
$989$	−40.0000	−1.27193
$990$	3.00000	0.0953463
$991$	32.0000	1.01651	0.508257	−	0.861206i	$-0.330290\pi$
$991$	32.0000	1.01651	0.508257	+	0.861206i	$0.330290\pi$
$992$	0	0
$993$	14.0000	0.444277
$994$	−80.0000	−2.53745
$995$	26.0000	0.824255
$996$	24.0000	0.760469
$997$	−28.0000	−0.886769	−0.443384	−	0.896332i	$-0.646222\pi$
$997$	−28.0000	−0.886769	−0.443384	+	0.896332i	$0.646222\pi$
$998$	37.0000	1.17121
$999$	−48.0000	−1.51865

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	122.2.a.a.1.1	✓	1
3.2	odd	2		1098.2.a.h.1.1		1
4.3	odd	2		976.2.a.c.1.1		1
5.2	odd	4		3050.2.b.e.1099.1		2
5.3	odd	4		3050.2.b.e.1099.2		2
5.4	even	2		3050.2.a.k.1.1		1
7.6	odd	2		5978.2.a.g.1.1		1
8.3	odd	2		3904.2.a.a.1.1		1
8.5	even	2		3904.2.a.h.1.1		1
12.11	even	2		8784.2.a.j.1.1		1
61.60	even	2		7442.2.a.b.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
122.2.a.a.1.1	✓	1	1.1	even	1	trivial
976.2.a.c.1.1		1	4.3	odd	2
1098.2.a.h.1.1		1	3.2	odd	2
3050.2.a.k.1.1		1	5.4	even	2
3050.2.b.e.1099.1		2	5.2	odd	4
3050.2.b.e.1099.2		2	5.3	odd	4
3904.2.a.a.1.1		1	8.3	odd	2
3904.2.a.h.1.1		1	8.5	even	2
5978.2.a.g.1.1		1	7.6	odd	2
7442.2.a.b.1.1		1	61.60	even	2
8784.2.a.j.1.1		1	12.11	even	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:	\( N \)	\(=\)	\( 122 = 2 \cdot 61 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	122.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more