LMFDB - Embedded newform 195.2.a.d.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [195,2,Mod(1,195)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("195.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(195, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 195 = 3 \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	195.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,2,1,2,1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$1.55708283941$
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	$\chi$	$=$	195.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

\(q+2.00000 q^{2} +1.00000 q^{3} +2.00000 q^{4} +1.00000 q^{5} +2.00000 q^{6} -3.00000 q^{7} +1.00000 q^{9} +2.00000 q^{10} -5.00000 q^{11} +2.00000 q^{12} +1.00000 q^{13} -6.00000 q^{14} +1.00000 q^{15} -4.00000 q^{16} +5.00000 q^{17} +2.00000 q^{18} +2.00000 q^{19} +2.00000 q^{20} -3.00000 q^{21} -10.0000 q^{22} -1.00000 q^{23} +1.00000 q^{25} +2.00000 q^{26} +1.00000 q^{27} -6.00000 q^{28} +10.0000 q^{29} +2.00000 q^{30} -2.00000 q^{31} -8.00000 q^{32} -5.00000 q^{33} +10.0000 q^{34} -3.00000 q^{35} +2.00000 q^{36} -3.00000 q^{37} +4.00000 q^{38} +1.00000 q^{39} -9.00000 q^{41} -6.00000 q^{42} -4.00000 q^{43} -10.0000 q^{44} +1.00000 q^{45} -2.00000 q^{46} +10.0000 q^{47} -4.00000 q^{48} +2.00000 q^{49} +2.00000 q^{50} +5.00000 q^{51} +2.00000 q^{52} +9.00000 q^{53} +2.00000 q^{54} -5.00000 q^{55} +2.00000 q^{57} +20.0000 q^{58} +2.00000 q^{60} -11.0000 q^{61} -4.00000 q^{62} -3.00000 q^{63} -8.00000 q^{64} +1.00000 q^{65} -10.0000 q^{66} -4.00000 q^{67} +10.0000 q^{68} -1.00000 q^{69} -6.00000 q^{70} +15.0000 q^{71} +6.00000 q^{73} -6.00000 q^{74} +1.00000 q^{75} +4.00000 q^{76} +15.0000 q^{77} +2.00000 q^{78} -11.0000 q^{79} -4.00000 q^{80} +1.00000 q^{81} -18.0000 q^{82} +8.00000 q^{83} -6.00000 q^{84} +5.00000 q^{85} -8.00000 q^{86} +10.0000 q^{87} -11.0000 q^{89} +2.00000 q^{90} -3.00000 q^{91} -2.00000 q^{92} -2.00000 q^{93} +20.0000 q^{94} +2.00000 q^{95} -8.00000 q^{96} -9.00000 q^{97} +4.00000 q^{98} -5.00000 q^{99} +O(q^{100})\)

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 $

$2$	2.00000	1.41421	0.707107	−	0.707107i	$-0.250000\pi$
$2$	2.00000	1.41421	0.707107	+	0.707107i	$0.250000\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	2.00000	1.00000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	2.00000	0.816497
$7$	−3.00000	−1.13389	−0.566947	−	0.823754i	$-0.691875\pi$
$7$	−3.00000	−1.13389	−0.566947	+	0.823754i	$0.691875\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	2.00000	0.632456
$11$	−5.00000	−1.50756	−0.753778	−	0.657129i	$-0.771771\pi$
$11$	−5.00000	−1.50756	−0.753778	+	0.657129i	$0.771771\pi$
$12$	2.00000	0.577350
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	−6.00000	−1.60357
$15$	1.00000	0.258199
$16$	−4.00000	−1.00000
$17$	5.00000	1.21268	0.606339	−	0.795206i	$-0.292637\pi$
$17$	5.00000	1.21268	0.606339	+	0.795206i	$0.292637\pi$
$18$	2.00000	0.471405
$19$	2.00000	0.458831	0.229416	−	0.973329i	$-0.426318\pi$
$19$	2.00000	0.458831	0.229416	+	0.973329i	$0.426318\pi$
$20$	2.00000	0.447214
$21$	−3.00000	−0.654654
$22$	−10.0000	−2.13201
$23$	−1.00000	−0.208514	−0.104257	−	0.994550i	$-0.533247\pi$
$23$	−1.00000	−0.208514	−0.104257	+	0.994550i	$0.533247\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	2.00000	0.392232
$27$	1.00000	0.192450
$28$	−6.00000	−1.13389
$29$	10.0000	1.85695	0.928477	−	0.371391i	$-0.121119\pi$
$29$	10.0000	1.85695	0.928477	+	0.371391i	$0.121119\pi$
$30$	2.00000	0.365148
$31$	−2.00000	−0.359211	−0.179605	−	0.983739i	$-0.557482\pi$
$31$	−2.00000	−0.359211	−0.179605	+	0.983739i	$0.557482\pi$
$32$	−8.00000	−1.41421
$33$	−5.00000	−0.870388
$34$	10.0000	1.71499
$35$	−3.00000	−0.507093
$36$	2.00000	0.333333
$37$	−3.00000	−0.493197	−0.246598	−	0.969118i	$-0.579313\pi$
$37$	−3.00000	−0.493197	−0.246598	+	0.969118i	$0.579313\pi$
$38$	4.00000	0.648886
$39$	1.00000	0.160128
$40$	0	0
$41$	−9.00000	−1.40556	−0.702782	−	0.711405i	$-0.748059\pi$
$41$	−9.00000	−1.40556	−0.702782	+	0.711405i	$0.748059\pi$
$42$	−6.00000	−0.925820
$43$	−4.00000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	−10.0000	−1.50756
$45$	1.00000	0.149071
$46$	−2.00000	−0.294884
$47$	10.0000	1.45865	0.729325	−	0.684167i	$-0.239834\pi$
$47$	10.0000	1.45865	0.729325	+	0.684167i	$0.239834\pi$
$48$	−4.00000	−0.577350
$49$	2.00000	0.285714
$50$	2.00000	0.282843
$51$	5.00000	0.700140
$52$	2.00000	0.277350
$53$	9.00000	1.23625	0.618123	−	0.786082i	$-0.287894\pi$
$53$	9.00000	1.23625	0.618123	+	0.786082i	$0.287894\pi$
$54$	2.00000	0.272166
$55$	−5.00000	−0.674200
$56$	0	0
$57$	2.00000	0.264906
$58$	20.0000	2.62613
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	2.00000	0.258199
$61$	−11.0000	−1.40841	−0.704203	−	0.709999i	$-0.748695\pi$
$61$	−11.0000	−1.40841	−0.704203	+	0.709999i	$0.748695\pi$
$62$	−4.00000	−0.508001
$63$	−3.00000	−0.377964
$64$	−8.00000	−1.00000
$65$	1.00000	0.124035
$66$	−10.0000	−1.23091
$67$	−4.00000	−0.488678	−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000	−0.488678	−0.244339	+	0.969690i	$0.578571\pi$
$68$	10.0000	1.21268
$69$	−1.00000	−0.120386
$70$	−6.00000	−0.717137
$71$	15.0000	1.78017	0.890086	−	0.455792i	$-0.150644\pi$
$71$	15.0000	1.78017	0.890086	+	0.455792i	$0.150644\pi$
$72$	0	0
$73$	6.00000	0.702247	0.351123	−	0.936329i	$-0.385800\pi$
$73$	6.00000	0.702247	0.351123	+	0.936329i	$0.385800\pi$
$74$	−6.00000	−0.697486
$75$	1.00000	0.115470
$76$	4.00000	0.458831
$77$	15.0000	1.70941
$78$	2.00000	0.226455
$79$	−11.0000	−1.23760	−0.618798	−	0.785550i	$-0.712380\pi$
$79$	−11.0000	−1.23760	−0.618798	+	0.785550i	$0.712380\pi$
$80$	−4.00000	−0.447214
$81$	1.00000	0.111111
$82$	−18.0000	−1.98777
$83$	8.00000	0.878114	0.439057	−	0.898459i	$-0.355313\pi$
$83$	8.00000	0.878114	0.439057	+	0.898459i	$0.355313\pi$
$84$	−6.00000	−0.654654
$85$	5.00000	0.542326
$86$	−8.00000	−0.862662
$87$	10.0000	1.07211
$88$	0	0
$89$	−11.0000	−1.16600	−0.582999	−	0.812473i	$-0.698121\pi$
$89$	−11.0000	−1.16600	−0.582999	+	0.812473i	$0.698121\pi$
$90$	2.00000	0.210819
$91$	−3.00000	−0.314485
$92$	−2.00000	−0.208514
$93$	−2.00000	−0.207390
$94$	20.0000	2.06284
$95$	2.00000	0.205196
$96$	−8.00000	−0.816497
$97$	−9.00000	−0.913812	−0.456906	−	0.889515i	$-0.651042\pi$
$97$	−9.00000	−0.913812	−0.456906	+	0.889515i	$0.651042\pi$
$98$	4.00000	0.404061
$99$	−5.00000	−0.502519
$100$	2.00000	0.200000
$101$	−12.0000	−1.19404	−0.597022	−	0.802225i	$-0.703650\pi$
$101$	−12.0000	−1.19404	−0.597022	+	0.802225i	$0.703650\pi$
$102$	10.0000	0.990148
$103$	4.00000	0.394132	0.197066	−	0.980390i	$-0.436859\pi$
$103$	4.00000	0.394132	0.197066	+	0.980390i	$0.436859\pi$
$104$	0	0
$105$	−3.00000	−0.292770
$106$	18.0000	1.74831
$107$	3.00000	0.290021	0.145010	−	0.989430i	$-0.453678\pi$
$107$	3.00000	0.290021	0.145010	+	0.989430i	$0.453678\pi$
$108$	2.00000	0.192450
$109$	16.0000	1.53252	0.766261	−	0.642529i	$-0.222115\pi$
$109$	16.0000	1.53252	0.766261	+	0.642529i	$0.222115\pi$
$110$	−10.0000	−0.953463
$111$	−3.00000	−0.284747
$112$	12.0000	1.13389
$113$	2.00000	0.188144	0.0940721	−	0.995565i	$-0.470012\pi$
$113$	2.00000	0.188144	0.0940721	+	0.995565i	$0.470012\pi$
$114$	4.00000	0.374634
$115$	−1.00000	−0.0932505
$116$	20.0000	1.85695
$117$	1.00000	0.0924500
$118$	0	0
$119$	−15.0000	−1.37505
$120$	0	0
$121$	14.0000	1.27273
$122$	−22.0000	−1.99179
$123$	−9.00000	−0.811503
$124$	−4.00000	−0.359211
$125$	1.00000	0.0894427
$126$	−6.00000	−0.534522
$127$	14.0000	1.24230	0.621150	−	0.783692i	$-0.286666\pi$
$127$	14.0000	1.24230	0.621150	+	0.783692i	$0.286666\pi$
$128$	0	0
$129$	−4.00000	−0.352180
$130$	2.00000	0.175412
$131$	6.00000	0.524222	0.262111	−	0.965038i	$-0.415581\pi$
$131$	6.00000	0.524222	0.262111	+	0.965038i	$0.415581\pi$
$132$	−10.0000	−0.870388
$133$	−6.00000	−0.520266
$134$	−8.00000	−0.691095
$135$	1.00000	0.0860663
$136$	0	0
$137$	6.00000	0.512615	0.256307	−	0.966595i	$-0.417494\pi$
$137$	6.00000	0.512615	0.256307	+	0.966595i	$0.417494\pi$
$138$	−2.00000	−0.170251
$139$	−17.0000	−1.44192	−0.720961	−	0.692976i	$-0.756299\pi$
$139$	−17.0000	−1.44192	−0.720961	+	0.692976i	$0.756299\pi$
$140$	−6.00000	−0.507093
$141$	10.0000	0.842152
$142$	30.0000	2.51754
$143$	−5.00000	−0.418121
$144$	−4.00000	−0.333333
$145$	10.0000	0.830455
$146$	12.0000	0.993127
$147$	2.00000	0.164957
$148$	−6.00000	−0.493197
$149$	−7.00000	−0.573462	−0.286731	−	0.958011i	$-0.592569\pi$
$149$	−7.00000	−0.573462	−0.286731	+	0.958011i	$0.592569\pi$
$150$	2.00000	0.163299
$151$	−12.0000	−0.976546	−0.488273	−	0.872691i	$-0.662373\pi$
$151$	−12.0000	−0.976546	−0.488273	+	0.872691i	$0.662373\pi$
$152$	0	0
$153$	5.00000	0.404226
$154$	30.0000	2.41747
$155$	−2.00000	−0.160644
$156$	2.00000	0.160128
$157$	22.0000	1.75579	0.877896	−	0.478852i	$-0.158947\pi$
$157$	22.0000	1.75579	0.877896	+	0.478852i	$0.158947\pi$
$158$	−22.0000	−1.75023
$159$	9.00000	0.713746
$160$	−8.00000	−0.632456
$161$	3.00000	0.236433
$162$	2.00000	0.157135
$163$	−11.0000	−0.861586	−0.430793	−	0.902451i	$-0.641766\pi$
$163$	−11.0000	−0.861586	−0.430793	+	0.902451i	$0.641766\pi$
$164$	−18.0000	−1.40556
$165$	−5.00000	−0.389249
$166$	16.0000	1.24184
$167$	−8.00000	−0.619059	−0.309529	−	0.950890i	$-0.600171\pi$
$167$	−8.00000	−0.619059	−0.309529	+	0.950890i	$0.600171\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	10.0000	0.766965
$171$	2.00000	0.152944
$172$	−8.00000	−0.609994
$173$	−2.00000	−0.152057	−0.0760286	−	0.997106i	$-0.524224\pi$
$173$	−2.00000	−0.152057	−0.0760286	+	0.997106i	$0.524224\pi$
$174$	20.0000	1.51620
$175$	−3.00000	−0.226779
$176$	20.0000	1.50756
$177$	0	0
$178$	−22.0000	−1.64897
$179$	−6.00000	−0.448461	−0.224231	−	0.974536i	$-0.571987\pi$
$179$	−6.00000	−0.448461	−0.224231	+	0.974536i	$0.571987\pi$
$180$	2.00000	0.149071
$181$	−23.0000	−1.70958	−0.854788	−	0.518977i	$-0.826313\pi$
$181$	−23.0000	−1.70958	−0.854788	+	0.518977i	$0.826313\pi$
$182$	−6.00000	−0.444750
$183$	−11.0000	−0.813143
$184$	0	0
$185$	−3.00000	−0.220564
$186$	−4.00000	−0.293294
$187$	−25.0000	−1.82818
$188$	20.0000	1.45865
$189$	−3.00000	−0.218218
$190$	4.00000	0.290191
$191$	−20.0000	−1.44715	−0.723575	−	0.690246i	$-0.757502\pi$
$191$	−20.0000	−1.44715	−0.723575	+	0.690246i	$0.757502\pi$
$192$	−8.00000	−0.577350
$193$	13.0000	0.935760	0.467880	−	0.883792i	$-0.345018\pi$
$193$	13.0000	0.935760	0.467880	+	0.883792i	$0.345018\pi$
$194$	−18.0000	−1.29232
$195$	1.00000	0.0716115
$196$	4.00000	0.285714
$197$	−12.0000	−0.854965	−0.427482	−	0.904024i	$-0.640599\pi$
$197$	−12.0000	−0.854965	−0.427482	+	0.904024i	$0.640599\pi$
$198$	−10.0000	−0.710669
$199$	4.00000	0.283552	0.141776	−	0.989899i	$-0.454719\pi$
$199$	4.00000	0.283552	0.141776	+	0.989899i	$0.454719\pi$
$200$	0	0
$201$	−4.00000	−0.282138
$202$	−24.0000	−1.68863
$203$	−30.0000	−2.10559
$204$	10.0000	0.700140
$205$	−9.00000	−0.628587
$206$	8.00000	0.557386
$207$	−1.00000	−0.0695048
$208$	−4.00000	−0.277350
$209$	−10.0000	−0.691714
$210$	−6.00000	−0.414039
$211$	−4.00000	−0.275371	−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000	−0.275371	−0.137686	+	0.990476i	$0.543966\pi$
$212$	18.0000	1.23625
$213$	15.0000	1.02778
$214$	6.00000	0.410152
$215$	−4.00000	−0.272798
$216$	0	0
$217$	6.00000	0.407307
$218$	32.0000	2.16731
$219$	6.00000	0.405442
$220$	−10.0000	−0.674200
$221$	5.00000	0.336336
$222$	−6.00000	−0.402694
$223$	0	0		−	1.00000i	$-0.5\pi$
$223$	0	0			1.00000i	$0.5\pi$
$224$	24.0000	1.60357
$225$	1.00000	0.0666667
$226$	4.00000	0.266076
$227$	−18.0000	−1.19470	−0.597351	−	0.801980i	$-0.703780\pi$
$227$	−18.0000	−1.19470	−0.597351	+	0.801980i	$0.703780\pi$
$228$	4.00000	0.264906
$229$	−14.0000	−0.925146	−0.462573	−	0.886581i	$-0.653074\pi$
$229$	−14.0000	−0.925146	−0.462573	+	0.886581i	$0.653074\pi$
$230$	−2.00000	−0.131876
$231$	15.0000	0.986928
$232$	0	0
$233$	−25.0000	−1.63780	−0.818902	−	0.573933i	$-0.805417\pi$
$233$	−25.0000	−1.63780	−0.818902	+	0.573933i	$0.805417\pi$
$234$	2.00000	0.130744
$235$	10.0000	0.652328
$236$	0	0
$237$	−11.0000	−0.714527
$238$	−30.0000	−1.94461
$239$	15.0000	0.970269	0.485135	−	0.874439i	$-0.338771\pi$
$239$	15.0000	0.970269	0.485135	+	0.874439i	$0.338771\pi$
$240$	−4.00000	−0.258199
$241$	−14.0000	−0.901819	−0.450910	−	0.892570i	$-0.648900\pi$
$241$	−14.0000	−0.901819	−0.450910	+	0.892570i	$0.648900\pi$
$242$	28.0000	1.79991
$243$	1.00000	0.0641500
$244$	−22.0000	−1.40841
$245$	2.00000	0.127775
$246$	−18.0000	−1.14764
$247$	2.00000	0.127257
$248$	0	0
$249$	8.00000	0.506979
$250$	2.00000	0.126491
$251$	20.0000	1.26239	0.631194	−	0.775625i	$-0.282565\pi$
$251$	20.0000	1.26239	0.631194	+	0.775625i	$0.282565\pi$
$252$	−6.00000	−0.377964
$253$	5.00000	0.314347
$254$	28.0000	1.75688
$255$	5.00000	0.313112
$256$	16.0000	1.00000
$257$	18.0000	1.12281	0.561405	−	0.827541i	$-0.310261\pi$
$257$	18.0000	1.12281	0.561405	+	0.827541i	$0.310261\pi$
$258$	−8.00000	−0.498058
$259$	9.00000	0.559233
$260$	2.00000	0.124035
$261$	10.0000	0.618984
$262$	12.0000	0.741362
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	0	0
$265$	9.00000	0.552866
$266$	−12.0000	−0.735767
$267$	−11.0000	−0.673189
$268$	−8.00000	−0.488678
$269$	32.0000	1.95107	0.975537	−	0.219834i	$-0.0705517\pi$
$269$	32.0000	1.95107	0.975537	+	0.219834i	$0.0705517\pi$
$270$	2.00000	0.121716
$271$	−2.00000	−0.121491	−0.0607457	−	0.998153i	$-0.519348\pi$
$271$	−2.00000	−0.121491	−0.0607457	+	0.998153i	$0.519348\pi$
$272$	−20.0000	−1.21268
$273$	−3.00000	−0.181568
$274$	12.0000	0.724947
$275$	−5.00000	−0.301511
$276$	−2.00000	−0.120386
$277$	26.0000	1.56219	0.781094	−	0.624413i	$-0.214662\pi$
$277$	26.0000	1.56219	0.781094	+	0.624413i	$0.214662\pi$
$278$	−34.0000	−2.03918
$279$	−2.00000	−0.119737
$280$	0	0
$281$	−10.0000	−0.596550	−0.298275	−	0.954480i	$-0.596411\pi$
$281$	−10.0000	−0.596550	−0.298275	+	0.954480i	$0.596411\pi$
$282$	20.0000	1.19098
$283$	−8.00000	−0.475551	−0.237775	−	0.971320i	$-0.576418\pi$
$283$	−8.00000	−0.475551	−0.237775	+	0.971320i	$0.576418\pi$
$284$	30.0000	1.78017
$285$	2.00000	0.118470
$286$	−10.0000	−0.591312
$287$	27.0000	1.59376
$288$	−8.00000	−0.471405
$289$	8.00000	0.470588
$290$	20.0000	1.17444
$291$	−9.00000	−0.527589
$292$	12.0000	0.702247
$293$	24.0000	1.40209	0.701047	−	0.713115i	$-0.252716\pi$
$293$	24.0000	1.40209	0.701047	+	0.713115i	$0.252716\pi$
$294$	4.00000	0.233285
$295$	0	0
$296$	0	0
$297$	−5.00000	−0.290129
$298$	−14.0000	−0.810998
$299$	−1.00000	−0.0578315
$300$	2.00000	0.115470
$301$	12.0000	0.691669
$302$	−24.0000	−1.38104
$303$	−12.0000	−0.689382
$304$	−8.00000	−0.458831
$305$	−11.0000	−0.629858
$306$	10.0000	0.571662
$307$	−19.0000	−1.08439	−0.542194	−	0.840254i	$-0.682406\pi$
$307$	−19.0000	−1.08439	−0.542194	+	0.840254i	$0.682406\pi$
$308$	30.0000	1.70941
$309$	4.00000	0.227552
$310$	−4.00000	−0.227185
$311$	24.0000	1.36092	0.680458	−	0.732787i	$-0.261781\pi$
$311$	24.0000	1.36092	0.680458	+	0.732787i	$0.261781\pi$
$312$	0	0
$313$	−10.0000	−0.565233	−0.282617	−	0.959233i	$-0.591202\pi$
$313$	−10.0000	−0.565233	−0.282617	+	0.959233i	$0.591202\pi$
$314$	44.0000	2.48306
$315$	−3.00000	−0.169031
$316$	−22.0000	−1.23760
$317$	12.0000	0.673987	0.336994	−	0.941507i	$-0.390590\pi$
$317$	12.0000	0.673987	0.336994	+	0.941507i	$0.390590\pi$
$318$	18.0000	1.00939
$319$	−50.0000	−2.79946
$320$	−8.00000	−0.447214
$321$	3.00000	0.167444
$322$	6.00000	0.334367
$323$	10.0000	0.556415
$324$	2.00000	0.111111
$325$	1.00000	0.0554700
$326$	−22.0000	−1.21847
$327$	16.0000	0.884802
$328$	0	0
$329$	−30.0000	−1.65395
$330$	−10.0000	−0.550482
$331$	32.0000	1.75888	0.879440	−	0.476011i	$-0.157918\pi$
$331$	32.0000	1.75888	0.879440	+	0.476011i	$0.157918\pi$
$332$	16.0000	0.878114
$333$	−3.00000	−0.164399
$334$	−16.0000	−0.875481
$335$	−4.00000	−0.218543
$336$	12.0000	0.654654
$337$	4.00000	0.217894	0.108947	−	0.994048i	$-0.465252\pi$
$337$	4.00000	0.217894	0.108947	+	0.994048i	$0.465252\pi$
$338$	2.00000	0.108786
$339$	2.00000	0.108625
$340$	10.0000	0.542326
$341$	10.0000	0.541530
$342$	4.00000	0.216295
$343$	15.0000	0.809924
$344$	0	0
$345$	−1.00000	−0.0538382
$346$	−4.00000	−0.215041
$347$	−1.00000	−0.0536828	−0.0268414	−	0.999640i	$-0.508545\pi$
$347$	−1.00000	−0.0536828	−0.0268414	+	0.999640i	$0.508545\pi$
$348$	20.0000	1.07211
$349$	20.0000	1.07058	0.535288	−	0.844670i	$-0.320203\pi$
$349$	20.0000	1.07058	0.535288	+	0.844670i	$0.320203\pi$
$350$	−6.00000	−0.320713
$351$	1.00000	0.0533761
$352$	40.0000	2.13201
$353$	−14.0000	−0.745145	−0.372572	−	0.928003i	$-0.621524\pi$
$353$	−14.0000	−0.745145	−0.372572	+	0.928003i	$0.621524\pi$
$354$	0	0
$355$	15.0000	0.796117
$356$	−22.0000	−1.16600
$357$	−15.0000	−0.793884
$358$	−12.0000	−0.634220
$359$	16.0000	0.844448	0.422224	−	0.906492i	$-0.361250\pi$
$359$	16.0000	0.844448	0.422224	+	0.906492i	$0.361250\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	−46.0000	−2.41771
$363$	14.0000	0.734809
$364$	−6.00000	−0.314485
$365$	6.00000	0.314054
$366$	−22.0000	−1.14996
$367$	8.00000	0.417597	0.208798	−	0.977959i	$-0.433045\pi$
$367$	8.00000	0.417597	0.208798	+	0.977959i	$0.433045\pi$
$368$	4.00000	0.208514
$369$	−9.00000	−0.468521
$370$	−6.00000	−0.311925
$371$	−27.0000	−1.40177
$372$	−4.00000	−0.207390
$373$	16.0000	0.828449	0.414224	−	0.910175i	$-0.364053\pi$
$373$	16.0000	0.828449	0.414224	+	0.910175i	$0.364053\pi$
$374$	−50.0000	−2.58544
$375$	1.00000	0.0516398
$376$	0	0
$377$	10.0000	0.515026
$378$	−6.00000	−0.308607
$379$	6.00000	0.308199	0.154100	−	0.988055i	$-0.450752\pi$
$379$	6.00000	0.308199	0.154100	+	0.988055i	$0.450752\pi$
$380$	4.00000	0.205196
$381$	14.0000	0.717242
$382$	−40.0000	−2.04658
$383$	−18.0000	−0.919757	−0.459879	−	0.887982i	$-0.652107\pi$
$383$	−18.0000	−0.919757	−0.459879	+	0.887982i	$0.652107\pi$
$384$	0	0
$385$	15.0000	0.764471
$386$	26.0000	1.32337
$387$	−4.00000	−0.203331
$388$	−18.0000	−0.913812
$389$	−24.0000	−1.21685	−0.608424	−	0.793612i	$-0.708198\pi$
$389$	−24.0000	−1.21685	−0.608424	+	0.793612i	$0.708198\pi$
$390$	2.00000	0.101274
$391$	−5.00000	−0.252861
$392$	0	0
$393$	6.00000	0.302660
$394$	−24.0000	−1.20910
$395$	−11.0000	−0.553470
$396$	−10.0000	−0.502519
$397$	−19.0000	−0.953583	−0.476791	−	0.879017i	$-0.658200\pi$
$397$	−19.0000	−0.953583	−0.476791	+	0.879017i	$0.658200\pi$
$398$	8.00000	0.401004
$399$	−6.00000	−0.300376
$400$	−4.00000	−0.200000
$401$	−18.0000	−0.898877	−0.449439	−	0.893311i	$-0.648376\pi$
$401$	−18.0000	−0.898877	−0.449439	+	0.893311i	$0.648376\pi$
$402$	−8.00000	−0.399004
$403$	−2.00000	−0.0996271
$404$	−24.0000	−1.19404
$405$	1.00000	0.0496904
$406$	−60.0000	−2.97775
$407$	15.0000	0.743522
$408$	0	0
$409$	−26.0000	−1.28562	−0.642809	−	0.766027i	$-0.722231\pi$
$409$	−26.0000	−1.28562	−0.642809	+	0.766027i	$0.722231\pi$
$410$	−18.0000	−0.888957
$411$	6.00000	0.295958
$412$	8.00000	0.394132
$413$	0	0
$414$	−2.00000	−0.0982946
$415$	8.00000	0.392705
$416$	−8.00000	−0.392232
$417$	−17.0000	−0.832494
$418$	−20.0000	−0.978232
$419$	26.0000	1.27018	0.635092	−	0.772437i	$-0.280962\pi$
$419$	26.0000	1.27018	0.635092	+	0.772437i	$0.280962\pi$
$420$	−6.00000	−0.292770
$421$	32.0000	1.55958	0.779792	−	0.626038i	$-0.215325\pi$
$421$	32.0000	1.55958	0.779792	+	0.626038i	$0.215325\pi$
$422$	−8.00000	−0.389434
$423$	10.0000	0.486217
$424$	0	0
$425$	5.00000	0.242536
$426$	30.0000	1.45350
$427$	33.0000	1.59698
$428$	6.00000	0.290021
$429$	−5.00000	−0.241402
$430$	−8.00000	−0.385794
$431$	−16.0000	−0.770693	−0.385346	−	0.922772i	$-0.625918\pi$
$431$	−16.0000	−0.770693	−0.385346	+	0.922772i	$0.625918\pi$
$432$	−4.00000	−0.192450
$433$	24.0000	1.15337	0.576683	−	0.816968i	$-0.304347\pi$
$433$	24.0000	1.15337	0.576683	+	0.816968i	$0.304347\pi$
$434$	12.0000	0.576018
$435$	10.0000	0.479463
$436$	32.0000	1.53252
$437$	−2.00000	−0.0956730
$438$	12.0000	0.573382
$439$	−33.0000	−1.57500	−0.787502	−	0.616312i	$-0.788626\pi$
$439$	−33.0000	−1.57500	−0.787502	+	0.616312i	$0.788626\pi$
$440$	0	0
$441$	2.00000	0.0952381
$442$	10.0000	0.475651
$443$	35.0000	1.66290	0.831450	−	0.555599i	$-0.187511\pi$
$443$	35.0000	1.66290	0.831450	+	0.555599i	$0.187511\pi$
$444$	−6.00000	−0.284747
$445$	−11.0000	−0.521450
$446$	0	0
$447$	−7.00000	−0.331089
$448$	24.0000	1.13389
$449$	15.0000	0.707894	0.353947	−	0.935266i	$-0.384839\pi$
$449$	15.0000	0.707894	0.353947	+	0.935266i	$0.384839\pi$
$450$	2.00000	0.0942809
$451$	45.0000	2.11897
$452$	4.00000	0.188144
$453$	−12.0000	−0.563809
$454$	−36.0000	−1.68956
$455$	−3.00000	−0.140642
$456$	0	0
$457$	−13.0000	−0.608114	−0.304057	−	0.952654i	$-0.598341\pi$
$457$	−13.0000	−0.608114	−0.304057	+	0.952654i	$0.598341\pi$
$458$	−28.0000	−1.30835
$459$	5.00000	0.233380
$460$	−2.00000	−0.0932505
$461$	3.00000	0.139724	0.0698620	−	0.997557i	$-0.477744\pi$
$461$	3.00000	0.139724	0.0698620	+	0.997557i	$0.477744\pi$
$462$	30.0000	1.39573
$463$	5.00000	0.232370	0.116185	−	0.993228i	$-0.462933\pi$
$463$	5.00000	0.232370	0.116185	+	0.993228i	$0.462933\pi$
$464$	−40.0000	−1.85695
$465$	−2.00000	−0.0927478
$466$	−50.0000	−2.31621
$467$	−29.0000	−1.34196	−0.670980	−	0.741475i	$-0.734126\pi$
$467$	−29.0000	−1.34196	−0.670980	+	0.741475i	$0.734126\pi$
$468$	2.00000	0.0924500
$469$	12.0000	0.554109
$470$	20.0000	0.922531
$471$	22.0000	1.01371
$472$	0	0
$473$	20.0000	0.919601
$474$	−22.0000	−1.01049
$475$	2.00000	0.0917663
$476$	−30.0000	−1.37505
$477$	9.00000	0.412082
$478$	30.0000	1.37217
$479$	5.00000	0.228456	0.114228	−	0.993455i	$-0.463561\pi$
$479$	5.00000	0.228456	0.114228	+	0.993455i	$0.463561\pi$
$480$	−8.00000	−0.365148
$481$	−3.00000	−0.136788
$482$	−28.0000	−1.27537
$483$	3.00000	0.136505
$484$	28.0000	1.27273
$485$	−9.00000	−0.408669
$486$	2.00000	0.0907218
$487$	7.00000	0.317200	0.158600	−	0.987343i	$-0.449302\pi$
$487$	7.00000	0.317200	0.158600	+	0.987343i	$0.449302\pi$
$488$	0	0
$489$	−11.0000	−0.497437
$490$	4.00000	0.180702
$491$	16.0000	0.722070	0.361035	−	0.932552i	$-0.382424\pi$
$491$	16.0000	0.722070	0.361035	+	0.932552i	$0.382424\pi$
$492$	−18.0000	−0.811503
$493$	50.0000	2.25189
$494$	4.00000	0.179969
$495$	−5.00000	−0.224733
$496$	8.00000	0.359211
$497$	−45.0000	−2.01853
$498$	16.0000	0.716977
$499$	34.0000	1.52205	0.761025	−	0.648723i	$-0.224697\pi$
$499$	34.0000	1.52205	0.761025	+	0.648723i	$0.224697\pi$
$500$	2.00000	0.0894427
$501$	−8.00000	−0.357414
$502$	40.0000	1.78529
$503$	12.0000	0.535054	0.267527	−	0.963550i	$-0.413794\pi$
$503$	12.0000	0.535054	0.267527	+	0.963550i	$0.413794\pi$
$504$	0	0
$505$	−12.0000	−0.533993
$506$	10.0000	0.444554
$507$	1.00000	0.0444116
$508$	28.0000	1.24230
$509$	21.0000	0.930809	0.465404	−	0.885098i	$-0.345909\pi$
$509$	21.0000	0.930809	0.465404	+	0.885098i	$0.345909\pi$
$510$	10.0000	0.442807
$511$	−18.0000	−0.796273
$512$	32.0000	1.41421
$513$	2.00000	0.0883022
$514$	36.0000	1.58789
$515$	4.00000	0.176261
$516$	−8.00000	−0.352180
$517$	−50.0000	−2.19900
$518$	18.0000	0.790875
$519$	−2.00000	−0.0877903
$520$	0	0
$521$	−22.0000	−0.963837	−0.481919	−	0.876216i	$-0.660060\pi$
$521$	−22.0000	−0.963837	−0.481919	+	0.876216i	$0.660060\pi$
$522$	20.0000	0.875376
$523$	−20.0000	−0.874539	−0.437269	−	0.899331i	$-0.644054\pi$
$523$	−20.0000	−0.874539	−0.437269	+	0.899331i	$0.644054\pi$
$524$	12.0000	0.524222
$525$	−3.00000	−0.130931
$526$	0	0
$527$	−10.0000	−0.435607
$528$	20.0000	0.870388
$529$	−22.0000	−0.956522
$530$	18.0000	0.781870
$531$	0	0
$532$	−12.0000	−0.520266
$533$	−9.00000	−0.389833
$534$	−22.0000	−0.952033
$535$	3.00000	0.129701
$536$	0	0
$537$	−6.00000	−0.258919
$538$	64.0000	2.75924
$539$	−10.0000	−0.430730
$540$	2.00000	0.0860663
$541$	38.0000	1.63375	0.816874	−	0.576816i	$-0.195705\pi$
$541$	38.0000	1.63375	0.816874	+	0.576816i	$0.195705\pi$
$542$	−4.00000	−0.171815
$543$	−23.0000	−0.987024
$544$	−40.0000	−1.71499
$545$	16.0000	0.685365
$546$	−6.00000	−0.256776
$547$	0	0		−	1.00000i	$-0.5\pi$
$547$	0	0			1.00000i	$0.5\pi$
$548$	12.0000	0.512615
$549$	−11.0000	−0.469469
$550$	−10.0000	−0.426401
$551$	20.0000	0.852029
$552$	0	0
$553$	33.0000	1.40330
$554$	52.0000	2.20927
$555$	−3.00000	−0.127343
$556$	−34.0000	−1.44192
$557$	−30.0000	−1.27114	−0.635570	−	0.772043i	$-0.719235\pi$
$557$	−30.0000	−1.27114	−0.635570	+	0.772043i	$0.719235\pi$
$558$	−4.00000	−0.169334
$559$	−4.00000	−0.169182
$560$	12.0000	0.507093
$561$	−25.0000	−1.05550
$562$	−20.0000	−0.843649
$563$	−41.0000	−1.72794	−0.863972	−	0.503540i	$-0.832031\pi$
$563$	−41.0000	−1.72794	−0.863972	+	0.503540i	$0.832031\pi$
$564$	20.0000	0.842152
$565$	2.00000	0.0841406
$566$	−16.0000	−0.672530
$567$	−3.00000	−0.125988
$568$	0	0
$569$	16.0000	0.670755	0.335377	−	0.942084i	$-0.391136\pi$
$569$	16.0000	0.670755	0.335377	+	0.942084i	$0.391136\pi$
$570$	4.00000	0.167542
$571$	17.0000	0.711428	0.355714	−	0.934595i	$-0.384238\pi$
$571$	17.0000	0.711428	0.355714	+	0.934595i	$0.384238\pi$
$572$	−10.0000	−0.418121
$573$	−20.0000	−0.835512
$574$	54.0000	2.25392
$575$	−1.00000	−0.0417029
$576$	−8.00000	−0.333333
$577$	−21.0000	−0.874241	−0.437121	−	0.899403i	$-0.644002\pi$
$577$	−21.0000	−0.874241	−0.437121	+	0.899403i	$0.644002\pi$
$578$	16.0000	0.665512
$579$	13.0000	0.540262
$580$	20.0000	0.830455
$581$	−24.0000	−0.995688
$582$	−18.0000	−0.746124
$583$	−45.0000	−1.86371
$584$	0	0
$585$	1.00000	0.0413449
$586$	48.0000	1.98286
$587$	42.0000	1.73353	0.866763	−	0.498721i	$-0.166197\pi$
$587$	42.0000	1.73353	0.866763	+	0.498721i	$0.166197\pi$
$588$	4.00000	0.164957
$589$	−4.00000	−0.164817
$590$	0	0
$591$	−12.0000	−0.493614
$592$	12.0000	0.493197
$593$	−36.0000	−1.47834	−0.739171	−	0.673517i	$-0.764783\pi$
$593$	−36.0000	−1.47834	−0.739171	+	0.673517i	$0.764783\pi$
$594$	−10.0000	−0.410305
$595$	−15.0000	−0.614940
$596$	−14.0000	−0.573462
$597$	4.00000	0.163709
$598$	−2.00000	−0.0817861
$599$	4.00000	0.163436	0.0817178	−	0.996656i	$-0.473959\pi$
$599$	4.00000	0.163436	0.0817178	+	0.996656i	$0.473959\pi$
$600$	0	0
$601$	−5.00000	−0.203954	−0.101977	−	0.994787i	$-0.532517\pi$
$601$	−5.00000	−0.203954	−0.101977	+	0.994787i	$0.532517\pi$
$602$	24.0000	0.978167
$603$	−4.00000	−0.162893
$604$	−24.0000	−0.976546
$605$	14.0000	0.569181
$606$	−24.0000	−0.974933
$607$	−40.0000	−1.62355	−0.811775	−	0.583970i	$-0.801498\pi$
$607$	−40.0000	−1.62355	−0.811775	+	0.583970i	$0.801498\pi$
$608$	−16.0000	−0.648886
$609$	−30.0000	−1.21566
$610$	−22.0000	−0.890754
$611$	10.0000	0.404557
$612$	10.0000	0.404226
$613$	3.00000	0.121169	0.0605844	−	0.998163i	$-0.480704\pi$
$613$	3.00000	0.121169	0.0605844	+	0.998163i	$0.480704\pi$
$614$	−38.0000	−1.53356
$615$	−9.00000	−0.362915
$616$	0	0
$617$	22.0000	0.885687	0.442843	−	0.896599i	$-0.353970\pi$
$617$	22.0000	0.885687	0.442843	+	0.896599i	$0.353970\pi$
$618$	8.00000	0.321807
$619$	−2.00000	−0.0803868	−0.0401934	−	0.999192i	$-0.512797\pi$
$619$	−2.00000	−0.0803868	−0.0401934	+	0.999192i	$0.512797\pi$
$620$	−4.00000	−0.160644
$621$	−1.00000	−0.0401286
$622$	48.0000	1.92462
$623$	33.0000	1.32212
$624$	−4.00000	−0.160128
$625$	1.00000	0.0400000
$626$	−20.0000	−0.799361
$627$	−10.0000	−0.399362
$628$	44.0000	1.75579
$629$	−15.0000	−0.598089
$630$	−6.00000	−0.239046
$631$	0	0		−	1.00000i	$-0.5\pi$
$631$	0	0			1.00000i	$0.5\pi$
$632$	0	0
$633$	−4.00000	−0.158986
$634$	24.0000	0.953162
$635$	14.0000	0.555573
$636$	18.0000	0.713746
$637$	2.00000	0.0792429
$638$	−100.000	−3.95904
$639$	15.0000	0.593391
$640$	0	0
$641$	−36.0000	−1.42191	−0.710957	−	0.703235i	$-0.751738\pi$
$641$	−36.0000	−1.42191	−0.710957	+	0.703235i	$0.751738\pi$
$642$	6.00000	0.236801
$643$	1.00000	0.0394362	0.0197181	−	0.999806i	$-0.493723\pi$
$643$	1.00000	0.0394362	0.0197181	+	0.999806i	$0.493723\pi$
$644$	6.00000	0.236433
$645$	−4.00000	−0.157500
$646$	20.0000	0.786889
$647$	21.0000	0.825595	0.412798	−	0.910823i	$-0.364552\pi$
$647$	21.0000	0.825595	0.412798	+	0.910823i	$0.364552\pi$
$648$	0	0
$649$	0	0
$650$	2.00000	0.0784465
$651$	6.00000	0.235159
$652$	−22.0000	−0.861586
$653$	−30.0000	−1.17399	−0.586995	−	0.809590i	$-0.699689\pi$
$653$	−30.0000	−1.17399	−0.586995	+	0.809590i	$0.699689\pi$
$654$	32.0000	1.25130
$655$	6.00000	0.234439
$656$	36.0000	1.40556
$657$	6.00000	0.234082
$658$	−60.0000	−2.33904
$659$	12.0000	0.467454	0.233727	−	0.972302i	$-0.424908\pi$
$659$	12.0000	0.467454	0.233727	+	0.972302i	$0.424908\pi$
$660$	−10.0000	−0.389249
$661$	−16.0000	−0.622328	−0.311164	−	0.950356i	$-0.600719\pi$
$661$	−16.0000	−0.622328	−0.311164	+	0.950356i	$0.600719\pi$
$662$	64.0000	2.48743
$663$	5.00000	0.194184
$664$	0	0
$665$	−6.00000	−0.232670
$666$	−6.00000	−0.232495
$667$	−10.0000	−0.387202
$668$	−16.0000	−0.619059
$669$	0	0
$670$	−8.00000	−0.309067
$671$	55.0000	2.12325
$672$	24.0000	0.925820
$673$	22.0000	0.848038	0.424019	−	0.905653i	$-0.360619\pi$
$673$	22.0000	0.848038	0.424019	+	0.905653i	$0.360619\pi$
$674$	8.00000	0.308148
$675$	1.00000	0.0384900
$676$	2.00000	0.0769231
$677$	−7.00000	−0.269032	−0.134516	−	0.990911i	$-0.542948\pi$
$677$	−7.00000	−0.269032	−0.134516	+	0.990911i	$0.542948\pi$
$678$	4.00000	0.153619
$679$	27.0000	1.03616
$680$	0	0
$681$	−18.0000	−0.689761
$682$	20.0000	0.765840
$683$	4.00000	0.153056	0.0765279	−	0.997067i	$-0.475617\pi$
$683$	4.00000	0.153056	0.0765279	+	0.997067i	$0.475617\pi$
$684$	4.00000	0.152944
$685$	6.00000	0.229248
$686$	30.0000	1.14541
$687$	−14.0000	−0.534133
$688$	16.0000	0.609994
$689$	9.00000	0.342873
$690$	−2.00000	−0.0761387
$691$	6.00000	0.228251	0.114125	−	0.993466i	$-0.463593\pi$
$691$	6.00000	0.228251	0.114125	+	0.993466i	$0.463593\pi$
$692$	−4.00000	−0.152057
$693$	15.0000	0.569803
$694$	−2.00000	−0.0759190
$695$	−17.0000	−0.644847
$696$	0	0
$697$	−45.0000	−1.70450
$698$	40.0000	1.51402
$699$	−25.0000	−0.945587
$700$	−6.00000	−0.226779
$701$	−4.00000	−0.151078	−0.0755390	−	0.997143i	$-0.524068\pi$
$701$	−4.00000	−0.151078	−0.0755390	+	0.997143i	$0.524068\pi$
$702$	2.00000	0.0754851
$703$	−6.00000	−0.226294
$704$	40.0000	1.50756
$705$	10.0000	0.376622
$706$	−28.0000	−1.05379
$707$	36.0000	1.35392
$708$	0	0
$709$	20.0000	0.751116	0.375558	−	0.926799i	$-0.377451\pi$
$709$	20.0000	0.751116	0.375558	+	0.926799i	$0.377451\pi$
$710$	30.0000	1.12588
$711$	−11.0000	−0.412532
$712$	0	0
$713$	2.00000	0.0749006
$714$	−30.0000	−1.12272
$715$	−5.00000	−0.186989
$716$	−12.0000	−0.448461
$717$	15.0000	0.560185
$718$	32.0000	1.19423
$719$	12.0000	0.447524	0.223762	−	0.974644i	$-0.428166\pi$
$719$	12.0000	0.447524	0.223762	+	0.974644i	$0.428166\pi$
$720$	−4.00000	−0.149071
$721$	−12.0000	−0.446903
$722$	−30.0000	−1.11648
$723$	−14.0000	−0.520666
$724$	−46.0000	−1.70958
$725$	10.0000	0.371391
$726$	28.0000	1.03918
$727$	6.00000	0.222528	0.111264	−	0.993791i	$-0.464510\pi$
$727$	6.00000	0.222528	0.111264	+	0.993791i	$0.464510\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	12.0000	0.444140
$731$	−20.0000	−0.739727
$732$	−22.0000	−0.813143
$733$	−15.0000	−0.554038	−0.277019	−	0.960864i	$-0.589346\pi$
$733$	−15.0000	−0.554038	−0.277019	+	0.960864i	$0.589346\pi$
$734$	16.0000	0.590571
$735$	2.00000	0.0737711
$736$	8.00000	0.294884
$737$	20.0000	0.736709
$738$	−18.0000	−0.662589
$739$	38.0000	1.39785	0.698926	−	0.715194i	$-0.253662\pi$
$739$	38.0000	1.39785	0.698926	+	0.715194i	$0.253662\pi$
$740$	−6.00000	−0.220564
$741$	2.00000	0.0734718
$742$	−54.0000	−1.98240
$743$	6.00000	0.220119	0.110059	−	0.993925i	$-0.464896\pi$
$743$	6.00000	0.220119	0.110059	+	0.993925i	$0.464896\pi$
$744$	0	0
$745$	−7.00000	−0.256460
$746$	32.0000	1.17160
$747$	8.00000	0.292705
$748$	−50.0000	−1.82818
$749$	−9.00000	−0.328853
$750$	2.00000	0.0730297
$751$	−45.0000	−1.64207	−0.821037	−	0.570875i	$-0.806604\pi$
$751$	−45.0000	−1.64207	−0.821037	+	0.570875i	$0.806604\pi$
$752$	−40.0000	−1.45865
$753$	20.0000	0.728841
$754$	20.0000	0.728357
$755$	−12.0000	−0.436725
$756$	−6.00000	−0.218218
$757$	36.0000	1.30844	0.654221	−	0.756303i	$-0.272997\pi$
$757$	36.0000	1.30844	0.654221	+	0.756303i	$0.272997\pi$
$758$	12.0000	0.435860
$759$	5.00000	0.181489
$760$	0	0
$761$	−14.0000	−0.507500	−0.253750	−	0.967270i	$-0.581664\pi$
$761$	−14.0000	−0.507500	−0.253750	+	0.967270i	$0.581664\pi$
$762$	28.0000	1.01433
$763$	−48.0000	−1.73772
$764$	−40.0000	−1.44715
$765$	5.00000	0.180775
$766$	−36.0000	−1.30073
$767$	0	0
$768$	16.0000	0.577350
$769$	−12.0000	−0.432731	−0.216366	−	0.976312i	$-0.569420\pi$
$769$	−12.0000	−0.432731	−0.216366	+	0.976312i	$0.569420\pi$
$770$	30.0000	1.08112
$771$	18.0000	0.648254
$772$	26.0000	0.935760
$773$	24.0000	0.863220	0.431610	−	0.902060i	$-0.357946\pi$
$773$	24.0000	0.863220	0.431610	+	0.902060i	$0.357946\pi$
$774$	−8.00000	−0.287554
$775$	−2.00000	−0.0718421
$776$	0	0
$777$	9.00000	0.322873
$778$	−48.0000	−1.72088
$779$	−18.0000	−0.644917
$780$	2.00000	0.0716115
$781$	−75.0000	−2.68371
$782$	−10.0000	−0.357599
$783$	10.0000	0.357371
$784$	−8.00000	−0.285714
$785$	22.0000	0.785214
$786$	12.0000	0.428026
$787$	44.0000	1.56843	0.784215	−	0.620489i	$-0.213066\pi$
$787$	44.0000	1.56843	0.784215	+	0.620489i	$0.213066\pi$
$788$	−24.0000	−0.854965
$789$	0	0
$790$	−22.0000	−0.782725
$791$	−6.00000	−0.213335
$792$	0	0
$793$	−11.0000	−0.390621
$794$	−38.0000	−1.34857
$795$	9.00000	0.319197
$796$	8.00000	0.283552
$797$	−5.00000	−0.177109	−0.0885545	−	0.996071i	$-0.528225\pi$
$797$	−5.00000	−0.177109	−0.0885545	+	0.996071i	$0.528225\pi$
$798$	−12.0000	−0.424795
$799$	50.0000	1.76887
$800$	−8.00000	−0.282843
$801$	−11.0000	−0.388666
$802$	−36.0000	−1.27120
$803$	−30.0000	−1.05868
$804$	−8.00000	−0.282138
$805$	3.00000	0.105736
$806$	−4.00000	−0.140894
$807$	32.0000	1.12645
$808$	0	0
$809$	−6.00000	−0.210949	−0.105474	−	0.994422i	$-0.533636\pi$
$809$	−6.00000	−0.210949	−0.105474	+	0.994422i	$0.533636\pi$
$810$	2.00000	0.0702728
$811$	−4.00000	−0.140459	−0.0702295	−	0.997531i	$-0.522373\pi$
$811$	−4.00000	−0.140459	−0.0702295	+	0.997531i	$0.522373\pi$
$812$	−60.0000	−2.10559
$813$	−2.00000	−0.0701431
$814$	30.0000	1.05150
$815$	−11.0000	−0.385313
$816$	−20.0000	−0.700140
$817$	−8.00000	−0.279885
$818$	−52.0000	−1.81814
$819$	−3.00000	−0.104828
$820$	−18.0000	−0.628587
$821$	−41.0000	−1.43091	−0.715455	−	0.698659i	$-0.753781\pi$
$821$	−41.0000	−1.43091	−0.715455	+	0.698659i	$0.753781\pi$
$822$	12.0000	0.418548
$823$	−48.0000	−1.67317	−0.836587	−	0.547833i	$-0.815453\pi$
$823$	−48.0000	−1.67317	−0.836587	+	0.547833i	$0.815453\pi$
$824$	0	0
$825$	−5.00000	−0.174078
$826$	0	0
$827$	−42.0000	−1.46048	−0.730242	−	0.683189i	$-0.760592\pi$
$827$	−42.0000	−1.46048	−0.730242	+	0.683189i	$0.760592\pi$
$828$	−2.00000	−0.0695048
$829$	14.0000	0.486240	0.243120	−	0.969996i	$-0.421829\pi$
$829$	14.0000	0.486240	0.243120	+	0.969996i	$0.421829\pi$
$830$	16.0000	0.555368
$831$	26.0000	0.901930
$832$	−8.00000	−0.277350
$833$	10.0000	0.346479
$834$	−34.0000	−1.17732
$835$	−8.00000	−0.276851
$836$	−20.0000	−0.691714
$837$	−2.00000	−0.0691301
$838$	52.0000	1.79631
$839$	7.00000	0.241667	0.120833	−	0.992673i	$-0.461443\pi$
$839$	7.00000	0.241667	0.120833	+	0.992673i	$0.461443\pi$
$840$	0	0
$841$	71.0000	2.44828
$842$	64.0000	2.20559
$843$	−10.0000	−0.344418
$844$	−8.00000	−0.275371
$845$	1.00000	0.0344010
$846$	20.0000	0.687614
$847$	−42.0000	−1.44314
$848$	−36.0000	−1.23625
$849$	−8.00000	−0.274559
$850$	10.0000	0.342997
$851$	3.00000	0.102839
$852$	30.0000	1.02778
$853$	51.0000	1.74621	0.873103	−	0.487535i	$-0.162104\pi$
$853$	51.0000	1.74621	0.873103	+	0.487535i	$0.162104\pi$
$854$	66.0000	2.25847
$855$	2.00000	0.0683986
$856$	0	0
$857$	−17.0000	−0.580709	−0.290354	−	0.956919i	$-0.593773\pi$
$857$	−17.0000	−0.580709	−0.290354	+	0.956919i	$0.593773\pi$
$858$	−10.0000	−0.341394
$859$	35.0000	1.19418	0.597092	−	0.802173i	$-0.296323\pi$
$859$	35.0000	1.19418	0.597092	+	0.802173i	$0.296323\pi$
$860$	−8.00000	−0.272798
$861$	27.0000	0.920158
$862$	−32.0000	−1.08992
$863$	−22.0000	−0.748889	−0.374444	−	0.927249i	$-0.622167\pi$
$863$	−22.0000	−0.748889	−0.374444	+	0.927249i	$0.622167\pi$
$864$	−8.00000	−0.272166
$865$	−2.00000	−0.0680020
$866$	48.0000	1.63111
$867$	8.00000	0.271694
$868$	12.0000	0.407307
$869$	55.0000	1.86575
$870$	20.0000	0.678064
$871$	−4.00000	−0.135535
$872$	0	0
$873$	−9.00000	−0.304604
$874$	−4.00000	−0.135302
$875$	−3.00000	−0.101419
$876$	12.0000	0.405442
$877$	22.0000	0.742887	0.371444	−	0.928456i	$-0.378863\pi$
$877$	22.0000	0.742887	0.371444	+	0.928456i	$0.378863\pi$
$878$	−66.0000	−2.22739
$879$	24.0000	0.809500
$880$	20.0000	0.674200
$881$	−14.0000	−0.471672	−0.235836	−	0.971793i	$-0.575783\pi$
$881$	−14.0000	−0.471672	−0.235836	+	0.971793i	$0.575783\pi$
$882$	4.00000	0.134687
$883$	12.0000	0.403832	0.201916	−	0.979403i	$-0.435283\pi$
$883$	12.0000	0.403832	0.201916	+	0.979403i	$0.435283\pi$
$884$	10.0000	0.336336
$885$	0	0
$886$	70.0000	2.35170
$887$	−15.0000	−0.503651	−0.251825	−	0.967773i	$-0.581031\pi$
$887$	−15.0000	−0.503651	−0.251825	+	0.967773i	$0.581031\pi$
$888$	0	0
$889$	−42.0000	−1.40863
$890$	−22.0000	−0.737442
$891$	−5.00000	−0.167506
$892$	0	0
$893$	20.0000	0.669274
$894$	−14.0000	−0.468230
$895$	−6.00000	−0.200558
$896$	0	0
$897$	−1.00000	−0.0333890
$898$	30.0000	1.00111
$899$	−20.0000	−0.667037
$900$	2.00000	0.0666667
$901$	45.0000	1.49917
$902$	90.0000	2.99667
$903$	12.0000	0.399335
$904$	0	0
$905$	−23.0000	−0.764546
$906$	−24.0000	−0.797347
$907$	2.00000	0.0664089	0.0332045	−	0.999449i	$-0.489429\pi$
$907$	2.00000	0.0664089	0.0332045	+	0.999449i	$0.489429\pi$
$908$	−36.0000	−1.19470
$909$	−12.0000	−0.398015
$910$	−6.00000	−0.198898
$911$	8.00000	0.265052	0.132526	−	0.991180i	$-0.457691\pi$
$911$	8.00000	0.265052	0.132526	+	0.991180i	$0.457691\pi$
$912$	−8.00000	−0.264906
$913$	−40.0000	−1.32381
$914$	−26.0000	−0.860004
$915$	−11.0000	−0.363649
$916$	−28.0000	−0.925146
$917$	−18.0000	−0.594412
$918$	10.0000	0.330049
$919$	29.0000	0.956622	0.478311	−	0.878191i	$-0.341249\pi$
$919$	29.0000	0.956622	0.478311	+	0.878191i	$0.341249\pi$
$920$	0	0
$921$	−19.0000	−0.626071
$922$	6.00000	0.197599
$923$	15.0000	0.493731
$924$	30.0000	0.986928
$925$	−3.00000	−0.0986394
$926$	10.0000	0.328620
$927$	4.00000	0.131377
$928$	−80.0000	−2.62613
$929$	21.0000	0.688988	0.344494	−	0.938789i	$-0.388051\pi$
$929$	21.0000	0.688988	0.344494	+	0.938789i	$0.388051\pi$
$930$	−4.00000	−0.131165
$931$	4.00000	0.131095
$932$	−50.0000	−1.63780
$933$	24.0000	0.785725
$934$	−58.0000	−1.89782
$935$	−25.0000	−0.817587
$936$	0	0
$937$	2.00000	0.0653372	0.0326686	−	0.999466i	$-0.489599\pi$
$937$	2.00000	0.0653372	0.0326686	+	0.999466i	$0.489599\pi$
$938$	24.0000	0.783628
$939$	−10.0000	−0.326338
$940$	20.0000	0.652328
$941$	−23.0000	−0.749779	−0.374889	−	0.927070i	$-0.622319\pi$
$941$	−23.0000	−0.749779	−0.374889	+	0.927070i	$0.622319\pi$
$942$	44.0000	1.43360
$943$	9.00000	0.293080
$944$	0	0
$945$	−3.00000	−0.0975900
$946$	40.0000	1.30051
$947$	36.0000	1.16984	0.584921	−	0.811090i	$-0.301125\pi$
$947$	36.0000	1.16984	0.584921	+	0.811090i	$0.301125\pi$
$948$	−22.0000	−0.714527
$949$	6.00000	0.194768
$950$	4.00000	0.129777
$951$	12.0000	0.389127
$952$	0	0
$953$	−11.0000	−0.356325	−0.178162	−	0.984001i	$-0.557015\pi$
$953$	−11.0000	−0.356325	−0.178162	+	0.984001i	$0.557015\pi$
$954$	18.0000	0.582772
$955$	−20.0000	−0.647185
$956$	30.0000	0.970269
$957$	−50.0000	−1.61627
$958$	10.0000	0.323085
$959$	−18.0000	−0.581250
$960$	−8.00000	−0.258199
$961$	−27.0000	−0.870968
$962$	−6.00000	−0.193448
$963$	3.00000	0.0966736
$964$	−28.0000	−0.901819
$965$	13.0000	0.418485
$966$	6.00000	0.193047
$967$	−40.0000	−1.28631	−0.643157	−	0.765735i	$-0.722376\pi$
$967$	−40.0000	−1.28631	−0.643157	+	0.765735i	$0.722376\pi$
$968$	0	0
$969$	10.0000	0.321246
$970$	−18.0000	−0.577945
$971$	0	0		−	1.00000i	$-0.5\pi$
$971$	0	0			1.00000i	$0.5\pi$
$972$	2.00000	0.0641500
$973$	51.0000	1.63498
$974$	14.0000	0.448589
$975$	1.00000	0.0320256
$976$	44.0000	1.40841
$977$	12.0000	0.383914	0.191957	−	0.981403i	$-0.438517\pi$
$977$	12.0000	0.383914	0.191957	+	0.981403i	$0.438517\pi$
$978$	−22.0000	−0.703482
$979$	55.0000	1.75781
$980$	4.00000	0.127775
$981$	16.0000	0.510841
$982$	32.0000	1.02116
$983$	36.0000	1.14822	0.574111	−	0.818778i	$-0.305348\pi$
$983$	36.0000	1.14822	0.574111	+	0.818778i	$0.305348\pi$
$984$	0	0
$985$	−12.0000	−0.382352
$986$	100.000	3.18465
$987$	−30.0000	−0.954911
$988$	4.00000	0.127257
$989$	4.00000	0.127193
$990$	−10.0000	−0.317821
$991$	39.0000	1.23888	0.619438	−	0.785046i	$-0.287361\pi$
$991$	39.0000	1.23888	0.619438	+	0.785046i	$0.287361\pi$
$992$	16.0000	0.508001
$993$	32.0000	1.01549
$994$	−90.0000	−2.85463
$995$	4.00000	0.126809
$996$	16.0000	0.506979
$997$	−16.0000	−0.506725	−0.253363	−	0.967371i	$-0.581537\pi$
$997$	−16.0000	−0.506725	−0.253363	+	0.967371i	$0.581537\pi$
$998$	68.0000	2.15250
$999$	−3.00000	−0.0949158

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	195.2.a.d.1.1	✓	1
3.2	odd	2		585.2.a.a.1.1		1
4.3	odd	2		3120.2.a.n.1.1		1
5.2	odd	4		975.2.c.b.274.2		2
5.3	odd	4		975.2.c.b.274.1		2
5.4	even	2		975.2.a.b.1.1		1
7.6	odd	2		9555.2.a.t.1.1		1
12.11	even	2		9360.2.a.w.1.1		1
13.12	even	2		2535.2.a.b.1.1		1
15.2	even	4		2925.2.c.d.2224.1		2
15.8	even	4		2925.2.c.d.2224.2		2
15.14	odd	2		2925.2.a.t.1.1		1
39.38	odd	2		7605.2.a.v.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
195.2.a.d.1.1	✓	1	1.1	even	1	trivial
585.2.a.a.1.1		1	3.2	odd	2
975.2.a.b.1.1		1	5.4	even	2
975.2.c.b.274.1		2	5.3	odd	4
975.2.c.b.274.2		2	5.2	odd	4
2535.2.a.b.1.1		1	13.12	even	2
2925.2.a.t.1.1		1	15.14	odd	2
2925.2.c.d.2224.1		2	15.2	even	4
2925.2.c.d.2224.2		2	15.8	even	4
3120.2.a.n.1.1		1	4.3	odd	2
7605.2.a.v.1.1		1	39.38	odd	2
9360.2.a.w.1.1		1	12.11	even	2
9555.2.a.t.1.1		1	7.6	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 195 = 3 \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	195.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more