LMFDB - Embedded newform 1155.2.a.m.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1155,2,Mod(1,1155)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1155, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1155.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1155 = 3 \cdot 5 \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1155.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:	$9.22272143346$
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	$\chi$	$=$	1155.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{2} +1.00000 q^{3} -1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} +1.00000 q^{7} -3.00000 q^{8} +1.00000 q^{9} +O(q^{10})$

\(q+1.00000 q^{2} +1.00000 q^{3} -1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} +1.00000 q^{7} -3.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -1.00000 q^{11} -1.00000 q^{12} +6.00000 q^{13} +1.00000 q^{14} +1.00000 q^{15} -1.00000 q^{16} +2.00000 q^{17} +1.00000 q^{18} +4.00000 q^{19} -1.00000 q^{20} +1.00000 q^{21} -1.00000 q^{22} -4.00000 q^{23} -3.00000 q^{24} +1.00000 q^{25} +6.00000 q^{26} +1.00000 q^{27} -1.00000 q^{28} -2.00000 q^{29} +1.00000 q^{30} +4.00000 q^{31} +5.00000 q^{32} -1.00000 q^{33} +2.00000 q^{34} +1.00000 q^{35} -1.00000 q^{36} -2.00000 q^{37} +4.00000 q^{38} +6.00000 q^{39} -3.00000 q^{40} +6.00000 q^{41} +1.00000 q^{42} +4.00000 q^{43} +1.00000 q^{44} +1.00000 q^{45} -4.00000 q^{46} +8.00000 q^{47} -1.00000 q^{48} +1.00000 q^{49} +1.00000 q^{50} +2.00000 q^{51} -6.00000 q^{52} -2.00000 q^{53} +1.00000 q^{54} -1.00000 q^{55} -3.00000 q^{56} +4.00000 q^{57} -2.00000 q^{58} -8.00000 q^{59} -1.00000 q^{60} +10.0000 q^{61} +4.00000 q^{62} +1.00000 q^{63} +7.00000 q^{64} +6.00000 q^{65} -1.00000 q^{66} -8.00000 q^{67} -2.00000 q^{68} -4.00000 q^{69} +1.00000 q^{70} -3.00000 q^{72} -14.0000 q^{73} -2.00000 q^{74} +1.00000 q^{75} -4.00000 q^{76} -1.00000 q^{77} +6.00000 q^{78} -16.0000 q^{79} -1.00000 q^{80} +1.00000 q^{81} +6.00000 q^{82} -4.00000 q^{83} -1.00000 q^{84} +2.00000 q^{85} +4.00000 q^{86} -2.00000 q^{87} +3.00000 q^{88} -6.00000 q^{89} +1.00000 q^{90} +6.00000 q^{91} +4.00000 q^{92} +4.00000 q^{93} +8.00000 q^{94} +4.00000 q^{95} +5.00000 q^{96} +18.0000 q^{97} +1.00000 q^{98} -1.00000 q^{99} +O(q^{100})\)

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	0.353553	−	0.935414i	$-0.384973\pi$
$2$	1.00000	0.707107	0.353553	+	0.935414i	$0.384973\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−1.00000	−0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	1.00000	0.408248
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−3.00000	−1.06066
$9$	1.00000	0.333333
$10$	1.00000	0.316228
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	−1.00000	−0.288675
$13$	6.00000	1.66410	0.832050	−	0.554700i	$-0.187167\pi$
$13$	6.00000	1.66410	0.832050	+	0.554700i	$0.187167\pi$
$14$	1.00000	0.267261
$15$	1.00000	0.258199
$16$	−1.00000	−0.250000
$17$	2.00000	0.485071	0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000	0.485071	0.242536	+	0.970143i	$0.422021\pi$
$18$	1.00000	0.235702
$19$	4.00000	0.917663	0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000	0.917663	0.458831	+	0.888523i	$0.348268\pi$
$20$	−1.00000	−0.223607
$21$	1.00000	0.218218
$22$	−1.00000	−0.213201
$23$	−4.00000	−0.834058	−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000	−0.834058	−0.417029	+	0.908893i	$0.636929\pi$
$24$	−3.00000	−0.612372
$25$	1.00000	0.200000
$26$	6.00000	1.17670
$27$	1.00000	0.192450
$28$	−1.00000	−0.188982
$29$	−2.00000	−0.371391	−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000	−0.371391	−0.185695	+	0.982607i	$0.559454\pi$
$30$	1.00000	0.182574
$31$	4.00000	0.718421	0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000	0.718421	0.359211	+	0.933257i	$0.383046\pi$
$32$	5.00000	0.883883
$33$	−1.00000	−0.174078
$34$	2.00000	0.342997
$35$	1.00000	0.169031
$36$	−1.00000	−0.166667
$37$	−2.00000	−0.328798	−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000	−0.328798	−0.164399	+	0.986394i	$0.552568\pi$
$38$	4.00000	0.648886
$39$	6.00000	0.960769
$40$	−3.00000	−0.474342
$41$	6.00000	0.937043	0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	1.00000	0.154303
$43$	4.00000	0.609994	0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000	0.609994	0.304997	+	0.952353i	$0.401344\pi$
$44$	1.00000	0.150756
$45$	1.00000	0.149071
$46$	−4.00000	−0.589768
$47$	8.00000	1.16692	0.583460	−	0.812142i	$-0.301699\pi$
$47$	8.00000	1.16692	0.583460	+	0.812142i	$0.301699\pi$
$48$	−1.00000	−0.144338
$49$	1.00000	0.142857
$50$	1.00000	0.141421
$51$	2.00000	0.280056
$52$	−6.00000	−0.832050
$53$	−2.00000	−0.274721	−0.137361	−	0.990521i	$-0.543862\pi$
$53$	−2.00000	−0.274721	−0.137361	+	0.990521i	$0.543862\pi$
$54$	1.00000	0.136083
$55$	−1.00000	−0.134840
$56$	−3.00000	−0.400892
$57$	4.00000	0.529813
$58$	−2.00000	−0.262613
$59$	−8.00000	−1.04151	−0.520756	−	0.853706i	$-0.674350\pi$
$59$	−8.00000	−1.04151	−0.520756	+	0.853706i	$0.674350\pi$
$60$	−1.00000	−0.129099
$61$	10.0000	1.28037	0.640184	−	0.768221i	$-0.278858\pi$
$61$	10.0000	1.28037	0.640184	+	0.768221i	$0.278858\pi$
$62$	4.00000	0.508001
$63$	1.00000	0.125988
$64$	7.00000	0.875000
$65$	6.00000	0.744208
$66$	−1.00000	−0.123091
$67$	−8.00000	−0.977356	−0.488678	−	0.872464i	$-0.662521\pi$
$67$	−8.00000	−0.977356	−0.488678	+	0.872464i	$0.662521\pi$
$68$	−2.00000	−0.242536
$69$	−4.00000	−0.481543
$70$	1.00000	0.119523
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	−3.00000	−0.353553
$73$	−14.0000	−1.63858	−0.819288	−	0.573382i	$-0.805631\pi$
$73$	−14.0000	−1.63858	−0.819288	+	0.573382i	$0.805631\pi$
$74$	−2.00000	−0.232495
$75$	1.00000	0.115470
$76$	−4.00000	−0.458831
$77$	−1.00000	−0.113961
$78$	6.00000	0.679366
$79$	−16.0000	−1.80014	−0.900070	−	0.435745i	$-0.856485\pi$
$79$	−16.0000	−1.80014	−0.900070	+	0.435745i	$0.856485\pi$
$80$	−1.00000	−0.111803
$81$	1.00000	0.111111
$82$	6.00000	0.662589
$83$	−4.00000	−0.439057	−0.219529	−	0.975606i	$-0.570452\pi$
$83$	−4.00000	−0.439057	−0.219529	+	0.975606i	$0.570452\pi$
$84$	−1.00000	−0.109109
$85$	2.00000	0.216930
$86$	4.00000	0.431331
$87$	−2.00000	−0.214423
$88$	3.00000	0.319801
$89$	−6.00000	−0.635999	−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000	−0.635999	−0.317999	+	0.948091i	$0.603011\pi$
$90$	1.00000	0.105409
$91$	6.00000	0.628971
$92$	4.00000	0.417029
$93$	4.00000	0.414781
$94$	8.00000	0.825137
$95$	4.00000	0.410391
$96$	5.00000	0.510310
$97$	18.0000	1.82762	0.913812	−	0.406138i	$-0.133125\pi$
$97$	18.0000	1.82762	0.913812	+	0.406138i	$0.133125\pi$
$98$	1.00000	0.101015
$99$	−1.00000	−0.100504
$100$	−1.00000	−0.100000
$101$	2.00000	0.199007	0.0995037	−	0.995037i	$-0.468274\pi$
$101$	2.00000	0.199007	0.0995037	+	0.995037i	$0.468274\pi$
$102$	2.00000	0.198030
$103$	−16.0000	−1.57653	−0.788263	−	0.615338i	$-0.789020\pi$
$103$	−16.0000	−1.57653	−0.788263	+	0.615338i	$0.789020\pi$
$104$	−18.0000	−1.76505
$105$	1.00000	0.0975900
$106$	−2.00000	−0.194257
$107$	−12.0000	−1.16008	−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−12.0000	−1.16008	−0.580042	+	0.814587i	$0.696964\pi$
$108$	−1.00000	−0.0962250
$109$	6.00000	0.574696	0.287348	−	0.957826i	$-0.407226\pi$
$109$	6.00000	0.574696	0.287348	+	0.957826i	$0.407226\pi$
$110$	−1.00000	−0.0953463
$111$	−2.00000	−0.189832
$112$	−1.00000	−0.0944911
$113$	−6.00000	−0.564433	−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000	−0.564433	−0.282216	+	0.959351i	$0.591070\pi$
$114$	4.00000	0.374634
$115$	−4.00000	−0.373002
$116$	2.00000	0.185695
$117$	6.00000	0.554700
$118$	−8.00000	−0.736460
$119$	2.00000	0.183340
$120$	−3.00000	−0.273861
$121$	1.00000	0.0909091
$122$	10.0000	0.905357
$123$	6.00000	0.541002
$124$	−4.00000	−0.359211
$125$	1.00000	0.0894427
$126$	1.00000	0.0890871
$127$	8.00000	0.709885	0.354943	−	0.934888i	$-0.384500\pi$
$127$	8.00000	0.709885	0.354943	+	0.934888i	$0.384500\pi$
$128$	−3.00000	−0.265165
$129$	4.00000	0.352180
$130$	6.00000	0.526235
$131$	20.0000	1.74741	0.873704	−	0.486458i	$-0.161711\pi$
$131$	20.0000	1.74741	0.873704	+	0.486458i	$0.161711\pi$
$132$	1.00000	0.0870388
$133$	4.00000	0.346844
$134$	−8.00000	−0.691095
$135$	1.00000	0.0860663
$136$	−6.00000	−0.514496
$137$	2.00000	0.170872	0.0854358	−	0.996344i	$-0.472772\pi$
$137$	2.00000	0.170872	0.0854358	+	0.996344i	$0.472772\pi$
$138$	−4.00000	−0.340503
$139$	−12.0000	−1.01783	−0.508913	−	0.860818i	$-0.669953\pi$
$139$	−12.0000	−1.01783	−0.508913	+	0.860818i	$0.669953\pi$
$140$	−1.00000	−0.0845154
$141$	8.00000	0.673722
$142$	0	0
$143$	−6.00000	−0.501745
$144$	−1.00000	−0.0833333
$145$	−2.00000	−0.166091
$146$	−14.0000	−1.15865
$147$	1.00000	0.0824786
$148$	2.00000	0.164399
$149$	−10.0000	−0.819232	−0.409616	−	0.912258i	$-0.634337\pi$
$149$	−10.0000	−0.819232	−0.409616	+	0.912258i	$0.634337\pi$
$150$	1.00000	0.0816497
$151$	−16.0000	−1.30206	−0.651031	−	0.759051i	$-0.725663\pi$
$151$	−16.0000	−1.30206	−0.651031	+	0.759051i	$0.725663\pi$
$152$	−12.0000	−0.973329
$153$	2.00000	0.161690
$154$	−1.00000	−0.0805823
$155$	4.00000	0.321288
$156$	−6.00000	−0.480384
$157$	−2.00000	−0.159617	−0.0798087	−	0.996810i	$-0.525431\pi$
$157$	−2.00000	−0.159617	−0.0798087	+	0.996810i	$0.525431\pi$
$158$	−16.0000	−1.27289
$159$	−2.00000	−0.158610
$160$	5.00000	0.395285
$161$	−4.00000	−0.315244
$162$	1.00000	0.0785674
$163$	24.0000	1.87983	0.939913	−	0.341415i	$-0.110906\pi$
$163$	24.0000	1.87983	0.939913	+	0.341415i	$0.110906\pi$
$164$	−6.00000	−0.468521
$165$	−1.00000	−0.0778499
$166$	−4.00000	−0.310460
$167$	−16.0000	−1.23812	−0.619059	−	0.785345i	$-0.712486\pi$
$167$	−16.0000	−1.23812	−0.619059	+	0.785345i	$0.712486\pi$
$168$	−3.00000	−0.231455
$169$	23.0000	1.76923
$170$	2.00000	0.153393
$171$	4.00000	0.305888
$172$	−4.00000	−0.304997
$173$	−18.0000	−1.36851	−0.684257	−	0.729241i	$-0.739873\pi$
$173$	−18.0000	−1.36851	−0.684257	+	0.729241i	$0.739873\pi$
$174$	−2.00000	−0.151620
$175$	1.00000	0.0755929
$176$	1.00000	0.0753778
$177$	−8.00000	−0.601317
$178$	−6.00000	−0.449719
$179$	−12.0000	−0.896922	−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000	−0.896922	−0.448461	+	0.893802i	$0.648028\pi$
$180$	−1.00000	−0.0745356
$181$	−2.00000	−0.148659	−0.0743294	−	0.997234i	$-0.523682\pi$
$181$	−2.00000	−0.148659	−0.0743294	+	0.997234i	$0.523682\pi$
$182$	6.00000	0.444750
$183$	10.0000	0.739221
$184$	12.0000	0.884652
$185$	−2.00000	−0.147043
$186$	4.00000	0.293294
$187$	−2.00000	−0.146254
$188$	−8.00000	−0.583460
$189$	1.00000	0.0727393
$190$	4.00000	0.290191
$191$	−8.00000	−0.578860	−0.289430	−	0.957199i	$-0.593466\pi$
$191$	−8.00000	−0.578860	−0.289430	+	0.957199i	$0.593466\pi$
$192$	7.00000	0.505181
$193$	−18.0000	−1.29567	−0.647834	−	0.761781i	$-0.724325\pi$
$193$	−18.0000	−1.29567	−0.647834	+	0.761781i	$0.724325\pi$
$194$	18.0000	1.29232
$195$	6.00000	0.429669
$196$	−1.00000	−0.0714286
$197$	−6.00000	−0.427482	−0.213741	−	0.976890i	$-0.568565\pi$
$197$	−6.00000	−0.427482	−0.213741	+	0.976890i	$0.568565\pi$
$198$	−1.00000	−0.0710669
$199$	20.0000	1.41776	0.708881	−	0.705328i	$-0.249200\pi$
$199$	20.0000	1.41776	0.708881	+	0.705328i	$0.249200\pi$
$200$	−3.00000	−0.212132
$201$	−8.00000	−0.564276
$202$	2.00000	0.140720
$203$	−2.00000	−0.140372
$204$	−2.00000	−0.140028
$205$	6.00000	0.419058
$206$	−16.0000	−1.11477
$207$	−4.00000	−0.278019
$208$	−6.00000	−0.416025
$209$	−4.00000	−0.276686
$210$	1.00000	0.0690066
$211$	4.00000	0.275371	0.137686	−	0.990476i	$-0.456034\pi$
$211$	4.00000	0.275371	0.137686	+	0.990476i	$0.456034\pi$
$212$	2.00000	0.137361
$213$	0	0
$214$	−12.0000	−0.820303
$215$	4.00000	0.272798
$216$	−3.00000	−0.204124
$217$	4.00000	0.271538
$218$	6.00000	0.406371
$219$	−14.0000	−0.946032
$220$	1.00000	0.0674200
$221$	12.0000	0.807207
$222$	−2.00000	−0.134231
$223$	0	0		−	1.00000i	$-0.5\pi$
$223$	0	0			1.00000i	$0.5\pi$
$224$	5.00000	0.334077
$225$	1.00000	0.0666667
$226$	−6.00000	−0.399114
$227$	−28.0000	−1.85843	−0.929213	−	0.369546i	$-0.879513\pi$
$227$	−28.0000	−1.85843	−0.929213	+	0.369546i	$0.879513\pi$
$228$	−4.00000	−0.264906
$229$	−2.00000	−0.132164	−0.0660819	−	0.997814i	$-0.521050\pi$
$229$	−2.00000	−0.132164	−0.0660819	+	0.997814i	$0.521050\pi$
$230$	−4.00000	−0.263752
$231$	−1.00000	−0.0657952
$232$	6.00000	0.393919
$233$	6.00000	0.393073	0.196537	−	0.980497i	$-0.437031\pi$
$233$	6.00000	0.393073	0.196537	+	0.980497i	$0.437031\pi$
$234$	6.00000	0.392232
$235$	8.00000	0.521862
$236$	8.00000	0.520756
$237$	−16.0000	−1.03931
$238$	2.00000	0.129641
$239$	8.00000	0.517477	0.258738	−	0.965947i	$-0.416693\pi$
$239$	8.00000	0.517477	0.258738	+	0.965947i	$0.416693\pi$
$240$	−1.00000	−0.0645497
$241$	−18.0000	−1.15948	−0.579741	−	0.814801i	$-0.696846\pi$
$241$	−18.0000	−1.15948	−0.579741	+	0.814801i	$0.696846\pi$
$242$	1.00000	0.0642824
$243$	1.00000	0.0641500
$244$	−10.0000	−0.640184
$245$	1.00000	0.0638877
$246$	6.00000	0.382546
$247$	24.0000	1.52708
$248$	−12.0000	−0.762001
$249$	−4.00000	−0.253490
$250$	1.00000	0.0632456
$251$	−24.0000	−1.51487	−0.757433	−	0.652913i	$-0.773547\pi$
$251$	−24.0000	−1.51487	−0.757433	+	0.652913i	$0.773547\pi$
$252$	−1.00000	−0.0629941
$253$	4.00000	0.251478
$254$	8.00000	0.501965
$255$	2.00000	0.125245
$256$	−17.0000	−1.06250
$257$	18.0000	1.12281	0.561405	−	0.827541i	$-0.310261\pi$
$257$	18.0000	1.12281	0.561405	+	0.827541i	$0.310261\pi$
$258$	4.00000	0.249029
$259$	−2.00000	−0.124274
$260$	−6.00000	−0.372104
$261$	−2.00000	−0.123797
$262$	20.0000	1.23560
$263$	16.0000	0.986602	0.493301	−	0.869859i	$-0.335790\pi$
$263$	16.0000	0.986602	0.493301	+	0.869859i	$0.335790\pi$
$264$	3.00000	0.184637
$265$	−2.00000	−0.122859
$266$	4.00000	0.245256
$267$	−6.00000	−0.367194
$268$	8.00000	0.488678
$269$	−10.0000	−0.609711	−0.304855	−	0.952399i	$-0.598608\pi$
$269$	−10.0000	−0.609711	−0.304855	+	0.952399i	$0.598608\pi$
$270$	1.00000	0.0608581
$271$	24.0000	1.45790	0.728948	−	0.684569i	$-0.240010\pi$
$271$	24.0000	1.45790	0.728948	+	0.684569i	$0.240010\pi$
$272$	−2.00000	−0.121268
$273$	6.00000	0.363137
$274$	2.00000	0.120824
$275$	−1.00000	−0.0603023
$276$	4.00000	0.240772
$277$	2.00000	0.120168	0.0600842	−	0.998193i	$-0.480863\pi$
$277$	2.00000	0.120168	0.0600842	+	0.998193i	$0.480863\pi$
$278$	−12.0000	−0.719712
$279$	4.00000	0.239474
$280$	−3.00000	−0.179284
$281$	−6.00000	−0.357930	−0.178965	−	0.983855i	$-0.557275\pi$
$281$	−6.00000	−0.357930	−0.178965	+	0.983855i	$0.557275\pi$
$282$	8.00000	0.476393
$283$	−4.00000	−0.237775	−0.118888	−	0.992908i	$-0.537933\pi$
$283$	−4.00000	−0.237775	−0.118888	+	0.992908i	$0.537933\pi$
$284$	0	0
$285$	4.00000	0.236940
$286$	−6.00000	−0.354787
$287$	6.00000	0.354169
$288$	5.00000	0.294628
$289$	−13.0000	−0.764706
$290$	−2.00000	−0.117444
$291$	18.0000	1.05518
$292$	14.0000	0.819288
$293$	−10.0000	−0.584206	−0.292103	−	0.956387i	$-0.594355\pi$
$293$	−10.0000	−0.584206	−0.292103	+	0.956387i	$0.594355\pi$
$294$	1.00000	0.0583212
$295$	−8.00000	−0.465778
$296$	6.00000	0.348743
$297$	−1.00000	−0.0580259
$298$	−10.0000	−0.579284
$299$	−24.0000	−1.38796
$300$	−1.00000	−0.0577350
$301$	4.00000	0.230556
$302$	−16.0000	−0.920697
$303$	2.00000	0.114897
$304$	−4.00000	−0.229416
$305$	10.0000	0.572598
$306$	2.00000	0.114332
$307$	12.0000	0.684876	0.342438	−	0.939540i	$-0.388747\pi$
$307$	12.0000	0.684876	0.342438	+	0.939540i	$0.388747\pi$
$308$	1.00000	0.0569803
$309$	−16.0000	−0.910208
$310$	4.00000	0.227185
$311$	12.0000	0.680458	0.340229	−	0.940343i	$-0.389495\pi$
$311$	12.0000	0.680458	0.340229	+	0.940343i	$0.389495\pi$
$312$	−18.0000	−1.01905
$313$	26.0000	1.46961	0.734803	−	0.678280i	$-0.237274\pi$
$313$	26.0000	1.46961	0.734803	+	0.678280i	$0.237274\pi$
$314$	−2.00000	−0.112867
$315$	1.00000	0.0563436
$316$	16.0000	0.900070
$317$	14.0000	0.786318	0.393159	−	0.919470i	$-0.371382\pi$
$317$	14.0000	0.786318	0.393159	+	0.919470i	$0.371382\pi$
$318$	−2.00000	−0.112154
$319$	2.00000	0.111979
$320$	7.00000	0.391312
$321$	−12.0000	−0.669775
$322$	−4.00000	−0.222911
$323$	8.00000	0.445132
$324$	−1.00000	−0.0555556
$325$	6.00000	0.332820
$326$	24.0000	1.32924
$327$	6.00000	0.331801
$328$	−18.0000	−0.993884
$329$	8.00000	0.441054
$330$	−1.00000	−0.0550482
$331$	−12.0000	−0.659580	−0.329790	−	0.944054i	$-0.606978\pi$
$331$	−12.0000	−0.659580	−0.329790	+	0.944054i	$0.606978\pi$
$332$	4.00000	0.219529
$333$	−2.00000	−0.109599
$334$	−16.0000	−0.875481
$335$	−8.00000	−0.437087
$336$	−1.00000	−0.0545545
$337$	22.0000	1.19842	0.599208	−	0.800593i	$-0.295482\pi$
$337$	22.0000	1.19842	0.599208	+	0.800593i	$0.295482\pi$
$338$	23.0000	1.25104
$339$	−6.00000	−0.325875
$340$	−2.00000	−0.108465
$341$	−4.00000	−0.216612
$342$	4.00000	0.216295
$343$	1.00000	0.0539949
$344$	−12.0000	−0.646997
$345$	−4.00000	−0.215353
$346$	−18.0000	−0.967686
$347$	28.0000	1.50312	0.751559	−	0.659665i	$-0.229302\pi$
$347$	28.0000	1.50312	0.751559	+	0.659665i	$0.229302\pi$
$348$	2.00000	0.107211
$349$	−22.0000	−1.17763	−0.588817	−	0.808267i	$-0.700406\pi$
$349$	−22.0000	−1.17763	−0.588817	+	0.808267i	$0.700406\pi$
$350$	1.00000	0.0534522
$351$	6.00000	0.320256
$352$	−5.00000	−0.266501
$353$	−30.0000	−1.59674	−0.798369	−	0.602168i	$-0.794304\pi$
$353$	−30.0000	−1.59674	−0.798369	+	0.602168i	$0.794304\pi$
$354$	−8.00000	−0.425195
$355$	0	0
$356$	6.00000	0.317999
$357$	2.00000	0.105851
$358$	−12.0000	−0.634220
$359$	−24.0000	−1.26667	−0.633336	−	0.773877i	$-0.718315\pi$
$359$	−24.0000	−1.26667	−0.633336	+	0.773877i	$0.718315\pi$
$360$	−3.00000	−0.158114
$361$	−3.00000	−0.157895
$362$	−2.00000	−0.105118
$363$	1.00000	0.0524864
$364$	−6.00000	−0.314485
$365$	−14.0000	−0.732793
$366$	10.0000	0.522708
$367$	−8.00000	−0.417597	−0.208798	−	0.977959i	$-0.566955\pi$
$367$	−8.00000	−0.417597	−0.208798	+	0.977959i	$0.566955\pi$
$368$	4.00000	0.208514
$369$	6.00000	0.312348
$370$	−2.00000	−0.103975
$371$	−2.00000	−0.103835
$372$	−4.00000	−0.207390
$373$	26.0000	1.34623	0.673114	−	0.739538i	$-0.264956\pi$
$373$	26.0000	1.34623	0.673114	+	0.739538i	$0.264956\pi$
$374$	−2.00000	−0.103418
$375$	1.00000	0.0516398
$376$	−24.0000	−1.23771
$377$	−12.0000	−0.618031
$378$	1.00000	0.0514344
$379$	−4.00000	−0.205466	−0.102733	−	0.994709i	$-0.532759\pi$
$379$	−4.00000	−0.205466	−0.102733	+	0.994709i	$0.532759\pi$
$380$	−4.00000	−0.205196
$381$	8.00000	0.409852
$382$	−8.00000	−0.409316
$383$	16.0000	0.817562	0.408781	−	0.912633i	$-0.365954\pi$
$383$	16.0000	0.817562	0.408781	+	0.912633i	$0.365954\pi$
$384$	−3.00000	−0.153093
$385$	−1.00000	−0.0509647
$386$	−18.0000	−0.916176
$387$	4.00000	0.203331
$388$	−18.0000	−0.913812
$389$	6.00000	0.304212	0.152106	−	0.988364i	$-0.451394\pi$
$389$	6.00000	0.304212	0.152106	+	0.988364i	$0.451394\pi$
$390$	6.00000	0.303822
$391$	−8.00000	−0.404577
$392$	−3.00000	−0.151523
$393$	20.0000	1.00887
$394$	−6.00000	−0.302276
$395$	−16.0000	−0.805047
$396$	1.00000	0.0502519
$397$	14.0000	0.702640	0.351320	−	0.936255i	$-0.385733\pi$
$397$	14.0000	0.702640	0.351320	+	0.936255i	$0.385733\pi$
$398$	20.0000	1.00251
$399$	4.00000	0.200250
$400$	−1.00000	−0.0500000
$401$	18.0000	0.898877	0.449439	−	0.893311i	$-0.351624\pi$
$401$	18.0000	0.898877	0.449439	+	0.893311i	$0.351624\pi$
$402$	−8.00000	−0.399004
$403$	24.0000	1.19553
$404$	−2.00000	−0.0995037
$405$	1.00000	0.0496904
$406$	−2.00000	−0.0992583
$407$	2.00000	0.0991363
$408$	−6.00000	−0.297044
$409$	−10.0000	−0.494468	−0.247234	−	0.968956i	$-0.579522\pi$
$409$	−10.0000	−0.494468	−0.247234	+	0.968956i	$0.579522\pi$
$410$	6.00000	0.296319
$411$	2.00000	0.0986527
$412$	16.0000	0.788263
$413$	−8.00000	−0.393654
$414$	−4.00000	−0.196589
$415$	−4.00000	−0.196352
$416$	30.0000	1.47087
$417$	−12.0000	−0.587643
$418$	−4.00000	−0.195646
$419$	−24.0000	−1.17248	−0.586238	−	0.810139i	$-0.699392\pi$
$419$	−24.0000	−1.17248	−0.586238	+	0.810139i	$0.699392\pi$
$420$	−1.00000	−0.0487950
$421$	−26.0000	−1.26716	−0.633581	−	0.773676i	$-0.718416\pi$
$421$	−26.0000	−1.26716	−0.633581	+	0.773676i	$0.718416\pi$
$422$	4.00000	0.194717
$423$	8.00000	0.388973
$424$	6.00000	0.291386
$425$	2.00000	0.0970143
$426$	0	0
$427$	10.0000	0.483934
$428$	12.0000	0.580042
$429$	−6.00000	−0.289683
$430$	4.00000	0.192897
$431$	−32.0000	−1.54139	−0.770693	−	0.637207i	$-0.780090\pi$
$431$	−32.0000	−1.54139	−0.770693	+	0.637207i	$0.780090\pi$
$432$	−1.00000	−0.0481125
$433$	34.0000	1.63394	0.816968	−	0.576683i	$-0.195653\pi$
$433$	34.0000	1.63394	0.816968	+	0.576683i	$0.195653\pi$
$434$	4.00000	0.192006
$435$	−2.00000	−0.0958927
$436$	−6.00000	−0.287348
$437$	−16.0000	−0.765384
$438$	−14.0000	−0.668946
$439$	−32.0000	−1.52728	−0.763638	−	0.645644i	$-0.776589\pi$
$439$	−32.0000	−1.52728	−0.763638	+	0.645644i	$0.776589\pi$
$440$	3.00000	0.143019
$441$	1.00000	0.0476190
$442$	12.0000	0.570782
$443$	−24.0000	−1.14027	−0.570137	−	0.821549i	$-0.693110\pi$
$443$	−24.0000	−1.14027	−0.570137	+	0.821549i	$0.693110\pi$
$444$	2.00000	0.0949158
$445$	−6.00000	−0.284427
$446$	0	0
$447$	−10.0000	−0.472984
$448$	7.00000	0.330719
$449$	18.0000	0.849473	0.424736	−	0.905317i	$-0.360367\pi$
$449$	18.0000	0.849473	0.424736	+	0.905317i	$0.360367\pi$
$450$	1.00000	0.0471405
$451$	−6.00000	−0.282529
$452$	6.00000	0.282216
$453$	−16.0000	−0.751746
$454$	−28.0000	−1.31411
$455$	6.00000	0.281284
$456$	−12.0000	−0.561951
$457$	30.0000	1.40334	0.701670	−	0.712502i	$-0.252438\pi$
$457$	30.0000	1.40334	0.701670	+	0.712502i	$0.252438\pi$
$458$	−2.00000	−0.0934539
$459$	2.00000	0.0933520
$460$	4.00000	0.186501
$461$	34.0000	1.58354	0.791769	−	0.610821i	$-0.209160\pi$
$461$	34.0000	1.58354	0.791769	+	0.610821i	$0.209160\pi$
$462$	−1.00000	−0.0465242
$463$	36.0000	1.67306	0.836531	−	0.547920i	$-0.184580\pi$
$463$	36.0000	1.67306	0.836531	+	0.547920i	$0.184580\pi$
$464$	2.00000	0.0928477
$465$	4.00000	0.185496
$466$	6.00000	0.277945
$467$	28.0000	1.29569	0.647843	−	0.761774i	$-0.275671\pi$
$467$	28.0000	1.29569	0.647843	+	0.761774i	$0.275671\pi$
$468$	−6.00000	−0.277350
$469$	−8.00000	−0.369406
$470$	8.00000	0.369012
$471$	−2.00000	−0.0921551
$472$	24.0000	1.10469
$473$	−4.00000	−0.183920
$474$	−16.0000	−0.734904
$475$	4.00000	0.183533
$476$	−2.00000	−0.0916698
$477$	−2.00000	−0.0915737
$478$	8.00000	0.365911
$479$	−16.0000	−0.731059	−0.365529	−	0.930800i	$-0.619112\pi$
$479$	−16.0000	−0.731059	−0.365529	+	0.930800i	$0.619112\pi$
$480$	5.00000	0.228218
$481$	−12.0000	−0.547153
$482$	−18.0000	−0.819878
$483$	−4.00000	−0.182006
$484$	−1.00000	−0.0454545
$485$	18.0000	0.817338
$486$	1.00000	0.0453609
$487$	−28.0000	−1.26880	−0.634401	−	0.773004i	$-0.718753\pi$
$487$	−28.0000	−1.26880	−0.634401	+	0.773004i	$0.718753\pi$
$488$	−30.0000	−1.35804
$489$	24.0000	1.08532
$490$	1.00000	0.0451754
$491$	12.0000	0.541552	0.270776	−	0.962642i	$-0.412720\pi$
$491$	12.0000	0.541552	0.270776	+	0.962642i	$0.412720\pi$
$492$	−6.00000	−0.270501
$493$	−4.00000	−0.180151
$494$	24.0000	1.07981
$495$	−1.00000	−0.0449467
$496$	−4.00000	−0.179605
$497$	0	0
$498$	−4.00000	−0.179244
$499$	20.0000	0.895323	0.447661	−	0.894203i	$-0.352257\pi$
$499$	20.0000	0.895323	0.447661	+	0.894203i	$0.352257\pi$
$500$	−1.00000	−0.0447214
$501$	−16.0000	−0.714827
$502$	−24.0000	−1.07117
$503$	16.0000	0.713405	0.356702	−	0.934218i	$-0.383901\pi$
$503$	16.0000	0.713405	0.356702	+	0.934218i	$0.383901\pi$
$504$	−3.00000	−0.133631
$505$	2.00000	0.0889988
$506$	4.00000	0.177822
$507$	23.0000	1.02147
$508$	−8.00000	−0.354943
$509$	22.0000	0.975133	0.487566	−	0.873086i	$-0.337885\pi$
$509$	22.0000	0.975133	0.487566	+	0.873086i	$0.337885\pi$
$510$	2.00000	0.0885615
$511$	−14.0000	−0.619324
$512$	−11.0000	−0.486136
$513$	4.00000	0.176604
$514$	18.0000	0.793946
$515$	−16.0000	−0.705044
$516$	−4.00000	−0.176090
$517$	−8.00000	−0.351840
$518$	−2.00000	−0.0878750
$519$	−18.0000	−0.790112
$520$	−18.0000	−0.789352
$521$	−14.0000	−0.613351	−0.306676	−	0.951814i	$-0.599217\pi$
$521$	−14.0000	−0.613351	−0.306676	+	0.951814i	$0.599217\pi$
$522$	−2.00000	−0.0875376
$523$	20.0000	0.874539	0.437269	−	0.899331i	$-0.355946\pi$
$523$	20.0000	0.874539	0.437269	+	0.899331i	$0.355946\pi$
$524$	−20.0000	−0.873704
$525$	1.00000	0.0436436
$526$	16.0000	0.697633
$527$	8.00000	0.348485
$528$	1.00000	0.0435194
$529$	−7.00000	−0.304348
$530$	−2.00000	−0.0868744
$531$	−8.00000	−0.347170
$532$	−4.00000	−0.173422
$533$	36.0000	1.55933
$534$	−6.00000	−0.259645
$535$	−12.0000	−0.518805
$536$	24.0000	1.03664
$537$	−12.0000	−0.517838
$538$	−10.0000	−0.431131
$539$	−1.00000	−0.0430730
$540$	−1.00000	−0.0430331
$541$	30.0000	1.28980	0.644900	−	0.764267i	$-0.276899\pi$
$541$	30.0000	1.28980	0.644900	+	0.764267i	$0.276899\pi$
$542$	24.0000	1.03089
$543$	−2.00000	−0.0858282
$544$	10.0000	0.428746
$545$	6.00000	0.257012
$546$	6.00000	0.256776
$547$	−12.0000	−0.513083	−0.256541	−	0.966533i	$-0.582583\pi$
$547$	−12.0000	−0.513083	−0.256541	+	0.966533i	$0.582583\pi$
$548$	−2.00000	−0.0854358
$549$	10.0000	0.426790
$550$	−1.00000	−0.0426401
$551$	−8.00000	−0.340811
$552$	12.0000	0.510754
$553$	−16.0000	−0.680389
$554$	2.00000	0.0849719
$555$	−2.00000	−0.0848953
$556$	12.0000	0.508913
$557$	2.00000	0.0847427	0.0423714	−	0.999102i	$-0.486509\pi$
$557$	2.00000	0.0847427	0.0423714	+	0.999102i	$0.486509\pi$
$558$	4.00000	0.169334
$559$	24.0000	1.01509
$560$	−1.00000	−0.0422577
$561$	−2.00000	−0.0844401
$562$	−6.00000	−0.253095
$563$	−44.0000	−1.85438	−0.927189	−	0.374593i	$-0.877783\pi$
$563$	−44.0000	−1.85438	−0.927189	+	0.374593i	$0.877783\pi$
$564$	−8.00000	−0.336861
$565$	−6.00000	−0.252422
$566$	−4.00000	−0.168133
$567$	1.00000	0.0419961
$568$	0	0
$569$	−30.0000	−1.25767	−0.628833	−	0.777541i	$-0.716467\pi$
$569$	−30.0000	−1.25767	−0.628833	+	0.777541i	$0.716467\pi$
$570$	4.00000	0.167542
$571$	−12.0000	−0.502184	−0.251092	−	0.967963i	$-0.580790\pi$
$571$	−12.0000	−0.502184	−0.251092	+	0.967963i	$0.580790\pi$
$572$	6.00000	0.250873
$573$	−8.00000	−0.334205
$574$	6.00000	0.250435
$575$	−4.00000	−0.166812
$576$	7.00000	0.291667
$577$	34.0000	1.41544	0.707719	−	0.706494i	$-0.249724\pi$
$577$	34.0000	1.41544	0.707719	+	0.706494i	$0.249724\pi$
$578$	−13.0000	−0.540729
$579$	−18.0000	−0.748054
$580$	2.00000	0.0830455
$581$	−4.00000	−0.165948
$582$	18.0000	0.746124
$583$	2.00000	0.0828315
$584$	42.0000	1.73797
$585$	6.00000	0.248069
$586$	−10.0000	−0.413096
$587$	−12.0000	−0.495293	−0.247647	−	0.968850i	$-0.579657\pi$
$587$	−12.0000	−0.495293	−0.247647	+	0.968850i	$0.579657\pi$
$588$	−1.00000	−0.0412393
$589$	16.0000	0.659269
$590$	−8.00000	−0.329355
$591$	−6.00000	−0.246807
$592$	2.00000	0.0821995
$593$	42.0000	1.72473	0.862367	−	0.506284i	$-0.168981\pi$
$593$	42.0000	1.72473	0.862367	+	0.506284i	$0.168981\pi$
$594$	−1.00000	−0.0410305
$595$	2.00000	0.0819920
$596$	10.0000	0.409616
$597$	20.0000	0.818546
$598$	−24.0000	−0.981433
$599$	32.0000	1.30748	0.653742	−	0.756717i	$-0.273198\pi$
$599$	32.0000	1.30748	0.653742	+	0.756717i	$0.273198\pi$
$600$	−3.00000	−0.122474
$601$	6.00000	0.244745	0.122373	−	0.992484i	$-0.460950\pi$
$601$	6.00000	0.244745	0.122373	+	0.992484i	$0.460950\pi$
$602$	4.00000	0.163028
$603$	−8.00000	−0.325785
$604$	16.0000	0.651031
$605$	1.00000	0.0406558
$606$	2.00000	0.0812444
$607$	8.00000	0.324710	0.162355	−	0.986732i	$-0.448091\pi$
$607$	8.00000	0.324710	0.162355	+	0.986732i	$0.448091\pi$
$608$	20.0000	0.811107
$609$	−2.00000	−0.0810441
$610$	10.0000	0.404888
$611$	48.0000	1.94187
$612$	−2.00000	−0.0808452
$613$	18.0000	0.727013	0.363507	−	0.931592i	$-0.381579\pi$
$613$	18.0000	0.727013	0.363507	+	0.931592i	$0.381579\pi$
$614$	12.0000	0.484281
$615$	6.00000	0.241943
$616$	3.00000	0.120873
$617$	18.0000	0.724653	0.362326	−	0.932051i	$-0.381983\pi$
$617$	18.0000	0.724653	0.362326	+	0.932051i	$0.381983\pi$
$618$	−16.0000	−0.643614
$619$	24.0000	0.964641	0.482321	−	0.875995i	$-0.339794\pi$
$619$	24.0000	0.964641	0.482321	+	0.875995i	$0.339794\pi$
$620$	−4.00000	−0.160644
$621$	−4.00000	−0.160514
$622$	12.0000	0.481156
$623$	−6.00000	−0.240385
$624$	−6.00000	−0.240192
$625$	1.00000	0.0400000
$626$	26.0000	1.03917
$627$	−4.00000	−0.159745
$628$	2.00000	0.0798087
$629$	−4.00000	−0.159490
$630$	1.00000	0.0398410
$631$	−24.0000	−0.955425	−0.477712	−	0.878516i	$-0.658534\pi$
$631$	−24.0000	−0.955425	−0.477712	+	0.878516i	$0.658534\pi$
$632$	48.0000	1.90934
$633$	4.00000	0.158986
$634$	14.0000	0.556011
$635$	8.00000	0.317470
$636$	2.00000	0.0793052
$637$	6.00000	0.237729
$638$	2.00000	0.0791808
$639$	0	0
$640$	−3.00000	−0.118585
$641$	18.0000	0.710957	0.355479	−	0.934684i	$-0.384318\pi$
$641$	18.0000	0.710957	0.355479	+	0.934684i	$0.384318\pi$
$642$	−12.0000	−0.473602
$643$	−20.0000	−0.788723	−0.394362	−	0.918955i	$-0.629034\pi$
$643$	−20.0000	−0.788723	−0.394362	+	0.918955i	$0.629034\pi$
$644$	4.00000	0.157622
$645$	4.00000	0.157500
$646$	8.00000	0.314756
$647$	0	0		−	1.00000i	$-0.5\pi$
$647$	0	0			1.00000i	$0.5\pi$
$648$	−3.00000	−0.117851
$649$	8.00000	0.314027
$650$	6.00000	0.235339
$651$	4.00000	0.156772
$652$	−24.0000	−0.939913
$653$	−10.0000	−0.391330	−0.195665	−	0.980671i	$-0.562687\pi$
$653$	−10.0000	−0.391330	−0.195665	+	0.980671i	$0.562687\pi$
$654$	6.00000	0.234619
$655$	20.0000	0.781465
$656$	−6.00000	−0.234261
$657$	−14.0000	−0.546192
$658$	8.00000	0.311872
$659$	12.0000	0.467454	0.233727	−	0.972302i	$-0.424908\pi$
$659$	12.0000	0.467454	0.233727	+	0.972302i	$0.424908\pi$
$660$	1.00000	0.0389249
$661$	−10.0000	−0.388955	−0.194477	−	0.980907i	$-0.562301\pi$
$661$	−10.0000	−0.388955	−0.194477	+	0.980907i	$0.562301\pi$
$662$	−12.0000	−0.466393
$663$	12.0000	0.466041
$664$	12.0000	0.465690
$665$	4.00000	0.155113
$666$	−2.00000	−0.0774984
$667$	8.00000	0.309761
$668$	16.0000	0.619059
$669$	0	0
$670$	−8.00000	−0.309067
$671$	−10.0000	−0.386046
$672$	5.00000	0.192879
$673$	−10.0000	−0.385472	−0.192736	−	0.981251i	$-0.561736\pi$
$673$	−10.0000	−0.385472	−0.192736	+	0.981251i	$0.561736\pi$
$674$	22.0000	0.847408
$675$	1.00000	0.0384900
$676$	−23.0000	−0.884615
$677$	−2.00000	−0.0768662	−0.0384331	−	0.999261i	$-0.512237\pi$
$677$	−2.00000	−0.0768662	−0.0384331	+	0.999261i	$0.512237\pi$
$678$	−6.00000	−0.230429
$679$	18.0000	0.690777
$680$	−6.00000	−0.230089
$681$	−28.0000	−1.07296
$682$	−4.00000	−0.153168
$683$	0	0		−	1.00000i	$-0.5\pi$
$683$	0	0			1.00000i	$0.5\pi$
$684$	−4.00000	−0.152944
$685$	2.00000	0.0764161
$686$	1.00000	0.0381802
$687$	−2.00000	−0.0763048
$688$	−4.00000	−0.152499
$689$	−12.0000	−0.457164
$690$	−4.00000	−0.152277
$691$	−24.0000	−0.913003	−0.456502	−	0.889723i	$-0.650898\pi$
$691$	−24.0000	−0.913003	−0.456502	+	0.889723i	$0.650898\pi$
$692$	18.0000	0.684257
$693$	−1.00000	−0.0379869
$694$	28.0000	1.06287
$695$	−12.0000	−0.455186
$696$	6.00000	0.227429
$697$	12.0000	0.454532
$698$	−22.0000	−0.832712
$699$	6.00000	0.226941
$700$	−1.00000	−0.0377964
$701$	−2.00000	−0.0755390	−0.0377695	−	0.999286i	$-0.512025\pi$
$701$	−2.00000	−0.0755390	−0.0377695	+	0.999286i	$0.512025\pi$
$702$	6.00000	0.226455
$703$	−8.00000	−0.301726
$704$	−7.00000	−0.263822
$705$	8.00000	0.301297
$706$	−30.0000	−1.12906
$707$	2.00000	0.0752177
$708$	8.00000	0.300658
$709$	−26.0000	−0.976450	−0.488225	−	0.872718i	$-0.662356\pi$
$709$	−26.0000	−0.976450	−0.488225	+	0.872718i	$0.662356\pi$
$710$	0	0
$711$	−16.0000	−0.600047
$712$	18.0000	0.674579
$713$	−16.0000	−0.599205
$714$	2.00000	0.0748481
$715$	−6.00000	−0.224387
$716$	12.0000	0.448461
$717$	8.00000	0.298765
$718$	−24.0000	−0.895672
$719$	44.0000	1.64092	0.820462	−	0.571702i	$-0.193717\pi$
$719$	44.0000	1.64092	0.820462	+	0.571702i	$0.193717\pi$
$720$	−1.00000	−0.0372678
$721$	−16.0000	−0.595871
$722$	−3.00000	−0.111648
$723$	−18.0000	−0.669427
$724$	2.00000	0.0743294
$725$	−2.00000	−0.0742781
$726$	1.00000	0.0371135
$727$	16.0000	0.593407	0.296704	−	0.954970i	$-0.404113\pi$
$727$	16.0000	0.593407	0.296704	+	0.954970i	$0.404113\pi$
$728$	−18.0000	−0.667124
$729$	1.00000	0.0370370
$730$	−14.0000	−0.518163
$731$	8.00000	0.295891
$732$	−10.0000	−0.369611
$733$	−10.0000	−0.369358	−0.184679	−	0.982799i	$-0.559125\pi$
$733$	−10.0000	−0.369358	−0.184679	+	0.982799i	$0.559125\pi$
$734$	−8.00000	−0.295285
$735$	1.00000	0.0368856
$736$	−20.0000	−0.737210
$737$	8.00000	0.294684
$738$	6.00000	0.220863
$739$	20.0000	0.735712	0.367856	−	0.929883i	$-0.380092\pi$
$739$	20.0000	0.735712	0.367856	+	0.929883i	$0.380092\pi$
$740$	2.00000	0.0735215
$741$	24.0000	0.881662
$742$	−2.00000	−0.0734223
$743$	−48.0000	−1.76095	−0.880475	−	0.474093i	$-0.842776\pi$
$743$	−48.0000	−1.76095	−0.880475	+	0.474093i	$0.842776\pi$
$744$	−12.0000	−0.439941
$745$	−10.0000	−0.366372
$746$	26.0000	0.951928
$747$	−4.00000	−0.146352
$748$	2.00000	0.0731272
$749$	−12.0000	−0.438470
$750$	1.00000	0.0365148
$751$	8.00000	0.291924	0.145962	−	0.989290i	$-0.453372\pi$
$751$	8.00000	0.291924	0.145962	+	0.989290i	$0.453372\pi$
$752$	−8.00000	−0.291730
$753$	−24.0000	−0.874609
$754$	−12.0000	−0.437014
$755$	−16.0000	−0.582300
$756$	−1.00000	−0.0363696
$757$	−10.0000	−0.363456	−0.181728	−	0.983349i	$-0.558169\pi$
$757$	−10.0000	−0.363456	−0.181728	+	0.983349i	$0.558169\pi$
$758$	−4.00000	−0.145287
$759$	4.00000	0.145191
$760$	−12.0000	−0.435286
$761$	−42.0000	−1.52250	−0.761249	−	0.648459i	$-0.775414\pi$
$761$	−42.0000	−1.52250	−0.761249	+	0.648459i	$0.775414\pi$
$762$	8.00000	0.289809
$763$	6.00000	0.217215
$764$	8.00000	0.289430
$765$	2.00000	0.0723102
$766$	16.0000	0.578103
$767$	−48.0000	−1.73318
$768$	−17.0000	−0.613435
$769$	−10.0000	−0.360609	−0.180305	−	0.983611i	$-0.557708\pi$
$769$	−10.0000	−0.360609	−0.180305	+	0.983611i	$0.557708\pi$
$770$	−1.00000	−0.0360375
$771$	18.0000	0.648254
$772$	18.0000	0.647834
$773$	54.0000	1.94225	0.971123	−	0.238581i	$-0.0766824\pi$
$773$	54.0000	1.94225	0.971123	+	0.238581i	$0.0766824\pi$
$774$	4.00000	0.143777
$775$	4.00000	0.143684
$776$	−54.0000	−1.93849
$777$	−2.00000	−0.0717496
$778$	6.00000	0.215110
$779$	24.0000	0.859889
$780$	−6.00000	−0.214834
$781$	0	0
$782$	−8.00000	−0.286079
$783$	−2.00000	−0.0714742
$784$	−1.00000	−0.0357143
$785$	−2.00000	−0.0713831
$786$	20.0000	0.713376
$787$	−44.0000	−1.56843	−0.784215	−	0.620489i	$-0.786934\pi$
$787$	−44.0000	−1.56843	−0.784215	+	0.620489i	$0.786934\pi$
$788$	6.00000	0.213741
$789$	16.0000	0.569615
$790$	−16.0000	−0.569254
$791$	−6.00000	−0.213335
$792$	3.00000	0.106600
$793$	60.0000	2.13066
$794$	14.0000	0.496841
$795$	−2.00000	−0.0709327
$796$	−20.0000	−0.708881
$797$	30.0000	1.06265	0.531327	−	0.847167i	$-0.321693\pi$
$797$	30.0000	1.06265	0.531327	+	0.847167i	$0.321693\pi$
$798$	4.00000	0.141598
$799$	16.0000	0.566039
$800$	5.00000	0.176777
$801$	−6.00000	−0.212000
$802$	18.0000	0.635602
$803$	14.0000	0.494049
$804$	8.00000	0.282138
$805$	−4.00000	−0.140981
$806$	24.0000	0.845364
$807$	−10.0000	−0.352017
$808$	−6.00000	−0.211079
$809$	−30.0000	−1.05474	−0.527372	−	0.849635i	$-0.676823\pi$
$809$	−30.0000	−1.05474	−0.527372	+	0.849635i	$0.676823\pi$
$810$	1.00000	0.0351364
$811$	−12.0000	−0.421377	−0.210688	−	0.977553i	$-0.567571\pi$
$811$	−12.0000	−0.421377	−0.210688	+	0.977553i	$0.567571\pi$
$812$	2.00000	0.0701862
$813$	24.0000	0.841717
$814$	2.00000	0.0701000
$815$	24.0000	0.840683
$816$	−2.00000	−0.0700140
$817$	16.0000	0.559769
$818$	−10.0000	−0.349642
$819$	6.00000	0.209657
$820$	−6.00000	−0.209529
$821$	−18.0000	−0.628204	−0.314102	−	0.949389i	$-0.601703\pi$
$821$	−18.0000	−0.628204	−0.314102	+	0.949389i	$0.601703\pi$
$822$	2.00000	0.0697580
$823$	−4.00000	−0.139431	−0.0697156	−	0.997567i	$-0.522209\pi$
$823$	−4.00000	−0.139431	−0.0697156	+	0.997567i	$0.522209\pi$
$824$	48.0000	1.67216
$825$	−1.00000	−0.0348155
$826$	−8.00000	−0.278356
$827$	−36.0000	−1.25184	−0.625921	−	0.779886i	$-0.715277\pi$
$827$	−36.0000	−1.25184	−0.625921	+	0.779886i	$0.715277\pi$
$828$	4.00000	0.139010
$829$	30.0000	1.04194	0.520972	−	0.853574i	$-0.325570\pi$
$829$	30.0000	1.04194	0.520972	+	0.853574i	$0.325570\pi$
$830$	−4.00000	−0.138842
$831$	2.00000	0.0693792
$832$	42.0000	1.45609
$833$	2.00000	0.0692959
$834$	−12.0000	−0.415526
$835$	−16.0000	−0.553703
$836$	4.00000	0.138343
$837$	4.00000	0.138260
$838$	−24.0000	−0.829066
$839$	−36.0000	−1.24286	−0.621429	−	0.783470i	$-0.713448\pi$
$839$	−36.0000	−1.24286	−0.621429	+	0.783470i	$0.713448\pi$
$840$	−3.00000	−0.103510
$841$	−25.0000	−0.862069
$842$	−26.0000	−0.896019
$843$	−6.00000	−0.206651
$844$	−4.00000	−0.137686
$845$	23.0000	0.791224
$846$	8.00000	0.275046
$847$	1.00000	0.0343604
$848$	2.00000	0.0686803
$849$	−4.00000	−0.137280
$850$	2.00000	0.0685994
$851$	8.00000	0.274236
$852$	0	0
$853$	54.0000	1.84892	0.924462	−	0.381273i	$-0.124514\pi$
$853$	54.0000	1.84892	0.924462	+	0.381273i	$0.124514\pi$
$854$	10.0000	0.342193
$855$	4.00000	0.136797
$856$	36.0000	1.23045
$857$	−30.0000	−1.02478	−0.512390	−	0.858753i	$-0.671240\pi$
$857$	−30.0000	−1.02478	−0.512390	+	0.858753i	$0.671240\pi$
$858$	−6.00000	−0.204837
$859$	8.00000	0.272956	0.136478	−	0.990643i	$-0.456422\pi$
$859$	8.00000	0.272956	0.136478	+	0.990643i	$0.456422\pi$
$860$	−4.00000	−0.136399
$861$	6.00000	0.204479
$862$	−32.0000	−1.08992
$863$	−12.0000	−0.408485	−0.204242	−	0.978920i	$-0.565473\pi$
$863$	−12.0000	−0.408485	−0.204242	+	0.978920i	$0.565473\pi$
$864$	5.00000	0.170103
$865$	−18.0000	−0.612018
$866$	34.0000	1.15537
$867$	−13.0000	−0.441503
$868$	−4.00000	−0.135769
$869$	16.0000	0.542763
$870$	−2.00000	−0.0678064
$871$	−48.0000	−1.62642
$872$	−18.0000	−0.609557
$873$	18.0000	0.609208
$874$	−16.0000	−0.541208
$875$	1.00000	0.0338062
$876$	14.0000	0.473016
$877$	−22.0000	−0.742887	−0.371444	−	0.928456i	$-0.621137\pi$
$877$	−22.0000	−0.742887	−0.371444	+	0.928456i	$0.621137\pi$
$878$	−32.0000	−1.07995
$879$	−10.0000	−0.337292
$880$	1.00000	0.0337100
$881$	42.0000	1.41502	0.707508	−	0.706705i	$-0.249819\pi$
$881$	42.0000	1.41502	0.707508	+	0.706705i	$0.249819\pi$
$882$	1.00000	0.0336718
$883$	16.0000	0.538443	0.269221	−	0.963078i	$-0.413234\pi$
$883$	16.0000	0.538443	0.269221	+	0.963078i	$0.413234\pi$
$884$	−12.0000	−0.403604
$885$	−8.00000	−0.268917
$886$	−24.0000	−0.806296
$887$	−8.00000	−0.268614	−0.134307	−	0.990940i	$-0.542881\pi$
$887$	−8.00000	−0.268614	−0.134307	+	0.990940i	$0.542881\pi$
$888$	6.00000	0.201347
$889$	8.00000	0.268311
$890$	−6.00000	−0.201120
$891$	−1.00000	−0.0335013
$892$	0	0
$893$	32.0000	1.07084
$894$	−10.0000	−0.334450
$895$	−12.0000	−0.401116
$896$	−3.00000	−0.100223
$897$	−24.0000	−0.801337
$898$	18.0000	0.600668
$899$	−8.00000	−0.266815
$900$	−1.00000	−0.0333333
$901$	−4.00000	−0.133259
$902$	−6.00000	−0.199778
$903$	4.00000	0.133112
$904$	18.0000	0.598671
$905$	−2.00000	−0.0664822
$906$	−16.0000	−0.531564
$907$	−32.0000	−1.06254	−0.531271	−	0.847202i	$-0.678286\pi$
$907$	−32.0000	−1.06254	−0.531271	+	0.847202i	$0.678286\pi$
$908$	28.0000	0.929213
$909$	2.00000	0.0663358
$910$	6.00000	0.198898
$911$	−48.0000	−1.59031	−0.795155	−	0.606406i	$-0.792611\pi$
$911$	−48.0000	−1.59031	−0.795155	+	0.606406i	$0.792611\pi$
$912$	−4.00000	−0.132453
$913$	4.00000	0.132381
$914$	30.0000	0.992312
$915$	10.0000	0.330590
$916$	2.00000	0.0660819
$917$	20.0000	0.660458
$918$	2.00000	0.0660098
$919$	−24.0000	−0.791687	−0.395843	−	0.918318i	$-0.629548\pi$
$919$	−24.0000	−0.791687	−0.395843	+	0.918318i	$0.629548\pi$
$920$	12.0000	0.395628
$921$	12.0000	0.395413
$922$	34.0000	1.11973
$923$	0	0
$924$	1.00000	0.0328976
$925$	−2.00000	−0.0657596
$926$	36.0000	1.18303
$927$	−16.0000	−0.525509
$928$	−10.0000	−0.328266
$929$	10.0000	0.328089	0.164045	−	0.986453i	$-0.447546\pi$
$929$	10.0000	0.328089	0.164045	+	0.986453i	$0.447546\pi$
$930$	4.00000	0.131165
$931$	4.00000	0.131095
$932$	−6.00000	−0.196537
$933$	12.0000	0.392862
$934$	28.0000	0.916188
$935$	−2.00000	−0.0654070
$936$	−18.0000	−0.588348
$937$	10.0000	0.326686	0.163343	−	0.986569i	$-0.447772\pi$
$937$	10.0000	0.326686	0.163343	+	0.986569i	$0.447772\pi$
$938$	−8.00000	−0.261209
$939$	26.0000	0.848478
$940$	−8.00000	−0.260931
$941$	18.0000	0.586783	0.293392	−	0.955992i	$-0.405216\pi$
$941$	18.0000	0.586783	0.293392	+	0.955992i	$0.405216\pi$
$942$	−2.00000	−0.0651635
$943$	−24.0000	−0.781548
$944$	8.00000	0.260378
$945$	1.00000	0.0325300
$946$	−4.00000	−0.130051
$947$	32.0000	1.03986	0.519930	−	0.854209i	$-0.325958\pi$
$947$	32.0000	1.03986	0.519930	+	0.854209i	$0.325958\pi$
$948$	16.0000	0.519656
$949$	−84.0000	−2.72676
$950$	4.00000	0.129777
$951$	14.0000	0.453981
$952$	−6.00000	−0.194461
$953$	46.0000	1.49009	0.745043	−	0.667016i	$-0.232429\pi$
$953$	46.0000	1.49009	0.745043	+	0.667016i	$0.232429\pi$
$954$	−2.00000	−0.0647524
$955$	−8.00000	−0.258874
$956$	−8.00000	−0.258738
$957$	2.00000	0.0646508
$958$	−16.0000	−0.516937
$959$	2.00000	0.0645834
$960$	7.00000	0.225924
$961$	−15.0000	−0.483871
$962$	−12.0000	−0.386896
$963$	−12.0000	−0.386695
$964$	18.0000	0.579741
$965$	−18.0000	−0.579441
$966$	−4.00000	−0.128698
$967$	−56.0000	−1.80084	−0.900419	−	0.435023i	$-0.856740\pi$
$967$	−56.0000	−1.80084	−0.900419	+	0.435023i	$0.856740\pi$
$968$	−3.00000	−0.0964237
$969$	8.00000	0.256997
$970$	18.0000	0.577945
$971$	32.0000	1.02693	0.513464	−	0.858111i	$-0.328362\pi$
$971$	32.0000	1.02693	0.513464	+	0.858111i	$0.328362\pi$
$972$	−1.00000	−0.0320750
$973$	−12.0000	−0.384702
$974$	−28.0000	−0.897178
$975$	6.00000	0.192154
$976$	−10.0000	−0.320092
$977$	26.0000	0.831814	0.415907	−	0.909407i	$-0.363464\pi$
$977$	26.0000	0.831814	0.415907	+	0.909407i	$0.363464\pi$
$978$	24.0000	0.767435
$979$	6.00000	0.191761
$980$	−1.00000	−0.0319438
$981$	6.00000	0.191565
$982$	12.0000	0.382935
$983$	−16.0000	−0.510321	−0.255160	−	0.966899i	$-0.582128\pi$
$983$	−16.0000	−0.510321	−0.255160	+	0.966899i	$0.582128\pi$
$984$	−18.0000	−0.573819
$985$	−6.00000	−0.191176
$986$	−4.00000	−0.127386
$987$	8.00000	0.254643
$988$	−24.0000	−0.763542
$989$	−16.0000	−0.508770
$990$	−1.00000	−0.0317821
$991$	0	0		−	1.00000i	$-0.5\pi$
$991$	0	0			1.00000i	$0.5\pi$
$992$	20.0000	0.635001
$993$	−12.0000	−0.380808
$994$	0	0
$995$	20.0000	0.634043
$996$	4.00000	0.126745
$997$	−10.0000	−0.316703	−0.158352	−	0.987383i	$-0.550618\pi$
$997$	−10.0000	−0.316703	−0.158352	+	0.987383i	$0.550618\pi$
$998$	20.0000	0.633089
$999$	−2.00000	−0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1155.2.a.m.1.1	✓	1
3.2	odd	2		3465.2.a.c.1.1		1
5.4	even	2		5775.2.a.d.1.1		1
7.6	odd	2		8085.2.a.r.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1155.2.a.m.1.1	✓	1	1.1	even	1	trivial
3465.2.a.c.1.1		1	3.2	odd	2
5775.2.a.d.1.1		1	5.4	even	2
8085.2.a.r.1.1		1	7.6	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1155 = 3 \cdot 5 \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1155.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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