LMFDB - Embedded newform 1155.2.a.k.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:	$1$
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	$\chi$	$=$	1155.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

\(q+1.00000 q^{2} -1.00000 q^{3} -1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} +1.00000 q^{7} -3.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -1.00000 q^{11} +1.00000 q^{12} -2.00000 q^{13} +1.00000 q^{14} -1.00000 q^{15} -1.00000 q^{16} -6.00000 q^{17} +1.00000 q^{18} +4.00000 q^{19} -1.00000 q^{20} -1.00000 q^{21} -1.00000 q^{22} -4.00000 q^{23} +3.00000 q^{24} +1.00000 q^{25} -2.00000 q^{26} -1.00000 q^{27} -1.00000 q^{28} -2.00000 q^{29} -1.00000 q^{30} -4.00000 q^{31} +5.00000 q^{32} +1.00000 q^{33} -6.00000 q^{34} +1.00000 q^{35} -1.00000 q^{36} -2.00000 q^{37} +4.00000 q^{38} +2.00000 q^{39} -3.00000 q^{40} -2.00000 q^{41} -1.00000 q^{42} -12.0000 q^{43} +1.00000 q^{44} +1.00000 q^{45} -4.00000 q^{46} +1.00000 q^{48} +1.00000 q^{49} +1.00000 q^{50} +6.00000 q^{51} +2.00000 q^{52} -2.00000 q^{53} -1.00000 q^{54} -1.00000 q^{55} -3.00000 q^{56} -4.00000 q^{57} -2.00000 q^{58} +1.00000 q^{60} -14.0000 q^{61} -4.00000 q^{62} +1.00000 q^{63} +7.00000 q^{64} -2.00000 q^{65} +1.00000 q^{66} -8.00000 q^{67} +6.00000 q^{68} +4.00000 q^{69} +1.00000 q^{70} -16.0000 q^{71} -3.00000 q^{72} +10.0000 q^{73} -2.00000 q^{74} -1.00000 q^{75} -4.00000 q^{76} -1.00000 q^{77} +2.00000 q^{78} +16.0000 q^{79} -1.00000 q^{80} +1.00000 q^{81} -2.00000 q^{82} -4.00000 q^{83} +1.00000 q^{84} -6.00000 q^{85} -12.0000 q^{86} +2.00000 q^{87} +3.00000 q^{88} -6.00000 q^{89} +1.00000 q^{90} -2.00000 q^{91} +4.00000 q^{92} +4.00000 q^{93} +4.00000 q^{95} -5.00000 q^{96} +2.00000 q^{97} +1.00000 q^{98} -1.00000 q^{99} +O(q^{100})\)

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
$13$	−2.00000	−0.554700	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	1.00000	0.267261
$15$	−1.00000	−0.258199
$16$	−1.00000	−0.250000
$17$	−6.00000	−1.45521	−0.727607	−	0.685994i	$-0.759367\pi$
$17$	−6.00000	−1.45521	−0.727607	+	0.685994i	$0.759367\pi$
$18$	1.00000	0.235702
$19$	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
$26$	−2.00000	−0.392232
$27$	−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	$-0.616954\pi$
$31$	−4.00000	−0.718421	−0.359211	+	0.933257i	$0.616954\pi$
$32$	5.00000	0.883883
$33$	1.00000	0.174078
$34$	−6.00000	−1.02899
$35$	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
$39$	2.00000	0.320256
$40$	−3.00000	−0.474342
$41$	−2.00000	−0.312348	−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000	−0.312348	−0.156174	+	0.987730i	$0.549916\pi$
$42$	−1.00000	−0.154303
$43$	−12.0000	−1.82998	−0.914991	−	0.403473i	$-0.867803\pi$
$43$	−12.0000	−1.82998	−0.914991	+	0.403473i	$0.867803\pi$
$44$	1.00000	0.150756
$45$	1.00000	0.149071
$46$	−4.00000	−0.589768
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	1.00000	0.144338
$49$	1.00000	0.142857
$50$	1.00000	0.141421
$51$	6.00000	0.840168
$52$	2.00000	0.277350
$53$	−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
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	1.00000	0.129099
$61$	−14.0000	−1.79252	−0.896258	−	0.443533i	$-0.853725\pi$
$61$	−14.0000	−1.79252	−0.896258	+	0.443533i	$0.853725\pi$
$62$	−4.00000	−0.508001
$63$	1.00000	0.125988
$64$	7.00000	0.875000
$65$	−2.00000	−0.248069
$66$	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	$0.662521\pi$
$68$	6.00000	0.727607
$69$	4.00000	0.481543
$70$	1.00000	0.119523
$71$	−16.0000	−1.89885	−0.949425	−	0.313993i	$-0.898333\pi$
$71$	−16.0000	−1.89885	−0.949425	+	0.313993i	$0.898333\pi$
$72$	−3.00000	−0.353553
$73$	10.0000	1.17041	0.585206	−	0.810885i	$-0.301014\pi$
$73$	10.0000	1.17041	0.585206	+	0.810885i	$0.301014\pi$
$74$	−2.00000	−0.232495
$75$	−1.00000	−0.115470
$76$	−4.00000	−0.458831
$77$	−1.00000	−0.113961
$78$	2.00000	0.226455
$79$	16.0000	1.80014	0.900070	−	0.435745i	$-0.143515\pi$
$79$	16.0000	1.80014	0.900070	+	0.435745i	$0.143515\pi$
$80$	−1.00000	−0.111803
$81$	1.00000	0.111111
$82$	−2.00000	−0.220863
$83$	−4.00000	−0.439057	−0.219529	−	0.975606i	$-0.570452\pi$
$83$	−4.00000	−0.439057	−0.219529	+	0.975606i	$0.570452\pi$
$84$	1.00000	0.109109
$85$	−6.00000	−0.650791
$86$	−12.0000	−1.29399
$87$	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
$91$	−2.00000	−0.209657
$92$	4.00000	0.417029
$93$	4.00000	0.414781
$94$	0	0
$95$	4.00000	0.410391
$96$	−5.00000	−0.510310
$97$	2.00000	0.203069	0.101535	−	0.994832i	$-0.467625\pi$
$97$	2.00000	0.203069	0.101535	+	0.994832i	$0.467625\pi$
$98$	1.00000	0.101015
$99$	−1.00000	−0.100504
$100$	−1.00000	−0.100000
$101$	10.0000	0.995037	0.497519	−	0.867453i	$-0.334245\pi$
$101$	10.0000	0.995037	0.497519	+	0.867453i	$0.334245\pi$
$102$	6.00000	0.594089
$103$	8.00000	0.788263	0.394132	−	0.919054i	$-0.371045\pi$
$103$	8.00000	0.788263	0.394132	+	0.919054i	$0.371045\pi$
$104$	6.00000	0.588348
$105$	−1.00000	−0.0975900
$106$	−2.00000	−0.194257
$107$	4.00000	0.386695	0.193347	−	0.981130i	$-0.438066\pi$
$107$	4.00000	0.386695	0.193347	+	0.981130i	$0.438066\pi$
$108$	1.00000	0.0962250
$109$	−10.0000	−0.957826	−0.478913	−	0.877862i	$-0.658969\pi$
$109$	−10.0000	−0.957826	−0.478913	+	0.877862i	$0.658969\pi$
$110$	−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
$117$	−2.00000	−0.184900
$118$	0	0
$119$	−6.00000	−0.550019
$120$	3.00000	0.273861
$121$	1.00000	0.0909091
$122$	−14.0000	−1.26750
$123$	2.00000	0.180334
$124$	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
$129$	12.0000	1.05654
$130$	−2.00000	−0.175412
$131$	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
$136$	18.0000	1.54349
$137$	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
$139$	20.0000	1.69638	0.848189	−	0.529694i	$-0.177693\pi$
$139$	20.0000	1.69638	0.848189	+	0.529694i	$0.177693\pi$
$140$	−1.00000	−0.0845154
$141$	0	0
$142$	−16.0000	−1.34269
$143$	2.00000	0.167248
$144$	−1.00000	−0.0833333
$145$	−2.00000	−0.166091
$146$	10.0000	0.827606
$147$	−1.00000	−0.0824786
$148$	2.00000	0.164399
$149$	6.00000	0.491539	0.245770	−	0.969328i	$-0.420959\pi$
$149$	6.00000	0.491539	0.245770	+	0.969328i	$0.420959\pi$
$150$	−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
$153$	−6.00000	−0.485071
$154$	−1.00000	−0.0805823
$155$	−4.00000	−0.321288
$156$	−2.00000	−0.160128
$157$	−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
$163$	−8.00000	−0.626608	−0.313304	−	0.949653i	$-0.601436\pi$
$163$	−8.00000	−0.626608	−0.313304	+	0.949653i	$0.601436\pi$
$164$	2.00000	0.156174
$165$	1.00000	0.0778499
$166$	−4.00000	−0.310460
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	3.00000	0.231455
$169$	−9.00000	−0.692308
$170$	−6.00000	−0.460179
$171$	4.00000	0.305888
$172$	12.0000	0.914991
$173$	−10.0000	−0.760286	−0.380143	−	0.924928i	$-0.624125\pi$
$173$	−10.0000	−0.760286	−0.380143	+	0.924928i	$0.624125\pi$
$174$	2.00000	0.151620
$175$	1.00000	0.0755929
$176$	1.00000	0.0753778
$177$	0	0
$178$	−6.00000	−0.449719
$179$	20.0000	1.49487	0.747435	−	0.664335i	$-0.231285\pi$
$179$	20.0000	1.49487	0.747435	+	0.664335i	$0.231285\pi$
$180$	−1.00000	−0.0745356
$181$	14.0000	1.04061	0.520306	−	0.853980i	$-0.325818\pi$
$181$	14.0000	1.04061	0.520306	+	0.853980i	$0.325818\pi$
$182$	−2.00000	−0.148250
$183$	14.0000	1.03491
$184$	12.0000	0.884652
$185$	−2.00000	−0.147043
$186$	4.00000	0.293294
$187$	6.00000	0.438763
$188$	0	0
$189$	−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
$193$	14.0000	1.00774	0.503871	−	0.863779i	$-0.331909\pi$
$193$	14.0000	1.00774	0.503871	+	0.863779i	$0.331909\pi$
$194$	2.00000	0.143592
$195$	2.00000	0.143223
$196$	−1.00000	−0.0714286
$197$	−22.0000	−1.56744	−0.783718	−	0.621117i	$-0.786679\pi$
$197$	−22.0000	−1.56744	−0.783718	+	0.621117i	$0.786679\pi$
$198$	−1.00000	−0.0710669
$199$	28.0000	1.98487	0.992434	−	0.122782i	$-0.0391815\pi$
$199$	28.0000	1.98487	0.992434	+	0.122782i	$0.0391815\pi$
$200$	−3.00000	−0.212132
$201$	8.00000	0.564276
$202$	10.0000	0.703598
$203$	−2.00000	−0.140372
$204$	−6.00000	−0.420084
$205$	−2.00000	−0.139686
$206$	8.00000	0.557386
$207$	−4.00000	−0.278019
$208$	2.00000	0.138675
$209$	−4.00000	−0.276686
$210$	−1.00000	−0.0690066
$211$	20.0000	1.37686	0.688428	−	0.725304i	$-0.258301\pi$
$211$	20.0000	1.37686	0.688428	+	0.725304i	$0.258301\pi$
$212$	2.00000	0.137361
$213$	16.0000	1.09630
$214$	4.00000	0.273434
$215$	−12.0000	−0.818393
$216$	3.00000	0.204124
$217$	−4.00000	−0.271538
$218$	−10.0000	−0.677285
$219$	−10.0000	−0.675737
$220$	1.00000	0.0674200
$221$	12.0000	0.807207
$222$	2.00000	0.134231
$223$	24.0000	1.60716	0.803579	−	0.595198i	$-0.202926\pi$
$223$	24.0000	1.60716	0.803579	+	0.595198i	$0.202926\pi$
$224$	5.00000	0.334077
$225$	1.00000	0.0666667
$226$	−6.00000	−0.399114
$227$	20.0000	1.32745	0.663723	−	0.747978i	$-0.268975\pi$
$227$	20.0000	1.32745	0.663723	+	0.747978i	$0.268975\pi$
$228$	4.00000	0.264906
$229$	14.0000	0.925146	0.462573	−	0.886581i	$-0.346926\pi$
$229$	14.0000	0.925146	0.462573	+	0.886581i	$0.346926\pi$
$230$	−4.00000	−0.263752
$231$	1.00000	0.0657952
$232$	6.00000	0.393919
$233$	−26.0000	−1.70332	−0.851658	−	0.524097i	$-0.824403\pi$
$233$	−26.0000	−1.70332	−0.851658	+	0.524097i	$0.824403\pi$
$234$	−2.00000	−0.130744
$235$	0	0
$236$	0	0
$237$	−16.0000	−1.03931
$238$	−6.00000	−0.388922
$239$	−24.0000	−1.55243	−0.776215	−	0.630468i	$-0.782863\pi$
$239$	−24.0000	−1.55243	−0.776215	+	0.630468i	$0.782863\pi$
$240$	1.00000	0.0645497
$241$	−10.0000	−0.644157	−0.322078	−	0.946713i	$-0.604381\pi$
$241$	−10.0000	−0.644157	−0.322078	+	0.946713i	$0.604381\pi$
$242$	1.00000	0.0642824
$243$	−1.00000	−0.0641500
$244$	14.0000	0.896258
$245$	1.00000	0.0638877
$246$	2.00000	0.127515
$247$	−8.00000	−0.509028
$248$	12.0000	0.762001
$249$	4.00000	0.253490
$250$	1.00000	0.0632456
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	−1.00000	−0.0629941
$253$	4.00000	0.251478
$254$	8.00000	0.501965
$255$	6.00000	0.375735
$256$	−17.0000	−1.06250
$257$	−14.0000	−0.873296	−0.436648	−	0.899632i	$-0.643834\pi$
$257$	−14.0000	−0.873296	−0.436648	+	0.899632i	$0.643834\pi$
$258$	12.0000	0.747087
$259$	−2.00000	−0.124274
$260$	2.00000	0.124035
$261$	−2.00000	−0.123797
$262$	20.0000	1.23560
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	−3.00000	−0.184637
$265$	−2.00000	−0.122859
$266$	4.00000	0.245256
$267$	6.00000	0.367194
$268$	8.00000	0.488678
$269$	−26.0000	−1.58525	−0.792624	−	0.609711i	$-0.791286\pi$
$269$	−26.0000	−1.58525	−0.792624	+	0.609711i	$0.791286\pi$
$270$	−1.00000	−0.0608581
$271$	−8.00000	−0.485965	−0.242983	−	0.970031i	$-0.578126\pi$
$271$	−8.00000	−0.485965	−0.242983	+	0.970031i	$0.578126\pi$
$272$	6.00000	0.363803
$273$	2.00000	0.121046
$274$	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	$0.480863\pi$
$278$	20.0000	1.19952
$279$	−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	$0.557275\pi$
$282$	0	0
$283$	−4.00000	−0.237775	−0.118888	−	0.992908i	$-0.537933\pi$
$283$	−4.00000	−0.237775	−0.118888	+	0.992908i	$0.537933\pi$
$284$	16.0000	0.949425
$285$	−4.00000	−0.236940
$286$	2.00000	0.118262
$287$	−2.00000	−0.118056
$288$	5.00000	0.294628
$289$	19.0000	1.11765
$290$	−2.00000	−0.117444
$291$	−2.00000	−0.117242
$292$	−10.0000	−0.585206
$293$	−18.0000	−1.05157	−0.525786	−	0.850617i	$-0.676229\pi$
$293$	−18.0000	−1.05157	−0.525786	+	0.850617i	$0.676229\pi$
$294$	−1.00000	−0.0583212
$295$	0	0
$296$	6.00000	0.348743
$297$	1.00000	0.0580259
$298$	6.00000	0.347571
$299$	8.00000	0.462652
$300$	1.00000	0.0577350
$301$	−12.0000	−0.691669
$302$	−16.0000	−0.920697
$303$	−10.0000	−0.574485
$304$	−4.00000	−0.229416
$305$	−14.0000	−0.801638
$306$	−6.00000	−0.342997
$307$	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
$309$	−8.00000	−0.455104
$310$	−4.00000	−0.227185
$311$	20.0000	1.13410	0.567048	−	0.823685i	$-0.308085\pi$
$311$	20.0000	1.13410	0.567048	+	0.823685i	$0.308085\pi$
$312$	−6.00000	−0.339683
$313$	−6.00000	−0.339140	−0.169570	−	0.985518i	$-0.554238\pi$
$313$	−6.00000	−0.339140	−0.169570	+	0.985518i	$0.554238\pi$
$314$	−2.00000	−0.112867
$315$	1.00000	0.0563436
$316$	−16.0000	−0.900070
$317$	−18.0000	−1.01098	−0.505490	−	0.862832i	$-0.668688\pi$
$317$	−18.0000	−1.01098	−0.505490	+	0.862832i	$0.668688\pi$
$318$	2.00000	0.112154
$319$	2.00000	0.111979
$320$	7.00000	0.391312
$321$	−4.00000	−0.223258
$322$	−4.00000	−0.222911
$323$	−24.0000	−1.33540
$324$	−1.00000	−0.0555556
$325$	−2.00000	−0.110940
$326$	−8.00000	−0.443079
$327$	10.0000	0.553001
$328$	6.00000	0.331295
$329$	0	0
$330$	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
$334$	0	0
$335$	−8.00000	−0.437087
$336$	1.00000	0.0545545
$337$	−26.0000	−1.41631	−0.708155	−	0.706057i	$-0.750472\pi$
$337$	−26.0000	−1.41631	−0.708155	+	0.706057i	$0.750472\pi$
$338$	−9.00000	−0.489535
$339$	6.00000	0.325875
$340$	6.00000	0.325396
$341$	4.00000	0.216612
$342$	4.00000	0.216295
$343$	1.00000	0.0539949
$344$	36.0000	1.94099
$345$	4.00000	0.215353
$346$	−10.0000	−0.537603
$347$	−4.00000	−0.214731	−0.107366	−	0.994220i	$-0.534242\pi$
$347$	−4.00000	−0.214731	−0.107366	+	0.994220i	$0.534242\pi$
$348$	−2.00000	−0.107211
$349$	18.0000	0.963518	0.481759	−	0.876304i	$-0.339998\pi$
$349$	18.0000	0.963518	0.481759	+	0.876304i	$0.339998\pi$
$350$	1.00000	0.0534522
$351$	2.00000	0.106752
$352$	−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	$0.794304\pi$
$354$	0	0
$355$	−16.0000	−0.849192
$356$	6.00000	0.317999
$357$	6.00000	0.317554
$358$	20.0000	1.05703
$359$	24.0000	1.26667	0.633336	−	0.773877i	$-0.281685\pi$
$359$	24.0000	1.26667	0.633336	+	0.773877i	$0.281685\pi$
$360$	−3.00000	−0.158114
$361$	−3.00000	−0.157895
$362$	14.0000	0.735824
$363$	−1.00000	−0.0524864
$364$	2.00000	0.104828
$365$	10.0000	0.523424
$366$	14.0000	0.731792
$367$	16.0000	0.835193	0.417597	−	0.908633i	$-0.362873\pi$
$367$	16.0000	0.835193	0.417597	+	0.908633i	$0.362873\pi$
$368$	4.00000	0.208514
$369$	−2.00000	−0.104116
$370$	−2.00000	−0.103975
$371$	−2.00000	−0.103835
$372$	−4.00000	−0.207390
$373$	−6.00000	−0.310668	−0.155334	−	0.987862i	$-0.549645\pi$
$373$	−6.00000	−0.310668	−0.155334	+	0.987862i	$0.549645\pi$
$374$	6.00000	0.310253
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	4.00000	0.206010
$378$	−1.00000	−0.0514344
$379$	−20.0000	−1.02733	−0.513665	−	0.857991i	$-0.671713\pi$
$379$	−20.0000	−1.02733	−0.513665	+	0.857991i	$0.671713\pi$
$380$	−4.00000	−0.205196
$381$	−8.00000	−0.409852
$382$	−8.00000	−0.409316
$383$	−8.00000	−0.408781	−0.204390	−	0.978889i	$-0.565521\pi$
$383$	−8.00000	−0.408781	−0.204390	+	0.978889i	$0.565521\pi$
$384$	3.00000	0.153093
$385$	−1.00000	−0.0509647
$386$	14.0000	0.712581
$387$	−12.0000	−0.609994
$388$	−2.00000	−0.101535
$389$	−26.0000	−1.31825	−0.659126	−	0.752032i	$-0.729074\pi$
$389$	−26.0000	−1.31825	−0.659126	+	0.752032i	$0.729074\pi$
$390$	2.00000	0.101274
$391$	24.0000	1.21373
$392$	−3.00000	−0.151523
$393$	−20.0000	−1.00887
$394$	−22.0000	−1.10834
$395$	16.0000	0.805047
$396$	1.00000	0.0502519
$397$	−18.0000	−0.903394	−0.451697	−	0.892171i	$-0.649181\pi$
$397$	−18.0000	−0.903394	−0.451697	+	0.892171i	$0.649181\pi$
$398$	28.0000	1.40351
$399$	−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
$403$	8.00000	0.398508
$404$	−10.0000	−0.497519
$405$	1.00000	0.0496904
$406$	−2.00000	−0.0992583
$407$	2.00000	0.0991363
$408$	−18.0000	−0.891133
$409$	14.0000	0.692255	0.346128	−	0.938187i	$-0.387496\pi$
$409$	14.0000	0.692255	0.346128	+	0.938187i	$0.387496\pi$
$410$	−2.00000	−0.0987730
$411$	−2.00000	−0.0986527
$412$	−8.00000	−0.394132
$413$	0	0
$414$	−4.00000	−0.196589
$415$	−4.00000	−0.196352
$416$	−10.0000	−0.490290
$417$	−20.0000	−0.979404
$418$	−4.00000	−0.195646
$419$	16.0000	0.781651	0.390826	−	0.920465i	$-0.372190\pi$
$419$	16.0000	0.781651	0.390826	+	0.920465i	$0.372190\pi$
$420$	1.00000	0.0487950
$421$	6.00000	0.292422	0.146211	−	0.989253i	$-0.453292\pi$
$421$	6.00000	0.292422	0.146211	+	0.989253i	$0.453292\pi$
$422$	20.0000	0.973585
$423$	0	0
$424$	6.00000	0.291386
$425$	−6.00000	−0.291043
$426$	16.0000	0.775203
$427$	−14.0000	−0.677507
$428$	−4.00000	−0.193347
$429$	−2.00000	−0.0965609
$430$	−12.0000	−0.578691
$431$	16.0000	0.770693	0.385346	−	0.922772i	$-0.374082\pi$
$431$	16.0000	0.770693	0.385346	+	0.922772i	$0.374082\pi$
$432$	1.00000	0.0481125
$433$	−14.0000	−0.672797	−0.336399	−	0.941720i	$-0.609209\pi$
$433$	−14.0000	−0.672797	−0.336399	+	0.941720i	$0.609209\pi$
$434$	−4.00000	−0.192006
$435$	2.00000	0.0958927
$436$	10.0000	0.478913
$437$	−16.0000	−0.765384
$438$	−10.0000	−0.477818
$439$	−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
$443$	−8.00000	−0.380091	−0.190046	−	0.981775i	$-0.560864\pi$
$443$	−8.00000	−0.380091	−0.190046	+	0.981775i	$0.560864\pi$
$444$	−2.00000	−0.0949158
$445$	−6.00000	−0.284427
$446$	24.0000	1.13643
$447$	−6.00000	−0.283790
$448$	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
$451$	2.00000	0.0941763
$452$	6.00000	0.282216
$453$	16.0000	0.751746
$454$	20.0000	0.938647
$455$	−2.00000	−0.0937614
$456$	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	$0.252438\pi$
$458$	14.0000	0.654177
$459$	6.00000	0.280056
$460$	4.00000	0.186501
$461$	−6.00000	−0.279448	−0.139724	−	0.990190i	$-0.544622\pi$
$461$	−6.00000	−0.279448	−0.139724	+	0.990190i	$0.544622\pi$
$462$	1.00000	0.0465242
$463$	4.00000	0.185896	0.0929479	−	0.995671i	$-0.470371\pi$
$463$	4.00000	0.185896	0.0929479	+	0.995671i	$0.470371\pi$
$464$	2.00000	0.0928477
$465$	4.00000	0.185496
$466$	−26.0000	−1.20443
$467$	−28.0000	−1.29569	−0.647843	−	0.761774i	$-0.724329\pi$
$467$	−28.0000	−1.29569	−0.647843	+	0.761774i	$0.724329\pi$
$468$	2.00000	0.0924500
$469$	−8.00000	−0.369406
$470$	0	0
$471$	2.00000	0.0921551
$472$	0	0
$473$	12.0000	0.551761
$474$	−16.0000	−0.734904
$475$	4.00000	0.183533
$476$	6.00000	0.275010
$477$	−2.00000	−0.0915737
$478$	−24.0000	−1.09773
$479$	−32.0000	−1.46212	−0.731059	−	0.682315i	$-0.760973\pi$
$479$	−32.0000	−1.46212	−0.731059	+	0.682315i	$0.760973\pi$
$480$	−5.00000	−0.228218
$481$	4.00000	0.182384
$482$	−10.0000	−0.455488
$483$	4.00000	0.182006
$484$	−1.00000	−0.0454545
$485$	2.00000	0.0908153
$486$	−1.00000	−0.0453609
$487$	4.00000	0.181257	0.0906287	−	0.995885i	$-0.471112\pi$
$487$	4.00000	0.181257	0.0906287	+	0.995885i	$0.471112\pi$
$488$	42.0000	1.90125
$489$	8.00000	0.361773
$490$	1.00000	0.0451754
$491$	−36.0000	−1.62466	−0.812329	−	0.583200i	$-0.801800\pi$
$491$	−36.0000	−1.62466	−0.812329	+	0.583200i	$0.801800\pi$
$492$	−2.00000	−0.0901670
$493$	12.0000	0.540453
$494$	−8.00000	−0.359937
$495$	−1.00000	−0.0449467
$496$	4.00000	0.179605
$497$	−16.0000	−0.717698
$498$	4.00000	0.179244
$499$	4.00000	0.179065	0.0895323	−	0.995984i	$-0.471463\pi$
$499$	4.00000	0.179065	0.0895323	+	0.995984i	$0.471463\pi$
$500$	−1.00000	−0.0447214
$501$	0	0
$502$	0	0
$503$	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
$505$	10.0000	0.444994
$506$	4.00000	0.177822
$507$	9.00000	0.399704
$508$	−8.00000	−0.354943
$509$	6.00000	0.265945	0.132973	−	0.991120i	$-0.457548\pi$
$509$	6.00000	0.265945	0.132973	+	0.991120i	$0.457548\pi$
$510$	6.00000	0.265684
$511$	10.0000	0.442374
$512$	−11.0000	−0.486136
$513$	−4.00000	−0.176604
$514$	−14.0000	−0.617514
$515$	8.00000	0.352522
$516$	−12.0000	−0.528271
$517$	0	0
$518$	−2.00000	−0.0878750
$519$	10.0000	0.438951
$520$	6.00000	0.263117
$521$	34.0000	1.48957	0.744784	−	0.667306i	$-0.232553\pi$
$521$	34.0000	1.48957	0.744784	+	0.667306i	$0.232553\pi$
$522$	−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
$526$	0	0
$527$	24.0000	1.04546
$528$	−1.00000	−0.0435194
$529$	−7.00000	−0.304348
$530$	−2.00000	−0.0868744
$531$	0	0
$532$	−4.00000	−0.173422
$533$	4.00000	0.173259
$534$	6.00000	0.259645
$535$	4.00000	0.172935
$536$	24.0000	1.03664
$537$	−20.0000	−0.863064
$538$	−26.0000	−1.12094
$539$	−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	$0.276899\pi$
$542$	−8.00000	−0.343629
$543$	−14.0000	−0.600798
$544$	−30.0000	−1.28624
$545$	−10.0000	−0.428353
$546$	2.00000	0.0855921
$547$	20.0000	0.855138	0.427569	−	0.903983i	$-0.359370\pi$
$547$	20.0000	0.855138	0.427569	+	0.903983i	$0.359370\pi$
$548$	−2.00000	−0.0854358
$549$	−14.0000	−0.597505
$550$	−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
$556$	−20.0000	−0.848189
$557$	18.0000	0.762684	0.381342	−	0.924434i	$-0.375462\pi$
$557$	18.0000	0.762684	0.381342	+	0.924434i	$0.375462\pi$
$558$	−4.00000	−0.169334
$559$	24.0000	1.01509
$560$	−1.00000	−0.0422577
$561$	−6.00000	−0.253320
$562$	−6.00000	−0.253095
$563$	−12.0000	−0.505740	−0.252870	−	0.967500i	$-0.581374\pi$
$563$	−12.0000	−0.505740	−0.252870	+	0.967500i	$0.581374\pi$
$564$	0	0
$565$	−6.00000	−0.252422
$566$	−4.00000	−0.168133
$567$	1.00000	0.0419961
$568$	48.0000	2.01404
$569$	34.0000	1.42535	0.712677	−	0.701492i	$-0.247483\pi$
$569$	34.0000	1.42535	0.712677	+	0.701492i	$0.247483\pi$
$570$	−4.00000	−0.167542
$571$	−28.0000	−1.17176	−0.585882	−	0.810397i	$-0.699252\pi$
$571$	−28.0000	−1.17176	−0.585882	+	0.810397i	$0.699252\pi$
$572$	−2.00000	−0.0836242
$573$	8.00000	0.334205
$574$	−2.00000	−0.0834784
$575$	−4.00000	−0.166812
$576$	7.00000	0.291667
$577$	−14.0000	−0.582828	−0.291414	−	0.956597i	$-0.594126\pi$
$577$	−14.0000	−0.582828	−0.291414	+	0.956597i	$0.594126\pi$
$578$	19.0000	0.790296
$579$	−14.0000	−0.581820
$580$	2.00000	0.0830455
$581$	−4.00000	−0.165948
$582$	−2.00000	−0.0829027
$583$	2.00000	0.0828315
$584$	−30.0000	−1.24141
$585$	−2.00000	−0.0826898
$586$	−18.0000	−0.743573
$587$	−36.0000	−1.48588	−0.742940	−	0.669359i	$-0.766569\pi$
$587$	−36.0000	−1.48588	−0.742940	+	0.669359i	$0.766569\pi$
$588$	1.00000	0.0412393
$589$	−16.0000	−0.659269
$590$	0	0
$591$	22.0000	0.904959
$592$	2.00000	0.0821995
$593$	−14.0000	−0.574911	−0.287456	−	0.957794i	$-0.592809\pi$
$593$	−14.0000	−0.574911	−0.287456	+	0.957794i	$0.592809\pi$
$594$	1.00000	0.0410305
$595$	−6.00000	−0.245976
$596$	−6.00000	−0.245770
$597$	−28.0000	−1.14596
$598$	8.00000	0.327144
$599$	16.0000	0.653742	0.326871	−	0.945069i	$-0.394006\pi$
$599$	16.0000	0.653742	0.326871	+	0.945069i	$0.394006\pi$
$600$	3.00000	0.122474
$601$	−18.0000	−0.734235	−0.367118	−	0.930175i	$-0.619655\pi$
$601$	−18.0000	−0.734235	−0.367118	+	0.930175i	$0.619655\pi$
$602$	−12.0000	−0.489083
$603$	−8.00000	−0.325785
$604$	16.0000	0.651031
$605$	1.00000	0.0406558
$606$	−10.0000	−0.406222
$607$	40.0000	1.62355	0.811775	−	0.583970i	$-0.198502\pi$
$607$	40.0000	1.62355	0.811775	+	0.583970i	$0.198502\pi$
$608$	20.0000	0.811107
$609$	2.00000	0.0810441
$610$	−14.0000	−0.566843
$611$	0	0
$612$	6.00000	0.242536
$613$	2.00000	0.0807792	0.0403896	−	0.999184i	$-0.487140\pi$
$613$	2.00000	0.0807792	0.0403896	+	0.999184i	$0.487140\pi$
$614$	12.0000	0.484281
$615$	2.00000	0.0806478
$616$	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	$0.381983\pi$
$618$	−8.00000	−0.321807
$619$	−48.0000	−1.92928	−0.964641	−	0.263566i	$-0.915101\pi$
$619$	−48.0000	−1.92928	−0.964641	+	0.263566i	$0.915101\pi$
$620$	4.00000	0.160644
$621$	4.00000	0.160514
$622$	20.0000	0.801927
$623$	−6.00000	−0.240385
$624$	−2.00000	−0.0800641
$625$	1.00000	0.0400000
$626$	−6.00000	−0.239808
$627$	4.00000	0.159745
$628$	2.00000	0.0798087
$629$	12.0000	0.478471
$630$	1.00000	0.0398410
$631$	8.00000	0.318475	0.159237	−	0.987240i	$-0.449096\pi$
$631$	8.00000	0.318475	0.159237	+	0.987240i	$0.449096\pi$
$632$	−48.0000	−1.90934
$633$	−20.0000	−0.794929
$634$	−18.0000	−0.714871
$635$	8.00000	0.317470
$636$	−2.00000	−0.0793052
$637$	−2.00000	−0.0792429
$638$	2.00000	0.0791808
$639$	−16.0000	−0.632950
$640$	−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	$0.384318\pi$
$642$	−4.00000	−0.157867
$643$	−12.0000	−0.473234	−0.236617	−	0.971603i	$-0.576039\pi$
$643$	−12.0000	−0.473234	−0.236617	+	0.971603i	$0.576039\pi$
$644$	4.00000	0.157622
$645$	12.0000	0.472500
$646$	−24.0000	−0.944267
$647$	−24.0000	−0.943537	−0.471769	−	0.881722i	$-0.656384\pi$
$647$	−24.0000	−0.943537	−0.471769	+	0.881722i	$0.656384\pi$
$648$	−3.00000	−0.117851
$649$	0	0
$650$	−2.00000	−0.0784465
$651$	4.00000	0.156772
$652$	8.00000	0.313304
$653$	−42.0000	−1.64359	−0.821794	−	0.569785i	$-0.807026\pi$
$653$	−42.0000	−1.64359	−0.821794	+	0.569785i	$0.807026\pi$
$654$	10.0000	0.391031
$655$	20.0000	0.781465
$656$	2.00000	0.0780869
$657$	10.0000	0.390137
$658$	0	0
$659$	−36.0000	−1.40236	−0.701180	−	0.712984i	$-0.747343\pi$
$659$	−36.0000	−1.40236	−0.701180	+	0.712984i	$0.747343\pi$
$660$	−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
$668$	0	0
$669$	−24.0000	−0.927894
$670$	−8.00000	−0.309067
$671$	14.0000	0.540464
$672$	−5.00000	−0.192879
$673$	−26.0000	−1.00223	−0.501113	−	0.865382i	$-0.667076\pi$
$673$	−26.0000	−1.00223	−0.501113	+	0.865382i	$0.667076\pi$
$674$	−26.0000	−1.00148
$675$	−1.00000	−0.0384900
$676$	9.00000	0.346154
$677$	6.00000	0.230599	0.115299	−	0.993331i	$-0.463217\pi$
$677$	6.00000	0.230599	0.115299	+	0.993331i	$0.463217\pi$
$678$	6.00000	0.230429
$679$	2.00000	0.0767530
$680$	18.0000	0.690268
$681$	−20.0000	−0.766402
$682$	4.00000	0.153168
$683$	−16.0000	−0.612223	−0.306111	−	0.951996i	$-0.599028\pi$
$683$	−16.0000	−0.612223	−0.306111	+	0.951996i	$0.599028\pi$
$684$	−4.00000	−0.152944
$685$	2.00000	0.0764161
$686$	1.00000	0.0381802
$687$	−14.0000	−0.534133
$688$	12.0000	0.457496
$689$	4.00000	0.152388
$690$	4.00000	0.152277
$691$	16.0000	0.608669	0.304334	−	0.952565i	$-0.401566\pi$
$691$	16.0000	0.608669	0.304334	+	0.952565i	$0.401566\pi$
$692$	10.0000	0.380143
$693$	−1.00000	−0.0379869
$694$	−4.00000	−0.151838
$695$	20.0000	0.758643
$696$	−6.00000	−0.227429
$697$	12.0000	0.454532
$698$	18.0000	0.681310
$699$	26.0000	0.983410
$700$	−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	$0.512025\pi$
$702$	2.00000	0.0754851
$703$	−8.00000	−0.301726
$704$	−7.00000	−0.263822
$705$	0	0
$706$	−30.0000	−1.12906
$707$	10.0000	0.376089
$708$	0	0
$709$	6.00000	0.225335	0.112667	−	0.993633i	$-0.464061\pi$
$709$	6.00000	0.225335	0.112667	+	0.993633i	$0.464061\pi$
$710$	−16.0000	−0.600469
$711$	16.0000	0.600047
$712$	18.0000	0.674579
$713$	16.0000	0.599205
$714$	6.00000	0.224544
$715$	2.00000	0.0747958
$716$	−20.0000	−0.747435
$717$	24.0000	0.896296
$718$	24.0000	0.895672
$719$	36.0000	1.34257	0.671287	−	0.741198i	$-0.265742\pi$
$719$	36.0000	1.34257	0.671287	+	0.741198i	$0.265742\pi$
$720$	−1.00000	−0.0372678
$721$	8.00000	0.297936
$722$	−3.00000	−0.111648
$723$	10.0000	0.371904
$724$	−14.0000	−0.520306
$725$	−2.00000	−0.0742781
$726$	−1.00000	−0.0371135
$727$	−8.00000	−0.296704	−0.148352	−	0.988935i	$-0.547397\pi$
$727$	−8.00000	−0.296704	−0.148352	+	0.988935i	$0.547397\pi$
$728$	6.00000	0.222375
$729$	1.00000	0.0370370
$730$	10.0000	0.370117
$731$	72.0000	2.66302
$732$	−14.0000	−0.517455
$733$	−34.0000	−1.25582	−0.627909	−	0.778287i	$-0.716089\pi$
$733$	−34.0000	−1.25582	−0.627909	+	0.778287i	$0.716089\pi$
$734$	16.0000	0.590571
$735$	−1.00000	−0.0368856
$736$	−20.0000	−0.737210
$737$	8.00000	0.294684
$738$	−2.00000	−0.0736210
$739$	36.0000	1.32428	0.662141	−	0.749380i	$-0.269648\pi$
$739$	36.0000	1.32428	0.662141	+	0.749380i	$0.269648\pi$
$740$	2.00000	0.0735215
$741$	8.00000	0.293887
$742$	−2.00000	−0.0734223
$743$	0	0		−	1.00000i	$-0.5\pi$
$743$	0	0			1.00000i	$0.5\pi$
$744$	−12.0000	−0.439941
$745$	6.00000	0.219823
$746$	−6.00000	−0.219676
$747$	−4.00000	−0.146352
$748$	−6.00000	−0.219382
$749$	4.00000	0.146157
$750$	−1.00000	−0.0365148
$751$	−40.0000	−1.45962	−0.729810	−	0.683650i	$-0.760392\pi$
$751$	−40.0000	−1.45962	−0.729810	+	0.683650i	$0.760392\pi$
$752$	0	0
$753$	0	0
$754$	4.00000	0.145671
$755$	−16.0000	−0.582300
$756$	1.00000	0.0363696
$757$	−42.0000	−1.52652	−0.763258	−	0.646094i	$-0.776401\pi$
$757$	−42.0000	−1.52652	−0.763258	+	0.646094i	$0.776401\pi$
$758$	−20.0000	−0.726433
$759$	−4.00000	−0.145191
$760$	−12.0000	−0.435286
$761$	30.0000	1.08750	0.543750	−	0.839248i	$-0.317004\pi$
$761$	30.0000	1.08750	0.543750	+	0.839248i	$0.317004\pi$
$762$	−8.00000	−0.289809
$763$	−10.0000	−0.362024
$764$	8.00000	0.289430
$765$	−6.00000	−0.216930
$766$	−8.00000	−0.289052
$767$	0	0
$768$	17.0000	0.613435
$769$	−18.0000	−0.649097	−0.324548	−	0.945869i	$-0.605212\pi$
$769$	−18.0000	−0.649097	−0.324548	+	0.945869i	$0.605212\pi$
$770$	−1.00000	−0.0360375
$771$	14.0000	0.504198
$772$	−14.0000	−0.503871
$773$	6.00000	0.215805	0.107903	−	0.994161i	$-0.465587\pi$
$773$	6.00000	0.215805	0.107903	+	0.994161i	$0.465587\pi$
$774$	−12.0000	−0.431331
$775$	−4.00000	−0.143684
$776$	−6.00000	−0.215387
$777$	2.00000	0.0717496
$778$	−26.0000	−0.932145
$779$	−8.00000	−0.286630
$780$	−2.00000	−0.0716115
$781$	16.0000	0.572525
$782$	24.0000	0.858238
$783$	2.00000	0.0714742
$784$	−1.00000	−0.0357143
$785$	−2.00000	−0.0713831
$786$	−20.0000	−0.713376
$787$	4.00000	0.142585	0.0712923	−	0.997455i	$-0.477288\pi$
$787$	4.00000	0.142585	0.0712923	+	0.997455i	$0.477288\pi$
$788$	22.0000	0.783718
$789$	0	0
$790$	16.0000	0.569254
$791$	−6.00000	−0.213335
$792$	3.00000	0.106600
$793$	28.0000	0.994309
$794$	−18.0000	−0.638796
$795$	2.00000	0.0709327
$796$	−28.0000	−0.992434
$797$	−34.0000	−1.20434	−0.602171	−	0.798367i	$-0.705697\pi$
$797$	−34.0000	−1.20434	−0.602171	+	0.798367i	$0.705697\pi$
$798$	−4.00000	−0.141598
$799$	0	0
$800$	5.00000	0.176777
$801$	−6.00000	−0.212000
$802$	18.0000	0.635602
$803$	−10.0000	−0.352892
$804$	−8.00000	−0.282138
$805$	−4.00000	−0.140981
$806$	8.00000	0.281788
$807$	26.0000	0.915243
$808$	−30.0000	−1.05540
$809$	18.0000	0.632846	0.316423	−	0.948618i	$-0.397518\pi$
$809$	18.0000	0.632846	0.316423	+	0.948618i	$0.397518\pi$
$810$	1.00000	0.0351364
$811$	−28.0000	−0.983213	−0.491606	−	0.870817i	$-0.663590\pi$
$811$	−28.0000	−0.983213	−0.491606	+	0.870817i	$0.663590\pi$
$812$	2.00000	0.0701862
$813$	8.00000	0.280572
$814$	2.00000	0.0701000
$815$	−8.00000	−0.280228
$816$	−6.00000	−0.210042
$817$	−48.0000	−1.67931
$818$	14.0000	0.489499
$819$	−2.00000	−0.0698857
$820$	2.00000	0.0698430
$821$	−2.00000	−0.0698005	−0.0349002	−	0.999391i	$-0.511111\pi$
$821$	−2.00000	−0.0698005	−0.0349002	+	0.999391i	$0.511111\pi$
$822$	−2.00000	−0.0697580
$823$	−20.0000	−0.697156	−0.348578	−	0.937280i	$-0.613335\pi$
$823$	−20.0000	−0.697156	−0.348578	+	0.937280i	$0.613335\pi$
$824$	−24.0000	−0.836080
$825$	1.00000	0.0348155
$826$	0	0
$827$	28.0000	0.973655	0.486828	−	0.873498i	$-0.338154\pi$
$827$	28.0000	0.973655	0.486828	+	0.873498i	$0.338154\pi$
$828$	4.00000	0.139010
$829$	−34.0000	−1.18087	−0.590434	−	0.807086i	$-0.701044\pi$
$829$	−34.0000	−1.18087	−0.590434	+	0.807086i	$0.701044\pi$
$830$	−4.00000	−0.138842
$831$	−2.00000	−0.0693792
$832$	−14.0000	−0.485363
$833$	−6.00000	−0.207888
$834$	−20.0000	−0.692543
$835$	0	0
$836$	4.00000	0.138343
$837$	4.00000	0.138260
$838$	16.0000	0.552711
$839$	−12.0000	−0.414286	−0.207143	−	0.978311i	$-0.566417\pi$
$839$	−12.0000	−0.414286	−0.207143	+	0.978311i	$0.566417\pi$
$840$	3.00000	0.103510
$841$	−25.0000	−0.862069
$842$	6.00000	0.206774
$843$	6.00000	0.206651
$844$	−20.0000	−0.688428
$845$	−9.00000	−0.309609
$846$	0	0
$847$	1.00000	0.0343604
$848$	2.00000	0.0686803
$849$	4.00000	0.137280
$850$	−6.00000	−0.205798
$851$	8.00000	0.274236
$852$	−16.0000	−0.548151
$853$	14.0000	0.479351	0.239675	−	0.970853i	$-0.422959\pi$
$853$	14.0000	0.479351	0.239675	+	0.970853i	$0.422959\pi$
$854$	−14.0000	−0.479070
$855$	4.00000	0.136797
$856$	−12.0000	−0.410152
$857$	10.0000	0.341593	0.170797	−	0.985306i	$-0.445366\pi$
$857$	10.0000	0.341593	0.170797	+	0.985306i	$0.445366\pi$
$858$	−2.00000	−0.0682789
$859$	0	0		−	1.00000i	$-0.5\pi$
$859$	0	0			1.00000i	$0.5\pi$
$860$	12.0000	0.409197
$861$	2.00000	0.0681598
$862$	16.0000	0.544962
$863$	4.00000	0.136162	0.0680808	−	0.997680i	$-0.478312\pi$
$863$	4.00000	0.136162	0.0680808	+	0.997680i	$0.478312\pi$
$864$	−5.00000	−0.170103
$865$	−10.0000	−0.340010
$866$	−14.0000	−0.475739
$867$	−19.0000	−0.645274
$868$	4.00000	0.135769
$869$	−16.0000	−0.542763
$870$	2.00000	0.0678064
$871$	16.0000	0.542139
$872$	30.0000	1.01593
$873$	2.00000	0.0676897
$874$	−16.0000	−0.541208
$875$	1.00000	0.0338062
$876$	10.0000	0.337869
$877$	−6.00000	−0.202606	−0.101303	−	0.994856i	$-0.532301\pi$
$877$	−6.00000	−0.202606	−0.101303	+	0.994856i	$0.532301\pi$
$878$	−32.0000	−1.07995
$879$	18.0000	0.607125
$880$	1.00000	0.0337100
$881$	26.0000	0.875962	0.437981	−	0.898984i	$-0.355694\pi$
$881$	26.0000	0.875962	0.437981	+	0.898984i	$0.355694\pi$
$882$	1.00000	0.0336718
$883$	0	0		−	1.00000i	$-0.5\pi$
$883$	0	0			1.00000i	$0.5\pi$
$884$	−12.0000	−0.403604
$885$	0	0
$886$	−8.00000	−0.268765
$887$	−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
$892$	−24.0000	−0.803579
$893$	0	0
$894$	−6.00000	−0.200670
$895$	20.0000	0.668526
$896$	−3.00000	−0.100223
$897$	−8.00000	−0.267112
$898$	18.0000	0.600668
$899$	8.00000	0.266815
$900$	−1.00000	−0.0333333
$901$	12.0000	0.399778
$902$	2.00000	0.0665927
$903$	12.0000	0.399335
$904$	18.0000	0.598671
$905$	14.0000	0.465376
$906$	16.0000	0.531564
$907$	0	0		−	1.00000i	$-0.5\pi$
$907$	0	0			1.00000i	$0.5\pi$
$908$	−20.0000	−0.663723
$909$	10.0000	0.331679
$910$	−2.00000	−0.0662994
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	4.00000	0.132453
$913$	4.00000	0.132381
$914$	30.0000	0.992312
$915$	14.0000	0.462826
$916$	−14.0000	−0.462573
$917$	20.0000	0.660458
$918$	6.00000	0.198030
$919$	56.0000	1.84727	0.923635	−	0.383274i	$-0.125203\pi$
$919$	56.0000	1.84727	0.923635	+	0.383274i	$0.125203\pi$
$920$	12.0000	0.395628
$921$	−12.0000	−0.395413
$922$	−6.00000	−0.197599
$923$	32.0000	1.05329
$924$	−1.00000	−0.0328976
$925$	−2.00000	−0.0657596
$926$	4.00000	0.131448
$927$	8.00000	0.262754
$928$	−10.0000	−0.328266
$929$	−6.00000	−0.196854	−0.0984268	−	0.995144i	$-0.531381\pi$
$929$	−6.00000	−0.196854	−0.0984268	+	0.995144i	$0.531381\pi$
$930$	4.00000	0.131165
$931$	4.00000	0.131095
$932$	26.0000	0.851658
$933$	−20.0000	−0.654771
$934$	−28.0000	−0.916188
$935$	6.00000	0.196221
$936$	6.00000	0.196116
$937$	34.0000	1.11073	0.555366	−	0.831606i	$-0.312578\pi$
$937$	34.0000	1.11073	0.555366	+	0.831606i	$0.312578\pi$
$938$	−8.00000	−0.261209
$939$	6.00000	0.195803
$940$	0	0
$941$	−22.0000	−0.717180	−0.358590	−	0.933495i	$-0.616742\pi$
$941$	−22.0000	−0.717180	−0.358590	+	0.933495i	$0.616742\pi$
$942$	2.00000	0.0651635
$943$	8.00000	0.260516
$944$	0	0
$945$	−1.00000	−0.0325300
$946$	12.0000	0.390154
$947$	−48.0000	−1.55979	−0.779895	−	0.625910i	$-0.784728\pi$
$947$	−48.0000	−1.55979	−0.779895	+	0.625910i	$0.784728\pi$
$948$	16.0000	0.519656
$949$	−20.0000	−0.649227
$950$	4.00000	0.129777
$951$	18.0000	0.583690
$952$	18.0000	0.583383
$953$	−34.0000	−1.10137	−0.550684	−	0.834714i	$-0.685633\pi$
$953$	−34.0000	−1.10137	−0.550684	+	0.834714i	$0.685633\pi$
$954$	−2.00000	−0.0647524
$955$	−8.00000	−0.258874
$956$	24.0000	0.776215
$957$	−2.00000	−0.0646508
$958$	−32.0000	−1.03387
$959$	2.00000	0.0645834
$960$	−7.00000	−0.225924
$961$	−15.0000	−0.483871
$962$	4.00000	0.128965
$963$	4.00000	0.128898
$964$	10.0000	0.322078
$965$	14.0000	0.450676
$966$	4.00000	0.128698
$967$	56.0000	1.80084	0.900419	−	0.435023i	$-0.143260\pi$
$967$	56.0000	1.80084	0.900419	+	0.435023i	$0.143260\pi$
$968$	−3.00000	−0.0964237
$969$	24.0000	0.770991
$970$	2.00000	0.0642161
$971$	−8.00000	−0.256732	−0.128366	−	0.991727i	$-0.540973\pi$
$971$	−8.00000	−0.256732	−0.128366	+	0.991727i	$0.540973\pi$
$972$	1.00000	0.0320750
$973$	20.0000	0.641171
$974$	4.00000	0.128168
$975$	2.00000	0.0640513
$976$	14.0000	0.448129
$977$	58.0000	1.85558	0.927792	−	0.373097i	$-0.121704\pi$
$977$	58.0000	1.85558	0.927792	+	0.373097i	$0.121704\pi$
$978$	8.00000	0.255812
$979$	6.00000	0.191761
$980$	−1.00000	−0.0319438
$981$	−10.0000	−0.319275
$982$	−36.0000	−1.14881
$983$	−56.0000	−1.78612	−0.893061	−	0.449935i	$-0.851447\pi$
$983$	−56.0000	−1.78612	−0.893061	+	0.449935i	$0.851447\pi$
$984$	−6.00000	−0.191273
$985$	−22.0000	−0.700978
$986$	12.0000	0.382158
$987$	0	0
$988$	8.00000	0.254514
$989$	48.0000	1.52631
$990$	−1.00000	−0.0317821
$991$	−16.0000	−0.508257	−0.254128	−	0.967170i	$-0.581789\pi$
$991$	−16.0000	−0.508257	−0.254128	+	0.967170i	$0.581789\pi$
$992$	−20.0000	−0.635001
$993$	12.0000	0.380808
$994$	−16.0000	−0.507489
$995$	28.0000	0.887660
$996$	−4.00000	−0.126745
$997$	46.0000	1.45683	0.728417	−	0.685134i	$-0.240256\pi$
$997$	46.0000	1.45683	0.728417	+	0.685134i	$0.240256\pi$
$998$	4.00000	0.126618
$999$	2.00000	0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1155.2.a.k.1.1	✓	1
3.2	odd	2		3465.2.a.b.1.1		1
5.4	even	2		5775.2.a.g.1.1		1
7.6	odd	2		8085.2.a.u.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1155.2.a.k.1.1	✓	1	1.1	even	1	trivial
3465.2.a.b.1.1		1	3.2	odd	2
5775.2.a.g.1.1		1	5.4	even	2
8085.2.a.u.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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