LMFDB - Embedded newform 1155.2.a.l.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)

$f(q)$

$=$

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

\(q+1.00000 q^{2} +1.00000 q^{3} -1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} -1.00000 q^{7} -3.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} +1.00000 q^{11} -1.00000 q^{12} -2.00000 q^{13} -1.00000 q^{14} -1.00000 q^{15} -1.00000 q^{16} -2.00000 q^{17} +1.00000 q^{18} -4.00000 q^{19} +1.00000 q^{20} -1.00000 q^{21} +1.00000 q^{22} -4.00000 q^{23} -3.00000 q^{24} +1.00000 q^{25} -2.00000 q^{26} +1.00000 q^{27} +1.00000 q^{28} -6.00000 q^{29} -1.00000 q^{30} -4.00000 q^{31} +5.00000 q^{32} +1.00000 q^{33} -2.00000 q^{34} +1.00000 q^{35} -1.00000 q^{36} -2.00000 q^{37} -4.00000 q^{38} -2.00000 q^{39} +3.00000 q^{40} -6.00000 q^{41} -1.00000 q^{42} -4.00000 q^{43} -1.00000 q^{44} -1.00000 q^{45} -4.00000 q^{46} +8.00000 q^{47} -1.00000 q^{48} +1.00000 q^{49} +1.00000 q^{50} -2.00000 q^{51} +2.00000 q^{52} +2.00000 q^{53} +1.00000 q^{54} -1.00000 q^{55} +3.00000 q^{56} -4.00000 q^{57} -6.00000 q^{58} -12.0000 q^{59} +1.00000 q^{60} -2.00000 q^{61} -4.00000 q^{62} -1.00000 q^{63} +7.00000 q^{64} +2.00000 q^{65} +1.00000 q^{66} -4.00000 q^{67} +2.00000 q^{68} -4.00000 q^{69} +1.00000 q^{70} +16.0000 q^{71} -3.00000 q^{72} +2.00000 q^{73} -2.00000 q^{74} +1.00000 q^{75} +4.00000 q^{76} -1.00000 q^{77} -2.00000 q^{78} -4.00000 q^{79} +1.00000 q^{80} +1.00000 q^{81} -6.00000 q^{82} +1.00000 q^{84} +2.00000 q^{85} -4.00000 q^{86} -6.00000 q^{87} -3.00000 q^{88} +6.00000 q^{89} -1.00000 q^{90} +2.00000 q^{91} +4.00000 q^{92} -4.00000 q^{93} +8.00000 q^{94} +4.00000 q^{95} +5.00000 q^{96} +14.0000 q^{97} +1.00000 q^{98} +1.00000 q^{99} +O(q^{100})\)

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
$13$	−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
$17$	−2.00000	−0.485071	−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000	−0.485071	−0.242536	+	0.970143i	$0.577979\pi$
$18$	1.00000	0.235702
$19$	−4.00000	−0.917663	−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−4.00000	−0.917663	−0.458831	+	0.888523i	$0.651732\pi$
$20$	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
$29$	−6.00000	−1.11417	−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000	−1.11417	−0.557086	+	0.830455i	$0.688081\pi$
$30$	−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
$34$	−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
$39$	−2.00000	−0.320256
$40$	3.00000	0.474342
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	−1.00000	−0.154303
$43$	−4.00000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	−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
$52$	2.00000	0.277350
$53$	2.00000	0.274721	0.137361	−	0.990521i	$-0.456138\pi$
$53$	2.00000	0.274721	0.137361	+	0.990521i	$0.456138\pi$
$54$	1.00000	0.136083
$55$	−1.00000	−0.134840
$56$	3.00000	0.400892
$57$	−4.00000	−0.529813
$58$	−6.00000	−0.787839
$59$	−12.0000	−1.56227	−0.781133	−	0.624364i	$-0.785358\pi$
$59$	−12.0000	−1.56227	−0.781133	+	0.624364i	$0.785358\pi$
$60$	1.00000	0.129099
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	−4.00000	−0.508001
$63$	−1.00000	−0.125988
$64$	7.00000	0.875000
$65$	2.00000	0.248069
$66$	1.00000	0.123091
$67$	−4.00000	−0.488678	−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000	−0.488678	−0.244339	+	0.969690i	$0.578571\pi$
$68$	2.00000	0.242536
$69$	−4.00000	−0.481543
$70$	1.00000	0.119523
$71$	16.0000	1.89885	0.949425	−	0.313993i	$-0.101667\pi$
$71$	16.0000	1.89885	0.949425	+	0.313993i	$0.101667\pi$
$72$	−3.00000	−0.353553
$73$	2.00000	0.234082	0.117041	−	0.993127i	$-0.462659\pi$
$73$	2.00000	0.234082	0.117041	+	0.993127i	$0.462659\pi$
$74$	−2.00000	−0.232495
$75$	1.00000	0.115470
$76$	4.00000	0.458831
$77$	−1.00000	−0.113961
$78$	−2.00000	−0.226455
$79$	−4.00000	−0.450035	−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000	−0.450035	−0.225018	+	0.974355i	$0.572244\pi$
$80$	1.00000	0.111803
$81$	1.00000	0.111111
$82$	−6.00000	−0.662589
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	1.00000	0.109109
$85$	2.00000	0.216930
$86$	−4.00000	−0.431331
$87$	−6.00000	−0.643268
$88$	−3.00000	−0.319801
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	−1.00000	−0.105409
$91$	2.00000	0.209657
$92$	4.00000	0.417029
$93$	−4.00000	−0.414781
$94$	8.00000	0.825137
$95$	4.00000	0.410391
$96$	5.00000	0.510310
$97$	14.0000	1.42148	0.710742	−	0.703452i	$-0.248359\pi$
$97$	14.0000	1.42148	0.710742	+	0.703452i	$0.248359\pi$
$98$	1.00000	0.101015
$99$	1.00000	0.100504
$100$	−1.00000	−0.100000
$101$	−2.00000	−0.199007	−0.0995037	−	0.995037i	$-0.531726\pi$
$101$	−2.00000	−0.199007	−0.0995037	+	0.995037i	$0.531726\pi$
$102$	−2.00000	−0.198030
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	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
$109$	−2.00000	−0.191565	−0.0957826	−	0.995402i	$-0.530535\pi$
$109$	−2.00000	−0.191565	−0.0957826	+	0.995402i	$0.530535\pi$
$110$	−1.00000	−0.0953463
$111$	−2.00000	−0.189832
$112$	1.00000	0.0944911
$113$	−10.0000	−0.940721	−0.470360	−	0.882474i	$-0.655876\pi$
$113$	−10.0000	−0.940721	−0.470360	+	0.882474i	$0.655876\pi$
$114$	−4.00000	−0.374634
$115$	4.00000	0.373002
$116$	6.00000	0.557086
$117$	−2.00000	−0.184900
$118$	−12.0000	−1.10469
$119$	2.00000	0.183340
$120$	3.00000	0.273861
$121$	1.00000	0.0909091
$122$	−2.00000	−0.181071
$123$	−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
$130$	2.00000	0.175412
$131$	12.0000	1.04844	0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000	1.04844	0.524222	+	0.851581i	$0.324356\pi$
$132$	−1.00000	−0.0870388
$133$	4.00000	0.346844
$134$	−4.00000	−0.345547
$135$	−1.00000	−0.0860663
$136$	6.00000	0.514496
$137$	14.0000	1.19610	0.598050	−	0.801459i	$-0.295942\pi$
$137$	14.0000	1.19610	0.598050	+	0.801459i	$0.295942\pi$
$138$	−4.00000	−0.340503
$139$	4.00000	0.339276	0.169638	−	0.985506i	$-0.445740\pi$
$139$	4.00000	0.339276	0.169638	+	0.985506i	$0.445740\pi$
$140$	−1.00000	−0.0845154
$141$	8.00000	0.673722
$142$	16.0000	1.34269
$143$	−2.00000	−0.167248
$144$	−1.00000	−0.0833333
$145$	6.00000	0.498273
$146$	2.00000	0.165521
$147$	1.00000	0.0824786
$148$	2.00000	0.164399
$149$	2.00000	0.163846	0.0819232	−	0.996639i	$-0.473894\pi$
$149$	2.00000	0.163846	0.0819232	+	0.996639i	$0.473894\pi$
$150$	1.00000	0.0816497
$151$	−12.0000	−0.976546	−0.488273	−	0.872691i	$-0.662373\pi$
$151$	−12.0000	−0.976546	−0.488273	+	0.872691i	$0.662373\pi$
$152$	12.0000	0.973329
$153$	−2.00000	−0.161690
$154$	−1.00000	−0.0805823
$155$	4.00000	0.321288
$156$	2.00000	0.160128
$157$	−22.0000	−1.75579	−0.877896	−	0.478852i	$-0.841053\pi$
$157$	−22.0000	−1.75579	−0.877896	+	0.478852i	$0.841053\pi$
$158$	−4.00000	−0.318223
$159$	2.00000	0.158610
$160$	−5.00000	−0.395285
$161$	4.00000	0.315244
$162$	1.00000	0.0785674
$163$	4.00000	0.313304	0.156652	−	0.987654i	$-0.449930\pi$
$163$	4.00000	0.313304	0.156652	+	0.987654i	$0.449930\pi$
$164$	6.00000	0.468521
$165$	−1.00000	−0.0778499
$166$	0	0
$167$	−20.0000	−1.54765	−0.773823	−	0.633402i	$-0.781658\pi$
$167$	−20.0000	−1.54765	−0.773823	+	0.633402i	$0.781658\pi$
$168$	3.00000	0.231455
$169$	−9.00000	−0.692308
$170$	2.00000	0.153393
$171$	−4.00000	−0.305888
$172$	4.00000	0.304997
$173$	2.00000	0.152057	0.0760286	−	0.997106i	$-0.475776\pi$
$173$	2.00000	0.152057	0.0760286	+	0.997106i	$0.475776\pi$
$174$	−6.00000	−0.454859
$175$	−1.00000	−0.0755929
$176$	−1.00000	−0.0753778
$177$	−12.0000	−0.901975
$178$	6.00000	0.449719
$179$	4.00000	0.298974	0.149487	−	0.988764i	$-0.452238\pi$
$179$	4.00000	0.298974	0.149487	+	0.988764i	$0.452238\pi$
$180$	1.00000	0.0745356
$181$	−10.0000	−0.743294	−0.371647	−	0.928374i	$-0.621207\pi$
$181$	−10.0000	−0.743294	−0.371647	+	0.928374i	$0.621207\pi$
$182$	2.00000	0.148250
$183$	−2.00000	−0.147844
$184$	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
$191$	24.0000	1.73658	0.868290	−	0.496058i	$-0.165220\pi$
$191$	24.0000	1.73658	0.868290	+	0.496058i	$0.165220\pi$
$192$	7.00000	0.505181
$193$	6.00000	0.431889	0.215945	−	0.976406i	$-0.430717\pi$
$193$	6.00000	0.431889	0.215945	+	0.976406i	$0.430717\pi$
$194$	14.0000	1.00514
$195$	2.00000	0.143223
$196$	−1.00000	−0.0714286
$197$	26.0000	1.85242	0.926212	−	0.377004i	$-0.123046\pi$
$197$	26.0000	1.85242	0.926212	+	0.377004i	$0.123046\pi$
$198$	1.00000	0.0710669
$199$	−28.0000	−1.98487	−0.992434	−	0.122782i	$-0.960818\pi$
$199$	−28.0000	−1.98487	−0.992434	+	0.122782i	$0.960818\pi$
$200$	−3.00000	−0.212132
$201$	−4.00000	−0.282138
$202$	−2.00000	−0.140720
$203$	6.00000	0.421117
$204$	2.00000	0.140028
$205$	6.00000	0.419058
$206$	0	0
$207$	−4.00000	−0.278019
$208$	2.00000	0.138675
$209$	−4.00000	−0.276686
$210$	1.00000	0.0690066
$211$	8.00000	0.550743	0.275371	−	0.961338i	$-0.411199\pi$
$211$	8.00000	0.550743	0.275371	+	0.961338i	$0.411199\pi$
$212$	−2.00000	−0.137361
$213$	16.0000	1.09630
$214$	4.00000	0.273434
$215$	4.00000	0.272798
$216$	−3.00000	−0.204124
$217$	4.00000	0.271538
$218$	−2.00000	−0.135457
$219$	2.00000	0.135147
$220$	1.00000	0.0674200
$221$	4.00000	0.269069
$222$	−2.00000	−0.134231
$223$	−16.0000	−1.07144	−0.535720	−	0.844396i	$-0.679960\pi$
$223$	−16.0000	−1.07144	−0.535720	+	0.844396i	$0.679960\pi$
$224$	−5.00000	−0.334077
$225$	1.00000	0.0666667
$226$	−10.0000	−0.665190
$227$	0	0		−	1.00000i	$-0.5\pi$
$227$	0	0			1.00000i	$0.5\pi$
$228$	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
$232$	18.0000	1.18176
$233$	−10.0000	−0.655122	−0.327561	−	0.944830i	$-0.606227\pi$
$233$	−10.0000	−0.655122	−0.327561	+	0.944830i	$0.606227\pi$
$234$	−2.00000	−0.130744
$235$	−8.00000	−0.521862
$236$	12.0000	0.781133
$237$	−4.00000	−0.259828
$238$	2.00000	0.129641
$239$	−16.0000	−1.03495	−0.517477	−	0.855697i	$-0.673129\pi$
$239$	−16.0000	−1.03495	−0.517477	+	0.855697i	$0.673129\pi$
$240$	1.00000	0.0645497
$241$	−14.0000	−0.901819	−0.450910	−	0.892570i	$-0.648900\pi$
$241$	−14.0000	−0.901819	−0.450910	+	0.892570i	$0.648900\pi$
$242$	1.00000	0.0642824
$243$	1.00000	0.0641500
$244$	2.00000	0.128037
$245$	−1.00000	−0.0638877
$246$	−6.00000	−0.382546
$247$	8.00000	0.509028
$248$	12.0000	0.762001
$249$	0	0
$250$	−1.00000	−0.0632456
$251$	12.0000	0.757433	0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000	0.757433	0.378717	+	0.925513i	$0.376365\pi$
$252$	1.00000	0.0629941
$253$	−4.00000	−0.251478
$254$	8.00000	0.501965
$255$	2.00000	0.125245
$256$	−17.0000	−1.06250
$257$	14.0000	0.873296	0.436648	−	0.899632i	$-0.356166\pi$
$257$	14.0000	0.873296	0.436648	+	0.899632i	$0.356166\pi$
$258$	−4.00000	−0.249029
$259$	2.00000	0.124274
$260$	−2.00000	−0.124035
$261$	−6.00000	−0.371391
$262$	12.0000	0.741362
$263$	8.00000	0.493301	0.246651	−	0.969104i	$-0.420670\pi$
$263$	8.00000	0.493301	0.246651	+	0.969104i	$0.420670\pi$
$264$	−3.00000	−0.184637
$265$	−2.00000	−0.122859
$266$	4.00000	0.245256
$267$	6.00000	0.367194
$268$	4.00000	0.244339
$269$	−22.0000	−1.34136	−0.670682	−	0.741745i	$-0.733998\pi$
$269$	−22.0000	−1.34136	−0.670682	+	0.741745i	$0.733998\pi$
$270$	−1.00000	−0.0608581
$271$	−32.0000	−1.94386	−0.971931	−	0.235267i	$-0.924404\pi$
$271$	−32.0000	−1.94386	−0.971931	+	0.235267i	$0.924404\pi$
$272$	2.00000	0.121268
$273$	2.00000	0.121046
$274$	14.0000	0.845771
$275$	1.00000	0.0603023
$276$	4.00000	0.240772
$277$	−22.0000	−1.32185	−0.660926	−	0.750451i	$-0.729836\pi$
$277$	−22.0000	−1.32185	−0.660926	+	0.750451i	$0.729836\pi$
$278$	4.00000	0.239904
$279$	−4.00000	−0.239474
$280$	−3.00000	−0.179284
$281$	30.0000	1.78965	0.894825	−	0.446417i	$-0.147300\pi$
$281$	30.0000	1.78965	0.894825	+	0.446417i	$0.147300\pi$
$282$	8.00000	0.476393
$283$	4.00000	0.237775	0.118888	−	0.992908i	$-0.462067\pi$
$283$	4.00000	0.237775	0.118888	+	0.992908i	$0.462067\pi$
$284$	−16.0000	−0.949425
$285$	4.00000	0.236940
$286$	−2.00000	−0.118262
$287$	6.00000	0.354169
$288$	5.00000	0.294628
$289$	−13.0000	−0.764706
$290$	6.00000	0.352332
$291$	14.0000	0.820695
$292$	−2.00000	−0.117041
$293$	−22.0000	−1.28525	−0.642627	−	0.766179i	$-0.722155\pi$
$293$	−22.0000	−1.28525	−0.642627	+	0.766179i	$0.722155\pi$
$294$	1.00000	0.0583212
$295$	12.0000	0.698667
$296$	6.00000	0.348743
$297$	1.00000	0.0580259
$298$	2.00000	0.115857
$299$	8.00000	0.462652
$300$	−1.00000	−0.0577350
$301$	4.00000	0.230556
$302$	−12.0000	−0.690522
$303$	−2.00000	−0.114897
$304$	4.00000	0.229416
$305$	2.00000	0.114520
$306$	−2.00000	−0.114332
$307$	−20.0000	−1.14146	−0.570730	−	0.821138i	$-0.693340\pi$
$307$	−20.0000	−1.14146	−0.570730	+	0.821138i	$0.693340\pi$
$308$	1.00000	0.0569803
$309$	0	0
$310$	4.00000	0.227185
$311$	−24.0000	−1.36092	−0.680458	−	0.732787i	$-0.738219\pi$
$311$	−24.0000	−1.36092	−0.680458	+	0.732787i	$0.738219\pi$
$312$	6.00000	0.339683
$313$	6.00000	0.339140	0.169570	−	0.985518i	$-0.445762\pi$
$313$	6.00000	0.339140	0.169570	+	0.985518i	$0.445762\pi$
$314$	−22.0000	−1.24153
$315$	1.00000	0.0563436
$316$	4.00000	0.225018
$317$	−6.00000	−0.336994	−0.168497	−	0.985702i	$-0.553891\pi$
$317$	−6.00000	−0.336994	−0.168497	+	0.985702i	$0.553891\pi$
$318$	2.00000	0.112154
$319$	−6.00000	−0.335936
$320$	−7.00000	−0.391312
$321$	4.00000	0.223258
$322$	4.00000	0.222911
$323$	8.00000	0.445132
$324$	−1.00000	−0.0555556
$325$	−2.00000	−0.110940
$326$	4.00000	0.221540
$327$	−2.00000	−0.110600
$328$	18.0000	0.993884
$329$	−8.00000	−0.441054
$330$	−1.00000	−0.0550482
$331$	4.00000	0.219860	0.109930	−	0.993939i	$-0.464937\pi$
$331$	4.00000	0.219860	0.109930	+	0.993939i	$0.464937\pi$
$332$	0	0
$333$	−2.00000	−0.109599
$334$	−20.0000	−1.09435
$335$	4.00000	0.218543
$336$	1.00000	0.0545545
$337$	14.0000	0.762629	0.381314	−	0.924445i	$-0.375472\pi$
$337$	14.0000	0.762629	0.381314	+	0.924445i	$0.375472\pi$
$338$	−9.00000	−0.489535
$339$	−10.0000	−0.543125
$340$	−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
$346$	2.00000	0.107521
$347$	20.0000	1.07366	0.536828	−	0.843692i	$-0.319622\pi$
$347$	20.0000	1.07366	0.536828	+	0.843692i	$0.319622\pi$
$348$	6.00000	0.321634
$349$	−2.00000	−0.107058	−0.0535288	−	0.998566i	$-0.517047\pi$
$349$	−2.00000	−0.107058	−0.0535288	+	0.998566i	$0.517047\pi$
$350$	−1.00000	−0.0534522
$351$	−2.00000	−0.106752
$352$	5.00000	0.266501
$353$	14.0000	0.745145	0.372572	−	0.928003i	$-0.378476\pi$
$353$	14.0000	0.745145	0.372572	+	0.928003i	$0.378476\pi$
$354$	−12.0000	−0.637793
$355$	−16.0000	−0.849192
$356$	−6.00000	−0.317999
$357$	2.00000	0.105851
$358$	4.00000	0.211407
$359$	16.0000	0.844448	0.422224	−	0.906492i	$-0.361250\pi$
$359$	16.0000	0.844448	0.422224	+	0.906492i	$0.361250\pi$
$360$	3.00000	0.158114
$361$	−3.00000	−0.157895
$362$	−10.0000	−0.525588
$363$	1.00000	0.0524864
$364$	−2.00000	−0.104828
$365$	−2.00000	−0.104685
$366$	−2.00000	−0.104542
$367$	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
$369$	−6.00000	−0.312348
$370$	2.00000	0.103975
$371$	−2.00000	−0.103835
$372$	4.00000	0.207390
$373$	2.00000	0.103556	0.0517780	−	0.998659i	$-0.483511\pi$
$373$	2.00000	0.103556	0.0517780	+	0.998659i	$0.483511\pi$
$374$	−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	$-0.467241\pi$
$379$	4.00000	0.205466	0.102733	+	0.994709i	$0.467241\pi$
$380$	−4.00000	−0.205196
$381$	8.00000	0.409852
$382$	24.0000	1.22795
$383$	24.0000	1.22634	0.613171	−	0.789950i	$-0.289894\pi$
$383$	24.0000	1.22634	0.613171	+	0.789950i	$0.289894\pi$
$384$	−3.00000	−0.153093
$385$	1.00000	0.0509647
$386$	6.00000	0.305392
$387$	−4.00000	−0.203331
$388$	−14.0000	−0.710742
$389$	−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
$391$	8.00000	0.404577
$392$	−3.00000	−0.151523
$393$	12.0000	0.605320
$394$	26.0000	1.30986
$395$	4.00000	0.201262
$396$	−1.00000	−0.0502519
$397$	2.00000	0.100377	0.0501886	−	0.998740i	$-0.484018\pi$
$397$	2.00000	0.100377	0.0501886	+	0.998740i	$0.484018\pi$
$398$	−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	$0.351624\pi$
$402$	−4.00000	−0.199502
$403$	8.00000	0.398508
$404$	2.00000	0.0995037
$405$	−1.00000	−0.0496904
$406$	6.00000	0.297775
$407$	−2.00000	−0.0991363
$408$	6.00000	0.297044
$409$	−38.0000	−1.87898	−0.939490	−	0.342578i	$-0.888700\pi$
$409$	−38.0000	−1.87898	−0.939490	+	0.342578i	$0.888700\pi$
$410$	6.00000	0.296319
$411$	14.0000	0.690569
$412$	0	0
$413$	12.0000	0.590481
$414$	−4.00000	−0.196589
$415$	0	0
$416$	−10.0000	−0.490290
$417$	4.00000	0.195881
$418$	−4.00000	−0.195646
$419$	20.0000	0.977064	0.488532	−	0.872546i	$-0.337533\pi$
$419$	20.0000	0.977064	0.488532	+	0.872546i	$0.337533\pi$
$420$	−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	$0.718416\pi$
$422$	8.00000	0.389434
$423$	8.00000	0.388973
$424$	−6.00000	−0.291386
$425$	−2.00000	−0.0970143
$426$	16.0000	0.775203
$427$	2.00000	0.0967868
$428$	−4.00000	−0.193347
$429$	−2.00000	−0.0965609
$430$	4.00000	0.192897
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	−1.00000	−0.0481125
$433$	22.0000	1.05725	0.528626	−	0.848855i	$-0.322707\pi$
$433$	22.0000	1.05725	0.528626	+	0.848855i	$0.322707\pi$
$434$	4.00000	0.192006
$435$	6.00000	0.287678
$436$	2.00000	0.0957826
$437$	16.0000	0.765384
$438$	2.00000	0.0955637
$439$	0	0		−	1.00000i	$-0.5\pi$
$439$	0	0			1.00000i	$0.5\pi$
$440$	3.00000	0.143019
$441$	1.00000	0.0476190
$442$	4.00000	0.190261
$443$	24.0000	1.14027	0.570137	−	0.821549i	$-0.306890\pi$
$443$	24.0000	1.14027	0.570137	+	0.821549i	$0.306890\pi$
$444$	2.00000	0.0949158
$445$	−6.00000	−0.284427
$446$	−16.0000	−0.757622
$447$	2.00000	0.0945968
$448$	−7.00000	−0.330719
$449$	−14.0000	−0.660701	−0.330350	−	0.943858i	$-0.607167\pi$
$449$	−14.0000	−0.660701	−0.330350	+	0.943858i	$0.607167\pi$
$450$	1.00000	0.0471405
$451$	−6.00000	−0.282529
$452$	10.0000	0.470360
$453$	−12.0000	−0.563809
$454$	0	0
$455$	−2.00000	−0.0937614
$456$	12.0000	0.561951
$457$	6.00000	0.280668	0.140334	−	0.990104i	$-0.455182\pi$
$457$	6.00000	0.280668	0.140334	+	0.990104i	$0.455182\pi$
$458$	−2.00000	−0.0934539
$459$	−2.00000	−0.0933520
$460$	−4.00000	−0.186501
$461$	−2.00000	−0.0931493	−0.0465746	−	0.998915i	$-0.514831\pi$
$461$	−2.00000	−0.0931493	−0.0465746	+	0.998915i	$0.514831\pi$
$462$	−1.00000	−0.0465242
$463$	−8.00000	−0.371792	−0.185896	−	0.982569i	$-0.559519\pi$
$463$	−8.00000	−0.371792	−0.185896	+	0.982569i	$0.559519\pi$
$464$	6.00000	0.278543
$465$	4.00000	0.185496
$466$	−10.0000	−0.463241
$467$	−12.0000	−0.555294	−0.277647	−	0.960683i	$-0.589555\pi$
$467$	−12.0000	−0.555294	−0.277647	+	0.960683i	$0.589555\pi$
$468$	2.00000	0.0924500
$469$	4.00000	0.184703
$470$	−8.00000	−0.369012
$471$	−22.0000	−1.01371
$472$	36.0000	1.65703
$473$	−4.00000	−0.183920
$474$	−4.00000	−0.183726
$475$	−4.00000	−0.183533
$476$	−2.00000	−0.0916698
$477$	2.00000	0.0915737
$478$	−16.0000	−0.731823
$479$	−8.00000	−0.365529	−0.182765	−	0.983157i	$-0.558505\pi$
$479$	−8.00000	−0.365529	−0.182765	+	0.983157i	$0.558505\pi$
$480$	−5.00000	−0.228218
$481$	4.00000	0.182384
$482$	−14.0000	−0.637683
$483$	4.00000	0.182006
$484$	−1.00000	−0.0454545
$485$	−14.0000	−0.635707
$486$	1.00000	0.0453609
$487$	16.0000	0.725029	0.362515	−	0.931978i	$-0.381918\pi$
$487$	16.0000	0.725029	0.362515	+	0.931978i	$0.381918\pi$
$488$	6.00000	0.271607
$489$	4.00000	0.180886
$490$	−1.00000	−0.0451754
$491$	−20.0000	−0.902587	−0.451294	−	0.892375i	$-0.649037\pi$
$491$	−20.0000	−0.902587	−0.451294	+	0.892375i	$0.649037\pi$
$492$	6.00000	0.270501
$493$	12.0000	0.540453
$494$	8.00000	0.359937
$495$	−1.00000	−0.0449467
$496$	4.00000	0.179605
$497$	−16.0000	−0.717698
$498$	0	0
$499$	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
$501$	−20.0000	−0.893534
$502$	12.0000	0.535586
$503$	−28.0000	−1.24846	−0.624229	−	0.781241i	$-0.714587\pi$
$503$	−28.0000	−1.24846	−0.624229	+	0.781241i	$0.714587\pi$
$504$	3.00000	0.133631
$505$	2.00000	0.0889988
$506$	−4.00000	−0.177822
$507$	−9.00000	−0.399704
$508$	−8.00000	−0.354943
$509$	−6.00000	−0.265945	−0.132973	−	0.991120i	$-0.542452\pi$
$509$	−6.00000	−0.265945	−0.132973	+	0.991120i	$0.542452\pi$
$510$	2.00000	0.0885615
$511$	−2.00000	−0.0884748
$512$	−11.0000	−0.486136
$513$	−4.00000	−0.176604
$514$	14.0000	0.617514
$515$	0	0
$516$	4.00000	0.176090
$517$	8.00000	0.351840
$518$	2.00000	0.0878750
$519$	2.00000	0.0877903
$520$	−6.00000	−0.263117
$521$	−10.0000	−0.438108	−0.219054	−	0.975713i	$-0.570297\pi$
$521$	−10.0000	−0.438108	−0.219054	+	0.975713i	$0.570297\pi$
$522$	−6.00000	−0.262613
$523$	12.0000	0.524723	0.262362	−	0.964970i	$-0.415499\pi$
$523$	12.0000	0.524723	0.262362	+	0.964970i	$0.415499\pi$
$524$	−12.0000	−0.524222
$525$	−1.00000	−0.0436436
$526$	8.00000	0.348817
$527$	8.00000	0.348485
$528$	−1.00000	−0.0435194
$529$	−7.00000	−0.304348
$530$	−2.00000	−0.0868744
$531$	−12.0000	−0.520756
$532$	−4.00000	−0.173422
$533$	12.0000	0.519778
$534$	6.00000	0.259645
$535$	−4.00000	−0.172935
$536$	12.0000	0.518321
$537$	4.00000	0.172613
$538$	−22.0000	−0.948487
$539$	1.00000	0.0430730
$540$	1.00000	0.0430331
$541$	−2.00000	−0.0859867	−0.0429934	−	0.999075i	$-0.513689\pi$
$541$	−2.00000	−0.0859867	−0.0429934	+	0.999075i	$0.513689\pi$
$542$	−32.0000	−1.37452
$543$	−10.0000	−0.429141
$544$	−10.0000	−0.428746
$545$	2.00000	0.0856706
$546$	2.00000	0.0855921
$547$	28.0000	1.19719	0.598597	−	0.801050i	$-0.295725\pi$
$547$	28.0000	1.19719	0.598597	+	0.801050i	$0.295725\pi$
$548$	−14.0000	−0.598050
$549$	−2.00000	−0.0853579
$550$	1.00000	0.0426401
$551$	24.0000	1.02243
$552$	12.0000	0.510754
$553$	4.00000	0.170097
$554$	−22.0000	−0.934690
$555$	2.00000	0.0848953
$556$	−4.00000	−0.169638
$557$	−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
$559$	8.00000	0.338364
$560$	−1.00000	−0.0422577
$561$	−2.00000	−0.0844401
$562$	30.0000	1.26547
$563$	−32.0000	−1.34864	−0.674320	−	0.738440i	$-0.735563\pi$
$563$	−32.0000	−1.34864	−0.674320	+	0.738440i	$0.735563\pi$
$564$	−8.00000	−0.336861
$565$	10.0000	0.420703
$566$	4.00000	0.168133
$567$	−1.00000	−0.0419961
$568$	−48.0000	−2.01404
$569$	−26.0000	−1.08998	−0.544988	−	0.838444i	$-0.683466\pi$
$569$	−26.0000	−1.08998	−0.544988	+	0.838444i	$0.683466\pi$
$570$	4.00000	0.167542
$571$	−32.0000	−1.33916	−0.669579	−	0.742741i	$-0.733526\pi$
$571$	−32.0000	−1.33916	−0.669579	+	0.742741i	$0.733526\pi$
$572$	2.00000	0.0836242
$573$	24.0000	1.00261
$574$	6.00000	0.250435
$575$	−4.00000	−0.166812
$576$	7.00000	0.291667
$577$	−42.0000	−1.74848	−0.874241	−	0.485491i	$-0.838641\pi$
$577$	−42.0000	−1.74848	−0.874241	+	0.485491i	$0.838641\pi$
$578$	−13.0000	−0.540729
$579$	6.00000	0.249351
$580$	−6.00000	−0.249136
$581$	0	0
$582$	14.0000	0.580319
$583$	2.00000	0.0828315
$584$	−6.00000	−0.248282
$585$	2.00000	0.0826898
$586$	−22.0000	−0.908812
$587$	−28.0000	−1.15568	−0.577842	−	0.816149i	$-0.696105\pi$
$587$	−28.0000	−1.15568	−0.577842	+	0.816149i	$0.696105\pi$
$588$	−1.00000	−0.0412393
$589$	16.0000	0.659269
$590$	12.0000	0.494032
$591$	26.0000	1.06950
$592$	2.00000	0.0821995
$593$	−10.0000	−0.410651	−0.205325	−	0.978694i	$-0.565825\pi$
$593$	−10.0000	−0.410651	−0.205325	+	0.978694i	$0.565825\pi$
$594$	1.00000	0.0410305
$595$	−2.00000	−0.0819920
$596$	−2.00000	−0.0819232
$597$	−28.0000	−1.14596
$598$	8.00000	0.327144
$599$	40.0000	1.63436	0.817178	−	0.576386i	$-0.195537\pi$
$599$	40.0000	1.63436	0.817178	+	0.576386i	$0.195537\pi$
$600$	−3.00000	−0.122474
$601$	−22.0000	−0.897399	−0.448699	−	0.893683i	$-0.648113\pi$
$601$	−22.0000	−0.897399	−0.448699	+	0.893683i	$0.648113\pi$
$602$	4.00000	0.163028
$603$	−4.00000	−0.162893
$604$	12.0000	0.488273
$605$	−1.00000	−0.0406558
$606$	−2.00000	−0.0812444
$607$	16.0000	0.649420	0.324710	−	0.945814i	$-0.394733\pi$
$607$	16.0000	0.649420	0.324710	+	0.945814i	$0.394733\pi$
$608$	−20.0000	−0.811107
$609$	6.00000	0.243132
$610$	2.00000	0.0809776
$611$	−16.0000	−0.647291
$612$	2.00000	0.0808452
$613$	−38.0000	−1.53481	−0.767403	−	0.641165i	$-0.778451\pi$
$613$	−38.0000	−1.53481	−0.767403	+	0.641165i	$0.778451\pi$
$614$	−20.0000	−0.807134
$615$	6.00000	0.241943
$616$	3.00000	0.120873
$617$	−10.0000	−0.402585	−0.201292	−	0.979531i	$-0.564514\pi$
$617$	−10.0000	−0.402585	−0.201292	+	0.979531i	$0.564514\pi$
$618$	0	0
$619$	−32.0000	−1.28619	−0.643094	−	0.765787i	$-0.722350\pi$
$619$	−32.0000	−1.28619	−0.643094	+	0.765787i	$0.722350\pi$
$620$	−4.00000	−0.160644
$621$	−4.00000	−0.160514
$622$	−24.0000	−0.962312
$623$	−6.00000	−0.240385
$624$	2.00000	0.0800641
$625$	1.00000	0.0400000
$626$	6.00000	0.239808
$627$	−4.00000	−0.159745
$628$	22.0000	0.877896
$629$	4.00000	0.159490
$630$	1.00000	0.0398410
$631$	−40.0000	−1.59237	−0.796187	−	0.605050i	$-0.793153\pi$
$631$	−40.0000	−1.59237	−0.796187	+	0.605050i	$0.793153\pi$
$632$	12.0000	0.477334
$633$	8.00000	0.317971
$634$	−6.00000	−0.238290
$635$	−8.00000	−0.317470
$636$	−2.00000	−0.0793052
$637$	−2.00000	−0.0792429
$638$	−6.00000	−0.237542
$639$	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	$-0.423961\pi$
$643$	12.0000	0.473234	0.236617	+	0.971603i	$0.423961\pi$
$644$	−4.00000	−0.157622
$645$	4.00000	0.157500
$646$	8.00000	0.314756
$647$	24.0000	0.943537	0.471769	−	0.881722i	$-0.343616\pi$
$647$	24.0000	0.943537	0.471769	+	0.881722i	$0.343616\pi$
$648$	−3.00000	−0.117851
$649$	−12.0000	−0.471041
$650$	−2.00000	−0.0784465
$651$	4.00000	0.156772
$652$	−4.00000	−0.156652
$653$	10.0000	0.391330	0.195665	−	0.980671i	$-0.437313\pi$
$653$	10.0000	0.391330	0.195665	+	0.980671i	$0.437313\pi$
$654$	−2.00000	−0.0782062
$655$	−12.0000	−0.468879
$656$	6.00000	0.234261
$657$	2.00000	0.0780274
$658$	−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	$0.562301\pi$
$662$	4.00000	0.155464
$663$	4.00000	0.155347
$664$	0	0
$665$	−4.00000	−0.155113
$666$	−2.00000	−0.0774984
$667$	24.0000	0.929284
$668$	20.0000	0.773823
$669$	−16.0000	−0.618596
$670$	4.00000	0.154533
$671$	−2.00000	−0.0772091
$672$	−5.00000	−0.192879
$673$	14.0000	0.539660	0.269830	−	0.962908i	$-0.413032\pi$
$673$	14.0000	0.539660	0.269830	+	0.962908i	$0.413032\pi$
$674$	14.0000	0.539260
$675$	1.00000	0.0384900
$676$	9.00000	0.346154
$677$	−14.0000	−0.538064	−0.269032	−	0.963131i	$-0.586704\pi$
$677$	−14.0000	−0.538064	−0.269032	+	0.963131i	$0.586704\pi$
$678$	−10.0000	−0.384048
$679$	−14.0000	−0.537271
$680$	−6.00000	−0.230089
$681$	0	0
$682$	−4.00000	−0.153168
$683$	8.00000	0.306111	0.153056	−	0.988218i	$-0.451089\pi$
$683$	8.00000	0.306111	0.153056	+	0.988218i	$0.451089\pi$
$684$	4.00000	0.152944
$685$	−14.0000	−0.534913
$686$	−1.00000	−0.0381802
$687$	−2.00000	−0.0763048
$688$	4.00000	0.152499
$689$	−4.00000	−0.152388
$690$	4.00000	0.152277
$691$	8.00000	0.304334	0.152167	−	0.988355i	$-0.451375\pi$
$691$	8.00000	0.304334	0.152167	+	0.988355i	$0.451375\pi$
$692$	−2.00000	−0.0760286
$693$	−1.00000	−0.0379869
$694$	20.0000	0.759190
$695$	−4.00000	−0.151729
$696$	18.0000	0.682288
$697$	12.0000	0.454532
$698$	−2.00000	−0.0757011
$699$	−10.0000	−0.378235
$700$	1.00000	0.0377964
$701$	−30.0000	−1.13308	−0.566542	−	0.824033i	$-0.691719\pi$
$701$	−30.0000	−1.13308	−0.566542	+	0.824033i	$0.691719\pi$
$702$	−2.00000	−0.0754851
$703$	8.00000	0.301726
$704$	7.00000	0.263822
$705$	−8.00000	−0.301297
$706$	14.0000	0.526897
$707$	2.00000	0.0752177
$708$	12.0000	0.450988
$709$	38.0000	1.42712	0.713560	−	0.700594i	$-0.247082\pi$
$709$	38.0000	1.42712	0.713560	+	0.700594i	$0.247082\pi$
$710$	−16.0000	−0.600469
$711$	−4.00000	−0.150012
$712$	−18.0000	−0.674579
$713$	16.0000	0.599205
$714$	2.00000	0.0748481
$715$	2.00000	0.0747958
$716$	−4.00000	−0.149487
$717$	−16.0000	−0.597531
$718$	16.0000	0.597115
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	1.00000	0.0372678
$721$	0	0
$722$	−3.00000	−0.111648
$723$	−14.0000	−0.520666
$724$	10.0000	0.371647
$725$	−6.00000	−0.222834
$726$	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	$0.404113\pi$
$728$	−6.00000	−0.222375
$729$	1.00000	0.0370370
$730$	−2.00000	−0.0740233
$731$	8.00000	0.295891
$732$	2.00000	0.0739221
$733$	22.0000	0.812589	0.406294	−	0.913742i	$-0.366821\pi$
$733$	22.0000	0.812589	0.406294	+	0.913742i	$0.366821\pi$
$734$	16.0000	0.590571
$735$	−1.00000	−0.0368856
$736$	−20.0000	−0.737210
$737$	−4.00000	−0.147342
$738$	−6.00000	−0.220863
$739$	−24.0000	−0.882854	−0.441427	−	0.897297i	$-0.645528\pi$
$739$	−24.0000	−0.882854	−0.441427	+	0.897297i	$0.645528\pi$
$740$	−2.00000	−0.0735215
$741$	8.00000	0.293887
$742$	−2.00000	−0.0734223
$743$	−40.0000	−1.46746	−0.733729	−	0.679442i	$-0.762222\pi$
$743$	−40.0000	−1.46746	−0.733729	+	0.679442i	$0.762222\pi$
$744$	12.0000	0.439941
$745$	−2.00000	−0.0732743
$746$	2.00000	0.0732252
$747$	0	0
$748$	2.00000	0.0731272
$749$	−4.00000	−0.146157
$750$	−1.00000	−0.0365148
$751$	−24.0000	−0.875772	−0.437886	−	0.899030i	$-0.644273\pi$
$751$	−24.0000	−0.875772	−0.437886	+	0.899030i	$0.644273\pi$
$752$	−8.00000	−0.291730
$753$	12.0000	0.437304
$754$	12.0000	0.437014
$755$	12.0000	0.436725
$756$	1.00000	0.0363696
$757$	46.0000	1.67190	0.835949	−	0.548807i	$-0.184918\pi$
$757$	46.0000	1.67190	0.835949	+	0.548807i	$0.184918\pi$
$758$	4.00000	0.145287
$759$	−4.00000	−0.145191
$760$	−12.0000	−0.435286
$761$	−6.00000	−0.217500	−0.108750	−	0.994069i	$-0.534685\pi$
$761$	−6.00000	−0.217500	−0.108750	+	0.994069i	$0.534685\pi$
$762$	8.00000	0.289809
$763$	2.00000	0.0724049
$764$	−24.0000	−0.868290
$765$	2.00000	0.0723102
$766$	24.0000	0.867155
$767$	24.0000	0.866590
$768$	−17.0000	−0.613435
$769$	18.0000	0.649097	0.324548	−	0.945869i	$-0.394788\pi$
$769$	18.0000	0.649097	0.324548	+	0.945869i	$0.394788\pi$
$770$	1.00000	0.0360375
$771$	14.0000	0.504198
$772$	−6.00000	−0.215945
$773$	−6.00000	−0.215805	−0.107903	−	0.994161i	$-0.534413\pi$
$773$	−6.00000	−0.215805	−0.107903	+	0.994161i	$0.534413\pi$
$774$	−4.00000	−0.143777
$775$	−4.00000	−0.143684
$776$	−42.0000	−1.50771
$777$	2.00000	0.0717496
$778$	−26.0000	−0.932145
$779$	24.0000	0.859889
$780$	−2.00000	−0.0716115
$781$	16.0000	0.572525
$782$	8.00000	0.286079
$783$	−6.00000	−0.214423
$784$	−1.00000	−0.0357143
$785$	22.0000	0.785214
$786$	12.0000	0.428026
$787$	−4.00000	−0.142585	−0.0712923	−	0.997455i	$-0.522712\pi$
$787$	−4.00000	−0.142585	−0.0712923	+	0.997455i	$0.522712\pi$
$788$	−26.0000	−0.926212
$789$	8.00000	0.284808
$790$	4.00000	0.142314
$791$	10.0000	0.355559
$792$	−3.00000	−0.106600
$793$	4.00000	0.142044
$794$	2.00000	0.0709773
$795$	−2.00000	−0.0709327
$796$	28.0000	0.992434
$797$	−30.0000	−1.06265	−0.531327	−	0.847167i	$-0.678307\pi$
$797$	−30.0000	−1.06265	−0.531327	+	0.847167i	$0.678307\pi$
$798$	4.00000	0.141598
$799$	−16.0000	−0.566039
$800$	5.00000	0.176777
$801$	6.00000	0.212000
$802$	18.0000	0.635602
$803$	2.00000	0.0705785
$804$	4.00000	0.141069
$805$	−4.00000	−0.140981
$806$	8.00000	0.281788
$807$	−22.0000	−0.774437
$808$	6.00000	0.211079
$809$	−18.0000	−0.632846	−0.316423	−	0.948618i	$-0.602482\pi$
$809$	−18.0000	−0.632846	−0.316423	+	0.948618i	$0.602482\pi$
$810$	−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	$0.663590\pi$
$812$	−6.00000	−0.210559
$813$	−32.0000	−1.12229
$814$	−2.00000	−0.0701000
$815$	−4.00000	−0.140114
$816$	2.00000	0.0700140
$817$	16.0000	0.559769
$818$	−38.0000	−1.32864
$819$	2.00000	0.0698857
$820$	−6.00000	−0.209529
$821$	−38.0000	−1.32621	−0.663105	−	0.748527i	$-0.730762\pi$
$821$	−38.0000	−1.32621	−0.663105	+	0.748527i	$0.730762\pi$
$822$	14.0000	0.488306
$823$	8.00000	0.278862	0.139431	−	0.990232i	$-0.455473\pi$
$823$	8.00000	0.278862	0.139431	+	0.990232i	$0.455473\pi$
$824$	0	0
$825$	1.00000	0.0348155
$826$	12.0000	0.417533
$827$	−28.0000	−0.973655	−0.486828	−	0.873498i	$-0.661846\pi$
$827$	−28.0000	−0.973655	−0.486828	+	0.873498i	$0.661846\pi$
$828$	4.00000	0.139010
$829$	−18.0000	−0.625166	−0.312583	−	0.949890i	$-0.601194\pi$
$829$	−18.0000	−0.625166	−0.312583	+	0.949890i	$0.601194\pi$
$830$	0	0
$831$	−22.0000	−0.763172
$832$	−14.0000	−0.485363
$833$	−2.00000	−0.0692959
$834$	4.00000	0.138509
$835$	20.0000	0.692129
$836$	4.00000	0.138343
$837$	−4.00000	−0.138260
$838$	20.0000	0.690889
$839$	48.0000	1.65714	0.828572	−	0.559883i	$-0.189154\pi$
$839$	48.0000	1.65714	0.828572	+	0.559883i	$0.189154\pi$
$840$	−3.00000	−0.103510
$841$	7.00000	0.241379
$842$	−26.0000	−0.896019
$843$	30.0000	1.03325
$844$	−8.00000	−0.275371
$845$	9.00000	0.309609
$846$	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
$852$	−16.0000	−0.548151
$853$	46.0000	1.57501	0.787505	−	0.616308i	$-0.211372\pi$
$853$	46.0000	1.57501	0.787505	+	0.616308i	$0.211372\pi$
$854$	2.00000	0.0684386
$855$	4.00000	0.136797
$856$	−12.0000	−0.410152
$857$	22.0000	0.751506	0.375753	−	0.926720i	$-0.377384\pi$
$857$	22.0000	0.751506	0.375753	+	0.926720i	$0.377384\pi$
$858$	−2.00000	−0.0682789
$859$	32.0000	1.09183	0.545913	−	0.837842i	$-0.316183\pi$
$859$	32.0000	1.09183	0.545913	+	0.837842i	$0.316183\pi$
$860$	−4.00000	−0.136399
$861$	6.00000	0.204479
$862$	0	0
$863$	12.0000	0.408485	0.204242	−	0.978920i	$-0.434527\pi$
$863$	12.0000	0.408485	0.204242	+	0.978920i	$0.434527\pi$
$864$	5.00000	0.170103
$865$	−2.00000	−0.0680020
$866$	22.0000	0.747590
$867$	−13.0000	−0.441503
$868$	−4.00000	−0.135769
$869$	−4.00000	−0.135691
$870$	6.00000	0.203419
$871$	8.00000	0.271070
$872$	6.00000	0.203186
$873$	14.0000	0.473828
$874$	16.0000	0.541208
$875$	1.00000	0.0338062
$876$	−2.00000	−0.0675737
$877$	2.00000	0.0675352	0.0337676	−	0.999430i	$-0.489249\pi$
$877$	2.00000	0.0675352	0.0337676	+	0.999430i	$0.489249\pi$
$878$	0	0
$879$	−22.0000	−0.742042
$880$	1.00000	0.0337100
$881$	−18.0000	−0.606435	−0.303218	−	0.952921i	$-0.598061\pi$
$881$	−18.0000	−0.606435	−0.303218	+	0.952921i	$0.598061\pi$
$882$	1.00000	0.0336718
$883$	44.0000	1.48072	0.740359	−	0.672212i	$-0.234656\pi$
$883$	44.0000	1.48072	0.740359	+	0.672212i	$0.234656\pi$
$884$	−4.00000	−0.134535
$885$	12.0000	0.403376
$886$	24.0000	0.806296
$887$	−36.0000	−1.20876	−0.604381	−	0.796696i	$-0.706579\pi$
$887$	−36.0000	−1.20876	−0.604381	+	0.796696i	$0.706579\pi$
$888$	6.00000	0.201347
$889$	−8.00000	−0.268311
$890$	−6.00000	−0.201120
$891$	1.00000	0.0335013
$892$	16.0000	0.535720
$893$	−32.0000	−1.07084
$894$	2.00000	0.0668900
$895$	−4.00000	−0.133705
$896$	3.00000	0.100223
$897$	8.00000	0.267112
$898$	−14.0000	−0.467186
$899$	24.0000	0.800445
$900$	−1.00000	−0.0333333
$901$	−4.00000	−0.133259
$902$	−6.00000	−0.199778
$903$	4.00000	0.133112
$904$	30.0000	0.997785
$905$	10.0000	0.332411
$906$	−12.0000	−0.398673
$907$	−28.0000	−0.929725	−0.464862	−	0.885383i	$-0.653896\pi$
$907$	−28.0000	−0.929725	−0.464862	+	0.885383i	$0.653896\pi$
$908$	0	0
$909$	−2.00000	−0.0663358
$910$	−2.00000	−0.0662994
$911$	56.0000	1.85536	0.927681	−	0.373373i	$-0.121799\pi$
$911$	56.0000	1.85536	0.927681	+	0.373373i	$0.121799\pi$
$912$	4.00000	0.132453
$913$	0	0
$914$	6.00000	0.198462
$915$	2.00000	0.0661180
$916$	2.00000	0.0660819
$917$	−12.0000	−0.396275
$918$	−2.00000	−0.0660098
$919$	44.0000	1.45143	0.725713	−	0.687998i	$-0.241510\pi$
$919$	44.0000	1.45143	0.725713	+	0.687998i	$0.241510\pi$
$920$	−12.0000	−0.395628
$921$	−20.0000	−0.659022
$922$	−2.00000	−0.0658665
$923$	−32.0000	−1.05329
$924$	1.00000	0.0328976
$925$	−2.00000	−0.0657596
$926$	−8.00000	−0.262896
$927$	0	0
$928$	−30.0000	−0.984798
$929$	6.00000	0.196854	0.0984268	−	0.995144i	$-0.468619\pi$
$929$	6.00000	0.196854	0.0984268	+	0.995144i	$0.468619\pi$
$930$	4.00000	0.131165
$931$	−4.00000	−0.131095
$932$	10.0000	0.327561
$933$	−24.0000	−0.785725
$934$	−12.0000	−0.392652
$935$	2.00000	0.0654070
$936$	6.00000	0.196116
$937$	−38.0000	−1.24141	−0.620703	−	0.784046i	$-0.713153\pi$
$937$	−38.0000	−1.24141	−0.620703	+	0.784046i	$0.713153\pi$
$938$	4.00000	0.130605
$939$	6.00000	0.195803
$940$	8.00000	0.260931
$941$	−50.0000	−1.62995	−0.814977	−	0.579494i	$-0.803250\pi$
$941$	−50.0000	−1.62995	−0.814977	+	0.579494i	$0.803250\pi$
$942$	−22.0000	−0.716799
$943$	24.0000	0.781548
$944$	12.0000	0.390567
$945$	1.00000	0.0325300
$946$	−4.00000	−0.130051
$947$	−8.00000	−0.259965	−0.129983	−	0.991516i	$-0.541492\pi$
$947$	−8.00000	−0.259965	−0.129983	+	0.991516i	$0.541492\pi$
$948$	4.00000	0.129914
$949$	−4.00000	−0.129845
$950$	−4.00000	−0.129777
$951$	−6.00000	−0.194563
$952$	−6.00000	−0.194461
$953$	−42.0000	−1.36051	−0.680257	−	0.732974i	$-0.738132\pi$
$953$	−42.0000	−1.36051	−0.680257	+	0.732974i	$0.738132\pi$
$954$	2.00000	0.0647524
$955$	−24.0000	−0.776622
$956$	16.0000	0.517477
$957$	−6.00000	−0.193952
$958$	−8.00000	−0.258468
$959$	−14.0000	−0.452084
$960$	−7.00000	−0.225924
$961$	−15.0000	−0.483871
$962$	4.00000	0.128965
$963$	4.00000	0.128898
$964$	14.0000	0.450910
$965$	−6.00000	−0.193147
$966$	4.00000	0.128698
$967$	8.00000	0.257263	0.128631	−	0.991692i	$-0.458942\pi$
$967$	8.00000	0.257263	0.128631	+	0.991692i	$0.458942\pi$
$968$	−3.00000	−0.0964237
$969$	8.00000	0.256997
$970$	−14.0000	−0.449513
$971$	20.0000	0.641831	0.320915	−	0.947108i	$-0.396010\pi$
$971$	20.0000	0.641831	0.320915	+	0.947108i	$0.396010\pi$
$972$	−1.00000	−0.0320750
$973$	−4.00000	−0.128234
$974$	16.0000	0.512673
$975$	−2.00000	−0.0640513
$976$	2.00000	0.0640184
$977$	−18.0000	−0.575871	−0.287936	−	0.957650i	$-0.592969\pi$
$977$	−18.0000	−0.575871	−0.287936	+	0.957650i	$0.592969\pi$
$978$	4.00000	0.127906
$979$	6.00000	0.191761
$980$	1.00000	0.0319438
$981$	−2.00000	−0.0638551
$982$	−20.0000	−0.638226
$983$	24.0000	0.765481	0.382741	−	0.923856i	$-0.374980\pi$
$983$	24.0000	0.765481	0.382741	+	0.923856i	$0.374980\pi$
$984$	18.0000	0.573819
$985$	−26.0000	−0.828429
$986$	12.0000	0.382158
$987$	−8.00000	−0.254643
$988$	−8.00000	−0.254514
$989$	16.0000	0.508770
$990$	−1.00000	−0.0317821
$991$	16.0000	0.508257	0.254128	−	0.967170i	$-0.418211\pi$
$991$	16.0000	0.508257	0.254128	+	0.967170i	$0.418211\pi$
$992$	−20.0000	−0.635001
$993$	4.00000	0.126936
$994$	−16.0000	−0.507489
$995$	28.0000	0.887660
$996$	0	0
$997$	46.0000	1.45683	0.728417	−	0.685134i	$-0.240256\pi$
$997$	46.0000	1.45683	0.728417	+	0.685134i	$0.240256\pi$
$998$	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.l.1.1	✓	1
3.2	odd	2		3465.2.a.d.1.1		1
5.4	even	2		5775.2.a.e.1.1		1
7.6	odd	2		8085.2.a.t.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1155.2.a.l.1.1	✓	1	1.1	even	1	trivial
3465.2.a.d.1.1		1	3.2	odd	2
5775.2.a.e.1.1		1	5.4	even	2
8085.2.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 $\F_{q}$

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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