LMFDB - Embedded newform 1110.2.a.e.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1110,2,Mod(1,1110)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1110, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1110.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1110 = 2 \cdot 3 \cdot 5 \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1110.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:	$8.86339462436$
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	$\chi$	$=$	1110.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} +3.00000 q^{11} +1.00000 q^{12} -7.00000 q^{13} +1.00000 q^{14} -1.00000 q^{15} +1.00000 q^{16} -3.00000 q^{17} -1.00000 q^{18} -1.00000 q^{19} -1.00000 q^{20} -1.00000 q^{21} -3.00000 q^{22} -3.00000 q^{23} -1.00000 q^{24} +1.00000 q^{25} +7.00000 q^{26} +1.00000 q^{27} -1.00000 q^{28} +6.00000 q^{29} +1.00000 q^{30} -10.0000 q^{31} -1.00000 q^{32} +3.00000 q^{33} +3.00000 q^{34} +1.00000 q^{35} +1.00000 q^{36} +1.00000 q^{37} +1.00000 q^{38} -7.00000 q^{39} +1.00000 q^{40} +1.00000 q^{42} -4.00000 q^{43} +3.00000 q^{44} -1.00000 q^{45} +3.00000 q^{46} -12.0000 q^{47} +1.00000 q^{48} -6.00000 q^{49} -1.00000 q^{50} -3.00000 q^{51} -7.00000 q^{52} +9.00000 q^{53} -1.00000 q^{54} -3.00000 q^{55} +1.00000 q^{56} -1.00000 q^{57} -6.00000 q^{58} +6.00000 q^{59} -1.00000 q^{60} -10.0000 q^{61} +10.0000 q^{62} -1.00000 q^{63} +1.00000 q^{64} +7.00000 q^{65} -3.00000 q^{66} -10.0000 q^{67} -3.00000 q^{68} -3.00000 q^{69} -1.00000 q^{70} -6.00000 q^{71} -1.00000 q^{72} +11.0000 q^{73} -1.00000 q^{74} +1.00000 q^{75} -1.00000 q^{76} -3.00000 q^{77} +7.00000 q^{78} +8.00000 q^{79} -1.00000 q^{80} +1.00000 q^{81} +9.00000 q^{83} -1.00000 q^{84} +3.00000 q^{85} +4.00000 q^{86} +6.00000 q^{87} -3.00000 q^{88} -9.00000 q^{89} +1.00000 q^{90} +7.00000 q^{91} -3.00000 q^{92} -10.0000 q^{93} +12.0000 q^{94} +1.00000 q^{95} -1.00000 q^{96} -16.0000 q^{97} +6.00000 q^{98} +3.00000 q^{99} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	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	−0.188982	−	0.981981i	$-0.560519\pi$
$7$	−1.00000	−0.377964	−0.188982	+	0.981981i	$0.560519\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	1.00000	0.316228
$11$	3.00000	0.904534	0.452267	−	0.891883i	$-0.350615\pi$
$11$	3.00000	0.904534	0.452267	+	0.891883i	$0.350615\pi$
$12$	1.00000	0.288675
$13$	−7.00000	−1.94145	−0.970725	−	0.240192i	$-0.922790\pi$
$13$	−7.00000	−1.94145	−0.970725	+	0.240192i	$0.922790\pi$
$14$	1.00000	0.267261
$15$	−1.00000	−0.258199
$16$	1.00000	0.250000
$17$	−3.00000	−0.727607	−0.363803	−	0.931476i	$-0.618522\pi$
$17$	−3.00000	−0.727607	−0.363803	+	0.931476i	$0.618522\pi$
$18$	−1.00000	−0.235702
$19$	−1.00000	−0.229416	−0.114708	−	0.993399i	$-0.536593\pi$
$19$	−1.00000	−0.229416	−0.114708	+	0.993399i	$0.536593\pi$
$20$	−1.00000	−0.223607
$21$	−1.00000	−0.218218
$22$	−3.00000	−0.639602
$23$	−3.00000	−0.625543	−0.312772	−	0.949828i	$-0.601257\pi$
$23$	−3.00000	−0.625543	−0.312772	+	0.949828i	$0.601257\pi$
$24$	−1.00000	−0.204124
$25$	1.00000	0.200000
$26$	7.00000	1.37281
$27$	1.00000	0.192450
$28$	−1.00000	−0.188982
$29$	6.00000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	1.00000	0.182574
$31$	−10.0000	−1.79605	−0.898027	−	0.439941i	$-0.854999\pi$
$31$	−10.0000	−1.79605	−0.898027	+	0.439941i	$0.854999\pi$
$32$	−1.00000	−0.176777
$33$	3.00000	0.522233
$34$	3.00000	0.514496
$35$	1.00000	0.169031
$36$	1.00000	0.166667
$37$	1.00000	0.164399
$37$	1.00000	0.164399
$38$	1.00000	0.162221
$39$	−7.00000	−1.12090
$40$	1.00000	0.158114
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	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	$0.598656\pi$
$44$	3.00000	0.452267
$45$	−1.00000	−0.149071
$46$	3.00000	0.442326
$47$	−12.0000	−1.75038	−0.875190	−	0.483779i	$-0.839264\pi$
$47$	−12.0000	−1.75038	−0.875190	+	0.483779i	$0.839264\pi$
$48$	1.00000	0.144338
$49$	−6.00000	−0.857143
$50$	−1.00000	−0.141421
$51$	−3.00000	−0.420084
$52$	−7.00000	−0.970725
$53$	9.00000	1.23625	0.618123	−	0.786082i	$-0.287894\pi$
$53$	9.00000	1.23625	0.618123	+	0.786082i	$0.287894\pi$
$54$	−1.00000	−0.136083
$55$	−3.00000	−0.404520
$56$	1.00000	0.133631
$57$	−1.00000	−0.132453
$58$	−6.00000	−0.787839
$59$	6.00000	0.781133	0.390567	−	0.920575i	$-0.372279\pi$
$59$	6.00000	0.781133	0.390567	+	0.920575i	$0.372279\pi$
$60$	−1.00000	−0.129099
$61$	−10.0000	−1.28037	−0.640184	−	0.768221i	$-0.721142\pi$
$61$	−10.0000	−1.28037	−0.640184	+	0.768221i	$0.721142\pi$
$62$	10.0000	1.27000
$63$	−1.00000	−0.125988
$64$	1.00000	0.125000
$65$	7.00000	0.868243
$66$	−3.00000	−0.369274
$67$	−10.0000	−1.22169	−0.610847	−	0.791748i	$-0.709171\pi$
$67$	−10.0000	−1.22169	−0.610847	+	0.791748i	$0.709171\pi$
$68$	−3.00000	−0.363803
$69$	−3.00000	−0.361158
$70$	−1.00000	−0.119523
$71$	−6.00000	−0.712069	−0.356034	−	0.934473i	$-0.615871\pi$
$71$	−6.00000	−0.712069	−0.356034	+	0.934473i	$0.615871\pi$
$72$	−1.00000	−0.117851
$73$	11.0000	1.28745	0.643726	−	0.765256i	$-0.277388\pi$
$73$	11.0000	1.28745	0.643726	+	0.765256i	$0.277388\pi$
$74$	−1.00000	−0.116248
$75$	1.00000	0.115470
$76$	−1.00000	−0.114708
$77$	−3.00000	−0.341882
$78$	7.00000	0.792594
$79$	8.00000	0.900070	0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000	0.900070	0.450035	+	0.893011i	$0.351411\pi$
$80$	−1.00000	−0.111803
$81$	1.00000	0.111111
$82$	0	0
$83$	9.00000	0.987878	0.493939	−	0.869496i	$-0.335557\pi$
$83$	9.00000	0.987878	0.493939	+	0.869496i	$0.335557\pi$
$84$	−1.00000	−0.109109
$85$	3.00000	0.325396
$86$	4.00000	0.431331
$87$	6.00000	0.643268
$88$	−3.00000	−0.319801
$89$	−9.00000	−0.953998	−0.476999	−	0.878904i	$-0.658275\pi$
$89$	−9.00000	−0.953998	−0.476999	+	0.878904i	$0.658275\pi$
$90$	1.00000	0.105409
$91$	7.00000	0.733799
$92$	−3.00000	−0.312772
$93$	−10.0000	−1.03695
$94$	12.0000	1.23771
$95$	1.00000	0.102598
$96$	−1.00000	−0.102062
$97$	−16.0000	−1.62455	−0.812277	−	0.583272i	$-0.801772\pi$
$97$	−16.0000	−1.62455	−0.812277	+	0.583272i	$0.801772\pi$
$98$	6.00000	0.606092
$99$	3.00000	0.301511
$100$	1.00000	0.100000
$101$	−6.00000	−0.597022	−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000	−0.597022	−0.298511	+	0.954406i	$0.596490\pi$
$102$	3.00000	0.297044
$103$	−4.00000	−0.394132	−0.197066	−	0.980390i	$-0.563141\pi$
$103$	−4.00000	−0.394132	−0.197066	+	0.980390i	$0.563141\pi$
$104$	7.00000	0.686406
$105$	1.00000	0.0975900
$106$	−9.00000	−0.874157
$107$	3.00000	0.290021	0.145010	−	0.989430i	$-0.453678\pi$
$107$	3.00000	0.290021	0.145010	+	0.989430i	$0.453678\pi$
$108$	1.00000	0.0962250
$109$	−7.00000	−0.670478	−0.335239	−	0.942133i	$-0.608817\pi$
$109$	−7.00000	−0.670478	−0.335239	+	0.942133i	$0.608817\pi$
$110$	3.00000	0.286039
$111$	1.00000	0.0949158
$112$	−1.00000	−0.0944911
$113$	18.0000	1.69330	0.846649	−	0.532152i	$-0.178617\pi$
$113$	18.0000	1.69330	0.846649	+	0.532152i	$0.178617\pi$
$114$	1.00000	0.0936586
$115$	3.00000	0.279751
$116$	6.00000	0.557086
$117$	−7.00000	−0.647150
$118$	−6.00000	−0.552345
$119$	3.00000	0.275010
$120$	1.00000	0.0912871
$121$	−2.00000	−0.181818
$122$	10.0000	0.905357
$123$	0	0
$124$	−10.0000	−0.898027
$125$	−1.00000	−0.0894427
$126$	1.00000	0.0890871
$127$	−7.00000	−0.621150	−0.310575	−	0.950549i	$-0.600522\pi$
$127$	−7.00000	−0.621150	−0.310575	+	0.950549i	$0.600522\pi$
$128$	−1.00000	−0.0883883
$129$	−4.00000	−0.352180
$130$	−7.00000	−0.613941
$131$	−12.0000	−1.04844	−0.524222	−	0.851581i	$-0.675644\pi$
$131$	−12.0000	−1.04844	−0.524222	+	0.851581i	$0.675644\pi$
$132$	3.00000	0.261116
$133$	1.00000	0.0867110
$134$	10.0000	0.863868
$135$	−1.00000	−0.0860663
$136$	3.00000	0.257248
$137$	−18.0000	−1.53784	−0.768922	−	0.639343i	$-0.779207\pi$
$137$	−18.0000	−1.53784	−0.768922	+	0.639343i	$0.779207\pi$
$138$	3.00000	0.255377
$139$	14.0000	1.18746	0.593732	−	0.804663i	$-0.297654\pi$
$139$	14.0000	1.18746	0.593732	+	0.804663i	$0.297654\pi$
$140$	1.00000	0.0845154
$141$	−12.0000	−1.01058
$142$	6.00000	0.503509
$143$	−21.0000	−1.75611
$144$	1.00000	0.0833333
$145$	−6.00000	−0.498273
$146$	−11.0000	−0.910366
$147$	−6.00000	−0.494872
$148$	1.00000	0.0821995
$149$	−6.00000	−0.491539	−0.245770	−	0.969328i	$-0.579041\pi$
$149$	−6.00000	−0.491539	−0.245770	+	0.969328i	$0.579041\pi$
$150$	−1.00000	−0.0816497
$151$	23.0000	1.87171	0.935857	−	0.352381i	$-0.114628\pi$
$151$	23.0000	1.87171	0.935857	+	0.352381i	$0.114628\pi$
$152$	1.00000	0.0811107
$153$	−3.00000	−0.242536
$154$	3.00000	0.241747
$155$	10.0000	0.803219
$156$	−7.00000	−0.560449
$157$	−4.00000	−0.319235	−0.159617	−	0.987179i	$-0.551026\pi$
$157$	−4.00000	−0.319235	−0.159617	+	0.987179i	$0.551026\pi$
$158$	−8.00000	−0.636446
$159$	9.00000	0.713746
$160$	1.00000	0.0790569
$161$	3.00000	0.236433
$162$	−1.00000	−0.0785674
$163$	−1.00000	−0.0783260	−0.0391630	−	0.999233i	$-0.512469\pi$
$163$	−1.00000	−0.0783260	−0.0391630	+	0.999233i	$0.512469\pi$
$164$	0	0
$165$	−3.00000	−0.233550
$166$	−9.00000	−0.698535
$167$	3.00000	0.232147	0.116073	−	0.993241i	$-0.462969\pi$
$167$	3.00000	0.232147	0.116073	+	0.993241i	$0.462969\pi$
$168$	1.00000	0.0771517
$169$	36.0000	2.76923
$170$	−3.00000	−0.230089
$171$	−1.00000	−0.0764719
$172$	−4.00000	−0.304997
$173$	−9.00000	−0.684257	−0.342129	−	0.939653i	$-0.611148\pi$
$173$	−9.00000	−0.684257	−0.342129	+	0.939653i	$0.611148\pi$
$174$	−6.00000	−0.454859
$175$	−1.00000	−0.0755929
$176$	3.00000	0.226134
$177$	6.00000	0.450988
$178$	9.00000	0.674579
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	−1.00000	−0.0745356
$181$	−22.0000	−1.63525	−0.817624	−	0.575753i	$-0.804709\pi$
$181$	−22.0000	−1.63525	−0.817624	+	0.575753i	$0.804709\pi$
$182$	−7.00000	−0.518875
$183$	−10.0000	−0.739221
$184$	3.00000	0.221163
$185$	−1.00000	−0.0735215
$186$	10.0000	0.733236
$187$	−9.00000	−0.658145
$188$	−12.0000	−0.875190
$189$	−1.00000	−0.0727393
$190$	−1.00000	−0.0725476
$191$	3.00000	0.217072	0.108536	−	0.994092i	$-0.465384\pi$
$191$	3.00000	0.217072	0.108536	+	0.994092i	$0.465384\pi$
$192$	1.00000	0.0721688
$193$	−10.0000	−0.719816	−0.359908	−	0.932988i	$-0.617192\pi$
$193$	−10.0000	−0.719816	−0.359908	+	0.932988i	$0.617192\pi$
$194$	16.0000	1.14873
$195$	7.00000	0.501280
$196$	−6.00000	−0.428571
$197$	3.00000	0.213741	0.106871	−	0.994273i	$-0.465917\pi$
$197$	3.00000	0.213741	0.106871	+	0.994273i	$0.465917\pi$
$198$	−3.00000	−0.213201
$199$	20.0000	1.41776	0.708881	−	0.705328i	$-0.249200\pi$
$199$	20.0000	1.41776	0.708881	+	0.705328i	$0.249200\pi$
$200$	−1.00000	−0.0707107
$201$	−10.0000	−0.705346
$202$	6.00000	0.422159
$203$	−6.00000	−0.421117
$204$	−3.00000	−0.210042
$205$	0	0
$206$	4.00000	0.278693
$207$	−3.00000	−0.208514
$208$	−7.00000	−0.485363
$209$	−3.00000	−0.207514
$210$	−1.00000	−0.0690066
$211$	2.00000	0.137686	0.0688428	−	0.997628i	$-0.478069\pi$
$211$	2.00000	0.137686	0.0688428	+	0.997628i	$0.478069\pi$
$212$	9.00000	0.618123
$213$	−6.00000	−0.411113
$214$	−3.00000	−0.205076
$215$	4.00000	0.272798
$216$	−1.00000	−0.0680414
$217$	10.0000	0.678844
$218$	7.00000	0.474100
$219$	11.0000	0.743311
$220$	−3.00000	−0.202260
$221$	21.0000	1.41261
$222$	−1.00000	−0.0671156
$223$	8.00000	0.535720	0.267860	−	0.963458i	$-0.413684\pi$
$223$	8.00000	0.535720	0.267860	+	0.963458i	$0.413684\pi$
$224$	1.00000	0.0668153
$225$	1.00000	0.0666667
$226$	−18.0000	−1.19734
$227$	24.0000	1.59294	0.796468	−	0.604681i	$-0.206699\pi$
$227$	24.0000	1.59294	0.796468	+	0.604681i	$0.206699\pi$
$228$	−1.00000	−0.0662266
$229$	−10.0000	−0.660819	−0.330409	−	0.943838i	$-0.607187\pi$
$229$	−10.0000	−0.660819	−0.330409	+	0.943838i	$0.607187\pi$
$230$	−3.00000	−0.197814
$231$	−3.00000	−0.197386
$232$	−6.00000	−0.393919
$233$	18.0000	1.17922	0.589610	−	0.807688i	$-0.299282\pi$
$233$	18.0000	1.17922	0.589610	+	0.807688i	$0.299282\pi$
$234$	7.00000	0.457604
$235$	12.0000	0.782794
$236$	6.00000	0.390567
$237$	8.00000	0.519656
$238$	−3.00000	−0.194461
$239$	12.0000	0.776215	0.388108	−	0.921614i	$-0.373129\pi$
$239$	12.0000	0.776215	0.388108	+	0.921614i	$0.373129\pi$
$240$	−1.00000	−0.0645497
$241$	26.0000	1.67481	0.837404	−	0.546585i	$-0.184072\pi$
$241$	26.0000	1.67481	0.837404	+	0.546585i	$0.184072\pi$
$242$	2.00000	0.128565
$243$	1.00000	0.0641500
$244$	−10.0000	−0.640184
$245$	6.00000	0.383326
$246$	0	0
$247$	7.00000	0.445399
$248$	10.0000	0.635001
$249$	9.00000	0.570352
$250$	1.00000	0.0632456
$251$	6.00000	0.378717	0.189358	−	0.981908i	$-0.439359\pi$
$251$	6.00000	0.378717	0.189358	+	0.981908i	$0.439359\pi$
$252$	−1.00000	−0.0629941
$253$	−9.00000	−0.565825
$254$	7.00000	0.439219
$255$	3.00000	0.187867
$256$	1.00000	0.0625000
$257$	9.00000	0.561405	0.280702	−	0.959795i	$-0.409433\pi$
$257$	9.00000	0.561405	0.280702	+	0.959795i	$0.409433\pi$
$258$	4.00000	0.249029
$259$	−1.00000	−0.0621370
$260$	7.00000	0.434122
$261$	6.00000	0.371391
$262$	12.0000	0.741362
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	−3.00000	−0.184637
$265$	−9.00000	−0.552866
$266$	−1.00000	−0.0613139
$267$	−9.00000	−0.550791
$268$	−10.0000	−0.610847
$269$	9.00000	0.548740	0.274370	−	0.961624i	$-0.411531\pi$
$269$	9.00000	0.548740	0.274370	+	0.961624i	$0.411531\pi$
$270$	1.00000	0.0608581
$271$	20.0000	1.21491	0.607457	−	0.794353i	$-0.292190\pi$
$271$	20.0000	1.21491	0.607457	+	0.794353i	$0.292190\pi$
$272$	−3.00000	−0.181902
$273$	7.00000	0.423659
$274$	18.0000	1.08742
$275$	3.00000	0.180907
$276$	−3.00000	−0.180579
$277$	23.0000	1.38194	0.690968	−	0.722885i	$-0.257185\pi$
$277$	23.0000	1.38194	0.690968	+	0.722885i	$0.257185\pi$
$278$	−14.0000	−0.839664
$279$	−10.0000	−0.598684
$280$	−1.00000	−0.0597614
$281$	27.0000	1.61068	0.805342	−	0.592810i	$-0.201981\pi$
$281$	27.0000	1.61068	0.805342	+	0.592810i	$0.201981\pi$
$282$	12.0000	0.714590
$283$	−13.0000	−0.772770	−0.386385	−	0.922338i	$-0.626276\pi$
$283$	−13.0000	−0.772770	−0.386385	+	0.922338i	$0.626276\pi$
$284$	−6.00000	−0.356034
$285$	1.00000	0.0592349
$286$	21.0000	1.24176
$287$	0	0
$288$	−1.00000	−0.0589256
$289$	−8.00000	−0.470588
$290$	6.00000	0.352332
$291$	−16.0000	−0.937937
$292$	11.0000	0.643726
$293$	−15.0000	−0.876309	−0.438155	−	0.898900i	$-0.644368\pi$
$293$	−15.0000	−0.876309	−0.438155	+	0.898900i	$0.644368\pi$
$294$	6.00000	0.349927
$295$	−6.00000	−0.349334
$296$	−1.00000	−0.0581238
$297$	3.00000	0.174078
$298$	6.00000	0.347571
$299$	21.0000	1.21446
$300$	1.00000	0.0577350
$301$	4.00000	0.230556
$302$	−23.0000	−1.32350
$303$	−6.00000	−0.344691
$304$	−1.00000	−0.0573539
$305$	10.0000	0.572598
$306$	3.00000	0.171499
$307$	−22.0000	−1.25561	−0.627803	−	0.778372i	$-0.716046\pi$
$307$	−22.0000	−1.25561	−0.627803	+	0.778372i	$0.716046\pi$
$308$	−3.00000	−0.170941
$309$	−4.00000	−0.227552
$310$	−10.0000	−0.567962
$311$	−24.0000	−1.36092	−0.680458	−	0.732787i	$-0.738219\pi$
$311$	−24.0000	−1.36092	−0.680458	+	0.732787i	$0.738219\pi$
$312$	7.00000	0.396297
$313$	8.00000	0.452187	0.226093	−	0.974106i	$-0.427405\pi$
$313$	8.00000	0.452187	0.226093	+	0.974106i	$0.427405\pi$
$314$	4.00000	0.225733
$315$	1.00000	0.0563436
$316$	8.00000	0.450035
$317$	−18.0000	−1.01098	−0.505490	−	0.862832i	$-0.668688\pi$
$317$	−18.0000	−1.01098	−0.505490	+	0.862832i	$0.668688\pi$
$318$	−9.00000	−0.504695
$319$	18.0000	1.00781
$320$	−1.00000	−0.0559017
$321$	3.00000	0.167444
$322$	−3.00000	−0.167183
$323$	3.00000	0.166924
$324$	1.00000	0.0555556
$325$	−7.00000	−0.388290
$326$	1.00000	0.0553849
$327$	−7.00000	−0.387101
$328$	0	0
$329$	12.0000	0.661581
$330$	3.00000	0.165145
$331$	−28.0000	−1.53902	−0.769510	−	0.638635i	$-0.779499\pi$
$331$	−28.0000	−1.53902	−0.769510	+	0.638635i	$0.779499\pi$
$332$	9.00000	0.493939
$333$	1.00000	0.0547997
$334$	−3.00000	−0.164153
$335$	10.0000	0.546358
$336$	−1.00000	−0.0545545
$337$	−13.0000	−0.708155	−0.354078	−	0.935216i	$-0.615205\pi$
$337$	−13.0000	−0.708155	−0.354078	+	0.935216i	$0.615205\pi$
$338$	−36.0000	−1.95814
$339$	18.0000	0.977626
$340$	3.00000	0.162698
$341$	−30.0000	−1.62459
$342$	1.00000	0.0540738
$343$	13.0000	0.701934
$344$	4.00000	0.215666
$345$	3.00000	0.161515
$346$	9.00000	0.483843
$347$	36.0000	1.93258	0.966291	−	0.257454i	$-0.0828835\pi$
$347$	36.0000	1.93258	0.966291	+	0.257454i	$0.0828835\pi$
$348$	6.00000	0.321634
$349$	8.00000	0.428230	0.214115	−	0.976808i	$-0.431313\pi$
$349$	8.00000	0.428230	0.214115	+	0.976808i	$0.431313\pi$
$350$	1.00000	0.0534522
$351$	−7.00000	−0.373632
$352$	−3.00000	−0.159901
$353$	18.0000	0.958043	0.479022	−	0.877803i	$-0.340992\pi$
$353$	18.0000	0.958043	0.479022	+	0.877803i	$0.340992\pi$
$354$	−6.00000	−0.318896
$355$	6.00000	0.318447
$356$	−9.00000	−0.476999
$357$	3.00000	0.158777
$358$	0	0
$359$	−12.0000	−0.633336	−0.316668	−	0.948536i	$-0.602564\pi$
$359$	−12.0000	−0.633336	−0.316668	+	0.948536i	$0.602564\pi$
$360$	1.00000	0.0527046
$361$	−18.0000	−0.947368
$362$	22.0000	1.15629
$363$	−2.00000	−0.104973
$364$	7.00000	0.366900
$365$	−11.0000	−0.575766
$366$	10.0000	0.522708
$367$	23.0000	1.20059	0.600295	−	0.799779i	$-0.295050\pi$
$367$	23.0000	1.20059	0.600295	+	0.799779i	$0.295050\pi$
$368$	−3.00000	−0.156386
$369$	0	0
$370$	1.00000	0.0519875
$371$	−9.00000	−0.467257
$372$	−10.0000	−0.518476
$373$	14.0000	0.724893	0.362446	−	0.932005i	$-0.381942\pi$
$373$	14.0000	0.724893	0.362446	+	0.932005i	$0.381942\pi$
$374$	9.00000	0.465379
$375$	−1.00000	−0.0516398
$376$	12.0000	0.618853
$377$	−42.0000	−2.16311
$378$	1.00000	0.0514344
$379$	26.0000	1.33553	0.667765	−	0.744372i	$-0.267251\pi$
$379$	26.0000	1.33553	0.667765	+	0.744372i	$0.267251\pi$
$380$	1.00000	0.0512989
$381$	−7.00000	−0.358621
$382$	−3.00000	−0.153493
$383$	−9.00000	−0.459879	−0.229939	−	0.973205i	$-0.573853\pi$
$383$	−9.00000	−0.459879	−0.229939	+	0.973205i	$0.573853\pi$
$384$	−1.00000	−0.0510310
$385$	3.00000	0.152894
$386$	10.0000	0.508987
$387$	−4.00000	−0.203331
$388$	−16.0000	−0.812277
$389$	18.0000	0.912636	0.456318	−	0.889817i	$-0.349168\pi$
$389$	18.0000	0.912636	0.456318	+	0.889817i	$0.349168\pi$
$390$	−7.00000	−0.354459
$391$	9.00000	0.455150
$392$	6.00000	0.303046
$393$	−12.0000	−0.605320
$394$	−3.00000	−0.151138
$395$	−8.00000	−0.402524
$396$	3.00000	0.150756
$397$	−4.00000	−0.200754	−0.100377	−	0.994949i	$-0.532005\pi$
$397$	−4.00000	−0.200754	−0.100377	+	0.994949i	$0.532005\pi$
$398$	−20.0000	−1.00251
$399$	1.00000	0.0500626
$400$	1.00000	0.0500000
$401$	15.0000	0.749064	0.374532	−	0.927214i	$-0.377803\pi$
$401$	15.0000	0.749064	0.374532	+	0.927214i	$0.377803\pi$
$402$	10.0000	0.498755
$403$	70.0000	3.48695
$404$	−6.00000	−0.298511
$405$	−1.00000	−0.0496904
$406$	6.00000	0.297775
$407$	3.00000	0.148704
$408$	3.00000	0.148522
$409$	−4.00000	−0.197787	−0.0988936	−	0.995098i	$-0.531530\pi$
$409$	−4.00000	−0.197787	−0.0988936	+	0.995098i	$0.531530\pi$
$410$	0	0
$411$	−18.0000	−0.887875
$412$	−4.00000	−0.197066
$413$	−6.00000	−0.295241
$414$	3.00000	0.147442
$415$	−9.00000	−0.441793
$416$	7.00000	0.343203
$417$	14.0000	0.685583
$418$	3.00000	0.146735
$419$	−27.0000	−1.31904	−0.659518	−	0.751689i	$-0.729240\pi$
$419$	−27.0000	−1.31904	−0.659518	+	0.751689i	$0.729240\pi$
$420$	1.00000	0.0487950
$421$	−10.0000	−0.487370	−0.243685	−	0.969854i	$-0.578356\pi$
$421$	−10.0000	−0.487370	−0.243685	+	0.969854i	$0.578356\pi$
$422$	−2.00000	−0.0973585
$423$	−12.0000	−0.583460
$424$	−9.00000	−0.437079
$425$	−3.00000	−0.145521
$426$	6.00000	0.290701
$427$	10.0000	0.483934
$428$	3.00000	0.145010
$429$	−21.0000	−1.01389
$430$	−4.00000	−0.192897
$431$	−3.00000	−0.144505	−0.0722525	−	0.997386i	$-0.523019\pi$
$431$	−3.00000	−0.144505	−0.0722525	+	0.997386i	$0.523019\pi$
$432$	1.00000	0.0481125
$433$	−31.0000	−1.48976	−0.744882	−	0.667196i	$-0.767494\pi$
$433$	−31.0000	−1.48976	−0.744882	+	0.667196i	$0.767494\pi$
$434$	−10.0000	−0.480015
$435$	−6.00000	−0.287678
$436$	−7.00000	−0.335239
$437$	3.00000	0.143509
$438$	−11.0000	−0.525600
$439$	20.0000	0.954548	0.477274	−	0.878755i	$-0.341625\pi$
$439$	20.0000	0.954548	0.477274	+	0.878755i	$0.341625\pi$
$440$	3.00000	0.143019
$441$	−6.00000	−0.285714
$442$	−21.0000	−0.998868
$443$	−12.0000	−0.570137	−0.285069	−	0.958507i	$-0.592016\pi$
$443$	−12.0000	−0.570137	−0.285069	+	0.958507i	$0.592016\pi$
$444$	1.00000	0.0474579
$445$	9.00000	0.426641
$446$	−8.00000	−0.378811
$447$	−6.00000	−0.283790
$448$	−1.00000	−0.0472456
$449$	−30.0000	−1.41579	−0.707894	−	0.706319i	$-0.750354\pi$
$449$	−30.0000	−1.41579	−0.707894	+	0.706319i	$0.750354\pi$
$450$	−1.00000	−0.0471405
$451$	0	0
$452$	18.0000	0.846649
$453$	23.0000	1.08063
$454$	−24.0000	−1.12638
$455$	−7.00000	−0.328165
$456$	1.00000	0.0468293
$457$	8.00000	0.374224	0.187112	−	0.982339i	$-0.440087\pi$
$457$	8.00000	0.374224	0.187112	+	0.982339i	$0.440087\pi$
$458$	10.0000	0.467269
$459$	−3.00000	−0.140028
$460$	3.00000	0.139876
$461$	18.0000	0.838344	0.419172	−	0.907907i	$-0.362320\pi$
$461$	18.0000	0.838344	0.419172	+	0.907907i	$0.362320\pi$
$462$	3.00000	0.139573
$463$	−4.00000	−0.185896	−0.0929479	−	0.995671i	$-0.529629\pi$
$463$	−4.00000	−0.185896	−0.0929479	+	0.995671i	$0.529629\pi$
$464$	6.00000	0.278543
$465$	10.0000	0.463739
$466$	−18.0000	−0.833834
$467$	0	0		−	1.00000i	$-0.5\pi$
$467$	0	0			1.00000i	$0.5\pi$
$468$	−7.00000	−0.323575
$469$	10.0000	0.461757
$470$	−12.0000	−0.553519
$471$	−4.00000	−0.184310
$472$	−6.00000	−0.276172
$473$	−12.0000	−0.551761
$474$	−8.00000	−0.367452
$475$	−1.00000	−0.0458831
$476$	3.00000	0.137505
$477$	9.00000	0.412082
$478$	−12.0000	−0.548867
$479$	−39.0000	−1.78196	−0.890978	−	0.454047i	$-0.849980\pi$
$479$	−39.0000	−1.78196	−0.890978	+	0.454047i	$0.849980\pi$
$480$	1.00000	0.0456435
$481$	−7.00000	−0.319173
$482$	−26.0000	−1.18427
$483$	3.00000	0.136505
$484$	−2.00000	−0.0909091
$485$	16.0000	0.726523
$486$	−1.00000	−0.0453609
$487$	−22.0000	−0.996915	−0.498458	−	0.866914i	$-0.666100\pi$
$487$	−22.0000	−0.996915	−0.498458	+	0.866914i	$0.666100\pi$
$488$	10.0000	0.452679
$489$	−1.00000	−0.0452216
$490$	−6.00000	−0.271052
$491$	−9.00000	−0.406164	−0.203082	−	0.979162i	$-0.565096\pi$
$491$	−9.00000	−0.406164	−0.203082	+	0.979162i	$0.565096\pi$
$492$	0	0
$493$	−18.0000	−0.810679
$494$	−7.00000	−0.314945
$495$	−3.00000	−0.134840
$496$	−10.0000	−0.449013
$497$	6.00000	0.269137
$498$	−9.00000	−0.403300
$499$	41.0000	1.83541	0.917706	−	0.397260i	$-0.130039\pi$
$499$	41.0000	1.83541	0.917706	+	0.397260i	$0.130039\pi$
$500$	−1.00000	−0.0447214
$501$	3.00000	0.134030
$502$	−6.00000	−0.267793
$503$	36.0000	1.60516	0.802580	−	0.596544i	$-0.203460\pi$
$503$	36.0000	1.60516	0.802580	+	0.596544i	$0.203460\pi$
$504$	1.00000	0.0445435
$505$	6.00000	0.266996
$506$	9.00000	0.400099
$507$	36.0000	1.59882
$508$	−7.00000	−0.310575
$509$	−33.0000	−1.46270	−0.731350	−	0.682003i	$-0.761109\pi$
$509$	−33.0000	−1.46270	−0.731350	+	0.682003i	$0.761109\pi$
$510$	−3.00000	−0.132842
$511$	−11.0000	−0.486611
$512$	−1.00000	−0.0441942
$513$	−1.00000	−0.0441511
$514$	−9.00000	−0.396973
$515$	4.00000	0.176261
$516$	−4.00000	−0.176090
$517$	−36.0000	−1.58328
$518$	1.00000	0.0439375
$519$	−9.00000	−0.395056
$520$	−7.00000	−0.306970
$521$	−12.0000	−0.525730	−0.262865	−	0.964833i	$-0.584667\pi$
$521$	−12.0000	−0.525730	−0.262865	+	0.964833i	$0.584667\pi$
$522$	−6.00000	−0.262613
$523$	44.0000	1.92399	0.961993	−	0.273075i	$-0.0880406\pi$
$523$	44.0000	1.92399	0.961993	+	0.273075i	$0.0880406\pi$
$524$	−12.0000	−0.524222
$525$	−1.00000	−0.0436436
$526$	0	0
$527$	30.0000	1.30682
$528$	3.00000	0.130558
$529$	−14.0000	−0.608696
$530$	9.00000	0.390935
$531$	6.00000	0.260378
$532$	1.00000	0.0433555
$533$	0	0
$534$	9.00000	0.389468
$535$	−3.00000	−0.129701
$536$	10.0000	0.431934
$537$	0	0
$538$	−9.00000	−0.388018
$539$	−18.0000	−0.775315
$540$	−1.00000	−0.0430331
$541$	−25.0000	−1.07483	−0.537417	−	0.843317i	$-0.680600\pi$
$541$	−25.0000	−1.07483	−0.537417	+	0.843317i	$0.680600\pi$
$542$	−20.0000	−0.859074
$543$	−22.0000	−0.944110
$544$	3.00000	0.128624
$545$	7.00000	0.299847
$546$	−7.00000	−0.299572
$547$	−37.0000	−1.58201	−0.791003	−	0.611812i	$-0.790441\pi$
$547$	−37.0000	−1.58201	−0.791003	+	0.611812i	$0.790441\pi$
$548$	−18.0000	−0.768922
$549$	−10.0000	−0.426790
$550$	−3.00000	−0.127920
$551$	−6.00000	−0.255609
$552$	3.00000	0.127688
$553$	−8.00000	−0.340195
$554$	−23.0000	−0.977176
$555$	−1.00000	−0.0424476
$556$	14.0000	0.593732
$557$	−24.0000	−1.01691	−0.508456	−	0.861088i	$-0.669784\pi$
$557$	−24.0000	−1.01691	−0.508456	+	0.861088i	$0.669784\pi$
$558$	10.0000	0.423334
$559$	28.0000	1.18427
$560$	1.00000	0.0422577
$561$	−9.00000	−0.379980
$562$	−27.0000	−1.13893
$563$	−18.0000	−0.758610	−0.379305	−	0.925272i	$-0.623837\pi$
$563$	−18.0000	−0.758610	−0.379305	+	0.925272i	$0.623837\pi$
$564$	−12.0000	−0.505291
$565$	−18.0000	−0.757266
$566$	13.0000	0.546431
$567$	−1.00000	−0.0419961
$568$	6.00000	0.251754
$569$	−39.0000	−1.63497	−0.817483	−	0.575953i	$-0.804631\pi$
$569$	−39.0000	−1.63497	−0.817483	+	0.575953i	$0.804631\pi$
$570$	−1.00000	−0.0418854
$571$	14.0000	0.585882	0.292941	−	0.956131i	$-0.405366\pi$
$571$	14.0000	0.585882	0.292941	+	0.956131i	$0.405366\pi$
$572$	−21.0000	−0.878054
$573$	3.00000	0.125327
$574$	0	0
$575$	−3.00000	−0.125109
$576$	1.00000	0.0416667
$577$	2.00000	0.0832611	0.0416305	−	0.999133i	$-0.486745\pi$
$577$	2.00000	0.0832611	0.0416305	+	0.999133i	$0.486745\pi$
$578$	8.00000	0.332756
$579$	−10.0000	−0.415586
$580$	−6.00000	−0.249136
$581$	−9.00000	−0.373383
$582$	16.0000	0.663221
$583$	27.0000	1.11823
$584$	−11.0000	−0.455183
$585$	7.00000	0.289414
$586$	15.0000	0.619644
$587$	−36.0000	−1.48588	−0.742940	−	0.669359i	$-0.766569\pi$
$587$	−36.0000	−1.48588	−0.742940	+	0.669359i	$0.766569\pi$
$588$	−6.00000	−0.247436
$589$	10.0000	0.412043
$590$	6.00000	0.247016
$591$	3.00000	0.123404
$592$	1.00000	0.0410997
$593$	−36.0000	−1.47834	−0.739171	−	0.673517i	$-0.764783\pi$
$593$	−36.0000	−1.47834	−0.739171	+	0.673517i	$0.764783\pi$
$594$	−3.00000	−0.123091
$595$	−3.00000	−0.122988
$596$	−6.00000	−0.245770
$597$	20.0000	0.818546
$598$	−21.0000	−0.858754
$599$	−12.0000	−0.490307	−0.245153	−	0.969484i	$-0.578838\pi$
$599$	−12.0000	−0.490307	−0.245153	+	0.969484i	$0.578838\pi$
$600$	−1.00000	−0.0408248
$601$	−37.0000	−1.50926	−0.754631	−	0.656150i	$-0.772184\pi$
$601$	−37.0000	−1.50926	−0.754631	+	0.656150i	$0.772184\pi$
$602$	−4.00000	−0.163028
$603$	−10.0000	−0.407231
$604$	23.0000	0.935857
$605$	2.00000	0.0813116
$606$	6.00000	0.243733
$607$	14.0000	0.568242	0.284121	−	0.958788i	$-0.408298\pi$
$607$	14.0000	0.568242	0.284121	+	0.958788i	$0.408298\pi$
$608$	1.00000	0.0405554
$609$	−6.00000	−0.243132
$610$	−10.0000	−0.404888
$611$	84.0000	3.39828
$612$	−3.00000	−0.121268
$613$	−34.0000	−1.37325	−0.686624	−	0.727013i	$-0.740908\pi$
$613$	−34.0000	−1.37325	−0.686624	+	0.727013i	$0.740908\pi$
$614$	22.0000	0.887848
$615$	0	0
$616$	3.00000	0.120873
$617$	42.0000	1.69086	0.845428	−	0.534089i	$-0.179345\pi$
$617$	42.0000	1.69086	0.845428	+	0.534089i	$0.179345\pi$
$618$	4.00000	0.160904
$619$	20.0000	0.803868	0.401934	−	0.915669i	$-0.368338\pi$
$619$	20.0000	0.803868	0.401934	+	0.915669i	$0.368338\pi$
$620$	10.0000	0.401610
$621$	−3.00000	−0.120386
$622$	24.0000	0.962312
$623$	9.00000	0.360577
$624$	−7.00000	−0.280224
$625$	1.00000	0.0400000
$626$	−8.00000	−0.319744
$627$	−3.00000	−0.119808
$628$	−4.00000	−0.159617
$629$	−3.00000	−0.119618
$630$	−1.00000	−0.0398410
$631$	−4.00000	−0.159237	−0.0796187	−	0.996825i	$-0.525370\pi$
$631$	−4.00000	−0.159237	−0.0796187	+	0.996825i	$0.525370\pi$
$632$	−8.00000	−0.318223
$633$	2.00000	0.0794929
$634$	18.0000	0.714871
$635$	7.00000	0.277787
$636$	9.00000	0.356873
$637$	42.0000	1.66410
$638$	−18.0000	−0.712627
$639$	−6.00000	−0.237356
$640$	1.00000	0.0395285
$641$	24.0000	0.947943	0.473972	−	0.880540i	$-0.342820\pi$
$641$	24.0000	0.947943	0.473972	+	0.880540i	$0.342820\pi$
$642$	−3.00000	−0.118401
$643$	−31.0000	−1.22252	−0.611260	−	0.791430i	$-0.709337\pi$
$643$	−31.0000	−1.22252	−0.611260	+	0.791430i	$0.709337\pi$
$644$	3.00000	0.118217
$645$	4.00000	0.157500
$646$	−3.00000	−0.118033
$647$	45.0000	1.76913	0.884566	−	0.466415i	$-0.154454\pi$
$647$	45.0000	1.76913	0.884566	+	0.466415i	$0.154454\pi$
$648$	−1.00000	−0.0392837
$649$	18.0000	0.706562
$650$	7.00000	0.274563
$651$	10.0000	0.391931
$652$	−1.00000	−0.0391630
$653$	−36.0000	−1.40879	−0.704394	−	0.709809i	$-0.748781\pi$
$653$	−36.0000	−1.40879	−0.704394	+	0.709809i	$0.748781\pi$
$654$	7.00000	0.273722
$655$	12.0000	0.468879
$656$	0	0
$657$	11.0000	0.429151
$658$	−12.0000	−0.467809
$659$	12.0000	0.467454	0.233727	−	0.972302i	$-0.424908\pi$
$659$	12.0000	0.467454	0.233727	+	0.972302i	$0.424908\pi$
$660$	−3.00000	−0.116775
$661$	−13.0000	−0.505641	−0.252821	−	0.967513i	$-0.581358\pi$
$661$	−13.0000	−0.505641	−0.252821	+	0.967513i	$0.581358\pi$
$662$	28.0000	1.08825
$663$	21.0000	0.815572
$664$	−9.00000	−0.349268
$665$	−1.00000	−0.0387783
$666$	−1.00000	−0.0387492
$667$	−18.0000	−0.696963
$668$	3.00000	0.116073
$669$	8.00000	0.309298
$670$	−10.0000	−0.386334
$671$	−30.0000	−1.15814
$672$	1.00000	0.0385758
$673$	−1.00000	−0.0385472	−0.0192736	−	0.999814i	$-0.506135\pi$
$673$	−1.00000	−0.0385472	−0.0192736	+	0.999814i	$0.506135\pi$
$674$	13.0000	0.500741
$675$	1.00000	0.0384900
$676$	36.0000	1.38462
$677$	3.00000	0.115299	0.0576497	−	0.998337i	$-0.481639\pi$
$677$	3.00000	0.115299	0.0576497	+	0.998337i	$0.481639\pi$
$678$	−18.0000	−0.691286
$679$	16.0000	0.614024
$680$	−3.00000	−0.115045
$681$	24.0000	0.919682
$682$	30.0000	1.14876
$683$	−36.0000	−1.37750	−0.688751	−	0.724998i	$-0.741841\pi$
$683$	−36.0000	−1.37750	−0.688751	+	0.724998i	$0.741841\pi$
$684$	−1.00000	−0.0382360
$685$	18.0000	0.687745
$686$	−13.0000	−0.496342
$687$	−10.0000	−0.381524
$688$	−4.00000	−0.152499
$689$	−63.0000	−2.40011
$690$	−3.00000	−0.114208
$691$	−10.0000	−0.380418	−0.190209	−	0.981744i	$-0.560917\pi$
$691$	−10.0000	−0.380418	−0.190209	+	0.981744i	$0.560917\pi$
$692$	−9.00000	−0.342129
$693$	−3.00000	−0.113961
$694$	−36.0000	−1.36654
$695$	−14.0000	−0.531050
$696$	−6.00000	−0.227429
$697$	0	0
$698$	−8.00000	−0.302804
$699$	18.0000	0.680823
$700$	−1.00000	−0.0377964
$701$	−48.0000	−1.81293	−0.906467	−	0.422276i	$-0.861231\pi$
$701$	−48.0000	−1.81293	−0.906467	+	0.422276i	$0.861231\pi$
$702$	7.00000	0.264198
$703$	−1.00000	−0.0377157
$704$	3.00000	0.113067
$705$	12.0000	0.451946
$706$	−18.0000	−0.677439
$707$	6.00000	0.225653
$708$	6.00000	0.225494
$709$	−25.0000	−0.938895	−0.469447	−	0.882960i	$-0.655547\pi$
$709$	−25.0000	−0.938895	−0.469447	+	0.882960i	$0.655547\pi$
$710$	−6.00000	−0.225176
$711$	8.00000	0.300023
$712$	9.00000	0.337289
$713$	30.0000	1.12351
$714$	−3.00000	−0.112272
$715$	21.0000	0.785355
$716$	0	0
$717$	12.0000	0.448148
$718$	12.0000	0.447836
$719$	6.00000	0.223762	0.111881	−	0.993722i	$-0.464312\pi$
$719$	6.00000	0.223762	0.111881	+	0.993722i	$0.464312\pi$
$720$	−1.00000	−0.0372678
$721$	4.00000	0.148968
$722$	18.0000	0.669891
$723$	26.0000	0.966950
$724$	−22.0000	−0.817624
$725$	6.00000	0.222834
$726$	2.00000	0.0742270
$727$	2.00000	0.0741759	0.0370879	−	0.999312i	$-0.488192\pi$
$727$	2.00000	0.0741759	0.0370879	+	0.999312i	$0.488192\pi$
$728$	−7.00000	−0.259437
$729$	1.00000	0.0370370
$730$	11.0000	0.407128
$731$	12.0000	0.443836
$732$	−10.0000	−0.369611
$733$	14.0000	0.517102	0.258551	−	0.965998i	$-0.416755\pi$
$733$	14.0000	0.517102	0.258551	+	0.965998i	$0.416755\pi$
$734$	−23.0000	−0.848945
$735$	6.00000	0.221313
$736$	3.00000	0.110581
$737$	−30.0000	−1.10506
$738$	0	0
$739$	−34.0000	−1.25071	−0.625355	−	0.780340i	$-0.715046\pi$
$739$	−34.0000	−1.25071	−0.625355	+	0.780340i	$0.715046\pi$
$740$	−1.00000	−0.0367607
$741$	7.00000	0.257151
$742$	9.00000	0.330400
$743$	6.00000	0.220119	0.110059	−	0.993925i	$-0.464896\pi$
$743$	6.00000	0.220119	0.110059	+	0.993925i	$0.464896\pi$
$744$	10.0000	0.366618
$745$	6.00000	0.219823
$746$	−14.0000	−0.512576
$747$	9.00000	0.329293
$748$	−9.00000	−0.329073
$749$	−3.00000	−0.109618
$750$	1.00000	0.0365148
$751$	−4.00000	−0.145962	−0.0729810	−	0.997333i	$-0.523251\pi$
$751$	−4.00000	−0.145962	−0.0729810	+	0.997333i	$0.523251\pi$
$752$	−12.0000	−0.437595
$753$	6.00000	0.218652
$754$	42.0000	1.52955
$755$	−23.0000	−0.837056
$756$	−1.00000	−0.0363696
$757$	−13.0000	−0.472493	−0.236247	−	0.971693i	$-0.575917\pi$
$757$	−13.0000	−0.472493	−0.236247	+	0.971693i	$0.575917\pi$
$758$	−26.0000	−0.944363
$759$	−9.00000	−0.326679
$760$	−1.00000	−0.0362738
$761$	6.00000	0.217500	0.108750	−	0.994069i	$-0.465315\pi$
$761$	6.00000	0.217500	0.108750	+	0.994069i	$0.465315\pi$
$762$	7.00000	0.253583
$763$	7.00000	0.253417
$764$	3.00000	0.108536
$765$	3.00000	0.108465
$766$	9.00000	0.325183
$767$	−42.0000	−1.51653
$768$	1.00000	0.0360844
$769$	−22.0000	−0.793340	−0.396670	−	0.917961i	$-0.629834\pi$
$769$	−22.0000	−0.793340	−0.396670	+	0.917961i	$0.629834\pi$
$770$	−3.00000	−0.108112
$771$	9.00000	0.324127
$772$	−10.0000	−0.359908
$773$	15.0000	0.539513	0.269756	−	0.962929i	$-0.413057\pi$
$773$	15.0000	0.539513	0.269756	+	0.962929i	$0.413057\pi$
$774$	4.00000	0.143777
$775$	−10.0000	−0.359211
$776$	16.0000	0.574367
$777$	−1.00000	−0.0358748
$778$	−18.0000	−0.645331
$779$	0	0
$780$	7.00000	0.250640
$781$	−18.0000	−0.644091
$782$	−9.00000	−0.321839
$783$	6.00000	0.214423
$784$	−6.00000	−0.214286
$785$	4.00000	0.142766
$786$	12.0000	0.428026
$787$	2.00000	0.0712923	0.0356462	−	0.999364i	$-0.488651\pi$
$787$	2.00000	0.0712923	0.0356462	+	0.999364i	$0.488651\pi$
$788$	3.00000	0.106871
$789$	0	0
$790$	8.00000	0.284627
$791$	−18.0000	−0.640006
$792$	−3.00000	−0.106600
$793$	70.0000	2.48577
$794$	4.00000	0.141955
$795$	−9.00000	−0.319197
$796$	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	$0.321693\pi$
$798$	−1.00000	−0.0353996
$799$	36.0000	1.27359
$800$	−1.00000	−0.0353553
$801$	−9.00000	−0.317999
$802$	−15.0000	−0.529668
$803$	33.0000	1.16454
$804$	−10.0000	−0.352673
$805$	−3.00000	−0.105736
$806$	−70.0000	−2.46564
$807$	9.00000	0.316815
$808$	6.00000	0.211079
$809$	−21.0000	−0.738321	−0.369160	−	0.929366i	$-0.620355\pi$
$809$	−21.0000	−0.738321	−0.369160	+	0.929366i	$0.620355\pi$
$810$	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
$813$	20.0000	0.701431
$814$	−3.00000	−0.105150
$815$	1.00000	0.0350285
$816$	−3.00000	−0.105021
$817$	4.00000	0.139942
$818$	4.00000	0.139857
$819$	7.00000	0.244600
$820$	0	0
$821$	−9.00000	−0.314102	−0.157051	−	0.987590i	$-0.550199\pi$
$821$	−9.00000	−0.314102	−0.157051	+	0.987590i	$0.550199\pi$
$822$	18.0000	0.627822
$823$	−7.00000	−0.244005	−0.122002	−	0.992530i	$-0.538932\pi$
$823$	−7.00000	−0.244005	−0.122002	+	0.992530i	$0.538932\pi$
$824$	4.00000	0.139347
$825$	3.00000	0.104447
$826$	6.00000	0.208767
$827$	−12.0000	−0.417281	−0.208640	−	0.977992i	$-0.566904\pi$
$827$	−12.0000	−0.417281	−0.208640	+	0.977992i	$0.566904\pi$
$828$	−3.00000	−0.104257
$829$	29.0000	1.00721	0.503606	−	0.863934i	$-0.332006\pi$
$829$	29.0000	1.00721	0.503606	+	0.863934i	$0.332006\pi$
$830$	9.00000	0.312395
$831$	23.0000	0.797861
$832$	−7.00000	−0.242681
$833$	18.0000	0.623663
$834$	−14.0000	−0.484780
$835$	−3.00000	−0.103819
$836$	−3.00000	−0.103757
$837$	−10.0000	−0.345651
$838$	27.0000	0.932700
$839$	30.0000	1.03572	0.517858	−	0.855467i	$-0.326730\pi$
$839$	30.0000	1.03572	0.517858	+	0.855467i	$0.326730\pi$
$840$	−1.00000	−0.0345033
$841$	7.00000	0.241379
$842$	10.0000	0.344623
$843$	27.0000	0.929929
$844$	2.00000	0.0688428
$845$	−36.0000	−1.23844
$846$	12.0000	0.412568
$847$	2.00000	0.0687208
$848$	9.00000	0.309061
$849$	−13.0000	−0.446159
$850$	3.00000	0.102899
$851$	−3.00000	−0.102839
$852$	−6.00000	−0.205557
$853$	−19.0000	−0.650548	−0.325274	−	0.945620i	$-0.605456\pi$
$853$	−19.0000	−0.650548	−0.325274	+	0.945620i	$0.605456\pi$
$854$	−10.0000	−0.342193
$855$	1.00000	0.0341993
$856$	−3.00000	−0.102538
$857$	−9.00000	−0.307434	−0.153717	−	0.988115i	$-0.549124\pi$
$857$	−9.00000	−0.307434	−0.153717	+	0.988115i	$0.549124\pi$
$858$	21.0000	0.716928
$859$	5.00000	0.170598	0.0852989	−	0.996355i	$-0.472815\pi$
$859$	5.00000	0.170598	0.0852989	+	0.996355i	$0.472815\pi$
$860$	4.00000	0.136399
$861$	0	0
$862$	3.00000	0.102180
$863$	0	0		−	1.00000i	$-0.5\pi$
$863$	0	0			1.00000i	$0.5\pi$
$864$	−1.00000	−0.0340207
$865$	9.00000	0.306009
$866$	31.0000	1.05342
$867$	−8.00000	−0.271694
$868$	10.0000	0.339422
$869$	24.0000	0.814144
$870$	6.00000	0.203419
$871$	70.0000	2.37186
$872$	7.00000	0.237050
$873$	−16.0000	−0.541518
$874$	−3.00000	−0.101477
$875$	1.00000	0.0338062
$876$	11.0000	0.371656
$877$	32.0000	1.08056	0.540282	−	0.841484i	$-0.318318\pi$
$877$	32.0000	1.08056	0.540282	+	0.841484i	$0.318318\pi$
$878$	−20.0000	−0.674967
$879$	−15.0000	−0.505937
$880$	−3.00000	−0.101130
$881$	−12.0000	−0.404290	−0.202145	−	0.979356i	$-0.564791\pi$
$881$	−12.0000	−0.404290	−0.202145	+	0.979356i	$0.564791\pi$
$882$	6.00000	0.202031
$883$	35.0000	1.17784	0.588922	−	0.808190i	$-0.299553\pi$
$883$	35.0000	1.17784	0.588922	+	0.808190i	$0.299553\pi$
$884$	21.0000	0.706306
$885$	−6.00000	−0.201688
$886$	12.0000	0.403148
$887$	−18.0000	−0.604381	−0.302190	−	0.953248i	$-0.597718\pi$
$887$	−18.0000	−0.604381	−0.302190	+	0.953248i	$0.597718\pi$
$888$	−1.00000	−0.0335578
$889$	7.00000	0.234772
$890$	−9.00000	−0.301681
$891$	3.00000	0.100504
$892$	8.00000	0.267860
$893$	12.0000	0.401565
$894$	6.00000	0.200670
$895$	0	0
$896$	1.00000	0.0334077
$897$	21.0000	0.701170
$898$	30.0000	1.00111
$899$	−60.0000	−2.00111
$900$	1.00000	0.0333333
$901$	−27.0000	−0.899500
$902$	0	0
$903$	4.00000	0.133112
$904$	−18.0000	−0.598671
$905$	22.0000	0.731305
$906$	−23.0000	−0.764124
$907$	−37.0000	−1.22856	−0.614282	−	0.789086i	$-0.710554\pi$
$907$	−37.0000	−1.22856	−0.614282	+	0.789086i	$0.710554\pi$
$908$	24.0000	0.796468
$909$	−6.00000	−0.199007
$910$	7.00000	0.232048
$911$	48.0000	1.59031	0.795155	−	0.606406i	$-0.207389\pi$
$911$	48.0000	1.59031	0.795155	+	0.606406i	$0.207389\pi$
$912$	−1.00000	−0.0331133
$913$	27.0000	0.893570
$914$	−8.00000	−0.264616
$915$	10.0000	0.330590
$916$	−10.0000	−0.330409
$917$	12.0000	0.396275
$918$	3.00000	0.0990148
$919$	−4.00000	−0.131948	−0.0659739	−	0.997821i	$-0.521015\pi$
$919$	−4.00000	−0.131948	−0.0659739	+	0.997821i	$0.521015\pi$
$920$	−3.00000	−0.0989071
$921$	−22.0000	−0.724925
$922$	−18.0000	−0.592798
$923$	42.0000	1.38245
$924$	−3.00000	−0.0986928
$925$	1.00000	0.0328798
$926$	4.00000	0.131448
$927$	−4.00000	−0.131377
$928$	−6.00000	−0.196960
$929$	−18.0000	−0.590561	−0.295280	−	0.955411i	$-0.595413\pi$
$929$	−18.0000	−0.590561	−0.295280	+	0.955411i	$0.595413\pi$
$930$	−10.0000	−0.327913
$931$	6.00000	0.196642
$932$	18.0000	0.589610
$933$	−24.0000	−0.785725
$934$	0	0
$935$	9.00000	0.294331
$936$	7.00000	0.228802
$937$	−10.0000	−0.326686	−0.163343	−	0.986569i	$-0.552228\pi$
$937$	−10.0000	−0.326686	−0.163343	+	0.986569i	$0.552228\pi$
$938$	−10.0000	−0.326512
$939$	8.00000	0.261070
$940$	12.0000	0.391397
$941$	18.0000	0.586783	0.293392	−	0.955992i	$-0.405216\pi$
$941$	18.0000	0.586783	0.293392	+	0.955992i	$0.405216\pi$
$942$	4.00000	0.130327
$943$	0	0
$944$	6.00000	0.195283
$945$	1.00000	0.0325300
$946$	12.0000	0.390154
$947$	−24.0000	−0.779895	−0.389948	−	0.920837i	$-0.627507\pi$
$947$	−24.0000	−0.779895	−0.389948	+	0.920837i	$0.627507\pi$
$948$	8.00000	0.259828
$949$	−77.0000	−2.49953
$950$	1.00000	0.0324443
$951$	−18.0000	−0.583690
$952$	−3.00000	−0.0972306
$953$	6.00000	0.194359	0.0971795	−	0.995267i	$-0.469018\pi$
$953$	6.00000	0.194359	0.0971795	+	0.995267i	$0.469018\pi$
$954$	−9.00000	−0.291386
$955$	−3.00000	−0.0970777
$956$	12.0000	0.388108
$957$	18.0000	0.581857
$958$	39.0000	1.26003
$959$	18.0000	0.581250
$960$	−1.00000	−0.0322749
$961$	69.0000	2.22581
$962$	7.00000	0.225689
$963$	3.00000	0.0966736
$964$	26.0000	0.837404
$965$	10.0000	0.321911
$966$	−3.00000	−0.0965234
$967$	−22.0000	−0.707472	−0.353736	−	0.935345i	$-0.615089\pi$
$967$	−22.0000	−0.707472	−0.353736	+	0.935345i	$0.615089\pi$
$968$	2.00000	0.0642824
$969$	3.00000	0.0963739
$970$	−16.0000	−0.513729
$971$	36.0000	1.15529	0.577647	−	0.816286i	$-0.303971\pi$
$971$	36.0000	1.15529	0.577647	+	0.816286i	$0.303971\pi$
$972$	1.00000	0.0320750
$973$	−14.0000	−0.448819
$974$	22.0000	0.704925
$975$	−7.00000	−0.224179
$976$	−10.0000	−0.320092
$977$	9.00000	0.287936	0.143968	−	0.989582i	$-0.454014\pi$
$977$	9.00000	0.287936	0.143968	+	0.989582i	$0.454014\pi$
$978$	1.00000	0.0319765
$979$	−27.0000	−0.862924
$980$	6.00000	0.191663
$981$	−7.00000	−0.223493
$982$	9.00000	0.287202
$983$	42.0000	1.33959	0.669796	−	0.742545i	$-0.266382\pi$
$983$	42.0000	1.33959	0.669796	+	0.742545i	$0.266382\pi$
$984$	0	0
$985$	−3.00000	−0.0955879
$986$	18.0000	0.573237
$987$	12.0000	0.381964
$988$	7.00000	0.222700
$989$	12.0000	0.381578
$990$	3.00000	0.0953463
$991$	−52.0000	−1.65183	−0.825917	−	0.563791i	$-0.809342\pi$
$991$	−52.0000	−1.65183	−0.825917	+	0.563791i	$0.809342\pi$
$992$	10.0000	0.317500
$993$	−28.0000	−0.888553
$994$	−6.00000	−0.190308
$995$	−20.0000	−0.634043
$996$	9.00000	0.285176
$997$	−1.00000	−0.0316703	−0.0158352	−	0.999875i	$-0.505041\pi$
$997$	−1.00000	−0.0316703	−0.0158352	+	0.999875i	$0.505041\pi$
$998$	−41.0000	−1.29783
$999$	1.00000	0.0316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1110.2.a.e.1.1	✓	1
3.2	odd	2		3330.2.a.u.1.1		1
4.3	odd	2		8880.2.a.f.1.1		1
5.4	even	2		5550.2.a.ba.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1110.2.a.e.1.1	✓	1	1.1	even	1	trivial
3330.2.a.u.1.1		1	3.2	odd	2
5550.2.a.ba.1.1		1	5.4	even	2
8880.2.a.f.1.1		1	4.3	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:	\( N \)	\(=\)	\( 1110 = 2 \cdot 3 \cdot 5 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1110.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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