LMFDB - Embedded newform 7406.2.a.g.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7406,2,Mod(1,7406)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7406, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("7406.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7406 = 2 \cdot 7 \cdot 23^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7406.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:	$59.1372077370$
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	$\chi$	$=$	7406.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -4.00000 q^{5} -1.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} -2.00000 q^{9} +O(q^{10})$

\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -4.00000 q^{5} -1.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} -2.00000 q^{9} -4.00000 q^{10} -1.00000 q^{12} +2.00000 q^{13} -1.00000 q^{14} +4.00000 q^{15} +1.00000 q^{16} +5.00000 q^{17} -2.00000 q^{18} -7.00000 q^{19} -4.00000 q^{20} +1.00000 q^{21} -1.00000 q^{24} +11.0000 q^{25} +2.00000 q^{26} +5.00000 q^{27} -1.00000 q^{28} +8.00000 q^{29} +4.00000 q^{30} -2.00000 q^{31} +1.00000 q^{32} +5.00000 q^{34} +4.00000 q^{35} -2.00000 q^{36} +8.00000 q^{37} -7.00000 q^{38} -2.00000 q^{39} -4.00000 q^{40} -2.00000 q^{41} +1.00000 q^{42} -3.00000 q^{43} +8.00000 q^{45} -1.00000 q^{48} +1.00000 q^{49} +11.0000 q^{50} -5.00000 q^{51} +2.00000 q^{52} -6.00000 q^{53} +5.00000 q^{54} -1.00000 q^{56} +7.00000 q^{57} +8.00000 q^{58} -9.00000 q^{59} +4.00000 q^{60} +6.00000 q^{61} -2.00000 q^{62} +2.00000 q^{63} +1.00000 q^{64} -8.00000 q^{65} +5.00000 q^{67} +5.00000 q^{68} +4.00000 q^{70} -2.00000 q^{71} -2.00000 q^{72} +17.0000 q^{73} +8.00000 q^{74} -11.0000 q^{75} -7.00000 q^{76} -2.00000 q^{78} -4.00000 q^{80} +1.00000 q^{81} -2.00000 q^{82} -13.0000 q^{83} +1.00000 q^{84} -20.0000 q^{85} -3.00000 q^{86} -8.00000 q^{87} +6.00000 q^{89} +8.00000 q^{90} -2.00000 q^{91} +2.00000 q^{93} +28.0000 q^{95} -1.00000 q^{96} -14.0000 q^{97} +1.00000 q^{98} +O(q^{100})\)

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	−0.288675	−	0.957427i	$-0.593215\pi$
$3$	−1.00000	−0.577350	−0.288675	+	0.957427i	$0.593215\pi$
$4$	1.00000	0.500000
$5$	−4.00000	−1.78885	−0.894427	−	0.447214i	$-0.852416\pi$
$5$	−4.00000	−1.78885	−0.894427	+	0.447214i	$0.852416\pi$
$6$	−1.00000	−0.408248
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	1.00000	0.353553
$9$	−2.00000	−0.666667
$10$	−4.00000	−1.26491
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	−1.00000	−0.288675
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	−1.00000	−0.267261
$15$	4.00000	1.03280
$16$	1.00000	0.250000
$17$	5.00000	1.21268	0.606339	−	0.795206i	$-0.292637\pi$
$17$	5.00000	1.21268	0.606339	+	0.795206i	$0.292637\pi$
$18$	−2.00000	−0.471405
$19$	−7.00000	−1.60591	−0.802955	−	0.596040i	$-0.796740\pi$
$19$	−7.00000	−1.60591	−0.802955	+	0.596040i	$0.796740\pi$
$20$	−4.00000	−0.894427
$21$	1.00000	0.218218
$22$	0	0
$23$	0	0
$23$	0	0
$24$	−1.00000	−0.204124
$25$	11.0000	2.20000
$26$	2.00000	0.392232
$27$	5.00000	0.962250
$28$	−1.00000	−0.188982
$29$	8.00000	1.48556	0.742781	−	0.669534i	$-0.233506\pi$
$29$	8.00000	1.48556	0.742781	+	0.669534i	$0.233506\pi$
$30$	4.00000	0.730297
$31$	−2.00000	−0.359211	−0.179605	−	0.983739i	$-0.557482\pi$
$31$	−2.00000	−0.359211	−0.179605	+	0.983739i	$0.557482\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	5.00000	0.857493
$35$	4.00000	0.676123
$36$	−2.00000	−0.333333
$37$	8.00000	1.31519	0.657596	−	0.753371i	$-0.271573\pi$
$37$	8.00000	1.31519	0.657596	+	0.753371i	$0.271573\pi$
$38$	−7.00000	−1.13555
$39$	−2.00000	−0.320256
$40$	−4.00000	−0.632456
$41$	−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
$43$	−3.00000	−0.457496	−0.228748	−	0.973486i	$-0.573463\pi$
$43$	−3.00000	−0.457496	−0.228748	+	0.973486i	$0.573463\pi$
$44$	0	0
$45$	8.00000	1.19257
$46$	0	0
$47$	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
$50$	11.0000	1.55563
$51$	−5.00000	−0.700140
$52$	2.00000	0.277350
$53$	−6.00000	−0.824163	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	5.00000	0.680414
$55$	0	0
$56$	−1.00000	−0.133631
$57$	7.00000	0.927173
$58$	8.00000	1.05045
$59$	−9.00000	−1.17170	−0.585850	−	0.810419i	$-0.699239\pi$
$59$	−9.00000	−1.17170	−0.585850	+	0.810419i	$0.699239\pi$
$60$	4.00000	0.516398
$61$	6.00000	0.768221	0.384111	−	0.923287i	$-0.374508\pi$
$61$	6.00000	0.768221	0.384111	+	0.923287i	$0.374508\pi$
$62$	−2.00000	−0.254000
$63$	2.00000	0.251976
$64$	1.00000	0.125000
$65$	−8.00000	−0.992278
$66$	0	0
$67$	5.00000	0.610847	0.305424	−	0.952217i	$-0.401202\pi$
$67$	5.00000	0.610847	0.305424	+	0.952217i	$0.401202\pi$
$68$	5.00000	0.606339
$69$	0	0
$70$	4.00000	0.478091
$71$	−2.00000	−0.237356	−0.118678	−	0.992933i	$-0.537866\pi$
$71$	−2.00000	−0.237356	−0.118678	+	0.992933i	$0.537866\pi$
$72$	−2.00000	−0.235702
$73$	17.0000	1.98970	0.994850	−	0.101361i	$-0.0323196\pi$
$73$	17.0000	1.98970	0.994850	+	0.101361i	$0.0323196\pi$
$74$	8.00000	0.929981
$75$	−11.0000	−1.27017
$76$	−7.00000	−0.802955
$77$	0	0
$78$	−2.00000	−0.226455
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	−4.00000	−0.447214
$81$	1.00000	0.111111
$82$	−2.00000	−0.220863
$83$	−13.0000	−1.42694	−0.713468	−	0.700688i	$-0.752876\pi$
$83$	−13.0000	−1.42694	−0.713468	+	0.700688i	$0.752876\pi$
$84$	1.00000	0.109109
$85$	−20.0000	−2.16930
$86$	−3.00000	−0.323498
$87$	−8.00000	−0.857690
$88$	0	0
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	8.00000	0.843274
$91$	−2.00000	−0.209657
$92$	0	0
$93$	2.00000	0.207390
$94$	0	0
$95$	28.0000	2.87274
$96$	−1.00000	−0.102062
$97$	−14.0000	−1.42148	−0.710742	−	0.703452i	$-0.751641\pi$
$97$	−14.0000	−1.42148	−0.710742	+	0.703452i	$0.751641\pi$
$98$	1.00000	0.101015
$99$	0	0
$100$	11.0000	1.10000
$101$	14.0000	1.39305	0.696526	−	0.717532i	$-0.254728\pi$
$101$	14.0000	1.39305	0.696526	+	0.717532i	$0.254728\pi$
$102$	−5.00000	−0.495074
$103$	−16.0000	−1.57653	−0.788263	−	0.615338i	$-0.789020\pi$
$103$	−16.0000	−1.57653	−0.788263	+	0.615338i	$0.789020\pi$
$104$	2.00000	0.196116
$105$	−4.00000	−0.390360
$106$	−6.00000	−0.582772
$107$	−12.0000	−1.16008	−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−12.0000	−1.16008	−0.580042	+	0.814587i	$0.696964\pi$
$108$	5.00000	0.481125
$109$	−4.00000	−0.383131	−0.191565	−	0.981480i	$-0.561356\pi$
$109$	−4.00000	−0.383131	−0.191565	+	0.981480i	$0.561356\pi$
$110$	0	0
$111$	−8.00000	−0.759326
$112$	−1.00000	−0.0944911
$113$	17.0000	1.59923	0.799613	−	0.600516i	$-0.205038\pi$
$113$	17.0000	1.59923	0.799613	+	0.600516i	$0.205038\pi$
$114$	7.00000	0.655610
$115$	0	0
$116$	8.00000	0.742781
$117$	−4.00000	−0.369800
$118$	−9.00000	−0.828517
$119$	−5.00000	−0.458349
$120$	4.00000	0.365148
$121$	−11.0000	−1.00000
$122$	6.00000	0.543214
$123$	2.00000	0.180334
$124$	−2.00000	−0.179605
$125$	−24.0000	−2.14663
$126$	2.00000	0.178174
$127$	−22.0000	−1.95218	−0.976092	−	0.217357i	$-0.930256\pi$
$127$	−22.0000	−1.95218	−0.976092	+	0.217357i	$0.930256\pi$
$128$	1.00000	0.0883883
$129$	3.00000	0.264135
$130$	−8.00000	−0.701646
$131$	1.00000	0.0873704	0.0436852	−	0.999045i	$-0.486090\pi$
$131$	1.00000	0.0873704	0.0436852	+	0.999045i	$0.486090\pi$
$132$	0	0
$133$	7.00000	0.606977
$134$	5.00000	0.431934
$135$	−20.0000	−1.72133
$136$	5.00000	0.428746
$137$	3.00000	0.256307	0.128154	−	0.991754i	$-0.459095\pi$
$137$	3.00000	0.256307	0.128154	+	0.991754i	$0.459095\pi$
$138$	0	0
$139$	−7.00000	−0.593732	−0.296866	−	0.954919i	$-0.595942\pi$
$139$	−7.00000	−0.593732	−0.296866	+	0.954919i	$0.595942\pi$
$140$	4.00000	0.338062
$141$	0	0
$142$	−2.00000	−0.167836
$143$	0	0
$144$	−2.00000	−0.166667
$145$	−32.0000	−2.65746
$146$	17.0000	1.40693
$147$	−1.00000	−0.0824786
$148$	8.00000	0.657596
$149$	10.0000	0.819232	0.409616	−	0.912258i	$-0.365663\pi$
$149$	10.0000	0.819232	0.409616	+	0.912258i	$0.365663\pi$
$150$	−11.0000	−0.898146
$151$	16.0000	1.30206	0.651031	−	0.759051i	$-0.274337\pi$
$151$	16.0000	1.30206	0.651031	+	0.759051i	$0.274337\pi$
$152$	−7.00000	−0.567775
$153$	−10.0000	−0.808452
$154$	0	0
$155$	8.00000	0.642575
$156$	−2.00000	−0.160128
$157$	−12.0000	−0.957704	−0.478852	−	0.877896i	$-0.658947\pi$
$157$	−12.0000	−0.957704	−0.478852	+	0.877896i	$0.658947\pi$
$158$	0	0
$159$	6.00000	0.475831
$160$	−4.00000	−0.316228
$161$	0	0
$162$	1.00000	0.0785674
$163$	20.0000	1.56652	0.783260	−	0.621694i	$-0.213555\pi$
$163$	20.0000	1.56652	0.783260	+	0.621694i	$0.213555\pi$
$164$	−2.00000	−0.156174
$165$	0	0
$166$	−13.0000	−1.00900
$167$	−24.0000	−1.85718	−0.928588	−	0.371113i	$-0.878976\pi$
$167$	−24.0000	−1.85718	−0.928588	+	0.371113i	$0.878976\pi$
$168$	1.00000	0.0771517
$169$	−9.00000	−0.692308
$170$	−20.0000	−1.53393
$171$	14.0000	1.07061
$172$	−3.00000	−0.228748
$173$	−6.00000	−0.456172	−0.228086	−	0.973641i	$-0.573247\pi$
$173$	−6.00000	−0.456172	−0.228086	+	0.973641i	$0.573247\pi$
$174$	−8.00000	−0.606478
$175$	−11.0000	−0.831522
$176$	0	0
$177$	9.00000	0.676481
$178$	6.00000	0.449719
$179$	−9.00000	−0.672692	−0.336346	−	0.941739i	$-0.609191\pi$
$179$	−9.00000	−0.672692	−0.336346	+	0.941739i	$0.609191\pi$
$180$	8.00000	0.596285
$181$	−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
$183$	−6.00000	−0.443533
$184$	0	0
$185$	−32.0000	−2.35269
$186$	2.00000	0.146647
$187$	0	0
$188$	0	0
$189$	−5.00000	−0.363696
$190$	28.0000	2.03133
$191$	−10.0000	−0.723575	−0.361787	−	0.932261i	$-0.617833\pi$
$191$	−10.0000	−0.723575	−0.361787	+	0.932261i	$0.617833\pi$
$192$	−1.00000	−0.0721688
$193$	−11.0000	−0.791797	−0.395899	−	0.918294i	$-0.629567\pi$
$193$	−11.0000	−0.791797	−0.395899	+	0.918294i	$0.629567\pi$
$194$	−14.0000	−1.00514
$195$	8.00000	0.572892
$196$	1.00000	0.0714286
$197$	8.00000	0.569976	0.284988	−	0.958531i	$-0.408010\pi$
$197$	8.00000	0.569976	0.284988	+	0.958531i	$0.408010\pi$
$198$	0	0
$199$	−8.00000	−0.567105	−0.283552	−	0.958957i	$-0.591513\pi$
$199$	−8.00000	−0.567105	−0.283552	+	0.958957i	$0.591513\pi$
$200$	11.0000	0.777817
$201$	−5.00000	−0.352673
$202$	14.0000	0.985037
$203$	−8.00000	−0.561490
$204$	−5.00000	−0.350070
$205$	8.00000	0.558744
$206$	−16.0000	−1.11477
$207$	0	0
$208$	2.00000	0.138675
$209$	0	0
$210$	−4.00000	−0.276026
$211$	−3.00000	−0.206529	−0.103264	−	0.994654i	$-0.532929\pi$
$211$	−3.00000	−0.206529	−0.103264	+	0.994654i	$0.532929\pi$
$212$	−6.00000	−0.412082
$213$	2.00000	0.137038
$214$	−12.0000	−0.820303
$215$	12.0000	0.818393
$216$	5.00000	0.340207
$217$	2.00000	0.135769
$218$	−4.00000	−0.270914
$219$	−17.0000	−1.14875
$220$	0	0
$221$	10.0000	0.672673
$222$	−8.00000	−0.536925
$223$	−2.00000	−0.133930	−0.0669650	−	0.997755i	$-0.521332\pi$
$223$	−2.00000	−0.133930	−0.0669650	+	0.997755i	$0.521332\pi$
$224$	−1.00000	−0.0668153
$225$	−22.0000	−1.46667
$226$	17.0000	1.13082
$227$	3.00000	0.199117	0.0995585	−	0.995032i	$-0.468257\pi$
$227$	3.00000	0.199117	0.0995585	+	0.995032i	$0.468257\pi$
$228$	7.00000	0.463586
$229$	−4.00000	−0.264327	−0.132164	−	0.991228i	$-0.542192\pi$
$229$	−4.00000	−0.264327	−0.132164	+	0.991228i	$0.542192\pi$
$230$	0	0
$231$	0	0
$232$	8.00000	0.525226
$233$	3.00000	0.196537	0.0982683	−	0.995160i	$-0.468670\pi$
$233$	3.00000	0.196537	0.0982683	+	0.995160i	$0.468670\pi$
$234$	−4.00000	−0.261488
$235$	0	0
$236$	−9.00000	−0.585850
$237$	0	0
$238$	−5.00000	−0.324102
$239$	30.0000	1.94054	0.970269	−	0.242028i	$-0.0778125\pi$
$239$	30.0000	1.94054	0.970269	+	0.242028i	$0.0778125\pi$
$240$	4.00000	0.258199
$241$	21.0000	1.35273	0.676364	−	0.736567i	$-0.263554\pi$
$241$	21.0000	1.35273	0.676364	+	0.736567i	$0.263554\pi$
$242$	−11.0000	−0.707107
$243$	−16.0000	−1.02640
$244$	6.00000	0.384111
$245$	−4.00000	−0.255551
$246$	2.00000	0.127515
$247$	−14.0000	−0.890799
$248$	−2.00000	−0.127000
$249$	13.0000	0.823842
$250$	−24.0000	−1.51789
$251$	−27.0000	−1.70422	−0.852112	−	0.523359i	$-0.824679\pi$
$251$	−27.0000	−1.70422	−0.852112	+	0.523359i	$0.824679\pi$
$252$	2.00000	0.125988
$253$	0	0
$254$	−22.0000	−1.38040
$255$	20.0000	1.25245
$256$	1.00000	0.0625000
$257$	−7.00000	−0.436648	−0.218324	−	0.975876i	$-0.570059\pi$
$257$	−7.00000	−0.436648	−0.218324	+	0.975876i	$0.570059\pi$
$258$	3.00000	0.186772
$259$	−8.00000	−0.497096
$260$	−8.00000	−0.496139
$261$	−16.0000	−0.990375
$262$	1.00000	0.0617802
$263$	20.0000	1.23325	0.616626	−	0.787256i	$-0.288499\pi$
$263$	20.0000	1.23325	0.616626	+	0.787256i	$0.288499\pi$
$264$	0	0
$265$	24.0000	1.47431
$266$	7.00000	0.429198
$267$	−6.00000	−0.367194
$268$	5.00000	0.305424
$269$	−24.0000	−1.46331	−0.731653	−	0.681677i	$-0.761251\pi$
$269$	−24.0000	−1.46331	−0.731653	+	0.681677i	$0.761251\pi$
$270$	−20.0000	−1.21716
$271$	30.0000	1.82237	0.911185	−	0.411997i	$-0.135169\pi$
$271$	30.0000	1.82237	0.911185	+	0.411997i	$0.135169\pi$
$272$	5.00000	0.303170
$273$	2.00000	0.121046
$274$	3.00000	0.181237
$275$	0	0
$276$	0	0
$277$	−26.0000	−1.56219	−0.781094	−	0.624413i	$-0.785338\pi$
$277$	−26.0000	−1.56219	−0.781094	+	0.624413i	$0.785338\pi$
$278$	−7.00000	−0.419832
$279$	4.00000	0.239474
$280$	4.00000	0.239046
$281$	25.0000	1.49137	0.745687	−	0.666296i	$-0.232121\pi$
$281$	25.0000	1.49137	0.745687	+	0.666296i	$0.232121\pi$
$282$	0	0
$283$	21.0000	1.24832	0.624160	−	0.781296i	$-0.285441\pi$
$283$	21.0000	1.24832	0.624160	+	0.781296i	$0.285441\pi$
$284$	−2.00000	−0.118678
$285$	−28.0000	−1.65858
$286$	0	0
$287$	2.00000	0.118056
$288$	−2.00000	−0.117851
$289$	8.00000	0.470588
$290$	−32.0000	−1.87910
$291$	14.0000	0.820695
$292$	17.0000	0.994850
$293$	14.0000	0.817889	0.408944	−	0.912559i	$-0.365897\pi$
$293$	14.0000	0.817889	0.408944	+	0.912559i	$0.365897\pi$
$294$	−1.00000	−0.0583212
$295$	36.0000	2.09600
$296$	8.00000	0.464991
$297$	0	0
$298$	10.0000	0.579284
$299$	0	0
$300$	−11.0000	−0.635085
$301$	3.00000	0.172917
$302$	16.0000	0.920697
$303$	−14.0000	−0.804279
$304$	−7.00000	−0.401478
$305$	−24.0000	−1.37424
$306$	−10.0000	−0.571662
$307$	−23.0000	−1.31268	−0.656340	−	0.754466i	$-0.727896\pi$
$307$	−23.0000	−1.31268	−0.656340	+	0.754466i	$0.727896\pi$
$308$	0	0
$309$	16.0000	0.910208
$310$	8.00000	0.454369
$311$	−8.00000	−0.453638	−0.226819	−	0.973937i	$-0.572833\pi$
$311$	−8.00000	−0.453638	−0.226819	+	0.973937i	$0.572833\pi$
$312$	−2.00000	−0.113228
$313$	−11.0000	−0.621757	−0.310878	−	0.950450i	$-0.600623\pi$
$313$	−11.0000	−0.621757	−0.310878	+	0.950450i	$0.600623\pi$
$314$	−12.0000	−0.677199
$315$	−8.00000	−0.450749
$316$	0	0
$317$	2.00000	0.112331	0.0561656	−	0.998421i	$-0.482113\pi$
$317$	2.00000	0.112331	0.0561656	+	0.998421i	$0.482113\pi$
$318$	6.00000	0.336463
$319$	0	0
$320$	−4.00000	−0.223607
$321$	12.0000	0.669775
$322$	0	0
$323$	−35.0000	−1.94745
$324$	1.00000	0.0555556
$325$	22.0000	1.22034
$326$	20.0000	1.10770
$327$	4.00000	0.221201
$328$	−2.00000	−0.110432
$329$	0	0
$330$	0	0
$331$	1.00000	0.0549650	0.0274825	−	0.999622i	$-0.491251\pi$
$331$	1.00000	0.0549650	0.0274825	+	0.999622i	$0.491251\pi$
$332$	−13.0000	−0.713468
$333$	−16.0000	−0.876795
$334$	−24.0000	−1.31322
$335$	−20.0000	−1.09272
$336$	1.00000	0.0545545
$337$	−27.0000	−1.47078	−0.735392	−	0.677642i	$-0.763002\pi$
$337$	−27.0000	−1.47078	−0.735392	+	0.677642i	$0.763002\pi$
$338$	−9.00000	−0.489535
$339$	−17.0000	−0.923313
$340$	−20.0000	−1.08465
$341$	0	0
$342$	14.0000	0.757033
$343$	−1.00000	−0.0539949
$344$	−3.00000	−0.161749
$345$	0	0
$346$	−6.00000	−0.322562
$347$	3.00000	0.161048	0.0805242	−	0.996753i	$-0.474341\pi$
$347$	3.00000	0.161048	0.0805242	+	0.996753i	$0.474341\pi$
$348$	−8.00000	−0.428845
$349$	−8.00000	−0.428230	−0.214115	−	0.976808i	$-0.568687\pi$
$349$	−8.00000	−0.428230	−0.214115	+	0.976808i	$0.568687\pi$
$350$	−11.0000	−0.587975
$351$	10.0000	0.533761
$352$	0	0
$353$	−11.0000	−0.585471	−0.292735	−	0.956193i	$-0.594566\pi$
$353$	−11.0000	−0.585471	−0.292735	+	0.956193i	$0.594566\pi$
$354$	9.00000	0.478345
$355$	8.00000	0.424596
$356$	6.00000	0.317999
$357$	5.00000	0.264628
$358$	−9.00000	−0.475665
$359$	−30.0000	−1.58334	−0.791670	−	0.610949i	$-0.790788\pi$
$359$	−30.0000	−1.58334	−0.791670	+	0.610949i	$0.790788\pi$
$360$	8.00000	0.421637
$361$	30.0000	1.57895
$362$	−10.0000	−0.525588
$363$	11.0000	0.577350
$364$	−2.00000	−0.104828
$365$	−68.0000	−3.55928
$366$	−6.00000	−0.313625
$367$	−22.0000	−1.14839	−0.574195	−	0.818718i	$-0.694685\pi$
$367$	−22.0000	−1.14839	−0.574195	+	0.818718i	$0.694685\pi$
$368$	0	0
$369$	4.00000	0.208232
$370$	−32.0000	−1.66360
$371$	6.00000	0.311504
$372$	2.00000	0.103695
$373$	−26.0000	−1.34623	−0.673114	−	0.739538i	$-0.735044\pi$
$373$	−26.0000	−1.34623	−0.673114	+	0.739538i	$0.735044\pi$
$374$	0	0
$375$	24.0000	1.23935
$376$	0	0
$377$	16.0000	0.824042
$378$	−5.00000	−0.257172
$379$	−35.0000	−1.79783	−0.898915	−	0.438124i	$-0.855643\pi$
$379$	−35.0000	−1.79783	−0.898915	+	0.438124i	$0.855643\pi$
$380$	28.0000	1.43637
$381$	22.0000	1.12709
$382$	−10.0000	−0.511645
$383$	−2.00000	−0.102195	−0.0510976	−	0.998694i	$-0.516272\pi$
$383$	−2.00000	−0.102195	−0.0510976	+	0.998694i	$0.516272\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	−11.0000	−0.559885
$387$	6.00000	0.304997
$388$	−14.0000	−0.710742
$389$	−14.0000	−0.709828	−0.354914	−	0.934899i	$-0.615490\pi$
$389$	−14.0000	−0.709828	−0.354914	+	0.934899i	$0.615490\pi$
$390$	8.00000	0.405096
$391$	0	0
$392$	1.00000	0.0505076
$393$	−1.00000	−0.0504433
$394$	8.00000	0.403034
$395$	0	0
$396$	0	0
$397$	−32.0000	−1.60603	−0.803017	−	0.595956i	$-0.796773\pi$
$397$	−32.0000	−1.60603	−0.803017	+	0.595956i	$0.796773\pi$
$398$	−8.00000	−0.401004
$399$	−7.00000	−0.350438
$400$	11.0000	0.550000
$401$	−25.0000	−1.24844	−0.624220	−	0.781248i	$-0.714583\pi$
$401$	−25.0000	−1.24844	−0.624220	+	0.781248i	$0.714583\pi$
$402$	−5.00000	−0.249377
$403$	−4.00000	−0.199254
$404$	14.0000	0.696526
$405$	−4.00000	−0.198762
$406$	−8.00000	−0.397033
$407$	0	0
$408$	−5.00000	−0.247537
$409$	−17.0000	−0.840596	−0.420298	−	0.907386i	$-0.638074\pi$
$409$	−17.0000	−0.840596	−0.420298	+	0.907386i	$0.638074\pi$
$410$	8.00000	0.395092
$411$	−3.00000	−0.147979
$412$	−16.0000	−0.788263
$413$	9.00000	0.442861
$414$	0	0
$415$	52.0000	2.55258
$416$	2.00000	0.0980581
$417$	7.00000	0.342791
$418$	0	0
$419$	9.00000	0.439679	0.219839	−	0.975536i	$-0.429447\pi$
$419$	9.00000	0.439679	0.219839	+	0.975536i	$0.429447\pi$
$420$	−4.00000	−0.195180
$421$	30.0000	1.46211	0.731055	−	0.682318i	$-0.239028\pi$
$421$	30.0000	1.46211	0.731055	+	0.682318i	$0.239028\pi$
$422$	−3.00000	−0.146038
$423$	0	0
$424$	−6.00000	−0.291386
$425$	55.0000	2.66789
$426$	2.00000	0.0969003
$427$	−6.00000	−0.290360
$428$	−12.0000	−0.580042
$429$	0	0
$430$	12.0000	0.578691
$431$	−12.0000	−0.578020	−0.289010	−	0.957326i	$-0.593326\pi$
$431$	−12.0000	−0.578020	−0.289010	+	0.957326i	$0.593326\pi$
$432$	5.00000	0.240563
$433$	10.0000	0.480569	0.240285	−	0.970702i	$-0.422759\pi$
$433$	10.0000	0.480569	0.240285	+	0.970702i	$0.422759\pi$
$434$	2.00000	0.0960031
$435$	32.0000	1.53428
$436$	−4.00000	−0.191565
$437$	0	0
$438$	−17.0000	−0.812291
$439$	32.0000	1.52728	0.763638	−	0.645644i	$-0.223411\pi$
$439$	32.0000	1.52728	0.763638	+	0.645644i	$0.223411\pi$
$440$	0	0
$441$	−2.00000	−0.0952381
$442$	10.0000	0.475651
$443$	7.00000	0.332580	0.166290	−	0.986077i	$-0.446821\pi$
$443$	7.00000	0.332580	0.166290	+	0.986077i	$0.446821\pi$
$444$	−8.00000	−0.379663
$445$	−24.0000	−1.13771
$446$	−2.00000	−0.0947027
$447$	−10.0000	−0.472984
$448$	−1.00000	−0.0472456
$449$	9.00000	0.424736	0.212368	−	0.977190i	$-0.431882\pi$
$449$	9.00000	0.424736	0.212368	+	0.977190i	$0.431882\pi$
$450$	−22.0000	−1.03709
$451$	0	0
$452$	17.0000	0.799613
$453$	−16.0000	−0.751746
$454$	3.00000	0.140797
$455$	8.00000	0.375046
$456$	7.00000	0.327805
$457$	−10.0000	−0.467780	−0.233890	−	0.972263i	$-0.575146\pi$
$457$	−10.0000	−0.467780	−0.233890	+	0.972263i	$0.575146\pi$
$458$	−4.00000	−0.186908
$459$	25.0000	1.16690
$460$	0	0
$461$	−10.0000	−0.465746	−0.232873	−	0.972507i	$-0.574813\pi$
$461$	−10.0000	−0.465746	−0.232873	+	0.972507i	$0.574813\pi$
$462$	0	0
$463$	4.00000	0.185896	0.0929479	−	0.995671i	$-0.470371\pi$
$463$	4.00000	0.185896	0.0929479	+	0.995671i	$0.470371\pi$
$464$	8.00000	0.371391
$465$	−8.00000	−0.370991
$466$	3.00000	0.138972
$467$	−16.0000	−0.740392	−0.370196	−	0.928954i	$-0.620709\pi$
$467$	−16.0000	−0.740392	−0.370196	+	0.928954i	$0.620709\pi$
$468$	−4.00000	−0.184900
$469$	−5.00000	−0.230879
$470$	0	0
$471$	12.0000	0.552931
$472$	−9.00000	−0.414259
$473$	0	0
$474$	0	0
$475$	−77.0000	−3.53300
$476$	−5.00000	−0.229175
$477$	12.0000	0.549442
$478$	30.0000	1.37217
$479$	20.0000	0.913823	0.456912	−	0.889512i	$-0.348956\pi$
$479$	20.0000	0.913823	0.456912	+	0.889512i	$0.348956\pi$
$480$	4.00000	0.182574
$481$	16.0000	0.729537
$482$	21.0000	0.956524
$483$	0	0
$484$	−11.0000	−0.500000
$485$	56.0000	2.54283
$486$	−16.0000	−0.725775
$487$	36.0000	1.63132	0.815658	−	0.578535i	$-0.196375\pi$
$487$	36.0000	1.63132	0.815658	+	0.578535i	$0.196375\pi$
$488$	6.00000	0.271607
$489$	−20.0000	−0.904431
$490$	−4.00000	−0.180702
$491$	4.00000	0.180517	0.0902587	−	0.995918i	$-0.471231\pi$
$491$	4.00000	0.180517	0.0902587	+	0.995918i	$0.471231\pi$
$492$	2.00000	0.0901670
$493$	40.0000	1.80151
$494$	−14.0000	−0.629890
$495$	0	0
$496$	−2.00000	−0.0898027
$497$	2.00000	0.0897123
$498$	13.0000	0.582544
$499$	15.0000	0.671492	0.335746	−	0.941953i	$-0.391012\pi$
$499$	15.0000	0.671492	0.335746	+	0.941953i	$0.391012\pi$
$500$	−24.0000	−1.07331
$501$	24.0000	1.07224
$502$	−27.0000	−1.20507
$503$	28.0000	1.24846	0.624229	−	0.781241i	$-0.285413\pi$
$503$	28.0000	1.24846	0.624229	+	0.781241i	$0.285413\pi$
$504$	2.00000	0.0890871
$505$	−56.0000	−2.49197
$506$	0	0
$507$	9.00000	0.399704
$508$	−22.0000	−0.976092
$509$	−16.0000	−0.709188	−0.354594	−	0.935020i	$-0.615381\pi$
$509$	−16.0000	−0.709188	−0.354594	+	0.935020i	$0.615381\pi$
$510$	20.0000	0.885615
$511$	−17.0000	−0.752036
$512$	1.00000	0.0441942
$513$	−35.0000	−1.54529
$514$	−7.00000	−0.308757
$515$	64.0000	2.82018
$516$	3.00000	0.132068
$517$	0	0
$518$	−8.00000	−0.351500
$519$	6.00000	0.263371
$520$	−8.00000	−0.350823
$521$	42.0000	1.84005	0.920027	−	0.391856i	$-0.128167\pi$
$521$	42.0000	1.84005	0.920027	+	0.391856i	$0.128167\pi$
$522$	−16.0000	−0.700301
$523$	−24.0000	−1.04945	−0.524723	−	0.851273i	$-0.675831\pi$
$523$	−24.0000	−1.04945	−0.524723	+	0.851273i	$0.675831\pi$
$524$	1.00000	0.0436852
$525$	11.0000	0.480079
$526$	20.0000	0.872041
$527$	−10.0000	−0.435607
$528$	0	0
$529$	0	0
$530$	24.0000	1.04249
$531$	18.0000	0.781133
$532$	7.00000	0.303488
$533$	−4.00000	−0.173259
$534$	−6.00000	−0.259645
$535$	48.0000	2.07522
$536$	5.00000	0.215967
$537$	9.00000	0.388379
$538$	−24.0000	−1.03471
$539$	0	0
$540$	−20.0000	−0.860663
$541$	−8.00000	−0.343947	−0.171973	−	0.985102i	$-0.555014\pi$
$541$	−8.00000	−0.343947	−0.171973	+	0.985102i	$0.555014\pi$
$542$	30.0000	1.28861
$543$	10.0000	0.429141
$544$	5.00000	0.214373
$545$	16.0000	0.685365
$546$	2.00000	0.0855921
$547$	−20.0000	−0.855138	−0.427569	−	0.903983i	$-0.640630\pi$
$547$	−20.0000	−0.855138	−0.427569	+	0.903983i	$0.640630\pi$
$548$	3.00000	0.128154
$549$	−12.0000	−0.512148
$550$	0	0
$551$	−56.0000	−2.38568
$552$	0	0
$553$	0	0
$554$	−26.0000	−1.10463
$555$	32.0000	1.35832
$556$	−7.00000	−0.296866
$557$	6.00000	0.254228	0.127114	−	0.991888i	$-0.459429\pi$
$557$	6.00000	0.254228	0.127114	+	0.991888i	$0.459429\pi$
$558$	4.00000	0.169334
$559$	−6.00000	−0.253773
$560$	4.00000	0.169031
$561$	0	0
$562$	25.0000	1.05456
$563$	21.0000	0.885044	0.442522	−	0.896758i	$-0.354084\pi$
$563$	21.0000	0.885044	0.442522	+	0.896758i	$0.354084\pi$
$564$	0	0
$565$	−68.0000	−2.86078
$566$	21.0000	0.882696
$567$	−1.00000	−0.0419961
$568$	−2.00000	−0.0839181
$569$	15.0000	0.628833	0.314416	−	0.949285i	$-0.398191\pi$
$569$	15.0000	0.628833	0.314416	+	0.949285i	$0.398191\pi$
$570$	−28.0000	−1.17279
$571$	13.0000	0.544033	0.272017	−	0.962293i	$-0.412309\pi$
$571$	13.0000	0.544033	0.272017	+	0.962293i	$0.412309\pi$
$572$	0	0
$573$	10.0000	0.417756
$574$	2.00000	0.0834784
$575$	0	0
$576$	−2.00000	−0.0833333
$577$	−27.0000	−1.12402	−0.562012	−	0.827129i	$-0.689973\pi$
$577$	−27.0000	−1.12402	−0.562012	+	0.827129i	$0.689973\pi$
$578$	8.00000	0.332756
$579$	11.0000	0.457144
$580$	−32.0000	−1.32873
$581$	13.0000	0.539331
$582$	14.0000	0.580319
$583$	0	0
$584$	17.0000	0.703465
$585$	16.0000	0.661519
$586$	14.0000	0.578335
$587$	−39.0000	−1.60970	−0.804851	−	0.593477i	$-0.797755\pi$
$587$	−39.0000	−1.60970	−0.804851	+	0.593477i	$0.797755\pi$
$588$	−1.00000	−0.0412393
$589$	14.0000	0.576860
$590$	36.0000	1.48210
$591$	−8.00000	−0.329076
$592$	8.00000	0.328798
$593$	21.0000	0.862367	0.431183	−	0.902264i	$-0.358096\pi$
$593$	21.0000	0.862367	0.431183	+	0.902264i	$0.358096\pi$
$594$	0	0
$595$	20.0000	0.819920
$596$	10.0000	0.409616
$597$	8.00000	0.327418
$598$	0	0
$599$	−12.0000	−0.490307	−0.245153	−	0.969484i	$-0.578838\pi$
$599$	−12.0000	−0.490307	−0.245153	+	0.969484i	$0.578838\pi$
$600$	−11.0000	−0.449073
$601$	2.00000	0.0815817	0.0407909	−	0.999168i	$-0.487012\pi$
$601$	2.00000	0.0815817	0.0407909	+	0.999168i	$0.487012\pi$
$602$	3.00000	0.122271
$603$	−10.0000	−0.407231
$604$	16.0000	0.651031
$605$	44.0000	1.78885
$606$	−14.0000	−0.568711
$607$	24.0000	0.974130	0.487065	−	0.873366i	$-0.338067\pi$
$607$	24.0000	0.974130	0.487065	+	0.873366i	$0.338067\pi$
$608$	−7.00000	−0.283887
$609$	8.00000	0.324176
$610$	−24.0000	−0.971732
$611$	0	0
$612$	−10.0000	−0.404226
$613$	−6.00000	−0.242338	−0.121169	−	0.992632i	$-0.538664\pi$
$613$	−6.00000	−0.242338	−0.121169	+	0.992632i	$0.538664\pi$
$614$	−23.0000	−0.928204
$615$	−8.00000	−0.322591
$616$	0	0
$617$	21.0000	0.845428	0.422714	−	0.906263i	$-0.361077\pi$
$617$	21.0000	0.845428	0.422714	+	0.906263i	$0.361077\pi$
$618$	16.0000	0.643614
$619$	4.00000	0.160774	0.0803868	−	0.996764i	$-0.474384\pi$
$619$	4.00000	0.160774	0.0803868	+	0.996764i	$0.474384\pi$
$620$	8.00000	0.321288
$621$	0	0
$622$	−8.00000	−0.320771
$623$	−6.00000	−0.240385
$624$	−2.00000	−0.0800641
$625$	41.0000	1.64000
$626$	−11.0000	−0.439648
$627$	0	0
$628$	−12.0000	−0.478852
$629$	40.0000	1.59490
$630$	−8.00000	−0.318728
$631$	20.0000	0.796187	0.398094	−	0.917345i	$-0.369672\pi$
$631$	20.0000	0.796187	0.398094	+	0.917345i	$0.369672\pi$
$632$	0	0
$633$	3.00000	0.119239
$634$	2.00000	0.0794301
$635$	88.0000	3.49217
$636$	6.00000	0.237915
$637$	2.00000	0.0792429
$638$	0	0
$639$	4.00000	0.158238
$640$	−4.00000	−0.158114
$641$	−45.0000	−1.77739	−0.888697	−	0.458496i	$-0.848388\pi$
$641$	−45.0000	−1.77739	−0.888697	+	0.458496i	$0.848388\pi$
$642$	12.0000	0.473602
$643$	−4.00000	−0.157745	−0.0788723	−	0.996885i	$-0.525132\pi$
$643$	−4.00000	−0.157745	−0.0788723	+	0.996885i	$0.525132\pi$
$644$	0	0
$645$	−12.0000	−0.472500
$646$	−35.0000	−1.37706
$647$	26.0000	1.02217	0.511083	−	0.859532i	$-0.329245\pi$
$647$	26.0000	1.02217	0.511083	+	0.859532i	$0.329245\pi$
$648$	1.00000	0.0392837
$649$	0	0
$650$	22.0000	0.862911
$651$	−2.00000	−0.0783862
$652$	20.0000	0.783260
$653$	−34.0000	−1.33052	−0.665261	−	0.746611i	$-0.731680\pi$
$653$	−34.0000	−1.33052	−0.665261	+	0.746611i	$0.731680\pi$
$654$	4.00000	0.156412
$655$	−4.00000	−0.156293
$656$	−2.00000	−0.0780869
$657$	−34.0000	−1.32647
$658$	0	0
$659$	29.0000	1.12968	0.564840	−	0.825201i	$-0.308938\pi$
$659$	29.0000	1.12968	0.564840	+	0.825201i	$0.308938\pi$
$660$	0	0
$661$	10.0000	0.388955	0.194477	−	0.980907i	$-0.437699\pi$
$661$	10.0000	0.388955	0.194477	+	0.980907i	$0.437699\pi$
$662$	1.00000	0.0388661
$663$	−10.0000	−0.388368
$664$	−13.0000	−0.504498
$665$	−28.0000	−1.08579
$666$	−16.0000	−0.619987
$667$	0	0
$668$	−24.0000	−0.928588
$669$	2.00000	0.0773245
$670$	−20.0000	−0.772667
$671$	0	0
$672$	1.00000	0.0385758
$673$	1.00000	0.0385472	0.0192736	−	0.999814i	$-0.493865\pi$
$673$	1.00000	0.0385472	0.0192736	+	0.999814i	$0.493865\pi$
$674$	−27.0000	−1.04000
$675$	55.0000	2.11695
$676$	−9.00000	−0.346154
$677$	8.00000	0.307465	0.153732	−	0.988113i	$-0.450871\pi$
$677$	8.00000	0.307465	0.153732	+	0.988113i	$0.450871\pi$
$678$	−17.0000	−0.652881
$679$	14.0000	0.537271
$680$	−20.0000	−0.766965
$681$	−3.00000	−0.114960
$682$	0	0
$683$	19.0000	0.727015	0.363507	−	0.931591i	$-0.381579\pi$
$683$	19.0000	0.727015	0.363507	+	0.931591i	$0.381579\pi$
$684$	14.0000	0.535303
$685$	−12.0000	−0.458496
$686$	−1.00000	−0.0381802
$687$	4.00000	0.152610
$688$	−3.00000	−0.114374
$689$	−12.0000	−0.457164
$690$	0	0
$691$	20.0000	0.760836	0.380418	−	0.924815i	$-0.375780\pi$
$691$	20.0000	0.760836	0.380418	+	0.924815i	$0.375780\pi$
$692$	−6.00000	−0.228086
$693$	0	0
$694$	3.00000	0.113878
$695$	28.0000	1.06210
$696$	−8.00000	−0.303239
$697$	−10.0000	−0.378777
$698$	−8.00000	−0.302804
$699$	−3.00000	−0.113470
$700$	−11.0000	−0.415761
$701$	22.0000	0.830929	0.415464	−	0.909610i	$-0.363619\pi$
$701$	22.0000	0.830929	0.415464	+	0.909610i	$0.363619\pi$
$702$	10.0000	0.377426
$703$	−56.0000	−2.11208
$704$	0	0
$705$	0	0
$706$	−11.0000	−0.413990
$707$	−14.0000	−0.526524
$708$	9.00000	0.338241
$709$	−32.0000	−1.20179	−0.600893	−	0.799330i	$-0.705188\pi$
$709$	−32.0000	−1.20179	−0.600893	+	0.799330i	$0.705188\pi$
$710$	8.00000	0.300235
$711$	0	0
$712$	6.00000	0.224860
$713$	0	0
$714$	5.00000	0.187120
$715$	0	0
$716$	−9.00000	−0.336346
$717$	−30.0000	−1.12037
$718$	−30.0000	−1.11959
$719$	−30.0000	−1.11881	−0.559406	−	0.828894i	$-0.688971\pi$
$719$	−30.0000	−1.11881	−0.559406	+	0.828894i	$0.688971\pi$
$720$	8.00000	0.298142
$721$	16.0000	0.595871
$722$	30.0000	1.11648
$723$	−21.0000	−0.780998
$724$	−10.0000	−0.371647
$725$	88.0000	3.26824
$726$	11.0000	0.408248
$727$	46.0000	1.70605	0.853023	−	0.521874i	$-0.174767\pi$
$727$	46.0000	1.70605	0.853023	+	0.521874i	$0.174767\pi$
$728$	−2.00000	−0.0741249
$729$	13.0000	0.481481
$730$	−68.0000	−2.51679
$731$	−15.0000	−0.554795
$732$	−6.00000	−0.221766
$733$	22.0000	0.812589	0.406294	−	0.913742i	$-0.366821\pi$
$733$	22.0000	0.812589	0.406294	+	0.913742i	$0.366821\pi$
$734$	−22.0000	−0.812035
$735$	4.00000	0.147542
$736$	0	0
$737$	0	0
$738$	4.00000	0.147242
$739$	−20.0000	−0.735712	−0.367856	−	0.929883i	$-0.619908\pi$
$739$	−20.0000	−0.735712	−0.367856	+	0.929883i	$0.619908\pi$
$740$	−32.0000	−1.17634
$741$	14.0000	0.514303
$742$	6.00000	0.220267
$743$	−30.0000	−1.10059	−0.550297	−	0.834969i	$-0.685485\pi$
$743$	−30.0000	−1.10059	−0.550297	+	0.834969i	$0.685485\pi$
$744$	2.00000	0.0733236
$745$	−40.0000	−1.46549
$746$	−26.0000	−0.951928
$747$	26.0000	0.951290
$748$	0	0
$749$	12.0000	0.438470
$750$	24.0000	0.876356
$751$	−26.0000	−0.948753	−0.474377	−	0.880322i	$-0.657327\pi$
$751$	−26.0000	−0.948753	−0.474377	+	0.880322i	$0.657327\pi$
$752$	0	0
$753$	27.0000	0.983935
$754$	16.0000	0.582686
$755$	−64.0000	−2.32920
$756$	−5.00000	−0.181848
$757$	28.0000	1.01768	0.508839	−	0.860862i	$-0.330075\pi$
$757$	28.0000	1.01768	0.508839	+	0.860862i	$0.330075\pi$
$758$	−35.0000	−1.27126
$759$	0	0
$760$	28.0000	1.01567
$761$	−33.0000	−1.19625	−0.598125	−	0.801403i	$-0.704087\pi$
$761$	−33.0000	−1.19625	−0.598125	+	0.801403i	$0.704087\pi$
$762$	22.0000	0.796976
$763$	4.00000	0.144810
$764$	−10.0000	−0.361787
$765$	40.0000	1.44620
$766$	−2.00000	−0.0722629
$767$	−18.0000	−0.649942
$768$	−1.00000	−0.0360844
$769$	−30.0000	−1.08183	−0.540914	−	0.841078i	$-0.681921\pi$
$769$	−30.0000	−1.08183	−0.540914	+	0.841078i	$0.681921\pi$
$770$	0	0
$771$	7.00000	0.252099
$772$	−11.0000	−0.395899
$773$	38.0000	1.36677	0.683383	−	0.730061i	$-0.260508\pi$
$773$	38.0000	1.36677	0.683383	+	0.730061i	$0.260508\pi$
$774$	6.00000	0.215666
$775$	−22.0000	−0.790263
$776$	−14.0000	−0.502571
$777$	8.00000	0.286998
$778$	−14.0000	−0.501924
$779$	14.0000	0.501602
$780$	8.00000	0.286446
$781$	0	0
$782$	0	0
$783$	40.0000	1.42948
$784$	1.00000	0.0357143
$785$	48.0000	1.71319
$786$	−1.00000	−0.0356688
$787$	5.00000	0.178231	0.0891154	−	0.996021i	$-0.471596\pi$
$787$	5.00000	0.178231	0.0891154	+	0.996021i	$0.471596\pi$
$788$	8.00000	0.284988
$789$	−20.0000	−0.712019
$790$	0	0
$791$	−17.0000	−0.604450
$792$	0	0
$793$	12.0000	0.426132
$794$	−32.0000	−1.13564
$795$	−24.0000	−0.851192
$796$	−8.00000	−0.283552
$797$	−30.0000	−1.06265	−0.531327	−	0.847167i	$-0.678307\pi$
$797$	−30.0000	−1.06265	−0.531327	+	0.847167i	$0.678307\pi$
$798$	−7.00000	−0.247797
$799$	0	0
$800$	11.0000	0.388909
$801$	−12.0000	−0.423999
$802$	−25.0000	−0.882781
$803$	0	0
$804$	−5.00000	−0.176336
$805$	0	0
$806$	−4.00000	−0.140894
$807$	24.0000	0.844840
$808$	14.0000	0.492518
$809$	17.0000	0.597688	0.298844	−	0.954302i	$-0.403399\pi$
$809$	17.0000	0.597688	0.298844	+	0.954302i	$0.403399\pi$
$810$	−4.00000	−0.140546
$811$	−47.0000	−1.65039	−0.825197	−	0.564846i	$-0.808936\pi$
$811$	−47.0000	−1.65039	−0.825197	+	0.564846i	$0.808936\pi$
$812$	−8.00000	−0.280745
$813$	−30.0000	−1.05215
$814$	0	0
$815$	−80.0000	−2.80228
$816$	−5.00000	−0.175035
$817$	21.0000	0.734697
$818$	−17.0000	−0.594391
$819$	4.00000	0.139771
$820$	8.00000	0.279372
$821$	−22.0000	−0.767805	−0.383903	−	0.923374i	$-0.625420\pi$
$821$	−22.0000	−0.767805	−0.383903	+	0.923374i	$0.625420\pi$
$822$	−3.00000	−0.104637
$823$	−50.0000	−1.74289	−0.871445	−	0.490493i	$-0.836817\pi$
$823$	−50.0000	−1.74289	−0.871445	+	0.490493i	$0.836817\pi$
$824$	−16.0000	−0.557386
$825$	0	0
$826$	9.00000	0.313150
$827$	−29.0000	−1.00843	−0.504214	−	0.863579i	$-0.668218\pi$
$827$	−29.0000	−1.00843	−0.504214	+	0.863579i	$0.668218\pi$
$828$	0	0
$829$	−38.0000	−1.31979	−0.659897	−	0.751356i	$-0.729400\pi$
$829$	−38.0000	−1.31979	−0.659897	+	0.751356i	$0.729400\pi$
$830$	52.0000	1.80495
$831$	26.0000	0.901930
$832$	2.00000	0.0693375
$833$	5.00000	0.173240
$834$	7.00000	0.242390
$835$	96.0000	3.32222
$836$	0	0
$837$	−10.0000	−0.345651
$838$	9.00000	0.310900
$839$	−28.0000	−0.966667	−0.483334	−	0.875436i	$-0.660574\pi$
$839$	−28.0000	−0.966667	−0.483334	+	0.875436i	$0.660574\pi$
$840$	−4.00000	−0.138013
$841$	35.0000	1.20690
$842$	30.0000	1.03387
$843$	−25.0000	−0.861046
$844$	−3.00000	−0.103264
$845$	36.0000	1.23844
$846$	0	0
$847$	11.0000	0.377964
$848$	−6.00000	−0.206041
$849$	−21.0000	−0.720718
$850$	55.0000	1.88648
$851$	0	0
$852$	2.00000	0.0685189
$853$	−36.0000	−1.23262	−0.616308	−	0.787505i	$-0.711372\pi$
$853$	−36.0000	−1.23262	−0.616308	+	0.787505i	$0.711372\pi$
$854$	−6.00000	−0.205316
$855$	−56.0000	−1.91516
$856$	−12.0000	−0.410152
$857$	−5.00000	−0.170797	−0.0853984	−	0.996347i	$-0.527216\pi$
$857$	−5.00000	−0.170797	−0.0853984	+	0.996347i	$0.527216\pi$
$858$	0	0
$859$	−11.0000	−0.375315	−0.187658	−	0.982235i	$-0.560090\pi$
$859$	−11.0000	−0.375315	−0.187658	+	0.982235i	$0.560090\pi$
$860$	12.0000	0.409197
$861$	−2.00000	−0.0681598
$862$	−12.0000	−0.408722
$863$	36.0000	1.22545	0.612727	−	0.790295i	$-0.290072\pi$
$863$	36.0000	1.22545	0.612727	+	0.790295i	$0.290072\pi$
$864$	5.00000	0.170103
$865$	24.0000	0.816024
$866$	10.0000	0.339814
$867$	−8.00000	−0.271694
$868$	2.00000	0.0678844
$869$	0	0
$870$	32.0000	1.08490
$871$	10.0000	0.338837
$872$	−4.00000	−0.135457
$873$	28.0000	0.947656
$874$	0	0
$875$	24.0000	0.811348
$876$	−17.0000	−0.574377
$877$	−34.0000	−1.14810	−0.574049	−	0.818821i	$-0.694628\pi$
$877$	−34.0000	−1.14810	−0.574049	+	0.818821i	$0.694628\pi$
$878$	32.0000	1.07995
$879$	−14.0000	−0.472208
$880$	0	0
$881$	−30.0000	−1.01073	−0.505363	−	0.862907i	$-0.668641\pi$
$881$	−30.0000	−1.01073	−0.505363	+	0.862907i	$0.668641\pi$
$882$	−2.00000	−0.0673435
$883$	11.0000	0.370179	0.185090	−	0.982722i	$-0.440742\pi$
$883$	11.0000	0.370179	0.185090	+	0.982722i	$0.440742\pi$
$884$	10.0000	0.336336
$885$	−36.0000	−1.21013
$886$	7.00000	0.235170
$887$	0	0		−	1.00000i	$-0.5\pi$
$887$	0	0			1.00000i	$0.5\pi$
$888$	−8.00000	−0.268462
$889$	22.0000	0.737856
$890$	−24.0000	−0.804482
$891$	0	0
$892$	−2.00000	−0.0669650
$893$	0	0
$894$	−10.0000	−0.334450
$895$	36.0000	1.20335
$896$	−1.00000	−0.0334077
$897$	0	0
$898$	9.00000	0.300334
$899$	−16.0000	−0.533630
$900$	−22.0000	−0.733333
$901$	−30.0000	−0.999445
$902$	0	0
$903$	−3.00000	−0.0998337
$904$	17.0000	0.565412
$905$	40.0000	1.32964
$906$	−16.0000	−0.531564
$907$	23.0000	0.763702	0.381851	−	0.924224i	$-0.375287\pi$
$907$	23.0000	0.763702	0.381851	+	0.924224i	$0.375287\pi$
$908$	3.00000	0.0995585
$909$	−28.0000	−0.928701
$910$	8.00000	0.265197
$911$	−28.0000	−0.927681	−0.463841	−	0.885919i	$-0.653529\pi$
$911$	−28.0000	−0.927681	−0.463841	+	0.885919i	$0.653529\pi$
$912$	7.00000	0.231793
$913$	0	0
$914$	−10.0000	−0.330771
$915$	24.0000	0.793416
$916$	−4.00000	−0.132164
$917$	−1.00000	−0.0330229
$918$	25.0000	0.825123
$919$	4.00000	0.131948	0.0659739	−	0.997821i	$-0.478985\pi$
$919$	4.00000	0.131948	0.0659739	+	0.997821i	$0.478985\pi$
$920$	0	0
$921$	23.0000	0.757876
$922$	−10.0000	−0.329332
$923$	−4.00000	−0.131662
$924$	0	0
$925$	88.0000	2.89342
$926$	4.00000	0.131448
$927$	32.0000	1.05102
$928$	8.00000	0.262613
$929$	50.0000	1.64045	0.820223	−	0.572043i	$-0.193849\pi$
$929$	50.0000	1.64045	0.820223	+	0.572043i	$0.193849\pi$
$930$	−8.00000	−0.262330
$931$	−7.00000	−0.229416
$932$	3.00000	0.0982683
$933$	8.00000	0.261908
$934$	−16.0000	−0.523536
$935$	0	0
$936$	−4.00000	−0.130744
$937$	19.0000	0.620703	0.310351	−	0.950622i	$-0.399553\pi$
$937$	19.0000	0.620703	0.310351	+	0.950622i	$0.399553\pi$
$938$	−5.00000	−0.163256
$939$	11.0000	0.358971
$940$	0	0
$941$	−6.00000	−0.195594	−0.0977972	−	0.995206i	$-0.531180\pi$
$941$	−6.00000	−0.195594	−0.0977972	+	0.995206i	$0.531180\pi$
$942$	12.0000	0.390981
$943$	0	0
$944$	−9.00000	−0.292925
$945$	20.0000	0.650600
$946$	0	0
$947$	27.0000	0.877382	0.438691	−	0.898638i	$-0.355442\pi$
$947$	27.0000	0.877382	0.438691	+	0.898638i	$0.355442\pi$
$948$	0	0
$949$	34.0000	1.10369
$950$	−77.0000	−2.49821
$951$	−2.00000	−0.0648544
$952$	−5.00000	−0.162051
$953$	14.0000	0.453504	0.226752	−	0.973952i	$-0.427189\pi$
$953$	14.0000	0.453504	0.226752	+	0.973952i	$0.427189\pi$
$954$	12.0000	0.388514
$955$	40.0000	1.29437
$956$	30.0000	0.970269
$957$	0	0
$958$	20.0000	0.646171
$959$	−3.00000	−0.0968751
$960$	4.00000	0.129099
$961$	−27.0000	−0.870968
$962$	16.0000	0.515861
$963$	24.0000	0.773389
$964$	21.0000	0.676364
$965$	44.0000	1.41641
$966$	0	0
$967$	56.0000	1.80084	0.900419	−	0.435023i	$-0.143260\pi$
$967$	56.0000	1.80084	0.900419	+	0.435023i	$0.143260\pi$
$968$	−11.0000	−0.353553
$969$	35.0000	1.12436
$970$	56.0000	1.79805
$971$	−1.00000	−0.0320915	−0.0160458	−	0.999871i	$-0.505108\pi$
$971$	−1.00000	−0.0320915	−0.0160458	+	0.999871i	$0.505108\pi$
$972$	−16.0000	−0.513200
$973$	7.00000	0.224410
$974$	36.0000	1.15351
$975$	−22.0000	−0.704564
$976$	6.00000	0.192055
$977$	30.0000	0.959785	0.479893	−	0.877327i	$-0.340676\pi$
$977$	30.0000	0.959785	0.479893	+	0.877327i	$0.340676\pi$
$978$	−20.0000	−0.639529
$979$	0	0
$980$	−4.00000	−0.127775
$981$	8.00000	0.255420
$982$	4.00000	0.127645
$983$	−22.0000	−0.701691	−0.350846	−	0.936433i	$-0.614106\pi$
$983$	−22.0000	−0.701691	−0.350846	+	0.936433i	$0.614106\pi$
$984$	2.00000	0.0637577
$985$	−32.0000	−1.01960
$986$	40.0000	1.27386
$987$	0	0
$988$	−14.0000	−0.445399
$989$	0	0
$990$	0	0
$991$	−30.0000	−0.952981	−0.476491	−	0.879180i	$-0.658091\pi$
$991$	−30.0000	−0.952981	−0.476491	+	0.879180i	$0.658091\pi$
$992$	−2.00000	−0.0635001
$993$	−1.00000	−0.0317340
$994$	2.00000	0.0634361
$995$	32.0000	1.01447
$996$	13.0000	0.411921
$997$	−54.0000	−1.71020	−0.855099	−	0.518465i	$-0.826503\pi$
$997$	−54.0000	−1.71020	−0.855099	+	0.518465i	$0.826503\pi$
$998$	15.0000	0.474817
$999$	40.0000	1.26554

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7406.2.a.g.1.1	✓	1
23.22	odd	2		7406.2.a.h.1.1	yes	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7406.2.a.g.1.1	✓	1	1.1	even	1	trivial
7406.2.a.h.1.1	yes	1	23.22	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:	\( N \)	\(=\)	\( 7406 = 2 \cdot 7 \cdot 23^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7406.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more