LMFDB - Embedded newform 8015.2.a.e.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8015,2,Mod(1,8015)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8015, 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("8015.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8015 = 5 \cdot 7 \cdot 229 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8015.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:	$64.0000972201$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	8015.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{2} +2.00000 q^{3} -1.00000 q^{4} -1.00000 q^{5} +2.00000 q^{6} -1.00000 q^{7} -3.00000 q^{8} +1.00000 q^{9} +O(q^{10})$

\(q+1.00000 q^{2} +2.00000 q^{3} -1.00000 q^{4} -1.00000 q^{5} +2.00000 q^{6} -1.00000 q^{7} -3.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} -1.00000 q^{11} -2.00000 q^{12} -5.00000 q^{13} -1.00000 q^{14} -2.00000 q^{15} -1.00000 q^{16} +2.00000 q^{17} +1.00000 q^{18} +4.00000 q^{19} +1.00000 q^{20} -2.00000 q^{21} -1.00000 q^{22} -4.00000 q^{23} -6.00000 q^{24} +1.00000 q^{25} -5.00000 q^{26} -4.00000 q^{27} +1.00000 q^{28} -4.00000 q^{29} -2.00000 q^{30} +9.00000 q^{31} +5.00000 q^{32} -2.00000 q^{33} +2.00000 q^{34} +1.00000 q^{35} -1.00000 q^{36} +7.00000 q^{37} +4.00000 q^{38} -10.0000 q^{39} +3.00000 q^{40} +1.00000 q^{41} -2.00000 q^{42} -3.00000 q^{43} +1.00000 q^{44} -1.00000 q^{45} -4.00000 q^{46} +3.00000 q^{47} -2.00000 q^{48} +1.00000 q^{49} +1.00000 q^{50} +4.00000 q^{51} +5.00000 q^{52} -5.00000 q^{53} -4.00000 q^{54} +1.00000 q^{55} +3.00000 q^{56} +8.00000 q^{57} -4.00000 q^{58} +4.00000 q^{59} +2.00000 q^{60} +2.00000 q^{61} +9.00000 q^{62} -1.00000 q^{63} +7.00000 q^{64} +5.00000 q^{65} -2.00000 q^{66} +12.0000 q^{67} -2.00000 q^{68} -8.00000 q^{69} +1.00000 q^{70} +9.00000 q^{71} -3.00000 q^{72} -14.0000 q^{73} +7.00000 q^{74} +2.00000 q^{75} -4.00000 q^{76} +1.00000 q^{77} -10.0000 q^{78} +12.0000 q^{79} +1.00000 q^{80} -11.0000 q^{81} +1.00000 q^{82} +6.00000 q^{83} +2.00000 q^{84} -2.00000 q^{85} -3.00000 q^{86} -8.00000 q^{87} +3.00000 q^{88} +9.00000 q^{89} -1.00000 q^{90} +5.00000 q^{91} +4.00000 q^{92} +18.0000 q^{93} +3.00000 q^{94} -4.00000 q^{95} +10.0000 q^{96} -10.0000 q^{97} +1.00000 q^{98} -1.00000 q^{99} +O(q^{100})\)

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	$0.384973\pi$
$3$	2.00000	1.15470	0.577350	−	0.816497i	$-0.304087\pi$
$3$	2.00000	1.15470	0.577350	+	0.816497i	$0.304087\pi$
$4$	−1.00000	−0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	2.00000	0.816497
$7$	−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	−0.150756	−	0.988571i	$-0.548171\pi$
$11$	−1.00000	−0.301511	−0.150756	+	0.988571i	$0.548171\pi$
$12$	−2.00000	−0.577350
$13$	−5.00000	−1.38675	−0.693375	−	0.720577i	$-0.743877\pi$
$13$	−5.00000	−1.38675	−0.693375	+	0.720577i	$0.743877\pi$
$14$	−1.00000	−0.267261
$15$	−2.00000	−0.516398
$16$	−1.00000	−0.250000
$17$	2.00000	0.485071	0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000	0.485071	0.242536	+	0.970143i	$0.422021\pi$
$18$	1.00000	0.235702
$19$	4.00000	0.917663	0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000	0.917663	0.458831	+	0.888523i	$0.348268\pi$
$20$	1.00000	0.223607
$21$	−2.00000	−0.436436
$22$	−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	$0.636929\pi$
$24$	−6.00000	−1.22474
$25$	1.00000	0.200000
$26$	−5.00000	−0.980581
$27$	−4.00000	−0.769800
$28$	1.00000	0.188982
$29$	−4.00000	−0.742781	−0.371391	−	0.928477i	$-0.621119\pi$
$29$	−4.00000	−0.742781	−0.371391	+	0.928477i	$0.621119\pi$
$30$	−2.00000	−0.365148
$31$	9.00000	1.61645	0.808224	−	0.588875i	$-0.200429\pi$
$31$	9.00000	1.61645	0.808224	+	0.588875i	$0.200429\pi$
$32$	5.00000	0.883883
$33$	−2.00000	−0.348155
$34$	2.00000	0.342997
$35$	1.00000	0.169031
$36$	−1.00000	−0.166667
$37$	7.00000	1.15079	0.575396	−	0.817875i	$-0.304848\pi$
$37$	7.00000	1.15079	0.575396	+	0.817875i	$0.304848\pi$
$38$	4.00000	0.648886
$39$	−10.0000	−1.60128
$40$	3.00000	0.474342
$41$	1.00000	0.156174	0.0780869	−	0.996947i	$-0.475119\pi$
$41$	1.00000	0.156174	0.0780869	+	0.996947i	$0.475119\pi$
$42$	−2.00000	−0.308607
$43$	−3.00000	−0.457496	−0.228748	−	0.973486i	$-0.573463\pi$
$43$	−3.00000	−0.457496	−0.228748	+	0.973486i	$0.573463\pi$
$44$	1.00000	0.150756
$45$	−1.00000	−0.149071
$46$	−4.00000	−0.589768
$47$	3.00000	0.437595	0.218797	−	0.975770i	$-0.429787\pi$
$47$	3.00000	0.437595	0.218797	+	0.975770i	$0.429787\pi$
$48$	−2.00000	−0.288675
$49$	1.00000	0.142857
$50$	1.00000	0.141421
$51$	4.00000	0.560112
$52$	5.00000	0.693375
$53$	−5.00000	−0.686803	−0.343401	−	0.939189i	$-0.611579\pi$
$53$	−5.00000	−0.686803	−0.343401	+	0.939189i	$0.611579\pi$
$54$	−4.00000	−0.544331
$55$	1.00000	0.134840
$56$	3.00000	0.400892
$57$	8.00000	1.05963
$58$	−4.00000	−0.525226
$59$	4.00000	0.520756	0.260378	−	0.965507i	$-0.416153\pi$
$59$	4.00000	0.520756	0.260378	+	0.965507i	$0.416153\pi$
$60$	2.00000	0.258199
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	9.00000	1.14300
$63$	−1.00000	−0.125988
$64$	7.00000	0.875000
$65$	5.00000	0.620174
$66$	−2.00000	−0.246183
$67$	12.0000	1.46603	0.733017	−	0.680211i	$-0.238112\pi$
$67$	12.0000	1.46603	0.733017	+	0.680211i	$0.238112\pi$
$68$	−2.00000	−0.242536
$69$	−8.00000	−0.963087
$70$	1.00000	0.119523
$71$	9.00000	1.06810	0.534052	−	0.845452i	$-0.320669\pi$
$71$	9.00000	1.06810	0.534052	+	0.845452i	$0.320669\pi$
$72$	−3.00000	−0.353553
$73$	−14.0000	−1.63858	−0.819288	−	0.573382i	$-0.805631\pi$
$73$	−14.0000	−1.63858	−0.819288	+	0.573382i	$0.805631\pi$
$74$	7.00000	0.813733
$75$	2.00000	0.230940
$76$	−4.00000	−0.458831
$77$	1.00000	0.113961
$78$	−10.0000	−1.13228
$79$	12.0000	1.35011	0.675053	−	0.737769i	$-0.264121\pi$
$79$	12.0000	1.35011	0.675053	+	0.737769i	$0.264121\pi$
$80$	1.00000	0.111803
$81$	−11.0000	−1.22222
$82$	1.00000	0.110432
$83$	6.00000	0.658586	0.329293	−	0.944228i	$-0.393190\pi$
$83$	6.00000	0.658586	0.329293	+	0.944228i	$0.393190\pi$
$84$	2.00000	0.218218
$85$	−2.00000	−0.216930
$86$	−3.00000	−0.323498
$87$	−8.00000	−0.857690
$88$	3.00000	0.319801
$89$	9.00000	0.953998	0.476999	−	0.878904i	$-0.341725\pi$
$89$	9.00000	0.953998	0.476999	+	0.878904i	$0.341725\pi$
$90$	−1.00000	−0.105409
$91$	5.00000	0.524142
$92$	4.00000	0.417029
$93$	18.0000	1.86651
$94$	3.00000	0.309426
$95$	−4.00000	−0.410391
$96$	10.0000	1.02062
$97$	−10.0000	−1.01535	−0.507673	−	0.861550i	$-0.669494\pi$
$97$	−10.0000	−1.01535	−0.507673	+	0.861550i	$0.669494\pi$
$98$	1.00000	0.101015
$99$	−1.00000	−0.100504
$100$	−1.00000	−0.100000
$101$	−14.0000	−1.39305	−0.696526	−	0.717532i	$-0.745272\pi$
$101$	−14.0000	−1.39305	−0.696526	+	0.717532i	$0.745272\pi$
$102$	4.00000	0.396059
$103$	12.0000	1.18240	0.591198	−	0.806527i	$-0.298655\pi$
$103$	12.0000	1.18240	0.591198	+	0.806527i	$0.298655\pi$
$104$	15.0000	1.47087
$105$	2.00000	0.195180
$106$	−5.00000	−0.485643
$107$	12.0000	1.16008	0.580042	−	0.814587i	$-0.303036\pi$
$107$	12.0000	1.16008	0.580042	+	0.814587i	$0.303036\pi$
$108$	4.00000	0.384900
$109$	−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
$111$	14.0000	1.32882
$112$	1.00000	0.0944911
$113$	14.0000	1.31701	0.658505	−	0.752577i	$-0.271189\pi$
$113$	14.0000	1.31701	0.658505	+	0.752577i	$0.271189\pi$
$114$	8.00000	0.749269
$115$	4.00000	0.373002
$116$	4.00000	0.371391
$117$	−5.00000	−0.462250
$118$	4.00000	0.368230
$119$	−2.00000	−0.183340
$120$	6.00000	0.547723
$121$	−10.0000	−0.909091
$122$	2.00000	0.181071
$123$	2.00000	0.180334
$124$	−9.00000	−0.808224
$125$	−1.00000	−0.0894427
$126$	−1.00000	−0.0890871
$127$	20.0000	1.77471	0.887357	−	0.461084i	$-0.152539\pi$
$127$	20.0000	1.77471	0.887357	+	0.461084i	$0.152539\pi$
$128$	−3.00000	−0.265165
$129$	−6.00000	−0.528271
$130$	5.00000	0.438529
$131$	−9.00000	−0.786334	−0.393167	−	0.919467i	$-0.628621\pi$
$131$	−9.00000	−0.786334	−0.393167	+	0.919467i	$0.628621\pi$
$132$	2.00000	0.174078
$133$	−4.00000	−0.346844
$134$	12.0000	1.03664
$135$	4.00000	0.344265
$136$	−6.00000	−0.514496
$137$	−12.0000	−1.02523	−0.512615	−	0.858619i	$-0.671323\pi$
$137$	−12.0000	−1.02523	−0.512615	+	0.858619i	$0.671323\pi$
$138$	−8.00000	−0.681005
$139$	19.0000	1.61156	0.805779	−	0.592216i	$-0.201747\pi$
$139$	19.0000	1.61156	0.805779	+	0.592216i	$0.201747\pi$
$140$	−1.00000	−0.0845154
$141$	6.00000	0.505291
$142$	9.00000	0.755263
$143$	5.00000	0.418121
$144$	−1.00000	−0.0833333
$145$	4.00000	0.332182
$146$	−14.0000	−1.15865
$147$	2.00000	0.164957
$148$	−7.00000	−0.575396
$149$	7.00000	0.573462	0.286731	−	0.958011i	$-0.407431\pi$
$149$	7.00000	0.573462	0.286731	+	0.958011i	$0.407431\pi$
$150$	2.00000	0.163299
$151$	−9.00000	−0.732410	−0.366205	−	0.930534i	$-0.619343\pi$
$151$	−9.00000	−0.732410	−0.366205	+	0.930534i	$0.619343\pi$
$152$	−12.0000	−0.973329
$153$	2.00000	0.161690
$154$	1.00000	0.0805823
$155$	−9.00000	−0.722897
$156$	10.0000	0.800641
$157$	−1.00000	−0.0798087	−0.0399043	−	0.999204i	$-0.512705\pi$
$157$	−1.00000	−0.0798087	−0.0399043	+	0.999204i	$0.512705\pi$
$158$	12.0000	0.954669
$159$	−10.0000	−0.793052
$160$	−5.00000	−0.395285
$161$	4.00000	0.315244
$162$	−11.0000	−0.864242
$163$	24.0000	1.87983	0.939913	−	0.341415i	$-0.110906\pi$
$163$	24.0000	1.87983	0.939913	+	0.341415i	$0.110906\pi$
$164$	−1.00000	−0.0780869
$165$	2.00000	0.155700
$166$	6.00000	0.465690
$167$	−12.0000	−0.928588	−0.464294	−	0.885681i	$-0.653692\pi$
$167$	−12.0000	−0.928588	−0.464294	+	0.885681i	$0.653692\pi$
$168$	6.00000	0.462910
$169$	12.0000	0.923077
$170$	−2.00000	−0.153393
$171$	4.00000	0.305888
$172$	3.00000	0.228748
$173$	12.0000	0.912343	0.456172	−	0.889892i	$-0.349220\pi$
$173$	12.0000	0.912343	0.456172	+	0.889892i	$0.349220\pi$
$174$	−8.00000	−0.606478
$175$	−1.00000	−0.0755929
$176$	1.00000	0.0753778
$177$	8.00000	0.601317
$178$	9.00000	0.674579
$179$	−26.0000	−1.94333	−0.971666	−	0.236360i	$-0.924046\pi$
$179$	−26.0000	−1.94333	−0.971666	+	0.236360i	$0.924046\pi$
$180$	1.00000	0.0745356
$181$	12.0000	0.891953	0.445976	−	0.895045i	$-0.352856\pi$
$181$	12.0000	0.891953	0.445976	+	0.895045i	$0.352856\pi$
$182$	5.00000	0.370625
$183$	4.00000	0.295689
$184$	12.0000	0.884652
$185$	−7.00000	−0.514650
$186$	18.0000	1.31982
$187$	−2.00000	−0.146254
$188$	−3.00000	−0.218797
$189$	4.00000	0.290957
$190$	−4.00000	−0.290191
$191$	−14.0000	−1.01300	−0.506502	−	0.862239i	$-0.669062\pi$
$191$	−14.0000	−1.01300	−0.506502	+	0.862239i	$0.669062\pi$
$192$	14.0000	1.01036
$193$	−3.00000	−0.215945	−0.107972	−	0.994154i	$-0.534436\pi$
$193$	−3.00000	−0.215945	−0.107972	+	0.994154i	$0.534436\pi$
$194$	−10.0000	−0.717958
$195$	10.0000	0.716115
$196$	−1.00000	−0.0714286
$197$	−6.00000	−0.427482	−0.213741	−	0.976890i	$-0.568565\pi$
$197$	−6.00000	−0.427482	−0.213741	+	0.976890i	$0.568565\pi$
$198$	−1.00000	−0.0710669
$199$	12.0000	0.850657	0.425329	−	0.905039i	$-0.360158\pi$
$199$	12.0000	0.850657	0.425329	+	0.905039i	$0.360158\pi$
$200$	−3.00000	−0.212132
$201$	24.0000	1.69283
$202$	−14.0000	−0.985037
$203$	4.00000	0.280745
$204$	−4.00000	−0.280056
$205$	−1.00000	−0.0698430
$206$	12.0000	0.836080
$207$	−4.00000	−0.278019
$208$	5.00000	0.346688
$209$	−4.00000	−0.276686
$210$	2.00000	0.138013
$211$	8.00000	0.550743	0.275371	−	0.961338i	$-0.411199\pi$
$211$	8.00000	0.550743	0.275371	+	0.961338i	$0.411199\pi$
$212$	5.00000	0.343401
$213$	18.0000	1.23334
$214$	12.0000	0.820303
$215$	3.00000	0.204598
$216$	12.0000	0.816497
$217$	−9.00000	−0.610960
$218$	−2.00000	−0.135457
$219$	−28.0000	−1.89206
$220$	−1.00000	−0.0674200
$221$	−10.0000	−0.672673
$222$	14.0000	0.939618
$223$	11.0000	0.736614	0.368307	−	0.929704i	$-0.379937\pi$
$223$	11.0000	0.736614	0.368307	+	0.929704i	$0.379937\pi$
$224$	−5.00000	−0.334077
$225$	1.00000	0.0666667
$226$	14.0000	0.931266
$227$	7.00000	0.464606	0.232303	−	0.972643i	$-0.425374\pi$
$227$	7.00000	0.464606	0.232303	+	0.972643i	$0.425374\pi$
$228$	−8.00000	−0.529813
$229$	1.00000	0.0660819
$229$	1.00000	0.0660819
$230$	4.00000	0.263752
$231$	2.00000	0.131590
$232$	12.0000	0.787839
$233$	23.0000	1.50678	0.753390	−	0.657574i	$-0.228417\pi$
$233$	23.0000	1.50678	0.753390	+	0.657574i	$0.228417\pi$
$234$	−5.00000	−0.326860
$235$	−3.00000	−0.195698
$236$	−4.00000	−0.260378
$237$	24.0000	1.55897
$238$	−2.00000	−0.129641
$239$	6.00000	0.388108	0.194054	−	0.980991i	$-0.437836\pi$
$239$	6.00000	0.388108	0.194054	+	0.980991i	$0.437836\pi$
$240$	2.00000	0.129099
$241$	20.0000	1.28831	0.644157	−	0.764894i	$-0.277208\pi$
$241$	20.0000	1.28831	0.644157	+	0.764894i	$0.277208\pi$
$242$	−10.0000	−0.642824
$243$	−10.0000	−0.641500
$244$	−2.00000	−0.128037
$245$	−1.00000	−0.0638877
$246$	2.00000	0.127515
$247$	−20.0000	−1.27257
$248$	−27.0000	−1.71450
$249$	12.0000	0.760469
$250$	−1.00000	−0.0632456
$251$	−3.00000	−0.189358	−0.0946792	−	0.995508i	$-0.530183\pi$
$251$	−3.00000	−0.189358	−0.0946792	+	0.995508i	$0.530183\pi$
$252$	1.00000	0.0629941
$253$	4.00000	0.251478
$254$	20.0000	1.25491
$255$	−4.00000	−0.250490
$256$	−17.0000	−1.06250
$257$	18.0000	1.12281	0.561405	−	0.827541i	$-0.310261\pi$
$257$	18.0000	1.12281	0.561405	+	0.827541i	$0.310261\pi$
$258$	−6.00000	−0.373544
$259$	−7.00000	−0.434959
$260$	−5.00000	−0.310087
$261$	−4.00000	−0.247594
$262$	−9.00000	−0.556022
$263$	6.00000	0.369976	0.184988	−	0.982741i	$-0.440775\pi$
$263$	6.00000	0.369976	0.184988	+	0.982741i	$0.440775\pi$
$264$	6.00000	0.369274
$265$	5.00000	0.307148
$266$	−4.00000	−0.245256
$267$	18.0000	1.10158
$268$	−12.0000	−0.733017
$269$	−6.00000	−0.365826	−0.182913	−	0.983129i	$-0.558553\pi$
$269$	−6.00000	−0.365826	−0.182913	+	0.983129i	$0.558553\pi$
$270$	4.00000	0.243432
$271$	−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
$273$	10.0000	0.605228
$274$	−12.0000	−0.724947
$275$	−1.00000	−0.0603023
$276$	8.00000	0.481543
$277$	19.0000	1.14160	0.570800	−	0.821089i	$-0.306633\pi$
$277$	19.0000	1.14160	0.570800	+	0.821089i	$0.306633\pi$
$278$	19.0000	1.13954
$279$	9.00000	0.538816
$280$	−3.00000	−0.179284
$281$	−20.0000	−1.19310	−0.596550	−	0.802576i	$-0.703462\pi$
$281$	−20.0000	−1.19310	−0.596550	+	0.802576i	$0.703462\pi$
$282$	6.00000	0.357295
$283$	27.0000	1.60498	0.802492	−	0.596663i	$-0.203507\pi$
$283$	27.0000	1.60498	0.802492	+	0.596663i	$0.203507\pi$
$284$	−9.00000	−0.534052
$285$	−8.00000	−0.473879
$286$	5.00000	0.295656
$287$	−1.00000	−0.0590281
$288$	5.00000	0.294628
$289$	−13.0000	−0.764706
$290$	4.00000	0.234888
$291$	−20.0000	−1.17242
$292$	14.0000	0.819288
$293$	−22.0000	−1.28525	−0.642627	−	0.766179i	$-0.722155\pi$
$293$	−22.0000	−1.28525	−0.642627	+	0.766179i	$0.722155\pi$
$294$	2.00000	0.116642
$295$	−4.00000	−0.232889
$296$	−21.0000	−1.22060
$297$	4.00000	0.232104
$298$	7.00000	0.405499
$299$	20.0000	1.15663
$300$	−2.00000	−0.115470
$301$	3.00000	0.172917
$302$	−9.00000	−0.517892
$303$	−28.0000	−1.60856
$304$	−4.00000	−0.229416
$305$	−2.00000	−0.114520
$306$	2.00000	0.114332
$307$	−2.00000	−0.114146	−0.0570730	−	0.998370i	$-0.518177\pi$
$307$	−2.00000	−0.114146	−0.0570730	+	0.998370i	$0.518177\pi$
$308$	−1.00000	−0.0569803
$309$	24.0000	1.36531
$310$	−9.00000	−0.511166
$311$	−20.0000	−1.13410	−0.567048	−	0.823685i	$-0.691915\pi$
$311$	−20.0000	−1.13410	−0.567048	+	0.823685i	$0.691915\pi$
$312$	30.0000	1.69842
$313$	1.00000	0.0565233	0.0282617	−	0.999601i	$-0.491003\pi$
$313$	1.00000	0.0565233	0.0282617	+	0.999601i	$0.491003\pi$
$314$	−1.00000	−0.0564333
$315$	1.00000	0.0563436
$316$	−12.0000	−0.675053
$317$	4.00000	0.224662	0.112331	−	0.993671i	$-0.464168\pi$
$317$	4.00000	0.224662	0.112331	+	0.993671i	$0.464168\pi$
$318$	−10.0000	−0.560772
$319$	4.00000	0.223957
$320$	−7.00000	−0.391312
$321$	24.0000	1.33955
$322$	4.00000	0.222911
$323$	8.00000	0.445132
$324$	11.0000	0.611111
$325$	−5.00000	−0.277350
$326$	24.0000	1.32924
$327$	−4.00000	−0.221201
$328$	−3.00000	−0.165647
$329$	−3.00000	−0.165395
$330$	2.00000	0.110096
$331$	−32.0000	−1.75888	−0.879440	−	0.476011i	$-0.842082\pi$
$331$	−32.0000	−1.75888	−0.879440	+	0.476011i	$0.842082\pi$
$332$	−6.00000	−0.329293
$333$	7.00000	0.383598
$334$	−12.0000	−0.656611
$335$	−12.0000	−0.655630
$336$	2.00000	0.109109
$337$	1.00000	0.0544735	0.0272367	−	0.999629i	$-0.491329\pi$
$337$	1.00000	0.0544735	0.0272367	+	0.999629i	$0.491329\pi$
$338$	12.0000	0.652714
$339$	28.0000	1.52075
$340$	2.00000	0.108465
$341$	−9.00000	−0.487377
$342$	4.00000	0.216295
$343$	−1.00000	−0.0539949
$344$	9.00000	0.485247
$345$	8.00000	0.430706
$346$	12.0000	0.645124
$347$	33.0000	1.77153	0.885766	−	0.464131i	$-0.153633\pi$
$347$	33.0000	1.77153	0.885766	+	0.464131i	$0.153633\pi$
$348$	8.00000	0.428845
$349$	−33.0000	−1.76645	−0.883225	−	0.468950i	$-0.844632\pi$
$349$	−33.0000	−1.76645	−0.883225	+	0.468950i	$0.844632\pi$
$350$	−1.00000	−0.0534522
$351$	20.0000	1.06752
$352$	−5.00000	−0.266501
$353$	2.00000	0.106449	0.0532246	−	0.998583i	$-0.483050\pi$
$353$	2.00000	0.106449	0.0532246	+	0.998583i	$0.483050\pi$
$354$	8.00000	0.425195
$355$	−9.00000	−0.477670
$356$	−9.00000	−0.476999
$357$	−4.00000	−0.211702
$358$	−26.0000	−1.37414
$359$	21.0000	1.10834	0.554169	−	0.832404i	$-0.313036\pi$
$359$	21.0000	1.10834	0.554169	+	0.832404i	$0.313036\pi$
$360$	3.00000	0.158114
$361$	−3.00000	−0.157895
$362$	12.0000	0.630706
$363$	−20.0000	−1.04973
$364$	−5.00000	−0.262071
$365$	14.0000	0.732793
$366$	4.00000	0.209083
$367$	10.0000	0.521996	0.260998	−	0.965339i	$-0.415948\pi$
$367$	10.0000	0.521996	0.260998	+	0.965339i	$0.415948\pi$
$368$	4.00000	0.208514
$369$	1.00000	0.0520579
$370$	−7.00000	−0.363913
$371$	5.00000	0.259587
$372$	−18.0000	−0.933257
$373$	35.0000	1.81223	0.906116	−	0.423030i	$-0.139034\pi$
$373$	35.0000	1.81223	0.906116	+	0.423030i	$0.139034\pi$
$374$	−2.00000	−0.103418
$375$	−2.00000	−0.103280
$376$	−9.00000	−0.464140
$377$	20.0000	1.03005
$378$	4.00000	0.205738
$379$	28.0000	1.43826	0.719132	−	0.694874i	$-0.244540\pi$
$379$	28.0000	1.43826	0.719132	+	0.694874i	$0.244540\pi$
$380$	4.00000	0.205196
$381$	40.0000	2.04926
$382$	−14.0000	−0.716302
$383$	2.00000	0.102195	0.0510976	−	0.998694i	$-0.483728\pi$
$383$	2.00000	0.102195	0.0510976	+	0.998694i	$0.483728\pi$
$384$	−6.00000	−0.306186
$385$	−1.00000	−0.0509647
$386$	−3.00000	−0.152696
$387$	−3.00000	−0.152499
$388$	10.0000	0.507673
$389$	−20.0000	−1.01404	−0.507020	−	0.861934i	$-0.669253\pi$
$389$	−20.0000	−1.01404	−0.507020	+	0.861934i	$0.669253\pi$
$390$	10.0000	0.506370
$391$	−8.00000	−0.404577
$392$	−3.00000	−0.151523
$393$	−18.0000	−0.907980
$394$	−6.00000	−0.302276
$395$	−12.0000	−0.603786
$396$	1.00000	0.0502519
$397$	−22.0000	−1.10415	−0.552074	−	0.833795i	$-0.686163\pi$
$397$	−22.0000	−1.10415	−0.552074	+	0.833795i	$0.686163\pi$
$398$	12.0000	0.601506
$399$	−8.00000	−0.400501
$400$	−1.00000	−0.0500000
$401$	38.0000	1.89763	0.948815	−	0.315833i	$-0.102284\pi$
$401$	38.0000	1.89763	0.948815	+	0.315833i	$0.102284\pi$
$402$	24.0000	1.19701
$403$	−45.0000	−2.24161
$404$	14.0000	0.696526
$405$	11.0000	0.546594
$406$	4.00000	0.198517
$407$	−7.00000	−0.346977
$408$	−12.0000	−0.594089
$409$	−4.00000	−0.197787	−0.0988936	−	0.995098i	$-0.531530\pi$
$409$	−4.00000	−0.197787	−0.0988936	+	0.995098i	$0.531530\pi$
$410$	−1.00000	−0.0493865
$411$	−24.0000	−1.18383
$412$	−12.0000	−0.591198
$413$	−4.00000	−0.196827
$414$	−4.00000	−0.196589
$415$	−6.00000	−0.294528
$416$	−25.0000	−1.22573
$417$	38.0000	1.86087
$418$	−4.00000	−0.195646
$419$	1.00000	0.0488532	0.0244266	−	0.999702i	$-0.492224\pi$
$419$	1.00000	0.0488532	0.0244266	+	0.999702i	$0.492224\pi$
$420$	−2.00000	−0.0975900
$421$	6.00000	0.292422	0.146211	−	0.989253i	$-0.453292\pi$
$421$	6.00000	0.292422	0.146211	+	0.989253i	$0.453292\pi$
$422$	8.00000	0.389434
$423$	3.00000	0.145865
$424$	15.0000	0.728464
$425$	2.00000	0.0970143
$426$	18.0000	0.872103
$427$	−2.00000	−0.0967868
$428$	−12.0000	−0.580042
$429$	10.0000	0.482805
$430$	3.00000	0.144673
$431$	−4.00000	−0.192673	−0.0963366	−	0.995349i	$-0.530713\pi$
$431$	−4.00000	−0.192673	−0.0963366	+	0.995349i	$0.530713\pi$
$432$	4.00000	0.192450
$433$	−18.0000	−0.865025	−0.432512	−	0.901628i	$-0.642373\pi$
$433$	−18.0000	−0.865025	−0.432512	+	0.901628i	$0.642373\pi$
$434$	−9.00000	−0.432014
$435$	8.00000	0.383571
$436$	2.00000	0.0957826
$437$	−16.0000	−0.765384
$438$	−28.0000	−1.33789
$439$	30.0000	1.43182	0.715911	−	0.698192i	$-0.246012\pi$
$439$	30.0000	1.43182	0.715911	+	0.698192i	$0.246012\pi$
$440$	−3.00000	−0.143019
$441$	1.00000	0.0476190
$442$	−10.0000	−0.475651
$443$	32.0000	1.52037	0.760183	−	0.649709i	$-0.225109\pi$
$443$	32.0000	1.52037	0.760183	+	0.649709i	$0.225109\pi$
$444$	−14.0000	−0.664411
$445$	−9.00000	−0.426641
$446$	11.0000	0.520865
$447$	14.0000	0.662177
$448$	−7.00000	−0.330719
$449$	−25.0000	−1.17982	−0.589911	−	0.807468i	$-0.700837\pi$
$449$	−25.0000	−1.17982	−0.589911	+	0.807468i	$0.700837\pi$
$450$	1.00000	0.0471405
$451$	−1.00000	−0.0470882
$452$	−14.0000	−0.658505
$453$	−18.0000	−0.845714
$454$	7.00000	0.328526
$455$	−5.00000	−0.234404
$456$	−24.0000	−1.12390
$457$	−34.0000	−1.59045	−0.795226	−	0.606313i	$-0.792648\pi$
$457$	−34.0000	−1.59045	−0.795226	+	0.606313i	$0.792648\pi$
$458$	1.00000	0.0467269
$459$	−8.00000	−0.373408
$460$	−4.00000	−0.186501
$461$	−6.00000	−0.279448	−0.139724	−	0.990190i	$-0.544622\pi$
$461$	−6.00000	−0.279448	−0.139724	+	0.990190i	$0.544622\pi$
$462$	2.00000	0.0930484
$463$	−23.0000	−1.06890	−0.534450	−	0.845200i	$-0.679481\pi$
$463$	−23.0000	−1.06890	−0.534450	+	0.845200i	$0.679481\pi$
$464$	4.00000	0.185695
$465$	−18.0000	−0.834730
$466$	23.0000	1.06545
$467$	−24.0000	−1.11059	−0.555294	−	0.831654i	$-0.687394\pi$
$467$	−24.0000	−1.11059	−0.555294	+	0.831654i	$0.687394\pi$
$468$	5.00000	0.231125
$469$	−12.0000	−0.554109
$470$	−3.00000	−0.138380
$471$	−2.00000	−0.0921551
$472$	−12.0000	−0.552345
$473$	3.00000	0.137940
$474$	24.0000	1.10236
$475$	4.00000	0.183533
$476$	2.00000	0.0916698
$477$	−5.00000	−0.228934
$478$	6.00000	0.274434
$479$	17.0000	0.776750	0.388375	−	0.921501i	$-0.373037\pi$
$479$	17.0000	0.776750	0.388375	+	0.921501i	$0.373037\pi$
$480$	−10.0000	−0.456435
$481$	−35.0000	−1.59586
$482$	20.0000	0.910975
$483$	8.00000	0.364013
$484$	10.0000	0.454545
$485$	10.0000	0.454077
$486$	−10.0000	−0.453609
$487$	20.0000	0.906287	0.453143	−	0.891438i	$-0.350303\pi$
$487$	20.0000	0.906287	0.453143	+	0.891438i	$0.350303\pi$
$488$	−6.00000	−0.271607
$489$	48.0000	2.17064
$490$	−1.00000	−0.0451754
$491$	−24.0000	−1.08310	−0.541552	−	0.840667i	$-0.682163\pi$
$491$	−24.0000	−1.08310	−0.541552	+	0.840667i	$0.682163\pi$
$492$	−2.00000	−0.0901670
$493$	−8.00000	−0.360302
$494$	−20.0000	−0.899843
$495$	1.00000	0.0449467
$496$	−9.00000	−0.404112
$497$	−9.00000	−0.403705
$498$	12.0000	0.537733
$499$	−10.0000	−0.447661	−0.223831	−	0.974628i	$-0.571856\pi$
$499$	−10.0000	−0.447661	−0.223831	+	0.974628i	$0.571856\pi$
$500$	1.00000	0.0447214
$501$	−24.0000	−1.07224
$502$	−3.00000	−0.133897
$503$	−36.0000	−1.60516	−0.802580	−	0.596544i	$-0.796540\pi$
$503$	−36.0000	−1.60516	−0.802580	+	0.596544i	$0.796540\pi$
$504$	3.00000	0.133631
$505$	14.0000	0.622992
$506$	4.00000	0.177822
$507$	24.0000	1.06588
$508$	−20.0000	−0.887357
$509$	−38.0000	−1.68432	−0.842160	−	0.539227i	$-0.818716\pi$
$509$	−38.0000	−1.68432	−0.842160	+	0.539227i	$0.818716\pi$
$510$	−4.00000	−0.177123
$511$	14.0000	0.619324
$512$	−11.0000	−0.486136
$513$	−16.0000	−0.706417
$514$	18.0000	0.793946
$515$	−12.0000	−0.528783
$516$	6.00000	0.264135
$517$	−3.00000	−0.131940
$518$	−7.00000	−0.307562
$519$	24.0000	1.05348
$520$	−15.0000	−0.657794
$521$	27.0000	1.18289	0.591446	−	0.806345i	$-0.298557\pi$
$521$	27.0000	1.18289	0.591446	+	0.806345i	$0.298557\pi$
$522$	−4.00000	−0.175075
$523$	21.0000	0.918266	0.459133	−	0.888368i	$-0.348160\pi$
$523$	21.0000	0.918266	0.459133	+	0.888368i	$0.348160\pi$
$524$	9.00000	0.393167
$525$	−2.00000	−0.0872872
$526$	6.00000	0.261612
$527$	18.0000	0.784092
$528$	2.00000	0.0870388
$529$	−7.00000	−0.304348
$530$	5.00000	0.217186
$531$	4.00000	0.173585
$532$	4.00000	0.173422
$533$	−5.00000	−0.216574
$534$	18.0000	0.778936
$535$	−12.0000	−0.518805
$536$	−36.0000	−1.55496
$537$	−52.0000	−2.24397
$538$	−6.00000	−0.258678
$539$	−1.00000	−0.0430730
$540$	−4.00000	−0.172133
$541$	−19.0000	−0.816874	−0.408437	−	0.912787i	$-0.633926\pi$
$541$	−19.0000	−0.816874	−0.408437	+	0.912787i	$0.633926\pi$
$542$	−32.0000	−1.37452
$543$	24.0000	1.02994
$544$	10.0000	0.428746
$545$	2.00000	0.0856706
$546$	10.0000	0.427960
$547$	2.00000	0.0855138	0.0427569	−	0.999086i	$-0.486386\pi$
$547$	2.00000	0.0855138	0.0427569	+	0.999086i	$0.486386\pi$
$548$	12.0000	0.512615
$549$	2.00000	0.0853579
$550$	−1.00000	−0.0426401
$551$	−16.0000	−0.681623
$552$	24.0000	1.02151
$553$	−12.0000	−0.510292
$554$	19.0000	0.807233
$555$	−14.0000	−0.594267
$556$	−19.0000	−0.805779
$557$	18.0000	0.762684	0.381342	−	0.924434i	$-0.375462\pi$
$557$	18.0000	0.762684	0.381342	+	0.924434i	$0.375462\pi$
$558$	9.00000	0.381000
$559$	15.0000	0.634432
$560$	−1.00000	−0.0422577
$561$	−4.00000	−0.168880
$562$	−20.0000	−0.843649
$563$	20.0000	0.842900	0.421450	−	0.906852i	$-0.361521\pi$
$563$	20.0000	0.842900	0.421450	+	0.906852i	$0.361521\pi$
$564$	−6.00000	−0.252646
$565$	−14.0000	−0.588984
$566$	27.0000	1.13489
$567$	11.0000	0.461957
$568$	−27.0000	−1.13289
$569$	11.0000	0.461144	0.230572	−	0.973055i	$-0.425940\pi$
$569$	11.0000	0.461144	0.230572	+	0.973055i	$0.425940\pi$
$570$	−8.00000	−0.335083
$571$	0	0		−	1.00000i	$-0.5\pi$
$571$	0	0			1.00000i	$0.5\pi$
$572$	−5.00000	−0.209061
$573$	−28.0000	−1.16972
$574$	−1.00000	−0.0417392
$575$	−4.00000	−0.166812
$576$	7.00000	0.291667
$577$	3.00000	0.124892	0.0624458	−	0.998048i	$-0.480110\pi$
$577$	3.00000	0.124892	0.0624458	+	0.998048i	$0.480110\pi$
$578$	−13.0000	−0.540729
$579$	−6.00000	−0.249351
$580$	−4.00000	−0.166091
$581$	−6.00000	−0.248922
$582$	−20.0000	−0.829027
$583$	5.00000	0.207079
$584$	42.0000	1.73797
$585$	5.00000	0.206725
$586$	−22.0000	−0.908812
$587$	12.0000	0.495293	0.247647	−	0.968850i	$-0.420343\pi$
$587$	12.0000	0.495293	0.247647	+	0.968850i	$0.420343\pi$
$588$	−2.00000	−0.0824786
$589$	36.0000	1.48335
$590$	−4.00000	−0.164677
$591$	−12.0000	−0.493614
$592$	−7.00000	−0.287698
$593$	0	0		−	1.00000i	$-0.5\pi$
$593$	0	0			1.00000i	$0.5\pi$
$594$	4.00000	0.164122
$595$	2.00000	0.0819920
$596$	−7.00000	−0.286731
$597$	24.0000	0.982255
$598$	20.0000	0.817861
$599$	46.0000	1.87951	0.939755	−	0.341850i	$-0.111053\pi$
$599$	46.0000	1.87951	0.939755	+	0.341850i	$0.111053\pi$
$600$	−6.00000	−0.244949
$601$	−10.0000	−0.407909	−0.203954	−	0.978980i	$-0.565379\pi$
$601$	−10.0000	−0.407909	−0.203954	+	0.978980i	$0.565379\pi$
$602$	3.00000	0.122271
$603$	12.0000	0.488678
$604$	9.00000	0.366205
$605$	10.0000	0.406558
$606$	−28.0000	−1.13742
$607$	−34.0000	−1.38002	−0.690009	−	0.723801i	$-0.742393\pi$
$607$	−34.0000	−1.38002	−0.690009	+	0.723801i	$0.742393\pi$
$608$	20.0000	0.811107
$609$	8.00000	0.324176
$610$	−2.00000	−0.0809776
$611$	−15.0000	−0.606835
$612$	−2.00000	−0.0808452
$613$	−6.00000	−0.242338	−0.121169	−	0.992632i	$-0.538664\pi$
$613$	−6.00000	−0.242338	−0.121169	+	0.992632i	$0.538664\pi$
$614$	−2.00000	−0.0807134
$615$	−2.00000	−0.0806478
$616$	−3.00000	−0.120873
$617$	41.0000	1.65060	0.825299	−	0.564696i	$-0.191007\pi$
$617$	41.0000	1.65060	0.825299	+	0.564696i	$0.191007\pi$
$618$	24.0000	0.965422
$619$	42.0000	1.68812	0.844061	−	0.536247i	$-0.180158\pi$
$619$	42.0000	1.68812	0.844061	+	0.536247i	$0.180158\pi$
$620$	9.00000	0.361449
$621$	16.0000	0.642058
$622$	−20.0000	−0.801927
$623$	−9.00000	−0.360577
$624$	10.0000	0.400320
$625$	1.00000	0.0400000
$626$	1.00000	0.0399680
$627$	−8.00000	−0.319489
$628$	1.00000	0.0399043
$629$	14.0000	0.558217
$630$	1.00000	0.0398410
$631$	7.00000	0.278666	0.139333	−	0.990246i	$-0.455504\pi$
$631$	7.00000	0.278666	0.139333	+	0.990246i	$0.455504\pi$
$632$	−36.0000	−1.43200
$633$	16.0000	0.635943
$634$	4.00000	0.158860
$635$	−20.0000	−0.793676
$636$	10.0000	0.396526
$637$	−5.00000	−0.198107
$638$	4.00000	0.158362
$639$	9.00000	0.356034
$640$	3.00000	0.118585
$641$	42.0000	1.65890	0.829450	−	0.558581i	$-0.188654\pi$
$641$	42.0000	1.65890	0.829450	+	0.558581i	$0.188654\pi$
$642$	24.0000	0.947204
$643$	42.0000	1.65632	0.828159	−	0.560493i	$-0.189388\pi$
$643$	42.0000	1.65632	0.828159	+	0.560493i	$0.189388\pi$
$644$	−4.00000	−0.157622
$645$	6.00000	0.236250
$646$	8.00000	0.314756
$647$	45.0000	1.76913	0.884566	−	0.466415i	$-0.154454\pi$
$647$	45.0000	1.76913	0.884566	+	0.466415i	$0.154454\pi$
$648$	33.0000	1.29636
$649$	−4.00000	−0.157014
$650$	−5.00000	−0.196116
$651$	−18.0000	−0.705476
$652$	−24.0000	−0.939913
$653$	6.00000	0.234798	0.117399	−	0.993085i	$-0.462544\pi$
$653$	6.00000	0.234798	0.117399	+	0.993085i	$0.462544\pi$
$654$	−4.00000	−0.156412
$655$	9.00000	0.351659
$656$	−1.00000	−0.0390434
$657$	−14.0000	−0.546192
$658$	−3.00000	−0.116952
$659$	−24.0000	−0.934907	−0.467454	−	0.884018i	$-0.654829\pi$
$659$	−24.0000	−0.934907	−0.467454	+	0.884018i	$0.654829\pi$
$660$	−2.00000	−0.0778499
$661$	−20.0000	−0.777910	−0.388955	−	0.921257i	$-0.627164\pi$
$661$	−20.0000	−0.777910	−0.388955	+	0.921257i	$0.627164\pi$
$662$	−32.0000	−1.24372
$663$	−20.0000	−0.776736
$664$	−18.0000	−0.698535
$665$	4.00000	0.155113
$666$	7.00000	0.271244
$667$	16.0000	0.619522
$668$	12.0000	0.464294
$669$	22.0000	0.850569
$670$	−12.0000	−0.463600
$671$	−2.00000	−0.0772091
$672$	−10.0000	−0.385758
$673$	22.0000	0.848038	0.424019	−	0.905653i	$-0.360619\pi$
$673$	22.0000	0.848038	0.424019	+	0.905653i	$0.360619\pi$
$674$	1.00000	0.0385186
$675$	−4.00000	−0.153960
$676$	−12.0000	−0.461538
$677$	−30.0000	−1.15299	−0.576497	−	0.817099i	$-0.695581\pi$
$677$	−30.0000	−1.15299	−0.576497	+	0.817099i	$0.695581\pi$
$678$	28.0000	1.07533
$679$	10.0000	0.383765
$680$	6.00000	0.230089
$681$	14.0000	0.536481
$682$	−9.00000	−0.344628
$683$	−19.0000	−0.727015	−0.363507	−	0.931591i	$-0.618421\pi$
$683$	−19.0000	−0.727015	−0.363507	+	0.931591i	$0.618421\pi$
$684$	−4.00000	−0.152944
$685$	12.0000	0.458496
$686$	−1.00000	−0.0381802
$687$	2.00000	0.0763048
$688$	3.00000	0.114374
$689$	25.0000	0.952424
$690$	8.00000	0.304555
$691$	50.0000	1.90209	0.951045	−	0.309053i	$-0.100012\pi$
$691$	50.0000	1.90209	0.951045	+	0.309053i	$0.100012\pi$
$692$	−12.0000	−0.456172
$693$	1.00000	0.0379869
$694$	33.0000	1.25266
$695$	−19.0000	−0.720711
$696$	24.0000	0.909718
$697$	2.00000	0.0757554
$698$	−33.0000	−1.24907
$699$	46.0000	1.73988
$700$	1.00000	0.0377964
$701$	−17.0000	−0.642081	−0.321041	−	0.947065i	$-0.604033\pi$
$701$	−17.0000	−0.642081	−0.321041	+	0.947065i	$0.604033\pi$
$702$	20.0000	0.754851
$703$	28.0000	1.05604
$704$	−7.00000	−0.263822
$705$	−6.00000	−0.225973
$706$	2.00000	0.0752710
$707$	14.0000	0.526524
$708$	−8.00000	−0.300658
$709$	34.0000	1.27690	0.638448	−	0.769665i	$-0.279577\pi$
$709$	34.0000	1.27690	0.638448	+	0.769665i	$0.279577\pi$
$710$	−9.00000	−0.337764
$711$	12.0000	0.450035
$712$	−27.0000	−1.01187
$713$	−36.0000	−1.34821
$714$	−4.00000	−0.149696
$715$	−5.00000	−0.186989
$716$	26.0000	0.971666
$717$	12.0000	0.448148
$718$	21.0000	0.783713
$719$	−41.0000	−1.52904	−0.764521	−	0.644599i	$-0.777024\pi$
$719$	−41.0000	−1.52904	−0.764521	+	0.644599i	$0.777024\pi$
$720$	1.00000	0.0372678
$721$	−12.0000	−0.446903
$722$	−3.00000	−0.111648
$723$	40.0000	1.48762
$724$	−12.0000	−0.445976
$725$	−4.00000	−0.148556
$726$	−20.0000	−0.742270
$727$	−16.0000	−0.593407	−0.296704	−	0.954970i	$-0.595887\pi$
$727$	−16.0000	−0.593407	−0.296704	+	0.954970i	$0.595887\pi$
$728$	−15.0000	−0.555937
$729$	13.0000	0.481481
$730$	14.0000	0.518163
$731$	−6.00000	−0.221918
$732$	−4.00000	−0.147844
$733$	−28.0000	−1.03420	−0.517102	−	0.855924i	$-0.672989\pi$
$733$	−28.0000	−1.03420	−0.517102	+	0.855924i	$0.672989\pi$
$734$	10.0000	0.369107
$735$	−2.00000	−0.0737711
$736$	−20.0000	−0.737210
$737$	−12.0000	−0.442026
$738$	1.00000	0.0368105
$739$	−30.0000	−1.10357	−0.551784	−	0.833987i	$-0.686053\pi$
$739$	−30.0000	−1.10357	−0.551784	+	0.833987i	$0.686053\pi$
$740$	7.00000	0.257325
$741$	−40.0000	−1.46944
$742$	5.00000	0.183556
$743$	8.00000	0.293492	0.146746	−	0.989174i	$-0.453120\pi$
$743$	8.00000	0.293492	0.146746	+	0.989174i	$0.453120\pi$
$744$	−54.0000	−1.97974
$745$	−7.00000	−0.256460
$746$	35.0000	1.28144
$747$	6.00000	0.219529
$748$	2.00000	0.0731272
$749$	−12.0000	−0.438470
$750$	−2.00000	−0.0730297
$751$	29.0000	1.05823	0.529113	−	0.848552i	$-0.322525\pi$
$751$	29.0000	1.05823	0.529113	+	0.848552i	$0.322525\pi$
$752$	−3.00000	−0.109399
$753$	−6.00000	−0.218652
$754$	20.0000	0.728357
$755$	9.00000	0.327544
$756$	−4.00000	−0.145479
$757$	−3.00000	−0.109037	−0.0545184	−	0.998513i	$-0.517362\pi$
$757$	−3.00000	−0.109037	−0.0545184	+	0.998513i	$0.517362\pi$
$758$	28.0000	1.01701
$759$	8.00000	0.290382
$760$	12.0000	0.435286
$761$	−47.0000	−1.70375	−0.851874	−	0.523746i	$-0.824534\pi$
$761$	−47.0000	−1.70375	−0.851874	+	0.523746i	$0.824534\pi$
$762$	40.0000	1.44905
$763$	2.00000	0.0724049
$764$	14.0000	0.506502
$765$	−2.00000	−0.0723102
$766$	2.00000	0.0722629
$767$	−20.0000	−0.722158
$768$	−34.0000	−1.22687
$769$	14.0000	0.504853	0.252426	−	0.967616i	$-0.418771\pi$
$769$	14.0000	0.504853	0.252426	+	0.967616i	$0.418771\pi$
$770$	−1.00000	−0.0360375
$771$	36.0000	1.29651
$772$	3.00000	0.107972
$773$	1.00000	0.0359675	0.0179838	−	0.999838i	$-0.494275\pi$
$773$	1.00000	0.0359675	0.0179838	+	0.999838i	$0.494275\pi$
$774$	−3.00000	−0.107833
$775$	9.00000	0.323290
$776$	30.0000	1.07694
$777$	−14.0000	−0.502247
$778$	−20.0000	−0.717035
$779$	4.00000	0.143315
$780$	−10.0000	−0.358057
$781$	−9.00000	−0.322045
$782$	−8.00000	−0.286079
$783$	16.0000	0.571793
$784$	−1.00000	−0.0357143
$785$	1.00000	0.0356915
$786$	−18.0000	−0.642039
$787$	22.0000	0.784215	0.392108	−	0.919919i	$-0.371746\pi$
$787$	22.0000	0.784215	0.392108	+	0.919919i	$0.371746\pi$
$788$	6.00000	0.213741
$789$	12.0000	0.427211
$790$	−12.0000	−0.426941
$791$	−14.0000	−0.497783
$792$	3.00000	0.106600
$793$	−10.0000	−0.355110
$794$	−22.0000	−0.780751
$795$	10.0000	0.354663
$796$	−12.0000	−0.425329
$797$	15.0000	0.531327	0.265664	−	0.964066i	$-0.414409\pi$
$797$	15.0000	0.531327	0.265664	+	0.964066i	$0.414409\pi$
$798$	−8.00000	−0.283197
$799$	6.00000	0.212265
$800$	5.00000	0.176777
$801$	9.00000	0.317999
$802$	38.0000	1.34183
$803$	14.0000	0.494049
$804$	−24.0000	−0.846415
$805$	−4.00000	−0.140981
$806$	−45.0000	−1.58506
$807$	−12.0000	−0.422420
$808$	42.0000	1.47755
$809$	0	0		−	1.00000i	$-0.5\pi$
$809$	0	0			1.00000i	$0.5\pi$
$810$	11.0000	0.386501
$811$	4.00000	0.140459	0.0702295	−	0.997531i	$-0.477627\pi$
$811$	4.00000	0.140459	0.0702295	+	0.997531i	$0.477627\pi$
$812$	−4.00000	−0.140372
$813$	−64.0000	−2.24458
$814$	−7.00000	−0.245350
$815$	−24.0000	−0.840683
$816$	−4.00000	−0.140028
$817$	−12.0000	−0.419827
$818$	−4.00000	−0.139857
$819$	5.00000	0.174714
$820$	1.00000	0.0349215
$821$	−10.0000	−0.349002	−0.174501	−	0.984657i	$-0.555831\pi$
$821$	−10.0000	−0.349002	−0.174501	+	0.984657i	$0.555831\pi$
$822$	−24.0000	−0.837096
$823$	−8.00000	−0.278862	−0.139431	−	0.990232i	$-0.544527\pi$
$823$	−8.00000	−0.278862	−0.139431	+	0.990232i	$0.544527\pi$
$824$	−36.0000	−1.25412
$825$	−2.00000	−0.0696311
$826$	−4.00000	−0.139178
$827$	24.0000	0.834562	0.417281	−	0.908778i	$-0.362983\pi$
$827$	24.0000	0.834562	0.417281	+	0.908778i	$0.362983\pi$
$828$	4.00000	0.139010
$829$	−42.0000	−1.45872	−0.729360	−	0.684130i	$-0.760182\pi$
$829$	−42.0000	−1.45872	−0.729360	+	0.684130i	$0.760182\pi$
$830$	−6.00000	−0.208263
$831$	38.0000	1.31821
$832$	−35.0000	−1.21341
$833$	2.00000	0.0692959
$834$	38.0000	1.31583
$835$	12.0000	0.415277
$836$	4.00000	0.138343
$837$	−36.0000	−1.24434
$838$	1.00000	0.0345444
$839$	−5.00000	−0.172619	−0.0863096	−	0.996268i	$-0.527507\pi$
$839$	−5.00000	−0.172619	−0.0863096	+	0.996268i	$0.527507\pi$
$840$	−6.00000	−0.207020
$841$	−13.0000	−0.448276
$842$	6.00000	0.206774
$843$	−40.0000	−1.37767
$844$	−8.00000	−0.275371
$845$	−12.0000	−0.412813
$846$	3.00000	0.103142
$847$	10.0000	0.343604
$848$	5.00000	0.171701
$849$	54.0000	1.85328
$850$	2.00000	0.0685994
$851$	−28.0000	−0.959828
$852$	−18.0000	−0.616670
$853$	−47.0000	−1.60925	−0.804625	−	0.593784i	$-0.797633\pi$
$853$	−47.0000	−1.60925	−0.804625	+	0.593784i	$0.797633\pi$
$854$	−2.00000	−0.0684386
$855$	−4.00000	−0.136797
$856$	−36.0000	−1.23045
$857$	39.0000	1.33221	0.666107	−	0.745856i	$-0.267959\pi$
$857$	39.0000	1.33221	0.666107	+	0.745856i	$0.267959\pi$
$858$	10.0000	0.341394
$859$	50.0000	1.70598	0.852989	−	0.521929i	$-0.174787\pi$
$859$	50.0000	1.70598	0.852989	+	0.521929i	$0.174787\pi$
$860$	−3.00000	−0.102299
$861$	−2.00000	−0.0681598
$862$	−4.00000	−0.136241
$863$	24.0000	0.816970	0.408485	−	0.912765i	$-0.366057\pi$
$863$	24.0000	0.816970	0.408485	+	0.912765i	$0.366057\pi$
$864$	−20.0000	−0.680414
$865$	−12.0000	−0.408012
$866$	−18.0000	−0.611665
$867$	−26.0000	−0.883006
$868$	9.00000	0.305480
$869$	−12.0000	−0.407072
$870$	8.00000	0.271225
$871$	−60.0000	−2.03302
$872$	6.00000	0.203186
$873$	−10.0000	−0.338449
$874$	−16.0000	−0.541208
$875$	1.00000	0.0338062
$876$	28.0000	0.946032
$877$	2.00000	0.0675352	0.0337676	−	0.999430i	$-0.489249\pi$
$877$	2.00000	0.0675352	0.0337676	+	0.999430i	$0.489249\pi$
$878$	30.0000	1.01245
$879$	−44.0000	−1.48408
$880$	−1.00000	−0.0337100
$881$	53.0000	1.78562	0.892808	−	0.450438i	$-0.148732\pi$
$881$	53.0000	1.78562	0.892808	+	0.450438i	$0.148732\pi$
$882$	1.00000	0.0336718
$883$	−33.0000	−1.11054	−0.555269	−	0.831671i	$-0.687385\pi$
$883$	−33.0000	−1.11054	−0.555269	+	0.831671i	$0.687385\pi$
$884$	10.0000	0.336336
$885$	−8.00000	−0.268917
$886$	32.0000	1.07506
$887$	3.00000	0.100730	0.0503651	−	0.998731i	$-0.483962\pi$
$887$	3.00000	0.100730	0.0503651	+	0.998731i	$0.483962\pi$
$888$	−42.0000	−1.40943
$889$	−20.0000	−0.670778
$890$	−9.00000	−0.301681
$891$	11.0000	0.368514
$892$	−11.0000	−0.368307
$893$	12.0000	0.401565
$894$	14.0000	0.468230
$895$	26.0000	0.869084
$896$	3.00000	0.100223
$897$	40.0000	1.33556
$898$	−25.0000	−0.834261
$899$	−36.0000	−1.20067
$900$	−1.00000	−0.0333333
$901$	−10.0000	−0.333148
$902$	−1.00000	−0.0332964
$903$	6.00000	0.199667
$904$	−42.0000	−1.39690
$905$	−12.0000	−0.398893
$906$	−18.0000	−0.598010
$907$	−12.0000	−0.398453	−0.199227	−	0.979953i	$-0.563843\pi$
$907$	−12.0000	−0.398453	−0.199227	+	0.979953i	$0.563843\pi$
$908$	−7.00000	−0.232303
$909$	−14.0000	−0.464351
$910$	−5.00000	−0.165748
$911$	−7.00000	−0.231920	−0.115960	−	0.993254i	$-0.536994\pi$
$911$	−7.00000	−0.231920	−0.115960	+	0.993254i	$0.536994\pi$
$912$	−8.00000	−0.264906
$913$	−6.00000	−0.198571
$914$	−34.0000	−1.12462
$915$	−4.00000	−0.132236
$916$	−1.00000	−0.0330409
$917$	9.00000	0.297206
$918$	−8.00000	−0.264039
$919$	−55.0000	−1.81428	−0.907141	−	0.420826i	$-0.861740\pi$
$919$	−55.0000	−1.81428	−0.907141	+	0.420826i	$0.861740\pi$
$920$	−12.0000	−0.395628
$921$	−4.00000	−0.131804
$922$	−6.00000	−0.197599
$923$	−45.0000	−1.48119
$924$	−2.00000	−0.0657952
$925$	7.00000	0.230159
$926$	−23.0000	−0.755827
$927$	12.0000	0.394132
$928$	−20.0000	−0.656532
$929$	30.0000	0.984268	0.492134	−	0.870519i	$-0.336217\pi$
$929$	30.0000	0.984268	0.492134	+	0.870519i	$0.336217\pi$
$930$	−18.0000	−0.590243
$931$	4.00000	0.131095
$932$	−23.0000	−0.753390
$933$	−40.0000	−1.30954
$934$	−24.0000	−0.785304
$935$	2.00000	0.0654070
$936$	15.0000	0.490290
$937$	−43.0000	−1.40475	−0.702374	−	0.711808i	$-0.747877\pi$
$937$	−43.0000	−1.40475	−0.702374	+	0.711808i	$0.747877\pi$
$938$	−12.0000	−0.391814
$939$	2.00000	0.0652675
$940$	3.00000	0.0978492
$941$	24.0000	0.782378	0.391189	−	0.920310i	$-0.372064\pi$
$941$	24.0000	0.782378	0.391189	+	0.920310i	$0.372064\pi$
$942$	−2.00000	−0.0651635
$943$	−4.00000	−0.130258
$944$	−4.00000	−0.130189
$945$	−4.00000	−0.130120
$946$	3.00000	0.0975384
$947$	38.0000	1.23483	0.617417	−	0.786636i	$-0.288179\pi$
$947$	38.0000	1.23483	0.617417	+	0.786636i	$0.288179\pi$
$948$	−24.0000	−0.779484
$949$	70.0000	2.27230
$950$	4.00000	0.129777
$951$	8.00000	0.259418
$952$	6.00000	0.194461
$953$	−2.00000	−0.0647864	−0.0323932	−	0.999475i	$-0.510313\pi$
$953$	−2.00000	−0.0647864	−0.0323932	+	0.999475i	$0.510313\pi$
$954$	−5.00000	−0.161881
$955$	14.0000	0.453029
$956$	−6.00000	−0.194054
$957$	8.00000	0.258603
$958$	17.0000	0.549245
$959$	12.0000	0.387500
$960$	−14.0000	−0.451848
$961$	50.0000	1.61290
$962$	−35.0000	−1.12845
$963$	12.0000	0.386695
$964$	−20.0000	−0.644157
$965$	3.00000	0.0965734
$966$	8.00000	0.257396
$967$	−1.00000	−0.0321578	−0.0160789	−	0.999871i	$-0.505118\pi$
$967$	−1.00000	−0.0321578	−0.0160789	+	0.999871i	$0.505118\pi$
$968$	30.0000	0.964237
$969$	16.0000	0.513994
$970$	10.0000	0.321081
$971$	2.00000	0.0641831	0.0320915	−	0.999485i	$-0.489783\pi$
$971$	2.00000	0.0641831	0.0320915	+	0.999485i	$0.489783\pi$
$972$	10.0000	0.320750
$973$	−19.0000	−0.609112
$974$	20.0000	0.640841
$975$	−10.0000	−0.320256
$976$	−2.00000	−0.0640184
$977$	59.0000	1.88758	0.943789	−	0.330550i	$-0.107234\pi$
$977$	59.0000	1.88758	0.943789	+	0.330550i	$0.107234\pi$
$978$	48.0000	1.53487
$979$	−9.00000	−0.287641
$980$	1.00000	0.0319438
$981$	−2.00000	−0.0638551
$982$	−24.0000	−0.765871
$983$	−20.0000	−0.637901	−0.318950	−	0.947771i	$-0.603330\pi$
$983$	−20.0000	−0.637901	−0.318950	+	0.947771i	$0.603330\pi$
$984$	−6.00000	−0.191273
$985$	6.00000	0.191176
$986$	−8.00000	−0.254772
$987$	−6.00000	−0.190982
$988$	20.0000	0.636285
$989$	12.0000	0.381578
$990$	1.00000	0.0317821
$991$	−25.0000	−0.794151	−0.397076	−	0.917786i	$-0.629975\pi$
$991$	−25.0000	−0.794151	−0.397076	+	0.917786i	$0.629975\pi$
$992$	45.0000	1.42875
$993$	−64.0000	−2.03098
$994$	−9.00000	−0.285463
$995$	−12.0000	−0.380426
$996$	−12.0000	−0.380235
$997$	40.0000	1.26681	0.633406	−	0.773819i	$-0.281656\pi$
$997$	40.0000	1.26681	0.633406	+	0.773819i	$0.281656\pi$
$998$	−10.0000	−0.316544
$999$	−28.0000	−0.885881

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8015.2.a.e.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8015.2.a.e.1.1	✓	1	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 8015 = 5 \cdot 7 \cdot 229 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8015.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more