LMFDB - Embedded newform 7942.2.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7942,2,Mod(1,7942)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7942, 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("7942.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7942 = 2 \cdot 11 \cdot 19^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7942.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:	$63.4171892853$
Analytic rank:	$2$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 418)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	7942.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

\(q-1.00000 q^{2} -2.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +2.00000 q^{6} -3.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} -1.00000 q^{11} -2.00000 q^{12} -4.00000 q^{13} +3.00000 q^{14} -2.00000 q^{15} +1.00000 q^{16} -4.00000 q^{17} -1.00000 q^{18} +1.00000 q^{20} +6.00000 q^{21} +1.00000 q^{22} +4.00000 q^{23} +2.00000 q^{24} -4.00000 q^{25} +4.00000 q^{26} +4.00000 q^{27} -3.00000 q^{28} -10.0000 q^{29} +2.00000 q^{30} -4.00000 q^{31} -1.00000 q^{32} +2.00000 q^{33} +4.00000 q^{34} -3.00000 q^{35} +1.00000 q^{36} -3.00000 q^{37} +8.00000 q^{39} -1.00000 q^{40} -6.00000 q^{41} -6.00000 q^{42} -4.00000 q^{43} -1.00000 q^{44} +1.00000 q^{45} -4.00000 q^{46} +6.00000 q^{47} -2.00000 q^{48} +2.00000 q^{49} +4.00000 q^{50} +8.00000 q^{51} -4.00000 q^{52} +1.00000 q^{53} -4.00000 q^{54} -1.00000 q^{55} +3.00000 q^{56} +10.0000 q^{58} -12.0000 q^{59} -2.00000 q^{60} +12.0000 q^{61} +4.00000 q^{62} -3.00000 q^{63} +1.00000 q^{64} -4.00000 q^{65} -2.00000 q^{66} -12.0000 q^{67} -4.00000 q^{68} -8.00000 q^{69} +3.00000 q^{70} -14.0000 q^{71} -1.00000 q^{72} +6.00000 q^{73} +3.00000 q^{74} +8.00000 q^{75} +3.00000 q^{77} -8.00000 q^{78} -17.0000 q^{79} +1.00000 q^{80} -11.0000 q^{81} +6.00000 q^{82} -13.0000 q^{83} +6.00000 q^{84} -4.00000 q^{85} +4.00000 q^{86} +20.0000 q^{87} +1.00000 q^{88} +2.00000 q^{89} -1.00000 q^{90} +12.0000 q^{91} +4.00000 q^{92} +8.00000 q^{93} -6.00000 q^{94} +2.00000 q^{96} -5.00000 q^{97} -2.00000 q^{98} -1.00000 q^{99} +O(q^{100})\)

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
$2$	−1.00000	−0.707107
$3$	−2.00000	−1.15470	−0.577350	−	0.816497i	$-0.695913\pi$
$3$	−2.00000	−1.15470	−0.577350	+	0.816497i	$0.695913\pi$
$4$	1.00000	0.500000
$5$	1.00000	0.447214	0.223607	−	0.974679i	$-0.428217\pi$
$5$	1.00000	0.447214	0.223607	+	0.974679i	$0.428217\pi$
$6$	2.00000	0.816497
$7$	−3.00000	−1.13389	−0.566947	−	0.823754i	$-0.691875\pi$
$7$	−3.00000	−1.13389	−0.566947	+	0.823754i	$0.691875\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	−1.00000	−0.316228
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	−2.00000	−0.577350
$13$	−4.00000	−1.10940	−0.554700	−	0.832050i	$-0.687167\pi$
$13$	−4.00000	−1.10940	−0.554700	+	0.832050i	$0.687167\pi$
$14$	3.00000	0.801784
$15$	−2.00000	−0.516398
$16$	1.00000	0.250000
$17$	−4.00000	−0.970143	−0.485071	−	0.874475i	$-0.661206\pi$
$17$	−4.00000	−0.970143	−0.485071	+	0.874475i	$0.661206\pi$
$18$	−1.00000	−0.235702
$19$	0	0
$19$	0	0
$20$	1.00000	0.223607
$21$	6.00000	1.30931
$22$	1.00000	0.213201
$23$	4.00000	0.834058	0.417029	−	0.908893i	$-0.363071\pi$
$23$	4.00000	0.834058	0.417029	+	0.908893i	$0.363071\pi$
$24$	2.00000	0.408248
$25$	−4.00000	−0.800000
$26$	4.00000	0.784465
$27$	4.00000	0.769800
$28$	−3.00000	−0.566947
$29$	−10.0000	−1.85695	−0.928477	−	0.371391i	$-0.878881\pi$
$29$	−10.0000	−1.85695	−0.928477	+	0.371391i	$0.878881\pi$
$30$	2.00000	0.365148
$31$	−4.00000	−0.718421	−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000	−0.718421	−0.359211	+	0.933257i	$0.616954\pi$
$32$	−1.00000	−0.176777
$33$	2.00000	0.348155
$34$	4.00000	0.685994
$35$	−3.00000	−0.507093
$36$	1.00000	0.166667
$37$	−3.00000	−0.493197	−0.246598	−	0.969118i	$-0.579313\pi$
$37$	−3.00000	−0.493197	−0.246598	+	0.969118i	$0.579313\pi$
$38$	0	0
$39$	8.00000	1.28103
$40$	−1.00000	−0.158114
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	−6.00000	−0.925820
$43$	−4.00000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	−1.00000	−0.150756
$45$	1.00000	0.149071
$46$	−4.00000	−0.589768
$47$	6.00000	0.875190	0.437595	−	0.899172i	$-0.355830\pi$
$47$	6.00000	0.875190	0.437595	+	0.899172i	$0.355830\pi$
$48$	−2.00000	−0.288675
$49$	2.00000	0.285714
$50$	4.00000	0.565685
$51$	8.00000	1.12022
$52$	−4.00000	−0.554700
$53$	1.00000	0.137361	0.0686803	−	0.997639i	$-0.478121\pi$
$53$	1.00000	0.137361	0.0686803	+	0.997639i	$0.478121\pi$
$54$	−4.00000	−0.544331
$55$	−1.00000	−0.134840
$56$	3.00000	0.400892
$57$	0	0
$58$	10.0000	1.31306
$59$	−12.0000	−1.56227	−0.781133	−	0.624364i	$-0.785358\pi$
$59$	−12.0000	−1.56227	−0.781133	+	0.624364i	$0.785358\pi$
$60$	−2.00000	−0.258199
$61$	12.0000	1.53644	0.768221	−	0.640184i	$-0.221142\pi$
$61$	12.0000	1.53644	0.768221	+	0.640184i	$0.221142\pi$
$62$	4.00000	0.508001
$63$	−3.00000	−0.377964
$64$	1.00000	0.125000
$65$	−4.00000	−0.496139
$66$	−2.00000	−0.246183
$67$	−12.0000	−1.46603	−0.733017	−	0.680211i	$-0.761888\pi$
$67$	−12.0000	−1.46603	−0.733017	+	0.680211i	$0.761888\pi$
$68$	−4.00000	−0.485071
$69$	−8.00000	−0.963087
$70$	3.00000	0.358569
$71$	−14.0000	−1.66149	−0.830747	−	0.556650i	$-0.812086\pi$
$71$	−14.0000	−1.66149	−0.830747	+	0.556650i	$0.812086\pi$
$72$	−1.00000	−0.117851
$73$	6.00000	0.702247	0.351123	−	0.936329i	$-0.385800\pi$
$73$	6.00000	0.702247	0.351123	+	0.936329i	$0.385800\pi$
$74$	3.00000	0.348743
$75$	8.00000	0.923760
$76$	0	0
$77$	3.00000	0.341882
$78$	−8.00000	−0.905822
$79$	−17.0000	−1.91265	−0.956325	−	0.292306i	$-0.905577\pi$
$79$	−17.0000	−1.91265	−0.956325	+	0.292306i	$0.905577\pi$
$80$	1.00000	0.111803
$81$	−11.0000	−1.22222
$82$	6.00000	0.662589
$83$	−13.0000	−1.42694	−0.713468	−	0.700688i	$-0.752876\pi$
$83$	−13.0000	−1.42694	−0.713468	+	0.700688i	$0.752876\pi$
$84$	6.00000	0.654654
$85$	−4.00000	−0.433861
$86$	4.00000	0.431331
$87$	20.0000	2.14423
$88$	1.00000	0.106600
$89$	2.00000	0.212000	0.106000	−	0.994366i	$-0.466196\pi$
$89$	2.00000	0.212000	0.106000	+	0.994366i	$0.466196\pi$
$90$	−1.00000	−0.105409
$91$	12.0000	1.25794
$92$	4.00000	0.417029
$93$	8.00000	0.829561
$94$	−6.00000	−0.618853
$95$	0	0
$96$	2.00000	0.204124
$97$	−5.00000	−0.507673	−0.253837	−	0.967247i	$-0.581693\pi$
$97$	−5.00000	−0.507673	−0.253837	+	0.967247i	$0.581693\pi$
$98$	−2.00000	−0.202031
$99$	−1.00000	−0.100504
$100$	−4.00000	−0.400000
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	−8.00000	−0.792118
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	4.00000	0.392232
$105$	6.00000	0.585540
$106$	−1.00000	−0.0971286
$107$	7.00000	0.676716	0.338358	−	0.941018i	$-0.390129\pi$
$107$	7.00000	0.676716	0.338358	+	0.941018i	$0.390129\pi$
$108$	4.00000	0.384900
$109$	4.00000	0.383131	0.191565	−	0.981480i	$-0.438644\pi$
$109$	4.00000	0.383131	0.191565	+	0.981480i	$0.438644\pi$
$110$	1.00000	0.0953463
$111$	6.00000	0.569495
$112$	−3.00000	−0.283473
$113$	6.00000	0.564433	0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000	0.564433	0.282216	+	0.959351i	$0.408930\pi$
$114$	0	0
$115$	4.00000	0.373002
$116$	−10.0000	−0.928477
$117$	−4.00000	−0.369800
$118$	12.0000	1.10469
$119$	12.0000	1.10004
$120$	2.00000	0.182574
$121$	1.00000	0.0909091
$122$	−12.0000	−1.08643
$123$	12.0000	1.08200
$124$	−4.00000	−0.359211
$125$	−9.00000	−0.804984
$126$	3.00000	0.267261
$127$	8.00000	0.709885	0.354943	−	0.934888i	$-0.384500\pi$
$127$	8.00000	0.709885	0.354943	+	0.934888i	$0.384500\pi$
$128$	−1.00000	−0.0883883
$129$	8.00000	0.704361
$130$	4.00000	0.350823
$131$	−16.0000	−1.39793	−0.698963	−	0.715158i	$-0.746355\pi$
$131$	−16.0000	−1.39793	−0.698963	+	0.715158i	$0.746355\pi$
$132$	2.00000	0.174078
$133$	0	0
$134$	12.0000	1.03664
$135$	4.00000	0.344265
$136$	4.00000	0.342997
$137$	1.00000	0.0854358	0.0427179	−	0.999087i	$-0.486398\pi$
$137$	1.00000	0.0854358	0.0427179	+	0.999087i	$0.486398\pi$
$138$	8.00000	0.681005
$139$	−17.0000	−1.44192	−0.720961	−	0.692976i	$-0.756299\pi$
$139$	−17.0000	−1.44192	−0.720961	+	0.692976i	$0.756299\pi$
$140$	−3.00000	−0.253546
$141$	−12.0000	−1.01058
$142$	14.0000	1.17485
$143$	4.00000	0.334497
$144$	1.00000	0.0833333
$145$	−10.0000	−0.830455
$146$	−6.00000	−0.496564
$147$	−4.00000	−0.329914
$148$	−3.00000	−0.246598
$149$	−6.00000	−0.491539	−0.245770	−	0.969328i	$-0.579041\pi$
$149$	−6.00000	−0.491539	−0.245770	+	0.969328i	$0.579041\pi$
$150$	−8.00000	−0.653197
$151$	−15.0000	−1.22068	−0.610341	−	0.792139i	$-0.708968\pi$
$151$	−15.0000	−1.22068	−0.610341	+	0.792139i	$0.708968\pi$
$152$	0	0
$153$	−4.00000	−0.323381
$154$	−3.00000	−0.241747
$155$	−4.00000	−0.321288
$156$	8.00000	0.640513
$157$	5.00000	0.399043	0.199522	−	0.979893i	$-0.436061\pi$
$157$	5.00000	0.399043	0.199522	+	0.979893i	$0.436061\pi$
$158$	17.0000	1.35245
$159$	−2.00000	−0.158610
$160$	−1.00000	−0.0790569
$161$	−12.0000	−0.945732
$162$	11.0000	0.864242
$163$	4.00000	0.313304	0.156652	−	0.987654i	$-0.449930\pi$
$163$	4.00000	0.313304	0.156652	+	0.987654i	$0.449930\pi$
$164$	−6.00000	−0.468521
$165$	2.00000	0.155700
$166$	13.0000	1.00900
$167$	−17.0000	−1.31550	−0.657750	−	0.753237i	$-0.728492\pi$
$167$	−17.0000	−1.31550	−0.657750	+	0.753237i	$0.728492\pi$
$168$	−6.00000	−0.462910
$169$	3.00000	0.230769
$170$	4.00000	0.306786
$171$	0	0
$172$	−4.00000	−0.304997
$173$	4.00000	0.304114	0.152057	−	0.988372i	$-0.451410\pi$
$173$	4.00000	0.304114	0.152057	+	0.988372i	$0.451410\pi$
$174$	−20.0000	−1.51620
$175$	12.0000	0.907115
$176$	−1.00000	−0.0753778
$177$	24.0000	1.80395
$178$	−2.00000	−0.149906
$179$	20.0000	1.49487	0.747435	−	0.664335i	$-0.231285\pi$
$179$	20.0000	1.49487	0.747435	+	0.664335i	$0.231285\pi$
$180$	1.00000	0.0745356
$181$	17.0000	1.26360	0.631800	−	0.775131i	$-0.282316\pi$
$181$	17.0000	1.26360	0.631800	+	0.775131i	$0.282316\pi$
$182$	−12.0000	−0.889499
$183$	−24.0000	−1.77413
$184$	−4.00000	−0.294884
$185$	−3.00000	−0.220564
$186$	−8.00000	−0.586588
$187$	4.00000	0.292509
$188$	6.00000	0.437595
$189$	−12.0000	−0.872872
$190$	0	0
$191$	18.0000	1.30243	0.651217	−	0.758891i	$-0.274259\pi$
$191$	18.0000	1.30243	0.651217	+	0.758891i	$0.274259\pi$
$192$	−2.00000	−0.144338
$193$	10.0000	0.719816	0.359908	−	0.932988i	$-0.382808\pi$
$193$	10.0000	0.719816	0.359908	+	0.932988i	$0.382808\pi$
$194$	5.00000	0.358979
$195$	8.00000	0.572892
$196$	2.00000	0.142857
$197$	−12.0000	−0.854965	−0.427482	−	0.904024i	$-0.640599\pi$
$197$	−12.0000	−0.854965	−0.427482	+	0.904024i	$0.640599\pi$
$198$	1.00000	0.0710669
$199$	−10.0000	−0.708881	−0.354441	−	0.935079i	$-0.615329\pi$
$199$	−10.0000	−0.708881	−0.354441	+	0.935079i	$0.615329\pi$
$200$	4.00000	0.282843
$201$	24.0000	1.69283
$202$	0	0
$203$	30.0000	2.10559
$204$	8.00000	0.560112
$205$	−6.00000	−0.419058
$206$	0	0
$207$	4.00000	0.278019
$208$	−4.00000	−0.277350
$209$	0	0
$210$	−6.00000	−0.414039
$211$	23.0000	1.58339	0.791693	−	0.610920i	$-0.209200\pi$
$211$	23.0000	1.58339	0.791693	+	0.610920i	$0.209200\pi$
$212$	1.00000	0.0686803
$213$	28.0000	1.91853
$214$	−7.00000	−0.478510
$215$	−4.00000	−0.272798
$216$	−4.00000	−0.272166
$217$	12.0000	0.814613
$218$	−4.00000	−0.270914
$219$	−12.0000	−0.810885
$220$	−1.00000	−0.0674200
$221$	16.0000	1.07628
$222$	−6.00000	−0.402694
$223$	−16.0000	−1.07144	−0.535720	−	0.844396i	$-0.679960\pi$
$223$	−16.0000	−1.07144	−0.535720	+	0.844396i	$0.679960\pi$
$224$	3.00000	0.200446
$225$	−4.00000	−0.266667
$226$	−6.00000	−0.399114
$227$	3.00000	0.199117	0.0995585	−	0.995032i	$-0.468257\pi$
$227$	3.00000	0.199117	0.0995585	+	0.995032i	$0.468257\pi$
$228$	0	0
$229$	−13.0000	−0.859064	−0.429532	−	0.903052i	$-0.641321\pi$
$229$	−13.0000	−0.859064	−0.429532	+	0.903052i	$0.641321\pi$
$230$	−4.00000	−0.263752
$231$	−6.00000	−0.394771
$232$	10.0000	0.656532
$233$	−16.0000	−1.04819	−0.524097	−	0.851658i	$-0.675597\pi$
$233$	−16.0000	−1.04819	−0.524097	+	0.851658i	$0.675597\pi$
$234$	4.00000	0.261488
$235$	6.00000	0.391397
$236$	−12.0000	−0.781133
$237$	34.0000	2.20854
$238$	−12.0000	−0.777844
$239$	−5.00000	−0.323423	−0.161712	−	0.986838i	$-0.551701\pi$
$239$	−5.00000	−0.323423	−0.161712	+	0.986838i	$0.551701\pi$
$240$	−2.00000	−0.129099
$241$	10.0000	0.644157	0.322078	−	0.946713i	$-0.395619\pi$
$241$	10.0000	0.644157	0.322078	+	0.946713i	$0.395619\pi$
$242$	−1.00000	−0.0642824
$243$	10.0000	0.641500
$244$	12.0000	0.768221
$245$	2.00000	0.127775
$246$	−12.0000	−0.765092
$247$	0	0
$248$	4.00000	0.254000
$249$	26.0000	1.64768
$250$	9.00000	0.569210
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	−3.00000	−0.188982
$253$	−4.00000	−0.251478
$254$	−8.00000	−0.501965
$255$	8.00000	0.500979
$256$	1.00000	0.0625000
$257$	1.00000	0.0623783	0.0311891	−	0.999514i	$-0.490071\pi$
$257$	1.00000	0.0623783	0.0311891	+	0.999514i	$0.490071\pi$
$258$	−8.00000	−0.498058
$259$	9.00000	0.559233
$260$	−4.00000	−0.248069
$261$	−10.0000	−0.618984
$262$	16.0000	0.988483
$263$	−23.0000	−1.41824	−0.709120	−	0.705087i	$-0.750908\pi$
$263$	−23.0000	−1.41824	−0.709120	+	0.705087i	$0.750908\pi$
$264$	−2.00000	−0.123091
$265$	1.00000	0.0614295
$266$	0	0
$267$	−4.00000	−0.244796
$268$	−12.0000	−0.733017
$269$	−15.0000	−0.914566	−0.457283	−	0.889321i	$-0.651177\pi$
$269$	−15.0000	−0.914566	−0.457283	+	0.889321i	$0.651177\pi$
$270$	−4.00000	−0.243432
$271$	−21.0000	−1.27566	−0.637830	−	0.770178i	$-0.720168\pi$
$271$	−21.0000	−1.27566	−0.637830	+	0.770178i	$0.720168\pi$
$272$	−4.00000	−0.242536
$273$	−24.0000	−1.45255
$274$	−1.00000	−0.0604122
$275$	4.00000	0.241209
$276$	−8.00000	−0.481543
$277$	−4.00000	−0.240337	−0.120168	−	0.992754i	$-0.538343\pi$
$277$	−4.00000	−0.240337	−0.120168	+	0.992754i	$0.538343\pi$
$278$	17.0000	1.01959
$279$	−4.00000	−0.239474
$280$	3.00000	0.179284
$281$	−4.00000	−0.238620	−0.119310	−	0.992857i	$-0.538068\pi$
$281$	−4.00000	−0.238620	−0.119310	+	0.992857i	$0.538068\pi$
$282$	12.0000	0.714590
$283$	13.0000	0.772770	0.386385	−	0.922338i	$-0.373724\pi$
$283$	13.0000	0.772770	0.386385	+	0.922338i	$0.373724\pi$
$284$	−14.0000	−0.830747
$285$	0	0
$286$	−4.00000	−0.236525
$287$	18.0000	1.06251
$288$	−1.00000	−0.0589256
$289$	−1.00000	−0.0588235
$290$	10.0000	0.587220
$291$	10.0000	0.586210
$292$	6.00000	0.351123
$293$	−30.0000	−1.75262	−0.876309	−	0.481749i	$-0.840002\pi$
$293$	−30.0000	−1.75262	−0.876309	+	0.481749i	$0.840002\pi$
$294$	4.00000	0.233285
$295$	−12.0000	−0.698667
$296$	3.00000	0.174371
$297$	−4.00000	−0.232104
$298$	6.00000	0.347571
$299$	−16.0000	−0.925304
$300$	8.00000	0.461880
$301$	12.0000	0.691669
$302$	15.0000	0.863153
$303$	0	0
$304$	0	0
$305$	12.0000	0.687118
$306$	4.00000	0.228665
$307$	−29.0000	−1.65512	−0.827559	−	0.561379i	$-0.810271\pi$
$307$	−29.0000	−1.65512	−0.827559	+	0.561379i	$0.810271\pi$
$308$	3.00000	0.170941
$309$	0	0
$310$	4.00000	0.227185
$311$	−18.0000	−1.02069	−0.510343	−	0.859971i	$-0.670482\pi$
$311$	−18.0000	−1.02069	−0.510343	+	0.859971i	$0.670482\pi$
$312$	−8.00000	−0.452911
$313$	−9.00000	−0.508710	−0.254355	−	0.967111i	$-0.581863\pi$
$313$	−9.00000	−0.508710	−0.254355	+	0.967111i	$0.581863\pi$
$314$	−5.00000	−0.282166
$315$	−3.00000	−0.169031
$316$	−17.0000	−0.956325
$317$	18.0000	1.01098	0.505490	−	0.862832i	$-0.331312\pi$
$317$	18.0000	1.01098	0.505490	+	0.862832i	$0.331312\pi$
$318$	2.00000	0.112154
$319$	10.0000	0.559893
$320$	1.00000	0.0559017
$321$	−14.0000	−0.781404
$322$	12.0000	0.668734
$323$	0	0
$324$	−11.0000	−0.611111
$325$	16.0000	0.887520
$326$	−4.00000	−0.221540
$327$	−8.00000	−0.442401
$328$	6.00000	0.331295
$329$	−18.0000	−0.992372
$330$	−2.00000	−0.110096
$331$	−20.0000	−1.09930	−0.549650	−	0.835395i	$-0.685239\pi$
$331$	−20.0000	−1.09930	−0.549650	+	0.835395i	$0.685239\pi$
$332$	−13.0000	−0.713468
$333$	−3.00000	−0.164399
$334$	17.0000	0.930199
$335$	−12.0000	−0.655630
$336$	6.00000	0.327327
$337$	−28.0000	−1.52526	−0.762629	−	0.646837i	$-0.776092\pi$
$337$	−28.0000	−1.52526	−0.762629	+	0.646837i	$0.776092\pi$
$338$	−3.00000	−0.163178
$339$	−12.0000	−0.651751
$340$	−4.00000	−0.216930
$341$	4.00000	0.216612
$342$	0	0
$343$	15.0000	0.809924
$344$	4.00000	0.215666
$345$	−8.00000	−0.430706
$346$	−4.00000	−0.215041
$347$	13.0000	0.697877	0.348938	−	0.937146i	$-0.386542\pi$
$347$	13.0000	0.697877	0.348938	+	0.937146i	$0.386542\pi$
$348$	20.0000	1.07211
$349$	−26.0000	−1.39175	−0.695874	−	0.718164i	$-0.744983\pi$
$349$	−26.0000	−1.39175	−0.695874	+	0.718164i	$0.744983\pi$
$350$	−12.0000	−0.641427
$351$	−16.0000	−0.854017
$352$	1.00000	0.0533002
$353$	19.0000	1.01127	0.505634	−	0.862748i	$-0.331259\pi$
$353$	19.0000	1.01127	0.505634	+	0.862748i	$0.331259\pi$
$354$	−24.0000	−1.27559
$355$	−14.0000	−0.743043
$356$	2.00000	0.106000
$357$	−24.0000	−1.27021
$358$	−20.0000	−1.05703
$359$	7.00000	0.369446	0.184723	−	0.982791i	$-0.440861\pi$
$359$	7.00000	0.369446	0.184723	+	0.982791i	$0.440861\pi$
$360$	−1.00000	−0.0527046
$361$	0	0
$362$	−17.0000	−0.893500
$363$	−2.00000	−0.104973
$364$	12.0000	0.628971
$365$	6.00000	0.314054
$366$	24.0000	1.25450
$367$	16.0000	0.835193	0.417597	−	0.908633i	$-0.362873\pi$
$367$	16.0000	0.835193	0.417597	+	0.908633i	$0.362873\pi$
$368$	4.00000	0.208514
$369$	−6.00000	−0.312348
$370$	3.00000	0.155963
$371$	−3.00000	−0.155752
$372$	8.00000	0.414781
$373$	28.0000	1.44979	0.724893	−	0.688862i	$-0.241889\pi$
$373$	28.0000	1.44979	0.724893	+	0.688862i	$0.241889\pi$
$374$	−4.00000	−0.206835
$375$	18.0000	0.929516
$376$	−6.00000	−0.309426
$377$	40.0000	2.06010
$378$	12.0000	0.617213
$379$	−4.00000	−0.205466	−0.102733	−	0.994709i	$-0.532759\pi$
$379$	−4.00000	−0.205466	−0.102733	+	0.994709i	$0.532759\pi$
$380$	0	0
$381$	−16.0000	−0.819705
$382$	−18.0000	−0.920960
$383$	−8.00000	−0.408781	−0.204390	−	0.978889i	$-0.565521\pi$
$383$	−8.00000	−0.408781	−0.204390	+	0.978889i	$0.565521\pi$
$384$	2.00000	0.102062
$385$	3.00000	0.152894
$386$	−10.0000	−0.508987
$387$	−4.00000	−0.203331
$388$	−5.00000	−0.253837
$389$	21.0000	1.06474	0.532371	−	0.846511i	$-0.321301\pi$
$389$	21.0000	1.06474	0.532371	+	0.846511i	$0.321301\pi$
$390$	−8.00000	−0.405096
$391$	−16.0000	−0.809155
$392$	−2.00000	−0.101015
$393$	32.0000	1.61419
$394$	12.0000	0.604551
$395$	−17.0000	−0.855363
$396$	−1.00000	−0.0502519
$397$	25.0000	1.25471	0.627357	−	0.778732i	$-0.284137\pi$
$397$	25.0000	1.25471	0.627357	+	0.778732i	$0.284137\pi$
$398$	10.0000	0.501255
$399$	0	0
$400$	−4.00000	−0.200000
$401$	−10.0000	−0.499376	−0.249688	−	0.968326i	$-0.580328\pi$
$401$	−10.0000	−0.499376	−0.249688	+	0.968326i	$0.580328\pi$
$402$	−24.0000	−1.19701
$403$	16.0000	0.797017
$404$	0	0
$405$	−11.0000	−0.546594
$406$	−30.0000	−1.48888
$407$	3.00000	0.148704
$408$	−8.00000	−0.396059
$409$	20.0000	0.988936	0.494468	−	0.869196i	$-0.335363\pi$
$409$	20.0000	0.988936	0.494468	+	0.869196i	$0.335363\pi$
$410$	6.00000	0.296319
$411$	−2.00000	−0.0986527
$412$	0	0
$413$	36.0000	1.77144
$414$	−4.00000	−0.196589
$415$	−13.0000	−0.638145
$416$	4.00000	0.196116
$417$	34.0000	1.66499
$418$	0	0
$419$	−38.0000	−1.85642	−0.928211	−	0.372055i	$-0.878653\pi$
$419$	−38.0000	−1.85642	−0.928211	+	0.372055i	$0.878653\pi$
$420$	6.00000	0.292770
$421$	3.00000	0.146211	0.0731055	−	0.997324i	$-0.476709\pi$
$421$	3.00000	0.146211	0.0731055	+	0.997324i	$0.476709\pi$
$422$	−23.0000	−1.11962
$423$	6.00000	0.291730
$424$	−1.00000	−0.0485643
$425$	16.0000	0.776114
$426$	−28.0000	−1.35660
$427$	−36.0000	−1.74216
$428$	7.00000	0.338358
$429$	−8.00000	−0.386244
$430$	4.00000	0.192897
$431$	−13.0000	−0.626188	−0.313094	−	0.949722i	$-0.601365\pi$
$431$	−13.0000	−0.626188	−0.313094	+	0.949722i	$0.601365\pi$
$432$	4.00000	0.192450
$433$	18.0000	0.865025	0.432512	−	0.901628i	$-0.357627\pi$
$433$	18.0000	0.865025	0.432512	+	0.901628i	$0.357627\pi$
$434$	−12.0000	−0.576018
$435$	20.0000	0.958927
$436$	4.00000	0.191565
$437$	0	0
$438$	12.0000	0.573382
$439$	−1.00000	−0.0477274	−0.0238637	−	0.999715i	$-0.507597\pi$
$439$	−1.00000	−0.0477274	−0.0238637	+	0.999715i	$0.507597\pi$
$440$	1.00000	0.0476731
$441$	2.00000	0.0952381
$442$	−16.0000	−0.761042
$443$	24.0000	1.14027	0.570137	−	0.821549i	$-0.306890\pi$
$443$	24.0000	1.14027	0.570137	+	0.821549i	$0.306890\pi$
$444$	6.00000	0.284747
$445$	2.00000	0.0948091
$446$	16.0000	0.757622
$447$	12.0000	0.567581
$448$	−3.00000	−0.141737
$449$	33.0000	1.55737	0.778683	−	0.627417i	$-0.215888\pi$
$449$	33.0000	1.55737	0.778683	+	0.627417i	$0.215888\pi$
$450$	4.00000	0.188562
$451$	6.00000	0.282529
$452$	6.00000	0.282216
$453$	30.0000	1.40952
$454$	−3.00000	−0.140797
$455$	12.0000	0.562569
$456$	0	0
$457$	38.0000	1.77757	0.888783	−	0.458329i	$-0.151552\pi$
$457$	38.0000	1.77757	0.888783	+	0.458329i	$0.151552\pi$
$458$	13.0000	0.607450
$459$	−16.0000	−0.746816
$460$	4.00000	0.186501
$461$	−2.00000	−0.0931493	−0.0465746	−	0.998915i	$-0.514831\pi$
$461$	−2.00000	−0.0931493	−0.0465746	+	0.998915i	$0.514831\pi$
$462$	6.00000	0.279145
$463$	4.00000	0.185896	0.0929479	−	0.995671i	$-0.470371\pi$
$463$	4.00000	0.185896	0.0929479	+	0.995671i	$0.470371\pi$
$464$	−10.0000	−0.464238
$465$	8.00000	0.370991
$466$	16.0000	0.741186
$467$	14.0000	0.647843	0.323921	−	0.946084i	$-0.394999\pi$
$467$	14.0000	0.647843	0.323921	+	0.946084i	$0.394999\pi$
$468$	−4.00000	−0.184900
$469$	36.0000	1.66233
$470$	−6.00000	−0.276759
$471$	−10.0000	−0.460776
$472$	12.0000	0.552345
$473$	4.00000	0.183920
$474$	−34.0000	−1.56167
$475$	0	0
$476$	12.0000	0.550019
$477$	1.00000	0.0457869
$478$	5.00000	0.228695
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	2.00000	0.0912871
$481$	12.0000	0.547153
$482$	−10.0000	−0.455488
$483$	24.0000	1.09204
$484$	1.00000	0.0454545
$485$	−5.00000	−0.227038
$486$	−10.0000	−0.453609
$487$	−18.0000	−0.815658	−0.407829	−	0.913058i	$-0.633714\pi$
$487$	−18.0000	−0.815658	−0.407829	+	0.913058i	$0.633714\pi$
$488$	−12.0000	−0.543214
$489$	−8.00000	−0.361773
$490$	−2.00000	−0.0903508
$491$	1.00000	0.0451294	0.0225647	−	0.999745i	$-0.492817\pi$
$491$	1.00000	0.0451294	0.0225647	+	0.999745i	$0.492817\pi$
$492$	12.0000	0.541002
$493$	40.0000	1.80151
$494$	0	0
$495$	−1.00000	−0.0449467
$496$	−4.00000	−0.179605
$497$	42.0000	1.88396
$498$	−26.0000	−1.16509
$499$	−40.0000	−1.79065	−0.895323	−	0.445418i	$-0.853055\pi$
$499$	−40.0000	−1.79065	−0.895323	+	0.445418i	$0.853055\pi$
$500$	−9.00000	−0.402492
$501$	34.0000	1.51901
$502$	0	0
$503$	8.00000	0.356702	0.178351	−	0.983967i	$-0.442924\pi$
$503$	8.00000	0.356702	0.178351	+	0.983967i	$0.442924\pi$
$504$	3.00000	0.133631
$505$	0	0
$506$	4.00000	0.177822
$507$	−6.00000	−0.266469
$508$	8.00000	0.354943
$509$	13.0000	0.576215	0.288107	−	0.957598i	$-0.406974\pi$
$509$	13.0000	0.576215	0.288107	+	0.957598i	$0.406974\pi$
$510$	−8.00000	−0.354246
$511$	−18.0000	−0.796273
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−1.00000	−0.0441081
$515$	0	0
$516$	8.00000	0.352180
$517$	−6.00000	−0.263880
$518$	−9.00000	−0.395437
$519$	−8.00000	−0.351161
$520$	4.00000	0.175412
$521$	−25.0000	−1.09527	−0.547635	−	0.836717i	$-0.684472\pi$
$521$	−25.0000	−1.09527	−0.547635	+	0.836717i	$0.684472\pi$
$522$	10.0000	0.437688
$523$	−9.00000	−0.393543	−0.196771	−	0.980449i	$-0.563046\pi$
$523$	−9.00000	−0.393543	−0.196771	+	0.980449i	$0.563046\pi$
$524$	−16.0000	−0.698963
$525$	−24.0000	−1.04745
$526$	23.0000	1.00285
$527$	16.0000	0.696971
$528$	2.00000	0.0870388
$529$	−7.00000	−0.304348
$530$	−1.00000	−0.0434372
$531$	−12.0000	−0.520756
$532$	0	0
$533$	24.0000	1.03956
$534$	4.00000	0.173097
$535$	7.00000	0.302636
$536$	12.0000	0.518321
$537$	−40.0000	−1.72613
$538$	15.0000	0.646696
$539$	−2.00000	−0.0861461
$540$	4.00000	0.172133
$541$	−22.0000	−0.945854	−0.472927	−	0.881102i	$-0.656803\pi$
$541$	−22.0000	−0.945854	−0.472927	+	0.881102i	$0.656803\pi$
$542$	21.0000	0.902027
$543$	−34.0000	−1.45908
$544$	4.00000	0.171499
$545$	4.00000	0.171341
$546$	24.0000	1.02711
$547$	−35.0000	−1.49649	−0.748246	−	0.663421i	$-0.769104\pi$
$547$	−35.0000	−1.49649	−0.748246	+	0.663421i	$0.769104\pi$
$548$	1.00000	0.0427179
$549$	12.0000	0.512148
$550$	−4.00000	−0.170561
$551$	0	0
$552$	8.00000	0.340503
$553$	51.0000	2.16874
$554$	4.00000	0.169944
$555$	6.00000	0.254686
$556$	−17.0000	−0.720961
$557$	−12.0000	−0.508456	−0.254228	−	0.967144i	$-0.581821\pi$
$557$	−12.0000	−0.508456	−0.254228	+	0.967144i	$0.581821\pi$
$558$	4.00000	0.169334
$559$	16.0000	0.676728
$560$	−3.00000	−0.126773
$561$	−8.00000	−0.337760
$562$	4.00000	0.168730
$563$	33.0000	1.39078	0.695392	−	0.718631i	$-0.255231\pi$
$563$	33.0000	1.39078	0.695392	+	0.718631i	$0.255231\pi$
$564$	−12.0000	−0.505291
$565$	6.00000	0.252422
$566$	−13.0000	−0.546431
$567$	33.0000	1.38587
$568$	14.0000	0.587427
$569$	36.0000	1.50920	0.754599	−	0.656186i	$-0.227831\pi$
$569$	36.0000	1.50920	0.754599	+	0.656186i	$0.227831\pi$
$570$	0	0
$571$	19.0000	0.795125	0.397563	−	0.917575i	$-0.369856\pi$
$571$	19.0000	0.795125	0.397563	+	0.917575i	$0.369856\pi$
$572$	4.00000	0.167248
$573$	−36.0000	−1.50392
$574$	−18.0000	−0.751305
$575$	−16.0000	−0.667246
$576$	1.00000	0.0416667
$577$	17.0000	0.707719	0.353860	−	0.935299i	$-0.384869\pi$
$577$	17.0000	0.707719	0.353860	+	0.935299i	$0.384869\pi$
$578$	1.00000	0.0415945
$579$	−20.0000	−0.831172
$580$	−10.0000	−0.415227
$581$	39.0000	1.61799
$582$	−10.0000	−0.414513
$583$	−1.00000	−0.0414158
$584$	−6.00000	−0.248282
$585$	−4.00000	−0.165380
$586$	30.0000	1.23929
$587$	24.0000	0.990586	0.495293	−	0.868726i	$-0.335061\pi$
$587$	24.0000	0.990586	0.495293	+	0.868726i	$0.335061\pi$
$588$	−4.00000	−0.164957
$589$	0	0
$590$	12.0000	0.494032
$591$	24.0000	0.987228
$592$	−3.00000	−0.123299
$593$	−12.0000	−0.492781	−0.246390	−	0.969171i	$-0.579245\pi$
$593$	−12.0000	−0.492781	−0.246390	+	0.969171i	$0.579245\pi$
$594$	4.00000	0.164122
$595$	12.0000	0.491952
$596$	−6.00000	−0.245770
$597$	20.0000	0.818546
$598$	16.0000	0.654289
$599$	−32.0000	−1.30748	−0.653742	−	0.756717i	$-0.726802\pi$
$599$	−32.0000	−1.30748	−0.653742	+	0.756717i	$0.726802\pi$
$600$	−8.00000	−0.326599
$601$	10.0000	0.407909	0.203954	−	0.978980i	$-0.434621\pi$
$601$	10.0000	0.407909	0.203954	+	0.978980i	$0.434621\pi$
$602$	−12.0000	−0.489083
$603$	−12.0000	−0.488678
$604$	−15.0000	−0.610341
$605$	1.00000	0.0406558
$606$	0	0
$607$	40.0000	1.62355	0.811775	−	0.583970i	$-0.198502\pi$
$607$	40.0000	1.62355	0.811775	+	0.583970i	$0.198502\pi$
$608$	0	0
$609$	−60.0000	−2.43132
$610$	−12.0000	−0.485866
$611$	−24.0000	−0.970936
$612$	−4.00000	−0.161690
$613$	−16.0000	−0.646234	−0.323117	−	0.946359i	$-0.604731\pi$
$613$	−16.0000	−0.646234	−0.323117	+	0.946359i	$0.604731\pi$
$614$	29.0000	1.17034
$615$	12.0000	0.483887
$616$	−3.00000	−0.120873
$617$	33.0000	1.32853	0.664265	−	0.747497i	$-0.268745\pi$
$617$	33.0000	1.32853	0.664265	+	0.747497i	$0.268745\pi$
$618$	0	0
$619$	−32.0000	−1.28619	−0.643094	−	0.765787i	$-0.722350\pi$
$619$	−32.0000	−1.28619	−0.643094	+	0.765787i	$0.722350\pi$
$620$	−4.00000	−0.160644
$621$	16.0000	0.642058
$622$	18.0000	0.721734
$623$	−6.00000	−0.240385
$624$	8.00000	0.320256
$625$	11.0000	0.440000
$626$	9.00000	0.359712
$627$	0	0
$628$	5.00000	0.199522
$629$	12.0000	0.478471
$630$	3.00000	0.119523
$631$	40.0000	1.59237	0.796187	−	0.605050i	$-0.206847\pi$
$631$	40.0000	1.59237	0.796187	+	0.605050i	$0.206847\pi$
$632$	17.0000	0.676224
$633$	−46.0000	−1.82834
$634$	−18.0000	−0.714871
$635$	8.00000	0.317470
$636$	−2.00000	−0.0793052
$637$	−8.00000	−0.316972
$638$	−10.0000	−0.395904
$639$	−14.0000	−0.553831
$640$	−1.00000	−0.0395285
$641$	41.0000	1.61940	0.809701	−	0.586842i	$-0.199629\pi$
$641$	41.0000	1.61940	0.809701	+	0.586842i	$0.199629\pi$
$642$	14.0000	0.552536
$643$	−26.0000	−1.02534	−0.512670	−	0.858586i	$-0.671344\pi$
$643$	−26.0000	−1.02534	−0.512670	+	0.858586i	$0.671344\pi$
$644$	−12.0000	−0.472866
$645$	8.00000	0.315000
$646$	0	0
$647$	42.0000	1.65119	0.825595	−	0.564263i	$-0.190840\pi$
$647$	42.0000	1.65119	0.825595	+	0.564263i	$0.190840\pi$
$648$	11.0000	0.432121
$649$	12.0000	0.471041
$650$	−16.0000	−0.627572
$651$	−24.0000	−0.940634
$652$	4.00000	0.156652
$653$	10.0000	0.391330	0.195665	−	0.980671i	$-0.437313\pi$
$653$	10.0000	0.391330	0.195665	+	0.980671i	$0.437313\pi$
$654$	8.00000	0.312825
$655$	−16.0000	−0.625172
$656$	−6.00000	−0.234261
$657$	6.00000	0.234082
$658$	18.0000	0.701713
$659$	31.0000	1.20759	0.603794	−	0.797140i	$-0.293655\pi$
$659$	31.0000	1.20759	0.603794	+	0.797140i	$0.293655\pi$
$660$	2.00000	0.0778499
$661$	−5.00000	−0.194477	−0.0972387	−	0.995261i	$-0.531001\pi$
$661$	−5.00000	−0.194477	−0.0972387	+	0.995261i	$0.531001\pi$
$662$	20.0000	0.777322
$663$	−32.0000	−1.24278
$664$	13.0000	0.504498
$665$	0	0
$666$	3.00000	0.116248
$667$	−40.0000	−1.54881
$668$	−17.0000	−0.657750
$669$	32.0000	1.23719
$670$	12.0000	0.463600
$671$	−12.0000	−0.463255
$672$	−6.00000	−0.231455
$673$	42.0000	1.61898	0.809491	−	0.587133i	$-0.199743\pi$
$673$	42.0000	1.61898	0.809491	+	0.587133i	$0.199743\pi$
$674$	28.0000	1.07852
$675$	−16.0000	−0.615840
$676$	3.00000	0.115385
$677$	38.0000	1.46046	0.730229	−	0.683202i	$-0.239413\pi$
$677$	38.0000	1.46046	0.730229	+	0.683202i	$0.239413\pi$
$678$	12.0000	0.460857
$679$	15.0000	0.575647
$680$	4.00000	0.153393
$681$	−6.00000	−0.229920
$682$	−4.00000	−0.153168
$683$	2.00000	0.0765279	0.0382639	−	0.999268i	$-0.487817\pi$
$683$	2.00000	0.0765279	0.0382639	+	0.999268i	$0.487817\pi$
$684$	0	0
$685$	1.00000	0.0382080
$686$	−15.0000	−0.572703
$687$	26.0000	0.991962
$688$	−4.00000	−0.152499
$689$	−4.00000	−0.152388
$690$	8.00000	0.304555
$691$	16.0000	0.608669	0.304334	−	0.952565i	$-0.401566\pi$
$691$	16.0000	0.608669	0.304334	+	0.952565i	$0.401566\pi$
$692$	4.00000	0.152057
$693$	3.00000	0.113961
$694$	−13.0000	−0.493473
$695$	−17.0000	−0.644847
$696$	−20.0000	−0.758098
$697$	24.0000	0.909065
$698$	26.0000	0.984115
$699$	32.0000	1.21035
$700$	12.0000	0.453557
$701$	12.0000	0.453234	0.226617	−	0.973984i	$-0.427233\pi$
$701$	12.0000	0.453234	0.226617	+	0.973984i	$0.427233\pi$
$702$	16.0000	0.603881
$703$	0	0
$704$	−1.00000	−0.0376889
$705$	−12.0000	−0.451946
$706$	−19.0000	−0.715074
$707$	0	0
$708$	24.0000	0.901975
$709$	19.0000	0.713560	0.356780	−	0.934188i	$-0.383875\pi$
$709$	19.0000	0.713560	0.356780	+	0.934188i	$0.383875\pi$
$710$	14.0000	0.525411
$711$	−17.0000	−0.637550
$712$	−2.00000	−0.0749532
$713$	−16.0000	−0.599205
$714$	24.0000	0.898177
$715$	4.00000	0.149592
$716$	20.0000	0.747435
$717$	10.0000	0.373457
$718$	−7.00000	−0.261238
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	1.00000	0.0372678
$721$	0	0
$722$	0	0
$723$	−20.0000	−0.743808
$724$	17.0000	0.631800
$725$	40.0000	1.48556
$726$	2.00000	0.0742270
$727$	14.0000	0.519231	0.259616	−	0.965712i	$-0.416404\pi$
$727$	14.0000	0.519231	0.259616	+	0.965712i	$0.416404\pi$
$728$	−12.0000	−0.444750
$729$	13.0000	0.481481
$730$	−6.00000	−0.222070
$731$	16.0000	0.591781
$732$	−24.0000	−0.887066
$733$	−32.0000	−1.18195	−0.590973	−	0.806691i	$-0.701256\pi$
$733$	−32.0000	−1.18195	−0.590973	+	0.806691i	$0.701256\pi$
$734$	−16.0000	−0.590571
$735$	−4.00000	−0.147542
$736$	−4.00000	−0.147442
$737$	12.0000	0.442026
$738$	6.00000	0.220863
$739$	−4.00000	−0.147142	−0.0735712	−	0.997290i	$-0.523440\pi$
$739$	−4.00000	−0.147142	−0.0735712	+	0.997290i	$0.523440\pi$
$740$	−3.00000	−0.110282
$741$	0	0
$742$	3.00000	0.110133
$743$	−9.00000	−0.330178	−0.165089	−	0.986279i	$-0.552791\pi$
$743$	−9.00000	−0.330178	−0.165089	+	0.986279i	$0.552791\pi$
$744$	−8.00000	−0.293294
$745$	−6.00000	−0.219823
$746$	−28.0000	−1.02515
$747$	−13.0000	−0.475645
$748$	4.00000	0.146254
$749$	−21.0000	−0.767323
$750$	−18.0000	−0.657267
$751$	−10.0000	−0.364905	−0.182453	−	0.983215i	$-0.558404\pi$
$751$	−10.0000	−0.364905	−0.182453	+	0.983215i	$0.558404\pi$
$752$	6.00000	0.218797
$753$	0	0
$754$	−40.0000	−1.45671
$755$	−15.0000	−0.545906
$756$	−12.0000	−0.436436
$757$	−22.0000	−0.799604	−0.399802	−	0.916602i	$-0.630921\pi$
$757$	−22.0000	−0.799604	−0.399802	+	0.916602i	$0.630921\pi$
$758$	4.00000	0.145287
$759$	8.00000	0.290382
$760$	0	0
$761$	22.0000	0.797499	0.398750	−	0.917060i	$-0.369444\pi$
$761$	22.0000	0.797499	0.398750	+	0.917060i	$0.369444\pi$
$762$	16.0000	0.579619
$763$	−12.0000	−0.434429
$764$	18.0000	0.651217
$765$	−4.00000	−0.144620
$766$	8.00000	0.289052
$767$	48.0000	1.73318
$768$	−2.00000	−0.0721688
$769$	−34.0000	−1.22607	−0.613036	−	0.790055i	$-0.710052\pi$
$769$	−34.0000	−1.22607	−0.613036	+	0.790055i	$0.710052\pi$
$770$	−3.00000	−0.108112
$771$	−2.00000	−0.0720282
$772$	10.0000	0.359908
$773$	26.0000	0.935155	0.467578	−	0.883952i	$-0.345127\pi$
$773$	26.0000	0.935155	0.467578	+	0.883952i	$0.345127\pi$
$774$	4.00000	0.143777
$775$	16.0000	0.574737
$776$	5.00000	0.179490
$777$	−18.0000	−0.645746
$778$	−21.0000	−0.752886
$779$	0	0
$780$	8.00000	0.286446
$781$	14.0000	0.500959
$782$	16.0000	0.572159
$783$	−40.0000	−1.42948
$784$	2.00000	0.0714286
$785$	5.00000	0.178458
$786$	−32.0000	−1.14140
$787$	−7.00000	−0.249523	−0.124762	−	0.992187i	$-0.539817\pi$
$787$	−7.00000	−0.249523	−0.124762	+	0.992187i	$0.539817\pi$
$788$	−12.0000	−0.427482
$789$	46.0000	1.63764
$790$	17.0000	0.604833
$791$	−18.0000	−0.640006
$792$	1.00000	0.0355335
$793$	−48.0000	−1.70453
$794$	−25.0000	−0.887217
$795$	−2.00000	−0.0709327
$796$	−10.0000	−0.354441
$797$	−37.0000	−1.31061	−0.655304	−	0.755366i	$-0.727459\pi$
$797$	−37.0000	−1.31061	−0.655304	+	0.755366i	$0.727459\pi$
$798$	0	0
$799$	−24.0000	−0.849059
$800$	4.00000	0.141421
$801$	2.00000	0.0706665
$802$	10.0000	0.353112
$803$	−6.00000	−0.211735
$804$	24.0000	0.846415
$805$	−12.0000	−0.422944
$806$	−16.0000	−0.563576
$807$	30.0000	1.05605
$808$	0	0
$809$	−28.0000	−0.984428	−0.492214	−	0.870474i	$-0.663812\pi$
$809$	−28.0000	−0.984428	−0.492214	+	0.870474i	$0.663812\pi$
$810$	11.0000	0.386501
$811$	−35.0000	−1.22902	−0.614508	−	0.788911i	$-0.710645\pi$
$811$	−35.0000	−1.22902	−0.614508	+	0.788911i	$0.710645\pi$
$812$	30.0000	1.05279
$813$	42.0000	1.47300
$814$	−3.00000	−0.105150
$815$	4.00000	0.140114
$816$	8.00000	0.280056
$817$	0	0
$818$	−20.0000	−0.699284
$819$	12.0000	0.419314
$820$	−6.00000	−0.209529
$821$	−52.0000	−1.81481	−0.907406	−	0.420255i	$-0.861941\pi$
$821$	−52.0000	−1.81481	−0.907406	+	0.420255i	$0.861941\pi$
$822$	2.00000	0.0697580
$823$	2.00000	0.0697156	0.0348578	−	0.999392i	$-0.488902\pi$
$823$	2.00000	0.0697156	0.0348578	+	0.999392i	$0.488902\pi$
$824$	0	0
$825$	−8.00000	−0.278524
$826$	−36.0000	−1.25260
$827$	44.0000	1.53003	0.765015	−	0.644013i	$-0.222732\pi$
$827$	44.0000	1.53003	0.765015	+	0.644013i	$0.222732\pi$
$828$	4.00000	0.139010
$829$	−46.0000	−1.59765	−0.798823	−	0.601566i	$-0.794544\pi$
$829$	−46.0000	−1.59765	−0.798823	+	0.601566i	$0.794544\pi$
$830$	13.0000	0.451237
$831$	8.00000	0.277517
$832$	−4.00000	−0.138675
$833$	−8.00000	−0.277184
$834$	−34.0000	−1.17732
$835$	−17.0000	−0.588309
$836$	0	0
$837$	−16.0000	−0.553041
$838$	38.0000	1.31269
$839$	56.0000	1.93333	0.966667	−	0.256036i	$-0.0824164\pi$
$839$	56.0000	1.93333	0.966667	+	0.256036i	$0.0824164\pi$
$840$	−6.00000	−0.207020
$841$	71.0000	2.44828
$842$	−3.00000	−0.103387
$843$	8.00000	0.275535
$844$	23.0000	0.791693
$845$	3.00000	0.103203
$846$	−6.00000	−0.206284
$847$	−3.00000	−0.103081
$848$	1.00000	0.0343401
$849$	−26.0000	−0.892318
$850$	−16.0000	−0.548795
$851$	−12.0000	−0.411355
$852$	28.0000	0.959264
$853$	42.0000	1.43805	0.719026	−	0.694983i	$-0.244588\pi$
$853$	42.0000	1.43805	0.719026	+	0.694983i	$0.244588\pi$
$854$	36.0000	1.23189
$855$	0	0
$856$	−7.00000	−0.239255
$857$	−48.0000	−1.63965	−0.819824	−	0.572615i	$-0.805929\pi$
$857$	−48.0000	−1.63965	−0.819824	+	0.572615i	$0.805929\pi$
$858$	8.00000	0.273115
$859$	−6.00000	−0.204717	−0.102359	−	0.994748i	$-0.532639\pi$
$859$	−6.00000	−0.204717	−0.102359	+	0.994748i	$0.532639\pi$
$860$	−4.00000	−0.136399
$861$	−36.0000	−1.22688
$862$	13.0000	0.442782
$863$	6.00000	0.204242	0.102121	−	0.994772i	$-0.467437\pi$
$863$	6.00000	0.204242	0.102121	+	0.994772i	$0.467437\pi$
$864$	−4.00000	−0.136083
$865$	4.00000	0.136004
$866$	−18.0000	−0.611665
$867$	2.00000	0.0679236
$868$	12.0000	0.407307
$869$	17.0000	0.576686
$870$	−20.0000	−0.678064
$871$	48.0000	1.62642
$872$	−4.00000	−0.135457
$873$	−5.00000	−0.169224
$874$	0	0
$875$	27.0000	0.912767
$876$	−12.0000	−0.405442
$877$	12.0000	0.405211	0.202606	−	0.979260i	$-0.435059\pi$
$877$	12.0000	0.405211	0.202606	+	0.979260i	$0.435059\pi$
$878$	1.00000	0.0337484
$879$	60.0000	2.02375
$880$	−1.00000	−0.0337100
$881$	−6.00000	−0.202145	−0.101073	−	0.994879i	$-0.532227\pi$
$881$	−6.00000	−0.202145	−0.101073	+	0.994879i	$0.532227\pi$
$882$	−2.00000	−0.0673435
$883$	−26.0000	−0.874970	−0.437485	−	0.899226i	$-0.644131\pi$
$883$	−26.0000	−0.874970	−0.437485	+	0.899226i	$0.644131\pi$
$884$	16.0000	0.538138
$885$	24.0000	0.806751
$886$	−24.0000	−0.806296
$887$	−29.0000	−0.973725	−0.486862	−	0.873479i	$-0.661859\pi$
$887$	−29.0000	−0.973725	−0.486862	+	0.873479i	$0.661859\pi$
$888$	−6.00000	−0.201347
$889$	−24.0000	−0.804934
$890$	−2.00000	−0.0670402
$891$	11.0000	0.368514
$892$	−16.0000	−0.535720
$893$	0	0
$894$	−12.0000	−0.401340
$895$	20.0000	0.668526
$896$	3.00000	0.100223
$897$	32.0000	1.06845
$898$	−33.0000	−1.10122
$899$	40.0000	1.33407
$900$	−4.00000	−0.133333
$901$	−4.00000	−0.133259
$902$	−6.00000	−0.199778
$903$	−24.0000	−0.798670
$904$	−6.00000	−0.199557
$905$	17.0000	0.565099
$906$	−30.0000	−0.996683
$907$	2.00000	0.0664089	0.0332045	−	0.999449i	$-0.489429\pi$
$907$	2.00000	0.0664089	0.0332045	+	0.999449i	$0.489429\pi$
$908$	3.00000	0.0995585
$909$	0	0
$910$	−12.0000	−0.397796
$911$	−30.0000	−0.993944	−0.496972	−	0.867766i	$-0.665555\pi$
$911$	−30.0000	−0.993944	−0.496972	+	0.867766i	$0.665555\pi$
$912$	0	0
$913$	13.0000	0.430237
$914$	−38.0000	−1.25693
$915$	−24.0000	−0.793416
$916$	−13.0000	−0.429532
$917$	48.0000	1.58510
$918$	16.0000	0.528079
$919$	45.0000	1.48441	0.742207	−	0.670171i	$-0.233779\pi$
$919$	45.0000	1.48441	0.742207	+	0.670171i	$0.233779\pi$
$920$	−4.00000	−0.131876
$921$	58.0000	1.91116
$922$	2.00000	0.0658665
$923$	56.0000	1.84326
$924$	−6.00000	−0.197386
$925$	12.0000	0.394558
$926$	−4.00000	−0.131448
$927$	0	0
$928$	10.0000	0.328266
$929$	14.0000	0.459325	0.229663	−	0.973270i	$-0.426238\pi$
$929$	14.0000	0.459325	0.229663	+	0.973270i	$0.426238\pi$
$930$	−8.00000	−0.262330
$931$	0	0
$932$	−16.0000	−0.524097
$933$	36.0000	1.17859
$934$	−14.0000	−0.458094
$935$	4.00000	0.130814
$936$	4.00000	0.130744
$937$	−38.0000	−1.24141	−0.620703	−	0.784046i	$-0.713153\pi$
$937$	−38.0000	−1.24141	−0.620703	+	0.784046i	$0.713153\pi$
$938$	−36.0000	−1.17544
$939$	18.0000	0.587408
$940$	6.00000	0.195698
$941$	24.0000	0.782378	0.391189	−	0.920310i	$-0.372064\pi$
$941$	24.0000	0.782378	0.391189	+	0.920310i	$0.372064\pi$
$942$	10.0000	0.325818
$943$	−24.0000	−0.781548
$944$	−12.0000	−0.390567
$945$	−12.0000	−0.390360
$946$	−4.00000	−0.130051
$947$	−30.0000	−0.974869	−0.487435	−	0.873160i	$-0.662067\pi$
$947$	−30.0000	−0.974869	−0.487435	+	0.873160i	$0.662067\pi$
$948$	34.0000	1.10427
$949$	−24.0000	−0.779073
$950$	0	0
$951$	−36.0000	−1.16738
$952$	−12.0000	−0.388922
$953$	0	0		−	1.00000i	$-0.5\pi$
$953$	0	0			1.00000i	$0.5\pi$
$954$	−1.00000	−0.0323762
$955$	18.0000	0.582466
$956$	−5.00000	−0.161712
$957$	−20.0000	−0.646508
$958$	0	0
$959$	−3.00000	−0.0968751
$960$	−2.00000	−0.0645497
$961$	−15.0000	−0.483871
$962$	−12.0000	−0.386896
$963$	7.00000	0.225572
$964$	10.0000	0.322078
$965$	10.0000	0.321911
$966$	−24.0000	−0.772187
$967$	−37.0000	−1.18984	−0.594920	−	0.803785i	$-0.702816\pi$
$967$	−37.0000	−1.18984	−0.594920	+	0.803785i	$0.702816\pi$
$968$	−1.00000	−0.0321412
$969$	0	0
$970$	5.00000	0.160540
$971$	50.0000	1.60458	0.802288	−	0.596937i	$-0.203616\pi$
$971$	50.0000	1.60458	0.802288	+	0.596937i	$0.203616\pi$
$972$	10.0000	0.320750
$973$	51.0000	1.63498
$974$	18.0000	0.576757
$975$	−32.0000	−1.02482
$976$	12.0000	0.384111
$977$	−9.00000	−0.287936	−0.143968	−	0.989582i	$-0.545986\pi$
$977$	−9.00000	−0.287936	−0.143968	+	0.989582i	$0.545986\pi$
$978$	8.00000	0.255812
$979$	−2.00000	−0.0639203
$980$	2.00000	0.0638877
$981$	4.00000	0.127710
$982$	−1.00000	−0.0319113
$983$	−38.0000	−1.21201	−0.606006	−	0.795460i	$-0.707229\pi$
$983$	−38.0000	−1.21201	−0.606006	+	0.795460i	$0.707229\pi$
$984$	−12.0000	−0.382546
$985$	−12.0000	−0.382352
$986$	−40.0000	−1.27386
$987$	36.0000	1.14589
$988$	0	0
$989$	−16.0000	−0.508770
$990$	1.00000	0.0317821
$991$	−38.0000	−1.20711	−0.603555	−	0.797321i	$-0.706250\pi$
$991$	−38.0000	−1.20711	−0.603555	+	0.797321i	$0.706250\pi$
$992$	4.00000	0.127000
$993$	40.0000	1.26936
$994$	−42.0000	−1.33216
$995$	−10.0000	−0.317021
$996$	26.0000	0.823842
$997$	44.0000	1.39349	0.696747	−	0.717317i	$-0.254630\pi$
$997$	44.0000	1.39349	0.696747	+	0.717317i	$0.254630\pi$
$998$	40.0000	1.26618
$999$	−12.0000	−0.379663

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7942.2.a.c.1.1		1
19.8	odd	6		418.2.e.a.45.1	✓	2
19.12	odd	6		418.2.e.a.353.1	yes	2
19.18	odd	2		7942.2.a.s.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
418.2.e.a.45.1	✓	2	19.8	odd	6
418.2.e.a.353.1	yes	2	19.12	odd	6
7942.2.a.c.1.1		1	1.1	even	1	trivial
7942.2.a.s.1.1		1	19.18	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:	\( N \)	\(=\)	\( 7942 = 2 \cdot 11 \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7942.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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