LMFDB - Embedded newform 5610.2.a.j.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5610,2,Mod(1,5610)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5610, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("5610.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5610 = 2 \cdot 3 \cdot 5 \cdot 11 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5610.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:	$44.7960755339$
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	$\chi$	$=$	5610.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

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

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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.00000	0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	1.00000	0.408248
$7$	3.00000	1.13389	0.566947	−	0.823754i	$-0.308125\pi$
$7$	3.00000	1.13389	0.566947	+	0.823754i	$0.308125\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	−1.00000	−0.316228
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	−1.00000	−0.288675
$13$	−1.00000	−0.277350	−0.138675	−	0.990338i	$-0.544284\pi$
$13$	−1.00000	−0.277350	−0.138675	+	0.990338i	$0.544284\pi$
$14$	−3.00000	−0.801784
$15$	−1.00000	−0.258199
$16$	1.00000	0.250000
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	−1.00000	−0.235702
$19$	−5.00000	−1.14708	−0.573539	−	0.819178i	$-0.694430\pi$
$19$	−5.00000	−1.14708	−0.573539	+	0.819178i	$0.694430\pi$
$20$	1.00000	0.223607
$21$	−3.00000	−0.654654
$22$	−1.00000	−0.213201
$23$	5.00000	1.04257	0.521286	−	0.853382i	$-0.325452\pi$
$23$	5.00000	1.04257	0.521286	+	0.853382i	$0.325452\pi$
$24$	1.00000	0.204124
$25$	1.00000	0.200000
$26$	1.00000	0.196116
$27$	−1.00000	−0.192450
$28$	3.00000	0.566947
$29$	−6.00000	−1.11417	−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000	−1.11417	−0.557086	+	0.830455i	$0.688081\pi$
$30$	1.00000	0.182574
$31$	1.00000	0.179605	0.0898027	−	0.995960i	$-0.471376\pi$
$31$	1.00000	0.179605	0.0898027	+	0.995960i	$0.471376\pi$
$32$	−1.00000	−0.176777
$33$	−1.00000	−0.174078
$34$	1.00000	0.171499
$35$	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	$0.579313\pi$
$38$	5.00000	0.811107
$39$	1.00000	0.160128
$40$	−1.00000	−0.158114
$41$	2.00000	0.312348	0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000	0.312348	0.156174	+	0.987730i	$0.450084\pi$
$42$	3.00000	0.462910
$43$	−8.00000	−1.21999	−0.609994	−	0.792406i	$-0.708828\pi$
$43$	−8.00000	−1.21999	−0.609994	+	0.792406i	$0.708828\pi$
$44$	1.00000	0.150756
$45$	1.00000	0.149071
$46$	−5.00000	−0.737210
$47$	−6.00000	−0.875190	−0.437595	−	0.899172i	$-0.644170\pi$
$47$	−6.00000	−0.875190	−0.437595	+	0.899172i	$0.644170\pi$
$48$	−1.00000	−0.144338
$49$	2.00000	0.285714
$50$	−1.00000	−0.141421
$51$	1.00000	0.140028
$52$	−1.00000	−0.138675
$53$	−10.0000	−1.37361	−0.686803	−	0.726844i	$-0.740986\pi$
$53$	−10.0000	−1.37361	−0.686803	+	0.726844i	$0.740986\pi$
$54$	1.00000	0.136083
$55$	1.00000	0.134840
$56$	−3.00000	−0.400892
$57$	5.00000	0.662266
$58$	6.00000	0.787839
$59$	−12.0000	−1.56227	−0.781133	−	0.624364i	$-0.785358\pi$
$59$	−12.0000	−1.56227	−0.781133	+	0.624364i	$0.785358\pi$
$60$	−1.00000	−0.129099
$61$	11.0000	1.40841	0.704203	−	0.709999i	$-0.251305\pi$
$61$	11.0000	1.40841	0.704203	+	0.709999i	$0.251305\pi$
$62$	−1.00000	−0.127000
$63$	3.00000	0.377964
$64$	1.00000	0.125000
$65$	−1.00000	−0.124035
$66$	1.00000	0.123091
$67$	−7.00000	−0.855186	−0.427593	−	0.903971i	$-0.640638\pi$
$67$	−7.00000	−0.855186	−0.427593	+	0.903971i	$0.640638\pi$
$68$	−1.00000	−0.121268
$69$	−5.00000	−0.601929
$70$	−3.00000	−0.358569
$71$	−10.0000	−1.18678	−0.593391	−	0.804914i	$-0.702211\pi$
$71$	−10.0000	−1.18678	−0.593391	+	0.804914i	$0.702211\pi$
$72$	−1.00000	−0.117851
$73$	0	0		−	1.00000i	$-0.5\pi$
$73$	0	0			1.00000i	$0.5\pi$
$74$	3.00000	0.348743
$75$	−1.00000	−0.115470
$76$	−5.00000	−0.573539
$77$	3.00000	0.341882
$78$	−1.00000	−0.113228
$79$	−2.00000	−0.225018	−0.112509	−	0.993651i	$-0.535889\pi$
$79$	−2.00000	−0.225018	−0.112509	+	0.993651i	$0.535889\pi$
$80$	1.00000	0.111803
$81$	1.00000	0.111111
$82$	−2.00000	−0.220863
$83$	7.00000	0.768350	0.384175	−	0.923260i	$-0.374486\pi$
$83$	7.00000	0.768350	0.384175	+	0.923260i	$0.374486\pi$
$84$	−3.00000	−0.327327
$85$	−1.00000	−0.108465
$86$	8.00000	0.862662
$87$	6.00000	0.643268
$88$	−1.00000	−0.106600
$89$	−10.0000	−1.06000	−0.529999	−	0.847998i	$-0.677808\pi$
$89$	−10.0000	−1.06000	−0.529999	+	0.847998i	$0.677808\pi$
$90$	−1.00000	−0.105409
$91$	−3.00000	−0.314485
$92$	5.00000	0.521286
$93$	−1.00000	−0.103695
$94$	6.00000	0.618853
$95$	−5.00000	−0.512989
$96$	1.00000	0.102062
$97$	−7.00000	−0.710742	−0.355371	−	0.934725i	$-0.615646\pi$
$97$	−7.00000	−0.710742	−0.355371	+	0.934725i	$0.615646\pi$
$98$	−2.00000	−0.202031
$99$	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	$0.745272\pi$
$102$	−1.00000	−0.0990148
$103$	1.00000	0.0985329	0.0492665	−	0.998786i	$-0.484312\pi$
$103$	1.00000	0.0985329	0.0492665	+	0.998786i	$0.484312\pi$
$104$	1.00000	0.0980581
$105$	−3.00000	−0.292770
$106$	10.0000	0.971286
$107$	−10.0000	−0.966736	−0.483368	−	0.875417i	$-0.660587\pi$
$107$	−10.0000	−0.966736	−0.483368	+	0.875417i	$0.660587\pi$
$108$	−1.00000	−0.0962250
$109$	−7.00000	−0.670478	−0.335239	−	0.942133i	$-0.608817\pi$
$109$	−7.00000	−0.670478	−0.335239	+	0.942133i	$0.608817\pi$
$110$	−1.00000	−0.0953463
$111$	3.00000	0.284747
$112$	3.00000	0.283473
$113$	10.0000	0.940721	0.470360	−	0.882474i	$-0.344124\pi$
$113$	10.0000	0.940721	0.470360	+	0.882474i	$0.344124\pi$
$114$	−5.00000	−0.468293
$115$	5.00000	0.466252
$116$	−6.00000	−0.557086
$117$	−1.00000	−0.0924500
$118$	12.0000	1.10469
$119$	−3.00000	−0.275010
$120$	1.00000	0.0912871
$121$	1.00000	0.0909091
$122$	−11.0000	−0.995893
$123$	−2.00000	−0.180334
$124$	1.00000	0.0898027
$125$	1.00000	0.0894427
$126$	−3.00000	−0.267261
$127$	12.0000	1.06483	0.532414	−	0.846484i	$-0.321285\pi$
$127$	12.0000	1.06483	0.532414	+	0.846484i	$0.321285\pi$
$128$	−1.00000	−0.0883883
$129$	8.00000	0.704361
$130$	1.00000	0.0877058
$131$	−9.00000	−0.786334	−0.393167	−	0.919467i	$-0.628621\pi$
$131$	−9.00000	−0.786334	−0.393167	+	0.919467i	$0.628621\pi$
$132$	−1.00000	−0.0870388
$133$	−15.0000	−1.30066
$134$	7.00000	0.604708
$135$	−1.00000	−0.0860663
$136$	1.00000	0.0857493
$137$	−7.00000	−0.598050	−0.299025	−	0.954245i	$-0.596661\pi$
$137$	−7.00000	−0.598050	−0.299025	+	0.954245i	$0.596661\pi$
$138$	5.00000	0.425628
$139$	22.0000	1.86602	0.933008	−	0.359856i	$-0.117174\pi$
$139$	22.0000	1.86602	0.933008	+	0.359856i	$0.117174\pi$
$140$	3.00000	0.253546
$141$	6.00000	0.505291
$142$	10.0000	0.839181
$143$	−1.00000	−0.0836242
$144$	1.00000	0.0833333
$145$	−6.00000	−0.498273
$146$	0	0
$147$	−2.00000	−0.164957
$148$	−3.00000	−0.246598
$149$	−5.00000	−0.409616	−0.204808	−	0.978802i	$-0.565657\pi$
$149$	−5.00000	−0.409616	−0.204808	+	0.978802i	$0.565657\pi$
$150$	1.00000	0.0816497
$151$	9.00000	0.732410	0.366205	−	0.930534i	$-0.380657\pi$
$151$	9.00000	0.732410	0.366205	+	0.930534i	$0.380657\pi$
$152$	5.00000	0.405554
$153$	−1.00000	−0.0808452
$154$	−3.00000	−0.241747
$155$	1.00000	0.0803219
$156$	1.00000	0.0800641
$157$	12.0000	0.957704	0.478852	−	0.877896i	$-0.341053\pi$
$157$	12.0000	0.957704	0.478852	+	0.877896i	$0.341053\pi$
$158$	2.00000	0.159111
$159$	10.0000	0.793052
$160$	−1.00000	−0.0790569
$161$	15.0000	1.18217
$162$	−1.00000	−0.0785674
$163$	24.0000	1.87983	0.939913	−	0.341415i	$-0.110906\pi$
$163$	24.0000	1.87983	0.939913	+	0.341415i	$0.110906\pi$
$164$	2.00000	0.156174
$165$	−1.00000	−0.0778499
$166$	−7.00000	−0.543305
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	3.00000	0.231455
$169$	−12.0000	−0.923077
$170$	1.00000	0.0766965
$171$	−5.00000	−0.382360
$172$	−8.00000	−0.609994
$173$	21.0000	1.59660	0.798300	−	0.602260i	$-0.205733\pi$
$173$	21.0000	1.59660	0.798300	+	0.602260i	$0.205733\pi$
$174$	−6.00000	−0.454859
$175$	3.00000	0.226779
$176$	1.00000	0.0753778
$177$	12.0000	0.901975
$178$	10.0000	0.749532
$179$	−21.0000	−1.56961	−0.784807	−	0.619740i	$-0.787238\pi$
$179$	−21.0000	−1.56961	−0.784807	+	0.619740i	$0.787238\pi$
$180$	1.00000	0.0745356
$181$	−20.0000	−1.48659	−0.743294	−	0.668965i	$-0.766738\pi$
$181$	−20.0000	−1.48659	−0.743294	+	0.668965i	$0.766738\pi$
$182$	3.00000	0.222375
$183$	−11.0000	−0.813143
$184$	−5.00000	−0.368605
$185$	−3.00000	−0.220564
$186$	1.00000	0.0733236
$187$	−1.00000	−0.0731272
$188$	−6.00000	−0.437595
$189$	−3.00000	−0.218218
$190$	5.00000	0.362738
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	−1.00000	−0.0721688
$193$	24.0000	1.72756	0.863779	−	0.503871i	$-0.168091\pi$
$193$	24.0000	1.72756	0.863779	+	0.503871i	$0.168091\pi$
$194$	7.00000	0.502571
$195$	1.00000	0.0716115
$196$	2.00000	0.142857
$197$	−15.0000	−1.06871	−0.534353	−	0.845262i	$-0.679445\pi$
$197$	−15.0000	−1.06871	−0.534353	+	0.845262i	$0.679445\pi$
$198$	−1.00000	−0.0710669
$199$	17.0000	1.20510	0.602549	−	0.798082i	$-0.294152\pi$
$199$	17.0000	1.20510	0.602549	+	0.798082i	$0.294152\pi$
$200$	−1.00000	−0.0707107
$201$	7.00000	0.493742
$202$	14.0000	0.985037
$203$	−18.0000	−1.26335
$204$	1.00000	0.0700140
$205$	2.00000	0.139686
$206$	−1.00000	−0.0696733
$207$	5.00000	0.347524
$208$	−1.00000	−0.0693375
$209$	−5.00000	−0.345857
$210$	3.00000	0.207020
$211$	18.0000	1.23917	0.619586	−	0.784929i	$-0.287301\pi$
$211$	18.0000	1.23917	0.619586	+	0.784929i	$0.287301\pi$
$212$	−10.0000	−0.686803
$213$	10.0000	0.685189
$214$	10.0000	0.683586
$215$	−8.00000	−0.545595
$216$	1.00000	0.0680414
$217$	3.00000	0.203653
$218$	7.00000	0.474100
$219$	0	0
$220$	1.00000	0.0674200
$221$	1.00000	0.0672673
$222$	−3.00000	−0.201347
$223$	−1.00000	−0.0669650	−0.0334825	−	0.999439i	$-0.510660\pi$
$223$	−1.00000	−0.0669650	−0.0334825	+	0.999439i	$0.510660\pi$
$224$	−3.00000	−0.200446
$225$	1.00000	0.0666667
$226$	−10.0000	−0.665190
$227$	−18.0000	−1.19470	−0.597351	−	0.801980i	$-0.703780\pi$
$227$	−18.0000	−1.19470	−0.597351	+	0.801980i	$0.703780\pi$
$228$	5.00000	0.331133
$229$	−21.0000	−1.38772	−0.693860	−	0.720110i	$-0.744091\pi$
$229$	−21.0000	−1.38772	−0.693860	+	0.720110i	$0.744091\pi$
$230$	−5.00000	−0.329690
$231$	−3.00000	−0.197386
$232$	6.00000	0.393919
$233$	26.0000	1.70332	0.851658	−	0.524097i	$-0.175597\pi$
$233$	26.0000	1.70332	0.851658	+	0.524097i	$0.175597\pi$
$234$	1.00000	0.0653720
$235$	−6.00000	−0.391397
$236$	−12.0000	−0.781133
$237$	2.00000	0.129914
$238$	3.00000	0.194461
$239$	18.0000	1.16432	0.582162	−	0.813073i	$-0.302207\pi$
$239$	18.0000	1.16432	0.582162	+	0.813073i	$0.302207\pi$
$240$	−1.00000	−0.0645497
$241$	−15.0000	−0.966235	−0.483117	−	0.875556i	$-0.660496\pi$
$241$	−15.0000	−0.966235	−0.483117	+	0.875556i	$0.660496\pi$
$242$	−1.00000	−0.0642824
$243$	−1.00000	−0.0641500
$244$	11.0000	0.704203
$245$	2.00000	0.127775
$246$	2.00000	0.127515
$247$	5.00000	0.318142
$248$	−1.00000	−0.0635001
$249$	−7.00000	−0.443607
$250$	−1.00000	−0.0632456
$251$	15.0000	0.946792	0.473396	−	0.880850i	$-0.343028\pi$
$251$	15.0000	0.946792	0.473396	+	0.880850i	$0.343028\pi$
$252$	3.00000	0.188982
$253$	5.00000	0.314347
$254$	−12.0000	−0.752947
$255$	1.00000	0.0626224
$256$	1.00000	0.0625000
$257$	−14.0000	−0.873296	−0.436648	−	0.899632i	$-0.643834\pi$
$257$	−14.0000	−0.873296	−0.436648	+	0.899632i	$0.643834\pi$
$258$	−8.00000	−0.498058
$259$	−9.00000	−0.559233
$260$	−1.00000	−0.0620174
$261$	−6.00000	−0.371391
$262$	9.00000	0.556022
$263$	11.0000	0.678289	0.339145	−	0.940734i	$-0.389862\pi$
$263$	11.0000	0.678289	0.339145	+	0.940734i	$0.389862\pi$
$264$	1.00000	0.0615457
$265$	−10.0000	−0.614295
$266$	15.0000	0.919709
$267$	10.0000	0.611990
$268$	−7.00000	−0.427593
$269$	5.00000	0.304855	0.152428	−	0.988315i	$-0.451291\pi$
$269$	5.00000	0.304855	0.152428	+	0.988315i	$0.451291\pi$
$270$	1.00000	0.0608581
$271$	−28.0000	−1.70088	−0.850439	−	0.526073i	$-0.823664\pi$
$271$	−28.0000	−1.70088	−0.850439	+	0.526073i	$0.823664\pi$
$272$	−1.00000	−0.0606339
$273$	3.00000	0.181568
$274$	7.00000	0.422885
$275$	1.00000	0.0603023
$276$	−5.00000	−0.300965
$277$	−8.00000	−0.480673	−0.240337	−	0.970690i	$-0.577258\pi$
$277$	−8.00000	−0.480673	−0.240337	+	0.970690i	$0.577258\pi$
$278$	−22.0000	−1.31947
$279$	1.00000	0.0598684
$280$	−3.00000	−0.179284
$281$	−30.0000	−1.78965	−0.894825	−	0.446417i	$-0.852700\pi$
$281$	−30.0000	−1.78965	−0.894825	+	0.446417i	$0.852700\pi$
$282$	−6.00000	−0.357295
$283$	4.00000	0.237775	0.118888	−	0.992908i	$-0.462067\pi$
$283$	4.00000	0.237775	0.118888	+	0.992908i	$0.462067\pi$
$284$	−10.0000	−0.593391
$285$	5.00000	0.296174
$286$	1.00000	0.0591312
$287$	6.00000	0.354169
$288$	−1.00000	−0.0589256
$289$	1.00000	0.0588235
$290$	6.00000	0.352332
$291$	7.00000	0.410347
$292$	0	0
$293$	0	0		−	1.00000i	$-0.5\pi$
$293$	0	0			1.00000i	$0.5\pi$
$294$	2.00000	0.116642
$295$	−12.0000	−0.698667
$296$	3.00000	0.174371
$297$	−1.00000	−0.0580259
$298$	5.00000	0.289642
$299$	−5.00000	−0.289157
$300$	−1.00000	−0.0577350
$301$	−24.0000	−1.38334
$302$	−9.00000	−0.517892
$303$	14.0000	0.804279
$304$	−5.00000	−0.286770
$305$	11.0000	0.629858
$306$	1.00000	0.0571662
$307$	−16.0000	−0.913168	−0.456584	−	0.889680i	$-0.650927\pi$
$307$	−16.0000	−0.913168	−0.456584	+	0.889680i	$0.650927\pi$
$308$	3.00000	0.170941
$309$	−1.00000	−0.0568880
$310$	−1.00000	−0.0567962
$311$	−6.00000	−0.340229	−0.170114	−	0.985424i	$-0.554414\pi$
$311$	−6.00000	−0.340229	−0.170114	+	0.985424i	$0.554414\pi$
$312$	−1.00000	−0.0566139
$313$	23.0000	1.30004	0.650018	−	0.759918i	$-0.274761\pi$
$313$	23.0000	1.30004	0.650018	+	0.759918i	$0.274761\pi$
$314$	−12.0000	−0.677199
$315$	3.00000	0.169031
$316$	−2.00000	−0.112509
$317$	26.0000	1.46031	0.730153	−	0.683284i	$-0.239449\pi$
$317$	26.0000	1.46031	0.730153	+	0.683284i	$0.239449\pi$
$318$	−10.0000	−0.560772
$319$	−6.00000	−0.335936
$320$	1.00000	0.0559017
$321$	10.0000	0.558146
$322$	−15.0000	−0.835917
$323$	5.00000	0.278207
$324$	1.00000	0.0555556
$325$	−1.00000	−0.0554700
$326$	−24.0000	−1.32924
$327$	7.00000	0.387101
$328$	−2.00000	−0.110432
$329$	−18.0000	−0.992372
$330$	1.00000	0.0550482
$331$	−32.0000	−1.75888	−0.879440	−	0.476011i	$-0.842082\pi$
$331$	−32.0000	−1.75888	−0.879440	+	0.476011i	$0.842082\pi$
$332$	7.00000	0.384175
$333$	−3.00000	−0.164399
$334$	0	0
$335$	−7.00000	−0.382451
$336$	−3.00000	−0.163663
$337$	36.0000	1.96104	0.980522	−	0.196407i	$-0.0629273\pi$
$337$	36.0000	1.96104	0.980522	+	0.196407i	$0.0629273\pi$
$338$	12.0000	0.652714
$339$	−10.0000	−0.543125
$340$	−1.00000	−0.0542326
$341$	1.00000	0.0541530
$342$	5.00000	0.270369
$343$	−15.0000	−0.809924
$344$	8.00000	0.431331
$345$	−5.00000	−0.269191
$346$	−21.0000	−1.12897
$347$	−18.0000	−0.966291	−0.483145	−	0.875540i	$-0.660506\pi$
$347$	−18.0000	−0.966291	−0.483145	+	0.875540i	$0.660506\pi$
$348$	6.00000	0.321634
$349$	14.0000	0.749403	0.374701	−	0.927146i	$-0.377745\pi$
$349$	14.0000	0.749403	0.374701	+	0.927146i	$0.377745\pi$
$350$	−3.00000	−0.160357
$351$	1.00000	0.0533761
$352$	−1.00000	−0.0533002
$353$	31.0000	1.64996	0.824982	−	0.565159i	$-0.191185\pi$
$353$	31.0000	1.64996	0.824982	+	0.565159i	$0.191185\pi$
$354$	−12.0000	−0.637793
$355$	−10.0000	−0.530745
$356$	−10.0000	−0.529999
$357$	3.00000	0.158777
$358$	21.0000	1.10988
$359$	−34.0000	−1.79445	−0.897226	−	0.441572i	$-0.854421\pi$
$359$	−34.0000	−1.79445	−0.897226	+	0.441572i	$0.854421\pi$
$360$	−1.00000	−0.0527046
$361$	6.00000	0.315789
$362$	20.0000	1.05118
$363$	−1.00000	−0.0524864
$364$	−3.00000	−0.157243
$365$	0	0
$366$	11.0000	0.574979
$367$	−8.00000	−0.417597	−0.208798	−	0.977959i	$-0.566955\pi$
$367$	−8.00000	−0.417597	−0.208798	+	0.977959i	$0.566955\pi$
$368$	5.00000	0.260643
$369$	2.00000	0.104116
$370$	3.00000	0.155963
$371$	−30.0000	−1.55752
$372$	−1.00000	−0.0518476
$373$	−14.0000	−0.724893	−0.362446	−	0.932005i	$-0.618058\pi$
$373$	−14.0000	−0.724893	−0.362446	+	0.932005i	$0.618058\pi$
$374$	1.00000	0.0517088
$375$	−1.00000	−0.0516398
$376$	6.00000	0.309426
$377$	6.00000	0.309016
$378$	3.00000	0.154303
$379$	−23.0000	−1.18143	−0.590715	−	0.806880i	$-0.701154\pi$
$379$	−23.0000	−1.18143	−0.590715	+	0.806880i	$0.701154\pi$
$380$	−5.00000	−0.256495
$381$	−12.0000	−0.614779
$382$	0	0
$383$	−34.0000	−1.73732	−0.868659	−	0.495410i	$-0.835018\pi$
$383$	−34.0000	−1.73732	−0.868659	+	0.495410i	$0.835018\pi$
$384$	1.00000	0.0510310
$385$	3.00000	0.152894
$386$	−24.0000	−1.22157
$387$	−8.00000	−0.406663
$388$	−7.00000	−0.355371
$389$	4.00000	0.202808	0.101404	−	0.994845i	$-0.467667\pi$
$389$	4.00000	0.202808	0.101404	+	0.994845i	$0.467667\pi$
$390$	−1.00000	−0.0506370
$391$	−5.00000	−0.252861
$392$	−2.00000	−0.101015
$393$	9.00000	0.453990
$394$	15.0000	0.755689
$395$	−2.00000	−0.100631
$396$	1.00000	0.0502519
$397$	18.0000	0.903394	0.451697	−	0.892171i	$-0.350819\pi$
$397$	18.0000	0.903394	0.451697	+	0.892171i	$0.350819\pi$
$398$	−17.0000	−0.852133
$399$	15.0000	0.750939
$400$	1.00000	0.0500000
$401$	15.0000	0.749064	0.374532	−	0.927214i	$-0.377803\pi$
$401$	15.0000	0.749064	0.374532	+	0.927214i	$0.377803\pi$
$402$	−7.00000	−0.349128
$403$	−1.00000	−0.0498135
$404$	−14.0000	−0.696526
$405$	1.00000	0.0496904
$406$	18.0000	0.893325
$407$	−3.00000	−0.148704
$408$	−1.00000	−0.0495074
$409$	26.0000	1.28562	0.642809	−	0.766027i	$-0.277769\pi$
$409$	26.0000	1.28562	0.642809	+	0.766027i	$0.277769\pi$
$410$	−2.00000	−0.0987730
$411$	7.00000	0.345285
$412$	1.00000	0.0492665
$413$	−36.0000	−1.77144
$414$	−5.00000	−0.245737
$415$	7.00000	0.343616
$416$	1.00000	0.0490290
$417$	−22.0000	−1.07734
$418$	5.00000	0.244558
$419$	−32.0000	−1.56330	−0.781651	−	0.623716i	$-0.785622\pi$
$419$	−32.0000	−1.56330	−0.781651	+	0.623716i	$0.785622\pi$
$420$	−3.00000	−0.146385
$421$	35.0000	1.70580	0.852898	−	0.522078i	$-0.174843\pi$
$421$	35.0000	1.70580	0.852898	+	0.522078i	$0.174843\pi$
$422$	−18.0000	−0.876226
$423$	−6.00000	−0.291730
$424$	10.0000	0.485643
$425$	−1.00000	−0.0485071
$426$	−10.0000	−0.484502
$427$	33.0000	1.59698
$428$	−10.0000	−0.483368
$429$	1.00000	0.0482805
$430$	8.00000	0.385794
$431$	32.0000	1.54139	0.770693	−	0.637207i	$-0.219910\pi$
$431$	32.0000	1.54139	0.770693	+	0.637207i	$0.219910\pi$
$432$	−1.00000	−0.0481125
$433$	6.00000	0.288342	0.144171	−	0.989553i	$-0.453949\pi$
$433$	6.00000	0.288342	0.144171	+	0.989553i	$0.453949\pi$
$434$	−3.00000	−0.144005
$435$	6.00000	0.287678
$436$	−7.00000	−0.335239
$437$	−25.0000	−1.19591
$438$	0	0
$439$	−34.0000	−1.62273	−0.811366	−	0.584539i	$-0.801275\pi$
$439$	−34.0000	−1.62273	−0.811366	+	0.584539i	$0.801275\pi$
$440$	−1.00000	−0.0476731
$441$	2.00000	0.0952381
$442$	−1.00000	−0.0475651
$443$	−26.0000	−1.23530	−0.617649	−	0.786454i	$-0.711915\pi$
$443$	−26.0000	−1.23530	−0.617649	+	0.786454i	$0.711915\pi$
$444$	3.00000	0.142374
$445$	−10.0000	−0.474045
$446$	1.00000	0.0473514
$447$	5.00000	0.236492
$448$	3.00000	0.141737
$449$	−19.0000	−0.896665	−0.448333	−	0.893867i	$-0.647982\pi$
$449$	−19.0000	−0.896665	−0.448333	+	0.893867i	$0.647982\pi$
$450$	−1.00000	−0.0471405
$451$	2.00000	0.0941763
$452$	10.0000	0.470360
$453$	−9.00000	−0.422857
$454$	18.0000	0.844782
$455$	−3.00000	−0.140642
$456$	−5.00000	−0.234146
$457$	−7.00000	−0.327446	−0.163723	−	0.986506i	$-0.552350\pi$
$457$	−7.00000	−0.327446	−0.163723	+	0.986506i	$0.552350\pi$
$458$	21.0000	0.981266
$459$	1.00000	0.0466760
$460$	5.00000	0.233126
$461$	−14.0000	−0.652045	−0.326023	−	0.945362i	$-0.605709\pi$
$461$	−14.0000	−0.652045	−0.326023	+	0.945362i	$0.605709\pi$
$462$	3.00000	0.139573
$463$	3.00000	0.139422	0.0697109	−	0.997567i	$-0.477792\pi$
$463$	3.00000	0.139422	0.0697109	+	0.997567i	$0.477792\pi$
$464$	−6.00000	−0.278543
$465$	−1.00000	−0.0463739
$466$	−26.0000	−1.20443
$467$	24.0000	1.11059	0.555294	−	0.831654i	$-0.312606\pi$
$467$	24.0000	1.11059	0.555294	+	0.831654i	$0.312606\pi$
$468$	−1.00000	−0.0462250
$469$	−21.0000	−0.969690
$470$	6.00000	0.276759
$471$	−12.0000	−0.552931
$472$	12.0000	0.552345
$473$	−8.00000	−0.367840
$474$	−2.00000	−0.0918630
$475$	−5.00000	−0.229416
$476$	−3.00000	−0.137505
$477$	−10.0000	−0.457869
$478$	−18.0000	−0.823301
$479$	5.00000	0.228456	0.114228	−	0.993455i	$-0.463561\pi$
$479$	5.00000	0.228456	0.114228	+	0.993455i	$0.463561\pi$
$480$	1.00000	0.0456435
$481$	3.00000	0.136788
$482$	15.0000	0.683231
$483$	−15.0000	−0.682524
$484$	1.00000	0.0454545
$485$	−7.00000	−0.317854
$486$	1.00000	0.0453609
$487$	−40.0000	−1.81257	−0.906287	−	0.422664i	$-0.861095\pi$
$487$	−40.0000	−1.81257	−0.906287	+	0.422664i	$0.861095\pi$
$488$	−11.0000	−0.497947
$489$	−24.0000	−1.08532
$490$	−2.00000	−0.0903508
$491$	2.00000	0.0902587	0.0451294	−	0.998981i	$-0.485630\pi$
$491$	2.00000	0.0902587	0.0451294	+	0.998981i	$0.485630\pi$
$492$	−2.00000	−0.0901670
$493$	6.00000	0.270226
$494$	−5.00000	−0.224961
$495$	1.00000	0.0449467
$496$	1.00000	0.0449013
$497$	−30.0000	−1.34568
$498$	7.00000	0.313678
$499$	−20.0000	−0.895323	−0.447661	−	0.894203i	$-0.647743\pi$
$499$	−20.0000	−0.895323	−0.447661	+	0.894203i	$0.647743\pi$
$500$	1.00000	0.0447214
$501$	0	0
$502$	−15.0000	−0.669483
$503$	16.0000	0.713405	0.356702	−	0.934218i	$-0.383901\pi$
$503$	16.0000	0.713405	0.356702	+	0.934218i	$0.383901\pi$
$504$	−3.00000	−0.133631
$505$	−14.0000	−0.622992
$506$	−5.00000	−0.222277
$507$	12.0000	0.532939
$508$	12.0000	0.532414
$509$	6.00000	0.265945	0.132973	−	0.991120i	$-0.457548\pi$
$509$	6.00000	0.265945	0.132973	+	0.991120i	$0.457548\pi$
$510$	−1.00000	−0.0442807
$511$	0	0
$512$	−1.00000	−0.0441942
$513$	5.00000	0.220755
$514$	14.0000	0.617514
$515$	1.00000	0.0440653
$516$	8.00000	0.352180
$517$	−6.00000	−0.263880
$518$	9.00000	0.395437
$519$	−21.0000	−0.921798
$520$	1.00000	0.0438529
$521$	15.0000	0.657162	0.328581	−	0.944476i	$-0.393430\pi$
$521$	15.0000	0.657162	0.328581	+	0.944476i	$0.393430\pi$
$522$	6.00000	0.262613
$523$	−20.0000	−0.874539	−0.437269	−	0.899331i	$-0.644054\pi$
$523$	−20.0000	−0.874539	−0.437269	+	0.899331i	$0.644054\pi$
$524$	−9.00000	−0.393167
$525$	−3.00000	−0.130931
$526$	−11.0000	−0.479623
$527$	−1.00000	−0.0435607
$528$	−1.00000	−0.0435194
$529$	2.00000	0.0869565
$530$	10.0000	0.434372
$531$	−12.0000	−0.520756
$532$	−15.0000	−0.650332
$533$	−2.00000	−0.0866296
$534$	−10.0000	−0.432742
$535$	−10.0000	−0.432338
$536$	7.00000	0.302354
$537$	21.0000	0.906217
$538$	−5.00000	−0.215565
$539$	2.00000	0.0861461
$540$	−1.00000	−0.0430331
$541$	2.00000	0.0859867	0.0429934	−	0.999075i	$-0.486311\pi$
$541$	2.00000	0.0859867	0.0429934	+	0.999075i	$0.486311\pi$
$542$	28.0000	1.20270
$543$	20.0000	0.858282
$544$	1.00000	0.0428746
$545$	−7.00000	−0.299847
$546$	−3.00000	−0.128388
$547$	17.0000	0.726868	0.363434	−	0.931620i	$-0.381604\pi$
$547$	17.0000	0.726868	0.363434	+	0.931620i	$0.381604\pi$
$548$	−7.00000	−0.299025
$549$	11.0000	0.469469
$550$	−1.00000	−0.0426401
$551$	30.0000	1.27804
$552$	5.00000	0.212814
$553$	−6.00000	−0.255146
$554$	8.00000	0.339887
$555$	3.00000	0.127343
$556$	22.0000	0.933008
$557$	−20.0000	−0.847427	−0.423714	−	0.905796i	$-0.639274\pi$
$557$	−20.0000	−0.847427	−0.423714	+	0.905796i	$0.639274\pi$
$558$	−1.00000	−0.0423334
$559$	8.00000	0.338364
$560$	3.00000	0.126773
$561$	1.00000	0.0422200
$562$	30.0000	1.26547
$563$	−3.00000	−0.126435	−0.0632175	−	0.998000i	$-0.520136\pi$
$563$	−3.00000	−0.126435	−0.0632175	+	0.998000i	$0.520136\pi$
$564$	6.00000	0.252646
$565$	10.0000	0.420703
$566$	−4.00000	−0.168133
$567$	3.00000	0.125988
$568$	10.0000	0.419591
$569$	13.0000	0.544988	0.272494	−	0.962157i	$-0.412151\pi$
$569$	13.0000	0.544988	0.272494	+	0.962157i	$0.412151\pi$
$570$	−5.00000	−0.209427
$571$	14.0000	0.585882	0.292941	−	0.956131i	$-0.405366\pi$
$571$	14.0000	0.585882	0.292941	+	0.956131i	$0.405366\pi$
$572$	−1.00000	−0.0418121
$573$	0	0
$574$	−6.00000	−0.250435
$575$	5.00000	0.208514
$576$	1.00000	0.0416667
$577$	38.0000	1.58196	0.790980	−	0.611842i	$-0.209571\pi$
$577$	38.0000	1.58196	0.790980	+	0.611842i	$0.209571\pi$
$578$	−1.00000	−0.0415945
$579$	−24.0000	−0.997406
$580$	−6.00000	−0.249136
$581$	21.0000	0.871227
$582$	−7.00000	−0.290159
$583$	−10.0000	−0.414158
$584$	0	0
$585$	−1.00000	−0.0413449
$586$	0	0
$587$	−18.0000	−0.742940	−0.371470	−	0.928445i	$-0.621146\pi$
$587$	−18.0000	−0.742940	−0.371470	+	0.928445i	$0.621146\pi$
$588$	−2.00000	−0.0824786
$589$	−5.00000	−0.206021
$590$	12.0000	0.494032
$591$	15.0000	0.617018
$592$	−3.00000	−0.123299
$593$	−6.00000	−0.246390	−0.123195	−	0.992382i	$-0.539314\pi$
$593$	−6.00000	−0.246390	−0.123195	+	0.992382i	$0.539314\pi$
$594$	1.00000	0.0410305
$595$	−3.00000	−0.122988
$596$	−5.00000	−0.204808
$597$	−17.0000	−0.695764
$598$	5.00000	0.204465
$599$	1.00000	0.0408589	0.0204294	−	0.999791i	$-0.493497\pi$
$599$	1.00000	0.0408589	0.0204294	+	0.999791i	$0.493497\pi$
$600$	1.00000	0.0408248
$601$	−7.00000	−0.285536	−0.142768	−	0.989756i	$-0.545600\pi$
$601$	−7.00000	−0.285536	−0.142768	+	0.989756i	$0.545600\pi$
$602$	24.0000	0.978167
$603$	−7.00000	−0.285062
$604$	9.00000	0.366205
$605$	1.00000	0.0406558
$606$	−14.0000	−0.568711
$607$	−29.0000	−1.17707	−0.588537	−	0.808470i	$-0.700296\pi$
$607$	−29.0000	−1.17707	−0.588537	+	0.808470i	$0.700296\pi$
$608$	5.00000	0.202777
$609$	18.0000	0.729397
$610$	−11.0000	−0.445377
$611$	6.00000	0.242734
$612$	−1.00000	−0.0404226
$613$	−26.0000	−1.05013	−0.525065	−	0.851062i	$-0.675959\pi$
$613$	−26.0000	−1.05013	−0.525065	+	0.851062i	$0.675959\pi$
$614$	16.0000	0.645707
$615$	−2.00000	−0.0806478
$616$	−3.00000	−0.120873
$617$	6.00000	0.241551	0.120775	−	0.992680i	$-0.461462\pi$
$617$	6.00000	0.241551	0.120775	+	0.992680i	$0.461462\pi$
$618$	1.00000	0.0402259
$619$	−25.0000	−1.00483	−0.502417	−	0.864625i	$-0.667556\pi$
$619$	−25.0000	−1.00483	−0.502417	+	0.864625i	$0.667556\pi$
$620$	1.00000	0.0401610
$621$	−5.00000	−0.200643
$622$	6.00000	0.240578
$623$	−30.0000	−1.20192
$624$	1.00000	0.0400320
$625$	1.00000	0.0400000
$626$	−23.0000	−0.919265
$627$	5.00000	0.199681
$628$	12.0000	0.478852
$629$	3.00000	0.119618
$630$	−3.00000	−0.119523
$631$	14.0000	0.557331	0.278666	−	0.960388i	$-0.410108\pi$
$631$	14.0000	0.557331	0.278666	+	0.960388i	$0.410108\pi$
$632$	2.00000	0.0795557
$633$	−18.0000	−0.715436
$634$	−26.0000	−1.03259
$635$	12.0000	0.476205
$636$	10.0000	0.396526
$637$	−2.00000	−0.0792429
$638$	6.00000	0.237542
$639$	−10.0000	−0.395594
$640$	−1.00000	−0.0395285
$641$	−30.0000	−1.18493	−0.592464	−	0.805597i	$-0.701845\pi$
$641$	−30.0000	−1.18493	−0.592464	+	0.805597i	$0.701845\pi$
$642$	−10.0000	−0.394669
$643$	20.0000	0.788723	0.394362	−	0.918955i	$-0.370966\pi$
$643$	20.0000	0.788723	0.394362	+	0.918955i	$0.370966\pi$
$644$	15.0000	0.591083
$645$	8.00000	0.315000
$646$	−5.00000	−0.196722
$647$	−14.0000	−0.550397	−0.275198	−	0.961387i	$-0.588744\pi$
$647$	−14.0000	−0.550397	−0.275198	+	0.961387i	$0.588744\pi$
$648$	−1.00000	−0.0392837
$649$	−12.0000	−0.471041
$650$	1.00000	0.0392232
$651$	−3.00000	−0.117579
$652$	24.0000	0.939913
$653$	12.0000	0.469596	0.234798	−	0.972044i	$-0.424557\pi$
$653$	12.0000	0.469596	0.234798	+	0.972044i	$0.424557\pi$
$654$	−7.00000	−0.273722
$655$	−9.00000	−0.351659
$656$	2.00000	0.0780869
$657$	0	0
$658$	18.0000	0.701713
$659$	−26.0000	−1.01282	−0.506408	−	0.862294i	$-0.669027\pi$
$659$	−26.0000	−1.01282	−0.506408	+	0.862294i	$0.669027\pi$
$660$	−1.00000	−0.0389249
$661$	29.0000	1.12797	0.563985	−	0.825785i	$-0.309268\pi$
$661$	29.0000	1.12797	0.563985	+	0.825785i	$0.309268\pi$
$662$	32.0000	1.24372
$663$	−1.00000	−0.0388368
$664$	−7.00000	−0.271653
$665$	−15.0000	−0.581675
$666$	3.00000	0.116248
$667$	−30.0000	−1.16160
$668$	0	0
$669$	1.00000	0.0386622
$670$	7.00000	0.270434
$671$	11.0000	0.424650
$672$	3.00000	0.115728
$673$	−20.0000	−0.770943	−0.385472	−	0.922720i	$-0.625961\pi$
$673$	−20.0000	−0.770943	−0.385472	+	0.922720i	$0.625961\pi$
$674$	−36.0000	−1.38667
$675$	−1.00000	−0.0384900
$676$	−12.0000	−0.461538
$677$	30.0000	1.15299	0.576497	−	0.817099i	$-0.304419\pi$
$677$	30.0000	1.15299	0.576497	+	0.817099i	$0.304419\pi$
$678$	10.0000	0.384048
$679$	−21.0000	−0.805906
$680$	1.00000	0.0383482
$681$	18.0000	0.689761
$682$	−1.00000	−0.0382920
$683$	−39.0000	−1.49229	−0.746147	−	0.665782i	$-0.768098\pi$
$683$	−39.0000	−1.49229	−0.746147	+	0.665782i	$0.768098\pi$
$684$	−5.00000	−0.191180
$685$	−7.00000	−0.267456
$686$	15.0000	0.572703
$687$	21.0000	0.801200
$688$	−8.00000	−0.304997
$689$	10.0000	0.380970
$690$	5.00000	0.190347
$691$	3.00000	0.114125	0.0570627	−	0.998371i	$-0.481827\pi$
$691$	3.00000	0.114125	0.0570627	+	0.998371i	$0.481827\pi$
$692$	21.0000	0.798300
$693$	3.00000	0.113961
$694$	18.0000	0.683271
$695$	22.0000	0.834508
$696$	−6.00000	−0.227429
$697$	−2.00000	−0.0757554
$698$	−14.0000	−0.529908
$699$	−26.0000	−0.983410
$700$	3.00000	0.113389
$701$	−14.0000	−0.528773	−0.264386	−	0.964417i	$-0.585169\pi$
$701$	−14.0000	−0.528773	−0.264386	+	0.964417i	$0.585169\pi$
$702$	−1.00000	−0.0377426
$703$	15.0000	0.565736
$704$	1.00000	0.0376889
$705$	6.00000	0.225973
$706$	−31.0000	−1.16670
$707$	−42.0000	−1.57957
$708$	12.0000	0.450988
$709$	−22.0000	−0.826227	−0.413114	−	0.910679i	$-0.635559\pi$
$709$	−22.0000	−0.826227	−0.413114	+	0.910679i	$0.635559\pi$
$710$	10.0000	0.375293
$711$	−2.00000	−0.0750059
$712$	10.0000	0.374766
$713$	5.00000	0.187251
$714$	−3.00000	−0.112272
$715$	−1.00000	−0.0373979
$716$	−21.0000	−0.784807
$717$	−18.0000	−0.672222
$718$	34.0000	1.26887
$719$	−30.0000	−1.11881	−0.559406	−	0.828894i	$-0.688971\pi$
$719$	−30.0000	−1.11881	−0.559406	+	0.828894i	$0.688971\pi$
$720$	1.00000	0.0372678
$721$	3.00000	0.111726
$722$	−6.00000	−0.223297
$723$	15.0000	0.557856
$724$	−20.0000	−0.743294
$725$	−6.00000	−0.222834
$726$	1.00000	0.0371135
$727$	−40.0000	−1.48352	−0.741759	−	0.670667i	$-0.766008\pi$
$727$	−40.0000	−1.48352	−0.741759	+	0.670667i	$0.766008\pi$
$728$	3.00000	0.111187
$729$	1.00000	0.0370370
$730$	0	0
$731$	8.00000	0.295891
$732$	−11.0000	−0.406572
$733$	−43.0000	−1.58824	−0.794121	−	0.607760i	$-0.792068\pi$
$733$	−43.0000	−1.58824	−0.794121	+	0.607760i	$0.792068\pi$
$734$	8.00000	0.295285
$735$	−2.00000	−0.0737711
$736$	−5.00000	−0.184302
$737$	−7.00000	−0.257848
$738$	−2.00000	−0.0736210
$739$	29.0000	1.06678	0.533391	−	0.845869i	$-0.320917\pi$
$739$	29.0000	1.06678	0.533391	+	0.845869i	$0.320917\pi$
$740$	−3.00000	−0.110282
$741$	−5.00000	−0.183680
$742$	30.0000	1.10133
$743$	8.00000	0.293492	0.146746	−	0.989174i	$-0.453120\pi$
$743$	8.00000	0.293492	0.146746	+	0.989174i	$0.453120\pi$
$744$	1.00000	0.0366618
$745$	−5.00000	−0.183186
$746$	14.0000	0.512576
$747$	7.00000	0.256117
$748$	−1.00000	−0.0365636
$749$	−30.0000	−1.09618
$750$	1.00000	0.0365148
$751$	24.0000	0.875772	0.437886	−	0.899030i	$-0.355727\pi$
$751$	24.0000	0.875772	0.437886	+	0.899030i	$0.355727\pi$
$752$	−6.00000	−0.218797
$753$	−15.0000	−0.546630
$754$	−6.00000	−0.218507
$755$	9.00000	0.327544
$756$	−3.00000	−0.109109
$757$	10.0000	0.363456	0.181728	−	0.983349i	$-0.441831\pi$
$757$	10.0000	0.363456	0.181728	+	0.983349i	$0.441831\pi$
$758$	23.0000	0.835398
$759$	−5.00000	−0.181489
$760$	5.00000	0.181369
$761$	−27.0000	−0.978749	−0.489375	−	0.872074i	$-0.662775\pi$
$761$	−27.0000	−0.978749	−0.489375	+	0.872074i	$0.662775\pi$
$762$	12.0000	0.434714
$763$	−21.0000	−0.760251
$764$	0	0
$765$	−1.00000	−0.0361551
$766$	34.0000	1.22847
$767$	12.0000	0.433295
$768$	−1.00000	−0.0360844
$769$	8.00000	0.288487	0.144244	−	0.989542i	$-0.453925\pi$
$769$	8.00000	0.288487	0.144244	+	0.989542i	$0.453925\pi$
$770$	−3.00000	−0.108112
$771$	14.0000	0.504198
$772$	24.0000	0.863779
$773$	3.00000	0.107903	0.0539513	−	0.998544i	$-0.482818\pi$
$773$	3.00000	0.107903	0.0539513	+	0.998544i	$0.482818\pi$
$774$	8.00000	0.287554
$775$	1.00000	0.0359211
$776$	7.00000	0.251285
$777$	9.00000	0.322873
$778$	−4.00000	−0.143407
$779$	−10.0000	−0.358287
$780$	1.00000	0.0358057
$781$	−10.0000	−0.357828
$782$	5.00000	0.178800
$783$	6.00000	0.214423
$784$	2.00000	0.0714286
$785$	12.0000	0.428298
$786$	−9.00000	−0.321019
$787$	51.0000	1.81795	0.908977	−	0.416847i	$-0.136865\pi$
$787$	51.0000	1.81795	0.908977	+	0.416847i	$0.136865\pi$
$788$	−15.0000	−0.534353
$789$	−11.0000	−0.391610
$790$	2.00000	0.0711568
$791$	30.0000	1.06668
$792$	−1.00000	−0.0355335
$793$	−11.0000	−0.390621
$794$	−18.0000	−0.638796
$795$	10.0000	0.354663
$796$	17.0000	0.602549
$797$	−31.0000	−1.09808	−0.549038	−	0.835797i	$-0.685006\pi$
$797$	−31.0000	−1.09808	−0.549038	+	0.835797i	$0.685006\pi$
$798$	−15.0000	−0.530994
$799$	6.00000	0.212265
$800$	−1.00000	−0.0353553
$801$	−10.0000	−0.353333
$802$	−15.0000	−0.529668
$803$	0	0
$804$	7.00000	0.246871
$805$	15.0000	0.528681
$806$	1.00000	0.0352235
$807$	−5.00000	−0.176008
$808$	14.0000	0.492518
$809$	30.0000	1.05474	0.527372	−	0.849635i	$-0.323177\pi$
$809$	30.0000	1.05474	0.527372	+	0.849635i	$0.323177\pi$
$810$	−1.00000	−0.0351364
$811$	38.0000	1.33436	0.667180	−	0.744896i	$-0.267501\pi$
$811$	38.0000	1.33436	0.667180	+	0.744896i	$0.267501\pi$
$812$	−18.0000	−0.631676
$813$	28.0000	0.982003
$814$	3.00000	0.105150
$815$	24.0000	0.840683
$816$	1.00000	0.0350070
$817$	40.0000	1.39942
$818$	−26.0000	−0.909069
$819$	−3.00000	−0.104828
$820$	2.00000	0.0698430
$821$	30.0000	1.04701	0.523504	−	0.852023i	$-0.324625\pi$
$821$	30.0000	1.04701	0.523504	+	0.852023i	$0.324625\pi$
$822$	−7.00000	−0.244153
$823$	22.0000	0.766872	0.383436	−	0.923567i	$-0.374741\pi$
$823$	22.0000	0.766872	0.383436	+	0.923567i	$0.374741\pi$
$824$	−1.00000	−0.0348367
$825$	−1.00000	−0.0348155
$826$	36.0000	1.25260
$827$	−22.0000	−0.765015	−0.382507	−	0.923952i	$-0.624939\pi$
$827$	−22.0000	−0.765015	−0.382507	+	0.923952i	$0.624939\pi$
$828$	5.00000	0.173762
$829$	−29.0000	−1.00721	−0.503606	−	0.863934i	$-0.667994\pi$
$829$	−29.0000	−1.00721	−0.503606	+	0.863934i	$0.667994\pi$
$830$	−7.00000	−0.242974
$831$	8.00000	0.277517
$832$	−1.00000	−0.0346688
$833$	−2.00000	−0.0692959
$834$	22.0000	0.761798
$835$	0	0
$836$	−5.00000	−0.172929
$837$	−1.00000	−0.0345651
$838$	32.0000	1.10542
$839$	12.0000	0.414286	0.207143	−	0.978311i	$-0.433583\pi$
$839$	12.0000	0.414286	0.207143	+	0.978311i	$0.433583\pi$
$840$	3.00000	0.103510
$841$	7.00000	0.241379
$842$	−35.0000	−1.20618
$843$	30.0000	1.03325
$844$	18.0000	0.619586
$845$	−12.0000	−0.412813
$846$	6.00000	0.206284
$847$	3.00000	0.103081
$848$	−10.0000	−0.343401
$849$	−4.00000	−0.137280
$850$	1.00000	0.0342997
$851$	−15.0000	−0.514193
$852$	10.0000	0.342594
$853$	24.0000	0.821744	0.410872	−	0.911693i	$-0.365224\pi$
$853$	24.0000	0.821744	0.410872	+	0.911693i	$0.365224\pi$
$854$	−33.0000	−1.12924
$855$	−5.00000	−0.170996
$856$	10.0000	0.341793
$857$	29.0000	0.990621	0.495311	−	0.868716i	$-0.335054\pi$
$857$	29.0000	0.990621	0.495311	+	0.868716i	$0.335054\pi$
$858$	−1.00000	−0.0341394
$859$	32.0000	1.09183	0.545913	−	0.837842i	$-0.316183\pi$
$859$	32.0000	1.09183	0.545913	+	0.837842i	$0.316183\pi$
$860$	−8.00000	−0.272798
$861$	−6.00000	−0.204479
$862$	−32.0000	−1.08992
$863$	−52.0000	−1.77010	−0.885050	−	0.465495i	$-0.845876\pi$
$863$	−52.0000	−1.77010	−0.885050	+	0.465495i	$0.845876\pi$
$864$	1.00000	0.0340207
$865$	21.0000	0.714021
$866$	−6.00000	−0.203888
$867$	−1.00000	−0.0339618
$868$	3.00000	0.101827
$869$	−2.00000	−0.0678454
$870$	−6.00000	−0.203419
$871$	7.00000	0.237186
$872$	7.00000	0.237050
$873$	−7.00000	−0.236914
$874$	25.0000	0.845638
$875$	3.00000	0.101419
$876$	0	0
$877$	52.0000	1.75592	0.877958	−	0.478738i	$-0.158906\pi$
$877$	52.0000	1.75592	0.877958	+	0.478738i	$0.158906\pi$
$878$	34.0000	1.14744
$879$	0	0
$880$	1.00000	0.0337100
$881$	−2.00000	−0.0673817	−0.0336909	−	0.999432i	$-0.510726\pi$
$881$	−2.00000	−0.0673817	−0.0336909	+	0.999432i	$0.510726\pi$
$882$	−2.00000	−0.0673435
$883$	−12.0000	−0.403832	−0.201916	−	0.979403i	$-0.564717\pi$
$883$	−12.0000	−0.403832	−0.201916	+	0.979403i	$0.564717\pi$
$884$	1.00000	0.0336336
$885$	12.0000	0.403376
$886$	26.0000	0.873487
$887$	12.0000	0.402921	0.201460	−	0.979497i	$-0.435431\pi$
$887$	12.0000	0.402921	0.201460	+	0.979497i	$0.435431\pi$
$888$	−3.00000	−0.100673
$889$	36.0000	1.20740
$890$	10.0000	0.335201
$891$	1.00000	0.0335013
$892$	−1.00000	−0.0334825
$893$	30.0000	1.00391
$894$	−5.00000	−0.167225
$895$	−21.0000	−0.701953
$896$	−3.00000	−0.100223
$897$	5.00000	0.166945
$898$	19.0000	0.634038
$899$	−6.00000	−0.200111
$900$	1.00000	0.0333333
$901$	10.0000	0.333148
$902$	−2.00000	−0.0665927
$903$	24.0000	0.798670
$904$	−10.0000	−0.332595
$905$	−20.0000	−0.664822
$906$	9.00000	0.299005
$907$	44.0000	1.46100	0.730498	−	0.682915i	$-0.239288\pi$
$907$	44.0000	1.46100	0.730498	+	0.682915i	$0.239288\pi$
$908$	−18.0000	−0.597351
$909$	−14.0000	−0.464351
$910$	3.00000	0.0994490
$911$	−44.0000	−1.45779	−0.728893	−	0.684628i	$-0.759965\pi$
$911$	−44.0000	−1.45779	−0.728893	+	0.684628i	$0.759965\pi$
$912$	5.00000	0.165567
$913$	7.00000	0.231666
$914$	7.00000	0.231539
$915$	−11.0000	−0.363649
$916$	−21.0000	−0.693860
$917$	−27.0000	−0.891619
$918$	−1.00000	−0.0330049
$919$	−11.0000	−0.362857	−0.181428	−	0.983404i	$-0.558072\pi$
$919$	−11.0000	−0.362857	−0.181428	+	0.983404i	$0.558072\pi$
$920$	−5.00000	−0.164845
$921$	16.0000	0.527218
$922$	14.0000	0.461065
$923$	10.0000	0.329154
$924$	−3.00000	−0.0986928
$925$	−3.00000	−0.0986394
$926$	−3.00000	−0.0985861
$927$	1.00000	0.0328443
$928$	6.00000	0.196960
$929$	−55.0000	−1.80449	−0.902246	−	0.431222i	$-0.858082\pi$
$929$	−55.0000	−1.80449	−0.902246	+	0.431222i	$0.858082\pi$
$930$	1.00000	0.0327913
$931$	−10.0000	−0.327737
$932$	26.0000	0.851658
$933$	6.00000	0.196431
$934$	−24.0000	−0.785304
$935$	−1.00000	−0.0327035
$936$	1.00000	0.0326860
$937$	−23.0000	−0.751377	−0.375689	−	0.926746i	$-0.622594\pi$
$937$	−23.0000	−0.751377	−0.375689	+	0.926746i	$0.622594\pi$
$938$	21.0000	0.685674
$939$	−23.0000	−0.750577
$940$	−6.00000	−0.195698
$941$	4.00000	0.130396	0.0651981	−	0.997872i	$-0.479232\pi$
$941$	4.00000	0.130396	0.0651981	+	0.997872i	$0.479232\pi$
$942$	12.0000	0.390981
$943$	10.0000	0.325645
$944$	−12.0000	−0.390567
$945$	−3.00000	−0.0975900
$946$	8.00000	0.260102
$947$	−12.0000	−0.389948	−0.194974	−	0.980808i	$-0.562462\pi$
$947$	−12.0000	−0.389948	−0.194974	+	0.980808i	$0.562462\pi$
$948$	2.00000	0.0649570
$949$	0	0
$950$	5.00000	0.162221
$951$	−26.0000	−0.843108
$952$	3.00000	0.0972306
$953$	52.0000	1.68445	0.842223	−	0.539130i	$-0.181247\pi$
$953$	52.0000	1.68445	0.842223	+	0.539130i	$0.181247\pi$
$954$	10.0000	0.323762
$955$	0	0
$956$	18.0000	0.582162
$957$	6.00000	0.193952
$958$	−5.00000	−0.161543
$959$	−21.0000	−0.678125
$960$	−1.00000	−0.0322749
$961$	−30.0000	−0.967742
$962$	−3.00000	−0.0967239
$963$	−10.0000	−0.322245
$964$	−15.0000	−0.483117
$965$	24.0000	0.772587
$966$	15.0000	0.482617
$967$	−18.0000	−0.578841	−0.289420	−	0.957202i	$-0.593463\pi$
$967$	−18.0000	−0.578841	−0.289420	+	0.957202i	$0.593463\pi$
$968$	−1.00000	−0.0321412
$969$	−5.00000	−0.160623
$970$	7.00000	0.224756
$971$	57.0000	1.82922	0.914609	−	0.404341i	$-0.132499\pi$
$971$	57.0000	1.82922	0.914609	+	0.404341i	$0.132499\pi$
$972$	−1.00000	−0.0320750
$973$	66.0000	2.11586
$974$	40.0000	1.28168
$975$	1.00000	0.0320256
$976$	11.0000	0.352101
$977$	11.0000	0.351921	0.175961	−	0.984397i	$-0.443697\pi$
$977$	11.0000	0.351921	0.175961	+	0.984397i	$0.443697\pi$
$978$	24.0000	0.767435
$979$	−10.0000	−0.319601
$980$	2.00000	0.0638877
$981$	−7.00000	−0.223493
$982$	−2.00000	−0.0638226
$983$	−44.0000	−1.40338	−0.701691	−	0.712481i	$-0.747571\pi$
$983$	−44.0000	−1.40338	−0.701691	+	0.712481i	$0.747571\pi$
$984$	2.00000	0.0637577
$985$	−15.0000	−0.477940
$986$	−6.00000	−0.191079
$987$	18.0000	0.572946
$988$	5.00000	0.159071
$989$	−40.0000	−1.27193
$990$	−1.00000	−0.0317821
$991$	−53.0000	−1.68360	−0.841800	−	0.539789i	$-0.818504\pi$
$991$	−53.0000	−1.68360	−0.841800	+	0.539789i	$0.818504\pi$
$992$	−1.00000	−0.0317500
$993$	32.0000	1.01549
$994$	30.0000	0.951542
$995$	17.0000	0.538936
$996$	−7.00000	−0.221803
$997$	46.0000	1.45683	0.728417	−	0.685134i	$-0.240256\pi$
$997$	46.0000	1.45683	0.728417	+	0.685134i	$0.240256\pi$
$998$	20.0000	0.633089
$999$	3.00000	0.0949158

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5610.2.a.j.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5610.2.a.j.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:	\( N \)	\(=\)	\( 5610 = 2 \cdot 3 \cdot 5 \cdot 11 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5610.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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