LMFDB - Embedded newform 186.2.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [186,2,Mod(1,186)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(186, 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("186.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 186 = 2 \cdot 3 \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	186.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:	$1.48521747760$
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	$\chi$	$=$	186.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +3.00000 q^{5} -1.00000 q^{6} -2.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -3.00000 q^{10} +5.00000 q^{11} +1.00000 q^{12} -7.00000 q^{13} +2.00000 q^{14} +3.00000 q^{15} +1.00000 q^{16} -1.00000 q^{17} -1.00000 q^{18} +7.00000 q^{19} +3.00000 q^{20} -2.00000 q^{21} -5.00000 q^{22} +4.00000 q^{23} -1.00000 q^{24} +4.00000 q^{25} +7.00000 q^{26} +1.00000 q^{27} -2.00000 q^{28} -8.00000 q^{29} -3.00000 q^{30} -1.00000 q^{31} -1.00000 q^{32} +5.00000 q^{33} +1.00000 q^{34} -6.00000 q^{35} +1.00000 q^{36} -6.00000 q^{37} -7.00000 q^{38} -7.00000 q^{39} -3.00000 q^{40} -2.00000 q^{41} +2.00000 q^{42} -10.0000 q^{43} +5.00000 q^{44} +3.00000 q^{45} -4.00000 q^{46} -1.00000 q^{47} +1.00000 q^{48} -3.00000 q^{49} -4.00000 q^{50} -1.00000 q^{51} -7.00000 q^{52} +6.00000 q^{53} -1.00000 q^{54} +15.0000 q^{55} +2.00000 q^{56} +7.00000 q^{57} +8.00000 q^{58} -10.0000 q^{59} +3.00000 q^{60} +1.00000 q^{61} +1.00000 q^{62} -2.00000 q^{63} +1.00000 q^{64} -21.0000 q^{65} -5.00000 q^{66} -3.00000 q^{67} -1.00000 q^{68} +4.00000 q^{69} +6.00000 q^{70} +3.00000 q^{71} -1.00000 q^{72} +14.0000 q^{73} +6.00000 q^{74} +4.00000 q^{75} +7.00000 q^{76} -10.0000 q^{77} +7.00000 q^{78} -11.0000 q^{79} +3.00000 q^{80} +1.00000 q^{81} +2.00000 q^{82} +7.00000 q^{83} -2.00000 q^{84} -3.00000 q^{85} +10.0000 q^{86} -8.00000 q^{87} -5.00000 q^{88} -6.00000 q^{89} -3.00000 q^{90} +14.0000 q^{91} +4.00000 q^{92} -1.00000 q^{93} +1.00000 q^{94} +21.0000 q^{95} -1.00000 q^{96} -3.00000 q^{97} +3.00000 q^{98} +5.00000 q^{99} +O(q^{100})\)

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
$5$	3.00000	1.34164	0.670820	−	0.741620i	$-0.265942\pi$
$5$	3.00000	1.34164	0.670820	+	0.741620i	$0.265942\pi$
$6$	−1.00000	−0.408248
$7$	−2.00000	−0.755929	−0.377964	−	0.925820i	$-0.623376\pi$
$7$	−2.00000	−0.755929	−0.377964	+	0.925820i	$0.623376\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	−3.00000	−0.948683
$11$	5.00000	1.50756	0.753778	−	0.657129i	$-0.228229\pi$
$11$	5.00000	1.50756	0.753778	+	0.657129i	$0.228229\pi$
$12$	1.00000	0.288675
$13$	−7.00000	−1.94145	−0.970725	−	0.240192i	$-0.922790\pi$
$13$	−7.00000	−1.94145	−0.970725	+	0.240192i	$0.922790\pi$
$14$	2.00000	0.534522
$15$	3.00000	0.774597
$16$	1.00000	0.250000
$17$	−1.00000	−0.242536	−0.121268	−	0.992620i	$-0.538696\pi$
$17$	−1.00000	−0.242536	−0.121268	+	0.992620i	$0.538696\pi$
$18$	−1.00000	−0.235702
$19$	7.00000	1.60591	0.802955	−	0.596040i	$-0.203260\pi$
$19$	7.00000	1.60591	0.802955	+	0.596040i	$0.203260\pi$
$20$	3.00000	0.670820
$21$	−2.00000	−0.436436
$22$	−5.00000	−1.06600
$23$	4.00000	0.834058	0.417029	−	0.908893i	$-0.363071\pi$
$23$	4.00000	0.834058	0.417029	+	0.908893i	$0.363071\pi$
$24$	−1.00000	−0.204124
$25$	4.00000	0.800000
$26$	7.00000	1.37281
$27$	1.00000	0.192450
$28$	−2.00000	−0.377964
$29$	−8.00000	−1.48556	−0.742781	−	0.669534i	$-0.766494\pi$
$29$	−8.00000	−1.48556	−0.742781	+	0.669534i	$0.766494\pi$
$30$	−3.00000	−0.547723
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	−1.00000	−0.176777
$33$	5.00000	0.870388
$34$	1.00000	0.171499
$35$	−6.00000	−1.01419
$36$	1.00000	0.166667
$37$	−6.00000	−0.986394	−0.493197	−	0.869918i	$-0.664172\pi$
$37$	−6.00000	−0.986394	−0.493197	+	0.869918i	$0.664172\pi$
$38$	−7.00000	−1.13555
$39$	−7.00000	−1.12090
$40$	−3.00000	−0.474342
$41$	−2.00000	−0.312348	−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000	−0.312348	−0.156174	+	0.987730i	$0.549916\pi$
$42$	2.00000	0.308607
$43$	−10.0000	−1.52499	−0.762493	−	0.646997i	$-0.776025\pi$
$43$	−10.0000	−1.52499	−0.762493	+	0.646997i	$0.776025\pi$
$44$	5.00000	0.753778
$45$	3.00000	0.447214
$46$	−4.00000	−0.589768
$47$	−1.00000	−0.145865	−0.0729325	−	0.997337i	$-0.523236\pi$
$47$	−1.00000	−0.145865	−0.0729325	+	0.997337i	$0.523236\pi$
$48$	1.00000	0.144338
$49$	−3.00000	−0.428571
$50$	−4.00000	−0.565685
$51$	−1.00000	−0.140028
$52$	−7.00000	−0.970725
$53$	6.00000	0.824163	0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000	0.824163	0.412082	+	0.911147i	$0.364802\pi$
$54$	−1.00000	−0.136083
$55$	15.0000	2.02260
$56$	2.00000	0.267261
$57$	7.00000	0.927173
$58$	8.00000	1.05045
$59$	−10.0000	−1.30189	−0.650945	−	0.759125i	$-0.725627\pi$
$59$	−10.0000	−1.30189	−0.650945	+	0.759125i	$0.725627\pi$
$60$	3.00000	0.387298
$61$	1.00000	0.128037	0.0640184	−	0.997949i	$-0.479608\pi$
$61$	1.00000	0.128037	0.0640184	+	0.997949i	$0.479608\pi$
$62$	1.00000	0.127000
$63$	−2.00000	−0.251976
$64$	1.00000	0.125000
$65$	−21.0000	−2.60473
$66$	−5.00000	−0.615457
$67$	−3.00000	−0.366508	−0.183254	−	0.983066i	$-0.558663\pi$
$67$	−3.00000	−0.366508	−0.183254	+	0.983066i	$0.558663\pi$
$68$	−1.00000	−0.121268
$69$	4.00000	0.481543
$70$	6.00000	0.717137
$71$	3.00000	0.356034	0.178017	−	0.984027i	$-0.443032\pi$
$71$	3.00000	0.356034	0.178017	+	0.984027i	$0.443032\pi$
$72$	−1.00000	−0.117851
$73$	14.0000	1.63858	0.819288	−	0.573382i	$-0.194369\pi$
$73$	14.0000	1.63858	0.819288	+	0.573382i	$0.194369\pi$
$74$	6.00000	0.697486
$75$	4.00000	0.461880
$76$	7.00000	0.802955
$77$	−10.0000	−1.13961
$78$	7.00000	0.792594
$79$	−11.0000	−1.23760	−0.618798	−	0.785550i	$-0.712380\pi$
$79$	−11.0000	−1.23760	−0.618798	+	0.785550i	$0.712380\pi$
$80$	3.00000	0.335410
$81$	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	$0.374486\pi$
$84$	−2.00000	−0.218218
$85$	−3.00000	−0.325396
$86$	10.0000	1.07833
$87$	−8.00000	−0.857690
$88$	−5.00000	−0.533002
$89$	−6.00000	−0.635999	−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000	−0.635999	−0.317999	+	0.948091i	$0.603011\pi$
$90$	−3.00000	−0.316228
$91$	14.0000	1.46760
$92$	4.00000	0.417029
$93$	−1.00000	−0.103695
$94$	1.00000	0.103142
$95$	21.0000	2.15455
$96$	−1.00000	−0.102062
$97$	−3.00000	−0.304604	−0.152302	−	0.988334i	$-0.548669\pi$
$97$	−3.00000	−0.304604	−0.152302	+	0.988334i	$0.548669\pi$
$98$	3.00000	0.303046
$99$	5.00000	0.502519
$100$	4.00000	0.400000
$101$	6.00000	0.597022	0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000	0.597022	0.298511	+	0.954406i	$0.403510\pi$
$102$	1.00000	0.0990148
$103$	10.0000	0.985329	0.492665	−	0.870219i	$-0.336023\pi$
$103$	10.0000	0.985329	0.492665	+	0.870219i	$0.336023\pi$
$104$	7.00000	0.686406
$105$	−6.00000	−0.585540
$106$	−6.00000	−0.582772
$107$	8.00000	0.773389	0.386695	−	0.922208i	$-0.373617\pi$
$107$	8.00000	0.773389	0.386695	+	0.922208i	$0.373617\pi$
$108$	1.00000	0.0962250
$109$	−6.00000	−0.574696	−0.287348	−	0.957826i	$-0.592774\pi$
$109$	−6.00000	−0.574696	−0.287348	+	0.957826i	$0.592774\pi$
$110$	−15.0000	−1.43019
$111$	−6.00000	−0.569495
$112$	−2.00000	−0.188982
$113$	6.00000	0.564433	0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000	0.564433	0.282216	+	0.959351i	$0.408930\pi$
$114$	−7.00000	−0.655610
$115$	12.0000	1.11901
$116$	−8.00000	−0.742781
$117$	−7.00000	−0.647150
$118$	10.0000	0.920575
$119$	2.00000	0.183340
$120$	−3.00000	−0.273861
$121$	14.0000	1.27273
$122$	−1.00000	−0.0905357
$123$	−2.00000	−0.180334
$124$	−1.00000	−0.0898027
$125$	−3.00000	−0.268328
$126$	2.00000	0.178174
$127$	4.00000	0.354943	0.177471	−	0.984126i	$-0.443208\pi$
$127$	4.00000	0.354943	0.177471	+	0.984126i	$0.443208\pi$
$128$	−1.00000	−0.0883883
$129$	−10.0000	−0.880451
$130$	21.0000	1.84182
$131$	10.0000	0.873704	0.436852	−	0.899533i	$-0.356093\pi$
$131$	10.0000	0.873704	0.436852	+	0.899533i	$0.356093\pi$
$132$	5.00000	0.435194
$133$	−14.0000	−1.21395
$134$	3.00000	0.259161
$135$	3.00000	0.258199
$136$	1.00000	0.0857493
$137$	−18.0000	−1.53784	−0.768922	−	0.639343i	$-0.779207\pi$
$137$	−18.0000	−1.53784	−0.768922	+	0.639343i	$0.779207\pi$
$138$	−4.00000	−0.340503
$139$	−8.00000	−0.678551	−0.339276	−	0.940687i	$-0.610182\pi$
$139$	−8.00000	−0.678551	−0.339276	+	0.940687i	$0.610182\pi$
$140$	−6.00000	−0.507093
$141$	−1.00000	−0.0842152
$142$	−3.00000	−0.251754
$143$	−35.0000	−2.92685
$144$	1.00000	0.0833333
$145$	−24.0000	−1.99309
$146$	−14.0000	−1.15865
$147$	−3.00000	−0.247436
$148$	−6.00000	−0.493197
$149$	15.0000	1.22885	0.614424	−	0.788976i	$-0.289388\pi$
$149$	15.0000	1.22885	0.614424	+	0.788976i	$0.289388\pi$
$150$	−4.00000	−0.326599
$151$	−1.00000	−0.0813788	−0.0406894	−	0.999172i	$-0.512955\pi$
$151$	−1.00000	−0.0813788	−0.0406894	+	0.999172i	$0.512955\pi$
$152$	−7.00000	−0.567775
$153$	−1.00000	−0.0808452
$154$	10.0000	0.805823
$155$	−3.00000	−0.240966
$156$	−7.00000	−0.560449
$157$	−4.00000	−0.319235	−0.159617	−	0.987179i	$-0.551026\pi$
$157$	−4.00000	−0.319235	−0.159617	+	0.987179i	$0.551026\pi$
$158$	11.0000	0.875113
$159$	6.00000	0.475831
$160$	−3.00000	−0.237171
$161$	−8.00000	−0.630488
$162$	−1.00000	−0.0785674
$163$	−9.00000	−0.704934	−0.352467	−	0.935824i	$-0.614657\pi$
$163$	−9.00000	−0.704934	−0.352467	+	0.935824i	$0.614657\pi$
$164$	−2.00000	−0.156174
$165$	15.0000	1.16775
$166$	−7.00000	−0.543305
$167$	10.0000	0.773823	0.386912	−	0.922117i	$-0.373542\pi$
$167$	10.0000	0.773823	0.386912	+	0.922117i	$0.373542\pi$
$168$	2.00000	0.154303
$169$	36.0000	2.76923
$170$	3.00000	0.230089
$171$	7.00000	0.535303
$172$	−10.0000	−0.762493
$173$	1.00000	0.0760286	0.0380143	−	0.999277i	$-0.487897\pi$
$173$	1.00000	0.0760286	0.0380143	+	0.999277i	$0.487897\pi$
$174$	8.00000	0.606478
$175$	−8.00000	−0.604743
$176$	5.00000	0.376889
$177$	−10.0000	−0.751646
$178$	6.00000	0.449719
$179$	19.0000	1.42013	0.710063	−	0.704138i	$-0.248666\pi$
$179$	19.0000	1.42013	0.710063	+	0.704138i	$0.248666\pi$
$180$	3.00000	0.223607
$181$	18.0000	1.33793	0.668965	−	0.743294i	$-0.266738\pi$
$181$	18.0000	1.33793	0.668965	+	0.743294i	$0.266738\pi$
$182$	−14.0000	−1.03775
$183$	1.00000	0.0739221
$184$	−4.00000	−0.294884
$185$	−18.0000	−1.32339
$186$	1.00000	0.0733236
$187$	−5.00000	−0.365636
$188$	−1.00000	−0.0729325
$189$	−2.00000	−0.145479
$190$	−21.0000	−1.52350
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	1.00000	0.0721688
$193$	19.0000	1.36765	0.683825	−	0.729646i	$-0.260315\pi$
$193$	19.0000	1.36765	0.683825	+	0.729646i	$0.260315\pi$
$194$	3.00000	0.215387
$195$	−21.0000	−1.50384
$196$	−3.00000	−0.214286
$197$	−2.00000	−0.142494	−0.0712470	−	0.997459i	$-0.522698\pi$
$197$	−2.00000	−0.142494	−0.0712470	+	0.997459i	$0.522698\pi$
$198$	−5.00000	−0.355335
$199$	−11.0000	−0.779769	−0.389885	−	0.920864i	$-0.627485\pi$
$199$	−11.0000	−0.779769	−0.389885	+	0.920864i	$0.627485\pi$
$200$	−4.00000	−0.282843
$201$	−3.00000	−0.211604
$202$	−6.00000	−0.422159
$203$	16.0000	1.12298
$204$	−1.00000	−0.0700140
$205$	−6.00000	−0.419058
$206$	−10.0000	−0.696733
$207$	4.00000	0.278019
$208$	−7.00000	−0.485363
$209$	35.0000	2.42100
$210$	6.00000	0.414039
$211$	0	0		−	1.00000i	$-0.5\pi$
$211$	0	0			1.00000i	$0.5\pi$
$212$	6.00000	0.412082
$213$	3.00000	0.205557
$214$	−8.00000	−0.546869
$215$	−30.0000	−2.04598
$216$	−1.00000	−0.0680414
$217$	2.00000	0.135769
$218$	6.00000	0.406371
$219$	14.0000	0.946032
$220$	15.0000	1.01130
$221$	7.00000	0.470871
$222$	6.00000	0.402694
$223$	−5.00000	−0.334825	−0.167412	−	0.985887i	$-0.553541\pi$
$223$	−5.00000	−0.334825	−0.167412	+	0.985887i	$0.553541\pi$
$224$	2.00000	0.133631
$225$	4.00000	0.266667
$226$	−6.00000	−0.399114
$227$	22.0000	1.46019	0.730096	−	0.683345i	$-0.239475\pi$
$227$	22.0000	1.46019	0.730096	+	0.683345i	$0.239475\pi$
$228$	7.00000	0.463586
$229$	13.0000	0.859064	0.429532	−	0.903052i	$-0.358679\pi$
$229$	13.0000	0.859064	0.429532	+	0.903052i	$0.358679\pi$
$230$	−12.0000	−0.791257
$231$	−10.0000	−0.657952
$232$	8.00000	0.525226
$233$	20.0000	1.31024	0.655122	−	0.755523i	$-0.272617\pi$
$233$	20.0000	1.31024	0.655122	+	0.755523i	$0.272617\pi$
$234$	7.00000	0.457604
$235$	−3.00000	−0.195698
$236$	−10.0000	−0.650945
$237$	−11.0000	−0.714527
$238$	−2.00000	−0.129641
$239$	−4.00000	−0.258738	−0.129369	−	0.991596i	$-0.541295\pi$
$239$	−4.00000	−0.258738	−0.129369	+	0.991596i	$0.541295\pi$
$240$	3.00000	0.193649
$241$	−14.0000	−0.901819	−0.450910	−	0.892570i	$-0.648900\pi$
$241$	−14.0000	−0.901819	−0.450910	+	0.892570i	$0.648900\pi$
$242$	−14.0000	−0.899954
$243$	1.00000	0.0641500
$244$	1.00000	0.0640184
$245$	−9.00000	−0.574989
$246$	2.00000	0.127515
$247$	−49.0000	−3.11780
$248$	1.00000	0.0635001
$249$	7.00000	0.443607
$250$	3.00000	0.189737
$251$	20.0000	1.26239	0.631194	−	0.775625i	$-0.282565\pi$
$251$	20.0000	1.26239	0.631194	+	0.775625i	$0.282565\pi$
$252$	−2.00000	−0.125988
$253$	20.0000	1.25739
$254$	−4.00000	−0.250982
$255$	−3.00000	−0.187867
$256$	1.00000	0.0625000
$257$	−12.0000	−0.748539	−0.374270	−	0.927320i	$-0.622107\pi$
$257$	−12.0000	−0.748539	−0.374270	+	0.927320i	$0.622107\pi$
$258$	10.0000	0.622573
$259$	12.0000	0.745644
$260$	−21.0000	−1.30236
$261$	−8.00000	−0.495188
$262$	−10.0000	−0.617802
$263$	12.0000	0.739952	0.369976	−	0.929041i	$-0.379366\pi$
$263$	12.0000	0.739952	0.369976	+	0.929041i	$0.379366\pi$
$264$	−5.00000	−0.307729
$265$	18.0000	1.10573
$266$	14.0000	0.858395
$267$	−6.00000	−0.367194
$268$	−3.00000	−0.183254
$269$	6.00000	0.365826	0.182913	−	0.983129i	$-0.441447\pi$
$269$	6.00000	0.365826	0.182913	+	0.983129i	$0.441447\pi$
$270$	−3.00000	−0.182574
$271$	−19.0000	−1.15417	−0.577084	−	0.816685i	$-0.695809\pi$
$271$	−19.0000	−1.15417	−0.577084	+	0.816685i	$0.695809\pi$
$272$	−1.00000	−0.0606339
$273$	14.0000	0.847319
$274$	18.0000	1.08742
$275$	20.0000	1.20605
$276$	4.00000	0.240772
$277$	−7.00000	−0.420589	−0.210295	−	0.977638i	$-0.567442\pi$
$277$	−7.00000	−0.420589	−0.210295	+	0.977638i	$0.567442\pi$
$278$	8.00000	0.479808
$279$	−1.00000	−0.0598684
$280$	6.00000	0.358569
$281$	−30.0000	−1.78965	−0.894825	−	0.446417i	$-0.852700\pi$
$281$	−30.0000	−1.78965	−0.894825	+	0.446417i	$0.852700\pi$
$282$	1.00000	0.0595491
$283$	5.00000	0.297219	0.148610	−	0.988896i	$-0.452520\pi$
$283$	5.00000	0.297219	0.148610	+	0.988896i	$0.452520\pi$
$284$	3.00000	0.178017
$285$	21.0000	1.24393
$286$	35.0000	2.06959
$287$	4.00000	0.236113
$288$	−1.00000	−0.0589256
$289$	−16.0000	−0.941176
$290$	24.0000	1.40933
$291$	−3.00000	−0.175863
$292$	14.0000	0.819288
$293$	−6.00000	−0.350524	−0.175262	−	0.984522i	$-0.556077\pi$
$293$	−6.00000	−0.350524	−0.175262	+	0.984522i	$0.556077\pi$
$294$	3.00000	0.174964
$295$	−30.0000	−1.74667
$296$	6.00000	0.348743
$297$	5.00000	0.290129
$298$	−15.0000	−0.868927
$299$	−28.0000	−1.61928
$300$	4.00000	0.230940
$301$	20.0000	1.15278
$302$	1.00000	0.0575435
$303$	6.00000	0.344691
$304$	7.00000	0.401478
$305$	3.00000	0.171780
$306$	1.00000	0.0571662
$307$	16.0000	0.913168	0.456584	−	0.889680i	$-0.349073\pi$
$307$	16.0000	0.913168	0.456584	+	0.889680i	$0.349073\pi$
$308$	−10.0000	−0.569803
$309$	10.0000	0.568880
$310$	3.00000	0.170389
$311$	21.0000	1.19080	0.595400	−	0.803429i	$-0.296993\pi$
$311$	21.0000	1.19080	0.595400	+	0.803429i	$0.296993\pi$
$312$	7.00000	0.396297
$313$	2.00000	0.113047	0.0565233	−	0.998401i	$-0.481998\pi$
$313$	2.00000	0.113047	0.0565233	+	0.998401i	$0.481998\pi$
$314$	4.00000	0.225733
$315$	−6.00000	−0.338062
$316$	−11.0000	−0.618798
$317$	5.00000	0.280828	0.140414	−	0.990093i	$-0.455157\pi$
$317$	5.00000	0.280828	0.140414	+	0.990093i	$0.455157\pi$
$318$	−6.00000	−0.336463
$319$	−40.0000	−2.23957
$320$	3.00000	0.167705
$321$	8.00000	0.446516
$322$	8.00000	0.445823
$323$	−7.00000	−0.389490
$324$	1.00000	0.0555556
$325$	−28.0000	−1.55316
$326$	9.00000	0.498464
$327$	−6.00000	−0.331801
$328$	2.00000	0.110432
$329$	2.00000	0.110264
$330$	−15.0000	−0.825723
$331$	0	0		−	1.00000i	$-0.5\pi$
$331$	0	0			1.00000i	$0.5\pi$
$332$	7.00000	0.384175
$333$	−6.00000	−0.328798
$334$	−10.0000	−0.547176
$335$	−9.00000	−0.491723
$336$	−2.00000	−0.109109
$337$	−8.00000	−0.435788	−0.217894	−	0.975972i	$-0.569919\pi$
$337$	−8.00000	−0.435788	−0.217894	+	0.975972i	$0.569919\pi$
$338$	−36.0000	−1.95814
$339$	6.00000	0.325875
$340$	−3.00000	−0.162698
$341$	−5.00000	−0.270765
$342$	−7.00000	−0.378517
$343$	20.0000	1.07990
$344$	10.0000	0.539164
$345$	12.0000	0.646058
$346$	−1.00000	−0.0537603
$347$	17.0000	0.912608	0.456304	−	0.889824i	$-0.349173\pi$
$347$	17.0000	0.912608	0.456304	+	0.889824i	$0.349173\pi$
$348$	−8.00000	−0.428845
$349$	24.0000	1.28469	0.642345	−	0.766415i	$-0.277962\pi$
$349$	24.0000	1.28469	0.642345	+	0.766415i	$0.277962\pi$
$350$	8.00000	0.427618
$351$	−7.00000	−0.373632
$352$	−5.00000	−0.266501
$353$	−11.0000	−0.585471	−0.292735	−	0.956193i	$-0.594566\pi$
$353$	−11.0000	−0.585471	−0.292735	+	0.956193i	$0.594566\pi$
$354$	10.0000	0.531494
$355$	9.00000	0.477670
$356$	−6.00000	−0.317999
$357$	2.00000	0.105851
$358$	−19.0000	−1.00418
$359$	−17.0000	−0.897226	−0.448613	−	0.893726i	$-0.648082\pi$
$359$	−17.0000	−0.897226	−0.448613	+	0.893726i	$0.648082\pi$
$360$	−3.00000	−0.158114
$361$	30.0000	1.57895
$362$	−18.0000	−0.946059
$363$	14.0000	0.734809
$364$	14.0000	0.733799
$365$	42.0000	2.19838
$366$	−1.00000	−0.0522708
$367$	−11.0000	−0.574195	−0.287098	−	0.957901i	$-0.592690\pi$
$367$	−11.0000	−0.574195	−0.287098	+	0.957901i	$0.592690\pi$
$368$	4.00000	0.208514
$369$	−2.00000	−0.104116
$370$	18.0000	0.935775
$371$	−12.0000	−0.623009
$372$	−1.00000	−0.0518476
$373$	−32.0000	−1.65690	−0.828449	−	0.560065i	$-0.810776\pi$
$373$	−32.0000	−1.65690	−0.828449	+	0.560065i	$0.810776\pi$
$374$	5.00000	0.258544
$375$	−3.00000	−0.154919
$376$	1.00000	0.0515711
$377$	56.0000	2.88415
$378$	2.00000	0.102869
$379$	1.00000	0.0513665	0.0256833	−	0.999670i	$-0.491824\pi$
$379$	1.00000	0.0513665	0.0256833	+	0.999670i	$0.491824\pi$
$380$	21.0000	1.07728
$381$	4.00000	0.204926
$382$	0	0
$383$	−10.0000	−0.510976	−0.255488	−	0.966812i	$-0.582236\pi$
$383$	−10.0000	−0.510976	−0.255488	+	0.966812i	$0.582236\pi$
$384$	−1.00000	−0.0510310
$385$	−30.0000	−1.52894
$386$	−19.0000	−0.967075
$387$	−10.0000	−0.508329
$388$	−3.00000	−0.152302
$389$	−34.0000	−1.72387	−0.861934	−	0.507020i	$-0.830747\pi$
$389$	−34.0000	−1.72387	−0.861934	+	0.507020i	$0.830747\pi$
$390$	21.0000	1.06338
$391$	−4.00000	−0.202289
$392$	3.00000	0.151523
$393$	10.0000	0.504433
$394$	2.00000	0.100759
$395$	−33.0000	−1.66041
$396$	5.00000	0.251259
$397$	−22.0000	−1.10415	−0.552074	−	0.833795i	$-0.686163\pi$
$397$	−22.0000	−1.10415	−0.552074	+	0.833795i	$0.686163\pi$
$398$	11.0000	0.551380
$399$	−14.0000	−0.700877
$400$	4.00000	0.200000
$401$	−23.0000	−1.14857	−0.574283	−	0.818657i	$-0.694719\pi$
$401$	−23.0000	−1.14857	−0.574283	+	0.818657i	$0.694719\pi$
$402$	3.00000	0.149626
$403$	7.00000	0.348695
$404$	6.00000	0.298511
$405$	3.00000	0.149071
$406$	−16.0000	−0.794067
$407$	−30.0000	−1.48704
$408$	1.00000	0.0495074
$409$	−30.0000	−1.48340	−0.741702	−	0.670729i	$-0.765981\pi$
$409$	−30.0000	−1.48340	−0.741702	+	0.670729i	$0.765981\pi$
$410$	6.00000	0.296319
$411$	−18.0000	−0.887875
$412$	10.0000	0.492665
$413$	20.0000	0.984136
$414$	−4.00000	−0.196589
$415$	21.0000	1.03085
$416$	7.00000	0.343203
$417$	−8.00000	−0.391762
$418$	−35.0000	−1.71191
$419$	−4.00000	−0.195413	−0.0977064	−	0.995215i	$-0.531151\pi$
$419$	−4.00000	−0.195413	−0.0977064	+	0.995215i	$0.531151\pi$
$420$	−6.00000	−0.292770
$421$	2.00000	0.0974740	0.0487370	−	0.998812i	$-0.484480\pi$
$421$	2.00000	0.0974740	0.0487370	+	0.998812i	$0.484480\pi$
$422$	0	0
$423$	−1.00000	−0.0486217
$424$	−6.00000	−0.291386
$425$	−4.00000	−0.194029
$426$	−3.00000	−0.145350
$427$	−2.00000	−0.0967868
$428$	8.00000	0.386695
$429$	−35.0000	−1.68982
$430$	30.0000	1.44673
$431$	24.0000	1.15604	0.578020	−	0.816023i	$-0.303826\pi$
$431$	24.0000	1.15604	0.578020	+	0.816023i	$0.303826\pi$
$432$	1.00000	0.0481125
$433$	28.0000	1.34559	0.672797	−	0.739827i	$-0.265093\pi$
$433$	28.0000	1.34559	0.672797	+	0.739827i	$0.265093\pi$
$434$	−2.00000	−0.0960031
$435$	−24.0000	−1.15071
$436$	−6.00000	−0.287348
$437$	28.0000	1.33942
$438$	−14.0000	−0.668946
$439$	32.0000	1.52728	0.763638	−	0.645644i	$-0.223411\pi$
$439$	32.0000	1.52728	0.763638	+	0.645644i	$0.223411\pi$
$440$	−15.0000	−0.715097
$441$	−3.00000	−0.142857
$442$	−7.00000	−0.332956
$443$	−14.0000	−0.665160	−0.332580	−	0.943075i	$-0.607919\pi$
$443$	−14.0000	−0.665160	−0.332580	+	0.943075i	$0.607919\pi$
$444$	−6.00000	−0.284747
$445$	−18.0000	−0.853282
$446$	5.00000	0.236757
$447$	15.0000	0.709476
$448$	−2.00000	−0.0944911
$449$	−11.0000	−0.519122	−0.259561	−	0.965727i	$-0.583578\pi$
$449$	−11.0000	−0.519122	−0.259561	+	0.965727i	$0.583578\pi$
$450$	−4.00000	−0.188562
$451$	−10.0000	−0.470882
$452$	6.00000	0.282216
$453$	−1.00000	−0.0469841
$454$	−22.0000	−1.03251
$455$	42.0000	1.96899
$456$	−7.00000	−0.327805
$457$	−42.0000	−1.96468	−0.982339	−	0.187112i	$-0.940087\pi$
$457$	−42.0000	−1.96468	−0.982339	+	0.187112i	$0.940087\pi$
$458$	−13.0000	−0.607450
$459$	−1.00000	−0.0466760
$460$	12.0000	0.559503
$461$	−34.0000	−1.58354	−0.791769	−	0.610821i	$-0.790840\pi$
$461$	−34.0000	−1.58354	−0.791769	+	0.610821i	$0.790840\pi$
$462$	10.0000	0.465242
$463$	33.0000	1.53364	0.766820	−	0.641862i	$-0.221838\pi$
$463$	33.0000	1.53364	0.766820	+	0.641862i	$0.221838\pi$
$464$	−8.00000	−0.371391
$465$	−3.00000	−0.139122
$466$	−20.0000	−0.926482
$467$	24.0000	1.11059	0.555294	−	0.831654i	$-0.312606\pi$
$467$	24.0000	1.11059	0.555294	+	0.831654i	$0.312606\pi$
$468$	−7.00000	−0.323575
$469$	6.00000	0.277054
$470$	3.00000	0.138380
$471$	−4.00000	−0.184310
$472$	10.0000	0.460287
$473$	−50.0000	−2.29900
$474$	11.0000	0.505247
$475$	28.0000	1.28473
$476$	2.00000	0.0916698
$477$	6.00000	0.274721
$478$	4.00000	0.182956
$479$	29.0000	1.32504	0.662522	−	0.749043i	$-0.269486\pi$
$479$	29.0000	1.32504	0.662522	+	0.749043i	$0.269486\pi$
$480$	−3.00000	−0.136931
$481$	42.0000	1.91504
$482$	14.0000	0.637683
$483$	−8.00000	−0.364013
$484$	14.0000	0.636364
$485$	−9.00000	−0.408669
$486$	−1.00000	−0.0453609
$487$	−27.0000	−1.22349	−0.611743	−	0.791056i	$-0.709531\pi$
$487$	−27.0000	−1.22349	−0.611743	+	0.791056i	$0.709531\pi$
$488$	−1.00000	−0.0452679
$489$	−9.00000	−0.406994
$490$	9.00000	0.406579
$491$	−20.0000	−0.902587	−0.451294	−	0.892375i	$-0.649037\pi$
$491$	−20.0000	−0.902587	−0.451294	+	0.892375i	$0.649037\pi$
$492$	−2.00000	−0.0901670
$493$	8.00000	0.360302
$494$	49.0000	2.20461
$495$	15.0000	0.674200
$496$	−1.00000	−0.0449013
$497$	−6.00000	−0.269137
$498$	−7.00000	−0.313678
$499$	−24.0000	−1.07439	−0.537194	−	0.843459i	$-0.680516\pi$
$499$	−24.0000	−1.07439	−0.537194	+	0.843459i	$0.680516\pi$
$500$	−3.00000	−0.134164
$501$	10.0000	0.446767
$502$	−20.0000	−0.892644
$503$	−33.0000	−1.47140	−0.735699	−	0.677309i	$-0.763146\pi$
$503$	−33.0000	−1.47140	−0.735699	+	0.677309i	$0.763146\pi$
$504$	2.00000	0.0890871
$505$	18.0000	0.800989
$506$	−20.0000	−0.889108
$507$	36.0000	1.59882
$508$	4.00000	0.177471
$509$	40.0000	1.77297	0.886484	−	0.462758i	$-0.153140\pi$
$509$	40.0000	1.77297	0.886484	+	0.462758i	$0.153140\pi$
$510$	3.00000	0.132842
$511$	−28.0000	−1.23865
$512$	−1.00000	−0.0441942
$513$	7.00000	0.309058
$514$	12.0000	0.529297
$515$	30.0000	1.32196
$516$	−10.0000	−0.440225
$517$	−5.00000	−0.219900
$518$	−12.0000	−0.527250
$519$	1.00000	0.0438951
$520$	21.0000	0.920911
$521$	12.0000	0.525730	0.262865	−	0.964833i	$-0.415333\pi$
$521$	12.0000	0.525730	0.262865	+	0.964833i	$0.415333\pi$
$522$	8.00000	0.350150
$523$	30.0000	1.31181	0.655904	−	0.754844i	$-0.272288\pi$
$523$	30.0000	1.31181	0.655904	+	0.754844i	$0.272288\pi$
$524$	10.0000	0.436852
$525$	−8.00000	−0.349149
$526$	−12.0000	−0.523225
$527$	1.00000	0.0435607
$528$	5.00000	0.217597
$529$	−7.00000	−0.304348
$530$	−18.0000	−0.781870
$531$	−10.0000	−0.433963
$532$	−14.0000	−0.606977
$533$	14.0000	0.606407
$534$	6.00000	0.259645
$535$	24.0000	1.03761
$536$	3.00000	0.129580
$537$	19.0000	0.819911
$538$	−6.00000	−0.258678
$539$	−15.0000	−0.646096
$540$	3.00000	0.129099
$541$	44.0000	1.89171	0.945854	−	0.324593i	$-0.105227\pi$
$541$	44.0000	1.89171	0.945854	+	0.324593i	$0.105227\pi$
$542$	19.0000	0.816120
$543$	18.0000	0.772454
$544$	1.00000	0.0428746
$545$	−18.0000	−0.771035
$546$	−14.0000	−0.599145
$547$	32.0000	1.36822	0.684111	−	0.729378i	$-0.260191\pi$
$547$	32.0000	1.36822	0.684111	+	0.729378i	$0.260191\pi$
$548$	−18.0000	−0.768922
$549$	1.00000	0.0426790
$550$	−20.0000	−0.852803
$551$	−56.0000	−2.38568
$552$	−4.00000	−0.170251
$553$	22.0000	0.935535
$554$	7.00000	0.297402
$555$	−18.0000	−0.764057
$556$	−8.00000	−0.339276
$557$	−24.0000	−1.01691	−0.508456	−	0.861088i	$-0.669784\pi$
$557$	−24.0000	−1.01691	−0.508456	+	0.861088i	$0.669784\pi$
$558$	1.00000	0.0423334
$559$	70.0000	2.96068
$560$	−6.00000	−0.253546
$561$	−5.00000	−0.211100
$562$	30.0000	1.26547
$563$	−24.0000	−1.01148	−0.505740	−	0.862686i	$-0.668780\pi$
$563$	−24.0000	−1.01148	−0.505740	+	0.862686i	$0.668780\pi$
$564$	−1.00000	−0.0421076
$565$	18.0000	0.757266
$566$	−5.00000	−0.210166
$567$	−2.00000	−0.0839921
$568$	−3.00000	−0.125877
$569$	26.0000	1.08998	0.544988	−	0.838444i	$-0.316534\pi$
$569$	26.0000	1.08998	0.544988	+	0.838444i	$0.316534\pi$
$570$	−21.0000	−0.879593
$571$	12.0000	0.502184	0.251092	−	0.967963i	$-0.419210\pi$
$571$	12.0000	0.502184	0.251092	+	0.967963i	$0.419210\pi$
$572$	−35.0000	−1.46342
$573$	0	0
$574$	−4.00000	−0.166957
$575$	16.0000	0.667246
$576$	1.00000	0.0416667
$577$	27.0000	1.12402	0.562012	−	0.827129i	$-0.310027\pi$
$577$	27.0000	1.12402	0.562012	+	0.827129i	$0.310027\pi$
$578$	16.0000	0.665512
$579$	19.0000	0.789613
$580$	−24.0000	−0.996546
$581$	−14.0000	−0.580818
$582$	3.00000	0.124354
$583$	30.0000	1.24247
$584$	−14.0000	−0.579324
$585$	−21.0000	−0.868243
$586$	6.00000	0.247858
$587$	−17.0000	−0.701665	−0.350833	−	0.936438i	$-0.614101\pi$
$587$	−17.0000	−0.701665	−0.350833	+	0.936438i	$0.614101\pi$
$588$	−3.00000	−0.123718
$589$	−7.00000	−0.288430
$590$	30.0000	1.23508
$591$	−2.00000	−0.0822690
$592$	−6.00000	−0.246598
$593$	−30.0000	−1.23195	−0.615976	−	0.787765i	$-0.711238\pi$
$593$	−30.0000	−1.23195	−0.615976	+	0.787765i	$0.711238\pi$
$594$	−5.00000	−0.205152
$595$	6.00000	0.245976
$596$	15.0000	0.614424
$597$	−11.0000	−0.450200
$598$	28.0000	1.14501
$599$	−39.0000	−1.59350	−0.796748	−	0.604311i	$-0.793448\pi$
$599$	−39.0000	−1.59350	−0.796748	+	0.604311i	$0.793448\pi$
$600$	−4.00000	−0.163299
$601$	−2.00000	−0.0815817	−0.0407909	−	0.999168i	$-0.512988\pi$
$601$	−2.00000	−0.0815817	−0.0407909	+	0.999168i	$0.512988\pi$
$602$	−20.0000	−0.815139
$603$	−3.00000	−0.122169
$604$	−1.00000	−0.0406894
$605$	42.0000	1.70754
$606$	−6.00000	−0.243733
$607$	10.0000	0.405887	0.202944	−	0.979190i	$-0.434949\pi$
$607$	10.0000	0.405887	0.202944	+	0.979190i	$0.434949\pi$
$608$	−7.00000	−0.283887
$609$	16.0000	0.648353
$610$	−3.00000	−0.121466
$611$	7.00000	0.283190
$612$	−1.00000	−0.0404226
$613$	5.00000	0.201948	0.100974	−	0.994889i	$-0.467804\pi$
$613$	5.00000	0.201948	0.100974	+	0.994889i	$0.467804\pi$
$614$	−16.0000	−0.645707
$615$	−6.00000	−0.241943
$616$	10.0000	0.402911
$617$	6.00000	0.241551	0.120775	−	0.992680i	$-0.461462\pi$
$617$	6.00000	0.241551	0.120775	+	0.992680i	$0.461462\pi$
$618$	−10.0000	−0.402259
$619$	−8.00000	−0.321547	−0.160774	−	0.986991i	$-0.551399\pi$
$619$	−8.00000	−0.321547	−0.160774	+	0.986991i	$0.551399\pi$
$620$	−3.00000	−0.120483
$621$	4.00000	0.160514
$622$	−21.0000	−0.842023
$623$	12.0000	0.480770
$624$	−7.00000	−0.280224
$625$	−29.0000	−1.16000
$626$	−2.00000	−0.0799361
$627$	35.0000	1.39777
$628$	−4.00000	−0.159617
$629$	6.00000	0.239236
$630$	6.00000	0.239046
$631$	−16.0000	−0.636950	−0.318475	−	0.947931i	$-0.603171\pi$
$631$	−16.0000	−0.636950	−0.318475	+	0.947931i	$0.603171\pi$
$632$	11.0000	0.437557
$633$	0	0
$634$	−5.00000	−0.198575
$635$	12.0000	0.476205
$636$	6.00000	0.237915
$637$	21.0000	0.832050
$638$	40.0000	1.58362
$639$	3.00000	0.118678
$640$	−3.00000	−0.118585
$641$	23.0000	0.908445	0.454223	−	0.890888i	$-0.349917\pi$
$641$	23.0000	0.908445	0.454223	+	0.890888i	$0.349917\pi$
$642$	−8.00000	−0.315735
$643$	−32.0000	−1.26196	−0.630978	−	0.775800i	$-0.717346\pi$
$643$	−32.0000	−1.26196	−0.630978	+	0.775800i	$0.717346\pi$
$644$	−8.00000	−0.315244
$645$	−30.0000	−1.18125
$646$	7.00000	0.275411
$647$	14.0000	0.550397	0.275198	−	0.961387i	$-0.411256\pi$
$647$	14.0000	0.550397	0.275198	+	0.961387i	$0.411256\pi$
$648$	−1.00000	−0.0392837
$649$	−50.0000	−1.96267
$650$	28.0000	1.09825
$651$	2.00000	0.0783862
$652$	−9.00000	−0.352467
$653$	27.0000	1.05659	0.528296	−	0.849060i	$-0.322831\pi$
$653$	27.0000	1.05659	0.528296	+	0.849060i	$0.322831\pi$
$654$	6.00000	0.234619
$655$	30.0000	1.17220
$656$	−2.00000	−0.0780869
$657$	14.0000	0.546192
$658$	−2.00000	−0.0779681
$659$	−26.0000	−1.01282	−0.506408	−	0.862294i	$-0.669027\pi$
$659$	−26.0000	−1.01282	−0.506408	+	0.862294i	$0.669027\pi$
$660$	15.0000	0.583874
$661$	4.00000	0.155582	0.0777910	−	0.996970i	$-0.475213\pi$
$661$	4.00000	0.155582	0.0777910	+	0.996970i	$0.475213\pi$
$662$	0	0
$663$	7.00000	0.271857
$664$	−7.00000	−0.271653
$665$	−42.0000	−1.62869
$666$	6.00000	0.232495
$667$	−32.0000	−1.23904
$668$	10.0000	0.386912
$669$	−5.00000	−0.193311
$670$	9.00000	0.347700
$671$	5.00000	0.193023
$672$	2.00000	0.0771517
$673$	8.00000	0.308377	0.154189	−	0.988041i	$-0.450724\pi$
$673$	8.00000	0.308377	0.154189	+	0.988041i	$0.450724\pi$
$674$	8.00000	0.308148
$675$	4.00000	0.153960
$676$	36.0000	1.38462
$677$	36.0000	1.38359	0.691796	−	0.722093i	$-0.256820\pi$
$677$	36.0000	1.38359	0.691796	+	0.722093i	$0.256820\pi$
$678$	−6.00000	−0.230429
$679$	6.00000	0.230259
$680$	3.00000	0.115045
$681$	22.0000	0.843042
$682$	5.00000	0.191460
$683$	16.0000	0.612223	0.306111	−	0.951996i	$-0.400972\pi$
$683$	16.0000	0.612223	0.306111	+	0.951996i	$0.400972\pi$
$684$	7.00000	0.267652
$685$	−54.0000	−2.06323
$686$	−20.0000	−0.763604
$687$	13.0000	0.495981
$688$	−10.0000	−0.381246
$689$	−42.0000	−1.60007
$690$	−12.0000	−0.456832
$691$	13.0000	0.494543	0.247272	−	0.968946i	$-0.420466\pi$
$691$	13.0000	0.494543	0.247272	+	0.968946i	$0.420466\pi$
$692$	1.00000	0.0380143
$693$	−10.0000	−0.379869
$694$	−17.0000	−0.645311
$695$	−24.0000	−0.910372
$696$	8.00000	0.303239
$697$	2.00000	0.0757554
$698$	−24.0000	−0.908413
$699$	20.0000	0.756469
$700$	−8.00000	−0.302372
$701$	35.0000	1.32193	0.660966	−	0.750416i	$-0.270147\pi$
$701$	35.0000	1.32193	0.660966	+	0.750416i	$0.270147\pi$
$702$	7.00000	0.264198
$703$	−42.0000	−1.58406
$704$	5.00000	0.188445
$705$	−3.00000	−0.112987
$706$	11.0000	0.413990
$707$	−12.0000	−0.451306
$708$	−10.0000	−0.375823
$709$	−21.0000	−0.788672	−0.394336	−	0.918966i	$-0.629025\pi$
$709$	−21.0000	−0.788672	−0.394336	+	0.918966i	$0.629025\pi$
$710$	−9.00000	−0.337764
$711$	−11.0000	−0.412532
$712$	6.00000	0.224860
$713$	−4.00000	−0.149801
$714$	−2.00000	−0.0748481
$715$	−105.000	−3.92678
$716$	19.0000	0.710063
$717$	−4.00000	−0.149383
$718$	17.0000	0.634434
$719$	−12.0000	−0.447524	−0.223762	−	0.974644i	$-0.571834\pi$
$719$	−12.0000	−0.447524	−0.223762	+	0.974644i	$0.571834\pi$
$720$	3.00000	0.111803
$721$	−20.0000	−0.744839
$722$	−30.0000	−1.11648
$723$	−14.0000	−0.520666
$724$	18.0000	0.668965
$725$	−32.0000	−1.18845
$726$	−14.0000	−0.519589
$727$	4.00000	0.148352	0.0741759	−	0.997245i	$-0.476367\pi$
$727$	4.00000	0.148352	0.0741759	+	0.997245i	$0.476367\pi$
$728$	−14.0000	−0.518875
$729$	1.00000	0.0370370
$730$	−42.0000	−1.55449
$731$	10.0000	0.369863
$732$	1.00000	0.0369611
$733$	−30.0000	−1.10808	−0.554038	−	0.832492i	$-0.686914\pi$
$733$	−30.0000	−1.10808	−0.554038	+	0.832492i	$0.686914\pi$
$734$	11.0000	0.406017
$735$	−9.00000	−0.331970
$736$	−4.00000	−0.147442
$737$	−15.0000	−0.552532
$738$	2.00000	0.0736210
$739$	−16.0000	−0.588570	−0.294285	−	0.955718i	$-0.595081\pi$
$739$	−16.0000	−0.588570	−0.294285	+	0.955718i	$0.595081\pi$
$740$	−18.0000	−0.661693
$741$	−49.0000	−1.80006
$742$	12.0000	0.440534
$743$	−16.0000	−0.586983	−0.293492	−	0.955962i	$-0.594817\pi$
$743$	−16.0000	−0.586983	−0.293492	+	0.955962i	$0.594817\pi$
$744$	1.00000	0.0366618
$745$	45.0000	1.64867
$746$	32.0000	1.17160
$747$	7.00000	0.256117
$748$	−5.00000	−0.182818
$749$	−16.0000	−0.584627
$750$	3.00000	0.109545
$751$	−38.0000	−1.38664	−0.693320	−	0.720630i	$-0.743853\pi$
$751$	−38.0000	−1.38664	−0.693320	+	0.720630i	$0.743853\pi$
$752$	−1.00000	−0.0364662
$753$	20.0000	0.728841
$754$	−56.0000	−2.03940
$755$	−3.00000	−0.109181
$756$	−2.00000	−0.0727393
$757$	18.0000	0.654221	0.327111	−	0.944986i	$-0.393925\pi$
$757$	18.0000	0.654221	0.327111	+	0.944986i	$0.393925\pi$
$758$	−1.00000	−0.0363216
$759$	20.0000	0.725954
$760$	−21.0000	−0.761750
$761$	1.00000	0.0362500	0.0181250	−	0.999836i	$-0.494230\pi$
$761$	1.00000	0.0362500	0.0181250	+	0.999836i	$0.494230\pi$
$762$	−4.00000	−0.144905
$763$	12.0000	0.434429
$764$	0	0
$765$	−3.00000	−0.108465
$766$	10.0000	0.361315
$767$	70.0000	2.52755
$768$	1.00000	0.0360844
$769$	−10.0000	−0.360609	−0.180305	−	0.983611i	$-0.557708\pi$
$769$	−10.0000	−0.360609	−0.180305	+	0.983611i	$0.557708\pi$
$770$	30.0000	1.08112
$771$	−12.0000	−0.432169
$772$	19.0000	0.683825
$773$	−16.0000	−0.575480	−0.287740	−	0.957709i	$-0.592904\pi$
$773$	−16.0000	−0.575480	−0.287740	+	0.957709i	$0.592904\pi$
$774$	10.0000	0.359443
$775$	−4.00000	−0.143684
$776$	3.00000	0.107694
$777$	12.0000	0.430498
$778$	34.0000	1.21896
$779$	−14.0000	−0.501602
$780$	−21.0000	−0.751921
$781$	15.0000	0.536742
$782$	4.00000	0.143040
$783$	−8.00000	−0.285897
$784$	−3.00000	−0.107143
$785$	−12.0000	−0.428298
$786$	−10.0000	−0.356688
$787$	40.0000	1.42585	0.712923	−	0.701242i	$-0.247371\pi$
$787$	40.0000	1.42585	0.712923	+	0.701242i	$0.247371\pi$
$788$	−2.00000	−0.0712470
$789$	12.0000	0.427211
$790$	33.0000	1.17409
$791$	−12.0000	−0.426671
$792$	−5.00000	−0.177667
$793$	−7.00000	−0.248577
$794$	22.0000	0.780751
$795$	18.0000	0.638394
$796$	−11.0000	−0.389885
$797$	0	0		−	1.00000i	$-0.5\pi$
$797$	0	0			1.00000i	$0.5\pi$
$798$	14.0000	0.495595
$799$	1.00000	0.0353775
$800$	−4.00000	−0.141421
$801$	−6.00000	−0.212000
$802$	23.0000	0.812158
$803$	70.0000	2.47025
$804$	−3.00000	−0.105802
$805$	−24.0000	−0.845889
$806$	−7.00000	−0.246564
$807$	6.00000	0.211210
$808$	−6.00000	−0.211079
$809$	−29.0000	−1.01959	−0.509793	−	0.860297i	$-0.670278\pi$
$809$	−29.0000	−1.01959	−0.509793	+	0.860297i	$0.670278\pi$
$810$	−3.00000	−0.105409
$811$	20.0000	0.702295	0.351147	−	0.936320i	$-0.385792\pi$
$811$	20.0000	0.702295	0.351147	+	0.936320i	$0.385792\pi$
$812$	16.0000	0.561490
$813$	−19.0000	−0.666359
$814$	30.0000	1.05150
$815$	−27.0000	−0.945769
$816$	−1.00000	−0.0350070
$817$	−70.0000	−2.44899
$818$	30.0000	1.04893
$819$	14.0000	0.489200
$820$	−6.00000	−0.209529
$821$	−10.0000	−0.349002	−0.174501	−	0.984657i	$-0.555831\pi$
$821$	−10.0000	−0.349002	−0.174501	+	0.984657i	$0.555831\pi$
$822$	18.0000	0.627822
$823$	43.0000	1.49889	0.749443	−	0.662069i	$-0.230321\pi$
$823$	43.0000	1.49889	0.749443	+	0.662069i	$0.230321\pi$
$824$	−10.0000	−0.348367
$825$	20.0000	0.696311
$826$	−20.0000	−0.695889
$827$	27.0000	0.938882	0.469441	−	0.882964i	$-0.344455\pi$
$827$	27.0000	0.938882	0.469441	+	0.882964i	$0.344455\pi$
$828$	4.00000	0.139010
$829$	26.0000	0.903017	0.451509	−	0.892267i	$-0.350886\pi$
$829$	26.0000	0.903017	0.451509	+	0.892267i	$0.350886\pi$
$830$	−21.0000	−0.728921
$831$	−7.00000	−0.242827
$832$	−7.00000	−0.242681
$833$	3.00000	0.103944
$834$	8.00000	0.277017
$835$	30.0000	1.03819
$836$	35.0000	1.21050
$837$	−1.00000	−0.0345651
$838$	4.00000	0.138178
$839$	12.0000	0.414286	0.207143	−	0.978311i	$-0.433583\pi$
$839$	12.0000	0.414286	0.207143	+	0.978311i	$0.433583\pi$
$840$	6.00000	0.207020
$841$	35.0000	1.20690
$842$	−2.00000	−0.0689246
$843$	−30.0000	−1.03325
$844$	0	0
$845$	108.000	3.71531
$846$	1.00000	0.0343807
$847$	−28.0000	−0.962091
$848$	6.00000	0.206041
$849$	5.00000	0.171600
$850$	4.00000	0.137199
$851$	−24.0000	−0.822709
$852$	3.00000	0.102778
$853$	−46.0000	−1.57501	−0.787505	−	0.616308i	$-0.788628\pi$
$853$	−46.0000	−1.57501	−0.787505	+	0.616308i	$0.788628\pi$
$854$	2.00000	0.0684386
$855$	21.0000	0.718185
$856$	−8.00000	−0.273434
$857$	−8.00000	−0.273275	−0.136637	−	0.990621i	$-0.543630\pi$
$857$	−8.00000	−0.273275	−0.136637	+	0.990621i	$0.543630\pi$
$858$	35.0000	1.19488
$859$	−26.0000	−0.887109	−0.443554	−	0.896248i	$-0.646283\pi$
$859$	−26.0000	−0.887109	−0.443554	+	0.896248i	$0.646283\pi$
$860$	−30.0000	−1.02299
$861$	4.00000	0.136320
$862$	−24.0000	−0.817443
$863$	10.0000	0.340404	0.170202	−	0.985409i	$-0.445558\pi$
$863$	10.0000	0.340404	0.170202	+	0.985409i	$0.445558\pi$
$864$	−1.00000	−0.0340207
$865$	3.00000	0.102003
$866$	−28.0000	−0.951479
$867$	−16.0000	−0.543388
$868$	2.00000	0.0678844
$869$	−55.0000	−1.86575
$870$	24.0000	0.813676
$871$	21.0000	0.711558
$872$	6.00000	0.203186
$873$	−3.00000	−0.101535
$874$	−28.0000	−0.947114
$875$	6.00000	0.202837
$876$	14.0000	0.473016
$877$	28.0000	0.945493	0.472746	−	0.881199i	$-0.343263\pi$
$877$	28.0000	0.945493	0.472746	+	0.881199i	$0.343263\pi$
$878$	−32.0000	−1.07995
$879$	−6.00000	−0.202375
$880$	15.0000	0.505650
$881$	51.0000	1.71823	0.859117	−	0.511780i	$-0.171014\pi$
$881$	51.0000	1.71823	0.859117	+	0.511780i	$0.171014\pi$
$882$	3.00000	0.101015
$883$	−46.0000	−1.54802	−0.774012	−	0.633171i	$-0.781753\pi$
$883$	−46.0000	−1.54802	−0.774012	+	0.633171i	$0.781753\pi$
$884$	7.00000	0.235435
$885$	−30.0000	−1.00844
$886$	14.0000	0.470339
$887$	16.0000	0.537227	0.268614	−	0.963248i	$-0.413434\pi$
$887$	16.0000	0.537227	0.268614	+	0.963248i	$0.413434\pi$
$888$	6.00000	0.201347
$889$	−8.00000	−0.268311
$890$	18.0000	0.603361
$891$	5.00000	0.167506
$892$	−5.00000	−0.167412
$893$	−7.00000	−0.234246
$894$	−15.0000	−0.501675
$895$	57.0000	1.90530
$896$	2.00000	0.0668153
$897$	−28.0000	−0.934893
$898$	11.0000	0.367075
$899$	8.00000	0.266815
$900$	4.00000	0.133333
$901$	−6.00000	−0.199889
$902$	10.0000	0.332964
$903$	20.0000	0.665558
$904$	−6.00000	−0.199557
$905$	54.0000	1.79502
$906$	1.00000	0.0332228
$907$	−1.00000	−0.0332045	−0.0166022	−	0.999862i	$-0.505285\pi$
$907$	−1.00000	−0.0332045	−0.0166022	+	0.999862i	$0.505285\pi$
$908$	22.0000	0.730096
$909$	6.00000	0.199007
$910$	−42.0000	−1.39229
$911$	−24.0000	−0.795155	−0.397578	−	0.917568i	$-0.630149\pi$
$911$	−24.0000	−0.795155	−0.397578	+	0.917568i	$0.630149\pi$
$912$	7.00000	0.231793
$913$	35.0000	1.15833
$914$	42.0000	1.38924
$915$	3.00000	0.0991769
$916$	13.0000	0.429532
$917$	−20.0000	−0.660458
$918$	1.00000	0.0330049
$919$	28.0000	0.923635	0.461817	−	0.886975i	$-0.347198\pi$
$919$	28.0000	0.923635	0.461817	+	0.886975i	$0.347198\pi$
$920$	−12.0000	−0.395628
$921$	16.0000	0.527218
$922$	34.0000	1.11973
$923$	−21.0000	−0.691223
$924$	−10.0000	−0.328976
$925$	−24.0000	−0.789115
$926$	−33.0000	−1.08445
$927$	10.0000	0.328443
$928$	8.00000	0.262613
$929$	14.0000	0.459325	0.229663	−	0.973270i	$-0.426238\pi$
$929$	14.0000	0.459325	0.229663	+	0.973270i	$0.426238\pi$
$930$	3.00000	0.0983739
$931$	−21.0000	−0.688247
$932$	20.0000	0.655122
$933$	21.0000	0.687509
$934$	−24.0000	−0.785304
$935$	−15.0000	−0.490552
$936$	7.00000	0.228802
$937$	1.00000	0.0326686	0.0163343	−	0.999867i	$-0.494800\pi$
$937$	1.00000	0.0326686	0.0163343	+	0.999867i	$0.494800\pi$
$938$	−6.00000	−0.195907
$939$	2.00000	0.0652675
$940$	−3.00000	−0.0978492
$941$	−48.0000	−1.56476	−0.782378	−	0.622804i	$-0.785993\pi$
$941$	−48.0000	−1.56476	−0.782378	+	0.622804i	$0.785993\pi$
$942$	4.00000	0.130327
$943$	−8.00000	−0.260516
$944$	−10.0000	−0.325472
$945$	−6.00000	−0.195180
$946$	50.0000	1.62564
$947$	−5.00000	−0.162478	−0.0812391	−	0.996695i	$-0.525888\pi$
$947$	−5.00000	−0.162478	−0.0812391	+	0.996695i	$0.525888\pi$
$948$	−11.0000	−0.357263
$949$	−98.0000	−3.18121
$950$	−28.0000	−0.908440
$951$	5.00000	0.162136
$952$	−2.00000	−0.0648204
$953$	−5.00000	−0.161966	−0.0809829	−	0.996715i	$-0.525806\pi$
$953$	−5.00000	−0.161966	−0.0809829	+	0.996715i	$0.525806\pi$
$954$	−6.00000	−0.194257
$955$	0	0
$956$	−4.00000	−0.129369
$957$	−40.0000	−1.29302
$958$	−29.0000	−0.936947
$959$	36.0000	1.16250
$960$	3.00000	0.0968246
$961$	1.00000	0.0322581
$962$	−42.0000	−1.35413
$963$	8.00000	0.257796
$964$	−14.0000	−0.450910
$965$	57.0000	1.83489
$966$	8.00000	0.257396
$967$	33.0000	1.06121	0.530604	−	0.847620i	$-0.321965\pi$
$967$	33.0000	1.06121	0.530604	+	0.847620i	$0.321965\pi$
$968$	−14.0000	−0.449977
$969$	−7.00000	−0.224872
$970$	9.00000	0.288973
$971$	34.0000	1.09111	0.545556	−	0.838074i	$-0.316319\pi$
$971$	34.0000	1.09111	0.545556	+	0.838074i	$0.316319\pi$
$972$	1.00000	0.0320750
$973$	16.0000	0.512936
$974$	27.0000	0.865136
$975$	−28.0000	−0.896718
$976$	1.00000	0.0320092
$977$	−22.0000	−0.703842	−0.351921	−	0.936030i	$-0.614471\pi$
$977$	−22.0000	−0.703842	−0.351921	+	0.936030i	$0.614471\pi$
$978$	9.00000	0.287788
$979$	−30.0000	−0.958804
$980$	−9.00000	−0.287494
$981$	−6.00000	−0.191565
$982$	20.0000	0.638226
$983$	24.0000	0.765481	0.382741	−	0.923856i	$-0.374980\pi$
$983$	24.0000	0.765481	0.382741	+	0.923856i	$0.374980\pi$
$984$	2.00000	0.0637577
$985$	−6.00000	−0.191176
$986$	−8.00000	−0.254772
$987$	2.00000	0.0636607
$988$	−49.0000	−1.55890
$989$	−40.0000	−1.27193
$990$	−15.0000	−0.476731
$991$	−20.0000	−0.635321	−0.317660	−	0.948205i	$-0.602897\pi$
$991$	−20.0000	−0.635321	−0.317660	+	0.948205i	$0.602897\pi$
$992$	1.00000	0.0317500
$993$	0	0
$994$	6.00000	0.190308
$995$	−33.0000	−1.04617
$996$	7.00000	0.221803
$997$	−42.0000	−1.33015	−0.665077	−	0.746775i	$-0.731601\pi$
$997$	−42.0000	−1.33015	−0.665077	+	0.746775i	$0.731601\pi$
$998$	24.0000	0.759707
$999$	−6.00000	−0.189832

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	186.2.a.b.1.1	✓	1
3.2	odd	2		558.2.a.f.1.1		1
4.3	odd	2		1488.2.a.h.1.1		1
5.2	odd	4		4650.2.d.k.3349.1		2
5.3	odd	4		4650.2.d.k.3349.2		2
5.4	even	2		4650.2.a.bh.1.1		1
7.6	odd	2		9114.2.a.b.1.1		1
8.3	odd	2		5952.2.a.s.1.1		1
8.5	even	2		5952.2.a.b.1.1		1
12.11	even	2		4464.2.a.c.1.1		1
31.30	odd	2		5766.2.a.c.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
186.2.a.b.1.1	✓	1	1.1	even	1	trivial
558.2.a.f.1.1		1	3.2	odd	2
1488.2.a.h.1.1		1	4.3	odd	2
4464.2.a.c.1.1		1	12.11	even	2
4650.2.a.bh.1.1		1	5.4	even	2
4650.2.d.k.3349.1		2	5.2	odd	4
4650.2.d.k.3349.2		2	5.3	odd	4
5766.2.a.c.1.1		1	31.30	odd	2
5952.2.a.b.1.1		1	8.5	even	2
5952.2.a.s.1.1		1	8.3	odd	2
9114.2.a.b.1.1		1	7.6	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:	\( N \)	\(=\)	\( 186 = 2 \cdot 3 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	186.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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