LMFDB - Embedded newform 6041.2.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6041,2,Mod(1,6041)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6041, 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("6041.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6041 = 7 \cdot 863 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6041.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:	$48.2376278611$
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	$\chi$	$=$	6041.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{2} +2.00000 q^{3} -1.00000 q^{4} -4.00000 q^{5} +2.00000 q^{6} +1.00000 q^{7} -3.00000 q^{8} +1.00000 q^{9} +O(q^{10})$

\(q+1.00000 q^{2} +2.00000 q^{3} -1.00000 q^{4} -4.00000 q^{5} +2.00000 q^{6} +1.00000 q^{7} -3.00000 q^{8} +1.00000 q^{9} -4.00000 q^{10} -4.00000 q^{11} -2.00000 q^{12} -4.00000 q^{13} +1.00000 q^{14} -8.00000 q^{15} -1.00000 q^{16} -6.00000 q^{17} +1.00000 q^{18} +6.00000 q^{19} +4.00000 q^{20} +2.00000 q^{21} -4.00000 q^{22} -8.00000 q^{23} -6.00000 q^{24} +11.0000 q^{25} -4.00000 q^{26} -4.00000 q^{27} -1.00000 q^{28} +6.00000 q^{29} -8.00000 q^{30} +2.00000 q^{31} +5.00000 q^{32} -8.00000 q^{33} -6.00000 q^{34} -4.00000 q^{35} -1.00000 q^{36} +10.0000 q^{37} +6.00000 q^{38} -8.00000 q^{39} +12.0000 q^{40} +6.00000 q^{41} +2.00000 q^{42} -4.00000 q^{43} +4.00000 q^{44} -4.00000 q^{45} -8.00000 q^{46} -2.00000 q^{48} +1.00000 q^{49} +11.0000 q^{50} -12.0000 q^{51} +4.00000 q^{52} +2.00000 q^{53} -4.00000 q^{54} +16.0000 q^{55} -3.00000 q^{56} +12.0000 q^{57} +6.00000 q^{58} +10.0000 q^{59} +8.00000 q^{60} -10.0000 q^{61} +2.00000 q^{62} +1.00000 q^{63} +7.00000 q^{64} +16.0000 q^{65} -8.00000 q^{66} -4.00000 q^{67} +6.00000 q^{68} -16.0000 q^{69} -4.00000 q^{70} +8.00000 q^{71} -3.00000 q^{72} +4.00000 q^{73} +10.0000 q^{74} +22.0000 q^{75} -6.00000 q^{76} -4.00000 q^{77} -8.00000 q^{78} +4.00000 q^{80} -11.0000 q^{81} +6.00000 q^{82} -4.00000 q^{83} -2.00000 q^{84} +24.0000 q^{85} -4.00000 q^{86} +12.0000 q^{87} +12.0000 q^{88} +4.00000 q^{89} -4.00000 q^{90} -4.00000 q^{91} +8.00000 q^{92} +4.00000 q^{93} -24.0000 q^{95} +10.0000 q^{96} -8.00000 q^{97} +1.00000 q^{98} -4.00000 q^{99} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	1.00000	0.707107	0.353553	−	0.935414i	$-0.384973\pi$
$2$	1.00000	0.707107	0.353553	+	0.935414i	$0.384973\pi$
$3$	2.00000	1.15470	0.577350	−	0.816497i	$-0.304087\pi$
$3$	2.00000	1.15470	0.577350	+	0.816497i	$0.304087\pi$
$4$	−1.00000	−0.500000
$5$	−4.00000	−1.78885	−0.894427	−	0.447214i	$-0.852416\pi$
$5$	−4.00000	−1.78885	−0.894427	+	0.447214i	$0.852416\pi$
$6$	2.00000	0.816497
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−3.00000	−1.06066
$9$	1.00000	0.333333
$10$	−4.00000	−1.26491
$11$	−4.00000	−1.20605	−0.603023	−	0.797724i	$-0.706037\pi$
$11$	−4.00000	−1.20605	−0.603023	+	0.797724i	$0.706037\pi$
$12$	−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	$0.687167\pi$
$14$	1.00000	0.267261
$15$	−8.00000	−2.06559
$16$	−1.00000	−0.250000
$17$	−6.00000	−1.45521	−0.727607	−	0.685994i	$-0.759367\pi$
$17$	−6.00000	−1.45521	−0.727607	+	0.685994i	$0.759367\pi$
$18$	1.00000	0.235702
$19$	6.00000	1.37649	0.688247	−	0.725476i	$-0.258380\pi$
$19$	6.00000	1.37649	0.688247	+	0.725476i	$0.258380\pi$
$20$	4.00000	0.894427
$21$	2.00000	0.436436
$22$	−4.00000	−0.852803
$23$	−8.00000	−1.66812	−0.834058	−	0.551677i	$-0.813988\pi$
$23$	−8.00000	−1.66812	−0.834058	+	0.551677i	$0.813988\pi$
$24$	−6.00000	−1.22474
$25$	11.0000	2.20000
$26$	−4.00000	−0.784465
$27$	−4.00000	−0.769800
$28$	−1.00000	−0.188982
$29$	6.00000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	−8.00000	−1.46059
$31$	2.00000	0.359211	0.179605	−	0.983739i	$-0.442518\pi$
$31$	2.00000	0.359211	0.179605	+	0.983739i	$0.442518\pi$
$32$	5.00000	0.883883
$33$	−8.00000	−1.39262
$34$	−6.00000	−1.02899
$35$	−4.00000	−0.676123
$36$	−1.00000	−0.166667
$37$	10.0000	1.64399	0.821995	−	0.569495i	$-0.192861\pi$
$37$	10.0000	1.64399	0.821995	+	0.569495i	$0.192861\pi$
$38$	6.00000	0.973329
$39$	−8.00000	−1.28103
$40$	12.0000	1.89737
$41$	6.00000	0.937043	0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	2.00000	0.308607
$43$	−4.00000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	4.00000	0.603023
$45$	−4.00000	−0.596285
$46$	−8.00000	−1.17954
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	−2.00000	−0.288675
$49$	1.00000	0.142857
$50$	11.0000	1.55563
$51$	−12.0000	−1.68034
$52$	4.00000	0.554700
$53$	2.00000	0.274721	0.137361	−	0.990521i	$-0.456138\pi$
$53$	2.00000	0.274721	0.137361	+	0.990521i	$0.456138\pi$
$54$	−4.00000	−0.544331
$55$	16.0000	2.15744
$56$	−3.00000	−0.400892
$57$	12.0000	1.58944
$58$	6.00000	0.787839
$59$	10.0000	1.30189	0.650945	−	0.759125i	$-0.274373\pi$
$59$	10.0000	1.30189	0.650945	+	0.759125i	$0.274373\pi$
$60$	8.00000	1.03280
$61$	−10.0000	−1.28037	−0.640184	−	0.768221i	$-0.721142\pi$
$61$	−10.0000	−1.28037	−0.640184	+	0.768221i	$0.721142\pi$
$62$	2.00000	0.254000
$63$	1.00000	0.125988
$64$	7.00000	0.875000
$65$	16.0000	1.98456
$66$	−8.00000	−0.984732
$67$	−4.00000	−0.488678	−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000	−0.488678	−0.244339	+	0.969690i	$0.578571\pi$
$68$	6.00000	0.727607
$69$	−16.0000	−1.92617
$70$	−4.00000	−0.478091
$71$	8.00000	0.949425	0.474713	−	0.880141i	$-0.342552\pi$
$71$	8.00000	0.949425	0.474713	+	0.880141i	$0.342552\pi$
$72$	−3.00000	−0.353553
$73$	4.00000	0.468165	0.234082	−	0.972217i	$-0.424791\pi$
$73$	4.00000	0.468165	0.234082	+	0.972217i	$0.424791\pi$
$74$	10.0000	1.16248
$75$	22.0000	2.54034
$76$	−6.00000	−0.688247
$77$	−4.00000	−0.455842
$78$	−8.00000	−0.905822
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	4.00000	0.447214
$81$	−11.0000	−1.22222
$82$	6.00000	0.662589
$83$	−4.00000	−0.439057	−0.219529	−	0.975606i	$-0.570452\pi$
$83$	−4.00000	−0.439057	−0.219529	+	0.975606i	$0.570452\pi$
$84$	−2.00000	−0.218218
$85$	24.0000	2.60317
$86$	−4.00000	−0.431331
$87$	12.0000	1.28654
$88$	12.0000	1.27920
$89$	4.00000	0.423999	0.212000	−	0.977270i	$-0.432002\pi$
$89$	4.00000	0.423999	0.212000	+	0.977270i	$0.432002\pi$
$90$	−4.00000	−0.421637
$91$	−4.00000	−0.419314
$92$	8.00000	0.834058
$93$	4.00000	0.414781
$94$	0	0
$95$	−24.0000	−2.46235
$96$	10.0000	1.02062
$97$	−8.00000	−0.812277	−0.406138	−	0.913812i	$-0.633125\pi$
$97$	−8.00000	−0.812277	−0.406138	+	0.913812i	$0.633125\pi$
$98$	1.00000	0.101015
$99$	−4.00000	−0.402015
$100$	−11.0000	−1.10000
$101$	8.00000	0.796030	0.398015	−	0.917379i	$-0.369699\pi$
$101$	8.00000	0.796030	0.398015	+	0.917379i	$0.369699\pi$
$102$	−12.0000	−1.18818
$103$	−14.0000	−1.37946	−0.689730	−	0.724066i	$-0.742271\pi$
$103$	−14.0000	−1.37946	−0.689730	+	0.724066i	$0.742271\pi$
$104$	12.0000	1.17670
$105$	−8.00000	−0.780720
$106$	2.00000	0.194257
$107$	20.0000	1.93347	0.966736	−	0.255774i	$-0.0823304\pi$
$107$	20.0000	1.93347	0.966736	+	0.255774i	$0.0823304\pi$
$108$	4.00000	0.384900
$109$	−10.0000	−0.957826	−0.478913	−	0.877862i	$-0.658969\pi$
$109$	−10.0000	−0.957826	−0.478913	+	0.877862i	$0.658969\pi$
$110$	16.0000	1.52554
$111$	20.0000	1.89832
$112$	−1.00000	−0.0944911
$113$	−18.0000	−1.69330	−0.846649	−	0.532152i	$-0.821383\pi$
$113$	−18.0000	−1.69330	−0.846649	+	0.532152i	$0.821383\pi$
$114$	12.0000	1.12390
$115$	32.0000	2.98402
$116$	−6.00000	−0.557086
$117$	−4.00000	−0.369800
$118$	10.0000	0.920575
$119$	−6.00000	−0.550019
$120$	24.0000	2.19089
$121$	5.00000	0.454545
$122$	−10.0000	−0.905357
$123$	12.0000	1.08200
$124$	−2.00000	−0.179605
$125$	−24.0000	−2.14663
$126$	1.00000	0.0890871
$127$	12.0000	1.06483	0.532414	−	0.846484i	$-0.321285\pi$
$127$	12.0000	1.06483	0.532414	+	0.846484i	$0.321285\pi$
$128$	−3.00000	−0.265165
$129$	−8.00000	−0.704361
$130$	16.0000	1.40329
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	8.00000	0.696311
$133$	6.00000	0.520266
$134$	−4.00000	−0.345547
$135$	16.0000	1.37706
$136$	18.0000	1.54349
$137$	−18.0000	−1.53784	−0.768922	−	0.639343i	$-0.779207\pi$
$137$	−18.0000	−1.53784	−0.768922	+	0.639343i	$0.779207\pi$
$138$	−16.0000	−1.36201
$139$	4.00000	0.339276	0.169638	−	0.985506i	$-0.445740\pi$
$139$	4.00000	0.339276	0.169638	+	0.985506i	$0.445740\pi$
$140$	4.00000	0.338062
$141$	0	0
$142$	8.00000	0.671345
$143$	16.0000	1.33799
$144$	−1.00000	−0.0833333
$145$	−24.0000	−1.99309
$146$	4.00000	0.331042
$147$	2.00000	0.164957
$148$	−10.0000	−0.821995
$149$	10.0000	0.819232	0.409616	−	0.912258i	$-0.365663\pi$
$149$	10.0000	0.819232	0.409616	+	0.912258i	$0.365663\pi$
$150$	22.0000	1.79629
$151$	16.0000	1.30206	0.651031	−	0.759051i	$-0.274337\pi$
$151$	16.0000	1.30206	0.651031	+	0.759051i	$0.274337\pi$
$152$	−18.0000	−1.45999
$153$	−6.00000	−0.485071
$154$	−4.00000	−0.322329
$155$	−8.00000	−0.642575
$156$	8.00000	0.640513
$157$	12.0000	0.957704	0.478852	−	0.877896i	$-0.341053\pi$
$157$	12.0000	0.957704	0.478852	+	0.877896i	$0.341053\pi$
$158$	0	0
$159$	4.00000	0.317221
$160$	−20.0000	−1.58114
$161$	−8.00000	−0.630488
$162$	−11.0000	−0.864242
$163$	20.0000	1.56652	0.783260	−	0.621694i	$-0.213555\pi$
$163$	20.0000	1.56652	0.783260	+	0.621694i	$0.213555\pi$
$164$	−6.00000	−0.468521
$165$	32.0000	2.49120
$166$	−4.00000	−0.310460
$167$	12.0000	0.928588	0.464294	−	0.885681i	$-0.346308\pi$
$167$	12.0000	0.928588	0.464294	+	0.885681i	$0.346308\pi$
$168$	−6.00000	−0.462910
$169$	3.00000	0.230769
$170$	24.0000	1.84072
$171$	6.00000	0.458831
$172$	4.00000	0.304997
$173$	0	0		−	1.00000i	$-0.5\pi$
$173$	0	0			1.00000i	$0.5\pi$
$174$	12.0000	0.909718
$175$	11.0000	0.831522
$176$	4.00000	0.301511
$177$	20.0000	1.50329
$178$	4.00000	0.299813
$179$	−12.0000	−0.896922	−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000	−0.896922	−0.448461	+	0.893802i	$0.648028\pi$
$180$	4.00000	0.298142
$181$	−6.00000	−0.445976	−0.222988	−	0.974821i	$-0.571581\pi$
$181$	−6.00000	−0.445976	−0.222988	+	0.974821i	$0.571581\pi$
$182$	−4.00000	−0.296500
$183$	−20.0000	−1.47844
$184$	24.0000	1.76930
$185$	−40.0000	−2.94086
$186$	4.00000	0.293294
$187$	24.0000	1.75505
$188$	0	0
$189$	−4.00000	−0.290957
$190$	−24.0000	−1.74114
$191$	−20.0000	−1.44715	−0.723575	−	0.690246i	$-0.757502\pi$
$191$	−20.0000	−1.44715	−0.723575	+	0.690246i	$0.757502\pi$
$192$	14.0000	1.01036
$193$	−6.00000	−0.431889	−0.215945	−	0.976406i	$-0.569283\pi$
$193$	−6.00000	−0.431889	−0.215945	+	0.976406i	$0.569283\pi$
$194$	−8.00000	−0.574367
$195$	32.0000	2.29157
$196$	−1.00000	−0.0714286
$197$	6.00000	0.427482	0.213741	−	0.976890i	$-0.431435\pi$
$197$	6.00000	0.427482	0.213741	+	0.976890i	$0.431435\pi$
$198$	−4.00000	−0.284268
$199$	−10.0000	−0.708881	−0.354441	−	0.935079i	$-0.615329\pi$
$199$	−10.0000	−0.708881	−0.354441	+	0.935079i	$0.615329\pi$
$200$	−33.0000	−2.33345
$201$	−8.00000	−0.564276
$202$	8.00000	0.562878
$203$	6.00000	0.421117
$204$	12.0000	0.840168
$205$	−24.0000	−1.67623
$206$	−14.0000	−0.975426
$207$	−8.00000	−0.556038
$208$	4.00000	0.277350
$209$	−24.0000	−1.66011
$210$	−8.00000	−0.552052
$211$	12.0000	0.826114	0.413057	−	0.910705i	$-0.364461\pi$
$211$	12.0000	0.826114	0.413057	+	0.910705i	$0.364461\pi$
$212$	−2.00000	−0.137361
$213$	16.0000	1.09630
$214$	20.0000	1.36717
$215$	16.0000	1.09119
$216$	12.0000	0.816497
$217$	2.00000	0.135769
$218$	−10.0000	−0.677285
$219$	8.00000	0.540590
$220$	−16.0000	−1.07872
$221$	24.0000	1.61441
$222$	20.0000	1.34231
$223$	14.0000	0.937509	0.468755	−	0.883328i	$-0.344703\pi$
$223$	14.0000	0.937509	0.468755	+	0.883328i	$0.344703\pi$
$224$	5.00000	0.334077
$225$	11.0000	0.733333
$226$	−18.0000	−1.19734
$227$	−24.0000	−1.59294	−0.796468	−	0.604681i	$-0.793301\pi$
$227$	−24.0000	−1.59294	−0.796468	+	0.604681i	$0.793301\pi$
$228$	−12.0000	−0.794719
$229$	2.00000	0.132164	0.0660819	−	0.997814i	$-0.478950\pi$
$229$	2.00000	0.132164	0.0660819	+	0.997814i	$0.478950\pi$
$230$	32.0000	2.11002
$231$	−8.00000	−0.526361
$232$	−18.0000	−1.18176
$233$	−26.0000	−1.70332	−0.851658	−	0.524097i	$-0.824403\pi$
$233$	−26.0000	−1.70332	−0.851658	+	0.524097i	$0.824403\pi$
$234$	−4.00000	−0.261488
$235$	0	0
$236$	−10.0000	−0.650945
$237$	0	0
$238$	−6.00000	−0.388922
$239$	−12.0000	−0.776215	−0.388108	−	0.921614i	$-0.626871\pi$
$239$	−12.0000	−0.776215	−0.388108	+	0.921614i	$0.626871\pi$
$240$	8.00000	0.516398
$241$	20.0000	1.28831	0.644157	−	0.764894i	$-0.277208\pi$
$241$	20.0000	1.28831	0.644157	+	0.764894i	$0.277208\pi$
$242$	5.00000	0.321412
$243$	−10.0000	−0.641500
$244$	10.0000	0.640184
$245$	−4.00000	−0.255551
$246$	12.0000	0.765092
$247$	−24.0000	−1.52708
$248$	−6.00000	−0.381000
$249$	−8.00000	−0.506979
$250$	−24.0000	−1.51789
$251$	−12.0000	−0.757433	−0.378717	−	0.925513i	$-0.623635\pi$
$251$	−12.0000	−0.757433	−0.378717	+	0.925513i	$0.623635\pi$
$252$	−1.00000	−0.0629941
$253$	32.0000	2.01182
$254$	12.0000	0.752947
$255$	48.0000	3.00588
$256$	−17.0000	−1.06250
$257$	18.0000	1.12281	0.561405	−	0.827541i	$-0.310261\pi$
$257$	18.0000	1.12281	0.561405	+	0.827541i	$0.310261\pi$
$258$	−8.00000	−0.498058
$259$	10.0000	0.621370
$260$	−16.0000	−0.992278
$261$	6.00000	0.371391
$262$	0	0
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	24.0000	1.47710
$265$	−8.00000	−0.491436
$266$	6.00000	0.367884
$267$	8.00000	0.489592
$268$	4.00000	0.244339
$269$	−6.00000	−0.365826	−0.182913	−	0.983129i	$-0.558553\pi$
$269$	−6.00000	−0.365826	−0.182913	+	0.983129i	$0.558553\pi$
$270$	16.0000	0.973729
$271$	−4.00000	−0.242983	−0.121491	−	0.992592i	$-0.538768\pi$
$271$	−4.00000	−0.242983	−0.121491	+	0.992592i	$0.538768\pi$
$272$	6.00000	0.363803
$273$	−8.00000	−0.484182
$274$	−18.0000	−1.08742
$275$	−44.0000	−2.65330
$276$	16.0000	0.963087
$277$	22.0000	1.32185	0.660926	−	0.750451i	$-0.270164\pi$
$277$	22.0000	1.32185	0.660926	+	0.750451i	$0.270164\pi$
$278$	4.00000	0.239904
$279$	2.00000	0.119737
$280$	12.0000	0.717137
$281$	6.00000	0.357930	0.178965	−	0.983855i	$-0.442725\pi$
$281$	6.00000	0.357930	0.178965	+	0.983855i	$0.442725\pi$
$282$	0	0
$283$	−6.00000	−0.356663	−0.178331	−	0.983970i	$-0.557070\pi$
$283$	−6.00000	−0.356663	−0.178331	+	0.983970i	$0.557070\pi$
$284$	−8.00000	−0.474713
$285$	−48.0000	−2.84327
$286$	16.0000	0.946100
$287$	6.00000	0.354169
$288$	5.00000	0.294628
$289$	19.0000	1.11765
$290$	−24.0000	−1.40933
$291$	−16.0000	−0.937937
$292$	−4.00000	−0.234082
$293$	2.00000	0.116841	0.0584206	−	0.998292i	$-0.481394\pi$
$293$	2.00000	0.116841	0.0584206	+	0.998292i	$0.481394\pi$
$294$	2.00000	0.116642
$295$	−40.0000	−2.32889
$296$	−30.0000	−1.74371
$297$	16.0000	0.928414
$298$	10.0000	0.579284
$299$	32.0000	1.85061
$300$	−22.0000	−1.27017
$301$	−4.00000	−0.230556
$302$	16.0000	0.920697
$303$	16.0000	0.919176
$304$	−6.00000	−0.344124
$305$	40.0000	2.29039
$306$	−6.00000	−0.342997
$307$	−2.00000	−0.114146	−0.0570730	−	0.998370i	$-0.518177\pi$
$307$	−2.00000	−0.114146	−0.0570730	+	0.998370i	$0.518177\pi$
$308$	4.00000	0.227921
$309$	−28.0000	−1.59286
$310$	−8.00000	−0.454369
$311$	18.0000	1.02069	0.510343	−	0.859971i	$-0.329518\pi$
$311$	18.0000	1.02069	0.510343	+	0.859971i	$0.329518\pi$
$312$	24.0000	1.35873
$313$	−22.0000	−1.24351	−0.621757	−	0.783210i	$-0.713581\pi$
$313$	−22.0000	−1.24351	−0.621757	+	0.783210i	$0.713581\pi$
$314$	12.0000	0.677199
$315$	−4.00000	−0.225374
$316$	0	0
$317$	−18.0000	−1.01098	−0.505490	−	0.862832i	$-0.668688\pi$
$317$	−18.0000	−1.01098	−0.505490	+	0.862832i	$0.668688\pi$
$318$	4.00000	0.224309
$319$	−24.0000	−1.34374
$320$	−28.0000	−1.56525
$321$	40.0000	2.23258
$322$	−8.00000	−0.445823
$323$	−36.0000	−2.00309
$324$	11.0000	0.611111
$325$	−44.0000	−2.44068
$326$	20.0000	1.10770
$327$	−20.0000	−1.10600
$328$	−18.0000	−0.993884
$329$	0	0
$330$	32.0000	1.76154
$331$	−28.0000	−1.53902	−0.769510	−	0.638635i	$-0.779499\pi$
$331$	−28.0000	−1.53902	−0.769510	+	0.638635i	$0.779499\pi$
$332$	4.00000	0.219529
$333$	10.0000	0.547997
$334$	12.0000	0.656611
$335$	16.0000	0.874173
$336$	−2.00000	−0.109109
$337$	18.0000	0.980522	0.490261	−	0.871576i	$-0.336901\pi$
$337$	18.0000	0.980522	0.490261	+	0.871576i	$0.336901\pi$
$338$	3.00000	0.163178
$339$	−36.0000	−1.95525
$340$	−24.0000	−1.30158
$341$	−8.00000	−0.433224
$342$	6.00000	0.324443
$343$	1.00000	0.0539949
$344$	12.0000	0.646997
$345$	64.0000	3.44564
$346$	0	0
$347$	20.0000	1.07366	0.536828	−	0.843692i	$-0.319622\pi$
$347$	20.0000	1.07366	0.536828	+	0.843692i	$0.319622\pi$
$348$	−12.0000	−0.643268
$349$	36.0000	1.92704	0.963518	−	0.267644i	$-0.0862451\pi$
$349$	36.0000	1.92704	0.963518	+	0.267644i	$0.0862451\pi$
$350$	11.0000	0.587975
$351$	16.0000	0.854017
$352$	−20.0000	−1.06600
$353$	−10.0000	−0.532246	−0.266123	−	0.963939i	$-0.585743\pi$
$353$	−10.0000	−0.532246	−0.266123	+	0.963939i	$0.585743\pi$
$354$	20.0000	1.06299
$355$	−32.0000	−1.69838
$356$	−4.00000	−0.212000
$357$	−12.0000	−0.635107
$358$	−12.0000	−0.634220
$359$	20.0000	1.05556	0.527780	−	0.849381i	$-0.323025\pi$
$359$	20.0000	1.05556	0.527780	+	0.849381i	$0.323025\pi$
$360$	12.0000	0.632456
$361$	17.0000	0.894737
$362$	−6.00000	−0.315353
$363$	10.0000	0.524864
$364$	4.00000	0.209657
$365$	−16.0000	−0.837478
$366$	−20.0000	−1.04542
$367$	4.00000	0.208798	0.104399	−	0.994535i	$-0.466708\pi$
$367$	4.00000	0.208798	0.104399	+	0.994535i	$0.466708\pi$
$368$	8.00000	0.417029
$369$	6.00000	0.312348
$370$	−40.0000	−2.07950
$371$	2.00000	0.103835
$372$	−4.00000	−0.207390
$373$	38.0000	1.96757	0.983783	−	0.179364i	$-0.0574041\pi$
$373$	38.0000	1.96757	0.983783	+	0.179364i	$0.0574041\pi$
$374$	24.0000	1.24101
$375$	−48.0000	−2.47871
$376$	0	0
$377$	−24.0000	−1.23606
$378$	−4.00000	−0.205738
$379$	28.0000	1.43826	0.719132	−	0.694874i	$-0.244540\pi$
$379$	28.0000	1.43826	0.719132	+	0.694874i	$0.244540\pi$
$380$	24.0000	1.23117
$381$	24.0000	1.22956
$382$	−20.0000	−1.02329
$383$	−6.00000	−0.306586	−0.153293	−	0.988181i	$-0.548988\pi$
$383$	−6.00000	−0.306586	−0.153293	+	0.988181i	$0.548988\pi$
$384$	−6.00000	−0.306186
$385$	16.0000	0.815436
$386$	−6.00000	−0.305392
$387$	−4.00000	−0.203331
$388$	8.00000	0.406138
$389$	6.00000	0.304212	0.152106	−	0.988364i	$-0.451394\pi$
$389$	6.00000	0.304212	0.152106	+	0.988364i	$0.451394\pi$
$390$	32.0000	1.62038
$391$	48.0000	2.42746
$392$	−3.00000	−0.151523
$393$	0	0
$394$	6.00000	0.302276
$395$	0	0
$396$	4.00000	0.201008
$397$	22.0000	1.10415	0.552074	−	0.833795i	$-0.313837\pi$
$397$	22.0000	1.10415	0.552074	+	0.833795i	$0.313837\pi$
$398$	−10.0000	−0.501255
$399$	12.0000	0.600751
$400$	−11.0000	−0.550000
$401$	6.00000	0.299626	0.149813	−	0.988714i	$-0.452133\pi$
$401$	6.00000	0.299626	0.149813	+	0.988714i	$0.452133\pi$
$402$	−8.00000	−0.399004
$403$	−8.00000	−0.398508
$404$	−8.00000	−0.398015
$405$	44.0000	2.18638
$406$	6.00000	0.297775
$407$	−40.0000	−1.98273
$408$	36.0000	1.78227
$409$	−22.0000	−1.08783	−0.543915	−	0.839140i	$-0.683059\pi$
$409$	−22.0000	−1.08783	−0.543915	+	0.839140i	$0.683059\pi$
$410$	−24.0000	−1.18528
$411$	−36.0000	−1.77575
$412$	14.0000	0.689730
$413$	10.0000	0.492068
$414$	−8.00000	−0.393179
$415$	16.0000	0.785409
$416$	−20.0000	−0.980581
$417$	8.00000	0.391762
$418$	−24.0000	−1.17388
$419$	−24.0000	−1.17248	−0.586238	−	0.810139i	$-0.699392\pi$
$419$	−24.0000	−1.17248	−0.586238	+	0.810139i	$0.699392\pi$
$420$	8.00000	0.390360
$421$	26.0000	1.26716	0.633581	−	0.773676i	$-0.281584\pi$
$421$	26.0000	1.26716	0.633581	+	0.773676i	$0.281584\pi$
$422$	12.0000	0.584151
$423$	0	0
$424$	−6.00000	−0.291386
$425$	−66.0000	−3.20147
$426$	16.0000	0.775203
$427$	−10.0000	−0.483934
$428$	−20.0000	−0.966736
$429$	32.0000	1.54497
$430$	16.0000	0.771589
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	4.00000	0.192450
$433$	26.0000	1.24948	0.624740	−	0.780833i	$-0.285205\pi$
$433$	26.0000	1.24948	0.624740	+	0.780833i	$0.285205\pi$
$434$	2.00000	0.0960031
$435$	−48.0000	−2.30142
$436$	10.0000	0.478913
$437$	−48.0000	−2.29615
$438$	8.00000	0.382255
$439$	8.00000	0.381819	0.190910	−	0.981608i	$-0.438856\pi$
$439$	8.00000	0.381819	0.190910	+	0.981608i	$0.438856\pi$
$440$	−48.0000	−2.28831
$441$	1.00000	0.0476190
$442$	24.0000	1.14156
$443$	12.0000	0.570137	0.285069	−	0.958507i	$-0.407984\pi$
$443$	12.0000	0.570137	0.285069	+	0.958507i	$0.407984\pi$
$444$	−20.0000	−0.949158
$445$	−16.0000	−0.758473
$446$	14.0000	0.662919
$447$	20.0000	0.945968
$448$	7.00000	0.330719
$449$	38.0000	1.79333	0.896665	−	0.442709i	$-0.145982\pi$
$449$	38.0000	1.79333	0.896665	+	0.442709i	$0.145982\pi$
$450$	11.0000	0.518545
$451$	−24.0000	−1.13012
$452$	18.0000	0.846649
$453$	32.0000	1.50349
$454$	−24.0000	−1.12638
$455$	16.0000	0.750092
$456$	−36.0000	−1.68585
$457$	10.0000	0.467780	0.233890	−	0.972263i	$-0.424854\pi$
$457$	10.0000	0.467780	0.233890	+	0.972263i	$0.424854\pi$
$458$	2.00000	0.0934539
$459$	24.0000	1.12022
$460$	−32.0000	−1.49201
$461$	−2.00000	−0.0931493	−0.0465746	−	0.998915i	$-0.514831\pi$
$461$	−2.00000	−0.0931493	−0.0465746	+	0.998915i	$0.514831\pi$
$462$	−8.00000	−0.372194
$463$	16.0000	0.743583	0.371792	−	0.928316i	$-0.378744\pi$
$463$	16.0000	0.743583	0.371792	+	0.928316i	$0.378744\pi$
$464$	−6.00000	−0.278543
$465$	−16.0000	−0.741982
$466$	−26.0000	−1.20443
$467$	−6.00000	−0.277647	−0.138823	−	0.990317i	$-0.544332\pi$
$467$	−6.00000	−0.277647	−0.138823	+	0.990317i	$0.544332\pi$
$468$	4.00000	0.184900
$469$	−4.00000	−0.184703
$470$	0	0
$471$	24.0000	1.10586
$472$	−30.0000	−1.38086
$473$	16.0000	0.735681
$474$	0	0
$475$	66.0000	3.02829
$476$	6.00000	0.275010
$477$	2.00000	0.0915737
$478$	−12.0000	−0.548867
$479$	40.0000	1.82765	0.913823	−	0.406112i	$-0.133116\pi$
$479$	40.0000	1.82765	0.913823	+	0.406112i	$0.133116\pi$
$480$	−40.0000	−1.82574
$481$	−40.0000	−1.82384
$482$	20.0000	0.910975
$483$	−16.0000	−0.728025
$484$	−5.00000	−0.227273
$485$	32.0000	1.45305
$486$	−10.0000	−0.453609
$487$	28.0000	1.26880	0.634401	−	0.773004i	$-0.281247\pi$
$487$	28.0000	1.26880	0.634401	+	0.773004i	$0.281247\pi$
$488$	30.0000	1.35804
$489$	40.0000	1.80886
$490$	−4.00000	−0.180702
$491$	20.0000	0.902587	0.451294	−	0.892375i	$-0.350963\pi$
$491$	20.0000	0.902587	0.451294	+	0.892375i	$0.350963\pi$
$492$	−12.0000	−0.541002
$493$	−36.0000	−1.62136
$494$	−24.0000	−1.07981
$495$	16.0000	0.719147
$496$	−2.00000	−0.0898027
$497$	8.00000	0.358849
$498$	−8.00000	−0.358489
$499$	12.0000	0.537194	0.268597	−	0.963253i	$-0.413440\pi$
$499$	12.0000	0.537194	0.268597	+	0.963253i	$0.413440\pi$
$500$	24.0000	1.07331
$501$	24.0000	1.07224
$502$	−12.0000	−0.535586
$503$	42.0000	1.87269	0.936344	−	0.351085i	$-0.114187\pi$
$503$	42.0000	1.87269	0.936344	+	0.351085i	$0.114187\pi$
$504$	−3.00000	−0.133631
$505$	−32.0000	−1.42398
$506$	32.0000	1.42257
$507$	6.00000	0.266469
$508$	−12.0000	−0.532414
$509$	16.0000	0.709188	0.354594	−	0.935020i	$-0.384619\pi$
$509$	16.0000	0.709188	0.354594	+	0.935020i	$0.384619\pi$
$510$	48.0000	2.12548
$511$	4.00000	0.176950
$512$	−11.0000	−0.486136
$513$	−24.0000	−1.05963
$514$	18.0000	0.793946
$515$	56.0000	2.46765
$516$	8.00000	0.352180
$517$	0	0
$518$	10.0000	0.439375
$519$	0	0
$520$	−48.0000	−2.10494
$521$	0	0		−	1.00000i	$-0.5\pi$
$521$	0	0			1.00000i	$0.5\pi$
$522$	6.00000	0.262613
$523$	−22.0000	−0.961993	−0.480996	−	0.876723i	$-0.659725\pi$
$523$	−22.0000	−0.961993	−0.480996	+	0.876723i	$0.659725\pi$
$524$	0	0
$525$	22.0000	0.960159
$526$	0	0
$527$	−12.0000	−0.522728
$528$	8.00000	0.348155
$529$	41.0000	1.78261
$530$	−8.00000	−0.347498
$531$	10.0000	0.433963
$532$	−6.00000	−0.260133
$533$	−24.0000	−1.03956
$534$	8.00000	0.346194
$535$	−80.0000	−3.45870
$536$	12.0000	0.518321
$537$	−24.0000	−1.03568
$538$	−6.00000	−0.258678
$539$	−4.00000	−0.172292
$540$	−16.0000	−0.688530
$541$	30.0000	1.28980	0.644900	−	0.764267i	$-0.276899\pi$
$541$	30.0000	1.28980	0.644900	+	0.764267i	$0.276899\pi$
$542$	−4.00000	−0.171815
$543$	−12.0000	−0.514969
$544$	−30.0000	−1.28624
$545$	40.0000	1.71341
$546$	−8.00000	−0.342368
$547$	−8.00000	−0.342055	−0.171028	−	0.985266i	$-0.554709\pi$
$547$	−8.00000	−0.342055	−0.171028	+	0.985266i	$0.554709\pi$
$548$	18.0000	0.768922
$549$	−10.0000	−0.426790
$550$	−44.0000	−1.87617
$551$	36.0000	1.53365
$552$	48.0000	2.04302
$553$	0	0
$554$	22.0000	0.934690
$555$	−80.0000	−3.39581
$556$	−4.00000	−0.169638
$557$	−42.0000	−1.77960	−0.889799	−	0.456354i	$-0.849155\pi$
$557$	−42.0000	−1.77960	−0.889799	+	0.456354i	$0.849155\pi$
$558$	2.00000	0.0846668
$559$	16.0000	0.676728
$560$	4.00000	0.169031
$561$	48.0000	2.02656
$562$	6.00000	0.253095
$563$	−4.00000	−0.168580	−0.0842900	−	0.996441i	$-0.526862\pi$
$563$	−4.00000	−0.168580	−0.0842900	+	0.996441i	$0.526862\pi$
$564$	0	0
$565$	72.0000	3.02906
$566$	−6.00000	−0.252199
$567$	−11.0000	−0.461957
$568$	−24.0000	−1.00702
$569$	−10.0000	−0.419222	−0.209611	−	0.977785i	$-0.567220\pi$
$569$	−10.0000	−0.419222	−0.209611	+	0.977785i	$0.567220\pi$
$570$	−48.0000	−2.01050
$571$	−40.0000	−1.67395	−0.836974	−	0.547243i	$-0.815677\pi$
$571$	−40.0000	−1.67395	−0.836974	+	0.547243i	$0.815677\pi$
$572$	−16.0000	−0.668994
$573$	−40.0000	−1.67102
$574$	6.00000	0.250435
$575$	−88.0000	−3.66985
$576$	7.00000	0.291667
$577$	32.0000	1.33218	0.666089	−	0.745873i	$-0.267967\pi$
$577$	32.0000	1.33218	0.666089	+	0.745873i	$0.267967\pi$
$578$	19.0000	0.790296
$579$	−12.0000	−0.498703
$580$	24.0000	0.996546
$581$	−4.00000	−0.165948
$582$	−16.0000	−0.663221
$583$	−8.00000	−0.331326
$584$	−12.0000	−0.496564
$585$	16.0000	0.661519
$586$	2.00000	0.0826192
$587$	−30.0000	−1.23823	−0.619116	−	0.785299i	$-0.712509\pi$
$587$	−30.0000	−1.23823	−0.619116	+	0.785299i	$0.712509\pi$
$588$	−2.00000	−0.0824786
$589$	12.0000	0.494451
$590$	−40.0000	−1.64677
$591$	12.0000	0.493614
$592$	−10.0000	−0.410997
$593$	6.00000	0.246390	0.123195	−	0.992382i	$-0.460686\pi$
$593$	6.00000	0.246390	0.123195	+	0.992382i	$0.460686\pi$
$594$	16.0000	0.656488
$595$	24.0000	0.983904
$596$	−10.0000	−0.409616
$597$	−20.0000	−0.818546
$598$	32.0000	1.30858
$599$	−12.0000	−0.490307	−0.245153	−	0.969484i	$-0.578838\pi$
$599$	−12.0000	−0.490307	−0.245153	+	0.969484i	$0.578838\pi$
$600$	−66.0000	−2.69444
$601$	22.0000	0.897399	0.448699	−	0.893683i	$-0.351887\pi$
$601$	22.0000	0.897399	0.448699	+	0.893683i	$0.351887\pi$
$602$	−4.00000	−0.163028
$603$	−4.00000	−0.162893
$604$	−16.0000	−0.651031
$605$	−20.0000	−0.813116
$606$	16.0000	0.649956
$607$	−44.0000	−1.78590	−0.892952	−	0.450151i	$-0.851370\pi$
$607$	−44.0000	−1.78590	−0.892952	+	0.450151i	$0.851370\pi$
$608$	30.0000	1.21666
$609$	12.0000	0.486265
$610$	40.0000	1.61955
$611$	0	0
$612$	6.00000	0.242536
$613$	−34.0000	−1.37325	−0.686624	−	0.727013i	$-0.740908\pi$
$613$	−34.0000	−1.37325	−0.686624	+	0.727013i	$0.740908\pi$
$614$	−2.00000	−0.0807134
$615$	−48.0000	−1.93555
$616$	12.0000	0.483494
$617$	−18.0000	−0.724653	−0.362326	−	0.932051i	$-0.618017\pi$
$617$	−18.0000	−0.724653	−0.362326	+	0.932051i	$0.618017\pi$
$618$	−28.0000	−1.12633
$619$	4.00000	0.160774	0.0803868	−	0.996764i	$-0.474384\pi$
$619$	4.00000	0.160774	0.0803868	+	0.996764i	$0.474384\pi$
$620$	8.00000	0.321288
$621$	32.0000	1.28412
$622$	18.0000	0.721734
$623$	4.00000	0.160257
$624$	8.00000	0.320256
$625$	41.0000	1.64000
$626$	−22.0000	−0.879297
$627$	−48.0000	−1.91694
$628$	−12.0000	−0.478852
$629$	−60.0000	−2.39236
$630$	−4.00000	−0.159364
$631$	40.0000	1.59237	0.796187	−	0.605050i	$-0.206847\pi$
$631$	40.0000	1.59237	0.796187	+	0.605050i	$0.206847\pi$
$632$	0	0
$633$	24.0000	0.953914
$634$	−18.0000	−0.714871
$635$	−48.0000	−1.90482
$636$	−4.00000	−0.158610
$637$	−4.00000	−0.158486
$638$	−24.0000	−0.950169
$639$	8.00000	0.316475
$640$	12.0000	0.474342
$641$	−26.0000	−1.02694	−0.513469	−	0.858108i	$-0.671640\pi$
$641$	−26.0000	−1.02694	−0.513469	+	0.858108i	$0.671640\pi$
$642$	40.0000	1.57867
$643$	−28.0000	−1.10421	−0.552106	−	0.833774i	$-0.686176\pi$
$643$	−28.0000	−1.10421	−0.552106	+	0.833774i	$0.686176\pi$
$644$	8.00000	0.315244
$645$	32.0000	1.26000
$646$	−36.0000	−1.41640
$647$	−28.0000	−1.10079	−0.550397	−	0.834903i	$-0.685524\pi$
$647$	−28.0000	−1.10079	−0.550397	+	0.834903i	$0.685524\pi$
$648$	33.0000	1.29636
$649$	−40.0000	−1.57014
$650$	−44.0000	−1.72582
$651$	4.00000	0.156772
$652$	−20.0000	−0.783260
$653$	−6.00000	−0.234798	−0.117399	−	0.993085i	$-0.537456\pi$
$653$	−6.00000	−0.234798	−0.117399	+	0.993085i	$0.537456\pi$
$654$	−20.0000	−0.782062
$655$	0	0
$656$	−6.00000	−0.234261
$657$	4.00000	0.156055
$658$	0	0
$659$	44.0000	1.71400	0.856998	−	0.515319i	$-0.172327\pi$
$659$	44.0000	1.71400	0.856998	+	0.515319i	$0.172327\pi$
$660$	−32.0000	−1.24560
$661$	−38.0000	−1.47803	−0.739014	−	0.673690i	$-0.764708\pi$
$661$	−38.0000	−1.47803	−0.739014	+	0.673690i	$0.764708\pi$
$662$	−28.0000	−1.08825
$663$	48.0000	1.86417
$664$	12.0000	0.465690
$665$	−24.0000	−0.930680
$666$	10.0000	0.387492
$667$	−48.0000	−1.85857
$668$	−12.0000	−0.464294
$669$	28.0000	1.08254
$670$	16.0000	0.618134
$671$	40.0000	1.54418
$672$	10.0000	0.385758
$673$	14.0000	0.539660	0.269830	−	0.962908i	$-0.413032\pi$
$673$	14.0000	0.539660	0.269830	+	0.962908i	$0.413032\pi$
$674$	18.0000	0.693334
$675$	−44.0000	−1.69356
$676$	−3.00000	−0.115385
$677$	−44.0000	−1.69106	−0.845529	−	0.533930i	$-0.820715\pi$
$677$	−44.0000	−1.69106	−0.845529	+	0.533930i	$0.820715\pi$
$678$	−36.0000	−1.38257
$679$	−8.00000	−0.307012
$680$	−72.0000	−2.76107
$681$	−48.0000	−1.83936
$682$	−8.00000	−0.306336
$683$	−24.0000	−0.918334	−0.459167	−	0.888350i	$-0.651852\pi$
$683$	−24.0000	−0.918334	−0.459167	+	0.888350i	$0.651852\pi$
$684$	−6.00000	−0.229416
$685$	72.0000	2.75098
$686$	1.00000	0.0381802
$687$	4.00000	0.152610
$688$	4.00000	0.152499
$689$	−8.00000	−0.304776
$690$	64.0000	2.43644
$691$	20.0000	0.760836	0.380418	−	0.924815i	$-0.375780\pi$
$691$	20.0000	0.760836	0.380418	+	0.924815i	$0.375780\pi$
$692$	0	0
$693$	−4.00000	−0.151947
$694$	20.0000	0.759190
$695$	−16.0000	−0.606915
$696$	−36.0000	−1.36458
$697$	−36.0000	−1.36360
$698$	36.0000	1.36262
$699$	−52.0000	−1.96682
$700$	−11.0000	−0.415761
$701$	−14.0000	−0.528773	−0.264386	−	0.964417i	$-0.585169\pi$
$701$	−14.0000	−0.528773	−0.264386	+	0.964417i	$0.585169\pi$
$702$	16.0000	0.603881
$703$	60.0000	2.26294
$704$	−28.0000	−1.05529
$705$	0	0
$706$	−10.0000	−0.376355
$707$	8.00000	0.300871
$708$	−20.0000	−0.751646
$709$	26.0000	0.976450	0.488225	−	0.872718i	$-0.337644\pi$
$709$	26.0000	0.976450	0.488225	+	0.872718i	$0.337644\pi$
$710$	−32.0000	−1.20094
$711$	0	0
$712$	−12.0000	−0.449719
$713$	−16.0000	−0.599205
$714$	−12.0000	−0.449089
$715$	−64.0000	−2.39346
$716$	12.0000	0.448461
$717$	−24.0000	−0.896296
$718$	20.0000	0.746393
$719$	−24.0000	−0.895049	−0.447524	−	0.894272i	$-0.647694\pi$
$719$	−24.0000	−0.895049	−0.447524	+	0.894272i	$0.647694\pi$
$720$	4.00000	0.149071
$721$	−14.0000	−0.521387
$722$	17.0000	0.632674
$723$	40.0000	1.48762
$724$	6.00000	0.222988
$725$	66.0000	2.45118
$726$	10.0000	0.371135
$727$	24.0000	0.890111	0.445055	−	0.895503i	$-0.353184\pi$
$727$	24.0000	0.890111	0.445055	+	0.895503i	$0.353184\pi$
$728$	12.0000	0.444750
$729$	13.0000	0.481481
$730$	−16.0000	−0.592187
$731$	24.0000	0.887672
$732$	20.0000	0.739221
$733$	16.0000	0.590973	0.295487	−	0.955347i	$-0.404518\pi$
$733$	16.0000	0.590973	0.295487	+	0.955347i	$0.404518\pi$
$734$	4.00000	0.147643
$735$	−8.00000	−0.295084
$736$	−40.0000	−1.47442
$737$	16.0000	0.589368
$738$	6.00000	0.220863
$739$	28.0000	1.03000	0.514998	−	0.857191i	$-0.327793\pi$
$739$	28.0000	1.03000	0.514998	+	0.857191i	$0.327793\pi$
$740$	40.0000	1.47043
$741$	−48.0000	−1.76332
$742$	2.00000	0.0734223
$743$	16.0000	0.586983	0.293492	−	0.955962i	$-0.405183\pi$
$743$	16.0000	0.586983	0.293492	+	0.955962i	$0.405183\pi$
$744$	−12.0000	−0.439941
$745$	−40.0000	−1.46549
$746$	38.0000	1.39128
$747$	−4.00000	−0.146352
$748$	−24.0000	−0.877527
$749$	20.0000	0.730784
$750$	−48.0000	−1.75271
$751$	32.0000	1.16770	0.583848	−	0.811863i	$-0.301546\pi$
$751$	32.0000	1.16770	0.583848	+	0.811863i	$0.301546\pi$
$752$	0	0
$753$	−24.0000	−0.874609
$754$	−24.0000	−0.874028
$755$	−64.0000	−2.32920
$756$	4.00000	0.145479
$757$	2.00000	0.0726912	0.0363456	−	0.999339i	$-0.488428\pi$
$757$	2.00000	0.0726912	0.0363456	+	0.999339i	$0.488428\pi$
$758$	28.0000	1.01701
$759$	64.0000	2.32305
$760$	72.0000	2.61171
$761$	48.0000	1.74000	0.869999	−	0.493053i	$-0.164119\pi$
$761$	48.0000	1.74000	0.869999	+	0.493053i	$0.164119\pi$
$762$	24.0000	0.869428
$763$	−10.0000	−0.362024
$764$	20.0000	0.723575
$765$	24.0000	0.867722
$766$	−6.00000	−0.216789
$767$	−40.0000	−1.44432
$768$	−34.0000	−1.22687
$769$	18.0000	0.649097	0.324548	−	0.945869i	$-0.394788\pi$
$769$	18.0000	0.649097	0.324548	+	0.945869i	$0.394788\pi$
$770$	16.0000	0.576600
$771$	36.0000	1.29651
$772$	6.00000	0.215945
$773$	−14.0000	−0.503545	−0.251773	−	0.967786i	$-0.581013\pi$
$773$	−14.0000	−0.503545	−0.251773	+	0.967786i	$0.581013\pi$
$774$	−4.00000	−0.143777
$775$	22.0000	0.790263
$776$	24.0000	0.861550
$777$	20.0000	0.717496
$778$	6.00000	0.215110
$779$	36.0000	1.28983
$780$	−32.0000	−1.14578
$781$	−32.0000	−1.14505
$782$	48.0000	1.71648
$783$	−24.0000	−0.857690
$784$	−1.00000	−0.0357143
$785$	−48.0000	−1.71319
$786$	0	0
$787$	−4.00000	−0.142585	−0.0712923	−	0.997455i	$-0.522712\pi$
$787$	−4.00000	−0.142585	−0.0712923	+	0.997455i	$0.522712\pi$
$788$	−6.00000	−0.213741
$789$	0	0
$790$	0	0
$791$	−18.0000	−0.640006
$792$	12.0000	0.426401
$793$	40.0000	1.42044
$794$	22.0000	0.780751
$795$	−16.0000	−0.567462
$796$	10.0000	0.354441
$797$	22.0000	0.779280	0.389640	−	0.920967i	$-0.372599\pi$
$797$	22.0000	0.779280	0.389640	+	0.920967i	$0.372599\pi$
$798$	12.0000	0.424795
$799$	0	0
$800$	55.0000	1.94454
$801$	4.00000	0.141333
$802$	6.00000	0.211867
$803$	−16.0000	−0.564628
$804$	8.00000	0.282138
$805$	32.0000	1.12785
$806$	−8.00000	−0.281788
$807$	−12.0000	−0.422420
$808$	−24.0000	−0.844317
$809$	18.0000	0.632846	0.316423	−	0.948618i	$-0.397518\pi$
$809$	18.0000	0.632846	0.316423	+	0.948618i	$0.397518\pi$
$810$	44.0000	1.54600
$811$	−22.0000	−0.772524	−0.386262	−	0.922389i	$-0.626234\pi$
$811$	−22.0000	−0.772524	−0.386262	+	0.922389i	$0.626234\pi$
$812$	−6.00000	−0.210559
$813$	−8.00000	−0.280572
$814$	−40.0000	−1.40200
$815$	−80.0000	−2.80228
$816$	12.0000	0.420084
$817$	−24.0000	−0.839654
$818$	−22.0000	−0.769212
$819$	−4.00000	−0.139771
$820$	24.0000	0.838116
$821$	2.00000	0.0698005	0.0349002	−	0.999391i	$-0.488889\pi$
$821$	2.00000	0.0698005	0.0349002	+	0.999391i	$0.488889\pi$
$822$	−36.0000	−1.25564
$823$	−20.0000	−0.697156	−0.348578	−	0.937280i	$-0.613335\pi$
$823$	−20.0000	−0.697156	−0.348578	+	0.937280i	$0.613335\pi$
$824$	42.0000	1.46314
$825$	−88.0000	−3.06377
$826$	10.0000	0.347945
$827$	−52.0000	−1.80822	−0.904109	−	0.427303i	$-0.859464\pi$
$827$	−52.0000	−1.80822	−0.904109	+	0.427303i	$0.859464\pi$
$828$	8.00000	0.278019
$829$	−56.0000	−1.94496	−0.972480	−	0.232986i	$-0.925151\pi$
$829$	−56.0000	−1.94496	−0.972480	+	0.232986i	$0.925151\pi$
$830$	16.0000	0.555368
$831$	44.0000	1.52634
$832$	−28.0000	−0.970725
$833$	−6.00000	−0.207888
$834$	8.00000	0.277017
$835$	−48.0000	−1.66111
$836$	24.0000	0.830057
$837$	−8.00000	−0.276520
$838$	−24.0000	−0.829066
$839$	44.0000	1.51905	0.759524	−	0.650479i	$-0.225432\pi$
$839$	44.0000	1.51905	0.759524	+	0.650479i	$0.225432\pi$
$840$	24.0000	0.828079
$841$	7.00000	0.241379
$842$	26.0000	0.896019
$843$	12.0000	0.413302
$844$	−12.0000	−0.413057
$845$	−12.0000	−0.412813
$846$	0	0
$847$	5.00000	0.171802
$848$	−2.00000	−0.0686803
$849$	−12.0000	−0.411839
$850$	−66.0000	−2.26378
$851$	−80.0000	−2.74236
$852$	−16.0000	−0.548151
$853$	−10.0000	−0.342393	−0.171197	−	0.985237i	$-0.554763\pi$
$853$	−10.0000	−0.342393	−0.171197	+	0.985237i	$0.554763\pi$
$854$	−10.0000	−0.342193
$855$	−24.0000	−0.820783
$856$	−60.0000	−2.05076
$857$	−12.0000	−0.409912	−0.204956	−	0.978771i	$-0.565705\pi$
$857$	−12.0000	−0.409912	−0.204956	+	0.978771i	$0.565705\pi$
$858$	32.0000	1.09246
$859$	−4.00000	−0.136478	−0.0682391	−	0.997669i	$-0.521738\pi$
$859$	−4.00000	−0.136478	−0.0682391	+	0.997669i	$0.521738\pi$
$860$	−16.0000	−0.545595
$861$	12.0000	0.408959
$862$	0	0
$863$	−1.00000	−0.0340404
$863$	−1.00000	−0.0340404
$864$	−20.0000	−0.680414
$865$	0	0
$866$	26.0000	0.883516
$867$	38.0000	1.29055
$868$	−2.00000	−0.0678844
$869$	0	0
$870$	−48.0000	−1.62735
$871$	16.0000	0.542139
$872$	30.0000	1.01593
$873$	−8.00000	−0.270759
$874$	−48.0000	−1.62362
$875$	−24.0000	−0.811348
$876$	−8.00000	−0.270295
$877$	10.0000	0.337676	0.168838	−	0.985644i	$-0.445999\pi$
$877$	10.0000	0.337676	0.168838	+	0.985644i	$0.445999\pi$
$878$	8.00000	0.269987
$879$	4.00000	0.134917
$880$	−16.0000	−0.539360
$881$	30.0000	1.01073	0.505363	−	0.862907i	$-0.331359\pi$
$881$	30.0000	1.01073	0.505363	+	0.862907i	$0.331359\pi$
$882$	1.00000	0.0336718
$883$	12.0000	0.403832	0.201916	−	0.979403i	$-0.435283\pi$
$883$	12.0000	0.403832	0.201916	+	0.979403i	$0.435283\pi$
$884$	−24.0000	−0.807207
$885$	−80.0000	−2.68917
$886$	12.0000	0.403148
$887$	−42.0000	−1.41022	−0.705111	−	0.709097i	$-0.749103\pi$
$887$	−42.0000	−1.41022	−0.705111	+	0.709097i	$0.749103\pi$
$888$	−60.0000	−2.01347
$889$	12.0000	0.402467
$890$	−16.0000	−0.536321
$891$	44.0000	1.47406
$892$	−14.0000	−0.468755
$893$	0	0
$894$	20.0000	0.668900
$895$	48.0000	1.60446
$896$	−3.00000	−0.100223
$897$	64.0000	2.13690
$898$	38.0000	1.26808
$899$	12.0000	0.400222
$900$	−11.0000	−0.366667
$901$	−12.0000	−0.399778
$902$	−24.0000	−0.799113
$903$	−8.00000	−0.266223
$904$	54.0000	1.79601
$905$	24.0000	0.797787
$906$	32.0000	1.06313
$907$	−28.0000	−0.929725	−0.464862	−	0.885383i	$-0.653896\pi$
$907$	−28.0000	−0.929725	−0.464862	+	0.885383i	$0.653896\pi$
$908$	24.0000	0.796468
$909$	8.00000	0.265343
$910$	16.0000	0.530395
$911$	32.0000	1.06021	0.530104	−	0.847933i	$-0.322153\pi$
$911$	32.0000	1.06021	0.530104	+	0.847933i	$0.322153\pi$
$912$	−12.0000	−0.397360
$913$	16.0000	0.529523
$914$	10.0000	0.330771
$915$	80.0000	2.64472
$916$	−2.00000	−0.0660819
$917$	0	0
$918$	24.0000	0.792118
$919$	24.0000	0.791687	0.395843	−	0.918318i	$-0.370452\pi$
$919$	24.0000	0.791687	0.395843	+	0.918318i	$0.370452\pi$
$920$	−96.0000	−3.16503
$921$	−4.00000	−0.131804
$922$	−2.00000	−0.0658665
$923$	−32.0000	−1.05329
$924$	8.00000	0.263181
$925$	110.000	3.61678
$926$	16.0000	0.525793
$927$	−14.0000	−0.459820
$928$	30.0000	0.984798
$929$	0	0		−	1.00000i	$-0.5\pi$
$929$	0	0			1.00000i	$0.5\pi$
$930$	−16.0000	−0.524661
$931$	6.00000	0.196642
$932$	26.0000	0.851658
$933$	36.0000	1.17859
$934$	−6.00000	−0.196326
$935$	−96.0000	−3.13954
$936$	12.0000	0.392232
$937$	14.0000	0.457360	0.228680	−	0.973502i	$-0.426559\pi$
$937$	14.0000	0.457360	0.228680	+	0.973502i	$0.426559\pi$
$938$	−4.00000	−0.130605
$939$	−44.0000	−1.43589
$940$	0	0
$941$	−52.0000	−1.69515	−0.847576	−	0.530674i	$-0.821939\pi$
$941$	−52.0000	−1.69515	−0.847576	+	0.530674i	$0.821939\pi$
$942$	24.0000	0.781962
$943$	−48.0000	−1.56310
$944$	−10.0000	−0.325472
$945$	16.0000	0.520480
$946$	16.0000	0.520205
$947$	36.0000	1.16984	0.584921	−	0.811090i	$-0.301125\pi$
$947$	36.0000	1.16984	0.584921	+	0.811090i	$0.301125\pi$
$948$	0	0
$949$	−16.0000	−0.519382
$950$	66.0000	2.14132
$951$	−36.0000	−1.16738
$952$	18.0000	0.583383
$953$	2.00000	0.0647864	0.0323932	−	0.999475i	$-0.489687\pi$
$953$	2.00000	0.0647864	0.0323932	+	0.999475i	$0.489687\pi$
$954$	2.00000	0.0647524
$955$	80.0000	2.58874
$956$	12.0000	0.388108
$957$	−48.0000	−1.55162
$958$	40.0000	1.29234
$959$	−18.0000	−0.581250
$960$	−56.0000	−1.80739
$961$	−27.0000	−0.870968
$962$	−40.0000	−1.28965
$963$	20.0000	0.644491
$964$	−20.0000	−0.644157
$965$	24.0000	0.772587
$966$	−16.0000	−0.514792
$967$	32.0000	1.02905	0.514525	−	0.857475i	$-0.327968\pi$
$967$	32.0000	1.02905	0.514525	+	0.857475i	$0.327968\pi$
$968$	−15.0000	−0.482118
$969$	−72.0000	−2.31297
$970$	32.0000	1.02746
$971$	−18.0000	−0.577647	−0.288824	−	0.957382i	$-0.593264\pi$
$971$	−18.0000	−0.577647	−0.288824	+	0.957382i	$0.593264\pi$
$972$	10.0000	0.320750
$973$	4.00000	0.128234
$974$	28.0000	0.897178
$975$	−88.0000	−2.81826
$976$	10.0000	0.320092
$977$	30.0000	0.959785	0.479893	−	0.877327i	$-0.340676\pi$
$977$	30.0000	0.959785	0.479893	+	0.877327i	$0.340676\pi$
$978$	40.0000	1.27906
$979$	−16.0000	−0.511362
$980$	4.00000	0.127775
$981$	−10.0000	−0.319275
$982$	20.0000	0.638226
$983$	−36.0000	−1.14822	−0.574111	−	0.818778i	$-0.694652\pi$
$983$	−36.0000	−1.14822	−0.574111	+	0.818778i	$0.694652\pi$
$984$	−36.0000	−1.14764
$985$	−24.0000	−0.764704
$986$	−36.0000	−1.14647
$987$	0	0
$988$	24.0000	0.763542
$989$	32.0000	1.01754
$990$	16.0000	0.508513
$991$	−44.0000	−1.39771	−0.698853	−	0.715265i	$-0.746306\pi$
$991$	−44.0000	−1.39771	−0.698853	+	0.715265i	$0.746306\pi$
$992$	10.0000	0.317500
$993$	−56.0000	−1.77711
$994$	8.00000	0.253745
$995$	40.0000	1.26809
$996$	8.00000	0.253490
$997$	28.0000	0.886769	0.443384	−	0.896332i	$-0.353778\pi$
$997$	28.0000	0.886769	0.443384	+	0.896332i	$0.353778\pi$
$998$	12.0000	0.379853
$999$	−40.0000	−1.26554

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6041.2.a.a.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6041.2.a.a.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:	\( N \)	\(=\)	\( 6041 = 7 \cdot 863 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6041.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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