LMFDB - Embedded newform 1369.2.b.d.1368.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1369,2,Mod(1368,1369)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1369, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1]))

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

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

chi := DirichletCharacter("1369.1368");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1369 = 37^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1369.b (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$10.9315200367$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 37)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1368.1
Root			$-0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	1369.1368
Dual form			1369.2.b.d.1368.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} -2.73205 q^{3} +1.00000 q^{4} -0.267949i q^{5} +2.73205i q^{6} +3.46410 q^{7} -3.00000i q^{8} +4.46410 q^{9} +O(q^{10})$	\(q-1.00000i q^{2} -2.73205 q^{3} +1.00000 q^{4} -0.267949i q^{5} +2.73205i q^{6} +3.46410 q^{7} -3.00000i q^{8} +4.46410 q^{9} -0.267949 q^{10} -4.73205 q^{11} -2.73205 q^{12} -3.46410i q^{13} -3.46410i q^{14} +0.732051i q^{15} -1.00000 q^{16} +3.73205i q^{17} -4.46410i q^{18} -1.26795i q^{19} -0.267949i q^{20} -9.46410 q^{21} +4.73205i q^{22} -0.732051i q^{23} +8.19615i q^{24} +4.92820 q^{25} -3.46410 q^{26} -4.00000 q^{27} +3.46410 q^{28} +0.267949i q^{29} +0.732051 q^{30} -8.19615i q^{31} -5.00000i q^{32} +12.9282 q^{33} +3.73205 q^{34} -0.928203i q^{35} +4.46410 q^{36} -1.26795 q^{38} +9.46410i q^{39} -0.803848 q^{40} +3.00000 q^{41} +9.46410i q^{42} +3.46410i q^{43} -4.73205 q^{44} -1.19615i q^{45} -0.732051 q^{46} -2.19615 q^{47} +2.73205 q^{48} +5.00000 q^{49} -4.92820i q^{50} -10.1962i q^{51} -3.46410i q^{52} -2.53590 q^{53} +4.00000i q^{54} +1.26795i q^{55} -10.3923i q^{56} +3.46410i q^{57} +0.267949 q^{58} -13.4641i q^{59} +0.732051i q^{60} -4.26795i q^{61} -8.19615 q^{62} +15.4641 q^{63} -7.00000 q^{64} -0.928203 q^{65} -12.9282i q^{66} -7.26795 q^{67} +3.73205i q^{68} +2.00000i q^{69} -0.928203 q^{70} +6.00000 q^{71} -13.3923i q^{72} -13.4641 q^{75} -1.26795i q^{76} -16.3923 q^{77} +9.46410 q^{78} -3.80385i q^{79} +0.267949i q^{80} -2.46410 q^{81} -3.00000i q^{82} -8.19615 q^{83} -9.46410 q^{84} +1.00000 q^{85} +3.46410 q^{86} -0.732051i q^{87} +14.1962i q^{88} +9.19615i q^{89} -1.19615 q^{90} -12.0000i q^{91} -0.732051i q^{92} +22.3923i q^{93} +2.19615i q^{94} -0.339746 q^{95} +13.6603i q^{96} -4.26795i q^{97} -5.00000i q^{98} -21.1244 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{3} + 4 q^{4} + 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^3 + 4 * q^4 + 4 * q^9	$ 4 q - 4 q^{3} + 4 q^{4} + 4 q^{9} - 8 q^{10} - 12 q^{11} - 4 q^{12} - 4 q^{16} - 24 q^{21} - 8 q^{25} - 16 q^{27} - 4 q^{30} + 24 q^{33} + 8 q^{34} + 4 q^{36} - 12 q^{38} - 24 q^{40} + 12 q^{41} - 12 q^{44} + 4 q^{46} + 12 q^{47} + 4 q^{48} + 20 q^{49} - 24 q^{53} + 8 q^{58} - 12 q^{62} + 48 q^{63} - 28 q^{64} + 24 q^{65} - 36 q^{67} + 24 q^{70} + 24 q^{71} - 40 q^{75} - 24 q^{77} + 24 q^{78} + 4 q^{81} - 12 q^{83} - 24 q^{84} + 4 q^{85} + 16 q^{90} - 36 q^{95} - 36 q^{99}+O(q^{100}) $ 4 * q - 4 * q^3 + 4 * q^4 + 4 * q^9 - 8 * q^10 - 12 * q^11 - 4 * q^12 - 4 * q^16 - 24 * q^21 - 8 * q^25 - 16 * q^27 - 4 * q^30 + 24 * q^33 + 8 * q^34 + 4 * q^36 - 12 * q^38 - 24 * q^40 + 12 * q^41 - 12 * q^44 + 4 * q^46 + 12 * q^47 + 4 * q^48 + 20 * q^49 - 24 * q^53 + 8 * q^58 - 12 * q^62 + 48 * q^63 - 28 * q^64 + 24 * q^65 - 36 * q^67 + 24 * q^70 + 24 * q^71 - 40 * q^75 - 24 * q^77 + 24 * q^78 + 4 * q^81 - 12 * q^83 - 24 * q^84 + 4 * q^85 + 16 * q^90 - 36 * q^95 - 36 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1369\mathbb{Z}\right)^\times$.

$n$	$2$
$\chi(n)$	$-1$

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 $

$2$	− 1.00000i	− 0.707107i	−0.935414	−	0.353553i	$-0.884973\pi$
$2$	− 1.00000i	− 0.707107i	0.935414	−	0.353553i	$-0.115027\pi$
$3$	−2.73205	−1.57735	−0.788675	−	0.614810i	$-0.789233\pi$
$3$	−2.73205	−1.57735	−0.788675	+	0.614810i	$0.789233\pi$
$4$	1.00000	0.500000
$5$	− 0.267949i	− 0.119831i	−0.998203	−	0.0599153i	$-0.980917\pi$
$5$	− 0.267949i	− 0.119831i	0.998203	−	0.0599153i	$-0.0190830\pi$
$6$	2.73205i	1.11536i
$7$	3.46410	1.30931	0.654654	−	0.755929i	$-0.272814\pi$
$7$	3.46410	1.30931	0.654654	+	0.755929i	$0.272814\pi$
$8$	− 3.00000i	− 1.06066i
$9$	4.46410	1.48803
$10$	−0.267949	−0.0847330
$11$	−4.73205	−1.42677	−0.713384	−	0.700774i	$-0.752838\pi$
$11$	−4.73205	−1.42677	−0.713384	+	0.700774i	$0.752838\pi$
$12$	−2.73205	−0.788675
$13$	− 3.46410i	− 0.960769i	−0.877058	−	0.480384i	$-0.840497\pi$
$13$	− 3.46410i	− 0.960769i	0.877058	−	0.480384i	$-0.159503\pi$
$14$	− 3.46410i	− 0.925820i
$15$	0.732051i	0.189015i
$16$	−1.00000	−0.250000
$17$	3.73205i	0.905155i	0.891725	+	0.452578i	$0.149495\pi$
$17$	3.73205i	0.905155i	−0.891725	+	0.452578i	$0.850505\pi$
$18$	− 4.46410i	− 1.05220i
$19$	− 1.26795i	− 0.290887i	−0.989367	−	0.145444i	$-0.953539\pi$
$19$	− 1.26795i	− 0.290887i	0.989367	−	0.145444i	$-0.0464610\pi$
$20$	− 0.267949i	− 0.0599153i
$21$	−9.46410	−2.06524
$22$	4.73205i	1.00888i
$23$	− 0.732051i	− 0.152643i	−0.997083	−	0.0763216i	$-0.975682\pi$
$23$	− 0.732051i	− 0.152643i	0.997083	−	0.0763216i	$-0.0243176\pi$
$24$	8.19615i	1.67303i
$25$	4.92820	0.985641
$26$	−3.46410	−0.679366
$27$	−4.00000	−0.769800
$28$	3.46410	0.654654
$29$	0.267949i	0.0497569i	0.999690	+	0.0248785i	$0.00791988\pi$
$29$	0.267949i	0.0497569i	−0.999690	+	0.0248785i	$0.992080\pi$
$30$	0.732051	0.133654
$31$	− 8.19615i	− 1.47207i	−0.676942	−	0.736036i	$-0.736695\pi$
$31$	− 8.19615i	− 1.47207i	0.676942	−	0.736036i	$-0.263305\pi$
$32$	− 5.00000i	− 0.883883i
$33$	12.9282	2.25051
$34$	3.73205	0.640041
$35$	− 0.928203i	− 0.156895i
$36$	4.46410	0.744017
$37$	0	0
$37$	0	0
$38$	−1.26795	−0.205689
$39$	9.46410i	1.51547i
$40$	−0.803848	−0.127099
$41$	3.00000	0.468521	0.234261	−	0.972174i	$-0.424733\pi$
$41$	3.00000	0.468521	0.234261	+	0.972174i	$0.424733\pi$
$42$	9.46410i	1.46034i
$43$	3.46410i	0.528271i	0.964486	+	0.264135i	$0.0850865\pi$
$43$	3.46410i	0.528271i	−0.964486	+	0.264135i	$0.914913\pi$
$44$	−4.73205	−0.713384
$45$	− 1.19615i	− 0.178312i
$46$	−0.732051	−0.107935
$47$	−2.19615	−0.320342	−0.160171	−	0.987089i	$-0.551205\pi$
$47$	−2.19615	−0.320342	−0.160171	+	0.987089i	$0.551205\pi$
$48$	2.73205	0.394338
$49$	5.00000	0.714286
$50$	− 4.92820i	− 0.696953i
$51$	− 10.1962i	− 1.42775i
$52$	− 3.46410i	− 0.480384i
$53$	−2.53590	−0.348332	−0.174166	−	0.984716i	$-0.555723\pi$
$53$	−2.53590	−0.348332	−0.174166	+	0.984716i	$0.555723\pi$
$54$	4.00000i	0.544331i
$55$	1.26795i	0.170970i
$56$	− 10.3923i	− 1.38873i
$57$	3.46410i	0.458831i
$58$	0.267949	0.0351835
$59$	− 13.4641i	− 1.75288i	−0.481514	−	0.876438i	$-0.659913\pi$
$59$	− 13.4641i	− 1.75288i	0.481514	−	0.876438i	$-0.340087\pi$
$60$	0.732051i	0.0945074i
$61$	− 4.26795i	− 0.546455i	−0.961949	−	0.273227i	$-0.911909\pi$
$61$	− 4.26795i	− 0.546455i	0.961949	−	0.273227i	$-0.0880912\pi$
$62$	−8.19615	−1.04091
$63$	15.4641	1.94829
$64$	−7.00000	−0.875000
$65$	−0.928203	−0.115129
$66$	− 12.9282i	− 1.59135i
$67$	−7.26795	−0.887921	−0.443961	−	0.896046i	$-0.646427\pi$
$67$	−7.26795	−0.887921	−0.443961	+	0.896046i	$0.646427\pi$
$68$	3.73205i	0.452578i
$69$	2.00000i	0.240772i
$70$	−0.928203	−0.110942
$71$	6.00000	0.712069	0.356034	−	0.934473i	$-0.384129\pi$
$71$	6.00000	0.712069	0.356034	+	0.934473i	$0.384129\pi$
$72$	− 13.3923i	− 1.57830i
$73$	0	0		−	1.00000i	$-0.5\pi$
$73$	0	0			1.00000i	$0.5\pi$
$74$	0	0
$75$	−13.4641	−1.55470
$76$	− 1.26795i	− 0.145444i
$77$	−16.3923	−1.86808
$78$	9.46410	1.07160
$79$	− 3.80385i	− 0.427966i	−0.976837	−	0.213983i	$-0.931356\pi$
$79$	− 3.80385i	− 0.427966i	0.976837	−	0.213983i	$-0.0686438\pi$
$80$	0.267949i	0.0299576i
$81$	−2.46410	−0.273789
$82$	− 3.00000i	− 0.331295i
$83$	−8.19615	−0.899645	−0.449822	−	0.893118i	$-0.648513\pi$
$83$	−8.19615	−0.899645	−0.449822	+	0.893118i	$0.648513\pi$
$84$	−9.46410	−1.03262
$85$	1.00000	0.108465
$86$	3.46410	0.373544
$87$	− 0.732051i	− 0.0784841i
$88$	14.1962i	1.51331i
$89$	9.19615i	0.974790i	0.873182	+	0.487395i	$0.162053\pi$
$89$	9.19615i	0.974790i	−0.873182	+	0.487395i	$0.837947\pi$
$90$	−1.19615	−0.126086
$91$	− 12.0000i	− 1.25794i
$92$	− 0.732051i	− 0.0763216i
$93$	22.3923i	2.32197i
$94$	2.19615i	0.226516i
$95$	−0.339746	−0.0348572
$96$	13.6603i	1.39419i
$97$	− 4.26795i	− 0.433345i	−0.976244	−	0.216672i	$-0.930480\pi$
$97$	− 4.26795i	− 0.433345i	0.976244	−	0.216672i	$-0.0695203\pi$
$98$	− 5.00000i	− 0.505076i
$99$	−21.1244	−2.12308
$100$	4.92820	0.492820
$101$	−9.00000	−0.895533	−0.447767	−	0.894150i	$-0.647781\pi$
$101$	−9.00000	−0.895533	−0.447767	+	0.894150i	$0.647781\pi$
$102$	−10.1962	−1.00957
$103$	− 12.9282i	− 1.27385i	−0.770924	−	0.636927i	$-0.780205\pi$
$103$	− 12.9282i	− 1.27385i	0.770924	−	0.636927i	$-0.219795\pi$
$104$	−10.3923	−1.01905
$105$	2.53590i	0.247478i
$106$	2.53590i	0.246308i
$107$	−12.9282	−1.24982	−0.624908	−	0.780698i	$-0.714864\pi$
$107$	−12.9282	−1.24982	−0.624908	+	0.780698i	$0.714864\pi$
$108$	−4.00000	−0.384900
$109$	− 6.80385i	− 0.651690i	−0.945423	−	0.325845i	$-0.894351\pi$
$109$	− 6.80385i	− 0.651690i	0.945423	−	0.325845i	$-0.105649\pi$
$110$	1.26795	0.120894
$111$	0	0
$112$	−3.46410	−0.327327
$113$	− 4.53590i	− 0.426701i	−0.976976	−	0.213351i	$-0.931562\pi$
$113$	− 4.53590i	− 0.426701i	0.976976	−	0.213351i	$-0.0684377\pi$
$114$	3.46410	0.324443
$115$	−0.196152	−0.0182913
$116$	0.267949i	0.0248785i
$117$	− 15.4641i	− 1.42966i
$118$	−13.4641	−1.23947
$119$	12.9282i	1.18513i
$120$	2.19615	0.200480
$121$	11.3923	1.03566
$122$	−4.26795	−0.386402
$123$	−8.19615	−0.739022
$124$	− 8.19615i	− 0.736036i
$125$	− 2.66025i	− 0.237940i
$126$	− 15.4641i	− 1.37765i
$127$	−12.1962	−1.08223	−0.541117	−	0.840947i	$-0.681998\pi$
$127$	−12.1962	−1.08223	−0.541117	+	0.840947i	$0.681998\pi$
$128$	− 3.00000i	− 0.265165i
$129$	− 9.46410i	− 0.833268i
$130$	0.928203i	0.0814088i
$131$	− 10.5885i	− 0.925118i	−0.886589	−	0.462559i	$-0.846931\pi$
$131$	− 10.5885i	− 0.925118i	0.886589	−	0.462559i	$-0.153069\pi$
$132$	12.9282	1.12526
$133$	− 4.39230i	− 0.380861i
$134$	7.26795i	0.627855i
$135$	1.07180i	0.0922456i
$136$	11.1962	0.960062
$137$	−6.46410	−0.552265	−0.276133	−	0.961120i	$-0.589053\pi$
$137$	−6.46410	−0.552265	−0.276133	+	0.961120i	$0.589053\pi$
$138$	2.00000	0.170251
$139$	12.5885	1.06774	0.533870	−	0.845567i	$-0.320737\pi$
$139$	12.5885	1.06774	0.533870	+	0.845567i	$0.320737\pi$
$140$	− 0.928203i	− 0.0784475i
$141$	6.00000	0.505291
$142$	− 6.00000i	− 0.503509i
$143$	16.3923i	1.37079i
$144$	−4.46410	−0.372008
$145$	0.0717968	0.00596240
$146$	0	0
$147$	−13.6603	−1.12668
$148$	0	0
$149$	5.53590	0.453518	0.226759	−	0.973951i	$-0.427187\pi$
$149$	5.53590	0.453518	0.226759	+	0.973951i	$0.427187\pi$
$150$	13.4641i	1.09934i
$151$	16.5885	1.34995	0.674975	−	0.737841i	$-0.264155\pi$
$151$	16.5885	1.34995	0.674975	+	0.737841i	$0.264155\pi$
$152$	−3.80385	−0.308533
$153$	16.6603i	1.34690i
$154$	16.3923i	1.32093i
$155$	−2.19615	−0.176399
$156$	9.46410i	0.757735i
$157$	−14.3205	−1.14290	−0.571450	−	0.820637i	$-0.693619\pi$
$157$	−14.3205	−1.14290	−0.571450	+	0.820637i	$0.693619\pi$
$158$	−3.80385	−0.302618
$159$	6.92820	0.549442
$160$	−1.33975	−0.105916
$161$	− 2.53590i	− 0.199857i
$162$	2.46410i	0.193598i
$163$	22.3923i	1.75390i	0.480581	+	0.876950i	$0.340426\pi$
$163$	22.3923i	1.75390i	−0.480581	+	0.876950i	$0.659574\pi$
$164$	3.00000	0.234261
$165$	− 3.46410i	− 0.269680i
$166$	8.19615i	0.636145i
$167$	− 10.9282i	− 0.845650i	−0.906211	−	0.422825i	$-0.861039\pi$
$167$	− 10.9282i	− 0.845650i	0.906211	−	0.422825i	$-0.138961\pi$
$168$	28.3923i	2.19051i
$169$	1.00000	0.0769231
$170$	− 1.00000i	− 0.0766965i
$171$	− 5.66025i	− 0.432850i
$172$	3.46410i	0.264135i
$173$	9.92820	0.754827	0.377414	−	0.926045i	$-0.376813\pi$
$173$	9.92820	0.754827	0.377414	+	0.926045i	$0.376813\pi$
$174$	−0.732051	−0.0554966
$175$	17.0718	1.29051
$176$	4.73205	0.356692
$177$	36.7846i	2.76490i
$178$	9.19615	0.689281
$179$	1.46410i	0.109432i	0.998502	+	0.0547160i	$0.0174254\pi$
$179$	1.46410i	0.109432i	−0.998502	+	0.0547160i	$0.982575\pi$
$180$	− 1.19615i	− 0.0891559i
$181$	9.92820	0.737958	0.368979	−	0.929438i	$-0.379708\pi$
$181$	9.92820	0.737958	0.368979	+	0.929438i	$0.379708\pi$
$182$	−12.0000	−0.889499
$183$	11.6603i	0.861951i
$184$	−2.19615	−0.161903
$185$	0	0
$186$	22.3923	1.64188
$187$	− 17.6603i	− 1.29145i
$188$	−2.19615	−0.160171
$189$	−13.8564	−1.00791
$190$	0.339746i	0.0246478i
$191$	− 6.19615i	− 0.448338i	−0.974550	−	0.224169i	$-0.928033\pi$
$191$	− 6.19615i	− 0.448338i	0.974550	−	0.224169i	$-0.0719667\pi$
$192$	19.1244	1.38018
$193$	− 14.6603i	− 1.05527i	−0.849472	−	0.527634i	$-0.823079\pi$
$193$	− 14.6603i	− 1.05527i	0.849472	−	0.527634i	$-0.176921\pi$
$194$	−4.26795	−0.306421
$195$	2.53590	0.181599
$196$	5.00000	0.357143
$197$	27.9282	1.98980	0.994901	−	0.100856i	$-0.0321581\pi$
$197$	27.9282	1.98980	0.994901	+	0.100856i	$0.0321581\pi$
$198$	21.1244i	1.50124i
$199$	− 11.0718i	− 0.784859i	−0.919782	−	0.392429i	$-0.871635\pi$
$199$	− 11.0718i	− 0.784859i	0.919782	−	0.392429i	$-0.128365\pi$
$200$	− 14.7846i	− 1.04543i
$201$	19.8564	1.40056
$202$	9.00000i	0.633238i
$203$	0.928203i	0.0651471i
$204$	− 10.1962i	− 0.713873i
$205$	− 0.803848i	− 0.0561432i
$206$	−12.9282	−0.900751
$207$	− 3.26795i	− 0.227138i
$208$	3.46410i	0.240192i
$209$	6.00000i	0.415029i
$210$	2.53590	0.174994
$211$	−10.0000	−0.688428	−0.344214	−	0.938891i	$-0.611855\pi$
$211$	−10.0000	−0.688428	−0.344214	+	0.938891i	$0.611855\pi$
$212$	−2.53590	−0.174166
$213$	−16.3923	−1.12318
$214$	12.9282i	0.883754i
$215$	0.928203	0.0633029
$216$	12.0000i	0.816497i
$217$	− 28.3923i	− 1.92740i
$218$	−6.80385	−0.460815
$219$	0	0
$220$	1.26795i	0.0854851i
$221$	12.9282	0.869645
$222$	0	0
$223$	4.58846	0.307266	0.153633	−	0.988128i	$-0.450903\pi$
$223$	4.58846	0.307266	0.153633	+	0.988128i	$0.450903\pi$
$224$	− 17.3205i	− 1.15728i
$225$	22.0000	1.46667
$226$	−4.53590	−0.301723
$227$	− 23.1244i	− 1.53482i	−0.641158	−	0.767409i	$-0.721546\pi$
$227$	− 23.1244i	− 1.53482i	0.641158	−	0.767409i	$-0.278454\pi$
$228$	3.46410i	0.229416i
$229$	7.00000	0.462573	0.231287	−	0.972886i	$-0.425707\pi$
$229$	7.00000	0.462573	0.231287	+	0.972886i	$0.425707\pi$
$230$	0.196152i	0.0129339i
$231$	44.7846	2.94661
$232$	0.803848	0.0527752
$233$	−10.8564	−0.711227	−0.355613	−	0.934633i	$-0.615728\pi$
$233$	−10.8564	−0.711227	−0.355613	+	0.934633i	$0.615728\pi$
$234$	−15.4641	−1.01092
$235$	0.588457i	0.0383867i
$236$	− 13.4641i	− 0.876438i
$237$	10.3923i	0.675053i
$238$	12.9282	0.838011
$239$	16.9282i	1.09499i	0.836808	+	0.547497i	$0.184419\pi$
$239$	16.9282i	1.09499i	−0.836808	+	0.547497i	$0.815581\pi$
$240$	− 0.732051i	− 0.0472537i
$241$	15.4641i	0.996130i	0.867140	+	0.498065i	$0.165956\pi$
$241$	15.4641i	0.996130i	−0.867140	+	0.498065i	$0.834044\pi$
$242$	− 11.3923i	− 0.732325i
$243$	18.7321	1.20166
$244$	− 4.26795i	− 0.273227i
$245$	− 1.33975i	− 0.0855932i
$246$	8.19615i	0.522568i
$247$	−4.39230	−0.279476
$248$	−24.5885	−1.56137
$249$	22.3923	1.41905
$250$	−2.66025	−0.168249
$251$	25.7128i	1.62298i	0.584367	+	0.811489i	$0.301343\pi$
$251$	25.7128i	1.62298i	−0.584367	+	0.811489i	$0.698657\pi$
$252$	15.4641	0.974147
$253$	3.46410i	0.217786i
$254$	12.1962i	0.765255i
$255$	−2.73205	−0.171088
$256$	−17.0000	−1.06250
$257$	19.5885i	1.22189i	0.791671	+	0.610947i	$0.209211\pi$
$257$	19.5885i	1.22189i	−0.791671	+	0.610947i	$0.790789\pi$
$258$	−9.46410	−0.589209
$259$	0	0
$260$	−0.928203	−0.0575647
$261$	1.19615i	0.0740400i
$262$	−10.5885	−0.654157
$263$	−7.60770	−0.469111	−0.234555	−	0.972103i	$-0.575363\pi$
$263$	−7.60770	−0.469111	−0.234555	+	0.972103i	$0.575363\pi$
$264$	− 38.7846i	− 2.38703i
$265$	0.679492i	0.0417409i
$266$	−4.39230	−0.269309
$267$	− 25.1244i	− 1.53759i
$268$	−7.26795	−0.443961
$269$	14.5359	0.886269	0.443135	−	0.896455i	$-0.353866\pi$
$269$	14.5359	0.886269	0.443135	+	0.896455i	$0.353866\pi$
$270$	1.07180	0.0652275
$271$	−11.6603	−0.708310	−0.354155	−	0.935187i	$-0.615231\pi$
$271$	−11.6603	−0.708310	−0.354155	+	0.935187i	$0.615231\pi$
$272$	− 3.73205i	− 0.226289i
$273$	32.7846i	1.98421i
$274$	6.46410i	0.390511i
$275$	−23.3205	−1.40628
$276$	2.00000i	0.120386i
$277$	10.2679i	0.616941i	0.951234	+	0.308471i	$0.0998172\pi$
$277$	10.2679i	0.616941i	−0.951234	+	0.308471i	$0.900183\pi$
$278$	− 12.5885i	− 0.755005i
$279$	− 36.5885i	− 2.19049i
$280$	−2.78461	−0.166412
$281$	33.1962i	1.98032i	0.139953	+	0.990158i	$0.455305\pi$
$281$	33.1962i	1.98032i	−0.139953	+	0.990158i	$0.544695\pi$
$282$	− 6.00000i	− 0.357295i
$283$	− 10.3923i	− 0.617758i	−0.951101	−	0.308879i	$-0.900046\pi$
$283$	− 10.3923i	− 0.617758i	0.951101	−	0.308879i	$-0.0999539\pi$
$284$	6.00000	0.356034
$285$	0.928203	0.0549820
$286$	16.3923	0.969297
$287$	10.3923	0.613438
$288$	− 22.3205i	− 1.31525i
$289$	3.07180	0.180694
$290$	− 0.0717968i	− 0.00421605i
$291$	11.6603i	0.683536i
$292$	0	0
$293$	28.8564	1.68581	0.842905	−	0.538063i	$-0.180844\pi$
$293$	28.8564	1.68581	0.842905	+	0.538063i	$0.180844\pi$
$294$	13.6603i	0.796682i
$295$	−3.60770	−0.210048
$296$	0	0
$297$	18.9282	1.09833
$298$	− 5.53590i	− 0.320686i
$299$	−2.53590	−0.146655
$300$	−13.4641	−0.777350
$301$	12.0000i	0.691669i
$302$	− 16.5885i	− 0.954558i
$303$	24.5885	1.41257
$304$	1.26795i	0.0727219i
$305$	−1.14359	−0.0654820
$306$	16.6603	0.952403
$307$	−14.3923	−0.821412	−0.410706	−	0.911768i	$-0.634718\pi$
$307$	−14.3923	−0.821412	−0.410706	+	0.911768i	$0.634718\pi$
$308$	−16.3923	−0.934038
$309$	35.3205i	2.00931i
$310$	2.19615i	0.124733i
$311$	− 30.3923i	− 1.72339i	−0.507427	−	0.861695i	$-0.669403\pi$
$311$	− 30.3923i	− 1.72339i	0.507427	−	0.861695i	$-0.330597\pi$
$312$	28.3923	1.60740
$313$	− 15.5885i	− 0.881112i	−0.897725	−	0.440556i	$-0.854781\pi$
$313$	− 15.5885i	− 0.881112i	0.897725	−	0.440556i	$-0.145219\pi$
$314$	14.3205i	0.808153i
$315$	− 4.14359i	− 0.233465i
$316$	− 3.80385i	− 0.213983i
$317$	−15.9282	−0.894617	−0.447309	−	0.894380i	$-0.647617\pi$
$317$	−15.9282	−0.894617	−0.447309	+	0.894380i	$0.647617\pi$
$318$	− 6.92820i	− 0.388514i
$319$	− 1.26795i	− 0.0709915i
$320$	1.87564i	0.104852i
$321$	35.3205	1.97140
$322$	−2.53590	−0.141320
$323$	4.73205	0.263298
$324$	−2.46410	−0.136895
$325$	− 17.0718i	− 0.946973i
$326$	22.3923	1.24020
$327$	18.5885i	1.02794i
$328$	− 9.00000i	− 0.496942i
$329$	−7.60770	−0.419426
$330$	−3.46410	−0.190693
$331$	32.7846i	1.80201i	0.433814	+	0.901003i	$0.357168\pi$
$331$	32.7846i	1.80201i	−0.433814	+	0.901003i	$0.642832\pi$
$332$	−8.19615	−0.449822
$333$	0	0
$334$	−10.9282	−0.597965
$335$	1.94744i	0.106400i
$336$	9.46410	0.516309
$337$	29.3923	1.60110	0.800550	−	0.599265i	$-0.204541\pi$
$337$	29.3923	1.60110	0.800550	+	0.599265i	$0.204541\pi$
$338$	− 1.00000i	− 0.0543928i
$339$	12.3923i	0.673058i
$340$	1.00000	0.0542326
$341$	38.7846i	2.10030i
$342$	−5.66025	−0.306071
$343$	−6.92820	−0.374088
$344$	10.3923	0.560316
$345$	0.535898	0.0288518
$346$	− 9.92820i	− 0.533744i
$347$	12.7321i	0.683492i	0.939792	+	0.341746i	$0.111018\pi$
$347$	12.7321i	0.683492i	−0.939792	+	0.341746i	$0.888982\pi$
$348$	− 0.732051i	− 0.0392420i
$349$	8.07180	0.432073	0.216037	−	0.976385i	$-0.430687\pi$
$349$	8.07180	0.432073	0.216037	+	0.976385i	$0.430687\pi$
$350$	− 17.0718i	− 0.912526i
$351$	13.8564i	0.739600i
$352$	23.6603i	1.26110i
$353$	12.2679i	0.652957i	0.945205	+	0.326479i	$0.105862\pi$
$353$	12.2679i	0.652957i	−0.945205	+	0.326479i	$0.894138\pi$
$354$	36.7846	1.95508
$355$	− 1.60770i	− 0.0853276i
$356$	9.19615i	0.487395i
$357$	− 35.3205i	− 1.86936i
$358$	1.46410	0.0773802
$359$	16.3923	0.865153	0.432576	−	0.901597i	$-0.357605\pi$
$359$	16.3923	0.865153	0.432576	+	0.901597i	$0.357605\pi$
$360$	−3.58846	−0.189128
$361$	17.3923	0.915384
$362$	− 9.92820i	− 0.521815i
$363$	−31.1244	−1.63361
$364$	− 12.0000i	− 0.628971i
$365$	0	0
$366$	11.6603	0.609491
$367$	−20.3923	−1.06447	−0.532235	−	0.846597i	$-0.678648\pi$
$367$	−20.3923	−1.06447	−0.532235	+	0.846597i	$0.678648\pi$
$368$	0.732051i	0.0381608i
$369$	13.3923	0.697176
$370$	0	0
$371$	−8.78461	−0.456074
$372$	22.3923i	1.16099i
$373$	17.5359	0.907974	0.453987	−	0.891008i	$-0.350001\pi$
$373$	17.5359	0.907974	0.453987	+	0.891008i	$0.350001\pi$
$374$	−17.6603	−0.913190
$375$	7.26795i	0.375315i
$376$	6.58846i	0.339774i
$377$	0.928203	0.0478049
$378$	13.8564i	0.712697i
$379$	8.53590	0.438460	0.219230	−	0.975673i	$-0.429646\pi$
$379$	8.53590	0.438460	0.219230	+	0.975673i	$0.429646\pi$
$380$	−0.339746	−0.0174286
$381$	33.3205	1.70706
$382$	−6.19615	−0.317023
$383$	15.8564i	0.810225i	0.914267	+	0.405112i	$0.132768\pi$
$383$	15.8564i	0.810225i	−0.914267	+	0.405112i	$0.867232\pi$
$384$	8.19615i	0.418258i
$385$	4.39230i	0.223853i
$386$	−14.6603	−0.746187
$387$	15.4641i	0.786084i
$388$	− 4.26795i	− 0.216672i
$389$	− 31.1962i	− 1.58171i	−0.612005	−	0.790854i	$-0.709637\pi$
$389$	− 31.1962i	− 1.58171i	0.612005	−	0.790854i	$-0.290363\pi$
$390$	− 2.53590i	− 0.128410i
$391$	2.73205	0.138166
$392$	− 15.0000i	− 0.757614i
$393$	28.9282i	1.45923i
$394$	− 27.9282i	− 1.40700i
$395$	−1.01924	−0.0512834
$396$	−21.1244	−1.06154
$397$	−21.2487	−1.06644	−0.533221	−	0.845976i	$-0.679019\pi$
$397$	−21.2487	−1.06644	−0.533221	+	0.845976i	$0.679019\pi$
$398$	−11.0718	−0.554979
$399$	12.0000i	0.600751i
$400$	−4.92820	−0.246410
$401$	− 5.60770i	− 0.280035i	−0.990149	−	0.140017i	$-0.955284\pi$
$401$	− 5.60770i	− 0.280035i	0.990149	−	0.140017i	$-0.0447159\pi$
$402$	− 19.8564i	− 0.990348i
$403$	−28.3923	−1.41432
$404$	−9.00000	−0.447767
$405$	0.660254i	0.0328083i
$406$	0.928203	0.0460660
$407$	0	0
$408$	−30.5885	−1.51435
$409$	8.66025i	0.428222i	0.976809	+	0.214111i	$0.0686854\pi$
$409$	8.66025i	0.428222i	−0.976809	+	0.214111i	$0.931315\pi$
$410$	−0.803848	−0.0396992
$411$	17.6603	0.871116
$412$	− 12.9282i	− 0.636927i
$413$	− 46.6410i	− 2.29505i
$414$	−3.26795	−0.160611
$415$	2.19615i	0.107805i
$416$	−17.3205	−0.849208
$417$	−34.3923	−1.68420
$418$	6.00000	0.293470
$419$	17.3205	0.846162	0.423081	−	0.906092i	$-0.360949\pi$
$419$	17.3205	0.846162	0.423081	+	0.906092i	$0.360949\pi$
$420$	2.53590i	0.123739i
$421$	16.2679i	0.792851i	0.918067	+	0.396426i	$0.129750\pi$
$421$	16.2679i	0.792851i	−0.918067	+	0.396426i	$0.870250\pi$
$422$	10.0000i	0.486792i
$423$	−9.80385	−0.476679
$424$	7.60770i	0.369462i
$425$	18.3923i	0.892158i
$426$	16.3923i	0.794210i
$427$	− 14.7846i	− 0.715477i
$428$	−12.9282	−0.624908
$429$	− 44.7846i	− 2.16222i
$430$	− 0.928203i	− 0.0447619i
$431$	11.5167i	0.554738i	0.960763	+	0.277369i	$0.0894625\pi$
$431$	11.5167i	0.554738i	−0.960763	+	0.277369i	$0.910538\pi$
$432$	4.00000	0.192450
$433$	27.0000	1.29754	0.648769	−	0.760986i	$-0.275284\pi$
$433$	27.0000	1.29754	0.648769	+	0.760986i	$0.275284\pi$
$434$	−28.3923	−1.36287
$435$	−0.196152	−0.00940479
$436$	− 6.80385i	− 0.325845i
$437$	−0.928203	−0.0444020
$438$	0	0
$439$	− 14.5359i	− 0.693761i	−0.937909	−	0.346880i	$-0.887241\pi$
$439$	− 14.5359i	− 0.693761i	0.937909	−	0.346880i	$-0.112759\pi$
$440$	3.80385	0.181341
$441$	22.3205	1.06288
$442$	− 12.9282i	− 0.614932i
$443$	33.4641	1.58993	0.794964	−	0.606657i	$-0.207490\pi$
$443$	33.4641	1.58993	0.794964	+	0.606657i	$0.207490\pi$
$444$	0	0
$445$	2.46410	0.116810
$446$	− 4.58846i	− 0.217270i
$447$	−15.1244	−0.715357
$448$	−24.2487	−1.14564
$449$	32.2487i	1.52191i	0.648804	+	0.760955i	$0.275269\pi$
$449$	32.2487i	1.52191i	−0.648804	+	0.760955i	$0.724731\pi$
$450$	− 22.0000i	− 1.03709i
$451$	−14.1962	−0.668471
$452$	− 4.53590i	− 0.213351i
$453$	−45.3205	−2.12934
$454$	−23.1244	−1.08528
$455$	−3.21539	−0.150740
$456$	10.3923	0.486664
$457$	36.1244i	1.68983i	0.534904	+	0.844913i	$0.320348\pi$
$457$	36.1244i	1.68983i	−0.534904	+	0.844913i	$0.679652\pi$
$458$	− 7.00000i	− 0.327089i
$459$	− 14.9282i	− 0.696789i
$460$	−0.196152	−0.00914565
$461$	− 16.5359i	− 0.770154i	−0.922885	−	0.385077i	$-0.874175\pi$
$461$	− 16.5359i	− 0.770154i	0.922885	−	0.385077i	$-0.125825\pi$
$462$	− 44.7846i	− 2.08357i
$463$	15.4641i	0.718678i	0.933207	+	0.359339i	$0.116998\pi$
$463$	15.4641i	0.718678i	−0.933207	+	0.359339i	$0.883002\pi$
$464$	− 0.267949i	− 0.0124392i
$465$	6.00000	0.278243
$466$	10.8564i	0.502913i
$467$	− 3.32051i	− 0.153655i	−0.997044	−	0.0768274i	$-0.975521\pi$
$467$	− 3.32051i	− 0.153655i	0.997044	−	0.0768274i	$-0.0244790\pi$
$468$	− 15.4641i	− 0.714828i
$469$	−25.1769	−1.16256
$470$	0.588457	0.0271435
$471$	39.1244	1.80276
$472$	−40.3923	−1.85921
$473$	− 16.3923i	− 0.753719i
$474$	10.3923	0.477334
$475$	− 6.24871i	− 0.286711i
$476$	12.9282i	0.592563i
$477$	−11.3205	−0.518330
$478$	16.9282	0.774278
$479$	− 39.3205i	− 1.79660i	−0.439383	−	0.898300i	$-0.644803\pi$
$479$	− 39.3205i	− 1.79660i	0.439383	−	0.898300i	$-0.355197\pi$
$480$	3.66025	0.167067
$481$	0	0
$482$	15.4641	0.704371
$483$	6.92820i	0.315244i
$484$	11.3923	0.517832
$485$	−1.14359	−0.0519279
$486$	− 18.7321i	− 0.849703i
$487$	32.4449i	1.47022i	0.677949	+	0.735109i	$0.262869\pi$
$487$	32.4449i	1.47022i	−0.677949	+	0.735109i	$0.737131\pi$
$488$	−12.8038	−0.579603
$489$	− 61.1769i	− 2.76652i
$490$	−1.33975	−0.0605236
$491$	31.2679	1.41110	0.705551	−	0.708659i	$-0.250699\pi$
$491$	31.2679	1.41110	0.705551	+	0.708659i	$0.250699\pi$
$492$	−8.19615	−0.369511
$493$	−1.00000	−0.0450377
$494$	4.39230i	0.197619i
$495$	5.66025i	0.254409i
$496$	8.19615i	0.368018i
$497$	20.7846	0.932317
$498$	− 22.3923i	− 1.00342i
$499$	− 12.5885i	− 0.563537i	−0.959482	−	0.281768i	$-0.909079\pi$
$499$	− 12.5885i	− 0.563537i	0.959482	−	0.281768i	$-0.0909210\pi$
$500$	− 2.66025i	− 0.118970i
$501$	29.8564i	1.33389i
$502$	25.7128	1.14762
$503$	− 8.98076i	− 0.400432i	−0.979752	−	0.200216i	$-0.935836\pi$
$503$	− 8.98076i	− 0.400432i	0.979752	−	0.200216i	$-0.0641645\pi$
$504$	− 46.3923i	− 2.06648i
$505$	2.41154i	0.107312i
$506$	3.46410	0.153998
$507$	−2.73205	−0.121335
$508$	−12.1962	−0.541117
$509$	−5.78461	−0.256398	−0.128199	−	0.991748i	$-0.540920\pi$
$509$	−5.78461	−0.256398	−0.128199	+	0.991748i	$0.540920\pi$
$510$	2.73205i	0.120977i
$511$	0	0
$512$	11.0000i	0.486136i
$513$	5.07180i	0.223925i
$514$	19.5885	0.864010
$515$	−3.46410	−0.152647
$516$	− 9.46410i	− 0.416634i
$517$	10.3923	0.457053
$518$	0	0
$519$	−27.1244	−1.19063
$520$	2.78461i	0.122113i
$521$	6.92820	0.303530	0.151765	−	0.988417i	$-0.451504\pi$
$521$	6.92820	0.303530	0.151765	+	0.988417i	$0.451504\pi$
$522$	1.19615	0.0523542
$523$	− 16.0526i	− 0.701929i	−0.936389	−	0.350965i	$-0.885854\pi$
$523$	− 16.0526i	− 0.701929i	0.936389	−	0.350965i	$-0.114146\pi$
$524$	− 10.5885i	− 0.462559i
$525$	−46.6410	−2.03558
$526$	7.60770i	0.331711i
$527$	30.5885	1.33245
$528$	−12.9282	−0.562628
$529$	22.4641	0.976700
$530$	0.679492	0.0295152
$531$	− 60.1051i	− 2.60834i
$532$	− 4.39230i	− 0.190431i
$533$	− 10.3923i	− 0.450141i
$534$	−25.1244	−1.08724
$535$	3.46410i	0.149766i
$536$	21.8038i	0.941783i
$537$	− 4.00000i	− 0.172613i
$538$	− 14.5359i	− 0.626687i
$539$	−23.6603	−1.01912
$540$	1.07180i	0.0461228i
$541$	− 4.26795i	− 0.183493i	−0.995782	−	0.0917467i	$-0.970755\pi$
$541$	− 4.26795i	− 0.183493i	0.995782	−	0.0917467i	$-0.0292450\pi$
$542$	11.6603i	0.500851i
$543$	−27.1244	−1.16402
$544$	18.6603	0.800052
$545$	−1.82309	−0.0780924
$546$	32.7846	1.40305
$547$	3.80385i	0.162641i	0.996688	+	0.0813204i	$0.0259137\pi$
$547$	3.80385i	0.162641i	−0.996688	+	0.0813204i	$0.974086\pi$
$548$	−6.46410	−0.276133
$549$	− 19.0526i	− 0.813143i
$550$	23.3205i	0.994390i
$551$	0.339746	0.0144737
$552$	6.00000	0.255377
$553$	− 13.1769i	− 0.560339i
$554$	10.2679	0.436243
$555$	0	0
$556$	12.5885	0.533870
$557$	− 17.7321i	− 0.751331i	−0.926755	−	0.375666i	$-0.877414\pi$
$557$	− 17.7321i	− 0.751331i	0.926755	−	0.375666i	$-0.122586\pi$
$558$	−36.5885	−1.54891
$559$	12.0000	0.507546
$560$	0.928203i	0.0392237i
$561$	48.2487i	2.03706i
$562$	33.1962	1.40030
$563$	34.5885i	1.45773i	0.684658	+	0.728865i	$0.259952\pi$
$563$	34.5885i	1.45773i	−0.684658	+	0.728865i	$0.740048\pi$
$564$	6.00000	0.252646
$565$	−1.21539	−0.0511319
$566$	−10.3923	−0.436821
$567$	−8.53590	−0.358474
$568$	− 18.0000i	− 0.755263i
$569$	− 36.5167i	− 1.53086i	−0.643520	−	0.765429i	$-0.722527\pi$
$569$	− 36.5167i	− 1.53086i	0.643520	−	0.765429i	$-0.277473\pi$
$570$	− 0.928203i	− 0.0388782i
$571$	−31.5167	−1.31893	−0.659466	−	0.751735i	$-0.729217\pi$
$571$	−31.5167	−1.31893	−0.659466	+	0.751735i	$0.729217\pi$
$572$	16.3923i	0.685397i
$573$	16.9282i	0.707186i
$574$	− 10.3923i	− 0.433766i
$575$	− 3.60770i	− 0.150451i
$576$	−31.2487	−1.30203
$577$	12.9282i	0.538208i	0.963111	+	0.269104i	$0.0867276\pi$
$577$	12.9282i	0.538208i	−0.963111	+	0.269104i	$0.913272\pi$
$578$	− 3.07180i	− 0.127770i
$579$	40.0526i	1.66453i
$580$	0.0717968	0.00298120
$581$	−28.3923	−1.17791
$582$	11.6603	0.483333
$583$	12.0000	0.496989
$584$	0	0
$585$	−4.14359	−0.171317
$586$	− 28.8564i	− 1.19205i
$587$	16.3397i	0.674413i	0.941431	+	0.337207i	$0.109482\pi$
$587$	16.3397i	0.674413i	−0.941431	+	0.337207i	$0.890518\pi$
$588$	−13.6603	−0.563339
$589$	−10.3923	−0.428207
$590$	3.60770i	0.148526i
$591$	−76.3013	−3.13861
$592$	0	0
$593$	17.7846	0.730326	0.365163	−	0.930944i	$-0.381013\pi$
$593$	17.7846	0.730326	0.365163	+	0.930944i	$0.381013\pi$
$594$	− 18.9282i	− 0.776634i
$595$	3.46410	0.142014
$596$	5.53590	0.226759
$597$	30.2487i	1.23800i
$598$	2.53590i	0.103701i
$599$	7.94744	0.324724	0.162362	−	0.986731i	$-0.448089\pi$
$599$	7.94744	0.324724	0.162362	+	0.986731i	$0.448089\pi$
$600$	40.3923i	1.64901i
$601$	31.3923	1.28052	0.640259	−	0.768159i	$-0.278827\pi$
$601$	31.3923	1.28052	0.640259	+	0.768159i	$0.278827\pi$
$602$	12.0000	0.489083
$603$	−32.4449	−1.32126
$604$	16.5885	0.674975
$605$	− 3.05256i	− 0.124104i
$606$	− 24.5885i	− 0.998838i
$607$	5.66025i	0.229743i	0.993380	+	0.114871i	$0.0366456\pi$
$607$	5.66025i	0.229743i	−0.993380	+	0.114871i	$0.963354\pi$
$608$	−6.33975	−0.257111
$609$	− 2.53590i	− 0.102760i
$610$	1.14359i	0.0463027i
$611$	7.60770i	0.307774i
$612$	16.6603i	0.673451i
$613$	3.39230	0.137014	0.0685070	−	0.997651i	$-0.478176\pi$
$613$	3.39230	0.137014	0.0685070	+	0.997651i	$0.478176\pi$
$614$	14.3923i	0.580826i
$615$	2.19615i	0.0885574i
$616$	49.1769i	1.98139i
$617$	−25.1769	−1.01358	−0.506792	−	0.862068i	$-0.669169\pi$
$617$	−25.1769	−1.01358	−0.506792	+	0.862068i	$0.669169\pi$
$618$	35.3205	1.42080
$619$	6.78461	0.272696	0.136348	−	0.990661i	$-0.456463\pi$
$619$	6.78461	0.272696	0.136348	+	0.990661i	$0.456463\pi$
$620$	−2.19615	−0.0881996
$621$	2.92820i	0.117505i
$622$	−30.3923	−1.21862
$623$	31.8564i	1.27630i
$624$	− 9.46410i	− 0.378867i
$625$	23.9282	0.957128
$626$	−15.5885	−0.623040
$627$	− 16.3923i	− 0.654646i
$628$	−14.3205	−0.571450
$629$	0	0
$630$	−4.14359	−0.165085
$631$	19.2679i	0.767045i	0.923532	+	0.383522i	$0.125289\pi$
$631$	19.2679i	0.767045i	−0.923532	+	0.383522i	$0.874711\pi$
$632$	−11.4115	−0.453927
$633$	27.3205	1.08589
$634$	15.9282i	0.632590i
$635$	3.26795i	0.129685i
$636$	6.92820	0.274721
$637$	− 17.3205i	− 0.686264i
$638$	−1.26795	−0.0501986
$639$	26.7846	1.05958
$640$	−0.803848	−0.0317749
$641$	−17.7846	−0.702450	−0.351225	−	0.936291i	$-0.614235\pi$
$641$	−17.7846	−0.702450	−0.351225	+	0.936291i	$0.614235\pi$
$642$	− 35.3205i	− 1.39399i
$643$	5.66025i	0.223219i	0.993752	+	0.111609i	$0.0356005\pi$
$643$	5.66025i	0.223219i	−0.993752	+	0.111609i	$0.964399\pi$
$644$	− 2.53590i	− 0.0999284i
$645$	−2.53590	−0.0998509
$646$	− 4.73205i	− 0.186180i
$647$	− 43.4641i	− 1.70875i	−0.519657	−	0.854375i	$-0.673940\pi$
$647$	− 43.4641i	− 1.70875i	0.519657	−	0.854375i	$-0.326060\pi$
$648$	7.39230i	0.290397i
$649$	63.7128i	2.50095i
$650$	−17.0718	−0.669611
$651$	77.5692i	3.04018i
$652$	22.3923i	0.876950i
$653$	2.80385i	0.109723i	0.998494	+	0.0548615i	$0.0174717\pi$
$653$	2.80385i	0.109723i	−0.998494	+	0.0548615i	$0.982528\pi$
$654$	18.5885	0.726866
$655$	−2.83717	−0.110857
$656$	−3.00000	−0.117130
$657$	0	0
$658$	7.60770i	0.296579i
$659$	9.46410	0.368669	0.184335	−	0.982864i	$-0.440987\pi$
$659$	9.46410	0.368669	0.184335	+	0.982864i	$0.440987\pi$
$660$	− 3.46410i	− 0.134840i
$661$	12.8038i	0.498012i	0.968502	+	0.249006i	$0.0801039\pi$
$661$	12.8038i	0.498012i	−0.968502	+	0.249006i	$0.919896\pi$
$662$	32.7846	1.27421
$663$	−35.3205	−1.37173
$664$	24.5885i	0.954217i
$665$	−1.17691	−0.0456388
$666$	0	0
$667$	0.196152	0.00759505
$668$	− 10.9282i	− 0.422825i
$669$	−12.5359	−0.484666
$670$	1.94744	0.0752362
$671$	20.1962i	0.779664i
$672$	47.3205i	1.82543i
$673$	15.7128	0.605684	0.302842	−	0.953041i	$-0.402064\pi$
$673$	15.7128	0.605684	0.302842	+	0.953041i	$0.402064\pi$
$674$	− 29.3923i	− 1.13215i
$675$	−19.7128	−0.758747
$676$	1.00000	0.0384615
$677$	−44.3205	−1.70338	−0.851688	−	0.524050i	$-0.824421\pi$
$677$	−44.3205	−1.70338	−0.851688	+	0.524050i	$0.824421\pi$
$678$	12.3923	0.475924
$679$	− 14.7846i	− 0.567381i
$680$	− 3.00000i	− 0.115045i
$681$	63.1769i	2.42094i
$682$	38.7846	1.48514
$683$	31.0718i	1.18893i	0.804122	+	0.594465i	$0.202636\pi$
$683$	31.0718i	1.18893i	−0.804122	+	0.594465i	$0.797364\pi$
$684$	− 5.66025i	− 0.216425i
$685$	1.73205i	0.0661783i
$686$	6.92820i	0.264520i
$687$	−19.1244	−0.729640
$688$	− 3.46410i	− 0.132068i
$689$	8.78461i	0.334667i
$690$	− 0.535898i	− 0.0204013i
$691$	16.0526	0.610668	0.305334	−	0.952245i	$-0.401232\pi$
$691$	16.0526	0.610668	0.305334	+	0.952245i	$0.401232\pi$
$692$	9.92820	0.377414
$693$	−73.1769	−2.77976
$694$	12.7321	0.483302
$695$	− 3.37307i	− 0.127948i
$696$	−2.19615	−0.0832449
$697$	11.1962i	0.424085i
$698$	− 8.07180i	− 0.305522i
$699$	29.6603	1.12185
$700$	17.0718	0.645253
$701$	− 5.85641i	− 0.221193i	−0.993865	−	0.110597i	$-0.964724\pi$
$701$	− 5.85641i	− 0.221193i	0.993865	−	0.110597i	$-0.0352762\pi$
$702$	13.8564	0.522976
$703$	0	0
$704$	33.1244	1.24842
$705$	− 1.60770i	− 0.0605493i
$706$	12.2679	0.461710
$707$	−31.1769	−1.17253
$708$	36.7846i	1.38245i
$709$	− 30.0000i	− 1.12667i	−0.826227	−	0.563337i	$-0.809517\pi$
$709$	− 30.0000i	− 1.12667i	0.826227	−	0.563337i	$-0.190483\pi$
$710$	−1.60770	−0.0603357
$711$	− 16.9808i	− 0.636828i
$712$	27.5885	1.03392
$713$	−6.00000	−0.224702
$714$	−35.3205	−1.32184
$715$	4.39230	0.164263
$716$	1.46410i	0.0547160i
$717$	− 46.2487i	− 1.72719i
$718$	− 16.3923i	− 0.611755i
$719$	26.4449	0.986227	0.493114	−	0.869965i	$-0.335859\pi$
$719$	26.4449	0.986227	0.493114	+	0.869965i	$0.335859\pi$
$720$	1.19615i	0.0445780i
$721$	− 44.7846i	− 1.66787i
$722$	− 17.3923i	− 0.647275i
$723$	− 42.2487i	− 1.57125i
$724$	9.92820	0.368979
$725$	1.32051i	0.0490424i
$726$	31.1244i	1.15513i
$727$	1.85641i	0.0688503i	0.999407	+	0.0344252i	$0.0109600\pi$
$727$	1.85641i	0.0688503i	−0.999407	+	0.0344252i	$0.989040\pi$
$728$	−36.0000	−1.33425
$729$	−43.7846	−1.62165
$730$	0	0
$731$	−12.9282	−0.478167
$732$	11.6603i	0.430975i
$733$	28.7846	1.06318	0.531592	−	0.847001i	$-0.321594\pi$
$733$	28.7846	1.06318	0.531592	+	0.847001i	$0.321594\pi$
$734$	20.3923i	0.752694i
$735$	3.66025i	0.135011i
$736$	−3.66025	−0.134919
$737$	34.3923	1.26686
$738$	− 13.3923i	− 0.492978i
$739$	−23.8038	−0.875639	−0.437819	−	0.899063i	$-0.644249\pi$
$739$	−23.8038	−0.875639	−0.437819	+	0.899063i	$0.644249\pi$
$740$	0	0
$741$	12.0000	0.440831
$742$	8.78461i	0.322493i
$743$	−43.2679	−1.58735	−0.793674	−	0.608344i	$-0.791834\pi$
$743$	−43.2679	−1.58735	−0.793674	+	0.608344i	$0.791834\pi$
$744$	67.1769	2.46283
$745$	− 1.48334i	− 0.0543454i
$746$	− 17.5359i	− 0.642035i
$747$	−36.5885	−1.33870
$748$	− 17.6603i	− 0.645723i
$749$	−44.7846	−1.63639
$750$	7.26795	0.265388
$751$	−9.21539	−0.336274	−0.168137	−	0.985764i	$-0.553775\pi$
$751$	−9.21539	−0.336274	−0.168137	+	0.985764i	$0.553775\pi$
$752$	2.19615	0.0800854
$753$	− 70.2487i	− 2.56001i
$754$	− 0.928203i	− 0.0338032i
$755$	− 4.44486i	− 0.161765i
$756$	−13.8564	−0.503953
$757$	42.3731i	1.54008i	0.637998	+	0.770038i	$0.279763\pi$
$757$	42.3731i	1.54008i	−0.637998	+	0.770038i	$0.720237\pi$
$758$	− 8.53590i	− 0.310038i
$759$	− 9.46410i	− 0.343525i
$760$	1.01924i	0.0369716i
$761$	−3.67949	−0.133381	−0.0666907	−	0.997774i	$-0.521244\pi$
$761$	−3.67949	−0.133381	−0.0666907	+	0.997774i	$0.521244\pi$
$762$	− 33.3205i	− 1.20707i
$763$	− 23.5692i	− 0.853263i
$764$	− 6.19615i	− 0.224169i
$765$	4.46410	0.161400
$766$	15.8564	0.572915
$767$	−46.6410	−1.68411
$768$	46.4449	1.67593
$769$	20.5359i	0.740543i	0.928923	+	0.370272i	$0.120735\pi$
$769$	20.5359i	0.740543i	−0.928923	+	0.370272i	$0.879265\pi$
$770$	4.39230	0.158288
$771$	− 53.5167i	− 1.92736i
$772$	− 14.6603i	− 0.527634i
$773$	−48.0333	−1.72764	−0.863819	−	0.503802i	$-0.831934\pi$
$773$	−48.0333	−1.72764	−0.863819	+	0.503802i	$0.831934\pi$
$774$	15.4641	0.555846
$775$	− 40.3923i	− 1.45093i
$776$	−12.8038	−0.459631
$777$	0	0
$778$	−31.1962	−1.11844
$779$	− 3.80385i	− 0.136287i
$780$	2.53590	0.0907997
$781$	−28.3923	−1.01596
$782$	− 2.73205i	− 0.0976979i
$783$	− 1.07180i	− 0.0383029i
$784$	−5.00000	−0.178571
$785$	3.83717i	0.136954i
$786$	28.9282	1.03183
$787$	0.392305	0.0139842	0.00699208	−	0.999976i	$-0.497774\pi$
$787$	0.392305	0.0139842	0.00699208	+	0.999976i	$0.497774\pi$
$788$	27.9282	0.994901
$789$	20.7846	0.739952
$790$	1.01924i	0.0362629i
$791$	− 15.7128i	− 0.558683i
$792$	63.3731i	2.25186i
$793$	−14.7846	−0.525017
$794$	21.2487i	0.754089i
$795$	− 1.85641i	− 0.0658400i
$796$	− 11.0718i	− 0.392429i
$797$	− 8.00000i	− 0.283375i	−0.989911	−	0.141687i	$-0.954747\pi$
$797$	− 8.00000i	− 0.283375i	0.989911	−	0.141687i	$-0.0452527\pi$
$798$	12.0000	0.424795
$799$	− 8.19615i	− 0.289959i
$800$	− 24.6410i	− 0.871191i
$801$	41.0526i	1.45052i
$802$	−5.60770	−0.198015
$803$	0	0
$804$	19.8564	0.700281
$805$	−0.679492	−0.0239489
$806$	28.3923i	1.00008i
$807$	−39.7128	−1.39796
$808$	27.0000i	0.949857i
$809$	− 33.8564i	− 1.19033i	−0.803604	−	0.595164i	$-0.797087\pi$
$809$	− 33.8564i	− 1.19033i	0.803604	−	0.595164i	$-0.202913\pi$
$810$	0.660254	0.0231990
$811$	18.2487	0.640799	0.320399	−	0.947283i	$-0.396183\pi$
$811$	18.2487	0.640799	0.320399	+	0.947283i	$0.396183\pi$
$812$	0.928203i	0.0325735i
$813$	31.8564	1.11725
$814$	0	0
$815$	6.00000	0.210171
$816$	10.1962i	0.356937i
$817$	4.39230	0.153667
$818$	8.66025	0.302799
$819$	− 53.5692i	− 1.87186i
$820$	− 0.803848i	− 0.0280716i
$821$	49.1769	1.71629	0.858143	−	0.513411i	$-0.171618\pi$
$821$	49.1769	1.71629	0.858143	+	0.513411i	$0.171618\pi$
$822$	− 17.6603i	− 0.615972i
$823$	−3.01924	−0.105244	−0.0526220	−	0.998615i	$-0.516758\pi$
$823$	−3.01924	−0.105244	−0.0526220	+	0.998615i	$0.516758\pi$
$824$	−38.7846	−1.35113
$825$	63.7128	2.21820
$826$	−46.6410	−1.62285
$827$	22.9282i	0.797292i	0.917105	+	0.398646i	$0.130520\pi$
$827$	22.9282i	0.797292i	−0.917105	+	0.398646i	$0.869480\pi$
$828$	− 3.26795i	− 0.113569i
$829$	− 44.5359i	− 1.54680i	−0.633921	−	0.773398i	$-0.718556\pi$
$829$	− 44.5359i	− 1.54680i	0.633921	−	0.773398i	$-0.281444\pi$
$830$	2.19615	0.0762296
$831$	− 28.0526i	− 0.973132i
$832$	24.2487i	0.840673i
$833$	18.6603i	0.646539i
$834$	34.3923i	1.19091i
$835$	−2.92820	−0.101335
$836$	6.00000i	0.207514i
$837$	32.7846i	1.13320i
$838$	− 17.3205i	− 0.598327i
$839$	−29.6603	−1.02399	−0.511993	−	0.858990i	$-0.671093\pi$
$839$	−29.6603	−1.02399	−0.511993	+	0.858990i	$0.671093\pi$
$840$	7.60770	0.262490
$841$	28.9282	0.997524
$842$	16.2679	0.560631
$843$	− 90.6936i	− 3.12365i
$844$	−10.0000	−0.344214
$845$	− 0.267949i	− 0.00921773i
$846$	9.80385i	0.337063i
$847$	39.4641	1.35600
$848$	2.53590	0.0870831
$849$	28.3923i	0.974421i
$850$	18.3923	0.630851
$851$	0	0
$852$	−16.3923	−0.561591
$853$	42.1244i	1.44231i	0.692774	+	0.721155i	$0.256389\pi$
$853$	42.1244i	1.44231i	−0.692774	+	0.721155i	$0.743611\pi$
$854$	−14.7846	−0.505919
$855$	−1.51666	−0.0518687
$856$	38.7846i	1.32563i
$857$	23.7321i	0.810671i	0.914168	+	0.405336i	$0.132845\pi$
$857$	23.7321i	0.810671i	−0.914168	+	0.405336i	$0.867155\pi$
$858$	−44.7846	−1.52892
$859$	− 8.53590i	− 0.291241i	−0.989341	−	0.145621i	$-0.953482\pi$
$859$	− 8.53590i	− 0.291241i	0.989341	−	0.145621i	$-0.0465179\pi$
$860$	0.928203	0.0316515
$861$	−28.3923	−0.967607
$862$	11.5167	0.392259
$863$	38.5359	1.31178	0.655889	−	0.754858i	$-0.272294\pi$
$863$	38.5359	1.31178	0.655889	+	0.754858i	$0.272294\pi$
$864$	20.0000i	0.680414i
$865$	− 2.66025i	− 0.0904514i
$866$	− 27.0000i	− 0.917497i
$867$	−8.39230	−0.285018
$868$	− 28.3923i	− 0.963698i
$869$	18.0000i	0.610608i
$870$	0.196152i	0.00665019i
$871$	25.1769i	0.853087i
$872$	−20.4115	−0.691222
$873$	− 19.0526i	− 0.644831i
$874$	0.928203i	0.0313969i
$875$	− 9.21539i	− 0.311537i
$876$	0	0
$877$	5.78461	0.195332	0.0976662	−	0.995219i	$-0.468862\pi$
$877$	5.78461	0.195332	0.0976662	+	0.995219i	$0.468862\pi$
$878$	−14.5359	−0.490563
$879$	−78.8372	−2.65911
$880$	− 1.26795i	− 0.0427426i
$881$	32.3205	1.08891	0.544453	−	0.838791i	$-0.316737\pi$
$881$	32.3205	1.08891	0.544453	+	0.838791i	$0.316737\pi$
$882$	− 22.3205i	− 0.751571i
$883$	35.6603i	1.20006i	0.799976	+	0.600032i	$0.204845\pi$
$883$	35.6603i	1.20006i	−0.799976	+	0.600032i	$0.795155\pi$
$884$	12.9282	0.434823
$885$	9.85641	0.331319
$886$	− 33.4641i	− 1.12425i
$887$	28.9808	0.973079	0.486539	−	0.873659i	$-0.338259\pi$
$887$	28.9808	0.973079	0.486539	+	0.873659i	$0.338259\pi$
$888$	0	0
$889$	−42.2487	−1.41698
$890$	− 2.46410i	− 0.0825969i
$891$	11.6603	0.390633
$892$	4.58846	0.153633
$893$	2.78461i	0.0931834i
$894$	15.1244i	0.505834i
$895$	0.392305	0.0131133
$896$	− 10.3923i	− 0.347183i
$897$	6.92820	0.231326
$898$	32.2487	1.07615
$899$	2.19615	0.0732458
$900$	22.0000	0.733333
$901$	− 9.46410i	− 0.315295i
$902$	14.1962i	0.472680i
$903$	− 32.7846i	− 1.09100i
$904$	−13.6077	−0.452585
$905$	− 2.66025i	− 0.0884298i
$906$	45.3205i	1.50567i
$907$	− 1.26795i	− 0.0421016i	−0.999778	−	0.0210508i	$-0.993299\pi$
$907$	− 1.26795i	− 0.0421016i	0.999778	−	0.0210508i	$-0.00670117\pi$
$908$	− 23.1244i	− 0.767409i
$909$	−40.1769	−1.33258
$910$	3.21539i	0.106589i
$911$	− 53.5167i	− 1.77309i	−0.462646	−	0.886543i	$-0.653100\pi$
$911$	− 53.5167i	− 1.77309i	0.462646	−	0.886543i	$-0.346900\pi$
$912$	− 3.46410i	− 0.114708i
$913$	38.7846	1.28358
$914$	36.1244	1.19489
$915$	3.12436	0.103288
$916$	7.00000	0.231287
$917$	− 36.6795i	− 1.21126i
$918$	−14.9282	−0.492704
$919$	4.05256i	0.133682i	0.997764	+	0.0668408i	$0.0212920\pi$
$919$	4.05256i	0.133682i	−0.997764	+	0.0668408i	$0.978708\pi$
$920$	0.588457i	0.0194009i
$921$	39.3205	1.29565
$922$	−16.5359	−0.544581
$923$	− 20.7846i	− 0.684134i
$924$	44.7846	1.47331
$925$	0	0
$926$	15.4641	0.508182
$927$	− 57.7128i	− 1.89554i
$928$	1.33975	0.0439793
$929$	−30.4641	−0.999495	−0.499747	−	0.866171i	$-0.666574\pi$
$929$	−30.4641	−0.999495	−0.499747	+	0.866171i	$0.666574\pi$
$930$	− 6.00000i	− 0.196748i
$931$	− 6.33975i	− 0.207777i
$932$	−10.8564	−0.355613
$933$	83.0333i	2.71839i
$934$	−3.32051	−0.108650
$935$	−4.73205	−0.154755
$936$	−46.3923	−1.51638
$937$	−12.6077	−0.411875	−0.205938	−	0.978565i	$-0.566024\pi$
$937$	−12.6077	−0.411875	−0.205938	+	0.978565i	$0.566024\pi$
$938$	25.1769i	0.822055i
$939$	42.5885i	1.38982i
$940$	0.588457i	0.0191934i
$941$	−4.17691	−0.136164	−0.0680818	−	0.997680i	$-0.521688\pi$
$941$	−4.17691	−0.136164	−0.0680818	+	0.997680i	$0.521688\pi$
$942$	− 39.1244i	− 1.27474i
$943$	− 2.19615i	− 0.0715166i
$944$	13.4641i	0.438219i
$945$	3.71281i	0.120778i
$946$	−16.3923	−0.532960
$947$	− 27.6077i	− 0.897130i	−0.893750	−	0.448565i	$-0.851935\pi$
$947$	− 27.6077i	− 0.897130i	0.893750	−	0.448565i	$-0.148065\pi$
$948$	10.3923i	0.337526i
$949$	0	0
$950$	−6.24871	−0.202735
$951$	43.5167	1.41112
$952$	38.7846	1.25702
$953$	−4.14359	−0.134224	−0.0671121	−	0.997745i	$-0.521379\pi$
$953$	−4.14359	−0.134224	−0.0671121	+	0.997745i	$0.521379\pi$
$954$	11.3205i	0.366515i
$955$	−1.66025	−0.0537246
$956$	16.9282i	0.547497i
$957$	3.46410i	0.111979i
$958$	−39.3205	−1.27039
$959$	−22.3923	−0.723085
$960$	− 5.12436i	− 0.165388i
$961$	−36.1769	−1.16700
$962$	0	0
$963$	−57.7128	−1.85977
$964$	15.4641i	0.498065i
$965$	−3.92820	−0.126453
$966$	6.92820	0.222911
$967$	− 32.7846i	− 1.05428i	−0.849778	−	0.527141i	$-0.823264\pi$
$967$	− 32.7846i	− 1.05428i	0.849778	−	0.527141i	$-0.176736\pi$
$968$	− 34.1769i	− 1.09849i
$969$	−12.9282	−0.415314
$970$	1.14359i	0.0367186i
$971$	−8.19615	−0.263027	−0.131514	−	0.991314i	$-0.541984\pi$
$971$	−8.19615	−0.263027	−0.131514	+	0.991314i	$0.541984\pi$
$972$	18.7321	0.600831
$973$	43.6077	1.39800
$974$	32.4449	1.03960
$975$	46.6410i	1.49371i
$976$	4.26795i	0.136614i
$977$	− 16.2487i	− 0.519842i	−0.965630	−	0.259921i	$-0.916303\pi$
$977$	− 16.2487i	− 0.519842i	0.965630	−	0.259921i	$-0.0836966\pi$
$978$	−61.1769	−1.95622
$979$	− 43.5167i	− 1.39080i
$980$	− 1.33975i	− 0.0427966i
$981$	− 30.3731i	− 0.969737i
$982$	− 31.2679i	− 0.997800i
$983$	57.1244	1.82198	0.910992	−	0.412424i	$-0.135318\pi$
$983$	57.1244	1.82198	0.910992	+	0.412424i	$0.135318\pi$
$984$	24.5885i	0.783851i
$985$	− 7.48334i	− 0.238439i
$986$	1.00000i	0.0318465i
$987$	20.7846	0.661581
$988$	−4.39230	−0.139738
$989$	2.53590	0.0806369
$990$	5.66025	0.179895
$991$	− 46.3923i	− 1.47370i	−0.676056	−	0.736850i	$-0.736312\pi$
$991$	− 46.3923i	− 1.47370i	0.676056	−	0.736850i	$-0.263688\pi$
$992$	−40.9808	−1.30114
$993$	− 89.5692i	− 2.84239i
$994$	− 20.7846i	− 0.659248i
$995$	−2.96668	−0.0940500
$996$	22.3923	0.709527
$997$	20.5359i	0.650378i	0.945649	+	0.325189i	$0.105428\pi$
$997$	20.5359i	0.650378i	−0.945649	+	0.325189i	$0.894572\pi$
$998$	−12.5885	−0.398481
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1369.2.b.d.1368.1		4
37.6	odd	4		1369.2.a.g.1.2		2
37.10	even	3		37.2.e.a.11.1	✓	4
37.11	even	6		37.2.e.a.27.1	yes	4
37.31	odd	4		1369.2.a.h.1.2		2
37.36	even	2	inner	1369.2.b.d.1368.3		4
111.11	odd	6		333.2.s.b.64.2		4
111.47	odd	6		333.2.s.b.307.2		4
148.11	odd	6		592.2.w.c.545.1		4
148.47	odd	6		592.2.w.c.529.1		4
185.47	odd	12		925.2.m.a.899.1		4
185.48	odd	12		925.2.m.a.249.1		4
185.84	even	6		925.2.n.a.751.2		4
185.122	odd	12		925.2.m.b.249.2		4
185.158	odd	12		925.2.m.b.899.2		4
185.159	even	6		925.2.n.a.101.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
37.2.e.a.11.1	✓	4	37.10	even	3
37.2.e.a.27.1	yes	4	37.11	even	6
333.2.s.b.64.2		4	111.11	odd	6
333.2.s.b.307.2		4	111.47	odd	6
592.2.w.c.529.1		4	148.47	odd	6
592.2.w.c.545.1		4	148.11	odd	6
925.2.m.a.249.1		4	185.48	odd	12
925.2.m.a.899.1		4	185.47	odd	12
925.2.m.b.249.2		4	185.122	odd	12
925.2.m.b.899.2		4	185.158	odd	12
925.2.n.a.101.2		4	185.159	even	6
925.2.n.a.751.2		4	185.84	even	6
1369.2.a.g.1.2		2	37.6	odd	4
1369.2.a.h.1.2		2	37.31	odd	4
1369.2.b.d.1368.1		4	1.1	even	1	trivial
1369.2.b.d.1368.3		4	37.36	even	2	inner

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:	\( N \)	\(=\)	\( 1369 = 37^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1369.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -2.73205 q^{3} +1.00000 q^{4} -0.267949i q^{5} +2.73205i q^{6} +3.46410 q^{7} -3.00000i q^{8} +4.46410 q^{9} +O(q^{10})\)	\(q-1.00000i q^{2} -2.73205 q^{3} +1.00000 q^{4} -0.267949i q^{5} +2.73205i q^{6} +3.46410 q^{7} -3.00000i q^{8} +4.46410 q^{9} -0.267949 q^{10} -4.73205 q^{11} -2.73205 q^{12} -3.46410i q^{13} -3.46410i q^{14} +0.732051i q^{15} -1.00000 q^{16} +3.73205i q^{17} -4.46410i q^{18} -1.26795i q^{19} -0.267949i q^{20} -9.46410 q^{21} +4.73205i q^{22} -0.732051i q^{23} +8.19615i q^{24} +4.92820 q^{25} -3.46410 q^{26} -4.00000 q^{27} +3.46410 q^{28} +0.267949i q^{29} +0.732051 q^{30} -8.19615i q^{31} -5.00000i q^{32} +12.9282 q^{33} +3.73205 q^{34} -0.928203i q^{35} +4.46410 q^{36} -1.26795 q^{38} +9.46410i q^{39} -0.803848 q^{40} +3.00000 q^{41} +9.46410i q^{42} +3.46410i q^{43} -4.73205 q^{44} -1.19615i q^{45} -0.732051 q^{46} -2.19615 q^{47} +2.73205 q^{48} +5.00000 q^{49} -4.92820i q^{50} -10.1962i q^{51} -3.46410i q^{52} -2.53590 q^{53} +4.00000i q^{54} +1.26795i q^{55} -10.3923i q^{56} +3.46410i q^{57} +0.267949 q^{58} -13.4641i q^{59} +0.732051i q^{60} -4.26795i q^{61} -8.19615 q^{62} +15.4641 q^{63} -7.00000 q^{64} -0.928203 q^{65} -12.9282i q^{66} -7.26795 q^{67} +3.73205i q^{68} +2.00000i q^{69} -0.928203 q^{70} +6.00000 q^{71} -13.3923i q^{72} -13.4641 q^{75} -1.26795i q^{76} -16.3923 q^{77} +9.46410 q^{78} -3.80385i q^{79} +0.267949i q^{80} -2.46410 q^{81} -3.00000i q^{82} -8.19615 q^{83} -9.46410 q^{84} +1.00000 q^{85} +3.46410 q^{86} -0.732051i q^{87} +14.1962i q^{88} +9.19615i q^{89} -1.19615 q^{90} -12.0000i q^{91} -0.732051i q^{92} +22.3923i q^{93} +2.19615i q^{94} -0.339746 q^{95} +13.6603i q^{96} -4.26795i q^{97} -5.00000i q^{98} -21.1244 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{3} + 4 q^{4} + 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^3 + 4 * q^4 + 4 * q^9	\( 4 q - 4 q^{3} + 4 q^{4} + 4 q^{9} - 8 q^{10} - 12 q^{11} - 4 q^{12} - 4 q^{16} - 24 q^{21} - 8 q^{25} - 16 q^{27} - 4 q^{30} + 24 q^{33} + 8 q^{34} + 4 q^{36} - 12 q^{38} - 24 q^{40} + 12 q^{41} - 12 q^{44} + 4 q^{46} + 12 q^{47} + 4 q^{48} + 20 q^{49} - 24 q^{53} + 8 q^{58} - 12 q^{62} + 48 q^{63} - 28 q^{64} + 24 q^{65} - 36 q^{67} + 24 q^{70} + 24 q^{71} - 40 q^{75} - 24 q^{77} + 24 q^{78} + 4 q^{81} - 12 q^{83} - 24 q^{84} + 4 q^{85} + 16 q^{90} - 36 q^{95} - 36 q^{99}+O(q^{100}) \) 4 * q - 4 * q^3 + 4 * q^4 + 4 * q^9 - 8 * q^10 - 12 * q^11 - 4 * q^12 - 4 * q^16 - 24 * q^21 - 8 * q^25 - 16 * q^27 - 4 * q^30 + 24 * q^33 + 8 * q^34 + 4 * q^36 - 12 * q^38 - 24 * q^40 + 12 * q^41 - 12 * q^44 + 4 * q^46 + 12 * q^47 + 4 * q^48 + 20 * q^49 - 24 * q^53 + 8 * q^58 - 12 * q^62 + 48 * q^63 - 28 * q^64 + 24 * q^65 - 36 * q^67 + 24 * q^70 + 24 * q^71 - 40 * q^75 - 24 * q^77 + 24 * q^78 + 4 * q^81 - 12 * q^83 - 24 * q^84 + 4 * q^85 + 16 * q^90 - 36 * q^95 - 36 * q^99

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