LMFDB - Embedded newform 1369.2.b.d.1368.2

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			1368.2
Root			$0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	1369.1368
Dual form			1369.2.b.d.1368.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} +0.732051 q^{3} +1.00000 q^{4} -3.73205i q^{5} -0.732051i q^{6} -3.46410 q^{7} -3.00000i q^{8} -2.46410 q^{9} +O(q^{10})$	\(q-1.00000i q^{2} +0.732051 q^{3} +1.00000 q^{4} -3.73205i q^{5} -0.732051i q^{6} -3.46410 q^{7} -3.00000i q^{8} -2.46410 q^{9} -3.73205 q^{10} -1.26795 q^{11} +0.732051 q^{12} +3.46410i q^{13} +3.46410i q^{14} -2.73205i q^{15} -1.00000 q^{16} +0.267949i q^{17} +2.46410i q^{18} -4.73205i q^{19} -3.73205i q^{20} -2.53590 q^{21} +1.26795i q^{22} +2.73205i q^{23} -2.19615i q^{24} -8.92820 q^{25} +3.46410 q^{26} -4.00000 q^{27} -3.46410 q^{28} +3.73205i q^{29} -2.73205 q^{30} +2.19615i q^{31} -5.00000i q^{32} -0.928203 q^{33} +0.267949 q^{34} +12.9282i q^{35} -2.46410 q^{36} -4.73205 q^{38} +2.53590i q^{39} -11.1962 q^{40} +3.00000 q^{41} +2.53590i q^{42} -3.46410i q^{43} -1.26795 q^{44} +9.19615i q^{45} +2.73205 q^{46} +8.19615 q^{47} -0.732051 q^{48} +5.00000 q^{49} +8.92820i q^{50} +0.196152i q^{51} +3.46410i q^{52} -9.46410 q^{53} +4.00000i q^{54} +4.73205i q^{55} +10.3923i q^{56} -3.46410i q^{57} +3.73205 q^{58} -6.53590i q^{59} -2.73205i q^{60} -7.73205i q^{61} +2.19615 q^{62} +8.53590 q^{63} -7.00000 q^{64} +12.9282 q^{65} +0.928203i q^{66} -10.7321 q^{67} +0.267949i q^{68} +2.00000i q^{69} +12.9282 q^{70} +6.00000 q^{71} +7.39230i q^{72} -6.53590 q^{75} -4.73205i q^{76} +4.39230 q^{77} +2.53590 q^{78} -14.1962i q^{79} +3.73205i q^{80} +4.46410 q^{81} -3.00000i q^{82} +2.19615 q^{83} -2.53590 q^{84} +1.00000 q^{85} -3.46410 q^{86} +2.73205i q^{87} +3.80385i q^{88} -1.19615i q^{89} +9.19615 q^{90} -12.0000i q^{91} +2.73205i q^{92} +1.60770i q^{93} -8.19615i q^{94} -17.6603 q^{95} -3.66025i q^{96} -7.73205i q^{97} -5.00000i q^{98} +3.12436 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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	$-0.115027\pi$
$3$	0.732051	0.422650	0.211325	−	0.977416i	$-0.432222\pi$
$3$	0.732051	0.422650	0.211325	+	0.977416i	$0.432222\pi$
$4$	1.00000	0.500000
$5$	− 3.73205i	− 1.66902i	−0.550990	−	0.834512i	$-0.685750\pi$
$5$	− 3.73205i	− 1.66902i	0.550990	−	0.834512i	$-0.314250\pi$
$6$	− 0.732051i	− 0.298858i
$7$	−3.46410	−1.30931	−0.654654	−	0.755929i	$-0.727186\pi$
$7$	−3.46410	−1.30931	−0.654654	+	0.755929i	$0.727186\pi$
$8$	− 3.00000i	− 1.06066i
$9$	−2.46410	−0.821367
$10$	−3.73205	−1.18018
$11$	−1.26795	−0.382301	−0.191151	−	0.981561i	$-0.561222\pi$
$11$	−1.26795	−0.382301	−0.191151	+	0.981561i	$0.561222\pi$
$12$	0.732051	0.211325
$13$	3.46410i	0.960769i	0.877058	+	0.480384i	$0.159503\pi$
$13$	3.46410i	0.960769i	−0.877058	+	0.480384i	$0.840497\pi$
$14$	3.46410i	0.925820i
$15$	− 2.73205i	− 0.705412i
$16$	−1.00000	−0.250000
$17$	0.267949i	0.0649872i	0.999472	+	0.0324936i	$0.0103449\pi$
$17$	0.267949i	0.0649872i	−0.999472	+	0.0324936i	$0.989655\pi$
$18$	2.46410i	0.580794i
$19$	− 4.73205i	− 1.08561i	−0.839860	−	0.542803i	$-0.817363\pi$
$19$	− 4.73205i	− 1.08561i	0.839860	−	0.542803i	$-0.182637\pi$
$20$	− 3.73205i	− 0.834512i
$21$	−2.53590	−0.553378
$22$	1.26795i	0.270328i
$23$	2.73205i	0.569672i	0.958576	+	0.284836i	$0.0919391\pi$
$23$	2.73205i	0.569672i	−0.958576	+	0.284836i	$0.908061\pi$
$24$	− 2.19615i	− 0.448288i
$25$	−8.92820	−1.78564
$26$	3.46410	0.679366
$27$	−4.00000	−0.769800
$28$	−3.46410	−0.654654
$29$	3.73205i	0.693024i	0.938045	+	0.346512i	$0.112634\pi$
$29$	3.73205i	0.693024i	−0.938045	+	0.346512i	$0.887366\pi$
$30$	−2.73205	−0.498802
$31$	2.19615i	0.394441i	0.980359	+	0.197220i	$0.0631914\pi$
$31$	2.19615i	0.394441i	−0.980359	+	0.197220i	$0.936809\pi$
$32$	− 5.00000i	− 0.883883i
$33$	−0.928203	−0.161579
$34$	0.267949	0.0459529
$35$	12.9282i	2.18527i
$36$	−2.46410	−0.410684
$37$	0	0
$37$	0	0
$38$	−4.73205	−0.767640
$39$	2.53590i	0.406069i
$40$	−11.1962	−1.77027
$41$	3.00000	0.468521	0.234261	−	0.972174i	$-0.424733\pi$
$41$	3.00000	0.468521	0.234261	+	0.972174i	$0.424733\pi$
$42$	2.53590i	0.391298i
$43$	− 3.46410i	− 0.528271i	−0.964486	−	0.264135i	$-0.914913\pi$
$43$	− 3.46410i	− 0.528271i	0.964486	−	0.264135i	$-0.0850865\pi$
$44$	−1.26795	−0.191151
$45$	9.19615i	1.37088i
$46$	2.73205	0.402819
$47$	8.19615	1.19553	0.597766	−	0.801671i	$-0.296055\pi$
$47$	8.19615	1.19553	0.597766	+	0.801671i	$0.296055\pi$
$48$	−0.732051	−0.105662
$49$	5.00000	0.714286
$50$	8.92820i	1.26264i
$51$	0.196152i	0.0274668i
$52$	3.46410i	0.480384i
$53$	−9.46410	−1.29999	−0.649997	−	0.759937i	$-0.725230\pi$
$53$	−9.46410	−1.29999	−0.649997	+	0.759937i	$0.725230\pi$
$54$	4.00000i	0.544331i
$55$	4.73205i	0.638070i
$56$	10.3923i	1.38873i
$57$	− 3.46410i	− 0.458831i
$58$	3.73205	0.490042
$59$	− 6.53590i	− 0.850901i	−0.904982	−	0.425451i	$-0.860116\pi$
$59$	− 6.53590i	− 0.850901i	0.904982	−	0.425451i	$-0.139884\pi$
$60$	− 2.73205i	− 0.352706i
$61$	− 7.73205i	− 0.989988i	−0.868896	−	0.494994i	$-0.835170\pi$
$61$	− 7.73205i	− 0.989988i	0.868896	−	0.494994i	$-0.164830\pi$
$62$	2.19615	0.278912
$63$	8.53590	1.07542
$64$	−7.00000	−0.875000
$65$	12.9282	1.60355
$66$	0.928203i	0.114254i
$67$	−10.7321	−1.31113	−0.655564	−	0.755139i	$-0.727569\pi$
$67$	−10.7321	−1.31113	−0.655564	+	0.755139i	$0.727569\pi$
$68$	0.267949i	0.0324936i
$69$	2.00000i	0.240772i
$70$	12.9282	1.54522
$71$	6.00000	0.712069	0.356034	−	0.934473i	$-0.384129\pi$
$71$	6.00000	0.712069	0.356034	+	0.934473i	$0.384129\pi$
$72$	7.39230i	0.871191i
$73$	0	0		−	1.00000i	$-0.5\pi$
$73$	0	0			1.00000i	$0.5\pi$
$74$	0	0
$75$	−6.53590	−0.754701
$76$	− 4.73205i	− 0.542803i
$77$	4.39230	0.500550
$78$	2.53590	0.287134
$79$	− 14.1962i	− 1.59719i	−0.601867	−	0.798596i	$-0.705576\pi$
$79$	− 14.1962i	− 1.59719i	0.601867	−	0.798596i	$-0.294424\pi$
$80$	3.73205i	0.417256i
$81$	4.46410	0.496011
$82$	− 3.00000i	− 0.331295i
$83$	2.19615	0.241059	0.120530	−	0.992710i	$-0.461541\pi$
$83$	2.19615	0.241059	0.120530	+	0.992710i	$0.461541\pi$
$84$	−2.53590	−0.276689
$85$	1.00000	0.108465
$86$	−3.46410	−0.373544
$87$	2.73205i	0.292907i
$88$	3.80385i	0.405492i
$89$	− 1.19615i	− 0.126792i	−0.997988	−	0.0633960i	$-0.979807\pi$
$89$	− 1.19615i	− 0.126792i	0.997988	−	0.0633960i	$-0.0201931\pi$
$90$	9.19615	0.969360
$91$	− 12.0000i	− 1.25794i
$92$	2.73205i	0.284836i
$93$	1.60770i	0.166710i
$94$	− 8.19615i	− 0.845369i
$95$	−17.6603	−1.81190
$96$	− 3.66025i	− 0.373573i
$97$	− 7.73205i	− 0.785071i	−0.919737	−	0.392535i	$-0.871598\pi$
$97$	− 7.73205i	− 0.785071i	0.919737	−	0.392535i	$-0.128402\pi$
$98$	− 5.00000i	− 0.505076i
$99$	3.12436	0.314010
$100$	−8.92820	−0.892820
$101$	−9.00000	−0.895533	−0.447767	−	0.894150i	$-0.647781\pi$
$101$	−9.00000	−0.895533	−0.447767	+	0.894150i	$0.647781\pi$
$102$	0.196152	0.0194220
$103$	0.928203i	0.0914586i	0.998954	+	0.0457293i	$0.0145612\pi$
$103$	0.928203i	0.0914586i	−0.998954	+	0.0457293i	$0.985439\pi$
$104$	10.3923	1.01905
$105$	9.46410i	0.923602i
$106$	9.46410i	0.919235i
$107$	0.928203	0.0897328	0.0448664	−	0.998993i	$-0.485714\pi$
$107$	0.928203	0.0897328	0.0448664	+	0.998993i	$0.485714\pi$
$108$	−4.00000	−0.384900
$109$	− 17.1962i	− 1.64709i	−0.567249	−	0.823546i	$-0.691992\pi$
$109$	− 17.1962i	− 1.64709i	0.567249	−	0.823546i	$-0.308008\pi$
$110$	4.73205	0.451183
$111$	0	0
$112$	3.46410	0.327327
$113$	− 11.4641i	− 1.07845i	−0.842161	−	0.539226i	$-0.818717\pi$
$113$	− 11.4641i	− 1.07845i	0.842161	−	0.539226i	$-0.181283\pi$
$114$	−3.46410	−0.324443
$115$	10.1962	0.950796
$116$	3.73205i	0.346512i
$117$	− 8.53590i	− 0.789144i
$118$	−6.53590	−0.601678
$119$	− 0.928203i	− 0.0850883i
$120$	−8.19615	−0.748203
$121$	−9.39230	−0.853846
$122$	−7.73205	−0.700027
$123$	2.19615	0.198020
$124$	2.19615i	0.197220i
$125$	14.6603i	1.31125i
$126$	− 8.53590i	− 0.760438i
$127$	−1.80385	−0.160066	−0.0800328	−	0.996792i	$-0.525503\pi$
$127$	−1.80385	−0.160066	−0.0800328	+	0.996792i	$0.525503\pi$
$128$	− 3.00000i	− 0.265165i
$129$	− 2.53590i	− 0.223273i
$130$	− 12.9282i	− 1.13388i
$131$	20.5885i	1.79882i	0.437104	+	0.899411i	$0.356004\pi$
$131$	20.5885i	1.79882i	−0.437104	+	0.899411i	$0.643996\pi$
$132$	−0.928203	−0.0807897
$133$	16.3923i	1.42139i
$134$	10.7321i	0.927108i
$135$	14.9282i	1.28482i
$136$	0.803848	0.0689294
$137$	0.464102	0.0396509	0.0198254	−	0.999803i	$-0.493689\pi$
$137$	0.464102	0.0396509	0.0198254	+	0.999803i	$0.493689\pi$
$138$	2.00000	0.170251
$139$	−18.5885	−1.57665	−0.788326	−	0.615258i	$-0.789052\pi$
$139$	−18.5885	−1.57665	−0.788326	+	0.615258i	$0.789052\pi$
$140$	12.9282i	1.09263i
$141$	6.00000	0.505291
$142$	− 6.00000i	− 0.503509i
$143$	− 4.39230i	− 0.367303i
$144$	2.46410	0.205342
$145$	13.9282	1.15667
$146$	0	0
$147$	3.66025	0.301893
$148$	0	0
$149$	12.4641	1.02110	0.510549	−	0.859848i	$-0.329442\pi$
$149$	12.4641	1.02110	0.510549	+	0.859848i	$0.329442\pi$
$150$	6.53590i	0.533654i
$151$	−14.5885	−1.18719	−0.593596	−	0.804763i	$-0.702292\pi$
$151$	−14.5885	−1.18719	−0.593596	+	0.804763i	$0.702292\pi$
$152$	−14.1962	−1.15146
$153$	− 0.660254i	− 0.0533784i
$154$	− 4.39230i	− 0.353942i
$155$	8.19615	0.658331
$156$	2.53590i	0.203034i
$157$	20.3205	1.62175	0.810877	−	0.585217i	$-0.198991\pi$
$157$	20.3205	1.62175	0.810877	+	0.585217i	$0.198991\pi$
$158$	−14.1962	−1.12939
$159$	−6.92820	−0.549442
$160$	−18.6603	−1.47522
$161$	− 9.46410i	− 0.745876i
$162$	− 4.46410i	− 0.350733i
$163$	1.60770i	0.125924i	0.998016	+	0.0629622i	$0.0200548\pi$
$163$	1.60770i	0.125924i	−0.998016	+	0.0629622i	$0.979945\pi$
$164$	3.00000	0.234261
$165$	3.46410i	0.269680i
$166$	− 2.19615i	− 0.170454i
$167$	2.92820i	0.226591i	0.993561	+	0.113296i	$0.0361407\pi$
$167$	2.92820i	0.226591i	−0.993561	+	0.113296i	$0.963859\pi$
$168$	7.60770i	0.586946i
$169$	1.00000	0.0769231
$170$	− 1.00000i	− 0.0766965i
$171$	11.6603i	0.891682i
$172$	− 3.46410i	− 0.264135i
$173$	−3.92820	−0.298656	−0.149328	−	0.988788i	$-0.547711\pi$
$173$	−3.92820	−0.298656	−0.149328	+	0.988788i	$0.547711\pi$
$174$	2.73205	0.207116
$175$	30.9282	2.33795
$176$	1.26795	0.0955753
$177$	− 4.78461i	− 0.359633i
$178$	−1.19615	−0.0896554
$179$	− 5.46410i	− 0.408406i	−0.978929	−	0.204203i	$-0.934540\pi$
$179$	− 5.46410i	− 0.408406i	0.978929	−	0.204203i	$-0.0654603\pi$
$180$	9.19615i	0.685441i
$181$	−3.92820	−0.291981	−0.145991	−	0.989286i	$-0.546637\pi$
$181$	−3.92820	−0.291981	−0.145991	+	0.989286i	$0.546637\pi$
$182$	−12.0000	−0.889499
$183$	− 5.66025i	− 0.418418i
$184$	8.19615	0.604228
$185$	0	0
$186$	1.60770	0.117882
$187$	− 0.339746i	− 0.0248447i
$188$	8.19615	0.597766
$189$	13.8564	1.00791
$190$	17.6603i	1.28121i
$191$	4.19615i	0.303623i	0.988409	+	0.151811i	$0.0485107\pi$
$191$	4.19615i	0.303623i	−0.988409	+	0.151811i	$0.951489\pi$
$192$	−5.12436	−0.369819
$193$	2.66025i	0.191489i	0.995406	+	0.0957446i	$0.0305232\pi$
$193$	2.66025i	0.191489i	−0.995406	+	0.0957446i	$0.969477\pi$
$194$	−7.73205	−0.555129
$195$	9.46410	0.677738
$196$	5.00000	0.357143
$197$	14.0718	1.00257	0.501287	−	0.865281i	$-0.332860\pi$
$197$	14.0718	1.00257	0.501287	+	0.865281i	$0.332860\pi$
$198$	− 3.12436i	− 0.222038i
$199$	− 24.9282i	− 1.76711i	−0.468324	−	0.883557i	$-0.655142\pi$
$199$	− 24.9282i	− 1.76711i	0.468324	−	0.883557i	$-0.344858\pi$
$200$	26.7846i	1.89396i
$201$	−7.85641	−0.554148
$202$	9.00000i	0.633238i
$203$	− 12.9282i	− 0.907382i
$204$	0.196152i	0.0137334i
$205$	− 11.1962i	− 0.781973i
$206$	0.928203	0.0646710
$207$	− 6.73205i	− 0.467910i
$208$	− 3.46410i	− 0.240192i
$209$	6.00000i	0.415029i
$210$	9.46410	0.653085
$211$	−10.0000	−0.688428	−0.344214	−	0.938891i	$-0.611855\pi$
$211$	−10.0000	−0.688428	−0.344214	+	0.938891i	$0.611855\pi$
$212$	−9.46410	−0.649997
$213$	4.39230	0.300956
$214$	− 0.928203i	− 0.0634507i
$215$	−12.9282	−0.881696
$216$	12.0000i	0.816497i
$217$	− 7.60770i	− 0.516444i
$218$	−17.1962	−1.16467
$219$	0	0
$220$	4.73205i	0.319035i
$221$	−0.928203	−0.0624377
$222$	0	0
$223$	−26.5885	−1.78049	−0.890247	−	0.455477i	$-0.849469\pi$
$223$	−26.5885	−1.78049	−0.890247	+	0.455477i	$0.849469\pi$
$224$	17.3205i	1.15728i
$225$	22.0000	1.46667
$226$	−11.4641	−0.762581
$227$	1.12436i	0.0746261i	0.999304	+	0.0373131i	$0.0118799\pi$
$227$	1.12436i	0.0746261i	−0.999304	+	0.0373131i	$0.988120\pi$
$228$	− 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	$0.425707\pi$
$230$	− 10.1962i	− 0.672314i
$231$	3.21539	0.211557
$232$	11.1962	0.735063
$233$	16.8564	1.10430	0.552150	−	0.833745i	$-0.313808\pi$
$233$	16.8564	1.10430	0.552150	+	0.833745i	$0.313808\pi$
$234$	−8.53590	−0.558009
$235$	− 30.5885i	− 1.99537i
$236$	− 6.53590i	− 0.425451i
$237$	− 10.3923i	− 0.675053i
$238$	−0.928203	−0.0601665
$239$	3.07180i	0.198698i	0.995053	+	0.0993490i	$0.0316760\pi$
$239$	3.07180i	0.198698i	−0.995053	+	0.0993490i	$0.968324\pi$
$240$	2.73205i	0.176353i
$241$	8.53590i	0.549846i	0.961466	+	0.274923i	$0.0886523\pi$
$241$	8.53590i	0.549846i	−0.961466	+	0.274923i	$0.911348\pi$
$242$	9.39230i	0.603760i
$243$	15.2679	0.979439
$244$	− 7.73205i	− 0.494994i
$245$	− 18.6603i	− 1.19216i
$246$	− 2.19615i	− 0.140022i
$247$	16.3923	1.04302
$248$	6.58846	0.418367
$249$	1.60770	0.101884
$250$	14.6603	0.927196
$251$	− 29.7128i	− 1.87546i	−0.347370	−	0.937728i	$-0.612925\pi$
$251$	− 29.7128i	− 1.87546i	0.347370	−	0.937728i	$-0.387075\pi$
$252$	8.53590	0.537711
$253$	− 3.46410i	− 0.217786i
$254$	1.80385i	0.113183i
$255$	0.732051	0.0458428
$256$	−17.0000	−1.06250
$257$	− 11.5885i	− 0.722868i	−0.932398	−	0.361434i	$-0.882287\pi$
$257$	− 11.5885i	− 0.722868i	0.932398	−	0.361434i	$-0.117713\pi$
$258$	−2.53590	−0.157878
$259$	0	0
$260$	12.9282	0.801773
$261$	− 9.19615i	− 0.569228i
$262$	20.5885	1.27196
$263$	−28.3923	−1.75074	−0.875372	−	0.483449i	$-0.839384\pi$
$263$	−28.3923	−1.75074	−0.875372	+	0.483449i	$0.839384\pi$
$264$	2.78461i	0.171381i
$265$	35.3205i	2.16972i
$266$	16.3923	1.00508
$267$	− 0.875644i	− 0.0535886i
$268$	−10.7321	−0.655564
$269$	21.4641	1.30869	0.654345	−	0.756196i	$-0.272945\pi$
$269$	21.4641	1.30869	0.654345	+	0.756196i	$0.272945\pi$
$270$	14.9282	0.908502
$271$	5.66025	0.343836	0.171918	−	0.985111i	$-0.445004\pi$
$271$	5.66025	0.343836	0.171918	+	0.985111i	$0.445004\pi$
$272$	− 0.267949i	− 0.0162468i
$273$	− 8.78461i	− 0.531669i
$274$	− 0.464102i	− 0.0280374i
$275$	11.3205	0.682652
$276$	2.00000i	0.120386i
$277$	13.7321i	0.825079i	0.910940	+	0.412539i	$0.135358\pi$
$277$	13.7321i	0.825079i	−0.910940	+	0.412539i	$0.864642\pi$
$278$	18.5885i	1.11486i
$279$	− 5.41154i	− 0.323981i
$280$	38.7846	2.31782
$281$	22.8038i	1.36036i	0.733044	+	0.680182i	$0.238099\pi$
$281$	22.8038i	1.36036i	−0.733044	+	0.680182i	$0.761901\pi$
$282$	− 6.00000i	− 0.357295i
$283$	10.3923i	0.617758i	0.951101	+	0.308879i	$0.0999539\pi$
$283$	10.3923i	0.617758i	−0.951101	+	0.308879i	$0.900046\pi$
$284$	6.00000	0.356034
$285$	−12.9282	−0.765801
$286$	−4.39230	−0.259722
$287$	−10.3923	−0.613438
$288$	12.3205i	0.725993i
$289$	16.9282	0.995777
$290$	− 13.9282i	− 0.817892i
$291$	− 5.66025i	− 0.331810i
$292$	0	0
$293$	1.14359	0.0668094	0.0334047	−	0.999442i	$-0.489365\pi$
$293$	1.14359	0.0668094	0.0334047	+	0.999442i	$0.489365\pi$
$294$	− 3.66025i	− 0.213470i
$295$	−24.3923	−1.42017
$296$	0	0
$297$	5.07180	0.294295
$298$	− 12.4641i	− 0.722026i
$299$	−9.46410	−0.547323
$300$	−6.53590	−0.377350
$301$	12.0000i	0.691669i
$302$	14.5885i	0.839471i
$303$	−6.58846	−0.378497
$304$	4.73205i	0.271402i
$305$	−28.8564	−1.65231
$306$	−0.660254	−0.0377442
$307$	6.39230	0.364828	0.182414	−	0.983222i	$-0.441609\pi$
$307$	6.39230	0.364828	0.182414	+	0.983222i	$0.441609\pi$
$308$	4.39230	0.250275
$309$	0.679492i	0.0386549i
$310$	− 8.19615i	− 0.465510i
$311$	− 9.60770i	− 0.544802i	−0.962184	−	0.272401i	$-0.912182\pi$
$311$	− 9.60770i	− 0.544802i	0.962184	−	0.272401i	$-0.0878178\pi$
$312$	7.60770	0.430701
$313$	15.5885i	0.881112i	0.897725	+	0.440556i	$0.145219\pi$
$313$	15.5885i	0.881112i	−0.897725	+	0.440556i	$0.854781\pi$
$314$	− 20.3205i	− 1.14675i
$315$	− 31.8564i	− 1.79491i
$316$	− 14.1962i	− 0.798596i
$317$	−2.07180	−0.116364	−0.0581818	−	0.998306i	$-0.518530\pi$
$317$	−2.07180	−0.116364	−0.0581818	+	0.998306i	$0.518530\pi$
$318$	6.92820i	0.388514i
$319$	− 4.73205i	− 0.264944i
$320$	26.1244i	1.46040i
$321$	0.679492	0.0379255
$322$	−9.46410	−0.527414
$323$	1.26795	0.0705506
$324$	4.46410	0.248006
$325$	− 30.9282i	− 1.71559i
$326$	1.60770	0.0890420
$327$	− 12.5885i	− 0.696143i
$328$	− 9.00000i	− 0.496942i
$329$	−28.3923	−1.56532
$330$	3.46410	0.190693
$331$	− 8.78461i	− 0.482846i	−0.970420	−	0.241423i	$-0.922386\pi$
$331$	− 8.78461i	− 0.482846i	0.970420	−	0.241423i	$-0.0776141\pi$
$332$	2.19615	0.120530
$333$	0	0
$334$	2.92820	0.160224
$335$	40.0526i	2.18831i
$336$	2.53590	0.138345
$337$	8.60770	0.468891	0.234446	−	0.972129i	$-0.424673\pi$
$337$	8.60770	0.468891	0.234446	+	0.972129i	$0.424673\pi$
$338$	− 1.00000i	− 0.0543928i
$339$	− 8.39230i	− 0.455807i
$340$	1.00000	0.0542326
$341$	− 2.78461i	− 0.150795i
$342$	11.6603	0.630514
$343$	6.92820	0.374088
$344$	−10.3923	−0.560316
$345$	7.46410	0.401854
$346$	3.92820i	0.211182i
$347$	9.26795i	0.497530i	0.968564	+	0.248765i	$0.0800246\pi$
$347$	9.26795i	0.497530i	−0.968564	+	0.248765i	$0.919975\pi$
$348$	2.73205i	0.146453i
$349$	21.9282	1.17379	0.586895	−	0.809663i	$-0.300350\pi$
$349$	21.9282	1.17379	0.586895	+	0.809663i	$0.300350\pi$
$350$	− 30.9282i	− 1.65318i
$351$	− 13.8564i	− 0.739600i
$352$	6.33975i	0.337910i
$353$	15.7321i	0.837333i	0.908140	+	0.418666i	$0.137502\pi$
$353$	15.7321i	0.837333i	−0.908140	+	0.418666i	$0.862498\pi$
$354$	−4.78461	−0.254299
$355$	− 22.3923i	− 1.18846i
$356$	− 1.19615i	− 0.0633960i
$357$	− 0.679492i	− 0.0359625i
$358$	−5.46410	−0.288787
$359$	−4.39230	−0.231817	−0.115908	−	0.993260i	$-0.536978\pi$
$359$	−4.39230	−0.231817	−0.115908	+	0.993260i	$0.536978\pi$
$360$	27.5885	1.45404
$361$	−3.39230	−0.178542
$362$	3.92820i	0.206462i
$363$	−6.87564	−0.360878
$364$	− 12.0000i	− 0.628971i
$365$	0	0
$366$	−5.66025	−0.295866
$367$	0.392305	0.0204781	0.0102391	−	0.999948i	$-0.496741\pi$
$367$	0.392305	0.0204781	0.0102391	+	0.999948i	$0.496741\pi$
$368$	− 2.73205i	− 0.142418i
$369$	−7.39230	−0.384828
$370$	0	0
$371$	32.7846	1.70209
$372$	1.60770i	0.0833551i
$373$	24.4641	1.26670	0.633352	−	0.773864i	$-0.281679\pi$
$373$	24.4641	1.26670	0.633352	+	0.773864i	$0.281679\pi$
$374$	−0.339746	−0.0175678
$375$	10.7321i	0.554201i
$376$	− 24.5885i	− 1.26805i
$377$	−12.9282	−0.665836
$378$	− 13.8564i	− 0.712697i
$379$	15.4641	0.794338	0.397169	−	0.917746i	$-0.369993\pi$
$379$	15.4641	0.794338	0.397169	+	0.917746i	$0.369993\pi$
$380$	−17.6603	−0.905952
$381$	−1.32051	−0.0676517
$382$	4.19615	0.214694
$383$	− 11.8564i	− 0.605834i	−0.953017	−	0.302917i	$-0.902039\pi$
$383$	− 11.8564i	− 0.605834i	0.953017	−	0.302917i	$-0.0979605\pi$
$384$	− 2.19615i	− 0.112072i
$385$	− 16.3923i	− 0.835429i
$386$	2.66025	0.135403
$387$	8.53590i	0.433904i
$388$	− 7.73205i	− 0.392535i
$389$	− 20.8038i	− 1.05480i	−0.849618	−	0.527398i	$-0.823168\pi$
$389$	− 20.8038i	− 1.05480i	0.849618	−	0.527398i	$-0.176832\pi$
$390$	− 9.46410i	− 0.479233i
$391$	−0.732051	−0.0370214
$392$	− 15.0000i	− 0.757614i
$393$	15.0718i	0.760272i
$394$	− 14.0718i	− 0.708927i
$395$	−52.9808	−2.66575
$396$	3.12436	0.157005
$397$	27.2487	1.36757	0.683787	−	0.729682i	$-0.260332\pi$
$397$	27.2487	1.36757	0.683787	+	0.729682i	$0.260332\pi$
$398$	−24.9282	−1.24954
$399$	12.0000i	0.600751i
$400$	8.92820	0.446410
$401$	− 26.3923i	− 1.31797i	−0.752157	−	0.658984i	$-0.770986\pi$
$401$	− 26.3923i	− 1.31797i	0.752157	−	0.658984i	$-0.229014\pi$
$402$	7.85641i	0.391842i
$403$	−7.60770	−0.378966
$404$	−9.00000	−0.447767
$405$	− 16.6603i	− 0.827855i
$406$	−12.9282	−0.641616
$407$	0	0
$408$	0.588457	0.0291330
$409$	− 8.66025i	− 0.428222i	−0.976809	−	0.214111i	$-0.931315\pi$
$409$	− 8.66025i	− 0.428222i	0.976809	−	0.214111i	$-0.0686854\pi$
$410$	−11.1962	−0.552939
$411$	0.339746	0.0167584
$412$	0.928203i	0.0457293i
$413$	22.6410i	1.11409i
$414$	−6.73205	−0.330862
$415$	− 8.19615i	− 0.402333i
$416$	17.3205	0.849208
$417$	−13.6077	−0.666372
$418$	6.00000	0.293470
$419$	−17.3205	−0.846162	−0.423081	−	0.906092i	$-0.639051\pi$
$419$	−17.3205	−0.846162	−0.423081	+	0.906092i	$0.639051\pi$
$420$	9.46410i	0.461801i
$421$	19.7321i	0.961681i	0.876808	+	0.480841i	$0.159668\pi$
$421$	19.7321i	0.961681i	−0.876808	+	0.480841i	$0.840332\pi$
$422$	10.0000i	0.486792i
$423$	−20.1962	−0.981971
$424$	28.3923i	1.37885i
$425$	− 2.39230i	− 0.116044i
$426$	− 4.39230i	− 0.212808i
$427$	26.7846i	1.29620i
$428$	0.928203	0.0448664
$429$	− 3.21539i	− 0.155241i
$430$	12.9282i	0.623453i
$431$	− 33.5167i	− 1.61444i	−0.590250	−	0.807220i	$-0.700971\pi$
$431$	− 33.5167i	− 1.61444i	0.590250	−	0.807220i	$-0.299029\pi$
$432$	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	$0.275284\pi$
$434$	−7.60770	−0.365181
$435$	10.1962	0.488868
$436$	− 17.1962i	− 0.823546i
$437$	12.9282	0.618440
$438$	0	0
$439$	− 21.4641i	− 1.02443i	−0.858859	−	0.512213i	$-0.828826\pi$
$439$	− 21.4641i	− 1.02443i	0.858859	−	0.512213i	$-0.171174\pi$
$440$	14.1962	0.676775
$441$	−12.3205	−0.586691
$442$	0.928203i	0.0441501i
$443$	26.5359	1.26076	0.630379	−	0.776287i	$-0.282899\pi$
$443$	26.5359	1.26076	0.630379	+	0.776287i	$0.282899\pi$
$444$	0	0
$445$	−4.46410	−0.211619
$446$	26.5885i	1.25900i
$447$	9.12436	0.431567
$448$	24.2487	1.14564
$449$	− 16.2487i	− 0.766824i	−0.923577	−	0.383412i	$-0.874749\pi$
$449$	− 16.2487i	− 0.766824i	0.923577	−	0.383412i	$-0.125251\pi$
$450$	− 22.0000i	− 1.03709i
$451$	−3.80385	−0.179116
$452$	− 11.4641i	− 0.539226i
$453$	−10.6795	−0.501766
$454$	1.12436	0.0527686
$455$	−44.7846	−2.09953
$456$	−10.3923	−0.486664
$457$	11.8756i	0.555519i	0.960651	+	0.277760i	$0.0895919\pi$
$457$	11.8756i	0.555519i	−0.960651	+	0.277760i	$0.910408\pi$
$458$	− 7.00000i	− 0.327089i
$459$	− 1.07180i	− 0.0500272i
$460$	10.1962	0.475398
$461$	− 23.4641i	− 1.09283i	−0.837514	−	0.546416i	$-0.815992\pi$
$461$	− 23.4641i	− 1.09283i	0.837514	−	0.546416i	$-0.184008\pi$
$462$	− 3.21539i	− 0.149593i
$463$	8.53590i	0.396697i	0.980132	+	0.198348i	$0.0635578\pi$
$463$	8.53590i	0.396697i	−0.980132	+	0.198348i	$0.936442\pi$
$464$	− 3.73205i	− 0.173256i
$465$	6.00000	0.278243
$466$	− 16.8564i	− 0.780858i
$467$	31.3205i	1.44934i	0.689096	+	0.724670i	$0.258008\pi$
$467$	31.3205i	1.44934i	−0.689096	+	0.724670i	$0.741992\pi$
$468$	− 8.53590i	− 0.394572i
$469$	37.1769	1.71667
$470$	−30.5885	−1.41094
$471$	14.8756	0.685434
$472$	−19.6077	−0.902517
$473$	4.39230i	0.201958i
$474$	−10.3923	−0.477334
$475$	42.2487i	1.93850i
$476$	− 0.928203i	− 0.0425441i
$477$	23.3205	1.06777
$478$	3.07180	0.140501
$479$	− 4.67949i	− 0.213811i	−0.994269	−	0.106906i	$-0.965906\pi$
$479$	− 4.67949i	− 0.213811i	0.994269	−	0.106906i	$-0.0340943\pi$
$480$	−13.6603	−0.623502
$481$	0	0
$482$	8.53590	0.388800
$483$	− 6.92820i	− 0.315244i
$484$	−9.39230	−0.426923
$485$	−28.8564	−1.31030
$486$	− 15.2679i	− 0.692568i
$487$	− 26.4449i	− 1.19833i	−0.800625	−	0.599166i	$-0.795499\pi$
$487$	− 26.4449i	− 1.19833i	0.800625	−	0.599166i	$-0.204501\pi$
$488$	−23.1962	−1.05004
$489$	1.17691i	0.0532219i
$490$	−18.6603	−0.842984
$491$	34.7321	1.56744	0.783718	−	0.621117i	$-0.213321\pi$
$491$	34.7321	1.56744	0.783718	+	0.621117i	$0.213321\pi$
$492$	2.19615	0.0990102
$493$	−1.00000	−0.0450377
$494$	− 16.3923i	− 0.737525i
$495$	− 11.6603i	− 0.524089i
$496$	− 2.19615i	− 0.0986102i
$497$	−20.7846	−0.932317
$498$	− 1.60770i	− 0.0720425i
$499$	18.5885i	0.832134i	0.909334	+	0.416067i	$0.136592\pi$
$499$	18.5885i	0.832134i	−0.909334	+	0.416067i	$0.863408\pi$
$500$	14.6603i	0.655626i
$501$	2.14359i	0.0957687i
$502$	−29.7128	−1.32615
$503$	42.9808i	1.91642i	0.286070	+	0.958209i	$0.407651\pi$
$503$	42.9808i	1.91642i	−0.286070	+	0.958209i	$0.592349\pi$
$504$	− 25.6077i	− 1.14066i
$505$	33.5885i	1.49467i
$506$	−3.46410	−0.153998
$507$	0.732051	0.0325115
$508$	−1.80385	−0.0800328
$509$	35.7846	1.58612	0.793062	−	0.609140i	$-0.208485\pi$
$509$	35.7846	1.58612	0.793062	+	0.609140i	$0.208485\pi$
$510$	− 0.732051i	− 0.0324158i
$511$	0	0
$512$	11.0000i	0.486136i
$513$	18.9282i	0.835701i
$514$	−11.5885	−0.511145
$515$	3.46410	0.152647
$516$	− 2.53590i	− 0.111637i
$517$	−10.3923	−0.457053
$518$	0	0
$519$	−2.87564	−0.126227
$520$	− 38.7846i	− 1.70082i
$521$	−6.92820	−0.303530	−0.151765	−	0.988417i	$-0.548496\pi$
$521$	−6.92820	−0.303530	−0.151765	+	0.988417i	$0.548496\pi$
$522$	−9.19615	−0.402505
$523$	22.0526i	0.964291i	0.876091	+	0.482146i	$0.160142\pi$
$523$	22.0526i	0.964291i	−0.876091	+	0.482146i	$0.839858\pi$
$524$	20.5885i	0.899411i
$525$	22.6410	0.988135
$526$	28.3923i	1.23796i
$527$	−0.588457	−0.0256336
$528$	0.928203	0.0403949
$529$	15.5359	0.675474
$530$	35.3205	1.53422
$531$	16.1051i	0.698903i
$532$	16.3923i	0.710697i
$533$	10.3923i	0.450141i
$534$	−0.875644	−0.0378928
$535$	− 3.46410i	− 0.149766i
$536$	32.1962i	1.39066i
$537$	− 4.00000i	− 0.172613i
$538$	− 21.4641i	− 0.925383i
$539$	−6.33975	−0.273072
$540$	14.9282i	0.642408i
$541$	− 7.73205i	− 0.332427i	−0.986090	−	0.166213i	$-0.946846\pi$
$541$	− 7.73205i	− 0.332427i	0.986090	−	0.166213i	$-0.0531541\pi$
$542$	− 5.66025i	− 0.243129i
$543$	−2.87564	−0.123406
$544$	1.33975	0.0574411
$545$	−64.1769	−2.74904
$546$	−8.78461	−0.375947
$547$	14.1962i	0.606984i	0.952834	+	0.303492i	$0.0981525\pi$
$547$	14.1962i	0.606984i	−0.952834	+	0.303492i	$0.901847\pi$
$548$	0.464102	0.0198254
$549$	19.0526i	0.813143i
$550$	− 11.3205i	− 0.482708i
$551$	17.6603	0.752352
$552$	6.00000	0.255377
$553$	49.1769i	2.09122i
$554$	13.7321	0.583419
$555$	0	0
$556$	−18.5885	−0.788326
$557$	− 14.2679i	− 0.604552i	−0.953220	−	0.302276i	$-0.902254\pi$
$557$	− 14.2679i	− 0.604552i	0.953220	−	0.302276i	$-0.0977465\pi$
$558$	−5.41154	−0.229089
$559$	12.0000	0.507546
$560$	− 12.9282i	− 0.546316i
$561$	− 0.248711i	− 0.0105006i
$562$	22.8038	0.961922
$563$	3.41154i	0.143779i	0.997413	+	0.0718897i	$0.0229030\pi$
$563$	3.41154i	0.143779i	−0.997413	+	0.0718897i	$0.977097\pi$
$564$	6.00000	0.252646
$565$	−42.7846	−1.79996
$566$	10.3923	0.436821
$567$	−15.4641	−0.649431
$568$	− 18.0000i	− 0.755263i
$569$	8.51666i	0.357037i	0.983937	+	0.178518i	$0.0571304\pi$
$569$	8.51666i	0.357037i	−0.983937	+	0.178518i	$0.942870\pi$
$570$	12.9282i	0.541503i
$571$	13.5167	0.565655	0.282827	−	0.959171i	$-0.408728\pi$
$571$	13.5167	0.565655	0.282827	+	0.959171i	$0.408728\pi$
$572$	− 4.39230i	− 0.183651i
$573$	3.07180i	0.128326i
$574$	10.3923i	0.433766i
$575$	− 24.3923i	− 1.01723i
$576$	17.2487	0.718696
$577$	− 0.928203i	− 0.0386416i	−0.999813	−	0.0193208i	$-0.993850\pi$
$577$	− 0.928203i	− 0.0386416i	0.999813	−	0.0193208i	$-0.00615039\pi$
$578$	− 16.9282i	− 0.704120i
$579$	1.94744i	0.0809329i
$580$	13.9282	0.578337
$581$	−7.60770	−0.315620
$582$	−5.66025	−0.234625
$583$	12.0000	0.496989
$584$	0	0
$585$	−31.8564	−1.31710
$586$	− 1.14359i	− 0.0472414i
$587$	33.6603i	1.38931i	0.719344	+	0.694654i	$0.244442\pi$
$587$	33.6603i	1.38931i	−0.719344	+	0.694654i	$0.755558\pi$
$588$	3.66025	0.150946
$589$	10.3923	0.428207
$590$	24.3923i	1.00422i
$591$	10.3013	0.423738
$592$	0	0
$593$	−23.7846	−0.976717	−0.488358	−	0.872643i	$-0.662404\pi$
$593$	−23.7846	−0.976717	−0.488358	+	0.872643i	$0.662404\pi$
$594$	− 5.07180i	− 0.208098i
$595$	−3.46410	−0.142014
$596$	12.4641	0.510549
$597$	− 18.2487i	− 0.746870i
$598$	9.46410i	0.387016i
$599$	46.0526	1.88166	0.940828	−	0.338884i	$-0.110049\pi$
$599$	46.0526	1.88166	0.940828	+	0.338884i	$0.110049\pi$
$600$	19.6077i	0.800481i
$601$	10.6077	0.432697	0.216348	−	0.976316i	$-0.430585\pi$
$601$	10.6077	0.432697	0.216348	+	0.976316i	$0.430585\pi$
$602$	12.0000	0.489083
$603$	26.4449	1.07692
$604$	−14.5885	−0.593596
$605$	35.0526i	1.42509i
$606$	6.58846i	0.267638i
$607$	− 11.6603i	− 0.473275i	−0.971598	−	0.236638i	$-0.923955\pi$
$607$	− 11.6603i	− 0.473275i	0.971598	−	0.236638i	$-0.0760454\pi$
$608$	−23.6603	−0.959550
$609$	− 9.46410i	− 0.383505i
$610$	28.8564i	1.16836i
$611$	28.3923i	1.14863i
$612$	− 0.660254i	− 0.0266892i
$613$	−17.3923	−0.702469	−0.351234	−	0.936288i	$-0.614238\pi$
$613$	−17.3923	−0.702469	−0.351234	+	0.936288i	$0.614238\pi$
$614$	− 6.39230i	− 0.257972i
$615$	− 8.19615i	− 0.330501i
$616$	− 13.1769i	− 0.530913i
$617$	37.1769	1.49669	0.748343	−	0.663312i	$-0.230850\pi$
$617$	37.1769	1.49669	0.748343	+	0.663312i	$0.230850\pi$
$618$	0.679492	0.0273332
$619$	−34.7846	−1.39811	−0.699056	−	0.715067i	$-0.746396\pi$
$619$	−34.7846	−1.39811	−0.699056	+	0.715067i	$0.746396\pi$
$620$	8.19615	0.329165
$621$	− 10.9282i	− 0.438534i
$622$	−9.60770	−0.385233
$623$	4.14359i	0.166010i
$624$	− 2.53590i	− 0.101517i
$625$	10.0718	0.402872
$626$	15.5885	0.623040
$627$	4.39230i	0.175412i
$628$	20.3205	0.810877
$629$	0	0
$630$	−31.8564	−1.26919
$631$	22.7321i	0.904949i	0.891777	+	0.452474i	$0.149459\pi$
$631$	22.7321i	0.904949i	−0.891777	+	0.452474i	$0.850541\pi$
$632$	−42.5885	−1.69408
$633$	−7.32051	−0.290964
$634$	2.07180i	0.0822816i
$635$	6.73205i	0.267153i
$636$	−6.92820	−0.274721
$637$	17.3205i	0.686264i
$638$	−4.73205	−0.187344
$639$	−14.7846	−0.584870
$640$	−11.1962	−0.442567
$641$	23.7846	0.939436	0.469718	−	0.882817i	$-0.344356\pi$
$641$	23.7846	0.939436	0.469718	+	0.882817i	$0.344356\pi$
$642$	− 0.679492i	− 0.0268174i
$643$	− 11.6603i	− 0.459836i	−0.973210	−	0.229918i	$-0.926154\pi$
$643$	− 11.6603i	− 0.459836i	0.973210	−	0.229918i	$-0.0738457\pi$
$644$	− 9.46410i	− 0.372938i
$645$	−9.46410	−0.372649
$646$	− 1.26795i	− 0.0498868i
$647$	− 36.5359i	− 1.43637i	−0.695850	−	0.718187i	$-0.744972\pi$
$647$	− 36.5359i	− 1.43637i	0.695850	−	0.718187i	$-0.255028\pi$
$648$	− 13.3923i	− 0.526099i
$649$	8.28719i	0.325301i
$650$	−30.9282	−1.21310
$651$	− 5.56922i	− 0.218275i
$652$	1.60770i	0.0629622i
$653$	13.1962i	0.516405i	0.966091	+	0.258203i	$0.0831302\pi$
$653$	13.1962i	0.516405i	−0.966091	+	0.258203i	$0.916870\pi$
$654$	−12.5885	−0.492248
$655$	76.8372	3.00228
$656$	−3.00000	−0.117130
$657$	0	0
$658$	28.3923i	1.10685i
$659$	2.53590	0.0987846	0.0493923	−	0.998779i	$-0.484272\pi$
$659$	2.53590	0.0987846	0.0493923	+	0.998779i	$0.484272\pi$
$660$	3.46410i	0.134840i
$661$	23.1962i	0.902226i	0.892467	+	0.451113i	$0.148973\pi$
$661$	23.1962i	0.902226i	−0.892467	+	0.451113i	$0.851027\pi$
$662$	−8.78461	−0.341424
$663$	−0.679492	−0.0263893
$664$	− 6.58846i	− 0.255682i
$665$	61.1769	2.37234
$666$	0	0
$667$	−10.1962	−0.394797
$668$	2.92820i	0.113296i
$669$	−19.4641	−0.752526
$670$	40.0526	1.54737
$671$	9.80385i	0.378473i
$672$	12.6795i	0.489122i
$673$	−39.7128	−1.53082	−0.765408	−	0.643545i	$-0.777463\pi$
$673$	−39.7128	−1.53082	−0.765408	+	0.643545i	$0.777463\pi$
$674$	− 8.60770i	− 0.331556i
$675$	35.7128	1.37459
$676$	1.00000	0.0384615
$677$	−9.67949	−0.372013	−0.186007	−	0.982549i	$-0.559555\pi$
$677$	−9.67949	−0.372013	−0.186007	+	0.982549i	$0.559555\pi$
$678$	−8.39230	−0.322305
$679$	26.7846i	1.02790i
$680$	− 3.00000i	− 0.115045i
$681$	0.823085i	0.0315407i
$682$	−2.78461	−0.106628
$683$	44.9282i	1.71913i	0.511026	+	0.859565i	$0.329265\pi$
$683$	44.9282i	1.71913i	−0.511026	+	0.859565i	$0.670735\pi$
$684$	11.6603i	0.445841i
$685$	− 1.73205i	− 0.0661783i
$686$	− 6.92820i	− 0.264520i
$687$	5.12436	0.195506
$688$	3.46410i	0.132068i
$689$	− 32.7846i	− 1.24899i
$690$	− 7.46410i	− 0.284153i
$691$	−22.0526	−0.838919	−0.419459	−	0.907774i	$-0.637780\pi$
$691$	−22.0526	−0.838919	−0.419459	+	0.907774i	$0.637780\pi$
$692$	−3.92820	−0.149328
$693$	−10.8231	−0.411135
$694$	9.26795	0.351807
$695$	69.3731i	2.63147i
$696$	8.19615	0.310674
$697$	0.803848i	0.0304479i
$698$	− 21.9282i	− 0.829995i
$699$	12.3397	0.466732
$700$	30.9282	1.16898
$701$	21.8564i	0.825505i	0.910843	+	0.412753i	$0.135433\pi$
$701$	21.8564i	0.825505i	−0.910843	+	0.412753i	$0.864567\pi$
$702$	−13.8564	−0.522976
$703$	0	0
$704$	8.87564	0.334513
$705$	− 22.3923i	− 0.843343i
$706$	15.7321	0.592084
$707$	31.1769	1.17253
$708$	− 4.78461i	− 0.179817i
$709$	− 30.0000i	− 1.12667i	−0.826227	−	0.563337i	$-0.809517\pi$
$709$	− 30.0000i	− 1.12667i	0.826227	−	0.563337i	$-0.190483\pi$
$710$	−22.3923	−0.840368
$711$	34.9808i	1.31188i
$712$	−3.58846	−0.134483
$713$	−6.00000	−0.224702
$714$	−0.679492	−0.0254293
$715$	−16.3923	−0.613037
$716$	− 5.46410i	− 0.204203i
$717$	2.24871i	0.0839797i
$718$	4.39230i	0.163919i
$719$	−32.4449	−1.20999	−0.604995	−	0.796230i	$-0.706825\pi$
$719$	−32.4449	−1.20999	−0.604995	+	0.796230i	$0.706825\pi$
$720$	− 9.19615i	− 0.342720i
$721$	− 3.21539i	− 0.119747i
$722$	3.39230i	0.126249i
$723$	6.24871i	0.232392i
$724$	−3.92820	−0.145991
$725$	− 33.3205i	− 1.23749i
$726$	6.87564i	0.255179i
$727$	− 25.8564i	− 0.958961i	−0.877553	−	0.479481i	$-0.840825\pi$
$727$	− 25.8564i	− 0.958961i	0.877553	−	0.479481i	$-0.159175\pi$
$728$	−36.0000	−1.33425
$729$	−2.21539	−0.0820515
$730$	0	0
$731$	0.928203	0.0343308
$732$	− 5.66025i	− 0.209209i
$733$	−12.7846	−0.472210	−0.236105	−	0.971728i	$-0.575871\pi$
$733$	−12.7846	−0.472210	−0.236105	+	0.971728i	$0.575871\pi$
$734$	− 0.392305i	− 0.0144802i
$735$	− 13.6603i	− 0.503866i
$736$	13.6603	0.503524
$737$	13.6077	0.501246
$738$	7.39230i	0.272115i
$739$	−34.1962	−1.25793	−0.628963	−	0.777435i	$-0.716520\pi$
$739$	−34.1962	−1.25793	−0.628963	+	0.777435i	$0.716520\pi$
$740$	0	0
$741$	12.0000	0.440831
$742$	− 32.7846i	− 1.20356i
$743$	−46.7321	−1.71443	−0.857216	−	0.514956i	$-0.827808\pi$
$743$	−46.7321	−1.71443	−0.857216	+	0.514956i	$0.827808\pi$
$744$	4.82309	0.176823
$745$	− 46.5167i	− 1.70424i
$746$	− 24.4641i	− 0.895694i
$747$	−5.41154	−0.197998
$748$	− 0.339746i	− 0.0124223i
$749$	−3.21539	−0.117488
$750$	10.7321	0.391879
$751$	−50.7846	−1.85316	−0.926578	−	0.376102i	$-0.877264\pi$
$751$	−50.7846	−1.85316	−0.926578	+	0.376102i	$0.877264\pi$
$752$	−8.19615	−0.298883
$753$	− 21.7513i	− 0.792661i
$754$	12.9282i	0.470817i
$755$	54.4449i	1.98145i
$756$	13.8564	0.503953
$757$	− 30.3731i	− 1.10393i	−0.833868	−	0.551964i	$-0.813879\pi$
$757$	− 30.3731i	− 1.10393i	0.833868	−	0.551964i	$-0.186121\pi$
$758$	− 15.4641i	− 0.561681i
$759$	− 2.53590i	− 0.0920473i
$760$	52.9808i	1.92181i
$761$	−38.3205	−1.38912	−0.694559	−	0.719436i	$-0.744400\pi$
$761$	−38.3205	−1.38912	−0.694559	+	0.719436i	$0.744400\pi$
$762$	1.32051i	0.0478370i
$763$	59.5692i	2.15655i
$764$	4.19615i	0.151811i
$765$	−2.46410	−0.0890898
$766$	−11.8564	−0.428389
$767$	22.6410	0.817520
$768$	−12.4449	−0.449065
$769$	27.4641i	0.990381i	0.868784	+	0.495190i	$0.164902\pi$
$769$	27.4641i	0.990381i	−0.868784	+	0.495190i	$0.835098\pi$
$770$	−16.3923	−0.590738
$771$	− 8.48334i	− 0.305520i
$772$	2.66025i	0.0957446i
$773$	42.0333	1.51183	0.755917	−	0.654668i	$-0.227191\pi$
$773$	42.0333	1.51183	0.755917	+	0.654668i	$0.227191\pi$
$774$	8.53590	0.306817
$775$	− 19.6077i	− 0.704329i
$776$	−23.1962	−0.832693
$777$	0	0
$778$	−20.8038	−0.745854
$779$	− 14.1962i	− 0.508630i
$780$	9.46410	0.338869
$781$	−7.60770	−0.272225
$782$	0.732051i	0.0261781i
$783$	− 14.9282i	− 0.533490i
$784$	−5.00000	−0.178571
$785$	− 75.8372i	− 2.70674i
$786$	15.0718	0.537593
$787$	−20.3923	−0.726907	−0.363454	−	0.931612i	$-0.618402\pi$
$787$	−20.3923	−0.726907	−0.363454	+	0.931612i	$0.618402\pi$
$788$	14.0718	0.501287
$789$	−20.7846	−0.739952
$790$	52.9808i	1.88497i
$791$	39.7128i	1.41203i
$792$	− 9.37307i	− 0.333057i
$793$	26.7846	0.951149
$794$	− 27.2487i	− 0.967021i
$795$	25.8564i	0.917032i
$796$	− 24.9282i	− 0.883557i
$797$	− 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
$799$	2.19615i	0.0776943i
$800$	44.6410i	1.57830i
$801$	2.94744i	0.104143i
$802$	−26.3923	−0.931945
$803$	0	0
$804$	−7.85641	−0.277074
$805$	−35.3205	−1.24488
$806$	7.60770i	0.267970i
$807$	15.7128	0.553117
$808$	27.0000i	0.949857i
$809$	− 6.14359i	− 0.215997i	−0.994151	−	0.107999i	$-0.965556\pi$
$809$	− 6.14359i	− 0.215997i	0.994151	−	0.107999i	$-0.0344442\pi$
$810$	−16.6603	−0.585382
$811$	−30.2487	−1.06218	−0.531088	−	0.847317i	$-0.678217\pi$
$811$	−30.2487	−1.06218	−0.531088	+	0.847317i	$0.678217\pi$
$812$	− 12.9282i	− 0.453691i
$813$	4.14359	0.145322
$814$	0	0
$815$	6.00000	0.210171
$816$	− 0.196152i	− 0.00686671i
$817$	−16.3923	−0.573494
$818$	−8.66025	−0.302799
$819$	29.5692i	1.03323i
$820$	− 11.1962i	− 0.390987i
$821$	−13.1769	−0.459877	−0.229939	−	0.973205i	$-0.573853\pi$
$821$	−13.1769	−0.459877	−0.229939	+	0.973205i	$0.573853\pi$
$822$	− 0.339746i	− 0.0118500i
$823$	−54.9808	−1.91651	−0.958254	−	0.285917i	$-0.907702\pi$
$823$	−54.9808	−1.91651	−0.958254	+	0.285917i	$0.907702\pi$
$824$	2.78461	0.0970065
$825$	8.28719	0.288523
$826$	22.6410	0.787782
$827$	9.07180i	0.315457i	0.987482	+	0.157729i	$0.0504171\pi$
$827$	9.07180i	0.315457i	−0.987482	+	0.157729i	$0.949583\pi$
$828$	− 6.73205i	− 0.233955i
$829$	− 51.4641i	− 1.78742i	−0.448643	−	0.893711i	$-0.648093\pi$
$829$	− 51.4641i	− 1.78742i	0.448643	−	0.893711i	$-0.351907\pi$
$830$	−8.19615	−0.284493
$831$	10.0526i	0.348719i
$832$	− 24.2487i	− 0.840673i
$833$	1.33975i	0.0464194i
$834$	13.6077i	0.471196i
$835$	10.9282	0.378186
$836$	6.00000i	0.207514i
$837$	− 8.78461i	− 0.303641i
$838$	17.3205i	0.598327i
$839$	−12.3397	−0.426015	−0.213008	−	0.977051i	$-0.568326\pi$
$839$	−12.3397	−0.426015	−0.213008	+	0.977051i	$0.568326\pi$
$840$	28.3923	0.979628
$841$	15.0718	0.519717
$842$	19.7321	0.680011
$843$	16.6936i	0.574957i
$844$	−10.0000	−0.344214
$845$	− 3.73205i	− 0.128386i
$846$	20.1962i	0.694358i
$847$	32.5359	1.11795
$848$	9.46410	0.324999
$849$	7.60770i	0.261095i
$850$	−2.39230	−0.0820554
$851$	0	0
$852$	4.39230	0.150478
$853$	17.8756i	0.612050i	0.952023	+	0.306025i	$0.0989992\pi$
$853$	17.8756i	0.612050i	−0.952023	+	0.306025i	$0.901001\pi$
$854$	26.7846	0.916550
$855$	43.5167	1.48824
$856$	− 2.78461i	− 0.0951760i
$857$	20.2679i	0.692340i	0.938172	+	0.346170i	$0.112518\pi$
$857$	20.2679i	0.692340i	−0.938172	+	0.346170i	$0.887482\pi$
$858$	−3.21539	−0.109772
$859$	− 15.4641i	− 0.527628i	−0.964574	−	0.263814i	$-0.915019\pi$
$859$	− 15.4641i	− 0.527628i	0.964574	−	0.263814i	$-0.0849806\pi$
$860$	−12.9282	−0.440848
$861$	−7.60770	−0.259270
$862$	−33.5167	−1.14158
$863$	45.4641	1.54762	0.773808	−	0.633420i	$-0.218350\pi$
$863$	45.4641	1.54762	0.773808	+	0.633420i	$0.218350\pi$
$864$	20.0000i	0.680414i
$865$	14.6603i	0.498464i
$866$	− 27.0000i	− 0.917497i
$867$	12.3923	0.420865
$868$	− 7.60770i	− 0.258222i
$869$	18.0000i	0.610608i
$870$	− 10.1962i	− 0.345682i
$871$	− 37.1769i	− 1.25969i
$872$	−51.5885	−1.74701
$873$	19.0526i	0.644831i
$874$	− 12.9282i	− 0.437303i
$875$	− 50.7846i	− 1.71683i
$876$	0	0
$877$	−35.7846	−1.20836	−0.604180	−	0.796848i	$-0.706499\pi$
$877$	−35.7846	−1.20836	−0.604180	+	0.796848i	$0.706499\pi$
$878$	−21.4641	−0.724378
$879$	0.837169	0.0282370
$880$	− 4.73205i	− 0.159517i
$881$	−2.32051	−0.0781799	−0.0390900	−	0.999236i	$-0.512446\pi$
$881$	−2.32051	−0.0781799	−0.0390900	+	0.999236i	$0.512446\pi$
$882$	12.3205i	0.414853i
$883$	18.3397i	0.617182i	0.951195	+	0.308591i	$0.0998574\pi$
$883$	18.3397i	0.617182i	−0.951195	+	0.308591i	$0.900143\pi$
$884$	−0.928203	−0.0312189
$885$	−17.8564	−0.600237
$886$	− 26.5359i	− 0.891491i
$887$	−22.9808	−0.771618	−0.385809	−	0.922579i	$-0.626078\pi$
$887$	−22.9808	−0.771618	−0.385809	+	0.922579i	$0.626078\pi$
$888$	0	0
$889$	6.24871	0.209575
$890$	4.46410i	0.149637i
$891$	−5.66025	−0.189626
$892$	−26.5885	−0.890247
$893$	− 38.7846i	− 1.29788i
$894$	− 9.12436i	− 0.305164i
$895$	−20.3923	−0.681640
$896$	10.3923i	0.347183i
$897$	−6.92820	−0.231326
$898$	−16.2487	−0.542227
$899$	−8.19615	−0.273357
$900$	22.0000	0.733333
$901$	− 2.53590i	− 0.0844830i
$902$	3.80385i	0.126654i
$903$	8.78461i	0.292334i
$904$	−34.3923	−1.14387
$905$	14.6603i	0.487323i
$906$	10.6795i	0.354802i
$907$	− 4.73205i	− 0.157125i	−0.996909	−	0.0785626i	$-0.974967\pi$
$907$	− 4.73205i	− 0.157125i	0.996909	−	0.0785626i	$-0.0250330\pi$
$908$	1.12436i	0.0373131i
$909$	22.1769	0.735562
$910$	44.7846i	1.48460i
$911$	− 8.48334i	− 0.281066i	−0.990076	−	0.140533i	$-0.955118\pi$
$911$	− 8.48334i	− 0.281066i	0.990076	−	0.140533i	$-0.0448815\pi$
$912$	3.46410i	0.114708i
$913$	−2.78461	−0.0921571
$914$	11.8756	0.392811
$915$	−21.1244	−0.698350
$916$	7.00000	0.231287
$917$	− 71.3205i	− 2.35521i
$918$	−1.07180	−0.0353746
$919$	− 34.0526i	− 1.12329i	−0.827378	−	0.561645i	$-0.810169\pi$
$919$	− 34.0526i	− 1.12329i	0.827378	−	0.561645i	$-0.189831\pi$
$920$	− 30.5885i	− 1.00847i
$921$	4.67949	0.154195
$922$	−23.4641	−0.772749
$923$	20.7846i	0.684134i
$924$	3.21539	0.105779
$925$	0	0
$926$	8.53590	0.280507
$927$	− 2.28719i	− 0.0751211i
$928$	18.6603	0.612553
$929$	−23.5359	−0.772188	−0.386094	−	0.922459i	$-0.626176\pi$
$929$	−23.5359	−0.772188	−0.386094	+	0.922459i	$0.626176\pi$
$930$	− 6.00000i	− 0.196748i
$931$	− 23.6603i	− 0.775434i
$932$	16.8564	0.552150
$933$	− 7.03332i	− 0.230261i
$934$	31.3205	1.02484
$935$	−1.26795	−0.0414664
$936$	−25.6077	−0.837014
$937$	−33.3923	−1.09088	−0.545440	−	0.838150i	$-0.683637\pi$
$937$	−33.3923	−1.09088	−0.545440	+	0.838150i	$0.683637\pi$
$938$	− 37.1769i	− 1.21387i
$939$	11.4115i	0.372402i
$940$	− 30.5885i	− 0.997685i
$941$	58.1769	1.89651	0.948257	−	0.317505i	$-0.102845\pi$
$941$	58.1769	1.89651	0.948257	+	0.317505i	$0.102845\pi$
$942$	− 14.8756i	− 0.484675i
$943$	8.19615i	0.266903i
$944$	6.53590i	0.212725i
$945$	− 51.7128i	− 1.68222i
$946$	4.39230	0.142806
$947$	− 48.3923i	− 1.57254i	−0.617884	−	0.786269i	$-0.712010\pi$
$947$	− 48.3923i	− 1.57254i	0.617884	−	0.786269i	$-0.287990\pi$
$948$	− 10.3923i	− 0.337526i
$949$	0	0
$950$	42.2487	1.37073
$951$	−1.51666	−0.0491811
$952$	−2.78461	−0.0902497
$953$	−31.8564	−1.03193	−0.515965	−	0.856610i	$-0.672567\pi$
$953$	−31.8564	−1.03193	−0.515965	+	0.856610i	$0.672567\pi$
$954$	− 23.3205i	− 0.755029i
$955$	15.6603	0.506754
$956$	3.07180i	0.0993490i
$957$	− 3.46410i	− 0.111979i
$958$	−4.67949	−0.151188
$959$	−1.60770	−0.0519152
$960$	19.1244i	0.617236i
$961$	26.1769	0.844417
$962$	0	0
$963$	−2.28719	−0.0737036
$964$	8.53590i	0.274923i
$965$	9.92820	0.319600
$966$	−6.92820	−0.222911
$967$	8.78461i	0.282494i	0.989974	+	0.141247i	$0.0451112\pi$
$967$	8.78461i	0.282494i	−0.989974	+	0.141247i	$0.954889\pi$
$968$	28.1769i	0.905640i
$969$	0.928203	0.0298182
$970$	28.8564i	0.926523i
$971$	2.19615	0.0704779	0.0352389	−	0.999379i	$-0.488781\pi$
$971$	2.19615	0.0704779	0.0352389	+	0.999379i	$0.488781\pi$
$972$	15.2679	0.489720
$973$	64.3923	2.06432
$974$	−26.4449	−0.847348
$975$	− 22.6410i	− 0.725093i
$976$	7.73205i	0.247497i
$977$	32.2487i	1.03173i	0.856671	+	0.515864i	$0.172529\pi$
$977$	32.2487i	1.03173i	−0.856671	+	0.515864i	$0.827471\pi$
$978$	1.17691	0.0376336
$979$	1.51666i	0.0484727i
$980$	− 18.6603i	− 0.596080i
$981$	42.3731i	1.35287i
$982$	− 34.7321i	− 1.10834i
$983$	32.8756	1.04857	0.524285	−	0.851543i	$-0.324333\pi$
$983$	32.8756	1.04857	0.524285	+	0.851543i	$0.324333\pi$
$984$	− 6.58846i	− 0.210032i
$985$	− 52.5167i	− 1.67332i
$986$	1.00000i	0.0318465i
$987$	−20.7846	−0.661581
$988$	16.3923	0.521509
$989$	9.46410	0.300941
$990$	−11.6603	−0.370587
$991$	− 25.6077i	− 0.813455i	−0.913549	−	0.406728i	$-0.866670\pi$
$991$	− 25.6077i	− 0.813455i	0.913549	−	0.406728i	$-0.133330\pi$
$992$	10.9808	0.348640
$993$	− 6.43078i	− 0.204075i
$994$	20.7846i	0.659248i
$995$	−93.0333	−2.94935
$996$	1.60770	0.0509418
$997$	27.4641i	0.869797i	0.900480	+	0.434898i	$0.143216\pi$
$997$	27.4641i	0.869797i	−0.900480	+	0.434898i	$0.856784\pi$
$998$	18.5885	0.588407
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1369.2.b.d.1368.2		4
37.6	odd	4		1369.2.a.g.1.1		2
37.26	even	3		37.2.e.a.27.2	yes	4
37.27	even	6		37.2.e.a.11.2	✓	4
37.31	odd	4		1369.2.a.h.1.1		2
37.36	even	2	inner	1369.2.b.d.1368.4		4
111.26	odd	6		333.2.s.b.64.1		4
111.101	odd	6		333.2.s.b.307.1		4
148.27	odd	6		592.2.w.c.529.2		4
148.63	odd	6		592.2.w.c.545.2		4
185.27	odd	12		925.2.m.b.899.1		4
185.63	odd	12		925.2.m.b.249.1		4
185.64	even	6		925.2.n.a.751.1		4
185.137	odd	12		925.2.m.a.249.2		4
185.138	odd	12		925.2.m.a.899.2		4
185.174	even	6		925.2.n.a.101.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
37.2.e.a.11.2	✓	4	37.27	even	6
37.2.e.a.27.2	yes	4	37.26	even	3
333.2.s.b.64.1		4	111.26	odd	6
333.2.s.b.307.1		4	111.101	odd	6
592.2.w.c.529.2		4	148.27	odd	6
592.2.w.c.545.2		4	148.63	odd	6
925.2.m.a.249.2		4	185.137	odd	12
925.2.m.a.899.2		4	185.138	odd	12
925.2.m.b.249.1		4	185.63	odd	12
925.2.m.b.899.1		4	185.27	odd	12
925.2.n.a.101.1		4	185.174	even	6
925.2.n.a.751.1		4	185.64	even	6
1369.2.a.g.1.1		2	37.6	odd	4
1369.2.a.h.1.1		2	37.31	odd	4
1369.2.b.d.1368.2		4	1.1	even	1	trivial
1369.2.b.d.1368.4		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)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} +0.732051 q^{3} +1.00000 q^{4} -3.73205i q^{5} -0.732051i q^{6} -3.46410 q^{7} -3.00000i q^{8} -2.46410 q^{9} +O(q^{10})\)	\(q-1.00000i q^{2} +0.732051 q^{3} +1.00000 q^{4} -3.73205i q^{5} -0.732051i q^{6} -3.46410 q^{7} -3.00000i q^{8} -2.46410 q^{9} -3.73205 q^{10} -1.26795 q^{11} +0.732051 q^{12} +3.46410i q^{13} +3.46410i q^{14} -2.73205i q^{15} -1.00000 q^{16} +0.267949i q^{17} +2.46410i q^{18} -4.73205i q^{19} -3.73205i q^{20} -2.53590 q^{21} +1.26795i q^{22} +2.73205i q^{23} -2.19615i q^{24} -8.92820 q^{25} +3.46410 q^{26} -4.00000 q^{27} -3.46410 q^{28} +3.73205i q^{29} -2.73205 q^{30} +2.19615i q^{31} -5.00000i q^{32} -0.928203 q^{33} +0.267949 q^{34} +12.9282i q^{35} -2.46410 q^{36} -4.73205 q^{38} +2.53590i q^{39} -11.1962 q^{40} +3.00000 q^{41} +2.53590i q^{42} -3.46410i q^{43} -1.26795 q^{44} +9.19615i q^{45} +2.73205 q^{46} +8.19615 q^{47} -0.732051 q^{48} +5.00000 q^{49} +8.92820i q^{50} +0.196152i q^{51} +3.46410i q^{52} -9.46410 q^{53} +4.00000i q^{54} +4.73205i q^{55} +10.3923i q^{56} -3.46410i q^{57} +3.73205 q^{58} -6.53590i q^{59} -2.73205i q^{60} -7.73205i q^{61} +2.19615 q^{62} +8.53590 q^{63} -7.00000 q^{64} +12.9282 q^{65} +0.928203i q^{66} -10.7321 q^{67} +0.267949i q^{68} +2.00000i q^{69} +12.9282 q^{70} +6.00000 q^{71} +7.39230i q^{72} -6.53590 q^{75} -4.73205i q^{76} +4.39230 q^{77} +2.53590 q^{78} -14.1962i q^{79} +3.73205i q^{80} +4.46410 q^{81} -3.00000i q^{82} +2.19615 q^{83} -2.53590 q^{84} +1.00000 q^{85} -3.46410 q^{86} +2.73205i q^{87} +3.80385i q^{88} -1.19615i q^{89} +9.19615 q^{90} -12.0000i q^{91} +2.73205i q^{92} +1.60770i q^{93} -8.19615i q^{94} -17.6603 q^{95} -3.66025i q^{96} -7.73205i q^{97} -5.00000i q^{98} +3.12436 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{3} + 4 q^{4} + 4 q^{9}+O(q^{10}) \) 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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