LMFDB - Embedded newform 38.3.b.a.37.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [38,3,Mod(37,38)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(38, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("38.37");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 38 = 2 \cdot 19 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	38.b (of order $2$, degree $1$, 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:	$1.03542500457$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-2}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 2 $ x^2 + 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			37.2
Root			$1.41421i$ of defining polynomial
Character	$\chi$	$=$	38.37
Dual form			38.3.b.a.37.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.41421i q^{2} +2.82843i q^{3} -2.00000 q^{4} -1.00000 q^{5} -4.00000 q^{6} +5.00000 q^{7} -2.82843i q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.41421i q^{2} +2.82843i q^{3} -2.00000 q^{4} -1.00000 q^{5} -4.00000 q^{6} +5.00000 q^{7} -2.82843i q^{8} +1.00000 q^{9} -1.41421i q^{10} +5.00000 q^{11} -5.65685i q^{12} -16.9706i q^{13} +7.07107i q^{14} -2.82843i q^{15} +4.00000 q^{16} -25.0000 q^{17} +1.41421i q^{18} +19.0000 q^{19} +2.00000 q^{20} +14.1421i q^{21} +7.07107i q^{22} -10.0000 q^{23} +8.00000 q^{24} -24.0000 q^{25} +24.0000 q^{26} +28.2843i q^{27} -10.0000 q^{28} +42.4264i q^{29} +4.00000 q^{30} -42.4264i q^{31} +5.65685i q^{32} +14.1421i q^{33} -35.3553i q^{34} -5.00000 q^{35} -2.00000 q^{36} -25.4558i q^{37} +26.8701i q^{38} +48.0000 q^{39} +2.82843i q^{40} -42.4264i q^{41} -20.0000 q^{42} +5.00000 q^{43} -10.0000 q^{44} -1.00000 q^{45} -14.1421i q^{46} +5.00000 q^{47} +11.3137i q^{48} -24.0000 q^{49} -33.9411i q^{50} -70.7107i q^{51} +33.9411i q^{52} -25.4558i q^{53} -40.0000 q^{54} -5.00000 q^{55} -14.1421i q^{56} +53.7401i q^{57} -60.0000 q^{58} +84.8528i q^{59} +5.65685i q^{60} +95.0000 q^{61} +60.0000 q^{62} +5.00000 q^{63} -8.00000 q^{64} +16.9706i q^{65} -20.0000 q^{66} +110.309i q^{67} +50.0000 q^{68} -28.2843i q^{69} -7.07107i q^{70} -2.82843i q^{72} -25.0000 q^{73} +36.0000 q^{74} -67.8823i q^{75} -38.0000 q^{76} +25.0000 q^{77} +67.8823i q^{78} +42.4264i q^{79} -4.00000 q^{80} -71.0000 q^{81} +60.0000 q^{82} -130.000 q^{83} -28.2843i q^{84} +25.0000 q^{85} +7.07107i q^{86} -120.000 q^{87} -14.1421i q^{88} -127.279i q^{89} -1.41421i q^{90} -84.8528i q^{91} +20.0000 q^{92} +120.000 q^{93} +7.07107i q^{94} -19.0000 q^{95} -16.0000 q^{96} +16.9706i q^{97} -33.9411i q^{98} +5.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{4} - 2 q^{5} - 8 q^{6} + 10 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q - 4 * q^4 - 2 * q^5 - 8 * q^6 + 10 * q^7 + 2 * q^9	$ 2 q - 4 q^{4} - 2 q^{5} - 8 q^{6} + 10 q^{7} + 2 q^{9} + 10 q^{11} + 8 q^{16} - 50 q^{17} + 38 q^{19} + 4 q^{20} - 20 q^{23} + 16 q^{24} - 48 q^{25} + 48 q^{26} - 20 q^{28} + 8 q^{30} - 10 q^{35} - 4 q^{36} + 96 q^{39} - 40 q^{42} + 10 q^{43} - 20 q^{44} - 2 q^{45} + 10 q^{47} - 48 q^{49} - 80 q^{54} - 10 q^{55} - 120 q^{58} + 190 q^{61} + 120 q^{62} + 10 q^{63} - 16 q^{64} - 40 q^{66} + 100 q^{68} - 50 q^{73} + 72 q^{74} - 76 q^{76} + 50 q^{77} - 8 q^{80} - 142 q^{81} + 120 q^{82} - 260 q^{83} + 50 q^{85} - 240 q^{87} + 40 q^{92} + 240 q^{93} - 38 q^{95} - 32 q^{96} + 10 q^{99}+O(q^{100}) $ 2 * q - 4 * q^4 - 2 * q^5 - 8 * q^6 + 10 * q^7 + 2 * q^9 + 10 * q^11 + 8 * q^16 - 50 * q^17 + 38 * q^19 + 4 * q^20 - 20 * q^23 + 16 * q^24 - 48 * q^25 + 48 * q^26 - 20 * q^28 + 8 * q^30 - 10 * q^35 - 4 * q^36 + 96 * q^39 - 40 * q^42 + 10 * q^43 - 20 * q^44 - 2 * q^45 + 10 * q^47 - 48 * q^49 - 80 * q^54 - 10 * q^55 - 120 * q^58 + 190 * q^61 + 120 * q^62 + 10 * q^63 - 16 * q^64 - 40 * q^66 + 100 * q^68 - 50 * q^73 + 72 * q^74 - 76 * q^76 + 50 * q^77 - 8 * q^80 - 142 * q^81 + 120 * q^82 - 260 * q^83 + 50 * q^85 - 240 * q^87 + 40 * q^92 + 240 * q^93 - 38 * q^95 - 32 * q^96 + 10 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/38\mathbb{Z}\right)^\times$.

$n$	$21$
$\chi(n)$	$-1$

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 $

$2$	1.41421i	0.707107i
$2$	1.41421i	0.707107i
$3$	2.82843i	0.942809i	0.881917	+	0.471405i	$0.156253\pi$
$3$	2.82843i	0.942809i	−0.881917	+	0.471405i	$0.843747\pi$
$4$	−2.00000	−0.500000
$5$	−1.00000	−0.200000	−0.100000	−	0.994987i	$-0.531884\pi$
$5$	−1.00000	−0.200000	−0.100000	+	0.994987i	$0.531884\pi$
$6$	−4.00000	−0.666667
$7$	5.00000	0.714286	0.357143	−	0.934050i	$-0.383751\pi$
$7$	5.00000	0.714286	0.357143	+	0.934050i	$0.383751\pi$
$8$	− 2.82843i	− 0.353553i
$9$	1.00000	0.111111
$10$	− 1.41421i	− 0.141421i
$11$	5.00000	0.454545	0.227273	−	0.973831i	$-0.427019\pi$
$11$	5.00000	0.454545	0.227273	+	0.973831i	$0.427019\pi$
$12$	− 5.65685i	− 0.471405i
$13$	− 16.9706i	− 1.30543i	−0.757604	−	0.652714i	$-0.773630\pi$
$13$	− 16.9706i	− 1.30543i	0.757604	−	0.652714i	$-0.226370\pi$
$14$	7.07107i	0.505076i
$15$	− 2.82843i	− 0.188562i
$16$	4.00000	0.250000
$17$	−25.0000	−1.47059	−0.735294	−	0.677748i	$-0.762956\pi$
$17$	−25.0000	−1.47059	−0.735294	+	0.677748i	$0.762956\pi$
$18$	1.41421i	0.0785674i
$19$	19.0000	1.00000
$19$	19.0000	1.00000
$20$	2.00000	0.100000
$21$	14.1421i	0.673435i
$22$	7.07107i	0.321412i
$23$	−10.0000	−0.434783	−0.217391	−	0.976085i	$-0.569755\pi$
$23$	−10.0000	−0.434783	−0.217391	+	0.976085i	$0.569755\pi$
$24$	8.00000	0.333333
$25$	−24.0000	−0.960000
$26$	24.0000	0.923077
$27$	28.2843i	1.04757i
$28$	−10.0000	−0.357143
$29$	42.4264i	1.46298i	0.681852	+	0.731490i	$0.261175\pi$
$29$	42.4264i	1.46298i	−0.681852	+	0.731490i	$0.738825\pi$
$30$	4.00000	0.133333
$31$	− 42.4264i	− 1.36859i	−0.729204	−	0.684297i	$-0.760109\pi$
$31$	− 42.4264i	− 1.36859i	0.729204	−	0.684297i	$-0.239891\pi$
$32$	5.65685i	0.176777i
$33$	14.1421i	0.428550i
$34$	− 35.3553i	− 1.03986i
$35$	−5.00000	−0.142857
$36$	−2.00000	−0.0555556
$37$	− 25.4558i	− 0.687996i	−0.938970	−	0.343998i	$-0.888219\pi$
$37$	− 25.4558i	− 0.687996i	0.938970	−	0.343998i	$-0.111781\pi$
$38$	26.8701i	0.707107i
$39$	48.0000	1.23077
$40$	2.82843i	0.0707107i
$41$	− 42.4264i	− 1.03479i	−0.855747	−	0.517395i	$-0.826902\pi$
$41$	− 42.4264i	− 1.03479i	0.855747	−	0.517395i	$-0.173098\pi$
$42$	−20.0000	−0.476190
$43$	5.00000	0.116279	0.0581395	−	0.998308i	$-0.481483\pi$
$43$	5.00000	0.116279	0.0581395	+	0.998308i	$0.481483\pi$
$44$	−10.0000	−0.227273
$45$	−1.00000	−0.0222222
$46$	− 14.1421i	− 0.307438i
$47$	5.00000	0.106383	0.0531915	−	0.998584i	$-0.483061\pi$
$47$	5.00000	0.106383	0.0531915	+	0.998584i	$0.483061\pi$
$48$	11.3137i	0.235702i
$49$	−24.0000	−0.489796
$50$	− 33.9411i	− 0.678823i
$51$	− 70.7107i	− 1.38648i
$52$	33.9411i	0.652714i
$53$	− 25.4558i	− 0.480299i	−0.970736	−	0.240149i	$-0.922804\pi$
$53$	− 25.4558i	− 0.480299i	0.970736	−	0.240149i	$-0.0771965\pi$
$54$	−40.0000	−0.740741
$55$	−5.00000	−0.0909091
$56$	− 14.1421i	− 0.252538i
$57$	53.7401i	0.942809i
$58$	−60.0000	−1.03448
$59$	84.8528i	1.43818i	0.694915	+	0.719092i	$0.255442\pi$
$59$	84.8528i	1.43818i	−0.694915	+	0.719092i	$0.744558\pi$
$60$	5.65685i	0.0942809i
$61$	95.0000	1.55738	0.778689	−	0.627411i	$-0.215885\pi$
$61$	95.0000	1.55738	0.778689	+	0.627411i	$0.215885\pi$
$62$	60.0000	0.967742
$63$	5.00000	0.0793651
$64$	−8.00000	−0.125000
$65$	16.9706i	0.261086i
$66$	−20.0000	−0.303030
$67$	110.309i	1.64640i	0.567753	+	0.823199i	$0.307813\pi$
$67$	110.309i	1.64640i	−0.567753	+	0.823199i	$0.692187\pi$
$68$	50.0000	0.735294
$69$	− 28.2843i	− 0.409917i
$70$	− 7.07107i	− 0.101015i
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	− 2.82843i	− 0.0392837i
$73$	−25.0000	−0.342466	−0.171233	−	0.985231i	$-0.554775\pi$
$73$	−25.0000	−0.342466	−0.171233	+	0.985231i	$0.554775\pi$
$74$	36.0000	0.486486
$75$	− 67.8823i	− 0.905097i
$76$	−38.0000	−0.500000
$77$	25.0000	0.324675
$78$	67.8823i	0.870285i
$79$	42.4264i	0.537043i	0.963274	+	0.268522i	$0.0865351\pi$
$79$	42.4264i	0.537043i	−0.963274	+	0.268522i	$0.913465\pi$
$80$	−4.00000	−0.0500000
$81$	−71.0000	−0.876543
$82$	60.0000	0.731707
$83$	−130.000	−1.56627	−0.783133	−	0.621855i	$-0.786379\pi$
$83$	−130.000	−1.56627	−0.783133	+	0.621855i	$0.786379\pi$
$84$	− 28.2843i	− 0.336718i
$85$	25.0000	0.294118
$86$	7.07107i	0.0822217i
$87$	−120.000	−1.37931
$88$	− 14.1421i	− 0.160706i
$89$	− 127.279i	− 1.43010i	−0.699071	−	0.715052i	$-0.746403\pi$
$89$	− 127.279i	− 1.43010i	0.699071	−	0.715052i	$-0.253597\pi$
$90$	− 1.41421i	− 0.0157135i
$91$	− 84.8528i	− 0.932449i
$92$	20.0000	0.217391
$93$	120.000	1.29032
$94$	7.07107i	0.0752241i
$95$	−19.0000	−0.200000
$96$	−16.0000	−0.166667
$97$	16.9706i	0.174954i	0.996167	+	0.0874771i	$0.0278805\pi$
$97$	16.9706i	0.174954i	−0.996167	+	0.0874771i	$0.972120\pi$
$98$	− 33.9411i	− 0.346338i
$99$	5.00000	0.0505051
$100$	48.0000	0.480000
$101$	50.0000	0.495050	0.247525	−	0.968882i	$-0.420383\pi$
$101$	50.0000	0.495050	0.247525	+	0.968882i	$0.420383\pi$
$102$	100.000	0.980392
$103$	− 16.9706i	− 0.164763i	−0.996601	−	0.0823814i	$-0.973747\pi$
$103$	− 16.9706i	− 0.164763i	0.996601	−	0.0823814i	$-0.0262526\pi$
$104$	−48.0000	−0.461538
$105$	− 14.1421i	− 0.134687i
$106$	36.0000	0.339623
$107$	− 101.823i	− 0.951620i	−0.879548	−	0.475810i	$-0.842155\pi$
$107$	− 101.823i	− 0.951620i	0.879548	−	0.475810i	$-0.157845\pi$
$108$	− 56.5685i	− 0.523783i
$109$	127.279i	1.16770i	0.811862	+	0.583850i	$0.198454\pi$
$109$	127.279i	1.16770i	−0.811862	+	0.583850i	$0.801546\pi$
$110$	− 7.07107i	− 0.0642824i
$111$	72.0000	0.648649
$112$	20.0000	0.178571
$113$	110.309i	0.976183i	0.872793	+	0.488091i	$0.162307\pi$
$113$	110.309i	0.976183i	−0.872793	+	0.488091i	$0.837693\pi$
$114$	−76.0000	−0.666667
$115$	10.0000	0.0869565
$116$	− 84.8528i	− 0.731490i
$117$	− 16.9706i	− 0.145048i
$118$	−120.000	−1.01695
$119$	−125.000	−1.05042
$120$	−8.00000	−0.0666667
$121$	−96.0000	−0.793388
$122$	134.350i	1.10123i
$123$	120.000	0.975610
$124$	84.8528i	0.684297i
$125$	49.0000	0.392000
$126$	7.07107i	0.0561196i
$127$	229.103i	1.80396i	0.431780	+	0.901979i	$0.357886\pi$
$127$	229.103i	1.80396i	−0.431780	+	0.901979i	$0.642114\pi$
$128$	− 11.3137i	− 0.0883883i
$129$	14.1421i	0.109629i
$130$	−24.0000	−0.184615
$131$	−163.000	−1.24427	−0.622137	−	0.782908i	$-0.713735\pi$
$131$	−163.000	−1.24427	−0.622137	+	0.782908i	$0.713735\pi$
$132$	− 28.2843i	− 0.214275i
$133$	95.0000	0.714286
$134$	−156.000	−1.16418
$135$	− 28.2843i	− 0.209513i
$136$	70.7107i	0.519931i
$137$	95.0000	0.693431	0.346715	−	0.937970i	$-0.387297\pi$
$137$	95.0000	0.693431	0.346715	+	0.937970i	$0.387297\pi$
$138$	40.0000	0.289855
$139$	125.000	0.899281	0.449640	−	0.893210i	$-0.351552\pi$
$139$	125.000	0.899281	0.449640	+	0.893210i	$0.351552\pi$
$140$	10.0000	0.0714286
$141$	14.1421i	0.100299i
$142$	0	0
$143$	− 84.8528i	− 0.593376i
$144$	4.00000	0.0277778
$145$	− 42.4264i	− 0.292596i
$146$	− 35.3553i	− 0.242160i
$147$	− 67.8823i	− 0.461784i
$148$	50.9117i	0.343998i
$149$	215.000	1.44295	0.721477	−	0.692439i	$-0.243464\pi$
$149$	215.000	1.44295	0.721477	+	0.692439i	$0.243464\pi$
$150$	96.0000	0.640000
$151$	− 84.8528i	− 0.561939i	−0.959717	−	0.280970i	$-0.909344\pi$
$151$	− 84.8528i	− 0.561939i	0.959717	−	0.280970i	$-0.0906560\pi$
$152$	− 53.7401i	− 0.353553i
$153$	−25.0000	−0.163399
$154$	35.3553i	0.229580i
$155$	42.4264i	0.273719i
$156$	−96.0000	−0.615385
$157$	−190.000	−1.21019	−0.605096	−	0.796153i	$-0.706865\pi$
$157$	−190.000	−1.21019	−0.605096	+	0.796153i	$0.706865\pi$
$158$	−60.0000	−0.379747
$159$	72.0000	0.452830
$160$	− 5.65685i	− 0.0353553i
$161$	−50.0000	−0.310559
$162$	− 100.409i	− 0.619810i
$163$	110.000	0.674847	0.337423	−	0.941353i	$-0.390445\pi$
$163$	110.000	0.674847	0.337423	+	0.941353i	$0.390445\pi$
$164$	84.8528i	0.517395i
$165$	− 14.1421i	− 0.0857099i
$166$	− 183.848i	− 1.10752i
$167$	59.3970i	0.355670i	0.984060	+	0.177835i	$0.0569094\pi$
$167$	59.3970i	0.355670i	−0.984060	+	0.177835i	$0.943091\pi$
$168$	40.0000	0.238095
$169$	−119.000	−0.704142
$170$	35.3553i	0.207973i
$171$	19.0000	0.111111
$172$	−10.0000	−0.0581395
$173$	− 186.676i	− 1.07905i	−0.841969	−	0.539527i	$-0.818603\pi$
$173$	− 186.676i	− 1.07905i	0.841969	−	0.539527i	$-0.181397\pi$
$174$	− 169.706i	− 0.975320i
$175$	−120.000	−0.685714
$176$	20.0000	0.113636
$177$	−240.000	−1.35593
$178$	180.000	1.01124
$179$	− 127.279i	− 0.711057i	−0.934665	−	0.355529i	$-0.884301\pi$
$179$	− 127.279i	− 0.711057i	0.934665	−	0.355529i	$-0.115699\pi$
$180$	2.00000	0.0111111
$181$	− 254.558i	− 1.40640i	−0.710992	−	0.703200i	$-0.751754\pi$
$181$	− 254.558i	− 1.40640i	0.710992	−	0.703200i	$-0.248246\pi$
$182$	120.000	0.659341
$183$	268.701i	1.46831i
$184$	28.2843i	0.153719i
$185$	25.4558i	0.137599i
$186$	169.706i	0.912396i
$187$	−125.000	−0.668449
$188$	−10.0000	−0.0531915
$189$	141.421i	0.748261i
$190$	− 26.8701i	− 0.141421i
$191$	293.000	1.53403	0.767016	−	0.641628i	$-0.221741\pi$
$191$	293.000	1.53403	0.767016	+	0.641628i	$0.221741\pi$
$192$	− 22.6274i	− 0.117851i
$193$	59.3970i	0.307756i	0.988090	+	0.153878i	$0.0491763\pi$
$193$	59.3970i	0.307756i	−0.988090	+	0.153878i	$0.950824\pi$
$194$	−24.0000	−0.123711
$195$	−48.0000	−0.246154
$196$	48.0000	0.244898
$197$	−70.0000	−0.355330	−0.177665	−	0.984091i	$-0.556854\pi$
$197$	−70.0000	−0.355330	−0.177665	+	0.984091i	$0.556854\pi$
$198$	7.07107i	0.0357125i
$199$	173.000	0.869347	0.434673	−	0.900588i	$-0.356864\pi$
$199$	173.000	0.869347	0.434673	+	0.900588i	$0.356864\pi$
$200$	67.8823i	0.339411i
$201$	−312.000	−1.55224
$202$	70.7107i	0.350053i
$203$	212.132i	1.04499i
$204$	141.421i	0.693242i
$205$	42.4264i	0.206958i
$206$	24.0000	0.116505
$207$	−10.0000	−0.0483092
$208$	− 67.8823i	− 0.326357i
$209$	95.0000	0.454545
$210$	20.0000	0.0952381
$211$	84.8528i	0.402146i	0.979576	+	0.201073i	$0.0644429\pi$
$211$	84.8528i	0.402146i	−0.979576	+	0.201073i	$0.935557\pi$
$212$	50.9117i	0.240149i
$213$	0	0
$214$	144.000	0.672897
$215$	−5.00000	−0.0232558
$216$	80.0000	0.370370
$217$	− 212.132i	− 0.977567i
$218$	−180.000	−0.825688
$219$	− 70.7107i	− 0.322880i
$220$	10.0000	0.0454545
$221$	424.264i	1.91975i
$222$	101.823i	0.458664i
$223$	− 364.867i	− 1.63618i	−0.575094	−	0.818088i	$-0.695034\pi$
$223$	− 364.867i	− 1.63618i	0.575094	−	0.818088i	$-0.304966\pi$
$224$	28.2843i	0.126269i
$225$	−24.0000	−0.106667
$226$	−156.000	−0.690265
$227$	67.8823i	0.299041i	0.988759	+	0.149520i	$0.0477730\pi$
$227$	67.8823i	0.299041i	−0.988759	+	0.149520i	$0.952227\pi$
$228$	− 107.480i	− 0.471405i
$229$	−145.000	−0.633188	−0.316594	−	0.948561i	$-0.602539\pi$
$229$	−145.000	−0.633188	−0.316594	+	0.948561i	$0.602539\pi$
$230$	14.1421i	0.0614875i
$231$	70.7107i	0.306107i
$232$	120.000	0.517241
$233$	335.000	1.43777	0.718884	−	0.695130i	$-0.244653\pi$
$233$	335.000	1.43777	0.718884	+	0.695130i	$0.244653\pi$
$234$	24.0000	0.102564
$235$	−5.00000	−0.0212766
$236$	− 169.706i	− 0.719092i
$237$	−120.000	−0.506329
$238$	− 176.777i	− 0.742759i
$239$	197.000	0.824268	0.412134	−	0.911123i	$-0.364784\pi$
$239$	197.000	0.824268	0.412134	+	0.911123i	$0.364784\pi$
$240$	− 11.3137i	− 0.0471405i
$241$	296.985i	1.23230i	0.787628	+	0.616151i	$0.211309\pi$
$241$	296.985i	1.23230i	−0.787628	+	0.616151i	$0.788691\pi$
$242$	− 135.765i	− 0.561010i
$243$	53.7401i	0.221153i
$244$	−190.000	−0.778689
$245$	24.0000	0.0979592
$246$	169.706i	0.689860i
$247$	− 322.441i	− 1.30543i
$248$	−120.000	−0.483871
$249$	− 367.696i	− 1.47669i
$250$	69.2965i	0.277186i
$251$	173.000	0.689243	0.344622	−	0.938742i	$-0.388007\pi$
$251$	173.000	0.689243	0.344622	+	0.938742i	$0.388007\pi$
$252$	−10.0000	−0.0396825
$253$	−50.0000	−0.197628
$254$	−324.000	−1.27559
$255$	70.7107i	0.277297i
$256$	16.0000	0.0625000
$257$	− 67.8823i	− 0.264133i	−0.991241	−	0.132067i	$-0.957839\pi$
$257$	− 67.8823i	− 0.264133i	0.991241	−	0.132067i	$-0.0421613\pi$
$258$	−20.0000	−0.0775194
$259$	− 127.279i	− 0.491426i
$260$	− 33.9411i	− 0.130543i
$261$	42.4264i	0.162553i
$262$	− 230.517i	− 0.879835i
$263$	−355.000	−1.34981	−0.674905	−	0.737905i	$-0.735815\pi$
$263$	−355.000	−1.34981	−0.674905	+	0.737905i	$0.735815\pi$
$264$	40.0000	0.151515
$265$	25.4558i	0.0960598i
$266$	134.350i	0.505076i
$267$	360.000	1.34831
$268$	− 220.617i	− 0.823199i
$269$	− 381.838i	− 1.41947i	−0.704468	−	0.709735i	$-0.748814\pi$
$269$	− 381.838i	− 1.41947i	0.704468	−	0.709735i	$-0.251186\pi$
$270$	40.0000	0.148148
$271$	110.000	0.405904	0.202952	−	0.979189i	$-0.434946\pi$
$271$	110.000	0.405904	0.202952	+	0.979189i	$0.434946\pi$
$272$	−100.000	−0.367647
$273$	240.000	0.879121
$274$	134.350i	0.490330i
$275$	−120.000	−0.436364
$276$	56.5685i	0.204958i
$277$	−265.000	−0.956679	−0.478339	−	0.878175i	$-0.658761\pi$
$277$	−265.000	−0.956679	−0.478339	+	0.878175i	$0.658761\pi$
$278$	176.777i	0.635887i
$279$	− 42.4264i	− 0.152066i
$280$	14.1421i	0.0505076i
$281$	424.264i	1.50984i	0.655819	+	0.754918i	$0.272323\pi$
$281$	424.264i	1.50984i	−0.655819	+	0.754918i	$0.727677\pi$
$282$	−20.0000	−0.0709220
$283$	125.000	0.441696	0.220848	−	0.975308i	$-0.429118\pi$
$283$	125.000	0.441696	0.220848	+	0.975308i	$0.429118\pi$
$284$	0	0
$285$	− 53.7401i	− 0.188562i
$286$	120.000	0.419580
$287$	− 212.132i	− 0.739136i
$288$	5.65685i	0.0196419i
$289$	336.000	1.16263
$290$	60.0000	0.206897
$291$	−48.0000	−0.164948
$292$	50.0000	0.171233
$293$	186.676i	0.637120i	0.947903	+	0.318560i	$0.103199\pi$
$293$	186.676i	0.637120i	−0.947903	+	0.318560i	$0.896801\pi$
$294$	96.0000	0.326531
$295$	− 84.8528i	− 0.287637i
$296$	−72.0000	−0.243243
$297$	141.421i	0.476166i
$298$	304.056i	1.02032i
$299$	169.706i	0.567577i
$300$	135.765i	0.452548i
$301$	25.0000	0.0830565
$302$	120.000	0.397351
$303$	141.421i	0.466737i
$304$	76.0000	0.250000
$305$	−95.0000	−0.311475
$306$	− 35.3553i	− 0.115540i
$307$	280.014i	0.912099i	0.889955	+	0.456049i	$0.150736\pi$
$307$	280.014i	0.912099i	−0.889955	+	0.456049i	$0.849264\pi$
$308$	−50.0000	−0.162338
$309$	48.0000	0.155340
$310$	−60.0000	−0.193548
$311$	−235.000	−0.755627	−0.377814	−	0.925882i	$-0.623324\pi$
$311$	−235.000	−0.755627	−0.377814	+	0.925882i	$0.623324\pi$
$312$	− 135.765i	− 0.435143i
$313$	−310.000	−0.990415	−0.495208	−	0.868775i	$-0.664908\pi$
$313$	−310.000	−0.990415	−0.495208	+	0.868775i	$0.664908\pi$
$314$	− 268.701i	− 0.855734i
$315$	−5.00000	−0.0158730
$316$	− 84.8528i	− 0.268522i
$317$	− 186.676i	− 0.588884i	−0.955669	−	0.294442i	$-0.904866\pi$
$317$	− 186.676i	− 0.588884i	0.955669	−	0.294442i	$-0.0951338\pi$
$318$	101.823i	0.320199i
$319$	212.132i	0.664991i
$320$	8.00000	0.0250000
$321$	288.000	0.897196
$322$	− 70.7107i	− 0.219598i
$323$	−475.000	−1.47059
$324$	142.000	0.438272
$325$	407.294i	1.25321i
$326$	155.563i	0.477189i
$327$	−360.000	−1.10092
$328$	−120.000	−0.365854
$329$	25.0000	0.0759878
$330$	20.0000	0.0606061
$331$	296.985i	0.897235i	0.893724	+	0.448618i	$0.148083\pi$
$331$	296.985i	0.897235i	−0.893724	+	0.448618i	$0.851917\pi$
$332$	260.000	0.783133
$333$	− 25.4558i	− 0.0764440i
$334$	−84.0000	−0.251497
$335$	− 110.309i	− 0.329280i
$336$	56.5685i	0.168359i
$337$	− 526.087i	− 1.56109i	−0.625099	−	0.780545i	$-0.714942\pi$
$337$	− 526.087i	− 1.56109i	0.625099	−	0.780545i	$-0.285058\pi$
$338$	− 168.291i	− 0.497904i
$339$	−312.000	−0.920354
$340$	−50.0000	−0.147059
$341$	− 212.132i	− 0.622088i
$342$	26.8701i	0.0785674i
$343$	−365.000	−1.06414
$344$	− 14.1421i	− 0.0411109i
$345$	28.2843i	0.0819834i
$346$	264.000	0.763006
$347$	125.000	0.360231	0.180115	−	0.983646i	$-0.442353\pi$
$347$	125.000	0.360231	0.180115	+	0.983646i	$0.442353\pi$
$348$	240.000	0.689655
$349$	23.0000	0.0659026	0.0329513	−	0.999457i	$-0.489509\pi$
$349$	23.0000	0.0659026	0.0329513	+	0.999457i	$0.489509\pi$
$350$	− 169.706i	− 0.484873i
$351$	480.000	1.36752
$352$	28.2843i	0.0803530i
$353$	410.000	1.16147	0.580737	−	0.814092i	$-0.302765\pi$
$353$	410.000	1.16147	0.580737	+	0.814092i	$0.302765\pi$
$354$	− 339.411i	− 0.958789i
$355$	0	0
$356$	254.558i	0.715052i
$357$	− 353.553i	− 0.990346i
$358$	180.000	0.502793
$359$	−475.000	−1.32312	−0.661560	−	0.749892i	$-0.730105\pi$
$359$	−475.000	−1.32312	−0.661560	+	0.749892i	$0.730105\pi$
$360$	2.82843i	0.00785674i
$361$	361.000	1.00000
$362$	360.000	0.994475
$363$	− 271.529i	− 0.748014i
$364$	169.706i	0.466224i
$365$	25.0000	0.0684932
$366$	−380.000	−1.03825
$367$	230.000	0.626703	0.313351	−	0.949637i	$-0.398548\pi$
$367$	230.000	0.626703	0.313351	+	0.949637i	$0.398548\pi$
$368$	−40.0000	−0.108696
$369$	− 42.4264i	− 0.114977i
$370$	−36.0000	−0.0972973
$371$	− 127.279i	− 0.343071i
$372$	−240.000	−0.645161
$373$	− 67.8823i	− 0.181990i	−0.995851	−	0.0909950i	$-0.970995\pi$
$373$	− 67.8823i	− 0.181990i	0.995851	−	0.0909950i	$-0.0290047\pi$
$374$	− 176.777i	− 0.472665i
$375$	138.593i	0.369581i
$376$	− 14.1421i	− 0.0376121i
$377$	720.000	1.90981
$378$	−200.000	−0.529101
$379$	− 254.558i	− 0.671658i	−0.941923	−	0.335829i	$-0.890984\pi$
$379$	− 254.558i	− 0.671658i	0.941923	−	0.335829i	$-0.109016\pi$
$380$	38.0000	0.100000
$381$	−648.000	−1.70079
$382$	414.365i	1.08472i
$383$	− 144.250i	− 0.376631i	−0.982109	−	0.188316i	$-0.939697\pi$
$383$	− 144.250i	− 0.376631i	0.982109	−	0.188316i	$-0.0603028\pi$
$384$	32.0000	0.0833333
$385$	−25.0000	−0.0649351
$386$	−84.0000	−0.217617
$387$	5.00000	0.0129199
$388$	− 33.9411i	− 0.0874771i
$389$	−553.000	−1.42159	−0.710797	−	0.703397i	$-0.751666\pi$
$389$	−553.000	−1.42159	−0.710797	+	0.703397i	$0.751666\pi$
$390$	− 67.8823i	− 0.174057i
$391$	250.000	0.639386
$392$	67.8823i	0.173169i
$393$	− 461.034i	− 1.17311i
$394$	− 98.9949i	− 0.251256i
$395$	− 42.4264i	− 0.107409i
$396$	−10.0000	−0.0252525
$397$	335.000	0.843829	0.421914	−	0.906636i	$-0.361358\pi$
$397$	335.000	0.843829	0.421914	+	0.906636i	$0.361358\pi$
$398$	244.659i	0.614721i
$399$	268.701i	0.673435i
$400$	−96.0000	−0.240000
$401$	212.132i	0.529008i	0.964385	+	0.264504i	$0.0852082\pi$
$401$	212.132i	0.529008i	−0.964385	+	0.264504i	$0.914792\pi$
$402$	− 441.235i	− 1.09760i
$403$	−720.000	−1.78660
$404$	−100.000	−0.247525
$405$	71.0000	0.175309
$406$	−300.000	−0.738916
$407$	− 127.279i	− 0.312725i
$408$	−200.000	−0.490196
$409$	721.249i	1.76344i	0.471769	+	0.881722i	$0.343616\pi$
$409$	721.249i	1.76344i	−0.471769	+	0.881722i	$0.656384\pi$
$410$	−60.0000	−0.146341
$411$	268.701i	0.653773i
$412$	33.9411i	0.0823814i
$413$	424.264i	1.02727i
$414$	− 14.1421i	− 0.0341597i
$415$	130.000	0.313253
$416$	96.0000	0.230769
$417$	353.553i	0.847850i
$418$	134.350i	0.321412i
$419$	62.0000	0.147971	0.0739857	−	0.997259i	$-0.476428\pi$
$419$	62.0000	0.147971	0.0739857	+	0.997259i	$0.476428\pi$
$420$	28.2843i	0.0673435i
$421$	296.985i	0.705427i	0.935731	+	0.352714i	$0.114741\pi$
$421$	296.985i	0.705427i	−0.935731	+	0.352714i	$0.885259\pi$
$422$	−120.000	−0.284360
$423$	5.00000	0.0118203
$424$	−72.0000	−0.169811
$425$	600.000	1.41176
$426$	0	0
$427$	475.000	1.11241
$428$	203.647i	0.475810i
$429$	240.000	0.559441
$430$	− 7.07107i	− 0.0164443i
$431$	− 509.117i	− 1.18125i	−0.806948	−	0.590623i	$-0.798882\pi$
$431$	− 509.117i	− 1.18125i	0.806948	−	0.590623i	$-0.201118\pi$
$432$	113.137i	0.261891i
$433$	− 229.103i	− 0.529105i	−0.964371	−	0.264553i	$-0.914776\pi$
$433$	− 229.103i	− 0.529105i	0.964371	−	0.264553i	$-0.0852243\pi$
$434$	300.000	0.691244
$435$	120.000	0.275862
$436$	− 254.558i	− 0.583850i
$437$	−190.000	−0.434783
$438$	100.000	0.228311
$439$	806.102i	1.83622i	0.396323	+	0.918111i	$0.370286\pi$
$439$	806.102i	1.83622i	−0.396323	+	0.918111i	$0.629714\pi$
$440$	14.1421i	0.0321412i
$441$	−24.0000	−0.0544218
$442$	−600.000	−1.35747
$443$	365.000	0.823928	0.411964	−	0.911200i	$-0.364843\pi$
$443$	365.000	0.823928	0.411964	+	0.911200i	$0.364843\pi$
$444$	−144.000	−0.324324
$445$	127.279i	0.286021i
$446$	516.000	1.15695
$447$	608.112i	1.36043i
$448$	−40.0000	−0.0892857
$449$	− 763.675i	− 1.70084i	−0.526108	−	0.850418i	$-0.676349\pi$
$449$	− 763.675i	− 1.70084i	0.526108	−	0.850418i	$-0.323651\pi$
$450$	− 33.9411i	− 0.0754247i
$451$	− 212.132i	− 0.470359i
$452$	− 220.617i	− 0.488091i
$453$	240.000	0.529801
$454$	−96.0000	−0.211454
$455$	84.8528i	0.186490i
$456$	152.000	0.333333
$457$	−265.000	−0.579869	−0.289934	−	0.957047i	$-0.593633\pi$
$457$	−265.000	−0.579869	−0.289934	+	0.957047i	$0.593633\pi$
$458$	− 205.061i	− 0.447731i
$459$	− 707.107i	− 1.54054i
$460$	−20.0000	−0.0434783
$461$	−553.000	−1.19957	−0.599783	−	0.800163i	$-0.704746\pi$
$461$	−553.000	−1.19957	−0.599783	+	0.800163i	$0.704746\pi$
$462$	−100.000	−0.216450
$463$	485.000	1.04752	0.523758	−	0.851867i	$-0.324530\pi$
$463$	485.000	1.04752	0.523758	+	0.851867i	$0.324530\pi$
$464$	169.706i	0.365745i
$465$	−120.000	−0.258065
$466$	473.762i	1.01666i
$467$	−115.000	−0.246253	−0.123126	−	0.992391i	$-0.539292\pi$
$467$	−115.000	−0.246253	−0.123126	+	0.992391i	$0.539292\pi$
$468$	33.9411i	0.0725238i
$469$	551.543i	1.17600i
$470$	− 7.07107i	− 0.0150448i
$471$	− 537.401i	− 1.14098i
$472$	240.000	0.508475
$473$	25.0000	0.0528541
$474$	− 169.706i	− 0.358029i
$475$	−456.000	−0.960000
$476$	250.000	0.525210
$477$	− 25.4558i	− 0.0533665i
$478$	278.600i	0.582845i
$479$	−490.000	−1.02296	−0.511482	−	0.859294i	$-0.670903\pi$
$479$	−490.000	−1.02296	−0.511482	+	0.859294i	$0.670903\pi$
$480$	16.0000	0.0333333
$481$	−432.000	−0.898129
$482$	−420.000	−0.871369
$483$	− 141.421i	− 0.292798i
$484$	192.000	0.396694
$485$	− 16.9706i	− 0.0349909i
$486$	−76.0000	−0.156379
$487$	− 610.940i	− 1.25450i	−0.778819	−	0.627249i	$-0.784181\pi$
$487$	− 610.940i	− 1.25450i	0.778819	−	0.627249i	$-0.215819\pi$
$488$	− 268.701i	− 0.550616i
$489$	311.127i	0.636252i
$490$	33.9411i	0.0692676i
$491$	−82.0000	−0.167006	−0.0835031	−	0.996508i	$-0.526611\pi$
$491$	−82.0000	−0.167006	−0.0835031	+	0.996508i	$0.526611\pi$
$492$	−240.000	−0.487805
$493$	− 1060.66i	− 2.15144i
$494$	456.000	0.923077
$495$	−5.00000	−0.0101010
$496$	− 169.706i	− 0.342148i
$497$	0	0
$498$	520.000	1.04418
$499$	485.000	0.971944	0.485972	−	0.873974i	$-0.338466\pi$
$499$	485.000	0.971944	0.485972	+	0.873974i	$0.338466\pi$
$500$	−98.0000	−0.196000
$501$	−168.000	−0.335329
$502$	244.659i	0.487368i
$503$	−250.000	−0.497018	−0.248509	−	0.968630i	$-0.579941\pi$
$503$	−250.000	−0.497018	−0.248509	+	0.968630i	$0.579941\pi$
$504$	− 14.1421i	− 0.0280598i
$505$	−50.0000	−0.0990099
$506$	− 70.7107i	− 0.139744i
$507$	− 336.583i	− 0.663871i
$508$	− 458.205i	− 0.901979i
$509$	− 169.706i	− 0.333410i	−0.986007	−	0.166705i	$-0.946687\pi$
$509$	− 169.706i	− 0.333410i	0.986007	−	0.166705i	$-0.0533127\pi$
$510$	−100.000	−0.196078
$511$	−125.000	−0.244618
$512$	22.6274i	0.0441942i
$513$	537.401i	1.04757i
$514$	96.0000	0.186770
$515$	16.9706i	0.0329525i
$516$	− 28.2843i	− 0.0548145i
$517$	25.0000	0.0483559
$518$	180.000	0.347490
$519$	528.000	1.01734
$520$	48.0000	0.0923077
$521$	− 127.279i	− 0.244298i	−0.992512	−	0.122149i	$-0.961021\pi$
$521$	− 127.279i	− 0.244298i	0.992512	−	0.122149i	$-0.0389786\pi$
$522$	−60.0000	−0.114943
$523$	356.382i	0.681418i	0.940169	+	0.340709i	$0.110667\pi$
$523$	356.382i	0.681418i	−0.940169	+	0.340709i	$0.889333\pi$
$524$	326.000	0.622137
$525$	− 339.411i	− 0.646498i
$526$	− 502.046i	− 0.954460i
$527$	1060.66i	2.01264i
$528$	56.5685i	0.107137i
$529$	−429.000	−0.810964
$530$	−36.0000	−0.0679245
$531$	84.8528i	0.159798i
$532$	−190.000	−0.357143
$533$	−720.000	−1.35084
$534$	509.117i	0.953402i
$535$	101.823i	0.190324i
$536$	312.000	0.582090
$537$	360.000	0.670391
$538$	540.000	1.00372
$539$	−120.000	−0.222635
$540$	56.5685i	0.104757i
$541$	−25.0000	−0.0462107	−0.0231054	−	0.999733i	$-0.507355\pi$
$541$	−25.0000	−0.0462107	−0.0231054	+	0.999733i	$0.507355\pi$
$542$	155.563i	0.287018i
$543$	720.000	1.32597
$544$	− 141.421i	− 0.259966i
$545$	− 127.279i	− 0.233540i
$546$	339.411i	0.621632i
$547$	16.9706i	0.0310248i	0.999880	+	0.0155124i	$0.00493795\pi$
$547$	16.9706i	0.0310248i	−0.999880	+	0.0155124i	$0.995062\pi$
$548$	−190.000	−0.346715
$549$	95.0000	0.173042
$550$	− 169.706i	− 0.308556i
$551$	806.102i	1.46298i
$552$	−80.0000	−0.144928
$553$	212.132i	0.383602i
$554$	− 374.767i	− 0.676474i
$555$	−72.0000	−0.129730
$556$	−250.000	−0.449640
$557$	−745.000	−1.33752	−0.668761	−	0.743477i	$-0.733175\pi$
$557$	−745.000	−1.33752	−0.668761	+	0.743477i	$0.733175\pi$
$558$	60.0000	0.107527
$559$	− 84.8528i	− 0.151794i
$560$	−20.0000	−0.0357143
$561$	− 353.553i	− 0.630220i
$562$	−600.000	−1.06762
$563$	313.955i	0.557647i	0.960342	+	0.278824i	$0.0899445\pi$
$563$	313.955i	0.557647i	−0.960342	+	0.278824i	$0.910056\pi$
$564$	− 28.2843i	− 0.0501494i
$565$	− 110.309i	− 0.195237i
$566$	176.777i	0.312326i
$567$	−355.000	−0.626102
$568$	0	0
$569$	424.264i	0.745631i	0.927905	+	0.372816i	$0.121608\pi$
$569$	424.264i	0.745631i	−0.927905	+	0.372816i	$0.878392\pi$
$570$	76.0000	0.133333
$571$	1070.00	1.87391	0.936953	−	0.349456i	$-0.113634\pi$
$571$	1070.00	1.87391	0.936953	+	0.349456i	$0.113634\pi$
$572$	169.706i	0.296688i
$573$	828.729i	1.44630i
$574$	300.000	0.522648
$575$	240.000	0.417391
$576$	−8.00000	−0.0138889
$577$	−25.0000	−0.0433276	−0.0216638	−	0.999765i	$-0.506896\pi$
$577$	−25.0000	−0.0433276	−0.0216638	+	0.999765i	$0.506896\pi$
$578$	475.176i	0.822103i
$579$	−168.000	−0.290155
$580$	84.8528i	0.146298i
$581$	−650.000	−1.11876
$582$	− 67.8823i	− 0.116636i
$583$	− 127.279i	− 0.218318i
$584$	70.7107i	0.121080i
$585$	16.9706i	0.0290095i
$586$	−264.000	−0.450512
$587$	725.000	1.23509	0.617547	−	0.786534i	$-0.288127\pi$
$587$	725.000	1.23509	0.617547	+	0.786534i	$0.288127\pi$
$588$	135.765i	0.230892i
$589$	− 806.102i	− 1.36859i
$590$	120.000	0.203390
$591$	− 197.990i	− 0.335008i
$592$	− 101.823i	− 0.171999i
$593$	650.000	1.09612	0.548061	−	0.836439i	$-0.315366\pi$
$593$	650.000	1.09612	0.548061	+	0.836439i	$0.315366\pi$
$594$	−200.000	−0.336700
$595$	125.000	0.210084
$596$	−430.000	−0.721477
$597$	489.318i	0.819628i
$598$	−240.000	−0.401338
$599$	− 296.985i	− 0.495801i	−0.968785	−	0.247901i	$-0.920259\pi$
$599$	− 296.985i	− 0.495801i	0.968785	−	0.247901i	$-0.0797406\pi$
$600$	−192.000	−0.320000
$601$	− 848.528i	− 1.41186i	−0.708281	−	0.705930i	$-0.750529\pi$
$601$	− 848.528i	− 1.41186i	0.708281	−	0.705930i	$-0.249471\pi$
$602$	35.3553i	0.0587298i
$603$	110.309i	0.182933i
$604$	169.706i	0.280970i
$605$	96.0000	0.158678
$606$	−200.000	−0.330033
$607$	− 271.529i	− 0.447329i	−0.974666	−	0.223665i	$-0.928198\pi$
$607$	− 271.529i	− 0.447329i	0.974666	−	0.223665i	$-0.0718021\pi$
$608$	107.480i	0.176777i
$609$	−600.000	−0.985222
$610$	− 134.350i	− 0.220246i
$611$	− 84.8528i	− 0.138875i
$612$	50.0000	0.0816993
$613$	1055.00	1.72104	0.860522	−	0.509413i	$-0.170138\pi$
$613$	1055.00	1.72104	0.860522	+	0.509413i	$0.170138\pi$
$614$	−396.000	−0.644951
$615$	−120.000	−0.195122
$616$	− 70.7107i	− 0.114790i
$617$	−505.000	−0.818476	−0.409238	−	0.912428i	$-0.634206\pi$
$617$	−505.000	−0.818476	−0.409238	+	0.912428i	$0.634206\pi$
$618$	67.8823i	0.109842i
$619$	−130.000	−0.210016	−0.105008	−	0.994471i	$-0.533487\pi$
$619$	−130.000	−0.210016	−0.105008	+	0.994471i	$0.533487\pi$
$620$	− 84.8528i	− 0.136859i
$621$	− 282.843i	− 0.455463i
$622$	− 332.340i	− 0.534309i
$623$	− 636.396i	− 1.02150i
$624$	192.000	0.307692
$625$	551.000	0.881600
$626$	− 438.406i	− 0.700329i
$627$	268.701i	0.428550i
$628$	380.000	0.605096
$629$	636.396i	1.01176i
$630$	− 7.07107i	− 0.0112239i
$631$	−475.000	−0.752773	−0.376387	−	0.926463i	$-0.622834\pi$
$631$	−475.000	−0.752773	−0.376387	+	0.926463i	$0.622834\pi$
$632$	120.000	0.189873
$633$	−240.000	−0.379147
$634$	264.000	0.416404
$635$	− 229.103i	− 0.360791i
$636$	−144.000	−0.226415
$637$	407.294i	0.639393i
$638$	−300.000	−0.470219
$639$	0	0
$640$	11.3137i	0.0176777i
$641$	− 848.528i	− 1.32376i	−0.749611	−	0.661878i	$-0.769760\pi$
$641$	− 848.528i	− 1.32376i	0.749611	−	0.661878i	$-0.230240\pi$
$642$	407.294i	0.634414i
$643$	−955.000	−1.48523	−0.742613	−	0.669721i	$-0.766414\pi$
$643$	−955.000	−1.48523	−0.742613	+	0.669721i	$0.766414\pi$
$644$	100.000	0.155280
$645$	− 14.1421i	− 0.0219258i
$646$	− 671.751i	− 1.03986i
$647$	965.000	1.49150	0.745750	−	0.666226i	$-0.232092\pi$
$647$	965.000	1.49150	0.745750	+	0.666226i	$0.232092\pi$
$648$	200.818i	0.309905i
$649$	424.264i	0.653720i
$650$	−576.000	−0.886154
$651$	600.000	0.921659
$652$	−220.000	−0.337423
$653$	935.000	1.43185	0.715926	−	0.698176i	$-0.246005\pi$
$653$	935.000	1.43185	0.715926	+	0.698176i	$0.246005\pi$
$654$	− 509.117i	− 0.778466i
$655$	163.000	0.248855
$656$	− 169.706i	− 0.258698i
$657$	−25.0000	−0.0380518
$658$	35.3553i	0.0537315i
$659$	− 84.8528i	− 0.128760i	−0.997925	−	0.0643800i	$-0.979493\pi$
$659$	− 84.8528i	− 0.128760i	0.997925	−	0.0643800i	$-0.0205070\pi$
$660$	28.2843i	0.0428550i
$661$	− 678.823i	− 1.02696i	−0.858101	−	0.513481i	$-0.828356\pi$
$661$	− 678.823i	− 1.02696i	0.858101	−	0.513481i	$-0.171644\pi$
$662$	−420.000	−0.634441
$663$	−1200.00	−1.80995
$664$	367.696i	0.553758i
$665$	−95.0000	−0.142857
$666$	36.0000	0.0540541
$667$	− 424.264i	− 0.636078i
$668$	− 118.794i	− 0.177835i
$669$	1032.00	1.54260
$670$	156.000	0.232836
$671$	475.000	0.707899
$672$	−80.0000	−0.119048
$673$	− 186.676i	− 0.277379i	−0.990336	−	0.138690i	$-0.955711\pi$
$673$	− 186.676i	− 0.277379i	0.990336	−	0.138690i	$-0.0442890\pi$
$674$	744.000	1.10386
$675$	− 678.823i	− 1.00566i
$676$	238.000	0.352071
$677$	907.925i	1.34110i	0.741864	+	0.670550i	$0.233942\pi$
$677$	907.925i	1.34110i	−0.741864	+	0.670550i	$0.766058\pi$
$678$	− 441.235i	− 0.650789i
$679$	84.8528i	0.124967i
$680$	− 70.7107i	− 0.103986i
$681$	−192.000	−0.281938
$682$	300.000	0.439883
$683$	1120.06i	1.63991i	0.572429	+	0.819954i	$0.306001\pi$
$683$	1120.06i	1.63991i	−0.572429	+	0.819954i	$0.693999\pi$
$684$	−38.0000	−0.0555556
$685$	−95.0000	−0.138686
$686$	− 516.188i	− 0.752461i
$687$	− 410.122i	− 0.596975i
$688$	20.0000	0.0290698
$689$	−432.000	−0.626996
$690$	−40.0000	−0.0579710
$691$	−715.000	−1.03473	−0.517366	−	0.855764i	$-0.673087\pi$
$691$	−715.000	−1.03473	−0.517366	+	0.855764i	$0.673087\pi$
$692$	373.352i	0.539527i
$693$	25.0000	0.0360750
$694$	176.777i	0.254721i
$695$	−125.000	−0.179856
$696$	339.411i	0.487660i
$697$	1060.66i	1.52175i
$698$	32.5269i	0.0466002i
$699$	947.523i	1.35554i
$700$	240.000	0.342857
$701$	−430.000	−0.613409	−0.306705	−	0.951805i	$-0.599226\pi$
$701$	−430.000	−0.613409	−0.306705	+	0.951805i	$0.599226\pi$
$702$	678.823i	0.966984i
$703$	− 483.661i	− 0.687996i
$704$	−40.0000	−0.0568182
$705$	− 14.1421i	− 0.0200598i
$706$	579.828i	0.821285i
$707$	250.000	0.353607
$708$	480.000	0.677966
$709$	−382.000	−0.538787	−0.269394	−	0.963030i	$-0.586823\pi$
$709$	−382.000	−0.538787	−0.269394	+	0.963030i	$0.586823\pi$
$710$	0	0
$711$	42.4264i	0.0596715i
$712$	−360.000	−0.505618
$713$	424.264i	0.595041i
$714$	500.000	0.700280
$715$	84.8528i	0.118675i
$716$	254.558i	0.355529i
$717$	557.200i	0.777127i
$718$	− 671.751i	− 0.935587i
$719$	−115.000	−0.159944	−0.0799722	−	0.996797i	$-0.525483\pi$
$719$	−115.000	−0.159944	−0.0799722	+	0.996797i	$0.525483\pi$
$720$	−4.00000	−0.00555556
$721$	− 84.8528i	− 0.117688i
$722$	510.531i	0.707107i
$723$	−840.000	−1.16183
$724$	509.117i	0.703200i
$725$	− 1018.23i	− 1.40446i
$726$	384.000	0.528926
$727$	−1075.00	−1.47868	−0.739340	−	0.673333i	$-0.764862\pi$
$727$	−1075.00	−1.47868	−0.739340	+	0.673333i	$0.764862\pi$
$728$	−240.000	−0.329670
$729$	−791.000	−1.08505
$730$	35.3553i	0.0484320i
$731$	−125.000	−0.170999
$732$	− 537.401i	− 0.734155i
$733$	530.000	0.723056	0.361528	−	0.932361i	$-0.382255\pi$
$733$	530.000	0.723056	0.361528	+	0.932361i	$0.382255\pi$
$734$	325.269i	0.443146i
$735$	67.8823i	0.0923568i
$736$	− 56.5685i	− 0.0768594i
$737$	551.543i	0.748363i
$738$	60.0000	0.0813008
$739$	−547.000	−0.740189	−0.370095	−	0.928994i	$-0.620675\pi$
$739$	−547.000	−0.740189	−0.370095	+	0.928994i	$0.620675\pi$
$740$	− 50.9117i	− 0.0687996i
$741$	912.000	1.23077
$742$	180.000	0.242588
$743$	− 958.837i	− 1.29049i	−0.763974	−	0.645247i	$-0.776755\pi$
$743$	− 958.837i	− 1.29049i	0.763974	−	0.645247i	$-0.223245\pi$
$744$	− 339.411i	− 0.456198i
$745$	−215.000	−0.288591
$746$	96.0000	0.128686
$747$	−130.000	−0.174029
$748$	250.000	0.334225
$749$	− 509.117i	− 0.679729i
$750$	−196.000	−0.261333
$751$	− 169.706i	− 0.225973i	−0.993597	−	0.112986i	$-0.963958\pi$
$751$	− 169.706i	− 0.225973i	0.993597	−	0.112986i	$-0.0360417\pi$
$752$	20.0000	0.0265957
$753$	489.318i	0.649825i
$754$	1018.23i	1.35044i
$755$	84.8528i	0.112388i
$756$	− 282.843i	− 0.374131i
$757$	1055.00	1.39366	0.696830	−	0.717237i	$-0.254593\pi$
$757$	1055.00	1.39366	0.696830	+	0.717237i	$0.254593\pi$
$758$	360.000	0.474934
$759$	− 141.421i	− 0.186326i
$760$	53.7401i	0.0707107i
$761$	215.000	0.282523	0.141261	−	0.989972i	$-0.454884\pi$
$761$	215.000	0.282523	0.141261	+	0.989972i	$0.454884\pi$
$762$	− 916.410i	− 1.20264i
$763$	636.396i	0.834071i
$764$	−586.000	−0.767016
$765$	25.0000	0.0326797
$766$	204.000	0.266319
$767$	1440.00	1.87744
$768$	45.2548i	0.0589256i
$769$	−145.000	−0.188557	−0.0942783	−	0.995546i	$-0.530054\pi$
$769$	−145.000	−0.188557	−0.0942783	+	0.995546i	$0.530054\pi$
$770$	− 35.3553i	− 0.0459160i
$771$	192.000	0.249027
$772$	− 118.794i	− 0.153878i
$773$	407.294i	0.526900i	0.964673	+	0.263450i	$0.0848604\pi$
$773$	407.294i	0.526900i	−0.964673	+	0.263450i	$0.915140\pi$
$774$	7.07107i	0.00913575i
$775$	1018.23i	1.31385i
$776$	48.0000	0.0618557
$777$	360.000	0.463320
$778$	− 782.060i	− 1.00522i
$779$	− 806.102i	− 1.03479i
$780$	96.0000	0.123077
$781$	0	0
$782$	353.553i	0.452114i
$783$	−1200.00	−1.53257
$784$	−96.0000	−0.122449
$785$	190.000	0.242038
$786$	652.000	0.829517
$787$	186.676i	0.237200i	0.992942	+	0.118600i	$0.0378406\pi$
$787$	186.676i	0.237200i	−0.992942	+	0.118600i	$0.962159\pi$
$788$	140.000	0.177665
$789$	− 1004.09i	− 1.27261i
$790$	60.0000	0.0759494
$791$	551.543i	0.697273i
$792$	− 14.1421i	− 0.0178562i
$793$	− 1612.20i	− 2.03304i
$794$	473.762i	0.596677i
$795$	−72.0000	−0.0905660
$796$	−346.000	−0.434673
$797$	− 704.278i	− 0.883662i	−0.897098	−	0.441831i	$-0.854329\pi$
$797$	− 704.278i	− 0.883662i	0.897098	−	0.441831i	$-0.145671\pi$
$798$	−380.000	−0.476190
$799$	−125.000	−0.156446
$800$	− 135.765i	− 0.169706i
$801$	− 127.279i	− 0.158900i
$802$	−300.000	−0.374065
$803$	−125.000	−0.155666
$804$	624.000	0.776119
$805$	50.0000	0.0621118
$806$	− 1018.23i	− 1.26332i
$807$	1080.00	1.33829
$808$	− 141.421i	− 0.175026i
$809$	−457.000	−0.564895	−0.282447	−	0.959283i	$-0.591146\pi$
$809$	−457.000	−0.564895	−0.282447	+	0.959283i	$0.591146\pi$
$810$	100.409i	0.123962i
$811$	509.117i	0.627764i	0.949462	+	0.313882i	$0.101630\pi$
$811$	509.117i	0.627764i	−0.949462	+	0.313882i	$0.898370\pi$
$812$	− 424.264i	− 0.522493i
$813$	311.127i	0.382690i
$814$	180.000	0.221130
$815$	−110.000	−0.134969
$816$	− 282.843i	− 0.346621i
$817$	95.0000	0.116279
$818$	−1020.00	−1.24694
$819$	− 84.8528i	− 0.103605i
$820$	− 84.8528i	− 0.103479i
$821$	167.000	0.203410	0.101705	−	0.994815i	$-0.467570\pi$
$821$	167.000	0.203410	0.101705	+	0.994815i	$0.467570\pi$
$822$	−380.000	−0.462287
$823$	−1315.00	−1.59781	−0.798906	−	0.601455i	$-0.794588\pi$
$823$	−1315.00	−1.59781	−0.798906	+	0.601455i	$0.794588\pi$
$824$	−48.0000	−0.0582524
$825$	− 339.411i	− 0.411408i
$826$	−600.000	−0.726392
$827$	− 534.573i	− 0.646400i	−0.946331	−	0.323200i	$-0.895241\pi$
$827$	− 534.573i	− 0.646400i	0.946331	−	0.323200i	$-0.104759\pi$
$828$	20.0000	0.0241546
$829$	763.675i	0.921201i	0.887608	+	0.460600i	$0.152366\pi$
$829$	763.675i	0.921201i	−0.887608	+	0.460600i	$0.847634\pi$
$830$	183.848i	0.221503i
$831$	− 749.533i	− 0.901965i
$832$	135.765i	0.163178i
$833$	600.000	0.720288
$834$	−500.000	−0.599520
$835$	− 59.3970i	− 0.0711341i
$836$	−190.000	−0.227273
$837$	1200.00	1.43369
$838$	87.6812i	0.104632i
$839$	− 339.411i	− 0.404543i	−0.979330	−	0.202271i	$-0.935168\pi$
$839$	− 339.411i	− 0.404543i	0.979330	−	0.202271i	$-0.0648323\pi$
$840$	−40.0000	−0.0476190
$841$	−959.000	−1.14031
$842$	−420.000	−0.498812
$843$	−1200.00	−1.42349
$844$	− 169.706i	− 0.201073i
$845$	119.000	0.140828
$846$	7.07107i	0.00835824i
$847$	−480.000	−0.566706
$848$	− 101.823i	− 0.120075i
$849$	353.553i	0.416435i
$850$	848.528i	0.998268i
$851$	254.558i	0.299129i
$852$	0	0
$853$	770.000	0.902696	0.451348	−	0.892348i	$-0.350943\pi$
$853$	770.000	0.902696	0.451348	+	0.892348i	$0.350943\pi$
$854$	671.751i	0.786594i
$855$	−19.0000	−0.0222222
$856$	−288.000	−0.336449
$857$	− 1255.82i	− 1.46537i	−0.680568	−	0.732685i	$-0.738267\pi$
$857$	− 1255.82i	− 1.46537i	0.680568	−	0.732685i	$-0.261733\pi$
$858$	339.411i	0.395584i
$859$	557.000	0.648428	0.324214	−	0.945984i	$-0.394900\pi$
$859$	557.000	0.648428	0.324214	+	0.945984i	$0.394900\pi$
$860$	10.0000	0.0116279
$861$	600.000	0.696864
$862$	720.000	0.835267
$863$	992.778i	1.15038i	0.818020	+	0.575190i	$0.195072\pi$
$863$	992.778i	1.15038i	−0.818020	+	0.575190i	$0.804928\pi$
$864$	−160.000	−0.185185
$865$	186.676i	0.215811i
$866$	324.000	0.374134
$867$	950.352i	1.09614i
$868$	424.264i	0.488783i
$869$	212.132i	0.244111i
$870$	169.706i	0.195064i
$871$	1872.00	2.14925
$872$	360.000	0.412844
$873$	16.9706i	0.0194394i
$874$	− 268.701i	− 0.307438i
$875$	245.000	0.280000
$876$	141.421i	0.161440i
$877$	186.676i	0.212858i	0.994320	+	0.106429i	$0.0339416\pi$
$877$	186.676i	0.212858i	−0.994320	+	0.106429i	$0.966058\pi$
$878$	−1140.00	−1.29841
$879$	−528.000	−0.600683
$880$	−20.0000	−0.0227273
$881$	−25.0000	−0.0283768	−0.0141884	−	0.999899i	$-0.504516\pi$
$881$	−25.0000	−0.0283768	−0.0141884	+	0.999899i	$0.504516\pi$
$882$	− 33.9411i	− 0.0384820i
$883$	965.000	1.09287	0.546433	−	0.837503i	$-0.315985\pi$
$883$	965.000	1.09287	0.546433	+	0.837503i	$0.315985\pi$
$884$	− 848.528i	− 0.959873i
$885$	240.000	0.271186
$886$	516.188i	0.582605i
$887$	− 780.646i	− 0.880097i	−0.897974	−	0.440048i	$-0.854961\pi$
$887$	− 780.646i	− 0.880097i	0.897974	−	0.440048i	$-0.145039\pi$
$888$	− 203.647i	− 0.229332i
$889$	1145.51i	1.28854i
$890$	−180.000	−0.202247
$891$	−355.000	−0.398429
$892$	729.734i	0.818088i
$893$	95.0000	0.106383
$894$	−860.000	−0.961969
$895$	127.279i	0.142211i
$896$	− 56.5685i	− 0.0631345i
$897$	−480.000	−0.535117
$898$	1080.00	1.20267
$899$	1800.00	2.00222
$900$	48.0000	0.0533333
$901$	636.396i	0.706322i
$902$	300.000	0.332594
$903$	70.7107i	0.0783064i
$904$	312.000	0.345133
$905$	254.558i	0.281280i
$906$	339.411i	0.374626i
$907$	− 313.955i	− 0.346147i	−0.984909	−	0.173074i	$-0.944630\pi$
$907$	− 313.955i	− 0.346147i	0.984909	−	0.173074i	$-0.0553698\pi$
$908$	− 135.765i	− 0.149520i
$909$	50.0000	0.0550055
$910$	−120.000	−0.131868
$911$	933.381i	1.02457i	0.858816	+	0.512284i	$0.171200\pi$
$911$	933.381i	1.02457i	−0.858816	+	0.512284i	$0.828800\pi$
$912$	214.960i	0.235702i
$913$	−650.000	−0.711939
$914$	− 374.767i	− 0.410029i
$915$	− 268.701i	− 0.293662i
$916$	290.000	0.316594
$917$	−815.000	−0.888768
$918$	1000.00	1.08932
$919$	−538.000	−0.585419	−0.292709	−	0.956201i	$-0.594557\pi$
$919$	−538.000	−0.585419	−0.292709	+	0.956201i	$0.594557\pi$
$920$	− 28.2843i	− 0.0307438i
$921$	−792.000	−0.859935
$922$	− 782.060i	− 0.848221i
$923$	0	0
$924$	− 141.421i	− 0.153053i
$925$	610.940i	0.660476i
$926$	685.894i	0.740706i
$927$	− 16.9706i	− 0.0183070i
$928$	−240.000	−0.258621
$929$	−742.000	−0.798708	−0.399354	−	0.916797i	$-0.630766\pi$
$929$	−742.000	−0.798708	−0.399354	+	0.916797i	$0.630766\pi$
$930$	− 169.706i	− 0.182479i
$931$	−456.000	−0.489796
$932$	−670.000	−0.718884
$933$	− 664.680i	− 0.712412i
$934$	− 162.635i	− 0.174127i
$935$	125.000	0.133690
$936$	−48.0000	−0.0512821
$937$	335.000	0.357524	0.178762	−	0.983892i	$-0.442791\pi$
$937$	335.000	0.357524	0.178762	+	0.983892i	$0.442791\pi$
$938$	−780.000	−0.831557
$939$	− 876.812i	− 0.933773i
$940$	10.0000	0.0106383
$941$	− 424.264i	− 0.450865i	−0.974259	−	0.225433i	$-0.927620\pi$
$941$	− 424.264i	− 0.450865i	0.974259	−	0.225433i	$-0.0723795\pi$
$942$	760.000	0.806794
$943$	424.264i	0.449909i
$944$	339.411i	0.359546i
$945$	− 141.421i	− 0.149652i
$946$	35.3553i	0.0373735i
$947$	−1210.00	−1.27772	−0.638860	−	0.769323i	$-0.720594\pi$
$947$	−1210.00	−1.27772	−0.638860	+	0.769323i	$0.720594\pi$
$948$	240.000	0.253165
$949$	424.264i	0.447064i
$950$	− 644.881i	− 0.678823i
$951$	528.000	0.555205
$952$	353.553i	0.371380i
$953$	992.778i	1.04174i	0.853636	+	0.520870i	$0.174392\pi$
$953$	992.778i	1.04174i	−0.853636	+	0.520870i	$0.825608\pi$
$954$	36.0000	0.0377358
$955$	−293.000	−0.306806
$956$	−394.000	−0.412134
$957$	−600.000	−0.626959
$958$	− 692.965i	− 0.723345i
$959$	475.000	0.495308
$960$	22.6274i	0.0235702i
$961$	−839.000	−0.873049
$962$	− 610.940i	− 0.635073i
$963$	− 101.823i	− 0.105736i
$964$	− 593.970i	− 0.616151i
$965$	− 59.3970i	− 0.0615513i
$966$	200.000	0.207039
$967$	350.000	0.361944	0.180972	−	0.983488i	$-0.442076\pi$
$967$	350.000	0.361944	0.180972	+	0.983488i	$0.442076\pi$
$968$	271.529i	0.280505i
$969$	− 1343.50i	− 1.38648i
$970$	24.0000	0.0247423
$971$	254.558i	0.262161i	0.991372	+	0.131081i	$0.0418447\pi$
$971$	254.558i	0.262161i	−0.991372	+	0.131081i	$0.958155\pi$
$972$	− 107.480i	− 0.110576i
$973$	625.000	0.642343
$974$	864.000	0.887064
$975$	−1152.00	−1.18154
$976$	380.000	0.389344
$977$	− 398.808i	− 0.408197i	−0.978950	−	0.204098i	$-0.934574\pi$
$977$	− 398.808i	− 0.408197i	0.978950	−	0.204098i	$-0.0654262\pi$
$978$	−440.000	−0.449898
$979$	− 636.396i	− 0.650047i
$980$	−48.0000	−0.0489796
$981$	127.279i	0.129744i
$982$	− 115.966i	− 0.118091i
$983$	− 695.793i	− 0.707826i	−0.935278	−	0.353913i	$-0.884851\pi$
$983$	− 695.793i	− 0.707826i	0.935278	−	0.353913i	$-0.115149\pi$
$984$	− 339.411i	− 0.344930i
$985$	70.0000	0.0710660
$986$	1500.00	1.52130
$987$	70.7107i	0.0716420i
$988$	644.881i	0.652714i
$989$	−50.0000	−0.0505561
$990$	− 7.07107i	− 0.00714249i
$991$	381.838i	0.385305i	0.981267	+	0.192653i	$0.0617091\pi$
$991$	381.838i	0.385305i	−0.981267	+	0.192653i	$0.938291\pi$
$992$	240.000	0.241935
$993$	−840.000	−0.845921
$994$	0	0
$995$	−173.000	−0.173869
$996$	735.391i	0.738344i
$997$	−265.000	−0.265797	−0.132899	−	0.991130i	$-0.542428\pi$
$997$	−265.000	−0.265797	−0.132899	+	0.991130i	$0.542428\pi$
$998$	685.894i	0.687268i
$999$	720.000	0.720721

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	38.3.b.a.37.2	yes	2
3.2	odd	2		342.3.d.a.37.1		2
4.3	odd	2		304.3.e.c.113.1		2
5.2	odd	4		950.3.d.a.949.2		4
5.3	odd	4		950.3.d.a.949.3		4
5.4	even	2		950.3.c.a.151.1		2
8.3	odd	2		1216.3.e.i.1025.2		2
8.5	even	2		1216.3.e.j.1025.1		2
12.11	even	2		2736.3.o.h.721.1		2
19.18	odd	2	inner	38.3.b.a.37.1	✓	2
57.56	even	2		342.3.d.a.37.2		2
76.75	even	2		304.3.e.c.113.2		2
95.18	even	4		950.3.d.a.949.1		4
95.37	even	4		950.3.d.a.949.4		4
95.94	odd	2		950.3.c.a.151.2		2
152.37	odd	2		1216.3.e.j.1025.2		2
152.75	even	2		1216.3.e.i.1025.1		2
228.227	odd	2		2736.3.o.h.721.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.3.b.a.37.1	✓	2	19.18	odd	2	inner
38.3.b.a.37.2	yes	2	1.1	even	1	trivial
304.3.e.c.113.1		2	4.3	odd	2
304.3.e.c.113.2		2	76.75	even	2
342.3.d.a.37.1		2	3.2	odd	2
342.3.d.a.37.2		2	57.56	even	2
950.3.c.a.151.1		2	5.4	even	2
950.3.c.a.151.2		2	95.94	odd	2
950.3.d.a.949.1		4	95.18	even	4
950.3.d.a.949.2		4	5.2	odd	4
950.3.d.a.949.3		4	5.3	odd	4
950.3.d.a.949.4		4	95.37	even	4
1216.3.e.i.1025.1		2	152.75	even	2
1216.3.e.i.1025.2		2	8.3	odd	2
1216.3.e.j.1025.1		2	8.5	even	2
1216.3.e.j.1025.2		2	152.37	odd	2
2736.3.o.h.721.1		2	12.11	even	2
2736.3.o.h.721.2		2	228.227	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 38 = 2 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	38.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.41421i q^{2} +2.82843i q^{3} -2.00000 q^{4} -1.00000 q^{5} -4.00000 q^{6} +5.00000 q^{7} -2.82843i q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.41421i q^{2} +2.82843i q^{3} -2.00000 q^{4} -1.00000 q^{5} -4.00000 q^{6} +5.00000 q^{7} -2.82843i q^{8} +1.00000 q^{9} -1.41421i q^{10} +5.00000 q^{11} -5.65685i q^{12} -16.9706i q^{13} +7.07107i q^{14} -2.82843i q^{15} +4.00000 q^{16} -25.0000 q^{17} +1.41421i q^{18} +19.0000 q^{19} +2.00000 q^{20} +14.1421i q^{21} +7.07107i q^{22} -10.0000 q^{23} +8.00000 q^{24} -24.0000 q^{25} +24.0000 q^{26} +28.2843i q^{27} -10.0000 q^{28} +42.4264i q^{29} +4.00000 q^{30} -42.4264i q^{31} +5.65685i q^{32} +14.1421i q^{33} -35.3553i q^{34} -5.00000 q^{35} -2.00000 q^{36} -25.4558i q^{37} +26.8701i q^{38} +48.0000 q^{39} +2.82843i q^{40} -42.4264i q^{41} -20.0000 q^{42} +5.00000 q^{43} -10.0000 q^{44} -1.00000 q^{45} -14.1421i q^{46} +5.00000 q^{47} +11.3137i q^{48} -24.0000 q^{49} -33.9411i q^{50} -70.7107i q^{51} +33.9411i q^{52} -25.4558i q^{53} -40.0000 q^{54} -5.00000 q^{55} -14.1421i q^{56} +53.7401i q^{57} -60.0000 q^{58} +84.8528i q^{59} +5.65685i q^{60} +95.0000 q^{61} +60.0000 q^{62} +5.00000 q^{63} -8.00000 q^{64} +16.9706i q^{65} -20.0000 q^{66} +110.309i q^{67} +50.0000 q^{68} -28.2843i q^{69} -7.07107i q^{70} -2.82843i q^{72} -25.0000 q^{73} +36.0000 q^{74} -67.8823i q^{75} -38.0000 q^{76} +25.0000 q^{77} +67.8823i q^{78} +42.4264i q^{79} -4.00000 q^{80} -71.0000 q^{81} +60.0000 q^{82} -130.000 q^{83} -28.2843i q^{84} +25.0000 q^{85} +7.07107i q^{86} -120.000 q^{87} -14.1421i q^{88} -127.279i q^{89} -1.41421i q^{90} -84.8528i q^{91} +20.0000 q^{92} +120.000 q^{93} +7.07107i q^{94} -19.0000 q^{95} -16.0000 q^{96} +16.9706i q^{97} -33.9411i q^{98} +5.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{4} - 2 q^{5} - 8 q^{6} + 10 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q - 4 * q^4 - 2 * q^5 - 8 * q^6 + 10 * q^7 + 2 * q^9	\( 2 q - 4 q^{4} - 2 q^{5} - 8 q^{6} + 10 q^{7} + 2 q^{9} + 10 q^{11} + 8 q^{16} - 50 q^{17} + 38 q^{19} + 4 q^{20} - 20 q^{23} + 16 q^{24} - 48 q^{25} + 48 q^{26} - 20 q^{28} + 8 q^{30} - 10 q^{35} - 4 q^{36} + 96 q^{39} - 40 q^{42} + 10 q^{43} - 20 q^{44} - 2 q^{45} + 10 q^{47} - 48 q^{49} - 80 q^{54} - 10 q^{55} - 120 q^{58} + 190 q^{61} + 120 q^{62} + 10 q^{63} - 16 q^{64} - 40 q^{66} + 100 q^{68} - 50 q^{73} + 72 q^{74} - 76 q^{76} + 50 q^{77} - 8 q^{80} - 142 q^{81} + 120 q^{82} - 260 q^{83} + 50 q^{85} - 240 q^{87} + 40 q^{92} + 240 q^{93} - 38 q^{95} - 32 q^{96} + 10 q^{99}+O(q^{100}) \) 2 * q - 4 * q^4 - 2 * q^5 - 8 * q^6 + 10 * q^7 + 2 * q^9 + 10 * q^11 + 8 * q^16 - 50 * q^17 + 38 * q^19 + 4 * q^20 - 20 * q^23 + 16 * q^24 - 48 * q^25 + 48 * q^26 - 20 * q^28 + 8 * q^30 - 10 * q^35 - 4 * q^36 + 96 * q^39 - 40 * q^42 + 10 * q^43 - 20 * q^44 - 2 * q^45 + 10 * q^47 - 48 * q^49 - 80 * q^54 - 10 * q^55 - 120 * q^58 + 190 * q^61 + 120 * q^62 + 10 * q^63 - 16 * q^64 - 40 * q^66 + 100 * q^68 - 50 * q^73 + 72 * q^74 - 76 * q^76 + 50 * q^77 - 8 * q^80 - 142 * q^81 + 120 * q^82 - 260 * q^83 + 50 * q^85 - 240 * q^87 + 40 * q^92 + 240 * q^93 - 38 * q^95 - 32 * q^96 + 10 * q^99

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