LMFDB - Embedded newform 1110.2.d.f.889.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1110,2,Mod(889,1110)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1110, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

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

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

chi := DirichletCharacter("1110.889");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1110 = 2 \cdot 3 \cdot 5 \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1110.d (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:	$8.86339462436$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 3x^{2} + 1 $ x^4 + 3*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			889.4
Root			$1.61803i$ of defining polynomial
Character	$\chi$	$=$	1110.889
Dual form			1110.2.d.f.889.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +2.23607i q^{5} -1.00000 q^{6} +2.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +2.23607i q^{5} -1.00000 q^{6} +2.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} -2.23607 q^{10} +5.23607 q^{11} -1.00000i q^{12} +4.47214i q^{13} -2.00000 q^{14} -2.23607 q^{15} +1.00000 q^{16} +4.47214i q^{17} -1.00000i q^{18} +7.23607 q^{19} -2.23607i q^{20} -2.00000 q^{21} +5.23607i q^{22} -4.00000i q^{23} +1.00000 q^{24} -5.00000 q^{25} -4.47214 q^{26} -1.00000i q^{27} -2.00000i q^{28} -4.00000 q^{29} -2.23607i q^{30} -2.47214 q^{31} +1.00000i q^{32} +5.23607i q^{33} -4.47214 q^{34} -4.47214 q^{35} +1.00000 q^{36} +1.00000i q^{37} +7.23607i q^{38} -4.47214 q^{39} +2.23607 q^{40} -4.47214 q^{41} -2.00000i q^{42} -10.4721i q^{43} -5.23607 q^{44} -2.23607i q^{45} +4.00000 q^{46} +9.70820i q^{47} +1.00000i q^{48} +3.00000 q^{49} -5.00000i q^{50} -4.47214 q^{51} -4.47214i q^{52} -12.4721i q^{53} +1.00000 q^{54} +11.7082i q^{55} +2.00000 q^{56} +7.23607i q^{57} -4.00000i q^{58} -6.94427 q^{59} +2.23607 q^{60} +5.23607 q^{61} -2.47214i q^{62} -2.00000i q^{63} -1.00000 q^{64} -10.0000 q^{65} -5.23607 q^{66} -10.4721i q^{67} -4.47214i q^{68} +4.00000 q^{69} -4.47214i q^{70} -2.47214 q^{71} +1.00000i q^{72} -10.4721i q^{73} -1.00000 q^{74} -5.00000i q^{75} -7.23607 q^{76} +10.4721i q^{77} -4.47214i q^{78} +2.47214 q^{79} +2.23607i q^{80} +1.00000 q^{81} -4.47214i q^{82} +15.4164i q^{83} +2.00000 q^{84} -10.0000 q^{85} +10.4721 q^{86} -4.00000i q^{87} -5.23607i q^{88} +7.52786 q^{89} +2.23607 q^{90} -8.94427 q^{91} +4.00000i q^{92} -2.47214i q^{93} -9.70820 q^{94} +16.1803i q^{95} -1.00000 q^{96} -8.18034i q^{97} +3.00000i q^{98} -5.23607 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{4} - 4 q^{6} - 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^4 - 4 * q^6 - 4 * q^9	$ 4 q - 4 q^{4} - 4 q^{6} - 4 q^{9} + 12 q^{11} - 8 q^{14} + 4 q^{16} + 20 q^{19} - 8 q^{21} + 4 q^{24} - 20 q^{25} - 16 q^{29} + 8 q^{31} + 4 q^{36} - 12 q^{44} + 16 q^{46} + 12 q^{49} + 4 q^{54} + 8 q^{56} + 8 q^{59} + 12 q^{61} - 4 q^{64} - 40 q^{65} - 12 q^{66} + 16 q^{69} + 8 q^{71} - 4 q^{74} - 20 q^{76} - 8 q^{79} + 4 q^{81} + 8 q^{84} - 40 q^{85} + 24 q^{86} + 48 q^{89} - 12 q^{94} - 4 q^{96} - 12 q^{99}+O(q^{100}) $ 4 * q - 4 * q^4 - 4 * q^6 - 4 * q^9 + 12 * q^11 - 8 * q^14 + 4 * q^16 + 20 * q^19 - 8 * q^21 + 4 * q^24 - 20 * q^25 - 16 * q^29 + 8 * q^31 + 4 * q^36 - 12 * q^44 + 16 * q^46 + 12 * q^49 + 4 * q^54 + 8 * q^56 + 8 * q^59 + 12 * q^61 - 4 * q^64 - 40 * q^65 - 12 * q^66 + 16 * q^69 + 8 * q^71 - 4 * q^74 - 20 * q^76 - 8 * q^79 + 4 * q^81 + 8 * q^84 - 40 * q^85 + 24 * q^86 + 48 * q^89 - 12 * q^94 - 4 * q^96 - 12 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1110\mathbb{Z}\right)^\times$.

$n$	$371$	$631$	$667$
$\chi(n)$	$1$	$1$	$-1$

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
$2$	1.00000i	0.707107i
$3$	1.00000i	0.577350i
$3$	1.00000i	0.577350i
$4$	−1.00000	−0.500000
$5$	2.23607i	1.00000i
$5$	2.23607i	1.00000i
$6$	−1.00000	−0.408248
$7$	2.00000i	0.755929i	0.925820	+	0.377964i	$0.123376\pi$
$7$	2.00000i	0.755929i	−0.925820	+	0.377964i	$0.876624\pi$
$8$	− 1.00000i	− 0.353553i
$9$	−1.00000	−0.333333
$10$	−2.23607	−0.707107
$11$	5.23607	1.57873	0.789367	−	0.613922i	$-0.210409\pi$
$11$	5.23607	1.57873	0.789367	+	0.613922i	$0.210409\pi$
$12$	− 1.00000i	− 0.288675i
$13$	4.47214i	1.24035i	0.784465	+	0.620174i	$0.212938\pi$
$13$	4.47214i	1.24035i	−0.784465	+	0.620174i	$0.787062\pi$
$14$	−2.00000	−0.534522
$15$	−2.23607	−0.577350
$16$	1.00000	0.250000
$17$	4.47214i	1.08465i	0.840168	+	0.542326i	$0.182456\pi$
$17$	4.47214i	1.08465i	−0.840168	+	0.542326i	$0.817544\pi$
$18$	− 1.00000i	− 0.235702i
$19$	7.23607	1.66007	0.830034	−	0.557713i	$-0.188321\pi$
$19$	7.23607	1.66007	0.830034	+	0.557713i	$0.188321\pi$
$20$	− 2.23607i	− 0.500000i
$21$	−2.00000	−0.436436
$22$	5.23607i	1.11633i
$23$	− 4.00000i	− 0.834058i	−0.908893	−	0.417029i	$-0.863071\pi$
$23$	− 4.00000i	− 0.834058i	0.908893	−	0.417029i	$-0.136929\pi$
$24$	1.00000	0.204124
$25$	−5.00000	−1.00000
$26$	−4.47214	−0.877058
$27$	− 1.00000i	− 0.192450i
$28$	− 2.00000i	− 0.377964i
$29$	−4.00000	−0.742781	−0.371391	−	0.928477i	$-0.621119\pi$
$29$	−4.00000	−0.742781	−0.371391	+	0.928477i	$0.621119\pi$
$30$	− 2.23607i	− 0.408248i
$31$	−2.47214	−0.444009	−0.222004	−	0.975046i	$-0.571260\pi$
$31$	−2.47214	−0.444009	−0.222004	+	0.975046i	$0.571260\pi$
$32$	1.00000i	0.176777i
$33$	5.23607i	0.911482i
$34$	−4.47214	−0.766965
$35$	−4.47214	−0.755929
$36$	1.00000	0.166667
$37$	1.00000i	0.164399i
$37$	1.00000i	0.164399i
$38$	7.23607i	1.17385i
$39$	−4.47214	−0.716115
$40$	2.23607	0.353553
$41$	−4.47214	−0.698430	−0.349215	−	0.937043i	$-0.613552\pi$
$41$	−4.47214	−0.698430	−0.349215	+	0.937043i	$0.613552\pi$
$42$	− 2.00000i	− 0.308607i
$43$	− 10.4721i	− 1.59699i	−0.602004	−	0.798493i	$-0.705631\pi$
$43$	− 10.4721i	− 1.59699i	0.602004	−	0.798493i	$-0.294369\pi$
$44$	−5.23607	−0.789367
$45$	− 2.23607i	− 0.333333i
$46$	4.00000	0.589768
$47$	9.70820i	1.41609i	0.706169	+	0.708044i	$0.250422\pi$
$47$	9.70820i	1.41609i	−0.706169	+	0.708044i	$0.749578\pi$
$48$	1.00000i	0.144338i
$49$	3.00000	0.428571
$50$	− 5.00000i	− 0.707107i
$51$	−4.47214	−0.626224
$52$	− 4.47214i	− 0.620174i
$53$	− 12.4721i	− 1.71318i	−0.515998	−	0.856590i	$-0.672579\pi$
$53$	− 12.4721i	− 1.71318i	0.515998	−	0.856590i	$-0.327421\pi$
$54$	1.00000	0.136083
$55$	11.7082i	1.57873i
$56$	2.00000	0.267261
$57$	7.23607i	0.958441i
$58$	− 4.00000i	− 0.525226i
$59$	−6.94427	−0.904067	−0.452034	−	0.892001i	$-0.649301\pi$
$59$	−6.94427	−0.904067	−0.452034	+	0.892001i	$0.649301\pi$
$60$	2.23607	0.288675
$61$	5.23607	0.670410	0.335205	−	0.942145i	$-0.391194\pi$
$61$	5.23607	0.670410	0.335205	+	0.942145i	$0.391194\pi$
$62$	− 2.47214i	− 0.313962i
$63$	− 2.00000i	− 0.251976i
$64$	−1.00000	−0.125000
$65$	−10.0000	−1.24035
$66$	−5.23607	−0.644515
$67$	− 10.4721i	− 1.27938i	−0.768635	−	0.639688i	$-0.779064\pi$
$67$	− 10.4721i	− 1.27938i	0.768635	−	0.639688i	$-0.220936\pi$
$68$	− 4.47214i	− 0.542326i
$69$	4.00000	0.481543
$70$	− 4.47214i	− 0.534522i
$71$	−2.47214	−0.293389	−0.146694	−	0.989182i	$-0.546863\pi$
$71$	−2.47214	−0.293389	−0.146694	+	0.989182i	$0.546863\pi$
$72$	1.00000i	0.117851i
$73$	− 10.4721i	− 1.22567i	−0.790211	−	0.612835i	$-0.790029\pi$
$73$	− 10.4721i	− 1.22567i	0.790211	−	0.612835i	$-0.209971\pi$
$74$	−1.00000	−0.116248
$75$	− 5.00000i	− 0.577350i
$76$	−7.23607	−0.830034
$77$	10.4721i	1.19341i
$78$	− 4.47214i	− 0.506370i
$79$	2.47214	0.278137	0.139069	−	0.990283i	$-0.455589\pi$
$79$	2.47214	0.278137	0.139069	+	0.990283i	$0.455589\pi$
$80$	2.23607i	0.250000i
$81$	1.00000	0.111111
$82$	− 4.47214i	− 0.493865i
$83$	15.4164i	1.69217i	0.533048	+	0.846085i	$0.321047\pi$
$83$	15.4164i	1.69217i	−0.533048	+	0.846085i	$0.678953\pi$
$84$	2.00000	0.218218
$85$	−10.0000	−1.08465
$86$	10.4721	1.12924
$87$	− 4.00000i	− 0.428845i
$88$	− 5.23607i	− 0.558167i
$89$	7.52786	0.797952	0.398976	−	0.916961i	$-0.369366\pi$
$89$	7.52786	0.797952	0.398976	+	0.916961i	$0.369366\pi$
$90$	2.23607	0.235702
$91$	−8.94427	−0.937614
$92$	4.00000i	0.417029i
$93$	− 2.47214i	− 0.256349i
$94$	−9.70820	−1.00132
$95$	16.1803i	1.66007i
$96$	−1.00000	−0.102062
$97$	− 8.18034i	− 0.830588i	−0.909687	−	0.415294i	$-0.863679\pi$
$97$	− 8.18034i	− 0.830588i	0.909687	−	0.415294i	$-0.136321\pi$
$98$	3.00000i	0.303046i
$99$	−5.23607	−0.526245
$100$	5.00000	0.500000
$101$	−4.76393	−0.474029	−0.237014	−	0.971506i	$-0.576169\pi$
$101$	−4.76393	−0.474029	−0.237014	+	0.971506i	$0.576169\pi$
$102$	− 4.47214i	− 0.442807i
$103$	19.7082i	1.94191i	0.239268	+	0.970954i	$0.423092\pi$
$103$	19.7082i	1.94191i	−0.239268	+	0.970954i	$0.576908\pi$
$104$	4.47214	0.438529
$105$	− 4.47214i	− 0.436436i
$106$	12.4721	1.21140
$107$	4.00000i	0.386695i	0.981130	+	0.193347i	$0.0619344\pi$
$107$	4.00000i	0.386695i	−0.981130	+	0.193347i	$0.938066\pi$
$108$	1.00000i	0.0962250i
$109$	14.1803	1.35823	0.679115	−	0.734032i	$-0.262364\pi$
$109$	14.1803	1.35823	0.679115	+	0.734032i	$0.262364\pi$
$110$	−11.7082	−1.11633
$111$	−1.00000	−0.0949158
$112$	2.00000i	0.188982i
$113$	− 10.0000i	− 0.940721i	−0.882474	−	0.470360i	$-0.844124\pi$
$113$	− 10.0000i	− 0.940721i	0.882474	−	0.470360i	$-0.155876\pi$
$114$	−7.23607	−0.677720
$115$	8.94427	0.834058
$116$	4.00000	0.371391
$117$	− 4.47214i	− 0.413449i
$118$	− 6.94427i	− 0.639272i
$119$	−8.94427	−0.819920
$120$	2.23607i	0.204124i
$121$	16.4164	1.49240
$122$	5.23607i	0.474051i
$123$	− 4.47214i	− 0.403239i
$124$	2.47214	0.222004
$125$	− 11.1803i	− 1.00000i
$126$	2.00000	0.178174
$127$	− 2.94427i	− 0.261262i	−0.991431	−	0.130631i	$-0.958300\pi$
$127$	− 2.94427i	− 0.261262i	0.991431	−	0.130631i	$-0.0417003\pi$
$128$	− 1.00000i	− 0.0883883i
$129$	10.4721	0.922020
$130$	− 10.0000i	− 0.877058i
$131$	−2.00000	−0.174741	−0.0873704	−	0.996176i	$-0.527846\pi$
$131$	−2.00000	−0.174741	−0.0873704	+	0.996176i	$0.527846\pi$
$132$	− 5.23607i	− 0.455741i
$133$	14.4721i	1.25489i
$134$	10.4721	0.904655
$135$	2.23607	0.192450
$136$	4.47214	0.383482
$137$	7.70820i	0.658556i	0.944233	+	0.329278i	$0.106805\pi$
$137$	7.70820i	0.658556i	−0.944233	+	0.329278i	$0.893195\pi$
$138$	4.00000i	0.340503i
$139$	−20.0000	−1.69638	−0.848189	−	0.529694i	$-0.822307\pi$
$139$	−20.0000	−1.69638	−0.848189	+	0.529694i	$0.822307\pi$
$140$	4.47214	0.377964
$141$	−9.70820	−0.817578
$142$	− 2.47214i	− 0.207457i
$143$	23.4164i	1.95818i
$144$	−1.00000	−0.0833333
$145$	− 8.94427i	− 0.742781i
$146$	10.4721	0.866680
$147$	3.00000i	0.247436i
$148$	− 1.00000i	− 0.0821995i
$149$	−18.6525	−1.52807	−0.764035	−	0.645175i	$-0.776785\pi$
$149$	−18.6525	−1.52807	−0.764035	+	0.645175i	$0.776785\pi$
$150$	5.00000	0.408248
$151$	8.94427	0.727875	0.363937	−	0.931423i	$-0.381432\pi$
$151$	8.94427	0.727875	0.363937	+	0.931423i	$0.381432\pi$
$152$	− 7.23607i	− 0.586923i
$153$	− 4.47214i	− 0.361551i
$154$	−10.4721	−0.843869
$155$	− 5.52786i	− 0.444009i
$156$	4.47214	0.358057
$157$	14.9443i	1.19268i	0.802731	+	0.596341i	$0.203380\pi$
$157$	14.9443i	1.19268i	−0.802731	+	0.596341i	$0.796620\pi$
$158$	2.47214i	0.196673i
$159$	12.4721	0.989105
$160$	−2.23607	−0.176777
$161$	8.00000	0.630488
$162$	1.00000i	0.0785674i
$163$	− 20.9443i	− 1.64048i	−0.572018	−	0.820241i	$-0.693839\pi$
$163$	− 20.9443i	− 1.64048i	0.572018	−	0.820241i	$-0.306161\pi$
$164$	4.47214	0.349215
$165$	−11.7082	−0.911482
$166$	−15.4164	−1.19655
$167$	2.47214i	0.191300i	0.995415	+	0.0956498i	$0.0304929\pi$
$167$	2.47214i	0.191300i	−0.995415	+	0.0956498i	$0.969507\pi$
$168$	2.00000i	0.154303i
$169$	−7.00000	−0.538462
$170$	− 10.0000i	− 0.766965i
$171$	−7.23607	−0.553356
$172$	10.4721i	0.798493i
$173$	− 18.0000i	− 1.36851i	−0.729241	−	0.684257i	$-0.760127\pi$
$173$	− 18.0000i	− 1.36851i	0.729241	−	0.684257i	$-0.239873\pi$
$174$	4.00000	0.303239
$175$	− 10.0000i	− 0.755929i
$176$	5.23607	0.394683
$177$	− 6.94427i	− 0.521963i
$178$	7.52786i	0.564237i
$179$	18.9443	1.41596	0.707981	−	0.706232i	$-0.249606\pi$
$179$	18.9443	1.41596	0.707981	+	0.706232i	$0.249606\pi$
$180$	2.23607i	0.166667i
$181$	6.94427	0.516164	0.258082	−	0.966123i	$-0.416910\pi$
$181$	6.94427	0.516164	0.258082	+	0.966123i	$0.416910\pi$
$182$	− 8.94427i	− 0.662994i
$183$	5.23607i	0.387061i
$184$	−4.00000	−0.294884
$185$	−2.23607	−0.164399
$186$	2.47214	0.181266
$187$	23.4164i	1.71238i
$188$	− 9.70820i	− 0.708044i
$189$	2.00000	0.145479
$190$	−16.1803	−1.17385
$191$	3.41641	0.247203	0.123601	−	0.992332i	$-0.460556\pi$
$191$	3.41641	0.247203	0.123601	+	0.992332i	$0.460556\pi$
$192$	− 1.00000i	− 0.0721688i
$193$	4.18034i	0.300907i	0.988617	+	0.150454i	$0.0480735\pi$
$193$	4.18034i	0.300907i	−0.988617	+	0.150454i	$0.951927\pi$
$194$	8.18034	0.587314
$195$	− 10.0000i	− 0.716115i
$196$	−3.00000	−0.214286
$197$	10.9443i	0.779747i	0.920868	+	0.389874i	$0.127481\pi$
$197$	10.9443i	0.779747i	−0.920868	+	0.389874i	$0.872519\pi$
$198$	− 5.23607i	− 0.372111i
$199$	5.52786	0.391860	0.195930	−	0.980618i	$-0.437227\pi$
$199$	5.52786	0.391860	0.195930	+	0.980618i	$0.437227\pi$
$200$	5.00000i	0.353553i
$201$	10.4721	0.738648
$202$	− 4.76393i	− 0.335189i
$203$	− 8.00000i	− 0.561490i
$204$	4.47214	0.313112
$205$	− 10.0000i	− 0.698430i
$206$	−19.7082	−1.37314
$207$	4.00000i	0.278019i
$208$	4.47214i	0.310087i
$209$	37.8885	2.62081
$210$	4.47214	0.308607
$211$	26.4721	1.82242	0.911208	−	0.411945i	$-0.135151\pi$
$211$	26.4721	1.82242	0.911208	+	0.411945i	$0.135151\pi$
$212$	12.4721i	0.856590i
$213$	− 2.47214i	− 0.169388i
$214$	−4.00000	−0.273434
$215$	23.4164	1.59699
$216$	−1.00000	−0.0680414
$217$	− 4.94427i	− 0.335639i
$218$	14.1803i	0.960414i
$219$	10.4721	0.707641
$220$	− 11.7082i	− 0.789367i
$221$	−20.0000	−1.34535
$222$	− 1.00000i	− 0.0671156i
$223$	− 10.0000i	− 0.669650i	−0.942280	−	0.334825i	$-0.891323\pi$
$223$	− 10.0000i	− 0.669650i	0.942280	−	0.334825i	$-0.108677\pi$
$224$	−2.00000	−0.133631
$225$	5.00000	0.333333
$226$	10.0000	0.665190
$227$	28.9443i	1.92110i	0.278109	+	0.960549i	$0.410292\pi$
$227$	28.9443i	1.92110i	−0.278109	+	0.960549i	$0.589708\pi$
$228$	− 7.23607i	− 0.479220i
$229$	13.4164	0.886581	0.443291	−	0.896378i	$-0.353811\pi$
$229$	13.4164	0.886581	0.443291	+	0.896378i	$0.353811\pi$
$230$	8.94427i	0.589768i
$231$	−10.4721	−0.689016
$232$	4.00000i	0.262613i
$233$	− 0.291796i	− 0.0191162i	−0.999954	−	0.00955810i	$-0.996958\pi$
$233$	− 0.291796i	− 0.0191162i	0.999954	−	0.00955810i	$-0.00304248\pi$
$234$	4.47214	0.292353
$235$	−21.7082	−1.41609
$236$	6.94427	0.452034
$237$	2.47214i	0.160582i
$238$	− 8.94427i	− 0.579771i
$239$	6.47214	0.418648	0.209324	−	0.977846i	$-0.432874\pi$
$239$	6.47214	0.418648	0.209324	+	0.977846i	$0.432874\pi$
$240$	−2.23607	−0.144338
$241$	12.4721	0.803401	0.401700	−	0.915771i	$-0.368419\pi$
$241$	12.4721	0.803401	0.401700	+	0.915771i	$0.368419\pi$
$242$	16.4164i	1.05529i
$243$	1.00000i	0.0641500i
$244$	−5.23607	−0.335205
$245$	6.70820i	0.428571i
$246$	4.47214	0.285133
$247$	32.3607i	2.05906i
$248$	2.47214i	0.156981i
$249$	−15.4164	−0.976975
$250$	11.1803	0.707107
$251$	−3.52786	−0.222677	−0.111338	−	0.993783i	$-0.535514\pi$
$251$	−3.52786	−0.222677	−0.111338	+	0.993783i	$0.535514\pi$
$252$	2.00000i	0.125988i
$253$	− 20.9443i	− 1.31676i
$254$	2.94427	0.184740
$255$	− 10.0000i	− 0.626224i
$256$	1.00000	0.0625000
$257$	5.05573i	0.315368i	0.987490	+	0.157684i	$0.0504027\pi$
$257$	5.05573i	0.315368i	−0.987490	+	0.157684i	$0.949597\pi$
$258$	10.4721i	0.651967i
$259$	−2.00000	−0.124274
$260$	10.0000	0.620174
$261$	4.00000	0.247594
$262$	− 2.00000i	− 0.123560i
$263$	10.6525i	0.656860i	0.944528	+	0.328430i	$0.106519\pi$
$263$	10.6525i	0.656860i	−0.944528	+	0.328430i	$0.893481\pi$
$264$	5.23607	0.322258
$265$	27.8885	1.71318
$266$	−14.4721	−0.887344
$267$	7.52786i	0.460698i
$268$	10.4721i	0.639688i
$269$	14.6525	0.893377	0.446689	−	0.894689i	$-0.352603\pi$
$269$	14.6525	0.893377	0.446689	+	0.894689i	$0.352603\pi$
$270$	2.23607i	0.136083i
$271$	−26.8328	−1.62998	−0.814989	−	0.579477i	$-0.803257\pi$
$271$	−26.8328	−1.62998	−0.814989	+	0.579477i	$0.803257\pi$
$272$	4.47214i	0.271163i
$273$	− 8.94427i	− 0.541332i
$274$	−7.70820	−0.465670
$275$	−26.1803	−1.57873
$276$	−4.00000	−0.240772
$277$	− 0.472136i	− 0.0283679i	−0.999899	−	0.0141840i	$-0.995485\pi$
$277$	− 0.472136i	− 0.0283679i	0.999899	−	0.0141840i	$-0.00451504\pi$
$278$	− 20.0000i	− 1.19952i
$279$	2.47214	0.148003
$280$	4.47214i	0.267261i
$281$	−18.3607	−1.09531	−0.547653	−	0.836705i	$-0.684479\pi$
$281$	−18.3607	−1.09531	−0.547653	+	0.836705i	$0.684479\pi$
$282$	− 9.70820i	− 0.578115i
$283$	− 20.9443i	− 1.24501i	−0.782617	−	0.622504i	$-0.786115\pi$
$283$	− 20.9443i	− 1.24501i	0.782617	−	0.622504i	$-0.213885\pi$
$284$	2.47214	0.146694
$285$	−16.1803	−0.958441
$286$	−23.4164	−1.38464
$287$	− 8.94427i	− 0.527964i
$288$	− 1.00000i	− 0.0589256i
$289$	−3.00000	−0.176471
$290$	8.94427	0.525226
$291$	8.18034	0.479540
$292$	10.4721i	0.612835i
$293$	26.0000i	1.51894i	0.650545	+	0.759468i	$0.274541\pi$
$293$	26.0000i	1.51894i	−0.650545	+	0.759468i	$0.725459\pi$
$294$	−3.00000	−0.174964
$295$	− 15.5279i	− 0.904067i
$296$	1.00000	0.0581238
$297$	− 5.23607i	− 0.303827i
$298$	− 18.6525i	− 1.08051i
$299$	17.8885	1.03452
$300$	5.00000i	0.288675i
$301$	20.9443	1.20721
$302$	8.94427i	0.514685i
$303$	− 4.76393i	− 0.273681i
$304$	7.23607	0.415017
$305$	11.7082i	0.670410i
$306$	4.47214	0.255655
$307$	27.4164i	1.56474i	0.622816	+	0.782369i	$0.285989\pi$
$307$	27.4164i	1.56474i	−0.622816	+	0.782369i	$0.714011\pi$
$308$	− 10.4721i	− 0.596705i
$309$	−19.7082	−1.12116
$310$	5.52786	0.313962
$311$	8.94427	0.507183	0.253592	−	0.967311i	$-0.418388\pi$
$311$	8.94427	0.507183	0.253592	+	0.967311i	$0.418388\pi$
$312$	4.47214i	0.253185i
$313$	2.29180i	0.129540i	0.997900	+	0.0647700i	$0.0206314\pi$
$313$	2.29180i	0.129540i	−0.997900	+	0.0647700i	$0.979369\pi$
$314$	−14.9443	−0.843354
$315$	4.47214	0.251976
$316$	−2.47214	−0.139069
$317$	− 2.94427i	− 0.165367i	−0.996576	−	0.0826834i	$-0.973651\pi$
$317$	− 2.94427i	− 0.165367i	0.996576	−	0.0826834i	$-0.0263490\pi$
$318$	12.4721i	0.699403i
$319$	−20.9443	−1.17265
$320$	− 2.23607i	− 0.125000i
$321$	−4.00000	−0.223258
$322$	8.00000i	0.445823i
$323$	32.3607i	1.80060i
$324$	−1.00000	−0.0555556
$325$	− 22.3607i	− 1.24035i
$326$	20.9443	1.16000
$327$	14.1803i	0.784175i
$328$	4.47214i	0.246932i
$329$	−19.4164	−1.07046
$330$	− 11.7082i	− 0.644515i
$331$	31.2361	1.71689	0.858445	−	0.512906i	$-0.171431\pi$
$331$	31.2361	1.71689	0.858445	+	0.512906i	$0.171431\pi$
$332$	− 15.4164i	− 0.846085i
$333$	− 1.00000i	− 0.0547997i
$334$	−2.47214	−0.135269
$335$	23.4164	1.27938
$336$	−2.00000	−0.109109
$337$	− 15.4164i	− 0.839785i	−0.907574	−	0.419893i	$-0.862068\pi$
$337$	− 15.4164i	− 0.839785i	0.907574	−	0.419893i	$-0.137932\pi$
$338$	− 7.00000i	− 0.380750i
$339$	10.0000	0.543125
$340$	10.0000	0.542326
$341$	−12.9443	−0.700972
$342$	− 7.23607i	− 0.391282i
$343$	20.0000i	1.07990i
$344$	−10.4721	−0.564620
$345$	8.94427i	0.481543i
$346$	18.0000	0.967686
$347$	3.05573i	0.164040i	0.996631	+	0.0820200i	$0.0261372\pi$
$347$	3.05573i	0.164040i	−0.996631	+	0.0820200i	$0.973863\pi$
$348$	4.00000i	0.214423i
$349$	−24.4721	−1.30996	−0.654982	−	0.755645i	$-0.727324\pi$
$349$	−24.4721	−1.30996	−0.654982	+	0.755645i	$0.727324\pi$
$350$	10.0000	0.534522
$351$	4.47214	0.238705
$352$	5.23607i	0.279083i
$353$	− 14.0000i	− 0.745145i	−0.928003	−	0.372572i	$-0.878476\pi$
$353$	− 14.0000i	− 0.745145i	0.928003	−	0.372572i	$-0.121524\pi$
$354$	6.94427	0.369084
$355$	− 5.52786i	− 0.293389i
$356$	−7.52786	−0.398976
$357$	− 8.94427i	− 0.473381i
$358$	18.9443i	1.00124i
$359$	−25.8885	−1.36635	−0.683173	−	0.730257i	$-0.739400\pi$
$359$	−25.8885	−1.36635	−0.683173	+	0.730257i	$0.739400\pi$
$360$	−2.23607	−0.117851
$361$	33.3607	1.75583
$362$	6.94427i	0.364983i
$363$	16.4164i	0.861638i
$364$	8.94427	0.468807
$365$	23.4164	1.22567
$366$	−5.23607	−0.273694
$367$	− 36.8328i	− 1.92266i	−0.275404	−	0.961329i	$-0.588812\pi$
$367$	− 36.8328i	− 1.92266i	0.275404	−	0.961329i	$-0.411188\pi$
$368$	− 4.00000i	− 0.208514i
$369$	4.47214	0.232810
$370$	− 2.23607i	− 0.116248i
$371$	24.9443	1.29504
$372$	2.47214i	0.128174i
$373$	− 2.94427i	− 0.152449i	−0.997091	−	0.0762243i	$-0.975713\pi$
$373$	− 2.94427i	− 0.152449i	0.997091	−	0.0762243i	$-0.0242865\pi$
$374$	−23.4164	−1.21083
$375$	11.1803	0.577350
$376$	9.70820	0.500662
$377$	− 17.8885i	− 0.921307i
$378$	2.00000i	0.102869i
$379$	13.8885	0.713407	0.356703	−	0.934218i	$-0.383901\pi$
$379$	13.8885	0.713407	0.356703	+	0.934218i	$0.383901\pi$
$380$	− 16.1803i	− 0.830034i
$381$	2.94427	0.150840
$382$	3.41641i	0.174799i
$383$	− 29.3050i	− 1.49741i	−0.662902	−	0.748707i	$-0.730675\pi$
$383$	− 29.3050i	− 1.49741i	0.662902	−	0.748707i	$-0.269325\pi$
$384$	1.00000	0.0510310
$385$	−23.4164	−1.19341
$386$	−4.18034	−0.212774
$387$	10.4721i	0.532329i
$388$	8.18034i	0.415294i
$389$	27.4164	1.39007	0.695034	−	0.718977i	$-0.255390\pi$
$389$	27.4164	1.39007	0.695034	+	0.718977i	$0.255390\pi$
$390$	10.0000	0.506370
$391$	17.8885	0.904663
$392$	− 3.00000i	− 0.151523i
$393$	− 2.00000i	− 0.100887i
$394$	−10.9443	−0.551364
$395$	5.52786i	0.278137i
$396$	5.23607	0.263122
$397$	3.88854i	0.195160i	0.995228	+	0.0975802i	$0.0311103\pi$
$397$	3.88854i	0.195160i	−0.995228	+	0.0975802i	$0.968890\pi$
$398$	5.52786i	0.277087i
$399$	−14.4721	−0.724513
$400$	−5.00000	−0.250000
$401$	−27.5279	−1.37468	−0.687338	−	0.726338i	$-0.741221\pi$
$401$	−27.5279	−1.37468	−0.687338	+	0.726338i	$0.741221\pi$
$402$	10.4721i	0.522303i
$403$	− 11.0557i	− 0.550725i
$404$	4.76393	0.237014
$405$	2.23607i	0.111111i
$406$	8.00000	0.397033
$407$	5.23607i	0.259542i
$408$	4.47214i	0.221404i
$409$	0.111456	0.00551115	0.00275558	−	0.999996i	$-0.499123\pi$
$409$	0.111456	0.00551115	0.00275558	+	0.999996i	$0.499123\pi$
$410$	10.0000	0.493865
$411$	−7.70820	−0.380218
$412$	− 19.7082i	− 0.970954i
$413$	− 13.8885i	− 0.683411i
$414$	−4.00000	−0.196589
$415$	−34.4721	−1.69217
$416$	−4.47214	−0.219265
$417$	− 20.0000i	− 0.979404i
$418$	37.8885i	1.85319i
$419$	6.76393	0.330440	0.165220	−	0.986257i	$-0.447167\pi$
$419$	6.76393	0.330440	0.165220	+	0.986257i	$0.447167\pi$
$420$	4.47214i	0.218218i
$421$	−11.7082	−0.570623	−0.285311	−	0.958435i	$-0.592097\pi$
$421$	−11.7082	−0.570623	−0.285311	+	0.958435i	$0.592097\pi$
$422$	26.4721i	1.28864i
$423$	− 9.70820i	− 0.472029i
$424$	−12.4721	−0.605700
$425$	− 22.3607i	− 1.08465i
$426$	2.47214	0.119775
$427$	10.4721i	0.506782i
$428$	− 4.00000i	− 0.193347i
$429$	−23.4164	−1.13055
$430$	23.4164i	1.12924i
$431$	−17.8885	−0.861661	−0.430830	−	0.902433i	$-0.641779\pi$
$431$	−17.8885	−0.861661	−0.430830	+	0.902433i	$0.641779\pi$
$432$	− 1.00000i	− 0.0481125i
$433$	21.5279i	1.03456i	0.855815	+	0.517282i	$0.173056\pi$
$433$	21.5279i	1.03456i	−0.855815	+	0.517282i	$0.826944\pi$
$434$	4.94427	0.237333
$435$	8.94427	0.428845
$436$	−14.1803	−0.679115
$437$	− 28.9443i	− 1.38459i
$438$	10.4721i	0.500378i
$439$	10.4721	0.499808	0.249904	−	0.968271i	$-0.419601\pi$
$439$	10.4721	0.499808	0.249904	+	0.968271i	$0.419601\pi$
$440$	11.7082	0.558167
$441$	−3.00000	−0.142857
$442$	− 20.0000i	− 0.951303i
$443$	− 26.8328i	− 1.27487i	−0.770506	−	0.637433i	$-0.779996\pi$
$443$	− 26.8328i	− 1.27487i	0.770506	−	0.637433i	$-0.220004\pi$
$444$	1.00000	0.0474579
$445$	16.8328i	0.797952i
$446$	10.0000	0.473514
$447$	− 18.6525i	− 0.882232i
$448$	− 2.00000i	− 0.0944911i
$449$	7.52786	0.355262	0.177631	−	0.984097i	$-0.443157\pi$
$449$	7.52786	0.355262	0.177631	+	0.984097i	$0.443157\pi$
$450$	5.00000i	0.235702i
$451$	−23.4164	−1.10264
$452$	10.0000i	0.470360i
$453$	8.94427i	0.420239i
$454$	−28.9443	−1.35842
$455$	− 20.0000i	− 0.937614i
$456$	7.23607	0.338860
$457$	− 13.1246i	− 0.613943i	−0.951719	−	0.306972i	$-0.900684\pi$
$457$	− 13.1246i	− 0.613943i	0.951719	−	0.306972i	$-0.0993157\pi$
$458$	13.4164i	0.626908i
$459$	4.47214	0.208741
$460$	−8.94427	−0.417029
$461$	6.47214	0.301437	0.150719	−	0.988577i	$-0.451841\pi$
$461$	6.47214	0.301437	0.150719	+	0.988577i	$0.451841\pi$
$462$	− 10.4721i	− 0.487208i
$463$	18.7639i	0.872034i	0.899938	+	0.436017i	$0.143611\pi$
$463$	18.7639i	0.872034i	−0.899938	+	0.436017i	$0.856389\pi$
$464$	−4.00000	−0.185695
$465$	5.52786	0.256349
$466$	0.291796	0.0135172
$467$	− 3.05573i	− 0.141402i	−0.997498	−	0.0707011i	$-0.977476\pi$
$467$	− 3.05573i	− 0.141402i	0.997498	−	0.0707011i	$-0.0225237\pi$
$468$	4.47214i	0.206725i
$469$	20.9443	0.967117
$470$	− 21.7082i	− 1.00132i
$471$	−14.9443	−0.688596
$472$	6.94427i	0.319636i
$473$	− 54.8328i	− 2.52122i
$474$	−2.47214	−0.113549
$475$	−36.1803	−1.66007
$476$	8.94427	0.409960
$477$	12.4721i	0.571060i
$478$	6.47214i	0.296029i
$479$	−30.4721	−1.39231	−0.696154	−	0.717893i	$-0.745107\pi$
$479$	−30.4721	−1.39231	−0.696154	+	0.717893i	$0.745107\pi$
$480$	− 2.23607i	− 0.102062i
$481$	−4.47214	−0.203912
$482$	12.4721i	0.568090i
$483$	8.00000i	0.364013i
$484$	−16.4164	−0.746200
$485$	18.2918	0.830588
$486$	−1.00000	−0.0453609
$487$	3.70820i	0.168035i	0.996464	+	0.0840174i	$0.0267751\pi$
$487$	3.70820i	0.168035i	−0.996464	+	0.0840174i	$0.973225\pi$
$488$	− 5.23607i	− 0.237026i
$489$	20.9443	0.947133
$490$	−6.70820	−0.303046
$491$	−21.2361	−0.958370	−0.479185	−	0.877714i	$-0.659068\pi$
$491$	−21.2361	−0.958370	−0.479185	+	0.877714i	$0.659068\pi$
$492$	4.47214i	0.201619i
$493$	− 17.8885i	− 0.805659i
$494$	−32.3607	−1.45598
$495$	− 11.7082i	− 0.526245i
$496$	−2.47214	−0.111002
$497$	− 4.94427i	− 0.221781i
$498$	− 15.4164i	− 0.690826i
$499$	−4.18034	−0.187138	−0.0935689	−	0.995613i	$-0.529828\pi$
$499$	−4.18034	−0.187138	−0.0935689	+	0.995613i	$0.529828\pi$
$500$	11.1803i	0.500000i
$501$	−2.47214	−0.110447
$502$	− 3.52786i	− 0.157456i
$503$	4.94427i	0.220454i	0.993906	+	0.110227i	$0.0351578\pi$
$503$	4.94427i	0.220454i	−0.993906	+	0.110227i	$0.964842\pi$
$504$	−2.00000	−0.0890871
$505$	− 10.6525i	− 0.474029i
$506$	20.9443	0.931086
$507$	− 7.00000i	− 0.310881i
$508$	2.94427i	0.130631i
$509$	−13.1246	−0.581738	−0.290869	−	0.956763i	$-0.593944\pi$
$509$	−13.1246	−0.581738	−0.290869	+	0.956763i	$0.593944\pi$
$510$	10.0000	0.442807
$511$	20.9443	0.926520
$512$	1.00000i	0.0441942i
$513$	− 7.23607i	− 0.319480i
$514$	−5.05573	−0.222999
$515$	−44.0689	−1.94191
$516$	−10.4721	−0.461010
$517$	50.8328i	2.23562i
$518$	− 2.00000i	− 0.0878750i
$519$	18.0000	0.790112
$520$	10.0000i	0.438529i
$521$	−24.4721	−1.07214	−0.536072	−	0.844172i	$-0.680092\pi$
$521$	−24.4721	−1.07214	−0.536072	+	0.844172i	$0.680092\pi$
$522$	4.00000i	0.175075i
$523$	22.8328i	0.998409i	0.866484	+	0.499205i	$0.166374\pi$
$523$	22.8328i	0.998409i	−0.866484	+	0.499205i	$0.833626\pi$
$524$	2.00000	0.0873704
$525$	10.0000	0.436436
$526$	−10.6525	−0.464470
$527$	− 11.0557i	− 0.481595i
$528$	5.23607i	0.227871i
$529$	7.00000	0.304348
$530$	27.8885i	1.21140i
$531$	6.94427	0.301356
$532$	− 14.4721i	− 0.627447i
$533$	− 20.0000i	− 0.866296i
$534$	−7.52786	−0.325763
$535$	−8.94427	−0.386695
$536$	−10.4721	−0.452327
$537$	18.9443i	0.817506i
$538$	14.6525i	0.631713i
$539$	15.7082	0.676600
$540$	−2.23607	−0.0962250
$541$	−7.70820	−0.331402	−0.165701	−	0.986176i	$-0.552989\pi$
$541$	−7.70820	−0.331402	−0.165701	+	0.986176i	$0.552989\pi$
$542$	− 26.8328i	− 1.15257i
$543$	6.94427i	0.298007i
$544$	−4.47214	−0.191741
$545$	31.7082i	1.35823i
$546$	8.94427	0.382780
$547$	− 12.3607i	− 0.528505i	−0.964454	−	0.264252i	$-0.914875\pi$
$547$	− 12.3607i	− 0.528505i	0.964454	−	0.264252i	$-0.0851251\pi$
$548$	− 7.70820i	− 0.329278i
$549$	−5.23607	−0.223470
$550$	− 26.1803i	− 1.11633i
$551$	−28.9443	−1.23307
$552$	− 4.00000i	− 0.170251i
$553$	4.94427i	0.210252i
$554$	0.472136	0.0200591
$555$	− 2.23607i	− 0.0949158i
$556$	20.0000	0.848189
$557$	− 4.11146i	− 0.174208i	−0.996199	−	0.0871040i	$-0.972239\pi$
$557$	− 4.11146i	− 0.174208i	0.996199	−	0.0871040i	$-0.0277612\pi$
$558$	2.47214i	0.104654i
$559$	46.8328	1.98082
$560$	−4.47214	−0.188982
$561$	−23.4164	−0.988642
$562$	− 18.3607i	− 0.774499i
$563$	13.8885i	0.585332i	0.956215	+	0.292666i	$0.0945425\pi$
$563$	13.8885i	0.585332i	−0.956215	+	0.292666i	$0.905458\pi$
$564$	9.70820	0.408789
$565$	22.3607	0.940721
$566$	20.9443	0.880353
$567$	2.00000i	0.0839921i
$568$	2.47214i	0.103729i
$569$	14.5836	0.611376	0.305688	−	0.952132i	$-0.401114\pi$
$569$	14.5836	0.611376	0.305688	+	0.952132i	$0.401114\pi$
$570$	− 16.1803i	− 0.677720i
$571$	−6.47214	−0.270850	−0.135425	−	0.990788i	$-0.543240\pi$
$571$	−6.47214	−0.270850	−0.135425	+	0.990788i	$0.543240\pi$
$572$	− 23.4164i	− 0.979089i
$573$	3.41641i	0.142722i
$574$	8.94427	0.373327
$575$	20.0000i	0.834058i
$576$	1.00000	0.0416667
$577$	− 32.5410i	− 1.35470i	−0.735661	−	0.677350i	$-0.763128\pi$
$577$	− 32.5410i	− 1.35470i	0.735661	−	0.677350i	$-0.236872\pi$
$578$	− 3.00000i	− 0.124784i
$579$	−4.18034	−0.173729
$580$	8.94427i	0.371391i
$581$	−30.8328	−1.27916
$582$	8.18034i	0.339086i
$583$	− 65.3050i	− 2.70465i
$584$	−10.4721	−0.433340
$585$	10.0000	0.413449
$586$	−26.0000	−1.07405
$587$	− 34.8328i	− 1.43770i	−0.695163	−	0.718852i	$-0.744668\pi$
$587$	− 34.8328i	− 1.43770i	0.695163	−	0.718852i	$-0.255332\pi$
$588$	− 3.00000i	− 0.123718i
$589$	−17.8885	−0.737085
$590$	15.5279	0.639272
$591$	−10.9443	−0.450187
$592$	1.00000i	0.0410997i
$593$	− 30.5410i	− 1.25417i	−0.778951	−	0.627085i	$-0.784248\pi$
$593$	− 30.5410i	− 1.25417i	0.778951	−	0.627085i	$-0.215752\pi$
$594$	5.23607	0.214838
$595$	− 20.0000i	− 0.819920i
$596$	18.6525	0.764035
$597$	5.52786i	0.226240i
$598$	17.8885i	0.731517i
$599$	25.8885	1.05778	0.528889	−	0.848691i	$-0.322609\pi$
$599$	25.8885	1.05778	0.528889	+	0.848691i	$0.322609\pi$
$600$	−5.00000	−0.204124
$601$	30.3607	1.23844	0.619219	−	0.785218i	$-0.287449\pi$
$601$	30.3607	1.23844	0.619219	+	0.785218i	$0.287449\pi$
$602$	20.9443i	0.853625i
$603$	10.4721i	0.426458i
$604$	−8.94427	−0.363937
$605$	36.7082i	1.49240i
$606$	4.76393	0.193522
$607$	10.1803i	0.413207i	0.978425	+	0.206604i	$0.0662411\pi$
$607$	10.1803i	0.413207i	−0.978425	+	0.206604i	$0.933759\pi$
$608$	7.23607i	0.293461i
$609$	8.00000	0.324176
$610$	−11.7082	−0.474051
$611$	−43.4164	−1.75644
$612$	4.47214i	0.180775i
$613$	26.9443i	1.08827i	0.838998	+	0.544134i	$0.183142\pi$
$613$	26.9443i	1.08827i	−0.838998	+	0.544134i	$0.816858\pi$
$614$	−27.4164	−1.10644
$615$	10.0000	0.403239
$616$	10.4721	0.421934
$617$	1.81966i	0.0732568i	0.999329	+	0.0366284i	$0.0116618\pi$
$617$	1.81966i	0.0732568i	−0.999329	+	0.0366284i	$0.988338\pi$
$618$	− 19.7082i	− 0.792780i
$619$	−12.3607	−0.496818	−0.248409	−	0.968655i	$-0.579908\pi$
$619$	−12.3607	−0.496818	−0.248409	+	0.968655i	$0.579908\pi$
$620$	5.52786i	0.222004i
$621$	−4.00000	−0.160514
$622$	8.94427i	0.358633i
$623$	15.0557i	0.603195i
$624$	−4.47214	−0.179029
$625$	25.0000	1.00000
$626$	−2.29180	−0.0915986
$627$	37.8885i	1.51312i
$628$	− 14.9443i	− 0.596341i
$629$	−4.47214	−0.178316
$630$	4.47214i	0.178174i
$631$	−21.8885	−0.871369	−0.435685	−	0.900099i	$-0.643494\pi$
$631$	−21.8885	−0.871369	−0.435685	+	0.900099i	$0.643494\pi$
$632$	− 2.47214i	− 0.0983363i
$633$	26.4721i	1.05217i
$634$	2.94427	0.116932
$635$	6.58359	0.261262
$636$	−12.4721	−0.494552
$637$	13.4164i	0.531577i
$638$	− 20.9443i	− 0.829192i
$639$	2.47214	0.0977962
$640$	2.23607	0.0883883
$641$	−9.41641	−0.371926	−0.185963	−	0.982557i	$-0.559540\pi$
$641$	−9.41641	−0.371926	−0.185963	+	0.982557i	$0.559540\pi$
$642$	− 4.00000i	− 0.157867i
$643$	38.8328i	1.53142i	0.643188	+	0.765708i	$0.277611\pi$
$643$	38.8328i	1.53142i	−0.643188	+	0.765708i	$0.722389\pi$
$644$	−8.00000	−0.315244
$645$	23.4164i	0.922020i
$646$	−32.3607	−1.27321
$647$	8.00000i	0.314512i	0.987558	+	0.157256i	$0.0502649\pi$
$647$	8.00000i	0.314512i	−0.987558	+	0.157256i	$0.949735\pi$
$648$	− 1.00000i	− 0.0392837i
$649$	−36.3607	−1.42728
$650$	22.3607	0.877058
$651$	4.94427	0.193781
$652$	20.9443i	0.820241i
$653$	− 46.7214i	− 1.82835i	−0.405322	−	0.914174i	$-0.632841\pi$
$653$	− 46.7214i	− 1.82835i	0.405322	−	0.914174i	$-0.367159\pi$
$654$	−14.1803	−0.554495
$655$	− 4.47214i	− 0.174741i
$656$	−4.47214	−0.174608
$657$	10.4721i	0.408557i
$658$	− 19.4164i	− 0.756930i
$659$	−4.29180	−0.167185	−0.0835923	−	0.996500i	$-0.526639\pi$
$659$	−4.29180	−0.167185	−0.0835923	+	0.996500i	$0.526639\pi$
$660$	11.7082	0.455741
$661$	−49.2361	−1.91506	−0.957531	−	0.288332i	$-0.906899\pi$
$661$	−49.2361	−1.91506	−0.957531	+	0.288332i	$0.906899\pi$
$662$	31.2361i	1.21402i
$663$	− 20.0000i	− 0.776736i
$664$	15.4164	0.598273
$665$	−32.3607	−1.25489
$666$	1.00000	0.0387492
$667$	16.0000i	0.619522i
$668$	− 2.47214i	− 0.0956498i
$669$	10.0000	0.386622
$670$	23.4164i	0.904655i
$671$	27.4164	1.05840
$672$	− 2.00000i	− 0.0771517i
$673$	2.11146i	0.0813907i	0.999172	+	0.0406953i	$0.0129573\pi$
$673$	2.11146i	0.0813907i	−0.999172	+	0.0406953i	$0.987043\pi$
$674$	15.4164	0.593818
$675$	5.00000i	0.192450i
$676$	7.00000	0.269231
$677$	− 25.4164i	− 0.976832i	−0.872611	−	0.488416i	$-0.837575\pi$
$677$	− 25.4164i	− 0.976832i	0.872611	−	0.488416i	$-0.162425\pi$
$678$	10.0000i	0.384048i
$679$	16.3607	0.627865
$680$	10.0000i	0.383482i
$681$	−28.9443	−1.10915
$682$	− 12.9443i	− 0.495662i
$683$	4.00000i	0.153056i	0.997067	+	0.0765279i	$0.0243834\pi$
$683$	4.00000i	0.153056i	−0.997067	+	0.0765279i	$0.975617\pi$
$684$	7.23607	0.276678
$685$	−17.2361	−0.658556
$686$	−20.0000	−0.763604
$687$	13.4164i	0.511868i
$688$	− 10.4721i	− 0.399246i
$689$	55.7771	2.12494
$690$	−8.94427	−0.340503
$691$	−29.3050	−1.11481	−0.557406	−	0.830240i	$-0.688203\pi$
$691$	−29.3050	−1.11481	−0.557406	+	0.830240i	$0.688203\pi$
$692$	18.0000i	0.684257i
$693$	− 10.4721i	− 0.397804i
$694$	−3.05573	−0.115994
$695$	− 44.7214i	− 1.69638i
$696$	−4.00000	−0.151620
$697$	− 20.0000i	− 0.757554i
$698$	− 24.4721i	− 0.926284i
$699$	0.291796	0.0110367
$700$	10.0000i	0.377964i
$701$	8.58359	0.324198	0.162099	−	0.986775i	$-0.448174\pi$
$701$	8.58359	0.324198	0.162099	+	0.986775i	$0.448174\pi$
$702$	4.47214i	0.168790i
$703$	7.23607i	0.272913i
$704$	−5.23607	−0.197342
$705$	− 21.7082i	− 0.817578i
$706$	14.0000	0.526897
$707$	− 9.52786i	− 0.358332i
$708$	6.94427i	0.260982i
$709$	−12.6525	−0.475174	−0.237587	−	0.971366i	$-0.576356\pi$
$709$	−12.6525	−0.475174	−0.237587	+	0.971366i	$0.576356\pi$
$710$	5.52786	0.207457
$711$	−2.47214	−0.0927123
$712$	− 7.52786i	− 0.282119i
$713$	9.88854i	0.370329i
$714$	8.94427	0.334731
$715$	−52.3607	−1.95818
$716$	−18.9443	−0.707981
$717$	6.47214i	0.241706i
$718$	− 25.8885i	− 0.966152i
$719$	−14.8328	−0.553171	−0.276585	−	0.960989i	$-0.589203\pi$
$719$	−14.8328	−0.553171	−0.276585	+	0.960989i	$0.589203\pi$
$720$	− 2.23607i	− 0.0833333i
$721$	−39.4164	−1.46794
$722$	33.3607i	1.24156i
$723$	12.4721i	0.463844i
$724$	−6.94427	−0.258082
$725$	20.0000	0.742781
$726$	−16.4164	−0.609270
$727$	− 7.70820i	− 0.285881i	−0.989731	−	0.142941i	$-0.954344\pi$
$727$	− 7.70820i	− 0.285881i	0.989731	−	0.142941i	$-0.0456558\pi$
$728$	8.94427i	0.331497i
$729$	−1.00000	−0.0370370
$730$	23.4164i	0.866680i
$731$	46.8328	1.73217
$732$	− 5.23607i	− 0.193531i
$733$	− 6.94427i	− 0.256493i	−0.991742	−	0.128246i	$-0.959065\pi$
$733$	− 6.94427i	− 0.256493i	0.991742	−	0.128246i	$-0.0409348\pi$
$734$	36.8328	1.35952
$735$	−6.70820	−0.247436
$736$	4.00000	0.147442
$737$	− 54.8328i	− 2.01979i
$738$	4.47214i	0.164622i
$739$	−8.36068	−0.307553	−0.153776	−	0.988106i	$-0.549144\pi$
$739$	−8.36068	−0.307553	−0.153776	+	0.988106i	$0.549144\pi$
$740$	2.23607	0.0821995
$741$	−32.3607	−1.18880
$742$	24.9443i	0.915733i
$743$	46.6525i	1.71151i	0.517379	+	0.855757i	$0.326908\pi$
$743$	46.6525i	1.71151i	−0.517379	+	0.855757i	$0.673092\pi$
$744$	−2.47214	−0.0906329
$745$	− 41.7082i	− 1.52807i
$746$	2.94427	0.107797
$747$	− 15.4164i	− 0.564057i
$748$	− 23.4164i	− 0.856189i
$749$	−8.00000	−0.292314
$750$	11.1803i	0.408248i
$751$	45.8885	1.67450	0.837248	−	0.546823i	$-0.184163\pi$
$751$	45.8885	1.67450	0.837248	+	0.546823i	$0.184163\pi$
$752$	9.70820i	0.354022i
$753$	− 3.52786i	− 0.128563i
$754$	17.8885	0.651462
$755$	20.0000i	0.727875i
$756$	−2.00000	−0.0727393
$757$	9.41641i	0.342245i	0.985250	+	0.171123i	$0.0547394\pi$
$757$	9.41641i	0.342245i	−0.985250	+	0.171123i	$0.945261\pi$
$758$	13.8885i	0.504455i
$759$	20.9443	0.760229
$760$	16.1803	0.586923
$761$	−37.4164	−1.35634	−0.678172	−	0.734903i	$-0.737228\pi$
$761$	−37.4164	−1.35634	−0.678172	+	0.734903i	$0.737228\pi$
$762$	2.94427i	0.106660i
$763$	28.3607i	1.02673i
$764$	−3.41641	−0.123601
$765$	10.0000	0.361551
$766$	29.3050	1.05883
$767$	− 31.0557i	− 1.12136i
$768$	1.00000i	0.0360844i
$769$	36.8328	1.32823	0.664113	−	0.747633i	$-0.268810\pi$
$769$	36.8328	1.32823	0.664113	+	0.747633i	$0.268810\pi$
$770$	− 23.4164i	− 0.843869i
$771$	−5.05573	−0.182078
$772$	− 4.18034i	− 0.150454i
$773$	20.8328i	0.749304i	0.927165	+	0.374652i	$0.122238\pi$
$773$	20.8328i	0.749304i	−0.927165	+	0.374652i	$0.877762\pi$
$774$	−10.4721	−0.376413
$775$	12.3607	0.444009
$776$	−8.18034	−0.293657
$777$	− 2.00000i	− 0.0717496i
$778$	27.4164i	0.982926i
$779$	−32.3607	−1.15944
$780$	10.0000i	0.358057i
$781$	−12.9443	−0.463182
$782$	17.8885i	0.639693i
$783$	4.00000i	0.142948i
$784$	3.00000	0.107143
$785$	−33.4164	−1.19268
$786$	2.00000	0.0713376
$787$	33.3050i	1.18719i	0.804763	+	0.593597i	$0.202293\pi$
$787$	33.3050i	1.18719i	−0.804763	+	0.593597i	$0.797707\pi$
$788$	− 10.9443i	− 0.389874i
$789$	−10.6525	−0.379238
$790$	−5.52786	−0.196673
$791$	20.0000	0.711118
$792$	5.23607i	0.186056i
$793$	23.4164i	0.831541i
$794$	−3.88854	−0.137999
$795$	27.8885i	0.989105i
$796$	−5.52786	−0.195930
$797$	15.8885i	0.562801i	0.959590	+	0.281401i	$0.0907990\pi$
$797$	15.8885i	0.562801i	−0.959590	+	0.281401i	$0.909201\pi$
$798$	− 14.4721i	− 0.512308i
$799$	−43.4164	−1.53596
$800$	− 5.00000i	− 0.176777i
$801$	−7.52786	−0.265984
$802$	− 27.5279i	− 0.972043i
$803$	− 54.8328i	− 1.93501i
$804$	−10.4721	−0.369324
$805$	17.8885i	0.630488i
$806$	11.0557	0.389421
$807$	14.6525i	0.515792i
$808$	4.76393i	0.167595i
$809$	26.3607	0.926792	0.463396	−	0.886151i	$-0.346631\pi$
$809$	26.3607	0.926792	0.463396	+	0.886151i	$0.346631\pi$
$810$	−2.23607	−0.0785674
$811$	2.11146	0.0741433	0.0370716	−	0.999313i	$-0.488197\pi$
$811$	2.11146	0.0741433	0.0370716	+	0.999313i	$0.488197\pi$
$812$	8.00000i	0.280745i
$813$	− 26.8328i	− 0.941068i
$814$	−5.23607	−0.183524
$815$	46.8328	1.64048
$816$	−4.47214	−0.156556
$817$	− 75.7771i	− 2.65110i
$818$	0.111456i	0.00389697i
$819$	8.94427	0.312538
$820$	10.0000i	0.349215i
$821$	43.9574	1.53412	0.767062	−	0.641573i	$-0.221718\pi$
$821$	43.9574	1.53412	0.767062	+	0.641573i	$0.221718\pi$
$822$	− 7.70820i	− 0.268854i
$823$	7.30495i	0.254635i	0.991862	+	0.127317i	$0.0406366\pi$
$823$	7.30495i	0.254635i	−0.991862	+	0.127317i	$0.959363\pi$
$824$	19.7082	0.686568
$825$	− 26.1803i	− 0.911482i
$826$	13.8885	0.483244
$827$	− 6.11146i	− 0.212516i	−0.994339	−	0.106258i	$-0.966113\pi$
$827$	− 6.11146i	− 0.212516i	0.994339	−	0.106258i	$-0.0338870\pi$
$828$	− 4.00000i	− 0.139010i
$829$	32.0689	1.11380	0.556899	−	0.830580i	$-0.311991\pi$
$829$	32.0689	1.11380	0.556899	+	0.830580i	$0.311991\pi$
$830$	− 34.4721i	− 1.19655i
$831$	0.472136	0.0163782
$832$	− 4.47214i	− 0.155043i
$833$	13.4164i	0.464851i
$834$	20.0000	0.692543
$835$	−5.52786	−0.191300
$836$	−37.8885	−1.31040
$837$	2.47214i	0.0854495i
$838$	6.76393i	0.233656i
$839$	6.11146	0.210991	0.105495	−	0.994420i	$-0.466357\pi$
$839$	6.11146	0.210991	0.105495	+	0.994420i	$0.466357\pi$
$840$	−4.47214	−0.154303
$841$	−13.0000	−0.448276
$842$	− 11.7082i	− 0.403491i
$843$	− 18.3607i	− 0.632375i
$844$	−26.4721	−0.911208
$845$	− 15.6525i	− 0.538462i
$846$	9.70820	0.333775
$847$	32.8328i	1.12815i
$848$	− 12.4721i	− 0.428295i
$849$	20.9443	0.718806
$850$	22.3607	0.766965
$851$	4.00000	0.137118
$852$	2.47214i	0.0846940i
$853$	− 4.47214i	− 0.153123i	−0.997065	−	0.0765615i	$-0.975606\pi$
$853$	− 4.47214i	− 0.153123i	0.997065	−	0.0765615i	$-0.0243942\pi$
$854$	−10.4721	−0.358349
$855$	− 16.1803i	− 0.553356i
$856$	4.00000	0.136717
$857$	17.0557i	0.582613i	0.956630	+	0.291306i	$0.0940899\pi$
$857$	17.0557i	0.582613i	−0.956630	+	0.291306i	$0.905910\pi$
$858$	− 23.4164i	− 0.799423i
$859$	−10.0689	−0.343546	−0.171773	−	0.985137i	$-0.554950\pi$
$859$	−10.0689	−0.343546	−0.171773	+	0.985137i	$0.554950\pi$
$860$	−23.4164	−0.798493
$861$	8.94427	0.304820
$862$	− 17.8885i	− 0.609286i
$863$	23.2361i	0.790965i	0.918474	+	0.395482i	$0.129423\pi$
$863$	23.2361i	0.790965i	−0.918474	+	0.395482i	$0.870577\pi$
$864$	1.00000	0.0340207
$865$	40.2492	1.36851
$866$	−21.5279	−0.731547
$867$	− 3.00000i	− 0.101885i
$868$	4.94427i	0.167820i
$869$	12.9443	0.439104
$870$	8.94427i	0.303239i
$871$	46.8328	1.58687
$872$	− 14.1803i	− 0.480207i
$873$	8.18034i	0.276863i
$874$	28.9443	0.979055
$875$	22.3607	0.755929
$876$	−10.4721	−0.353821
$877$	− 24.8328i	− 0.838545i	−0.907861	−	0.419272i	$-0.862285\pi$
$877$	− 24.8328i	− 0.838545i	0.907861	−	0.419272i	$-0.137715\pi$
$878$	10.4721i	0.353417i
$879$	−26.0000	−0.876958
$880$	11.7082i	0.394683i
$881$	−1.63932	−0.0552301	−0.0276151	−	0.999619i	$-0.508791\pi$
$881$	−1.63932	−0.0552301	−0.0276151	+	0.999619i	$0.508791\pi$
$882$	− 3.00000i	− 0.101015i
$883$	− 22.8328i	− 0.768385i	−0.923253	−	0.384193i	$-0.874480\pi$
$883$	− 22.8328i	− 0.768385i	0.923253	−	0.384193i	$-0.125520\pi$
$884$	20.0000	0.672673
$885$	15.5279	0.521963
$886$	26.8328	0.901466
$887$	34.2918i	1.15141i	0.817659	+	0.575703i	$0.195272\pi$
$887$	34.2918i	1.15141i	−0.817659	+	0.575703i	$0.804728\pi$
$888$	1.00000i	0.0335578i
$889$	5.88854	0.197495
$890$	−16.8328	−0.564237
$891$	5.23607	0.175415
$892$	10.0000i	0.334825i
$893$	70.2492i	2.35080i
$894$	18.6525	0.623832
$895$	42.3607i	1.41596i
$896$	2.00000	0.0668153
$897$	17.8885i	0.597281i
$898$	7.52786i	0.251208i
$899$	9.88854	0.329801
$900$	−5.00000	−0.166667
$901$	55.7771	1.85820
$902$	− 23.4164i	− 0.779681i
$903$	20.9443i	0.696982i
$904$	−10.0000	−0.332595
$905$	15.5279i	0.516164i
$906$	−8.94427	−0.297154
$907$	47.4164i	1.57444i	0.616675	+	0.787218i	$0.288479\pi$
$907$	47.4164i	1.57444i	−0.616675	+	0.787218i	$0.711521\pi$
$908$	− 28.9443i	− 0.960549i
$909$	4.76393	0.158010
$910$	20.0000	0.662994
$911$	30.2492	1.00220	0.501101	−	0.865389i	$-0.332929\pi$
$911$	30.2492	1.00220	0.501101	+	0.865389i	$0.332929\pi$
$912$	7.23607i	0.239610i
$913$	80.7214i	2.67149i
$914$	13.1246	0.434124
$915$	−11.7082	−0.387061
$916$	−13.4164	−0.443291
$917$	− 4.00000i	− 0.132092i
$918$	4.47214i	0.147602i
$919$	50.8328	1.67682	0.838410	−	0.545040i	$-0.183486\pi$
$919$	50.8328	1.67682	0.838410	+	0.545040i	$0.183486\pi$
$920$	− 8.94427i	− 0.294884i
$921$	−27.4164	−0.903401
$922$	6.47214i	0.213148i
$923$	− 11.0557i	− 0.363904i
$924$	10.4721	0.344508
$925$	− 5.00000i	− 0.164399i
$926$	−18.7639	−0.616621
$927$	− 19.7082i	− 0.647302i
$928$	− 4.00000i	− 0.131306i
$929$	−47.8885	−1.57117	−0.785586	−	0.618752i	$-0.787638\pi$
$929$	−47.8885	−1.57117	−0.785586	+	0.618752i	$0.787638\pi$
$930$	5.52786i	0.181266i
$931$	21.7082	0.711458
$932$	0.291796i	0.00955810i
$933$	8.94427i	0.292822i
$934$	3.05573	0.0999865
$935$	−52.3607	−1.71238
$936$	−4.47214	−0.146176
$937$	− 51.4164i	− 1.67970i	−0.542818	−	0.839850i	$-0.682643\pi$
$937$	− 51.4164i	− 1.67970i	0.542818	−	0.839850i	$-0.317357\pi$
$938$	20.9443i	0.683855i
$939$	−2.29180	−0.0747899
$940$	21.7082	0.708044
$941$	7.23607	0.235889	0.117945	−	0.993020i	$-0.462369\pi$
$941$	7.23607	0.235889	0.117945	+	0.993020i	$0.462369\pi$
$942$	− 14.9443i	− 0.486911i
$943$	17.8885i	0.582531i
$944$	−6.94427	−0.226017
$945$	4.47214i	0.145479i
$946$	54.8328	1.78277
$947$	5.88854i	0.191352i	0.995413	+	0.0956760i	$0.0305013\pi$
$947$	5.88854i	0.191352i	−0.995413	+	0.0956760i	$0.969499\pi$
$948$	− 2.47214i	− 0.0802912i
$949$	46.8328	1.52026
$950$	− 36.1803i	− 1.17385i
$951$	2.94427	0.0954746
$952$	8.94427i	0.289886i
$953$	− 7.70820i	− 0.249693i	−0.992176	−	0.124847i	$-0.960156\pi$
$953$	− 7.70820i	− 0.249693i	0.992176	−	0.124847i	$-0.0398439\pi$
$954$	−12.4721	−0.403800
$955$	7.63932i	0.247203i
$956$	−6.47214	−0.209324
$957$	− 20.9443i	− 0.677032i
$958$	− 30.4721i	− 0.984510i
$959$	−15.4164	−0.497822
$960$	2.23607	0.0721688
$961$	−24.8885	−0.802856
$962$	− 4.47214i	− 0.144187i
$963$	− 4.00000i	− 0.128898i
$964$	−12.4721	−0.401700
$965$	−9.34752	−0.300907
$966$	−8.00000	−0.257396
$967$	48.4296i	1.55739i	0.627403	+	0.778695i	$0.284118\pi$
$967$	48.4296i	1.55739i	−0.627403	+	0.778695i	$0.715882\pi$
$968$	− 16.4164i	− 0.527643i
$969$	−32.3607	−1.03957
$970$	18.2918i	0.587314i
$971$	26.7639	0.858895	0.429448	−	0.903092i	$-0.358708\pi$
$971$	26.7639	0.858895	0.429448	+	0.903092i	$0.358708\pi$
$972$	− 1.00000i	− 0.0320750i
$973$	− 40.0000i	− 1.28234i
$974$	−3.70820	−0.118819
$975$	22.3607	0.716115
$976$	5.23607	0.167602
$977$	− 26.3607i	− 0.843353i	−0.906746	−	0.421676i	$-0.861442\pi$
$977$	− 26.3607i	− 0.843353i	0.906746	−	0.421676i	$-0.138558\pi$
$978$	20.9443i	0.669724i
$979$	39.4164	1.25975
$980$	− 6.70820i	− 0.214286i
$981$	−14.1803	−0.452743
$982$	− 21.2361i	− 0.677670i
$983$	− 21.1246i	− 0.673770i	−0.941546	−	0.336885i	$-0.890627\pi$
$983$	− 21.1246i	− 0.673770i	0.941546	−	0.336885i	$-0.109373\pi$
$984$	−4.47214	−0.142566
$985$	−24.4721	−0.779747
$986$	17.8885	0.569687
$987$	− 19.4164i	− 0.618031i
$988$	− 32.3607i	− 1.02953i
$989$	−41.8885	−1.33198
$990$	11.7082	0.372111
$991$	47.7771	1.51769	0.758845	−	0.651272i	$-0.225764\pi$
$991$	47.7771	1.51769	0.758845	+	0.651272i	$0.225764\pi$
$992$	− 2.47214i	− 0.0784904i
$993$	31.2361i	0.991247i
$994$	4.94427	0.156823
$995$	12.3607i	0.391860i
$996$	15.4164	0.488488
$997$	− 12.1115i	− 0.383574i	−0.981437	−	0.191787i	$-0.938572\pi$
$997$	− 12.1115i	− 0.383574i	0.981437	−	0.191787i	$-0.0614282\pi$
$998$	− 4.18034i	− 0.132326i
$999$	1.00000	0.0316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1110.2.d.f.889.4	yes	4
3.2	odd	2		3330.2.d.j.1999.1		4
5.2	odd	4		5550.2.a.bu.1.2		2
5.3	odd	4		5550.2.a.bz.1.2		2
5.4	even	2	inner	1110.2.d.f.889.1	✓	4
15.14	odd	2		3330.2.d.j.1999.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1110.2.d.f.889.1	✓	4	5.4	even	2	inner
1110.2.d.f.889.4	yes	4	1.1	even	1	trivial
3330.2.d.j.1999.1		4	3.2	odd	2
3330.2.d.j.1999.4		4	15.14	odd	2
5550.2.a.bu.1.2		2	5.2	odd	4
5550.2.a.bz.1.2		2	5.3	odd	4

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:	\( N \)	\(=\)	\( 1110 = 2 \cdot 3 \cdot 5 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1110.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +2.23607i q^{5} -1.00000 q^{6} +2.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +2.23607i q^{5} -1.00000 q^{6} +2.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} -2.23607 q^{10} +5.23607 q^{11} -1.00000i q^{12} +4.47214i q^{13} -2.00000 q^{14} -2.23607 q^{15} +1.00000 q^{16} +4.47214i q^{17} -1.00000i q^{18} +7.23607 q^{19} -2.23607i q^{20} -2.00000 q^{21} +5.23607i q^{22} -4.00000i q^{23} +1.00000 q^{24} -5.00000 q^{25} -4.47214 q^{26} -1.00000i q^{27} -2.00000i q^{28} -4.00000 q^{29} -2.23607i q^{30} -2.47214 q^{31} +1.00000i q^{32} +5.23607i q^{33} -4.47214 q^{34} -4.47214 q^{35} +1.00000 q^{36} +1.00000i q^{37} +7.23607i q^{38} -4.47214 q^{39} +2.23607 q^{40} -4.47214 q^{41} -2.00000i q^{42} -10.4721i q^{43} -5.23607 q^{44} -2.23607i q^{45} +4.00000 q^{46} +9.70820i q^{47} +1.00000i q^{48} +3.00000 q^{49} -5.00000i q^{50} -4.47214 q^{51} -4.47214i q^{52} -12.4721i q^{53} +1.00000 q^{54} +11.7082i q^{55} +2.00000 q^{56} +7.23607i q^{57} -4.00000i q^{58} -6.94427 q^{59} +2.23607 q^{60} +5.23607 q^{61} -2.47214i q^{62} -2.00000i q^{63} -1.00000 q^{64} -10.0000 q^{65} -5.23607 q^{66} -10.4721i q^{67} -4.47214i q^{68} +4.00000 q^{69} -4.47214i q^{70} -2.47214 q^{71} +1.00000i q^{72} -10.4721i q^{73} -1.00000 q^{74} -5.00000i q^{75} -7.23607 q^{76} +10.4721i q^{77} -4.47214i q^{78} +2.47214 q^{79} +2.23607i q^{80} +1.00000 q^{81} -4.47214i q^{82} +15.4164i q^{83} +2.00000 q^{84} -10.0000 q^{85} +10.4721 q^{86} -4.00000i q^{87} -5.23607i q^{88} +7.52786 q^{89} +2.23607 q^{90} -8.94427 q^{91} +4.00000i q^{92} -2.47214i q^{93} -9.70820 q^{94} +16.1803i q^{95} -1.00000 q^{96} -8.18034i q^{97} +3.00000i q^{98} -5.23607 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{4} - 4 q^{6} - 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^4 - 4 * q^6 - 4 * q^9	\( 4 q - 4 q^{4} - 4 q^{6} - 4 q^{9} + 12 q^{11} - 8 q^{14} + 4 q^{16} + 20 q^{19} - 8 q^{21} + 4 q^{24} - 20 q^{25} - 16 q^{29} + 8 q^{31} + 4 q^{36} - 12 q^{44} + 16 q^{46} + 12 q^{49} + 4 q^{54} + 8 q^{56} + 8 q^{59} + 12 q^{61} - 4 q^{64} - 40 q^{65} - 12 q^{66} + 16 q^{69} + 8 q^{71} - 4 q^{74} - 20 q^{76} - 8 q^{79} + 4 q^{81} + 8 q^{84} - 40 q^{85} + 24 q^{86} + 48 q^{89} - 12 q^{94} - 4 q^{96} - 12 q^{99}+O(q^{100}) \) 4 * q - 4 * q^4 - 4 * q^6 - 4 * q^9 + 12 * q^11 - 8 * q^14 + 4 * q^16 + 20 * q^19 - 8 * q^21 + 4 * q^24 - 20 * q^25 - 16 * q^29 + 8 * q^31 + 4 * q^36 - 12 * q^44 + 16 * q^46 + 12 * q^49 + 4 * q^54 + 8 * q^56 + 8 * q^59 + 12 * q^61 - 4 * q^64 - 40 * q^65 - 12 * q^66 + 16 * q^69 + 8 * q^71 - 4 * q^74 - 20 * q^76 - 8 * q^79 + 4 * q^81 + 8 * q^84 - 40 * q^85 + 24 * q^86 + 48 * q^89 - 12 * q^94 - 4 * q^96 - 12 * q^99

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