LMFDB - Embedded newform 2890.2.b.b.2311.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2890,2,Mod(2311,2890)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2890, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2890.2311");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2890 = 2 \cdot 5 \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2890.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$23.0767661842$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 170)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2311.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	2890.2311
Dual form			2890.2.b.b.2311.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -1.00000i q^{3} +1.00000 q^{4} +1.00000i q^{5} +1.00000i q^{6} +2.00000i q^{7} -1.00000 q^{8} +2.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -1.00000i q^{3} +1.00000 q^{4} +1.00000i q^{5} +1.00000i q^{6} +2.00000i q^{7} -1.00000 q^{8} +2.00000 q^{9} -1.00000i q^{10} -1.00000i q^{12} -1.00000 q^{13} -2.00000i q^{14} +1.00000 q^{15} +1.00000 q^{16} -2.00000 q^{18} +1.00000 q^{19} +1.00000i q^{20} +2.00000 q^{21} -6.00000i q^{23} +1.00000i q^{24} -1.00000 q^{25} +1.00000 q^{26} -5.00000i q^{27} +2.00000i q^{28} +3.00000i q^{29} -1.00000 q^{30} -5.00000i q^{31} -1.00000 q^{32} -2.00000 q^{35} +2.00000 q^{36} -8.00000i q^{37} -1.00000 q^{38} +1.00000i q^{39} -1.00000i q^{40} +6.00000i q^{41} -2.00000 q^{42} +10.0000 q^{43} +2.00000i q^{45} +6.00000i q^{46} -3.00000 q^{47} -1.00000i q^{48} +3.00000 q^{49} +1.00000 q^{50} -1.00000 q^{52} +3.00000 q^{53} +5.00000i q^{54} -2.00000i q^{56} -1.00000i q^{57} -3.00000i q^{58} -3.00000 q^{59} +1.00000 q^{60} +11.0000i q^{61} +5.00000i q^{62} +4.00000i q^{63} +1.00000 q^{64} -1.00000i q^{65} +2.00000 q^{67} -6.00000 q^{69} +2.00000 q^{70} -9.00000i q^{71} -2.00000 q^{72} -11.0000i q^{73} +8.00000i q^{74} +1.00000i q^{75} +1.00000 q^{76} -1.00000i q^{78} +8.00000i q^{79} +1.00000i q^{80} +1.00000 q^{81} -6.00000i q^{82} +12.0000 q^{83} +2.00000 q^{84} -10.0000 q^{86} +3.00000 q^{87} +15.0000 q^{89} -2.00000i q^{90} -2.00000i q^{91} -6.00000i q^{92} -5.00000 q^{93} +3.00000 q^{94} +1.00000i q^{95} +1.00000i q^{96} +7.00000i q^{97} -3.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{4} - 2 q^{8} + 4 q^{9}+O(q^{10}) $ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^8 + 4 * q^9	$ 2 q - 2 q^{2} + 2 q^{4} - 2 q^{8} + 4 q^{9} - 2 q^{13} + 2 q^{15} + 2 q^{16} - 4 q^{18} + 2 q^{19} + 4 q^{21} - 2 q^{25} + 2 q^{26} - 2 q^{30} - 2 q^{32} - 4 q^{35} + 4 q^{36} - 2 q^{38} - 4 q^{42} + 20 q^{43} - 6 q^{47} + 6 q^{49} + 2 q^{50} - 2 q^{52} + 6 q^{53} - 6 q^{59} + 2 q^{60} + 2 q^{64} + 4 q^{67} - 12 q^{69} + 4 q^{70} - 4 q^{72} + 2 q^{76} + 2 q^{81} + 24 q^{83} + 4 q^{84} - 20 q^{86} + 6 q^{87} + 30 q^{89} - 10 q^{93} + 6 q^{94} - 6 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^8 + 4 * q^9 - 2 * q^13 + 2 * q^15 + 2 * q^16 - 4 * q^18 + 2 * q^19 + 4 * q^21 - 2 * q^25 + 2 * q^26 - 2 * q^30 - 2 * q^32 - 4 * q^35 + 4 * q^36 - 2 * q^38 - 4 * q^42 + 20 * q^43 - 6 * q^47 + 6 * q^49 + 2 * q^50 - 2 * q^52 + 6 * q^53 - 6 * q^59 + 2 * q^60 + 2 * q^64 + 4 * q^67 - 12 * q^69 + 4 * q^70 - 4 * q^72 + 2 * q^76 + 2 * q^81 + 24 * q^83 + 4 * q^84 - 20 * q^86 + 6 * q^87 + 30 * q^89 - 10 * q^93 + 6 * q^94 - 6 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2890\mathbb{Z}\right)^\times$.

$n$	$581$	$1157$
$\chi(n)$	$-1$	$1$

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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	− 1.00000i	− 0.577350i	−0.957427	−	0.288675i	$-0.906785\pi$
$3$	− 1.00000i	− 0.577350i	0.957427	−	0.288675i	$-0.0932147\pi$
$4$	1.00000	0.500000
$5$	1.00000i	0.447214i
$5$	1.00000i	0.447214i
$6$	1.00000i	0.408248i
$7$	2.00000i	0.755929i	0.925820	+	0.377964i	$0.123376\pi$
$7$	2.00000i	0.755929i	−0.925820	+	0.377964i	$0.876624\pi$
$8$	−1.00000	−0.353553
$9$	2.00000	0.666667
$10$	− 1.00000i	− 0.316228i
$11$	0	0	1.00000			$0$
$11$	0	0	−1.00000			$\pi$
$12$	− 1.00000i	− 0.288675i
$13$	−1.00000	−0.277350	−0.138675	−	0.990338i	$-0.544284\pi$
$13$	−1.00000	−0.277350	−0.138675	+	0.990338i	$0.544284\pi$
$14$	− 2.00000i	− 0.534522i
$15$	1.00000	0.258199
$16$	1.00000	0.250000
$17$	0	0
$17$	0	0
$18$	−2.00000	−0.471405
$19$	1.00000	0.229416	0.114708	−	0.993399i	$-0.463407\pi$
$19$	1.00000	0.229416	0.114708	+	0.993399i	$0.463407\pi$
$20$	1.00000i	0.223607i
$21$	2.00000	0.436436
$22$	0	0
$23$	− 6.00000i	− 1.25109i	−0.780189	−	0.625543i	$-0.784877\pi$
$23$	− 6.00000i	− 1.25109i	0.780189	−	0.625543i	$-0.215123\pi$
$24$	1.00000i	0.204124i
$25$	−1.00000	−0.200000
$26$	1.00000	0.196116
$27$	− 5.00000i	− 0.962250i
$28$	2.00000i	0.377964i
$29$	3.00000i	0.557086i	0.960424	+	0.278543i	$0.0898515\pi$
$29$	3.00000i	0.557086i	−0.960424	+	0.278543i	$0.910149\pi$
$30$	−1.00000	−0.182574
$31$	− 5.00000i	− 0.898027i	−0.893525	−	0.449013i	$-0.851776\pi$
$31$	− 5.00000i	− 0.898027i	0.893525	−	0.449013i	$-0.148224\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	0	0
$35$	−2.00000	−0.338062
$36$	2.00000	0.333333
$37$	− 8.00000i	− 1.31519i	−0.753371	−	0.657596i	$-0.771573\pi$
$37$	− 8.00000i	− 1.31519i	0.753371	−	0.657596i	$-0.228427\pi$
$38$	−1.00000	−0.162221
$39$	1.00000i	0.160128i
$40$	− 1.00000i	− 0.158114i
$41$	6.00000i	0.937043i	0.883452	+	0.468521i	$0.155213\pi$
$41$	6.00000i	0.937043i	−0.883452	+	0.468521i	$0.844787\pi$
$42$	−2.00000	−0.308607
$43$	10.0000	1.52499	0.762493	−	0.646997i	$-0.223975\pi$
$43$	10.0000	1.52499	0.762493	+	0.646997i	$0.223975\pi$
$44$	0	0
$45$	2.00000i	0.298142i
$46$	6.00000i	0.884652i
$47$	−3.00000	−0.437595	−0.218797	−	0.975770i	$-0.570213\pi$
$47$	−3.00000	−0.437595	−0.218797	+	0.975770i	$0.570213\pi$
$48$	− 1.00000i	− 0.144338i
$49$	3.00000	0.428571
$50$	1.00000	0.141421
$51$	0	0
$52$	−1.00000	−0.138675
$53$	3.00000	0.412082	0.206041	−	0.978543i	$-0.433942\pi$
$53$	3.00000	0.412082	0.206041	+	0.978543i	$0.433942\pi$
$54$	5.00000i	0.680414i
$55$	0	0
$56$	− 2.00000i	− 0.267261i
$57$	− 1.00000i	− 0.132453i
$58$	− 3.00000i	− 0.393919i
$59$	−3.00000	−0.390567	−0.195283	−	0.980747i	$-0.562563\pi$
$59$	−3.00000	−0.390567	−0.195283	+	0.980747i	$0.562563\pi$
$60$	1.00000	0.129099
$61$	11.0000i	1.40841i	0.709999	+	0.704203i	$0.248695\pi$
$61$	11.0000i	1.40841i	−0.709999	+	0.704203i	$0.751305\pi$
$62$	5.00000i	0.635001i
$63$	4.00000i	0.503953i
$64$	1.00000	0.125000
$65$	− 1.00000i	− 0.124035i
$66$	0	0
$67$	2.00000	0.244339	0.122169	−	0.992509i	$-0.461015\pi$
$67$	2.00000	0.244339	0.122169	+	0.992509i	$0.461015\pi$
$68$	0	0
$69$	−6.00000	−0.722315
$70$	2.00000	0.239046
$71$	− 9.00000i	− 1.06810i	−0.845452	−	0.534052i	$-0.820669\pi$
$71$	− 9.00000i	− 1.06810i	0.845452	−	0.534052i	$-0.179331\pi$
$72$	−2.00000	−0.235702
$73$	− 11.0000i	− 1.28745i	−0.765256	−	0.643726i	$-0.777388\pi$
$73$	− 11.0000i	− 1.28745i	0.765256	−	0.643726i	$-0.222612\pi$
$74$	8.00000i	0.929981i
$75$	1.00000i	0.115470i
$76$	1.00000	0.114708
$77$	0	0
$78$	− 1.00000i	− 0.113228i
$79$	8.00000i	0.900070i	0.893011	+	0.450035i	$0.148589\pi$
$79$	8.00000i	0.900070i	−0.893011	+	0.450035i	$0.851411\pi$
$80$	1.00000i	0.111803i
$81$	1.00000	0.111111
$82$	− 6.00000i	− 0.662589i
$83$	12.0000	1.31717	0.658586	−	0.752506i	$-0.271155\pi$
$83$	12.0000	1.31717	0.658586	+	0.752506i	$0.271155\pi$
$84$	2.00000	0.218218
$85$	0	0
$86$	−10.0000	−1.07833
$87$	3.00000	0.321634
$88$	0	0
$89$	15.0000	1.59000	0.794998	−	0.606612i	$-0.207472\pi$
$89$	15.0000	1.59000	0.794998	+	0.606612i	$0.207472\pi$
$90$	− 2.00000i	− 0.210819i
$91$	− 2.00000i	− 0.209657i
$92$	− 6.00000i	− 0.625543i
$93$	−5.00000	−0.518476
$94$	3.00000	0.309426
$95$	1.00000i	0.102598i
$96$	1.00000i	0.102062i
$97$	7.00000i	0.710742i	0.934725	+	0.355371i	$0.115646\pi$
$97$	7.00000i	0.710742i	−0.934725	+	0.355371i	$0.884354\pi$
$98$	−3.00000	−0.303046
$99$	0	0
$100$	−1.00000	−0.100000
$101$	6.00000	0.597022	0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000	0.597022	0.298511	+	0.954406i	$0.403510\pi$
$102$	0	0
$103$	8.00000	0.788263	0.394132	−	0.919054i	$-0.371045\pi$
$103$	8.00000	0.788263	0.394132	+	0.919054i	$0.371045\pi$
$104$	1.00000	0.0980581
$105$	2.00000i	0.195180i
$106$	−3.00000	−0.291386
$107$	12.0000i	1.16008i	0.814587	+	0.580042i	$0.196964\pi$
$107$	12.0000i	1.16008i	−0.814587	+	0.580042i	$0.803036\pi$
$108$	− 5.00000i	− 0.481125i
$109$	11.0000i	1.05361i	0.849987	+	0.526804i	$0.176610\pi$
$109$	11.0000i	1.05361i	−0.849987	+	0.526804i	$0.823390\pi$
$110$	0	0
$111$	−8.00000	−0.759326
$112$	2.00000i	0.188982i
$113$	9.00000i	0.846649i	0.905978	+	0.423324i	$0.139137\pi$
$113$	9.00000i	0.846649i	−0.905978	+	0.423324i	$0.860863\pi$
$114$	1.00000i	0.0936586i
$115$	6.00000	0.559503
$116$	3.00000i	0.278543i
$117$	−2.00000	−0.184900
$118$	3.00000	0.276172
$119$	0	0
$120$	−1.00000	−0.0912871
$121$	11.0000	1.00000
$122$	− 11.0000i	− 0.995893i
$123$	6.00000	0.541002
$124$	− 5.00000i	− 0.449013i
$125$	− 1.00000i	− 0.0894427i
$126$	− 4.00000i	− 0.356348i
$127$	−11.0000	−0.976092	−0.488046	−	0.872818i	$-0.662290\pi$
$127$	−11.0000	−0.976092	−0.488046	+	0.872818i	$0.662290\pi$
$128$	−1.00000	−0.0883883
$129$	− 10.0000i	− 0.880451i
$130$	1.00000i	0.0877058i
$131$	18.0000i	1.57267i	0.617802	+	0.786334i	$0.288023\pi$
$131$	18.0000i	1.57267i	−0.617802	+	0.786334i	$0.711977\pi$
$132$	0	0
$133$	2.00000i	0.173422i
$134$	−2.00000	−0.172774
$135$	5.00000	0.430331
$136$	0	0
$137$	−18.0000	−1.53784	−0.768922	−	0.639343i	$-0.779207\pi$
$137$	−18.0000	−1.53784	−0.768922	+	0.639343i	$0.779207\pi$
$138$	6.00000	0.510754
$139$	− 8.00000i	− 0.678551i	−0.940687	−	0.339276i	$-0.889818\pi$
$139$	− 8.00000i	− 0.678551i	0.940687	−	0.339276i	$-0.110182\pi$
$140$	−2.00000	−0.169031
$141$	3.00000i	0.252646i
$142$	9.00000i	0.755263i
$143$	0	0
$144$	2.00000	0.166667
$145$	−3.00000	−0.249136
$146$	11.0000i	0.910366i
$147$	− 3.00000i	− 0.247436i
$148$	− 8.00000i	− 0.657596i
$149$	−12.0000	−0.983078	−0.491539	−	0.870855i	$-0.663566\pi$
$149$	−12.0000	−0.983078	−0.491539	+	0.870855i	$0.663566\pi$
$150$	− 1.00000i	− 0.0816497i
$151$	22.0000	1.79033	0.895167	−	0.445730i	$-0.147056\pi$
$151$	22.0000	1.79033	0.895167	+	0.445730i	$0.147056\pi$
$152$	−1.00000	−0.0811107
$153$	0	0
$154$	0	0
$155$	5.00000	0.401610
$156$	1.00000i	0.0800641i
$157$	14.0000	1.11732	0.558661	−	0.829396i	$-0.311315\pi$
$157$	14.0000	1.11732	0.558661	+	0.829396i	$0.311315\pi$
$158$	− 8.00000i	− 0.636446i
$159$	− 3.00000i	− 0.237915i
$160$	− 1.00000i	− 0.0790569i
$161$	12.0000	0.945732
$162$	−1.00000	−0.0785674
$163$	− 16.0000i	− 1.25322i	−0.779334	−	0.626608i	$-0.784443\pi$
$163$	− 16.0000i	− 1.25322i	0.779334	−	0.626608i	$-0.215557\pi$
$164$	6.00000i	0.468521i
$165$	0	0
$166$	−12.0000	−0.931381
$167$	18.0000i	1.39288i	0.717614	+	0.696441i	$0.245234\pi$
$167$	18.0000i	1.39288i	−0.717614	+	0.696441i	$0.754766\pi$
$168$	−2.00000	−0.154303
$169$	−12.0000	−0.923077
$170$	0	0
$171$	2.00000	0.152944
$172$	10.0000	0.762493
$173$	− 18.0000i	− 1.36851i	−0.729241	−	0.684257i	$-0.760127\pi$
$173$	− 18.0000i	− 1.36851i	0.729241	−	0.684257i	$-0.239873\pi$
$174$	−3.00000	−0.227429
$175$	− 2.00000i	− 0.151186i
$176$	0	0
$177$	3.00000i	0.225494i
$178$	−15.0000	−1.12430
$179$	12.0000	0.896922	0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000	0.896922	0.448461	+	0.893802i	$0.351972\pi$
$180$	2.00000i	0.149071i
$181$	− 10.0000i	− 0.743294i	−0.928374	−	0.371647i	$-0.878793\pi$
$181$	− 10.0000i	− 0.743294i	0.928374	−	0.371647i	$-0.121207\pi$
$182$	2.00000i	0.148250i
$183$	11.0000	0.813143
$184$	6.00000i	0.442326i
$185$	8.00000	0.588172
$186$	5.00000	0.366618
$187$	0	0
$188$	−3.00000	−0.218797
$189$	10.0000	0.727393
$190$	− 1.00000i	− 0.0725476i
$191$	12.0000	0.868290	0.434145	−	0.900843i	$-0.357051\pi$
$191$	12.0000	0.868290	0.434145	+	0.900843i	$0.357051\pi$
$192$	− 1.00000i	− 0.0721688i
$193$	2.00000i	0.143963i	0.997406	+	0.0719816i	$0.0229323\pi$
$193$	2.00000i	0.143963i	−0.997406	+	0.0719816i	$0.977068\pi$
$194$	− 7.00000i	− 0.502571i
$195$	−1.00000	−0.0716115
$196$	3.00000	0.214286
$197$	− 24.0000i	− 1.70993i	−0.518686	−	0.854965i	$-0.673579\pi$
$197$	− 24.0000i	− 1.70993i	0.518686	−	0.854965i	$-0.326421\pi$
$198$	0	0
$199$	− 11.0000i	− 0.779769i	−0.920864	−	0.389885i	$-0.872515\pi$
$199$	− 11.0000i	− 0.779769i	0.920864	−	0.389885i	$-0.127485\pi$
$200$	1.00000	0.0707107
$201$	− 2.00000i	− 0.141069i
$202$	−6.00000	−0.422159
$203$	−6.00000	−0.421117
$204$	0	0
$205$	−6.00000	−0.419058
$206$	−8.00000	−0.557386
$207$	− 12.0000i	− 0.834058i
$208$	−1.00000	−0.0693375
$209$	0	0
$210$	− 2.00000i	− 0.138013i
$211$	2.00000i	0.137686i	0.997628	+	0.0688428i	$0.0219307\pi$
$211$	2.00000i	0.137686i	−0.997628	+	0.0688428i	$0.978069\pi$
$212$	3.00000	0.206041
$213$	−9.00000	−0.616670
$214$	− 12.0000i	− 0.820303i
$215$	10.0000i	0.681994i
$216$	5.00000i	0.340207i
$217$	10.0000	0.678844
$218$	− 11.0000i	− 0.745014i
$219$	−11.0000	−0.743311
$220$	0	0
$221$	0	0
$222$	8.00000	0.536925
$223$	1.00000	0.0669650	0.0334825	−	0.999439i	$-0.489340\pi$
$223$	1.00000	0.0669650	0.0334825	+	0.999439i	$0.489340\pi$
$224$	− 2.00000i	− 0.133631i
$225$	−2.00000	−0.133333
$226$	− 9.00000i	− 0.598671i
$227$	3.00000i	0.199117i	0.995032	+	0.0995585i	$0.0317430\pi$
$227$	3.00000i	0.199117i	−0.995032	+	0.0995585i	$0.968257\pi$
$228$	− 1.00000i	− 0.0662266i
$229$	22.0000	1.45380	0.726900	−	0.686743i	$-0.240960\pi$
$229$	22.0000	1.45380	0.726900	+	0.686743i	$0.240960\pi$
$230$	−6.00000	−0.395628
$231$	0	0
$232$	− 3.00000i	− 0.196960i
$233$	− 9.00000i	− 0.589610i	−0.955557	−	0.294805i	$-0.904745\pi$
$233$	− 9.00000i	− 0.589610i	0.955557	−	0.294805i	$-0.0952546\pi$
$234$	2.00000	0.130744
$235$	− 3.00000i	− 0.195698i
$236$	−3.00000	−0.195283
$237$	8.00000	0.519656
$238$	0	0
$239$	6.00000	0.388108	0.194054	−	0.980991i	$-0.437836\pi$
$239$	6.00000	0.388108	0.194054	+	0.980991i	$0.437836\pi$
$240$	1.00000	0.0645497
$241$	10.0000i	0.644157i	0.946713	+	0.322078i	$0.104381\pi$
$241$	10.0000i	0.644157i	−0.946713	+	0.322078i	$0.895619\pi$
$242$	−11.0000	−0.707107
$243$	− 16.0000i	− 1.02640i
$244$	11.0000i	0.704203i
$245$	3.00000i	0.191663i
$246$	−6.00000	−0.382546
$247$	−1.00000	−0.0636285
$248$	5.00000i	0.317500i
$249$	− 12.0000i	− 0.760469i
$250$	1.00000i	0.0632456i
$251$	−12.0000	−0.757433	−0.378717	−	0.925513i	$-0.623635\pi$
$251$	−12.0000	−0.757433	−0.378717	+	0.925513i	$0.623635\pi$
$252$	4.00000i	0.251976i
$253$	0	0
$254$	11.0000	0.690201
$255$	0	0
$256$	1.00000	0.0625000
$257$	−6.00000	−0.374270	−0.187135	−	0.982334i	$-0.559920\pi$
$257$	−6.00000	−0.374270	−0.187135	+	0.982334i	$0.559920\pi$
$258$	10.0000i	0.622573i
$259$	16.0000	0.994192
$260$	− 1.00000i	− 0.0620174i
$261$	6.00000i	0.371391i
$262$	− 18.0000i	− 1.11204i
$263$	−9.00000	−0.554964	−0.277482	−	0.960731i	$-0.589500\pi$
$263$	−9.00000	−0.554964	−0.277482	+	0.960731i	$0.589500\pi$
$264$	0	0
$265$	3.00000i	0.184289i
$266$	− 2.00000i	− 0.122628i
$267$	− 15.0000i	− 0.917985i
$268$	2.00000	0.122169
$269$	− 15.0000i	− 0.914566i	−0.889321	−	0.457283i	$-0.848823\pi$
$269$	− 15.0000i	− 0.914566i	0.889321	−	0.457283i	$-0.151177\pi$
$270$	−5.00000	−0.304290
$271$	32.0000	1.94386	0.971931	−	0.235267i	$-0.0755965\pi$
$271$	32.0000	1.94386	0.971931	+	0.235267i	$0.0755965\pi$
$272$	0	0
$273$	−2.00000	−0.121046
$274$	18.0000	1.08742
$275$	0	0
$276$	−6.00000	−0.361158
$277$	10.0000i	0.600842i	0.953807	+	0.300421i	$0.0971271\pi$
$277$	10.0000i	0.600842i	−0.953807	+	0.300421i	$0.902873\pi$
$278$	8.00000i	0.479808i
$279$	− 10.0000i	− 0.598684i
$280$	2.00000	0.119523
$281$	−27.0000	−1.61068	−0.805342	−	0.592810i	$-0.798019\pi$
$281$	−27.0000	−1.61068	−0.805342	+	0.592810i	$0.798019\pi$
$282$	− 3.00000i	− 0.178647i
$283$	− 13.0000i	− 0.772770i	−0.922338	−	0.386385i	$-0.873724\pi$
$283$	− 13.0000i	− 0.772770i	0.922338	−	0.386385i	$-0.126276\pi$
$284$	− 9.00000i	− 0.534052i
$285$	1.00000	0.0592349
$286$	0	0
$287$	−12.0000	−0.708338
$288$	−2.00000	−0.117851
$289$	0	0
$290$	3.00000	0.176166
$291$	7.00000	0.410347
$292$	− 11.0000i	− 0.643726i
$293$	−3.00000	−0.175262	−0.0876309	−	0.996153i	$-0.527930\pi$
$293$	−3.00000	−0.175262	−0.0876309	+	0.996153i	$0.527930\pi$
$294$	3.00000i	0.174964i
$295$	− 3.00000i	− 0.174667i
$296$	8.00000i	0.464991i
$297$	0	0
$298$	12.0000	0.695141
$299$	6.00000i	0.346989i
$300$	1.00000i	0.0577350i
$301$	20.0000i	1.15278i
$302$	−22.0000	−1.26596
$303$	− 6.00000i	− 0.344691i
$304$	1.00000	0.0573539
$305$	−11.0000	−0.629858
$306$	0	0
$307$	20.0000	1.14146	0.570730	−	0.821138i	$-0.306660\pi$
$307$	20.0000	1.14146	0.570730	+	0.821138i	$0.306660\pi$
$308$	0	0
$309$	− 8.00000i	− 0.455104i
$310$	−5.00000	−0.283981
$311$	0	0	1.00000			$0$
$311$	0	0	−1.00000			$\pi$
$312$	− 1.00000i	− 0.0566139i
$313$	14.0000i	0.791327i	0.918396	+	0.395663i	$0.129485\pi$
$313$	14.0000i	0.791327i	−0.918396	+	0.395663i	$0.870515\pi$
$314$	−14.0000	−0.790066
$315$	−4.00000	−0.225374
$316$	8.00000i	0.450035i
$317$	0	0	1.00000			$0$
$317$	0	0	−1.00000			$\pi$
$318$	3.00000i	0.168232i
$319$	0	0
$320$	1.00000i	0.0559017i
$321$	12.0000	0.669775
$322$	−12.0000	−0.668734
$323$	0	0
$324$	1.00000	0.0555556
$325$	1.00000	0.0554700
$326$	16.0000i	0.886158i
$327$	11.0000	0.608301
$328$	− 6.00000i	− 0.331295i
$329$	− 6.00000i	− 0.330791i
$330$	0	0
$331$	31.0000	1.70391	0.851957	−	0.523612i	$-0.175416\pi$
$331$	31.0000	1.70391	0.851957	+	0.523612i	$0.175416\pi$
$332$	12.0000	0.658586
$333$	− 16.0000i	− 0.876795i
$334$	− 18.0000i	− 0.984916i
$335$	2.00000i	0.109272i
$336$	2.00000	0.109109
$337$	13.0000i	0.708155i	0.935216	+	0.354078i	$0.115205\pi$
$337$	13.0000i	0.708155i	−0.935216	+	0.354078i	$0.884795\pi$
$338$	12.0000	0.652714
$339$	9.00000	0.488813
$340$	0	0
$341$	0	0
$342$	−2.00000	−0.108148
$343$	20.0000i	1.07990i
$344$	−10.0000	−0.539164
$345$	− 6.00000i	− 0.323029i
$346$	18.0000i	0.967686i
$347$	− 15.0000i	− 0.805242i	−0.915367	−	0.402621i	$-0.868099\pi$
$347$	− 15.0000i	− 0.805242i	0.915367	−	0.402621i	$-0.131901\pi$
$348$	3.00000	0.160817
$349$	28.0000	1.49881	0.749403	−	0.662114i	$-0.230341\pi$
$349$	28.0000	1.49881	0.749403	+	0.662114i	$0.230341\pi$
$350$	2.00000i	0.106904i
$351$	5.00000i	0.266880i
$352$	0	0
$353$	18.0000	0.958043	0.479022	−	0.877803i	$-0.340992\pi$
$353$	18.0000	0.958043	0.479022	+	0.877803i	$0.340992\pi$
$354$	− 3.00000i	− 0.159448i
$355$	9.00000	0.477670
$356$	15.0000	0.794998
$357$	0	0
$358$	−12.0000	−0.634220
$359$	−30.0000	−1.58334	−0.791670	−	0.610949i	$-0.790788\pi$
$359$	−30.0000	−1.58334	−0.791670	+	0.610949i	$0.790788\pi$
$360$	− 2.00000i	− 0.105409i
$361$	−18.0000	−0.947368
$362$	10.0000i	0.525588i
$363$	− 11.0000i	− 0.577350i
$364$	− 2.00000i	− 0.104828i
$365$	11.0000	0.575766
$366$	−11.0000	−0.574979
$367$	− 22.0000i	− 1.14839i	−0.818718	−	0.574195i	$-0.805315\pi$
$367$	− 22.0000i	− 1.14839i	0.818718	−	0.574195i	$-0.194685\pi$
$368$	− 6.00000i	− 0.312772i
$369$	12.0000i	0.624695i
$370$	−8.00000	−0.415900
$371$	6.00000i	0.311504i
$372$	−5.00000	−0.259238
$373$	−22.0000	−1.13912	−0.569558	−	0.821951i	$-0.692886\pi$
$373$	−22.0000	−1.13912	−0.569558	+	0.821951i	$0.692886\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	3.00000	0.154713
$377$	− 3.00000i	− 0.154508i
$378$	−10.0000	−0.514344
$379$	− 8.00000i	− 0.410932i	−0.978664	−	0.205466i	$-0.934129\pi$
$379$	− 8.00000i	− 0.410932i	0.978664	−	0.205466i	$-0.0658711\pi$
$380$	1.00000i	0.0512989i
$381$	11.0000i	0.563547i
$382$	−12.0000	−0.613973
$383$	9.00000	0.459879	0.229939	−	0.973205i	$-0.426147\pi$
$383$	9.00000	0.459879	0.229939	+	0.973205i	$0.426147\pi$
$384$	1.00000i	0.0510310i
$385$	0	0
$386$	− 2.00000i	− 0.101797i
$387$	20.0000	1.01666
$388$	7.00000i	0.355371i
$389$	−12.0000	−0.608424	−0.304212	−	0.952604i	$-0.598393\pi$
$389$	−12.0000	−0.608424	−0.304212	+	0.952604i	$0.598393\pi$
$390$	1.00000	0.0506370
$391$	0	0
$392$	−3.00000	−0.151523
$393$	18.0000	0.907980
$394$	24.0000i	1.20910i
$395$	−8.00000	−0.402524
$396$	0	0
$397$	20.0000i	1.00377i	0.864934	+	0.501886i	$0.167360\pi$
$397$	20.0000i	1.00377i	−0.864934	+	0.501886i	$0.832640\pi$
$398$	11.0000i	0.551380i
$399$	2.00000	0.100125
$400$	−1.00000	−0.0500000
$401$	36.0000i	1.79775i	0.438201	+	0.898877i	$0.355616\pi$
$401$	36.0000i	1.79775i	−0.438201	+	0.898877i	$0.644384\pi$
$402$	2.00000i	0.0997509i
$403$	5.00000i	0.249068i
$404$	6.00000	0.298511
$405$	1.00000i	0.0496904i
$406$	6.00000	0.297775
$407$	0	0
$408$	0	0
$409$	29.0000	1.43396	0.716979	−	0.697095i	$-0.245524\pi$
$409$	29.0000	1.43396	0.716979	+	0.697095i	$0.245524\pi$
$410$	6.00000	0.296319
$411$	18.0000i	0.887875i
$412$	8.00000	0.394132
$413$	− 6.00000i	− 0.295241i
$414$	12.0000i	0.589768i
$415$	12.0000i	0.589057i
$416$	1.00000	0.0490290
$417$	−8.00000	−0.391762
$418$	0	0
$419$	6.00000i	0.293119i	0.989202	+	0.146560i	$0.0468200\pi$
$419$	6.00000i	0.293119i	−0.989202	+	0.146560i	$0.953180\pi$
$420$	2.00000i	0.0975900i
$421$	−4.00000	−0.194948	−0.0974740	−	0.995238i	$-0.531076\pi$
$421$	−4.00000	−0.194948	−0.0974740	+	0.995238i	$0.531076\pi$
$422$	− 2.00000i	− 0.0973585i
$423$	−6.00000	−0.291730
$424$	−3.00000	−0.145693
$425$	0	0
$426$	9.00000	0.436051
$427$	−22.0000	−1.06465
$428$	12.0000i	0.580042i
$429$	0	0
$430$	− 10.0000i	− 0.482243i
$431$	− 24.0000i	− 1.15604i	−0.816023	−	0.578020i	$-0.803826\pi$
$431$	− 24.0000i	− 1.15604i	0.816023	−	0.578020i	$-0.196174\pi$
$432$	− 5.00000i	− 0.240563i
$433$	34.0000	1.63394	0.816968	−	0.576683i	$-0.195653\pi$
$433$	34.0000	1.63394	0.816968	+	0.576683i	$0.195653\pi$
$434$	−10.0000	−0.480015
$435$	3.00000i	0.143839i
$436$	11.0000i	0.526804i
$437$	− 6.00000i	− 0.287019i
$438$	11.0000	0.525600
$439$	− 8.00000i	− 0.381819i	−0.981608	−	0.190910i	$-0.938856\pi$
$439$	− 8.00000i	− 0.381819i	0.981608	−	0.190910i	$-0.0611437\pi$
$440$	0	0
$441$	6.00000	0.285714
$442$	0	0
$443$	12.0000	0.570137	0.285069	−	0.958507i	$-0.407984\pi$
$443$	12.0000	0.570137	0.285069	+	0.958507i	$0.407984\pi$
$444$	−8.00000	−0.379663
$445$	15.0000i	0.711068i
$446$	−1.00000	−0.0473514
$447$	12.0000i	0.567581i
$448$	2.00000i	0.0944911i
$449$	0	0	1.00000			$0$
$449$	0	0	−1.00000			$\pi$
$450$	2.00000	0.0942809
$451$	0	0
$452$	9.00000i	0.423324i
$453$	− 22.0000i	− 1.03365i
$454$	− 3.00000i	− 0.140797i
$455$	2.00000	0.0937614
$456$	1.00000i	0.0468293i
$457$	−26.0000	−1.21623	−0.608114	−	0.793849i	$-0.708074\pi$
$457$	−26.0000	−1.21623	−0.608114	+	0.793849i	$0.708074\pi$
$458$	−22.0000	−1.02799
$459$	0	0
$460$	6.00000	0.279751
$461$	−30.0000	−1.39724	−0.698620	−	0.715493i	$-0.746202\pi$
$461$	−30.0000	−1.39724	−0.698620	+	0.715493i	$0.746202\pi$
$462$	0	0
$463$	−19.0000	−0.883005	−0.441502	−	0.897260i	$-0.645554\pi$
$463$	−19.0000	−0.883005	−0.441502	+	0.897260i	$0.645554\pi$
$464$	3.00000i	0.139272i
$465$	− 5.00000i	− 0.231869i
$466$	9.00000i	0.416917i
$467$	−24.0000	−1.11059	−0.555294	−	0.831654i	$-0.687394\pi$
$467$	−24.0000	−1.11059	−0.555294	+	0.831654i	$0.687394\pi$
$468$	−2.00000	−0.0924500
$469$	4.00000i	0.184703i
$470$	3.00000i	0.138380i
$471$	− 14.0000i	− 0.645086i
$472$	3.00000	0.138086
$473$	0	0
$474$	−8.00000	−0.367452
$475$	−1.00000	−0.0458831
$476$	0	0
$477$	6.00000	0.274721
$478$	−6.00000	−0.274434
$479$	− 27.0000i	− 1.23366i	−0.787096	−	0.616831i	$-0.788416\pi$
$479$	− 27.0000i	− 1.23366i	0.787096	−	0.616831i	$-0.211584\pi$
$480$	−1.00000	−0.0456435
$481$	8.00000i	0.364769i
$482$	− 10.0000i	− 0.455488i
$483$	− 12.0000i	− 0.546019i
$484$	11.0000	0.500000
$485$	−7.00000	−0.317854
$486$	16.0000i	0.725775i
$487$	− 22.0000i	− 0.996915i	−0.866914	−	0.498458i	$-0.833900\pi$
$487$	− 22.0000i	− 0.996915i	0.866914	−	0.498458i	$-0.166100\pi$
$488$	− 11.0000i	− 0.497947i
$489$	−16.0000	−0.723545
$490$	− 3.00000i	− 0.135526i
$491$	39.0000	1.76005	0.880023	−	0.474932i	$-0.157527\pi$
$491$	39.0000	1.76005	0.880023	+	0.474932i	$0.157527\pi$
$492$	6.00000	0.270501
$493$	0	0
$494$	1.00000	0.0449921
$495$	0	0
$496$	− 5.00000i	− 0.224507i
$497$	18.0000	0.807410
$498$	12.0000i	0.537733i
$499$	− 22.0000i	− 0.984855i	−0.870353	−	0.492428i	$-0.836110\pi$
$499$	− 22.0000i	− 0.984855i	0.870353	−	0.492428i	$-0.163890\pi$
$500$	− 1.00000i	− 0.0447214i
$501$	18.0000	0.804181
$502$	12.0000	0.535586
$503$	− 12.0000i	− 0.535054i	−0.963550	−	0.267527i	$-0.913794\pi$
$503$	− 12.0000i	− 0.535054i	0.963550	−	0.267527i	$-0.0862064\pi$
$504$	− 4.00000i	− 0.178174i
$505$	6.00000i	0.266996i
$506$	0	0
$507$	12.0000i	0.532939i
$508$	−11.0000	−0.488046
$509$	−36.0000	−1.59567	−0.797836	−	0.602875i	$-0.794022\pi$
$509$	−36.0000	−1.59567	−0.797836	+	0.602875i	$0.794022\pi$
$510$	0	0
$511$	22.0000	0.973223
$512$	−1.00000	−0.0441942
$513$	− 5.00000i	− 0.220755i
$514$	6.00000	0.264649
$515$	8.00000i	0.352522i
$516$	− 10.0000i	− 0.440225i
$517$	0	0
$518$	−16.0000	−0.703000
$519$	−18.0000	−0.790112
$520$	1.00000i	0.0438529i
$521$	36.0000i	1.57719i	0.614914	+	0.788594i	$0.289191\pi$
$521$	36.0000i	1.57719i	−0.614914	+	0.788594i	$0.710809\pi$
$522$	− 6.00000i	− 0.262613i
$523$	2.00000	0.0874539	0.0437269	−	0.999044i	$-0.486077\pi$
$523$	2.00000	0.0874539	0.0437269	+	0.999044i	$0.486077\pi$
$524$	18.0000i	0.786334i
$525$	−2.00000	−0.0872872
$526$	9.00000	0.392419
$527$	0	0
$528$	0	0
$529$	−13.0000	−0.565217
$530$	− 3.00000i	− 0.130312i
$531$	−6.00000	−0.260378
$532$	2.00000i	0.0867110i
$533$	− 6.00000i	− 0.259889i
$534$	15.0000i	0.649113i
$535$	−12.0000	−0.518805
$536$	−2.00000	−0.0863868
$537$	− 12.0000i	− 0.517838i
$538$	15.0000i	0.646696i
$539$	0	0
$540$	5.00000	0.215166
$541$	− 2.00000i	− 0.0859867i	−0.999075	−	0.0429934i	$-0.986311\pi$
$541$	− 2.00000i	− 0.0859867i	0.999075	−	0.0429934i	$-0.0136894\pi$
$542$	−32.0000	−1.37452
$543$	−10.0000	−0.429141
$544$	0	0
$545$	−11.0000	−0.471188
$546$	2.00000	0.0855921
$547$	31.0000i	1.32546i	0.748857	+	0.662732i	$0.230603\pi$
$547$	31.0000i	1.32546i	−0.748857	+	0.662732i	$0.769397\pi$
$548$	−18.0000	−0.768922
$549$	22.0000i	0.938937i
$550$	0	0
$551$	3.00000i	0.127804i
$552$	6.00000	0.255377
$553$	−16.0000	−0.680389
$554$	− 10.0000i	− 0.424859i
$555$	− 8.00000i	− 0.339581i
$556$	− 8.00000i	− 0.339276i
$557$	39.0000	1.65248	0.826242	−	0.563316i	$-0.190475\pi$
$557$	39.0000	1.65248	0.826242	+	0.563316i	$0.190475\pi$
$558$	10.0000i	0.423334i
$559$	−10.0000	−0.422955
$560$	−2.00000	−0.0845154
$561$	0	0
$562$	27.0000	1.13893
$563$	−18.0000	−0.758610	−0.379305	−	0.925272i	$-0.623837\pi$
$563$	−18.0000	−0.758610	−0.379305	+	0.925272i	$0.623837\pi$
$564$	3.00000i	0.126323i
$565$	−9.00000	−0.378633
$566$	13.0000i	0.546431i
$567$	2.00000i	0.0839921i
$568$	9.00000i	0.377632i
$569$	−27.0000	−1.13190	−0.565949	−	0.824440i	$-0.691490\pi$
$569$	−27.0000	−1.13190	−0.565949	+	0.824440i	$0.691490\pi$
$570$	−1.00000	−0.0418854
$571$	38.0000i	1.59025i	0.606445	+	0.795125i	$0.292595\pi$
$571$	38.0000i	1.59025i	−0.606445	+	0.795125i	$0.707405\pi$
$572$	0	0
$573$	− 12.0000i	− 0.501307i
$574$	12.0000	0.500870
$575$	6.00000i	0.250217i
$576$	2.00000	0.0833333
$577$	−28.0000	−1.16566	−0.582828	−	0.812596i	$-0.698054\pi$
$577$	−28.0000	−1.16566	−0.582828	+	0.812596i	$0.698054\pi$
$578$	0	0
$579$	2.00000	0.0831172
$580$	−3.00000	−0.124568
$581$	24.0000i	0.995688i
$582$	−7.00000	−0.290159
$583$	0	0
$584$	11.0000i	0.455183i
$585$	− 2.00000i	− 0.0826898i
$586$	3.00000	0.123929
$587$	42.0000	1.73353	0.866763	−	0.498721i	$-0.166197\pi$
$587$	42.0000	1.73353	0.866763	+	0.498721i	$0.166197\pi$
$588$	− 3.00000i	− 0.123718i
$589$	− 5.00000i	− 0.206021i
$590$	3.00000i	0.123508i
$591$	−24.0000	−0.987228
$592$	− 8.00000i	− 0.328798i
$593$	−30.0000	−1.23195	−0.615976	−	0.787765i	$-0.711238\pi$
$593$	−30.0000	−1.23195	−0.615976	+	0.787765i	$0.711238\pi$
$594$	0	0
$595$	0	0
$596$	−12.0000	−0.491539
$597$	−11.0000	−0.450200
$598$	− 6.00000i	− 0.245358i
$599$	−18.0000	−0.735460	−0.367730	−	0.929933i	$-0.619865\pi$
$599$	−18.0000	−0.735460	−0.367730	+	0.929933i	$0.619865\pi$
$600$	− 1.00000i	− 0.0408248i
$601$	8.00000i	0.326327i	0.986599	+	0.163163i	$0.0521698\pi$
$601$	8.00000i	0.326327i	−0.986599	+	0.163163i	$0.947830\pi$
$602$	− 20.0000i	− 0.815139i
$603$	4.00000	0.162893
$604$	22.0000	0.895167
$605$	11.0000i	0.447214i
$606$	6.00000i	0.243733i
$607$	− 26.0000i	− 1.05531i	−0.849460	−	0.527654i	$-0.823072\pi$
$607$	− 26.0000i	− 1.05531i	0.849460	−	0.527654i	$-0.176928\pi$
$608$	−1.00000	−0.0405554
$609$	6.00000i	0.243132i
$610$	11.0000	0.445377
$611$	3.00000	0.121367
$612$	0	0
$613$	−1.00000	−0.0403896	−0.0201948	−	0.999796i	$-0.506429\pi$
$613$	−1.00000	−0.0403896	−0.0201948	+	0.999796i	$0.506429\pi$
$614$	−20.0000	−0.807134
$615$	6.00000i	0.241943i
$616$	0	0
$617$	39.0000i	1.57008i	0.619445	+	0.785040i	$0.287358\pi$
$617$	39.0000i	1.57008i	−0.619445	+	0.785040i	$0.712642\pi$
$618$	8.00000i	0.321807i
$619$	− 46.0000i	− 1.84890i	−0.381308	−	0.924448i	$-0.624526\pi$
$619$	− 46.0000i	− 1.84890i	0.381308	−	0.924448i	$-0.375474\pi$
$620$	5.00000	0.200805
$621$	−30.0000	−1.20386
$622$	0	0
$623$	30.0000i	1.20192i
$624$	1.00000i	0.0400320i
$625$	1.00000	0.0400000
$626$	− 14.0000i	− 0.559553i
$627$	0	0
$628$	14.0000	0.558661
$629$	0	0
$630$	4.00000	0.159364
$631$	−32.0000	−1.27390	−0.636950	−	0.770905i	$-0.719804\pi$
$631$	−32.0000	−1.27390	−0.636950	+	0.770905i	$0.719804\pi$
$632$	− 8.00000i	− 0.318223i
$633$	2.00000	0.0794929
$634$	0	0
$635$	− 11.0000i	− 0.436522i
$636$	− 3.00000i	− 0.118958i
$637$	−3.00000	−0.118864
$638$	0	0
$639$	− 18.0000i	− 0.712069i
$640$	− 1.00000i	− 0.0395285i
$641$	30.0000i	1.18493i	0.805597	+	0.592464i	$0.201845\pi$
$641$	30.0000i	1.18493i	−0.805597	+	0.592464i	$0.798155\pi$
$642$	−12.0000	−0.473602
$643$	4.00000i	0.157745i	0.996885	+	0.0788723i	$0.0251319\pi$
$643$	4.00000i	0.157745i	−0.996885	+	0.0788723i	$0.974868\pi$
$644$	12.0000	0.472866
$645$	10.0000	0.393750
$646$	0	0
$647$	−33.0000	−1.29736	−0.648682	−	0.761060i	$-0.724679\pi$
$647$	−33.0000	−1.29736	−0.648682	+	0.761060i	$0.724679\pi$
$648$	−1.00000	−0.0392837
$649$	0	0
$650$	−1.00000	−0.0392232
$651$	− 10.0000i	− 0.391931i
$652$	− 16.0000i	− 0.626608i
$653$	6.00000i	0.234798i	0.993085	+	0.117399i	$0.0374557\pi$
$653$	6.00000i	0.234798i	−0.993085	+	0.117399i	$0.962544\pi$
$654$	−11.0000	−0.430134
$655$	−18.0000	−0.703318
$656$	6.00000i	0.234261i
$657$	− 22.0000i	− 0.858302i
$658$	6.00000i	0.233904i
$659$	9.00000	0.350590	0.175295	−	0.984516i	$-0.443912\pi$
$659$	9.00000	0.350590	0.175295	+	0.984516i	$0.443912\pi$
$660$	0	0
$661$	−20.0000	−0.777910	−0.388955	−	0.921257i	$-0.627164\pi$
$661$	−20.0000	−0.777910	−0.388955	+	0.921257i	$0.627164\pi$
$662$	−31.0000	−1.20485
$663$	0	0
$664$	−12.0000	−0.465690
$665$	−2.00000	−0.0775567
$666$	16.0000i	0.619987i
$667$	18.0000	0.696963
$668$	18.0000i	0.696441i
$669$	− 1.00000i	− 0.0386622i
$670$	− 2.00000i	− 0.0772667i
$671$	0	0
$672$	−2.00000	−0.0771517
$673$	23.0000i	0.886585i	0.896377	+	0.443292i	$0.146190\pi$
$673$	23.0000i	0.886585i	−0.896377	+	0.443292i	$0.853810\pi$
$674$	− 13.0000i	− 0.500741i
$675$	5.00000i	0.192450i
$676$	−12.0000	−0.461538
$677$	− 18.0000i	− 0.691796i	−0.938272	−	0.345898i	$-0.887574\pi$
$677$	− 18.0000i	− 0.691796i	0.938272	−	0.345898i	$-0.112426\pi$
$678$	−9.00000	−0.345643
$679$	−14.0000	−0.537271
$680$	0	0
$681$	3.00000	0.114960
$682$	0	0
$683$	3.00000i	0.114792i	0.998351	+	0.0573959i	$0.0182797\pi$
$683$	3.00000i	0.114792i	−0.998351	+	0.0573959i	$0.981720\pi$
$684$	2.00000	0.0764719
$685$	− 18.0000i	− 0.687745i
$686$	− 20.0000i	− 0.763604i
$687$	− 22.0000i	− 0.839352i
$688$	10.0000	0.381246
$689$	−3.00000	−0.114291
$690$	6.00000i	0.228416i
$691$	− 28.0000i	− 1.06517i	−0.846376	−	0.532585i	$-0.821221\pi$
$691$	− 28.0000i	− 1.06517i	0.846376	−	0.532585i	$-0.178779\pi$
$692$	− 18.0000i	− 0.684257i
$693$	0	0
$694$	15.0000i	0.569392i
$695$	8.00000	0.303457
$696$	−3.00000	−0.113715
$697$	0	0
$698$	−28.0000	−1.05982
$699$	−9.00000	−0.340411
$700$	− 2.00000i	− 0.0755929i
$701$	−36.0000	−1.35970	−0.679851	−	0.733351i	$-0.737955\pi$
$701$	−36.0000	−1.35970	−0.679851	+	0.733351i	$0.737955\pi$
$702$	− 5.00000i	− 0.188713i
$703$	− 8.00000i	− 0.301726i
$704$	0	0
$705$	−3.00000	−0.112987
$706$	−18.0000	−0.677439
$707$	12.0000i	0.451306i
$708$	3.00000i	0.112747i
$709$	− 41.0000i	− 1.53979i	−0.638172	−	0.769894i	$-0.720309\pi$
$709$	− 41.0000i	− 1.53979i	0.638172	−	0.769894i	$-0.279691\pi$
$710$	−9.00000	−0.337764
$711$	16.0000i	0.600047i
$712$	−15.0000	−0.562149
$713$	−30.0000	−1.12351
$714$	0	0
$715$	0	0
$716$	12.0000	0.448461
$717$	− 6.00000i	− 0.224074i
$718$	30.0000	1.11959
$719$	39.0000i	1.45445i	0.686397	+	0.727227i	$0.259191\pi$
$719$	39.0000i	1.45445i	−0.686397	+	0.727227i	$0.740809\pi$
$720$	2.00000i	0.0745356i
$721$	16.0000i	0.595871i
$722$	18.0000	0.669891
$723$	10.0000	0.371904
$724$	− 10.0000i	− 0.371647i
$725$	− 3.00000i	− 0.111417i
$726$	11.0000i	0.408248i
$727$	−19.0000	−0.704671	−0.352335	−	0.935874i	$-0.614612\pi$
$727$	−19.0000	−0.704671	−0.352335	+	0.935874i	$0.614612\pi$
$728$	2.00000i	0.0741249i
$729$	−13.0000	−0.481481
$730$	−11.0000	−0.407128
$731$	0	0
$732$	11.0000	0.406572
$733$	−14.0000	−0.517102	−0.258551	−	0.965998i	$-0.583245\pi$
$733$	−14.0000	−0.517102	−0.258551	+	0.965998i	$0.583245\pi$
$734$	22.0000i	0.812035i
$735$	3.00000	0.110657
$736$	6.00000i	0.221163i
$737$	0	0
$738$	− 12.0000i	− 0.441726i
$739$	−29.0000	−1.06678	−0.533391	−	0.845869i	$-0.679083\pi$
$739$	−29.0000	−1.06678	−0.533391	+	0.845869i	$0.679083\pi$
$740$	8.00000	0.294086
$741$	1.00000i	0.0367359i
$742$	− 6.00000i	− 0.220267i
$743$	− 30.0000i	− 1.10059i	−0.834969	−	0.550297i	$-0.814515\pi$
$743$	− 30.0000i	− 1.10059i	0.834969	−	0.550297i	$-0.185485\pi$
$744$	5.00000	0.183309
$745$	− 12.0000i	− 0.439646i
$746$	22.0000	0.805477
$747$	24.0000	0.878114
$748$	0	0
$749$	−24.0000	−0.876941
$750$	1.00000	0.0365148
$751$	25.0000i	0.912263i	0.889912	+	0.456131i	$0.150765\pi$
$751$	25.0000i	0.912263i	−0.889912	+	0.456131i	$0.849235\pi$
$752$	−3.00000	−0.109399
$753$	12.0000i	0.437304i
$754$	3.00000i	0.109254i
$755$	22.0000i	0.800662i
$756$	10.0000	0.363696
$757$	13.0000	0.472493	0.236247	−	0.971693i	$-0.424083\pi$
$757$	13.0000	0.472493	0.236247	+	0.971693i	$0.424083\pi$
$758$	8.00000i	0.290573i
$759$	0	0
$760$	− 1.00000i	− 0.0362738i
$761$	6.00000	0.217500	0.108750	−	0.994069i	$-0.465315\pi$
$761$	6.00000	0.217500	0.108750	+	0.994069i	$0.465315\pi$
$762$	− 11.0000i	− 0.398488i
$763$	−22.0000	−0.796453
$764$	12.0000	0.434145
$765$	0	0
$766$	−9.00000	−0.325183
$767$	3.00000	0.108324
$768$	− 1.00000i	− 0.0360844i
$769$	−31.0000	−1.11789	−0.558944	−	0.829205i	$-0.688793\pi$
$769$	−31.0000	−1.11789	−0.558944	+	0.829205i	$0.688793\pi$
$770$	0	0
$771$	6.00000i	0.216085i
$772$	2.00000i	0.0719816i
$773$	−42.0000	−1.51064	−0.755318	−	0.655359i	$-0.772517\pi$
$773$	−42.0000	−1.51064	−0.755318	+	0.655359i	$0.772517\pi$
$774$	−20.0000	−0.718885
$775$	5.00000i	0.179605i
$776$	− 7.00000i	− 0.251285i
$777$	− 16.0000i	− 0.573997i
$778$	12.0000	0.430221
$779$	6.00000i	0.214972i
$780$	−1.00000	−0.0358057
$781$	0	0
$782$	0	0
$783$	15.0000	0.536056
$784$	3.00000	0.107143
$785$	14.0000i	0.499681i
$786$	−18.0000	−0.642039
$787$	7.00000i	0.249523i	0.992187	+	0.124762i	$0.0398166\pi$
$787$	7.00000i	0.249523i	−0.992187	+	0.124762i	$0.960183\pi$
$788$	− 24.0000i	− 0.854965i
$789$	9.00000i	0.320408i
$790$	8.00000	0.284627
$791$	−18.0000	−0.640006
$792$	0	0
$793$	− 11.0000i	− 0.390621i
$794$	− 20.0000i	− 0.709773i
$795$	3.00000	0.106399
$796$	− 11.0000i	− 0.389885i
$797$	−6.00000	−0.212531	−0.106265	−	0.994338i	$-0.533889\pi$
$797$	−6.00000	−0.212531	−0.106265	+	0.994338i	$0.533889\pi$
$798$	−2.00000	−0.0707992
$799$	0	0
$800$	1.00000	0.0353553
$801$	30.0000	1.06000
$802$	− 36.0000i	− 1.27120i
$803$	0	0
$804$	− 2.00000i	− 0.0705346i
$805$	12.0000i	0.422944i
$806$	− 5.00000i	− 0.176117i
$807$	−15.0000	−0.528025
$808$	−6.00000	−0.211079
$809$	42.0000i	1.47664i	0.674450	+	0.738321i	$0.264381\pi$
$809$	42.0000i	1.47664i	−0.674450	+	0.738321i	$0.735619\pi$
$810$	− 1.00000i	− 0.0351364i
$811$	− 2.00000i	− 0.0702295i	−0.999383	−	0.0351147i	$-0.988820\pi$
$811$	− 2.00000i	− 0.0702295i	0.999383	−	0.0351147i	$-0.0111797\pi$
$812$	−6.00000	−0.210559
$813$	− 32.0000i	− 1.12229i
$814$	0	0
$815$	16.0000	0.560456
$816$	0	0
$817$	10.0000	0.349856
$818$	−29.0000	−1.01396
$819$	− 4.00000i	− 0.139771i
$820$	−6.00000	−0.209529
$821$	− 15.0000i	− 0.523504i	−0.965135	−	0.261752i	$-0.915700\pi$
$821$	− 15.0000i	− 0.523504i	0.965135	−	0.261752i	$-0.0843002\pi$
$822$	− 18.0000i	− 0.627822i
$823$	− 10.0000i	− 0.348578i	−0.984695	−	0.174289i	$-0.944237\pi$
$823$	− 10.0000i	− 0.348578i	0.984695	−	0.174289i	$-0.0557627\pi$
$824$	−8.00000	−0.278693
$825$	0	0
$826$	6.00000i	0.208767i
$827$	12.0000i	0.417281i	0.977992	+	0.208640i	$0.0669038\pi$
$827$	12.0000i	0.417281i	−0.977992	+	0.208640i	$0.933096\pi$
$828$	− 12.0000i	− 0.417029i
$829$	32.0000	1.11141	0.555703	−	0.831381i	$-0.312449\pi$
$829$	32.0000	1.11141	0.555703	+	0.831381i	$0.312449\pi$
$830$	− 12.0000i	− 0.416526i
$831$	10.0000	0.346896
$832$	−1.00000	−0.0346688
$833$	0	0
$834$	8.00000	0.277017
$835$	−18.0000	−0.622916
$836$	0	0
$837$	−25.0000	−0.864126
$838$	− 6.00000i	− 0.207267i
$839$	− 21.0000i	− 0.725001i	−0.931984	−	0.362500i	$-0.881923\pi$
$839$	− 21.0000i	− 0.725001i	0.931984	−	0.362500i	$-0.118077\pi$
$840$	− 2.00000i	− 0.0690066i
$841$	20.0000	0.689655
$842$	4.00000	0.137849
$843$	27.0000i	0.929929i
$844$	2.00000i	0.0688428i
$845$	− 12.0000i	− 0.412813i
$846$	6.00000	0.206284
$847$	22.0000i	0.755929i
$848$	3.00000	0.103020
$849$	−13.0000	−0.446159
$850$	0	0
$851$	−48.0000	−1.64542
$852$	−9.00000	−0.308335
$853$	− 2.00000i	− 0.0684787i	−0.999414	−	0.0342393i	$-0.989099\pi$
$853$	− 2.00000i	− 0.0684787i	0.999414	−	0.0342393i	$-0.0109009\pi$
$854$	22.0000	0.752825
$855$	2.00000i	0.0683986i
$856$	− 12.0000i	− 0.410152i
$857$	33.0000i	1.12726i	0.826028	+	0.563629i	$0.190595\pi$
$857$	33.0000i	1.12726i	−0.826028	+	0.563629i	$0.809405\pi$
$858$	0	0
$859$	1.00000	0.0341196	0.0170598	−	0.999854i	$-0.494569\pi$
$859$	1.00000	0.0341196	0.0170598	+	0.999854i	$0.494569\pi$
$860$	10.0000i	0.340997i
$861$	12.0000i	0.408959i
$862$	24.0000i	0.817443i
$863$	−48.0000	−1.63394	−0.816970	−	0.576681i	$-0.804348\pi$
$863$	−48.0000	−1.63394	−0.816970	+	0.576681i	$0.804348\pi$
$864$	5.00000i	0.170103i
$865$	18.0000	0.612018
$866$	−34.0000	−1.15537
$867$	0	0
$868$	10.0000	0.339422
$869$	0	0
$870$	− 3.00000i	− 0.101710i
$871$	−2.00000	−0.0677674
$872$	− 11.0000i	− 0.372507i
$873$	14.0000i	0.473828i
$874$	6.00000i	0.202953i
$875$	2.00000	0.0676123
$876$	−11.0000	−0.371656
$877$	− 40.0000i	− 1.35070i	−0.737496	−	0.675352i	$-0.763992\pi$
$877$	− 40.0000i	− 1.35070i	0.737496	−	0.675352i	$-0.236008\pi$
$878$	8.00000i	0.269987i
$879$	3.00000i	0.101187i
$880$	0	0
$881$	− 18.0000i	− 0.606435i	−0.952921	−	0.303218i	$-0.901939\pi$
$881$	− 18.0000i	− 0.606435i	0.952921	−	0.303218i	$-0.0980609\pi$
$882$	−6.00000	−0.202031
$883$	38.0000	1.27880	0.639401	−	0.768874i	$-0.279182\pi$
$883$	38.0000	1.27880	0.639401	+	0.768874i	$0.279182\pi$
$884$	0	0
$885$	−3.00000	−0.100844
$886$	−12.0000	−0.403148
$887$	30.0000i	1.00730i	0.863907	+	0.503651i	$0.168010\pi$
$887$	30.0000i	1.00730i	−0.863907	+	0.503651i	$0.831990\pi$
$888$	8.00000	0.268462
$889$	− 22.0000i	− 0.737856i
$890$	− 15.0000i	− 0.502801i
$891$	0	0
$892$	1.00000	0.0334825
$893$	−3.00000	−0.100391
$894$	− 12.0000i	− 0.401340i
$895$	12.0000i	0.401116i
$896$	− 2.00000i	− 0.0668153i
$897$	6.00000	0.200334
$898$	0	0
$899$	15.0000	0.500278
$900$	−2.00000	−0.0666667
$901$	0	0
$902$	0	0
$903$	20.0000	0.665558
$904$	− 9.00000i	− 0.299336i
$905$	10.0000	0.332411
$906$	22.0000i	0.730901i
$907$	47.0000i	1.56061i	0.625400	+	0.780305i	$0.284936\pi$
$907$	47.0000i	1.56061i	−0.625400	+	0.780305i	$0.715064\pi$
$908$	3.00000i	0.0995585i
$909$	12.0000	0.398015
$910$	−2.00000	−0.0662994
$911$	48.0000i	1.59031i	0.606406	+	0.795155i	$0.292611\pi$
$911$	48.0000i	1.59031i	−0.606406	+	0.795155i	$0.707389\pi$
$912$	− 1.00000i	− 0.0331133i
$913$	0	0
$914$	26.0000	0.860004
$915$	11.0000i	0.363649i
$916$	22.0000	0.726900
$917$	−36.0000	−1.18882
$918$	0	0
$919$	2.00000	0.0659739	0.0329870	−	0.999456i	$-0.489498\pi$
$919$	2.00000	0.0659739	0.0329870	+	0.999456i	$0.489498\pi$
$920$	−6.00000	−0.197814
$921$	− 20.0000i	− 0.659022i
$922$	30.0000	0.987997
$923$	9.00000i	0.296239i
$924$	0	0
$925$	8.00000i	0.263038i
$926$	19.0000	0.624379
$927$	16.0000	0.525509
$928$	− 3.00000i	− 0.0984798i
$929$	− 30.0000i	− 0.984268i	−0.870519	−	0.492134i	$-0.836217\pi$
$929$	− 30.0000i	− 0.984268i	0.870519	−	0.492134i	$-0.163783\pi$
$930$	5.00000i	0.163956i
$931$	3.00000	0.0983210
$932$	− 9.00000i	− 0.294805i
$933$	0	0
$934$	24.0000	0.785304
$935$	0	0
$936$	2.00000	0.0653720
$937$	−44.0000	−1.43742	−0.718709	−	0.695311i	$-0.755266\pi$
$937$	−44.0000	−1.43742	−0.718709	+	0.695311i	$0.755266\pi$
$938$	− 4.00000i	− 0.130605i
$939$	14.0000	0.456873
$940$	− 3.00000i	− 0.0978492i
$941$	− 15.0000i	− 0.488986i	−0.969651	−	0.244493i	$-0.921378\pi$
$941$	− 15.0000i	− 0.488986i	0.969651	−	0.244493i	$-0.0786215\pi$
$942$	14.0000i	0.456145i
$943$	36.0000	1.17232
$944$	−3.00000	−0.0976417
$945$	10.0000i	0.325300i
$946$	0	0
$947$	− 51.0000i	− 1.65728i	−0.559784	−	0.828639i	$-0.689116\pi$
$947$	− 51.0000i	− 1.65728i	0.559784	−	0.828639i	$-0.310884\pi$
$948$	8.00000	0.259828
$949$	11.0000i	0.357075i
$950$	1.00000	0.0324443
$951$	0	0
$952$	0	0
$953$	−24.0000	−0.777436	−0.388718	−	0.921357i	$-0.627082\pi$
$953$	−24.0000	−0.777436	−0.388718	+	0.921357i	$0.627082\pi$
$954$	−6.00000	−0.194257
$955$	12.0000i	0.388311i
$956$	6.00000	0.194054
$957$	0	0
$958$	27.0000i	0.872330i
$959$	− 36.0000i	− 1.16250i
$960$	1.00000	0.0322749
$961$	6.00000	0.193548
$962$	− 8.00000i	− 0.257930i
$963$	24.0000i	0.773389i
$964$	10.0000i	0.322078i
$965$	−2.00000	−0.0643823
$966$	12.0000i	0.386094i
$967$	40.0000	1.28631	0.643157	−	0.765735i	$-0.277624\pi$
$967$	40.0000	1.28631	0.643157	+	0.765735i	$0.277624\pi$
$968$	−11.0000	−0.353553
$969$	0	0
$970$	7.00000	0.224756
$971$	21.0000	0.673922	0.336961	−	0.941519i	$-0.390601\pi$
$971$	21.0000	0.673922	0.336961	+	0.941519i	$0.390601\pi$
$972$	− 16.0000i	− 0.513200i
$973$	16.0000	0.512936
$974$	22.0000i	0.704925i
$975$	− 1.00000i	− 0.0320256i
$976$	11.0000i	0.352101i
$977$	12.0000	0.383914	0.191957	−	0.981403i	$-0.438517\pi$
$977$	12.0000	0.383914	0.191957	+	0.981403i	$0.438517\pi$
$978$	16.0000	0.511624
$979$	0	0
$980$	3.00000i	0.0958315i
$981$	22.0000i	0.702406i
$982$	−39.0000	−1.24454
$983$	− 6.00000i	− 0.191370i	−0.995412	−	0.0956851i	$-0.969496\pi$
$983$	− 6.00000i	− 0.191370i	0.995412	−	0.0956851i	$-0.0305042\pi$
$984$	−6.00000	−0.191273
$985$	24.0000	0.764704
$986$	0	0
$987$	−6.00000	−0.190982
$988$	−1.00000	−0.0318142
$989$	− 60.0000i	− 1.90789i
$990$	0	0
$991$	− 29.0000i	− 0.921215i	−0.887604	−	0.460608i	$-0.847632\pi$
$991$	− 29.0000i	− 0.921215i	0.887604	−	0.460608i	$-0.152368\pi$
$992$	5.00000i	0.158750i
$993$	− 31.0000i	− 0.983755i
$994$	−18.0000	−0.570925
$995$	11.0000	0.348723
$996$	− 12.0000i	− 0.380235i
$997$	− 46.0000i	− 1.45683i	−0.685134	−	0.728417i	$-0.740256\pi$
$997$	− 46.0000i	− 1.45683i	0.685134	−	0.728417i	$-0.259744\pi$
$998$	22.0000i	0.696398i
$999$	−40.0000	−1.26554

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2890.2.b.b.2311.1		2
17.4	even	4		2890.2.a.n.1.1		1
17.13	even	4		170.2.a.e.1.1	✓	1
17.16	even	2	inner	2890.2.b.b.2311.2		2
51.47	odd	4		1530.2.a.g.1.1		1
68.47	odd	4		1360.2.a.d.1.1		1
85.13	odd	4		850.2.c.e.749.1		2
85.47	odd	4		850.2.c.e.749.2		2
85.64	even	4		850.2.a.b.1.1		1
119.13	odd	4		8330.2.a.q.1.1		1
136.13	even	4		5440.2.a.k.1.1		1
136.115	odd	4		5440.2.a.r.1.1		1
255.149	odd	4		7650.2.a.bo.1.1		1
340.319	odd	4		6800.2.a.t.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
170.2.a.e.1.1	✓	1	17.13	even	4
850.2.a.b.1.1		1	85.64	even	4
850.2.c.e.749.1		2	85.13	odd	4
850.2.c.e.749.2		2	85.47	odd	4
1360.2.a.d.1.1		1	68.47	odd	4
1530.2.a.g.1.1		1	51.47	odd	4
2890.2.a.n.1.1		1	17.4	even	4
2890.2.b.b.2311.1		2	1.1	even	1	trivial
2890.2.b.b.2311.2		2	17.16	even	2	inner
5440.2.a.k.1.1		1	136.13	even	4
5440.2.a.r.1.1		1	136.115	odd	4
6800.2.a.t.1.1		1	340.319	odd	4
7650.2.a.bo.1.1		1	255.149	odd	4
8330.2.a.q.1.1		1	119.13	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:	\( N \)	\(=\)	\( 2890 = 2 \cdot 5 \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2890.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -1.00000i q^{3} +1.00000 q^{4} +1.00000i q^{5} +1.00000i q^{6} +2.00000i q^{7} -1.00000 q^{8} +2.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -1.00000i q^{3} +1.00000 q^{4} +1.00000i q^{5} +1.00000i q^{6} +2.00000i q^{7} -1.00000 q^{8} +2.00000 q^{9} -1.00000i q^{10} -1.00000i q^{12} -1.00000 q^{13} -2.00000i q^{14} +1.00000 q^{15} +1.00000 q^{16} -2.00000 q^{18} +1.00000 q^{19} +1.00000i q^{20} +2.00000 q^{21} -6.00000i q^{23} +1.00000i q^{24} -1.00000 q^{25} +1.00000 q^{26} -5.00000i q^{27} +2.00000i q^{28} +3.00000i q^{29} -1.00000 q^{30} -5.00000i q^{31} -1.00000 q^{32} -2.00000 q^{35} +2.00000 q^{36} -8.00000i q^{37} -1.00000 q^{38} +1.00000i q^{39} -1.00000i q^{40} +6.00000i q^{41} -2.00000 q^{42} +10.0000 q^{43} +2.00000i q^{45} +6.00000i q^{46} -3.00000 q^{47} -1.00000i q^{48} +3.00000 q^{49} +1.00000 q^{50} -1.00000 q^{52} +3.00000 q^{53} +5.00000i q^{54} -2.00000i q^{56} -1.00000i q^{57} -3.00000i q^{58} -3.00000 q^{59} +1.00000 q^{60} +11.0000i q^{61} +5.00000i q^{62} +4.00000i q^{63} +1.00000 q^{64} -1.00000i q^{65} +2.00000 q^{67} -6.00000 q^{69} +2.00000 q^{70} -9.00000i q^{71} -2.00000 q^{72} -11.0000i q^{73} +8.00000i q^{74} +1.00000i q^{75} +1.00000 q^{76} -1.00000i q^{78} +8.00000i q^{79} +1.00000i q^{80} +1.00000 q^{81} -6.00000i q^{82} +12.0000 q^{83} +2.00000 q^{84} -10.0000 q^{86} +3.00000 q^{87} +15.0000 q^{89} -2.00000i q^{90} -2.00000i q^{91} -6.00000i q^{92} -5.00000 q^{93} +3.00000 q^{94} +1.00000i q^{95} +1.00000i q^{96} +7.00000i q^{97} -3.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{8} + 4 q^{9}+O(q^{10}) \) 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^8 + 4 * q^9	\( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{8} + 4 q^{9} - 2 q^{13} + 2 q^{15} + 2 q^{16} - 4 q^{18} + 2 q^{19} + 4 q^{21} - 2 q^{25} + 2 q^{26} - 2 q^{30} - 2 q^{32} - 4 q^{35} + 4 q^{36} - 2 q^{38} - 4 q^{42} + 20 q^{43} - 6 q^{47} + 6 q^{49} + 2 q^{50} - 2 q^{52} + 6 q^{53} - 6 q^{59} + 2 q^{60} + 2 q^{64} + 4 q^{67} - 12 q^{69} + 4 q^{70} - 4 q^{72} + 2 q^{76} + 2 q^{81} + 24 q^{83} + 4 q^{84} - 20 q^{86} + 6 q^{87} + 30 q^{89} - 10 q^{93} + 6 q^{94} - 6 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^8 + 4 * q^9 - 2 * q^13 + 2 * q^15 + 2 * q^16 - 4 * q^18 + 2 * q^19 + 4 * q^21 - 2 * q^25 + 2 * q^26 - 2 * q^30 - 2 * q^32 - 4 * q^35 + 4 * q^36 - 2 * q^38 - 4 * q^42 + 20 * q^43 - 6 * q^47 + 6 * q^49 + 2 * q^50 - 2 * q^52 + 6 * q^53 - 6 * q^59 + 2 * q^60 + 2 * q^64 + 4 * q^67 - 12 * q^69 + 4 * q^70 - 4 * q^72 + 2 * q^76 + 2 * q^81 + 24 * q^83 + 4 * q^84 - 20 * q^86 + 6 * q^87 + 30 * q^89 - 10 * q^93 + 6 * q^94 - 6 * q^98

Introduction

L-functions

Modular forms

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Database

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