LMFDB - Embedded newform 2550.2.d.j.2449.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2550,2,Mod(2449,2550)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2550, 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("2550.2449");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2550 = 2 \cdot 3 \cdot 5^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2550.d (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:	$20.3618525154$
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:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 510)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2449.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	2550.2449
Dual form			2550.2.d.j.2449.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} -2.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} -2.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +4.00000 q^{11} +1.00000i q^{12} -2.00000 q^{14} +1.00000 q^{16} +1.00000i q^{17} +1.00000i q^{18} -4.00000 q^{19} -2.00000 q^{21} -4.00000i q^{22} -4.00000i q^{23} +1.00000 q^{24} +1.00000i q^{27} +2.00000i q^{28} -6.00000 q^{29} -8.00000 q^{31} -1.00000i q^{32} -4.00000i q^{33} +1.00000 q^{34} +1.00000 q^{36} -6.00000i q^{37} +4.00000i q^{38} +8.00000 q^{41} +2.00000i q^{42} -2.00000i q^{43} -4.00000 q^{44} -4.00000 q^{46} -8.00000i q^{47} -1.00000i q^{48} +3.00000 q^{49} +1.00000 q^{51} -14.0000i q^{53} +1.00000 q^{54} +2.00000 q^{56} +4.00000i q^{57} +6.00000i q^{58} -6.00000 q^{59} +2.00000 q^{61} +8.00000i q^{62} +2.00000i q^{63} -1.00000 q^{64} -4.00000 q^{66} +2.00000i q^{67} -1.00000i q^{68} -4.00000 q^{69} -10.0000 q^{71} -1.00000i q^{72} -4.00000i q^{73} -6.00000 q^{74} +4.00000 q^{76} -8.00000i q^{77} -4.00000 q^{79} +1.00000 q^{81} -8.00000i q^{82} +16.0000i q^{83} +2.00000 q^{84} -2.00000 q^{86} +6.00000i q^{87} +4.00000i q^{88} -6.00000 q^{89} +4.00000i q^{92} +8.00000i q^{93} -8.00000 q^{94} -1.00000 q^{96} -8.00000i q^{97} -3.00000i q^{98} -4.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4} - 2 q^{6} - 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^4 - 2 * q^6 - 2 * q^9	$ 2 q - 2 q^{4} - 2 q^{6} - 2 q^{9} + 8 q^{11} - 4 q^{14} + 2 q^{16} - 8 q^{19} - 4 q^{21} + 2 q^{24} - 12 q^{29} - 16 q^{31} + 2 q^{34} + 2 q^{36} + 16 q^{41} - 8 q^{44} - 8 q^{46} + 6 q^{49} + 2 q^{51} + 2 q^{54} + 4 q^{56} - 12 q^{59} + 4 q^{61} - 2 q^{64} - 8 q^{66} - 8 q^{69} - 20 q^{71} - 12 q^{74} + 8 q^{76} - 8 q^{79} + 2 q^{81} + 4 q^{84} - 4 q^{86} - 12 q^{89} - 16 q^{94} - 2 q^{96} - 8 q^{99}+O(q^{100}) $ 2 * q - 2 * q^4 - 2 * q^6 - 2 * q^9 + 8 * q^11 - 4 * q^14 + 2 * q^16 - 8 * q^19 - 4 * q^21 + 2 * q^24 - 12 * q^29 - 16 * q^31 + 2 * q^34 + 2 * q^36 + 16 * q^41 - 8 * q^44 - 8 * q^46 + 6 * q^49 + 2 * q^51 + 2 * q^54 + 4 * q^56 - 12 * q^59 + 4 * q^61 - 2 * q^64 - 8 * q^66 - 8 * q^69 - 20 * q^71 - 12 * q^74 + 8 * q^76 - 8 * q^79 + 2 * q^81 + 4 * q^84 - 4 * q^86 - 12 * q^89 - 16 * q^94 - 2 * q^96 - 8 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2550\mathbb{Z}\right)^\times$.

$n$	$751$	$851$	$1327$
$\chi(n)$	$1$	$1$	$-1$

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
$5$	0	0
$5$	0	0
$6$	−1.00000	−0.408248
$7$	− 2.00000i	− 0.755929i	−0.925820	−	0.377964i	$-0.876624\pi$
$7$	− 2.00000i	− 0.755929i	0.925820	−	0.377964i	$-0.123376\pi$
$8$	1.00000i	0.353553i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	4.00000	1.20605	0.603023	−	0.797724i	$-0.293963\pi$
$11$	4.00000	1.20605	0.603023	+	0.797724i	$0.293963\pi$
$12$	1.00000i	0.288675i
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	−2.00000	−0.534522
$15$	0	0
$16$	1.00000	0.250000
$17$	1.00000i	0.242536i
$17$	1.00000i	0.242536i
$18$	1.00000i	0.235702i
$19$	−4.00000	−0.917663	−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−4.00000	−0.917663	−0.458831	+	0.888523i	$0.651732\pi$
$20$	0	0
$21$	−2.00000	−0.436436
$22$	− 4.00000i	− 0.852803i
$23$	− 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
$25$	0	0
$26$	0	0
$27$	1.00000i	0.192450i
$28$	2.00000i	0.377964i
$29$	−6.00000	−1.11417	−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000	−1.11417	−0.557086	+	0.830455i	$0.688081\pi$
$30$	0	0
$31$	−8.00000	−1.43684	−0.718421	−	0.695608i	$-0.755135\pi$
$31$	−8.00000	−1.43684	−0.718421	+	0.695608i	$0.755135\pi$
$32$	− 1.00000i	− 0.176777i
$33$	− 4.00000i	− 0.696311i
$34$	1.00000	0.171499
$35$	0	0
$36$	1.00000	0.166667
$37$	− 6.00000i	− 0.986394i	−0.869918	−	0.493197i	$-0.835828\pi$
$37$	− 6.00000i	− 0.986394i	0.869918	−	0.493197i	$-0.164172\pi$
$38$	4.00000i	0.648886i
$39$	0	0
$40$	0	0
$41$	8.00000	1.24939	0.624695	−	0.780869i	$-0.285223\pi$
$41$	8.00000	1.24939	0.624695	+	0.780869i	$0.285223\pi$
$42$	2.00000i	0.308607i
$43$	− 2.00000i	− 0.304997i	−0.988304	−	0.152499i	$-0.951268\pi$
$43$	− 2.00000i	− 0.304997i	0.988304	−	0.152499i	$-0.0487319\pi$
$44$	−4.00000	−0.603023
$45$	0	0
$46$	−4.00000	−0.589768
$47$	− 8.00000i	− 1.16692i	−0.812142	−	0.583460i	$-0.801699\pi$
$47$	− 8.00000i	− 1.16692i	0.812142	−	0.583460i	$-0.198301\pi$
$48$	− 1.00000i	− 0.144338i
$49$	3.00000	0.428571
$50$	0	0
$51$	1.00000	0.140028
$52$	0	0
$53$	− 14.0000i	− 1.92305i	−0.274721	−	0.961524i	$-0.588586\pi$
$53$	− 14.0000i	− 1.92305i	0.274721	−	0.961524i	$-0.411414\pi$
$54$	1.00000	0.136083
$55$	0	0
$56$	2.00000	0.267261
$57$	4.00000i	0.529813i
$58$	6.00000i	0.787839i
$59$	−6.00000	−0.781133	−0.390567	−	0.920575i	$-0.627721\pi$
$59$	−6.00000	−0.781133	−0.390567	+	0.920575i	$0.627721\pi$
$60$	0	0
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	8.00000i	1.01600i
$63$	2.00000i	0.251976i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	−4.00000	−0.492366
$67$	2.00000i	0.244339i	0.992509	+	0.122169i	$0.0389851\pi$
$67$	2.00000i	0.244339i	−0.992509	+	0.122169i	$0.961015\pi$
$68$	− 1.00000i	− 0.121268i
$69$	−4.00000	−0.481543
$70$	0	0
$71$	−10.0000	−1.18678	−0.593391	−	0.804914i	$-0.702211\pi$
$71$	−10.0000	−1.18678	−0.593391	+	0.804914i	$0.702211\pi$
$72$	− 1.00000i	− 0.117851i
$73$	− 4.00000i	− 0.468165i	−0.972217	−	0.234082i	$-0.924791\pi$
$73$	− 4.00000i	− 0.468165i	0.972217	−	0.234082i	$-0.0752085\pi$
$74$	−6.00000	−0.697486
$75$	0	0
$76$	4.00000	0.458831
$77$	− 8.00000i	− 0.911685i
$78$	0	0
$79$	−4.00000	−0.450035	−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000	−0.450035	−0.225018	+	0.974355i	$0.572244\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	− 8.00000i	− 0.883452i
$83$	16.0000i	1.75623i	0.478451	+	0.878114i	$0.341198\pi$
$83$	16.0000i	1.75623i	−0.478451	+	0.878114i	$0.658802\pi$
$84$	2.00000	0.218218
$85$	0	0
$86$	−2.00000	−0.215666
$87$	6.00000i	0.643268i
$88$	4.00000i	0.426401i
$89$	−6.00000	−0.635999	−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000	−0.635999	−0.317999	+	0.948091i	$0.603011\pi$
$90$	0	0
$91$	0	0
$92$	4.00000i	0.417029i
$93$	8.00000i	0.829561i
$94$	−8.00000	−0.825137
$95$	0	0
$96$	−1.00000	−0.102062
$97$	− 8.00000i	− 0.812277i	−0.913812	−	0.406138i	$-0.866875\pi$
$97$	− 8.00000i	− 0.812277i	0.913812	−	0.406138i	$-0.133125\pi$
$98$	− 3.00000i	− 0.303046i
$99$	−4.00000	−0.402015
$100$	0	0
$101$	−12.0000	−1.19404	−0.597022	−	0.802225i	$-0.703650\pi$
$101$	−12.0000	−1.19404	−0.597022	+	0.802225i	$0.703650\pi$
$102$	− 1.00000i	− 0.0990148i
$103$	20.0000i	1.97066i	0.170664	+	0.985329i	$0.445409\pi$
$103$	20.0000i	1.97066i	−0.170664	+	0.985329i	$0.554591\pi$
$104$	0	0
$105$	0	0
$106$	−14.0000	−1.35980
$107$	− 4.00000i	− 0.386695i	−0.981130	−	0.193347i	$-0.938066\pi$
$107$	− 4.00000i	− 0.386695i	0.981130	−	0.193347i	$-0.0619344\pi$
$108$	− 1.00000i	− 0.0962250i
$109$	2.00000	0.191565	0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000	0.191565	0.0957826	+	0.995402i	$0.469465\pi$
$110$	0	0
$111$	−6.00000	−0.569495
$112$	− 2.00000i	− 0.188982i
$113$	18.0000i	1.69330i	0.532152	+	0.846649i	$0.321383\pi$
$113$	18.0000i	1.69330i	−0.532152	+	0.846649i	$0.678617\pi$
$114$	4.00000	0.374634
$115$	0	0
$116$	6.00000	0.557086
$117$	0	0
$118$	6.00000i	0.552345i
$119$	2.00000	0.183340
$120$	0	0
$121$	5.00000	0.454545
$122$	− 2.00000i	− 0.181071i
$123$	− 8.00000i	− 0.721336i
$124$	8.00000	0.718421
$125$	0	0
$126$	2.00000	0.178174
$127$	4.00000i	0.354943i	0.984126	+	0.177471i	$0.0567917\pi$
$127$	4.00000i	0.354943i	−0.984126	+	0.177471i	$0.943208\pi$
$128$	1.00000i	0.0883883i
$129$	−2.00000	−0.176090
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	4.00000i	0.348155i
$133$	8.00000i	0.693688i
$134$	2.00000	0.172774
$135$	0	0
$136$	−1.00000	−0.0857493
$137$	− 6.00000i	− 0.512615i	−0.966595	−	0.256307i	$-0.917494\pi$
$137$	− 6.00000i	− 0.512615i	0.966595	−	0.256307i	$-0.0825059\pi$
$138$	4.00000i	0.340503i
$139$	4.00000	0.339276	0.169638	−	0.985506i	$-0.445740\pi$
$139$	4.00000	0.339276	0.169638	+	0.985506i	$0.445740\pi$
$140$	0	0
$141$	−8.00000	−0.673722
$142$	10.0000i	0.839181i
$143$	0	0
$144$	−1.00000	−0.0833333
$145$	0	0
$146$	−4.00000	−0.331042
$147$	− 3.00000i	− 0.247436i
$148$	6.00000i	0.493197i
$149$	−20.0000	−1.63846	−0.819232	−	0.573462i	$-0.805600\pi$
$149$	−20.0000	−1.63846	−0.819232	+	0.573462i	$0.805600\pi$
$150$	0	0
$151$	−16.0000	−1.30206	−0.651031	−	0.759051i	$-0.725663\pi$
$151$	−16.0000	−1.30206	−0.651031	+	0.759051i	$0.725663\pi$
$152$	− 4.00000i	− 0.324443i
$153$	− 1.00000i	− 0.0808452i
$154$	−8.00000	−0.644658
$155$	0	0
$156$	0	0
$157$	0	0	1.00000			$0$
$157$	0	0	−1.00000			$\pi$
$158$	4.00000i	0.318223i
$159$	−14.0000	−1.11027
$160$	0	0
$161$	−8.00000	−0.630488
$162$	− 1.00000i	− 0.0785674i
$163$	− 8.00000i	− 0.626608i	−0.949653	−	0.313304i	$-0.898564\pi$
$163$	− 8.00000i	− 0.626608i	0.949653	−	0.313304i	$-0.101436\pi$
$164$	−8.00000	−0.624695
$165$	0	0
$166$	16.0000	1.24184
$167$	− 16.0000i	− 1.23812i	−0.785345	−	0.619059i	$-0.787514\pi$
$167$	− 16.0000i	− 1.23812i	0.785345	−	0.619059i	$-0.212486\pi$
$168$	− 2.00000i	− 0.154303i
$169$	13.0000	1.00000
$170$	0	0
$171$	4.00000	0.305888
$172$	2.00000i	0.152499i
$173$	− 2.00000i	− 0.152057i	−0.997106	−	0.0760286i	$-0.975776\pi$
$173$	− 2.00000i	− 0.152057i	0.997106	−	0.0760286i	$-0.0242240\pi$
$174$	6.00000	0.454859
$175$	0	0
$176$	4.00000	0.301511
$177$	6.00000i	0.450988i
$178$	6.00000i	0.449719i
$179$	−2.00000	−0.149487	−0.0747435	−	0.997203i	$-0.523814\pi$
$179$	−2.00000	−0.149487	−0.0747435	+	0.997203i	$0.523814\pi$
$180$	0	0
$181$	14.0000	1.04061	0.520306	−	0.853980i	$-0.325818\pi$
$181$	14.0000	1.04061	0.520306	+	0.853980i	$0.325818\pi$
$182$	0	0
$183$	− 2.00000i	− 0.147844i
$184$	4.00000	0.294884
$185$	0	0
$186$	8.00000	0.586588
$187$	4.00000i	0.292509i
$188$	8.00000i	0.583460i
$189$	2.00000	0.145479
$190$	0	0
$191$	−20.0000	−1.44715	−0.723575	−	0.690246i	$-0.757502\pi$
$191$	−20.0000	−1.44715	−0.723575	+	0.690246i	$0.757502\pi$
$192$	1.00000i	0.0721688i
$193$	− 16.0000i	− 1.15171i	−0.817554	−	0.575853i	$-0.804670\pi$
$193$	− 16.0000i	− 1.15171i	0.817554	−	0.575853i	$-0.195330\pi$
$194$	−8.00000	−0.574367
$195$	0	0
$196$	−3.00000	−0.214286
$197$	− 6.00000i	− 0.427482i	−0.976890	−	0.213741i	$-0.931435\pi$
$197$	− 6.00000i	− 0.427482i	0.976890	−	0.213741i	$-0.0685649\pi$
$198$	4.00000i	0.284268i
$199$	8.00000	0.567105	0.283552	−	0.958957i	$-0.408487\pi$
$199$	8.00000	0.567105	0.283552	+	0.958957i	$0.408487\pi$
$200$	0	0
$201$	2.00000	0.141069
$202$	12.0000i	0.844317i
$203$	12.0000i	0.842235i
$204$	−1.00000	−0.0700140
$205$	0	0
$206$	20.0000	1.39347
$207$	4.00000i	0.278019i
$208$	0	0
$209$	−16.0000	−1.10674
$210$	0	0
$211$	4.00000	0.275371	0.137686	−	0.990476i	$-0.456034\pi$
$211$	4.00000	0.275371	0.137686	+	0.990476i	$0.456034\pi$
$212$	14.0000i	0.961524i
$213$	10.0000i	0.685189i
$214$	−4.00000	−0.273434
$215$	0	0
$216$	−1.00000	−0.0680414
$217$	16.0000i	1.08615i
$218$	− 2.00000i	− 0.135457i
$219$	−4.00000	−0.270295
$220$	0	0
$221$	0	0
$222$	6.00000i	0.402694i
$223$	− 4.00000i	− 0.267860i	−0.990991	−	0.133930i	$-0.957240\pi$
$223$	− 4.00000i	− 0.267860i	0.990991	−	0.133930i	$-0.0427597\pi$
$224$	−2.00000	−0.133631
$225$	0	0
$226$	18.0000	1.19734
$227$	− 12.0000i	− 0.796468i	−0.917284	−	0.398234i	$-0.869623\pi$
$227$	− 12.0000i	− 0.796468i	0.917284	−	0.398234i	$-0.130377\pi$
$228$	− 4.00000i	− 0.264906i
$229$	26.0000	1.71813	0.859064	−	0.511868i	$-0.171046\pi$
$229$	26.0000	1.71813	0.859064	+	0.511868i	$0.171046\pi$
$230$	0	0
$231$	−8.00000	−0.526361
$232$	− 6.00000i	− 0.393919i
$233$	− 10.0000i	− 0.655122i	−0.944830	−	0.327561i	$-0.893773\pi$
$233$	− 10.0000i	− 0.655122i	0.944830	−	0.327561i	$-0.106227\pi$
$234$	0	0
$235$	0	0
$236$	6.00000	0.390567
$237$	4.00000i	0.259828i
$238$	− 2.00000i	− 0.129641i
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	0	0
$241$	10.0000	0.644157	0.322078	−	0.946713i	$-0.395619\pi$
$241$	10.0000	0.644157	0.322078	+	0.946713i	$0.395619\pi$
$242$	− 5.00000i	− 0.321412i
$243$	− 1.00000i	− 0.0641500i
$244$	−2.00000	−0.128037
$245$	0	0
$246$	−8.00000	−0.510061
$247$	0	0
$248$	− 8.00000i	− 0.508001i
$249$	16.0000	1.01396
$250$	0	0
$251$	−2.00000	−0.126239	−0.0631194	−	0.998006i	$-0.520105\pi$
$251$	−2.00000	−0.126239	−0.0631194	+	0.998006i	$0.520105\pi$
$252$	− 2.00000i	− 0.125988i
$253$	− 16.0000i	− 1.00591i
$254$	4.00000	0.250982
$255$	0	0
$256$	1.00000	0.0625000
$257$	30.0000i	1.87135i	0.352865	+	0.935674i	$0.385208\pi$
$257$	30.0000i	1.87135i	−0.352865	+	0.935674i	$0.614792\pi$
$258$	2.00000i	0.124515i
$259$	−12.0000	−0.745644
$260$	0	0
$261$	6.00000	0.371391
$262$	0	0
$263$	− 16.0000i	− 0.986602i	−0.869859	−	0.493301i	$-0.835790\pi$
$263$	− 16.0000i	− 0.986602i	0.869859	−	0.493301i	$-0.164210\pi$
$264$	4.00000	0.246183
$265$	0	0
$266$	8.00000	0.490511
$267$	6.00000i	0.367194i
$268$	− 2.00000i	− 0.122169i
$269$	2.00000	0.121942	0.0609711	−	0.998140i	$-0.480580\pi$
$269$	2.00000	0.121942	0.0609711	+	0.998140i	$0.480580\pi$
$270$	0	0
$271$	16.0000	0.971931	0.485965	−	0.873978i	$-0.338468\pi$
$271$	16.0000	0.971931	0.485965	+	0.873978i	$0.338468\pi$
$272$	1.00000i	0.0606339i
$273$	0	0
$274$	−6.00000	−0.362473
$275$	0	0
$276$	4.00000	0.240772
$277$	6.00000i	0.360505i	0.983620	+	0.180253i	$0.0576915\pi$
$277$	6.00000i	0.360505i	−0.983620	+	0.180253i	$0.942309\pi$
$278$	− 4.00000i	− 0.239904i
$279$	8.00000	0.478947
$280$	0	0
$281$	−10.0000	−0.596550	−0.298275	−	0.954480i	$-0.596411\pi$
$281$	−10.0000	−0.596550	−0.298275	+	0.954480i	$0.596411\pi$
$282$	8.00000i	0.476393i
$283$	12.0000i	0.713326i	0.934233	+	0.356663i	$0.116086\pi$
$283$	12.0000i	0.713326i	−0.934233	+	0.356663i	$0.883914\pi$
$284$	10.0000	0.593391
$285$	0	0
$286$	0	0
$287$	− 16.0000i	− 0.944450i
$288$	1.00000i	0.0589256i
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	−8.00000	−0.468968
$292$	4.00000i	0.234082i
$293$	− 18.0000i	− 1.05157i	−0.850617	−	0.525786i	$-0.823771\pi$
$293$	− 18.0000i	− 1.05157i	0.850617	−	0.525786i	$-0.176229\pi$
$294$	−3.00000	−0.174964
$295$	0	0
$296$	6.00000	0.348743
$297$	4.00000i	0.232104i
$298$	20.0000i	1.15857i
$299$	0	0
$300$	0	0
$301$	−4.00000	−0.230556
$302$	16.0000i	0.920697i
$303$	12.0000i	0.689382i
$304$	−4.00000	−0.229416
$305$	0	0
$306$	−1.00000	−0.0571662
$307$	2.00000i	0.114146i	0.998370	+	0.0570730i	$0.0181768\pi$
$307$	2.00000i	0.114146i	−0.998370	+	0.0570730i	$0.981823\pi$
$308$	8.00000i	0.455842i
$309$	20.0000	1.13776
$310$	0	0
$311$	10.0000	0.567048	0.283524	−	0.958965i	$-0.408496\pi$
$311$	10.0000	0.567048	0.283524	+	0.958965i	$0.408496\pi$
$312$	0	0
$313$	8.00000i	0.452187i	0.974106	+	0.226093i	$0.0725954\pi$
$313$	8.00000i	0.452187i	−0.974106	+	0.226093i	$0.927405\pi$
$314$	0	0
$315$	0	0
$316$	4.00000	0.225018
$317$	18.0000i	1.01098i	0.862832	+	0.505490i	$0.168688\pi$
$317$	18.0000i	1.01098i	−0.862832	+	0.505490i	$0.831312\pi$
$318$	14.0000i	0.785081i
$319$	−24.0000	−1.34374
$320$	0	0
$321$	−4.00000	−0.223258
$322$	8.00000i	0.445823i
$323$	− 4.00000i	− 0.222566i
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	−8.00000	−0.443079
$327$	− 2.00000i	− 0.110600i
$328$	8.00000i	0.441726i
$329$	−16.0000	−0.882109
$330$	0	0
$331$	0	0		−	1.00000i	$-0.5\pi$
$331$	0	0			1.00000i	$0.5\pi$
$332$	− 16.0000i	− 0.878114i
$333$	6.00000i	0.328798i
$334$	−16.0000	−0.875481
$335$	0	0
$336$	−2.00000	−0.109109
$337$	20.0000i	1.08947i	0.838608	+	0.544735i	$0.183370\pi$
$337$	20.0000i	1.08947i	−0.838608	+	0.544735i	$0.816630\pi$
$338$	− 13.0000i	− 0.707107i
$339$	18.0000	0.977626
$340$	0	0
$341$	−32.0000	−1.73290
$342$	− 4.00000i	− 0.216295i
$343$	− 20.0000i	− 1.07990i
$344$	2.00000	0.107833
$345$	0	0
$346$	−2.00000	−0.107521
$347$	− 20.0000i	− 1.07366i	−0.843692	−	0.536828i	$-0.819622\pi$
$347$	− 20.0000i	− 1.07366i	0.843692	−	0.536828i	$-0.180378\pi$
$348$	− 6.00000i	− 0.321634i
$349$	14.0000	0.749403	0.374701	−	0.927146i	$-0.377745\pi$
$349$	14.0000	0.749403	0.374701	+	0.927146i	$0.377745\pi$
$350$	0	0
$351$	0	0
$352$	− 4.00000i	− 0.213201i
$353$	− 6.00000i	− 0.319348i	−0.987170	−	0.159674i	$-0.948956\pi$
$353$	− 6.00000i	− 0.319348i	0.987170	−	0.159674i	$-0.0510443\pi$
$354$	6.00000	0.318896
$355$	0	0
$356$	6.00000	0.317999
$357$	− 2.00000i	− 0.105851i
$358$	2.00000i	0.105703i
$359$	4.00000	0.211112	0.105556	−	0.994413i	$-0.466338\pi$
$359$	4.00000	0.211112	0.105556	+	0.994413i	$0.466338\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	− 14.0000i	− 0.735824i
$363$	− 5.00000i	− 0.262432i
$364$	0	0
$365$	0	0
$366$	−2.00000	−0.104542
$367$	− 38.0000i	− 1.98358i	−0.127862	−	0.991792i	$-0.540812\pi$
$367$	− 38.0000i	− 1.98358i	0.127862	−	0.991792i	$-0.459188\pi$
$368$	− 4.00000i	− 0.208514i
$369$	−8.00000	−0.416463
$370$	0	0
$371$	−28.0000	−1.45369
$372$	− 8.00000i	− 0.414781i
$373$	− 36.0000i	− 1.86401i	−0.362446	−	0.932005i	$-0.618058\pi$
$373$	− 36.0000i	− 1.86401i	0.362446	−	0.932005i	$-0.381942\pi$
$374$	4.00000	0.206835
$375$	0	0
$376$	8.00000	0.412568
$377$	0	0
$378$	− 2.00000i	− 0.102869i
$379$	28.0000	1.43826	0.719132	−	0.694874i	$-0.244540\pi$
$379$	28.0000	1.43826	0.719132	+	0.694874i	$0.244540\pi$
$380$	0	0
$381$	4.00000	0.204926
$382$	20.0000i	1.02329i
$383$	16.0000i	0.817562i	0.912633	+	0.408781i	$0.134046\pi$
$383$	16.0000i	0.817562i	−0.912633	+	0.408781i	$0.865954\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	−16.0000	−0.814379
$387$	2.00000i	0.101666i
$388$	8.00000i	0.406138i
$389$	36.0000	1.82527	0.912636	−	0.408773i	$-0.134043\pi$
$389$	36.0000	1.82527	0.912636	+	0.408773i	$0.134043\pi$
$390$	0	0
$391$	4.00000	0.202289
$392$	3.00000i	0.151523i
$393$	0	0
$394$	−6.00000	−0.302276
$395$	0	0
$396$	4.00000	0.201008
$397$	34.0000i	1.70641i	0.521575	+	0.853206i	$0.325345\pi$
$397$	34.0000i	1.70641i	−0.521575	+	0.853206i	$0.674655\pi$
$398$	− 8.00000i	− 0.401004i
$399$	8.00000	0.400501
$400$	0	0
$401$	20.0000	0.998752	0.499376	−	0.866385i	$-0.333563\pi$
$401$	20.0000	0.998752	0.499376	+	0.866385i	$0.333563\pi$
$402$	− 2.00000i	− 0.0997509i
$403$	0	0
$404$	12.0000	0.597022
$405$	0	0
$406$	12.0000	0.595550
$407$	− 24.0000i	− 1.18964i
$408$	1.00000i	0.0495074i
$409$	26.0000	1.28562	0.642809	−	0.766027i	$-0.277769\pi$
$409$	26.0000	1.28562	0.642809	+	0.766027i	$0.277769\pi$
$410$	0	0
$411$	−6.00000	−0.295958
$412$	− 20.0000i	− 0.985329i
$413$	12.0000i	0.590481i
$414$	4.00000	0.196589
$415$	0	0
$416$	0	0
$417$	− 4.00000i	− 0.195881i
$418$	16.0000i	0.782586i
$419$	−32.0000	−1.56330	−0.781651	−	0.623716i	$-0.785622\pi$
$419$	−32.0000	−1.56330	−0.781651	+	0.623716i	$0.785622\pi$
$420$	0	0
$421$	−10.0000	−0.487370	−0.243685	−	0.969854i	$-0.578356\pi$
$421$	−10.0000	−0.487370	−0.243685	+	0.969854i	$0.578356\pi$
$422$	− 4.00000i	− 0.194717i
$423$	8.00000i	0.388973i
$424$	14.0000	0.679900
$425$	0	0
$426$	10.0000	0.484502
$427$	− 4.00000i	− 0.193574i
$428$	4.00000i	0.193347i
$429$	0	0
$430$	0	0
$431$	10.0000	0.481683	0.240842	−	0.970564i	$-0.422577\pi$
$431$	10.0000	0.481683	0.240842	+	0.970564i	$0.422577\pi$
$432$	1.00000i	0.0481125i
$433$	− 6.00000i	− 0.288342i	−0.989553	−	0.144171i	$-0.953949\pi$
$433$	− 6.00000i	− 0.288342i	0.989553	−	0.144171i	$-0.0460515\pi$
$434$	16.0000	0.768025
$435$	0	0
$436$	−2.00000	−0.0957826
$437$	16.0000i	0.765384i
$438$	4.00000i	0.191127i
$439$	28.0000	1.33637	0.668184	−	0.743996i	$-0.267072\pi$
$439$	28.0000	1.33637	0.668184	+	0.743996i	$0.267072\pi$
$440$	0	0
$441$	−3.00000	−0.142857
$442$	0	0
$443$	0	0	1.00000			$0$
$443$	0	0	−1.00000			$\pi$
$444$	6.00000	0.284747
$445$	0	0
$446$	−4.00000	−0.189405
$447$	20.0000i	0.945968i
$448$	2.00000i	0.0944911i
$449$	12.0000	0.566315	0.283158	−	0.959073i	$-0.408618\pi$
$449$	12.0000	0.566315	0.283158	+	0.959073i	$0.408618\pi$
$450$	0	0
$451$	32.0000	1.50682
$452$	− 18.0000i	− 0.846649i
$453$	16.0000i	0.751746i
$454$	−12.0000	−0.563188
$455$	0	0
$456$	−4.00000	−0.187317
$457$	− 18.0000i	− 0.842004i	−0.907060	−	0.421002i	$-0.861678\pi$
$457$	− 18.0000i	− 0.842004i	0.907060	−	0.421002i	$-0.138322\pi$
$458$	− 26.0000i	− 1.21490i
$459$	−1.00000	−0.0466760
$460$	0	0
$461$	24.0000	1.11779	0.558896	−	0.829238i	$-0.311225\pi$
$461$	24.0000	1.11779	0.558896	+	0.829238i	$0.311225\pi$
$462$	8.00000i	0.372194i
$463$	4.00000i	0.185896i	0.995671	+	0.0929479i	$0.0296290\pi$
$463$	4.00000i	0.185896i	−0.995671	+	0.0929479i	$0.970371\pi$
$464$	−6.00000	−0.278543
$465$	0	0
$466$	−10.0000	−0.463241
$467$	− 28.0000i	− 1.29569i	−0.761774	−	0.647843i	$-0.775671\pi$
$467$	− 28.0000i	− 1.29569i	0.761774	−	0.647843i	$-0.224329\pi$
$468$	0	0
$469$	4.00000	0.184703
$470$	0	0
$471$	0	0
$472$	− 6.00000i	− 0.276172i
$473$	− 8.00000i	− 0.367840i
$474$	4.00000	0.183726
$475$	0	0
$476$	−2.00000	−0.0916698
$477$	14.0000i	0.641016i
$478$	0	0
$479$	−30.0000	−1.37073	−0.685367	−	0.728197i	$-0.740358\pi$
$479$	−30.0000	−1.37073	−0.685367	+	0.728197i	$0.740358\pi$
$480$	0	0
$481$	0	0
$482$	− 10.0000i	− 0.455488i
$483$	8.00000i	0.364013i
$484$	−5.00000	−0.227273
$485$	0	0
$486$	−1.00000	−0.0453609
$487$	30.0000i	1.35943i	0.733476	+	0.679715i	$0.237896\pi$
$487$	30.0000i	1.35943i	−0.733476	+	0.679715i	$0.762104\pi$
$488$	2.00000i	0.0905357i
$489$	−8.00000	−0.361773
$490$	0	0
$491$	−6.00000	−0.270776	−0.135388	−	0.990793i	$-0.543228\pi$
$491$	−6.00000	−0.270776	−0.135388	+	0.990793i	$0.543228\pi$
$492$	8.00000i	0.360668i
$493$	− 6.00000i	− 0.270226i
$494$	0	0
$495$	0	0
$496$	−8.00000	−0.359211
$497$	20.0000i	0.897123i
$498$	− 16.0000i	− 0.716977i
$499$	−36.0000	−1.61158	−0.805791	−	0.592200i	$-0.798259\pi$
$499$	−36.0000	−1.61158	−0.805791	+	0.592200i	$0.798259\pi$
$500$	0	0
$501$	−16.0000	−0.714827
$502$	2.00000i	0.0892644i
$503$	− 44.0000i	− 1.96186i	−0.194354	−	0.980932i	$-0.562261\pi$
$503$	− 44.0000i	− 1.96186i	0.194354	−	0.980932i	$-0.437739\pi$
$504$	−2.00000	−0.0890871
$505$	0	0
$506$	−16.0000	−0.711287
$507$	− 13.0000i	− 0.577350i
$508$	− 4.00000i	− 0.177471i
$509$	36.0000	1.59567	0.797836	−	0.602875i	$-0.205978\pi$
$509$	36.0000	1.59567	0.797836	+	0.602875i	$0.205978\pi$
$510$	0	0
$511$	−8.00000	−0.353899
$512$	− 1.00000i	− 0.0441942i
$513$	− 4.00000i	− 0.176604i
$514$	30.0000	1.32324
$515$	0	0
$516$	2.00000	0.0880451
$517$	− 32.0000i	− 1.40736i
$518$	12.0000i	0.527250i
$519$	−2.00000	−0.0877903
$520$	0	0
$521$	12.0000	0.525730	0.262865	−	0.964833i	$-0.415333\pi$
$521$	12.0000	0.525730	0.262865	+	0.964833i	$0.415333\pi$
$522$	− 6.00000i	− 0.262613i
$523$	− 30.0000i	− 1.31181i	−0.754844	−	0.655904i	$-0.772288\pi$
$523$	− 30.0000i	− 1.31181i	0.754844	−	0.655904i	$-0.227712\pi$
$524$	0	0
$525$	0	0
$526$	−16.0000	−0.697633
$527$	− 8.00000i	− 0.348485i
$528$	− 4.00000i	− 0.174078i
$529$	7.00000	0.304348
$530$	0	0
$531$	6.00000	0.260378
$532$	− 8.00000i	− 0.346844i
$533$	0	0
$534$	6.00000	0.259645
$535$	0	0
$536$	−2.00000	−0.0863868
$537$	2.00000i	0.0863064i
$538$	− 2.00000i	− 0.0862261i
$539$	12.0000	0.516877
$540$	0	0
$541$	−30.0000	−1.28980	−0.644900	−	0.764267i	$-0.723101\pi$
$541$	−30.0000	−1.28980	−0.644900	+	0.764267i	$0.723101\pi$
$542$	− 16.0000i	− 0.687259i
$543$	− 14.0000i	− 0.600798i
$544$	1.00000	0.0428746
$545$	0	0
$546$	0	0
$547$	36.0000i	1.53925i	0.638497	+	0.769624i	$0.279557\pi$
$547$	36.0000i	1.53925i	−0.638497	+	0.769624i	$0.720443\pi$
$548$	6.00000i	0.256307i
$549$	−2.00000	−0.0853579
$550$	0	0
$551$	24.0000	1.02243
$552$	− 4.00000i	− 0.170251i
$553$	8.00000i	0.340195i
$554$	6.00000	0.254916
$555$	0	0
$556$	−4.00000	−0.169638
$557$	− 18.0000i	− 0.762684i	−0.924434	−	0.381342i	$-0.875462\pi$
$557$	− 18.0000i	− 0.762684i	0.924434	−	0.381342i	$-0.124538\pi$
$558$	− 8.00000i	− 0.338667i
$559$	0	0
$560$	0	0
$561$	4.00000	0.168880
$562$	10.0000i	0.421825i
$563$	− 40.0000i	− 1.68580i	−0.538071	−	0.842900i	$-0.680847\pi$
$563$	− 40.0000i	− 1.68580i	0.538071	−	0.842900i	$-0.319153\pi$
$564$	8.00000	0.336861
$565$	0	0
$566$	12.0000	0.504398
$567$	− 2.00000i	− 0.0839921i
$568$	− 10.0000i	− 0.419591i
$569$	−34.0000	−1.42535	−0.712677	−	0.701492i	$-0.752517\pi$
$569$	−34.0000	−1.42535	−0.712677	+	0.701492i	$0.752517\pi$
$570$	0	0
$571$	−28.0000	−1.17176	−0.585882	−	0.810397i	$-0.699252\pi$
$571$	−28.0000	−1.17176	−0.585882	+	0.810397i	$0.699252\pi$
$572$	0	0
$573$	20.0000i	0.835512i
$574$	−16.0000	−0.667827
$575$	0	0
$576$	1.00000	0.0416667
$577$	− 10.0000i	− 0.416305i	−0.978096	−	0.208153i	$-0.933255\pi$
$577$	− 10.0000i	− 0.416305i	0.978096	−	0.208153i	$-0.0667451\pi$
$578$	1.00000i	0.0415945i
$579$	−16.0000	−0.664937
$580$	0	0
$581$	32.0000	1.32758
$582$	8.00000i	0.331611i
$583$	− 56.0000i	− 2.31928i
$584$	4.00000	0.165521
$585$	0	0
$586$	−18.0000	−0.743573
$587$	− 32.0000i	− 1.32078i	−0.750922	−	0.660391i	$-0.770391\pi$
$587$	− 32.0000i	− 1.32078i	0.750922	−	0.660391i	$-0.229609\pi$
$588$	3.00000i	0.123718i
$589$	32.0000	1.31854
$590$	0	0
$591$	−6.00000	−0.246807
$592$	− 6.00000i	− 0.246598i
$593$	− 46.0000i	− 1.88899i	−0.328521	−	0.944497i	$-0.606550\pi$
$593$	− 46.0000i	− 1.88899i	0.328521	−	0.944497i	$-0.393450\pi$
$594$	4.00000	0.164122
$595$	0	0
$596$	20.0000	0.819232
$597$	− 8.00000i	− 0.327418i
$598$	0	0
$599$	−36.0000	−1.47092	−0.735460	−	0.677568i	$-0.763034\pi$
$599$	−36.0000	−1.47092	−0.735460	+	0.677568i	$0.763034\pi$
$600$	0	0
$601$	14.0000	0.571072	0.285536	−	0.958368i	$-0.407828\pi$
$601$	14.0000	0.571072	0.285536	+	0.958368i	$0.407828\pi$
$602$	4.00000i	0.163028i
$603$	− 2.00000i	− 0.0814463i
$604$	16.0000	0.651031
$605$	0	0
$606$	12.0000	0.487467
$607$	22.0000i	0.892952i	0.894795	+	0.446476i	$0.147321\pi$
$607$	22.0000i	0.892952i	−0.894795	+	0.446476i	$0.852679\pi$
$608$	4.00000i	0.162221i
$609$	12.0000	0.486265
$610$	0	0
$611$	0	0
$612$	1.00000i	0.0404226i
$613$	36.0000i	1.45403i	0.686624	+	0.727013i	$0.259092\pi$
$613$	36.0000i	1.45403i	−0.686624	+	0.727013i	$0.740908\pi$
$614$	2.00000	0.0807134
$615$	0	0
$616$	8.00000	0.322329
$617$	2.00000i	0.0805170i	0.999189	+	0.0402585i	$0.0128181\pi$
$617$	2.00000i	0.0805170i	−0.999189	+	0.0402585i	$0.987182\pi$
$618$	− 20.0000i	− 0.804518i
$619$	12.0000	0.482321	0.241160	−	0.970485i	$-0.422472\pi$
$619$	12.0000	0.482321	0.241160	+	0.970485i	$0.422472\pi$
$620$	0	0
$621$	4.00000	0.160514
$622$	− 10.0000i	− 0.400963i
$623$	12.0000i	0.480770i
$624$	0	0
$625$	0	0
$626$	8.00000	0.319744
$627$	16.0000i	0.638978i
$628$	0	0
$629$	6.00000	0.239236
$630$	0	0
$631$	−24.0000	−0.955425	−0.477712	−	0.878516i	$-0.658534\pi$
$631$	−24.0000	−0.955425	−0.477712	+	0.878516i	$0.658534\pi$
$632$	− 4.00000i	− 0.159111i
$633$	− 4.00000i	− 0.158986i
$634$	18.0000	0.714871
$635$	0	0
$636$	14.0000	0.555136
$637$	0	0
$638$	24.0000i	0.950169i
$639$	10.0000	0.395594
$640$	0	0
$641$	28.0000	1.10593	0.552967	−	0.833203i	$-0.313496\pi$
$641$	28.0000	1.10593	0.552967	+	0.833203i	$0.313496\pi$
$642$	4.00000i	0.157867i
$643$	− 16.0000i	− 0.630978i	−0.948929	−	0.315489i	$-0.897831\pi$
$643$	− 16.0000i	− 0.630978i	0.948929	−	0.315489i	$-0.102169\pi$
$644$	8.00000	0.315244
$645$	0	0
$646$	−4.00000	−0.157378
$647$	− 24.0000i	− 0.943537i	−0.881722	−	0.471769i	$-0.843616\pi$
$647$	− 24.0000i	− 0.943537i	0.881722	−	0.471769i	$-0.156384\pi$
$648$	1.00000i	0.0392837i
$649$	−24.0000	−0.942082
$650$	0	0
$651$	16.0000	0.627089
$652$	8.00000i	0.313304i
$653$	− 14.0000i	− 0.547862i	−0.961749	−	0.273931i	$-0.911676\pi$
$653$	− 14.0000i	− 0.547862i	0.961749	−	0.273931i	$-0.0883240\pi$
$654$	−2.00000	−0.0782062
$655$	0	0
$656$	8.00000	0.312348
$657$	4.00000i	0.156055i
$658$	16.0000i	0.623745i
$659$	30.0000	1.16863	0.584317	−	0.811525i	$-0.301362\pi$
$659$	30.0000	1.16863	0.584317	+	0.811525i	$0.301362\pi$
$660$	0	0
$661$	−30.0000	−1.16686	−0.583432	−	0.812162i	$-0.698291\pi$
$661$	−30.0000	−1.16686	−0.583432	+	0.812162i	$0.698291\pi$
$662$	0	0
$663$	0	0
$664$	−16.0000	−0.620920
$665$	0	0
$666$	6.00000	0.232495
$667$	24.0000i	0.929284i
$668$	16.0000i	0.619059i
$669$	−4.00000	−0.154649
$670$	0	0
$671$	8.00000	0.308837
$672$	2.00000i	0.0771517i
$673$	0	0	1.00000			$0$
$673$	0	0	−1.00000			$\pi$
$674$	20.0000	0.770371
$675$	0	0
$676$	−13.0000	−0.500000
$677$	− 6.00000i	− 0.230599i	−0.993331	−	0.115299i	$-0.963217\pi$
$677$	− 6.00000i	− 0.230599i	0.993331	−	0.115299i	$-0.0367827\pi$
$678$	− 18.0000i	− 0.691286i
$679$	−16.0000	−0.614024
$680$	0	0
$681$	−12.0000	−0.459841
$682$	32.0000i	1.22534i
$683$	4.00000i	0.153056i	0.997067	+	0.0765279i	$0.0243834\pi$
$683$	4.00000i	0.153056i	−0.997067	+	0.0765279i	$0.975617\pi$
$684$	−4.00000	−0.152944
$685$	0	0
$686$	−20.0000	−0.763604
$687$	− 26.0000i	− 0.991962i
$688$	− 2.00000i	− 0.0762493i
$689$	0	0
$690$	0	0
$691$	28.0000	1.06517	0.532585	−	0.846376i	$-0.321221\pi$
$691$	28.0000	1.06517	0.532585	+	0.846376i	$0.321221\pi$
$692$	2.00000i	0.0760286i
$693$	8.00000i	0.303895i
$694$	−20.0000	−0.759190
$695$	0	0
$696$	−6.00000	−0.227429
$697$	8.00000i	0.303022i
$698$	− 14.0000i	− 0.529908i
$699$	−10.0000	−0.378235
$700$	0	0
$701$	−8.00000	−0.302156	−0.151078	−	0.988522i	$-0.548274\pi$
$701$	−8.00000	−0.302156	−0.151078	+	0.988522i	$0.548274\pi$
$702$	0	0
$703$	24.0000i	0.905177i
$704$	−4.00000	−0.150756
$705$	0	0
$706$	−6.00000	−0.225813
$707$	24.0000i	0.902613i
$708$	− 6.00000i	− 0.225494i
$709$	−2.00000	−0.0751116	−0.0375558	−	0.999295i	$-0.511957\pi$
$709$	−2.00000	−0.0751116	−0.0375558	+	0.999295i	$0.511957\pi$
$710$	0	0
$711$	4.00000	0.150012
$712$	− 6.00000i	− 0.224860i
$713$	32.0000i	1.19841i
$714$	−2.00000	−0.0748481
$715$	0	0
$716$	2.00000	0.0747435
$717$	0	0
$718$	− 4.00000i	− 0.149279i
$719$	−14.0000	−0.522112	−0.261056	−	0.965324i	$-0.584071\pi$
$719$	−14.0000	−0.522112	−0.261056	+	0.965324i	$0.584071\pi$
$720$	0	0
$721$	40.0000	1.48968
$722$	3.00000i	0.111648i
$723$	− 10.0000i	− 0.371904i
$724$	−14.0000	−0.520306
$725$	0	0
$726$	−5.00000	−0.185567
$727$	− 44.0000i	− 1.63187i	−0.578144	−	0.815935i	$-0.696223\pi$
$727$	− 44.0000i	− 1.63187i	0.578144	−	0.815935i	$-0.303777\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	2.00000	0.0739727
$732$	2.00000i	0.0739221i
$733$	44.0000i	1.62518i	0.582838	+	0.812589i	$0.301942\pi$
$733$	44.0000i	1.62518i	−0.582838	+	0.812589i	$0.698058\pi$
$734$	−38.0000	−1.40261
$735$	0	0
$736$	−4.00000	−0.147442
$737$	8.00000i	0.294684i
$738$	8.00000i	0.294484i
$739$	32.0000	1.17714	0.588570	−	0.808447i	$-0.299691\pi$
$739$	32.0000	1.17714	0.588570	+	0.808447i	$0.299691\pi$
$740$	0	0
$741$	0	0
$742$	28.0000i	1.02791i
$743$	40.0000i	1.46746i	0.679442	+	0.733729i	$0.262222\pi$
$743$	40.0000i	1.46746i	−0.679442	+	0.733729i	$0.737778\pi$
$744$	−8.00000	−0.293294
$745$	0	0
$746$	−36.0000	−1.31805
$747$	− 16.0000i	− 0.585409i
$748$	− 4.00000i	− 0.146254i
$749$	−8.00000	−0.292314
$750$	0	0
$751$	36.0000	1.31366	0.656829	−	0.754039i	$-0.271897\pi$
$751$	36.0000	1.31366	0.656829	+	0.754039i	$0.271897\pi$
$752$	− 8.00000i	− 0.291730i
$753$	2.00000i	0.0728841i
$754$	0	0
$755$	0	0
$756$	−2.00000	−0.0727393
$757$	8.00000i	0.290765i	0.989376	+	0.145382i	$0.0464413\pi$
$757$	8.00000i	0.290765i	−0.989376	+	0.145382i	$0.953559\pi$
$758$	− 28.0000i	− 1.01701i
$759$	−16.0000	−0.580763
$760$	0	0
$761$	−46.0000	−1.66750	−0.833749	−	0.552143i	$-0.813810\pi$
$761$	−46.0000	−1.66750	−0.833749	+	0.552143i	$0.813810\pi$
$762$	− 4.00000i	− 0.144905i
$763$	− 4.00000i	− 0.144810i
$764$	20.0000	0.723575
$765$	0	0
$766$	16.0000	0.578103
$767$	0	0
$768$	− 1.00000i	− 0.0360844i
$769$	50.0000	1.80305	0.901523	−	0.432731i	$-0.142450\pi$
$769$	50.0000	1.80305	0.901523	+	0.432731i	$0.142450\pi$
$770$	0	0
$771$	30.0000	1.08042
$772$	16.0000i	0.575853i
$773$	− 14.0000i	− 0.503545i	−0.967786	−	0.251773i	$-0.918987\pi$
$773$	− 14.0000i	− 0.503545i	0.967786	−	0.251773i	$-0.0810135\pi$
$774$	2.00000	0.0718885
$775$	0	0
$776$	8.00000	0.287183
$777$	12.0000i	0.430498i
$778$	− 36.0000i	− 1.29066i
$779$	−32.0000	−1.14652
$780$	0	0
$781$	−40.0000	−1.43131
$782$	− 4.00000i	− 0.143040i
$783$	− 6.00000i	− 0.214423i
$784$	3.00000	0.107143
$785$	0	0
$786$	0	0
$787$	8.00000i	0.285169i	0.989783	+	0.142585i	$0.0455413\pi$
$787$	8.00000i	0.285169i	−0.989783	+	0.142585i	$0.954459\pi$
$788$	6.00000i	0.213741i
$789$	−16.0000	−0.569615
$790$	0	0
$791$	36.0000	1.28001
$792$	− 4.00000i	− 0.142134i
$793$	0	0
$794$	34.0000	1.20661
$795$	0	0
$796$	−8.00000	−0.283552
$797$	42.0000i	1.48772i	0.668338	+	0.743858i	$0.267006\pi$
$797$	42.0000i	1.48772i	−0.668338	+	0.743858i	$0.732994\pi$
$798$	− 8.00000i	− 0.283197i
$799$	8.00000	0.283020
$800$	0	0
$801$	6.00000	0.212000
$802$	− 20.0000i	− 0.706225i
$803$	− 16.0000i	− 0.564628i
$804$	−2.00000	−0.0705346
$805$	0	0
$806$	0	0
$807$	− 2.00000i	− 0.0704033i
$808$	− 12.0000i	− 0.422159i
$809$	0	0		−	1.00000i	$-0.5\pi$
$809$	0	0			1.00000i	$0.5\pi$
$810$	0	0
$811$	−4.00000	−0.140459	−0.0702295	−	0.997531i	$-0.522373\pi$
$811$	−4.00000	−0.140459	−0.0702295	+	0.997531i	$0.522373\pi$
$812$	− 12.0000i	− 0.421117i
$813$	− 16.0000i	− 0.561144i
$814$	−24.0000	−0.841200
$815$	0	0
$816$	1.00000	0.0350070
$817$	8.00000i	0.279885i
$818$	− 26.0000i	− 0.909069i
$819$	0	0
$820$	0	0
$821$	−6.00000	−0.209401	−0.104701	−	0.994504i	$-0.533388\pi$
$821$	−6.00000	−0.209401	−0.104701	+	0.994504i	$0.533388\pi$
$822$	6.00000i	0.209274i
$823$	14.0000i	0.488009i	0.969774	+	0.244005i	$0.0784612\pi$
$823$	14.0000i	0.488009i	−0.969774	+	0.244005i	$0.921539\pi$
$824$	−20.0000	−0.696733
$825$	0	0
$826$	12.0000	0.417533
$827$	12.0000i	0.417281i	0.977992	+	0.208640i	$0.0669038\pi$
$827$	12.0000i	0.417281i	−0.977992	+	0.208640i	$0.933096\pi$
$828$	− 4.00000i	− 0.139010i
$829$	38.0000	1.31979	0.659897	−	0.751356i	$-0.270600\pi$
$829$	38.0000	1.31979	0.659897	+	0.751356i	$0.270600\pi$
$830$	0	0
$831$	6.00000	0.208138
$832$	0	0
$833$	3.00000i	0.103944i
$834$	−4.00000	−0.138509
$835$	0	0
$836$	16.0000	0.553372
$837$	− 8.00000i	− 0.276520i
$838$	32.0000i	1.10542i
$839$	6.00000	0.207143	0.103572	−	0.994622i	$-0.466973\pi$
$839$	6.00000	0.207143	0.103572	+	0.994622i	$0.466973\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	10.0000i	0.344623i
$843$	10.0000i	0.344418i
$844$	−4.00000	−0.137686
$845$	0	0
$846$	8.00000	0.275046
$847$	− 10.0000i	− 0.343604i
$848$	− 14.0000i	− 0.480762i
$849$	12.0000	0.411839
$850$	0	0
$851$	−24.0000	−0.822709
$852$	− 10.0000i	− 0.342594i
$853$	− 6.00000i	− 0.205436i	−0.994711	−	0.102718i	$-0.967246\pi$
$853$	− 6.00000i	− 0.205436i	0.994711	−	0.102718i	$-0.0327539\pi$
$854$	−4.00000	−0.136877
$855$	0	0
$856$	4.00000	0.136717
$857$	26.0000i	0.888143i	0.895991	+	0.444072i	$0.146466\pi$
$857$	26.0000i	0.888143i	−0.895991	+	0.444072i	$0.853534\pi$
$858$	0	0
$859$	−40.0000	−1.36478	−0.682391	−	0.730987i	$-0.739060\pi$
$859$	−40.0000	−1.36478	−0.682391	+	0.730987i	$0.739060\pi$
$860$	0	0
$861$	−16.0000	−0.545279
$862$	− 10.0000i	− 0.340601i
$863$	− 8.00000i	− 0.272323i	−0.990687	−	0.136162i	$-0.956523\pi$
$863$	− 8.00000i	− 0.272323i	0.990687	−	0.136162i	$-0.0434766\pi$
$864$	1.00000	0.0340207
$865$	0	0
$866$	−6.00000	−0.203888
$867$	1.00000i	0.0339618i
$868$	− 16.0000i	− 0.543075i
$869$	−16.0000	−0.542763
$870$	0	0
$871$	0	0
$872$	2.00000i	0.0677285i
$873$	8.00000i	0.270759i
$874$	16.0000	0.541208
$875$	0	0
$876$	4.00000	0.135147
$877$	− 18.0000i	− 0.607817i	−0.952701	−	0.303908i	$-0.901708\pi$
$877$	− 18.0000i	− 0.607817i	0.952701	−	0.303908i	$-0.0982917\pi$
$878$	− 28.0000i	− 0.944954i
$879$	−18.0000	−0.607125
$880$	0	0
$881$	44.0000	1.48240	0.741199	−	0.671286i	$-0.234258\pi$
$881$	44.0000	1.48240	0.741199	+	0.671286i	$0.234258\pi$
$882$	3.00000i	0.101015i
$883$	2.00000i	0.0673054i	0.999434	+	0.0336527i	$0.0107140\pi$
$883$	2.00000i	0.0673054i	−0.999434	+	0.0336527i	$0.989286\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 12.0000i	− 0.402921i	−0.979497	−	0.201460i	$-0.935431\pi$
$887$	− 12.0000i	− 0.402921i	0.979497	−	0.201460i	$-0.0645687\pi$
$888$	− 6.00000i	− 0.201347i
$889$	8.00000	0.268311
$890$	0	0
$891$	4.00000	0.134005
$892$	4.00000i	0.133930i
$893$	32.0000i	1.07084i
$894$	20.0000	0.668900
$895$	0	0
$896$	2.00000	0.0668153
$897$	0	0
$898$	− 12.0000i	− 0.400445i
$899$	48.0000	1.60089
$900$	0	0
$901$	14.0000	0.466408
$902$	− 32.0000i	− 1.06548i
$903$	4.00000i	0.133112i
$904$	−18.0000	−0.598671
$905$	0	0
$906$	16.0000	0.531564
$907$	− 16.0000i	− 0.531271i	−0.964073	−	0.265636i	$-0.914418\pi$
$907$	− 16.0000i	− 0.531271i	0.964073	−	0.265636i	$-0.0855818\pi$
$908$	12.0000i	0.398234i
$909$	12.0000	0.398015
$910$	0	0
$911$	38.0000	1.25900	0.629498	−	0.777002i	$-0.283261\pi$
$911$	38.0000	1.25900	0.629498	+	0.777002i	$0.283261\pi$
$912$	4.00000i	0.132453i
$913$	64.0000i	2.11809i
$914$	−18.0000	−0.595387
$915$	0	0
$916$	−26.0000	−0.859064
$917$	0	0
$918$	1.00000i	0.0330049i
$919$	−16.0000	−0.527791	−0.263896	−	0.964551i	$-0.585007\pi$
$919$	−16.0000	−0.527791	−0.263896	+	0.964551i	$0.585007\pi$
$920$	0	0
$921$	2.00000	0.0659022
$922$	− 24.0000i	− 0.790398i
$923$	0	0
$924$	8.00000	0.263181
$925$	0	0
$926$	4.00000	0.131448
$927$	− 20.0000i	− 0.656886i
$928$	6.00000i	0.196960i
$929$	32.0000	1.04989	0.524943	−	0.851137i	$-0.324087\pi$
$929$	32.0000	1.04989	0.524943	+	0.851137i	$0.324087\pi$
$930$	0	0
$931$	−12.0000	−0.393284
$932$	10.0000i	0.327561i
$933$	− 10.0000i	− 0.327385i
$934$	−28.0000	−0.916188
$935$	0	0
$936$	0	0
$937$	22.0000i	0.718709i	0.933201	+	0.359354i	$0.117003\pi$
$937$	22.0000i	0.718709i	−0.933201	+	0.359354i	$0.882997\pi$
$938$	− 4.00000i	− 0.130605i
$939$	8.00000	0.261070
$940$	0	0
$941$	−6.00000	−0.195594	−0.0977972	−	0.995206i	$-0.531180\pi$
$941$	−6.00000	−0.195594	−0.0977972	+	0.995206i	$0.531180\pi$
$942$	0	0
$943$	− 32.0000i	− 1.04206i
$944$	−6.00000	−0.195283
$945$	0	0
$946$	−8.00000	−0.260102
$947$	12.0000i	0.389948i	0.980808	+	0.194974i	$0.0624622\pi$
$947$	12.0000i	0.389948i	−0.980808	+	0.194974i	$0.937538\pi$
$948$	− 4.00000i	− 0.129914i
$949$	0	0
$950$	0	0
$951$	18.0000	0.583690
$952$	2.00000i	0.0648204i
$953$	54.0000i	1.74923i	0.484817	+	0.874616i	$0.338886\pi$
$953$	54.0000i	1.74923i	−0.484817	+	0.874616i	$0.661114\pi$
$954$	14.0000	0.453267
$955$	0	0
$956$	0	0
$957$	24.0000i	0.775810i
$958$	30.0000i	0.969256i
$959$	−12.0000	−0.387500
$960$	0	0
$961$	33.0000	1.06452
$962$	0	0
$963$	4.00000i	0.128898i
$964$	−10.0000	−0.322078
$965$	0	0
$966$	8.00000	0.257396
$967$	− 8.00000i	− 0.257263i	−0.991692	−	0.128631i	$-0.958942\pi$
$967$	− 8.00000i	− 0.257263i	0.991692	−	0.128631i	$-0.0410584\pi$
$968$	5.00000i	0.160706i
$969$	−4.00000	−0.128499
$970$	0	0
$971$	50.0000	1.60458	0.802288	−	0.596937i	$-0.203616\pi$
$971$	50.0000	1.60458	0.802288	+	0.596937i	$0.203616\pi$
$972$	1.00000i	0.0320750i
$973$	− 8.00000i	− 0.256468i
$974$	30.0000	0.961262
$975$	0	0
$976$	2.00000	0.0640184
$977$	− 42.0000i	− 1.34370i	−0.740688	−	0.671850i	$-0.765500\pi$
$977$	− 42.0000i	− 1.34370i	0.740688	−	0.671850i	$-0.234500\pi$
$978$	8.00000i	0.255812i
$979$	−24.0000	−0.767043
$980$	0	0
$981$	−2.00000	−0.0638551
$982$	6.00000i	0.191468i
$983$	48.0000i	1.53096i	0.643458	+	0.765481i	$0.277499\pi$
$983$	48.0000i	1.53096i	−0.643458	+	0.765481i	$0.722501\pi$
$984$	8.00000	0.255031
$985$	0	0
$986$	−6.00000	−0.191079
$987$	16.0000i	0.509286i
$988$	0	0
$989$	−8.00000	−0.254385
$990$	0	0
$991$	48.0000	1.52477	0.762385	−	0.647124i	$-0.224028\pi$
$991$	48.0000	1.52477	0.762385	+	0.647124i	$0.224028\pi$
$992$	8.00000i	0.254000i
$993$	0	0
$994$	20.0000	0.634361
$995$	0	0
$996$	−16.0000	−0.506979
$997$	− 62.0000i	− 1.96356i	−0.190022	−	0.981780i	$-0.560856\pi$
$997$	− 62.0000i	− 1.96356i	0.190022	−	0.981780i	$-0.439144\pi$
$998$	36.0000i	1.13956i
$999$	6.00000	0.189832

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2550.2.d.j.2449.1		2
5.2	odd	4		2550.2.a.y.1.1		1
5.3	odd	4		510.2.a.b.1.1	✓	1
5.4	even	2	inner	2550.2.d.j.2449.2		2
15.2	even	4		7650.2.a.y.1.1		1
15.8	even	4		1530.2.a.i.1.1		1
20.3	even	4		4080.2.a.n.1.1		1
85.33	odd	4		8670.2.a.c.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
510.2.a.b.1.1	✓	1	5.3	odd	4
1530.2.a.i.1.1		1	15.8	even	4
2550.2.a.y.1.1		1	5.2	odd	4
2550.2.d.j.2449.1		2	1.1	even	1	trivial
2550.2.d.j.2449.2		2	5.4	even	2	inner
4080.2.a.n.1.1		1	20.3	even	4
7650.2.a.y.1.1		1	15.2	even	4
8670.2.a.c.1.1		1	85.33	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:	\( N \)	\(=\)	\( 2550 = 2 \cdot 3 \cdot 5^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2550.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} -2.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} -2.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +4.00000 q^{11} +1.00000i q^{12} -2.00000 q^{14} +1.00000 q^{16} +1.00000i q^{17} +1.00000i q^{18} -4.00000 q^{19} -2.00000 q^{21} -4.00000i q^{22} -4.00000i q^{23} +1.00000 q^{24} +1.00000i q^{27} +2.00000i q^{28} -6.00000 q^{29} -8.00000 q^{31} -1.00000i q^{32} -4.00000i q^{33} +1.00000 q^{34} +1.00000 q^{36} -6.00000i q^{37} +4.00000i q^{38} +8.00000 q^{41} +2.00000i q^{42} -2.00000i q^{43} -4.00000 q^{44} -4.00000 q^{46} -8.00000i q^{47} -1.00000i q^{48} +3.00000 q^{49} +1.00000 q^{51} -14.0000i q^{53} +1.00000 q^{54} +2.00000 q^{56} +4.00000i q^{57} +6.00000i q^{58} -6.00000 q^{59} +2.00000 q^{61} +8.00000i q^{62} +2.00000i q^{63} -1.00000 q^{64} -4.00000 q^{66} +2.00000i q^{67} -1.00000i q^{68} -4.00000 q^{69} -10.0000 q^{71} -1.00000i q^{72} -4.00000i q^{73} -6.00000 q^{74} +4.00000 q^{76} -8.00000i q^{77} -4.00000 q^{79} +1.00000 q^{81} -8.00000i q^{82} +16.0000i q^{83} +2.00000 q^{84} -2.00000 q^{86} +6.00000i q^{87} +4.00000i q^{88} -6.00000 q^{89} +4.00000i q^{92} +8.00000i q^{93} -8.00000 q^{94} -1.00000 q^{96} -8.00000i q^{97} -3.00000i q^{98} -4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} - 2 q^{6} - 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^4 - 2 * q^6 - 2 * q^9	\( 2 q - 2 q^{4} - 2 q^{6} - 2 q^{9} + 8 q^{11} - 4 q^{14} + 2 q^{16} - 8 q^{19} - 4 q^{21} + 2 q^{24} - 12 q^{29} - 16 q^{31} + 2 q^{34} + 2 q^{36} + 16 q^{41} - 8 q^{44} - 8 q^{46} + 6 q^{49} + 2 q^{51} + 2 q^{54} + 4 q^{56} - 12 q^{59} + 4 q^{61} - 2 q^{64} - 8 q^{66} - 8 q^{69} - 20 q^{71} - 12 q^{74} + 8 q^{76} - 8 q^{79} + 2 q^{81} + 4 q^{84} - 4 q^{86} - 12 q^{89} - 16 q^{94} - 2 q^{96} - 8 q^{99}+O(q^{100}) \) 2 * q - 2 * q^4 - 2 * q^6 - 2 * q^9 + 8 * q^11 - 4 * q^14 + 2 * q^16 - 8 * q^19 - 4 * q^21 + 2 * q^24 - 12 * q^29 - 16 * q^31 + 2 * q^34 + 2 * q^36 + 16 * q^41 - 8 * q^44 - 8 * q^46 + 6 * q^49 + 2 * q^51 + 2 * q^54 + 4 * q^56 - 12 * q^59 + 4 * q^61 - 2 * q^64 - 8 * q^66 - 8 * q^69 - 20 * q^71 - 12 * q^74 + 8 * q^76 - 8 * q^79 + 2 * q^81 + 4 * q^84 - 4 * q^86 - 12 * q^89 - 16 * q^94 - 2 * q^96 - 8 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

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