LMFDB - Embedded newform 2550.2.d.g.2449.2

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 102)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

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

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} -2.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} -2.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} -1.00000i q^{12} +2.00000i q^{13} +2.00000 q^{14} +1.00000 q^{16} +1.00000i q^{17} -1.00000i q^{18} +4.00000 q^{19} +2.00000 q^{21} -6.00000i q^{23} +1.00000 q^{24} -2.00000 q^{26} -1.00000i q^{27} +2.00000i q^{28} -10.0000 q^{31} +1.00000i q^{32} -1.00000 q^{34} +1.00000 q^{36} -8.00000i q^{37} +4.00000i q^{38} -2.00000 q^{39} +6.00000 q^{41} +2.00000i q^{42} -4.00000i q^{43} +6.00000 q^{46} -12.0000i q^{47} +1.00000i q^{48} +3.00000 q^{49} -1.00000 q^{51} -2.00000i q^{52} +6.00000i q^{53} +1.00000 q^{54} -2.00000 q^{56} +4.00000i q^{57} +12.0000 q^{59} +8.00000 q^{61} -10.0000i q^{62} +2.00000i q^{63} -1.00000 q^{64} +4.00000i q^{67} -1.00000i q^{68} +6.00000 q^{69} +6.00000 q^{71} +1.00000i q^{72} +2.00000i q^{73} +8.00000 q^{74} -4.00000 q^{76} -2.00000i q^{78} +10.0000 q^{79} +1.00000 q^{81} +6.00000i q^{82} +12.0000i q^{83} -2.00000 q^{84} +4.00000 q^{86} +18.0000 q^{89} +4.00000 q^{91} +6.00000i q^{92} -10.0000i q^{93} +12.0000 q^{94} -1.00000 q^{96} -14.0000i q^{97} +3.00000i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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} + 4 q^{14} + 2 q^{16} + 8 q^{19} + 4 q^{21} + 2 q^{24} - 4 q^{26} - 20 q^{31} - 2 q^{34} + 2 q^{36} - 4 q^{39} + 12 q^{41} + 12 q^{46} + 6 q^{49} - 2 q^{51} + 2 q^{54} - 4 q^{56} + 24 q^{59} + 16 q^{61} - 2 q^{64} + 12 q^{69} + 12 q^{71} + 16 q^{74} - 8 q^{76} + 20 q^{79} + 2 q^{81} - 4 q^{84} + 8 q^{86} + 36 q^{89} + 8 q^{91} + 24 q^{94} - 2 q^{96}+O(q^{100}) $ 2 * q - 2 * q^4 - 2 * q^6 - 2 * q^9 + 4 * q^14 + 2 * q^16 + 8 * q^19 + 4 * q^21 + 2 * q^24 - 4 * q^26 - 20 * q^31 - 2 * q^34 + 2 * q^36 - 4 * q^39 + 12 * q^41 + 12 * q^46 + 6 * q^49 - 2 * q^51 + 2 * q^54 - 4 * q^56 + 24 * q^59 + 16 * q^61 - 2 * q^64 + 12 * q^69 + 12 * q^71 + 16 * q^74 - 8 * q^76 + 20 * q^79 + 2 * q^81 - 4 * q^84 + 8 * q^86 + 36 * q^89 + 8 * q^91 + 24 * q^94 - 2 * q^96

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
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	− 1.00000i	− 0.288675i
$13$	2.00000i	0.554700i	0.960769	+	0.277350i	$0.0894562\pi$
$13$	2.00000i	0.554700i	−0.960769	+	0.277350i	$0.910544\pi$
$14$	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	$-0.348268\pi$
$19$	4.00000	0.917663	0.458831	+	0.888523i	$0.348268\pi$
$20$	0	0
$21$	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	$-0.215123\pi$
$24$	1.00000	0.204124
$25$	0	0
$26$	−2.00000	−0.392232
$27$	− 1.00000i	− 0.192450i
$28$	2.00000i	0.377964i
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	−10.0000	−1.79605	−0.898027	−	0.439941i	$-0.854999\pi$
$31$	−10.0000	−1.79605	−0.898027	+	0.439941i	$0.854999\pi$
$32$	1.00000i	0.176777i
$33$	0	0
$34$	−1.00000	−0.171499
$35$	0	0
$36$	1.00000	0.166667
$37$	− 8.00000i	− 1.31519i	−0.753371	−	0.657596i	$-0.771573\pi$
$37$	− 8.00000i	− 1.31519i	0.753371	−	0.657596i	$-0.228427\pi$
$38$	4.00000i	0.648886i
$39$	−2.00000	−0.320256
$40$	0	0
$41$	6.00000	0.937043	0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	2.00000i	0.308607i
$43$	− 4.00000i	− 0.609994i	−0.952353	−	0.304997i	$-0.901344\pi$
$43$	− 4.00000i	− 0.609994i	0.952353	−	0.304997i	$-0.0986555\pi$
$44$	0	0
$45$	0	0
$46$	6.00000	0.884652
$47$	− 12.0000i	− 1.75038i	−0.483779	−	0.875190i	$-0.660736\pi$
$47$	− 12.0000i	− 1.75038i	0.483779	−	0.875190i	$-0.339264\pi$
$48$	1.00000i	0.144338i
$49$	3.00000	0.428571
$50$	0	0
$51$	−1.00000	−0.140028
$52$	− 2.00000i	− 0.277350i
$53$	6.00000i	0.824163i	0.911147	+	0.412082i	$0.135198\pi$
$53$	6.00000i	0.824163i	−0.911147	+	0.412082i	$0.864802\pi$
$54$	1.00000	0.136083
$55$	0	0
$56$	−2.00000	−0.267261
$57$	4.00000i	0.529813i
$58$	0	0
$59$	12.0000	1.56227	0.781133	−	0.624364i	$-0.214642\pi$
$59$	12.0000	1.56227	0.781133	+	0.624364i	$0.214642\pi$
$60$	0	0
$61$	8.00000	1.02430	0.512148	−	0.858898i	$-0.328850\pi$
$61$	8.00000	1.02430	0.512148	+	0.858898i	$0.328850\pi$
$62$	− 10.0000i	− 1.27000i
$63$	2.00000i	0.251976i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	0	0
$67$	4.00000i	0.488678i	0.969690	+	0.244339i	$0.0785709\pi$
$67$	4.00000i	0.488678i	−0.969690	+	0.244339i	$0.921429\pi$
$68$	− 1.00000i	− 0.121268i
$69$	6.00000	0.722315
$70$	0	0
$71$	6.00000	0.712069	0.356034	−	0.934473i	$-0.384129\pi$
$71$	6.00000	0.712069	0.356034	+	0.934473i	$0.384129\pi$
$72$	1.00000i	0.117851i
$73$	2.00000i	0.234082i	0.993127	+	0.117041i	$0.0373409\pi$
$73$	2.00000i	0.234082i	−0.993127	+	0.117041i	$0.962659\pi$
$74$	8.00000	0.929981
$75$	0	0
$76$	−4.00000	−0.458831
$77$	0	0
$78$	− 2.00000i	− 0.226455i
$79$	10.0000	1.12509	0.562544	−	0.826767i	$-0.309823\pi$
$79$	10.0000	1.12509	0.562544	+	0.826767i	$0.309823\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	6.00000i	0.662589i
$83$	12.0000i	1.31717i	0.752506	+	0.658586i	$0.228845\pi$
$83$	12.0000i	1.31717i	−0.752506	+	0.658586i	$0.771155\pi$
$84$	−2.00000	−0.218218
$85$	0	0
$86$	4.00000	0.431331
$87$	0	0
$88$	0	0
$89$	18.0000	1.90800	0.953998	−	0.299813i	$-0.0969242\pi$
$89$	18.0000	1.90800	0.953998	+	0.299813i	$0.0969242\pi$
$90$	0	0
$91$	4.00000	0.419314
$92$	6.00000i	0.625543i
$93$	− 10.0000i	− 1.03695i
$94$	12.0000	1.23771
$95$	0	0
$96$	−1.00000	−0.102062
$97$	− 14.0000i	− 1.42148i	−0.703452	−	0.710742i	$-0.748359\pi$
$97$	− 14.0000i	− 1.42148i	0.703452	−	0.710742i	$-0.251641\pi$
$98$	3.00000i	0.303046i
$99$	0	0
$100$	0	0
$101$	6.00000	0.597022	0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000	0.597022	0.298511	+	0.954406i	$0.403510\pi$
$102$	− 1.00000i	− 0.0990148i
$103$	− 4.00000i	− 0.394132i	−0.980390	−	0.197066i	$-0.936859\pi$
$103$	− 4.00000i	− 0.394132i	0.980390	−	0.197066i	$-0.0631413\pi$
$104$	2.00000	0.196116
$105$	0	0
$106$	−6.00000	−0.582772
$107$	0	0	1.00000			$0$
$107$	0	0	−1.00000			$\pi$
$108$	1.00000i	0.0962250i
$109$	−20.0000	−1.91565	−0.957826	−	0.287348i	$-0.907226\pi$
$109$	−20.0000	−1.91565	−0.957826	+	0.287348i	$0.907226\pi$
$110$	0	0
$111$	8.00000	0.759326
$112$	− 2.00000i	− 0.188982i
$113$	− 6.00000i	− 0.564433i	−0.959351	−	0.282216i	$-0.908930\pi$
$113$	− 6.00000i	− 0.564433i	0.959351	−	0.282216i	$-0.0910696\pi$
$114$	−4.00000	−0.374634
$115$	0	0
$116$	0	0
$117$	− 2.00000i	− 0.184900i
$118$	12.0000i	1.10469i
$119$	2.00000	0.183340
$120$	0	0
$121$	−11.0000	−1.00000
$122$	8.00000i	0.724286i
$123$	6.00000i	0.541002i
$124$	10.0000	0.898027
$125$	0	0
$126$	−2.00000	−0.178174
$127$	− 8.00000i	− 0.709885i	−0.934888	−	0.354943i	$-0.884500\pi$
$127$	− 8.00000i	− 0.709885i	0.934888	−	0.354943i	$-0.115500\pi$
$128$	− 1.00000i	− 0.0883883i
$129$	4.00000	0.352180
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	− 8.00000i	− 0.693688i
$134$	−4.00000	−0.345547
$135$	0	0
$136$	1.00000	0.0857493
$137$	6.00000i	0.512615i	0.966595	+	0.256307i	$0.0825059\pi$
$137$	6.00000i	0.512615i	−0.966595	+	0.256307i	$0.917494\pi$
$138$	6.00000i	0.510754i
$139$	−8.00000	−0.678551	−0.339276	−	0.940687i	$-0.610182\pi$
$139$	−8.00000	−0.678551	−0.339276	+	0.940687i	$0.610182\pi$
$140$	0	0
$141$	12.0000	1.01058
$142$	6.00000i	0.503509i
$143$	0	0
$144$	−1.00000	−0.0833333
$145$	0	0
$146$	−2.00000	−0.165521
$147$	3.00000i	0.247436i
$148$	8.00000i	0.657596i
$149$	6.00000	0.491539	0.245770	−	0.969328i	$-0.420959\pi$
$149$	6.00000	0.491539	0.245770	+	0.969328i	$0.420959\pi$
$150$	0	0
$151$	8.00000	0.651031	0.325515	−	0.945537i	$-0.394462\pi$
$151$	8.00000	0.651031	0.325515	+	0.945537i	$0.394462\pi$
$152$	− 4.00000i	− 0.324443i
$153$	− 1.00000i	− 0.0808452i
$154$	0	0
$155$	0	0
$156$	2.00000	0.160128
$157$	10.0000i	0.798087i	0.916932	+	0.399043i	$0.130658\pi$
$157$	10.0000i	0.798087i	−0.916932	+	0.399043i	$0.869342\pi$
$158$	10.0000i	0.795557i
$159$	−6.00000	−0.475831
$160$	0	0
$161$	−12.0000	−0.945732
$162$	1.00000i	0.0785674i
$163$	20.0000i	1.56652i	0.621694	+	0.783260i	$0.286445\pi$
$163$	20.0000i	1.56652i	−0.621694	+	0.783260i	$0.713555\pi$
$164$	−6.00000	−0.468521
$165$	0	0
$166$	−12.0000	−0.931381
$167$	− 18.0000i	− 1.39288i	−0.717614	−	0.696441i	$-0.754766\pi$
$167$	− 18.0000i	− 1.39288i	0.717614	−	0.696441i	$-0.245234\pi$
$168$	− 2.00000i	− 0.154303i
$169$	9.00000	0.692308
$170$	0	0
$171$	−4.00000	−0.305888
$172$	4.00000i	0.304997i
$173$	0	0	1.00000			$0$
$173$	0	0	−1.00000			$\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	12.0000i	0.901975i
$178$	18.0000i	1.34916i
$179$	12.0000	0.896922	0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000	0.896922	0.448461	+	0.893802i	$0.351972\pi$
$180$	0	0
$181$	8.00000	0.594635	0.297318	−	0.954779i	$-0.403908\pi$
$181$	8.00000	0.594635	0.297318	+	0.954779i	$0.403908\pi$
$182$	4.00000i	0.296500i
$183$	8.00000i	0.591377i
$184$	−6.00000	−0.442326
$185$	0	0
$186$	10.0000	0.733236
$187$	0	0
$188$	12.0000i	0.875190i
$189$	−2.00000	−0.145479
$190$	0	0
$191$	−12.0000	−0.868290	−0.434145	−	0.900843i	$-0.642949\pi$
$191$	−12.0000	−0.868290	−0.434145	+	0.900843i	$0.642949\pi$
$192$	− 1.00000i	− 0.0721688i
$193$	14.0000i	1.00774i	0.863779	+	0.503871i	$0.168091\pi$
$193$	14.0000i	1.00774i	−0.863779	+	0.503871i	$0.831909\pi$
$194$	14.0000	1.00514
$195$	0	0
$196$	−3.00000	−0.214286
$197$	12.0000i	0.854965i	0.904024	+	0.427482i	$0.140599\pi$
$197$	12.0000i	0.854965i	−0.904024	+	0.427482i	$0.859401\pi$
$198$	0	0
$199$	−2.00000	−0.141776	−0.0708881	−	0.997484i	$-0.522583\pi$
$199$	−2.00000	−0.141776	−0.0708881	+	0.997484i	$0.522583\pi$
$200$	0	0
$201$	−4.00000	−0.282138
$202$	6.00000i	0.422159i
$203$	0	0
$204$	1.00000	0.0700140
$205$	0	0
$206$	4.00000	0.278693
$207$	6.00000i	0.417029i
$208$	2.00000i	0.138675i
$209$	0	0
$210$	0	0
$211$	8.00000	0.550743	0.275371	−	0.961338i	$-0.411199\pi$
$211$	8.00000	0.550743	0.275371	+	0.961338i	$0.411199\pi$
$212$	− 6.00000i	− 0.412082i
$213$	6.00000i	0.411113i
$214$	0	0
$215$	0	0
$216$	−1.00000	−0.0680414
$217$	20.0000i	1.35769i
$218$	− 20.0000i	− 1.35457i
$219$	−2.00000	−0.135147
$220$	0	0
$221$	−2.00000	−0.134535
$222$	8.00000i	0.536925i
$223$	− 28.0000i	− 1.87502i	−0.347960	−	0.937509i	$-0.613126\pi$
$223$	− 28.0000i	− 1.87502i	0.347960	−	0.937509i	$-0.386874\pi$
$224$	2.00000	0.133631
$225$	0	0
$226$	6.00000	0.399114
$227$	− 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
$229$	−14.0000	−0.925146	−0.462573	−	0.886581i	$-0.653074\pi$
$229$	−14.0000	−0.925146	−0.462573	+	0.886581i	$0.653074\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 30.0000i	− 1.96537i	−0.185296	−	0.982683i	$-0.559325\pi$
$233$	− 30.0000i	− 1.96537i	0.185296	−	0.982683i	$-0.440675\pi$
$234$	2.00000	0.130744
$235$	0	0
$236$	−12.0000	−0.781133
$237$	10.0000i	0.649570i
$238$	2.00000i	0.129641i
$239$	12.0000	0.776215	0.388108	−	0.921614i	$-0.373129\pi$
$239$	12.0000	0.776215	0.388108	+	0.921614i	$0.373129\pi$
$240$	0	0
$241$	−10.0000	−0.644157	−0.322078	−	0.946713i	$-0.604381\pi$
$241$	−10.0000	−0.644157	−0.322078	+	0.946713i	$0.604381\pi$
$242$	− 11.0000i	− 0.707107i
$243$	1.00000i	0.0641500i
$244$	−8.00000	−0.512148
$245$	0	0
$246$	−6.00000	−0.382546
$247$	8.00000i	0.509028i
$248$	10.0000i	0.635001i
$249$	−12.0000	−0.760469
$250$	0	0
$251$	12.0000	0.757433	0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000	0.757433	0.378717	+	0.925513i	$0.376365\pi$
$252$	− 2.00000i	− 0.125988i
$253$	0	0
$254$	8.00000	0.501965
$255$	0	0
$256$	1.00000	0.0625000
$257$	− 18.0000i	− 1.12281i	−0.827541	−	0.561405i	$-0.810261\pi$
$257$	− 18.0000i	− 1.12281i	0.827541	−	0.561405i	$-0.189739\pi$
$258$	4.00000i	0.249029i
$259$	−16.0000	−0.994192
$260$	0	0
$261$	0	0
$262$	0	0
$263$	12.0000i	0.739952i	0.929041	+	0.369976i	$0.120634\pi$
$263$	12.0000i	0.739952i	−0.929041	+	0.369976i	$0.879366\pi$
$264$	0	0
$265$	0	0
$266$	8.00000	0.490511
$267$	18.0000i	1.10158i
$268$	− 4.00000i	− 0.244339i
$269$	−24.0000	−1.46331	−0.731653	−	0.681677i	$-0.761251\pi$
$269$	−24.0000	−1.46331	−0.731653	+	0.681677i	$0.761251\pi$
$270$	0	0
$271$	−16.0000	−0.971931	−0.485965	−	0.873978i	$-0.661532\pi$
$271$	−16.0000	−0.971931	−0.485965	+	0.873978i	$0.661532\pi$
$272$	1.00000i	0.0606339i
$273$	4.00000i	0.242091i
$274$	−6.00000	−0.362473
$275$	0	0
$276$	−6.00000	−0.361158
$277$	4.00000i	0.240337i	0.992754	+	0.120168i	$0.0383434\pi$
$277$	4.00000i	0.240337i	−0.992754	+	0.120168i	$0.961657\pi$
$278$	− 8.00000i	− 0.479808i
$279$	10.0000	0.598684
$280$	0	0
$281$	30.0000	1.78965	0.894825	−	0.446417i	$-0.147300\pi$
$281$	30.0000	1.78965	0.894825	+	0.446417i	$0.147300\pi$
$282$	12.0000i	0.714590i
$283$	− 16.0000i	− 0.951101i	−0.879688	−	0.475551i	$-0.842249\pi$
$283$	− 16.0000i	− 0.951101i	0.879688	−	0.475551i	$-0.157751\pi$
$284$	−6.00000	−0.356034
$285$	0	0
$286$	0	0
$287$	− 12.0000i	− 0.708338i
$288$	− 1.00000i	− 0.0589256i
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	14.0000	0.820695
$292$	− 2.00000i	− 0.117041i
$293$	18.0000i	1.05157i	0.850617	+	0.525786i	$0.176229\pi$
$293$	18.0000i	1.05157i	−0.850617	+	0.525786i	$0.823771\pi$
$294$	−3.00000	−0.174964
$295$	0	0
$296$	−8.00000	−0.464991
$297$	0	0
$298$	6.00000i	0.347571i
$299$	12.0000	0.693978
$300$	0	0
$301$	−8.00000	−0.461112
$302$	8.00000i	0.460348i
$303$	6.00000i	0.344691i
$304$	4.00000	0.229416
$305$	0	0
$306$	1.00000	0.0571662
$307$	− 20.0000i	− 1.14146i	−0.821138	−	0.570730i	$-0.806660\pi$
$307$	− 20.0000i	− 1.14146i	0.821138	−	0.570730i	$-0.193340\pi$
$308$	0	0
$309$	4.00000	0.227552
$310$	0	0
$311$	18.0000	1.02069	0.510343	−	0.859971i	$-0.329518\pi$
$311$	18.0000	1.02069	0.510343	+	0.859971i	$0.329518\pi$
$312$	2.00000i	0.113228i
$313$	14.0000i	0.791327i	0.918396	+	0.395663i	$0.129485\pi$
$313$	14.0000i	0.791327i	−0.918396	+	0.395663i	$0.870515\pi$
$314$	−10.0000	−0.564333
$315$	0	0
$316$	−10.0000	−0.562544
$317$	− 12.0000i	− 0.673987i	−0.941507	−	0.336994i	$-0.890590\pi$
$317$	− 12.0000i	− 0.673987i	0.941507	−	0.336994i	$-0.109410\pi$
$318$	− 6.00000i	− 0.336463i
$319$	0	0
$320$	0	0
$321$	0	0
$322$	− 12.0000i	− 0.668734i
$323$	4.00000i	0.222566i
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	−20.0000	−1.10770
$327$	− 20.0000i	− 1.10600i
$328$	− 6.00000i	− 0.331295i
$329$	−24.0000	−1.32316
$330$	0	0
$331$	20.0000	1.09930	0.549650	−	0.835395i	$-0.314761\pi$
$331$	20.0000	1.09930	0.549650	+	0.835395i	$0.314761\pi$
$332$	− 12.0000i	− 0.658586i
$333$	8.00000i	0.438397i
$334$	18.0000	0.984916
$335$	0	0
$336$	2.00000	0.109109
$337$	22.0000i	1.19842i	0.800593	+	0.599208i	$0.204518\pi$
$337$	22.0000i	1.19842i	−0.800593	+	0.599208i	$0.795482\pi$
$338$	9.00000i	0.489535i
$339$	6.00000	0.325875
$340$	0	0
$341$	0	0
$342$	− 4.00000i	− 0.216295i
$343$	− 20.0000i	− 1.07990i
$344$	−4.00000	−0.215666
$345$	0	0
$346$	0	0
$347$	12.0000i	0.644194i	0.946707	+	0.322097i	$0.104388\pi$
$347$	12.0000i	0.644194i	−0.946707	+	0.322097i	$0.895612\pi$
$348$	0	0
$349$	−26.0000	−1.39175	−0.695874	−	0.718164i	$-0.744983\pi$
$349$	−26.0000	−1.39175	−0.695874	+	0.718164i	$0.744983\pi$
$350$	0	0
$351$	2.00000	0.106752
$352$	0	0
$353$	− 18.0000i	− 0.958043i	−0.877803	−	0.479022i	$-0.840992\pi$
$353$	− 18.0000i	− 0.958043i	0.877803	−	0.479022i	$-0.159008\pi$
$354$	−12.0000	−0.637793
$355$	0	0
$356$	−18.0000	−0.953998
$357$	2.00000i	0.105851i
$358$	12.0000i	0.634220i
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	8.00000i	0.420471i
$363$	− 11.0000i	− 0.577350i
$364$	−4.00000	−0.209657
$365$	0	0
$366$	−8.00000	−0.418167
$367$	10.0000i	0.521996i	0.965339	+	0.260998i	$0.0840516\pi$
$367$	10.0000i	0.521996i	−0.965339	+	0.260998i	$0.915948\pi$
$368$	− 6.00000i	− 0.312772i
$369$	−6.00000	−0.312348
$370$	0	0
$371$	12.0000	0.623009
$372$	10.0000i	0.518476i
$373$	− 22.0000i	− 1.13912i	−0.821951	−	0.569558i	$-0.807114\pi$
$373$	− 22.0000i	− 1.13912i	0.821951	−	0.569558i	$-0.192886\pi$
$374$	0	0
$375$	0	0
$376$	−12.0000	−0.618853
$377$	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
$381$	8.00000	0.409852
$382$	− 12.0000i	− 0.613973i
$383$	− 12.0000i	− 0.613171i	−0.951843	−	0.306586i	$-0.900813\pi$
$383$	− 12.0000i	− 0.613171i	0.951843	−	0.306586i	$-0.0991866\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	−14.0000	−0.712581
$387$	4.00000i	0.203331i
$388$	14.0000i	0.710742i
$389$	30.0000	1.52106	0.760530	−	0.649303i	$-0.224939\pi$
$389$	30.0000	1.52106	0.760530	+	0.649303i	$0.224939\pi$
$390$	0	0
$391$	6.00000	0.303433
$392$	− 3.00000i	− 0.151523i
$393$	0	0
$394$	−12.0000	−0.604551
$395$	0	0
$396$	0	0
$397$	16.0000i	0.803017i	0.915855	+	0.401508i	$0.131514\pi$
$397$	16.0000i	0.803017i	−0.915855	+	0.401508i	$0.868486\pi$
$398$	− 2.00000i	− 0.100251i
$399$	8.00000	0.400501
$400$	0	0
$401$	−18.0000	−0.898877	−0.449439	−	0.893311i	$-0.648376\pi$
$401$	−18.0000	−0.898877	−0.449439	+	0.893311i	$0.648376\pi$
$402$	− 4.00000i	− 0.199502i
$403$	− 20.0000i	− 0.996271i
$404$	−6.00000	−0.298511
$405$	0	0
$406$	0	0
$407$	0	0
$408$	1.00000i	0.0495074i
$409$	−38.0000	−1.87898	−0.939490	−	0.342578i	$-0.888700\pi$
$409$	−38.0000	−1.87898	−0.939490	+	0.342578i	$0.888700\pi$
$410$	0	0
$411$	−6.00000	−0.295958
$412$	4.00000i	0.197066i
$413$	− 24.0000i	− 1.18096i
$414$	−6.00000	−0.294884
$415$	0	0
$416$	−2.00000	−0.0980581
$417$	− 8.00000i	− 0.391762i
$418$	0	0
$419$	36.0000	1.75872	0.879358	−	0.476162i	$-0.157972\pi$
$419$	36.0000	1.75872	0.879358	+	0.476162i	$0.157972\pi$
$420$	0	0
$421$	−22.0000	−1.07221	−0.536107	−	0.844150i	$-0.680106\pi$
$421$	−22.0000	−1.07221	−0.536107	+	0.844150i	$0.680106\pi$
$422$	8.00000i	0.389434i
$423$	12.0000i	0.583460i
$424$	6.00000	0.291386
$425$	0	0
$426$	−6.00000	−0.290701
$427$	− 16.0000i	− 0.774294i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−30.0000	−1.44505	−0.722525	−	0.691345i	$-0.757018\pi$
$431$	−30.0000	−1.44505	−0.722525	+	0.691345i	$0.757018\pi$
$432$	− 1.00000i	− 0.0481125i
$433$	14.0000i	0.672797i	0.941720	+	0.336399i	$0.109209\pi$
$433$	14.0000i	0.672797i	−0.941720	+	0.336399i	$0.890791\pi$
$434$	−20.0000	−0.960031
$435$	0	0
$436$	20.0000	0.957826
$437$	− 24.0000i	− 1.14808i
$438$	− 2.00000i	− 0.0955637i
$439$	−26.0000	−1.24091	−0.620456	−	0.784241i	$-0.713053\pi$
$439$	−26.0000	−1.24091	−0.620456	+	0.784241i	$0.713053\pi$
$440$	0	0
$441$	−3.00000	−0.142857
$442$	− 2.00000i	− 0.0951303i
$443$	− 36.0000i	− 1.71041i	−0.518289	−	0.855206i	$-0.673431\pi$
$443$	− 36.0000i	− 1.71041i	0.518289	−	0.855206i	$-0.326569\pi$
$444$	−8.00000	−0.379663
$445$	0	0
$446$	28.0000	1.32584
$447$	6.00000i	0.283790i
$448$	2.00000i	0.0944911i
$449$	−6.00000	−0.283158	−0.141579	−	0.989927i	$-0.545218\pi$
$449$	−6.00000	−0.283158	−0.141579	+	0.989927i	$0.545218\pi$
$450$	0	0
$451$	0	0
$452$	6.00000i	0.282216i
$453$	8.00000i	0.375873i
$454$	12.0000	0.563188
$455$	0	0
$456$	4.00000	0.187317
$457$	10.0000i	0.467780i	0.972263	+	0.233890i	$0.0751456\pi$
$457$	10.0000i	0.467780i	−0.972263	+	0.233890i	$0.924854\pi$
$458$	− 14.0000i	− 0.654177i
$459$	1.00000	0.0466760
$460$	0	0
$461$	30.0000	1.39724	0.698620	−	0.715493i	$-0.253798\pi$
$461$	30.0000	1.39724	0.698620	+	0.715493i	$0.253798\pi$
$462$	0	0
$463$	20.0000i	0.929479i	0.885448	+	0.464739i	$0.153852\pi$
$463$	20.0000i	0.929479i	−0.885448	+	0.464739i	$0.846148\pi$
$464$	0	0
$465$	0	0
$466$	30.0000	1.38972
$467$	− 12.0000i	− 0.555294i	−0.960683	−	0.277647i	$-0.910445\pi$
$467$	− 12.0000i	− 0.555294i	0.960683	−	0.277647i	$-0.0895545\pi$
$468$	2.00000i	0.0924500i
$469$	8.00000	0.369406
$470$	0	0
$471$	−10.0000	−0.460776
$472$	− 12.0000i	− 0.552345i
$473$	0	0
$474$	−10.0000	−0.459315
$475$	0	0
$476$	−2.00000	−0.0916698
$477$	− 6.00000i	− 0.274721i
$478$	12.0000i	0.548867i
$479$	−6.00000	−0.274147	−0.137073	−	0.990561i	$-0.543770\pi$
$479$	−6.00000	−0.274147	−0.137073	+	0.990561i	$0.543770\pi$
$480$	0	0
$481$	16.0000	0.729537
$482$	− 10.0000i	− 0.455488i
$483$	− 12.0000i	− 0.546019i
$484$	11.0000	0.500000
$485$	0	0
$486$	−1.00000	−0.0453609
$487$	10.0000i	0.453143i	0.973995	+	0.226572i	$0.0727517\pi$
$487$	10.0000i	0.453143i	−0.973995	+	0.226572i	$0.927248\pi$
$488$	− 8.00000i	− 0.362143i
$489$	−20.0000	−0.904431
$490$	0	0
$491$	−36.0000	−1.62466	−0.812329	−	0.583200i	$-0.801800\pi$
$491$	−36.0000	−1.62466	−0.812329	+	0.583200i	$0.801800\pi$
$492$	− 6.00000i	− 0.270501i
$493$	0	0
$494$	−8.00000	−0.359937
$495$	0	0
$496$	−10.0000	−0.449013
$497$	− 12.0000i	− 0.538274i
$498$	− 12.0000i	− 0.537733i
$499$	−32.0000	−1.43252	−0.716258	−	0.697835i	$-0.754147\pi$
$499$	−32.0000	−1.43252	−0.716258	+	0.697835i	$0.754147\pi$
$500$	0	0
$501$	18.0000	0.804181
$502$	12.0000i	0.535586i
$503$	− 6.00000i	− 0.267527i	−0.991013	−	0.133763i	$-0.957294\pi$
$503$	− 6.00000i	− 0.267527i	0.991013	−	0.133763i	$-0.0427062\pi$
$504$	2.00000	0.0890871
$505$	0	0
$506$	0	0
$507$	9.00000i	0.399704i
$508$	8.00000i	0.354943i
$509$	6.00000	0.265945	0.132973	−	0.991120i	$-0.457548\pi$
$509$	6.00000	0.265945	0.132973	+	0.991120i	$0.457548\pi$
$510$	0	0
$511$	4.00000	0.176950
$512$	1.00000i	0.0441942i
$513$	− 4.00000i	− 0.176604i
$514$	18.0000	0.793946
$515$	0	0
$516$	−4.00000	−0.176090
$517$	0	0
$518$	− 16.0000i	− 0.703000i
$519$	0	0
$520$	0	0
$521$	6.00000	0.262865	0.131432	−	0.991325i	$-0.458042\pi$
$521$	6.00000	0.262865	0.131432	+	0.991325i	$0.458042\pi$
$522$	0	0
$523$	20.0000i	0.874539i	0.899331	+	0.437269i	$0.144054\pi$
$523$	20.0000i	0.874539i	−0.899331	+	0.437269i	$0.855946\pi$
$524$	0	0
$525$	0	0
$526$	−12.0000	−0.523225
$527$	− 10.0000i	− 0.435607i
$528$	0	0
$529$	−13.0000	−0.565217
$530$	0	0
$531$	−12.0000	−0.520756
$532$	8.00000i	0.346844i
$533$	12.0000i	0.519778i
$534$	−18.0000	−0.778936
$535$	0	0
$536$	4.00000	0.172774
$537$	12.0000i	0.517838i
$538$	− 24.0000i	− 1.03471i
$539$	0	0
$540$	0	0
$541$	−16.0000	−0.687894	−0.343947	−	0.938989i	$-0.611764\pi$
$541$	−16.0000	−0.687894	−0.343947	+	0.938989i	$0.611764\pi$
$542$	− 16.0000i	− 0.687259i
$543$	8.00000i	0.343313i
$544$	−1.00000	−0.0428746
$545$	0	0
$546$	−4.00000	−0.171184
$547$	− 8.00000i	− 0.342055i	−0.985266	−	0.171028i	$-0.945291\pi$
$547$	− 8.00000i	− 0.342055i	0.985266	−	0.171028i	$-0.0547087\pi$
$548$	− 6.00000i	− 0.256307i
$549$	−8.00000	−0.341432
$550$	0	0
$551$	0	0
$552$	− 6.00000i	− 0.255377i
$553$	− 20.0000i	− 0.850487i
$554$	−4.00000	−0.169944
$555$	0	0
$556$	8.00000	0.339276
$557$	42.0000i	1.77960i	0.456354	+	0.889799i	$0.349155\pi$
$557$	42.0000i	1.77960i	−0.456354	+	0.889799i	$0.650845\pi$
$558$	10.0000i	0.423334i
$559$	8.00000	0.338364
$560$	0	0
$561$	0	0
$562$	30.0000i	1.26547i
$563$	− 12.0000i	− 0.505740i	−0.967500	−	0.252870i	$-0.918626\pi$
$563$	− 12.0000i	− 0.505740i	0.967500	−	0.252870i	$-0.0813744\pi$
$564$	−12.0000	−0.505291
$565$	0	0
$566$	16.0000	0.672530
$567$	− 2.00000i	− 0.0839921i
$568$	− 6.00000i	− 0.251754i
$569$	6.00000	0.251533	0.125767	−	0.992060i	$-0.459861\pi$
$569$	6.00000	0.251533	0.125767	+	0.992060i	$0.459861\pi$
$570$	0	0
$571$	20.0000	0.836974	0.418487	−	0.908223i	$-0.362561\pi$
$571$	20.0000	0.836974	0.418487	+	0.908223i	$0.362561\pi$
$572$	0	0
$573$	− 12.0000i	− 0.501307i
$574$	12.0000	0.500870
$575$	0	0
$576$	1.00000	0.0416667
$577$	− 2.00000i	− 0.0832611i	−0.999133	−	0.0416305i	$-0.986745\pi$
$577$	− 2.00000i	− 0.0832611i	0.999133	−	0.0416305i	$-0.0132552\pi$
$578$	− 1.00000i	− 0.0415945i
$579$	−14.0000	−0.581820
$580$	0	0
$581$	24.0000	0.995688
$582$	14.0000i	0.580319i
$583$	0	0
$584$	2.00000	0.0827606
$585$	0	0
$586$	−18.0000	−0.743573
$587$	12.0000i	0.495293i	0.968850	+	0.247647i	$0.0796572\pi$
$587$	12.0000i	0.495293i	−0.968850	+	0.247647i	$0.920343\pi$
$588$	− 3.00000i	− 0.123718i
$589$	−40.0000	−1.64817
$590$	0	0
$591$	−12.0000	−0.493614
$592$	− 8.00000i	− 0.328798i
$593$	18.0000i	0.739171i	0.929197	+	0.369586i	$0.120500\pi$
$593$	18.0000i	0.739171i	−0.929197	+	0.369586i	$0.879500\pi$
$594$	0	0
$595$	0	0
$596$	−6.00000	−0.245770
$597$	− 2.00000i	− 0.0818546i
$598$	12.0000i	0.490716i
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	0	0
$601$	−22.0000	−0.897399	−0.448699	−	0.893683i	$-0.648113\pi$
$601$	−22.0000	−0.897399	−0.448699	+	0.893683i	$0.648113\pi$
$602$	− 8.00000i	− 0.326056i
$603$	− 4.00000i	− 0.162893i
$604$	−8.00000	−0.325515
$605$	0	0
$606$	−6.00000	−0.243733
$607$	10.0000i	0.405887i	0.979190	+	0.202944i	$0.0650509\pi$
$607$	10.0000i	0.405887i	−0.979190	+	0.202944i	$0.934949\pi$
$608$	4.00000i	0.162221i
$609$	0	0
$610$	0	0
$611$	24.0000	0.970936
$612$	1.00000i	0.0404226i
$613$	− 34.0000i	− 1.37325i	−0.727013	−	0.686624i	$-0.759092\pi$
$613$	− 34.0000i	− 1.37325i	0.727013	−	0.686624i	$-0.240908\pi$
$614$	20.0000	0.807134
$615$	0	0
$616$	0	0
$617$	− 18.0000i	− 0.724653i	−0.932051	−	0.362326i	$-0.881983\pi$
$617$	− 18.0000i	− 0.724653i	0.932051	−	0.362326i	$-0.118017\pi$
$618$	4.00000i	0.160904i
$619$	28.0000	1.12542	0.562708	−	0.826656i	$-0.309760\pi$
$619$	28.0000	1.12542	0.562708	+	0.826656i	$0.309760\pi$
$620$	0	0
$621$	−6.00000	−0.240772
$622$	18.0000i	0.721734i
$623$	− 36.0000i	− 1.44231i
$624$	−2.00000	−0.0800641
$625$	0	0
$626$	−14.0000	−0.559553
$627$	0	0
$628$	− 10.0000i	− 0.399043i
$629$	8.00000	0.318981
$630$	0	0
$631$	−4.00000	−0.159237	−0.0796187	−	0.996825i	$-0.525370\pi$
$631$	−4.00000	−0.159237	−0.0796187	+	0.996825i	$0.525370\pi$
$632$	− 10.0000i	− 0.397779i
$633$	8.00000i	0.317971i
$634$	12.0000	0.476581
$635$	0	0
$636$	6.00000	0.237915
$637$	6.00000i	0.237729i
$638$	0	0
$639$	−6.00000	−0.237356
$640$	0	0
$641$	30.0000	1.18493	0.592464	−	0.805597i	$-0.298155\pi$
$641$	30.0000	1.18493	0.592464	+	0.805597i	$0.298155\pi$
$642$	0	0
$643$	− 4.00000i	− 0.157745i	−0.996885	−	0.0788723i	$-0.974868\pi$
$643$	− 4.00000i	− 0.157745i	0.996885	−	0.0788723i	$-0.0251319\pi$
$644$	12.0000	0.472866
$645$	0	0
$646$	−4.00000	−0.157378
$647$	− 36.0000i	− 1.41531i	−0.706560	−	0.707653i	$-0.749754\pi$
$647$	− 36.0000i	− 1.41531i	0.706560	−	0.707653i	$-0.250246\pi$
$648$	− 1.00000i	− 0.0392837i
$649$	0	0
$650$	0	0
$651$	−20.0000	−0.783862
$652$	− 20.0000i	− 0.783260i
$653$	− 24.0000i	− 0.939193i	−0.882881	−	0.469596i	$-0.844399\pi$
$653$	− 24.0000i	− 0.939193i	0.882881	−	0.469596i	$-0.155601\pi$
$654$	20.0000	0.782062
$655$	0	0
$656$	6.00000	0.234261
$657$	− 2.00000i	− 0.0780274i
$658$	− 24.0000i	− 0.935617i
$659$	−36.0000	−1.40236	−0.701180	−	0.712984i	$-0.747343\pi$
$659$	−36.0000	−1.40236	−0.701180	+	0.712984i	$0.747343\pi$
$660$	0	0
$661$	−22.0000	−0.855701	−0.427850	−	0.903850i	$-0.640729\pi$
$661$	−22.0000	−0.855701	−0.427850	+	0.903850i	$0.640729\pi$
$662$	20.0000i	0.777322i
$663$	− 2.00000i	− 0.0776736i
$664$	12.0000	0.465690
$665$	0	0
$666$	−8.00000	−0.309994
$667$	0	0
$668$	18.0000i	0.696441i
$669$	28.0000	1.08254
$670$	0	0
$671$	0	0
$672$	2.00000i	0.0771517i
$673$	− 34.0000i	− 1.31060i	−0.755367	−	0.655302i	$-0.772541\pi$
$673$	− 34.0000i	− 1.31060i	0.755367	−	0.655302i	$-0.227459\pi$
$674$	−22.0000	−0.847408
$675$	0	0
$676$	−9.00000	−0.346154
$677$	12.0000i	0.461197i	0.973049	+	0.230599i	$0.0740685\pi$
$677$	12.0000i	0.461197i	−0.973049	+	0.230599i	$0.925932\pi$
$678$	6.00000i	0.230429i
$679$	−28.0000	−1.07454
$680$	0	0
$681$	12.0000	0.459841
$682$	0	0
$683$	− 12.0000i	− 0.459167i	−0.973289	−	0.229584i	$-0.926264\pi$
$683$	− 12.0000i	− 0.459167i	0.973289	−	0.229584i	$-0.0737364\pi$
$684$	4.00000	0.152944
$685$	0	0
$686$	20.0000	0.763604
$687$	− 14.0000i	− 0.534133i
$688$	− 4.00000i	− 0.152499i
$689$	−12.0000	−0.457164
$690$	0	0
$691$	32.0000	1.21734	0.608669	−	0.793424i	$-0.291704\pi$
$691$	32.0000	1.21734	0.608669	+	0.793424i	$0.291704\pi$
$692$	0	0
$693$	0	0
$694$	−12.0000	−0.455514
$695$	0	0
$696$	0	0
$697$	6.00000i	0.227266i
$698$	− 26.0000i	− 0.984115i
$699$	30.0000	1.13470
$700$	0	0
$701$	−42.0000	−1.58632	−0.793159	−	0.609015i	$-0.791565\pi$
$701$	−42.0000	−1.58632	−0.793159	+	0.609015i	$0.791565\pi$
$702$	2.00000i	0.0754851i
$703$	− 32.0000i	− 1.20690i
$704$	0	0
$705$	0	0
$706$	18.0000	0.677439
$707$	− 12.0000i	− 0.451306i
$708$	− 12.0000i	− 0.450988i
$709$	−20.0000	−0.751116	−0.375558	−	0.926799i	$-0.622549\pi$
$709$	−20.0000	−0.751116	−0.375558	+	0.926799i	$0.622549\pi$
$710$	0	0
$711$	−10.0000	−0.375029
$712$	− 18.0000i	− 0.674579i
$713$	60.0000i	2.24702i
$714$	−2.00000	−0.0748481
$715$	0	0
$716$	−12.0000	−0.448461
$717$	12.0000i	0.448148i
$718$	0	0
$719$	−6.00000	−0.223762	−0.111881	−	0.993722i	$-0.535688\pi$
$719$	−6.00000	−0.223762	−0.111881	+	0.993722i	$0.535688\pi$
$720$	0	0
$721$	−8.00000	−0.297936
$722$	− 3.00000i	− 0.111648i
$723$	− 10.0000i	− 0.371904i
$724$	−8.00000	−0.297318
$725$	0	0
$726$	11.0000	0.408248
$727$	− 8.00000i	− 0.296704i	−0.988935	−	0.148352i	$-0.952603\pi$
$727$	− 8.00000i	− 0.296704i	0.988935	−	0.148352i	$-0.0473968\pi$
$728$	− 4.00000i	− 0.148250i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	4.00000	0.147945
$732$	− 8.00000i	− 0.295689i
$733$	26.0000i	0.960332i	0.877178	+	0.480166i	$0.159424\pi$
$733$	26.0000i	0.960332i	−0.877178	+	0.480166i	$0.840576\pi$
$734$	−10.0000	−0.369107
$735$	0	0
$736$	6.00000	0.221163
$737$	0	0
$738$	− 6.00000i	− 0.220863i
$739$	4.00000	0.147142	0.0735712	−	0.997290i	$-0.476560\pi$
$739$	4.00000	0.147142	0.0735712	+	0.997290i	$0.476560\pi$
$740$	0	0
$741$	−8.00000	−0.293887
$742$	12.0000i	0.440534i
$743$	42.0000i	1.54083i	0.637542	+	0.770415i	$0.279951\pi$
$743$	42.0000i	1.54083i	−0.637542	+	0.770415i	$0.720049\pi$
$744$	−10.0000	−0.366618
$745$	0	0
$746$	22.0000	0.805477
$747$	− 12.0000i	− 0.439057i
$748$	0	0
$749$	0	0
$750$	0	0
$751$	2.00000	0.0729810	0.0364905	−	0.999334i	$-0.488382\pi$
$751$	2.00000	0.0729810	0.0364905	+	0.999334i	$0.488382\pi$
$752$	− 12.0000i	− 0.437595i
$753$	12.0000i	0.437304i
$754$	0	0
$755$	0	0
$756$	2.00000	0.0727393
$757$	34.0000i	1.23575i	0.786276	+	0.617876i	$0.212006\pi$
$757$	34.0000i	1.23575i	−0.786276	+	0.617876i	$0.787994\pi$
$758$	28.0000i	1.01701i
$759$	0	0
$760$	0	0
$761$	−54.0000	−1.95750	−0.978749	−	0.205061i	$-0.934261\pi$
$761$	−54.0000	−1.95750	−0.978749	+	0.205061i	$0.934261\pi$
$762$	8.00000i	0.289809i
$763$	40.0000i	1.44810i
$764$	12.0000	0.434145
$765$	0	0
$766$	12.0000	0.433578
$767$	24.0000i	0.866590i
$768$	1.00000i	0.0360844i
$769$	22.0000	0.793340	0.396670	−	0.917961i	$-0.370166\pi$
$769$	22.0000	0.793340	0.396670	+	0.917961i	$0.370166\pi$
$770$	0	0
$771$	18.0000	0.648254
$772$	− 14.0000i	− 0.503871i
$773$	− 6.00000i	− 0.215805i	−0.994161	−	0.107903i	$-0.965587\pi$
$773$	− 6.00000i	− 0.215805i	0.994161	−	0.107903i	$-0.0344134\pi$
$774$	−4.00000	−0.143777
$775$	0	0
$776$	−14.0000	−0.502571
$777$	− 16.0000i	− 0.573997i
$778$	30.0000i	1.07555i
$779$	24.0000	0.859889
$780$	0	0
$781$	0	0
$782$	6.00000i	0.214560i
$783$	0	0
$784$	3.00000	0.107143
$785$	0	0
$786$	0	0
$787$	− 32.0000i	− 1.14068i	−0.821410	−	0.570338i	$-0.806812\pi$
$787$	− 32.0000i	− 1.14068i	0.821410	−	0.570338i	$-0.193188\pi$
$788$	− 12.0000i	− 0.427482i
$789$	−12.0000	−0.427211
$790$	0	0
$791$	−12.0000	−0.426671
$792$	0	0
$793$	16.0000i	0.568177i
$794$	−16.0000	−0.567819
$795$	0	0
$796$	2.00000	0.0708881
$797$	− 6.00000i	− 0.212531i	−0.994338	−	0.106265i	$-0.966111\pi$
$797$	− 6.00000i	− 0.212531i	0.994338	−	0.106265i	$-0.0338893\pi$
$798$	8.00000i	0.283197i
$799$	12.0000	0.424529
$800$	0	0
$801$	−18.0000	−0.635999
$802$	− 18.0000i	− 0.635602i
$803$	0	0
$804$	4.00000	0.141069
$805$	0	0
$806$	20.0000	0.704470
$807$	− 24.0000i	− 0.844840i
$808$	− 6.00000i	− 0.211079i
$809$	−6.00000	−0.210949	−0.105474	−	0.994422i	$-0.533636\pi$
$809$	−6.00000	−0.210949	−0.105474	+	0.994422i	$0.533636\pi$
$810$	0	0
$811$	−4.00000	−0.140459	−0.0702295	−	0.997531i	$-0.522373\pi$
$811$	−4.00000	−0.140459	−0.0702295	+	0.997531i	$0.522373\pi$
$812$	0	0
$813$	− 16.0000i	− 0.561144i
$814$	0	0
$815$	0	0
$816$	−1.00000	−0.0350070
$817$	− 16.0000i	− 0.559769i
$818$	− 38.0000i	− 1.32864i
$819$	−4.00000	−0.139771
$820$	0	0
$821$	−36.0000	−1.25641	−0.628204	−	0.778048i	$-0.716210\pi$
$821$	−36.0000	−1.25641	−0.628204	+	0.778048i	$0.716210\pi$
$822$	− 6.00000i	− 0.209274i
$823$	− 34.0000i	− 1.18517i	−0.805510	−	0.592583i	$-0.798108\pi$
$823$	− 34.0000i	− 1.18517i	0.805510	−	0.592583i	$-0.201892\pi$
$824$	−4.00000	−0.139347
$825$	0	0
$826$	24.0000	0.835067
$827$	− 48.0000i	− 1.66912i	−0.550914	−	0.834562i	$-0.685721\pi$
$827$	− 48.0000i	− 1.66912i	0.550914	−	0.834562i	$-0.314279\pi$
$828$	− 6.00000i	− 0.208514i
$829$	10.0000	0.347314	0.173657	−	0.984806i	$-0.444442\pi$
$829$	10.0000	0.347314	0.173657	+	0.984806i	$0.444442\pi$
$830$	0	0
$831$	−4.00000	−0.138758
$832$	− 2.00000i	− 0.0693375i
$833$	3.00000i	0.103944i
$834$	8.00000	0.277017
$835$	0	0
$836$	0	0
$837$	10.0000i	0.345651i
$838$	36.0000i	1.24360i
$839$	18.0000	0.621429	0.310715	−	0.950503i	$-0.399432\pi$
$839$	18.0000	0.621429	0.310715	+	0.950503i	$0.399432\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	− 22.0000i	− 0.758170i
$843$	30.0000i	1.03325i
$844$	−8.00000	−0.275371
$845$	0	0
$846$	−12.0000	−0.412568
$847$	22.0000i	0.755929i
$848$	6.00000i	0.206041i
$849$	16.0000	0.549119
$850$	0	0
$851$	−48.0000	−1.64542
$852$	− 6.00000i	− 0.205557i
$853$	− 28.0000i	− 0.958702i	−0.877623	−	0.479351i	$-0.840872\pi$
$853$	− 28.0000i	− 0.958702i	0.877623	−	0.479351i	$-0.159128\pi$
$854$	16.0000	0.547509
$855$	0	0
$856$	0	0
$857$	− 6.00000i	− 0.204956i	−0.994735	−	0.102478i	$-0.967323\pi$
$857$	− 6.00000i	− 0.204956i	0.994735	−	0.102478i	$-0.0326771\pi$
$858$	0	0
$859$	28.0000	0.955348	0.477674	−	0.878537i	$-0.341480\pi$
$859$	28.0000	0.955348	0.477674	+	0.878537i	$0.341480\pi$
$860$	0	0
$861$	12.0000	0.408959
$862$	− 30.0000i	− 1.02180i
$863$	48.0000i	1.63394i	0.576681	+	0.816970i	$0.304348\pi$
$863$	48.0000i	1.63394i	−0.576681	+	0.816970i	$0.695652\pi$
$864$	1.00000	0.0340207
$865$	0	0
$866$	−14.0000	−0.475739
$867$	− 1.00000i	− 0.0339618i
$868$	− 20.0000i	− 0.678844i
$869$	0	0
$870$	0	0
$871$	−8.00000	−0.271070
$872$	20.0000i	0.677285i
$873$	14.0000i	0.473828i
$874$	24.0000	0.811812
$875$	0	0
$876$	2.00000	0.0675737
$877$	4.00000i	0.135070i	0.997717	+	0.0675352i	$0.0215135\pi$
$877$	4.00000i	0.135070i	−0.997717	+	0.0675352i	$0.978487\pi$
$878$	− 26.0000i	− 0.877457i
$879$	−18.0000	−0.607125
$880$	0	0
$881$	−54.0000	−1.81931	−0.909653	−	0.415369i	$-0.863653\pi$
$881$	−54.0000	−1.81931	−0.909653	+	0.415369i	$0.863653\pi$
$882$	− 3.00000i	− 0.101015i
$883$	20.0000i	0.673054i	0.941674	+	0.336527i	$0.109252\pi$
$883$	20.0000i	0.673054i	−0.941674	+	0.336527i	$0.890748\pi$
$884$	2.00000	0.0672673
$885$	0	0
$886$	36.0000	1.20944
$887$	18.0000i	0.604381i	0.953248	+	0.302190i	$0.0977178\pi$
$887$	18.0000i	0.604381i	−0.953248	+	0.302190i	$0.902282\pi$
$888$	− 8.00000i	− 0.268462i
$889$	−16.0000	−0.536623
$890$	0	0
$891$	0	0
$892$	28.0000i	0.937509i
$893$	− 48.0000i	− 1.60626i
$894$	−6.00000	−0.200670
$895$	0	0
$896$	−2.00000	−0.0668153
$897$	12.0000i	0.400668i
$898$	− 6.00000i	− 0.200223i
$899$	0	0
$900$	0	0
$901$	−6.00000	−0.199889
$902$	0	0
$903$	− 8.00000i	− 0.266223i
$904$	−6.00000	−0.199557
$905$	0	0
$906$	−8.00000	−0.265782
$907$	40.0000i	1.32818i	0.747653	+	0.664089i	$0.231180\pi$
$907$	40.0000i	1.32818i	−0.747653	+	0.664089i	$0.768820\pi$
$908$	12.0000i	0.398234i
$909$	−6.00000	−0.199007
$910$	0	0
$911$	6.00000	0.198789	0.0993944	−	0.995048i	$-0.468309\pi$
$911$	6.00000	0.198789	0.0993944	+	0.995048i	$0.468309\pi$
$912$	4.00000i	0.132453i
$913$	0	0
$914$	−10.0000	−0.330771
$915$	0	0
$916$	14.0000	0.462573
$917$	0	0
$918$	1.00000i	0.0330049i
$919$	16.0000	0.527791	0.263896	−	0.964551i	$-0.414993\pi$
$919$	16.0000	0.527791	0.263896	+	0.964551i	$0.414993\pi$
$920$	0	0
$921$	20.0000	0.659022
$922$	30.0000i	0.987997i
$923$	12.0000i	0.394985i
$924$	0	0
$925$	0	0
$926$	−20.0000	−0.657241
$927$	4.00000i	0.131377i
$928$	0	0
$929$	18.0000	0.590561	0.295280	−	0.955411i	$-0.404587\pi$
$929$	18.0000	0.590561	0.295280	+	0.955411i	$0.404587\pi$
$930$	0	0
$931$	12.0000	0.393284
$932$	30.0000i	0.982683i
$933$	18.0000i	0.589294i
$934$	12.0000	0.392652
$935$	0	0
$936$	−2.00000	−0.0653720
$937$	10.0000i	0.326686i	0.986569	+	0.163343i	$0.0522277\pi$
$937$	10.0000i	0.326686i	−0.986569	+	0.163343i	$0.947772\pi$
$938$	8.00000i	0.261209i
$939$	−14.0000	−0.456873
$940$	0	0
$941$	12.0000	0.391189	0.195594	−	0.980685i	$-0.437336\pi$
$941$	12.0000	0.391189	0.195594	+	0.980685i	$0.437336\pi$
$942$	− 10.0000i	− 0.325818i
$943$	− 36.0000i	− 1.17232i
$944$	12.0000	0.390567
$945$	0	0
$946$	0	0
$947$	− 24.0000i	− 0.779895i	−0.920837	−	0.389948i	$-0.872493\pi$
$947$	− 24.0000i	− 0.779895i	0.920837	−	0.389948i	$-0.127507\pi$
$948$	− 10.0000i	− 0.324785i
$949$	−4.00000	−0.129845
$950$	0	0
$951$	12.0000	0.389127
$952$	− 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	$0.661114\pi$
$954$	6.00000	0.194257
$955$	0	0
$956$	−12.0000	−0.388108
$957$	0	0
$958$	− 6.00000i	− 0.193851i
$959$	12.0000	0.387500
$960$	0	0
$961$	69.0000	2.22581
$962$	16.0000i	0.515861i
$963$	0	0
$964$	10.0000	0.322078
$965$	0	0
$966$	12.0000	0.386094
$967$	4.00000i	0.128631i	0.997930	+	0.0643157i	$0.0204865\pi$
$967$	4.00000i	0.128631i	−0.997930	+	0.0643157i	$0.979514\pi$
$968$	11.0000i	0.353553i
$969$	−4.00000	−0.128499
$970$	0	0
$971$	12.0000	0.385098	0.192549	−	0.981287i	$-0.438325\pi$
$971$	12.0000	0.385098	0.192549	+	0.981287i	$0.438325\pi$
$972$	− 1.00000i	− 0.0320750i
$973$	16.0000i	0.512936i
$974$	−10.0000	−0.320421
$975$	0	0
$976$	8.00000	0.256074
$977$	30.0000i	0.959785i	0.877327	+	0.479893i	$0.159324\pi$
$977$	30.0000i	0.959785i	−0.877327	+	0.479893i	$0.840676\pi$
$978$	− 20.0000i	− 0.639529i
$979$	0	0
$980$	0	0
$981$	20.0000	0.638551
$982$	− 36.0000i	− 1.14881i
$983$	6.00000i	0.191370i	0.995412	+	0.0956851i	$0.0305042\pi$
$983$	6.00000i	0.191370i	−0.995412	+	0.0956851i	$0.969496\pi$
$984$	6.00000	0.191273
$985$	0	0
$986$	0	0
$987$	− 24.0000i	− 0.763928i
$988$	− 8.00000i	− 0.254514i
$989$	−24.0000	−0.763156
$990$	0	0
$991$	2.00000	0.0635321	0.0317660	−	0.999495i	$-0.489887\pi$
$991$	2.00000	0.0635321	0.0317660	+	0.999495i	$0.489887\pi$
$992$	− 10.0000i	− 0.317500i
$993$	20.0000i	0.634681i
$994$	12.0000	0.380617
$995$	0	0
$996$	12.0000	0.380235
$997$	− 8.00000i	− 0.253363i	−0.991943	−	0.126681i	$-0.959567\pi$
$997$	− 8.00000i	− 0.253363i	0.991943	−	0.126681i	$-0.0404325\pi$
$998$	− 32.0000i	− 1.01294i
$999$	−8.00000	−0.253109

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2550.2.d.g.2449.2		2
5.2	odd	4		102.2.a.b.1.1	✓	1
5.3	odd	4		2550.2.a.u.1.1		1
5.4	even	2	inner	2550.2.d.g.2449.1		2
15.2	even	4		306.2.a.c.1.1		1
15.8	even	4		7650.2.a.j.1.1		1
20.7	even	4		816.2.a.d.1.1		1
35.27	even	4		4998.2.a.d.1.1		1
40.27	even	4		3264.2.a.w.1.1		1
40.37	odd	4		3264.2.a.i.1.1		1
60.47	odd	4		2448.2.a.i.1.1		1
85.2	odd	8		1734.2.f.b.1483.2		4
85.32	odd	8		1734.2.f.b.1483.1		4
85.42	odd	8		1734.2.f.b.829.1		4
85.47	odd	4		1734.2.b.f.577.2		2
85.67	odd	4		1734.2.a.b.1.1		1
85.72	odd	4		1734.2.b.f.577.1		2
85.77	odd	8		1734.2.f.b.829.2		4
120.77	even	4		9792.2.a.bg.1.1		1
120.107	odd	4		9792.2.a.ba.1.1		1
255.152	even	4		5202.2.a.j.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
102.2.a.b.1.1	✓	1	5.2	odd	4
306.2.a.c.1.1		1	15.2	even	4
816.2.a.d.1.1		1	20.7	even	4
1734.2.a.b.1.1		1	85.67	odd	4
1734.2.b.f.577.1		2	85.72	odd	4
1734.2.b.f.577.2		2	85.47	odd	4
1734.2.f.b.829.1		4	85.42	odd	8
1734.2.f.b.829.2		4	85.77	odd	8
1734.2.f.b.1483.1		4	85.32	odd	8
1734.2.f.b.1483.2		4	85.2	odd	8
2448.2.a.i.1.1		1	60.47	odd	4
2550.2.a.u.1.1		1	5.3	odd	4
2550.2.d.g.2449.1		2	5.4	even	2	inner
2550.2.d.g.2449.2		2	1.1	even	1	trivial
3264.2.a.i.1.1		1	40.37	odd	4
3264.2.a.w.1.1		1	40.27	even	4
4998.2.a.d.1.1		1	35.27	even	4
5202.2.a.j.1.1		1	255.152	even	4
7650.2.a.j.1.1		1	15.8	even	4
9792.2.a.ba.1.1		1	120.107	odd	4
9792.2.a.bg.1.1		1	120.77	even	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)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} -2.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} -2.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} -1.00000i q^{12} +2.00000i q^{13} +2.00000 q^{14} +1.00000 q^{16} +1.00000i q^{17} -1.00000i q^{18} +4.00000 q^{19} +2.00000 q^{21} -6.00000i q^{23} +1.00000 q^{24} -2.00000 q^{26} -1.00000i q^{27} +2.00000i q^{28} -10.0000 q^{31} +1.00000i q^{32} -1.00000 q^{34} +1.00000 q^{36} -8.00000i q^{37} +4.00000i q^{38} -2.00000 q^{39} +6.00000 q^{41} +2.00000i q^{42} -4.00000i q^{43} +6.00000 q^{46} -12.0000i q^{47} +1.00000i q^{48} +3.00000 q^{49} -1.00000 q^{51} -2.00000i q^{52} +6.00000i q^{53} +1.00000 q^{54} -2.00000 q^{56} +4.00000i q^{57} +12.0000 q^{59} +8.00000 q^{61} -10.0000i q^{62} +2.00000i q^{63} -1.00000 q^{64} +4.00000i q^{67} -1.00000i q^{68} +6.00000 q^{69} +6.00000 q^{71} +1.00000i q^{72} +2.00000i q^{73} +8.00000 q^{74} -4.00000 q^{76} -2.00000i q^{78} +10.0000 q^{79} +1.00000 q^{81} +6.00000i q^{82} +12.0000i q^{83} -2.00000 q^{84} +4.00000 q^{86} +18.0000 q^{89} +4.00000 q^{91} +6.00000i q^{92} -10.0000i q^{93} +12.0000 q^{94} -1.00000 q^{96} -14.0000i q^{97} +3.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} - 2 q^{6} - 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^4 - 2 * q^6 - 2 * q^9	\( 2 q - 2 q^{4} - 2 q^{6} - 2 q^{9} + 4 q^{14} + 2 q^{16} + 8 q^{19} + 4 q^{21} + 2 q^{24} - 4 q^{26} - 20 q^{31} - 2 q^{34} + 2 q^{36} - 4 q^{39} + 12 q^{41} + 12 q^{46} + 6 q^{49} - 2 q^{51} + 2 q^{54} - 4 q^{56} + 24 q^{59} + 16 q^{61} - 2 q^{64} + 12 q^{69} + 12 q^{71} + 16 q^{74} - 8 q^{76} + 20 q^{79} + 2 q^{81} - 4 q^{84} + 8 q^{86} + 36 q^{89} + 8 q^{91} + 24 q^{94} - 2 q^{96}+O(q^{100}) \) 2 * q - 2 * q^4 - 2 * q^6 - 2 * q^9 + 4 * q^14 + 2 * q^16 + 8 * q^19 + 4 * q^21 + 2 * q^24 - 4 * q^26 - 20 * q^31 - 2 * q^34 + 2 * q^36 - 4 * q^39 + 12 * q^41 + 12 * q^46 + 6 * q^49 - 2 * q^51 + 2 * q^54 - 4 * q^56 + 24 * q^59 + 16 * q^61 - 2 * q^64 + 12 * q^69 + 12 * q^71 + 16 * q^74 - 8 * q^76 + 20 * q^79 + 2 * q^81 - 4 * q^84 + 8 * q^86 + 36 * q^89 + 8 * q^91 + 24 * q^94 - 2 * q^96

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

Properties

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