LMFDB - Embedded newform 3450.2.d.n.2899.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3450,2,Mod(2899,3450)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3450, 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("3450.2899");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3450 = 2 \cdot 3 \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3450.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:	$27.5483886973$
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 690)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2899.2
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	3450.2899
Dual form			3450.2.d.n.2899.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -4.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -4.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} -2.00000 q^{11} +1.00000i q^{12} +4.00000 q^{14} +1.00000 q^{16} -2.00000i q^{17} -1.00000i q^{18} -4.00000 q^{21} -2.00000i q^{22} +1.00000i q^{23} -1.00000 q^{24} +1.00000i q^{27} +4.00000i q^{28} +4.00000 q^{29} +1.00000i q^{32} +2.00000i q^{33} +2.00000 q^{34} +1.00000 q^{36} -10.0000i q^{37} +6.00000 q^{41} -4.00000i q^{42} +2.00000i q^{43} +2.00000 q^{44} -1.00000 q^{46} -12.0000i q^{47} -1.00000i q^{48} -9.00000 q^{49} -2.00000 q^{51} +6.00000i q^{53} -1.00000 q^{54} -4.00000 q^{56} +4.00000i q^{58} -12.0000 q^{59} -14.0000 q^{61} +4.00000i q^{63} -1.00000 q^{64} -2.00000 q^{66} -2.00000i q^{67} +2.00000i q^{68} +1.00000 q^{69} -2.00000 q^{71} +1.00000i q^{72} +6.00000i q^{73} +10.0000 q^{74} +8.00000i q^{77} -8.00000 q^{79} +1.00000 q^{81} +6.00000i q^{82} +8.00000i q^{83} +4.00000 q^{84} -2.00000 q^{86} -4.00000i q^{87} +2.00000i q^{88} +8.00000 q^{89} -1.00000i q^{92} +12.0000 q^{94} +1.00000 q^{96} -9.00000i q^{98} +2.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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} - 4 q^{11} + 8 q^{14} + 2 q^{16} - 8 q^{21} - 2 q^{24} + 8 q^{29} + 4 q^{34} + 2 q^{36} + 12 q^{41} + 4 q^{44} - 2 q^{46} - 18 q^{49} - 4 q^{51} - 2 q^{54} - 8 q^{56} - 24 q^{59} - 28 q^{61} - 2 q^{64} - 4 q^{66} + 2 q^{69} - 4 q^{71} + 20 q^{74} - 16 q^{79} + 2 q^{81} + 8 q^{84} - 4 q^{86} + 16 q^{89} + 24 q^{94} + 2 q^{96} + 4 q^{99}+O(q^{100}) $ 2 * q - 2 * q^4 + 2 * q^6 - 2 * q^9 - 4 * q^11 + 8 * q^14 + 2 * q^16 - 8 * q^21 - 2 * q^24 + 8 * q^29 + 4 * q^34 + 2 * q^36 + 12 * q^41 + 4 * q^44 - 2 * q^46 - 18 * q^49 - 4 * q^51 - 2 * q^54 - 8 * q^56 - 24 * q^59 - 28 * q^61 - 2 * q^64 - 4 * q^66 + 2 * q^69 - 4 * q^71 + 20 * q^74 - 16 * q^79 + 2 * q^81 + 8 * q^84 - 4 * q^86 + 16 * q^89 + 24 * q^94 + 2 * q^96 + 4 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3450\mathbb{Z}\right)^\times$.

$n$	$277$	$1151$	$1201$
$\chi(n)$	$-1$	$1$	$1$

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
$7$	− 4.00000i	− 1.51186i	−0.654654	−	0.755929i	$-0.727186\pi$
$7$	− 4.00000i	− 1.51186i	0.654654	−	0.755929i	$-0.272814\pi$
$8$	− 1.00000i	− 0.353553i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	−2.00000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	1.00000i	0.288675i
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	4.00000	1.06904
$15$	0	0
$16$	1.00000	0.250000
$17$	− 2.00000i	− 0.485071i	−0.970143	−	0.242536i	$-0.922021\pi$
$17$	− 2.00000i	− 0.485071i	0.970143	−	0.242536i	$-0.0779791\pi$
$18$	− 1.00000i	− 0.235702i
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	−4.00000	−0.872872
$22$	− 2.00000i	− 0.426401i
$23$	1.00000i	0.208514i
$23$	1.00000i	0.208514i
$24$	−1.00000	−0.204124
$25$	0	0
$26$	0	0
$27$	1.00000i	0.192450i
$28$	4.00000i	0.755929i
$29$	4.00000	0.742781	0.371391	−	0.928477i	$-0.378881\pi$
$29$	4.00000	0.742781	0.371391	+	0.928477i	$0.378881\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	1.00000i	0.176777i
$33$	2.00000i	0.348155i
$34$	2.00000	0.342997
$35$	0	0
$36$	1.00000	0.166667
$37$	− 10.0000i	− 1.64399i	−0.569495	−	0.821995i	$-0.692861\pi$
$37$	− 10.0000i	− 1.64399i	0.569495	−	0.821995i	$-0.307139\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	6.00000	0.937043	0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	− 4.00000i	− 0.617213i
$43$	2.00000i	0.304997i	0.988304	+	0.152499i	$0.0487319\pi$
$43$	2.00000i	0.304997i	−0.988304	+	0.152499i	$0.951268\pi$
$44$	2.00000	0.301511
$45$	0	0
$46$	−1.00000	−0.147442
$47$	− 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
$49$	−9.00000	−1.28571
$50$	0	0
$51$	−2.00000	−0.280056
$52$	0	0
$53$	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
$56$	−4.00000	−0.534522
$57$	0	0
$58$	4.00000i	0.525226i
$59$	−12.0000	−1.56227	−0.781133	−	0.624364i	$-0.785358\pi$
$59$	−12.0000	−1.56227	−0.781133	+	0.624364i	$0.785358\pi$
$60$	0	0
$61$	−14.0000	−1.79252	−0.896258	−	0.443533i	$-0.853725\pi$
$61$	−14.0000	−1.79252	−0.896258	+	0.443533i	$0.853725\pi$
$62$	0	0
$63$	4.00000i	0.503953i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	−2.00000	−0.246183
$67$	− 2.00000i	− 0.244339i	−0.992509	−	0.122169i	$-0.961015\pi$
$67$	− 2.00000i	− 0.244339i	0.992509	−	0.122169i	$-0.0389851\pi$
$68$	2.00000i	0.242536i
$69$	1.00000	0.120386
$70$	0	0
$71$	−2.00000	−0.237356	−0.118678	−	0.992933i	$-0.537866\pi$
$71$	−2.00000	−0.237356	−0.118678	+	0.992933i	$0.537866\pi$
$72$	1.00000i	0.117851i
$73$	6.00000i	0.702247i	0.936329	+	0.351123i	$0.114200\pi$
$73$	6.00000i	0.702247i	−0.936329	+	0.351123i	$0.885800\pi$
$74$	10.0000	1.16248
$75$	0	0
$76$	0	0
$77$	8.00000i	0.911685i
$78$	0	0
$79$	−8.00000	−0.900070	−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000	−0.900070	−0.450035	+	0.893011i	$0.648589\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	6.00000i	0.662589i
$83$	8.00000i	0.878114i	0.898459	+	0.439057i	$0.144687\pi$
$83$	8.00000i	0.878114i	−0.898459	+	0.439057i	$0.855313\pi$
$84$	4.00000	0.436436
$85$	0	0
$86$	−2.00000	−0.215666
$87$	− 4.00000i	− 0.428845i
$88$	2.00000i	0.213201i
$89$	8.00000	0.847998	0.423999	−	0.905663i	$-0.360626\pi$
$89$	8.00000	0.847998	0.423999	+	0.905663i	$0.360626\pi$
$90$	0	0
$91$	0	0
$92$	− 1.00000i	− 0.104257i
$93$	0	0
$94$	12.0000	1.23771
$95$	0	0
$96$	1.00000	0.102062
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	− 9.00000i	− 0.909137i
$99$	2.00000	0.201008
$100$	0	0
$101$	−12.0000	−1.19404	−0.597022	−	0.802225i	$-0.703650\pi$
$101$	−12.0000	−1.19404	−0.597022	+	0.802225i	$0.703650\pi$
$102$	− 2.00000i	− 0.198030i
$103$	4.00000i	0.394132i	0.980390	+	0.197066i	$0.0631413\pi$
$103$	4.00000i	0.394132i	−0.980390	+	0.197066i	$0.936859\pi$
$104$	0	0
$105$	0	0
$106$	−6.00000	−0.582772
$107$	12.0000i	1.16008i	0.814587	+	0.580042i	$0.196964\pi$
$107$	12.0000i	1.16008i	−0.814587	+	0.580042i	$0.803036\pi$
$108$	− 1.00000i	− 0.0962250i
$109$	−14.0000	−1.34096	−0.670478	−	0.741929i	$-0.733911\pi$
$109$	−14.0000	−1.34096	−0.670478	+	0.741929i	$0.733911\pi$
$110$	0	0
$111$	−10.0000	−0.949158
$112$	− 4.00000i	− 0.377964i
$113$	6.00000i	0.564433i	0.959351	+	0.282216i	$0.0910696\pi$
$113$	6.00000i	0.564433i	−0.959351	+	0.282216i	$0.908930\pi$
$114$	0	0
$115$	0	0
$116$	−4.00000	−0.371391
$117$	0	0
$118$	− 12.0000i	− 1.10469i
$119$	−8.00000	−0.733359
$120$	0	0
$121$	−7.00000	−0.636364
$122$	− 14.0000i	− 1.26750i
$123$	− 6.00000i	− 0.541002i
$124$	0	0
$125$	0	0
$126$	−4.00000	−0.356348
$127$	− 6.00000i	− 0.532414i	−0.963916	−	0.266207i	$-0.914230\pi$
$127$	− 6.00000i	− 0.532414i	0.963916	−	0.266207i	$-0.0857705\pi$
$128$	− 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	$0.5\pi$
$132$	− 2.00000i	− 0.174078i
$133$	0	0
$134$	2.00000	0.172774
$135$	0	0
$136$	−2.00000	−0.171499
$137$	2.00000i	0.170872i	0.996344	+	0.0854358i	$0.0272282\pi$
$137$	2.00000i	0.170872i	−0.996344	+	0.0854358i	$0.972772\pi$
$138$	1.00000i	0.0851257i
$139$	−20.0000	−1.69638	−0.848189	−	0.529694i	$-0.822307\pi$
$139$	−20.0000	−1.69638	−0.848189	+	0.529694i	$0.822307\pi$
$140$	0	0
$141$	−12.0000	−1.01058
$142$	− 2.00000i	− 0.167836i
$143$	0	0
$144$	−1.00000	−0.0833333
$145$	0	0
$146$	−6.00000	−0.496564
$147$	9.00000i	0.742307i
$148$	10.0000i	0.821995i
$149$	14.0000	1.14692	0.573462	−	0.819232i	$-0.305600\pi$
$149$	14.0000	1.14692	0.573462	+	0.819232i	$0.305600\pi$
$150$	0	0
$151$	−12.0000	−0.976546	−0.488273	−	0.872691i	$-0.662373\pi$
$151$	−12.0000	−0.976546	−0.488273	+	0.872691i	$0.662373\pi$
$152$	0	0
$153$	2.00000i	0.161690i
$154$	−8.00000	−0.644658
$155$	0	0
$156$	0	0
$157$	22.0000i	1.75579i	0.478852	+	0.877896i	$0.341053\pi$
$157$	22.0000i	1.75579i	−0.478852	+	0.877896i	$0.658947\pi$
$158$	− 8.00000i	− 0.636446i
$159$	6.00000	0.475831
$160$	0	0
$161$	4.00000	0.315244
$162$	1.00000i	0.0785674i
$163$	− 16.0000i	− 1.25322i	−0.779334	−	0.626608i	$-0.784443\pi$
$163$	− 16.0000i	− 1.25322i	0.779334	−	0.626608i	$-0.215557\pi$
$164$	−6.00000	−0.468521
$165$	0	0
$166$	−8.00000	−0.620920
$167$	0	0	1.00000			$0$
$167$	0	0	−1.00000			$\pi$
$168$	4.00000i	0.308607i
$169$	13.0000	1.00000
$170$	0	0
$171$	0	0
$172$	− 2.00000i	− 0.152499i
$173$	− 6.00000i	− 0.456172i	−0.973641	−	0.228086i	$-0.926753\pi$
$173$	− 6.00000i	− 0.456172i	0.973641	−	0.228086i	$-0.0732467\pi$
$174$	4.00000	0.303239
$175$	0	0
$176$	−2.00000	−0.150756
$177$	12.0000i	0.901975i
$178$	8.00000i	0.599625i
$179$	−4.00000	−0.298974	−0.149487	−	0.988764i	$-0.547762\pi$
$179$	−4.00000	−0.298974	−0.149487	+	0.988764i	$0.547762\pi$
$180$	0	0
$181$	−6.00000	−0.445976	−0.222988	−	0.974821i	$-0.571581\pi$
$181$	−6.00000	−0.445976	−0.222988	+	0.974821i	$0.571581\pi$
$182$	0	0
$183$	14.0000i	1.03491i
$184$	1.00000	0.0737210
$185$	0	0
$186$	0	0
$187$	4.00000i	0.292509i
$188$	12.0000i	0.875190i
$189$	4.00000	0.290957
$190$	0	0
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	1.00000i	0.0721688i
$193$	− 14.0000i	− 1.00774i	−0.863779	−	0.503871i	$-0.831909\pi$
$193$	− 14.0000i	− 1.00774i	0.863779	−	0.503871i	$-0.168091\pi$
$194$	0	0
$195$	0	0
$196$	9.00000	0.642857
$197$	6.00000i	0.427482i	0.976890	+	0.213741i	$0.0685649\pi$
$197$	6.00000i	0.427482i	−0.976890	+	0.213741i	$0.931435\pi$
$198$	2.00000i	0.142134i
$199$	−16.0000	−1.13421	−0.567105	−	0.823646i	$-0.691937\pi$
$199$	−16.0000	−1.13421	−0.567105	+	0.823646i	$0.691937\pi$
$200$	0	0
$201$	−2.00000	−0.141069
$202$	− 12.0000i	− 0.844317i
$203$	− 16.0000i	− 1.12298i
$204$	2.00000	0.140028
$205$	0	0
$206$	−4.00000	−0.278693
$207$	− 1.00000i	− 0.0695048i
$208$	0	0
$209$	0	0
$210$	0	0
$211$	4.00000	0.275371	0.137686	−	0.990476i	$-0.456034\pi$
$211$	4.00000	0.275371	0.137686	+	0.990476i	$0.456034\pi$
$212$	− 6.00000i	− 0.412082i
$213$	2.00000i	0.137038i
$214$	−12.0000	−0.820303
$215$	0	0
$216$	1.00000	0.0680414
$217$	0	0
$218$	− 14.0000i	− 0.948200i
$219$	6.00000	0.405442
$220$	0	0
$221$	0	0
$222$	− 10.0000i	− 0.671156i
$223$	2.00000i	0.133930i	0.997755	+	0.0669650i	$0.0213316\pi$
$223$	2.00000i	0.133930i	−0.997755	+	0.0669650i	$0.978668\pi$
$224$	4.00000	0.267261
$225$	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	$-0.130377\pi$
$228$	0	0
$229$	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
$232$	− 4.00000i	− 0.262613i
$233$	10.0000i	0.655122i	0.944830	+	0.327561i	$0.106227\pi$
$233$	10.0000i	0.655122i	−0.944830	+	0.327561i	$0.893773\pi$
$234$	0	0
$235$	0	0
$236$	12.0000	0.781133
$237$	8.00000i	0.519656i
$238$	− 8.00000i	− 0.518563i
$239$	−22.0000	−1.42306	−0.711531	−	0.702655i	$-0.751998\pi$
$239$	−22.0000	−1.42306	−0.711531	+	0.702655i	$0.751998\pi$
$240$	0	0
$241$	−14.0000	−0.901819	−0.450910	−	0.892570i	$-0.648900\pi$
$241$	−14.0000	−0.901819	−0.450910	+	0.892570i	$0.648900\pi$
$242$	− 7.00000i	− 0.449977i
$243$	− 1.00000i	− 0.0641500i
$244$	14.0000	0.896258
$245$	0	0
$246$	6.00000	0.382546
$247$	0	0
$248$	0	0
$249$	8.00000	0.506979
$250$	0	0
$251$	18.0000	1.13615	0.568075	−	0.822977i	$-0.307688\pi$
$251$	18.0000	1.13615	0.568075	+	0.822977i	$0.307688\pi$
$252$	− 4.00000i	− 0.251976i
$253$	− 2.00000i	− 0.125739i
$254$	6.00000	0.376473
$255$	0	0
$256$	1.00000	0.0625000
$257$	− 2.00000i	− 0.124757i	−0.998053	−	0.0623783i	$-0.980131\pi$
$257$	− 2.00000i	− 0.124757i	0.998053	−	0.0623783i	$-0.0198685\pi$
$258$	2.00000i	0.124515i
$259$	−40.0000	−2.48548
$260$	0	0
$261$	−4.00000	−0.247594
$262$	0	0
$263$	− 32.0000i	− 1.97320i	−0.163144	−	0.986602i	$-0.552164\pi$
$263$	− 32.0000i	− 1.97320i	0.163144	−	0.986602i	$-0.447836\pi$
$264$	2.00000	0.123091
$265$	0	0
$266$	0	0
$267$	− 8.00000i	− 0.489592i
$268$	2.00000i	0.122169i
$269$	−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
$271$	8.00000	0.485965	0.242983	−	0.970031i	$-0.421874\pi$
$271$	8.00000	0.485965	0.242983	+	0.970031i	$0.421874\pi$
$272$	− 2.00000i	− 0.121268i
$273$	0	0
$274$	−2.00000	−0.120824
$275$	0	0
$276$	−1.00000	−0.0601929
$277$	− 8.00000i	− 0.480673i	−0.970690	−	0.240337i	$-0.922742\pi$
$277$	− 8.00000i	− 0.480673i	0.970690	−	0.240337i	$-0.0772579\pi$
$278$	− 20.0000i	− 1.19952i
$279$	0	0
$280$	0	0
$281$	32.0000	1.90896	0.954480	−	0.298275i	$-0.0964112\pi$
$281$	32.0000	1.90896	0.954480	+	0.298275i	$0.0964112\pi$
$282$	− 12.0000i	− 0.714590i
$283$	− 10.0000i	− 0.594438i	−0.954809	−	0.297219i	$-0.903941\pi$
$283$	− 10.0000i	− 0.594438i	0.954809	−	0.297219i	$-0.0960592\pi$
$284$	2.00000	0.118678
$285$	0	0
$286$	0	0
$287$	− 24.0000i	− 1.41668i
$288$	− 1.00000i	− 0.0589256i
$289$	13.0000	0.764706
$290$	0	0
$291$	0	0
$292$	− 6.00000i	− 0.351123i
$293$	14.0000i	0.817889i	0.912559	+	0.408944i	$0.134103\pi$
$293$	14.0000i	0.817889i	−0.912559	+	0.408944i	$0.865897\pi$
$294$	−9.00000	−0.524891
$295$	0	0
$296$	−10.0000	−0.581238
$297$	− 2.00000i	− 0.116052i
$298$	14.0000i	0.810998i
$299$	0	0
$300$	0	0
$301$	8.00000	0.461112
$302$	− 12.0000i	− 0.690522i
$303$	12.0000i	0.689382i
$304$	0	0
$305$	0	0
$306$	−2.00000	−0.114332
$307$	− 4.00000i	− 0.228292i	−0.993464	−	0.114146i	$-0.963587\pi$
$307$	− 4.00000i	− 0.228292i	0.993464	−	0.114146i	$-0.0364132\pi$
$308$	− 8.00000i	− 0.455842i
$309$	4.00000	0.227552
$310$	0	0
$311$	−2.00000	−0.113410	−0.0567048	−	0.998391i	$-0.518059\pi$
$311$	−2.00000	−0.113410	−0.0567048	+	0.998391i	$0.518059\pi$
$312$	0	0
$313$	20.0000i	1.13047i	0.824931	+	0.565233i	$0.191214\pi$
$313$	20.0000i	1.13047i	−0.824931	+	0.565233i	$0.808786\pi$
$314$	−22.0000	−1.24153
$315$	0	0
$316$	8.00000	0.450035
$317$	10.0000i	0.561656i	0.959758	+	0.280828i	$0.0906090\pi$
$317$	10.0000i	0.561656i	−0.959758	+	0.280828i	$0.909391\pi$
$318$	6.00000i	0.336463i
$319$	−8.00000	−0.447914
$320$	0	0
$321$	12.0000	0.669775
$322$	4.00000i	0.222911i
$323$	0	0
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	16.0000	0.886158
$327$	14.0000i	0.774202i
$328$	− 6.00000i	− 0.331295i
$329$	−48.0000	−2.64633
$330$	0	0
$331$	−28.0000	−1.53902	−0.769510	−	0.638635i	$-0.779499\pi$
$331$	−28.0000	−1.53902	−0.769510	+	0.638635i	$0.779499\pi$
$332$	− 8.00000i	− 0.439057i
$333$	10.0000i	0.547997i
$334$	0	0
$335$	0	0
$336$	−4.00000	−0.218218
$337$	4.00000i	0.217894i	0.994048	+	0.108947i	$0.0347479\pi$
$337$	4.00000i	0.217894i	−0.994048	+	0.108947i	$0.965252\pi$
$338$	13.0000i	0.707107i
$339$	6.00000	0.325875
$340$	0	0
$341$	0	0
$342$	0	0
$343$	8.00000i	0.431959i
$344$	2.00000	0.107833
$345$	0	0
$346$	6.00000	0.322562
$347$	− 4.00000i	− 0.214731i	−0.994220	−	0.107366i	$-0.965758\pi$
$347$	− 4.00000i	− 0.214731i	0.994220	−	0.107366i	$-0.0342415\pi$
$348$	4.00000i	0.214423i
$349$	30.0000	1.60586	0.802932	−	0.596071i	$-0.203272\pi$
$349$	30.0000	1.60586	0.802932	+	0.596071i	$0.203272\pi$
$350$	0	0
$351$	0	0
$352$	− 2.00000i	− 0.106600i
$353$	− 34.0000i	− 1.80964i	−0.425797	−	0.904819i	$-0.640006\pi$
$353$	− 34.0000i	− 1.80964i	0.425797	−	0.904819i	$-0.359994\pi$
$354$	−12.0000	−0.637793
$355$	0	0
$356$	−8.00000	−0.423999
$357$	8.00000i	0.423405i
$358$	− 4.00000i	− 0.211407i
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	− 6.00000i	− 0.315353i
$363$	7.00000i	0.367405i
$364$	0	0
$365$	0	0
$366$	−14.0000	−0.731792
$367$	− 4.00000i	− 0.208798i	−0.994535	−	0.104399i	$-0.966708\pi$
$367$	− 4.00000i	− 0.208798i	0.994535	−	0.104399i	$-0.0332919\pi$
$368$	1.00000i	0.0521286i
$369$	−6.00000	−0.312348
$370$	0	0
$371$	24.0000	1.24602
$372$	0	0
$373$	− 22.0000i	− 1.13912i	−0.821951	−	0.569558i	$-0.807114\pi$
$373$	− 22.0000i	− 1.13912i	0.821951	−	0.569558i	$-0.192886\pi$
$374$	−4.00000	−0.206835
$375$	0	0
$376$	−12.0000	−0.618853
$377$	0	0
$378$	4.00000i	0.205738i
$379$	−4.00000	−0.205466	−0.102733	−	0.994709i	$-0.532759\pi$
$379$	−4.00000	−0.205466	−0.102733	+	0.994709i	$0.532759\pi$
$380$	0	0
$381$	−6.00000	−0.307389
$382$	0	0
$383$	− 24.0000i	− 1.22634i	−0.789950	−	0.613171i	$-0.789894\pi$
$383$	− 24.0000i	− 1.22634i	0.789950	−	0.613171i	$-0.210106\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	14.0000	0.712581
$387$	− 2.00000i	− 0.101666i
$388$	0	0
$389$	−22.0000	−1.11544	−0.557722	−	0.830028i	$-0.688325\pi$
$389$	−22.0000	−1.11544	−0.557722	+	0.830028i	$0.688325\pi$
$390$	0	0
$391$	2.00000	0.101144
$392$	9.00000i	0.454569i
$393$	0	0
$394$	−6.00000	−0.302276
$395$	0	0
$396$	−2.00000	−0.100504
$397$	− 32.0000i	− 1.60603i	−0.595956	−	0.803017i	$-0.703227\pi$
$397$	− 32.0000i	− 1.60603i	0.595956	−	0.803017i	$-0.296773\pi$
$398$	− 16.0000i	− 0.802008i
$399$	0	0
$400$	0	0
$401$	−12.0000	−0.599251	−0.299626	−	0.954057i	$-0.596862\pi$
$401$	−12.0000	−0.599251	−0.299626	+	0.954057i	$0.596862\pi$
$402$	− 2.00000i	− 0.0997509i
$403$	0	0
$404$	12.0000	0.597022
$405$	0	0
$406$	16.0000	0.794067
$407$	20.0000i	0.991363i
$408$	2.00000i	0.0990148i
$409$	10.0000	0.494468	0.247234	−	0.968956i	$-0.420478\pi$
$409$	10.0000	0.494468	0.247234	+	0.968956i	$0.420478\pi$
$410$	0	0
$411$	2.00000	0.0986527
$412$	− 4.00000i	− 0.197066i
$413$	48.0000i	2.36193i
$414$	1.00000	0.0491473
$415$	0	0
$416$	0	0
$417$	20.0000i	0.979404i
$418$	0	0
$419$	26.0000	1.27018	0.635092	−	0.772437i	$-0.280962\pi$
$419$	26.0000	1.27018	0.635092	+	0.772437i	$0.280962\pi$
$420$	0	0
$421$	−6.00000	−0.292422	−0.146211	−	0.989253i	$-0.546708\pi$
$421$	−6.00000	−0.292422	−0.146211	+	0.989253i	$0.546708\pi$
$422$	4.00000i	0.194717i
$423$	12.0000i	0.583460i
$424$	6.00000	0.291386
$425$	0	0
$426$	−2.00000	−0.0969003
$427$	56.0000i	2.71003i
$428$	− 12.0000i	− 0.580042i
$429$	0	0
$430$	0	0
$431$	24.0000	1.15604	0.578020	−	0.816023i	$-0.303826\pi$
$431$	24.0000	1.15604	0.578020	+	0.816023i	$0.303826\pi$
$432$	1.00000i	0.0481125i
$433$	24.0000i	1.15337i	0.816968	+	0.576683i	$0.195653\pi$
$433$	24.0000i	1.15337i	−0.816968	+	0.576683i	$0.804347\pi$
$434$	0	0
$435$	0	0
$436$	14.0000	0.670478
$437$	0	0
$438$	6.00000i	0.286691i
$439$	−20.0000	−0.954548	−0.477274	−	0.878755i	$-0.658375\pi$
$439$	−20.0000	−0.954548	−0.477274	+	0.878755i	$0.658375\pi$
$440$	0	0
$441$	9.00000	0.428571
$442$	0	0
$443$	− 20.0000i	− 0.950229i	−0.879924	−	0.475114i	$-0.842407\pi$
$443$	− 20.0000i	− 0.950229i	0.879924	−	0.475114i	$-0.157593\pi$
$444$	10.0000	0.474579
$445$	0	0
$446$	−2.00000	−0.0947027
$447$	− 14.0000i	− 0.662177i
$448$	4.00000i	0.188982i
$449$	34.0000	1.60456	0.802280	−	0.596948i	$-0.203620\pi$
$449$	34.0000	1.60456	0.802280	+	0.596948i	$0.203620\pi$
$450$	0	0
$451$	−12.0000	−0.565058
$452$	− 6.00000i	− 0.282216i
$453$	12.0000i	0.563809i
$454$	12.0000	0.563188
$455$	0	0
$456$	0	0
$457$	24.0000i	1.12267i	0.827588	+	0.561336i	$0.189713\pi$
$457$	24.0000i	1.12267i	−0.827588	+	0.561336i	$0.810287\pi$
$458$	26.0000i	1.21490i
$459$	2.00000	0.0933520
$460$	0	0
$461$	20.0000	0.931493	0.465746	−	0.884918i	$-0.345786\pi$
$461$	20.0000	0.931493	0.465746	+	0.884918i	$0.345786\pi$
$462$	8.00000i	0.372194i
$463$	18.0000i	0.836531i	0.908325	+	0.418265i	$0.137362\pi$
$463$	18.0000i	0.836531i	−0.908325	+	0.418265i	$0.862638\pi$
$464$	4.00000	0.185695
$465$	0	0
$466$	−10.0000	−0.463241
$467$	− 24.0000i	− 1.11059i	−0.831654	−	0.555294i	$-0.812606\pi$
$467$	− 24.0000i	− 1.11059i	0.831654	−	0.555294i	$-0.187394\pi$
$468$	0	0
$469$	−8.00000	−0.369406
$470$	0	0
$471$	22.0000	1.01371
$472$	12.0000i	0.552345i
$473$	− 4.00000i	− 0.183920i
$474$	−8.00000	−0.367452
$475$	0	0
$476$	8.00000	0.366679
$477$	− 6.00000i	− 0.274721i
$478$	− 22.0000i	− 1.00626i
$479$	40.0000	1.82765	0.913823	−	0.406112i	$-0.133116\pi$
$479$	40.0000	1.82765	0.913823	+	0.406112i	$0.133116\pi$
$480$	0	0
$481$	0	0
$482$	− 14.0000i	− 0.637683i
$483$	− 4.00000i	− 0.182006i
$484$	7.00000	0.318182
$485$	0	0
$486$	1.00000	0.0453609
$487$	− 18.0000i	− 0.815658i	−0.913058	−	0.407829i	$-0.866286\pi$
$487$	− 18.0000i	− 0.815658i	0.913058	−	0.407829i	$-0.133714\pi$
$488$	14.0000i	0.633750i
$489$	−16.0000	−0.723545
$490$	0	0
$491$	−24.0000	−1.08310	−0.541552	−	0.840667i	$-0.682163\pi$
$491$	−24.0000	−1.08310	−0.541552	+	0.840667i	$0.682163\pi$
$492$	6.00000i	0.270501i
$493$	− 8.00000i	− 0.360302i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	8.00000i	0.358849i
$498$	8.00000i	0.358489i
$499$	28.0000	1.25345	0.626726	−	0.779240i	$-0.284395\pi$
$499$	28.0000	1.25345	0.626726	+	0.779240i	$0.284395\pi$
$500$	0	0
$501$	0	0
$502$	18.0000i	0.803379i
$503$	− 8.00000i	− 0.356702i	−0.983967	−	0.178351i	$-0.942924\pi$
$503$	− 8.00000i	− 0.356702i	0.983967	−	0.178351i	$-0.0570763\pi$
$504$	4.00000	0.178174
$505$	0	0
$506$	2.00000	0.0889108
$507$	− 13.0000i	− 0.577350i
$508$	6.00000i	0.266207i
$509$	−28.0000	−1.24108	−0.620539	−	0.784176i	$-0.713086\pi$
$509$	−28.0000	−1.24108	−0.620539	+	0.784176i	$0.713086\pi$
$510$	0	0
$511$	24.0000	1.06170
$512$	1.00000i	0.0441942i
$513$	0	0
$514$	2.00000	0.0882162
$515$	0	0
$516$	−2.00000	−0.0880451
$517$	24.0000i	1.05552i
$518$	− 40.0000i	− 1.75750i
$519$	−6.00000	−0.263371
$520$	0	0
$521$	32.0000	1.40195	0.700973	−	0.713188i	$-0.252749\pi$
$521$	32.0000	1.40195	0.700973	+	0.713188i	$0.252749\pi$
$522$	− 4.00000i	− 0.175075i
$523$	22.0000i	0.961993i	0.876723	+	0.480996i	$0.159725\pi$
$523$	22.0000i	0.961993i	−0.876723	+	0.480996i	$0.840275\pi$
$524$	0	0
$525$	0	0
$526$	32.0000	1.39527
$527$	0	0
$528$	2.00000i	0.0870388i
$529$	−1.00000	−0.0434783
$530$	0	0
$531$	12.0000	0.520756
$532$	0	0
$533$	0	0
$534$	8.00000	0.346194
$535$	0	0
$536$	−2.00000	−0.0863868
$537$	4.00000i	0.172613i
$538$	− 24.0000i	− 1.03471i
$539$	18.0000	0.775315
$540$	0	0
$541$	−22.0000	−0.945854	−0.472927	−	0.881102i	$-0.656803\pi$
$541$	−22.0000	−0.945854	−0.472927	+	0.881102i	$0.656803\pi$
$542$	8.00000i	0.343629i
$543$	6.00000i	0.257485i
$544$	2.00000	0.0857493
$545$	0	0
$546$	0	0
$547$	− 12.0000i	− 0.513083i	−0.966533	−	0.256541i	$-0.917417\pi$
$547$	− 12.0000i	− 0.513083i	0.966533	−	0.256541i	$-0.0825830\pi$
$548$	− 2.00000i	− 0.0854358i
$549$	14.0000	0.597505
$550$	0	0
$551$	0	0
$552$	− 1.00000i	− 0.0425628i
$553$	32.0000i	1.36078i
$554$	8.00000	0.339887
$555$	0	0
$556$	20.0000	0.848189
$557$	− 26.0000i	− 1.10166i	−0.834619	−	0.550828i	$-0.814312\pi$
$557$	− 26.0000i	− 1.10166i	0.834619	−	0.550828i	$-0.185688\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	4.00000	0.168880
$562$	32.0000i	1.34984i
$563$	8.00000i	0.337160i	0.985688	+	0.168580i	$0.0539181\pi$
$563$	8.00000i	0.337160i	−0.985688	+	0.168580i	$0.946082\pi$
$564$	12.0000	0.505291
$565$	0	0
$566$	10.0000	0.420331
$567$	− 4.00000i	− 0.167984i
$568$	2.00000i	0.0839181i
$569$	12.0000	0.503066	0.251533	−	0.967849i	$-0.419065\pi$
$569$	12.0000	0.503066	0.251533	+	0.967849i	$0.419065\pi$
$570$	0	0
$571$	−36.0000	−1.50655	−0.753277	−	0.657704i	$-0.771528\pi$
$571$	−36.0000	−1.50655	−0.753277	+	0.657704i	$0.771528\pi$
$572$	0	0
$573$	0	0
$574$	24.0000	1.00174
$575$	0	0
$576$	1.00000	0.0416667
$577$	10.0000i	0.416305i	0.978096	+	0.208153i	$0.0667451\pi$
$577$	10.0000i	0.416305i	−0.978096	+	0.208153i	$0.933255\pi$
$578$	13.0000i	0.540729i
$579$	−14.0000	−0.581820
$580$	0	0
$581$	32.0000	1.32758
$582$	0	0
$583$	− 12.0000i	− 0.496989i
$584$	6.00000	0.248282
$585$	0	0
$586$	−14.0000	−0.578335
$587$	4.00000i	0.165098i	0.996587	+	0.0825488i	$0.0263060\pi$
$587$	4.00000i	0.165098i	−0.996587	+	0.0825488i	$0.973694\pi$
$588$	− 9.00000i	− 0.371154i
$589$	0	0
$590$	0	0
$591$	6.00000	0.246807
$592$	− 10.0000i	− 0.410997i
$593$	− 30.0000i	− 1.23195i	−0.787765	−	0.615976i	$-0.788762\pi$
$593$	− 30.0000i	− 1.23195i	0.787765	−	0.615976i	$-0.211238\pi$
$594$	2.00000	0.0820610
$595$	0	0
$596$	−14.0000	−0.573462
$597$	16.0000i	0.654836i
$598$	0	0
$599$	30.0000	1.22577	0.612883	−	0.790173i	$-0.290010\pi$
$599$	30.0000	1.22577	0.612883	+	0.790173i	$0.290010\pi$
$600$	0	0
$601$	−10.0000	−0.407909	−0.203954	−	0.978980i	$-0.565379\pi$
$601$	−10.0000	−0.407909	−0.203954	+	0.978980i	$0.565379\pi$
$602$	8.00000i	0.326056i
$603$	2.00000i	0.0814463i
$604$	12.0000	0.488273
$605$	0	0
$606$	−12.0000	−0.487467
$607$	14.0000i	0.568242i	0.958788	+	0.284121i	$0.0917018\pi$
$607$	14.0000i	0.568242i	−0.958788	+	0.284121i	$0.908298\pi$
$608$	0	0
$609$	−16.0000	−0.648353
$610$	0	0
$611$	0	0
$612$	− 2.00000i	− 0.0808452i
$613$	14.0000i	0.565455i	0.959200	+	0.282727i	$0.0912392\pi$
$613$	14.0000i	0.565455i	−0.959200	+	0.282727i	$0.908761\pi$
$614$	4.00000	0.161427
$615$	0	0
$616$	8.00000	0.322329
$617$	14.0000i	0.563619i	0.959470	+	0.281809i	$0.0909346\pi$
$617$	14.0000i	0.563619i	−0.959470	+	0.281809i	$0.909065\pi$
$618$	4.00000i	0.160904i
$619$	24.0000	0.964641	0.482321	−	0.875995i	$-0.339794\pi$
$619$	24.0000	0.964641	0.482321	+	0.875995i	$0.339794\pi$
$620$	0	0
$621$	−1.00000	−0.0401286
$622$	− 2.00000i	− 0.0801927i
$623$	− 32.0000i	− 1.28205i
$624$	0	0
$625$	0	0
$626$	−20.0000	−0.799361
$627$	0	0
$628$	− 22.0000i	− 0.877896i
$629$	−20.0000	−0.797452
$630$	0	0
$631$	−16.0000	−0.636950	−0.318475	−	0.947931i	$-0.603171\pi$
$631$	−16.0000	−0.636950	−0.318475	+	0.947931i	$0.603171\pi$
$632$	8.00000i	0.318223i
$633$	− 4.00000i	− 0.158986i
$634$	−10.0000	−0.397151
$635$	0	0
$636$	−6.00000	−0.237915
$637$	0	0
$638$	− 8.00000i	− 0.316723i
$639$	2.00000	0.0791188
$640$	0	0
$641$	−8.00000	−0.315981	−0.157991	−	0.987441i	$-0.550502\pi$
$641$	−8.00000	−0.315981	−0.157991	+	0.987441i	$0.550502\pi$
$642$	12.0000i	0.473602i
$643$	14.0000i	0.552106i	0.961142	+	0.276053i	$0.0890266\pi$
$643$	14.0000i	0.552106i	−0.961142	+	0.276053i	$0.910973\pi$
$644$	−4.00000	−0.157622
$645$	0	0
$646$	0	0
$647$	− 12.0000i	− 0.471769i	−0.971781	−	0.235884i	$-0.924201\pi$
$647$	− 12.0000i	− 0.471769i	0.971781	−	0.235884i	$-0.0757987\pi$
$648$	− 1.00000i	− 0.0392837i
$649$	24.0000	0.942082
$650$	0	0
$651$	0	0
$652$	16.0000i	0.626608i
$653$	− 26.0000i	− 1.01746i	−0.860927	−	0.508729i	$-0.830115\pi$
$653$	− 26.0000i	− 1.01746i	0.860927	−	0.508729i	$-0.169885\pi$
$654$	−14.0000	−0.547443
$655$	0	0
$656$	6.00000	0.234261
$657$	− 6.00000i	− 0.234082i
$658$	− 48.0000i	− 1.87123i
$659$	18.0000	0.701180	0.350590	−	0.936529i	$-0.385981\pi$
$659$	18.0000	0.701180	0.350590	+	0.936529i	$0.385981\pi$
$660$	0	0
$661$	6.00000	0.233373	0.116686	−	0.993169i	$-0.462773\pi$
$661$	6.00000	0.233373	0.116686	+	0.993169i	$0.462773\pi$
$662$	− 28.0000i	− 1.08825i
$663$	0	0
$664$	8.00000	0.310460
$665$	0	0
$666$	−10.0000	−0.387492
$667$	4.00000i	0.154881i
$668$	0	0
$669$	2.00000	0.0773245
$670$	0	0
$671$	28.0000	1.08093
$672$	− 4.00000i	− 0.154303i
$673$	− 18.0000i	− 0.693849i	−0.937893	−	0.346925i	$-0.887226\pi$
$673$	− 18.0000i	− 0.693849i	0.937893	−	0.346925i	$-0.112774\pi$
$674$	−4.00000	−0.154074
$675$	0	0
$676$	−13.0000	−0.500000
$677$	22.0000i	0.845529i	0.906240	+	0.422764i	$0.138940\pi$
$677$	22.0000i	0.845529i	−0.906240	+	0.422764i	$0.861060\pi$
$678$	6.00000i	0.230429i
$679$	0	0
$680$	0	0
$681$	−12.0000	−0.459841
$682$	0	0
$683$	− 36.0000i	− 1.37750i	−0.724998	−	0.688751i	$-0.758159\pi$
$683$	− 36.0000i	− 1.37750i	0.724998	−	0.688751i	$-0.241841\pi$
$684$	0	0
$685$	0	0
$686$	−8.00000	−0.305441
$687$	− 26.0000i	− 0.991962i
$688$	2.00000i	0.0762493i
$689$	0	0
$690$	0	0
$691$	−20.0000	−0.760836	−0.380418	−	0.924815i	$-0.624220\pi$
$691$	−20.0000	−0.760836	−0.380418	+	0.924815i	$0.624220\pi$
$692$	6.00000i	0.228086i
$693$	− 8.00000i	− 0.303895i
$694$	4.00000	0.151838
$695$	0	0
$696$	−4.00000	−0.151620
$697$	− 12.0000i	− 0.454532i
$698$	30.0000i	1.13552i
$699$	10.0000	0.378235
$700$	0	0
$701$	6.00000	0.226617	0.113308	−	0.993560i	$-0.463855\pi$
$701$	6.00000	0.226617	0.113308	+	0.993560i	$0.463855\pi$
$702$	0	0
$703$	0	0
$704$	2.00000	0.0753778
$705$	0	0
$706$	34.0000	1.27961
$707$	48.0000i	1.80523i
$708$	− 12.0000i	− 0.450988i
$709$	−22.0000	−0.826227	−0.413114	−	0.910679i	$-0.635559\pi$
$709$	−22.0000	−0.826227	−0.413114	+	0.910679i	$0.635559\pi$
$710$	0	0
$711$	8.00000	0.300023
$712$	− 8.00000i	− 0.299813i
$713$	0	0
$714$	−8.00000	−0.299392
$715$	0	0
$716$	4.00000	0.149487
$717$	22.0000i	0.821605i
$718$	0	0
$719$	−18.0000	−0.671287	−0.335643	−	0.941989i	$-0.608954\pi$
$719$	−18.0000	−0.671287	−0.335643	+	0.941989i	$0.608954\pi$
$720$	0	0
$721$	16.0000	0.595871
$722$	− 19.0000i	− 0.707107i
$723$	14.0000i	0.520666i
$724$	6.00000	0.222988
$725$	0	0
$726$	−7.00000	−0.259794
$727$	− 8.00000i	− 0.296704i	−0.988935	−	0.148352i	$-0.952603\pi$
$727$	− 8.00000i	− 0.296704i	0.988935	−	0.148352i	$-0.0473968\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	4.00000	0.147945
$732$	− 14.0000i	− 0.517455i
$733$	− 30.0000i	− 1.10808i	−0.832492	−	0.554038i	$-0.813086\pi$
$733$	− 30.0000i	− 1.10808i	0.832492	−	0.554038i	$-0.186914\pi$
$734$	4.00000	0.147643
$735$	0	0
$736$	−1.00000	−0.0368605
$737$	4.00000i	0.147342i
$738$	− 6.00000i	− 0.220863i
$739$	−36.0000	−1.32428	−0.662141	−	0.749380i	$-0.730352\pi$
$739$	−36.0000	−1.32428	−0.662141	+	0.749380i	$0.730352\pi$
$740$	0	0
$741$	0	0
$742$	24.0000i	0.881068i
$743$	16.0000i	0.586983i	0.955962	+	0.293492i	$0.0948173\pi$
$743$	16.0000i	0.586983i	−0.955962	+	0.293492i	$0.905183\pi$
$744$	0	0
$745$	0	0
$746$	22.0000	0.805477
$747$	− 8.00000i	− 0.292705i
$748$	− 4.00000i	− 0.146254i
$749$	48.0000	1.75388
$750$	0	0
$751$	40.0000	1.45962	0.729810	−	0.683650i	$-0.239608\pi$
$751$	40.0000	1.45962	0.729810	+	0.683650i	$0.239608\pi$
$752$	− 12.0000i	− 0.437595i
$753$	− 18.0000i	− 0.655956i
$754$	0	0
$755$	0	0
$756$	−4.00000	−0.145479
$757$	− 22.0000i	− 0.799604i	−0.916602	−	0.399802i	$-0.869079\pi$
$757$	− 22.0000i	− 0.799604i	0.916602	−	0.399802i	$-0.130921\pi$
$758$	− 4.00000i	− 0.145287i
$759$	−2.00000	−0.0725954
$760$	0	0
$761$	2.00000	0.0724999	0.0362500	−	0.999343i	$-0.488459\pi$
$761$	2.00000	0.0724999	0.0362500	+	0.999343i	$0.488459\pi$
$762$	− 6.00000i	− 0.217357i
$763$	56.0000i	2.02734i
$764$	0	0
$765$	0	0
$766$	24.0000	0.867155
$767$	0	0
$768$	− 1.00000i	− 0.0360844i
$769$	−6.00000	−0.216366	−0.108183	−	0.994131i	$-0.534503\pi$
$769$	−6.00000	−0.216366	−0.108183	+	0.994131i	$0.534503\pi$
$770$	0	0
$771$	−2.00000	−0.0720282
$772$	14.0000i	0.503871i
$773$	6.00000i	0.215805i	0.994161	+	0.107903i	$0.0344134\pi$
$773$	6.00000i	0.215805i	−0.994161	+	0.107903i	$0.965587\pi$
$774$	2.00000	0.0718885
$775$	0	0
$776$	0	0
$777$	40.0000i	1.43499i
$778$	− 22.0000i	− 0.788738i
$779$	0	0
$780$	0	0
$781$	4.00000	0.143131
$782$	2.00000i	0.0715199i
$783$	4.00000i	0.142948i
$784$	−9.00000	−0.321429
$785$	0	0
$786$	0	0
$787$	34.0000i	1.21197i	0.795476	+	0.605985i	$0.207221\pi$
$787$	34.0000i	1.21197i	−0.795476	+	0.605985i	$0.792779\pi$
$788$	− 6.00000i	− 0.213741i
$789$	−32.0000	−1.13923
$790$	0	0
$791$	24.0000	0.853342
$792$	− 2.00000i	− 0.0710669i
$793$	0	0
$794$	32.0000	1.13564
$795$	0	0
$796$	16.0000	0.567105
$797$	− 50.0000i	− 1.77109i	−0.464553	−	0.885545i	$-0.653785\pi$
$797$	− 50.0000i	− 1.77109i	0.464553	−	0.885545i	$-0.346215\pi$
$798$	0	0
$799$	−24.0000	−0.849059
$800$	0	0
$801$	−8.00000	−0.282666
$802$	− 12.0000i	− 0.423735i
$803$	− 12.0000i	− 0.423471i
$804$	2.00000	0.0705346
$805$	0	0
$806$	0	0
$807$	24.0000i	0.844840i
$808$	12.0000i	0.422159i
$809$	30.0000	1.05474	0.527372	−	0.849635i	$-0.323177\pi$
$809$	30.0000	1.05474	0.527372	+	0.849635i	$0.323177\pi$
$810$	0	0
$811$	52.0000	1.82597	0.912983	−	0.407997i	$-0.133772\pi$
$811$	52.0000	1.82597	0.912983	+	0.407997i	$0.133772\pi$
$812$	16.0000i	0.561490i
$813$	− 8.00000i	− 0.280572i
$814$	−20.0000	−0.701000
$815$	0	0
$816$	−2.00000	−0.0700140
$817$	0	0
$818$	10.0000i	0.349642i
$819$	0	0
$820$	0	0
$821$	−16.0000	−0.558404	−0.279202	−	0.960232i	$-0.590070\pi$
$821$	−16.0000	−0.558404	−0.279202	+	0.960232i	$0.590070\pi$
$822$	2.00000i	0.0697580i
$823$	− 10.0000i	− 0.348578i	−0.984695	−	0.174289i	$-0.944237\pi$
$823$	− 10.0000i	− 0.348578i	0.984695	−	0.174289i	$-0.0557627\pi$
$824$	4.00000	0.139347
$825$	0	0
$826$	−48.0000	−1.67013
$827$	− 44.0000i	− 1.53003i	−0.644013	−	0.765015i	$-0.722732\pi$
$827$	− 44.0000i	− 1.53003i	0.644013	−	0.765015i	$-0.277268\pi$
$828$	1.00000i	0.0347524i
$829$	−10.0000	−0.347314	−0.173657	−	0.984806i	$-0.555558\pi$
$829$	−10.0000	−0.347314	−0.173657	+	0.984806i	$0.555558\pi$
$830$	0	0
$831$	−8.00000	−0.277517
$832$	0	0
$833$	18.0000i	0.623663i
$834$	−20.0000	−0.692543
$835$	0	0
$836$	0	0
$837$	0	0
$838$	26.0000i	0.898155i
$839$	20.0000	0.690477	0.345238	−	0.938515i	$-0.387798\pi$
$839$	20.0000	0.690477	0.345238	+	0.938515i	$0.387798\pi$
$840$	0	0
$841$	−13.0000	−0.448276
$842$	− 6.00000i	− 0.206774i
$843$	− 32.0000i	− 1.10214i
$844$	−4.00000	−0.137686
$845$	0	0
$846$	−12.0000	−0.412568
$847$	28.0000i	0.962091i
$848$	6.00000i	0.206041i
$849$	−10.0000	−0.343199
$850$	0	0
$851$	10.0000	0.342796
$852$	− 2.00000i	− 0.0685189i
$853$	56.0000i	1.91740i	0.284413	+	0.958702i	$0.408201\pi$
$853$	56.0000i	1.91740i	−0.284413	+	0.958702i	$0.591799\pi$
$854$	−56.0000	−1.91628
$855$	0	0
$856$	12.0000	0.410152
$857$	42.0000i	1.43469i	0.696717	+	0.717346i	$0.254643\pi$
$857$	42.0000i	1.43469i	−0.696717	+	0.717346i	$0.745357\pi$
$858$	0	0
$859$	−36.0000	−1.22830	−0.614152	−	0.789188i	$-0.710502\pi$
$859$	−36.0000	−1.22830	−0.614152	+	0.789188i	$0.710502\pi$
$860$	0	0
$861$	−24.0000	−0.817918
$862$	24.0000i	0.817443i
$863$	36.0000i	1.22545i	0.790295	+	0.612727i	$0.209928\pi$
$863$	36.0000i	1.22545i	−0.790295	+	0.612727i	$0.790072\pi$
$864$	−1.00000	−0.0340207
$865$	0	0
$866$	−24.0000	−0.815553
$867$	− 13.0000i	− 0.441503i
$868$	0	0
$869$	16.0000	0.542763
$870$	0	0
$871$	0	0
$872$	14.0000i	0.474100i
$873$	0	0
$874$	0	0
$875$	0	0
$876$	−6.00000	−0.202721
$877$	56.0000i	1.89099i	0.325643	+	0.945493i	$0.394419\pi$
$877$	56.0000i	1.89099i	−0.325643	+	0.945493i	$0.605581\pi$
$878$	− 20.0000i	− 0.674967i
$879$	14.0000	0.472208
$880$	0	0
$881$	40.0000	1.34763	0.673817	−	0.738898i	$-0.264654\pi$
$881$	40.0000	1.34763	0.673817	+	0.738898i	$0.264654\pi$
$882$	9.00000i	0.303046i
$883$	− 48.0000i	− 1.61533i	−0.589643	−	0.807664i	$-0.700731\pi$
$883$	− 48.0000i	− 1.61533i	0.589643	−	0.807664i	$-0.299269\pi$
$884$	0	0
$885$	0	0
$886$	20.0000	0.671913
$887$	36.0000i	1.20876i	0.796696	+	0.604381i	$0.206579\pi$
$887$	36.0000i	1.20876i	−0.796696	+	0.604381i	$0.793421\pi$
$888$	10.0000i	0.335578i
$889$	−24.0000	−0.804934
$890$	0	0
$891$	−2.00000	−0.0670025
$892$	− 2.00000i	− 0.0669650i
$893$	0	0
$894$	14.0000	0.468230
$895$	0	0
$896$	−4.00000	−0.133631
$897$	0	0
$898$	34.0000i	1.13459i
$899$	0	0
$900$	0	0
$901$	12.0000	0.399778
$902$	− 12.0000i	− 0.399556i
$903$	− 8.00000i	− 0.266223i
$904$	6.00000	0.199557
$905$	0	0
$906$	−12.0000	−0.398673
$907$	− 42.0000i	− 1.39459i	−0.716786	−	0.697294i	$-0.754387\pi$
$907$	− 42.0000i	− 1.39459i	0.716786	−	0.697294i	$-0.245613\pi$
$908$	12.0000i	0.398234i
$909$	12.0000	0.398015
$910$	0	0
$911$	−12.0000	−0.397578	−0.198789	−	0.980042i	$-0.563701\pi$
$911$	−12.0000	−0.397578	−0.198789	+	0.980042i	$0.563701\pi$
$912$	0	0
$913$	− 16.0000i	− 0.529523i
$914$	−24.0000	−0.793849
$915$	0	0
$916$	−26.0000	−0.859064
$917$	0	0
$918$	2.00000i	0.0660098i
$919$	32.0000	1.05558	0.527791	−	0.849374i	$-0.323020\pi$
$919$	32.0000	1.05558	0.527791	+	0.849374i	$0.323020\pi$
$920$	0	0
$921$	−4.00000	−0.131804
$922$	20.0000i	0.658665i
$923$	0	0
$924$	−8.00000	−0.263181
$925$	0	0
$926$	−18.0000	−0.591517
$927$	− 4.00000i	− 0.131377i
$928$	4.00000i	0.131306i
$929$	−30.0000	−0.984268	−0.492134	−	0.870519i	$-0.663783\pi$
$929$	−30.0000	−0.984268	−0.492134	+	0.870519i	$0.663783\pi$
$930$	0	0
$931$	0	0
$932$	− 10.0000i	− 0.327561i
$933$	2.00000i	0.0654771i
$934$	24.0000	0.785304
$935$	0	0
$936$	0	0
$937$	− 48.0000i	− 1.56809i	−0.620703	−	0.784046i	$-0.713153\pi$
$937$	− 48.0000i	− 1.56809i	0.620703	−	0.784046i	$-0.286847\pi$
$938$	− 8.00000i	− 0.261209i
$939$	20.0000	0.652675
$940$	0	0
$941$	26.0000	0.847576	0.423788	−	0.905761i	$-0.360700\pi$
$941$	26.0000	0.847576	0.423788	+	0.905761i	$0.360700\pi$
$942$	22.0000i	0.716799i
$943$	6.00000i	0.195387i
$944$	−12.0000	−0.390567
$945$	0	0
$946$	4.00000	0.130051
$947$	52.0000i	1.68977i	0.534946	+	0.844886i	$0.320332\pi$
$947$	52.0000i	1.68977i	−0.534946	+	0.844886i	$0.679668\pi$
$948$	− 8.00000i	− 0.259828i
$949$	0	0
$950$	0	0
$951$	10.0000	0.324272
$952$	8.00000i	0.259281i
$953$	14.0000i	0.453504i	0.973952	+	0.226752i	$0.0728108\pi$
$953$	14.0000i	0.453504i	−0.973952	+	0.226752i	$0.927189\pi$
$954$	6.00000	0.194257
$955$	0	0
$956$	22.0000	0.711531
$957$	8.00000i	0.258603i
$958$	40.0000i	1.29234i
$959$	8.00000	0.258333
$960$	0	0
$961$	−31.0000	−1.00000
$962$	0	0
$963$	− 12.0000i	− 0.386695i
$964$	14.0000	0.450910
$965$	0	0
$966$	4.00000	0.128698
$967$	− 58.0000i	− 1.86515i	−0.360971	−	0.932577i	$-0.617555\pi$
$967$	− 58.0000i	− 1.86515i	0.360971	−	0.932577i	$-0.382445\pi$
$968$	7.00000i	0.224989i
$969$	0	0
$970$	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
$973$	80.0000i	2.56468i
$974$	18.0000	0.576757
$975$	0	0
$976$	−14.0000	−0.448129
$977$	− 18.0000i	− 0.575871i	−0.957650	−	0.287936i	$-0.907031\pi$
$977$	− 18.0000i	− 0.575871i	0.957650	−	0.287936i	$-0.0929689\pi$
$978$	− 16.0000i	− 0.511624i
$979$	−16.0000	−0.511362
$980$	0	0
$981$	14.0000	0.446986
$982$	− 24.0000i	− 0.765871i
$983$	24.0000i	0.765481i	0.923856	+	0.382741i	$0.125020\pi$
$983$	24.0000i	0.765481i	−0.923856	+	0.382741i	$0.874980\pi$
$984$	−6.00000	−0.191273
$985$	0	0
$986$	8.00000	0.254772
$987$	48.0000i	1.52786i
$988$	0	0
$989$	−2.00000	−0.0635963
$990$	0	0
$991$	52.0000	1.65183	0.825917	−	0.563791i	$-0.190658\pi$
$991$	52.0000	1.65183	0.825917	+	0.563791i	$0.190658\pi$
$992$	0	0
$993$	28.0000i	0.888553i
$994$	−8.00000	−0.253745
$995$	0	0
$996$	−8.00000	−0.253490
$997$	− 40.0000i	− 1.26681i	−0.773819	−	0.633406i	$-0.781656\pi$
$997$	− 40.0000i	− 1.26681i	0.773819	−	0.633406i	$-0.218344\pi$
$998$	28.0000i	0.886325i
$999$	10.0000	0.316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3450.2.d.n.2899.2		2
5.2	odd	4		690.2.a.b.1.1	✓	1
5.3	odd	4		3450.2.a.t.1.1		1
5.4	even	2	inner	3450.2.d.n.2899.1		2
15.2	even	4		2070.2.a.s.1.1		1
20.7	even	4		5520.2.a.r.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
690.2.a.b.1.1	✓	1	5.2	odd	4
2070.2.a.s.1.1		1	15.2	even	4
3450.2.a.t.1.1		1	5.3	odd	4
3450.2.d.n.2899.1		2	5.4	even	2	inner
3450.2.d.n.2899.2		2	1.1	even	1	trivial
5520.2.a.r.1.1		1	20.7	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:	\( N \)	\(=\)	\( 3450 = 2 \cdot 3 \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3450.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -4.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -4.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} -2.00000 q^{11} +1.00000i q^{12} +4.00000 q^{14} +1.00000 q^{16} -2.00000i q^{17} -1.00000i q^{18} -4.00000 q^{21} -2.00000i q^{22} +1.00000i q^{23} -1.00000 q^{24} +1.00000i q^{27} +4.00000i q^{28} +4.00000 q^{29} +1.00000i q^{32} +2.00000i q^{33} +2.00000 q^{34} +1.00000 q^{36} -10.0000i q^{37} +6.00000 q^{41} -4.00000i q^{42} +2.00000i q^{43} +2.00000 q^{44} -1.00000 q^{46} -12.0000i q^{47} -1.00000i q^{48} -9.00000 q^{49} -2.00000 q^{51} +6.00000i q^{53} -1.00000 q^{54} -4.00000 q^{56} +4.00000i q^{58} -12.0000 q^{59} -14.0000 q^{61} +4.00000i q^{63} -1.00000 q^{64} -2.00000 q^{66} -2.00000i q^{67} +2.00000i q^{68} +1.00000 q^{69} -2.00000 q^{71} +1.00000i q^{72} +6.00000i q^{73} +10.0000 q^{74} +8.00000i q^{77} -8.00000 q^{79} +1.00000 q^{81} +6.00000i q^{82} +8.00000i q^{83} +4.00000 q^{84} -2.00000 q^{86} -4.00000i q^{87} +2.00000i q^{88} +8.00000 q^{89} -1.00000i q^{92} +12.0000 q^{94} +1.00000 q^{96} -9.00000i q^{98} +2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} + 2 q^{6} - 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^4 + 2 * q^6 - 2 * q^9	\( 2 q - 2 q^{4} + 2 q^{6} - 2 q^{9} - 4 q^{11} + 8 q^{14} + 2 q^{16} - 8 q^{21} - 2 q^{24} + 8 q^{29} + 4 q^{34} + 2 q^{36} + 12 q^{41} + 4 q^{44} - 2 q^{46} - 18 q^{49} - 4 q^{51} - 2 q^{54} - 8 q^{56} - 24 q^{59} - 28 q^{61} - 2 q^{64} - 4 q^{66} + 2 q^{69} - 4 q^{71} + 20 q^{74} - 16 q^{79} + 2 q^{81} + 8 q^{84} - 4 q^{86} + 16 q^{89} + 24 q^{94} + 2 q^{96} + 4 q^{99}+O(q^{100}) \) 2 * q - 2 * q^4 + 2 * q^6 - 2 * q^9 - 4 * q^11 + 8 * q^14 + 2 * q^16 - 8 * q^21 - 2 * q^24 + 8 * q^29 + 4 * q^34 + 2 * q^36 + 12 * q^41 + 4 * q^44 - 2 * q^46 - 18 * q^49 - 4 * q^51 - 2 * q^54 - 8 * q^56 - 24 * q^59 - 28 * q^61 - 2 * q^64 - 4 * q^66 + 2 * q^69 - 4 * q^71 + 20 * q^74 - 16 * q^79 + 2 * q^81 + 8 * q^84 - 4 * q^86 + 16 * q^89 + 24 * q^94 + 2 * q^96 + 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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