LMFDB - Embedded newform 3450.2.d.j.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 138)
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.j.2899.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +2.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +2.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} -6.00000 q^{11} +1.00000i q^{12} -2.00000i q^{13} -2.00000 q^{14} +1.00000 q^{16} -1.00000i q^{18} +2.00000 q^{21} -6.00000i q^{22} -1.00000i q^{23} -1.00000 q^{24} +2.00000 q^{26} +1.00000i q^{27} -2.00000i q^{28} -6.00000 q^{29} +8.00000 q^{31} +1.00000i q^{32} +6.00000i q^{33} +1.00000 q^{36} -2.00000 q^{39} +10.0000 q^{41} +2.00000i q^{42} -12.0000i q^{43} +6.00000 q^{44} +1.00000 q^{46} +8.00000i q^{47} -1.00000i q^{48} +3.00000 q^{49} +2.00000i q^{52} +2.00000i q^{53} -1.00000 q^{54} +2.00000 q^{56} -6.00000i q^{58} +12.0000 q^{59} +4.00000 q^{61} +8.00000i q^{62} -2.00000i q^{63} -1.00000 q^{64} -6.00000 q^{66} +12.0000i q^{67} -1.00000 q^{69} +1.00000i q^{72} -10.0000i q^{73} -12.0000i q^{77} -2.00000i q^{78} +6.00000 q^{79} +1.00000 q^{81} +10.0000i q^{82} +14.0000i q^{83} -2.00000 q^{84} +12.0000 q^{86} +6.00000i q^{87} +6.00000i q^{88} +4.00000 q^{91} +1.00000i q^{92} -8.00000i q^{93} -8.00000 q^{94} +1.00000 q^{96} +6.00000i q^{97} +3.00000i q^{98} +6.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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} - 12 q^{11} - 4 q^{14} + 2 q^{16} + 4 q^{21} - 2 q^{24} + 4 q^{26} - 12 q^{29} + 16 q^{31} + 2 q^{36} - 4 q^{39} + 20 q^{41} + 12 q^{44} + 2 q^{46} + 6 q^{49} - 2 q^{54} + 4 q^{56} + 24 q^{59} + 8 q^{61} - 2 q^{64} - 12 q^{66} - 2 q^{69} + 12 q^{79} + 2 q^{81} - 4 q^{84} + 24 q^{86} + 8 q^{91} - 16 q^{94} + 2 q^{96} + 12 q^{99}+O(q^{100}) $ 2 * q - 2 * q^4 + 2 * q^6 - 2 * q^9 - 12 * q^11 - 4 * q^14 + 2 * q^16 + 4 * q^21 - 2 * q^24 + 4 * q^26 - 12 * q^29 + 16 * q^31 + 2 * q^36 - 4 * q^39 + 20 * q^41 + 12 * q^44 + 2 * q^46 + 6 * q^49 - 2 * q^54 + 4 * q^56 + 24 * q^59 + 8 * q^61 - 2 * q^64 - 12 * q^66 - 2 * q^69 + 12 * q^79 + 2 * q^81 - 4 * q^84 + 24 * q^86 + 8 * q^91 - 16 * q^94 + 2 * q^96 + 12 * 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
$7$	2.00000i	0.755929i	0.925820	+	0.377964i	$0.123376\pi$
$7$	2.00000i	0.755929i	−0.925820	+	0.377964i	$0.876624\pi$
$8$	− 1.00000i	− 0.353553i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	−6.00000	−1.80907	−0.904534	−	0.426401i	$-0.859781\pi$
$11$	−6.00000	−1.80907	−0.904534	+	0.426401i	$0.859781\pi$
$12$	1.00000i	0.288675i
$13$	− 2.00000i	− 0.554700i	−0.960769	−	0.277350i	$-0.910544\pi$
$13$	− 2.00000i	− 0.554700i	0.960769	−	0.277350i	$-0.0894562\pi$
$14$	−2.00000	−0.534522
$15$	0	0
$16$	1.00000	0.250000
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	− 1.00000i	− 0.235702i
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	2.00000	0.436436
$22$	− 6.00000i	− 1.27920i
$23$	− 1.00000i	− 0.208514i
$23$	− 1.00000i	− 0.208514i
$24$	−1.00000	−0.204124
$25$	0	0
$26$	2.00000	0.392232
$27$	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	$-0.244865\pi$
$31$	8.00000	1.43684	0.718421	+	0.695608i	$0.244865\pi$
$32$	1.00000i	0.176777i
$33$	6.00000i	1.04447i
$34$	0	0
$35$	0	0
$36$	1.00000	0.166667
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	0	0
$39$	−2.00000	−0.320256
$40$	0	0
$41$	10.0000	1.56174	0.780869	−	0.624695i	$-0.214777\pi$
$41$	10.0000	1.56174	0.780869	+	0.624695i	$0.214777\pi$
$42$	2.00000i	0.308607i
$43$	− 12.0000i	− 1.82998i	−0.403473	−	0.914991i	$-0.632197\pi$
$43$	− 12.0000i	− 1.82998i	0.403473	−	0.914991i	$-0.367803\pi$
$44$	6.00000	0.904534
$45$	0	0
$46$	1.00000	0.147442
$47$	8.00000i	1.16692i	0.812142	+	0.583460i	$0.198301\pi$
$47$	8.00000i	1.16692i	−0.812142	+	0.583460i	$0.801699\pi$
$48$	− 1.00000i	− 0.144338i
$49$	3.00000	0.428571
$50$	0	0
$51$	0	0
$52$	2.00000i	0.277350i
$53$	2.00000i	0.274721i	0.990521	+	0.137361i	$0.0438619\pi$
$53$	2.00000i	0.274721i	−0.990521	+	0.137361i	$0.956138\pi$
$54$	−1.00000	−0.136083
$55$	0	0
$56$	2.00000	0.267261
$57$	0	0
$58$	− 6.00000i	− 0.787839i
$59$	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
$61$	4.00000	0.512148	0.256074	−	0.966657i	$-0.417571\pi$
$61$	4.00000	0.512148	0.256074	+	0.966657i	$0.417571\pi$
$62$	8.00000i	1.01600i
$63$	− 2.00000i	− 0.251976i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	−6.00000	−0.738549
$67$	12.0000i	1.46603i	0.680211	+	0.733017i	$0.261888\pi$
$67$	12.0000i	1.46603i	−0.680211	+	0.733017i	$0.738112\pi$
$68$	0	0
$69$	−1.00000	−0.120386
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	1.00000i	0.117851i
$73$	− 10.0000i	− 1.17041i	−0.810885	−	0.585206i	$-0.801014\pi$
$73$	− 10.0000i	− 1.17041i	0.810885	−	0.585206i	$-0.198986\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 12.0000i	− 1.36753i
$78$	− 2.00000i	− 0.226455i
$79$	6.00000	0.675053	0.337526	−	0.941316i	$-0.390410\pi$
$79$	6.00000	0.675053	0.337526	+	0.941316i	$0.390410\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	10.0000i	1.10432i
$83$	14.0000i	1.53670i	0.640030	+	0.768350i	$0.278922\pi$
$83$	14.0000i	1.53670i	−0.640030	+	0.768350i	$0.721078\pi$
$84$	−2.00000	−0.218218
$85$	0	0
$86$	12.0000	1.29399
$87$	6.00000i	0.643268i
$88$	6.00000i	0.639602i
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	0	0
$91$	4.00000	0.419314
$92$	1.00000i	0.104257i
$93$	− 8.00000i	− 0.829561i
$94$	−8.00000	−0.825137
$95$	0	0
$96$	1.00000	0.102062
$97$	6.00000i	0.609208i	0.952479	+	0.304604i	$0.0985241\pi$
$97$	6.00000i	0.609208i	−0.952479	+	0.304604i	$0.901476\pi$
$98$	3.00000i	0.303046i
$99$	6.00000	0.603023
$100$	0	0
$101$	−6.00000	−0.597022	−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000	−0.597022	−0.298511	+	0.954406i	$0.596490\pi$
$102$	0	0
$103$	14.0000i	1.37946i	0.724066	+	0.689730i	$0.242271\pi$
$103$	14.0000i	1.37946i	−0.724066	+	0.689730i	$0.757729\pi$
$104$	−2.00000	−0.196116
$105$	0	0
$106$	−2.00000	−0.194257
$107$	− 14.0000i	− 1.35343i	−0.736245	−	0.676716i	$-0.763403\pi$
$107$	− 14.0000i	− 1.35343i	0.736245	−	0.676716i	$-0.236597\pi$
$108$	− 1.00000i	− 0.0962250i
$109$	16.0000	1.53252	0.766261	−	0.642529i	$-0.222115\pi$
$109$	16.0000	1.53252	0.766261	+	0.642529i	$0.222115\pi$
$110$	0	0
$111$	0	0
$112$	2.00000i	0.188982i
$113$	− 8.00000i	− 0.752577i	−0.926503	−	0.376288i	$-0.877200\pi$
$113$	− 8.00000i	− 0.752577i	0.926503	−	0.376288i	$-0.122800\pi$
$114$	0	0
$115$	0	0
$116$	6.00000	0.557086
$117$	2.00000i	0.184900i
$118$	12.0000i	1.10469i
$119$	0	0
$120$	0	0
$121$	25.0000	2.27273
$122$	4.00000i	0.362143i
$123$	− 10.0000i	− 0.901670i
$124$	−8.00000	−0.718421
$125$	0	0
$126$	2.00000	0.178174
$127$	− 12.0000i	− 1.06483i	−0.846484	−	0.532414i	$-0.821285\pi$
$127$	− 12.0000i	− 1.06483i	0.846484	−	0.532414i	$-0.178715\pi$
$128$	− 1.00000i	− 0.0883883i
$129$	−12.0000	−1.05654
$130$	0	0
$131$	−8.00000	−0.698963	−0.349482	−	0.936943i	$-0.613642\pi$
$131$	−8.00000	−0.698963	−0.349482	+	0.936943i	$0.613642\pi$
$132$	− 6.00000i	− 0.522233i
$133$	0	0
$134$	−12.0000	−1.03664
$135$	0	0
$136$	0	0
$137$	12.0000i	1.02523i	0.858619	+	0.512615i	$0.171323\pi$
$137$	12.0000i	1.02523i	−0.858619	+	0.512615i	$0.828677\pi$
$138$	− 1.00000i	− 0.0851257i
$139$	−12.0000	−1.01783	−0.508913	−	0.860818i	$-0.669953\pi$
$139$	−12.0000	−1.01783	−0.508913	+	0.860818i	$0.669953\pi$
$140$	0	0
$141$	8.00000	0.673722
$142$	0	0
$143$	12.0000i	1.00349i
$144$	−1.00000	−0.0833333
$145$	0	0
$146$	10.0000	0.827606
$147$	− 3.00000i	− 0.247436i
$148$	0	0
$149$	18.0000	1.47462	0.737309	−	0.675556i	$-0.236096\pi$
$149$	18.0000	1.47462	0.737309	+	0.675556i	$0.236096\pi$
$150$	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
$153$	0	0
$154$	12.0000	0.966988
$155$	0	0
$156$	2.00000	0.160128
$157$	− 16.0000i	− 1.27694i	−0.769647	−	0.638470i	$-0.779568\pi$
$157$	− 16.0000i	− 1.27694i	0.769647	−	0.638470i	$-0.220432\pi$
$158$	6.00000i	0.477334i
$159$	2.00000	0.158610
$160$	0	0
$161$	2.00000	0.157622
$162$	1.00000i	0.0785674i
$163$	− 12.0000i	− 0.939913i	−0.882690	−	0.469956i	$-0.844270\pi$
$163$	− 12.0000i	− 0.939913i	0.882690	−	0.469956i	$-0.155730\pi$
$164$	−10.0000	−0.780869
$165$	0	0
$166$	−14.0000	−1.08661
$167$	− 8.00000i	− 0.619059i	−0.950890	−	0.309529i	$-0.899829\pi$
$167$	− 8.00000i	− 0.619059i	0.950890	−	0.309529i	$-0.100171\pi$
$168$	− 2.00000i	− 0.154303i
$169$	9.00000	0.692308
$170$	0	0
$171$	0	0
$172$	12.0000i	0.914991i
$173$	− 18.0000i	− 1.36851i	−0.729241	−	0.684257i	$-0.760127\pi$
$173$	− 18.0000i	− 1.36851i	0.729241	−	0.684257i	$-0.239873\pi$
$174$	−6.00000	−0.454859
$175$	0	0
$176$	−6.00000	−0.452267
$177$	− 12.0000i	− 0.901975i
$178$	0	0
$179$	16.0000	1.19590	0.597948	−	0.801535i	$-0.295983\pi$
$179$	16.0000	1.19590	0.597948	+	0.801535i	$0.295983\pi$
$180$	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
$183$	− 4.00000i	− 0.295689i
$184$	−1.00000	−0.0737210
$185$	0	0
$186$	8.00000	0.586588
$187$	0	0
$188$	− 8.00000i	− 0.583460i
$189$	−2.00000	−0.145479
$190$	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	$0.168091\pi$
$193$	14.0000i	1.00774i	−0.863779	+	0.503871i	$0.831909\pi$
$194$	−6.00000	−0.430775
$195$	0	0
$196$	−3.00000	−0.214286
$197$	6.00000i	0.427482i	0.976890	+	0.213741i	$0.0685649\pi$
$197$	6.00000i	0.427482i	−0.976890	+	0.213741i	$0.931435\pi$
$198$	6.00000i	0.426401i
$199$	−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
$201$	12.0000	0.846415
$202$	− 6.00000i	− 0.422159i
$203$	− 12.0000i	− 0.842235i
$204$	0	0
$205$	0	0
$206$	−14.0000	−0.975426
$207$	1.00000i	0.0695048i
$208$	− 2.00000i	− 0.138675i
$209$	0	0
$210$	0	0
$211$	20.0000	1.37686	0.688428	−	0.725304i	$-0.258301\pi$
$211$	20.0000	1.37686	0.688428	+	0.725304i	$0.258301\pi$
$212$	− 2.00000i	− 0.137361i
$213$	0	0
$214$	14.0000	0.957020
$215$	0	0
$216$	1.00000	0.0680414
$217$	16.0000i	1.08615i
$218$	16.0000i	1.08366i
$219$	−10.0000	−0.675737
$220$	0	0
$221$	0	0
$222$	0	0
$223$	12.0000i	0.803579i	0.915732	+	0.401790i	$0.131612\pi$
$223$	12.0000i	0.803579i	−0.915732	+	0.401790i	$0.868388\pi$
$224$	−2.00000	−0.133631
$225$	0	0
$226$	8.00000	0.532152
$227$	10.0000i	0.663723i	0.943328	+	0.331862i	$0.107677\pi$
$227$	10.0000i	0.663723i	−0.943328	+	0.331862i	$0.892323\pi$
$228$	0	0
$229$	24.0000	1.58596	0.792982	−	0.609245i	$-0.208527\pi$
$229$	24.0000	1.58596	0.792982	+	0.609245i	$0.208527\pi$
$230$	0	0
$231$	−12.0000	−0.789542
$232$	6.00000i	0.393919i
$233$	10.0000i	0.655122i	0.944830	+	0.327561i	$0.106227\pi$
$233$	10.0000i	0.655122i	−0.944830	+	0.327561i	$0.893773\pi$
$234$	−2.00000	−0.130744
$235$	0	0
$236$	−12.0000	−0.781133
$237$	− 6.00000i	− 0.389742i
$238$	0	0
$239$	16.0000	1.03495	0.517477	−	0.855697i	$-0.326871\pi$
$239$	16.0000	1.03495	0.517477	+	0.855697i	$0.326871\pi$
$240$	0	0
$241$	−6.00000	−0.386494	−0.193247	−	0.981150i	$-0.561902\pi$
$241$	−6.00000	−0.386494	−0.193247	+	0.981150i	$0.561902\pi$
$242$	25.0000i	1.60706i
$243$	− 1.00000i	− 0.0641500i
$244$	−4.00000	−0.256074
$245$	0	0
$246$	10.0000	0.637577
$247$	0	0
$248$	− 8.00000i	− 0.508001i
$249$	14.0000	0.887214
$250$	0	0
$251$	−6.00000	−0.378717	−0.189358	−	0.981908i	$-0.560641\pi$
$251$	−6.00000	−0.378717	−0.189358	+	0.981908i	$0.560641\pi$
$252$	2.00000i	0.125988i
$253$	6.00000i	0.377217i
$254$	12.0000	0.752947
$255$	0	0
$256$	1.00000	0.0625000
$257$	6.00000i	0.374270i	0.982334	+	0.187135i	$0.0599201\pi$
$257$	6.00000i	0.374270i	−0.982334	+	0.187135i	$0.940080\pi$
$258$	− 12.0000i	− 0.747087i
$259$	0	0
$260$	0	0
$261$	6.00000	0.371391
$262$	− 8.00000i	− 0.494242i
$263$	4.00000i	0.246651i	0.992366	+	0.123325i	$0.0393559\pi$
$263$	4.00000i	0.246651i	−0.992366	+	0.123325i	$0.960644\pi$
$264$	6.00000	0.369274
$265$	0	0
$266$	0	0
$267$	0	0
$268$	− 12.0000i	− 0.733017i
$269$	−18.0000	−1.09748	−0.548740	−	0.835993i	$-0.684892\pi$
$269$	−18.0000	−1.09748	−0.548740	+	0.835993i	$0.684892\pi$
$270$	0	0
$271$	28.0000	1.70088	0.850439	−	0.526073i	$-0.176336\pi$
$271$	28.0000	1.70088	0.850439	+	0.526073i	$0.176336\pi$
$272$	0	0
$273$	− 4.00000i	− 0.242091i
$274$	−12.0000	−0.724947
$275$	0	0
$276$	1.00000	0.0601929
$277$	− 26.0000i	− 1.56219i	−0.624413	−	0.781094i	$-0.714662\pi$
$277$	− 26.0000i	− 1.56219i	0.624413	−	0.781094i	$-0.285338\pi$
$278$	− 12.0000i	− 0.719712i
$279$	−8.00000	−0.478947
$280$	0	0
$281$	16.0000	0.954480	0.477240	−	0.878773i	$-0.341637\pi$
$281$	16.0000	0.954480	0.477240	+	0.878773i	$0.341637\pi$
$282$	8.00000i	0.476393i
$283$	24.0000i	1.42665i	0.700832	+	0.713326i	$0.252812\pi$
$283$	24.0000i	1.42665i	−0.700832	+	0.713326i	$0.747188\pi$
$284$	0	0
$285$	0	0
$286$	−12.0000	−0.709575
$287$	20.0000i	1.18056i
$288$	− 1.00000i	− 0.0589256i
$289$	17.0000	1.00000
$290$	0	0
$291$	6.00000	0.351726
$292$	10.0000i	0.585206i
$293$	− 6.00000i	− 0.350524i	−0.984522	−	0.175262i	$-0.943923\pi$
$293$	− 6.00000i	− 0.350524i	0.984522	−	0.175262i	$-0.0560772\pi$
$294$	3.00000	0.174964
$295$	0	0
$296$	0	0
$297$	− 6.00000i	− 0.348155i
$298$	18.0000i	1.04271i
$299$	−2.00000	−0.115663
$300$	0	0
$301$	24.0000	1.38334
$302$	− 12.0000i	− 0.690522i
$303$	6.00000i	0.344691i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	20.0000i	1.14146i	0.821138	+	0.570730i	$0.193340\pi$
$307$	20.0000i	1.14146i	−0.821138	+	0.570730i	$0.806660\pi$
$308$	12.0000i	0.683763i
$309$	14.0000	0.796432
$310$	0	0
$311$	16.0000	0.907277	0.453638	−	0.891186i	$-0.350126\pi$
$311$	16.0000	0.907277	0.453638	+	0.891186i	$0.350126\pi$
$312$	2.00000i	0.113228i
$313$	− 26.0000i	− 1.46961i	−0.678280	−	0.734803i	$-0.737274\pi$
$313$	− 26.0000i	− 1.46961i	0.678280	−	0.734803i	$-0.262726\pi$
$314$	16.0000	0.902932
$315$	0	0
$316$	−6.00000	−0.337526
$317$	− 22.0000i	− 1.23564i	−0.786318	−	0.617822i	$-0.788015\pi$
$317$	− 22.0000i	− 1.23564i	0.786318	−	0.617822i	$-0.211985\pi$
$318$	2.00000i	0.112154i
$319$	36.0000	2.01561
$320$	0	0
$321$	−14.0000	−0.781404
$322$	2.00000i	0.111456i
$323$	0	0
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	12.0000	0.664619
$327$	− 16.0000i	− 0.884802i
$328$	− 10.0000i	− 0.552158i
$329$	−16.0000	−0.882109
$330$	0	0
$331$	4.00000	0.219860	0.109930	−	0.993939i	$-0.464937\pi$
$331$	4.00000	0.219860	0.109930	+	0.993939i	$0.464937\pi$
$332$	− 14.0000i	− 0.768350i
$333$	0	0
$334$	8.00000	0.437741
$335$	0	0
$336$	2.00000	0.109109
$337$	10.0000i	0.544735i	0.962193	+	0.272367i	$0.0878066\pi$
$337$	10.0000i	0.544735i	−0.962193	+	0.272367i	$0.912193\pi$
$338$	9.00000i	0.489535i
$339$	−8.00000	−0.434500
$340$	0	0
$341$	−48.0000	−2.59935
$342$	0	0
$343$	20.0000i	1.07990i
$344$	−12.0000	−0.646997
$345$	0	0
$346$	18.0000	0.967686
$347$	8.00000i	0.429463i	0.976673	+	0.214731i	$0.0688876\pi$
$347$	8.00000i	0.429463i	−0.976673	+	0.214731i	$0.931112\pi$
$348$	− 6.00000i	− 0.321634i
$349$	−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
$352$	− 6.00000i	− 0.319801i
$353$	− 2.00000i	− 0.106449i	−0.998583	−	0.0532246i	$-0.983050\pi$
$353$	− 2.00000i	− 0.106449i	0.998583	−	0.0532246i	$-0.0169499\pi$
$354$	12.0000	0.637793
$355$	0	0
$356$	0	0
$357$	0	0
$358$	16.0000i	0.845626i
$359$	−24.0000	−1.26667	−0.633336	−	0.773877i	$-0.718315\pi$
$359$	−24.0000	−1.26667	−0.633336	+	0.773877i	$0.718315\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	8.00000i	0.420471i
$363$	− 25.0000i	− 1.31216i
$364$	−4.00000	−0.209657
$365$	0	0
$366$	4.00000	0.209083
$367$	14.0000i	0.730794i	0.930852	+	0.365397i	$0.119067\pi$
$367$	14.0000i	0.730794i	−0.930852	+	0.365397i	$0.880933\pi$
$368$	− 1.00000i	− 0.0521286i
$369$	−10.0000	−0.520579
$370$	0	0
$371$	−4.00000	−0.207670
$372$	8.00000i	0.414781i
$373$	− 32.0000i	− 1.65690i	−0.560065	−	0.828449i	$-0.689224\pi$
$373$	− 32.0000i	− 1.65690i	0.560065	−	0.828449i	$-0.310776\pi$
$374$	0	0
$375$	0	0
$376$	8.00000	0.412568
$377$	12.0000i	0.618031i
$378$	− 2.00000i	− 0.102869i
$379$	0	0		−	1.00000i	$-0.5\pi$
$379$	0	0			1.00000i	$0.5\pi$
$380$	0	0
$381$	−12.0000	−0.614779
$382$	0	0
$383$	4.00000i	0.204390i	0.994764	+	0.102195i	$0.0325866\pi$
$383$	4.00000i	0.204390i	−0.994764	+	0.102195i	$0.967413\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	−14.0000	−0.712581
$387$	12.0000i	0.609994i
$388$	− 6.00000i	− 0.304604i
$389$	−6.00000	−0.304212	−0.152106	−	0.988364i	$-0.548606\pi$
$389$	−6.00000	−0.304212	−0.152106	+	0.988364i	$0.548606\pi$
$390$	0	0
$391$	0	0
$392$	− 3.00000i	− 0.151523i
$393$	8.00000i	0.403547i
$394$	−6.00000	−0.302276
$395$	0	0
$396$	−6.00000	−0.301511
$397$	− 6.00000i	− 0.301131i	−0.988600	−	0.150566i	$-0.951890\pi$
$397$	− 6.00000i	− 0.301131i	0.988600	−	0.150566i	$-0.0481095\pi$
$398$	− 2.00000i	− 0.100251i
$399$	0	0
$400$	0	0
$401$	−32.0000	−1.59800	−0.799002	−	0.601329i	$-0.794638\pi$
$401$	−32.0000	−1.59800	−0.799002	+	0.601329i	$0.794638\pi$
$402$	12.0000i	0.598506i
$403$	− 16.0000i	− 0.797017i
$404$	6.00000	0.298511
$405$	0	0
$406$	12.0000	0.595550
$407$	0	0
$408$	0	0
$409$	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
$411$	12.0000	0.591916
$412$	− 14.0000i	− 0.689730i
$413$	24.0000i	1.18096i
$414$	−1.00000	−0.0491473
$415$	0	0
$416$	2.00000	0.0980581
$417$	12.0000i	0.587643i
$418$	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
$421$	4.00000	0.194948	0.0974740	−	0.995238i	$-0.468924\pi$
$421$	4.00000	0.194948	0.0974740	+	0.995238i	$0.468924\pi$
$422$	20.0000i	0.973585i
$423$	− 8.00000i	− 0.388973i
$424$	2.00000	0.0971286
$425$	0	0
$426$	0	0
$427$	8.00000i	0.387147i
$428$	14.0000i	0.676716i
$429$	12.0000	0.579365
$430$	0	0
$431$	−12.0000	−0.578020	−0.289010	−	0.957326i	$-0.593326\pi$
$431$	−12.0000	−0.578020	−0.289010	+	0.957326i	$0.593326\pi$
$432$	1.00000i	0.0481125i
$433$	− 2.00000i	− 0.0961139i	−0.998845	−	0.0480569i	$-0.984697\pi$
$433$	− 2.00000i	− 0.0961139i	0.998845	−	0.0480569i	$-0.0153029\pi$
$434$	−16.0000	−0.768025
$435$	0	0
$436$	−16.0000	−0.766261
$437$	0	0
$438$	− 10.0000i	− 0.477818i
$439$	16.0000	0.763638	0.381819	−	0.924237i	$-0.375298\pi$
$439$	16.0000	0.763638	0.381819	+	0.924237i	$0.375298\pi$
$440$	0	0
$441$	−3.00000	−0.142857
$442$	0	0
$443$	36.0000i	1.71041i	0.518289	+	0.855206i	$0.326569\pi$
$443$	36.0000i	1.71041i	−0.518289	+	0.855206i	$0.673431\pi$
$444$	0	0
$445$	0	0
$446$	−12.0000	−0.568216
$447$	− 18.0000i	− 0.851371i
$448$	− 2.00000i	− 0.0944911i
$449$	−2.00000	−0.0943858	−0.0471929	−	0.998886i	$-0.515028\pi$
$449$	−2.00000	−0.0943858	−0.0471929	+	0.998886i	$0.515028\pi$
$450$	0	0
$451$	−60.0000	−2.82529
$452$	8.00000i	0.376288i
$453$	12.0000i	0.563809i
$454$	−10.0000	−0.469323
$455$	0	0
$456$	0	0
$457$	− 26.0000i	− 1.21623i	−0.793849	−	0.608114i	$-0.791926\pi$
$457$	− 26.0000i	− 1.21623i	0.793849	−	0.608114i	$-0.208074\pi$
$458$	24.0000i	1.12145i
$459$	0	0
$460$	0	0
$461$	−30.0000	−1.39724	−0.698620	−	0.715493i	$-0.746202\pi$
$461$	−30.0000	−1.39724	−0.698620	+	0.715493i	$0.746202\pi$
$462$	− 12.0000i	− 0.558291i
$463$	− 40.0000i	− 1.85896i	−0.368875	−	0.929479i	$-0.620257\pi$
$463$	− 40.0000i	− 1.85896i	0.368875	−	0.929479i	$-0.379743\pi$
$464$	−6.00000	−0.278543
$465$	0	0
$466$	−10.0000	−0.463241
$467$	− 18.0000i	− 0.832941i	−0.909149	−	0.416470i	$-0.863267\pi$
$467$	− 18.0000i	− 0.832941i	0.909149	−	0.416470i	$-0.136733\pi$
$468$	− 2.00000i	− 0.0924500i
$469$	−24.0000	−1.10822
$470$	0	0
$471$	−16.0000	−0.737241
$472$	− 12.0000i	− 0.552345i
$473$	72.0000i	3.31056i
$474$	6.00000	0.275589
$475$	0	0
$476$	0	0
$477$	− 2.00000i	− 0.0915737i
$478$	16.0000i	0.731823i
$479$	−24.0000	−1.09659	−0.548294	−	0.836286i	$-0.684723\pi$
$479$	−24.0000	−1.09659	−0.548294	+	0.836286i	$0.684723\pi$
$480$	0	0
$481$	0	0
$482$	− 6.00000i	− 0.273293i
$483$	− 2.00000i	− 0.0910032i
$484$	−25.0000	−1.13636
$485$	0	0
$486$	1.00000	0.0453609
$487$	− 8.00000i	− 0.362515i	−0.983436	−	0.181257i	$-0.941983\pi$
$487$	− 8.00000i	− 0.362515i	0.983436	−	0.181257i	$-0.0580167\pi$
$488$	− 4.00000i	− 0.181071i
$489$	−12.0000	−0.542659
$490$	0	0
$491$	−32.0000	−1.44414	−0.722070	−	0.691820i	$-0.756809\pi$
$491$	−32.0000	−1.44414	−0.722070	+	0.691820i	$0.756809\pi$
$492$	10.0000i	0.450835i
$493$	0	0
$494$	0	0
$495$	0	0
$496$	8.00000	0.359211
$497$	0	0
$498$	14.0000i	0.627355i
$499$	20.0000	0.895323	0.447661	−	0.894203i	$-0.352257\pi$
$499$	20.0000	0.895323	0.447661	+	0.894203i	$0.352257\pi$
$500$	0	0
$501$	−8.00000	−0.357414
$502$	− 6.00000i	− 0.267793i
$503$	− 12.0000i	− 0.535054i	−0.963550	−	0.267527i	$-0.913794\pi$
$503$	− 12.0000i	− 0.535054i	0.963550	−	0.267527i	$-0.0862064\pi$
$504$	−2.00000	−0.0890871
$505$	0	0
$506$	−6.00000	−0.266733
$507$	− 9.00000i	− 0.399704i
$508$	12.0000i	0.532414i
$509$	10.0000	0.443242	0.221621	−	0.975133i	$-0.428865\pi$
$509$	10.0000	0.443242	0.221621	+	0.975133i	$0.428865\pi$
$510$	0	0
$511$	20.0000	0.884748
$512$	1.00000i	0.0441942i
$513$	0	0
$514$	−6.00000	−0.264649
$515$	0	0
$516$	12.0000	0.528271
$517$	− 48.0000i	− 2.11104i
$518$	0	0
$519$	−18.0000	−0.790112
$520$	0	0
$521$	4.00000	0.175243	0.0876216	−	0.996154i	$-0.472073\pi$
$521$	4.00000	0.175243	0.0876216	+	0.996154i	$0.472073\pi$
$522$	6.00000i	0.262613i
$523$	− 4.00000i	− 0.174908i	−0.996169	−	0.0874539i	$-0.972127\pi$
$523$	− 4.00000i	− 0.174908i	0.996169	−	0.0874539i	$-0.0278730\pi$
$524$	8.00000	0.349482
$525$	0	0
$526$	−4.00000	−0.174408
$527$	0	0
$528$	6.00000i	0.261116i
$529$	−1.00000	−0.0434783
$530$	0	0
$531$	−12.0000	−0.520756
$532$	0	0
$533$	− 20.0000i	− 0.866296i
$534$	0	0
$535$	0	0
$536$	12.0000	0.518321
$537$	− 16.0000i	− 0.690451i
$538$	− 18.0000i	− 0.776035i
$539$	−18.0000	−0.775315
$540$	0	0
$541$	26.0000	1.11783	0.558914	−	0.829226i	$-0.311218\pi$
$541$	26.0000	1.11783	0.558914	+	0.829226i	$0.311218\pi$
$542$	28.0000i	1.20270i
$543$	− 8.00000i	− 0.343313i
$544$	0	0
$545$	0	0
$546$	4.00000	0.171184
$547$	− 28.0000i	− 1.19719i	−0.801050	−	0.598597i	$-0.795725\pi$
$547$	− 28.0000i	− 1.19719i	0.801050	−	0.598597i	$-0.204275\pi$
$548$	− 12.0000i	− 0.512615i
$549$	−4.00000	−0.170716
$550$	0	0
$551$	0	0
$552$	1.00000i	0.0425628i
$553$	12.0000i	0.510292i
$554$	26.0000	1.10463
$555$	0	0
$556$	12.0000	0.508913
$557$	− 6.00000i	− 0.254228i	−0.991888	−	0.127114i	$-0.959429\pi$
$557$	− 6.00000i	− 0.254228i	0.991888	−	0.127114i	$-0.0405714\pi$
$558$	− 8.00000i	− 0.338667i
$559$	−24.0000	−1.01509
$560$	0	0
$561$	0	0
$562$	16.0000i	0.674919i
$563$	14.0000i	0.590030i	0.955493	+	0.295015i	$0.0953246\pi$
$563$	14.0000i	0.590030i	−0.955493	+	0.295015i	$0.904675\pi$
$564$	−8.00000	−0.336861
$565$	0	0
$566$	−24.0000	−1.00880
$567$	2.00000i	0.0839921i
$568$	0	0
$569$	−16.0000	−0.670755	−0.335377	−	0.942084i	$-0.608864\pi$
$569$	−16.0000	−0.670755	−0.335377	+	0.942084i	$0.608864\pi$
$570$	0	0
$571$	4.00000	0.167395	0.0836974	−	0.996491i	$-0.473327\pi$
$571$	4.00000	0.167395	0.0836974	+	0.996491i	$0.473327\pi$
$572$	− 12.0000i	− 0.501745i
$573$	0	0
$574$	−20.0000	−0.834784
$575$	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	$-0.0132552\pi$
$578$	17.0000i	0.707107i
$579$	14.0000	0.581820
$580$	0	0
$581$	−28.0000	−1.16164
$582$	6.00000i	0.248708i
$583$	− 12.0000i	− 0.496989i
$584$	−10.0000	−0.413803
$585$	0	0
$586$	6.00000	0.247858
$587$	− 16.0000i	− 0.660391i	−0.943913	−	0.330195i	$-0.892885\pi$
$587$	− 16.0000i	− 0.660391i	0.943913	−	0.330195i	$-0.107115\pi$
$588$	3.00000i	0.123718i
$589$	0	0
$590$	0	0
$591$	6.00000	0.246807
$592$	0	0
$593$	− 6.00000i	− 0.246390i	−0.992382	−	0.123195i	$-0.960686\pi$
$593$	− 6.00000i	− 0.246390i	0.992382	−	0.123195i	$-0.0393141\pi$
$594$	6.00000	0.246183
$595$	0	0
$596$	−18.0000	−0.737309
$597$	2.00000i	0.0818546i
$598$	− 2.00000i	− 0.0817861i
$599$	32.0000	1.30748	0.653742	−	0.756717i	$-0.273198\pi$
$599$	32.0000	1.30748	0.653742	+	0.756717i	$0.273198\pi$
$600$	0	0
$601$	10.0000	0.407909	0.203954	−	0.978980i	$-0.434621\pi$
$601$	10.0000	0.407909	0.203954	+	0.978980i	$0.434621\pi$
$602$	24.0000i	0.978167i
$603$	− 12.0000i	− 0.488678i
$604$	12.0000	0.488273
$605$	0	0
$606$	−6.00000	−0.243733
$607$	4.00000i	0.162355i	0.996700	+	0.0811775i	$0.0258681\pi$
$607$	4.00000i	0.162355i	−0.996700	+	0.0811775i	$0.974132\pi$
$608$	0	0
$609$	−12.0000	−0.486265
$610$	0	0
$611$	16.0000	0.647291
$612$	0	0
$613$	− 4.00000i	− 0.161558i	−0.996732	−	0.0807792i	$-0.974259\pi$
$613$	− 4.00000i	− 0.161558i	0.996732	−	0.0807792i	$-0.0257409\pi$
$614$	−20.0000	−0.807134
$615$	0	0
$616$	−12.0000	−0.483494
$617$	32.0000i	1.28827i	0.764911	+	0.644136i	$0.222783\pi$
$617$	32.0000i	1.28827i	−0.764911	+	0.644136i	$0.777217\pi$
$618$	14.0000i	0.563163i
$619$	0	0		−	1.00000i	$-0.5\pi$
$619$	0	0			1.00000i	$0.5\pi$
$620$	0	0
$621$	1.00000	0.0401286
$622$	16.0000i	0.641542i
$623$	0	0
$624$	−2.00000	−0.0800641
$625$	0	0
$626$	26.0000	1.03917
$627$	0	0
$628$	16.0000i	0.638470i
$629$	0	0
$630$	0	0
$631$	−2.00000	−0.0796187	−0.0398094	−	0.999207i	$-0.512675\pi$
$631$	−2.00000	−0.0796187	−0.0398094	+	0.999207i	$0.512675\pi$
$632$	− 6.00000i	− 0.238667i
$633$	− 20.0000i	− 0.794929i
$634$	22.0000	0.873732
$635$	0	0
$636$	−2.00000	−0.0793052
$637$	− 6.00000i	− 0.237729i
$638$	36.0000i	1.42525i
$639$	0	0
$640$	0	0
$641$	16.0000	0.631962	0.315981	−	0.948766i	$-0.397666\pi$
$641$	16.0000	0.631962	0.315981	+	0.948766i	$0.397666\pi$
$642$	− 14.0000i	− 0.552536i
$643$	44.0000i	1.73519i	0.497271	+	0.867595i	$0.334335\pi$
$643$	44.0000i	1.73519i	−0.497271	+	0.867595i	$0.665665\pi$
$644$	−2.00000	−0.0788110
$645$	0	0
$646$	0	0
$647$	40.0000i	1.57256i	0.617869	+	0.786281i	$0.287996\pi$
$647$	40.0000i	1.57256i	−0.617869	+	0.786281i	$0.712004\pi$
$648$	− 1.00000i	− 0.0392837i
$649$	−72.0000	−2.82625
$650$	0	0
$651$	16.0000	0.627089
$652$	12.0000i	0.469956i
$653$	− 22.0000i	− 0.860927i	−0.902608	−	0.430463i	$-0.858350\pi$
$653$	− 22.0000i	− 0.860927i	0.902608	−	0.430463i	$-0.141650\pi$
$654$	16.0000	0.625650
$655$	0	0
$656$	10.0000	0.390434
$657$	10.0000i	0.390137i
$658$	− 16.0000i	− 0.623745i
$659$	−42.0000	−1.63609	−0.818044	−	0.575156i	$-0.804941\pi$
$659$	−42.0000	−1.63609	−0.818044	+	0.575156i	$0.804941\pi$
$660$	0	0
$661$	−4.00000	−0.155582	−0.0777910	−	0.996970i	$-0.524787\pi$
$661$	−4.00000	−0.155582	−0.0777910	+	0.996970i	$0.524787\pi$
$662$	4.00000i	0.155464i
$663$	0	0
$664$	14.0000	0.543305
$665$	0	0
$666$	0	0
$667$	6.00000i	0.232321i
$668$	8.00000i	0.309529i
$669$	12.0000	0.463947
$670$	0	0
$671$	−24.0000	−0.926510
$672$	2.00000i	0.0771517i
$673$	− 46.0000i	− 1.77317i	−0.462566	−	0.886585i	$-0.653071\pi$
$673$	− 46.0000i	− 1.77317i	0.462566	−	0.886585i	$-0.346929\pi$
$674$	−10.0000	−0.385186
$675$	0	0
$676$	−9.00000	−0.346154
$677$	− 6.00000i	− 0.230599i	−0.993331	−	0.115299i	$-0.963217\pi$
$677$	− 6.00000i	− 0.230599i	0.993331	−	0.115299i	$-0.0367827\pi$
$678$	− 8.00000i	− 0.307238i
$679$	−12.0000	−0.460518
$680$	0	0
$681$	10.0000	0.383201
$682$	− 48.0000i	− 1.83801i
$683$	− 24.0000i	− 0.918334i	−0.888350	−	0.459167i	$-0.848148\pi$
$683$	− 24.0000i	− 0.918334i	0.888350	−	0.459167i	$-0.151852\pi$
$684$	0	0
$685$	0	0
$686$	−20.0000	−0.763604
$687$	− 24.0000i	− 0.915657i
$688$	− 12.0000i	− 0.457496i
$689$	4.00000	0.152388
$690$	0	0
$691$	−4.00000	−0.152167	−0.0760836	−	0.997101i	$-0.524242\pi$
$691$	−4.00000	−0.152167	−0.0760836	+	0.997101i	$0.524242\pi$
$692$	18.0000i	0.684257i
$693$	12.0000i	0.455842i
$694$	−8.00000	−0.303676
$695$	0	0
$696$	6.00000	0.227429
$697$	0	0
$698$	− 26.0000i	− 0.984115i
$699$	10.0000	0.378235
$700$	0	0
$701$	2.00000	0.0755390	0.0377695	−	0.999286i	$-0.487975\pi$
$701$	2.00000	0.0755390	0.0377695	+	0.999286i	$0.487975\pi$
$702$	2.00000i	0.0754851i
$703$	0	0
$704$	6.00000	0.226134
$705$	0	0
$706$	2.00000	0.0752710
$707$	− 12.0000i	− 0.451306i
$708$	12.0000i	0.450988i
$709$	−4.00000	−0.150223	−0.0751116	−	0.997175i	$-0.523931\pi$
$709$	−4.00000	−0.150223	−0.0751116	+	0.997175i	$0.523931\pi$
$710$	0	0
$711$	−6.00000	−0.225018
$712$	0	0
$713$	− 8.00000i	− 0.299602i
$714$	0	0
$715$	0	0
$716$	−16.0000	−0.597948
$717$	− 16.0000i	− 0.597531i
$718$	− 24.0000i	− 0.895672i
$719$	48.0000	1.79010	0.895049	−	0.445968i	$-0.147140\pi$
$719$	48.0000	1.79010	0.895049	+	0.445968i	$0.147140\pi$
$720$	0	0
$721$	−28.0000	−1.04277
$722$	− 19.0000i	− 0.707107i
$723$	6.00000i	0.223142i
$724$	−8.00000	−0.297318
$725$	0	0
$726$	25.0000	0.927837
$727$	18.0000i	0.667583i	0.942647	+	0.333792i	$0.108328\pi$
$727$	18.0000i	0.667583i	−0.942647	+	0.333792i	$0.891672\pi$
$728$	− 4.00000i	− 0.148250i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	0	0
$732$	4.00000i	0.147844i
$733$	36.0000i	1.32969i	0.746981	+	0.664845i	$0.231502\pi$
$733$	36.0000i	1.32969i	−0.746981	+	0.664845i	$0.768498\pi$
$734$	−14.0000	−0.516749
$735$	0	0
$736$	1.00000	0.0368605
$737$	− 72.0000i	− 2.65215i
$738$	− 10.0000i	− 0.368105i
$739$	20.0000	0.735712	0.367856	−	0.929883i	$-0.380092\pi$
$739$	20.0000	0.735712	0.367856	+	0.929883i	$0.380092\pi$
$740$	0	0
$741$	0	0
$742$	− 4.00000i	− 0.146845i
$743$	4.00000i	0.146746i	0.997305	+	0.0733729i	$0.0233763\pi$
$743$	4.00000i	0.146746i	−0.997305	+	0.0733729i	$0.976624\pi$
$744$	−8.00000	−0.293294
$745$	0	0
$746$	32.0000	1.17160
$747$	− 14.0000i	− 0.512233i
$748$	0	0
$749$	28.0000	1.02310
$750$	0	0
$751$	−38.0000	−1.38664	−0.693320	−	0.720630i	$-0.743853\pi$
$751$	−38.0000	−1.38664	−0.693320	+	0.720630i	$0.743853\pi$
$752$	8.00000i	0.291730i
$753$	6.00000i	0.218652i
$754$	−12.0000	−0.437014
$755$	0	0
$756$	2.00000	0.0727393
$757$	32.0000i	1.16306i	0.813525	+	0.581530i	$0.197546\pi$
$757$	32.0000i	1.16306i	−0.813525	+	0.581530i	$0.802454\pi$
$758$	0	0
$759$	6.00000	0.217786
$760$	0	0
$761$	30.0000	1.08750	0.543750	−	0.839248i	$-0.317004\pi$
$761$	30.0000	1.08750	0.543750	+	0.839248i	$0.317004\pi$
$762$	− 12.0000i	− 0.434714i
$763$	32.0000i	1.15848i
$764$	0	0
$765$	0	0
$766$	−4.00000	−0.144526
$767$	− 24.0000i	− 0.866590i
$768$	− 1.00000i	− 0.0360844i
$769$	26.0000	0.937584	0.468792	−	0.883309i	$-0.344689\pi$
$769$	26.0000	0.937584	0.468792	+	0.883309i	$0.344689\pi$
$770$	0	0
$771$	6.00000	0.216085
$772$	− 14.0000i	− 0.503871i
$773$	− 30.0000i	− 1.07903i	−0.841978	−	0.539513i	$-0.818609\pi$
$773$	− 30.0000i	− 1.07903i	0.841978	−	0.539513i	$-0.181391\pi$
$774$	−12.0000	−0.431331
$775$	0	0
$776$	6.00000	0.215387
$777$	0	0
$778$	− 6.00000i	− 0.215110i
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	− 6.00000i	− 0.214423i
$784$	3.00000	0.107143
$785$	0	0
$786$	−8.00000	−0.285351
$787$	28.0000i	0.998092i	0.866575	+	0.499046i	$0.166316\pi$
$787$	28.0000i	0.998092i	−0.866575	+	0.499046i	$0.833684\pi$
$788$	− 6.00000i	− 0.213741i
$789$	4.00000	0.142404
$790$	0	0
$791$	16.0000	0.568895
$792$	− 6.00000i	− 0.213201i
$793$	− 8.00000i	− 0.284088i
$794$	6.00000	0.212932
$795$	0	0
$796$	2.00000	0.0708881
$797$	18.0000i	0.637593i	0.947823	+	0.318796i	$0.103279\pi$
$797$	18.0000i	0.637593i	−0.947823	+	0.318796i	$0.896721\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	0	0
$802$	− 32.0000i	− 1.12996i
$803$	60.0000i	2.11735i
$804$	−12.0000	−0.423207
$805$	0	0
$806$	16.0000	0.563576
$807$	18.0000i	0.633630i
$808$	6.00000i	0.211079i
$809$	2.00000	0.0703163	0.0351581	−	0.999382i	$-0.488807\pi$
$809$	2.00000	0.0703163	0.0351581	+	0.999382i	$0.488807\pi$
$810$	0	0
$811$	28.0000	0.983213	0.491606	−	0.870817i	$-0.336410\pi$
$811$	28.0000	0.983213	0.491606	+	0.870817i	$0.336410\pi$
$812$	12.0000i	0.421117i
$813$	− 28.0000i	− 0.982003i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	10.0000i	0.349642i
$819$	−4.00000	−0.139771
$820$	0	0
$821$	−6.00000	−0.209401	−0.104701	−	0.994504i	$-0.533388\pi$
$821$	−6.00000	−0.209401	−0.104701	+	0.994504i	$0.533388\pi$
$822$	12.0000i	0.418548i
$823$	− 24.0000i	− 0.836587i	−0.908312	−	0.418294i	$-0.862628\pi$
$823$	− 24.0000i	− 0.836587i	0.908312	−	0.418294i	$-0.137372\pi$
$824$	14.0000	0.487713
$825$	0	0
$826$	−24.0000	−0.835067
$827$	− 18.0000i	− 0.625921i	−0.949766	−	0.312961i	$-0.898679\pi$
$827$	− 18.0000i	− 0.625921i	0.949766	−	0.312961i	$-0.101321\pi$
$828$	− 1.00000i	− 0.0347524i
$829$	−42.0000	−1.45872	−0.729360	−	0.684130i	$-0.760182\pi$
$829$	−42.0000	−1.45872	−0.729360	+	0.684130i	$0.760182\pi$
$830$	0	0
$831$	−26.0000	−0.901930
$832$	2.00000i	0.0693375i
$833$	0	0
$834$	−12.0000	−0.415526
$835$	0	0
$836$	0	0
$837$	8.00000i	0.276520i
$838$	26.0000i	0.898155i
$839$	12.0000	0.414286	0.207143	−	0.978311i	$-0.433583\pi$
$839$	12.0000	0.414286	0.207143	+	0.978311i	$0.433583\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	4.00000i	0.137849i
$843$	− 16.0000i	− 0.551069i
$844$	−20.0000	−0.688428
$845$	0	0
$846$	8.00000	0.275046
$847$	50.0000i	1.71802i
$848$	2.00000i	0.0686803i
$849$	24.0000	0.823678
$850$	0	0
$851$	0	0
$852$	0	0
$853$	6.00000i	0.205436i	0.994711	+	0.102718i	$0.0327539\pi$
$853$	6.00000i	0.205436i	−0.994711	+	0.102718i	$0.967246\pi$
$854$	−8.00000	−0.273754
$855$	0	0
$856$	−14.0000	−0.478510
$857$	− 2.00000i	− 0.0683187i	−0.999416	−	0.0341593i	$-0.989125\pi$
$857$	− 2.00000i	− 0.0683187i	0.999416	−	0.0341593i	$-0.0108754\pi$
$858$	12.0000i	0.409673i
$859$	4.00000	0.136478	0.0682391	−	0.997669i	$-0.478262\pi$
$859$	4.00000	0.136478	0.0682391	+	0.997669i	$0.478262\pi$
$860$	0	0
$861$	20.0000	0.681598
$862$	− 12.0000i	− 0.408722i
$863$	32.0000i	1.08929i	0.838666	+	0.544646i	$0.183336\pi$
$863$	32.0000i	1.08929i	−0.838666	+	0.544646i	$0.816664\pi$
$864$	−1.00000	−0.0340207
$865$	0	0
$866$	2.00000	0.0679628
$867$	− 17.0000i	− 0.577350i
$868$	− 16.0000i	− 0.543075i
$869$	−36.0000	−1.22122
$870$	0	0
$871$	24.0000	0.813209
$872$	− 16.0000i	− 0.541828i
$873$	− 6.00000i	− 0.203069i
$874$	0	0
$875$	0	0
$876$	10.0000	0.337869
$877$	− 10.0000i	− 0.337676i	−0.985644	−	0.168838i	$-0.945999\pi$
$877$	− 10.0000i	− 0.337676i	0.985644	−	0.168838i	$-0.0540015\pi$
$878$	16.0000i	0.539974i
$879$	−6.00000	−0.202375
$880$	0	0
$881$	−32.0000	−1.07811	−0.539054	−	0.842271i	$-0.681218\pi$
$881$	−32.0000	−1.07811	−0.539054	+	0.842271i	$0.681218\pi$
$882$	− 3.00000i	− 0.101015i
$883$	36.0000i	1.21150i	0.795656	+	0.605748i	$0.207126\pi$
$883$	36.0000i	1.21150i	−0.795656	+	0.605748i	$0.792874\pi$
$884$	0	0
$885$	0	0
$886$	−36.0000	−1.20944
$887$	16.0000i	0.537227i	0.963248	+	0.268614i	$0.0865655\pi$
$887$	16.0000i	0.537227i	−0.963248	+	0.268614i	$0.913434\pi$
$888$	0	0
$889$	24.0000	0.804934
$890$	0	0
$891$	−6.00000	−0.201008
$892$	− 12.0000i	− 0.401790i
$893$	0	0
$894$	18.0000	0.602010
$895$	0	0
$896$	2.00000	0.0668153
$897$	2.00000i	0.0667781i
$898$	− 2.00000i	− 0.0667409i
$899$	−48.0000	−1.60089
$900$	0	0
$901$	0	0
$902$	− 60.0000i	− 1.99778i
$903$	− 24.0000i	− 0.798670i
$904$	−8.00000	−0.266076
$905$	0	0
$906$	−12.0000	−0.398673
$907$	− 16.0000i	− 0.531271i	−0.964073	−	0.265636i	$-0.914418\pi$
$907$	− 16.0000i	− 0.531271i	0.964073	−	0.265636i	$-0.0855818\pi$
$908$	− 10.0000i	− 0.331862i
$909$	6.00000	0.199007
$910$	0	0
$911$	−20.0000	−0.662630	−0.331315	−	0.943520i	$-0.607492\pi$
$911$	−20.0000	−0.662630	−0.331315	+	0.943520i	$0.607492\pi$
$912$	0	0
$913$	− 84.0000i	− 2.77999i
$914$	26.0000	0.860004
$915$	0	0
$916$	−24.0000	−0.792982
$917$	− 16.0000i	− 0.528367i
$918$	0	0
$919$	−10.0000	−0.329870	−0.164935	−	0.986304i	$-0.552741\pi$
$919$	−10.0000	−0.329870	−0.164935	+	0.986304i	$0.552741\pi$
$920$	0	0
$921$	20.0000	0.659022
$922$	− 30.0000i	− 0.987997i
$923$	0	0
$924$	12.0000	0.394771
$925$	0	0
$926$	40.0000	1.31448
$927$	− 14.0000i	− 0.459820i
$928$	− 6.00000i	− 0.196960i
$929$	54.0000	1.77168	0.885841	−	0.463988i	$-0.153582\pi$
$929$	54.0000	1.77168	0.885841	+	0.463988i	$0.153582\pi$
$930$	0	0
$931$	0	0
$932$	− 10.0000i	− 0.327561i
$933$	− 16.0000i	− 0.523816i
$934$	18.0000	0.588978
$935$	0	0
$936$	2.00000	0.0653720
$937$	− 2.00000i	− 0.0653372i	−0.999466	−	0.0326686i	$-0.989599\pi$
$937$	− 2.00000i	− 0.0653372i	0.999466	−	0.0326686i	$-0.0104006\pi$
$938$	− 24.0000i	− 0.783628i
$939$	−26.0000	−0.848478
$940$	0	0
$941$	54.0000	1.76035	0.880175	−	0.474650i	$-0.157425\pi$
$941$	54.0000	1.76035	0.880175	+	0.474650i	$0.157425\pi$
$942$	− 16.0000i	− 0.521308i
$943$	− 10.0000i	− 0.325645i
$944$	12.0000	0.390567
$945$	0	0
$946$	−72.0000	−2.34092
$947$	− 8.00000i	− 0.259965i	−0.991516	−	0.129983i	$-0.958508\pi$
$947$	− 8.00000i	− 0.259965i	0.991516	−	0.129983i	$-0.0414921\pi$
$948$	6.00000i	0.194871i
$949$	−20.0000	−0.649227
$950$	0	0
$951$	−22.0000	−0.713399
$952$	0	0
$953$	− 24.0000i	− 0.777436i	−0.921357	−	0.388718i	$-0.872918\pi$
$953$	− 24.0000i	− 0.777436i	0.921357	−	0.388718i	$-0.127082\pi$
$954$	2.00000	0.0647524
$955$	0	0
$956$	−16.0000	−0.517477
$957$	− 36.0000i	− 1.16371i
$958$	− 24.0000i	− 0.775405i
$959$	−24.0000	−0.775000
$960$	0	0
$961$	33.0000	1.06452
$962$	0	0
$963$	14.0000i	0.451144i
$964$	6.00000	0.193247
$965$	0	0
$966$	2.00000	0.0643489
$967$	8.00000i	0.257263i	0.991692	+	0.128631i	$0.0410584\pi$
$967$	8.00000i	0.257263i	−0.991692	+	0.128631i	$0.958942\pi$
$968$	− 25.0000i	− 0.803530i
$969$	0	0
$970$	0	0
$971$	−10.0000	−0.320915	−0.160458	−	0.987043i	$-0.551297\pi$
$971$	−10.0000	−0.320915	−0.160458	+	0.987043i	$0.551297\pi$
$972$	1.00000i	0.0320750i
$973$	− 24.0000i	− 0.769405i
$974$	8.00000	0.256337
$975$	0	0
$976$	4.00000	0.128037
$977$	− 12.0000i	− 0.383914i	−0.981403	−	0.191957i	$-0.938517\pi$
$977$	− 12.0000i	− 0.383914i	0.981403	−	0.191957i	$-0.0614834\pi$
$978$	− 12.0000i	− 0.383718i
$979$	0	0
$980$	0	0
$981$	−16.0000	−0.510841
$982$	− 32.0000i	− 1.02116i
$983$	52.0000i	1.65854i	0.558846	+	0.829271i	$0.311244\pi$
$983$	52.0000i	1.65854i	−0.558846	+	0.829271i	$0.688756\pi$
$984$	−10.0000	−0.318788
$985$	0	0
$986$	0	0
$987$	16.0000i	0.509286i
$988$	0	0
$989$	−12.0000	−0.381578
$990$	0	0
$991$	−16.0000	−0.508257	−0.254128	−	0.967170i	$-0.581789\pi$
$991$	−16.0000	−0.508257	−0.254128	+	0.967170i	$0.581789\pi$
$992$	8.00000i	0.254000i
$993$	− 4.00000i	− 0.126936i
$994$	0	0
$995$	0	0
$996$	−14.0000	−0.443607
$997$	− 10.0000i	− 0.316703i	−0.987383	−	0.158352i	$-0.949382\pi$
$997$	− 10.0000i	− 0.316703i	0.987383	−	0.158352i	$-0.0506179\pi$
$998$	20.0000i	0.633089i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3450.2.d.j.2899.2		2
5.2	odd	4		138.2.a.a.1.1	✓	1
5.3	odd	4		3450.2.a.y.1.1		1
5.4	even	2	inner	3450.2.d.j.2899.1		2
15.2	even	4		414.2.a.d.1.1		1
20.7	even	4		1104.2.a.e.1.1		1
35.27	even	4		6762.2.a.q.1.1		1
40.27	even	4		4416.2.a.m.1.1		1
40.37	odd	4		4416.2.a.z.1.1		1
60.47	odd	4		3312.2.a.n.1.1		1
115.22	even	4		3174.2.a.b.1.1		1
345.137	odd	4		9522.2.a.i.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
138.2.a.a.1.1	✓	1	5.2	odd	4
414.2.a.d.1.1		1	15.2	even	4
1104.2.a.e.1.1		1	20.7	even	4
3174.2.a.b.1.1		1	115.22	even	4
3312.2.a.n.1.1		1	60.47	odd	4
3450.2.a.y.1.1		1	5.3	odd	4
3450.2.d.j.2899.1		2	5.4	even	2	inner
3450.2.d.j.2899.2		2	1.1	even	1	trivial
4416.2.a.m.1.1		1	40.27	even	4
4416.2.a.z.1.1		1	40.37	odd	4
6762.2.a.q.1.1		1	35.27	even	4
9522.2.a.i.1.1		1	345.137	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:	\( N \)	\(=\)	\( 3450 = 2 \cdot 3 \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3450.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +2.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +2.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} -6.00000 q^{11} +1.00000i q^{12} -2.00000i q^{13} -2.00000 q^{14} +1.00000 q^{16} -1.00000i q^{18} +2.00000 q^{21} -6.00000i q^{22} -1.00000i q^{23} -1.00000 q^{24} +2.00000 q^{26} +1.00000i q^{27} -2.00000i q^{28} -6.00000 q^{29} +8.00000 q^{31} +1.00000i q^{32} +6.00000i q^{33} +1.00000 q^{36} -2.00000 q^{39} +10.0000 q^{41} +2.00000i q^{42} -12.0000i q^{43} +6.00000 q^{44} +1.00000 q^{46} +8.00000i q^{47} -1.00000i q^{48} +3.00000 q^{49} +2.00000i q^{52} +2.00000i q^{53} -1.00000 q^{54} +2.00000 q^{56} -6.00000i q^{58} +12.0000 q^{59} +4.00000 q^{61} +8.00000i q^{62} -2.00000i q^{63} -1.00000 q^{64} -6.00000 q^{66} +12.0000i q^{67} -1.00000 q^{69} +1.00000i q^{72} -10.0000i q^{73} -12.0000i q^{77} -2.00000i q^{78} +6.00000 q^{79} +1.00000 q^{81} +10.0000i q^{82} +14.0000i q^{83} -2.00000 q^{84} +12.0000 q^{86} +6.00000i q^{87} +6.00000i q^{88} +4.00000 q^{91} +1.00000i q^{92} -8.00000i q^{93} -8.00000 q^{94} +1.00000 q^{96} +6.00000i q^{97} +3.00000i q^{98} +6.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} + 2 q^{6} - 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^4 + 2 * q^6 - 2 * q^9	\( 2 q - 2 q^{4} + 2 q^{6} - 2 q^{9} - 12 q^{11} - 4 q^{14} + 2 q^{16} + 4 q^{21} - 2 q^{24} + 4 q^{26} - 12 q^{29} + 16 q^{31} + 2 q^{36} - 4 q^{39} + 20 q^{41} + 12 q^{44} + 2 q^{46} + 6 q^{49} - 2 q^{54} + 4 q^{56} + 24 q^{59} + 8 q^{61} - 2 q^{64} - 12 q^{66} - 2 q^{69} + 12 q^{79} + 2 q^{81} - 4 q^{84} + 24 q^{86} + 8 q^{91} - 16 q^{94} + 2 q^{96} + 12 q^{99}+O(q^{100}) \) 2 * q - 2 * q^4 + 2 * q^6 - 2 * q^9 - 12 * q^11 - 4 * q^14 + 2 * q^16 + 4 * q^21 - 2 * q^24 + 4 * q^26 - 12 * q^29 + 16 * q^31 + 2 * q^36 - 4 * q^39 + 20 * q^41 + 12 * q^44 + 2 * q^46 + 6 * q^49 - 2 * q^54 + 4 * q^56 + 24 * q^59 + 8 * q^61 - 2 * q^64 - 12 * q^66 - 2 * q^69 + 12 * q^79 + 2 * q^81 - 4 * q^84 + 24 * q^86 + 8 * q^91 - 16 * q^94 + 2 * q^96 + 12 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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