LMFDB - Embedded newform 3450.2.d.r.2899.1

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			2899.1
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	3450.2899
Dual form			3450.2.d.r.2899.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$	$q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +1.00000i q^{8} -1.00000 q^{9} +2.00000 q^{11} -1.00000i q^{12} +1.00000 q^{16} -6.00000i q^{17} +1.00000i q^{18} -4.00000 q^{19} -2.00000i q^{22} +1.00000i q^{23} -1.00000 q^{24} -1.00000i q^{27} -8.00000 q^{31} -1.00000i q^{32} +2.00000i q^{33} -6.00000 q^{34} +1.00000 q^{36} +6.00000i q^{37} +4.00000i q^{38} -2.00000 q^{41} -2.00000i q^{43} -2.00000 q^{44} +1.00000 q^{46} -4.00000i q^{47} +1.00000i q^{48} +7.00000 q^{49} +6.00000 q^{51} -2.00000i q^{53} -1.00000 q^{54} -4.00000i q^{57} -2.00000 q^{61} +8.00000i q^{62} -1.00000 q^{64} +2.00000 q^{66} +2.00000i q^{67} +6.00000i q^{68} -1.00000 q^{69} -10.0000 q^{71} -1.00000i q^{72} -10.0000i q^{73} +6.00000 q^{74} +4.00000 q^{76} +1.00000 q^{81} +2.00000i q^{82} -4.00000i q^{83} -2.00000 q^{86} +2.00000i q^{88} -4.00000 q^{89} -1.00000i q^{92} -8.00000i q^{93} -4.00000 q^{94} +1.00000 q^{96} -16.0000i q^{97} -7.00000i q^{98} -2.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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} + 2 q^{16} - 8 q^{19} - 2 q^{24} - 16 q^{31} - 12 q^{34} + 2 q^{36} - 4 q^{41} - 4 q^{44} + 2 q^{46} + 14 q^{49} + 12 q^{51} - 2 q^{54} - 4 q^{61} - 2 q^{64} + 4 q^{66} - 2 q^{69} - 20 q^{71} + 12 q^{74} + 8 q^{76} + 2 q^{81} - 4 q^{86} - 8 q^{89} - 8 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 + 2 * q^16 - 8 * q^19 - 2 * q^24 - 16 * q^31 - 12 * q^34 + 2 * q^36 - 4 * q^41 - 4 * q^44 + 2 * q^46 + 14 * q^49 + 12 * q^51 - 2 * q^54 - 4 * q^61 - 2 * q^64 + 4 * q^66 - 2 * q^69 - 20 * q^71 + 12 * q^74 + 8 * q^76 + 2 * q^81 - 4 * q^86 - 8 * q^89 - 8 * 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
$7$	0	0	1.00000			$0$
$7$	0	0	−1.00000			$\pi$
$8$	1.00000i	0.353553i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	2.00000	0.603023	0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000	0.603023	0.301511	+	0.953463i	$0.402509\pi$
$12$	− 1.00000i	− 0.288675i
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	− 6.00000i	− 1.45521i	−0.685994	−	0.727607i	$-0.740633\pi$
$17$	− 6.00000i	− 1.45521i	0.685994	−	0.727607i	$-0.259367\pi$
$18$	1.00000i	0.235702i
$19$	−4.00000	−0.917663	−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−4.00000	−0.917663	−0.458831	+	0.888523i	$0.651732\pi$
$20$	0	0
$21$	0	0
$22$	− 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
$28$	0	0
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	−8.00000	−1.43684	−0.718421	−	0.695608i	$-0.755135\pi$
$31$	−8.00000	−1.43684	−0.718421	+	0.695608i	$0.755135\pi$
$32$	− 1.00000i	− 0.176777i
$33$	2.00000i	0.348155i
$34$	−6.00000	−1.02899
$35$	0	0
$36$	1.00000	0.166667
$37$	6.00000i	0.986394i	0.869918	+	0.493197i	$0.164172\pi$
$37$	6.00000i	0.986394i	−0.869918	+	0.493197i	$0.835828\pi$
$38$	4.00000i	0.648886i
$39$	0	0
$40$	0	0
$41$	−2.00000	−0.312348	−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000	−0.312348	−0.156174	+	0.987730i	$0.549916\pi$
$42$	0	0
$43$	− 2.00000i	− 0.304997i	−0.988304	−	0.152499i	$-0.951268\pi$
$43$	− 2.00000i	− 0.304997i	0.988304	−	0.152499i	$-0.0487319\pi$
$44$	−2.00000	−0.301511
$45$	0	0
$46$	1.00000	0.147442
$47$	− 4.00000i	− 0.583460i	−0.956501	−	0.291730i	$-0.905769\pi$
$47$	− 4.00000i	− 0.583460i	0.956501	−	0.291730i	$-0.0942309\pi$
$48$	1.00000i	0.144338i
$49$	7.00000	1.00000
$50$	0	0
$51$	6.00000	0.840168
$52$	0	0
$53$	− 2.00000i	− 0.274721i	−0.990521	−	0.137361i	$-0.956138\pi$
$53$	− 2.00000i	− 0.274721i	0.990521	−	0.137361i	$-0.0438619\pi$
$54$	−1.00000	−0.136083
$55$	0	0
$56$	0	0
$57$	− 4.00000i	− 0.529813i
$58$	0	0
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	8.00000i	1.01600i
$63$	0	0
$64$	−1.00000	−0.125000
$65$	0	0
$66$	2.00000	0.246183
$67$	2.00000i	0.244339i	0.992509	+	0.122169i	$0.0389851\pi$
$67$	2.00000i	0.244339i	−0.992509	+	0.122169i	$0.961015\pi$
$68$	6.00000i	0.727607i
$69$	−1.00000	−0.120386
$70$	0	0
$71$	−10.0000	−1.18678	−0.593391	−	0.804914i	$-0.702211\pi$
$71$	−10.0000	−1.18678	−0.593391	+	0.804914i	$0.702211\pi$
$72$	− 1.00000i	− 0.117851i
$73$	− 10.0000i	− 1.17041i	−0.810885	−	0.585206i	$-0.801014\pi$
$73$	− 10.0000i	− 1.17041i	0.810885	−	0.585206i	$-0.198986\pi$
$74$	6.00000	0.697486
$75$	0	0
$76$	4.00000	0.458831
$77$	0	0
$78$	0	0
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	2.00000i	0.220863i
$83$	− 4.00000i	− 0.439057i	−0.975606	−	0.219529i	$-0.929548\pi$
$83$	− 4.00000i	− 0.439057i	0.975606	−	0.219529i	$-0.0704519\pi$
$84$	0	0
$85$	0	0
$86$	−2.00000	−0.215666
$87$	0	0
$88$	2.00000i	0.213201i
$89$	−4.00000	−0.423999	−0.212000	−	0.977270i	$-0.567998\pi$
$89$	−4.00000	−0.423999	−0.212000	+	0.977270i	$0.567998\pi$
$90$	0	0
$91$	0	0
$92$	− 1.00000i	− 0.104257i
$93$	− 8.00000i	− 0.829561i
$94$	−4.00000	−0.412568
$95$	0	0
$96$	1.00000	0.102062
$97$	− 16.0000i	− 1.62455i	−0.583272	−	0.812277i	$-0.698228\pi$
$97$	− 16.0000i	− 1.62455i	0.583272	−	0.812277i	$-0.301772\pi$
$98$	− 7.00000i	− 0.707107i
$99$	−2.00000	−0.201008
$100$	0	0
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	− 6.00000i	− 0.594089i
$103$	− 16.0000i	− 1.57653i	−0.615338	−	0.788263i	$-0.710980\pi$
$103$	− 16.0000i	− 1.57653i	0.615338	−	0.788263i	$-0.289020\pi$
$104$	0	0
$105$	0	0
$106$	−2.00000	−0.194257
$107$	− 8.00000i	− 0.773389i	−0.922208	−	0.386695i	$-0.873617\pi$
$107$	− 8.00000i	− 0.773389i	0.922208	−	0.386695i	$-0.126383\pi$
$108$	1.00000i	0.0962250i
$109$	−2.00000	−0.191565	−0.0957826	−	0.995402i	$-0.530535\pi$
$109$	−2.00000	−0.191565	−0.0957826	+	0.995402i	$0.530535\pi$
$110$	0	0
$111$	−6.00000	−0.569495
$112$	0	0
$113$	10.0000i	0.940721i	0.882474	+	0.470360i	$0.155876\pi$
$113$	10.0000i	0.940721i	−0.882474	+	0.470360i	$0.844124\pi$
$114$	−4.00000	−0.374634
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−7.00000	−0.636364
$122$	2.00000i	0.181071i
$123$	− 2.00000i	− 0.180334i
$124$	8.00000	0.718421
$125$	0	0
$126$	0	0
$127$	− 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
$131$	−4.00000	−0.349482	−0.174741	−	0.984614i	$-0.555909\pi$
$131$	−4.00000	−0.349482	−0.174741	+	0.984614i	$0.555909\pi$
$132$	− 2.00000i	− 0.174078i
$133$	0	0
$134$	2.00000	0.172774
$135$	0	0
$136$	6.00000	0.514496
$137$	− 2.00000i	− 0.170872i	−0.996344	−	0.0854358i	$-0.972772\pi$
$137$	− 2.00000i	− 0.170872i	0.996344	−	0.0854358i	$-0.0272282\pi$
$138$	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
$141$	4.00000	0.336861
$142$	10.0000i	0.839181i
$143$	0	0
$144$	−1.00000	−0.0833333
$145$	0	0
$146$	−10.0000	−0.827606
$147$	7.00000i	0.577350i
$148$	− 6.00000i	− 0.493197i
$149$	−10.0000	−0.819232	−0.409616	−	0.912258i	$-0.634337\pi$
$149$	−10.0000	−0.819232	−0.409616	+	0.912258i	$0.634337\pi$
$150$	0	0
$151$	−20.0000	−1.62758	−0.813788	−	0.581161i	$-0.802599\pi$
$151$	−20.0000	−1.62758	−0.813788	+	0.581161i	$0.802599\pi$
$152$	− 4.00000i	− 0.324443i
$153$	6.00000i	0.485071i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	6.00000i	0.478852i	0.970915	+	0.239426i	$0.0769593\pi$
$157$	6.00000i	0.478852i	−0.970915	+	0.239426i	$0.923041\pi$
$158$	0	0
$159$	2.00000	0.158610
$160$	0	0
$161$	0	0
$162$	− 1.00000i	− 0.0785674i
$163$	− 8.00000i	− 0.626608i	−0.949653	−	0.313304i	$-0.898564\pi$
$163$	− 8.00000i	− 0.626608i	0.949653	−	0.313304i	$-0.101436\pi$
$164$	2.00000	0.156174
$165$	0	0
$166$	−4.00000	−0.310460
$167$	− 8.00000i	− 0.619059i	−0.950890	−	0.309529i	$-0.899829\pi$
$167$	− 8.00000i	− 0.619059i	0.950890	−	0.309529i	$-0.100171\pi$
$168$	0	0
$169$	13.0000	1.00000
$170$	0	0
$171$	4.00000	0.305888
$172$	2.00000i	0.152499i
$173$	6.00000i	0.456172i	0.973641	+	0.228086i	$0.0732467\pi$
$173$	6.00000i	0.456172i	−0.973641	+	0.228086i	$0.926753\pi$
$174$	0	0
$175$	0	0
$176$	2.00000	0.150756
$177$	0	0
$178$	4.00000i	0.299813i
$179$	24.0000	1.79384	0.896922	−	0.442189i	$-0.145798\pi$
$179$	24.0000	1.79384	0.896922	+	0.442189i	$0.145798\pi$
$180$	0	0
$181$	6.00000	0.445976	0.222988	−	0.974821i	$-0.428419\pi$
$181$	6.00000	0.445976	0.222988	+	0.974821i	$0.428419\pi$
$182$	0	0
$183$	− 2.00000i	− 0.147844i
$184$	−1.00000	−0.0737210
$185$	0	0
$186$	−8.00000	−0.586588
$187$	− 12.0000i	− 0.877527i
$188$	4.00000i	0.291730i
$189$	0	0
$190$	0	0
$191$	−8.00000	−0.578860	−0.289430	−	0.957199i	$-0.593466\pi$
$191$	−8.00000	−0.578860	−0.289430	+	0.957199i	$0.593466\pi$
$192$	− 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	$-0.168091\pi$
$194$	−16.0000	−1.14873
$195$	0	0
$196$	−7.00000	−0.500000
$197$	− 14.0000i	− 0.997459i	−0.866758	−	0.498729i	$-0.833800\pi$
$197$	− 14.0000i	− 0.997459i	0.866758	−	0.498729i	$-0.166200\pi$
$198$	2.00000i	0.142134i
$199$	−24.0000	−1.70131	−0.850657	−	0.525720i	$-0.823796\pi$
$199$	−24.0000	−1.70131	−0.850657	+	0.525720i	$0.823796\pi$
$200$	0	0
$201$	−2.00000	−0.141069
$202$	0	0
$203$	0	0
$204$	−6.00000	−0.420084
$205$	0	0
$206$	−16.0000	−1.11477
$207$	− 1.00000i	− 0.0695048i
$208$	0	0
$209$	−8.00000	−0.553372
$210$	0	0
$211$	−20.0000	−1.37686	−0.688428	−	0.725304i	$-0.741699\pi$
$211$	−20.0000	−1.37686	−0.688428	+	0.725304i	$0.741699\pi$
$212$	2.00000i	0.137361i
$213$	− 10.0000i	− 0.685189i
$214$	−8.00000	−0.546869
$215$	0	0
$216$	1.00000	0.0680414
$217$	0	0
$218$	2.00000i	0.135457i
$219$	10.0000	0.675737
$220$	0	0
$221$	0	0
$222$	6.00000i	0.402694i
$223$	− 14.0000i	− 0.937509i	−0.883328	−	0.468755i	$-0.844703\pi$
$223$	− 14.0000i	− 0.937509i	0.883328	−	0.468755i	$-0.155297\pi$
$224$	0	0
$225$	0	0
$226$	10.0000	0.665190
$227$	− 8.00000i	− 0.530979i	−0.964114	−	0.265489i	$-0.914466\pi$
$227$	− 8.00000i	− 0.530979i	0.964114	−	0.265489i	$-0.0855335\pi$
$228$	4.00000i	0.264906i
$229$	14.0000	0.925146	0.462573	−	0.886581i	$-0.346926\pi$
$229$	14.0000	0.925146	0.462573	+	0.886581i	$0.346926\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	22.0000i	1.44127i	0.693316	+	0.720634i	$0.256149\pi$
$233$	22.0000i	1.44127i	−0.693316	+	0.720634i	$0.743851\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	26.0000	1.68180	0.840900	−	0.541190i	$-0.182026\pi$
$239$	26.0000	1.68180	0.840900	+	0.541190i	$0.182026\pi$
$240$	0	0
$241$	2.00000	0.128831	0.0644157	−	0.997923i	$-0.479482\pi$
$241$	2.00000	0.128831	0.0644157	+	0.997923i	$0.479482\pi$
$242$	7.00000i	0.449977i
$243$	1.00000i	0.0641500i
$244$	2.00000	0.128037
$245$	0	0
$246$	−2.00000	−0.127515
$247$	0	0
$248$	− 8.00000i	− 0.508001i
$249$	4.00000	0.253490
$250$	0	0
$251$	−18.0000	−1.13615	−0.568075	−	0.822977i	$-0.692312\pi$
$251$	−18.0000	−1.13615	−0.568075	+	0.822977i	$0.692312\pi$
$252$	0	0
$253$	2.00000i	0.125739i
$254$	−6.00000	−0.376473
$255$	0	0
$256$	1.00000	0.0625000
$257$	− 6.00000i	− 0.374270i	−0.982334	−	0.187135i	$-0.940080\pi$
$257$	− 6.00000i	− 0.374270i	0.982334	−	0.187135i	$-0.0599201\pi$
$258$	− 2.00000i	− 0.124515i
$259$	0	0
$260$	0	0
$261$	0	0
$262$	4.00000i	0.247121i
$263$	16.0000i	0.986602i	0.869859	+	0.493301i	$0.164210\pi$
$263$	16.0000i	0.986602i	−0.869859	+	0.493301i	$0.835790\pi$
$264$	−2.00000	−0.123091
$265$	0	0
$266$	0	0
$267$	− 4.00000i	− 0.244796i
$268$	− 2.00000i	− 0.122169i
$269$	−28.0000	−1.70719	−0.853595	−	0.520937i	$-0.825583\pi$
$269$	−28.0000	−1.70719	−0.853595	+	0.520937i	$0.825583\pi$
$270$	0	0
$271$	−8.00000	−0.485965	−0.242983	−	0.970031i	$-0.578126\pi$
$271$	−8.00000	−0.485965	−0.242983	+	0.970031i	$0.578126\pi$
$272$	− 6.00000i	− 0.363803i
$273$	0	0
$274$	−2.00000	−0.120824
$275$	0	0
$276$	1.00000	0.0601929
$277$	− 24.0000i	− 1.44202i	−0.692925	−	0.721010i	$-0.743678\pi$
$277$	− 24.0000i	− 1.44202i	0.692925	−	0.721010i	$-0.256322\pi$
$278$	12.0000i	0.719712i
$279$	8.00000	0.478947
$280$	0	0
$281$	12.0000	0.715860	0.357930	−	0.933748i	$-0.383483\pi$
$281$	12.0000	0.715860	0.357930	+	0.933748i	$0.383483\pi$
$282$	− 4.00000i	− 0.238197i
$283$	10.0000i	0.594438i	0.954809	+	0.297219i	$0.0960592\pi$
$283$	10.0000i	0.594438i	−0.954809	+	0.297219i	$0.903941\pi$
$284$	10.0000	0.593391
$285$	0	0
$286$	0	0
$287$	0	0
$288$	1.00000i	0.0589256i
$289$	−19.0000	−1.11765
$290$	0	0
$291$	16.0000	0.937937
$292$	10.0000i	0.585206i
$293$	22.0000i	1.28525i	0.766179	+	0.642627i	$0.222155\pi$
$293$	22.0000i	1.28525i	−0.766179	+	0.642627i	$0.777845\pi$
$294$	7.00000	0.408248
$295$	0	0
$296$	−6.00000	−0.348743
$297$	− 2.00000i	− 0.116052i
$298$	10.0000i	0.579284i
$299$	0	0
$300$	0	0
$301$	0	0
$302$	20.0000i	1.15087i
$303$	0	0
$304$	−4.00000	−0.229416
$305$	0	0
$306$	6.00000	0.342997
$307$	− 12.0000i	− 0.684876i	−0.939540	−	0.342438i	$-0.888747\pi$
$307$	− 12.0000i	− 0.684876i	0.939540	−	0.342438i	$-0.111253\pi$
$308$	0	0
$309$	16.0000	0.910208
$310$	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	$-0.808786\pi$
$313$	− 20.0000i	− 1.13047i	0.824931	−	0.565233i	$-0.191214\pi$
$314$	6.00000	0.338600
$315$	0	0
$316$	0	0
$317$	6.00000i	0.336994i	0.985702	+	0.168497i	$0.0538913\pi$
$317$	6.00000i	0.336994i	−0.985702	+	0.168497i	$0.946109\pi$
$318$	− 2.00000i	− 0.112154i
$319$	0	0
$320$	0	0
$321$	8.00000	0.446516
$322$	0	0
$323$	24.0000i	1.33540i
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	−8.00000	−0.443079
$327$	− 2.00000i	− 0.110600i
$328$	− 2.00000i	− 0.110432i
$329$	0	0
$330$	0	0
$331$	4.00000	0.219860	0.109930	−	0.993939i	$-0.464937\pi$
$331$	4.00000	0.219860	0.109930	+	0.993939i	$0.464937\pi$
$332$	4.00000i	0.219529i
$333$	− 6.00000i	− 0.328798i
$334$	−8.00000	−0.437741
$335$	0	0
$336$	0	0
$337$	− 28.0000i	− 1.52526i	−0.646837	−	0.762629i	$-0.723908\pi$
$337$	− 28.0000i	− 1.52526i	0.646837	−	0.762629i	$-0.276092\pi$
$338$	− 13.0000i	− 0.707107i
$339$	−10.0000	−0.543125
$340$	0	0
$341$	−16.0000	−0.866449
$342$	− 4.00000i	− 0.216295i
$343$	0	0
$344$	2.00000	0.107833
$345$	0	0
$346$	6.00000	0.322562
$347$	4.00000i	0.214731i	0.994220	+	0.107366i	$0.0342415\pi$
$347$	4.00000i	0.214731i	−0.994220	+	0.107366i	$0.965758\pi$
$348$	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
$351$	0	0
$352$	− 2.00000i	− 0.106600i
$353$	− 6.00000i	− 0.319348i	−0.987170	−	0.159674i	$-0.948956\pi$
$353$	− 6.00000i	− 0.319348i	0.987170	−	0.159674i	$-0.0510443\pi$
$354$	0	0
$355$	0	0
$356$	4.00000	0.212000
$357$	0	0
$358$	− 24.0000i	− 1.26844i
$359$	32.0000	1.68890	0.844448	−	0.535638i	$-0.179929\pi$
$359$	32.0000	1.68890	0.844448	+	0.535638i	$0.179929\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	− 6.00000i	− 0.315353i
$363$	− 7.00000i	− 0.367405i
$364$	0	0
$365$	0	0
$366$	−2.00000	−0.104542
$367$	− 16.0000i	− 0.835193i	−0.908633	−	0.417597i	$-0.862873\pi$
$367$	− 16.0000i	− 0.835193i	0.908633	−	0.417597i	$-0.137127\pi$
$368$	1.00000i	0.0521286i
$369$	2.00000	0.104116
$370$	0	0
$371$	0	0
$372$	8.00000i	0.414781i
$373$	− 6.00000i	− 0.310668i	−0.987862	−	0.155334i	$-0.950355\pi$
$373$	− 6.00000i	− 0.310668i	0.987862	−	0.155334i	$-0.0496454\pi$
$374$	−12.0000	−0.620505
$375$	0	0
$376$	4.00000	0.206284
$377$	0	0
$378$	0	0
$379$	8.00000	0.410932	0.205466	−	0.978664i	$-0.434129\pi$
$379$	8.00000	0.410932	0.205466	+	0.978664i	$0.434129\pi$
$380$	0	0
$381$	6.00000	0.307389
$382$	8.00000i	0.409316i
$383$	16.0000i	0.817562i	0.912633	+	0.408781i	$0.134046\pi$
$383$	16.0000i	0.817562i	−0.912633	+	0.408781i	$0.865954\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	−14.0000	−0.712581
$387$	2.00000i	0.101666i
$388$	16.0000i	0.812277i
$389$	18.0000	0.912636	0.456318	−	0.889817i	$-0.349168\pi$
$389$	18.0000	0.912636	0.456318	+	0.889817i	$0.349168\pi$
$390$	0	0
$391$	6.00000	0.303433
$392$	7.00000i	0.353553i
$393$	− 4.00000i	− 0.201773i
$394$	−14.0000	−0.705310
$395$	0	0
$396$	2.00000	0.100504
$397$	− 8.00000i	− 0.401508i	−0.979642	−	0.200754i	$-0.935661\pi$
$397$	− 8.00000i	− 0.401508i	0.979642	−	0.200754i	$-0.0643393\pi$
$398$	24.0000i	1.20301i
$399$	0	0
$400$	0	0
$401$	0	0		−	1.00000i	$-0.5\pi$
$401$	0	0			1.00000i	$0.5\pi$
$402$	2.00000i	0.0997509i
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	12.0000i	0.594818i
$408$	6.00000i	0.297044i
$409$	−22.0000	−1.08783	−0.543915	−	0.839140i	$-0.683059\pi$
$409$	−22.0000	−1.08783	−0.543915	+	0.839140i	$0.683059\pi$
$410$	0	0
$411$	2.00000	0.0986527
$412$	16.0000i	0.788263i
$413$	0	0
$414$	−1.00000	−0.0491473
$415$	0	0
$416$	0	0
$417$	− 12.0000i	− 0.587643i
$418$	8.00000i	0.391293i
$419$	6.00000	0.293119	0.146560	−	0.989202i	$-0.453180\pi$
$419$	6.00000	0.293119	0.146560	+	0.989202i	$0.453180\pi$
$420$	0	0
$421$	22.0000	1.07221	0.536107	−	0.844150i	$-0.319894\pi$
$421$	22.0000	1.07221	0.536107	+	0.844150i	$0.319894\pi$
$422$	20.0000i	0.973585i
$423$	4.00000i	0.194487i
$424$	2.00000	0.0971286
$425$	0	0
$426$	−10.0000	−0.484502
$427$	0	0
$428$	8.00000i	0.386695i
$429$	0	0
$430$	0	0
$431$	−40.0000	−1.92673	−0.963366	−	0.268190i	$-0.913575\pi$
$431$	−40.0000	−1.92673	−0.963366	+	0.268190i	$0.913575\pi$
$432$	− 1.00000i	− 0.0481125i
$433$	32.0000i	1.53782i	0.639356	+	0.768911i	$0.279201\pi$
$433$	32.0000i	1.53782i	−0.639356	+	0.768911i	$0.720799\pi$
$434$	0	0
$435$	0	0
$436$	2.00000	0.0957826
$437$	− 4.00000i	− 0.191346i
$438$	− 10.0000i	− 0.477818i
$439$	12.0000	0.572729	0.286364	−	0.958121i	$-0.407553\pi$
$439$	12.0000	0.572729	0.286364	+	0.958121i	$0.407553\pi$
$440$	0	0
$441$	−7.00000	−0.333333
$442$	0	0
$443$	36.0000i	1.71041i	0.518289	+	0.855206i	$0.326569\pi$
$443$	36.0000i	1.71041i	−0.518289	+	0.855206i	$0.673431\pi$
$444$	6.00000	0.284747
$445$	0	0
$446$	−14.0000	−0.662919
$447$	− 10.0000i	− 0.472984i
$448$	0	0
$449$	2.00000	0.0943858	0.0471929	−	0.998886i	$-0.484972\pi$
$449$	2.00000	0.0943858	0.0471929	+	0.998886i	$0.484972\pi$
$450$	0	0
$451$	−4.00000	−0.188353
$452$	− 10.0000i	− 0.470360i
$453$	− 20.0000i	− 0.939682i
$454$	−8.00000	−0.375459
$455$	0	0
$456$	4.00000	0.187317
$457$	8.00000i	0.374224i	0.982339	+	0.187112i	$0.0599128\pi$
$457$	8.00000i	0.374224i	−0.982339	+	0.187112i	$0.940087\pi$
$458$	− 14.0000i	− 0.654177i
$459$	−6.00000	−0.280056
$460$	0	0
$461$	32.0000	1.49039	0.745194	−	0.666847i	$-0.232357\pi$
$461$	32.0000	1.49039	0.745194	+	0.666847i	$0.232357\pi$
$462$	0	0
$463$	34.0000i	1.58011i	0.613033	+	0.790057i	$0.289949\pi$
$463$	34.0000i	1.58011i	−0.613033	+	0.790057i	$0.710051\pi$
$464$	0	0
$465$	0	0
$466$	22.0000	1.01913
$467$	4.00000i	0.185098i	0.995708	+	0.0925490i	$0.0295015\pi$
$467$	4.00000i	0.185098i	−0.995708	+	0.0925490i	$0.970499\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−6.00000	−0.276465
$472$	0	0
$473$	− 4.00000i	− 0.183920i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	2.00000i	0.0915737i
$478$	− 26.0000i	− 1.18921i
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	0	0
$481$	0	0
$482$	− 2.00000i	− 0.0910975i
$483$	0	0
$484$	7.00000	0.318182
$485$	0	0
$486$	1.00000	0.0453609
$487$	− 2.00000i	− 0.0906287i	−0.998973	−	0.0453143i	$-0.985571\pi$
$487$	− 2.00000i	− 0.0906287i	0.998973	−	0.0453143i	$-0.0144289\pi$
$488$	− 2.00000i	− 0.0905357i
$489$	8.00000	0.361773
$490$	0	0
$491$	−12.0000	−0.541552	−0.270776	−	0.962642i	$-0.587280\pi$
$491$	−12.0000	−0.541552	−0.270776	+	0.962642i	$0.587280\pi$
$492$	2.00000i	0.0901670i
$493$	0	0
$494$	0	0
$495$	0	0
$496$	−8.00000	−0.359211
$497$	0	0
$498$	− 4.00000i	− 0.179244i
$499$	36.0000	1.61158	0.805791	−	0.592200i	$-0.201741\pi$
$499$	36.0000	1.61158	0.805791	+	0.592200i	$0.201741\pi$
$500$	0	0
$501$	8.00000	0.357414
$502$	18.0000i	0.803379i
$503$	− 24.0000i	− 1.07011i	−0.844818	−	0.535054i	$-0.820291\pi$
$503$	− 24.0000i	− 1.07011i	0.844818	−	0.535054i	$-0.179709\pi$
$504$	0	0
$505$	0	0
$506$	2.00000	0.0889108
$507$	13.0000i	0.577350i
$508$	6.00000i	0.266207i
$509$	0	0		−	1.00000i	$-0.5\pi$
$509$	0	0			1.00000i	$0.5\pi$
$510$	0	0
$511$	0	0
$512$	− 1.00000i	− 0.0441942i
$513$	4.00000i	0.176604i
$514$	−6.00000	−0.264649
$515$	0	0
$516$	−2.00000	−0.0880451
$517$	− 8.00000i	− 0.351840i
$518$	0	0
$519$	−6.00000	−0.263371
$520$	0	0
$521$	36.0000	1.57719	0.788594	−	0.614914i	$-0.210809\pi$
$521$	36.0000	1.57719	0.788594	+	0.614914i	$0.210809\pi$
$522$	0	0
$523$	34.0000i	1.48672i	0.668894	+	0.743358i	$0.266768\pi$
$523$	34.0000i	1.48672i	−0.668894	+	0.743358i	$0.733232\pi$
$524$	4.00000	0.174741
$525$	0	0
$526$	16.0000	0.697633
$527$	48.0000i	2.09091i
$528$	2.00000i	0.0870388i
$529$	−1.00000	−0.0434783
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0	0
$534$	−4.00000	−0.173097
$535$	0	0
$536$	−2.00000	−0.0863868
$537$	24.0000i	1.03568i
$538$	28.0000i	1.20717i
$539$	14.0000	0.603023
$540$	0	0
$541$	26.0000	1.11783	0.558914	−	0.829226i	$-0.311218\pi$
$541$	26.0000	1.11783	0.558914	+	0.829226i	$0.311218\pi$
$542$	8.00000i	0.343629i
$543$	6.00000i	0.257485i
$544$	−6.00000	−0.257248
$545$	0	0
$546$	0	0
$547$	− 20.0000i	− 0.855138i	−0.903983	−	0.427569i	$-0.859370\pi$
$547$	− 20.0000i	− 0.855138i	0.903983	−	0.427569i	$-0.140630\pi$
$548$	2.00000i	0.0854358i
$549$	2.00000	0.0853579
$550$	0	0
$551$	0	0
$552$	− 1.00000i	− 0.0425628i
$553$	0	0
$554$	−24.0000	−1.01966
$555$	0	0
$556$	12.0000	0.508913
$557$	30.0000i	1.27114i	0.772043	+	0.635570i	$0.219235\pi$
$557$	30.0000i	1.27114i	−0.772043	+	0.635570i	$0.780765\pi$
$558$	− 8.00000i	− 0.338667i
$559$	0	0
$560$	0	0
$561$	12.0000	0.506640
$562$	− 12.0000i	− 0.506189i
$563$	4.00000i	0.168580i	0.996441	+	0.0842900i	$0.0268622\pi$
$563$	4.00000i	0.168580i	−0.996441	+	0.0842900i	$0.973138\pi$
$564$	−4.00000	−0.168430
$565$	0	0
$566$	10.0000	0.420331
$567$	0	0
$568$	− 10.0000i	− 0.419591i
$569$	0	0		−	1.00000i	$-0.5\pi$
$569$	0	0			1.00000i	$0.5\pi$
$570$	0	0
$571$	−16.0000	−0.669579	−0.334790	−	0.942293i	$-0.608665\pi$
$571$	−16.0000	−0.669579	−0.334790	+	0.942293i	$0.608665\pi$
$572$	0	0
$573$	− 8.00000i	− 0.334205i
$574$	0	0
$575$	0	0
$576$	1.00000	0.0416667
$577$	2.00000i	0.0832611i	0.999133	+	0.0416305i	$0.0132552\pi$
$577$	2.00000i	0.0832611i	−0.999133	+	0.0416305i	$0.986745\pi$
$578$	19.0000i	0.790296i
$579$	14.0000	0.581820
$580$	0	0
$581$	0	0
$582$	− 16.0000i	− 0.663221i
$583$	− 4.00000i	− 0.165663i
$584$	10.0000	0.413803
$585$	0	0
$586$	22.0000	0.908812
$587$	28.0000i	1.15568i	0.816149	+	0.577842i	$0.196105\pi$
$587$	28.0000i	1.15568i	−0.816149	+	0.577842i	$0.803895\pi$
$588$	− 7.00000i	− 0.288675i
$589$	32.0000	1.31854
$590$	0	0
$591$	14.0000	0.575883
$592$	6.00000i	0.246598i
$593$	− 42.0000i	− 1.72473i	−0.506284	−	0.862367i	$-0.668981\pi$
$593$	− 42.0000i	− 1.72473i	0.506284	−	0.862367i	$-0.331019\pi$
$594$	−2.00000	−0.0820610
$595$	0	0
$596$	10.0000	0.409616
$597$	− 24.0000i	− 0.982255i
$598$	0	0
$599$	−26.0000	−1.06233	−0.531166	−	0.847268i	$-0.678246\pi$
$599$	−26.0000	−1.06233	−0.531166	+	0.847268i	$0.678246\pi$
$600$	0	0
$601$	22.0000	0.897399	0.448699	−	0.893683i	$-0.351887\pi$
$601$	22.0000	0.897399	0.448699	+	0.893683i	$0.351887\pi$
$602$	0	0
$603$	− 2.00000i	− 0.0814463i
$604$	20.0000	0.813788
$605$	0	0
$606$	0	0
$607$	22.0000i	0.892952i	0.894795	+	0.446476i	$0.147321\pi$
$607$	22.0000i	0.892952i	−0.894795	+	0.446476i	$0.852679\pi$
$608$	4.00000i	0.162221i
$609$	0	0
$610$	0	0
$611$	0	0
$612$	− 6.00000i	− 0.242536i
$613$	− 10.0000i	− 0.403896i	−0.979396	−	0.201948i	$-0.935273\pi$
$613$	− 10.0000i	− 0.403896i	0.979396	−	0.201948i	$-0.0647272\pi$
$614$	−12.0000	−0.484281
$615$	0	0
$616$	0	0
$617$	42.0000i	1.69086i	0.534089	+	0.845428i	$0.320655\pi$
$617$	42.0000i	1.69086i	−0.534089	+	0.845428i	$0.679345\pi$
$618$	− 16.0000i	− 0.643614i
$619$	−20.0000	−0.803868	−0.401934	−	0.915669i	$-0.631662\pi$
$619$	−20.0000	−0.803868	−0.401934	+	0.915669i	$0.631662\pi$
$620$	0	0
$621$	1.00000	0.0401286
$622$	2.00000i	0.0801927i
$623$	0	0
$624$	0	0
$625$	0	0
$626$	−20.0000	−0.799361
$627$	− 8.00000i	− 0.319489i
$628$	− 6.00000i	− 0.239426i
$629$	36.0000	1.43541
$630$	0	0
$631$	−16.0000	−0.636950	−0.318475	−	0.947931i	$-0.603171\pi$
$631$	−16.0000	−0.636950	−0.318475	+	0.947931i	$0.603171\pi$
$632$	0	0
$633$	− 20.0000i	− 0.794929i
$634$	6.00000	0.238290
$635$	0	0
$636$	−2.00000	−0.0793052
$637$	0	0
$638$	0	0
$639$	10.0000	0.395594
$640$	0	0
$641$	4.00000	0.157991	0.0789953	−	0.996875i	$-0.474829\pi$
$641$	4.00000	0.157991	0.0789953	+	0.996875i	$0.474829\pi$
$642$	− 8.00000i	− 0.315735i
$643$	− 14.0000i	− 0.552106i	−0.961142	−	0.276053i	$-0.910973\pi$
$643$	− 14.0000i	− 0.552106i	0.961142	−	0.276053i	$-0.0890266\pi$
$644$	0	0
$645$	0	0
$646$	24.0000	0.944267
$647$	36.0000i	1.41531i	0.706560	+	0.707653i	$0.250246\pi$
$647$	36.0000i	1.41531i	−0.706560	+	0.707653i	$0.749754\pi$
$648$	1.00000i	0.0392837i
$649$	0	0
$650$	0	0
$651$	0	0
$652$	8.00000i	0.313304i
$653$	− 14.0000i	− 0.547862i	−0.961749	−	0.273931i	$-0.911676\pi$
$653$	− 14.0000i	− 0.547862i	0.961749	−	0.273931i	$-0.0883240\pi$
$654$	−2.00000	−0.0782062
$655$	0	0
$656$	−2.00000	−0.0780869
$657$	10.0000i	0.390137i
$658$	0	0
$659$	−18.0000	−0.701180	−0.350590	−	0.936529i	$-0.614019\pi$
$659$	−18.0000	−0.701180	−0.350590	+	0.936529i	$0.614019\pi$
$660$	0	0
$661$	10.0000	0.388955	0.194477	−	0.980907i	$-0.437699\pi$
$661$	10.0000	0.388955	0.194477	+	0.980907i	$0.437699\pi$
$662$	− 4.00000i	− 0.155464i
$663$	0	0
$664$	4.00000	0.155230
$665$	0	0
$666$	−6.00000	−0.232495
$667$	0	0
$668$	8.00000i	0.309529i
$669$	14.0000	0.541271
$670$	0	0
$671$	−4.00000	−0.154418
$672$	0	0
$673$	− 10.0000i	− 0.385472i	−0.981251	−	0.192736i	$-0.938264\pi$
$673$	− 10.0000i	− 0.385472i	0.981251	−	0.192736i	$-0.0617360\pi$
$674$	−28.0000	−1.07852
$675$	0	0
$676$	−13.0000	−0.500000
$677$	6.00000i	0.230599i	0.993331	+	0.115299i	$0.0367827\pi$
$677$	6.00000i	0.230599i	−0.993331	+	0.115299i	$0.963217\pi$
$678$	10.0000i	0.384048i
$679$	0	0
$680$	0	0
$681$	8.00000	0.306561
$682$	16.0000i	0.612672i
$683$	20.0000i	0.765279i	0.923898	+	0.382639i	$0.124985\pi$
$683$	20.0000i	0.765279i	−0.923898	+	0.382639i	$0.875015\pi$
$684$	−4.00000	−0.152944
$685$	0	0
$686$	0	0
$687$	14.0000i	0.534133i
$688$	− 2.00000i	− 0.0762493i
$689$	0	0
$690$	0	0
$691$	20.0000	0.760836	0.380418	−	0.924815i	$-0.375780\pi$
$691$	20.0000	0.760836	0.380418	+	0.924815i	$0.375780\pi$
$692$	− 6.00000i	− 0.228086i
$693$	0	0
$694$	4.00000	0.151838
$695$	0	0
$696$	0	0
$697$	12.0000i	0.454532i
$698$	26.0000i	0.984115i
$699$	−22.0000	−0.832116
$700$	0	0
$701$	−18.0000	−0.679851	−0.339925	−	0.940452i	$-0.610402\pi$
$701$	−18.0000	−0.679851	−0.339925	+	0.940452i	$0.610402\pi$
$702$	0	0
$703$	− 24.0000i	− 0.905177i
$704$	−2.00000	−0.0753778
$705$	0	0
$706$	−6.00000	−0.225813
$707$	0	0
$708$	0	0
$709$	38.0000	1.42712	0.713560	−	0.700594i	$-0.247082\pi$
$709$	38.0000	1.42712	0.713560	+	0.700594i	$0.247082\pi$
$710$	0	0
$711$	0	0
$712$	− 4.00000i	− 0.149906i
$713$	− 8.00000i	− 0.299602i
$714$	0	0
$715$	0	0
$716$	−24.0000	−0.896922
$717$	26.0000i	0.970988i
$718$	− 32.0000i	− 1.19423i
$719$	22.0000	0.820462	0.410231	−	0.911982i	$-0.365448\pi$
$719$	22.0000	0.820462	0.410231	+	0.911982i	$0.365448\pi$
$720$	0	0
$721$	0	0
$722$	3.00000i	0.111648i
$723$	2.00000i	0.0743808i
$724$	−6.00000	−0.222988
$725$	0	0
$726$	−7.00000	−0.259794
$727$	28.0000i	1.03846i	0.854634	+	0.519231i	$0.173782\pi$
$727$	28.0000i	1.03846i	−0.854634	+	0.519231i	$0.826218\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	−12.0000	−0.443836
$732$	2.00000i	0.0739221i
$733$	− 30.0000i	− 1.10808i	−0.832492	−	0.554038i	$-0.813086\pi$
$733$	− 30.0000i	− 1.10808i	0.832492	−	0.554038i	$-0.186914\pi$
$734$	−16.0000	−0.590571
$735$	0	0
$736$	1.00000	0.0368605
$737$	4.00000i	0.147342i
$738$	− 2.00000i	− 0.0736210i
$739$	−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
$742$	0	0
$743$	− 24.0000i	− 0.880475i	−0.897881	−	0.440237i	$-0.854894\pi$
$743$	− 24.0000i	− 0.880475i	0.897881	−	0.440237i	$-0.145106\pi$
$744$	8.00000	0.293294
$745$	0	0
$746$	−6.00000	−0.219676
$747$	4.00000i	0.146352i
$748$	12.0000i	0.438763i
$749$	0	0
$750$	0	0
$751$	32.0000	1.16770	0.583848	−	0.811863i	$-0.301546\pi$
$751$	32.0000	1.16770	0.583848	+	0.811863i	$0.301546\pi$
$752$	− 4.00000i	− 0.145865i
$753$	− 18.0000i	− 0.655956i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	42.0000i	1.52652i	0.646094	+	0.763258i	$0.276401\pi$
$757$	42.0000i	1.52652i	−0.646094	+	0.763258i	$0.723599\pi$
$758$	− 8.00000i	− 0.290573i
$759$	−2.00000	−0.0725954
$760$	0	0
$761$	−14.0000	−0.507500	−0.253750	−	0.967270i	$-0.581664\pi$
$761$	−14.0000	−0.507500	−0.253750	+	0.967270i	$0.581664\pi$
$762$	− 6.00000i	− 0.217357i
$763$	0	0
$764$	8.00000	0.289430
$765$	0	0
$766$	16.0000	0.578103
$767$	0	0
$768$	1.00000i	0.0360844i
$769$	42.0000	1.51456	0.757279	−	0.653091i	$-0.226528\pi$
$769$	42.0000	1.51456	0.757279	+	0.653091i	$0.226528\pi$
$770$	0	0
$771$	6.00000	0.216085
$772$	14.0000i	0.503871i
$773$	38.0000i	1.36677i	0.730061	+	0.683383i	$0.239492\pi$
$773$	38.0000i	1.36677i	−0.730061	+	0.683383i	$0.760508\pi$
$774$	2.00000	0.0718885
$775$	0	0
$776$	16.0000	0.574367
$777$	0	0
$778$	− 18.0000i	− 0.645331i
$779$	8.00000	0.286630
$780$	0	0
$781$	−20.0000	−0.715656
$782$	− 6.00000i	− 0.214560i
$783$	0	0
$784$	7.00000	0.250000
$785$	0	0
$786$	−4.00000	−0.142675
$787$	− 18.0000i	− 0.641631i	−0.947142	−	0.320815i	$-0.896043\pi$
$787$	− 18.0000i	− 0.641631i	0.947142	−	0.320815i	$-0.103957\pi$
$788$	14.0000i	0.498729i
$789$	−16.0000	−0.569615
$790$	0	0
$791$	0	0
$792$	− 2.00000i	− 0.0710669i
$793$	0	0
$794$	−8.00000	−0.283909
$795$	0	0
$796$	24.0000	0.850657
$797$	− 42.0000i	− 1.48772i	−0.668338	−	0.743858i	$-0.732994\pi$
$797$	− 42.0000i	− 1.48772i	0.668338	−	0.743858i	$-0.267006\pi$
$798$	0	0
$799$	−24.0000	−0.849059
$800$	0	0
$801$	4.00000	0.141333
$802$	0	0
$803$	− 20.0000i	− 0.705785i
$804$	2.00000	0.0705346
$805$	0	0
$806$	0	0
$807$	− 28.0000i	− 0.985647i
$808$	0	0
$809$	6.00000	0.210949	0.105474	−	0.994422i	$-0.466364\pi$
$809$	6.00000	0.210949	0.105474	+	0.994422i	$0.466364\pi$
$810$	0	0
$811$	20.0000	0.702295	0.351147	−	0.936320i	$-0.385792\pi$
$811$	20.0000	0.702295	0.351147	+	0.936320i	$0.385792\pi$
$812$	0	0
$813$	− 8.00000i	− 0.280572i
$814$	12.0000	0.420600
$815$	0	0
$816$	6.00000	0.210042
$817$	8.00000i	0.279885i
$818$	22.0000i	0.769212i
$819$	0	0
$820$	0	0
$821$	44.0000	1.53561	0.767805	−	0.640683i	$-0.221349\pi$
$821$	44.0000	1.53561	0.767805	+	0.640683i	$0.221349\pi$
$822$	− 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	$-0.0557627\pi$
$824$	16.0000	0.557386
$825$	0	0
$826$	0	0
$827$	32.0000i	1.11275i	0.830932	+	0.556375i	$0.187808\pi$
$827$	32.0000i	1.11275i	−0.830932	+	0.556375i	$0.812192\pi$
$828$	1.00000i	0.0347524i
$829$	6.00000	0.208389	0.104194	−	0.994557i	$-0.466774\pi$
$829$	6.00000	0.208389	0.104194	+	0.994557i	$0.466774\pi$
$830$	0	0
$831$	24.0000	0.832551
$832$	0	0
$833$	− 42.0000i	− 1.45521i
$834$	−12.0000	−0.415526
$835$	0	0
$836$	8.00000	0.276686
$837$	8.00000i	0.276520i
$838$	− 6.00000i	− 0.207267i
$839$	−12.0000	−0.414286	−0.207143	−	0.978311i	$-0.566417\pi$
$839$	−12.0000	−0.414286	−0.207143	+	0.978311i	$0.566417\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	− 22.0000i	− 0.758170i
$843$	12.0000i	0.413302i
$844$	20.0000	0.688428
$845$	0	0
$846$	4.00000	0.137523
$847$	0	0
$848$	− 2.00000i	− 0.0686803i
$849$	−10.0000	−0.343199
$850$	0	0
$851$	−6.00000	−0.205677
$852$	10.0000i	0.342594i
$853$	− 16.0000i	− 0.547830i	−0.961754	−	0.273915i	$-0.911681\pi$
$853$	− 16.0000i	− 0.547830i	0.961754	−	0.273915i	$-0.0883186\pi$
$854$	0	0
$855$	0	0
$856$	8.00000	0.273434
$857$	− 18.0000i	− 0.614868i	−0.951569	−	0.307434i	$-0.900530\pi$
$857$	− 18.0000i	− 0.614868i	0.951569	−	0.307434i	$-0.0994704\pi$
$858$	0	0
$859$	20.0000	0.682391	0.341196	−	0.939992i	$-0.389168\pi$
$859$	20.0000	0.682391	0.341196	+	0.939992i	$0.389168\pi$
$860$	0	0
$861$	0	0
$862$	40.0000i	1.36241i
$863$	− 12.0000i	− 0.408485i	−0.978920	−	0.204242i	$-0.934527\pi$
$863$	− 12.0000i	− 0.408485i	0.978920	−	0.204242i	$-0.0654731\pi$
$864$	−1.00000	−0.0340207
$865$	0	0
$866$	32.0000	1.08740
$867$	− 19.0000i	− 0.645274i
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	− 2.00000i	− 0.0677285i
$873$	16.0000i	0.541518i
$874$	−4.00000	−0.135302
$875$	0	0
$876$	−10.0000	−0.337869
$877$	− 32.0000i	− 1.08056i	−0.841484	−	0.540282i	$-0.818318\pi$
$877$	− 32.0000i	− 1.08056i	0.841484	−	0.540282i	$-0.181682\pi$
$878$	− 12.0000i	− 0.404980i
$879$	−22.0000	−0.742042
$880$	0	0
$881$	36.0000	1.21287	0.606435	−	0.795133i	$-0.292599\pi$
$881$	36.0000	1.21287	0.606435	+	0.795133i	$0.292599\pi$
$882$	7.00000i	0.235702i
$883$	8.00000i	0.269221i	0.990899	+	0.134611i	$0.0429784\pi$
$883$	8.00000i	0.269221i	−0.990899	+	0.134611i	$0.957022\pi$
$884$	0	0
$885$	0	0
$886$	36.0000	1.20944
$887$	28.0000i	0.940148i	0.882627	+	0.470074i	$0.155773\pi$
$887$	28.0000i	0.940148i	−0.882627	+	0.470074i	$0.844227\pi$
$888$	− 6.00000i	− 0.201347i
$889$	0	0
$890$	0	0
$891$	2.00000	0.0670025
$892$	14.0000i	0.468755i
$893$	16.0000i	0.535420i
$894$	−10.0000	−0.334450
$895$	0	0
$896$	0	0
$897$	0	0
$898$	− 2.00000i	− 0.0667409i
$899$	0	0
$900$	0	0
$901$	−12.0000	−0.399778
$902$	4.00000i	0.133185i
$903$	0	0
$904$	−10.0000	−0.332595
$905$	0	0
$906$	−20.0000	−0.664455
$907$	− 38.0000i	− 1.26177i	−0.775877	−	0.630885i	$-0.782692\pi$
$907$	− 38.0000i	− 1.26177i	0.775877	−	0.630885i	$-0.217308\pi$
$908$	8.00000i	0.265489i
$909$	0	0
$910$	0	0
$911$	20.0000	0.662630	0.331315	−	0.943520i	$-0.392508\pi$
$911$	20.0000	0.662630	0.331315	+	0.943520i	$0.392508\pi$
$912$	− 4.00000i	− 0.132453i
$913$	− 8.00000i	− 0.264761i
$914$	8.00000	0.264616
$915$	0	0
$916$	−14.0000	−0.462573
$917$	0	0
$918$	6.00000i	0.198030i
$919$	−48.0000	−1.58337	−0.791687	−	0.610927i	$-0.790797\pi$
$919$	−48.0000	−1.58337	−0.791687	+	0.610927i	$0.790797\pi$
$920$	0	0
$921$	12.0000	0.395413
$922$	− 32.0000i	− 1.05386i
$923$	0	0
$924$	0	0
$925$	0	0
$926$	34.0000	1.11731
$927$	16.0000i	0.525509i
$928$	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
$931$	−28.0000	−0.917663
$932$	− 22.0000i	− 0.720634i
$933$	− 2.00000i	− 0.0654771i
$934$	4.00000	0.130884
$935$	0	0
$936$	0	0
$937$	− 40.0000i	− 1.30674i	−0.757037	−	0.653372i	$-0.773354\pi$
$937$	− 40.0000i	− 1.30674i	0.757037	−	0.653372i	$-0.226646\pi$
$938$	0	0
$939$	20.0000	0.652675
$940$	0	0
$941$	−38.0000	−1.23876	−0.619382	−	0.785090i	$-0.712617\pi$
$941$	−38.0000	−1.23876	−0.619382	+	0.785090i	$0.712617\pi$
$942$	6.00000i	0.195491i
$943$	− 2.00000i	− 0.0651290i
$944$	0	0
$945$	0	0
$946$	−4.00000	−0.130051
$947$	− 52.0000i	− 1.68977i	−0.534946	−	0.844886i	$-0.679668\pi$
$947$	− 52.0000i	− 1.68977i	0.534946	−	0.844886i	$-0.320332\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	−6.00000	−0.194563
$952$	0	0
$953$	− 22.0000i	− 0.712650i	−0.934362	−	0.356325i	$-0.884030\pi$
$953$	− 22.0000i	− 0.712650i	0.934362	−	0.356325i	$-0.115970\pi$
$954$	2.00000	0.0647524
$955$	0	0
$956$	−26.0000	−0.840900
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	33.0000	1.06452
$962$	0	0
$963$	8.00000i	0.257796i
$964$	−2.00000	−0.0644157
$965$	0	0
$966$	0	0
$967$	38.0000i	1.22200i	0.791632	+	0.610999i	$0.209232\pi$
$967$	38.0000i	1.22200i	−0.791632	+	0.610999i	$0.790768\pi$
$968$	− 7.00000i	− 0.224989i
$969$	−24.0000	−0.770991
$970$	0	0
$971$	6.00000	0.192549	0.0962746	−	0.995355i	$-0.469307\pi$
$971$	6.00000	0.192549	0.0962746	+	0.995355i	$0.469307\pi$
$972$	− 1.00000i	− 0.0320750i
$973$	0	0
$974$	−2.00000	−0.0640841
$975$	0	0
$976$	−2.00000	−0.0640184
$977$	− 38.0000i	− 1.21573i	−0.794041	−	0.607864i	$-0.792027\pi$
$977$	− 38.0000i	− 1.21573i	0.794041	−	0.607864i	$-0.207973\pi$
$978$	− 8.00000i	− 0.255812i
$979$	−8.00000	−0.255681
$980$	0	0
$981$	2.00000	0.0638551
$982$	12.0000i	0.382935i
$983$	24.0000i	0.765481i	0.923856	+	0.382741i	$0.125020\pi$
$983$	24.0000i	0.765481i	−0.923856	+	0.382741i	$0.874980\pi$
$984$	2.00000	0.0637577
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	2.00000	0.0635963
$990$	0	0
$991$	−36.0000	−1.14358	−0.571789	−	0.820401i	$-0.693750\pi$
$991$	−36.0000	−1.14358	−0.571789	+	0.820401i	$0.693750\pi$
$992$	8.00000i	0.254000i
$993$	4.00000i	0.126936i
$994$	0	0
$995$	0	0
$996$	−4.00000	−0.126745
$997$	16.0000i	0.506725i	0.967371	+	0.253363i	$0.0815366\pi$
$997$	16.0000i	0.506725i	−0.967371	+	0.253363i	$0.918463\pi$
$998$	− 36.0000i	− 1.13956i
$999$	6.00000	0.189832

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
690.2.a.i.1.1	✓	1	5.2	odd	4
2070.2.a.g.1.1		1	15.2	even	4
3450.2.a.c.1.1		1	5.3	odd	4
3450.2.d.r.2899.1		2	1.1	even	1	trivial
3450.2.d.r.2899.2		2	5.4	even	2	inner
5520.2.a.d.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)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +1.00000i q^{8} -1.00000 q^{9} +2.00000 q^{11} -1.00000i q^{12} +1.00000 q^{16} -6.00000i q^{17} +1.00000i q^{18} -4.00000 q^{19} -2.00000i q^{22} +1.00000i q^{23} -1.00000 q^{24} -1.00000i q^{27} -8.00000 q^{31} -1.00000i q^{32} +2.00000i q^{33} -6.00000 q^{34} +1.00000 q^{36} +6.00000i q^{37} +4.00000i q^{38} -2.00000 q^{41} -2.00000i q^{43} -2.00000 q^{44} +1.00000 q^{46} -4.00000i q^{47} +1.00000i q^{48} +7.00000 q^{49} +6.00000 q^{51} -2.00000i q^{53} -1.00000 q^{54} -4.00000i q^{57} -2.00000 q^{61} +8.00000i q^{62} -1.00000 q^{64} +2.00000 q^{66} +2.00000i q^{67} +6.00000i q^{68} -1.00000 q^{69} -10.0000 q^{71} -1.00000i q^{72} -10.0000i q^{73} +6.00000 q^{74} +4.00000 q^{76} +1.00000 q^{81} +2.00000i q^{82} -4.00000i q^{83} -2.00000 q^{86} +2.00000i q^{88} -4.00000 q^{89} -1.00000i q^{92} -8.00000i q^{93} -4.00000 q^{94} +1.00000 q^{96} -16.0000i q^{97} -7.00000i q^{98} -2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} + 2 q^{6} - 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^4 + 2 * q^6 - 2 * q^9	\( 2 q - 2 q^{4} + 2 q^{6} - 2 q^{9} + 4 q^{11} + 2 q^{16} - 8 q^{19} - 2 q^{24} - 16 q^{31} - 12 q^{34} + 2 q^{36} - 4 q^{41} - 4 q^{44} + 2 q^{46} + 14 q^{49} + 12 q^{51} - 2 q^{54} - 4 q^{61} - 2 q^{64} + 4 q^{66} - 2 q^{69} - 20 q^{71} + 12 q^{74} + 8 q^{76} + 2 q^{81} - 4 q^{86} - 8 q^{89} - 8 q^{94} + 2 q^{96} - 4 q^{99}+O(q^{100}) \) 2 * q - 2 * q^4 + 2 * q^6 - 2 * q^9 + 4 * q^11 + 2 * q^16 - 8 * q^19 - 2 * q^24 - 16 * q^31 - 12 * q^34 + 2 * q^36 - 4 * q^41 - 4 * q^44 + 2 * q^46 + 14 * q^49 + 12 * q^51 - 2 * q^54 - 4 * q^61 - 2 * q^64 + 4 * q^66 - 2 * q^69 - 20 * q^71 + 12 * q^74 + 8 * q^76 + 2 * q^81 - 4 * q^86 - 8 * q^89 - 8 * q^94 + 2 * q^96 - 4 * q^99

Introduction

L-functions

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