LMFDB - Embedded newform 3450.2.d.t.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.t.2899.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +1.00000i q^{8} -1.00000 q^{9} +4.00000 q^{11} -1.00000i q^{12} -2.00000i q^{13} +1.00000 q^{16} +6.00000i q^{17} +1.00000i q^{18} -4.00000 q^{19} -4.00000i q^{22} -1.00000i q^{23} -1.00000 q^{24} -2.00000 q^{26} -1.00000i q^{27} +2.00000 q^{29} -1.00000i q^{32} +4.00000i q^{33} +6.00000 q^{34} +1.00000 q^{36} +2.00000i q^{37} +4.00000i q^{38} +2.00000 q^{39} +10.0000 q^{41} -4.00000i q^{43} -4.00000 q^{44} -1.00000 q^{46} +1.00000i q^{48} +7.00000 q^{49} -6.00000 q^{51} +2.00000i q^{52} +6.00000i q^{53} -1.00000 q^{54} -4.00000i q^{57} -2.00000i q^{58} +4.00000 q^{59} -10.0000 q^{61} -1.00000 q^{64} +4.00000 q^{66} +12.0000i q^{67} -6.00000i q^{68} +1.00000 q^{69} -8.00000 q^{71} -1.00000i q^{72} +10.0000i q^{73} +2.00000 q^{74} +4.00000 q^{76} -2.00000i q^{78} +8.00000 q^{79} +1.00000 q^{81} -10.0000i q^{82} -4.00000i q^{83} -4.00000 q^{86} +2.00000i q^{87} +4.00000i q^{88} -18.0000 q^{89} +1.00000i q^{92} +1.00000 q^{96} -2.00000i q^{97} -7.00000i q^{98} -4.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4} + 2 q^{6} - 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^4 + 2 * q^6 - 2 * q^9	$ 2 q - 2 q^{4} + 2 q^{6} - 2 q^{9} + 8 q^{11} + 2 q^{16} - 8 q^{19} - 2 q^{24} - 4 q^{26} + 4 q^{29} + 12 q^{34} + 2 q^{36} + 4 q^{39} + 20 q^{41} - 8 q^{44} - 2 q^{46} + 14 q^{49} - 12 q^{51} - 2 q^{54} + 8 q^{59} - 20 q^{61} - 2 q^{64} + 8 q^{66} + 2 q^{69} - 16 q^{71} + 4 q^{74} + 8 q^{76} + 16 q^{79} + 2 q^{81} - 8 q^{86} - 36 q^{89} + 2 q^{96} - 8 q^{99}+O(q^{100}) $ 2 * q - 2 * q^4 + 2 * q^6 - 2 * q^9 + 8 * q^11 + 2 * q^16 - 8 * q^19 - 2 * q^24 - 4 * q^26 + 4 * q^29 + 12 * q^34 + 2 * q^36 + 4 * q^39 + 20 * q^41 - 8 * q^44 - 2 * q^46 + 14 * q^49 - 12 * q^51 - 2 * q^54 + 8 * q^59 - 20 * q^61 - 2 * q^64 + 8 * q^66 + 2 * q^69 - 16 * q^71 + 4 * q^74 + 8 * q^76 + 16 * q^79 + 2 * q^81 - 8 * q^86 - 36 * q^89 + 2 * q^96 - 8 * q^99

Display 100 coefficients 10 coefficients

Character values

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

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

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
$11$	4.00000	1.20605	0.603023	−	0.797724i	$-0.293963\pi$
$11$	4.00000	1.20605	0.603023	+	0.797724i	$0.293963\pi$
$12$	− 1.00000i	− 0.288675i
$13$	− 2.00000i	− 0.554700i	−0.960769	−	0.277350i	$-0.910544\pi$
$13$	− 2.00000i	− 0.554700i	0.960769	−	0.277350i	$-0.0894562\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	6.00000i	1.45521i	0.685994	+	0.727607i	$0.259367\pi$
$17$	6.00000i	1.45521i	−0.685994	+	0.727607i	$0.740633\pi$
$18$	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
$22$	− 4.00000i	− 0.852803i
$23$	− 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
$28$	0	0
$29$	2.00000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	− 1.00000i	− 0.176777i
$33$	4.00000i	0.696311i
$34$	6.00000	1.02899
$35$	0	0
$36$	1.00000	0.166667
$37$	2.00000i	0.328798i	0.986394	+	0.164399i	$0.0525685\pi$
$37$	2.00000i	0.328798i	−0.986394	+	0.164399i	$0.947432\pi$
$38$	4.00000i	0.648886i
$39$	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	$0.214777\pi$
$42$	0	0
$43$	− 4.00000i	− 0.609994i	−0.952353	−	0.304997i	$-0.901344\pi$
$43$	− 4.00000i	− 0.609994i	0.952353	−	0.304997i	$-0.0986555\pi$
$44$	−4.00000	−0.603023
$45$	0	0
$46$	−1.00000	−0.147442
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	1.00000i	0.144338i
$49$	7.00000	1.00000
$50$	0	0
$51$	−6.00000	−0.840168
$52$	2.00000i	0.277350i
$53$	6.00000i	0.824163i	0.911147	+	0.412082i	$0.135198\pi$
$53$	6.00000i	0.824163i	−0.911147	+	0.412082i	$0.864802\pi$
$54$	−1.00000	−0.136083
$55$	0	0
$56$	0	0
$57$	− 4.00000i	− 0.529813i
$58$	− 2.00000i	− 0.262613i
$59$	4.00000	0.520756	0.260378	−	0.965507i	$-0.416153\pi$
$59$	4.00000	0.520756	0.260378	+	0.965507i	$0.416153\pi$
$60$	0	0
$61$	−10.0000	−1.28037	−0.640184	−	0.768221i	$-0.721142\pi$
$61$	−10.0000	−1.28037	−0.640184	+	0.768221i	$0.721142\pi$
$62$	0	0
$63$	0	0
$64$	−1.00000	−0.125000
$65$	0	0
$66$	4.00000	0.492366
$67$	12.0000i	1.46603i	0.680211	+	0.733017i	$0.261888\pi$
$67$	12.0000i	1.46603i	−0.680211	+	0.733017i	$0.738112\pi$
$68$	− 6.00000i	− 0.727607i
$69$	1.00000	0.120386
$70$	0	0
$71$	−8.00000	−0.949425	−0.474713	−	0.880141i	$-0.657448\pi$
$71$	−8.00000	−0.949425	−0.474713	+	0.880141i	$0.657448\pi$
$72$	− 1.00000i	− 0.117851i
$73$	10.0000i	1.17041i	0.810885	+	0.585206i	$0.198986\pi$
$73$	10.0000i	1.17041i	−0.810885	+	0.585206i	$0.801014\pi$
$74$	2.00000	0.232495
$75$	0	0
$76$	4.00000	0.458831
$77$	0	0
$78$	− 2.00000i	− 0.226455i
$79$	8.00000	0.900070	0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000	0.900070	0.450035	+	0.893011i	$0.351411\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	− 10.0000i	− 1.10432i
$83$	− 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
$86$	−4.00000	−0.431331
$87$	2.00000i	0.214423i
$88$	4.00000i	0.426401i
$89$	−18.0000	−1.90800	−0.953998	−	0.299813i	$-0.903076\pi$
$89$	−18.0000	−1.90800	−0.953998	+	0.299813i	$0.903076\pi$
$90$	0	0
$91$	0	0
$92$	1.00000i	0.104257i
$93$	0	0
$94$	0	0
$95$	0	0
$96$	1.00000	0.102062
$97$	− 2.00000i	− 0.203069i	−0.994832	−	0.101535i	$-0.967625\pi$
$97$	− 2.00000i	− 0.203069i	0.994832	−	0.101535i	$-0.0323753\pi$
$98$	− 7.00000i	− 0.707107i
$99$	−4.00000	−0.402015
$100$	0	0
$101$	6.00000	0.597022	0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000	0.597022	0.298511	+	0.954406i	$0.403510\pi$
$102$	6.00000i	0.594089i
$103$	0	0	1.00000			$0$
$103$	0	0	−1.00000			$\pi$
$104$	2.00000	0.196116
$105$	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
$109$	10.0000	0.957826	0.478913	−	0.877862i	$-0.341031\pi$
$109$	10.0000	0.957826	0.478913	+	0.877862i	$0.341031\pi$
$110$	0	0
$111$	−2.00000	−0.189832
$112$	0	0
$113$	− 6.00000i	− 0.564433i	−0.959351	−	0.282216i	$-0.908930\pi$
$113$	− 6.00000i	− 0.564433i	0.959351	−	0.282216i	$-0.0910696\pi$
$114$	−4.00000	−0.374634
$115$	0	0
$116$	−2.00000	−0.185695
$117$	2.00000i	0.184900i
$118$	− 4.00000i	− 0.368230i
$119$	0	0
$120$	0	0
$121$	5.00000	0.454545
$122$	10.0000i	0.905357i
$123$	10.0000i	0.901670i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	16.0000i	1.41977i	0.704317	+	0.709885i	$0.251253\pi$
$127$	16.0000i	1.41977i	−0.704317	+	0.709885i	$0.748747\pi$
$128$	1.00000i	0.0883883i
$129$	4.00000	0.352180
$130$	0	0
$131$	20.0000	1.74741	0.873704	−	0.486458i	$-0.161711\pi$
$131$	20.0000	1.74741	0.873704	+	0.486458i	$0.161711\pi$
$132$	− 4.00000i	− 0.348155i
$133$	0	0
$134$	12.0000	1.03664
$135$	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
$139$	20.0000	1.69638	0.848189	−	0.529694i	$-0.177693\pi$
$139$	20.0000	1.69638	0.848189	+	0.529694i	$0.177693\pi$
$140$	0	0
$141$	0	0
$142$	8.00000i	0.671345i
$143$	− 8.00000i	− 0.668994i
$144$	−1.00000	−0.0833333
$145$	0	0
$146$	10.0000	0.827606
$147$	7.00000i	0.577350i
$148$	− 2.00000i	− 0.164399i
$149$	10.0000	0.819232	0.409616	−	0.912258i	$-0.365663\pi$
$149$	10.0000	0.819232	0.409616	+	0.912258i	$0.365663\pi$
$150$	0	0
$151$	8.00000	0.651031	0.325515	−	0.945537i	$-0.394462\pi$
$151$	8.00000	0.651031	0.325515	+	0.945537i	$0.394462\pi$
$152$	− 4.00000i	− 0.324443i
$153$	− 6.00000i	− 0.485071i
$154$	0	0
$155$	0	0
$156$	−2.00000	−0.160128
$157$	10.0000i	0.798087i	0.916932	+	0.399043i	$0.130658\pi$
$157$	10.0000i	0.798087i	−0.916932	+	0.399043i	$0.869342\pi$
$158$	− 8.00000i	− 0.636446i
$159$	−6.00000	−0.475831
$160$	0	0
$161$	0	0
$162$	− 1.00000i	− 0.0785674i
$163$	20.0000i	1.56652i	0.621694	+	0.783260i	$0.286445\pi$
$163$	20.0000i	1.56652i	−0.621694	+	0.783260i	$0.713555\pi$
$164$	−10.0000	−0.780869
$165$	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
$169$	9.00000	0.692308
$170$	0	0
$171$	4.00000	0.305888
$172$	4.00000i	0.304997i
$173$	14.0000i	1.06440i	0.846619	+	0.532200i	$0.178635\pi$
$173$	14.0000i	1.06440i	−0.846619	+	0.532200i	$0.821365\pi$
$174$	2.00000	0.151620
$175$	0	0
$176$	4.00000	0.301511
$177$	4.00000i	0.300658i
$178$	18.0000i	1.34916i
$179$	−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
$181$	−2.00000	−0.148659	−0.0743294	−	0.997234i	$-0.523682\pi$
$181$	−2.00000	−0.148659	−0.0743294	+	0.997234i	$0.523682\pi$
$182$	0	0
$183$	− 10.0000i	− 0.739221i
$184$	1.00000	0.0737210
$185$	0	0
$186$	0	0
$187$	24.0000i	1.75505i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	− 1.00000i	− 0.0721688i
$193$	2.00000i	0.143963i	0.997406	+	0.0719816i	$0.0229323\pi$
$193$	2.00000i	0.143963i	−0.997406	+	0.0719816i	$0.977068\pi$
$194$	−2.00000	−0.143592
$195$	0	0
$196$	−7.00000	−0.500000
$197$	− 6.00000i	− 0.427482i	−0.976890	−	0.213741i	$-0.931435\pi$
$197$	− 6.00000i	− 0.427482i	0.976890	−	0.213741i	$-0.0685649\pi$
$198$	4.00000i	0.284268i
$199$	16.0000	1.13421	0.567105	−	0.823646i	$-0.308063\pi$
$199$	16.0000	1.13421	0.567105	+	0.823646i	$0.308063\pi$
$200$	0	0
$201$	−12.0000	−0.846415
$202$	− 6.00000i	− 0.422159i
$203$	0	0
$204$	6.00000	0.420084
$205$	0	0
$206$	0	0
$207$	1.00000i	0.0695048i
$208$	− 2.00000i	− 0.138675i
$209$	−16.0000	−1.10674
$210$	0	0
$211$	4.00000	0.275371	0.137686	−	0.990476i	$-0.456034\pi$
$211$	4.00000	0.275371	0.137686	+	0.990476i	$0.456034\pi$
$212$	− 6.00000i	− 0.412082i
$213$	− 8.00000i	− 0.548151i
$214$	12.0000	0.820303
$215$	0	0
$216$	1.00000	0.0680414
$217$	0	0
$218$	− 10.0000i	− 0.677285i
$219$	−10.0000	−0.675737
$220$	0	0
$221$	12.0000	0.807207
$222$	2.00000i	0.134231i
$223$	16.0000i	1.07144i	0.844396	+	0.535720i	$0.179960\pi$
$223$	16.0000i	1.07144i	−0.844396	+	0.535720i	$0.820040\pi$
$224$	0	0
$225$	0	0
$226$	−6.00000	−0.399114
$227$	20.0000i	1.32745i	0.747978	+	0.663723i	$0.231025\pi$
$227$	20.0000i	1.32745i	−0.747978	+	0.663723i	$0.768975\pi$
$228$	4.00000i	0.264906i
$229$	−14.0000	−0.925146	−0.462573	−	0.886581i	$-0.653074\pi$
$229$	−14.0000	−0.925146	−0.462573	+	0.886581i	$0.653074\pi$
$230$	0	0
$231$	0	0
$232$	2.00000i	0.131306i
$233$	− 6.00000i	− 0.393073i	−0.980497	−	0.196537i	$-0.937031\pi$
$233$	− 6.00000i	− 0.393073i	0.980497	−	0.196537i	$-0.0629694\pi$
$234$	2.00000	0.130744
$235$	0	0
$236$	−4.00000	−0.260378
$237$	8.00000i	0.519656i
$238$	0	0
$239$	−16.0000	−1.03495	−0.517477	−	0.855697i	$-0.673129\pi$
$239$	−16.0000	−1.03495	−0.517477	+	0.855697i	$0.673129\pi$
$240$	0	0
$241$	18.0000	1.15948	0.579741	−	0.814801i	$-0.303154\pi$
$241$	18.0000	1.15948	0.579741	+	0.814801i	$0.303154\pi$
$242$	− 5.00000i	− 0.321412i
$243$	1.00000i	0.0641500i
$244$	10.0000	0.640184
$245$	0	0
$246$	10.0000	0.637577
$247$	8.00000i	0.509028i
$248$	0	0
$249$	4.00000	0.253490
$250$	0	0
$251$	20.0000	1.26239	0.631194	−	0.775625i	$-0.282565\pi$
$251$	20.0000	1.26239	0.631194	+	0.775625i	$0.282565\pi$
$252$	0	0
$253$	− 4.00000i	− 0.251478i
$254$	16.0000	1.00393
$255$	0	0
$256$	1.00000	0.0625000
$257$	14.0000i	0.873296i	0.899632	+	0.436648i	$0.143834\pi$
$257$	14.0000i	0.873296i	−0.899632	+	0.436648i	$0.856166\pi$
$258$	− 4.00000i	− 0.249029i
$259$	0	0
$260$	0	0
$261$	−2.00000	−0.123797
$262$	− 20.0000i	− 1.23560i
$263$	− 24.0000i	− 1.47990i	−0.672660	−	0.739952i	$-0.734848\pi$
$263$	− 24.0000i	− 1.47990i	0.672660	−	0.739952i	$-0.265152\pi$
$264$	−4.00000	−0.246183
$265$	0	0
$266$	0	0
$267$	− 18.0000i	− 1.10158i
$268$	− 12.0000i	− 0.733017i
$269$	−14.0000	−0.853595	−0.426798	−	0.904347i	$-0.640358\pi$
$269$	−14.0000	−0.853595	−0.426798	+	0.904347i	$0.640358\pi$
$270$	0	0
$271$	0	0		−	1.00000i	$-0.5\pi$
$271$	0	0			1.00000i	$0.5\pi$
$272$	6.00000i	0.363803i
$273$	0	0
$274$	−2.00000	−0.120824
$275$	0	0
$276$	−1.00000	−0.0601929
$277$	10.0000i	0.600842i	0.953807	+	0.300421i	$0.0971271\pi$
$277$	10.0000i	0.600842i	−0.953807	+	0.300421i	$0.902873\pi$
$278$	− 20.0000i	− 1.19952i
$279$	0	0
$280$	0	0
$281$	18.0000	1.07379	0.536895	−	0.843649i	$-0.319597\pi$
$281$	18.0000	1.07379	0.536895	+	0.843649i	$0.319597\pi$
$282$	0	0
$283$	− 4.00000i	− 0.237775i	−0.992908	−	0.118888i	$-0.962067\pi$
$283$	− 4.00000i	− 0.237775i	0.992908	−	0.118888i	$-0.0379328\pi$
$284$	8.00000	0.474713
$285$	0	0
$286$	−8.00000	−0.473050
$287$	0	0
$288$	1.00000i	0.0589256i
$289$	−19.0000	−1.11765
$290$	0	0
$291$	2.00000	0.117242
$292$	− 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
$296$	−2.00000	−0.116248
$297$	− 4.00000i	− 0.232104i
$298$	− 10.0000i	− 0.579284i
$299$	−2.00000	−0.115663
$300$	0	0
$301$	0	0
$302$	− 8.00000i	− 0.460348i
$303$	6.00000i	0.344691i
$304$	−4.00000	−0.229416
$305$	0	0
$306$	−6.00000	−0.342997
$307$	− 4.00000i	− 0.228292i	−0.993464	−	0.114146i	$-0.963587\pi$
$307$	− 4.00000i	− 0.228292i	0.993464	−	0.114146i	$-0.0364132\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	8.00000	0.453638	0.226819	−	0.973937i	$-0.427167\pi$
$311$	8.00000	0.453638	0.226819	+	0.973937i	$0.427167\pi$
$312$	2.00000i	0.113228i
$313$	10.0000i	0.565233i	0.959233	+	0.282617i	$0.0912024\pi$
$313$	10.0000i	0.565233i	−0.959233	+	0.282617i	$0.908798\pi$
$314$	10.0000	0.564333
$315$	0	0
$316$	−8.00000	−0.450035
$317$	− 30.0000i	− 1.68497i	−0.538721	−	0.842484i	$-0.681092\pi$
$317$	− 30.0000i	− 1.68497i	0.538721	−	0.842484i	$-0.318908\pi$
$318$	6.00000i	0.336463i
$319$	8.00000	0.447914
$320$	0	0
$321$	−12.0000	−0.669775
$322$	0	0
$323$	− 24.0000i	− 1.33540i
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	20.0000	1.10770
$327$	10.0000i	0.553001i
$328$	10.0000i	0.552158i
$329$	0	0
$330$	0	0
$331$	28.0000	1.53902	0.769510	−	0.638635i	$-0.220501\pi$
$331$	28.0000	1.53902	0.769510	+	0.638635i	$0.220501\pi$
$332$	4.00000i	0.219529i
$333$	− 2.00000i	− 0.109599i
$334$	−8.00000	−0.437741
$335$	0	0
$336$	0	0
$337$	− 2.00000i	− 0.108947i	−0.998515	−	0.0544735i	$-0.982652\pi$
$337$	− 2.00000i	− 0.108947i	0.998515	−	0.0544735i	$-0.0173480\pi$
$338$	− 9.00000i	− 0.489535i
$339$	6.00000	0.325875
$340$	0	0
$341$	0	0
$342$	− 4.00000i	− 0.216295i
$343$	0	0
$344$	4.00000	0.215666
$345$	0	0
$346$	14.0000	0.752645
$347$	− 12.0000i	− 0.644194i	−0.946707	−	0.322097i	$-0.895612\pi$
$347$	− 12.0000i	− 0.644194i	0.946707	−	0.322097i	$-0.104388\pi$
$348$	− 2.00000i	− 0.107211i
$349$	−14.0000	−0.749403	−0.374701	−	0.927146i	$-0.622255\pi$
$349$	−14.0000	−0.749403	−0.374701	+	0.927146i	$0.622255\pi$
$350$	0	0
$351$	−2.00000	−0.106752
$352$	− 4.00000i	− 0.213201i
$353$	34.0000i	1.80964i	0.425797	+	0.904819i	$0.359994\pi$
$353$	34.0000i	1.80964i	−0.425797	+	0.904819i	$0.640006\pi$
$354$	4.00000	0.212598
$355$	0	0
$356$	18.0000	0.953998
$357$	0	0
$358$	4.00000i	0.211407i
$359$	−8.00000	−0.422224	−0.211112	−	0.977462i	$-0.567708\pi$
$359$	−8.00000	−0.422224	−0.211112	+	0.977462i	$0.567708\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	2.00000i	0.105118i
$363$	5.00000i	0.262432i
$364$	0	0
$365$	0	0
$366$	−10.0000	−0.522708
$367$	8.00000i	0.417597i	0.977959	+	0.208798i	$0.0669552\pi$
$367$	8.00000i	0.417597i	−0.977959	+	0.208798i	$0.933045\pi$
$368$	− 1.00000i	− 0.0521286i
$369$	−10.0000	−0.520579
$370$	0	0
$371$	0	0
$372$	0	0
$373$	− 18.0000i	− 0.932005i	−0.884783	−	0.466002i	$-0.845694\pi$
$373$	− 18.0000i	− 0.932005i	0.884783	−	0.466002i	$-0.154306\pi$
$374$	24.0000	1.24101
$375$	0	0
$376$	0	0
$377$	− 4.00000i	− 0.206010i
$378$	0	0
$379$	−12.0000	−0.616399	−0.308199	−	0.951322i	$-0.599726\pi$
$379$	−12.0000	−0.616399	−0.308199	+	0.951322i	$0.599726\pi$
$380$	0	0
$381$	−16.0000	−0.819705
$382$	0	0
$383$	− 32.0000i	− 1.63512i	−0.575841	−	0.817562i	$-0.695325\pi$
$383$	− 32.0000i	− 1.63512i	0.575841	−	0.817562i	$-0.304675\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	2.00000	0.101797
$387$	4.00000i	0.203331i
$388$	2.00000i	0.101535i
$389$	−38.0000	−1.92668	−0.963338	−	0.268290i	$-0.913542\pi$
$389$	−38.0000	−1.92668	−0.963338	+	0.268290i	$0.913542\pi$
$390$	0	0
$391$	6.00000	0.303433
$392$	7.00000i	0.353553i
$393$	20.0000i	1.00887i
$394$	−6.00000	−0.302276
$395$	0	0
$396$	4.00000	0.201008
$397$	2.00000i	0.100377i	0.998740	+	0.0501886i	$0.0159822\pi$
$397$	2.00000i	0.100377i	−0.998740	+	0.0501886i	$0.984018\pi$
$398$	− 16.0000i	− 0.802008i
$399$	0	0
$400$	0	0
$401$	10.0000	0.499376	0.249688	−	0.968326i	$-0.419672\pi$
$401$	10.0000	0.499376	0.249688	+	0.968326i	$0.419672\pi$
$402$	12.0000i	0.598506i
$403$	0	0
$404$	−6.00000	−0.298511
$405$	0	0
$406$	0	0
$407$	8.00000i	0.396545i
$408$	− 6.00000i	− 0.297044i
$409$	−26.0000	−1.28562	−0.642809	−	0.766027i	$-0.722231\pi$
$409$	−26.0000	−1.28562	−0.642809	+	0.766027i	$0.722231\pi$
$410$	0	0
$411$	2.00000	0.0986527
$412$	0	0
$413$	0	0
$414$	1.00000	0.0491473
$415$	0	0
$416$	−2.00000	−0.0980581
$417$	20.0000i	0.979404i
$418$	16.0000i	0.782586i
$419$	−12.0000	−0.586238	−0.293119	−	0.956076i	$-0.594693\pi$
$419$	−12.0000	−0.586238	−0.293119	+	0.956076i	$0.594693\pi$
$420$	0	0
$421$	−2.00000	−0.0974740	−0.0487370	−	0.998812i	$-0.515520\pi$
$421$	−2.00000	−0.0974740	−0.0487370	+	0.998812i	$0.515520\pi$
$422$	− 4.00000i	− 0.194717i
$423$	0	0
$424$	−6.00000	−0.291386
$425$	0	0
$426$	−8.00000	−0.387601
$427$	0	0
$428$	− 12.0000i	− 0.580042i
$429$	8.00000	0.386244
$430$	0	0
$431$	−32.0000	−1.54139	−0.770693	−	0.637207i	$-0.780090\pi$
$431$	−32.0000	−1.54139	−0.770693	+	0.637207i	$0.780090\pi$
$432$	− 1.00000i	− 0.0481125i
$433$	34.0000i	1.63394i	0.576683	+	0.816968i	$0.304347\pi$
$433$	34.0000i	1.63394i	−0.576683	+	0.816968i	$0.695653\pi$
$434$	0	0
$435$	0	0
$436$	−10.0000	−0.478913
$437$	4.00000i	0.191346i
$438$	10.0000i	0.477818i
$439$	8.00000	0.381819	0.190910	−	0.981608i	$-0.438856\pi$
$439$	8.00000	0.381819	0.190910	+	0.981608i	$0.438856\pi$
$440$	0	0
$441$	−7.00000	−0.333333
$442$	− 12.0000i	− 0.570782i
$443$	− 4.00000i	− 0.190046i	−0.995475	−	0.0950229i	$-0.969708\pi$
$443$	− 4.00000i	− 0.190046i	0.995475	−	0.0950229i	$-0.0302924\pi$
$444$	2.00000	0.0949158
$445$	0	0
$446$	16.0000	0.757622
$447$	10.0000i	0.472984i
$448$	0	0
$449$	30.0000	1.41579	0.707894	−	0.706319i	$-0.249646\pi$
$449$	30.0000	1.41579	0.707894	+	0.706319i	$0.249646\pi$
$450$	0	0
$451$	40.0000	1.88353
$452$	6.00000i	0.282216i
$453$	8.00000i	0.375873i
$454$	20.0000	0.938647
$455$	0	0
$456$	4.00000	0.187317
$457$	− 10.0000i	− 0.467780i	−0.972263	−	0.233890i	$-0.924854\pi$
$457$	− 10.0000i	− 0.467780i	0.972263	−	0.233890i	$-0.0751456\pi$
$458$	14.0000i	0.654177i
$459$	6.00000	0.280056
$460$	0	0
$461$	−18.0000	−0.838344	−0.419172	−	0.907907i	$-0.637680\pi$
$461$	−18.0000	−0.838344	−0.419172	+	0.907907i	$0.637680\pi$
$462$	0	0
$463$	16.0000i	0.743583i	0.928316	+	0.371792i	$0.121256\pi$
$463$	16.0000i	0.743583i	−0.928316	+	0.371792i	$0.878744\pi$
$464$	2.00000	0.0928477
$465$	0	0
$466$	−6.00000	−0.277945
$467$	− 12.0000i	− 0.555294i	−0.960683	−	0.277647i	$-0.910445\pi$
$467$	− 12.0000i	− 0.555294i	0.960683	−	0.277647i	$-0.0895545\pi$
$468$	− 2.00000i	− 0.0924500i
$469$	0	0
$470$	0	0
$471$	−10.0000	−0.460776
$472$	4.00000i	0.184115i
$473$	− 16.0000i	− 0.735681i
$474$	8.00000	0.367452
$475$	0	0
$476$	0	0
$477$	− 6.00000i	− 0.274721i
$478$	16.0000i	0.731823i
$479$	−32.0000	−1.46212	−0.731059	−	0.682315i	$-0.760973\pi$
$479$	−32.0000	−1.46212	−0.731059	+	0.682315i	$0.760973\pi$
$480$	0	0
$481$	4.00000	0.182384
$482$	− 18.0000i	− 0.819878i
$483$	0	0
$484$	−5.00000	−0.227273
$485$	0	0
$486$	1.00000	0.0453609
$487$	− 40.0000i	− 1.81257i	−0.422664	−	0.906287i	$-0.638905\pi$
$487$	− 40.0000i	− 1.81257i	0.422664	−	0.906287i	$-0.361095\pi$
$488$	− 10.0000i	− 0.452679i
$489$	−20.0000	−0.904431
$490$	0	0
$491$	−4.00000	−0.180517	−0.0902587	−	0.995918i	$-0.528769\pi$
$491$	−4.00000	−0.180517	−0.0902587	+	0.995918i	$0.528769\pi$
$492$	− 10.0000i	− 0.450835i
$493$	12.0000i	0.540453i
$494$	8.00000	0.359937
$495$	0	0
$496$	0	0
$497$	0	0
$498$	− 4.00000i	− 0.179244i
$499$	−20.0000	−0.895323	−0.447661	−	0.894203i	$-0.647743\pi$
$499$	−20.0000	−0.895323	−0.447661	+	0.894203i	$0.647743\pi$
$500$	0	0
$501$	8.00000	0.357414
$502$	− 20.0000i	− 0.892644i
$503$	− 40.0000i	− 1.78351i	−0.452517	−	0.891756i	$-0.649474\pi$
$503$	− 40.0000i	− 1.78351i	0.452517	−	0.891756i	$-0.350526\pi$
$504$	0	0
$505$	0	0
$506$	−4.00000	−0.177822
$507$	9.00000i	0.399704i
$508$	− 16.0000i	− 0.709885i
$509$	34.0000	1.50702	0.753512	−	0.657434i	$-0.228358\pi$
$509$	34.0000	1.50702	0.753512	+	0.657434i	$0.228358\pi$
$510$	0	0
$511$	0	0
$512$	− 1.00000i	− 0.0441942i
$513$	4.00000i	0.176604i
$514$	14.0000	0.617514
$515$	0	0
$516$	−4.00000	−0.176090
$517$	0	0
$518$	0	0
$519$	−14.0000	−0.614532
$520$	0	0
$521$	−14.0000	−0.613351	−0.306676	−	0.951814i	$-0.599217\pi$
$521$	−14.0000	−0.613351	−0.306676	+	0.951814i	$0.599217\pi$
$522$	2.00000i	0.0875376i
$523$	− 36.0000i	− 1.57417i	−0.616844	−	0.787085i	$-0.711589\pi$
$523$	− 36.0000i	− 1.57417i	0.616844	−	0.787085i	$-0.288411\pi$
$524$	−20.0000	−0.873704
$525$	0	0
$526$	−24.0000	−1.04645
$527$	0	0
$528$	4.00000i	0.174078i
$529$	−1.00000	−0.0434783
$530$	0	0
$531$	−4.00000	−0.173585
$532$	0	0
$533$	− 20.0000i	− 0.866296i
$534$	−18.0000	−0.778936
$535$	0	0
$536$	−12.0000	−0.518321
$537$	− 4.00000i	− 0.172613i
$538$	14.0000i	0.603583i
$539$	28.0000	1.20605
$540$	0	0
$541$	14.0000	0.601907	0.300954	−	0.953639i	$-0.402695\pi$
$541$	14.0000	0.601907	0.300954	+	0.953639i	$0.402695\pi$
$542$	0	0
$543$	− 2.00000i	− 0.0858282i
$544$	6.00000	0.257248
$545$	0	0
$546$	0	0
$547$	− 4.00000i	− 0.171028i	−0.996337	−	0.0855138i	$-0.972747\pi$
$547$	− 4.00000i	− 0.171028i	0.996337	−	0.0855138i	$-0.0272532\pi$
$548$	2.00000i	0.0854358i
$549$	10.0000	0.426790
$550$	0	0
$551$	−8.00000	−0.340811
$552$	1.00000i	0.0425628i
$553$	0	0
$554$	10.0000	0.424859
$555$	0	0
$556$	−20.0000	−0.848189
$557$	− 14.0000i	− 0.593199i	−0.955002	−	0.296600i	$-0.904147\pi$
$557$	− 14.0000i	− 0.593199i	0.955002	−	0.296600i	$-0.0958526\pi$
$558$	0	0
$559$	−8.00000	−0.338364
$560$	0	0
$561$	−24.0000	−1.01328
$562$	− 18.0000i	− 0.759284i
$563$	− 36.0000i	− 1.51722i	−0.651546	−	0.758610i	$-0.725879\pi$
$563$	− 36.0000i	− 1.51722i	0.651546	−	0.758610i	$-0.274121\pi$
$564$	0	0
$565$	0	0
$566$	−4.00000	−0.168133
$567$	0	0
$568$	− 8.00000i	− 0.335673i
$569$	−18.0000	−0.754599	−0.377300	−	0.926091i	$-0.623147\pi$
$569$	−18.0000	−0.754599	−0.377300	+	0.926091i	$0.623147\pi$
$570$	0	0
$571$	−4.00000	−0.167395	−0.0836974	−	0.996491i	$-0.526673\pi$
$571$	−4.00000	−0.167395	−0.0836974	+	0.996491i	$0.526673\pi$
$572$	8.00000i	0.334497i
$573$	0	0
$574$	0	0
$575$	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	$-0.0132552\pi$
$578$	19.0000i	0.790296i
$579$	−2.00000	−0.0831172
$580$	0	0
$581$	0	0
$582$	− 2.00000i	− 0.0829027i
$583$	24.0000i	0.993978i
$584$	−10.0000	−0.413803
$585$	0	0
$586$	22.0000	0.908812
$587$	− 28.0000i	− 1.15568i	−0.816149	−	0.577842i	$-0.803895\pi$
$587$	− 28.0000i	− 1.15568i	0.816149	−	0.577842i	$-0.196105\pi$
$588$	− 7.00000i	− 0.288675i
$589$	0	0
$590$	0	0
$591$	6.00000	0.246807
$592$	2.00000i	0.0821995i
$593$	− 30.0000i	− 1.23195i	−0.787765	−	0.615976i	$-0.788762\pi$
$593$	− 30.0000i	− 1.23195i	0.787765	−	0.615976i	$-0.211238\pi$
$594$	−4.00000	−0.164122
$595$	0	0
$596$	−10.0000	−0.409616
$597$	16.0000i	0.654836i
$598$	2.00000i	0.0817861i
$599$	40.0000	1.63436	0.817178	−	0.576386i	$-0.195537\pi$
$599$	40.0000	1.63436	0.817178	+	0.576386i	$0.195537\pi$
$600$	0	0
$601$	−38.0000	−1.55005	−0.775026	−	0.631929i	$-0.782263\pi$
$601$	−38.0000	−1.55005	−0.775026	+	0.631929i	$0.782263\pi$
$602$	0	0
$603$	− 12.0000i	− 0.488678i
$604$	−8.00000	−0.325515
$605$	0	0
$606$	6.00000	0.243733
$607$	16.0000i	0.649420i	0.945814	+	0.324710i	$0.105267\pi$
$607$	16.0000i	0.649420i	−0.945814	+	0.324710i	$0.894733\pi$
$608$	4.00000i	0.162221i
$609$	0	0
$610$	0	0
$611$	0	0
$612$	6.00000i	0.242536i
$613$	− 18.0000i	− 0.727013i	−0.931592	−	0.363507i	$-0.881579\pi$
$613$	− 18.0000i	− 0.727013i	0.931592	−	0.363507i	$-0.118421\pi$
$614$	−4.00000	−0.161427
$615$	0	0
$616$	0	0
$617$	− 18.0000i	− 0.724653i	−0.932051	−	0.362326i	$-0.881983\pi$
$617$	− 18.0000i	− 0.724653i	0.932051	−	0.362326i	$-0.118017\pi$
$618$	0	0
$619$	4.00000	0.160774	0.0803868	−	0.996764i	$-0.474384\pi$
$619$	4.00000	0.160774	0.0803868	+	0.996764i	$0.474384\pi$
$620$	0	0
$621$	−1.00000	−0.0401286
$622$	− 8.00000i	− 0.320771i
$623$	0	0
$624$	2.00000	0.0800641
$625$	0	0
$626$	10.0000	0.399680
$627$	− 16.0000i	− 0.638978i
$628$	− 10.0000i	− 0.399043i
$629$	−12.0000	−0.478471
$630$	0	0
$631$	16.0000	0.636950	0.318475	−	0.947931i	$-0.396829\pi$
$631$	16.0000	0.636950	0.318475	+	0.947931i	$0.396829\pi$
$632$	8.00000i	0.318223i
$633$	4.00000i	0.158986i
$634$	−30.0000	−1.19145
$635$	0	0
$636$	6.00000	0.237915
$637$	− 14.0000i	− 0.554700i
$638$	− 8.00000i	− 0.316723i
$639$	8.00000	0.316475
$640$	0	0
$641$	26.0000	1.02694	0.513469	−	0.858108i	$-0.328360\pi$
$641$	26.0000	1.02694	0.513469	+	0.858108i	$0.328360\pi$
$642$	12.0000i	0.473602i
$643$	20.0000i	0.788723i	0.918955	+	0.394362i	$0.129034\pi$
$643$	20.0000i	0.788723i	−0.918955	+	0.394362i	$0.870966\pi$
$644$	0	0
$645$	0	0
$646$	−24.0000	−0.944267
$647$	− 8.00000i	− 0.314512i	−0.987558	−	0.157256i	$-0.949735\pi$
$647$	− 8.00000i	− 0.314512i	0.987558	−	0.157256i	$-0.0502649\pi$
$648$	1.00000i	0.0392837i
$649$	16.0000	0.628055
$650$	0	0
$651$	0	0
$652$	− 20.0000i	− 0.783260i
$653$	14.0000i	0.547862i	0.961749	+	0.273931i	$0.0883240\pi$
$653$	14.0000i	0.547862i	−0.961749	+	0.273931i	$0.911676\pi$
$654$	10.0000	0.391031
$655$	0	0
$656$	10.0000	0.390434
$657$	− 10.0000i	− 0.390137i
$658$	0	0
$659$	20.0000	0.779089	0.389545	−	0.921008i	$-0.372632\pi$
$659$	20.0000	0.779089	0.389545	+	0.921008i	$0.372632\pi$
$660$	0	0
$661$	−50.0000	−1.94477	−0.972387	−	0.233373i	$-0.925024\pi$
$661$	−50.0000	−1.94477	−0.972387	+	0.233373i	$0.925024\pi$
$662$	− 28.0000i	− 1.08825i
$663$	12.0000i	0.466041i
$664$	4.00000	0.155230
$665$	0	0
$666$	−2.00000	−0.0774984
$667$	− 2.00000i	− 0.0774403i
$668$	8.00000i	0.309529i
$669$	−16.0000	−0.618596
$670$	0	0
$671$	−40.0000	−1.54418
$672$	0	0
$673$	2.00000i	0.0770943i	0.999257	+	0.0385472i	$0.0122730\pi$
$673$	2.00000i	0.0770943i	−0.999257	+	0.0385472i	$0.987727\pi$
$674$	−2.00000	−0.0770371
$675$	0	0
$676$	−9.00000	−0.346154
$677$	− 38.0000i	− 1.46046i	−0.683202	−	0.730229i	$-0.739413\pi$
$677$	− 38.0000i	− 1.46046i	0.683202	−	0.730229i	$-0.260587\pi$
$678$	− 6.00000i	− 0.230429i
$679$	0	0
$680$	0	0
$681$	−20.0000	−0.766402
$682$	0	0
$683$	− 36.0000i	− 1.37750i	−0.724998	−	0.688751i	$-0.758159\pi$
$683$	− 36.0000i	− 1.37750i	0.724998	−	0.688751i	$-0.241841\pi$
$684$	−4.00000	−0.152944
$685$	0	0
$686$	0	0
$687$	− 14.0000i	− 0.534133i
$688$	− 4.00000i	− 0.152499i
$689$	12.0000	0.457164
$690$	0	0
$691$	−28.0000	−1.06517	−0.532585	−	0.846376i	$-0.678779\pi$
$691$	−28.0000	−1.06517	−0.532585	+	0.846376i	$0.678779\pi$
$692$	− 14.0000i	− 0.532200i
$693$	0	0
$694$	−12.0000	−0.455514
$695$	0	0
$696$	−2.00000	−0.0758098
$697$	60.0000i	2.27266i
$698$	14.0000i	0.529908i
$699$	6.00000	0.226941
$700$	0	0
$701$	−50.0000	−1.88847	−0.944237	−	0.329267i	$-0.893198\pi$
$701$	−50.0000	−1.88847	−0.944237	+	0.329267i	$0.893198\pi$
$702$	2.00000i	0.0754851i
$703$	− 8.00000i	− 0.301726i
$704$	−4.00000	−0.150756
$705$	0	0
$706$	34.0000	1.27961
$707$	0	0
$708$	− 4.00000i	− 0.150329i
$709$	34.0000	1.27690	0.638448	−	0.769665i	$-0.279577\pi$
$709$	34.0000	1.27690	0.638448	+	0.769665i	$0.279577\pi$
$710$	0	0
$711$	−8.00000	−0.300023
$712$	− 18.0000i	− 0.674579i
$713$	0	0
$714$	0	0
$715$	0	0
$716$	4.00000	0.149487
$717$	− 16.0000i	− 0.597531i
$718$	8.00000i	0.298557i
$719$	32.0000	1.19340	0.596699	−	0.802465i	$-0.296479\pi$
$719$	32.0000	1.19340	0.596699	+	0.802465i	$0.296479\pi$
$720$	0	0
$721$	0	0
$722$	3.00000i	0.111648i
$723$	18.0000i	0.669427i
$724$	2.00000	0.0743294
$725$	0	0
$726$	5.00000	0.185567
$727$	− 16.0000i	− 0.593407i	−0.954970	−	0.296704i	$-0.904113\pi$
$727$	− 16.0000i	− 0.593407i	0.954970	−	0.296704i	$-0.0958873\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	24.0000	0.887672
$732$	10.0000i	0.369611i
$733$	6.00000i	0.221615i	0.993842	+	0.110808i	$0.0353437\pi$
$733$	6.00000i	0.221615i	−0.993842	+	0.110808i	$0.964656\pi$
$734$	8.00000	0.295285
$735$	0	0
$736$	−1.00000	−0.0368605
$737$	48.0000i	1.76810i
$738$	10.0000i	0.368105i
$739$	12.0000	0.441427	0.220714	−	0.975339i	$-0.429161\pi$
$739$	12.0000	0.441427	0.220714	+	0.975339i	$0.429161\pi$
$740$	0	0
$741$	−8.00000	−0.293887
$742$	0	0
$743$	− 24.0000i	− 0.880475i	−0.897881	−	0.440237i	$-0.854894\pi$
$743$	− 24.0000i	− 0.880475i	0.897881	−	0.440237i	$-0.145106\pi$
$744$	0	0
$745$	0	0
$746$	−18.0000	−0.659027
$747$	4.00000i	0.146352i
$748$	− 24.0000i	− 0.877527i
$749$	0	0
$750$	0	0
$751$	−40.0000	−1.45962	−0.729810	−	0.683650i	$-0.760392\pi$
$751$	−40.0000	−1.45962	−0.729810	+	0.683650i	$0.760392\pi$
$752$	0	0
$753$	20.0000i	0.728841i
$754$	−4.00000	−0.145671
$755$	0	0
$756$	0	0
$757$	18.0000i	0.654221i	0.944986	+	0.327111i	$0.106075\pi$
$757$	18.0000i	0.654221i	−0.944986	+	0.327111i	$0.893925\pi$
$758$	12.0000i	0.435860i
$759$	4.00000	0.145191
$760$	0	0
$761$	−22.0000	−0.797499	−0.398750	−	0.917060i	$-0.630556\pi$
$761$	−22.0000	−0.797499	−0.398750	+	0.917060i	$0.630556\pi$
$762$	16.0000i	0.579619i
$763$	0	0
$764$	0	0
$765$	0	0
$766$	−32.0000	−1.15621
$767$	− 8.00000i	− 0.288863i
$768$	1.00000i	0.0360844i
$769$	14.0000	0.504853	0.252426	−	0.967616i	$-0.418771\pi$
$769$	14.0000	0.504853	0.252426	+	0.967616i	$0.418771\pi$
$770$	0	0
$771$	−14.0000	−0.504198
$772$	− 2.00000i	− 0.0719816i
$773$	54.0000i	1.94225i	0.238581	+	0.971123i	$0.423318\pi$
$773$	54.0000i	1.94225i	−0.238581	+	0.971123i	$0.576682\pi$
$774$	4.00000	0.143777
$775$	0	0
$776$	2.00000	0.0717958
$777$	0	0
$778$	38.0000i	1.36237i
$779$	−40.0000	−1.43315
$780$	0	0
$781$	−32.0000	−1.14505
$782$	− 6.00000i	− 0.214560i
$783$	− 2.00000i	− 0.0714742i
$784$	7.00000	0.250000
$785$	0	0
$786$	20.0000	0.713376
$787$	− 4.00000i	− 0.142585i	−0.997455	−	0.0712923i	$-0.977288\pi$
$787$	− 4.00000i	− 0.142585i	0.997455	−	0.0712923i	$-0.0227123\pi$
$788$	6.00000i	0.213741i
$789$	24.0000	0.854423
$790$	0	0
$791$	0	0
$792$	− 4.00000i	− 0.142134i
$793$	20.0000i	0.710221i
$794$	2.00000	0.0709773
$795$	0	0
$796$	−16.0000	−0.567105
$797$	34.0000i	1.20434i	0.798367	+	0.602171i	$0.205697\pi$
$797$	34.0000i	1.20434i	−0.798367	+	0.602171i	$0.794303\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	18.0000	0.635999
$802$	− 10.0000i	− 0.353112i
$803$	40.0000i	1.41157i
$804$	12.0000	0.423207
$805$	0	0
$806$	0	0
$807$	− 14.0000i	− 0.492823i
$808$	6.00000i	0.211079i
$809$	−26.0000	−0.914111	−0.457056	−	0.889438i	$-0.651096\pi$
$809$	−26.0000	−0.914111	−0.457056	+	0.889438i	$0.651096\pi$
$810$	0	0
$811$	28.0000	0.983213	0.491606	−	0.870817i	$-0.336410\pi$
$811$	28.0000	0.983213	0.491606	+	0.870817i	$0.336410\pi$
$812$	0	0
$813$	0	0
$814$	8.00000	0.280400
$815$	0	0
$816$	−6.00000	−0.210042
$817$	16.0000i	0.559769i
$818$	26.0000i	0.909069i
$819$	0	0
$820$	0	0
$821$	−42.0000	−1.46581	−0.732905	−	0.680331i	$-0.761836\pi$
$821$	−42.0000	−1.46581	−0.732905	+	0.680331i	$0.761836\pi$
$822$	− 2.00000i	− 0.0697580i
$823$	− 40.0000i	− 1.39431i	−0.716919	−	0.697156i	$-0.754448\pi$
$823$	− 40.0000i	− 1.39431i	0.716919	−	0.697156i	$-0.245552\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	12.0000i	0.417281i	0.977992	+	0.208640i	$0.0669038\pi$
$827$	12.0000i	0.417281i	−0.977992	+	0.208640i	$0.933096\pi$
$828$	− 1.00000i	− 0.0347524i
$829$	−46.0000	−1.59765	−0.798823	−	0.601566i	$-0.794544\pi$
$829$	−46.0000	−1.59765	−0.798823	+	0.601566i	$0.794544\pi$
$830$	0	0
$831$	−10.0000	−0.346896
$832$	2.00000i	0.0693375i
$833$	42.0000i	1.45521i
$834$	20.0000	0.692543
$835$	0	0
$836$	16.0000	0.553372
$837$	0	0
$838$	12.0000i	0.414533i
$839$	−24.0000	−0.828572	−0.414286	−	0.910147i	$-0.635969\pi$
$839$	−24.0000	−0.828572	−0.414286	+	0.910147i	$0.635969\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	2.00000i	0.0689246i
$843$	18.0000i	0.619953i
$844$	−4.00000	−0.137686
$845$	0	0
$846$	0	0
$847$	0	0
$848$	6.00000i	0.206041i
$849$	4.00000	0.137280
$850$	0	0
$851$	2.00000	0.0685591
$852$	8.00000i	0.274075i
$853$	6.00000i	0.205436i	0.994711	+	0.102718i	$0.0327539\pi$
$853$	6.00000i	0.205436i	−0.994711	+	0.102718i	$0.967246\pi$
$854$	0	0
$855$	0	0
$856$	−12.0000	−0.410152
$857$	− 42.0000i	− 1.43469i	−0.696717	−	0.717346i	$-0.745357\pi$
$857$	− 42.0000i	− 1.43469i	0.696717	−	0.717346i	$-0.254643\pi$
$858$	− 8.00000i	− 0.273115i
$859$	−44.0000	−1.50126	−0.750630	−	0.660722i	$-0.770250\pi$
$859$	−44.0000	−1.50126	−0.750630	+	0.660722i	$0.770250\pi$
$860$	0	0
$861$	0	0
$862$	32.0000i	1.08992i
$863$	− 32.0000i	− 1.08929i	−0.838666	−	0.544646i	$-0.816664\pi$
$863$	− 32.0000i	− 1.08929i	0.838666	−	0.544646i	$-0.183336\pi$
$864$	−1.00000	−0.0340207
$865$	0	0
$866$	34.0000	1.15537
$867$	− 19.0000i	− 0.645274i
$868$	0	0
$869$	32.0000	1.08553
$870$	0	0
$871$	24.0000	0.813209
$872$	10.0000i	0.338643i
$873$	2.00000i	0.0676897i
$874$	4.00000	0.135302
$875$	0	0
$876$	10.0000	0.337869
$877$	50.0000i	1.68838i	0.536044	+	0.844190i	$0.319918\pi$
$877$	50.0000i	1.68838i	−0.536044	+	0.844190i	$0.680082\pi$
$878$	− 8.00000i	− 0.269987i
$879$	−22.0000	−0.742042
$880$	0	0
$881$	10.0000	0.336909	0.168454	−	0.985709i	$-0.446122\pi$
$881$	10.0000	0.336909	0.168454	+	0.985709i	$0.446122\pi$
$882$	7.00000i	0.235702i
$883$	20.0000i	0.673054i	0.941674	+	0.336527i	$0.109252\pi$
$883$	20.0000i	0.673054i	−0.941674	+	0.336527i	$0.890748\pi$
$884$	−12.0000	−0.403604
$885$	0	0
$886$	−4.00000	−0.134383
$887$	24.0000i	0.805841i	0.915235	+	0.402921i	$0.132005\pi$
$887$	24.0000i	0.805841i	−0.915235	+	0.402921i	$0.867995\pi$
$888$	− 2.00000i	− 0.0671156i
$889$	0	0
$890$	0	0
$891$	4.00000	0.134005
$892$	− 16.0000i	− 0.535720i
$893$	0	0
$894$	10.0000	0.334450
$895$	0	0
$896$	0	0
$897$	− 2.00000i	− 0.0667781i
$898$	− 30.0000i	− 1.00111i
$899$	0	0
$900$	0	0
$901$	−36.0000	−1.19933
$902$	− 40.0000i	− 1.33185i
$903$	0	0
$904$	6.00000	0.199557
$905$	0	0
$906$	8.00000	0.265782
$907$	− 44.0000i	− 1.46100i	−0.682915	−	0.730498i	$-0.739288\pi$
$907$	− 44.0000i	− 1.46100i	0.682915	−	0.730498i	$-0.260712\pi$
$908$	− 20.0000i	− 0.663723i
$909$	−6.00000	−0.199007
$910$	0	0
$911$	32.0000	1.06021	0.530104	−	0.847933i	$-0.322153\pi$
$911$	32.0000	1.06021	0.530104	+	0.847933i	$0.322153\pi$
$912$	− 4.00000i	− 0.132453i
$913$	− 16.0000i	− 0.529523i
$914$	−10.0000	−0.330771
$915$	0	0
$916$	14.0000	0.462573
$917$	0	0
$918$	− 6.00000i	− 0.198030i
$919$	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
$922$	18.0000i	0.592798i
$923$	16.0000i	0.526646i
$924$	0	0
$925$	0	0
$926$	16.0000	0.525793
$927$	0	0
$928$	− 2.00000i	− 0.0656532i
$929$	30.0000	0.984268	0.492134	−	0.870519i	$-0.336217\pi$
$929$	30.0000	0.984268	0.492134	+	0.870519i	$0.336217\pi$
$930$	0	0
$931$	−28.0000	−0.917663
$932$	6.00000i	0.196537i
$933$	8.00000i	0.261908i
$934$	−12.0000	−0.392652
$935$	0	0
$936$	−2.00000	−0.0653720
$937$	22.0000i	0.718709i	0.933201	+	0.359354i	$0.117003\pi$
$937$	22.0000i	0.718709i	−0.933201	+	0.359354i	$0.882997\pi$
$938$	0	0
$939$	−10.0000	−0.326338
$940$	0	0
$941$	14.0000	0.456387	0.228193	−	0.973616i	$-0.426718\pi$
$941$	14.0000	0.456387	0.228193	+	0.973616i	$0.426718\pi$
$942$	10.0000i	0.325818i
$943$	− 10.0000i	− 0.325645i
$944$	4.00000	0.130189
$945$	0	0
$946$	−16.0000	−0.520205
$947$	− 4.00000i	− 0.129983i	−0.997886	−	0.0649913i	$-0.979298\pi$
$947$	− 4.00000i	− 0.129983i	0.997886	−	0.0649913i	$-0.0207020\pi$
$948$	− 8.00000i	− 0.259828i
$949$	20.0000	0.649227
$950$	0	0
$951$	30.0000	0.972817
$952$	0	0
$953$	2.00000i	0.0647864i	0.999475	+	0.0323932i	$0.0103129\pi$
$953$	2.00000i	0.0647864i	−0.999475	+	0.0323932i	$0.989687\pi$
$954$	−6.00000	−0.194257
$955$	0	0
$956$	16.0000	0.517477
$957$	8.00000i	0.258603i
$958$	32.0000i	1.03387i
$959$	0	0
$960$	0	0
$961$	−31.0000	−1.00000
$962$	− 4.00000i	− 0.128965i
$963$	− 12.0000i	− 0.386695i
$964$	−18.0000	−0.579741
$965$	0	0
$966$	0	0
$967$	24.0000i	0.771788i	0.922543	+	0.385894i	$0.126107\pi$
$967$	24.0000i	0.771788i	−0.922543	+	0.385894i	$0.873893\pi$
$968$	5.00000i	0.160706i
$969$	24.0000	0.770991
$970$	0	0
$971$	52.0000	1.66876	0.834380	−	0.551190i	$-0.185826\pi$
$971$	52.0000	1.66876	0.834380	+	0.551190i	$0.185826\pi$
$972$	− 1.00000i	− 0.0320750i
$973$	0	0
$974$	−40.0000	−1.28168
$975$	0	0
$976$	−10.0000	−0.320092
$977$	− 42.0000i	− 1.34370i	−0.740688	−	0.671850i	$-0.765500\pi$
$977$	− 42.0000i	− 1.34370i	0.740688	−	0.671850i	$-0.234500\pi$
$978$	20.0000i	0.639529i
$979$	−72.0000	−2.30113
$980$	0	0
$981$	−10.0000	−0.319275
$982$	4.00000i	0.127645i
$983$	24.0000i	0.765481i	0.923856	+	0.382741i	$0.125020\pi$
$983$	24.0000i	0.765481i	−0.923856	+	0.382741i	$0.874980\pi$
$984$	−10.0000	−0.318788
$985$	0	0
$986$	12.0000	0.382158
$987$	0	0
$988$	− 8.00000i	− 0.254514i
$989$	−4.00000	−0.127193
$990$	0	0
$991$	0	0		−	1.00000i	$-0.5\pi$
$991$	0	0			1.00000i	$0.5\pi$
$992$	0	0
$993$	28.0000i	0.888553i
$994$	0	0
$995$	0	0
$996$	−4.00000	−0.126745
$997$	− 38.0000i	− 1.20347i	−0.798695	−	0.601736i	$-0.794476\pi$
$997$	− 38.0000i	− 1.20347i	0.798695	−	0.601736i	$-0.205524\pi$
$998$	20.0000i	0.633089i
$999$	2.00000	0.0632772

Twists

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

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

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +1.00000i q^{8} -1.00000 q^{9} +4.00000 q^{11} -1.00000i q^{12} -2.00000i q^{13} +1.00000 q^{16} +6.00000i q^{17} +1.00000i q^{18} -4.00000 q^{19} -4.00000i q^{22} -1.00000i q^{23} -1.00000 q^{24} -2.00000 q^{26} -1.00000i q^{27} +2.00000 q^{29} -1.00000i q^{32} +4.00000i q^{33} +6.00000 q^{34} +1.00000 q^{36} +2.00000i q^{37} +4.00000i q^{38} +2.00000 q^{39} +10.0000 q^{41} -4.00000i q^{43} -4.00000 q^{44} -1.00000 q^{46} +1.00000i q^{48} +7.00000 q^{49} -6.00000 q^{51} +2.00000i q^{52} +6.00000i q^{53} -1.00000 q^{54} -4.00000i q^{57} -2.00000i q^{58} +4.00000 q^{59} -10.0000 q^{61} -1.00000 q^{64} +4.00000 q^{66} +12.0000i q^{67} -6.00000i q^{68} +1.00000 q^{69} -8.00000 q^{71} -1.00000i q^{72} +10.0000i q^{73} +2.00000 q^{74} +4.00000 q^{76} -2.00000i q^{78} +8.00000 q^{79} +1.00000 q^{81} -10.0000i q^{82} -4.00000i q^{83} -4.00000 q^{86} +2.00000i q^{87} +4.00000i q^{88} -18.0000 q^{89} +1.00000i q^{92} +1.00000 q^{96} -2.00000i q^{97} -7.00000i q^{98} -4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} + 2 q^{6} - 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^4 + 2 * q^6 - 2 * q^9	\( 2 q - 2 q^{4} + 2 q^{6} - 2 q^{9} + 8 q^{11} + 2 q^{16} - 8 q^{19} - 2 q^{24} - 4 q^{26} + 4 q^{29} + 12 q^{34} + 2 q^{36} + 4 q^{39} + 20 q^{41} - 8 q^{44} - 2 q^{46} + 14 q^{49} - 12 q^{51} - 2 q^{54} + 8 q^{59} - 20 q^{61} - 2 q^{64} + 8 q^{66} + 2 q^{69} - 16 q^{71} + 4 q^{74} + 8 q^{76} + 16 q^{79} + 2 q^{81} - 8 q^{86} - 36 q^{89} + 2 q^{96} - 8 q^{99}+O(q^{100}) \) 2 * q - 2 * q^4 + 2 * q^6 - 2 * q^9 + 8 * q^11 + 2 * q^16 - 8 * q^19 - 2 * q^24 - 4 * q^26 + 4 * q^29 + 12 * q^34 + 2 * q^36 + 4 * q^39 + 20 * q^41 - 8 * q^44 - 2 * q^46 + 14 * q^49 - 12 * q^51 - 2 * q^54 + 8 * q^59 - 20 * q^61 - 2 * q^64 + 8 * q^66 + 2 * q^69 - 16 * q^71 + 4 * q^74 + 8 * q^76 + 16 * q^79 + 2 * q^81 - 8 * q^86 - 36 * q^89 + 2 * q^96 - 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more