LMFDB - Embedded newform 4650.2.d.bh.3349.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4650,2,Mod(3349,4650)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4650.3349"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4650, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 4650 = 2 \cdot 3 \cdot 5^{2} \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4650.d (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,-4,0,4,0,0,-4,0,14,0,0,-2,0,4,0,0,14] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$37.1304369399$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{33})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 17x^{2} + 64 $ x^4 + 17*x^2 + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 930)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3349.4
Root			$3.37228i$ of defining polynomial
Character	$\chi$	$=$	4650.3349
Dual form			4650.2.d.bh.3349.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +3.37228i q^{7} -1.00000i q^{8} -1.00000 q^{9} +0.627719 q^{11} +1.00000i q^{12} +2.00000i q^{13} -3.37228 q^{14} +1.00000 q^{16} -4.74456i q^{17} -1.00000i q^{18} +0.627719 q^{19} +3.37228 q^{21} +0.627719i q^{22} -3.37228i q^{23} -1.00000 q^{24} -2.00000 q^{26} +1.00000i q^{27} -3.37228i q^{28} +8.74456 q^{29} -1.00000 q^{31} +1.00000i q^{32} -0.627719i q^{33} +4.74456 q^{34} +1.00000 q^{36} -0.744563i q^{37} +0.627719i q^{38} +2.00000 q^{39} +0.744563 q^{41} +3.37228i q^{42} -0.627719i q^{43} -0.627719 q^{44} +3.37228 q^{46} -6.74456i q^{47} -1.00000i q^{48} -4.37228 q^{49} -4.74456 q^{51} -2.00000i q^{52} -1.37228i q^{53} -1.00000 q^{54} +3.37228 q^{56} -0.627719i q^{57} +8.74456i q^{58} -2.74456 q^{59} +11.4891 q^{61} -1.00000i q^{62} -3.37228i q^{63} -1.00000 q^{64} +0.627719 q^{66} +10.7446i q^{67} +4.74456i q^{68} -3.37228 q^{69} +3.37228 q^{71} +1.00000i q^{72} +8.11684i q^{73} +0.744563 q^{74} -0.627719 q^{76} +2.11684i q^{77} +2.00000i q^{78} +4.62772 q^{79} +1.00000 q^{81} +0.744563i q^{82} +12.0000i q^{83} -3.37228 q^{84} +0.627719 q^{86} -8.74456i q^{87} -0.627719i q^{88} +1.37228 q^{89} -6.74456 q^{91} +3.37228i q^{92} +1.00000i q^{93} +6.74456 q^{94} +1.00000 q^{96} +2.00000i q^{97} -4.37228i q^{98} -0.627719 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{4} + 4 q^{6} - 4 q^{9} + 14 q^{11} - 2 q^{14} + 4 q^{16} + 14 q^{19} + 2 q^{21} - 4 q^{24} - 8 q^{26} + 12 q^{29} - 4 q^{31} - 4 q^{34} + 4 q^{36} + 8 q^{39} - 20 q^{41} - 14 q^{44} + 2 q^{46}+ \cdots - 14 q^{99}+O(q^{100}) $ 4 * q - 4 * q^4 + 4 * q^6 - 4 * q^9 + 14 * q^11 - 2 * q^14 + 4 * q^16 + 14 * q^19 + 2 * q^21 - 4 * q^24 - 8 * q^26 + 12 * q^29 - 4 * q^31 - 4 * q^34 + 4 * q^36 + 8 * q^39 - 20 * q^41 - 14 * q^44 + 2 * q^46 - 6 * q^49 + 4 * q^51 - 4 * q^54 + 2 * q^56 + 12 * q^59 - 4 * q^64 + 14 * q^66 - 2 * q^69 + 2 * q^71 - 20 * q^74 - 14 * q^76 + 30 * q^79 + 4 * q^81 - 2 * q^84 + 14 * q^86 - 6 * q^89 - 4 * q^91 + 4 * q^94 + 4 * q^96 - 14 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4650\mathbb{Z}\right)^\times$.

$n$	$1801$	$2977$	$3101$
$\chi(n)$	$1$	$-1$	$1$

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
$7$	3.37228i	1.27460i	0.770615	+	0.637301i	$0.219949\pi$
$7$	3.37228i	1.27460i	−0.770615	+	0.637301i	$0.780051\pi$
$8$	− 1.00000i	− 0.353553i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	0.627719	0.189264	0.0946322	−	0.995512i	$-0.469833\pi$
$11$	0.627719	0.189264	0.0946322	+	0.995512i	$0.469833\pi$
$12$	1.00000i	0.288675i
$13$	2.00000i	0.554700i	0.960769	+	0.277350i	$0.0894562\pi$
$13$	2.00000i	0.554700i	−0.960769	+	0.277350i	$0.910544\pi$
$14$	−3.37228	−0.901280
$15$	0	0
$16$	1.00000	0.250000
$17$	− 4.74456i	− 1.15073i	−0.817898	−	0.575363i	$-0.804861\pi$
$17$	− 4.74456i	− 1.15073i	0.817898	−	0.575363i	$-0.195139\pi$
$18$	− 1.00000i	− 0.235702i
$19$	0.627719	0.144009	0.0720043	−	0.997404i	$-0.477060\pi$
$19$	0.627719	0.144009	0.0720043	+	0.997404i	$0.477060\pi$
$20$	0	0
$21$	3.37228	0.735892
$22$	0.627719i	0.133830i
$23$	− 3.37228i	− 0.703169i	−0.936156	−	0.351585i	$-0.885643\pi$
$23$	− 3.37228i	− 0.703169i	0.936156	−	0.351585i	$-0.114357\pi$
$24$	−1.00000	−0.204124
$25$	0	0
$26$	−2.00000	−0.392232
$27$	1.00000i	0.192450i
$28$	− 3.37228i	− 0.637301i
$29$	8.74456	1.62382	0.811912	−	0.583779i	$-0.198427\pi$
$29$	8.74456	1.62382	0.811912	+	0.583779i	$0.198427\pi$
$30$	0	0
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	1.00000i	0.176777i
$33$	− 0.627719i	− 0.109272i
$34$	4.74456	0.813686
$35$	0	0
$36$	1.00000	0.166667
$37$	− 0.744563i	− 0.122405i	−0.998125	−	0.0612027i	$-0.980506\pi$
$37$	− 0.744563i	− 0.122405i	0.998125	−	0.0612027i	$-0.0194936\pi$
$38$	0.627719i	0.101829i
$39$	2.00000	0.320256
$40$	0	0
$41$	0.744563	0.116281	0.0581406	−	0.998308i	$-0.481483\pi$
$41$	0.744563	0.116281	0.0581406	+	0.998308i	$0.481483\pi$
$42$	3.37228i	0.520354i
$43$	− 0.627719i	− 0.0957262i	−0.998854	−	0.0478631i	$-0.984759\pi$
$43$	− 0.627719i	− 0.0957262i	0.998854	−	0.0478631i	$-0.0152411\pi$
$44$	−0.627719	−0.0946322
$45$	0	0
$46$	3.37228	0.497216
$47$	− 6.74456i	− 0.983796i	−0.870653	−	0.491898i	$-0.836303\pi$
$47$	− 6.74456i	− 0.983796i	0.870653	−	0.491898i	$-0.163697\pi$
$48$	− 1.00000i	− 0.144338i
$49$	−4.37228	−0.624612
$50$	0	0
$51$	−4.74456	−0.664372
$52$	− 2.00000i	− 0.277350i
$53$	− 1.37228i	− 0.188497i	−0.995549	−	0.0942487i	$-0.969955\pi$
$53$	− 1.37228i	− 0.188497i	0.995549	−	0.0942487i	$-0.0300449\pi$
$54$	−1.00000	−0.136083
$55$	0	0
$56$	3.37228	0.450640
$57$	− 0.627719i	− 0.0831434i
$58$	8.74456i	1.14822i
$59$	−2.74456	−0.357312	−0.178656	−	0.983912i	$-0.557175\pi$
$59$	−2.74456	−0.357312	−0.178656	+	0.983912i	$0.557175\pi$
$60$	0	0
$61$	11.4891	1.47103	0.735516	−	0.677507i	$-0.236940\pi$
$61$	11.4891	1.47103	0.735516	+	0.677507i	$0.236940\pi$
$62$	− 1.00000i	− 0.127000i
$63$	− 3.37228i	− 0.424868i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	0.627719	0.0772668
$67$	10.7446i	1.31266i	0.754475	+	0.656329i	$0.227892\pi$
$67$	10.7446i	1.31266i	−0.754475	+	0.656329i	$0.772108\pi$
$68$	4.74456i	0.575363i
$69$	−3.37228	−0.405975
$70$	0	0
$71$	3.37228	0.400216	0.200108	−	0.979774i	$-0.435871\pi$
$71$	3.37228	0.400216	0.200108	+	0.979774i	$0.435871\pi$
$72$	1.00000i	0.117851i
$73$	8.11684i	0.950005i	0.879985	+	0.475002i	$0.157553\pi$
$73$	8.11684i	0.950005i	−0.879985	+	0.475002i	$0.842447\pi$
$74$	0.744563	0.0865536
$75$	0	0
$76$	−0.627719	−0.0720043
$77$	2.11684i	0.241237i
$78$	2.00000i	0.226455i
$79$	4.62772	0.520659	0.260330	−	0.965520i	$-0.416169\pi$
$79$	4.62772	0.520659	0.260330	+	0.965520i	$0.416169\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0.744563i	0.0822232i
$83$	12.0000i	1.31717i	0.752506	+	0.658586i	$0.228845\pi$
$83$	12.0000i	1.31717i	−0.752506	+	0.658586i	$0.771155\pi$
$84$	−3.37228	−0.367946
$85$	0	0
$86$	0.627719	0.0676886
$87$	− 8.74456i	− 0.937516i
$88$	− 0.627719i	− 0.0669150i
$89$	1.37228	0.145462	0.0727308	−	0.997352i	$-0.476829\pi$
$89$	1.37228	0.145462	0.0727308	+	0.997352i	$0.476829\pi$
$90$	0	0
$91$	−6.74456	−0.707022
$92$	3.37228i	0.351585i
$93$	1.00000i	0.103695i
$94$	6.74456	0.695649
$95$	0	0
$96$	1.00000	0.102062
$97$	2.00000i	0.203069i	0.994832	+	0.101535i	$0.0323753\pi$
$97$	2.00000i	0.203069i	−0.994832	+	0.101535i	$0.967625\pi$
$98$	− 4.37228i	− 0.441667i
$99$	−0.627719	−0.0630881
$100$	0	0
$101$	8.11684	0.807656	0.403828	−	0.914835i	$-0.367679\pi$
$101$	8.11684	0.807656	0.403828	+	0.914835i	$0.367679\pi$
$102$	− 4.74456i	− 0.469782i
$103$	8.00000i	0.788263i	0.919054	+	0.394132i	$0.128955\pi$
$103$	8.00000i	0.788263i	−0.919054	+	0.394132i	$0.871045\pi$
$104$	2.00000	0.196116
$105$	0	0
$106$	1.37228	0.133288
$107$	0.627719i	0.0606839i	0.999540	+	0.0303419i	$0.00965962\pi$
$107$	0.627719i	0.0606839i	−0.999540	+	0.0303419i	$0.990340\pi$
$108$	− 1.00000i	− 0.0962250i
$109$	−4.74456	−0.454447	−0.227223	−	0.973843i	$-0.572965\pi$
$109$	−4.74456	−0.454447	−0.227223	+	0.973843i	$0.572965\pi$
$110$	0	0
$111$	−0.744563	−0.0706708
$112$	3.37228i	0.318651i
$113$	2.62772i	0.247195i	0.992332	+	0.123597i	$0.0394432\pi$
$113$	2.62772i	0.247195i	−0.992332	+	0.123597i	$0.960557\pi$
$114$	0.627719	0.0587912
$115$	0	0
$116$	−8.74456	−0.811912
$117$	− 2.00000i	− 0.184900i
$118$	− 2.74456i	− 0.252657i
$119$	16.0000	1.46672
$120$	0	0
$121$	−10.6060	−0.964179
$122$	11.4891i	1.04018i
$123$	− 0.744563i	− 0.0671350i
$124$	1.00000	0.0898027
$125$	0	0
$126$	3.37228	0.300427
$127$	13.4891i	1.19697i	0.801135	+	0.598483i	$0.204230\pi$
$127$	13.4891i	1.19697i	−0.801135	+	0.598483i	$0.795770\pi$
$128$	− 1.00000i	− 0.0883883i
$129$	−0.627719	−0.0552675
$130$	0	0
$131$	−16.2337	−1.41834	−0.709172	−	0.705036i	$-0.750931\pi$
$131$	−16.2337	−1.41834	−0.709172	+	0.705036i	$0.750931\pi$
$132$	0.627719i	0.0546359i
$133$	2.11684i	0.183554i
$134$	−10.7446	−0.928189
$135$	0	0
$136$	−4.74456	−0.406843
$137$	− 3.48913i	− 0.298096i	−0.988830	−	0.149048i	$-0.952379\pi$
$137$	− 3.48913i	− 0.298096i	0.988830	−	0.149048i	$-0.0476209\pi$
$138$	− 3.37228i	− 0.287068i
$139$	10.7446	0.911342	0.455671	−	0.890148i	$-0.349399\pi$
$139$	10.7446	0.911342	0.455671	+	0.890148i	$0.349399\pi$
$140$	0	0
$141$	−6.74456	−0.567995
$142$	3.37228i	0.282996i
$143$	1.25544i	0.104985i
$144$	−1.00000	−0.0833333
$145$	0	0
$146$	−8.11684	−0.671755
$147$	4.37228i	0.360620i
$148$	0.744563i	0.0612027i
$149$	12.1168	0.992651	0.496325	−	0.868137i	$-0.334682\pi$
$149$	12.1168	0.992651	0.496325	+	0.868137i	$0.334682\pi$
$150$	0	0
$151$	−8.00000	−0.651031	−0.325515	−	0.945537i	$-0.605538\pi$
$151$	−8.00000	−0.651031	−0.325515	+	0.945537i	$0.605538\pi$
$152$	− 0.627719i	− 0.0509147i
$153$	4.74456i	0.383575i
$154$	−2.11684	−0.170580
$155$	0	0
$156$	−2.00000	−0.160128
$157$	9.37228i	0.747989i	0.927431	+	0.373995i	$0.122012\pi$
$157$	9.37228i	0.747989i	−0.927431	+	0.373995i	$0.877988\pi$
$158$	4.62772i	0.368162i
$159$	−1.37228	−0.108829
$160$	0	0
$161$	11.3723	0.896261
$162$	1.00000i	0.0785674i
$163$	− 24.2337i	− 1.89813i	−0.315082	−	0.949064i	$-0.602032\pi$
$163$	− 24.2337i	− 1.89813i	0.315082	−	0.949064i	$-0.397968\pi$
$164$	−0.744563	−0.0581406
$165$	0	0
$166$	−12.0000	−0.931381
$167$	− 12.6277i	− 0.977162i	−0.872518	−	0.488581i	$-0.837515\pi$
$167$	− 12.6277i	− 0.977162i	0.872518	−	0.488581i	$-0.162485\pi$
$168$	− 3.37228i	− 0.260177i
$169$	9.00000	0.692308
$170$	0	0
$171$	−0.627719	−0.0480028
$172$	0.627719i	0.0478631i
$173$	18.0000i	1.36851i	0.729241	+	0.684257i	$0.239873\pi$
$173$	18.0000i	1.36851i	−0.729241	+	0.684257i	$0.760127\pi$
$174$	8.74456	0.662924
$175$	0	0
$176$	0.627719	0.0473161
$177$	2.74456i	0.206294i
$178$	1.37228i	0.102857i
$179$	12.0000	0.896922	0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000	0.896922	0.448461	+	0.893802i	$0.351972\pi$
$180$	0	0
$181$	14.8614	1.10464	0.552320	−	0.833632i	$-0.313743\pi$
$181$	14.8614	1.10464	0.552320	+	0.833632i	$0.313743\pi$
$182$	− 6.74456i	− 0.499940i
$183$	− 11.4891i	− 0.849301i
$184$	−3.37228	−0.248608
$185$	0	0
$186$	−1.00000	−0.0733236
$187$	− 2.97825i	− 0.217791i
$188$	6.74456i	0.491898i
$189$	−3.37228	−0.245297
$190$	0	0
$191$	16.0000	1.15772	0.578860	−	0.815427i	$-0.303498\pi$
$191$	16.0000	1.15772	0.578860	+	0.815427i	$0.303498\pi$
$192$	1.00000i	0.0721688i
$193$	− 15.4891i	− 1.11493i	−0.830200	−	0.557466i	$-0.811774\pi$
$193$	− 15.4891i	− 1.11493i	0.830200	−	0.557466i	$-0.188226\pi$
$194$	−2.00000	−0.143592
$195$	0	0
$196$	4.37228	0.312306
$197$	19.4891i	1.38854i	0.719713	+	0.694271i	$0.244273\pi$
$197$	19.4891i	1.38854i	−0.719713	+	0.694271i	$0.755727\pi$
$198$	− 0.627719i	− 0.0446100i
$199$	12.6277	0.895155	0.447578	−	0.894245i	$-0.352287\pi$
$199$	12.6277	0.895155	0.447578	+	0.894245i	$0.352287\pi$
$200$	0	0
$201$	10.7446	0.757863
$202$	8.11684i	0.571099i
$203$	29.4891i	2.06973i
$204$	4.74456	0.332186
$205$	0	0
$206$	−8.00000	−0.557386
$207$	3.37228i	0.234390i
$208$	2.00000i	0.138675i
$209$	0.394031	0.0272557
$210$	0	0
$211$	−0.627719	−0.0432139	−0.0216070	−	0.999767i	$-0.506878\pi$
$211$	−0.627719	−0.0432139	−0.0216070	+	0.999767i	$0.506878\pi$
$212$	1.37228i	0.0942487i
$213$	− 3.37228i	− 0.231065i
$214$	−0.627719	−0.0429100
$215$	0	0
$216$	1.00000	0.0680414
$217$	− 3.37228i	− 0.228925i
$218$	− 4.74456i	− 0.321342i
$219$	8.11684	0.548485
$220$	0	0
$221$	9.48913	0.638308
$222$	− 0.744563i	− 0.0499718i
$223$	− 9.25544i	− 0.619790i	−0.950771	−	0.309895i	$-0.899706\pi$
$223$	− 9.25544i	− 0.619790i	0.950771	−	0.309895i	$-0.100294\pi$
$224$	−3.37228	−0.225320
$225$	0	0
$226$	−2.62772	−0.174793
$227$	6.11684i	0.405989i	0.979180	+	0.202995i	$0.0650674\pi$
$227$	6.11684i	0.405989i	−0.979180	+	0.202995i	$0.934933\pi$
$228$	0.627719i	0.0415717i
$229$	−3.88316	−0.256606	−0.128303	−	0.991735i	$-0.540953\pi$
$229$	−3.88316	−0.256606	−0.128303	+	0.991735i	$0.540953\pi$
$230$	0	0
$231$	2.11684	0.139278
$232$	− 8.74456i	− 0.574109i
$233$	14.8614i	0.973603i	0.873513	+	0.486802i	$0.161837\pi$
$233$	14.8614i	0.973603i	−0.873513	+	0.486802i	$0.838163\pi$
$234$	2.00000	0.130744
$235$	0	0
$236$	2.74456	0.178656
$237$	− 4.62772i	− 0.300603i
$238$	16.0000i	1.03713i
$239$	29.4891	1.90749	0.953746	−	0.300612i	$-0.0971910\pi$
$239$	29.4891	1.90749	0.953746	+	0.300612i	$0.0971910\pi$
$240$	0	0
$241$	4.51087	0.290571	0.145285	−	0.989390i	$-0.453590\pi$
$241$	4.51087	0.290571	0.145285	+	0.989390i	$0.453590\pi$
$242$	− 10.6060i	− 0.681778i
$243$	− 1.00000i	− 0.0641500i
$244$	−11.4891	−0.735516
$245$	0	0
$246$	0.744563	0.0474716
$247$	1.25544i	0.0798816i
$248$	1.00000i	0.0635001i
$249$	12.0000	0.760469
$250$	0	0
$251$	−4.00000	−0.252478	−0.126239	−	0.992000i	$-0.540291\pi$
$251$	−4.00000	−0.252478	−0.126239	+	0.992000i	$0.540291\pi$
$252$	3.37228i	0.212434i
$253$	− 2.11684i	− 0.133085i
$254$	−13.4891	−0.846383
$255$	0	0
$256$	1.00000	0.0625000
$257$	13.3723i	0.834140i	0.908874	+	0.417070i	$0.136943\pi$
$257$	13.3723i	0.834140i	−0.908874	+	0.417070i	$0.863057\pi$
$258$	− 0.627719i	− 0.0390801i
$259$	2.51087	0.156018
$260$	0	0
$261$	−8.74456	−0.541275
$262$	− 16.2337i	− 1.00292i
$263$	18.9783i	1.17025i	0.810943	+	0.585125i	$0.198954\pi$
$263$	18.9783i	1.17025i	−0.810943	+	0.585125i	$0.801046\pi$
$264$	−0.627719	−0.0386334
$265$	0	0
$266$	−2.11684	−0.129792
$267$	− 1.37228i	− 0.0839823i
$268$	− 10.7446i	− 0.656329i
$269$	2.00000	0.121942	0.0609711	−	0.998140i	$-0.480580\pi$
$269$	2.00000	0.121942	0.0609711	+	0.998140i	$0.480580\pi$
$270$	0	0
$271$	−13.8832	−0.843342	−0.421671	−	0.906749i	$-0.638556\pi$
$271$	−13.8832	−0.843342	−0.421671	+	0.906749i	$0.638556\pi$
$272$	− 4.74456i	− 0.287681i
$273$	6.74456i	0.408200i
$274$	3.48913	0.210786
$275$	0	0
$276$	3.37228	0.202987
$277$	15.2554i	0.916610i	0.888795	+	0.458305i	$0.151543\pi$
$277$	15.2554i	0.916610i	−0.888795	+	0.458305i	$0.848457\pi$
$278$	10.7446i	0.644416i
$279$	1.00000	0.0598684
$280$	0	0
$281$	16.7446	0.998897	0.499448	−	0.866344i	$-0.333536\pi$
$281$	16.7446	0.998897	0.499448	+	0.866344i	$0.333536\pi$
$282$	− 6.74456i	− 0.401633i
$283$	− 12.0000i	− 0.713326i	−0.934233	−	0.356663i	$-0.883914\pi$
$283$	− 12.0000i	− 0.713326i	0.934233	−	0.356663i	$-0.116086\pi$
$284$	−3.37228	−0.200108
$285$	0	0
$286$	−1.25544	−0.0742356
$287$	2.51087i	0.148212i
$288$	− 1.00000i	− 0.0589256i
$289$	−5.51087	−0.324169
$290$	0	0
$291$	2.00000	0.117242
$292$	− 8.11684i	− 0.475002i
$293$	7.48913i	0.437519i	0.975779	+	0.218760i	$0.0702011\pi$
$293$	7.48913i	0.437519i	−0.975779	+	0.218760i	$0.929799\pi$
$294$	−4.37228	−0.254997
$295$	0	0
$296$	−0.744563	−0.0432768
$297$	0.627719i	0.0364239i
$298$	12.1168i	0.701910i
$299$	6.74456	0.390048
$300$	0	0
$301$	2.11684	0.122013
$302$	− 8.00000i	− 0.460348i
$303$	− 8.11684i	− 0.466301i
$304$	0.627719	0.0360021
$305$	0	0
$306$	−4.74456	−0.271229
$307$	30.9783i	1.76802i	0.467466	+	0.884011i	$0.345167\pi$
$307$	30.9783i	1.76802i	−0.467466	+	0.884011i	$0.654833\pi$
$308$	− 2.11684i	− 0.120618i
$309$	8.00000	0.455104
$310$	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	$-0.908798\pi$
$313$	− 10.0000i	− 0.565233i	0.959233	−	0.282617i	$-0.0912024\pi$
$314$	−9.37228	−0.528908
$315$	0	0
$316$	−4.62772	−0.260330
$317$	− 24.7446i	− 1.38979i	−0.719110	−	0.694897i	$-0.755450\pi$
$317$	− 24.7446i	− 1.38979i	0.719110	−	0.694897i	$-0.244550\pi$
$318$	− 1.37228i	− 0.0769537i
$319$	5.48913	0.307332
$320$	0	0
$321$	0.627719	0.0350358
$322$	11.3723i	0.633752i
$323$	− 2.97825i	− 0.165714i
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	24.2337	1.34218
$327$	4.74456i	0.262375i
$328$	− 0.744563i	− 0.0411116i
$329$	22.7446	1.25395
$330$	0	0
$331$	−1.48913	−0.0818497	−0.0409249	−	0.999162i	$-0.513030\pi$
$331$	−1.48913	−0.0818497	−0.0409249	+	0.999162i	$0.513030\pi$
$332$	− 12.0000i	− 0.658586i
$333$	0.744563i	0.0408018i
$334$	12.6277	0.690958
$335$	0	0
$336$	3.37228	0.183973
$337$	28.9783i	1.57855i	0.614043	+	0.789273i	$0.289542\pi$
$337$	28.9783i	1.57855i	−0.614043	+	0.789273i	$0.710458\pi$
$338$	9.00000i	0.489535i
$339$	2.62772	0.142718
$340$	0	0
$341$	−0.627719	−0.0339929
$342$	− 0.627719i	− 0.0339431i
$343$	8.86141i	0.478471i
$344$	−0.627719	−0.0338443
$345$	0	0
$346$	−18.0000	−0.967686
$347$	36.4674i	1.95767i	0.204648	+	0.978836i	$0.434395\pi$
$347$	36.4674i	1.95767i	−0.204648	+	0.978836i	$0.565605\pi$
$348$	8.74456i	0.468758i
$349$	−7.25544	−0.388375	−0.194187	−	0.980964i	$-0.562207\pi$
$349$	−7.25544	−0.388375	−0.194187	+	0.980964i	$0.562207\pi$
$350$	0	0
$351$	−2.00000	−0.106752
$352$	0.627719i	0.0334575i
$353$	− 15.4891i	− 0.824403i	−0.911093	−	0.412201i	$-0.864760\pi$
$353$	− 15.4891i	− 0.824403i	0.911093	−	0.412201i	$-0.135240\pi$
$354$	−2.74456	−0.145872
$355$	0	0
$356$	−1.37228	−0.0727308
$357$	− 16.0000i	− 0.846810i
$358$	12.0000i	0.634220i
$359$	−3.37228	−0.177982	−0.0889911	−	0.996032i	$-0.528364\pi$
$359$	−3.37228	−0.177982	−0.0889911	+	0.996032i	$0.528364\pi$
$360$	0	0
$361$	−18.6060	−0.979262
$362$	14.8614i	0.781098i
$363$	10.6060i	0.556669i
$364$	6.74456	0.353511
$365$	0	0
$366$	11.4891	0.600546
$367$	− 20.2337i	− 1.05619i	−0.849185	−	0.528095i	$-0.822907\pi$
$367$	− 20.2337i	− 1.05619i	0.849185	−	0.528095i	$-0.177093\pi$
$368$	− 3.37228i	− 0.175792i
$369$	−0.744563	−0.0387604
$370$	0	0
$371$	4.62772	0.240259
$372$	− 1.00000i	− 0.0518476i
$373$	− 14.8614i	− 0.769494i	−0.923022	−	0.384747i	$-0.874289\pi$
$373$	− 14.8614i	− 0.769494i	0.923022	−	0.384747i	$-0.125711\pi$
$374$	2.97825	0.154002
$375$	0	0
$376$	−6.74456	−0.347824
$377$	17.4891i	0.900736i
$378$	− 3.37228i	− 0.173451i
$379$	28.8614	1.48251	0.741255	−	0.671223i	$-0.234231\pi$
$379$	28.8614	1.48251	0.741255	+	0.671223i	$0.234231\pi$
$380$	0	0
$381$	13.4891	0.691069
$382$	16.0000i	0.818631i
$383$	− 13.4891i	− 0.689262i	−0.938738	−	0.344631i	$-0.888004\pi$
$383$	− 13.4891i	− 0.689262i	0.938738	−	0.344631i	$-0.111996\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	15.4891	0.788376
$387$	0.627719i	0.0319087i
$388$	− 2.00000i	− 0.101535i
$389$	−16.9783	−0.860831	−0.430416	−	0.902631i	$-0.641633\pi$
$389$	−16.9783	−0.860831	−0.430416	+	0.902631i	$0.641633\pi$
$390$	0	0
$391$	−16.0000	−0.809155
$392$	4.37228i	0.220834i
$393$	16.2337i	0.818881i
$394$	−19.4891	−0.981848
$395$	0	0
$396$	0.627719	0.0315441
$397$	− 1.60597i	− 0.0806013i	−0.999188	−	0.0403006i	$-0.987168\pi$
$397$	− 1.60597i	− 0.0806013i	0.999188	−	0.0403006i	$-0.0128316\pi$
$398$	12.6277i	0.632970i
$399$	2.11684	0.105975
$400$	0	0
$401$	22.6277	1.12997	0.564987	−	0.825100i	$-0.308881\pi$
$401$	22.6277	1.12997	0.564987	+	0.825100i	$0.308881\pi$
$402$	10.7446i	0.535890i
$403$	− 2.00000i	− 0.0996271i
$404$	−8.11684	−0.403828
$405$	0	0
$406$	−29.4891	−1.46352
$407$	− 0.467376i	− 0.0231670i
$408$	4.74456i	0.234891i
$409$	10.2337	0.506023	0.253012	−	0.967463i	$-0.418579\pi$
$409$	10.2337	0.506023	0.253012	+	0.967463i	$0.418579\pi$
$410$	0	0
$411$	−3.48913	−0.172106
$412$	− 8.00000i	− 0.394132i
$413$	− 9.25544i	− 0.455430i
$414$	−3.37228	−0.165739
$415$	0	0
$416$	−2.00000	−0.0980581
$417$	− 10.7446i	− 0.526163i
$418$	0.394031i	0.0192727i
$419$	12.0000	0.586238	0.293119	−	0.956076i	$-0.405307\pi$
$419$	12.0000	0.586238	0.293119	+	0.956076i	$0.405307\pi$
$420$	0	0
$421$	19.4891	0.949842	0.474921	−	0.880028i	$-0.342477\pi$
$421$	19.4891	0.949842	0.474921	+	0.880028i	$0.342477\pi$
$422$	− 0.627719i	− 0.0305569i
$423$	6.74456i	0.327932i
$424$	−1.37228	−0.0666439
$425$	0	0
$426$	3.37228	0.163388
$427$	38.7446i	1.87498i
$428$	− 0.627719i	− 0.0303419i
$429$	1.25544	0.0606131
$430$	0	0
$431$	−26.9783	−1.29950	−0.649748	−	0.760149i	$-0.725126\pi$
$431$	−26.9783	−1.29950	−0.649748	+	0.760149i	$0.725126\pi$
$432$	1.00000i	0.0481125i
$433$	− 36.1168i	− 1.73566i	−0.496857	−	0.867832i	$-0.665513\pi$
$433$	− 36.1168i	− 1.73566i	0.496857	−	0.867832i	$-0.334487\pi$
$434$	3.37228	0.161875
$435$	0	0
$436$	4.74456	0.227223
$437$	− 2.11684i	− 0.101262i
$438$	8.11684i	0.387838i
$439$	−14.7446	−0.703720	−0.351860	−	0.936053i	$-0.614451\pi$
$439$	−14.7446	−0.703720	−0.351860	+	0.936053i	$0.614451\pi$
$440$	0	0
$441$	4.37228	0.208204
$442$	9.48913i	0.451352i
$443$	− 34.3505i	− 1.63204i	−0.578022	−	0.816022i	$-0.696175\pi$
$443$	− 34.3505i	− 1.63204i	0.578022	−	0.816022i	$-0.303825\pi$
$444$	0.744563	0.0353354
$445$	0	0
$446$	9.25544	0.438258
$447$	− 12.1168i	− 0.573107i
$448$	− 3.37228i	− 0.159325i
$449$	8.97825	0.423710	0.211855	−	0.977301i	$-0.432050\pi$
$449$	8.97825	0.423710	0.211855	+	0.977301i	$0.432050\pi$
$450$	0	0
$451$	0.467376	0.0220079
$452$	− 2.62772i	− 0.123597i
$453$	8.00000i	0.375873i
$454$	−6.11684	−0.287078
$455$	0	0
$456$	−0.627719	−0.0293956
$457$	34.4674i	1.61232i	0.591700	+	0.806158i	$0.298457\pi$
$457$	34.4674i	1.61232i	−0.591700	+	0.806158i	$0.701543\pi$
$458$	− 3.88316i	− 0.181448i
$459$	4.74456	0.221457
$460$	0	0
$461$	−19.7228	−0.918583	−0.459291	−	0.888286i	$-0.651897\pi$
$461$	−19.7228	−0.918583	−0.459291	+	0.888286i	$0.651897\pi$
$462$	2.11684i	0.0984845i
$463$	− 2.51087i	− 0.116690i	−0.998296	−	0.0583451i	$-0.981418\pi$
$463$	− 2.51087i	− 0.116690i	0.998296	−	0.0583451i	$-0.0185824\pi$
$464$	8.74456	0.405956
$465$	0	0
$466$	−14.8614	−0.688441
$467$	6.51087i	0.301287i	0.988588	+	0.150644i	$0.0481346\pi$
$467$	6.51087i	0.301287i	−0.988588	+	0.150644i	$0.951865\pi$
$468$	2.00000i	0.0924500i
$469$	−36.2337	−1.67312
$470$	0	0
$471$	9.37228	0.431852
$472$	2.74456i	0.126329i
$473$	− 0.394031i	− 0.0181176i
$474$	4.62772	0.212558
$475$	0	0
$476$	−16.0000	−0.733359
$477$	1.37228i	0.0628324i
$478$	29.4891i	1.34880i
$479$	−8.86141	−0.404888	−0.202444	−	0.979294i	$-0.564888\pi$
$479$	−8.86141	−0.404888	−0.202444	+	0.979294i	$0.564888\pi$
$480$	0	0
$481$	1.48913	0.0678983
$482$	4.51087i	0.205465i
$483$	− 11.3723i	− 0.517457i
$484$	10.6060	0.482090
$485$	0	0
$486$	1.00000	0.0453609
$487$	− 37.4891i	− 1.69879i	−0.527754	−	0.849397i	$-0.676966\pi$
$487$	− 37.4891i	− 1.69879i	0.527754	−	0.849397i	$-0.323034\pi$
$488$	− 11.4891i	− 0.520088i
$489$	−24.2337	−1.09589
$490$	0	0
$491$	27.6060	1.24584	0.622920	−	0.782286i	$-0.285946\pi$
$491$	27.6060	1.24584	0.622920	+	0.782286i	$0.285946\pi$
$492$	0.744563i	0.0335675i
$493$	− 41.4891i	− 1.86858i
$494$	−1.25544	−0.0564848
$495$	0	0
$496$	−1.00000	−0.0449013
$497$	11.3723i	0.510117i
$498$	12.0000i	0.537733i
$499$	−33.4891	−1.49918	−0.749590	−	0.661903i	$-0.769749\pi$
$499$	−33.4891	−1.49918	−0.749590	+	0.661903i	$0.769749\pi$
$500$	0	0
$501$	−12.6277	−0.564165
$502$	− 4.00000i	− 0.178529i
$503$	− 5.48913i	− 0.244748i	−0.992484	−	0.122374i	$-0.960949\pi$
$503$	− 5.48913i	− 0.244748i	0.992484	−	0.122374i	$-0.0390507\pi$
$504$	−3.37228	−0.150213
$505$	0	0
$506$	2.11684	0.0941052
$507$	− 9.00000i	− 0.399704i
$508$	− 13.4891i	− 0.598483i
$509$	27.2554	1.20808	0.604038	−	0.796956i	$-0.293558\pi$
$509$	27.2554	1.20808	0.604038	+	0.796956i	$0.293558\pi$
$510$	0	0
$511$	−27.3723	−1.21088
$512$	1.00000i	0.0441942i
$513$	0.627719i	0.0277145i
$514$	−13.3723	−0.589826
$515$	0	0
$516$	0.627719	0.0276338
$517$	− 4.23369i	− 0.186197i
$518$	2.51087i	0.110322i
$519$	18.0000	0.790112
$520$	0	0
$521$	36.9783	1.62005	0.810023	−	0.586398i	$-0.199454\pi$
$521$	36.9783	1.62005	0.810023	+	0.586398i	$0.199454\pi$
$522$	− 8.74456i	− 0.382739i
$523$	12.8614i	0.562390i	0.959651	+	0.281195i	$0.0907308\pi$
$523$	12.8614i	0.562390i	−0.959651	+	0.281195i	$0.909269\pi$
$524$	16.2337	0.709172
$525$	0	0
$526$	−18.9783	−0.827491
$527$	4.74456i	0.206676i
$528$	− 0.627719i	− 0.0273179i
$529$	11.6277	0.505553
$530$	0	0
$531$	2.74456	0.119104
$532$	− 2.11684i	− 0.0917768i
$533$	1.48913i	0.0645012i
$534$	1.37228	0.0593844
$535$	0	0
$536$	10.7446	0.464094
$537$	− 12.0000i	− 0.517838i
$538$	2.00000i	0.0862261i
$539$	−2.74456	−0.118217
$540$	0	0
$541$	−15.4891	−0.665930	−0.332965	−	0.942939i	$-0.608049\pi$
$541$	−15.4891	−0.665930	−0.332965	+	0.942939i	$0.608049\pi$
$542$	− 13.8832i	− 0.596333i
$543$	− 14.8614i	− 0.637764i
$544$	4.74456	0.203421
$545$	0	0
$546$	−6.74456	−0.288641
$547$	− 2.74456i	− 0.117349i	−0.998277	−	0.0586745i	$-0.981313\pi$
$547$	− 2.74456i	− 0.117349i	0.998277	−	0.0586745i	$-0.0186874\pi$
$548$	3.48913i	0.149048i
$549$	−11.4891	−0.490344
$550$	0	0
$551$	5.48913	0.233845
$552$	3.37228i	0.143534i
$553$	15.6060i	0.663633i
$554$	−15.2554	−0.648141
$555$	0	0
$556$	−10.7446	−0.455671
$557$	11.8832i	0.503505i	0.967792	+	0.251753i	$0.0810070\pi$
$557$	11.8832i	0.503505i	−0.967792	+	0.251753i	$0.918993\pi$
$558$	1.00000i	0.0423334i
$559$	1.25544	0.0530993
$560$	0	0
$561$	−2.97825	−0.125742
$562$	16.7446i	0.706327i
$563$	6.97825i	0.294098i	0.989129	+	0.147049i	$0.0469775\pi$
$563$	6.97825i	0.294098i	−0.989129	+	0.147049i	$0.953022\pi$
$564$	6.74456	0.283997
$565$	0	0
$566$	12.0000	0.504398
$567$	3.37228i	0.141623i
$568$	− 3.37228i	− 0.141498i
$569$	1.37228	0.0575290	0.0287645	−	0.999586i	$-0.490843\pi$
$569$	1.37228	0.0575290	0.0287645	+	0.999586i	$0.490843\pi$
$570$	0	0
$571$	−26.7446	−1.11923	−0.559613	−	0.828754i	$-0.689050\pi$
$571$	−26.7446	−1.11923	−0.559613	+	0.828754i	$0.689050\pi$
$572$	− 1.25544i	− 0.0524925i
$573$	− 16.0000i	− 0.668410i
$574$	−2.51087	−0.104802
$575$	0	0
$576$	1.00000	0.0416667
$577$	− 2.23369i	− 0.0929896i	−0.998919	−	0.0464948i	$-0.985195\pi$
$577$	− 2.23369i	− 0.0929896i	0.998919	−	0.0464948i	$-0.0148051\pi$
$578$	− 5.51087i	− 0.229222i
$579$	−15.4891	−0.643706
$580$	0	0
$581$	−40.4674	−1.67887
$582$	2.00000i	0.0829027i
$583$	− 0.861407i	− 0.0356758i
$584$	8.11684	0.335877
$585$	0	0
$586$	−7.48913	−0.309373
$587$	12.0000i	0.495293i	0.968850	+	0.247647i	$0.0796572\pi$
$587$	12.0000i	0.495293i	−0.968850	+	0.247647i	$0.920343\pi$
$588$	− 4.37228i	− 0.180310i
$589$	−0.627719	−0.0258647
$590$	0	0
$591$	19.4891	0.801675
$592$	− 0.744563i	− 0.0306013i
$593$	− 44.9783i	− 1.84704i	−0.383556	−	0.923518i	$-0.625301\pi$
$593$	− 44.9783i	− 1.84704i	0.383556	−	0.923518i	$-0.374699\pi$
$594$	−0.627719	−0.0257556
$595$	0	0
$596$	−12.1168	−0.496325
$597$	− 12.6277i	− 0.516818i
$598$	6.74456i	0.275806i
$599$	−10.1168	−0.413363	−0.206682	−	0.978408i	$-0.566266\pi$
$599$	−10.1168	−0.413363	−0.206682	+	0.978408i	$0.566266\pi$
$600$	0	0
$601$	12.5109	0.510329	0.255165	−	0.966898i	$-0.417870\pi$
$601$	12.5109	0.510329	0.255165	+	0.966898i	$0.417870\pi$
$602$	2.11684i	0.0862761i
$603$	− 10.7446i	− 0.437552i
$604$	8.00000	0.325515
$605$	0	0
$606$	8.11684	0.329724
$607$	− 4.62772i	− 0.187833i	−0.995580	−	0.0939167i	$-0.970061\pi$
$607$	− 4.62772i	− 0.187833i	0.995580	−	0.0939167i	$-0.0299387\pi$
$608$	0.627719i	0.0254574i
$609$	29.4891	1.19496
$610$	0	0
$611$	13.4891	0.545712
$612$	− 4.74456i	− 0.191788i
$613$	− 35.4891i	− 1.43339i	−0.697386	−	0.716696i	$-0.745653\pi$
$613$	− 35.4891i	− 1.43339i	0.697386	−	0.716696i	$-0.254347\pi$
$614$	−30.9783	−1.25018
$615$	0	0
$616$	2.11684	0.0852901
$617$	7.88316i	0.317364i	0.987330	+	0.158682i	$0.0507244\pi$
$617$	7.88316i	0.317364i	−0.987330	+	0.158682i	$0.949276\pi$
$618$	8.00000i	0.321807i
$619$	−16.2337	−0.652487	−0.326244	−	0.945286i	$-0.605783\pi$
$619$	−16.2337	−0.652487	−0.326244	+	0.945286i	$0.605783\pi$
$620$	0	0
$621$	3.37228	0.135325
$622$	8.00000i	0.320771i
$623$	4.62772i	0.185406i
$624$	2.00000	0.0800641
$625$	0	0
$626$	10.0000	0.399680
$627$	− 0.394031i	− 0.0157361i
$628$	− 9.37228i	− 0.373995i
$629$	−3.53262	−0.140855
$630$	0	0
$631$	−46.3505	−1.84519	−0.922593	−	0.385775i	$-0.873934\pi$
$631$	−46.3505	−1.84519	−0.922593	+	0.385775i	$0.873934\pi$
$632$	− 4.62772i	− 0.184081i
$633$	0.627719i	0.0249496i
$634$	24.7446	0.982732
$635$	0	0
$636$	1.37228	0.0544145
$637$	− 8.74456i	− 0.346472i
$638$	5.48913i	0.217317i
$639$	−3.37228	−0.133405
$640$	0	0
$641$	18.0000	0.710957	0.355479	−	0.934684i	$-0.384318\pi$
$641$	18.0000	0.710957	0.355479	+	0.934684i	$0.384318\pi$
$642$	0.627719i	0.0247741i
$643$	25.0951i	0.989654i	0.868992	+	0.494827i	$0.164769\pi$
$643$	25.0951i	0.989654i	−0.868992	+	0.494827i	$0.835231\pi$
$644$	−11.3723	−0.448131
$645$	0	0
$646$	2.97825	0.117178
$647$	50.5842i	1.98867i	0.106287	+	0.994335i	$0.466104\pi$
$647$	50.5842i	1.98867i	−0.106287	+	0.994335i	$0.533896\pi$
$648$	− 1.00000i	− 0.0392837i
$649$	−1.72281	−0.0676263
$650$	0	0
$651$	−3.37228	−0.132170
$652$	24.2337i	0.949064i
$653$	− 40.9783i	− 1.60360i	−0.597591	−	0.801801i	$-0.703875\pi$
$653$	− 40.9783i	− 1.60360i	0.597591	−	0.801801i	$-0.296125\pi$
$654$	−4.74456	−0.185527
$655$	0	0
$656$	0.744563	0.0290703
$657$	− 8.11684i	− 0.316668i
$658$	22.7446i	0.886675i
$659$	−36.0000	−1.40236	−0.701180	−	0.712984i	$-0.747343\pi$
$659$	−36.0000	−1.40236	−0.701180	+	0.712984i	$0.747343\pi$
$660$	0	0
$661$	−4.97825	−0.193632	−0.0968158	−	0.995302i	$-0.530866\pi$
$661$	−4.97825	−0.193632	−0.0968158	+	0.995302i	$0.530866\pi$
$662$	− 1.48913i	− 0.0578765i
$663$	− 9.48913i	− 0.368527i
$664$	12.0000	0.465690
$665$	0	0
$666$	−0.744563	−0.0288512
$667$	− 29.4891i	− 1.14182i
$668$	12.6277i	0.488581i
$669$	−9.25544	−0.357836
$670$	0	0
$671$	7.21194	0.278414
$672$	3.37228i	0.130089i
$673$	0.510875i	0.0196928i	0.999952	+	0.00984639i	$0.00313425\pi$
$673$	0.510875i	0.0196928i	−0.999952	+	0.00984639i	$0.996866\pi$
$674$	−28.9783	−1.11620
$675$	0	0
$676$	−9.00000	−0.346154
$677$	− 41.6060i	− 1.59905i	−0.600635	−	0.799524i	$-0.705085\pi$
$677$	− 41.6060i	− 1.59905i	0.600635	−	0.799524i	$-0.294915\pi$
$678$	2.62772i	0.100917i
$679$	−6.74456	−0.258833
$680$	0	0
$681$	6.11684	0.234398
$682$	− 0.627719i	− 0.0240366i
$683$	12.8614i	0.492128i	0.969254	+	0.246064i	$0.0791373\pi$
$683$	12.8614i	0.492128i	−0.969254	+	0.246064i	$0.920863\pi$
$684$	0.627719	0.0240014
$685$	0	0
$686$	−8.86141	−0.338330
$687$	3.88316i	0.148152i
$688$	− 0.627719i	− 0.0239316i
$689$	2.74456	0.104560
$690$	0	0
$691$	−19.1386	−0.728066	−0.364033	−	0.931386i	$-0.618601\pi$
$691$	−19.1386	−0.728066	−0.364033	+	0.931386i	$0.618601\pi$
$692$	− 18.0000i	− 0.684257i
$693$	− 2.11684i	− 0.0804123i
$694$	−36.4674	−1.38428
$695$	0	0
$696$	−8.74456	−0.331462
$697$	− 3.53262i	− 0.133808i
$698$	− 7.25544i	− 0.274622i
$699$	14.8614	0.562110
$700$	0	0
$701$	45.6060	1.72251	0.861257	−	0.508170i	$-0.169678\pi$
$701$	45.6060	1.72251	0.861257	+	0.508170i	$0.169678\pi$
$702$	− 2.00000i	− 0.0754851i
$703$	− 0.467376i	− 0.0176274i
$704$	−0.627719	−0.0236580
$705$	0	0
$706$	15.4891	0.582941
$707$	27.3723i	1.02944i
$708$	− 2.74456i	− 0.103147i
$709$	−40.1168	−1.50662	−0.753310	−	0.657666i	$-0.771544\pi$
$709$	−40.1168	−1.50662	−0.753310	+	0.657666i	$0.771544\pi$
$710$	0	0
$711$	−4.62772	−0.173553
$712$	− 1.37228i	− 0.0514284i
$713$	3.37228i	0.126293i
$714$	16.0000	0.598785
$715$	0	0
$716$	−12.0000	−0.448461
$717$	− 29.4891i	− 1.10129i
$718$	− 3.37228i	− 0.125852i
$719$	−38.7446	−1.44493	−0.722464	−	0.691408i	$-0.756991\pi$
$719$	−38.7446	−1.44493	−0.722464	+	0.691408i	$0.756991\pi$
$720$	0	0
$721$	−26.9783	−1.00472
$722$	− 18.6060i	− 0.692442i
$723$	− 4.51087i	− 0.167761i
$724$	−14.8614	−0.552320
$725$	0	0
$726$	−10.6060	−0.393624
$727$	19.3723i	0.718478i	0.933246	+	0.359239i	$0.116964\pi$
$727$	19.3723i	0.718478i	−0.933246	+	0.359239i	$0.883036\pi$
$728$	6.74456i	0.249970i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	−2.97825	−0.110155
$732$	11.4891i	0.424650i
$733$	34.0000i	1.25582i	0.778287	+	0.627909i	$0.216089\pi$
$733$	34.0000i	1.25582i	−0.778287	+	0.627909i	$0.783911\pi$
$734$	20.2337	0.746839
$735$	0	0
$736$	3.37228	0.124304
$737$	6.74456i	0.248439i
$738$	− 0.744563i	− 0.0274077i
$739$	22.9783	0.845269	0.422634	−	0.906300i	$-0.361105\pi$
$739$	22.9783	0.845269	0.422634	+	0.906300i	$0.361105\pi$
$740$	0	0
$741$	1.25544	0.0461196
$742$	4.62772i	0.169889i
$743$	35.3723i	1.29768i	0.760924	+	0.648842i	$0.224746\pi$
$743$	35.3723i	1.29768i	−0.760924	+	0.648842i	$0.775254\pi$
$744$	1.00000	0.0366618
$745$	0	0
$746$	14.8614	0.544115
$747$	− 12.0000i	− 0.439057i
$748$	2.97825i	0.108896i
$749$	−2.11684	−0.0773478
$750$	0	0
$751$	−38.7446	−1.41381	−0.706905	−	0.707309i	$-0.749909\pi$
$751$	−38.7446	−1.41381	−0.706905	+	0.707309i	$0.749909\pi$
$752$	− 6.74456i	− 0.245949i
$753$	4.00000i	0.145768i
$754$	−17.4891	−0.636916
$755$	0	0
$756$	3.37228	0.122649
$757$	− 26.0000i	− 0.944986i	−0.881334	−	0.472493i	$-0.843354\pi$
$757$	− 26.0000i	− 0.944986i	0.881334	−	0.472493i	$-0.156646\pi$
$758$	28.8614i	1.04829i
$759$	−2.11684	−0.0768366
$760$	0	0
$761$	−46.8614	−1.69872	−0.849362	−	0.527810i	$-0.823013\pi$
$761$	−46.8614	−1.69872	−0.849362	+	0.527810i	$0.823013\pi$
$762$	13.4891i	0.488659i
$763$	− 16.0000i	− 0.579239i
$764$	−16.0000	−0.578860
$765$	0	0
$766$	13.4891	0.487382
$767$	− 5.48913i	− 0.198201i
$768$	− 1.00000i	− 0.0360844i
$769$	9.37228	0.337973	0.168987	−	0.985618i	$-0.445951\pi$
$769$	9.37228	0.337973	0.168987	+	0.985618i	$0.445951\pi$
$770$	0	0
$771$	13.3723	0.481591
$772$	15.4891i	0.557466i
$773$	− 21.6060i	− 0.777113i	−0.921425	−	0.388556i	$-0.872974\pi$
$773$	− 21.6060i	− 0.777113i	0.921425	−	0.388556i	$-0.127026\pi$
$774$	−0.627719	−0.0225629
$775$	0	0
$776$	2.00000	0.0717958
$777$	− 2.51087i	− 0.0900771i
$778$	− 16.9783i	− 0.608700i
$779$	0.467376	0.0167455
$780$	0	0
$781$	2.11684	0.0757466
$782$	− 16.0000i	− 0.572159i
$783$	8.74456i	0.312505i
$784$	−4.37228	−0.156153
$785$	0	0
$786$	−16.2337	−0.579036
$787$	− 9.88316i	− 0.352296i	−0.984364	−	0.176148i	$-0.943636\pi$
$787$	− 9.88316i	− 0.352296i	0.984364	−	0.176148i	$-0.0563638\pi$
$788$	− 19.4891i	− 0.694271i
$789$	18.9783	0.675644
$790$	0	0
$791$	−8.86141	−0.315075
$792$	0.627719i	0.0223050i
$793$	22.9783i	0.815982i
$794$	1.60597	0.0569937
$795$	0	0
$796$	−12.6277	−0.447578
$797$	− 20.5109i	− 0.726532i	−0.931685	−	0.363266i	$-0.881662\pi$
$797$	− 20.5109i	− 0.726532i	0.931685	−	0.363266i	$-0.118338\pi$
$798$	2.11684i	0.0749355i
$799$	−32.0000	−1.13208
$800$	0	0
$801$	−1.37228	−0.0484872
$802$	22.6277i	0.799013i
$803$	5.09509i	0.179802i
$804$	−10.7446	−0.378932
$805$	0	0
$806$	2.00000	0.0704470
$807$	− 2.00000i	− 0.0704033i
$808$	− 8.11684i	− 0.285550i
$809$	−9.60597	−0.337728	−0.168864	−	0.985639i	$-0.554010\pi$
$809$	−9.60597	−0.337728	−0.168864	+	0.985639i	$0.554010\pi$
$810$	0	0
$811$	−40.6277	−1.42663	−0.713316	−	0.700842i	$-0.752808\pi$
$811$	−40.6277	−1.42663	−0.713316	+	0.700842i	$0.752808\pi$
$812$	− 29.4891i	− 1.03487i
$813$	13.8832i	0.486904i
$814$	0.467376	0.0163815
$815$	0	0
$816$	−4.74456	−0.166093
$817$	− 0.394031i	− 0.0137854i
$818$	10.2337i	0.357813i
$819$	6.74456	0.235674
$820$	0	0
$821$	15.2554	0.532418	0.266209	−	0.963915i	$-0.414229\pi$
$821$	15.2554	0.532418	0.266209	+	0.963915i	$0.414229\pi$
$822$	− 3.48913i	− 0.121697i
$823$	17.2554i	0.601487i	0.953705	+	0.300743i	$0.0972348\pi$
$823$	17.2554i	0.601487i	−0.953705	+	0.300743i	$0.902765\pi$
$824$	8.00000	0.278693
$825$	0	0
$826$	9.25544	0.322038
$827$	− 17.4891i	− 0.608156i	−0.952647	−	0.304078i	$-0.901652\pi$
$827$	− 17.4891i	− 0.608156i	0.952647	−	0.304078i	$-0.0983484\pi$
$828$	− 3.37228i	− 0.117195i
$829$	31.0951	1.07998	0.539989	−	0.841672i	$-0.318429\pi$
$829$	31.0951	1.07998	0.539989	+	0.841672i	$0.318429\pi$
$830$	0	0
$831$	15.2554	0.529205
$832$	− 2.00000i	− 0.0693375i
$833$	20.7446i	0.718756i
$834$	10.7446	0.372054
$835$	0	0
$836$	−0.394031	−0.0136278
$837$	− 1.00000i	− 0.0345651i
$838$	12.0000i	0.414533i
$839$	−16.8614	−0.582120	−0.291060	−	0.956705i	$-0.594008\pi$
$839$	−16.8614	−0.582120	−0.291060	+	0.956705i	$0.594008\pi$
$840$	0	0
$841$	47.4674	1.63681
$842$	19.4891i	0.671640i
$843$	− 16.7446i	− 0.576713i
$844$	0.627719	0.0216070
$845$	0	0
$846$	−6.74456	−0.231883
$847$	− 35.7663i	− 1.22895i
$848$	− 1.37228i	− 0.0471243i
$849$	−12.0000	−0.411839
$850$	0	0
$851$	−2.51087	−0.0860717
$852$	3.37228i	0.115532i
$853$	39.0951i	1.33859i	0.742997	+	0.669295i	$0.233404\pi$
$853$	39.0951i	1.33859i	−0.742997	+	0.669295i	$0.766596\pi$
$854$	−38.7446	−1.32581
$855$	0	0
$856$	0.627719	0.0214550
$857$	7.48913i	0.255824i	0.991786	+	0.127912i	$0.0408275\pi$
$857$	7.48913i	0.255824i	−0.991786	+	0.127912i	$0.959173\pi$
$858$	1.25544i	0.0428599i
$859$	17.4891	0.596721	0.298361	−	0.954453i	$-0.403560\pi$
$859$	17.4891	0.596721	0.298361	+	0.954453i	$0.403560\pi$
$860$	0	0
$861$	2.51087	0.0855704
$862$	− 26.9783i	− 0.918883i
$863$	6.35053i	0.216175i	0.994141	+	0.108087i	$0.0344726\pi$
$863$	6.35053i	0.216175i	−0.994141	+	0.108087i	$0.965527\pi$
$864$	−1.00000	−0.0340207
$865$	0	0
$866$	36.1168	1.22730
$867$	5.51087i	0.187159i
$868$	3.37228i	0.114463i
$869$	2.90491	0.0985422
$870$	0	0
$871$	−21.4891	−0.728131
$872$	4.74456i	0.160671i
$873$	− 2.00000i	− 0.0676897i
$874$	2.11684	0.0716033
$875$	0	0
$876$	−8.11684	−0.274243
$877$	− 44.9783i	− 1.51881i	−0.650620	−	0.759404i	$-0.725491\pi$
$877$	− 44.9783i	− 1.51881i	0.650620	−	0.759404i	$-0.274509\pi$
$878$	− 14.7446i	− 0.497605i
$879$	7.48913	0.252602
$880$	0	0
$881$	−3.02175	−0.101805	−0.0509027	−	0.998704i	$-0.516210\pi$
$881$	−3.02175	−0.101805	−0.0509027	+	0.998704i	$0.516210\pi$
$882$	4.37228i	0.147222i
$883$	− 44.8614i	− 1.50971i	−0.655894	−	0.754853i	$-0.727708\pi$
$883$	− 44.8614i	− 1.50971i	0.655894	−	0.754853i	$-0.272292\pi$
$884$	−9.48913	−0.319154
$885$	0	0
$886$	34.3505	1.15403
$887$	− 30.7446i	− 1.03230i	−0.856498	−	0.516151i	$-0.827364\pi$
$887$	− 30.7446i	− 1.03230i	0.856498	−	0.516151i	$-0.172636\pi$
$888$	0.744563i	0.0249859i
$889$	−45.4891	−1.52566
$890$	0	0
$891$	0.627719	0.0210294
$892$	9.25544i	0.309895i
$893$	− 4.23369i	− 0.141675i
$894$	12.1168	0.405248
$895$	0	0
$896$	3.37228	0.112660
$897$	− 6.74456i	− 0.225194i
$898$	8.97825i	0.299608i
$899$	−8.74456	−0.291647
$900$	0	0
$901$	−6.51087	−0.216909
$902$	0.467376i	0.0155619i
$903$	− 2.11684i	− 0.0704442i
$904$	2.62772	0.0873966
$905$	0	0
$906$	−8.00000	−0.265782
$907$	− 17.4891i	− 0.580717i	−0.956918	−	0.290358i	$-0.906225\pi$
$907$	− 17.4891i	− 0.580717i	0.956918	−	0.290358i	$-0.0937745\pi$
$908$	− 6.11684i	− 0.202995i
$909$	−8.11684	−0.269219
$910$	0	0
$911$	−2.51087	−0.0831890	−0.0415945	−	0.999135i	$-0.513244\pi$
$911$	−2.51087	−0.0831890	−0.0415945	+	0.999135i	$0.513244\pi$
$912$	− 0.627719i	− 0.0207858i
$913$	7.53262i	0.249293i
$914$	−34.4674	−1.14008
$915$	0	0
$916$	3.88316	0.128303
$917$	− 54.7446i	− 1.80782i
$918$	4.74456i	0.156594i
$919$	12.2337	0.403552	0.201776	−	0.979432i	$-0.435329\pi$
$919$	12.2337	0.403552	0.201776	+	0.979432i	$0.435329\pi$
$920$	0	0
$921$	30.9783	1.02077
$922$	− 19.7228i	− 0.649536i
$923$	6.74456i	0.222000i
$924$	−2.11684	−0.0696391
$925$	0	0
$926$	2.51087	0.0825125
$927$	− 8.00000i	− 0.262754i
$928$	8.74456i	0.287054i
$929$	16.1168	0.528776	0.264388	−	0.964416i	$-0.414830\pi$
$929$	16.1168	0.528776	0.264388	+	0.964416i	$0.414830\pi$
$930$	0	0
$931$	−2.74456	−0.0899494
$932$	− 14.8614i	− 0.486802i
$933$	− 8.00000i	− 0.261908i
$934$	−6.51087	−0.213042
$935$	0	0
$936$	−2.00000	−0.0653720
$937$	− 31.2554i	− 1.02107i	−0.859857	−	0.510535i	$-0.829447\pi$
$937$	− 31.2554i	− 1.02107i	0.859857	−	0.510535i	$-0.170553\pi$
$938$	− 36.2337i	− 1.18307i
$939$	−10.0000	−0.326338
$940$	0	0
$941$	−19.7228	−0.642945	−0.321473	−	0.946919i	$-0.604178\pi$
$941$	−19.7228	−0.642945	−0.321473	+	0.946919i	$0.604178\pi$
$942$	9.37228i	0.305365i
$943$	− 2.51087i	− 0.0817653i
$944$	−2.74456	−0.0893279
$945$	0	0
$946$	0.394031	0.0128110
$947$	− 2.74456i	− 0.0891863i	−0.999005	−	0.0445932i	$-0.985801\pi$
$947$	− 2.74456i	− 0.0891863i	0.999005	−	0.0445932i	$-0.0141991\pi$
$948$	4.62772i	0.150301i
$949$	−16.2337	−0.526968
$950$	0	0
$951$	−24.7446	−0.802397
$952$	− 16.0000i	− 0.518563i
$953$	− 11.7228i	− 0.379739i	−0.981809	−	0.189870i	$-0.939193\pi$
$953$	− 11.7228i	− 0.379739i	0.981809	−	0.189870i	$-0.0608065\pi$
$954$	−1.37228	−0.0444292
$955$	0	0
$956$	−29.4891	−0.953746
$957$	− 5.48913i	− 0.177438i
$958$	− 8.86141i	− 0.286299i
$959$	11.7663	0.379954
$960$	0	0
$961$	1.00000	0.0322581
$962$	1.48913i	0.0480113i
$963$	− 0.627719i	− 0.0202280i
$964$	−4.51087	−0.145285
$965$	0	0
$966$	11.3723	0.365897
$967$	8.00000i	0.257263i	0.991692	+	0.128631i	$0.0410584\pi$
$967$	8.00000i	0.257263i	−0.991692	+	0.128631i	$0.958942\pi$
$968$	10.6060i	0.340889i
$969$	−2.97825	−0.0956752
$970$	0	0
$971$	−48.7011	−1.56289	−0.781446	−	0.623973i	$-0.785517\pi$
$971$	−48.7011	−1.56289	−0.781446	+	0.623973i	$0.785517\pi$
$972$	1.00000i	0.0320750i
$973$	36.2337i	1.16160i
$974$	37.4891	1.20123
$975$	0	0
$976$	11.4891	0.367758
$977$	− 46.0000i	− 1.47167i	−0.677161	−	0.735835i	$-0.736790\pi$
$977$	− 46.0000i	− 1.47167i	0.677161	−	0.735835i	$-0.263210\pi$
$978$	− 24.2337i	− 0.774908i
$979$	0.861407	0.0275307
$980$	0	0
$981$	4.74456	0.151482
$982$	27.6060i	0.880942i
$983$	2.97825i	0.0949914i	0.998871	+	0.0474957i	$0.0151240\pi$
$983$	2.97825i	0.0949914i	−0.998871	+	0.0474957i	$0.984876\pi$
$984$	−0.744563	−0.0237358
$985$	0	0
$986$	41.4891	1.32128
$987$	− 22.7446i	− 0.723967i
$988$	− 1.25544i	− 0.0399408i
$989$	−2.11684	−0.0673117
$990$	0	0
$991$	−47.6060	−1.51225	−0.756127	−	0.654425i	$-0.772911\pi$
$991$	−47.6060	−1.51225	−0.756127	+	0.654425i	$0.772911\pi$
$992$	− 1.00000i	− 0.0317500i
$993$	1.48913i	0.0472560i
$994$	−11.3723	−0.360707
$995$	0	0
$996$	−12.0000	−0.380235
$997$	6.00000i	0.190022i	0.995476	+	0.0950110i	$0.0302886\pi$
$997$	6.00000i	0.190022i	−0.995476	+	0.0950110i	$0.969711\pi$
$998$	− 33.4891i	− 1.06008i
$999$	0.744563	0.0235569

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4650.2.d.bh.3349.4		4
5.2	odd	4		4650.2.a.by.1.1		2
5.3	odd	4		930.2.a.r.1.2	✓	2
5.4	even	2	inner	4650.2.d.bh.3349.1		4
15.8	even	4		2790.2.a.bd.1.2		2
20.3	even	4		7440.2.a.bg.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
930.2.a.r.1.2	✓	2	5.3	odd	4
2790.2.a.bd.1.2		2	15.8	even	4
4650.2.a.by.1.1		2	5.2	odd	4
4650.2.d.bh.3349.1		4	5.4	even	2	inner
4650.2.d.bh.3349.4		4	1.1	even	1	trivial
7440.2.a.bg.1.1		2	20.3	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 4650 = 2 \cdot 3 \cdot 5^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4650.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +3.37228i q^{7} -1.00000i q^{8} -1.00000 q^{9} +0.627719 q^{11} +1.00000i q^{12} +2.00000i q^{13} -3.37228 q^{14} +1.00000 q^{16} -4.74456i q^{17} -1.00000i q^{18} +0.627719 q^{19} +3.37228 q^{21} +0.627719i q^{22} -3.37228i q^{23} -1.00000 q^{24} -2.00000 q^{26} +1.00000i q^{27} -3.37228i q^{28} +8.74456 q^{29} -1.00000 q^{31} +1.00000i q^{32} -0.627719i q^{33} +4.74456 q^{34} +1.00000 q^{36} -0.744563i q^{37} +0.627719i q^{38} +2.00000 q^{39} +0.744563 q^{41} +3.37228i q^{42} -0.627719i q^{43} -0.627719 q^{44} +3.37228 q^{46} -6.74456i q^{47} -1.00000i q^{48} -4.37228 q^{49} -4.74456 q^{51} -2.00000i q^{52} -1.37228i q^{53} -1.00000 q^{54} +3.37228 q^{56} -0.627719i q^{57} +8.74456i q^{58} -2.74456 q^{59} +11.4891 q^{61} -1.00000i q^{62} -3.37228i q^{63} -1.00000 q^{64} +0.627719 q^{66} +10.7446i q^{67} +4.74456i q^{68} -3.37228 q^{69} +3.37228 q^{71} +1.00000i q^{72} +8.11684i q^{73} +0.744563 q^{74} -0.627719 q^{76} +2.11684i q^{77} +2.00000i q^{78} +4.62772 q^{79} +1.00000 q^{81} +0.744563i q^{82} +12.0000i q^{83} -3.37228 q^{84} +0.627719 q^{86} -8.74456i q^{87} -0.627719i q^{88} +1.37228 q^{89} -6.74456 q^{91} +3.37228i q^{92} +1.00000i q^{93} +6.74456 q^{94} +1.00000 q^{96} +2.00000i q^{97} -4.37228i q^{98} -0.627719 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{4} + 4 q^{6} - 4 q^{9} + 14 q^{11} - 2 q^{14} + 4 q^{16} + 14 q^{19} + 2 q^{21} - 4 q^{24} - 8 q^{26} + 12 q^{29} - 4 q^{31} - 4 q^{34} + 4 q^{36} + 8 q^{39} - 20 q^{41} - 14 q^{44} + 2 q^{46}+ \cdots - 14 q^{99}+O(q^{100}) \) 4 * q - 4 * q^4 + 4 * q^6 - 4 * q^9 + 14 * q^11 - 2 * q^14 + 4 * q^16 + 14 * q^19 + 2 * q^21 - 4 * q^24 - 8 * q^26 + 12 * q^29 - 4 * q^31 - 4 * q^34 + 4 * q^36 + 8 * q^39 - 20 * q^41 - 14 * q^44 + 2 * q^46 - 6 * q^49 + 4 * q^51 - 4 * q^54 + 2 * q^56 + 12 * q^59 - 4 * q^64 + 14 * q^66 - 2 * q^69 + 2 * q^71 - 20 * q^74 - 14 * q^76 + 30 * q^79 + 4 * q^81 - 2 * q^84 + 14 * q^86 - 6 * q^89 - 4 * q^91 + 4 * q^94 + 4 * q^96 - 14 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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