LMFDB - Embedded newform 4650.2.d.bd.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,-4,0,2,0,0,-2,0,4,0,0,6] 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:	$\Q(i, \sqrt{17})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 9x^{2} + 16 $ x^4 + 9*x^2 + 16
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			$2.56155i$ of defining polynomial
Character	$\chi$	$=$	4650.3349
Dual form			4650.2.d.bd.3349.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +2.56155i q^{7} -1.00000i q^{8} -1.00000 q^{9} +2.56155 q^{11} -1.00000i q^{12} -2.00000i q^{13} -2.56155 q^{14} +1.00000 q^{16} -3.12311i q^{17} -1.00000i q^{18} +7.68466 q^{19} -2.56155 q^{21} +2.56155i q^{22} -1.43845i q^{23} +1.00000 q^{24} +2.00000 q^{26} -1.00000i q^{27} -2.56155i q^{28} -7.12311 q^{29} +1.00000 q^{31} +1.00000i q^{32} +2.56155i q^{33} +3.12311 q^{34} +1.00000 q^{36} -3.12311i q^{37} +7.68466i q^{38} +2.00000 q^{39} +7.12311 q^{41} -2.56155i q^{42} -12.8078i q^{43} -2.56155 q^{44} +1.43845 q^{46} +5.12311i q^{47} +1.00000i q^{48} +0.438447 q^{49} +3.12311 q^{51} +2.00000i q^{52} -7.43845i q^{53} +1.00000 q^{54} +2.56155 q^{56} +7.68466i q^{57} -7.12311i q^{58} +13.1231 q^{59} +6.00000 q^{61} +1.00000i q^{62} -2.56155i q^{63} -1.00000 q^{64} -2.56155 q^{66} -15.3693i q^{67} +3.12311i q^{68} +1.43845 q^{69} -7.68466 q^{71} +1.00000i q^{72} +10.8078i q^{73} +3.12311 q^{74} -7.68466 q^{76} +6.56155i q^{77} +2.00000i q^{78} +4.31534 q^{79} +1.00000 q^{81} +7.12311i q^{82} -14.2462i q^{83} +2.56155 q^{84} +12.8078 q^{86} -7.12311i q^{87} -2.56155i q^{88} +13.6847 q^{89} +5.12311 q^{91} +1.43845i q^{92} +1.00000i q^{93} -5.12311 q^{94} -1.00000 q^{96} -6.00000i q^{97} +0.438447i q^{98} -2.56155 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{4} - 4 q^{6} - 4 q^{9} + 2 q^{11} - 2 q^{14} + 4 q^{16} + 6 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} + 12 q^{41} - 2 q^{44} + 14 q^{46}+ \cdots - 2 q^{99}+O(q^{100}) $ 4 * q - 4 * q^4 - 4 * q^6 - 4 * q^9 + 2 * q^11 - 2 * q^14 + 4 * q^16 + 6 * 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 + 12 * q^41 - 2 * q^44 + 14 * q^46 + 10 * q^49 - 4 * q^51 + 4 * q^54 + 2 * q^56 + 36 * q^59 + 24 * q^61 - 4 * q^64 - 2 * q^66 + 14 * q^69 - 6 * q^71 - 4 * q^74 - 6 * q^76 + 42 * q^79 + 4 * q^81 + 2 * q^84 + 10 * q^86 + 30 * q^89 + 4 * q^91 - 4 * q^94 - 4 * q^96 - 2 * 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
$7$	2.56155i	0.968176i	0.875019	+	0.484088i	$0.160849\pi$
$7$	2.56155i	0.968176i	−0.875019	+	0.484088i	$0.839151\pi$
$8$	− 1.00000i	− 0.353553i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	2.56155	0.772337	0.386169	−	0.922428i	$-0.373798\pi$
$11$	2.56155	0.772337	0.386169	+	0.922428i	$0.373798\pi$
$12$	− 1.00000i	− 0.288675i
$13$	− 2.00000i	− 0.554700i	−0.960769	−	0.277350i	$-0.910544\pi$
$13$	− 2.00000i	− 0.554700i	0.960769	−	0.277350i	$-0.0894562\pi$
$14$	−2.56155	−0.684604
$15$	0	0
$16$	1.00000	0.250000
$17$	− 3.12311i	− 0.757464i	−0.925506	−	0.378732i	$-0.876360\pi$
$17$	− 3.12311i	− 0.757464i	0.925506	−	0.378732i	$-0.123640\pi$
$18$	− 1.00000i	− 0.235702i
$19$	7.68466	1.76298	0.881491	−	0.472201i	$-0.156540\pi$
$19$	7.68466	1.76298	0.881491	+	0.472201i	$0.156540\pi$
$20$	0	0
$21$	−2.56155	−0.558977
$22$	2.56155i	0.546125i
$23$	− 1.43845i	− 0.299937i	−0.988691	−	0.149968i	$-0.952083\pi$
$23$	− 1.43845i	− 0.299937i	0.988691	−	0.149968i	$-0.0479172\pi$
$24$	1.00000	0.204124
$25$	0	0
$26$	2.00000	0.392232
$27$	− 1.00000i	− 0.192450i
$28$	− 2.56155i	− 0.484088i
$29$	−7.12311	−1.32273	−0.661364	−	0.750065i	$-0.730022\pi$
$29$	−7.12311	−1.32273	−0.661364	+	0.750065i	$0.730022\pi$
$30$	0	0
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	1.00000i	0.176777i
$33$	2.56155i	0.445909i
$34$	3.12311	0.535608
$35$	0	0
$36$	1.00000	0.166667
$37$	− 3.12311i	− 0.513435i	−0.966486	−	0.256718i	$-0.917359\pi$
$37$	− 3.12311i	− 0.513435i	0.966486	−	0.256718i	$-0.0826411\pi$
$38$	7.68466i	1.24662i
$39$	2.00000	0.320256
$40$	0	0
$41$	7.12311	1.11244	0.556221	−	0.831034i	$-0.312251\pi$
$41$	7.12311	1.11244	0.556221	+	0.831034i	$0.312251\pi$
$42$	− 2.56155i	− 0.395256i
$43$	− 12.8078i	− 1.95317i	−0.215142	−	0.976583i	$-0.569021\pi$
$43$	− 12.8078i	− 1.95317i	0.215142	−	0.976583i	$-0.430979\pi$
$44$	−2.56155	−0.386169
$45$	0	0
$46$	1.43845	0.212087
$47$	5.12311i	0.747282i	0.927573	+	0.373641i	$0.121891\pi$
$47$	5.12311i	0.747282i	−0.927573	+	0.373641i	$0.878109\pi$
$48$	1.00000i	0.144338i
$49$	0.438447	0.0626353
$50$	0	0
$51$	3.12311	0.437322
$52$	2.00000i	0.277350i
$53$	− 7.43845i	− 1.02175i	−0.859655	−	0.510875i	$-0.829322\pi$
$53$	− 7.43845i	− 1.02175i	0.859655	−	0.510875i	$-0.170678\pi$
$54$	1.00000	0.136083
$55$	0	0
$56$	2.56155	0.342302
$57$	7.68466i	1.01786i
$58$	− 7.12311i	− 0.935310i
$59$	13.1231	1.70848	0.854241	−	0.519877i	$-0.174022\pi$
$59$	13.1231	1.70848	0.854241	+	0.519877i	$0.174022\pi$
$60$	0	0
$61$	6.00000	0.768221	0.384111	−	0.923287i	$-0.374508\pi$
$61$	6.00000	0.768221	0.384111	+	0.923287i	$0.374508\pi$
$62$	1.00000i	0.127000i
$63$	− 2.56155i	− 0.322725i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	−2.56155	−0.315305
$67$	− 15.3693i	− 1.87766i	−0.344380	−	0.938830i	$-0.611911\pi$
$67$	− 15.3693i	− 1.87766i	0.344380	−	0.938830i	$-0.388089\pi$
$68$	3.12311i	0.378732i
$69$	1.43845	0.173169
$70$	0	0
$71$	−7.68466	−0.912001	−0.456001	−	0.889979i	$-0.650719\pi$
$71$	−7.68466	−0.912001	−0.456001	+	0.889979i	$0.650719\pi$
$72$	1.00000i	0.117851i
$73$	10.8078i	1.26495i	0.774580	+	0.632477i	$0.217962\pi$
$73$	10.8078i	1.26495i	−0.774580	+	0.632477i	$0.782038\pi$
$74$	3.12311	0.363054
$75$	0	0
$76$	−7.68466	−0.881491
$77$	6.56155i	0.747758i
$78$	2.00000i	0.226455i
$79$	4.31534	0.485514	0.242757	−	0.970087i	$-0.421948\pi$
$79$	4.31534	0.485514	0.242757	+	0.970087i	$0.421948\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	7.12311i	0.786615i
$83$	− 14.2462i	− 1.56372i	−0.623451	−	0.781862i	$-0.714270\pi$
$83$	− 14.2462i	− 1.56372i	0.623451	−	0.781862i	$-0.285730\pi$
$84$	2.56155	0.279488
$85$	0	0
$86$	12.8078	1.38110
$87$	− 7.12311i	− 0.763677i
$88$	− 2.56155i	− 0.273062i
$89$	13.6847	1.45057	0.725285	−	0.688448i	$-0.241708\pi$
$89$	13.6847	1.45057	0.725285	+	0.688448i	$0.241708\pi$
$90$	0	0
$91$	5.12311	0.537047
$92$	1.43845i	0.149968i
$93$	1.00000i	0.103695i
$94$	−5.12311	−0.528408
$95$	0	0
$96$	−1.00000	−0.102062
$97$	− 6.00000i	− 0.609208i	−0.952479	−	0.304604i	$-0.901476\pi$
$97$	− 6.00000i	− 0.609208i	0.952479	−	0.304604i	$-0.0985241\pi$
$98$	0.438447i	0.0442899i
$99$	−2.56155	−0.257446
$100$	0	0
$101$	−0.561553	−0.0558766	−0.0279383	−	0.999610i	$-0.508894\pi$
$101$	−0.561553	−0.0558766	−0.0279383	+	0.999610i	$0.508894\pi$
$102$	3.12311i	0.309234i
$103$	1.75379i	0.172806i	0.996260	+	0.0864030i	$0.0275373\pi$
$103$	1.75379i	0.172806i	−0.996260	+	0.0864030i	$0.972463\pi$
$104$	−2.00000	−0.196116
$105$	0	0
$106$	7.43845	0.722486
$107$	− 2.56155i	− 0.247635i	−0.992305	−	0.123817i	$-0.960486\pi$
$107$	− 2.56155i	− 0.247635i	0.992305	−	0.123817i	$-0.0395137\pi$
$108$	1.00000i	0.0962250i
$109$	−5.36932	−0.514287	−0.257144	−	0.966373i	$-0.582781\pi$
$109$	−5.36932	−0.514287	−0.257144	+	0.966373i	$0.582781\pi$
$110$	0	0
$111$	3.12311	0.296432
$112$	2.56155i	0.242044i
$113$	1.68466i	0.158479i	0.996856	+	0.0792397i	$0.0252492\pi$
$113$	1.68466i	0.158479i	−0.996856	+	0.0792397i	$0.974751\pi$
$114$	−7.68466	−0.719734
$115$	0	0
$116$	7.12311	0.661364
$117$	2.00000i	0.184900i
$118$	13.1231i	1.20808i
$119$	8.00000	0.733359
$120$	0	0
$121$	−4.43845	−0.403495
$122$	6.00000i	0.543214i
$123$	7.12311i	0.642269i
$124$	−1.00000	−0.0898027
$125$	0	0
$126$	2.56155	0.228201
$127$	0	0	1.00000			$0$
$127$	0	0	−1.00000			$\pi$
$128$	− 1.00000i	− 0.0883883i
$129$	12.8078	1.12766
$130$	0	0
$131$	−15.3693	−1.34282	−0.671412	−	0.741085i	$-0.734312\pi$
$131$	−15.3693	−1.34282	−0.671412	+	0.741085i	$0.734312\pi$
$132$	− 2.56155i	− 0.222955i
$133$	19.6847i	1.70688i
$134$	15.3693	1.32771
$135$	0	0
$136$	−3.12311	−0.267804
$137$	20.2462i	1.72975i	0.501987	+	0.864875i	$0.332603\pi$
$137$	20.2462i	1.72975i	−0.501987	+	0.864875i	$0.667397\pi$
$138$	1.43845i	0.122449i
$139$	17.1231	1.45236	0.726181	−	0.687503i	$-0.241293\pi$
$139$	17.1231	1.45236	0.726181	+	0.687503i	$0.241293\pi$
$140$	0	0
$141$	−5.12311	−0.431443
$142$	− 7.68466i	− 0.644882i
$143$	− 5.12311i	− 0.428416i
$144$	−1.00000	−0.0833333
$145$	0	0
$146$	−10.8078	−0.894457
$147$	0.438447i	0.0361625i
$148$	3.12311i	0.256718i
$149$	−17.0540	−1.39712	−0.698558	−	0.715553i	$-0.746175\pi$
$149$	−17.0540	−1.39712	−0.698558	+	0.715553i	$0.746175\pi$
$150$	0	0
$151$	0	0		−	1.00000i	$-0.5\pi$
$151$	0	0			1.00000i	$0.5\pi$
$152$	− 7.68466i	− 0.623308i
$153$	3.12311i	0.252488i
$154$	−6.56155	−0.528745
$155$	0	0
$156$	−2.00000	−0.160128
$157$	17.0540i	1.36106i	0.732722	+	0.680528i	$0.238249\pi$
$157$	17.0540i	1.36106i	−0.732722	+	0.680528i	$0.761751\pi$
$158$	4.31534i	0.343310i
$159$	7.43845	0.589907
$160$	0	0
$161$	3.68466	0.290392
$162$	1.00000i	0.0785674i
$163$	15.3693i	1.20382i	0.798565	+	0.601909i	$0.205593\pi$
$163$	15.3693i	1.20382i	−0.798565	+	0.601909i	$0.794407\pi$
$164$	−7.12311	−0.556221
$165$	0	0
$166$	14.2462	1.10572
$167$	− 6.56155i	− 0.507748i	−0.967237	−	0.253874i	$-0.918295\pi$
$167$	− 6.56155i	− 0.507748i	0.967237	−	0.253874i	$-0.0817049\pi$
$168$	2.56155i	0.197628i
$169$	9.00000	0.692308
$170$	0	0
$171$	−7.68466	−0.587661
$172$	12.8078i	0.976583i
$173$	− 8.24621i	− 0.626948i	−0.949597	−	0.313474i	$-0.898507\pi$
$173$	− 8.24621i	− 0.626948i	0.949597	−	0.313474i	$-0.101493\pi$
$174$	7.12311	0.540001
$175$	0	0
$176$	2.56155	0.193084
$177$	13.1231i	0.986393i
$178$	13.6847i	1.02571i
$179$	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
$181$	−13.0540	−0.970294	−0.485147	−	0.874433i	$-0.661234\pi$
$181$	−13.0540	−0.970294	−0.485147	+	0.874433i	$0.661234\pi$
$182$	5.12311i	0.379750i
$183$	6.00000i	0.443533i
$184$	−1.43845	−0.106044
$185$	0	0
$186$	−1.00000	−0.0733236
$187$	− 8.00000i	− 0.585018i
$188$	− 5.12311i	− 0.373641i
$189$	2.56155	0.186326
$190$	0	0
$191$	−14.2462	−1.03082	−0.515410	−	0.856944i	$-0.672360\pi$
$191$	−14.2462	−1.03082	−0.515410	+	0.856944i	$0.672360\pi$
$192$	− 1.00000i	− 0.0721688i
$193$	− 14.4924i	− 1.04319i	−0.853194	−	0.521594i	$-0.825338\pi$
$193$	− 14.4924i	− 1.04319i	0.853194	−	0.521594i	$-0.174662\pi$
$194$	6.00000	0.430775
$195$	0	0
$196$	−0.438447	−0.0313177
$197$	6.00000i	0.427482i	0.976890	+	0.213741i	$0.0685649\pi$
$197$	6.00000i	0.427482i	−0.976890	+	0.213741i	$0.931435\pi$
$198$	− 2.56155i	− 0.182042i
$199$	16.8078	1.19147	0.595735	−	0.803181i	$-0.296861\pi$
$199$	16.8078	1.19147	0.595735	+	0.803181i	$0.296861\pi$
$200$	0	0
$201$	15.3693	1.08407
$202$	− 0.561553i	− 0.0395107i
$203$	− 18.2462i	− 1.28063i
$204$	−3.12311	−0.218661
$205$	0	0
$206$	−1.75379	−0.122192
$207$	1.43845i	0.0999790i
$208$	− 2.00000i	− 0.138675i
$209$	19.6847	1.36162
$210$	0	0
$211$	−23.6847	−1.63052	−0.815260	−	0.579096i	$-0.803406\pi$
$211$	−23.6847	−1.63052	−0.815260	+	0.579096i	$0.803406\pi$
$212$	7.43845i	0.510875i
$213$	− 7.68466i	− 0.526544i
$214$	2.56155	0.175104
$215$	0	0
$216$	−1.00000	−0.0680414
$217$	2.56155i	0.173890i
$218$	− 5.36932i	− 0.363656i
$219$	−10.8078	−0.730321
$220$	0	0
$221$	−6.24621	−0.420166
$222$	3.12311i	0.209609i
$223$	21.1231i	1.41451i	0.706960	+	0.707254i	$0.250066\pi$
$223$	21.1231i	1.41451i	−0.706960	+	0.707254i	$0.749934\pi$
$224$	−2.56155	−0.171151
$225$	0	0
$226$	−1.68466	−0.112062
$227$	7.68466i	0.510049i	0.966935	+	0.255024i	$0.0820835\pi$
$227$	7.68466i	0.510049i	−0.966935	+	0.255024i	$0.917917\pi$
$228$	− 7.68466i	− 0.508929i
$229$	16.5616	1.09442	0.547209	−	0.836996i	$-0.315690\pi$
$229$	16.5616	1.09442	0.547209	+	0.836996i	$0.315690\pi$
$230$	0	0
$231$	−6.56155	−0.431718
$232$	7.12311i	0.467655i
$233$	17.0540i	1.11724i	0.829423	+	0.558622i	$0.188670\pi$
$233$	17.0540i	1.11724i	−0.829423	+	0.558622i	$0.811330\pi$
$234$	−2.00000	−0.130744
$235$	0	0
$236$	−13.1231	−0.854241
$237$	4.31534i	0.280312i
$238$	8.00000i	0.518563i
$239$	−24.0000	−1.55243	−0.776215	−	0.630468i	$-0.782863\pi$
$239$	−24.0000	−1.55243	−0.776215	+	0.630468i	$0.782863\pi$
$240$	0	0
$241$	12.2462	0.788848	0.394424	−	0.918929i	$-0.370944\pi$
$241$	12.2462	0.788848	0.394424	+	0.918929i	$0.370944\pi$
$242$	− 4.43845i	− 0.285314i
$243$	1.00000i	0.0641500i
$244$	−6.00000	−0.384111
$245$	0	0
$246$	−7.12311	−0.454153
$247$	− 15.3693i	− 0.977926i
$248$	− 1.00000i	− 0.0635001i
$249$	14.2462	0.902817
$250$	0	0
$251$	16.4924	1.04099	0.520496	−	0.853864i	$-0.325747\pi$
$251$	16.4924	1.04099	0.520496	+	0.853864i	$0.325747\pi$
$252$	2.56155i	0.161363i
$253$	− 3.68466i	− 0.231652i
$254$	0	0
$255$	0	0
$256$	1.00000	0.0625000
$257$	− 1.68466i	− 0.105086i	−0.998619	−	0.0525431i	$-0.983267\pi$
$257$	− 1.68466i	− 0.105086i	0.998619	−	0.0525431i	$-0.0167327\pi$
$258$	12.8078i	0.797377i
$259$	8.00000	0.497096
$260$	0	0
$261$	7.12311	0.440909
$262$	− 15.3693i	− 0.949520i
$263$	20.4924i	1.26362i	0.775125	+	0.631808i	$0.217687\pi$
$263$	20.4924i	1.26362i	−0.775125	+	0.631808i	$0.782313\pi$
$264$	2.56155	0.157653
$265$	0	0
$266$	−19.6847	−1.20694
$267$	13.6847i	0.837487i
$268$	15.3693i	0.938830i
$269$	0.246211	0.0150118	0.00750588	−	0.999972i	$-0.497611\pi$
$269$	0.246211	0.0150118	0.00750588	+	0.999972i	$0.497611\pi$
$270$	0	0
$271$	27.0540	1.64341	0.821706	−	0.569912i	$-0.193023\pi$
$271$	27.0540	1.64341	0.821706	+	0.569912i	$0.193023\pi$
$272$	− 3.12311i	− 0.189366i
$273$	5.12311i	0.310064i
$274$	−20.2462	−1.22312
$275$	0	0
$276$	−1.43845	−0.0865843
$277$	− 5.36932i	− 0.322611i	−0.986905	−	0.161305i	$-0.948430\pi$
$277$	− 5.36932i	− 0.322611i	0.986905	−	0.161305i	$-0.0515704\pi$
$278$	17.1231i	1.02698i
$279$	−1.00000	−0.0598684
$280$	0	0
$281$	4.87689	0.290931	0.145466	−	0.989363i	$-0.453532\pi$
$281$	4.87689	0.290931	0.145466	+	0.989363i	$0.453532\pi$
$282$	− 5.12311i	− 0.305077i
$283$	0	0	1.00000			$0$
$283$	0	0	−1.00000			$\pi$
$284$	7.68466	0.456001
$285$	0	0
$286$	5.12311	0.302936
$287$	18.2462i	1.07704i
$288$	− 1.00000i	− 0.0589256i
$289$	7.24621	0.426248
$290$	0	0
$291$	6.00000	0.351726
$292$	− 10.8078i	− 0.632477i
$293$	− 26.4924i	− 1.54770i	−0.633367	−	0.773852i	$-0.718327\pi$
$293$	− 26.4924i	− 1.54770i	0.633367	−	0.773852i	$-0.281673\pi$
$294$	−0.438447	−0.0255708
$295$	0	0
$296$	−3.12311	−0.181527
$297$	− 2.56155i	− 0.148636i
$298$	− 17.0540i	− 0.987910i
$299$	−2.87689	−0.166375
$300$	0	0
$301$	32.8078	1.89101
$302$	0	0
$303$	− 0.561553i	− 0.0322604i
$304$	7.68466	0.440745
$305$	0	0
$306$	−3.12311	−0.178536
$307$	0	0	1.00000			$0$
$307$	0	0	−1.00000			$\pi$
$308$	− 6.56155i	− 0.373879i
$309$	−1.75379	−0.0997696
$310$	0	0
$311$	−1.75379	−0.0994482	−0.0497241	−	0.998763i	$-0.515834\pi$
$311$	−1.75379	−0.0994482	−0.0497241	+	0.998763i	$0.515834\pi$
$312$	− 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	$0.908798\pi$
$314$	−17.0540	−0.962412
$315$	0	0
$316$	−4.31534	−0.242757
$317$	16.8769i	0.947901i	0.880552	+	0.473950i	$0.157172\pi$
$317$	16.8769i	0.947901i	−0.880552	+	0.473950i	$0.842828\pi$
$318$	7.43845i	0.417127i
$319$	−18.2462	−1.02159
$320$	0	0
$321$	2.56155	0.142972
$322$	3.68466i	0.205338i
$323$	− 24.0000i	− 1.33540i
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	−15.3693	−0.851228
$327$	− 5.36932i	− 0.296924i
$328$	− 7.12311i	− 0.393308i
$329$	−13.1231	−0.723500
$330$	0	0
$331$	−6.24621	−0.343323	−0.171661	−	0.985156i	$-0.554914\pi$
$331$	−6.24621	−0.343323	−0.171661	+	0.985156i	$0.554914\pi$
$332$	14.2462i	0.781862i
$333$	3.12311i	0.171145i
$334$	6.56155	0.359032
$335$	0	0
$336$	−2.56155	−0.139744
$337$	− 10.0000i	− 0.544735i	−0.962193	−	0.272367i	$-0.912193\pi$
$337$	− 10.0000i	− 0.544735i	0.962193	−	0.272367i	$-0.0878066\pi$
$338$	9.00000i	0.489535i
$339$	−1.68466	−0.0914981
$340$	0	0
$341$	2.56155	0.138716
$342$	− 7.68466i	− 0.415539i
$343$	19.0540i	1.02882i
$344$	−12.8078	−0.690548
$345$	0	0
$346$	8.24621	0.443319
$347$	14.2462i	0.764777i	0.924002	+	0.382388i	$0.124898\pi$
$347$	14.2462i	0.764777i	−0.924002	+	0.382388i	$0.875102\pi$
$348$	7.12311i	0.381839i
$349$	−5.36932	−0.287413	−0.143706	−	0.989620i	$-0.545902\pi$
$349$	−5.36932	−0.287413	−0.143706	+	0.989620i	$0.545902\pi$
$350$	0	0
$351$	−2.00000	−0.106752
$352$	2.56155i	0.136531i
$353$	0.246211i	0.0131045i	0.999979	+	0.00655225i	$0.00208566\pi$
$353$	0.246211i	0.0131045i	−0.999979	+	0.00655225i	$0.997914\pi$
$354$	−13.1231	−0.697485
$355$	0	0
$356$	−13.6847	−0.725285
$357$	8.00000i	0.423405i
$358$	12.0000i	0.634220i
$359$	31.6847	1.67225	0.836126	−	0.548537i	$-0.184815\pi$
$359$	31.6847	1.67225	0.836126	+	0.548537i	$0.184815\pi$
$360$	0	0
$361$	40.0540	2.10810
$362$	− 13.0540i	− 0.686102i
$363$	− 4.43845i	− 0.232958i
$364$	−5.12311	−0.268524
$365$	0	0
$366$	−6.00000	−0.313625
$367$	15.3693i	0.802272i	0.916019	+	0.401136i	$0.131384\pi$
$367$	15.3693i	0.802272i	−0.916019	+	0.401136i	$0.868616\pi$
$368$	− 1.43845i	− 0.0749842i
$369$	−7.12311	−0.370814
$370$	0	0
$371$	19.0540	0.989233
$372$	− 1.00000i	− 0.0518476i
$373$	5.68466i	0.294340i	0.989111	+	0.147170i	$0.0470165\pi$
$373$	5.68466i	0.294340i	−0.989111	+	0.147170i	$0.952983\pi$
$374$	8.00000	0.413670
$375$	0	0
$376$	5.12311	0.264204
$377$	14.2462i	0.733717i
$378$	2.56155i	0.131752i
$379$	7.05398	0.362338	0.181169	−	0.983452i	$-0.442012\pi$
$379$	7.05398	0.362338	0.181169	+	0.983452i	$0.442012\pi$
$380$	0	0
$381$	0	0
$382$	− 14.2462i	− 0.728900i
$383$	10.2462i	0.523557i	0.965128	+	0.261778i	$0.0843090\pi$
$383$	10.2462i	0.523557i	−0.965128	+	0.261778i	$0.915691\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	14.4924	0.737645
$387$	12.8078i	0.651055i
$388$	6.00000i	0.304604i
$389$	−7.75379	−0.393133	−0.196566	−	0.980491i	$-0.562979\pi$
$389$	−7.75379	−0.393133	−0.196566	+	0.980491i	$0.562979\pi$
$390$	0	0
$391$	−4.49242	−0.227192
$392$	− 0.438447i	− 0.0221449i
$393$	− 15.3693i	− 0.775279i
$394$	−6.00000	−0.302276
$395$	0	0
$396$	2.56155	0.128723
$397$	9.05398i	0.454406i	0.973847	+	0.227203i	$0.0729581\pi$
$397$	9.05398i	0.454406i	−0.973847	+	0.227203i	$0.927042\pi$
$398$	16.8078i	0.842497i
$399$	−19.6847	−0.985466
$400$	0	0
$401$	−13.6847	−0.683379	−0.341690	−	0.939813i	$-0.610999\pi$
$401$	−13.6847	−0.683379	−0.341690	+	0.939813i	$0.610999\pi$
$402$	15.3693i	0.766552i
$403$	− 2.00000i	− 0.0996271i
$404$	0.561553	0.0279383
$405$	0	0
$406$	18.2462	0.905544
$407$	− 8.00000i	− 0.396545i
$408$	− 3.12311i	− 0.154617i
$409$	37.3693	1.84779	0.923897	−	0.382642i	$-0.124986\pi$
$409$	37.3693	1.84779	0.923897	+	0.382642i	$0.124986\pi$
$410$	0	0
$411$	−20.2462	−0.998672
$412$	− 1.75379i	− 0.0864030i
$413$	33.6155i	1.65411i
$414$	−1.43845	−0.0706958
$415$	0	0
$416$	2.00000	0.0980581
$417$	17.1231i	0.838522i
$418$	19.6847i	0.962808i
$419$	−5.75379	−0.281091	−0.140545	−	0.990074i	$-0.544886\pi$
$419$	−5.75379	−0.281091	−0.140545	+	0.990074i	$0.544886\pi$
$420$	0	0
$421$	−14.4924	−0.706317	−0.353159	−	0.935563i	$-0.614892\pi$
$421$	−14.4924	−0.706317	−0.353159	+	0.935563i	$0.614892\pi$
$422$	− 23.6847i	− 1.15295i
$423$	− 5.12311i	− 0.249094i
$424$	−7.43845	−0.361243
$425$	0	0
$426$	7.68466	0.372323
$427$	15.3693i	0.743773i
$428$	2.56155i	0.123817i
$429$	5.12311	0.247346
$430$	0	0
$431$	−30.2462	−1.45691	−0.728454	−	0.685094i	$-0.759761\pi$
$431$	−30.2462	−1.45691	−0.728454	+	0.685094i	$0.759761\pi$
$432$	− 1.00000i	− 0.0481125i
$433$	− 35.3002i	− 1.69642i	−0.529661	−	0.848209i	$-0.677681\pi$
$433$	− 35.3002i	− 1.69642i	0.529661	−	0.848209i	$-0.322319\pi$
$434$	−2.56155	−0.122958
$435$	0	0
$436$	5.36932	0.257144
$437$	− 11.0540i	− 0.528783i
$438$	− 10.8078i	− 0.516415i
$439$	−9.61553	−0.458924	−0.229462	−	0.973318i	$-0.573697\pi$
$439$	−9.61553	−0.458924	−0.229462	+	0.973318i	$0.573697\pi$
$440$	0	0
$441$	−0.438447	−0.0208784
$442$	− 6.24621i	− 0.297102i
$443$	− 23.0540i	− 1.09533i	−0.836699	−	0.547664i	$-0.815517\pi$
$443$	− 23.0540i	− 1.09533i	0.836699	−	0.547664i	$-0.184483\pi$
$444$	−3.12311	−0.148216
$445$	0	0
$446$	−21.1231	−1.00021
$447$	− 17.0540i	− 0.806625i
$448$	− 2.56155i	− 0.121022i
$449$	−10.4924	−0.495168	−0.247584	−	0.968866i	$-0.579637\pi$
$449$	−10.4924	−0.495168	−0.247584	+	0.968866i	$0.579637\pi$
$450$	0	0
$451$	18.2462	0.859181
$452$	− 1.68466i	− 0.0792397i
$453$	0	0
$454$	−7.68466	−0.360659
$455$	0	0
$456$	7.68466	0.359867
$457$	− 24.7386i	− 1.15722i	−0.815603	−	0.578612i	$-0.803594\pi$
$457$	− 24.7386i	− 1.15722i	0.815603	−	0.578612i	$-0.196406\pi$
$458$	16.5616i	0.773871i
$459$	−3.12311	−0.145774
$460$	0	0
$461$	9.36932	0.436373	0.218186	−	0.975907i	$-0.429986\pi$
$461$	9.36932	0.436373	0.218186	+	0.975907i	$0.429986\pi$
$462$	− 6.56155i	− 0.305271i
$463$	− 30.7386i	− 1.42855i	−0.699867	−	0.714273i	$-0.746758\pi$
$463$	− 30.7386i	− 1.42855i	0.699867	−	0.714273i	$-0.253242\pi$
$464$	−7.12311	−0.330682
$465$	0	0
$466$	−17.0540	−0.790010
$467$	− 9.75379i	− 0.451352i	−0.974202	−	0.225676i	$-0.927541\pi$
$467$	− 9.75379i	− 0.451352i	0.974202	−	0.225676i	$-0.0724590\pi$
$468$	− 2.00000i	− 0.0924500i
$469$	39.3693	1.81791
$470$	0	0
$471$	−17.0540	−0.785806
$472$	− 13.1231i	− 0.604040i
$473$	− 32.8078i	− 1.50850i
$474$	−4.31534	−0.198210
$475$	0	0
$476$	−8.00000	−0.366679
$477$	7.43845i	0.340583i
$478$	− 24.0000i	− 1.09773i
$479$	−10.5616	−0.482570	−0.241285	−	0.970454i	$-0.577569\pi$
$479$	−10.5616	−0.482570	−0.241285	+	0.970454i	$0.577569\pi$
$480$	0	0
$481$	−6.24621	−0.284803
$482$	12.2462i	0.557800i
$483$	3.68466i	0.167658i
$484$	4.43845	0.201748
$485$	0	0
$486$	−1.00000	−0.0453609
$487$	24.0000i	1.08754i	0.839233	+	0.543772i	$0.183004\pi$
$487$	24.0000i	1.08754i	−0.839233	+	0.543772i	$0.816996\pi$
$488$	− 6.00000i	− 0.271607i
$489$	−15.3693	−0.695025
$490$	0	0
$491$	−25.9309	−1.17024	−0.585122	−	0.810945i	$-0.698953\pi$
$491$	−25.9309	−1.17024	−0.585122	+	0.810945i	$0.698953\pi$
$492$	− 7.12311i	− 0.321134i
$493$	22.2462i	1.00192i
$494$	15.3693	0.691498
$495$	0	0
$496$	1.00000	0.0449013
$497$	− 19.6847i	− 0.882978i
$498$	14.2462i	0.638388i
$499$	−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
$501$	6.56155	0.293149
$502$	16.4924i	0.736093i
$503$	− 26.2462i	− 1.17026i	−0.810939	−	0.585130i	$-0.801043\pi$
$503$	− 26.2462i	− 1.17026i	0.810939	−	0.585130i	$-0.198957\pi$
$504$	−2.56155	−0.114101
$505$	0	0
$506$	3.68466	0.163803
$507$	9.00000i	0.399704i
$508$	0	0
$509$	−19.6155	−0.869443	−0.434721	−	0.900565i	$-0.643153\pi$
$509$	−19.6155	−0.869443	−0.434721	+	0.900565i	$0.643153\pi$
$510$	0	0
$511$	−27.6847	−1.22470
$512$	1.00000i	0.0441942i
$513$	− 7.68466i	− 0.339286i
$514$	1.68466	0.0743071
$515$	0	0
$516$	−12.8078	−0.563830
$517$	13.1231i	0.577154i
$518$	8.00000i	0.351500i
$519$	8.24621	0.361968
$520$	0	0
$521$	−32.2462	−1.41273	−0.706366	−	0.707847i	$-0.749667\pi$
$521$	−32.2462	−1.41273	−0.706366	+	0.707847i	$0.749667\pi$
$522$	7.12311i	0.311770i
$523$	22.4233i	0.980502i	0.871581	+	0.490251i	$0.163095\pi$
$523$	22.4233i	0.980502i	−0.871581	+	0.490251i	$0.836905\pi$
$524$	15.3693	0.671412
$525$	0	0
$526$	−20.4924	−0.893512
$527$	− 3.12311i	− 0.136045i
$528$	2.56155i	0.111477i
$529$	20.9309	0.910038
$530$	0	0
$531$	−13.1231	−0.569494
$532$	− 19.6847i	− 0.853438i
$533$	− 14.2462i	− 0.617072i
$534$	−13.6847	−0.592193
$535$	0	0
$536$	−15.3693	−0.663853
$537$	12.0000i	0.517838i
$538$	0.246211i	0.0106149i
$539$	1.12311	0.0483756
$540$	0	0
$541$	19.7538	0.849282	0.424641	−	0.905362i	$-0.360400\pi$
$541$	19.7538	0.849282	0.424641	+	0.905362i	$0.360400\pi$
$542$	27.0540i	1.16207i
$543$	− 13.0540i	− 0.560200i
$544$	3.12311	0.133902
$545$	0	0
$546$	−5.12311	−0.219249
$547$	26.8769i	1.14917i	0.818444	+	0.574587i	$0.194837\pi$
$547$	26.8769i	1.14917i	−0.818444	+	0.574587i	$0.805163\pi$
$548$	− 20.2462i	− 0.864875i
$549$	−6.00000	−0.256074
$550$	0	0
$551$	−54.7386	−2.33194
$552$	− 1.43845i	− 0.0612244i
$553$	11.0540i	0.470063i
$554$	5.36932	0.228120
$555$	0	0
$556$	−17.1231	−0.726181
$557$	− 26.8078i	− 1.13588i	−0.823069	−	0.567941i	$-0.807740\pi$
$557$	− 26.8078i	− 1.13588i	0.823069	−	0.567941i	$-0.192260\pi$
$558$	− 1.00000i	− 0.0423334i
$559$	−25.6155	−1.08342
$560$	0	0
$561$	8.00000	0.337760
$562$	4.87689i	0.205719i
$563$	− 16.4924i	− 0.695073i	−0.937667	−	0.347536i	$-0.887018\pi$
$563$	− 16.4924i	− 0.695073i	0.937667	−	0.347536i	$-0.112982\pi$
$564$	5.12311	0.215722
$565$	0	0
$566$	0	0
$567$	2.56155i	0.107575i
$568$	7.68466i	0.322441i
$569$	−30.8078	−1.29153	−0.645764	−	0.763537i	$-0.723461\pi$
$569$	−30.8078	−1.29153	−0.645764	+	0.763537i	$0.723461\pi$
$570$	0	0
$571$	41.1231	1.72095	0.860474	−	0.509494i	$-0.170167\pi$
$571$	41.1231	1.72095	0.860474	+	0.509494i	$0.170167\pi$
$572$	5.12311i	0.214208i
$573$	− 14.2462i	− 0.595144i
$574$	−18.2462	−0.761582
$575$	0	0
$576$	1.00000	0.0416667
$577$	15.1231i	0.629583i	0.949161	+	0.314792i	$0.101935\pi$
$577$	15.1231i	0.629583i	−0.949161	+	0.314792i	$0.898065\pi$
$578$	7.24621i	0.301403i
$579$	14.4924	0.602285
$580$	0	0
$581$	36.4924	1.51396
$582$	6.00000i	0.248708i
$583$	− 19.0540i	− 0.789135i
$584$	10.8078	0.447228
$585$	0	0
$586$	26.4924	1.09439
$587$	− 0.492423i	− 0.0203245i	−0.999948	−	0.0101622i	$-0.996765\pi$
$587$	− 0.492423i	− 0.0203245i	0.999948	−	0.0101622i	$-0.00323479\pi$
$588$	− 0.438447i	− 0.0180813i
$589$	7.68466	0.316641
$590$	0	0
$591$	−6.00000	−0.246807
$592$	− 3.12311i	− 0.128359i
$593$	30.0000i	1.23195i	0.787765	+	0.615976i	$0.211238\pi$
$593$	30.0000i	1.23195i	−0.787765	+	0.615976i	$0.788762\pi$
$594$	2.56155	0.105102
$595$	0	0
$596$	17.0540	0.698558
$597$	16.8078i	0.687896i
$598$	− 2.87689i	− 0.117645i
$599$	−38.4233	−1.56993	−0.784967	−	0.619538i	$-0.787320\pi$
$599$	−38.4233	−1.56993	−0.784967	+	0.619538i	$0.787320\pi$
$600$	0	0
$601$	2.00000	0.0815817	0.0407909	−	0.999168i	$-0.487012\pi$
$601$	2.00000	0.0815817	0.0407909	+	0.999168i	$0.487012\pi$
$602$	32.8078i	1.33714i
$603$	15.3693i	0.625887i
$604$	0	0
$605$	0	0
$606$	0.561553	0.0228115
$607$	7.05398i	0.286312i	0.989700	+	0.143156i	$0.0457251\pi$
$607$	7.05398i	0.286312i	−0.989700	+	0.143156i	$0.954275\pi$
$608$	7.68466i	0.311654i
$609$	18.2462	0.739374
$610$	0	0
$611$	10.2462	0.414517
$612$	− 3.12311i	− 0.126244i
$613$	− 2.00000i	− 0.0807792i	−0.999184	−	0.0403896i	$-0.987140\pi$
$613$	− 2.00000i	− 0.0807792i	0.999184	−	0.0403896i	$-0.0128599\pi$
$614$	0	0
$615$	0	0
$616$	6.56155	0.264372
$617$	− 15.4384i	− 0.621528i	−0.950487	−	0.310764i	$-0.899415\pi$
$617$	− 15.4384i	− 0.621528i	0.950487	−	0.310764i	$-0.100585\pi$
$618$	− 1.75379i	− 0.0705477i
$619$	−11.3693	−0.456971	−0.228486	−	0.973547i	$-0.573377\pi$
$619$	−11.3693	−0.456971	−0.228486	+	0.973547i	$0.573377\pi$
$620$	0	0
$621$	−1.43845	−0.0577229
$622$	− 1.75379i	− 0.0703205i
$623$	35.0540i	1.40441i
$624$	2.00000	0.0800641
$625$	0	0
$626$	−10.0000	−0.399680
$627$	19.6847i	0.786130i
$628$	− 17.0540i	− 0.680528i
$629$	−9.75379	−0.388909
$630$	0	0
$631$	29.3002	1.16642	0.583211	−	0.812321i	$-0.301796\pi$
$631$	29.3002	1.16642	0.583211	+	0.812321i	$0.301796\pi$
$632$	− 4.31534i	− 0.171655i
$633$	− 23.6847i	− 0.941381i
$634$	−16.8769	−0.670267
$635$	0	0
$636$	−7.43845	−0.294954
$637$	− 0.876894i	− 0.0347438i
$638$	− 18.2462i	− 0.722374i
$639$	7.68466	0.304000
$640$	0	0
$641$	−5.50758	−0.217536	−0.108768	−	0.994067i	$-0.534691\pi$
$641$	−5.50758	−0.217536	−0.108768	+	0.994067i	$0.534691\pi$
$642$	2.56155i	0.101096i
$643$	7.05398i	0.278182i	0.990280	+	0.139091i	$0.0444180\pi$
$643$	7.05398i	0.278182i	−0.990280	+	0.139091i	$0.955582\pi$
$644$	−3.68466	−0.145196
$645$	0	0
$646$	24.0000	0.944267
$647$	− 45.9309i	− 1.80573i	−0.429925	−	0.902864i	$-0.641460\pi$
$647$	− 45.9309i	− 1.80573i	0.429925	−	0.902864i	$-0.358540\pi$
$648$	− 1.00000i	− 0.0392837i
$649$	33.6155	1.31952
$650$	0	0
$651$	−2.56155	−0.100395
$652$	− 15.3693i	− 0.601909i
$653$	− 40.2462i	− 1.57496i	−0.616343	−	0.787478i	$-0.711386\pi$
$653$	− 40.2462i	− 1.57496i	0.616343	−	0.787478i	$-0.288614\pi$
$654$	5.36932	0.209957
$655$	0	0
$656$	7.12311	0.278111
$657$	− 10.8078i	− 0.421651i
$658$	− 13.1231i	− 0.511592i
$659$	30.7386	1.19741	0.598704	−	0.800971i	$-0.295683\pi$
$659$	30.7386	1.19741	0.598704	+	0.800971i	$0.295683\pi$
$660$	0	0
$661$	26.4924	1.03044	0.515218	−	0.857059i	$-0.327711\pi$
$661$	26.4924	1.03044	0.515218	+	0.857059i	$0.327711\pi$
$662$	− 6.24621i	− 0.242766i
$663$	− 6.24621i	− 0.242583i
$664$	−14.2462	−0.552860
$665$	0	0
$666$	−3.12311	−0.121018
$667$	10.2462i	0.396735i
$668$	6.56155i	0.253874i
$669$	−21.1231	−0.816666
$670$	0	0
$671$	15.3693	0.593326
$672$	− 2.56155i	− 0.0988140i
$673$	− 11.7538i	− 0.453075i	−0.974002	−	0.226538i	$-0.927259\pi$
$673$	− 11.7538i	− 0.453075i	0.974002	−	0.226538i	$-0.0727406\pi$
$674$	10.0000	0.385186
$675$	0	0
$676$	−9.00000	−0.346154
$677$	32.4233i	1.24613i	0.782171	+	0.623064i	$0.214112\pi$
$677$	32.4233i	1.24613i	−0.782171	+	0.623064i	$0.785888\pi$
$678$	− 1.68466i	− 0.0646989i
$679$	15.3693	0.589820
$680$	0	0
$681$	−7.68466	−0.294477
$682$	2.56155i	0.0980869i
$683$	− 31.6847i	− 1.21238i	−0.795320	−	0.606190i	$-0.792697\pi$
$683$	− 31.6847i	− 1.21238i	0.795320	−	0.606190i	$-0.207303\pi$
$684$	7.68466	0.293830
$685$	0	0
$686$	−19.0540	−0.727484
$687$	16.5616i	0.631863i
$688$	− 12.8078i	− 0.488291i
$689$	−14.8769	−0.566765
$690$	0	0
$691$	15.0540	0.572680	0.286340	−	0.958128i	$-0.407561\pi$
$691$	15.0540	0.572680	0.286340	+	0.958128i	$0.407561\pi$
$692$	8.24621i	0.313474i
$693$	− 6.56155i	− 0.249253i
$694$	−14.2462	−0.540779
$695$	0	0
$696$	−7.12311	−0.270001
$697$	− 22.2462i	− 0.842635i
$698$	− 5.36932i	− 0.203232i
$699$	−17.0540	−0.645041
$700$	0	0
$701$	5.19224	0.196108	0.0980540	−	0.995181i	$-0.468738\pi$
$701$	5.19224	0.196108	0.0980540	+	0.995181i	$0.468738\pi$
$702$	− 2.00000i	− 0.0754851i
$703$	− 24.0000i	− 0.905177i
$704$	−2.56155	−0.0965422
$705$	0	0
$706$	−0.246211	−0.00926628
$707$	− 1.43845i	− 0.0540984i
$708$	− 13.1231i	− 0.493197i
$709$	−17.0540	−0.640475	−0.320238	−	0.947337i	$-0.603763\pi$
$709$	−17.0540	−0.640475	−0.320238	+	0.947337i	$0.603763\pi$
$710$	0	0
$711$	−4.31534	−0.161838
$712$	− 13.6847i	− 0.512854i
$713$	− 1.43845i	− 0.0538703i
$714$	−8.00000	−0.299392
$715$	0	0
$716$	−12.0000	−0.448461
$717$	− 24.0000i	− 0.896296i
$718$	31.6847i	1.18246i
$719$	34.8769	1.30069	0.650344	−	0.759640i	$-0.274625\pi$
$719$	34.8769	1.30069	0.650344	+	0.759640i	$0.274625\pi$
$720$	0	0
$721$	−4.49242	−0.167307
$722$	40.0540i	1.49065i
$723$	12.2462i	0.455441i
$724$	13.0540	0.485147
$725$	0	0
$726$	4.43845	0.164726
$727$	23.0540i	0.855025i	0.904010	+	0.427512i	$0.140610\pi$
$727$	23.0540i	0.855025i	−0.904010	+	0.427512i	$0.859390\pi$
$728$	− 5.12311i	− 0.189875i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	−40.0000	−1.47945
$732$	− 6.00000i	− 0.221766i
$733$	6.49242i	0.239803i	0.992786	+	0.119902i	$0.0382579\pi$
$733$	6.49242i	0.239803i	−0.992786	+	0.119902i	$0.961742\pi$
$734$	−15.3693	−0.567292
$735$	0	0
$736$	1.43845	0.0530219
$737$	− 39.3693i	− 1.45019i
$738$	− 7.12311i	− 0.262205i
$739$	−52.9848	−1.94908	−0.974540	−	0.224216i	$-0.928018\pi$
$739$	−52.9848	−1.94908	−0.974540	+	0.224216i	$0.928018\pi$
$740$	0	0
$741$	15.3693	0.564606
$742$	19.0540i	0.699493i
$743$	50.4233i	1.84985i	0.380148	+	0.924926i	$0.375873\pi$
$743$	50.4233i	1.84985i	−0.380148	+	0.924926i	$0.624127\pi$
$744$	1.00000	0.0366618
$745$	0	0
$746$	−5.68466	−0.208130
$747$	14.2462i	0.521242i
$748$	8.00000i	0.292509i
$749$	6.56155	0.239754
$750$	0	0
$751$	45.1231	1.64657	0.823283	−	0.567631i	$-0.192140\pi$
$751$	45.1231	1.64657	0.823283	+	0.567631i	$0.192140\pi$
$752$	5.12311i	0.186820i
$753$	16.4924i	0.601017i
$754$	−14.2462	−0.518816
$755$	0	0
$756$	−2.56155	−0.0931628
$757$	10.0000i	0.363456i	0.983349	+	0.181728i	$0.0581691\pi$
$757$	10.0000i	0.363456i	−0.983349	+	0.181728i	$0.941831\pi$
$758$	7.05398i	0.256212i
$759$	3.68466	0.133745
$760$	0	0
$761$	−20.4233	−0.740344	−0.370172	−	0.928963i	$-0.620701\pi$
$761$	−20.4233	−0.740344	−0.370172	+	0.928963i	$0.620701\pi$
$762$	0	0
$763$	− 13.7538i	− 0.497921i
$764$	14.2462	0.515410
$765$	0	0
$766$	−10.2462	−0.370211
$767$	− 26.2462i	− 0.947696i
$768$	1.00000i	0.0360844i
$769$	20.5616	0.741469	0.370734	−	0.928739i	$-0.379106\pi$
$769$	20.5616	0.741469	0.370734	+	0.928739i	$0.379106\pi$
$770$	0	0
$771$	1.68466	0.0606715
$772$	14.4924i	0.521594i
$773$	11.4384i	0.411412i	0.978614	+	0.205706i	$0.0659491\pi$
$773$	11.4384i	0.411412i	−0.978614	+	0.205706i	$0.934051\pi$
$774$	−12.8078	−0.460366
$775$	0	0
$776$	−6.00000	−0.215387
$777$	8.00000i	0.286998i
$778$	− 7.75379i	− 0.277987i
$779$	54.7386	1.96122
$780$	0	0
$781$	−19.6847	−0.704372
$782$	− 4.49242i	− 0.160649i
$783$	7.12311i	0.254559i
$784$	0.438447	0.0156588
$785$	0	0
$786$	15.3693	0.548205
$787$	− 17.9309i	− 0.639166i	−0.947558	−	0.319583i	$-0.896457\pi$
$787$	− 17.9309i	− 0.639166i	0.947558	−	0.319583i	$-0.103543\pi$
$788$	− 6.00000i	− 0.213741i
$789$	−20.4924	−0.729550
$790$	0	0
$791$	−4.31534	−0.153436
$792$	2.56155i	0.0910208i
$793$	− 12.0000i	− 0.426132i
$794$	−9.05398	−0.321314
$795$	0	0
$796$	−16.8078	−0.595735
$797$	− 26.9848i	− 0.955852i	−0.878400	−	0.477926i	$-0.841389\pi$
$797$	− 26.9848i	− 0.955852i	0.878400	−	0.477926i	$-0.158611\pi$
$798$	− 19.6847i	− 0.696829i
$799$	16.0000	0.566039
$800$	0	0
$801$	−13.6847	−0.483524
$802$	− 13.6847i	− 0.483222i
$803$	27.6847i	0.976970i
$804$	−15.3693	−0.542034
$805$	0	0
$806$	2.00000	0.0704470
$807$	0.246211i	0.00866705i
$808$	0.561553i	0.0197554i
$809$	37.6847	1.32492	0.662461	−	0.749096i	$-0.269512\pi$
$809$	37.6847	1.32492	0.662461	+	0.749096i	$0.269512\pi$
$810$	0	0
$811$	−24.6695	−0.866263	−0.433132	−	0.901331i	$-0.642592\pi$
$811$	−24.6695	−0.866263	−0.433132	+	0.901331i	$0.642592\pi$
$812$	18.2462i	0.640316i
$813$	27.0540i	0.948824i
$814$	8.00000	0.280400
$815$	0	0
$816$	3.12311	0.109331
$817$	− 98.4233i	− 3.44340i
$818$	37.3693i	1.30659i
$819$	−5.12311	−0.179016
$820$	0	0
$821$	19.6155	0.684587	0.342293	−	0.939593i	$-0.388796\pi$
$821$	19.6155	0.684587	0.342293	+	0.939593i	$0.388796\pi$
$822$	− 20.2462i	− 0.706168i
$823$	25.6155i	0.892901i	0.894808	+	0.446451i	$0.147312\pi$
$823$	25.6155i	0.892901i	−0.894808	+	0.446451i	$0.852688\pi$
$824$	1.75379	0.0610961
$825$	0	0
$826$	−33.6155	−1.16963
$827$	1.75379i	0.0609852i	0.999535	+	0.0304926i	$0.00970760\pi$
$827$	1.75379i	0.0609852i	−0.999535	+	0.0304926i	$0.990292\pi$
$828$	− 1.43845i	− 0.0499895i
$829$	−24.4233	−0.848256	−0.424128	−	0.905602i	$-0.639419\pi$
$829$	−24.4233	−0.848256	−0.424128	+	0.905602i	$0.639419\pi$
$830$	0	0
$831$	5.36932	0.186260
$832$	2.00000i	0.0693375i
$833$	− 1.36932i	− 0.0474440i
$834$	−17.1231	−0.592925
$835$	0	0
$836$	−19.6847	−0.680808
$837$	− 1.00000i	− 0.0345651i
$838$	− 5.75379i	− 0.198761i
$839$	−6.06913	−0.209530	−0.104765	−	0.994497i	$-0.533409\pi$
$839$	−6.06913	−0.209530	−0.104765	+	0.994497i	$0.533409\pi$
$840$	0	0
$841$	21.7386	0.749608
$842$	− 14.4924i	− 0.499442i
$843$	4.87689i	0.167969i
$844$	23.6847	0.815260
$845$	0	0
$846$	5.12311	0.176136
$847$	− 11.3693i	− 0.390654i
$848$	− 7.43845i	− 0.255437i
$849$	0	0
$850$	0	0
$851$	−4.49242	−0.153998
$852$	7.68466i	0.263272i
$853$	− 47.7926i	− 1.63639i	−0.574942	−	0.818194i	$-0.694976\pi$
$853$	− 47.7926i	− 1.63639i	0.574942	−	0.818194i	$-0.305024\pi$
$854$	−15.3693	−0.525927
$855$	0	0
$856$	−2.56155	−0.0875521
$857$	29.2311i	0.998514i	0.866454	+	0.499257i	$0.166394\pi$
$857$	29.2311i	0.998514i	−0.866454	+	0.499257i	$0.833606\pi$
$858$	5.12311i	0.174900i
$859$	−26.7386	−0.912310	−0.456155	−	0.889900i	$-0.650774\pi$
$859$	−26.7386	−0.912310	−0.456155	+	0.889900i	$0.650774\pi$
$860$	0	0
$861$	−18.2462	−0.621829
$862$	− 30.2462i	− 1.03019i
$863$	− 39.5464i	− 1.34618i	−0.739563	−	0.673088i	$-0.764968\pi$
$863$	− 39.5464i	− 1.34618i	0.739563	−	0.673088i	$-0.235032\pi$
$864$	1.00000	0.0340207
$865$	0	0
$866$	35.3002	1.19955
$867$	7.24621i	0.246094i
$868$	− 2.56155i	− 0.0869448i
$869$	11.0540	0.374980
$870$	0	0
$871$	−30.7386	−1.04154
$872$	5.36932i	0.181828i
$873$	6.00000i	0.203069i
$874$	11.0540	0.373906
$875$	0	0
$876$	10.8078	0.365161
$877$	− 50.0000i	− 1.68838i	−0.536044	−	0.844190i	$-0.680082\pi$
$877$	− 50.0000i	− 1.68838i	0.536044	−	0.844190i	$-0.319918\pi$
$878$	− 9.61553i	− 0.324508i
$879$	26.4924	0.893567
$880$	0	0
$881$	9.50758	0.320318	0.160159	−	0.987091i	$-0.448799\pi$
$881$	9.50758	0.320318	0.160159	+	0.987091i	$0.448799\pi$
$882$	− 0.438447i	− 0.0147633i
$883$	− 30.4233i	− 1.02383i	−0.859038	−	0.511913i	$-0.828937\pi$
$883$	− 30.4233i	− 1.02383i	0.859038	−	0.511913i	$-0.171063\pi$
$884$	6.24621	0.210083
$885$	0	0
$886$	23.0540	0.774513
$887$	− 32.6307i	− 1.09563i	−0.836599	−	0.547816i	$-0.815460\pi$
$887$	− 32.6307i	− 1.09563i	0.836599	−	0.547816i	$-0.184540\pi$
$888$	− 3.12311i	− 0.104805i
$889$	0	0
$890$	0	0
$891$	2.56155	0.0858152
$892$	− 21.1231i	− 0.707254i
$893$	39.3693i	1.31744i
$894$	17.0540	0.570370
$895$	0	0
$896$	2.56155	0.0855755
$897$	− 2.87689i	− 0.0960567i
$898$	− 10.4924i	− 0.350137i
$899$	−7.12311	−0.237569
$900$	0	0
$901$	−23.2311	−0.773939
$902$	18.2462i	0.607532i
$903$	32.8078i	1.09177i
$904$	1.68466	0.0560309
$905$	0	0
$906$	0	0
$907$	24.0000i	0.796907i	0.917189	+	0.398453i	$0.130453\pi$
$907$	24.0000i	0.796907i	−0.917189	+	0.398453i	$0.869547\pi$
$908$	− 7.68466i	− 0.255024i
$909$	0.561553	0.0186255
$910$	0	0
$911$	−11.5076	−0.381263	−0.190632	−	0.981662i	$-0.561054\pi$
$911$	−11.5076	−0.381263	−0.190632	+	0.981662i	$0.561054\pi$
$912$	7.68466i	0.254464i
$913$	− 36.4924i	− 1.20772i
$914$	24.7386	0.818281
$915$	0	0
$916$	−16.5616	−0.547209
$917$	− 39.3693i	− 1.30009i
$918$	− 3.12311i	− 0.103078i
$919$	1.61553	0.0532914	0.0266457	−	0.999645i	$-0.491517\pi$
$919$	1.61553	0.0532914	0.0266457	+	0.999645i	$0.491517\pi$
$920$	0	0
$921$	0	0
$922$	9.36932i	0.308562i
$923$	15.3693i	0.505887i
$924$	6.56155	0.215859
$925$	0	0
$926$	30.7386	1.01013
$927$	− 1.75379i	− 0.0576020i
$928$	− 7.12311i	− 0.233827i
$929$	−7.43845	−0.244048	−0.122024	−	0.992527i	$-0.538938\pi$
$929$	−7.43845	−0.244048	−0.122024	+	0.992527i	$0.538938\pi$
$930$	0	0
$931$	3.36932	0.110425
$932$	− 17.0540i	− 0.558622i
$933$	− 1.75379i	− 0.0574165i
$934$	9.75379	0.319154
$935$	0	0
$936$	2.00000	0.0653720
$937$	41.3693i	1.35148i	0.737142	+	0.675738i	$0.236175\pi$
$937$	41.3693i	1.35148i	−0.737142	+	0.675738i	$0.763825\pi$
$938$	39.3693i	1.28545i
$939$	−10.0000	−0.326338
$940$	0	0
$941$	2.63068	0.0857578	0.0428789	−	0.999080i	$-0.486347\pi$
$941$	2.63068	0.0857578	0.0428789	+	0.999080i	$0.486347\pi$
$942$	− 17.0540i	− 0.555649i
$943$	− 10.2462i	− 0.333663i
$944$	13.1231	0.427121
$945$	0	0
$946$	32.8078	1.06667
$947$	− 9.12311i	− 0.296461i	−0.988953	−	0.148231i	$-0.952642\pi$
$947$	− 9.12311i	− 0.296461i	0.988953	−	0.148231i	$-0.0473578\pi$
$948$	− 4.31534i	− 0.140156i
$949$	21.6155	0.701670
$950$	0	0
$951$	−16.8769	−0.547271
$952$	− 8.00000i	− 0.259281i
$953$	− 16.1080i	− 0.521788i	−0.965367	−	0.260894i	$-0.915983\pi$
$953$	− 16.1080i	− 0.521788i	0.965367	−	0.260894i	$-0.0840173\pi$
$954$	−7.43845	−0.240829
$955$	0	0
$956$	24.0000	0.776215
$957$	− 18.2462i	− 0.589816i
$958$	− 10.5616i	− 0.341228i
$959$	−51.8617	−1.67470
$960$	0	0
$961$	1.00000	0.0322581
$962$	− 6.24621i	− 0.201386i
$963$	2.56155i	0.0825449i
$964$	−12.2462	−0.394424
$965$	0	0
$966$	−3.68466	−0.118552
$967$	− 42.2462i	− 1.35855i	−0.733885	−	0.679273i	$-0.762295\pi$
$967$	− 42.2462i	− 1.35855i	0.733885	−	0.679273i	$-0.237705\pi$
$968$	4.43845i	0.142657i
$969$	24.0000	0.770991
$970$	0	0
$971$	−37.1231	−1.19134	−0.595669	−	0.803230i	$-0.703113\pi$
$971$	−37.1231	−1.19134	−0.595669	+	0.803230i	$0.703113\pi$
$972$	− 1.00000i	− 0.0320750i
$973$	43.8617i	1.40614i
$974$	−24.0000	−0.769010
$975$	0	0
$976$	6.00000	0.192055
$977$	42.9848i	1.37521i	0.726086	+	0.687604i	$0.241337\pi$
$977$	42.9848i	1.37521i	−0.726086	+	0.687604i	$0.758663\pi$
$978$	− 15.3693i	− 0.491457i
$979$	35.0540	1.12033
$980$	0	0
$981$	5.36932	0.171429
$982$	− 25.9309i	− 0.827487i
$983$	− 40.9848i	− 1.30721i	−0.756834	−	0.653607i	$-0.773255\pi$
$983$	− 40.9848i	− 1.30721i	0.756834	−	0.653607i	$-0.226745\pi$
$984$	7.12311	0.227076
$985$	0	0
$986$	−22.2462	−0.708464
$987$	− 13.1231i	− 0.417713i
$988$	15.3693i	0.488963i
$989$	−18.4233	−0.585827
$990$	0	0
$991$	35.6847	1.13356	0.566780	−	0.823869i	$-0.308189\pi$
$991$	35.6847	1.13356	0.566780	+	0.823869i	$0.308189\pi$
$992$	1.00000i	0.0317500i
$993$	− 6.24621i	− 0.198218i
$994$	19.6847	0.624359
$995$	0	0
$996$	−14.2462	−0.451408
$997$	− 29.5076i	− 0.934514i	−0.884121	−	0.467257i	$-0.845242\pi$
$997$	− 29.5076i	− 0.934514i	0.884121	−	0.467257i	$-0.154758\pi$
$998$	− 20.0000i	− 0.633089i
$999$	−3.12311	−0.0988107

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
930.2.a.p.1.2	✓	2	5.3	odd	4
2790.2.a.be.1.2		2	15.8	even	4
4650.2.a.ce.1.1		2	5.2	odd	4
4650.2.d.bd.3349.1		4	5.4	even	2	inner
4650.2.d.bd.3349.4		4	1.1	even	1	trivial
7440.2.a.bl.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)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +2.56155i q^{7} -1.00000i q^{8} -1.00000 q^{9} +2.56155 q^{11} -1.00000i q^{12} -2.00000i q^{13} -2.56155 q^{14} +1.00000 q^{16} -3.12311i q^{17} -1.00000i q^{18} +7.68466 q^{19} -2.56155 q^{21} +2.56155i q^{22} -1.43845i q^{23} +1.00000 q^{24} +2.00000 q^{26} -1.00000i q^{27} -2.56155i q^{28} -7.12311 q^{29} +1.00000 q^{31} +1.00000i q^{32} +2.56155i q^{33} +3.12311 q^{34} +1.00000 q^{36} -3.12311i q^{37} +7.68466i q^{38} +2.00000 q^{39} +7.12311 q^{41} -2.56155i q^{42} -12.8078i q^{43} -2.56155 q^{44} +1.43845 q^{46} +5.12311i q^{47} +1.00000i q^{48} +0.438447 q^{49} +3.12311 q^{51} +2.00000i q^{52} -7.43845i q^{53} +1.00000 q^{54} +2.56155 q^{56} +7.68466i q^{57} -7.12311i q^{58} +13.1231 q^{59} +6.00000 q^{61} +1.00000i q^{62} -2.56155i q^{63} -1.00000 q^{64} -2.56155 q^{66} -15.3693i q^{67} +3.12311i q^{68} +1.43845 q^{69} -7.68466 q^{71} +1.00000i q^{72} +10.8078i q^{73} +3.12311 q^{74} -7.68466 q^{76} +6.56155i q^{77} +2.00000i q^{78} +4.31534 q^{79} +1.00000 q^{81} +7.12311i q^{82} -14.2462i q^{83} +2.56155 q^{84} +12.8078 q^{86} -7.12311i q^{87} -2.56155i q^{88} +13.6847 q^{89} +5.12311 q^{91} +1.43845i q^{92} +1.00000i q^{93} -5.12311 q^{94} -1.00000 q^{96} -6.00000i q^{97} +0.438447i q^{98} -2.56155 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{4} - 4 q^{6} - 4 q^{9} + 2 q^{11} - 2 q^{14} + 4 q^{16} + 6 q^{19} - 2 q^{21} + 4 q^{24} + 8 q^{26} - 12 q^{29} + 4 q^{31} - 4 q^{34} + 4 q^{36} + 8 q^{39} + 12 q^{41} - 2 q^{44} + 14 q^{46}+ \cdots - 2 q^{99}+O(q^{100}) \) 4 * q - 4 * q^4 - 4 * q^6 - 4 * q^9 + 2 * q^11 - 2 * q^14 + 4 * q^16 + 6 * 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 + 12 * q^41 - 2 * q^44 + 14 * q^46 + 10 * q^49 - 4 * q^51 + 4 * q^54 + 2 * q^56 + 36 * q^59 + 24 * q^61 - 4 * q^64 - 2 * q^66 + 14 * q^69 - 6 * q^71 - 4 * q^74 - 6 * q^76 + 42 * q^79 + 4 * q^81 + 2 * q^84 + 10 * q^86 + 30 * q^89 + 4 * q^91 - 4 * q^94 - 4 * q^96 - 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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