LMFDB - Embedded newform 510.2.d.b.409.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [510,2,Mod(409,510)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(510, 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("510.409");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 510 = 2 \cdot 3 \cdot 5 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	510.d (of order $2$, degree $1$, 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:	$4.07237050309$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 3x^{2} + 1 $ x^4 + 3*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			409.4
Root			$-1.61803i$ of defining polynomial
Character	$\chi$	$=$	510.409
Dual form			510.2.d.b.409.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +2.23607i q^{5} -1.00000 q^{6} +3.23607i q^{7} -1.00000i q^{8} -1.00000 q^{9} -2.23607 q^{10} +2.00000 q^{11} -1.00000i q^{12} -0.763932i q^{13} -3.23607 q^{14} -2.23607 q^{15} +1.00000 q^{16} +1.00000i q^{17} -1.00000i q^{18} -2.47214 q^{19} -2.23607i q^{20} -3.23607 q^{21} +2.00000i q^{22} -4.00000i q^{23} +1.00000 q^{24} -5.00000 q^{25} +0.763932 q^{26} -1.00000i q^{27} -3.23607i q^{28} +4.00000 q^{29} -2.23607i q^{30} -6.47214 q^{31} +1.00000i q^{32} +2.00000i q^{33} -1.00000 q^{34} -7.23607 q^{35} +1.00000 q^{36} +2.00000i q^{37} -2.47214i q^{38} +0.763932 q^{39} +2.23607 q^{40} +5.70820 q^{41} -3.23607i q^{42} +10.1803i q^{43} -2.00000 q^{44} -2.23607i q^{45} +4.00000 q^{46} +1.52786i q^{47} +1.00000i q^{48} -3.47214 q^{49} -5.00000i q^{50} -1.00000 q^{51} +0.763932i q^{52} -6.94427i q^{53} +1.00000 q^{54} +4.47214i q^{55} +3.23607 q^{56} -2.47214i q^{57} +4.00000i q^{58} +1.70820 q^{59} +2.23607 q^{60} -4.47214 q^{61} -6.47214i q^{62} -3.23607i q^{63} -1.00000 q^{64} +1.70820 q^{65} -2.00000 q^{66} +11.7082i q^{67} -1.00000i q^{68} +4.00000 q^{69} -7.23607i q^{70} +6.76393 q^{71} +1.00000i q^{72} -13.2361i q^{73} -2.00000 q^{74} -5.00000i q^{75} +2.47214 q^{76} +6.47214i q^{77} +0.763932i q^{78} +16.9443 q^{79} +2.23607i q^{80} +1.00000 q^{81} +5.70820i q^{82} -1.52786i q^{83} +3.23607 q^{84} -2.23607 q^{85} -10.1803 q^{86} +4.00000i q^{87} -2.00000i q^{88} -3.52786 q^{89} +2.23607 q^{90} +2.47214 q^{91} +4.00000i q^{92} -6.47214i q^{93} -1.52786 q^{94} -5.52786i q^{95} -1.00000 q^{96} +11.7082i q^{97} -3.47214i q^{98} -2.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{4} - 4 q^{6} - 4 q^{9} + 8 q^{11} - 4 q^{14} + 4 q^{16} + 8 q^{19} - 4 q^{21} + 4 q^{24} - 20 q^{25} + 12 q^{26} + 16 q^{29} - 8 q^{31} - 4 q^{34} - 20 q^{35} + 4 q^{36} + 12 q^{39} - 4 q^{41}+ \cdots - 8 q^{99}+O(q^{100}) $ 4 * q - 4 * q^4 - 4 * q^6 - 4 * q^9 + 8 * q^11 - 4 * q^14 + 4 * q^16 + 8 * q^19 - 4 * q^21 + 4 * q^24 - 20 * q^25 + 12 * q^26 + 16 * q^29 - 8 * q^31 - 4 * q^34 - 20 * q^35 + 4 * q^36 + 12 * q^39 - 4 * q^41 - 8 * q^44 + 16 * q^46 + 4 * q^49 - 4 * q^51 + 4 * q^54 + 4 * q^56 - 20 * q^59 - 4 * q^64 - 20 * q^65 - 8 * q^66 + 16 * q^69 + 36 * q^71 - 8 * q^74 - 8 * q^76 + 32 * q^79 + 4 * q^81 + 4 * q^84 + 4 * q^86 - 32 * q^89 - 8 * q^91 - 24 * q^94 - 4 * q^96 - 8 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/510\mathbb{Z}\right)^\times$.

$n$	$241$	$307$	$341$
$\chi(n)$	$1$	$-1$	$1$

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
$5$	2.23607i	1.00000i
$5$	2.23607i	1.00000i
$6$	−1.00000	−0.408248
$7$	3.23607i	1.22312i	0.791199	+	0.611559i	$0.209457\pi$
$7$	3.23607i	1.22312i	−0.791199	+	0.611559i	$0.790543\pi$
$8$	− 1.00000i	− 0.353553i
$9$	−1.00000	−0.333333
$10$	−2.23607	−0.707107
$11$	2.00000	0.603023	0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000	0.603023	0.301511	+	0.953463i	$0.402509\pi$
$12$	− 1.00000i	− 0.288675i
$13$	− 0.763932i	− 0.211877i	−0.994373	−	0.105938i	$-0.966215\pi$
$13$	− 0.763932i	− 0.211877i	0.994373	−	0.105938i	$-0.0337846\pi$
$14$	−3.23607	−0.864876
$15$	−2.23607	−0.577350
$16$	1.00000	0.250000
$17$	1.00000i	0.242536i
$17$	1.00000i	0.242536i
$18$	− 1.00000i	− 0.235702i
$19$	−2.47214	−0.567147	−0.283573	−	0.958951i	$-0.591520\pi$
$19$	−2.47214	−0.567147	−0.283573	+	0.958951i	$0.591520\pi$
$20$	− 2.23607i	− 0.500000i
$21$	−3.23607	−0.706168
$22$	2.00000i	0.426401i
$23$	− 4.00000i	− 0.834058i	−0.908893	−	0.417029i	$-0.863071\pi$
$23$	− 4.00000i	− 0.834058i	0.908893	−	0.417029i	$-0.136929\pi$
$24$	1.00000	0.204124
$25$	−5.00000	−1.00000
$26$	0.763932	0.149819
$27$	− 1.00000i	− 0.192450i
$28$	− 3.23607i	− 0.611559i
$29$	4.00000	0.742781	0.371391	−	0.928477i	$-0.378881\pi$
$29$	4.00000	0.742781	0.371391	+	0.928477i	$0.378881\pi$
$30$	− 2.23607i	− 0.408248i
$31$	−6.47214	−1.16243	−0.581215	−	0.813750i	$-0.697422\pi$
$31$	−6.47214	−1.16243	−0.581215	+	0.813750i	$0.697422\pi$
$32$	1.00000i	0.176777i
$33$	2.00000i	0.348155i
$34$	−1.00000	−0.171499
$35$	−7.23607	−1.22312
$36$	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	$0.947432\pi$
$38$	− 2.47214i	− 0.401033i
$39$	0.763932	0.122327
$40$	2.23607	0.353553
$41$	5.70820	0.891472	0.445736	−	0.895165i	$-0.352942\pi$
$41$	5.70820	0.891472	0.445736	+	0.895165i	$0.352942\pi$
$42$	− 3.23607i	− 0.499336i
$43$	10.1803i	1.55249i	0.630433	+	0.776244i	$0.282877\pi$
$43$	10.1803i	1.55249i	−0.630433	+	0.776244i	$0.717123\pi$
$44$	−2.00000	−0.301511
$45$	− 2.23607i	− 0.333333i
$46$	4.00000	0.589768
$47$	1.52786i	0.222862i	0.993772	+	0.111431i	$0.0355434\pi$
$47$	1.52786i	0.222862i	−0.993772	+	0.111431i	$0.964457\pi$
$48$	1.00000i	0.144338i
$49$	−3.47214	−0.496019
$50$	− 5.00000i	− 0.707107i
$51$	−1.00000	−0.140028
$52$	0.763932i	0.105938i
$53$	− 6.94427i	− 0.953869i	−0.878939	−	0.476935i	$-0.841748\pi$
$53$	− 6.94427i	− 0.953869i	0.878939	−	0.476935i	$-0.158252\pi$
$54$	1.00000	0.136083
$55$	4.47214i	0.603023i
$56$	3.23607	0.432438
$57$	− 2.47214i	− 0.327442i
$58$	4.00000i	0.525226i
$59$	1.70820	0.222389	0.111195	−	0.993799i	$-0.464532\pi$
$59$	1.70820	0.222389	0.111195	+	0.993799i	$0.464532\pi$
$60$	2.23607	0.288675
$61$	−4.47214	−0.572598	−0.286299	−	0.958140i	$-0.592425\pi$
$61$	−4.47214	−0.572598	−0.286299	+	0.958140i	$0.592425\pi$
$62$	− 6.47214i	− 0.821962i
$63$	− 3.23607i	− 0.407706i
$64$	−1.00000	−0.125000
$65$	1.70820	0.211877
$66$	−2.00000	−0.246183
$67$	11.7082i	1.43038i	0.698928	+	0.715192i	$0.253661\pi$
$67$	11.7082i	1.43038i	−0.698928	+	0.715192i	$0.746339\pi$
$68$	− 1.00000i	− 0.121268i
$69$	4.00000	0.481543
$70$	− 7.23607i	− 0.864876i
$71$	6.76393	0.802731	0.401366	−	0.915918i	$-0.368536\pi$
$71$	6.76393	0.802731	0.401366	+	0.915918i	$0.368536\pi$
$72$	1.00000i	0.117851i
$73$	− 13.2361i	− 1.54916i	−0.632473	−	0.774582i	$-0.717960\pi$
$73$	− 13.2361i	− 1.54916i	0.632473	−	0.774582i	$-0.282040\pi$
$74$	−2.00000	−0.232495
$75$	− 5.00000i	− 0.577350i
$76$	2.47214	0.283573
$77$	6.47214i	0.737568i
$78$	0.763932i	0.0864983i
$79$	16.9443	1.90638	0.953190	−	0.302373i	$-0.0977787\pi$
$79$	16.9443	1.90638	0.953190	+	0.302373i	$0.0977787\pi$
$80$	2.23607i	0.250000i
$81$	1.00000	0.111111
$82$	5.70820i	0.630366i
$83$	− 1.52786i	− 0.167705i	−0.996478	−	0.0838524i	$-0.973278\pi$
$83$	− 1.52786i	− 0.167705i	0.996478	−	0.0838524i	$-0.0267224\pi$
$84$	3.23607	0.353084
$85$	−2.23607	−0.242536
$86$	−10.1803	−1.09777
$87$	4.00000i	0.428845i
$88$	− 2.00000i	− 0.213201i
$89$	−3.52786	−0.373953	−0.186976	−	0.982364i	$-0.559869\pi$
$89$	−3.52786	−0.373953	−0.186976	+	0.982364i	$0.559869\pi$
$90$	2.23607	0.235702
$91$	2.47214	0.259150
$92$	4.00000i	0.417029i
$93$	− 6.47214i	− 0.671129i
$94$	−1.52786	−0.157587
$95$	− 5.52786i	− 0.567147i
$96$	−1.00000	−0.102062
$97$	11.7082i	1.18879i	0.804174	+	0.594394i	$0.202608\pi$
$97$	11.7082i	1.18879i	−0.804174	+	0.594394i	$0.797392\pi$
$98$	− 3.47214i	− 0.350739i
$99$	−2.00000	−0.201008
$100$	5.00000	0.500000
$101$	11.7082	1.16501	0.582505	−	0.812827i	$-0.302073\pi$
$101$	11.7082	1.16501	0.582505	+	0.812827i	$0.302073\pi$
$102$	− 1.00000i	− 0.0990148i
$103$	6.94427i	0.684239i	0.939656	+	0.342120i	$0.111145\pi$
$103$	6.94427i	0.684239i	−0.939656	+	0.342120i	$0.888855\pi$
$104$	−0.763932	−0.0749097
$105$	− 7.23607i	− 0.706168i
$106$	6.94427	0.674487
$107$	8.00000i	0.773389i	0.922208	+	0.386695i	$0.126383\pi$
$107$	8.00000i	0.773389i	−0.922208	+	0.386695i	$0.873617\pi$
$108$	1.00000i	0.0962250i
$109$	16.4721	1.57774	0.788872	−	0.614557i	$-0.210665\pi$
$109$	16.4721	1.57774	0.788872	+	0.614557i	$0.210665\pi$
$110$	−4.47214	−0.426401
$111$	−2.00000	−0.189832
$112$	3.23607i	0.305780i
$113$	10.0000i	0.940721i	0.882474	+	0.470360i	$0.155876\pi$
$113$	10.0000i	0.940721i	−0.882474	+	0.470360i	$0.844124\pi$
$114$	2.47214	0.231537
$115$	8.94427	0.834058
$116$	−4.00000	−0.371391
$117$	0.763932i	0.0706255i
$118$	1.70820i	0.157253i
$119$	−3.23607	−0.296650
$120$	2.23607i	0.204124i
$121$	−7.00000	−0.636364
$122$	− 4.47214i	− 0.404888i
$123$	5.70820i	0.514691i
$124$	6.47214	0.581215
$125$	− 11.1803i	− 1.00000i
$126$	3.23607	0.288292
$127$	14.0000i	1.24230i	0.783692	+	0.621150i	$0.213334\pi$
$127$	14.0000i	1.24230i	−0.783692	+	0.621150i	$0.786666\pi$
$128$	− 1.00000i	− 0.0883883i
$129$	−10.1803	−0.896329
$130$	1.70820i	0.149819i
$131$	3.52786	0.308231	0.154115	−	0.988053i	$-0.450747\pi$
$131$	3.52786	0.308231	0.154115	+	0.988053i	$0.450747\pi$
$132$	− 2.00000i	− 0.174078i
$133$	− 8.00000i	− 0.693688i
$134$	−11.7082	−1.01143
$135$	2.23607	0.192450
$136$	1.00000	0.0857493
$137$	− 2.00000i	− 0.170872i	−0.996344	−	0.0854358i	$-0.972772\pi$
$137$	− 2.00000i	− 0.170872i	0.996344	−	0.0854358i	$-0.0272282\pi$
$138$	4.00000i	0.340503i
$139$	−10.4721	−0.888235	−0.444117	−	0.895969i	$-0.646483\pi$
$139$	−10.4721	−0.888235	−0.444117	+	0.895969i	$0.646483\pi$
$140$	7.23607	0.611559
$141$	−1.52786	−0.128669
$142$	6.76393i	0.567617i
$143$	− 1.52786i	− 0.127766i
$144$	−1.00000	−0.0833333
$145$	8.94427i	0.742781i
$146$	13.2361	1.09542
$147$	− 3.47214i	− 0.286377i
$148$	− 2.00000i	− 0.164399i
$149$	10.1803	0.834006	0.417003	−	0.908905i	$-0.363080\pi$
$149$	10.1803	0.834006	0.417003	+	0.908905i	$0.363080\pi$
$150$	5.00000	0.408248
$151$	−4.00000	−0.325515	−0.162758	−	0.986666i	$-0.552039\pi$
$151$	−4.00000	−0.325515	−0.162758	+	0.986666i	$0.552039\pi$
$152$	2.47214i	0.200517i
$153$	− 1.00000i	− 0.0808452i
$154$	−6.47214	−0.521540
$155$	− 14.4721i	− 1.16243i
$156$	−0.763932	−0.0611635
$157$	− 12.1803i	− 0.972097i	−0.873932	−	0.486048i	$-0.838438\pi$
$157$	− 12.1803i	− 0.972097i	0.873932	−	0.486048i	$-0.161562\pi$
$158$	16.9443i	1.34801i
$159$	6.94427	0.550717
$160$	−2.23607	−0.176777
$161$	12.9443	1.02015
$162$	1.00000i	0.0785674i
$163$	13.5279i	1.05958i	0.848128	+	0.529792i	$0.177730\pi$
$163$	13.5279i	1.05958i	−0.848128	+	0.529792i	$0.822270\pi$
$164$	−5.70820	−0.445736
$165$	−4.47214	−0.348155
$166$	1.52786	0.118585
$167$	21.8885i	1.69379i	0.531763	+	0.846893i	$0.321530\pi$
$167$	21.8885i	1.69379i	−0.531763	+	0.846893i	$0.678470\pi$
$168$	3.23607i	0.249668i
$169$	12.4164	0.955108
$170$	− 2.23607i	− 0.171499i
$171$	2.47214	0.189049
$172$	− 10.1803i	− 0.776244i
$173$	− 19.8885i	− 1.51210i	−0.654515	−	0.756049i	$-0.727127\pi$
$173$	− 19.8885i	− 1.51210i	0.654515	−	0.756049i	$-0.272873\pi$
$174$	−4.00000	−0.303239
$175$	− 16.1803i	− 1.22312i
$176$	2.00000	0.150756
$177$	1.70820i	0.128396i
$178$	− 3.52786i	− 0.264425i
$179$	−5.70820	−0.426651	−0.213326	−	0.976981i	$-0.568430\pi$
$179$	−5.70820	−0.426651	−0.213326	+	0.976981i	$0.568430\pi$
$180$	2.23607i	0.166667i
$181$	−8.47214	−0.629729	−0.314864	−	0.949137i	$-0.601959\pi$
$181$	−8.47214	−0.629729	−0.314864	+	0.949137i	$0.601959\pi$
$182$	2.47214i	0.183247i
$183$	− 4.47214i	− 0.330590i
$184$	−4.00000	−0.294884
$185$	−4.47214	−0.328798
$186$	6.47214	0.474560
$187$	2.00000i	0.146254i
$188$	− 1.52786i	− 0.111431i
$189$	3.23607	0.235389
$190$	5.52786	0.401033
$191$	14.4721	1.04717	0.523584	−	0.851974i	$-0.324595\pi$
$191$	14.4721	1.04717	0.523584	+	0.851974i	$0.324595\pi$
$192$	− 1.00000i	− 0.0721688i
$193$	− 25.5967i	− 1.84249i	−0.388978	−	0.921247i	$-0.627172\pi$
$193$	− 25.5967i	− 1.84249i	0.388978	−	0.921247i	$-0.372828\pi$
$194$	−11.7082	−0.840600
$195$	1.70820i	0.122327i
$196$	3.47214	0.248010
$197$	− 19.8885i	− 1.41700i	−0.705711	−	0.708500i	$-0.749372\pi$
$197$	− 19.8885i	− 1.41700i	0.705711	−	0.708500i	$-0.250628\pi$
$198$	− 2.00000i	− 0.142134i
$199$	−8.94427	−0.634043	−0.317021	−	0.948418i	$-0.602683\pi$
$199$	−8.94427	−0.634043	−0.317021	+	0.948418i	$0.602683\pi$
$200$	5.00000i	0.353553i
$201$	−11.7082	−0.825833
$202$	11.7082i	0.823786i
$203$	12.9443i	0.908510i
$204$	1.00000	0.0700140
$205$	12.7639i	0.891472i
$206$	−6.94427	−0.483830
$207$	4.00000i	0.278019i
$208$	− 0.763932i	− 0.0529692i
$209$	−4.94427	−0.342002
$210$	7.23607	0.499336
$211$	−7.05573	−0.485736	−0.242868	−	0.970059i	$-0.578088\pi$
$211$	−7.05573	−0.485736	−0.242868	+	0.970059i	$0.578088\pi$
$212$	6.94427i	0.476935i
$213$	6.76393i	0.463457i
$214$	−8.00000	−0.546869
$215$	−22.7639	−1.55249
$216$	−1.00000	−0.0680414
$217$	− 20.9443i	− 1.42179i
$218$	16.4721i	1.11563i
$219$	13.2361	0.894411
$220$	− 4.47214i	− 0.301511i
$221$	0.763932	0.0513876
$222$	− 2.00000i	− 0.134231i
$223$	− 26.9443i	− 1.80432i	−0.431400	−	0.902161i	$-0.641980\pi$
$223$	− 26.9443i	− 1.80432i	0.431400	−	0.902161i	$-0.358020\pi$
$224$	−3.23607	−0.216219
$225$	5.00000	0.333333
$226$	−10.0000	−0.665190
$227$	− 25.8885i	− 1.71828i	−0.511738	−	0.859142i	$-0.670998\pi$
$227$	− 25.8885i	− 1.71828i	0.511738	−	0.859142i	$-0.329002\pi$
$228$	2.47214i	0.163721i
$229$	−5.41641	−0.357926	−0.178963	−	0.983856i	$-0.557274\pi$
$229$	−5.41641	−0.357926	−0.178963	+	0.983856i	$0.557274\pi$
$230$	8.94427i	0.589768i
$231$	−6.47214	−0.425835
$232$	− 4.00000i	− 0.262613i
$233$	21.4164i	1.40304i	0.712652	+	0.701518i	$0.247494\pi$
$233$	21.4164i	1.40304i	−0.712652	+	0.701518i	$0.752506\pi$
$234$	−0.763932	−0.0499398
$235$	−3.41641	−0.222862
$236$	−1.70820	−0.111195
$237$	16.9443i	1.10065i
$238$	− 3.23607i	− 0.209763i
$239$	−13.8885	−0.898375	−0.449188	−	0.893437i	$-0.648287\pi$
$239$	−13.8885	−0.898375	−0.449188	+	0.893437i	$0.648287\pi$
$240$	−2.23607	−0.144338
$241$	−17.4164	−1.12189	−0.560945	−	0.827853i	$-0.689562\pi$
$241$	−17.4164	−1.12189	−0.560945	+	0.827853i	$0.689562\pi$
$242$	− 7.00000i	− 0.449977i
$243$	1.00000i	0.0641500i
$244$	4.47214	0.286299
$245$	− 7.76393i	− 0.496019i
$246$	−5.70820	−0.363942
$247$	1.88854i	0.120165i
$248$	6.47214i	0.410981i
$249$	1.52786	0.0968244
$250$	11.1803	0.707107
$251$	12.1803	0.768816	0.384408	−	0.923163i	$-0.374406\pi$
$251$	12.1803	0.768816	0.384408	+	0.923163i	$0.374406\pi$
$252$	3.23607i	0.203853i
$253$	− 8.00000i	− 0.502956i
$254$	−14.0000	−0.878438
$255$	− 2.23607i	− 0.140028i
$256$	1.00000	0.0625000
$257$	18.0000i	1.12281i	0.827541	+	0.561405i	$0.189739\pi$
$257$	18.0000i	1.12281i	−0.827541	+	0.561405i	$0.810261\pi$
$258$	− 10.1803i	− 0.633800i
$259$	−6.47214	−0.402159
$260$	−1.70820	−0.105938
$261$	−4.00000	−0.247594
$262$	3.52786i	0.217952i
$263$	4.00000i	0.246651i	0.992366	+	0.123325i	$0.0393559\pi$
$263$	4.00000i	0.246651i	−0.992366	+	0.123325i	$0.960644\pi$
$264$	2.00000	0.123091
$265$	15.5279	0.953869
$266$	8.00000	0.490511
$267$	− 3.52786i	− 0.215902i
$268$	− 11.7082i	− 0.715192i
$269$	24.9443	1.52088	0.760440	−	0.649409i	$-0.224984\pi$
$269$	24.9443	1.52088	0.760440	+	0.649409i	$0.224984\pi$
$270$	2.23607i	0.136083i
$271$	4.00000	0.242983	0.121491	−	0.992592i	$-0.461232\pi$
$271$	4.00000	0.242983	0.121491	+	0.992592i	$0.461232\pi$
$272$	1.00000i	0.0606339i
$273$	2.47214i	0.149620i
$274$	2.00000	0.120824
$275$	−10.0000	−0.603023
$276$	−4.00000	−0.240772
$277$	− 18.3607i	− 1.10319i	−0.834113	−	0.551593i	$-0.814020\pi$
$277$	− 18.3607i	− 1.10319i	0.834113	−	0.551593i	$-0.185980\pi$
$278$	− 10.4721i	− 0.628077i
$279$	6.47214	0.387477
$280$	7.23607i	0.432438i
$281$	−25.4164	−1.51622	−0.758108	−	0.652129i	$-0.773876\pi$
$281$	−25.4164	−1.51622	−0.758108	+	0.652129i	$0.773876\pi$
$282$	− 1.52786i	− 0.0909830i
$283$	− 9.88854i	− 0.587813i	−0.955834	−	0.293906i	$-0.905045\pi$
$283$	− 9.88854i	− 0.587813i	0.955834	−	0.293906i	$-0.0949554\pi$
$284$	−6.76393	−0.401366
$285$	5.52786	0.327442
$286$	1.52786	0.0903445
$287$	18.4721i	1.09038i
$288$	− 1.00000i	− 0.0589256i
$289$	−1.00000	−0.0588235
$290$	−8.94427	−0.525226
$291$	−11.7082	−0.686347
$292$	13.2361i	0.774582i
$293$	− 2.58359i	− 0.150935i	−0.997148	−	0.0754675i	$-0.975955\pi$
$293$	− 2.58359i	− 0.150935i	0.997148	−	0.0754675i	$-0.0240449\pi$
$294$	3.47214	0.202499
$295$	3.81966i	0.222389i
$296$	2.00000	0.116248
$297$	− 2.00000i	− 0.116052i
$298$	10.1803i	0.589731i
$299$	−3.05573	−0.176717
$300$	5.00000i	0.288675i
$301$	−32.9443	−1.89888
$302$	− 4.00000i	− 0.230174i
$303$	11.7082i	0.672619i
$304$	−2.47214	−0.141787
$305$	− 10.0000i	− 0.572598i
$306$	1.00000	0.0571662
$307$	− 18.1803i	− 1.03761i	−0.854894	−	0.518803i	$-0.826378\pi$
$307$	− 18.1803i	− 1.03761i	0.854894	−	0.518803i	$-0.173622\pi$
$308$	− 6.47214i	− 0.368784i
$309$	−6.94427	−0.395046
$310$	14.4721	0.821962
$311$	−29.5967	−1.67828	−0.839139	−	0.543917i	$-0.816941\pi$
$311$	−29.5967	−1.67828	−0.839139	+	0.543917i	$0.816941\pi$
$312$	− 0.763932i	− 0.0432491i
$313$	8.29180i	0.468680i	0.972155	+	0.234340i	$0.0752929\pi$
$313$	8.29180i	0.468680i	−0.972155	+	0.234340i	$0.924707\pi$
$314$	12.1803	0.687376
$315$	7.23607	0.407706
$316$	−16.9443	−0.953190
$317$	− 23.8885i	− 1.34171i	−0.741587	−	0.670857i	$-0.765926\pi$
$317$	− 23.8885i	− 1.34171i	0.741587	−	0.670857i	$-0.234074\pi$
$318$	6.94427i	0.389415i
$319$	8.00000	0.447914
$320$	− 2.23607i	− 0.125000i
$321$	−8.00000	−0.446516
$322$	12.9443i	0.721356i
$323$	− 2.47214i	− 0.137553i
$324$	−1.00000	−0.0555556
$325$	3.81966i	0.211877i
$326$	−13.5279	−0.749239
$327$	16.4721i	0.910911i
$328$	− 5.70820i	− 0.315183i
$329$	−4.94427	−0.272587
$330$	− 4.47214i	− 0.246183i
$331$	23.4164	1.28708	0.643541	−	0.765412i	$-0.277465\pi$
$331$	23.4164	1.28708	0.643541	+	0.765412i	$0.277465\pi$
$332$	1.52786i	0.0838524i
$333$	− 2.00000i	− 0.109599i
$334$	−21.8885	−1.19769
$335$	−26.1803	−1.43038
$336$	−3.23607	−0.176542
$337$	2.18034i	0.118771i	0.998235	+	0.0593853i	$0.0189141\pi$
$337$	2.18034i	0.118771i	−0.998235	+	0.0593853i	$0.981086\pi$
$338$	12.4164i	0.675364i
$339$	−10.0000	−0.543125
$340$	2.23607	0.121268
$341$	−12.9443	−0.700972
$342$	2.47214i	0.133678i
$343$	11.4164i	0.616428i
$344$	10.1803	0.548887
$345$	8.94427i	0.481543i
$346$	19.8885	1.06921
$347$	30.8328i	1.65519i	0.561324	+	0.827596i	$0.310292\pi$
$347$	30.8328i	1.65519i	−0.561324	+	0.827596i	$0.689708\pi$
$348$	− 4.00000i	− 0.214423i
$349$	19.5279	1.04530	0.522651	−	0.852547i	$-0.324943\pi$
$349$	19.5279	1.04530	0.522651	+	0.852547i	$0.324943\pi$
$350$	16.1803	0.864876
$351$	−0.763932	−0.0407757
$352$	2.00000i	0.106600i
$353$	− 2.94427i	− 0.156708i	−0.996926	−	0.0783539i	$-0.975034\pi$
$353$	− 2.94427i	− 0.156708i	0.996926	−	0.0783539i	$-0.0249664\pi$
$354$	−1.70820	−0.0907900
$355$	15.1246i	0.802731i
$356$	3.52786	0.186976
$357$	− 3.23607i	− 0.171271i
$358$	− 5.70820i	− 0.301688i
$359$	18.4721	0.974922	0.487461	−	0.873145i	$-0.337923\pi$
$359$	18.4721	0.974922	0.487461	+	0.873145i	$0.337923\pi$
$360$	−2.23607	−0.117851
$361$	−12.8885	−0.678344
$362$	− 8.47214i	− 0.445286i
$363$	− 7.00000i	− 0.367405i
$364$	−2.47214	−0.129575
$365$	29.5967	1.54916
$366$	4.47214	0.233762
$367$	− 28.1803i	− 1.47100i	−0.677524	−	0.735501i	$-0.736947\pi$
$367$	− 28.1803i	− 1.47100i	0.677524	−	0.735501i	$-0.263053\pi$
$368$	− 4.00000i	− 0.208514i
$369$	−5.70820	−0.297157
$370$	− 4.47214i	− 0.232495i
$371$	22.4721	1.16670
$372$	6.47214i	0.335565i
$373$	− 3.23607i	− 0.167557i	−0.996484	−	0.0837786i	$-0.973301\pi$
$373$	− 3.23607i	− 0.167557i	0.996484	−	0.0837786i	$-0.0266989\pi$
$374$	−2.00000	−0.103418
$375$	11.1803	0.577350
$376$	1.52786	0.0787936
$377$	− 3.05573i	− 0.157378i
$378$	3.23607i	0.166445i
$379$	10.4721	0.537917	0.268959	−	0.963152i	$-0.413320\pi$
$379$	10.4721	0.537917	0.268959	+	0.963152i	$0.413320\pi$
$380$	5.52786i	0.283573i
$381$	−14.0000	−0.717242
$382$	14.4721i	0.740459i
$383$	17.8885i	0.914062i	0.889451	+	0.457031i	$0.151087\pi$
$383$	17.8885i	0.914062i	−0.889451	+	0.457031i	$0.848913\pi$
$384$	1.00000	0.0510310
$385$	−14.4721	−0.737568
$386$	25.5967	1.30284
$387$	− 10.1803i	− 0.517496i
$388$	− 11.7082i	− 0.594394i
$389$	2.18034	0.110548	0.0552738	−	0.998471i	$-0.482397\pi$
$389$	2.18034	0.110548	0.0552738	+	0.998471i	$0.482397\pi$
$390$	−1.70820	−0.0864983
$391$	4.00000	0.202289
$392$	3.47214i	0.175369i
$393$	3.52786i	0.177957i
$394$	19.8885	1.00197
$395$	37.8885i	1.90638i
$396$	2.00000	0.100504
$397$	− 31.8885i	− 1.60044i	−0.599706	−	0.800220i	$-0.704716\pi$
$397$	− 31.8885i	− 1.60044i	0.599706	−	0.800220i	$-0.295284\pi$
$398$	− 8.94427i	− 0.448336i
$399$	8.00000	0.400501
$400$	−5.00000	−0.250000
$401$	22.0689	1.10207	0.551034	−	0.834483i	$-0.314234\pi$
$401$	22.0689	1.10207	0.551034	+	0.834483i	$0.314234\pi$
$402$	− 11.7082i	− 0.583952i
$403$	4.94427i	0.246292i
$404$	−11.7082	−0.582505
$405$	2.23607i	0.111111i
$406$	−12.9443	−0.642413
$407$	4.00000i	0.198273i
$408$	1.00000i	0.0495074i
$409$	18.3607	0.907877	0.453939	−	0.891033i	$-0.350019\pi$
$409$	18.3607	0.907877	0.453939	+	0.891033i	$0.350019\pi$
$410$	−12.7639	−0.630366
$411$	2.00000	0.0986527
$412$	− 6.94427i	− 0.342120i
$413$	5.52786i	0.272008i
$414$	−4.00000	−0.196589
$415$	3.41641	0.167705
$416$	0.763932	0.0374548
$417$	− 10.4721i	− 0.512823i
$418$	− 4.94427i	− 0.241832i
$419$	38.3607	1.87404	0.937021	−	0.349273i	$-0.113571\pi$
$419$	38.3607	1.87404	0.937021	+	0.349273i	$0.113571\pi$
$420$	7.23607i	0.353084i
$421$	−10.5836	−0.515813	−0.257906	−	0.966170i	$-0.583033\pi$
$421$	−10.5836	−0.515813	−0.257906	+	0.966170i	$0.583033\pi$
$422$	− 7.05573i	− 0.343467i
$423$	− 1.52786i	− 0.0742873i
$424$	−6.94427	−0.337244
$425$	− 5.00000i	− 0.242536i
$426$	−6.76393	−0.327714
$427$	− 14.4721i	− 0.700356i
$428$	− 8.00000i	− 0.386695i
$429$	1.52786	0.0737660
$430$	− 22.7639i	− 1.09777i
$431$	−17.5967	−0.847606	−0.423803	−	0.905755i	$-0.639305\pi$
$431$	−17.5967	−0.847606	−0.423803	+	0.905755i	$0.639305\pi$
$432$	− 1.00000i	− 0.0481125i
$433$	16.3607i	0.786244i	0.919486	+	0.393122i	$0.128605\pi$
$433$	16.3607i	0.786244i	−0.919486	+	0.393122i	$0.871395\pi$
$434$	20.9443	1.00536
$435$	−8.94427	−0.428845
$436$	−16.4721	−0.788872
$437$	9.88854i	0.473033i
$438$	13.2361i	0.632444i
$439$	−22.4721	−1.07254	−0.536268	−	0.844048i	$-0.680166\pi$
$439$	−22.4721	−1.07254	−0.536268	+	0.844048i	$0.680166\pi$
$440$	4.47214	0.213201
$441$	3.47214	0.165340
$442$	0.763932i	0.0363365i
$443$	− 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
$445$	− 7.88854i	− 0.373953i
$446$	26.9443	1.27585
$447$	10.1803i	0.481514i
$448$	− 3.23607i	− 0.152890i
$449$	31.5967	1.49114	0.745571	−	0.666426i	$-0.232177\pi$
$449$	31.5967	1.49114	0.745571	+	0.666426i	$0.232177\pi$
$450$	5.00000i	0.235702i
$451$	11.4164	0.537578
$452$	− 10.0000i	− 0.470360i
$453$	− 4.00000i	− 0.187936i
$454$	25.8885	1.21501
$455$	5.52786i	0.259150i
$456$	−2.47214	−0.115768
$457$	28.3607i	1.32666i	0.748328	+	0.663328i	$0.230857\pi$
$457$	28.3607i	1.32666i	−0.748328	+	0.663328i	$0.769143\pi$
$458$	− 5.41641i	− 0.253092i
$459$	1.00000	0.0466760
$460$	−8.94427	−0.417029
$461$	28.0689	1.30730	0.653649	−	0.756798i	$-0.273237\pi$
$461$	28.0689	1.30730	0.653649	+	0.756798i	$0.273237\pi$
$462$	− 6.47214i	− 0.301111i
$463$	1.05573i	0.0490638i	0.999699	+	0.0245319i	$0.00780954\pi$
$463$	1.05573i	0.0490638i	−0.999699	+	0.0245319i	$0.992190\pi$
$464$	4.00000	0.185695
$465$	14.4721	0.671129
$466$	−21.4164	−0.992096
$467$	− 16.9443i	− 0.784087i	−0.919947	−	0.392044i	$-0.871768\pi$
$467$	− 16.9443i	− 0.784087i	0.919947	−	0.392044i	$-0.128232\pi$
$468$	− 0.763932i	− 0.0353128i
$469$	−37.8885	−1.74953
$470$	− 3.41641i	− 0.157587i
$471$	12.1803	0.561240
$472$	− 1.70820i	− 0.0786265i
$473$	20.3607i	0.936185i
$474$	−16.9443	−0.778276
$475$	12.3607	0.567147
$476$	3.23607	0.148325
$477$	6.94427i	0.317956i
$478$	− 13.8885i	− 0.635247i
$479$	−13.5967	−0.621251	−0.310626	−	0.950532i	$-0.600539\pi$
$479$	−13.5967	−0.621251	−0.310626	+	0.950532i	$0.600539\pi$
$480$	− 2.23607i	− 0.102062i
$481$	1.52786	0.0696646
$482$	− 17.4164i	− 0.793296i
$483$	12.9443i	0.588985i
$484$	7.00000	0.318182
$485$	−26.1803	−1.18879
$486$	−1.00000	−0.0453609
$487$	− 29.7082i	− 1.34621i	−0.739548	−	0.673104i	$-0.764961\pi$
$487$	− 29.7082i	− 1.34621i	0.739548	−	0.673104i	$-0.235039\pi$
$488$	4.47214i	0.202444i
$489$	−13.5279	−0.611751
$490$	7.76393	0.350739
$491$	25.7082	1.16020	0.580098	−	0.814547i	$-0.303014\pi$
$491$	25.7082	1.16020	0.580098	+	0.814547i	$0.303014\pi$
$492$	− 5.70820i	− 0.257346i
$493$	4.00000i	0.180151i
$494$	−1.88854	−0.0849696
$495$	− 4.47214i	− 0.201008i
$496$	−6.47214	−0.290607
$497$	21.8885i	0.981835i
$498$	1.52786i	0.0684652i
$499$	16.0000	0.716258	0.358129	−	0.933672i	$-0.383415\pi$
$499$	16.0000	0.716258	0.358129	+	0.933672i	$0.383415\pi$
$500$	11.1803i	0.500000i
$501$	−21.8885	−0.977908
$502$	12.1803i	0.543635i
$503$	0.583592i	0.0260211i	0.999915	+	0.0130105i	$0.00414150\pi$
$503$	0.583592i	0.0260211i	−0.999915	+	0.0130105i	$0.995858\pi$
$504$	−3.23607	−0.144146
$505$	26.1803i	1.16501i
$506$	8.00000	0.355643
$507$	12.4164i	0.551432i
$508$	− 14.0000i	− 0.621150i
$509$	−32.6525	−1.44730	−0.723648	−	0.690169i	$-0.757536\pi$
$509$	−32.6525	−1.44730	−0.723648	+	0.690169i	$0.757536\pi$
$510$	2.23607	0.0990148
$511$	42.8328	1.89481
$512$	1.00000i	0.0441942i
$513$	2.47214i	0.109147i
$514$	−18.0000	−0.793946
$515$	−15.5279	−0.684239
$516$	10.1803	0.448164
$517$	3.05573i	0.134391i
$518$	− 6.47214i	− 0.284369i
$519$	19.8885	0.873010
$520$	− 1.70820i	− 0.0749097i
$521$	28.1803	1.23460	0.617302	−	0.786727i	$-0.288226\pi$
$521$	28.1803	1.23460	0.617302	+	0.786727i	$0.288226\pi$
$522$	− 4.00000i	− 0.175075i
$523$	27.7082i	1.21160i	0.795619	+	0.605798i	$0.207146\pi$
$523$	27.7082i	1.21160i	−0.795619	+	0.605798i	$0.792854\pi$
$524$	−3.52786	−0.154115
$525$	16.1803	0.706168
$526$	−4.00000	−0.174408
$527$	− 6.47214i	− 0.281931i
$528$	2.00000i	0.0870388i
$529$	7.00000	0.304348
$530$	15.5279i	0.674487i
$531$	−1.70820	−0.0741297
$532$	8.00000i	0.346844i
$533$	− 4.36068i	− 0.188882i
$534$	3.52786	0.152666
$535$	−17.8885	−0.773389
$536$	11.7082	0.505717
$537$	− 5.70820i	− 0.246327i
$538$	24.9443i	1.07542i
$539$	−6.94427	−0.299111
$540$	−2.23607	−0.0962250
$541$	31.3050	1.34590	0.672952	−	0.739686i	$-0.265026\pi$
$541$	31.3050	1.34590	0.672952	+	0.739686i	$0.265026\pi$
$542$	4.00000i	0.171815i
$543$	− 8.47214i	− 0.363574i
$544$	−1.00000	−0.0428746
$545$	36.8328i	1.57774i
$546$	−2.47214	−0.105798
$547$	− 16.9443i	− 0.724485i	−0.932084	−	0.362242i	$-0.882011\pi$
$547$	− 16.9443i	− 0.724485i	0.932084	−	0.362242i	$-0.117989\pi$
$548$	2.00000i	0.0854358i
$549$	4.47214	0.190866
$550$	− 10.0000i	− 0.426401i
$551$	−9.88854	−0.421266
$552$	− 4.00000i	− 0.170251i
$553$	54.8328i	2.33173i
$554$	18.3607	0.780071
$555$	− 4.47214i	− 0.189832i
$556$	10.4721	0.444117
$557$	− 42.9443i	− 1.81961i	−0.415039	−	0.909804i	$-0.636232\pi$
$557$	− 42.9443i	− 1.81961i	0.415039	−	0.909804i	$-0.363768\pi$
$558$	6.47214i	0.273987i
$559$	7.77709	0.328936
$560$	−7.23607	−0.305780
$561$	−2.00000	−0.0844401
$562$	− 25.4164i	− 1.07213i
$563$	− 37.8885i	− 1.59681i	−0.602120	−	0.798406i	$-0.705677\pi$
$563$	− 37.8885i	− 1.59681i	0.602120	−	0.798406i	$-0.294323\pi$
$564$	1.52786	0.0643347
$565$	−22.3607	−0.940721
$566$	9.88854	0.415646
$567$	3.23607i	0.135902i
$568$	− 6.76393i	− 0.283808i
$569$	−6.00000	−0.251533	−0.125767	−	0.992060i	$-0.540139\pi$
$569$	−6.00000	−0.251533	−0.125767	+	0.992060i	$0.540139\pi$
$570$	5.52786i	0.231537i
$571$	−8.36068	−0.349884	−0.174942	−	0.984579i	$-0.555974\pi$
$571$	−8.36068	−0.349884	−0.174942	+	0.984579i	$0.555974\pi$
$572$	1.52786i	0.0638832i
$573$	14.4721i	0.604582i
$574$	−18.4721	−0.771012
$575$	20.0000i	0.834058i
$576$	1.00000	0.0416667
$577$	− 8.58359i	− 0.357340i	−0.983909	−	0.178670i	$-0.942821\pi$
$577$	− 8.58359i	− 0.357340i	0.983909	−	0.178670i	$-0.0571794\pi$
$578$	− 1.00000i	− 0.0415945i
$579$	25.5967	1.06376
$580$	− 8.94427i	− 0.371391i
$581$	4.94427	0.205123
$582$	− 11.7082i	− 0.485321i
$583$	− 13.8885i	− 0.575205i
$584$	−13.2361	−0.547712
$585$	−1.70820	−0.0706255
$586$	2.58359	0.106727
$587$	− 14.4721i	− 0.597329i	−0.954358	−	0.298664i	$-0.903459\pi$
$587$	− 14.4721i	− 0.597329i	0.954358	−	0.298664i	$-0.0965411\pi$
$588$	3.47214i	0.143188i
$589$	16.0000	0.659269
$590$	−3.81966	−0.157253
$591$	19.8885	0.818105
$592$	2.00000i	0.0821995i
$593$	− 13.0557i	− 0.536134i	−0.963400	−	0.268067i	$-0.913615\pi$
$593$	− 13.0557i	− 0.536134i	0.963400	−	0.268067i	$-0.0863849\pi$
$594$	2.00000	0.0820610
$595$	− 7.23607i	− 0.296650i
$596$	−10.1803	−0.417003
$597$	− 8.94427i	− 0.366065i
$598$	− 3.05573i	− 0.124958i
$599$	−34.4721	−1.40849	−0.704247	−	0.709955i	$-0.748715\pi$
$599$	−34.4721	−1.40849	−0.704247	+	0.709955i	$0.748715\pi$
$600$	−5.00000	−0.204124
$601$	−19.5279	−0.796558	−0.398279	−	0.917264i	$-0.630392\pi$
$601$	−19.5279	−0.796558	−0.398279	+	0.917264i	$0.630392\pi$
$602$	− 32.9443i	− 1.34271i
$603$	− 11.7082i	− 0.476795i
$604$	4.00000	0.162758
$605$	− 15.6525i	− 0.636364i
$606$	−11.7082	−0.475613
$607$	9.12461i	0.370357i	0.982705	+	0.185178i	$0.0592863\pi$
$607$	9.12461i	0.370357i	−0.982705	+	0.185178i	$0.940714\pi$
$608$	− 2.47214i	− 0.100258i
$609$	−12.9443	−0.524528
$610$	10.0000	0.404888
$611$	1.16718	0.0472192
$612$	1.00000i	0.0404226i
$613$	19.8197i	0.800509i	0.916404	+	0.400254i	$0.131078\pi$
$613$	19.8197i	0.800509i	−0.916404	+	0.400254i	$0.868922\pi$
$614$	18.1803	0.733699
$615$	−12.7639	−0.514691
$616$	6.47214	0.260770
$617$	− 11.5279i	− 0.464094i	−0.972705	−	0.232047i	$-0.925458\pi$
$617$	− 11.5279i	− 0.464094i	0.972705	−	0.232047i	$-0.0745424\pi$
$618$	− 6.94427i	− 0.279340i
$619$	31.0557	1.24824	0.624118	−	0.781330i	$-0.285459\pi$
$619$	31.0557	1.24824	0.624118	+	0.781330i	$0.285459\pi$
$620$	14.4721i	0.581215i
$621$	−4.00000	−0.160514
$622$	− 29.5967i	− 1.18672i
$623$	− 11.4164i	− 0.457389i
$624$	0.763932	0.0305818
$625$	25.0000	1.00000
$626$	−8.29180	−0.331407
$627$	− 4.94427i	− 0.197455i
$628$	12.1803i	0.486048i
$629$	−2.00000	−0.0797452
$630$	7.23607i	0.288292i
$631$	25.8885	1.03061	0.515303	−	0.857008i	$-0.327679\pi$
$631$	25.8885	1.03061	0.515303	+	0.857008i	$0.327679\pi$
$632$	− 16.9443i	− 0.674007i
$633$	− 7.05573i	− 0.280440i
$634$	23.8885	0.948735
$635$	−31.3050	−1.24230
$636$	−6.94427	−0.275358
$637$	2.65248i	0.105095i
$638$	8.00000i	0.316723i
$639$	−6.76393	−0.267577
$640$	2.23607	0.0883883
$641$	−1.70820	−0.0674700	−0.0337350	−	0.999431i	$-0.510740\pi$
$641$	−1.70820	−0.0674700	−0.0337350	+	0.999431i	$0.510740\pi$
$642$	− 8.00000i	− 0.315735i
$643$	0.583592i	0.0230146i	0.999934	+	0.0115073i	$0.00366297\pi$
$643$	0.583592i	0.0230146i	−0.999934	+	0.0115073i	$0.996337\pi$
$644$	−12.9443	−0.510076
$645$	− 22.7639i	− 0.896329i
$646$	2.47214	0.0972649
$647$	− 3.05573i	− 0.120133i	−0.998194	−	0.0600665i	$-0.980869\pi$
$647$	− 3.05573i	− 0.120133i	0.998194	−	0.0600665i	$-0.0191313\pi$
$648$	− 1.00000i	− 0.0392837i
$649$	3.41641	0.134106
$650$	−3.81966	−0.149819
$651$	20.9443	0.820871
$652$	− 13.5279i	− 0.529792i
$653$	12.4721i	0.488072i	0.969766	+	0.244036i	$0.0784716\pi$
$653$	12.4721i	0.488072i	−0.969766	+	0.244036i	$0.921528\pi$
$654$	−16.4721	−0.644111
$655$	7.88854i	0.308231i
$656$	5.70820	0.222868
$657$	13.2361i	0.516388i
$658$	− 4.94427i	− 0.192748i
$659$	−41.7082	−1.62472	−0.812360	−	0.583156i	$-0.801818\pi$
$659$	−41.7082	−1.62472	−0.812360	+	0.583156i	$0.801818\pi$
$660$	4.47214	0.174078
$661$	−13.4164	−0.521838	−0.260919	−	0.965361i	$-0.584026\pi$
$661$	−13.4164	−0.521838	−0.260919	+	0.965361i	$0.584026\pi$
$662$	23.4164i	0.910105i
$663$	0.763932i	0.0296687i
$664$	−1.52786	−0.0592926
$665$	17.8885	0.693688
$666$	2.00000	0.0774984
$667$	− 16.0000i	− 0.619522i
$668$	− 21.8885i	− 0.846893i
$669$	26.9443	1.04173
$670$	− 26.1803i	− 1.01143i
$671$	−8.94427	−0.345290
$672$	− 3.23607i	− 0.124834i
$673$	18.1803i	0.700801i	0.936600	+	0.350400i	$0.113954\pi$
$673$	18.1803i	0.700801i	−0.936600	+	0.350400i	$0.886046\pi$
$674$	−2.18034	−0.0839836
$675$	5.00000i	0.192450i
$676$	−12.4164	−0.477554
$677$	− 26.9443i	− 1.03555i	−0.855516	−	0.517776i	$-0.826760\pi$
$677$	− 26.9443i	− 1.03555i	0.855516	−	0.517776i	$-0.173240\pi$
$678$	− 10.0000i	− 0.384048i
$679$	−37.8885	−1.45403
$680$	2.23607i	0.0857493i
$681$	25.8885	0.992051
$682$	− 12.9443i	− 0.495662i
$683$	15.0557i	0.576091i	0.957617	+	0.288046i	$0.0930055\pi$
$683$	15.0557i	0.576091i	−0.957617	+	0.288046i	$0.906994\pi$
$684$	−2.47214	−0.0945245
$685$	4.47214	0.170872
$686$	−11.4164	−0.435880
$687$	− 5.41641i	− 0.206649i
$688$	10.1803i	0.388122i
$689$	−5.30495	−0.202103
$690$	−8.94427	−0.340503
$691$	12.9443	0.492423	0.246212	−	0.969216i	$-0.420814\pi$
$691$	12.9443	0.492423	0.246212	+	0.969216i	$0.420814\pi$
$692$	19.8885i	0.756049i
$693$	− 6.47214i	− 0.245856i
$694$	−30.8328	−1.17040
$695$	− 23.4164i	− 0.888235i
$696$	4.00000	0.151620
$697$	5.70820i	0.216214i
$698$	19.5279i	0.739141i
$699$	−21.4164	−0.810043
$700$	16.1803i	0.611559i
$701$	−37.5967	−1.42001	−0.710005	−	0.704197i	$-0.751307\pi$
$701$	−37.5967	−1.42001	−0.710005	+	0.704197i	$0.751307\pi$
$702$	− 0.763932i	− 0.0288328i
$703$	− 4.94427i	− 0.186477i
$704$	−2.00000	−0.0753778
$705$	− 3.41641i	− 0.128669i
$706$	2.94427	0.110809
$707$	37.8885i	1.42495i
$708$	− 1.70820i	− 0.0641982i
$709$	48.4721	1.82041	0.910205	−	0.414159i	$-0.135924\pi$
$709$	48.4721	1.82041	0.910205	+	0.414159i	$0.135924\pi$
$710$	−15.1246	−0.567617
$711$	−16.9443	−0.635460
$712$	3.52786i	0.132212i
$713$	25.8885i	0.969534i
$714$	3.23607	0.121107
$715$	3.41641	0.127766
$716$	5.70820	0.213326
$717$	− 13.8885i	− 0.518677i
$718$	18.4721i	0.689374i
$719$	−30.5410	−1.13899	−0.569494	−	0.821996i	$-0.692861\pi$
$719$	−30.5410	−1.13899	−0.569494	+	0.821996i	$0.692861\pi$
$720$	− 2.23607i	− 0.0833333i
$721$	−22.4721	−0.836906
$722$	− 12.8885i	− 0.479662i
$723$	− 17.4164i	− 0.647723i
$724$	8.47214	0.314864
$725$	−20.0000	−0.742781
$726$	7.00000	0.259794
$727$	11.8885i	0.440922i	0.975396	+	0.220461i	$0.0707561\pi$
$727$	11.8885i	0.440922i	−0.975396	+	0.220461i	$0.929244\pi$
$728$	− 2.47214i	− 0.0916235i
$729$	−1.00000	−0.0370370
$730$	29.5967i	1.09542i
$731$	−10.1803	−0.376533
$732$	4.47214i	0.165295i
$733$	28.5410i	1.05419i	0.849807	+	0.527093i	$0.176718\pi$
$733$	28.5410i	1.05419i	−0.849807	+	0.527093i	$0.823282\pi$
$734$	28.1803	1.04016
$735$	7.76393	0.286377
$736$	4.00000	0.147442
$737$	23.4164i	0.862554i
$738$	− 5.70820i	− 0.210122i
$739$	−42.4721	−1.56236	−0.781181	−	0.624304i	$-0.785383\pi$
$739$	−42.4721	−1.56236	−0.781181	+	0.624304i	$0.785383\pi$
$740$	4.47214	0.164399
$741$	−1.88854	−0.0693774
$742$	22.4721i	0.824978i
$743$	− 42.4721i	− 1.55815i	−0.626930	−	0.779076i	$-0.715689\pi$
$743$	− 42.4721i	− 1.55815i	0.626930	−	0.779076i	$-0.284311\pi$
$744$	−6.47214	−0.237280
$745$	22.7639i	0.834006i
$746$	3.23607	0.118481
$747$	1.52786i	0.0559016i
$748$	− 2.00000i	− 0.0731272i
$749$	−25.8885	−0.945947
$750$	11.1803i	0.408248i
$751$	−5.88854	−0.214876	−0.107438	−	0.994212i	$-0.534265\pi$
$751$	−5.88854	−0.214876	−0.107438	+	0.994212i	$0.534265\pi$
$752$	1.52786i	0.0557155i
$753$	12.1803i	0.443876i
$754$	3.05573	0.111283
$755$	− 8.94427i	− 0.325515i
$756$	−3.23607	−0.117695
$757$	11.5967i	0.421491i	0.977541	+	0.210745i	$0.0675891\pi$
$757$	11.5967i	0.421491i	−0.977541	+	0.210745i	$0.932411\pi$
$758$	10.4721i	0.380365i
$759$	8.00000	0.290382
$760$	−5.52786	−0.200517
$761$	−0.832816	−0.0301895	−0.0150948	−	0.999886i	$-0.504805\pi$
$761$	−0.832816	−0.0301895	−0.0150948	+	0.999886i	$0.504805\pi$
$762$	− 14.0000i	− 0.507166i
$763$	53.3050i	1.92977i
$764$	−14.4721	−0.523584
$765$	2.23607	0.0808452
$766$	−17.8885	−0.646339
$767$	− 1.30495i	− 0.0471191i
$768$	1.00000i	0.0360844i
$769$	−10.3607	−0.373616	−0.186808	−	0.982396i	$-0.559814\pi$
$769$	−10.3607	−0.373616	−0.186808	+	0.982396i	$0.559814\pi$
$770$	− 14.4721i	− 0.521540i
$771$	−18.0000	−0.648254
$772$	25.5967i	0.921247i
$773$	− 6.58359i	− 0.236795i	−0.992966	−	0.118398i	$-0.962224\pi$
$773$	− 6.58359i	− 0.236795i	0.992966	−	0.118398i	$-0.0377758\pi$
$774$	10.1803	0.365925
$775$	32.3607	1.16243
$776$	11.7082	0.420300
$777$	− 6.47214i	− 0.232187i
$778$	2.18034i	0.0781690i
$779$	−14.1115	−0.505595
$780$	− 1.70820i	− 0.0611635i
$781$	13.5279	0.484065
$782$	4.00000i	0.143040i
$783$	− 4.00000i	− 0.142948i
$784$	−3.47214	−0.124005
$785$	27.2361	0.972097
$786$	−3.52786	−0.125835
$787$	37.5279i	1.33772i	0.743387	+	0.668862i	$0.233218\pi$
$787$	37.5279i	1.33772i	−0.743387	+	0.668862i	$0.766782\pi$
$788$	19.8885i	0.708500i
$789$	−4.00000	−0.142404
$790$	−37.8885	−1.34801
$791$	−32.3607	−1.15061
$792$	2.00000i	0.0710669i
$793$	3.41641i	0.121320i
$794$	31.8885	1.13168
$795$	15.5279i	0.550717i
$796$	8.94427	0.317021
$797$	46.0000i	1.62940i	0.579880	+	0.814702i	$0.303099\pi$
$797$	46.0000i	1.62940i	−0.579880	+	0.814702i	$0.696901\pi$
$798$	8.00000i	0.283197i
$799$	−1.52786	−0.0540519
$800$	− 5.00000i	− 0.176777i
$801$	3.52786	0.124651
$802$	22.0689i	0.779279i
$803$	− 26.4721i	− 0.934181i
$804$	11.7082	0.412917
$805$	28.9443i	1.02015i
$806$	−4.94427	−0.174155
$807$	24.9443i	0.878080i
$808$	− 11.7082i	− 0.411893i
$809$	−6.87539	−0.241726	−0.120863	−	0.992669i	$-0.538566\pi$
$809$	−6.87539	−0.241726	−0.120863	+	0.992669i	$0.538566\pi$
$810$	−2.23607	−0.0785674
$811$	−38.2492	−1.34311	−0.671556	−	0.740954i	$-0.734374\pi$
$811$	−38.2492	−1.34311	−0.671556	+	0.740954i	$0.734374\pi$
$812$	− 12.9443i	− 0.454255i
$813$	4.00000i	0.140286i
$814$	−4.00000	−0.140200
$815$	−30.2492	−1.05958
$816$	−1.00000	−0.0350070
$817$	− 25.1672i	− 0.880488i
$818$	18.3607i	0.641966i
$819$	−2.47214	−0.0863834
$820$	− 12.7639i	− 0.445736i
$821$	−11.4164	−0.398435	−0.199218	−	0.979955i	$-0.563840\pi$
$821$	−11.4164	−0.398435	−0.199218	+	0.979955i	$0.563840\pi$
$822$	2.00000i	0.0697580i
$823$	− 15.8197i	− 0.551439i	−0.961238	−	0.275719i	$-0.911084\pi$
$823$	− 15.8197i	− 0.551439i	0.961238	−	0.275719i	$-0.0889160\pi$
$824$	6.94427	0.241915
$825$	− 10.0000i	− 0.348155i
$826$	−5.52786	−0.192339
$827$	− 40.9443i	− 1.42377i	−0.702295	−	0.711886i	$-0.747841\pi$
$827$	− 40.9443i	− 1.42377i	0.702295	−	0.711886i	$-0.252159\pi$
$828$	− 4.00000i	− 0.139010i
$829$	−34.0000	−1.18087	−0.590434	−	0.807086i	$-0.701044\pi$
$829$	−34.0000	−1.18087	−0.590434	+	0.807086i	$0.701044\pi$
$830$	3.41641i	0.118585i
$831$	18.3607	0.636925
$832$	0.763932i	0.0264846i
$833$	− 3.47214i	− 0.120302i
$834$	10.4721	0.362620
$835$	−48.9443	−1.69379
$836$	4.94427	0.171001
$837$	6.47214i	0.223710i
$838$	38.3607i	1.32515i
$839$	−36.0689	−1.24524	−0.622618	−	0.782526i	$-0.713931\pi$
$839$	−36.0689	−1.24524	−0.622618	+	0.782526i	$0.713931\pi$
$840$	−7.23607	−0.249668
$841$	−13.0000	−0.448276
$842$	− 10.5836i	− 0.364735i
$843$	− 25.4164i	− 0.875388i
$844$	7.05573	0.242868
$845$	27.7639i	0.955108i
$846$	1.52786	0.0525290
$847$	− 22.6525i	− 0.778348i
$848$	− 6.94427i	− 0.238467i
$849$	9.88854	0.339374
$850$	5.00000	0.171499
$851$	8.00000	0.274236
$852$	− 6.76393i	− 0.231728i
$853$	− 38.3607i	− 1.31344i	−0.754132	−	0.656722i	$-0.771942\pi$
$853$	− 38.3607i	− 1.31344i	0.754132	−	0.656722i	$-0.228058\pi$
$854$	14.4721	0.495226
$855$	5.52786i	0.189049i
$856$	8.00000	0.273434
$857$	5.63932i	0.192636i	0.995351	+	0.0963178i	$0.0307065\pi$
$857$	5.63932i	0.192636i	−0.995351	+	0.0963178i	$0.969294\pi$
$858$	1.52786i	0.0521604i
$859$	−24.9443	−0.851088	−0.425544	−	0.904938i	$-0.639917\pi$
$859$	−24.9443	−0.851088	−0.425544	+	0.904938i	$0.639917\pi$
$860$	22.7639	0.776244
$861$	−18.4721	−0.629529
$862$	− 17.5967i	− 0.599348i
$863$	− 41.3050i	− 1.40604i	−0.711172	−	0.703018i	$-0.751835\pi$
$863$	− 41.3050i	− 1.40604i	0.711172	−	0.703018i	$-0.248165\pi$
$864$	1.00000	0.0340207
$865$	44.4721	1.51210
$866$	−16.3607	−0.555959
$867$	− 1.00000i	− 0.0339618i
$868$	20.9443i	0.710895i
$869$	33.8885	1.14959
$870$	− 8.94427i	− 0.303239i
$871$	8.94427	0.303065
$872$	− 16.4721i	− 0.557817i
$873$	− 11.7082i	− 0.396263i
$874$	−9.88854	−0.334485
$875$	36.1803	1.22312
$876$	−13.2361	−0.447205
$877$	− 31.5279i	− 1.06462i	−0.846549	−	0.532310i	$-0.821324\pi$
$877$	− 31.5279i	− 1.06462i	0.846549	−	0.532310i	$-0.178676\pi$
$878$	− 22.4721i	− 0.758398i
$879$	2.58359	0.0871424
$880$	4.47214i	0.150756i
$881$	33.1246	1.11600	0.557998	−	0.829842i	$-0.311570\pi$
$881$	33.1246	1.11600	0.557998	+	0.829842i	$0.311570\pi$
$882$	3.47214i	0.116913i
$883$	22.7639i	0.766067i	0.923734	+	0.383034i	$0.125121\pi$
$883$	22.7639i	0.766067i	−0.923734	+	0.383034i	$0.874879\pi$
$884$	−0.763932	−0.0256938
$885$	−3.81966	−0.128396
$886$	4.00000	0.134383
$887$	− 8.94427i	− 0.300319i	−0.988662	−	0.150160i	$-0.952021\pi$
$887$	− 8.94427i	− 0.300319i	0.988662	−	0.150160i	$-0.0479788\pi$
$888$	2.00000i	0.0671156i
$889$	−45.3050	−1.51948
$890$	7.88854	0.264425
$891$	2.00000	0.0670025
$892$	26.9443i	0.902161i
$893$	− 3.77709i	− 0.126395i
$894$	−10.1803	−0.340481
$895$	− 12.7639i	− 0.426651i
$896$	3.23607	0.108109
$897$	− 3.05573i	− 0.102028i
$898$	31.5967i	1.05440i
$899$	−25.8885	−0.863431
$900$	−5.00000	−0.166667
$901$	6.94427	0.231347
$902$	11.4164i	0.380125i
$903$	− 32.9443i	− 1.09632i
$904$	10.0000	0.332595
$905$	− 18.9443i	− 0.629729i
$906$	4.00000	0.132891
$907$	− 40.3607i	− 1.34015i	−0.742291	−	0.670077i	$-0.766261\pi$
$907$	− 40.3607i	− 1.34015i	0.742291	−	0.670077i	$-0.233739\pi$
$908$	25.8885i	0.859142i
$909$	−11.7082	−0.388337
$910$	−5.52786	−0.183247
$911$	54.5410	1.80702	0.903512	−	0.428562i	$-0.140980\pi$
$911$	54.5410	1.80702	0.903512	+	0.428562i	$0.140980\pi$
$912$	− 2.47214i	− 0.0818606i
$913$	− 3.05573i	− 0.101130i
$914$	−28.3607	−0.938088
$915$	10.0000	0.330590
$916$	5.41641	0.178963
$917$	11.4164i	0.377003i
$918$	1.00000i	0.0330049i
$919$	36.7214	1.21133	0.605663	−	0.795721i	$-0.292908\pi$
$919$	36.7214	1.21133	0.605663	+	0.795721i	$0.292908\pi$
$920$	− 8.94427i	− 0.294884i
$921$	18.1803	0.599063
$922$	28.0689i	0.924399i
$923$	− 5.16718i	− 0.170080i
$924$	6.47214	0.212918
$925$	− 10.0000i	− 0.328798i
$926$	−1.05573	−0.0346934
$927$	− 6.94427i	− 0.228080i
$928$	4.00000i	0.131306i
$929$	34.0689	1.11776	0.558882	−	0.829247i	$-0.311231\pi$
$929$	34.0689	1.11776	0.558882	+	0.829247i	$0.311231\pi$
$930$	14.4721i	0.474560i
$931$	8.58359	0.281316
$932$	− 21.4164i	− 0.701518i
$933$	− 29.5967i	− 0.968954i
$934$	16.9443	0.554434
$935$	−4.47214	−0.146254
$936$	0.763932	0.0249699
$937$	26.4721i	0.864807i	0.901680	+	0.432403i	$0.142334\pi$
$937$	26.4721i	0.864807i	−0.901680	+	0.432403i	$0.857666\pi$
$938$	− 37.8885i	− 1.23710i
$939$	−8.29180	−0.270593
$940$	3.41641	0.111431
$941$	−49.3050	−1.60730	−0.803648	−	0.595105i	$-0.797110\pi$
$941$	−49.3050	−1.60730	−0.803648	+	0.595105i	$0.797110\pi$
$942$	12.1803i	0.396857i
$943$	− 22.8328i	− 0.743539i
$944$	1.70820	0.0555973
$945$	7.23607i	0.235389i
$946$	−20.3607	−0.661983
$947$	− 40.7214i	− 1.32327i	−0.749828	−	0.661633i	$-0.769864\pi$
$947$	− 40.7214i	− 1.32327i	0.749828	−	0.661633i	$-0.230136\pi$
$948$	− 16.9443i	− 0.550324i
$949$	−10.1115	−0.328232
$950$	12.3607i	0.401033i
$951$	23.8885	0.774639
$952$	3.23607i	0.104882i
$953$	− 9.05573i	− 0.293344i	−0.989185	−	0.146672i	$-0.953144\pi$
$953$	− 9.05573i	− 0.293344i	0.989185	−	0.146672i	$-0.0468562\pi$
$954$	−6.94427	−0.224829
$955$	32.3607i	1.04717i
$956$	13.8885	0.449188
$957$	8.00000i	0.258603i
$958$	− 13.5967i	− 0.439291i
$959$	6.47214	0.208996
$960$	2.23607	0.0721688
$961$	10.8885	0.351243
$962$	1.52786i	0.0492603i
$963$	− 8.00000i	− 0.257796i
$964$	17.4164	0.560945
$965$	57.2361	1.84249
$966$	−12.9443	−0.416475
$967$	8.47214i	0.272446i	0.990678	+	0.136223i	$0.0434963\pi$
$967$	8.47214i	0.272446i	−0.990678	+	0.136223i	$0.956504\pi$
$968$	7.00000i	0.224989i
$969$	2.47214	0.0794164
$970$	− 26.1803i	− 0.840600i
$971$	−27.2361	−0.874047	−0.437024	−	0.899450i	$-0.643967\pi$
$971$	−27.2361	−0.874047	−0.437024	+	0.899450i	$0.643967\pi$
$972$	− 1.00000i	− 0.0320750i
$973$	− 33.8885i	− 1.08642i
$974$	29.7082	0.951912
$975$	−3.81966	−0.122327
$976$	−4.47214	−0.143150
$977$	9.05573i	0.289718i	0.989452	+	0.144859i	$0.0462729\pi$
$977$	9.05573i	0.289718i	−0.989452	+	0.144859i	$0.953727\pi$
$978$	− 13.5279i	− 0.432573i
$979$	−7.05573	−0.225502
$980$	7.76393i	0.248010i
$981$	−16.4721	−0.525915
$982$	25.7082i	0.820382i
$983$	15.4164i	0.491707i	0.969307	+	0.245854i	$0.0790682\pi$
$983$	15.4164i	0.491707i	−0.969307	+	0.245854i	$0.920932\pi$
$984$	5.70820	0.181971
$985$	44.4721	1.41700
$986$	−4.00000	−0.127386
$987$	− 4.94427i	− 0.157378i
$988$	− 1.88854i	− 0.0600826i
$989$	40.7214	1.29486
$990$	4.47214	0.142134
$991$	42.8328	1.36063	0.680315	−	0.732920i	$-0.261843\pi$
$991$	42.8328	1.36063	0.680315	+	0.732920i	$0.261843\pi$
$992$	− 6.47214i	− 0.205491i
$993$	23.4164i	0.743097i
$994$	−21.8885	−0.694262
$995$	− 20.0000i	− 0.634043i
$996$	−1.52786	−0.0484122
$997$	58.9443i	1.86678i	0.358859	+	0.933392i	$0.383166\pi$
$997$	58.9443i	1.86678i	−0.358859	+	0.933392i	$0.616834\pi$
$998$	16.0000i	0.506471i
$999$	2.00000	0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	510.2.d.b.409.4	yes	4
3.2	odd	2		1530.2.d.f.919.1		4
4.3	odd	2		4080.2.m.m.2449.2		4
5.2	odd	4		2550.2.a.bh.1.1		2
5.3	odd	4		2550.2.a.bk.1.2		2
5.4	even	2	inner	510.2.d.b.409.1	✓	4
15.2	even	4		7650.2.a.da.1.1		2
15.8	even	4		7650.2.a.cx.1.2		2
15.14	odd	2		1530.2.d.f.919.4		4
20.19	odd	2		4080.2.m.m.2449.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
510.2.d.b.409.1	✓	4	5.4	even	2	inner
510.2.d.b.409.4	yes	4	1.1	even	1	trivial
1530.2.d.f.919.1		4	3.2	odd	2
1530.2.d.f.919.4		4	15.14	odd	2
2550.2.a.bh.1.1		2	5.2	odd	4
2550.2.a.bk.1.2		2	5.3	odd	4
4080.2.m.m.2449.2		4	4.3	odd	2
4080.2.m.m.2449.3		4	20.19	odd	2
7650.2.a.cx.1.2		2	15.8	even	4
7650.2.a.da.1.1		2	15.2	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:	\( N \)	\(=\)	\( 510 = 2 \cdot 3 \cdot 5 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	510.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +2.23607i q^{5} -1.00000 q^{6} +3.23607i q^{7} -1.00000i q^{8} -1.00000 q^{9} -2.23607 q^{10} +2.00000 q^{11} -1.00000i q^{12} -0.763932i q^{13} -3.23607 q^{14} -2.23607 q^{15} +1.00000 q^{16} +1.00000i q^{17} -1.00000i q^{18} -2.47214 q^{19} -2.23607i q^{20} -3.23607 q^{21} +2.00000i q^{22} -4.00000i q^{23} +1.00000 q^{24} -5.00000 q^{25} +0.763932 q^{26} -1.00000i q^{27} -3.23607i q^{28} +4.00000 q^{29} -2.23607i q^{30} -6.47214 q^{31} +1.00000i q^{32} +2.00000i q^{33} -1.00000 q^{34} -7.23607 q^{35} +1.00000 q^{36} +2.00000i q^{37} -2.47214i q^{38} +0.763932 q^{39} +2.23607 q^{40} +5.70820 q^{41} -3.23607i q^{42} +10.1803i q^{43} -2.00000 q^{44} -2.23607i q^{45} +4.00000 q^{46} +1.52786i q^{47} +1.00000i q^{48} -3.47214 q^{49} -5.00000i q^{50} -1.00000 q^{51} +0.763932i q^{52} -6.94427i q^{53} +1.00000 q^{54} +4.47214i q^{55} +3.23607 q^{56} -2.47214i q^{57} +4.00000i q^{58} +1.70820 q^{59} +2.23607 q^{60} -4.47214 q^{61} -6.47214i q^{62} -3.23607i q^{63} -1.00000 q^{64} +1.70820 q^{65} -2.00000 q^{66} +11.7082i q^{67} -1.00000i q^{68} +4.00000 q^{69} -7.23607i q^{70} +6.76393 q^{71} +1.00000i q^{72} -13.2361i q^{73} -2.00000 q^{74} -5.00000i q^{75} +2.47214 q^{76} +6.47214i q^{77} +0.763932i q^{78} +16.9443 q^{79} +2.23607i q^{80} +1.00000 q^{81} +5.70820i q^{82} -1.52786i q^{83} +3.23607 q^{84} -2.23607 q^{85} -10.1803 q^{86} +4.00000i q^{87} -2.00000i q^{88} -3.52786 q^{89} +2.23607 q^{90} +2.47214 q^{91} +4.00000i q^{92} -6.47214i q^{93} -1.52786 q^{94} -5.52786i q^{95} -1.00000 q^{96} +11.7082i q^{97} -3.47214i q^{98} -2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{4} - 4 q^{6} - 4 q^{9} + 8 q^{11} - 4 q^{14} + 4 q^{16} + 8 q^{19} - 4 q^{21} + 4 q^{24} - 20 q^{25} + 12 q^{26} + 16 q^{29} - 8 q^{31} - 4 q^{34} - 20 q^{35} + 4 q^{36} + 12 q^{39} - 4 q^{41}+ \cdots - 8 q^{99}+O(q^{100}) \) 4 * q - 4 * q^4 - 4 * q^6 - 4 * q^9 + 8 * q^11 - 4 * q^14 + 4 * q^16 + 8 * q^19 - 4 * q^21 + 4 * q^24 - 20 * q^25 + 12 * q^26 + 16 * q^29 - 8 * q^31 - 4 * q^34 - 20 * q^35 + 4 * q^36 + 12 * q^39 - 4 * q^41 - 8 * q^44 + 16 * q^46 + 4 * q^49 - 4 * q^51 + 4 * q^54 + 4 * q^56 - 20 * q^59 - 4 * q^64 - 20 * q^65 - 8 * q^66 + 16 * q^69 + 36 * q^71 - 8 * q^74 - 8 * q^76 + 32 * q^79 + 4 * q^81 + 4 * q^84 + 4 * q^86 - 32 * q^89 - 8 * q^91 - 24 * q^94 - 4 * q^96 - 8 * q^99

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