LMFDB - Embedded newform 390.2.e.e.79.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [390,2,Mod(79,390)] 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("390.79"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(390, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 390 = 2 \cdot 3 \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	390.e (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,-4,0,-4,0,0,-4,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(10)] == traces)

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

Self dual:	no
Analytic conductor:	$3.11416567883$
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			79.2
Root			$-0.618034i$ of defining polynomial
Character	$\chi$	$=$	390.79
Dual form			390.2.e.e.79.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +2.23607i q^{5} -1.00000 q^{6} +3.23607i q^{7} +1.00000i q^{8} -1.00000 q^{9} +2.23607 q^{10} +4.47214 q^{11} +1.00000i q^{12} -1.00000i q^{13} +3.23607 q^{14} +2.23607 q^{15} +1.00000 q^{16} +7.23607i q^{17} +1.00000i q^{18} +2.76393 q^{19} -2.23607i q^{20} +3.23607 q^{21} -4.47214i q^{22} -2.76393i q^{23} +1.00000 q^{24} -5.00000 q^{25} -1.00000 q^{26} +1.00000i q^{27} -3.23607i q^{28} +3.70820 q^{29} -2.23607i q^{30} -4.00000 q^{31} -1.00000i q^{32} -4.47214i q^{33} +7.23607 q^{34} -7.23607 q^{35} +1.00000 q^{36} -10.9443i q^{37} -2.76393i q^{38} -1.00000 q^{39} -2.23607 q^{40} +3.52786 q^{41} -3.23607i q^{42} +2.47214i q^{43} -4.47214 q^{44} -2.23607i q^{45} -2.76393 q^{46} +12.9443i q^{47} -1.00000i q^{48} -3.47214 q^{49} +5.00000i q^{50} +7.23607 q^{51} +1.00000i q^{52} +0.472136i q^{53} +1.00000 q^{54} +10.0000i q^{55} -3.23607 q^{56} -2.76393i q^{57} -3.70820i q^{58} +8.47214 q^{59} -2.23607 q^{60} -10.9443 q^{61} +4.00000i q^{62} -3.23607i q^{63} -1.00000 q^{64} +2.23607 q^{65} -4.47214 q^{66} -7.23607i q^{68} -2.76393 q^{69} +7.23607i q^{70} -2.47214 q^{71} -1.00000i q^{72} -13.2361i q^{73} -10.9443 q^{74} +5.00000i q^{75} -2.76393 q^{76} +14.4721i q^{77} +1.00000i q^{78} +4.00000 q^{79} +2.23607i q^{80} +1.00000 q^{81} -3.52786i q^{82} +4.94427i q^{83} -3.23607 q^{84} -16.1803 q^{85} +2.47214 q^{86} -3.70820i q^{87} +4.47214i q^{88} -0.472136 q^{89} -2.23607 q^{90} +3.23607 q^{91} +2.76393i q^{92} +4.00000i q^{93} +12.9443 q^{94} +6.18034i q^{95} -1.00000 q^{96} -3.70820i q^{97} +3.47214i q^{98} -4.47214 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{4} - 4 q^{6} - 4 q^{9} + 4 q^{14} + 4 q^{16} + 20 q^{19} + 4 q^{21} + 4 q^{24} - 20 q^{25} - 4 q^{26} - 12 q^{29} - 16 q^{31} + 20 q^{34} - 20 q^{35} + 4 q^{36} - 4 q^{39} + 32 q^{41} - 20 q^{46}+ \cdots - 4 q^{96}+O(q^{100}) $ 4 * q - 4 * q^4 - 4 * q^6 - 4 * q^9 + 4 * q^14 + 4 * q^16 + 20 * q^19 + 4 * q^21 + 4 * q^24 - 20 * q^25 - 4 * q^26 - 12 * q^29 - 16 * q^31 + 20 * q^34 - 20 * q^35 + 4 * q^36 - 4 * q^39 + 32 * q^41 - 20 * q^46 + 4 * q^49 + 20 * q^51 + 4 * q^54 - 4 * q^56 + 16 * q^59 - 8 * q^61 - 4 * q^64 - 20 * q^69 + 8 * q^71 - 8 * q^74 - 20 * q^76 + 16 * q^79 + 4 * q^81 - 4 * q^84 - 20 * q^85 - 8 * q^86 + 16 * q^89 + 4 * q^91 + 16 * q^94 - 4 * q^96

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/390\mathbb{Z}\right)^\times$.

$n$	$131$	$157$	$301$
$\chi(n)$	$1$	$-1$	$1$

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
$11$	4.47214	1.34840	0.674200	−	0.738549i	$-0.264489\pi$
$11$	4.47214	1.34840	0.674200	+	0.738549i	$0.264489\pi$
$12$	1.00000i	0.288675i
$13$	− 1.00000i	− 0.277350i
$13$	− 1.00000i	− 0.277350i
$14$	3.23607	0.864876
$15$	2.23607	0.577350
$16$	1.00000	0.250000
$17$	7.23607i	1.75500i	0.479573	+	0.877502i	$0.340792\pi$
$17$	7.23607i	1.75500i	−0.479573	+	0.877502i	$0.659208\pi$
$18$	1.00000i	0.235702i
$19$	2.76393	0.634089	0.317045	−	0.948411i	$-0.397309\pi$
$19$	2.76393	0.634089	0.317045	+	0.948411i	$0.397309\pi$
$20$	− 2.23607i	− 0.500000i
$21$	3.23607	0.706168
$22$	− 4.47214i	− 0.953463i
$23$	− 2.76393i	− 0.576320i	−0.957582	−	0.288160i	$-0.906957\pi$
$23$	− 2.76393i	− 0.576320i	0.957582	−	0.288160i	$-0.0930434\pi$
$24$	1.00000	0.204124
$25$	−5.00000	−1.00000
$26$	−1.00000	−0.196116
$27$	1.00000i	0.192450i
$28$	− 3.23607i	− 0.611559i
$29$	3.70820	0.688596	0.344298	−	0.938860i	$-0.388117\pi$
$29$	3.70820	0.688596	0.344298	+	0.938860i	$0.388117\pi$
$30$	− 2.23607i	− 0.408248i
$31$	−4.00000	−0.718421	−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000	−0.718421	−0.359211	+	0.933257i	$0.616954\pi$
$32$	− 1.00000i	− 0.176777i
$33$	− 4.47214i	− 0.778499i
$34$	7.23607	1.24098
$35$	−7.23607	−1.22312
$36$	1.00000	0.166667
$37$	− 10.9443i	− 1.79923i	−0.436687	−	0.899614i	$-0.643848\pi$
$37$	− 10.9443i	− 1.79923i	0.436687	−	0.899614i	$-0.356152\pi$
$38$	− 2.76393i	− 0.448369i
$39$	−1.00000	−0.160128
$40$	−2.23607	−0.353553
$41$	3.52786	0.550960	0.275480	−	0.961307i	$-0.411163\pi$
$41$	3.52786	0.550960	0.275480	+	0.961307i	$0.411163\pi$
$42$	− 3.23607i	− 0.499336i
$43$	2.47214i	0.376997i	0.982073	+	0.188499i	$0.0603621\pi$
$43$	2.47214i	0.376997i	−0.982073	+	0.188499i	$0.939638\pi$
$44$	−4.47214	−0.674200
$45$	− 2.23607i	− 0.333333i
$46$	−2.76393	−0.407520
$47$	12.9443i	1.88812i	0.329779	+	0.944058i	$0.393026\pi$
$47$	12.9443i	1.88812i	−0.329779	+	0.944058i	$0.606974\pi$
$48$	− 1.00000i	− 0.144338i
$49$	−3.47214	−0.496019
$50$	5.00000i	0.707107i
$51$	7.23607	1.01325
$52$	1.00000i	0.138675i
$53$	0.472136i	0.0648529i	0.999474	+	0.0324264i	$0.0103235\pi$
$53$	0.472136i	0.0648529i	−0.999474	+	0.0324264i	$0.989677\pi$
$54$	1.00000	0.136083
$55$	10.0000i	1.34840i
$56$	−3.23607	−0.432438
$57$	− 2.76393i	− 0.366092i
$58$	− 3.70820i	− 0.486911i
$59$	8.47214	1.10298	0.551489	−	0.834182i	$-0.314060\pi$
$59$	8.47214	1.10298	0.551489	+	0.834182i	$0.314060\pi$
$60$	−2.23607	−0.288675
$61$	−10.9443	−1.40127	−0.700635	−	0.713520i	$-0.747100\pi$
$61$	−10.9443	−1.40127	−0.700635	+	0.713520i	$0.747100\pi$
$62$	4.00000i	0.508001i
$63$	− 3.23607i	− 0.407706i
$64$	−1.00000	−0.125000
$65$	2.23607	0.277350
$66$	−4.47214	−0.550482
$67$	0	0	1.00000			$0$
$67$	0	0	−1.00000			$\pi$
$68$	− 7.23607i	− 0.877502i
$69$	−2.76393	−0.332738
$70$	7.23607i	0.864876i
$71$	−2.47214	−0.293389	−0.146694	−	0.989182i	$-0.546863\pi$
$71$	−2.47214	−0.293389	−0.146694	+	0.989182i	$0.546863\pi$
$72$	− 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	$-0.282040\pi$
$74$	−10.9443	−1.27225
$75$	5.00000i	0.577350i
$76$	−2.76393	−0.317045
$77$	14.4721i	1.64925i
$78$	1.00000i	0.113228i
$79$	4.00000	0.450035	0.225018	−	0.974355i	$-0.427756\pi$
$79$	4.00000	0.450035	0.225018	+	0.974355i	$0.427756\pi$
$80$	2.23607i	0.250000i
$81$	1.00000	0.111111
$82$	− 3.52786i	− 0.389587i
$83$	4.94427i	0.542704i	0.962480	+	0.271352i	$0.0874708\pi$
$83$	4.94427i	0.542704i	−0.962480	+	0.271352i	$0.912529\pi$
$84$	−3.23607	−0.353084
$85$	−16.1803	−1.75500
$86$	2.47214	0.266577
$87$	− 3.70820i	− 0.397561i
$88$	4.47214i	0.476731i
$89$	−0.472136	−0.0500463	−0.0250232	−	0.999687i	$-0.507966\pi$
$89$	−0.472136	−0.0500463	−0.0250232	+	0.999687i	$0.507966\pi$
$90$	−2.23607	−0.235702
$91$	3.23607	0.339232
$92$	2.76393i	0.288160i
$93$	4.00000i	0.414781i
$94$	12.9443	1.33510
$95$	6.18034i	0.634089i
$96$	−1.00000	−0.102062
$97$	− 3.70820i	− 0.376511i	−0.982120	−	0.188256i	$-0.939717\pi$
$97$	− 3.70820i	− 0.376511i	0.982120	−	0.188256i	$-0.0602833\pi$
$98$	3.47214i	0.350739i
$99$	−4.47214	−0.449467
$100$	5.00000	0.500000
$101$	−9.23607	−0.919023	−0.459512	−	0.888172i	$-0.651976\pi$
$101$	−9.23607	−0.919023	−0.459512	+	0.888172i	$0.651976\pi$
$102$	− 7.23607i	− 0.716477i
$103$	− 8.47214i	− 0.834784i	−0.908726	−	0.417392i	$-0.862944\pi$
$103$	− 8.47214i	− 0.834784i	0.908726	−	0.417392i	$-0.137056\pi$
$104$	1.00000	0.0980581
$105$	7.23607i	0.706168i
$106$	0.472136	0.0458579
$107$	− 1.52786i	− 0.147704i	−0.997269	−	0.0738521i	$-0.976471\pi$
$107$	− 1.52786i	− 0.147704i	0.997269	−	0.0738521i	$-0.0235293\pi$
$108$	− 1.00000i	− 0.0962250i
$109$	2.29180	0.219514	0.109757	−	0.993958i	$-0.464993\pi$
$109$	2.29180	0.219514	0.109757	+	0.993958i	$0.464993\pi$
$110$	10.0000	0.953463
$111$	−10.9443	−1.03878
$112$	3.23607i	0.305780i
$113$	− 18.6525i	− 1.75468i	−0.479872	−	0.877339i	$-0.659317\pi$
$113$	− 18.6525i	− 1.75468i	0.479872	−	0.877339i	$-0.340683\pi$
$114$	−2.76393	−0.258866
$115$	6.18034	0.576320
$116$	−3.70820	−0.344298
$117$	1.00000i	0.0924500i
$118$	− 8.47214i	− 0.779923i
$119$	−23.4164	−2.14658
$120$	2.23607i	0.204124i
$121$	9.00000	0.818182
$122$	10.9443i	0.990848i
$123$	− 3.52786i	− 0.318097i
$124$	4.00000	0.359211
$125$	− 11.1803i	− 1.00000i
$126$	−3.23607	−0.288292
$127$	21.4164i	1.90040i	0.311642	+	0.950199i	$0.399121\pi$
$127$	21.4164i	1.90040i	−0.311642	+	0.950199i	$0.600879\pi$
$128$	1.00000i	0.0883883i
$129$	2.47214	0.217659
$130$	− 2.23607i	− 0.196116i
$131$	9.70820	0.848210	0.424105	−	0.905613i	$-0.360589\pi$
$131$	9.70820	0.848210	0.424105	+	0.905613i	$0.360589\pi$
$132$	4.47214i	0.389249i
$133$	8.94427i	0.775567i
$134$	0	0
$135$	−2.23607	−0.192450
$136$	−7.23607	−0.620488
$137$	− 19.8885i	− 1.69919i	−0.527433	−	0.849596i	$-0.676846\pi$
$137$	− 19.8885i	− 1.69919i	0.527433	−	0.849596i	$-0.323154\pi$
$138$	2.76393i	0.235282i
$139$	6.47214	0.548959	0.274480	−	0.961593i	$-0.411494\pi$
$139$	6.47214	0.548959	0.274480	+	0.961593i	$0.411494\pi$
$140$	7.23607	0.611559
$141$	12.9443	1.09010
$142$	2.47214i	0.207457i
$143$	− 4.47214i	− 0.373979i
$144$	−1.00000	−0.0833333
$145$	8.29180i	0.688596i
$146$	−13.2361	−1.09542
$147$	3.47214i	0.286377i
$148$	10.9443i	0.899614i
$149$	−13.5279	−1.10825	−0.554123	−	0.832435i	$-0.686946\pi$
$149$	−13.5279	−1.10825	−0.554123	+	0.832435i	$0.686946\pi$
$150$	5.00000	0.408248
$151$	−19.4164	−1.58008	−0.790042	−	0.613052i	$-0.789942\pi$
$151$	−19.4164	−1.58008	−0.790042	+	0.613052i	$0.789942\pi$
$152$	2.76393i	0.224184i
$153$	− 7.23607i	− 0.585001i
$154$	14.4721	1.16620
$155$	− 8.94427i	− 0.718421i
$156$	1.00000	0.0800641
$157$	− 18.9443i	− 1.51192i	−0.654619	−	0.755959i	$-0.727171\pi$
$157$	− 18.9443i	− 1.51192i	0.654619	−	0.755959i	$-0.272829\pi$
$158$	− 4.00000i	− 0.318223i
$159$	0.472136	0.0374428
$160$	2.23607	0.176777
$161$	8.94427	0.704907
$162$	− 1.00000i	− 0.0785674i
$163$	− 12.9443i	− 1.01387i	−0.861983	−	0.506937i	$-0.830778\pi$
$163$	− 12.9443i	− 1.01387i	0.861983	−	0.506937i	$-0.169222\pi$
$164$	−3.52786	−0.275480
$165$	10.0000	0.778499
$166$	4.94427	0.383750
$167$	− 19.4164i	− 1.50249i	−0.660025	−	0.751243i	$-0.729454\pi$
$167$	− 19.4164i	− 1.50249i	0.660025	−	0.751243i	$-0.270546\pi$
$168$	3.23607i	0.249668i
$169$	−1.00000	−0.0769231
$170$	16.1803i	1.24098i
$171$	−2.76393	−0.211363
$172$	− 2.47214i	− 0.188499i
$173$	22.9443i	1.74442i	0.489131	+	0.872210i	$0.337314\pi$
$173$	22.9443i	1.74442i	−0.489131	+	0.872210i	$0.662686\pi$
$174$	−3.70820	−0.281118
$175$	− 16.1803i	− 1.22312i
$176$	4.47214	0.337100
$177$	− 8.47214i	− 0.636805i
$178$	0.472136i	0.0353881i
$179$	−8.18034	−0.611427	−0.305714	−	0.952124i	$-0.598895\pi$
$179$	−8.18034	−0.611427	−0.305714	+	0.952124i	$0.598895\pi$
$180$	2.23607i	0.166667i
$181$	17.4164	1.29455	0.647276	−	0.762256i	$-0.275908\pi$
$181$	17.4164	1.29455	0.647276	+	0.762256i	$0.275908\pi$
$182$	− 3.23607i	− 0.239873i
$183$	10.9443i	0.809024i
$184$	2.76393	0.203760
$185$	24.4721	1.79923
$186$	4.00000	0.293294
$187$	32.3607i	2.36645i
$188$	− 12.9443i	− 0.944058i
$189$	−3.23607	−0.235389
$190$	6.18034	0.448369
$191$	22.4721	1.62603	0.813013	−	0.582245i	$-0.197826\pi$
$191$	22.4721	1.62603	0.813013	+	0.582245i	$0.197826\pi$
$192$	1.00000i	0.0721688i
$193$	8.29180i	0.596857i	0.954432	+	0.298428i	$0.0964624\pi$
$193$	8.29180i	0.596857i	−0.954432	+	0.298428i	$0.903538\pi$
$194$	−3.70820	−0.266234
$195$	− 2.23607i	− 0.160128i
$196$	3.47214	0.248010
$197$	− 14.9443i	− 1.06474i	−0.846513	−	0.532368i	$-0.821302\pi$
$197$	− 14.9443i	− 1.06474i	0.846513	−	0.532368i	$-0.178698\pi$
$198$	4.47214i	0.317821i
$199$	0.944272	0.0669377	0.0334688	−	0.999440i	$-0.489345\pi$
$199$	0.944272	0.0669377	0.0334688	+	0.999440i	$0.489345\pi$
$200$	− 5.00000i	− 0.353553i
$201$	0	0
$202$	9.23607i	0.649847i
$203$	12.0000i	0.842235i
$204$	−7.23607	−0.506626
$205$	7.88854i	0.550960i
$206$	−8.47214	−0.590282
$207$	2.76393i	0.192107i
$208$	− 1.00000i	− 0.0693375i
$209$	12.3607	0.855006
$210$	7.23607	0.499336
$211$	13.8885	0.956127	0.478063	−	0.878325i	$-0.341339\pi$
$211$	13.8885	0.956127	0.478063	+	0.878325i	$0.341339\pi$
$212$	− 0.472136i	− 0.0324264i
$213$	2.47214i	0.169388i
$214$	−1.52786	−0.104443
$215$	−5.52786	−0.376997
$216$	−1.00000	−0.0680414
$217$	− 12.9443i	− 0.878714i
$218$	− 2.29180i	− 0.155220i
$219$	−13.2361	−0.894411
$220$	− 10.0000i	− 0.674200i
$221$	7.23607	0.486751
$222$	10.9443i	0.734531i
$223$	− 16.7639i	− 1.12260i	−0.827614	−	0.561298i	$-0.810302\pi$
$223$	− 16.7639i	− 1.12260i	0.827614	−	0.561298i	$-0.189698\pi$
$224$	3.23607	0.216219
$225$	5.00000	0.333333
$226$	−18.6525	−1.24074
$227$	7.05573i	0.468305i	0.972200	+	0.234153i	$0.0752315\pi$
$227$	7.05573i	0.468305i	−0.972200	+	0.234153i	$0.924768\pi$
$228$	2.76393i	0.183046i
$229$	−9.70820	−0.641536	−0.320768	−	0.947158i	$-0.603941\pi$
$229$	−9.70820	−0.641536	−0.320768	+	0.947158i	$0.603941\pi$
$230$	− 6.18034i	− 0.407520i
$231$	14.4721	0.952197
$232$	3.70820i	0.243456i
$233$	− 20.1803i	− 1.32206i	−0.750360	−	0.661029i	$-0.770120\pi$
$233$	− 20.1803i	− 1.32206i	0.750360	−	0.661029i	$-0.229880\pi$
$234$	1.00000	0.0653720
$235$	−28.9443	−1.88812
$236$	−8.47214	−0.551489
$237$	− 4.00000i	− 0.259828i
$238$	23.4164i	1.51786i
$239$	21.8885	1.41585	0.707926	−	0.706287i	$-0.249631\pi$
$239$	21.8885	1.41585	0.707926	+	0.706287i	$0.249631\pi$
$240$	2.23607	0.144338
$241$	3.52786	0.227250	0.113625	−	0.993524i	$-0.463754\pi$
$241$	3.52786	0.227250	0.113625	+	0.993524i	$0.463754\pi$
$242$	− 9.00000i	− 0.578542i
$243$	− 1.00000i	− 0.0641500i
$244$	10.9443	0.700635
$245$	− 7.76393i	− 0.496019i
$246$	−3.52786	−0.224928
$247$	− 2.76393i	− 0.175865i
$248$	− 4.00000i	− 0.254000i
$249$	4.94427	0.313331
$250$	−11.1803	−0.707107
$251$	2.29180	0.144657	0.0723284	−	0.997381i	$-0.476957\pi$
$251$	2.29180	0.144657	0.0723284	+	0.997381i	$0.476957\pi$
$252$	3.23607i	0.203853i
$253$	− 12.3607i	− 0.777109i
$254$	21.4164	1.34378
$255$	16.1803i	1.01325i
$256$	1.00000	0.0625000
$257$	0.763932i	0.0476528i	0.999716	+	0.0238264i	$0.00758489\pi$
$257$	0.763932i	0.0476528i	−0.999716	+	0.0238264i	$0.992415\pi$
$258$	− 2.47214i	− 0.153908i
$259$	35.4164	2.20067
$260$	−2.23607	−0.138675
$261$	−3.70820	−0.229532
$262$	− 9.70820i	− 0.599775i
$263$	− 10.1803i	− 0.627747i	−0.949465	−	0.313873i	$-0.898373\pi$
$263$	− 10.1803i	− 0.627747i	0.949465	−	0.313873i	$-0.101627\pi$
$264$	4.47214	0.275241
$265$	−1.05573	−0.0648529
$266$	8.94427	0.548408
$267$	0.472136i	0.0288943i
$268$	0	0
$269$	−6.76393	−0.412404	−0.206202	−	0.978509i	$-0.566110\pi$
$269$	−6.76393	−0.412404	−0.206202	+	0.978509i	$0.566110\pi$
$270$	2.23607i	0.136083i
$271$	−11.4164	−0.693497	−0.346749	−	0.937958i	$-0.612714\pi$
$271$	−11.4164	−0.693497	−0.346749	+	0.937958i	$0.612714\pi$
$272$	7.23607i	0.438751i
$273$	− 3.23607i	− 0.195856i
$274$	−19.8885	−1.20151
$275$	−22.3607	−1.34840
$276$	2.76393	0.166369
$277$	20.8328i	1.25172i	0.779934	+	0.625861i	$0.215252\pi$
$277$	20.8328i	1.25172i	−0.779934	+	0.625861i	$0.784748\pi$
$278$	− 6.47214i	− 0.388173i
$279$	4.00000	0.239474
$280$	− 7.23607i	− 0.432438i
$281$	24.4721	1.45989	0.729943	−	0.683508i	$-0.239547\pi$
$281$	24.4721	1.45989	0.729943	+	0.683508i	$0.239547\pi$
$282$	− 12.9443i	− 0.770820i
$283$	11.4164i	0.678635i	0.940672	+	0.339318i	$0.110196\pi$
$283$	11.4164i	0.678635i	−0.940672	+	0.339318i	$0.889804\pi$
$284$	2.47214	0.146694
$285$	6.18034	0.366092
$286$	−4.47214	−0.264443
$287$	11.4164i	0.673889i
$288$	1.00000i	0.0589256i
$289$	−35.3607	−2.08004
$290$	8.29180	0.486911
$291$	−3.70820	−0.217379
$292$	13.2361i	0.774582i
$293$	− 2.94427i	− 0.172006i	−0.996295	−	0.0860031i	$-0.972591\pi$
$293$	− 2.94427i	− 0.172006i	0.996295	−	0.0860031i	$-0.0274095\pi$
$294$	3.47214	0.202499
$295$	18.9443i	1.10298i
$296$	10.9443	0.636123
$297$	4.47214i	0.259500i
$298$	13.5279i	0.783648i
$299$	−2.76393	−0.159842
$300$	− 5.00000i	− 0.288675i
$301$	−8.00000	−0.461112
$302$	19.4164i	1.11729i
$303$	9.23607i	0.530598i
$304$	2.76393	0.158522
$305$	− 24.4721i	− 1.40127i
$306$	−7.23607	−0.413658
$307$	− 24.3607i	− 1.39034i	−0.718847	−	0.695169i	$-0.755330\pi$
$307$	− 24.3607i	− 1.39034i	0.718847	−	0.695169i	$-0.244670\pi$
$308$	− 14.4721i	− 0.824626i
$309$	−8.47214	−0.481963
$310$	−8.94427	−0.508001
$311$	−22.4721	−1.27428	−0.637139	−	0.770749i	$-0.719882\pi$
$311$	−22.4721	−1.27428	−0.637139	+	0.770749i	$0.719882\pi$
$312$	− 1.00000i	− 0.0566139i
$313$	19.4164i	1.09748i	0.835993	+	0.548740i	$0.184892\pi$
$313$	19.4164i	1.09748i	−0.835993	+	0.548740i	$0.815108\pi$
$314$	−18.9443	−1.06909
$315$	7.23607	0.407706
$316$	−4.00000	−0.225018
$317$	13.4164i	0.753541i	0.926307	+	0.376770i	$0.122965\pi$
$317$	13.4164i	0.753541i	−0.926307	+	0.376770i	$0.877035\pi$
$318$	− 0.472136i	− 0.0264761i
$319$	16.5836	0.928503
$320$	− 2.23607i	− 0.125000i
$321$	−1.52786	−0.0852771
$322$	− 8.94427i	− 0.498445i
$323$	20.0000i	1.11283i
$324$	−1.00000	−0.0555556
$325$	5.00000i	0.277350i
$326$	−12.9443	−0.716917
$327$	− 2.29180i	− 0.126737i
$328$	3.52786i	0.194794i
$329$	−41.8885	−2.30939
$330$	− 10.0000i	− 0.550482i
$331$	−2.76393	−0.151919	−0.0759597	−	0.997111i	$-0.524202\pi$
$331$	−2.76393	−0.151919	−0.0759597	+	0.997111i	$0.524202\pi$
$332$	− 4.94427i	− 0.271352i
$333$	10.9443i	0.599742i
$334$	−19.4164	−1.06242
$335$	0	0
$336$	3.23607	0.176542
$337$	− 6.47214i	− 0.352560i	−0.984340	−	0.176280i	$-0.943594\pi$
$337$	− 6.47214i	− 0.352560i	0.984340	−	0.176280i	$-0.0564064\pi$
$338$	1.00000i	0.0543928i
$339$	−18.6525	−1.01306
$340$	16.1803	0.877502
$341$	−17.8885	−0.968719
$342$	2.76393i	0.149456i
$343$	11.4164i	0.616428i
$344$	−2.47214	−0.133289
$345$	− 6.18034i	− 0.332738i
$346$	22.9443	1.23349
$347$	1.52786i	0.0820200i	0.999159	+	0.0410100i	$0.0130576\pi$
$347$	1.52786i	0.0820200i	−0.999159	+	0.0410100i	$0.986942\pi$
$348$	3.70820i	0.198781i
$349$	15.2361	0.815568	0.407784	−	0.913078i	$-0.366302\pi$
$349$	15.2361	0.815568	0.407784	+	0.913078i	$0.366302\pi$
$350$	−16.1803	−0.864876
$351$	1.00000	0.0533761
$352$	− 4.47214i	− 0.238366i
$353$	26.9443i	1.43410i	0.697022	+	0.717049i	$0.254508\pi$
$353$	26.9443i	1.43410i	−0.697022	+	0.717049i	$0.745492\pi$
$354$	−8.47214	−0.450289
$355$	− 5.52786i	− 0.293389i
$356$	0.472136	0.0250232
$357$	23.4164i	1.23933i
$358$	8.18034i	0.432344i
$359$	−6.47214	−0.341586	−0.170793	−	0.985307i	$-0.554633\pi$
$359$	−6.47214	−0.341586	−0.170793	+	0.985307i	$0.554633\pi$
$360$	2.23607	0.117851
$361$	−11.3607	−0.597931
$362$	− 17.4164i	− 0.915386i
$363$	− 9.00000i	− 0.472377i
$364$	−3.23607	−0.169616
$365$	29.5967	1.54916
$366$	10.9443	0.572066
$367$	0.472136i	0.0246453i	0.999924	+	0.0123226i	$0.00392252\pi$
$367$	0.472136i	0.0246453i	−0.999924	+	0.0123226i	$0.996077\pi$
$368$	− 2.76393i	− 0.144080i
$369$	−3.52786	−0.183653
$370$	− 24.4721i	− 1.27225i
$371$	−1.52786	−0.0793227
$372$	− 4.00000i	− 0.207390i
$373$	− 2.00000i	− 0.103556i	−0.998659	−	0.0517780i	$-0.983511\pi$
$373$	− 2.00000i	− 0.103556i	0.998659	−	0.0517780i	$-0.0164888\pi$
$374$	32.3607	1.67333
$375$	−11.1803	−0.577350
$376$	−12.9443	−0.667550
$377$	− 3.70820i	− 0.190982i
$378$	3.23607i	0.166445i
$379$	35.1246	1.80423	0.902115	−	0.431496i	$-0.142014\pi$
$379$	35.1246	1.80423	0.902115	+	0.431496i	$0.142014\pi$
$380$	− 6.18034i	− 0.317045i
$381$	21.4164	1.09720
$382$	− 22.4721i	− 1.14977i
$383$	− 5.52786i	− 0.282461i	−0.989977	−	0.141230i	$-0.954894\pi$
$383$	− 5.52786i	− 0.282461i	0.989977	−	0.141230i	$-0.0451058\pi$
$384$	1.00000	0.0510310
$385$	−32.3607	−1.64925
$386$	8.29180	0.422041
$387$	− 2.47214i	− 0.125666i
$388$	3.70820i	0.188256i
$389$	−12.6525	−0.641506	−0.320753	−	0.947163i	$-0.603936\pi$
$389$	−12.6525	−0.641506	−0.320753	+	0.947163i	$0.603936\pi$
$390$	−2.23607	−0.113228
$391$	20.0000	1.01144
$392$	− 3.47214i	− 0.175369i
$393$	− 9.70820i	− 0.489714i
$394$	−14.9443	−0.752882
$395$	8.94427i	0.450035i
$396$	4.47214	0.224733
$397$	17.4164i	0.874104i	0.899436	+	0.437052i	$0.143978\pi$
$397$	17.4164i	0.874104i	−0.899436	+	0.437052i	$0.856022\pi$
$398$	− 0.944272i	− 0.0473321i
$399$	8.94427	0.447774
$400$	−5.00000	−0.250000
$401$	−24.4721	−1.22208	−0.611040	−	0.791600i	$-0.709249\pi$
$401$	−24.4721	−1.22208	−0.611040	+	0.791600i	$0.709249\pi$
$402$	0	0
$403$	4.00000i	0.199254i
$404$	9.23607	0.459512
$405$	2.23607i	0.111111i
$406$	12.0000	0.595550
$407$	− 48.9443i	− 2.42608i
$408$	7.23607i	0.358239i
$409$	−6.00000	−0.296681	−0.148340	−	0.988936i	$-0.547393\pi$
$409$	−6.00000	−0.296681	−0.148340	+	0.988936i	$0.547393\pi$
$410$	7.88854	0.389587
$411$	−19.8885	−0.981030
$412$	8.47214i	0.417392i
$413$	27.4164i	1.34907i
$414$	2.76393	0.135840
$415$	−11.0557	−0.542704
$416$	−1.00000	−0.0490290
$417$	− 6.47214i	− 0.316942i
$418$	− 12.3607i	− 0.604581i
$419$	−27.5967	−1.34819	−0.674095	−	0.738645i	$-0.735466\pi$
$419$	−27.5967	−1.34819	−0.674095	+	0.738645i	$0.735466\pi$
$420$	− 7.23607i	− 0.353084i
$421$	−13.7082	−0.668097	−0.334048	−	0.942556i	$-0.608415\pi$
$421$	−13.7082	−0.668097	−0.334048	+	0.942556i	$0.608415\pi$
$422$	− 13.8885i	− 0.676084i
$423$	− 12.9443i	− 0.629372i
$424$	−0.472136	−0.0229289
$425$	− 36.1803i	− 1.75500i
$426$	2.47214	0.119775
$427$	− 35.4164i	− 1.71392i
$428$	1.52786i	0.0738521i
$429$	−4.47214	−0.215917
$430$	5.52786i	0.266577i
$431$	−14.8328	−0.714472	−0.357236	−	0.934014i	$-0.616281\pi$
$431$	−14.8328	−0.714472	−0.357236	+	0.934014i	$0.616281\pi$
$432$	1.00000i	0.0481125i
$433$	9.52786i	0.457880i	0.973441	+	0.228940i	$0.0735259\pi$
$433$	9.52786i	0.457880i	−0.973441	+	0.228940i	$0.926474\pi$
$434$	−12.9443	−0.621345
$435$	8.29180	0.397561
$436$	−2.29180	−0.109757
$437$	− 7.63932i	− 0.365438i
$438$	13.2361i	0.632444i
$439$	4.94427	0.235977	0.117989	−	0.993015i	$-0.462355\pi$
$439$	4.94427	0.235977	0.117989	+	0.993015i	$0.462355\pi$
$440$	−10.0000	−0.476731
$441$	3.47214	0.165340
$442$	− 7.23607i	− 0.344185i
$443$	22.4721i	1.06768i	0.845584	+	0.533842i	$0.179252\pi$
$443$	22.4721i	1.06768i	−0.845584	+	0.533842i	$0.820748\pi$
$444$	10.9443	0.519392
$445$	− 1.05573i	− 0.0500463i
$446$	−16.7639	−0.793795
$447$	13.5279i	0.639846i
$448$	− 3.23607i	− 0.152890i
$449$	−0.472136	−0.0222815	−0.0111407	−	0.999938i	$-0.503546\pi$
$449$	−0.472136	−0.0222815	−0.0111407	+	0.999938i	$0.503546\pi$
$450$	− 5.00000i	− 0.235702i
$451$	15.7771	0.742914
$452$	18.6525i	0.877339i
$453$	19.4164i	0.912262i
$454$	7.05573	0.331142
$455$	7.23607i	0.339232i
$456$	2.76393	0.129433
$457$	− 3.70820i	− 0.173462i	−0.996232	−	0.0867312i	$-0.972358\pi$
$457$	− 3.70820i	− 0.173462i	0.996232	−	0.0867312i	$-0.0276421\pi$
$458$	9.70820i	0.453635i
$459$	−7.23607	−0.337751
$460$	−6.18034	−0.288160
$461$	41.8885	1.95094	0.975472	−	0.220124i	$-0.0706461\pi$
$461$	41.8885	1.95094	0.975472	+	0.220124i	$0.0706461\pi$
$462$	− 14.4721i	− 0.673305i
$463$	22.2918i	1.03599i	0.855384	+	0.517994i	$0.173321\pi$
$463$	22.2918i	1.03599i	−0.855384	+	0.517994i	$0.826679\pi$
$464$	3.70820	0.172149
$465$	−8.94427	−0.414781
$466$	−20.1803	−0.934836
$467$	6.47214i	0.299495i	0.988724	+	0.149747i	$0.0478460\pi$
$467$	6.47214i	0.299495i	−0.988724	+	0.149747i	$0.952154\pi$
$468$	− 1.00000i	− 0.0462250i
$469$	0	0
$470$	28.9443i	1.33510i
$471$	−18.9443	−0.872906
$472$	8.47214i	0.389962i
$473$	11.0557i	0.508343i
$474$	−4.00000	−0.183726
$475$	−13.8197	−0.634089
$476$	23.4164	1.07329
$477$	− 0.472136i	− 0.0216176i
$478$	− 21.8885i	− 1.00116i
$479$	36.0000	1.64488	0.822441	−	0.568850i	$-0.192612\pi$
$479$	36.0000	1.64488	0.822441	+	0.568850i	$0.192612\pi$
$480$	− 2.23607i	− 0.102062i
$481$	−10.9443	−0.499016
$482$	− 3.52786i	− 0.160690i
$483$	− 8.94427i	− 0.406978i
$484$	−9.00000	−0.409091
$485$	8.29180	0.376511
$486$	−1.00000	−0.0453609
$487$	− 23.5967i	− 1.06927i	−0.845083	−	0.534635i	$-0.820449\pi$
$487$	− 23.5967i	− 1.06927i	0.845083	−	0.534635i	$-0.179551\pi$
$488$	− 10.9443i	− 0.495424i
$489$	−12.9443	−0.585360
$490$	−7.76393	−0.350739
$491$	−16.1803	−0.730209	−0.365104	−	0.930967i	$-0.618967\pi$
$491$	−16.1803	−0.730209	−0.365104	+	0.930967i	$0.618967\pi$
$492$	3.52786i	0.159048i
$493$	26.8328i	1.20849i
$494$	−2.76393	−0.124355
$495$	− 10.0000i	− 0.449467i
$496$	−4.00000	−0.179605
$497$	− 8.00000i	− 0.358849i
$498$	− 4.94427i	− 0.221558i
$499$	39.1246	1.75146	0.875729	−	0.482803i	$-0.160381\pi$
$499$	39.1246	1.75146	0.875729	+	0.482803i	$0.160381\pi$
$500$	11.1803i	0.500000i
$501$	−19.4164	−0.867461
$502$	− 2.29180i	− 0.102288i
$503$	30.7639i	1.37170i	0.727745	+	0.685848i	$0.240569\pi$
$503$	30.7639i	1.37170i	−0.727745	+	0.685848i	$0.759431\pi$
$504$	3.23607	0.144146
$505$	− 20.6525i	− 0.919023i
$506$	−12.3607	−0.549499
$507$	1.00000i	0.0444116i
$508$	− 21.4164i	− 0.950199i
$509$	15.0557	0.667333	0.333667	−	0.942691i	$-0.391714\pi$
$509$	15.0557	0.667333	0.333667	+	0.942691i	$0.391714\pi$
$510$	16.1803	0.716477
$511$	42.8328	1.89481
$512$	− 1.00000i	− 0.0441942i
$513$	2.76393i	0.122031i
$514$	0.763932	0.0336956
$515$	18.9443	0.834784
$516$	−2.47214	−0.108830
$517$	57.8885i	2.54594i
$518$	− 35.4164i	− 1.55611i
$519$	22.9443	1.00714
$520$	2.23607i	0.0980581i
$521$	20.8328	0.912702	0.456351	−	0.889800i	$-0.349156\pi$
$521$	20.8328	0.912702	0.456351	+	0.889800i	$0.349156\pi$
$522$	3.70820i	0.162304i
$523$	− 8.00000i	− 0.349816i	−0.984585	−	0.174908i	$-0.944037\pi$
$523$	− 8.00000i	− 0.349816i	0.984585	−	0.174908i	$-0.0559627\pi$
$524$	−9.70820	−0.424105
$525$	−16.1803	−0.706168
$526$	−10.1803	−0.443884
$527$	− 28.9443i	− 1.26083i
$528$	− 4.47214i	− 0.194625i
$529$	15.3607	0.667856
$530$	1.05573i	0.0458579i
$531$	−8.47214	−0.367659
$532$	− 8.94427i	− 0.387783i
$533$	− 3.52786i	− 0.152809i
$534$	0.472136	0.0204313
$535$	3.41641	0.147704
$536$	0	0
$537$	8.18034i	0.353008i
$538$	6.76393i	0.291614i
$539$	−15.5279	−0.668832
$540$	2.23607	0.0962250
$541$	−4.76393	−0.204817	−0.102409	−	0.994742i	$-0.532655\pi$
$541$	−4.76393	−0.204817	−0.102409	+	0.994742i	$0.532655\pi$
$542$	11.4164i	0.490377i
$543$	− 17.4164i	− 0.747410i
$544$	7.23607	0.310244
$545$	5.12461i	0.219514i
$546$	−3.23607	−0.138491
$547$	12.0000i	0.513083i	0.966533	+	0.256541i	$0.0825830\pi$
$547$	12.0000i	0.513083i	−0.966533	+	0.256541i	$0.917417\pi$
$548$	19.8885i	0.849596i
$549$	10.9443	0.467090
$550$	22.3607i	0.953463i
$551$	10.2492	0.436632
$552$	− 2.76393i	− 0.117641i
$553$	12.9443i	0.550446i
$554$	20.8328	0.885102
$555$	− 24.4721i	− 1.03878i
$556$	−6.47214	−0.274480
$557$	19.3050i	0.817977i	0.912540	+	0.408989i	$0.134118\pi$
$557$	19.3050i	0.817977i	−0.912540	+	0.408989i	$0.865882\pi$
$558$	− 4.00000i	− 0.169334i
$559$	2.47214	0.104560
$560$	−7.23607	−0.305780
$561$	32.3607	1.36627
$562$	− 24.4721i	− 1.03229i
$563$	19.4164i	0.818304i	0.912466	+	0.409152i	$0.134175\pi$
$563$	19.4164i	0.818304i	−0.912466	+	0.409152i	$0.865825\pi$
$564$	−12.9443	−0.545052
$565$	41.7082	1.75468
$566$	11.4164	0.479867
$567$	3.23607i	0.135902i
$568$	− 2.47214i	− 0.103729i
$569$	−34.9443	−1.46494	−0.732470	−	0.680799i	$-0.761633\pi$
$569$	−34.9443	−1.46494	−0.732470	+	0.680799i	$0.761633\pi$
$570$	− 6.18034i	− 0.258866i
$571$	−8.58359	−0.359212	−0.179606	−	0.983739i	$-0.557482\pi$
$571$	−8.58359	−0.359212	−0.179606	+	0.983739i	$0.557482\pi$
$572$	4.47214i	0.186989i
$573$	− 22.4721i	− 0.938787i
$574$	11.4164	0.476512
$575$	13.8197i	0.576320i
$576$	1.00000	0.0416667
$577$	− 43.1246i	− 1.79530i	−0.440708	−	0.897651i	$-0.645273\pi$
$577$	− 43.1246i	− 1.79530i	0.440708	−	0.897651i	$-0.354727\pi$
$578$	35.3607i	1.47081i
$579$	8.29180	0.344595
$580$	− 8.29180i	− 0.344298i
$581$	−16.0000	−0.663792
$582$	3.70820i	0.153710i
$583$	2.11146i	0.0874476i
$584$	13.2361	0.547712
$585$	−2.23607	−0.0924500
$586$	−2.94427	−0.121627
$587$	− 2.11146i	− 0.0871491i	−0.999050	−	0.0435746i	$-0.986125\pi$
$587$	− 2.11146i	− 0.0871491i	0.999050	−	0.0435746i	$-0.0138746\pi$
$588$	− 3.47214i	− 0.143188i
$589$	−11.0557	−0.455543
$590$	18.9443	0.779923
$591$	−14.9443	−0.614725
$592$	− 10.9443i	− 0.449807i
$593$	− 3.52786i	− 0.144872i	−0.997373	−	0.0724360i	$-0.976923\pi$
$593$	− 3.52786i	− 0.144872i	0.997373	−	0.0724360i	$-0.0230773\pi$
$594$	4.47214	0.183494
$595$	− 52.3607i	− 2.14658i
$596$	13.5279	0.554123
$597$	− 0.944272i	− 0.0386465i
$598$	2.76393i	0.113026i
$599$	6.11146	0.249707	0.124854	−	0.992175i	$-0.460154\pi$
$599$	6.11146	0.249707	0.124854	+	0.992175i	$0.460154\pi$
$600$	−5.00000	−0.204124
$601$	13.4164	0.547267	0.273633	−	0.961834i	$-0.411775\pi$
$601$	13.4164	0.547267	0.273633	+	0.961834i	$0.411775\pi$
$602$	8.00000i	0.326056i
$603$	0	0
$604$	19.4164	0.790042
$605$	20.1246i	0.818182i
$606$	9.23607	0.375190
$607$	1.41641i	0.0574902i	0.999587	+	0.0287451i	$0.00915111\pi$
$607$	1.41641i	0.0574902i	−0.999587	+	0.0287451i	$0.990849\pi$
$608$	− 2.76393i	− 0.112092i
$609$	12.0000	0.486265
$610$	−24.4721	−0.990848
$611$	12.9443	0.523669
$612$	7.23607i	0.292501i
$613$	− 18.0000i	− 0.727013i	−0.931592	−	0.363507i	$-0.881579\pi$
$613$	− 18.0000i	− 0.727013i	0.931592	−	0.363507i	$-0.118421\pi$
$614$	−24.3607	−0.983117
$615$	7.88854	0.318097
$616$	−14.4721	−0.583099
$617$	35.8885i	1.44482i	0.691466	+	0.722409i	$0.256965\pi$
$617$	35.8885i	1.44482i	−0.691466	+	0.722409i	$0.743035\pi$
$618$	8.47214i	0.340799i
$619$	25.2361	1.01432	0.507162	−	0.861851i	$-0.330695\pi$
$619$	25.2361	1.01432	0.507162	+	0.861851i	$0.330695\pi$
$620$	8.94427i	0.359211i
$621$	2.76393	0.110913
$622$	22.4721i	0.901051i
$623$	− 1.52786i	− 0.0612126i
$624$	−1.00000	−0.0400320
$625$	25.0000	1.00000
$626$	19.4164	0.776036
$627$	− 12.3607i	− 0.493638i
$628$	18.9443i	0.755959i
$629$	79.1935	3.15765
$630$	− 7.23607i	− 0.288292i
$631$	−12.5836	−0.500945	−0.250472	−	0.968124i	$-0.580586\pi$
$631$	−12.5836	−0.500945	−0.250472	+	0.968124i	$0.580586\pi$
$632$	4.00000i	0.159111i
$633$	− 13.8885i	− 0.552020i
$634$	13.4164	0.532834
$635$	−47.8885	−1.90040
$636$	−0.472136	−0.0187214
$637$	3.47214i	0.137571i
$638$	− 16.5836i	− 0.656551i
$639$	2.47214	0.0977962
$640$	−2.23607	−0.0883883
$641$	−23.3050	−0.920490	−0.460245	−	0.887792i	$-0.652238\pi$
$641$	−23.3050	−0.920490	−0.460245	+	0.887792i	$0.652238\pi$
$642$	1.52786i	0.0603000i
$643$	− 19.0557i	− 0.751485i	−0.926724	−	0.375742i	$-0.877388\pi$
$643$	− 19.0557i	− 0.751485i	0.926724	−	0.375742i	$-0.122612\pi$
$644$	−8.94427	−0.352454
$645$	5.52786i	0.217659i
$646$	20.0000	0.786889
$647$	− 3.70820i	− 0.145785i	−0.997340	−	0.0728923i	$-0.976777\pi$
$647$	− 3.70820i	− 0.145785i	0.997340	−	0.0728923i	$-0.0232229\pi$
$648$	1.00000i	0.0392837i
$649$	37.8885	1.48726
$650$	5.00000	0.196116
$651$	−12.9443	−0.507326
$652$	12.9443i	0.506937i
$653$	30.3607i	1.18811i	0.804426	+	0.594053i	$0.202473\pi$
$653$	30.3607i	1.18811i	−0.804426	+	0.594053i	$0.797527\pi$
$654$	−2.29180	−0.0896163
$655$	21.7082i	0.848210i
$656$	3.52786	0.137740
$657$	13.2361i	0.516388i
$658$	41.8885i	1.63299i
$659$	10.0689	0.392228	0.196114	−	0.980581i	$-0.437168\pi$
$659$	10.0689	0.392228	0.196114	+	0.980581i	$0.437168\pi$
$660$	−10.0000	−0.389249
$661$	−41.7082	−1.62226	−0.811131	−	0.584865i	$-0.801147\pi$
$661$	−41.7082	−1.62226	−0.811131	+	0.584865i	$0.801147\pi$
$662$	2.76393i	0.107423i
$663$	− 7.23607i	− 0.281026i
$664$	−4.94427	−0.191875
$665$	−20.0000	−0.775567
$666$	10.9443	0.424082
$667$	− 10.2492i	− 0.396852i
$668$	19.4164i	0.751243i
$669$	−16.7639	−0.648131
$670$	0	0
$671$	−48.9443	−1.88947
$672$	− 3.23607i	− 0.124834i
$673$	− 44.0000i	− 1.69608i	−0.529936	−	0.848038i	$-0.677784\pi$
$673$	− 44.0000i	− 1.69608i	0.529936	−	0.848038i	$-0.322216\pi$
$674$	−6.47214	−0.249297
$675$	− 5.00000i	− 0.192450i
$676$	1.00000	0.0384615
$677$	27.5279i	1.05798i	0.848628	+	0.528991i	$0.177429\pi$
$677$	27.5279i	1.05798i	−0.848628	+	0.528991i	$0.822571\pi$
$678$	18.6525i	0.716344i
$679$	12.0000	0.460518
$680$	− 16.1803i	− 0.620488i
$681$	7.05573	0.270376
$682$	17.8885i	0.684988i
$683$	41.8885i	1.60282i	0.598115	+	0.801410i	$0.295917\pi$
$683$	41.8885i	1.60282i	−0.598115	+	0.801410i	$0.704083\pi$
$684$	2.76393	0.105682
$685$	44.4721	1.69919
$686$	11.4164	0.435880
$687$	9.70820i	0.370391i
$688$	2.47214i	0.0942493i
$689$	0.472136	0.0179869
$690$	−6.18034	−0.235282
$691$	0.291796	0.0111004	0.00555022	−	0.999985i	$-0.498233\pi$
$691$	0.291796	0.0111004	0.00555022	+	0.999985i	$0.498233\pi$
$692$	− 22.9443i	− 0.872210i
$693$	− 14.4721i	− 0.549751i
$694$	1.52786	0.0579969
$695$	14.4721i	0.548959i
$696$	3.70820	0.140559
$697$	25.5279i	0.966937i
$698$	− 15.2361i	− 0.576694i
$699$	−20.1803	−0.763291
$700$	16.1803i	0.611559i
$701$	−24.0689	−0.909069	−0.454535	−	0.890729i	$-0.650194\pi$
$701$	−24.0689	−0.909069	−0.454535	+	0.890729i	$0.650194\pi$
$702$	− 1.00000i	− 0.0377426i
$703$	− 30.2492i	− 1.14087i
$704$	−4.47214	−0.168550
$705$	28.9443i	1.09010i
$706$	26.9443	1.01406
$707$	− 29.8885i	− 1.12407i
$708$	8.47214i	0.318402i
$709$	−32.5410	−1.22210	−0.611052	−	0.791591i	$-0.709253\pi$
$709$	−32.5410	−1.22210	−0.611052	+	0.791591i	$0.709253\pi$
$710$	−5.52786	−0.207457
$711$	−4.00000	−0.150012
$712$	− 0.472136i	− 0.0176940i
$713$	11.0557i	0.414040i
$714$	23.4164	0.876337
$715$	10.0000	0.373979
$716$	8.18034	0.305714
$717$	− 21.8885i	− 0.817443i
$718$	6.47214i	0.241538i
$719$	−20.9443	−0.781090	−0.390545	−	0.920584i	$-0.627713\pi$
$719$	−20.9443	−0.781090	−0.390545	+	0.920584i	$0.627713\pi$
$720$	− 2.23607i	− 0.0833333i
$721$	27.4164	1.02104
$722$	11.3607i	0.422801i
$723$	− 3.52786i	− 0.131203i
$724$	−17.4164	−0.647276
$725$	−18.5410	−0.688596
$726$	−9.00000	−0.334021
$727$	18.0000i	0.667583i	0.942647	+	0.333792i	$0.108328\pi$
$727$	18.0000i	0.667583i	−0.942647	+	0.333792i	$0.891672\pi$
$728$	3.23607i	0.119937i
$729$	−1.00000	−0.0370370
$730$	− 29.5967i	− 1.09542i
$731$	−17.8885	−0.661632
$732$	− 10.9443i	− 0.404512i
$733$	− 33.4164i	− 1.23426i	−0.786860	−	0.617132i	$-0.788295\pi$
$733$	− 33.4164i	− 1.23426i	0.786860	−	0.617132i	$-0.211705\pi$
$734$	0.472136	0.0174269
$735$	−7.76393	−0.286377
$736$	−2.76393	−0.101880
$737$	0	0
$738$	3.52786i	0.129862i
$739$	0.291796	0.0107339	0.00536695	−	0.999986i	$-0.498292\pi$
$739$	0.291796	0.0107339	0.00536695	+	0.999986i	$0.498292\pi$
$740$	−24.4721	−0.899614
$741$	−2.76393	−0.101536
$742$	1.52786i	0.0560897i
$743$	− 28.3607i	− 1.04045i	−0.854029	−	0.520226i	$-0.825848\pi$
$743$	− 28.3607i	− 1.04045i	0.854029	−	0.520226i	$-0.174152\pi$
$744$	−4.00000	−0.146647
$745$	− 30.2492i	− 1.10825i
$746$	−2.00000	−0.0732252
$747$	− 4.94427i	− 0.180901i
$748$	− 32.3607i	− 1.18322i
$749$	4.94427	0.180660
$750$	11.1803i	0.408248i
$751$	−46.8328	−1.70895	−0.854477	−	0.519489i	$-0.826122\pi$
$751$	−46.8328	−1.70895	−0.854477	+	0.519489i	$0.826122\pi$
$752$	12.9443i	0.472029i
$753$	− 2.29180i	− 0.0835177i
$754$	−3.70820	−0.135045
$755$	− 43.4164i	− 1.58008i
$756$	3.23607	0.117695
$757$	− 38.0000i	− 1.38113i	−0.723269	−	0.690567i	$-0.757361\pi$
$757$	− 38.0000i	− 1.38113i	0.723269	−	0.690567i	$-0.242639\pi$
$758$	− 35.1246i	− 1.27578i
$759$	−12.3607	−0.448664
$760$	−6.18034	−0.224184
$761$	7.52786	0.272885	0.136442	−	0.990648i	$-0.456433\pi$
$761$	7.52786	0.272885	0.136442	+	0.990648i	$0.456433\pi$
$762$	− 21.4164i	− 0.775835i
$763$	7.41641i	0.268492i
$764$	−22.4721	−0.813013
$765$	16.1803	0.585001
$766$	−5.52786	−0.199730
$767$	− 8.47214i	− 0.305911i
$768$	− 1.00000i	− 0.0360844i
$769$	4.83282	0.174276	0.0871379	−	0.996196i	$-0.472228\pi$
$769$	4.83282	0.174276	0.0871379	+	0.996196i	$0.472228\pi$
$770$	32.3607i	1.16620i
$771$	0.763932	0.0275123
$772$	− 8.29180i	− 0.298428i
$773$	2.94427i	0.105898i	0.998597	+	0.0529491i	$0.0168621\pi$
$773$	2.94427i	0.105898i	−0.998597	+	0.0529491i	$0.983138\pi$
$774$	−2.47214	−0.0888591
$775$	20.0000	0.718421
$776$	3.70820	0.133117
$777$	− 35.4164i	− 1.27056i
$778$	12.6525i	0.453613i
$779$	9.75078	0.349358
$780$	2.23607i	0.0800641i
$781$	−11.0557	−0.395605
$782$	− 20.0000i	− 0.715199i
$783$	3.70820i	0.132520i
$784$	−3.47214	−0.124005
$785$	42.3607	1.51192
$786$	−9.70820	−0.346280
$787$	28.9443i	1.03175i	0.856663	+	0.515876i	$0.172533\pi$
$787$	28.9443i	1.03175i	−0.856663	+	0.515876i	$0.827467\pi$
$788$	14.9443i	0.532368i
$789$	−10.1803	−0.362430
$790$	8.94427	0.318223
$791$	60.3607	2.14618
$792$	− 4.47214i	− 0.158910i
$793$	10.9443i	0.388642i
$794$	17.4164	0.618085
$795$	1.05573i	0.0374428i
$796$	−0.944272	−0.0334688
$797$	9.05573i	0.320770i	0.987055	+	0.160385i	$0.0512736\pi$
$797$	9.05573i	0.320770i	−0.987055	+	0.160385i	$0.948726\pi$
$798$	− 8.94427i	− 0.316624i
$799$	−93.6656	−3.31365
$800$	5.00000i	0.176777i
$801$	0.472136	0.0166821
$802$	24.4721i	0.864141i
$803$	− 59.1935i	− 2.08889i
$804$	0	0
$805$	20.0000i	0.704907i
$806$	4.00000	0.140894
$807$	6.76393i	0.238102i
$808$	− 9.23607i	− 0.324924i
$809$	−31.5279	−1.10846	−0.554230	−	0.832363i	$-0.686987\pi$
$809$	−31.5279	−1.10846	−0.554230	+	0.832363i	$0.686987\pi$
$810$	2.23607	0.0785674
$811$	4.29180	0.150705	0.0753527	−	0.997157i	$-0.475992\pi$
$811$	4.29180	0.150705	0.0753527	+	0.997157i	$0.475992\pi$
$812$	− 12.0000i	− 0.421117i
$813$	11.4164i	0.400391i
$814$	−48.9443	−1.71550
$815$	28.9443	1.01387
$816$	7.23607	0.253313
$817$	6.83282i	0.239050i
$818$	6.00000i	0.209785i
$819$	−3.23607	−0.113077
$820$	− 7.88854i	− 0.275480i
$821$	−30.4721	−1.06348	−0.531742	−	0.846906i	$-0.678463\pi$
$821$	−30.4721	−1.06348	−0.531742	+	0.846906i	$0.678463\pi$
$822$	19.8885i	0.693693i
$823$	− 48.2492i	− 1.68186i	−0.541142	−	0.840931i	$-0.682008\pi$
$823$	− 48.2492i	− 1.68186i	0.541142	−	0.840931i	$-0.317992\pi$
$824$	8.47214	0.295141
$825$	22.3607i	0.778499i
$826$	27.4164	0.953939
$827$	− 41.8885i	− 1.45661i	−0.685254	−	0.728304i	$-0.740309\pi$
$827$	− 41.8885i	− 1.45661i	0.685254	−	0.728304i	$-0.259691\pi$
$828$	− 2.76393i	− 0.0960533i
$829$	−42.7214	−1.48377	−0.741887	−	0.670525i	$-0.766069\pi$
$829$	−42.7214	−1.48377	−0.741887	+	0.670525i	$0.766069\pi$
$830$	11.0557i	0.383750i
$831$	20.8328	0.722682
$832$	1.00000i	0.0346688i
$833$	− 25.1246i	− 0.870516i
$834$	−6.47214	−0.224112
$835$	43.4164	1.50249
$836$	−12.3607	−0.427503
$837$	− 4.00000i	− 0.138260i
$838$	27.5967i	0.953314i
$839$	51.7771	1.78754	0.893772	−	0.448522i	$-0.148049\pi$
$839$	51.7771	1.78754	0.893772	+	0.448522i	$0.148049\pi$
$840$	−7.23607	−0.249668
$841$	−15.2492	−0.525835
$842$	13.7082i	0.472416i
$843$	− 24.4721i	− 0.842865i
$844$	−13.8885	−0.478063
$845$	− 2.23607i	− 0.0769231i
$846$	−12.9443	−0.445033
$847$	29.1246i	1.00073i
$848$	0.472136i	0.0162132i
$849$	11.4164	0.391810
$850$	−36.1803	−1.24098
$851$	−30.2492	−1.03693
$852$	− 2.47214i	− 0.0846940i
$853$	− 41.4164i	− 1.41807i	−0.705173	−	0.709035i	$-0.749131\pi$
$853$	− 41.4164i	− 1.41807i	0.705173	−	0.709035i	$-0.250869\pi$
$854$	−35.4164	−1.21192
$855$	− 6.18034i	− 0.211363i
$856$	1.52786	0.0522213
$857$	− 0.180340i	− 0.00616029i	−0.999995	−	0.00308015i	$-0.999020\pi$
$857$	− 0.180340i	− 0.00616029i	0.999995	−	0.00308015i	$-0.000980443\pi$
$858$	4.47214i	0.152676i
$859$	−36.0000	−1.22830	−0.614152	−	0.789188i	$-0.710502\pi$
$859$	−36.0000	−1.22830	−0.614152	+	0.789188i	$0.710502\pi$
$860$	5.52786	0.188499
$861$	11.4164	0.389070
$862$	14.8328i	0.505208i
$863$	− 22.4721i	− 0.764960i	−0.923964	−	0.382480i	$-0.875070\pi$
$863$	− 22.4721i	− 0.764960i	0.923964	−	0.382480i	$-0.124930\pi$
$864$	1.00000	0.0340207
$865$	−51.3050	−1.74442
$866$	9.52786	0.323770
$867$	35.3607i	1.20091i
$868$	12.9443i	0.439357i
$869$	17.8885	0.606827
$870$	− 8.29180i	− 0.281118i
$871$	0	0
$872$	2.29180i	0.0776100i
$873$	3.70820i	0.125504i
$874$	−7.63932	−0.258404
$875$	36.1803	1.22312
$876$	13.2361	0.447205
$877$	24.4721i	0.826365i	0.910648	+	0.413183i	$0.135583\pi$
$877$	24.4721i	0.826365i	−0.910648	+	0.413183i	$0.864417\pi$
$878$	− 4.94427i	− 0.166861i
$879$	−2.94427	−0.0993078
$880$	10.0000i	0.337100i
$881$	34.3607	1.15764	0.578820	−	0.815455i	$-0.303513\pi$
$881$	34.3607	1.15764	0.578820	+	0.815455i	$0.303513\pi$
$882$	− 3.47214i	− 0.116913i
$883$	− 25.8885i	− 0.871219i	−0.900136	−	0.435609i	$-0.856533\pi$
$883$	− 25.8885i	− 0.871219i	0.900136	−	0.435609i	$-0.143467\pi$
$884$	−7.23607	−0.243375
$885$	18.9443	0.636805
$886$	22.4721	0.754966
$887$	− 10.1803i	− 0.341822i	−0.985286	−	0.170911i	$-0.945329\pi$
$887$	− 10.1803i	− 0.341822i	0.985286	−	0.170911i	$-0.0546711\pi$
$888$	− 10.9443i	− 0.367266i
$889$	−69.3050	−2.32441
$890$	−1.05573	−0.0353881
$891$	4.47214	0.149822
$892$	16.7639i	0.561298i
$893$	35.7771i	1.19723i
$894$	13.5279	0.452439
$895$	− 18.2918i	− 0.611427i
$896$	−3.23607	−0.108109
$897$	2.76393i	0.0922850i
$898$	0.472136i	0.0157554i
$899$	−14.8328	−0.494702
$900$	−5.00000	−0.166667
$901$	−3.41641	−0.113817
$902$	− 15.7771i	− 0.525320i
$903$	8.00000i	0.266223i
$904$	18.6525	0.620372
$905$	38.9443i	1.29455i
$906$	19.4164	0.645067
$907$	45.5279i	1.51173i	0.654729	+	0.755864i	$0.272783\pi$
$907$	45.5279i	1.51173i	−0.654729	+	0.755864i	$0.727217\pi$
$908$	− 7.05573i	− 0.234153i
$909$	9.23607	0.306341
$910$	7.23607	0.239873
$911$	−11.0557	−0.366293	−0.183146	−	0.983086i	$-0.558628\pi$
$911$	−11.0557	−0.366293	−0.183146	+	0.983086i	$0.558628\pi$
$912$	− 2.76393i	− 0.0915229i
$913$	22.1115i	0.731782i
$914$	−3.70820	−0.122656
$915$	−24.4721	−0.809024
$916$	9.70820	0.320768
$917$	31.4164i	1.03746i
$918$	7.23607i	0.238826i
$919$	0	0		−	1.00000i	$-0.5\pi$
$919$	0	0			1.00000i	$0.5\pi$
$920$	6.18034i	0.203760i
$921$	−24.3607	−0.802712
$922$	− 41.8885i	− 1.37953i
$923$	2.47214i	0.0813713i
$924$	−14.4721	−0.476098
$925$	54.7214i	1.79923i
$926$	22.2918	0.732554
$927$	8.47214i	0.278261i
$928$	− 3.70820i	− 0.121728i
$929$	−32.4721	−1.06538	−0.532688	−	0.846312i	$-0.678818\pi$
$929$	−32.4721	−1.06538	−0.532688	+	0.846312i	$0.678818\pi$
$930$	8.94427i	0.293294i
$931$	−9.59675	−0.314521
$932$	20.1803i	0.661029i
$933$	22.4721i	0.735705i
$934$	6.47214	0.211775
$935$	−72.3607	−2.36645
$936$	−1.00000	−0.0326860
$937$	19.7771i	0.646089i	0.946384	+	0.323045i	$0.104706\pi$
$937$	19.7771i	0.646089i	−0.946384	+	0.323045i	$0.895294\pi$
$938$	0	0
$939$	19.4164	0.633631
$940$	28.9443	0.944058
$941$	−37.5279	−1.22337	−0.611687	−	0.791100i	$-0.709509\pi$
$941$	−37.5279	−1.22337	−0.611687	+	0.791100i	$0.709509\pi$
$942$	18.9443i	0.617238i
$943$	− 9.75078i	− 0.317529i
$944$	8.47214	0.275745
$945$	− 7.23607i	− 0.235389i
$946$	11.0557	0.359453
$947$	4.94427i	0.160667i	0.996768	+	0.0803336i	$0.0255986\pi$
$947$	4.94427i	0.160667i	−0.996768	+	0.0803336i	$0.974401\pi$
$948$	4.00000i	0.129914i
$949$	−13.2361	−0.429661
$950$	13.8197i	0.448369i
$951$	13.4164	0.435057
$952$	− 23.4164i	− 0.758930i
$953$	− 29.1246i	− 0.943439i	−0.881749	−	0.471719i	$-0.843634\pi$
$953$	− 29.1246i	− 0.943439i	0.881749	−	0.471719i	$-0.156366\pi$
$954$	−0.472136	−0.0152860
$955$	50.2492i	1.62603i
$956$	−21.8885	−0.707926
$957$	− 16.5836i	− 0.536071i
$958$	− 36.0000i	− 1.16311i
$959$	64.3607	2.07831
$960$	−2.23607	−0.0721688
$961$	−15.0000	−0.483871
$962$	10.9443i	0.352857i
$963$	1.52786i	0.0492347i
$964$	−3.52786	−0.113625
$965$	−18.5410	−0.596857
$966$	−8.94427	−0.287777
$967$	− 56.5410i	− 1.81824i	−0.416538	−	0.909118i	$-0.636757\pi$
$967$	− 56.5410i	− 1.81824i	0.416538	−	0.909118i	$-0.363243\pi$
$968$	9.00000i	0.289271i
$969$	20.0000	0.642493
$970$	− 8.29180i	− 0.266234i
$971$	23.2361	0.745681	0.372840	−	0.927895i	$-0.378384\pi$
$971$	23.2361	0.745681	0.372840	+	0.927895i	$0.378384\pi$
$972$	1.00000i	0.0320750i
$973$	20.9443i	0.671443i
$974$	−23.5967	−0.756089
$975$	5.00000	0.160128
$976$	−10.9443	−0.350318
$977$	− 10.3607i	− 0.331468i	−0.986171	−	0.165734i	$-0.947001\pi$
$977$	− 10.3607i	− 0.331468i	0.986171	−	0.165734i	$-0.0529992\pi$
$978$	12.9443i	0.413912i
$979$	−2.11146	−0.0674824
$980$	7.76393i	0.248010i
$981$	−2.29180	−0.0731714
$982$	16.1803i	0.516335i
$983$	7.41641i	0.236547i	0.992981	+	0.118273i	$0.0377359\pi$
$983$	7.41641i	0.236547i	−0.992981	+	0.118273i	$0.962264\pi$
$984$	3.52786	0.112464
$985$	33.4164	1.06474
$986$	26.8328	0.854531
$987$	41.8885i	1.33333i
$988$	2.76393i	0.0879324i
$989$	6.83282	0.217271
$990$	−10.0000	−0.317821
$991$	24.9443	0.792381	0.396190	−	0.918168i	$-0.370332\pi$
$991$	24.9443	0.792381	0.396190	+	0.918168i	$0.370332\pi$
$992$	4.00000i	0.127000i
$993$	2.76393i	0.0877107i
$994$	−8.00000	−0.253745
$995$	2.11146i	0.0669377i
$996$	−4.94427	−0.156665
$997$	43.8885i	1.38996i	0.719027	+	0.694982i	$0.244588\pi$
$997$	43.8885i	1.38996i	−0.719027	+	0.694982i	$0.755412\pi$
$998$	− 39.1246i	− 1.23847i
$999$	10.9443	0.346261

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	390.2.e.e.79.2	✓	4
3.2	odd	2		1170.2.e.e.469.3		4
4.3	odd	2		3120.2.l.k.1249.4		4
5.2	odd	4		1950.2.a.bf.1.1		2
5.3	odd	4		1950.2.a.be.1.2		2
5.4	even	2	inner	390.2.e.e.79.3	yes	4
15.2	even	4		5850.2.a.cf.1.1		2
15.8	even	4		5850.2.a.cm.1.2		2
15.14	odd	2		1170.2.e.e.469.2		4
20.19	odd	2		3120.2.l.k.1249.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
390.2.e.e.79.2	✓	4	1.1	even	1	trivial
390.2.e.e.79.3	yes	4	5.4	even	2	inner
1170.2.e.e.469.2		4	15.14	odd	2
1170.2.e.e.469.3		4	3.2	odd	2
1950.2.a.be.1.2		2	5.3	odd	4
1950.2.a.bf.1.1		2	5.2	odd	4
3120.2.l.k.1249.1		4	20.19	odd	2
3120.2.l.k.1249.4		4	4.3	odd	2
5850.2.a.cf.1.1		2	15.2	even	4
5850.2.a.cm.1.2		2	15.8	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:	\( N \)	\(=\)	\( 390 = 2 \cdot 3 \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	390.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +2.23607i q^{5} -1.00000 q^{6} +3.23607i q^{7} +1.00000i q^{8} -1.00000 q^{9} +2.23607 q^{10} +4.47214 q^{11} +1.00000i q^{12} -1.00000i q^{13} +3.23607 q^{14} +2.23607 q^{15} +1.00000 q^{16} +7.23607i q^{17} +1.00000i q^{18} +2.76393 q^{19} -2.23607i q^{20} +3.23607 q^{21} -4.47214i q^{22} -2.76393i q^{23} +1.00000 q^{24} -5.00000 q^{25} -1.00000 q^{26} +1.00000i q^{27} -3.23607i q^{28} +3.70820 q^{29} -2.23607i q^{30} -4.00000 q^{31} -1.00000i q^{32} -4.47214i q^{33} +7.23607 q^{34} -7.23607 q^{35} +1.00000 q^{36} -10.9443i q^{37} -2.76393i q^{38} -1.00000 q^{39} -2.23607 q^{40} +3.52786 q^{41} -3.23607i q^{42} +2.47214i q^{43} -4.47214 q^{44} -2.23607i q^{45} -2.76393 q^{46} +12.9443i q^{47} -1.00000i q^{48} -3.47214 q^{49} +5.00000i q^{50} +7.23607 q^{51} +1.00000i q^{52} +0.472136i q^{53} +1.00000 q^{54} +10.0000i q^{55} -3.23607 q^{56} -2.76393i q^{57} -3.70820i q^{58} +8.47214 q^{59} -2.23607 q^{60} -10.9443 q^{61} +4.00000i q^{62} -3.23607i q^{63} -1.00000 q^{64} +2.23607 q^{65} -4.47214 q^{66} -7.23607i q^{68} -2.76393 q^{69} +7.23607i q^{70} -2.47214 q^{71} -1.00000i q^{72} -13.2361i q^{73} -10.9443 q^{74} +5.00000i q^{75} -2.76393 q^{76} +14.4721i q^{77} +1.00000i q^{78} +4.00000 q^{79} +2.23607i q^{80} +1.00000 q^{81} -3.52786i q^{82} +4.94427i q^{83} -3.23607 q^{84} -16.1803 q^{85} +2.47214 q^{86} -3.70820i q^{87} +4.47214i q^{88} -0.472136 q^{89} -2.23607 q^{90} +3.23607 q^{91} +2.76393i q^{92} +4.00000i q^{93} +12.9443 q^{94} +6.18034i q^{95} -1.00000 q^{96} -3.70820i q^{97} +3.47214i q^{98} -4.47214 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{4} - 4 q^{6} - 4 q^{9} + 4 q^{14} + 4 q^{16} + 20 q^{19} + 4 q^{21} + 4 q^{24} - 20 q^{25} - 4 q^{26} - 12 q^{29} - 16 q^{31} + 20 q^{34} - 20 q^{35} + 4 q^{36} - 4 q^{39} + 32 q^{41} - 20 q^{46}+ \cdots - 4 q^{96}+O(q^{100}) \) 4 * q - 4 * q^4 - 4 * q^6 - 4 * q^9 + 4 * q^14 + 4 * q^16 + 20 * q^19 + 4 * q^21 + 4 * q^24 - 20 * q^25 - 4 * q^26 - 12 * q^29 - 16 * q^31 + 20 * q^34 - 20 * q^35 + 4 * q^36 - 4 * q^39 + 32 * q^41 - 20 * q^46 + 4 * q^49 + 20 * q^51 + 4 * q^54 - 4 * q^56 + 16 * q^59 - 8 * q^61 - 4 * q^64 - 20 * q^69 + 8 * q^71 - 8 * q^74 - 20 * q^76 + 16 * q^79 + 4 * q^81 - 4 * q^84 - 20 * q^85 - 8 * q^86 + 16 * q^89 + 4 * q^91 + 16 * q^94 - 4 * q^96

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more