LMFDB - Embedded newform 1287.2.b.c.298.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1287,2,Mod(298,1287)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1287, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1287.298");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1287 = 3^{2} \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1287.b (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:	$10.2767467401$
Analytic rank:	$0$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} + 23x^{12} + 201x^{10} + 835x^{8} + 1695x^{6} + 1565x^{4} + 511x^{2} + 49 $ x^14 + 23x^12 + 201x^10 + 835x^8 + 1695x^6 + 1565x^4 + 511x^2 + 49
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	no (minimal twist has level 429)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			298.14
Root			$2.73878i$ of defining polynomial
Character	$\chi$	$=$	1287.298
Dual form			1287.2.b.c.298.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.73878i q^{2} -5.50093 q^{4} -2.84154i q^{5} +3.93129i q^{7} -9.58828i q^{8} +O(q^{10})$	\(q+2.73878i q^{2} -5.50093 q^{4} -2.84154i q^{5} +3.93129i q^{7} -9.58828i q^{8} +7.78236 q^{10} -1.00000i q^{11} +(1.78444 - 3.13301i) q^{13} -10.7670 q^{14} +15.2584 q^{16} +3.81818 q^{17} +2.94082i q^{19} +15.6311i q^{20} +2.73878 q^{22} -1.89484 q^{23} -3.07435 q^{25} +(8.58064 + 4.88720i) q^{26} -21.6258i q^{28} +2.09928 q^{29} +6.16032i q^{31} +22.6127i q^{32} +10.4572i q^{34} +11.1709 q^{35} +8.34404i q^{37} -8.05426 q^{38} -27.2455 q^{40} +6.35787i q^{41} +11.7275 q^{43} +5.50093i q^{44} -5.18956i q^{46} -5.31137i q^{47} -8.45506 q^{49} -8.41997i q^{50} +(-9.81609 + 17.2345i) q^{52} +2.37985 q^{53} -2.84154 q^{55} +37.6943 q^{56} +5.74947i q^{58} +5.38624i q^{59} -2.34857 q^{61} -16.8718 q^{62} -31.4147 q^{64} +(-8.90258 - 5.07056i) q^{65} -10.4731i q^{67} -21.0035 q^{68} +30.5947i q^{70} +10.0939i q^{71} -15.1079i q^{73} -22.8525 q^{74} -16.1772i q^{76} +3.93129 q^{77} +1.57120 q^{79} -43.3572i q^{80} -17.4128 q^{82} +10.6736i q^{83} -10.8495i q^{85} +32.1190i q^{86} -9.58828 q^{88} -3.23647i q^{89} +(12.3168 + 7.01516i) q^{91} +10.4234 q^{92} +14.5467 q^{94} +8.35645 q^{95} +17.7914i q^{97} -23.1566i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q - 18 q^{4}+O(q^{10}) $ 14 * q - 18 * q^4	$ 14 q - 18 q^{4} - 16 q^{14} + 34 q^{16} - 4 q^{17} + 6 q^{22} + 8 q^{23} - 26 q^{25} + 6 q^{26} + 24 q^{29} + 8 q^{35} + 32 q^{38} - 20 q^{40} + 32 q^{43} - 46 q^{49} + 4 q^{52} - 20 q^{53} + 12 q^{55} + 32 q^{56} - 20 q^{61} - 72 q^{62} - 58 q^{64} - 12 q^{65} + 20 q^{68} + 12 q^{77} + 12 q^{79} + 20 q^{82} - 30 q^{88} + 16 q^{91} + 24 q^{92} + 64 q^{94} + 36 q^{95}+O(q^{100}) $ 14 * q - 18 * q^4 - 16 * q^14 + 34 * q^16 - 4 * q^17 + 6 * q^22 + 8 * q^23 - 26 * q^25 + 6 * q^26 + 24 * q^29 + 8 * q^35 + 32 * q^38 - 20 * q^40 + 32 * q^43 - 46 * q^49 + 4 * q^52 - 20 * q^53 + 12 * q^55 + 32 * q^56 - 20 * q^61 - 72 * q^62 - 58 * q^64 - 12 * q^65 + 20 * q^68 + 12 * q^77 + 12 * q^79 + 20 * q^82 - 30 * q^88 + 16 * q^91 + 24 * q^92 + 64 * q^94 + 36 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1287\mathbb{Z}\right)^\times$.

$n$	$496$	$937$	$1145$
$\chi(n)$	$-1$	$1$	$1$

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 $

$2$			2.73878i			1.93661i	0.249768	+	0.968306i	$0.419646\pi$
$2$			2.73878i			1.93661i	−0.249768	+	0.968306i	$0.580354\pi$
$3$		0			0
$3$		0			0
$4$	−5.50093			−2.75046
$5$		−	2.84154i		−	1.27078i	−0.772193	−	0.635388i	$-0.780840\pi$
$5$		−	2.84154i		−	1.27078i	0.772193	−	0.635388i	$-0.219160\pi$
$6$		0			0
$7$			3.93129i			1.48589i	0.669353	+	0.742944i	$0.266571\pi$
$7$			3.93129i			1.48589i	−0.669353	+	0.742944i	$0.733429\pi$
$8$		−	9.58828i		−	3.38997i
$9$		0			0
$10$	7.78236			2.46100
$11$		−	1.00000i		−	0.301511i
$11$		−	1.00000i		−	0.301511i
$12$		0			0
$13$	1.78444	−	3.13301i	0.494915	−	0.868941i
$13$	1.78444	−	3.13301i	0.494915	−	0.868941i
$14$	−10.7670			−2.87759
$15$		0			0
$16$	15.2584			3.81459
$17$	3.81818			0.926044			0.463022	−	0.886347i	$-0.346765\pi$
$17$	3.81818			0.926044			0.463022	+	0.886347i	$0.346765\pi$
$18$		0			0
$19$			2.94082i			0.674670i	0.941385	+	0.337335i	$0.109526\pi$
$19$			2.94082i			0.674670i	−0.941385	+	0.337335i	$0.890474\pi$
$20$			15.6311i			3.49522i
$21$		0			0
$22$	2.73878			0.583910
$23$	−1.89484			−0.395102			−0.197551	−	0.980293i	$-0.563299\pi$
$23$	−1.89484			−0.395102			−0.197551	+	0.980293i	$0.563299\pi$
$24$		0			0
$25$	−3.07435			−0.614869
$26$	8.58064	+	4.88720i	1.68280	+	0.958459i
$27$		0			0
$28$		−	21.6258i		−	4.08688i
$29$	2.09928			0.389826			0.194913	−	0.980821i	$-0.437557\pi$
$29$	2.09928			0.389826			0.194913	+	0.980821i	$0.437557\pi$
$30$		0			0
$31$			6.16032i			1.10643i	0.833040	+	0.553213i	$0.186598\pi$
$31$			6.16032i			1.10643i	−0.833040	+	0.553213i	$0.813402\pi$
$32$			22.6127i			3.99741i
$33$		0			0
$34$			10.4572i			1.79339i
$35$	11.1709			1.88823
$36$		0			0
$37$			8.34404i			1.37175i	0.727719	+	0.685876i	$0.240581\pi$
$37$			8.34404i			1.37175i	−0.727719	+	0.685876i	$0.759419\pi$
$38$	−8.05426			−1.30657
$39$		0			0
$40$	−27.2455			−4.30789
$41$			6.35787i			0.992933i	0.868056	+	0.496466i	$0.165369\pi$
$41$			6.35787i			0.992933i	−0.868056	+	0.496466i	$0.834631\pi$
$42$		0			0
$43$	11.7275			1.78842			0.894212	−	0.447643i	$-0.147736\pi$
$43$	11.7275			1.78842			0.894212	+	0.447643i	$0.147736\pi$
$44$			5.50093i			0.829296i
$45$		0			0
$46$		−	5.18956i		−	0.765159i
$47$		−	5.31137i		−	0.774743i	−0.921924	−	0.387371i	$-0.873383\pi$
$47$		−	5.31137i		−	0.774743i	0.921924	−	0.387371i	$-0.126617\pi$
$48$		0			0
$49$	−8.45506			−1.20787
$50$		−	8.41997i		−	1.19076i
$51$		0			0
$52$	−9.81609	+	17.2345i	−1.36125	+	2.38999i
$53$	2.37985			0.326898			0.163449	−	0.986552i	$-0.447738\pi$
$53$	2.37985			0.326898			0.163449	+	0.986552i	$0.447738\pi$
$54$		0			0
$55$	−2.84154			−0.383153
$56$	37.6943			5.03712
$57$		0			0
$58$			5.74947i			0.754942i
$59$			5.38624i			0.701229i	0.936520	+	0.350615i	$0.114027\pi$
$59$			5.38624i			0.701229i	−0.936520	+	0.350615i	$0.885973\pi$
$60$		0			0
$61$	−2.34857			−0.300703			−0.150352	−	0.988633i	$-0.548041\pi$
$61$	−2.34857			−0.300703			−0.150352	+	0.988633i	$0.548041\pi$
$62$	−16.8718			−2.14272
$63$		0			0
$64$	−31.4147			−3.92684
$65$	−8.90258	−	5.07056i	−1.10423	−	0.628926i
$66$		0			0
$67$		−	10.4731i		−	1.27949i	−0.768588	−	0.639744i	$-0.779040\pi$
$67$		−	10.4731i		−	1.27949i	0.768588	−	0.639744i	$-0.220960\pi$
$68$	−21.0035			−2.54705
$69$		0			0
$70$			30.5947i			3.65677i
$71$			10.0939i			1.19793i	0.800776	+	0.598964i	$0.204421\pi$
$71$			10.0939i			1.19793i	−0.800776	+	0.598964i	$0.795579\pi$
$72$		0			0
$73$		−	15.1079i		−	1.76824i	−0.467260	−	0.884120i	$-0.654759\pi$
$73$		−	15.1079i		−	1.76824i	0.467260	−	0.884120i	$-0.345241\pi$
$74$	−22.8525			−2.65655
$75$		0			0
$76$		−	16.1772i		−	1.85566i
$77$	3.93129			0.448012
$78$		0			0
$79$	1.57120			0.176774			0.0883872	−	0.996086i	$-0.471829\pi$
$79$	1.57120			0.176774			0.0883872	+	0.996086i	$0.471829\pi$
$80$		−	43.3572i		−	4.84748i
$81$		0			0
$82$	−17.4128			−1.92293
$83$			10.6736i			1.17157i	0.810465	+	0.585787i	$0.199215\pi$
$83$			10.6736i			1.17157i	−0.810465	+	0.585787i	$0.800785\pi$
$84$		0			0
$85$		−	10.8495i		−	1.17679i
$86$			32.1190i			3.46348i
$87$		0			0
$88$	−9.58828			−1.02211
$89$		−	3.23647i		−	0.343065i	−0.985178	−	0.171533i	$-0.945128\pi$
$89$		−	3.23647i		−	0.343065i	0.985178	−	0.171533i	$-0.0548719\pi$
$90$		0			0
$91$	12.3168	+	7.01516i	1.29115	+	0.735389i
$92$	10.4234			1.08671
$93$		0			0
$94$	14.5467			1.50038
$95$	8.35645			0.857354
$96$		0			0
$97$			17.7914i			1.80645i	0.429172	+	0.903223i	$0.358805\pi$
$97$			17.7914i			1.80645i	−0.429172	+	0.903223i	$0.641195\pi$
$98$		−	23.1566i		−	2.33917i
$99$		0			0
$100$	16.9118			1.69118
$101$	−12.1229			−1.20627			−0.603136	−	0.797638i	$-0.706083\pi$
$101$	−12.1229			−1.20627			−0.603136	+	0.797638i	$0.706083\pi$
$102$		0			0
$103$	3.40983			0.335981			0.167990	−	0.985789i	$-0.446272\pi$
$103$	3.40983			0.335981			0.167990	+	0.985789i	$0.446272\pi$
$104$	−30.0402	−	17.1097i	−2.94568	−	1.67775i
$105$		0			0
$106$			6.51790i			0.633075i
$107$	12.2352			1.18282			0.591411	−	0.806370i	$-0.298571\pi$
$107$	12.2352			1.18282			0.591411	+	0.806370i	$0.298571\pi$
$108$		0			0
$109$			8.72763i			0.835956i	0.908457	+	0.417978i	$0.137261\pi$
$109$			8.72763i			0.835956i	−0.908457	+	0.417978i	$0.862739\pi$
$110$		−	7.78236i		−	0.742019i
$111$		0			0
$112$			59.9850i			5.66805i
$113$	−0.0155525			−0.00146305			−0.000731527	−	1.00000i	$-0.500233\pi$
$113$	−0.0155525			−0.00146305			−0.000731527		1.00000i	$0.500233\pi$
$114$		0			0
$115$			5.38427i			0.502086i
$116$	−11.5480			−1.07220
$117$		0			0
$118$	−14.7517			−1.35801
$119$			15.0104i			1.37600i
$120$		0			0
$121$	−1.00000			−0.0909091
$122$		−	6.43222i		−	0.582346i
$123$		0			0
$124$		−	33.8875i		−	3.04318i
$125$		−	5.47182i		−	0.489414i
$126$		0			0
$127$	5.91514			0.524883			0.262442	−	0.964948i	$-0.415472\pi$
$127$	5.91514			0.524883			0.262442	+	0.964948i	$0.415472\pi$
$128$		−	40.8125i		−	3.60735i
$129$		0			0
$130$	13.8872	−	24.3822i	1.21799	−	2.13846i
$131$	1.32191			0.115496			0.0577481	−	0.998331i	$-0.481608\pi$
$131$	1.32191			0.115496			0.0577481	+	0.998331i	$0.481608\pi$
$132$		0			0
$133$	−11.5612			−1.00248
$134$	28.6835			2.47787
$135$		0			0
$136$		−	36.6097i		−	3.13926i
$137$		−	4.82127i		−	0.411909i	−0.978562	−	0.205955i	$-0.933970\pi$
$137$		−	4.82127i		−	0.411909i	0.978562	−	0.205955i	$-0.0660299\pi$
$138$		0			0
$139$	17.1877			1.45784			0.728919	−	0.684600i	$-0.240023\pi$
$139$	17.1877			1.45784			0.728919	+	0.684600i	$0.240023\pi$
$140$	−61.4504			−5.19351
$141$		0			0
$142$	−27.6451			−2.31992
$143$	−3.13301	−	1.78444i	−0.261996	−	0.149223i
$144$		0			0
$145$		−	5.96518i		−	0.495382i
$146$	41.3771			3.42439
$147$		0			0
$148$		−	45.8999i		−	3.77295i
$149$		−	5.31754i		−	0.435629i	−0.975990	−	0.217815i	$-0.930107\pi$
$149$		−	5.31754i		−	0.435629i	0.975990	−	0.217815i	$-0.0698929\pi$
$150$		0			0
$151$		−	6.77573i		−	0.551401i	−0.961244	−	0.275700i	$-0.911090\pi$
$151$		−	6.77573i		−	0.551401i	0.961244	−	0.275700i	$-0.0889098\pi$
$152$	28.1974			2.28711
$153$		0			0
$154$			10.7670i			0.867626i
$155$	17.5048			1.40602
$156$		0			0
$157$	16.2884			1.29995			0.649977	−	0.759954i	$-0.274779\pi$
$157$	16.2884			1.29995			0.649977	+	0.759954i	$0.274779\pi$
$158$			4.30319i			0.342343i
$159$		0			0
$160$	64.2550			5.07981
$161$		−	7.44918i		−	0.587077i
$162$		0			0
$163$			11.4032i			0.893170i	0.894741	+	0.446585i	$0.147360\pi$
$163$			11.4032i			0.893170i	−0.894741	+	0.446585i	$0.852640\pi$
$164$		−	34.9742i		−	2.73103i
$165$		0			0
$166$	−29.2325			−2.26888
$167$			14.7302i			1.13985i	0.821695	+	0.569927i	$0.193029\pi$
$167$			14.7302i			1.13985i	−0.821695	+	0.569927i	$0.806971\pi$
$168$		0			0
$169$	−6.63153	−	11.1814i	−0.510118	−	0.860105i
$170$	29.7144			2.27899
$171$		0			0
$172$	−64.5121			−4.91900
$173$	0.186647			0.0141905			0.00709527	−	0.999975i	$-0.497741\pi$
$173$	0.186647			0.0141905			0.00709527	+	0.999975i	$0.497741\pi$
$174$		0			0
$175$		−	12.0862i		−	0.913627i
$176$		−	15.2584i		−	1.15014i
$177$		0			0
$178$	8.86399			0.664384
$179$	10.9555			0.818856			0.409428	−	0.912342i	$-0.365728\pi$
$179$	10.9555			0.818856			0.409428	+	0.912342i	$0.365728\pi$
$180$		0			0
$181$	8.52034			0.633312			0.316656	−	0.948540i	$-0.397440\pi$
$181$	8.52034			0.633312			0.316656	+	0.948540i	$0.397440\pi$
$182$	−19.2130	+	33.7330i	−1.42416	+	2.50046i
$183$		0			0
$184$			18.1683i			1.33938i
$185$	23.7099			1.74319
$186$		0			0
$187$		−	3.81818i		−	0.279213i
$188$			29.2175i			2.13090i
$189$		0			0
$190$			22.8865i			1.66036i
$191$	11.8003			0.853840			0.426920	−	0.904289i	$-0.359599\pi$
$191$	11.8003			0.853840			0.426920	+	0.904289i	$0.359599\pi$
$192$		0			0
$193$			10.9405i			0.787518i	0.919214	+	0.393759i	$0.128826\pi$
$193$			10.9405i			0.787518i	−0.919214	+	0.393759i	$0.871174\pi$
$194$	−48.7268			−3.49838
$195$		0			0
$196$	46.5107			3.32219
$197$			9.43307i			0.672079i	0.941848	+	0.336039i	$0.109088\pi$
$197$			9.43307i			0.672079i	−0.941848	+	0.336039i	$0.890912\pi$
$198$		0			0
$199$	15.3198			1.08599			0.542996	−	0.839735i	$-0.317290\pi$
$199$	15.3198			1.08599			0.542996	+	0.839735i	$0.317290\pi$
$200$			29.4777i			2.08439i
$201$		0			0
$202$		−	33.2020i		−	2.33608i
$203$			8.25288i			0.579238i
$204$		0			0
$205$	18.0661			1.26179
$206$			9.33878i			0.650664i
$207$		0			0
$208$	27.2277	−	47.8046i	1.88790	−	3.31465i
$209$	2.94082			0.203421
$210$		0			0
$211$	2.91243			0.200500			0.100250	−	0.994962i	$-0.468036\pi$
$211$	2.91243			0.200500			0.100250	+	0.994962i	$0.468036\pi$
$212$	−13.0914			−0.899122
$213$		0			0
$214$			33.5096i			2.29067i
$215$		−	33.3241i		−	2.27269i
$216$		0			0
$217$	−24.2180			−1.64403
$218$	−23.9031			−1.61892
$219$		0			0
$220$	15.6311			1.05385
$221$	6.81332	−	11.9624i	0.458313	−	0.804677i
$222$		0			0
$223$		−	10.8511i		−	0.726641i	−0.931664	−	0.363320i	$-0.881643\pi$
$223$		−	10.8511i		−	0.726641i	0.931664	−	0.363320i	$-0.118357\pi$
$224$	−88.8973			−5.93970
$225$		0			0
$226$		−	0.0425948i		−	0.00283337i
$227$			0.322636i			0.0214141i	0.999943	+	0.0107070i	$0.00340822\pi$
$227$			0.322636i			0.0214141i	−0.999943	+	0.0107070i	$0.996592\pi$
$228$		0			0
$229$		−	5.94060i		−	0.392566i	−0.980547	−	0.196283i	$-0.937113\pi$
$229$		−	5.94060i		−	0.392566i	0.980547	−	0.196283i	$-0.0628871\pi$
$230$	−14.7463			−0.972345
$231$		0			0
$232$		−	20.1285i		−	1.32150i
$233$	−12.6202			−0.826774			−0.413387	−	0.910555i	$-0.635654\pi$
$233$	−12.6202			−0.826774			−0.413387	+	0.910555i	$0.635654\pi$
$234$		0			0
$235$	−15.0925			−0.984524
$236$		−	29.6293i		−	1.92871i
$237$		0			0
$238$	−41.1101			−2.66477
$239$			20.7706i			1.34354i	0.740762	+	0.671768i	$0.234465\pi$
$239$			20.7706i			1.34354i	−0.740762	+	0.671768i	$0.765535\pi$
$240$		0			0
$241$		−	18.2163i		−	1.17341i	−0.809800	−	0.586706i	$-0.800424\pi$
$241$		−	18.2163i		−	1.17341i	0.809800	−	0.586706i	$-0.199576\pi$
$242$		−	2.73878i		−	0.176056i
$243$		0			0
$244$	12.9193			0.827074
$245$			24.0254i			1.53492i
$246$		0			0
$247$	9.21362	+	5.24772i	0.586249	+	0.333905i
$248$	59.0668			3.75075
$249$		0			0
$250$	14.9861			0.947806
$251$	−22.6662			−1.43068			−0.715340	−	0.698776i	$-0.753728\pi$
$251$	−22.6662			−1.43068			−0.715340	+	0.698776i	$0.753728\pi$
$252$		0			0
$253$			1.89484i			0.119128i
$254$			16.2003i			1.01650i
$255$		0			0
$256$	48.9471			3.05920
$257$	17.7333			1.10617			0.553087	−	0.833124i	$-0.313450\pi$
$257$	17.7333			1.10617			0.553087	+	0.833124i	$0.313450\pi$
$258$		0			0
$259$	−32.8028			−2.03827
$260$	48.9724	+	27.8928i	3.03714	+	1.72984i
$261$		0			0
$262$			3.62044i			0.223671i
$263$	−29.5107			−1.81971			−0.909853	−	0.414930i	$-0.863806\pi$
$263$	−29.5107			−1.81971			−0.909853	+	0.414930i	$0.863806\pi$
$264$		0			0
$265$		−	6.76245i		−	0.415414i
$266$		−	31.6637i		−	1.94142i
$267$		0			0
$268$			57.6116i			3.51919i
$269$	9.02091			0.550015			0.275007	−	0.961442i	$-0.411320\pi$
$269$	9.02091			0.550015			0.275007	+	0.961442i	$0.411320\pi$
$270$		0			0
$271$			4.58320i			0.278410i	0.990264	+	0.139205i	$0.0444547\pi$
$271$			4.58320i			0.278410i	−0.990264	+	0.139205i	$0.955545\pi$
$272$	58.2591			3.53248
$273$		0			0
$274$	13.2044			0.797708
$275$			3.07435i			0.185390i
$276$		0			0
$277$	−20.9859			−1.26092			−0.630461	−	0.776221i	$-0.717134\pi$
$277$	−20.9859			−1.26092			−0.630461	+	0.776221i	$0.717134\pi$
$278$			47.0732i			2.82327i
$279$		0			0
$280$		−	107.110i		−	6.40104i
$281$		−	10.5360i		−	0.628524i	−0.949336	−	0.314262i	$-0.898243\pi$
$281$		−	10.5360i		−	0.628524i	0.949336	−	0.314262i	$-0.101757\pi$
$282$		0			0
$283$	2.57169			0.152871			0.0764355	−	0.997075i	$-0.475646\pi$
$283$	2.57169			0.152871			0.0764355	+	0.997075i	$0.475646\pi$
$284$		−	55.5260i		−	3.29486i
$285$		0			0
$286$	4.88720	−	8.58064i	0.288986	−	0.507384i
$287$	−24.9947			−1.47539
$288$		0			0
$289$	−2.42153			−0.142443
$290$	16.3373			0.959362
$291$		0			0
$292$			83.1072i			4.86348i
$293$		−	5.16162i		−	0.301545i	−0.988568	−	0.150772i	$-0.951824\pi$
$293$		−	5.16162i		−	0.301545i	0.988568	−	0.150772i	$-0.0481761\pi$
$294$		0			0
$295$	15.3052			0.891105
$296$	80.0050			4.65019
$297$		0			0
$298$	14.5636			0.843645
$299$	−3.38124	+	5.93656i	−0.195542	+	0.343320i
$300$		0			0
$301$			46.1042i			2.65740i
$302$	18.5572			1.06785
$303$		0			0
$304$			44.8720i			2.57359i
$305$			6.67355i			0.382126i
$306$		0			0
$307$		−	30.0561i		−	1.71539i	−0.514156	−	0.857697i	$-0.671895\pi$
$307$		−	30.0561i		−	1.71539i	0.514156	−	0.857697i	$-0.328105\pi$
$308$	−21.6258			−1.23224
$309$		0			0
$310$			47.9418i			2.72291i
$311$	−21.7462			−1.23311			−0.616557	−	0.787310i	$-0.711473\pi$
$311$	−21.7462			−1.23311			−0.616557	+	0.787310i	$0.711473\pi$
$312$		0			0
$313$	−24.4361			−1.38121			−0.690606	−	0.723231i	$-0.742656\pi$
$313$	−24.4361			−1.38121			−0.690606	+	0.723231i	$0.742656\pi$
$314$			44.6103i			2.51751i
$315$		0			0
$316$	−8.64309			−0.486212
$317$			21.1040i			1.18532i	0.805454	+	0.592658i	$0.201921\pi$
$317$			21.1040i			1.18532i	−0.805454	+	0.592658i	$0.798079\pi$
$318$		0			0
$319$		−	2.09928i		−	0.117537i
$320$			89.2661i			4.99013i
$321$		0			0
$322$	20.4017			1.13694
$323$			11.2286i			0.624774i
$324$		0			0
$325$	−5.48600	+	9.63196i	−0.304308	+	0.534285i
$326$	−31.2310			−1.72972
$327$		0			0
$328$	60.9611			3.36601
$329$	20.8805			1.15118
$330$		0			0
$331$		−	0.560518i		−	0.0308089i	−0.999881	−	0.0154044i	$-0.995096\pi$
$331$		−	0.560518i		−	0.0308089i	0.999881	−	0.0154044i	$-0.00490358\pi$
$332$		−	58.7144i		−	3.22237i
$333$		0			0
$334$	−40.3427			−2.20745
$335$	−29.7596			−1.62594
$336$		0			0
$337$	7.76850			0.423177			0.211589	−	0.977359i	$-0.432136\pi$
$337$	7.76850			0.423177			0.211589	+	0.977359i	$0.432136\pi$
$338$	30.6233	−	18.1623i	1.66569	−	0.987900i
$339$		0			0
$340$			59.6823i			3.23673i
$341$	6.16032			0.333600
$342$		0			0
$343$		−	5.72025i		−	0.308864i
$344$		−	112.446i		−	6.06270i
$345$		0			0
$346$			0.511186i			0.0274816i
$347$	10.1357			0.544110			0.272055	−	0.962282i	$-0.412297\pi$
$347$	10.1357			0.544110			0.272055	+	0.962282i	$0.412297\pi$
$348$		0			0
$349$		−	20.0030i		−	1.07074i	−0.844619	−	0.535369i	$-0.820173\pi$
$349$		−	20.0030i		−	1.07074i	0.844619	−	0.535369i	$-0.179827\pi$
$350$	33.1013			1.76934
$351$		0			0
$352$	22.6127			1.20526
$353$		−	26.7905i		−	1.42591i	−0.701208	−	0.712957i	$-0.747355\pi$
$353$		−	26.7905i		−	1.42591i	0.701208	−	0.712957i	$-0.252645\pi$
$354$		0			0
$355$	28.6823			1.52230
$356$			17.8036i			0.943589i
$357$		0			0
$358$			30.0049i			1.58581i
$359$		−	26.7347i		−	1.41101i	−0.708707	−	0.705503i	$-0.750721\pi$
$359$		−	26.7347i		−	1.41101i	0.708707	−	0.705503i	$-0.249279\pi$
$360$		0			0
$361$	10.3516			0.544820
$362$			23.3354i			1.22648i
$363$		0			0
$364$	−67.7537	−	38.5899i	−3.55126	−	2.02266i
$365$	−42.9296			−2.24704
$366$		0			0
$367$	1.89563			0.0989512			0.0494756	−	0.998775i	$-0.484245\pi$
$367$	1.89563			0.0989512			0.0494756	+	0.998775i	$0.484245\pi$
$368$	−28.9122			−1.50715
$369$		0			0
$370$			64.9363i			3.37588i
$371$			9.35590i			0.485734i
$372$		0			0
$373$	−36.7517			−1.90293			−0.951464	−	0.307759i	$-0.900421\pi$
$373$	−36.7517			−1.90293			−0.951464	+	0.307759i	$0.900421\pi$
$374$	10.4572			0.540726
$375$		0			0
$376$	−50.9269			−2.62635
$377$	3.74604	−	6.57707i	0.192931	−	0.338736i
$378$		0			0
$379$			26.1328i			1.34235i	0.741298	+	0.671176i	$0.234211\pi$
$379$			26.1328i			1.34235i	−0.741298	+	0.671176i	$0.765789\pi$
$380$	−45.9682			−2.35812
$381$		0			0
$382$			32.3185i			1.65356i
$383$			16.9211i			0.864629i	0.901723	+	0.432315i	$0.142303\pi$
$383$			16.9211i			0.864629i	−0.901723	+	0.432315i	$0.857697\pi$
$384$		0			0
$385$		−	11.1709i		−	0.569323i
$386$	−29.9638			−1.52512
$387$		0			0
$388$		−	97.8693i		−	4.96856i
$389$	−5.05607			−0.256353			−0.128177	−	0.991751i	$-0.540912\pi$
$389$	−5.05607			−0.256353			−0.128177	+	0.991751i	$0.540912\pi$
$390$		0			0
$391$	−7.23484			−0.365882
$392$			81.0694i			4.09462i
$393$		0			0
$394$	−25.8351			−1.30156
$395$		−	4.46464i		−	0.224640i
$396$		0			0
$397$			15.7731i			0.791631i	0.918330	+	0.395815i	$0.129538\pi$
$397$			15.7731i			0.791631i	−0.918330	+	0.395815i	$0.870462\pi$
$398$			41.9576i			2.10314i
$399$		0			0
$400$	−46.9095			−2.34547
$401$		−	8.42218i		−	0.420584i	−0.977639	−	0.210292i	$-0.932559\pi$
$401$		−	8.42218i		−	0.420584i	0.977639	−	0.210292i	$-0.0674414\pi$
$402$		0			0
$403$	19.3003	+	10.9927i	0.961419	+	0.547587i
$404$	66.6872			3.31781
$405$		0			0
$406$	−22.6028			−1.12176
$407$	8.34404			0.413599
$408$		0			0
$409$		−	31.3861i		−	1.55194i	−0.630768	−	0.775971i	$-0.717260\pi$
$409$		−	31.3861i		−	1.55194i	0.630768	−	0.775971i	$-0.282740\pi$
$410$			49.4792i			2.44361i
$411$		0			0
$412$	−18.7572			−0.924102
$413$	−21.1749			−1.04195
$414$		0			0
$415$	30.3293			1.48881
$416$	70.8460	+	40.3512i	3.47351	+	1.97838i
$417$		0			0
$418$			8.05426i			0.393947i
$419$	6.62928			0.323861			0.161931	−	0.986802i	$-0.448228\pi$
$419$	6.62928			0.323861			0.161931	+	0.986802i	$0.448228\pi$
$420$		0			0
$421$			2.61327i			0.127363i	0.997970	+	0.0636816i	$0.0202842\pi$
$421$			2.61327i			0.127363i	−0.997970	+	0.0636816i	$0.979716\pi$
$422$			7.97651i			0.388290i
$423$		0			0
$424$		−	22.8187i		−	1.10817i
$425$	−11.7384			−0.569396
$426$		0			0
$427$		−	9.23291i		−	0.446812i
$428$	−67.3050			−3.25331
$429$		0			0
$430$	91.2675			4.40131
$431$		−	24.8745i		−	1.19816i	−0.800688	−	0.599081i	$-0.795533\pi$
$431$		−	24.8745i		−	1.19816i	0.800688	−	0.599081i	$-0.204467\pi$
$432$		0			0
$433$	−6.55556			−0.315040			−0.157520	−	0.987516i	$-0.550350\pi$
$433$	−6.55556			−0.315040			−0.157520	+	0.987516i	$0.550350\pi$
$434$		−	66.3278i		−	3.18384i
$435$		0			0
$436$		−	48.0101i		−	2.29927i
$437$		−	5.57239i		−	0.266563i
$438$		0			0
$439$	38.2180			1.82405			0.912023	−	0.410138i	$-0.134520\pi$
$439$	38.2180			1.82405			0.912023	+	0.410138i	$0.134520\pi$
$440$			27.2455i			1.29888i
$441$		0			0
$442$	32.7624	+	18.6602i	1.55835	+	0.887575i
$443$	12.7407			0.605330			0.302665	−	0.953097i	$-0.402124\pi$
$443$	12.7407			0.605330			0.302665	+	0.953097i	$0.402124\pi$
$444$		0			0
$445$	−9.19656			−0.435959
$446$	29.7187			1.40722
$447$		0			0
$448$		−	123.500i		−	5.83484i
$449$		−	25.9513i		−	1.22472i	−0.790581	−	0.612358i	$-0.790221\pi$
$449$		−	25.9513i		−	1.22472i	0.790581	−	0.612358i	$-0.209779\pi$
$450$		0			0
$451$	6.35787			0.299381
$452$	0.0855531			0.00402408
$453$		0			0
$454$	−0.883629			−0.0414708
$455$	19.9339	−	34.9986i	0.934514	−	1.64076i
$456$		0			0
$457$		−	16.0372i		−	0.750190i	−0.926986	−	0.375095i	$-0.877610\pi$
$457$		−	16.0372i		−	0.750190i	0.926986	−	0.375095i	$-0.122390\pi$
$458$	16.2700			0.760247
$459$		0			0
$460$		−	29.6185i		−	1.38097i
$461$			3.48566i			0.162343i	0.996700	+	0.0811716i	$0.0258662\pi$
$461$			3.48566i			0.162343i	−0.996700	+	0.0811716i	$0.974134\pi$
$462$		0			0
$463$		−	27.0784i		−	1.25844i	−0.777227	−	0.629220i	$-0.783374\pi$
$463$		−	27.0784i		−	1.25844i	0.777227	−	0.629220i	$-0.216626\pi$
$464$	32.0315			1.48703
$465$		0			0
$466$		−	34.5639i		−	1.60114i
$467$	−15.6899			−0.726041			−0.363020	−	0.931781i	$-0.618254\pi$
$467$	−15.6899			−0.726041			−0.363020	+	0.931781i	$0.618254\pi$
$468$		0			0
$469$	41.1727			1.90118
$470$		−	41.3350i		−	1.90664i
$471$		0			0
$472$	51.6448			2.37715
$473$		−	11.7275i		−	0.539230i
$474$		0			0
$475$		−	9.04109i		−	0.414834i
$476$		−	82.5709i		−	3.78463i
$477$		0			0
$478$	−56.8860			−2.60191
$479$		−	33.0061i		−	1.50809i	−0.656825	−	0.754043i	$-0.728101\pi$
$479$		−	33.0061i		−	1.50809i	0.656825	−	0.754043i	$-0.271899\pi$
$480$		0			0
$481$	26.1420	+	14.8895i	1.19197	+	0.678901i
$482$	49.8904			2.27244
$483$		0			0
$484$	5.50093			0.250042
$485$	50.5550			2.29559
$486$		0			0
$487$		−	18.6684i		−	0.845945i	−0.906142	−	0.422973i	$-0.860987\pi$
$487$		−	18.6684i		−	0.845945i	0.906142	−	0.422973i	$-0.139013\pi$
$488$			22.5187i			1.01938i
$489$		0			0
$490$	−65.8003			−2.97255
$491$	−6.78238			−0.306084			−0.153042	−	0.988220i	$-0.548907\pi$
$491$	−6.78238			−0.306084			−0.153042	+	0.988220i	$0.548907\pi$
$492$		0			0
$493$	8.01542			0.360996
$494$	−14.3724	+	25.2341i	−0.646643	+	1.13534i
$495$		0			0
$496$			93.9963i			4.22056i
$497$	−39.6822			−1.77999
$498$		0			0
$499$			0.212900i			0.00953070i	0.999989	+	0.00476535i	$0.00151686\pi$
$499$			0.212900i			0.00953070i	−0.999989	+	0.00476535i	$0.998483\pi$
$500$			30.1001i			1.34612i
$501$		0			0
$502$		−	62.0779i		−	2.77067i
$503$	10.5282			0.469430			0.234715	−	0.972064i	$-0.424584\pi$
$503$	10.5282			0.469430			0.234715	+	0.972064i	$0.424584\pi$
$504$		0			0
$505$			34.4477i			1.53290i
$506$	−5.18956			−0.230704
$507$		0			0
$508$	−32.5387			−1.44367
$509$		−	26.1335i		−	1.15835i	−0.815204	−	0.579175i	$-0.803375\pi$
$509$		−	26.1335i		−	1.15835i	0.815204	−	0.579175i	$-0.196625\pi$
$510$		0			0
$511$	59.3934			2.62741
$512$			52.4306i			2.31713i
$513$		0			0
$514$			48.5677i			2.14223i
$515$		−	9.68917i		−	0.426956i
$516$		0			0
$517$	−5.31137			−0.233594
$518$		−	89.8398i		−	3.94734i
$519$		0			0
$520$	−48.6180	+	85.3604i	−2.13204	+	3.74330i
$521$	−44.5983			−1.95389			−0.976944	−	0.213495i	$-0.931515\pi$
$521$	−44.5983			−1.95389			−0.976944	+	0.213495i	$0.931515\pi$
$522$		0			0
$523$	−35.9788			−1.57324			−0.786622	−	0.617435i	$-0.788172\pi$
$523$	−35.9788			−1.57324			−0.786622	+	0.617435i	$0.788172\pi$
$524$	−7.27176			−0.317668
$525$		0			0
$526$		−	80.8233i		−	3.52406i
$527$			23.5212i			1.02460i
$528$		0			0
$529$	−19.4096			−0.843895
$530$	18.5209			0.804495
$531$		0			0
$532$	63.5974			2.75730
$533$	19.9193	+	11.3453i	0.862800	+	0.491418i
$534$		0			0
$535$		−	34.7668i		−	1.50310i
$536$	−100.419			−4.33743
$537$		0			0
$538$			24.7063i			1.06516i
$539$			8.45506i			0.364185i
$540$		0			0
$541$			19.2806i			0.828937i	0.910064	+	0.414469i	$0.136033\pi$
$541$			19.2806i			0.828937i	−0.910064	+	0.414469i	$0.863967\pi$
$542$	−12.5524			−0.539172
$543$		0			0
$544$			86.3395i			3.70177i
$545$	24.7999			1.06231
$546$		0			0
$547$	−18.3680			−0.785359			−0.392679	−	0.919675i	$-0.628452\pi$
$547$	−18.3680			−0.785359			−0.392679	+	0.919675i	$0.628452\pi$
$548$			26.5215i			1.13294i
$549$		0			0
$550$	−8.41997			−0.359029
$551$			6.17360i			0.263004i
$552$		0			0
$553$			6.17686i			0.262667i
$554$		−	57.4759i		−	2.44192i
$555$		0			0
$556$	−94.5481			−4.00973
$557$			20.9920i			0.889459i	0.895665	+	0.444730i	$0.146700\pi$
$557$			20.9920i			0.889459i	−0.895665	+	0.444730i	$0.853300\pi$
$558$		0			0
$559$	20.9270	−	36.7424i	0.885119	−	1.55404i
$560$	170.450			7.20282
$561$		0			0
$562$	28.8558			1.21721
$563$	3.18924			0.134411			0.0672053	−	0.997739i	$-0.478592\pi$
$563$	3.18924			0.134411			0.0672053	+	0.997739i	$0.478592\pi$
$564$		0			0
$565$			0.0441930i			0.00185921i
$566$			7.04329i			0.296052i
$567$		0			0
$568$	96.7834			4.06094
$569$	19.0570			0.798909			0.399455	−	0.916753i	$-0.369200\pi$
$569$	19.0570			0.798909			0.399455	+	0.916753i	$0.369200\pi$
$570$		0			0
$571$	−13.1418			−0.549967			−0.274984	−	0.961449i	$-0.588672\pi$
$571$	−13.1418			−0.549967			−0.274984	+	0.961449i	$0.588672\pi$
$572$	17.2345	+	9.81609i	0.720610	+	0.410431i
$573$		0			0
$574$		−	68.4549i		−	2.85725i
$575$	5.82540			0.242936
$576$		0			0
$577$		−	6.38825i		−	0.265946i	−0.991120	−	0.132973i	$-0.957548\pi$
$577$		−	6.38825i		−	0.265946i	0.991120	−	0.132973i	$-0.0424524\pi$
$578$		−	6.63205i		−	0.275857i
$579$		0			0
$580$			32.8140i			1.36253i
$581$	−41.9608			−1.74083
$582$		0			0
$583$		−	2.37985i		−	0.0985635i
$584$	−144.858			−5.99428
$585$		0			0
$586$	14.1365			0.583975
$587$		−	8.66483i		−	0.357636i	−0.983882	−	0.178818i	$-0.942773\pi$
$587$		−	8.66483i		−	0.357636i	0.983882	−	0.178818i	$-0.0572273\pi$
$588$		0			0
$589$	−18.1164			−0.746472
$590$			41.9177i			1.72572i
$591$		0			0
$592$			127.316i			5.23267i
$593$		−	5.41552i		−	0.222389i	−0.993799	−	0.111194i	$-0.964532\pi$
$593$		−	5.41552i		−	0.222389i	0.993799	−	0.111194i	$-0.0354676\pi$
$594$		0			0
$595$	42.6525			1.74858
$596$			29.2514i			1.19818i
$597$		0			0
$598$	−16.2590	−	9.26047i	−0.664878	−	0.378689i
$599$	−17.1821			−0.702041			−0.351021	−	0.936368i	$-0.614165\pi$
$599$	−17.1821			−0.702041			−0.351021	+	0.936368i	$0.614165\pi$
$600$		0			0
$601$	4.44618			0.181364			0.0906818	−	0.995880i	$-0.471095\pi$
$601$	4.44618			0.181364			0.0906818	+	0.995880i	$0.471095\pi$
$602$	−126.269			−5.14635
$603$		0			0
$604$			37.2728i			1.51661i
$605$			2.84154i			0.115525i
$606$		0			0
$607$	12.1499			0.493148			0.246574	−	0.969124i	$-0.420695\pi$
$607$	12.1499			0.493148			0.246574	+	0.969124i	$0.420695\pi$
$608$	−66.5000			−2.69693
$609$		0			0
$610$	−18.2774			−0.740031
$611$	−16.6406	−	9.47783i	−0.673206	−	0.383432i
$612$		0			0
$613$			30.6073i			1.23622i	0.786093	+	0.618108i	$0.212101\pi$
$613$			30.6073i			1.23622i	−0.786093	+	0.618108i	$0.787899\pi$
$614$	82.3172			3.32205
$615$		0			0
$616$		−	37.6943i		−	1.51875i
$617$			15.3414i			0.617622i	0.951123	+	0.308811i	$0.0999311\pi$
$617$			15.3414i			0.617622i	−0.951123	+	0.308811i	$0.900069\pi$
$618$		0			0
$619$		−	10.0391i		−	0.403505i	−0.979437	−	0.201752i	$-0.935336\pi$
$619$		−	10.0391i		−	0.403505i	0.979437	−	0.201752i	$-0.0646636\pi$
$620$	−96.2926			−3.86720
$621$		0			0
$622$		−	59.5581i		−	2.38806i
$623$	12.7235			0.509757
$624$		0			0
$625$	−30.9201			−1.23681
$626$		−	66.9253i		−	2.67487i
$627$		0			0
$628$	−89.6012			−3.57548
$629$			31.8590i			1.27030i
$630$		0			0
$631$			20.8523i			0.830119i	0.909794	+	0.415059i	$0.136239\pi$
$631$			20.8523i			0.830119i	−0.909794	+	0.415059i	$0.863761\pi$
$632$		−	15.0652i		−	0.599260i
$633$		0			0
$634$	−57.7991			−2.29550
$635$		−	16.8081i		−	0.667009i
$636$		0			0
$637$	−15.0876	+	26.4898i	−0.597791	+	1.04956i
$638$	5.74947			0.227624
$639$		0			0
$640$	−115.970			−4.58413
$641$	29.3934			1.16097			0.580484	−	0.814272i	$-0.302863\pi$
$641$	29.3934			1.16097			0.580484	+	0.814272i	$0.302863\pi$
$642$		0			0
$643$			7.23812i			0.285444i	0.989763	+	0.142722i	$0.0455854\pi$
$643$			7.23812i			0.285444i	−0.989763	+	0.142722i	$0.954415\pi$
$644$			40.9774i			1.61474i
$645$		0			0
$646$	−30.7526			−1.20994
$647$	−25.4069			−0.998848			−0.499424	−	0.866358i	$-0.666455\pi$
$647$	−25.4069			−0.998848			−0.499424	+	0.866358i	$0.666455\pi$
$648$		0			0
$649$	5.38624			0.211429
$650$	−26.3799	−	15.0249i	−1.03470	−	0.589327i
$651$		0			0
$652$		−	62.7284i		−	2.45663i
$653$	10.1486			0.397146			0.198573	−	0.980086i	$-0.436369\pi$
$653$	10.1486			0.397146			0.198573	+	0.980086i	$0.436369\pi$
$654$		0			0
$655$		−	3.75627i		−	0.146770i
$656$			97.0107i			3.78763i
$657$		0			0
$658$			57.1873i			2.22939i
$659$	18.8648			0.734869			0.367434	−	0.930049i	$-0.380236\pi$
$659$	18.8648			0.734869			0.367434	+	0.930049i	$0.380236\pi$
$660$		0			0
$661$		−	7.39059i		−	0.287461i	−0.989617	−	0.143730i	$-0.954090\pi$
$661$		−	7.39059i		−	0.287461i	0.989617	−	0.143730i	$-0.0459098\pi$
$662$	1.53514			0.0596648
$663$		0			0
$664$	102.341			3.97160
$665$			32.8516i			1.27393i
$666$		0			0
$667$	−3.97780			−0.154021
$668$		−	81.0296i		−	3.13513i
$669$		0			0
$670$		−	81.5052i		−	3.14882i
$671$			2.34857i			0.0906655i
$672$		0			0
$673$	13.8241			0.532879			0.266439	−	0.963852i	$-0.414153\pi$
$673$	13.8241			0.532879			0.266439	+	0.963852i	$0.414153\pi$
$674$			21.2762i			0.819530i
$675$		0			0
$676$	36.4796	+	61.5079i	1.40306	+	2.36569i
$677$	28.1714			1.08272			0.541358	−	0.840792i	$-0.317910\pi$
$677$	28.1714			1.08272			0.541358	+	0.840792i	$0.317910\pi$
$678$		0			0
$679$	−69.9433			−2.68418
$680$	−104.028			−3.98929
$681$		0			0
$682$			16.8718i			0.646053i
$683$			20.1458i			0.770856i	0.922738	+	0.385428i	$0.125946\pi$
$683$			20.1458i			0.770856i	−0.922738	+	0.385428i	$0.874054\pi$
$684$		0			0
$685$	−13.6998			−0.523444
$686$	15.6665			0.598150
$687$		0			0
$688$	178.942			6.82211
$689$	4.24671	−	7.45611i	0.161787	−	0.284055i
$690$		0			0
$691$			4.60090i			0.175026i	0.996163	+	0.0875132i	$0.0278920\pi$
$691$			4.60090i			0.175026i	−0.996163	+	0.0875132i	$0.972108\pi$
$692$	−1.02673			−0.0390306
$693$		0			0
$694$			27.7593i			1.05373i
$695$		−	48.8394i		−	1.85258i
$696$		0			0
$697$			24.2755i			0.919499i
$698$	54.7839			2.07360
$699$		0			0
$700$			66.4851i			2.51290i
$701$	−2.14477			−0.0810067			−0.0405033	−	0.999179i	$-0.512896\pi$
$701$	−2.14477			−0.0810067			−0.0405033	+	0.999179i	$0.512896\pi$
$702$		0			0
$703$	−24.5383			−0.925479
$704$			31.4147i			1.18399i
$705$		0			0
$706$	73.3733			2.76144
$707$		−	47.6586i		−	1.79239i
$708$		0			0
$709$		−	18.0751i		−	0.678826i	−0.940637	−	0.339413i	$-0.889772\pi$
$709$		−	18.0751i		−	0.678826i	0.940637	−	0.339413i	$-0.110228\pi$
$710$			78.5546i			2.94810i
$711$		0			0
$712$	−31.0322			−1.16298
$713$		−	11.6728i		−	0.437151i
$714$		0			0
$715$	−5.07056	+	8.90258i	−0.189628	+	0.332938i
$716$	−60.2657			−2.25223
$717$		0			0
$718$	73.2206			2.73257
$719$	−5.67258			−0.211552			−0.105776	−	0.994390i	$-0.533733\pi$
$719$	−5.67258			−0.211552			−0.105776	+	0.994390i	$0.533733\pi$
$720$		0			0
$721$			13.4050i			0.499230i
$722$			28.3507i			1.05511i
$723$		0			0
$724$	−46.8698			−1.74190
$725$	−6.45391			−0.239692
$726$		0			0
$727$	−20.7279			−0.768756			−0.384378	−	0.923176i	$-0.625584\pi$
$727$	−20.7279			−0.768756			−0.384378	+	0.923176i	$0.625584\pi$
$728$	67.2634	−	118.097i	2.49295	−	4.37696i
$729$		0			0
$730$		−	117.575i		−	4.35163i
$731$	44.7776			1.65616
$732$		0			0
$733$		−	13.8875i		−	0.512945i	−0.966552	−	0.256473i	$-0.917440\pi$
$733$		−	13.8875i		−	0.512945i	0.966552	−	0.256473i	$-0.0825603\pi$
$734$			5.19172i			0.191630i
$735$		0			0
$736$		−	42.8476i		−	1.57938i
$737$	−10.4731			−0.385780
$738$		0			0
$739$		−	2.10062i		−	0.0772725i	−0.999253	−	0.0386362i	$-0.987699\pi$
$739$		−	2.10062i		−	0.0772725i	0.999253	−	0.0386362i	$-0.0123014\pi$
$740$	−130.427			−4.79457
$741$		0			0
$742$	−25.6238			−0.940678
$743$			39.5255i			1.45005i	0.688722	+	0.725025i	$0.258172\pi$
$743$			39.5255i			1.45005i	−0.688722	+	0.725025i	$0.741828\pi$
$744$		0			0
$745$	−15.1100			−0.553587
$746$		−	100.655i		−	3.68523i
$747$		0			0
$748$			21.0035i			0.767964i
$749$			48.1002i			1.75754i
$750$		0			0
$751$	7.96076			0.290492			0.145246	−	0.989396i	$-0.453603\pi$
$751$	7.96076			0.290492			0.145246	+	0.989396i	$0.453603\pi$
$752$		−	81.0427i		−	2.95532i
$753$		0			0
$754$	18.0132	+	10.2596i	0.656000	+	0.373632i
$755$	−19.2535			−0.700707
$756$		0			0
$757$	−21.7720			−0.791316			−0.395658	−	0.918398i	$-0.629484\pi$
$757$	−21.7720			−0.791316			−0.395658	+	0.918398i	$0.629484\pi$
$758$	−71.5721			−2.59962
$759$		0			0
$760$		−	80.1240i		−	2.90640i
$761$			25.3978i			0.920668i	0.887746	+	0.460334i	$0.152270\pi$
$761$			25.3978i			0.920668i	−0.887746	+	0.460334i	$0.847730\pi$
$762$		0			0
$763$	−34.3109			−1.24214
$764$	−64.9126			−2.34846
$765$		0			0
$766$	−46.3433			−1.67445
$767$	16.8752	+	9.61144i	0.609327	+	0.347049i
$768$		0			0
$769$			17.5471i			0.632763i	0.948632	+	0.316382i	$0.102468\pi$
$769$			17.5471i			0.632763i	−0.948632	+	0.316382i	$0.897532\pi$
$770$	30.5947			1.10256
$771$		0			0
$772$		−	60.1832i		−	2.16604i
$773$			16.9181i			0.608503i	0.952592	+	0.304251i	$0.0984063\pi$
$773$			16.9181i			0.608503i	−0.952592	+	0.304251i	$0.901594\pi$
$774$		0			0
$775$		−	18.9389i		−	0.680307i
$776$	170.589			6.12379
$777$		0			0
$778$		−	13.8475i		−	0.496456i
$779$	−18.6974			−0.669902
$780$		0			0
$781$	10.0939			0.361189
$782$		−	19.8147i		−	0.708570i
$783$		0			0
$784$	−129.010			−4.60751
$785$		−	46.2841i		−	1.65195i
$786$		0			0
$787$			27.1659i			0.968361i	0.874968	+	0.484181i	$0.160882\pi$
$787$			27.1659i			0.968361i	−0.874968	+	0.484181i	$0.839118\pi$
$788$		−	51.8907i		−	1.84853i
$789$		0			0
$790$	12.2277			0.435041
$791$		−	0.0611413i		−	0.00217394i
$792$		0			0
$793$	−4.19089	+	7.35810i	−0.148823	+	0.261294i
$794$	−43.1992			−1.53308
$795$		0			0
$796$	−84.2731			−2.98698
$797$	12.3253			0.436583			0.218292	−	0.975884i	$-0.429952\pi$
$797$	12.3253			0.436583			0.218292	+	0.975884i	$0.429952\pi$
$798$		0			0
$799$		−	20.2797i		−	0.717446i
$800$		−	69.5194i		−	2.45788i
$801$		0			0
$802$	23.0665			0.814507
$803$	−15.1079			−0.533144
$804$		0			0
$805$	−21.1671			−0.746043
$806$	−30.1067	+	52.8594i	−1.06046	+	1.86189i
$807$		0			0
$808$			116.238i			4.08923i
$809$	−51.6306			−1.81523			−0.907617	−	0.419798i	$-0.862101\pi$
$809$	−51.6306			−1.81523			−0.907617	+	0.419798i	$0.862101\pi$
$810$		0			0
$811$		−	22.5443i		−	0.791637i	−0.918329	−	0.395818i	$-0.870461\pi$
$811$		−	22.5443i		−	0.791637i	0.918329	−	0.395818i	$-0.129539\pi$
$812$		−	45.3985i		−	1.59317i
$813$		0			0
$814$			22.8525i			0.800980i
$815$	32.4027			1.13502
$816$		0			0
$817$			34.4884i			1.20660i
$818$	85.9597			3.00551
$819$		0			0
$820$	−99.3806			−3.47052
$821$		−	18.1907i		−	0.634861i	−0.948282	−	0.317430i	$-0.897180\pi$
$821$		−	18.1907i		−	0.634861i	0.948282	−	0.317430i	$-0.102820\pi$
$822$		0			0
$823$	−44.4631			−1.54989			−0.774944	−	0.632030i	$-0.782222\pi$
$823$	−44.4631			−1.54989			−0.774944	+	0.632030i	$0.782222\pi$
$824$		−	32.6944i		−	1.13896i
$825$		0			0
$826$		−	57.9934i		−	2.01785i
$827$			6.33341i			0.220234i	0.993919	+	0.110117i	$0.0351226\pi$
$827$			6.33341i			0.220234i	−0.993919	+	0.110117i	$0.964877\pi$
$828$		0			0
$829$	−7.54455			−0.262033			−0.131017	−	0.991380i	$-0.541824\pi$
$829$	−7.54455			−0.262033			−0.131017	+	0.991380i	$0.541824\pi$
$830$			83.0654i			2.88324i
$831$		0			0
$832$	−56.0577	+	98.4226i	−1.94345	+	3.41219i
$833$	−32.2829			−1.11854
$834$		0			0
$835$	41.8563			1.44850
$836$	−16.1772			−0.559501
$837$		0			0
$838$			18.1561i			0.627194i
$839$		−	30.1703i		−	1.04159i	−0.853681	−	0.520797i	$-0.825635\pi$
$839$		−	30.1703i		−	1.04159i	0.853681	−	0.520797i	$-0.174365\pi$
$840$		0			0
$841$	−24.5930			−0.848035
$842$	−7.15719			−0.246653
$843$		0			0
$844$	−16.0211			−0.551468
$845$	−31.7723	+	18.8438i	−1.09300	+	0.648245i
$846$		0			0
$847$		−	3.93129i		−	0.135081i
$848$	36.3127			1.24698
$849$		0			0
$850$		−	32.1489i		−	1.10270i
$851$		−	15.8106i		−	0.541981i
$852$		0			0
$853$		−	14.6801i		−	0.502637i	−0.967904	−	0.251319i	$-0.919136\pi$
$853$		−	14.6801i		−	0.502637i	0.967904	−	0.251319i	$-0.0808642\pi$
$854$	25.2869			0.865301
$855$		0			0
$856$		−	117.315i		−	4.00973i
$857$	−21.1257			−0.721641			−0.360821	−	0.932635i	$-0.617503\pi$
$857$	−21.1257			−0.721641			−0.360821	+	0.932635i	$0.617503\pi$
$858$		0			0
$859$	29.4462			1.00469			0.502346	−	0.864667i	$-0.332470\pi$
$859$	29.4462			1.00469			0.502346	+	0.864667i	$0.332470\pi$
$860$			183.314i			6.25094i
$861$		0			0
$862$	68.1258			2.32038
$863$			33.8929i			1.15373i	0.816840	+	0.576865i	$0.195724\pi$
$863$			33.8929i			1.15373i	−0.816840	+	0.576865i	$0.804276\pi$
$864$		0			0
$865$		−	0.530366i		−	0.0180330i
$866$		−	17.9543i		−	0.610111i
$867$		0			0
$868$	133.221			4.52183
$869$		−	1.57120i		−	0.0532995i
$870$		0			0
$871$	−32.8122	−	18.6886i	−1.11180	−	0.633239i
$872$	83.6830			2.83386
$873$		0			0
$874$	15.2616			0.516230
$875$	21.5113			0.727215
$876$		0			0
$877$			6.62356i			0.223662i	0.993727	+	0.111831i	$0.0356715\pi$
$877$			6.62356i			0.223662i	−0.993727	+	0.111831i	$0.964329\pi$
$878$			104.671i			3.53247i
$879$		0			0
$880$	−43.3572			−1.46157
$881$	18.7993			0.633366			0.316683	−	0.948531i	$-0.397431\pi$
$881$	18.7993			0.633366			0.316683	+	0.948531i	$0.397431\pi$
$882$		0			0
$883$	35.0323			1.17893			0.589465	−	0.807794i	$-0.299339\pi$
$883$	35.0323			1.17893			0.589465	+	0.807794i	$0.299339\pi$
$884$	−37.4796	+	65.8043i	−1.26057	+	2.21324i
$885$		0			0
$886$			34.8941i			1.17229i
$887$	−12.8619			−0.431860			−0.215930	−	0.976409i	$-0.569278\pi$
$887$	−12.8619			−0.431860			−0.215930	+	0.976409i	$0.569278\pi$
$888$		0			0
$889$			23.2541i			0.779918i
$890$		−	25.1874i		−	0.844283i
$891$		0			0
$892$			59.6909i			1.99860i
$893$	15.6198			0.522696
$894$		0			0
$895$		−	31.1306i		−	1.04058i
$896$	160.446			5.36012
$897$		0			0
$898$	71.0749			2.37180
$899$			12.9322i			0.431314i
$900$		0			0
$901$	9.08670			0.302722
$902$			17.4128i			0.579784i
$903$		0			0
$904$			0.149121i			0.00495971i
$905$		−	24.2109i		−	0.804797i
$906$		0			0
$907$	−47.0378			−1.56187			−0.780933	−	0.624615i	$-0.785256\pi$
$907$	−47.0378			−1.56187			−0.780933	+	0.624615i	$0.785256\pi$
$908$		−	1.77480i		−	0.0588987i
$909$		0			0
$910$	95.8536	+	54.5945i	3.17752	+	1.80979i
$911$	−40.3206			−1.33588			−0.667940	−	0.744215i	$-0.732824\pi$
$911$	−40.3206			−1.33588			−0.667940	+	0.744215i	$0.732824\pi$
$912$		0			0
$913$	10.6736			0.353243
$914$	43.9225			1.45283
$915$		0			0
$916$			32.6788i			1.07974i
$917$			5.19683i			0.171614i
$918$		0			0
$919$	−1.80179			−0.0594356			−0.0297178	−	0.999558i	$-0.509461\pi$
$919$	−1.80179			−0.0594356			−0.0297178	+	0.999558i	$0.509461\pi$
$920$	51.6259			1.70205
$921$		0			0
$922$	−9.54646			−0.314396
$923$	31.6244	+	18.0120i	1.04093	+	0.592873i
$924$		0			0
$925$		−	25.6525i		−	0.843448i
$926$	74.1619			2.43711
$927$		0			0
$928$			47.4705i			1.55829i
$929$			26.3015i			0.862924i	0.902131	+	0.431462i	$0.142002\pi$
$929$			26.3015i			0.862924i	−0.902131	+	0.431462i	$0.857998\pi$
$930$		0			0
$931$		−	24.8648i		−	0.814910i
$932$	69.4226			2.27401
$933$		0			0
$934$		−	42.9711i		−	1.40606i
$935$	−10.8495			−0.354817
$936$		0			0
$937$	8.45280			0.276141			0.138070	−	0.990422i	$-0.455910\pi$
$937$	8.45280			0.276141			0.138070	+	0.990422i	$0.455910\pi$
$938$			112.763i			3.68184i
$939$		0			0
$940$	83.0226			2.70790
$941$		−	8.34188i		−	0.271937i	−0.990713	−	0.135969i	$-0.956585\pi$
$941$		−	8.34188i		−	0.271937i	0.990713	−	0.135969i	$-0.0434147\pi$
$942$		0			0
$943$		−	12.0472i		−	0.392310i
$944$			82.1852i			2.67490i
$945$		0			0
$946$	32.1190			1.04428
$947$			21.8593i			0.710330i	0.934804	+	0.355165i	$0.115575\pi$
$947$			21.8593i			0.710330i	−0.934804	+	0.355165i	$0.884425\pi$
$948$		0			0
$949$	−47.3331	−	26.9591i	−1.53650	−	0.875129i
$950$	24.7616			0.803372
$951$		0			0
$952$	143.924			4.66459
$953$	57.6509			1.86750			0.933748	−	0.357932i	$-0.116518\pi$
$953$	57.6509			1.86750			0.933748	+	0.357932i	$0.116518\pi$
$954$		0			0
$955$		−	33.5310i		−	1.08504i
$956$		−	114.257i		−	3.69535i
$957$		0			0
$958$	90.3964			2.92058
$959$	18.9538			0.612051
$960$		0			0
$961$	−6.94950			−0.224177
$962$	−40.7790	+	71.5972i	−1.31477	+	2.30838i
$963$		0			0
$964$			100.206i			3.22743i
$965$	31.0880			1.00076
$966$		0			0
$967$		−	41.2637i		−	1.32695i	−0.748197	−	0.663476i	$-0.769080\pi$
$967$		−	41.2637i		−	1.32695i	0.748197	−	0.663476i	$-0.230920\pi$
$968$			9.58828i			0.308179i
$969$		0			0
$970$			138.459i			4.44566i
$971$	−10.2876			−0.330146			−0.165073	−	0.986281i	$-0.552786\pi$
$971$	−10.2876			−0.330146			−0.165073	+	0.986281i	$0.552786\pi$
$972$		0			0
$973$			67.5697i			2.16618i
$974$	51.1286			1.63827
$975$		0			0
$976$	−35.8353			−1.14706
$977$		−	54.0502i		−	1.72922i	−0.502445	−	0.864609i	$-0.667566\pi$
$977$		−	54.0502i		−	1.72922i	0.502445	−	0.864609i	$-0.332434\pi$
$978$		0			0
$979$	−3.23647			−0.103438
$980$		−	132.162i		−	4.22176i
$981$		0			0
$982$		−	18.5755i		−	0.592766i
$983$			33.3316i			1.06311i	0.847022	+	0.531557i	$0.178393\pi$
$983$			33.3316i			1.06311i	−0.847022	+	0.531557i	$0.821607\pi$
$984$		0			0
$985$	26.8045			0.854061
$986$			21.9525i			0.699109i
$987$		0			0
$988$	−50.6835	−	28.8673i	−1.61246	−	0.918392i
$989$	−22.2217			−0.706610
$990$		0			0
$991$	15.1783			0.482155			0.241077	−	0.970506i	$-0.422499\pi$
$991$	15.1783			0.482155			0.241077	+	0.970506i	$0.422499\pi$
$992$	−139.302			−4.42283
$993$		0			0
$994$		−	108.681i		−	3.44715i
$995$		−	43.5318i		−	1.38005i
$996$		0			0
$997$	−23.6131			−0.747834			−0.373917	−	0.927462i	$-0.621986\pi$
$997$	−23.6131			−0.747834			−0.373917	+	0.927462i	$0.621986\pi$
$998$	−0.583086			−0.0184573
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1287.2.b.c.298.14		14
3.2	odd	2		429.2.b.b.298.1	✓	14
13.12	even	2	inner	1287.2.b.c.298.1		14
39.5	even	4		5577.2.a.x.1.1		7
39.8	even	4		5577.2.a.y.1.7		7
39.38	odd	2		429.2.b.b.298.14	yes	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
429.2.b.b.298.1	✓	14	3.2	odd	2
429.2.b.b.298.14	yes	14	39.38	odd	2
1287.2.b.c.298.1		14	13.12	even	2	inner
1287.2.b.c.298.14		14	1.1	even	1	trivial
5577.2.a.x.1.1		7	39.5	even	4
5577.2.a.y.1.7		7	39.8	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 1287 = 3^{2} \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1287.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.73878i q^{2} -5.50093 q^{4} -2.84154i q^{5} +3.93129i q^{7} -9.58828i q^{8} +O(q^{10})\)	\(q+2.73878i q^{2} -5.50093 q^{4} -2.84154i q^{5} +3.93129i q^{7} -9.58828i q^{8} +7.78236 q^{10} -1.00000i q^{11} +(1.78444 - 3.13301i) q^{13} -10.7670 q^{14} +15.2584 q^{16} +3.81818 q^{17} +2.94082i q^{19} +15.6311i q^{20} +2.73878 q^{22} -1.89484 q^{23} -3.07435 q^{25} +(8.58064 + 4.88720i) q^{26} -21.6258i q^{28} +2.09928 q^{29} +6.16032i q^{31} +22.6127i q^{32} +10.4572i q^{34} +11.1709 q^{35} +8.34404i q^{37} -8.05426 q^{38} -27.2455 q^{40} +6.35787i q^{41} +11.7275 q^{43} +5.50093i q^{44} -5.18956i q^{46} -5.31137i q^{47} -8.45506 q^{49} -8.41997i q^{50} +(-9.81609 + 17.2345i) q^{52} +2.37985 q^{53} -2.84154 q^{55} +37.6943 q^{56} +5.74947i q^{58} +5.38624i q^{59} -2.34857 q^{61} -16.8718 q^{62} -31.4147 q^{64} +(-8.90258 - 5.07056i) q^{65} -10.4731i q^{67} -21.0035 q^{68} +30.5947i q^{70} +10.0939i q^{71} -15.1079i q^{73} -22.8525 q^{74} -16.1772i q^{76} +3.93129 q^{77} +1.57120 q^{79} -43.3572i q^{80} -17.4128 q^{82} +10.6736i q^{83} -10.8495i q^{85} +32.1190i q^{86} -9.58828 q^{88} -3.23647i q^{89} +(12.3168 + 7.01516i) q^{91} +10.4234 q^{92} +14.5467 q^{94} +8.35645 q^{95} +17.7914i q^{97} -23.1566i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 18 q^{4}+O(q^{10}) \) 14 * q - 18 * q^4	\( 14 q - 18 q^{4} - 16 q^{14} + 34 q^{16} - 4 q^{17} + 6 q^{22} + 8 q^{23} - 26 q^{25} + 6 q^{26} + 24 q^{29} + 8 q^{35} + 32 q^{38} - 20 q^{40} + 32 q^{43} - 46 q^{49} + 4 q^{52} - 20 q^{53} + 12 q^{55} + 32 q^{56} - 20 q^{61} - 72 q^{62} - 58 q^{64} - 12 q^{65} + 20 q^{68} + 12 q^{77} + 12 q^{79} + 20 q^{82} - 30 q^{88} + 16 q^{91} + 24 q^{92} + 64 q^{94} + 36 q^{95}+O(q^{100}) \) 14 * q - 18 * q^4 - 16 * q^14 + 34 * q^16 - 4 * q^17 + 6 * q^22 + 8 * q^23 - 26 * q^25 + 6 * q^26 + 24 * q^29 + 8 * q^35 + 32 * q^38 - 20 * q^40 + 32 * q^43 - 46 * q^49 + 4 * q^52 - 20 * q^53 + 12 * q^55 + 32 * q^56 - 20 * q^61 - 72 * q^62 - 58 * q^64 - 12 * q^65 + 20 * q^68 + 12 * q^77 + 12 * q^79 + 20 * q^82 - 30 * q^88 + 16 * q^91 + 24 * q^92 + 64 * q^94 + 36 * q^95

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