LMFDB - Embedded newform 475.3.d.c.474.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [475,3,Mod(474,475)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(475, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("475.474");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 475 = 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	475.d (of order $2$, degree $1$, not 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:	$12.9428125571$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	no (minimal twist has level 95)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			474.2
Character	$\chi$	$=$	475.474
Dual form			475.3.d.c.474.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.38208 q^{2} +4.08851 q^{3} +7.43849 q^{4} -13.8277 q^{6} -4.56923i q^{7} -11.6293 q^{8} +7.71590 q^{9} +O(q^{10})$	\(q-3.38208 q^{2} +4.08851 q^{3} +7.43849 q^{4} -13.8277 q^{6} -4.56923i q^{7} -11.6293 q^{8} +7.71590 q^{9} +6.24529 q^{11} +30.4124 q^{12} -14.4666 q^{13} +15.4535i q^{14} +9.57723 q^{16} -26.6967i q^{17} -26.0958 q^{18} +(18.2451 + 5.30257i) q^{19} -18.6813i q^{21} -21.1221 q^{22} -33.4334i q^{23} -47.5464 q^{24} +48.9271 q^{26} -5.25004 q^{27} -33.9882i q^{28} -10.3640i q^{29} -30.5308i q^{31} +14.1261 q^{32} +25.5339 q^{33} +90.2906i q^{34} +57.3947 q^{36} -18.1542 q^{37} +(-61.7064 - 17.9338i) q^{38} -59.1466 q^{39} +65.5629i q^{41} +63.1819i q^{42} +71.3965i q^{43} +46.4556 q^{44} +113.075i q^{46} -53.1345i q^{47} +39.1566 q^{48} +28.1221 q^{49} -109.150i q^{51} -107.609 q^{52} +21.9762 q^{53} +17.7561 q^{54} +53.1369i q^{56} +(74.5951 + 21.6796i) q^{57} +35.0518i q^{58} -92.9158i q^{59} -27.8117 q^{61} +103.258i q^{62} -35.2557i q^{63} -86.0847 q^{64} -86.3579 q^{66} +60.7093 q^{67} -198.583i q^{68} -136.693i q^{69} -103.561i q^{71} -89.7304 q^{72} -11.9203i q^{73} +61.3990 q^{74} +(135.716 + 39.4432i) q^{76} -28.5362i q^{77} +200.039 q^{78} -9.41895i q^{79} -90.9080 q^{81} -221.739i q^{82} -15.2174i q^{83} -138.961i q^{84} -241.469i q^{86} -42.3732i q^{87} -72.6282 q^{88} -11.8378i q^{89} +66.1010i q^{91} -248.694i q^{92} -124.826i q^{93} +179.705i q^{94} +57.7548 q^{96} -62.0200 q^{97} -95.1114 q^{98} +48.1881 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 24 q^{4} - 56 q^{6} + 96 q^{9}+O(q^{10}) $ 24 * q + 24 * q^4 - 56 * q^6 + 96 * q^9	$ 24 q + 24 q^{4} - 56 q^{6} + 96 q^{9} + 64 q^{11} - 88 q^{16} - 16 q^{19} - 200 q^{24} + 216 q^{26} - 160 q^{36} - 152 q^{39} + 512 q^{44} - 144 q^{49} + 152 q^{54} - 592 q^{61} - 376 q^{64} + 304 q^{66} - 272 q^{74} + 496 q^{76} - 744 q^{81} - 88 q^{96} + 624 q^{99}+O(q^{100}) $ 24 * q + 24 * q^4 - 56 * q^6 + 96 * q^9 + 64 * q^11 - 88 * q^16 - 16 * q^19 - 200 * q^24 + 216 * q^26 - 160 * q^36 - 152 * q^39 + 512 * q^44 - 144 * q^49 + 152 * q^54 - 592 * q^61 - 376 * q^64 + 304 * q^66 - 272 * q^74 + 496 * q^76 - 744 * q^81 - 88 * q^96 + 624 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/475\mathbb{Z}\right)^\times$.

$n$	$77$	$401$
$\chi(n)$	$-1$	$-1$

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 $

$2$	−3.38208			−1.69104			−0.845521	−	0.533942i	$-0.820710\pi$
$2$	−3.38208			−1.69104			−0.845521	+	0.533942i	$0.820710\pi$
$3$	4.08851			1.36284			0.681418	−	0.731894i	$-0.261364\pi$
$3$	4.08851			1.36284			0.681418	+	0.731894i	$0.261364\pi$
$4$	7.43849			1.85962
$5$		0			0
$5$		0			0
$6$	−13.8277			−2.30461
$7$		−	4.56923i		−	0.652747i	−0.945241	−	0.326374i	$-0.894173\pi$
$7$		−	4.56923i		−	0.652747i	0.945241	−	0.326374i	$-0.105827\pi$
$8$	−11.6293			−1.45366
$9$	7.71590			0.857323
$10$		0			0
$11$	6.24529			0.567754			0.283877	−	0.958861i	$-0.408379\pi$
$11$	6.24529			0.567754			0.283877	+	0.958861i	$0.408379\pi$
$12$	30.4124			2.53436
$13$	−14.4666			−1.11281			−0.556406	−	0.830911i	$-0.687820\pi$
$13$	−14.4666			−1.11281			−0.556406	+	0.830911i	$0.687820\pi$
$14$			15.4535i			1.10382i
$15$		0			0
$16$	9.57723			0.598577
$17$		−	26.6967i		−	1.57040i	−0.619245	−	0.785198i	$-0.712561\pi$
$17$		−	26.6967i		−	1.57040i	0.619245	−	0.785198i	$-0.287439\pi$
$18$	−26.0958			−1.44977
$19$	18.2451	+	5.30257i	0.960267	+	0.279083i
$19$	18.2451	+	5.30257i	0.960267	+	0.279083i
$20$		0			0
$21$		−	18.6813i		−	0.889588i
$22$	−21.1221			−0.960095
$23$		−	33.4334i		−	1.45363i	−0.686836	−	0.726813i	$-0.741001\pi$
$23$		−	33.4334i		−	1.45363i	0.686836	−	0.726813i	$-0.258999\pi$
$24$	−47.5464			−1.98110
$25$		0			0
$26$	48.9271			1.88181
$27$	−5.25004			−0.194446
$28$		−	33.9882i		−	1.21386i
$29$		−	10.3640i		−	0.357378i	−0.983906	−	0.178689i	$-0.942814\pi$
$29$		−	10.3640i		−	0.357378i	0.983906	−	0.178689i	$-0.0571856\pi$
$30$		0			0
$31$		−	30.5308i		−	0.984865i	−0.870351	−	0.492433i	$-0.836108\pi$
$31$		−	30.5308i		−	0.984865i	0.870351	−	0.492433i	$-0.163892\pi$
$32$	14.1261			0.441442
$33$	25.5339			0.773755
$34$			90.2906i			2.65561i
$35$		0			0
$36$	57.3947			1.59430
$37$	−18.1542			−0.490653			−0.245327	−	0.969440i	$-0.578895\pi$
$37$	−18.1542			−0.490653			−0.245327	+	0.969440i	$0.578895\pi$
$38$	−61.7064	−	17.9338i	−1.62385	−	0.471941i
$39$	−59.1466			−1.51658
$40$		0			0
$41$			65.5629i			1.59909i	0.600603	+	0.799547i	$0.294927\pi$
$41$			65.5629i			1.59909i	−0.600603	+	0.799547i	$0.705073\pi$
$42$			63.1819i			1.50433i
$43$			71.3965i			1.66038i	0.557479	+	0.830191i	$0.311769\pi$
$43$			71.3965i			1.66038i	−0.557479	+	0.830191i	$0.688231\pi$
$44$	46.4556			1.05581
$45$		0			0
$46$			113.075i			2.45814i
$47$		−	53.1345i		−	1.13052i	−0.824912	−	0.565261i	$-0.808776\pi$
$47$		−	53.1345i		−	1.13052i	0.824912	−	0.565261i	$-0.191224\pi$
$48$	39.1566			0.815762
$49$	28.1221			0.573921
$50$		0			0
$51$		−	109.150i		−	2.14019i
$52$	−107.609			−2.06941
$53$	21.9762			0.414646			0.207323	−	0.978273i	$-0.433525\pi$
$53$	21.9762			0.414646			0.207323	+	0.978273i	$0.433525\pi$
$54$	17.7561			0.328816
$55$		0			0
$56$			53.1369i			0.948873i
$57$	74.5951	+	21.6796i	1.30869	+	0.380344i
$58$			35.0518i			0.604341i
$59$		−	92.9158i		−	1.57484i	−0.616414	−	0.787422i	$-0.711415\pi$
$59$		−	92.9158i		−	1.57484i	0.616414	−	0.787422i	$-0.288585\pi$
$60$		0			0
$61$	−27.8117			−0.455929			−0.227964	−	0.973669i	$-0.573207\pi$
$61$	−27.8117			−0.455929			−0.227964	+	0.973669i	$0.573207\pi$
$62$			103.258i			1.66545i
$63$		−	35.2557i		−	0.559615i
$64$	−86.0847			−1.34507
$65$		0			0
$66$	−86.3579			−1.30845
$67$	60.7093			0.906109			0.453055	−	0.891483i	$-0.350334\pi$
$67$	60.7093			0.906109			0.453055	+	0.891483i	$0.350334\pi$
$68$		−	198.583i		−	2.92035i
$69$		−	136.693i		−	1.98105i
$70$		0			0
$71$		−	103.561i		−	1.45861i	−0.684188	−	0.729306i	$-0.739843\pi$
$71$		−	103.561i		−	1.45861i	0.684188	−	0.729306i	$-0.260157\pi$
$72$	−89.7304			−1.24626
$73$		−	11.9203i		−	0.163292i	−0.996661	−	0.0816461i	$-0.973982\pi$
$73$		−	11.9203i		−	0.163292i	0.996661	−	0.0816461i	$-0.0260177\pi$
$74$	61.3990			0.829716
$75$		0			0
$76$	135.716	+	39.4432i	1.78574	+	0.518989i
$77$		−	28.5362i		−	0.370600i
$78$	200.039			2.56460
$79$		−	9.41895i		−	0.119227i	−0.998222	−	0.0596136i	$-0.981013\pi$
$79$		−	9.41895i		−	0.119227i	0.998222	−	0.0596136i	$-0.0189869\pi$
$80$		0			0
$81$	−90.9080			−1.12232
$82$		−	221.739i		−	2.70414i
$83$		−	15.2174i		−	0.183343i	−0.995789	−	0.0916714i	$-0.970779\pi$
$83$		−	15.2174i		−	0.183343i	0.995789	−	0.0916714i	$-0.0292209\pi$
$84$		−	138.961i		−	1.65430i
$85$		0			0
$86$		−	241.469i		−	2.80778i
$87$		−	42.3732i		−	0.487048i
$88$	−72.6282			−0.825321
$89$		−	11.8378i		−	0.133009i	−0.997786	−	0.0665047i	$-0.978815\pi$
$89$		−	11.8378i		−	0.133009i	0.997786	−	0.0665047i	$-0.0211847\pi$
$90$		0			0
$91$			66.1010i			0.726385i
$92$		−	248.694i		−	2.70320i
$93$		−	124.826i		−	1.34221i
$94$			179.705i			1.91176i
$95$		0			0
$96$	57.7548			0.601613
$97$	−62.0200			−0.639381			−0.319691	−	0.947522i	$-0.603579\pi$
$97$	−62.0200			−0.639381			−0.319691	+	0.947522i	$0.603579\pi$
$98$	−95.1114			−0.970525
$99$	48.1881			0.486748
$100$		0			0
$101$	139.132			1.37755			0.688773	−	0.724977i	$-0.258150\pi$
$101$	139.132			1.37755			0.688773	+	0.724977i	$0.258150\pi$
$102$			369.154i			3.61916i
$103$	137.668			1.33658			0.668291	−	0.743900i	$-0.267026\pi$
$103$	137.668			1.33658			0.668291	+	0.743900i	$0.267026\pi$
$104$	168.236			1.61765
$105$		0			0
$106$	−74.3254			−0.701183
$107$	126.783			1.18488			0.592442	−	0.805613i	$-0.298164\pi$
$107$	126.783			1.18488			0.592442	+	0.805613i	$0.298164\pi$
$108$	−39.0524			−0.361596
$109$		−	18.4688i		−	0.169438i	−0.996405	−	0.0847191i	$-0.973001\pi$
$109$		−	18.4688i		−	0.169438i	0.996405	−	0.0847191i	$-0.0269993\pi$
$110$		0			0
$111$	−74.2235			−0.668680
$112$		−	43.7606i		−	0.390719i
$113$	−134.320			−1.18867			−0.594337	−	0.804216i	$-0.702586\pi$
$113$	−134.320			−1.18867			−0.594337	+	0.804216i	$0.702586\pi$
$114$	−252.287	−	73.3223i	−2.21304	−	0.643178i
$115$		0			0
$116$		−	77.0923i		−	0.664589i
$117$	−111.623			−0.954039
$118$			314.249i			2.66313i
$119$	−121.984			−1.02507
$120$		0			0
$121$	−81.9964			−0.677656
$122$	94.0614			0.770995
$123$			268.054i			2.17930i
$124$		−	227.103i		−	1.83148i
$125$		0			0
$126$			119.238i			0.946333i
$127$	76.6711			0.603709			0.301855	−	0.953354i	$-0.402394\pi$
$127$	76.6711			0.603709			0.301855	+	0.953354i	$0.402394\pi$
$128$	234.641			1.83313
$129$			291.905i			2.26283i
$130$		0			0
$131$	−82.1678			−0.627235			−0.313617	−	0.949549i	$-0.601541\pi$
$131$	−82.1678			−0.627235			−0.313617	+	0.949549i	$0.601541\pi$
$132$	189.934			1.43889
$133$	24.2287	−	83.3659i	0.182171	−	0.626812i
$134$	−205.324			−1.53227
$135$		0			0
$136$			310.464i			2.28282i
$137$			38.0930i			0.278051i	0.990289	+	0.139026i	$0.0443970\pi$
$137$			38.0930i			0.278051i	−0.990289	+	0.139026i	$0.955603\pi$
$138$			462.306i			3.35004i
$139$	46.5503			0.334894			0.167447	−	0.985881i	$-0.446448\pi$
$139$	46.5503			0.334894			0.167447	+	0.985881i	$0.446448\pi$
$140$		0			0
$141$		−	217.241i		−	1.54072i
$142$			350.253i			2.46657i
$143$	−90.3478			−0.631803
$144$	73.8970			0.513173
$145$		0			0
$146$			40.3156i			0.276134i
$147$	114.978			0.782160
$148$	−135.040			−0.912431
$149$	210.319			1.41153			0.705767	−	0.708444i	$-0.250603\pi$
$149$	210.319			1.41153			0.705767	+	0.708444i	$0.250603\pi$
$150$		0			0
$151$		−	120.965i		−	0.801093i	−0.916276	−	0.400546i	$-0.868820\pi$
$151$		−	120.965i		−	0.801093i	0.916276	−	0.400546i	$-0.131180\pi$
$152$	−212.177	−	61.6651i	−1.39590	−	0.405692i
$153$		−	205.989i		−	1.34634i
$154$			96.5117i			0.626700i
$155$		0			0
$156$	−439.962			−2.82027
$157$			248.160i			1.58064i	0.612695	+	0.790319i	$0.290085\pi$
$157$			248.160i			1.58064i	−0.612695	+	0.790319i	$0.709915\pi$
$158$			31.8557i			0.201618i
$159$	89.8500			0.565094
$160$		0			0
$161$	−152.765			−0.948850
$162$	307.458			1.89789
$163$			26.4077i			0.162010i	0.996714	+	0.0810051i	$0.0258130\pi$
$163$			26.4077i			0.162010i	−0.996714	+	0.0810051i	$0.974187\pi$
$164$			487.689i			2.97371i
$165$		0			0
$166$			51.4667i			0.310040i
$167$	−256.876			−1.53818			−0.769089	−	0.639141i	$-0.779290\pi$
$167$	−256.876			−1.53818			−0.769089	+	0.639141i	$0.779290\pi$
$168$			217.251i			1.29316i
$169$	40.2813			0.238351
$170$		0			0
$171$	140.777	+	40.9142i	0.823259	+	0.239264i
$172$			531.082i			3.08769i
$173$	−135.679			−0.784271			−0.392136	−	0.919907i	$-0.628264\pi$
$173$	−135.679			−0.784271			−0.392136	+	0.919907i	$0.628264\pi$
$174$			143.310i			0.823618i
$175$		0			0
$176$	59.8126			0.339844
$177$		−	379.887i		−	2.14626i
$178$			40.0366i			0.224925i
$179$		−	43.6655i		−	0.243941i	−0.992534	−	0.121971i	$-0.961079\pi$
$179$		−	43.6655i		−	0.243941i	0.992534	−	0.121971i	$-0.0389214\pi$
$180$		0			0
$181$			247.032i			1.36482i	0.730970	+	0.682410i	$0.239068\pi$
$181$			247.032i			1.36482i	−0.730970	+	0.682410i	$0.760932\pi$
$182$		−	223.559i		−	1.22835i
$183$	−113.708			−0.621356
$184$			388.806i			2.11308i
$185$		0			0
$186$			422.170i			2.26973i
$187$		−	166.729i		−	0.891598i
$188$		−	395.241i		−	2.10234i
$189$			23.9886i			0.126924i
$190$		0			0
$191$	134.807			0.705795			0.352897	−	0.935662i	$-0.385197\pi$
$191$	134.807			0.705795			0.352897	+	0.935662i	$0.385197\pi$
$192$	−351.958			−1.83311
$193$	41.0916			0.212910			0.106455	−	0.994318i	$-0.466050\pi$
$193$	41.0916			0.212910			0.106455	+	0.994318i	$0.466050\pi$
$194$	209.757			1.08122
$195$		0			0
$196$	209.186			1.06728
$197$		−	100.288i		−	0.509076i	−0.967063	−	0.254538i	$-0.918077\pi$
$197$		−	100.288i		−	0.509076i	0.967063	−	0.254538i	$-0.0819235\pi$
$198$	−162.976			−0.823111
$199$	229.156			1.15154			0.575770	−	0.817612i	$-0.304703\pi$
$199$	229.156			1.15154			0.575770	+	0.817612i	$0.304703\pi$
$200$		0			0
$201$	248.211			1.23488
$202$	−470.557			−2.32949
$203$	−47.3553			−0.233278
$204$		−	811.910i		−	3.97995i
$205$		0			0
$206$	−465.605			−2.26022
$207$		−	257.969i		−	1.24623i
$208$	−138.550			−0.666103
$209$	113.946	+	33.1161i	0.545195	+	0.158450i
$210$		0			0
$211$			194.492i			0.921765i	0.887461	+	0.460882i	$0.152467\pi$
$211$			194.492i			0.921765i	−0.887461	+	0.460882i	$0.847533\pi$
$212$	163.470			0.771085
$213$		−	423.412i		−	1.98785i
$214$	−428.789			−2.00369
$215$		0			0
$216$	61.0542			0.282658
$217$	−139.502			−0.642868
$218$			62.4629i			0.286527i
$219$		−	48.7364i		−	0.222541i
$220$		0			0
$221$			386.210i			1.74756i
$222$	251.030			1.13077
$223$	29.2071			0.130974			0.0654868	−	0.997853i	$-0.479140\pi$
$223$	29.2071			0.130974			0.0654868	+	0.997853i	$0.479140\pi$
$224$		−	64.5456i		−	0.288150i
$225$		0			0
$226$	454.282			2.01010
$227$	313.596			1.38148			0.690740	−	0.723104i	$-0.257285\pi$
$227$	313.596			1.38148			0.690740	+	0.723104i	$0.257285\pi$
$228$	554.876	+	161.264i	2.43366	+	0.707297i
$229$	−37.9407			−0.165680			−0.0828400	−	0.996563i	$-0.526399\pi$
$229$	−37.9407			−0.165680			−0.0828400	+	0.996563i	$0.526399\pi$
$230$		0			0
$231$		−	116.670i		−	0.505067i
$232$			120.525i			0.519506i
$233$			416.600i			1.78798i	0.448086	+	0.893990i	$0.352106\pi$
$233$			416.600i			1.78798i	−0.448086	+	0.893990i	$0.647894\pi$
$234$	377.517			1.61332
$235$		0			0
$236$		−	691.154i		−	2.92862i
$237$		−	38.5095i		−	0.162487i
$238$	412.559			1.73344
$239$	98.2483			0.411081			0.205540	−	0.978649i	$-0.434105\pi$
$239$	98.2483			0.411081			0.205540	+	0.978649i	$0.434105\pi$
$240$		0			0
$241$			85.1333i			0.353250i	0.984278	+	0.176625i	$0.0565180\pi$
$241$			85.1333i			0.353250i	−0.984278	+	0.176625i	$0.943482\pi$
$242$	277.319			1.14594
$243$	−324.428			−1.33509
$244$	−206.877			−0.847856
$245$		0			0
$246$		−	906.582i		−	3.68529i
$247$	−263.943	−	76.7100i	−1.06860	−	0.310567i
$248$			355.051i			1.43166i
$249$		−	62.2167i		−	0.249866i
$250$		0			0
$251$	118.230			0.471036			0.235518	−	0.971870i	$-0.424321\pi$
$251$	118.230			0.471036			0.235518	+	0.971870i	$0.424321\pi$
$252$		−	262.250i		−	1.04067i
$253$		−	208.801i		−	0.825301i
$254$	−259.308			−1.02090
$255$		0			0
$256$	−449.237			−1.75483
$257$	−290.346			−1.12975			−0.564875	−	0.825176i	$-0.691076\pi$
$257$	−290.346			−1.12975			−0.564875	+	0.825176i	$0.691076\pi$
$258$		−	987.247i		−	3.82654i
$259$			82.9506i			0.320273i
$260$		0			0
$261$		−	79.9674i		−	0.306388i
$262$	277.898			1.06068
$263$			451.882i			1.71818i	0.511822	+	0.859092i	$0.328971\pi$
$263$			451.882i			1.71818i	−0.511822	+	0.859092i	$0.671029\pi$
$264$	−296.941			−1.12478
$265$		0			0
$266$	−81.9435	+	281.951i	−0.308058	+	1.05996i
$267$		−	48.3991i		−	0.181270i
$268$	451.586			1.68502
$269$			110.442i			0.410567i	0.978703	+	0.205283i	$0.0658116\pi$
$269$			110.442i			0.410567i	−0.978703	+	0.205283i	$0.934188\pi$
$270$		0			0
$271$	−502.913			−1.85577			−0.927883	−	0.372871i	$-0.878373\pi$
$271$	−502.913			−1.85577			−0.927883	+	0.372871i	$0.878373\pi$
$272$		−	255.681i		−	0.940002i
$273$			270.255i			0.989944i
$274$		−	128.834i		−	0.470196i
$275$		0			0
$276$		−	1016.79i		−	3.68401i
$277$			180.350i			0.651084i	0.945528	+	0.325542i	$0.105547\pi$
$277$			180.350i			0.651084i	−0.945528	+	0.325542i	$0.894453\pi$
$278$	−157.437			−0.566320
$279$		−	235.573i		−	0.844347i
$280$		0			0
$281$			361.878i			1.28782i	0.765100	+	0.643912i	$0.222690\pi$
$281$			361.878i			1.28782i	−0.765100	+	0.643912i	$0.777310\pi$
$282$			734.727i			2.60541i
$283$			39.8981i			0.140983i	0.997512	+	0.0704914i	$0.0224567\pi$
$283$			39.8981i			0.140983i	−0.997512	+	0.0704914i	$0.977543\pi$
$284$		−	770.341i		−	2.71247i
$285$		0			0
$286$	305.564			1.06841
$287$	299.572			1.04380
$288$	108.996			0.378458
$289$	−423.715			−1.46614
$290$		0			0
$291$	−253.569			−0.871372
$292$		−	88.6693i		−	0.303662i
$293$	335.148			1.14385			0.571925	−	0.820306i	$-0.306197\pi$
$293$	335.148			1.14385			0.571925	+	0.820306i	$0.306197\pi$
$294$	−388.864			−1.32267
$295$		0			0
$296$	211.120			0.713243
$297$	−32.7880			−0.110397
$298$	−711.315			−2.38696
$299$			483.666i			1.61761i
$300$		0			0
$301$	326.227			1.08381
$302$			409.114i			1.35468i
$303$	568.843			1.87737
$304$	174.737	+	50.7840i	0.574793	+	0.167053i
$305$		0			0
$306$			696.674i			2.27671i
$307$	−531.973			−1.73281			−0.866406	−	0.499341i	$-0.833576\pi$
$307$	−531.973			−1.73281			−0.866406	+	0.499341i	$0.833576\pi$
$308$		−	212.266i		−	0.689176i
$309$	562.857			1.82154
$310$		0			0
$311$	−144.171			−0.463572			−0.231786	−	0.972767i	$-0.574457\pi$
$311$	−144.171			−0.463572			−0.231786	+	0.972767i	$0.574457\pi$
$312$	687.833			2.20459
$313$		−	238.517i		−	0.762036i	−0.924568	−	0.381018i	$-0.875574\pi$
$313$		−	238.517i		−	0.762036i	0.924568	−	0.381018i	$-0.124426\pi$
$314$		−	839.299i		−	2.67293i
$315$		0			0
$316$		−	70.0628i		−	0.221718i
$317$	−9.36886			−0.0295548			−0.0147774	−	0.999891i	$-0.504704\pi$
$317$	−9.36886			−0.0295548			−0.0147774	+	0.999891i	$0.504704\pi$
$318$	−303.880			−0.955598
$319$		−	64.7260i		−	0.202903i
$320$		0			0
$321$	518.352			1.61480
$322$	516.664			1.60455
$323$	141.561	−	487.084i	0.438271	−	1.50800i
$324$	−676.218			−2.08709
$325$		0			0
$326$		−	89.3130i		−	0.273966i
$327$		−	75.5097i		−	0.230917i
$328$		−	762.449i		−	2.32454i
$329$	−242.784			−0.737945
$330$		0			0
$331$		−	255.663i		−	0.772396i	−0.922416	−	0.386198i	$-0.873788\pi$
$331$		−	255.663i		−	0.772396i	0.922416	−	0.386198i	$-0.126212\pi$
$332$		−	113.195i		−	0.340948i
$333$	−140.076			−0.420648
$334$	868.776			2.60113
$335$		0			0
$336$		−	178.915i		−	0.532486i
$337$	−59.1258			−0.175448			−0.0877238	−	0.996145i	$-0.527959\pi$
$337$	−59.1258			−0.175448			−0.0877238	+	0.996145i	$0.527959\pi$
$338$	−136.235			−0.403061
$339$	−549.169			−1.61997
$340$		0			0
$341$		−	190.674i		−	0.559161i
$342$	−476.120	−	138.375i	−1.39217	−	0.404606i
$343$		−	352.389i		−	1.02737i
$344$		−	830.289i		−	2.41363i
$345$		0			0
$346$	458.878			1.32624
$347$			152.643i			0.439894i	0.975512	+	0.219947i	$0.0705885\pi$
$347$			152.643i			0.439894i	−0.975512	+	0.219947i	$0.929411\pi$
$348$		−	315.193i		−	0.905726i
$349$	136.603			0.391411			0.195706	−	0.980663i	$-0.437300\pi$
$349$	136.603			0.391411			0.195706	+	0.980663i	$0.437300\pi$
$350$		0			0
$351$	75.9500			0.216382
$352$	88.2218			0.250630
$353$			220.194i			0.623780i	0.950118	+	0.311890i	$0.100962\pi$
$353$			220.194i			0.623780i	−0.950118	+	0.311890i	$0.899038\pi$
$354$			1284.81i			3.62941i
$355$		0			0
$356$		−	88.0557i		−	0.247348i
$357$	−498.731			−1.39700
$358$			147.680i			0.412515i
$359$	−217.423			−0.605635			−0.302817	−	0.953049i	$-0.597927\pi$
$359$	−217.423			−0.605635			−0.302817	+	0.953049i	$0.597927\pi$
$360$		0			0
$361$	304.765	+	193.492i	0.844225	+	0.535988i
$362$		−	835.484i		−	2.30797i
$363$	−335.243			−0.923534
$364$			491.692i			1.35080i
$365$		0			0
$366$	384.571			1.05074
$367$			363.227i			0.989718i	0.868973	+	0.494859i	$0.164780\pi$
$367$			363.227i			0.989718i	−0.868973	+	0.494859i	$0.835220\pi$
$368$		−	320.199i		−	0.870106i
$369$			505.877i			1.37094i
$370$		0			0
$371$		−	100.414i		−	0.270659i
$372$		−	928.514i		−	2.49601i
$373$	14.8850			0.0399061			0.0199531	−	0.999801i	$-0.493648\pi$
$373$	14.8850			0.0399061			0.0199531	+	0.999801i	$0.493648\pi$
$374$			563.891i			1.50773i
$375$		0			0
$376$			617.916i			1.64339i
$377$			149.931i			0.397695i
$378$		−	81.1316i		−	0.214634i
$379$		−	214.679i		−	0.566436i	−0.959056	−	0.283218i	$-0.908598\pi$
$379$		−	214.679i		−	0.566436i	0.959056	−	0.283218i	$-0.0914020\pi$
$380$		0			0
$381$	313.470			0.822757
$382$	−455.928			−1.19353
$383$	556.024			1.45176			0.725880	−	0.687822i	$-0.241433\pi$
$383$	556.024			1.45176			0.725880	+	0.687822i	$0.241433\pi$
$384$	959.332			2.49826
$385$		0			0
$386$	−138.975			−0.360040
$387$			550.888i			1.42348i
$388$	−461.335			−1.18901
$389$	−64.2081			−0.165059			−0.0825296	−	0.996589i	$-0.526300\pi$
$389$	−64.2081			−0.165059			−0.0825296	+	0.996589i	$0.526300\pi$
$390$		0			0
$391$	−892.562			−2.28277
$392$	−327.040			−0.834286
$393$	−335.944			−0.854818
$394$			339.183i			0.860870i
$395$		0			0
$396$	358.447			0.905168
$397$			5.41669i			0.0136441i	0.999977	+	0.00682203i	$0.00217154\pi$
$397$			5.41669i			0.0136441i	−0.999977	+	0.00682203i	$0.997828\pi$
$398$	−775.026			−1.94730
$399$	99.0592	−	340.842i	0.248269	−	0.854242i
$400$		0			0
$401$			130.654i			0.325820i	0.986641	+	0.162910i	$0.0520881\pi$
$401$			130.654i			0.325820i	−0.986641	+	0.162910i	$0.947912\pi$
$402$	−839.469			−2.08823
$403$			441.676i			1.09597i
$404$	1034.93			2.56172
$405$		0			0
$406$	160.160			0.394482
$407$	−113.378			−0.278570
$408$			1269.33i			3.11111i
$409$		−	473.522i		−	1.15776i	−0.815414	−	0.578878i	$-0.803491\pi$
$409$		−	473.522i		−	1.15776i	0.815414	−	0.578878i	$-0.196509\pi$
$410$		0			0
$411$			155.744i			0.378938i
$412$	1024.04			2.48554
$413$	−424.554			−1.02798
$414$			872.472i			2.10742i
$415$		0			0
$416$	−204.356			−0.491242
$417$	190.321			0.456406
$418$	−385.374	−	112.001i	−0.921948	−	0.267946i
$419$	414.091			0.988284			0.494142	−	0.869381i	$-0.335482\pi$
$419$	414.091			0.988284			0.494142	+	0.869381i	$0.335482\pi$
$420$		0			0
$421$			238.371i			0.566201i	0.959090	+	0.283100i	$0.0913630\pi$
$421$			238.371i			0.566201i	−0.959090	+	0.283100i	$0.908637\pi$
$422$		−	657.790i		−	1.55874i
$423$		−	409.981i		−	0.969221i
$424$	−255.568			−0.602754
$425$		0			0
$426$			1432.01i			3.36154i
$427$			127.078i			0.297606i
$428$	943.072			2.20344
$429$	−369.388			−0.861044
$430$		0			0
$431$		−	432.728i		−	1.00401i	−0.864865	−	0.502005i	$-0.832596\pi$
$431$		−	432.728i		−	1.00401i	0.864865	−	0.502005i	$-0.167404\pi$
$432$	−50.2808			−0.116391
$433$	711.686			1.64362			0.821808	−	0.569764i	$-0.192965\pi$
$433$	711.686			1.64362			0.821808	+	0.569764i	$0.192965\pi$
$434$	471.809			1.08712
$435$		0			0
$436$		−	137.380i		−	0.315091i
$437$	177.283	−	609.994i	0.405682	−	1.39587i
$438$			164.831i			0.376325i
$439$			381.873i			0.869870i	0.900462	+	0.434935i	$0.143229\pi$
$439$			381.873i			0.869870i	−0.900462	+	0.434935i	$0.856771\pi$
$440$		0			0
$441$	216.988			0.492036
$442$		−	1306.19i		−	2.95519i
$443$			409.840i			0.925147i	0.886581	+	0.462573i	$0.153074\pi$
$443$			409.840i			0.925147i	−0.886581	+	0.462573i	$0.846926\pi$
$444$	−552.111			−1.24349
$445$		0			0
$446$	−98.7809			−0.221482
$447$	859.889			1.92369
$448$			393.341i			0.877993i
$449$		−	234.199i		−	0.521601i	−0.965393	−	0.260800i	$-0.916014\pi$
$449$		−	234.199i		−	0.521601i	0.965393	−	0.260800i	$-0.0839864\pi$
$450$		0			0
$451$			409.459i			0.907891i
$452$	−999.140			−2.21049
$453$		−	494.567i		−	1.09176i
$454$	−1060.61			−2.33614
$455$		0			0
$456$	−867.488	−	252.118i	−1.90239	−	0.552891i
$457$			614.393i			1.34440i	0.740367	+	0.672202i	$0.234652\pi$
$457$			614.393i			1.34440i	−0.740367	+	0.672202i	$0.765348\pi$
$458$	128.319			0.280172
$459$			140.159i			0.305357i
$460$		0			0
$461$	−51.6808			−0.112106			−0.0560530	−	0.998428i	$-0.517852\pi$
$461$	−51.6808			−0.112106			−0.0560530	+	0.998428i	$0.517852\pi$
$462$			394.589i			0.854089i
$463$			364.290i			0.786804i	0.919367	+	0.393402i	$0.128702\pi$
$463$			364.290i			0.786804i	−0.919367	+	0.393402i	$0.871298\pi$
$464$		−	99.2580i		−	0.213918i
$465$		0			0
$466$		−	1408.97i		−	3.02355i
$467$			358.002i			0.766599i	0.923624	+	0.383300i	$0.125212\pi$
$467$			358.002i			0.766599i	−0.923624	+	0.383300i	$0.874788\pi$
$468$	−830.304			−1.77415
$469$		−	277.395i		−	0.591460i
$470$		0			0
$471$			1014.61i			2.15415i
$472$			1080.54i			2.28929i
$473$			445.892i			0.942688i
$474$			130.242i			0.274773i
$475$		0			0
$476$	−907.374			−1.90625
$477$	169.566			0.355485
$478$	−332.284			−0.695155
$479$	246.301			0.514198			0.257099	−	0.966385i	$-0.417233\pi$
$479$	246.301			0.514198			0.257099	+	0.966385i	$0.417233\pi$
$480$		0			0
$481$	262.628			0.546005
$482$		−	287.928i		−	0.597361i
$483$	−624.580			−1.29313
$484$	−609.930			−1.26018
$485$		0			0
$486$	1097.24			2.25770
$487$	732.073			1.50323			0.751614	−	0.659603i	$-0.229275\pi$
$487$	732.073			1.50323			0.751614	+	0.659603i	$0.229275\pi$
$488$	323.430			0.662766
$489$			107.968i			0.220793i
$490$		0			0
$491$	284.304			0.579030			0.289515	−	0.957173i	$-0.406506\pi$
$491$	284.304			0.579030			0.289515	+	0.957173i	$0.406506\pi$
$492$			1993.92i			4.05268i
$493$	−276.684			−0.561225
$494$	892.679	+	259.440i	1.80704	+	0.525182i
$495$		0			0
$496$		−	292.401i		−	0.589517i
$497$	−473.196			−0.952105
$498$			210.422i			0.422534i
$499$	161.653			0.323955			0.161977	−	0.986794i	$-0.448213\pi$
$499$	161.653			0.323955			0.161977	+	0.986794i	$0.448213\pi$
$500$		0			0
$501$	−1050.24			−2.09629
$502$	−399.864			−0.796542
$503$		−	232.214i		−	0.461658i	−0.972994	−	0.230829i	$-0.925856\pi$
$503$		−	232.214i		−	0.461658i	0.972994	−	0.230829i	$-0.0741439\pi$
$504$			409.999i			0.813490i
$505$		0			0
$506$			706.183i			1.39562i
$507$	164.690			0.324833
$508$	570.317			1.12267
$509$			164.016i			0.322232i	0.986935	+	0.161116i	$0.0515094\pi$
$509$			164.016i			0.322232i	−0.986935	+	0.161116i	$0.948491\pi$
$510$		0			0
$511$	−54.4667			−0.106589
$512$	580.794			1.13436
$513$	−95.7873	−	27.8387i	−0.186720	−	0.0542665i
$514$	981.974			1.91046
$515$		0			0
$516$			2171.33i			4.20801i
$517$		−	331.840i		−	0.641857i
$518$		−	280.546i		−	0.541595i
$519$	−554.725			−1.06883
$520$		0			0
$521$		−	917.279i		−	1.76061i	−0.474407	−	0.880306i	$-0.657337\pi$
$521$		−	917.279i		−	1.76061i	0.474407	−	0.880306i	$-0.342663\pi$
$522$			270.456i			0.518116i
$523$	552.428			1.05627			0.528134	−	0.849161i	$-0.322892\pi$
$523$	552.428			1.05627			0.528134	+	0.849161i	$0.322892\pi$
$524$	−611.204			−1.16642
$525$		0			0
$526$		−	1528.30i		−	2.90552i
$527$	−815.073			−1.54663
$528$	244.544			0.463152
$529$	−588.791			−1.11303
$530$		0			0
$531$		−	716.930i		−	1.35015i
$532$	180.225	−	620.117i	0.338769	−	1.16563i
$533$		−	948.469i		−	1.77949i
$534$			163.690i			0.306535i
$535$		0			0
$536$	−706.006			−1.31717
$537$		−	178.527i		−	0.332452i
$538$		−	373.526i		−	0.694286i
$539$	175.631			0.325846
$540$		0			0
$541$	629.744			1.16404			0.582018	−	0.813176i	$-0.302263\pi$
$541$	629.744			1.16404			0.582018	+	0.813176i	$0.302263\pi$
$542$	1700.89			3.13818
$543$			1009.99i			1.86003i
$544$		−	377.122i		−	0.693238i
$545$		0			0
$546$		−	914.024i		−	1.67404i
$547$	290.138			0.530416			0.265208	−	0.964191i	$-0.414559\pi$
$547$	290.138			0.530416			0.265208	+	0.964191i	$0.414559\pi$
$548$			283.354i			0.517070i
$549$	−214.592			−0.390878
$550$		0			0
$551$	54.9557	−	189.091i	0.0997381	−	0.343178i
$552$			1589.64i			2.87978i
$553$	−43.0374			−0.0778253
$554$		−	609.960i		−	1.10101i
$555$		0			0
$556$	346.264			0.622777
$557$		−	2.72609i		−	0.00489423i	−0.999997	−	0.00244712i	$-0.999221\pi$
$557$		−	2.72609i		−	0.00489423i	0.999997	−	0.00244712i	$-0.000778942\pi$
$558$			796.727i			1.42783i
$559$		−	1032.86i		−	1.84769i
$560$		0			0
$561$		−	681.672i		−	1.21510i
$562$		−	1223.90i		−	2.17776i
$563$	−92.3767			−0.164079			−0.0820397	−	0.996629i	$-0.526143\pi$
$563$	−92.3767			−0.164079			−0.0820397	+	0.996629i	$0.526143\pi$
$564$		−	1615.94i		−	2.86515i
$565$		0			0
$566$		−	134.939i		−	0.238408i
$567$			415.379i			0.732592i
$568$			1204.34i			2.12033i
$569$			675.420i			1.18703i	0.804823	+	0.593515i	$0.202260\pi$
$569$			675.420i			1.18703i	−0.804823	+	0.593515i	$0.797740\pi$
$570$		0			0
$571$	−673.447			−1.17942			−0.589708	−	0.807616i	$-0.700757\pi$
$571$	−673.447			−1.17942			−0.589708	+	0.807616i	$0.700757\pi$
$572$	−672.052			−1.17492
$573$	551.159			0.961882
$574$	−1013.18			−1.76512
$575$		0			0
$576$	−664.221			−1.15316
$577$		−	380.054i		−	0.658672i	−0.944213	−	0.329336i	$-0.893175\pi$
$577$		−	380.054i		−	0.658672i	0.944213	−	0.329336i	$-0.106825\pi$
$578$	1433.04			2.47931
$579$	168.003			0.290161
$580$		0			0
$581$	−69.5320			−0.119676
$582$	857.593			1.47353
$583$	137.248			0.235417
$584$			138.625i			0.237371i
$585$		0			0
$586$	−1133.50			−1.93430
$587$			934.331i			1.59171i	0.605490	+	0.795853i	$0.292977\pi$
$587$			934.331i			1.59171i	−0.605490	+	0.795853i	$0.707023\pi$
$588$	855.260			1.45452
$589$	161.892	−	557.037i	0.274859	−	0.945734i
$590$		0			0
$591$		−	410.029i		−	0.693788i
$592$	−173.867			−0.293694
$593$		−	1086.40i		−	1.83204i	−0.401133	−	0.916020i	$-0.631384\pi$
$593$		−	1086.40i		−	1.83204i	0.401133	−	0.916020i	$-0.368616\pi$
$594$	110.892			0.186687
$595$		0			0
$596$	1564.45			2.62492
$597$	936.907			1.56936
$598$		−	1635.80i		−	2.73545i
$599$		−	639.332i		−	1.06733i	−0.845695	−	0.533666i	$-0.820814\pi$
$599$		−	639.332i		−	1.06733i	0.845695	−	0.533666i	$-0.179186\pi$
$600$		0			0
$601$		−	1185.57i		−	1.97265i	−0.164800	−	0.986327i	$-0.552698\pi$
$601$		−	1185.57i		−	1.97265i	0.164800	−	0.986327i	$-0.447302\pi$
$602$	−1103.33			−1.83277
$603$	468.427			0.776828
$604$		−	899.798i		−	1.48973i
$605$		0			0
$606$	−1923.88			−3.17471
$607$	−468.485			−0.771804			−0.385902	−	0.922540i	$-0.626110\pi$
$607$	−468.485			−0.771804			−0.385902	+	0.922540i	$0.626110\pi$
$608$	257.732	+	74.9049i	0.423902	+	0.123199i
$609$	−193.613			−0.317919
$610$		0			0
$611$			768.673i			1.25806i
$612$		−	1532.25i		−	2.50368i
$613$			489.776i			0.798982i	0.916737	+	0.399491i	$0.130813\pi$
$613$			489.776i			0.798982i	−0.916737	+	0.399491i	$0.869187\pi$
$614$	1799.18			2.93026
$615$		0			0
$616$			331.855i			0.538726i
$617$		−	916.972i		−	1.48618i	−0.669193	−	0.743089i	$-0.733360\pi$
$617$		−	916.972i		−	1.48618i	0.669193	−	0.743089i	$-0.266640\pi$
$618$	−1903.63			−3.08031
$619$	248.052			0.400730			0.200365	−	0.979721i	$-0.435787\pi$
$619$	248.052			0.400730			0.200365	+	0.979721i	$0.435787\pi$
$620$		0			0
$621$			175.527i			0.282651i
$622$	487.598			0.783919
$623$	−54.0898			−0.0868216
$624$	−566.461			−0.907790
$625$		0			0
$626$			806.686i			1.28864i
$627$	465.868	+	135.396i	0.743012	+	0.215942i
$628$			1845.94i			2.93939i
$629$			484.657i			0.770520i
$630$		0			0
$631$	198.320			0.314295			0.157148	−	0.987575i	$-0.449770\pi$
$631$	198.320			0.314295			0.157148	+	0.987575i	$0.449770\pi$
$632$			109.536i			0.173316i
$633$			795.184i			1.25621i
$634$	31.6863			0.0499784
$635$		0			0
$636$	668.348			1.05086
$637$	−406.830			−0.638666
$638$			218.909i			0.343117i
$639$		−	799.070i		−	1.25050i
$640$		0			0
$641$			669.775i			1.04489i	0.852673	+	0.522445i	$0.174980\pi$
$641$			669.775i			1.04489i	−0.852673	+	0.522445i	$0.825020\pi$
$642$	−1753.11			−2.73070
$643$		−	104.564i		−	0.162619i	−0.996689	−	0.0813096i	$-0.974090\pi$
$643$		−	104.564i		−	0.162619i	0.996689	−	0.0813096i	$-0.0259103\pi$
$644$	−1136.34			−1.76450
$645$		0			0
$646$	−478.773	+	1647.36i	−0.741134	+	2.55009i
$647$		−	504.773i		−	0.780175i	−0.920778	−	0.390087i	$-0.872445\pi$
$647$		−	504.773i		−	0.780175i	0.920778	−	0.390087i	$-0.127555\pi$
$648$	1057.19			1.63147
$649$		−	580.286i		−	0.894124i
$650$		0			0
$651$	−570.357			−0.876124
$652$			196.433i			0.301278i
$653$		−	799.353i		−	1.22412i	−0.790810	−	0.612062i	$-0.790340\pi$
$653$		−	799.353i		−	1.22412i	0.790810	−	0.612062i	$-0.209660\pi$
$654$			255.380i			0.390490i
$655$		0			0
$656$			627.910i			0.957180i
$657$		−	91.9761i		−	0.139994i
$658$	821.115			1.24790
$659$			1147.64i			1.74149i	0.491732	+	0.870747i	$0.336364\pi$
$659$			1147.64i			1.74149i	−0.491732	+	0.870747i	$0.663636\pi$
$660$		0			0
$661$		−	989.094i		−	1.49636i	−0.663496	−	0.748180i	$-0.730928\pi$
$661$		−	989.094i		−	1.49636i	0.663496	−	0.748180i	$-0.269072\pi$
$662$			864.675i			1.30616i
$663$			1579.02i			2.38163i
$664$			176.968i			0.266518i
$665$		0			0
$666$	473.748			0.711334
$667$	−346.502			−0.519494
$668$	−1910.77			−2.86043
$669$	119.414			0.178496
$670$		0			0
$671$	−173.692			−0.258855
$672$		−	263.895i		−	0.392701i
$673$	502.564			0.746752			0.373376	−	0.927680i	$-0.378200\pi$
$673$	502.564			0.746752			0.373376	+	0.927680i	$0.378200\pi$
$674$	199.969			0.296689
$675$		0			0
$676$	299.632			0.443243
$677$	−825.425			−1.21924			−0.609620	−	0.792694i	$-0.708678\pi$
$677$	−825.425			−1.21924			−0.609620	+	0.792694i	$0.708678\pi$
$678$	1857.34			2.73944
$679$			283.384i			0.417354i
$680$		0			0
$681$	1282.14			1.88273
$682$			644.875i			0.945564i
$683$	−627.488			−0.918723			−0.459362	−	0.888249i	$-0.651922\pi$
$683$	−627.488			−0.918723			−0.459362	+	0.888249i	$0.651922\pi$
$684$	1047.17	+	304.340i	1.53095	+	0.444941i
$685$		0			0
$686$			1191.81i			1.73733i
$687$	−155.121			−0.225795
$688$			683.780i			0.993866i
$689$	−317.920			−0.461423
$690$		0			0
$691$	−545.404			−0.789297			−0.394649	−	0.918832i	$-0.629134\pi$
$691$	−545.404			−0.789297			−0.394649	+	0.918832i	$0.629134\pi$
$692$	−1009.25			−1.45845
$693$		−	220.182i		−	0.317723i
$694$		−	516.253i		−	0.743880i
$695$		0			0
$696$			492.769i			0.708002i
$697$	1750.31			2.51121
$698$	−462.001			−0.661893
$699$			1703.27i			2.43673i
$700$		0			0
$701$	−544.518			−0.776774			−0.388387	−	0.921496i	$-0.626968\pi$
$701$	−544.518			−0.776774			−0.388387	+	0.921496i	$0.626968\pi$
$702$	−256.869			−0.365911
$703$	−331.224	−	96.2639i	−0.471158	−	0.136933i
$704$	−537.624			−0.763670
$705$		0			0
$706$		−	744.716i		−	1.05484i
$707$		−	635.727i		−	0.899190i
$708$		−	2825.79i		−	3.99123i
$709$	191.235			0.269725			0.134863	−	0.990864i	$-0.456941\pi$
$709$	191.235			0.269725			0.134863	+	0.990864i	$0.456941\pi$
$710$		0			0
$711$		−	72.6757i		−	0.102216i
$712$			137.666i			0.193351i
$713$	−1020.75			−1.43162
$714$	1686.75			2.36239
$715$		0			0
$716$		−	324.805i		−	0.453639i
$717$	401.689			0.560236
$718$	735.343			1.02415
$719$	−996.287			−1.38566			−0.692828	−	0.721102i	$-0.743636\pi$
$719$	−996.287			−1.38566			−0.692828	+	0.721102i	$0.743636\pi$
$720$		0			0
$721$		−	629.037i		−	0.872450i
$722$	−1030.74	−	654.405i	−1.42762	−	0.906379i
$723$			348.068i			0.481422i
$724$			1837.55i			2.53805i
$725$		0			0
$726$	1133.82			1.56173
$727$			623.938i			0.858237i	0.903248	+	0.429119i	$0.141176\pi$
$727$			623.938i			0.858237i	−0.903248	+	0.429119i	$0.858824\pi$
$728$		−	768.707i		−	1.05592i
$729$	−508.254			−0.697193
$730$		0			0
$731$	1906.05			2.60746
$732$	−845.818			−1.15549
$733$			331.044i			0.451629i	0.974170	+	0.225814i	$0.0725043\pi$
$733$			331.044i			0.451629i	−0.974170	+	0.225814i	$0.927496\pi$
$734$		−	1228.46i		−	1.67366i
$735$		0			0
$736$		−	472.284i		−	0.641691i
$737$	379.147			0.514447
$738$		−	1710.92i		−	2.31832i
$739$	835.615			1.13074			0.565368	−	0.824838i	$-0.308734\pi$
$739$	835.615			1.13074			0.565368	+	0.824838i	$0.308734\pi$
$740$		0			0
$741$	−1079.13	−	313.630i	−1.45632	−	0.423252i
$742$			339.610i			0.457695i
$743$	209.741			0.282290			0.141145	−	0.989989i	$-0.454922\pi$
$743$	209.741			0.282290			0.141145	+	0.989989i	$0.454922\pi$
$744$			1451.63i			1.95112i
$745$		0			0
$746$	−50.3423			−0.0674829
$747$		−	117.416i		−	0.157184i
$748$		−	1240.21i		−	1.65804i
$749$		−	579.299i		−	0.773430i
$750$		0			0
$751$			1097.59i			1.46150i	0.682643	+	0.730752i	$0.260831\pi$
$751$			1097.59i			1.46150i	−0.682643	+	0.730752i	$0.739169\pi$
$752$		−	508.881i		−	0.676704i
$753$	483.385			0.641945
$754$		−	507.079i		−	0.672519i
$755$		0			0
$756$			178.439i			0.236031i
$757$		−	551.713i		−	0.728816i	−0.931239	−	0.364408i	$-0.881271\pi$
$757$		−	551.713i		−	0.728816i	0.931239	−	0.364408i	$-0.118729\pi$
$758$			726.063i			0.957867i
$759$		−	853.685i		−	1.12475i
$760$		0			0
$761$	1115.81			1.46624			0.733119	−	0.680101i	$-0.238064\pi$
$761$	1115.81			1.46624			0.733119	+	0.680101i	$0.238064\pi$
$762$	−1060.18			−1.39132
$763$	−84.3880			−0.110600
$764$	1002.76			1.31251
$765$		0			0
$766$	−1880.52			−2.45499
$767$			1344.17i			1.75251i
$768$	−1836.71			−2.39155
$769$	−1432.18			−1.86239			−0.931195	−	0.364522i	$-0.881232\pi$
$769$	−1432.18			−1.86239			−0.931195	+	0.364522i	$0.881232\pi$
$770$		0			0
$771$	−1187.08			−1.53967
$772$	305.660			0.395932
$773$	−753.633			−0.974946			−0.487473	−	0.873138i	$-0.662081\pi$
$773$	−753.633			−0.974946			−0.487473	+	0.873138i	$0.662081\pi$
$774$		−	1863.15i		−	2.40717i
$775$		0			0
$776$	721.248			0.929443
$777$			339.144i			0.436479i
$778$	217.157			0.279122
$779$	−347.652	+	1196.20i	−0.446280	+	1.53556i
$780$		0			0
$781$		−	646.771i		−	0.828132i
$782$	3018.72			3.86026
$783$			54.4112i			0.0694907i
$784$	269.332			0.343536
$785$		0			0
$786$	1136.19			1.44553
$787$	1196.41			1.52022			0.760110	−	0.649795i	$-0.225145\pi$
$787$	1196.41			1.52022			0.760110	+	0.649795i	$0.225145\pi$
$788$		−	745.992i		−	0.946691i
$789$			1847.52i			2.34160i
$790$		0			0
$791$			613.740i			0.775904i
$792$	−560.392			−0.707566
$793$	402.339			0.507363
$794$		−	18.3197i		−	0.0230727i
$795$		0			0
$796$	1704.58			2.14143
$797$	1345.94			1.68876			0.844379	−	0.535747i	$-0.179970\pi$
$797$	1345.94			1.68876			0.844379	+	0.535747i	$0.179970\pi$
$798$	−335.027	+	1152.76i	−0.419833	+	1.44456i
$799$	−1418.52			−1.77537
$800$		0			0
$801$		−	91.3397i		−	0.114032i
$802$		−	441.883i		−	0.550976i
$803$		−	74.4459i		−	0.0927097i
$804$	1846.31			2.29641
$805$		0			0
$806$		−	1493.79i		−	1.85333i
$807$			451.545i			0.559535i
$808$	−1618.01			−2.00248
$809$	−73.4294			−0.0907657			−0.0453828	−	0.998970i	$-0.514451\pi$
$809$	−73.4294			−0.0907657			−0.0453828	+	0.998970i	$0.514451\pi$
$810$		0			0
$811$			420.830i			0.518903i	0.965756	+	0.259451i	$0.0835418\pi$
$811$			420.830i			0.518903i	−0.965756	+	0.259451i	$0.916458\pi$
$812$	−352.253			−0.433809
$813$	−2056.16			−2.52911
$814$	383.454			0.471074
$815$		0			0
$816$		−	1045.35i		−	1.28107i
$817$	−378.585	+	1302.63i	−0.463384	+	1.59441i
$818$			1601.49i			1.95781i
$819$			510.029i			0.622746i
$820$		0			0
$821$	−80.2743			−0.0977762			−0.0488881	−	0.998804i	$-0.515568\pi$
$821$	−80.2743			−0.0977762			−0.0488881	+	0.998804i	$0.515568\pi$
$822$		−	526.738i		−	0.640800i
$823$		−	444.054i		−	0.539555i	−0.962923	−	0.269778i	$-0.913050\pi$
$823$		−	444.054i		−	0.539555i	0.962923	−	0.269778i	$-0.0869502\pi$
$824$	−1600.98			−1.94294
$825$		0			0
$826$	1435.88			1.73835
$827$	−257.906			−0.311857			−0.155929	−	0.987768i	$-0.549837\pi$
$827$	−257.906			−0.311857			−0.155929	+	0.987768i	$0.549837\pi$
$828$		−	1918.90i		−	2.31751i
$829$		−	610.791i		−	0.736781i	−0.929671	−	0.368391i	$-0.879909\pi$
$829$		−	610.791i		−	0.736781i	0.929671	−	0.368391i	$-0.120091\pi$
$830$		0			0
$831$			737.364i			0.887321i
$832$	1245.35			1.49681
$833$		−	750.769i		−	0.901283i
$834$	−643.683			−0.771802
$835$		0			0
$836$	847.585	+	246.334i	1.01386	+	0.294658i
$837$			160.288i			0.191503i
$838$	−1400.49			−1.67123
$839$			168.780i			0.201168i	0.994929	+	0.100584i	$0.0320711\pi$
$839$			168.780i			0.201168i	−0.994929	+	0.100584i	$0.967929\pi$
$840$		0			0
$841$	733.588			0.872281
$842$		−	806.189i		−	0.957470i
$843$			1479.54i			1.75509i
$844$			1446.73i			1.71414i
$845$		0			0
$846$			1386.59i			1.63899i
$847$			374.660i			0.442338i
$848$	210.471			0.248197
$849$			163.124i			0.192136i
$850$		0			0
$851$			606.955i			0.713226i
$852$		−	3149.55i		−	3.69665i
$853$			932.527i			1.09323i	0.837383	+	0.546616i	$0.184084\pi$
$853$			932.527i			1.09323i	−0.837383	+	0.546616i	$0.815916\pi$
$854$		−	429.788i		−	0.503265i
$855$		0			0
$856$	−1474.39			−1.72242
$857$	658.085			0.767893			0.383947	−	0.923355i	$-0.374565\pi$
$857$	658.085			0.767893			0.383947	+	0.923355i	$0.374565\pi$
$858$	1249.30			1.45606
$859$	634.851			0.739058			0.369529	−	0.929219i	$-0.379519\pi$
$859$	634.851			0.739058			0.369529	+	0.929219i	$0.379519\pi$
$860$		0			0
$861$	1224.80			1.42253
$862$			1463.52i			1.69782i
$863$	−168.480			−0.195226			−0.0976130	−	0.995224i	$-0.531121\pi$
$863$	−168.480			−0.195226			−0.0976130	+	0.995224i	$0.531121\pi$
$864$	−74.1627			−0.0858365
$865$		0			0
$866$	−2406.98			−2.77943
$867$	−1732.36			−1.99811
$868$	−1037.69			−1.19549
$869$		−	58.8241i		−	0.0676917i
$870$		0			0
$871$	−878.255			−1.00833
$872$			214.778i			0.246306i
$873$	−478.540			−0.548156
$874$	−599.586	+	2063.05i	−0.686025	+	2.36047i
$875$		0			0
$876$		−	362.525i		−	0.413842i
$877$	352.865			0.402355			0.201177	−	0.979555i	$-0.435523\pi$
$877$	352.865			0.402355			0.201177	+	0.979555i	$0.435523\pi$
$878$		−	1291.53i		−	1.47099i
$879$	1370.26			1.55888
$880$		0			0
$881$	868.676			0.986011			0.493006	−	0.870026i	$-0.335898\pi$
$881$	868.676			0.986011			0.493006	+	0.870026i	$0.335898\pi$
$882$	−733.871			−0.832053
$883$		−	220.851i		−	0.250114i	−0.992150	−	0.125057i	$-0.960089\pi$
$883$		−	220.851i		−	0.250114i	0.992150	−	0.125057i	$-0.0399114\pi$
$884$			2872.82i			3.24980i
$885$		0			0
$886$		−	1386.11i		−	1.56446i
$887$	−1308.29			−1.47496			−0.737482	−	0.675367i	$-0.763985\pi$
$887$	−1308.29			−1.47496			−0.737482	+	0.675367i	$0.763985\pi$
$888$	863.166			0.972034
$889$		−	350.328i		−	0.394070i
$890$		0			0
$891$	−567.747			−0.637202
$892$	217.257			0.243562
$893$	281.750	−	969.443i	0.315509	−	1.08560i
$894$	−2908.22			−3.25304
$895$		0			0
$896$		−	1072.13i		−	1.19657i
$897$			1977.47i			2.20454i
$898$			792.080i			0.882049i
$899$	−316.420			−0.351969
$900$		0			0
$901$		−	586.693i		−	0.651158i
$902$		−	1384.82i		−	1.53528i
$903$	1333.78			1.47706
$904$	1562.05			1.72793
$905$		0			0
$906$			1672.67i			1.84621i
$907$	−723.418			−0.797594			−0.398797	−	0.917039i	$-0.630572\pi$
$907$	−723.418			−0.797594			−0.398797	+	0.917039i	$0.630572\pi$
$908$	2332.68			2.56903
$909$	1073.53			1.18100
$910$		0			0
$911$		−	572.989i		−	0.628967i	−0.949263	−	0.314483i	$-0.898169\pi$
$911$		−	572.989i		−	0.628967i	0.949263	−	0.314483i	$-0.101831\pi$
$912$	714.415	+	207.631i	0.783349	+	0.227665i
$913$		−	95.0373i		−	0.104093i
$914$		−	2077.93i		−	2.27345i
$915$		0			0
$916$	−282.222			−0.308102
$917$			375.443i			0.409426i
$918$		−	474.029i		−	0.516372i
$919$	−11.3971			−0.0124016			−0.00620079	−	0.999981i	$-0.501974\pi$
$919$	−11.3971			−0.0124016			−0.00620079	+	0.999981i	$0.501974\pi$
$920$		0			0
$921$	−2174.98			−2.36154
$922$	174.789			0.189576
$923$			1498.18i			1.62316i
$924$		−	867.852i		−	0.939234i
$925$		0			0
$926$		−	1232.06i		−	1.33052i
$927$	1062.23			1.14588
$928$		−	146.403i		−	0.157762i
$929$	−81.6994			−0.0879434			−0.0439717	−	0.999033i	$-0.514001\pi$
$929$	−81.6994			−0.0879434			−0.0439717	+	0.999033i	$0.514001\pi$
$930$		0			0
$931$	513.090	+	149.120i	0.551117	+	0.160172i
$932$			3098.87i			3.32497i
$933$	−589.443			−0.631772
$934$		−	1210.79i		−	1.29635i
$935$		0			0
$936$	1298.09			1.38685
$937$		−	1797.25i		−	1.91809i	−0.283257	−	0.959044i	$-0.591415\pi$
$937$		−	1797.25i		−	1.91809i	0.283257	−	0.959044i	$-0.408585\pi$
$938$			938.173i			1.00018i
$939$		−	975.180i		−	1.03853i
$940$		0			0
$941$			265.809i			0.282475i	0.989976	+	0.141237i	$0.0451081\pi$
$941$			265.809i			0.282475i	−0.989976	+	0.141237i	$0.954892\pi$
$942$		−	3431.48i		−	3.64276i
$943$	2191.99			2.32448
$944$		−	889.876i		−	0.942665i
$945$		0			0
$946$		−	1508.04i		−	1.59413i
$947$		−	1177.82i		−	1.24373i	−0.783123	−	0.621867i	$-0.786374\pi$
$947$		−	1177.82i		−	1.24373i	0.783123	−	0.621867i	$-0.213626\pi$
$948$		−	286.453i		−	0.302165i
$949$			172.446i			0.181714i
$950$		0			0
$951$	−38.3047			−0.0402783
$952$	1418.58			1.49011
$953$	−181.390			−0.190336			−0.0951681	−	0.995461i	$-0.530339\pi$
$953$	−181.390			−0.190336			−0.0951681	+	0.995461i	$0.530339\pi$
$954$	−573.488			−0.601140
$955$		0			0
$956$	730.820			0.764456
$957$		−	264.633i		−	0.276523i
$958$	−833.009			−0.869530
$959$	174.056			0.181497
$960$		0			0
$961$	28.8690			0.0300405
$962$	−888.231			−0.923318
$963$	978.242			1.01583
$964$			633.264i			0.656912i
$965$		0			0
$966$	2112.38			2.18673
$967$		−	1536.16i		−	1.58859i	−0.607535	−	0.794293i	$-0.707841\pi$
$967$		−	1536.16i		−	1.58859i	0.607535	−	0.794293i	$-0.292159\pi$
$968$	953.559			0.985081
$969$	578.775	−	1991.45i	0.597291	−	2.05516i
$970$		0			0
$971$		−	54.9334i		−	0.0565741i	−0.999600	−	0.0282870i	$-0.990995\pi$
$971$		−	54.9334i		−	0.0565741i	0.999600	−	0.0282870i	$-0.00900524\pi$
$972$	−2413.25			−2.48277
$973$		−	212.699i		−	0.218601i
$974$	−2475.93			−2.54202
$975$		0			0
$976$	−266.359			−0.272908
$977$	259.857			0.265975			0.132987	−	0.991118i	$-0.457543\pi$
$977$	259.857			0.265975			0.132987	+	0.991118i	$0.457543\pi$
$978$		−	365.157i		−	0.373371i
$979$		−	73.9308i		−	0.0755166i
$980$		0			0
$981$		−	142.503i		−	0.145263i
$982$	−961.540			−0.979165
$983$	−895.533			−0.911020			−0.455510	−	0.890231i	$-0.650543\pi$
$983$	−895.533			−0.911020			−0.455510	+	0.890231i	$0.650543\pi$
$984$		−	3117.28i		−	3.16797i
$985$		0			0
$986$	935.769			0.949055
$987$	−992.624			−1.00570
$988$	−1963.34	−	570.607i	−1.98719	−	0.577537i
$989$	2387.02			2.41357
$990$		0			0
$991$		−	1321.52i		−	1.33352i	−0.745272	−	0.666761i	$-0.767680\pi$
$991$		−	1321.52i		−	1.33352i	0.745272	−	0.666761i	$-0.232320\pi$
$992$		−	431.282i		−	0.434760i
$993$		−	1045.28i		−	1.05265i
$994$	1600.39			1.61005
$995$		0			0
$996$		−	462.798i		−	0.464657i
$997$		−	1012.20i		−	1.01525i	−0.861579	−	0.507624i	$-0.830524\pi$
$997$		−	1012.20i		−	1.01525i	0.861579	−	0.507624i	$-0.169476\pi$
$998$	−546.726			−0.547821
$999$	95.3101			0.0954055

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	475.3.d.c.474.2		24
5.2	odd	4		95.3.c.a.56.1	✓	12
5.3	odd	4		475.3.c.g.151.12		12
5.4	even	2	inner	475.3.d.c.474.23		24
15.2	even	4		855.3.e.a.721.12		12
19.18	odd	2	inner	475.3.d.c.474.24		24
20.7	even	4		1520.3.h.a.721.10		12
95.18	even	4		475.3.c.g.151.1		12
95.37	even	4		95.3.c.a.56.12	yes	12
95.94	odd	2	inner	475.3.d.c.474.1		24
285.227	odd	4		855.3.e.a.721.1		12
380.227	odd	4		1520.3.h.a.721.3		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
95.3.c.a.56.1	✓	12	5.2	odd	4
95.3.c.a.56.12	yes	12	95.37	even	4
475.3.c.g.151.1		12	95.18	even	4
475.3.c.g.151.12		12	5.3	odd	4
475.3.d.c.474.1		24	95.94	odd	2	inner
475.3.d.c.474.2		24	1.1	even	1	trivial
475.3.d.c.474.23		24	5.4	even	2	inner
475.3.d.c.474.24		24	19.18	odd	2	inner
855.3.e.a.721.1		12	285.227	odd	4
855.3.e.a.721.12		12	15.2	even	4
1520.3.h.a.721.3		12	380.227	odd	4
1520.3.h.a.721.10		12	20.7	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:	\( N \)	\(=\)	\( 475 = 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	475.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-3.38208 q^{2} +4.08851 q^{3} +7.43849 q^{4} -13.8277 q^{6} -4.56923i q^{7} -11.6293 q^{8} +7.71590 q^{9} +O(q^{10})\)	\(q-3.38208 q^{2} +4.08851 q^{3} +7.43849 q^{4} -13.8277 q^{6} -4.56923i q^{7} -11.6293 q^{8} +7.71590 q^{9} +6.24529 q^{11} +30.4124 q^{12} -14.4666 q^{13} +15.4535i q^{14} +9.57723 q^{16} -26.6967i q^{17} -26.0958 q^{18} +(18.2451 + 5.30257i) q^{19} -18.6813i q^{21} -21.1221 q^{22} -33.4334i q^{23} -47.5464 q^{24} +48.9271 q^{26} -5.25004 q^{27} -33.9882i q^{28} -10.3640i q^{29} -30.5308i q^{31} +14.1261 q^{32} +25.5339 q^{33} +90.2906i q^{34} +57.3947 q^{36} -18.1542 q^{37} +(-61.7064 - 17.9338i) q^{38} -59.1466 q^{39} +65.5629i q^{41} +63.1819i q^{42} +71.3965i q^{43} +46.4556 q^{44} +113.075i q^{46} -53.1345i q^{47} +39.1566 q^{48} +28.1221 q^{49} -109.150i q^{51} -107.609 q^{52} +21.9762 q^{53} +17.7561 q^{54} +53.1369i q^{56} +(74.5951 + 21.6796i) q^{57} +35.0518i q^{58} -92.9158i q^{59} -27.8117 q^{61} +103.258i q^{62} -35.2557i q^{63} -86.0847 q^{64} -86.3579 q^{66} +60.7093 q^{67} -198.583i q^{68} -136.693i q^{69} -103.561i q^{71} -89.7304 q^{72} -11.9203i q^{73} +61.3990 q^{74} +(135.716 + 39.4432i) q^{76} -28.5362i q^{77} +200.039 q^{78} -9.41895i q^{79} -90.9080 q^{81} -221.739i q^{82} -15.2174i q^{83} -138.961i q^{84} -241.469i q^{86} -42.3732i q^{87} -72.6282 q^{88} -11.8378i q^{89} +66.1010i q^{91} -248.694i q^{92} -124.826i q^{93} +179.705i q^{94} +57.7548 q^{96} -62.0200 q^{97} -95.1114 q^{98} +48.1881 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 24 q^{4} - 56 q^{6} + 96 q^{9}+O(q^{10}) \) 24 * q + 24 * q^4 - 56 * q^6 + 96 * q^9	\( 24 q + 24 q^{4} - 56 q^{6} + 96 q^{9} + 64 q^{11} - 88 q^{16} - 16 q^{19} - 200 q^{24} + 216 q^{26} - 160 q^{36} - 152 q^{39} + 512 q^{44} - 144 q^{49} + 152 q^{54} - 592 q^{61} - 376 q^{64} + 304 q^{66} - 272 q^{74} + 496 q^{76} - 744 q^{81} - 88 q^{96} + 624 q^{99}+O(q^{100}) \) 24 * q + 24 * q^4 - 56 * q^6 + 96 * q^9 + 64 * q^11 - 88 * q^16 - 16 * q^19 - 200 * q^24 + 216 * q^26 - 160 * q^36 - 152 * q^39 + 512 * q^44 - 144 * q^49 + 152 * q^54 - 592 * q^61 - 376 * q^64 + 304 * q^66 - 272 * q^74 + 496 * q^76 - 744 * q^81 - 88 * q^96 + 624 * q^99

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