LMFDB - Embedded newform 475.3.d.c.474.16

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			474.16
Character	$\chi$	$=$	475.474
Dual form			475.3.d.c.474.15

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.14149 q^{2} -4.13699 q^{3} -2.69701 q^{4} -4.72232 q^{6} -7.54444i q^{7} -7.64455 q^{8} +8.11468 q^{9} +O(q^{10})$	\(q+1.14149 q^{2} -4.13699 q^{3} -2.69701 q^{4} -4.72232 q^{6} -7.54444i q^{7} -7.64455 q^{8} +8.11468 q^{9} -18.6624 q^{11} +11.1575 q^{12} +16.9074 q^{13} -8.61188i q^{14} +2.06188 q^{16} -2.51694i q^{17} +9.26281 q^{18} +(-8.62635 - 16.9289i) q^{19} +31.2113i q^{21} -21.3029 q^{22} +41.9708i q^{23} +31.6254 q^{24} +19.2995 q^{26} +3.66254 q^{27} +20.3474i q^{28} +57.0573i q^{29} -30.5634i q^{31} +32.9318 q^{32} +77.2063 q^{33} -2.87306i q^{34} -21.8854 q^{36} +24.2954 q^{37} +(-9.84687 - 19.3241i) q^{38} -69.9456 q^{39} +34.9984i q^{41} +35.6273i q^{42} -3.51369i q^{43} +50.3327 q^{44} +47.9091i q^{46} +9.97353i q^{47} -8.52996 q^{48} -7.91856 q^{49} +10.4126i q^{51} -45.5993 q^{52} +37.3445 q^{53} +4.18074 q^{54} +57.6738i q^{56} +(35.6871 + 70.0345i) q^{57} +65.1302i q^{58} +26.0785i q^{59} -8.46643 q^{61} -34.8877i q^{62} -61.2207i q^{63} +29.3437 q^{64} +88.1299 q^{66} +12.5009 q^{67} +6.78822i q^{68} -173.633i q^{69} -21.6862i q^{71} -62.0331 q^{72} -77.1277i q^{73} +27.7328 q^{74} +(23.2653 + 45.6572i) q^{76} +140.798i q^{77} -79.8420 q^{78} +85.3703i q^{79} -88.1841 q^{81} +39.9502i q^{82} -32.5378i q^{83} -84.1770i q^{84} -4.01083i q^{86} -236.046i q^{87} +142.666 q^{88} -15.2618i q^{89} -127.557i q^{91} -113.196i q^{92} +126.440i q^{93} +11.3847i q^{94} -136.238 q^{96} +55.5679 q^{97} -9.03893 q^{98} -151.440 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	1.14149			0.570744			0.285372	−	0.958417i	$-0.407883\pi$
$2$	1.14149			0.570744			0.285372	+	0.958417i	$0.407883\pi$
$3$	−4.13699			−1.37900			−0.689498	−	0.724287i	$-0.742169\pi$
$3$	−4.13699			−1.37900			−0.689498	+	0.724287i	$0.742169\pi$
$4$	−2.69701			−0.674252
$5$		0			0
$5$		0			0
$6$	−4.72232			−0.787053
$7$		−	7.54444i		−	1.07778i	−0.842377	−	0.538888i	$-0.818844\pi$
$7$		−	7.54444i		−	1.07778i	0.842377	−	0.538888i	$-0.181156\pi$
$8$	−7.64455			−0.955568
$9$	8.11468			0.901632
$10$		0			0
$11$	−18.6624			−1.69658			−0.848292	−	0.529529i	$-0.822369\pi$
$11$	−18.6624			−1.69658			−0.848292	+	0.529529i	$0.822369\pi$
$12$	11.1575			0.929791
$13$	16.9074			1.30057			0.650283	−	0.759692i	$-0.274650\pi$
$13$	16.9074			1.30057			0.650283	+	0.759692i	$0.274650\pi$
$14$		−	8.61188i		−	0.615134i
$15$		0			0
$16$	2.06188			0.128867
$17$		−	2.51694i		−	0.148056i	−0.997256	−	0.0740278i	$-0.976415\pi$
$17$		−	2.51694i		−	0.148056i	0.997256	−	0.0740278i	$-0.0235853\pi$
$18$	9.26281			0.514600
$19$	−8.62635	−	16.9289i	−0.454019	−	0.890992i
$19$	−8.62635	−	16.9289i	−0.454019	−	0.890992i
$20$		0			0
$21$			31.2113i			1.48625i
$22$	−21.3029			−0.968314
$23$			41.9708i			1.82482i	0.409281	+	0.912408i	$0.365780\pi$
$23$			41.9708i			1.82482i	−0.409281	+	0.912408i	$0.634220\pi$
$24$	31.6254			1.31773
$25$		0			0
$26$	19.2995			0.742290
$27$	3.66254			0.135650
$28$			20.3474i			0.726693i
$29$			57.0573i			1.96749i	0.179561	+	0.983747i	$0.442532\pi$
$29$			57.0573i			1.96749i	−0.179561	+	0.983747i	$0.557468\pi$
$30$		0			0
$31$		−	30.5634i		−	0.985916i	−0.870053	−	0.492958i	$-0.835916\pi$
$31$		−	30.5634i		−	0.985916i	0.870053	−	0.492958i	$-0.164084\pi$
$32$	32.9318			1.02912
$33$	77.2063			2.33958
$34$		−	2.87306i		−	0.0845017i
$35$		0			0
$36$	−21.8854			−0.607927
$37$	24.2954			0.656631			0.328316	−	0.944568i	$-0.393519\pi$
$37$	24.2954			0.656631			0.328316	+	0.944568i	$0.393519\pi$
$38$	−9.84687	−	19.3241i	−0.259128	−	0.508528i
$39$	−69.9456			−1.79348
$40$		0			0
$41$			34.9984i			0.853618i	0.904342	+	0.426809i	$0.140362\pi$
$41$			34.9984i			0.853618i	−0.904342	+	0.426809i	$0.859638\pi$
$42$			35.6273i			0.848268i
$43$		−	3.51369i		−	0.0817137i	−0.999165	−	0.0408568i	$-0.986991\pi$
$43$		−	3.51369i		−	0.0817137i	0.999165	−	0.0408568i	$-0.0130088\pi$
$44$	50.3327			1.14392
$45$		0			0
$46$			47.9091i			1.04150i
$47$			9.97353i			0.212203i	0.994355	+	0.106101i	$0.0338368\pi$
$47$			9.97353i			0.212203i	−0.994355	+	0.106101i	$0.966163\pi$
$48$	−8.52996			−0.177708
$49$	−7.91856			−0.161603
$50$		0			0
$51$			10.4126i			0.204168i
$52$	−45.5993			−0.876909
$53$	37.3445			0.704612			0.352306	−	0.935885i	$-0.385398\pi$
$53$	37.3445			0.704612			0.352306	+	0.935885i	$0.385398\pi$
$54$	4.18074			0.0774212
$55$		0			0
$56$			57.6738i			1.02989i
$57$	35.6871	+	70.0345i	0.626090	+	1.22868i
$58$			65.1302i			1.12293i
$59$			26.0785i			0.442009i	0.975273	+	0.221004i	$0.0709335\pi$
$59$			26.0785i			0.442009i	−0.975273	+	0.221004i	$0.929066\pi$
$60$		0			0
$61$	−8.46643			−0.138794			−0.0693970	−	0.997589i	$-0.522108\pi$
$61$	−8.46643			−0.138794			−0.0693970	+	0.997589i	$0.522108\pi$
$62$		−	34.8877i		−	0.562705i
$63$		−	61.2207i		−	0.971758i
$64$	29.3437			0.458496
$65$		0			0
$66$	88.1299			1.33530
$67$	12.5009			0.186581			0.0932906	−	0.995639i	$-0.470261\pi$
$67$	12.5009			0.186581			0.0932906	+	0.995639i	$0.470261\pi$
$68$			6.78822i			0.0998267i
$69$		−	173.633i		−	2.51642i
$70$		0			0
$71$		−	21.6862i		−	0.305439i	−0.988270	−	0.152720i	$-0.951197\pi$
$71$		−	21.6862i		−	0.305439i	0.988270	−	0.152720i	$-0.0488032\pi$
$72$	−62.0331			−0.861571
$73$		−	77.1277i		−	1.05654i	−0.849075	−	0.528272i	$-0.822840\pi$
$73$		−	77.1277i		−	1.05654i	0.849075	−	0.528272i	$-0.177160\pi$
$74$	27.7328			0.374768
$75$		0			0
$76$	23.2653	+	45.6572i	0.306123	+	0.600753i
$77$			140.798i			1.82854i
$78$	−79.8420			−1.02361
$79$			85.3703i			1.08064i	0.841461	+	0.540319i	$0.181696\pi$
$79$			85.3703i			1.08064i	−0.841461	+	0.540319i	$0.818304\pi$
$80$		0			0
$81$	−88.1841			−1.08869
$82$			39.9502i			0.487197i
$83$		−	32.5378i		−	0.392021i	−0.980602	−	0.196011i	$-0.937201\pi$
$83$		−	32.5378i		−	0.392021i	0.980602	−	0.196011i	$-0.0627987\pi$
$84$		−	84.1770i		−	1.00211i
$85$		0			0
$86$		−	4.01083i		−	0.0466375i
$87$		−	236.046i		−	2.71317i
$88$	142.666			1.62120
$89$		−	15.2618i		−	0.171481i	−0.996318	−	0.0857403i	$-0.972674\pi$
$89$		−	15.2618i		−	0.171481i	0.996318	−	0.0857403i	$-0.0273255\pi$
$90$		0			0
$91$		−	127.557i		−	1.40172i
$92$		−	113.196i		−	1.23039i
$93$			126.440i			1.35957i
$94$			11.3847i			0.121113i
$95$		0			0
$96$	−136.238			−1.41915
$97$	55.5679			0.572865			0.286433	−	0.958100i	$-0.407531\pi$
$97$	55.5679			0.572865			0.286433	+	0.958100i	$0.407531\pi$
$98$	−9.03893			−0.0922340
$99$	−151.440			−1.52969
$100$		0			0
$101$	−82.9964			−0.821747			−0.410873	−	0.911692i	$-0.634776\pi$
$101$	−82.9964			−0.821747			−0.410873	+	0.911692i	$0.634776\pi$
$102$			11.8858i			0.116528i
$103$	27.8765			0.270646			0.135323	−	0.990802i	$-0.456793\pi$
$103$	27.8765			0.270646			0.135323	+	0.990802i	$0.456793\pi$
$104$	−129.249			−1.24278
$105$		0			0
$106$	42.6282			0.402153
$107$	14.6484			0.136901			0.0684506	−	0.997655i	$-0.478194\pi$
$107$	14.6484			0.136901			0.0684506	+	0.997655i	$0.478194\pi$
$108$	−9.87790			−0.0914620
$109$			34.0962i			0.312809i	0.987693	+	0.156405i	$0.0499904\pi$
$109$			34.0962i			0.312809i	−0.987693	+	0.156405i	$0.950010\pi$
$110$		0			0
$111$	−100.510			−0.905492
$112$		−	15.5557i		−	0.138890i
$113$	−23.0375			−0.203872			−0.101936	−	0.994791i	$-0.532504\pi$
$113$	−23.0375			−0.203872			−0.101936	+	0.994791i	$0.532504\pi$
$114$	40.7364	+	79.9435i	0.357337	+	0.701258i
$115$		0			0
$116$		−	153.884i		−	1.32659i
$117$	137.198			1.17263
$118$			29.7683i			0.252274i
$119$	−18.9889			−0.159571
$120$		0			0
$121$	227.286			1.87840
$122$	−9.66432			−0.0792157
$123$		−	144.788i		−	1.17714i
$124$			82.4297i			0.664756i
$125$		0			0
$126$		−	69.8827i		−	0.554625i
$127$	−157.848			−1.24290			−0.621448	−	0.783455i	$-0.713455\pi$
$127$	−157.848			−1.24290			−0.621448	+	0.783455i	$0.713455\pi$
$128$	−98.2317			−0.767435
$129$			14.5361i			0.112683i
$130$		0			0
$131$	−44.6611			−0.340925			−0.170462	−	0.985364i	$-0.554526\pi$
$131$	−44.6611			−0.340925			−0.170462	+	0.985364i	$0.554526\pi$
$132$	−208.226			−1.57747
$133$	−127.719	+	65.0810i	−0.960291	+	0.489331i
$134$	14.2697			0.106490
$135$		0			0
$136$			19.2409i			0.141477i
$137$		−	111.423i		−	0.813303i	−0.913583	−	0.406652i	$-0.866696\pi$
$137$		−	111.423i		−	0.813303i	0.913583	−	0.406652i	$-0.133304\pi$
$138$		−	198.200i		−	1.43623i
$139$	235.986			1.69774			0.848869	−	0.528603i	$-0.177284\pi$
$139$	235.986			1.69774			0.848869	+	0.528603i	$0.177284\pi$
$140$		0			0
$141$		−	41.2604i		−	0.292627i
$142$		−	24.7545i		−	0.174328i
$143$	−315.532			−2.20652
$144$	16.7315			0.116191
$145$		0			0
$146$		−	88.0403i		−	0.603016i
$147$	32.7590			0.222850
$148$	−65.5248			−0.442735
$149$	35.6000			0.238926			0.119463	−	0.992839i	$-0.461883\pi$
$149$	35.6000			0.238926			0.119463	+	0.992839i	$0.461883\pi$
$150$		0			0
$151$		−	7.98568i		−	0.0528853i	−0.999650	−	0.0264426i	$-0.991582\pi$
$151$		−	7.98568i		−	0.0528853i	0.999650	−	0.0264426i	$-0.00841794\pi$
$152$	65.9446	+	129.413i	0.433846	+	0.851404i
$153$		−	20.4242i		−	0.133492i
$154$			160.719i			1.04363i
$155$		0			0
$156$	188.644			1.20925
$157$			171.708i			1.09368i	0.837237	+	0.546840i	$0.184169\pi$
$157$			171.708i			1.09368i	−0.837237	+	0.546840i	$0.815831\pi$
$158$			97.4491i			0.616767i
$159$	−154.494			−0.971658
$160$		0			0
$161$	316.646			1.96675
$162$	−100.661			−0.621364
$163$			238.048i			1.46041i	0.683226	+	0.730207i	$0.260577\pi$
$163$			238.048i			1.46041i	−0.683226	+	0.730207i	$0.739423\pi$
$164$		−	94.3908i		−	0.575554i
$165$		0			0
$166$		−	37.1415i		−	0.223744i
$167$	257.824			1.54386			0.771929	−	0.635709i	$-0.219292\pi$
$167$	257.824			1.54386			0.771929	+	0.635709i	$0.219292\pi$
$168$		−	238.596i		−	1.42021i
$169$	116.859			0.691471
$170$		0			0
$171$	−70.0001	−	137.372i	−0.409358	−	0.803347i
$172$			9.47644i			0.0550956i
$173$	25.1930			0.145624			0.0728122	−	0.997346i	$-0.476803\pi$
$173$	25.1930			0.145624			0.0728122	+	0.997346i	$0.476803\pi$
$174$		−	269.443i		−	1.54852i
$175$		0			0
$176$	−38.4796			−0.218634
$177$		−	107.887i		−	0.609529i
$178$		−	17.4211i		−	0.0978715i
$179$			305.181i			1.70492i	0.522790	+	0.852462i	$0.324891\pi$
$179$			305.181i			1.70492i	−0.522790	+	0.852462i	$0.675109\pi$
$180$		0			0
$181$			297.130i			1.64160i	0.571216	+	0.820800i	$0.306472\pi$
$181$			297.130i			1.64160i	−0.571216	+	0.820800i	$0.693528\pi$
$182$		−	145.604i		−	0.800023i
$183$	35.0255			0.191396
$184$		−	320.848i		−	1.74374i
$185$		0			0
$186$			144.330i			0.775969i
$187$			46.9723i			0.251189i
$188$		−	26.8987i		−	0.143078i
$189$		−	27.6318i		−	0.146200i
$190$		0			0
$191$	112.461			0.588799			0.294399	−	0.955682i	$-0.404880\pi$
$191$	112.461			0.588799			0.294399	+	0.955682i	$0.404880\pi$
$192$	−121.395			−0.632264
$193$	−115.955			−0.600801			−0.300401	−	0.953813i	$-0.597120\pi$
$193$	−115.955			−0.600801			−0.300401	+	0.953813i	$0.597120\pi$
$194$	63.4301			0.326959
$195$		0			0
$196$	21.3564			0.108961
$197$			255.746i			1.29820i	0.760702	+	0.649101i	$0.224855\pi$
$197$			255.746i			1.29820i	−0.760702	+	0.649101i	$0.775145\pi$
$198$	−172.866			−0.873063
$199$	140.348			0.705264			0.352632	−	0.935762i	$-0.385287\pi$
$199$	140.348			0.705264			0.352632	+	0.935762i	$0.385287\pi$
$200$		0			0
$201$	−51.7163			−0.257295
$202$	−94.7393			−0.469007
$203$	430.465			2.12052
$204$		−	28.0828i		−	0.137661i
$205$		0			0
$206$	31.8207			0.154469
$207$			340.580i			1.64531i
$208$	34.8609			0.167600
$209$	160.989	+	315.933i	0.770281	+	1.51164i
$210$		0			0
$211$			185.863i			0.880869i	0.897785	+	0.440434i	$0.145176\pi$
$211$			185.863i			0.880869i	−0.897785	+	0.440434i	$0.854824\pi$
$212$	−100.718			−0.475086
$213$			89.7156i			0.421200i
$214$	16.7210			0.0781355
$215$		0			0
$216$	−27.9985			−0.129623
$217$	−230.584			−1.06260
$218$			38.9204i			0.178534i
$219$			319.077i			1.45697i
$220$		0			0
$221$		−	42.5549i		−	0.192556i
$222$	−114.730			−0.516804
$223$	−387.193			−1.73629			−0.868146	−	0.496310i	$-0.834688\pi$
$223$	−387.193			−1.73629			−0.868146	+	0.496310i	$0.834688\pi$
$224$		−	248.452i		−	1.10916i
$225$		0			0
$226$	−26.2970			−0.116358
$227$	25.8306			0.113791			0.0568956	−	0.998380i	$-0.481880\pi$
$227$	25.8306			0.113791			0.0568956	+	0.998380i	$0.481880\pi$
$228$	−96.2485	−	188.884i	−0.422142	−	0.828436i
$229$	−198.554			−0.867047			−0.433523	−	0.901142i	$-0.642730\pi$
$229$	−198.554			−0.867047			−0.433523	+	0.901142i	$0.642730\pi$
$230$		0			0
$231$		−	582.478i		−	2.52155i
$232$		−	436.177i		−	1.88007i
$233$			47.0260i			0.201828i	0.994895	+	0.100914i	$0.0321768\pi$
$233$			47.0260i			0.201828i	−0.994895	+	0.100914i	$0.967823\pi$
$234$	156.610			0.669272
$235$		0			0
$236$		−	70.3340i		−	0.298025i
$237$		−	353.176i		−	1.49019i
$238$	−21.6756			−0.0910740
$239$	−274.382			−1.14804			−0.574022	−	0.818840i	$-0.694618\pi$
$239$	−274.382			−1.14804			−0.574022	+	0.818840i	$0.694618\pi$
$240$		0			0
$241$			17.9173i			0.0743457i	0.999309	+	0.0371729i	$0.0118352\pi$
$241$			17.9173i			0.0743457i	−0.999309	+	0.0371729i	$0.988165\pi$
$242$	259.444			1.07208
$243$	331.854			1.36565
$244$	22.8340			0.0935821
$245$		0			0
$246$		−	165.273i		−	0.671843i
$247$	−145.849	−	286.222i	−0.590481	−	1.15879i
$248$			233.643i			0.942110i
$249$			134.608i			0.540596i
$250$		0			0
$251$	5.24025			0.0208775			0.0104387	−	0.999946i	$-0.496677\pi$
$251$	5.24025			0.0208775			0.0104387	+	0.999946i	$0.496677\pi$
$252$			165.113i			0.655209i
$253$		−	783.277i		−	3.09596i
$254$	−180.181			−0.709375
$255$		0			0
$256$	−229.505			−0.896504
$257$	225.273			0.876548			0.438274	−	0.898841i	$-0.355590\pi$
$257$	225.273			0.876548			0.438274	+	0.898841i	$0.355590\pi$
$258$			16.5928i			0.0643130i
$259$		−	183.295i		−	0.707702i
$260$		0			0
$261$			463.002i			1.77395i
$262$	−50.9801			−0.194581
$263$			162.013i			0.616020i	0.951383	+	0.308010i	$0.0996630\pi$
$263$			162.013i			0.616020i	−0.951383	+	0.308010i	$0.900337\pi$
$264$	−590.207			−2.23563
$265$		0			0
$266$	−145.789	+	74.2891i	−0.548080	+	0.279282i
$267$			63.1378i			0.236471i
$268$	−33.7151			−0.125803
$269$		−	198.513i		−	0.737967i	−0.929436	−	0.368983i	$-0.879706\pi$
$269$		−	198.513i		−	0.737967i	0.929436	−	0.368983i	$-0.120294\pi$
$270$		0			0
$271$	350.818			1.29453			0.647265	−	0.762265i	$-0.275913\pi$
$271$	350.818			1.29453			0.647265	+	0.762265i	$0.275913\pi$
$272$		−	5.18963i		−	0.0190795i
$273$			527.700i			1.93297i
$274$		−	127.187i		−	0.464187i
$275$		0			0
$276$			468.289i			1.69670i
$277$			344.086i			1.24219i	0.783737	+	0.621093i	$0.213311\pi$
$277$			344.086i			1.24219i	−0.783737	+	0.621093i	$0.786689\pi$
$278$	269.375			0.968973
$279$		−	248.012i		−	0.888933i
$280$		0			0
$281$		−	474.794i		−	1.68966i	−0.535037	−	0.844828i	$-0.679702\pi$
$281$		−	474.794i		−	1.68966i	0.535037	−	0.844828i	$-0.320298\pi$
$282$		−	47.0982i		−	0.167015i
$283$			297.732i			1.05206i	0.850467	+	0.526028i	$0.176319\pi$
$283$			297.732i			1.05206i	−0.850467	+	0.526028i	$0.823681\pi$
$284$			58.4878i			0.205943i
$285$		0			0
$286$	−360.176			−1.25936
$287$	264.043			0.920010
$288$	267.231			0.927886
$289$	282.665			0.978080
$290$		0			0
$291$	−229.884			−0.789979
$292$			208.014i			0.712377i
$293$	379.347			1.29470			0.647350	−	0.762193i	$-0.275877\pi$
$293$	379.347			1.29470			0.647350	+	0.762193i	$0.275877\pi$
$294$	37.3940			0.127190
$295$		0			0
$296$	−185.727			−0.627456
$297$	−68.3519			−0.230141
$298$	40.6370			0.136366
$299$			709.615i			2.37329i
$300$		0			0
$301$	−26.5088			−0.0880691
$302$		−	9.11555i		−	0.0301839i
$303$	343.355			1.13319
$304$	−17.7865	−	34.9052i	−0.0585081	−	0.114820i
$305$		0			0
$306$		−	23.3140i		−	0.0761894i
$307$	296.656			0.966306			0.483153	−	0.875536i	$-0.339491\pi$
$307$	296.656			0.966306			0.483153	+	0.875536i	$0.339491\pi$
$308$		−	379.732i		−	1.23290i
$309$	−115.325			−0.373220
$310$		0			0
$311$	−466.827			−1.50105			−0.750525	−	0.660842i	$-0.770199\pi$
$311$	−466.827			−1.50105			−0.750525	+	0.660842i	$0.770199\pi$
$312$	534.702			1.71379
$313$			180.002i			0.575087i	0.957768	+	0.287543i	$0.0928386\pi$
$313$			180.002i			0.575087i	−0.957768	+	0.287543i	$0.907161\pi$
$314$			196.002i			0.624210i
$315$		0			0
$316$		−	230.244i		−	0.728621i
$317$	156.703			0.494331			0.247165	−	0.968973i	$-0.420501\pi$
$317$	156.703			0.494331			0.247165	+	0.968973i	$0.420501\pi$
$318$	−176.352			−0.554568
$319$		−	1064.83i		−	3.33802i
$320$		0			0
$321$	−60.6004			−0.188786
$322$	361.447			1.12251
$323$	−42.6090	+	21.7121i	−0.131916	+	0.0672200i
$324$	237.833			0.734053
$325$		0			0
$326$			271.728i			0.833522i
$327$		−	141.056i		−	0.431363i
$328$		−	267.547i		−	0.815691i
$329$	75.2447			0.228707
$330$		0			0
$331$			314.141i			0.949065i	0.880238	+	0.474533i	$0.157383\pi$
$331$			314.141i			0.949065i	−0.880238	+	0.474533i	$0.842617\pi$
$332$			87.7546i			0.264321i
$333$	197.149			0.592040
$334$	294.303			0.881146
$335$		0			0
$336$			64.3538i			0.191529i
$337$	−248.342			−0.736921			−0.368461	−	0.929643i	$-0.620115\pi$
$337$	−248.342			−0.736921			−0.368461	+	0.929643i	$0.620115\pi$
$338$	133.393			0.394653
$339$	95.3059			0.281138
$340$		0			0
$341$			570.387i			1.67269i
$342$	−79.9043	−	156.809i	−0.233638	−	0.458505i
$343$		−	309.936i		−	0.903605i
$344$			26.8605i			0.0780830i
$345$		0			0
$346$	28.7575			0.0831142
$347$		−	323.239i		−	0.931525i	−0.884910	−	0.465762i	$-0.845780\pi$
$347$		−	323.239i		−	0.931525i	0.884910	−	0.465762i	$-0.154220\pi$
$348$			636.616i			1.82936i
$349$	−470.656			−1.34858			−0.674292	−	0.738465i	$-0.735551\pi$
$349$	−470.656			−1.34858			−0.674292	+	0.738465i	$0.735551\pi$
$350$		0			0
$351$	61.9239			0.176421
$352$	−614.587			−1.74599
$353$			32.7215i			0.0926954i	0.998925	+	0.0463477i	$0.0147582\pi$
$353$			32.7215i			0.0926954i	−0.998925	+	0.0463477i	$0.985242\pi$
$354$		−	123.151i		−	0.347885i
$355$		0			0
$356$			41.1611i			0.115621i
$357$	78.5570			0.220048
$358$			348.361i			0.973074i
$359$	−332.156			−0.925225			−0.462613	−	0.886561i	$-0.653088\pi$
$359$	−332.156			−0.925225			−0.462613	+	0.886561i	$0.653088\pi$
$360$		0			0
$361$	−212.172	+	292.069i	−0.587734	+	0.809054i
$362$			339.170i			0.936932i
$363$	−940.280			−2.59030
$364$			344.021i			0.945112i
$365$		0			0
$366$	39.9812			0.109238
$367$		−	219.826i		−	0.598981i	−0.954099	−	0.299491i	$-0.903183\pi$
$367$		−	219.826i		−	0.598981i	0.954099	−	0.299491i	$-0.0968168\pi$
$368$			86.5386i			0.235159i
$369$			284.001i			0.769649i
$370$		0			0
$371$		−	281.743i		−	0.759415i
$372$		−	341.011i		−	0.916696i
$373$	416.517			1.11667			0.558334	−	0.829616i	$-0.311441\pi$
$373$	416.517			1.11667			0.558334	+	0.829616i	$0.311441\pi$
$374$			53.6182i			0.143364i
$375$		0			0
$376$		−	76.2431i		−	0.202774i
$377$			964.688i			2.55885i
$378$		−	31.5414i		−	0.0834428i
$379$		−	396.483i		−	1.04613i	−0.852293	−	0.523064i	$-0.824789\pi$
$379$		−	396.483i		−	1.04613i	0.852293	−	0.523064i	$-0.175211\pi$
$380$		0			0
$381$	653.015			1.71395
$382$	128.372			0.336053
$383$	332.639			0.868510			0.434255	−	0.900790i	$-0.357012\pi$
$383$	332.639			0.868510			0.434255	+	0.900790i	$0.357012\pi$
$384$	406.384			1.05829
$385$		0			0
$386$	−132.361			−0.342903
$387$		−	28.5125i		−	0.0736756i
$388$	−149.867			−0.386255
$389$	−120.491			−0.309745			−0.154873	−	0.987934i	$-0.549497\pi$
$389$	−120.491			−0.309745			−0.154873	+	0.987934i	$0.549497\pi$
$390$		0			0
$391$	105.638			0.270174
$392$	60.5338			0.154423
$393$	184.763			0.470134
$394$			291.931i			0.740941i
$395$		0			0
$396$	408.434			1.03140
$397$			569.599i			1.43476i	0.696683	+	0.717380i	$0.254659\pi$
$397$			569.599i			1.43476i	−0.696683	+	0.717380i	$0.745341\pi$
$398$	160.205			0.402525
$399$	528.371	−	269.239i	1.32424	−	0.674786i
$400$		0			0
$401$			354.490i			0.884016i	0.897011	+	0.442008i	$0.145734\pi$
$401$			354.490i			0.884016i	−0.897011	+	0.442008i	$0.854266\pi$
$402$	−59.0335			−0.146849
$403$		−	516.746i		−	1.28225i
$404$	223.842			0.554064
$405$		0			0
$406$	491.371			1.21027
$407$	−453.410			−1.11403
$408$		−	79.5994i		−	0.195097i
$409$			362.568i			0.886474i	0.896404	+	0.443237i	$0.146170\pi$
$409$			362.568i			0.886474i	−0.896404	+	0.443237i	$0.853830\pi$
$410$		0			0
$411$			460.954i			1.12154i
$412$	−75.1832			−0.182484
$413$	196.748			0.476387
$414$			388.767i			0.939052i
$415$		0			0
$416$	556.790			1.33844
$417$	−976.270			−2.34118
$418$	183.767	+	360.634i	0.439633	+	0.862761i
$419$	−422.158			−1.00754			−0.503769	−	0.863838i	$-0.668054\pi$
$419$	−422.158			−1.00754			−0.503769	+	0.863838i	$0.668054\pi$
$420$		0			0
$421$		−	219.243i		−	0.520767i	−0.965505	−	0.260384i	$-0.916151\pi$
$421$		−	219.243i		−	0.520767i	0.965505	−	0.260384i	$-0.0838490\pi$
$422$			212.161i			0.502750i
$423$			80.9320i			0.191329i
$424$	−285.481			−0.673305
$425$		0			0
$426$			102.409i			0.240397i
$427$			63.8745i			0.149589i
$428$	−39.5069			−0.0923059
$429$	1305.35			3.04278
$430$		0			0
$431$			251.672i			0.583925i	0.956430	+	0.291962i	$0.0943082\pi$
$431$			251.672i			0.583925i	−0.956430	+	0.291962i	$0.905692\pi$
$432$	7.55171			0.0174808
$433$	29.8124			0.0688509			0.0344255	−	0.999407i	$-0.489040\pi$
$433$	29.8124			0.0688509			0.0344255	+	0.999407i	$0.489040\pi$
$434$	−263.208			−0.606471
$435$		0			0
$436$		−	91.9577i		−	0.210912i
$437$	710.517	−	362.055i	1.62590	−	0.828501i
$438$			364.222i			0.831557i
$439$			349.367i			0.795826i	0.917423	+	0.397913i	$0.130265\pi$
$439$			349.367i			0.795826i	−0.917423	+	0.397913i	$0.869735\pi$
$440$		0			0
$441$	−64.2566			−0.145707
$442$		−	48.5758i		−	0.109900i
$443$			395.503i			0.892784i	0.894838	+	0.446392i	$0.147291\pi$
$443$			395.503i			0.892784i	−0.894838	+	0.446392i	$0.852709\pi$
$444$	271.075			0.610530
$445$		0			0
$446$	−441.976			−0.990977
$447$	−147.277			−0.329479
$448$		−	221.382i		−	0.494156i
$449$		−	76.7191i		−	0.170867i	−0.996344	−	0.0854333i	$-0.972773\pi$
$449$		−	76.7191i		−	0.170867i	0.996344	−	0.0854333i	$-0.0272275\pi$
$450$		0			0
$451$		−	653.154i		−	1.44824i
$452$	62.1323			0.137461
$453$			33.0367i			0.0729286i
$454$	29.4853			0.0649456
$455$		0			0
$456$	−272.812	−	535.382i	−0.598272	−	1.17408i
$457$		−	10.3059i		−	0.0225512i	−0.999936	−	0.0112756i	$-0.996411\pi$
$457$		−	10.3059i		−	0.0225512i	0.999936	−	0.0112756i	$-0.00358922\pi$
$458$	−226.647			−0.494861
$459$		−	9.21841i		−	0.0200837i
$460$		0			0
$461$	34.7399			0.0753577			0.0376789	−	0.999290i	$-0.488004\pi$
$461$	34.7399			0.0753577			0.0376789	+	0.999290i	$0.488004\pi$
$462$		−	664.891i		−	1.43916i
$463$		−	321.502i		−	0.694389i	−0.937793	−	0.347194i	$-0.887134\pi$
$463$		−	321.502i		−	0.694389i	0.937793	−	0.347194i	$-0.112866\pi$
$464$			117.645i			0.253546i
$465$		0			0
$466$			53.6796i			0.115192i
$467$		−	297.489i		−	0.637022i	−0.947919	−	0.318511i	$-0.896817\pi$
$467$		−	297.489i		−	0.637022i	0.947919	−	0.318511i	$-0.103183\pi$
$468$	−370.024			−0.790649
$469$		−	94.3126i		−	0.201093i
$470$		0			0
$471$		−	710.353i		−	1.50818i
$472$		−	199.359i		−	0.422370i
$473$			65.5739i			0.138634i
$474$		−	403.146i		−	0.850519i
$475$		0			0
$476$	51.2133			0.107591
$477$	303.038			0.635301
$478$	−313.204			−0.655238
$479$	−425.338			−0.887971			−0.443986	−	0.896034i	$-0.646436\pi$
$479$	−425.338			−0.887971			−0.443986	+	0.896034i	$0.646436\pi$
$480$		0			0
$481$	410.770			0.853992
$482$			20.4524i			0.0424324i
$483$	−1309.96			−2.71214
$484$	−612.992			−1.26651
$485$		0			0
$486$	378.807			0.779438
$487$	313.568			0.643876			0.321938	−	0.946761i	$-0.395666\pi$
$487$	313.568			0.643876			0.321938	+	0.946761i	$0.395666\pi$
$488$	64.7220			0.132627
$489$		−	984.801i		−	2.01391i
$490$		0			0
$491$	96.7180			0.196982			0.0984908	−	0.995138i	$-0.468598\pi$
$491$	96.7180			0.196982			0.0984908	+	0.995138i	$0.468598\pi$
$492$			390.494i			0.793687i
$493$	143.610			0.291298
$494$	−166.485	−	326.719i	−0.337013	−	0.661374i
$495$		0			0
$496$		−	63.0180i		−	0.127052i
$497$	−163.610			−0.329195
$498$			153.654i			0.308542i
$499$	442.389			0.886551			0.443275	−	0.896385i	$-0.353816\pi$
$499$	442.389			0.886551			0.443275	+	0.896385i	$0.353816\pi$
$500$		0			0
$501$	−1066.62			−2.12897
$502$	5.98168			0.0119157
$503$		−	824.040i		−	1.63825i	−0.573615	−	0.819125i	$-0.694459\pi$
$503$		−	824.040i		−	1.63825i	0.573615	−	0.819125i	$-0.305541\pi$
$504$			468.005i			0.928581i
$505$		0			0
$506$		−	894.100i		−	1.76700i
$507$	−483.443			−0.953537
$508$	425.717			0.838025
$509$			400.790i			0.787407i	0.919237	+	0.393704i	$0.128806\pi$
$509$			400.790i			0.787407i	−0.919237	+	0.393704i	$0.871194\pi$
$510$		0			0
$511$	−581.885			−1.13872
$512$	130.950			0.255761
$513$	−31.5944	−	62.0026i	−0.0615875	−	0.120863i
$514$	257.146			0.500284
$515$		0			0
$516$		−	39.2039i		−	0.0759766i
$517$		−	186.130i		−	0.360020i
$518$		−	209.229i		−	0.403916i
$519$	−104.223			−0.200816
$520$		0			0
$521$			672.304i			1.29041i	0.764009	+	0.645205i	$0.223228\pi$
$521$			672.304i			1.29041i	−0.764009	+	0.645205i	$0.776772\pi$
$522$			528.511i			1.01247i
$523$	−822.554			−1.57276			−0.786380	−	0.617743i	$-0.788047\pi$
$523$	−822.554			−1.57276			−0.786380	+	0.617743i	$0.788047\pi$
$524$	120.451			0.229869
$525$		0			0
$526$			184.936i			0.351589i
$527$	−76.9264			−0.145970
$528$	159.190			0.301496
$529$	−1232.55			−2.32996
$530$		0			0
$531$			211.619i			0.398529i
$532$	344.458	−	175.524i	0.647478	−	0.329932i
$533$			591.730i			1.11019i
$534$			72.0710i			0.134964i
$535$		0			0
$536$	−95.5640			−0.178291
$537$		−	1262.53i		−	2.35108i
$538$		−	226.600i		−	0.421190i
$539$	147.779			0.274173
$540$		0			0
$541$	−665.318			−1.22979			−0.614897	−	0.788608i	$-0.710802\pi$
$541$	−665.318			−1.22979			−0.614897	+	0.788608i	$0.710802\pi$
$542$	400.454			0.738845
$543$		−	1229.22i		−	2.26376i
$544$		−	82.8875i		−	0.152367i
$545$		0			0
$546$			602.363i			1.10323i
$547$	267.766			0.489516			0.244758	−	0.969584i	$-0.421291\pi$
$547$	267.766			0.489516			0.244758	+	0.969584i	$0.421291\pi$
$548$			300.507i			0.548371i
$549$	−68.7024			−0.125141
$550$		0			0
$551$	965.915	−	492.197i	1.75302	−	0.893279i
$552$			1327.34i			2.40461i
$553$	644.071			1.16469
$554$			392.769i			0.708970i
$555$		0			0
$556$	−636.455			−1.14470
$557$		−	524.902i		−	0.942374i	−0.882033	−	0.471187i	$-0.843826\pi$
$557$		−	524.902i		−	0.942374i	0.882033	−	0.471187i	$-0.156174\pi$
$558$		−	283.103i		−	0.507353i
$559$		−	59.4072i		−	0.106274i
$560$		0			0
$561$		−	194.324i		−	0.346388i
$562$		−	541.971i		−	0.964361i
$563$	−427.553			−0.759419			−0.379710	−	0.925106i	$-0.623976\pi$
$563$	−427.553			−0.759419			−0.379710	+	0.925106i	$0.623976\pi$
$564$			111.280i			0.197304i
$565$		0			0
$566$			339.857i			0.600454i
$567$			665.299i			1.17337i
$568$			165.781i			0.291868i
$569$		−	70.9006i		−	0.124606i	−0.998057	−	0.0623028i	$-0.980156\pi$
$569$		−	70.9006i		−	0.124606i	0.998057	−	0.0623028i	$-0.0198445\pi$
$570$		0			0
$571$	236.621			0.414397			0.207199	−	0.978299i	$-0.433565\pi$
$571$	236.621			0.414397			0.207199	+	0.978299i	$0.433565\pi$
$572$	850.993			1.48775
$573$	−465.248			−0.811952
$574$	301.402			0.525090
$575$		0			0
$576$	238.115			0.413394
$577$		−	444.876i		−	0.771016i	−0.922704	−	0.385508i	$-0.874026\pi$
$577$		−	444.876i		−	0.771016i	0.922704	−	0.385508i	$-0.125974\pi$
$578$	322.658			0.558233
$579$	479.703			0.828503
$580$		0			0
$581$	−245.479			−0.422512
$582$	−262.409			−0.450875
$583$	−696.938			−1.19543
$584$			589.607i			1.00960i
$585$		0			0
$586$	433.020			0.738942
$587$			441.274i			0.751745i	0.926671	+	0.375872i	$0.122657\pi$
$587$			441.274i			0.751745i	−0.926671	+	0.375872i	$0.877343\pi$
$588$	−88.3512			−0.150257
$589$	−517.403	+	263.651i	−0.878444	+	0.447624i
$590$		0			0
$591$		−	1058.02i		−	1.79022i
$592$	50.0940			0.0846183
$593$			163.929i			0.276440i	0.990402	+	0.138220i	$0.0441380\pi$
$593$			163.929i			0.276440i	−0.990402	+	0.138220i	$0.955862\pi$
$594$	−78.0228			−0.131352
$595$		0			0
$596$	−96.0135			−0.161097
$597$	−580.617			−0.972557
$598$			810.016i			1.35454i
$599$		−	1005.09i		−	1.67795i	−0.544170	−	0.838975i	$-0.683155\pi$
$599$		−	1005.09i		−	1.67795i	0.544170	−	0.838975i	$-0.316845\pi$
$600$		0			0
$601$		−	823.540i		−	1.37028i	−0.728410	−	0.685142i	$-0.759740\pi$
$601$		−	823.540i		−	1.37028i	0.728410	−	0.685142i	$-0.240260\pi$
$602$	−30.2595			−0.0502649
$603$	101.441			0.168228
$604$			21.5374i			0.0356580i
$605$		0			0
$606$	391.936			0.646758
$607$	−533.092			−0.878241			−0.439121	−	0.898428i	$-0.644710\pi$
$607$	−533.092			−0.878241			−0.439121	+	0.898428i	$0.644710\pi$
$608$	−284.081	−	557.497i	−0.467239	−	0.916937i
$609$	−1780.83			−2.92419
$610$		0			0
$611$			168.626i			0.275984i
$612$			55.0842i			0.0900069i
$613$			633.246i			1.03303i	0.856279	+	0.516514i	$0.172771\pi$
$613$			633.246i			1.03303i	−0.856279	+	0.516514i	$0.827229\pi$
$614$	338.629			0.551513
$615$		0			0
$616$		−	1076.33i		−	1.74729i
$617$			553.380i			0.896888i	0.893811	+	0.448444i	$0.148022\pi$
$617$			553.380i			0.896888i	−0.893811	+	0.448444i	$0.851978\pi$
$618$	−131.642			−0.213013
$619$	811.036			1.31024			0.655118	−	0.755527i	$-0.272619\pi$
$619$	811.036			1.31024			0.655118	+	0.755527i	$0.272619\pi$
$620$		0			0
$621$			153.720i			0.247536i
$622$	−532.877			−0.856715
$623$	−115.142			−0.184818
$624$	−144.219			−0.231120
$625$		0			0
$626$			205.470i			0.328227i
$627$	−666.009	−	1307.01i	−1.06221	−	2.08455i
$628$		−	463.097i		−	0.737415i
$629$		−	61.1501i		−	0.0972179i
$630$		0			0
$631$	−636.837			−1.00925			−0.504626	−	0.863338i	$-0.668369\pi$
$631$	−636.837			−1.00925			−0.504626	+	0.863338i	$0.668369\pi$
$632$		−	652.617i		−	1.03262i
$633$		−	768.914i		−	1.21471i
$634$	178.874			0.282136
$635$		0			0
$636$	416.670			0.655142
$637$	−133.882			−0.210176
$638$		−	1215.49i		−	1.90515i
$639$		−	175.977i		−	0.275394i
$640$		0			0
$641$		−	706.078i		−	1.10153i	−0.834662	−	0.550763i	$-0.814337\pi$
$641$		−	706.078i		−	1.10153i	0.834662	−	0.550763i	$-0.185663\pi$
$642$	−69.1746			−0.107749
$643$		−	5.05767i		−	0.00786574i	−0.999992	−	0.00393287i	$-0.998748\pi$
$643$		−	5.05767i		−	0.00786574i	0.999992	−	0.00393287i	$-0.00125187\pi$
$644$	−853.997			−1.32608
$645$		0			0
$646$	−48.6376	+	24.7840i	−0.0752904	+	0.0383654i
$647$			566.847i			0.876116i	0.898947	+	0.438058i	$0.144334\pi$
$647$			566.847i			0.876116i	−0.898947	+	0.438058i	$0.855666\pi$
$648$	674.127			1.04032
$649$		−	486.688i		−	0.749905i
$650$		0			0
$651$	953.922			1.46532
$652$		−	642.016i		−	0.984687i
$653$		−	354.083i		−	0.542240i	−0.962546	−	0.271120i	$-0.912606\pi$
$653$		−	354.083i		−	0.542240i	0.962546	−	0.271120i	$-0.0873940\pi$
$654$		−	161.013i		−	0.246198i
$655$		0			0
$656$			72.1623i			0.110003i
$657$		−	625.867i		−	0.952614i
$658$	85.8908			0.130533
$659$			768.905i			1.16677i	0.812194	+	0.583387i	$0.198273\pi$
$659$			768.905i			1.16677i	−0.812194	+	0.583387i	$0.801727\pi$
$660$		0			0
$661$		−	135.070i		−	0.204342i	−0.994767	−	0.102171i	$-0.967421\pi$
$661$		−	135.070i		−	0.204342i	0.994767	−	0.102171i	$-0.0325790\pi$
$662$			358.587i			0.541673i
$663$			176.049i			0.265534i
$664$			248.737i			0.374603i
$665$		0			0
$666$	225.043			0.337903
$667$	−2394.74			−3.59032
$668$	−695.354			−1.04095
$669$	1601.81			2.39434
$670$		0			0
$671$	158.004			0.235476
$672$			1027.84i			1.52953i
$673$	1177.68			1.74989			0.874945	−	0.484223i	$-0.160898\pi$
$673$	1177.68			1.74989			0.874945	+	0.484223i	$0.160898\pi$
$674$	−283.480			−0.420593
$675$		0			0
$676$	−315.169			−0.466226
$677$	547.098			0.808121			0.404061	−	0.914732i	$-0.367599\pi$
$677$	547.098			0.808121			0.404061	+	0.914732i	$0.367599\pi$
$678$	108.790			0.160458
$679$		−	419.229i		−	0.617421i
$680$		0			0
$681$	−106.861			−0.156918
$682$			651.090i			0.954677i
$683$	−722.363			−1.05763			−0.528816	−	0.848736i	$-0.677364\pi$
$683$	−722.363			−1.05763			−0.528816	+	0.848736i	$0.677364\pi$
$684$	188.791	+	370.494i	0.276010	+	0.541658i
$685$		0			0
$686$		−	353.788i		−	0.515727i
$687$	821.415			1.19565
$688$		−	7.24479i		−	0.0105302i
$689$	631.396			0.916395
$690$		0			0
$691$	1170.23			1.69354			0.846769	−	0.531961i	$-0.178545\pi$
$691$	1170.23			1.69354			0.846769	+	0.531961i	$0.178545\pi$
$692$	−67.9458			−0.0981876
$693$			1142.53i			1.64867i
$694$		−	368.973i		−	0.531662i
$695$		0			0
$696$			1804.46i			2.59262i
$697$	88.0889			0.126383
$698$	−537.248			−0.769696
$699$		−	194.546i		−	0.278321i
$700$		0			0
$701$	61.2853			0.0874255			0.0437128	−	0.999044i	$-0.486081\pi$
$701$	61.2853			0.0874255			0.0437128	+	0.999044i	$0.486081\pi$
$702$	70.6853			0.100691
$703$	−209.580	−	411.292i	−0.298123	−	0.585053i
$704$	−547.625			−0.777876
$705$		0			0
$706$			37.3511i			0.0529053i
$707$			626.161i			0.885659i
$708$			290.971i			0.410976i
$709$	323.882			0.456815			0.228408	−	0.973566i	$-0.426648\pi$
$709$	323.882			0.456815			0.228408	+	0.973566i	$0.426648\pi$
$710$		0			0
$711$			692.753i			0.974336i
$712$			116.669i			0.163861i
$713$	1282.77			1.79912
$714$	89.6718			0.125591
$715$		0			0
$716$		−	823.076i		−	1.14955i
$717$	1135.12			1.58315
$718$	−379.152			−0.528066
$719$	727.469			1.01178			0.505889	−	0.862598i	$-0.331164\pi$
$719$	727.469			1.01178			0.505889	+	0.862598i	$0.331164\pi$
$720$		0			0
$721$		−	210.313i		−	0.291696i
$722$	−242.192	+	333.392i	−0.335445	+	0.461762i
$723$		−	74.1238i		−	0.102523i
$724$		−	801.360i		−	1.10685i
$725$		0			0
$726$	−1073.32			−1.47840
$727$			249.400i			0.343054i	0.985179	+	0.171527i	$0.0548700\pi$
$727$			249.400i			0.343054i	−0.985179	+	0.171527i	$0.945130\pi$
$728$			975.112i			1.33944i
$729$	−579.219			−0.794539
$730$		0			0
$731$	−8.84375			−0.0120982
$732$	−94.4641			−0.129049
$733$			728.881i			0.994381i	0.867641	+	0.497190i	$0.165635\pi$
$733$			728.881i			0.994381i	−0.867641	+	0.497190i	$0.834365\pi$
$734$		−	250.929i		−	0.341865i
$735$		0			0
$736$			1382.17i			1.87795i
$737$	−233.298			−0.316551
$738$			324.183i			0.439272i
$739$	−927.565			−1.25516			−0.627581	−	0.778551i	$-0.715955\pi$
$739$	−927.565			−1.25516			−0.627581	+	0.778551i	$0.715955\pi$
$740$		0			0
$741$	603.375	+	1184.10i	0.814271	+	1.59797i
$742$		−	321.606i		−	0.433431i
$743$	743.549			1.00074			0.500370	−	0.865812i	$-0.333197\pi$
$743$	743.549			1.00074			0.500370	+	0.865812i	$0.333197\pi$
$744$		−	966.580i		−	1.29917i
$745$		0			0
$746$	475.449			0.637331
$747$		−	264.034i		−	0.353459i
$748$		−	126.685i		−	0.169364i
$749$		−	110.514i		−	0.147549i
$750$		0			0
$751$			1328.55i			1.76904i	0.466499	+	0.884522i	$0.345515\pi$
$751$			1328.55i			1.76904i	−0.466499	+	0.884522i	$0.654485\pi$
$752$			20.5642i			0.0273460i
$753$	−21.6789			−0.0287900
$754$			1101.18i			1.46045i
$755$		0			0
$756$			74.5232i			0.0985757i
$757$		−	776.561i		−	1.02584i	−0.858436	−	0.512920i	$-0.828564\pi$
$757$		−	776.561i		−	1.02584i	0.858436	−	0.512920i	$-0.171436\pi$
$758$		−	452.580i		−	0.597071i
$759$			3240.41i			4.26931i
$760$		0			0
$761$	−26.9260			−0.0353824			−0.0176912	−	0.999843i	$-0.505632\pi$
$761$	−26.9260			−0.0353824			−0.0176912	+	0.999843i	$0.505632\pi$
$762$	745.408			0.978226
$763$	257.237			0.337139
$764$	−303.307			−0.396999
$765$		0			0
$766$	379.704			0.495697
$767$			440.919i			0.574862i
$768$	949.460			1.23628
$769$	1433.78			1.86447			0.932236	−	0.361852i	$-0.117855\pi$
$769$	1433.78			1.86447			0.932236	+	0.361852i	$0.117855\pi$
$770$		0			0
$771$	−931.952			−1.20876
$772$	312.731			0.405091
$773$	−528.633			−0.683871			−0.341936	−	0.939723i	$-0.611083\pi$
$773$	−528.633			−0.683871			−0.341936	+	0.939723i	$0.611083\pi$
$774$		−	32.5466i		−	0.0420499i
$775$		0			0
$776$	−424.792			−0.547412
$777$			758.289i			0.975919i
$778$	−137.539			−0.176785
$779$	592.482	−	301.908i	0.760567	−	0.387559i
$780$		0			0
$781$			404.717i			0.518203i
$782$	120.585			0.154200
$783$			208.975i			0.266890i
$784$	−16.3271			−0.0208254
$785$		0			0
$786$	210.904			0.268326
$787$	1152.91			1.46494			0.732469	−	0.680800i	$-0.238368\pi$
$787$	1152.91			1.46494			0.732469	+	0.680800i	$0.238368\pi$
$788$		−	689.749i		−	0.875316i
$789$		−	670.247i		−	0.849489i
$790$		0			0
$791$			173.805i			0.219728i
$792$	1157.69			1.46173
$793$	−143.145			−0.180511
$794$			650.190i			0.818880i
$795$		0			0
$796$	−378.519			−0.475526
$797$	797.879			1.00110			0.500552	−	0.865707i	$-0.333131\pi$
$797$	797.879			1.00110			0.500552	+	0.865707i	$0.333131\pi$
$798$	603.129	−	307.333i	0.755800	−	0.385130i
$799$	25.1028			0.0314178
$800$		0			0
$801$		−	123.845i		−	0.154612i
$802$			404.646i			0.504546i
$803$			1439.39i			1.79252i
$804$	139.479			0.173482
$805$		0			0
$806$		−	589.859i		−	0.731835i
$807$			821.246i			1.01765i
$808$	634.470			0.785235
$809$	587.649			0.726390			0.363195	−	0.931713i	$-0.381686\pi$
$809$	587.649			0.726390			0.363195	+	0.931713i	$0.381686\pi$
$810$		0			0
$811$			81.5557i			0.100562i	0.998735	+	0.0502810i	$0.0160117\pi$
$811$			81.5557i			0.100562i	−0.998735	+	0.0502810i	$0.983988\pi$
$812$	−1160.97			−1.42976
$813$	−1451.33			−1.78515
$814$	−517.562			−0.635826
$815$		0			0
$816$			21.4694i			0.0263106i
$817$	−59.4827	+	30.3103i	−0.0728062	+	0.0370995i
$818$			413.867i			0.505949i
$819$		−	1035.08i		−	1.26383i
$820$		0			0
$821$	−304.834			−0.371296			−0.185648	−	0.982616i	$-0.559438\pi$
$821$	−304.834			−0.371296			−0.185648	+	0.982616i	$0.559438\pi$
$822$			526.173i			0.640113i
$823$		−	867.997i		−	1.05467i	−0.849656	−	0.527337i	$-0.823190\pi$
$823$		−	867.997i		−	1.05467i	0.849656	−	0.527337i	$-0.176810\pi$
$824$	−213.104			−0.258621
$825$		0			0
$826$	224.585			0.271895
$827$	374.568			0.452924			0.226462	−	0.974020i	$-0.427284\pi$
$827$	374.568			0.452924			0.226462	+	0.974020i	$0.427284\pi$
$828$		−	918.546i		−	1.10935i
$829$			1197.29i			1.44426i	0.691759	+	0.722128i	$0.256836\pi$
$829$			1197.29i			1.44426i	−0.691759	+	0.722128i	$0.743164\pi$
$830$		0			0
$831$		−	1423.48i		−	1.71297i
$832$	496.125			0.596304
$833$			19.9306i			0.0239262i
$834$	−1114.40			−1.33621
$835$		0			0
$836$	−434.188	−	852.075i	−0.519363	−	1.01923i
$837$		−	111.940i		−	0.133739i
$838$	−481.888			−0.575046
$839$			213.064i			0.253950i	0.991906	+	0.126975i	$0.0405268\pi$
$839$			213.064i			0.253950i	−0.991906	+	0.126975i	$0.959473\pi$
$840$		0			0
$841$	−2414.54			−2.87103
$842$		−	250.263i		−	0.297225i
$843$			1964.22i			2.33003i
$844$		−	501.275i		−	0.593927i
$845$		0			0
$846$			92.3829i			0.109200i
$847$		−	1714.75i		−	2.02449i
$848$	76.9996			0.0908015
$849$		−	1231.71i		−	1.45078i
$850$		0			0
$851$			1019.70i			1.19823i
$852$		−	241.963i		−	0.283995i
$853$			448.797i			0.526139i	0.964777	+	0.263070i	$0.0847349\pi$
$853$			448.797i			0.526139i	−0.964777	+	0.263070i	$0.915265\pi$
$854$			72.9119i			0.0853769i
$855$		0			0
$856$	−111.981			−0.130818
$857$	−541.680			−0.632065			−0.316032	−	0.948748i	$-0.602351\pi$
$857$	−541.680			−0.632065			−0.316032	+	0.948748i	$0.602351\pi$
$858$	1490.04			1.73665
$859$	529.227			0.616097			0.308048	−	0.951371i	$-0.400324\pi$
$859$	529.227			0.616097			0.308048	+	0.951371i	$0.400324\pi$
$860$		0			0
$861$	−1092.34			−1.26869
$862$			287.280i			0.333271i
$863$	507.027			0.587517			0.293758	−	0.955880i	$-0.405094\pi$
$863$	507.027			0.587517			0.293758	+	0.955880i	$0.405094\pi$
$864$	120.614			0.139600
$865$		0			0
$866$	34.0305			0.0392962
$867$	−1169.38			−1.34877
$868$	621.886			0.716458
$869$		−	1593.22i		−	1.83339i
$870$		0			0
$871$	211.358			0.242661
$872$		−	260.650i		−	0.298911i
$873$	450.916			0.516513
$874$	811.046	−	413.281i	0.927971	−	0.472862i
$875$		0			0
$876$		−	860.552i		−	0.982365i
$877$	80.9460			0.0922988			0.0461494	−	0.998935i	$-0.485305\pi$
$877$	80.9460			0.0922988			0.0461494	+	0.998935i	$0.485305\pi$
$878$			398.798i			0.454212i
$879$	−1569.36			−1.78539
$880$		0			0
$881$	844.217			0.958249			0.479125	−	0.877747i	$-0.340954\pi$
$881$	844.217			0.958249			0.479125	+	0.877747i	$0.340954\pi$
$882$	−73.3481			−0.0831611
$883$		−	43.2611i		−	0.0489933i	−0.999700	−	0.0244967i	$-0.992202\pi$
$883$		−	43.2611i		−	0.0489933i	0.999700	−	0.0244967i	$-0.00779831\pi$
$884$			114.771i			0.129831i
$885$		0			0
$886$			451.462i			0.509551i
$887$	−1549.50			−1.74690			−0.873450	−	0.486913i	$-0.838123\pi$
$887$	−1549.50			−1.74690			−0.873450	+	0.486913i	$0.838123\pi$
$888$	768.351			0.865260
$889$			1190.87i			1.33956i
$890$		0			0
$891$	1645.73			1.84706
$892$	1044.26			1.17070
$893$	168.840	−	86.0352i	0.189071	−	0.0963440i
$894$	−168.115			−0.188048
$895$		0			0
$896$			741.103i			0.827124i
$897$		−	2935.67i		−	3.27276i
$898$		−	87.5739i		−	0.0975210i
$899$	1743.87			1.93978
$900$		0			0
$901$		−	93.9939i		−	0.104322i
$902$		−	745.567i		−	0.826571i
$903$	109.667			0.121447
$904$	176.111			0.194813
$905$		0			0
$906$			37.7109i			0.0416235i
$907$	1314.64			1.44944			0.724721	−	0.689042i	$-0.241968\pi$
$907$	1314.64			1.44944			0.724721	+	0.689042i	$0.241968\pi$
$908$	−69.6653			−0.0767239
$909$	−673.490			−0.740913
$910$		0			0
$911$			970.388i			1.06519i	0.846370	+	0.532595i	$0.178783\pi$
$911$			970.388i			1.06519i	−0.846370	+	0.532595i	$0.821217\pi$
$912$	73.5825	+	144.402i	0.0806825	+	0.158336i
$913$			607.234i			0.665097i
$914$		−	11.7641i		−	0.0128710i
$915$		0			0
$916$	535.501			0.584608
$917$			336.943i			0.367441i
$918$		−	10.5227i		−	0.0114626i
$919$	−1762.43			−1.91777			−0.958886	−	0.283793i	$-0.908407\pi$
$919$	−1762.43			−1.91777			−0.958886	+	0.283793i	$0.908407\pi$
$920$		0			0
$921$	−1227.26			−1.33253
$922$	39.6552			0.0430099
$923$		−	366.656i		−	0.397244i
$924$			1570.95i			1.70016i
$925$		0			0
$926$		−	366.990i		−	0.396318i
$927$	226.209			0.244023
$928$			1879.00i			2.02478i
$929$	652.176			0.702020			0.351010	−	0.936372i	$-0.385838\pi$
$929$	652.176			0.702020			0.351010	+	0.936372i	$0.385838\pi$
$930$		0			0
$931$	68.3083	+	134.052i	0.0733709	+	0.143987i
$932$		−	126.830i		−	0.136083i
$933$	1931.26			2.06994
$934$		−	339.580i		−	0.363576i
$935$		0			0
$936$	−1048.82			−1.12053
$937$		−	958.489i		−	1.02293i	−0.859303	−	0.511467i	$-0.829102\pi$
$937$		−	958.489i		−	1.02293i	0.859303	−	0.511467i	$-0.170898\pi$
$938$		−	107.657i		−	0.114772i
$939$		−	744.667i		−	0.793043i
$940$		0			0
$941$		−	1645.01i		−	1.74815i	−0.485792	−	0.874074i	$-0.661469\pi$
$941$		−	1645.01i		−	1.74815i	0.485792	−	0.874074i	$-0.338531\pi$
$942$		−	810.859i		−	0.860784i
$943$	−1468.91			−1.55770
$944$			53.7707i			0.0569605i
$945$		0			0
$946$			74.8518i			0.0791245i
$947$		−	1108.58i		−	1.17062i	−0.810810	−	0.585310i	$-0.800973\pi$
$947$		−	1108.58i		−	1.17062i	0.810810	−	0.585310i	$-0.199027\pi$
$948$			952.519i			1.00477i
$949$		−	1304.03i		−	1.37411i
$950$		0			0
$951$	−648.278			−0.681680
$952$	145.162			0.152481
$953$	−509.195			−0.534307			−0.267154	−	0.963654i	$-0.586083\pi$
$953$	−509.195			−0.534307			−0.267154	+	0.963654i	$0.586083\pi$
$954$	345.915			0.362594
$955$		0			0
$956$	740.011			0.774070
$957$			4405.18i			4.60312i
$958$	−485.518			−0.506804
$959$	−840.620			−0.876559
$960$		0			0
$961$	26.8786			0.0279694
$962$	468.889			0.487411
$963$	118.867			0.123434
$964$		−	48.3232i		−	0.0501278i
$965$		0			0
$966$	−1495.30			−1.54793
$967$			1773.83i			1.83436i	0.398471	+	0.917181i	$0.369541\pi$
$967$			1773.83i			1.83436i	−0.398471	+	0.917181i	$0.630459\pi$
$968$	−1737.50			−1.79494
$969$	176.273	−	89.8225i	0.181912	−	0.0926961i
$970$		0			0
$971$			1200.09i			1.23593i	0.786205	+	0.617966i	$0.212043\pi$
$971$			1200.09i			1.23593i	−0.786205	+	0.617966i	$0.787957\pi$
$972$	−895.012			−0.920794
$973$		−	1780.38i		−	1.82978i
$974$	357.933			0.367488
$975$		0			0
$976$	−17.4567			−0.0178860
$977$	−396.951			−0.406296			−0.203148	−	0.979148i	$-0.565117\pi$
$977$	−396.951			−0.406296			−0.203148	+	0.979148i	$0.565117\pi$
$978$		−	1124.14i		−	1.14942i
$979$			284.822i			0.290931i
$980$		0			0
$981$			276.680i			0.282039i
$982$	110.402			0.112426
$983$	1736.52			1.76655			0.883273	−	0.468858i	$-0.155334\pi$
$983$	1736.52			1.76655			0.883273	+	0.468858i	$0.155334\pi$
$984$			1106.84i			1.12483i
$985$		0			0
$986$	163.929			0.166257
$987$	−311.286			−0.315386
$988$	393.355	+	771.943i	0.398133	+	0.781319i
$989$	147.472			0.149112
$990$		0			0
$991$		−	1213.47i		−	1.22449i	−0.790669	−	0.612244i	$-0.790267\pi$
$991$		−	1213.47i		−	1.22449i	0.790669	−	0.612244i	$-0.209733\pi$
$992$		−	1006.51i		−	1.01462i
$993$		−	1299.60i		−	1.30876i
$994$	−186.759			−0.187886
$995$		0			0
$996$		−	363.040i		−	0.364498i
$997$		−	134.929i		−	0.135335i	−0.997708	−	0.0676677i	$-0.978444\pi$
$997$		−	134.929i		−	0.135335i	0.997708	−	0.0676677i	$-0.0215558\pi$
$998$	504.981			0.505993
$999$	88.9827			0.0890718

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	475.3.d.c.474.16		24
5.2	odd	4		95.3.c.a.56.8	yes	12
5.3	odd	4		475.3.c.g.151.5		12
5.4	even	2	inner	475.3.d.c.474.9		24
15.2	even	4		855.3.e.a.721.5		12
19.18	odd	2	inner	475.3.d.c.474.10		24
20.7	even	4		1520.3.h.a.721.2		12
95.18	even	4		475.3.c.g.151.8		12
95.37	even	4		95.3.c.a.56.5	✓	12
95.94	odd	2	inner	475.3.d.c.474.15		24
285.227	odd	4		855.3.e.a.721.8		12
380.227	odd	4		1520.3.h.a.721.11		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
95.3.c.a.56.5	✓	12	95.37	even	4
95.3.c.a.56.8	yes	12	5.2	odd	4
475.3.c.g.151.5		12	5.3	odd	4
475.3.c.g.151.8		12	95.18	even	4
475.3.d.c.474.9		24	5.4	even	2	inner
475.3.d.c.474.10		24	19.18	odd	2	inner
475.3.d.c.474.15		24	95.94	odd	2	inner
475.3.d.c.474.16		24	1.1	even	1	trivial
855.3.e.a.721.5		12	15.2	even	4
855.3.e.a.721.8		12	285.227	odd	4
1520.3.h.a.721.2		12	20.7	even	4
1520.3.h.a.721.11		12	380.227	odd	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)

\(f(q)\)	\(=\)	\(q+1.14149 q^{2} -4.13699 q^{3} -2.69701 q^{4} -4.72232 q^{6} -7.54444i q^{7} -7.64455 q^{8} +8.11468 q^{9} +O(q^{10})\)	\(q+1.14149 q^{2} -4.13699 q^{3} -2.69701 q^{4} -4.72232 q^{6} -7.54444i q^{7} -7.64455 q^{8} +8.11468 q^{9} -18.6624 q^{11} +11.1575 q^{12} +16.9074 q^{13} -8.61188i q^{14} +2.06188 q^{16} -2.51694i q^{17} +9.26281 q^{18} +(-8.62635 - 16.9289i) q^{19} +31.2113i q^{21} -21.3029 q^{22} +41.9708i q^{23} +31.6254 q^{24} +19.2995 q^{26} +3.66254 q^{27} +20.3474i q^{28} +57.0573i q^{29} -30.5634i q^{31} +32.9318 q^{32} +77.2063 q^{33} -2.87306i q^{34} -21.8854 q^{36} +24.2954 q^{37} +(-9.84687 - 19.3241i) q^{38} -69.9456 q^{39} +34.9984i q^{41} +35.6273i q^{42} -3.51369i q^{43} +50.3327 q^{44} +47.9091i q^{46} +9.97353i q^{47} -8.52996 q^{48} -7.91856 q^{49} +10.4126i q^{51} -45.5993 q^{52} +37.3445 q^{53} +4.18074 q^{54} +57.6738i q^{56} +(35.6871 + 70.0345i) q^{57} +65.1302i q^{58} +26.0785i q^{59} -8.46643 q^{61} -34.8877i q^{62} -61.2207i q^{63} +29.3437 q^{64} +88.1299 q^{66} +12.5009 q^{67} +6.78822i q^{68} -173.633i q^{69} -21.6862i q^{71} -62.0331 q^{72} -77.1277i q^{73} +27.7328 q^{74} +(23.2653 + 45.6572i) q^{76} +140.798i q^{77} -79.8420 q^{78} +85.3703i q^{79} -88.1841 q^{81} +39.9502i q^{82} -32.5378i q^{83} -84.1770i q^{84} -4.01083i q^{86} -236.046i q^{87} +142.666 q^{88} -15.2618i q^{89} -127.557i q^{91} -113.196i q^{92} +126.440i q^{93} +11.3847i q^{94} -136.238 q^{96} +55.5679 q^{97} -9.03893 q^{98} -151.440 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 24 q^{4} - 56 q^{6} + 96 q^{9}+O(q^{10}) \) 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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