LMFDB - Embedded newform 475.3.d.c.474.10

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			474.10
Character	$\chi$	$=$	475.474
Dual form			475.3.d.c.474.9

$q$-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	$-0.592117\pi$
$2$	−1.14149			−0.570744			−0.285372	+	0.958417i	$0.592117\pi$
$3$	4.13699			1.37900			0.689498	−	0.724287i	$-0.257831\pi$
$3$	4.13699			1.37900			0.689498	+	0.724287i	$0.257831\pi$
$4$	−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	$-0.725350\pi$
$13$	−16.9074			−1.30057			−0.650283	+	0.759692i	$0.725350\pi$
$14$			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	$-0.557468\pi$
$29$		−	57.0573i		−	1.96749i	0.179561	−	0.983747i	$-0.442532\pi$
$30$		0			0
$31$			30.5634i			0.985916i	0.870053	+	0.492958i	$0.164084\pi$
$31$			30.5634i			0.985916i	−0.870053	+	0.492958i	$0.835916\pi$
$32$	−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	$-0.606481\pi$
$37$	−24.2954			−0.656631			−0.328316	+	0.944568i	$0.606481\pi$
$38$	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	$-0.859638\pi$
$41$		−	34.9984i		−	0.853618i	0.904342	−	0.426809i	$-0.140362\pi$
$42$			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	$-0.614602\pi$
$53$	−37.3445			−0.704612			−0.352306	+	0.935885i	$0.614602\pi$
$54$	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	$-0.929066\pi$
$59$		−	26.0785i		−	0.442009i	0.975273	−	0.221004i	$-0.0709335\pi$
$60$		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	$-0.529739\pi$
$67$	−12.5009			−0.186581			−0.0932906	+	0.995639i	$0.529739\pi$
$68$			6.78822i			0.0998267i
$69$			173.633i			2.51642i
$70$		0			0
$71$			21.6862i			0.305439i	0.988270	+	0.152720i	$0.0488032\pi$
$71$			21.6862i			0.305439i	−0.988270	+	0.152720i	$0.951197\pi$
$72$	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	$-0.818304\pi$
$79$		−	85.3703i		−	1.08064i	0.841461	−	0.540319i	$-0.181696\pi$
$80$		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	$0.0273255\pi$
$89$			15.2618i			0.171481i	−0.996318	+	0.0857403i	$0.972674\pi$
$90$		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	$-0.592469\pi$
$97$	−55.5679			−0.572865			−0.286433	+	0.958100i	$0.592469\pi$
$98$	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	$-0.543207\pi$
$103$	−27.8765			−0.270646			−0.135323	+	0.990802i	$0.543207\pi$
$104$	−129.249			−1.24278
$105$		0			0
$106$	42.6282			0.402153
$107$	−14.6484			−0.136901			−0.0684506	−	0.997655i	$-0.521806\pi$
$107$	−14.6484			−0.136901			−0.0684506	+	0.997655i	$0.521806\pi$
$108$	9.87790			0.0914620
$109$		−	34.0962i		−	0.312809i	−0.987693	−	0.156405i	$-0.950010\pi$
$109$		−	34.0962i		−	0.312809i	0.987693	−	0.156405i	$-0.0499904\pi$
$110$		0			0
$111$	−100.510			−0.905492
$112$		−	15.5557i		−	0.138890i
$113$	23.0375			0.203872			0.101936	−	0.994791i	$-0.467496\pi$
$113$	23.0375			0.203872			0.101936	+	0.994791i	$0.467496\pi$
$114$	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	$-0.286545\pi$
$127$	157.848			1.24290			0.621448	+	0.783455i	$0.286545\pi$
$128$	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	$0.00841794\pi$
$151$			7.98568i			0.0528853i	−0.999650	+	0.0264426i	$0.991582\pi$
$152$	−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	$-0.780708\pi$
$167$	−257.824			−1.54386			−0.771929	+	0.635709i	$0.780708\pi$
$168$		−	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	$-0.523197\pi$
$173$	−25.1930			−0.145624			−0.0728122	+	0.997346i	$0.523197\pi$
$174$			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	$-0.675109\pi$
$179$		−	305.181i		−	1.70492i	0.522790	−	0.852462i	$-0.324891\pi$
$180$		0			0
$181$		−	297.130i		−	1.64160i	−0.571216	−	0.820800i	$-0.693528\pi$
$181$		−	297.130i		−	1.64160i	0.571216	−	0.820800i	$-0.306472\pi$
$182$		−	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	$-0.402880\pi$
$193$	115.955			0.600801			0.300401	+	0.953813i	$0.402880\pi$
$194$	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	$-0.854824\pi$
$211$		−	185.863i		−	0.880869i	0.897785	−	0.440434i	$-0.145176\pi$
$212$	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	$-0.165312\pi$
$223$	387.193			1.73629			0.868146	+	0.496310i	$0.165312\pi$
$224$			248.452i			1.10916i
$225$		0			0
$226$	−26.2970			−0.116358
$227$	−25.8306			−0.113791			−0.0568956	−	0.998380i	$-0.518120\pi$
$227$	−25.8306			−0.113791			−0.0568956	+	0.998380i	$0.518120\pi$
$228$	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	$-0.988165\pi$
$241$		−	17.9173i		−	0.0743457i	0.999309	−	0.0371729i	$-0.0118352\pi$
$242$	−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	$-0.644410\pi$
$257$	−225.273			−0.876548			−0.438274	+	0.898841i	$0.644410\pi$
$258$			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	$0.120294\pi$
$269$			198.513i			0.737967i	−0.929436	+	0.368983i	$0.879706\pi$
$270$		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	$0.320298\pi$
$281$			474.794i			1.68966i	−0.535037	+	0.844828i	$0.679702\pi$
$282$		−	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	$-0.724123\pi$
$293$	−379.347			−1.29470			−0.647350	+	0.762193i	$0.724123\pi$
$294$	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	$-0.660509\pi$
$307$	−296.656			−0.966306			−0.483153	+	0.875536i	$0.660509\pi$
$308$		−	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	$-0.579499\pi$
$317$	−156.703			−0.494331			−0.247165	+	0.968973i	$0.579499\pi$
$318$	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	$-0.842617\pi$
$331$		−	314.141i		−	0.949065i	0.880238	−	0.474533i	$-0.157383\pi$
$332$			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	$-0.379885\pi$
$337$	248.342			0.736921			0.368461	+	0.929643i	$0.379885\pi$
$338$	−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	$-0.688559\pi$
$373$	−416.517			−1.11667			−0.558334	+	0.829616i	$0.688559\pi$
$374$		−	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	$0.175211\pi$
$379$			396.483i			1.04613i	−0.852293	+	0.523064i	$0.824789\pi$
$380$		0			0
$381$	653.015			1.71395
$382$	−128.372			−0.336053
$383$	−332.639			−0.868510			−0.434255	−	0.900790i	$-0.642988\pi$
$383$	−332.639			−0.868510			−0.434255	+	0.900790i	$0.642988\pi$
$384$	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	$-0.854266\pi$
$401$		−	354.490i		−	0.884016i	0.897011	−	0.442008i	$-0.145734\pi$
$402$	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	$-0.853830\pi$
$409$		−	362.568i		−	0.886474i	0.896404	−	0.443237i	$-0.146170\pi$
$410$		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	$0.0838490\pi$
$421$			219.243i			0.520767i	−0.965505	+	0.260384i	$0.916151\pi$
$422$			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	$-0.905692\pi$
$431$		−	251.672i		−	0.583925i	0.956430	−	0.291962i	$-0.0943082\pi$
$432$	−7.55171			−0.0174808
$433$	−29.8124			−0.0688509			−0.0344255	−	0.999407i	$-0.510960\pi$
$433$	−29.8124			−0.0688509			−0.0344255	+	0.999407i	$0.510960\pi$
$434$	−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	$-0.869735\pi$
$439$		−	349.367i		−	0.795826i	0.917423	−	0.397913i	$-0.130265\pi$
$440$		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	$0.0272275\pi$
$449$			76.7191i			0.170867i	−0.996344	+	0.0854333i	$0.972773\pi$
$450$		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	$-0.604334\pi$
$487$	−313.568			−0.643876			−0.321938	+	0.946761i	$0.604334\pi$
$488$	−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	$-0.871194\pi$
$509$		−	400.790i		−	0.787407i	0.919237	−	0.393704i	$-0.128806\pi$
$510$		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	$-0.776772\pi$
$521$		−	672.304i		−	1.29041i	0.764009	−	0.645205i	$-0.223228\pi$
$522$			528.511i			1.01247i
$523$	822.554			1.57276			0.786380	−	0.617743i	$-0.211953\pi$
$523$	822.554			1.57276			0.786380	+	0.617743i	$0.211953\pi$
$524$	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	$-0.578709\pi$
$547$	−267.766			−0.489516			−0.244758	+	0.969584i	$0.578709\pi$
$548$			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	$-0.376024\pi$
$563$	427.553			0.759419			0.379710	+	0.925106i	$0.376024\pi$
$564$		−	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	$0.0198445\pi$
$569$			70.9006i			0.124606i	−0.998057	+	0.0623028i	$0.980156\pi$
$570$		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	$0.316845\pi$
$599$			1005.09i			1.67795i	−0.544170	+	0.838975i	$0.683155\pi$
$600$		0			0
$601$			823.540i			1.37028i	0.728410	+	0.685142i	$0.240260\pi$
$601$			823.540i			1.37028i	−0.728410	+	0.685142i	$0.759740\pi$
$602$	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	$-0.355290\pi$
$607$	533.092			0.878241			0.439121	+	0.898428i	$0.355290\pi$
$608$	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	$0.185663\pi$
$641$			706.078i			1.10153i	−0.834662	+	0.550763i	$0.814337\pi$
$642$	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	$-0.801727\pi$
$659$		−	768.905i		−	1.16677i	0.812194	−	0.583387i	$-0.198273\pi$
$660$		0			0
$661$			135.070i			0.204342i	0.994767	+	0.102171i	$0.0325790\pi$
$661$			135.070i			0.204342i	−0.994767	+	0.102171i	$0.967421\pi$
$662$			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	$-0.839102\pi$
$673$	−1177.68			−1.74989			−0.874945	+	0.484223i	$0.839102\pi$
$674$	−283.480			−0.420593
$675$		0			0
$676$	−315.169			−0.466226
$677$	−547.098			−0.808121			−0.404061	−	0.914732i	$-0.632401\pi$
$677$	−547.098			−0.808121			−0.404061	+	0.914732i	$0.632401\pi$
$678$	−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	$-0.322636\pi$
$683$	722.363			1.05763			0.528816	+	0.848736i	$0.322636\pi$
$684$	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	$-0.666803\pi$
$743$	−743.549			−1.00074			−0.500370	+	0.865812i	$0.666803\pi$
$744$			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	$-0.654485\pi$
$751$		−	1328.55i		−	1.76904i	0.466499	−	0.884522i	$-0.345515\pi$
$752$			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	$-0.388917\pi$
$773$	528.633			0.683871			0.341936	+	0.939723i	$0.388917\pi$
$774$			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	$-0.761632\pi$
$787$	−1152.91			−1.46494			−0.732469	+	0.680800i	$0.761632\pi$
$788$		−	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	$-0.666869\pi$
$797$	−797.879			−1.00110			−0.500552	+	0.865707i	$0.666869\pi$
$798$	−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	$-0.983988\pi$
$811$		−	81.5557i		−	0.100562i	0.998735	−	0.0502810i	$-0.0160117\pi$
$812$	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	$-0.572716\pi$
$827$	−374.568			−0.452924			−0.226462	+	0.974020i	$0.572716\pi$
$828$		−	918.546i		−	1.10935i
$829$		−	1197.29i		−	1.44426i	−0.691759	−	0.722128i	$-0.743164\pi$
$829$		−	1197.29i		−	1.44426i	0.691759	−	0.722128i	$-0.256836\pi$
$830$		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	$-0.959473\pi$
$839$		−	213.064i		−	0.253950i	0.991906	−	0.126975i	$-0.0405268\pi$
$840$		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	$-0.397649\pi$
$857$	541.680			0.632065			0.316032	+	0.948748i	$0.397649\pi$
$858$	−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	$-0.594906\pi$
$863$	−507.027			−0.587517			−0.293758	+	0.955880i	$0.594906\pi$
$864$	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	$-0.514695\pi$
$877$	−80.9460			−0.0922988			−0.0461494	+	0.998935i	$0.514695\pi$
$878$			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	$-0.161877\pi$
$887$	1549.50			1.74690			0.873450	+	0.486913i	$0.161877\pi$
$888$	−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	$-0.758032\pi$
$907$	−1314.64			−1.44944			−0.724721	+	0.689042i	$0.758032\pi$
$908$	69.6653			0.0767239
$909$	−673.490			−0.740913
$910$		0			0
$911$		−	970.388i		−	1.06519i	−0.846370	−	0.532595i	$-0.821217\pi$
$911$		−	970.388i		−	1.06519i	0.846370	−	0.532595i	$-0.178783\pi$
$912$	−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	$0.338531\pi$
$941$			1645.01i			1.74815i	−0.485792	+	0.874074i	$0.661469\pi$
$942$		−	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	$-0.413917\pi$
$953$	509.195			0.534307			0.267154	+	0.963654i	$0.413917\pi$
$954$	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	$-0.787957\pi$
$971$		−	1200.09i		−	1.23593i	0.786205	−	0.617966i	$-0.212043\pi$
$972$	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	$-0.434883\pi$
$977$	396.951			0.406296			0.203148	+	0.979148i	$0.434883\pi$
$978$		−	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	$-0.844666\pi$
$983$	−1736.52			−1.76655			−0.883273	+	0.468858i	$0.844666\pi$
$984$		−	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	$0.209733\pi$
$991$			1213.47i			1.22449i	−0.790669	+	0.612244i	$0.790267\pi$
$992$		−	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.474.10		24
5.2	odd	4		95.3.c.a.56.5	✓	12
5.3	odd	4		475.3.c.g.151.8		12
5.4	even	2	inner	475.3.d.c.474.15		24
15.2	even	4		855.3.e.a.721.8		12
19.18	odd	2	inner	475.3.d.c.474.16		24
20.7	even	4		1520.3.h.a.721.11		12
95.18	even	4		475.3.c.g.151.5		12
95.37	even	4		95.3.c.a.56.8	yes	12
95.94	odd	2	inner	475.3.d.c.474.9		24
285.227	odd	4		855.3.e.a.721.5		12
380.227	odd	4		1520.3.h.a.721.2		12

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

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