LMFDB - Embedded newform 475.3.d.d.474.12

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:	$28$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			474.12
Character	$\chi$	$=$	475.474
Dual form			475.3.d.d.474.11

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.40632 q^{2} -2.91656 q^{3} -2.02225 q^{4} +4.10162 q^{6} +11.7208i q^{7} +8.46924 q^{8} -0.493705 q^{9} +O(q^{10})$	\(q-1.40632 q^{2} -2.91656 q^{3} -2.02225 q^{4} +4.10162 q^{6} +11.7208i q^{7} +8.46924 q^{8} -0.493705 q^{9} -6.36259 q^{11} +5.89801 q^{12} +17.7815 q^{13} -16.4833i q^{14} -3.82148 q^{16} +17.5868i q^{17} +0.694309 q^{18} +(16.8066 - 8.86223i) q^{19} -34.1845i q^{21} +8.94786 q^{22} +1.53826i q^{23} -24.7010 q^{24} -25.0065 q^{26} +27.6889 q^{27} -23.7025i q^{28} +39.9749i q^{29} +9.24369i q^{31} -28.5027 q^{32} +18.5568 q^{33} -24.7328i q^{34} +0.998396 q^{36} -56.4183 q^{37} +(-23.6355 + 12.4632i) q^{38} -51.8607 q^{39} -39.5382i q^{41} +48.0744i q^{42} +18.4360i q^{43} +12.8668 q^{44} -2.16329i q^{46} +61.5720i q^{47} +11.1456 q^{48} -88.3780 q^{49} -51.2930i q^{51} -35.9587 q^{52} -66.5116 q^{53} -38.9396 q^{54} +99.2665i q^{56} +(-49.0173 + 25.8472i) q^{57} -56.2177i q^{58} +9.42839i q^{59} -36.5472 q^{61} -12.9996i q^{62} -5.78663i q^{63} +55.3700 q^{64} -26.0969 q^{66} -2.55637 q^{67} -35.5650i q^{68} -4.48642i q^{69} -53.8192i q^{71} -4.18130 q^{72} -115.725i q^{73} +79.3424 q^{74} +(-33.9871 + 17.9217i) q^{76} -74.5748i q^{77} +72.9329 q^{78} -95.9222i q^{79} -76.3129 q^{81} +55.6035i q^{82} +143.935i q^{83} +69.1296i q^{84} -25.9270i q^{86} -116.589i q^{87} -53.8863 q^{88} +19.4615i q^{89} +208.414i q^{91} -3.11075i q^{92} -26.9597i q^{93} -86.5902i q^{94} +83.1297 q^{96} +163.156 q^{97} +124.288 q^{98} +3.14124 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 28 q + 56 q^{4} - 8 q^{6} + 72 q^{9}+O(q^{10}) $ 28 * q + 56 * q^4 - 8 * q^6 + 72 * q^9	$ 28 q + 56 q^{4} - 8 q^{6} + 72 q^{9} - 8 q^{11} + 72 q^{16} - 78 q^{19} + 88 q^{24} + 60 q^{26} + 8 q^{36} + 64 q^{39} + 104 q^{44} - 468 q^{49} - 196 q^{54} + 444 q^{61} + 436 q^{64} + 184 q^{66} + 184 q^{74} - 702 q^{76} + 804 q^{81} + 380 q^{96} + 360 q^{99}+O(q^{100}) $ 28 * q + 56 * q^4 - 8 * q^6 + 72 * q^9 - 8 * q^11 + 72 * q^16 - 78 * q^19 + 88 * q^24 + 60 * q^26 + 8 * q^36 + 64 * q^39 + 104 * q^44 - 468 * q^49 - 196 * q^54 + 444 * q^61 + 436 * q^64 + 184 * q^66 + 184 * q^74 - 702 * q^76 + 804 * q^81 + 380 * q^96 + 360 * 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 $

$2$	−1.40632			−0.703162			−0.351581	−	0.936157i	$-0.614356\pi$
$2$	−1.40632			−0.703162			−0.351581	+	0.936157i	$0.614356\pi$
$3$	−2.91656			−0.972185			−0.486093	−	0.873907i	$-0.661578\pi$
$3$	−2.91656			−0.972185			−0.486093	+	0.873907i	$0.661578\pi$
$4$	−2.02225			−0.505563
$5$		0			0
$5$		0			0
$6$	4.10162			0.683604
$7$			11.7208i			1.67441i	0.546893	+	0.837203i	$0.315810\pi$
$7$			11.7208i			1.67441i	−0.546893	+	0.837203i	$0.684190\pi$
$8$	8.46924			1.05865
$9$	−0.493705			−0.0548561
$10$		0			0
$11$	−6.36259			−0.578417			−0.289209	−	0.957266i	$-0.593392\pi$
$11$	−6.36259			−0.578417			−0.289209	+	0.957266i	$0.593392\pi$
$12$	5.89801			0.491501
$13$	17.7815			1.36781			0.683903	−	0.729573i	$-0.260281\pi$
$13$	17.7815			1.36781			0.683903	+	0.729573i	$0.260281\pi$
$14$		−	16.4833i		−	1.17738i
$15$		0			0
$16$	−3.82148			−0.238842
$17$			17.5868i			1.03452i	0.855828	+	0.517260i	$0.173048\pi$
$17$			17.5868i			1.03452i	−0.855828	+	0.517260i	$0.826952\pi$
$18$	0.694309			0.0385727
$19$	16.8066	−	8.86223i	0.884557	−	0.466433i
$19$	16.8066	−	8.86223i	0.884557	−	0.466433i
$20$		0			0
$21$		−	34.1845i		−	1.62783i
$22$	8.94786			0.406721
$23$			1.53826i			0.0668809i	0.999441	+	0.0334404i	$0.0106464\pi$
$23$			1.53826i			0.0668809i	−0.999441	+	0.0334404i	$0.989354\pi$
$24$	−24.7010			−1.02921
$25$		0			0
$26$	−25.0065			−0.961789
$27$	27.6889			1.02552
$28$		−	23.7025i		−	0.846518i
$29$			39.9749i			1.37845i	0.724549	+	0.689223i	$0.242048\pi$
$29$			39.9749i			1.37845i	−0.724549	+	0.689223i	$0.757952\pi$
$30$		0			0
$31$			9.24369i			0.298184i	0.988823	+	0.149092i	$0.0476350\pi$
$31$			9.24369i			0.298184i	−0.988823	+	0.149092i	$0.952365\pi$
$32$	−28.5027			−0.890710
$33$	18.5568			0.562328
$34$		−	24.7328i		−	0.727435i
$35$		0			0
$36$	0.998396			0.0277332
$37$	−56.4183			−1.52482			−0.762410	−	0.647095i	$-0.775984\pi$
$37$	−56.4183			−1.52482			−0.762410	+	0.647095i	$0.775984\pi$
$38$	−23.6355	+	12.4632i	−0.621986	+	0.327978i
$39$	−51.8607			−1.32976
$40$		0			0
$41$		−	39.5382i		−	0.964347i	−0.876076	−	0.482173i	$-0.839848\pi$
$41$		−	39.5382i		−	0.964347i	0.876076	−	0.482173i	$-0.160152\pi$
$42$			48.0744i			1.14463i
$43$			18.4360i			0.428744i	0.976752	+	0.214372i	$0.0687704\pi$
$43$			18.4360i			0.428744i	−0.976752	+	0.214372i	$0.931230\pi$
$44$	12.8668			0.292426
$45$		0			0
$46$		−	2.16329i		−	0.0470281i
$47$			61.5720i			1.31004i	0.755610	+	0.655021i	$0.227340\pi$
$47$			61.5720i			1.31004i	−0.755610	+	0.655021i	$0.772660\pi$
$48$	11.1456			0.232199
$49$	−88.3780			−1.80363
$50$		0			0
$51$		−	51.2930i		−	1.00574i
$52$	−35.9587			−0.691513
$53$	−66.5116			−1.25494			−0.627468	−	0.778642i	$-0.715909\pi$
$53$	−66.5116			−1.25494			−0.627468	+	0.778642i	$0.715909\pi$
$54$	−38.9396			−0.721103
$55$		0			0
$56$			99.2665i			1.77262i
$57$	−49.0173	+	25.8472i	−0.859953	+	0.453459i
$58$		−	56.2177i		−	0.969270i
$59$			9.42839i			0.159803i	0.996803	+	0.0799016i	$0.0254606\pi$
$59$			9.42839i			0.159803i	−0.996803	+	0.0799016i	$0.974539\pi$
$60$		0			0
$61$	−36.5472			−0.599134			−0.299567	−	0.954075i	$-0.596842\pi$
$61$	−36.5472			−0.599134			−0.299567	+	0.954075i	$0.596842\pi$
$62$		−	12.9996i		−	0.209671i
$63$		−	5.78663i		−	0.0918513i
$64$	55.3700			0.865156
$65$		0			0
$66$	−26.0969			−0.395408
$67$	−2.55637			−0.0381548			−0.0190774	−	0.999818i	$-0.506073\pi$
$67$	−2.55637			−0.0381548			−0.0190774	+	0.999818i	$0.506073\pi$
$68$		−	35.5650i		−	0.523015i
$69$		−	4.48642i		−	0.0650206i
$70$		0			0
$71$		−	53.8192i		−	0.758017i	−0.925393	−	0.379008i	$-0.876265\pi$
$71$		−	53.8192i		−	0.758017i	0.925393	−	0.379008i	$-0.123735\pi$
$72$	−4.18130			−0.0580737
$73$		−	115.725i		−	1.58527i	−0.609695	−	0.792636i	$-0.708708\pi$
$73$		−	115.725i		−	1.58527i	0.609695	−	0.792636i	$-0.291292\pi$
$74$	79.3424			1.07219
$75$		0			0
$76$	−33.9871	+	17.9217i	−0.447199	+	0.235811i
$77$		−	74.5748i		−	0.968504i
$78$	72.9329			0.935037
$79$		−	95.9222i		−	1.21421i	−0.794623	−	0.607103i	$-0.792332\pi$
$79$		−	95.9222i		−	1.21421i	0.794623	−	0.607103i	$-0.207668\pi$
$80$		0			0
$81$	−76.3129			−0.942135
$82$			55.6035i			0.678092i
$83$			143.935i			1.73416i	0.498168	+	0.867081i	$0.334006\pi$
$83$			143.935i			1.73416i	−0.498168	+	0.867081i	$0.665994\pi$
$84$			69.1296i			0.822972i
$85$		0			0
$86$		−	25.9270i		−	0.301476i
$87$		−	116.589i		−	1.34010i
$88$	−53.8863			−0.612344
$89$			19.4615i			0.218668i	0.994005	+	0.109334i	$0.0348718\pi$
$89$			19.4615i			0.218668i	−0.994005	+	0.109334i	$0.965128\pi$
$90$		0			0
$91$			208.414i			2.29026i
$92$		−	3.11075i		−	0.0338125i
$93$		−	26.9597i		−	0.289890i
$94$		−	86.5902i		−	0.921172i
$95$		0			0
$96$	83.1297			0.865935
$97$	163.156			1.68202			0.841012	−	0.541017i	$-0.181960\pi$
$97$	163.156			1.68202			0.841012	+	0.541017i	$0.181960\pi$
$98$	124.288			1.26825
$99$	3.14124			0.0317297
$100$		0			0
$101$	−57.2111			−0.566447			−0.283223	−	0.959054i	$-0.591404\pi$
$101$	−57.2111			−0.566447			−0.283223	+	0.959054i	$0.591404\pi$
$102$			72.1346i			0.707202i
$103$	22.5345			0.218782			0.109391	−	0.993999i	$-0.465110\pi$
$103$	22.5345			0.218782			0.109391	+	0.993999i	$0.465110\pi$
$104$	150.596			1.44803
$105$		0			0
$106$	93.5369			0.882424
$107$	−68.7799			−0.642803			−0.321401	−	0.946943i	$-0.604154\pi$
$107$	−68.7799			−0.642803			−0.321401	+	0.946943i	$0.604154\pi$
$108$	−55.9940			−0.518463
$109$			79.5462i			0.729781i	0.931050	+	0.364891i	$0.118894\pi$
$109$			79.5462i			0.729781i	−0.931050	+	0.364891i	$0.881106\pi$
$110$		0			0
$111$	164.547			1.48241
$112$		−	44.7909i		−	0.399919i
$113$	−188.929			−1.67193			−0.835967	−	0.548780i	$-0.815093\pi$
$113$	−188.929			−1.67193			−0.835967	+	0.548780i	$0.815093\pi$
$114$	68.9342	−	36.3495i	0.604686	−	0.318855i
$115$		0			0
$116$		−	80.8394i		−	0.696891i
$117$	−8.77880			−0.0750325
$118$		−	13.2594i		−	0.112368i
$119$	−206.132			−1.73221
$120$		0			0
$121$	−80.5175			−0.665434
$122$	51.3972			0.421288
$123$			115.315i			0.937524i
$124$		−	18.6931i		−	0.150751i
$125$		0			0
$126$			8.13788i			0.0645864i
$127$	−12.5720			−0.0989923			−0.0494961	−	0.998774i	$-0.515762\pi$
$127$	−12.5720			−0.0989923			−0.0494961	+	0.998774i	$0.515762\pi$
$128$	36.1427			0.282365
$129$		−	53.7695i		−	0.416818i
$130$		0			0
$131$	96.6711			0.737947			0.368974	−	0.929440i	$-0.379709\pi$
$131$	96.6711			0.737947			0.368974	+	0.929440i	$0.379709\pi$
$132$	−37.5266			−0.284293
$133$	103.873	+	196.987i	0.780998	+	1.48111i
$134$	3.59508			0.0268290
$135$		0			0
$136$			148.947i			1.09520i
$137$		−	219.187i		−	1.59991i	−0.600061	−	0.799954i	$-0.704857\pi$
$137$		−	219.187i		−	1.59991i	0.600061	−	0.799954i	$-0.295143\pi$
$138$			6.30936i			0.0457200i
$139$	188.602			1.35685			0.678425	−	0.734669i	$-0.262663\pi$
$139$	188.602			1.35685			0.678425	+	0.734669i	$0.262663\pi$
$140$		0			0
$141$		−	179.578i		−	1.27360i
$142$			75.6872i			0.533008i
$143$	−113.136			−0.791163
$144$	1.88668			0.0131020
$145$		0			0
$146$			162.747i			1.11470i
$147$	257.759			1.75346
$148$	114.092			0.770893
$149$	−177.432			−1.19082			−0.595410	−	0.803422i	$-0.703010\pi$
$149$	−177.432			−1.19082			−0.595410	+	0.803422i	$0.703010\pi$
$150$		0			0
$151$		−	9.49631i		−	0.0628895i	−0.999505	−	0.0314447i	$-0.989989\pi$
$151$		−	9.49631i		−	0.0628895i	0.999505	−	0.0314447i	$-0.0100108\pi$
$152$	142.339	−	75.0563i	0.936440	−	0.493792i
$153$		−	8.68271i		−	0.0567497i
$154$			104.876i			0.681016i
$155$		0			0
$156$	104.875			0.672278
$157$		−	27.3090i		−	0.173943i	−0.996211	−	0.0869715i	$-0.972281\pi$
$157$		−	27.3090i		−	0.173943i	0.996211	−	0.0869715i	$-0.0277189\pi$
$158$			134.898i			0.853783i
$159$	193.985			1.22003
$160$		0			0
$161$	−18.0297			−0.111986
$162$	107.321			0.662473
$163$			128.144i			0.786161i	0.919504	+	0.393081i	$0.128591\pi$
$163$			128.144i			0.786161i	−0.919504	+	0.393081i	$0.871409\pi$
$164$			79.9563i			0.487538i
$165$		0			0
$166$		−	202.420i		−	1.21940i
$167$	−83.5190			−0.500114			−0.250057	−	0.968231i	$-0.580449\pi$
$167$	−83.5190			−0.500114			−0.250057	+	0.968231i	$0.580449\pi$
$168$		−	289.516i		−	1.72331i
$169$	147.181			0.870894
$170$		0			0
$171$	−8.29749	+	4.37532i	−0.0485233	+	0.0255867i
$172$		−	37.2822i		−	0.216757i
$173$	125.617			0.726110			0.363055	−	0.931768i	$-0.381734\pi$
$173$	125.617			0.726110			0.363055	+	0.931768i	$0.381734\pi$
$174$			163.962i			0.942310i
$175$		0			0
$176$	24.3145			0.138151
$177$		−	27.4984i		−	0.155358i
$178$		−	27.3691i		−	0.153759i
$179$		−	86.1103i		−	0.481063i	−0.970641	−	0.240532i	$-0.922678\pi$
$179$		−	86.1103i		−	0.481063i	0.970641	−	0.240532i	$-0.0773218\pi$
$180$		0			0
$181$		−	299.094i		−	1.65246i	−0.563336	−	0.826228i	$-0.690482\pi$
$181$		−	299.094i		−	1.65246i	0.563336	−	0.826228i	$-0.309518\pi$
$182$		−	293.097i		−	1.61043i
$183$	106.592			0.582469
$184$			13.0279i			0.0708038i
$185$		0			0
$186$			37.9141i			0.203839i
$187$		−	111.898i		−	0.598384i
$188$		−	124.514i		−	0.662309i
$189$			324.537i			1.71713i
$190$		0			0
$191$	−6.80277			−0.0356166			−0.0178083	−	0.999841i	$-0.505669\pi$
$191$	−6.80277			−0.0356166			−0.0178083	+	0.999841i	$0.505669\pi$
$192$	−161.490			−0.841092
$193$	−235.945			−1.22251			−0.611256	−	0.791433i	$-0.709335\pi$
$193$	−235.945			−1.22251			−0.611256	+	0.791433i	$0.709335\pi$
$194$	−229.451			−1.18274
$195$		0			0
$196$	178.723			0.911850
$197$		−	29.4014i		−	0.149245i	−0.997212	−	0.0746227i	$-0.976225\pi$
$197$		−	29.4014i		−	0.149245i	0.997212	−	0.0746227i	$-0.0237752\pi$
$198$	−4.41760			−0.0223111
$199$	65.6370			0.329834			0.164917	−	0.986307i	$-0.447264\pi$
$199$	65.6370			0.329834			0.164917	+	0.986307i	$0.447264\pi$
$200$		0			0
$201$	7.45579			0.0370935
$202$	80.4573			0.398304
$203$	−468.539			−2.30808
$204$			103.727i			0.508468i
$205$		0			0
$206$	−31.6909			−0.153839
$207$		−	0.759447i		−	0.00366882i
$208$	−67.9516			−0.326690
$209$	−106.933	+	56.3867i	−0.511643	+	0.269793i
$210$		0			0
$211$		−	45.4463i		−	0.215385i	−0.994184	−	0.107693i	$-0.965654\pi$
$211$		−	45.4463i		−	0.215385i	0.994184	−	0.107693i	$-0.0343463\pi$
$212$	134.503			0.634450
$213$			156.967i			0.736933i
$214$	96.7268			0.451995
$215$		0			0
$216$	234.504			1.08567
$217$	−108.344			−0.499280
$218$		−	111.868i		−	0.513154i
$219$			337.518i			1.54118i
$220$		0			0
$221$			312.720i			1.41502i
$222$	−231.407			−1.04237
$223$	236.447			1.06030			0.530149	−	0.847904i	$-0.322136\pi$
$223$	236.447			1.06030			0.530149	+	0.847904i	$0.322136\pi$
$224$		−	334.076i		−	1.49141i
$225$		0			0
$226$	265.695			1.17564
$227$	−143.537			−0.632323			−0.316162	−	0.948705i	$-0.602394\pi$
$227$	−143.537			−0.632323			−0.316162	+	0.948705i	$0.602394\pi$
$228$	99.1254	−	52.2695i	0.434761	−	0.229252i
$229$	102.645			0.448230			0.224115	−	0.974563i	$-0.428051\pi$
$229$	102.645			0.448230			0.224115	+	0.974563i	$0.428051\pi$
$230$		0			0
$231$			217.502i			0.941566i
$232$			338.557i			1.45930i
$233$			170.820i			0.733132i	0.930392	+	0.366566i	$0.119467\pi$
$233$			170.820i			0.733132i	−0.930392	+	0.366566i	$0.880533\pi$
$234$	12.3458			0.0527600
$235$		0			0
$236$		−	19.0666i		−	0.0807907i
$237$			279.762i			1.18043i
$238$	289.889			1.21802
$239$	−194.352			−0.813186			−0.406593	−	0.913609i	$-0.633283\pi$
$239$	−194.352			−0.813186			−0.406593	+	0.913609i	$0.633283\pi$
$240$		0			0
$241$			457.159i			1.89693i	0.316888	+	0.948463i	$0.397362\pi$
$241$			457.159i			1.89693i	−0.316888	+	0.948463i	$0.602638\pi$
$242$	113.234			0.467908
$243$	−26.6294			−0.109586
$244$	73.9076			0.302900
$245$		0			0
$246$		−	162.171i		−	0.659231i
$247$	298.846	−	157.584i	1.20990	−	0.637990i
$248$			78.2870i			0.315674i
$249$		−	419.796i		−	1.68593i
$250$		0			0
$251$	−345.070			−1.37478			−0.687391	−	0.726288i	$-0.741244\pi$
$251$	−345.070			−1.37478			−0.687391	+	0.726288i	$0.741244\pi$
$252$			11.7020i			0.0464367i
$253$		−	9.78732i		−	0.0386850i
$254$	17.6803			0.0696076
$255$		0			0
$256$	−272.308			−1.06370
$257$	−286.104			−1.11324			−0.556622	−	0.830766i	$-0.687903\pi$
$257$	−286.104			−1.11324			−0.556622	+	0.830766i	$0.687903\pi$
$258$			75.6174i			0.293091i
$259$		−	661.270i		−	2.55317i
$260$		0			0
$261$		−	19.7358i		−	0.0756161i
$262$	−135.951			−0.518896
$263$			203.484i			0.773703i	0.922142	+	0.386851i	$0.126437\pi$
$263$			203.484i			0.773703i	−0.922142	+	0.386851i	$0.873563\pi$
$264$	157.162			0.595312
$265$		0			0
$266$	−146.079	−	277.028i	−0.549168	−	1.04146i
$267$		−	56.7604i		−	0.212586i
$268$	5.16963			0.0192896
$269$		−	420.928i		−	1.56479i	−0.622783	−	0.782395i	$-0.713998\pi$
$269$		−	420.928i		−	1.56479i	0.622783	−	0.782395i	$-0.286002\pi$
$270$		0			0
$271$	−127.006			−0.468655			−0.234328	−	0.972158i	$-0.575289\pi$
$271$	−127.006			−0.468655			−0.234328	+	0.972158i	$0.575289\pi$
$272$		−	67.2078i		−	0.247087i
$273$		−	607.850i		−	2.22656i
$274$			308.249i			1.12499i
$275$		0			0
$276$			9.07268i			0.0328720i
$277$			342.553i			1.23665i	0.785921	+	0.618327i	$0.212189\pi$
$277$			342.553i			1.23665i	−0.785921	+	0.618327i	$0.787811\pi$
$278$	−265.236			−0.954086
$279$		−	4.56366i		−	0.0163572i
$280$		0			0
$281$		−	151.942i		−	0.540718i	−0.962760	−	0.270359i	$-0.912858\pi$
$281$		−	151.942i		−	0.540718i	0.962760	−	0.270359i	$-0.0871424\pi$
$282$			252.545i			0.895550i
$283$		−	240.982i		−	0.851527i	−0.904834	−	0.425764i	$-0.860006\pi$
$283$		−	240.982i		−	0.851527i	0.904834	−	0.425764i	$-0.139994\pi$
$284$			108.836i			0.383225i
$285$		0			0
$286$	159.106			0.556315
$287$	463.421			1.61471
$288$	14.0719			0.0488609
$289$	−20.2970			−0.0702317
$290$		0			0
$291$	−475.854			−1.63524
$292$			234.025i			0.801455i
$293$	−7.05363			−0.0240738			−0.0120369	−	0.999928i	$-0.503832\pi$
$293$	−7.05363			−0.0240738			−0.0120369	+	0.999928i	$0.503832\pi$
$294$	−362.493			−1.23297
$295$		0			0
$296$	−477.820			−1.61426
$297$	−176.173			−0.593176
$298$	249.527			0.837339
$299$			27.3526i			0.0914801i
$300$		0			0
$301$	−216.085			−0.717891
$302$			13.3549i			0.0442215i
$303$	166.859			0.550691
$304$	−64.2260	+	33.8668i	−0.211270	+	0.111404i
$305$		0			0
$306$			12.2107i			0.0399042i
$307$	−44.7370			−0.145723			−0.0728616	−	0.997342i	$-0.523213\pi$
$307$	−44.7370			−0.145723			−0.0728616	+	0.997342i	$0.523213\pi$
$308$			150.809i			0.489640i
$309$	−65.7232			−0.212696
$310$		0			0
$311$	−110.679			−0.355883			−0.177941	−	0.984041i	$-0.556944\pi$
$311$	−110.679			−0.355883			−0.177941	+	0.984041i	$0.556944\pi$
$312$	−439.220			−1.40776
$313$		−	131.862i		−	0.421286i	−0.977563	−	0.210643i	$-0.932444\pi$
$313$		−	131.862i		−	0.421286i	0.977563	−	0.210643i	$-0.0675557\pi$
$314$			38.4054i			0.122310i
$315$		0			0
$316$			193.979i			0.613858i
$317$	−309.486			−0.976295			−0.488148	−	0.872761i	$-0.662327\pi$
$317$	−309.486			−0.976295			−0.488148	+	0.872761i	$0.662327\pi$
$318$	−272.806			−0.857879
$319$		−	254.344i		−	0.797316i
$320$		0			0
$321$	200.600			0.624923
$322$	25.3556			0.0787441
$323$	155.859	+	295.575i	0.482534	+	0.915091i
$324$	154.324			0.476309
$325$		0			0
$326$		−	180.212i		−	0.552799i
$327$		−	232.001i		−	0.709483i
$328$		−	334.859i		−	1.02091i
$329$	−721.675			−2.19354
$330$		0			0
$331$			200.092i			0.604508i	0.953227	+	0.302254i	$0.0977390\pi$
$331$			200.092i			0.604508i	−0.953227	+	0.302254i	$0.902261\pi$
$332$		−	291.074i		−	0.876728i
$333$	27.8540			0.0836456
$334$	117.455			0.351661
$335$		0			0
$336$			130.635i			0.388795i
$337$	−425.356			−1.26218			−0.631092	−	0.775708i	$-0.717393\pi$
$337$	−425.356			−1.26218			−0.631092	+	0.775708i	$0.717393\pi$
$338$	−206.984			−0.612380
$339$	551.021			1.62543
$340$		0			0
$341$		−	58.8138i		−	0.172475i
$342$	11.6690	−	6.15312i	0.0341198	−	0.0179916i
$343$		−	461.543i		−	1.34561i
$344$			156.139i			0.453892i
$345$		0			0
$346$	−176.658			−0.510573
$347$			553.742i			1.59580i	0.602790	+	0.797900i	$0.294056\pi$
$347$			553.742i			1.59580i	−0.602790	+	0.797900i	$0.705944\pi$
$348$			235.773i			0.677508i
$349$	−273.499			−0.783666			−0.391833	−	0.920036i	$-0.628159\pi$
$349$	−273.499			−0.783666			−0.391833	+	0.920036i	$0.628159\pi$
$350$		0			0
$351$	492.350			1.40271
$352$	181.351			0.515202
$353$			545.758i			1.54606i	0.634372	+	0.773028i	$0.281259\pi$
$353$			545.758i			1.54606i	−0.634372	+	0.773028i	$0.718741\pi$
$354$			38.6717i			0.109242i
$355$		0			0
$356$		−	39.3560i		−	0.110551i
$357$	601.197			1.68402
$358$			121.099i			0.338265i
$359$	73.9842			0.206084			0.103042	−	0.994677i	$-0.467142\pi$
$359$	73.9842			0.206084			0.103042	+	0.994677i	$0.467142\pi$
$360$		0			0
$361$	203.922	−	297.887i	0.564881	−	0.825173i
$362$			420.624i			1.16194i
$363$	234.834			0.646925
$364$		−	421.466i		−	1.15787i
$365$		0			0
$366$	−149.903			−0.409570
$367$		−	397.670i		−	1.08357i	−0.840517	−	0.541785i	$-0.817749\pi$
$367$		−	397.670i		−	1.08357i	0.840517	−	0.541785i	$-0.182251\pi$
$368$		−	5.87843i		−	0.0159740i
$369$			19.5202i			0.0529003i
$370$		0			0
$371$		−	779.572i		−	2.10127i
$372$			54.5194i			0.146558i
$373$	−284.800			−0.763538			−0.381769	−	0.924258i	$-0.624685\pi$
$373$	−284.800			−0.763538			−0.381769	+	0.924258i	$0.624685\pi$
$374$			157.365i			0.420761i
$375$		0			0
$376$			521.468i			1.38688i
$377$			710.813i			1.88545i
$378$		−	456.404i		−	1.20742i
$379$			543.499i			1.43403i	0.697055	+	0.717017i	$0.254493\pi$
$379$			543.499i			1.43403i	−0.697055	+	0.717017i	$0.745507\pi$
$380$		0			0
$381$	36.6670			0.0962388
$382$	9.56690			0.0250442
$383$	598.496			1.56265			0.781326	−	0.624123i	$-0.214544\pi$
$383$	598.496			1.56265			0.781326	+	0.624123i	$0.214544\pi$
$384$	−105.412			−0.274511
$385$		0			0
$386$	331.815			0.859624
$387$		−	9.10193i		−	0.0235192i
$388$	−329.943			−0.850369
$389$	−714.756			−1.83742			−0.918709	−	0.394935i	$-0.870767\pi$
$389$	−714.756			−1.83742			−0.918709	+	0.394935i	$0.870767\pi$
$390$		0			0
$391$	−27.0531			−0.0691896
$392$	−748.494			−1.90942
$393$	−281.947			−0.717421
$394$			41.3478i			0.104944i
$395$		0			0
$396$	−6.35238			−0.0160414
$397$			375.385i			0.945555i	0.881182	+	0.472777i	$0.156749\pi$
$397$			375.385i			0.945555i	−0.881182	+	0.472777i	$0.843251\pi$
$398$	−92.3068			−0.231927
$399$	−302.950	−	574.524i	−0.759274	−	1.43991i
$400$		0			0
$401$		−	261.219i		−	0.651419i	−0.945470	−	0.325709i	$-0.894397\pi$
$401$		−	261.219i		−	0.651419i	0.945470	−	0.325709i	$-0.105603\pi$
$402$	−10.4853			−0.0260827
$403$			164.367i			0.407858i
$404$	115.695			0.286375
$405$		0			0
$406$	658.918			1.62295
$407$	358.966			0.881982
$408$		−	434.413i		−	1.06474i
$409$			723.501i			1.76895i	0.466587	+	0.884475i	$0.345484\pi$
$409$			723.501i			1.76895i	−0.466587	+	0.884475i	$0.654516\pi$
$410$		0			0
$411$			639.272i			1.55541i
$412$	−45.5705			−0.110608
$413$	−110.509			−0.267575
$414$			1.06803i			0.00257978i
$415$		0			0
$416$	−506.821			−1.21832
$417$	−550.069			−1.31911
$418$	150.383	−	79.2980i	0.359768	−	0.189708i
$419$	667.084			1.59209			0.796043	−	0.605240i	$-0.206923\pi$
$419$	667.084			1.59209			0.796043	+	0.605240i	$0.206923\pi$
$420$		0			0
$421$		−	358.717i		−	0.852059i	−0.904709	−	0.426030i	$-0.859912\pi$
$421$		−	358.717i		−	0.852059i	0.904709	−	0.426030i	$-0.140088\pi$
$422$			63.9123i			0.151451i
$423$		−	30.3984i		−	0.0718638i
$424$	−563.303			−1.32854
$425$		0			0
$426$		−	220.746i		−	0.518183i
$427$		−	428.363i		−	1.00319i
$428$	139.090			0.324978
$429$	329.968			0.769156
$430$		0			0
$431$		−	742.801i		−	1.72344i	−0.507387	−	0.861718i	$-0.669389\pi$
$431$		−	742.801i		−	1.72344i	0.507387	−	0.861718i	$-0.330611\pi$
$432$	−105.813			−0.244937
$433$	802.697			1.85380			0.926902	−	0.375304i	$-0.122462\pi$
$433$	802.697			1.85380			0.926902	+	0.375304i	$0.122462\pi$
$434$	152.366			0.351075
$435$		0			0
$436$		−	160.862i		−	0.368951i
$437$	13.6324	+	25.8529i	0.0311955	+	0.0591599i
$438$		−	474.660i		−	1.08370i
$439$			89.6015i			0.204104i	0.994779	+	0.102052i	$0.0325408\pi$
$439$			89.6015i			0.204104i	−0.994779	+	0.102052i	$0.967459\pi$
$440$		0			0
$441$	43.6326			0.0989402
$442$		−	439.786i		−	0.994990i
$443$			541.058i			1.22135i	0.791881	+	0.610675i	$0.209102\pi$
$443$			541.058i			1.22135i	−0.791881	+	0.610675i	$0.790898\pi$
$444$	−332.756			−0.749450
$445$		0			0
$446$	−332.520			−0.745561
$447$	517.490			1.15770
$448$			648.982i			1.44862i
$449$			462.639i			1.03038i	0.857077	+	0.515188i	$0.172278\pi$
$449$			462.639i			1.03038i	−0.857077	+	0.515188i	$0.827722\pi$
$450$		0			0
$451$			251.565i			0.557795i
$452$	382.061			0.845269
$453$			27.6965i			0.0611402i
$454$	201.860			0.444626
$455$		0			0
$456$	−415.139	+	218.906i	−0.910393	+	0.480057i
$457$			668.815i			1.46349i	0.681578	+	0.731745i	$0.261294\pi$
$457$			668.815i			1.46349i	−0.681578	+	0.731745i	$0.738706\pi$
$458$	−144.352			−0.315178
$459$			486.961i			1.06092i
$460$		0			0
$461$	317.165			0.687993			0.343996	−	0.938971i	$-0.388219\pi$
$461$	317.165			0.687993			0.343996	+	0.938971i	$0.388219\pi$
$462$		−	305.878i		−	0.662073i
$463$		−	888.893i		−	1.91985i	−0.280250	−	0.959927i	$-0.590417\pi$
$463$		−	888.893i		−	1.91985i	0.280250	−	0.959927i	$-0.409583\pi$
$464$		−	152.763i		−	0.329231i
$465$		0			0
$466$		−	240.228i		−	0.515510i
$467$		−	564.676i		−	1.20916i	−0.796546	−	0.604578i	$-0.793342\pi$
$467$		−	564.676i		−	1.20916i	0.796546	−	0.604578i	$-0.206658\pi$
$468$	17.7530			0.0379337
$469$		−	29.9628i		−	0.0638865i
$470$		0			0
$471$			79.6483i			0.169105i
$472$			79.8513i			0.169176i
$473$		−	117.301i		−	0.247993i
$474$		−	393.437i		−	0.830035i
$475$		0			0
$476$	416.852			0.875740
$477$	32.8371			0.0688409
$478$	273.321			0.571802
$479$	535.419			1.11778			0.558892	−	0.829240i	$-0.311227\pi$
$479$	535.419			1.11778			0.558892	+	0.829240i	$0.311227\pi$
$480$		0			0
$481$	−1003.20			−2.08566
$482$		−	642.914i		−	1.33385i
$483$	52.5846			0.108871
$484$	162.827			0.336419
$485$		0			0
$486$	37.4496			0.0770567
$487$	396.046			0.813236			0.406618	−	0.913598i	$-0.366708\pi$
$487$	396.046			0.813236			0.406618	+	0.913598i	$0.366708\pi$
$488$	−309.527			−0.634276
$489$		−	373.740i		−	0.764294i
$490$		0			0
$491$	14.9147			0.0303763			0.0151881	−	0.999885i	$-0.495165\pi$
$491$	14.9147			0.0303763			0.0151881	+	0.999885i	$0.495165\pi$
$492$		−	233.197i		−	0.473977i
$493$	−703.033			−1.42603
$494$	−420.274	+	221.613i	−0.850757	+	0.448610i
$495$		0			0
$496$		−	35.3246i		−	0.0712189i
$497$	630.806			1.26923
$498$			590.369i			1.18548i
$499$	437.113			0.875978			0.437989	−	0.898980i	$-0.355691\pi$
$499$	437.113			0.875978			0.437989	+	0.898980i	$0.355691\pi$
$500$		0			0
$501$	243.588			0.486203
$502$	485.281			0.966694
$503$			391.808i			0.778942i	0.921039	+	0.389471i	$0.127342\pi$
$503$			391.808i			0.778942i	−0.921039	+	0.389471i	$0.872658\pi$
$504$		−	49.0084i		−	0.0972388i
$505$		0			0
$506$			13.7641i			0.0272019i
$507$	−429.262			−0.846671
$508$	25.4238			0.0500469
$509$		−	83.4859i		−	0.164020i	−0.996632	−	0.0820098i	$-0.973866\pi$
$509$		−	83.4859i		−	0.164020i	0.996632	−	0.0820098i	$-0.0261339\pi$
$510$		0			0
$511$	1356.39			2.65439
$512$	238.383			0.465591
$513$	465.356	−	245.385i	0.907126	−	0.478334i
$514$	402.354			0.782791
$515$		0			0
$516$			108.736i			0.210728i
$517$		−	391.757i		−	0.757751i
$518$			929.959i			1.79529i
$519$	−366.369			−0.705914
$520$		0			0
$521$			561.658i			1.07804i	0.842294	+	0.539019i	$0.181205\pi$
$521$			561.658i			1.07804i	−0.842294	+	0.539019i	$0.818795\pi$
$522$			27.7549i			0.0531704i
$523$	−941.358			−1.79992			−0.899960	−	0.435973i	$-0.856404\pi$
$523$	−941.358			−1.79992			−0.899960	+	0.435973i	$0.856404\pi$
$524$	−195.493			−0.373079
$525$		0			0
$526$		−	286.164i		−	0.544038i
$527$	−162.567			−0.308477
$528$	−70.9146			−0.134308
$529$	526.634			0.995527
$530$		0			0
$531$		−	4.65484i		−	0.00876618i
$532$	−210.057	−	398.358i	−0.394844	−	0.748793i
$533$		−	703.048i		−	1.31904i
$534$			79.8236i			0.149482i
$535$		0			0
$536$	−21.6505			−0.0403927
$537$			251.145i			0.467682i
$538$			591.962i			1.10030i
$539$	562.313			1.04325
$540$		0			0
$541$	890.685			1.64637			0.823184	−	0.567774i	$-0.192195\pi$
$541$	890.685			1.64637			0.823184	+	0.567774i	$0.192195\pi$
$542$	178.611			0.329541
$543$			872.326i			1.60649i
$544$		−	501.273i		−	0.921457i
$545$		0			0
$546$			854.835i			1.56563i
$547$	−374.919			−0.685410			−0.342705	−	0.939443i	$-0.611343\pi$
$547$	−374.919			−0.685410			−0.342705	+	0.939443i	$0.611343\pi$
$548$			443.252i			0.808855i
$549$	18.0435			0.0328662
$550$		0			0
$551$	354.267	+	671.841i	0.642952	+	1.21931i
$552$		−	37.9966i		−	0.0688344i
$553$	1124.29			2.03307
$554$		−	481.740i		−	0.869568i
$555$		0			0
$556$	−381.402			−0.685974
$557$		−	93.1828i		−	0.167294i	−0.996495	−	0.0836470i	$-0.973343\pi$
$557$		−	93.1828i		−	0.167294i	0.996495	−	0.0836470i	$-0.0266568\pi$
$558$			6.41798i			0.0115018i
$559$			327.819i			0.586438i
$560$		0			0
$561$			326.356i			0.581740i
$562$			213.679i			0.380213i
$563$	−526.142			−0.934533			−0.467267	−	0.884116i	$-0.654761\pi$
$563$	−526.142			−0.934533			−0.467267	+	0.884116i	$0.654761\pi$
$564$			363.152i			0.643887i
$565$		0			0
$566$			338.899i			0.598762i
$567$		−	894.451i		−	1.57752i
$568$		−	455.807i		−	0.802478i
$569$		−	320.890i		−	0.563955i	−0.959421	−	0.281977i	$-0.909010\pi$
$569$		−	320.890i		−	0.563955i	0.959421	−	0.281977i	$-0.0909903\pi$
$570$		0			0
$571$	649.237			1.13702			0.568508	−	0.822677i	$-0.307521\pi$
$571$	649.237			1.13702			0.568508	+	0.822677i	$0.307521\pi$
$572$	228.790			0.399983
$573$	19.8407			0.0346259
$574$	−651.720			−1.13540
$575$		0			0
$576$	−27.3364			−0.0474591
$577$			49.2538i			0.0853618i	0.999089	+	0.0426809i	$0.0135899\pi$
$577$			49.2538i			0.0853618i	−0.999089	+	0.0426809i	$0.986410\pi$
$578$	28.5441			0.0493842
$579$	688.146			1.18851
$580$		0			0
$581$	−1687.04			−2.90369
$582$	669.205			1.14984
$583$	423.186			0.725877
$584$		−	980.102i		−	1.67826i
$585$		0			0
$586$	9.91968			0.0169278
$587$			194.238i			0.330900i	0.986218	+	0.165450i	$0.0529076\pi$
$587$			194.238i			0.330900i	−0.986218	+	0.165450i	$0.947092\pi$
$588$	−521.254			−0.886487
$589$	81.9197	+	155.355i	0.139083	+	0.263760i
$590$		0			0
$591$			85.7507i			0.145094i
$592$	215.601			0.364192
$593$		−	284.628i		−	0.479980i	−0.970775	−	0.239990i	$-0.922856\pi$
$593$		−	284.628i		−	0.479980i	0.970775	−	0.239990i	$-0.0771442\pi$
$594$	247.757			0.417099
$595$		0			0
$596$	358.813			0.602035
$597$	−191.434			−0.320660
$598$		−	38.4665i		−	0.0643253i
$599$			723.173i			1.20730i	0.797249	+	0.603650i	$0.206288\pi$
$599$			723.173i			1.20730i	−0.797249	+	0.603650i	$0.793712\pi$
$600$		0			0
$601$			194.647i			0.323872i	0.986801	+	0.161936i	$0.0517739\pi$
$601$			194.647i			0.323872i	−0.986801	+	0.161936i	$0.948226\pi$
$602$	303.886			0.504793
$603$	1.26209			0.00209302
$604$			19.2039i			0.0317946i
$605$		0			0
$606$	−234.658			−0.387225
$607$	69.2294			0.114052			0.0570259	−	0.998373i	$-0.481838\pi$
$607$	69.2294			0.114052			0.0570259	+	0.998373i	$0.481838\pi$
$608$	−479.033	+	252.598i	−0.787883	+	0.415456i
$609$	1366.52			2.24388
$610$		0			0
$611$			1094.84i			1.79188i
$612$			17.5586i			0.0286906i
$613$		−	220.182i		−	0.359187i	−0.983741	−	0.179593i	$-0.942522\pi$
$613$		−	220.182i		−	0.359187i	0.983741	−	0.179593i	$-0.0574782\pi$
$614$	62.9147			0.102467
$615$		0			0
$616$		−	631.592i		−	1.02531i
$617$		−	399.663i		−	0.647753i	−0.946099	−	0.323876i	$-0.895014\pi$
$617$		−	399.663i		−	0.647753i	0.946099	−	0.323876i	$-0.104986\pi$
$618$	92.4281			0.149560
$619$	−1061.05			−1.71413			−0.857067	−	0.515206i	$-0.827716\pi$
$619$	−1061.05			−1.71413			−0.857067	+	0.515206i	$0.827716\pi$
$620$		0			0
$621$			42.5928i			0.0685874i
$622$	155.651			0.250243
$623$	−228.105			−0.366139
$624$	198.185			0.317603
$625$		0			0
$626$			185.441i			0.296232i
$627$	311.877	−	164.455i	0.497411	−	0.262289i
$628$			55.2258i			0.0879392i
$629$		−	992.220i		−	1.57746i
$630$		0			0
$631$	374.486			0.593480			0.296740	−	0.954958i	$-0.404101\pi$
$631$	374.486			0.593480			0.296740	+	0.954958i	$0.404101\pi$
$632$		−	812.388i		−	1.28542i
$633$			132.547i			0.209395i
$634$	435.237			0.686494
$635$		0			0
$636$	−392.286			−0.616803
$637$	−1571.49			−2.46702
$638$			357.690i			0.560643i
$639$			26.5708i			0.0415818i
$640$		0			0
$641$		−	1147.29i		−	1.78984i	−0.446227	−	0.894920i	$-0.647233\pi$
$641$		−	1147.29i		−	1.78984i	0.446227	−	0.894920i	$-0.352767\pi$
$642$	−282.109			−0.439422
$643$		−	824.922i		−	1.28293i	−0.767154	−	0.641463i	$-0.778328\pi$
$643$		−	824.922i		−	1.28293i	0.767154	−	0.641463i	$-0.221672\pi$
$644$	36.4606			0.0566159
$645$		0			0
$646$	−219.188	−	415.674i	−0.339300	−	0.643457i
$647$		−	153.175i		−	0.236746i	−0.992969	−	0.118373i	$-0.962232\pi$
$647$		−	153.175i		−	0.236746i	0.992969	−	0.118373i	$-0.0377679\pi$
$648$	−646.312			−0.997395
$649$		−	59.9890i		−	0.0924329i
$650$		0			0
$651$	315.991			0.485393
$652$		−	259.140i		−	0.397454i
$653$		−	1120.33i		−	1.71567i	−0.513928	−	0.857833i	$-0.671810\pi$
$653$		−	1120.33i		−	1.71567i	0.513928	−	0.857833i	$-0.328190\pi$
$654$			326.268i			0.498881i
$655$		0			0
$656$			151.095i			0.230327i
$657$			57.1339i			0.0869618i
$658$	1014.91			1.54242
$659$			875.929i			1.32918i	0.747208	+	0.664590i	$0.231394\pi$
$659$			875.929i			1.32918i	−0.747208	+	0.664590i	$0.768606\pi$
$660$		0			0
$661$		−	283.488i		−	0.428878i	−0.976737	−	0.214439i	$-0.931208\pi$
$661$		−	283.488i		−	0.428878i	0.976737	−	0.214439i	$-0.0687923\pi$
$662$		−	281.394i		−	0.425067i
$663$		−	912.066i		−	1.37566i
$664$			1219.02i			1.83588i
$665$		0			0
$666$	−39.1717			−0.0588164
$667$	−61.4918			−0.0921917
$668$	168.897			0.252839
$669$	−689.609			−1.03081
$670$		0			0
$671$	232.535			0.346549
$672$			974.350i			1.44993i
$673$	−373.637			−0.555182			−0.277591	−	0.960699i	$-0.589536\pi$
$673$	−373.637			−0.555182			−0.277591	+	0.960699i	$0.589536\pi$
$674$	598.188			0.887520
$675$		0			0
$676$	−297.638			−0.440292
$677$	−215.956			−0.318990			−0.159495	−	0.987199i	$-0.550987\pi$
$677$	−215.956			−0.318990			−0.159495	+	0.987199i	$0.550987\pi$
$678$	−774.913			−1.14294
$679$			1912.33i			2.81639i
$680$		0			0
$681$	418.635			0.614735
$682$			82.7113i			0.121278i
$683$	469.374			0.687224			0.343612	−	0.939112i	$-0.388350\pi$
$683$	469.374			0.687224			0.343612	+	0.939112i	$0.388350\pi$
$684$	16.7796	−	8.84801i	0.0245316	−	0.0129357i
$685$		0			0
$686$			649.079i			0.946179i
$687$	−299.369			−0.435762
$688$		−	70.4527i		−	0.102402i
$689$	−1182.68			−1.71651
$690$		0			0
$691$	−80.4567			−0.116435			−0.0582176	−	0.998304i	$-0.518542\pi$
$691$	−80.4567			−0.116435			−0.0582176	+	0.998304i	$0.518542\pi$
$692$	−254.030			−0.367095
$693$			36.8180i			0.0531284i
$694$		−	778.741i		−	1.12211i
$695$		0			0
$696$		−	987.421i		−	1.41871i
$697$	695.352			0.997636
$698$	384.629			0.551044
$699$		−	498.205i		−	0.712740i
$700$		0			0
$701$	144.654			0.206353			0.103177	−	0.994663i	$-0.467099\pi$
$701$	144.654			0.206353			0.103177	+	0.994663i	$0.467099\pi$
$702$	−692.404			−0.986330
$703$	−948.199	+	499.992i	−1.34879	+	0.711226i
$704$	−352.296			−0.500421
$705$		0			0
$706$		−	767.512i		−	1.08713i
$707$		−	670.562i		−	0.948461i
$708$			55.6088i			0.0785435i
$709$	108.558			0.153115			0.0765573	−	0.997065i	$-0.475607\pi$
$709$	108.558			0.153115			0.0765573	+	0.997065i	$0.475607\pi$
$710$		0			0
$711$			47.3573i			0.0666066i
$712$			164.824i			0.231494i
$713$	−14.2192			−0.0199428
$714$	−845.477			−1.18414
$715$		0			0
$716$			174.137i			0.243208i
$717$	566.837			0.790568
$718$	−104.046			−0.144911
$719$	−804.887			−1.11945			−0.559727	−	0.828677i	$-0.689094\pi$
$719$	−804.887			−1.11945			−0.559727	+	0.828677i	$0.689094\pi$
$720$		0			0
$721$			264.124i			0.366329i
$722$	−286.780	+	418.926i	−0.397202	+	0.580230i
$723$		−	1333.33i		−	1.84416i
$724$			604.845i			0.835421i
$725$		0			0
$726$	−330.252			−0.454893
$727$		−	242.633i		−	0.333746i	−0.985978	−	0.166873i	$-0.946633\pi$
$727$		−	242.633i		−	0.333746i	0.985978	−	0.166873i	$-0.0533670\pi$
$728$			1765.11i			2.42460i
$729$	764.482			1.04867
$730$		0			0
$731$	−324.231			−0.443544
$732$	−215.556			−0.294475
$733$			577.444i			0.787782i	0.919157	+	0.393891i	$0.128871\pi$
$733$			577.444i			0.787782i	−0.919157	+	0.393891i	$0.871129\pi$
$734$			559.253i			0.761925i
$735$		0			0
$736$		−	43.8446i		−	0.0595715i
$737$	16.2651			0.0220694
$738$		−	27.4517i		−	0.0371975i
$739$	482.046			0.652295			0.326148	−	0.945319i	$-0.394249\pi$
$739$	482.046			0.652295			0.326148	+	0.945319i	$0.394249\pi$
$740$		0			0
$741$	−871.600	+	459.601i	−1.17625	+	0.620244i
$742$			1096.33i			1.47753i
$743$	−713.547			−0.960359			−0.480179	−	0.877170i	$-0.659428\pi$
$743$	−713.547			−0.960359			−0.480179	+	0.877170i	$0.659428\pi$
$744$		−	228.328i		−	0.306893i
$745$		0			0
$746$	400.521			0.536891
$747$		−	71.0616i		−	0.0951293i
$748$			226.286i			0.302521i
$749$		−	806.158i		−	1.07631i
$750$		0			0
$751$			1049.66i			1.39768i	0.715276	+	0.698842i	$0.246301\pi$
$751$			1049.66i			1.39768i	−0.715276	+	0.698842i	$0.753699\pi$
$752$		−	235.296i		−	0.312894i
$753$	1006.42			1.33654
$754$		−	999.634i		−	1.32577i
$755$		0			0
$756$		−	656.296i		−	0.868117i
$757$			87.6677i			0.115809i	0.998322	+	0.0579047i	$0.0184420\pi$
$757$			87.6677i			0.115809i	−0.998322	+	0.0579047i	$0.981558\pi$
$758$		−	764.336i		−	1.00836i
$759$			28.5453i			0.0376090i
$760$		0			0
$761$	706.869			0.928869			0.464435	−	0.885607i	$-0.346258\pi$
$761$	706.869			0.928869			0.464435	+	0.885607i	$0.346258\pi$
$762$	−51.5657			−0.0676715
$763$	−932.348			−1.22195
$764$	13.7569			0.0180064
$765$		0			0
$766$	−841.679			−1.09880
$767$			167.651i			0.218580i
$768$	794.202			1.03412
$769$	69.6679			0.0905955			0.0452977	−	0.998974i	$-0.485576\pi$
$769$	69.6679			0.0905955			0.0452977	+	0.998974i	$0.485576\pi$
$770$		0			0
$771$	834.437			1.08228
$772$	477.140			0.618057
$773$	1236.62			1.59977			0.799886	−	0.600153i	$-0.204893\pi$
$773$	1236.62			1.59977			0.799886	+	0.600153i	$0.204893\pi$
$774$			12.8003i			0.0165378i
$775$		0			0
$776$	1381.81			1.78068
$777$			1928.63i			2.48215i
$778$	1005.18			1.29200
$779$	−350.397	−	664.502i	−0.449803	−	0.853019i
$780$		0			0
$781$			342.429i			0.438450i
$782$	38.0455			0.0486515
$783$			1106.86i			1.41362i
$784$	337.735			0.430784
$785$		0			0
$786$	396.508			0.504463
$787$	−962.806			−1.22339			−0.611693	−	0.791095i	$-0.709511\pi$
$787$	−962.806			−1.22339			−0.611693	+	0.791095i	$0.709511\pi$
$788$			59.4570i			0.0754530i
$789$		−	593.472i		−	0.752182i
$790$		0			0
$791$		−	2214.40i		−	2.79950i
$792$	26.6039			0.0335908
$793$	−649.863			−0.819499
$794$		−	527.913i		−	0.664878i
$795$		0			0
$796$	−132.735			−0.166752
$797$	412.057			0.517010			0.258505	−	0.966010i	$-0.416770\pi$
$797$	412.057			0.517010			0.258505	+	0.966010i	$0.416770\pi$
$798$	426.047	+	807.966i	0.533893	+	1.01249i
$799$	−1082.86			−1.35527
$800$		0			0
$801$		−	9.60822i		−	0.0119953i
$802$			367.358i			0.458053i
$803$			736.310i			0.916949i
$804$	−15.0775			−0.0187531
$805$		0			0
$806$		−	231.153i		−	0.286790i
$807$			1227.66i			1.52126i
$808$	−484.535			−0.599671
$809$	−445.562			−0.550756			−0.275378	−	0.961336i	$-0.588803\pi$
$809$	−445.562			−0.550756			−0.275378	+	0.961336i	$0.588803\pi$
$810$		0			0
$811$			558.857i			0.689096i	0.938769	+	0.344548i	$0.111968\pi$
$811$			558.857i			0.689096i	−0.938769	+	0.344548i	$0.888032\pi$
$812$	947.505			1.16688
$813$	370.419			0.455620
$814$	−504.823			−0.620176
$815$		0			0
$816$			196.015i			0.240215i
$817$	163.384	+	309.846i	0.199980	+	0.379248i
$818$		−	1017.48i		−	1.24386i
$819$		−	102.895i		−	0.125635i
$820$		0			0
$821$	−1469.94			−1.79043			−0.895215	−	0.445634i	$-0.852978\pi$
$821$	−1469.94			−1.79043			−0.895215	+	0.445634i	$0.852978\pi$
$822$		−	899.024i		−	1.09370i
$823$		−	165.659i		−	0.201287i	−0.994923	−	0.100643i	$-0.967910\pi$
$823$		−	165.659i		−	0.201287i	0.994923	−	0.100643i	$-0.0320901\pi$
$824$	190.850			0.231614
$825$		0			0
$826$	155.411			0.188149
$827$	−503.503			−0.608831			−0.304415	−	0.952539i	$-0.598461\pi$
$827$	−503.503			−0.608831			−0.304415	+	0.952539i	$0.598461\pi$
$828$			1.53579i			0.00185482i
$829$			156.053i			0.188242i	0.995561	+	0.0941211i	$0.0300041\pi$
$829$			156.053i			0.188242i	−0.995561	+	0.0941211i	$0.969996\pi$
$830$		0			0
$831$		−	999.075i		−	1.20226i
$832$	984.560			1.18337
$833$		−	1554.29i		−	1.86589i
$834$	773.575			0.927548
$835$		0			0
$836$	216.246	−	114.028i	0.258668	−	0.136397i
$837$			255.948i			0.305792i
$838$	−938.136			−1.11949
$839$		−	543.442i		−	0.647725i	−0.946104	−	0.323863i	$-0.895018\pi$
$839$		−	543.442i		−	0.647725i	0.946104	−	0.323863i	$-0.104982\pi$
$840$		0			0
$841$	−756.994			−0.900112
$842$			504.472i			0.599136i
$843$			443.147i			0.525678i
$844$			91.9040i			0.108891i
$845$		0			0
$846$			42.7500i			0.0505319i
$847$		−	943.732i		−	1.11421i
$848$	254.173			0.299732
$849$			702.838i			0.827842i
$850$		0			0
$851$		−	86.7861i		−	0.101981i
$852$		−	317.426i		−	0.372566i
$853$		−	788.764i		−	0.924694i	−0.886699	−	0.462347i	$-0.847007\pi$
$853$		−	788.764i		−	0.924694i	0.886699	−	0.462347i	$-0.152993\pi$
$854$			602.418i			0.705407i
$855$		0			0
$856$	−582.514			−0.680506
$857$	−387.309			−0.451935			−0.225968	−	0.974135i	$-0.572554\pi$
$857$	−387.309			−0.451935			−0.225968	+	0.974135i	$0.572554\pi$
$858$	−464.042			−0.540842
$859$	47.4552			0.0552447			0.0276224	−	0.999618i	$-0.491206\pi$
$859$	47.4552			0.0552447			0.0276224	+	0.999618i	$0.491206\pi$
$860$		0			0
$861$	−1351.59			−1.56979
$862$			1044.62i			1.21185i
$863$	852.591			0.987938			0.493969	−	0.869479i	$-0.335546\pi$
$863$	852.591			0.987938			0.493969	+	0.869479i	$0.335546\pi$
$864$	−789.209			−0.913437
$865$		0			0
$866$	−1128.85			−1.30352
$867$	59.1972			0.0682782
$868$	219.099			0.252418
$869$			610.313i			0.702317i
$870$		0			0
$871$	−45.4560			−0.0521883
$872$			673.695i			0.772587i
$873$	−80.5511			−0.0922693
$874$	−19.1716	−	36.3575i	−0.0219355	−	0.0415990i
$875$		0			0
$876$		−	682.547i		−	0.779163i
$877$	−832.003			−0.948692			−0.474346	−	0.880339i	$-0.657315\pi$
$877$	−832.003			−0.948692			−0.474346	+	0.880339i	$0.657315\pi$
$878$		−	126.009i		−	0.143518i
$879$	20.5723			0.0234042
$880$		0			0
$881$	−450.264			−0.511083			−0.255541	−	0.966798i	$-0.582254\pi$
$881$	−450.264			−0.511083			−0.255541	+	0.966798i	$0.582254\pi$
$882$	−61.3616			−0.0695710
$883$			1035.05i			1.17219i	0.810241	+	0.586096i	$0.199336\pi$
$883$			1035.05i			1.17219i	−0.810241	+	0.586096i	$0.800664\pi$
$884$		−	632.399i		−	0.715384i
$885$		0			0
$886$		−	760.903i		−	0.858807i
$887$	296.394			0.334153			0.167077	−	0.985944i	$-0.446567\pi$
$887$	296.394			0.334153			0.167077	+	0.985944i	$0.446567\pi$
$888$	1393.59			1.56936
$889$		−	147.355i		−	0.165753i
$890$		0			0
$891$	485.548			0.544947
$892$	−478.155			−0.536048
$893$	545.665	+	1034.81i	0.611047	+	1.15881i
$894$	−727.759			−0.814048
$895$		0			0
$896$			423.623i			0.472794i
$897$		−	79.7752i		−	0.0889356i
$898$		−	650.620i		−	0.724521i
$899$	−369.516			−0.411030
$900$		0			0
$901$		−	1169.73i		−	1.29826i
$902$		−	353.782i		−	0.392220i
$903$	630.224			0.697922
$904$	−1600.08			−1.77000
$905$		0			0
$906$		−	38.9503i		−	0.0429915i
$907$	−775.691			−0.855227			−0.427614	−	0.903962i	$-0.640646\pi$
$907$	−775.691			−0.855227			−0.427614	+	0.903962i	$0.640646\pi$
$908$	290.269			0.319679
$909$	28.2454			0.0310730
$910$		0			0
$911$		−	1211.86i		−	1.33025i	−0.746731	−	0.665126i	$-0.768378\pi$
$911$		−	1211.86i		−	1.33025i	0.746731	−	0.665126i	$-0.231622\pi$
$912$	187.319	−	98.7745i	0.205393	−	0.108305i
$913$		−	915.802i		−	1.00307i
$914$		−	940.570i		−	1.02907i
$915$		0			0
$916$	−207.573			−0.226609
$917$			1133.07i			1.23562i
$918$		−	684.824i		−	0.745996i
$919$	1413.02			1.53757			0.768784	−	0.639509i	$-0.220862\pi$
$919$	1413.02			1.53757			0.768784	+	0.639509i	$0.220862\pi$
$920$		0			0
$921$	130.478			0.141670
$922$	−446.036			−0.483770
$923$		−	956.985i		−	1.03682i
$924$		−	439.843i		−	0.476021i
$925$		0			0
$926$			1250.07i			1.34997i
$927$	−11.1254			−0.0120015
$928$		−	1139.39i		−	1.22780i
$929$	775.131			0.834371			0.417185	−	0.908821i	$-0.363017\pi$
$929$	775.131			0.834371			0.417185	+	0.908821i	$0.363017\pi$
$930$		0			0
$931$	−1485.33	+	783.226i	−1.59541	+	0.841274i
$932$		−	345.441i		−	0.370644i
$933$	322.803			0.345984
$934$			794.117i			0.850233i
$935$		0			0
$936$	−74.3498			−0.0794335
$937$			808.931i			0.863320i	0.902036	+	0.431660i	$0.142072\pi$
$937$			808.931i			0.863320i	−0.902036	+	0.431660i	$0.857928\pi$
$938$			42.1374i			0.0449226i
$939$			384.584i			0.409568i
$940$		0			0
$941$			1692.92i			1.79906i	0.436856	+	0.899532i	$0.356092\pi$
$941$			1692.92i			1.79906i	−0.436856	+	0.899532i	$0.643908\pi$
$942$		−	112.011i		−	0.118908i
$943$	60.8201			0.0644964
$944$		−	36.0304i		−	0.0381678i
$945$		0			0
$946$			164.963i			0.174379i
$947$			1511.42i			1.59601i	0.602654	+	0.798003i	$0.294110\pi$
$947$			1511.42i			1.59601i	−0.602654	+	0.798003i	$0.705890\pi$
$948$		−	565.750i		−	0.596783i
$949$		−	2057.76i		−	2.16835i
$950$		0			0
$951$	902.632			0.949140
$952$	−1745.78			−1.83381
$953$	−305.820			−0.320902			−0.160451	−	0.987044i	$-0.551295\pi$
$953$	−305.820			−0.320902			−0.160451	+	0.987044i	$0.551295\pi$
$954$	−46.1796			−0.0484063
$955$		0			0
$956$	393.028			0.411117
$957$			741.808i			0.775139i
$958$	−752.972			−0.785983
$959$	2569.06			2.67889
$960$		0			0
$961$	875.554			0.911087
$962$	1410.83			1.46656
$963$	33.9570			0.0352617
$964$		−	924.491i		−	0.959016i
$965$		0			0
$966$	−73.9510			−0.0765538
$967$		−	188.075i		−	0.194493i	−0.995260	−	0.0972466i	$-0.968996\pi$
$967$		−	188.075i		−	0.194493i	0.995260	−	0.0972466i	$-0.0310035\pi$
$968$	−681.922			−0.704465
$969$	−454.570	−	862.059i	−0.469113	−	0.889638i
$970$		0			0
$971$			1108.38i			1.14148i	0.821129	+	0.570742i	$0.193345\pi$
$971$			1108.38i			1.14148i	−0.821129	+	0.570742i	$0.806655\pi$
$972$	53.8514			0.0554027
$973$			2210.58i			2.27192i
$974$	−556.969			−0.571836
$975$		0			0
$976$	139.664			0.143099
$977$	559.011			0.572171			0.286086	−	0.958204i	$-0.407646\pi$
$977$	559.011			0.572171			0.286086	+	0.958204i	$0.407646\pi$
$978$			525.599i			0.537423i
$979$		−	123.825i		−	0.126481i
$980$		0			0
$981$		−	39.2723i		−	0.0400330i
$982$	−20.9750			−0.0213594
$983$	−1006.84			−1.02425			−0.512126	−	0.858910i	$-0.671142\pi$
$983$	−1006.84			−1.02425			−0.512126	+	0.858910i	$0.671142\pi$
$984$			976.634i			0.992514i
$985$		0			0
$986$	988.692			1.00273
$987$	2104.81			2.13253
$988$	−604.342	+	318.674i	−0.611682	+	0.322544i
$989$	−28.3593			−0.0286748
$990$		0			0
$991$			999.726i			1.00881i	0.863468	+	0.504403i	$0.168287\pi$
$991$			999.726i			1.00881i	−0.863468	+	0.504403i	$0.831713\pi$
$992$		−	263.470i		−	0.265595i
$993$		−	583.580i		−	0.587693i
$994$	−887.117			−0.892472
$995$		0			0
$996$			848.933i			0.852342i
$997$		−	1250.46i		−	1.25422i	−0.778929	−	0.627112i	$-0.784237\pi$
$997$		−	1250.46i		−	1.25422i	0.778929	−	0.627112i	$-0.215763\pi$
$998$	−614.722			−0.615954
$999$	−1562.16			−1.56373

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	475.3.d.d.474.12		28
5.2	odd	4		475.3.c.h.151.6	✓	14
5.3	odd	4		475.3.c.i.151.9	yes	14
5.4	even	2	inner	475.3.d.d.474.17		28
19.18	odd	2	inner	475.3.d.d.474.18		28
95.18	even	4		475.3.c.i.151.6	yes	14
95.37	even	4		475.3.c.h.151.9	yes	14
95.94	odd	2	inner	475.3.d.d.474.11		28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
475.3.c.h.151.6	✓	14	5.2	odd	4
475.3.c.h.151.9	yes	14	95.37	even	4
475.3.c.i.151.6	yes	14	95.18	even	4
475.3.c.i.151.9	yes	14	5.3	odd	4
475.3.d.d.474.11		28	95.94	odd	2	inner
475.3.d.d.474.12		28	1.1	even	1	trivial
475.3.d.d.474.17		28	5.4	even	2	inner
475.3.d.d.474.18		28	19.18	odd	2	inner

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)

\(f(q)\)	\(=\)	\(q-1.40632 q^{2} -2.91656 q^{3} -2.02225 q^{4} +4.10162 q^{6} +11.7208i q^{7} +8.46924 q^{8} -0.493705 q^{9} +O(q^{10})\)	\(q-1.40632 q^{2} -2.91656 q^{3} -2.02225 q^{4} +4.10162 q^{6} +11.7208i q^{7} +8.46924 q^{8} -0.493705 q^{9} -6.36259 q^{11} +5.89801 q^{12} +17.7815 q^{13} -16.4833i q^{14} -3.82148 q^{16} +17.5868i q^{17} +0.694309 q^{18} +(16.8066 - 8.86223i) q^{19} -34.1845i q^{21} +8.94786 q^{22} +1.53826i q^{23} -24.7010 q^{24} -25.0065 q^{26} +27.6889 q^{27} -23.7025i q^{28} +39.9749i q^{29} +9.24369i q^{31} -28.5027 q^{32} +18.5568 q^{33} -24.7328i q^{34} +0.998396 q^{36} -56.4183 q^{37} +(-23.6355 + 12.4632i) q^{38} -51.8607 q^{39} -39.5382i q^{41} +48.0744i q^{42} +18.4360i q^{43} +12.8668 q^{44} -2.16329i q^{46} +61.5720i q^{47} +11.1456 q^{48} -88.3780 q^{49} -51.2930i q^{51} -35.9587 q^{52} -66.5116 q^{53} -38.9396 q^{54} +99.2665i q^{56} +(-49.0173 + 25.8472i) q^{57} -56.2177i q^{58} +9.42839i q^{59} -36.5472 q^{61} -12.9996i q^{62} -5.78663i q^{63} +55.3700 q^{64} -26.0969 q^{66} -2.55637 q^{67} -35.5650i q^{68} -4.48642i q^{69} -53.8192i q^{71} -4.18130 q^{72} -115.725i q^{73} +79.3424 q^{74} +(-33.9871 + 17.9217i) q^{76} -74.5748i q^{77} +72.9329 q^{78} -95.9222i q^{79} -76.3129 q^{81} +55.6035i q^{82} +143.935i q^{83} +69.1296i q^{84} -25.9270i q^{86} -116.589i q^{87} -53.8863 q^{88} +19.4615i q^{89} +208.414i q^{91} -3.11075i q^{92} -26.9597i q^{93} -86.5902i q^{94} +83.1297 q^{96} +163.156 q^{97} +124.288 q^{98} +3.14124 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q + 56 q^{4} - 8 q^{6} + 72 q^{9}+O(q^{10}) \) 28 * q + 56 * q^4 - 8 * q^6 + 72 * q^9	\( 28 q + 56 q^{4} - 8 q^{6} + 72 q^{9} - 8 q^{11} + 72 q^{16} - 78 q^{19} + 88 q^{24} + 60 q^{26} + 8 q^{36} + 64 q^{39} + 104 q^{44} - 468 q^{49} - 196 q^{54} + 444 q^{61} + 436 q^{64} + 184 q^{66} + 184 q^{74} - 702 q^{76} + 804 q^{81} + 380 q^{96} + 360 q^{99}+O(q^{100}) \) 28 * q + 56 * q^4 - 8 * q^6 + 72 * q^9 - 8 * q^11 + 72 * q^16 - 78 * q^19 + 88 * q^24 + 60 * q^26 + 8 * q^36 + 64 * q^39 + 104 * q^44 - 468 * q^49 - 196 * q^54 + 444 * q^61 + 436 * q^64 + 184 * q^66 + 184 * q^74 - 702 * q^76 + 804 * q^81 + 380 * q^96 + 360 * q^99

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