LMFDB - Embedded newform 475.3.d.d.474.3

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			474.3
Character	$\chi$	$=$	475.474
Dual form			475.3.d.d.474.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.01757 q^{2} -4.83172 q^{3} +5.10575 q^{4} +14.5801 q^{6} -6.15539i q^{7} -3.33669 q^{8} +14.3455 q^{9} +O(q^{10})$	\(q-3.01757 q^{2} -4.83172 q^{3} +5.10575 q^{4} +14.5801 q^{6} -6.15539i q^{7} -3.33669 q^{8} +14.3455 q^{9} +5.09534 q^{11} -24.6696 q^{12} +4.20326 q^{13} +18.5743i q^{14} -10.3543 q^{16} +1.29941i q^{17} -43.2887 q^{18} +(-18.3714 - 4.84663i) q^{19} +29.7411i q^{21} -15.3756 q^{22} +37.1355i q^{23} +16.1219 q^{24} -12.6837 q^{26} -25.8282 q^{27} -31.4279i q^{28} -42.9209i q^{29} +52.1594i q^{31} +44.5916 q^{32} -24.6193 q^{33} -3.92107i q^{34} +73.2448 q^{36} -41.8600 q^{37} +(55.4372 + 14.6251i) q^{38} -20.3090 q^{39} +7.13498i q^{41} -89.7461i q^{42} +59.2867i q^{43} +26.0155 q^{44} -112.059i q^{46} -57.1573i q^{47} +50.0291 q^{48} +11.1112 q^{49} -6.27840i q^{51} +21.4608 q^{52} +61.6167 q^{53} +77.9384 q^{54} +20.5386i q^{56} +(88.7657 + 23.4176i) q^{57} +129.517i q^{58} -54.5223i q^{59} +38.4303 q^{61} -157.395i q^{62} -88.3024i q^{63} -93.1413 q^{64} +74.2904 q^{66} +44.7851 q^{67} +6.63448i q^{68} -179.428i q^{69} -13.0537i q^{71} -47.8666 q^{72} -134.087i q^{73} +126.316 q^{74} +(-93.8001 - 24.7457i) q^{76} -31.3638i q^{77} +61.2839 q^{78} +128.146i q^{79} -4.31527 q^{81} -21.5303i q^{82} +23.2187i q^{83} +151.851i q^{84} -178.902i q^{86} +207.382i q^{87} -17.0015 q^{88} +21.0659i q^{89} -25.8727i q^{91} +189.605i q^{92} -252.020i q^{93} +172.476i q^{94} -215.454 q^{96} +75.9288 q^{97} -33.5288 q^{98} +73.0954 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−3.01757			−1.50879			−0.754393	−	0.656423i	$-0.772069\pi$
$2$	−3.01757			−1.50879			−0.754393	+	0.656423i	$0.772069\pi$
$3$	−4.83172			−1.61057			−0.805287	−	0.592885i	$-0.797989\pi$
$3$	−4.83172			−1.61057			−0.805287	+	0.592885i	$0.797989\pi$
$4$	5.10575			1.27644
$5$		0			0
$5$		0			0
$6$	14.5801			2.43001
$7$		−	6.15539i		−	0.879341i	−0.898159	−	0.439671i	$-0.855095\pi$
$7$		−	6.15539i		−	0.879341i	0.898159	−	0.439671i	$-0.144905\pi$
$8$	−3.33669			−0.417086
$9$	14.3455			1.59395
$10$		0			0
$11$	5.09534			0.463213			0.231606	−	0.972810i	$-0.425602\pi$
$11$	5.09534			0.463213			0.231606	+	0.972810i	$0.425602\pi$
$12$	−24.6696			−2.05580
$13$	4.20326			0.323328			0.161664	−	0.986846i	$-0.448314\pi$
$13$	4.20326			0.323328			0.161664	+	0.986846i	$0.448314\pi$
$14$			18.5743i			1.32674i
$15$		0			0
$16$	−10.3543			−0.647144
$17$			1.29941i			0.0764361i	0.999269	+	0.0382180i	$0.0121681\pi$
$17$			1.29941i			0.0764361i	−0.999269	+	0.0382180i	$0.987832\pi$
$18$	−43.2887			−2.40493
$19$	−18.3714	−	4.84663i	−0.966918	−	0.255086i
$19$	−18.3714	−	4.84663i	−0.966918	−	0.255086i
$20$		0			0
$21$			29.7411i			1.41624i
$22$	−15.3756			−0.698889
$23$			37.1355i			1.61459i	0.590151	+	0.807293i	$0.299068\pi$
$23$			37.1355i			1.61459i	−0.590151	+	0.807293i	$0.700932\pi$
$24$	16.1219			0.671748
$25$		0			0
$26$	−12.6837			−0.487833
$27$	−25.8282			−0.956599
$28$		−	31.4279i		−	1.12242i
$29$		−	42.9209i		−	1.48003i	−0.672590	−	0.740015i	$-0.734818\pi$
$29$		−	42.9209i		−	1.48003i	0.672590	−	0.740015i	$-0.265182\pi$
$30$		0			0
$31$			52.1594i			1.68256i	0.540599	+	0.841280i	$0.318198\pi$
$31$			52.1594i			1.68256i	−0.540599	+	0.841280i	$0.681802\pi$
$32$	44.5916			1.39349
$33$	−24.6193			−0.746038
$34$		−	3.92107i		−	0.115326i
$35$		0			0
$36$	73.2448			2.03458
$37$	−41.8600			−1.13135			−0.565676	−	0.824628i	$-0.691385\pi$
$37$	−41.8600			−1.13135			−0.565676	+	0.824628i	$0.691385\pi$
$38$	55.4372	+	14.6251i	1.45887	+	0.384870i
$39$	−20.3090			−0.520744
$40$		0			0
$41$			7.13498i			0.174024i	0.996207	+	0.0870119i	$0.0277318\pi$
$41$			7.13498i			0.174024i	−0.996207	+	0.0870119i	$0.972268\pi$
$42$		−	89.7461i		−	2.13681i
$43$			59.2867i			1.37876i	0.724400	+	0.689380i	$0.242117\pi$
$43$			59.2867i			1.37876i	−0.724400	+	0.689380i	$0.757883\pi$
$44$	26.0155			0.591262
$45$		0			0
$46$		−	112.059i		−	2.43607i
$47$		−	57.1573i		−	1.21611i	−0.793894	−	0.608057i	$-0.791949\pi$
$47$		−	57.1573i		−	1.21611i	0.793894	−	0.608057i	$-0.208051\pi$
$48$	50.0291			1.04227
$49$	11.1112			0.226759
$50$		0			0
$51$		−	6.27840i		−	0.123106i
$52$	21.4608			0.412708
$53$	61.6167			1.16258			0.581290	−	0.813697i	$-0.302548\pi$
$53$	61.6167			1.16258			0.581290	+	0.813697i	$0.302548\pi$
$54$	77.9384			1.44330
$55$		0			0
$56$			20.5386i			0.366761i
$57$	88.7657	+	23.4176i	1.55729	+	0.410835i
$58$			129.517i			2.23305i
$59$		−	54.5223i		−	0.924106i	−0.886852	−	0.462053i	$-0.847113\pi$
$59$		−	54.5223i		−	0.924106i	0.886852	−	0.462053i	$-0.152887\pi$
$60$		0			0
$61$	38.4303			0.630006			0.315003	−	0.949091i	$-0.397995\pi$
$61$	38.4303			0.630006			0.315003	+	0.949091i	$0.397995\pi$
$62$		−	157.395i		−	2.53863i
$63$		−	88.3024i		−	1.40163i
$64$	−93.1413			−1.45533
$65$		0			0
$66$	74.2904			1.12561
$67$	44.7851			0.668435			0.334217	−	0.942496i	$-0.391528\pi$
$67$	44.7851			0.668435			0.334217	+	0.942496i	$0.391528\pi$
$68$			6.63448i			0.0975659i
$69$		−	179.428i		−	2.60041i
$70$		0			0
$71$		−	13.0537i		−	0.183855i	−0.995766	−	0.0919277i	$-0.970697\pi$
$71$		−	13.0537i		−	0.183855i	0.995766	−	0.0919277i	$-0.0293029\pi$
$72$	−47.8666			−0.664814
$73$		−	134.087i		−	1.83681i	−0.395638	−	0.918406i	$-0.629477\pi$
$73$		−	134.087i		−	1.83681i	0.395638	−	0.918406i	$-0.370523\pi$
$74$	126.316			1.70697
$75$		0			0
$76$	−93.8001	−	24.7457i	−1.23421	−	0.325601i
$77$		−	31.3638i		−	0.407322i
$78$	61.2839			0.785691
$79$			128.146i			1.62210i	0.584977	+	0.811050i	$0.301103\pi$
$79$			128.146i			1.62210i	−0.584977	+	0.811050i	$0.698897\pi$
$80$		0			0
$81$	−4.31527			−0.0532750
$82$		−	21.5303i		−	0.262565i
$83$			23.2187i			0.279743i	0.990170	+	0.139872i	$0.0446690\pi$
$83$			23.2187i			0.279743i	−0.990170	+	0.139872i	$0.955331\pi$
$84$			151.851i			1.80775i
$85$		0			0
$86$		−	178.902i		−	2.08026i
$87$			207.382i			2.38370i
$88$	−17.0015			−0.193199
$89$			21.0659i			0.236695i	0.992972	+	0.118348i	$0.0377597\pi$
$89$			21.0659i			0.236695i	−0.992972	+	0.118348i	$0.962240\pi$
$90$		0			0
$91$		−	25.8727i		−	0.284316i
$92$			189.605i			2.06092i
$93$		−	252.020i		−	2.70989i
$94$			172.476i			1.83486i
$95$		0			0
$96$	−215.454			−2.24432
$97$	75.9288			0.782771			0.391385	−	0.920227i	$-0.371996\pi$
$97$	75.9288			0.782771			0.391385	+	0.920227i	$0.371996\pi$
$98$	−33.5288			−0.342130
$99$	73.0954			0.738337
$100$		0			0
$101$	137.524			1.36162			0.680810	−	0.732460i	$-0.261628\pi$
$101$	137.524			1.36162			0.680810	+	0.732460i	$0.261628\pi$
$102$			18.9455i			0.185741i
$103$	−13.7280			−0.133281			−0.0666407	−	0.997777i	$-0.521228\pi$
$103$	−13.7280			−0.133281			−0.0666407	+	0.997777i	$0.521228\pi$
$104$	−14.0250			−0.134856
$105$		0			0
$106$	−185.933			−1.75408
$107$	64.4608			0.602437			0.301219	−	0.953555i	$-0.402607\pi$
$107$	64.4608			0.602437			0.301219	+	0.953555i	$0.402607\pi$
$108$	−131.872			−1.22104
$109$			46.6388i			0.427879i	0.976847	+	0.213939i	$0.0686295\pi$
$109$			46.6388i			0.427879i	−0.976847	+	0.213939i	$0.931371\pi$
$110$		0			0
$111$	202.256			1.82213
$112$			63.7348i			0.569061i
$113$	−56.7151			−0.501904			−0.250952	−	0.968000i	$-0.580744\pi$
$113$	−56.7151			−0.501904			−0.250952	+	0.968000i	$0.580744\pi$
$114$	−267.857	−	70.6643i	−2.34962	−	0.619862i
$115$		0			0
$116$		−	219.143i		−	1.88917i
$117$	60.2981			0.515368
$118$			164.525i			1.39428i
$119$	7.99839			0.0672134
$120$		0			0
$121$	−95.0375			−0.785434
$122$	−115.966			−0.950544
$123$		−	34.4742i		−	0.280278i
$124$			266.313i			2.14768i
$125$		0			0
$126$			266.459i			2.11475i
$127$	−238.179			−1.87542			−0.937711	−	0.347417i	$-0.887059\pi$
$127$	−238.179			−1.87542			−0.937711	+	0.347417i	$0.887059\pi$
$128$	102.694			0.802299
$129$		−	286.457i		−	2.22060i
$130$		0			0
$131$	−41.0781			−0.313573			−0.156787	−	0.987633i	$-0.550113\pi$
$131$	−41.0781			−0.313573			−0.156787	+	0.987633i	$0.550113\pi$
$132$	−125.700			−0.952271
$133$	−29.8329	+	113.083i	−0.224308	+	0.850251i
$134$	−135.142			−1.00853
$135$		0			0
$136$		−	4.33573i		−	0.0318804i
$137$		−	141.611i		−	1.03366i	−0.856088	−	0.516829i	$-0.827112\pi$
$137$		−	141.611i		−	1.03366i	0.856088	−	0.516829i	$-0.172888\pi$
$138$			541.438i			3.92347i
$139$	133.167			0.958038			0.479019	−	0.877805i	$-0.340993\pi$
$139$	133.167			0.958038			0.479019	+	0.877805i	$0.340993\pi$
$140$		0			0
$141$			276.168i			1.95864i
$142$			39.3906i			0.277398i
$143$	21.4171			0.149770
$144$	−148.538			−1.03152
$145$		0			0
$146$			404.618i			2.77136i
$147$	−53.6861			−0.365211
$148$	−213.727			−1.44410
$149$	275.472			1.84881			0.924403	−	0.381417i	$-0.124564\pi$
$149$	275.472			1.84881			0.924403	+	0.381417i	$0.124564\pi$
$150$		0			0
$151$		−	141.099i		−	0.934430i	−0.884144	−	0.467215i	$-0.845257\pi$
$151$		−	141.099i		−	0.934430i	0.884144	−	0.467215i	$-0.154743\pi$
$152$	61.2998	+	16.1717i	0.403288	+	0.106393i
$153$			18.6408i			0.121835i
$154$			94.6426i			0.614562i
$155$		0			0
$156$	−103.693			−0.664697
$157$		−	134.179i		−	0.854644i	−0.904099	−	0.427322i	$-0.859457\pi$
$157$		−	134.179i		−	0.854644i	0.904099	−	0.427322i	$-0.140543\pi$
$158$		−	386.690i		−	2.44740i
$159$	−297.715			−1.87242
$160$		0			0
$161$	228.583			1.41977
$162$	13.0217			0.0803806
$163$		−	176.426i		−	1.08237i	−0.840905	−	0.541183i	$-0.817976\pi$
$163$		−	176.426i		−	1.08237i	0.840905	−	0.541183i	$-0.182024\pi$
$164$			36.4294i			0.222131i
$165$		0			0
$166$		−	70.0641i		−	0.422073i
$167$	174.834			1.04691			0.523455	−	0.852053i	$-0.324643\pi$
$167$	174.834			1.04691			0.523455	+	0.852053i	$0.324643\pi$
$168$		−	99.2369i		−	0.590696i
$169$	−151.333			−0.895459
$170$		0			0
$171$	−263.548	−	69.5276i	−1.54122	−	0.406594i
$172$			302.703i			1.75990i
$173$	254.288			1.46987			0.734937	−	0.678135i	$-0.237212\pi$
$173$	254.288			1.46987			0.734937	+	0.678135i	$0.237212\pi$
$174$		−	625.790i		−	3.59649i
$175$		0			0
$176$	−52.7587			−0.299765
$177$			263.437i			1.48834i
$178$		−	63.5678i		−	0.357122i
$179$		−	207.084i		−	1.15689i	−0.815720	−	0.578447i	$-0.803659\pi$
$179$		−	207.084i		−	1.15689i	0.815720	−	0.578447i	$-0.196341\pi$
$180$		0			0
$181$		−	137.706i		−	0.760805i	−0.924821	−	0.380402i	$-0.875786\pi$
$181$		−	137.706i		−	0.760805i	0.924821	−	0.380402i	$-0.124214\pi$
$182$			78.0729i			0.428972i
$183$	−185.685			−1.01467
$184$		−	123.909i		−	0.673421i
$185$		0			0
$186$			760.488i			4.08864i
$187$			6.62095i			0.0354061i
$188$		−	291.831i		−	1.55229i
$189$			158.983i			0.841178i
$190$		0			0
$191$	28.0282			0.146745			0.0733723	−	0.997305i	$-0.476624\pi$
$191$	28.0282			0.146745			0.0733723	+	0.997305i	$0.476624\pi$
$192$	450.033			2.34392
$193$	250.152			1.29612			0.648062	−	0.761588i	$-0.275580\pi$
$193$	250.152			1.29612			0.648062	+	0.761588i	$0.275580\pi$
$194$	−229.121			−1.18103
$195$		0			0
$196$	56.7309			0.289443
$197$			48.2838i			0.245096i	0.992463	+	0.122548i	$0.0391065\pi$
$197$			48.2838i			0.245096i	−0.992463	+	0.122548i	$0.960894\pi$
$198$	−220.571			−1.11399
$199$	265.858			1.33597			0.667984	−	0.744175i	$-0.267157\pi$
$199$	265.858			1.33597			0.667984	+	0.744175i	$0.267157\pi$
$200$		0			0
$201$	−216.389			−1.07656
$202$	−414.988			−2.05440
$203$	−264.195			−1.30145
$204$		−	32.0560i		−	0.157137i
$205$		0			0
$206$	41.4252			0.201093
$207$			532.729i			2.57357i
$208$	−43.5219			−0.209240
$209$	−93.6087	−	24.6952i	−0.447889	−	0.118159i
$210$		0			0
$211$		−	182.771i		−	0.866213i	−0.901343	−	0.433107i	$-0.857417\pi$
$211$		−	182.771i		−	0.866213i	0.901343	−	0.433107i	$-0.142583\pi$
$212$	314.600			1.48396
$213$			63.0720i			0.296113i
$214$	−194.515			−0.908949
$215$		0			0
$216$	86.1806			0.398984
$217$	321.061			1.47955
$218$		−	140.736i		−	0.645577i
$219$			647.873i			2.95832i
$220$		0			0
$221$			5.46178i			0.0247139i
$222$	−610.323			−2.74920
$223$	−106.620			−0.478118			−0.239059	−	0.971005i	$-0.576839\pi$
$223$	−106.620			−0.478118			−0.239059	+	0.971005i	$0.576839\pi$
$224$		−	274.479i		−	1.22535i
$225$		0			0
$226$	171.142			0.757266
$227$	−35.0010			−0.154189			−0.0770947	−	0.997024i	$-0.524564\pi$
$227$	−35.0010			−0.154189			−0.0770947	+	0.997024i	$0.524564\pi$
$228$	453.216	+	119.564i	1.98779	+	0.524405i
$229$	77.3701			0.337861			0.168930	−	0.985628i	$-0.445969\pi$
$229$	77.3701			0.337861			0.168930	+	0.985628i	$0.445969\pi$
$230$		0			0
$231$			151.541i			0.656022i
$232$			143.214i			0.617300i
$233$		−	89.0034i		−	0.381989i	−0.981591	−	0.190994i	$-0.938829\pi$
$233$		−	89.0034i		−	0.381989i	0.981591	−	0.190994i	$-0.0611712\pi$
$234$	−181.954			−0.777581
$235$		0			0
$236$		−	278.377i		−	1.17956i
$237$		−	619.165i		−	2.61251i
$238$	−24.1357			−0.101411
$239$	−47.2189			−0.197569			−0.0987843	−	0.995109i	$-0.531495\pi$
$239$	−47.2189			−0.197569			−0.0987843	+	0.995109i	$0.531495\pi$
$240$		0			0
$241$		−	73.1493i		−	0.303524i	−0.988417	−	0.151762i	$-0.951505\pi$
$241$		−	73.1493i		−	0.303524i	0.988417	−	0.151762i	$-0.0484947\pi$
$242$	286.783			1.18505
$243$	253.304			1.04240
$244$	196.216			0.804163
$245$		0			0
$246$			104.029i			0.422880i
$247$	−77.2200	−	20.3717i	−0.312632	−	0.0824765i
$248$		−	174.039i		−	0.701772i
$249$		−	112.186i		−	0.450547i
$250$		0			0
$251$	266.248			1.06075			0.530374	−	0.847764i	$-0.322051\pi$
$251$	266.248			1.06075			0.530374	+	0.847764i	$0.322051\pi$
$252$		−	450.850i		−	1.78909i
$253$			189.218i			0.747896i
$254$	718.721			2.82961
$255$		0			0
$256$	62.6778			0.244835
$257$	−76.0071			−0.295747			−0.147874	−	0.989006i	$-0.547243\pi$
$257$	−76.0071			−0.295747			−0.147874	+	0.989006i	$0.547243\pi$
$258$			864.405i			3.35041i
$259$			257.665i			0.994845i
$260$		0			0
$261$		−	615.723i		−	2.35909i
$262$	123.956			0.473115
$263$		−	291.378i		−	1.10790i	−0.832549	−	0.553951i	$-0.813119\pi$
$263$		−	291.378i		−	1.10790i	0.832549	−	0.553951i	$-0.186881\pi$
$264$	82.1468			0.311162
$265$		0			0
$266$	90.0231	−	341.238i	0.338433	−	1.28285i
$267$		−	101.784i		−	0.381215i
$268$	228.662			0.853215
$269$		−	406.215i		−	1.51009i	−0.655672	−	0.755046i	$-0.727615\pi$
$269$		−	406.215i		−	1.51009i	0.655672	−	0.755046i	$-0.272385\pi$
$270$		0			0
$271$	−289.751			−1.06919			−0.534596	−	0.845108i	$-0.679536\pi$
$271$	−289.751			−1.06919			−0.534596	+	0.845108i	$0.679536\pi$
$272$		−	13.4545i		−	0.0494652i
$273$			125.010i			0.457912i
$274$			427.322i			1.55957i
$275$		0			0
$276$		−	916.116i		−	3.31926i
$277$			102.717i			0.370820i	0.982661	+	0.185410i	$0.0593613\pi$
$277$			102.717i			0.370820i	−0.982661	+	0.185410i	$0.940639\pi$
$278$	−401.842			−1.44547
$279$			748.255i			2.68192i
$280$		0			0
$281$			0.126667i			0.000450774i	1.00000		0.000225387i	$7.17429e-5\pi$
$281$			0.126667i			0.000450774i	−1.00000		0.000225387i	$0.999928\pi$
$282$		−	833.358i		−	2.95517i
$283$		−	84.2579i		−	0.297731i	−0.988857	−	0.148866i	$-0.952438\pi$
$283$		−	84.2579i		−	0.297731i	0.988857	−	0.148866i	$-0.0475622\pi$
$284$		−	66.6491i		−	0.234680i
$285$		0			0
$286$	−64.6275			−0.225970
$287$	43.9186			0.153026
$288$	639.691			2.22115
$289$	287.312			0.994158
$290$		0			0
$291$	−366.867			−1.26071
$292$		−	684.617i		−	2.34458i
$293$	404.118			1.37924			0.689621	−	0.724171i	$-0.257777\pi$
$293$	404.118			1.37924			0.689621	+	0.724171i	$0.257777\pi$
$294$	162.002			0.551026
$295$		0			0
$296$	139.674			0.471871
$297$	−131.603			−0.443109
$298$	−831.258			−2.78946
$299$			156.090i			0.522041i
$300$		0			0
$301$	364.933			1.21240
$302$			425.777i			1.40986i
$303$	−664.476			−2.19299
$304$	190.224	+	50.1835i	0.625736	+	0.165077i
$305$		0			0
$306$		−	56.2499i		−	0.183823i
$307$	−158.822			−0.517335			−0.258667	−	0.965966i	$-0.583283\pi$
$307$	−158.822			−0.517335			−0.258667	+	0.965966i	$0.583283\pi$
$308$		−	160.136i		−	0.519921i
$309$	66.3298			0.214659
$310$		0			0
$311$	256.822			0.825793			0.412897	−	0.910778i	$-0.364517\pi$
$311$	256.822			0.825793			0.412897	+	0.910778i	$0.364517\pi$
$312$	67.7648			0.217195
$313$			429.290i			1.37153i	0.727822	+	0.685766i	$0.240533\pi$
$313$			429.290i			1.37153i	−0.727822	+	0.685766i	$0.759467\pi$
$314$			404.895i			1.28948i
$315$		0			0
$316$			654.281i			2.07051i
$317$	−230.765			−0.727966			−0.363983	−	0.931406i	$-0.618583\pi$
$317$	−230.765			−0.727966			−0.363983	+	0.931406i	$0.618583\pi$
$318$	898.376			2.82508
$319$		−	218.696i		−	0.685569i
$320$		0			0
$321$	−311.457			−0.970270
$322$	−689.767			−2.14213
$323$	6.29778	−	23.8721i	0.0194978	−	0.0739074i
$324$	−22.0327			−0.0680022
$325$		0			0
$326$			532.378i			1.63306i
$327$		−	225.346i		−	0.689130i
$328$		−	23.8072i		−	0.0725829i
$329$	−351.826			−1.06938
$330$		0			0
$331$		−	528.588i		−	1.59694i	−0.602032	−	0.798472i	$-0.705642\pi$
$331$		−	528.588i		−	1.59694i	0.602032	−	0.798472i	$-0.294358\pi$
$332$			118.549i			0.357075i
$333$	−600.505			−1.80332
$334$	−527.575			−1.57957
$335$		0			0
$336$		−	307.949i		−	0.916515i
$337$	61.4407			0.182317			0.0911583	−	0.995836i	$-0.470943\pi$
$337$	61.4407			0.182317			0.0911583	+	0.995836i	$0.470943\pi$
$338$	456.657			1.35106
$339$	274.032			0.808353
$340$		0			0
$341$			265.770i			0.779383i
$342$	795.277	+	209.805i	2.32537	+	0.613464i
$343$		−	370.008i		−	1.07874i
$344$		−	197.821i		−	0.575061i
$345$		0			0
$346$	−767.333			−2.21773
$347$			555.017i			1.59947i	0.600351	+	0.799737i	$0.295028\pi$
$347$			555.017i			1.59947i	−0.600351	+	0.799737i	$0.704972\pi$
$348$			1058.84i			3.04264i
$349$	−34.6618			−0.0993176			−0.0496588	−	0.998766i	$-0.515813\pi$
$349$	−34.6618			−0.0993176			−0.0496588	+	0.998766i	$0.515813\pi$
$350$		0			0
$351$	−108.563			−0.309295
$352$	227.209			0.645481
$353$		−	493.002i		−	1.39661i	−0.715802	−	0.698303i	$-0.753939\pi$
$353$		−	493.002i		−	1.39661i	0.715802	−	0.698303i	$-0.246061\pi$
$354$		−	794.939i		−	2.24559i
$355$		0			0
$356$			107.557i			0.302127i
$357$	−38.6460			−0.108252
$358$			624.892i			1.74551i
$359$	−145.520			−0.405349			−0.202675	−	0.979246i	$-0.564963\pi$
$359$	−145.520			−0.405349			−0.202675	+	0.979246i	$0.564963\pi$
$360$		0			0
$361$	314.020	+	178.079i	0.869862	+	0.493295i
$362$			415.537i			1.14789i
$363$	459.195			1.26500
$364$		−	132.100i		−	0.362911i
$365$		0			0
$366$	560.317			1.53092
$367$		−	342.929i		−	0.934413i	−0.884148	−	0.467206i	$-0.845261\pi$
$367$		−	342.929i		−	0.934413i	0.884148	−	0.467206i	$-0.154739\pi$
$368$		−	384.512i		−	1.04487i
$369$			102.355i			0.277385i
$370$		0			0
$371$		−	379.275i		−	1.02230i
$372$		−	1286.75i		−	3.45900i
$373$	−16.6464			−0.0446283			−0.0223141	−	0.999751i	$-0.507103\pi$
$373$	−16.6464			−0.0446283			−0.0223141	+	0.999751i	$0.507103\pi$
$374$		−	19.9792i		−	0.0534203i
$375$		0			0
$376$			190.716i			0.507224i
$377$		−	180.408i		−	0.478535i
$378$		−	479.742i		−	1.26916i
$379$			312.804i			0.825341i	0.910880	+	0.412671i	$0.135404\pi$
$379$			312.804i			0.825341i	−0.910880	+	0.412671i	$0.864596\pi$
$380$		0			0
$381$	1150.81			3.02051
$382$	−84.5772			−0.221406
$383$	−493.251			−1.28786			−0.643931	−	0.765083i	$-0.722698\pi$
$383$	−493.251			−1.28786			−0.643931	+	0.765083i	$0.722698\pi$
$384$	−496.190			−1.29216
$385$		0			0
$386$	−754.852			−1.95557
$387$			850.500i			2.19767i
$388$	387.673			0.999158
$389$	407.055			1.04641			0.523207	−	0.852205i	$-0.324735\pi$
$389$	407.055			1.04641			0.523207	+	0.852205i	$0.324735\pi$
$390$		0			0
$391$	−48.2543			−0.123413
$392$	−37.0745			−0.0945778
$393$	198.478			0.505033
$394$		−	145.700i		−	0.369797i
$395$		0			0
$396$	373.207			0.942442
$397$		−	208.967i		−	0.526366i	−0.964746	−	0.263183i	$-0.915228\pi$
$397$		−	208.967i		−	0.526366i	0.964746	−	0.263183i	$-0.0847723\pi$
$398$	−802.245			−2.01569
$399$	144.144	−	546.388i	0.361264	−	1.36939i
$400$		0			0
$401$			240.966i			0.600913i	0.953796	+	0.300456i	$0.0971390\pi$
$401$			240.966i			0.600913i	−0.953796	+	0.300456i	$0.902861\pi$
$402$	652.971			1.62430
$403$			219.240i			0.544019i
$404$	702.162			1.73802
$405$		0			0
$406$	797.227			1.96361
$407$	−213.291			−0.524056
$408$			20.9491i			0.0513457i
$409$			259.859i			0.635352i	0.948199	+	0.317676i	$0.102902\pi$
$409$			259.859i			0.635352i	−0.948199	+	0.317676i	$0.897098\pi$
$410$		0			0
$411$			684.226i			1.66478i
$412$	−70.0916			−0.170125
$413$	−335.606			−0.812605
$414$		−	1607.55i		−	3.88297i
$415$		0			0
$416$	187.430			0.450554
$417$	−643.427			−1.54299
$418$	282.471	+	74.5197i	0.675769	+	0.178277i
$419$	729.115			1.74013			0.870066	−	0.492935i	$-0.164076\pi$
$419$	729.115			1.74013			0.870066	+	0.492935i	$0.164076\pi$
$420$		0			0
$421$			608.641i			1.44570i	0.691003	+	0.722851i	$0.257169\pi$
$421$			608.641i			1.44570i	−0.691003	+	0.722851i	$0.742831\pi$
$422$			551.525i			1.30693i
$423$		−	819.953i		−	1.93842i
$424$	−205.596			−0.484895
$425$		0			0
$426$		−	190.324i		−	0.446771i
$427$		−	236.554i		−	0.553990i
$428$	329.121			0.768973
$429$	−103.481			−0.241215
$430$		0			0
$431$			537.936i			1.24811i	0.781380	+	0.624056i	$0.214516\pi$
$431$			537.936i			1.24811i	−0.781380	+	0.624056i	$0.785484\pi$
$432$	267.433			0.619058
$433$	−255.532			−0.590144			−0.295072	−	0.955475i	$-0.595344\pi$
$433$	−255.532			−0.590144			−0.295072	+	0.955475i	$0.595344\pi$
$434$	−968.826			−2.23232
$435$		0			0
$436$			238.126i			0.546160i
$437$	179.982	−	682.233i	0.411858	−	1.56117i
$438$		−	1955.00i		−	4.46348i
$439$		−	699.710i		−	1.59387i	−0.604063	−	0.796937i	$-0.706452\pi$
$439$		−	699.710i		−	1.59387i	0.604063	−	0.796937i	$-0.293548\pi$
$440$		0			0
$441$	159.396			0.361442
$442$		−	16.4813i		−	0.0372880i
$443$			682.143i			1.53983i	0.638149	+	0.769913i	$0.279701\pi$
$443$			682.143i			1.53983i	−0.638149	+	0.769913i	$0.720299\pi$
$444$	1032.67			2.32583
$445$		0			0
$446$	321.735			0.721378
$447$	−1331.01			−2.97764
$448$			573.321i			1.27973i
$449$		−	675.642i		−	1.50477i	−0.658723	−	0.752386i	$-0.728903\pi$
$449$		−	675.642i		−	1.50477i	0.658723	−	0.752386i	$-0.271097\pi$
$450$		0			0
$451$			36.3551i			0.0806100i
$452$	−289.573			−0.640649
$453$			681.751i			1.50497i
$454$	105.618			0.232639
$455$		0			0
$456$	−296.183	−	78.1372i	−0.649525	−	0.171353i
$457$		−	124.772i		−	0.273024i	−0.990638	−	0.136512i	$-0.956411\pi$
$457$		−	124.772i		−	0.273024i	0.990638	−	0.136512i	$-0.0435893\pi$
$458$	−233.470			−0.509759
$459$		−	33.5615i		−	0.0731187i
$460$		0			0
$461$	328.840			0.713320			0.356660	−	0.934234i	$-0.383916\pi$
$461$	328.840			0.713320			0.356660	+	0.934234i	$0.383916\pi$
$462$		−	457.287i		−	0.989798i
$463$			122.860i			0.265356i	0.991159	+	0.132678i	$0.0423577\pi$
$463$			122.860i			0.265356i	−0.991159	+	0.132678i	$0.957642\pi$
$464$			444.416i			0.957793i
$465$		0			0
$466$			268.574i			0.576340i
$467$			653.452i			1.39925i	0.714508	+	0.699627i	$0.246651\pi$
$467$			653.452i			1.39925i	−0.714508	+	0.699627i	$0.753349\pi$
$468$	307.867			0.657836
$469$		−	275.670i		−	0.587782i
$470$		0			0
$471$			648.316i			1.37647i
$472$			181.924i			0.385432i
$473$			302.086i			0.638659i
$474$			1868.38i			3.94172i
$475$		0			0
$476$	40.8378			0.0857937
$477$	883.925			1.85309
$478$	142.487			0.298089
$479$	−360.148			−0.751874			−0.375937	−	0.926645i	$-0.622679\pi$
$479$	−360.148			−0.751874			−0.375937	+	0.926645i	$0.622679\pi$
$480$		0			0
$481$	−175.949			−0.365798
$482$			220.733i			0.457953i
$483$	−1104.45			−2.28665
$484$	−485.238			−1.00256
$485$		0			0
$486$	−764.363			−1.57276
$487$	47.1304			0.0967770			0.0483885	−	0.998829i	$-0.484591\pi$
$487$	47.1304			0.0967770			0.0483885	+	0.998829i	$0.484591\pi$
$488$	−128.230			−0.262766
$489$			852.440i			1.74323i
$490$		0			0
$491$	−744.260			−1.51580			−0.757902	−	0.652369i	$-0.773775\pi$
$491$	−744.260			−1.51580			−0.757902	+	0.652369i	$0.773775\pi$
$492$		−	176.017i		−	0.357758i
$493$	55.7720			0.113128
$494$	233.017	+	61.4731i	0.471695	+	0.124439i
$495$		0			0
$496$		−	540.074i		−	1.08886i
$497$	−80.3508			−0.161672
$498$			338.530i			0.679779i
$499$	−814.737			−1.63274			−0.816370	−	0.577529i	$-0.804017\pi$
$499$	−814.737			−1.63274			−0.816370	+	0.577529i	$0.804017\pi$
$500$		0			0
$501$	−844.750			−1.68613
$502$	−803.423			−1.60044
$503$			618.711i			1.23004i	0.788511	+	0.615020i	$0.210852\pi$
$503$			618.711i			1.23004i	−0.788511	+	0.615020i	$0.789148\pi$
$504$			294.637i			0.584598i
$505$		0			0
$506$		−	570.979i		−	1.12842i
$507$	731.197			1.44220
$508$	−1216.08			−2.39386
$509$			309.300i			0.607661i	0.952726	+	0.303831i	$0.0982657\pi$
$509$			309.300i			0.607661i	−0.952726	+	0.303831i	$0.901734\pi$
$510$		0			0
$511$	−825.360			−1.61519
$512$	−599.912			−1.17170
$513$	474.501	+	125.180i	0.924954	+	0.244015i
$514$	229.357			0.446220
$515$		0			0
$516$		−	1462.58i		−	2.83445i
$517$		−	291.236i		−	0.563319i
$518$		−	777.523i		−	1.50101i
$519$	−1228.65			−2.36734
$520$		0			0
$521$		−	163.664i		−	0.314134i	−0.987588	−	0.157067i	$-0.949796\pi$
$521$		−	163.664i		−	0.314134i	0.987588	−	0.157067i	$-0.0502039\pi$
$522$			1857.99i			3.55937i
$523$	579.994			1.10897			0.554487	−	0.832192i	$-0.312914\pi$
$523$	579.994			1.10897			0.554487	+	0.832192i	$0.312914\pi$
$524$	−209.734			−0.400257
$525$		0			0
$526$			879.256i			1.67159i
$527$	−67.7766			−0.128608
$528$	254.915			0.482794
$529$	−850.044			−1.60689
$530$		0			0
$531$		−	782.152i		−	1.47298i
$532$	−152.320	+	577.376i	−0.286315	+	1.08529i
$533$			29.9902i			0.0562668i
$534$			307.142i			0.575172i
$535$		0			0
$536$	−149.434			−0.278795
$537$			1000.57i			1.86327i
$538$			1225.78i			2.27841i
$539$	56.6152			0.105037
$540$		0			0
$541$	−706.641			−1.30618			−0.653088	−	0.757282i	$-0.726527\pi$
$541$	−706.641			−1.30618			−0.653088	+	0.757282i	$0.726527\pi$
$542$	874.345			1.61318
$543$			665.356i			1.22533i
$544$			57.9429i			0.106513i
$545$		0			0
$546$		−	377.226i		−	0.690891i
$547$	726.986			1.32904			0.664521	−	0.747269i	$-0.268636\pi$
$547$	726.986			1.32904			0.664521	+	0.747269i	$0.268636\pi$
$548$		−	723.032i		−	1.31940i
$549$	551.304			1.00420
$550$		0			0
$551$	−208.022	+	788.519i	−0.377535	+	1.43107i
$552$			598.696i			1.08459i
$553$	788.788			1.42638
$554$		−	309.957i		−	0.559489i
$555$		0			0
$556$	679.919			1.22288
$557$		−	12.5711i		−	0.0225692i	−0.999936	−	0.0112846i	$-0.996408\pi$
$557$		−	12.5711i		−	0.0225692i	0.999936	−	0.0112846i	$-0.00359208\pi$
$558$		−	2257.91i		−	4.04644i
$559$			249.198i			0.445792i
$560$		0			0
$561$		−	31.9906i		−	0.0570242i
$562$		−	0.382228i		−	0.000680122i
$563$	635.525			1.12882			0.564410	−	0.825495i	$-0.309104\pi$
$563$	635.525			1.12882			0.564410	+	0.825495i	$0.309104\pi$
$564$			1410.05i			2.50008i
$565$		0			0
$566$			254.255i			0.449213i
$567$			26.5622i			0.0468469i
$568$			43.5562i			0.0766834i
$569$			306.296i			0.538306i	0.963097	+	0.269153i	$0.0867438\pi$
$569$			306.296i			0.538306i	−0.963097	+	0.269153i	$0.913256\pi$
$570$		0			0
$571$	127.895			0.223984			0.111992	−	0.993709i	$-0.464277\pi$
$571$	127.895			0.223984			0.111992	+	0.993709i	$0.464277\pi$
$572$	109.350			0.191172
$573$	−135.425			−0.236343
$574$	−132.528			−0.230884
$575$		0			0
$576$	−1336.16			−2.31973
$577$			558.970i			0.968752i	0.874860	+	0.484376i	$0.160953\pi$
$577$			558.970i			0.968752i	−0.874860	+	0.484376i	$0.839047\pi$
$578$	−866.984			−1.49997
$579$	−1208.66			−2.08750
$580$		0			0
$581$	142.920			0.245990
$582$	1107.05			1.90214
$583$	313.958			0.538521
$584$			447.407i			0.766109i
$585$		0			0
$586$	−1219.46			−2.08098
$587$		−	140.817i		−	0.239893i	−0.992780	−	0.119946i	$-0.961728\pi$
$587$		−	140.817i		−	0.239893i	0.992780	−	0.119946i	$-0.0382723\pi$
$588$	−274.108			−0.466170
$589$	252.797	−	958.243i	0.429198	−	1.62690i
$590$		0			0
$591$		−	233.294i		−	0.394745i
$592$	433.432			0.732148
$593$			340.511i			0.574218i	0.957898	+	0.287109i	$0.0926942\pi$
$593$			340.511i			0.574218i	−0.957898	+	0.287109i	$0.907306\pi$
$594$	397.123			0.668557
$595$		0			0
$596$	1406.49			2.35989
$597$	−1284.55			−2.15168
$598$		−	471.014i		−	0.787648i
$599$			694.681i			1.15973i	0.814711	+	0.579867i	$0.196896\pi$
$599$			694.681i			1.15973i	−0.814711	+	0.579867i	$0.803104\pi$
$600$		0			0
$601$		−	243.254i		−	0.404749i	−0.979308	−	0.202375i	$-0.935134\pi$
$601$		−	243.254i		−	0.404749i	0.979308	−	0.202375i	$-0.0648658\pi$
$602$	−1101.21			−1.82925
$603$	642.467			1.06545
$604$		−	720.416i		−	1.19274i
$605$		0			0
$606$	2005.11			3.30876
$607$	942.849			1.55329			0.776647	−	0.629936i	$-0.216919\pi$
$607$	942.849			1.55329			0.776647	+	0.629936i	$0.216919\pi$
$608$	−819.213	−	216.119i	−1.34739	−	0.355459i
$609$	1276.52			2.09609
$610$		0			0
$611$		−	240.247i		−	0.393203i
$612$			95.1752i			0.155515i
$613$		−	402.930i		−	0.657308i	−0.944450	−	0.328654i	$-0.893405\pi$
$613$		−	402.930i		−	0.657308i	0.944450	−	0.328654i	$-0.106595\pi$
$614$	479.256			0.780548
$615$		0			0
$616$			104.651i			0.169888i
$617$		−	964.671i		−	1.56349i	−0.623600	−	0.781743i	$-0.714331\pi$
$617$		−	964.671i		−	1.56349i	0.623600	−	0.781743i	$-0.285669\pi$
$618$	−200.155			−0.323875
$619$	−830.004			−1.34088			−0.670439	−	0.741964i	$-0.733894\pi$
$619$	−830.004			−1.34088			−0.670439	+	0.741964i	$0.733894\pi$
$620$		0			0
$621$		−	959.142i		−	1.54451i
$622$	−774.978			−1.24595
$623$	129.669			0.208136
$624$	210.286			0.336996
$625$		0			0
$626$		−	1295.41i		−	2.06935i
$627$	452.291	+	119.321i	0.721358	+	0.190304i
$628$		−	685.085i		−	1.09090i
$629$		−	54.3935i		−	0.0864761i
$630$		0			0
$631$	775.784			1.22945			0.614726	−	0.788741i	$-0.289267\pi$
$631$	775.784			1.22945			0.614726	+	0.788741i	$0.289267\pi$
$632$		−	427.583i		−	0.676555i
$633$			883.099i			1.39510i
$634$	696.351			1.09834
$635$		0			0
$636$	−1520.06			−2.39003
$637$	46.7032			0.0733174
$638$			659.933i			1.03438i
$639$		−	187.263i		−	0.293056i
$640$		0			0
$641$			277.516i			0.432943i	0.976289	+	0.216471i	$0.0694548\pi$
$641$			277.516i			0.432943i	−0.976289	+	0.216471i	$0.930545\pi$
$642$	939.843			1.46393
$643$		−	617.315i		−	0.960055i	−0.877254	−	0.480027i	$-0.840627\pi$
$643$		−	617.315i		−	0.960055i	0.877254	−	0.480027i	$-0.159373\pi$
$644$	1167.09			1.81225
$645$		0			0
$646$	−19.0040	+	72.0358i	−0.0294180	+	0.111511i
$647$		−	642.107i		−	0.992437i	−0.868198	−	0.496218i	$-0.834722\pi$
$647$		−	642.107i		−	0.992437i	0.868198	−	0.496218i	$-0.165278\pi$
$648$	14.3987			0.0222202
$649$		−	277.809i		−	0.428058i
$650$		0			0
$651$	−1551.28			−2.38292
$652$		−	900.786i		−	1.38157i
$653$		−	803.876i		−	1.23105i	−0.788117	−	0.615525i	$-0.788944\pi$
$653$		−	803.876i		−	1.23105i	0.788117	−	0.615525i	$-0.211056\pi$
$654$			679.997i			1.03975i
$655$		0			0
$656$		−	73.8777i		−	0.112619i
$657$		−	1923.56i		−	2.92779i
$658$	1061.66			1.61346
$659$		−	1024.56i		−	1.55472i	−0.629054	−	0.777362i	$-0.716558\pi$
$659$		−	1024.56i		−	1.55472i	0.629054	−	0.777362i	$-0.283442\pi$
$660$		0			0
$661$		−	1059.45i		−	1.60281i	−0.598125	−	0.801403i	$-0.704087\pi$
$661$		−	1059.45i		−	1.60281i	0.598125	−	0.801403i	$-0.295913\pi$
$662$			1595.05i			2.40945i
$663$		−	26.3898i		−	0.0398036i
$664$		−	77.4734i		−	0.116677i
$665$		0			0
$666$	1812.07			2.72082
$667$	1593.89			2.38964
$668$	892.659			1.33632
$669$	515.160			0.770045
$670$		0			0
$671$	195.816			0.291826
$672$			1326.21i			1.97352i
$673$	766.057			1.13827			0.569136	−	0.822243i	$-0.307278\pi$
$673$	766.057			1.13827			0.569136	+	0.822243i	$0.307278\pi$
$674$	−185.402			−0.275077
$675$		0			0
$676$	−772.666			−1.14300
$677$	−386.540			−0.570960			−0.285480	−	0.958385i	$-0.592153\pi$
$677$	−386.540			−0.570960			−0.285480	+	0.958385i	$0.592153\pi$
$678$	−826.911			−1.21963
$679$		−	467.371i		−	0.688323i
$680$		0			0
$681$	169.115			0.248334
$682$		−	801.979i		−	1.17592i
$683$	1160.28			1.69880			0.849402	−	0.527746i	$-0.176963\pi$
$683$	1160.28			1.69880			0.849402	+	0.527746i	$0.176963\pi$
$684$	−1345.61	−	354.991i	−1.96727	−	0.518992i
$685$		0			0
$686$			1116.53i			1.62759i
$687$	−373.831			−0.544149
$688$		−	613.873i		−	0.892257i
$689$	258.991			0.375894
$690$		0			0
$691$	−180.805			−0.261657			−0.130828	−	0.991405i	$-0.541764\pi$
$691$	−180.805			−0.261657			−0.130828	+	0.991405i	$0.541764\pi$
$692$	1298.33			1.87620
$693$		−	449.931i		−	0.649251i
$694$		−	1674.81i		−	2.41327i
$695$		0			0
$696$		−	691.968i		−	0.994207i
$697$	−9.27128			−0.0133017
$698$	104.595			0.149849
$699$			430.040i			0.615221i
$700$		0			0
$701$	1195.12			1.70488			0.852442	−	0.522821i	$-0.175121\pi$
$701$	1195.12			1.70488			0.852442	+	0.522821i	$0.175121\pi$
$702$	327.596			0.466661
$703$	769.029	+	202.880i	1.09393	+	0.288592i
$704$	−474.586			−0.674128
$705$		0			0
$706$			1487.67i			2.10718i
$707$		−	846.512i		−	1.19733i
$708$			1345.04i			1.89978i
$709$	−352.953			−0.497818			−0.248909	−	0.968527i	$-0.580072\pi$
$709$	−352.953			−0.497818			−0.248909	+	0.968527i	$0.580072\pi$
$710$		0			0
$711$			1838.32i			2.58554i
$712$		−	70.2902i		−	0.0987221i
$713$	−1936.96			−2.71664
$714$	116.617			0.163329
$715$		0			0
$716$		−	1057.32i		−	1.47670i
$717$	228.149			0.318199
$718$	439.118			0.611585
$719$	−388.649			−0.540541			−0.270271	−	0.962784i	$-0.587113\pi$
$719$	−388.649			−0.540541			−0.270271	+	0.962784i	$0.587113\pi$
$720$		0			0
$721$			84.5010i			0.117200i
$722$	−947.579	−	537.368i	−1.31244	−	0.744277i
$723$			353.437i			0.488848i
$724$		−	703.091i		−	0.971120i
$725$		0			0
$726$	−1385.65			−1.90862
$727$			1159.46i			1.59486i	0.603410	+	0.797431i	$0.293808\pi$
$727$			1159.46i			1.59486i	−0.603410	+	0.797431i	$0.706192\pi$
$728$			86.3292i			0.118584i
$729$	−1185.06			−1.62559
$730$		0			0
$731$	−77.0379			−0.105387
$732$	−948.060			−1.29516
$733$			420.415i			0.573554i	0.957997	+	0.286777i	$0.0925838\pi$
$733$			420.415i			0.573554i	−0.957997	+	0.286777i	$0.907416\pi$
$734$			1034.81i			1.40983i
$735$		0			0
$736$			1655.93i			2.24991i
$737$	228.195			0.309627
$738$		−	308.864i		−	0.418515i
$739$	−117.628			−0.159173			−0.0795863	−	0.996828i	$-0.525360\pi$
$739$	−117.628			−0.159173			−0.0795863	+	0.996828i	$0.525360\pi$
$740$		0			0
$741$	373.106	+	98.4303i	0.503517	+	0.132834i
$742$			1144.49i			1.54244i
$743$	131.785			0.177368			0.0886842	−	0.996060i	$-0.471734\pi$
$743$	131.785			0.177368			0.0886842	+	0.996060i	$0.471734\pi$
$744$			840.911i			1.13026i
$745$		0			0
$746$	50.2316			0.0673346
$747$			333.084i			0.445896i
$748$			33.8049i			0.0451937i
$749$		−	396.781i		−	0.529748i
$750$		0			0
$751$			162.721i			0.216672i	0.994114	+	0.108336i	$0.0345522\pi$
$751$			162.721i			0.216672i	−0.994114	+	0.108336i	$0.965448\pi$
$752$			591.824i			0.787001i
$753$	−1286.44			−1.70841
$754$			544.394i			0.722008i
$755$		0			0
$756$			811.725i			1.07371i
$757$			1220.77i			1.61264i	0.591477	+	0.806322i	$0.298545\pi$
$757$			1220.77i			1.61264i	−0.591477	+	0.806322i	$0.701455\pi$
$758$		−	943.910i		−	1.24526i
$759$		−	914.248i		−	1.20454i
$760$		0			0
$761$	−164.940			−0.216741			−0.108370	−	0.994111i	$-0.534563\pi$
$761$	−164.940			−0.216741			−0.108370	+	0.994111i	$0.534563\pi$
$762$	−3472.66			−4.55730
$763$	287.080			0.376251
$764$	143.105			0.187310
$765$		0			0
$766$	1488.42			1.94311
$767$		−	229.172i		−	0.298789i
$768$	−302.842			−0.394325
$769$	1070.57			1.39216			0.696079	−	0.717965i	$-0.254926\pi$
$769$	1070.57			1.39216			0.696079	+	0.717965i	$0.254926\pi$
$770$		0			0
$771$	367.245			0.476323
$772$	1277.21			1.65442
$773$	−181.214			−0.234430			−0.117215	−	0.993107i	$-0.537397\pi$
$773$	−181.214			−0.234430			−0.117215	+	0.993107i	$0.537397\pi$
$774$		−	2566.45i		−	3.31582i
$775$		0			0
$776$	−253.350			−0.326483
$777$		−	1244.96i		−	1.60227i
$778$	−1228.32			−1.57882
$779$	34.5806	−	131.080i	0.0443910	−	0.168267i
$780$		0			0
$781$		−	66.5132i		−	0.0851641i
$782$	145.611			0.186203
$783$			1108.57i			1.41580i
$784$	−115.048			−0.146745
$785$		0			0
$786$	−598.922			−0.761987
$787$	1339.67			1.70225			0.851123	−	0.524966i	$-0.175922\pi$
$787$	1339.67			1.70225			0.851123	+	0.524966i	$0.175922\pi$
$788$			246.525i			0.312849i
$789$			1407.86i			1.78436i
$790$		0			0
$791$			349.104i			0.441345i
$792$	−243.896			−0.307950
$793$	161.533			0.203698
$794$			630.574i			0.794174i
$795$		0			0
$796$	1357.40			1.70528
$797$	−456.100			−0.572272			−0.286136	−	0.958189i	$-0.592371\pi$
$797$	−456.100			−0.572272			−0.286136	+	0.958189i	$0.592371\pi$
$798$	−434.966	+	1648.77i	−0.545071	+	2.06612i
$799$	74.2710			0.0929549
$800$		0			0
$801$			302.201i			0.377280i
$802$		−	727.133i		−	0.906649i
$803$		−	683.220i		−	0.850835i
$804$	−1104.83			−1.37417
$805$		0			0
$806$		−	661.572i		−	0.820809i
$807$			1962.72i			2.43212i
$808$	−458.873			−0.567913
$809$	415.566			0.513678			0.256839	−	0.966454i	$-0.417319\pi$
$809$	415.566			0.513678			0.256839	+	0.966454i	$0.417319\pi$
$810$		0			0
$811$		−	891.629i		−	1.09942i	−0.835356	−	0.549710i	$-0.814738\pi$
$811$		−	891.629i		−	1.09942i	0.835356	−	0.549710i	$-0.185262\pi$
$812$	−1348.91			−1.66122
$813$	1400.00			1.72201
$814$	643.621			0.790690
$815$		0			0
$816$			65.0085i			0.0796673i
$817$	287.341	−	1089.18i	0.351702	−	1.33315i
$818$		−	784.143i		−	0.958611i
$819$		−	371.158i		−	0.453185i
$820$		0			0
$821$	−241.578			−0.294249			−0.147124	−	0.989118i	$-0.547002\pi$
$821$	−241.578			−0.294249			−0.147124	+	0.989118i	$0.547002\pi$
$822$		−	2064.70i		−	2.51180i
$823$		−	585.888i		−	0.711894i	−0.934506	−	0.355947i	$-0.884158\pi$
$823$		−	585.888i		−	0.711894i	0.934506	−	0.355947i	$-0.115842\pi$
$824$	45.8060			0.0555897
$825$		0			0
$826$	1012.72			1.22605
$827$	−61.7508			−0.0746684			−0.0373342	−	0.999303i	$-0.511887\pi$
$827$	−61.7508			−0.0746684			−0.0373342	+	0.999303i	$0.511887\pi$
$828$			2719.98i			3.28500i
$829$			776.948i			0.937211i	0.883408	+	0.468606i	$0.155243\pi$
$829$			776.948i			0.937211i	−0.883408	+	0.468606i	$0.844757\pi$
$830$		0			0
$831$		−	496.301i		−	0.597233i
$832$	−391.498			−0.470550
$833$			14.4380i			0.0173325i
$834$	1941.59			2.32804
$835$		0			0
$836$	−477.943	−	126.088i	−0.571702	−	0.150823i
$837$		−	1347.18i		−	1.60954i
$838$	−2200.16			−2.62549
$839$		−	997.057i		−	1.18839i	−0.804322	−	0.594194i	$-0.797471\pi$
$839$		−	997.057i		−	1.18839i	0.804322	−	0.594194i	$-0.202529\pi$
$840$		0			0
$841$	−1001.20			−1.19049
$842$		−	1836.62i		−	2.18126i
$843$		−	0.612022i		−	0.000726005i
$844$		−	933.183i		−	1.10567i
$845$		0			0
$846$			2474.27i			2.92467i
$847$			584.993i			0.690665i
$848$	−637.998			−0.752357
$849$			407.111i			0.479518i
$850$		0			0
$851$		−	1554.49i		−	1.82667i
$852$			322.030i			0.377969i
$853$		−	56.0425i		−	0.0657004i	−0.999460	−	0.0328502i	$-0.989542\pi$
$853$		−	56.0425i		−	0.0657004i	0.999460	−	0.0328502i	$-0.0104584\pi$
$854$			713.818i			0.835853i
$855$		0			0
$856$	−215.085			−0.251268
$857$	−926.670			−1.08129			−0.540647	−	0.841249i	$-0.681821\pi$
$857$	−926.670			−1.08129			−0.540647	+	0.841249i	$0.681821\pi$
$858$	312.262			0.363942
$859$	−83.5317			−0.0972429			−0.0486215	−	0.998817i	$-0.515483\pi$
$859$	−83.5317			−0.0972429			−0.0486215	+	0.998817i	$0.515483\pi$
$860$		0			0
$861$	−212.202			−0.246460
$862$		−	1623.26i		−	1.88313i
$863$	187.981			0.217823			0.108911	−	0.994051i	$-0.465264\pi$
$863$	187.981			0.217823			0.108911	+	0.994051i	$0.465264\pi$
$864$	−1151.72			−1.33301
$865$		0			0
$866$	771.088			0.890402
$867$	−1388.21			−1.60116
$868$	1639.26			1.88855
$869$			652.946i			0.751377i
$870$		0			0
$871$	188.244			0.216124
$872$		−	155.619i		−	0.178462i
$873$	1089.24			1.24770
$874$	−543.109	+	2058.69i	−0.621406	+	2.35548i
$875$		0			0
$876$			3307.88i			3.77612i
$877$	−732.648			−0.835402			−0.417701	−	0.908585i	$-0.637164\pi$
$877$	−732.648			−0.835402			−0.417701	+	0.908585i	$0.637164\pi$
$878$			2111.43i			2.40482i
$879$	−1952.59			−2.22137
$880$		0			0
$881$	1157.50			1.31385			0.656925	−	0.753956i	$-0.271857\pi$
$881$	1157.50			1.31385			0.656925	+	0.753956i	$0.271857\pi$
$882$	−480.988			−0.545338
$883$		−	1565.09i		−	1.77247i	−0.463237	−	0.886234i	$-0.653312\pi$
$883$		−	1565.09i		−	1.77247i	0.463237	−	0.886234i	$-0.346688\pi$
$884$			27.8865i			0.0315458i
$885$		0			0
$886$		−	2058.42i		−	2.32327i
$887$	618.630			0.697440			0.348720	−	0.937227i	$-0.386616\pi$
$887$	618.630			0.697440			0.348720	+	0.937227i	$0.386616\pi$
$888$	−674.865			−0.759983
$889$			1466.08i			1.64914i
$890$		0			0
$891$	−21.9878			−0.0246776
$892$	−544.377			−0.610288
$893$	−277.021	+	1050.06i	−0.310213	+	1.17588i
$894$	4016.41			4.49262
$895$		0			0
$896$		−	632.123i		−	0.705495i
$897$		−	754.185i		−	0.840786i
$898$			2038.80i			2.27038i
$899$	2238.73			2.49024
$900$		0			0
$901$			80.0655i			0.0888630i
$902$		−	109.704i		−	0.121623i
$903$	−1763.25			−1.95266
$904$	189.241			0.209337
$905$		0			0
$906$		−	2057.23i		−	2.27068i
$907$	−289.512			−0.319197			−0.159598	−	0.987182i	$-0.551020\pi$
$907$	−289.512			−0.319197			−0.159598	+	0.987182i	$0.551020\pi$
$908$	−178.706			−0.196813
$909$	1972.85			2.17035
$910$		0			0
$911$			1095.19i			1.20219i	0.799179	+	0.601093i	$0.205268\pi$
$911$			1095.19i			1.20219i	−0.799179	+	0.601093i	$0.794732\pi$
$912$	−919.108	−	242.473i	−1.00779	−	0.265870i
$913$			118.307i			0.129580i
$914$			376.509i			0.411935i
$915$		0			0
$916$	395.032			0.431258
$917$			252.852i			0.275738i
$918$			101.274i			0.110321i
$919$	393.630			0.428324			0.214162	−	0.976798i	$-0.431298\pi$
$919$	393.630			0.428324			0.214162	+	0.976798i	$0.431298\pi$
$920$		0			0
$921$	767.382			0.833206
$922$	−992.300			−1.07625
$923$		−	54.8683i		−	0.0594456i
$924$			773.731i			0.837372i
$925$		0			0
$926$		−	370.739i		−	0.400366i
$927$	−196.935			−0.212444
$928$		−	1913.91i		−	2.06241i
$929$	826.818			0.890008			0.445004	−	0.895529i	$-0.353202\pi$
$929$	826.818			0.890008			0.445004	+	0.895529i	$0.353202\pi$
$930$		0			0
$931$	−204.128	−	53.8518i	−0.219257	−	0.0578429i
$932$		−	454.429i		−	0.487585i
$933$	−1240.89			−1.33000
$934$		−	1971.84i		−	2.11118i
$935$		0			0
$936$	−201.196			−0.214953
$937$			360.005i			0.384210i	0.981374	+	0.192105i	$0.0615315\pi$
$937$			360.005i			0.384210i	−0.981374	+	0.192105i	$0.938469\pi$
$938$			831.854i			0.886838i
$939$		−	2074.21i		−	2.20896i
$940$		0			0
$941$		−	499.144i		−	0.530440i	−0.964188	−	0.265220i	$-0.914555\pi$
$941$		−	499.144i		−	0.530440i	0.964188	−	0.265220i	$-0.0854445\pi$
$942$		−	1956.34i		−	2.07680i
$943$	−264.961			−0.280976
$944$			564.540i			0.598030i
$945$		0			0
$946$		−	911.566i		−	0.963600i
$947$			404.548i			0.427189i	0.976922	+	0.213594i	$0.0685171\pi$
$947$			404.548i			0.427189i	−0.976922	+	0.213594i	$0.931483\pi$
$948$		−	3161.30i		−	3.33471i
$949$		−	563.604i		−	0.593893i
$950$		0			0
$951$	1114.99			1.17244
$952$	−26.6881			−0.0280338
$953$	−441.098			−0.462852			−0.231426	−	0.972852i	$-0.574339\pi$
$953$	−441.098			−0.462852			−0.231426	+	0.972852i	$0.574339\pi$
$954$	−2667.31			−2.79592
$955$		0			0
$956$	−241.088			−0.252184
$957$			1056.68i			1.10416i
$958$	1086.77			1.13442
$959$	−871.673			−0.908939
$960$		0			0
$961$	−1759.60			−1.83101
$962$	530.938			0.551911
$963$	924.725			0.960254
$964$		−	373.482i		−	0.387429i
$965$		0			0
$966$	3332.76			3.45007
$967$			1146.66i			1.18579i	0.805279	+	0.592897i	$0.202016\pi$
$967$			1146.66i			1.18579i	−0.805279	+	0.592897i	$0.797984\pi$
$968$	317.110			0.327593
$969$	−30.4291	+	115.343i	−0.0314026	+	0.119033i
$970$		0			0
$971$		−	37.3147i		−	0.0384292i	−0.999815	−	0.0192146i	$-0.993883\pi$
$971$		−	37.3147i		−	0.0384292i	0.999815	−	0.0192146i	$-0.00611657\pi$
$972$	1293.31			1.33056
$973$		−	819.696i		−	0.842442i
$974$	−142.219			−0.146016
$975$		0			0
$976$	−397.920			−0.407705
$977$	892.499			0.913509			0.456755	−	0.889593i	$-0.349012\pi$
$977$	892.499			0.913509			0.456755	+	0.889593i	$0.349012\pi$
$978$		−	2572.30i		−	2.63017i
$979$			107.338i			0.109640i
$980$		0			0
$981$			669.058i			0.682017i
$982$	2245.86			2.28702
$983$	−336.089			−0.341902			−0.170951	−	0.985280i	$-0.554684\pi$
$983$	−336.089			−0.341902			−0.170951	+	0.985280i	$0.554684\pi$
$984$			115.030i			0.116900i
$985$		0			0
$986$	−168.296			−0.170686
$987$	1699.92			1.72231
$988$	−394.266	−	104.013i	−0.399055	−	0.105276i
$989$	−2201.64			−2.22613
$990$		0			0
$991$			552.478i			0.557495i	0.960364	+	0.278747i	$0.0899192\pi$
$991$			552.478i			0.557495i	−0.960364	+	0.278747i	$0.910081\pi$
$992$			2325.87i			2.34463i
$993$			2553.99i			2.57200i
$994$	242.464			0.243928
$995$		0			0
$996$		−	572.795i		−	0.575095i
$997$		−	459.052i		−	0.460433i	−0.973139	−	0.230217i	$-0.926057\pi$
$997$		−	459.052i		−	0.460433i	0.973139	−	0.230217i	$-0.0739435\pi$
$998$	2458.53			2.46346
$999$	1081.17			1.08225

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	475.3.d.d.474.3		28
5.2	odd	4		475.3.c.h.151.2	✓	14
5.3	odd	4		475.3.c.i.151.13	yes	14
5.4	even	2	inner	475.3.d.d.474.26		28
19.18	odd	2	inner	475.3.d.d.474.25		28
95.18	even	4		475.3.c.i.151.2	yes	14
95.37	even	4		475.3.c.h.151.13	yes	14
95.94	odd	2	inner	475.3.d.d.474.4		28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
475.3.c.h.151.2	✓	14	5.2	odd	4
475.3.c.h.151.13	yes	14	95.37	even	4
475.3.c.i.151.2	yes	14	95.18	even	4
475.3.c.i.151.13	yes	14	5.3	odd	4
475.3.d.d.474.3		28	1.1	even	1	trivial
475.3.d.d.474.4		28	95.94	odd	2	inner
475.3.d.d.474.25		28	19.18	odd	2	inner
475.3.d.d.474.26		28	5.4	even	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)

\(f(q)\)	\(=\)	\(q-3.01757 q^{2} -4.83172 q^{3} +5.10575 q^{4} +14.5801 q^{6} -6.15539i q^{7} -3.33669 q^{8} +14.3455 q^{9} +O(q^{10})\)	\(q-3.01757 q^{2} -4.83172 q^{3} +5.10575 q^{4} +14.5801 q^{6} -6.15539i q^{7} -3.33669 q^{8} +14.3455 q^{9} +5.09534 q^{11} -24.6696 q^{12} +4.20326 q^{13} +18.5743i q^{14} -10.3543 q^{16} +1.29941i q^{17} -43.2887 q^{18} +(-18.3714 - 4.84663i) q^{19} +29.7411i q^{21} -15.3756 q^{22} +37.1355i q^{23} +16.1219 q^{24} -12.6837 q^{26} -25.8282 q^{27} -31.4279i q^{28} -42.9209i q^{29} +52.1594i q^{31} +44.5916 q^{32} -24.6193 q^{33} -3.92107i q^{34} +73.2448 q^{36} -41.8600 q^{37} +(55.4372 + 14.6251i) q^{38} -20.3090 q^{39} +7.13498i q^{41} -89.7461i q^{42} +59.2867i q^{43} +26.0155 q^{44} -112.059i q^{46} -57.1573i q^{47} +50.0291 q^{48} +11.1112 q^{49} -6.27840i q^{51} +21.4608 q^{52} +61.6167 q^{53} +77.9384 q^{54} +20.5386i q^{56} +(88.7657 + 23.4176i) q^{57} +129.517i q^{58} -54.5223i q^{59} +38.4303 q^{61} -157.395i q^{62} -88.3024i q^{63} -93.1413 q^{64} +74.2904 q^{66} +44.7851 q^{67} +6.63448i q^{68} -179.428i q^{69} -13.0537i q^{71} -47.8666 q^{72} -134.087i q^{73} +126.316 q^{74} +(-93.8001 - 24.7457i) q^{76} -31.3638i q^{77} +61.2839 q^{78} +128.146i q^{79} -4.31527 q^{81} -21.5303i q^{82} +23.2187i q^{83} +151.851i q^{84} -178.902i q^{86} +207.382i q^{87} -17.0015 q^{88} +21.0659i q^{89} -25.8727i q^{91} +189.605i q^{92} -252.020i q^{93} +172.476i q^{94} -215.454 q^{96} +75.9288 q^{97} -33.5288 q^{98} +73.0954 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q + 56 q^{4} - 8 q^{6} + 72 q^{9}+O(q^{10}) \) 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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