LMFDB - Embedded newform 260.2.x.a.121.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [260,2,Mod(101,260)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(260, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 5]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("260.101");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 260 = 2^{2} \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	260.x (of order $6$, degree $2$, 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:	$2.07611045255$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{6})$
Coefficient field:	8.0.22581504.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} + 5x^{6} + 2x^{5} - 11x^{4} + 4x^{3} + 20x^{2} - 32x + 16 $ x^8 - 4x^7 + 5x^6 + 2x^5 - 11x^4 + 4x^3 + 20x^2 - 32*x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			121.4
Root			$1.40994 + 0.109843i$ of defining polynomial
Character	$\chi$	$=$	260.121
Dual form			260.2.x.a.101.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.16612 - 2.01978i) q^{3} -1.00000i q^{5} +(0.346241 - 0.199902i) q^{7} +(-1.21969 - 2.11256i) q^{9} +O(q^{10})$	\(q+(1.16612 - 2.01978i) q^{3} -1.00000i q^{5} +(0.346241 - 0.199902i) q^{7} +(-1.21969 - 2.11256i) q^{9} +(1.50000 + 0.866025i) q^{11} +(-0.619491 - 3.55193i) q^{13} +(-2.01978 - 1.16612i) q^{15} +(0.346241 + 0.599706i) q^{17} +(-4.65213 + 2.68591i) q^{19} -0.932442i q^{21} +(-0.0535636 + 0.0927749i) q^{23} -1.00000 q^{25} +1.30752 q^{27} +(2.45174 - 4.24653i) q^{29} +7.86488i q^{31} +(3.49837 - 2.01978i) q^{33} +(-0.199902 - 0.346241i) q^{35} +(1.96128 + 1.13234i) q^{37} +(-7.89654 - 2.89075i) q^{39} +(6.69615 + 3.86603i) q^{41} +(3.00530 + 5.20533i) q^{43} +(-2.11256 + 1.21969i) q^{45} -3.46410i q^{47} +(-3.42008 + 5.92375i) q^{49} +1.61504 q^{51} +11.7189 q^{53} +(0.866025 - 1.50000i) q^{55} +12.5284i q^{57} +(-6.30059 + 3.63765i) q^{59} +(-4.34461 - 7.52509i) q^{61} +(-0.844610 - 0.487636i) q^{63} +(-3.55193 + 0.619491i) q^{65} +(-1.15009 - 0.664004i) q^{67} +(0.124924 + 0.216374i) q^{69} +(-3.35847 + 1.93902i) q^{71} +10.2251i q^{73} +(-1.16612 + 2.01978i) q^{75} +0.692481 q^{77} -13.1533 q^{79} +(5.18379 - 8.97859i) q^{81} +14.0791i q^{83} +(0.599706 - 0.346241i) q^{85} +(-5.71806 - 9.90396i) q^{87} +(0.300587 + 0.173544i) q^{89} +(-0.924532 - 1.10599i) q^{91} +(15.8854 + 9.17142i) q^{93} +(2.68591 + 4.65213i) q^{95} +(-7.66436 + 4.42502i) q^{97} -4.22512i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 2 q^{3} + 6 q^{7} - 4 q^{9}+O(q^{10}) $ 8 * q - 2 * q^3 + 6 * q^7 - 4 * q^9	$ 8 q - 2 q^{3} + 6 q^{7} - 4 q^{9} + 12 q^{11} - 8 q^{13} - 6 q^{15} + 6 q^{17} - 6 q^{23} - 8 q^{25} + 4 q^{27} - 6 q^{33} - 6 q^{35} + 6 q^{37} - 4 q^{39} + 12 q^{41} + 10 q^{43} - 4 q^{49} + 24 q^{53} - 24 q^{59} - 4 q^{61} + 24 q^{63} - 54 q^{67} - 24 q^{69} - 36 q^{71} + 2 q^{75} + 12 q^{77} - 16 q^{79} + 8 q^{81} + 18 q^{85} - 6 q^{87} - 24 q^{89} + 24 q^{93} - 30 q^{97}+O(q^{100}) $ 8 * q - 2 * q^3 + 6 * q^7 - 4 * q^9 + 12 * q^11 - 8 * q^13 - 6 * q^15 + 6 * q^17 - 6 * q^23 - 8 * q^25 + 4 * q^27 - 6 * q^33 - 6 * q^35 + 6 * q^37 - 4 * q^39 + 12 * q^41 + 10 * q^43 - 4 * q^49 + 24 * q^53 - 24 * q^59 - 4 * q^61 + 24 * q^63 - 54 * q^67 - 24 * q^69 - 36 * q^71 + 2 * q^75 + 12 * q^77 - 16 * q^79 + 8 * q^81 + 18 * q^85 - 6 * q^87 - 24 * q^89 + 24 * q^93 - 30 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/260\mathbb{Z}\right)^\times$.

$n$	$41$	$131$	$157$
$\chi(n)$	$e\left(\frac{1}{6}\right)$	$1$	$1$

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 $

$2$		0			0
$2$		0			0
$3$	1.16612	−	2.01978i	0.673262	−	1.16612i	−0.303712	−	0.952764i	$-0.598226\pi$
$3$	1.16612	−	2.01978i	0.673262	−	1.16612i	0.976974	−	0.213359i	$-0.0684405\pi$
$4$		0			0
$5$		−	1.00000i		−	0.447214i
$5$		−	1.00000i		−	0.447214i
$6$		0			0
$7$	0.346241	−	0.199902i	0.130867	−	0.0755559i	−0.433137	−	0.901328i	$-0.642594\pi$
$7$	0.346241	−	0.199902i	0.130867	−	0.0755559i	0.564004	+	0.825772i	$0.309260\pi$
$8$		0			0
$9$	−1.21969	−	2.11256i	−0.406562	−	0.704187i
$10$		0			0
$11$	1.50000	+	0.866025i	0.452267	+	0.261116i	0.708787	−	0.705422i	$-0.249243\pi$
$11$	1.50000	+	0.866025i	0.452267	+	0.261116i	−0.256520	+	0.966539i	$0.582576\pi$
$12$		0			0
$13$	−0.619491	−	3.55193i	−0.171816	−	0.985129i
$13$	−0.619491	−	3.55193i	−0.171816	−	0.985129i
$14$		0			0
$15$	−2.01978	−	1.16612i	−0.521506	−	0.301092i
$16$		0			0
$17$	0.346241	+	0.599706i	0.0839757	+	0.145450i	0.904954	−	0.425509i	$-0.139905\pi$
$17$	0.346241	+	0.599706i	0.0839757	+	0.145450i	−0.820979	+	0.570959i	$0.806572\pi$
$18$		0			0
$19$	−4.65213	+	2.68591i	−1.06727	+	0.616190i	−0.927435	−	0.373985i	$-0.877991\pi$
$19$	−4.65213	+	2.68591i	−1.06727	+	0.616190i	−0.139837	+	0.990175i	$0.544658\pi$
$20$		0			0
$21$		−	0.932442i		−	0.203476i
$22$		0			0
$23$	−0.0535636	+	0.0927749i	−0.0111688	+	0.0193449i	−0.871556	−	0.490296i	$-0.836889\pi$
$23$	−0.0535636	+	0.0927749i	−0.0111688	+	0.0193449i	0.860387	+	0.509641i	$0.170222\pi$
$24$		0			0
$25$	−1.00000			−0.200000
$26$		0			0
$27$	1.30752			0.251632
$28$		0			0
$29$	2.45174	−	4.24653i	0.455276	−	0.788562i	−0.543428	−	0.839456i	$-0.682874\pi$
$29$	2.45174	−	4.24653i	0.455276	−	0.788562i	0.998704	+	0.0508943i	$0.0162072\pi$
$30$		0			0
$31$			7.86488i			1.41257i	0.707925	+	0.706287i	$0.249631\pi$
$31$			7.86488i			1.41257i	−0.707925	+	0.706287i	$0.750369\pi$
$32$		0			0
$33$	3.49837	−	2.01978i	0.608988	−	0.351599i
$34$		0			0
$35$	−0.199902	−	0.346241i	−0.0337896	−	0.0585254i
$36$		0			0
$37$	1.96128	+	1.13234i	0.322432	+	0.186156i	0.652476	−	0.757809i	$-0.273730\pi$
$37$	1.96128	+	1.13234i	0.322432	+	0.186156i	−0.330044	+	0.943966i	$0.607064\pi$
$38$		0			0
$39$	−7.89654	−	2.89075i	−1.26446	−	0.462891i
$40$		0			0
$41$	6.69615	+	3.86603i	1.04576	+	0.603772i	0.921460	−	0.388473i	$-0.126997\pi$
$41$	6.69615	+	3.86603i	1.04576	+	0.603772i	0.124303	+	0.992244i	$0.460331\pi$
$42$		0			0
$43$	3.00530	+	5.20533i	0.458304	+	0.793806i	0.998871	−	0.0474947i	$-0.0151237\pi$
$43$	3.00530	+	5.20533i	0.458304	+	0.793806i	−0.540567	+	0.841301i	$0.681790\pi$
$44$		0			0
$45$	−2.11256	+	1.21969i	−0.314922	+	0.181820i
$46$		0			0
$47$		−	3.46410i		−	0.505291i	−0.967559	−	0.252646i	$-0.918699\pi$
$47$		−	3.46410i		−	0.505291i	0.967559	−	0.252646i	$-0.0813007\pi$
$48$		0			0
$49$	−3.42008	+	5.92375i	−0.488583	+	0.846250i
$50$		0			0
$51$	1.61504			0.226150
$52$		0			0
$53$	11.7189			1.60972			0.804858	−	0.593468i	$-0.202242\pi$
$53$	11.7189			1.60972			0.804858	+	0.593468i	$0.202242\pi$
$54$		0			0
$55$	0.866025	−	1.50000i	0.116775	−	0.202260i
$56$		0			0
$57$			12.5284i			1.65943i
$58$		0			0
$59$	−6.30059	+	3.63765i	−0.820267	+	0.473581i	−0.850508	−	0.525961i	$-0.823706\pi$
$59$	−6.30059	+	3.63765i	−0.820267	+	0.473581i	0.0302418	+	0.999543i	$0.490372\pi$
$60$		0			0
$61$	−4.34461	−	7.52509i	−0.556270	−	0.963489i	−0.997803	−	0.0662436i	$-0.978899\pi$
$61$	−4.34461	−	7.52509i	−0.556270	−	0.963489i	0.441533	−	0.897245i	$-0.354435\pi$
$62$		0			0
$63$	−0.844610	−	0.487636i	−0.106411	−	0.0614364i
$64$		0			0
$65$	−3.55193	+	0.619491i	−0.440563	+	0.0768384i
$66$		0			0
$67$	−1.15009	−	0.664004i	−0.140506	−	0.0811210i	0.428099	−	0.903732i	$-0.359183\pi$
$67$	−1.15009	−	0.664004i	−0.140506	−	0.0811210i	−0.568605	+	0.822611i	$0.692517\pi$
$68$		0			0
$69$	0.124924	+	0.216374i	0.0150390	+	0.0260484i
$70$		0			0
$71$	−3.35847	+	1.93902i	−0.398577	+	0.230119i	−0.685870	−	0.727724i	$-0.740578\pi$
$71$	−3.35847	+	1.93902i	−0.398577	+	0.230119i	0.287293	+	0.957843i	$0.407245\pi$
$72$		0			0
$73$			10.2251i			1.19676i	0.801213	+	0.598380i	$0.204189\pi$
$73$			10.2251i			1.19676i	−0.801213	+	0.598380i	$0.795811\pi$
$74$		0			0
$75$	−1.16612	+	2.01978i	−0.134652	+	0.233225i
$76$		0			0
$77$	0.692481			0.0789156
$78$		0			0
$79$	−13.1533			−1.47986			−0.739932	−	0.672681i	$-0.765142\pi$
$79$	−13.1533			−1.47986			−0.739932	+	0.672681i	$0.765142\pi$
$80$		0			0
$81$	5.18379	−	8.97859i	0.575976	−	0.997621i
$82$		0			0
$83$			14.0791i			1.54539i	0.634780	+	0.772693i	$0.281091\pi$
$83$			14.0791i			1.54539i	−0.634780	+	0.772693i	$0.718909\pi$
$84$		0			0
$85$	0.599706	−	0.346241i	0.0650473	−	0.0375551i
$86$		0			0
$87$	−5.71806	−	9.90396i	−0.613040	−	1.06182i
$88$		0			0
$89$	0.300587	+	0.173544i	0.0318622	+	0.0183956i	0.515846	−	0.856681i	$-0.327477\pi$
$89$	0.300587	+	0.173544i	0.0318622	+	0.0183956i	−0.483984	+	0.875077i	$0.660811\pi$
$90$		0			0
$91$	−0.924532	−	1.10599i	−0.0969173	−	0.115939i
$92$		0			0
$93$	15.8854	+	9.17142i	1.64724	+	0.951032i
$94$		0			0
$95$	2.68591	+	4.65213i	0.275568	+	0.477298i
$96$		0			0
$97$	−7.66436	+	4.42502i	−0.778198	+	0.449293i	−0.835791	−	0.549047i	$-0.814991\pi$
$97$	−7.66436	+	4.42502i	−0.778198	+	0.449293i	0.0575932	+	0.998340i	$0.481657\pi$
$98$		0			0
$99$		−	4.22512i		−	0.424640i
$100$		0			0
$101$	2.05193	−	3.55405i	0.204175	−	0.353641i	−0.745695	−	0.666288i	$-0.767882\pi$
$101$	2.05193	−	3.55405i	0.204175	−	0.353641i	0.949870	+	0.312646i	$0.101216\pi$
$102$		0			0
$103$	−11.2325			−1.10677			−0.553384	−	0.832926i	$-0.686664\pi$
$103$	−11.2325			−1.10677			−0.553384	+	0.832926i	$0.686664\pi$
$104$		0			0
$105$	−0.932442			−0.0909970
$106$		0			0
$107$	8.80165	−	15.2449i	0.850888	−	1.47378i	−0.0295208	−	0.999564i	$-0.509398\pi$
$107$	8.80165	−	15.2449i	0.850888	−	1.47378i	0.880408	−	0.474216i	$-0.157269\pi$
$108$		0			0
$109$		−	15.1830i		−	1.45427i	−0.686495	−	0.727134i	$-0.740852\pi$
$109$		−	15.1830i		−	1.45427i	0.686495	−	0.727134i	$-0.259148\pi$
$110$		0			0
$111$	4.57418	−	2.64091i	0.434162	−	0.250664i
$112$		0			0
$113$	−4.45011	−	7.70781i	−0.418631	−	0.725090i	0.577171	−	0.816623i	$-0.304157\pi$
$113$	−4.45011	−	7.70781i	−0.418631	−	0.725090i	−0.995802	+	0.0915332i	$0.970823\pi$
$114$		0			0
$115$	0.0927749	+	0.0535636i	0.00865131	+	0.00499483i
$116$		0			0
$117$	−6.74809	+	5.64096i	−0.623861	+	0.521507i
$118$		0			0
$119$	0.239765	+	0.138429i	0.0219792	+	0.0126897i
$120$		0			0
$121$	−4.00000	−	6.92820i	−0.363636	−	0.629837i
$122$		0			0
$123$	15.6171	−	9.01652i	1.40814	−	0.812993i
$124$		0			0
$125$			1.00000i			0.0894427i
$126$		0			0
$127$	6.07829	−	10.5279i	0.539361	−	0.934201i	−0.459577	−	0.888138i	$-0.651999\pi$
$127$	6.07829	−	10.5279i	0.539361	−	0.934201i	0.998939	−	0.0460632i	$-0.0146676\pi$
$128$		0			0
$129$	14.0182			1.23423
$130$		0			0
$131$	−2.11773			−0.185027			−0.0925135	−	0.995711i	$-0.529490\pi$
$131$	−2.11773			−0.185027			−0.0925135	+	0.995711i	$0.529490\pi$
$132$		0			0
$133$	−1.07384	+	1.85994i	−0.0931135	+	0.161277i
$134$		0			0
$135$		−	1.30752i		−	0.112533i
$136$		0			0
$137$	15.6171	−	9.01652i	1.33426	−	0.770334i	0.348308	−	0.937380i	$-0.386756\pi$
$137$	15.6171	−	9.01652i	1.33426	−	0.770334i	0.985949	+	0.167046i	$0.0534230\pi$
$138$		0			0
$139$	4.92008	+	8.52183i	0.417316	+	0.722812i	0.995668	−	0.0929749i	$-0.0296376\pi$
$139$	4.92008	+	8.52183i	0.417316	+	0.722812i	−0.578353	+	0.815787i	$0.696304\pi$
$140$		0			0
$141$	−6.99674	−	4.03957i	−0.589232	−	0.340193i
$142$		0			0
$143$	2.14683	−	5.86440i	0.179527	−	0.490405i
$144$		0			0
$145$	−4.24653	−	2.45174i	−0.352655	−	0.203606i
$146$		0			0
$147$	7.97647	+	13.8156i	0.657888	+	1.13950i
$148$		0			0
$149$	−7.69289	+	4.44149i	−0.630226	+	0.363861i	−0.780840	−	0.624731i	$-0.785208\pi$
$149$	−7.69289	+	4.44149i	−0.630226	+	0.363861i	0.150613	+	0.988593i	$0.451875\pi$
$150$		0			0
$151$			4.43937i			0.361271i	0.983550	+	0.180636i	$0.0578155\pi$
$151$			4.43937i			0.361271i	−0.983550	+	0.180636i	$0.942185\pi$
$152$		0			0
$153$	0.844610	−	1.46291i	0.0682827	−	0.118269i
$154$		0			0
$155$	7.86488			0.631723
$156$		0			0
$157$	−4.16719			−0.332578			−0.166289	−	0.986077i	$-0.553178\pi$
$157$	−4.16719			−0.332578			−0.166289	+	0.986077i	$0.553178\pi$
$158$		0			0
$159$	13.6657	−	23.6697i	1.08376	−	1.87713i
$160$		0			0
$161$			0.0428299i			0.00337547i
$162$		0			0
$163$	3.20145	−	1.84836i	0.250757	−	0.144775i	−0.369354	−	0.929289i	$-0.620421\pi$
$163$	3.20145	−	1.84836i	0.250757	−	0.144775i	0.620111	+	0.784514i	$0.287088\pi$
$164$		0			0
$165$	−2.01978	−	3.49837i	−0.157240	−	0.272348i
$166$		0			0
$167$	−18.6171	−	10.7486i	−1.44063	−	0.831750i	−0.442741	−	0.896650i	$-0.645994\pi$
$167$	−18.6171	−	10.7486i	−1.44063	−	0.831750i	−0.997892	+	0.0648999i	$0.979327\pi$
$168$		0			0
$169$	−12.2325	+	4.40078i	−0.940959	+	0.338522i
$170$		0			0
$171$	11.3483	+	6.55193i	0.867825	+	0.501039i
$172$		0			0
$173$	−5.80589	−	10.0561i	−0.441414	−	0.764551i	0.556381	−	0.830927i	$-0.312189\pi$
$173$	−5.80589	−	10.0561i	−0.441414	−	0.764551i	−0.997795	+	0.0663766i	$0.978856\pi$
$174$		0			0
$175$	−0.346241	+	0.199902i	−0.0261733	+	0.0151112i
$176$		0			0
$177$			16.9678i			1.27538i
$178$		0			0
$179$	−2.48516	+	4.30442i	−0.185749	+	0.321728i	−0.943829	−	0.330435i	$-0.892805\pi$
$179$	−2.48516	+	4.30442i	−0.185749	+	0.321728i	0.758079	+	0.652162i	$0.226138\pi$
$180$		0			0
$181$	−17.3695			−1.29107			−0.645534	−	0.763732i	$-0.723365\pi$
$181$	−17.3695			−1.29107			−0.645534	+	0.763732i	$0.723365\pi$
$182$		0			0
$183$	−20.2654			−1.49806
$184$		0			0
$185$	1.13234	−	1.96128i	0.0832516	−	0.144196i
$186$		0			0
$187$			1.19941i			0.0877098i
$188$		0			0
$189$	0.452716	−	0.261376i	0.0329303	−	0.0190123i
$190$		0			0
$191$	10.2523	+	17.7575i	0.741832	+	1.28489i	0.951660	+	0.307153i	$0.0993762\pi$
$191$	10.2523	+	17.7575i	0.741832	+	1.28489i	−0.209828	+	0.977738i	$0.567290\pi$
$192$		0			0
$193$	−23.0428	−	13.3038i	−1.65866	−	0.957626i	−0.973336	−	0.229386i	$-0.926328\pi$
$193$	−23.0428	−	13.3038i	−1.65866	−	0.957626i	−0.685322	−	0.728240i	$-0.740338\pi$
$194$		0			0
$195$	−2.89075	+	7.89654i	−0.207011	+	0.565483i
$196$		0			0
$197$	0.353583	+	0.204141i	0.0251917	+	0.0145445i	0.512543	−	0.858662i	$-0.328704\pi$
$197$	0.353583	+	0.204141i	0.0251917	+	0.0145445i	−0.487351	+	0.873206i	$0.662037\pi$
$198$		0			0
$199$	12.1998	+	21.1307i	0.864823	+	1.49792i	0.867223	+	0.497919i	$0.165902\pi$
$199$	12.1998	+	21.1307i	0.864823	+	1.49792i	−0.00240070	+	0.999997i	$0.500764\pi$
$200$		0			0
$201$	−2.68229	+	1.54862i	−0.189194	+	0.109231i
$202$		0			0
$203$		−	1.96043i		−	0.137595i
$204$		0			0
$205$	3.86603	−	6.69615i	0.270015	−	0.467680i
$206$		0			0
$207$	0.261323			0.0181632
$208$		0			0
$209$	−9.30426			−0.643589
$210$		0			0
$211$	−0.880509	+	1.52509i	−0.0606167	+	0.104991i	−0.894741	−	0.446585i	$-0.852640\pi$
$211$	−0.880509	+	1.52509i	−0.0606167	+	0.104991i	0.834125	+	0.551576i	$0.185973\pi$
$212$		0			0
$213$			9.04452i			0.619721i
$214$		0			0
$215$	5.20533	−	3.00530i	0.355001	−	0.204960i
$216$		0			0
$217$	1.57221	+	2.72314i	0.106728	+	0.184859i
$218$		0			0
$219$	20.6525	+	11.9237i	1.39557	+	0.805732i
$220$		0			0
$221$	1.91562	−	1.60134i	0.128859	−	0.107718i
$222$		0			0
$223$	9.80263	+	5.65955i	0.656433	+	0.378991i	0.790916	−	0.611924i	$-0.209604\pi$
$223$	9.80263	+	5.65955i	0.656433	+	0.378991i	−0.134484	+	0.990916i	$0.542938\pi$
$224$		0			0
$225$	1.21969	+	2.11256i	0.0813125	+	0.140837i
$226$		0			0
$227$	−6.98084	+	4.03039i	−0.463334	+	0.267506i	−0.713445	−	0.700711i	$-0.752866\pi$
$227$	−6.98084	+	4.03039i	−0.463334	+	0.267506i	0.250111	+	0.968217i	$0.419533\pi$
$228$		0			0
$229$			11.5715i			0.764666i	0.924025	+	0.382333i	$0.124879\pi$
$229$			11.5715i			0.764666i	−0.924025	+	0.382333i	$0.875121\pi$
$230$		0			0
$231$	0.807519	−	1.39866i	0.0531308	−	0.0920253i
$232$		0			0
$233$	−24.0900			−1.57819			−0.789094	−	0.614272i	$-0.789450\pi$
$233$	−24.0900			−1.57819			−0.789094	+	0.614272i	$0.789450\pi$
$234$		0			0
$235$	−3.46410			−0.225973
$236$		0			0
$237$	−15.3384	+	26.5669i	−0.996336	+	1.72570i
$238$		0			0
$239$		−	30.7089i		−	1.98639i	−0.116459	−	0.993196i	$-0.537154\pi$
$239$		−	30.7089i		−	1.98639i	0.116459	−	0.993196i	$-0.462846\pi$
$240$		0			0
$241$	6.86541	−	3.96374i	0.442240	−	0.255327i	−0.262308	−	0.964984i	$-0.584483\pi$
$241$	6.86541	−	3.96374i	0.442240	−	0.255327i	0.704547	+	0.709657i	$0.251150\pi$
$242$		0			0
$243$	−10.1286	−	17.5432i	−0.649750	−	1.12540i
$244$		0			0
$245$	5.92375	+	3.42008i	0.378454	+	0.218501i
$246$		0			0
$247$	12.4221	+	14.8602i	0.790401	+	0.945529i
$248$		0			0
$249$	28.4368	+	16.4180i	1.80211	+	1.04045i
$250$		0			0
$251$	11.3112	+	19.5916i	0.713956	+	1.23661i	0.963361	+	0.268209i	$0.0864317\pi$
$251$	11.3112	+	19.5916i	0.713956	+	1.23661i	−0.249405	+	0.968399i	$0.580235\pi$
$252$		0			0
$253$	−0.160691	+	0.0927749i	−0.0101025	+	0.00583271i
$254$		0			0
$255$		−	1.61504i		−	0.101138i
$256$		0			0
$257$	12.8982	−	22.3403i	0.804566	−	1.39355i	−0.112018	−	0.993706i	$-0.535731\pi$
$257$	12.8982	−	22.3403i	0.804566	−	1.39355i	0.916584	−	0.399843i	$-0.130935\pi$
$258$		0			0
$259$	0.905432			0.0562608
$260$		0			0
$261$	−11.9614			−0.740393
$262$		0			0
$263$	−0.795286	+	1.37748i	−0.0490394	+	0.0849388i	−0.889503	−	0.456929i	$-0.848949\pi$
$263$	−0.795286	+	1.37748i	−0.0490394	+	0.0849388i	0.840464	+	0.541868i	$0.182283\pi$
$264$		0			0
$265$		−	11.7189i		−	0.719887i
$266$		0			0
$267$	0.701043	−	0.404747i	0.0429031	−	0.0247701i
$268$		0			0
$269$	13.9114	+	24.0952i	0.848192	+	1.46911i	0.882820	+	0.469712i	$0.155642\pi$
$269$	13.9114	+	24.0952i	0.848192	+	1.46911i	−0.0346278	+	0.999400i	$0.511025\pi$
$270$		0			0
$271$	−20.3520	−	11.7502i	−1.23629	−	0.713774i	−0.267959	−	0.963430i	$-0.586349\pi$
$271$	−20.3520	−	11.7502i	−1.23629	−	0.713774i	−0.968335	+	0.249656i	$0.919682\pi$
$272$		0			0
$273$	−3.31197	+	0.577640i	−0.200450	+	0.0349603i
$274$		0			0
$275$	−1.50000	−	0.866025i	−0.0904534	−	0.0522233i
$276$		0			0
$277$	−1.49837	−	2.59525i	−0.0900283	−	0.155934i	0.817494	−	0.575936i	$-0.195362\pi$
$277$	−1.49837	−	2.59525i	−0.0900283	−	0.155934i	−0.907523	+	0.420003i	$0.862029\pi$
$278$		0			0
$279$	16.6150	−	9.59270i	0.994716	−	0.574300i
$280$		0			0
$281$			24.6085i			1.46802i	0.679138	+	0.734011i	$0.262354\pi$
$281$			24.6085i			1.46802i	−0.679138	+	0.734011i	$0.737646\pi$
$282$		0			0
$283$	4.08444	−	7.07446i	0.242795	−	0.420533i	−0.718715	−	0.695305i	$-0.755269\pi$
$283$	4.08444	−	7.07446i	0.242795	−	0.420533i	0.961509	+	0.274772i	$0.0886025\pi$
$284$		0			0
$285$	12.5284			0.742118
$286$		0			0
$287$	3.09131			0.182474
$288$		0			0
$289$	8.26023	−	14.3071i	0.485896	−	0.841597i
$290$		0			0
$291$			20.6405i			1.20997i
$292$		0			0
$293$	−6.49837	+	3.75184i	−0.379639	+	0.219185i	−0.677661	−	0.735374i	$-0.737006\pi$
$293$	−6.49837	+	3.75184i	−0.379639	+	0.219185i	0.298022	+	0.954559i	$0.403673\pi$
$294$		0			0
$295$	3.63765	+	6.30059i	0.211792	+	0.366834i
$296$		0			0
$297$	1.96128	+	1.13234i	0.113805	+	0.0657053i
$298$		0			0
$299$	0.362712	+	0.132781i	0.0209762	+	0.00767893i
$300$		0			0
$301$	2.08112	+	1.20153i	0.119953	+	0.0692552i
$302$		0			0
$303$	−4.78561	−	8.28893i	−0.274926	−	0.476186i
$304$		0			0
$305$	−7.52509	+	4.34461i	−0.430885	+	0.248772i
$306$		0			0
$307$			5.95293i			0.339752i	0.985465	+	0.169876i	$0.0543367\pi$
$307$			5.95293i			0.339752i	−0.985465	+	0.169876i	$0.945663\pi$
$308$		0			0
$309$	−13.0984	+	22.6872i	−0.745144	+	1.29063i
$310$		0			0
$311$	−17.9247			−1.01642			−0.508208	−	0.861235i	$-0.669692\pi$
$311$	−17.9247			−1.01642			−0.508208	+	0.861235i	$0.669692\pi$
$312$		0			0
$313$	−17.0073			−0.961312			−0.480656	−	0.876909i	$-0.659601\pi$
$313$	−17.0073			−0.961312			−0.480656	+	0.876909i	$0.659601\pi$
$314$		0			0
$315$	−0.487636	+	0.844610i	−0.0274752	+	0.0475884i
$316$		0			0
$317$			3.09300i			0.173720i	0.996221	+	0.0868601i	$0.0276833\pi$
$317$			3.09300i			0.173720i	−0.996221	+	0.0868601i	$0.972317\pi$
$318$		0			0
$319$	7.35521	−	4.24653i	0.411813	−	0.237760i
$320$		0			0
$321$	−20.5276	−	35.5549i	−1.14574	−	1.98448i
$322$		0			0
$323$	−3.22151	−	1.85994i	−0.179250	−	0.103490i
$324$		0			0
$325$	0.619491	+	3.55193i	0.0343632	+	0.197026i
$326$		0			0
$327$	−30.6664	−	17.7053i	−1.69586	−	0.979103i
$328$		0			0
$329$	−0.692481	−	1.19941i	−0.0381777	−	0.0661258i
$330$		0			0
$331$	19.5481	−	11.2861i	1.07446	−	0.620340i	0.145064	−	0.989422i	$-0.453661\pi$
$331$	19.5481	−	11.2861i	1.07446	−	0.620340i	0.929397	+	0.369082i	$0.120328\pi$
$332$		0			0
$333$		−	5.52442i		−	0.302736i
$334$		0			0
$335$	−0.664004	+	1.15009i	−0.0362784	+	0.0628360i
$336$		0			0
$337$	18.0603			0.983808			0.491904	−	0.870649i	$-0.336301\pi$
$337$	18.0603			0.983808			0.491904	+	0.870649i	$0.336301\pi$
$338$		0			0
$339$	−20.7575			−1.12739
$340$		0			0
$341$	−6.81119	+	11.7973i	−0.368847	+	0.638861i
$342$		0			0
$343$			5.53335i			0.298773i
$344$		0			0
$345$	0.216374	−	0.124924i	0.0116492	−	0.00672566i
$346$		0			0
$347$	−13.2359	−	22.9252i	−0.710538	−	1.23069i	−0.964655	−	0.263514i	$-0.915118\pi$
$347$	−13.2359	−	22.9252i	−0.710538	−	1.23069i	0.254118	−	0.967173i	$-0.418215\pi$
$348$		0			0
$349$	10.7190	+	6.18860i	0.573773	+	0.331268i	0.758655	−	0.651493i	$-0.225857\pi$
$349$	10.7190	+	6.18860i	0.573773	+	0.331268i	−0.184882	+	0.982761i	$0.559190\pi$
$350$		0			0
$351$	−0.809996	−	4.64422i	−0.0432344	−	0.247890i
$352$		0			0
$353$	16.6978	+	9.64047i	0.888733	+	0.513110i	0.873528	−	0.486774i	$-0.161826\pi$
$353$	16.6978	+	9.64047i	0.888733	+	0.513110i	0.0152053	+	0.999884i	$0.495160\pi$
$354$		0			0
$355$	1.93902	+	3.35847i	0.102912	+	0.178249i
$356$		0			0
$357$	0.559192	−	0.322849i	0.0295956	−	0.0170870i
$358$		0			0
$359$		−	26.5506i		−	1.40129i	−0.713512	−	0.700643i	$-0.752897\pi$
$359$		−	26.5506i		−	1.40129i	0.713512	−	0.700643i	$-0.247103\pi$
$360$		0			0
$361$	4.92820	−	8.53590i	0.259379	−	0.449258i
$362$		0			0
$363$	−18.6580			−0.979290
$364$		0			0
$365$	10.2251			0.535207
$366$		0			0
$367$	−3.68718	+	6.38638i	−0.192469	+	0.333366i	−0.946068	−	0.323968i	$-0.894983\pi$
$367$	−3.68718	+	6.38638i	−0.192469	+	0.333366i	0.753599	+	0.657335i	$0.228316\pi$
$368$		0			0
$369$		−	18.8614i		−	0.981883i
$370$		0			0
$371$	4.05756	−	2.34263i	0.210658	−	0.121624i
$372$		0			0
$373$	14.1574	+	24.5214i	0.733044	+	1.26967i	0.955576	+	0.294744i	$0.0952344\pi$
$373$	14.1574	+	24.5214i	0.733044	+	1.26967i	−0.222532	+	0.974925i	$0.571432\pi$
$374$		0			0
$375$	2.01978	+	1.16612i	0.104301	+	0.0602183i
$376$		0			0
$377$	−16.6022	−	6.07772i	−0.855059	−	0.313018i
$378$		0			0
$379$	9.02975	+	5.21333i	0.463827	+	0.267791i	0.713652	−	0.700500i	$-0.247040\pi$
$379$	9.02975	+	5.21333i	0.463827	+	0.267791i	−0.249825	+	0.968291i	$0.580373\pi$
$380$		0			0
$381$	−14.1761	−	24.5537i	−0.726262	−	1.25792i
$382$		0			0
$383$	−9.39811	+	5.42600i	−0.480221	+	0.277256i	−0.720509	−	0.693446i	$-0.756092\pi$
$383$	−9.39811	+	5.42600i	−0.480221	+	0.277256i	0.240288	+	0.970702i	$0.422758\pi$
$384$		0			0
$385$		−	0.692481i		−	0.0352921i
$386$		0			0
$387$	7.33105	−	12.6978i	0.372658	−	0.645463i
$388$		0			0
$389$	20.2893			1.02871			0.514353	−	0.857578i	$-0.328032\pi$
$389$	20.2893			1.02871			0.514353	+	0.857578i	$0.328032\pi$
$390$		0			0
$391$	−0.0741836			−0.00375163
$392$		0			0
$393$	−2.46953	+	4.27736i	−0.124572	+	0.215764i
$394$		0			0
$395$			13.1533i			0.661815i
$396$		0			0
$397$	21.8695	−	12.6263i	1.09760	−	0.633698i	0.162008	−	0.986789i	$-0.448203\pi$
$397$	21.8695	−	12.6263i	1.09760	−	0.633698i	0.935589	+	0.353091i	$0.114870\pi$
$398$		0			0
$399$	2.50445	+	4.33784i	0.125380	+	0.217164i
$400$		0			0
$401$	10.8377	+	6.25714i	0.541208	+	0.312467i	0.745568	−	0.666429i	$-0.232178\pi$
$401$	10.8377	+	6.25714i	0.541208	+	0.312467i	−0.204360	+	0.978896i	$0.565511\pi$
$402$		0			0
$403$	27.9355	−	4.87223i	1.39157	−	0.242703i
$404$		0			0
$405$	−8.97859	−	5.18379i	−0.446149	−	0.257585i
$406$		0			0
$407$	1.96128	+	3.39703i	0.0972169	+	0.168385i
$408$		0			0
$409$	5.19248	−	2.99788i	0.256752	−	0.148236i	−0.366100	−	0.930575i	$-0.619307\pi$
$409$	5.19248	−	2.99788i	0.256752	−	0.148236i	0.622852	+	0.782340i	$0.285974\pi$
$410$		0			0
$411$		−	42.0575i		−	2.07454i
$412$		0			0
$413$	−1.45435	+	2.51900i	−0.0715637	+	0.123952i
$414$		0			0
$415$	14.0791			0.691118
$416$		0			0
$417$	22.9497			1.12385
$418$		0			0
$419$	5.48516	−	9.50057i	0.267968	−	0.464133i	−0.700369	−	0.713781i	$-0.746981\pi$
$419$	5.48516	−	9.50057i	0.267968	−	0.464133i	0.968337	+	0.249647i	$0.0803147\pi$
$420$		0			0
$421$		−	36.6085i		−	1.78419i	−0.451848	−	0.892095i	$-0.649235\pi$
$421$		−	36.6085i		−	1.78419i	0.451848	−	0.892095i	$-0.350765\pi$
$422$		0			0
$423$	−7.31812	+	4.22512i	−0.355819	+	0.205432i
$424$		0			0
$425$	−0.346241	−	0.599706i	−0.0167951	−	0.0290900i
$426$		0			0
$427$	−3.00856	−	1.73699i	−0.145595	−	0.0840590i
$428$		0			0
$429$	−9.34135	−	11.1747i	−0.451005	−	0.539521i
$430$		0			0
$431$	−10.4873	−	6.05484i	−0.505155	−	0.291651i	0.225685	−	0.974200i	$-0.427538\pi$
$431$	−10.4873	−	6.05484i	−0.505155	−	0.291651i	−0.730840	+	0.682549i	$0.760871\pi$
$432$		0			0
$433$	−4.50897	−	7.80977i	−0.216687	−	0.375314i	0.737106	−	0.675777i	$-0.236192\pi$
$433$	−4.50897	−	7.80977i	−0.216687	−	0.375314i	−0.953793	+	0.300464i	$0.902859\pi$
$434$		0			0
$435$	−9.90396	+	5.71806i	−0.474859	+	0.274160i
$436$		0			0
$437$		−	0.575468i		−	0.0275284i
$438$		0			0
$439$	−6.07547	+	10.5230i	−0.289966	+	0.502236i	−0.973801	−	0.227400i	$-0.926977\pi$
$439$	−6.07547	+	10.5230i	−0.289966	+	0.502236i	0.683835	+	0.729637i	$0.260311\pi$
$440$		0			0
$441$	16.6857			0.794557
$442$		0			0
$443$	15.3116			0.727476			0.363738	−	0.931501i	$-0.381500\pi$
$443$	15.3116			0.727476			0.363738	+	0.931501i	$0.381500\pi$
$444$		0			0
$445$	0.173544	−	0.300587i	0.00822678	−	0.0142492i
$446$		0			0
$447$			20.7173i			0.979895i
$448$		0			0
$449$	−19.6929	+	11.3697i	−0.929365	+	0.536569i	−0.886611	−	0.462516i	$-0.846947\pi$
$449$	−19.6929	+	11.3697i	−0.929365	+	0.536569i	−0.0427543	+	0.999086i	$0.513613\pi$
$450$		0			0
$451$	6.69615	+	11.5981i	0.315310	+	0.546132i
$452$		0			0
$453$	8.96658	+	5.17686i	0.421287	+	0.243230i
$454$		0			0
$455$	−1.10599	+	0.924532i	−0.0518494	+	0.0433427i
$456$		0			0
$457$	−9.30548	−	5.37252i	−0.435292	−	0.251316i	0.266307	−	0.963888i	$-0.414197\pi$
$457$	−9.30548	−	5.37252i	−0.435292	−	0.251316i	−0.701599	+	0.712572i	$0.747530\pi$
$458$		0			0
$459$	0.452716	+	0.784127i	0.0211310	+	0.0365999i
$460$		0			0
$461$	10.2973	−	5.94516i	0.479594	−	0.276894i	−0.240653	−	0.970611i	$-0.577362\pi$
$461$	10.2973	−	5.94516i	0.479594	−	0.276894i	0.720247	+	0.693717i	$0.244028\pi$
$462$		0			0
$463$			3.39726i			0.157884i	0.996879	+	0.0789420i	$0.0251542\pi$
$463$			3.39726i			0.157884i	−0.996879	+	0.0789420i	$0.974846\pi$
$464$		0			0
$465$	9.17142	−	15.8854i	0.425315	−	0.736667i
$466$		0			0
$467$	−6.39426			−0.295891			−0.147946	−	0.988996i	$-0.547266\pi$
$467$	−6.39426			−0.295891			−0.147946	+	0.988996i	$0.547266\pi$
$468$		0			0
$469$	−0.530943			−0.0245167
$470$		0			0
$471$	−4.85945	+	8.41682i	−0.223912	+	0.387826i
$472$		0			0
$473$			10.4107i			0.478683i
$474$		0			0
$475$	4.65213	−	2.68591i	0.213454	−	0.123238i
$476$		0			0
$477$	−14.2934	−	24.7569i	−0.654450	−	1.13354i
$478$		0			0
$479$	14.1330	+	8.15968i	0.645752	+	0.372825i	0.786827	−	0.617174i	$-0.211722\pi$
$479$	14.1330	+	8.15968i	0.645752	+	0.372825i	−0.141075	+	0.989999i	$0.545056\pi$
$480$		0			0
$481$	2.80702	−	7.66781i	0.127989	−	0.349622i
$482$		0			0
$483$	0.0865072	+	0.0499450i	0.00393622	+	0.00227258i
$484$		0			0
$485$	4.42502	+	7.66436i	0.200930	+	0.348021i
$486$		0			0
$487$	−10.3356	+	5.96728i	−0.468352	+	0.270403i	−0.715550	−	0.698562i	$-0.753824\pi$
$487$	−10.3356	+	5.96728i	−0.468352	+	0.270403i	0.247197	+	0.968965i	$0.420490\pi$
$488$		0			0
$489$		−	8.62166i		−	0.389885i
$490$		0			0
$491$	−17.0259	+	29.4896i	−0.768366	+	1.33085i	0.170082	+	0.985430i	$0.445597\pi$
$491$	−17.0259	+	29.4896i	−0.768366	+	1.33085i	−0.938448	+	0.345419i	$0.887737\pi$
$492$		0			0
$493$	3.39557			0.152929
$494$		0			0
$495$	−4.22512			−0.189905
$496$		0			0
$497$	−0.775227	+	1.34273i	−0.0347737	+	0.0602298i
$498$		0			0
$499$		−	12.5854i		−	0.563398i	−0.959503	−	0.281699i	$-0.909102\pi$
$499$		−	12.5854i		−	0.563398i	0.959503	−	0.281699i	$-0.0908979\pi$
$500$		0			0
$501$	−43.4196	+	25.0683i	−1.93985	+	1.11997i
$502$		0			0
$503$	9.09433	+	15.7518i	0.405496	+	0.702340i	0.994379	−	0.105879i	$-0.0337655\pi$
$503$	9.09433	+	15.7518i	0.405496	+	0.702340i	−0.588883	+	0.808218i	$0.700432\pi$
$504$		0			0
$505$	−3.55405	−	2.05193i	−0.158153	−	0.0913098i
$506$		0			0
$507$	−5.37592	+	29.8388i	−0.238753	+	1.32519i
$508$		0			0
$509$	−25.1265	−	14.5068i	−1.11371	−	0.643001i	−0.173923	−	0.984759i	$-0.555644\pi$
$509$	−25.1265	−	14.5068i	−1.11371	−	0.643001i	−0.939788	+	0.341758i	$0.888978\pi$
$510$		0			0
$511$	2.04402	+	3.54035i	0.0904223	+	0.156616i
$512$		0			0
$513$	−6.08275	+	3.51187i	−0.268560	+	0.155053i
$514$		0			0
$515$			11.2325i			0.494961i
$516$		0			0
$517$	3.00000	−	5.19615i	0.131940	−	0.228527i
$518$		0			0
$519$	−27.0815			−1.18875
$520$		0			0
$521$	35.0240			1.53443			0.767214	−	0.641391i	$-0.221642\pi$
$521$	35.0240			1.53443			0.767214	+	0.641391i	$0.221642\pi$
$522$		0			0
$523$	4.63870	−	8.03447i	0.202836	−	0.351323i	−0.746605	−	0.665268i	$-0.768317\pi$
$523$	4.63870	−	8.03447i	0.202836	−	0.351323i	0.949441	+	0.313945i	$0.101651\pi$
$524$		0			0
$525$			0.932442i			0.0406951i
$526$		0			0
$527$	−4.71662	+	2.72314i	−0.205459	+	0.118622i
$528$		0			0
$529$	11.4943	+	19.9086i	0.499751	+	0.865593i
$530$		0			0
$531$	15.3695	+	8.87358i	0.666979	+	0.385080i
$532$		0			0
$533$	9.58366	−	26.1793i	0.415114	−	1.13395i
$534$		0			0
$535$	−15.2449	−	8.80165i	−0.659095	−	0.380528i
$536$		0			0
$537$	5.79600	+	10.0390i	0.250116	+	0.433214i
$538$		0			0
$539$	−10.2602	+	5.92375i	−0.441940	+	0.255154i
$540$		0			0
$541$			24.3814i			1.04824i	0.851644	+	0.524120i	$0.175606\pi$
$541$			24.3814i			1.04824i	−0.851644	+	0.524120i	$0.824394\pi$
$542$		0			0
$543$	−20.2550	+	35.0827i	−0.869226	+	1.50554i
$544$		0			0
$545$	−15.1830			−0.650369
$546$		0			0
$547$	26.5270			1.13421			0.567106	−	0.823645i	$-0.308063\pi$
$547$	26.5270			1.13421			0.567106	+	0.823645i	$0.308063\pi$
$548$		0			0
$549$	−10.5981	+	18.3565i	−0.452317	+	0.783436i
$550$		0			0
$551$			26.3406i			1.12215i
$552$		0			0
$553$	−4.55422	+	2.62938i	−0.193665	+	0.111813i
$554$		0			0
$555$	−2.64091	−	4.57418i	−0.112100	−	0.194163i
$556$		0			0
$557$	−2.78142	−	1.60586i	−0.117853	−	0.0680423i	0.439915	−	0.898039i	$-0.355009\pi$
$557$	−2.78142	−	1.60586i	−0.117853	−	0.0680423i	−0.557768	+	0.829997i	$0.688342\pi$
$558$		0			0
$559$	16.6272	−	13.8993i	0.703257	−	0.587877i
$560$		0			0
$561$	2.42256	+	1.39866i	0.102280	+	0.0590516i
$562$		0			0
$563$	−17.1426	−	29.6918i	−0.722474	−	1.25136i	−0.960005	−	0.279982i	$-0.909672\pi$
$563$	−17.1426	−	29.6918i	−0.722474	−	1.25136i	0.237531	−	0.971380i	$-0.423662\pi$
$564$		0			0
$565$	−7.70781	+	4.45011i	−0.324270	+	0.187217i
$566$		0			0
$567$		−	4.14500i		−	0.174074i
$568$		0			0
$569$	−17.8228	+	30.8701i	−0.747172	+	1.29414i	0.202001	+	0.979385i	$0.435256\pi$
$569$	−17.8228	+	30.8701i	−0.747172	+	1.29414i	−0.949173	+	0.314755i	$0.898078\pi$
$570$		0			0
$571$	−4.53590			−0.189821			−0.0949107	−	0.995486i	$-0.530257\pi$
$571$	−4.53590			−0.189821			−0.0949107	+	0.995486i	$0.530257\pi$
$572$		0			0
$573$	47.8219			1.99779
$574$		0			0
$575$	0.0535636	−	0.0927749i	0.00223376	−	0.00386898i
$576$		0			0
$577$			18.4475i			0.767981i	0.923337	+	0.383991i	$0.125450\pi$
$577$			18.4475i			0.767981i	−0.923337	+	0.383991i	$0.874550\pi$
$578$		0			0
$579$	−53.7415	+	31.0277i	−2.23342	+	1.28947i
$580$		0			0
$581$	2.81445	+	4.87477i	0.116763	+	0.202240i
$582$		0			0
$583$	17.5784	+	10.1489i	0.728021	+	0.420323i
$584$		0			0
$585$	5.64096	+	6.74809i	0.233225	+	0.278999i
$586$		0			0
$587$	−37.2316	−	21.4957i	−1.53671	−	0.887222i	−0.999028	−	0.0440760i	$-0.985966\pi$
$587$	−37.2316	−	21.4957i	−1.53671	−	0.887222i	−0.537685	−	0.843146i	$-0.680701\pi$
$588$		0			0
$589$	−21.1244	−	36.5885i	−0.870414	−	1.50760i
$590$		0			0
$591$	0.824642	−	0.476107i	0.0339212	−	0.0195844i
$592$		0			0
$593$		−	12.8614i		−	0.528153i	−0.964502	−	0.264076i	$-0.914933\pi$
$593$		−	12.8614i		−	0.528153i	0.964502	−	0.264076i	$-0.0850671\pi$
$594$		0			0
$595$	0.138429	−	0.239765i	0.00567502	−	0.00982942i
$596$		0			0
$597$	56.9060			2.32901
$598$		0			0
$599$	28.3170			1.15700			0.578500	−	0.815682i	$-0.303638\pi$
$599$	28.3170			1.15700			0.578500	+	0.815682i	$0.303638\pi$
$600$		0			0
$601$	3.56734	−	6.17882i	0.145515	−	0.252039i	−0.784050	−	0.620698i	$-0.786849\pi$
$601$	3.56734	−	6.17882i	0.145515	−	0.252039i	0.929565	+	0.368658i	$0.120183\pi$
$602$		0			0
$603$			3.23951i			0.131923i
$604$		0			0
$605$	−6.92820	+	4.00000i	−0.281672	+	0.162623i
$606$		0			0
$607$	11.0901	+	19.2086i	0.450133	+	0.779653i	0.998394	−	0.0566544i	$-0.0180433\pi$
$607$	11.0901	+	19.2086i	0.450133	+	0.779653i	−0.548261	+	0.836307i	$0.684710\pi$
$608$		0			0
$609$	−3.95965	−	2.28610i	−0.160453	−	0.0926376i
$610$		0			0
$611$	−12.3043	+	2.14598i	−0.497777	+	0.0868171i
$612$		0			0
$613$	−0.279399	−	0.161311i	−0.0112848	−	0.00651530i	0.494347	−	0.869265i	$-0.335407\pi$
$613$	−0.279399	−	0.161311i	−0.0112848	−	0.00651530i	−0.505632	+	0.862749i	$0.668741\pi$
$614$		0			0
$615$	−9.01652	−	15.6171i	−0.363581	−	0.629741i
$616$		0			0
$617$	−32.2150	+	18.5993i	−1.29693	+	0.748781i	−0.979872	−	0.199626i	$-0.936027\pi$
$617$	−32.2150	+	18.5993i	−1.29693	+	0.748781i	−0.317055	+	0.948407i	$0.602694\pi$
$618$		0			0
$619$		−	3.94911i		−	0.158728i	−0.996846	−	0.0793641i	$-0.974711\pi$
$619$		−	3.94911i		−	0.158728i	0.996846	−	0.0793641i	$-0.0252890\pi$
$620$		0			0
$621$	−0.0700354	+	0.121305i	−0.00281043	+	0.00486780i
$622$		0			0
$623$	0.138767			0.00555959
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$	−10.8499	+	18.7926i	−0.433304	+	0.750504i
$628$		0			0
$629$			1.56825i			0.0625304i
$630$		0			0
$631$	29.0824	−	16.7908i	1.15775	−	0.668429i	0.206989	−	0.978343i	$-0.433634\pi$
$631$	29.0824	−	16.7908i	1.15775	−	0.668429i	0.950765	+	0.309914i	$0.100300\pi$
$632$		0			0
$633$	2.05356	+	3.55688i	0.0816218	+	0.141373i
$634$		0			0
$635$	−10.5279	−	6.07829i	−0.417787	−	0.241210i
$636$		0			0
$637$	23.1595	+	8.47818i	0.917612	+	0.335918i
$638$		0			0
$639$	8.19257	+	4.72998i	0.324093	+	0.187115i
$640$		0			0
$641$	16.5900	+	28.7347i	0.655266	+	1.13495i	0.981827	+	0.189778i	$0.0607767\pi$
$641$	16.5900	+	28.7347i	0.655266	+	1.13495i	−0.326561	+	0.945176i	$0.605890\pi$
$642$		0			0
$643$	27.1643	−	15.6833i	1.07125	−	0.618489i	0.142730	−	0.989762i	$-0.454412\pi$
$643$	27.1643	−	15.6833i	1.07125	−	0.618489i	0.928524	+	0.371273i	$0.121078\pi$
$644$		0			0
$645$		−	14.0182i		−	0.551966i
$646$		0			0
$647$	9.79529	−	16.9659i	0.385092	−	0.667000i	−0.606690	−	0.794939i	$-0.707503\pi$
$647$	9.79529	−	16.9659i	0.385092	−	0.667000i	0.991782	+	0.127939i	$0.0408362\pi$
$648$		0			0
$649$	−12.6012			−0.494639
$650$		0			0
$651$	7.33355			0.287424
$652$		0			0
$653$	−3.55626	+	6.15962i	−0.139167	+	0.241044i	−0.927182	−	0.374612i	$-0.877776\pi$
$653$	−3.55626	+	6.15962i	−0.139167	+	0.241044i	0.788015	+	0.615657i	$0.211109\pi$
$654$		0			0
$655$			2.11773i			0.0827466i
$656$		0			0
$657$	21.6012	−	12.4714i	0.842742	−	0.486557i
$658$		0			0
$659$	−9.29211	−	16.0944i	−0.361969	−	0.626949i	0.626316	−	0.779570i	$-0.284562\pi$
$659$	−9.29211	−	16.0944i	−0.361969	−	0.626949i	−0.988285	+	0.152621i	$0.951229\pi$
$660$		0			0
$661$	14.5413	+	8.39540i	0.565590	+	0.326543i	0.755386	−	0.655280i	$-0.227449\pi$
$661$	14.5413	+	8.39540i	0.565590	+	0.326543i	−0.189796	+	0.981823i	$0.560783\pi$
$662$		0			0
$663$	−1.00050	−	5.73650i	−0.0388563	−	0.222787i
$664$		0			0
$665$	1.85994	+	1.07384i	0.0721254	+	0.0416416i
$666$		0			0
$667$	0.262648	+	0.454919i	0.0101698	+	0.0176146i
$668$		0			0
$669$	22.8621	−	13.1995i	0.883902	−	0.510321i
$670$		0			0
$671$		−	15.0502i		−	0.581005i
$672$		0			0
$673$	11.2957	−	19.5647i	0.435417	−	0.754165i	−0.561912	−	0.827197i	$-0.689934\pi$
$673$	11.2957	−	19.5647i	0.435417	−	0.754165i	0.997330	+	0.0730322i	$0.0232676\pi$
$674$		0			0
$675$	−1.30752			−0.0503264
$676$		0			0
$677$	5.31616			0.204317			0.102158	−	0.994768i	$-0.467425\pi$
$677$	5.31616			0.204317			0.102158	+	0.994768i	$0.467425\pi$
$678$		0			0
$679$	−1.76914	+	3.06424i	−0.0678935	+	0.117595i
$680$		0			0
$681$			18.7997i			0.720407i
$682$		0			0
$683$	3.42419	−	1.97695i	0.131023	−	0.0756461i	−0.433056	−	0.901367i	$-0.642565\pi$
$683$	3.42419	−	1.97695i	0.131023	−	0.0756461i	0.564079	+	0.825721i	$0.309231\pi$
$684$		0			0
$685$	−9.01652	−	15.6171i	−0.344504	−	0.596698i
$686$		0			0
$687$	23.3719	+	13.4938i	0.891694	+	0.514820i
$688$		0			0
$689$	−7.25976	−	41.6248i	−0.276575	−	1.58578i
$690$		0			0
$691$	−11.1493	−	6.43704i	−0.424139	−	0.244877i	0.272708	−	0.962097i	$-0.412081\pi$
$691$	−11.1493	−	6.43704i	−0.424139	−	0.244877i	−0.696846	+	0.717220i	$0.745414\pi$
$692$		0			0
$693$	−0.844610	−	1.46291i	−0.0320841	−	0.0555713i
$694$		0			0
$695$	8.52183	−	4.92008i	0.323251	−	0.186629i
$696$		0			0
$697$			5.35430i			0.202809i
$698$		0			0
$699$	−28.0919	+	48.6566i	−1.06253	+	1.84036i
$700$		0			0
$701$	34.9777			1.32109			0.660544	−	0.750787i	$-0.270326\pi$
$701$	34.9777			1.32109			0.660544	+	0.750787i	$0.270326\pi$
$702$		0			0
$703$	−12.1655			−0.458830
$704$		0			0
$705$	−4.03957	+	6.99674i	−0.152139	+	0.263512i
$706$		0			0
$707$		−	1.64074i		−	0.0617065i
$708$		0			0
$709$	−17.1183	+	9.88325i	−0.642891	+	0.371173i	−0.785727	−	0.618573i	$-0.787711\pi$
$709$	−17.1183	+	9.88325i	−0.642891	+	0.371173i	0.142836	+	0.989746i	$0.454378\pi$
$710$		0			0
$711$	16.0429	+	27.7872i	0.601657	+	1.04210i
$712$		0			0
$713$	−0.729664	−	0.421272i	−0.0273261	−	0.0157767i
$714$		0			0
$715$	−5.86440	−	2.14683i	−0.219316	−	0.0802868i
$716$		0			0
$717$	−62.0253	−	35.8103i	−2.31638	−	1.33736i
$718$		0			0
$719$	−22.1234	−	38.3188i	−0.825062	−	1.42905i	−0.901872	−	0.432004i	$-0.857807\pi$
$719$	−22.1234	−	38.3188i	−0.825062	−	1.42905i	0.0768099	−	0.997046i	$-0.475527\pi$
$720$		0			0
$721$	−3.88914	+	2.24539i	−0.144839	+	0.0836228i
$722$		0			0
$723$		−	18.4889i		−	0.687608i
$724$		0			0
$725$	−2.45174	+	4.24653i	−0.0910553	+	0.157712i
$726$		0			0
$727$	−31.4877			−1.16781			−0.583907	−	0.811821i	$-0.698477\pi$
$727$	−31.4877			−1.16781			−0.583907	+	0.811821i	$0.698477\pi$
$728$		0			0
$729$	−16.1420			−0.597853
$730$		0			0
$731$	−2.08112	+	3.60460i	−0.0769728	+	0.133321i
$732$		0			0
$733$		−	42.4714i		−	1.56872i	−0.620307	−	0.784359i	$-0.712992\pi$
$733$		−	42.4714i		−	1.56872i	0.620307	−	0.784359i	$-0.287008\pi$
$734$		0			0
$735$	13.8156	−	7.97647i	0.509598	−	0.294216i
$736$		0			0
$737$	−1.15009	−	1.99201i	−0.0423640	−	0.0733767i
$738$		0			0
$739$	−11.9368	−	6.89173i	−0.439103	−	0.253516i	0.264114	−	0.964492i	$-0.414921\pi$
$739$	−11.9368	−	6.89173i	−0.439103	−	0.253516i	−0.703217	+	0.710975i	$0.748254\pi$
$740$		0			0
$741$	44.5000	−	7.76123i	1.63475	−	0.285116i
$742$		0			0
$743$	−13.8038	−	7.96961i	−0.506411	−	0.292377i	0.224946	−	0.974371i	$-0.427779\pi$
$743$	−13.8038	−	7.96961i	−0.506411	−	0.292377i	−0.731357	+	0.681995i	$0.761113\pi$
$744$		0			0
$745$	4.44149	+	7.69289i	0.162724	+	0.281846i
$746$		0			0
$747$	29.7430	−	17.1721i	1.08824	−	0.628296i
$748$		0			0
$749$		−	7.03787i		−	0.257158i
$750$		0			0
$751$	−0.758540	+	1.31383i	−0.0276795	+	0.0479423i	−0.879533	−	0.475837i	$-0.842145\pi$
$751$	−0.758540	+	1.31383i	−0.0276795	+	0.0479423i	0.851854	+	0.523780i	$0.175478\pi$
$752$		0			0
$753$	52.7610			1.92272
$754$		0			0
$755$	4.43937			0.161565
$756$		0			0
$757$	−23.7414	+	41.1214i	−0.862897	+	1.49458i	0.00622310	+	0.999981i	$0.498019\pi$
$757$	−23.7414	+	41.1214i	−0.862897	+	1.49458i	−0.869120	+	0.494601i	$0.835314\pi$
$758$		0			0
$759$			0.432748i			0.0157078i
$760$		0			0
$761$	−33.3805	+	19.2722i	−1.21004	+	0.698618i	−0.962768	−	0.270327i	$-0.912868\pi$
$761$	−33.3805	+	19.2722i	−1.21004	+	0.698618i	−0.247274	+	0.968946i	$0.579535\pi$
$762$		0			0
$763$	−3.03512	−	5.25697i	−0.109879	−	0.190315i
$764$		0			0
$765$	−1.46291	−	0.844610i	−0.0528916	−	0.0305370i
$766$		0			0
$767$	16.8238	+	20.1258i	0.607473	+	0.726700i
$768$		0			0
$769$	34.5236	+	19.9322i	1.24495	+	0.718775i	0.970099	−	0.242711i	$-0.0780366\pi$
$769$	34.5236	+	19.9322i	1.24495	+	0.718775i	0.274856	+	0.961486i	$0.411370\pi$
$770$		0			0
$771$	−30.0817	−	52.1031i	−1.08337	−	1.87645i
$772$		0			0
$773$	28.5396	−	16.4774i	1.02650	−	0.592650i	0.110519	−	0.993874i	$-0.464749\pi$
$773$	28.5396	−	16.4774i	1.02650	−	0.592650i	0.915980	+	0.401224i	$0.131415\pi$
$774$		0			0
$775$		−	7.86488i		−	0.282515i
$776$		0			0
$777$	1.05585	−	1.82878i	0.0378783	−	0.0656071i
$778$		0			0
$779$	−41.5352			−1.48815
$780$		0			0
$781$	−6.71695			−0.240351
$782$		0			0
$783$	3.20569	−	5.55242i	0.114562	−	0.198427i
$784$		0			0
$785$			4.16719i			0.148733i
$786$		0			0
$787$	−0.934698	+	0.539648i	−0.0333184	+	0.0192364i	−0.516567	−	0.856247i	$-0.672790\pi$
$787$	−0.934698	+	0.539648i	−0.0333184	+	0.0192364i	0.483248	+	0.875483i	$0.339457\pi$
$788$		0			0
$789$	1.85480	+	3.21261i	0.0660327	+	0.114372i
$790$		0			0
$791$	−3.08162	−	1.77

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:	\( N \)	\(=\)	\( 260 = 2^{2} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	260.x (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(1.16612 - 2.01978i) q^{3} -1.00000i q^{5} +(0.346241 - 0.199902i) q^{7} +(-1.21969 - 2.11256i) q^{9} +O(q^{10})\)	\(q+(1.16612 - 2.01978i) q^{3} -1.00000i q^{5} +(0.346241 - 0.199902i) q^{7} +(-1.21969 - 2.11256i) q^{9} +(1.50000 + 0.866025i) q^{11} +(-0.619491 - 3.55193i) q^{13} +(-2.01978 - 1.16612i) q^{15} +(0.346241 + 0.599706i) q^{17} +(-4.65213 + 2.68591i) q^{19} -0.932442i q^{21} +(-0.0535636 + 0.0927749i) q^{23} -1.00000 q^{25} +1.30752 q^{27} +(2.45174 - 4.24653i) q^{29} +7.86488i q^{31} +(3.49837 - 2.01978i) q^{33} +(-0.199902 - 0.346241i) q^{35} +(1.96128 + 1.13234i) q^{37} +(-7.89654 - 2.89075i) q^{39} +(6.69615 + 3.86603i) q^{41} +(3.00530 + 5.20533i) q^{43} +(-2.11256 + 1.21969i) q^{45} -3.46410i q^{47} +(-3.42008 + 5.92375i) q^{49} +1.61504 q^{51} +11.7189 q^{53} +(0.866025 - 1.50000i) q^{55} +12.5284i q^{57} +(-6.30059 + 3.63765i) q^{59} +(-4.34461 - 7.52509i) q^{61} +(-0.844610 - 0.487636i) q^{63} +(-3.55193 + 0.619491i) q^{65} +(-1.15009 - 0.664004i) q^{67} +(0.124924 + 0.216374i) q^{69} +(-3.35847 + 1.93902i) q^{71} +10.2251i q^{73} +(-1.16612 + 2.01978i) q^{75} +0.692481 q^{77} -13.1533 q^{79} +(5.18379 - 8.97859i) q^{81} +14.0791i q^{83} +(0.599706 - 0.346241i) q^{85} +(-5.71806 - 9.90396i) q^{87} +(0.300587 + 0.173544i) q^{89} +(-0.924532 - 1.10599i) q^{91} +(15.8854 + 9.17142i) q^{93} +(2.68591 + 4.65213i) q^{95} +(-7.66436 + 4.42502i) q^{97} -4.22512i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 2 q^{3} + 6 q^{7} - 4 q^{9}+O(q^{10}) \) 8 * q - 2 * q^3 + 6 * q^7 - 4 * q^9	\( 8 q - 2 q^{3} + 6 q^{7} - 4 q^{9} + 12 q^{11} - 8 q^{13} - 6 q^{15} + 6 q^{17} - 6 q^{23} - 8 q^{25} + 4 q^{27} - 6 q^{33} - 6 q^{35} + 6 q^{37} - 4 q^{39} + 12 q^{41} + 10 q^{43} - 4 q^{49} + 24 q^{53} - 24 q^{59} - 4 q^{61} + 24 q^{63} - 54 q^{67} - 24 q^{69} - 36 q^{71} + 2 q^{75} + 12 q^{77} - 16 q^{79} + 8 q^{81} + 18 q^{85} - 6 q^{87} - 24 q^{89} + 24 q^{93} - 30 q^{97}+O(q^{100}) \) 8 * q - 2 * q^3 + 6 * q^7 - 4 * q^9 + 12 * q^11 - 8 * q^13 - 6 * q^15 + 6 * q^17 - 6 * q^23 - 8 * q^25 + 4 * q^27 - 6 * q^33 - 6 * q^35 + 6 * q^37 - 4 * q^39 + 12 * q^41 + 10 * q^43 - 4 * q^49 + 24 * q^53 - 24 * q^59 - 4 * q^61 + 24 * q^63 - 54 * q^67 - 24 * q^69 - 36 * q^71 + 2 * q^75 + 12 * q^77 - 16 * q^79 + 8 * q^81 + 18 * q^85 - 6 * q^87 - 24 * q^89 + 24 * q^93 - 30 * q^97

Introduction

L-functions

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