LMFDB - Embedded newform 93.2.p.a.53.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [93,2,Mod(11,93)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(93, base_ring=CyclotomicField(30))

chi = DirichletCharacter(H, H._module([15, 23]))

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

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

chi := DirichletCharacter("93.11");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 93 = 3 \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	93.p (of order $30$, degree $8$, 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:	$0.742608738798$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\Q(\zeta_{15})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} + x^{5} - x^{4} + x^{3} - x + 1 $ x^8 - x^7 + x^5 - x^4 + x^3 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{30}]$

Embedding invariants

Embedding label			53.1
Root			$-0.104528 + 0.994522i$ of defining polynomial
Character	$\chi$	$=$	93.53
Dual form			93.2.p.a.86.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.360114 - 1.69420i) q^{3} +(-1.61803 - 1.17557i) q^{4} +(1.88052 - 0.837263i) q^{7} +(-2.74064 - 1.22021i) q^{9} +O(q^{10})$	$q+(0.360114 - 1.69420i) q^{3} +(-1.61803 - 1.17557i) q^{4} +(1.88052 - 0.837263i) q^{7} +(-2.74064 - 1.22021i) q^{9} +(-2.57433 + 2.31794i) q^{12} +(5.30806 + 4.77940i) q^{13} +(1.23607 + 3.80423i) q^{16} +(-1.74569 - 1.93879i) q^{19} +(-0.741290 - 3.48749i) q^{21} +(-2.50000 + 4.33013i) q^{25} +(-3.05422 + 4.20378i) q^{27} +(-4.02701 - 0.855968i) q^{28} +(5.55987 + 0.296414i) q^{31} +(3.00000 + 5.19615i) q^{36} +(-10.4506 - 6.03366i) q^{37} +(10.0088 - 7.27180i) q^{39} +(6.31815 - 5.68889i) q^{43} +(6.89025 - 0.724194i) q^{48} +(-1.84856 + 2.05303i) q^{49} +(-2.97010 - 13.9732i) q^{52} +(-3.91335 + 2.25937i) q^{57} +2.62396i q^{61} -6.17547 q^{63} +(2.47214 - 7.60845i) q^{64} +(7.49021 + 12.9734i) q^{67} +(-15.5471 - 1.63406i) q^{73} +(6.43582 + 5.79484i) q^{75} +(0.545408 + 5.18921i) q^{76} +(-11.5976 + 1.21895i) q^{79} +(6.02218 + 6.68830i) q^{81} +(-2.90036 + 6.51432i) q^{84} +(13.9835 + 4.54353i) q^{91} +(2.50437 - 9.31279i) q^{93} +(-5.78749 - 4.20486i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 3 q^{3} - 4 q^{4} - 4 q^{7} - 3 q^{9}+O(q^{10}) $ 8 * q + 3 * q^3 - 4 * q^4 - 4 * q^7 - 3 * q^9	$ 8 q + 3 q^{3} - 4 q^{4} - 4 q^{7} - 3 q^{9} + 6 q^{12} + 9 q^{13} - 8 q^{16} + 7 q^{19} - 33 q^{21} - 20 q^{25} - 18 q^{28} - 4 q^{31} + 24 q^{36} + 9 q^{37} + 27 q^{39} + 44 q^{43} + 12 q^{48} + 9 q^{49} + 18 q^{52} - 21 q^{57} - 24 q^{63} - 16 q^{64} + 16 q^{67} + 3 q^{73} - 15 q^{75} - 66 q^{76} - 35 q^{79} + 9 q^{81} + 24 q^{84} - 5 q^{91} + 15 q^{93} + 15 q^{97}+O(q^{100}) $ 8 * q + 3 * q^3 - 4 * q^4 - 4 * q^7 - 3 * q^9 + 6 * q^12 + 9 * q^13 - 8 * q^16 + 7 * q^19 - 33 * q^21 - 20 * q^25 - 18 * q^28 - 4 * q^31 + 24 * q^36 + 9 * q^37 + 27 * q^39 + 44 * q^43 + 12 * q^48 + 9 * q^49 + 18 * q^52 - 21 * q^57 - 24 * q^63 - 16 * q^64 + 16 * q^67 + 3 * q^73 - 15 * q^75 - 66 * q^76 - 35 * q^79 + 9 * q^81 + 24 * q^84 - 5 * q^91 + 15 * q^93 + 15 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/93\mathbb{Z}\right)^\times$.

$n$	$32$	$34$
$\chi(n)$	$-1$	$e\left(\frac{17}{30}\right)$

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		0.309017	−	0.951057i	$-0.400000\pi$
$2$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$3$	0.360114	−	1.69420i	0.207912	−	0.978148i
$3$	0.360114	−	1.69420i	0.207912	−	0.978148i
$4$	−1.61803	−	1.17557i	−0.809017	−	0.587785i
$5$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$5$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$6$		0			0
$7$	1.88052	−	0.837263i	0.710771	−	0.316456i	−0.0193127	−	0.999813i	$-0.506148\pi$
$7$	1.88052	−	0.837263i	0.710771	−	0.316456i	0.730084	+	0.683358i	$0.239481\pi$
$8$		0			0
$9$	−2.74064	−	1.22021i	−0.913545	−	0.406737i
$10$		0			0
$11$		0			0		−0.994522	−	0.104528i	$-0.966667\pi$
$11$		0			0		0.994522	+	0.104528i	$0.0333333\pi$
$12$	−2.57433	+	2.31794i	−0.743145	+	0.669131i
$13$	5.30806	+	4.77940i	1.47219	+	1.32557i	0.828040	+	0.560670i	$0.189456\pi$
$13$	5.30806	+	4.77940i	1.47219	+	1.32557i	0.644152	+	0.764898i	$0.277210\pi$
$14$		0			0
$15$		0			0
$16$	1.23607	+	3.80423i	0.309017	+	0.951057i
$17$		0			0		0.994522	−	0.104528i	$-0.0333333\pi$
$17$		0			0		−0.994522	+	0.104528i	$0.966667\pi$
$18$		0			0
$19$	−1.74569	−	1.93879i	−0.400489	−	0.444789i	0.508843	−	0.860859i	$-0.330073\pi$
$19$	−1.74569	−	1.93879i	−0.400489	−	0.444789i	−0.909332	+	0.416071i	$0.863407\pi$
$20$		0			0
$21$	−0.741290	−	3.48749i	−0.161763	−	0.761034i
$22$		0			0
$23$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$23$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$24$		0			0
$25$	−2.50000	+	4.33013i	−0.500000	+	0.866025i
$26$		0			0
$27$	−3.05422	+	4.20378i	−0.587785	+	0.809017i
$28$	−4.02701	−	0.855968i	−0.761034	−	0.161763i
$29$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$29$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$30$		0			0
$31$	5.55987	+	0.296414i	0.998582	+	0.0532375i
$31$	5.55987	+	0.296414i	0.998582	+	0.0532375i
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$	3.00000	+	5.19615i	0.500000	+	0.866025i
$37$	−10.4506	−	6.03366i	−1.71807	−	0.991928i	−0.922437	−	0.386148i	$-0.873806\pi$
$37$	−10.4506	−	6.03366i	−1.71807	−	0.991928i	−0.795632	−	0.605780i	$-0.792861\pi$
$38$		0			0
$39$	10.0088	−	7.27180i	1.60269	−	1.16442i
$40$		0			0
$41$		0			0		0.978148	−	0.207912i	$-0.0666667\pi$
$41$		0			0		−0.978148	+	0.207912i	$0.933333\pi$
$42$		0			0
$43$	6.31815	−	5.68889i	0.963509	−	0.867548i	−0.0277313	−	0.999615i	$-0.508828\pi$
$43$	6.31815	−	5.68889i	0.963509	−	0.867548i	0.991241	+	0.132068i	$0.0421616\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$47$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$48$	6.89025	−	0.724194i	0.994522	−	0.104528i
$49$	−1.84856	+	2.05303i	−0.264080	+	0.293290i
$50$		0			0
$51$		0			0
$52$	−2.97010	−	13.9732i	−0.411879	−	1.93774i
$53$		0			0		0.406737	−	0.913545i	$-0.366667\pi$
$53$		0			0		−0.406737	+	0.913545i	$0.633333\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−3.91335	+	2.25937i	−0.518335	+	0.299261i
$58$		0			0
$59$		0			0		−0.978148	−	0.207912i	$-0.933333\pi$
$59$		0			0		0.978148	+	0.207912i	$0.0666667\pi$
$60$		0			0
$61$			2.62396i			0.335964i	0.985790	+	0.167982i	$0.0537250\pi$
$61$			2.62396i			0.335964i	−0.985790	+	0.167982i	$0.946275\pi$
$62$		0			0
$63$	−6.17547			−0.778035
$64$	2.47214	−	7.60845i	0.309017	−	0.951057i
$65$		0			0
$66$		0			0
$67$	7.49021	+	12.9734i	0.915075	+	1.58496i	0.806792	+	0.590836i	$0.201202\pi$
$67$	7.49021	+	12.9734i	0.915075	+	1.58496i	0.108283	+	0.994120i	$0.465465\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		−0.913545	−	0.406737i	$-0.866667\pi$
$71$		0			0		0.913545	+	0.406737i	$0.133333\pi$
$72$		0			0
$73$	−15.5471	−	1.63406i	−1.81965	−	0.191253i	−0.867613	−	0.497239i	$-0.834347\pi$
$73$	−15.5471	−	1.63406i	−1.81965	−	0.191253i	−0.952036	+	0.305987i	$0.901014\pi$
$74$		0			0
$75$	6.43582	+	5.79484i	0.743145	+	0.669131i
$76$	0.545408	+	5.18921i	0.0625626	+	0.595243i
$77$		0			0
$78$		0			0
$79$	−11.5976	+	1.21895i	−1.30483	+	0.137143i	−0.731307	−	0.682048i	$-0.761089\pi$
$79$	−11.5976	+	1.21895i	−1.30483	+	0.137143i	−0.573521	+	0.819191i	$0.694423\pi$
$80$		0			0
$81$	6.02218	+	6.68830i	0.669131	+	0.743145i
$82$		0			0
$83$		0			0		−0.207912	−	0.978148i	$-0.566667\pi$
$83$		0			0		0.207912	+	0.978148i	$0.433333\pi$
$84$	−2.90036	+	6.51432i	−0.316456	+	0.710771i
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$89$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$90$		0			0
$91$	13.9835	+	4.54353i	1.46587	+	0.476291i
$92$		0			0
$93$	2.50437	−	9.31279i	0.259691	−	0.965692i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−5.78749	−	4.20486i	−0.587630	−	0.426938i	0.253837	−	0.967247i	$-0.418307\pi$
$97$	−5.78749	−	4.20486i	−0.587630	−	0.426938i	−0.841467	+	0.540309i	$0.818307\pi$
$98$		0			0
$99$		0			0
$100$	9.13545	−	4.06737i	0.913545	−	0.406737i
$101$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$101$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$102$		0			0
$103$	−18.4310	+	3.91762i	−1.81606	+	0.386015i	−0.985329	−	0.170664i	$-0.945409\pi$
$103$	−18.4310	+	3.91762i	−1.81606	+	0.386015i	−0.830728	+	0.556679i	$0.812075\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		−0.104528	−	0.994522i	$-0.533333\pi$
$107$		0			0		0.104528	+	0.994522i	$0.466667\pi$
$108$	9.88367	−	3.21140i	0.951057	−	0.309017i
$109$	5.18661	+	15.9628i	0.496788	+	1.52895i	0.814152	+	0.580651i	$0.197202\pi$
$109$	5.18661	+	15.9628i	0.496788	+	1.52895i	−0.317365	+	0.948304i	$0.602798\pi$
$110$		0			0
$111$	−13.9856	+	15.5326i	−1.32746	+	1.47429i
$112$	5.50959	+	6.11902i	0.520607	+	0.578193i
$113$		0			0		0.104528	−	0.994522i	$-0.466667\pi$
$113$		0			0		−0.104528	+	0.994522i	$0.533333\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−8.71560	−	19.5755i	−0.805757	−	1.80976i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	10.7596	+	2.28703i	0.978148	+	0.207912i
$122$		0			0
$123$		0			0
$124$	−8.64760	−	7.01562i	−0.776578	−	0.630022i
$125$		0			0
$126$		0			0
$127$	3.90960	−	18.3932i	0.346921	−	1.63214i	−0.365804	−	0.930692i	$-0.619206\pi$
$127$	3.90960	−	18.3932i	0.346921	−	1.63214i	0.712725	−	0.701443i	$-0.247461\pi$
$128$		0			0
$129$	−7.36287	−	12.7529i	−0.648265	−	1.12283i
$130$		0			0
$131$		0			0		0.913545	−	0.406737i	$-0.133333\pi$
$131$		0			0		−0.913545	+	0.406737i	$0.866667\pi$
$132$		0			0
$133$	−4.90609	−	2.18433i	−0.425412	−	0.189406i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		−0.743145	−	0.669131i	$-0.766667\pi$
$137$		0			0		0.743145	+	0.669131i	$0.233333\pi$
$138$		0			0
$139$	19.5625	−	6.35624i	1.65927	−	0.539130i	0.678551	−	0.734553i	$-0.262608\pi$
$139$	19.5625	−	6.35624i	1.65927	−	0.539130i	0.980719	+	0.195424i	$0.0626082\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	1.25434	−	11.9343i	0.104528	−	0.994522i
$145$		0			0
$146$		0			0
$147$	2.81256	+	3.87115i	0.231976	+	0.319287i
$148$	9.81644	+	22.0481i	0.806907	+	1.81234i
$149$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$149$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$150$		0			0
$151$	11.2844	−	15.5316i	0.918310	−	1.26395i	−0.0459384	−	0.998944i	$-0.514628\pi$
$151$	11.2844	−	15.5316i	0.918310	−	1.26395i	0.964248	−	0.265001i	$-0.0853722\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	−24.7430			−1.98103
$157$	6.83983	−	21.0508i	0.545878	−	1.68004i	−0.173016	−	0.984919i	$-0.555351\pi$
$157$	6.83983	−	21.0508i	0.545878	−	1.68004i	0.718894	−	0.695120i	$-0.244649\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	10.2480	−	7.44563i	0.802688	−	0.583187i	−0.109014	−	0.994040i	$-0.534769\pi$
$163$	10.2480	−	7.44563i	0.802688	−	0.583187i	0.911701	+	0.410854i	$0.134769\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.743145	−	0.669131i	$-0.233333\pi$
$167$		0			0		−0.743145	+	0.669131i	$0.766667\pi$
$168$		0			0
$169$	3.97398	+	37.8099i	0.305691	+	2.90846i
$170$		0			0
$171$	2.41858	+	7.44363i	0.184953	+	0.569228i
$172$	−16.9107	+	1.77738i	−1.28943	+	0.135524i
$173$		0			0		0.669131	−	0.743145i	$-0.266667\pi$
$173$		0			0		−0.669131	+	0.743145i	$0.733333\pi$
$174$		0			0
$175$	−1.07585	+	10.2361i	−0.0813269	+	0.773773i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		−0.406737	−	0.913545i	$-0.633333\pi$
$179$		0			0		0.406737	+	0.913545i	$0.366667\pi$
$180$		0			0
$181$	1.18741	−	0.685551i	0.0882594	−	0.0509566i	−0.455221	−	0.890379i	$-0.650440\pi$
$181$	1.18741	−	0.685551i	0.0882594	−	0.0509566i	0.543480	+	0.839422i	$0.317106\pi$
$182$		0			0
$183$	4.44552	+	0.944923i	0.328622	+	0.0698508i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	−2.22387	+	10.4625i	−0.161763	+	0.761034i
$190$		0			0
$191$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$191$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$192$	−12.0000	−	6.92820i	−0.866025	−	0.500000i
$193$	−24.9872	+	11.1250i	−1.79862	+	0.800798i	−0.827788	+	0.561041i	$0.810401\pi$
$193$	−24.9872	+	11.1250i	−1.79862	+	0.800798i	−0.970833	+	0.239757i	$0.922932\pi$
$194$		0			0
$195$		0			0
$196$	5.40451	−	1.14876i	0.386036	−	0.0820546i
$197$		0			0		−0.994522	−	0.104528i	$-0.966667\pi$
$197$		0			0		0.994522	+	0.104528i	$0.0333333\pi$
$198$		0			0
$199$	−20.5851	−	18.5350i	−1.45924	−	1.31391i	−0.856693	−	0.515826i	$-0.827485\pi$
$199$	−20.5851	−	18.5350i	−1.45924	−	1.31391i	−0.602549	−	0.798082i	$-0.705848\pi$
$200$		0			0
$201$	24.6769	−	8.01802i	1.74058	−	0.565547i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	−11.6208	+	26.1007i	−0.805757	+	1.80976i
$209$		0			0
$210$		0			0
$211$	−5.30439	+	9.18747i	−0.365169	+	0.632491i	−0.988803	−	0.149225i	$-0.952322\pi$
$211$	−5.30439	+	9.18747i	−0.365169	+	0.632491i	0.623634	+	0.781716i	$0.285655\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	10.7036	−	4.09766i	0.726610	−	0.278167i
$218$		0			0
$219$	−8.36715	+	25.7515i	−0.565400	+	1.74012i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−1.64093	−	0.947392i	−0.109885	−	0.0634420i	0.444050	−	0.896002i	$-0.353541\pi$
$223$	−1.64093	−	0.947392i	−0.109885	−	0.0634420i	−0.553935	+	0.832560i	$0.686874\pi$
$224$		0			0
$225$	12.1353	−	8.81678i	0.809017	−	0.587785i
$226$		0			0
$227$		0			0		0.978148	−	0.207912i	$-0.0666667\pi$
$227$		0			0		−0.978148	+	0.207912i	$0.933333\pi$
$228$	8.98798	+	0.944674i	0.595243	+	0.0625626i
$229$	17.8742	−	16.0940i	1.18116	−	1.06352i	0.184418	−	0.982848i	$-0.440960\pi$
$229$	17.8742	−	16.0940i	1.18116	−	1.06352i	0.996741	−	0.0806717i	$-0.0257065\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$233$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−2.11129	+	20.0876i	−0.137143	+	1.30483i
$238$		0			0
$239$		0			0		0.406737	−	0.913545i	$-0.366667\pi$
$239$		0			0		−0.406737	+	0.913545i	$0.633333\pi$
$240$		0			0
$241$	0.704489	+	1.58231i	0.0453801	+	0.101925i	0.934813	−	0.355141i	$-0.115567\pi$
$241$	0.704489	+	1.58231i	0.0453801	+	0.101925i	−0.889433	+	0.457066i	$0.848900\pi$
$242$		0			0
$243$	13.5000	−	7.79423i	0.866025	−	0.500000i
$244$	3.08465	−	4.24566i	0.197474	−	0.271800i
$245$		0			0
$246$		0			0
$247$		−	18.6346i		−	1.18569i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		0.207912	−	0.978148i	$-0.433333\pi$
$251$		0			0		−0.207912	+	0.978148i	$0.566667\pi$
$252$	9.99211	+	7.25969i	0.629444	+	0.457318i
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−12.9443	+	9.40456i	−0.809017	+	0.587785i
$257$		0			0		−0.913545	−	0.406737i	$-0.866667\pi$
$257$		0			0		0.913545	+	0.406737i	$0.133333\pi$
$258$		0			0
$259$	−24.7044	−	2.59653i	−1.53505	−	0.161341i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$263$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	3.13176	−	29.7967i	0.191303	−	1.82012i
$269$		0			0		−0.207912	−	0.978148i	$-0.566667\pi$
$269$		0			0		0.207912	+	0.978148i	$0.433333\pi$
$270$		0			0
$271$	−13.3801	−	18.4161i	−0.812782	−	1.11870i	−0.990888	−	0.134687i	$-0.956997\pi$
$271$	−13.3801	−	18.4161i	−0.812782	−	1.11870i	0.178107	−	0.984011i	$-0.443003\pi$
$272$		0			0
$273$	12.7333	−	22.0548i	0.770655	−	1.33481i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	11.5309	+	3.74663i	0.692827	+	0.225113i	0.634203	−	0.773167i	$-0.281328\pi$
$277$	11.5309	+	3.74663i	0.692827	+	0.225113i	0.0586248	+	0.998280i	$0.481328\pi$
$278$		0			0
$279$	−14.8759	−	7.59657i	−0.890596	−	0.454795i
$280$		0			0
$281$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$281$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$282$		0			0
$283$	−23.5708	−	17.1252i	−1.40114	−	1.01799i	−0.994537	−	0.104384i	$-0.966713\pi$
$283$	−23.5708	−	17.1252i	−1.40114	−	1.01799i	−0.406604	−	0.913604i	$-0.633287\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	16.6285	−	3.53450i	0.978148	−	0.207912i
$290$		0			0
$291$	−9.20803	+	8.29094i	−0.539784	+	0.486024i
$292$	23.2348	+	20.9207i	1.35971	+	1.22429i
$293$		0			0		−0.104528	−	0.994522i	$-0.533333\pi$
$293$		0			0		0.104528	+	0.994522i	$0.466667\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$	−3.60114	−	16.9420i	−0.207912	−	0.978148i
$301$	7.11834	−	15.9880i	0.410294	−	0.921536i
$302$		0			0
$303$		0			0
$304$	5.21779	−	9.03748i	0.299261	−	0.518335i
$305$		0			0
$306$		0			0
$307$	−1.89363	−	0.402503i	−0.108075	−	0.0229721i	0.153557	−	0.988140i	$-0.450927\pi$
$307$	−1.89363	−	0.402503i	−0.108075	−	0.0229721i	−0.261632	+	0.965168i	$0.584261\pi$
$308$		0			0
$309$			32.6366i			1.85663i
$310$		0			0
$311$		0			0		1.00000			$0$
$311$		0			0		−1.00000			$\pi$
$312$		0			0
$313$	0.456222	−	2.14635i	0.0257872	−	0.121319i	−0.963371	−	0.268171i	$-0.913581\pi$
$313$	0.456222	−	2.14635i	0.0257872	−	0.121319i	0.989158	+	0.146852i	$0.0469141\pi$
$314$		0			0
$315$		0			0
$316$	20.1982	+	11.6614i	1.13624	+	0.656007i
$317$		0			0		0.913545	−	0.406737i	$-0.133333\pi$
$317$		0			0		−0.913545	+	0.406737i	$0.866667\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	−1.88151	−	17.9014i	−0.104528	−	0.994522i
$325$	−33.9656	+	11.0361i	−1.88407	+	0.612172i
$326$		0			0
$327$	28.9119	−	3.03876i	1.59883	−	0.168044i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	3.96125	+	18.6362i	0.217730	+	1.02434i	0.942205	+	0.335038i	$0.108749\pi$
$331$	3.96125	+	18.6362i	0.217730	+	1.02434i	−0.724475	+	0.689301i	$0.757918\pi$
$332$		0			0
$333$	21.2790	+	29.2880i	1.16608	+	1.60497i
$334$		0			0
$335$		0			0
$336$	12.3509	−	7.13081i	0.673799	−	0.389018i
$337$	−18.3358	+	25.2371i	−0.998816	+	1.37475i	−0.0727670	+	0.997349i	$0.523183\pi$
$337$	−18.3358	+	25.2371i	−0.998816	+	1.37475i	−0.926049	+	0.377403i	$0.876817\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−6.21008	+	19.1127i	−0.335313	+	1.03199i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$347$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$348$		0			0
$349$	−5.25197	+	3.81578i	−0.281132	+	0.204254i	−0.719411	−	0.694585i	$-0.755588\pi$
$349$	−5.25197	+	3.81578i	−0.281132	+	0.204254i	0.438279	+	0.898839i	$0.355588\pi$
$350$		0			0
$351$	−36.3035	+	7.71655i	−1.93774	+	0.411879i
$352$		0			0
$353$		0			0		0.743145	−	0.669131i	$-0.233333\pi$
$353$		0			0		−0.743145	+	0.669131i	$0.766667\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		0.669131	−	0.743145i	$-0.266667\pi$
$359$		0			0		−0.669131	+	0.743145i	$0.733333\pi$
$360$		0			0
$361$	1.27458	−	12.1269i	0.0670834	−	0.638256i
$362$		0			0
$363$	7.74937	−	17.4054i	0.406737	−	0.913545i
$364$	−17.2846	−	23.7902i	−0.905960	−	1.24695i
$365$		0			0
$366$		0			0
$367$	−19.5000	+	11.2583i	−1.01789	+	0.587680i	−0.913493	−	0.406855i	$-0.866625\pi$
$367$	−19.5000	+	11.2583i	−1.01789	+	0.587680i	−0.104399	+	0.994535i	$0.533292\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$	−15.0000	+	12.1244i	−0.777714	+	0.628619i
$373$	31.8968			1.65155			0.825776	−	0.563998i	$-0.190737\pi$
$373$	31.8968			1.65155			0.825776	+	0.563998i	$0.190737\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	20.8353	−	9.27649i	1.07024	−	0.476501i	0.205466	−	0.978664i	$-0.434129\pi$
$379$	20.8353	−	9.27649i	1.07024	−	0.476501i	0.864773	+	0.502163i	$0.167462\pi$
$380$		0			0
$381$	−29.7539	−	13.2473i	−1.52434	−	0.678680i
$382$		0			0
$383$		0			0		−0.994522	−	0.104528i	$-0.966667\pi$
$383$		0			0		0.994522	+	0.104528i	$0.0333333\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−24.2574	+	7.88171i	−1.23307	+	0.400650i
$388$	4.42125	+	13.6072i	0.224455	+	0.690801i
$389$		0			0		0.994522	−	0.104528i	$-0.0333333\pi$
$389$		0			0		−0.994522	+	0.104528i	$0.966667\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−19.8617	+	34.4015i	−0.996832	+	1.72656i	−0.429533	+	0.903051i	$0.641322\pi$
$397$	−19.8617	+	34.4015i	−0.996832	+	1.72656i	−0.567298	+	0.823512i	$0.692011\pi$
$398$		0			0
$399$	−5.46745	+	7.52530i	−0.273715	+	0.376736i
$400$	−19.5630	−	4.15823i	−0.978148	−	0.207912i
$401$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$401$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$402$		0			0
$403$	28.0954	+	28.1462i	1.39953	+	1.40206i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	22.5000	+	12.9904i	1.11255	+	0.642333i	0.939490	−	0.342578i	$-0.111300\pi$
$409$	22.5000	+	12.9904i	1.11255	+	0.642333i	0.173064	+	0.984911i	$0.444633\pi$
$410$		0			0
$411$		0			0
$412$	34.4274	+	15.3281i	1.69611	+	0.755159i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−3.72403	−	35.4318i	−0.182367	−	1.73510i
$418$		0			0
$419$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$419$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$420$		0			0
$421$	26.5939	−	29.5355i	1.29611	−	1.43947i	0.463002	−	0.886357i	$-0.346772\pi$
$421$	26.5939	−	29.5355i	1.29611	−	1.43947i	0.833105	−	0.553115i	$-0.186561\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	2.19694	+	4.93442i	0.106318	+	0.238793i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		−0.978148	−	0.207912i	$-0.933333\pi$
$431$		0			0		0.978148	+	0.207912i	$0.0666667\pi$
$432$	−19.7673	−	6.42280i	−0.951057	−	0.309017i
$433$		−	26.3289i		−	1.26529i	−0.774443	−	0.632644i	$-0.781970\pi$
$433$		−	26.3289i		−	1.26529i	0.774443	−	0.632644i	$-0.218030\pi$
$434$		0			0
$435$		0			0
$436$	10.3732	−	31.9255i	0.496788	−	1.52895i
$437$		0			0
$438$		0			0
$439$	2.21665	+	3.83936i	0.105795	+	0.183242i	0.914063	−	0.405573i	$-0.132928\pi$
$439$	2.21665	+	3.83936i	0.105795	+	0.183242i	−0.808268	+	0.588815i	$0.799595\pi$
$440$		0			0
$441$	7.57135	−	3.37098i	0.360541	−	0.160523i
$442$		0			0
$443$		0			0		−0.913545	−	0.406737i	$-0.866667\pi$
$443$		0			0		0.913545	+	0.406737i	$0.133333\pi$
$444$	40.8889	−	8.69121i	1.94050	−	0.412467i
$445$		0			0
$446$		0			0
$447$		0			0
$448$	−1.72137	−	16.3777i	−0.0813269	−	0.773773i
$449$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$449$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	−22.2500	−	24.7112i	−1.04540	−	1.16103i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	17.3073	+	23.8214i	0.809599	+	1.11432i	0.991385	+	0.130980i	$0.0418123\pi$
$457$	17.3073	+	23.8214i	0.809599	+	1.11432i	−0.181786	+	0.983338i	$0.558188\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$461$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$462$		0			0
$463$	−25.3704	−	8.24334i	−1.17906	−	0.383101i	−0.347044	−	0.937849i	$-0.612815\pi$
$463$	−25.3704	−	8.24334i	−1.17906	−	0.383101i	−0.832018	+	0.554748i	$0.812815\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$467$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$468$	−8.91031	+	41.9197i	−0.411879	+	1.93774i
$469$	24.9477	+	18.1255i	1.15198	+	0.836960i
$470$		0			0
$471$	−33.2012	−	19.1687i	−1.52983	−	0.883249i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	12.7594	−	2.71210i	0.585443	−	0.124440i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		−0.104528	−	0.994522i	$-0.533333\pi$
$479$		0			0		0.104528	+	0.994522i	$0.466667\pi$
$480$		0			0
$481$	−26.6352	−	81.9747i	−1.21446	−	3.73772i
$482$		0			0
$483$		0			0
$484$	−14.7209	−	16.3492i	−0.669131	−	0.743145i
$485$		0			0
$486$		0			0
$487$	2.34269	−	5.26176i	0.106157	−	0.238433i	−0.852655	−	0.522475i	$-0.825009\pi$
$487$	2.34269	−	5.26176i	0.106157	−	0.238433i	0.958812	+	0.284042i	$0.0916755\pi$
$488$		0			0
$489$	−8.92394	−	20.0435i	−0.403554	−	0.906398i
$490$		0			0
$491$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$491$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	5.74475	+	21.5174i	0.257947	+	0.966159i
$497$		0			0
$498$		0			0
$499$	−1.33345	+	6.27337i	−0.0596933	+	0.280835i	−0.997863	−	0.0653408i	$-0.979187\pi$
$499$	−1.33345	+	6.27337i	−0.0596933	+	0.280835i	0.938170	+	0.346176i	$0.112520\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		0.913545	−	0.406737i	$-0.133333\pi$
$503$		0			0		−0.913545	+	0.406737i	$0.866667\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	65.4887	+	6.88314i	2.90846	+	0.305691i
$508$	−27.9484	+	25.1649i	−1.24001	+	1.11651i
$509$		0			0		−0.743145	−	0.669131i	$-0.766667\pi$
$509$		0			0		0.743145	+	0.669131i	$0.233333\pi$
$510$		0			0
$511$	−30.6048	+	9.94410i	−1.35388	+	0.439901i
$512$		0			0
$513$	13.4820	−	1.41701i	0.595243	−	0.0625626i
$514$		0			0
$515$		0			0
$516$	−3.07852	+	29.2902i	−0.135524	+	1.28943i
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$521$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$522$		0			0
$523$	3.70752	−	5.10296i	0.162119	−	0.223137i	−0.720228	−	0.693738i	$-0.755963\pi$
$523$	3.70752	−	5.10296i	0.162119	−	0.223137i	0.882346	+	0.470601i	$0.155963\pi$
$524$		0			0
$525$	16.9545	+	5.50886i	0.739956	+	0.240426i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−7.10739	+	21.8743i	−0.309017	+	0.951057i
$530$		0			0
$531$		0			0
$532$	5.37038	+	9.30178i	0.232836	+	0.403283i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	1.22283	+	11.6345i	0.0525737	+	0.500205i	0.988847	+	0.148933i	$0.0475840\pi$
$541$	1.22283	+	11.6345i	0.0525737	+	0.500205i	−0.936274	+	0.351272i	$0.885749\pi$
$542$		0			0
$543$	−0.733859	−	2.25859i	−0.0314929	−	0.0969252i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−2.78871	+	26.5328i	−0.119237	+	1.13446i	0.757280	+	0.653090i	$0.226528\pi$
$547$	−2.78871	+	26.5328i	−0.119237	+	1.13446i	−0.876517	+	0.481371i	$0.840139\pi$
$548$		0			0
$549$	3.20178	−	7.19132i	0.136649	−	0.306918i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−20.7889	+	12.0025i	−0.884034	+	0.510397i
$554$		0			0
$555$		0			0
$556$	−39.1250	−	12.7125i	−1.65927	−	0.539130i
$557$		0			0			−	1.00000i	$-0.5\pi$
$557$		0			0				1.00000i	$0.5\pi$
$558$		0			0
$559$	60.7266			2.56846
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$563$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	16.9247	+	7.53536i	0.710771	+	0.316456i
$568$		0			0
$569$		0			0		−0.994522	−	0.104528i	$-0.966667\pi$
$569$		0			0		0.994522	+	0.104528i	$0.0333333\pi$
$570$		0			0
$571$	−27.0305	−	24.3383i	−1.13119	−	1.01853i	−0.999629	−	0.0272193i	$-0.991335\pi$
$571$	−27.0305	−	24.3383i	−1.13119	−	1.01853i	−0.131560	−	0.991308i	$-0.541999\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−16.0591	+	17.8355i	−0.669131	+	0.743145i
$577$	18.3295	+	20.3570i	0.763067	+	0.847472i	0.992035	−	0.125962i	$-0.0402018\pi$
$577$	18.3295	+	20.3570i	0.763067	+	0.847472i	−0.228968	+	0.973434i	$0.573535\pi$
$578$		0			0
$579$	9.84981	+	46.3397i	0.409344	+	1.92581i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$587$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$588$		−	9.57001i		−	0.394661i
$589$	−9.13114	−	11.2969i	−0.376242	−	0.465479i
$590$		0			0
$591$		0			0
$592$	10.0357	−	47.2145i	0.412467	−	1.94050i
$593$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$593$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−38.8149	+	28.2007i	−1.58859	+	1.15418i
$598$		0			0
$599$		0			0		0.978148	−	0.207912i	$-0.0666667\pi$
$599$		0			0		−0.978148	+	0.207912i	$0.933333\pi$
$600$		0			0
$601$	−27.9222	+	25.1413i	−1.13897	+	1.02553i	−0.139596	+	0.990209i	$0.544580\pi$
$601$	−27.9222	+	25.1413i	−1.13897	+	1.02553i	−0.999376	+	0.0353259i	$0.988753\pi$
$602$		0			0
$603$	−4.69764	−	44.6951i	−0.191303	−	1.82012i
$604$	−36.5170	+	11.8651i	−1.48586	+	0.482784i
$605$		0			0
$606$		0			0
$607$	31.3670	−	34.8365i	1.27315	−	1.41397i	0.407585	−	0.913167i	$-0.366371\pi$
$607$	31.3670	−	34.8365i	1.27315	−	1.41397i	0.865560	−	0.500805i	$-0.166962\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	6.10698	+	13.7165i	0.246659	+	0.554004i	0.993867	−	0.110586i	$-0.0352728\pi$
$613$	6.10698	+	13.7165i	0.246659	+	0.554004i	−0.747208	+	0.664590i	$0.768606\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		−0.978148	−	0.207912i	$-0.933333\pi$
$617$		0			0		0.978148	+	0.207912i	$0.0666667\pi$
$618$		0			0
$619$			27.8416i			1.11905i	0.828814	+	0.559525i	$0.189016\pi$
$619$			27.8416i			1.11905i	−0.828814	+	0.559525i	$0.810984\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$	40.0351	+	29.0872i	1.60269	+	1.16442i
$625$	−12.5000	−	21.6506i	−0.500000	−	0.866025i
$626$		0			0
$627$		0			0
$628$	−35.8138	+	26.0203i	−1.42913	+	1.03832i
$629$		0			0
$630$		0			0
$631$	45.2310	+	4.75397i	1.80062	+	0.189253i	0.944717	−	0.327886i	$-0.106336\pi$
$631$	45.2310	+	4.75397i	1.80062	+	0.189253i	0.855901	+	0.517139i	$0.173003\pi$
$632$		0			0
$633$	13.6552	+	12.2952i	0.542747	+	0.488691i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−19.6245	+	2.06262i	−0.777551	+	0.0817240i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		−0.207912	−	0.978148i	$-0.566667\pi$
$641$		0			0		0.207912	+	0.978148i	$0.433333\pi$
$642$		0			0
$643$	−26.3040	−	36.2043i	−1.03733	−	1.42776i	−0.899301	−	0.437330i	$-0.855924\pi$
$643$	−26.3040	−	36.2043i	−1.03733	−	1.42776i	−0.138027	−	0.990429i	$-0.544076\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$647$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	−3.08773	−	19.6097i	−0.121018	−	0.768566i
$652$	−25.3345			−0.992176
$653$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$653$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	40.6150	+	23.4491i	1.58454	+	0.914836i
$658$		0			0
$659$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$659$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$660$		0			0
$661$	0.309416	−	0.0657685i	0.0120349	−	0.00255810i	−0.201890	−	0.979408i	$-0.564708\pi$
$661$	0.309416	−	0.0657685i	0.0120349	−	0.00255810i	0.213925	+	0.976850i	$0.431375\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	−2.19599	+	2.43890i	−0.0849020	+	0.0942932i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	21.0831	−	47.3533i	0.812693	−	1.82534i	0.366332	−	0.930484i	$-0.380614\pi$
$673$	21.0831	−	47.3533i	0.812693	−	1.82534i	0.446361	−	0.894853i	$-0.352720\pi$
$674$		0			0
$675$	−10.5673	−	23.7346i	−0.406737	−	0.913545i
$676$	38.0182	−	65.8494i	1.46224	−	2.53267i
$677$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$677$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$678$		0			0
$679$	−14.4041	−	3.06168i	−0.552778	−	0.117496i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		1.00000			$0$
$683$		0			0		−1.00000			$\pi$
$684$	4.83716	−	14.8873i	0.184953	−	0.569228i
$685$		0			0
$686$		0			0
$687$	−20.8297	−	36.0781i	−0.794703	−	1.37647i
$688$	29.4515	+	17.0038i	1.12283	+	0.648265i
$689$		0			0
$690$		0			0
$691$	21.9891	+	9.79018i	0.836505	+	0.372436i	0.779857	−	0.625958i	$-0.215292\pi$
$691$	21.9891	+	9.79018i	0.836505	+	0.372436i	0.0566486	+	0.998394i	$0.481959\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	13.7740	−	15.2976i	0.520607	−	0.578193i
$701$		0			0		−0.669131	−	0.743145i	$-0.733333\pi$
$701$		0			0		0.669131	+	0.743145i	$0.266667\pi$
$702$		0			0
$703$	6.54556	+	30.7944i	0.246870	+	1.16143i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	25.4518	−	35.0315i	0.955864	−	1.31563i	0.00699141	−	0.999976i	$-0.497775\pi$
$709$	25.4518	−	35.0315i	0.955864	−	1.31563i	0.948873	−	0.315659i	$-0.102225\pi$
$710$		0			0
$711$	33.2721	+	10.8108i	1.24780	+	0.405435i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$719$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$720$		0			0
$721$	−31.3798	+	22.7987i	−1.16864	+	0.849069i
$722$		0			0
$723$	2.93444	−	0.623735i	0.109133	−	0.0231970i
$724$	−2.72718	−	0.286638i	−0.101355	−	0.0106528i
$725$		0			0
$726$		0			0
$727$	−0.522642	−	4.97261i	−0.0193837	−	0.184424i	0.980546	−	0.196290i	$-0.0628894\pi$
$727$	−0.522642	−	4.97261i	−0.0193837	−	0.184424i	−0.999930	+	0.0118662i	$0.996223\pi$
$728$		0			0
$729$	−8.34346	−	25.6785i	−0.309017	−	0.951057i
$730$		0			0
$731$		0			0
$732$	−6.08217	−	6.75494i	−0.224804	−	0.249670i
$733$	−0.731699	+	6.96165i	−0.0270259	+	0.257135i	0.972664	+	0.232219i	$0.0745985\pi$
$733$	−0.731699	+	6.96165i	−0.0270259	+	0.257135i	−0.999690	+	0.0249159i	$0.992068\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−35.4736	+	20.4807i	−1.30492	+	0.753395i	−0.981243	−	0.192773i	$-0.938252\pi$
$739$	−35.4736	+	20.4807i	−1.30492	+	0.753395i	−0.323675	+	0.946168i	$0.604919\pi$
$740$		0			0
$741$	−31.5707	−	6.71056i	−1.15978	−	0.246519i
$742$		0			0
$743$		0			0			−	1.00000i	$-0.5\pi$
$743$		0			0				1.00000i	$0.5\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−10.0490	+	4.47410i	−0.366693	+	0.163262i	−0.581807	−	0.813327i	$-0.697654\pi$
$751$	−10.0490	+	4.47410i	−0.366693	+	0.163262i	0.215114	+	0.976589i	$0.430988\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$	15.8977	−	14.3143i	0.578193	−	0.520607i
$757$	−40.5145	−	36.4794i	−1.47253	−	1.32587i	−0.827117	−	0.562029i	$-0.810021\pi$
$757$	−40.5145	−	36.4794i	−1.47253	−	1.32587i	−0.645408	−	0.763838i	$-0.723312\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		0.994522	−	0.104528i	$-0.0333333\pi$
$761$		0			0		−0.994522	+	0.104528i	$0.966667\pi$
$762$		0			0
$763$	23.1186	+	25.6758i	0.836948	+	0.929525i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	11.2718	+	25.3169i	0.406737	+	0.913545i
$769$	−26.0474	+	45.1155i	−0.939295	+	1.62691i	−0.172504	+	0.985009i	$0.555186\pi$
$769$	−26.0474	+	45.1155i	−0.939295	+	1.62691i	−0.766791	+	0.641897i	$0.778148\pi$
$770$		0			0
$771$		0			0
$772$	53.5085	+	11.3736i	1.92581	+	0.409344i
$773$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$773$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$774$		0			0
$775$	−15.1832	+	23.3339i	−0.545396	+	0.838179i
$776$		0			0
$777$	−13.2954	+	40.9191i	−0.476971	+	1.46797i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−10.0951	−	4.49464i	−0.360541	−	0.160523i
$785$		0			0
$786$		0			0
$787$	34.7534	−	31.2921i	1.23883	−	1.11544i	0.249708	−	0.968321i	$-0.419665\pi$
$787$	34.7534	−	31.2921i	1.23883	−	1.11544i	0.989118	−	0.147123i	$-0.0470014\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−12.5410	+	13.9281i	−0.445342	+	0.494603i
$794$		0			0
$795$		0			0
$796$	11.5183	+	54.1895i	0.408256	+	1.92069i
$797$		0			0		0.406737	−	0.913545i	$-0.366667\pi$
$797$		0			0		−0.406737	+	0.913545i	$0.633333\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$	−49.3538	−	16.0360i	−1.74058	−	0.565547i
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		0.207912	−	0.978148i	$-0.433333\pi$
$809$		0			0		−0.207912	+	0.978148i	$0.566667\pi$
$810$		0			0
$811$	−9.50000	−	16.4545i	−0.333590	−	0.577795i	0.649623	−	0.760257i	$-0.274927\pi$
$811$	−9.50000	−	16.4545i	−0.333590	−	0.577795i	−0.983213	+	0.182462i	$0.941593\pi$
$812$		0			0
$813$	−36.0189	+	16.0367i	−1.26324	+	0.562430i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−22.0591	−	2.31851i	−0.771751	−	0.0811143i
$818$		0			0
$819$	−32.7798	−	29.5150i	−1.14542	−	1.03134i
$820$		0			0
$821$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$821$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$822$		0			0
$823$	41.7318	−	4.38619i	1.45468	−	0.152893i	0.656051	−	0.754717i	$-0.272226\pi$
$823$	41.7318	−	4.38619i	1.45468	−	0.152893i	0.798629	+	0.601824i	$0.205559\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		−0.207912	−	0.978148i	$-0.566667\pi$
$827$		0			0		0.207912	+	0.978148i	$0.433333\pi$
$828$		0			0
$829$	24.7616	+	34.0815i	0.860008	+	1.18370i	0.981568	+	0.191115i	$0.0612103\pi$
$829$	24.7616	+	34.0815i	0.860008	+	1.18370i	−0.121560	+	0.992584i	$0.538790\pi$
$830$		0			0
$831$	10.5000	−	18.1865i	0.364241	−	0.630884i
$832$	49.4861	−	28.5708i	1.71562	−	0.990515i
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−18.2271	+	22.4671i	−0.630022	+	0.776578i
$838$		0			0
$839$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$839$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$840$		0			0
$841$	23.4615	+	17.0458i	0.809017	+	0.587785i
$842$		0			0
$843$		0			0
$844$	19.3832	−	8.62995i	0.667197	−	0.297055i
$845$		0			0
$846$		0			0
$847$	22.1486	−	4.70782i	0.761034	−	0.161763i
$848$		0			0
$849$	−37.5018	+	33.7667i	−1.28706	+	1.15887i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−7.10739	−	21.8743i	−0.243352	−	0.748962i	−0.995903	−	0.0904274i	$-0.971177\pi$
$853$	−7.10739	−	21.8743i	−0.243352	−	0.748962i	0.752551	−	0.658534i	$-0.228823\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		0.104528	−	0.994522i	$-0.466667\pi$
$857$		0			0		−0.104528	+	0.994522i	$0.533333\pi$
$858$		0			0
$859$	−21.6407	+	48.6059i	−0.738373	+	1.65841i	0.0141966	+	0.999899i	$0.495481\pi$
$859$	−21.6407	+	48.6059i	−0.738373	+	1.65841i	−0.752569	+	0.658513i	$0.771186\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$863$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		−	29.4449i		−	1.00000i
$868$	−22.1359	−	5.95273i	−0.751342	−	0.202049i
$869$		0			0
$870$		0			0
$871$	−22.2467	+	104.662i	−0.753800	+	3.54635i
$872$		0			0
$873$	10.7306	+	18.5859i	0.363176	+	0.629038i
$874$		0			0
$875$		0			0
$876$	43.8110	−	31.8305i	1.48024	−	1.07545i
$877$	22.8386	+	10.1684i	0.771206	+	0.343363i	0.754331	−	0.656495i	$-0.227962\pi$
$877$	22.8386	+	10.1684i	0.771206	+	0.343363i	0.0168752	+	0.999858i	$0.494628\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		−0.743145	−	0.669131i	$-0.766667\pi$
$881$		0			0		0.743145	+	0.669131i	$0.233333\pi$
$882$		0			0
$883$	31.8221	−	10.3396i	1.07090	−	0.347956i	0.280062	−	0.959982i	$-0.409645\pi$
$883$	31.8221	−	10.3396i	1.07090	−	0.347956i	0.790838	+	0.612026i	$0.209645\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		−0.669131	−	0.743145i	$-0.733333\pi$
$887$		0			0		0.669131	+	0.743145i	$0.266667\pi$
$888$		0			0
$889$	−8.04787	−	37.8623i	−0.269917	−	1.26986i
$890$		0			0
$891$		0			0
$892$	1.54136	+	3.46194i	0.0516084	+	0.115914i
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	−30.0000			−1.00000
$901$		0			0
$902$		0			0
$903$	−24.5236	−	17.8174i	−0.816093	−	0.592926i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	4.76572	−	3.46250i	0.158243	−	0.114970i	−0.505846	−	0.862624i	$-0.668820\pi$
$907$	4.76572	−	3.46250i	0.158243	−	0.114970i	0.664089	+	0.747653i	$0.268820\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		0.743145	−	0.669131i	$-0.233333\pi$
$911$		0			0		−0.743145	+	0.669131i	$0.766667\pi$
$912$	−13.4323	−	12.0945i	−0.444789	−	0.400489i
$913$		0			0
$914$		0			0
$915$		0			0
$916$	−47.8406	+	5.02825i	−1.58070	+	0.166138i
$917$		0			0
$918$		0			0
$919$	3.80918	−	36.2420i	0.125653	−	1.19551i	−0.732007	−	0.681297i	$-0.761416\pi$
$919$	3.80918	−	36.2420i	0.125653	−	1.19551i	0.857661	−	0.514216i	$-0.171917\pi$
$920$		0			0
$921$	−1.36384	+	3.06324i	−0.0449402	+	0.100937i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	52.2530	−	30.1683i	1.71807	−	0.991928i
$926$		0			0
$927$	55.2929	+	11.7529i	1.81606	+	0.386015i
$928$		0			0
$929$		0			0			−	1.00000i	$-0.5\pi$
$929$		0			0				1.00000i	$0.5\pi$
$930$		0			0
$931$	7.20741			0.236213
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−53.5216	+	23.8293i	−1.74847	+	0.778471i	−0.756209	+	0.654330i	$0.772951\pi$
$937$	−53.5216	+	23.8293i	−1.74847	+	0.778471i	−0.992265	+	0.124141i	$0.960383\pi$
$938$		0			0
$939$	−3.47206	−	1.54586i	−0.113307	−	0.0504473i
$940$		0			0
$941$		0			0		−0.994522	−	0.104528i	$-0.966667\pi$
$941$		0			0		0.994522	+	0.104528i	$0.0333333\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		0.994522	−	0.104528i	$-0.0333333\pi$
$947$		0			0		−0.994522	+	0.104528i	$0.966667\pi$
$948$	27.0305	−	30.0204i	0.877909	−	0.975017i
$949$	−74.7151	−	82.9795i	−2.42535	−	2.69363i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$953$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	30.8243	+	3.29604i	0.994332	+	0.106324i
$962$		0			0
$963$		0			0
$964$	0.720227	−	3.38840i	0.0231970	−	0.109133i
$965$		0			0
$966$		0			0
$967$	0.712915	+	0.411602i	0.0229258	+	0.0132362i	0.511419	−	0.859331i	$-0.329120\pi$
$967$	0.712915	+	0.411602i	0.0229258	+	0.0132362i	−0.488493	+	0.872568i	$0.662453\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		0.978148	−	0.207912i	$-0.0666667\pi$
$971$		0			0		−0.978148	+	0.207912i	$0.933333\pi$
$972$	−31.0061	−	3.25887i	−0.994522	−	0.104528i
$973$	31.4659	−	28.3320i	1.00875	−	0.908283i
$974$		0			0
$975$	6.46588	+	61.5188i	0.207074	+	1.97018i
$976$	−9.98213	+	3.24339i	−0.319520	+	0.103818i
$977$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$977$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	5.26329	−	50.0769i	0.168044	−	1.59883i
$982$		0			0
$983$		0			0		0.406737	−	0.913545i	$-0.366667\pi$
$983$		0			0		−0.406737	+	0.913545i	$0.633333\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	−21.9063	+	30.1514i	−0.696931	+	0.959243i
$989$		0			0
$990$		0			0
$991$		−	59.0364i		−	1.87535i	−0.347509	−	0.937676i	$-0.612973\pi$
$991$		−	59.0364i		−	1.87535i	0.347509	−	0.937676i	$-0.387027\pi$
$992$		0			0
$993$	33.0000			1.04722
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−28.1059	−	48.6809i	−0.890124	−	1.54174i	−0.839726	−	0.543010i	$-0.817285\pi$
$997$	−28.1059	−	48.6809i	−0.890124	−	1.54174i	−0.0503972	−	0.998729i	$-0.516049\pi$
$998$		0			0
$999$	57.2826	−	25.5039i	1.81234	−	0.806907i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	93.2.p.a.53.1	✓	8
3.2	odd	2	CM	93.2.p.a.53.1	✓	8
31.24	odd	30	inner	93.2.p.a.86.1	yes	8
93.86	even	30	inner	93.2.p.a.86.1	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
93.2.p.a.53.1	✓	8	1.1	even	1	trivial
93.2.p.a.53.1	✓	8	3.2	odd	2	CM
93.2.p.a.86.1	yes	8	31.24	odd	30	inner
93.2.p.a.86.1	yes	8	93.86	even	30	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:	\( N \)	\(=\)	\( 93 = 3 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	93.p (of order \(30\), degree \(8\), minimal)

\(f(q)\)	\(=\)	\(q+(0.360114 - 1.69420i) q^{3} +(-1.61803 - 1.17557i) q^{4} +(1.88052 - 0.837263i) q^{7} +(-2.74064 - 1.22021i) q^{9} +O(q^{10})\)	\(q+(0.360114 - 1.69420i) q^{3} +(-1.61803 - 1.17557i) q^{4} +(1.88052 - 0.837263i) q^{7} +(-2.74064 - 1.22021i) q^{9} +(-2.57433 + 2.31794i) q^{12} +(5.30806 + 4.77940i) q^{13} +(1.23607 + 3.80423i) q^{16} +(-1.74569 - 1.93879i) q^{19} +(-0.741290 - 3.48749i) q^{21} +(-2.50000 + 4.33013i) q^{25} +(-3.05422 + 4.20378i) q^{27} +(-4.02701 - 0.855968i) q^{28} +(5.55987 + 0.296414i) q^{31} +(3.00000 + 5.19615i) q^{36} +(-10.4506 - 6.03366i) q^{37} +(10.0088 - 7.27180i) q^{39} +(6.31815 - 5.68889i) q^{43} +(6.89025 - 0.724194i) q^{48} +(-1.84856 + 2.05303i) q^{49} +(-2.97010 - 13.9732i) q^{52} +(-3.91335 + 2.25937i) q^{57} +2.62396i q^{61} -6.17547 q^{63} +(2.47214 - 7.60845i) q^{64} +(7.49021 + 12.9734i) q^{67} +(-15.5471 - 1.63406i) q^{73} +(6.43582 + 5.79484i) q^{75} +(0.545408 + 5.18921i) q^{76} +(-11.5976 + 1.21895i) q^{79} +(6.02218 + 6.68830i) q^{81} +(-2.90036 + 6.51432i) q^{84} +(13.9835 + 4.54353i) q^{91} +(2.50437 - 9.31279i) q^{93} +(-5.78749 - 4.20486i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 3 q^{3} - 4 q^{4} - 4 q^{7} - 3 q^{9}+O(q^{10}) \) 8 * q + 3 * q^3 - 4 * q^4 - 4 * q^7 - 3 * q^9	\( 8 q + 3 q^{3} - 4 q^{4} - 4 q^{7} - 3 q^{9} + 6 q^{12} + 9 q^{13} - 8 q^{16} + 7 q^{19} - 33 q^{21} - 20 q^{25} - 18 q^{28} - 4 q^{31} + 24 q^{36} + 9 q^{37} + 27 q^{39} + 44 q^{43} + 12 q^{48} + 9 q^{49} + 18 q^{52} - 21 q^{57} - 24 q^{63} - 16 q^{64} + 16 q^{67} + 3 q^{73} - 15 q^{75} - 66 q^{76} - 35 q^{79} + 9 q^{81} + 24 q^{84} - 5 q^{91} + 15 q^{93} + 15 q^{97}+O(q^{100}) \) 8 * q + 3 * q^3 - 4 * q^4 - 4 * q^7 - 3 * q^9 + 6 * q^12 + 9 * q^13 - 8 * q^16 + 7 * q^19 - 33 * q^21 - 20 * q^25 - 18 * q^28 - 4 * q^31 + 24 * q^36 + 9 * q^37 + 27 * q^39 + 44 * q^43 + 12 * q^48 + 9 * q^49 + 18 * q^52 - 21 * q^57 - 24 * q^63 - 16 * q^64 + 16 * q^67 + 3 * q^73 - 15 * q^75 - 66 * q^76 - 35 * q^79 + 9 * q^81 + 24 * q^84 - 5 * q^91 + 15 * q^93 + 15 * q^97

Introduction

L-functions

Modular forms

Varieties

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Database

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