LMFDB - Embedded newform 1350.2.e.l.451.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1350,2,Mod(451,1350)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1350, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([4, 0]))

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

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

chi := DirichletCharacter("1350.451");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1350 = 2 \cdot 3^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1350.e (of order $3$, degree $2$, 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:	$10.7798042729$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{3})$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{-11})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 2x^{2} - 3x + 9 $ x^4 - x^3 - 2x^2 - 3x + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 3 $
Twist minimal:	no (minimal twist has level 90)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			451.2
Root			$1.68614 - 0.396143i$ of defining polynomial
Character	$\chi$	$=$	1350.451
Dual form			1350.2.e.l.901.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.500000 - 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{4} +(1.68614 - 2.92048i) q^{7} -1.00000 q^{8} +O(q^{10})$	\(q+(0.500000 - 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{4} +(1.68614 - 2.92048i) q^{7} -1.00000 q^{8} +(-2.18614 + 3.78651i) q^{11} +(-3.37228 - 5.84096i) q^{13} +(-1.68614 - 2.92048i) q^{14} +(-0.500000 + 0.866025i) q^{16} -1.62772 q^{17} -2.37228 q^{19} +(2.18614 + 3.78651i) q^{22} +(-0.686141 - 1.18843i) q^{23} -6.74456 q^{26} -3.37228 q^{28} +(0.686141 - 1.18843i) q^{29} +(-2.37228 - 4.10891i) q^{31} +(0.500000 + 0.866025i) q^{32} +(-0.813859 + 1.40965i) q^{34} +4.00000 q^{37} +(-1.18614 + 2.05446i) q^{38} +(-1.50000 - 2.59808i) q^{41} +(-2.81386 + 4.87375i) q^{43} +4.37228 q^{44} -1.37228 q^{46} +(3.68614 - 6.38458i) q^{47} +(-2.18614 - 3.78651i) q^{49} +(-3.37228 + 5.84096i) q^{52} -11.4891 q^{53} +(-1.68614 + 2.92048i) q^{56} +(-0.686141 - 1.18843i) q^{58} +(2.18614 + 3.78651i) q^{59} +(4.05842 - 7.02939i) q^{61} -4.74456 q^{62} +1.00000 q^{64} +(-3.50000 - 6.06218i) q^{67} +(0.813859 + 1.40965i) q^{68} +6.00000 q^{71} -3.11684 q^{73} +(2.00000 - 3.46410i) q^{74} +(1.18614 + 2.05446i) q^{76} +(7.37228 + 12.7692i) q^{77} +(-1.00000 + 1.73205i) q^{79} -3.00000 q^{82} +(-3.68614 + 6.38458i) q^{83} +(2.81386 + 4.87375i) q^{86} +(2.18614 - 3.78651i) q^{88} -16.1168 q^{89} -22.7446 q^{91} +(-0.686141 + 1.18843i) q^{92} +(-3.68614 - 6.38458i) q^{94} +(-4.18614 + 7.25061i) q^{97} -4.37228 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{2} - 2 q^{4} + q^{7} - 4 q^{8}+O(q^{10}) $ 4 * q + 2 * q^2 - 2 * q^4 + q^7 - 4 * q^8	$ 4 q + 2 q^{2} - 2 q^{4} + q^{7} - 4 q^{8} - 3 q^{11} - 2 q^{13} - q^{14} - 2 q^{16} - 18 q^{17} + 2 q^{19} + 3 q^{22} + 3 q^{23} - 4 q^{26} - 2 q^{28} - 3 q^{29} + 2 q^{31} + 2 q^{32} - 9 q^{34} + 16 q^{37} + q^{38} - 6 q^{41} - 17 q^{43} + 6 q^{44} + 6 q^{46} + 9 q^{47} - 3 q^{49} - 2 q^{52} - q^{56} + 3 q^{58} + 3 q^{59} - q^{61} + 4 q^{62} + 4 q^{64} - 14 q^{67} + 9 q^{68} + 24 q^{71} + 22 q^{73} + 8 q^{74} - q^{76} + 18 q^{77} - 4 q^{79} - 12 q^{82} - 9 q^{83} + 17 q^{86} + 3 q^{88} - 30 q^{89} - 68 q^{91} + 3 q^{92} - 9 q^{94} - 11 q^{97} - 6 q^{98}+O(q^{100}) $ 4 * q + 2 * q^2 - 2 * q^4 + q^7 - 4 * q^8 - 3 * q^11 - 2 * q^13 - q^14 - 2 * q^16 - 18 * q^17 + 2 * q^19 + 3 * q^22 + 3 * q^23 - 4 * q^26 - 2 * q^28 - 3 * q^29 + 2 * q^31 + 2 * q^32 - 9 * q^34 + 16 * q^37 + q^38 - 6 * q^41 - 17 * q^43 + 6 * q^44 + 6 * q^46 + 9 * q^47 - 3 * q^49 - 2 * q^52 - q^56 + 3 * q^58 + 3 * q^59 - q^61 + 4 * q^62 + 4 * q^64 - 14 * q^67 + 9 * q^68 + 24 * q^71 + 22 * q^73 + 8 * q^74 - q^76 + 18 * q^77 - 4 * q^79 - 12 * q^82 - 9 * q^83 + 17 * q^86 + 3 * q^88 - 30 * q^89 - 68 * q^91 + 3 * q^92 - 9 * q^94 - 11 * q^97 - 6 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1350\mathbb{Z}\right)^\times$.

$n$	$1001$	$1027$
$\chi(n)$	$e\left(\frac{2}{3}\right)$	$1$

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 $

$2$	0.500000	−	0.866025i	0.353553	−	0.612372i
$2$	0.500000	−	0.866025i	0.353553	−	0.612372i
$3$		0			0
$3$		0			0
$4$	−0.500000	−	0.866025i	−0.250000	−	0.433013i
$5$		0			0
$5$		0			0
$6$		0			0
$7$	1.68614	−	2.92048i	0.637301	−	1.10384i	−0.348721	−	0.937226i	$-0.613384\pi$
$7$	1.68614	−	2.92048i	0.637301	−	1.10384i	0.986023	−	0.166612i	$-0.0532826\pi$
$8$	−1.00000			−0.353553
$9$		0			0
$10$		0			0
$11$	−2.18614	+	3.78651i	−0.659146	+	1.14167i	0.321691	+	0.946845i	$0.395749\pi$
$11$	−2.18614	+	3.78651i	−0.659146	+	1.14167i	−0.980837	+	0.194830i	$0.937584\pi$
$12$		0			0
$13$	−3.37228	−	5.84096i	−0.935303	−	1.61999i	−0.774094	−	0.633071i	$-0.781794\pi$
$13$	−3.37228	−	5.84096i	−0.935303	−	1.61999i	−0.161209	−	0.986920i	$-0.551539\pi$
$14$	−1.68614	−	2.92048i	−0.450640	−	0.780531i
$15$		0			0
$16$	−0.500000	+	0.866025i	−0.125000	+	0.216506i
$17$	−1.62772			−0.394780			−0.197390	−	0.980325i	$-0.563246\pi$
$17$	−1.62772			−0.394780			−0.197390	+	0.980325i	$0.563246\pi$
$18$		0			0
$19$	−2.37228			−0.544239			−0.272119	−	0.962264i	$-0.587725\pi$
$19$	−2.37228			−0.544239			−0.272119	+	0.962264i	$0.587725\pi$
$20$		0			0
$21$		0			0
$22$	2.18614	+	3.78651i	0.466087	+	0.807286i
$23$	−0.686141	−	1.18843i	−0.143070	−	0.247805i	0.785581	−	0.618759i	$-0.212364\pi$
$23$	−0.686141	−	1.18843i	−0.143070	−	0.247805i	−0.928651	+	0.370954i	$0.879031\pi$
$24$		0			0
$25$		0			0
$26$	−6.74456			−1.32272
$27$		0			0
$28$	−3.37228			−0.637301
$29$	0.686141	−	1.18843i	0.127413	−	0.220686i	−0.795261	−	0.606268i	$-0.792666\pi$
$29$	0.686141	−	1.18843i	0.127413	−	0.220686i	0.922674	+	0.385582i	$0.125999\pi$
$30$		0			0
$31$	−2.37228	−	4.10891i	−0.426074	−	0.737982i	0.570446	−	0.821335i	$-0.306770\pi$
$31$	−2.37228	−	4.10891i	−0.426074	−	0.737982i	−0.996520	+	0.0833529i	$0.973437\pi$
$32$	0.500000	+	0.866025i	0.0883883	+	0.153093i
$33$		0			0
$34$	−0.813859	+	1.40965i	−0.139576	+	0.241752i
$35$		0			0
$36$		0			0
$37$	4.00000			0.657596			0.328798	−	0.944400i	$-0.393356\pi$
$37$	4.00000			0.657596			0.328798	+	0.944400i	$0.393356\pi$
$38$	−1.18614	+	2.05446i	−0.192417	+	0.333277i
$39$		0			0
$40$		0			0
$41$	−1.50000	−	2.59808i	−0.234261	−	0.405751i	0.724797	−	0.688963i	$-0.241934\pi$
$41$	−1.50000	−	2.59808i	−0.234261	−	0.405751i	−0.959058	+	0.283211i	$0.908600\pi$
$42$		0			0
$43$	−2.81386	+	4.87375i	−0.429110	+	0.743240i	−0.996794	−	0.0800065i	$-0.974506\pi$
$43$	−2.81386	+	4.87375i	−0.429110	+	0.743240i	0.567685	+	0.823246i	$0.307839\pi$
$44$	4.37228			0.659146
$45$		0			0
$46$	−1.37228			−0.202332
$47$	3.68614	−	6.38458i	0.537679	−	0.931287i	−0.461350	−	0.887218i	$-0.652635\pi$
$47$	3.68614	−	6.38458i	0.537679	−	0.931287i	0.999029	−	0.0440687i	$-0.0140321\pi$
$48$		0			0
$49$	−2.18614	−	3.78651i	−0.312306	−	0.540930i
$50$		0			0
$51$		0			0
$52$	−3.37228	+	5.84096i	−0.467651	+	0.809996i
$53$	−11.4891			−1.57815			−0.789076	−	0.614295i	$-0.789440\pi$
$53$	−11.4891			−1.57815			−0.789076	+	0.614295i	$0.789440\pi$
$54$		0			0
$55$		0			0
$56$	−1.68614	+	2.92048i	−0.225320	+	0.390266i
$57$		0			0
$58$	−0.686141	−	1.18843i	−0.0900947	−	0.156049i
$59$	2.18614	+	3.78651i	0.284611	+	0.492961i	0.972515	−	0.232841i	$-0.0748021\pi$
$59$	2.18614	+	3.78651i	0.284611	+	0.492961i	−0.687904	+	0.725802i	$0.741469\pi$
$60$		0			0
$61$	4.05842	−	7.02939i	0.519628	−	0.900022i	−0.480112	−	0.877207i	$-0.659404\pi$
$61$	4.05842	−	7.02939i	0.519628	−	0.900022i	0.999740	−	0.0228144i	$-0.00726267\pi$
$62$	−4.74456			−0.602560
$63$		0			0
$64$	1.00000			0.125000
$65$		0			0
$66$		0			0
$67$	−3.50000	−	6.06218i	−0.427593	−	0.740613i	0.569066	−	0.822292i	$-0.307305\pi$
$67$	−3.50000	−	6.06218i	−0.427593	−	0.740613i	−0.996659	+	0.0816792i	$0.973972\pi$
$68$	0.813859	+	1.40965i	0.0986949	+	0.170945i
$69$		0			0
$70$		0			0
$71$	6.00000			0.712069			0.356034	−	0.934473i	$-0.384129\pi$
$71$	6.00000			0.712069			0.356034	+	0.934473i	$0.384129\pi$
$72$		0			0
$73$	−3.11684			−0.364799			−0.182399	−	0.983225i	$-0.558386\pi$
$73$	−3.11684			−0.364799			−0.182399	+	0.983225i	$0.558386\pi$
$74$	2.00000	−	3.46410i	0.232495	−	0.402694i
$75$		0			0
$76$	1.18614	+	2.05446i	0.136060	+	0.235662i
$77$	7.37228	+	12.7692i	0.840149	+	1.45518i
$78$		0			0
$79$	−1.00000	+	1.73205i	−0.112509	+	0.194871i	−0.916781	−	0.399390i	$-0.869222\pi$
$79$	−1.00000	+	1.73205i	−0.112509	+	0.194871i	0.804272	+	0.594261i	$0.202555\pi$
$80$		0			0
$81$		0			0
$82$	−3.00000			−0.331295
$83$	−3.68614	+	6.38458i	−0.404607	+	0.700799i	−0.994276	−	0.106846i	$-0.965925\pi$
$83$	−3.68614	+	6.38458i	−0.404607	+	0.700799i	0.589669	+	0.807645i	$0.299258\pi$
$84$		0			0
$85$		0			0
$86$	2.81386	+	4.87375i	0.303426	+	0.525550i
$87$		0			0
$88$	2.18614	−	3.78651i	0.233043	−	0.403643i
$89$	−16.1168			−1.70838			−0.854191	−	0.519959i	$-0.825947\pi$
$89$	−16.1168			−1.70838			−0.854191	+	0.519959i	$0.825947\pi$
$90$		0			0
$91$	−22.7446			−2.38428
$92$	−0.686141	+	1.18843i	−0.0715351	+	0.123902i
$93$		0			0
$94$	−3.68614	−	6.38458i	−0.380196	−	0.658519i
$95$		0			0
$96$		0			0
$97$	−4.18614	+	7.25061i	−0.425038	+	0.736188i	−0.996424	−	0.0844938i	$-0.973073\pi$
$97$	−4.18614	+	7.25061i	−0.425038	+	0.736188i	0.571386	+	0.820682i	$0.306406\pi$
$98$	−4.37228			−0.441667
$99$		0			0
$100$		0			0
$101$	1.37228	−	2.37686i	0.136547	−	0.236507i	−0.789640	−	0.613570i	$-0.789733\pi$
$101$	1.37228	−	2.37686i	0.136547	−	0.236507i	0.926187	+	0.377064i	$0.123066\pi$
$102$		0			0
$103$	−8.00000	−	13.8564i	−0.788263	−	1.36531i	−0.927030	−	0.374987i	$-0.877647\pi$
$103$	−8.00000	−	13.8564i	−0.788263	−	1.36531i	0.138767	−	0.990325i	$-0.455686\pi$
$104$	3.37228	+	5.84096i	0.330679	+	0.572754i
$105$		0			0
$106$	−5.74456	+	9.94987i	−0.557961	+	0.966417i
$107$	8.48913			0.820675			0.410337	−	0.911934i	$-0.365411\pi$
$107$	8.48913			0.820675			0.410337	+	0.911934i	$0.365411\pi$
$108$		0			0
$109$	15.3723			1.47240			0.736199	−	0.676765i	$-0.236619\pi$
$109$	15.3723			1.47240			0.736199	+	0.676765i	$0.236619\pi$
$110$		0			0
$111$		0			0
$112$	1.68614	+	2.92048i	0.159325	+	0.275960i
$113$	−1.62772	−	2.81929i	−0.153123	−	0.265217i	0.779251	−	0.626712i	$-0.215600\pi$
$113$	−1.62772	−	2.81929i	−0.153123	−	0.265217i	−0.932374	+	0.361495i	$0.882266\pi$
$114$		0			0
$115$		0			0
$116$	−1.37228			−0.127413
$117$		0			0
$118$	4.37228			0.402501
$119$	−2.74456	+	4.75372i	−0.251594	+	0.435773i
$120$		0			0
$121$	−4.05842	−	7.02939i	−0.368947	−	0.639036i
$122$	−4.05842	−	7.02939i	−0.367432	−	0.636411i
$123$		0			0
$124$	−2.37228	+	4.10891i	−0.213037	+	0.368991i
$125$		0			0
$126$		0			0
$127$	8.11684			0.720253			0.360127	−	0.932903i	$-0.382733\pi$
$127$	8.11684			0.720253			0.360127	+	0.932903i	$0.382733\pi$
$128$	0.500000	−	0.866025i	0.0441942	−	0.0765466i
$129$		0			0
$130$		0			0
$131$	−1.37228	−	2.37686i	−0.119897	−	0.207667i	0.799830	−	0.600227i	$-0.204923\pi$
$131$	−1.37228	−	2.37686i	−0.119897	−	0.207667i	−0.919727	+	0.392560i	$0.871590\pi$
$132$		0			0
$133$	−4.00000	+	6.92820i	−0.346844	+	0.600751i
$134$	−7.00000			−0.604708
$135$		0			0
$136$	1.62772			0.139576
$137$	9.55842	−	16.5557i	0.816631	−	1.41445i	−0.0915197	−	0.995803i	$-0.529172\pi$
$137$	9.55842	−	16.5557i	0.816631	−	1.41445i	0.908151	−	0.418643i	$-0.137494\pi$
$138$		0			0
$139$	−0.441578	−	0.764836i	−0.0374542	−	0.0648725i	0.846691	−	0.532085i	$-0.178591\pi$
$139$	−0.441578	−	0.764836i	−0.0374542	−	0.0648725i	−0.884145	+	0.467213i	$0.845258\pi$
$140$		0			0
$141$		0			0
$142$	3.00000	−	5.19615i	0.251754	−	0.436051i
$143$	29.4891			2.46600
$144$		0			0
$145$		0			0
$146$	−1.55842	+	2.69927i	−0.128976	+	0.223393i
$147$		0			0
$148$	−2.00000	−	3.46410i	−0.164399	−	0.284747i
$149$	0.941578	+	1.63086i	0.0771371	+	0.133605i	0.902014	−	0.431708i	$-0.142089\pi$
$149$	0.941578	+	1.63086i	0.0771371	+	0.133605i	−0.824877	+	0.565313i	$0.808755\pi$
$150$		0			0
$151$	5.00000	−	8.66025i	0.406894	−	0.704761i	−0.587646	−	0.809118i	$-0.699945\pi$
$151$	5.00000	−	8.66025i	0.406894	−	0.704761i	0.994540	+	0.104357i	$0.0332784\pi$
$152$	2.37228			0.192417
$153$		0			0
$154$	14.7446			1.18815
$155$		0			0
$156$		0			0
$157$	−3.37228	−	5.84096i	−0.269137	−	0.466160i	0.699502	−	0.714631i	$-0.253405\pi$
$157$	−3.37228	−	5.84096i	−0.269137	−	0.466160i	−0.968639	+	0.248471i	$0.920072\pi$
$158$	1.00000	+	1.73205i	0.0795557	+	0.137795i
$159$		0			0
$160$		0			0
$161$	−4.62772			−0.364715
$162$		0			0
$163$	21.4891			1.68316			0.841579	−	0.540134i	$-0.181626\pi$
$163$	21.4891			1.68316			0.841579	+	0.540134i	$0.181626\pi$
$164$	−1.50000	+	2.59808i	−0.117130	+	0.202876i
$165$		0			0
$166$	3.68614	+	6.38458i	0.286100	+	0.495540i
$167$	−6.68614	−	11.5807i	−0.517389	−	0.896144i	−0.999796	−	0.0201970i	$-0.993571\pi$
$167$	−6.68614	−	11.5807i	−0.517389	−	0.896144i	0.482407	−	0.875947i	$-0.339763\pi$
$168$		0			0
$169$	−16.2446	+	28.1364i	−1.24958	+	2.16434i
$170$		0			0
$171$		0			0
$172$	5.62772			0.429110
$173$	−10.3723	+	17.9653i	−0.788590	+	1.36588i	0.138241	+	0.990399i	$0.455855\pi$
$173$	−10.3723	+	17.9653i	−0.788590	+	1.36588i	−0.926831	+	0.375479i	$0.877478\pi$
$174$		0			0
$175$		0			0
$176$	−2.18614	−	3.78651i	−0.164787	−	0.285419i
$177$		0			0
$178$	−8.05842	+	13.9576i	−0.604004	+	1.04617i
$179$	14.7446			1.10206			0.551030	−	0.834485i	$-0.314235\pi$
$179$	14.7446			1.10206			0.551030	+	0.834485i	$0.314235\pi$
$180$		0			0
$181$	20.8614			1.55062			0.775308	−	0.631583i	$-0.217595\pi$
$181$	20.8614			1.55062			0.775308	+	0.631583i	$0.217595\pi$
$182$	−11.3723	+	19.6974i	−0.842970	+	1.46007i
$183$		0			0
$184$	0.686141	+	1.18843i	0.0505830	+	0.0876123i
$185$		0			0
$186$		0			0
$187$	3.55842	−	6.16337i	0.260218	−	0.450710i
$188$	−7.37228			−0.537679
$189$		0			0
$190$		0			0
$191$	8.74456	−	15.1460i	0.632734	−	1.09593i	−0.354256	−	0.935148i	$-0.615266\pi$
$191$	8.74456	−	15.1460i	0.632734	−	1.09593i	0.986990	−	0.160780i	$-0.0514008\pi$
$192$		0			0
$193$	10.5584	+	18.2877i	0.760012	+	1.31638i	0.942844	+	0.333235i	$0.108140\pi$
$193$	10.5584	+	18.2877i	0.760012	+	1.31638i	−0.182832	+	0.983144i	$0.558526\pi$
$194$	4.18614	+	7.25061i	0.300547	+	0.520563i
$195$		0			0
$196$	−2.18614	+	3.78651i	−0.156153	+	0.270465i
$197$	−5.48913			−0.391084			−0.195542	−	0.980695i	$-0.562647\pi$
$197$	−5.48913			−0.391084			−0.195542	+	0.980695i	$0.562647\pi$
$198$		0			0
$199$	13.4891			0.956219			0.478109	−	0.878300i	$-0.341322\pi$
$199$	13.4891			0.956219			0.478109	+	0.878300i	$0.341322\pi$
$200$		0			0
$201$		0			0
$202$	−1.37228	−	2.37686i	−0.0965534	−	0.167235i
$203$	−2.31386	−	4.00772i	−0.162401	−	0.281287i
$204$		0			0
$205$		0			0
$206$	−16.0000			−1.11477
$207$		0			0
$208$	6.74456			0.467651
$209$	5.18614	−	8.98266i	0.358733	−	0.621344i
$210$		0			0
$211$	9.37228	+	16.2333i	0.645214	+	1.11754i	0.984252	+	0.176771i	$0.0565653\pi$
$211$	9.37228	+	16.2333i	0.645214	+	1.11754i	−0.339037	+	0.940773i	$0.610101\pi$
$212$	5.74456	+	9.94987i	0.394538	+	0.683360i
$213$		0			0
$214$	4.24456	−	7.35180i	0.290152	−	0.502559i
$215$		0			0
$216$		0			0
$217$	−16.0000			−1.08615
$218$	7.68614	−	13.3128i	0.520571	−	0.901656i
$219$		0			0
$220$		0			0
$221$	5.48913	+	9.50744i	0.369239	+	0.639540i
$222$		0			0
$223$	3.31386	−	5.73977i	0.221912	−	0.384364i	−0.733476	−	0.679715i	$-0.762103\pi$
$223$	3.31386	−	5.73977i	0.221912	−	0.384364i	0.955389	+	0.295351i	$0.0954368\pi$
$224$	3.37228			0.225320
$225$		0			0
$226$	−3.25544			−0.216548
$227$	−9.55842	+	16.5557i	−0.634415	+	1.09884i	0.352224	+	0.935916i	$0.385426\pi$
$227$	−9.55842	+	16.5557i	−0.634415	+	1.09884i	−0.986639	+	0.162923i	$0.947908\pi$
$228$		0			0
$229$	−6.31386	−	10.9359i	−0.417232	−	0.722666i	0.578428	−	0.815733i	$-0.303666\pi$
$229$	−6.31386	−	10.9359i	−0.417232	−	0.722666i	−0.995660	+	0.0930670i	$0.970333\pi$
$230$		0			0
$231$		0			0
$232$	−0.686141	+	1.18843i	−0.0450473	+	0.0780243i
$233$	7.11684			0.466240			0.233120	−	0.972448i	$-0.425107\pi$
$233$	7.11684			0.466240			0.233120	+	0.972448i	$0.425107\pi$
$234$		0			0
$235$		0			0
$236$	2.18614	−	3.78651i	0.142306	−	0.246481i
$237$		0			0
$238$	2.74456	+	4.75372i	0.177904	+	0.308138i
$239$	−1.62772	−	2.81929i	−0.105288	−	0.182365i	0.808568	−	0.588403i	$-0.200243\pi$
$239$	−1.62772	−	2.81929i	−0.105288	−	0.182365i	−0.913856	+	0.406038i	$0.866910\pi$
$240$		0			0
$241$	6.24456	−	10.8159i	0.402248	−	0.696713i	−0.591749	−	0.806122i	$-0.701562\pi$
$241$	6.24456	−	10.8159i	0.402248	−	0.696713i	0.993997	+	0.109409i	$0.0348958\pi$
$242$	−8.11684			−0.521770
$243$		0			0
$244$	−8.11684			−0.519628
$245$		0			0
$246$		0			0
$247$	8.00000	+	13.8564i	0.509028	+	0.881662i
$248$	2.37228	+	4.10891i	0.150640	+	0.260916i
$249$		0			0
$250$		0			0
$251$	24.6060			1.55312			0.776558	−	0.630046i	$-0.216964\pi$
$251$	24.6060			1.55312			0.776558	+	0.630046i	$0.216964\pi$
$252$		0			0
$253$	6.00000			0.377217
$254$	4.05842	−	7.02939i	0.254648	−	0.441063i
$255$		0			0
$256$	−0.500000	−	0.866025i	−0.0312500	−	0.0541266i
$257$	−2.18614	−	3.78651i	−0.136368	−	0.236196i	0.789751	−	0.613427i	$-0.210210\pi$
$257$	−2.18614	−	3.78651i	−0.136368	−	0.236196i	−0.926119	+	0.377231i	$0.876876\pi$
$258$		0			0
$259$	6.74456	−	11.6819i	0.419087	−	0.725880i
$260$		0			0
$261$		0			0
$262$	−2.74456			−0.169560
$263$	8.74456	−	15.1460i	0.539213	−	0.933944i	−0.459734	−	0.888057i	$-0.652055\pi$
$263$	8.74456	−	15.1460i	0.539213	−	0.933944i	0.998947	−	0.0458872i	$-0.0146115\pi$
$264$		0			0
$265$		0			0
$266$	4.00000	+	6.92820i	0.245256	+	0.424795i
$267$		0			0
$268$	−3.50000	+	6.06218i	−0.213797	+	0.370306i
$269$	1.37228			0.0836695			0.0418347	−	0.999125i	$-0.486680\pi$
$269$	1.37228			0.0836695			0.0418347	+	0.999125i	$0.486680\pi$
$270$		0			0
$271$	8.00000			0.485965			0.242983	−	0.970031i	$-0.421874\pi$
$271$	8.00000			0.485965			0.242983	+	0.970031i	$0.421874\pi$
$272$	0.813859	−	1.40965i	0.0493475	−	0.0854723i
$273$		0			0
$274$	−9.55842	−	16.5557i	−0.577445	−	1.00016i
$275$		0			0
$276$		0			0
$277$	8.37228	−	14.5012i	0.503042	−	0.871294i	−0.496952	−	0.867778i	$-0.665548\pi$
$277$	8.37228	−	14.5012i	0.503042	−	0.871294i	0.999994	−	0.00351574i	$-0.00111910\pi$
$278$	−0.883156			−0.0529682
$279$		0			0
$280$		0			0
$281$	−0.686141	+	1.18843i	−0.0409317	+	0.0708958i	−0.885765	−	0.464133i	$-0.846366\pi$
$281$	−0.686141	+	1.18843i	−0.0409317	+	0.0708958i	0.844834	+	0.535029i	$0.179699\pi$
$282$		0			0
$283$	−1.56930	−	2.71810i	−0.0932850	−	0.161574i	0.815607	−	0.578607i	$-0.196403\pi$
$283$	−1.56930	−	2.71810i	−0.0932850	−	0.161574i	−0.908892	+	0.417033i	$0.863070\pi$
$284$	−3.00000	−	5.19615i	−0.178017	−	0.308335i
$285$		0			0
$286$	14.7446	−	25.5383i	0.871864	−	1.51011i
$287$	−10.1168			−0.597178
$288$		0			0
$289$	−14.3505			−0.844149
$290$		0			0
$291$		0			0
$292$	1.55842	+	2.69927i	0.0911997	+	0.157963i
$293$	−13.1168	−	22.7190i	−0.766294	−	1.32726i	−0.939560	−	0.342385i	$-0.888765\pi$
$293$	−13.1168	−	22.7190i	−0.766294	−	1.32726i	0.173265	−	0.984875i	$-0.444568\pi$
$294$		0			0
$295$		0			0
$296$	−4.00000			−0.232495
$297$		0			0
$298$	1.88316			0.109088
$299$	−4.62772	+	8.01544i	−0.267628	+	0.463545i
$300$		0			0
$301$	9.48913	+	16.4356i	0.546944	+	0.947335i
$302$	−5.00000	−	8.66025i	−0.287718	−	0.498342i
$303$		0			0
$304$	1.18614	−	2.05446i	0.0680298	−	0.117831i
$305$		0			0
$306$		0			0
$307$	−1.23369			−0.0704103			−0.0352051	−	0.999380i	$-0.511208\pi$
$307$	−1.23369			−0.0704103			−0.0352051	+	0.999380i	$0.511208\pi$
$308$	7.37228	−	12.7692i	0.420075	−	0.727591i
$309$		0			0
$310$		0			0
$311$	−10.3723	−	17.9653i	−0.588158	−	1.01872i	−0.994474	−	0.104987i	$-0.966520\pi$
$311$	−10.3723	−	17.9653i	−0.588158	−	1.01872i	0.406315	−	0.913733i	$-0.366813\pi$
$312$		0			0
$313$	1.81386	−	3.14170i	0.102525	−	0.177579i	−0.810199	−	0.586155i	$-0.800641\pi$
$313$	1.81386	−	3.14170i	0.102525	−	0.177579i	0.912724	+	0.408576i	$0.133974\pi$
$314$	−6.74456			−0.380618
$315$		0			0
$316$	2.00000			0.112509
$317$	−1.37228	+	2.37686i	−0.0770750	+	0.133498i	−0.901987	−	0.431764i	$-0.857891\pi$
$317$	−1.37228	+	2.37686i	−0.0770750	+	0.133498i	0.824912	+	0.565262i	$0.191225\pi$
$318$		0			0
$319$	3.00000	+	5.19615i	0.167968	+	0.290929i
$320$		0			0
$321$		0			0
$322$	−2.31386	+	4.00772i	−0.128946	+	0.223342i
$323$	3.86141			0.214854
$324$		0			0
$325$		0			0
$326$	10.7446	−	18.6101i	0.595086	−	1.03072i
$327$		0			0
$328$	1.50000	+	2.59808i	0.0828236	+	0.143455i
$329$	−12.4307	−	21.5306i	−0.685327	−	1.18702i
$330$		0			0
$331$	−8.11684	+	14.0588i	−0.446142	+	0.772741i	−0.998131	−	0.0611107i	$-0.980536\pi$
$331$	−8.11684	+	14.0588i	−0.446142	+	0.772741i	0.551989	+	0.833851i	$0.313869\pi$
$332$	7.37228			0.404607
$333$		0			0
$334$	−13.3723			−0.731699
$335$		0			0
$336$		0			0
$337$	−1.18614	−	2.05446i	−0.0646132	−	0.111913i	0.831909	−	0.554912i	$-0.187248\pi$
$337$	−1.18614	−	2.05446i	−0.0646132	−	0.111913i	−0.896522	+	0.442999i	$0.853915\pi$
$338$	16.2446	+	28.1364i	0.883588	+	1.53042i
$339$		0			0
$340$		0			0
$341$	20.7446			1.12338
$342$		0			0
$343$	8.86141			0.478471
$344$	2.81386	−	4.87375i	0.151713	−	0.262775i
$345$		0			0
$346$	10.3723	+	17.9653i	0.557617	+	0.965821i
$347$	2.44158	+	4.22894i	0.131071	+	0.227021i	0.924090	−	0.382176i	$-0.124825\pi$
$347$	2.44158	+	4.22894i	0.131071	+	0.227021i	−0.793019	+	0.609197i	$0.791492\pi$
$348$		0			0
$349$	−9.05842	+	15.6896i	−0.484886	+	0.839848i	−0.999849	−	0.0173648i	$-0.994472\pi$
$349$	−9.05842	+	15.6896i	−0.484886	+	0.839848i	0.514963	+	0.857212i	$0.327806\pi$
$350$		0			0
$351$		0			0
$352$	−4.37228			−0.233043
$353$	10.6753	−	18.4901i	0.568187	−	0.984129i	−0.428558	−	0.903514i	$-0.640978\pi$
$353$	10.6753	−	18.4901i	0.568187	−	0.984129i	0.996745	−	0.0806147i	$-0.0256883\pi$
$354$		0			0
$355$		0			0
$356$	8.05842	+	13.9576i	0.427096	+	0.739751i
$357$		0			0
$358$	7.37228	−	12.7692i	0.389637	−	0.674871i
$359$	17.4891			0.923041			0.461520	−	0.887130i	$-0.347304\pi$
$359$	17.4891			0.923041			0.461520	+	0.887130i	$0.347304\pi$
$360$		0			0
$361$	−13.3723			−0.703804
$362$	10.4307	−	18.0665i	0.548226	−	0.949555i
$363$		0			0
$364$	11.3723	+	19.6974i	0.596070	+	1.03242i
$365$		0			0
$366$		0			0
$367$	−8.00000	+	13.8564i	−0.417597	+	0.723299i	−0.995697	−	0.0926670i	$-0.970461\pi$
$367$	−8.00000	+	13.8564i	−0.417597	+	0.723299i	0.578101	+	0.815966i	$0.303794\pi$
$368$	1.37228			0.0715351
$369$		0			0
$370$		0			0
$371$	−19.3723	+	33.5538i	−1.00576	+	1.74203i
$372$		0			0
$373$	−7.74456	−	13.4140i	−0.400998	−	0.694549i	0.592848	−	0.805314i	$-0.298003\pi$
$373$	−7.74456	−	13.4140i	−0.400998	−	0.694549i	−0.993847	+	0.110765i	$0.964670\pi$
$374$	−3.55842	−	6.16337i	−0.184002	−	0.318700i
$375$		0			0
$376$	−3.68614	+	6.38458i	−0.190098	+	0.329260i
$377$	−9.25544			−0.476679
$378$		0			0
$379$	17.8614			0.917479			0.458739	−	0.888571i	$-0.348301\pi$
$379$	17.8614			0.917479			0.458739	+	0.888571i	$0.348301\pi$
$380$		0			0
$381$		0			0
$382$	−8.74456	−	15.1460i	−0.447411	−	0.774938i
$383$	11.4891	+	19.8997i	0.587067	+	1.01683i	0.994614	+	0.103646i	$0.0330508\pi$
$383$	11.4891	+	19.8997i	0.587067	+	1.01683i	−0.407547	+	0.913184i	$0.633616\pi$
$384$		0			0
$385$		0			0
$386$	21.1168			1.07482
$387$		0			0
$388$	8.37228			0.425038
$389$	−2.31386	+	4.00772i	−0.117317	+	0.203200i	−0.918704	−	0.394947i	$-0.870763\pi$
$389$	−2.31386	+	4.00772i	−0.117317	+	0.203200i	0.801386	+	0.598147i	$0.204096\pi$
$390$		0			0
$391$	1.11684	+	1.93443i	0.0564812	+	0.0978284i
$392$	2.18614	+	3.78651i	0.110417	+	0.191247i
$393$		0			0
$394$	−2.74456	+	4.75372i	−0.138269	+	0.239489i
$395$		0			0
$396$		0			0
$397$	−22.7446			−1.14152			−0.570758	−	0.821118i	$-0.693351\pi$
$397$	−22.7446			−1.14152			−0.570758	+	0.821118i	$0.693351\pi$
$398$	6.74456	−	11.6819i	0.338074	−	0.585562i
$399$		0			0
$400$		0			0
$401$	−0.558422	−	0.967215i	−0.0278863	−	0.0483004i	0.851745	−	0.523956i	$-0.175544\pi$
$401$	−0.558422	−	0.967215i	−0.0278863	−	0.0483004i	−0.879632	+	0.475655i	$0.842211\pi$
$402$		0			0
$403$	−16.0000	+	27.7128i	−0.797017	+	1.38047i
$404$	−2.74456			−0.136547
$405$		0			0
$406$	−4.62772			−0.229670
$407$	−8.74456	+	15.1460i	−0.433452	+	0.750761i
$408$		0			0
$409$	−8.93070	−	15.4684i	−0.441595	−	0.764865i	0.556213	−	0.831040i	$-0.312254\pi$
$409$	−8.93070	−	15.4684i	−0.441595	−	0.764865i	−0.997808	+	0.0661749i	$0.978920\pi$
$410$		0			0
$411$		0			0
$412$	−8.00000	+	13.8564i	−0.394132	+	0.682656i
$413$	14.7446			0.725532
$414$		0			0
$415$		0			0
$416$	3.37228	−	5.84096i	0.165340	−	0.286377i
$417$		0			0
$418$	−5.18614	−	8.98266i	−0.253662	−	0.439356i
$419$	12.8614	+	22.2766i	0.628321	+	1.08828i	0.987889	+	0.155165i	$0.0495908\pi$
$419$	12.8614	+	22.2766i	0.628321	+	1.08828i	−0.359568	+	0.933119i	$0.617076\pi$
$420$		0			0
$421$	−15.2337	+	26.3855i	−0.742445	+	1.28595i	0.208935	+	0.977930i	$0.433000\pi$
$421$	−15.2337	+	26.3855i	−0.742445	+	1.28595i	−0.951379	+	0.308022i	$0.900333\pi$
$422$	18.7446			0.912471
$423$		0			0
$424$	11.4891			0.557961
$425$		0			0
$426$		0			0
$427$	−13.6861	−	23.7051i	−0.662319	−	1.14717i
$428$	−4.24456	−	7.35180i	−0.205169	−	0.355363i
$429$		0			0
$430$		0			0
$431$	−8.23369			−0.396603			−0.198301	−	0.980141i	$-0.563542\pi$
$431$	−8.23369			−0.396603			−0.198301	+	0.980141i	$0.563542\pi$
$432$		0			0
$433$	−6.37228			−0.306232			−0.153116	−	0.988208i	$-0.548931\pi$
$433$	−6.37228			−0.306232			−0.153116	+	0.988208i	$0.548931\pi$
$434$	−8.00000	+	13.8564i	−0.384012	+	0.665129i
$435$		0			0
$436$	−7.68614	−	13.3128i	−0.368099	−	0.637567i
$437$	1.62772	+	2.81929i	0.0778643	+	0.134865i
$438$		0			0
$439$	9.11684	−	15.7908i	0.435123	−	0.753656i	−0.562182	−	0.827013i	$-0.690038\pi$
$439$	9.11684	−	15.7908i	0.435123	−	0.753656i	0.997306	+	0.0733577i	$0.0233715\pi$
$440$		0			0
$441$		0			0
$442$	10.9783			0.522182
$443$	1.75544	−	3.04051i	0.0834033	−	0.144459i	−0.821306	−	0.570487i	$-0.806754\pi$
$443$	1.75544	−	3.04051i	0.0834033	−	0.144459i	0.904710	+	0.426029i	$0.140088\pi$
$444$		0			0
$445$		0			0
$446$	−3.31386	−	5.73977i	−0.156916	−	0.271786i
$447$		0			0
$448$	1.68614	−	2.92048i	0.0796627	−	0.137980i
$449$	−9.86141			−0.465389			−0.232694	−	0.972550i	$-0.574754\pi$
$449$	−9.86141			−0.465389			−0.232694	+	0.972550i	$0.574754\pi$
$450$		0			0
$451$	13.1168			0.617648
$452$	−1.62772	+	2.81929i	−0.0765614	+	0.132608i
$453$		0			0
$454$	9.55842	+	16.5557i	0.448599	+	0.776996i
$455$		0			0
$456$		0			0
$457$	6.44158	−	11.1571i	0.301324	−	0.521909i	−0.675112	−	0.737715i	$-0.735905\pi$
$457$	6.44158	−	11.1571i	0.301324	−	0.521909i	0.976436	+	0.215806i	$0.0692380\pi$
$458$	−12.6277			−0.590055
$459$		0			0
$460$		0			0
$461$	0.941578	−	1.63086i	0.0438537	−	0.0759568i	−0.843265	−	0.537497i	$-0.819370\pi$
$461$	0.941578	−	1.63086i	0.0438537	−	0.0759568i	0.887119	+	0.461541i	$0.152703\pi$
$462$		0			0
$463$	10.0000	+	17.3205i	0.464739	+	0.804952i	0.999190	−	0.0402476i	$-0.0128147\pi$
$463$	10.0000	+	17.3205i	0.464739	+	0.804952i	−0.534450	+	0.845200i	$0.679481\pi$
$464$	0.686141	+	1.18843i	0.0318533	+	0.0551715i
$465$		0			0
$466$	3.55842	−	6.16337i	0.164841	−	0.285512i
$467$	43.1168			1.99521			0.997605	−	0.0691713i	$-0.0220355\pi$
$467$	43.1168			1.99521			0.997605	+	0.0691713i	$0.0220355\pi$
$468$		0			0
$469$	−23.6060			−1.09002
$470$		0			0
$471$		0			0
$472$	−2.18614	−	3.78651i	−0.100625	−	0.174288i
$473$	−12.3030	−	21.3094i	−0.565692	−	0.979807i
$474$		0			0
$475$		0			0
$476$	5.48913			0.251594
$477$		0			0
$478$	−3.25544			−0.148900
$479$	−0.255437	+	0.442430i	−0.0116712	+	0.0202152i	−0.871802	−	0.489858i	$-0.837048\pi$
$479$	−0.255437	+	0.442430i	−0.0116712	+	0.0202152i	0.860131	+	0.510074i	$0.170382\pi$
$480$		0			0
$481$	−13.4891	−	23.3639i	−0.615051	−	1.06530i
$482$	−6.24456	−	10.8159i	−0.284432	−	0.492651i
$483$		0			0
$484$	−4.05842	+	7.02939i	−0.184474	+	0.319518i
$485$		0			0
$486$		0			0
$487$	12.7446			0.577511			0.288756	−	0.957403i	$-0.406758\pi$
$487$	12.7446			0.577511			0.288756	+	0.957403i	$0.406758\pi$
$488$	−4.05842	+	7.02939i	−0.183716	+	0.318206i
$489$		0			0
$490$		0			0
$491$	−18.3030	−	31.7017i	−0.826002	−	1.43068i	−0.901151	−	0.433505i	$-0.857277\pi$
$491$	−18.3030	−	31.7017i	−0.826002	−	1.43068i	0.0751489	−	0.997172i	$-0.476057\pi$
$492$		0			0
$493$	−1.11684	+	1.93443i	−0.0503001	+	0.0871224i
$494$	16.0000			0.719874
$495$		0			0
$496$	4.74456			0.213037
$497$	10.1168	−	17.5229i	0.453802	−	0.786009i
$498$		0			0
$499$	−7.55842	−	13.0916i	−0.338361	−	0.586059i	0.645763	−	0.763538i	$-0.276539\pi$
$499$	−7.55842	−	13.0916i	−0.338361	−	0.586059i	−0.984125	+	0.177479i	$0.943206\pi$
$500$		0			0
$501$		0			0
$502$	12.3030	−	21.3094i	0.549109	−	0.951085i
$503$	−6.86141			−0.305935			−0.152967	−	0.988231i	$-0.548883\pi$
$503$	−6.86141			−0.305935			−0.152967	+	0.988231i	$0.548883\pi$
$504$		0			0
$505$		0			0
$506$	3.00000	−	5.19615i	0.133366	−	0.230997i
$507$		0			0
$508$	−4.05842	−	7.02939i	−0.180063	−	0.311879i
$509$	−21.1753	−	36.6766i	−0.938577	−	1.62566i	−0.768127	−	0.640297i	$-0.778811\pi$
$509$	−21.1753	−	36.6766i	−0.938577	−	1.62566i	−0.170450	−	0.985366i	$-0.554522\pi$
$510$		0			0
$511$	−5.25544	+	9.10268i	−0.232487	+	0.402679i
$512$	−1.00000			−0.0441942
$513$		0			0
$514$	−4.37228			−0.192853
$515$		0			0
$516$		0			0
$517$	16.1168	+	27.9152i	0.708818	+	1.22771i
$518$	−6.74456	−	11.6819i	−0.296339	−	0.513274i
$519$		0			0
$520$		0			0
$521$	6.76631			0.296438			0.148219	−	0.988955i	$-0.452646\pi$
$521$	6.76631			0.296438			0.148219	+	0.988955i	$0.452646\pi$
$522$		0			0
$523$	−6.11684			−0.267471			−0.133735	−	0.991017i	$-0.542697\pi$
$523$	−6.11684			−0.267471			−0.133735	+	0.991017i	$0.542697\pi$
$524$	−1.37228	+	2.37686i	−0.0599484	+	0.103834i
$525$		0			0
$526$	−8.74456	−	15.1460i	−0.381281	−	0.660398i
$527$	3.86141	+	6.68815i	0.168206	+	0.291340i
$528$		0			0
$529$	10.5584	−	18.2877i	0.459062	−	0.795118i
$530$		0			0
$531$		0			0
$532$	8.00000			0.346844
$533$	−10.1168	+	17.5229i	−0.438209	+	0.759001i
$534$		0			0
$535$		0			0
$536$	3.50000	+	6.06218i	0.151177	+	0.261846i
$537$		0			0
$538$	0.686141	−	1.18843i	0.0295816	−	0.0512369i
$539$	19.1168			0.823421
$540$		0			0
$541$	27.3723			1.17683			0.588413	−	0.808560i	$-0.299753\pi$
$541$	27.3723			1.17683			0.588413	+	0.808560i	$0.299753\pi$
$542$	4.00000	−	6.92820i	0.171815	−	0.297592i
$543$		0			0
$544$	−0.813859	−	1.40965i	−0.0348939	−	0.0604381i
$545$		0			0
$546$		0			0
$547$	−14.7337	+	25.5195i	−0.629967	+	1.09113i	0.357591	+	0.933878i	$0.383598\pi$
$547$	−14.7337	+	25.5195i	−0.629967	+	1.09113i	−0.987558	+	0.157256i	$0.949735\pi$
$548$	−19.1168			−0.816631
$549$		0			0
$550$		0			0
$551$	−1.62772	+	2.81929i	−0.0693431	+	0.120106i
$552$		0			0
$553$	3.37228	+	5.84096i	0.143404	+	0.248383i
$554$	−8.37228	−	14.5012i	−0.355704	−	0.616098i
$555$		0			0
$556$	−0.441578	+	0.764836i	−0.0187271	+	0.0324363i
$557$	−44.2337			−1.87424			−0.937121	−	0.349005i	$-0.886520\pi$
$557$	−44.2337			−1.87424			−0.937121	+	0.349005i	$0.886520\pi$
$558$		0			0
$559$	37.9565			1.60539
$560$		0			0
$561$		0			0
$562$	0.686141	+	1.18843i	0.0289431	+	0.0501309i
$563$	20.3614	+	35.2670i	0.858131	+	1.48633i	0.873710	+	0.486448i	$0.161708\pi$
$563$	20.3614	+	35.2670i	0.858131	+	1.48633i	−0.0155787	+	0.999879i	$0.504959\pi$
$564$		0			0
$565$		0			0
$566$	−3.13859			−0.131925
$567$		0			0
$568$	−6.00000			−0.251754
$569$	−15.3030	+	26.5055i	−0.641534	+	1.11117i	0.343556	+	0.939132i	$0.388369\pi$
$569$	−15.3030	+	26.5055i	−0.641534	+	1.11117i	−0.985090	+	0.172038i	$0.944965\pi$
$570$		0			0
$571$	−4.30298	−	7.45299i	−0.180074	−	0.311898i	0.761831	−	0.647775i	$-0.224300\pi$
$571$	−4.30298	−	7.45299i	−0.180074	−	0.311898i	−0.941906	+	0.335878i	$0.890967\pi$
$572$	−14.7446	−	25.5383i	−0.616501	−	1.06781i
$573$		0			0
$574$	−5.05842	+	8.76144i	−0.211134	+	0.365696i
$575$		0			0
$576$		0			0
$577$	41.1168			1.71172			0.855858	−	0.517210i	$-0.173030\pi$
$577$	41.1168			1.71172			0.855858	+	0.517210i	$0.173030\pi$
$578$	−7.17527	+	12.4279i	−0.298452	+	0.516934i
$579$		0			0
$580$		0			0
$581$	12.4307	+	21.5306i	0.515712	+	0.893240i
$582$		0			0
$583$	25.1168	−	43.5036i	1.04023	−	1.80174i
$584$	3.11684			0.128976
$585$		0			0
$586$	−26.2337			−1.08370
$587$	−13.5000	+	23.3827i	−0.557205	+	0.965107i	0.440524	+	0.897741i	$0.354793\pi$
$587$	−13.5000	+	23.3827i	−0.557205	+	0.965107i	−0.997728	+	0.0673658i	$0.978541\pi$
$588$		0			0
$589$	5.62772	+	9.74749i	0.231886	+	0.401639i
$590$		0			0
$591$		0			0
$592$	−2.00000	+	3.46410i	−0.0821995	+	0.142374i
$593$	−19.7228			−0.809919			−0.404959	−	0.914335i	$-0.632714\pi$
$593$	−19.7228			−0.809919			−0.404959	+	0.914335i	$0.632714\pi$
$594$		0			0
$595$		0			0
$596$	0.941578	−	1.63086i	0.0385685	−	0.0668027i
$597$		0			0
$598$	4.62772	+	8.01544i	0.189241	+	0.327776i
$599$	−1.88316	−	3.26172i	−0.0769437	−	0.133270i	0.824986	−	0.565153i	$-0.191183\pi$
$599$	−1.88316	−	3.26172i	−0.0769437	−	0.133270i	−0.901930	+	0.431883i	$0.857849\pi$
$600$		0			0
$601$	0.930703	−	1.61203i	0.0379642	−	0.0657559i	−0.846419	−	0.532517i	$-0.821246\pi$
$601$	0.930703	−	1.61203i	0.0379642	−	0.0657559i	0.884383	+	0.466762i	$0.154579\pi$
$602$	18.9783			0.773496
$603$		0			0
$604$	−10.0000			−0.406894
$605$		0			0
$606$		0			0
$607$	9.05842	+	15.6896i	0.367670	+	0.636823i	0.989201	−	0.146567i	$-0.0468223\pi$
$607$	9.05842	+	15.6896i	0.367670	+	0.636823i	−0.621531	+	0.783390i	$0.713489\pi$
$608$	−1.18614	−	2.05446i	−0.0481044	−	0.0833192i
$609$		0			0
$610$		0			0
$611$	−49.7228			−2.01157
$612$		0			0
$613$	−34.2337			−1.38269			−0.691343	−	0.722527i	$-0.742981\pi$
$613$	−34.2337			−1.38269			−0.691343	+	0.722527i	$0.742981\pi$
$614$	−0.616844	+	1.06841i	−0.0248938	+	0.0431173i
$615$		0			0
$616$	−7.37228	−	12.7692i	−0.297038	−	0.514484i
$617$	2.44158	+	4.22894i	0.0982942	+	0.170251i	0.910979	−	0.412453i	$-0.135328\pi$
$617$	2.44158	+	4.22894i	0.0982942	+	0.170251i	−0.812684	+	0.582704i	$0.801995\pi$
$618$		0			0
$619$	10.4416	−	18.0853i	0.419682	−	0.726911i	−0.576225	−	0.817291i	$-0.695475\pi$
$619$	10.4416	−	18.0853i	0.419682	−	0.726911i	0.995907	+	0.0903798i	$0.0288081\pi$
$620$		0			0
$621$		0			0
$622$	−20.7446			−0.831781
$623$	−27.1753	+	47.0689i	−1.08875	+	1.88578i
$624$		0			0
$625$		0			0
$626$	−1.81386	−	3.14170i	−0.0724964	−	0.125567i
$627$		0			0
$628$	−3.37228	+	5.84096i	−0.134569	+	0.233080i
$629$	−6.51087			−0.259606
$630$		0			0
$631$	−23.7228			−0.944390			−0.472195	−	0.881494i	$-0.656538\pi$
$631$	−23.7228			−0.944390			−0.472195	+	0.881494i	$0.656538\pi$
$632$	1.00000	−	1.73205i	0.0397779	−	0.0688973i
$633$		0			0
$634$	1.37228	+	2.37686i	0.0545003	+	0.0943972i
$635$		0			0
$636$		0			0
$637$	−14.7446	+	25.5383i	−0.584201	+	1.01187i
$638$	6.00000			0.237542
$639$		0			0
$640$		0			0
$641$	−19.5000	+	33.7750i	−0.770204	+	1.33403i	0.167247	+	0.985915i	$0.446512\pi$
$641$	−19.5000	+	33.7750i	−0.770204	+	1.33403i	−0.937451	+	0.348117i	$0.886821\pi$
$642$		0			0
$643$	5.50000	+	9.52628i	0.216899	+	0.375680i	0.953858	−	0.300257i	$-0.0970725\pi$
$643$	5.50000	+	9.52628i	0.216899	+	0.375680i	−0.736959	+	0.675937i	$0.763739\pi$
$644$	2.31386	+	4.00772i	0.0911788	+	0.157926i
$645$		0			0
$646$	1.93070	−	3.34408i	0.0759625	−	0.131571i
$647$	−39.0951			−1.53699			−0.768493	−	0.639858i	$-0.778993\pi$
$647$	−39.0951			−1.53699			−0.768493	+	0.639858i	$0.778993\pi$
$648$		0			0
$649$	−19.1168			−0.750402
$650$		0			0
$651$		0			0
$652$	−10.7446	−	18.6101i	−0.420790	−	0.728829i
$653$	−9.86141	−	17.0805i	−0.385907	−	0.668410i	0.605988	−	0.795474i	$-0.292778\pi$
$653$	−9.86141	−	17.0805i	−0.385907	−	0.668410i	−0.991895	+	0.127064i	$0.959445\pi$
$654$		0			0
$655$		0			0
$656$	3.00000			0.117130
$657$		0			0
$658$	−24.8614			−0.969199
$659$	8.74456	−	15.1460i	0.340640	−	0.590005i	−0.643912	−	0.765100i	$-0.722690\pi$
$659$	8.74456	−	15.1460i	0.340640	−	0.590005i	0.984552	+	0.175094i	$0.0560230\pi$
$660$		0			0
$661$	6.11684	+	10.5947i	0.237918	+	0.412085i	0.960117	−	0.279600i	$-0.0902018\pi$
$661$	6.11684	+	10.5947i	0.237918	+	0.412085i	−0.722199	+	0.691685i	$0.756868\pi$
$662$	8.11684	+	14.0588i	0.315470	+	0.546410i
$663$		0			0
$664$	3.68614	−	6.38458i	0.143050	−	0.247770i
$665$		0			0
$666$		0			0
$667$	−1.88316			−0.0729161
$668$	−6.68614	+	11.5807i	−0.258695	+	0.448072i
$669$		0			0
$670$		0			0
$671$	17.7446	+	30.7345i	0.685021	+	1.18649i
$672$		0			0
$673$	−5.00000	+	8.66025i	−0.192736	+	0.333828i	−0.946156	−	0.323711i	$-0.895069\pi$
$673$	−5.00000	+	8.66025i	−0.192736	+	0.333828i	0.753420	+	0.657539i	$0.228403\pi$
$674$	−2.37228			−0.0913769
$675$		0			0
$676$	32.4891			1.24958
$677$	−6.86141	+	11.8843i	−0.263705	+	0.456751i	−0.967224	−	0.253926i	$-0.918278\pi$
$677$	−6.86141	+	11.8843i	−0.263705	+	0.456751i	0.703518	+	0.710677i	$0.251611\pi$
$678$		0			0
$679$	14.1168	+	24.4511i	0.541755	+	0.938347i
$680$		0			0
$681$		0			0
$682$	10.3723	−	17.9653i	0.397175	−	0.687928i
$683$	−30.0951			−1.15156			−0.575778	−	0.817606i	$-0.695301\pi$
$683$	−30.0951			−1.15156			−0.575778	+	0.817606i	$0.695301\pi$
$684$		0			0
$685$		0			0
$686$	4.43070	−	7.67420i	0.169165	−	0.293002i
$687$		0			0
$688$	−2.81386	−	4.87375i	−0.107277	−	0.185810i
$689$	38.7446	+	67.1076i	1.47605	+	2.55659i
$690$		0			0
$691$	18.1168	−	31.3793i	0.689197	−	1.19372i	−0.282901	−	0.959149i	$-0.591297\pi$
$691$	18.1168	−	31.3793i	0.689197	−	1.19372i	0.972098	−	0.234575i	$-0.0753700\pi$
$692$	20.7446			0.788590
$693$		0			0
$694$	4.88316			0.185362
$695$		0			0
$696$		0			0
$697$	2.44158	+	4.22894i	0.0924814	+	0.160182i
$698$	9.05842	+	15.6896i	0.342866	+	0.593862i
$699$		0			0
$700$		0			0
$701$	−42.8614			−1.61885			−0.809426	−	0.587221i	$-0.800222\pi$
$701$	−42.8614			−1.61885			−0.809426	+	0.587221i	$0.800222\pi$
$702$		0			0
$703$	−9.48913			−0.357889
$704$	−2.18614	+	3.78651i	−0.0823933	+	0.142709i
$705$		0			0
$706$	−10.6753	−	18.4901i	−0.401769	−	0.695884i
$707$	−4.62772	−	8.01544i	−0.174043	−	0.301452i
$708$		0			0
$709$	−1.43070	+	2.47805i	−0.0537312	+	0.0930652i	−0.891640	−	0.452745i	$-0.850445\pi$
$709$	−1.43070	+	2.47805i	−0.0537312	+	0.0930652i	0.837909	+	0.545810i	$0.183778\pi$
$710$		0			0
$711$		0			0
$712$	16.1168			0.604004
$713$	−3.25544	+	5.63858i	−0.121917	+	0.211167i
$714$		0			0
$715$		0			0
$716$	−7.37228	−	12.7692i	−0.275515	−	0.477206i
$717$		0			0
$718$	8.74456	−	15.1460i	0.326344	−	0.565245i
$719$	3.76631			0.140460			0.0702299	−	0.997531i	$-0.477627\pi$
$719$	3.76631			0.140460			0.0702299	+	0.997531i	$0.477627\pi$
$720$		0			0
$721$	−53.9565			−2.00945
$722$	−6.68614	+	11.5807i	−0.248832	+	0.430990i
$723$		0			0
$724$	−10.4307	−	18.0665i	−0.387654	−	0.671436i
$725$		0			0
$726$		0			0
$727$	9.05842	−	15.6896i	0.335958	−	0.581897i	−0.647710	−	0.761887i	$-0.724273\pi$
$727$	9.05842	−	15.6896i	0.335958	−	0.581897i	0.983668	+	0.179990i	$0.0576066\pi$
$728$	22.7446			0.842970
$729$		0			0
$730$		0			0
$731$	4.58017	−	7.93309i	0.169404	−	0.293416i
$732$		0			0
$733$	−0.116844	−	0.202380i	−0.00431573	−	0.00747506i	0.863859	−	0.503733i	$-0.168040\pi$
$733$	−0.116844	−	0.202380i	−0.00431573	−	0.00747506i	−0.868175	+	0.496258i	$0.834707\pi$
$734$	8.00000	+	13.8564i	0.295285	+	0.511449i
$735$		0			0
$736$	0.686141	−	1.18843i	0.0252915	−	0.0438061i
$737$	30.6060			1.12739
$738$		0			0
$739$	−41.1168			−1.51251			−0.756254	−	0.654278i	$-0.772972\pi$
$739$	−41.1168			−1.51251			−0.756254	+	0.654278i	$0.772972\pi$
$740$		0			0
$741$		0			0
$742$	19.3723	+	33.5538i	0.711179	+	1.23180i
$743$	−3.94158	−	6.82701i	−0.144602	−	0.250459i	0.784622	−	0.619974i	$-0.212857\pi$
$743$	−3.94158	−	6.82701i	−0.144602	−	0.250459i	−0.929225	+	0.369516i	$0.879524\pi$
$744$		0			0
$745$		0			0
$746$	−15.4891			−0.567097
$747$		0			0
$748$	−7.11684			−0.260218
$749$	14.3139	−	24.7923i	0.523017	−	0.905892i
$750$		0			0
$751$	−8.11684	−	14.0588i	−0.296188	−	0.513012i	0.679073	−	0.734071i	$-0.262382\pi$
$751$	−8.11684	−	14.0588i	−0.296188	−	0.513012i	−0.975261	+	0.221059i	$0.929049\pi$
$752$	3.68614	+	6.38458i	0.134420	+	0.232822i
$753$		0			0
$754$	−4.62772	+	8.01544i	−0.168532	+	0.291905i
$755$		0			0
$756$		0			0
$757$	10.0000			0.363456			0.181728	−	0.983349i	$-0.441831\pi$
$757$	10.0000			0.363456			0.181728	+	0.983349i	$0.441831\pi$
$758$	8.93070	−	15.4684i	0.324378	−	0.561839i
$759$		0			0
$760$		0			0
$761$	−25.5475	−	44.2496i	−0.926098	−	1.60405i	−0.789786	−	0.613383i	$-0.789808\pi$
$761$	−25.5475	−	44.2496i	−0.926098	−	1.60405i	−0.136312	−	0.990666i	$-0.543525\pi$
$762$		0			0
$763$	25.9198	−	44.8945i	0.938361	−	1.62529i
$764$	−17.4891			−0.632734
$765$		0			0
$766$	22.9783			0.830238
$767$	14.7446	−	25.5383i	0.532395	−	0.922136i
$768$		0			0
$769$	23.4307	+	40.5832i	0.844933	+	1.46347i	0.885680	+	0.464297i	$0.153693\pi$
$769$	23.4307	+	40.5832i	0.844933	+	1.46347i	−0.0407468	+	0.999170i	$0.512974\pi$
$770$		0			0
$771$		0			0
$772$	10.5584	−	18.2877i	0.380006	−	0.658190i
$773$	3.25544			0.117090			0.0585450	−	0.998285i	$-0.481354\pi$
$773$	3.25544			0.117090			0.0585450	+	0.998285i	$0.481354\pi$
$774$		0			0
$775$		0			0
$776$	4.18614	−	7.25061i	0.150274	−	0.260282i
$777$		0			0
$778$	2.31386	+	4.00772i	0.0829559	+	0.143684i
$779$	3.55842	+	6.16337i	0.127494	+	0.220826i
$780$		0			0
$781$	−13.1168	+	22.7190i	−0.469358	+	0.812951i
$782$	2.23369			0.0798765
$783$		0			0
$784$	4.37228			0.156153
$785$		0			0
$786$		0			0
$787$	−14.0000	−	24.2487i	−0.499046	−	0.864373i	0.500953	−	0.865474i	$-0.332983\pi$
$787$	−14.0000	−	24.2487i	−0.499046	−	0.864373i	−0.999999	+	0.00110111i	$0.999650\pi$
$788$	2.74456	+	4.75372i	0.0977710	+	0.169344i

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:	\( N \)	\(=\)	\( 1350 = 2 \cdot 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1350.e (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.500000 - 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{4} +(1.68614 - 2.92048i) q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q+(0.500000 - 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{4} +(1.68614 - 2.92048i) q^{7} -1.00000 q^{8} +(-2.18614 + 3.78651i) q^{11} +(-3.37228 - 5.84096i) q^{13} +(-1.68614 - 2.92048i) q^{14} +(-0.500000 + 0.866025i) q^{16} -1.62772 q^{17} -2.37228 q^{19} +(2.18614 + 3.78651i) q^{22} +(-0.686141 - 1.18843i) q^{23} -6.74456 q^{26} -3.37228 q^{28} +(0.686141 - 1.18843i) q^{29} +(-2.37228 - 4.10891i) q^{31} +(0.500000 + 0.866025i) q^{32} +(-0.813859 + 1.40965i) q^{34} +4.00000 q^{37} +(-1.18614 + 2.05446i) q^{38} +(-1.50000 - 2.59808i) q^{41} +(-2.81386 + 4.87375i) q^{43} +4.37228 q^{44} -1.37228 q^{46} +(3.68614 - 6.38458i) q^{47} +(-2.18614 - 3.78651i) q^{49} +(-3.37228 + 5.84096i) q^{52} -11.4891 q^{53} +(-1.68614 + 2.92048i) q^{56} +(-0.686141 - 1.18843i) q^{58} +(2.18614 + 3.78651i) q^{59} +(4.05842 - 7.02939i) q^{61} -4.74456 q^{62} +1.00000 q^{64} +(-3.50000 - 6.06218i) q^{67} +(0.813859 + 1.40965i) q^{68} +6.00000 q^{71} -3.11684 q^{73} +(2.00000 - 3.46410i) q^{74} +(1.18614 + 2.05446i) q^{76} +(7.37228 + 12.7692i) q^{77} +(-1.00000 + 1.73205i) q^{79} -3.00000 q^{82} +(-3.68614 + 6.38458i) q^{83} +(2.81386 + 4.87375i) q^{86} +(2.18614 - 3.78651i) q^{88} -16.1168 q^{89} -22.7446 q^{91} +(-0.686141 + 1.18843i) q^{92} +(-3.68614 - 6.38458i) q^{94} +(-4.18614 + 7.25061i) q^{97} -4.37228 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{2} - 2 q^{4} + q^{7} - 4 q^{8}+O(q^{10}) \) 4 * q + 2 * q^2 - 2 * q^4 + q^7 - 4 * q^8	\( 4 q + 2 q^{2} - 2 q^{4} + q^{7} - 4 q^{8} - 3 q^{11} - 2 q^{13} - q^{14} - 2 q^{16} - 18 q^{17} + 2 q^{19} + 3 q^{22} + 3 q^{23} - 4 q^{26} - 2 q^{28} - 3 q^{29} + 2 q^{31} + 2 q^{32} - 9 q^{34} + 16 q^{37} + q^{38} - 6 q^{41} - 17 q^{43} + 6 q^{44} + 6 q^{46} + 9 q^{47} - 3 q^{49} - 2 q^{52} - q^{56} + 3 q^{58} + 3 q^{59} - q^{61} + 4 q^{62} + 4 q^{64} - 14 q^{67} + 9 q^{68} + 24 q^{71} + 22 q^{73} + 8 q^{74} - q^{76} + 18 q^{77} - 4 q^{79} - 12 q^{82} - 9 q^{83} + 17 q^{86} + 3 q^{88} - 30 q^{89} - 68 q^{91} + 3 q^{92} - 9 q^{94} - 11 q^{97} - 6 q^{98}+O(q^{100}) \) 4 * q + 2 * q^2 - 2 * q^4 + q^7 - 4 * q^8 - 3 * q^11 - 2 * q^13 - q^14 - 2 * q^16 - 18 * q^17 + 2 * q^19 + 3 * q^22 + 3 * q^23 - 4 * q^26 - 2 * q^28 - 3 * q^29 + 2 * q^31 + 2 * q^32 - 9 * q^34 + 16 * q^37 + q^38 - 6 * q^41 - 17 * q^43 + 6 * q^44 + 6 * q^46 + 9 * q^47 - 3 * q^49 - 2 * q^52 - q^56 + 3 * q^58 + 3 * q^59 - q^61 + 4 * q^62 + 4 * q^64 - 14 * q^67 + 9 * q^68 + 24 * q^71 + 22 * q^73 + 8 * q^74 - q^76 + 18 * q^77 - 4 * q^79 - 12 * q^82 - 9 * q^83 + 17 * q^86 + 3 * q^88 - 30 * q^89 - 68 * q^91 + 3 * q^92 - 9 * q^94 - 11 * q^97 - 6 * q^98

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