LMFDB - Embedded newform 243.2.c.d.82.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [243,2,Mod(82,243)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(243, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([2]))

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

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

chi := DirichletCharacter("243.82");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 243 = 3^{5} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	243.c (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:	$1.94036476912$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{3})$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 3 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			82.2
Root			$-0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	243.82
Dual form			243.2.c.d.163.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.866025 + 1.50000i) q^{2} +(-0.500000 + 0.866025i) q^{4} +(1.73205 - 3.00000i) q^{5} +(0.500000 + 0.866025i) q^{7} +1.73205 q^{8} +O(q^{10})$	\(q+(0.866025 + 1.50000i) q^{2} +(-0.500000 + 0.866025i) q^{4} +(1.73205 - 3.00000i) q^{5} +(0.500000 + 0.866025i) q^{7} +1.73205 q^{8} +6.00000 q^{10} +(-1.73205 - 3.00000i) q^{11} +(-2.50000 + 4.33013i) q^{13} +(-0.866025 + 1.50000i) q^{14} +(2.50000 + 4.33013i) q^{16} -1.00000 q^{19} +(1.73205 + 3.00000i) q^{20} +(3.00000 - 5.19615i) q^{22} +(-3.46410 + 6.00000i) q^{23} +(-3.50000 - 6.06218i) q^{25} -8.66025 q^{26} -1.00000 q^{28} +(-1.73205 - 3.00000i) q^{29} +(-2.50000 + 4.33013i) q^{31} +(-2.59808 + 4.50000i) q^{32} +3.46410 q^{35} -1.00000 q^{37} +(-0.866025 - 1.50000i) q^{38} +(3.00000 - 5.19615i) q^{40} +(1.73205 - 3.00000i) q^{41} +(0.500000 + 0.866025i) q^{43} +3.46410 q^{44} -12.0000 q^{46} +(-1.73205 - 3.00000i) q^{47} +(3.00000 - 5.19615i) q^{49} +(6.06218 - 10.5000i) q^{50} +(-2.50000 - 4.33013i) q^{52} +10.3923 q^{53} -12.0000 q^{55} +(0.866025 + 1.50000i) q^{56} +(3.00000 - 5.19615i) q^{58} +(1.73205 - 3.00000i) q^{59} +(-1.00000 - 1.73205i) q^{61} -8.66025 q^{62} +1.00000 q^{64} +(8.66025 + 15.0000i) q^{65} +(-4.00000 + 6.92820i) q^{67} +(3.00000 + 5.19615i) q^{70} -10.3923 q^{71} +2.00000 q^{73} +(-0.866025 - 1.50000i) q^{74} +(0.500000 - 0.866025i) q^{76} +(1.73205 - 3.00000i) q^{77} +(0.500000 + 0.866025i) q^{79} +17.3205 q^{80} +6.00000 q^{82} +(3.46410 + 6.00000i) q^{83} +(-0.866025 + 1.50000i) q^{86} +(-3.00000 - 5.19615i) q^{88} -10.3923 q^{89} -5.00000 q^{91} +(-3.46410 - 6.00000i) q^{92} +(3.00000 - 5.19615i) q^{94} +(-1.73205 + 3.00000i) q^{95} +(-8.50000 - 14.7224i) q^{97} +10.3923 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{4} + 2 q^{7}+O(q^{10}) $ 4 * q - 2 * q^4 + 2 * q^7	$ 4 q - 2 q^{4} + 2 q^{7} + 24 q^{10} - 10 q^{13} + 10 q^{16} - 4 q^{19} + 12 q^{22} - 14 q^{25} - 4 q^{28} - 10 q^{31} - 4 q^{37} + 12 q^{40} + 2 q^{43} - 48 q^{46} + 12 q^{49} - 10 q^{52} - 48 q^{55} + 12 q^{58} - 4 q^{61} + 4 q^{64} - 16 q^{67} + 12 q^{70} + 8 q^{73} + 2 q^{76} + 2 q^{79} + 24 q^{82} - 12 q^{88} - 20 q^{91} + 12 q^{94} - 34 q^{97}+O(q^{100}) $ 4 * q - 2 * q^4 + 2 * q^7 + 24 * q^10 - 10 * q^13 + 10 * q^16 - 4 * q^19 + 12 * q^22 - 14 * q^25 - 4 * q^28 - 10 * q^31 - 4 * q^37 + 12 * q^40 + 2 * q^43 - 48 * q^46 + 12 * q^49 - 10 * q^52 - 48 * q^55 + 12 * q^58 - 4 * q^61 + 4 * q^64 - 16 * q^67 + 12 * q^70 + 8 * q^73 + 2 * q^76 + 2 * q^79 + 24 * q^82 - 12 * q^88 - 20 * q^91 + 12 * q^94 - 34 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/243\mathbb{Z}\right)^\times$.

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

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 $

$2$	0.866025	+	1.50000i	0.612372	+	1.06066i	0.990839	+	0.135045i	$0.0431180\pi$
$2$	0.866025	+	1.50000i	0.612372	+	1.06066i	−0.378467	+	0.925615i	$0.623549\pi$
$3$		0			0
$3$		0			0
$4$	−0.500000	+	0.866025i	−0.250000	+	0.433013i
$5$	1.73205	−	3.00000i	0.774597	−	1.34164i	−0.160424	−	0.987048i	$-0.551286\pi$
$5$	1.73205	−	3.00000i	0.774597	−	1.34164i	0.935021	−	0.354593i	$-0.115380\pi$
$6$		0			0
$7$	0.500000	+	0.866025i	0.188982	+	0.327327i	0.944911	−	0.327327i	$-0.106148\pi$
$7$	0.500000	+	0.866025i	0.188982	+	0.327327i	−0.755929	+	0.654654i	$0.772814\pi$
$8$	1.73205			0.612372
$9$		0			0
$10$	6.00000			1.89737
$11$	−1.73205	−	3.00000i	−0.522233	−	0.904534i	−0.999665	−	0.0258656i	$-0.991766\pi$
$11$	−1.73205	−	3.00000i	−0.522233	−	0.904534i	0.477432	−	0.878668i	$-0.341568\pi$
$12$		0			0
$13$	−2.50000	+	4.33013i	−0.693375	+	1.20096i	0.277350	+	0.960769i	$0.410544\pi$
$13$	−2.50000	+	4.33013i	−0.693375	+	1.20096i	−0.970725	+	0.240192i	$0.922790\pi$
$14$	−0.866025	+	1.50000i	−0.231455	+	0.400892i
$15$		0			0
$16$	2.50000	+	4.33013i	0.625000	+	1.08253i
$17$		0			0			−	1.00000i	$-0.5\pi$
$17$		0			0				1.00000i	$0.5\pi$
$18$		0			0
$19$	−1.00000			−0.229416			−0.114708	−	0.993399i	$-0.536593\pi$
$19$	−1.00000			−0.229416			−0.114708	+	0.993399i	$0.536593\pi$
$20$	1.73205	+	3.00000i	0.387298	+	0.670820i
$21$		0			0
$22$	3.00000	−	5.19615i	0.639602	−	1.10782i
$23$	−3.46410	+	6.00000i	−0.722315	+	1.25109i	0.237754	+	0.971325i	$0.423589\pi$
$23$	−3.46410	+	6.00000i	−0.722315	+	1.25109i	−0.960070	+	0.279761i	$0.909745\pi$
$24$		0			0
$25$	−3.50000	−	6.06218i	−0.700000	−	1.21244i
$26$	−8.66025			−1.69842
$27$		0			0
$28$	−1.00000			−0.188982
$29$	−1.73205	−	3.00000i	−0.321634	−	0.557086i	0.659192	−	0.751975i	$-0.270899\pi$
$29$	−1.73205	−	3.00000i	−0.321634	−	0.557086i	−0.980825	+	0.194889i	$0.937565\pi$
$30$		0			0
$31$	−2.50000	+	4.33013i	−0.449013	+	0.777714i	−0.998322	−	0.0579057i	$-0.981558\pi$
$31$	−2.50000	+	4.33013i	−0.449013	+	0.777714i	0.549309	+	0.835619i	$0.314891\pi$
$32$	−2.59808	+	4.50000i	−0.459279	+	0.795495i
$33$		0			0
$34$		0			0
$35$	3.46410			0.585540
$36$		0			0
$37$	−1.00000			−0.164399			−0.0821995	−	0.996616i	$-0.526194\pi$
$37$	−1.00000			−0.164399			−0.0821995	+	0.996616i	$0.526194\pi$
$38$	−0.866025	−	1.50000i	−0.140488	−	0.243332i
$39$		0			0
$40$	3.00000	−	5.19615i	0.474342	−	0.821584i
$41$	1.73205	−	3.00000i	0.270501	−	0.468521i	−0.698489	−	0.715621i	$-0.746144\pi$
$41$	1.73205	−	3.00000i	0.270501	−	0.468521i	0.968990	+	0.247099i	$0.0794774\pi$
$42$		0			0
$43$	0.500000	+	0.866025i	0.0762493	+	0.132068i	0.901629	−	0.432511i	$-0.142372\pi$
$43$	0.500000	+	0.866025i	0.0762493	+	0.132068i	−0.825380	+	0.564578i	$0.809039\pi$
$44$	3.46410			0.522233
$45$		0			0
$46$	−12.0000			−1.76930
$47$	−1.73205	−	3.00000i	−0.252646	−	0.437595i	0.711608	−	0.702577i	$-0.247967\pi$
$47$	−1.73205	−	3.00000i	−0.252646	−	0.437595i	−0.964253	+	0.264982i	$0.914634\pi$
$48$		0			0
$49$	3.00000	−	5.19615i	0.428571	−	0.742307i
$50$	6.06218	−	10.5000i	0.857321	−	1.48492i
$51$		0			0
$52$	−2.50000	−	4.33013i	−0.346688	−	0.600481i
$53$	10.3923			1.42749			0.713746	−	0.700404i	$-0.246997\pi$
$53$	10.3923			1.42749			0.713746	+	0.700404i	$0.246997\pi$
$54$		0			0
$55$	−12.0000			−1.61808
$56$	0.866025	+	1.50000i	0.115728	+	0.200446i
$57$		0			0
$58$	3.00000	−	5.19615i	0.393919	−	0.682288i
$59$	1.73205	−	3.00000i	0.225494	−	0.390567i	−0.730974	−	0.682406i	$-0.760934\pi$
$59$	1.73205	−	3.00000i	0.225494	−	0.390567i	0.956467	+	0.291839i	$0.0942671\pi$
$60$		0			0
$61$	−1.00000	−	1.73205i	−0.128037	−	0.221766i	0.794879	−	0.606768i	$-0.207534\pi$
$61$	−1.00000	−	1.73205i	−0.128037	−	0.221766i	−0.922916	+	0.385002i	$0.874201\pi$
$62$	−8.66025			−1.09985
$63$		0			0
$64$	1.00000			0.125000
$65$	8.66025	+	15.0000i	1.07417	+	1.86052i
$66$		0			0
$67$	−4.00000	+	6.92820i	−0.488678	+	0.846415i	−0.999915	−	0.0130248i	$-0.995854\pi$
$67$	−4.00000	+	6.92820i	−0.488678	+	0.846415i	0.511237	+	0.859440i	$0.329187\pi$
$68$		0			0
$69$		0			0
$70$	3.00000	+	5.19615i	0.358569	+	0.621059i
$71$	−10.3923			−1.23334			−0.616670	−	0.787222i	$-0.711519\pi$
$71$	−10.3923			−1.23334			−0.616670	+	0.787222i	$0.711519\pi$
$72$		0			0
$73$	2.00000			0.234082			0.117041	−	0.993127i	$-0.462659\pi$
$73$	2.00000			0.234082			0.117041	+	0.993127i	$0.462659\pi$
$74$	−0.866025	−	1.50000i	−0.100673	−	0.174371i
$75$		0			0
$76$	0.500000	−	0.866025i	0.0573539	−	0.0993399i
$77$	1.73205	−	3.00000i	0.197386	−	0.341882i
$78$		0			0
$79$	0.500000	+	0.866025i	0.0562544	+	0.0974355i	0.892781	−	0.450490i	$-0.148751\pi$
$79$	0.500000	+	0.866025i	0.0562544	+	0.0974355i	−0.836527	+	0.547926i	$0.815418\pi$
$80$	17.3205			1.93649
$81$		0			0
$82$	6.00000			0.662589
$83$	3.46410	+	6.00000i	0.380235	+	0.658586i	0.991096	−	0.133152i	$-0.0425099\pi$
$83$	3.46410	+	6.00000i	0.380235	+	0.658586i	−0.610861	+	0.791738i	$0.709177\pi$
$84$		0			0
$85$		0			0
$86$	−0.866025	+	1.50000i	−0.0933859	+	0.161749i
$87$		0			0
$88$	−3.00000	−	5.19615i	−0.319801	−	0.553912i
$89$	−10.3923			−1.10158			−0.550791	−	0.834643i	$-0.685674\pi$
$89$	−10.3923			−1.10158			−0.550791	+	0.834643i	$0.685674\pi$
$90$		0			0
$91$	−5.00000			−0.524142
$92$	−3.46410	−	6.00000i	−0.361158	−	0.625543i
$93$		0			0
$94$	3.00000	−	5.19615i	0.309426	−	0.535942i
$95$	−1.73205	+	3.00000i	−0.177705	+	0.307794i
$96$		0			0
$97$	−8.50000	−	14.7224i	−0.863044	−	1.49484i	−0.868976	−	0.494854i	$-0.835222\pi$
$97$	−8.50000	−	14.7224i	−0.863044	−	1.49484i	0.00593185	−	0.999982i	$-0.498112\pi$
$98$	10.3923			1.04978
$99$		0			0
$100$	7.00000			0.700000
$101$	−6.92820	−	12.0000i	−0.689382	−	1.19404i	−0.972038	−	0.234823i	$-0.924549\pi$
$101$	−6.92820	−	12.0000i	−0.689382	−	1.19404i	0.282656	−	0.959221i	$-0.408784\pi$
$102$		0			0
$103$	−4.00000	+	6.92820i	−0.394132	+	0.682656i	−0.992990	−	0.118199i	$-0.962288\pi$
$103$	−4.00000	+	6.92820i	−0.394132	+	0.682656i	0.598858	+	0.800855i	$0.295621\pi$
$104$	−4.33013	+	7.50000i	−0.424604	+	0.735436i
$105$		0			0
$106$	9.00000	+	15.5885i	0.874157	+	1.51408i
$107$	−10.3923			−1.00466			−0.502331	−	0.864675i	$-0.667524\pi$
$107$	−10.3923			−1.00466			−0.502331	+	0.864675i	$0.667524\pi$
$108$		0			0
$109$	17.0000			1.62830			0.814152	−	0.580651i	$-0.197202\pi$
$109$	17.0000			1.62830			0.814152	+	0.580651i	$0.197202\pi$
$110$	−10.3923	−	18.0000i	−0.990867	−	1.71623i
$111$		0			0
$112$	−2.50000	+	4.33013i	−0.236228	+	0.409159i
$113$	−8.66025	+	15.0000i	−0.814688	+	1.41108i	0.0948634	+	0.995490i	$0.469759\pi$
$113$	−8.66025	+	15.0000i	−0.814688	+	1.41108i	−0.909552	+	0.415591i	$0.863575\pi$
$114$		0			0
$115$	12.0000	+	20.7846i	1.11901	+	1.93817i
$116$	3.46410			0.321634
$117$		0			0
$118$	6.00000			0.552345
$119$		0			0
$120$		0			0
$121$	−0.500000	+	0.866025i	−0.0454545	+	0.0787296i
$122$	1.73205	−	3.00000i	0.156813	−	0.271607i
$123$		0			0
$124$	−2.50000	−	4.33013i	−0.224507	−	0.388857i
$125$	−6.92820			−0.619677
$126$		0			0
$127$	17.0000			1.50851			0.754253	−	0.656584i	$-0.227999\pi$
$127$	17.0000			1.50851			0.754253	+	0.656584i	$0.227999\pi$
$128$	6.06218	+	10.5000i	0.535826	+	0.928078i
$129$		0			0
$130$	−15.0000	+	25.9808i	−1.31559	+	2.27866i
$131$	1.73205	−	3.00000i	0.151330	−	0.262111i	−0.780387	−	0.625297i	$-0.784978\pi$
$131$	1.73205	−	3.00000i	0.151330	−	0.262111i	0.931717	+	0.363186i	$0.118311\pi$
$132$		0			0
$133$	−0.500000	−	0.866025i	−0.0433555	−	0.0750939i
$134$	−13.8564			−1.19701
$135$		0			0
$136$		0			0
$137$	3.46410	+	6.00000i	0.295958	+	0.512615i	0.975207	−	0.221293i	$-0.0710278\pi$
$137$	3.46410	+	6.00000i	0.295958	+	0.512615i	−0.679249	+	0.733908i	$0.737694\pi$
$138$		0			0
$139$	6.50000	−	11.2583i	0.551323	−	0.954919i	−0.446857	−	0.894606i	$-0.647457\pi$
$139$	6.50000	−	11.2583i	0.551323	−	0.954919i	0.998179	−	0.0603135i	$-0.0192101\pi$
$140$	−1.73205	+	3.00000i	−0.146385	+	0.253546i
$141$		0			0
$142$	−9.00000	−	15.5885i	−0.755263	−	1.30815i
$143$	17.3205			1.44841
$144$		0			0
$145$	−12.0000			−0.996546
$146$	1.73205	+	3.00000i	0.143346	+	0.248282i
$147$		0			0
$148$	0.500000	−	0.866025i	0.0410997	−	0.0711868i
$149$	−3.46410	+	6.00000i	−0.283790	+	0.491539i	−0.972315	−	0.233674i	$-0.924925\pi$
$149$	−3.46410	+	6.00000i	−0.283790	+	0.491539i	0.688525	+	0.725213i	$0.258259\pi$
$150$		0			0
$151$	8.00000	+	13.8564i	0.651031	+	1.12762i	0.982873	+	0.184284i	$0.0589965\pi$
$151$	8.00000	+	13.8564i	0.651031	+	1.12762i	−0.331842	+	0.943335i	$0.607670\pi$
$152$	−1.73205			−0.140488
$153$		0			0
$154$	6.00000			0.483494
$155$	8.66025	+	15.0000i	0.695608	+	1.20483i
$156$		0			0
$157$	6.50000	−	11.2583i	0.518756	−	0.898513i	−0.481006	−	0.876717i	$-0.659728\pi$
$157$	6.50000	−	11.2583i	0.518756	−	0.898513i	0.999762	−	0.0217953i	$-0.00693820\pi$
$158$	−0.866025	+	1.50000i	−0.0688973	+	0.119334i
$159$		0			0
$160$	9.00000	+	15.5885i	0.711512	+	1.23238i
$161$	−6.92820			−0.546019
$162$		0			0
$163$	−1.00000			−0.0783260			−0.0391630	−	0.999233i	$-0.512469\pi$
$163$	−1.00000			−0.0783260			−0.0391630	+	0.999233i	$0.512469\pi$
$164$	1.73205	+	3.00000i	0.135250	+	0.234261i
$165$		0			0
$166$	−6.00000	+	10.3923i	−0.465690	+	0.806599i
$167$	12.1244	−	21.0000i	0.938211	−	1.62503i	0.169405	−	0.985547i	$-0.445815\pi$
$167$	12.1244	−	21.0000i	0.938211	−	1.62503i	0.768806	−	0.639482i	$-0.220851\pi$
$168$		0			0
$169$	−6.00000	−	10.3923i	−0.461538	−	0.799408i
$170$		0			0
$171$		0			0
$172$	−1.00000			−0.0762493
$173$	−6.92820	−	12.0000i	−0.526742	−	0.912343i	−0.999514	−	0.0311588i	$-0.990080\pi$
$173$	−6.92820	−	12.0000i	−0.526742	−	0.912343i	0.472773	−	0.881184i	$-0.343253\pi$
$174$		0			0
$175$	3.50000	−	6.06218i	0.264575	−	0.458258i
$176$	8.66025	−	15.0000i	0.652791	−	1.13067i
$177$		0			0
$178$	−9.00000	−	15.5885i	−0.674579	−	1.16840i
$179$	20.7846			1.55351			0.776757	−	0.629800i	$-0.216863\pi$
$179$	20.7846			1.55351			0.776757	+	0.629800i	$0.216863\pi$
$180$		0			0
$181$	17.0000			1.26360			0.631800	−	0.775131i	$-0.282316\pi$
$181$	17.0000			1.26360			0.631800	+	0.775131i	$0.282316\pi$
$182$	−4.33013	−	7.50000i	−0.320970	−	0.555937i
$183$		0			0
$184$	−6.00000	+	10.3923i	−0.442326	+	0.766131i
$185$	−1.73205	+	3.00000i	−0.127343	+	0.220564i
$186$		0			0
$187$		0			0
$188$	3.46410			0.252646
$189$		0			0
$190$	−6.00000			−0.435286
$191$	3.46410	+	6.00000i	0.250654	+	0.434145i	0.963706	−	0.266966i	$-0.0860212\pi$
$191$	3.46410	+	6.00000i	0.250654	+	0.434145i	−0.713052	+	0.701111i	$0.752688\pi$
$192$		0			0
$193$	5.00000	−	8.66025i	0.359908	−	0.623379i	−0.628037	−	0.778183i	$-0.716141\pi$
$193$	5.00000	−	8.66025i	0.359908	−	0.623379i	0.987945	+	0.154805i	$0.0494748\pi$
$194$	14.7224	−	25.5000i	1.05701	−	1.83079i
$195$		0			0
$196$	3.00000	+	5.19615i	0.214286	+	0.371154i
$197$	10.3923			0.740421			0.370211	−	0.928948i	$-0.379286\pi$
$197$	10.3923			0.740421			0.370211	+	0.928948i	$0.379286\pi$
$198$		0			0
$199$	−19.0000			−1.34687			−0.673437	−	0.739244i	$-0.735183\pi$
$199$	−19.0000			−1.34687			−0.673437	+	0.739244i	$0.735183\pi$
$200$	−6.06218	−	10.5000i	−0.428661	−	0.742462i
$201$		0			0
$202$	12.0000	−	20.7846i	0.844317	−	1.46240i
$203$	1.73205	−	3.00000i	0.121566	−	0.210559i
$204$		0			0
$205$	−6.00000	−	10.3923i	−0.419058	−	0.725830i
$206$	−13.8564			−0.965422
$207$		0			0
$208$	−25.0000			−1.73344
$209$	1.73205	+	3.00000i	0.119808	+	0.207514i
$210$		0			0
$211$	−2.50000	+	4.33013i	−0.172107	+	0.298098i	−0.939156	−	0.343490i	$-0.888391\pi$
$211$	−2.50000	+	4.33013i	−0.172107	+	0.298098i	0.767049	+	0.641588i	$0.221724\pi$
$212$	−5.19615	+	9.00000i	−0.356873	+	0.618123i
$213$		0			0
$214$	−9.00000	−	15.5885i	−0.615227	−	1.06561i
$215$	3.46410			0.236250
$216$		0			0
$217$	−5.00000			−0.339422
$218$	14.7224	+	25.5000i	0.997129	+	1.72708i
$219$		0			0
$220$	6.00000	−	10.3923i	0.404520	−	0.700649i
$221$		0			0
$222$		0			0
$223$	9.50000	+	16.4545i	0.636167	+	1.10187i	0.986267	+	0.165161i	$0.0528144\pi$
$223$	9.50000	+	16.4545i	0.636167	+	1.10187i	−0.350100	+	0.936713i	$0.613852\pi$
$224$	−5.19615			−0.347183
$225$		0			0
$226$	−30.0000			−1.99557
$227$	−6.92820	−	12.0000i	−0.459841	−	0.796468i	0.539111	−	0.842235i	$-0.318760\pi$
$227$	−6.92820	−	12.0000i	−0.459841	−	0.796468i	−0.998952	+	0.0457666i	$0.985427\pi$
$228$		0			0
$229$	−2.50000	+	4.33013i	−0.165205	+	0.286143i	−0.936728	−	0.350058i	$-0.886162\pi$
$229$	−2.50000	+	4.33013i	−0.165205	+	0.286143i	0.771523	+	0.636201i	$0.219495\pi$
$230$	−20.7846	+	36.0000i	−1.37050	+	2.37377i
$231$		0			0
$232$	−3.00000	−	5.19615i	−0.196960	−	0.341144i
$233$		0			0			−	1.00000i	$-0.5\pi$
$233$		0			0				1.00000i	$0.5\pi$
$234$		0			0
$235$	−12.0000			−0.782794
$236$	1.73205	+	3.00000i	0.112747	+	0.195283i
$237$		0			0
$238$		0			0
$239$	−3.46410	+	6.00000i	−0.224074	+	0.388108i	−0.956041	−	0.293232i	$-0.905269\pi$
$239$	−3.46410	+	6.00000i	−0.224074	+	0.388108i	0.731967	+	0.681340i	$0.238602\pi$
$240$		0			0
$241$	9.50000	+	16.4545i	0.611949	+	1.05993i	0.990912	+	0.134515i	$0.0429475\pi$
$241$	9.50000	+	16.4545i	0.611949	+	1.05993i	−0.378963	+	0.925412i	$0.623719\pi$
$242$	−1.73205			−0.111340
$243$		0			0
$244$	2.00000			0.128037
$245$	−10.3923	−	18.0000i	−0.663940	−	1.14998i
$246$		0			0
$247$	2.50000	−	4.33013i	0.159071	−	0.275519i
$248$	−4.33013	+	7.50000i	−0.274963	+	0.476250i
$249$		0			0
$250$	−6.00000	−	10.3923i	−0.379473	−	0.657267i
$251$	−20.7846			−1.31191			−0.655956	−	0.754799i	$-0.727735\pi$
$251$	−20.7846			−1.31191			−0.655956	+	0.754799i	$0.727735\pi$
$252$		0			0
$253$	24.0000			1.50887
$254$	14.7224	+	25.5000i	0.923768	+	1.60001i
$255$		0			0
$256$	−9.50000	+	16.4545i	−0.593750	+	1.02841i
$257$	1.73205	−	3.00000i	0.108042	−	0.187135i	−0.806935	−	0.590641i	$-0.798875\pi$
$257$	1.73205	−	3.00000i	0.108042	−	0.187135i	0.914977	+	0.403506i	$0.132208\pi$
$258$		0			0
$259$	−0.500000	−	0.866025i	−0.0310685	−	0.0538122i
$260$	−17.3205			−1.07417
$261$		0			0
$262$	6.00000			0.370681
$263$	−6.92820	−	12.0000i	−0.427211	−	0.739952i	0.569413	−	0.822052i	$-0.307171\pi$
$263$	−6.92820	−	12.0000i	−0.427211	−	0.739952i	−0.996624	+	0.0821001i	$0.973837\pi$
$264$		0			0
$265$	18.0000	−	31.1769i	1.10573	−	1.91518i
$266$	0.866025	−	1.50000i	0.0530994	−	0.0919709i
$267$		0			0
$268$	−4.00000	−	6.92820i	−0.244339	−	0.423207i
$269$		0			0			−	1.00000i	$-0.5\pi$
$269$		0			0				1.00000i	$0.5\pi$
$270$		0			0
$271$	−16.0000			−0.971931			−0.485965	−	0.873978i	$-0.661532\pi$
$271$	−16.0000			−0.971931			−0.485965	+	0.873978i	$0.661532\pi$
$272$		0			0
$273$		0			0
$274$	−6.00000	+	10.3923i	−0.362473	+	0.627822i
$275$	−12.1244	+	21.0000i	−0.731126	+	1.26635i
$276$		0			0
$277$	−8.50000	−	14.7224i	−0.510716	−	0.884585i	−0.999923	−	0.0124177i	$-0.996047\pi$
$277$	−8.50000	−	14.7224i	−0.510716	−	0.884585i	0.489207	−	0.872167i	$-0.337286\pi$
$278$	22.5167			1.35046
$279$		0			0
$280$	6.00000			0.358569
$281$	−6.92820	−	12.0000i	−0.413302	−	0.715860i	0.581947	−	0.813227i	$-0.302291\pi$
$281$	−6.92820	−	12.0000i	−0.413302	−	0.715860i	−0.995249	+	0.0973670i	$0.968958\pi$
$282$		0			0
$283$	6.50000	−	11.2583i	0.386385	−	0.669238i	−0.605575	−	0.795788i	$-0.707057\pi$
$283$	6.50000	−	11.2583i	0.386385	−	0.669238i	0.991960	+	0.126550i	$0.0403903\pi$
$284$	5.19615	−	9.00000i	0.308335	−	0.534052i
$285$		0			0
$286$	15.0000	+	25.9808i	0.886969	+	1.53627i
$287$	3.46410			0.204479
$288$		0			0
$289$	−17.0000			−1.00000
$290$	−10.3923	−	18.0000i	−0.610257	−	1.05700i
$291$		0			0
$292$	−1.00000	+	1.73205i	−0.0585206	+	0.101361i
$293$	6.92820	−	12.0000i	0.404750	−	0.701047i	−0.589542	−	0.807737i	$-0.700692\pi$
$293$	6.92820	−	12.0000i	0.404750	−	0.701047i	0.994292	+	0.106690i	$0.0340252\pi$
$294$		0			0
$295$	−6.00000	−	10.3923i	−0.349334	−	0.605063i
$296$	−1.73205			−0.100673
$297$		0			0
$298$	−12.0000			−0.695141
$299$	−17.3205	−	30.0000i	−1.00167	−	1.73494i
$300$		0			0
$301$	−0.500000	+	0.866025i	−0.0288195	+	0.0499169i
$302$	−13.8564	+	24.0000i	−0.797347	+	1.38104i
$303$		0			0
$304$	−2.50000	−	4.33013i	−0.143385	−	0.248350i
$305$	−6.92820			−0.396708
$306$		0			0
$307$	20.0000			1.14146			0.570730	−	0.821138i	$-0.306660\pi$
$307$	20.0000			1.14146			0.570730	+	0.821138i	$0.306660\pi$
$308$	1.73205	+	3.00000i	0.0986928	+	0.170941i
$309$		0			0
$310$	−15.0000	+	25.9808i	−0.851943	+	1.47561i
$311$	6.92820	−	12.0000i	0.392862	−	0.680458i	−0.599963	−	0.800027i	$-0.704818\pi$
$311$	6.92820	−	12.0000i	0.392862	−	0.680458i	0.992826	+	0.119570i	$0.0381515\pi$
$312$		0			0
$313$	0.500000	+	0.866025i	0.0282617	+	0.0489506i	0.879810	−	0.475325i	$-0.157669\pi$
$313$	0.500000	+	0.866025i	0.0282617	+	0.0489506i	−0.851549	+	0.524276i	$0.824336\pi$
$314$	22.5167			1.27069
$315$		0			0
$316$	−1.00000			−0.0562544
$317$	13.8564	+	24.0000i	0.778253	+	1.34797i	0.932948	+	0.360012i	$0.117227\pi$
$317$	13.8564	+	24.0000i	0.778253	+	1.34797i	−0.154694	+	0.987962i	$0.549439\pi$
$318$		0			0
$319$	−6.00000	+	10.3923i	−0.335936	+	0.581857i
$320$	1.73205	−	3.00000i	0.0968246	−	0.167705i
$321$		0			0
$322$	−6.00000	−	10.3923i	−0.334367	−	0.579141i
$323$		0			0
$324$		0			0
$325$	35.0000			1.94145
$326$	−0.866025	−	1.50000i	−0.0479647	−	0.0830773i
$327$		0			0
$328$	3.00000	−	5.19615i	0.165647	−	0.286910i
$329$	1.73205	−	3.00000i	0.0954911	−	0.165395i
$330$		0			0
$331$	9.50000	+	16.4545i	0.522167	+	0.904420i	0.999667	+	0.0257885i	$0.00820965\pi$
$331$	9.50000	+	16.4545i	0.522167	+	0.904420i	−0.477500	+	0.878632i	$0.658457\pi$
$332$	−6.92820			−0.380235
$333$		0			0
$334$	42.0000			2.29814
$335$	13.8564	+	24.0000i	0.757056	+	1.31126i
$336$		0			0
$337$	−2.50000	+	4.33013i	−0.136184	+	0.235877i	−0.926049	−	0.377403i	$-0.876817\pi$
$337$	−2.50000	+	4.33013i	−0.136184	+	0.235877i	0.789865	+	0.613280i	$0.210150\pi$
$338$	10.3923	−	18.0000i	0.565267	−	0.979071i
$339$		0			0
$340$		0			0
$341$	17.3205			0.937958
$342$		0			0
$343$	13.0000			0.701934
$344$	0.866025	+	1.50000i	0.0466930	+	0.0808746i
$345$		0			0
$346$	12.0000	−	20.7846i	0.645124	−	1.11739i
$347$	12.1244	−	21.0000i	0.650870	−	1.12734i	−0.332043	−	0.943264i	$-0.607738\pi$
$347$	12.1244	−	21.0000i	0.650870	−	1.12734i	0.982912	−	0.184075i	$-0.0589288\pi$
$348$		0			0
$349$	0.500000	+	0.866025i	0.0267644	+	0.0463573i	0.879097	−	0.476642i	$-0.158146\pi$
$349$	0.500000	+	0.866025i	0.0267644	+	0.0463573i	−0.852333	+	0.523000i	$0.824813\pi$
$350$	12.1244			0.648074
$351$		0			0
$352$	18.0000			0.959403
$353$	8.66025	+	15.0000i	0.460939	+	0.798369i	0.999008	−	0.0445312i	$-0.0141794\pi$
$353$	8.66025	+	15.0000i	0.460939	+	0.798369i	−0.538069	+	0.842901i	$0.680846\pi$
$354$		0			0
$355$	−18.0000	+	31.1769i	−0.955341	+	1.65470i
$356$	5.19615	−	9.00000i	0.275396	−	0.476999i
$357$		0			0
$358$	18.0000	+	31.1769i	0.951330	+	1.64775i
$359$	−31.1769			−1.64545			−0.822727	−	0.568436i	$-0.807549\pi$
$359$	−31.1769			−1.64545			−0.822727	+	0.568436i	$0.807549\pi$
$360$		0			0
$361$	−18.0000			−0.947368
$362$	14.7224	+	25.5000i	0.773794	+	1.34025i
$363$		0			0
$364$	2.50000	−	4.33013i	0.131036	−	0.226960i
$365$	3.46410	−	6.00000i	0.181319	−	0.314054i
$366$		0			0
$367$	8.00000	+	13.8564i	0.417597	+	0.723299i	0.995697	−	0.0926670i	$-0.0295392\pi$
$367$	8.00000	+	13.8564i	0.417597	+	0.723299i	−0.578101	+	0.815966i	$0.696206\pi$
$368$	−34.6410			−1.80579
$369$		0			0
$370$	−6.00000			−0.311925
$371$	5.19615	+	9.00000i	0.269771	+	0.467257i
$372$		0			0
$373$	−11.5000	+	19.9186i	−0.595447	+	1.03135i	0.398036	+	0.917370i	$0.369692\pi$
$373$	−11.5000	+	19.9186i	−0.595447	+	1.03135i	−0.993484	+	0.113975i	$0.963641\pi$
$374$		0			0
$375$		0			0
$376$	−3.00000	−	5.19615i	−0.154713	−	0.267971i
$377$	17.3205			0.892052
$378$		0			0
$379$	−19.0000			−0.975964			−0.487982	−	0.872854i	$-0.662267\pi$
$379$	−19.0000			−0.975964			−0.487982	+	0.872854i	$0.662267\pi$
$380$	−1.73205	−	3.00000i	−0.0888523	−	0.153897i
$381$		0			0
$382$	−6.00000	+	10.3923i	−0.306987	+	0.531717i
$383$	−8.66025	+	15.0000i	−0.442518	+	0.766464i	−0.997876	−	0.0651476i	$-0.979248\pi$
$383$	−8.66025	+	15.0000i	−0.442518	+	0.766464i	0.555357	+	0.831612i	$0.312581\pi$
$384$		0			0
$385$	−6.00000	−	10.3923i	−0.305788	−	0.529641i
$386$	17.3205			0.881591
$387$		0			0
$388$	17.0000			0.863044
$389$	3.46410	+	6.00000i	0.175637	+	0.304212i	0.940382	−	0.340121i	$-0.110468\pi$
$389$	3.46410	+	6.00000i	0.175637	+	0.304212i	−0.764745	+	0.644334i	$0.777135\pi$
$390$		0			0
$391$		0			0
$392$	5.19615	−	9.00000i	0.262445	−	0.454569i
$393$		0			0
$394$	9.00000	+	15.5885i	0.453413	+	0.785335i
$395$	3.46410			0.174298
$396$		0			0
$397$	−1.00000			−0.0501886			−0.0250943	−	0.999685i	$-0.507989\pi$
$397$	−1.00000			−0.0501886			−0.0250943	+	0.999685i	$0.507989\pi$
$398$	−16.4545	−	28.5000i	−0.824789	−	1.42858i
$399$		0			0
$400$	17.5000	−	30.3109i	0.875000	−	1.51554i
$401$	1.73205	−	3.00000i	0.0864945	−	0.149813i	−0.819533	−	0.573033i	$-0.805767\pi$
$401$	1.73205	−	3.00000i	0.0864945	−	0.149813i	0.906027	+	0.423220i	$0.139100\pi$
$402$		0			0
$403$	−12.5000	−	21.6506i	−0.622669	−	1.07849i
$404$	13.8564			0.689382
$405$		0			0
$406$	6.00000			0.297775
$407$	1.73205	+	3.00000i	0.0858546	+	0.148704i
$408$		0			0
$409$	−2.50000	+	4.33013i	−0.123617	+	0.214111i	−0.921192	−	0.389109i	$-0.872783\pi$
$409$	−2.50000	+	4.33013i	−0.123617	+	0.214111i	0.797574	+	0.603220i	$0.206116\pi$
$410$	10.3923	−	18.0000i	0.513239	−	0.888957i
$411$		0			0
$412$	−4.00000	−	6.92820i	−0.197066	−	0.341328i
$413$	3.46410			0.170457
$414$		0			0
$415$	24.0000			1.17811
$416$	−12.9904	−	22.5000i	−0.636906	−	1.10315i
$417$		0			0
$418$	−3.00000	+	5.19615i	−0.146735	+	0.254152i
$419$	−19.0526	+	33.0000i	−0.930778	+	1.61216i	−0.148784	+	0.988870i	$0.547536\pi$
$419$	−19.0526	+	33.0000i	−0.930778	+	1.61216i	−0.781994	+	0.623286i	$0.785797\pi$
$420$		0			0
$421$	9.50000	+	16.4545i	0.463002	+	0.801942i	0.999109	−	0.0422075i	$-0.0134391\pi$
$421$	9.50000	+	16.4545i	0.463002	+	0.801942i	−0.536107	+	0.844150i	$0.680106\pi$
$422$	−8.66025			−0.421575
$423$		0			0
$424$	18.0000			0.874157
$425$		0			0
$426$		0			0
$427$	1.00000	−	1.73205i	0.0483934	−	0.0838198i
$428$	5.19615	−	9.00000i	0.251166	−	0.435031i
$429$		0			0
$430$	3.00000	+	5.19615i	0.144673	+	0.250581i
$431$	20.7846			1.00116			0.500580	−	0.865690i	$-0.333120\pi$
$431$	20.7846			1.00116			0.500580	+	0.865690i	$0.333120\pi$
$432$		0			0
$433$	−1.00000			−0.0480569			−0.0240285	−	0.999711i	$-0.507649\pi$
$433$	−1.00000			−0.0480569			−0.0240285	+	0.999711i	$0.507649\pi$
$434$	−4.33013	−	7.50000i	−0.207853	−	0.360012i
$435$		0			0
$436$	−8.50000	+	14.7224i	−0.407076	+	0.705077i
$437$	3.46410	−	6.00000i	0.165710	−	0.287019i
$438$		0			0
$439$	−10.0000	−	17.3205i	−0.477274	−	0.826663i	0.522387	−	0.852709i	$-0.325042\pi$
$439$	−10.0000	−	17.3205i	−0.477274	−	0.826663i	−0.999661	+	0.0260459i	$0.991708\pi$
$440$	−20.7846			−0.990867
$441$		0			0
$442$		0			0
$443$	−6.92820	−	12.0000i	−0.329169	−	0.570137i	0.653178	−	0.757204i	$-0.273435\pi$
$443$	−6.92820	−	12.0000i	−0.329169	−	0.570137i	−0.982347	+	0.187067i	$0.940102\pi$
$444$		0			0
$445$	−18.0000	+	31.1769i	−0.853282	+	1.47793i
$446$	−16.4545	+	28.5000i	−0.779142	+	1.34951i
$447$		0			0
$448$	0.500000	+	0.866025i	0.0236228	+	0.0409159i
$449$	10.3923			0.490443			0.245222	−	0.969467i	$-0.421139\pi$
$449$	10.3923			0.490443			0.245222	+	0.969467i	$0.421139\pi$
$450$		0			0
$451$	−12.0000			−0.565058
$452$	−8.66025	−	15.0000i	−0.407344	−	0.705541i
$453$		0			0
$454$	12.0000	−	20.7846i	0.563188	−	0.975470i
$455$	−8.66025	+	15.0000i	−0.405999	+	0.703211i
$456$		0			0
$457$	−8.50000	−	14.7224i	−0.397613	−	0.688686i	0.595818	−	0.803120i	$-0.296828\pi$
$457$	−8.50000	−	14.7224i	−0.397613	−	0.688686i	−0.993431	+	0.114433i	$0.963495\pi$
$458$	−8.66025			−0.404667
$459$		0			0
$460$	−24.0000			−1.11901
$461$	13.8564	+	24.0000i	0.645357	+	1.11779i	0.984219	+	0.176955i	$0.0566248\pi$
$461$	13.8564	+	24.0000i	0.645357	+	1.11779i	−0.338862	+	0.940836i	$0.610042\pi$
$462$		0			0
$463$	15.5000	−	26.8468i	0.720346	−	1.24768i	−0.240515	−	0.970645i	$-0.577316\pi$
$463$	15.5000	−	26.8468i	0.720346	−	1.24768i	0.960861	−	0.277031i	$-0.0893503\pi$
$464$	8.66025	−	15.0000i	0.402042	−	0.696358i
$465$		0			0
$466$		0			0
$467$	10.3923			0.480899			0.240449	−	0.970662i	$-0.422705\pi$
$467$	10.3923			0.480899			0.240449	+	0.970662i	$0.422705\pi$
$468$		0			0
$469$	−8.00000			−0.369406
$470$	−10.3923	−	18.0000i	−0.479361	−	0.830278i
$471$		0			0
$472$	3.00000	−	5.19615i	0.138086	−	0.239172i
$473$	1.73205	−	3.00000i	0.0796398	−	0.137940i
$474$		0			0
$475$	3.50000	+	6.06218i	0.160591	+	0.278152i
$476$		0			0
$477$		0			0
$478$	−12.0000			−0.548867
$479$	−6.92820	−	12.0000i	−0.316558	−	0.548294i	0.663210	−	0.748434i	$-0.269194\pi$
$479$	−6.92820	−	12.0000i	−0.316558	−	0.548294i	−0.979767	+	0.200140i	$0.935860\pi$
$480$		0			0
$481$	2.50000	−	4.33013i	0.113990	−	0.197437i
$482$	−16.4545	+	28.5000i	−0.749481	+	1.29814i
$483$		0			0
$484$	−0.500000	−	0.866025i	−0.0227273	−	0.0393648i
$485$	−58.8897			−2.67404
$486$		0			0
$487$	−19.0000			−0.860972			−0.430486	−	0.902597i	$-0.641658\pi$
$487$	−19.0000			−0.860972			−0.430486	+	0.902597i	$0.641658\pi$
$488$	−1.73205	−	3.00000i	−0.0784063	−	0.135804i
$489$		0			0
$490$	18.0000	−	31.1769i	0.813157	−	1.40843i
$491$	−19.0526	+	33.0000i	−0.859830	+	1.48927i	0.0122607	+	0.999925i	$0.496097\pi$
$491$	−19.0526	+	33.0000i	−0.859830	+	1.48927i	−0.872091	+	0.489344i	$0.837236\pi$
$492$		0			0
$493$		0			0
$494$	8.66025			0.389643
$495$		0			0
$496$	−25.0000			−1.12253
$497$	−5.19615	−	9.00000i	−0.233079	−	0.403705i
$498$		0			0
$499$	14.0000	−	24.2487i	0.626726	−	1.08552i	−0.361478	−	0.932381i	$-0.617728\pi$
$499$	14.0000	−	24.2487i	0.626726	−	1.08552i	0.988204	−	0.153141i	$-0.0489388\pi$
$500$	3.46410	−	6.00000i	0.154919	−	0.268328i
$501$		0			0
$502$	−18.0000	−	31.1769i	−0.803379	−	1.39149i
$503$	−41.5692			−1.85348			−0.926740	−	0.375703i	$-0.877401\pi$
$503$	−41.5692			−1.85348			−0.926740	+	0.375703i	$0.877401\pi$
$504$		0			0
$505$	−48.0000			−2.13597
$506$	20.7846	+	36.0000i	0.923989	+	1.60040i
$507$		0			0
$508$	−8.50000	+	14.7224i	−0.377127	+	0.653202i
$509$	−13.8564	+	24.0000i	−0.614174	+	1.06378i	0.376354	+	0.926476i	$0.377178\pi$
$509$	−13.8564	+	24.0000i	−0.614174	+	1.06378i	−0.990529	+	0.137305i	$0.956156\pi$
$510$		0			0
$511$	1.00000	+	1.73205i	0.0442374	+	0.0766214i
$512$	−8.66025			−0.382733
$513$		0			0
$514$	6.00000			0.264649
$515$	13.8564	+	24.0000i	0.610586	+	1.05757i
$516$		0			0
$517$	−6.00000	+	10.3923i	−0.263880	+	0.457053i
$518$	0.866025	−	1.50000i	0.0380510	−	0.0659062i
$519$		0			0
$520$	15.0000	+	25.9808i	0.657794	+	1.13933i
$521$	20.7846			0.910590			0.455295	−	0.890341i	$-0.349534\pi$
$521$	20.7846			0.910590			0.455295	+	0.890341i	$0.349534\pi$
$522$		0			0
$523$	20.0000			0.874539			0.437269	−	0.899331i	$-0.355946\pi$
$523$	20.0000			0.874539			0.437269	+	0.899331i	$0.355946\pi$
$524$	1.73205	+	3.00000i	0.0756650	+	0.131056i
$525$		0			0
$526$	12.0000	−	20.7846i	0.523225	−	0.906252i
$527$		0			0
$528$		0			0
$529$	−12.5000	−	21.6506i	−0.543478	−	0.941332i
$530$	62.3538			2.70848
$531$		0			0
$532$	1.00000			0.0433555
$533$	8.66025	+	15.0000i	0.375117	+	0.649722i
$534$		0			0
$535$	−18.0000	+	31.1769i	−0.778208	+	1.34790i
$536$	−6.92820	+	12.0000i	−0.299253	+	0.518321i
$537$		0			0
$538$		0			0
$539$	−20.7846			−0.895257
$540$		0			0
$541$	17.0000			0.730887			0.365444	−	0.930834i	$-0.380917\pi$
$541$	17.0000			0.730887			0.365444	+	0.930834i	$0.380917\pi$
$542$	−13.8564	−	24.0000i	−0.595184	−	1.03089i
$543$		0			0
$544$		0			0
$545$	29.4449	−	51.0000i	1.26128	−	2.18460i
$546$		0			0
$547$	−10.0000	−	17.3205i	−0.427569	−	0.740571i	0.569087	−	0.822277i	$-0.307297\pi$
$547$	−10.0000	−	17.3205i	−0.427569	−	0.740571i	−0.996657	+	0.0817056i	$0.973963\pi$
$548$	−6.92820			−0.295958
$549$		0			0
$550$	−42.0000			−1.79089
$551$	1.73205	+	3.00000i	0.0737878	+	0.127804i
$552$		0			0
$553$	−0.500000	+	0.866025i	−0.0212622	+	0.0368271i
$554$	14.7224	−	25.5000i	0.625496	−	1.08339i
$555$		0			0
$556$	6.50000	+	11.2583i	0.275661	+	0.477460i
$557$	−10.3923			−0.440336			−0.220168	−	0.975462i	$-0.570661\pi$
$557$	−10.3923			−0.440336			−0.220168	+	0.975462i	$0.570661\pi$
$558$		0			0
$559$	−5.00000			−0.211477
$560$	8.66025	+	15.0000i	0.365963	+	0.633866i
$561$		0			0
$562$	12.0000	−	20.7846i	0.506189	−	0.876746i
$563$	17.3205	−	30.0000i	0.729972	−	1.26435i	−0.226922	−	0.973913i	$-0.572866\pi$
$563$	17.3205	−	30.0000i	0.729972	−	1.26435i	0.956894	−	0.290436i	$-0.0938004\pi$
$564$		0			0
$565$	30.0000	+	51.9615i	1.26211	+	2.18604i
$566$	22.5167			0.946446
$567$		0			0
$568$	−18.0000			−0.755263
$569$	−12.1244	−	21.0000i	−0.508279	−	0.880366i	−0.999954	−	0.00958679i	$-0.996948\pi$
$569$	−12.1244	−	21.0000i	−0.508279	−	0.880366i	0.491675	−	0.870779i	$-0.336385\pi$
$570$		0			0
$571$	−20.5000	+	35.5070i	−0.857898	+	1.48592i	0.0160316	+	0.999871i	$0.494897\pi$
$571$	−20.5000	+	35.5070i	−0.857898	+	1.48592i	−0.873930	+	0.486052i	$0.838437\pi$
$572$	−8.66025	+	15.0000i	−0.362103	+	0.627182i
$573$		0			0
$574$	3.00000	+	5.19615i	0.125218	+	0.216883i
$575$	48.4974			2.02248
$576$		0			0
$577$	−34.0000			−1.41544			−0.707719	−	0.706494i	$-0.750276\pi$
$577$	−34.0000			−1.41544			−0.707719	+	0.706494i	$0.750276\pi$
$578$	−14.7224	−	25.5000i	−0.612372	−	1.06066i
$579$		0			0
$580$	6.00000	−	10.3923i	0.249136	−	0.431517i
$581$	−3.46410	+	6.00000i	−0.143715	+	0.248922i
$582$		0			0
$583$	−18.0000	−	31.1769i	−0.745484	−	1.29122i
$584$	3.46410			0.143346
$585$		0			0
$586$	24.0000			0.991431
$587$	3.46410	+	6.00000i	0.142979	+	0.247647i	0.928617	−	0.371040i	$-0.120999\pi$
$587$	3.46410	+	6.00000i	0.142979	+	0.247647i	−0.785638	+	0.618686i	$0.787665\pi$
$588$		0			0
$589$	2.50000	−	4.33013i	0.103011	−	0.178420i
$590$	10.3923	−	18.0000i	0.427844	−	0.741048i
$591$		0			0
$592$	−2.50000	−	4.33013i	−0.102749	−	0.177967i
$593$	−20.7846			−0.853522			−0.426761	−	0.904365i	$-0.640345\pi$
$593$	−20.7846			−0.853522			−0.426761	+	0.904365i	$0.640345\pi$
$594$		0			0
$595$		0			0
$596$	−3.46410	−	6.00000i	−0.141895	−	0.245770i
$597$		0			0
$598$	30.0000	−	51.9615i	1.22679	−	2.12486i
$599$	12.1244	−	21.0000i	0.495388	−	0.858037i	−0.504598	−	0.863354i	$-0.668359\pi$
$599$	12.1244	−	21.0000i	0.495388	−	0.858037i	0.999986	+	0.00531761i	$0.00169266\pi$
$600$		0			0
$601$	−17.5000	−	30.3109i	−0.713840	−	1.23641i	−0.963405	−	0.268049i	$-0.913621\pi$
$601$	−17.5000	−	30.3109i	−0.713840	−	1.23641i	0.249565	−	0.968358i	$-0.419712\pi$
$602$	−1.73205			−0.0705931
$603$		0			0
$604$	−16.0000			−0.651031
$605$	1.73205	+	3.00000i	0.0704179	+	0.121967i
$606$		0			0
$607$	6.50000	−	11.2583i	0.263827	−	0.456962i	−0.703429	−	0.710766i	$-0.748349\pi$
$607$	6.50000	−	11.2583i	0.263827	−	0.456962i	0.967256	+	0.253804i	$0.0816819\pi$
$608$	2.59808	−	4.50000i	0.105366	−	0.182499i
$609$		0			0
$610$	−6.00000	−	10.3923i	−0.242933	−	0.420772i
$611$	17.3205			0.700713
$612$		0			0
$613$	2.00000			0.0807792			0.0403896	−	0.999184i	$-0.487140\pi$
$613$	2.00000			0.0807792			0.0403896	+	0.999184i	$0.487140\pi$
$614$	17.3205	+	30.0000i	0.698999	+	1.21070i
$615$		0			0
$616$	3.00000	−	5.19615i	0.120873	−	0.209359i
$617$	−3.46410	+	6.00000i	−0.139459	+	0.241551i	−0.927292	−	0.374338i	$-0.877870\pi$
$617$	−3.46410	+	6.00000i	−0.139459	+	0.241551i	0.787833	+	0.615889i	$0.211203\pi$
$618$		0			0
$619$	−10.0000	−	17.3205i	−0.401934	−	0.696170i	0.592025	−	0.805919i	$-0.298329\pi$
$619$	−10.0000	−	17.3205i	−0.401934	−	0.696170i	−0.993959	+	0.109749i	$0.964995\pi$
$620$	−17.3205			−0.695608
$621$		0			0
$622$	24.0000			0.962312
$623$	−5.19615	−	9.00000i	−0.208179	−	0.360577i
$624$		0			0
$625$	5.50000	−	9.52628i	0.220000	−	0.381051i
$626$	−0.866025	+	1.50000i	−0.0346133	+	0.0599521i
$627$		0			0
$628$	6.50000	+	11.2583i	0.259378	+	0.449256i
$629$		0			0
$630$		0			0
$631$	17.0000			0.676759			0.338380	−	0.941010i	$-0.390121\pi$
$631$	17.0000			0.676759			0.338380	+	0.941010i	$0.390121\pi$
$632$	0.866025	+	1.50000i	0.0344486	+	0.0596668i
$633$		0			0
$634$	−24.0000	+	41.5692i	−0.953162	+	1.65092i
$635$	29.4449	−	51.0000i	1.16848	−	2.02387i
$636$		0			0
$637$	15.0000	+	25.9808i	0.594322	+	1.02940i
$638$	−20.7846			−0.822871
$639$		0			0
$640$	42.0000			1.66020
$641$	−22.5167	−	39.0000i	−0.889355	−	1.54041i	−0.840640	−	0.541595i	$-0.817821\pi$
$641$	−22.5167	−	39.0000i	−0.889355	−	1.54041i	−0.0487148	−	0.998813i	$-0.515513\pi$
$642$		0			0
$643$	−22.0000	+	38.1051i	−0.867595	+	1.50272i	−0.00314839	+	0.999995i	$0.501002\pi$
$643$	−22.0000	+	38.1051i	−0.867595	+	1.50272i	−0.864447	+	0.502724i	$0.832331\pi$
$644$	3.46410	−	6.00000i	0.136505	−	0.236433i
$645$		0			0
$646$		0			0
$647$	20.7846			0.817127			0.408564	−	0.912730i	$-0.366030\pi$
$647$	20.7846			0.817127			0.408564	+	0.912730i	$0.366030\pi$
$648$		0			0
$649$	−12.0000			−0.471041
$650$	30.3109	+	52.5000i	1.18889	+	2.05922i
$651$		0			0
$652$	0.500000	−	0.866025i	0.0195815	−	0.0339162i
$653$	−8.66025	+	15.0000i	−0.338902	+	0.586995i	−0.984226	−	0.176913i	$-0.943389\pi$
$653$	−8.66025	+	15.0000i	−0.338902	+	0.586995i	0.645325	+	0.763909i	$0.276722\pi$
$654$		0			0
$655$	−6.00000	−	10.3923i	−0.234439	−	0.406061i
$656$	17.3205			0.676252
$657$		0			0
$658$	6.00000			0.233904
$659$	13.8564	+	24.0000i	0.539769	+	0.934907i	0.998916	+	0.0465470i	$0.0148217\pi$
$659$	13.8564	+	24.0000i	0.539769	+	0.934907i	−0.459147	+	0.888360i	$0.651845\pi$
$660$		0			0
$661$	5.00000	−	8.66025i	0.194477	−	0.336845i	−0.752252	−	0.658876i	$-0.771032\pi$
$661$	5.00000	−	8.66025i	0.194477	−	0.336845i	0.946729	+	0.322031i	$0.104366\pi$
$662$	−16.4545	+	28.5000i	−0.639522	+	1.10768i
$663$		0			0
$664$	6.00000	+	10.3923i	0.232845	+	0.403300i
$665$	−3.46410			−0.134332
$666$		0			0
$667$	24.0000			0.929284
$668$	12.1244	+	21.0000i	0.469105	+	0.812514i
$669$		0			0
$670$	−24.0000	+	41.5692i	−0.927201	+	1.60596i
$671$	−3.46410	+	6.00000i	−0.133730	+	0.231627i
$672$		0			0
$673$	18.5000	+	32.0429i	0.713123	+	1.23516i	0.963679	+	0.267063i	$0.0860531\pi$
$673$	18.5000	+	32.0429i	0.713123	+	1.23516i	−0.250557	+	0.968102i	$0.580614\pi$
$674$	−8.66025			−0.333581
$675$		0			0
$676$	12.0000			0.461538
$677$	8.66025	+	15.0000i	0.332841	+	0.576497i	0.983068	−	0.183243i	$-0.0586596\pi$
$677$	8.66025	+	15.0000i	0.332841	+	0.576497i	−0.650227	+	0.759740i	$0.725326\pi$
$678$		0			0
$679$	8.50000	−	14.7224i	0.326200	−	0.564995i
$680$		0			0
$681$		0			0
$682$	15.0000	+	25.9808i	0.574380	+	0.994855i
$683$	−41.5692			−1.59060			−0.795301	−	0.606215i	$-0.792687\pi$
$683$	−41.5692			−1.59060			−0.795301	+	0.606215i	$0.792687\pi$
$684$		0			0
$685$	24.0000			0.916993
$686$	11.2583	+	19.5000i	0.429845	+	0.744513i
$687$		0			0
$688$	−2.50000	+	4.33013i	−0.0953116	+	0.165085i
$689$	−25.9808	+	45.0000i	−0.989788	+	1.71436i
$690$		0			0
$691$	−8.50000	−	14.7224i	−0.323355	−	0.560068i	0.657823	−	0.753173i	$-0.271478\pi$
$691$	−8.50000	−	14.7224i	−0.323355	−	0.560068i	−0.981178	+	0.193105i	$0.938144\pi$
$692$	13.8564			0.526742
$693$		0			0
$694$	42.0000			1.59430
$695$	−22.5167	−	39.0000i	−0.854106	−	1.47935i
$696$		0			0
$697$		0			0
$698$	−0.866025	+	1.50000i	−0.0327795	+	0.0567758i
$699$		0			0
$700$	3.50000	+	6.06218i	0.132288	+	0.229129i
$701$	41.5692			1.57005			0.785024	−	0.619466i	$-0.212651\pi$
$701$	41.5692			1.57005			0.785024	+	0.619466i	$0.212651\pi$
$702$		0			0
$703$	1.00000			0.0377157
$704$	−1.73205	−	3.00000i	−0.0652791	−	0.113067i
$705$		0			0
$706$	−15.0000	+	25.9808i	−0.564532	+	0.977799i
$707$	6.92820	−	12.0000i	0.260562	−	0.451306i
$708$		0			0
$709$	9.50000	+	16.4545i	0.356780	+	0.617961i	0.987421	−	0.158114i	$-0.0505412\pi$
$709$	9.50000	+	16.4545i	0.356780	+	0.617961i	−0.630641	+	0.776075i	$0.717208\pi$
$710$	−62.3538			−2.34010
$711$		0			0
$712$	−18.0000			−0.674579
$713$	−17.3205	−	30.0000i	−0.648658	−	1.12351i
$714$		0			0
$715$	30.0000	−	51.9615i	1.12194	−	1.94325i
$716$	−10.3923	+	18.0000i	−0.388379	+	0.672692i
$717$		0			0
$718$	−27.0000	−	46.7654i	−1.00763	−	1.74527i
$719$	−10.3923			−0.387568			−0.193784	−	0.981044i	$-0.562076\pi$
$719$	−10.3923			−0.387568			−0.193784	+	0.981044i	$0.562076\pi$
$720$		0			0
$721$	−8.00000			−0.297936
$722$	−15.5885	−	27.0000i	−0.580142	−	1.00484i
$723$		0			0
$724$	−8.50000	+	14.7224i	−0.315900	+	0.547155i
$725$	−12.1244	+	21.0000i	−0.450287	+	0.779920i
$726$		0			0
$727$	8.00000	+	13.8564i	0.296704	+	0.513906i	0.975380	−	0.220532i	$-0.0707793\pi$
$727$	8.00000	+	13.8564i	0.296704	+	0.513906i	−0.678676	+	0.734438i	$0.737446\pi$
$728$	−8.66025			−0.320970
$729$		0			0
$730$	12.0000			0.444140
$731$		0			0
$732$		0			0
$733$	−20.5000	+	35.5070i	−0.757185	+	1.31148i	0.187096	+	0.982342i	$0.440092\pi$
$733$	−20.5000	+	35.5070i	−0.757185	+	1.31148i	−0.944281	+	0.329141i	$0.893241\pi$
$734$	−13.8564	+	24.0000i	−0.511449	+	0.885856i
$735$		0			0
$736$	−18.0000	−	31.1769i	−0.663489	−	1.14920i
$737$	27.7128			1.02081
$738$		0			0
$739$	−19.0000			−0.698926			−0.349463	−	0.936950i	$-0.613636\pi$
$739$	−19.0000			−0.698926			−0.349463	+	0.936950i	$0.613636\pi$
$740$	−1.73205	−	3.00000i	−0.0636715	−	0.110282i
$741$		0			0
$742$	−9.00000	+	15.5885i	−0.330400	+	0.572270i
$743$	−3.46410	+	6.00000i	−0.127086	+	0.220119i	−0.922546	−	0.385887i	$-0.873896\pi$
$743$	−3.46410	+	6.00000i	−0.127086	+	0.220119i	0.795461	+	0.606005i	$0.207229\pi$
$744$		0			0
$745$	12.0000	+	20.7846i	0.439646	+	0.761489i
$746$	−39.8372			−1.45854
$747$		0			0
$748$		0			0
$749$	−5.19615	−	9.00000i	−0.189863	−	0.328853i
$750$		0			0
$751$	−2.50000	+	4.33013i	−0.0912263	+	0.158009i	−0.908027	−	0.418911i	$-0.862412\pi$
$751$	−2.50000	+	4.33013i	−0.0912263	+	0.158009i	0.816801	+	0.576919i	$0.195745\pi$
$752$	8.66025	−	15.0000i	0.315807	−	0.546994i
$753$		0			0
$754$	15.0000	+	25.9808i	0.546268	+	0.946164i
$755$	55.4256			2.01715
$756$		0			0
$757$	38.0000			1.38113			0.690567	−	0.723269i	$-0.257361\pi$
$757$	38.0000			1.38113			0.690567	+	0.723269i	$0.257361\pi$
$758$	−16.4545	−	28.5000i	−0.597654	−	1.03517i
$759$		0			0
$760$	−3.00000	+	5.19615i	−0.108821	+	0.188484i
$761$	−13.8564	+	24.0000i	−0.502294	+	0.869999i	0.497702	+	0.867348i	$0.334177\pi$
$761$	−13.8564	+	24.0000i	−0.502294	+	0.869999i	−0.999996	+	0.00265131i	$0.999156\pi$
$762$		0			0
$763$	8.50000	+	14.7224i	0.307721	+	0.532988i
$764$	−6.92820			−0.250654
$765$		0			0
$766$	−30.0000			−1.08394
$767$	8.66025	+	15.0000i	0.312704	+	0.541619i
$768$		0			0
$769$	6.50000	−	11.2583i	0.234396	−	0.405986i	−0.724701	−	0.689063i	$-0.758022\pi$
$769$	6.50000	−	11.2583i	0.234396	−	0.405986i	0.959097	+	0.283078i	$0.0913554\pi$
$770$	10.3923	−	18.0000i	0.374513	−	0.648675i
$771$		0			0
$772$	5.00000	+	8.66025i	0.179954	+	0.311689i
$773$	−20.7846			−0.747570			−0.373785	−	0.927515i	$-0.621940\pi$
$773$	−20.7846			−0.747570			−0.373785	+	0.927515i	$0.621940\pi$
$774$		0			0
$775$	35.0000			1.25724
$776$	−14.7224	−	25.5000i	−0.528505	−	0.915397i
$777$		0			0
$778$	−6.00000	+	10.3923i	−0.215110	+	0.372582i
$779$	−1.73205	+	3.00000i	−0.0620572	+	0.107486i
$780$		0			0
$781$	18.0000	+	31.1769i	0.644091	+	1.11560i
$782$		0			0
$783$		0			0
$784$	30.0000			1.07143
$785$	−22.5167	−	39.0000i	−0.803654	−	1.39197i

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:	\( N \)	\(=\)	\( 243 = 3^{5} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	243.c (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.866025 + 1.50000i) q^{2} +(-0.500000 + 0.866025i) q^{4} +(1.73205 - 3.00000i) q^{5} +(0.500000 + 0.866025i) q^{7} +1.73205 q^{8} +O(q^{10})\)	\(q+(0.866025 + 1.50000i) q^{2} +(-0.500000 + 0.866025i) q^{4} +(1.73205 - 3.00000i) q^{5} +(0.500000 + 0.866025i) q^{7} +1.73205 q^{8} +6.00000 q^{10} +(-1.73205 - 3.00000i) q^{11} +(-2.50000 + 4.33013i) q^{13} +(-0.866025 + 1.50000i) q^{14} +(2.50000 + 4.33013i) q^{16} -1.00000 q^{19} +(1.73205 + 3.00000i) q^{20} +(3.00000 - 5.19615i) q^{22} +(-3.46410 + 6.00000i) q^{23} +(-3.50000 - 6.06218i) q^{25} -8.66025 q^{26} -1.00000 q^{28} +(-1.73205 - 3.00000i) q^{29} +(-2.50000 + 4.33013i) q^{31} +(-2.59808 + 4.50000i) q^{32} +3.46410 q^{35} -1.00000 q^{37} +(-0.866025 - 1.50000i) q^{38} +(3.00000 - 5.19615i) q^{40} +(1.73205 - 3.00000i) q^{41} +(0.500000 + 0.866025i) q^{43} +3.46410 q^{44} -12.0000 q^{46} +(-1.73205 - 3.00000i) q^{47} +(3.00000 - 5.19615i) q^{49} +(6.06218 - 10.5000i) q^{50} +(-2.50000 - 4.33013i) q^{52} +10.3923 q^{53} -12.0000 q^{55} +(0.866025 + 1.50000i) q^{56} +(3.00000 - 5.19615i) q^{58} +(1.73205 - 3.00000i) q^{59} +(-1.00000 - 1.73205i) q^{61} -8.66025 q^{62} +1.00000 q^{64} +(8.66025 + 15.0000i) q^{65} +(-4.00000 + 6.92820i) q^{67} +(3.00000 + 5.19615i) q^{70} -10.3923 q^{71} +2.00000 q^{73} +(-0.866025 - 1.50000i) q^{74} +(0.500000 - 0.866025i) q^{76} +(1.73205 - 3.00000i) q^{77} +(0.500000 + 0.866025i) q^{79} +17.3205 q^{80} +6.00000 q^{82} +(3.46410 + 6.00000i) q^{83} +(-0.866025 + 1.50000i) q^{86} +(-3.00000 - 5.19615i) q^{88} -10.3923 q^{89} -5.00000 q^{91} +(-3.46410 - 6.00000i) q^{92} +(3.00000 - 5.19615i) q^{94} +(-1.73205 + 3.00000i) q^{95} +(-8.50000 - 14.7224i) q^{97} +10.3923 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{4} + 2 q^{7}+O(q^{10}) \) 4 * q - 2 * q^4 + 2 * q^7	\( 4 q - 2 q^{4} + 2 q^{7} + 24 q^{10} - 10 q^{13} + 10 q^{16} - 4 q^{19} + 12 q^{22} - 14 q^{25} - 4 q^{28} - 10 q^{31} - 4 q^{37} + 12 q^{40} + 2 q^{43} - 48 q^{46} + 12 q^{49} - 10 q^{52} - 48 q^{55} + 12 q^{58} - 4 q^{61} + 4 q^{64} - 16 q^{67} + 12 q^{70} + 8 q^{73} + 2 q^{76} + 2 q^{79} + 24 q^{82} - 12 q^{88} - 20 q^{91} + 12 q^{94} - 34 q^{97}+O(q^{100}) \) 4 * q - 2 * q^4 + 2 * q^7 + 24 * q^10 - 10 * q^13 + 10 * q^16 - 4 * q^19 + 12 * q^22 - 14 * q^25 - 4 * q^28 - 10 * q^31 - 4 * q^37 + 12 * q^40 + 2 * q^43 - 48 * q^46 + 12 * q^49 - 10 * q^52 - 48 * q^55 + 12 * q^58 - 4 * q^61 + 4 * q^64 - 16 * q^67 + 12 * q^70 + 8 * q^73 + 2 * q^76 + 2 * q^79 + 24 * q^82 - 12 * q^88 - 20 * q^91 + 12 * q^94 - 34 * q^97

Introduction

L-functions

Modular forms

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