LMFDB - Embedded newform 3240.2.q.v.1081.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3240,2,Mod(1081,3240)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3240, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 2, 0]))

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

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

chi := DirichletCharacter("3240.1081");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3240 = 2^{3} \cdot 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3240.q (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:	$25.8715302549$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1080)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			1081.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	3240.1081
Dual form			3240.2.q.v.2161.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.500000 - 0.866025i) q^{5} +(1.00000 + 1.73205i) q^{7} +O(q^{10})$	$q+(0.500000 - 0.866025i) q^{5} +(1.00000 + 1.73205i) q^{7} +(3.00000 - 5.19615i) q^{13} +7.00000 q^{17} +7.00000 q^{19} +(-3.50000 + 6.06218i) q^{23} +(-0.500000 - 0.866025i) q^{25} +(-3.00000 - 5.19615i) q^{29} +(-1.50000 + 2.59808i) q^{31} +2.00000 q^{35} -6.00000 q^{37} +(-2.00000 + 3.46410i) q^{41} +(-4.00000 - 6.92820i) q^{43} +(2.00000 + 3.46410i) q^{47} +(1.50000 - 2.59808i) q^{49} -5.00000 q^{53} +(-3.00000 + 5.19615i) q^{59} +(1.50000 + 2.59808i) q^{61} +(-3.00000 - 5.19615i) q^{65} +(5.00000 - 8.66025i) q^{67} +12.0000 q^{71} +16.0000 q^{73} +(-0.500000 - 0.866025i) q^{79} +(-4.50000 - 7.79423i) q^{83} +(3.50000 - 6.06218i) q^{85} -4.00000 q^{89} +12.0000 q^{91} +(3.50000 - 6.06218i) q^{95} +(8.00000 + 13.8564i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{5} + 2 q^{7}+O(q^{10}) $ 2 * q + q^5 + 2 * q^7	$ 2 q + q^{5} + 2 q^{7} + 6 q^{13} + 14 q^{17} + 14 q^{19} - 7 q^{23} - q^{25} - 6 q^{29} - 3 q^{31} + 4 q^{35} - 12 q^{37} - 4 q^{41} - 8 q^{43} + 4 q^{47} + 3 q^{49} - 10 q^{53} - 6 q^{59} + 3 q^{61} - 6 q^{65} + 10 q^{67} + 24 q^{71} + 32 q^{73} - q^{79} - 9 q^{83} + 7 q^{85} - 8 q^{89} + 24 q^{91} + 7 q^{95} + 16 q^{97}+O(q^{100}) $ 2 * q + q^5 + 2 * q^7 + 6 * q^13 + 14 * q^17 + 14 * q^19 - 7 * q^23 - q^25 - 6 * q^29 - 3 * q^31 + 4 * q^35 - 12 * q^37 - 4 * q^41 - 8 * q^43 + 4 * q^47 + 3 * q^49 - 10 * q^53 - 6 * q^59 + 3 * q^61 - 6 * q^65 + 10 * q^67 + 24 * q^71 + 32 * q^73 - q^79 - 9 * q^83 + 7 * q^85 - 8 * q^89 + 24 * q^91 + 7 * q^95 + 16 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3240\mathbb{Z}\right)^\times$.

$n$	$1297$	$1621$	$2431$	$3161$
$\chi(n)$	$1$	$1$	$1$	$e\left(\frac{1}{3}\right)$

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			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	0.500000	−	0.866025i	0.223607	−	0.387298i
$5$	0.500000	−	0.866025i	0.223607	−	0.387298i
$6$		0			0
$7$	1.00000	+	1.73205i	0.377964	+	0.654654i	0.990766	−	0.135583i	$-0.0432908\pi$
$7$	1.00000	+	1.73205i	0.377964	+	0.654654i	−0.612801	+	0.790237i	$0.709957\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$11$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$12$		0			0
$13$	3.00000	−	5.19615i	0.832050	−	1.44115i	−0.0643593	−	0.997927i	$-0.520500\pi$
$13$	3.00000	−	5.19615i	0.832050	−	1.44115i	0.896410	−	0.443227i	$-0.146166\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	7.00000			1.69775			0.848875	−	0.528594i	$-0.177281\pi$
$17$	7.00000			1.69775			0.848875	+	0.528594i	$0.177281\pi$
$18$		0			0
$19$	7.00000			1.60591			0.802955	−	0.596040i	$-0.203260\pi$
$19$	7.00000			1.60591			0.802955	+	0.596040i	$0.203260\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−3.50000	+	6.06218i	−0.729800	+	1.26405i	0.227167	+	0.973856i	$0.427054\pi$
$23$	−3.50000	+	6.06218i	−0.729800	+	1.26405i	−0.956967	+	0.290196i	$0.906280\pi$
$24$		0			0
$25$	−0.500000	−	0.866025i	−0.100000	−	0.173205i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−3.00000	−	5.19615i	−0.557086	−	0.964901i	−0.997738	−	0.0672232i	$-0.978586\pi$
$29$	−3.00000	−	5.19615i	−0.557086	−	0.964901i	0.440652	−	0.897678i	$-0.354747\pi$
$30$		0			0
$31$	−1.50000	+	2.59808i	−0.269408	+	0.466628i	−0.968709	−	0.248199i	$-0.920161\pi$
$31$	−1.50000	+	2.59808i	−0.269408	+	0.466628i	0.699301	+	0.714827i	$0.253495\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	2.00000			0.338062
$36$		0			0
$37$	−6.00000			−0.986394			−0.493197	−	0.869918i	$-0.664172\pi$
$37$	−6.00000			−0.986394			−0.493197	+	0.869918i	$0.664172\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−2.00000	+	3.46410i	−0.312348	+	0.541002i	−0.978870	−	0.204483i	$-0.934449\pi$
$41$	−2.00000	+	3.46410i	−0.312348	+	0.541002i	0.666523	+	0.745485i	$0.267782\pi$
$42$		0			0
$43$	−4.00000	−	6.92820i	−0.609994	−	1.05654i	−0.991241	−	0.132068i	$-0.957838\pi$
$43$	−4.00000	−	6.92820i	−0.609994	−	1.05654i	0.381246	−	0.924473i	$-0.375495\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	2.00000	+	3.46410i	0.291730	+	0.505291i	0.974219	−	0.225605i	$-0.0724358\pi$
$47$	2.00000	+	3.46410i	0.291730	+	0.505291i	−0.682489	+	0.730896i	$0.739102\pi$
$48$		0			0
$49$	1.50000	−	2.59808i	0.214286	−	0.371154i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−5.00000			−0.686803			−0.343401	−	0.939189i	$-0.611579\pi$
$53$	−5.00000			−0.686803			−0.343401	+	0.939189i	$0.611579\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−3.00000	+	5.19615i	−0.390567	+	0.676481i	−0.992524	−	0.122047i	$-0.961054\pi$
$59$	−3.00000	+	5.19615i	−0.390567	+	0.676481i	0.601958	+	0.798528i	$0.294388\pi$
$60$		0			0
$61$	1.50000	+	2.59808i	0.192055	+	0.332650i	0.945931	−	0.324367i	$-0.105151\pi$
$61$	1.50000	+	2.59808i	0.192055	+	0.332650i	−0.753876	+	0.657017i	$0.771818\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−3.00000	−	5.19615i	−0.372104	−	0.644503i
$66$		0			0
$67$	5.00000	−	8.66025i	0.610847	−	1.05802i	−0.380251	−	0.924883i	$-0.624162\pi$
$67$	5.00000	−	8.66025i	0.610847	−	1.05802i	0.991098	−	0.133135i	$-0.0425044\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	12.0000			1.42414			0.712069	−	0.702109i	$-0.247758\pi$
$71$	12.0000			1.42414			0.712069	+	0.702109i	$0.247758\pi$
$72$		0			0
$73$	16.0000			1.87266			0.936329	−	0.351123i	$-0.114200\pi$
$73$	16.0000			1.87266			0.936329	+	0.351123i	$0.114200\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−0.500000	−	0.866025i	−0.0562544	−	0.0974355i	0.836527	−	0.547926i	$-0.184582\pi$
$79$	−0.500000	−	0.866025i	−0.0562544	−	0.0974355i	−0.892781	+	0.450490i	$0.851249\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−4.50000	−	7.79423i	−0.493939	−	0.855528i	0.506036	−	0.862512i	$-0.331110\pi$
$83$	−4.50000	−	7.79423i	−0.493939	−	0.855528i	−0.999976	+	0.00698436i	$0.997777\pi$
$84$		0			0
$85$	3.50000	−	6.06218i	0.379628	−	0.657536i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−4.00000			−0.423999			−0.212000	−	0.977270i	$-0.567998\pi$
$89$	−4.00000			−0.423999			−0.212000	+	0.977270i	$0.567998\pi$
$90$		0			0
$91$	12.0000			1.25794
$92$		0			0
$93$		0			0
$94$		0			0
$95$	3.50000	−	6.06218i	0.359092	−	0.621966i
$96$		0			0
$97$	8.00000	+	13.8564i	0.812277	+	1.40690i	0.911267	+	0.411816i	$0.135106\pi$
$97$	8.00000	+	13.8564i	0.812277	+	1.40690i	−0.0989899	+	0.995088i	$0.531561\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−2.00000	−	3.46410i	−0.199007	−	0.344691i	0.749199	−	0.662344i	$-0.230438\pi$
$101$	−2.00000	−	3.46410i	−0.199007	−	0.344691i	−0.948207	+	0.317653i	$0.897105\pi$
$102$		0			0
$103$	−2.00000	+	3.46410i	−0.197066	+	0.341328i	−0.947576	−	0.319531i	$-0.896475\pi$
$103$	−2.00000	+	3.46410i	−0.197066	+	0.341328i	0.750510	+	0.660859i	$0.229808\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	4.00000			0.386695			0.193347	−	0.981130i	$-0.438066\pi$
$107$	4.00000			0.386695			0.193347	+	0.981130i	$0.438066\pi$
$108$		0			0
$109$	5.00000			0.478913			0.239457	−	0.970907i	$-0.423031\pi$
$109$	5.00000			0.478913			0.239457	+	0.970907i	$0.423031\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	3.00000	−	5.19615i	0.282216	−	0.488813i	−0.689714	−	0.724082i	$-0.742264\pi$
$113$	3.00000	−	5.19615i	0.282216	−	0.488813i	0.971930	+	0.235269i	$0.0755971\pi$
$114$		0			0
$115$	3.50000	+	6.06218i	0.326377	+	0.565301i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	7.00000	+	12.1244i	0.641689	+	1.11144i
$120$		0			0
$121$	5.50000	−	9.52628i	0.500000	−	0.866025i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−1.00000			−0.0894427
$126$		0			0
$127$	14.0000			1.24230			0.621150	−	0.783692i	$-0.286666\pi$
$127$	14.0000			1.24230			0.621150	+	0.783692i	$0.286666\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	7.00000	−	12.1244i	0.611593	−	1.05931i	−0.379379	−	0.925241i	$-0.623862\pi$
$131$	7.00000	−	12.1244i	0.611593	−	1.05931i	0.990972	−	0.134069i	$-0.0428042\pi$
$132$		0			0
$133$	7.00000	+	12.1244i	0.606977	+	1.05131i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	1.50000	+	2.59808i	0.128154	+	0.221969i	0.922961	−	0.384893i	$-0.125762\pi$
$137$	1.50000	+	2.59808i	0.128154	+	0.221969i	−0.794808	+	0.606861i	$0.792428\pi$
$138$		0			0
$139$	−6.00000	+	10.3923i	−0.508913	+	0.881464i	0.491033	+	0.871141i	$0.336619\pi$
$139$	−6.00000	+	10.3923i	−0.508913	+	0.881464i	−0.999947	+	0.0103230i	$0.996714\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$	−6.00000			−0.498273
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−6.00000	+	10.3923i	−0.491539	+	0.851371i	−0.999953	−	0.00974235i	$-0.996899\pi$
$149$	−6.00000	+	10.3923i	−0.491539	+	0.851371i	0.508413	+	0.861113i	$0.330232\pi$
$150$		0			0
$151$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$151$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	1.50000	+	2.59808i	0.120483	+	0.208683i
$156$		0			0
$157$	10.0000	−	17.3205i	0.798087	−	1.38233i	−0.122774	−	0.992435i	$-0.539179\pi$
$157$	10.0000	−	17.3205i	0.798087	−	1.38233i	0.920860	−	0.389892i	$-0.127488\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−14.0000			−1.10335
$162$		0			0
$163$	2.00000			0.156652			0.0783260	−	0.996928i	$-0.475042\pi$
$163$	2.00000			0.156652			0.0783260	+	0.996928i	$0.475042\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−9.50000	+	16.4545i	−0.735132	+	1.27329i	0.219533	+	0.975605i	$0.429547\pi$
$167$	−9.50000	+	16.4545i	−0.735132	+	1.27329i	−0.954665	+	0.297681i	$0.903787\pi$
$168$		0			0
$169$	−11.5000	−	19.9186i	−0.884615	−	1.53220i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−4.50000	−	7.79423i	−0.342129	−	0.592584i	0.642699	−	0.766119i	$-0.277815\pi$
$173$	−4.50000	−	7.79423i	−0.342129	−	0.592584i	−0.984828	+	0.173534i	$0.944481\pi$
$174$		0			0
$175$	1.00000	−	1.73205i	0.0755929	−	0.130931i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−4.00000			−0.298974			−0.149487	−	0.988764i	$-0.547762\pi$
$179$	−4.00000			−0.298974			−0.149487	+	0.988764i	$0.547762\pi$
$180$		0			0
$181$	5.00000			0.371647			0.185824	−	0.982583i	$-0.440505\pi$
$181$	5.00000			0.371647			0.185824	+	0.982583i	$0.440505\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−3.00000	+	5.19615i	−0.220564	+	0.382029i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−1.00000	−	1.73205i	−0.0723575	−	0.125327i	0.827577	−	0.561353i	$-0.189719\pi$
$191$	−1.00000	−	1.73205i	−0.0723575	−	0.125327i	−0.899934	+	0.436026i	$0.856386\pi$
$192$		0			0
$193$	−3.00000	+	5.19615i	−0.215945	+	0.374027i	−0.953564	−	0.301189i	$-0.902616\pi$
$193$	−3.00000	+	5.19615i	−0.215945	+	0.374027i	0.737620	+	0.675216i	$0.235950\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−23.0000			−1.63868			−0.819341	−	0.573306i	$-0.805660\pi$
$197$	−23.0000			−1.63868			−0.819341	+	0.573306i	$0.805660\pi$
$198$		0			0
$199$	24.0000			1.70131			0.850657	−	0.525720i	$-0.176204\pi$
$199$	24.0000			1.70131			0.850657	+	0.525720i	$0.176204\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	6.00000	−	10.3923i	0.421117	−	0.729397i
$204$		0			0
$205$	2.00000	+	3.46410i	0.139686	+	0.241943i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$	7.50000	−	12.9904i	0.516321	−	0.894295i	−0.483499	−	0.875345i	$-0.660634\pi$
$211$	7.50000	−	12.9904i	0.516321	−	0.894295i	0.999820	−	0.0189499i	$-0.00603229\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−8.00000			−0.545595
$216$		0			0
$217$	−6.00000			−0.407307
$218$		0			0
$219$		0			0
$220$		0			0
$221$	21.0000	−	36.3731i	1.41261	−	2.44672i
$222$		0			0
$223$	−3.00000	−	5.19615i	−0.200895	−	0.347960i	0.747922	−	0.663786i	$-0.231052\pi$
$223$	−3.00000	−	5.19615i	−0.200895	−	0.347960i	−0.948817	+	0.315826i	$0.897718\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	13.5000	+	23.3827i	0.896026	+	1.55196i	0.832529	+	0.553981i	$0.186892\pi$
$227$	13.5000	+	23.3827i	0.896026	+	1.55196i	0.0634974	+	0.997982i	$0.479775\pi$
$228$		0			0
$229$	−1.50000	+	2.59808i	−0.0991228	+	0.171686i	−0.911322	−	0.411695i	$-0.864937\pi$
$229$	−1.50000	+	2.59808i	−0.0991228	+	0.171686i	0.812199	+	0.583380i	$0.198270\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−22.0000			−1.44127			−0.720634	−	0.693316i	$-0.756149\pi$
$233$	−22.0000			−1.44127			−0.720634	+	0.693316i	$0.756149\pi$
$234$		0			0
$235$	4.00000			0.260931
$236$		0			0
$237$		0			0
$238$		0			0
$239$	9.00000	−	15.5885i	0.582162	−	1.00833i	−0.413061	−	0.910703i	$-0.635540\pi$
$239$	9.00000	−	15.5885i	0.582162	−	1.00833i	0.995223	−	0.0976302i	$-0.0311262\pi$
$240$		0			0
$241$	−5.50000	−	9.52628i	−0.354286	−	0.613642i	0.632709	−	0.774389i	$-0.281943\pi$
$241$	−5.50000	−	9.52628i	−0.354286	−	0.613642i	−0.986996	+	0.160748i	$0.948609\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−1.50000	−	2.59808i	−0.0958315	−	0.165985i
$246$		0			0
$247$	21.0000	−	36.3731i	1.33620	−	2.31436i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	16.0000			1.00991			0.504956	−	0.863145i	$-0.331509\pi$
$251$	16.0000			1.00991			0.504956	+	0.863145i	$0.331509\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−4.50000	+	7.79423i	−0.280702	+	0.486191i	−0.971558	−	0.236802i	$-0.923901\pi$
$257$	−4.50000	+	7.79423i	−0.280702	+	0.486191i	0.690856	+	0.722993i	$0.257234\pi$
$258$		0			0
$259$	−6.00000	−	10.3923i	−0.372822	−	0.645746i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	12.0000	+	20.7846i	0.739952	+	1.28163i	0.952517	+	0.304487i	$0.0984850\pi$
$263$	12.0000	+	20.7846i	0.739952	+	1.28163i	−0.212565	+	0.977147i	$0.568182\pi$
$264$		0			0
$265$	−2.50000	+	4.33013i	−0.153574	+	0.265998i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	18.0000			1.09748			0.548740	−	0.835993i	$-0.315108\pi$
$269$	18.0000			1.09748			0.548740	+	0.835993i	$0.315108\pi$
$270$		0			0
$271$	−13.0000			−0.789694			−0.394847	−	0.918747i	$-0.629202\pi$
$271$	−13.0000			−0.789694			−0.394847	+	0.918747i	$0.629202\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	4.00000	+	6.92820i	0.240337	+	0.416275i	0.960810	−	0.277207i	$-0.0894088\pi$
$277$	4.00000	+	6.92820i	0.240337	+	0.416275i	−0.720473	+	0.693482i	$0.756075\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−13.0000	−	22.5167i	−0.775515	−	1.34323i	−0.934505	−	0.355951i	$-0.884157\pi$
$281$	−13.0000	−	22.5167i	−0.775515	−	1.34323i	0.158990	−	0.987280i	$-0.449176\pi$
$282$		0			0
$283$	−2.00000	+	3.46410i	−0.118888	+	0.205919i	−0.919327	−	0.393494i	$-0.871266\pi$
$283$	−2.00000	+	3.46410i	−0.118888	+	0.205919i	0.800439	+	0.599414i	$0.204600\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−8.00000			−0.472225
$288$		0			0
$289$	32.0000			1.88235
$290$		0			0
$291$		0			0
$292$		0			0
$293$	5.50000	−	9.52628i	0.321313	−	0.556531i	−0.659446	−	0.751752i	$-0.729209\pi$
$293$	5.50000	−	9.52628i	0.321313	−	0.556531i	0.980759	+	0.195221i	$0.0625424\pi$
$294$		0			0
$295$	3.00000	+	5.19615i	0.174667	+	0.302532i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	21.0000	+	36.3731i	1.21446	+	2.10351i
$300$		0			0
$301$	8.00000	−	13.8564i	0.461112	−	0.798670i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	3.00000			0.171780
$306$		0			0
$307$	−26.0000			−1.48390			−0.741949	−	0.670456i	$-0.766098\pi$
$307$	−26.0000			−1.48390			−0.741949	+	0.670456i	$0.766098\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−5.00000	+	8.66025i	−0.283524	+	0.491078i	−0.972250	−	0.233944i	$-0.924837\pi$
$311$	−5.00000	+	8.66025i	−0.283524	+	0.491078i	0.688726	+	0.725022i	$0.258170\pi$
$312$		0			0
$313$	−8.00000	−	13.8564i	−0.452187	−	0.783210i	0.546335	−	0.837567i	$-0.316023\pi$
$313$	−8.00000	−	13.8564i	−0.452187	−	0.783210i	−0.998522	+	0.0543564i	$0.982689\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	0.500000	+	0.866025i	0.0280828	+	0.0486408i	0.879725	−	0.475482i	$-0.157726\pi$
$317$	0.500000	+	0.866025i	0.0280828	+	0.0486408i	−0.851642	+	0.524123i	$0.824393\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$	49.0000			2.72643
$324$		0			0
$325$	−6.00000			−0.332820
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−4.00000	+	6.92820i	−0.220527	+	0.381964i
$330$		0			0
$331$	−2.00000	−	3.46410i	−0.109930	−	0.190404i	0.805812	−	0.592172i	$-0.201729\pi$
$331$	−2.00000	−	3.46410i	−0.109930	−	0.190404i	−0.915742	+	0.401768i	$0.868396\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−5.00000	−	8.66025i	−0.273179	−	0.473160i
$336$		0			0
$337$	−17.0000	+	29.4449i	−0.926049	+	1.60396i	−0.136184	+	0.990684i	$0.543484\pi$
$337$	−17.0000	+	29.4449i	−0.926049	+	1.60396i	−0.789865	+	0.613280i	$0.789850\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	20.0000			1.07990
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−6.00000	+	10.3923i	−0.322097	+	0.557888i	−0.980921	−	0.194409i	$-0.937721\pi$
$347$	−6.00000	+	10.3923i	−0.322097	+	0.557888i	0.658824	+	0.752297i	$0.271054\pi$
$348$		0			0
$349$	17.5000	+	30.3109i	0.936754	+	1.62250i	0.771477	+	0.636257i	$0.219518\pi$
$349$	17.5000	+	30.3109i	0.936754	+	1.62250i	0.165277	+	0.986247i	$0.447148\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	9.00000	+	15.5885i	0.479022	+	0.829690i	0.999711	−	0.0240566i	$-0.00765819\pi$
$353$	9.00000	+	15.5885i	0.479022	+	0.829690i	−0.520689	+	0.853746i	$0.674325\pi$
$354$		0			0
$355$	6.00000	−	10.3923i	0.318447	−	0.551566i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−10.0000			−0.527780			−0.263890	−	0.964553i	$-0.585006\pi$
$359$	−10.0000			−0.527780			−0.263890	+	0.964553i	$0.585006\pi$
$360$		0			0
$361$	30.0000			1.57895
$362$		0			0
$363$		0			0
$364$		0			0
$365$	8.00000	−	13.8564i	0.418739	−	0.725277i
$366$		0			0
$367$	13.0000	+	22.5167i	0.678594	+	1.17536i	0.975404	+	0.220423i	$0.0707439\pi$
$367$	13.0000	+	22.5167i	0.678594	+	1.17536i	−0.296810	+	0.954937i	$0.595923\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−5.00000	−	8.66025i	−0.259587	−	0.449618i
$372$		0			0
$373$	−4.00000	+	6.92820i	−0.207112	+	0.358729i	−0.950804	−	0.309794i	$-0.899740\pi$
$373$	−4.00000	+	6.92820i	−0.207112	+	0.358729i	0.743691	+	0.668523i	$0.233073\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−36.0000			−1.85409
$378$		0			0
$379$	−35.0000			−1.79783			−0.898915	−	0.438124i	$-0.855643\pi$
$379$	−35.0000			−1.79783			−0.898915	+	0.438124i	$0.855643\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−17.5000	+	30.3109i	−0.894208	+	1.54881i	−0.0594268	+	0.998233i	$0.518927\pi$
$383$	−17.5000	+	30.3109i	−0.894208	+	1.54881i	−0.834781	+	0.550581i	$0.814406\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−12.0000	−	20.7846i	−0.608424	−	1.05382i	−0.991500	−	0.130105i	$-0.958469\pi$
$389$	−12.0000	−	20.7846i	−0.608424	−	1.05382i	0.383076	−	0.923717i	$-0.374865\pi$
$390$		0			0
$391$	−24.5000	+	42.4352i	−1.23902	+	2.14604i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−1.00000			−0.0503155
$396$		0			0
$397$	−34.0000			−1.70641			−0.853206	−	0.521575i	$-0.825345\pi$
$397$	−34.0000			−1.70641			−0.853206	+	0.521575i	$0.825345\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	9.00000	−	15.5885i	0.449439	−	0.778450i	−0.548911	−	0.835881i	$-0.684957\pi$
$401$	9.00000	−	15.5885i	0.449439	−	0.778450i	0.998350	+	0.0574304i	$0.0182907\pi$
$402$		0			0
$403$	9.00000	+	15.5885i	0.448322	+	0.776516i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−0.500000	+	0.866025i	−0.0247234	+	0.0428222i	−0.878122	−	0.478436i	$-0.841204\pi$
$409$	−0.500000	+	0.866025i	−0.0247234	+	0.0428222i	0.853399	+	0.521258i	$0.174537\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−12.0000			−0.590481
$414$		0			0
$415$	−9.00000			−0.441793
$416$		0			0
$417$		0			0
$418$		0			0
$419$	8.00000	−	13.8564i	0.390826	−	0.676930i	−0.601733	−	0.798697i	$-0.705523\pi$
$419$	8.00000	−	13.8564i	0.390826	−	0.676930i	0.992559	+	0.121768i	$0.0388562\pi$
$420$		0			0
$421$	−1.50000	−	2.59808i	−0.0731055	−	0.126622i	0.827155	−	0.561973i	$-0.189958\pi$
$421$	−1.50000	−	2.59808i	−0.0731055	−	0.126622i	−0.900261	+	0.435351i	$0.856624\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−3.50000	−	6.06218i	−0.169775	−	0.294059i
$426$		0			0
$427$	−3.00000	+	5.19615i	−0.145180	+	0.251459i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−26.0000			−1.25238			−0.626188	−	0.779672i	$-0.715386\pi$
$431$	−26.0000			−1.25238			−0.626188	+	0.779672i	$0.715386\pi$
$432$		0			0
$433$	−2.00000			−0.0961139			−0.0480569	−	0.998845i	$-0.515303\pi$
$433$	−2.00000			−0.0961139			−0.0480569	+	0.998845i	$0.515303\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−24.5000	+	42.4352i	−1.17199	+	2.02995i
$438$		0			0
$439$	−9.50000	−	16.4545i	−0.453410	−	0.785330i	0.545185	−	0.838316i	$-0.316459\pi$
$439$	−9.50000	−	16.4545i	−0.453410	−	0.785330i	−0.998595	+	0.0529862i	$0.983126\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−4.50000	−	7.79423i	−0.213801	−	0.370315i	0.739100	−	0.673596i	$-0.235251\pi$
$443$	−4.50000	−	7.79423i	−0.213801	−	0.370315i	−0.952901	+	0.303281i	$0.901918\pi$
$444$		0			0
$445$	−2.00000	+	3.46410i	−0.0948091	+	0.164214i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	14.0000			0.660701			0.330350	−	0.943858i	$-0.392833\pi$
$449$	14.0000			0.660701			0.330350	+	0.943858i	$0.392833\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$	6.00000	−	10.3923i	0.281284	−	0.487199i
$456$		0			0
$457$	−2.00000	−	3.46410i	−0.0935561	−	0.162044i	0.815449	−	0.578829i	$-0.196490\pi$
$457$	−2.00000	−	3.46410i	−0.0935561	−	0.162044i	−0.909005	+	0.416785i	$0.863157\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−7.00000	−	12.1244i	−0.326023	−	0.564688i	0.655696	−	0.755025i	$-0.272375\pi$
$461$	−7.00000	−	12.1244i	−0.326023	−	0.564688i	−0.981719	+	0.190337i	$0.939042\pi$
$462$		0			0
$463$	13.0000	−	22.5167i	0.604161	−	1.04644i	−0.388022	−	0.921650i	$-0.626842\pi$
$463$	13.0000	−	22.5167i	0.604161	−	1.04644i	0.992183	−	0.124788i	$-0.0398251\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	37.0000			1.71216			0.856078	−	0.516847i	$-0.172894\pi$
$467$	37.0000			1.71216			0.856078	+	0.516847i	$0.172894\pi$
$468$		0			0
$469$	20.0000			0.923514
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−3.50000	−	6.06218i	−0.160591	−	0.278152i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−12.0000	−	20.7846i	−0.548294	−	0.949673i	−0.998392	−	0.0566937i	$-0.981944\pi$
$479$	−12.0000	−	20.7846i	−0.548294	−	0.949673i	0.450098	−	0.892979i	$-0.351389\pi$
$480$		0			0
$481$	−18.0000	+	31.1769i	−0.820729	+	1.42154i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	16.0000			0.726523
$486$		0			0
$487$	12.0000			0.543772			0.271886	−	0.962329i	$-0.412353\pi$
$487$	12.0000			0.543772			0.271886	+	0.962329i	$0.412353\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	17.0000	−	29.4449i	0.767199	−	1.32883i	−0.171877	−	0.985118i	$-0.554983\pi$
$491$	17.0000	−	29.4449i	0.767199	−	1.32883i	0.939076	−	0.343710i	$-0.111684\pi$
$492$		0			0
$493$	−21.0000	−	36.3731i	−0.945792	−	1.63816i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	12.0000	+	20.7846i	0.538274	+	0.932317i
$498$		0			0
$499$	−5.50000	+	9.52628i	−0.246214	+	0.426455i	−0.962472	−	0.271380i	$-0.912520\pi$
$499$	−5.50000	+	9.52628i	−0.246214	+	0.426455i	0.716258	+	0.697835i	$0.245853\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−9.00000			−0.401290			−0.200645	−	0.979664i	$-0.564304\pi$
$503$	−9.00000			−0.401290			−0.200645	+	0.979664i	$0.564304\pi$
$504$		0			0
$505$	−4.00000			−0.177998
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−18.0000	+	31.1769i	−0.797836	+	1.38189i	0.123187	+	0.992384i	$0.460689\pi$
$509$	−18.0000	+	31.1769i	−0.797836	+	1.38189i	−0.921023	+	0.389509i	$0.872645\pi$
$510$		0			0
$511$	16.0000	+	27.7128i	0.707798	+	1.22594i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	2.00000	+	3.46410i	0.0881305	+	0.152647i
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	28.0000			1.22670			0.613351	−	0.789810i	$-0.289821\pi$
$521$	28.0000			1.22670			0.613351	+	0.789810i	$0.289821\pi$
$522$		0			0
$523$	−18.0000			−0.787085			−0.393543	−	0.919306i	$-0.628751\pi$
$523$	−18.0000			−0.787085			−0.393543	+	0.919306i	$0.628751\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−10.5000	+	18.1865i	−0.457387	+	0.792218i
$528$		0			0
$529$	−13.0000	−	22.5167i	−0.565217	−	0.978985i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	12.0000	+	20.7846i	0.519778	+	0.900281i
$534$		0			0
$535$	2.00000	−	3.46410i	0.0864675	−	0.149766i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−34.0000			−1.46177			−0.730887	−	0.682498i	$-0.760893\pi$
$541$	−34.0000			−1.46177			−0.730887	+	0.682498i	$0.760893\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	2.50000	−	4.33013i	0.107088	−	0.185482i
$546$		0			0
$547$	4.00000	+	6.92820i	0.171028	+	0.296229i	0.938779	−	0.344519i	$-0.111958\pi$
$547$	4.00000	+	6.92820i	0.171028	+	0.296229i	−0.767752	+	0.640747i	$0.778625\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−21.0000	−	36.3731i	−0.894630	−	1.54954i
$552$		0			0
$553$	1.00000	−	1.73205i	0.0425243	−	0.0736543i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	34.0000			1.44063			0.720313	−	0.693649i	$-0.243998\pi$
$557$	34.0000			1.44063			0.720313	+	0.693649i	$0.243998\pi$
$558$		0			0
$559$	−48.0000			−2.03018
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−18.0000	+	31.1769i	−0.758610	+	1.31395i	0.184950	+	0.982748i	$0.440788\pi$
$563$	−18.0000	+	31.1769i	−0.758610	+	1.31395i	−0.943560	+	0.331202i	$0.892546\pi$
$564$		0			0
$565$	−3.00000	−	5.19615i	−0.126211	−	0.218604i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−11.0000	−	19.0526i	−0.461144	−	0.798725i	0.537874	−	0.843025i	$-0.319228\pi$
$569$	−11.0000	−	19.0526i	−0.461144	−	0.798725i	−0.999018	+	0.0443003i	$0.985894\pi$
$570$		0			0
$571$	8.50000	−	14.7224i	0.355714	−	0.616115i	−0.631526	−	0.775355i	$-0.717571\pi$
$571$	8.50000	−	14.7224i	0.355714	−	0.616115i	0.987240	+	0.159240i	$0.0509044\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	7.00000			0.291920
$576$		0			0
$577$	−20.0000			−0.832611			−0.416305	−	0.909225i	$-0.636675\pi$
$577$	−20.0000			−0.832611			−0.416305	+	0.909225i	$0.636675\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	9.00000	−	15.5885i	0.373383	−	0.646718i
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$	4.50000	+	7.79423i	0.185735	+	0.321702i	0.943824	−	0.330449i	$-0.107200\pi$
$587$	4.50000	+	7.79423i	0.185735	+	0.321702i	−0.758089	+	0.652151i	$0.773867\pi$
$588$		0			0
$589$	−10.5000	+	18.1865i	−0.432645	+	0.749363i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	13.0000			0.533846			0.266923	−	0.963718i	$-0.413993\pi$
$593$	13.0000			0.533846			0.266923	+	0.963718i	$0.413993\pi$
$594$		0			0
$595$	14.0000			0.573944
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−13.0000	+	22.5167i	−0.531166	+	0.920006i	0.468173	+	0.883637i	$0.344912\pi$
$599$	−13.0000	+	22.5167i	−0.531166	+	0.920006i	−0.999338	+	0.0363689i	$0.988421\pi$
$600$		0			0
$601$	−2.50000	−	4.33013i	−0.101977	−	0.176630i	0.810522	−	0.585708i	$-0.199184\pi$
$601$	−2.50000	−	4.33013i	−0.101977	−	0.176630i	−0.912499	+	0.409079i	$0.865850\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−5.50000	−	9.52628i	−0.223607	−	0.387298i
$606$		0			0
$607$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$607$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	24.0000			0.970936
$612$		0			0
$613$	−16.0000			−0.646234			−0.323117	−	0.946359i	$-0.604731\pi$
$613$	−16.0000			−0.646234			−0.323117	+	0.946359i	$0.604731\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	14.5000	−	25.1147i	0.583748	−	1.01108i	−0.411282	−	0.911508i	$-0.634919\pi$
$617$	14.5000	−	25.1147i	0.583748	−	1.01108i	0.995030	−	0.0995732i	$-0.0317477\pi$
$618$		0			0
$619$	−4.00000	−	6.92820i	−0.160774	−	0.278468i	0.774373	−	0.632730i	$-0.218066\pi$
$619$	−4.00000	−	6.92820i	−0.160774	−	0.278468i	−0.935146	+	0.354262i	$0.884732\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−4.00000	−	6.92820i	−0.160257	−	0.277573i
$624$		0			0
$625$	−0.500000	+	0.866025i	−0.0200000	+	0.0346410i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−42.0000			−1.67465
$630$		0			0
$631$	−47.0000			−1.87104			−0.935520	−	0.353273i	$-0.885069\pi$
$631$	−47.0000			−1.87104			−0.935520	+	0.353273i	$0.885069\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	7.00000	−	12.1244i	0.277787	−	0.481140i
$636$		0			0
$637$	−9.00000	−	15.5885i	−0.356593	−	0.617637i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−24.0000	−	41.5692i	−0.947943	−	1.64189i	−0.749749	−	0.661723i	$-0.769826\pi$
$641$	−24.0000	−	41.5692i	−0.947943	−	1.64189i	−0.198194	−	0.980163i	$-0.563508\pi$
$642$		0			0
$643$	−21.0000	+	36.3731i	−0.828159	+	1.43441i	0.0713216	+	0.997453i	$0.477278\pi$
$643$	−21.0000	+	36.3731i	−0.828159	+	1.43441i	−0.899481	+	0.436960i	$0.856055\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−9.00000			−0.353827			−0.176913	−	0.984226i	$-0.556611\pi$
$647$	−9.00000			−0.353827			−0.176913	+	0.984226i	$0.556611\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−1.50000	+	2.59808i	−0.0586995	+	0.101671i	−0.893882	−	0.448303i	$-0.852029\pi$
$653$	−1.50000	+	2.59808i	−0.0586995	+	0.101671i	0.835182	+	0.549973i	$0.185362\pi$
$654$		0			0
$655$	−7.00000	−	12.1244i	−0.273513	−	0.473738i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	23.0000	+	39.8372i	0.895953	+	1.55184i	0.832621	+	0.553843i	$0.186839\pi$
$659$	23.0000	+	39.8372i	0.895953	+	1.55184i	0.0633318	+	0.997993i	$0.479827\pi$
$660$		0			0
$661$	1.00000	−	1.73205i	0.0388955	−	0.0673690i	−0.845922	−	0.533306i	$-0.820949\pi$
$661$	1.00000	−	1.73205i	0.0388955	−	0.0673690i	0.884818	+	0.465937i	$0.154283\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	14.0000			0.542897
$666$		0			0
$667$	42.0000			1.62625
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	11.0000	+	19.0526i	0.424019	+	0.734422i	0.996328	−	0.0856156i	$-0.0272857\pi$
$673$	11.0000	+	19.0526i	0.424019	+	0.734422i	−0.572309	+	0.820038i	$0.693952\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−9.00000	−	15.5885i	−0.345898	−	0.599113i	0.639618	−	0.768693i	$-0.279092\pi$
$677$	−9.00000	−	15.5885i	−0.345898	−	0.599113i	−0.985517	+	0.169580i	$0.945759\pi$
$678$		0			0
$679$	−16.0000	+	27.7128i	−0.614024	+	1.06352i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	9.00000			0.344375			0.172188	−	0.985064i	$-0.444916\pi$
$683$	9.00000			0.344375			0.172188	+	0.985064i	$0.444916\pi$
$684$		0			0
$685$	3.00000			0.114624
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−15.0000	+	25.9808i	−0.571454	+	0.989788i
$690$		0			0
$691$	−0.500000	−	0.866025i	−0.0190209	−	0.0329452i	0.856358	−	0.516382i	$-0.172722\pi$
$691$	−0.500000	−	0.866025i	−0.0190209	−	0.0329452i	−0.875379	+	0.483437i	$0.839388\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	6.00000	+	10.3923i	0.227593	+	0.394203i
$696$		0			0
$697$	−14.0000	+	24.2487i	−0.530288	+	0.918485i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−38.0000			−1.43524			−0.717620	−	0.696435i	$-0.754769\pi$
$701$	−38.0000			−1.43524			−0.717620	+	0.696435i	$0.754769\pi$
$702$		0			0
$703$	−42.0000			−1.58406
$704$		0			0
$705$		0			0
$706$		0			0
$707$	4.00000	−	6.92820i	0.150435	−	0.260562i
$708$		0			0
$709$	7.00000	+	12.1244i	0.262891	+	0.455340i	0.967009	−	0.254743i	$-0.0819909\pi$
$709$	7.00000	+	12.1244i	0.262891	+	0.455340i	−0.704118	+	0.710083i	$0.748658\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−10.5000	−	18.1865i	−0.393228	−	0.681091i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−30.0000			−1.11881			−0.559406	−	0.828894i	$-0.688971\pi$
$719$	−30.0000			−1.11881			−0.559406	+	0.828894i	$0.688971\pi$
$720$		0			0
$721$	−8.00000			−0.297936
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−3.00000	+	5.19615i	−0.111417	+	0.192980i
$726$		0			0
$727$	−4.00000	−	6.92820i	−0.148352	−	0.256953i	0.782267	−	0.622944i	$-0.214063\pi$
$727$	−4.00000	−	6.92820i	−0.148352	−	0.256953i	−0.930618	+	0.365991i	$0.880730\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−28.0000	−	48.4974i	−1.03562	−	1.79374i
$732$		0			0
$733$	8.00000	−	13.8564i	0.295487	−	0.511798i	−0.679611	−	0.733572i	$-0.737852\pi$
$733$	8.00000	−	13.8564i	0.295487	−	0.511798i	0.975098	+	0.221774i	$0.0711849\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−35.0000			−1.28750			−0.643748	−	0.765238i	$-0.722621\pi$
$739$	−35.0000			−1.28750			−0.643748	+	0.765238i	$0.722621\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$743$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$744$		0			0
$745$	6.00000	+	10.3923i	0.219823	+	0.380745i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	4.00000	+	6.92820i	0.146157	+	0.253151i
$750$		0			0
$751$	7.50000	−	12.9904i	0.273679	−	0.474026i	−0.696122	−	0.717923i	$-0.745093\pi$
$751$	7.50000	−	12.9904i	0.273679	−	0.474026i	0.969801	+	0.243898i	$0.0784261\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−6.00000			−0.218074			−0.109037	−	0.994038i	$-0.534777\pi$
$757$	−6.00000			−0.218074			−0.109037	+	0.994038i	$0.534777\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	11.0000	−	19.0526i	0.398750	−	0.690655i	−0.594822	−	0.803857i	$-0.702778\pi$
$761$	11.0000	−	19.0526i	0.398750	−	0.690655i	0.993572	+	0.113203i	$0.0361109\pi$
$762$		0			0
$763$	5.00000	+	8.66025i	0.181012	+	0.313522i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	18.0000	+	31.1769i	0.649942	+	1.12573i
$768$		0			0
$769$	13.5000	−	23.3827i	0.486822	−	0.843201i	−0.513063	−	0.858351i	$-0.671489\pi$
$769$	13.5000	−	23.3827i	0.486822	−	0.843201i	0.999885	+	0.0151499i	$0.00482254\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−5.00000			−0.179838			−0.0899188	−	0.995949i	$-0.528661\pi$
$773$	−5.00000			−0.179838			−0.0899188	+	0.995949i	$0.528661\pi$
$774$		0			0
$775$	3.00000			0.107763
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−14.0000	+	24.2487i	−0.501602	+	0.868800i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−10.0000	−	17.3205i	−0.356915	−	0.618195i
$786$		0			0
$787$	6.00000	−	10.3923i	0.213877	−	0.370446i	−0.739048	−	0.673653i	$-0.764724\pi$
$787$	6.00000	−	10.3923i	0.213877	−	0.370446i	0.952925	+	0.303207i	$0.0980575\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	12.0000			0.426671
$792$		0			0
$793$	18.0000			0.639199
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−12.5000	+	21.6506i	−0.442773	+	0.766905i	−0.997894	−	0.0648645i	$-0.979338\pi$
$797$	−12.5000	+	21.6506i	−0.442773	+	0.766905i	0.555121	+

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:	\( N \)	\(=\)	\( 3240 = 2^{3} \cdot 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3240.q (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.500000 - 0.866025i) q^{5} +(1.00000 + 1.73205i) q^{7} +O(q^{10})\)	\(q+(0.500000 - 0.866025i) q^{5} +(1.00000 + 1.73205i) q^{7} +(3.00000 - 5.19615i) q^{13} +7.00000 q^{17} +7.00000 q^{19} +(-3.50000 + 6.06218i) q^{23} +(-0.500000 - 0.866025i) q^{25} +(-3.00000 - 5.19615i) q^{29} +(-1.50000 + 2.59808i) q^{31} +2.00000 q^{35} -6.00000 q^{37} +(-2.00000 + 3.46410i) q^{41} +(-4.00000 - 6.92820i) q^{43} +(2.00000 + 3.46410i) q^{47} +(1.50000 - 2.59808i) q^{49} -5.00000 q^{53} +(-3.00000 + 5.19615i) q^{59} +(1.50000 + 2.59808i) q^{61} +(-3.00000 - 5.19615i) q^{65} +(5.00000 - 8.66025i) q^{67} +12.0000 q^{71} +16.0000 q^{73} +(-0.500000 - 0.866025i) q^{79} +(-4.50000 - 7.79423i) q^{83} +(3.50000 - 6.06218i) q^{85} -4.00000 q^{89} +12.0000 q^{91} +(3.50000 - 6.06218i) q^{95} +(8.00000 + 13.8564i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{5} + 2 q^{7}+O(q^{10}) \) 2 * q + q^5 + 2 * q^7	\( 2 q + q^{5} + 2 q^{7} + 6 q^{13} + 14 q^{17} + 14 q^{19} - 7 q^{23} - q^{25} - 6 q^{29} - 3 q^{31} + 4 q^{35} - 12 q^{37} - 4 q^{41} - 8 q^{43} + 4 q^{47} + 3 q^{49} - 10 q^{53} - 6 q^{59} + 3 q^{61} - 6 q^{65} + 10 q^{67} + 24 q^{71} + 32 q^{73} - q^{79} - 9 q^{83} + 7 q^{85} - 8 q^{89} + 24 q^{91} + 7 q^{95} + 16 q^{97}+O(q^{100}) \) 2 * q + q^5 + 2 * q^7 + 6 * q^13 + 14 * q^17 + 14 * q^19 - 7 * q^23 - q^25 - 6 * q^29 - 3 * q^31 + 4 * q^35 - 12 * q^37 - 4 * q^41 - 8 * q^43 + 4 * q^47 + 3 * q^49 - 10 * q^53 - 6 * q^59 + 3 * q^61 - 6 * q^65 + 10 * q^67 + 24 * q^71 + 32 * q^73 - q^79 - 9 * q^83 + 7 * q^85 - 8 * q^89 + 24 * q^91 + 7 * q^95 + 16 * q^97

Introduction

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