LMFDB - Embedded newform 3240.2.q.bd.1081.2

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:	$4$
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:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			1081.2
Root			$1.68614 + 0.396143i$ of defining polynomial
Character	$\chi$	$=$	3240.1081
Dual form			3240.2.q.bd.2161.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.500000 - 0.866025i) q^{5} +(1.18614 + 2.05446i) q^{7} +O(q^{10})$	$q+(0.500000 - 0.866025i) q^{5} +(1.18614 + 2.05446i) q^{7} +(-1.68614 - 2.92048i) q^{11} +(-1.18614 + 2.05446i) q^{13} -4.74456 q^{17} +1.00000 q^{19} +(-0.186141 + 0.322405i) q^{23} +(-0.500000 - 0.866025i) q^{25} +(-1.68614 - 2.92048i) q^{29} +(3.05842 - 5.29734i) q^{31} +2.37228 q^{35} +6.00000 q^{37} +(5.87228 - 10.1711i) q^{41} +(-3.37228 - 5.84096i) q^{43} +(-1.81386 - 3.14170i) q^{47} +(0.686141 - 1.18843i) q^{49} +7.11684 q^{53} -3.37228 q^{55} +(2.50000 - 4.33013i) q^{59} +(-0.627719 - 1.08724i) q^{61} +(1.18614 + 2.05446i) q^{65} +(-5.37228 + 9.30506i) q^{67} -1.37228 q^{71} +3.25544 q^{73} +(4.00000 - 6.92820i) q^{77} +(-4.37228 - 7.57301i) q^{79} +(-5.00000 - 8.66025i) q^{83} +(-2.37228 + 4.10891i) q^{85} -1.37228 q^{89} -5.62772 q^{91} +(0.500000 - 0.866025i) q^{95} +(-3.37228 - 5.84096i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{5} - q^{7}+O(q^{10}) $ 4 * q + 2 * q^5 - q^7	$ 4 q + 2 q^{5} - q^{7} - q^{11} + q^{13} + 4 q^{17} + 4 q^{19} + 5 q^{23} - 2 q^{25} - q^{29} - 5 q^{31} - 2 q^{35} + 24 q^{37} + 12 q^{41} - 2 q^{43} - 13 q^{47} - 3 q^{49} - 6 q^{53} - 2 q^{55} + 10 q^{59} - 14 q^{61} - q^{65} - 10 q^{67} + 6 q^{71} + 36 q^{73} + 16 q^{77} - 6 q^{79} - 20 q^{83} + 2 q^{85} + 6 q^{89} - 34 q^{91} + 2 q^{95} - 2 q^{97}+O(q^{100}) $ 4 * q + 2 * q^5 - q^7 - q^11 + q^13 + 4 * q^17 + 4 * q^19 + 5 * q^23 - 2 * q^25 - q^29 - 5 * q^31 - 2 * q^35 + 24 * q^37 + 12 * q^41 - 2 * q^43 - 13 * q^47 - 3 * q^49 - 6 * q^53 - 2 * q^55 + 10 * q^59 - 14 * q^61 - q^65 - 10 * q^67 + 6 * q^71 + 36 * q^73 + 16 * q^77 - 6 * q^79 - 20 * q^83 + 2 * q^85 + 6 * q^89 - 34 * q^91 + 2 * q^95 - 2 * 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
$7$	1.18614	+	2.05446i	0.448319	+	0.776511i	0.998277	−	0.0586811i	$-0.0186895\pi$
$7$	1.18614	+	2.05446i	0.448319	+	0.776511i	−0.549958	+	0.835192i	$0.685356\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−1.68614	−	2.92048i	−0.508391	−	0.880558i	−0.999953	−	0.00971581i	$-0.996907\pi$
$11$	−1.68614	−	2.92048i	−0.508391	−	0.880558i	0.491562	−	0.870842i	$-0.336426\pi$
$12$		0			0
$13$	−1.18614	+	2.05446i	−0.328976	+	0.569804i	−0.982309	−	0.187267i	$-0.940037\pi$
$13$	−1.18614	+	2.05446i	−0.328976	+	0.569804i	0.653333	+	0.757071i	$0.273370\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−4.74456			−1.15073			−0.575363	−	0.817898i	$-0.695139\pi$
$17$	−4.74456			−1.15073			−0.575363	+	0.817898i	$0.695139\pi$
$18$		0			0
$19$	1.00000			0.229416			0.114708	−	0.993399i	$-0.463407\pi$
$19$	1.00000			0.229416			0.114708	+	0.993399i	$0.463407\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−0.186141	+	0.322405i	−0.0388130	+	0.0672261i	−0.884779	−	0.466010i	$-0.845691\pi$
$23$	−0.186141	+	0.322405i	−0.0388130	+	0.0672261i	0.845966	+	0.533236i	$0.179024\pi$
$24$		0			0
$25$	−0.500000	−	0.866025i	−0.100000	−	0.173205i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−1.68614	−	2.92048i	−0.313108	−	0.542320i	0.665925	−	0.746019i	$-0.268037\pi$
$29$	−1.68614	−	2.92048i	−0.313108	−	0.542320i	−0.979034	+	0.203699i	$0.934704\pi$
$30$		0			0
$31$	3.05842	−	5.29734i	0.549309	−	0.951431i	−0.449013	−	0.893525i	$-0.648224\pi$
$31$	3.05842	−	5.29734i	0.549309	−	0.951431i	0.998322	−	0.0579057i	$-0.0184423\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	2.37228			0.400989
$36$		0			0
$37$	6.00000			0.986394			0.493197	−	0.869918i	$-0.335828\pi$
$37$	6.00000			0.986394			0.493197	+	0.869918i	$0.335828\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	5.87228	−	10.1711i	0.917096	−	1.58846i	0.113293	−	0.993562i	$-0.463860\pi$
$41$	5.87228	−	10.1711i	0.917096	−	1.58846i	0.803803	−	0.594896i	$-0.202807\pi$
$42$		0			0
$43$	−3.37228	−	5.84096i	−0.514268	−	0.890738i	−0.999863	−	0.0165545i	$-0.994730\pi$
$43$	−3.37228	−	5.84096i	−0.514268	−	0.890738i	0.485595	−	0.874184i	$-0.338603\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−1.81386	−	3.14170i	−0.264579	−	0.458264i	0.702875	−	0.711314i	$-0.251900\pi$
$47$	−1.81386	−	3.14170i	−0.264579	−	0.458264i	−0.967453	+	0.253050i	$0.918566\pi$
$48$		0			0
$49$	0.686141	−	1.18843i	0.0980201	−	0.169776i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	7.11684			0.977574			0.488787	−	0.872403i	$-0.337440\pi$
$53$	7.11684			0.977574			0.488787	+	0.872403i	$0.337440\pi$
$54$		0			0
$55$	−3.37228			−0.454718
$56$		0			0
$57$		0			0
$58$		0			0
$59$	2.50000	−	4.33013i	0.325472	−	0.563735i	−0.656136	−	0.754643i	$-0.727810\pi$
$59$	2.50000	−	4.33013i	0.325472	−	0.563735i	0.981608	+	0.190909i	$0.0611434\pi$
$60$		0			0
$61$	−0.627719	−	1.08724i	−0.0803711	−	0.139207i	0.823038	−	0.567986i	$-0.192277\pi$
$61$	−0.627719	−	1.08724i	−0.0803711	−	0.139207i	−0.903409	+	0.428779i	$0.858944\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	1.18614	+	2.05446i	0.147123	+	0.254824i
$66$		0			0
$67$	−5.37228	+	9.30506i	−0.656329	+	1.13679i	0.325230	+	0.945635i	$0.394558\pi$
$67$	−5.37228	+	9.30506i	−0.656329	+	1.13679i	−0.981559	+	0.191160i	$0.938775\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−1.37228			−0.162860			−0.0814299	−	0.996679i	$-0.525949\pi$
$71$	−1.37228			−0.162860			−0.0814299	+	0.996679i	$0.525949\pi$
$72$		0			0
$73$	3.25544			0.381020			0.190510	−	0.981685i	$-0.438986\pi$
$73$	3.25544			0.381020			0.190510	+	0.981685i	$0.438986\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	4.00000	−	6.92820i	0.455842	−	0.789542i
$78$		0			0
$79$	−4.37228	−	7.57301i	−0.491920	−	0.852031i	0.508037	−	0.861335i	$-0.330371\pi$
$79$	−4.37228	−	7.57301i	−0.491920	−	0.852031i	−0.999957	+	0.00930489i	$0.997038\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−5.00000	−	8.66025i	−0.548821	−	0.950586i	−0.998356	−	0.0573233i	$-0.981743\pi$
$83$	−5.00000	−	8.66025i	−0.548821	−	0.950586i	0.449534	−	0.893263i	$-0.351590\pi$
$84$		0			0
$85$	−2.37228	+	4.10891i	−0.257310	+	0.445674i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−1.37228			−0.145462			−0.0727308	−	0.997352i	$-0.523171\pi$
$89$	−1.37228			−0.145462			−0.0727308	+	0.997352i	$0.523171\pi$
$90$		0			0
$91$	−5.62772			−0.589945
$92$		0			0
$93$		0			0
$94$		0			0
$95$	0.500000	−	0.866025i	0.0512989	−	0.0888523i
$96$		0			0
$97$	−3.37228	−	5.84096i	−0.342403	−	0.593060i	0.642475	−	0.766306i	$-0.277908\pi$
$97$	−3.37228	−	5.84096i	−0.342403	−	0.593060i	−0.984878	+	0.173247i	$0.944574\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	6.43070	+	11.1383i	0.639879	+	1.10830i	0.985459	+	0.169913i	$0.0543488\pi$
$101$	6.43070	+	11.1383i	0.639879	+	1.10830i	−0.345580	+	0.938389i	$0.612318\pi$
$102$		0			0
$103$	−8.55842	+	14.8236i	−0.843286	+	1.46061i	0.0438147	+	0.999040i	$0.486049\pi$
$103$	−8.55842	+	14.8236i	−0.843286	+	1.46061i	−0.887101	+	0.461575i	$0.847284\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	6.00000			0.580042			0.290021	−	0.957020i	$-0.406338\pi$
$107$	6.00000			0.580042			0.290021	+	0.957020i	$0.406338\pi$
$108$		0			0
$109$	18.1168			1.73528			0.867639	−	0.497194i	$-0.165636\pi$
$109$	18.1168			1.73528			0.867639	+	0.497194i	$0.165636\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	1.62772	−	2.81929i	0.153123	−	0.265217i	−0.779251	−	0.626712i	$-0.784400\pi$
$113$	1.62772	−	2.81929i	0.153123	−	0.265217i	0.932374	+	0.361495i	$0.117734\pi$
$114$		0			0
$115$	0.186141	+	0.322405i	0.0173577	+	0.0300644i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−5.62772	−	9.74749i	−0.515892	−	0.893551i
$120$		0			0
$121$	−0.186141	+	0.322405i	−0.0169219	+	0.0293096i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−1.00000			−0.0894427
$126$		0			0
$127$	15.8614			1.40747			0.703736	−	0.710461i	$-0.251514\pi$
$127$	15.8614			1.40747			0.703736	+	0.710461i	$0.251514\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	8.87228	−	15.3672i	0.775175	−	1.34264i	−0.159521	−	0.987194i	$-0.550995\pi$
$131$	8.87228	−	15.3672i	0.775175	−	1.34264i	0.934696	−	0.355448i	$-0.115672\pi$
$132$		0			0
$133$	1.18614	+	2.05446i	0.102851	+	0.178144i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−8.37228	−	14.5012i	−0.715292	−	1.23892i	−0.962847	−	0.270049i	$-0.912960\pi$
$137$	−8.37228	−	14.5012i	−0.715292	−	1.23892i	0.247554	−	0.968874i	$-0.420373\pi$
$138$		0			0
$139$	−7.50000	+	12.9904i	−0.636142	+	1.10183i	0.350130	+	0.936701i	$0.386137\pi$
$139$	−7.50000	+	12.9904i	−0.636142	+	1.10183i	−0.986272	+	0.165129i	$0.947196\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	8.00000			0.668994
$144$		0			0
$145$	−3.37228			−0.280053
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−1.74456	+	3.02167i	−0.142920	+	0.247545i	−0.928595	−	0.371095i	$-0.878983\pi$
$149$	−1.74456	+	3.02167i	−0.142920	+	0.247545i	0.785675	+	0.618640i	$0.212316\pi$
$150$		0			0
$151$	−9.31386	−	16.1321i	−0.757951	−	1.31281i	−0.943894	−	0.330249i	$-0.892867\pi$
$151$	−9.31386	−	16.1321i	−0.757951	−	1.31281i	0.185943	−	0.982561i	$-0.440466\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−3.05842	−	5.29734i	−0.245658	−	0.425493i
$156$		0			0
$157$	2.93070	−	5.07613i	0.233896	−	0.405119i	−0.725056	−	0.688690i	$-0.758186\pi$
$157$	2.93070	−	5.07613i	0.233896	−	0.405119i	0.958951	+	0.283571i	$0.0915193\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−0.883156			−0.0696024
$162$		0			0
$163$	−13.4891			−1.05655			−0.528275	−	0.849073i	$-0.677161\pi$
$163$	−13.4891			−1.05655			−0.528275	+	0.849073i	$0.677161\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−6.00000	+	10.3923i	−0.464294	+	0.804181i	−0.999169	−	0.0407502i	$-0.987025\pi$
$167$	−6.00000	+	10.3923i	−0.464294	+	0.804181i	0.534875	+	0.844931i	$0.320359\pi$
$168$		0			0
$169$	3.68614	+	6.38458i	0.283549	+	0.491122i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−6.93070	−	12.0043i	−0.526932	−	0.912672i	−0.999507	−	0.0313823i	$-0.990009\pi$
$173$	−6.93070	−	12.0043i	−0.526932	−	0.912672i	0.472576	−	0.881290i	$-0.343324\pi$
$174$		0			0
$175$	1.18614	−	2.05446i	0.0896638	−	0.155302i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	15.7446			1.17680			0.588402	−	0.808569i	$-0.299757\pi$
$179$	15.7446			1.17680			0.588402	+	0.808569i	$0.299757\pi$
$180$		0			0
$181$	14.8614			1.10464			0.552320	−	0.833632i	$-0.313743\pi$
$181$	14.8614			1.10464			0.552320	+	0.833632i	$0.313743\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	3.00000	−	5.19615i	0.220564	−	0.382029i
$186$		0			0
$187$	8.00000	+	13.8564i	0.585018	+	1.01328i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−5.68614	−	9.84868i	−0.411435	−	0.712626i	0.583612	−	0.812033i	$-0.301639\pi$
$191$	−5.68614	−	9.84868i	−0.411435	−	0.712626i	−0.995047	+	0.0994067i	$0.968306\pi$
$192$		0			0
$193$	5.37228	−	9.30506i	0.386705	−	0.669793i	−0.605299	−	0.795998i	$-0.706946\pi$
$193$	5.37228	−	9.30506i	0.386705	−	0.669793i	0.992004	+	0.126205i	$0.0402797\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	6.37228			0.454006			0.227003	−	0.973894i	$-0.427107\pi$
$197$	6.37228			0.454006			0.227003	+	0.973894i	$0.427107\pi$
$198$		0			0
$199$	−9.48913			−0.672666			−0.336333	−	0.941743i	$-0.609187\pi$
$199$	−9.48913			−0.672666			−0.336333	+	0.941743i	$0.609187\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	4.00000	−	6.92820i	0.280745	−	0.486265i
$204$		0			0
$205$	−5.87228	−	10.1711i	−0.410138	−	0.710380i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−1.68614	−	2.92048i	−0.116633	−	0.202014i
$210$		0			0
$211$	−8.50000	+	14.7224i	−0.585164	+	1.01353i	0.409691	+	0.912224i	$0.365637\pi$
$211$	−8.50000	+	14.7224i	−0.585164	+	1.01353i	−0.994855	+	0.101310i	$0.967697\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−6.74456			−0.459975
$216$		0			0
$217$	14.5109			0.985062
$218$		0			0
$219$		0			0
$220$		0			0
$221$	5.62772	−	9.74749i	0.378561	−	0.655687i
$222$		0			0
$223$	−1.25544	−	2.17448i	−0.0840703	−	0.145614i	0.820924	−	0.571037i	$-0.193459\pi$
$223$	−1.25544	−	2.17448i	−0.0840703	−	0.145614i	−0.904995	+	0.425423i	$0.860125\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	2.74456	+	4.75372i	0.182163	+	0.315516i	0.942617	−	0.333876i	$-0.108357\pi$
$227$	2.74456	+	4.75372i	0.182163	+	0.315516i	−0.760454	+	0.649392i	$0.775023\pi$
$228$		0			0
$229$	2.62772	−	4.55134i	0.173645	−	0.300761i	−0.766047	−	0.642785i	$-0.777779\pi$
$229$	2.62772	−	4.55134i	0.173645	−	0.300761i	0.939691	+	0.342024i	$0.111112\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−20.2337			−1.32555			−0.662776	−	0.748817i	$-0.730622\pi$
$233$	−20.2337			−1.32555			−0.662776	+	0.748817i	$0.730622\pi$
$234$		0			0
$235$	−3.62772			−0.236646
$236$		0			0
$237$		0			0
$238$		0			0
$239$	7.11684	−	12.3267i	0.460350	−	0.797350i	−0.538628	−	0.842544i	$-0.681057\pi$
$239$	7.11684	−	12.3267i	0.460350	−	0.797350i	0.998978	+	0.0451935i	$0.0143905\pi$
$240$		0			0
$241$	−14.8030	−	25.6395i	−0.953544	−	1.65159i	−0.737665	−	0.675167i	$-0.764072\pi$
$241$	−14.8030	−	25.6395i	−0.953544	−	1.65159i	−0.215879	−	0.976420i	$-0.569262\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−0.686141	−	1.18843i	−0.0438359	−	0.0759260i
$246$		0			0
$247$	−1.18614	+	2.05446i	−0.0754723	+	0.130722i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−15.8614			−1.00116			−0.500582	−	0.865689i	$-0.666880\pi$
$251$	−15.8614			−1.00116			−0.500582	+	0.865689i	$0.666880\pi$
$252$		0			0
$253$	1.25544			0.0789287
$254$		0			0
$255$		0			0
$256$		0			0
$257$	11.7446	−	20.3422i	0.732606	−	1.26891i	−0.223160	−	0.974782i	$-0.571637\pi$
$257$	11.7446	−	20.3422i	0.732606	−	1.26891i	0.955766	−	0.294128i	$-0.0950294\pi$
$258$		0			0
$259$	7.11684	+	12.3267i	0.442219	+	0.765946i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−6.55842	−	11.3595i	−0.404410	−	0.700458i	0.589843	−	0.807518i	$-0.299190\pi$
$263$	−6.55842	−	11.3595i	−0.404410	−	0.700458i	−0.994253	+	0.107060i	$0.965856\pi$
$264$		0			0
$265$	3.55842	−	6.16337i	0.218592	−	0.378613i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−12.1168			−0.738777			−0.369389	−	0.929275i	$-0.620433\pi$
$269$	−12.1168			−0.738777			−0.369389	+	0.929275i	$0.620433\pi$
$270$		0			0
$271$	−20.7446			−1.26014			−0.630071	−	0.776537i	$-0.716974\pi$
$271$	−20.7446			−1.26014			−0.630071	+	0.776537i	$0.716974\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−1.68614	+	2.92048i	−0.101678	+	0.176112i
$276$		0			0
$277$	8.55842	+	14.8236i	0.514226	+	0.890665i	0.999864	+	0.0165051i	$0.00525398\pi$
$277$	8.55842	+	14.8236i	0.514226	+	0.890665i	−0.485638	+	0.874160i	$0.661413\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	5.81386	+	10.0699i	0.346826	+	0.600720i	0.985684	−	0.168605i	$-0.0539262\pi$
$281$	5.81386	+	10.0699i	0.346826	+	0.600720i	−0.638858	+	0.769325i	$0.720593\pi$
$282$		0			0
$283$	−2.62772	+	4.55134i	−0.156202	+	0.270549i	−0.933496	−	0.358588i	$-0.883258\pi$
$283$	−2.62772	+	4.55134i	−0.156202	+	0.270549i	0.777294	+	0.629137i	$0.216592\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	27.8614			1.64461
$288$		0			0
$289$	5.51087			0.324169
$290$		0			0
$291$		0			0
$292$		0			0
$293$	6.93070	−	12.0043i	0.404896	−	0.701300i	−0.589413	−	0.807832i	$-0.700641\pi$
$293$	6.93070	−	12.0043i	0.404896	−	0.701300i	0.994309	+	0.106531i	$0.0339744\pi$
$294$		0			0
$295$	−2.50000	−	4.33013i	−0.145556	−	0.252110i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−0.441578	−	0.764836i	−0.0255371	−	0.0442316i
$300$		0			0
$301$	8.00000	−	13.8564i	0.461112	−	0.798670i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−1.25544			−0.0718861
$306$		0			0
$307$	−10.7446			−0.613225			−0.306612	−	0.951834i	$-0.599195\pi$
$307$	−10.7446			−0.613225			−0.306612	+	0.951834i	$0.599195\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−5.43070	+	9.40625i	−0.307947	+	0.533380i	−0.977913	−	0.209012i	$-0.932975\pi$
$311$	−5.43070	+	9.40625i	−0.307947	+	0.533380i	0.669966	+	0.742392i	$0.266309\pi$
$312$		0			0
$313$	2.00000	+	3.46410i	0.113047	+	0.195803i	0.916997	−	0.398894i	$-0.130606\pi$
$313$	2.00000	+	3.46410i	0.113047	+	0.195803i	−0.803951	+	0.594696i	$0.797272\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−8.18614	−	14.1788i	−0.459779	−	0.796361i	0.539170	−	0.842197i	$-0.318738\pi$
$317$	−8.18614	−	14.1788i	−0.459779	−	0.796361i	−0.998949	+	0.0458359i	$0.985405\pi$
$318$		0			0
$319$	−5.68614	+	9.84868i	−0.318363	+	0.551420i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−4.74456			−0.263995
$324$		0			0
$325$	2.37228			0.131590
$326$		0			0
$327$		0			0
$328$		0			0
$329$	4.30298	−	7.45299i	0.237231	−	0.410897i
$330$		0			0
$331$	−9.68614	−	16.7769i	−0.532398	−	0.922141i	−0.999284	−	0.0378236i	$-0.987957\pi$
$331$	−9.68614	−	16.7769i	−0.532398	−	0.922141i	0.466886	−	0.884318i	$-0.345376\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	5.37228	+	9.30506i	0.293519	+	0.508390i
$336$		0			0
$337$	−0.372281	+	0.644810i	−0.0202795	+	0.0351250i	−0.875987	−	0.482335i	$-0.839789\pi$
$337$	−0.372281	+	0.644810i	−0.0202795	+	0.0351250i	0.855708	+	0.517460i	$0.173122\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−20.6277			−1.11705
$342$		0			0
$343$	19.8614			1.07242
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−15.1168	+	26.1831i	−0.811515	+	1.40558i	0.100289	+	0.994958i	$0.468023\pi$
$347$	−15.1168	+	26.1831i	−0.811515	+	1.40558i	−0.911804	+	0.410626i	$0.865310\pi$
$348$		0			0
$349$	12.6861	+	21.9730i	0.679074	+	1.17619i	0.975260	+	0.221060i	$0.0709516\pi$
$349$	12.6861	+	21.9730i	0.679074	+	1.17619i	−0.296187	+	0.955130i	$0.595715\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	12.4891	+	21.6318i	0.664729	+	1.15134i	0.979359	+	0.202130i	$0.0647864\pi$
$353$	12.4891	+	21.6318i	0.664729	+	1.15134i	−0.314630	+	0.949215i	$0.601880\pi$
$354$		0			0
$355$	−0.686141	+	1.18843i	−0.0364166	+	0.0630753i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	10.6277			0.560910			0.280455	−	0.959867i	$-0.409515\pi$
$359$	10.6277			0.560910			0.280455	+	0.959867i	$0.409515\pi$
$360$		0			0
$361$	−18.0000			−0.947368
$362$		0			0
$363$		0			0
$364$		0			0
$365$	1.62772	−	2.81929i	0.0851987	−	0.147568i
$366$		0			0
$367$	−2.11684	−	3.66648i	−0.110498	−	0.191389i	0.805473	−	0.592633i	$-0.201911\pi$
$367$	−2.11684	−	3.66648i	−0.110498	−	0.191389i	−0.915971	+	0.401244i	$0.868578\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	8.44158	+	14.6212i	0.438265	+	0.759097i
$372$		0			0
$373$	−0.372281	+	0.644810i	−0.0192760	+	0.0333870i	−0.875502	−	0.483214i	$-0.839469\pi$
$373$	−0.372281	+	0.644810i	−0.0192760	+	0.0333870i	0.856226	+	0.516601i	$0.172803\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	8.00000			0.412021
$378$		0			0
$379$	25.3505			1.30217			0.651085	−	0.759005i	$-0.274314\pi$
$379$	25.3505			1.30217			0.651085	+	0.759005i	$0.274314\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−6.81386	+	11.8020i	−0.348172	+	0.603052i	−0.985925	−	0.167190i	$-0.946531\pi$
$383$	−6.81386	+	11.8020i	−0.348172	+	0.603052i	0.637753	+	0.770241i	$0.279864\pi$
$384$		0			0
$385$	−4.00000	−	6.92820i	−0.203859	−	0.353094i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	4.62772	+	8.01544i	0.234635	+	0.406399i	0.959166	−	0.282842i	$-0.0912773\pi$
$389$	4.62772	+	8.01544i	0.234635	+	0.406399i	−0.724532	+	0.689241i	$0.757944\pi$
$390$		0			0
$391$	0.883156	−	1.52967i	0.0446631	−	0.0773588i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−8.74456			−0.439987
$396$		0			0
$397$	34.0000			1.70641			0.853206	−	0.521575i	$-0.174655\pi$
$397$	34.0000			1.70641			0.853206	+	0.521575i	$0.174655\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−7.67527	+	13.2940i	−0.383284	+	0.663868i	−0.991530	−	0.129881i	$-0.958540\pi$
$401$	−7.67527	+	13.2940i	−0.383284	+	0.663868i	0.608245	+	0.793749i	$0.291874\pi$
$402$		0			0
$403$	7.25544	+	12.5668i	0.361419	+	0.625996i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−10.1168	−	17.5229i	−0.501473	−	0.868577i
$408$		0			0
$409$	7.81386	−	13.5340i	0.386370	−	0.669213i	−0.605588	−	0.795778i	$-0.707062\pi$
$409$	7.81386	−	13.5340i	0.386370	−	0.669213i	0.991958	+	0.126565i	$0.0403953\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	11.8614			0.583662
$414$		0			0
$415$	−10.0000			−0.490881
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−5.48913	+	9.50744i	−0.268161	+	0.464469i	−0.968387	−	0.249452i	$-0.919749\pi$
$419$	−5.48913	+	9.50744i	−0.268161	+	0.464469i	0.700226	+	0.713922i	$0.253083\pi$
$420$		0			0
$421$	−11.3139	−	19.5962i	−0.551404	−	0.955059i	−0.998174	−	0.0604104i	$-0.980759\pi$
$421$	−11.3139	−	19.5962i	−0.551404	−	0.955059i	0.446770	−	0.894649i	$-0.352574\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	2.37228	+	4.10891i	0.115073	+	0.199311i
$426$		0			0
$427$	1.48913	−	2.57924i	0.0720638	−	0.124818i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−15.3723			−0.740457			−0.370228	−	0.928941i	$-0.620721\pi$
$431$	−15.3723			−0.740457			−0.370228	+	0.928941i	$0.620721\pi$
$432$		0			0
$433$	39.2119			1.88441			0.942203	−	0.335043i	$-0.108751\pi$
$433$	39.2119			1.88441			0.942203	+	0.335043i	$0.108751\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−0.186141	+	0.322405i	−0.00890432	+	0.0154227i
$438$		0			0
$439$	−15.5475	−	26.9291i	−0.742044	−	1.28526i	−0.951563	−	0.307453i	$-0.900523\pi$
$439$	−15.5475	−	26.9291i	−0.742044	−	1.28526i	0.209519	−	0.977805i	$-0.432810\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	8.48913	+	14.7036i	0.403331	+	0.698589i	0.994126	−	0.108233i	$-0.0345192\pi$
$443$	8.48913	+	14.7036i	0.403331	+	0.698589i	−0.590795	+	0.806822i	$0.701186\pi$
$444$		0			0
$445$	−0.686141	+	1.18843i	−0.0325262	+	0.0563370i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	22.7228			1.07236			0.536178	−	0.844105i	$-0.319868\pi$
$449$	22.7228			1.07236			0.536178	+	0.844105i	$0.319868\pi$
$450$		0			0
$451$	−39.6060			−1.86497
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−2.81386	+	4.87375i	−0.131916	+	0.228485i
$456$		0			0
$457$	20.1168	+	34.8434i	0.941026	+	1.62991i	0.763519	+	0.645786i	$0.223470\pi$
$457$	20.1168	+	34.8434i	0.941026	+	1.62991i	0.177508	+	0.984119i	$0.443197\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	6.94158	+	12.0232i	0.323302	+	0.559975i	0.981167	−	0.193161i	$-0.0618739\pi$
$461$	6.94158	+	12.0232i	0.323302	+	0.559975i	−0.657866	+	0.753135i	$0.728541\pi$
$462$		0			0
$463$	10.9307	−	18.9325i	0.507993	−	0.879869i	−0.491964	−	0.870615i	$-0.663721\pi$
$463$	10.9307	−	18.9325i	0.507993	−	0.879869i	0.999957	−	0.00925409i	$-0.00294571\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	18.2337			0.843754			0.421877	−	0.906653i	$-0.361371\pi$
$467$	18.2337			0.843754			0.421877	+	0.906653i	$0.361371\pi$
$468$		0			0
$469$	−25.4891			−1.17698
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−11.3723	+	19.6974i	−0.522898	+	0.905686i
$474$		0			0
$475$	−0.500000	−	0.866025i	−0.0229416	−	0.0397360i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−12.0584	−	20.8858i	−0.550963	−	0.954297i	−0.998205	−	0.0598836i	$-0.980927\pi$
$479$	−12.0584	−	20.8858i	−0.550963	−	0.954297i	0.447242	−	0.894413i	$-0.352406\pi$
$480$		0			0
$481$	−7.11684	+	12.3267i	−0.324500	+	0.562051i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−6.74456			−0.306255
$486$		0			0
$487$	−20.6060			−0.933746			−0.466873	−	0.884324i	$-0.654619\pi$
$487$	−20.6060			−0.933746			−0.466873	+	0.884324i	$0.654619\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	9.98913	−	17.3017i	0.450803	−	0.780814i	−0.547633	−	0.836719i	$-0.684471\pi$
$491$	9.98913	−	17.3017i	0.450803	−	0.780814i	0.998436	+	0.0559050i	$0.0178044\pi$
$492$		0			0
$493$	8.00000	+	13.8564i	0.360302	+	0.624061i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−1.62772	−	2.81929i	−0.0730132	−	0.126463i
$498$		0			0
$499$	−4.12772	+	7.14942i	−0.184782	+	0.320052i	−0.943503	−	0.331364i	$-0.892491\pi$
$499$	−4.12772	+	7.14942i	−0.184782	+	0.320052i	0.758721	+	0.651416i	$0.225825\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	34.9783			1.55960			0.779802	−	0.626027i	$-0.215320\pi$
$503$	34.9783			1.55960			0.779802	+	0.626027i	$0.215320\pi$
$504$		0			0
$505$	12.8614			0.572325
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−19.0000	+	32.9090i	−0.842160	+	1.45866i	0.0459045	+	0.998946i	$0.485383\pi$
$509$	−19.0000	+	32.9090i	−0.842160	+	1.45866i	−0.888065	+	0.459718i	$0.847950\pi$
$510$		0			0
$511$	3.86141	+	6.68815i	0.170819	+	0.295866i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	8.55842	+	14.8236i	0.377129	+	0.653207i
$516$		0			0
$517$	−6.11684	+	10.5947i	−0.269018	+	0.465954i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−17.1168			−0.749903			−0.374951	−	0.927045i	$-0.622341\pi$
$521$	−17.1168			−0.749903			−0.374951	+	0.927045i	$0.622341\pi$
$522$		0			0
$523$	−5.76631			−0.252143			−0.126072	−	0.992021i	$-0.540237\pi$
$523$	−5.76631			−0.252143			−0.126072	+	0.992021i	$0.540237\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−14.5109	+	25.1336i	−0.632104	+	1.09484i
$528$		0			0
$529$	11.4307	+	19.7986i	0.496987	+	0.860807i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	13.9307	+	24.1287i	0.603406	+	1.04513i
$534$		0			0
$535$	3.00000	−	5.19615i	0.129701	−	0.224649i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−4.62772			−0.199330
$540$		0			0
$541$	−38.3505			−1.64882			−0.824409	−	0.565994i	$-0.808492\pi$
$541$	−38.3505			−1.64882			−0.824409	+	0.565994i	$0.808492\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	9.05842	−	15.6896i	0.388020	−	0.672071i
$546$		0			0
$547$	20.0000	+	34.6410i	0.855138	+	1.48114i	0.876517	+	0.481371i	$0.159861\pi$
$547$	20.0000	+	34.6410i	0.855138	+	1.48114i	−0.0213785	+	0.999771i	$0.506805\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−1.68614	−	2.92048i	−0.0718320	−	0.124417i
$552$		0			0
$553$	10.3723	−	17.9653i	0.441074	−	0.763963i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−8.60597			−0.364647			−0.182323	−	0.983239i	$-0.558362\pi$
$557$	−8.60597			−0.364647			−0.182323	+	0.983239i	$0.558362\pi$
$558$		0			0
$559$	16.0000			0.676728
$560$		0			0
$561$		0			0
$562$		0			0
$563$	12.1168	−	20.9870i	0.510664	−	0.884496i	−0.489260	−	0.872138i	$-0.662733\pi$
$563$	12.1168	−	20.9870i	0.510664	−	0.884496i	0.999924	−	0.0123579i	$-0.00393375\pi$
$564$		0			0
$565$	−1.62772	−	2.81929i	−0.0684786	−	0.118608i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	21.6168	+	37.4415i	0.906225	+	1.56963i	0.819264	+	0.573416i	$0.194382\pi$
$569$	21.6168	+	37.4415i	0.906225	+	1.56963i	0.0869612	+	0.996212i	$0.472284\pi$
$570$		0			0
$571$	−7.05842	+	12.2255i	−0.295386	+	0.511623i	−0.975075	−	0.221877i	$-0.928782\pi$
$571$	−7.05842	+	12.2255i	−0.295386	+	0.511623i	0.679689	+	0.733501i	$0.262115\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	0.372281			0.0155252
$576$		0			0
$577$	13.4891			0.561560			0.280780	−	0.959772i	$-0.409407\pi$
$577$	13.4891			0.561560			0.280780	+	0.959772i	$0.409407\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	11.8614	−	20.5446i	0.492094	−	0.852332i
$582$		0			0
$583$	−12.0000	−	20.7846i	−0.496989	−	0.860811i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	15.0000	+	25.9808i	0.619116	+	1.07234i	0.989647	+	0.143521i	$0.0458424\pi$
$587$	15.0000	+	25.9808i	0.619116	+	1.07234i	−0.370531	+	0.928820i	$0.620824\pi$
$588$		0			0
$589$	3.05842	−	5.29734i	0.126020	−	0.218273i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−16.9783			−0.697213			−0.348607	−	0.937269i	$-0.613345\pi$
$593$	−16.9783			−0.697213			−0.348607	+	0.937269i	$0.613345\pi$
$594$		0			0
$595$	−11.2554			−0.461428
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−18.1753	+	31.4805i	−0.742621	+	1.28626i	0.208677	+	0.977985i	$0.433084\pi$
$599$	−18.1753	+	31.4805i	−0.742621	+	1.28626i	−0.951298	+	0.308273i	$0.900249\pi$
$600$		0			0
$601$	−6.50000	−	11.2583i	−0.265141	−	0.459237i	0.702460	−	0.711723i	$-0.252085\pi$
$601$	−6.50000	−	11.2583i	−0.265141	−	0.459237i	−0.967600	+	0.252486i	$0.918752\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	0.186141	+	0.322405i	0.00756769	+	0.0131076i
$606$		0			0
$607$	16.6277	−	28.8001i	0.674898	−	1.16896i	−0.301600	−	0.953434i	$-0.597521\pi$
$607$	16.6277	−	28.8001i	0.674898	−	1.16896i	0.976499	−	0.215524i	$-0.0691458\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	8.60597			0.348160
$612$		0			0
$613$	−33.3505			−1.34702			−0.673508	−	0.739180i	$-0.735213\pi$
$613$	−33.3505			−1.34702			−0.673508	+	0.739180i	$0.735213\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−21.8614	+	37.8651i	−0.880107	+	1.52439i	−0.0288861	+	0.999583i	$0.509196\pi$
$617$	−21.8614	+	37.8651i	−0.880107	+	1.52439i	−0.851221	+	0.524807i	$0.824137\pi$
$618$		0			0
$619$	19.1861	+	33.2314i	0.771156	+	1.33568i	0.936930	+	0.349518i	$0.113655\pi$
$619$	19.1861	+	33.2314i	0.771156	+	1.33568i	−0.165774	+	0.986164i	$0.553012\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−1.62772	−	2.81929i	−0.0652132	−	0.112953i
$624$		0			0
$625$	−0.500000	+	0.866025i	−0.0200000	+	0.0346410i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−28.4674			−1.13507
$630$		0			0
$631$	−36.6277			−1.45813			−0.729063	−	0.684446i	$-0.760044\pi$
$631$	−36.6277			−1.45813			−0.729063	+	0.684446i	$0.760044\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	7.93070	−	13.7364i	0.314720	−	0.545112i
$636$		0			0
$637$	1.62772	+	2.81929i	0.0644926	+	0.111704i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	1.19702	+	2.07329i	0.0472793	+	0.0818901i	0.888697	−	0.458496i	$-0.151612\pi$
$641$	1.19702	+	2.07329i	0.0472793	+	0.0818901i	−0.841417	+	0.540386i	$0.818278\pi$
$642$		0			0
$643$	−10.6277	+	18.4077i	−0.419116	+	0.725931i	−0.995851	−	0.0910009i	$-0.970993\pi$
$643$	−10.6277	+	18.4077i	−0.419116	+	0.725931i	0.576734	+	0.816932i	$0.304327\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−25.7228			−1.01127			−0.505634	−	0.862748i	$-0.668741\pi$
$647$	−25.7228			−1.01127			−0.505634	+	0.862748i	$0.668741\pi$
$648$		0			0
$649$	−16.8614			−0.661868
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−12.3723	+	21.4294i	−0.484165	+	0.838598i	−0.999835	−	0.0181894i	$-0.994210\pi$
$653$	−12.3723	+	21.4294i	−0.484165	+	0.838598i	0.515670	+	0.856787i	$0.327543\pi$
$654$		0			0
$655$	−8.87228	−	15.3672i	−0.346669	−	0.600448i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−23.7921	−	41.2091i	−0.926809	−	1.60528i	−0.788626	−	0.614874i	$-0.789207\pi$
$659$	−23.7921	−	41.2091i	−0.926809	−	1.60528i	−0.138183	−	0.990407i	$-0.544126\pi$
$660$		0			0
$661$	22.5475	−	39.0535i	0.876998	−	1.51901i	0.0223789	−	0.999750i	$-0.492876\pi$
$661$	22.5475	−	39.0535i	0.876998	−	1.51901i	0.854619	−	0.519255i	$-0.173791\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	2.37228			0.0919931
$666$		0			0
$667$	1.25544			0.0486107
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−2.11684	+	3.66648i	−0.0817199	+	0.141543i
$672$		0			0
$673$	0.255437	+	0.442430i	0.00984639	+	0.0170544i	0.870907	−	0.491449i	$-0.163532\pi$
$673$	0.255437	+	0.442430i	0.00984639	+	0.0170544i	−0.861060	+	0.508503i	$0.830199\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−10.5584	−	18.2877i	−0.405793	−	0.702854i	0.588620	−	0.808410i	$-0.299671\pi$
$677$	−10.5584	−	18.2877i	−0.405793	−	0.702854i	−0.994413	+	0.105555i	$0.966338\pi$
$678$		0			0
$679$	8.00000	−	13.8564i	0.307012	−	0.531760i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−38.2337			−1.46297			−0.731486	−	0.681857i	$-0.761173\pi$
$683$	−38.2337			−1.46297			−0.731486	+	0.681857i	$0.761173\pi$
$684$		0			0
$685$	−16.7446			−0.639777
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−8.44158	+	14.6212i	−0.321599	+	0.557025i
$690$		0			0
$691$	−20.6753	−	35.8106i	−0.786524	−	1.36230i	−0.928084	−	0.372370i	$-0.878545\pi$
$691$	−20.6753	−	35.8106i	−0.786524	−	1.36230i	0.141560	−	0.989930i	$-0.454788\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	7.50000	+	12.9904i	0.284491	+	0.492753i
$696$		0			0
$697$	−27.8614	+	48.2574i	−1.05533	+	1.82788i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	8.11684			0.306569			0.153284	−	0.988182i	$-0.451015\pi$
$701$	8.11684			0.306569			0.153284	+	0.988182i	$0.451015\pi$
$702$		0			0
$703$	6.00000			0.226294
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−15.2554	+	26.4232i	−0.573740	+	0.993746i
$708$		0			0
$709$	−9.23369	−	15.9932i	−0.346778	−	0.600638i	0.638897	−	0.769292i	$-0.279391\pi$
$709$	−9.23369	−	15.9932i	−0.346778	−	0.600638i	−0.985675	+	0.168655i	$0.946058\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	1.13859	+	1.97210i	0.0426407	+	0.0738558i
$714$		0			0
$715$	4.00000	−	6.92820i	0.149592	−	0.259100i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	10.3940			0.387632			0.193816	−	0.981038i	$-0.437914\pi$
$719$	10.3940			0.387632			0.193816	+	0.981038i	$0.437914\pi$
$720$		0			0
$721$	−40.6060			−1.51225
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−1.68614	+	2.92048i	−0.0626217	+	0.108464i
$726$		0			0
$727$	23.5584	+	40.8044i	0.873734	+	1.51335i	0.858106	+	0.513473i	$0.171641\pi$
$727$	23.5584	+	40.8044i	0.873734	+	1.51335i	0.0156277	+	0.999878i	$0.495025\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	16.0000	+	27.7128i	0.591781	+	1.02500i
$732$		0			0
$733$	14.4891	−	25.0959i	0.535168	−	0.926938i	−0.463987	−	0.885842i	$-0.653582\pi$
$733$	14.4891	−	25.0959i	0.535168	−	0.926938i	0.999155	−	0.0410963i	$-0.0130851\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	36.2337			1.33469
$738$		0			0
$739$	−12.8614			−0.473114			−0.236557	−	0.971618i	$-0.576019\pi$
$739$	−12.8614			−0.473114			−0.236557	+	0.971618i	$0.576019\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−14.7446	+	25.5383i	−0.540926	+	0.936911i	0.457926	+	0.888991i	$0.348593\pi$
$743$	−14.7446	+	25.5383i	−0.540926	+	0.936911i	−0.998851	+	0.0479200i	$0.984741\pi$
$744$		0			0
$745$	1.74456	+	3.02167i	0.0639158	+	0.110705i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	7.11684	+	12.3267i	0.260044	+	0.450409i
$750$		0			0
$751$	19.1168	−	33.1113i	0.697584	−	1.20825i	−0.271718	−	0.962377i	$-0.587592\pi$
$751$	19.1168	−	33.1113i	0.697584	−	1.20825i	0.969302	−	0.245873i	$-0.0790747\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−18.6277			−0.677932
$756$		0			0
$757$	27.1168			0.985578			0.492789	−	0.870149i	$-0.335977\pi$
$757$	27.1168			0.985578			0.492789	+	0.870149i	$0.335977\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−8.61684	+	14.9248i	−0.312360	+	0.541024i	−0.978873	−	0.204470i	$-0.934453\pi$
$761$	−8.61684	+	14.9248i	−0.312360	+	0.541024i	0.666513	+	0.745494i	$0.267786\pi$
$762$		0			0
$763$	21.4891	+	37.2203i	0.777959	+	1.34746i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	5.93070	+	10.2723i	0.214145	+	0.370911i
$768$		0			0
$769$	16.1753	−	28.0164i	0.583295	−	1.01030i	−0.411791	−	0.911278i	$-0.635096\pi$
$769$	16.1753	−	28.0164i	0.583295	−	1.01030i	0.995086	−	0.0990181i	$-0.0315702\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−18.0000			−0.647415			−0.323708	−	0.946157i	$-0.604929\pi$
$773$	−18.0000			−0.647415			−0.323708	+	0.946157i	$0.604929\pi$
$774$		0			0
$775$	−6.11684			−0.219724
$776$		0			0
$777$		0			0
$778$		0			0
$779$	5.87228	−	10.1711i	0.210396	−	0.364417i
$780$		0			0
$781$	2.31386	+	4.00772i	0.0827964	+	0.143408i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−2.93070	−	5.07613i	−0.104601	−	0.181175i
$786$		0			0
$787$	−7.86141	+	13.6164i	−0.280229	+	0.485371i	−0.971441	−	0.237281i	$-0.923744\pi$
$787$	−7.86141	+	13.6164i	−0.280229	+	0.485371i	0.691212	+	0.722652i	$0.257077\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	7.72281			0.274592
$792$		0			0
$793$	2.97825			0.105761
$794$		0			0
$795$		0			0
$796$		0			0
$797$	7.11684	−	12.3267i	0.252092	−	0.436635i	−0.712010	−	0.702169i	$-0.752215\pi$
$797$	7.11684	−	12.3267i	0.252092	−	0.436635i	0.964101	+	0.265534i	$0.0855483\pi$

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)

\(f(q)\)	\(=\)	\(q+(0.500000 - 0.866025i) q^{5} +(1.18614 + 2.05446i) q^{7} +O(q^{10})\)	\(q+(0.500000 - 0.866025i) q^{5} +(1.18614 + 2.05446i) q^{7} +(-1.68614 - 2.92048i) q^{11} +(-1.18614 + 2.05446i) q^{13} -4.74456 q^{17} +1.00000 q^{19} +(-0.186141 + 0.322405i) q^{23} +(-0.500000 - 0.866025i) q^{25} +(-1.68614 - 2.92048i) q^{29} +(3.05842 - 5.29734i) q^{31} +2.37228 q^{35} +6.00000 q^{37} +(5.87228 - 10.1711i) q^{41} +(-3.37228 - 5.84096i) q^{43} +(-1.81386 - 3.14170i) q^{47} +(0.686141 - 1.18843i) q^{49} +7.11684 q^{53} -3.37228 q^{55} +(2.50000 - 4.33013i) q^{59} +(-0.627719 - 1.08724i) q^{61} +(1.18614 + 2.05446i) q^{65} +(-5.37228 + 9.30506i) q^{67} -1.37228 q^{71} +3.25544 q^{73} +(4.00000 - 6.92820i) q^{77} +(-4.37228 - 7.57301i) q^{79} +(-5.00000 - 8.66025i) q^{83} +(-2.37228 + 4.10891i) q^{85} -1.37228 q^{89} -5.62772 q^{91} +(0.500000 - 0.866025i) q^{95} +(-3.37228 - 5.84096i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{5} - q^{7}+O(q^{10}) \) 4 * q + 2 * q^5 - q^7	\( 4 q + 2 q^{5} - q^{7} - q^{11} + q^{13} + 4 q^{17} + 4 q^{19} + 5 q^{23} - 2 q^{25} - q^{29} - 5 q^{31} - 2 q^{35} + 24 q^{37} + 12 q^{41} - 2 q^{43} - 13 q^{47} - 3 q^{49} - 6 q^{53} - 2 q^{55} + 10 q^{59} - 14 q^{61} - q^{65} - 10 q^{67} + 6 q^{71} + 36 q^{73} + 16 q^{77} - 6 q^{79} - 20 q^{83} + 2 q^{85} + 6 q^{89} - 34 q^{91} + 2 q^{95} - 2 q^{97}+O(q^{100}) \) 4 * q + 2 * q^5 - q^7 - q^11 + q^13 + 4 * q^17 + 4 * q^19 + 5 * q^23 - 2 * q^25 - q^29 - 5 * q^31 - 2 * q^35 + 24 * q^37 + 12 * q^41 - 2 * q^43 - 13 * q^47 - 3 * q^49 - 6 * q^53 - 2 * q^55 + 10 * q^59 - 14 * q^61 - q^65 - 10 * q^67 + 6 * q^71 + 36 * q^73 + 16 * q^77 - 6 * q^79 - 20 * q^83 + 2 * q^85 + 6 * q^89 - 34 * q^91 + 2 * q^95 - 2 * q^97

Introduction

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