LMFDB - Embedded newform 3240.2.q.bh.2161.4

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:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{3})$
Coefficient field:	8.0.3887771904.9
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 26x^{6} + 217x^{4} + 672x^{2} + 576 $ x^8 + 26x^6 + 217x^4 + 672*x^2 + 576
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			2161.4
Root			$3.61675i$ of defining polynomial
Character	$\chi$	$=$	3240.2161
Dual form			3240.2.q.bh.1081.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.500000 + 0.866025i) q^{5} +(2.17440 - 3.76617i) q^{7} +(1.35058 - 2.33927i) q^{11} +(3.13219 + 5.42512i) q^{13} +5.37969 q^{17} -6.23349 q^{19} +(2.80837 + 4.86424i) q^{23} +(-0.500000 + 0.866025i) q^{25} +(0.381473 - 0.660731i) q^{29} +(-3.30837 - 5.73027i) q^{31} +4.34880 q^{35} +11.1491 q^{37} +(4.51367 + 7.81790i) q^{41} +(-4.84249 + 8.38744i) q^{43} +(2.61352 - 4.52676i) q^{47} +(-5.95601 - 10.3161i) q^{49} +10.6294 q^{53} +2.70115 q^{55} +(0.866025 + 1.50000i) q^{59} +(-0.816460 + 1.41415i) q^{61} +(-3.13219 + 5.42512i) q^{65} +(-4.76295 - 8.24967i) q^{67} -14.3017 q^{71} -10.6132 q^{73} +(-5.87339 - 10.1730i) q^{77} +(-0.267949 + 0.464102i) q^{79} +(0.0731007 - 0.126614i) q^{83} +(2.68985 + 4.65895i) q^{85} +7.85997 q^{89} +27.2425 q^{91} +(-3.11675 - 5.39836i) q^{95} +(0.194848 - 0.337487i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 4 q^{5} - 2 q^{7} + 4 q^{11} + 4 q^{17} + 10 q^{23} - 4 q^{25} - 4 q^{29} - 14 q^{31} - 4 q^{35} + 28 q^{37} + 4 q^{41} - 22 q^{43} - 10 q^{49} + 16 q^{53} + 8 q^{55} - 4 q^{61} - 24 q^{67} - 56 q^{71}+ \cdots + 10 q^{97}+O(q^{100}) $ 8 * q + 4 * q^5 - 2 * q^7 + 4 * q^11 + 4 * q^17 + 10 * q^23 - 4 * q^25 - 4 * q^29 - 14 * q^31 - 4 * q^35 + 28 * q^37 + 4 * q^41 - 22 * q^43 - 10 * q^49 + 16 * q^53 + 8 * q^55 - 4 * q^61 - 24 * q^67 - 56 * q^71 + 4 * q^73 - 30 * q^77 - 16 * q^79 + 6 * q^83 + 2 * q^85 + 16 * q^89 + 60 * q^91 + 10 * q^97

Character values

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

$n$	$1297$	$1621$	$2431$	$3161$
$\chi(n)$	$1$	$1$	$1$	$e\left(\frac{2}{3}\right)$

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
$7$	2.17440	−	3.76617i	0.821845	−	1.42348i	−0.0824614	−	0.996594i	$-0.526278\pi$
$7$	2.17440	−	3.76617i	0.821845	−	1.42348i	0.904307	−	0.426883i	$-0.140389\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	1.35058	−	2.33927i	0.407214	−	0.705316i	−0.587362	−	0.809324i	$-0.699834\pi$
$11$	1.35058	−	2.33927i	0.407214	−	0.705316i	0.994576	+	0.104008i	$0.0331668\pi$
$12$		0			0
$13$	3.13219	+	5.42512i	0.868714	+	1.50466i	0.863311	+	0.504672i	$0.168386\pi$
$13$	3.13219	+	5.42512i	0.868714	+	1.50466i	0.00540274	+	0.999985i	$0.498280\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	5.37969			1.30477			0.652384	−	0.757889i	$-0.273769\pi$
$17$	5.37969			1.30477			0.652384	+	0.757889i	$0.273769\pi$
$18$		0			0
$19$	−6.23349			−1.43006			−0.715030	−	0.699093i	$-0.753587\pi$
$19$	−6.23349			−1.43006			−0.715030	+	0.699093i	$0.753587\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	2.80837	+	4.86424i	0.585586	+	1.01427i	0.994802	+	0.101827i	$0.0324690\pi$
$23$	2.80837	+	4.86424i	0.585586	+	1.01427i	−0.409216	+	0.912438i	$0.634198\pi$
$24$		0			0
$25$	−0.500000	+	0.866025i	−0.100000	+	0.173205i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	0.381473	−	0.660731i	0.0708378	−	0.122695i	−0.828431	−	0.560091i	$-0.810766\pi$
$29$	0.381473	−	0.660731i	0.0708378	−	0.122695i	0.899269	+	0.437397i	$0.144099\pi$
$30$		0			0
$31$	−3.30837	−	5.73027i	−0.594201	−	1.02919i	−0.993659	−	0.112435i	$-0.964135\pi$
$31$	−3.30837	−	5.73027i	−0.594201	−	1.02919i	0.399458	−	0.916752i	$-0.369198\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	4.34880			0.735081
$36$		0			0
$37$	11.1491			1.83290			0.916449	−	0.400152i	$-0.131043\pi$
$37$	11.1491			1.83290			0.916449	+	0.400152i	$0.131043\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	4.51367	+	7.81790i	0.704916	+	1.22095i	0.966722	+	0.255830i	$0.0823488\pi$
$41$	4.51367	+	7.81790i	0.704916	+	1.22095i	−0.261805	+	0.965121i	$0.584318\pi$
$42$		0			0
$43$	−4.84249	+	8.38744i	−0.738473	+	1.27907i	0.214710	+	0.976678i	$0.431119\pi$
$43$	−4.84249	+	8.38744i	−0.738473	+	1.27907i	−0.953183	+	0.302395i	$0.902214\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	2.61352	−	4.52676i	0.381222	−	0.660295i	−0.610015	−	0.792389i	$-0.708837\pi$
$47$	2.61352	−	4.52676i	0.381222	−	0.660295i	0.991237	+	0.132094i	$0.0421701\pi$
$48$		0			0
$49$	−5.95601	−	10.3161i	−0.850859	−	1.47373i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	10.6294			1.46005			0.730027	−	0.683418i	$-0.239507\pi$
$53$	10.6294			1.46005			0.730027	+	0.683418i	$0.239507\pi$
$54$		0			0
$55$	2.70115			0.364224
$56$		0			0
$57$		0			0
$58$		0			0
$59$	0.866025	+	1.50000i	0.112747	+	0.195283i	0.916877	−	0.399170i	$-0.130702\pi$
$59$	0.866025	+	1.50000i	0.112747	+	0.195283i	−0.804130	+	0.594454i	$0.797368\pi$
$60$		0			0
$61$	−0.816460	+	1.41415i	−0.104537	+	0.181063i	−0.913549	−	0.406729i	$-0.866669\pi$
$61$	−0.816460	+	1.41415i	−0.104537	+	0.181063i	0.809012	+	0.587792i	$0.200003\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−3.13219	+	5.42512i	−0.388501	+	0.672903i
$66$		0			0
$67$	−4.76295	−	8.24967i	−0.581887	−	1.00786i	−0.995256	−	0.0972936i	$-0.968981\pi$
$67$	−4.76295	−	8.24967i	−0.581887	−	1.00786i	0.413369	−	0.910564i	$-0.364352\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−14.3017			−1.69730			−0.848651	−	0.528953i	$-0.822585\pi$
$71$	−14.3017			−1.69730			−0.848651	+	0.528953i	$0.822585\pi$
$72$		0			0
$73$	−10.6132			−1.24218			−0.621090	−	0.783740i	$-0.713310\pi$
$73$	−10.6132			−1.24218			−0.621090	+	0.783740i	$0.713310\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−5.87339	−	10.1730i	−0.669334	−	1.15932i
$78$		0			0
$79$	−0.267949	+	0.464102i	−0.0301466	+	0.0522155i	−0.880705	−	0.473665i	$-0.842931\pi$
$79$	−0.267949	+	0.464102i	−0.0301466	+	0.0522155i	0.850558	+	0.525880i	$0.176264\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	0.0731007	−	0.126614i	0.00802385	−	0.0138977i	−0.861986	−	0.506933i	$-0.830779\pi$
$83$	0.0731007	−	0.126614i	0.00802385	−	0.0138977i	0.870009	+	0.493035i	$0.164113\pi$
$84$		0			0
$85$	2.68985	+	4.65895i	0.291755	+	0.505334i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	7.85997			0.833155			0.416578	−	0.909100i	$-0.363229\pi$
$89$	7.85997			0.833155			0.416578	+	0.909100i	$0.363229\pi$
$90$		0			0
$91$	27.2425			2.85579
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−3.11675	−	5.39836i	−0.319771	−	0.553860i
$96$		0			0
$97$	0.194848	−	0.337487i	0.0197839	−	0.0342667i	−0.855964	−	0.517035i	$-0.827036\pi$
$97$	0.194848	−	0.337487i	0.0197839	−	0.0342667i	0.875748	+	0.482769i	$0.160369\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	6.43142	−	11.1396i	0.639951	−	1.10843i	−0.345492	−	0.938422i	$-0.612288\pi$
$101$	6.43142	−	11.1396i	0.639951	−	1.10843i	0.985443	−	0.170006i	$-0.0543786\pi$
$102$		0			0
$103$	−2.94865	−	5.10722i	−0.290539	−	0.503229i	0.683398	−	0.730046i	$-0.260501\pi$
$103$	−2.94865	−	5.10722i	−0.290539	−	0.503229i	−0.973937	+	0.226817i	$0.927168\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	14.3923			1.39136			0.695678	−	0.718353i	$-0.255104\pi$
$107$	14.3923			1.39136			0.695678	+	0.718353i	$0.255104\pi$
$108$		0			0
$109$	5.15264			0.493534			0.246767	−	0.969075i	$-0.420632\pi$
$109$	5.15264			0.493534			0.246767	+	0.969075i	$0.420632\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−9.19129	−	15.9198i	−0.864643	−	1.49761i	−0.867401	−	0.497610i	$-0.834211\pi$
$113$	−9.19129	−	15.9198i	−0.864643	−	1.49761i	0.00275732	−	0.999996i	$-0.499122\pi$
$114$		0			0
$115$	−2.80837	+	4.86424i	−0.261882	+	0.453593i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	11.6976	−	20.2608i	1.07232	−	1.85731i
$120$		0			0
$121$	1.85188	+	3.20755i	0.168353	+	0.291596i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−1.00000			−0.0894427
$126$		0			0
$127$	−7.27700			−0.645729			−0.322865	−	0.946445i	$-0.604646\pi$
$127$	−7.27700			−0.645729			−0.322865	+	0.946445i	$0.604646\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	2.68249	+	4.64620i	0.234370	+	0.405941i	0.959089	−	0.283103i	$-0.0913640\pi$
$131$	2.68249	+	4.64620i	0.234370	+	0.405941i	−0.724720	+	0.689044i	$0.758031\pi$
$132$		0			0
$133$	−13.5541	+	23.4764i	−1.17529	+	2.03566i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	2.34880	−	4.06823i	0.200671	−	0.347573i	−0.748074	−	0.663616i	$-0.769021\pi$
$137$	2.34880	−	4.06823i	0.200671	−	0.347573i	0.948745	+	0.316043i	$0.102354\pi$
$138$		0			0
$139$	0.331181	+	0.573622i	0.0280904	+	0.0486540i	0.879729	−	0.475476i	$-0.157724\pi$
$139$	0.331181	+	0.573622i	0.0280904	+	0.0486540i	−0.851638	+	0.524130i	$0.824391\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	16.9211			1.41501
$144$		0			0
$145$	0.762947			0.0633593
$146$		0			0
$147$		0			0
$148$		0			0
$149$	1.08085	+	1.87208i	0.0885464	+	0.153367i	0.906897	−	0.421353i	$-0.138444\pi$
$149$	1.08085	+	1.87208i	0.0885464	+	0.153367i	−0.818350	+	0.574719i	$0.805111\pi$
$150$		0			0
$151$	4.22396	−	7.31612i	0.343741	−	0.595377i	−0.641383	−	0.767221i	$-0.721639\pi$
$151$	4.22396	−	7.31612i	0.343741	−	0.595377i	0.985124	+	0.171844i	$0.0549724\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	3.30837	−	5.73027i	0.265735	−	0.460266i
$156$		0			0
$157$	−2.87555	−	4.98060i	−0.229494	−	0.397495i	0.728164	−	0.685403i	$-0.240374\pi$
$157$	−2.87555	−	4.98060i	−0.229494	−	0.397495i	−0.957658	+	0.287907i	$0.907040\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	24.4261			1.92504
$162$		0			0
$163$	12.8438			1.00600			0.503002	−	0.864285i	$-0.332229\pi$
$163$	12.8438			1.00600			0.503002	+	0.864285i	$0.332229\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	10.9542	+	18.9733i	0.847664	+	1.46820i	0.883287	+	0.468832i	$0.155325\pi$
$167$	10.9542	+	18.9733i	0.847664	+	1.46820i	−0.0356235	+	0.999365i	$0.511342\pi$
$168$		0			0
$169$	−13.1213	+	22.7267i	−1.00933	+	1.74821i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−1.34427	+	2.32835i	−0.102203	+	0.177021i	−0.912592	−	0.408871i	$-0.865922\pi$
$173$	−1.34427	+	2.32835i	−0.102203	+	0.177021i	0.810389	+	0.585892i	$0.199256\pi$
$174$		0			0
$175$	2.17440	+	3.76617i	0.164369	+	0.284696i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	9.02733			0.674735			0.337367	−	0.941373i	$-0.390464\pi$
$179$	9.02733			0.674735			0.337367	+	0.941373i	$0.390464\pi$
$180$		0			0
$181$	−5.85024			−0.434845			−0.217422	−	0.976078i	$-0.569765\pi$
$181$	−5.85024			−0.434845			−0.217422	+	0.976078i	$0.569765\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	5.57454	+	9.65539i	0.409848	+	0.709878i
$186$		0			0
$187$	7.26569	−	12.5845i	0.531320	−	0.920273i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	6.49678	−	11.2528i	0.470090	−	0.814220i	−0.529325	−	0.848419i	$-0.677555\pi$
$191$	6.49678	−	11.2528i	0.470090	−	0.814220i	0.999415	+	0.0341989i	$0.0108880\pi$
$192$		0			0
$193$	−2.25664	−	3.90862i	−0.162437	−	0.281348i	0.773305	−	0.634034i	$-0.218602\pi$
$193$	−2.25664	−	3.90862i	−0.162437	−	0.281348i	−0.935742	+	0.352685i	$0.885269\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−16.8431			−1.20002			−0.600011	−	0.799992i	$-0.704837\pi$
$197$	−16.8431			−1.20002			−0.600011	+	0.799992i	$0.704837\pi$
$198$		0			0
$199$	11.6006			0.822343			0.411171	−	0.911558i	$-0.365120\pi$
$199$	11.6006			0.822343			0.411171	+	0.911558i	$0.365120\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−1.65895	−	2.87339i	−0.116435	−	0.201672i
$204$		0			0
$205$	−4.51367	+	7.81790i	−0.315248	+	0.546026i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−8.41881	+	14.5818i	−0.582341	+	1.00864i
$210$		0			0
$211$	0.431766	+	0.747840i	0.0297240	+	0.0514834i	0.880505	−	0.474037i	$-0.157204\pi$
$211$	0.431766	+	0.747840i	0.0297240	+	0.0514834i	−0.850781	+	0.525521i	$0.823871\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−9.68498			−0.660510
$216$		0			0
$217$	−28.7749			−1.95337
$218$		0			0
$219$		0			0
$220$		0			0
$221$	16.8502	+	29.1855i	1.13347	+	1.96323i
$222$		0			0
$223$	4.43321	−	7.67854i	0.296869	−	0.514193i	−0.678549	−	0.734555i	$-0.737391\pi$
$223$	4.43321	−	7.67854i	0.296869	−	0.514193i	0.975418	+	0.220363i	$0.0707241\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−6.84249	+	11.8515i	−0.454152	+	0.786614i	−0.998639	−	0.0521549i	$-0.983391\pi$
$227$	−6.84249	+	11.8515i	−0.454152	+	0.786614i	0.544487	+	0.838769i	$0.316724\pi$
$228$		0			0
$229$	1.23349	+	2.13647i	0.0815114	+	0.141182i	0.903899	−	0.427745i	$-0.140692\pi$
$229$	1.23349	+	2.13647i	0.0815114	+	0.141182i	−0.822388	+	0.568927i	$0.807359\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	12.8438			0.841425			0.420712	−	0.907194i	$-0.361780\pi$
$233$	12.8438			0.841425			0.420712	+	0.907194i	$0.361780\pi$
$234$		0			0
$235$	5.22705			0.340975
$236$		0			0
$237$		0			0
$238$		0			0
$239$	4.26795	+	7.39230i	0.276071	+	0.478168i	0.970405	−	0.241484i	$-0.0776343\pi$
$239$	4.26795	+	7.39230i	0.276071	+	0.478168i	−0.694334	+	0.719653i	$0.744301\pi$
$240$		0			0
$241$	−2.56988	+	4.45116i	−0.165540	+	0.286725i	−0.936847	−	0.349739i	$-0.886270\pi$
$241$	−2.56988	+	4.45116i	−0.165540	+	0.286725i	0.771307	+	0.636464i	$0.219604\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	5.95601	−	10.3161i	0.380516	−	0.659073i
$246$		0			0
$247$	−19.5245	−	33.8174i	−1.24231	−	2.15175i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	1.13080			0.0713753			0.0356877	−	0.999363i	$-0.488638\pi$
$251$	1.13080			0.0713753			0.0356877	+	0.999363i	$0.488638\pi$
$252$		0			0
$253$	15.1717			0.953837
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−5.08441	−	8.80646i	−0.317157	−	0.549332i	0.662737	−	0.748852i	$-0.269395\pi$
$257$	−5.08441	−	8.80646i	−0.317157	−	0.549332i	−0.979894	+	0.199521i	$0.936061\pi$
$258$		0			0
$259$	24.2425	−	41.9893i	1.50636	−	2.60909i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−4.10366	+	7.10774i	−0.253042	+	0.438282i	−0.964362	−	0.264587i	$-0.914764\pi$
$263$	−4.10366	+	7.10774i	−0.253042	+	0.438282i	0.711320	+	0.702869i	$0.248098\pi$
$264$		0			0
$265$	5.31468	+	9.20529i	0.326478	+	0.565477i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	26.9246			1.64162			0.820812	−	0.571198i	$-0.193521\pi$
$269$	26.9246			1.64162			0.820812	+	0.571198i	$0.193521\pi$
$270$		0			0
$271$	−4.30529			−0.261528			−0.130764	−	0.991414i	$-0.541743\pi$
$271$	−4.30529			−0.261528			−0.130764	+	0.991414i	$0.541743\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	1.35058	+	2.33927i	0.0814429	+	0.141063i
$276$		0			0
$277$	−10.1090	+	17.5094i	−0.607394	+	1.05204i	0.384275	+	0.923219i	$0.374452\pi$
$277$	−10.1090	+	17.5094i	−0.607394	+	1.05204i	−0.991668	+	0.128818i	$0.958882\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−0.0704014	+	0.121939i	−0.00419979	+	0.00727425i	−0.868118	−	0.496358i	$-0.834670\pi$
$281$	−0.0704014	+	0.121939i	−0.00419979	+	0.00727425i	0.863918	+	0.503633i	$0.168004\pi$
$282$		0			0
$283$	−0.115305	−	0.199715i	−0.00685420	−	0.0118718i	0.862578	−	0.505924i	$-0.168848\pi$
$283$	−0.115305	−	0.199715i	−0.00685420	−	0.0118718i	−0.869432	+	0.494052i	$0.835515\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	39.2580			2.31733
$288$		0			0
$289$	11.9411			0.702417
$290$		0			0
$291$		0			0
$292$		0			0
$293$	4.06145	+	7.03464i	0.237272	+	0.410968i	0.959931	−	0.280238i	$-0.0904132\pi$
$293$	4.06145	+	7.03464i	0.237272	+	0.410968i	−0.722658	+	0.691206i	$0.757080\pi$
$294$		0			0
$295$	−0.866025	+	1.50000i	−0.0504219	+	0.0873334i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−17.5927	+	30.4715i	−1.01741	+	1.76221i
$300$		0			0
$301$	21.0590	+	36.4753i	1.21382	+	2.10240i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−1.63292			−0.0935008
$306$		0			0
$307$	−0.444364			−0.0253612			−0.0126806	−	0.999920i	$-0.504036\pi$
$307$	−0.444364			−0.0253612			−0.0126806	+	0.999920i	$0.504036\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−3.26261	−	5.65100i	−0.185005	−	0.320439i	0.758573	−	0.651588i	$-0.225897\pi$
$311$	−3.26261	−	5.65100i	−0.185005	−	0.320439i	−0.943578	+	0.331149i	$0.892564\pi$
$312$		0			0
$313$	8.47882	−	14.6858i	0.479251	−	0.830088i	−0.520465	−	0.853883i	$-0.674242\pi$
$313$	8.47882	−	14.6858i	0.479251	−	0.830088i	0.999717	+	0.0237950i	$0.00757489\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−6.99678	+	12.1188i	−0.392978	+	0.680658i	−0.992841	−	0.119445i	$-0.961889\pi$
$317$	−6.99678	+	12.1188i	−0.392978	+	0.680658i	0.599863	+	0.800103i	$0.295222\pi$
$318$		0			0
$319$	−1.03042	−	1.78474i	−0.0576924	−	0.0999261i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−33.5343			−1.86590
$324$		0			0
$325$	−6.26439			−0.347486
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−11.3657	−	19.6859i	−0.626610	−	1.08532i
$330$		0			0
$331$	−9.64099	+	16.6987i	−0.529917	+	0.917843i	0.469474	+	0.882946i	$0.344444\pi$
$331$	−9.64099	+	16.6987i	−0.529917	+	0.917843i	−0.999391	+	0.0348968i	$0.988890\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	4.76295	−	8.24967i	0.260228	−	0.450727i
$336$		0			0
$337$	−8.62936	−	14.9465i	−0.470071	−	0.814187i	0.529343	−	0.848408i	$-0.322438\pi$
$337$	−8.62936	−	14.9465i	−0.470071	−	0.814187i	−0.999414	+	0.0342208i	$0.989105\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−17.8729			−0.967869
$342$		0			0
$343$	−21.3614			−1.15341
$344$		0			0
$345$		0			0
$346$		0			0
$347$	6.80159	+	11.7807i	0.365128	+	0.632421i	0.988797	−	0.149268i	$-0.0476917\pi$
$347$	6.80159	+	11.7807i	0.365128	+	0.632421i	−0.623668	+	0.781689i	$0.714358\pi$
$348$		0			0
$349$	0.809813	−	1.40264i	0.0433483	−	0.0750814i	−0.843537	−	0.537071i	$-0.819531\pi$
$349$	0.809813	−	1.40264i	0.0433483	−	0.0750814i	0.886885	+	0.461989i	$0.152864\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	14.5401	−	25.1842i	0.773890	−	1.34042i	−0.161525	−	0.986869i	$-0.551641\pi$
$353$	14.5401	−	25.1842i	0.773890	−	1.34042i	0.935416	−	0.353549i	$-0.115025\pi$
$354$		0			0
$355$	−7.15086	−	12.3857i	−0.379528	−	0.657362i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−13.9218			−0.734762			−0.367381	−	0.930070i	$-0.619746\pi$
$359$	−13.9218			−0.734762			−0.367381	+	0.930070i	$0.619746\pi$
$360$		0			0
$361$	19.8564			1.04507
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−5.30659	−	9.19129i	−0.277760	−	0.481094i
$366$		0			0
$367$	−9.33262	+	16.1646i	−0.487159	+	0.843784i	−0.999891	−	0.0147647i	$-0.995300\pi$
$367$	−9.33262	+	16.1646i	−0.487159	+	0.843784i	0.512732	+	0.858549i	$0.328633\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	23.1125	−	40.0319i	1.19994	−	2.07836i
$372$		0			0
$373$	−9.59360	−	16.6166i	−0.496738	−	0.860375i	0.503255	−	0.864138i	$-0.332136\pi$
$373$	−9.59360	−	16.6166i	−0.496738	−	0.860375i	−0.999993	+	0.00376306i	$0.998802\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	4.77939			0.246151
$378$		0			0
$379$	−1.82213			−0.0935966			−0.0467983	−	0.998904i	$-0.514902\pi$
$379$	−1.82213			−0.0935966			−0.0467983	+	0.998904i	$0.514902\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	1.97719	+	3.42460i	0.101030	+	0.174989i	0.912109	−	0.409947i	$-0.134453\pi$
$383$	1.97719	+	3.42460i	0.101030	+	0.174989i	−0.811079	+	0.584936i	$0.801120\pi$
$384$		0			0
$385$	5.87339	−	10.1730i	0.299335	−	0.518464i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	12.5823	−	21.7932i	0.637947	−	1.10496i	−0.347935	−	0.937519i	$-0.613117\pi$
$389$	12.5823	−	21.7932i	0.637947	−	1.10496i	0.985883	−	0.167439i	$-0.0535496\pi$
$390$		0			0
$391$	15.1082	+	26.1681i	0.764054	+	1.32338i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−0.535898			−0.0269640
$396$		0			0
$397$	−13.8635			−0.695791			−0.347895	−	0.937533i	$-0.613104\pi$
$397$	−13.8635			−0.695791			−0.347895	+	0.937533i	$0.613104\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−18.2095	−	31.5397i	−0.909338	−	1.57502i	−0.814986	−	0.579481i	$-0.803255\pi$
$401$	−18.2095	−	31.5397i	−0.909338	−	1.57502i	−0.0943519	−	0.995539i	$-0.530078\pi$
$402$		0			0
$403$	20.7249	−	35.8966i	1.03238	−	1.78814i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	15.0577	−	26.0807i	0.746382	−	1.29277i
$408$		0			0
$409$	−15.1762	−	26.2860i	−0.750416	−	1.29976i	−0.947621	−	0.319396i	$-0.896520\pi$
$409$	−15.1762	−	26.2860i	−0.750416	−	1.29976i	0.197205	−	0.980362i	$-0.436813\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	7.53234			0.370642
$414$		0			0
$415$	0.146201			0.00717675
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−5.11531	−	8.85997i	−0.249899	−	0.432838i	0.713599	−	0.700555i	$-0.247064\pi$
$419$	−5.11531	−	8.85997i	−0.249899	−	0.432838i	−0.963498	+	0.267717i	$0.913731\pi$
$420$		0			0
$421$	−5.15217	+	8.92381i	−0.251101	+	0.434920i	−0.963829	−	0.266520i	$-0.914126\pi$
$421$	−5.15217	+	8.92381i	−0.251101	+	0.434920i	0.712728	+	0.701440i	$0.247459\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−2.68985	+	4.65895i	−0.130477	+	0.225992i
$426$		0			0
$427$	3.55062	+	6.14985i	0.171827	+	0.297612i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−41.0317			−1.97643			−0.988213	−	0.153086i	$-0.951079\pi$
$431$	−41.0317			−1.97643			−0.988213	+	0.153086i	$0.951079\pi$
$432$		0			0
$433$	16.7270			0.803850			0.401925	−	0.915673i	$-0.368341\pi$
$433$	16.7270			0.803850			0.401925	+	0.915673i	$0.368341\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−17.5060	−	30.3212i	−0.837424	−	1.45046i
$438$		0			0
$439$	−6.24302	+	10.8132i	−0.297963	+	0.516087i	−0.975670	−	0.219245i	$-0.929641\pi$
$439$	−6.24302	+	10.8132i	−0.297963	+	0.516087i	0.677707	+	0.735332i	$0.262974\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	3.80803	−	6.59570i	0.180925	−	0.313371i	−0.761271	−	0.648434i	$-0.775424\pi$
$443$	3.80803	−	6.59570i	0.180925	−	0.313371i	0.942196	+	0.335063i	$0.108758\pi$
$444$		0			0
$445$	3.92998	+	6.80693i	0.186299	+	0.322680i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−12.8585			−0.606831			−0.303415	−	0.952858i	$-0.598127\pi$
$449$	−12.8585			−0.606831			−0.303415	+	0.952858i	$0.598127\pi$
$450$		0			0
$451$	24.3842			1.14821
$452$		0			0
$453$		0			0
$454$		0			0
$455$	13.6213	+	23.5927i	0.638575	+	1.10604i
$456$		0			0
$457$	−6.30659	+	10.9233i	−0.295010	+	0.510972i	−0.974987	−	0.222261i	$-0.928656\pi$
$457$	−6.30659	+	10.9233i	−0.295010	+	0.510972i	0.679977	+	0.733233i	$0.261990\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−15.6078	+	27.0336i	−0.726930	+	1.25908i	0.231245	+	0.972895i	$0.425720\pi$
$461$	−15.6078	+	27.0336i	−0.726930	+	1.25908i	−0.958175	+	0.286183i	$0.907613\pi$
$462$		0			0
$463$	−1.97955	−	3.42868i	−0.0919975	−	0.159344i	0.816354	−	0.577552i	$-0.195992\pi$
$463$	−1.97955	−	3.42868i	−0.0919975	−	0.159344i	−0.908352	+	0.418207i	$0.862659\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−20.7260			−0.959083			−0.479542	−	0.877519i	$-0.659197\pi$
$467$	−20.7260			−0.959083			−0.479542	+	0.877519i	$0.659197\pi$
$468$		0			0
$469$	−41.4262			−1.91288
$470$		0			0
$471$		0			0
$472$		0			0
$473$	13.0803	+	22.6558i	0.601434	+	1.04171i
$474$		0			0
$475$	3.11675	−	5.39836i	0.143006	−	0.247694i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−14.8611	+	25.7401i	−0.679020	+	1.17610i	0.296257	+	0.955108i	$0.404262\pi$
$479$	−14.8611	+	25.7401i	−0.679020	+	1.17610i	−0.975277	+	0.220988i	$0.929072\pi$
$480$		0			0
$481$	34.9211	+	60.4851i	1.59226	+	2.75788i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	0.389697			0.0176952
$486$		0			0
$487$	−0.0532416			−0.00241260			−0.00120630	−	0.999999i	$-0.500384\pi$
$487$	−0.0532416			−0.00241260			−0.00120630	+	0.999999i	$0.500384\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	6.42926	+	11.1358i	0.290148	+	0.502552i	0.973845	−	0.227215i	$-0.0729620\pi$
$491$	6.42926	+	11.1358i	0.290148	+	0.502552i	−0.683696	+	0.729767i	$0.739629\pi$
$492$		0			0
$493$	2.05221	−	3.55453i	0.0924269	−	0.160088i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−31.0976	+	53.8627i	−1.39492	+	2.41607i
$498$		0			0
$499$	−19.1096	−	33.0988i	−0.855464	−	1.48171i	−0.876214	−	0.481923i	$-0.839939\pi$
$499$	−19.1096	−	33.0988i	−0.855464	−	1.48171i	0.0207497	−	0.999785i	$-0.493395\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−31.0828			−1.38591			−0.692956	−	0.720980i	$-0.743692\pi$
$503$	−31.0828			−1.38591			−0.692956	+	0.720980i	$0.743692\pi$
$504$		0			0
$505$	12.8628			0.572389
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−17.4670	−	30.2537i	−0.774210	−	1.34097i	−0.935237	−	0.354022i	$-0.884814\pi$
$509$	−17.4670	−	30.2537i	−0.774210	−	1.34097i	0.161027	−	0.986950i	$-0.448519\pi$
$510$		0			0
$511$	−23.0773	+	39.9710i	−1.02088	+	1.76821i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	2.94865	−	5.10722i	0.129933	−	0.225051i
$516$		0			0
$517$	−7.05953	−	12.2275i	−0.310478	−	0.537764i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−16.3880			−0.717970			−0.358985	−	0.933343i	$-0.616877\pi$
$521$	−16.3880			−0.717970			−0.358985	+	0.933343i	$0.616877\pi$
$522$		0			0
$523$	14.6682			0.641393			0.320697	−	0.947182i	$-0.396083\pi$
$523$	14.6682			0.641393			0.320697	+	0.947182i	$0.396083\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−17.7980	−	30.8271i	−0.775294	−	1.34285i
$528$		0			0
$529$	−4.27391	+	7.40264i	−0.185822	+	0.321854i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−28.2754	+	48.9743i	−1.22474	+	2.12131i
$534$		0			0
$535$	7.19615	+	12.4641i	0.311117	+	0.538870i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−32.1762			−1.38593
$540$		0			0
$541$	29.9301			1.28680			0.643398	−	0.765532i	$-0.277524\pi$
$541$	29.9301			1.28680			0.643398	+	0.765532i	$0.277524\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	2.57632	+	4.46232i	0.110358	+	0.191145i
$546$		0			0
$547$	20.2116	−	35.0076i	0.864188	−	1.49682i	−0.00366412	−	0.999993i	$-0.501166\pi$
$547$	20.2116	−	35.0076i	0.864188	−	1.49682i	0.867852	−	0.496823i	$-0.165500\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−2.37791	+	4.11866i	−0.101302	+	0.175461i
$552$		0			0
$553$	1.16526	+	2.01828i	0.0495517	+	0.0858261i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−5.62909			−0.238512			−0.119256	−	0.992864i	$-0.538051\pi$
$557$	−5.62909			−0.238512			−0.119256	+	0.992864i	$0.538051\pi$
$558$		0			0
$559$	−60.6705			−2.56609
$560$		0			0
$561$		0			0
$562$		0			0
$563$	0.642775	+	1.11332i	0.0270897	+	0.0469208i	0.879252	−	0.476356i	$-0.158043\pi$
$563$	0.642775	+	1.11332i	0.0270897	+	0.0469208i	−0.852163	+	0.523277i	$0.824709\pi$
$564$		0			0
$565$	9.19129	−	15.9198i	0.386680	−	0.669750i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−2.70077	+	4.67787i	−0.113222	+	0.196106i	−0.917068	−	0.398732i	$-0.869451\pi$
$569$	−2.70077	+	4.67787i	−0.113222	+	0.196106i	0.803846	+	0.594838i	$0.202784\pi$
$570$		0			0
$571$	11.9004	+	20.6121i	0.498016	+	0.862589i	0.999997	−	0.00228950i	$-0.000728772\pi$
$571$	11.9004	+	20.6121i	0.498016	+	0.862589i	−0.501981	+	0.864878i	$0.667395\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−5.61675			−0.234234
$576$		0			0
$577$	−26.9926			−1.12372			−0.561858	−	0.827233i	$-0.689913\pi$
$577$	−26.9926			−1.12372			−0.561858	+	0.827233i	$0.689913\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−0.317900	−	0.550619i	−0.0131887	−	0.0228435i
$582$		0			0
$583$	14.3558	−	24.8649i	0.594555	−	1.02980i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−13.1604	+	22.7945i	−0.543187	+	0.940828i	0.455531	+	0.890220i	$0.349449\pi$
$587$	−13.1604	+	22.7945i	−0.543187	+	0.940828i	−0.998719	+	0.0506084i	$0.983884\pi$
$588$		0			0
$589$	20.6227	+	35.7196i	0.849744	+	1.47180i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−27.4951			−1.12909			−0.564544	−	0.825403i	$-0.690948\pi$
$593$	−27.4951			−1.12909			−0.564544	+	0.825403i	$0.690948\pi$
$594$		0			0
$595$	23.3952			0.959109
$596$		0			0
$597$		0			0
$598$		0			0
$599$	4.98204	+	8.62915i	0.203561	+	0.352578i	0.949673	−	0.313242i	$-0.101415\pi$
$599$	4.98204	+	8.62915i	0.203561	+	0.352578i	−0.746112	+	0.665820i	$0.768082\pi$
$600$		0			0
$601$	−15.6391	+	27.0877i	−0.637931	+	1.10493i	0.347955	+	0.937511i	$0.386876\pi$
$601$	−15.6391	+	27.0877i	−0.637931	+	1.10493i	−0.985886	+	0.167418i	$0.946457\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−1.85188	+	3.20755i	−0.0752897	+	0.130406i
$606$		0			0
$607$	−13.1666	−	22.8052i	−0.534414	−	0.925633i	−0.999191	−	0.0402049i	$-0.987199\pi$
$607$	−13.1666	−	22.8052i	−0.534414	−	0.925633i	0.464777	−	0.885428i	$-0.346134\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	32.7443			1.32469
$612$		0			0
$613$	−18.6020			−0.751329			−0.375664	−	0.926756i	$-0.622585\pi$
$613$	−18.6020			−0.751329			−0.375664	+	0.926756i	$0.622585\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−8.66382	−	15.0062i	−0.348792	−	0.604126i	0.637243	−	0.770663i	$-0.280075\pi$
$617$	−8.66382	−	15.0062i	−0.348792	−	0.604126i	−0.986035	+	0.166537i	$0.946741\pi$
$618$		0			0
$619$	−10.7577	+	18.6330i	−0.432390	+	0.748922i	−0.997079	−	0.0763826i	$-0.975663\pi$
$619$	−10.7577	+	18.6330i	−0.432390	+	0.748922i	0.564689	+	0.825304i	$0.308996\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	17.0907	−	29.6020i	0.684724	−	1.18598i
$624$		0			0
$625$	−0.500000	−	0.866025i	−0.0200000	−	0.0346410i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	59.9786			2.39150
$630$		0			0
$631$	−20.9696			−0.834786			−0.417393	−	0.908726i	$-0.637056\pi$
$631$	−20.9696			−0.834786			−0.417393	+	0.908726i	$0.637056\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−3.63850	−	6.30207i	−0.144389	−	0.250090i
$636$		0			0
$637$	37.3108	−	64.6242i	1.47831	−	2.56050i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	7.92025	−	13.7183i	0.312831	−	0.541839i	−0.666143	−	0.745824i	$-0.732056\pi$
$641$	7.92025	−	13.7183i	0.312831	−	0.541839i	0.978974	+	0.203985i	$0.0653893\pi$
$642$		0			0
$643$	8.47541	+	14.6798i	0.334238	+	0.578916i	0.983338	−	0.181786i	$-0.0581879\pi$
$643$	8.47541	+	14.6798i	0.334238	+	0.578916i	−0.649101	+	0.760703i	$0.724855\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	41.0504			1.61386			0.806929	−	0.590648i	$-0.201128\pi$
$647$	41.0504			1.61386			0.806929	+	0.590648i	$0.201128\pi$
$648$		0			0
$649$	4.67854			0.183649
$650$		0			0
$651$		0			0
$652$		0			0
$653$	15.3635	+	26.6104i	0.601221	+	1.04135i	0.992637	+	0.121131i	$0.0386521\pi$
$653$	15.3635	+	26.6104i	0.601221	+	1.04135i	−0.391416	+	0.920214i	$0.628015\pi$
$654$		0			0
$655$	−2.68249	+	4.64620i	−0.104813	+	0.181542i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−12.1175	+	20.9881i	−0.472030	+	0.817579i	−0.999488	−	0.0320016i	$-0.989812\pi$
$659$	−12.1175	+	20.9881i	−0.472030	+	0.817579i	0.527458	+	0.849581i	$0.323145\pi$
$660$		0			0
$661$	−18.1424	−	31.4236i	−0.705659	−	1.22224i	−0.966453	−	0.256843i	$-0.917318\pi$
$661$	−18.1424	−	31.4236i	−0.705659	−	1.22224i	0.260794	−	0.965394i	$-0.416016\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−27.1082			−1.05121
$666$		0			0
$667$	4.28528			0.165927
$668$		0			0
$669$		0			0
$670$		0			0
$671$	2.20539	+	3.81984i	0.0851380	+	0.147463i
$672$		0			0
$673$	−4.20616	+	7.28528i	−0.162135	+	0.280827i	−0.935634	−	0.352971i	$-0.885171\pi$
$673$	−4.20616	+	7.28528i	−0.162135	+	0.280827i	0.773499	+	0.633798i	$0.218505\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−21.0545	+	36.4674i	−0.809189	+	1.40156i	0.104237	+	0.994552i	$0.466760\pi$
$677$	−21.0545	+	36.4674i	−0.809189	+	1.40156i	−0.913426	+	0.407004i	$0.866573\pi$
$678$		0			0
$679$	−0.847356	−	1.46766i	−0.0325185	−	0.0563238i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	23.2140			0.888260			0.444130	−	0.895962i	$-0.353513\pi$
$683$	23.2140			0.888260			0.444130	+	0.895962i	$0.353513\pi$
$684$		0			0
$685$	4.69759			0.179486
$686$		0			0
$687$		0			0
$688$		0			0
$689$	33.2932	+	57.6655i	1.26837	+	2.19688i
$690$		0			0
$691$	15.3716	−	26.6244i	0.584763	−	1.01284i	−0.410141	−	0.912022i	$-0.634521\pi$
$691$	15.3716	−	26.6244i	0.584763	−	1.01284i	0.994905	−	0.100818i	$-0.0321460\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−0.331181	+	0.573622i	−0.0125624	+	0.0217587i
$696$		0			0
$697$	24.2821	+	42.0579i	0.919752	+	1.59306i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−15.1624			−0.572675			−0.286338	−	0.958129i	$-0.592438\pi$
$701$	−15.1624			−0.572675			−0.286338	+	0.958129i	$0.592438\pi$
$702$		0			0
$703$	−69.4977			−2.62116
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−27.9690	−	48.4436i	−1.05188	−	1.82191i
$708$		0			0
$709$	19.8058	−	34.3046i	0.743821	−	1.28834i	−0.206922	−	0.978357i	$-0.566345\pi$
$709$	19.8058	−	34.3046i	0.743821	−	1.28834i	0.950743	−	0.309979i	$-0.100322\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	18.5823	−	32.1855i	0.695912	−	1.20536i
$714$		0			0
$715$	8.46054	+	14.6541i	0.316406	+	0.548032i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−35.7093			−1.33173			−0.665865	−	0.746072i	$-0.731937\pi$
$719$	−35.7093			−1.33173			−0.665865	+	0.746072i	$0.731937\pi$
$720$		0			0
$721$	−25.6462			−0.955114
$722$		0			0
$723$		0			0
$724$		0			0
$725$	0.381473	+	0.660731i	0.0141676	+	0.0245389i
$726$		0			0
$727$	−12.8269	+	22.2169i	−0.475724	+	0.823977i	−0.999613	−	0.0278087i	$-0.991147\pi$
$727$	−12.8269	+	22.2169i	−0.475724	+	0.823977i	0.523890	+	0.851786i	$0.324480\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−26.0511	+	45.1218i	−0.963535	+	1.66889i
$732$		0			0
$733$	−1.15264	−	1.99644i	−0.0425739	−	0.0737401i	0.843953	−	0.536417i	$-0.180222\pi$
$733$	−1.15264	−	1.99644i	−0.0425739	−	0.0737401i	−0.886527	+	0.462677i	$0.846889\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−25.7309			−0.947810
$738$		0			0
$739$	7.86679			0.289385			0.144692	−	0.989477i	$-0.453781\pi$
$739$	7.86679			0.289385			0.144692	+	0.989477i	$0.453781\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	3.78253	+	6.55154i	0.138768	+	0.240353i	0.927030	−	0.374986i	$-0.122353\pi$
$743$	3.78253	+	6.55154i	0.138768	+	0.240353i	−0.788263	+	0.615339i	$0.789019\pi$
$744$		0			0
$745$	−1.08085	+	1.87208i	−0.0395992	+	0.0685878i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	31.2946	−	54.2038i	1.14348	−	1.98057i
$750$		0			0
$751$	25.3043	+	43.8284i	0.923368	+	1.59932i	0.794165	+	0.607703i	$0.207909\pi$
$751$	25.3043	+	43.8284i	0.923368	+	1.59932i	0.129204	+	0.991618i	$0.458758\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	8.44793			0.307452
$756$		0			0
$757$	−54.7114			−1.98852			−0.994259	−	0.106999i	$-0.965876\pi$
$757$	−54.7114			−1.98852			−0.994259	+	0.106999i	$0.965876\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	17.9778	+	31.1384i	0.651694	+	1.12877i	0.982712	+	0.185142i	$0.0592745\pi$
$761$	17.9778	+	31.1384i	0.651694	+	1.12877i	−0.331018	+	0.943624i	$0.607392\pi$
$762$		0			0
$763$	11.2039	−	19.4057i	0.405608	−	0.702534i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−5.42512	+	9.39658i	−0.195890	+	0.339291i
$768$		0			0
$769$	1.01164	+	1.75220i	0.0364805	+	0.0631861i	0.883689	−	0.468074i	$-0.155052\pi$
$769$	1.01164	+	1.75220i	0.0364805	+	0.0631861i	−0.847209	+	0.531260i	$0.821719\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−19.5708			−0.703914			−0.351957	−	0.936016i	$-0.614484\pi$
$773$	−19.5708			−0.703914			−0.351957	+	0.936016i	$0.614484\pi$
$774$		0			0
$775$	6.61675			0.237681
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−28.1359	−	48.7328i	−1.00807	−	1.74603i
$780$		0			0
$781$	−19.3156	+	33.4556i	−0.691166	+	1.19713i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	2.87555	−	4.98060i	0.102633	−	0.177765i
$786$		0			0
$787$	0.490131	+	0.848932i	0.0174713	+	0.0302612i	0.874629	−	0.484793i	$-0.161105\pi$
$787$	0.490131	+	0.848932i	0.0174713	+	0.0302612i	−0.857158	+	0.515054i	$0.827772\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−79.9421			−2.84241
$792$		0			0
$793$	−10.2292			−0.363251
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−6.09346	−	10.5542i	−0.215841	−	0.373848i	0.737691	−	0.675138i	$-0.235916\pi$
$797$	−6.09346	−	10.5542i	−0.215841	−	0.373848i	−0.953533	+	0.301290i	$0.902583\pi$
$798$		0			0
$799$	14.0600	−	24.3526i	0.497405	−	0.861532i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−14.3339	+	24.8271i	−0.505833	+	0.876129i
$804$		0			0
$805$	12.2130	+	21.1536i	0.430453	+	0.745567i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−13.6760			−0.480823			−0.240412	−	0.970671i	$-0.577282\pi$
$809$	−13.6760			−0.480823			−0.240412	+	0.970671i	$0.577282\pi$
$810$		0			0
$811$	−13.4256			−0.471436			−0.235718	−	0.971822i	$-0.575744\pi$
$811$	−13.4256			−0.471436			−0.235718	+	0.971822i	$0.575744\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	6.42190	+	11.1231i	0.224949	+	0.389624i
$816$		0			0
$817$	30.1856	−	52.2830i	1.05606	−	1.82915i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	9.96732	−	17.2639i	0.347862	−	0.602515i	−0.638007	−	0.770030i	$-0.720241\pi$
$821$	9.96732	−	17.2639i	0.347862	−	0.602515i	0.985869	+	0.167516i	$0.0535745\pi$
$822$		0			0
$823$	−26.2763	−	45.5119i	−0.915935	−	1.58645i	−0.805527	−	0.592559i	$-0.798118\pi$
$823$	−26.2763	−	45.5119i	−0.915935	−	1.58645i	−0.110408	−	0.993886i	$-0.535216\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	50.1251			1.74302			0.871511	−	0.490376i	$-0.163141\pi$
$827$	50.1251			1.74302			0.871511	+	0.490376i	$0.163141\pi$
$828$		0			0
$829$	−13.1526			−0.456810			−0.228405	−	0.973566i	$-0.573351\pi$
$829$	−13.1526			−0.456810			−0.228405	+	0.973566i	$0.573351\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−32.0415	−	55.4975i	−1.11017	−	1.92288i
$834$		0			0
$835$	−10.9542	+	18.9733i	−0.379087	+	0.656598i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−12.1797	+	21.0958i	−0.420488	+	0.728307i	−0.995987	−	0.0894955i	$-0.971475\pi$
$839$	−12.1797	+	21.0958i	−0.420488	+	0.728307i	0.575499	+	0.817802i	$0.304808\pi$
$840$		0			0
$841$	14.2090	+	24.6106i	0.489964	+	0.848643i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−26.2425			−0.902771
$846$		0			0
$847$	16.1069			0.553440
$848$		0			0
$849$		0			0
$850$		0			0
$851$	31.3108	+	54.2319i	1.07332	+	1.85904i
$852$		0			0
$853$	−2.78898	+	4.83065i	−0.0954927	+	0.165398i	−0.909814	−	0.415016i	$-0.863776\pi$
$853$	−2.78898	+	4.83065i	−0.0954927	+	0.165398i	0.814321	+	0.580414i	$0.197109\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−2.01618	+	3.49212i	−0.0688712	+	0.119288i	−0.898405	−	0.439169i	$-0.855273\pi$
$857$	−2.01618	+	3.49212i	−0.0688712	+	0.119288i	0.829533	+	0.558457i	$0.188606\pi$
$858$		0			0
$859$	19.2775	+	33.3896i	0.657739	+	1.13924i	0.981200	+	0.192996i	$0.0618204\pi$
$859$	19.2775	+	33.3896i	0.657739	+	1.13924i	−0.323461	+	0.946242i	$0.604846\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−31.3959			−1.06873			−0.534364	−	0.845255i	$-0.679449\pi$
$863$	−31.3959			−1.06873			−0.534364	+	0.845255i	$0.679449\pi$
$864$		0			0
$865$	−2.68854			−0.0914132
$866$		0			0
$867$		0			0
$868$		0			0
$869$	0.723772	+	1.25361i	0.0245523	+	0.0425258i
$870$		0			0
$871$	29.8369	−	51.6791i	1.01099	−	1.75108i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−2.17440	+	3.76617i	−0.0735081	+	0.127320i
$876$		0			0
$877$	−12.0604	−	20.8892i	−0.407251	−	0.705379i	0.587330	−	0.809348i	$-0.300179\pi$
$877$	−12.0604	−	20.8892i	−0.407251	−	0.705379i	−0.994581	+	0.103969i	$0.966846\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−42.3270			−1.42603			−0.713016	−	0.701148i	$-0.752671\pi$
$881$	−42.3270			−1.42603			−0.713016	+	0.701148i	$0.752671\pi$
$882$		0			0
$883$	44.6242			1.50172			0.750861	−	0.660460i	$-0.229639\pi$
$883$	44.6242			1.50172			0.750861	+	0.660460i	$0.229639\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	12.7978	+	22.1665i	0.429709	+	0.744278i	0.996847	−	0.0793445i	$-0.0252827\pi$
$887$	12.7978	+	22.1665i	0.429709	+	0.744278i	−0.567138	+	0.823623i	$0.691949\pi$
$888$		0			0
$889$	−15.8231	+	27.4064i	−0.530690	+	0.919181i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−16.2914	+	28.2175i	−0.545170	+	0.944262i
$894$		0			0
$895$	4.51367	+	7.81790i	0.150875	+	0.261324i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−5.04822			−0.168368
$900$		0			0
$901$	57.1827			1.90503
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−2.92512	−	5.06645i	−0.0972342	−	0.168415i
$906$		0			0
$907$	−14.7005	+	25.4620i	−0.488121	+	0.845451i	−0.999907	−	0.0136626i	$-0.995651\pi$
$907$	−14.7005	+	25.4620i	−0.488121	+	0.845451i	0.511785	+	0.859113i	$0.328984\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−7.34345	+	12.7192i	−0.243299	+	0.421407i	−0.961652	−	0.274272i	$-0.911563\pi$
$911$	−7.34345	+	12.7192i	−0.243299	+	0.421407i	0.718353	+	0.695679i	$0.244896\pi$
$912$		0			0
$913$	−0.197456	−	0.342004i	−0.00653485	−	0.0113187i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	23.3312			0.770463
$918$		0			0
$919$	0.353784			0.0116703			0.00583513	−	0.999983i	$-0.498143\pi$
$919$	0.353784			0.0116703			0.00583513	+	0.999983i	$0.498143\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−44.7958	−	77.5885i	−1.47447	−	2.55386i
$924$		0			0
$925$	−5.57454	+	9.65539i	−0.183290	+	0.317467i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−13.6670	+	23.6720i	−0.448400	+	0.776652i	−0.998282	−	0.0585902i	$-0.981339\pi$
$929$	−13.6670	+	23.6720i	−0.448400	+	0.776652i	0.549882	+	0.835243i	$0.314673\pi$
$930$		0			0
$931$	37.1268	+	64.3054i	1.21678	+	2.10753i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	14.5314			0.475227
$936$		0			0
$937$	−4.17594			−0.136422			−0.0682111	−	0.997671i	$-0.521729\pi$
$937$	−4.17594			−0.136422			−0.0682111	+	0.997671i	$0.521729\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	8.59846	+	14.8930i	0.280302	+	0.485497i	0.971459	−	0.237207i	$-0.0762321\pi$
$941$	8.59846	+	14.8930i	0.280302	+	0.485497i	−0.691157	+	0.722704i	$0.742899\pi$
$942$		0			0
$943$	−25.3521	+	43.9112i	−0.825579	+	1.42994i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	16.3543	−	28.3265i	0.531443	−	0.920486i	−0.467884	−	0.883790i	$-0.654983\pi$
$947$	16.3543	−	28.3265i	0.531443	−	0.920486i	0.999326	−	0.0366959i	$-0.0116833\pi$
$948$		0			0
$949$	−33.2425	−	57.5778i	−1.07910	−	1.86905i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	33.1122			1.07261			0.536305	−	0.844024i	$-0.319820\pi$
$953$	33.1122			1.07261			0.536305	+	0.844024i	$0.319820\pi$
$954$		0			0
$955$	12.9936			0.420462
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−10.2144	−	17.6919i	−0.329841	−	0.571302i
$960$		0			0
$961$	−6.39066	+	11.0689i	−0.206150	+	0.357063i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	2.25664	−	3.90862i	0.0726438	−	0.125823i
$966$		0			0
$967$	−24.7636	−	42.8918i	−0.796343	−	1.37931i	−0.921983	−	0.387230i	$-0.873432\pi$
$967$	−24.7636	−	42.8918i	−0.796343	−	1.37931i	0.125640	−	0.992076i	$-0.459901\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	4.82907			0.154972			0.0774862	−	0.996993i	$-0.475311\pi$
$971$	4.82907			0.154972			0.0774862	+	0.996993i	$0.475311\pi$
$972$		0			0
$973$	2.88048			0.0923439
$974$		0			0
$975$		0			0
$976$		0			0
$977$	7.47054	+	12.9394i	0.239004	+	0.413967i	0.960429	−	0.278526i	$-0.0898458\pi$
$977$	7.47054	+	12.9394i	0.239004	+	0.413967i	−0.721425	+	0.692493i	$0.756512\pi$
$978$		0			0
$979$	10.6155	−	18.3866i	0.339273	−	0.587638i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−25.2847	+	43.7945i	−0.806458	+	1.39683i	0.108844	+	0.994059i	$0.465285\pi$
$983$	−25.2847	+	43.7945i	−0.806458	+	1.39683i	−0.915302	+	0.402768i	$0.868048\pi$
$984$		0			0
$985$	−8.42156	−	14.5866i	−0.268333	−	0.464767i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−54.3981			−1.72976
$990$		0			0
$991$	37.4566			1.18985			0.594924	−	0.803782i	$-0.297182\pi$
$991$	37.4566			1.18985			0.594924	+	0.803782i	$0.297182\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	5.80029	+	10.0464i	0.183881	+	0.318492i
$996$		0			0
$997$	3.86911	−	6.70150i	0.122536	−	0.212239i	−0.798231	−	0.602351i	$-0.794231\pi$
$997$	3.86911	−	6.70150i	0.122536	−	0.212239i	0.920767	+	0.390113i	$0.127564\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3240.2.q.bh.2161.4		8
3.2	odd	2		3240.2.q.bg.2161.4		8
9.2	odd	6		3240.2.a.v.1.1	yes	4
9.4	even	3	inner	3240.2.q.bh.1081.4		8
9.5	odd	6		3240.2.q.bg.1081.4		8
9.7	even	3		3240.2.a.t.1.1	✓	4
36.7	odd	6		6480.2.a.by.1.4		4
36.11	even	6		6480.2.a.ca.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3240.2.a.t.1.1	✓	4	9.7	even	3
3240.2.a.v.1.1	yes	4	9.2	odd	6
3240.2.q.bg.1081.4		8	9.5	odd	6
3240.2.q.bg.2161.4		8	3.2	odd	2
3240.2.q.bh.1081.4		8	9.4	even	3	inner
3240.2.q.bh.2161.4		8	1.1	even	1	trivial
6480.2.a.by.1.4		4	36.7	odd	6
6480.2.a.ca.1.4		4	36.11	even	6

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)

\(f(q)\)	\(=\)	\(q+(0.500000 + 0.866025i) q^{5} +(2.17440 - 3.76617i) q^{7} +(1.35058 - 2.33927i) q^{11} +(3.13219 + 5.42512i) q^{13} +5.37969 q^{17} -6.23349 q^{19} +(2.80837 + 4.86424i) q^{23} +(-0.500000 + 0.866025i) q^{25} +(0.381473 - 0.660731i) q^{29} +(-3.30837 - 5.73027i) q^{31} +4.34880 q^{35} +11.1491 q^{37} +(4.51367 + 7.81790i) q^{41} +(-4.84249 + 8.38744i) q^{43} +(2.61352 - 4.52676i) q^{47} +(-5.95601 - 10.3161i) q^{49} +10.6294 q^{53} +2.70115 q^{55} +(0.866025 + 1.50000i) q^{59} +(-0.816460 + 1.41415i) q^{61} +(-3.13219 + 5.42512i) q^{65} +(-4.76295 - 8.24967i) q^{67} -14.3017 q^{71} -10.6132 q^{73} +(-5.87339 - 10.1730i) q^{77} +(-0.267949 + 0.464102i) q^{79} +(0.0731007 - 0.126614i) q^{83} +(2.68985 + 4.65895i) q^{85} +7.85997 q^{89} +27.2425 q^{91} +(-3.11675 - 5.39836i) q^{95} +(0.194848 - 0.337487i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 4 q^{5} - 2 q^{7} + 4 q^{11} + 4 q^{17} + 10 q^{23} - 4 q^{25} - 4 q^{29} - 14 q^{31} - 4 q^{35} + 28 q^{37} + 4 q^{41} - 22 q^{43} - 10 q^{49} + 16 q^{53} + 8 q^{55} - 4 q^{61} - 24 q^{67} - 56 q^{71}+ \cdots + 10 q^{97}+O(q^{100}) \) 8 * q + 4 * q^5 - 2 * q^7 + 4 * q^11 + 4 * q^17 + 10 * q^23 - 4 * q^25 - 4 * q^29 - 14 * q^31 - 4 * q^35 + 28 * q^37 + 4 * q^41 - 22 * q^43 - 10 * q^49 + 16 * q^53 + 8 * q^55 - 4 * q^61 - 24 * q^67 - 56 * q^71 + 4 * q^73 - 30 * q^77 - 16 * q^79 + 6 * q^83 + 2 * q^85 + 16 * q^89 + 60 * q^91 + 10 * q^97

Introduction

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