LMFDB - Embedded newform 64.3.f.a.47.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [64,3,Mod(15,64)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(64, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 1]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("64.15");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$N$	$=$	$64 = 2^{6}$
Weight:	$k$	$=$	$3$
Character orbit:	$[\chi]$	$=$	64.f (of order $4$ , degree $2$ , not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$1.74387369191$
Analytic rank:	$0$
Dimension:	$6$
Relative dimension:	$3$ over $\Q(i)$
Coefficient field:	6.0.399424.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{6} - 2x^{5} + 3x^{4} - 6x^{3} + 6x^{2} - 8x + 8$ x^6 - 2x^5 + 3x^4 - 6x^3 + 6x^2 - 8*x + 8
Coefficient ring:	$\Z[a_1, \ldots, a_{9}]$
Coefficient ring index:	$2^{6}$
Twist minimal:	no (minimal twist has level 16)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			47.2
Root			$-0.671462 + 1.24464i$ of defining polynomial
Character	$\chi$	$=$	64.47
Dual form			64.3.f.a.15.2

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.146365 + 0.146365i) q^{3} +(3.68585 - 3.68585i) q^{5} +9.66442 q^{7} +8.95715i q^{9} +(-5.51806 - 5.51806i) q^{11} +(-6.27131 - 6.27131i) q^{13} +1.07896i q^{15} -6.78623 q^{17} +(-13.5181 + 13.5181i) q^{19} +(-1.41454 + 1.41454i) q^{21} -17.0790 q^{23} -2.17092i q^{25} +(-2.62831 - 2.62831i) q^{27} +(4.85677 + 4.85677i) q^{29} +5.25662i q^{31} +1.61531 q^{33} +(35.6216 - 35.6216i) q^{35} +(-18.1856 + 18.1856i) q^{37} +1.83581 q^{39} -48.2302i q^{41} +(54.5113 + 54.5113i) q^{43} +(33.0147 + 33.0147i) q^{45} -40.4015i q^{47} +44.4011 q^{49} +(0.993270 - 0.993270i) q^{51} +(10.8996 - 10.8996i) q^{53} -40.6774 q^{55} -3.95715i q^{57} +(-50.8898 - 50.8898i) q^{59} +(-17.0147 - 17.0147i) q^{61} +86.5657i q^{63} -46.2302 q^{65} +(-22.9191 + 22.9191i) q^{67} +(2.49977 - 2.49977i) q^{69} +51.6047 q^{71} +78.5032i q^{73} +(0.317748 + 0.317748i) q^{75} +(-53.3288 - 53.3288i) q^{77} -108.512i q^{79} -79.8450 q^{81} +(-57.3173 + 57.3173i) q^{83} +(-25.0130 + 25.0130i) q^{85} -1.42173 q^{87} +44.1276i q^{89} +(-60.6086 - 60.6086i) q^{91} +(-0.769387 - 0.769387i) q^{93} +99.6510i q^{95} +112.700 q^{97} +(49.4261 - 49.4261i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$6 q + 2 q^{3} - 2 q^{5} + 4 q^{7} + 18 q^{11} - 2 q^{13} - 4 q^{17} - 30 q^{19} - 20 q^{21} - 60 q^{23} - 64 q^{27} - 18 q^{29} - 4 q^{33} + 100 q^{35} + 46 q^{37} + 196 q^{39} + 114 q^{43} + 66 q^{45}+ \cdots + 226 q^{99}+O(q^{100})$ 6 * q + 2 * q^3 - 2 * q^5 + 4 * q^7 + 18 * q^11 - 2 * q^13 - 4 * q^17 - 30 * q^19 - 20 * q^21 - 60 * q^23 - 64 * q^27 - 18 * q^29 - 4 * q^33 + 100 * q^35 + 46 * q^37 + 196 * q^39 + 114 * q^43 + 66 * q^45 - 46 * q^49 - 156 * q^51 + 78 * q^53 - 252 * q^55 - 206 * q^59 + 30 * q^61 + 12 * q^65 + 226 * q^67 - 116 * q^69 + 260 * q^71 + 238 * q^75 - 212 * q^77 + 86 * q^81 - 318 * q^83 - 212 * q^85 - 444 * q^87 - 188 * q^91 - 32 * q^93 - 4 * q^97 + 226 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/64\mathbb{Z}\right)^\times$ .

$n$	$5$	$63$
$\chi(n)$	$e\left(\frac{3}{4}\right)$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$ -expansion $a_n$ , the Satake parameters $\alpha_p$ , and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$ .

Display

a_p

with

p

up to: 50 250 1000 (See

a_n

instead) (See

a_n

instead) (See

a_n

instead) Display

a_n

with

n

up to: 50 250 1000 (See only

a_p

) (See only

a_p

) (See only

a_p

)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$\alpha_n$			$\theta_n$
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$\alpha_p$			$\theta_p$

$2$		0			0
$2$		0			0
$3$	−0.146365	+	0.146365i	−0.0487885	+	0.0487885i	−0.731080	−	0.682292i	$-0.760983\pi$
$3$	−0.146365	+	0.146365i	−0.0487885	+	0.0487885i	0.682292	+	0.731080i	$0.260983\pi$
$4$		0			0
$5$	3.68585	−	3.68585i	0.737169	−	0.737169i	−0.234860	−	0.972029i	$-0.575463\pi$
$5$	3.68585	−	3.68585i	0.737169	−	0.737169i	0.972029	+	0.234860i	$0.0754632\pi$
$6$		0			0
$7$	9.66442			1.38063			0.690316	−	0.723508i	$-0.257472\pi$
$7$	9.66442			1.38063			0.690316	+	0.723508i	$0.257472\pi$
$8$		0			0
$9$			8.95715i			0.995239i
$10$		0			0
$11$	−5.51806	−	5.51806i	−0.501642	−	0.501642i	0.410306	−	0.911948i	$-0.365422\pi$
$11$	−5.51806	−	5.51806i	−0.501642	−	0.501642i	−0.911948	+	0.410306i	$0.865422\pi$
$12$		0			0
$13$	−6.27131	−	6.27131i	−0.482408	−	0.482408i	0.423492	−	0.905900i	$-0.360804\pi$
$13$	−6.27131	−	6.27131i	−0.482408	−	0.482408i	−0.905900	+	0.423492i	$0.860804\pi$
$14$		0			0
$15$			1.07896i			0.0719308i
$16$		0			0
$17$	−6.78623			−0.399190			−0.199595	−	0.979878i	$-0.563963\pi$
$17$	−6.78623			−0.399190			−0.199595	+	0.979878i	$0.563963\pi$
$18$		0			0
$19$	−13.5181	+	13.5181i	−0.711477	+	0.711477i	−0.966844	−	0.255367i	$-0.917804\pi$
$19$	−13.5181	+	13.5181i	−0.711477	+	0.711477i	0.255367	+	0.966844i	$0.417804\pi$
$20$		0			0
$21$	−1.41454	+	1.41454i	−0.0673590	+	0.0673590i
$22$		0			0
$23$	−17.0790			−0.742564			−0.371282	−	0.928520i	$-0.621082\pi$
$23$	−17.0790			−0.742564			−0.371282	+	0.928520i	$0.621082\pi$
$24$		0			0
$25$		−	2.17092i		−	0.0868370i
$26$		0			0
$27$	−2.62831	−	2.62831i	−0.0973447	−	0.0973447i
$28$		0			0
$29$	4.85677	+	4.85677i	0.167475	+	0.167475i	0.785868	−	0.618394i	$-0.212216\pi$
$29$	4.85677	+	4.85677i	0.167475	+	0.167475i	−0.618394	+	0.785868i	$0.712216\pi$
$30$		0			0
$31$			5.25662i			0.169568i	0.996399	+	0.0847841i	$0.0270201\pi$
$31$			5.25662i			0.169568i	−0.996399	+	0.0847841i	$0.972980\pi$
$32$		0			0
$33$	1.61531			0.0489487
$34$		0			0
$35$	35.6216	−	35.6216i	1.01776	−	1.01776i
$36$		0			0
$37$	−18.1856	+	18.1856i	−0.491503	+	0.491503i	−0.908780	−	0.417276i	$-0.862985\pi$
$37$	−18.1856	+	18.1856i	−0.491503	+	0.491503i	0.417276	+	0.908780i	$0.362985\pi$
$38$		0			0
$39$	1.83581			0.0470720
$40$		0			0
$41$		−	48.2302i		−	1.17635i	−0.808735	−	0.588173i	$-0.799848\pi$
$41$		−	48.2302i		−	1.17635i	0.808735	−	0.588173i	$-0.200152\pi$
$42$		0			0
$43$	54.5113	+	54.5113i	1.26771	+	1.26771i	0.947271	+	0.320435i	$0.103829\pi$
$43$	54.5113	+	54.5113i	1.26771	+	1.26771i	0.320435	+	0.947271i	$0.396171\pi$
$44$		0			0
$45$	33.0147	+	33.0147i	0.733660	+	0.733660i
$46$		0			0
$47$		−	40.4015i		−	0.859607i	−0.902922	−	0.429804i	$-0.858583\pi$
$47$		−	40.4015i		−	0.859607i	0.902922	−	0.429804i	$-0.141417\pi$
$48$		0			0
$49$	44.4011			0.906144
$50$		0			0
$51$	0.993270	−	0.993270i	0.0194759	−	0.0194759i
$52$		0			0
$53$	10.8996	−	10.8996i	0.205653	−	0.205653i	−0.596764	−	0.802417i	$-0.703547\pi$
$53$	10.8996	−	10.8996i	0.205653	−	0.205653i	0.802417	+	0.596764i	$0.203547\pi$
$54$		0			0
$55$	−40.6774			−0.739590
$56$		0			0
$57$		−	3.95715i		−	0.0694238i
$58$		0			0
$59$	−50.8898	−	50.8898i	−0.862538	−	0.862538i	0.129094	−	0.991632i	$-0.458793\pi$
$59$	−50.8898	−	50.8898i	−0.862538	−	0.862538i	−0.991632	+	0.129094i	$0.958793\pi$
$60$		0			0
$61$	−17.0147	−	17.0147i	−0.278929	−	0.278929i	0.553752	−	0.832682i	$-0.313196\pi$
$61$	−17.0147	−	17.0147i	−0.278929	−	0.278929i	−0.832682	+	0.553752i	$0.813196\pi$
$62$		0			0
$63$			86.5657i			1.37406i
$64$		0			0
$65$	−46.2302			−0.711233
$66$		0			0
$67$	−22.9191	+	22.9191i	−0.342077	+	0.342077i	−0.857148	−	0.515071i	$-0.827766\pi$
$67$	−22.9191	+	22.9191i	−0.342077	+	0.342077i	0.515071	+	0.857148i	$0.327766\pi$
$68$		0			0
$69$	2.49977	−	2.49977i	0.0362286	−	0.0362286i
$70$		0			0
$71$	51.6047			0.726827			0.363414	−	0.931628i	$-0.381611\pi$
$71$	51.6047			0.726827			0.363414	+	0.931628i	$0.381611\pi$
$72$		0			0
$73$			78.5032i			1.07539i	0.843141	+	0.537693i	$0.180704\pi$
$73$			78.5032i			1.07539i	−0.843141	+	0.537693i	$0.819296\pi$
$74$		0			0
$75$	0.317748	+	0.317748i	0.00423664	+	0.00423664i
$76$		0			0
$77$	−53.3288	−	53.3288i	−0.692582	−	0.692582i
$78$		0			0
$79$		−	108.512i		−	1.37357i	−0.726859	−	0.686787i	$-0.759021\pi$
$79$		−	108.512i		−	1.37357i	0.726859	−	0.686787i	$-0.240979\pi$
$80$		0			0
$81$	−79.8450			−0.985741
$82$		0			0
$83$	−57.3173	+	57.3173i	−0.690570	+	0.690570i	−0.962357	−	0.271788i	$-0.912385\pi$
$83$	−57.3173	+	57.3173i	−0.690570	+	0.690570i	0.271788	+	0.962357i	$0.412385\pi$
$84$		0			0
$85$	−25.0130	+	25.0130i	−0.294271	+	0.294271i
$86$		0			0
$87$	−1.42173			−0.0163417
$88$		0			0
$89$			44.1276i			0.495816i	0.968784	+	0.247908i	$0.0797431\pi$
$89$			44.1276i			0.495816i	−0.968784	+	0.247908i	$0.920257\pi$
$90$		0			0
$91$	−60.6086	−	60.6086i	−0.666028	−	0.666028i
$92$		0			0
$93$	−0.769387	−	0.769387i	−0.00827298	−	0.00827298i
$94$		0			0
$95$			99.6510i			1.04896i
$96$		0			0
$97$	112.700			1.16185			0.580926	−	0.813956i	$-0.302691\pi$
$97$	112.700			1.16185			0.580926	+	0.813956i	$0.302691\pi$
$98$		0			0
$99$	49.4261	−	49.4261i	0.499253	−	0.499253i
$100$		0			0
$101$	97.3859	−	97.3859i	0.964217	−	0.964217i	−0.0351644	−	0.999382i	$-0.511195\pi$
$101$	97.3859	−	97.3859i	0.964217	−	0.964217i	0.999382	+	0.0351644i	$0.0111955\pi$
$102$		0			0
$103$	138.698			1.34658			0.673290	−	0.739379i	$-0.264881\pi$
$103$	138.698			1.34658			0.673290	+	0.739379i	$0.264881\pi$
$104$		0			0
$105$			10.4275i			0.0993099i
$106$		0			0
$107$	31.7386	+	31.7386i	0.296622	+	0.296622i	0.839689	−	0.543067i	$-0.182737\pi$
$107$	31.7386	+	31.7386i	0.296622	+	0.296622i	−0.543067	+	0.839689i	$0.682737\pi$
$108$		0			0
$109$	0.712308	+	0.712308i	0.00653493	+	0.00653493i	0.710367	−	0.703832i	$-0.248529\pi$
$109$	0.712308	+	0.712308i	0.00653493	+	0.00653493i	−0.703832	+	0.710367i	$0.748529\pi$
$110$		0			0
$111$		−	5.32349i		−	0.0479594i
$112$		0			0
$113$	14.8888			0.131759			0.0658795	−	0.997828i	$-0.479015\pi$
$113$	14.8888			0.131759			0.0658795	+	0.997828i	$0.479015\pi$
$114$		0			0
$115$	−62.9504	+	62.9504i	−0.547395	+	0.547395i
$116$		0			0
$117$	56.1731	−	56.1731i	0.480112	−	0.480112i
$118$		0			0
$119$	−65.5850			−0.551134
$120$		0			0
$121$		−	60.1021i		−	0.496711i
$122$		0			0
$123$	7.05923	+	7.05923i	0.0573921	+	0.0573921i
$124$		0			0
$125$	84.1445	+	84.1445i	0.673156	+	0.673156i
$126$		0			0
$127$		−	106.861i		−	0.841425i	−0.907194	−	0.420712i	$-0.861780\pi$
$127$		−	106.861i		−	0.841425i	0.907194	−	0.420712i	$-0.138220\pi$
$128$		0			0
$129$	−15.9572			−0.123699
$130$		0			0
$131$	153.198	−	153.198i	1.16945	−	1.16945i	0.187116	−	0.982338i	$-0.440086\pi$
$131$	153.198	−	153.198i	1.16945	−	1.16945i	0.982338	−	0.187116i	$-0.0599139\pi$
$132$		0			0
$133$	−130.644	+	130.644i	−0.982287	+	0.982287i
$134$		0			0
$135$	−19.3751			−0.143519
$136$		0			0
$137$		−	75.1700i		−	0.548686i	−0.961632	−	0.274343i	$-0.911540\pi$
$137$		−	75.1700i		−	0.548686i	0.961632	−	0.274343i	$-0.0884604\pi$
$138$		0			0
$139$	107.425	+	107.425i	0.772843	+	0.772843i	0.978603	−	0.205760i	$-0.0659665\pi$
$139$	107.425	+	107.425i	0.772843	+	0.772843i	−0.205760	+	0.978603i	$0.565966\pi$
$140$		0			0
$141$	5.91339	+	5.91339i	0.0419389	+	0.0419389i
$142$		0			0
$143$			69.2109i			0.483992i
$144$		0			0
$145$	35.8026			0.246915
$146$		0			0
$147$	−6.49879	+	6.49879i	−0.0442094	+	0.0442094i
$148$		0			0
$149$	−146.031	+	146.031i	−0.980074	+	0.980074i	−0.999805	−	0.0197310i	$-0.993719\pi$
$149$	−146.031	+	146.031i	−0.980074	+	0.980074i	0.0197310	+	0.999805i	$0.493719\pi$
$150$		0			0
$151$	−220.513			−1.46035			−0.730175	−	0.683260i	$-0.760561\pi$
$151$	−220.513			−1.46035			−0.730175	+	0.683260i	$0.760561\pi$
$152$		0			0
$153$		−	60.7853i		−	0.397290i
$154$		0			0
$155$	19.3751	+	19.3751i	0.125000	+	0.125000i
$156$		0			0
$157$	−109.561	−	109.561i	−0.697839	−	0.697839i	0.266105	−	0.963944i	$-0.414263\pi$
$157$	−109.561	−	109.561i	−0.697839	−	0.697839i	−0.963944	+	0.266105i	$0.914263\pi$
$158$		0			0
$159$			3.19066i			0.0200670i
$160$		0			0
$161$	−165.058			−1.02521
$162$		0			0
$163$	−56.7781	+	56.7781i	−0.348332	+	0.348332i	−0.859488	−	0.511156i	$-0.829217\pi$
$163$	−56.7781	+	56.7781i	−0.348332	+	0.348332i	0.511156	+	0.859488i	$0.329217\pi$
$164$		0			0
$165$	5.95377	−	5.95377i	0.0360835	−	0.0360835i
$166$		0			0
$167$	−106.677			−0.638781			−0.319391	−	0.947623i	$-0.603478\pi$
$167$	−106.677			−0.638781			−0.319391	+	0.947623i	$0.603478\pi$
$168$		0			0
$169$		−	90.3414i		−	0.534564i
$170$		0			0
$171$	−121.083	−	121.083i	−0.708090	−	0.708090i
$172$		0			0
$173$	178.360	+	178.360i	1.03098	+	1.03098i	0.999504	+	0.0314805i	$0.0100222\pi$
$173$	178.360	+	178.360i	1.03098	+	1.03098i	0.0314805	+	0.999504i	$0.489978\pi$
$174$		0			0
$175$		−	20.9807i		−	0.119890i
$176$		0			0
$177$	14.8970			0.0841639
$178$		0			0
$179$	−60.4622	+	60.4622i	−0.337778	+	0.337778i	−0.855530	−	0.517753i	$-0.826769\pi$
$179$	−60.4622	+	60.4622i	−0.337778	+	0.337778i	0.517753	+	0.855530i	$0.326769\pi$
$180$		0			0
$181$	147.113	−	147.113i	0.812779	−	0.812779i	−0.172271	−	0.985050i	$-0.555110\pi$
$181$	147.113	−	147.113i	0.812779	−	0.812779i	0.985050	+	0.172271i	$0.0551105\pi$
$182$		0			0
$183$	4.98073			0.0272171
$184$		0			0
$185$			134.059i			0.724642i
$186$		0			0
$187$	37.4468	+	37.4468i	0.200250	+	0.200250i
$188$		0			0
$189$	−25.4011	−	25.4011i	−0.134397	−	0.134397i
$190$		0			0
$191$			106.861i			0.559481i	0.960076	+	0.279741i	$0.0902485\pi$
$191$			106.861i			0.559481i	−0.960076	+	0.279741i	$0.909752\pi$
$192$		0			0
$193$	68.1873			0.353302			0.176651	−	0.984274i	$-0.443474\pi$
$193$	68.1873			0.353302			0.176651	+	0.984274i	$0.443474\pi$
$194$		0			0
$195$	6.76650	−	6.76650i	0.0347000	−	0.0347000i
$196$		0			0
$197$	−61.8529	+	61.8529i	−0.313974	+	0.313974i	−0.846447	−	0.532473i	$-0.821263\pi$
$197$	−61.8529	+	61.8529i	−0.313974	+	0.313974i	0.532473	+	0.846447i	$0.321263\pi$
$198$		0			0
$199$	158.466			0.796310			0.398155	−	0.917318i	$-0.369651\pi$
$199$	158.466			0.796310			0.398155	+	0.917318i	$0.369651\pi$
$200$		0			0
$201$		−	6.70914i		−	0.0333788i
$202$		0			0
$203$	46.9379	+	46.9379i	0.231221	+	0.231221i
$204$		0			0
$205$	−177.769	−	177.769i	−0.867166	−	0.867166i
$206$		0			0
$207$		−	152.979i		−	0.739028i
$208$		0			0
$209$	149.187			0.713813
$210$		0			0
$211$	197.031	−	197.031i	0.933798	−	0.933798i	−0.0641430	−	0.997941i	$-0.520431\pi$
$211$	197.031	−	197.031i	0.933798	−	0.933798i	0.997941	+	0.0641430i	$0.0204314\pi$
$212$		0			0
$213$	−7.55315	+	7.55315i	−0.0354608	+	0.0354608i
$214$		0			0
$215$	401.841			1.86903
$216$		0			0
$217$			50.8022i			0.234111i
$218$		0			0
$219$	−11.4902	−	11.4902i	−0.0524664	−	0.0524664i
$220$		0			0
$221$	42.5585	+	42.5585i	0.192573	+	0.192573i
$222$		0			0
$223$		−	15.7698i		−	0.0707168i	−0.999375	−	0.0353584i	$-0.988743\pi$
$223$		−	15.7698i		−	0.0707168i	0.999375	−	0.0353584i	$-0.0112573\pi$
$224$		0			0
$225$	19.4453			0.0864236
$226$		0			0
$227$	−199.289	+	199.289i	−0.877927	+	0.877927i	−0.993320	−	0.115393i	$-0.963187\pi$
$227$	−199.289	+	199.289i	−0.877927	+	0.877927i	0.115393	+	0.993320i	$0.463187\pi$
$228$		0			0
$229$	230.522	−	230.522i	1.00664	−	1.00664i	0.00666715	−	0.999978i	$-0.497878\pi$
$229$	230.522	−	230.522i	1.00664	−	1.00664i	0.999978	−	0.00666715i	$-0.00212224\pi$
$230$		0			0
$231$	15.6110			0.0675801
$232$		0			0
$233$			344.791i			1.47979i	0.672722	+	0.739895i	$0.265125\pi$
$233$			344.791i			1.47979i	−0.672722	+	0.739895i	$0.734875\pi$
$234$		0			0
$235$	−148.914	−	148.914i	−0.633676	−	0.633676i
$236$		0			0
$237$	15.8825	+	15.8825i	0.0670146	+	0.0670146i
$238$		0			0
$239$			77.1978i			0.323004i	0.986872	+	0.161502i	$0.0516337\pi$
$239$			77.1978i			0.323004i	−0.986872	+	0.161502i	$0.948366\pi$
$240$		0			0
$241$	−293.483			−1.21777			−0.608885	−	0.793259i	$-0.708383\pi$
$241$	−293.483			−1.21777			−0.608885	+	0.793259i	$0.708383\pi$
$242$		0			0
$243$	35.3413	−	35.3413i	0.145438	−	0.145438i
$244$		0			0
$245$	163.656	−	163.656i	0.667982	−	0.667982i
$246$		0			0
$247$	169.552			0.686445
$248$		0			0
$249$		−	16.7785i		−	0.0673837i
$250$		0			0
$251$	−79.6322	−	79.6322i	−0.317260	−	0.317260i	0.530454	−	0.847714i	$-0.322021\pi$
$251$	−79.6322	−	79.6322i	−0.317260	−	0.317260i	−0.847714	+	0.530454i	$0.822021\pi$
$252$		0			0
$253$	94.2427	+	94.2427i	0.372501	+	0.372501i
$254$		0			0
$255$		−	7.32208i		−	0.0287140i
$256$		0			0
$257$	221.860			0.863270			0.431635	−	0.902048i	$-0.357937\pi$
$257$	221.860			0.863270			0.431635	+	0.902048i	$0.357937\pi$
$258$		0			0
$259$	−175.753	+	175.753i	−0.678585	+	0.678585i
$260$		0			0
$261$	−43.5028	+	43.5028i	−0.166678	+	0.166678i
$262$		0			0
$263$	−374.223			−1.42290			−0.711451	−	0.702736i	$-0.751961\pi$
$263$	−374.223			−1.42290			−0.711451	+	0.702736i	$0.751961\pi$
$264$		0			0
$265$		−	80.3486i		−	0.303202i
$266$		0			0
$267$	−6.45876	−	6.45876i	−0.0241901	−	0.0241901i
$268$		0			0
$269$	−357.970	−	357.970i	−1.33075	−	1.33075i	−0.904704	−	0.426042i	$-0.859908\pi$
$269$	−357.970	−	357.970i	−1.33075	−	1.33075i	−0.426042	−	0.904704i	$-0.640092\pi$
$270$		0			0
$271$			359.030i			1.32484i	0.749135	+	0.662418i	$0.230470\pi$
$271$			359.030i			1.32484i	−0.749135	+	0.662418i	$0.769530\pi$
$272$		0			0
$273$	17.7420			0.0649890
$274$		0			0
$275$	−11.9793	+	11.9793i	−0.0435610	+	0.0435610i
$276$		0			0
$277$	−351.765	+	351.765i	−1.26991	+	1.26991i	−0.323775	+	0.946134i	$0.604952\pi$
$277$	−351.765	+	351.765i	−1.26991	+	1.26991i	−0.946134	+	0.323775i	$0.895048\pi$
$278$		0			0
$279$	−47.0843			−0.168761
$280$		0			0
$281$		−	191.390i		−	0.681103i	−0.940226	−	0.340552i	$-0.889386\pi$
$281$		−	191.390i		−	0.681103i	0.940226	−	0.340552i	$-0.110614\pi$
$282$		0			0
$283$	31.3119	+	31.3119i	0.110643	+	0.110643i	0.760261	−	0.649618i	$-0.225071\pi$
$283$	31.3119	+	31.3119i	0.110643	+	0.110643i	−0.649618	+	0.760261i	$0.725071\pi$
$284$		0			0
$285$	−14.5855	−	14.5855i	−0.0511771	−	0.0511771i
$286$		0			0
$287$		−	466.117i		−	1.62410i
$288$		0			0
$289$	−242.947			−0.840647
$290$		0			0
$291$	−16.4953	+	16.4953i	−0.0566850	+	0.0566850i
$292$		0			0
$293$	−92.0889	+	92.0889i	−0.314297	+	0.314297i	−0.846572	−	0.532275i	$-0.821337\pi$
$293$	−92.0889	+	92.0889i	−0.314297	+	0.314297i	0.532275	+	0.846572i	$0.321337\pi$
$294$		0			0
$295$	−375.144			−1.27167
$296$		0			0
$297$			29.0063i			0.0976643i
$298$		0			0
$299$	107.107	+	107.107i	0.358219	+	0.358219i
$300$		0			0
$301$	526.821	+	526.821i	1.75023	+	1.75023i
$302$		0			0
$303$			28.5079i			0.0940854i
$304$		0			0
$305$	−125.427			−0.411236
$306$		0			0
$307$	257.566	−	257.566i	0.838978	−	0.838978i	−0.149746	−	0.988724i	$-0.547846\pi$
$307$	257.566	−	257.566i	0.838978	−	0.838978i	0.988724	+	0.149746i	$0.0478457\pi$
$308$		0			0
$309$	−20.3005	+	20.3005i	−0.0656976	+	0.0656976i
$310$		0			0
$311$	130.914			0.420946			0.210473	−	0.977600i	$-0.432500\pi$
$311$	130.914			0.420946			0.210473	+	0.977600i	$0.432500\pi$
$312$		0			0
$313$		−	51.8354i		−	0.165608i	−0.996566	−	0.0828041i	$-0.973612\pi$
$313$		−	51.8354i		−	0.165608i	0.996566	−	0.0828041i	$-0.0263876\pi$
$314$		0			0
$315$	319.068	+	319.068i	1.01291	+	1.01291i
$316$		0			0
$317$	−109.636	−	109.636i	−0.345856	−	0.345856i	0.512707	−	0.858563i	$-0.328643\pi$
$317$	−109.636	−	109.636i	−0.345856	−	0.345856i	−0.858563	+	0.512707i	$0.828643\pi$
$318$		0			0
$319$		−	53.5999i		−	0.168025i
$320$		0			0
$321$	−9.29086			−0.0289435
$322$		0			0
$323$	91.7367	−	91.7367i	0.284014	−	0.284014i
$324$		0			0
$325$	−13.6145	+	13.6145i	−0.0418909	+	0.0418909i
$326$		0			0
$327$	−0.208514			−0.000637659
$328$		0			0
$329$		−	390.458i		−	1.18680i
$330$		0			0
$331$	−323.226	−	323.226i	−0.976515	−	0.976515i	0.0232157	−	0.999730i	$-0.492610\pi$
$331$	−323.226	−	323.226i	−0.976515	−	0.976515i	−0.999730	+	0.0232157i	$0.992610\pi$
$332$		0			0
$333$	−162.891	−	162.891i	−0.489163	−	0.489163i
$334$		0			0
$335$			168.953i			0.504337i
$336$		0			0
$337$	315.159			0.935191			0.467596	−	0.883943i	$-0.345120\pi$
$337$	315.159			0.935191			0.467596	+	0.883943i	$0.345120\pi$
$338$		0			0
$339$	−2.17920	+	2.17920i	−0.00642832	+	0.00642832i
$340$		0			0
$341$	29.0063	−	29.0063i	0.0850625	−	0.0850625i
$342$		0			0
$343$	−44.4459			−0.129580
$344$		0			0
$345$		−	18.4275i		−	0.0534132i
$346$		0			0
$347$	307.568	+	307.568i	0.886363	+	0.886363i	0.994172	−	0.107809i	$-0.0343835\pi$
$347$	307.568	+	307.568i	0.886363	+	0.886363i	−0.107809	+	0.994172i	$0.534384\pi$
$348$		0			0
$349$	170.461	+	170.461i	0.488427	+	0.488427i	0.907810	−	0.419382i	$-0.137753\pi$
$349$	170.461	+	170.461i	0.488427	+	0.488427i	−0.419382	+	0.907810i	$0.637753\pi$
$350$		0			0
$351$			32.9659i			0.0939198i
$352$		0			0
$353$	238.136			0.674606			0.337303	−	0.941396i	$-0.390485\pi$
$353$	238.136			0.674606			0.337303	+	0.941396i	$0.390485\pi$
$354$		0			0
$355$	190.207	−	190.207i	0.535795	−	0.535795i
$356$		0			0
$357$	9.59938	−	9.59938i	0.0268890	−	0.0268890i
$358$		0			0
$359$	−33.6470			−0.0937241			−0.0468620	−	0.998901i	$-0.514922\pi$
$359$	−33.6470			−0.0937241			−0.0468620	+	0.998901i	$0.514922\pi$
$360$		0			0
$361$		−	4.47577i		−	0.0123983i
$362$		0			0
$363$	8.79687	+	8.79687i	0.0242338	+	0.0242338i
$364$		0			0
$365$	289.351	+	289.351i	0.792741	+	0.792741i
$366$		0			0
$367$			240.758i			0.656016i	0.944675	+	0.328008i	$0.106377\pi$
$367$			240.758i			0.656016i	−0.944675	+	0.328008i	$0.893623\pi$
$368$		0			0
$369$	432.005			1.17075
$370$		0			0
$371$	105.339	−	105.339i	0.283931	−	0.283931i
$372$		0			0
$373$	432.504	−	432.504i	1.15953	−	1.15953i	0.174951	−	0.984577i	$-0.444023\pi$
$373$	432.504	−	432.504i	1.15953	−	1.15953i	0.984577	−	0.174951i	$-0.0559766\pi$
$374$		0			0
$375$	−24.6317			−0.0656845
$376$		0			0
$377$		−	60.9166i		−	0.161583i
$378$		0			0
$379$	174.716	+	174.716i	0.460993	+	0.460993i	0.898981	−	0.437988i	$-0.144309\pi$
$379$	174.716	+	174.716i	0.460993	+	0.460993i	−0.437988	+	0.898981i	$0.644309\pi$
$380$		0			0
$381$	15.6408	+	15.6408i	0.0410518	+	0.0410518i
$382$		0			0
$383$		−	673.381i		−	1.75817i	−0.476661	−	0.879087i	$-0.658153\pi$
$383$		−	673.381i		−	1.75817i	0.476661	−	0.879087i	$-0.341847\pi$
$384$		0			0
$385$	−393.124			−1.02110
$386$		0			0
$387$	−488.266	+	488.266i	−1.26167	+	1.26167i
$388$		0			0
$389$	−274.646	+	274.646i	−0.706031	+	0.706031i	−0.965698	−	0.259667i	$-0.916387\pi$
$389$	−274.646	+	274.646i	−0.706031	+	0.706031i	0.259667	+	0.965698i	$0.416387\pi$
$390$		0			0
$391$	115.902			0.296424
$392$		0			0
$393$			44.8459i			0.114112i
$394$		0			0
$395$	−399.960	−	399.960i	−1.01256	−	1.01256i
$396$		0			0
$397$	−271.254	−	271.254i	−0.683259	−	0.683259i	0.277474	−	0.960733i	$-0.410503\pi$
$397$	−271.254	−	271.254i	−0.683259	−	0.683259i	−0.960733	+	0.277474i	$0.910503\pi$
$398$		0			0
$399$		−	38.2436i		−	0.0958487i
$400$		0			0
$401$	−415.193			−1.03539			−0.517697	−	0.855564i	$-0.673210\pi$
$401$	−415.193			−1.03539			−0.517697	+	0.855564i	$0.673210\pi$
$402$		0			0
$403$	32.9659	−	32.9659i	0.0818011	−	0.0818011i
$404$		0			0
$405$	−294.296	+	294.296i	−0.726658	+	0.726658i
$406$		0			0
$407$	200.699			0.493117
$408$		0			0
$409$			634.686i			1.55180i	0.630856	+	0.775900i	$0.282704\pi$
$409$			634.686i			1.55180i	−0.630856	+	0.775900i	$0.717296\pi$
$410$		0			0
$411$	11.0023	+	11.0023i	0.0267696	+	0.0267696i
$412$		0			0
$413$	−491.820	−	491.820i	−1.19085	−	1.19085i
$414$		0			0
$415$			422.525i			1.01813i
$416$		0			0
$417$	−31.4467			−0.0754117
$418$		0			0
$419$	−19.2687	+	19.2687i	−0.0459873	+	0.0459873i	−0.729726	−	0.683739i	$-0.760353\pi$
$419$	−19.2687	+	19.2687i	−0.0459873	+	0.0459873i	0.683739	+	0.729726i	$0.260353\pi$
$420$		0			0
$421$	244.505	−	244.505i	0.580773	−	0.580773i	−0.354343	−	0.935116i	$-0.615295\pi$
$421$	244.505	−	244.505i	0.580773	−	0.580773i	0.935116	+	0.354343i	$0.115295\pi$
$422$		0			0
$423$	361.883			0.855515
$424$		0			0
$425$			14.7324i			0.0346644i
$426$		0			0
$427$	−164.437	−	164.437i	−0.385099	−	0.385099i
$428$		0			0
$429$	−10.1301	−	10.1301i	−0.0236133	−	0.0236133i
$430$		0			0
$431$			337.331i			0.782670i	0.920248	+	0.391335i	$0.127987\pi$
$431$			337.331i			0.782670i	−0.920248	+	0.391335i	$0.872013\pi$
$432$		0			0
$433$	−424.560			−0.980508			−0.490254	−	0.871580i	$-0.663096\pi$
$433$	−424.560			−0.980508			−0.490254	+	0.871580i	$0.663096\pi$
$434$		0			0
$435$	−5.24027	+	5.24027i	−0.0120466	+	0.0120466i
$436$		0			0
$437$	230.874	−	230.874i	0.528317	−	0.528317i
$438$		0			0
$439$	162.004			0.369029			0.184514	−	0.982830i	$-0.440929\pi$
$439$	162.004			0.369029			0.184514	+	0.982830i	$0.440929\pi$
$440$		0			0
$441$			397.707i			0.901831i
$442$		0			0
$443$	−492.189	−	492.189i	−1.11104	−	1.11104i	−0.993010	−	0.118026i	$-0.962343\pi$
$443$	−492.189	−	492.189i	−1.11104	−	1.11104i	−0.118026	−	0.993010i	$-0.537657\pi$
$444$		0			0
$445$	162.648	+	162.648i	0.365500	+	0.365500i
$446$		0			0
$447$		−	42.7478i		−	0.0956327i
$448$		0			0
$449$	195.434			0.435266			0.217633	−	0.976031i	$-0.430166\pi$
$449$	195.434			0.435266			0.217633	+	0.976031i	$0.430166\pi$
$450$		0			0
$451$	−266.137	+	266.137i	−0.590104	+	0.590104i
$452$		0			0
$453$	32.2755	−	32.2755i	0.0712483	−	0.0712483i
$454$		0			0
$455$	−446.788			−0.981951
$456$		0			0
$457$			386.874i			0.846552i	0.906001	+	0.423276i	$0.139120\pi$
$457$			386.874i			0.846552i	−0.906001	+	0.423276i	$0.860880\pi$
$458$		0			0
$459$	17.8363	+	17.8363i	0.0388590	+	0.0388590i
$460$		0			0
$461$	−174.401	−	174.401i	−0.378310	−	0.378310i	0.492182	−	0.870492i	$-0.336199\pi$
$461$	−174.401	−	174.401i	−0.378310	−	0.378310i	−0.870492	+	0.492182i	$0.836199\pi$
$462$		0			0
$463$		−	60.5295i		−	0.130733i	−0.997861	−	0.0653666i	$-0.979178\pi$
$463$		−	60.5295i		−	0.130733i	0.997861	−	0.0653666i	$-0.0208217\pi$
$464$		0			0
$465$	−5.67168			−0.0121972
$466$		0			0
$467$	−306.482	+	306.482i	−0.656279	+	0.656279i	−0.954497	−	0.298219i	$-0.903607\pi$
$467$	−306.482	+	306.482i	−0.656279	+	0.656279i	0.298219	+	0.954497i	$0.403607\pi$
$468$		0			0
$469$	−221.500	+	221.500i	−0.472282	+	0.472282i
$470$		0			0
$471$	32.0718			0.0680930
$472$		0			0
$473$		−	601.593i		−	1.27187i
$474$		0			0
$475$	29.3467	+	29.3467i	0.0617825	+	0.0617825i
$476$		0			0
$477$	97.6295	+	97.6295i	0.204674	+	0.204674i
$478$		0			0
$479$		−	376.452i		−	0.785912i	−0.919557	−	0.392956i	$-0.871453\pi$
$479$		−	376.452i		−	0.785912i	0.919557	−	0.392956i	$-0.128547\pi$
$480$		0			0
$481$	228.095			0.474210
$482$		0			0
$483$	24.1588	−	24.1588i	0.0500183	−	0.0500183i
$484$		0			0
$485$	415.393	−	415.393i	0.856481	−	0.856481i
$486$		0			0
$487$	−77.2033			−0.158528			−0.0792641	−	0.996854i	$-0.525257\pi$
$487$	−77.2033			−0.158528			−0.0792641	+	0.996854i	$0.525257\pi$
$488$		0			0
$489$		−	16.6207i		−	0.0339892i
$490$		0			0
$491$	581.438	+	581.438i	1.18419	+	1.18419i	0.978648	+	0.205543i	$0.0658960\pi$
$491$	581.438	+	581.438i	1.18419	+	1.18419i	0.205543	+	0.978648i	$0.434104\pi$
$492$		0			0
$493$	−32.9592	−	32.9592i	−0.0668543	−	0.0668543i
$494$		0			0
$495$		−	364.354i		−	0.736069i
$496$		0			0
$497$	498.730			1.00348
$498$		0			0
$499$	174.006	−	174.006i	0.348709	−	0.348709i	−0.510920	−	0.859629i	$-0.670695\pi$
$499$	174.006	−	174.006i	0.348709	−	0.348709i	0.859629	+	0.510920i	$0.170695\pi$
$500$		0			0
$501$	15.6138	−	15.6138i	0.0311652	−	0.0311652i
$502$		0			0
$503$	355.262			0.706286			0.353143	−	0.935569i	$-0.385113\pi$
$503$	355.262			0.706286			0.353143	+	0.935569i	$0.385113\pi$
$504$		0			0
$505$		−	717.899i		−	1.42158i
$506$		0			0
$507$	13.2229	+	13.2229i	0.0260806	+	0.0260806i
$508$		0			0
$509$	279.667	+	279.667i	0.549444	+	0.549444i	0.926280	−	0.376836i	$-0.122988\pi$
$509$	279.667	+	279.667i	0.549444	+	0.549444i	−0.376836	+	0.926280i	$0.622988\pi$
$510$		0			0
$511$			758.688i			1.48471i
$512$		0			0
$513$	71.0592			0.138517
$514$		0			0
$515$	511.218	−	511.218i	0.992657	−	0.992657i
$516$		0			0
$517$	−222.938	+	222.938i	−0.431215	+	0.431215i
$518$		0			0
$519$	−52.2116			−0.100600
$520$		0			0
$521$			705.745i			1.35460i	0.735708	+	0.677299i	$0.236849\pi$
$521$			705.745i			1.35460i	−0.735708	+	0.677299i	$0.763151\pi$
$522$		0			0
$523$	186.762	+	186.762i	0.357098	+	0.357098i	0.862742	−	0.505644i	$-0.168745\pi$
$523$	186.762	+	186.762i	0.357098	+	0.357098i	−0.505644	+	0.862742i	$0.668745\pi$
$524$		0			0
$525$	3.07085	+	3.07085i	0.00584925	+	0.00584925i
$526$		0			0
$527$		−	35.6726i		−	0.0676899i
$528$		0			0
$529$	−237.309			−0.448599
$530$		0			0
$531$	455.827	−	455.827i	0.858432	−	0.858432i
$532$		0			0
$533$	−302.466	+	302.466i	−0.567479	+	0.567479i
$534$		0			0
$535$	233.967			0.437321
$536$		0			0
$537$		−	17.6992i		−	0.0329593i
$538$		0			0
$539$	−245.008	−	245.008i	−0.454560	−	0.454560i
$540$		0			0
$541$	119.274	+	119.274i	0.220470	+	0.220470i	0.808696	−	0.588226i	$-0.200174\pi$
$541$	119.274	+	119.274i	0.220470	+	0.220470i	−0.588226	+	0.808696i	$0.700174\pi$
$542$		0			0
$543$			43.0645i			0.0793085i
$544$		0			0
$545$	5.25091			0.00963470
$546$		0			0
$547$	141.472	−	141.472i	0.258632	−	0.258632i	−0.565865	−	0.824498i	$-0.691458\pi$
$547$	141.472	−	141.472i	0.258632	−	0.258632i	0.824498	+	0.565865i	$0.191458\pi$
$548$		0			0
$549$	152.403	−	152.403i	0.277602	−	0.277602i
$550$		0			0
$551$	−131.308			−0.238309
$552$		0			0
$553$		−	1048.71i		−	1.89640i
$554$		0			0
$555$	−19.6216	−	19.6216i	−0.0353542	−	0.0353542i
$556$		0			0
$557$	375.881	+	375.881i	0.674831	+	0.674831i	0.958826	−	0.283995i	$-0.0916599\pi$
$557$	375.881	+	375.881i	0.674831	+	0.674831i	−0.283995	+	0.958826i	$0.591660\pi$
$558$		0			0
$559$		−	683.715i		−	1.22310i
$560$		0			0
$561$	−10.9618			−0.0195398
$562$		0			0
$563$	−305.349	+	305.349i	−0.542360	+	0.542360i	−0.924220	−	0.381860i	$-0.875284\pi$
$563$	−305.349	+	305.349i	−0.542360	+	0.542360i	0.381860	+	0.924220i	$0.375284\pi$
$564$		0			0
$565$	54.8777	−	54.8777i	0.0971287	−	0.0971287i
$566$		0			0
$567$	−771.656			−1.36095
$568$		0			0
$569$			296.778i			0.521578i	0.965396	+	0.260789i	$0.0839827\pi$
$569$			296.778i			0.521578i	−0.965396	+	0.260789i	$0.916017\pi$
$570$		0			0
$571$	347.717	+	347.717i	0.608961	+	0.608961i	0.942674	−	0.333714i	$-0.108302\pi$
$571$	347.717	+	347.717i	0.608961	+	0.608961i	−0.333714	+	0.942674i	$0.608302\pi$
$572$		0			0
$573$	−15.6408	−	15.6408i	−0.0272963	−	0.0272963i
$574$		0			0
$575$			37.0771i			0.0644820i
$576$		0			0
$577$	−189.382			−0.328218			−0.164109	−	0.986442i	$-0.552475\pi$
$577$	−189.382			−0.328218			−0.164109	+	0.986442i	$0.552475\pi$
$578$		0			0
$579$	−9.98027	+	9.98027i	−0.0172371	+	0.0172371i
$580$		0			0
$581$	−553.939	+	553.939i	−0.953423	+	0.953423i
$582$		0			0
$583$	−120.289			−0.206328
$584$		0			0
$585$		−	414.091i		−	0.707847i
$586$		0			0
$587$	641.187	+	641.187i	1.09231	+	1.09231i	0.995281	+	0.0970301i	$0.0309343\pi$
$587$	641.187	+	641.187i	1.09231	+	1.09231i	0.0970301	+	0.995281i	$0.469066\pi$
$588$		0			0
$589$	−71.0592	−	71.0592i	−0.120644	−	0.120644i
$590$		0			0
$591$		−	18.1063i		−	0.0306367i
$592$		0			0
$593$	−127.909			−0.215697			−0.107849	−	0.994167i	$-0.534396\pi$
$593$	−127.909			−0.215697			−0.107849	+	0.994167i	$0.534396\pi$
$594$		0			0
$595$	−241.736	+	241.736i	−0.406279	+	0.406279i
$596$		0			0
$597$	−23.1939	+	23.1939i	−0.0388508	+	0.0388508i
$598$		0			0
$599$	−794.804			−1.32688			−0.663442	−	0.748227i	$-0.730905\pi$
$599$	−794.804			−1.32688			−0.663442	+	0.748227i	$0.730905\pi$
$600$		0			0
$601$		−	89.2746i		−	0.148543i	−0.997238	−	0.0742717i	$-0.976337\pi$
$601$		−	89.2746i		−	0.148543i	0.997238	−	0.0742717i	$-0.0236632\pi$
$602$		0			0
$603$	−205.290	−	205.290i	−0.340448	−	0.340448i
$604$		0			0
$605$	−221.527	−	221.527i	−0.366160	−	0.366160i
$606$		0			0
$607$			316.002i			0.520596i	0.965528	+	0.260298i	$0.0838208\pi$
$607$			316.002i			0.520596i	−0.965528	+	0.260298i	$0.916179\pi$
$608$		0			0
$609$	−13.7402			−0.0225619
$610$		0			0
$611$	−253.370	+	253.370i	−0.414682	+	0.414682i
$612$		0			0
$613$	−192.003	+	192.003i	−0.313219	+	0.313219i	−0.846155	−	0.532936i	$-0.821089\pi$
$613$	−192.003	+	192.003i	−0.313219	+	0.313219i	0.532936	+	0.846155i	$0.321089\pi$
$614$		0			0
$615$	52.0385			0.0846154
$616$		0			0
$617$		−	105.762i		−	0.171413i	−0.996320	−	0.0857066i	$-0.972685\pi$
$617$		−	105.762i		−	0.171413i	0.996320	−	0.0857066i	$-0.0273148\pi$
$618$		0			0
$619$	−553.819	−	553.819i	−0.894699	−	0.894699i	0.100262	−	0.994961i	$-0.468032\pi$
$619$	−553.819	−	553.819i	−0.894699	−	0.894699i	−0.994961	+	0.100262i	$0.968032\pi$
$620$		0			0
$621$	44.8888	+	44.8888i	0.0722846	+	0.0722846i
$622$		0			0
$623$			426.468i			0.684539i
$624$		0			0
$625$	674.560			1.07930
$626$		0			0
$627$	−21.8358	+	21.8358i	−0.0348258	+	0.0348258i
$628$		0			0
$629$	123.412	−	123.412i	0.196203	−	0.196203i
$630$		0			0
$631$	−762.907			−1.20904			−0.604522	−	0.796589i	$-0.706636\pi$
$631$	−762.907			−1.20904			−0.604522	+	0.796589i	$0.706636\pi$
$632$		0			0
$633$			57.6772i			0.0911172i
$634$		0			0
$635$	−393.873	−	393.873i	−0.620272	−	0.620272i
$636$		0			0
$637$	−278.453	−	278.453i	−0.437132	−	0.437132i
$638$		0			0
$639$			462.232i			0.723367i
$640$		0			0
$641$	−412.834			−0.644046			−0.322023	−	0.946732i	$-0.604363\pi$
$641$	−412.834			−0.644046			−0.322023	+	0.946732i	$0.604363\pi$
$642$		0			0
$643$	372.515	−	372.515i	0.579339	−	0.579339i	−0.355382	−	0.934721i	$-0.615649\pi$
$643$	372.515	−	372.515i	0.579339	−	0.579339i	0.934721	+	0.355382i	$0.115649\pi$
$644$		0			0
$645$	−58.8156	+	58.8156i	−0.0911870	+	0.0911870i
$646$		0			0
$647$	1170.94			1.80980			0.904899	−	0.425627i	$-0.139946\pi$
$647$	1170.94			1.80980			0.904899	+	0.425627i	$0.139946\pi$
$648$		0			0
$649$			561.625i			0.865370i
$650$		0			0
$651$	−7.43568	−	7.43568i	−0.0114219	−	0.0114219i
$652$		0			0
$653$	13.7523	+	13.7523i	0.0210602	+	0.0210602i	0.717558	−	0.696498i	$-0.245260\pi$
$653$	13.7523	+	13.7523i	0.0210602	+	0.0210602i	−0.696498	+	0.717558i	$0.745260\pi$
$654$		0			0
$655$		−	1129.33i		−	1.72417i
$656$		0			0
$657$	−703.165			−1.07027
$658$		0			0
$659$	−283.149	+	283.149i	−0.429664	+	0.429664i	−0.888514	−	0.458850i	$-0.848262\pi$
$659$	−283.149	+	283.149i	−0.429664	+	0.429664i	0.458850	+	0.888514i	$0.348262\pi$
$660$		0			0
$661$	287.535	−	287.535i	0.435000	−	0.435000i	−0.455325	−	0.890325i	$-0.650477\pi$
$661$	287.535	−	287.535i	0.435000	−	0.435000i	0.890325	+	0.455325i	$0.150477\pi$
$662$		0			0
$663$	−12.4582			−0.0187907
$664$		0			0
$665$			963.069i			1.44822i
$666$		0			0
$667$	−82.9486	−	82.9486i	−0.124361	−	0.124361i
$668$		0			0
$669$	2.30816	+	2.30816i	0.00345017	+	0.00345017i
$670$		0			0
$671$			187.776i			0.279845i
$672$		0			0
$673$	−45.5265			−0.0676471			−0.0338236	−	0.999428i	$-0.510768\pi$
$673$	−45.5265			−0.0676471			−0.0338236	+	0.999428i	$0.510768\pi$
$674$		0			0
$675$	−5.70586	+	5.70586i	−0.00845312	+	0.00845312i
$676$		0			0
$677$	208.341	−	208.341i	0.307742	−	0.307742i	−0.536291	−	0.844033i	$-0.680175\pi$
$677$	208.341	−	208.341i	0.307742	−	0.307742i	0.844033	+	0.536291i	$0.180175\pi$
$678$		0			0
$679$	1089.18			1.60409
$680$		0			0
$681$		−	58.3382i		−	0.0856655i
$682$		0			0
$683$	219.645	+	219.645i	0.321589	+	0.321589i	0.849377	−	0.527787i	$-0.176978\pi$
$683$	219.645	+	219.645i	0.321589	+	0.321589i	−0.527787	+	0.849377i	$0.676978\pi$
$684$		0			0
$685$	−277.065	−	277.065i	−0.404475	−	0.404475i
$686$		0			0
$687$			67.4808i			0.0982254i
$688$		0			0
$689$	−136.710			−0.198418
$690$		0			0
$691$	−692.991	+	692.991i	−1.00288	+	1.00288i	−0.00288571	+	0.999996i	$0.500919\pi$
$691$	−692.991	+	692.991i	−1.00288	+	1.00288i	−0.999996	+	0.00288571i	$0.999081\pi$
$692$		0			0
$693$	477.675	−	477.675i	0.689285	−	0.689285i
$694$		0			0
$695$	791.905			1.13943
$696$		0			0
$697$			327.301i			0.469585i
$698$		0			0
$699$	−50.4655	−	50.4655i	−0.0721967	−	0.0721967i
$700$		0			0
$701$	195.377	+	195.377i	0.278712	+	0.278712i	0.832595	−	0.553883i	$-0.186854\pi$
$701$	195.377	+	195.377i	0.278712	+	0.278712i	−0.553883	+	0.832595i	$0.686854\pi$
$702$		0			0
$703$		−	491.668i		−	0.699386i
$704$		0			0
$705$	43.5917			0.0618322
$706$		0			0
$707$	941.179	−	941.179i	1.33123	−	1.33123i
$708$		0			0
$709$	318.083	−	318.083i	0.448636	−	0.448636i	−0.446265	−	0.894901i	$-0.647246\pi$
$709$	318.083	−	318.083i	0.448636	−	0.448636i	0.894901	+	0.446265i	$0.147246\pi$
$710$		0			0
$711$	971.962			1.36703
$712$		0			0
$713$		−	89.7775i		−	0.125915i
$714$		0			0
$715$	255.101	+	255.101i	0.356784	+	0.356784i
$716$		0			0
$717$	−11.2991	−	11.2991i	−0.0157589	−	0.0157589i
$718$		0			0
$719$			1122.38i			1.56103i	0.625139	+	0.780514i	$0.285042\pi$
$719$			1122.38i			1.56103i	−0.625139	+	0.780514i	$0.714958\pi$
$720$		0			0
$721$	1340.43			1.85913
$722$		0			0
$723$	42.9557	−	42.9557i	0.0594132	−	0.0594132i
$724$		0			0
$725$	10.5437	−	10.5437i	0.0145430	−	0.0145430i
$726$		0			0
$727$	−529.192			−0.727911			−0.363956	−	0.931416i	$-0.618574\pi$
$727$	−529.192			−0.727911			−0.363956	+	0.931416i	$0.618574\pi$
$728$		0			0
$729$		−	708.260i		−	0.971549i
$730$		0			0
$731$	−369.926	−	369.926i	−0.506055	−	0.506055i
$732$		0			0
$733$	−263.121	−	263.121i	−0.358965	−	0.358965i	0.504466	−	0.863431i	$-0.331689\pi$
$733$	−263.121	−	263.121i	−0.358965	−	0.358965i	−0.863431	+	0.504466i	$0.831689\pi$
$734$		0			0
$735$			47.9070i			0.0651797i
$736$		0			0
$737$	252.938			0.343200
$738$		0			0
$739$	44.5459	−	44.5459i	0.0602787	−	0.0602787i	−0.676325	−	0.736603i	$-0.736428\pi$
$739$	44.5459	−	44.5459i	0.0602787	−	0.0602787i	0.736603	+	0.676325i	$0.236428\pi$
$740$		0			0
$741$	−24.8165	+	24.8165i	−0.0334906	+	0.0334906i
$742$		0			0
$743$	−762.894			−1.02678			−0.513388	−	0.858157i	$-0.671610\pi$
$743$	−762.894			−1.02678			−0.513388	+	0.858157i	$0.671610\pi$
$744$		0			0
$745$			1076.50i			1.44496i
$746$		0			0
$747$	−513.400	−	513.400i	−0.687282	−	0.687282i
$748$		0			0
$749$	306.735	+	306.735i	0.409526	+	0.409526i
$750$		0			0
$751$		−	1342.93i		−	1.78819i	−0.447876	−	0.894095i	$-0.647820\pi$
$751$		−	1342.93i		−	1.78819i	0.447876	−	0.894095i	$-0.352180\pi$
$752$		0			0
$753$	23.3108			0.0309573
$754$		0			0
$755$	−812.776	+	812.776i	−1.07652	+	1.07652i
$756$		0			0
$757$	−394.830	+	394.830i	−0.521573	+	0.521573i	−0.918046	−	0.396474i	$-0.870234\pi$
$757$	−394.830	+	394.830i	−0.521573	+	0.521573i	0.396474	+	0.918046i	$0.370234\pi$
$758$		0			0
$759$	−27.5878			−0.0363475
$760$		0			0
$761$		−	480.213i		−	0.631029i	−0.948921	−	0.315514i	$-0.897823\pi$
$761$		−	480.213i		−	0.631029i	0.948921	−	0.315514i	$-0.102177\pi$
$762$		0			0
$763$	6.88404	+	6.88404i	0.00902233	+	0.00902233i
$764$		0			0
$765$	−224.045	−	224.045i	−0.292870	−	0.292870i
$766$		0			0
$767$			638.291i			0.832191i
$768$		0			0
$769$	472.763			0.614777			0.307388	−	0.951584i	$-0.400545\pi$
$769$	472.763			0.614777			0.307388	+	0.951584i	$0.400545\pi$
$770$		0			0
$771$	−32.4727	+	32.4727i	−0.0421177	+	0.0421177i
$772$		0			0
$773$	857.735	−	857.735i	1.10962	−	1.10962i	0.116418	−	0.993200i	$-0.462859\pi$
$773$	857.735	−	857.735i	1.10962	−	1.10962i	0.993200	−	0.116418i	$-0.0371412\pi$
$774$		0			0
$775$	11.4117			0.0147248
$776$		0			0
$777$		−	51.4485i		−	0.0662143i
$778$		0			0
$779$	651.978	+	651.978i	0.836942	+	0.836942i
$780$		0			0
$781$	−284.758	−	284.758i	−0.364607	−	0.364607i
$782$		0			0
$783$		−	25.5302i		−	0.0326056i
$784$		0			0
$785$	−807.648			−1.02885
$786$		0			0
$787$	−170.355	+	170.355i	−0.216462	+	0.216462i	−0.807006	−	0.590544i	$-0.798913\pi$
$787$	−170.355	+	170.355i	−0.216462	+	0.216462i	0.590544	+	0.807006i	$0.298913\pi$
$788$		0			0
$789$	54.7733	−	54.7733i	0.0694212	−	0.0694212i
$790$		0			0
$791$	143.891			0.181911
$792$		0			0
$793$			213.409i			0.269116i
$794$		0			0
$795$	11.7603	+	11.7603i	0.0147928	+	0.0147928i
$796$		0			0
$797$	835.571	+	835.571i	1.04840	+	1.04840i	0.998768	+	0.0496277i	$0.0158035\pi$
$797$	835.571	+	835.571i	1.04840	+	1.04840i	0.0496277	+	0.998768i	$0.484197\pi$
$798$		0			0
$799$			274.174i			0.343147i
$800$		0			0
$801$	−395.258			−0.493456
$802$		0			0
$803$	433.185	−	433.185i	0.539458	−	0.539458i
$804$		0			0
$805$	−608.380	+	608.380i	−0.755751	+	0.755751i
$806$		0			0
$807$	104.789			0.129850
$808$		0			0
$809$		−	371.926i		−	0.459735i	−0.973222	−	0.229868i	$-0.926171\pi$
$809$		−	371.926i		−	0.459735i	0.973222	−	0.229868i	$-0.0738293\pi$
$810$		0			0
$811$	275.629	+	275.629i	0.339863	+	0.339863i	0.856316	−	0.516453i	$-0.172748\pi$
$811$	275.629	+	275.629i	0.339863	+	0.339863i	−0.516453	+	0.856316i	$0.672748\pi$
$812$		0			0
$813$	−52.5497	−	52.5497i	−0.0646367	−	0.0646367i
$814$		0			0
$815$			418.550i			0.513559i
$816$		0			0
$817$	−1473.77			−1.80389
$818$		0			0
$819$	542.880	−	542.880i	0.662858	−	0.662858i
$820$		0			0
$821$	−904.923	+	904.923i	−1.10222	+	1.10222i	−0.108079	+	0.994142i	$0.534470\pi$
$821$	−904.923	+	904.923i	−1.10222	+	1.10222i	−0.994142	+	0.108079i	$0.965530\pi$
$822$		0			0
$823$	−523.237			−0.635768			−0.317884	−	0.948130i	$-0.602972\pi$
$823$	−523.237			−0.635768			−0.317884	+	0.948130i	$0.602972\pi$
$824$		0			0
$825$		−	3.50671i		−	0.00425055i
$826$		0			0
$827$	−722.805	−	722.805i	−0.874008	−	0.874008i	0.118898	−	0.992906i	$-0.462064\pi$
$827$	−722.805	−	722.805i	−0.874008	−	0.874008i	−0.992906	+	0.118898i	$0.962064\pi$
$828$		0			0
$829$	−286.380	−	286.380i	−0.345453	−	0.345453i	0.512960	−	0.858413i	$-0.328549\pi$
$829$	−286.380	−	286.380i	−0.345453	−	0.345453i	−0.858413	+	0.512960i	$0.828549\pi$
$830$		0			0
$831$		−	102.972i		−	0.123914i
$832$		0			0
$833$	−301.316			−0.361724
$834$		0			0
$835$	−393.193	+	393.193i	−0.470890	+	0.470890i
$836$		0			0
$837$	13.8160	−	13.8160i	0.0165066	−	0.0165066i
$838$		0			0
$839$	−1353.58			−1.61333			−0.806666	−	0.591008i	$-0.798730\pi$
$839$	−1353.58			−1.61333			−0.806666	+	0.591008i	$0.798730\pi$
$840$		0			0
$841$		−	793.824i		−	0.943904i
$842$		0			0
$843$	28.0129	+	28.0129i	0.0332300	+	0.0332300i
$844$		0			0
$845$	−332.984	−	332.984i	−0.394064	−	0.394064i
$846$		0			0
$847$		−	580.852i		−	0.685776i
$848$		0			0
$849$	−9.16597			−0.0107962
$850$		0			0
$851$	310.591	−	310.591i	0.364972	−	0.364972i
$852$		0			0
$853$	−668.253	+	668.253i	−0.783415	+	0.783415i	−0.980405	−	0.196990i	$-0.936883\pi$
$853$	−668.253	+	668.253i	−0.783415	+	0.783415i	0.196990	+	0.980405i	$0.436883\pi$
$854$		0			0
$855$	−892.589			−1.04396
$856$		0			0
$857$		−	488.688i		−	0.570230i	−0.958493	−	0.285115i	$-0.907968\pi$
$857$		−	488.688i		−	0.570230i	0.958493	−	0.285115i	$-0.0920319\pi$
$858$		0			0
$859$	268.818	+	268.818i	0.312943	+	0.312943i	0.846048	−	0.533106i	$-0.178975\pi$
$859$	268.818	+	268.818i	0.312943	+	0.312943i	−0.533106	+	0.846048i	$0.678975\pi$
$860$		0			0
$861$	68.2234	+	68.2234i	0.0792374	+	0.0792374i
$862$		0			0
$863$		−	152.667i		−	0.176903i	−0.996080	−	0.0884514i	$-0.971808\pi$
$863$		−	152.667i		−	0.176903i	0.996080	−	0.0884514i	$-0.0281918\pi$
$864$		0			0
$865$	1314.82			1.52002
$866$		0			0
$867$	35.5591	−	35.5591i	0.0410139	−	0.0410139i
$868$		0			0
$869$	−598.777	+	598.777i	−0.689042	+	0.689042i
$870$		0			0
$871$	287.466			0.330041
$872$		0			0
$873$			1009.47i			1.15632i
$874$		0			0
$875$	813.208	+	813.208i	0.929380	+	0.929380i
$876$		0			0
$877$	162.637	+	162.637i	0.185447	+	0.185447i	0.793725	−	0.608277i	$-0.208139\pi$
$877$	162.637	+	162.637i	0.185447	+	0.185447i	−0.608277	+	0.793725i	$0.708139\pi$
$878$		0			0
$879$		−	26.9573i		−	0.0306681i
$880$		0			0
$881$	873.243			0.991196			0.495598	−	0.868552i	$-0.334949\pi$
$881$	873.243			0.991196			0.495598	+	0.868552i	$0.334949\pi$
$882$		0			0
$883$	230.025	−	230.025i	0.260504	−	0.260504i	−0.564755	−	0.825259i	$-0.691029\pi$
$883$	230.025	−	230.025i	0.260504	−	0.260504i	0.825259	+	0.564755i	$0.191029\pi$
$884$		0			0
$885$	54.9081	−	54.9081i	0.0620430	−	0.0620430i
$886$		0			0
$887$	−430.685			−0.485552			−0.242776	−	0.970082i	$-0.578058\pi$
$887$	−430.685			−0.485552			−0.242776	+	0.970082i	$0.578058\pi$
$888$		0			0
$889$		−	1032.75i		−	1.16170i
$890$		0			0
$891$	440.589	+	440.589i	0.494489	+	0.494489i
$892$		0			0
$893$	546.150	+	546.150i	0.611590	+	0.611590i
$894$		0			0
$895$			445.709i			0.497999i
$896$		0			0
$897$	−31.3537			−0.0349539
$898$		0			0
$899$	−25.5302	+	25.5302i	−0.0283984	+	0.0283984i
$900$		0			0
$901$	−73.9673	+	73.9673i	−0.0820947	+	0.0820947i
$902$		0			0
$903$	−154.217			−0.170783
$904$		0			0
$905$		−	1084.47i		−	1.19831i
$906$		0			0
$907$	−22.2262	−	22.2262i	−0.0245052	−	0.0245052i	0.694748	−	0.719253i	$-0.255516\pi$
$907$	−22.2262	−	22.2262i	−0.0245052	−	0.0245052i	−0.719253	+	0.694748i	$0.755516\pi$
$908$		0			0
$909$	872.301	+	872.301i	0.959627	+	0.959627i
$910$		0			0
$911$		−	1399.85i		−	1.53661i	−0.640083	−	0.768306i	$-0.721100\pi$
$911$		−	1399.85i		−	1.53661i	0.640083	−	0.768306i	$-0.278900\pi$
$912$		0			0
$913$	632.560			0.692837
$914$		0			0
$915$	18.3582	−	18.3582i	0.0200636	−	0.0200636i
$916$		0			0
$917$	1480.57	−	1480.57i	1.61458	−	1.61458i
$918$		0			0
$919$	806.944			0.878068			0.439034	−	0.898470i	$-0.355321\pi$
$919$	806.944			0.878068			0.439034	+	0.898470i	$0.355321\pi$
$920$		0			0
$921$			75.3976i			0.0818650i
$922$		0			0
$923$	−323.629	−	323.629i	−0.350628	−	0.350628i
$924$		0			0
$925$	39.4796	+	39.4796i	0.0426806	+	0.0426806i
$926$		0			0
$927$			1242.34i			1.34017i
$928$		0			0
$929$	−1620.69			−1.74455			−0.872276	−	0.489013i	$-0.837357\pi$
$929$	−1620.69			−1.74455			−0.872276	+	0.489013i	$0.837357\pi$
$930$		0			0
$931$	−600.216	+	600.216i	−0.644701	+	0.644701i
$932$		0			0
$933$	−19.1613	+	19.1613i	−0.0205373	+	0.0205373i
$934$		0			0
$935$	276.046			0.295237
$936$		0			0
$937$		−	598.181i		−	0.638400i	−0.947687	−	0.319200i	$-0.896586\pi$
$937$		−	598.181i		−	0.638400i	0.947687	−	0.319200i	$-0.103414\pi$
$938$		0			0
$939$	7.58691	+	7.58691i	0.00807978	+	0.00807978i
$940$		0			0
$941$	−977.842	−	977.842i	−1.03915	−	1.03915i	−0.999202	−	0.0399498i	$-0.987280\pi$
$941$	−977.842	−	977.842i	−1.03915	−	1.03915i	−0.0399498	−	0.999202i	$-0.512720\pi$
$942$		0			0
$943$			823.721i			0.873511i
$944$		0			0
$945$	−187.249			−0.198147
$946$		0			0
$947$	827.881	−	827.881i	0.874215	−	0.874215i	−0.118714	−	0.992929i	$-0.537877\pi$
$947$	827.881	−	827.881i	0.874215	−	0.874215i	0.992929	+	0.118714i	$0.0378771\pi$
$948$		0			0
$949$	492.317	−	492.317i	0.518775	−	0.518775i
$950$		0			0
$951$	32.0939			0.0337476
$952$		0			0
$953$			1846.78i			1.93786i	0.247333	+	0.968930i	$0.420446\pi$
$953$			1846.78i			1.93786i	−0.247333	+	0.968930i	$0.579554\pi$
$954$		0			0
$955$	393.873	+	393.873i	0.412432	+	0.412432i
$956$		0			0
$957$	7.84517	+	7.84517i	0.00819767	+	0.00819767i
$958$		0			0
$959$		−	726.475i		−	0.757534i
$960$		0			0
$961$	933.368			0.971247
$962$		0			0
$963$	−284.287	+	284.287i	−0.295210	+	0.295210i
$964$		0			0
$965$	251.328	−	251.328i	0.260443	−	0.260443i
$966$		0			0
$967$	363.922			0.376341			0.188170	−	0.982136i	$-0.439744\pi$
$967$	363.922			0.376341			0.188170	+	0.982136i	$0.439744\pi$
$968$		0			0
$969$			26.8542i			0.0277133i
$970$		0			0
$971$	−1161.30	−	1161.30i	−1.19598	−	1.19598i	−0.975360	−	0.220619i	$-0.929192\pi$
$971$	−1161.30	−	1161.30i	−1.19598	−	1.19598i	−0.220619	−	0.975360i	$-0.570808\pi$
$972$		0			0
$973$	1038.20	+	1038.20i	1.06701	+	1.06701i
$974$		0			0
$975$		−	3.98540i		−	0.00408759i
$976$		0			0
$977$	1159.63			1.18693			0.593467	−	0.804858i	$-0.297759\pi$
$977$	1159.63			1.18693			0.593467	+	0.804858i	$0.297759\pi$
$978$		0			0
$979$	243.499	−	243.499i	0.248722	−	0.248722i
$980$		0			0
$981$	−6.38025	+	6.38025i	−0.00650382	+	0.00650382i
$982$		0			0
$983$	1780.51			1.81131			0.905653	−	0.424020i	$-0.139382\pi$
$983$	1780.51			1.81131			0.905653	+	0.424020i	$0.139382\pi$
$984$		0			0
$985$			455.961i			0.462904i
$986$		0			0
$987$	57.1495	+	57.1495i	0.0579022	+	0.0579022i
$988$		0			0
$989$	−930.997	−	930.997i	−0.941352	−	0.941352i
$990$		0			0
$991$		−	675.783i		−	0.681920i	−0.940078	−	0.340960i	$-0.889248\pi$
$991$		−	675.783i		−	0.681920i	0.940078	−	0.340960i	$-0.110752\pi$
$992$		0			0
$993$	94.6184			0.0952854
$994$		0			0
$995$	584.080	−	584.080i	0.587015	−	0.587015i
$996$		0			0
$997$	9.44963	−	9.44963i	0.00947806	−	0.00947806i	−0.702352	−	0.711830i	$-0.747867\pi$
$997$	9.44963	−	9.44963i	0.00947806	−	0.00947806i	0.711830	+	0.702352i	$0.247867\pi$
$998$		0			0
$999$	95.5948			0.0956905

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	64.3.f.a.47.2		6
3.2	odd	2		576.3.m.a.559.1		6
4.3	odd	2		16.3.f.a.3.2	✓	6
8.3	odd	2		128.3.f.b.95.2		6
8.5	even	2		128.3.f.a.95.2		6
12.11	even	2		144.3.m.a.19.2		6
16.3	odd	4		128.3.f.a.31.2		6
16.5	even	4		16.3.f.a.11.2	yes	6
16.11	odd	4	inner	64.3.f.a.15.2		6
16.13	even	4		128.3.f.b.31.2		6
20.3	even	4		400.3.k.c.99.3		6
20.7	even	4		400.3.k.d.99.1		6
20.19	odd	2		400.3.r.c.51.2		6
24.5	odd	2		1152.3.m.b.991.3		6
24.11	even	2		1152.3.m.a.991.3		6
32.3	odd	8		1024.3.d.k.511.8		12
32.5	even	8		1024.3.c.j.1023.6		12
32.11	odd	8		1024.3.c.j.1023.5		12
32.13	even	8		1024.3.d.k.511.7		12
32.19	odd	8		1024.3.d.k.511.5		12
32.21	even	8		1024.3.c.j.1023.7		12
32.27	odd	8		1024.3.c.j.1023.8		12
32.29	even	8		1024.3.d.k.511.6		12
48.5	odd	4		144.3.m.a.91.2		6
48.11	even	4		576.3.m.a.271.1		6
48.29	odd	4		1152.3.m.a.415.3		6
48.35	even	4		1152.3.m.b.415.3		6
80.37	odd	4		400.3.k.c.299.3		6
80.53	odd	4		400.3.k.d.299.1		6
80.69	even	4		400.3.r.c.251.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
16.3.f.a.3.2	✓	6	4.3	odd	2
16.3.f.a.11.2	yes	6	16.5	even	4
64.3.f.a.15.2		6	16.11	odd	4	inner
64.3.f.a.47.2		6	1.1	even	1	trivial
128.3.f.a.31.2		6	16.3	odd	4
128.3.f.a.95.2		6	8.5	even	2
128.3.f.b.31.2		6	16.13	even	4
128.3.f.b.95.2		6	8.3	odd	2
144.3.m.a.19.2		6	12.11	even	2
144.3.m.a.91.2		6	48.5	odd	4
400.3.k.c.99.3		6	20.3	even	4
400.3.k.c.299.3		6	80.37	odd	4
400.3.k.d.99.1		6	20.7	even	4
400.3.k.d.299.1		6	80.53	odd	4
400.3.r.c.51.2		6	20.19	odd	2
400.3.r.c.251.2		6	80.69	even	4
576.3.m.a.271.1		6	48.11	even	4
576.3.m.a.559.1		6	3.2	odd	2
1024.3.c.j.1023.5		12	32.11	odd	8
1024.3.c.j.1023.6		12	32.5	even	8
1024.3.c.j.1023.7		12	32.21	even	8
1024.3.c.j.1023.8		12	32.27	odd	8
1024.3.d.k.511.5		12	32.19	odd	8
1024.3.d.k.511.6		12	32.29	even	8
1024.3.d.k.511.7		12	32.13	even	8
1024.3.d.k.511.8		12	32.3	odd	8
1152.3.m.a.415.3		6	48.29	odd	4
1152.3.m.a.991.3		6	24.11	even	2
1152.3.m.b.415.3		6	48.35	even	4
1152.3.m.b.991.3		6	24.5	odd	2

Overview	Random
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Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Character values

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	64.3.f.a.47.2

Level	$64$
Weight	$3$
Character	64.47
Analytic conductor	$1.744$
Analytic rank	$0$
Dimension	$6$
Inner twists	$2$