LMFDB - Embedded newform 1700.2.m.c.1449.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1700,2,Mod(149,1700)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1700.149"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1700, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 2, 1])) N = Newforms(chi, 2, names="a")

Level:	$N$	$=$	$1700 = 2^{2} \cdot 5^{2} \cdot 17$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	1700.m (of order $4$ , degree $2$ , not minimal)

Newform invariants

comment:select newform

sage:traces = [12,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	no
Analytic conductor:	$13.5745683436$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$6$ over $\Q(i)$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{12} + 6 x^{10} - 16 x^{9} + 9 x^{8} - 72 x^{7} + 114 x^{6} - 144 x^{5} + 391 x^{4} - 484 x^{3} + \cdots + 121$ x^12 + 6x^10 - 16x^9 + 9x^8 - 72x^7 + 114x^6 - 144x^5 + 391x^4 - 484x^3 + 504x^2 - 396x + 121
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$2^{2}$
Twist minimal:	no (minimal twist has level 340)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			1449.4
Root			$-0.411629 - 1.88205i$ of defining polynomial
Character	$\chi$	$=$	1700.1449
Dual form			1700.2.m.c.149.4

$q$ -expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.0789919 + 0.0789919i) q^{3} +(-1.97356 + 1.97356i) q^{7} -2.98752i q^{9} +(-0.864802 - 0.864802i) q^{11} +3.40173i q^{13} +(-1.04677 + 3.98802i) q^{17} -3.42309i q^{19} -0.311790 q^{21} +(6.74584 - 6.74584i) q^{23} +(0.472965 - 0.472965i) q^{27} +(-7.08106 + 7.08106i) q^{29} +(-5.64295 + 5.64295i) q^{31} -0.136625i q^{33} +(-4.04367 - 4.04367i) q^{37} +(-0.268709 + 0.268709i) q^{39} +(-8.23755 - 8.23755i) q^{41} -7.49527 q^{43} +8.61396i q^{47} -0.789873i q^{49} +(-0.397707 + 0.232335i) q^{51} -3.58108 q^{53} +(0.270396 - 0.270396i) q^{57} -4.23376i q^{59} +(-7.67312 - 7.67312i) q^{61} +(5.89605 + 5.89605i) q^{63} +11.1125i q^{67} +1.06573 q^{69} +(0.266369 - 0.266369i) q^{71} +(-6.97393 - 6.97393i) q^{73} +3.41348 q^{77} +(10.2529 + 10.2529i) q^{79} -8.88784 q^{81} +1.52005 q^{83} -1.11869 q^{87} -4.28461 q^{89} +(-6.71352 - 6.71352i) q^{91} -0.891494 q^{93} +(-1.80991 - 1.80991i) q^{97} +(-2.58361 + 2.58361i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$12 q - 4 q^{3} - 12 q^{11} + 8 q^{17} + 16 q^{21} + 12 q^{23} + 20 q^{27} - 4 q^{29} - 8 q^{31} - 20 q^{37} - 36 q^{39} + 8 q^{41} + 8 q^{43} + 24 q^{51} - 16 q^{53} - 8 q^{61} - 4 q^{63} + 32 q^{69} + 28 q^{71}+ \cdots + 40 q^{99}+O(q^{100})$ 12 * q - 4 * q^3 - 12 * q^11 + 8 * q^17 + 16 * q^21 + 12 * q^23 + 20 * q^27 - 4 * q^29 - 8 * q^31 - 20 * q^37 - 36 * q^39 + 8 * q^41 + 8 * q^43 + 24 * q^51 - 16 * q^53 - 8 * q^61 - 4 * q^63 + 32 * q^69 + 28 * q^71 - 48 * q^77 + 24 * q^79 - 4 * q^81 + 16 * q^83 + 56 * q^89 - 4 * q^91 + 24 * q^93 - 44 * q^97 + 40 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1700\mathbb{Z}\right)^\times$ .

$n$	$477$	$851$	$1601$
$\chi(n)$	$-1$	$1$	$e\left(\frac{3}{4}\right)$

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
$3$	0.0789919	+	0.0789919i	0.0456060	+	0.0456060i	0.729542	−	0.683936i	$-0.239733\pi$
$3$	0.0789919	+	0.0789919i	0.0456060	+	0.0456060i	−0.683936	+	0.729542i	$0.739733\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−1.97356	+	1.97356i	−0.745935	+	0.745935i	−0.973713	−	0.227778i	$-0.926854\pi$
$7$	−1.97356	+	1.97356i	−0.745935	+	0.745935i	0.227778	+	0.973713i	$0.426854\pi$
$8$		0			0
$9$		−	2.98752i		−	0.995840i
$10$		0			0
$11$	−0.864802	−	0.864802i	−0.260748	−	0.260748i	0.564610	−	0.825358i	$-0.309027\pi$
$11$	−0.864802	−	0.864802i	−0.260748	−	0.260748i	−0.825358	+	0.564610i	$0.809027\pi$
$12$		0			0
$13$			3.40173i			0.943471i	0.881740	+	0.471736i	$0.156372\pi$
$13$			3.40173i			0.943471i	−0.881740	+	0.471736i	$0.843628\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−1.04677	+	3.98802i	−0.253879	+	0.967236i
$17$	−1.04677	+	3.98802i	−0.253879	+	0.967236i
$18$		0			0
$19$		−	3.42309i		−	0.785311i	−0.919686	−	0.392656i	$-0.871556\pi$
$19$		−	3.42309i		−	0.785311i	0.919686	−	0.392656i	$-0.128444\pi$
$20$		0			0
$21$	−0.311790			−0.0680382
$22$		0			0
$23$	6.74584	−	6.74584i	1.40660	−	1.40660i	0.630047	−	0.776557i	$-0.283036\pi$
$23$	6.74584	−	6.74584i	1.40660	−	1.40660i	0.776557	−	0.630047i	$-0.216964\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	0.472965	−	0.472965i	0.0910222	−	0.0910222i
$28$		0			0
$29$	−7.08106	+	7.08106i	−1.31492	+	1.31492i	−0.397178	+	0.917742i	$0.630010\pi$
$29$	−7.08106	+	7.08106i	−1.31492	+	1.31492i	−0.917742	+	0.397178i	$0.869990\pi$
$30$		0			0
$31$	−5.64295	+	5.64295i	−1.01350	+	1.01350i	−0.0135959	+	0.999908i	$0.504328\pi$
$31$	−5.64295	+	5.64295i	−1.01350	+	1.01350i	−0.999908	+	0.0135959i	$0.995672\pi$
$32$		0			0
$33$		−	0.136625i		−	0.0237833i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−4.04367	−	4.04367i	−0.664776	−	0.664776i	0.291726	−	0.956502i	$-0.405770\pi$
$37$	−4.04367	−	4.04367i	−0.664776	−	0.664776i	−0.956502	+	0.291726i	$0.905770\pi$
$38$		0			0
$39$	−0.268709	+	0.268709i	−0.0430279	+	0.0430279i
$40$		0			0
$41$	−8.23755	−	8.23755i	−1.28649	−	1.28649i	−0.936905	−	0.349584i	$-0.886323\pi$
$41$	−8.23755	−	8.23755i	−1.28649	−	1.28649i	−0.349584	−	0.936905i	$-0.613677\pi$
$42$		0			0
$43$	−7.49527			−1.14302			−0.571509	−	0.820596i	$-0.693642\pi$
$43$	−7.49527			−1.14302			−0.571509	+	0.820596i	$0.693642\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			8.61396i			1.25648i	0.778021	+	0.628238i	$0.216223\pi$
$47$			8.61396i			1.25648i	−0.778021	+	0.628238i	$0.783777\pi$
$48$		0			0
$49$		−	0.789873i		−	0.112839i
$50$		0			0
$51$	−0.397707	+	0.232335i	−0.0556901	+	0.0325334i
$52$		0			0
$53$	−3.58108			−0.491899			−0.245949	−	0.969283i	$-0.579100\pi$
$53$	−3.58108			−0.491899			−0.245949	+	0.969283i	$0.579100\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	0.270396	−	0.270396i	0.0358149	−	0.0358149i
$58$		0			0
$59$		−	4.23376i		−	0.551189i	−0.961274	−	0.275594i	$-0.911125\pi$
$59$		−	4.23376i		−	0.551189i	0.961274	−	0.275594i	$-0.0888747\pi$
$60$		0			0
$61$	−7.67312	−	7.67312i	−0.982443	−	0.982443i	0.0174058	−	0.999849i	$-0.494459\pi$
$61$	−7.67312	−	7.67312i	−0.982443	−	0.982443i	−0.999849	+	0.0174058i	$0.994459\pi$
$62$		0			0
$63$	5.89605	+	5.89605i	0.742832	+	0.742832i
$64$		0			0
$65$		0			0
$66$		0			0
$67$			11.1125i			1.35760i	0.734322	+	0.678802i	$0.237500\pi$
$67$			11.1125i			1.35760i	−0.734322	+	0.678802i	$0.762500\pi$
$68$		0			0
$69$	1.06573			0.128299
$70$		0			0
$71$	0.266369	−	0.266369i	0.0316122	−	0.0316122i	−0.691124	−	0.722736i	$-0.742884\pi$
$71$	0.266369	−	0.266369i	0.0316122	−	0.0316122i	0.722736	+	0.691124i	$0.242884\pi$
$72$		0			0
$73$	−6.97393	−	6.97393i	−0.816237	−	0.816237i	0.169324	−	0.985561i	$-0.445842\pi$
$73$	−6.97393	−	6.97393i	−0.816237	−	0.816237i	−0.985561	+	0.169324i	$0.945842\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	3.41348			0.389002
$78$		0			0
$79$	10.2529	+	10.2529i	1.15355	+	1.15355i	0.985836	+	0.167710i	$0.0536373\pi$
$79$	10.2529	+	10.2529i	1.15355	+	1.15355i	0.167710	+	0.985836i	$0.446363\pi$
$80$		0			0
$81$	−8.88784			−0.987538
$82$		0			0
$83$	1.52005			0.166848			0.0834238	−	0.996514i	$-0.473414\pi$
$83$	1.52005			0.166848			0.0834238	+	0.996514i	$0.473414\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−1.11869			−0.119936
$88$		0			0
$89$	−4.28461			−0.454168			−0.227084	−	0.973875i	$-0.572919\pi$
$89$	−4.28461			−0.454168			−0.227084	+	0.973875i	$0.572919\pi$
$90$		0			0
$91$	−6.71352	−	6.71352i	−0.703769	−	0.703769i
$92$		0			0
$93$	−0.891494			−0.0924436
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−1.80991	−	1.80991i	−0.183769	−	0.183769i	0.609227	−	0.792996i	$-0.291480\pi$
$97$	−1.80991	−	1.80991i	−0.183769	−	0.183769i	−0.792996	+	0.609227i	$0.791480\pi$
$98$		0			0
$99$	−2.58361	+	2.58361i	−0.259663	+	0.259663i
$100$		0			0
$101$	2.83260			0.281855			0.140927	−	0.990020i	$-0.454992\pi$
$101$	2.83260			0.281855			0.140927	+	0.990020i	$0.454992\pi$
$102$		0			0
$103$		−	5.02114i		−	0.494748i	−0.968920	−	0.247374i	$-0.920432\pi$
$103$		−	5.02114i		−	0.494748i	0.968920	−	0.247374i	$-0.0795676\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	0.442925	+	0.442925i	0.0428192	+	0.0428192i	0.728192	−	0.685373i	$-0.240361\pi$
$107$	0.442925	+	0.442925i	0.0428192	+	0.0428192i	−0.685373	+	0.728192i	$0.740361\pi$
$108$		0			0
$109$	−10.0151	−	10.0151i	−0.959274	−	0.959274i	0.0399281	−	0.999203i	$-0.487287\pi$
$109$	−10.0151	−	10.0151i	−0.959274	−	0.959274i	−0.999203	+	0.0399281i	$0.987287\pi$
$110$		0			0
$111$		−	0.638834i		−	0.0606355i
$112$		0			0
$113$	−7.12246	+	7.12246i	−0.670024	+	0.670024i	−0.957721	−	0.287697i	$-0.907110\pi$
$113$	−7.12246	+	7.12246i	−0.670024	+	0.670024i	0.287697	+	0.957721i	$0.407110\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	10.1628			0.939547
$118$		0			0
$119$	−5.80473	−	9.93645i	−0.532119	−	0.910873i
$120$		0			0
$121$		−	9.50424i		−	0.864021i
$122$		0			0
$123$		−	1.30140i		−	0.117343i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−15.7863			−1.40081			−0.700403	−	0.713748i	$-0.746996\pi$
$127$	−15.7863			−1.40081			−0.700403	+	0.713748i	$0.746996\pi$
$128$		0			0
$129$	−0.592065	−	0.592065i	−0.0521284	−	0.0521284i
$130$		0			0
$131$	8.21894	−	8.21894i	0.718092	−	0.718092i	−0.250122	−	0.968214i	$-0.580471\pi$
$131$	8.21894	−	8.21894i	0.718092	−	0.718092i	0.968214	+	0.250122i	$0.0804707\pi$
$132$		0			0
$133$	6.75568	+	6.75568i	0.585792	+	0.585792i
$134$		0			0
$135$		0			0
$136$		0			0
$137$			3.40173i			0.290630i	0.989385	+	0.145315i	$0.0464195\pi$
$137$			3.40173i			0.290630i	−0.989385	+	0.145315i	$0.953580\pi$
$138$		0			0
$139$	5.46988	−	5.46988i	0.463950	−	0.463950i	−0.435998	−	0.899948i	$-0.643605\pi$
$139$	5.46988	−	5.46988i	0.463950	−	0.463950i	0.899948	+	0.435998i	$0.143605\pi$
$140$		0			0
$141$	−0.680433	+	0.680433i	−0.0573028	+	0.0573028i
$142$		0			0
$143$	2.94183	−	2.94183i	0.246008	−	0.246008i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	0.0623936	−	0.0623936i	0.00514613	−	0.00514613i
$148$		0			0
$149$	−4.56165			−0.373705			−0.186853	−	0.982388i	$-0.559829\pi$
$149$	−4.56165			−0.373705			−0.186853	+	0.982388i	$0.559829\pi$
$150$		0			0
$151$			12.7622i			1.03857i	0.854601	+	0.519285i	$0.173802\pi$
$151$			12.7622i			1.03857i	−0.854601	+	0.519285i	$0.826198\pi$
$152$		0			0
$153$	11.9143	+	3.12724i	0.963213	+	0.252823i
$154$		0			0
$155$		0			0
$156$		0			0
$157$			22.4488i			1.79161i	0.444452	+	0.895803i	$0.353399\pi$
$157$			22.4488i			1.79161i	−0.444452	+	0.895803i	$0.646601\pi$
$158$		0			0
$159$	−0.282876	−	0.282876i	−0.0224335	−	0.0224335i
$160$		0			0
$161$			26.6266i			2.09847i
$162$		0			0
$163$	2.02135	−	2.02135i	0.158324	−	0.158324i	−0.623499	−	0.781824i	$-0.714290\pi$
$163$	2.02135	−	2.02135i	0.158324	−	0.158324i	0.781824	+	0.623499i	$0.214290\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	0.0463087	+	0.0463087i	0.00358347	+	0.00358347i	0.708896	−	0.705313i	$-0.249193\pi$
$167$	0.0463087	+	0.0463087i	0.00358347	+	0.00358347i	−0.705313	+	0.708896i	$0.749193\pi$
$168$		0			0
$169$	1.42820			0.109862
$170$		0			0
$171$	−10.2266			−0.782045
$172$		0			0
$173$	10.2763	+	10.2763i	0.781294	+	0.781294i	0.980049	−	0.198755i	$-0.0636899\pi$
$173$	10.2763	+	10.2763i	0.781294	+	0.781294i	−0.198755	+	0.980049i	$0.563690\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	0.334433	−	0.334433i	0.0251375	−	0.0251375i
$178$		0			0
$179$			21.5967i			1.61421i	0.590408	+	0.807105i	$0.298967\pi$
$179$			21.5967i			1.61421i	−0.590408	+	0.807105i	$0.701033\pi$
$180$		0			0
$181$	−2.58484	−	2.58484i	−0.192130	−	0.192130i	0.604486	−	0.796616i	$-0.293379\pi$
$181$	−2.58484	−	2.58484i	−0.192130	−	0.192130i	−0.796616	+	0.604486i	$0.793379\pi$
$182$		0			0
$183$		−	1.21223i		−	0.0896105i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	4.35409	−	2.54360i	0.318403	−	0.186006i
$188$		0			0
$189$			1.86685i			0.135793i
$190$		0			0
$191$	22.8681			1.65468			0.827340	−	0.561702i	$-0.189853\pi$
$191$	22.8681			1.65468			0.827340	+	0.561702i	$0.189853\pi$
$192$		0			0
$193$	−5.09713	+	5.09713i	−0.366899	+	0.366899i	−0.866345	−	0.499446i	$-0.833537\pi$
$193$	−5.09713	+	5.09713i	−0.366899	+	0.366899i	0.499446	+	0.866345i	$0.333537\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−11.3698	+	11.3698i	−0.810065	+	0.810065i	−0.984643	−	0.174578i	$-0.944144\pi$
$197$	−11.3698	+	11.3698i	−0.810065	+	0.810065i	0.174578	+	0.984643i	$0.444144\pi$
$198$		0			0
$199$	7.48293	−	7.48293i	0.530451	−	0.530451i	−0.390255	−	0.920707i	$-0.627613\pi$
$199$	7.48293	−	7.48293i	0.530451	−	0.530451i	0.920707	+	0.390255i	$0.127613\pi$
$200$		0			0
$201$	−0.877794	+	0.877794i	−0.0619148	+	0.0619148i
$202$		0			0
$203$		−	27.9498i		−	1.96169i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−20.1533	−	20.1533i	−1.40075	−	1.40075i
$208$		0			0
$209$	−2.96030	+	2.96030i	−0.204768	+	0.204768i
$210$		0			0
$211$	−3.45096	−	3.45096i	−0.237574	−	0.237574i	0.578271	−	0.815845i	$-0.303728\pi$
$211$	−3.45096	−	3.45096i	−0.237574	−	0.237574i	−0.815845	+	0.578271i	$0.803728\pi$
$212$		0			0
$213$	0.0420820			0.00288341
$214$		0			0
$215$		0			0
$216$		0			0
$217$		−	22.2734i		−	1.51202i
$218$		0			0
$219$		−	1.10177i		−	0.0744505i
$220$		0			0
$221$	−13.5662	−	3.56083i	−0.912559	−	0.239527i
$222$		0			0
$223$	8.91984			0.597317			0.298658	−	0.954360i	$-0.403461\pi$
$223$	8.91984			0.597317			0.298658	+	0.954360i	$0.403461\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	9.14513	−	9.14513i	0.606984	−	0.606984i	−0.335173	−	0.942157i	$-0.608795\pi$
$227$	9.14513	−	9.14513i	0.606984	−	0.606984i	0.942157	+	0.335173i	$0.108795\pi$
$228$		0			0
$229$		−	13.9331i		−	0.920726i	−0.887731	−	0.460363i	$-0.847719\pi$
$229$		−	13.9331i		−	0.920726i	0.887731	−	0.460363i	$-0.152281\pi$
$230$		0			0
$231$	0.269637	+	0.269637i	0.0177408	+	0.0177408i
$232$		0			0
$233$	−11.0316	−	11.0316i	−0.722702	−	0.722702i	0.246452	−	0.969155i	$-0.420735\pi$
$233$	−11.0316	−	11.0316i	−0.722702	−	0.722702i	−0.969155	+	0.246452i	$0.920735\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$			1.61980i			0.105217i
$238$		0			0
$239$	−14.8296			−0.959246			−0.479623	−	0.877475i	$-0.659227\pi$
$239$	−14.8296			−0.959246			−0.479623	+	0.877475i	$0.659227\pi$
$240$		0			0
$241$	0.224326	−	0.224326i	0.0144501	−	0.0144501i	−0.699845	−	0.714295i	$-0.746748\pi$
$241$	0.224326	−	0.224326i	0.0144501	−	0.0144501i	0.714295	+	0.699845i	$0.246748\pi$
$242$		0			0
$243$	−2.12096	−	2.12096i	−0.136060	−	0.136060i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	11.6445			0.740919
$248$		0			0
$249$	0.120072	+	0.120072i	0.00760925	+	0.00760925i
$250$		0			0
$251$	−6.74154			−0.425522			−0.212761	−	0.977104i	$-0.568246\pi$
$251$	−6.74154			−0.425522			−0.212761	+	0.977104i	$0.568246\pi$
$252$		0			0
$253$	−11.6676			−0.733537
$254$		0			0
$255$		0			0
$256$		0			0
$257$	4.80913			0.299985			0.149993	−	0.988687i	$-0.452075\pi$
$257$	4.80913			0.299985			0.149993	+	0.988687i	$0.452075\pi$
$258$		0			0
$259$	15.9609			0.991759
$260$		0			0
$261$	21.1548	+	21.1548i	1.30945	+	1.30945i
$262$		0			0
$263$	−6.06324			−0.373876			−0.186938	−	0.982372i	$-0.559856\pi$
$263$	−6.06324			−0.373876			−0.186938	+	0.982372i	$0.559856\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−0.338449	−	0.338449i	−0.0207128	−	0.0207128i
$268$		0			0
$269$	12.1126	−	12.1126i	0.738517	−	0.738517i	−0.233774	−	0.972291i	$-0.575108\pi$
$269$	12.1126	−	12.1126i	0.738517	−	0.738517i	0.972291	+	0.233774i	$0.0751076\pi$
$270$		0			0
$271$	26.0008			1.57944			0.789719	−	0.613468i	$-0.210226\pi$
$271$	26.0008			1.57944			0.789719	+	0.613468i	$0.210226\pi$
$272$		0			0
$273$		−	1.06063i		−	0.0641921i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−4.51774	−	4.51774i	−0.271445	−	0.271445i	0.558237	−	0.829682i	$-0.311478\pi$
$277$	−4.51774	−	4.51774i	−0.271445	−	0.271445i	−0.829682	+	0.558237i	$0.811478\pi$
$278$		0			0
$279$	16.8584	+	16.8584i	1.00929	+	1.00929i
$280$		0			0
$281$		−	1.36275i		−	0.0812946i	−0.999174	−	0.0406473i	$-0.987058\pi$
$281$		−	1.36275i		−	0.0812946i	0.999174	−	0.0406473i	$-0.0129420\pi$
$282$		0			0
$283$	−17.6020	+	17.6020i	−1.04633	+	1.04633i	−0.0474589	+	0.998873i	$0.515112\pi$
$283$	−17.6020	+	17.6020i	−1.04633	+	1.04633i	−0.998873	+	0.0474589i	$0.984888\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	32.5146			1.91928
$288$		0			0
$289$	−14.8086	−	8.34906i	−0.871091	−	0.491121i
$290$		0			0
$291$		−	0.285936i		−	0.0167619i
$292$		0			0
$293$			13.2673i			0.775081i	0.921853	+	0.387541i	$0.126675\pi$
$293$			13.2673i			0.775081i	−0.921853	+	0.387541i	$0.873325\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	−0.818043			−0.0474676
$298$		0			0
$299$	22.9475	+	22.9475i	1.32709	+	1.32709i
$300$		0			0
$301$	14.7924	−	14.7924i	0.852618	−	0.852618i
$302$		0			0
$303$	0.223753	+	0.223753i	0.0128543	+	0.0128543i
$304$		0			0
$305$		0			0
$306$		0			0
$307$		−	20.9807i		−	1.19743i	−0.800960	−	0.598717i	$-0.795677\pi$
$307$		−	20.9807i		−	1.19743i	0.800960	−	0.598717i	$-0.204323\pi$
$308$		0			0
$309$	0.396629	−	0.396629i	0.0225635	−	0.0225635i
$310$		0			0
$311$	−0.507265	+	0.507265i	−0.0287644	+	0.0287644i	−0.721343	−	0.692578i	$-0.756475\pi$
$311$	−0.507265	+	0.507265i	−0.0287644	+	0.0287644i	0.692578	+	0.721343i	$0.256475\pi$
$312$		0			0
$313$	7.50076	−	7.50076i	0.423968	−	0.423968i	−0.462599	−	0.886567i	$-0.653083\pi$
$313$	7.50076	−	7.50076i	0.423968	−	0.423968i	0.886567	+	0.462599i	$0.153083\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	15.0951	−	15.0951i	0.847823	−	0.847823i	−0.142038	−	0.989861i	$-0.545365\pi$
$317$	15.0951	−	15.0951i	0.847823	−	0.847823i	0.989861	+	0.142038i	$0.0453655\pi$
$318$		0			0
$319$	12.2474			0.685724
$320$		0			0
$321$			0.0699749i			0.00390562i
$322$		0			0
$323$	13.6514	+	3.58319i	0.759582	+	0.199374i
$324$		0			0
$325$		0			0
$326$		0			0
$327$		−	1.58223i		−	0.0874973i
$328$		0			0
$329$	−17.0002	−	17.0002i	−0.937250	−	0.937250i
$330$		0			0
$331$		−	12.4420i		−	0.683872i	−0.939723	−	0.341936i	$-0.888917\pi$
$331$		−	12.4420i		−	0.683872i	0.939723	−	0.341936i	$-0.111083\pi$
$332$		0			0
$333$	−12.0806	+	12.0806i	−0.662010	+	0.662010i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	7.22692	+	7.22692i	0.393675	+	0.393675i	0.875995	−	0.482320i	$-0.160206\pi$
$337$	7.22692	+	7.22692i	0.393675	+	0.393675i	−0.482320	+	0.875995i	$0.660206\pi$
$338$		0			0
$339$	−1.12523			−0.0611142
$340$		0			0
$341$	9.76006			0.528537
$342$		0			0
$343$	−12.2561	−	12.2561i	−0.661765	−	0.661765i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−12.3917	+	12.3917i	−0.665223	+	0.665223i	−0.956606	−	0.291383i	$-0.905885\pi$
$347$	−12.3917	+	12.3917i	−0.665223	+	0.665223i	0.291383	+	0.956606i	$0.405885\pi$
$348$		0			0
$349$			20.4572i			1.09505i	0.836789	+	0.547525i	$0.184430\pi$
$349$			20.4572i			1.09505i	−0.836789	+	0.547525i	$0.815570\pi$
$350$		0			0
$351$	1.60890	+	1.60890i	0.0858769	+	0.0858769i
$352$		0			0
$353$			20.0292i			1.06605i	0.846101	+	0.533023i	$0.178944\pi$
$353$			20.0292i			1.06605i	−0.846101	+	0.533023i	$0.821056\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	0.326372	−	1.24342i	0.0172734	−	0.0658090i
$358$		0			0
$359$			19.4670i			1.02743i	0.857961	+	0.513715i	$0.171731\pi$
$359$			19.4670i			1.02743i	−0.857961	+	0.513715i	$0.828269\pi$
$360$		0			0
$361$	7.28243			0.383286
$362$		0			0
$363$	0.750757	−	0.750757i	0.0394045	−	0.0394045i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	16.4806	−	16.4806i	0.860278	−	0.860278i	−0.131092	−	0.991370i	$-0.541848\pi$
$367$	16.4806	−	16.4806i	0.860278	−	0.860278i	0.991370	+	0.131092i	$0.0418484\pi$
$368$		0			0
$369$	−24.6098	+	24.6098i	−1.28114	+	1.28114i
$370$		0			0
$371$	7.06747	−	7.06747i	0.366925	−	0.366925i
$372$		0			0
$373$			15.4630i			0.800642i	0.916375	+	0.400321i	$0.131101\pi$
$373$			15.4630i			0.800642i	−0.916375	+	0.400321i	$0.868899\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−24.0879	−	24.0879i	−1.24059	−	1.24059i
$378$		0			0
$379$	5.34520	−	5.34520i	0.274564	−	0.274564i	−0.556370	−	0.830935i	$-0.687806\pi$
$379$	5.34520	−	5.34520i	0.274564	−	0.274564i	0.830935	+	0.556370i	$0.187806\pi$
$380$		0			0
$381$	−1.24699	−	1.24699i	−0.0638851	−	0.0638851i
$382$		0			0
$383$	10.1244			0.517333			0.258666	−	0.965967i	$-0.416717\pi$
$383$	10.1244			0.517333			0.258666	+	0.965967i	$0.416717\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$			22.3923i			1.13826i
$388$		0			0
$389$		−	12.6889i		−	0.643355i	−0.946849	−	0.321677i	$-0.895753\pi$
$389$		−	12.6889i		−	0.643355i	0.946849	−	0.321677i	$-0.104247\pi$
$390$		0			0
$391$	19.8412	+	33.9638i	1.00341	+	1.71762i
$392$		0			0
$393$	1.29846			0.0654986
$394$		0			0
$395$		0			0
$396$		0			0
$397$	16.5441	−	16.5441i	0.830323	−	0.830323i	−0.157238	−	0.987561i	$-0.550259\pi$
$397$	16.5441	−	16.5441i	0.830323	−	0.830323i	0.987561	+	0.157238i	$0.0502590\pi$
$398$		0			0
$399$			1.06729i			0.0534312i
$400$		0			0
$401$	7.99397	+	7.99397i	0.399200	+	0.399200i	0.877951	−	0.478751i	$-0.158910\pi$
$401$	7.99397	+	7.99397i	0.399200	+	0.399200i	−0.478751	+	0.877951i	$0.658910\pi$
$402$		0			0
$403$	−19.1958	−	19.1958i	−0.956211	−	0.956211i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			6.99395i			0.346677i
$408$		0			0
$409$	21.1081			1.04373			0.521864	−	0.853029i	$-0.325237\pi$
$409$	21.1081			1.04373			0.521864	+	0.853029i	$0.325237\pi$
$410$		0			0
$411$	−0.268709	+	0.268709i	−0.0132545	+	0.0132545i
$412$		0			0
$413$	8.35558	+	8.35558i	0.411151	+	0.411151i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	0.864153			0.0423177
$418$		0			0
$419$	14.2384	+	14.2384i	0.695589	+	0.695589i	0.963456	−	0.267867i	$-0.0863187\pi$
$419$	14.2384	+	14.2384i	0.695589	+	0.695589i	−0.267867	+	0.963456i	$0.586319\pi$
$420$		0			0
$421$	11.6969			0.570072			0.285036	−	0.958517i	$-0.407995\pi$
$421$	11.6969			0.570072			0.285036	+	0.958517i	$0.407995\pi$
$422$		0			0
$423$	25.7344			1.25125
$424$		0			0
$425$		0			0
$426$		0			0
$427$	30.2867			1.46568
$428$		0			0
$429$	0.464761			0.0224388
$430$		0			0
$431$	2.04736	+	2.04736i	0.0986177	+	0.0986177i	0.754694	−	0.656077i	$-0.227785\pi$
$431$	2.04736	+	2.04736i	0.0986177	+	0.0986177i	−0.656077	+	0.754694i	$0.727785\pi$
$432$		0			0
$433$	19.7898			0.951038			0.475519	−	0.879705i	$-0.342260\pi$
$433$	19.7898			0.951038			0.475519	+	0.879705i	$0.342260\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−23.0916	−	23.0916i	−1.10462	−	1.10462i
$438$		0			0
$439$	25.0903	−	25.0903i	1.19749	−	1.19749i	0.222580	−	0.974914i	$-0.428552\pi$
$439$	25.0903	−	25.0903i	1.19749	−	1.19749i	0.974914	−	0.222580i	$-0.0714478\pi$
$440$		0			0
$441$	−2.35976			−0.112370
$442$		0			0
$443$			14.7915i			0.702763i	0.936232	+	0.351382i	$0.114288\pi$
$443$			14.7915i			0.702763i	−0.936232	+	0.351382i	$0.885712\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	−0.360333	−	0.360333i	−0.0170432	−	0.0170432i
$448$		0			0
$449$	12.2258	+	12.2258i	0.576973	+	0.576973i	0.934068	−	0.357095i	$-0.116233\pi$
$449$	12.2258	+	12.2258i	0.576973	+	0.576973i	−0.357095	+	0.934068i	$0.616233\pi$
$450$		0			0
$451$			14.2477i			0.670898i
$452$		0			0
$453$	−1.00811	+	1.00811i	−0.0473650	+	0.0473650i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−4.94745			−0.231432			−0.115716	−	0.993282i	$-0.536916\pi$
$457$	−4.94745			−0.231432			−0.115716	+	0.993282i	$0.536916\pi$
$458$		0			0
$459$	1.39111	+	2.38128i	0.0649314	+	0.111149i
$460$		0			0
$461$			9.14971i			0.426145i	0.977036	+	0.213072i	$0.0683470\pi$
$461$			9.14971i			0.426145i	−0.977036	+	0.213072i	$0.931653\pi$
$462$		0			0
$463$		−	9.60773i		−	0.446509i	−0.974760	−	0.223255i	$-0.928332\pi$
$463$		−	9.60773i		−	0.446509i	0.974760	−	0.223255i	$-0.0716681\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−15.2164			−0.704132			−0.352066	−	0.935975i	$-0.614521\pi$
$467$	−15.2164			−0.704132			−0.352066	+	0.935975i	$0.614521\pi$
$468$		0			0
$469$	−21.9311	−	21.9311i	−1.01268	−	1.01268i
$470$		0			0
$471$	−1.77327	+	1.77327i	−0.0817079	+	0.0817079i
$472$		0			0
$473$	6.48192	+	6.48192i	0.298039	+	0.298039i
$474$		0			0
$475$		0			0
$476$		0			0
$477$			10.6985i			0.489853i
$478$		0			0
$479$	−11.1721	+	11.1721i	−0.510466	+	0.510466i	−0.914669	−	0.404203i	$-0.867549\pi$
$479$	−11.1721	+	11.1721i	−0.510466	+	0.510466i	0.404203	+	0.914669i	$0.367549\pi$
$480$		0			0
$481$	13.7555	−	13.7555i	0.627197	−	0.627197i
$482$		0			0
$483$	−2.10329	+	2.10329i	−0.0957028	+	0.0957028i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−8.29075	+	8.29075i	−0.375690	+	0.375690i	−0.869545	−	0.493855i	$-0.835588\pi$
$487$	−8.29075	+	8.29075i	−0.375690	+	0.375690i	0.493855	+	0.869545i	$0.335588\pi$
$488$		0			0
$489$	0.319340			0.0144411
$490$		0			0
$491$			6.73370i			0.303888i	0.988389	+	0.151944i	$0.0485533\pi$
$491$			6.73370i			0.303888i	−0.988389	+	0.151944i	$0.951447\pi$
$492$		0			0
$493$	−20.8271	−	35.6516i	−0.938008	−	1.60567i
$494$		0			0
$495$		0			0
$496$		0			0
$497$			1.05139i			0.0471613i
$498$		0			0
$499$	−19.5901	−	19.5901i	−0.876972	−	0.876972i	0.116248	−	0.993220i	$-0.462913\pi$
$499$	−19.5901	−	19.5901i	−0.876972	−	0.876972i	−0.993220	+	0.116248i	$0.962913\pi$
$500$		0			0
$501$			0.00731601i			0.000326855i
$502$		0			0
$503$	−7.00563	+	7.00563i	−0.312366	+	0.312366i	−0.845825	−	0.533460i	$-0.820892\pi$
$503$	−7.00563	+	7.00563i	−0.312366	+	0.312366i	0.533460	+	0.845825i	$0.320892\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	0.112817	+	0.112817i	0.00501036	+	0.00501036i
$508$		0			0
$509$	−10.2748			−0.455423			−0.227712	−	0.973729i	$-0.573124\pi$
$509$	−10.2748			−0.455423			−0.227712	+	0.973729i	$0.573124\pi$
$510$		0			0
$511$	27.5269			1.21772
$512$		0			0
$513$	−1.61900	−	1.61900i	−0.0714808	−	0.0714808i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	7.44937	−	7.44937i	0.327623	−	0.327623i
$518$		0			0
$519$			1.62349i			0.0712633i
$520$		0			0
$521$	−6.83069	−	6.83069i	−0.299258	−	0.299258i	0.541465	−	0.840723i	$-0.317870\pi$
$521$	−6.83069	−	6.83069i	−0.299258	−	0.299258i	−0.840723	+	0.541465i	$0.817870\pi$
$522$		0			0
$523$		−	8.21602i		−	0.359262i	−0.983734	−	0.179631i	$-0.942510\pi$
$523$		−	8.21602i		−	0.359262i	0.983734	−	0.179631i	$-0.0574903\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−16.5973	−	28.4110i	−0.722990	−	1.23760i
$528$		0			0
$529$		−	68.0126i		−	2.95707i
$530$		0			0
$531$	−12.6484			−0.548896
$532$		0			0
$533$	28.0220	−	28.0220i	1.21377	−	1.21377i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−1.70596	+	1.70596i	−0.0736176	+	0.0736176i
$538$		0			0
$539$	−0.683084	+	0.683084i	−0.0294225	+	0.0294225i
$540$		0			0
$541$	−24.1876	+	24.1876i	−1.03991	+	1.03991i	−0.0407372	+	0.999170i	$0.512971\pi$
$541$	−24.1876	+	24.1876i	−1.03991	+	1.03991i	−0.999170	+	0.0407372i	$0.987029\pi$
$542$		0			0
$543$		−	0.408363i		−	0.0175245i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	15.8269	+	15.8269i	0.676711	+	0.676711i	0.959254	−	0.282544i	$-0.0911782\pi$
$547$	15.8269	+	15.8269i	0.676711	+	0.676711i	−0.282544	+	0.959254i	$0.591178\pi$
$548$		0			0
$549$	−22.9236	+	22.9236i	−0.978356	+	0.978356i
$550$		0			0
$551$	24.2391	+	24.2391i	1.03262	+	1.03262i
$552$		0			0
$553$	−40.4696			−1.72094
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	26.3875i		−	1.11807i	−0.829143	−	0.559036i	$-0.811171\pi$
$557$		−	26.3875i		−	1.11807i	0.829143	−	0.559036i	$-0.188829\pi$
$558$		0			0
$559$		−	25.4969i		−	1.07840i
$560$		0			0
$561$	0.544861	+	0.143014i	0.0230041	+	0.00603807i
$562$		0			0
$563$	3.35687			0.141475			0.0707376	−	0.997495i	$-0.477465\pi$
$563$	3.35687			0.141475			0.0707376	+	0.997495i	$0.477465\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	17.5407	−	17.5407i	0.736639	−	0.736639i
$568$		0			0
$569$			42.4229i			1.77846i	0.457458	+	0.889231i	$0.348760\pi$
$569$			42.4229i			1.77846i	−0.457458	+	0.889231i	$0.651240\pi$
$570$		0			0
$571$	−9.57415	−	9.57415i	−0.400666	−	0.400666i	0.477802	−	0.878468i	$-0.341434\pi$
$571$	−9.57415	−	9.57415i	−0.400666	−	0.400666i	−0.878468	+	0.477802i	$0.841434\pi$
$572$		0			0
$573$	1.80640	+	1.80640i	0.0754632	+	0.0754632i
$574$		0			0
$575$		0			0
$576$		0			0
$577$			20.7010i			0.861792i	0.902402	+	0.430896i	$0.141802\pi$
$577$			20.7010i			0.861792i	−0.902402	+	0.430896i	$0.858198\pi$
$578$		0			0
$579$	−0.805263			−0.0334656
$580$		0			0
$581$	−2.99992	+	2.99992i	−0.124458	+	0.124458i
$582$		0			0
$583$	3.09692	+	3.09692i	0.128261	+	0.128261i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−33.1051			−1.36639			−0.683197	−	0.730235i	$-0.739411\pi$
$587$	−33.1051			−1.36639			−0.683197	+	0.730235i	$0.739411\pi$
$588$		0			0
$589$	19.3163	+	19.3163i	0.795916	+	0.795916i
$590$		0			0
$591$	−1.79624			−0.0738876
$592$		0			0
$593$	−11.1110			−0.456276			−0.228138	−	0.973629i	$-0.573264\pi$
$593$	−11.1110			−0.456276			−0.228138	+	0.973629i	$0.573264\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	1.18218			0.0483835
$598$		0			0
$599$	38.6224			1.57807			0.789034	−	0.614349i	$-0.210581\pi$
$599$	38.6224			1.57807			0.789034	+	0.614349i	$0.210581\pi$
$600$		0			0
$601$	−10.5195	−	10.5195i	−0.429099	−	0.429099i	0.459223	−	0.888321i	$-0.348128\pi$
$601$	−10.5195	−	10.5195i	−0.429099	−	0.429099i	−0.888321	+	0.459223i	$0.848128\pi$
$602$		0			0
$603$	33.1987			1.35196
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−22.2976	−	22.2976i	−0.905031	−	0.905031i	0.0908348	−	0.995866i	$-0.471046\pi$
$607$	−22.2976	−	22.2976i	−0.905031	−	0.905031i	−0.995866	+	0.0908348i	$0.971046\pi$
$608$		0			0
$609$	2.20780	−	2.20780i	0.0894648	−	0.0894648i
$610$		0			0
$611$	−29.3024			−1.18545
$612$		0			0
$613$		−	4.65730i		−	0.188107i	−0.995567	−	0.0940533i	$-0.970018\pi$
$613$		−	4.65730i		−	0.188107i	0.995567	−	0.0940533i	$-0.0299824\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	29.1461	+	29.1461i	1.17338	+	1.17338i	0.981399	+	0.191978i	$0.0614902\pi$
$617$	29.1461	+	29.1461i	1.17338	+	1.17338i	0.191978	+	0.981399i	$0.438510\pi$
$618$		0			0
$619$	−6.07033	−	6.07033i	−0.243987	−	0.243987i	0.574510	−	0.818497i	$-0.305193\pi$
$619$	−6.07033	−	6.07033i	−0.243987	−	0.243987i	−0.818497	+	0.574510i	$0.805193\pi$
$620$		0			0
$621$		−	6.38109i		−	0.256064i
$622$		0			0
$623$	8.45594	−	8.45594i	0.338780	−	0.338780i
$624$		0			0
$625$		0			0
$626$		0			0
$627$	−0.467679			−0.0186773
$628$		0			0
$629$	20.3590	−	11.8934i	0.811767	−	0.474223i
$630$		0			0
$631$		−	17.3123i		−	0.689192i	−0.938751	−	0.344596i	$-0.888016\pi$
$631$		−	17.3123i		−	0.689192i	0.938751	−	0.344596i	$-0.111984\pi$
$632$		0			0
$633$		−	0.545195i		−	0.0216696i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	2.68694			0.106460
$638$		0			0
$639$	−0.795784	−	0.795784i	−0.0314807	−	0.0314807i
$640$		0			0
$641$	−11.0134	+	11.0134i	−0.435002	+	0.435002i	−0.890326	−	0.455324i	$-0.849524\pi$
$641$	−11.0134	+	11.0134i	−0.435002	+	0.435002i	0.455324	+	0.890326i	$0.349524\pi$
$642$		0			0
$643$	−7.87307	−	7.87307i	−0.310484	−	0.310484i	0.534613	−	0.845097i	$-0.320457\pi$
$643$	−7.87307	−	7.87307i	−0.310484	−	0.310484i	−0.845097	+	0.534613i	$0.820457\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			15.7228i			0.618129i	0.951041	+	0.309064i	$0.100016\pi$
$647$			15.7228i			0.618129i	−0.951041	+	0.309064i	$0.899984\pi$
$648$		0			0
$649$	−3.66136	+	3.66136i	−0.143721	+	0.143721i
$650$		0			0
$651$	1.75942	−	1.75942i	0.0689570	−	0.0689570i
$652$		0			0
$653$	−18.7020	+	18.7020i	−0.731866	+	0.731866i	−0.970989	−	0.239123i	$-0.923140\pi$
$653$	−18.7020	+	18.7020i	−0.731866	+	0.731866i	0.239123	+	0.970989i	$0.423140\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−20.8348	+	20.8348i	−0.812842	+	0.812842i
$658$		0			0
$659$	−31.7204			−1.23565			−0.617826	−	0.786315i	$-0.711986\pi$
$659$	−31.7204			−1.23565			−0.617826	+	0.786315i	$0.711986\pi$
$660$		0			0
$661$		−	41.6769i		−	1.62105i	−0.585707	−	0.810523i	$-0.699183\pi$
$661$		−	41.6769i		−	1.62105i	0.585707	−	0.810523i	$-0.300817\pi$
$662$		0			0
$663$	−0.790341	−	1.35289i	−0.0306943	−	0.0525420i
$664$		0			0
$665$		0			0
$666$		0			0
$667$			95.5353i			3.69914i
$668$		0			0
$669$	0.704595	+	0.704595i	0.0272412	+	0.0272412i
$670$		0			0
$671$			13.2715i			0.512339i
$672$		0			0
$673$	−1.85204	+	1.85204i	−0.0713907	+	0.0713907i	−0.741901	−	0.670510i	$-0.766075\pi$
$673$	−1.85204	+	1.85204i	−0.0713907	+	0.0713907i	0.670510	+	0.741901i	$0.266075\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	8.46007	+	8.46007i	0.325147	+	0.325147i	0.850738	−	0.525591i	$-0.176156\pi$
$677$	8.46007	+	8.46007i	0.325147	+	0.325147i	−0.525591	+	0.850738i	$0.676156\pi$
$678$		0			0
$679$	7.14393			0.274159
$680$		0			0
$681$	1.44478			0.0553642
$682$		0			0
$683$	14.2066	+	14.2066i	0.543599	+	0.543599i	0.924582	−	0.380983i	$-0.124414\pi$
$683$	14.2066	+	14.2066i	0.543599	+	0.543599i	−0.380983	+	0.924582i	$0.624414\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	1.10060	−	1.10060i	0.0419906	−	0.0419906i
$688$		0			0
$689$		−	12.1819i		−	0.464092i
$690$		0			0
$691$	−14.9314	−	14.9314i	−0.568017	−	0.568017i	0.363556	−	0.931572i	$-0.381563\pi$
$691$	−14.9314	−	14.9314i	−0.568017	−	0.568017i	−0.931572	+	0.363556i	$0.881563\pi$
$692$		0			0
$693$		−	10.1978i		−	0.387383i
$694$		0			0
$695$		0			0
$696$		0			0
$697$	41.4743	−	24.2287i	1.57095	−	0.917727i
$698$		0			0
$699$		−	1.74281i		−	0.0659191i
$700$		0			0
$701$	−35.6241			−1.34551			−0.672753	−	0.739868i	$-0.734888\pi$
$701$	−35.6241			−1.34551			−0.672753	+	0.739868i	$0.734888\pi$
$702$		0			0
$703$	−13.8419	+	13.8419i	−0.522056	+	0.522056i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−5.59031	+	5.59031i	−0.210245	+	0.210245i
$708$		0			0
$709$	−27.4579	+	27.4579i	−1.03120	+	1.03120i	−0.0317069	+	0.999497i	$0.510094\pi$
$709$	−27.4579	+	27.4579i	−1.03120	+	1.03120i	−0.999497	+	0.0317069i	$0.989906\pi$
$710$		0			0
$711$	30.6309	−	30.6309i	1.14875	−	1.14875i
$712$		0			0
$713$			76.1328i			2.85120i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−1.17142	−	1.17142i	−0.0437473	−	0.0437473i
$718$		0			0
$719$	7.29926	−	7.29926i	0.272217	−	0.272217i	−0.557775	−	0.829992i	$-0.688345\pi$
$719$	7.29926	−	7.29926i	0.272217	−	0.272217i	0.829992	+	0.557775i	$0.188345\pi$
$720$		0			0
$721$	9.90953	+	9.90953i	0.369050	+	0.369050i
$722$		0			0
$723$	0.0354399			0.00131802
$724$		0			0
$725$		0			0
$726$		0			0
$727$		−	36.4252i		−	1.35094i	−0.737389	−	0.675468i	$-0.763941\pi$
$727$		−	36.4252i		−	1.35094i	0.737389	−	0.675468i	$-0.236059\pi$
$728$		0			0
$729$			26.3284i			0.975128i
$730$		0			0
$731$	7.84581	−	29.8913i	0.290188	−	1.10557i
$732$		0			0
$733$	−33.2356			−1.22759			−0.613793	−	0.789467i	$-0.710357\pi$
$733$	−33.2356			−1.22759			−0.613793	+	0.789467i	$0.710357\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	9.61008	−	9.61008i	0.353992	−	0.353992i
$738$		0			0
$739$			11.8455i			0.435745i	0.975977	+	0.217872i	$0.0699116\pi$
$739$			11.8455i			0.435745i	−0.975977	+	0.217872i	$0.930088\pi$
$740$		0			0
$741$	0.919817	+	0.919817i	0.0337903	+	0.0337903i
$742$		0			0
$743$	−12.4200	−	12.4200i	−0.455645	−	0.455645i	0.441578	−	0.897223i	$-0.354419\pi$
$743$	−12.4200	−	12.4200i	−0.455645	−	0.455645i	−0.897223	+	0.441578i	$0.854419\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		−	4.54119i		−	0.166154i
$748$		0			0
$749$	−1.74828			−0.0638807
$750$		0			0
$751$	−11.8641	+	11.8641i	−0.432927	+	0.432927i	−0.889623	−	0.456696i	$-0.849033\pi$
$751$	−11.8641	+	11.8641i	−0.432927	+	0.432927i	0.456696	+	0.889623i	$0.349033\pi$
$752$		0			0
$753$	−0.532527	−	0.532527i	−0.0194064	−	0.0194064i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−45.4038			−1.65023			−0.825114	−	0.564966i	$-0.808889\pi$
$757$	−45.4038			−1.65023			−0.825114	+	0.564966i	$0.808889\pi$
$758$		0			0
$759$	−0.921647	−	0.921647i	−0.0334537	−	0.0334537i
$760$		0			0
$761$	−8.72830			−0.316401			−0.158200	−	0.987407i	$-0.550569\pi$
$761$	−8.72830			−0.316401			−0.158200	+	0.987407i	$0.550569\pi$
$762$		0			0
$763$	39.5309			1.43111
$764$		0			0
$765$		0			0
$766$		0			0
$767$	14.4021			0.520031
$768$		0			0
$769$	0.280789			0.0101255			0.00506276	−	0.999987i	$-0.498388\pi$
$769$	0.280789			0.0101255			0.00506276	+	0.999987i	$0.498388\pi$
$770$		0			0
$771$	0.379882	+	0.379882i	0.0136811	+	0.0136811i
$772$		0			0
$773$	14.6221			0.525922			0.262961	−	0.964807i	$-0.415301\pi$
$773$	14.6221			0.525922			0.262961	+	0.964807i	$0.415301\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	1.26078	+	1.26078i	0.0452301	+	0.0452301i
$778$		0			0
$779$	−28.1979	+	28.1979i	−1.01029	+	1.01029i
$780$		0			0
$781$	−0.460713			−0.0164856
$782$		0			0
$783$			6.69819i			0.239374i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−30.0405	−	30.0405i	−1.07083	−	1.07083i	−0.997293	−	0.0735341i	$-0.976572\pi$
$787$	−30.0405	−	30.0405i	−1.07083	−	1.07083i	−0.0735341	−	0.997293i	$-0.523428\pi$
$788$		0			0
$789$	−0.478947	−	0.478947i	−0.0170510	−	0.0170510i
$790$		0			0
$791$		−	28.1132i		−	0.999590i
$792$		0			0
$793$	26.1019	−	26.1019i	0.926906	−	0.926906i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	39.4840			1.39860			0.699298	−	0.714831i	$-0.253496\pi$
$797$	39.4840			1.39860			0.699298	+	0.714831i	$0.253496\pi$
$798$		0			0
$799$	−34.3526	−	9.01682i	−1.21531	−	0.318992i
$800$		0			0
$801$			12.8004i			0.452279i
$802$		0			0
$803$			12.0621i			0.425664i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	1.91359			0.0673616
$808$		0			0
$809$	22.5276	+	22.5276i	0.792030	+	0.792030i	0.981824	−	0.189794i	$-0.0607820\pi$
$809$	22.5276	+	22.5276i	0.792030	+	0.792030i	−0.189794	+	0.981824i	$0.560782\pi$
$810$		0			0
$811$	−30.7078	+	30.7078i	−1.07830	+	1.07830i	−0.0816334	+	0.996662i	$0.526014\pi$
$811$	−30.7078	+	30.7078i	−1.07830	+	1.07830i	−0.996662	+	0.0816334i	$0.973986\pi$
$812$		0			0
$813$	2.05385	+	2.05385i	0.0720318	+	0.0720318i
$814$		0			0
$815$		0			0
$816$		0			0
$817$			25.6570i			0.897625i
$818$		0			0
$819$	−20.0568	+	20.0568i	−0.700841	+	0.700841i
$820$		0			0
$821$	−6.72368	+	6.72368i	−0.234658	+	0.234658i	−0.814634	−	0.579976i	$-0.803062\pi$
$821$	−6.72368	+	6.72368i	−0.234658	+	0.234658i	0.579976	+	0.814634i	$0.303062\pi$
$822$		0			0
$823$	−15.5249	+	15.5249i	−0.541165	+	0.541165i	−0.923870	−	0.382706i	$-0.874992\pi$
$823$	−15.5249	+	15.5249i	−0.541165	+	0.541165i	0.382706	+	0.923870i	$0.374992\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−27.8026	+	27.8026i	−0.966791	+	0.966791i	−0.999466	−	0.0326746i	$-0.989598\pi$
$827$	−27.8026	+	27.8026i	−0.966791	+	0.966791i	0.0326746	+	0.999466i	$0.489598\pi$
$828$		0			0
$829$	7.75039			0.269182			0.134591	−	0.990901i	$-0.457028\pi$
$829$	7.75039			0.269182			0.134591	+	0.990901i	$0.457028\pi$
$830$		0			0
$831$		−	0.713729i		−	0.0247590i
$832$		0			0
$833$	3.15003	+	0.826814i	0.109142	+	0.0286474i
$834$		0			0
$835$		0			0
$836$		0			0
$837$			5.33784i			0.184503i
$838$		0			0
$839$	−8.10722	−	8.10722i	−0.279892	−	0.279892i	0.553174	−	0.833066i	$-0.313417\pi$
$839$	−8.10722	−	8.10722i	−0.279892	−	0.279892i	−0.833066	+	0.553174i	$0.813417\pi$
$840$		0			0
$841$		−	71.2827i		−	2.45803i
$842$		0			0
$843$	0.107646	−	0.107646i	0.00370752	−	0.00370752i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	18.7572	+	18.7572i	0.644504	+	0.644504i
$848$		0			0
$849$	−2.78083			−0.0954380
$850$		0			0
$851$	−54.5559			−1.87015
$852$		0			0
$853$	−19.5663	−	19.5663i	−0.669936	−	0.669936i	0.287765	−	0.957701i	$-0.407088\pi$
$853$	−19.5663	−	19.5663i	−0.669936	−	0.669936i	−0.957701	+	0.287765i	$0.907088\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−3.79301	+	3.79301i	−0.129567	+	0.129567i	−0.768916	−	0.639349i	$-0.779204\pi$
$857$	−3.79301	+	3.79301i	−0.129567	+	0.129567i	0.639349	+	0.768916i	$0.279204\pi$
$858$		0			0
$859$		−	8.60568i		−	0.293622i	−0.989165	−	0.146811i	$-0.953099\pi$
$859$		−	8.60568i		−	0.293622i	0.989165	−	0.146811i	$-0.0469009\pi$
$860$		0			0
$861$	2.56839	+	2.56839i	0.0875304	+	0.0875304i
$862$		0			0
$863$		−	54.1029i		−	1.84168i	−0.389937	−	0.920841i	$-0.627503\pi$
$863$		−	54.1029i		−	1.84168i	0.389937	−	0.920841i	$-0.372497\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−0.510247	−	1.82926i	−0.0173289	−	0.0621250i
$868$		0			0
$869$		−	17.7335i		−	0.601569i
$870$		0			0
$871$	−37.8016			−1.28086
$872$		0			0
$873$	−5.40715	+	5.40715i	−0.183004	+	0.183004i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−37.1480	+	37.1480i	−1.25440	+	1.25440i	−0.300671	+	0.953728i	$0.597211\pi$
$877$	−37.1480	+	37.1480i	−1.25440	+	1.25440i	−0.953728	+	0.300671i	$0.902789\pi$
$878$		0			0
$879$	−1.04801	+	1.04801i	−0.0353483	+	0.0353483i
$880$		0			0
$881$	−21.8952	+	21.8952i	−0.737667	+	0.737667i	−0.972126	−	0.234459i	$-0.924668\pi$
$881$	−21.8952	+	21.8952i	−0.737667	+	0.737667i	0.234459	+	0.972126i	$0.424668\pi$
$882$		0			0
$883$		−	38.5522i		−	1.29739i	−0.761051	−	0.648693i	$-0.775316\pi$
$883$		−	38.5522i		−	1.29739i	0.761051	−	0.648693i	$-0.224684\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	14.4598	+	14.4598i	0.485512	+	0.485512i	0.906887	−	0.421375i	$-0.138452\pi$
$887$	14.4598	+	14.4598i	0.485512	+	0.485512i	−0.421375	+	0.906887i	$0.638452\pi$
$888$		0			0
$889$	31.1552	−	31.1552i	1.04491	−	1.04491i
$890$		0			0
$891$	7.68622	+	7.68622i	0.257498	+	0.257498i
$892$		0			0
$893$	29.4864			0.986725
$894$		0			0
$895$		0			0
$896$		0			0
$897$			3.62534i			0.121046i
$898$		0			0
$899$		−	79.9161i		−	2.66535i
$900$		0			0
$901$	3.74856	−	14.2814i	0.124883	−	0.475782i
$902$		0			0
$903$	2.33695			0.0777689
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−2.10137	+	2.10137i	−0.0697748	+	0.0697748i	−0.741133	−	0.671358i	$-0.765711\pi$
$907$	−2.10137	+	2.10137i	−0.0697748	+	0.0697748i	0.671358	+	0.741133i	$0.265711\pi$
$908$		0			0
$909$		−	8.46246i		−	0.280682i
$910$		0			0
$911$	−18.3384	−	18.3384i	−0.607579	−	0.607579i	0.334733	−	0.942313i	$-0.391354\pi$
$911$	−18.3384	−	18.3384i	−0.607579	−	0.607579i	−0.942313	+	0.334733i	$0.891354\pi$
$912$		0			0
$913$	−1.31455	−	1.31455i	−0.0435051	−	0.0435051i
$914$		0			0
$915$		0			0
$916$		0			0
$917$			32.4411i			1.07130i
$918$		0			0
$919$	13.2875			0.438313			0.219157	−	0.975690i	$-0.429669\pi$
$919$	13.2875			0.438313			0.219157	+	0.975690i	$0.429669\pi$
$920$		0			0
$921$	1.65731	−	1.65731i	0.0546102	−	0.0546102i
$922$		0			0
$923$	0.906118	+	0.906118i	0.0298252	+	0.0298252i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	−15.0008			−0.492690
$928$		0			0
$929$	−3.35997	−	3.35997i	−0.110237	−	0.110237i	0.649837	−	0.760074i	$-0.274837\pi$
$929$	−3.35997	−	3.35997i	−0.110237	−	0.110237i	−0.760074	+	0.649837i	$0.774837\pi$
$930$		0			0
$931$	−2.70381			−0.0886138
$932$		0			0
$933$	−0.0801396			−0.00262365
$934$		0			0
$935$		0			0
$936$		0			0
$937$	13.3082			0.434759			0.217380	−	0.976087i	$-0.430249\pi$
$937$	13.3082			0.434759			0.217380	+	0.976087i	$0.430249\pi$
$938$		0			0
$939$	1.18500			0.0386710
$940$		0			0
$941$	−3.04861	−	3.04861i	−0.0993817	−	0.0993817i	0.655668	−	0.755050i	$-0.272387\pi$
$941$	−3.04861	−	3.04861i	−0.0993817	−	0.0993817i	−0.755050	+	0.655668i	$0.772387\pi$
$942$		0			0
$943$	−111.138			−3.61916
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−38.5353	−	38.5353i	−1.25223	−	1.25223i	−0.954719	−	0.297510i	$-0.903844\pi$
$947$	−38.5353	−	38.5353i	−1.25223	−	1.25223i	−0.297510	−	0.954719i	$-0.596156\pi$
$948$		0			0
$949$	23.7235	−	23.7235i	0.770096	−	0.770096i
$950$		0			0
$951$	2.38477			0.0773316
$952$		0			0
$953$		−	31.4405i		−	1.01846i	−0.860631	−	0.509229i	$-0.829931\pi$
$953$		−	31.4405i		−	1.01846i	0.860631	−	0.509229i	$-0.170069\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	0.967447	+	0.967447i	0.0312731	+	0.0312731i
$958$		0			0
$959$	−6.71352	−	6.71352i	−0.216791	−	0.216791i
$960$		0			0
$961$		−	32.6857i		−	1.05438i
$962$		0			0
$963$	1.32325	−	1.32325i	0.0426411	−	0.0426411i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	28.4642			0.915347			0.457674	−	0.889120i	$-0.348683\pi$
$967$	28.4642			0.915347			0.457674	+	0.889120i	$0.348683\pi$
$968$		0			0
$969$	0.795303	+	1.36139i	0.0255488	+	0.0437341i
$970$		0			0
$971$			44.9224i			1.44163i	0.693128	+	0.720815i	$0.256232\pi$
$971$			44.9224i			1.44163i	−0.693128	+	0.720815i	$0.743768\pi$
$972$		0			0
$973$			21.5903i			0.692153i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−37.8272			−1.21020			−0.605100	−	0.796150i	$-0.706867\pi$
$977$	−37.8272			−1.21020			−0.605100	+	0.796150i	$0.706867\pi$
$978$		0			0
$979$	3.70534	+	3.70534i	0.118423	+	0.118423i
$980$		0			0
$981$	−29.9204	+	29.9204i	−0.955284	+	0.955284i
$982$		0			0
$983$	26.7685	+	26.7685i	0.853783	+	0.853783i	0.990597	−	0.136814i	$-0.0436862\pi$
$983$	26.7685	+	26.7685i	0.853783	+	0.853783i	−0.136814	+	0.990597i	$0.543686\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		−	2.68575i		−	0.0854883i
$988$		0			0
$989$	−50.5619	+	50.5619i	−1.60777	+	1.60777i
$990$		0			0
$991$	−12.6185	+	12.6185i	−0.400838	+	0.400838i	−0.878528	−	0.477690i	$-0.841474\pi$
$991$	−12.6185	+	12.6185i	−0.400838	+	0.400838i	0.477690	+	0.878528i	$0.341474\pi$
$992$		0			0
$993$	0.982814	−	0.982814i	0.0311887	−	0.0311887i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−36.3754	+	36.3754i	−1.15202	+	1.15202i	−0.165872	+	0.986147i	$0.553044\pi$
$997$	−36.3754	+	36.3754i	−1.15202	+	1.15202i	−0.986147	+	0.165872i	$0.946956\pi$
$998$		0			0
$999$	−3.82503			−0.121019

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	1700.2.m.c.1449.4		12
5.2	odd	4		1700.2.o.d.701.4		12
5.3	odd	4		340.2.o.a.21.3	✓	12
5.4	even	2		1700.2.m.f.1449.3		12
15.8	even	4		3060.2.be.b.361.6		12
17.13	even	4		1700.2.m.f.149.3		12
20.3	even	4		1360.2.bt.c.1041.4		12
85.8	odd	8		5780.2.a.m.1.3		6
85.13	odd	4		340.2.o.a.81.3	yes	12
85.43	odd	8		5780.2.a.n.1.4		6
85.47	odd	4		1700.2.o.d.1101.4		12
85.53	odd	8		5780.2.c.h.5201.7		12
85.64	even	4	inner	1700.2.m.c.149.4		12
85.83	odd	8		5780.2.c.h.5201.6		12
255.98	even	4		3060.2.be.b.1441.6		12
340.183	even	4		1360.2.bt.c.81.4		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
340.2.o.a.21.3	✓	12	5.3	odd	4
340.2.o.a.81.3	yes	12	85.13	odd	4
1360.2.bt.c.81.4		12	340.183	even	4
1360.2.bt.c.1041.4		12	20.3	even	4
1700.2.m.c.149.4		12	85.64	even	4	inner
1700.2.m.c.1449.4		12	1.1	even	1	trivial
1700.2.m.f.149.3		12	17.13	even	4
1700.2.m.f.1449.3		12	5.4	even	2
1700.2.o.d.701.4		12	5.2	odd	4
1700.2.o.d.1101.4		12	85.47	odd	4
3060.2.be.b.361.6		12	15.8	even	4
3060.2.be.b.1441.6		12	255.98	even	4
5780.2.a.m.1.3		6	85.8	odd	8
5780.2.a.n.1.4		6	85.43	odd	8
5780.2.c.h.5201.6		12	85.83	odd	8
5780.2.c.h.5201.7		12	85.53	odd	8

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

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Character values

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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	1700.2.m.c.1449.4

Level	$1700$
Weight	$2$
Character	1700.1449
Analytic conductor	$13.575$
Analytic rank	$0$
Dimension	$12$
Inner twists	$2$