LMFDB - Embedded newform 2640.2.d.i.529.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2640,2,Mod(529,2640)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2640, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 1, 0]))

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

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

chi := DirichletCharacter("2640.529");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2640 = 2^{4} \cdot 3 \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2640.d (of order $2$, degree $1$, 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:	$21.0805061336$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.350464.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 $ x^6 - 2x^5 + 2x^4 + 2x^3 + 4x^2 - 4*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 165)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			529.1
Root			$0.403032 - 0.403032i$ of defining polynomial
Character	$\chi$	$=$	2640.529
Dual form			2640.2.d.i.529.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} +(-1.48119 - 1.67513i) q^{5} -2.80606i q^{7} -1.00000 q^{9} +O(q^{10})$	$q-1.00000i q^{3} +(-1.48119 - 1.67513i) q^{5} -2.80606i q^{7} -1.00000 q^{9} +1.00000 q^{11} +5.11871i q^{13} +(-1.67513 + 1.48119i) q^{15} +4.54420i q^{17} -4.57452 q^{19} -2.80606 q^{21} +4.00000i q^{23} +(-0.612127 + 4.96239i) q^{25} +1.00000i q^{27} +2.38787 q^{29} +0.962389 q^{31} -1.00000i q^{33} +(-4.70052 + 4.15633i) q^{35} +1.61213i q^{37} +5.11871 q^{39} -2.38787 q^{41} +2.80606i q^{43} +(1.48119 + 1.67513i) q^{45} +4.31265i q^{47} -0.873992 q^{49} +4.54420 q^{51} +6.57452i q^{53} +(-1.48119 - 1.67513i) q^{55} +4.57452i q^{57} -13.2750 q^{59} +7.92478 q^{61} +2.80606i q^{63} +(8.57452 - 7.58181i) q^{65} -10.7005i q^{67} +4.00000 q^{69} +7.35026 q^{71} -6.41819i q^{73} +(4.96239 + 0.612127i) q^{75} -2.80606i q^{77} +1.35026 q^{79} +1.00000 q^{81} -0.806063i q^{83} +(7.61213 - 6.73084i) q^{85} -2.38787i q^{87} +2.96239 q^{89} +14.3634 q^{91} -0.962389i q^{93} +(6.77575 + 7.66291i) q^{95} -9.92478i q^{97} -1.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 2 q^{5} - 6 q^{9}+O(q^{10}) $ 6 * q + 2 * q^5 - 6 * q^9	$ 6 q + 2 q^{5} - 6 q^{9} + 6 q^{11} - 4 q^{19} - 16 q^{21} - 2 q^{25} + 16 q^{29} - 16 q^{31} + 12 q^{35} - 12 q^{39} - 16 q^{41} - 2 q^{45} - 22 q^{49} + 8 q^{51} + 2 q^{55} - 16 q^{59} + 4 q^{61} + 28 q^{65} + 24 q^{69} + 24 q^{71} + 8 q^{75} - 12 q^{79} + 6 q^{81} + 44 q^{85} - 4 q^{89} - 16 q^{91} + 44 q^{95} - 6 q^{99}+O(q^{100}) $ 6 * q + 2 * q^5 - 6 * q^9 + 6 * q^11 - 4 * q^19 - 16 * q^21 - 2 * q^25 + 16 * q^29 - 16 * q^31 + 12 * q^35 - 12 * q^39 - 16 * q^41 - 2 * q^45 - 22 * q^49 + 8 * q^51 + 2 * q^55 - 16 * q^59 + 4 * q^61 + 28 * q^65 + 24 * q^69 + 24 * q^71 + 8 * q^75 - 12 * q^79 + 6 * q^81 + 44 * q^85 - 4 * q^89 - 16 * q^91 + 44 * q^95 - 6 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2640\mathbb{Z}\right)^\times$.

$n$	$661$	$881$	$991$	$1057$	$1201$
$\chi(n)$	$1$	$1$	$1$	$-1$	$1$

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
$3$		−	1.00000i		−	0.577350i
$3$		−	1.00000i		−	0.577350i
$4$		0			0
$5$	−1.48119	−	1.67513i	−0.662410	−	0.749141i
$5$	−1.48119	−	1.67513i	−0.662410	−	0.749141i
$6$		0			0
$7$		−	2.80606i		−	1.06059i	−0.847812	−	0.530296i	$-0.822081\pi$
$7$		−	2.80606i		−	1.06059i	0.847812	−	0.530296i	$-0.177919\pi$
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$	1.00000			0.301511
$11$	1.00000			0.301511
$12$		0			0
$13$			5.11871i			1.41968i	0.704365	+	0.709838i	$0.251232\pi$
$13$			5.11871i			1.41968i	−0.704365	+	0.709838i	$0.748768\pi$
$14$		0			0
$15$	−1.67513	+	1.48119i	−0.432517	+	0.382443i
$16$		0			0
$17$			4.54420i			1.10213i	0.834462	+	0.551065i	$0.185778\pi$
$17$			4.54420i			1.10213i	−0.834462	+	0.551065i	$0.814222\pi$
$18$		0			0
$19$	−4.57452			−1.04947			−0.524733	−	0.851267i	$-0.675835\pi$
$19$	−4.57452			−1.04947			−0.524733	+	0.851267i	$0.675835\pi$
$20$		0			0
$21$	−2.80606			−0.612333
$22$		0			0
$23$			4.00000i			0.834058i	0.908893	+	0.417029i	$0.136929\pi$
$23$			4.00000i			0.834058i	−0.908893	+	0.417029i	$0.863071\pi$
$24$		0			0
$25$	−0.612127	+	4.96239i	−0.122425	+	0.992478i
$26$		0			0
$27$			1.00000i			0.192450i
$28$		0			0
$29$	2.38787			0.443417			0.221708	−	0.975113i	$-0.428837\pi$
$29$	2.38787			0.443417			0.221708	+	0.975113i	$0.428837\pi$
$30$		0			0
$31$	0.962389			0.172850			0.0864250	−	0.996258i	$-0.472456\pi$
$31$	0.962389			0.172850			0.0864250	+	0.996258i	$0.472456\pi$
$32$		0			0
$33$		−	1.00000i		−	0.174078i
$34$		0			0
$35$	−4.70052	+	4.15633i	−0.794533	+	0.702547i
$36$		0			0
$37$			1.61213i			0.265032i	0.991181	+	0.132516i	$0.0423056\pi$
$37$			1.61213i			0.265032i	−0.991181	+	0.132516i	$0.957694\pi$
$38$		0			0
$39$	5.11871			0.819650
$40$		0			0
$41$	−2.38787			−0.372923			−0.186462	−	0.982462i	$-0.559702\pi$
$41$	−2.38787			−0.372923			−0.186462	+	0.982462i	$0.559702\pi$
$42$		0			0
$43$			2.80606i			0.427921i	0.976842	+	0.213960i	$0.0686363\pi$
$43$			2.80606i			0.427921i	−0.976842	+	0.213960i	$0.931364\pi$
$44$		0			0
$45$	1.48119	+	1.67513i	0.220803	+	0.249714i
$46$		0			0
$47$			4.31265i			0.629065i	0.949247	+	0.314532i	$0.101848\pi$
$47$			4.31265i			0.629065i	−0.949247	+	0.314532i	$0.898152\pi$
$48$		0			0
$49$	−0.873992			−0.124856
$50$		0			0
$51$	4.54420			0.636315
$52$		0			0
$53$			6.57452i			0.903079i	0.892251	+	0.451540i	$0.149125\pi$
$53$			6.57452i			0.903079i	−0.892251	+	0.451540i	$0.850875\pi$
$54$		0			0
$55$	−1.48119	−	1.67513i	−0.199724	−	0.225875i
$56$		0			0
$57$			4.57452i			0.605909i
$58$		0			0
$59$	−13.2750			−1.72826			−0.864131	−	0.503266i	$-0.832132\pi$
$59$	−13.2750			−1.72826			−0.864131	+	0.503266i	$0.832132\pi$
$60$		0			0
$61$	7.92478			1.01466			0.507332	−	0.861751i	$-0.330632\pi$
$61$	7.92478			1.01466			0.507332	+	0.861751i	$0.330632\pi$
$62$		0			0
$63$			2.80606i			0.353531i
$64$		0			0
$65$	8.57452	−	7.58181i	1.06354	−	0.940408i
$66$		0			0
$67$		−	10.7005i		−	1.30728i	−0.756807	−	0.653639i	$-0.773242\pi$
$67$		−	10.7005i		−	1.30728i	0.756807	−	0.653639i	$-0.226758\pi$
$68$		0			0
$69$	4.00000			0.481543
$70$		0			0
$71$	7.35026			0.872316			0.436158	−	0.899870i	$-0.356339\pi$
$71$	7.35026			0.872316			0.436158	+	0.899870i	$0.356339\pi$
$72$		0			0
$73$		−	6.41819i		−	0.751192i	−0.926783	−	0.375596i	$-0.877438\pi$
$73$		−	6.41819i		−	0.751192i	0.926783	−	0.375596i	$-0.122562\pi$
$74$		0			0
$75$	4.96239	+	0.612127i	0.573007	+	0.0706823i
$76$		0			0
$77$		−	2.80606i		−	0.319781i
$78$		0			0
$79$	1.35026			0.151916			0.0759582	−	0.997111i	$-0.475798\pi$
$79$	1.35026			0.151916			0.0759582	+	0.997111i	$0.475798\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$		−	0.806063i		−	0.0884770i	−0.999021	−	0.0442385i	$-0.985914\pi$
$83$		−	0.806063i		−	0.0884770i	0.999021	−	0.0442385i	$-0.0140861\pi$
$84$		0			0
$85$	7.61213	−	6.73084i	0.825651	−	0.730062i
$86$		0			0
$87$		−	2.38787i		−	0.256007i
$88$		0			0
$89$	2.96239			0.314013			0.157006	−	0.987598i	$-0.449816\pi$
$89$	2.96239			0.314013			0.157006	+	0.987598i	$0.449816\pi$
$90$		0			0
$91$	14.3634			1.50570
$92$		0			0
$93$		−	0.962389i		−	0.0997950i
$94$		0			0
$95$	6.77575	+	7.66291i	0.695177	+	0.786198i
$96$		0			0
$97$		−	9.92478i		−	1.00771i	−0.863789	−	0.503854i	$-0.831915\pi$
$97$		−	9.92478i		−	1.00771i	0.863789	−	0.503854i	$-0.168085\pi$
$98$		0			0
$99$	−1.00000			−0.100504
$100$		0			0
$101$	−13.6121			−1.35446			−0.677229	−	0.735773i	$-0.736819\pi$
$101$	−13.6121			−1.35446			−0.677229	+	0.735773i	$0.736819\pi$
$102$		0			0
$103$			16.3127i			1.60733i	0.595080	+	0.803667i	$0.297120\pi$
$103$			16.3127i			1.60733i	−0.595080	+	0.803667i	$0.702880\pi$
$104$		0			0
$105$	4.15633	+	4.70052i	0.405616	+	0.458724i
$106$		0			0
$107$			9.43136i			0.911764i	0.890040	+	0.455882i	$0.150676\pi$
$107$			9.43136i			0.911764i	−0.890040	+	0.455882i	$0.849324\pi$
$108$		0			0
$109$	15.4010			1.47515			0.737576	−	0.675264i	$-0.235970\pi$
$109$	15.4010			1.47515			0.737576	+	0.675264i	$0.235970\pi$
$110$		0			0
$111$	1.61213			0.153016
$112$		0			0
$113$			13.7381i			1.29238i	0.763179	+	0.646188i	$0.223638\pi$
$113$			13.7381i			1.29238i	−0.763179	+	0.646188i	$0.776362\pi$
$114$		0			0
$115$	6.70052	−	5.92478i	0.624827	−	0.552488i
$116$		0			0
$117$		−	5.11871i		−	0.473225i
$118$		0			0
$119$	12.7513			1.16891
$120$		0			0
$121$	1.00000			0.0909091
$122$		0			0
$123$			2.38787i			0.215307i
$124$		0			0
$125$	9.21933	−	6.32487i	0.824602	−	0.565713i
$126$		0			0
$127$		−	12.4182i		−	1.10194i	−0.834526	−	0.550968i	$-0.814259\pi$
$127$		−	12.4182i		−	1.10194i	0.834526	−	0.550968i	$-0.185741\pi$
$128$		0			0
$129$	2.80606			0.247060
$130$		0			0
$131$	−5.92478			−0.517650			−0.258825	−	0.965924i	$-0.583335\pi$
$131$	−5.92478			−0.517650			−0.258825	+	0.965924i	$0.583335\pi$
$132$		0			0
$133$			12.8364i			1.11306i
$134$		0			0
$135$	1.67513	−	1.48119i	0.144172	−	0.127481i
$136$		0			0
$137$			9.79877i			0.837165i	0.908179	+	0.418583i	$0.137473\pi$
$137$			9.79877i			0.837165i	−0.908179	+	0.418583i	$0.862527\pi$
$138$		0			0
$139$	−2.12601			−0.180326			−0.0901628	−	0.995927i	$-0.528739\pi$
$139$	−2.12601			−0.180326			−0.0901628	+	0.995927i	$0.528739\pi$
$140$		0			0
$141$	4.31265			0.363191
$142$		0			0
$143$			5.11871i			0.428048i
$144$		0			0
$145$	−3.53690	−	4.00000i	−0.293724	−	0.332182i
$146$		0			0
$147$			0.873992i			0.0720856i
$148$		0			0
$149$	−8.31265			−0.680999			−0.340499	−	0.940245i	$-0.610596\pi$
$149$	−8.31265			−0.680999			−0.340499	+	0.940245i	$0.610596\pi$
$150$		0			0
$151$	15.2750			1.24307			0.621533	−	0.783388i	$-0.286510\pi$
$151$	15.2750			1.24307			0.621533	+	0.783388i	$0.286510\pi$
$152$		0			0
$153$		−	4.54420i		−	0.367377i
$154$		0			0
$155$	−1.42548	−	1.61213i	−0.114498	−	0.129489i
$156$		0			0
$157$			7.01317i			0.559712i	0.960042	+	0.279856i	$0.0902868\pi$
$157$			7.01317i			0.559712i	−0.960042	+	0.279856i	$0.909713\pi$
$158$		0			0
$159$	6.57452			0.521393
$160$		0			0
$161$	11.2243			0.884595
$162$		0			0
$163$			6.76116i			0.529575i	0.964307	+	0.264787i	$0.0853018\pi$
$163$			6.76116i			0.529575i	−0.964307	+	0.264787i	$0.914698\pi$
$164$		0			0
$165$	−1.67513	+	1.48119i	−0.130409	+	0.115311i
$166$		0			0
$167$			11.8192i			0.914600i	0.889312	+	0.457300i	$0.151183\pi$
$167$			11.8192i			0.914600i	−0.889312	+	0.457300i	$0.848817\pi$
$168$		0			0
$169$	−13.2012			−1.01548
$170$		0			0
$171$	4.57452			0.349822
$172$		0			0
$173$		−	6.99271i		−	0.531646i	−0.964022	−	0.265823i	$-0.914356\pi$
$173$		−	6.99271i		−	0.531646i	0.964022	−	0.265823i	$-0.0856436\pi$
$174$		0			0
$175$	13.9248	+	1.71767i	1.05261	+	0.129843i
$176$		0			0
$177$			13.2750i			0.997813i
$178$		0			0
$179$	−2.70052			−0.201847			−0.100923	−	0.994894i	$-0.532180\pi$
$179$	−2.70052			−0.201847			−0.100923	+	0.994894i	$0.532180\pi$
$180$		0			0
$181$	−21.1998			−1.57577			−0.787885	−	0.615822i	$-0.788824\pi$
$181$	−21.1998			−1.57577			−0.787885	+	0.615822i	$0.788824\pi$
$182$		0			0
$183$		−	7.92478i		−	0.585816i
$184$		0			0
$185$	2.70052	−	2.38787i	0.198546	−	0.175560i
$186$		0			0
$187$			4.54420i			0.332305i
$188$		0			0
$189$	2.80606			0.204111
$190$		0			0
$191$	−13.1490			−0.951430			−0.475715	−	0.879599i	$-0.657811\pi$
$191$	−13.1490			−0.951430			−0.475715	+	0.879599i	$0.657811\pi$
$192$		0			0
$193$		−	5.89446i		−	0.424293i	−0.977238	−	0.212146i	$-0.931955\pi$
$193$		−	5.89446i		−	0.424293i	0.977238	−	0.212146i	$-0.0680453\pi$
$194$		0			0
$195$	−7.58181	−	8.57452i	−0.542945	−	0.614034i
$196$		0			0
$197$			20.3938i			1.45299i	0.687169	+	0.726497i	$0.258853\pi$
$197$			20.3938i			1.45299i	−0.687169	+	0.726497i	$0.741147\pi$
$198$		0			0
$199$	17.4010			1.23353			0.616764	−	0.787148i	$-0.288443\pi$
$199$	17.4010			1.23353			0.616764	+	0.787148i	$0.288443\pi$
$200$		0			0
$201$	−10.7005			−0.754757
$202$		0			0
$203$		−	6.70052i		−	0.470285i
$204$		0			0
$205$	3.53690	+	4.00000i	0.247028	+	0.279372i
$206$		0			0
$207$		−	4.00000i		−	0.278019i
$208$		0			0
$209$	−4.57452			−0.316426
$210$		0			0
$211$	−18.1260			−1.24785			−0.623923	−	0.781486i	$-0.714462\pi$
$211$	−18.1260			−1.24785			−0.623923	+	0.781486i	$0.714462\pi$
$212$		0			0
$213$		−	7.35026i		−	0.503632i
$214$		0			0
$215$	4.70052	−	4.15633i	0.320573	−	0.283459i
$216$		0			0
$217$		−	2.70052i		−	0.183323i
$218$		0			0
$219$	−6.41819			−0.433701
$220$		0			0
$221$	−23.2605			−1.56467
$222$		0			0
$223$			23.6385i			1.58295i	0.611202	+	0.791475i	$0.290686\pi$
$223$			23.6385i			1.58295i	−0.611202	+	0.791475i	$0.709314\pi$
$224$		0			0
$225$	0.612127	−	4.96239i	0.0408085	−	0.330826i
$226$		0			0
$227$			1.26916i			0.0842371i	0.999113	+	0.0421185i	$0.0134107\pi$
$227$			1.26916i			0.0842371i	−0.999113	+	0.0421185i	$0.986589\pi$
$228$		0			0
$229$	−16.1768			−1.06899			−0.534496	−	0.845171i	$-0.679499\pi$
$229$	−16.1768			−1.06899			−0.534496	+	0.845171i	$0.679499\pi$
$230$		0			0
$231$	−2.80606			−0.184625
$232$		0			0
$233$			18.4690i			1.20994i	0.796247	+	0.604971i	$0.206815\pi$
$233$			18.4690i			1.20994i	−0.796247	+	0.604971i	$0.793185\pi$
$234$		0			0
$235$	7.22425	−	6.38787i	0.471258	−	0.416699i
$236$		0			0
$237$		−	1.35026i		−	0.0877089i
$238$		0			0
$239$	19.3258			1.25008			0.625042	−	0.780591i	$-0.285082\pi$
$239$	19.3258			1.25008			0.625042	+	0.780591i	$0.285082\pi$
$240$		0			0
$241$	28.5501			1.83907			0.919536	−	0.393006i	$-0.128565\pi$
$241$	28.5501			1.83907			0.919536	+	0.393006i	$0.128565\pi$
$242$		0			0
$243$		−	1.00000i		−	0.0641500i
$244$		0			0
$245$	1.29455	+	1.46405i	0.0827059	+	0.0935348i
$246$		0			0
$247$		−	23.4156i		−	1.48990i
$248$		0			0
$249$	−0.806063			−0.0510822
$250$		0			0
$251$	−23.1998			−1.46436			−0.732180	−	0.681112i	$-0.761497\pi$
$251$	−23.1998			−1.46436			−0.732180	+	0.681112i	$0.761497\pi$
$252$		0			0
$253$			4.00000i			0.251478i
$254$		0			0
$255$	−6.73084	−	7.61213i	−0.421502	−	0.476690i
$256$		0			0
$257$		−	10.8872i		−	0.679123i	−0.940584	−	0.339561i	$-0.889721\pi$
$257$		−	10.8872i		−	0.679123i	0.940584	−	0.339561i	$-0.110279\pi$
$258$		0			0
$259$	4.52373			0.281091
$260$		0			0
$261$	−2.38787			−0.147806
$262$		0			0
$263$			9.11871i			0.562284i	0.959666	+	0.281142i	$0.0907132\pi$
$263$			9.11871i			0.562284i	−0.959666	+	0.281142i	$0.909287\pi$
$264$		0			0
$265$	11.0132	−	9.73813i	0.676534	−	0.598209i
$266$		0			0
$267$		−	2.96239i		−	0.181295i
$268$		0			0
$269$	11.2995			0.688941			0.344471	−	0.938797i	$-0.388058\pi$
$269$	11.2995			0.688941			0.344471	+	0.938797i	$0.388058\pi$
$270$		0			0
$271$	−25.1998			−1.53078			−0.765390	−	0.643567i	$-0.777454\pi$
$271$	−25.1998			−1.53078			−0.765390	+	0.643567i	$0.777454\pi$
$272$		0			0
$273$		−	14.3634i		−	0.869315i
$274$		0			0
$275$	−0.612127	+	4.96239i	−0.0369126	+	0.299243i
$276$		0			0
$277$		−	2.41819i		−	0.145295i	−0.997358	−	0.0726475i	$-0.976855\pi$
$277$		−	2.41819i		−	0.145295i	0.997358	−	0.0726475i	$-0.0231448\pi$
$278$		0			0
$279$	−0.962389			−0.0576167
$280$		0			0
$281$	30.4894			1.81885			0.909424	−	0.415870i	$-0.136523\pi$
$281$	30.4894			1.81885			0.909424	+	0.415870i	$0.136523\pi$
$282$		0			0
$283$			8.35756i			0.496805i	0.968657	+	0.248403i	$0.0799056\pi$
$283$			8.35756i			0.496805i	−0.968657	+	0.248403i	$0.920094\pi$
$284$		0			0
$285$	7.66291	−	6.77575i	0.453912	−	0.401361i
$286$		0			0
$287$			6.70052i			0.395519i
$288$		0			0
$289$	−3.64974			−0.214690
$290$		0			0
$291$	−9.92478			−0.581801
$292$		0			0
$293$			23.0943i			1.34918i	0.738192	+	0.674591i	$0.235680\pi$
$293$			23.0943i			1.34918i	−0.738192	+	0.674591i	$0.764320\pi$
$294$		0			0
$295$	19.6629	+	22.2374i	1.14482	+	1.29471i
$296$		0			0
$297$			1.00000i			0.0580259i
$298$		0			0
$299$	−20.4749			−1.18409
$300$		0			0
$301$	7.87399			0.453849
$302$		0			0
$303$			13.6121i			0.781996i
$304$		0			0
$305$	−11.7381	−	13.2750i	−0.672124	−	0.760127i
$306$		0			0
$307$		−	2.65562i		−	0.151564i	−0.997124	−	0.0757821i	$-0.975855\pi$
$307$		−	2.65562i		−	0.151564i	0.997124	−	0.0757821i	$-0.0241453\pi$
$308$		0			0
$309$	16.3127			0.927994
$310$		0			0
$311$	15.9756			0.905891			0.452946	−	0.891538i	$-0.350373\pi$
$311$	15.9756			0.905891			0.452946	+	0.891538i	$0.350373\pi$
$312$		0			0
$313$			0.0606343i			0.00342726i	0.999999	+	0.00171363i	$0.000545465\pi$
$313$			0.0606343i			0.00342726i	−0.999999	+	0.00171363i	$0.999455\pi$
$314$		0			0
$315$	4.70052	−	4.15633i	0.264844	−	0.234182i
$316$		0			0
$317$			3.81336i			0.214180i	0.994249	+	0.107090i	$0.0341532\pi$
$317$			3.81336i			0.214180i	−0.994249	+	0.107090i	$0.965847\pi$
$318$		0			0
$319$	2.38787			0.133695
$320$		0			0
$321$	9.43136			0.526407
$322$		0			0
$323$		−	20.7875i		−	1.15665i
$324$		0			0
$325$	−25.4010	−	3.13330i	−1.40900	−	0.173804i
$326$		0			0
$327$		−	15.4010i		−	0.851680i
$328$		0			0
$329$	12.1016			0.667181
$330$		0			0
$331$	−24.2882			−1.33500			−0.667500	−	0.744609i	$-0.732636\pi$
$331$	−24.2882			−1.33500			−0.667500	+	0.744609i	$0.732636\pi$
$332$		0			0
$333$		−	1.61213i		−	0.0883440i
$334$		0			0
$335$	−17.9248	+	15.8496i	−0.979335	+	0.865954i
$336$		0			0
$337$		−	22.5804i		−	1.23003i	−0.788514	−	0.615016i	$-0.789149\pi$
$337$		−	22.5804i		−	1.23003i	0.788514	−	0.615016i	$-0.210851\pi$
$338$		0			0
$339$	13.7381			0.746153
$340$		0			0
$341$	0.962389			0.0521163
$342$		0			0
$343$		−	17.1900i		−	0.928171i
$344$		0			0
$345$	−5.92478	−	6.70052i	−0.318979	−	0.360744i
$346$		0			0
$347$		−	20.1925i		−	1.08399i	−0.840381	−	0.541996i	$-0.817669\pi$
$347$		−	20.1925i		−	1.08399i	0.840381	−	0.541996i	$-0.182331\pi$
$348$		0			0
$349$	27.2506			1.45869			0.729346	−	0.684145i	$-0.239825\pi$
$349$	27.2506			1.45869			0.729346	+	0.684145i	$0.239825\pi$
$350$		0			0
$351$	−5.11871			−0.273217
$352$		0			0
$353$		−	5.02302i		−	0.267349i	−0.991025	−	0.133674i	$-0.957322\pi$
$353$		−	5.02302i		−	0.267349i	0.991025	−	0.133674i	$-0.0426776\pi$
$354$		0			0
$355$	−10.8872	−	12.3127i	−0.577831	−	0.653488i
$356$		0			0
$357$		−	12.7513i		−	0.674871i
$358$		0			0
$359$	−18.1768			−0.959334			−0.479667	−	0.877450i	$-0.659243\pi$
$359$	−18.1768			−0.959334			−0.479667	+	0.877450i	$0.659243\pi$
$360$		0			0
$361$	1.92619			0.101379
$362$		0			0
$363$		−	1.00000i		−	0.0524864i
$364$		0			0
$365$	−10.7513	+	9.50659i	−0.562749	+	0.497598i
$366$		0			0
$367$			8.56467i			0.447072i	0.974696	+	0.223536i	$0.0717600\pi$
$367$			8.56467i			0.447072i	−0.974696	+	0.223536i	$0.928240\pi$
$368$		0			0
$369$	2.38787			0.124308
$370$		0			0
$371$	18.4485			0.957799
$372$		0			0
$373$		−	7.81924i		−	0.404865i	−0.979296	−	0.202432i	$-0.935115\pi$
$373$		−	7.81924i		−	0.404865i	0.979296	−	0.202432i	$-0.0648846\pi$
$374$		0			0
$375$	−6.32487	−	9.21933i	−0.326615	−	0.476084i
$376$		0			0
$377$			12.2228i			0.629508i
$378$		0			0
$379$	1.14903			0.0590218			0.0295109	−	0.999564i	$-0.490605\pi$
$379$	1.14903			0.0590218			0.0295109	+	0.999564i	$0.490605\pi$
$380$		0			0
$381$	−12.4182			−0.636203
$382$		0			0
$383$			34.0870i			1.74176i	0.491493	+	0.870882i	$0.336451\pi$
$383$			34.0870i			1.74176i	−0.491493	+	0.870882i	$0.663549\pi$
$384$		0			0
$385$	−4.70052	+	4.15633i	−0.239561	+	0.211826i
$386$		0			0
$387$		−	2.80606i		−	0.142640i
$388$		0			0
$389$	−23.7743			−1.20541			−0.602703	−	0.797965i	$-0.705910\pi$
$389$	−23.7743			−1.20541			−0.602703	+	0.797965i	$0.705910\pi$
$390$		0			0
$391$	−18.1768			−0.919240
$392$		0			0
$393$			5.92478i			0.298865i
$394$		0			0
$395$	−2.00000	−	2.26187i	−0.100631	−	0.113807i
$396$		0			0
$397$		−	4.15045i		−	0.208305i	−0.994561	−	0.104152i	$-0.966787\pi$
$397$		−	4.15045i		−	0.208305i	0.994561	−	0.104152i	$-0.0332130\pi$
$398$		0			0
$399$	12.8364			0.642623
$400$		0			0
$401$	−33.9149			−1.69363			−0.846815	−	0.531887i	$-0.821483\pi$
$401$	−33.9149			−1.69363			−0.846815	+	0.531887i	$0.821483\pi$
$402$		0			0
$403$			4.92619i			0.245391i
$404$		0			0
$405$	−1.48119	−	1.67513i	−0.0736011	−	0.0832379i
$406$		0			0
$407$			1.61213i			0.0799102i
$408$		0			0
$409$	−8.55008			−0.422774			−0.211387	−	0.977402i	$-0.567798\pi$
$409$	−8.55008			−0.422774			−0.211387	+	0.977402i	$0.567798\pi$
$410$		0			0
$411$	9.79877			0.483338
$412$		0			0
$413$			37.2506i			1.83298i
$414$		0			0
$415$	−1.35026	+	1.19394i	−0.0662817	+	0.0586080i
$416$		0			0
$417$			2.12601i			0.104111i
$418$		0			0
$419$	13.2995			0.649722			0.324861	−	0.945762i	$-0.394682\pi$
$419$	13.2995			0.649722			0.324861	+	0.945762i	$0.394682\pi$
$420$		0			0
$421$	33.3014			1.62301			0.811505	−	0.584345i	$-0.198649\pi$
$421$	33.3014			1.62301			0.811505	+	0.584345i	$0.198649\pi$
$422$		0			0
$423$		−	4.31265i		−	0.209688i
$424$		0			0
$425$	−22.5501	−	2.78163i	−1.09384	−	0.134929i
$426$		0			0
$427$		−	22.2374i		−	1.07614i
$428$		0			0
$429$	5.11871			0.247134
$430$		0			0
$431$	−2.07522			−0.0999600			−0.0499800	−	0.998750i	$-0.515916\pi$
$431$	−2.07522			−0.0999600			−0.0499800	+	0.998750i	$0.515916\pi$
$432$		0			0
$433$			37.4010i			1.79738i	0.438585	+	0.898690i	$0.355480\pi$
$433$			37.4010i			1.79738i	−0.438585	+	0.898690i	$0.644520\pi$
$434$		0			0
$435$	−4.00000	+	3.53690i	−0.191785	+	0.169582i
$436$		0			0
$437$		−	18.2981i		−	0.875315i
$438$		0			0
$439$	−6.02444			−0.287531			−0.143765	−	0.989612i	$-0.545921\pi$
$439$	−6.02444			−0.287531			−0.143765	+	0.989612i	$0.545921\pi$
$440$		0			0
$441$	0.873992			0.0416187
$442$		0			0
$443$		−	10.1359i		−	0.481569i	−0.970579	−	0.240785i	$-0.922595\pi$
$443$		−	10.1359i		−	0.481569i	0.970579	−	0.240785i	$-0.0774047\pi$
$444$		0			0
$445$	−4.38787	−	4.96239i	−0.208005	−	0.235240i
$446$		0			0
$447$			8.31265i			0.393175i
$448$		0			0
$449$	11.4861			0.542063			0.271032	−	0.962570i	$-0.412635\pi$
$449$	11.4861			0.542063			0.271032	+	0.962570i	$0.412635\pi$
$450$		0			0
$451$	−2.38787			−0.112441
$452$		0			0
$453$		−	15.2750i		−	0.717684i
$454$		0			0
$455$	−21.2750	−	24.0606i	−0.997389	−	1.12798i
$456$		0			0
$457$		−	15.0435i		−	0.703705i	−0.936055	−	0.351852i	$-0.885552\pi$
$457$		−	15.0435i		−	0.703705i	0.936055	−	0.351852i	$-0.114448\pi$
$458$		0			0
$459$	−4.54420			−0.212105
$460$		0			0
$461$	−26.2374			−1.22200			−0.610999	−	0.791631i	$-0.709232\pi$
$461$	−26.2374			−1.22200			−0.610999	+	0.791631i	$0.709232\pi$
$462$		0			0
$463$		−	5.46168i		−	0.253826i	−0.991914	−	0.126913i	$-0.959493\pi$
$463$		−	5.46168i		−	0.253826i	0.991914	−	0.126913i	$-0.0405069\pi$
$464$		0			0
$465$	−1.61213	+	1.42548i	−0.0747606	+	0.0661053i
$466$		0			0
$467$			2.70052i			0.124965i	0.998046	+	0.0624827i	$0.0199018\pi$
$467$			2.70052i			0.124965i	−0.998046	+	0.0624827i	$0.980098\pi$
$468$		0			0
$469$	−30.0263			−1.38649
$470$		0			0
$471$	7.01317			0.323150
$472$		0			0
$473$			2.80606i			0.129023i
$474$		0			0
$475$	2.80018	−	22.7005i	0.128481	−	1.04157i
$476$		0			0
$477$		−	6.57452i		−	0.301026i
$478$		0			0
$479$	16.7757			0.766503			0.383252	−	0.923644i	$-0.374804\pi$
$479$	16.7757			0.766503			0.383252	+	0.923644i	$0.374804\pi$
$480$		0			0
$481$	−8.25202			−0.376260
$482$		0			0
$483$		−	11.2243i		−	0.510721i
$484$		0			0
$485$	−16.6253	+	14.7005i	−0.754916	+	0.667516i
$486$		0			0
$487$			34.3996i			1.55880i	0.626529	+	0.779398i	$0.284475\pi$
$487$			34.3996i			1.55880i	−0.626529	+	0.779398i	$0.715525\pi$
$488$		0			0
$489$	6.76116			0.305750
$490$		0			0
$491$	−14.9525			−0.674799			−0.337399	−	0.941362i	$-0.609547\pi$
$491$	−14.9525			−0.674799			−0.337399	+	0.941362i	$0.609547\pi$
$492$		0			0
$493$			10.8510i			0.488703i
$494$		0			0
$495$	1.48119	+	1.67513i	0.0665747	+	0.0752915i
$496$		0			0
$497$		−	20.6253i		−	0.925171i
$498$		0			0
$499$	−2.85097			−0.127627			−0.0638135	−	0.997962i	$-0.520326\pi$
$499$	−2.85097			−0.127627			−0.0638135	+	0.997962i	$0.520326\pi$
$500$		0			0
$501$	11.8192			0.528045
$502$		0			0
$503$			1.48024i			0.0660006i	0.999455	+	0.0330003i	$0.0105062\pi$
$503$			1.48024i			0.0660006i	−0.999455	+	0.0330003i	$0.989494\pi$
$504$		0			0
$505$	20.1622	+	22.8021i	0.897206	+	1.01468i
$506$		0			0
$507$			13.2012i			0.586287i
$508$		0			0
$509$	23.2995			1.03273			0.516366	−	0.856368i	$-0.327285\pi$
$509$	23.2995			1.03273			0.516366	+	0.856368i	$0.327285\pi$
$510$		0			0
$511$	−18.0098			−0.796709
$512$		0			0
$513$		−	4.57452i		−	0.201970i
$514$		0			0
$515$	27.3258	−	24.1622i	1.20412	−	1.06471i
$516$		0			0
$517$			4.31265i			0.189670i
$518$		0			0
$519$	−6.99271			−0.306946
$520$		0			0
$521$	−6.81194			−0.298437			−0.149218	−	0.988804i	$-0.547676\pi$
$521$	−6.81194			−0.298437			−0.149218	+	0.988804i	$0.547676\pi$
$522$		0			0
$523$		−	16.0567i		−	0.702109i	−0.936355	−	0.351054i	$-0.885823\pi$
$523$		−	16.0567i		−	0.702109i	0.936355	−	0.351054i	$-0.114177\pi$
$524$		0			0
$525$	1.71767	−	13.9248i	0.0749651	−	0.607727i
$526$		0			0
$527$			4.37328i			0.190503i
$528$		0			0
$529$	7.00000			0.304348
$530$		0			0
$531$	13.2750			0.576088
$532$		0			0
$533$		−	12.2228i		−	0.529430i
$534$		0			0
$535$	15.7988	−	13.9697i	0.683040	−	0.603962i
$536$		0			0
$537$			2.70052i			0.116536i
$538$		0			0
$539$	−0.873992			−0.0376455
$540$		0			0
$541$	6.62530			0.284844			0.142422	−	0.989806i	$-0.454511\pi$
$541$	6.62530			0.284844			0.142422	+	0.989806i	$0.454511\pi$
$542$		0			0
$543$			21.1998i			0.909771i
$544$		0			0
$545$	−22.8119	−	25.7988i	−0.977156	−	1.10510i
$546$		0			0
$547$			5.75860i			0.246220i	0.992393	+	0.123110i	$0.0392868\pi$
$547$			5.75860i			0.246220i	−0.992393	+	0.123110i	$0.960713\pi$
$548$		0			0
$549$	−7.92478			−0.338221
$550$		0			0
$551$	−10.9234			−0.465351
$552$		0			0
$553$		−	3.78892i		−	0.161121i
$554$		0			0
$555$	−2.38787	−	2.70052i	−0.101360	−	0.114631i
$556$		0			0
$557$			19.8700i			0.841920i	0.907079	+	0.420960i	$0.138307\pi$
$557$			19.8700i			0.841920i	−0.907079	+	0.420960i	$0.861693\pi$
$558$		0			0
$559$	−14.3634			−0.607509
$560$		0			0
$561$	4.54420			0.191856
$562$		0			0
$563$			5.83383i			0.245866i	0.992415	+	0.122933i	$0.0392301\pi$
$563$			5.83383i			0.245866i	−0.992415	+	0.122933i	$0.960770\pi$
$564$		0			0
$565$	23.0132	−	20.3488i	0.968171	−	0.856082i
$566$		0			0
$567$		−	2.80606i		−	0.117844i
$568$		0			0
$569$	−46.7123			−1.95828			−0.979140	−	0.203185i	$-0.934871\pi$
$569$	−46.7123			−1.95828			−0.979140	+	0.203185i	$0.934871\pi$
$570$		0			0
$571$	24.4241			1.02212			0.511058	−	0.859546i	$-0.329254\pi$
$571$	24.4241			1.02212			0.511058	+	0.859546i	$0.329254\pi$
$572$		0			0
$573$			13.1490i			0.549309i
$574$		0			0
$575$	−19.8496	−	2.44851i	−0.827784	−	0.102110i
$576$		0			0
$577$			16.5647i			0.689596i	0.938677	+	0.344798i	$0.112053\pi$
$577$			16.5647i			0.689596i	−0.938677	+	0.344798i	$0.887947\pi$
$578$		0			0
$579$	−5.89446			−0.244965
$580$		0			0
$581$	−2.26187			−0.0938380
$582$		0			0
$583$			6.57452i			0.272289i
$584$		0			0
$585$	−8.57452	+	7.58181i	−0.354513	+	0.313469i
$586$		0			0
$587$		−	27.3258i		−	1.12786i	−0.825823	−	0.563929i	$-0.809289\pi$
$587$		−	27.3258i		−	1.12786i	0.825823	−	0.563929i	$-0.190711\pi$
$588$		0			0
$589$	−4.40246			−0.181400
$590$		0			0
$591$	20.3938			0.838887
$592$		0			0
$593$			12.6048i			0.517618i	0.965928	+	0.258809i	$0.0833301\pi$
$593$			12.6048i			0.517618i	−0.965928	+	0.258809i	$0.916670\pi$
$594$		0			0
$595$	−18.8872	−	21.3601i	−0.774298	−	0.875679i
$596$		0			0
$597$		−	17.4010i		−	0.712177i
$598$		0			0
$599$	10.5990			0.433061			0.216531	−	0.976276i	$-0.430526\pi$
$599$	10.5990			0.433061			0.216531	+	0.976276i	$0.430526\pi$
$600$		0			0
$601$	−22.4749			−0.916768			−0.458384	−	0.888754i	$-0.651572\pi$
$601$	−22.4749			−0.916768			−0.458384	+	0.888754i	$0.651572\pi$
$602$		0			0
$603$			10.7005i			0.435759i
$604$		0			0
$605$	−1.48119	−	1.67513i	−0.0602191	−	0.0681038i
$606$		0			0
$607$		−	33.5183i		−	1.36047i	−0.732995	−	0.680234i	$-0.761878\pi$
$607$		−	33.5183i		−	1.36047i	0.732995	−	0.680234i	$-0.238122\pi$
$608$		0			0
$609$	−6.70052			−0.271519
$610$		0			0
$611$	−22.0752			−0.893068
$612$		0			0
$613$		−	11.5672i		−	0.467196i	−0.972333	−	0.233598i	$-0.924950\pi$
$613$		−	11.5672i		−	0.467196i	0.972333	−	0.233598i	$-0.0750499\pi$
$614$		0			0
$615$	4.00000	−	3.53690i	0.161296	−	0.142622i
$616$		0			0
$617$		−	33.3357i		−	1.34204i	−0.741438	−	0.671022i	$-0.765856\pi$
$617$		−	33.3357i		−	1.34204i	0.741438	−	0.671022i	$-0.234144\pi$
$618$		0			0
$619$	4.43866			0.178405			0.0892024	−	0.996014i	$-0.471568\pi$
$619$	4.43866			0.178405			0.0892024	+	0.996014i	$0.471568\pi$
$620$		0			0
$621$	−4.00000			−0.160514
$622$		0			0
$623$		−	8.31265i		−	0.333039i
$624$		0			0
$625$	−24.2506	−	6.07522i	−0.970024	−	0.243009i
$626$		0			0
$627$			4.57452i			0.182689i
$628$		0			0
$629$	−7.32582			−0.292100
$630$		0			0
$631$	39.8496			1.58639			0.793193	−	0.608971i	$-0.208417\pi$
$631$	39.8496			1.58639			0.793193	+	0.608971i	$0.208417\pi$
$632$		0			0
$633$			18.1260i			0.720444i
$634$		0			0
$635$	−20.8021	+	18.3938i	−0.825506	+	0.729934i
$636$		0			0
$637$		−	4.47371i		−	0.177255i
$638$		0			0
$639$	−7.35026			−0.290772
$640$		0			0
$641$	−3.88858			−0.153590			−0.0767948	−	0.997047i	$-0.524469\pi$
$641$	−3.88858			−0.153590			−0.0767948	+	0.997047i	$0.524469\pi$
$642$		0			0
$643$			40.9380i			1.61444i	0.590254	+	0.807218i	$0.299028\pi$
$643$			40.9380i			1.61444i	−0.590254	+	0.807218i	$0.700972\pi$
$644$		0			0
$645$	−4.15633	−	4.70052i	−0.163655	−	0.185083i
$646$		0			0
$647$		−	1.76257i		−	0.0692939i	−0.999400	−	0.0346469i	$-0.988969\pi$
$647$		−	1.76257i		−	0.0692939i	0.999400	−	0.0346469i	$-0.0110307\pi$
$648$		0			0
$649$	−13.2750			−0.521091
$650$		0			0
$651$	−2.70052			−0.105842
$652$		0			0
$653$			7.03761i			0.275403i	0.990474	+	0.137702i	$0.0439715\pi$
$653$			7.03761i			0.275403i	−0.990474	+	0.137702i	$0.956029\pi$
$654$		0			0
$655$	8.77575	+	9.92478i	0.342897	+	0.387793i
$656$		0			0
$657$			6.41819i			0.250397i
$658$		0			0
$659$	−12.6253			−0.491812			−0.245906	−	0.969294i	$-0.579085\pi$
$659$	−12.6253			−0.491812			−0.245906	+	0.969294i	$0.579085\pi$
$660$		0			0
$661$	13.2243			0.514364			0.257182	−	0.966363i	$-0.417206\pi$
$661$	13.2243			0.514364			0.257182	+	0.966363i	$0.417206\pi$
$662$		0			0
$663$			23.2605i			0.903361i
$664$		0			0
$665$	21.5026	−	19.0132i	0.833836	−	0.737299i
$666$		0			0
$667$			9.55149i			0.369835i
$668$		0			0
$669$	23.6385			0.913916
$670$		0			0
$671$	7.92478			0.305933
$672$		0			0
$673$		−	18.2677i		−	0.704170i	−0.935968	−	0.352085i	$-0.885473\pi$
$673$		−	18.2677i		−	0.704170i	0.935968	−	0.352085i	$-0.114527\pi$
$674$		0			0
$675$	−4.96239	−	0.612127i	−0.191002	−	0.0235608i
$676$		0			0
$677$		−	33.0191i		−	1.26903i	−0.772912	−	0.634513i	$-0.781201\pi$
$677$		−	33.0191i		−	1.26903i	0.772912	−	0.634513i	$-0.218799\pi$
$678$		0			0
$679$	−27.8496			−1.06877
$680$		0			0
$681$	1.26916			0.0486343
$682$		0			0
$683$			30.8627i			1.18093i	0.807063	+	0.590465i	$0.201056\pi$
$683$			30.8627i			1.18093i	−0.807063	+	0.590465i	$0.798944\pi$
$684$		0			0
$685$	16.4142	−	14.5139i	0.627155	−	0.554547i
$686$		0			0
$687$			16.1768i			0.617183i
$688$		0			0
$689$	−33.6531			−1.28208
$690$		0			0
$691$	−27.6991			−1.05372			−0.526862	−	0.849951i	$-0.676631\pi$
$691$	−27.6991			−1.05372			−0.526862	+	0.849951i	$0.676631\pi$
$692$		0			0
$693$			2.80606i			0.106594i
$694$		0			0
$695$	3.14903	+	3.56134i	0.119450	+	0.135089i
$696$		0			0
$697$		−	10.8510i		−	0.411010i
$698$		0			0
$699$	18.4690			0.698561
$700$		0			0
$701$	−38.9643			−1.47166			−0.735831	−	0.677166i	$-0.763208\pi$
$701$	−38.9643			−1.47166			−0.735831	+	0.677166i	$0.763208\pi$
$702$		0			0
$703$		−	7.37470i		−	0.278142i
$704$		0			0
$705$	−6.38787	−	7.22425i	−0.240581	−	0.272081i
$706$		0			0
$707$			38.1965i			1.43653i
$708$		0			0
$709$	4.32250			0.162335			0.0811674	−	0.996700i	$-0.474135\pi$
$709$	4.32250			0.162335			0.0811674	+	0.996700i	$0.474135\pi$
$710$		0			0
$711$	−1.35026			−0.0506388
$712$		0			0
$713$			3.84955i			0.144167i
$714$		0			0
$715$	8.57452	−	7.58181i	0.320669	−	0.283544i
$716$		0			0
$717$		−	19.3258i		−	0.721736i
$718$		0			0
$719$	−38.3996			−1.43206			−0.716032	−	0.698067i	$-0.754044\pi$
$719$	−38.3996			−1.43206			−0.716032	+	0.698067i	$0.754044\pi$
$720$		0			0
$721$	45.7743			1.70473
$722$		0			0
$723$		−	28.5501i		−	1.06179i
$724$		0			0
$725$	−1.46168	+	11.8496i	−0.0542855	+	0.440081i
$726$		0			0
$727$			21.6728i			0.803798i	0.915684	+	0.401899i	$0.131650\pi$
$727$			21.6728i			0.803798i	−0.915684	+	0.401899i	$0.868350\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$	−12.7513			−0.471624
$732$		0			0
$733$		−	25.0698i		−	0.925976i	−0.886365	−	0.462988i	$-0.846777\pi$
$733$		−	25.0698i		−	0.925976i	0.886365	−	0.462988i	$-0.153223\pi$
$734$		0			0
$735$	1.46405	−	1.29455i	0.0540023	−	0.0477503i
$736$		0			0
$737$		−	10.7005i		−	0.394159i
$738$		0			0
$739$	−18.9018			−0.695312			−0.347656	−	0.937622i	$-0.613022\pi$
$739$	−18.9018			−0.695312			−0.347656	+	0.937622i	$0.613022\pi$
$740$		0			0
$741$	−23.4156			−0.860195
$742$		0			0
$743$			43.6688i			1.60205i	0.598629	+	0.801026i	$0.295712\pi$
$743$			43.6688i			1.60205i	−0.598629	+	0.801026i	$0.704288\pi$
$744$		0			0
$745$	12.3127	+	13.9248i	0.451101	+	0.510164i
$746$		0			0
$747$			0.806063i			0.0294923i
$748$		0			0
$749$	26.4650			0.967010
$750$		0			0
$751$	47.0541			1.71703			0.858514	−	0.512789i	$-0.171388\pi$
$751$	47.0541			1.71703			0.858514	+	0.512789i	$0.171388\pi$
$752$		0			0
$753$			23.1998i			0.845448i
$754$		0			0
$755$	−22.6253	−	25.5877i	−0.823419	−	0.931231i
$756$		0			0
$757$		−	26.9116i		−	0.978119i	−0.872250	−	0.489059i	$-0.837340\pi$
$757$		−	26.9116i		−	0.978119i	0.872250	−	0.489059i	$-0.162660\pi$
$758$		0			0
$759$	4.00000			0.145191
$760$		0			0
$761$	16.6859			0.604865			0.302432	−	0.953171i	$-0.402201\pi$
$761$	16.6859			0.604865			0.302432	+	0.953171i	$0.402201\pi$
$762$		0			0
$763$		−	43.2163i		−	1.56454i
$764$		0			0
$765$	−7.61213	+	6.73084i	−0.275217	+	0.243354i
$766$		0			0
$767$		−	67.9511i		−	2.45357i
$768$		0			0
$769$	−23.2995			−0.840201			−0.420100	−	0.907478i	$-0.638005\pi$
$769$	−23.2995			−0.840201			−0.420100	+	0.907478i	$0.638005\pi$
$770$		0			0
$771$	−10.8872			−0.392092
$772$		0			0
$773$			1.63656i			0.0588631i	0.999567	+	0.0294316i	$0.00936971\pi$
$773$			1.63656i			0.0588631i	−0.999567	+	0.0294316i	$0.990630\pi$
$774$		0			0
$775$	−0.589104	+	4.77575i	−0.0211612	+	0.171550i
$776$		0			0
$777$		−	4.52373i		−	0.162288i
$778$		0			0
$779$	10.9234			0.391370
$780$		0			0
$781$	7.35026			0.263013
$782$		0			0
$783$			2.38787i			0.0853356i
$784$		0			0
$785$	11.7480	−	10.3879i	0.419304	−	0.370759i
$786$		0			0
$787$			2.09095i			0.0745344i	0.999305	+	0.0372672i	$0.0118653\pi$
$787$			2.09095i			0.0745344i	−0.999305	+	0.0372672i	$0.988135\pi$
$788$		0			0
$789$	9.11871			0.324635
$790$		0			0
$791$	38.5501			1.37068
$792$		0			0
$793$			40.5647i			1.44049i
$794$		0			0
$795$	−9.73813	−	11.0132i	−0.345376	−	0.390597i
$796$		0			0
$797$		−	26.8872i		−	0.952392i	−0.879339	−	0.476196i	$-0.842015\pi$
$797$		−	26.8872i		−	0.952392i	0.879339	−	0.476196i	$-0.157985\pi$
$798$		0			0
$799$	−19.5975			−0.693311
$800$		0			0
$801$	−2.96239			−0.104671
$802$		0			0
$803$		−	6.41819i		−	0.226493i
$804$		0			0
$805$	−16.6253	−	18.8021i	−0.585965	−	0.662687i
$806$		0			0
$807$		−	11.2995i		−	0.397760i
$808$		0			0
$809$	45.4128			1.59663			0.798315	−	0.602241i	$-0.205725\pi$
$809$	45.4128			1.59663			0.798315	+	0.602241i	$0.205725\pi$
$810$		0			0
$811$	−34.3488			−1.20615			−0.603076	−	0.797684i	$-0.706058\pi$
$811$	−34.3488			−1.20615			−0.603076	+	0.797684i	$0.706058\pi$
$812$		0			0
$813$			25.1998i			0.883796i
$814$		0			0
$815$	11.3258	−	10.0146i	0.396726	−	0.350796i
$816$		0			0
$817$		−	12.8364i		−	0.449088i
$818$		0			0
$819$	−14.3634			−0.501899
$820$		0			0
$821$	18.7612			0.654769			0.327384	−	0.944891i	$-0.393833\pi$
$821$	18.7612			0.654769			0.327384	+	0.944891i	$0.393833\pi$
$822$		0			0
$823$		−	33.0592i		−	1.15237i	−0.817319	−	0.576186i	$-0.804540\pi$
$823$		−	33.0592i		−	1.15237i	0.817319	−	0.576186i	$-0.195460\pi$
$824$		0			0
$825$	4.96239	+	0.612127i	0.172768	+	0.0213115i
$826$		0			0
$827$			10.8813i			0.378379i	0.981941	+	0.189190i	$0.0605861\pi$
$827$			10.8813i			0.378379i	−0.981941	+	0.189190i	$0.939414\pi$
$828$		0			0
$829$	−42.7221			−1.48380			−0.741900	−	0.670510i	$-0.766075\pi$
$829$	−42.7221			−1.48380			−0.741900	+	0.670510i	$0.766075\pi$
$830$		0			0
$831$	−2.41819			−0.0838861
$832$		0			0
$833$		−	3.97159i		−	0.137608i
$834$		0			0
$835$	19.7988	−	17.5066i	0.685165	−	0.605840i
$836$		0			0
$837$			0.962389i			0.0332650i
$838$		0			0
$839$	−21.1735			−0.730989			−0.365495	−	0.930813i	$-0.619100\pi$
$839$	−21.1735			−0.730989			−0.365495	+	0.930813i	$0.619100\pi$
$840$		0			0
$841$	−23.2981			−0.803381
$842$		0			0
$843$		−	30.4894i		−	1.05011i
$844$		0			0
$845$	19.5536	+	22.1138i	0.672664	+	0.760737i
$846$		0			0
$847$		−	2.80606i		−	0.0964175i
$848$		0			0
$849$	8.35756			0.286831
$850$		0			0
$851$	−6.44851			−0.221052
$852$		0			0
$853$		−	54.4709i		−	1.86505i	−0.361109	−	0.932524i	$-0.617602\pi$
$853$		−	54.4709i		−	1.86505i	0.361109	−	0.932524i	$-0.382398\pi$
$854$		0			0
$855$	−6.77575	−	7.66291i	−0.231726	−	0.262066i
$856$		0			0
$857$			38.5705i			1.31754i	0.752342	+	0.658772i	$0.228924\pi$
$857$			38.5705i			1.31754i	−0.752342	+	0.658772i	$0.771076\pi$
$858$		0			0
$859$	−18.5139			−0.631685			−0.315843	−	0.948812i	$-0.602287\pi$
$859$	−18.5139			−0.631685			−0.315843	+	0.948812i	$0.602287\pi$
$860$		0			0
$861$	6.70052			0.228353
$862$		0			0
$863$			22.5383i			0.767213i	0.923497	+	0.383607i	$0.125318\pi$
$863$			22.5383i			0.767213i	−0.923497	+	0.383607i	$0.874682\pi$
$864$		0			0
$865$	−11.7137	+	10.3576i	−0.398278	+	0.352167i
$866$		0			0
$867$			3.64974i			0.123952i
$868$		0			0
$869$	1.35026			0.0458045
$870$		0			0
$871$	54.7729			1.85591
$872$		0			0
$873$			9.92478i			0.335903i
$874$		0			0
$875$	−17.7480	−	25.8700i	−0.599991	−	0.874566i
$876$		0			0
$877$			18.9419i			0.639623i	0.947481	+	0.319812i	$0.103620\pi$
$877$			18.9419i			0.639623i	−0.947481	+	0.319812i	$0.896380\pi$
$878$		0			0
$879$	23.0943			0.778951
$880$		0			0
$881$	−20.0263			−0.674705			−0.337352	−	0.941378i	$-0.609531\pi$
$881$	−20.0263			−0.674705			−0.337352	+	0.941378i	$0.609531\pi$
$882$		0			0
$883$			22.2981i			0.750390i	0.926946	+	0.375195i	$0.122424\pi$
$883$			22.2981i			0.750390i	−0.926946	+	0.375195i	$0.877576\pi$
$884$		0			0
$885$	22.2374	−	19.6629i	0.747503	−	0.660962i
$886$		0			0
$887$		−	3.10413i		−	0.104226i	−0.998641	−	0.0521132i	$-0.983404\pi$
$887$		−	3.10413i		−	0.104226i	0.998641	−	0.0521132i	$-0.0165957\pi$
$888$		0			0
$889$	−34.8462			−1.16871
$890$		0			0
$891$	1.00000			0.0335013
$892$		0			0
$893$		−	19.7283i		−	0.660182i
$894$		0			0
$895$	4.00000	+	4.52373i	0.133705	+	0.151212i
$896$		0			0
$897$			20.4749i			0.683636i
$898$		0			0
$899$	2.29806			0.0766447
$900$		0			0
$901$	−29.8759			−0.995311
$902$		0			0
$903$		−	7.87399i		−	0.262030i
$904$		0			0
$905$	31.4010	+	35.5125i	1.04381	+	1.18047i
$906$		0			0
$907$		−	10.8627i		−	0.360691i	−0.983603	−	0.180345i	$-0.942278\pi$
$907$		−	10.8627i		−	0.360691i	0.983603	−	0.180345i	$-0.0577216\pi$
$908$		0			0
$909$	13.6121			0.451486
$910$		0			0
$911$	5.82321			0.192931			0.0964657	−	0.995336i	$-0.469246\pi$
$911$	5.82321			0.192931			0.0964657	+	0.995336i	$0.469246\pi$
$912$		0			0
$913$		−	0.806063i		−	0.0266768i
$914$		0			0
$915$	−13.2750	+	11.7381i	−0.438859	+	0.388051i
$916$		0			0
$917$			16.6253i			0.549016i
$918$		0			0
$919$	−31.2750			−1.03167			−0.515834	−	0.856688i	$-0.672518\pi$
$919$	−31.2750			−1.03167			−0.515834	+	0.856688i	$0.672518\pi$
$920$		0			0
$921$	−2.65562			−0.0875056
$922$		0			0
$923$			37.6239i			1.23841i
$924$		0			0
$925$	−8.00000	−	0.986826i	−0.263038	−	0.0324466i
$926$		0			0
$927$		−	16.3127i		−	0.535778i
$928$		0			0
$929$	21.2243			0.696345			0.348173	−	0.937430i	$-0.386802\pi$
$929$	21.2243			0.696345			0.348173	+	0.937430i	$0.386802\pi$
$930$		0			0
$931$	3.99809			0.131032
$932$		0			0
$933$		−	15.9756i		−	0.523016i
$934$		0			0
$935$	7.61213	−	6.73084i	0.248943	−	0.220122i
$936$		0			0
$937$			16.4328i			0.536835i	0.963303	+	0.268418i	$0.0865007\pi$
$937$			16.4328i			0.536835i	−0.963303	+	0.268418i	$0.913499\pi$
$938$		0			0
$939$	0.0606343			0.00197873
$940$		0			0
$941$	5.86414			0.191166			0.0955828	−	0.995421i	$-0.469529\pi$
$941$	5.86414			0.191166			0.0955828	+	0.995421i	$0.469529\pi$
$942$		0			0
$943$		−	9.55149i		−	0.311039i
$944$		0			0
$945$	−4.15633	−	4.70052i	−0.135205	−	0.152908i
$946$		0			0
$947$			38.7415i			1.25893i	0.777030	+	0.629464i	$0.216726\pi$
$947$			38.7415i			1.25893i	−0.777030	+	0.629464i	$0.783274\pi$
$948$		0			0
$949$	32.8529			1.06645
$950$		0			0
$951$	3.81336			0.123657
$952$		0			0
$953$			6.09569i			0.197459i	0.995114	+	0.0987294i	$0.0314778\pi$
$953$			6.09569i			0.197459i	−0.995114	+	0.0987294i	$0.968522\pi$
$954$		0			0
$955$	19.4763	+	22.0263i	0.630237	+	0.712756i
$956$		0			0
$957$		−	2.38787i		−	0.0771890i
$958$		0			0
$959$	27.4960			0.887891
$960$		0			0
$961$	−30.0738			−0.970123
$962$		0			0
$963$		−	9.43136i		−	0.303921i
$964$		0			0
$965$	−9.87399	+	8.73084i	−0.317855	+	0.281056i
$966$		0			0
$967$		−	37.9697i		−	1.22102i	−0.792008	−	0.610511i	$-0.790964\pi$
$967$		−	37.9697i		−	1.22102i	0.792008	−	0.610511i	$-0.209036\pi$
$968$		0			0
$969$	−20.7875			−0.667791
$970$		0			0
$971$	60.8773			1.95365			0.976823	−	0.214049i	$-0.0686651\pi$
$971$	60.8773			1.95365			0.976823	+	0.214049i	$0.0686651\pi$
$972$		0			0
$973$			5.96571i			0.191252i
$974$		0			0
$975$	−3.13330	+	25.4010i	−0.100346	+	0.813485i
$976$		0			0
$977$			21.3963i			0.684529i	0.939604	+	0.342264i	$0.111194\pi$
$977$			21.3963i			0.684529i	−0.939604	+	0.342264i	$0.888806\pi$
$978$		0			0
$979$	2.96239			0.0946784
$980$		0			0
$981$	−15.4010			−0.491718
$982$		0			0
$983$			18.8919i			0.602558i	0.953536	+	0.301279i	$0.0974136\pi$
$983$			18.8919i			0.602558i	−0.953536	+	0.301279i	$0.902586\pi$
$984$		0			0
$985$	34.1622	−	30.2071i	1.08850	−	0.962479i
$986$		0			0
$987$		−	12.1016i		−	0.385197i
$988$		0			0
$989$	−11.2243			−0.356911
$990$		0			0
$991$	1.33567			0.0424291			0.0212145	−	0.999775i	$-0.493247\pi$
$991$	1.33567			0.0424291			0.0212145	+	0.999775i	$0.493247\pi$
$992$		0			0
$993$			24.2882i			0.770763i
$994$		0			0
$995$	−25.7743	−	29.1490i	−0.817101	−	0.924086i
$996$		0			0
$997$		−	48.9076i		−	1.54892i	−0.632623	−	0.774460i	$-0.718022\pi$
$997$		−	48.9076i		−	1.54892i	0.632623	−	0.774460i	$-0.281978\pi$
$998$		0			0
$999$	−1.61213			−0.0510054

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2640.2.d.i.529.1		6
4.3	odd	2		165.2.c.a.34.6	yes	6
5.4	even	2	inner	2640.2.d.i.529.4		6
12.11	even	2		495.2.c.d.199.1		6
20.3	even	4		825.2.a.n.1.3		3
20.7	even	4		825.2.a.h.1.1		3
20.19	odd	2		165.2.c.a.34.1	✓	6
44.43	even	2		1815.2.c.d.364.1		6
60.23	odd	4		2475.2.a.y.1.1		3
60.47	odd	4		2475.2.a.be.1.3		3
60.59	even	2		495.2.c.d.199.6		6
220.43	odd	4		9075.2.a.cc.1.1		3
220.87	odd	4		9075.2.a.ck.1.3		3
220.219	even	2		1815.2.c.d.364.6		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.2.c.a.34.1	✓	6	20.19	odd	2
165.2.c.a.34.6	yes	6	4.3	odd	2
495.2.c.d.199.1		6	12.11	even	2
495.2.c.d.199.6		6	60.59	even	2
825.2.a.h.1.1		3	20.7	even	4
825.2.a.n.1.3		3	20.3	even	4
1815.2.c.d.364.1		6	44.43	even	2
1815.2.c.d.364.6		6	220.219	even	2
2475.2.a.y.1.1		3	60.23	odd	4
2475.2.a.be.1.3		3	60.47	odd	4
2640.2.d.i.529.1		6	1.1	even	1	trivial
2640.2.d.i.529.4		6	5.4	even	2	inner
9075.2.a.cc.1.1		3	220.43	odd	4
9075.2.a.ck.1.3		3	220.87	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2640 = 2^{4} \cdot 3 \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2640.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} +(-1.48119 - 1.67513i) q^{5} -2.80606i q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{3} +(-1.48119 - 1.67513i) q^{5} -2.80606i q^{7} -1.00000 q^{9} +1.00000 q^{11} +5.11871i q^{13} +(-1.67513 + 1.48119i) q^{15} +4.54420i q^{17} -4.57452 q^{19} -2.80606 q^{21} +4.00000i q^{23} +(-0.612127 + 4.96239i) q^{25} +1.00000i q^{27} +2.38787 q^{29} +0.962389 q^{31} -1.00000i q^{33} +(-4.70052 + 4.15633i) q^{35} +1.61213i q^{37} +5.11871 q^{39} -2.38787 q^{41} +2.80606i q^{43} +(1.48119 + 1.67513i) q^{45} +4.31265i q^{47} -0.873992 q^{49} +4.54420 q^{51} +6.57452i q^{53} +(-1.48119 - 1.67513i) q^{55} +4.57452i q^{57} -13.2750 q^{59} +7.92478 q^{61} +2.80606i q^{63} +(8.57452 - 7.58181i) q^{65} -10.7005i q^{67} +4.00000 q^{69} +7.35026 q^{71} -6.41819i q^{73} +(4.96239 + 0.612127i) q^{75} -2.80606i q^{77} +1.35026 q^{79} +1.00000 q^{81} -0.806063i q^{83} +(7.61213 - 6.73084i) q^{85} -2.38787i q^{87} +2.96239 q^{89} +14.3634 q^{91} -0.962389i q^{93} +(6.77575 + 7.66291i) q^{95} -9.92478i q^{97} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 2 q^{5} - 6 q^{9}+O(q^{10}) \) 6 * q + 2 * q^5 - 6 * q^9	\( 6 q + 2 q^{5} - 6 q^{9} + 6 q^{11} - 4 q^{19} - 16 q^{21} - 2 q^{25} + 16 q^{29} - 16 q^{31} + 12 q^{35} - 12 q^{39} - 16 q^{41} - 2 q^{45} - 22 q^{49} + 8 q^{51} + 2 q^{55} - 16 q^{59} + 4 q^{61} + 28 q^{65} + 24 q^{69} + 24 q^{71} + 8 q^{75} - 12 q^{79} + 6 q^{81} + 44 q^{85} - 4 q^{89} - 16 q^{91} + 44 q^{95} - 6 q^{99}+O(q^{100}) \) 6 * q + 2 * q^5 - 6 * q^9 + 6 * q^11 - 4 * q^19 - 16 * q^21 - 2 * q^25 + 16 * q^29 - 16 * q^31 + 12 * q^35 - 12 * q^39 - 16 * q^41 - 2 * q^45 - 22 * q^49 + 8 * q^51 + 2 * q^55 - 16 * q^59 + 4 * q^61 + 28 * q^65 + 24 * q^69 + 24 * q^71 + 8 * q^75 - 12 * q^79 + 6 * q^81 + 44 * q^85 - 4 * q^89 - 16 * q^91 + 44 * q^95 - 6 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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