LMFDB - Embedded newform 1600.2.n.f.1343.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1600,2,Mod(1343,1600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1600, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 0, 3]))

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

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

chi := DirichletCharacter("1600.1343");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1600 = 2^{6} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1600.n (of order $4$, degree $2$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$12.7760643234$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 800)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			1343.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	1600.1343
Dual form			1600.2.n.f.1407.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.00000 + 1.00000i) q^{3} +(1.00000 + 1.00000i) q^{7} +1.00000i q^{9} +O(q^{10})$	$q+(-1.00000 + 1.00000i) q^{3} +(1.00000 + 1.00000i) q^{7} +1.00000i q^{9} +4.00000i q^{11} +(4.00000 + 4.00000i) q^{13} +(4.00000 - 4.00000i) q^{17} -4.00000 q^{19} -2.00000 q^{21} +(5.00000 - 5.00000i) q^{23} +(-4.00000 - 4.00000i) q^{27} -2.00000i q^{29} +8.00000i q^{31} +(-4.00000 - 4.00000i) q^{33} -8.00000 q^{39} -4.00000 q^{41} +(-7.00000 + 7.00000i) q^{43} +(3.00000 + 3.00000i) q^{47} -5.00000i q^{49} +8.00000i q^{51} +(4.00000 + 4.00000i) q^{53} +(4.00000 - 4.00000i) q^{57} -4.00000 q^{59} +8.00000 q^{61} +(-1.00000 + 1.00000i) q^{63} +(3.00000 + 3.00000i) q^{67} +10.0000i q^{69} +16.0000i q^{71} +(-4.00000 - 4.00000i) q^{73} +(-4.00000 + 4.00000i) q^{77} -8.00000 q^{79} +5.00000 q^{81} +(5.00000 - 5.00000i) q^{83} +(2.00000 + 2.00000i) q^{87} +10.0000i q^{89} +8.00000i q^{91} +(-8.00000 - 8.00000i) q^{93} +(-12.0000 + 12.0000i) q^{97} -4.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} + 2 q^{7}+O(q^{10}) $ 2 * q - 2 * q^3 + 2 * q^7	$ 2 q - 2 q^{3} + 2 q^{7} + 8 q^{13} + 8 q^{17} - 8 q^{19} - 4 q^{21} + 10 q^{23} - 8 q^{27} - 8 q^{33} - 16 q^{39} - 8 q^{41} - 14 q^{43} + 6 q^{47} + 8 q^{53} + 8 q^{57} - 8 q^{59} + 16 q^{61} - 2 q^{63} + 6 q^{67} - 8 q^{73} - 8 q^{77} - 16 q^{79} + 10 q^{81} + 10 q^{83} + 4 q^{87} - 16 q^{93} - 24 q^{97} - 8 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 + 2 * q^7 + 8 * q^13 + 8 * q^17 - 8 * q^19 - 4 * q^21 + 10 * q^23 - 8 * q^27 - 8 * q^33 - 16 * q^39 - 8 * q^41 - 14 * q^43 + 6 * q^47 + 8 * q^53 + 8 * q^57 - 8 * q^59 + 16 * q^61 - 2 * q^63 + 6 * q^67 - 8 * q^73 - 8 * q^77 - 16 * q^79 + 10 * q^81 + 10 * q^83 + 4 * q^87 - 16 * q^93 - 24 * q^97 - 8 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1600\mathbb{Z}\right)^\times$.

$n$	$577$	$901$	$1151$
$\chi(n)$	$e\left(\frac{3}{4}\right)$	$1$	$-1$

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
$3$	−1.00000	+	1.00000i	−0.577350	+	0.577350i	−0.934172	−	0.356822i	$-0.883860\pi$
$3$	−1.00000	+	1.00000i	−0.577350	+	0.577350i	0.356822	+	0.934172i	$0.383860\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	1.00000	+	1.00000i	0.377964	+	0.377964i	0.870367	−	0.492403i	$-0.163881\pi$
$7$	1.00000	+	1.00000i	0.377964	+	0.377964i	−0.492403	+	0.870367i	$0.663881\pi$
$8$		0			0
$9$			1.00000i			0.333333i
$10$		0			0
$11$			4.00000i			1.20605i	0.797724	+	0.603023i	$0.206037\pi$
$11$			4.00000i			1.20605i	−0.797724	+	0.603023i	$0.793963\pi$
$12$		0			0
$13$	4.00000	+	4.00000i	1.10940	+	1.10940i	0.993229	+	0.116171i	$0.0370621\pi$
$13$	4.00000	+	4.00000i	1.10940	+	1.10940i	0.116171	+	0.993229i	$0.462938\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	4.00000	−	4.00000i	0.970143	−	0.970143i	−0.0294245	−	0.999567i	$-0.509367\pi$
$17$	4.00000	−	4.00000i	0.970143	−	0.970143i	0.999567	+	0.0294245i	$0.00936746\pi$
$18$		0			0
$19$	−4.00000			−0.917663			−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−4.00000			−0.917663			−0.458831	+	0.888523i	$0.651732\pi$
$20$		0			0
$21$	−2.00000			−0.436436
$22$		0			0
$23$	5.00000	−	5.00000i	1.04257	−	1.04257i	0.0435195	−	0.999053i	$-0.486143\pi$
$23$	5.00000	−	5.00000i	1.04257	−	1.04257i	0.999053	−	0.0435195i	$-0.0138571\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	−4.00000	−	4.00000i	−0.769800	−	0.769800i
$28$		0			0
$29$		−	2.00000i		−	0.371391i	−0.982607	−	0.185695i	$-0.940546\pi$
$29$		−	2.00000i		−	0.371391i	0.982607	−	0.185695i	$-0.0594537\pi$
$30$		0			0
$31$			8.00000i			1.43684i	0.695608	+	0.718421i	$0.255135\pi$
$31$			8.00000i			1.43684i	−0.695608	+	0.718421i	$0.744865\pi$
$32$		0			0
$33$	−4.00000	−	4.00000i	−0.696311	−	0.696311i
$34$		0			0
$35$		0			0
$36$		0			0
$37$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$37$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$38$		0			0
$39$	−8.00000			−1.28103
$40$		0			0
$41$	−4.00000			−0.624695			−0.312348	−	0.949968i	$-0.601115\pi$
$41$	−4.00000			−0.624695			−0.312348	+	0.949968i	$0.601115\pi$
$42$		0			0
$43$	−7.00000	+	7.00000i	−1.06749	+	1.06749i	−0.0699387	+	0.997551i	$0.522280\pi$
$43$	−7.00000	+	7.00000i	−1.06749	+	1.06749i	−0.997551	+	0.0699387i	$0.977720\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	3.00000	+	3.00000i	0.437595	+	0.437595i	0.891202	−	0.453607i	$-0.149863\pi$
$47$	3.00000	+	3.00000i	0.437595	+	0.437595i	−0.453607	+	0.891202i	$0.649863\pi$
$48$		0			0
$49$		−	5.00000i		−	0.714286i
$50$		0			0
$51$			8.00000i			1.12022i
$52$		0			0
$53$	4.00000	+	4.00000i	0.549442	+	0.549442i	0.926279	−	0.376837i	$-0.122988\pi$
$53$	4.00000	+	4.00000i	0.549442	+	0.549442i	−0.376837	+	0.926279i	$0.622988\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	4.00000	−	4.00000i	0.529813	−	0.529813i
$58$		0			0
$59$	−4.00000			−0.520756			−0.260378	−	0.965507i	$-0.583847\pi$
$59$	−4.00000			−0.520756			−0.260378	+	0.965507i	$0.583847\pi$
$60$		0			0
$61$	8.00000			1.02430			0.512148	−	0.858898i	$-0.328850\pi$
$61$	8.00000			1.02430			0.512148	+	0.858898i	$0.328850\pi$
$62$		0			0
$63$	−1.00000	+	1.00000i	−0.125988	+	0.125988i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	3.00000	+	3.00000i	0.366508	+	0.366508i	0.866202	−	0.499694i	$-0.166554\pi$
$67$	3.00000	+	3.00000i	0.366508	+	0.366508i	−0.499694	+	0.866202i	$0.666554\pi$
$68$		0			0
$69$			10.0000i			1.20386i
$70$		0			0
$71$			16.0000i			1.89885i	0.313993	+	0.949425i	$0.398333\pi$
$71$			16.0000i			1.89885i	−0.313993	+	0.949425i	$0.601667\pi$
$72$		0			0
$73$	−4.00000	−	4.00000i	−0.468165	−	0.468165i	0.433155	−	0.901319i	$-0.357400\pi$
$73$	−4.00000	−	4.00000i	−0.468165	−	0.468165i	−0.901319	+	0.433155i	$0.857400\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−4.00000	+	4.00000i	−0.455842	+	0.455842i
$78$		0			0
$79$	−8.00000			−0.900070			−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000			−0.900070			−0.450035	+	0.893011i	$0.648589\pi$
$80$		0			0
$81$	5.00000			0.555556
$82$		0			0
$83$	5.00000	−	5.00000i	0.548821	−	0.548821i	−0.377279	−	0.926100i	$-0.623140\pi$
$83$	5.00000	−	5.00000i	0.548821	−	0.548821i	0.926100	+	0.377279i	$0.123140\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	2.00000	+	2.00000i	0.214423	+	0.214423i
$88$		0			0
$89$			10.0000i			1.06000i	0.847998	+	0.529999i	$0.177808\pi$
$89$			10.0000i			1.06000i	−0.847998	+	0.529999i	$0.822192\pi$
$90$		0			0
$91$			8.00000i			0.838628i
$92$		0			0
$93$	−8.00000	−	8.00000i	−0.829561	−	0.829561i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−12.0000	+	12.0000i	−1.21842	+	1.21842i	−0.250229	+	0.968187i	$0.580506\pi$
$97$	−12.0000	+	12.0000i	−1.21842	+	1.21842i	−0.968187	+	0.250229i	$0.919494\pi$
$98$		0			0
$99$	−4.00000			−0.402015
$100$		0			0
$101$	−6.00000			−0.597022			−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000			−0.597022			−0.298511	+	0.954406i	$0.596490\pi$
$102$		0			0
$103$	3.00000	−	3.00000i	0.295599	−	0.295599i	−0.543688	−	0.839287i	$-0.682973\pi$
$103$	3.00000	−	3.00000i	0.295599	−	0.295599i	0.839287	+	0.543688i	$0.182973\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−3.00000	−	3.00000i	−0.290021	−	0.290021i	0.547068	−	0.837088i	$-0.315744\pi$
$107$	−3.00000	−	3.00000i	−0.290021	−	0.290021i	−0.837088	+	0.547068i	$0.815744\pi$
$108$		0			0
$109$		−	16.0000i		−	1.53252i	−0.642529	−	0.766261i	$-0.722115\pi$
$109$		−	16.0000i		−	1.53252i	0.642529	−	0.766261i	$-0.277885\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	8.00000	+	8.00000i	0.752577	+	0.752577i	0.974959	−	0.222383i	$-0.0713835\pi$
$113$	8.00000	+	8.00000i	0.752577	+	0.752577i	−0.222383	+	0.974959i	$0.571383\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−4.00000	+	4.00000i	−0.369800	+	0.369800i
$118$		0			0
$119$	8.00000			0.733359
$120$		0			0
$121$	−5.00000			−0.454545
$122$		0			0
$123$	4.00000	−	4.00000i	0.360668	−	0.360668i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−15.0000	−	15.0000i	−1.33103	−	1.33103i	−0.904445	−	0.426589i	$-0.859715\pi$
$127$	−15.0000	−	15.0000i	−1.33103	−	1.33103i	−0.426589	−	0.904445i	$-0.640285\pi$
$128$		0			0
$129$		−	14.0000i		−	1.23263i
$130$		0			0
$131$			12.0000i			1.04844i	0.851581	+	0.524222i	$0.175644\pi$
$131$			12.0000i			1.04844i	−0.851581	+	0.524222i	$0.824356\pi$
$132$		0			0
$133$	−4.00000	−	4.00000i	−0.346844	−	0.346844i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−8.00000	+	8.00000i	−0.683486	+	0.683486i	−0.960784	−	0.277298i	$-0.910561\pi$
$137$	−8.00000	+	8.00000i	−0.683486	+	0.683486i	0.277298	+	0.960784i	$0.410561\pi$
$138$		0			0
$139$	−12.0000			−1.01783			−0.508913	−	0.860818i	$-0.669953\pi$
$139$	−12.0000			−1.01783			−0.508913	+	0.860818i	$0.669953\pi$
$140$		0			0
$141$	−6.00000			−0.505291
$142$		0			0
$143$	−16.0000	+	16.0000i	−1.33799	+	1.33799i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	5.00000	+	5.00000i	0.412393	+	0.412393i
$148$		0			0
$149$			8.00000i			0.655386i	0.944784	+	0.327693i	$0.106271\pi$
$149$			8.00000i			0.655386i	−0.944784	+	0.327693i	$0.893729\pi$
$150$		0			0
$151$		−	8.00000i		−	0.651031i	−0.945537	−	0.325515i	$-0.894462\pi$
$151$		−	8.00000i		−	0.651031i	0.945537	−	0.325515i	$-0.105538\pi$
$152$		0			0
$153$	4.00000	+	4.00000i	0.323381	+	0.323381i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	8.00000	−	8.00000i	0.638470	−	0.638470i	−0.311708	−	0.950178i	$-0.600901\pi$
$157$	8.00000	−	8.00000i	0.638470	−	0.638470i	0.950178	+	0.311708i	$0.100901\pi$
$158$		0			0
$159$	−8.00000			−0.634441
$160$		0			0
$161$	10.0000			0.788110
$162$		0			0
$163$	9.00000	−	9.00000i	0.704934	−	0.704934i	−0.260531	−	0.965465i	$-0.583898\pi$
$163$	9.00000	−	9.00000i	0.704934	−	0.704934i	0.965465	+	0.260531i	$0.0838976\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	5.00000	+	5.00000i	0.386912	+	0.386912i	0.873584	−	0.486673i	$-0.161790\pi$
$167$	5.00000	+	5.00000i	0.386912	+	0.386912i	−0.486673	+	0.873584i	$0.661790\pi$
$168$		0			0
$169$			19.0000i			1.46154i
$170$		0			0
$171$		−	4.00000i		−	0.305888i
$172$		0			0
$173$	−8.00000	−	8.00000i	−0.608229	−	0.608229i	0.334254	−	0.942483i	$-0.391516\pi$
$173$	−8.00000	−	8.00000i	−0.608229	−	0.608229i	−0.942483	+	0.334254i	$0.891516\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	4.00000	−	4.00000i	0.300658	−	0.300658i
$178$		0			0
$179$	12.0000			0.896922			0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000			0.896922			0.448461	+	0.893802i	$0.351972\pi$
$180$		0			0
$181$	−10.0000			−0.743294			−0.371647	−	0.928374i	$-0.621207\pi$
$181$	−10.0000			−0.743294			−0.371647	+	0.928374i	$0.621207\pi$
$182$		0			0
$183$	−8.00000	+	8.00000i	−0.591377	+	0.591377i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	16.0000	+	16.0000i	1.17004	+	1.17004i
$188$		0			0
$189$		−	8.00000i		−	0.581914i
$190$		0			0
$191$		0			0		1.00000			$0$
$191$		0			0		−1.00000			$\pi$
$192$		0			0
$193$	4.00000	+	4.00000i	0.287926	+	0.287926i	0.836260	−	0.548333i	$-0.184738\pi$
$193$	4.00000	+	4.00000i	0.287926	+	0.287926i	−0.548333	+	0.836260i	$0.684738\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	4.00000	−	4.00000i	0.284988	−	0.284988i	−0.550106	−	0.835095i	$-0.685413\pi$
$197$	4.00000	−	4.00000i	0.284988	−	0.284988i	0.835095	+	0.550106i	$0.185413\pi$
$198$		0			0
$199$		0			0			−	1.00000i	$-0.5\pi$
$199$		0			0				1.00000i	$0.5\pi$
$200$		0			0
$201$	−6.00000			−0.423207
$202$		0			0
$203$	2.00000	−	2.00000i	0.140372	−	0.140372i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	5.00000	+	5.00000i	0.347524	+	0.347524i
$208$		0			0
$209$		−	16.0000i		−	1.10674i
$210$		0			0
$211$		−	20.0000i		−	1.37686i	−0.725304	−	0.688428i	$-0.758301\pi$
$211$		−	20.0000i		−	1.37686i	0.725304	−	0.688428i	$-0.241699\pi$
$212$		0			0
$213$	−16.0000	−	16.0000i	−1.09630	−	1.09630i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−8.00000	+	8.00000i	−0.543075	+	0.543075i
$218$		0			0
$219$	8.00000			0.540590
$220$		0			0
$221$	32.0000			2.15255
$222$		0			0
$223$	9.00000	−	9.00000i	0.602685	−	0.602685i	−0.338340	−	0.941024i	$-0.609865\pi$
$223$	9.00000	−	9.00000i	0.602685	−	0.602685i	0.941024	+	0.338340i	$0.109865\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−7.00000	−	7.00000i	−0.464606	−	0.464606i	0.435556	−	0.900162i	$-0.356552\pi$
$227$	−7.00000	−	7.00000i	−0.464606	−	0.464606i	−0.900162	+	0.435556i	$0.856552\pi$
$228$		0			0
$229$			26.0000i			1.71813i	0.511868	+	0.859064i	$0.328954\pi$
$229$			26.0000i			1.71813i	−0.511868	+	0.859064i	$0.671046\pi$
$230$		0			0
$231$		−	8.00000i		−	0.526361i
$232$		0			0
$233$	12.0000	+	12.0000i	0.786146	+	0.786146i	0.980860	−	0.194714i	$-0.0623778\pi$
$233$	12.0000	+	12.0000i	0.786146	+	0.786146i	−0.194714	+	0.980860i	$0.562378\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	8.00000	−	8.00000i	0.519656	−	0.519656i
$238$		0			0
$239$	24.0000			1.55243			0.776215	−	0.630468i	$-0.217137\pi$
$239$	24.0000			1.55243			0.776215	+	0.630468i	$0.217137\pi$
$240$		0			0
$241$	4.00000			0.257663			0.128831	−	0.991667i	$-0.458877\pi$
$241$	4.00000			0.257663			0.128831	+	0.991667i	$0.458877\pi$
$242$		0			0
$243$	7.00000	−	7.00000i	0.449050	−	0.449050i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−16.0000	−	16.0000i	−1.01806	−	1.01806i
$248$		0			0
$249$			10.0000i			0.633724i
$250$		0			0
$251$		−	12.0000i		−	0.757433i	−0.925513	−	0.378717i	$-0.876365\pi$
$251$		−	12.0000i		−	0.757433i	0.925513	−	0.378717i	$-0.123635\pi$
$252$		0			0
$253$	20.0000	+	20.0000i	1.25739	+	1.25739i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	16.0000	−	16.0000i	0.998053	−	0.998053i	−0.00194553	−	0.999998i	$-0.500619\pi$
$257$	16.0000	−	16.0000i	0.998053	−	0.998053i	0.999998	+	0.00194553i	$0.000619281\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	2.00000			0.123797
$262$		0			0
$263$	−9.00000	+	9.00000i	−0.554964	+	0.554964i	−0.927869	−	0.372906i	$-0.878362\pi$
$263$	−9.00000	+	9.00000i	−0.554964	+	0.554964i	0.372906	+	0.927869i	$0.378362\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−10.0000	−	10.0000i	−0.611990	−	0.611990i
$268$		0			0
$269$		−	24.0000i		−	1.46331i	−0.681677	−	0.731653i	$-0.738749\pi$
$269$		−	24.0000i		−	1.46331i	0.681677	−	0.731653i	$-0.261251\pi$
$270$		0			0
$271$		−	8.00000i		−	0.485965i	−0.970031	−	0.242983i	$-0.921874\pi$
$271$		−	8.00000i		−	0.485965i	0.970031	−	0.242983i	$-0.0781258\pi$
$272$		0			0
$273$	−8.00000	−	8.00000i	−0.484182	−	0.484182i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	16.0000	−	16.0000i	0.961347	−	0.961347i	−0.0379334	−	0.999280i	$-0.512077\pi$
$277$	16.0000	−	16.0000i	0.961347	−	0.961347i	0.999280	+	0.0379334i	$0.0120775\pi$
$278$		0			0
$279$	−8.00000			−0.478947
$280$		0			0
$281$	−20.0000			−1.19310			−0.596550	−	0.802576i	$-0.703462\pi$
$281$	−20.0000			−1.19310			−0.596550	+	0.802576i	$0.703462\pi$
$282$		0			0
$283$	−5.00000	+	5.00000i	−0.297219	+	0.297219i	−0.839924	−	0.542705i	$-0.817400\pi$
$283$	−5.00000	+	5.00000i	−0.297219	+	0.297219i	0.542705	+	0.839924i	$0.317400\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−4.00000	−	4.00000i	−0.236113	−	0.236113i
$288$		0			0
$289$		−	15.0000i		−	0.882353i
$290$		0			0
$291$		−	24.0000i		−	1.40690i
$292$		0			0
$293$	8.00000	+	8.00000i	0.467365	+	0.467365i	0.901060	−	0.433695i	$-0.142790\pi$
$293$	8.00000	+	8.00000i	0.467365	+	0.467365i	−0.433695	+	0.901060i	$0.642790\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	16.0000	−	16.0000i	0.928414	−	0.928414i
$298$		0			0
$299$	40.0000			2.31326
$300$		0			0
$301$	−14.0000			−0.806947
$302$		0			0
$303$	6.00000	−	6.00000i	0.344691	−	0.344691i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	3.00000	+	3.00000i	0.171219	+	0.171219i	0.787515	−	0.616296i	$-0.211367\pi$
$307$	3.00000	+	3.00000i	0.171219	+	0.171219i	−0.616296	+	0.787515i	$0.711367\pi$
$308$		0			0
$309$			6.00000i			0.341328i
$310$		0			0
$311$		−	8.00000i		−	0.453638i	−0.973937	−	0.226819i	$-0.927167\pi$
$311$		−	8.00000i		−	0.453638i	0.973937	−	0.226819i	$-0.0728326\pi$
$312$		0			0
$313$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$313$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−20.0000	+	20.0000i	−1.12331	+	1.12331i	−0.132072	+	0.991240i	$0.542163\pi$
$317$	−20.0000	+	20.0000i	−1.12331	+	1.12331i	−0.991240	+	0.132072i	$0.957837\pi$
$318$		0			0
$319$	8.00000			0.447914
$320$		0			0
$321$	6.00000			0.334887
$322$		0			0
$323$	−16.0000	+	16.0000i	−0.890264	+	0.890264i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	16.0000	+	16.0000i	0.884802	+	0.884802i
$328$		0			0
$329$			6.00000i			0.330791i
$330$		0			0
$331$		−	12.0000i		−	0.659580i	−0.944054	−	0.329790i	$-0.893022\pi$
$331$		−	12.0000i		−	0.659580i	0.944054	−	0.329790i	$-0.106978\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−16.0000	+	16.0000i	−0.871576	+	0.871576i	−0.992644	−	0.121069i	$-0.961368\pi$
$337$	−16.0000	+	16.0000i	−0.871576	+	0.871576i	0.121069	+	0.992644i	$0.461368\pi$
$338$		0			0
$339$	−16.0000			−0.869001
$340$		0			0
$341$	−32.0000			−1.73290
$342$		0			0
$343$	12.0000	−	12.0000i	0.647939	−	0.647939i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−1.00000	−	1.00000i	−0.0536828	−	0.0536828i	0.679756	−	0.733439i	$-0.262086\pi$
$347$	−1.00000	−	1.00000i	−0.0536828	−	0.0536828i	−0.733439	+	0.679756i	$0.762086\pi$
$348$		0			0
$349$			2.00000i			0.107058i	0.998566	+	0.0535288i	$0.0170469\pi$
$349$			2.00000i			0.107058i	−0.998566	+	0.0535288i	$0.982953\pi$
$350$		0			0
$351$		−	32.0000i		−	1.70803i
$352$		0			0
$353$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$353$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	−8.00000	+	8.00000i	−0.423405	+	0.423405i
$358$		0			0
$359$	24.0000			1.26667			0.633336	−	0.773877i	$-0.281685\pi$
$359$	24.0000			1.26667			0.633336	+	0.773877i	$0.281685\pi$
$360$		0			0
$361$	−3.00000			−0.157895
$362$		0			0
$363$	5.00000	−	5.00000i	0.262432	−	0.262432i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	23.0000	+	23.0000i	1.20059	+	1.20059i	0.973987	+	0.226603i	$0.0727620\pi$
$367$	23.0000	+	23.0000i	1.20059	+	1.20059i	0.226603	+	0.973987i	$0.427238\pi$
$368$		0			0
$369$		−	4.00000i		−	0.208232i
$370$		0			0
$371$			8.00000i			0.415339i
$372$		0			0
$373$	24.0000	+	24.0000i	1.24267	+	1.24267i	0.958888	+	0.283785i	$0.0915902\pi$
$373$	24.0000	+	24.0000i	1.24267	+	1.24267i	0.283785	+	0.958888i	$0.408410\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	8.00000	−	8.00000i	0.412021	−	0.412021i
$378$		0			0
$379$	−4.00000			−0.205466			−0.102733	−	0.994709i	$-0.532759\pi$
$379$	−4.00000			−0.205466			−0.102733	+	0.994709i	$0.532759\pi$
$380$		0			0
$381$	30.0000			1.53695
$382$		0			0
$383$	17.0000	−	17.0000i	0.868659	−	0.868659i	−0.123665	−	0.992324i	$-0.539465\pi$
$383$	17.0000	−	17.0000i	0.868659	−	0.868659i	0.992324	+	0.123665i	$0.0394647\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−7.00000	−	7.00000i	−0.355830	−	0.355830i
$388$		0			0
$389$			8.00000i			0.405616i	0.979219	+	0.202808i	$0.0650067\pi$
$389$			8.00000i			0.405616i	−0.979219	+	0.202808i	$0.934993\pi$
$390$		0			0
$391$		−	40.0000i		−	2.02289i
$392$		0			0
$393$	−12.0000	−	12.0000i	−0.605320	−	0.605320i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	4.00000	−	4.00000i	0.200754	−	0.200754i	−0.599569	−	0.800323i	$-0.704661\pi$
$397$	4.00000	−	4.00000i	0.200754	−	0.200754i	0.800323	+	0.599569i	$0.204661\pi$
$398$		0			0
$399$	8.00000			0.400501
$400$		0			0
$401$	30.0000			1.49813			0.749064	−	0.662497i	$-0.230503\pi$
$401$	30.0000			1.49813			0.749064	+	0.662497i	$0.230503\pi$
$402$		0			0
$403$	−32.0000	+	32.0000i	−1.59403	+	1.59403i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$			20.0000i			0.988936i	0.869196	+	0.494468i	$0.164637\pi$
$409$			20.0000i			0.988936i	−0.869196	+	0.494468i	$0.835363\pi$
$410$		0			0
$411$		−	16.0000i		−	0.789222i
$412$		0			0
$413$	−4.00000	−	4.00000i	−0.196827	−	0.196827i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	12.0000	−	12.0000i	0.587643	−	0.587643i
$418$		0			0
$419$	28.0000			1.36789			0.683945	−	0.729534i	$-0.260263\pi$
$419$	28.0000			1.36789			0.683945	+	0.729534i	$0.260263\pi$
$420$		0			0
$421$	8.00000			0.389896			0.194948	−	0.980814i	$-0.437546\pi$
$421$	8.00000			0.389896			0.194948	+	0.980814i	$0.437546\pi$
$422$		0			0
$423$	−3.00000	+	3.00000i	−0.145865	+	0.145865i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	8.00000	+	8.00000i	0.387147	+	0.387147i
$428$		0			0
$429$		−	32.0000i		−	1.54497i
$430$		0			0
$431$			8.00000i			0.385346i	0.981263	+	0.192673i	$0.0617157\pi$
$431$			8.00000i			0.385346i	−0.981263	+	0.192673i	$0.938284\pi$
$432$		0			0
$433$	−20.0000	−	20.0000i	−0.961139	−	0.961139i	0.0381340	−	0.999273i	$-0.487859\pi$
$433$	−20.0000	−	20.0000i	−0.961139	−	0.961139i	−0.999273	+	0.0381340i	$0.987859\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−20.0000	+	20.0000i	−0.956730	+	0.956730i
$438$		0			0
$439$	−8.00000			−0.381819			−0.190910	−	0.981608i	$-0.561144\pi$
$439$	−8.00000			−0.381819			−0.190910	+	0.981608i	$0.561144\pi$
$440$		0			0
$441$	5.00000			0.238095
$442$		0			0
$443$	13.0000	−	13.0000i	0.617649	−	0.617649i	−0.327279	−	0.944928i	$-0.606132\pi$
$443$	13.0000	−	13.0000i	0.617649	−	0.617649i	0.944928	+	0.327279i	$0.106132\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	−8.00000	−	8.00000i	−0.378387	−	0.378387i
$448$		0			0
$449$		−	4.00000i		−	0.188772i	−0.995536	−	0.0943858i	$-0.969911\pi$
$449$		−	4.00000i		−	0.188772i	0.995536	−	0.0943858i	$-0.0300887\pi$
$450$		0			0
$451$		−	16.0000i		−	0.753411i
$452$		0			0
$453$	8.00000	+	8.00000i	0.375873	+	0.375873i
$454$		0			0
$455$		0			0
$456$		0			0
$457$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$457$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$458$		0			0
$459$	−32.0000			−1.49363
$460$		0			0
$461$	−18.0000			−0.838344			−0.419172	−	0.907907i	$-0.637680\pi$
$461$	−18.0000			−0.838344			−0.419172	+	0.907907i	$0.637680\pi$
$462$		0			0
$463$	3.00000	−	3.00000i	0.139422	−	0.139422i	−0.633951	−	0.773373i	$-0.718568\pi$
$463$	3.00000	−	3.00000i	0.139422	−	0.139422i	0.773373	+	0.633951i	$0.218568\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−25.0000	−	25.0000i	−1.15686	−	1.15686i	−0.985147	−	0.171715i	$-0.945069\pi$
$467$	−25.0000	−	25.0000i	−1.15686	−	1.15686i	−0.171715	−	0.985147i	$-0.554931\pi$
$468$		0			0
$469$			6.00000i			0.277054i
$470$		0			0
$471$			16.0000i			0.737241i
$472$		0			0
$473$	−28.0000	−	28.0000i	−1.28744	−	1.28744i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−4.00000	+	4.00000i	−0.183147	+	0.183147i
$478$		0			0
$479$	16.0000			0.731059			0.365529	−	0.930800i	$-0.380888\pi$
$479$	16.0000			0.731059			0.365529	+	0.930800i	$0.380888\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$	−10.0000	+	10.0000i	−0.455016	+	0.455016i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−25.0000	−	25.0000i	−1.13286	−	1.13286i	−0.989700	−	0.143158i	$-0.954274\pi$
$487$	−25.0000	−	25.0000i	−1.13286	−	1.13286i	−0.143158	−	0.989700i	$-0.545726\pi$
$488$		0			0
$489$			18.0000i			0.813988i
$490$		0			0
$491$			12.0000i			0.541552i	0.962642	+	0.270776i	$0.0872803\pi$
$491$			12.0000i			0.541552i	−0.962642	+	0.270776i	$0.912720\pi$
$492$		0			0
$493$	−8.00000	−	8.00000i	−0.360302	−	0.360302i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−16.0000	+	16.0000i	−0.717698	+	0.717698i
$498$		0			0
$499$	20.0000			0.895323			0.447661	−	0.894203i	$-0.352257\pi$
$499$	20.0000			0.895323			0.447661	+	0.894203i	$0.352257\pi$
$500$		0			0
$501$	−10.0000			−0.446767
$502$		0			0
$503$	11.0000	−	11.0000i	0.490466	−	0.490466i	−0.417987	−	0.908453i	$-0.637264\pi$
$503$	11.0000	−	11.0000i	0.490466	−	0.490466i	0.908453	+	0.417987i	$0.137264\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−19.0000	−	19.0000i	−0.843820	−	0.843820i
$508$		0			0
$509$			30.0000i			1.32973i	0.746965	+	0.664863i	$0.231510\pi$
$509$			30.0000i			1.32973i	−0.746965	+	0.664863i	$0.768490\pi$
$510$		0			0
$511$		−	8.00000i		−	0.353899i
$512$		0			0
$513$	16.0000	+	16.0000i	0.706417	+	0.706417i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−12.0000	+	12.0000i	−0.527759	+	0.527759i
$518$		0			0
$519$	16.0000			0.702322
$520$		0			0
$521$	−22.0000			−0.963837			−0.481919	−	0.876216i	$-0.660060\pi$
$521$	−22.0000			−0.963837			−0.481919	+	0.876216i	$0.660060\pi$
$522$		0			0
$523$	−5.00000	+	5.00000i	−0.218635	+	0.218635i	−0.807923	−	0.589288i	$-0.799408\pi$
$523$	−5.00000	+	5.00000i	−0.218635	+	0.218635i	0.589288	+	0.807923i	$0.299408\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	32.0000	+	32.0000i	1.39394	+	1.39394i
$528$		0			0
$529$		−	27.0000i		−	1.17391i
$530$		0			0
$531$		−	4.00000i		−	0.173585i
$532$		0			0
$533$	−16.0000	−	16.0000i	−0.693037	−	0.693037i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−12.0000	+	12.0000i	−0.517838	+	0.517838i
$538$		0			0
$539$	20.0000			0.861461
$540$		0			0
$541$	30.0000			1.28980			0.644900	−	0.764267i	$-0.276899\pi$
$541$	30.0000			1.28980			0.644900	+	0.764267i	$0.276899\pi$
$542$		0			0
$543$	10.0000	−	10.0000i	0.429141	−	0.429141i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	17.0000	+	17.0000i	0.726868	+	0.726868i	0.969994	−	0.243127i	$-0.0781732\pi$
$547$	17.0000	+	17.0000i	0.726868	+	0.726868i	−0.243127	+	0.969994i	$0.578173\pi$
$548$		0			0
$549$			8.00000i			0.341432i
$550$		0			0
$551$			8.00000i			0.340811i
$552$		0			0
$553$	−8.00000	−	8.00000i	−0.340195	−	0.340195i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$557$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$558$		0			0
$559$	−56.0000			−2.36855
$560$		0			0
$561$	−32.0000			−1.35104
$562$		0			0
$563$	15.0000	−	15.0000i	0.632175	−	0.632175i	−0.316438	−	0.948613i	$-0.602487\pi$
$563$	15.0000	−	15.0000i	0.632175	−	0.632175i	0.948613	+	0.316438i	$0.102487\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	5.00000	+	5.00000i	0.209980	+	0.209980i
$568$		0			0
$569$		−	20.0000i		−	0.838444i	−0.907884	−	0.419222i	$-0.862303\pi$
$569$		−	20.0000i		−	0.838444i	0.907884	−	0.419222i	$-0.137697\pi$
$570$		0			0
$571$		−	4.00000i		−	0.167395i	−0.996491	−	0.0836974i	$-0.973327\pi$
$571$		−	4.00000i		−	0.167395i	0.996491	−	0.0836974i	$-0.0266729\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	8.00000	−	8.00000i	0.333044	−	0.333044i	−0.520697	−	0.853741i	$-0.674328\pi$
$577$	8.00000	−	8.00000i	0.333044	−	0.333044i	0.853741	+	0.520697i	$0.174328\pi$
$578$		0			0
$579$	−8.00000			−0.332469
$580$		0			0
$581$	10.0000			0.414870
$582$		0			0
$583$	−16.0000	+	16.0000i	−0.662652	+	0.662652i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	25.0000	+	25.0000i	1.03186	+	1.03186i	0.999475	+	0.0323850i	$0.0103103\pi$
$587$	25.0000	+	25.0000i	1.03186	+	1.03186i	0.0323850	+	0.999475i	$0.489690\pi$
$588$		0			0
$589$		−	32.0000i		−	1.31854i
$590$		0			0
$591$			8.00000i			0.329076i
$592$		0			0
$593$	8.00000	+	8.00000i	0.328521	+	0.328521i	0.852024	−	0.523503i	$-0.175375\pi$
$593$	8.00000	+	8.00000i	0.328521	+	0.328521i	−0.523503	+	0.852024i	$0.675375\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	16.0000			0.653742			0.326871	−	0.945069i	$-0.394006\pi$
$599$	16.0000			0.653742			0.326871	+	0.945069i	$0.394006\pi$
$600$		0			0
$601$	4.00000			0.163163			0.0815817	−	0.996667i	$-0.474003\pi$
$601$	4.00000			0.163163			0.0815817	+	0.996667i	$0.474003\pi$
$602$		0			0
$603$	−3.00000	+	3.00000i	−0.122169	+	0.122169i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−5.00000	−	5.00000i	−0.202944	−	0.202944i	0.598316	−	0.801260i	$-0.295837\pi$
$607$	−5.00000	−	5.00000i	−0.202944	−	0.202944i	−0.801260	+	0.598316i	$0.795837\pi$
$608$		0			0
$609$			4.00000i			0.162088i
$610$		0			0
$611$			24.0000i			0.970936i
$612$		0			0
$613$	−12.0000	−	12.0000i	−0.484675	−	0.484675i	0.421946	−	0.906621i	$-0.361347\pi$
$613$	−12.0000	−	12.0000i	−0.484675	−	0.484675i	−0.906621	+	0.421946i	$0.861347\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−4.00000	+	4.00000i	−0.161034	+	0.161034i	−0.783025	−	0.621991i	$-0.786324\pi$
$617$	−4.00000	+	4.00000i	−0.161034	+	0.161034i	0.621991	+	0.783025i	$0.286324\pi$
$618$		0			0
$619$	−12.0000			−0.482321			−0.241160	−	0.970485i	$-0.577528\pi$
$619$	−12.0000			−0.482321			−0.241160	+	0.970485i	$0.577528\pi$
$620$		0			0
$621$	−40.0000			−1.60514
$622$		0			0
$623$	−10.0000	+	10.0000i	−0.400642	+	0.400642i
$624$		0			0
$625$		0			0
$626$		0			0
$627$	16.0000	+	16.0000i	0.638978	+	0.638978i
$628$		0			0
$629$		0			0
$630$		0			0
$631$			40.0000i			1.59237i	0.605050	+	0.796187i	$0.293153\pi$
$631$			40.0000i			1.59237i	−0.605050	+	0.796187i	$0.706847\pi$
$632$		0			0
$633$	20.0000	+	20.0000i	0.794929	+	0.794929i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	20.0000	−	20.0000i	0.792429	−	0.792429i
$638$		0			0
$639$	−16.0000			−0.632950
$640$		0			0
$641$	12.0000			0.473972			0.236986	−	0.971513i	$-0.423841\pi$
$641$	12.0000			0.473972			0.236986	+	0.971513i	$0.423841\pi$
$642$		0			0
$643$	11.0000	−	11.0000i	0.433798	−	0.433798i	−0.456120	−	0.889918i	$-0.650761\pi$
$643$	11.0000	−	11.0000i	0.433798	−	0.433798i	0.889918	+	0.456120i	$0.150761\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−1.00000	−	1.00000i	−0.0393141	−	0.0393141i	0.687176	−	0.726491i	$-0.258850\pi$
$647$	−1.00000	−	1.00000i	−0.0393141	−	0.0393141i	−0.726491	+	0.687176i	$0.758850\pi$
$648$		0			0
$649$		−	16.0000i		−	0.628055i
$650$		0			0
$651$		−	16.0000i		−	0.627089i
$652$		0			0
$653$	20.0000	+	20.0000i	0.782660	+	0.782660i	0.980279	−	0.197619i	$-0.0633207\pi$
$653$	20.0000	+	20.0000i	0.782660	+	0.782660i	−0.197619	+	0.980279i	$0.563321\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	4.00000	−	4.00000i	0.156055	−	0.156055i
$658$		0			0
$659$	−4.00000			−0.155818			−0.0779089	−	0.996960i	$-0.524824\pi$
$659$	−4.00000			−0.155818			−0.0779089	+	0.996960i	$0.524824\pi$
$660$		0			0
$661$		0			0			−	1.00000i	$-0.5\pi$
$661$		0			0				1.00000i	$0.5\pi$
$662$		0			0
$663$	−32.0000	+	32.0000i	−1.24278	+	1.24278i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−10.0000	−	10.0000i	−0.387202	−	0.387202i
$668$		0			0
$669$			18.0000i			0.695920i
$670$		0			0
$671$			32.0000i			1.23535i
$672$		0			0
$673$	−4.00000	−	4.00000i	−0.154189	−	0.154189i	0.625797	−	0.779986i	$-0.284774\pi$
$673$	−4.00000	−	4.00000i	−0.154189	−	0.154189i	−0.779986	+	0.625797i	$0.784774\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	20.0000	−	20.0000i	0.768662	−	0.768662i	−0.209209	−	0.977871i	$-0.567089\pi$
$677$	20.0000	−	20.0000i	0.768662	−	0.768662i	0.977871	+	0.209209i	$0.0670888\pi$
$678$		0			0
$679$	−24.0000			−0.921035
$680$		0			0
$681$	14.0000			0.536481
$682$		0			0
$683$	17.0000	−	17.0000i	0.650487	−	0.650487i	−0.302623	−	0.953110i	$-0.597862\pi$
$683$	17.0000	−	17.0000i	0.650487	−	0.650487i	0.953110	+	0.302623i	$0.0978624\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	−26.0000	−	26.0000i	−0.991962	−	0.991962i
$688$		0			0
$689$			32.0000i			1.21910i
$690$		0			0
$691$			20.0000i			0.760836i	0.924815	+	0.380418i	$0.124220\pi$
$691$			20.0000i			0.760836i	−0.924815	+	0.380418i	$0.875780\pi$
$692$		0			0
$693$	−4.00000	−	4.00000i	−0.151947	−	0.151947i
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−16.0000	+	16.0000i	−0.606043	+	0.606043i
$698$		0			0
$699$	−24.0000			−0.907763
$700$		0			0
$701$	48.0000			1.81293			0.906467	−	0.422276i	$-0.138769\pi$
$701$	48.0000			1.81293			0.906467	+	0.422276i	$0.138769\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−6.00000	−	6.00000i	−0.225653	−	0.225653i
$708$		0			0
$709$			6.00000i			0.225335i	0.993633	+	0.112667i	$0.0359394\pi$
$709$			6.00000i			0.225335i	−0.993633	+	0.112667i	$0.964061\pi$
$710$		0			0
$711$		−	8.00000i		−	0.300023i
$712$		0			0
$713$	40.0000	+	40.0000i	1.49801	+	1.49801i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−24.0000	+	24.0000i	−0.896296	+	0.896296i
$718$		0			0
$719$	24.0000			0.895049			0.447524	−	0.894272i	$-0.352306\pi$
$719$	24.0000			0.895049			0.447524	+	0.894272i	$0.352306\pi$
$720$		0			0
$721$	6.00000			0.223452
$722$		0			0
$723$	−4.00000	+	4.00000i	−0.148762	+	0.148762i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−15.0000	−	15.0000i	−0.556319	−	0.556319i	0.371938	−	0.928257i	$-0.378693\pi$
$727$	−15.0000	−	15.0000i	−0.556319	−	0.556319i	−0.928257	+	0.371938i	$0.878693\pi$
$728$		0			0
$729$			29.0000i			1.07407i
$730$		0			0
$731$			56.0000i			2.07123i
$732$		0			0
$733$	−24.0000	−	24.0000i	−0.886460	−	0.886460i	0.107721	−	0.994181i	$-0.465645\pi$
$733$	−24.0000	−	24.0000i	−0.886460	−	0.886460i	−0.994181	+	0.107721i	$0.965645\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−12.0000	+	12.0000i	−0.442026	+	0.442026i
$738$		0			0
$739$	−52.0000			−1.91285			−0.956425	−	0.291977i	$-0.905687\pi$
$739$	−52.0000			−1.91285			−0.956425	+	0.291977i	$0.905687\pi$
$740$		0			0
$741$	32.0000			1.17555
$742$		0			0
$743$	1.00000	−	1.00000i	0.0366864	−	0.0366864i	−0.688526	−	0.725212i	$-0.741742\pi$
$743$	1.00000	−	1.00000i	0.0366864	−	0.0366864i	0.725212	+	0.688526i	$0.241742\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	5.00000	+	5.00000i	0.182940	+	0.182940i
$748$		0			0
$749$		−	6.00000i		−	0.219235i
$750$		0			0
$751$		−	16.0000i		−	0.583848i	−0.956441	−	0.291924i	$-0.905705\pi$
$751$		−	16.0000i		−	0.583848i	0.956441	−	0.291924i	$-0.0942955\pi$
$752$		0			0
$753$	12.0000	+	12.0000i	0.437304	+	0.437304i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	24.0000	−	24.0000i	0.872295	−	0.872295i	−0.120427	−	0.992722i	$-0.538426\pi$
$757$	24.0000	−	24.0000i	0.872295	−	0.872295i	0.992722	+	0.120427i	$0.0384265\pi$
$758$		0			0
$759$	−40.0000			−1.45191
$760$		0			0
$761$	10.0000			0.362500			0.181250	−	0.983437i	$-0.441986\pi$
$761$	10.0000			0.362500			0.181250	+	0.983437i	$0.441986\pi$
$762$		0			0
$763$	16.0000	−	16.0000i	0.579239	−	0.579239i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−16.0000	−	16.0000i	−0.577727	−	0.577727i
$768$		0			0
$769$		−	2.00000i		−	0.0721218i	−0.999350	−	0.0360609i	$-0.988519\pi$
$769$		−	2.00000i		−	0.0721218i	0.999350	−	0.0360609i	$-0.0114810\pi$
$770$		0			0
$771$			32.0000i			1.15245i
$772$		0			0
$773$	−20.0000	−	20.0000i	−0.719350	−	0.719350i	0.249122	−	0.968472i	$-0.419858\pi$
$773$	−20.0000	−	20.0000i	−0.719350	−	0.719350i	−0.968472	+	0.249122i	$0.919858\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$	16.0000			0.573259
$780$		0			0
$781$	−64.0000			−2.29010
$782$		0			0
$783$	−8.00000	+	8.00000i	−0.285897	+	0.285897i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−11.0000	−	11.0000i	−0.392108	−	0.392108i	0.483330	−	0.875438i	$-0.339427\pi$
$787$	−11.0000	−	11.0000i	−0.392108	−	0.392108i	−0.875438	+	0.483330i	$0.839427\pi$
$788$		0			0
$789$		−	18.0000i		−	0.640817i
$790$		0			0
$791$			16.0000i			0.568895i
$792$		0			0
$793$	32.0000	+	32.0000i	1.13635	+	1.13635i
$794$		0			0
$795$		0			0

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:	\( N \)	\(=\)	\( 1600 = 2^{6} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1600.n (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.00000 + 1.00000i) q^{3} +(1.00000 + 1.00000i) q^{7} +1.00000i q^{9} +O(q^{10})\)	\(q+(-1.00000 + 1.00000i) q^{3} +(1.00000 + 1.00000i) q^{7} +1.00000i q^{9} +4.00000i q^{11} +(4.00000 + 4.00000i) q^{13} +(4.00000 - 4.00000i) q^{17} -4.00000 q^{19} -2.00000 q^{21} +(5.00000 - 5.00000i) q^{23} +(-4.00000 - 4.00000i) q^{27} -2.00000i q^{29} +8.00000i q^{31} +(-4.00000 - 4.00000i) q^{33} -8.00000 q^{39} -4.00000 q^{41} +(-7.00000 + 7.00000i) q^{43} +(3.00000 + 3.00000i) q^{47} -5.00000i q^{49} +8.00000i q^{51} +(4.00000 + 4.00000i) q^{53} +(4.00000 - 4.00000i) q^{57} -4.00000 q^{59} +8.00000 q^{61} +(-1.00000 + 1.00000i) q^{63} +(3.00000 + 3.00000i) q^{67} +10.0000i q^{69} +16.0000i q^{71} +(-4.00000 - 4.00000i) q^{73} +(-4.00000 + 4.00000i) q^{77} -8.00000 q^{79} +5.00000 q^{81} +(5.00000 - 5.00000i) q^{83} +(2.00000 + 2.00000i) q^{87} +10.0000i q^{89} +8.00000i q^{91} +(-8.00000 - 8.00000i) q^{93} +(-12.0000 + 12.0000i) q^{97} -4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + 2 q^{7}+O(q^{10}) \) 2 * q - 2 * q^3 + 2 * q^7	\( 2 q - 2 q^{3} + 2 q^{7} + 8 q^{13} + 8 q^{17} - 8 q^{19} - 4 q^{21} + 10 q^{23} - 8 q^{27} - 8 q^{33} - 16 q^{39} - 8 q^{41} - 14 q^{43} + 6 q^{47} + 8 q^{53} + 8 q^{57} - 8 q^{59} + 16 q^{61} - 2 q^{63} + 6 q^{67} - 8 q^{73} - 8 q^{77} - 16 q^{79} + 10 q^{81} + 10 q^{83} + 4 q^{87} - 16 q^{93} - 24 q^{97} - 8 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 + 2 * q^7 + 8 * q^13 + 8 * q^17 - 8 * q^19 - 4 * q^21 + 10 * q^23 - 8 * q^27 - 8 * q^33 - 16 * q^39 - 8 * q^41 - 14 * q^43 + 6 * q^47 + 8 * q^53 + 8 * q^57 - 8 * q^59 + 16 * q^61 - 2 * q^63 + 6 * q^67 - 8 * q^73 - 8 * q^77 - 16 * q^79 + 10 * q^81 + 10 * q^83 + 4 * q^87 - 16 * q^93 - 24 * q^97 - 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more