LMFDB - Embedded newform 1440.2.x.p.127.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1440,2,Mod(127,1440)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1440, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 0, 0, 1]))

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

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

chi := DirichletCharacter("1440.127");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1440 = 2^{5} \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1440.x (of order $4$, degree $2$, 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:	$11.4984578911$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(i)$
Coefficient field:	$\Q(i, \sqrt{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 9 $ x^4 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			127.1
Root			$1.22474 - 1.22474i$ of defining polynomial
Character	$\chi$	$=$	1440.127
Dual form			1440.2.x.p.703.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.224745 + 2.22474i) q^{5} +(3.44949 - 3.44949i) q^{7} +O(q^{10})$	$q+(-0.224745 + 2.22474i) q^{5} +(3.44949 - 3.44949i) q^{7} -2.44949i q^{11} +(2.00000 - 2.00000i) q^{13} +(-4.44949 - 4.44949i) q^{17} +2.00000 q^{19} +(-4.44949 - 4.44949i) q^{23} +(-4.89898 - 1.00000i) q^{25} +4.44949i q^{29} -0.898979i q^{31} +(6.89898 + 8.44949i) q^{35} +(-6.89898 - 6.89898i) q^{37} +4.89898 q^{41} +(6.00000 + 6.00000i) q^{43} +(-5.34847 + 5.34847i) q^{47} -16.7980i q^{49} +(6.89898 - 6.89898i) q^{53} +(5.44949 + 0.550510i) q^{55} +3.34847 q^{59} -7.79796 q^{61} +(4.00000 + 4.89898i) q^{65} +(2.89898 - 2.89898i) q^{67} -12.0000i q^{71} +(-1.89898 + 1.89898i) q^{73} +(-8.44949 - 8.44949i) q^{77} +12.8990 q^{79} +(10.4495 + 10.4495i) q^{83} +(10.8990 - 8.89898i) q^{85} +12.0000i q^{89} -13.7980i q^{91} +(-0.449490 + 4.44949i) q^{95} +(3.00000 + 3.00000i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{5} + 4 q^{7}+O(q^{10}) $ 4 * q + 4 * q^5 + 4 * q^7	$ 4 q + 4 q^{5} + 4 q^{7} + 8 q^{13} - 8 q^{17} + 8 q^{19} - 8 q^{23} + 8 q^{35} - 8 q^{37} + 24 q^{43} + 8 q^{47} + 8 q^{53} + 12 q^{55} - 16 q^{59} + 8 q^{61} + 16 q^{65} - 8 q^{67} + 12 q^{73} - 24 q^{77} + 32 q^{79} + 32 q^{83} + 24 q^{85} + 8 q^{95} + 12 q^{97}+O(q^{100}) $ 4 * q + 4 * q^5 + 4 * q^7 + 8 * q^13 - 8 * q^17 + 8 * q^19 - 8 * q^23 + 8 * q^35 - 8 * q^37 + 24 * q^43 + 8 * q^47 + 8 * q^53 + 12 * q^55 - 16 * q^59 + 8 * q^61 + 16 * q^65 - 8 * q^67 + 12 * q^73 - 24 * q^77 + 32 * q^79 + 32 * q^83 + 24 * q^85 + 8 * q^95 + 12 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1440\mathbb{Z}\right)^\times$.

$n$	$577$	$641$	$901$	$991$
$\chi(n)$	$e\left(\frac{1}{4}\right)$	$1$	$1$	$-1$

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
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−0.224745	+	2.22474i	−0.100509	+	0.994936i
$5$	−0.224745	+	2.22474i	−0.100509	+	0.994936i
$6$		0			0
$7$	3.44949	−	3.44949i	1.30378	−	1.30378i	0.377964	−	0.925820i	$-0.376624\pi$
$7$	3.44949	−	3.44949i	1.30378	−	1.30378i	0.925820	−	0.377964i	$-0.123376\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$		−	2.44949i		−	0.738549i	−0.929320	−	0.369274i	$-0.879606\pi$
$11$		−	2.44949i		−	0.738549i	0.929320	−	0.369274i	$-0.120394\pi$
$12$		0			0
$13$	2.00000	−	2.00000i	0.554700	−	0.554700i	−0.373094	−	0.927794i	$-0.621703\pi$
$13$	2.00000	−	2.00000i	0.554700	−	0.554700i	0.927794	+	0.373094i	$0.121703\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−4.44949	−	4.44949i	−1.07916	−	1.07916i	−0.996585	−	0.0825749i	$-0.973686\pi$
$17$	−4.44949	−	4.44949i	−1.07916	−	1.07916i	−0.0825749	−	0.996585i	$-0.526314\pi$
$18$		0			0
$19$	2.00000			0.458831			0.229416	−	0.973329i	$-0.426318\pi$
$19$	2.00000			0.458831			0.229416	+	0.973329i	$0.426318\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−4.44949	−	4.44949i	−0.927783	−	0.927783i	0.0697797	−	0.997562i	$-0.477770\pi$
$23$	−4.44949	−	4.44949i	−0.927783	−	0.927783i	−0.997562	+	0.0697797i	$0.977770\pi$
$24$		0			0
$25$	−4.89898	−	1.00000i	−0.979796	−	0.200000i
$26$		0			0
$27$		0			0
$28$		0			0
$29$			4.44949i			0.826250i	0.910674	+	0.413125i	$0.135563\pi$
$29$			4.44949i			0.826250i	−0.910674	+	0.413125i	$0.864437\pi$
$30$		0			0
$31$		−	0.898979i		−	0.161461i	−0.996736	−	0.0807307i	$-0.974275\pi$
$31$		−	0.898979i		−	0.161461i	0.996736	−	0.0807307i	$-0.0257254\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	6.89898	+	8.44949i	1.16614	+	1.42822i
$36$		0			0
$37$	−6.89898	−	6.89898i	−1.13419	−	1.13419i	−0.989474	−	0.144711i	$-0.953775\pi$
$37$	−6.89898	−	6.89898i	−1.13419	−	1.13419i	−0.144711	−	0.989474i	$-0.546225\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	4.89898			0.765092			0.382546	−	0.923936i	$-0.375047\pi$
$41$	4.89898			0.765092			0.382546	+	0.923936i	$0.375047\pi$
$42$		0			0
$43$	6.00000	+	6.00000i	0.914991	+	0.914991i	0.996660	−	0.0816682i	$-0.0260248\pi$
$43$	6.00000	+	6.00000i	0.914991	+	0.914991i	−0.0816682	+	0.996660i	$0.526025\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−5.34847	+	5.34847i	−0.780154	+	0.780154i	−0.979857	−	0.199702i	$-0.936003\pi$
$47$	−5.34847	+	5.34847i	−0.780154	+	0.780154i	0.199702	+	0.979857i	$0.436003\pi$
$48$		0			0
$49$		−	16.7980i		−	2.39971i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	6.89898	−	6.89898i	0.947648	−	0.947648i	−0.0510485	−	0.998696i	$-0.516256\pi$
$53$	6.89898	−	6.89898i	0.947648	−	0.947648i	0.998696	+	0.0510485i	$0.0162563\pi$
$54$		0			0
$55$	5.44949	+	0.550510i	0.734809	+	0.0742308i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	3.34847			0.435934			0.217967	−	0.975956i	$-0.430058\pi$
$59$	3.34847			0.435934			0.217967	+	0.975956i	$0.430058\pi$
$60$		0			0
$61$	−7.79796			−0.998426			−0.499213	−	0.866479i	$-0.666378\pi$
$61$	−7.79796			−0.998426			−0.499213	+	0.866479i	$0.666378\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	4.00000	+	4.89898i	0.496139	+	0.607644i
$66$		0			0
$67$	2.89898	−	2.89898i	0.354167	−	0.354167i	−0.507491	−	0.861657i	$-0.669427\pi$
$67$	2.89898	−	2.89898i	0.354167	−	0.354167i	0.861657	+	0.507491i	$0.169427\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	12.0000i		−	1.42414i	−0.702109	−	0.712069i	$-0.747758\pi$
$71$		−	12.0000i		−	1.42414i	0.702109	−	0.712069i	$-0.252242\pi$
$72$		0			0
$73$	−1.89898	+	1.89898i	−0.222259	+	0.222259i	−0.809449	−	0.587190i	$-0.800234\pi$
$73$	−1.89898	+	1.89898i	−0.222259	+	0.222259i	0.587190	+	0.809449i	$0.300234\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−8.44949	−	8.44949i	−0.962909	−	0.962909i
$78$		0			0
$79$	12.8990			1.45125			0.725624	−	0.688091i	$-0.241551\pi$
$79$	12.8990			1.45125			0.725624	+	0.688091i	$0.241551\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	10.4495	+	10.4495i	1.14698	+	1.14698i	0.987143	+	0.159837i	$0.0510969\pi$
$83$	10.4495	+	10.4495i	1.14698	+	1.14698i	0.159837	+	0.987143i	$0.448903\pi$
$84$		0			0
$85$	10.8990	−	8.89898i	1.18216	−	0.965230i
$86$		0			0
$87$		0			0
$88$		0			0
$89$			12.0000i			1.27200i	0.771690	+	0.635999i	$0.219412\pi$
$89$			12.0000i			1.27200i	−0.771690	+	0.635999i	$0.780588\pi$
$90$		0			0
$91$		−	13.7980i		−	1.44642i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−0.449490	+	4.44949i	−0.0461167	+	0.456508i
$96$		0			0
$97$	3.00000	+	3.00000i	0.304604	+	0.304604i	0.842812	−	0.538208i	$-0.180899\pi$
$97$	3.00000	+	3.00000i	0.304604	+	0.304604i	−0.538208	+	0.842812i	$0.680899\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−3.55051			−0.353289			−0.176644	−	0.984275i	$-0.556524\pi$
$101$	−3.55051			−0.353289			−0.176644	+	0.984275i	$0.556524\pi$
$102$		0			0
$103$	−5.44949	−	5.44949i	−0.536954	−	0.536954i	0.385679	−	0.922633i	$-0.373967\pi$
$103$	−5.44949	−	5.44949i	−0.536954	−	0.536954i	−0.922633	+	0.385679i	$0.873967\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	7.34847	−	7.34847i	0.710403	−	0.710403i	−0.256216	−	0.966620i	$-0.582476\pi$
$107$	7.34847	−	7.34847i	0.710403	−	0.710403i	0.966620	+	0.256216i	$0.0824759\pi$
$108$		0			0
$109$			7.79796i			0.746909i	0.927648	+	0.373455i	$0.121827\pi$
$109$			7.79796i			0.746909i	−0.927648	+	0.373455i	$0.878173\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	3.55051	−	3.55051i	0.334004	−	0.334004i	−0.520101	−	0.854105i	$-0.674106\pi$
$113$	3.55051	−	3.55051i	0.334004	−	0.334004i	0.854105	+	0.520101i	$0.174106\pi$
$114$		0			0
$115$	10.8990	−	8.89898i	1.01634	−	0.829834i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−30.6969			−2.81398
$120$		0			0
$121$	5.00000			0.454545
$122$		0			0
$123$		0			0
$124$		0			0
$125$	3.32577	−	10.6742i	0.297465	−	0.954733i
$126$		0			0
$127$	2.55051	−	2.55051i	0.226321	−	0.226321i	−0.584833	−	0.811154i	$-0.698840\pi$
$127$	2.55051	−	2.55051i	0.226321	−	0.226321i	0.811154	+	0.584833i	$0.198840\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			1.55051i			0.135469i	0.997703	+	0.0677344i	$0.0215770\pi$
$131$			1.55051i			0.135469i	−0.997703	+	0.0677344i	$0.978423\pi$
$132$		0			0
$133$	6.89898	−	6.89898i	0.598217	−	0.598217i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	0.449490	+	0.449490i	0.0384025	+	0.0384025i	0.726047	−	0.687645i	$-0.241355\pi$
$137$	0.449490	+	0.449490i	0.0384025	+	0.0384025i	−0.687645	+	0.726047i	$0.741355\pi$
$138$		0			0
$139$	3.79796			0.322139			0.161069	−	0.986943i	$-0.448506\pi$
$139$	3.79796			0.322139			0.161069	+	0.986943i	$0.448506\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−4.89898	−	4.89898i	−0.409673	−	0.409673i
$144$		0			0
$145$	−9.89898	−	1.00000i	−0.822066	−	0.0830455i
$146$		0			0
$147$		0			0
$148$		0			0
$149$			15.5505i			1.27395i	0.770886	+	0.636974i	$0.219814\pi$
$149$			15.5505i			1.27395i	−0.770886	+	0.636974i	$0.780186\pi$
$150$		0			0
$151$			15.7980i			1.28562i	0.766026	+	0.642810i	$0.222231\pi$
$151$			15.7980i			1.28562i	−0.766026	+	0.642810i	$0.777769\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	2.00000	+	0.202041i	0.160644	+	0.0162283i
$156$		0			0
$157$	−4.89898	−	4.89898i	−0.390981	−	0.390981i	0.484056	−	0.875037i	$-0.339163\pi$
$157$	−4.89898	−	4.89898i	−0.390981	−	0.390981i	−0.875037	+	0.484056i	$0.839163\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−30.6969			−2.41926
$162$		0			0
$163$	−0.898979	−	0.898979i	−0.0704135	−	0.0704135i	0.671023	−	0.741437i	$-0.265855\pi$
$163$	−0.898979	−	0.898979i	−0.0704135	−	0.0704135i	−0.741437	+	0.671023i	$0.765855\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	16.4495	−	16.4495i	1.27290	−	1.27290i	0.328340	−	0.944560i	$-0.393511\pi$
$167$	16.4495	−	16.4495i	1.27290	−	1.27290i	0.944560	−	0.328340i	$-0.106489\pi$
$168$		0			0
$169$			5.00000i			0.384615i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−8.24745	+	8.24745i	−0.627042	+	0.627042i	−0.947323	−	0.320281i	$-0.896223\pi$
$173$	−8.24745	+	8.24745i	−0.627042	+	0.627042i	0.320281	+	0.947323i	$0.396223\pi$
$174$		0			0
$175$	−20.3485	+	13.4495i	−1.53820	+	1.01669i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	0.247449			0.0184952			0.00924759	−	0.999957i	$-0.497056\pi$
$179$	0.247449			0.0184952			0.00924759	+	0.999957i	$0.497056\pi$
$180$		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
$183$		0			0
$184$		0			0
$185$	16.8990	−	13.7980i	1.24244	−	1.01445i
$186$		0			0
$187$	−10.8990	+	10.8990i	−0.797012	+	0.797012i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	11.1010i		−	0.803242i	−0.915806	−	0.401621i	$-0.868447\pi$
$191$		−	11.1010i		−	0.803242i	0.915806	−	0.401621i	$-0.131553\pi$
$192$		0			0
$193$	−0.101021	+	0.101021i	−0.00727162	+	0.00727162i	−0.710733	−	0.703462i	$-0.751637\pi$
$193$	−0.101021	+	0.101021i	−0.00727162	+	0.00727162i	0.703462	+	0.710733i	$0.251637\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	11.7980	+	11.7980i	0.840570	+	0.840570i	0.988933	−	0.148363i	$-0.0474005\pi$
$197$	11.7980	+	11.7980i	0.840570	+	0.840570i	−0.148363	+	0.988933i	$0.547400\pi$
$198$		0			0
$199$	−3.79796			−0.269230			−0.134615	−	0.990898i	$-0.542980\pi$
$199$	−3.79796			−0.269230			−0.134615	+	0.990898i	$0.542980\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	15.3485	+	15.3485i	1.07725	+	1.07725i
$204$		0			0
$205$	−1.10102	+	10.8990i	−0.0768986	+	0.761218i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		−	4.89898i		−	0.338869i
$210$		0			0
$211$			21.5959i			1.48672i	0.668889	+	0.743362i	$0.266770\pi$
$211$			21.5959i			1.48672i	−0.668889	+	0.743362i	$0.733230\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−14.6969	+	12.0000i	−1.00232	+	0.818393i
$216$		0			0
$217$	−3.10102	−	3.10102i	−0.210511	−	0.210511i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−17.7980			−1.19722
$222$		0			0
$223$	−3.65153	−	3.65153i	−0.244525	−	0.244525i	0.574194	−	0.818719i	$-0.305315\pi$
$223$	−3.65153	−	3.65153i	−0.244525	−	0.244525i	−0.818719	+	0.574194i	$0.805315\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	12.8990	−	12.8990i	0.856135	−	0.856135i	−0.134745	−	0.990880i	$-0.543022\pi$
$227$	12.8990	−	12.8990i	0.856135	−	0.856135i	0.990880	+	0.134745i	$0.0430215\pi$
$228$		0			0
$229$			10.0000i			0.660819i	0.943838	+	0.330409i	$0.107187\pi$
$229$			10.0000i			0.660819i	−0.943838	+	0.330409i	$0.892813\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	5.34847	−	5.34847i	0.350390	−	0.350390i	−0.509865	−	0.860255i	$-0.670305\pi$
$233$	5.34847	−	5.34847i	0.350390	−	0.350390i	0.860255	+	0.509865i	$0.170305\pi$
$234$		0			0
$235$	−10.6969	−	13.1010i	−0.697791	−	0.854616i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−29.7980			−1.92747			−0.963735	−	0.266862i	$-0.914013\pi$
$239$	−29.7980			−1.92747			−0.963735	+	0.266862i	$0.914013\pi$
$240$		0			0
$241$	5.79796			0.373479			0.186740	−	0.982409i	$-0.440208\pi$
$241$	5.79796			0.373479			0.186740	+	0.982409i	$0.440208\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	37.3712	+	3.77526i	2.38756	+	0.241192i
$246$		0			0
$247$	4.00000	−	4.00000i	0.254514	−	0.254514i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			17.1464i			1.08227i	0.840935	+	0.541136i	$0.182006\pi$
$251$			17.1464i			1.08227i	−0.840935	+	0.541136i	$0.817994\pi$
$252$		0			0
$253$	−10.8990	+	10.8990i	−0.685213	+	0.685213i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−6.65153	−	6.65153i	−0.414911	−	0.414911i	0.468534	−	0.883445i	$-0.344782\pi$
$257$	−6.65153	−	6.65153i	−0.414911	−	0.414911i	−0.883445	+	0.468534i	$0.844782\pi$
$258$		0			0
$259$	−47.5959			−2.95747
$260$		0			0
$261$		0			0
$262$		0			0
$263$	18.2474	+	18.2474i	1.12519	+	1.12519i	0.990949	+	0.134237i	$0.0428582\pi$
$263$	18.2474	+	18.2474i	1.12519	+	1.12519i	0.134237	+	0.990949i	$0.457142\pi$
$264$		0			0
$265$	13.7980	+	16.8990i	0.847602	+	1.03810i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	0.449490i		−	0.0274059i	−0.999906	−	0.0137029i	$-0.995638\pi$
$269$		−	0.449490i		−	0.0274059i	0.999906	−	0.0137029i	$-0.00436192\pi$
$270$		0			0
$271$		−	23.7980i		−	1.44562i	−0.691045	−	0.722812i	$-0.742849\pi$
$271$		−	23.7980i		−	1.44562i	0.691045	−	0.722812i	$-0.257151\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−2.44949	+	12.0000i	−0.147710	+	0.723627i
$276$		0			0
$277$	20.6969	+	20.6969i	1.24356	+	1.24356i	0.958514	+	0.285044i	$0.0920083\pi$
$277$	20.6969	+	20.6969i	1.24356	+	1.24356i	0.285044	+	0.958514i	$0.407992\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	10.6969			0.638126			0.319063	−	0.947734i	$-0.396632\pi$
$281$	10.6969			0.638126			0.319063	+	0.947734i	$0.396632\pi$
$282$		0			0
$283$	20.6969	+	20.6969i	1.23031	+	1.23031i	0.963846	+	0.266459i	$0.0858537\pi$
$283$	20.6969	+	20.6969i	1.23031	+	1.23031i	0.266459	+	0.963846i	$0.414146\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	16.8990	−	16.8990i	0.997515	−	0.997515i
$288$		0			0
$289$			22.5959i			1.32917i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	2.89898	−	2.89898i	0.169360	−	0.169360i	−0.617338	−	0.786698i	$-0.711789\pi$
$293$	2.89898	−	2.89898i	0.169360	−	0.169360i	0.786698	+	0.617338i	$0.211789\pi$
$294$		0			0
$295$	−0.752551	+	7.44949i	−0.0438152	+	0.433726i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−17.7980			−1.02928
$300$		0			0
$301$	41.3939			2.38590
$302$		0			0
$303$		0			0
$304$		0			0
$305$	1.75255	−	17.3485i	0.100351	−	0.993370i
$306$		0			0
$307$	−11.7980	+	11.7980i	−0.673345	+	0.673345i	−0.958486	−	0.285141i	$-0.907960\pi$
$307$	−11.7980	+	11.7980i	−0.673345	+	0.673345i	0.285141	+	0.958486i	$0.407960\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			13.7980i			0.782410i	0.920303	+	0.391205i	$0.127942\pi$
$311$			13.7980i			0.782410i	−0.920303	+	0.391205i	$0.872058\pi$
$312$		0			0
$313$	1.89898	−	1.89898i	0.107337	−	0.107337i	−0.651399	−	0.758736i	$-0.725818\pi$
$313$	1.89898	−	1.89898i	0.107337	−	0.107337i	0.758736	+	0.651399i	$0.225818\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−23.3485	−	23.3485i	−1.31138	−	1.31138i	−0.920398	−	0.390983i	$-0.872135\pi$
$317$	−23.3485	−	23.3485i	−1.31138	−	1.31138i	−0.390983	−	0.920398i	$-0.627865\pi$
$318$		0			0
$319$	10.8990			0.610226
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−8.89898	−	8.89898i	−0.495152	−	0.495152i
$324$		0			0
$325$	−11.7980	+	7.79796i	−0.654433	+	0.432553i
$326$		0			0
$327$		0			0
$328$		0			0
$329$			36.8990i			2.03431i
$330$		0			0
$331$		−	18.0000i		−	0.989369i	−0.869072	−	0.494685i	$-0.835284\pi$
$331$		−	18.0000i		−	0.989369i	0.869072	−	0.494685i	$-0.164716\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	5.79796	+	7.10102i	0.316776	+	0.387970i
$336$		0			0
$337$	17.8990	+	17.8990i	0.975020	+	0.975020i	0.999696	−	0.0246760i	$-0.00785540\pi$
$337$	17.8990	+	17.8990i	0.975020	+	0.975020i	−0.0246760	+	0.999696i	$0.507855\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−2.20204			−0.119247
$342$		0			0
$343$	−33.7980	−	33.7980i	−1.82492	−	1.82492i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	12.0000	−	12.0000i	0.644194	−	0.644194i	−0.307390	−	0.951584i	$-0.599456\pi$
$347$	12.0000	−	12.0000i	0.644194	−	0.644194i	0.951584	+	0.307390i	$0.0994556\pi$
$348$		0			0
$349$		−	17.5959i		−	0.941888i	−0.882163	−	0.470944i	$-0.843913\pi$
$349$		−	17.5959i		−	0.941888i	0.882163	−	0.470944i	$-0.156087\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−6.24745	+	6.24745i	−0.332518	+	0.332518i	−0.853542	−	0.521024i	$-0.825550\pi$
$353$	−6.24745	+	6.24745i	−0.332518	+	0.332518i	0.521024	+	0.853542i	$0.325550\pi$
$354$		0			0
$355$	26.6969	+	2.69694i	1.41693	+	0.143139i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	18.6969			0.986787			0.493393	−	0.869806i	$-0.335756\pi$
$359$	18.6969			0.986787			0.493393	+	0.869806i	$0.335756\pi$
$360$		0			0
$361$	−15.0000			−0.789474
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−3.79796	−	4.65153i	−0.198794	−	0.243472i
$366$		0			0
$367$	−6.34847	+	6.34847i	−0.331387	+	0.331387i	−0.853113	−	0.521726i	$-0.825288\pi$
$367$	−6.34847	+	6.34847i	−0.331387	+	0.331387i	0.521726	+	0.853113i	$0.325288\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		−	47.5959i		−	2.47106i
$372$		0			0
$373$	−16.8990	+	16.8990i	−0.874996	+	0.874996i	−0.993012	−	0.118016i	$-0.962347\pi$
$373$	−16.8990	+	16.8990i	−0.874996	+	0.874996i	0.118016	+	0.993012i	$0.462347\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	8.89898	+	8.89898i	0.458321	+	0.458321i
$378$		0			0
$379$	−19.7980			−1.01695			−0.508476	−	0.861076i	$-0.669791\pi$
$379$	−19.7980			−1.01695			−0.508476	+	0.861076i	$0.669791\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−5.34847	−	5.34847i	−0.273294	−	0.273294i	0.557131	−	0.830425i	$-0.311902\pi$
$383$	−5.34847	−	5.34847i	−0.273294	−	0.273294i	−0.830425	+	0.557131i	$0.811902\pi$
$384$		0			0
$385$	20.6969	−	16.8990i	1.05481	−	0.861252i
$386$		0			0
$387$		0			0
$388$		0			0
$389$			0.853572i			0.0432778i	0.999766	+	0.0216389i	$0.00688841\pi$
$389$			0.853572i			0.0432778i	−0.999766	+	0.0216389i	$0.993112\pi$
$390$		0			0
$391$			39.5959i			2.00245i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−2.89898	+	28.6969i	−0.145863	+	1.44390i
$396$		0			0
$397$	11.5959	+	11.5959i	0.581982	+	0.581982i	0.935448	−	0.353465i	$-0.114997\pi$
$397$	11.5959	+	11.5959i	0.581982	+	0.581982i	−0.353465	+	0.935448i	$0.614997\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−15.5959			−0.778823			−0.389411	−	0.921064i	$-0.627322\pi$
$401$	−15.5959			−0.778823			−0.389411	+	0.921064i	$0.627322\pi$
$402$		0			0
$403$	−1.79796	−	1.79796i	−0.0895627	−	0.0895627i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−16.8990	+	16.8990i	−0.837651	+	0.837651i
$408$		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
$411$		0			0
$412$		0			0
$413$	11.5505	−	11.5505i	0.568363	−	0.568363i
$414$		0			0
$415$	−25.5959	+	20.8990i	−1.25645	+	1.02589i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−36.2474			−1.77080			−0.885402	−	0.464826i	$-0.846117\pi$
$419$	−36.2474			−1.77080			−0.885402	+	0.464826i	$0.846117\pi$
$420$		0			0
$421$	−10.0000			−0.487370			−0.243685	−	0.969854i	$-0.578356\pi$
$421$	−10.0000			−0.487370			−0.243685	+	0.969854i	$0.578356\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	17.3485	+	26.2474i	0.841524	+	1.27319i
$426$		0			0
$427$	−26.8990	+	26.8990i	−1.30173	+	1.30173i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	28.4949i		−	1.37255i	−0.727342	−	0.686275i	$-0.759244\pi$
$431$		−	28.4949i		−	1.37255i	0.727342	−	0.686275i	$-0.240756\pi$
$432$		0			0
$433$	−16.7980	+	16.7980i	−0.807258	+	0.807258i	−0.984218	−	0.176960i	$-0.943374\pi$
$433$	−16.7980	+	16.7980i	−0.807258	+	0.807258i	0.176960	+	0.984218i	$0.443374\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−8.89898	−	8.89898i	−0.425696	−	0.425696i
$438$		0			0
$439$	0.202041			0.00964289			0.00482145	−	0.999988i	$-0.498465\pi$
$439$	0.202041			0.00964289			0.00482145	+	0.999988i	$0.498465\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	3.10102	+	3.10102i	0.147334	+	0.147334i	0.776926	−	0.629592i	$-0.216778\pi$
$443$	3.10102	+	3.10102i	0.147334	+	0.147334i	−0.629592	+	0.776926i	$0.716778\pi$
$444$		0			0
$445$	−26.6969	−	2.69694i	−1.26556	−	0.127847i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			28.4949i			1.34476i	0.740207	+	0.672379i	$0.234727\pi$
$449$			28.4949i			1.34476i	−0.740207	+	0.672379i	$0.765273\pi$
$450$		0			0
$451$		−	12.0000i		−	0.565058i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	30.6969	+	3.10102i	1.43909	+	0.145378i
$456$		0			0
$457$	−18.7980	−	18.7980i	−0.879331	−	0.879331i	0.114134	−	0.993465i	$-0.463591\pi$
$457$	−18.7980	−	18.7980i	−0.879331	−	0.879331i	−0.993465	+	0.114134i	$0.963591\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−26.2474			−1.22247			−0.611233	−	0.791451i	$-0.709326\pi$
$461$	−26.2474			−1.22247			−0.611233	+	0.791451i	$0.709326\pi$
$462$		0			0
$463$	1.24745	+	1.24745i	0.0579739	+	0.0579739i	0.735499	−	0.677525i	$-0.236948\pi$
$463$	1.24745	+	1.24745i	0.0579739	+	0.0579739i	−0.677525	+	0.735499i	$0.736948\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−13.1464	+	13.1464i	−0.608344	+	0.608344i	−0.942513	−	0.334169i	$-0.891544\pi$
$467$	−13.1464	+	13.1464i	−0.608344	+	0.608344i	0.334169	+	0.942513i	$0.391544\pi$
$468$		0			0
$469$		−	20.0000i		−	0.923514i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	14.6969	−	14.6969i	0.675766	−	0.675766i
$474$		0			0
$475$	−9.79796	−	2.00000i	−0.449561	−	0.0917663i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	17.3939			0.794747			0.397373	−	0.917657i	$-0.369922\pi$
$479$	17.3939			0.794747			0.397373	+	0.917657i	$0.369922\pi$
$480$		0			0
$481$	−27.5959			−1.25827
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−7.34847	+	6.00000i	−0.333677	+	0.272446i
$486$		0			0
$487$	18.5505	−	18.5505i	0.840604	−	0.840604i	−0.148334	−	0.988937i	$-0.547391\pi$
$487$	18.5505	−	18.5505i	0.840604	−	0.840604i	0.988937	+	0.148334i	$0.0473909\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	27.3485i		−	1.23422i	−0.786877	−	0.617110i	$-0.788304\pi$
$491$		−	27.3485i		−	1.23422i	0.786877	−	0.617110i	$-0.211696\pi$
$492$		0			0
$493$	19.7980	−	19.7980i	0.891655	−	0.891655i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−41.3939	−	41.3939i	−1.85677	−	1.85677i
$498$		0			0
$499$	9.59592			0.429572			0.214786	−	0.976661i	$-0.431095\pi$
$499$	9.59592			0.429572			0.214786	+	0.976661i	$0.431095\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	1.75255	+	1.75255i	0.0781424	+	0.0781424i	0.745098	−	0.666955i	$-0.232403\pi$
$503$	1.75255	+	1.75255i	0.0781424	+	0.0781424i	−0.666955	+	0.745098i	$0.732403\pi$
$504$		0			0
$505$	0.797959	−	7.89898i	0.0355087	−	0.351500i
$506$		0			0
$507$		0			0
$508$		0			0
$509$			28.0454i			1.24309i	0.783378	+	0.621545i	$0.213495\pi$
$509$			28.0454i			1.24309i	−0.783378	+	0.621545i	$0.786505\pi$
$510$		0			0
$511$			13.1010i			0.579555i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	13.3485	−	10.8990i	0.588204	−	0.480266i
$516$		0			0
$517$	13.1010	+	13.1010i	0.576182	+	0.576182i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	12.4949			0.547411			0.273706	−	0.961813i	$-0.411751\pi$
$521$	12.4949			0.547411			0.273706	+	0.961813i	$0.411751\pi$
$522$		0			0
$523$	17.1010	+	17.1010i	0.747775	+	0.747775i	0.974061	−	0.226286i	$-0.0726583\pi$
$523$	17.1010	+	17.1010i	0.747775	+	0.747775i	−0.226286	+	0.974061i	$0.572658\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−4.00000	+	4.00000i	−0.174243	+	0.174243i
$528$		0			0
$529$			16.5959i			0.721562i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	9.79796	−	9.79796i	0.424397	−	0.424397i
$534$		0			0
$535$	14.6969	+	18.0000i	0.635404	+	0.778208i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−41.1464			−1.77230
$540$		0			0
$541$	−3.79796			−0.163287			−0.0816435	−	0.996662i	$-0.526017\pi$
$541$	−3.79796			−0.163287			−0.0816435	+	0.996662i	$0.526017\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−17.3485	−	1.75255i	−0.743127	−	0.0750710i
$546$		0			0
$547$	13.5959	−	13.5959i	0.581319	−	0.581319i	−0.353946	−	0.935266i	$-0.615160\pi$
$547$	13.5959	−	13.5959i	0.581319	−	0.581319i	0.935266	+	0.353946i	$0.115160\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$			8.89898i			0.379109i
$552$		0			0
$553$	44.4949	−	44.4949i	1.89212	−	1.89212i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−10.8990	−	10.8990i	−0.461805	−	0.461805i	0.437442	−	0.899247i	$-0.355885\pi$
$557$	−10.8990	−	10.8990i	−0.461805	−	0.461805i	−0.899247	+	0.437442i	$0.855885\pi$
$558$		0			0
$559$	24.0000			1.01509
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−21.7980	−	21.7980i	−0.918674	−	0.918674i	0.0782586	−	0.996933i	$-0.475064\pi$
$563$	−21.7980	−	21.7980i	−0.918674	−	0.918674i	−0.996933	+	0.0782586i	$0.975064\pi$
$564$		0			0
$565$	7.10102	+	8.69694i	0.298742	+	0.365883i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	25.3939i		−	1.06457i	−0.846566	−	0.532283i	$-0.821334\pi$
$569$		−	25.3939i		−	1.06457i	0.846566	−	0.532283i	$-0.178666\pi$
$570$		0			0
$571$		−	33.5959i		−	1.40595i	−0.711217	−	0.702973i	$-0.751856\pi$
$571$		−	33.5959i		−	1.40595i	0.711217	−	0.702973i	$-0.248144\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	17.3485	+	26.2474i	0.723481	+	1.09459i
$576$		0			0
$577$	−17.6969	−	17.6969i	−0.736733	−	0.736733i	0.235211	−	0.971944i	$-0.424422\pi$
$577$	−17.6969	−	17.6969i	−0.736733	−	0.736733i	−0.971944	+	0.235211i	$0.924422\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	72.0908			2.99083
$582$		0			0
$583$	−16.8990	−	16.8990i	−0.699884	−	0.699884i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−9.14643	+	9.14643i	−0.377513	+	0.377513i	−0.870204	−	0.492691i	$-0.836013\pi$
$587$	−9.14643	+	9.14643i	−0.377513	+	0.377513i	0.492691	+	0.870204i	$0.336013\pi$
$588$		0			0
$589$		−	1.79796i		−	0.0740836i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	24.4495	−	24.4495i	1.00402	−	1.00402i	0.00402832	−	0.999992i	$-0.498718\pi$
$593$	24.4495	−	24.4495i	1.00402	−	1.00402i	0.999992	−	0.00402832i	$-0.00128226\pi$
$594$		0			0
$595$	6.89898	−	68.2929i	0.282831	−	2.79973i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	20.4949			0.837399			0.418699	−	0.908125i	$-0.362486\pi$
$599$	20.4949			0.837399			0.418699	+	0.908125i	$0.362486\pi$
$600$		0			0
$601$	34.0000			1.38689			0.693444	−	0.720510i	$-0.256092\pi$
$601$	34.0000			1.38689			0.693444	+	0.720510i	$0.256092\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−1.12372	+	11.1237i	−0.0456859	+	0.452244i
$606$		0			0
$607$	11.4495	−	11.4495i	0.464720	−	0.464720i	−0.435479	−	0.900199i	$-0.643421\pi$
$607$	11.4495	−	11.4495i	0.464720	−	0.464720i	0.900199	+	0.435479i	$0.143421\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			21.3939i			0.865504i
$612$		0			0
$613$	−19.5959	+	19.5959i	−0.791472	+	0.791472i	−0.981733	−	0.190262i	$-0.939066\pi$
$613$	−19.5959	+	19.5959i	−0.791472	+	0.791472i	0.190262	+	0.981733i	$0.439066\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	5.34847	+	5.34847i	0.215321	+	0.215321i	0.806523	−	0.591202i	$-0.201346\pi$
$617$	5.34847	+	5.34847i	0.215321	+	0.215321i	−0.591202	+	0.806523i	$0.701346\pi$
$618$		0			0
$619$	27.3939			1.10105			0.550526	−	0.834818i	$-0.314427\pi$
$619$	27.3939			1.10105			0.550526	+	0.834818i	$0.314427\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	41.3939	+	41.3939i	1.65841	+	1.65841i
$624$		0			0
$625$	23.0000	+	9.79796i	0.920000	+	0.391918i
$626$		0			0
$627$		0			0
$628$		0			0
$629$			61.3939i			2.44793i
$630$		0			0
$631$			3.10102i			0.123450i	0.998093	+	0.0617248i	$0.0196601\pi$
$631$			3.10102i			0.123450i	−0.998093	+	0.0617248i	$0.980340\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	5.10102	+	6.24745i	0.202428	+	0.247922i
$636$		0			0
$637$	−33.5959	−	33.5959i	−1.33112	−	1.33112i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−14.2020			−0.560947			−0.280473	−	0.959862i	$-0.590491\pi$
$641$	−14.2020			−0.560947			−0.280473	+	0.959862i	$0.590491\pi$
$642$		0			0
$643$	32.8990	+	32.8990i	1.29741	+	1.29741i	0.930098	+	0.367311i	$0.119721\pi$
$643$	32.8990	+	32.8990i	1.29741	+	1.29741i	0.367311	+	0.930098i	$0.380279\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	14.2474	−	14.2474i	0.560125	−	0.560125i	−0.369218	−	0.929343i	$-0.620374\pi$
$647$	14.2474	−	14.2474i	0.560125	−	0.560125i	0.929343	+	0.369218i	$0.120374\pi$
$648$		0			0
$649$		−	8.20204i		−	0.321958i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	6.44949	−	6.44949i	0.252388	−	0.252388i	−0.569561	−	0.821949i	$-0.692887\pi$
$653$	6.44949	−	6.44949i	0.252388	−	0.252388i	0.821949	+	0.569561i	$0.192887\pi$
$654$		0			0
$655$	−3.44949	−	0.348469i	−0.134783	−	0.0136158i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	15.7526			0.613632			0.306816	−	0.951769i	$-0.400736\pi$
$659$	15.7526			0.613632			0.306816	+	0.951769i	$0.400736\pi$
$660$		0			0
$661$	27.7980			1.08122			0.540608	−	0.841275i	$-0.318194\pi$
$661$	27.7980			1.08122			0.540608	+	0.841275i	$0.318194\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	13.7980	+	16.8990i	0.535062	+	0.655314i
$666$		0			0
$667$	19.7980	−	19.7980i	0.766580	−	0.766580i
$668$		0			0
$669$		0			0
$670$		0			0
$671$			19.1010i			0.737387i
$672$		0			0
$673$	−10.7980	+	10.7980i	−0.416231	+	0.416231i	−0.883902	−	0.467672i	$-0.845093\pi$
$673$	−10.7980	+	10.7980i	−0.416231	+	0.416231i	0.467672	+	0.883902i	$0.345093\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	19.3485	+	19.3485i	0.743622	+	0.743622i	0.973273	−	0.229651i	$-0.0737585\pi$
$677$	19.3485	+	19.3485i	0.743622	+	0.743622i	−0.229651	+	0.973273i	$0.573758\pi$
$678$		0			0
$679$	20.6969			0.794276
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−27.3485	−	27.3485i	−1.04646	−	1.04646i	−0.998867	−	0.0475933i	$-0.984845\pi$
$683$	−27.3485	−	27.3485i	−1.04646	−	1.04646i	−0.0475933	−	0.998867i	$-0.515155\pi$
$684$		0			0
$685$	−1.10102	+	0.898979i	−0.0420678	+	0.0343482i
$686$		0			0
$687$		0			0
$688$		0			0
$689$		−	27.5959i		−	1.05132i
$690$		0			0
$691$			11.7980i			0.448816i	0.974495	+	0.224408i	$0.0720447\pi$
$691$			11.7980i			0.448816i	−0.974495	+	0.224408i	$0.927955\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−0.853572	+	8.44949i	−0.0323778	+	0.320507i
$696$		0			0
$697$	−21.7980	−	21.7980i	−0.825657	−	0.825657i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−46.7423			−1.76543			−0.882717	−	0.469905i	$-0.844288\pi$
$701$	−46.7423			−1.76543			−0.882717	+	0.469905i	$0.844288\pi$
$702$		0			0
$703$	−13.7980	−	13.7980i	−0.520400	−	0.520400i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−12.2474	+	12.2474i	−0.460613	+	0.460613i
$708$		0			0
$709$		−	27.7980i		−	1.04397i	−0.852953	−	0.521987i	$-0.825191\pi$
$709$		−	27.7980i		−	1.04397i	0.852953	−	0.521987i	$-0.174809\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−4.00000	+	4.00000i	−0.149801	+	0.149801i
$714$		0			0
$715$	12.0000	−	9.79796i	0.448775	−	0.366423i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	6.20204			0.231297			0.115649	−	0.993290i	$-0.463105\pi$
$719$	6.20204			0.231297			0.115649	+	0.993290i	$0.463105\pi$
$720$		0			0
$721$	−37.5959			−1.40015
$722$		0			0
$723$		0			0
$724$		0			0
$725$	4.44949	−	21.7980i	0.165250	−	0.809556i
$726$		0			0
$727$	26.1464	−	26.1464i	0.969717	−	0.969717i	−0.0298374	−	0.999555i	$-0.509499\pi$
$727$	26.1464	−	26.1464i	0.969717	−	0.969717i	0.999555	+	0.0298374i	$0.00949896\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		−	53.3939i		−	1.97484i
$732$		0			0
$733$	22.8990	−	22.8990i	0.845793	−	0.845793i	−0.143812	−	0.989605i	$-0.545936\pi$
$733$	22.8990	−	22.8990i	0.845793	−	0.845793i	0.989605	+	0.143812i	$0.0459360\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−7.10102	−	7.10102i	−0.261569	−	0.261569i
$738$		0			0
$739$	−31.3939			−1.15484			−0.577421	−	0.816446i	$-0.695941\pi$
$739$	−31.3939			−1.15484			−0.577421	+	0.816446i	$0.695941\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−3.55051	−	3.55051i	−0.130256	−	0.130256i	0.638973	−	0.769229i	$-0.279359\pi$
$743$	−3.55051	−	3.55051i	−0.130256	−	0.130256i	−0.769229	+	0.638973i	$0.779359\pi$
$744$		0			0
$745$	−34.5959	−	3.49490i	−1.26750	−	0.128043i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		−	50.6969i		−	1.85243i
$750$		0			0
$751$			13.3031i			0.485436i	0.970097	+	0.242718i	$0.0780389\pi$
$751$			13.3031i			0.485436i	−0.970097	+	0.242718i	$0.921961\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−35.1464	−	3.55051i	−1.27911	−	0.129216i
$756$		0			0
$757$	−12.6969	−	12.6969i	−0.461478	−	0.461478i	0.437662	−	0.899140i	$-0.355807\pi$
$757$	−12.6969	−	12.6969i	−0.461478	−	0.461478i	−0.899140	+	0.437662i	$0.855807\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	5.30306			0.192236			0.0961179	−	0.995370i	$-0.469357\pi$
$761$	5.30306			0.192236			0.0961179	+	0.995370i	$0.469357\pi$
$762$		0			0
$763$	26.8990	+	26.8990i	0.973808	+	0.973808i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	6.69694	−	6.69694i	0.241812	−	0.241812i
$768$		0			0
$769$		−	53.1918i		−	1.91815i	−0.283158	−	0.959073i	$-0.591382\pi$
$769$		−	53.1918i		−	1.91815i	0.283158	−	0.959073i	$-0.408618\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−8.65153	+	8.65153i	−0.311174	+	0.311174i	−0.845364	−	0.534190i	$-0.820617\pi$
$773$	−8.65153	+	8.65153i	−0.311174	+	0.311174i	0.534190	+	0.845364i	$0.320617\pi$
$774$		0			0
$775$	−0.898979	+	4.40408i	−0.0322923	+	0.158199i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	9.79796			0.351048
$780$		0			0
$781$	−29.3939			−1.05180
$782$		0			0
$783$		0			0
$784$		0			0
$785$	12.0000	−	9.79796i	0.428298	−	0.349704i
$786$		0			0
$787$	−34.8990	+	34.8990i	−1.24401	+	1.24401i	−0.285693	+	0.958321i	$0.592224\pi$
$787$	−34.8990	+	34.8990i	−1.24401	+	1.24401i	−0.958321	+	0.285693i	$0.907776\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	24.4949i		−	0.870938i
$792$		0			0
$793$	−15.5959	+	15.5959i	−0.553827	+	0.553827i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−5.55051	−	5.55051i	−0.196609	−	0.196609i	0.601936	−	0.798545i	$-0.294396\pi$
$797$	−5.55051	−	5.55051i	−0.196609	−	0.196609i	−0.798545	+	0.601936i	$0.794396\pi$
$798$		0			0
$799$	47.5959			1.68382
$800$		0			0
$801$		0			0
$802$		0			0
$803$	4.65153	+	4.65153i	0.164149	+	0.164149i
$804$		0			0
$805$	6.89898	−	68.2929i	0.243157	−	2.40701i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			9.79796i			0.344478i	0.985055	+	0.172239i	$0.0551001\pi$
$809$			9.79796i			0.344478i	−0.985055	+	0.172239i	$0.944900\pi$
$810$		0			0
$811$		−	3.79796i		−	0.133364i	−0.997774	−	0.0666822i	$-0.978759\pi$
$811$		−	3.79796i		−	0.133364i	0.997774	−	0.0666822i	$-0.0212414\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	2.20204	−	1.79796i	0.0771341	−	0.0629798i
$816$		0			0
$817$	12.0000	+	12.0000i	0.419827	+	0.419827i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−12.4495			−0.434490			−0.217245	−	0.976117i	$-0.569707\pi$
$821$	−12.4495			−0.434490			−0.217245	+	0.976117i	$0.569707\pi$
$822$		0			0
$823$	4.34847	+	4.34847i	0.151578	+	0.151578i	0.778822	−	0.627244i	$-0.215817\pi$
$823$	4.34847	+	4.34847i	0.151578	+	0.151578i	−0.627244	+	0.778822i	$0.715817\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−8.00000	+	8.00000i	−0.278187	+	0.278187i	−0.832385	−	0.554198i	$-0.813025\pi$
$827$	−8.00000	+	8.00000i	−0.278187	+	0.278187i	0.554198	+	0.832385i	$0.313025\pi$
$828$		0			0
$829$			15.7980i			0.548686i	0.961632	+	0.274343i	$0.0884603\pi$
$829$			15.7980i			0.548686i	−0.961632	+	0.274343i	$0.911540\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−74.7423	+	74.7423i	−2.58967	+	2.58967i
$834$		0			0
$835$	32.8990	+	40.2929i	1.13852	+	1.39439i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−26.6969			−0.921681			−0.460840	−	0.887483i	$-0.652452\pi$
$839$	−26.6969			−0.921681			−0.460840	+	0.887483i	$0.652452\pi$
$840$		0			0
$841$	9.20204			0.317312
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−11.1237	−	1.12372i	−0.382668	−	0.0386573i
$846$		0			0
$847$	17.2474	−	17.2474i	0.592629	−	0.592629i
$848$		0			0
$849$		0			0
$850$		0			0
$851$			61.3939i			2.10455i
$852$		0			0
$853$	−2.20204	+	2.20204i	−0.0753964	+	0.0753964i	−0.743799	−	0.668403i	$-0.766978\pi$
$853$	−2.20204	+	2.20204i	−0.0753964	+	0.0753964i	0.668403	+	0.743799i	$0.266978\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	28.4495	+	28.4495i	0.971816	+	0.971816i	0.999614	−	0.0277975i	$-0.00884936\pi$
$857$	28.4495	+	28.4495i	0.971816	+	0.971816i	−0.0277975	+	0.999614i	$0.508849\pi$
$858$		0			0
$859$	−23.7980			−0.811976			−0.405988	−	0.913878i	$-0.633073\pi$
$859$	−23.7980			−0.811976			−0.405988	+	0.913878i	$0.633073\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	6.65153	+	6.65153i	0.226421	+	0.226421i	0.811196	−	0.584775i	$-0.198817\pi$
$863$	6.65153	+	6.65153i	0.226421	+	0.226421i	−0.584775	+	0.811196i	$0.698817\pi$
$864$		0			0
$865$	−16.4949	−	20.2020i	−0.560843	−	0.686890i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	31.5959i		−	1.07182i
$870$		0			0
$871$		−	11.5959i		−	0.392913i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−25.3485	−	48.2929i	−0.856935	−	1.63260i
$876$		0			0
$877$	−16.8990	−	16.8990i	−0.570638	−	0.570638i	0.361669	−	0.932307i	$-0.382207\pi$
$877$	−16.8990	−	16.8990i	−0.570638	−	0.570638i	−0.932307	+	0.361669i	$0.882207\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	34.2020			1.15230			0.576148	−	0.817345i	$-0.304555\pi$
$881$	34.2020			1.15230			0.576148	+	0.817345i	$0.304555\pi$
$882$		0			0
$883$	−6.69694	−	6.69694i	−0.225370	−	0.225370i	0.585385	−	0.810755i	$-0.300943\pi$
$883$	−6.69694	−	6.69694i	−0.225370	−	0.225370i	−0.810755	+	0.585385i	$0.800943\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	31.1464	−	31.1464i	1.04579	−	1.04579i	0.0468949	−	0.998900i	$-0.485067\pi$
$887$	31.1464	−	31.1464i	1.04579	−	1.04579i	0.998900	−	0.0468949i	$-0.0149326\pi$
$888$		0			0
$889$		−	17.5959i		−	0.590148i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−10.6969	+	10.6969i	−0.357959	+	0.357959i
$894$		0			0
$895$	−0.0556128	+	0.550510i	−0.00185893	+	0.0184015i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	4.00000			0.133407
$900$		0			0
$901$	−61.3939			−2.04533
$902$		0			0
$903$		0			0
$904$		0			0
$905$	2.24745	−	22.2474i	0.0747077	−	0.739530i
$906$		0			0
$907$	−9.79796	+	9.79796i	−0.325336	+	0.325336i	−0.850810	−	0.525474i	$-0.823888\pi$
$907$	−9.79796	+	9.79796i	−0.325336	+	0.325336i	0.525474	+	0.850810i	$0.323888\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			48.0908i			1.59332i	0.604428	+	0.796660i	$0.293402\pi$
$911$			48.0908i			1.59332i	−0.604428	+	0.796660i	$0.706598\pi$
$912$		0			0
$913$	25.5959	−	25.5959i	0.847101	−	0.847101i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	5.34847	+	5.34847i	0.176622	+	0.176622i
$918$		0			0
$919$	31.1010			1.02593			0.512964	−	0.858410i	$-0.328547\pi$
$919$	31.1010			1.02593			0.512964	+	0.858410i	$0.328547\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−24.0000	−	24.0000i	−0.789970	−	0.789970i
$924$		0			0
$925$	26.8990	+	40.6969i	0.884433	+	1.33811i
$926$		0			0
$927$		0			0
$928$		0			0
$929$			32.0908i			1.05287i	0.850217	+	0.526433i	$0.176471\pi$
$929$			32.0908i			1.05287i	−0.850217	+	0.526433i	$0.823529\pi$
$930$		0			0
$931$		−	33.5959i		−	1.10106i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−21.7980	−	26.6969i	−0.712869	−	0.873083i
$936$		0			0
$937$	16.5959	+	16.5959i	0.542165	+	0.542165i	0.924163	−	0.381998i	$-0.124764\pi$
$937$	16.5959	+	16.5959i	0.542165	+	0.542165i	−0.381998	+	0.924163i	$0.624764\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−7.14643			−0.232967			−0.116483	−	0.993193i	$-0.537162\pi$
$941$	−7.14643			−0.232967			−0.116483	+	0.993193i	$0.537162\pi$
$942$		0			0
$943$	−21.7980	−	21.7980i	−0.709839	−	0.709839i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	28.4949	−	28.4949i	0.925960	−	0.925960i	−0.0714821	−	0.997442i	$-0.522773\pi$
$947$	28.4949	−	28.4949i	0.925960	−	0.925960i	0.997442	+	0.0714821i	$0.0227729\pi$
$948$		0			0
$949$			7.59592i			0.246574i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	13.3485	−	13.3485i	0.432399	−	0.432399i	−0.457044	−	0.889444i	$-0.651092\pi$
$953$	13.3485	−	13.3485i	0.432399	−	0.432399i	0.889444	+	0.457044i	$0.151092\pi$
$954$		0			0
$955$	24.6969	+	2.49490i	0.799174	+	0.0807330i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	3.10102			0.100137
$960$		0			0
$961$	30.1918			0.973930
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−0.202041	−	0.247449i	−0.00650393	−	0.00796566i
$966$		0			0
$967$	8.34847	−	8.34847i	0.268469	−	0.268469i	−0.560014	−	0.828483i	$-0.689204\pi$
$967$	8.34847	−	8.34847i	0.268469	−	0.268469i	0.828483	+	0.560014i	$0.189204\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			38.0454i			1.22094i	0.792041	+	0.610468i	$0.209018\pi$
$971$			38.0454i			1.22094i	−0.792041	+	0.610468i	$0.790982\pi$
$972$		0			0
$973$	13.1010	−	13.1010i	0.419999	−	0.419999i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	15.5505	+	15.5505i	0.497505	+	0.497505i	0.910660	−	0.413156i	$-0.135573\pi$
$977$	15.5505	+	15.5505i	0.497505	+	0.497505i	−0.413156	+	0.910660i	$0.635573\pi$
$978$		0			0
$979$	29.3939			0.939432
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−1.75255	−	1.75255i	−0.0558977	−	0.0558977i	0.678605	−	0.734503i	$-0.262585\pi$
$983$	−1.75255	−	1.75255i	−0.0558977	−	0.0558977i	−0.734503	+	0.678605i	$0.762585\pi$
$984$		0			0
$985$	−28.8990	+	23.5959i	−0.920798	+	0.751828i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	53.3939i		−	1.69783i
$990$		0			0
$991$		−	43.7980i		−	1.39129i	−0.718387	−	0.695644i	$-0.755119\pi$
$991$		−	43.7980i		−	1.39129i	0.718387	−	0.695644i	$-0.244881\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	0.853572	−	8.44949i	0.0270600	−	0.267867i
$996$		0			0
$997$	39.7980	+	39.7980i	1.26041	+	1.26041i	0.950892	+	0.309522i	$0.100169\pi$
$997$	39.7980	+	39.7980i	1.26041	+	1.26041i	0.309522	+	0.950892i	$0.399831\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1440.2.x.p.127.1	yes	4
3.2	odd	2		1440.2.x.l.127.2	yes	4
4.3	odd	2		1440.2.x.o.127.1	yes	4
5.3	odd	4		1440.2.x.o.703.1	yes	4
12.11	even	2		1440.2.x.k.127.2	✓	4
15.8	even	4		1440.2.x.k.703.2	yes	4
20.3	even	4	inner	1440.2.x.p.703.1	yes	4
60.23	odd	4		1440.2.x.l.703.2	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1440.2.x.k.127.2	✓	4	12.11	even	2
1440.2.x.k.703.2	yes	4	15.8	even	4
1440.2.x.l.127.2	yes	4	3.2	odd	2
1440.2.x.l.703.2	yes	4	60.23	odd	4
1440.2.x.o.127.1	yes	4	4.3	odd	2
1440.2.x.o.703.1	yes	4	5.3	odd	4
1440.2.x.p.127.1	yes	4	1.1	even	1	trivial
1440.2.x.p.703.1	yes	4	20.3	even	4	inner

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:	\( N \)	\(=\)	\( 1440 = 2^{5} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1440.x (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.224745 + 2.22474i) q^{5} +(3.44949 - 3.44949i) q^{7} +O(q^{10})\)	\(q+(-0.224745 + 2.22474i) q^{5} +(3.44949 - 3.44949i) q^{7} -2.44949i q^{11} +(2.00000 - 2.00000i) q^{13} +(-4.44949 - 4.44949i) q^{17} +2.00000 q^{19} +(-4.44949 - 4.44949i) q^{23} +(-4.89898 - 1.00000i) q^{25} +4.44949i q^{29} -0.898979i q^{31} +(6.89898 + 8.44949i) q^{35} +(-6.89898 - 6.89898i) q^{37} +4.89898 q^{41} +(6.00000 + 6.00000i) q^{43} +(-5.34847 + 5.34847i) q^{47} -16.7980i q^{49} +(6.89898 - 6.89898i) q^{53} +(5.44949 + 0.550510i) q^{55} +3.34847 q^{59} -7.79796 q^{61} +(4.00000 + 4.89898i) q^{65} +(2.89898 - 2.89898i) q^{67} -12.0000i q^{71} +(-1.89898 + 1.89898i) q^{73} +(-8.44949 - 8.44949i) q^{77} +12.8990 q^{79} +(10.4495 + 10.4495i) q^{83} +(10.8990 - 8.89898i) q^{85} +12.0000i q^{89} -13.7980i q^{91} +(-0.449490 + 4.44949i) q^{95} +(3.00000 + 3.00000i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{5} + 4 q^{7}+O(q^{10}) \) 4 * q + 4 * q^5 + 4 * q^7	\( 4 q + 4 q^{5} + 4 q^{7} + 8 q^{13} - 8 q^{17} + 8 q^{19} - 8 q^{23} + 8 q^{35} - 8 q^{37} + 24 q^{43} + 8 q^{47} + 8 q^{53} + 12 q^{55} - 16 q^{59} + 8 q^{61} + 16 q^{65} - 8 q^{67} + 12 q^{73} - 24 q^{77} + 32 q^{79} + 32 q^{83} + 24 q^{85} + 8 q^{95} + 12 q^{97}+O(q^{100}) \) 4 * q + 4 * q^5 + 4 * q^7 + 8 * q^13 - 8 * q^17 + 8 * q^19 - 8 * q^23 + 8 * q^35 - 8 * q^37 + 24 * q^43 + 8 * q^47 + 8 * q^53 + 12 * q^55 - 16 * q^59 + 8 * q^61 + 16 * q^65 - 8 * q^67 + 12 * q^73 - 24 * q^77 + 32 * q^79 + 32 * q^83 + 24 * q^85 + 8 * q^95 + 12 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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