LMFDB - Embedded newform 880.2.bf.b.287.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [880,2,Mod(287,880)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(880, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 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("880.287");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 880 = 2^{4} \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	880.bf (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:	$7.02683537787$
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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			287.1
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	880.287
Dual form			880.2.bf.b.463.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.00000 - 2.00000i) q^{5} +(-3.00000 + 3.00000i) q^{7} -3.00000i q^{9} +O(q^{10})$	$q+(-1.00000 - 2.00000i) q^{5} +(-3.00000 + 3.00000i) q^{7} -3.00000i q^{9} +1.00000i q^{11} +(-2.00000 + 2.00000i) q^{13} +(4.00000 + 4.00000i) q^{17} +6.00000 q^{19} +(6.00000 + 6.00000i) q^{23} +(-3.00000 + 4.00000i) q^{25} +2.00000i q^{29} +(9.00000 + 3.00000i) q^{35} +(-1.00000 - 1.00000i) q^{37} +2.00000 q^{41} +(3.00000 + 3.00000i) q^{43} +(-6.00000 + 3.00000i) q^{45} +(-6.00000 + 6.00000i) q^{47} -11.0000i q^{49} +(5.00000 - 5.00000i) q^{53} +(2.00000 - 1.00000i) q^{55} -12.0000 q^{59} +6.00000 q^{61} +(9.00000 + 9.00000i) q^{63} +(6.00000 + 2.00000i) q^{65} +(6.00000 - 6.00000i) q^{67} +12.0000i q^{71} +(-4.00000 + 4.00000i) q^{73} +(-3.00000 - 3.00000i) q^{77} -6.00000 q^{79} -9.00000 q^{81} +(3.00000 + 3.00000i) q^{83} +(4.00000 - 12.0000i) q^{85} +10.0000i q^{89} -12.0000i q^{91} +(-6.00000 - 12.0000i) q^{95} +(-7.00000 - 7.00000i) q^{97} +3.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{5} - 6 q^{7}+O(q^{10}) $ 2 * q - 2 * q^5 - 6 * q^7	$ 2 q - 2 q^{5} - 6 q^{7} - 4 q^{13} + 8 q^{17} + 12 q^{19} + 12 q^{23} - 6 q^{25} + 18 q^{35} - 2 q^{37} + 4 q^{41} + 6 q^{43} - 12 q^{45} - 12 q^{47} + 10 q^{53} + 4 q^{55} - 24 q^{59} + 12 q^{61} + 18 q^{63} + 12 q^{65} + 12 q^{67} - 8 q^{73} - 6 q^{77} - 12 q^{79} - 18 q^{81} + 6 q^{83} + 8 q^{85} - 12 q^{95} - 14 q^{97} + 6 q^{99}+O(q^{100}) $ 2 * q - 2 * q^5 - 6 * q^7 - 4 * q^13 + 8 * q^17 + 12 * q^19 + 12 * q^23 - 6 * q^25 + 18 * q^35 - 2 * q^37 + 4 * q^41 + 6 * q^43 - 12 * q^45 - 12 * q^47 + 10 * q^53 + 4 * q^55 - 24 * q^59 + 12 * q^61 + 18 * q^63 + 12 * q^65 + 12 * q^67 - 8 * q^73 - 6 * q^77 - 12 * q^79 - 18 * q^81 + 6 * q^83 + 8 * q^85 - 12 * q^95 - 14 * q^97 + 6 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/880\mathbb{Z}\right)^\times$.

$n$	$111$	$177$	$321$	$661$
$\chi(n)$	$-1$	$e\left(\frac{1}{4}\right)$	$1$	$1$

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
$3$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$3$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$4$		0			0
$5$	−1.00000	−	2.00000i	−0.447214	−	0.894427i
$5$	−1.00000	−	2.00000i	−0.447214	−	0.894427i
$6$		0			0
$7$	−3.00000	+	3.00000i	−1.13389	+	1.13389i	−0.144370	+	0.989524i	$0.546115\pi$
$7$	−3.00000	+	3.00000i	−1.13389	+	1.13389i	−0.989524	+	0.144370i	$0.953885\pi$
$8$		0			0
$9$		−	3.00000i		−	1.00000i
$10$		0			0
$11$			1.00000i			0.301511i
$11$			1.00000i			0.301511i
$12$		0			0
$13$	−2.00000	+	2.00000i	−0.554700	+	0.554700i	−0.927794	−	0.373094i	$-0.878297\pi$
$13$	−2.00000	+	2.00000i	−0.554700	+	0.554700i	0.373094	+	0.927794i	$0.378297\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	4.00000	+	4.00000i	0.970143	+	0.970143i	0.999567	−	0.0294245i	$-0.00936746\pi$
$17$	4.00000	+	4.00000i	0.970143	+	0.970143i	−0.0294245	+	0.999567i	$0.509367\pi$
$18$		0			0
$19$	6.00000			1.37649			0.688247	−	0.725476i	$-0.258380\pi$
$19$	6.00000			1.37649			0.688247	+	0.725476i	$0.258380\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	6.00000	+	6.00000i	1.25109	+	1.25109i	0.955233	+	0.295853i	$0.0956039\pi$
$23$	6.00000	+	6.00000i	1.25109	+	1.25109i	0.295853	+	0.955233i	$0.404396\pi$
$24$		0			0
$25$	−3.00000	+	4.00000i	−0.600000	+	0.800000i
$26$		0			0
$27$		0			0
$28$		0			0
$29$			2.00000i			0.371391i	0.982607	+	0.185695i	$0.0594537\pi$
$29$			2.00000i			0.371391i	−0.982607	+	0.185695i	$0.940546\pi$
$30$		0			0
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	9.00000	+	3.00000i	1.52128	+	0.507093i
$36$		0			0
$37$	−1.00000	−	1.00000i	−0.164399	−	0.164399i	0.620113	−	0.784512i	$-0.287087\pi$
$37$	−1.00000	−	1.00000i	−0.164399	−	0.164399i	−0.784512	+	0.620113i	$0.787087\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	2.00000			0.312348			0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000			0.312348			0.156174	+	0.987730i	$0.450084\pi$
$42$		0			0
$43$	3.00000	+	3.00000i	0.457496	+	0.457496i	0.897833	−	0.440337i	$-0.145141\pi$
$43$	3.00000	+	3.00000i	0.457496	+	0.457496i	−0.440337	+	0.897833i	$0.645141\pi$
$44$		0			0
$45$	−6.00000	+	3.00000i	−0.894427	+	0.447214i
$46$		0			0
$47$	−6.00000	+	6.00000i	−0.875190	+	0.875190i	−0.993032	−	0.117842i	$-0.962402\pi$
$47$	−6.00000	+	6.00000i	−0.875190	+	0.875190i	0.117842	+	0.993032i	$0.462402\pi$
$48$		0			0
$49$		−	11.0000i		−	1.57143i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	5.00000	−	5.00000i	0.686803	−	0.686803i	−0.274721	−	0.961524i	$-0.588586\pi$
$53$	5.00000	−	5.00000i	0.686803	−	0.686803i	0.961524	+	0.274721i	$0.0885855\pi$
$54$		0			0
$55$	2.00000	−	1.00000i	0.269680	−	0.134840i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−12.0000			−1.56227			−0.781133	−	0.624364i	$-0.785358\pi$
$59$	−12.0000			−1.56227			−0.781133	+	0.624364i	$0.785358\pi$
$60$		0			0
$61$	6.00000			0.768221			0.384111	−	0.923287i	$-0.374508\pi$
$61$	6.00000			0.768221			0.384111	+	0.923287i	$0.374508\pi$
$62$		0			0
$63$	9.00000	+	9.00000i	1.13389	+	1.13389i
$64$		0			0
$65$	6.00000	+	2.00000i	0.744208	+	0.248069i
$66$		0			0
$67$	6.00000	−	6.00000i	0.733017	−	0.733017i	−0.238200	−	0.971216i	$-0.576557\pi$
$67$	6.00000	−	6.00000i	0.733017	−	0.733017i	0.971216	+	0.238200i	$0.0765572\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			12.0000i			1.42414i	0.702109	+	0.712069i	$0.252242\pi$
$71$			12.0000i			1.42414i	−0.702109	+	0.712069i	$0.747758\pi$
$72$		0			0
$73$	−4.00000	+	4.00000i	−0.468165	+	0.468165i	−0.901319	−	0.433155i	$-0.857400\pi$
$73$	−4.00000	+	4.00000i	−0.468165	+	0.468165i	0.433155	+	0.901319i	$0.357400\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−3.00000	−	3.00000i	−0.341882	−	0.341882i
$78$		0			0
$79$	−6.00000			−0.675053			−0.337526	−	0.941316i	$-0.609590\pi$
$79$	−6.00000			−0.675053			−0.337526	+	0.941316i	$0.609590\pi$
$80$		0			0
$81$	−9.00000			−1.00000
$82$		0			0
$83$	3.00000	+	3.00000i	0.329293	+	0.329293i	0.852318	−	0.523025i	$-0.175196\pi$
$83$	3.00000	+	3.00000i	0.329293	+	0.329293i	−0.523025	+	0.852318i	$0.675196\pi$
$84$		0			0
$85$	4.00000	−	12.0000i	0.433861	−	1.30158i
$86$		0			0
$87$		0			0
$88$		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
$91$		−	12.0000i		−	1.25794i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−6.00000	−	12.0000i	−0.615587	−	1.23117i
$96$		0			0
$97$	−7.00000	−	7.00000i	−0.710742	−	0.710742i	0.255948	−	0.966691i	$-0.417612\pi$
$97$	−7.00000	−	7.00000i	−0.710742	−	0.710742i	−0.966691	+	0.255948i	$0.917612\pi$
$98$		0			0
$99$	3.00000			0.301511
$100$		0			0
$101$	2.00000			0.199007			0.0995037	−	0.995037i	$-0.468274\pi$
$101$	2.00000			0.199007			0.0995037	+	0.995037i	$0.468274\pi$
$102$		0			0
$103$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$103$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−3.00000	+	3.00000i	−0.290021	+	0.290021i	−0.837088	−	0.547068i	$-0.815744\pi$
$107$	−3.00000	+	3.00000i	−0.290021	+	0.290021i	0.547068	+	0.837088i	$0.315744\pi$
$108$		0			0
$109$			18.0000i			1.72409i	0.506834	+	0.862044i	$0.330816\pi$
$109$			18.0000i			1.72409i	−0.506834	+	0.862044i	$0.669184\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	3.00000	−	3.00000i	0.282216	−	0.282216i	−0.551776	−	0.833992i	$-0.686050\pi$
$113$	3.00000	−	3.00000i	0.282216	−	0.282216i	0.833992	+	0.551776i	$0.186050\pi$
$114$		0			0
$115$	6.00000	−	18.0000i	0.559503	−	1.67851i
$116$		0			0
$117$	6.00000	+	6.00000i	0.554700	+	0.554700i
$118$		0			0
$119$	−24.0000			−2.20008
$120$		0			0
$121$	−1.00000			−0.0909091
$122$		0			0
$123$		0			0
$124$		0			0
$125$	11.0000	+	2.00000i	0.983870	+	0.178885i
$126$		0			0
$127$	−9.00000	+	9.00000i	−0.798621	+	0.798621i	−0.982878	−	0.184257i	$-0.941012\pi$
$127$	−9.00000	+	9.00000i	−0.798621	+	0.798621i	0.184257	+	0.982878i	$0.441012\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			6.00000i			0.524222i	0.965038	+	0.262111i	$0.0844187\pi$
$131$			6.00000i			0.524222i	−0.965038	+	0.262111i	$0.915581\pi$
$132$		0			0
$133$	−18.0000	+	18.0000i	−1.56080	+	1.56080i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	3.00000	+	3.00000i	0.256307	+	0.256307i	0.823550	−	0.567243i	$-0.191990\pi$
$137$	3.00000	+	3.00000i	0.256307	+	0.256307i	−0.567243	+	0.823550i	$0.691990\pi$
$138$		0			0
$139$	12.0000			1.01783			0.508913	−	0.860818i	$-0.330047\pi$
$139$	12.0000			1.01783			0.508913	+	0.860818i	$0.330047\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−2.00000	−	2.00000i	−0.167248	−	0.167248i
$144$		0			0
$145$	4.00000	−	2.00000i	0.332182	−	0.166091i
$146$		0			0
$147$		0			0
$148$		0			0
$149$		−	10.0000i		−	0.819232i	−0.912258	−	0.409616i	$-0.865663\pi$
$149$		−	10.0000i		−	0.819232i	0.912258	−	0.409616i	$-0.134337\pi$
$150$		0			0
$151$		−	18.0000i		−	1.46482i	−0.680864	−	0.732410i	$-0.738396\pi$
$151$		−	18.0000i		−	1.46482i	0.680864	−	0.732410i	$-0.261604\pi$
$152$		0			0
$153$	12.0000	−	12.0000i	0.970143	−	0.970143i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−5.00000	−	5.00000i	−0.399043	−	0.399043i	0.478852	−	0.877896i	$-0.341053\pi$
$157$	−5.00000	−	5.00000i	−0.399043	−	0.399043i	−0.877896	+	0.478852i	$0.841053\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−36.0000			−2.83720
$162$		0			0
$163$	12.0000	+	12.0000i	0.939913	+	0.939913i	0.998294	−	0.0583818i	$-0.0185941\pi$
$163$	12.0000	+	12.0000i	0.939913	+	0.939913i	−0.0583818	+	0.998294i	$0.518594\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−3.00000	+	3.00000i	−0.232147	+	0.232147i	−0.813588	−	0.581441i	$-0.802489\pi$
$167$	−3.00000	+	3.00000i	−0.232147	+	0.232147i	0.581441	+	0.813588i	$0.302489\pi$
$168$		0			0
$169$			5.00000i			0.384615i
$170$		0			0
$171$		−	18.0000i		−	1.37649i
$172$		0			0
$173$	12.0000	−	12.0000i	0.912343	−	0.912343i	−0.0841131	−	0.996456i	$-0.526806\pi$
$173$	12.0000	−	12.0000i	0.912343	−	0.912343i	0.996456	+	0.0841131i	$0.0268057\pi$
$174$		0			0
$175$	−3.00000	−	21.0000i	−0.226779	−	1.58745i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$		0			0
$181$	24.0000			1.78391			0.891953	−	0.452128i	$-0.149335\pi$
$181$	24.0000			1.78391			0.891953	+	0.452128i	$0.149335\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−1.00000	+	3.00000i	−0.0735215	+	0.220564i
$186$		0			0
$187$	−4.00000	+	4.00000i	−0.292509	+	0.292509i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	12.0000i		−	0.868290i	−0.900843	−	0.434145i	$-0.857051\pi$
$191$		−	12.0000i		−	0.868290i	0.900843	−	0.434145i	$-0.142949\pi$
$192$		0			0
$193$	16.0000	−	16.0000i	1.15171	−	1.15171i	0.165494	−	0.986211i	$-0.447078\pi$
$193$	16.0000	−	16.0000i	1.15171	−	1.15171i	0.986211	−	0.165494i	$-0.0529220\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−12.0000	−	12.0000i	−0.854965	−	0.854965i	0.135775	−	0.990740i	$-0.456648\pi$
$197$	−12.0000	−	12.0000i	−0.854965	−	0.854965i	−0.990740	+	0.135775i	$0.956648\pi$
$198$		0			0
$199$	−24.0000			−1.70131			−0.850657	−	0.525720i	$-0.823796\pi$
$199$	−24.0000			−1.70131			−0.850657	+	0.525720i	$0.823796\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−6.00000	−	6.00000i	−0.421117	−	0.421117i
$204$		0			0
$205$	−2.00000	−	4.00000i	−0.139686	−	0.279372i
$206$		0			0
$207$	18.0000	−	18.0000i	1.25109	−	1.25109i
$208$		0			0
$209$			6.00000i			0.415029i
$210$		0			0
$211$		−	18.0000i		−	1.23917i	−0.784929	−	0.619586i	$-0.787301\pi$
$211$		−	18.0000i		−	1.23917i	0.784929	−	0.619586i	$-0.212699\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	3.00000	−	9.00000i	0.204598	−	0.613795i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−16.0000			−1.07628
$222$		0			0
$223$	6.00000	+	6.00000i	0.401790	+	0.401790i	0.878863	−	0.477074i	$-0.158302\pi$
$223$	6.00000	+	6.00000i	0.401790	+	0.401790i	−0.477074	+	0.878863i	$0.658302\pi$
$224$		0			0
$225$	12.0000	+	9.00000i	0.800000	+	0.600000i
$226$		0			0
$227$	−15.0000	+	15.0000i	−0.995585	+	0.995585i	−0.999990	−	0.00440533i	$-0.998598\pi$
$227$	−15.0000	+	15.0000i	−0.995585	+	0.995585i	0.00440533	+	0.999990i	$0.498598\pi$
$228$		0			0
$229$		−	4.00000i		−	0.264327i	−0.991228	−	0.132164i	$-0.957808\pi$
$229$		−	4.00000i		−	0.264327i	0.991228	−	0.132164i	$-0.0421925\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−14.0000	+	14.0000i	−0.917170	+	0.917170i	−0.996823	−	0.0796522i	$-0.974619\pi$
$233$	−14.0000	+	14.0000i	−0.917170	+	0.917170i	0.0796522	+	0.996823i	$0.474619\pi$
$234$		0			0
$235$	18.0000	+	6.00000i	1.17419	+	0.391397i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−30.0000			−1.94054			−0.970269	−	0.242028i	$-0.922188\pi$
$239$	−30.0000			−1.94054			−0.970269	+	0.242028i	$0.922188\pi$
$240$		0			0
$241$	26.0000			1.67481			0.837404	−	0.546585i	$-0.184072\pi$
$241$	26.0000			1.67481			0.837404	+	0.546585i	$0.184072\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−22.0000	+	11.0000i	−1.40553	+	0.702764i
$246$		0			0
$247$	−12.0000	+	12.0000i	−0.763542	+	0.763542i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	24.0000i		−	1.51487i	−0.652913	−	0.757433i	$-0.726453\pi$
$251$		−	24.0000i		−	1.51487i	0.652913	−	0.757433i	$-0.273547\pi$
$252$		0			0
$253$	−6.00000	+	6.00000i	−0.377217	+	0.377217i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	9.00000	+	9.00000i	0.561405	+	0.561405i	0.929706	−	0.368302i	$-0.120061\pi$
$257$	9.00000	+	9.00000i	0.561405	+	0.561405i	−0.368302	+	0.929706i	$0.620061\pi$
$258$		0			0
$259$	6.00000			0.372822
$260$		0			0
$261$	6.00000			0.371391
$262$		0			0
$263$	−3.00000	−	3.00000i	−0.184988	−	0.184988i	0.608537	−	0.793525i	$-0.291757\pi$
$263$	−3.00000	−	3.00000i	−0.184988	−	0.184988i	−0.793525	+	0.608537i	$0.791757\pi$
$264$		0			0
$265$	−15.0000	−	5.00000i	−0.921443	−	0.307148i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	14.0000i		−	0.853595i	−0.904347	−	0.426798i	$-0.859642\pi$
$269$		−	14.0000i		−	0.853595i	0.904347	−	0.426798i	$-0.140358\pi$
$270$		0			0
$271$			18.0000i			1.09342i	0.837321	+	0.546711i	$0.184120\pi$
$271$			18.0000i			1.09342i	−0.837321	+	0.546711i	$0.815880\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−4.00000	−	3.00000i	−0.241209	−	0.180907i
$276$		0			0
$277$	10.0000	+	10.0000i	0.600842	+	0.600842i	0.940536	−	0.339694i	$-0.110324\pi$
$277$	10.0000	+	10.0000i	0.600842	+	0.600842i	−0.339694	+	0.940536i	$0.610324\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	22.0000			1.31241			0.656205	−	0.754583i	$-0.272161\pi$
$281$	22.0000			1.31241			0.656205	+	0.754583i	$0.272161\pi$
$282$		0			0
$283$	−15.0000	−	15.0000i	−0.891657	−	0.891657i	0.103022	−	0.994679i	$-0.467149\pi$
$283$	−15.0000	−	15.0000i	−0.891657	−	0.891657i	−0.994679	+	0.103022i	$0.967149\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−6.00000	+	6.00000i	−0.354169	+	0.354169i
$288$		0			0
$289$			15.0000i			0.882353i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−18.0000	+	18.0000i	−1.05157	+	1.05157i	−0.0529754	+	0.998596i	$0.516870\pi$
$293$	−18.0000	+	18.0000i	−1.05157	+	1.05157i	−0.998596	+	0.0529754i	$0.983130\pi$
$294$		0			0
$295$	12.0000	+	24.0000i	0.698667	+	1.39733i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−24.0000			−1.38796
$300$		0			0
$301$	−18.0000			−1.03750
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−6.00000	−	12.0000i	−0.343559	−	0.687118i
$306$		0			0
$307$	−15.0000	+	15.0000i	−0.856095	+	0.856095i	−0.990876	−	0.134780i	$-0.956967\pi$
$307$	−15.0000	+	15.0000i	−0.856095	+	0.856095i	0.134780	+	0.990876i	$0.456967\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			12.0000i			0.680458i	0.940343	+	0.340229i	$0.110505\pi$
$311$			12.0000i			0.680458i	−0.940343	+	0.340229i	$0.889495\pi$
$312$		0			0
$313$	−1.00000	+	1.00000i	−0.0565233	+	0.0565233i	−0.734803	−	0.678280i	$-0.762726\pi$
$313$	−1.00000	+	1.00000i	−0.0565233	+	0.0565233i	0.678280	+	0.734803i	$0.262726\pi$
$314$		0			0
$315$	9.00000	−	27.0000i	0.507093	−	1.52128i
$316$		0			0
$317$	11.0000	+	11.0000i	0.617822	+	0.617822i	0.944972	−	0.327151i	$-0.106088\pi$
$317$	11.0000	+	11.0000i	0.617822	+	0.617822i	−0.327151	+	0.944972i	$0.606088\pi$
$318$		0			0
$319$	−2.00000			−0.111979
$320$		0			0
$321$		0			0
$322$		0			0
$323$	24.0000	+	24.0000i	1.33540	+	1.33540i
$324$		0			0
$325$	−2.00000	−	14.0000i	−0.110940	−	0.776580i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		−	36.0000i		−	1.98474i
$330$		0			0
$331$		−	24.0000i		−	1.31916i	−0.751635	−	0.659580i	$-0.770734\pi$
$331$		−	24.0000i		−	1.31916i	0.751635	−	0.659580i	$-0.229266\pi$
$332$		0			0
$333$	−3.00000	+	3.00000i	−0.164399	+	0.164399i
$334$		0			0
$335$	−18.0000	−	6.00000i	−0.983445	−	0.327815i
$336$		0			0
$337$	4.00000	+	4.00000i	0.217894	+	0.217894i	0.807610	−	0.589716i	$-0.200761\pi$
$337$	4.00000	+	4.00000i	0.217894	+	0.217894i	−0.589716	+	0.807610i	$0.700761\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	12.0000	+	12.0000i	0.647939	+	0.647939i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	3.00000	−	3.00000i	0.161048	−	0.161048i	−0.621983	−	0.783031i	$-0.713673\pi$
$347$	3.00000	−	3.00000i	0.161048	−	0.161048i	0.783031	+	0.621983i	$0.213673\pi$
$348$		0			0
$349$			6.00000i			0.321173i	0.987022	+	0.160586i	$0.0513385\pi$
$349$			6.00000i			0.321173i	−0.987022	+	0.160586i	$0.948662\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−11.0000	+	11.0000i	−0.585471	+	0.585471i	−0.936401	−	0.350931i	$-0.885865\pi$
$353$	−11.0000	+	11.0000i	−0.585471	+	0.585471i	0.350931	+	0.936401i	$0.385865\pi$
$354$		0			0
$355$	24.0000	−	12.0000i	1.27379	−	0.636894i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	30.0000			1.58334			0.791670	−	0.610949i	$-0.209212\pi$
$359$	30.0000			1.58334			0.791670	+	0.610949i	$0.209212\pi$
$360$		0			0
$361$	17.0000			0.894737
$362$		0			0
$363$		0			0
$364$		0			0
$365$	12.0000	+	4.00000i	0.628109	+	0.209370i
$366$		0			0
$367$	18.0000	−	18.0000i	0.939592	−	0.939592i	−0.0586842	−	0.998277i	$-0.518691\pi$
$367$	18.0000	−	18.0000i	0.939592	−	0.939592i	0.998277	+	0.0586842i	$0.0186905\pi$
$368$		0			0
$369$		−	6.00000i		−	0.312348i
$370$		0			0
$371$			30.0000i			1.55752i
$372$		0			0
$373$	16.0000	−	16.0000i	0.828449	−	0.828449i	−0.158854	−	0.987302i	$-0.550780\pi$
$373$	16.0000	−	16.0000i	0.828449	−	0.828449i	0.987302	+	0.158854i	$0.0507798\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−4.00000	−	4.00000i	−0.206010	−	0.206010i
$378$		0			0
$379$	36.0000			1.84920			0.924598	−	0.380945i	$-0.124401\pi$
$379$	36.0000			1.84920			0.924598	+	0.380945i	$0.124401\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	12.0000	+	12.0000i	0.613171	+	0.613171i	0.943771	−	0.330600i	$-0.107251\pi$
$383$	12.0000	+	12.0000i	0.613171	+	0.613171i	−0.330600	+	0.943771i	$0.607251\pi$
$384$		0			0
$385$	−3.00000	+	9.00000i	−0.152894	+	0.458682i
$386$		0			0
$387$	9.00000	−	9.00000i	0.457496	−	0.457496i
$388$		0			0
$389$		−	26.0000i		−	1.31825i	−0.752032	−	0.659126i	$-0.770926\pi$
$389$		−	26.0000i		−	1.31825i	0.752032	−	0.659126i	$-0.229074\pi$
$390$		0			0
$391$			48.0000i			2.42746i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	6.00000	+	12.0000i	0.301893	+	0.603786i
$396$		0			0
$397$	−17.0000	−	17.0000i	−0.853206	−	0.853206i	0.137321	−	0.990527i	$-0.456151\pi$
$397$	−17.0000	−	17.0000i	−0.853206	−	0.853206i	−0.990527	+	0.137321i	$0.956151\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	8.00000			0.399501			0.199750	−	0.979847i	$-0.435987\pi$
$401$	8.00000			0.399501			0.199750	+	0.979847i	$0.435987\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	9.00000	+	18.0000i	0.447214	+	0.894427i
$406$		0			0
$407$	1.00000	−	1.00000i	0.0495682	−	0.0495682i
$408$		0			0
$409$		−	30.0000i		−	1.48340i	−0.670729	−	0.741702i	$-0.734019\pi$
$409$		−	30.0000i		−	1.48340i	0.670729	−	0.741702i	$-0.265981\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	36.0000	−	36.0000i	1.77144	−	1.77144i
$414$		0			0
$415$	3.00000	−	9.00000i	0.147264	−	0.441793i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−12.0000			−0.586238			−0.293119	−	0.956076i	$-0.594693\pi$
$419$	−12.0000			−0.586238			−0.293119	+	0.956076i	$0.594693\pi$
$420$		0			0
$421$	−26.0000			−1.26716			−0.633581	−	0.773676i	$-0.718416\pi$
$421$	−26.0000			−1.26716			−0.633581	+	0.773676i	$0.718416\pi$
$422$		0			0
$423$	18.0000	+	18.0000i	0.875190	+	0.875190i
$424$		0			0
$425$	−28.0000	+	4.00000i	−1.35820	+	0.194029i
$426$		0			0
$427$	−18.0000	+	18.0000i	−0.871081	+	0.871081i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		1.00000			$0$
$431$		0			0		−1.00000			$\pi$
$432$		0			0
$433$	−25.0000	+	25.0000i	−1.20142	+	1.20142i	−0.227690	+	0.973734i	$0.573117\pi$
$433$	−25.0000	+	25.0000i	−1.20142	+	1.20142i	−0.973734	+	0.227690i	$0.926883\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	36.0000	+	36.0000i	1.72211	+	1.72211i
$438$		0			0
$439$	24.0000			1.14546			0.572729	−	0.819745i	$-0.305885\pi$
$439$	24.0000			1.14546			0.572729	+	0.819745i	$0.305885\pi$
$440$		0			0
$441$	−33.0000			−1.57143
$442$		0			0
$443$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$443$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$444$		0			0
$445$	20.0000	−	10.0000i	0.948091	−	0.474045i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			20.0000i			0.943858i	0.881636	+	0.471929i	$0.156442\pi$
$449$			20.0000i			0.943858i	−0.881636	+	0.471929i	$0.843558\pi$
$450$		0			0
$451$			2.00000i			0.0941763i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−24.0000	+	12.0000i	−1.12514	+	0.562569i
$456$		0			0
$457$	14.0000	+	14.0000i	0.654892	+	0.654892i	0.954167	−	0.299275i	$-0.0967447\pi$
$457$	14.0000	+	14.0000i	0.654892	+	0.654892i	−0.299275	+	0.954167i	$0.596745\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	10.0000			0.465746			0.232873	−	0.972507i	$-0.425187\pi$
$461$	10.0000			0.465746			0.232873	+	0.972507i	$0.425187\pi$
$462$		0			0
$463$	6.00000	+	6.00000i	0.278844	+	0.278844i	0.832647	−	0.553804i	$-0.186824\pi$
$463$	6.00000	+	6.00000i	0.278844	+	0.278844i	−0.553804	+	0.832647i	$0.686824\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−18.0000	+	18.0000i	−0.832941	+	0.832941i	−0.987918	−	0.154977i	$-0.950470\pi$
$467$	−18.0000	+	18.0000i	−0.832941	+	0.832941i	0.154977	+	0.987918i	$0.450470\pi$
$468$		0			0
$469$			36.0000i			1.66233i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−3.00000	+	3.00000i	−0.137940	+	0.137940i
$474$		0			0
$475$	−18.0000	+	24.0000i	−0.825897	+	1.10120i
$476$		0			0
$477$	−15.0000	−	15.0000i	−0.686803	−	0.686803i
$478$		0			0
$479$	−18.0000			−0.822441			−0.411220	−	0.911536i	$-0.634897\pi$
$479$	−18.0000			−0.822441			−0.411220	+	0.911536i	$0.634897\pi$
$480$		0			0
$481$	4.00000			0.182384
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−7.00000	+	21.0000i	−0.317854	+	0.953561i
$486$		0			0
$487$	18.0000	−	18.0000i	0.815658	−	0.815658i	−0.169818	−	0.985476i	$-0.554318\pi$
$487$	18.0000	−	18.0000i	0.815658	−	0.815658i	0.985476	+	0.169818i	$0.0543179\pi$
$488$		0			0
$489$		0			0
$490$		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
$495$	−3.00000	−	6.00000i	−0.134840	−	0.269680i
$496$		0			0
$497$	−36.0000	−	36.0000i	−1.61482	−	1.61482i
$498$		0			0
$499$	−12.0000			−0.537194			−0.268597	−	0.963253i	$-0.586560\pi$
$499$	−12.0000			−0.537194			−0.268597	+	0.963253i	$0.586560\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	9.00000	+	9.00000i	0.401290	+	0.401290i	0.878688	−	0.477397i	$-0.158420\pi$
$503$	9.00000	+	9.00000i	0.401290	+	0.401290i	−0.477397	+	0.878688i	$0.658420\pi$
$504$		0			0
$505$	−2.00000	−	4.00000i	−0.0889988	−	0.177998i
$506$		0			0
$507$		0			0
$508$		0			0
$509$			28.0000i			1.24108i	0.784176	+	0.620539i	$0.213086\pi$
$509$			28.0000i			1.24108i	−0.784176	+	0.620539i	$0.786914\pi$
$510$		0			0
$511$		−	24.0000i		−	1.06170i
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−6.00000	−	6.00000i	−0.263880	−	0.263880i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	40.0000			1.75243			0.876216	−	0.481919i	$-0.160060\pi$
$521$	40.0000			1.75243			0.876216	+	0.481919i	$0.160060\pi$
$522$		0			0
$523$	3.00000	+	3.00000i	0.131181	+	0.131181i	0.769649	−	0.638468i	$-0.220431\pi$
$523$	3.00000	+	3.00000i	0.131181	+	0.131181i	−0.638468	+	0.769649i	$0.720431\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$			49.0000i			2.13043i
$530$		0			0
$531$			36.0000i			1.56227i
$532$		0			0
$533$	−4.00000	+	4.00000i	−0.173259	+	0.173259i
$534$		0			0
$535$	9.00000	+	3.00000i	0.389104	+	0.129701i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	11.0000			0.473804
$540$		0			0
$541$	6.00000			0.257960			0.128980	−	0.991647i	$-0.458830\pi$
$541$	6.00000			0.257960			0.128980	+	0.991647i	$0.458830\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	36.0000	−	18.0000i	1.54207	−	0.771035i
$546$		0			0
$547$	3.00000	−	3.00000i	0.128271	−	0.128271i	−0.640057	−	0.768328i	$-0.721089\pi$
$547$	3.00000	−	3.00000i	0.128271	−	0.128271i	0.768328	+	0.640057i	$0.221089\pi$
$548$		0			0
$549$		−	18.0000i		−	0.768221i
$550$		0			0
$551$			12.0000i			0.511217i
$552$		0			0
$553$	18.0000	−	18.0000i	0.765438	−	0.765438i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−4.00000	−	4.00000i	−0.169485	−	0.169485i	0.617268	−	0.786753i	$-0.288240\pi$
$557$	−4.00000	−	4.00000i	−0.169485	−	0.169485i	−0.786753	+	0.617268i	$0.788240\pi$
$558$		0			0
$559$	−12.0000			−0.507546
$560$		0			0
$561$		0			0
$562$		0			0
$563$	15.0000	+	15.0000i	0.632175	+	0.632175i	0.948613	−	0.316438i	$-0.102487\pi$
$563$	15.0000	+	15.0000i	0.632175	+	0.632175i	−0.316438	+	0.948613i	$0.602487\pi$
$564$		0			0
$565$	−9.00000	−	3.00000i	−0.378633	−	0.126211i
$566$		0			0
$567$	27.0000	−	27.0000i	1.13389	−	1.13389i
$568$		0			0
$569$			10.0000i			0.419222i	0.977785	+	0.209611i	$0.0672197\pi$
$569$			10.0000i			0.419222i	−0.977785	+	0.209611i	$0.932780\pi$
$570$		0			0
$571$		−	12.0000i		−	0.502184i	−0.967963	−	0.251092i	$-0.919210\pi$
$571$		−	12.0000i		−	0.502184i	0.967963	−	0.251092i	$-0.0807897\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−42.0000	+	6.00000i	−1.75152	+	0.250217i
$576$		0			0
$577$	−31.0000	−	31.0000i	−1.29055	−	1.29055i	−0.934448	−	0.356098i	$-0.884107\pi$
$577$	−31.0000	−	31.0000i	−1.29055	−	1.29055i	−0.356098	−	0.934448i	$-0.615893\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−18.0000			−0.746766
$582$		0			0
$583$	5.00000	+	5.00000i	0.207079	+	0.207079i
$584$		0			0
$585$	6.00000	−	18.0000i	0.248069	−	0.744208i
$586$		0			0
$587$	−18.0000	+	18.0000i	−0.742940	+	0.742940i	−0.973143	−	0.230203i	$-0.926061\pi$
$587$	−18.0000	+	18.0000i	−0.742940	+	0.742940i	0.230203	+	0.973143i	$0.426061\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$593$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$594$		0			0
$595$	24.0000	+	48.0000i	0.983904	+	1.96781i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	24.0000			0.980613			0.490307	−	0.871550i	$-0.336885\pi$
$599$	24.0000			0.980613			0.490307	+	0.871550i	$0.336885\pi$
$600$		0			0
$601$	−30.0000			−1.22373			−0.611863	−	0.790964i	$-0.709580\pi$
$601$	−30.0000			−1.22373			−0.611863	+	0.790964i	$0.709580\pi$
$602$		0			0
$603$	−18.0000	−	18.0000i	−0.733017	−	0.733017i
$604$		0			0
$605$	1.00000	+	2.00000i	0.0406558	+	0.0813116i
$606$		0			0
$607$	15.0000	−	15.0000i	0.608831	−	0.608831i	−0.333809	−	0.942641i	$-0.608334\pi$
$607$	15.0000	−	15.0000i	0.608831	−	0.608831i	0.942641	+	0.333809i	$0.108334\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	24.0000i		−	0.970936i
$612$		0			0
$613$	32.0000	−	32.0000i	1.29247	−	1.29247i	0.359212	−	0.933256i	$-0.383046\pi$
$613$	32.0000	−	32.0000i	1.29247	−	1.29247i	0.933256	−	0.359212i	$-0.116954\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−1.00000	−	1.00000i	−0.0402585	−	0.0402585i	0.686691	−	0.726949i	$-0.259063\pi$
$617$	−1.00000	−	1.00000i	−0.0402585	−	0.0402585i	−0.726949	+	0.686691i	$0.759063\pi$
$618$		0			0
$619$	−24.0000			−0.964641			−0.482321	−	0.875995i	$-0.660206\pi$
$619$	−24.0000			−0.964641			−0.482321	+	0.875995i	$0.660206\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−30.0000	−	30.0000i	−1.20192	−	1.20192i
$624$		0			0
$625$	−7.00000	−	24.0000i	−0.280000	−	0.960000i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	8.00000i		−	0.318981i
$630$		0			0
$631$		−	12.0000i		−	0.477712i	−0.971055	−	0.238856i	$-0.923228\pi$
$631$		−	12.0000i		−	0.477712i	0.971055	−	0.238856i	$-0.0767725\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	27.0000	+	9.00000i	1.07146	+	0.357154i
$636$		0			0
$637$	22.0000	+	22.0000i	0.871672	+	0.871672i
$638$		0			0
$639$	36.0000			1.42414
$640$		0			0
$641$	2.00000			0.0789953			0.0394976	−	0.999220i	$-0.487424\pi$
$641$	2.00000			0.0789953			0.0394976	+	0.999220i	$0.487424\pi$
$642$		0			0
$643$	−24.0000	−	24.0000i	−0.946468	−	0.946468i	0.0521706	−	0.998638i	$-0.483386\pi$
$643$	−24.0000	−	24.0000i	−0.946468	−	0.946468i	−0.998638	+	0.0521706i	$0.983386\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	12.0000	−	12.0000i	0.471769	−	0.471769i	−0.430718	−	0.902487i	$-0.641740\pi$
$647$	12.0000	−	12.0000i	0.471769	−	0.471769i	0.902487	+	0.430718i	$0.141740\pi$
$648$		0			0
$649$		−	12.0000i		−	0.471041i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	1.00000	−	1.00000i	0.0391330	−	0.0391330i	−0.687270	−	0.726403i	$-0.741191\pi$
$653$	1.00000	−	1.00000i	0.0391330	−	0.0391330i	0.726403	+	0.687270i	$0.241191\pi$
$654$		0			0
$655$	12.0000	−	6.00000i	0.468879	−	0.234439i
$656$		0			0
$657$	12.0000	+	12.0000i	0.468165	+	0.468165i
$658$		0			0
$659$	−12.0000			−0.467454			−0.233727	−	0.972302i	$-0.575092\pi$
$659$	−12.0000			−0.467454			−0.233727	+	0.972302i	$0.575092\pi$
$660$		0			0
$661$	−24.0000			−0.933492			−0.466746	−	0.884391i	$-0.654574\pi$
$661$	−24.0000			−0.933492			−0.466746	+	0.884391i	$0.654574\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	54.0000	+	18.0000i	2.09403	+	0.698010i
$666$		0			0
$667$	−12.0000	+	12.0000i	−0.464642	+	0.464642i
$668$		0			0
$669$		0			0
$670$		0			0
$671$			6.00000i			0.231627i
$672$		0			0
$673$	−8.00000	+	8.00000i	−0.308377	+	0.308377i	−0.844280	−	0.535903i	$-0.819971\pi$
$673$	−8.00000	+	8.00000i	−0.308377	+	0.308377i	0.535903	+	0.844280i	$0.319971\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−2.00000	−	2.00000i	−0.0768662	−	0.0768662i	0.667628	−	0.744495i	$-0.267310\pi$
$677$	−2.00000	−	2.00000i	−0.0768662	−	0.0768662i	−0.744495	+	0.667628i	$0.767310\pi$
$678$		0			0
$679$	42.0000			1.61181
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−18.0000	−	18.0000i	−0.688751	−	0.688751i	0.273205	−	0.961956i	$-0.411916\pi$
$683$	−18.0000	−	18.0000i	−0.688751	−	0.688751i	−0.961956	+	0.273205i	$0.911916\pi$
$684$		0			0
$685$	3.00000	−	9.00000i	0.114624	−	0.343872i
$686$		0			0
$687$		0			0
$688$		0			0
$689$			20.0000i			0.761939i
$690$		0			0
$691$		0			0		1.00000			$0$
$691$		0			0		−1.00000			$\pi$
$692$		0			0
$693$	−9.00000	+	9.00000i	−0.341882	+	0.341882i
$694$		0			0
$695$	−12.0000	−	24.0000i	−0.455186	−	0.910372i
$696$		0			0
$697$	8.00000	+	8.00000i	0.303022	+	0.303022i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−10.0000			−0.377695			−0.188847	−	0.982006i	$-0.560475\pi$
$701$	−10.0000			−0.377695			−0.188847	+	0.982006i	$0.560475\pi$
$702$		0			0
$703$	−6.00000	−	6.00000i	−0.226294	−	0.226294i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−6.00000	+	6.00000i	−0.225653	+	0.225653i
$708$		0			0
$709$		−	26.0000i		−	0.976450i	−0.872718	−	0.488225i	$-0.837644\pi$
$709$		−	26.0000i		−	0.976450i	0.872718	−	0.488225i	$-0.162356\pi$
$710$		0			0
$711$			18.0000i			0.675053i
$712$		0			0
$713$		0			0
$714$		0			0
$715$	−2.00000	+	6.00000i	−0.0747958	+	0.224387i
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−8.00000	−	6.00000i	−0.297113	−	0.222834i
$726$		0			0
$727$	12.0000	−	12.0000i	0.445055	−	0.445055i	−0.448651	−	0.893707i	$-0.648096\pi$
$727$	12.0000	−	12.0000i	0.445055	−	0.445055i	0.893707	+	0.448651i	$0.148096\pi$
$728$		0			0
$729$			27.0000i			1.00000i
$730$		0			0
$731$			24.0000i			0.887672i
$732$		0			0
$733$	−8.00000	+	8.00000i	−0.295487	+	0.295487i	−0.839243	−	0.543756i	$-0.817002\pi$
$733$	−8.00000	+	8.00000i	−0.295487	+	0.295487i	0.543756	+	0.839243i	$0.317002\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	6.00000	+	6.00000i	0.221013	+	0.221013i
$738$		0			0
$739$	36.0000			1.32428			0.662141	−	0.749380i	$-0.269648\pi$
$739$	36.0000			1.32428			0.662141	+	0.749380i	$0.269648\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−21.0000	−	21.0000i	−0.770415	−	0.770415i	0.207764	−	0.978179i	$-0.433381\pi$
$743$	−21.0000	−	21.0000i	−0.770415	−	0.770415i	−0.978179	+	0.207764i	$0.933381\pi$
$744$		0			0
$745$	−20.0000	+	10.0000i	−0.732743	+	0.366372i
$746$		0			0
$747$	9.00000	−	9.00000i	0.329293	−	0.329293i
$748$		0			0
$749$		−	18.0000i		−	0.657706i
$750$		0			0
$751$		−	48.0000i		−	1.75154i	−0.482724	−	0.875772i	$-0.660353\pi$
$751$		−	48.0000i		−	1.75154i	0.482724	−	0.875772i	$-0.339647\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−36.0000	+	18.0000i	−1.31017	+	0.655087i
$756$		0			0
$757$	−7.00000	−	7.00000i	−0.254419	−	0.254419i	0.568360	−	0.822780i	$-0.307578\pi$
$757$	−7.00000	−	7.00000i	−0.254419	−	0.254419i	−0.822780	+	0.568360i	$0.807578\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−14.0000			−0.507500			−0.253750	−	0.967270i	$-0.581664\pi$
$761$	−14.0000			−0.507500			−0.253750	+	0.967270i	$0.581664\pi$
$762$		0			0
$763$	−54.0000	−	54.0000i	−1.95493	−	1.95493i
$764$		0			0
$765$	−36.0000	−	12.0000i	−1.30158	−	0.433861i
$766$		0			0
$767$	24.0000	−	24.0000i	0.866590	−	0.866590i
$768$		0			0
$769$		−	18.0000i		−	0.649097i	−0.945869	−	0.324548i	$-0.894788\pi$
$769$		−	18.0000i		−	0.649097i	0.945869	−	0.324548i	$-0.105212\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−23.0000	+	23.0000i	−0.827253	+	0.827253i	−0.987136	−	0.159883i	$-0.948888\pi$
$773$	−23.0000	+	23.0000i	−0.827253	+	0.827253i	0.159883	+	0.987136i	$0.448888\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$	12.0000			0.429945
$780$		0			0
$781$	−12.0000			−0.429394
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−5.00000	+	15.0000i	−0.178458	+	0.535373i
$786$		0			0
$787$	−39.0000	+	39.0000i	−1.39020	+	1.39020i	−0.565346	+	0.824854i	$0.691257\pi$
$787$	−39.0000	+	39.0000i	−1.39020	+	1.39020i	−0.824854	+	0.565346i	$0.808743\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			18.0000i			0.640006i
$792$		0			0
$793$	−12.0000	+	12.0000i	−0.426132	+	0.426132i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−3.00000	−	3.00000i	−0.106265	−	0.106265i	0.651975	−	0.758240i	$-0.273941\pi$
$797$	−3.00000	−	3.00000i	−0.106265	−	0.106265i	−0.758240	+	0.651975i	$0.773941\pi$
$798$		0			0
$799$	−48.0000			−1.69812
$800$		0			0
$801$	30.0000			1.06000
$802$		0			0
$803$	−4.00000	−	4.00000i	−0.141157	−	0.141157i
$804$		0			0
$805$	36.0000	+	72.0000i	1.26883	+	2.53767i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			38.0000i			1.33601i	0.744157	+	0.668004i	$0.232851\pi$
$809$			38.0000i			1.33601i	−0.744157	+	0.668004i	$0.767149\pi$
$810$		0			0
$811$		−	12.0000i		−	0.421377i	−0.977553	−	0.210688i	$-0.932429\pi$
$811$		−	12.0000i		−	0.421377i	0.977553	−	0.210688i	$-0.0675706\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	12.0000	−	36.0000i	0.420342	−	1.26102i
$816$		0			0
$817$	18.0000	+	18.0000i	0.629740	+	0.629740i
$818$		0			0
$819$	−36.0000			−1.25794
$820$		0			0
$821$	10.0000			0.349002			0.174501	−	0.984657i	$-0.444169\pi$
$821$	10.0000			0.349002			0.174501	+	0.984657i	$0.444169\pi$
$822$		0			0
$823$	−36.0000	−	36.0000i	−1.25488	−	1.25488i	−0.953506	−	0.301376i	$-0.902554\pi$
$823$	−36.0000	−	36.0000i	−1.25488	−	1.25488i	−0.301376	−	0.953506i	$-0.597446\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	9.00000	−	9.00000i	0.312961	−	0.312961i	−0.533095	−	0.846055i	$-0.678971\pi$
$827$	9.00000	−	9.00000i	0.312961	−	0.312961i	0.846055	+	0.533095i	$0.178971\pi$
$828$		0			0
$829$		−	12.0000i		−	0.416777i	−0.978046	−	0.208389i	$-0.933178\pi$
$829$		−	12.0000i		−	0.416777i	0.978046	−	0.208389i	$-0.0668219\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	44.0000	−	44.0000i	1.52451	−	1.52451i
$834$		0			0
$835$	9.00000	+	3.00000i	0.311458	+	0.103819i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	24.0000			0.828572			0.414286	−	0.910147i	$-0.364031\pi$
$839$	24.0000			0.828572			0.414286	+	0.910147i	$0.364031\pi$
$840$		0			0
$841$	25.0000			0.862069
$842$		0			0
$843$		0			0
$844$		0			0
$845$	10.0000	−	5.00000i	0.344010	−	0.172005i
$846$		0			0
$847$	3.00000	−	3.00000i	0.103081	−	0.103081i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		−	12.0000i		−	0.411355i
$852$		0			0
$853$	28.0000	−	28.0000i	0.958702	−	0.958702i	−0.0404787	−	0.999180i	$-0.512888\pi$
$853$	28.0000	−	28.0000i	0.958702	−	0.958702i	0.999180	+	0.0404787i	$0.0128883\pi$
$854$		0			0
$855$	−36.0000	+	18.0000i	−1.23117	+	0.615587i
$856$		0			0
$857$	16.0000	+	16.0000i	0.546550	+	0.546550i	0.925441	−	0.378892i	$-0.123695\pi$
$857$	16.0000	+	16.0000i	0.546550	+	0.546550i	−0.378892	+	0.925441i	$0.623695\pi$
$858$		0			0
$859$	24.0000			0.818869			0.409435	−	0.912339i	$-0.365726\pi$
$859$	24.0000			0.818869			0.409435	+	0.912339i	$0.365726\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−12.0000	−	12.0000i	−0.408485	−	0.408485i	0.472725	−	0.881210i	$-0.343270\pi$
$863$	−12.0000	−	12.0000i	−0.408485	−	0.408485i	−0.881210	+	0.472725i	$0.843270\pi$
$864$		0			0
$865$	−36.0000	−	12.0000i	−1.22404	−	0.408012i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	6.00000i		−	0.203536i
$870$		0			0
$871$			24.0000i			0.813209i
$872$		0			0
$873$	−21.0000	+	21.0000i	−0.710742	+	0.710742i
$874$		0			0
$875$	−39.0000	+	27.0000i	−1.31844	+	0.912767i
$876$		0			0
$877$	−16.0000	−	16.0000i	−0.540282	−	0.540282i	0.383330	−	0.923611i	$-0.374777\pi$
$877$	−16.0000	−	16.0000i	−0.540282	−	0.540282i	−0.923611	+	0.383330i	$0.874777\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−20.0000			−0.673817			−0.336909	−	0.941537i	$-0.609381\pi$
$881$	−20.0000			−0.673817			−0.336909	+	0.941537i	$0.609381\pi$
$882$		0			0
$883$	−36.0000	−	36.0000i	−1.21150	−	1.21150i	−0.970535	−	0.240962i	$-0.922537\pi$
$883$	−36.0000	−	36.0000i	−1.21150	−	1.21150i	−0.240962	−	0.970535i	$-0.577463\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	33.0000	−	33.0000i	1.10803	−	1.10803i	0.114622	−	0.993409i	$-0.463434\pi$
$887$	33.0000	−	33.0000i	1.10803	−	1.10803i	0.993409	−	0.114622i	$-0.0365658\pi$
$888$		0			0
$889$		−	54.0000i		−	1.81110i
$890$		0			0
$891$		−	9.00000i		−	0.301511i
$892$		0			0
$893$	−36.0000	+	36.0000i	−1.20469	+	1.20469i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$	40.0000			1.33259
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−24.0000	−	48.0000i	−0.797787	−	1.59557i
$906$		0			0
$907$	18.0000	−	18.0000i	0.597680	−	0.597680i	−0.342014	−	0.939695i	$-0.611109\pi$
$907$	18.0000	−	18.0000i	0.597680	−	0.597680i	0.939695	+	0.342014i	$0.111109\pi$
$908$		0			0
$909$		−	6.00000i		−	0.199007i
$910$		0			0
$911$			12.0000i			0.397578i	0.980042	+	0.198789i	$0.0637008\pi$
$911$			12.0000i			0.397578i	−0.980042	+	0.198789i	$0.936299\pi$
$912$		0			0
$913$	−3.00000	+	3.00000i	−0.0992855	+	0.0992855i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−18.0000	−	18.0000i	−0.594412	−	0.594412i
$918$		0			0
$919$	24.0000			0.791687			0.395843	−	0.918318i	$-0.370452\pi$
$919$	24.0000			0.791687			0.395843	+	0.918318i	$0.370452\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−24.0000	−	24.0000i	−0.789970	−	0.789970i
$924$		0			0
$925$	7.00000	−	1.00000i	0.230159	−	0.0328798i
$926$		0			0
$927$		0			0
$928$		0			0
$929$		−	34.0000i		−	1.11550i	−0.830008	−	0.557752i	$-0.811664\pi$
$929$		−	34.0000i		−	1.11550i	0.830008	−	0.557752i	$-0.188336\pi$
$930$		0			0
$931$		−	66.0000i		−	2.16306i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	12.0000	+	4.00000i	0.392442	+	0.130814i
$936$		0			0
$937$	10.0000	+	10.0000i	0.326686	+	0.326686i	0.851325	−	0.524639i	$-0.175800\pi$
$937$	10.0000	+	10.0000i	0.326686	+	0.326686i	−0.524639	+	0.851325i	$0.675800\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	2.00000			0.0651981			0.0325991	−	0.999469i	$-0.489622\pi$
$941$	2.00000			0.0651981			0.0325991	+	0.999469i	$0.489622\pi$
$942$		0			0
$943$	12.0000	+	12.0000i	0.390774	+	0.390774i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−6.00000	+	6.00000i	−0.194974	+	0.194974i	−0.797841	−	0.602868i	$-0.794025\pi$
$947$	−6.00000	+	6.00000i	−0.194974	+	0.194974i	0.602868	+	0.797841i	$0.294025\pi$
$948$		0			0
$949$		−	16.0000i		−	0.519382i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	30.0000	−	30.0000i	0.971795	−	0.971795i	−0.0278177	−	0.999613i	$-0.508856\pi$
$953$	30.0000	−	30.0000i	0.971795	−	0.971795i	0.999613	+	0.0278177i	$0.00885579\pi$
$954$		0			0
$955$	−24.0000	+	12.0000i	−0.776622	+	0.388311i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−18.0000			−0.581250
$960$		0			0
$961$	31.0000			1.00000
$962$		0			0
$963$	9.00000	+	9.00000i	0.290021	+	0.290021i
$964$		0			0
$965$	−48.0000	−	16.0000i	−1.54517	−	0.515058i
$966$		0			0
$967$	−3.00000	+	3.00000i	−0.0964735	+	0.0964735i	−0.753696	−	0.657223i	$-0.771731\pi$
$967$	−3.00000	+	3.00000i	−0.0964735	+	0.0964735i	0.657223	+	0.753696i	$0.271731\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			12.0000i			0.385098i	0.981287	+	0.192549i	$0.0616755\pi$
$971$			12.0000i			0.385098i	−0.981287	+	0.192549i	$0.938325\pi$
$972$		0			0
$973$	−36.0000	+	36.0000i	−1.15411	+	1.15411i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−35.0000	−	35.0000i	−1.11975	−	1.11975i	−0.991778	−	0.127971i	$-0.959153\pi$
$977$	−35.0000	−	35.0000i	−1.11975	−	1.11975i	−0.127971	−	0.991778i	$-0.540847\pi$
$978$		0			0
$979$	−10.0000			−0.319601
$980$		0			0
$981$	54.0000			1.72409
$982$		0			0
$983$	12.0000	+	12.0000i	0.382741	+	0.382741i	0.872089	−	0.489348i	$-0.162765\pi$
$983$	12.0000	+	12.0000i	0.382741	+	0.382741i	−0.489348	+	0.872089i	$0.662765\pi$
$984$		0			0
$985$	−12.0000	+	36.0000i	−0.382352	+	1.14706i
$986$		0			0
$987$		0			0
$988$		0			0
$989$			36.0000i			1.14473i
$990$		0			0
$991$		−	12.0000i		−	0.381193i	−0.981669	−	0.190596i	$-0.938958\pi$
$991$		−	12.0000i		−	0.381193i	0.981669	−	0.190596i	$-0.0610421\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	24.0000	+	48.0000i	0.760851	+	1.52170i
$996$		0			0
$997$	8.00000	+	8.00000i	0.253363	+	0.253363i	0.822348	−	0.568985i	$-0.192664\pi$
$997$	8.00000	+	8.00000i	0.253363	+	0.253363i	−0.568985	+	0.822348i	$0.692664\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	880.2.bf.b.287.1	✓	2
4.3	odd	2		880.2.bf.c.287.1	yes	2
5.3	odd	4		880.2.bf.c.463.1	yes	2
20.3	even	4	inner	880.2.bf.b.463.1	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
880.2.bf.b.287.1	✓	2	1.1	even	1	trivial
880.2.bf.b.463.1	yes	2	20.3	even	4	inner
880.2.bf.c.287.1	yes	2	4.3	odd	2
880.2.bf.c.463.1	yes	2	5.3	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:	\( N \)	\(=\)	\( 880 = 2^{4} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	880.bf (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.00000 - 2.00000i) q^{5} +(-3.00000 + 3.00000i) q^{7} -3.00000i q^{9} +O(q^{10})\)	\(q+(-1.00000 - 2.00000i) q^{5} +(-3.00000 + 3.00000i) q^{7} -3.00000i q^{9} +1.00000i q^{11} +(-2.00000 + 2.00000i) q^{13} +(4.00000 + 4.00000i) q^{17} +6.00000 q^{19} +(6.00000 + 6.00000i) q^{23} +(-3.00000 + 4.00000i) q^{25} +2.00000i q^{29} +(9.00000 + 3.00000i) q^{35} +(-1.00000 - 1.00000i) q^{37} +2.00000 q^{41} +(3.00000 + 3.00000i) q^{43} +(-6.00000 + 3.00000i) q^{45} +(-6.00000 + 6.00000i) q^{47} -11.0000i q^{49} +(5.00000 - 5.00000i) q^{53} +(2.00000 - 1.00000i) q^{55} -12.0000 q^{59} +6.00000 q^{61} +(9.00000 + 9.00000i) q^{63} +(6.00000 + 2.00000i) q^{65} +(6.00000 - 6.00000i) q^{67} +12.0000i q^{71} +(-4.00000 + 4.00000i) q^{73} +(-3.00000 - 3.00000i) q^{77} -6.00000 q^{79} -9.00000 q^{81} +(3.00000 + 3.00000i) q^{83} +(4.00000 - 12.0000i) q^{85} +10.0000i q^{89} -12.0000i q^{91} +(-6.00000 - 12.0000i) q^{95} +(-7.00000 - 7.00000i) q^{97} +3.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{5} - 6 q^{7}+O(q^{10}) \) 2 * q - 2 * q^5 - 6 * q^7	\( 2 q - 2 q^{5} - 6 q^{7} - 4 q^{13} + 8 q^{17} + 12 q^{19} + 12 q^{23} - 6 q^{25} + 18 q^{35} - 2 q^{37} + 4 q^{41} + 6 q^{43} - 12 q^{45} - 12 q^{47} + 10 q^{53} + 4 q^{55} - 24 q^{59} + 12 q^{61} + 18 q^{63} + 12 q^{65} + 12 q^{67} - 8 q^{73} - 6 q^{77} - 12 q^{79} - 18 q^{81} + 6 q^{83} + 8 q^{85} - 12 q^{95} - 14 q^{97} + 6 q^{99}+O(q^{100}) \) 2 * q - 2 * q^5 - 6 * q^7 - 4 * q^13 + 8 * q^17 + 12 * q^19 + 12 * q^23 - 6 * q^25 + 18 * q^35 - 2 * q^37 + 4 * q^41 + 6 * q^43 - 12 * q^45 - 12 * q^47 + 10 * q^53 + 4 * q^55 - 24 * q^59 + 12 * q^61 + 18 * q^63 + 12 * q^65 + 12 * q^67 - 8 * q^73 - 6 * q^77 - 12 * q^79 - 18 * q^81 + 6 * q^83 + 8 * q^85 - 12 * q^95 - 14 * q^97 + 6 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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