LMFDB - Embedded newform 260.2.p.b.207.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [260,2,Mod(103,260)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(260, base_ring=CyclotomicField(4))

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

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

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

chi := DirichletCharacter("260.103");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 260 = 2^{2} \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	260.p (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:	$2.07611045255$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{4}]$

Embedding invariants

Embedding label			207.1
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	260.207
Dual form			260.2.p.b.103.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.00000 - 1.00000i) q^{2} -2.00000i q^{4} +(-2.00000 - 1.00000i) q^{5} +(-2.00000 - 2.00000i) q^{8} -3.00000i q^{9} +O(q^{10})$	\(q+(1.00000 - 1.00000i) q^{2} -2.00000i q^{4} +(-2.00000 - 1.00000i) q^{5} +(-2.00000 - 2.00000i) q^{8} -3.00000i q^{9} +(-3.00000 + 1.00000i) q^{10} +(2.00000 - 3.00000i) q^{13} -4.00000 q^{16} +(5.00000 + 5.00000i) q^{17} +(-3.00000 - 3.00000i) q^{18} +(-2.00000 + 4.00000i) q^{20} +(3.00000 + 4.00000i) q^{25} +(-1.00000 - 5.00000i) q^{26} +4.00000i q^{29} +(-4.00000 + 4.00000i) q^{32} +10.0000 q^{34} -6.00000 q^{36} +(7.00000 - 7.00000i) q^{37} +(2.00000 + 6.00000i) q^{40} +10.0000i q^{41} +(-3.00000 + 6.00000i) q^{45} -7.00000i q^{49} +(7.00000 + 1.00000i) q^{50} +(-6.00000 - 4.00000i) q^{52} +(5.00000 - 5.00000i) q^{53} +(4.00000 + 4.00000i) q^{58} -12.0000 q^{61} +8.00000i q^{64} +(-7.00000 + 4.00000i) q^{65} +(10.0000 - 10.0000i) q^{68} +(-6.00000 + 6.00000i) q^{72} +(11.0000 + 11.0000i) q^{73} -14.0000i q^{74} +(8.00000 + 4.00000i) q^{80} -9.00000 q^{81} +(10.0000 + 10.0000i) q^{82} +(-5.00000 - 15.0000i) q^{85} +10.0000 q^{89} +(3.00000 + 9.00000i) q^{90} +(13.0000 - 13.0000i) q^{97} +(-7.00000 - 7.00000i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} - 4 q^{5} - 4 q^{8}+O(q^{10}) $ 2 * q + 2 * q^2 - 4 * q^5 - 4 * q^8	$ 2 q + 2 q^{2} - 4 q^{5} - 4 q^{8} - 6 q^{10} + 4 q^{13} - 8 q^{16} + 10 q^{17} - 6 q^{18} - 4 q^{20} + 6 q^{25} - 2 q^{26} - 8 q^{32} + 20 q^{34} - 12 q^{36} + 14 q^{37} + 4 q^{40} - 6 q^{45} + 14 q^{50} - 12 q^{52} + 10 q^{53} + 8 q^{58} - 24 q^{61} - 14 q^{65} + 20 q^{68} - 12 q^{72} + 22 q^{73} + 16 q^{80} - 18 q^{81} + 20 q^{82} - 10 q^{85} + 20 q^{89} + 6 q^{90} + 26 q^{97} - 14 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 - 4 * q^5 - 4 * q^8 - 6 * q^10 + 4 * q^13 - 8 * q^16 + 10 * q^17 - 6 * q^18 - 4 * q^20 + 6 * q^25 - 2 * q^26 - 8 * q^32 + 20 * q^34 - 12 * q^36 + 14 * q^37 + 4 * q^40 - 6 * q^45 + 14 * q^50 - 12 * q^52 + 10 * q^53 + 8 * q^58 - 24 * q^61 - 14 * q^65 + 20 * q^68 - 12 * q^72 + 22 * q^73 + 16 * q^80 - 18 * q^81 + 20 * q^82 - 10 * q^85 + 20 * q^89 + 6 * q^90 + 26 * q^97 - 14 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/260\mathbb{Z}\right)^\times$.

$n$	$41$	$131$	$157$
$\chi(n)$	$-1$	$-1$	$e\left(\frac{1}{4}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	1.00000	−	1.00000i	0.707107	−	0.707107i
$2$	1.00000	−	1.00000i	0.707107	−	0.707107i
$3$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$3$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$4$		−	2.00000i		−	1.00000i
$5$	−2.00000	−	1.00000i	−0.894427	−	0.447214i
$5$	−2.00000	−	1.00000i	−0.894427	−	0.447214i
$6$		0			0
$7$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$7$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$8$	−2.00000	−	2.00000i	−0.707107	−	0.707107i
$9$		−	3.00000i		−	1.00000i
$10$	−3.00000	+	1.00000i	−0.948683	+	0.316228i
$11$		0			0			−	1.00000i	$-0.5\pi$
$11$		0			0				1.00000i	$0.5\pi$
$12$		0			0
$13$	2.00000	−	3.00000i	0.554700	−	0.832050i
$13$	2.00000	−	3.00000i	0.554700	−	0.832050i
$14$		0			0
$15$		0			0
$16$	−4.00000			−1.00000
$17$	5.00000	+	5.00000i	1.21268	+	1.21268i	0.970143	+	0.242536i	$0.0779791\pi$
$17$	5.00000	+	5.00000i	1.21268	+	1.21268i	0.242536	+	0.970143i	$0.422021\pi$
$18$	−3.00000	−	3.00000i	−0.707107	−	0.707107i
$19$		0			0		1.00000			$0$
$19$		0			0		−1.00000			$\pi$
$20$	−2.00000	+	4.00000i	−0.447214	+	0.894427i
$21$		0			0
$22$		0			0
$23$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$23$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$24$		0			0
$25$	3.00000	+	4.00000i	0.600000	+	0.800000i
$26$	−1.00000	−	5.00000i	−0.196116	−	0.980581i
$27$		0			0
$28$		0			0
$29$			4.00000i			0.742781i	0.928477	+	0.371391i	$0.121119\pi$
$29$			4.00000i			0.742781i	−0.928477	+	0.371391i	$0.878881\pi$
$30$		0			0
$31$		0			0			−	1.00000i	$-0.5\pi$
$31$		0			0				1.00000i	$0.5\pi$
$32$	−4.00000	+	4.00000i	−0.707107	+	0.707107i
$33$		0			0
$34$	10.0000			1.71499
$35$		0			0
$36$	−6.00000			−1.00000
$37$	7.00000	−	7.00000i	1.15079	−	1.15079i	0.164399	−	0.986394i	$-0.447432\pi$
$37$	7.00000	−	7.00000i	1.15079	−	1.15079i	0.986394	−	0.164399i	$-0.0525685\pi$
$38$		0			0
$39$		0			0
$40$	2.00000	+	6.00000i	0.316228	+	0.948683i
$41$			10.0000i			1.56174i	0.624695	+	0.780869i	$0.285223\pi$
$41$			10.0000i			1.56174i	−0.624695	+	0.780869i	$0.714777\pi$
$42$		0			0
$43$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$43$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$44$		0			0
$45$	−3.00000	+	6.00000i	−0.447214	+	0.894427i
$46$		0			0
$47$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$47$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$48$		0			0
$49$		−	7.00000i		−	1.00000i
$50$	7.00000	+	1.00000i	0.989949	+	0.141421i
$51$		0			0
$52$	−6.00000	−	4.00000i	−0.832050	−	0.554700i
$53$	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
$55$		0			0
$56$		0			0
$57$		0			0
$58$	4.00000	+	4.00000i	0.525226	+	0.525226i
$59$		0			0		1.00000			$0$
$59$		0			0		−1.00000			$\pi$
$60$		0			0
$61$	−12.0000			−1.53644			−0.768221	−	0.640184i	$-0.778858\pi$
$61$	−12.0000			−1.53644			−0.768221	+	0.640184i	$0.778858\pi$
$62$		0			0
$63$		0			0
$64$			8.00000i			1.00000i
$65$	−7.00000	+	4.00000i	−0.868243	+	0.496139i
$66$		0			0
$67$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$67$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$68$	10.0000	−	10.0000i	1.21268	−	1.21268i
$69$		0			0
$70$		0			0
$71$		0			0			−	1.00000i	$-0.5\pi$
$71$		0			0				1.00000i	$0.5\pi$
$72$	−6.00000	+	6.00000i	−0.707107	+	0.707107i
$73$	11.0000	+	11.0000i	1.28745	+	1.28745i	0.936329	+	0.351123i	$0.114200\pi$
$73$	11.0000	+	11.0000i	1.28745	+	1.28745i	0.351123	+	0.936329i	$0.385800\pi$
$74$		−	14.0000i		−	1.62747i
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0			−	1.00000i	$-0.5\pi$
$79$		0			0				1.00000i	$0.5\pi$
$80$	8.00000	+	4.00000i	0.894427	+	0.447214i
$81$	−9.00000			−1.00000
$82$	10.0000	+	10.0000i	1.10432	+	1.10432i
$83$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$83$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$84$		0			0
$85$	−5.00000	−	15.0000i	−0.542326	−	1.62698i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	10.0000			1.06000			0.529999	−	0.847998i	$-0.322192\pi$
$89$	10.0000			1.06000			0.529999	+	0.847998i	$0.322192\pi$
$90$	3.00000	+	9.00000i	0.316228	+	0.948683i
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	13.0000	−	13.0000i	1.31995	−	1.31995i	0.406138	−	0.913812i	$-0.366875\pi$
$97$	13.0000	−	13.0000i	1.31995	−	1.31995i	0.913812	−	0.406138i	$-0.133125\pi$
$98$	−7.00000	−	7.00000i	−0.707107	−	0.707107i
$99$		0			0
$100$	8.00000	−	6.00000i	0.800000	−	0.600000i
$101$	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	$0.750000\pi$
$104$	−10.0000	+	2.00000i	−0.980581	+	0.196116i
$105$		0			0
$106$		−	10.0000i		−	0.971286i
$107$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$107$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$108$		0			0
$109$	−20.0000			−1.91565			−0.957826	−	0.287348i	$-0.907226\pi$
$109$	−20.0000			−1.91565			−0.957826	+	0.287348i	$0.907226\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−15.0000	+	15.0000i	−1.41108	+	1.41108i	−0.658505	+	0.752577i	$0.728811\pi$
$113$	−15.0000	+	15.0000i	−1.41108	+	1.41108i	−0.752577	+	0.658505i	$0.771189\pi$
$114$		0			0
$115$		0			0
$116$	8.00000			0.742781
$117$	−9.00000	−	6.00000i	−0.832050	−	0.554700i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−11.0000			−1.00000
$122$	−12.0000	+	12.0000i	−1.08643	+	1.08643i
$123$		0			0
$124$		0			0
$125$	−2.00000	−	11.0000i	−0.178885	−	0.983870i
$126$		0			0
$127$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$127$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$128$	8.00000	+	8.00000i	0.707107	+	0.707107i
$129$		0			0
$130$	−3.00000	+	11.0000i	−0.263117	+	0.964764i
$131$		0			0		1.00000			$0$
$131$		0			0		−1.00000			$\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		−	20.0000i		−	1.71499i
$137$	−7.00000	+	7.00000i	−0.598050	+	0.598050i	−0.939793	−	0.341743i	$-0.888983\pi$
$137$	−7.00000	+	7.00000i	−0.598050	+	0.598050i	0.341743	+	0.939793i	$0.388983\pi$
$138$		0			0
$139$		0			0			−	1.00000i	$-0.5\pi$
$139$		0			0				1.00000i	$0.5\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$			12.0000i			1.00000i
$145$	4.00000	−	8.00000i	0.332182	−	0.664364i
$146$	22.0000			1.82073
$147$		0			0
$148$	−14.0000	−	14.0000i	−1.15079	−	1.15079i
$149$	−20.0000			−1.63846			−0.819232	−	0.573462i	$-0.805600\pi$
$149$	−20.0000			−1.63846			−0.819232	+	0.573462i	$0.805600\pi$
$150$		0			0
$151$		0			0			−	1.00000i	$-0.5\pi$
$151$		0			0				1.00000i	$0.5\pi$
$152$		0			0
$153$	15.0000	−	15.0000i	1.21268	−	1.21268i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	5.00000	+	5.00000i	0.399043	+	0.399043i	0.877896	−	0.478852i	$-0.158947\pi$
$157$	5.00000	+	5.00000i	0.399043	+	0.399043i	−0.478852	+	0.877896i	$0.658947\pi$
$158$		0			0
$159$		0			0
$160$	12.0000	−	4.00000i	0.948683	−	0.316228i
$161$		0			0
$162$	−9.00000	+	9.00000i	−0.707107	+	0.707107i
$163$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$163$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$164$	20.0000			1.56174
$165$		0			0
$166$		0			0
$167$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$167$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$168$		0			0
$169$	−5.00000	−	12.0000i	−0.384615	−	0.923077i
$170$	−20.0000	−	10.0000i	−1.53393	−	0.766965i
$171$		0			0
$172$		0			0
$173$	−15.0000	+	15.0000i	−1.14043	+	1.14043i	−0.152057	+	0.988372i	$0.548590\pi$
$173$	−15.0000	+	15.0000i	−1.14043	+	1.14043i	−0.988372	+	0.152057i	$0.951410\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$	10.0000	−	10.0000i	0.749532	−	0.749532i
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$	12.0000	+	6.00000i	0.894427	+	0.447214i
$181$	18.0000			1.33793			0.668965	−	0.743294i	$-0.266738\pi$
$181$	18.0000			1.33793			0.668965	+	0.743294i	$0.266738\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−21.0000	+	7.00000i	−1.54395	+	0.514650i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		1.00000			$0$
$191$		0			0		−1.00000			$\pi$
$192$		0			0
$193$	19.0000	+	19.0000i	1.36765	+	1.36765i	0.863779	+	0.503871i	$0.168091\pi$
$193$	19.0000	+	19.0000i	1.36765	+	1.36765i	0.503871	+	0.863779i	$0.331909\pi$
$194$		−	26.0000i		−	1.86669i
$195$		0			0
$196$	−14.0000			−1.00000
$197$	13.0000	−	13.0000i	0.926212	−	0.926212i	−0.0712470	−	0.997459i	$-0.522698\pi$
$197$	13.0000	−	13.0000i	0.926212	−	0.926212i	0.997459	+	0.0712470i	$0.0226979\pi$
$198$		0			0
$199$		0			0			−	1.00000i	$-0.5\pi$
$199$		0			0				1.00000i	$0.5\pi$
$200$	2.00000	−	14.0000i	0.141421	−	0.989949i
$201$		0			0
$202$	2.00000	−	2.00000i	0.140720	−	0.140720i
$203$		0			0
$204$		0			0
$205$	10.0000	−	20.0000i	0.698430	−	1.39686i
$206$		0			0
$207$		0			0
$208$	−8.00000	+	12.0000i	−0.554700	+	0.832050i
$209$		0			0
$210$		0			0
$211$		0			0		1.00000			$0$
$211$		0			0		−1.00000			$\pi$
$212$	−10.0000	−	10.0000i	−0.686803	−	0.686803i
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$	−20.0000	+	20.0000i	−1.35457	+	1.35457i
$219$		0			0
$220$		0			0
$221$	25.0000	−	5.00000i	1.68168	−	0.336336i
$222$		0			0
$223$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$223$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$224$		0			0
$225$	12.0000	−	9.00000i	0.800000	−	0.600000i
$226$			30.0000i			1.99557i
$227$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$227$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$228$		0			0
$229$	30.0000			1.98246			0.991228	−	0.132164i	$-0.0421925\pi$
$229$	30.0000			1.98246			0.991228	+	0.132164i	$0.0421925\pi$
$230$		0			0
$231$		0			0
$232$	8.00000	−	8.00000i	0.525226	−	0.525226i
$233$	−5.00000	+	5.00000i	−0.327561	+	0.327561i	−0.851658	−	0.524097i	$-0.824403\pi$
$233$	−5.00000	+	5.00000i	−0.327561	+	0.327561i	0.524097	+	0.851658i	$0.324403\pi$
$234$	−15.0000	+	3.00000i	−0.980581	+	0.196116i
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		1.00000			$0$
$239$		0			0		−1.00000			$\pi$
$240$		0			0
$241$			30.0000i			1.93247i	0.257663	+	0.966235i	$0.417048\pi$
$241$			30.0000i			1.93247i	−0.257663	+	0.966235i	$0.582952\pi$
$242$	−11.0000	+	11.0000i	−0.707107	+	0.707107i
$243$		0			0
$244$			24.0000i			1.53644i
$245$	−7.00000	+	14.0000i	−0.447214	+	0.894427i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$	−13.0000	−	9.00000i	−0.822192	−	0.569210i
$251$		0			0		1.00000			$0$
$251$		0			0		−1.00000			$\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	16.0000			1.00000
$257$	15.0000	+	15.0000i	0.935674	+	0.935674i	0.998053	−	0.0623783i	$-0.0198685\pi$
$257$	15.0000	+	15.0000i	0.935674	+	0.935674i	−0.0623783	+	0.998053i	$0.519869\pi$
$258$		0			0
$259$		0			0
$260$	8.00000	+	14.0000i	0.496139	+	0.868243i
$261$	12.0000			0.742781
$262$		0			0
$263$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$263$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$264$		0			0
$265$	−15.0000	+	5.00000i	−0.921443	+	0.307148i
$266$		0			0
$267$		0			0
$268$		0			0
$269$			26.0000i			1.58525i	0.609711	+	0.792624i	$0.291286\pi$
$269$			26.0000i			1.58525i	−0.609711	+	0.792624i	$0.708714\pi$
$270$		0			0
$271$		0			0			−	1.00000i	$-0.5\pi$
$271$		0			0				1.00000i	$0.5\pi$
$272$	−20.0000	−	20.0000i	−1.21268	−	1.21268i
$273$		0			0
$274$			14.0000i			0.845771i
$275$		0			0
$276$		0			0
$277$	−5.00000	−	5.00000i	−0.300421	−	0.300421i	0.540758	−	0.841178i	$-0.318138\pi$
$277$	−5.00000	−	5.00000i	−0.300421	−	0.300421i	−0.841178	+	0.540758i	$0.818138\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			10.0000i			0.596550i	0.954480	+	0.298275i	$0.0964112\pi$
$281$			10.0000i			0.596550i	−0.954480	+	0.298275i	$0.903589\pi$
$282$		0			0
$283$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$283$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$	12.0000	+	12.0000i	0.707107	+	0.707107i
$289$			33.0000i			1.94118i
$290$	−4.00000	−	12.0000i	−0.234888	−	0.704664i
$291$		0			0
$292$	22.0000	−	22.0000i	1.28745	−	1.28745i
$293$	−19.0000	−	19.0000i	−1.10999	−	1.10999i	−0.993151	−	0.116841i	$-0.962723\pi$
$293$	−19.0000	−	19.0000i	−1.10999	−	1.10999i	−0.116841	−	0.993151i	$-0.537277\pi$
$294$		0			0
$295$		0			0
$296$	−28.0000			−1.62747
$297$		0			0
$298$	−20.0000	+	20.0000i	−1.15857	+	1.15857i
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	24.0000	+	12.0000i	1.37424	+	0.687118i
$306$		−	30.0000i		−	1.71499i
$307$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$307$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		1.00000			$0$
$311$		0			0		−1.00000			$\pi$
$312$		0			0
$313$	25.0000	−	25.0000i	1.41308	−	1.41308i	0.678280	−	0.734803i	$-0.262726\pi$
$313$	25.0000	−	25.0000i	1.41308	−	1.41308i	0.734803	−	0.678280i	$-0.237274\pi$
$314$	10.0000			0.564333
$315$		0			0
$316$		0			0
$317$	−3.00000	+	3.00000i	−0.168497	+	0.168497i	−0.786318	−	0.617822i	$-0.788015\pi$
$317$	−3.00000	+	3.00000i	−0.168497	+	0.168497i	0.617822	+	0.786318i	$0.288015\pi$
$318$		0			0
$319$		0			0
$320$	8.00000	−	16.0000i	0.447214	−	0.894427i
$321$		0			0
$322$		0			0
$323$		0			0
$324$			18.0000i			1.00000i
$325$	18.0000	−	1.00000i	0.998460	−	0.0554700i
$326$		0			0
$327$		0			0
$328$	20.0000	−	20.0000i	1.10432	−	1.10432i
$329$		0			0
$330$		0			0
$331$		0			0			−	1.00000i	$-0.5\pi$
$331$		0			0				1.00000i	$0.5\pi$
$332$		0			0
$333$	−21.0000	−	21.0000i	−1.15079	−	1.15079i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−25.0000	−	25.0000i	−1.36184	−	1.36184i	−0.871576	−	0.490261i	$-0.836901\pi$
$337$	−25.0000	−	25.0000i	−1.36184	−	1.36184i	−0.490261	−	0.871576i	$-0.663099\pi$
$338$	−17.0000	−	7.00000i	−0.924678	−	0.380750i
$339$		0			0
$340$	−30.0000	+	10.0000i	−1.62698	+	0.542326i
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$			30.0000i			1.61281i
$347$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$347$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$348$		0			0
$349$	−10.0000			−0.535288			−0.267644	−	0.963518i	$-0.586245\pi$
$349$	−10.0000			−0.535288			−0.267644	+	0.963518i	$0.586245\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−9.00000	−	9.00000i	−0.479022	−	0.479022i	0.425797	−	0.904819i	$-0.359994\pi$
$353$	−9.00000	−	9.00000i	−0.479022	−	0.479022i	−0.904819	+	0.425797i	$0.859994\pi$
$354$		0			0
$355$		0			0
$356$		−	20.0000i		−	1.06000i
$357$		0			0
$358$		0			0
$359$		0			0		1.00000			$0$
$359$		0			0		−1.00000			$\pi$
$360$	18.0000	−	6.00000i	0.948683	−	0.316228i
$361$	19.0000			1.00000
$362$	18.0000	−	18.0000i	0.946059	−	0.946059i
$363$		0			0
$364$		0			0
$365$	−11.0000	−	33.0000i	−0.575766	−	1.72730i
$366$		0			0
$367$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$367$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$368$		0			0
$369$	30.0000			1.56174
$370$	−14.0000	+	28.0000i	−0.727825	+	1.45565i
$371$		0			0
$372$		0			0
$373$	−25.0000	+	25.0000i	−1.29445	+	1.29445i	−0.362446	+	0.932005i	$0.618058\pi$
$373$	−25.0000	+	25.0000i	−1.29445	+	1.29445i	−0.932005	+	0.362446i	$0.881942\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	12.0000	+	8.00000i	0.618031	+	0.412021i
$378$		0			0
$379$		0			0		1.00000			$0$
$379$		0			0		−1.00000			$\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$383$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$384$		0			0
$385$		0			0
$386$	38.0000			1.93415
$387$		0			0
$388$	−26.0000	−	26.0000i	−1.31995	−	1.31995i
$389$		−	34.0000i		−	1.72387i	−0.507020	−	0.861934i	$-0.669253\pi$
$389$		−	34.0000i		−	1.72387i	0.507020	−	0.861934i	$-0.330747\pi$
$390$		0			0
$391$		0			0
$392$	−14.0000	+	14.0000i	−0.707107	+	0.707107i
$393$		0			0
$394$		−	26.0000i		−	1.30986i
$395$		0			0
$396$		0			0
$397$	−13.0000	+	13.0000i	−0.652451	+	0.652451i	−0.953583	−	0.301131i	$-0.902636\pi$
$397$	−13.0000	+	13.0000i	−0.652451	+	0.652451i	0.301131	+	0.953583i	$0.402636\pi$
$398$		0			0
$399$		0			0
$400$	−12.0000	−	16.0000i	−0.600000	−	0.800000i
$401$		−	40.0000i		−	1.99750i	−0.0499376	−	0.998752i	$-0.515902\pi$
$401$		−	40.0000i		−	1.99750i	0.0499376	−	0.998752i	$-0.484098\pi$
$402$		0			0
$403$		0			0
$404$		−	4.00000i		−	0.199007i
$405$	18.0000	+	9.00000i	0.894427	+	0.447214i
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−40.0000			−1.97787			−0.988936	−	0.148340i	$-0.952607\pi$
$409$	−40.0000			−1.97787			−0.988936	+	0.148340i	$0.952607\pi$
$410$	−10.0000	−	30.0000i	−0.493865	−	1.48159i
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$	4.00000	+	20.0000i	0.196116	+	0.980581i
$417$		0			0
$418$		0			0
$419$		0			0			−	1.00000i	$-0.5\pi$
$419$		0			0				1.00000i	$0.5\pi$
$420$		0			0
$421$		−	30.0000i		−	1.46211i	−0.682318	−	0.731055i	$-0.739028\pi$
$421$		−	30.0000i		−	1.46211i	0.682318	−	0.731055i	$-0.260972\pi$
$422$		0			0
$423$		0			0
$424$	−20.0000			−0.971286
$425$	−5.00000	+	35.0000i	−0.242536	+	1.69775i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0			−	1.00000i	$-0.5\pi$
$431$		0			0				1.00000i	$0.5\pi$
$432$		0			0
$433$	5.00000	−	5.00000i	0.240285	−	0.240285i	−0.576683	−	0.816968i	$-0.695653\pi$
$433$	5.00000	−	5.00000i	0.240285	−	0.240285i	0.816968	+	0.576683i	$0.195653\pi$
$434$		0			0
$435$		0			0
$436$			40.0000i			1.91565i
$437$		0			0
$438$		0			0
$439$		0			0			−	1.00000i	$-0.5\pi$
$439$		0			0				1.00000i	$0.5\pi$
$440$		0			0
$441$	−21.0000			−1.00000
$442$	20.0000	−	30.0000i	0.951303	−	1.42695i
$443$		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
$449$	40.0000			1.88772			0.943858	−	0.330350i	$-0.107167\pi$
$449$	40.0000			1.88772			0.943858	+	0.330350i	$0.107167\pi$
$450$	3.00000	−	21.0000i	0.141421	−	0.989949i
$451$		0			0
$452$	30.0000	+	30.0000i	1.41108	+	1.41108i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−17.0000	+	17.0000i	−0.795226	+	0.795226i	−0.982339	−	0.187112i	$-0.940087\pi$
$457$	−17.0000	+	17.0000i	−0.795226	+	0.795226i	0.187112	+	0.982339i	$0.440087\pi$
$458$	30.0000	−	30.0000i	1.40181	−	1.40181i
$459$		0			0
$460$		0			0
$461$			20.0000i			0.931493i	0.884918	+	0.465746i	$0.154214\pi$
$461$			20.0000i			0.931493i	−0.884918	+	0.465746i	$0.845786\pi$
$462$		0			0
$463$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$463$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$464$		−	16.0000i		−	0.742781i
$465$		0			0
$466$			10.0000i			0.463241i
$467$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$467$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$468$	−12.0000	+	18.0000i	−0.554700	+	0.832050i
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−15.0000	−	15.0000i	−0.686803	−	0.686803i
$478$		0			0
$479$		0			0		1.00000			$0$
$479$		0			0		−1.00000			$\pi$
$480$		0			0
$481$	−7.00000	−	35.0000i	−0.319173	−	1.59586i
$482$	30.0000	+	30.0000i	1.36646	+	1.36646i
$483$		0			0
$484$			22.0000i			1.00000i
$485$	−39.0000	+	13.0000i	−1.77090	+	0.590300i
$486$		0			0
$487$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$487$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$488$	24.0000	+	24.0000i	1.08643	+	1.08643i
$489$		0			0
$490$	7.00000	+	21.0000i	0.316228	+	0.948683i
$491$		0			0		1.00000			$0$
$491$		0			0		−1.00000			$\pi$
$492$		0			0
$493$	−20.0000	+	20.0000i	−0.900755	+	0.900755i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$		0			0		1.00000			$0$
$499$		0			0		−1.00000			$\pi$
$500$	−22.0000	+	4.00000i	−0.983870	+	0.178885i
$501$		0			0
$502$		0			0
$503$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$503$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$504$		0			0
$505$	−4.00000	−	2.00000i	−0.177998	−	0.0889988i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−10.0000			−0.443242			−0.221621	−	0.975133i	$-0.571135\pi$
$509$	−10.0000			−0.443242			−0.221621	+	0.975133i	$0.571135\pi$
$510$		0			0
$511$		0			0
$512$	16.0000	−	16.0000i	0.707107	−	0.707107i
$513$		0			0
$514$	30.0000			1.32324
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$	22.0000	+	6.00000i	0.964764	+	0.263117i
$521$	−22.0000			−0.963837			−0.481919	−	0.876216i	$-0.660060\pi$
$521$	−22.0000			−0.963837			−0.481919	+	0.876216i	$0.660060\pi$
$522$	12.0000	−	12.0000i	0.525226	−	0.525226i
$523$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$523$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$		−	23.0000i		−	1.00000i
$530$	−10.0000	+	20.0000i	−0.434372	+	0.868744i
$531$		0			0
$532$		0			0
$533$	30.0000	+	20.0000i	1.29944	+	0.866296i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$	26.0000	+	26.0000i	1.12094	+	1.12094i
$539$		0			0
$540$		0			0
$541$			20.0000i			0.859867i	0.902861	+	0.429934i	$0.141463\pi$
$541$			20.0000i			0.859867i	−0.902861	+	0.429934i	$0.858537\pi$
$542$		0			0
$543$		0			0
$544$	−40.0000			−1.71499
$545$	40.0000	+	20.0000i	1.71341	+	0.856706i
$546$		0			0
$547$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$547$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$548$	14.0000	+	14.0000i	0.598050	+	0.598050i
$549$			36.0000i			1.53644i
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$	−10.0000			−0.424859
$555$		0			0
$556$		0			0
$557$	−33.0000	+	33.0000i	−1.39825	+	1.39825i	−0.593199	+	0.805056i	$0.702135\pi$
$557$	−33.0000	+	33.0000i	−1.39825	+	1.39825i	−0.805056	+	0.593199i	$0.797865\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	10.0000	+	10.0000i	0.421825	+	0.421825i
$563$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$563$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$564$		0			0
$565$	45.0000	−	15.0000i	1.89316	−	0.631055i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	26.0000i		−	1.08998i	−0.838444	−	0.544988i	$-0.816534\pi$
$569$		−	26.0000i		−	1.08998i	0.838444	−	0.544988i	$-0.183466\pi$
$570$		0			0
$571$		0			0		1.00000			$0$
$571$		0			0		−1.00000			$\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	24.0000			1.00000
$577$	−23.0000	+	23.0000i	−0.957503	+	0.957503i	−0.999133	−	0.0416305i	$-0.986745\pi$
$577$	−23.0000	+	23.0000i	−0.957503	+	0.957503i	0.0416305	+	0.999133i	$0.486745\pi$
$578$	33.0000	+	33.0000i	1.37262	+	1.37262i
$579$		0			0
$580$	−16.0000	−	8.00000i	−0.664364	−	0.332182i
$581$		0			0
$582$		0			0
$583$		0			0
$584$		−	44.0000i		−	1.82073i
$585$	12.0000	+	21.0000i	0.496139	+	0.868243i
$586$	−38.0000			−1.56977
$587$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$587$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$	−28.0000	+	28.0000i	−1.15079	+	1.15079i
$593$	−31.0000	−	31.0000i	−1.27302	−	1.27302i	−0.944497	−	0.328521i	$-0.893450\pi$
$593$	−31.0000	−	31.0000i	−1.27302	−	1.27302i	−0.328521	−	0.944497i	$-0.606550\pi$
$594$		0			0
$595$		0			0
$596$			40.0000i			1.63846i
$597$		0			0
$598$		0			0
$599$		0			0			−	1.00000i	$-0.5\pi$
$599$		0			0				1.00000i	$0.5\pi$
$600$		0			0
$601$	−48.0000			−1.95796			−0.978980	−	0.203954i	$-0.934621\pi$
$601$	−48.0000			−1.95796			−0.978980	+	0.203954i	$0.934621\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	22.0000	+	11.0000i	0.894427	+	0.447214i
$606$		0			0
$607$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$607$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$608$		0			0
$609$		0			0
$610$	36.0000	−	12.0000i	1.45760	−	0.485866i
$611$		0			0
$612$	−30.0000	−	30.0000i	−1.21268	−	1.21268i
$613$	−1.00000	−	1.00000i	−0.0403896	−	0.0403896i	0.686624	−	0.727013i	$-0.259092\pi$
$613$	−1.00000	−	1.00000i	−0.0403896	−	0.0403896i	−0.727013	+	0.686624i	$0.759092\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	3.00000	−	3.00000i	0.120775	−	0.120775i	−0.644136	−	0.764911i	$-0.722783\pi$
$617$	3.00000	−	3.00000i	0.120775	−	0.120775i	0.764911	+	0.644136i	$0.222783\pi$
$618$		0			0
$619$		0			0		1.00000			$0$
$619$		0			0		−1.00000			$\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−7.00000	+	24.0000i	−0.280000	+	0.960000i
$626$		−	50.0000i		−	1.99840i
$627$		0			0
$628$	10.0000	−	10.0000i	0.399043	−	0.399043i
$629$	70.0000			2.79108
$630$		0			0
$631$		0			0			−	1.00000i	$-0.5\pi$
$631$		0			0				1.00000i	$0.5\pi$
$632$		0			0
$633$		0			0
$634$			6.00000i			0.238290i
$635$		0			0
$636$		0			0
$637$	−21.0000	−	14.0000i	−0.832050	−	0.554700i
$638$		0			0
$639$		0			0
$640$	−8.00000	−	24.0000i	−0.316228	−	0.948683i
$641$	−8.00000			−0.315981			−0.157991	−	0.987441i	$-0.550502\pi$
$641$	−8.00000			−0.315981			−0.157991	+	0.987441i	$0.550502\pi$
$642$		0			0
$643$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$643$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$647$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$648$	18.0000	+	18.0000i	0.707107	+	0.707107i
$649$		0			0
$650$	17.0000	−	19.0000i	0.666795	−	0.745241i
$651$		0			0
$652$		0			0
$653$	35.0000	−	35.0000i	1.36966	−	1.36966i	0.508729	−	0.860927i	$-0.330115\pi$
$653$	35.0000	−	35.0000i	1.36966	−	1.36966i	0.860927	−	0.508729i	$-0.169885\pi$
$654$		0			0
$655$		0			0
$656$		−	40.0000i		−	1.56174i
$657$	33.0000	−	33.0000i	1.28745	−	1.28745i
$658$		0			0
$659$		0			0			−	1.00000i	$-0.5\pi$
$659$		0			0				1.00000i	$0.5\pi$
$660$		0			0
$661$		−	50.0000i		−	1.94477i	−0.233373	−	0.972387i	$-0.574976\pi$
$661$		−	50.0000i		−	1.94477i	0.233373	−	0.972387i	$-0.425024\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$	−42.0000			−1.62747
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	35.0000	−	35.0000i	1.34915	−	1.34915i	0.462566	−	0.886585i	$-0.346929\pi$
$673$	35.0000	−	35.0000i	1.34915	−	1.34915i	0.886585	−	0.462566i	$-0.153071\pi$
$674$	−50.0000			−1.92593
$675$		0			0
$676$	−24.0000	+	10.0000i	−0.923077	+	0.384615i
$677$	25.0000	+	25.0000i	0.960828	+	0.960828i	0.999261	−	0.0384331i	$-0.0122367\pi$
$677$	25.0000	+	25.0000i	0.960828	+	0.960828i	−0.0384331	+	0.999261i	$0.512237\pi$
$678$		0			0
$679$		0			0
$680$	−20.0000	+	40.0000i	−0.766965	+	1.53393i
$681$		0			0
$682$		0			0
$683$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$683$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$684$		0			0
$685$	21.0000	−	7.00000i	0.802369	−	0.267456i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−5.00000	−	25.0000i	−0.190485	−	0.952424i
$690$		0			0
$691$		0			0			−	1.00000i	$-0.5\pi$
$691$		0			0				1.00000i	$0.5\pi$
$692$	30.0000	+	30.0000i	1.14043	+	1.14043i
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−50.0000	+	50.0000i	−1.89389	+	1.89389i
$698$	−10.0000	+	10.0000i	−0.378506	+	0.378506i
$699$		0			0
$700$		0			0
$701$	52.0000			1.96401			0.982006	−	0.188847i	$-0.0604752\pi$
$701$	52.0000			1.96401			0.982006	+	0.188847i	$0.0604752\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$	−18.0000			−0.677439
$707$		0			0
$708$		0			0
$709$	−30.0000			−1.12667			−0.563337	−	0.826227i	$-0.690483\pi$
$709$	−30.0000			−1.12667			−0.563337	+	0.826227i	$0.690483\pi$
$710$		0			0
$711$		0			0
$712$	−20.0000	−	20.0000i	−0.749532	−	0.749532i
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$	12.0000	−	24.0000i	0.447214	−	0.894427i
$721$		0			0
$722$	19.0000	−	19.0000i	0.707107	−	0.707107i
$723$		0			0
$724$		−	36.0000i		−	1.33793i
$725$	−16.0000	+	12.0000i	−0.594225	+	0.445669i
$726$		0			0
$727$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$727$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$728$		0			0
$729$			27.0000i			1.00000i
$730$	−44.0000	−	22.0000i	−1.62851	−	0.814257i
$731$		0			0
$732$		0			0
$733$	29.0000	+	29.0000i	1.07114	+	1.07114i	0.997268	+	0.0738717i	$0.0235355\pi$
$733$	29.0000	+	29.0000i	1.07114	+	1.07114i	0.0738717	+	0.997268i	$0.476464\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$	30.0000	−	30.0000i	1.10432	−	1.10432i
$739$		0			0		1.00000			$0$
$739$		0			0		−1.00000			$\pi$
$740$	14.0000	+	42.0000i	0.514650	+	1.54395i
$741$		0			0
$742$		0			0
$743$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$743$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$744$		0			0
$745$	40.0000	+	20.0000i	1.46549	+	0.732743i
$746$			50.0000i			1.83063i
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		1.00000			$0$
$751$		0			0		−1.00000			$\pi$
$752$		0			0
$753$		0			0
$754$	20.0000	−	4.00000i	0.728357	−	0.145671i
$755$		0			0
$756$		0			0
$757$	35.0000	+	35.0000i	1.27210	+	1.27210i	0.944986	+	0.327111i	$0.106075\pi$
$757$	35.0000	+	35.0000i	1.27210	+	1.27210i	0.327111	+	0.944986i	$0.393925\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	40.0000i		−	1.45000i	−0.688749	−	0.724999i	$-0.741840\pi$
$761$		−	40.0000i		−	1.45000i	0.688749	−	0.724999i	$-0.258160\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	−45.0000	+	15.0000i	−1.62698	+	0.542326i
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−50.0000			−1.80305			−0.901523	−	0.432731i	$-0.857550\pi$
$769$	−50.0000			−1.80305			−0.901523	+	0.432731i	$0.857550\pi$
$770$		0			0
$771$		0			0
$772$	38.0000	−	38.0000i	1.36765	−	1.36765i
$773$	−39.0000	−	39.0000i	−1.40273	−	1.40273i	−0.791285	−	0.611448i	$-0.790588\pi$
$773$	−39.0000	−	39.0000i	−1.40273	−	1.40273i	−0.611448	−	0.791285i	$-0.709412\pi$
$774$		0			0
$775$		0			0
$776$	−52.0000			−1.86669
$777$		0			0
$778$	−34.0000	−	34.0000i	−1.21896	−	1.21896i
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$			28.0000i			1.00000i
$785$	−5.00000	−	15.0000i	−0.178458	−	0.535373i
$786$		0			0
$787$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$787$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$788$	−26.0000	−	26.0000i	−0.926212	−	0.926212i
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−24.0000	+	36.0000i	−0.852265	+	1.27840i
$794$			26.0000i			0.922705i
$795$		0			0
$796$		0			0
$797$	−15.0000	−	15.0000i	−0.531327	−	0.531327i	0.389640	−	0.920967i	$-0.372599\pi$
$797$	−15.0000	−	15.0000i	−0.531327	−	0.531327i	−0.920967	+	0.389640i	$0.872599\pi$
$798$		0			0
$799$		0			0
$800$	−28.0000	−	4.00000i	−0.989949	−	0.141421i
$801$		−	30.0000i		−	1.06000i
$802$	−40.0000	−	40.0000i	−1.41245	−	1.41245i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$	−4.00000	−	4.00000i	−0.140720	−	0.140720i
$809$		−	56.0000i		−	1.96886i	−0.175791	−	0.984428i	$-0.556248\pi$
$809$		−	56.0000i		−	1.96886i	0.175791	−	0.984428i	$-0.443752\pi$
$810$	27.0000	−	9.00000i	0.948683	−	0.316228i
$811$		0			0			−	1.00000i	$-0.5\pi$
$811$		0			0				1.00000i	$0.5\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$	−40.0000	+	40.0000i	−1.39857	+	1.39857i
$819$		0			0
$820$	−40.0000	−	20.0000i	−1.39686	−	0.698430i
$821$			50.0000i			1.74501i	0.488603	+	0.872506i	$0.337507\pi$
$821$			50.0000i			1.74501i	−0.488603	+	0.872506i	$0.662493\pi$
$822$		0			0
$823$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$823$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$827$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$828$		0			0
$829$			54.0000i			1.87550i	0.347314	+	0.937749i	$0.387094\pi$
$829$			54.0000i			1.87550i	−0.347314	+	0.937749i	$0.612906\pi$
$830$		0			0
$831$		0			0
$832$	24.0000	+	16.0000i	0.832050	+	0.554700i
$833$	35.0000	−	35.0000i	1.21268	−	1.21268i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		1.00000			$0$
$839$		0			0		−1.00000			$\pi$
$840$		0			0
$841$	13.0000			0.448276
$842$	−30.0000	−	30.0000i	−1.03387	−	1.03387i
$843$		0			0
$844$		0			0
$845$	−2.00000	+	29.0000i	−0.0688021	+	0.997630i
$846$		0			0
$847$		0			0
$848$	−20.0000	+	20.0000i	−0.686803	+	0.686803i
$849$		0			0
$850$	30.0000	+	40.0000i	1.02899	+	1.37199i
$851$		0			0
$852$		0			0
$853$	41.0000	+	41.0000i	1.40381	+	1.40381i	0.787505	+	0.616308i	$0.211372\pi$
$853$	41.0000	+	41.0000i	1.40381	+	1.40381i	0.616308	+	0.787505i	$0.288628\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	25.0000	+	25.0000i	0.853984	+	0.853984i	0.990621	−	0.136637i	$-0.0436295\pi$
$857$	25.0000	+	25.0000i	0.853984	+	0.853984i	−0.136637	+	0.990621i	$0.543630\pi$
$858$		0			0
$859$		0			0			−	1.00000i	$-0.5\pi$
$859$		0			0				1.00000i	$0.5\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$863$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$864$		0			0
$865$	45.0000	−	15.0000i	1.53005	−	0.510015i
$866$		−	10.0000i		−	0.339814i
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$	40.0000	+	40.0000i	1.35457	+	1.35457i
$873$	−39.0000	−	39.0000i	−1.31995	−	1.31995i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	23.0000	−	23.0000i	0.776655	−	0.776655i	−0.202606	−	0.979260i	$-0.564941\pi$
$877$	23.0000	−	23.0000i	0.776655	−	0.776655i	0.979260	+	0.202606i	$0.0649409\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−32.0000			−1.07811			−0.539054	−	0.842271i	$-0.681218\pi$
$881$	−32.0000			−1.07811			−0.539054	+	0.842271i	$0.681218\pi$
$882$	−21.0000	+	21.0000i	−0.707107	+	0.707107i
$883$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$883$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$884$	−10.0000	−	50.0000i	−0.336336	−	1.68168i
$885$		0			0
$886$		0			0
$887$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$887$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$888$		0			0
$889$		0			0
$890$	−30.0000	+	10.0000i	−1.00560	+	0.335201i
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	40.0000	−	40.0000i	1.33482	−	1.33482i
$899$		0			0
$900$	−18.0000	−	24.0000i	−0.600000	−	0.800000i
$901$	50.0000			1.66574
$902$		0			0
$903$		0			0
$904$	60.0000			1.99557
$905$	−36.0000	−	18.0000i	−1.19668	−	0.598340i
$906$		0			0
$907$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$907$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$908$		0			0
$909$		−	6.00000i		−	0.199007i
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$		0			0
$913$		0			0
$914$			34.0000i			1.12462i
$915$		0			0
$916$		−	60.0000i		−	1.98246i
$917$		0			0
$918$		0			0
$919$		0			0			−	1.00000i	$-0.5\pi$
$919$		0			0				1.00000i	$0.5\pi$
$920$		0			0
$921$		0			0
$922$	20.0000	+	20.0000i	0.658665	+	0.658665i
$923$		0			0
$924$		0			0
$925$	49.0000	+	7.00000i	1.61111	+	0.230159i
$926$		0			0
$927$		0			0
$928$	−16.0000	−	16.0000i	−0.525226	−	0.525226i
$929$	40.0000			1.31236			0.656179	−	0.754606i	$-0.272172\pi$
$929$	40.0000			1.31236			0.656179	+	0.754606i	$0.272172\pi$
$930$		0			0
$931$		0			0
$932$	10.0000	+	10.0000i	0.327561	+	0.327561i
$933$		0			0
$934$		0			0
$935$		0			0
$936$	6.00000	+	30.0000i	0.196116	+	0.980581i
$937$	−5.00000	−	5.00000i	−0.163343	−	0.163343i	0.620703	−	0.784046i	$-0.286847\pi$
$937$	−5.00000	−	5.00000i	−0.163343	−	0.163343i	−0.784046	+	0.620703i	$0.786847\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$			20.0000i			0.651981i	0.945373	+	0.325991i	$0.105698\pi$
$941$			20.0000i			0.651981i	−0.945373	+	0.325991i	$0.894302\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$947$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$948$		0			0
$949$	55.0000	−	11.0000i	1.78538	−	0.357075i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	15.0000	−	15.0000i	0.485898	−	0.485898i	−0.421111	−	0.907009i	$-0.638360\pi$
$953$	15.0000	−	15.0000i	0.485898	−	0.485898i	0.907009	+	0.421111i	$0.138360\pi$
$954$	−30.0000			−0.971286
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−31.0000			−1.00000
$962$	−42.0000	−	28.0000i	−1.35413	−	0.902756i
$963$		0			0
$964$	60.0000			1.93247
$965$	−19.0000	−	57.0000i	−0.611632	−	1.83489i
$966$		0			0
$967$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$967$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$968$	22.0000	+	22.0000i	0.707107	+	0.707107i
$969$		0			0
$970$	−26.0000	+	52.0000i	−0.834810	+	1.66962i
$971$		0			0		1.00000			$0$
$971$		0			0		−1.00000			$\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$	48.0000			1.53644
$977$	27.0000	−	27.0000i	0.863807	−	0.863807i	−0.127971	−	0.991778i	$-0.540847\pi$
$977$	27.0000	−	27.0000i	0.863807	−	0.863807i	0.991778	+	0.127971i	$0.0408466\pi$
$978$		0			0
$979$		0			0
$980$	28.0000	+	14.0000i	0.894427	+	0.447214i
$981$			60.0000i			1.91565i
$982$		0			0
$983$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$983$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$984$		0			0
$985$	−39.0000	+	13.0000i	−1.24264	+	0.414214i
$986$			40.0000i			1.27386i
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		1.00000			$0$
$991$		0			0		−1.00000			$\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−25.0000	−	25.0000i	−0.791758	−	0.791758i	0.190022	−	0.981780i	$-0.439144\pi$
$997$	−25.0000	−	25.0000i	−0.791758	−	0.791758i	−0.981780	+	0.190022i	$0.939144\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	260.2.p.b.207.1	yes	2
4.3	odd	2	CM	260.2.p.b.207.1	yes	2
5.3	odd	4		260.2.p.a.103.1	✓	2
13.12	even	2		260.2.p.a.207.1	yes	2
20.3	even	4		260.2.p.a.103.1	✓	2
52.51	odd	2		260.2.p.a.207.1	yes	2
65.38	odd	4	inner	260.2.p.b.103.1	yes	2
260.103	even	4	inner	260.2.p.b.103.1	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
260.2.p.a.103.1	✓	2	5.3	odd	4
260.2.p.a.103.1	✓	2	20.3	even	4
260.2.p.a.207.1	yes	2	13.12	even	2
260.2.p.a.207.1	yes	2	52.51	odd	2
260.2.p.b.103.1	yes	2	65.38	odd	4	inner
260.2.p.b.103.1	yes	2	260.103	even	4	inner
260.2.p.b.207.1	yes	2	1.1	even	1	trivial
260.2.p.b.207.1	yes	2	4.3	odd	2	CM

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:	\( N \)	\(=\)	\( 260 = 2^{2} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	260.p (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(1.00000 - 1.00000i) q^{2} -2.00000i q^{4} +(-2.00000 - 1.00000i) q^{5} +(-2.00000 - 2.00000i) q^{8} -3.00000i q^{9} +O(q^{10})\)	\(q+(1.00000 - 1.00000i) q^{2} -2.00000i q^{4} +(-2.00000 - 1.00000i) q^{5} +(-2.00000 - 2.00000i) q^{8} -3.00000i q^{9} +(-3.00000 + 1.00000i) q^{10} +(2.00000 - 3.00000i) q^{13} -4.00000 q^{16} +(5.00000 + 5.00000i) q^{17} +(-3.00000 - 3.00000i) q^{18} +(-2.00000 + 4.00000i) q^{20} +(3.00000 + 4.00000i) q^{25} +(-1.00000 - 5.00000i) q^{26} +4.00000i q^{29} +(-4.00000 + 4.00000i) q^{32} +10.0000 q^{34} -6.00000 q^{36} +(7.00000 - 7.00000i) q^{37} +(2.00000 + 6.00000i) q^{40} +10.0000i q^{41} +(-3.00000 + 6.00000i) q^{45} -7.00000i q^{49} +(7.00000 + 1.00000i) q^{50} +(-6.00000 - 4.00000i) q^{52} +(5.00000 - 5.00000i) q^{53} +(4.00000 + 4.00000i) q^{58} -12.0000 q^{61} +8.00000i q^{64} +(-7.00000 + 4.00000i) q^{65} +(10.0000 - 10.0000i) q^{68} +(-6.00000 + 6.00000i) q^{72} +(11.0000 + 11.0000i) q^{73} -14.0000i q^{74} +(8.00000 + 4.00000i) q^{80} -9.00000 q^{81} +(10.0000 + 10.0000i) q^{82} +(-5.00000 - 15.0000i) q^{85} +10.0000 q^{89} +(3.00000 + 9.00000i) q^{90} +(13.0000 - 13.0000i) q^{97} +(-7.00000 - 7.00000i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} - 4 q^{5} - 4 q^{8}+O(q^{10}) \) 2 * q + 2 * q^2 - 4 * q^5 - 4 * q^8	\( 2 q + 2 q^{2} - 4 q^{5} - 4 q^{8} - 6 q^{10} + 4 q^{13} - 8 q^{16} + 10 q^{17} - 6 q^{18} - 4 q^{20} + 6 q^{25} - 2 q^{26} - 8 q^{32} + 20 q^{34} - 12 q^{36} + 14 q^{37} + 4 q^{40} - 6 q^{45} + 14 q^{50} - 12 q^{52} + 10 q^{53} + 8 q^{58} - 24 q^{61} - 14 q^{65} + 20 q^{68} - 12 q^{72} + 22 q^{73} + 16 q^{80} - 18 q^{81} + 20 q^{82} - 10 q^{85} + 20 q^{89} + 6 q^{90} + 26 q^{97} - 14 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 - 4 * q^5 - 4 * q^8 - 6 * q^10 + 4 * q^13 - 8 * q^16 + 10 * q^17 - 6 * q^18 - 4 * q^20 + 6 * q^25 - 2 * q^26 - 8 * q^32 + 20 * q^34 - 12 * q^36 + 14 * q^37 + 4 * q^40 - 6 * q^45 + 14 * q^50 - 12 * q^52 + 10 * q^53 + 8 * q^58 - 24 * q^61 - 14 * q^65 + 20 * q^68 - 12 * q^72 + 22 * q^73 + 16 * q^80 - 18 * q^81 + 20 * q^82 - 10 * q^85 + 20 * q^89 + 6 * q^90 + 26 * q^97 - 14 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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