LMFDB - Embedded newform 260.2.u.b.239.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [260,2,Mod(99,260)]

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, 2, 1]))

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

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

chi := DirichletCharacter("260.99");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 260 = 2^{2} \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	260.u (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			239.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	260.239
Dual form			260.2.u.b.99.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.00000 + 1.00000i) q^{2} +2.00000i q^{4} +(1.00000 + 2.00000i) q^{5} +(-2.00000 + 2.00000i) q^{8} -3.00000 q^{9} +O(q^{10})$	\(q+(1.00000 + 1.00000i) q^{2} +2.00000i q^{4} +(1.00000 + 2.00000i) q^{5} +(-2.00000 + 2.00000i) q^{8} -3.00000 q^{9} +(-1.00000 + 3.00000i) q^{10} +(2.00000 - 3.00000i) q^{13} -4.00000 q^{16} +8.00000 q^{17} +(-3.00000 - 3.00000i) q^{18} +(-4.00000 + 2.00000i) q^{20} +(-3.00000 + 4.00000i) q^{25} +(5.00000 - 1.00000i) q^{26} -4.00000 q^{29} +(-4.00000 - 4.00000i) q^{32} +(8.00000 + 8.00000i) q^{34} -6.00000i q^{36} +(5.00000 - 5.00000i) q^{37} +(-6.00000 - 2.00000i) q^{40} +(9.00000 - 9.00000i) q^{41} +(-3.00000 - 6.00000i) q^{45} -7.00000i q^{49} +(-7.00000 + 1.00000i) q^{50} +(6.00000 + 4.00000i) q^{52} +4.00000i q^{53} +(-4.00000 - 4.00000i) q^{58} -10.0000 q^{61} -8.00000i q^{64} +(8.00000 + 1.00000i) q^{65} +16.0000i q^{68} +(6.00000 - 6.00000i) q^{72} +(-11.0000 + 11.0000i) q^{73} +10.0000 q^{74} +(-4.00000 - 8.00000i) q^{80} +9.00000 q^{81} +18.0000 q^{82} +(8.00000 + 16.0000i) q^{85} +(-13.0000 - 13.0000i) q^{89} +(3.00000 - 9.00000i) q^{90} +(-5.00000 - 5.00000i) q^{97} +(7.00000 - 7.00000i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{5} - 4 q^{8} - 6 q^{9}+O(q^{10}) $ 2 * q + 2 * q^2 + 2 * q^5 - 4 * q^8 - 6 * q^9	$ 2 q + 2 q^{2} + 2 q^{5} - 4 q^{8} - 6 q^{9} - 2 q^{10} + 4 q^{13} - 8 q^{16} + 16 q^{17} - 6 q^{18} - 8 q^{20} - 6 q^{25} + 10 q^{26} - 8 q^{29} - 8 q^{32} + 16 q^{34} + 10 q^{37} - 12 q^{40} + 18 q^{41} - 6 q^{45} - 14 q^{50} + 12 q^{52} - 8 q^{58} - 20 q^{61} + 16 q^{65} + 12 q^{72} - 22 q^{73} + 20 q^{74} - 8 q^{80} + 18 q^{81} + 36 q^{82} + 16 q^{85} - 26 q^{89} + 6 q^{90} - 10 q^{97} + 14 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^5 - 4 * q^8 - 6 * q^9 - 2 * q^10 + 4 * q^13 - 8 * q^16 + 16 * q^17 - 6 * q^18 - 8 * q^20 - 6 * q^25 + 10 * q^26 - 8 * q^29 - 8 * q^32 + 16 * q^34 + 10 * q^37 - 12 * q^40 + 18 * q^41 - 6 * q^45 - 14 * q^50 + 12 * q^52 - 8 * q^58 - 20 * q^61 + 16 * q^65 + 12 * q^72 - 22 * q^73 + 20 * q^74 - 8 * q^80 + 18 * q^81 + 36 * q^82 + 16 * q^85 - 26 * q^89 + 6 * q^90 - 10 * q^97 + 14 * q^98

Display 100 coefficients 10 coefficients

Character values

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

$n$	$41$	$131$	$157$
$\chi(n)$	$e\left(\frac{3}{4}\right)$	$-1$	$-1$

Coefficient data

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

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

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

$2$	1.00000	+	1.00000i	0.707107	+	0.707107i
$2$	1.00000	+	1.00000i	0.707107	+	0.707107i
$3$		0			0			−	1.00000i	$-0.5\pi$
$3$		0			0				1.00000i	$0.5\pi$
$4$			2.00000i			1.00000i
$5$	1.00000	+	2.00000i	0.447214	+	0.894427i
$5$	1.00000	+	2.00000i	0.447214	+	0.894427i
$6$		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
$9$	−3.00000			−1.00000
$10$	−1.00000	+	3.00000i	−0.316228	+	0.948683i
$11$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$11$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$12$		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
$17$	8.00000			1.94029			0.970143	−	0.242536i	$-0.0779791\pi$
$17$	8.00000			1.94029			0.970143	+	0.242536i	$0.0779791\pi$
$18$	−3.00000	−	3.00000i	−0.707107	−	0.707107i
$19$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$19$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$20$	−4.00000	+	2.00000i	−0.894427	+	0.447214i
$21$		0			0
$22$		0			0
$23$		0			0		1.00000			$0$
$23$		0			0		−1.00000			$\pi$
$24$		0			0
$25$	−3.00000	+	4.00000i	−0.600000	+	0.800000i
$26$	5.00000	−	1.00000i	0.980581	−	0.196116i
$27$		0			0
$28$		0			0
$29$	−4.00000			−0.742781			−0.371391	−	0.928477i	$-0.621119\pi$
$29$	−4.00000			−0.742781			−0.371391	+	0.928477i	$0.621119\pi$
$30$		0			0
$31$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$31$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$32$	−4.00000	−	4.00000i	−0.707107	−	0.707107i
$33$		0			0
$34$	8.00000	+	8.00000i	1.37199	+	1.37199i
$35$		0			0
$36$		−	6.00000i		−	1.00000i
$37$	5.00000	−	5.00000i	0.821995	−	0.821995i	−0.164399	−	0.986394i	$-0.552568\pi$
$37$	5.00000	−	5.00000i	0.821995	−	0.821995i	0.986394	+	0.164399i	$0.0525685\pi$
$38$		0			0
$39$		0			0
$40$	−6.00000	−	2.00000i	−0.948683	−	0.316228i
$41$	9.00000	−	9.00000i	1.40556	−	1.40556i	0.624695	−	0.780869i	$-0.285223\pi$
$41$	9.00000	−	9.00000i	1.40556	−	1.40556i	0.780869	−	0.624695i	$-0.214777\pi$
$42$		0			0
$43$		0			0		1.00000			$0$
$43$		0			0		−1.00000			$\pi$
$44$		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
$53$			4.00000i			0.549442i	0.961524	+	0.274721i	$0.0885855\pi$
$53$			4.00000i			0.549442i	−0.961524	+	0.274721i	$0.911414\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$	−4.00000	−	4.00000i	−0.525226	−	0.525226i
$59$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$59$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$60$		0			0
$61$	−10.0000			−1.28037			−0.640184	−	0.768221i	$-0.721142\pi$
$61$	−10.0000			−1.28037			−0.640184	+	0.768221i	$0.721142\pi$
$62$		0			0
$63$		0			0
$64$		−	8.00000i		−	1.00000i
$65$	8.00000	+	1.00000i	0.992278	+	0.124035i
$66$		0			0
$67$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$67$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$68$			16.0000i			1.94029i
$69$		0			0
$70$		0			0
$71$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$71$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$72$	6.00000	−	6.00000i	0.707107	−	0.707107i
$73$	−11.0000	+	11.0000i	−1.28745	+	1.28745i	−0.351123	+	0.936329i	$0.614200\pi$
$73$	−11.0000	+	11.0000i	−1.28745	+	1.28745i	−0.936329	+	0.351123i	$0.885800\pi$
$74$	10.0000			1.16248
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		1.00000			$0$
$79$		0			0		−1.00000			$\pi$
$80$	−4.00000	−	8.00000i	−0.447214	−	0.894427i
$81$	9.00000			1.00000
$82$	18.0000			1.98777
$83$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$83$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$84$		0			0
$85$	8.00000	+	16.0000i	0.867722	+	1.73544i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−13.0000	−	13.0000i	−1.37800	−	1.37800i	−0.847998	−	0.529999i	$-0.822192\pi$
$89$	−13.0000	−	13.0000i	−1.37800	−	1.37800i	−0.529999	−	0.847998i	$-0.677808\pi$
$90$	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
$97$	−5.00000	−	5.00000i	−0.507673	−	0.507673i	0.406138	−	0.913812i	$-0.366875\pi$
$97$	−5.00000	−	5.00000i	−0.507673	−	0.507673i	−0.913812	+	0.406138i	$0.866875\pi$
$98$	7.00000	−	7.00000i	0.707107	−	0.707107i
$99$		0			0
$100$	−8.00000	−	6.00000i	−0.800000	−	0.600000i
$101$			20.0000i			1.99007i	0.0995037	+	0.995037i	$0.468274\pi$
$101$			20.0000i			1.99007i	−0.0995037	+	0.995037i	$0.531726\pi$
$102$		0			0
$103$		0			0		1.00000			$0$
$103$		0			0		−1.00000			$\pi$
$104$	2.00000	+	10.0000i	0.196116	+	0.980581i
$105$		0			0
$106$	−4.00000	+	4.00000i	−0.388514	+	0.388514i
$107$		0			0			−	1.00000i	$-0.5\pi$
$107$		0			0				1.00000i	$0.5\pi$
$108$		0			0
$109$	−13.0000	+	13.0000i	−1.24517	+	1.24517i	−0.287348	+	0.957826i	$0.592774\pi$
$109$	−13.0000	+	13.0000i	−1.24517	+	1.24517i	−0.957826	+	0.287348i	$0.907226\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	14.0000i		−	1.31701i	−0.752577	−	0.658505i	$-0.771189\pi$
$113$		−	14.0000i		−	1.31701i	0.752577	−	0.658505i	$-0.228811\pi$
$114$		0			0
$115$		0			0
$116$		−	8.00000i		−	0.742781i
$117$	−6.00000	+	9.00000i	−0.554700	+	0.832050i
$118$		0			0
$119$		0			0
$120$		0			0
$121$			11.0000i			1.00000i
$122$	−10.0000	−	10.0000i	−0.905357	−	0.905357i
$123$		0			0
$124$		0			0
$125$	−11.0000	−	2.00000i	−0.983870	−	0.178885i
$126$		0			0
$127$		0			0		1.00000			$0$
$127$		0			0		−1.00000			$\pi$
$128$	8.00000	−	8.00000i	0.707107	−	0.707107i
$129$		0			0
$130$	7.00000	+	9.00000i	0.613941	+	0.789352i
$131$		0			0		1.00000			$0$
$131$		0			0		−1.00000			$\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$	−16.0000	+	16.0000i	−1.37199	+	1.37199i
$137$	7.00000	−	7.00000i	0.598050	−	0.598050i	−0.341743	−	0.939793i	$-0.611017\pi$
$137$	7.00000	−	7.00000i	0.598050	−	0.598050i	0.939793	+	0.341743i	$0.111017\pi$
$138$		0			0
$139$		0			0		1.00000			$0$
$139$		0			0		−1.00000			$\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	12.0000			1.00000
$145$	−4.00000	−	8.00000i	−0.332182	−	0.664364i
$146$	−22.0000			−1.82073
$147$		0			0
$148$	10.0000	+	10.0000i	0.821995	+	0.821995i
$149$	3.00000	−	3.00000i	0.245770	−	0.245770i	−0.573462	−	0.819232i	$-0.694400\pi$
$149$	3.00000	−	3.00000i	0.245770	−	0.245770i	0.819232	+	0.573462i	$0.194400\pi$
$150$		0			0
$151$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$151$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$152$		0			0
$153$	−24.0000			−1.94029
$154$		0			0
$155$		0			0
$156$		0			0
$157$			22.0000i			1.75579i	0.478852	+	0.877896i	$0.341053\pi$
$157$			22.0000i			1.75579i	−0.478852	+	0.877896i	$0.658947\pi$
$158$		0			0
$159$		0			0
$160$	4.00000	−	12.0000i	0.316228	−	0.948683i
$161$		0			0
$162$	9.00000	+	9.00000i	0.707107	+	0.707107i
$163$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$163$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$164$	18.0000	+	18.0000i	1.40556	+	1.40556i
$165$		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
$170$	−8.00000	+	24.0000i	−0.613572	+	1.84072i
$171$		0			0
$172$		0			0
$173$	26.0000			1.97674			0.988372	−	0.152057i	$-0.0485898\pi$
$173$	26.0000			1.97674			0.988372	+	0.152057i	$0.0485898\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		−	26.0000i		−	1.94878i
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$	12.0000	−	6.00000i	0.894427	−	0.447214i
$181$			18.0000i			1.33793i	0.743294	+	0.668965i	$0.233262\pi$
$181$			18.0000i			1.33793i	−0.743294	+	0.668965i	$0.766738\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	15.0000	+	5.00000i	1.10282	+	0.367607i
$186$		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
$193$	−5.00000	+	5.00000i	−0.359908	+	0.359908i	−0.863779	−	0.503871i	$-0.831909\pi$
$193$	−5.00000	+	5.00000i	−0.359908	+	0.359908i	0.503871	+	0.863779i	$0.331909\pi$
$194$		−	10.0000i		−	0.717958i
$195$		0			0
$196$	14.0000			1.00000
$197$	−15.0000	−	15.0000i	−1.06871	−	1.06871i	−0.997459	−	0.0712470i	$-0.977302\pi$
$197$	−15.0000	−	15.0000i	−1.06871	−	1.06871i	−0.0712470	−	0.997459i	$-0.522698\pi$
$198$		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
$202$	−20.0000	+	20.0000i	−1.40720	+	1.40720i
$203$		0			0
$204$		0			0
$205$	27.0000	+	9.00000i	1.88576	+	0.628587i
$206$		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			$\pi$
$212$	−8.00000			−0.549442
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$	−26.0000			−1.76094
$219$		0			0
$220$		0			0
$221$	16.0000	−	24.0000i	1.07628	−	1.61441i
$222$		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
$225$	9.00000	−	12.0000i	0.600000	−	0.800000i
$226$	14.0000	−	14.0000i	0.931266	−	0.931266i
$227$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$227$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$228$		0			0
$229$	−13.0000	−	13.0000i	−0.859064	−	0.859064i	0.132164	−	0.991228i	$-0.457808\pi$
$229$	−13.0000	−	13.0000i	−0.859064	−	0.859064i	−0.991228	+	0.132164i	$0.957808\pi$
$230$		0			0
$231$		0			0
$232$	8.00000	−	8.00000i	0.525226	−	0.525226i
$233$	26.0000			1.70332			0.851658	−	0.524097i	$-0.175597\pi$
$233$	26.0000			1.70332			0.851658	+	0.524097i	$0.175597\pi$
$234$	−15.0000	+	3.00000i	−0.980581	+	0.196116i
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$239$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$240$		0			0
$241$	−11.0000	−	11.0000i	−0.708572	−	0.708572i	0.257663	−	0.966235i	$-0.417048\pi$
$241$	−11.0000	−	11.0000i	−0.708572	−	0.708572i	−0.966235	+	0.257663i	$0.917048\pi$
$242$	−11.0000	+	11.0000i	−0.707107	+	0.707107i
$243$		0			0
$244$		−	20.0000i		−	1.28037i
$245$	14.0000	−	7.00000i	0.894427	−	0.447214i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$	−9.00000	−	13.0000i	−0.569210	−	0.822192i
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	16.0000			1.00000
$257$	−2.00000			−0.124757			−0.0623783	−	0.998053i	$-0.519869\pi$
$257$	−2.00000			−0.124757			−0.0623783	+	0.998053i	$0.519869\pi$
$258$		0			0
$259$		0			0
$260$	−2.00000	+	16.0000i	−0.124035	+	0.992278i
$261$	12.0000			0.742781
$262$		0			0
$263$		0			0			−	1.00000i	$-0.5\pi$
$263$		0			0				1.00000i	$0.5\pi$
$264$		0			0
$265$	−8.00000	+	4.00000i	−0.491436	+	0.245718i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−20.0000			−1.21942			−0.609711	−	0.792624i	$-0.708714\pi$
$269$	−20.0000			−1.21942			−0.609711	+	0.792624i	$0.708714\pi$
$270$		0			0
$271$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$271$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$272$	−32.0000			−1.94029
$273$		0			0
$274$	14.0000			0.845771
$275$		0			0
$276$		0			0
$277$	−18.0000			−1.08152			−0.540758	−	0.841178i	$-0.681862\pi$
$277$	−18.0000			−1.08152			−0.540758	+	0.841178i	$0.681862\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−21.0000	−	21.0000i	−1.25275	−	1.25275i	−0.954480	−	0.298275i	$-0.903589\pi$
$281$	−21.0000	−	21.0000i	−1.25275	−	1.25275i	−0.298275	−	0.954480i	$-0.596411\pi$
$282$		0			0
$283$		0			0		1.00000			$0$
$283$		0			0		−1.00000			$\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$	12.0000	+	12.0000i	0.707107	+	0.707107i
$289$	47.0000			2.76471
$290$	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	0.116841	−	0.993151i	$-0.462723\pi$
$293$	19.0000	−	19.0000i	1.10999	−	1.10999i	0.993151	−	0.116841i	$-0.0372769\pi$
$294$		0			0
$295$		0			0
$296$			20.0000i			1.16248i
$297$		0			0
$298$	6.00000			0.347571
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−10.0000	−	20.0000i	−0.572598	−	1.14520i
$306$	−24.0000	−	24.0000i	−1.37199	−	1.37199i
$307$		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			−	1.00000i	$-0.5\pi$
$311$		0			0				1.00000i	$0.5\pi$
$312$		0			0
$313$		−	26.0000i		−	1.46961i	−0.678280	−	0.734803i	$-0.737274\pi$
$313$		−	26.0000i		−	1.46961i	0.678280	−	0.734803i	$-0.262726\pi$
$314$	−22.0000	+	22.0000i	−1.24153	+	1.24153i
$315$		0			0
$316$		0			0
$317$	−3.00000	−	3.00000i	−0.168497	−	0.168497i	0.617822	−	0.786318i	$-0.288015\pi$
$317$	−3.00000	−	3.00000i	−0.168497	−	0.168497i	−0.786318	+	0.617822i	$0.788015\pi$
$318$		0			0
$319$		0			0
$320$	16.0000	−	8.00000i	0.894427	−	0.447214i
$321$		0			0
$322$		0			0
$323$		0			0
$324$			18.0000i			1.00000i
$325$	6.00000	+	17.0000i	0.332820	+	0.942990i
$326$		0			0
$327$		0			0
$328$			36.0000i			1.98777i
$329$		0			0
$330$		0			0
$331$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$331$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$332$		0			0
$333$	−15.0000	+	15.0000i	−0.821995	+	0.821995i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	32.0000			1.74315			0.871576	−	0.490261i	$-0.163099\pi$
$337$	32.0000			1.74315			0.871576	+	0.490261i	$0.163099\pi$
$338$	7.00000	−	17.0000i	0.380750	−	0.924678i
$339$		0			0
$340$	−32.0000	+	16.0000i	−1.73544	+	0.867722i
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$	26.0000	+	26.0000i	1.39777	+	1.39777i
$347$		0			0			−	1.00000i	$-0.5\pi$
$347$		0			0				1.00000i	$0.5\pi$
$348$		0			0
$349$	13.0000	+	13.0000i	0.695874	+	0.695874i	0.963518	−	0.267644i	$-0.0862451\pi$
$349$	13.0000	+	13.0000i	0.695874	+	0.695874i	−0.267644	+	0.963518i	$0.586245\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	25.0000	+	25.0000i	1.33062	+	1.33062i	0.904819	+	0.425797i	$0.140006\pi$
$353$	25.0000	+	25.0000i	1.33062	+	1.33062i	0.425797	+	0.904819i	$0.359994\pi$
$354$		0			0
$355$		0			0
$356$	26.0000	−	26.0000i	1.37800	−	1.37800i
$357$		0			0
$358$		0			0
$359$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$359$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$360$	18.0000	+	6.00000i	0.948683	+	0.316228i
$361$		−	19.0000i		−	1.00000i
$362$	−18.0000	+	18.0000i	−0.946059	+	0.946059i
$363$		0			0
$364$		0			0
$365$	−33.0000	−	11.0000i	−1.72730	−	0.575766i
$366$		0			0
$367$		0			0			−	1.00000i	$-0.5\pi$
$367$		0			0				1.00000i	$0.5\pi$
$368$		0			0
$369$	−27.0000	+	27.0000i	−1.40556	+	1.40556i
$370$	10.0000	+	20.0000i	0.519875	+	1.03975i
$371$		0			0
$372$		0			0
$373$			14.0000i			0.724893i	0.932005	+	0.362446i	$0.118058\pi$
$373$			14.0000i			0.724893i	−0.932005	+	0.362446i	$0.881942\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−8.00000	+	12.0000i	−0.412021	+	0.618031i
$378$		0			0
$379$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$379$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$380$		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
$386$	−10.0000			−0.508987
$387$		0			0
$388$	10.0000	−	10.0000i	0.507673	−	0.507673i
$389$			34.0000i			1.72387i	0.507020	+	0.861934i	$0.330747\pi$
$389$			34.0000i			1.72387i	−0.507020	+	0.861934i	$0.669253\pi$
$390$		0			0
$391$		0			0
$392$	14.0000	+	14.0000i	0.707107	+	0.707107i
$393$		0			0
$394$		−	30.0000i		−	1.51138i
$395$		0			0
$396$		0			0
$397$	−25.0000	+	25.0000i	−1.25471	+	1.25471i	−0.301131	+	0.953583i	$0.597364\pi$
$397$	−25.0000	+	25.0000i	−1.25471	+	1.25471i	−0.953583	+	0.301131i	$0.902636\pi$
$398$		0			0
$399$		0			0
$400$	12.0000	−	16.0000i	0.600000	−	0.800000i
$401$	19.0000	+	19.0000i	0.948815	+	0.948815i	0.998752	−	0.0499376i	$-0.0159023\pi$
$401$	19.0000	+	19.0000i	0.948815	+	0.948815i	−0.0499376	+	0.998752i	$0.515902\pi$
$402$		0			0
$403$		0			0
$404$	−40.0000			−1.99007
$405$	9.00000	+	18.0000i	0.447214	+	0.894427i
$406$		0			0
$407$		0			0
$408$		0			0
$409$	23.0000	−	23.0000i	1.13728	−	1.13728i	0.148340	−	0.988936i	$-0.452607\pi$
$409$	23.0000	−	23.0000i	1.13728	−	1.13728i	0.988936	−	0.148340i	$-0.0473931\pi$
$410$	18.0000	+	36.0000i	0.888957	+	1.77791i
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$	−20.0000	+	4.00000i	−0.980581	+	0.196116i
$417$		0			0
$418$		0			0
$419$		0			0		1.00000			$0$
$419$		0			0		−1.00000			$\pi$
$420$		0			0
$421$	1.00000	−	1.00000i	0.0487370	−	0.0487370i	−0.682318	−	0.731055i	$-0.739028\pi$
$421$	1.00000	−	1.00000i	0.0487370	−	0.0487370i	0.731055	+	0.682318i	$0.239028\pi$
$422$		0			0
$423$		0			0
$424$	−8.00000	−	8.00000i	−0.388514	−	0.388514i
$425$	−24.0000	+	32.0000i	−1.16417	+	1.55223i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$431$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$432$		0			0
$433$	−34.0000			−1.63394			−0.816968	−	0.576683i	$-0.804347\pi$
$433$	−34.0000			−1.63394			−0.816968	+	0.576683i	$0.804347\pi$
$434$		0			0
$435$		0			0
$436$	−26.0000	−	26.0000i	−1.24517	−	1.24517i
$437$		0			0
$438$		0			0
$439$		0			0			−	1.00000i	$-0.5\pi$
$439$		0			0				1.00000i	$0.5\pi$
$440$		0			0
$441$			21.0000i			1.00000i
$442$	40.0000	−	8.00000i	1.90261	−	0.380521i
$443$		0			0			−	1.00000i	$-0.5\pi$
$443$		0			0				1.00000i	$0.5\pi$
$444$		0			0
$445$	13.0000	−	39.0000i	0.616259	−	1.84878i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	13.0000	+	13.0000i	0.613508	+	0.613508i	0.943858	−	0.330350i	$-0.107167\pi$
$449$	13.0000	+	13.0000i	0.613508	+	0.613508i	−0.330350	+	0.943858i	$0.607167\pi$
$450$	21.0000	−	3.00000i	0.989949	−	0.141421i
$451$		0			0
$452$	28.0000			1.31701
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−25.0000	−	25.0000i	−1.16945	−	1.16945i	−0.982339	−	0.187112i	$-0.940087\pi$
$457$	−25.0000	−	25.0000i	−1.16945	−	1.16945i	−0.187112	−	0.982339i	$-0.559913\pi$
$458$		−	26.0000i		−	1.21490i
$459$		0			0
$460$		0			0
$461$	−29.0000	+	29.0000i	−1.35066	+	1.35066i	−0.465746	+	0.884918i	$0.654214\pi$
$461$	−29.0000	+	29.0000i	−1.35066	+	1.35066i	−0.884918	+	0.465746i	$0.845786\pi$
$462$		0			0
$463$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$463$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$464$	16.0000			0.742781
$465$		0			0
$466$	26.0000	+	26.0000i	1.20443	+	1.20443i
$467$		0			0		1.00000			$0$
$467$		0			0		−1.00000			$\pi$
$468$	−18.0000	−	12.0000i	−0.832050	−	0.554700i
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$		−	12.0000i		−	0.549442i
$478$		0			0
$479$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$479$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$480$		0			0
$481$	−5.00000	−	25.0000i	−0.227980	−	1.13990i
$482$		−	22.0000i		−	1.00207i
$483$		0			0
$484$	−22.0000			−1.00000
$485$	5.00000	−	15.0000i	0.227038	−	0.681115i
$486$		0			0
$487$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$487$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$488$	20.0000	−	20.0000i	0.905357	−	0.905357i
$489$		0			0
$490$	21.0000	+	7.00000i	0.948683	+	0.316228i
$491$		0			0			−	1.00000i	$-0.5\pi$
$491$		0			0				1.00000i	$0.5\pi$
$492$		0			0
$493$	−32.0000			−1.44121
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$499$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$500$	4.00000	−	22.0000i	0.178885	−	0.983870i
$501$		0			0
$502$		0			0
$503$		0			0			−	1.00000i	$-0.5\pi$
$503$		0			0				1.00000i	$0.5\pi$
$504$		0			0
$505$	−40.0000	+	20.0000i	−1.77998	+	0.889988i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	17.0000	−	17.0000i	0.753512	−	0.753512i	−0.221621	−	0.975133i	$-0.571135\pi$
$509$	17.0000	−	17.0000i	0.753512	−	0.753512i	0.975133	+	0.221621i	$0.0711348\pi$
$510$		0			0
$511$		0			0
$512$	16.0000	+	16.0000i	0.707107	+	0.707107i
$513$		0			0
$514$	−2.00000	−	2.00000i	−0.0882162	−	0.0882162i
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$	−18.0000	+	14.0000i	−0.789352	+	0.613941i
$521$	40.0000			1.75243			0.876216	−	0.481919i	$-0.160060\pi$
$521$	40.0000			1.75243			0.876216	+	0.481919i	$0.160060\pi$
$522$	12.0000	+	12.0000i	0.525226	+	0.525226i
$523$		0			0			−	1.00000i	$-0.5\pi$
$523$		0			0				1.00000i	$0.5\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	23.0000			1.00000
$530$	−12.0000	−	4.00000i	−0.521247	−	0.173749i
$531$		0			0
$532$		0			0
$533$	−9.00000	−	45.0000i	−0.389833	−	1.94917i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$	−20.0000	−	20.0000i	−0.862261	−	0.862261i
$539$		0			0
$540$		0			0
$541$	−31.0000	−	31.0000i	−1.33279	−	1.33279i	−0.902861	−	0.429934i	$-0.858537\pi$
$541$	−31.0000	−	31.0000i	−1.33279	−	1.33279i	−0.429934	−	0.902861i	$-0.641463\pi$
$542$		0			0
$543$		0			0
$544$	−32.0000	−	32.0000i	−1.37199	−	1.37199i
$545$	−39.0000	−	13.0000i	−1.67058	−	0.556859i
$546$		0			0
$547$		0			0			−	1.00000i	$-0.5\pi$
$547$		0			0				1.00000i	$0.5\pi$
$548$	14.0000	+	14.0000i	0.598050	+	0.598050i
$549$	30.0000			1.28037
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$	−18.0000	−	18.0000i	−0.764747	−	0.764747i
$555$		0			0
$556$		0			0
$557$	5.00000	−	5.00000i	0.211857	−	0.211857i	−0.593199	−	0.805056i	$-0.702135\pi$
$557$	5.00000	−	5.00000i	0.211857	−	0.211857i	0.805056	+	0.593199i	$0.202135\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		−	42.0000i		−	1.77166i
$563$		0			0		1.00000			$0$
$563$		0			0		−1.00000			$\pi$
$564$		0			0
$565$	28.0000	−	14.0000i	1.17797	−	0.588984i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			40.0000i			1.67689i	0.544988	+	0.838444i	$0.316534\pi$
$569$			40.0000i			1.67689i	−0.544988	+	0.838444i	$0.683466\pi$
$570$		0			0
$571$		0			0			−	1.00000i	$-0.5\pi$
$571$		0			0				1.00000i	$0.5\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$			24.0000i			1.00000i
$577$	23.0000	+	23.0000i	0.957503	+	0.957503i	0.999133	−	0.0416305i	$-0.0132552\pi$
$577$	23.0000	+	23.0000i	0.957503	+	0.957503i	−0.0416305	+	0.999133i	$0.513255\pi$
$578$	47.0000	+	47.0000i	1.95494	+	1.95494i
$579$		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
$585$	−24.0000	−	3.00000i	−0.992278	−	0.124035i
$586$	38.0000			1.56977
$587$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$587$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$	−20.0000	+	20.0000i	−0.821995	+	0.821995i
$593$	−15.0000	+	15.0000i	−0.615976	+	0.615976i	−0.944497	−	0.328521i	$-0.893450\pi$
$593$	−15.0000	+	15.0000i	−0.615976	+	0.615976i	0.328521	+	0.944497i	$0.393450\pi$
$594$		0			0
$595$		0			0
$596$	6.00000	+	6.00000i	0.245770	+	0.245770i
$597$		0			0
$598$		0			0
$599$		0			0		1.00000			$0$
$599$		0			0		−1.00000			$\pi$
$600$		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			−	1.00000i	$-0.5\pi$
$607$		0			0				1.00000i	$0.5\pi$
$608$		0			0
$609$		0			0
$610$	10.0000	−	30.0000i	0.404888	−	1.21466i
$611$		0			0
$612$		−	48.0000i		−	1.94029i
$613$	1.00000	+	1.00000i	0.0403896	+	0.0403896i	0.727013	−	0.686624i	$-0.240908\pi$
$613$	1.00000	+	1.00000i	0.0403896	+	0.0403896i	−0.686624	+	0.727013i	$0.740908\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	35.0000	+	35.0000i	1.40905	+	1.40905i	0.764911	+	0.644136i	$0.222783\pi$
$617$	35.0000	+	35.0000i	1.40905	+	1.40905i	0.644136	+	0.764911i	$0.277217\pi$
$618$		0			0
$619$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$619$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−7.00000	−	24.0000i	−0.280000	−	0.960000i
$626$	26.0000	−	26.0000i	1.03917	−	1.03917i
$627$		0			0
$628$	−44.0000			−1.75579
$629$	40.0000	−	40.0000i	1.59490	−	1.59490i
$630$		0			0
$631$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$631$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$632$		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
$640$	24.0000	+	8.00000i	0.948683	+	0.316228i
$641$			50.0000i			1.97488i	0.157991	+	0.987441i	$0.449498\pi$
$641$			50.0000i			1.97488i	−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		1.00000			$0$
$647$		0			0		−1.00000			$\pi$
$648$	−18.0000	+	18.0000i	−0.707107	+	0.707107i
$649$		0			0
$650$	−11.0000	+	23.0000i	−0.431455	+	0.902134i
$651$		0			0
$652$		0			0
$653$			26.0000i			1.01746i	0.860927	+	0.508729i	$0.169885\pi$
$653$			26.0000i			1.01746i	−0.860927	+	0.508729i	$0.830115\pi$
$654$		0			0
$655$		0			0
$656$	−36.0000	+	36.0000i	−1.40556	+	1.40556i
$657$	33.0000	−	33.0000i	1.28745	−	1.28745i
$658$		0			0
$659$		0			0		1.00000			$0$
$659$		0			0		−1.00000			$\pi$
$660$		0			0
$661$	−19.0000	−	19.0000i	−0.739014	−	0.739014i	0.233373	−	0.972387i	$-0.425024\pi$
$661$	−19.0000	−	19.0000i	−0.739014	−	0.739014i	−0.972387	+	0.233373i	$0.925024\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$	−30.0000			−1.16248
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	24.0000			0.925132			0.462566	−	0.886585i	$-0.346929\pi$
$673$	24.0000			0.925132			0.462566	+	0.886585i	$0.346929\pi$
$674$	32.0000	+	32.0000i	1.23259	+	1.23259i
$675$		0			0
$676$	24.0000	−	10.0000i	0.923077	−	0.384615i
$677$			52.0000i			1.99852i	0.0384331	+	0.999261i	$0.487763\pi$
$677$			52.0000i			1.99852i	−0.0384331	+	0.999261i	$0.512237\pi$
$678$		0			0
$679$		0			0
$680$	−48.0000	−	16.0000i	−1.84072	−	0.613572i
$681$		0			0
$682$		0			0
$683$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$683$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$684$		0			0
$685$	21.0000	+	7.00000i	0.802369	+	0.267456i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	12.0000	+	8.00000i	0.457164	+	0.304776i
$690$		0			0
$691$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$691$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$692$			52.0000i			1.97674i
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	72.0000	−	72.0000i	2.72719	−	2.72719i
$698$			26.0000i			0.984115i
$699$		0			0
$700$		0			0
$701$			10.0000i			0.377695i	0.982006	+	0.188847i	$0.0604752\pi$
$701$			10.0000i			0.377695i	−0.982006	+	0.188847i	$0.939525\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$			50.0000i			1.88177i
$707$		0			0
$708$		0			0
$709$	37.0000	+	37.0000i	1.38956	+	1.38956i	0.826227	+	0.563337i	$0.190483\pi$
$709$	37.0000	+	37.0000i	1.38956	+	1.38956i	0.563337	+	0.826227i	$0.309517\pi$
$710$		0			0
$711$		0			0
$712$	52.0000			1.94878
$713$		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
$724$	−36.0000			−1.33793
$725$	12.0000	−	16.0000i	0.445669	−	0.594225i
$726$		0			0
$727$		0			0		1.00000			$0$
$727$		0			0		−1.00000			$\pi$
$728$		0			0
$729$	−27.0000			−1.00000
$730$	−22.0000	−	44.0000i	−0.814257	−	1.62851i
$731$		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
$738$	−54.0000			−1.98777
$739$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$739$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$740$	−10.0000	+	30.0000i	−0.367607	+	1.10282i
$741$		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
$745$	9.00000	+	3.00000i	0.329734	+	0.109911i
$746$	−14.0000	+	14.0000i	−0.512576	+	0.512576i
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0			−	1.00000i	$-0.5\pi$
$751$		0			0				1.00000i	$0.5\pi$
$752$		0			0
$753$		0			0
$754$	−20.0000	+	4.00000i	−0.728357	+	0.145671i
$755$		0			0
$756$		0			0
$757$		−	52.0000i		−	1.88997i	−0.327111	−	0.944986i	$-0.606075\pi$
$757$		−	52.0000i		−	1.88997i	0.327111	−	0.944986i	$-0.393925\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−39.0000	−	39.0000i	−1.41375	−	1.41375i	−0.724999	−	0.688749i	$-0.758160\pi$
$761$	−39.0000	−	39.0000i	−1.41375	−	1.41375i	−0.688749	−	0.724999i	$-0.741840\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	−24.0000	−	48.0000i	−0.867722	−	1.73544i
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−13.0000	+	13.0000i	−0.468792	+	0.468792i	−0.901523	−	0.432731i	$-0.857550\pi$
$769$	−13.0000	+	13.0000i	−0.468792	+	0.468792i	0.432731	+	0.901523i	$0.357550\pi$
$770$		0			0
$771$		0			0
$772$	−10.0000	−	10.0000i	−0.359908	−	0.359908i
$773$	5.00000	+	5.00000i	0.179838	+	0.179838i	0.791285	−	0.611448i	$-0.209412\pi$
$773$	5.00000	+	5.00000i	0.179838	+	0.179838i	−0.611448	+	0.791285i	$0.709412\pi$
$774$		0			0
$775$		0			0
$776$	20.0000			0.717958
$777$		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
$785$	−44.0000	+	22.0000i	−1.57043	+	0.785214i
$786$		0			0
$787$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$787$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$788$	30.0000	−	30.0000i	1.06871	−	1.06871i
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−20.0000	+	30.0000i	−0.710221	+	1.06533i
$794$	−50.0000			−1.77443
$795$		0			0
$796$		0			0
$797$	−52.0000			−1.84193			−0.920967	−	0.389640i	$-0.872599\pi$
$797$	−52.0000			−1.84193			−0.920967	+	0.389640i	$0.872599\pi$
$798$		0			0
$799$		0			0
$800$	28.0000	−	4.00000i	0.989949	−	0.141421i
$801$	39.0000	+	39.0000i	1.37800	+	1.37800i
$802$			38.0000i			1.34183i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$	−40.0000	−	40.0000i	−1.40720	−	1.40720i
$809$	−56.0000			−1.96886			−0.984428	−	0.175791i	$-0.943752\pi$
$809$	−56.0000			−1.96886			−0.984428	+	0.175791i	$0.943752\pi$
$810$	−9.00000	+	27.0000i	−0.316228	+	0.948683i
$811$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$811$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$	46.0000			1.60835
$819$		0			0
$820$	−18.0000	+	54.0000i	−0.628587	+	1.88576i
$821$	39.0000	−	39.0000i	1.36111	−	1.36111i	0.488603	−	0.872506i	$-0.337507\pi$
$821$	39.0000	−	39.0000i	1.36111	−	1.36111i	0.872506	−	0.488603i	$-0.162493\pi$
$822$		0			0
$823$		0			0		1.00000			$0$
$823$		0			0		−1.00000			$\pi$
$824$		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
$829$		−	20.0000i		−	0.694629i	−0.937749	−	0.347314i	$-0.887094\pi$
$829$		−	20.0000i		−	0.694629i	0.937749	−	0.347314i	$-0.112906\pi$
$830$		0			0
$831$		0			0
$832$	−24.0000	−	16.0000i	−0.832050	−	0.554700i
$833$		−	56.0000i		−	1.94029i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$839$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$840$		0			0
$841$	−13.0000			−0.448276
$842$	2.00000			0.0689246
$843$		0			0
$844$		0			0
$845$	19.0000	−	22.0000i	0.653620	−	0.756823i
$846$		0			0
$847$		0			0
$848$		−	16.0000i		−	0.549442i
$849$		0			0
$850$	−56.0000	+	8.00000i	−1.92078	+	0.274398i
$851$		0			0
$852$		0			0
$853$	41.0000	−	41.0000i	1.40381	−	1.40381i	0.616308	−	0.787505i	$-0.288628\pi$
$853$	41.0000	−	41.0000i	1.40381	−	1.40381i	0.787505	−	0.616308i	$-0.211372\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	58.0000			1.98124			0.990621	−	0.136637i	$-0.0436295\pi$
$857$	58.0000			1.98124			0.990621	+	0.136637i	$0.0436295\pi$
$858$		0			0
$859$		0			0		1.00000			$0$
$859$		0			0		−1.00000			$\pi$
$860$		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
$865$	26.0000	+	52.0000i	0.884027	+	1.76805i
$866$	−34.0000	−	34.0000i	−1.15537	−	1.15537i
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		−	52.0000i		−	1.76094i
$873$	15.0000	+	15.0000i	0.507673	+	0.507673i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	35.0000	+	35.0000i	1.18187	+	1.18187i	0.979260	+	0.202606i	$0.0649409\pi$
$877$	35.0000	+	35.0000i	1.18187	+	1.18187i	0.202606	+	0.979260i	$0.435059\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			32.0000i			1.07811i	0.842271	+	0.539054i	$0.181218\pi$
$881$			32.0000i			1.07811i	−0.842271	+	0.539054i	$0.818782\pi$
$882$	−21.0000	+	21.0000i	−0.707107	+	0.707107i
$883$		0			0		1.00000			$0$
$883$		0			0		−1.00000			$\pi$
$884$	48.0000	+	32.0000i	1.61441	+	1.07628i
$885$		0			0
$886$		0			0
$887$		0			0			−	1.00000i	$-0.5\pi$
$887$		0			0				1.00000i	$0.5\pi$
$888$		0			0
$889$		0			0
$890$	52.0000	−	26.0000i	1.74304	−	0.871522i
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$			26.0000i			0.867631i
$899$		0			0
$900$	24.0000	+	18.0000i	0.800000	+	0.600000i
$901$			32.0000i			1.06607i
$902$		0			0
$903$		0			0
$904$	28.0000	+	28.0000i	0.931266	+	0.931266i
$905$	−36.0000	+	18.0000i	−1.19668	+	0.598340i
$906$		0			0
$907$		0			0		1.00000			$0$
$907$		0			0		−1.00000			$\pi$
$908$		0			0
$909$		−	60.0000i		−	1.99007i
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$		0			0
$913$		0			0
$914$		−	50.0000i		−	1.65385i
$915$		0			0
$916$	26.0000	−	26.0000i	0.859064	−	0.859064i
$917$		0			0
$918$		0			0
$919$		0			0		1.00000			$0$
$919$		0			0		−1.00000			$\pi$
$920$		0			0
$921$		0			0
$922$	−58.0000			−1.91013
$923$		0			0
$924$		0			0
$925$	5.00000	+	35.0000i	0.164399	+	1.15079i
$926$		0			0
$927$		0			0
$928$	16.0000	+	16.0000i	0.525226	+	0.525226i
$929$	−3.00000	+	3.00000i	−0.0984268	+	0.0984268i	−0.754606	−	0.656179i	$-0.772172\pi$
$929$	−3.00000	+	3.00000i	−0.0984268	+	0.0984268i	0.656179	+	0.754606i	$0.272172\pi$
$930$		0			0
$931$		0			0
$932$			52.0000i			1.70332i
$933$		0			0
$934$		0			0
$935$		0			0
$936$	−6.00000	−	30.0000i	−0.196116	−	0.980581i
$937$			48.0000i			1.56809i	0.620703	+	0.784046i	$0.286847\pi$
$937$			48.0000i			1.56809i	−0.620703	+	0.784046i	$0.713153\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	39.0000	−	39.0000i	1.27136	−	1.27136i	0.325991	−	0.945373i	$-0.394302\pi$
$941$	39.0000	−	39.0000i	1.27136	−	1.27136i	0.945373	−	0.325991i	$-0.105698\pi$
$942$		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
$949$	11.0000	+	55.0000i	0.357075	+	1.78538i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−26.0000			−0.842223			−0.421111	−	0.907009i	$-0.638360\pi$
$953$	−26.0000			−0.842223			−0.421111	+	0.907009i	$0.638360\pi$
$954$	12.0000	−	12.0000i	0.388514	−	0.388514i
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$		−	31.0000i		−	1.00000i
$962$	20.0000	−	30.0000i	0.644826	−	0.967239i
$963$		0			0
$964$	22.0000	−	22.0000i	0.708572	−	0.708572i
$965$	−15.0000	−	5.00000i	−0.482867	−	0.160956i
$966$		0			0
$967$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$967$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$968$	−22.0000	−	22.0000i	−0.707107	−	0.707107i
$969$		0			0
$970$	20.0000	−	10.0000i	0.642161	−	0.321081i
$971$		0			0		1.00000			$0$
$971$		0			0		−1.00000			$\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$	40.0000			1.28037
$977$	−27.0000	−	27.0000i	−0.863807	−	0.863807i	0.127971	−	0.991778i	$-0.459153\pi$
$977$	−27.0000	−	27.0000i	−0.863807	−	0.863807i	−0.991778	+	0.127971i	$0.959153\pi$
$978$		0			0
$979$		0			0
$980$	14.0000	+	28.0000i	0.447214	+	0.894427i
$981$	39.0000	−	39.0000i	1.24517	−	1.24517i
$982$		0			0
$983$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$983$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$984$		0			0
$985$	15.0000	−	45.0000i	0.477940	−	1.43382i
$986$	−32.0000	−	32.0000i	−1.01909	−	1.01909i
$987$		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
$997$			12.0000i			0.380044i	0.981780	+	0.190022i	$0.0608559\pi$
$997$			12.0000i			0.380044i	−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.u.b.239.1	yes	2
4.3	odd	2	CM	260.2.u.b.239.1	yes	2
5.4	even	2		260.2.u.a.239.1	yes	2
13.8	odd	4		260.2.u.a.99.1	✓	2
20.19	odd	2		260.2.u.a.239.1	yes	2
52.47	even	4		260.2.u.a.99.1	✓	2
65.34	odd	4	inner	260.2.u.b.99.1	yes	2
260.99	even	4	inner	260.2.u.b.99.1	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
260.2.u.a.99.1	✓	2	13.8	odd	4
260.2.u.a.99.1	✓	2	52.47	even	4
260.2.u.a.239.1	yes	2	5.4	even	2
260.2.u.a.239.1	yes	2	20.19	odd	2
260.2.u.b.99.1	yes	2	65.34	odd	4	inner
260.2.u.b.99.1	yes	2	260.99	even	4	inner
260.2.u.b.239.1	yes	2	1.1	even	1	trivial
260.2.u.b.239.1	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.u (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(1.00000 + 1.00000i) q^{2} +2.00000i q^{4} +(1.00000 + 2.00000i) q^{5} +(-2.00000 + 2.00000i) q^{8} -3.00000 q^{9} +O(q^{10})\)	\(q+(1.00000 + 1.00000i) q^{2} +2.00000i q^{4} +(1.00000 + 2.00000i) q^{5} +(-2.00000 + 2.00000i) q^{8} -3.00000 q^{9} +(-1.00000 + 3.00000i) q^{10} +(2.00000 - 3.00000i) q^{13} -4.00000 q^{16} +8.00000 q^{17} +(-3.00000 - 3.00000i) q^{18} +(-4.00000 + 2.00000i) q^{20} +(-3.00000 + 4.00000i) q^{25} +(5.00000 - 1.00000i) q^{26} -4.00000 q^{29} +(-4.00000 - 4.00000i) q^{32} +(8.00000 + 8.00000i) q^{34} -6.00000i q^{36} +(5.00000 - 5.00000i) q^{37} +(-6.00000 - 2.00000i) q^{40} +(9.00000 - 9.00000i) q^{41} +(-3.00000 - 6.00000i) q^{45} -7.00000i q^{49} +(-7.00000 + 1.00000i) q^{50} +(6.00000 + 4.00000i) q^{52} +4.00000i q^{53} +(-4.00000 - 4.00000i) q^{58} -10.0000 q^{61} -8.00000i q^{64} +(8.00000 + 1.00000i) q^{65} +16.0000i q^{68} +(6.00000 - 6.00000i) q^{72} +(-11.0000 + 11.0000i) q^{73} +10.0000 q^{74} +(-4.00000 - 8.00000i) q^{80} +9.00000 q^{81} +18.0000 q^{82} +(8.00000 + 16.0000i) q^{85} +(-13.0000 - 13.0000i) q^{89} +(3.00000 - 9.00000i) q^{90} +(-5.00000 - 5.00000i) q^{97} +(7.00000 - 7.00000i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{5} - 4 q^{8} - 6 q^{9}+O(q^{10}) \) 2 * q + 2 * q^2 + 2 * q^5 - 4 * q^8 - 6 * q^9	\( 2 q + 2 q^{2} + 2 q^{5} - 4 q^{8} - 6 q^{9} - 2 q^{10} + 4 q^{13} - 8 q^{16} + 16 q^{17} - 6 q^{18} - 8 q^{20} - 6 q^{25} + 10 q^{26} - 8 q^{29} - 8 q^{32} + 16 q^{34} + 10 q^{37} - 12 q^{40} + 18 q^{41} - 6 q^{45} - 14 q^{50} + 12 q^{52} - 8 q^{58} - 20 q^{61} + 16 q^{65} + 12 q^{72} - 22 q^{73} + 20 q^{74} - 8 q^{80} + 18 q^{81} + 36 q^{82} + 16 q^{85} - 26 q^{89} + 6 q^{90} - 10 q^{97} + 14 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^5 - 4 * q^8 - 6 * q^9 - 2 * q^10 + 4 * q^13 - 8 * q^16 + 16 * q^17 - 6 * q^18 - 8 * q^20 - 6 * q^25 + 10 * q^26 - 8 * q^29 - 8 * q^32 + 16 * q^34 + 10 * q^37 - 12 * q^40 + 18 * q^41 - 6 * q^45 - 14 * q^50 + 12 * q^52 - 8 * q^58 - 20 * q^61 + 16 * q^65 + 12 * q^72 - 22 * q^73 + 20 * q^74 - 8 * q^80 + 18 * q^81 + 36 * q^82 + 16 * q^85 - 26 * q^89 + 6 * q^90 - 10 * q^97 + 14 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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