LMFDB - Embedded newform 260.3.l.a.47.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [260,3,Mod(47,260)]

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, 1, 1]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("260.47");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 260 = 2^{2} \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	260.l (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:	$7.08448687337$
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:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{4}]$

Embedding invariants

Embedding label			47.1
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	260.47
Dual form			260.3.l.a.83.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.00000i q^{2} -4.00000 q^{4} +(-4.00000 + 3.00000i) q^{5} -8.00000i q^{8} -9.00000i q^{9} +O(q^{10})$	$q+2.00000i q^{2} -4.00000 q^{4} +(-4.00000 + 3.00000i) q^{5} -8.00000i q^{8} -9.00000i q^{9} +(-6.00000 - 8.00000i) q^{10} +(-12.0000 - 5.00000i) q^{13} +16.0000 q^{16} +(7.00000 - 7.00000i) q^{17} +18.0000 q^{18} +(16.0000 - 12.0000i) q^{20} +(7.00000 - 24.0000i) q^{25} +(10.0000 - 24.0000i) q^{26} -42.0000i q^{29} +32.0000i q^{32} +(14.0000 + 14.0000i) q^{34} +36.0000i q^{36} -70.0000 q^{37} +(24.0000 + 32.0000i) q^{40} +(-31.0000 - 31.0000i) q^{41} +(27.0000 + 36.0000i) q^{45} +49.0000 q^{49} +(48.0000 + 14.0000i) q^{50} +(48.0000 + 20.0000i) q^{52} +(-17.0000 + 17.0000i) q^{53} +84.0000 q^{58} -120.000 q^{61} -64.0000 q^{64} +(63.0000 - 16.0000i) q^{65} +(-28.0000 + 28.0000i) q^{68} -72.0000 q^{72} +96.0000i q^{73} -140.000i q^{74} +(-64.0000 + 48.0000i) q^{80} -81.0000 q^{81} +(62.0000 - 62.0000i) q^{82} +(-7.00000 + 49.0000i) q^{85} +(-119.000 - 119.000i) q^{89} +(-72.0000 + 54.0000i) q^{90} +130.000i q^{97} +98.0000i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 8 q^{4} - 8 q^{5}+O(q^{10}) $ 2 * q - 8 * q^4 - 8 * q^5	$ 2 q - 8 q^{4} - 8 q^{5} - 12 q^{10} - 24 q^{13} + 32 q^{16} + 14 q^{17} + 36 q^{18} + 32 q^{20} + 14 q^{25} + 20 q^{26} + 28 q^{34} - 140 q^{37} + 48 q^{40} - 62 q^{41} + 54 q^{45} + 98 q^{49} + 96 q^{50} + 96 q^{52} - 34 q^{53} + 168 q^{58} - 240 q^{61} - 128 q^{64} + 126 q^{65} - 56 q^{68} - 144 q^{72} - 128 q^{80} - 162 q^{81} + 124 q^{82} - 14 q^{85} - 238 q^{89} - 144 q^{90}+O(q^{100}) $ 2 * q - 8 * q^4 - 8 * q^5 - 12 * q^10 - 24 * q^13 + 32 * q^16 + 14 * q^17 + 36 * q^18 + 32 * q^20 + 14 * q^25 + 20 * q^26 + 28 * q^34 - 140 * q^37 + 48 * q^40 - 62 * q^41 + 54 * q^45 + 98 * q^49 + 96 * q^50 + 96 * q^52 - 34 * q^53 + 168 * q^58 - 240 * q^61 - 128 * q^64 + 126 * q^65 - 56 * q^68 - 144 * q^72 - 128 * q^80 - 162 * q^81 + 124 * q^82 - 14 * q^85 - 238 * q^89 - 144 * q^90

Display 100 coefficients 10 coefficients

Character values

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

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

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 $

$2$			2.00000i			1.00000i
$2$			2.00000i			1.00000i
$3$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$3$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$4$	−4.00000			−1.00000
$5$	−4.00000	+	3.00000i	−0.800000	+	0.600000i
$5$	−4.00000	+	3.00000i	−0.800000	+	0.600000i
$6$		0			0
$7$		0			0		1.00000			$0$
$7$		0			0		−1.00000			$\pi$
$8$		−	8.00000i		−	1.00000i
$9$		−	9.00000i		−	1.00000i
$10$	−6.00000	−	8.00000i	−0.600000	−	0.800000i
$11$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$11$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$12$		0			0
$13$	−12.0000	−	5.00000i	−0.923077	−	0.384615i
$13$	−12.0000	−	5.00000i	−0.923077	−	0.384615i
$14$		0			0
$15$		0			0
$16$	16.0000			1.00000
$17$	7.00000	−	7.00000i	0.411765	−	0.411765i	−0.470588	−	0.882353i	$-0.655958\pi$
$17$	7.00000	−	7.00000i	0.411765	−	0.411765i	0.882353	+	0.470588i	$0.155958\pi$
$18$	18.0000			1.00000
$19$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$19$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$20$	16.0000	−	12.0000i	0.800000	−	0.600000i
$21$		0			0
$22$		0			0
$23$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$23$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$24$		0			0
$25$	7.00000	−	24.0000i	0.280000	−	0.960000i
$26$	10.0000	−	24.0000i	0.384615	−	0.923077i
$27$		0			0
$28$		0			0
$29$		−	42.0000i		−	1.44828i	−0.689655	−	0.724138i	$-0.742238\pi$
$29$		−	42.0000i		−	1.44828i	0.689655	−	0.724138i	$-0.257762\pi$
$30$		0			0
$31$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$31$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$32$			32.0000i			1.00000i
$33$		0			0
$34$	14.0000	+	14.0000i	0.411765	+	0.411765i
$35$		0			0
$36$			36.0000i			1.00000i
$37$	−70.0000			−1.89189			−0.945946	−	0.324324i	$-0.894863\pi$
$37$	−70.0000			−1.89189			−0.945946	+	0.324324i	$0.894863\pi$
$38$		0			0
$39$		0			0
$40$	24.0000	+	32.0000i	0.600000	+	0.800000i
$41$	−31.0000	−	31.0000i	−0.756098	−	0.756098i	0.219512	−	0.975610i	$-0.429553\pi$
$41$	−31.0000	−	31.0000i	−0.756098	−	0.756098i	−0.975610	+	0.219512i	$0.929553\pi$
$42$		0			0
$43$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$43$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$44$		0			0
$45$	27.0000	+	36.0000i	0.600000	+	0.800000i
$46$		0			0
$47$		0			0		1.00000			$0$
$47$		0			0		−1.00000			$\pi$
$48$		0			0
$49$	49.0000			1.00000
$50$	48.0000	+	14.0000i	0.960000	+	0.280000i
$51$		0			0
$52$	48.0000	+	20.0000i	0.923077	+	0.384615i
$53$	−17.0000	+	17.0000i	−0.320755	+	0.320755i	−0.849057	−	0.528302i	$-0.822829\pi$
$53$	−17.0000	+	17.0000i	−0.320755	+	0.320755i	0.528302	+	0.849057i	$0.322829\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$	84.0000			1.44828
$59$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$59$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$60$		0			0
$61$	−120.000			−1.96721			−0.983607	−	0.180328i	$-0.942284\pi$
$61$	−120.000			−1.96721			−0.983607	+	0.180328i	$0.942284\pi$
$62$		0			0
$63$		0			0
$64$	−64.0000			−1.00000
$65$	63.0000	−	16.0000i	0.969231	−	0.246154i
$66$		0			0
$67$		0			0			−	1.00000i	$-0.5\pi$
$67$		0			0				1.00000i	$0.5\pi$
$68$	−28.0000	+	28.0000i	−0.411765	+	0.411765i
$69$		0			0
$70$		0			0
$71$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$71$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$72$	−72.0000			−1.00000
$73$			96.0000i			1.31507i	0.753425	+	0.657534i	$0.228401\pi$
$73$			96.0000i			1.31507i	−0.753425	+	0.657534i	$0.771599\pi$
$74$		−	140.000i		−	1.89189i
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0			−	1.00000i	$-0.5\pi$
$79$		0			0				1.00000i	$0.5\pi$
$80$	−64.0000	+	48.0000i	−0.800000	+	0.600000i
$81$	−81.0000			−1.00000
$82$	62.0000	−	62.0000i	0.756098	−	0.756098i
$83$		0			0		1.00000			$0$
$83$		0			0		−1.00000			$\pi$
$84$		0			0
$85$	−7.00000	+	49.0000i	−0.0823529	+	0.576471i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−119.000	−	119.000i	−1.33708	−	1.33708i	−0.898876	−	0.438202i	$-0.855615\pi$
$89$	−119.000	−	119.000i	−1.33708	−	1.33708i	−0.438202	−	0.898876i	$-0.644385\pi$
$90$	−72.0000	+	54.0000i	−0.800000	+	0.600000i
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$			130.000i			1.34021i	0.742268	+	0.670103i	$0.233750\pi$
$97$			130.000i			1.34021i	−0.742268	+	0.670103i	$0.766250\pi$
$98$			98.0000i			1.00000i
$99$		0			0
$100$	−28.0000	+	96.0000i	−0.280000	+	0.960000i
$101$		−	40.0000i		−	0.396040i	−0.980198	−	0.198020i	$-0.936549\pi$
$101$		−	40.0000i		−	0.396040i	0.980198	−	0.198020i	$-0.0634510\pi$
$102$		0			0
$103$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$103$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$104$	−40.0000	+	96.0000i	−0.384615	+	0.923077i
$105$		0			0
$106$	−34.0000	−	34.0000i	−0.320755	−	0.320755i
$107$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$107$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$108$		0			0
$109$	−31.0000	+	31.0000i	−0.284404	+	0.284404i	−0.834862	−	0.550459i	$-0.814453\pi$
$109$	−31.0000	+	31.0000i	−0.284404	+	0.284404i	0.550459	+	0.834862i	$0.314453\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	127.000	−	127.000i	1.12389	−	1.12389i	0.132743	−	0.991150i	$-0.457621\pi$
$113$	127.000	−	127.000i	1.12389	−	1.12389i	0.991150	−	0.132743i	$-0.0423786\pi$
$114$		0			0
$115$		0			0
$116$			168.000i			1.44828i
$117$	−45.0000	+	108.000i	−0.384615	+	0.923077i
$118$		0			0
$119$		0			0
$120$		0			0
$121$		−	121.000i		−	1.00000i
$122$		−	240.000i		−	1.96721i
$123$		0			0
$124$		0			0
$125$	44.0000	+	117.000i	0.352000	+	0.936000i
$126$		0			0
$127$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$127$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$128$		−	128.000i		−	1.00000i
$129$		0			0
$130$	32.0000	+	126.000i	0.246154	+	0.969231i
$131$		0			0		1.00000			$0$
$131$		0			0		−1.00000			$\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$	−56.0000	−	56.0000i	−0.411765	−	0.411765i
$137$	176.000			1.28467			0.642336	−	0.766423i	$-0.277965\pi$
$137$	176.000			1.28467			0.642336	+	0.766423i	$0.277965\pi$
$138$		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
$144$		−	144.000i		−	1.00000i
$145$	126.000	+	168.000i	0.868966	+	1.15862i
$146$	−192.000			−1.31507
$147$		0			0
$148$	280.000			1.89189
$149$	191.000	−	191.000i	1.28188	−	1.28188i	0.342282	−	0.939597i	$-0.388800\pi$
$149$	191.000	−	191.000i	1.28188	−	1.28188i	0.939597	−	0.342282i	$-0.111200\pi$
$150$		0			0
$151$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$151$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$152$		0			0
$153$	−63.0000	−	63.0000i	−0.411765	−	0.411765i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	217.000	+	217.000i	1.38217	+	1.38217i	0.840764	+	0.541401i	$0.182106\pi$
$157$	217.000	+	217.000i	1.38217	+	1.38217i	0.541401	+	0.840764i	$0.317894\pi$
$158$		0			0
$159$		0			0
$160$	−96.0000	−	128.000i	−0.600000	−	0.800000i
$161$		0			0
$162$		−	162.000i		−	1.00000i
$163$		0			0			−	1.00000i	$-0.5\pi$
$163$		0			0				1.00000i	$0.5\pi$
$164$	124.000	+	124.000i	0.756098	+	0.756098i
$165$		0			0
$166$		0			0
$167$		0			0		1.00000			$0$
$167$		0			0		−1.00000			$\pi$
$168$		0			0
$169$	119.000	+	120.000i	0.704142	+	0.710059i
$170$	−98.0000	−	14.0000i	−0.576471	−	0.0823529i
$171$		0			0
$172$		0			0
$173$	−113.000	−	113.000i	−0.653179	−	0.653179i	0.300578	−	0.953757i	$-0.402820\pi$
$173$	−113.000	−	113.000i	−0.653179	−	0.653179i	−0.953757	+	0.300578i	$0.902820\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$	238.000	−	238.000i	1.33708	−	1.33708i
$179$		0			0		1.00000			$0$
$179$		0			0		−1.00000			$\pi$
$180$	−108.000	−	144.000i	−0.600000	−	0.800000i
$181$		−	38.0000i		−	0.209945i	−0.994475	−	0.104972i	$-0.966525\pi$
$181$		−	38.0000i		−	0.209945i	0.994475	−	0.104972i	$-0.0334754\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	280.000	−	210.000i	1.51351	−	1.13514i
$186$		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
$193$		−	190.000i		−	0.984456i	−0.870466	−	0.492228i	$-0.836183\pi$
$193$		−	190.000i		−	0.984456i	0.870466	−	0.492228i	$-0.163817\pi$
$194$	−260.000			−1.34021
$195$		0			0
$196$	−196.000			−1.00000
$197$			390.000i			1.97970i	0.142132	+	0.989848i	$0.454604\pi$
$197$			390.000i			1.97970i	−0.142132	+	0.989848i	$0.545396\pi$
$198$		0			0
$199$		0			0		1.00000			$0$
$199$		0			0		−1.00000			$\pi$
$200$	−192.000	−	56.0000i	−0.960000	−	0.280000i
$201$		0			0
$202$	80.0000			0.396040
$203$		0			0
$204$		0			0
$205$	217.000	+	31.0000i	1.05854	+	0.151220i
$206$		0			0
$207$		0			0
$208$	−192.000	−	80.0000i	−0.923077	−	0.384615i
$209$		0			0
$210$		0			0
$211$		0			0		1.00000			$0$
$211$		0			0		−1.00000			$\pi$
$212$	68.0000	−	68.0000i	0.320755	−	0.320755i
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$	−62.0000	−	62.0000i	−0.284404	−	0.284404i
$219$		0			0
$220$		0			0
$221$	−119.000	+	49.0000i	−0.538462	+	0.221719i
$222$		0			0
$223$		0			0		1.00000			$0$
$223$		0			0		−1.00000			$\pi$
$224$		0			0
$225$	−216.000	−	63.0000i	−0.960000	−	0.280000i
$226$	254.000	+	254.000i	1.12389	+	1.12389i
$227$		0			0			−	1.00000i	$-0.5\pi$
$227$		0			0				1.00000i	$0.5\pi$
$228$		0			0
$229$	281.000	+	281.000i	1.22707	+	1.22707i	0.965066	+	0.262009i	$0.0843849\pi$
$229$	281.000	+	281.000i	1.22707	+	1.22707i	0.262009	+	0.965066i	$0.415615\pi$
$230$		0			0
$231$		0			0
$232$	−336.000			−1.44828
$233$	−313.000	−	313.000i	−1.34335	−	1.34335i	−0.892704	−	0.450644i	$-0.851194\pi$
$233$	−313.000	−	313.000i	−1.34335	−	1.34335i	−0.450644	−	0.892704i	$-0.648806\pi$
$234$	−216.000	−	90.0000i	−0.923077	−	0.384615i
$235$		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
$241$	329.000	−	329.000i	1.36515	−	1.36515i	0.497925	−	0.867220i	$-0.334095\pi$
$241$	329.000	−	329.000i	1.36515	−	1.36515i	0.867220	−	0.497925i	$-0.165905\pi$
$242$	242.000			1.00000
$243$		0			0
$244$	480.000			1.96721
$245$	−196.000	+	147.000i	−0.800000	+	0.600000i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$	−234.000	+	88.0000i	−0.936000	+	0.352000i
$251$		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
$256$	256.000			1.00000
$257$	−223.000	+	223.000i	−0.867704	+	0.867704i	−0.992218	−	0.124514i	$-0.960263\pi$
$257$	−223.000	+	223.000i	−0.867704	+	0.867704i	0.124514	+	0.992218i	$0.460263\pi$
$258$		0			0
$259$		0			0
$260$	−252.000	+	64.0000i	−0.969231	+	0.246154i
$261$	−378.000			−1.44828
$262$		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
$265$	17.0000	−	119.000i	0.0641509	−	0.449057i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	520.000i		−	1.93309i	−0.256506	−	0.966543i	$-0.582571\pi$
$269$		−	520.000i		−	1.93309i	0.256506	−	0.966543i	$-0.417429\pi$
$270$		0			0
$271$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$271$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$272$	112.000	−	112.000i	0.411765	−	0.411765i
$273$		0			0
$274$			352.000i			1.28467i
$275$		0			0
$276$		0			0
$277$	−137.000	+	137.000i	−0.494585	+	0.494585i	−0.909747	−	0.415162i	$-0.863725\pi$
$277$	−137.000	+	137.000i	−0.494585	+	0.494585i	0.415162	+	0.909747i	$0.363725\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−391.000	+	391.000i	−1.39146	+	1.39146i	−0.569395	+	0.822064i	$0.692822\pi$
$281$	−391.000	+	391.000i	−1.39146	+	1.39146i	−0.822064	+	0.569395i	$0.807178\pi$
$282$		0			0
$283$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$283$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$	288.000			1.00000
$289$			191.000i			0.660900i
$290$	−336.000	+	252.000i	−1.15862	+	0.868966i
$291$		0			0
$292$		−	384.000i		−	1.31507i
$293$			136.000i			0.464164i	0.972696	+	0.232082i	$0.0745537\pi$
$293$			136.000i			0.464164i	−0.972696	+	0.232082i	$0.925446\pi$
$294$		0			0
$295$		0			0
$296$			560.000i			1.89189i
$297$		0			0
$298$	382.000	+	382.000i	1.28188	+	1.28188i
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	480.000	−	360.000i	1.57377	−	1.18033i
$306$	126.000	−	126.000i	0.411765	−	0.411765i
$307$		0			0		1.00000			$0$
$307$		0			0		−1.00000			$\pi$
$308$		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
$313$	−287.000	+	287.000i	−0.916933	+	0.916933i	−0.996805	−	0.0798722i	$-0.974549\pi$
$313$	−287.000	+	287.000i	−0.916933	+	0.916933i	0.0798722	+	0.996805i	$0.474549\pi$
$314$	−434.000	+	434.000i	−1.38217	+	1.38217i
$315$		0			0
$316$		0			0
$317$		−	616.000i		−	1.94322i	−0.236593	−	0.971609i	$-0.576031\pi$
$317$		−	616.000i		−	1.94322i	0.236593	−	0.971609i	$-0.423969\pi$
$318$		0			0
$319$		0			0
$320$	256.000	−	192.000i	0.800000	−	0.600000i
$321$		0			0
$322$		0			0
$323$		0			0
$324$	324.000			1.00000
$325$	−204.000	+	253.000i	−0.627692	+	0.778462i
$326$		0			0
$327$		0			0
$328$	−248.000	+	248.000i	−0.756098	+	0.756098i
$329$		0			0
$330$		0			0
$331$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$331$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$332$		0			0
$333$			630.000i			1.89189i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	113.000	−	113.000i	0.335312	−	0.335312i	−0.519288	−	0.854599i	$-0.673803\pi$
$337$	113.000	−	113.000i	0.335312	−	0.335312i	0.854599	+	0.519288i	$0.173803\pi$
$338$	−240.000	+	238.000i	−0.710059	+	0.704142i
$339$		0			0
$340$	28.0000	−	196.000i	0.0823529	−	0.576471i
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$	226.000	−	226.000i	0.653179	−	0.653179i
$347$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$347$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$348$		0			0
$349$	119.000	+	119.000i	0.340974	+	0.340974i	0.856734	−	0.515759i	$-0.172490\pi$
$349$	119.000	+	119.000i	0.340974	+	0.340974i	−0.515759	+	0.856734i	$0.672490\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	450.000			1.27479			0.637394	−	0.770538i	$-0.280012\pi$
$353$	450.000			1.27479			0.637394	+	0.770538i	$0.280012\pi$
$354$		0			0
$355$		0			0
$356$	476.000	+	476.000i	1.33708	+	1.33708i
$357$		0			0
$358$		0			0
$359$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$359$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$360$	288.000	−	216.000i	0.800000	−	0.600000i
$361$		−	361.000i		−	1.00000i
$362$	76.0000			0.209945
$363$		0			0
$364$		0			0
$365$	−288.000	−	384.000i	−0.789041	−	1.05205i
$366$		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
$369$	−279.000	+	279.000i	−0.756098	+	0.756098i
$370$	420.000	+	560.000i	1.13514	+	1.51351i
$371$		0			0
$372$		0			0
$373$	−527.000	+	527.000i	−1.41287	+	1.41287i	−0.675603	+	0.737265i	$0.736117\pi$
$373$	−527.000	+	527.000i	−1.41287	+	1.41287i	−0.737265	+	0.675603i	$0.763883\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−210.000	+	504.000i	−0.557029	+	1.33687i
$378$		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		1.00000			$0$
$383$		0			0		−1.00000			$\pi$
$384$		0			0
$385$		0			0
$386$	380.000			0.984456
$387$		0			0
$388$		−	520.000i		−	1.34021i
$389$	−378.000			−0.971722			−0.485861	−	0.874036i	$-0.661494\pi$
$389$	−378.000			−0.971722			−0.485861	+	0.874036i	$0.661494\pi$
$390$		0			0
$391$		0			0
$392$		−	392.000i		−	1.00000i
$393$		0			0
$394$	−780.000			−1.97970
$395$		0			0
$396$		0			0
$397$	−650.000			−1.63728			−0.818640	−	0.574307i	$-0.805271\pi$
$397$	−650.000			−1.63728			−0.818640	+	0.574307i	$0.805271\pi$
$398$		0			0
$399$		0			0
$400$	112.000	−	384.000i	0.280000	−	0.960000i
$401$	439.000	−	439.000i	1.09476	−	1.09476i	0.0997506	−	0.995012i	$-0.468195\pi$
$401$	439.000	−	439.000i	1.09476	−	1.09476i	0.995012	−	0.0997506i	$-0.0318045\pi$
$402$		0			0
$403$		0			0
$404$			160.000i			0.396040i
$405$	324.000	−	243.000i	0.800000	−	0.600000i
$406$		0			0
$407$		0			0
$408$		0			0
$409$	271.000	−	271.000i	0.662592	−	0.662592i	−0.293399	−	0.955990i	$-0.594786\pi$
$409$	271.000	−	271.000i	0.662592	−	0.662592i	0.955990	+	0.293399i	$0.0947863\pi$
$410$	−62.0000	+	434.000i	−0.151220	+	1.05854i
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$	160.000	−	384.000i	0.384615	−	0.923077i
$417$		0			0
$418$		0			0
$419$		0			0			−	1.00000i	$-0.5\pi$
$419$		0			0				1.00000i	$0.5\pi$
$420$		0			0
$421$	−449.000	−	449.000i	−1.06651	−	1.06651i	−0.997625	−	0.0688836i	$-0.978056\pi$
$421$	−449.000	−	449.000i	−1.06651	−	1.06651i	−0.0688836	−	0.997625i	$-0.521944\pi$
$422$		0			0
$423$		0			0
$424$	136.000	+	136.000i	0.320755	+	0.320755i
$425$	−119.000	−	217.000i	−0.280000	−	0.510588i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$431$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$432$		0			0
$433$	−553.000	−	553.000i	−1.27714	−	1.27714i	−0.942263	−	0.334873i	$-0.891307\pi$
$433$	−553.000	−	553.000i	−1.27714	−	1.27714i	−0.334873	−	0.942263i	$-0.608693\pi$
$434$		0			0
$435$		0			0
$436$	124.000	−	124.000i	0.284404	−	0.284404i
$437$		0			0
$438$		0			0
$439$		0			0		1.00000			$0$
$439$		0			0		−1.00000			$\pi$
$440$		0			0
$441$		−	441.000i		−	1.00000i
$442$	−98.0000	−	238.000i	−0.221719	−	0.538462i
$443$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$443$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$444$		0			0
$445$	833.000	+	119.000i	1.87191	+	0.267416i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−631.000	−	631.000i	−1.40535	−	1.40535i	−0.781737	−	0.623608i	$-0.785666\pi$
$449$	−631.000	−	631.000i	−1.40535	−	1.40535i	−0.623608	−	0.781737i	$-0.714334\pi$
$450$	126.000	−	432.000i	0.280000	−	0.960000i
$451$		0			0
$452$	−508.000	+	508.000i	−1.12389	+	1.12389i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$		−	850.000i		−	1.85996i	−0.367615	−	0.929978i	$-0.619826\pi$
$457$		−	850.000i		−	1.85996i	0.367615	−	0.929978i	$-0.380174\pi$
$458$	−562.000	+	562.000i	−1.22707	+	1.22707i
$459$		0			0
$460$		0			0
$461$	641.000	+	641.000i	1.39046	+	1.39046i	0.824295	+	0.566161i	$0.191572\pi$
$461$	641.000	+	641.000i	1.39046	+	1.39046i	0.566161	+	0.824295i	$0.308428\pi$
$462$		0			0
$463$		0			0			−	1.00000i	$-0.5\pi$
$463$		0			0				1.00000i	$0.5\pi$
$464$		−	672.000i		−	1.44828i
$465$		0			0
$466$	626.000	−	626.000i	1.34335	−	1.34335i
$467$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$467$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$468$	180.000	−	432.000i	0.384615	−	0.923077i
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$	153.000	+	153.000i	0.320755	+	0.320755i
$478$		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
$481$	840.000	+	350.000i	1.74636	+	0.727651i
$482$	658.000	+	658.000i	1.36515	+	1.36515i
$483$		0			0
$484$			484.000i			1.00000i
$485$	−390.000	−	520.000i	−0.804124	−	1.07216i
$486$		0			0
$487$		0			0			−	1.00000i	$-0.5\pi$
$487$		0			0				1.00000i	$0.5\pi$
$488$			960.000i			1.96721i
$489$		0			0
$490$	−294.000	−	392.000i	−0.600000	−	0.800000i
$491$		0			0			−	1.00000i	$-0.5\pi$
$491$		0			0				1.00000i	$0.5\pi$
$492$		0			0
$493$	−294.000	−	294.000i	−0.596349	−	0.596349i
$494$		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	$0.750000\pi$
$500$	−176.000	−	468.000i	−0.352000	−	0.936000i
$501$		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
$505$	120.000	+	160.000i	0.237624	+	0.316832i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	239.000	−	239.000i	0.469548	−	0.469548i	−0.432220	−	0.901768i	$-0.642270\pi$
$509$	239.000	−	239.000i	0.469548	−	0.469548i	0.901768	+	0.432220i	$0.142270\pi$
$510$		0			0
$511$		0			0
$512$			512.000i			1.00000i
$513$		0			0
$514$	−446.000	−	446.000i	−0.867704	−	0.867704i
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$	−128.000	−	504.000i	−0.246154	−	0.969231i
$521$	880.000			1.68906			0.844530	−	0.535509i	$-0.179880\pi$
$521$	880.000			1.68906			0.844530	+	0.535509i	$0.179880\pi$
$522$		−	756.000i		−	1.44828i
$523$		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
$529$			529.000i			1.00000i
$530$	238.000	+	34.0000i	0.449057	+	0.0641509i
$531$		0			0
$532$		0			0
$533$	217.000	+	527.000i	0.407129	+	0.988743i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$	1040.00			1.93309
$539$		0			0
$540$		0			0
$541$	−761.000	+	761.000i	−1.40665	+	1.40665i	−0.630314	+	0.776340i	$0.717074\pi$
$541$	−761.000	+	761.000i	−1.40665	+	1.40665i	−0.776340	+	0.630314i	$0.782926\pi$
$542$		0			0
$543$		0			0
$544$	224.000	+	224.000i	0.411765	+	0.411765i
$545$	31.0000	−	217.000i	0.0568807	−	0.398165i
$546$		0			0
$547$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$547$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$548$	−704.000			−1.28467
$549$			1080.00i			1.96721i
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$	−274.000	−	274.000i	−0.494585	−	0.494585i
$555$		0			0
$556$		0			0
$557$	330.000			0.592460			0.296230	−	0.955117i	$-0.404271\pi$
$557$	330.000			0.592460			0.296230	+	0.955117i	$0.404271\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	−782.000	−	782.000i	−1.39146	−	1.39146i
$563$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$563$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$564$		0			0
$565$	−127.000	+	889.000i	−0.224779	+	1.57345i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	1040.00			1.82777			0.913884	−	0.405975i	$-0.133068\pi$
$569$	1040.00			1.82777			0.913884	+	0.405975i	$0.133068\pi$
$570$		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
$576$			576.000i			1.00000i
$577$			96.0000i			0.166378i	0.996534	+	0.0831889i	$0.0265105\pi$
$577$			96.0000i			0.166378i	−0.996534	+	0.0831889i	$0.973490\pi$
$578$	−382.000			−0.660900
$579$		0			0
$580$	−504.000	−	672.000i	−0.868966	−	1.15862i
$581$		0			0
$582$		0			0
$583$		0			0
$584$	768.000			1.31507
$585$	−144.000	−	567.000i	−0.246154	−	0.969231i
$586$	−272.000			−0.464164
$587$		0			0			−	1.00000i	$-0.5\pi$
$587$		0			0				1.00000i	$0.5\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$	−1120.00			−1.89189
$593$			930.000i			1.56830i	0.620573	+	0.784148i	$0.286900\pi$
$593$			930.000i			1.56830i	−0.620573	+	0.784148i	$0.713100\pi$
$594$		0			0
$595$		0			0
$596$	−764.000	+	764.000i	−1.28188	+	1.28188i
$597$		0			0
$598$		0			0
$599$		0			0			−	1.00000i	$-0.5\pi$
$599$		0			0				1.00000i	$0.5\pi$
$600$		0			0
$601$	1102.00			1.83361			0.916805	−	0.399334i	$-0.130759\pi$
$601$	1102.00			1.83361			0.916805	+	0.399334i	$0.130759\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	363.000	+	484.000i	0.600000	+	0.800000i
$606$		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
$610$	720.000	+	960.000i	1.18033	+	1.57377i
$611$		0			0
$612$	252.000	+	252.000i	0.411765	+	0.411765i
$613$	1224.00			1.99674			0.998369	−	0.0570962i	$-0.0181842\pi$
$613$	1224.00			1.99674			0.998369	+	0.0570962i	$0.0181842\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	210.000i		−	0.340357i	−0.985413	−	0.170178i	$-0.945566\pi$
$617$		−	210.000i		−	0.340357i	0.985413	−	0.170178i	$-0.0544344\pi$
$618$		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
$625$	−527.000	−	336.000i	−0.843200	−	0.537600i
$626$	−574.000	−	574.000i	−0.916933	−	0.916933i
$627$		0			0
$628$	−868.000	−	868.000i	−1.38217	−	1.38217i
$629$	−490.000	+	490.000i	−0.779014	+	0.779014i
$630$		0			0
$631$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$631$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$632$		0			0
$633$		0			0
$634$	1232.00			1.94322
$635$		0			0
$636$		0			0
$637$	−588.000	−	245.000i	−0.923077	−	0.384615i
$638$		0			0
$639$		0			0
$640$	384.000	+	512.000i	0.600000	+	0.800000i
$641$			400.000i			0.624025i	0.950078	+	0.312012i	$0.101003\pi$
$641$			400.000i			0.624025i	−0.950078	+	0.312012i	$0.898997\pi$
$642$		0			0
$643$		0			0		1.00000			$0$
$643$		0			0		−1.00000			$\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$647$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$648$			648.000i			1.00000i
$649$		0			0
$650$	−506.000	−	408.000i	−0.778462	−	0.627692i
$651$		0			0
$652$		0			0
$653$	887.000	−	887.000i	1.35835	−	1.35835i	0.482389	−	0.875957i	$-0.339769\pi$
$653$	887.000	−	887.000i	1.35835	−	1.35835i	0.875957	−	0.482389i	$-0.160231\pi$
$654$		0			0
$655$		0			0
$656$	−496.000	−	496.000i	−0.756098	−	0.756098i
$657$	864.000			1.31507
$658$		0			0
$659$		0			0			−	1.00000i	$-0.5\pi$
$659$		0			0				1.00000i	$0.5\pi$
$660$		0			0
$661$	−889.000	+	889.000i	−1.34493	+	1.34493i	−0.453858	+	0.891074i	$0.649953\pi$
$661$	−889.000	+	889.000i	−1.34493	+	1.34493i	−0.891074	+	0.453858i	$0.850047\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$	−1260.00			−1.89189
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−937.000	−	937.000i	−1.39227	−	1.39227i	−0.820208	−	0.572065i	$-0.806142\pi$
$673$	−937.000	−	937.000i	−1.39227	−	1.39227i	−0.572065	−	0.820208i	$-0.693858\pi$
$674$	226.000	+	226.000i	0.335312	+	0.335312i
$675$		0			0
$676$	−476.000	−	480.000i	−0.704142	−	0.710059i
$677$	727.000	+	727.000i	1.07386	+	1.07386i	0.997046	+	0.0768095i	$0.0244733\pi$
$677$	727.000	+	727.000i	1.07386	+	1.07386i	0.0768095	+	0.997046i	$0.475527\pi$
$678$		0			0
$679$		0			0
$680$	392.000	+	56.0000i	0.576471	+	0.0823529i
$681$		0			0
$682$		0			0
$683$		0			0			−	1.00000i	$-0.5\pi$
$683$		0			0				1.00000i	$0.5\pi$
$684$		0			0
$685$	−704.000	+	528.000i	−1.02774	+	0.770803i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	289.000	−	119.000i	0.419448	−	0.172714i
$690$		0			0
$691$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$691$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$692$	452.000	+	452.000i	0.653179	+	0.653179i
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−434.000			−0.622669
$698$	−238.000	+	238.000i	−0.340974	+	0.340974i
$699$		0			0
$700$		0			0
$701$		−	520.000i		−	0.741797i	−0.928673	−	0.370899i	$-0.879050\pi$
$701$		−	520.000i		−	0.741797i	0.928673	−	0.370899i	$-0.120950\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$			900.000i			1.27479i
$707$		0			0
$708$		0			0
$709$	−919.000	−	919.000i	−1.29619	−	1.29619i	−0.930889	−	0.365303i	$-0.880965\pi$
$709$	−919.000	−	919.000i	−1.29619	−	1.29619i	−0.365303	−	0.930889i	$-0.619035\pi$
$710$		0			0
$711$		0			0
$712$	−952.000	+	952.000i	−1.33708	+	1.33708i
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		1.00000			$0$
$719$		0			0		−1.00000			$\pi$
$720$	432.000	+	576.000i	0.600000	+	0.800000i
$721$		0			0
$722$	722.000			1.00000
$723$		0			0
$724$			152.000i			0.209945i
$725$	−1008.00	−	294.000i	−1.39034	−	0.405517i
$726$		0			0
$727$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$727$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$728$		0			0
$729$			729.000i			1.00000i
$730$	768.000	−	576.000i	1.05205	−	0.789041i
$731$		0			0
$732$		0			0
$733$	216.000			0.294679			0.147340	−	0.989086i	$-0.452929\pi$
$733$	216.000			0.294679			0.147340	+	0.989086i	$0.452929\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$	−558.000	−	558.000i	−0.756098	−	0.756098i
$739$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$739$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$740$	−1120.00	+	840.000i	−1.51351	+	1.13514i
$741$		0			0
$742$		0			0
$743$		0			0		1.00000			$0$
$743$		0			0		−1.00000			$\pi$
$744$		0			0
$745$	−191.000	+	1337.00i	−0.256376	+	1.79463i
$746$	−1054.00	−	1054.00i	−1.41287	−	1.41287i
$747$		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
$754$	−1008.00	−	420.000i	−1.33687	−	0.557029i
$755$		0			0
$756$		0			0
$757$	−127.000	−	127.000i	−0.167768	−	0.167768i	0.618230	−	0.785997i	$-0.287850\pi$
$757$	−127.000	−	127.000i	−0.167768	−	0.167768i	−0.785997	+	0.618230i	$0.787850\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	721.000	−	721.000i	0.947438	−	0.947438i	−0.0512484	−	0.998686i	$-0.516320\pi$
$761$	721.000	−	721.000i	0.947438	−	0.947438i	0.998686	+	0.0512484i	$0.0163200\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	441.000	+	63.0000i	0.576471	+	0.0823529i
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−1081.00	+	1081.00i	−1.40572	+	1.40572i	−0.625488	+	0.780234i	$0.715100\pi$
$769$	−1081.00	+	1081.00i	−1.40572	+	1.40572i	−0.780234	+	0.625488i	$0.784900\pi$
$770$		0			0
$771$		0			0
$772$			760.000i			0.984456i
$773$	390.000			0.504528			0.252264	−	0.967658i	$-0.418825\pi$
$773$	390.000			0.504528			0.252264	+	0.967658i	$0.418825\pi$
$774$		0			0
$775$		0			0
$776$	1040.00			1.34021
$777$		0			0
$778$		−	756.000i		−	0.971722i
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	784.000			1.00000
$785$	−1519.00	−	217.000i	−1.93503	−	0.276433i
$786$		0			0
$787$		0			0		1.00000			$0$
$787$		0			0		−1.00000			$\pi$
$788$		−	1560.00i		−	1.97970i
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	1440.00	+	600.000i	1.81589	+	0.756620i
$794$		−	1300.00i		−	1.63728i
$795$		0			0
$796$		0			0
$797$	1127.00	−	1127.00i	1.41405	−	1.41405i	0.696361	−	0.717691i	$-0.254801\pi$
$797$	1127.00	−	1127.00i	1.41405	−	1.41405i	0.717691	−	0.696361i	$-0.245199\pi$
$798$		0			0
$799$		0			0
$800$	768.000	+	224.000i	0.960000	+	0.280000i
$801$	−1071.00	+	1071.00i	−1.33708	+	1.33708i
$802$	878.000	+	878.000i	1.09476	+	1.09476i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$	−320.000			−0.396040
$809$		−	1518.00i		−	1.87639i	−0.346106	−	0.938195i	$-0.612496\pi$
$809$		−	1518.00i		−	1.87639i	0.346106	−	0.938195i	$-0.387504\pi$
$810$	486.000	+	648.000i	0.600000	+	0.800000i
$811$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$811$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$	542.000	+	542.000i	0.662592	+	0.662592i
$819$		0			0
$820$	−868.000	−	124.000i	−1.05854	−	0.151220i
$821$	−271.000	−	271.000i	−0.330085	−	0.330085i	0.522533	−	0.852619i	$-0.324987\pi$
$821$	−271.000	−	271.000i	−0.330085	−	0.330085i	−0.852619	+	0.522533i	$0.824987\pi$
$822$		0			0
$823$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$823$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		1.00000			$0$
$827$		0			0		−1.00000			$\pi$
$828$		0			0
$829$	1080.00			1.30277			0.651387	−	0.758745i	$-0.274187\pi$
$829$	1080.00			1.30277			0.651387	+	0.758745i	$0.274187\pi$
$830$		0			0
$831$		0			0
$832$	768.000	+	320.000i	0.923077	+	0.384615i
$833$	343.000	−	343.000i	0.411765	−	0.411765i
$834$		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
$841$	−923.000			−1.09750
$842$	898.000	−	898.000i	1.06651	−	1.06651i
$843$		0			0
$844$		0			0
$845$	−836.000	−	123.000i	−0.989349	−	0.145562i
$846$		0			0
$847$		0			0
$848$	−272.000	+	272.000i	−0.320755	+	0.320755i
$849$		0			0
$850$	434.000	−	238.000i	0.510588	−	0.280000i
$851$		0			0
$852$		0			0
$853$		−	1656.00i		−	1.94138i	−0.240328	−	0.970692i	$-0.577255\pi$
$853$		−	1656.00i		−	1.94138i	0.240328	−	0.970692i	$-0.422745\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	1057.00	−	1057.00i	1.23337	−	1.23337i	0.270712	−	0.962660i	$-0.412741\pi$
$857$	1057.00	−	1057.00i	1.23337	−	1.23337i	0.962660	−	0.270712i	$-0.0872590\pi$
$858$		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		1.00000			$0$
$863$		0			0		−1.00000			$\pi$
$864$		0			0
$865$	791.000	+	113.000i	0.914451	+	0.130636i
$866$	1106.00	−	1106.00i	1.27714	−	1.27714i
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$	248.000	+	248.000i	0.284404	+	0.284404i
$873$	1170.00			1.34021
$874$		0			0
$875$		0			0
$876$		0			0
$877$		−	1610.00i		−	1.83580i	−0.396807	−	0.917902i	$-0.629882\pi$
$877$		−	1610.00i		−	1.83580i	0.396807	−	0.917902i	$-0.370118\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			738.000i			0.837684i	0.908059	+	0.418842i	$0.137564\pi$
$881$			738.000i			0.837684i	−0.908059	+	0.418842i	$0.862436\pi$
$882$	882.000			1.00000
$883$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$883$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$884$	476.000	−	196.000i	0.538462	−	0.221719i
$885$		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
$890$	−238.000	+	1666.00i	−0.267416	+	1.87191i
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	1262.00	−	1262.00i	1.40535	−	1.40535i
$899$		0			0
$900$	864.000	+	252.000i	0.960000	+	0.280000i
$901$			238.000i			0.264151i
$902$		0			0
$903$		0			0
$904$	−1016.00	−	1016.00i	−1.12389	−	1.12389i
$905$	114.000	+	152.000i	0.125967	+	0.167956i
$906$		0			0
$907$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$907$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$908$		0			0
$909$	−360.000			−0.396040
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$		0			0
$913$		0			0
$914$	1700.00			1.85996
$915$		0			0
$916$	−1124.00	−	1124.00i	−1.22707	−	1.22707i
$917$		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
$922$	−1282.00	+	1282.00i	−1.39046	+	1.39046i
$923$		0			0
$924$		0			0
$925$	−490.000	+	1680.00i	−0.529730	+	1.81622i
$926$		0			0
$927$		0			0
$928$	1344.00			1.44828
$929$	−791.000	+	791.000i	−0.851453	+	0.851453i	−0.990312	−	0.138859i	$-0.955657\pi$
$929$	−791.000	+	791.000i	−0.851453	+	0.851453i	0.138859	+	0.990312i	$0.455657\pi$
$930$		0			0
$931$		0			0
$932$	1252.00	+	1252.00i	1.34335	+	1.34335i
$933$		0			0
$934$		0			0
$935$		0			0
$936$	864.000	+	360.000i	0.923077	+	0.384615i
$937$	−1127.00	−	1127.00i	−1.20277	−	1.20277i	−0.973319	−	0.229456i	$-0.926305\pi$
$937$	−1127.00	−	1127.00i	−1.20277	−	1.20277i	−0.229456	−	0.973319i	$-0.573695\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−1321.00	−	1321.00i	−1.40383	−	1.40383i	−0.787460	−	0.616366i	$-0.788604\pi$
$941$	−1321.00	−	1321.00i	−1.40383	−	1.40383i	−0.616366	−	0.787460i	$-0.711396\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		1.00000			$0$
$947$		0			0		−1.00000			$\pi$
$948$		0			0
$949$	480.000	−	1152.00i	0.505796	−	1.21391i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	113.000	+	113.000i	0.118573	+	0.118573i	0.763903	−	0.645331i	$-0.223280\pi$
$953$	113.000	+	113.000i	0.118573	+	0.118573i	−0.645331	+	0.763903i	$0.723280\pi$
$954$	−306.000	+	306.000i	−0.320755	+	0.320755i
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$			961.000i			1.00000i
$962$	−700.000	+	1680.00i	−0.727651	+	1.74636i
$963$		0			0
$964$	−1316.00	+	1316.00i	−1.36515	+	1.36515i
$965$	570.000	+	760.000i	0.590674	+	0.787565i
$966$		0			0
$967$		0			0			−	1.00000i	$-0.5\pi$
$967$		0			0				1.00000i	$0.5\pi$
$968$	−968.000			−1.00000
$969$		0			0
$970$	1040.00	−	780.000i	1.07216	−	0.804124i
$971$		0			0		1.00000			$0$
$971$		0			0		−1.00000			$\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$	−1920.00			−1.96721
$977$			496.000i			0.507677i	0.967247	+	0.253838i	$0.0816931\pi$
$977$			496.000i			0.507677i	−0.967247	+	0.253838i	$0.918307\pi$
$978$		0			0
$979$		0			0
$980$	784.000	−	588.000i	0.800000	−	0.600000i
$981$	279.000	+	279.000i	0.284404	+	0.284404i
$982$		0			0
$983$		0			0			−	1.00000i	$-0.5\pi$
$983$		0			0				1.00000i	$0.5\pi$
$984$		0			0
$985$	−1170.00	−	1560.00i	−1.18782	−	1.58376i
$986$	588.000	−	588.000i	0.596349	−	0.596349i
$987$		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
$997$	−553.000	−	553.000i	−0.554664	−	0.554664i	0.373119	−	0.927783i	$-0.378288\pi$
$997$	−553.000	−	553.000i	−0.554664	−	0.554664i	−0.927783	+	0.373119i	$0.878288\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	260.3.l.a.47.1	✓	2
4.3	odd	2	CM	260.3.l.a.47.1	✓	2
5.3	odd	4		260.3.s.b.203.1	yes	2
13.5	odd	4		260.3.s.b.187.1	yes	2
20.3	even	4		260.3.s.b.203.1	yes	2
52.31	even	4		260.3.s.b.187.1	yes	2
65.18	even	4	inner	260.3.l.a.83.1	yes	2
260.83	odd	4	inner	260.3.l.a.83.1	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
260.3.l.a.47.1	✓	2	1.1	even	1	trivial
260.3.l.a.47.1	✓	2	4.3	odd	2	CM
260.3.l.a.83.1	yes	2	65.18	even	4	inner
260.3.l.a.83.1	yes	2	260.83	odd	4	inner
260.3.s.b.187.1	yes	2	13.5	odd	4
260.3.s.b.187.1	yes	2	52.31	even	4
260.3.s.b.203.1	yes	2	5.3	odd	4
260.3.s.b.203.1	yes	2	20.3	even	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:	\( N \)	\(=\)	\( 260 = 2^{2} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	260.l (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+2.00000i q^{2} -4.00000 q^{4} +(-4.00000 + 3.00000i) q^{5} -8.00000i q^{8} -9.00000i q^{9} +O(q^{10})\)	\(q+2.00000i q^{2} -4.00000 q^{4} +(-4.00000 + 3.00000i) q^{5} -8.00000i q^{8} -9.00000i q^{9} +(-6.00000 - 8.00000i) q^{10} +(-12.0000 - 5.00000i) q^{13} +16.0000 q^{16} +(7.00000 - 7.00000i) q^{17} +18.0000 q^{18} +(16.0000 - 12.0000i) q^{20} +(7.00000 - 24.0000i) q^{25} +(10.0000 - 24.0000i) q^{26} -42.0000i q^{29} +32.0000i q^{32} +(14.0000 + 14.0000i) q^{34} +36.0000i q^{36} -70.0000 q^{37} +(24.0000 + 32.0000i) q^{40} +(-31.0000 - 31.0000i) q^{41} +(27.0000 + 36.0000i) q^{45} +49.0000 q^{49} +(48.0000 + 14.0000i) q^{50} +(48.0000 + 20.0000i) q^{52} +(-17.0000 + 17.0000i) q^{53} +84.0000 q^{58} -120.000 q^{61} -64.0000 q^{64} +(63.0000 - 16.0000i) q^{65} +(-28.0000 + 28.0000i) q^{68} -72.0000 q^{72} +96.0000i q^{73} -140.000i q^{74} +(-64.0000 + 48.0000i) q^{80} -81.0000 q^{81} +(62.0000 - 62.0000i) q^{82} +(-7.00000 + 49.0000i) q^{85} +(-119.000 - 119.000i) q^{89} +(-72.0000 + 54.0000i) q^{90} +130.000i q^{97} +98.0000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 8 q^{4} - 8 q^{5}+O(q^{10}) \) 2 * q - 8 * q^4 - 8 * q^5	\( 2 q - 8 q^{4} - 8 q^{5} - 12 q^{10} - 24 q^{13} + 32 q^{16} + 14 q^{17} + 36 q^{18} + 32 q^{20} + 14 q^{25} + 20 q^{26} + 28 q^{34} - 140 q^{37} + 48 q^{40} - 62 q^{41} + 54 q^{45} + 98 q^{49} + 96 q^{50} + 96 q^{52} - 34 q^{53} + 168 q^{58} - 240 q^{61} - 128 q^{64} + 126 q^{65} - 56 q^{68} - 144 q^{72} - 128 q^{80} - 162 q^{81} + 124 q^{82} - 14 q^{85} - 238 q^{89} - 144 q^{90}+O(q^{100}) \) 2 * q - 8 * q^4 - 8 * q^5 - 12 * q^10 - 24 * q^13 + 32 * q^16 + 14 * q^17 + 36 * q^18 + 32 * q^20 + 14 * q^25 + 20 * q^26 + 28 * q^34 - 140 * q^37 + 48 * q^40 - 62 * q^41 + 54 * q^45 + 98 * q^49 + 96 * q^50 + 96 * q^52 - 34 * q^53 + 168 * q^58 - 240 * q^61 - 128 * q^64 + 126 * q^65 - 56 * q^68 - 144 * q^72 - 128 * q^80 - 162 * q^81 + 124 * q^82 - 14 * q^85 - 238 * q^89 - 144 * q^90

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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