LMFDB - Newform orbit 510.2.u.b

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [510,2,Mod(121,510)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(510, base_ring=CyclotomicField(8))

chi = DirichletCharacter(H, H._module([0, 0, 7]))

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

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

chi := DirichletCharacter("510.121");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 510 = 2 \cdot 3 \cdot 5 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	510.u (of order $8$, degree $4$, 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:	$4.07237050309$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$2$ over $\Q(\zeta_{8})$
Coefficient field:	$\Q(\zeta_{16})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 1 $ x^8 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{8}]$

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

Coefficients of the $q$-expansion are expressed in terms of a primitive root of unity $\zeta_{16}$. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + \zeta_{16}^{6} q^{2} + \zeta_{16} q^{3} - \zeta_{16}^{4} q^{4} - \zeta_{16}^{5} q^{5} + \zeta_{16}^{7} q^{6} + (\zeta_{16}^{6} - \zeta_{16}^{4} + \cdots + 1) q^{7} + \cdots + \zeta_{16}^{2} q^{9} +O(q^{10}) $ q + z^6 * q^2 + z * q^3 - z^4 * q^4 - z^5 * q^5 + z^7 * q^6 + (z^6 - z^4 - z^2 + 1) * q^7 + z^2 * q^8 + z^2 * q^9	$ q + \zeta_{16}^{6} q^{2} + \zeta_{16} q^{3} - \zeta_{16}^{4} q^{4} - \zeta_{16}^{5} q^{5} + \zeta_{16}^{7} q^{6} + (\zeta_{16}^{6} - \zeta_{16}^{4} + \cdots + 1) q^{7} + \cdots + ( - \zeta_{16}^{7} + \zeta_{16}^{3}) q^{99} +O(q^{100}) $ q + z^6 * q^2 + z * q^3 - z^4 * q^4 - z^5 * q^5 + z^7 * q^6 + (z^6 - z^4 - z^2 + 1) * q^7 + z^2 * q^8 + z^2 * q^9 + z^3 * q^10 + (-z^5 + z) * q^11 - z^5 * q^12 + (-z^5 - 2z^4 - z^3) q^13 + (z^6 - z^4 + z^2 + 1) * q^14 - z^6 * q^15 - q^16 + (-z^7 - z^5 + 2z^4 + 3z^3 - z^2 + z) * q^17 - q^18 + (-2z^4 + 2) q^19 - z * q^20 + (z^7 - z^5 - z^3 + z) * q^21 + (z^7 + z^3) * q^22 + (-2z^7 + z^6 - 2z^5 + 2z - 1) q^23 + z^3 * q^24 - z^2 * q^25 + (z^3 + 2z^2 + z) q^26 + z^3 * q^27 + (z^6 - z^4 + z^2 - 1) * q^28 + (-z^7 - 2z^6 - 2z^5 - 2z^4 - z^3 - 2z^2 + 2) * q^29 + z^4 * q^30 + (-2z^7 + 2z^6 - 2z^4 + 2z^3 - z^2 + 4z - 1) q^31 - z^6 * q^32 + (-z^6 + z^2) * q^33 + (z^7 + z^5 + z^3 - 2z^2 - 3z + 1) * q^34 + (z^7 - z^5 + z^3 - z) * q^35 - z^6 * q^36 + (-4z^2 - 4) q^37 + (2z^6 + 2z^2) * q^38 + (-z^6 - 2z^5 - z^4) q^39 - z^7 * q^40 + (3z^6 + 3z^4 + 2z^3 + 3z^2 + 3) * q^41 + (z^7 - z^5 + z^3 + z) * q^42 + (-2z^7 + 2z^5 - 4z^4 + z^3 - 2z^2 + z - 4) * q^43 + (-z^5 - z) * q^44 - z^7 * q^45 + (2z^7 - z^6 + 2z^5 - z^4 + 2z^3) q^46 + (4z^5 + 2z^4 + 4z^3) q^47 - z * q^48 + (-3z^6 - 2z^4 + 2) * q^49 + q^50 + (-z^6 + 2z^5 + 3z^4 - z^3 + z^2 + 1) * q^51 + (z^7 - z - 2) * q^52 + (2z^6 + 2z^4 - 2z^3 + 2z - 2) * q^53 - z * q^54 + (-z^6 - z^2) * q^55 + (-z^6 - z^4 + z^2 - 1) * q^56 + (-2z^5 + 2z) * q^57 + (2z^6 + z^5 + 2z^4 + 2z^3 + 2z^2 + z + 2) * q^58 + (-5z^7 + 5z^5 + 4z^4 + 2z^3 + 2z + 4) q^59 - z^2 * q^60 + (2z^6 - 4z^3 + 2) * q^61 + (4z^7 - z^6 + 2z^5 - 2z^4 + 2z^2 - 2z + 1) q^62 + (-z^6 - z^4 + z^2 - 1) * q^63 + z^4 * q^64 + (-z^2 - 2z - 1) q^65 + (z^4 - 1) * q^66 + (-2z^7 + 2z^6 + 5z^5 - 5z^3 - 2z^2 + 2z - 2) * q^67 + (-3z^7 + z^6 - z^5 - z^3 - z + 2) q^68 + (z^7 - 2z^6 + 2z^2 - z + 2) * q^69 + (-z^7 - z^5 + z^3 - z) * q^70 + (-3z^6 + 3z^4 - 3z^2 - 3) q^71 + z^4 * q^72 + (-2z^7 + 3z^6 + 2z^5 + 3z^4 - 2z^3 + 5z^2 - 5) * q^73 + (-4z^6 + 4) q^74 - z^3 * q^75 + (-2z^4 - 2) q^76 + (2z^7 - 2z^5) * q^77 + (z^4 + 2z^3 + z^2) q^78 + (-4z^6 + 6z^5 - z^4 + z^2 - 6z + 4) q^79 + z^5 * q^80 + z^4 * q^81 + (3z^6 - 3z^4 - 3z^2 - 2z - 3) * q^82 + (2z^7 - 4z^6 + 2z^5 + 2z^4 - 4z^3 + 4z - 2) * q^83 + (z^7 - z^5 + z^3 - z) * q^84 + (z^7 - z^6 - z^4 - z^2 + 2z + 3) q^85 + (z^7 - 4z^6 + 2z^5 - 2z^3 + 4z^2 - z + 2) * q^86 + (-2z^7 - 2z^6 - 2z^5 - z^4 - 2z^3 + 2z + 1) q^87 + (-z^7 + z^3) * q^88 + (-6z^6 - 6z^5 + 2z^4 - 6z^3 - 6z^2) q^89 + z^5 * q^90 + (2z^7 + 2z^6 - 2z^4 + 2z^2 - 2) * q^91 + (-2z^5 + z^4 - 2z^3 + z^2 - 2z) q^92 + (2z^7 - 2z^5 + 2z^4 - z^3 + 4z^2 - z + 2) * q^93 + (-4z^3 - 2z^2 - 4z) q^94 + (-2z^5 - 2z) * q^95 - z^7 * q^96 + (-2z^7 + z^6 + 2z^5 + z^4 - 2z^3 - z^2 + 1) q^97 + (2z^6 + 3z^4 + 2z^2) q^98 + (-z^7 + z^3) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{7}+O(q^{10}) $ 8 * q + 8 * q^7	$ 8 q + 8 q^{7} + 8 q^{14} - 8 q^{16} - 8 q^{18} + 16 q^{19} - 8 q^{23} - 8 q^{28} + 16 q^{29} - 8 q^{31} + 8 q^{34} - 32 q^{37} + 24 q^{41} - 32 q^{43} + 16 q^{49} + 8 q^{50} + 8 q^{51} - 16 q^{52} - 16 q^{53} - 8 q^{56} + 16 q^{58} + 32 q^{59} + 16 q^{61} + 8 q^{62} - 8 q^{63} - 8 q^{65} - 8 q^{66} - 16 q^{67} + 16 q^{68} + 16 q^{69} - 24 q^{71} - 40 q^{73} + 32 q^{74} - 16 q^{76} + 32 q^{79} - 24 q^{82} - 16 q^{83} + 24 q^{85} + 16 q^{86} + 8 q^{87} - 16 q^{91} + 16 q^{93} + 8 q^{97}+O(q^{100}) $ 8 * q + 8 * q^7 + 8 * q^14 - 8 * q^16 - 8 * q^18 + 16 * q^19 - 8 * q^23 - 8 * q^28 + 16 * q^29 - 8 * q^31 + 8 * q^34 - 32 * q^37 + 24 * q^41 - 32 * q^43 + 16 * q^49 + 8 * q^50 + 8 * q^51 - 16 * q^52 - 16 * q^53 - 8 * q^56 + 16 * q^58 + 32 * q^59 + 16 * q^61 + 8 * q^62 - 8 * q^63 - 8 * q^65 - 8 * q^66 - 16 * q^67 + 16 * q^68 + 16 * q^69 - 24 * q^71 - 40 * q^73 + 32 * q^74 - 16 * q^76 + 32 * q^79 - 24 * q^82 - 16 * q^83 + 24 * q^85 + 16 * q^86 + 8 * q^87 - 16 * q^91 + 16 * q^93 + 8 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/510\mathbb{Z}\right)^\times$.

$n$	$241$	$307$	$341$
$\chi(n)$	$\zeta_{16}^{2}$	$1$	$1$

Embeddings

For each embedding $\iota_m$ of the coefficient field, the values $\iota_m(a_n)$ are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label

$\iota_m(\nu)$

$ a_{2} $

$ a_{3} $

$ a_{4} $

$ a_{5} $

$ a_{6} $

$ a_{7} $

$ a_{8} $

$ a_{9} $

$ a_{10} $

121.1

−0.923880	+	0.382683i
0.923880	−	0.382683i
−0.382683	−	0.923880i
0.382683	+	0.923880i
−0.382683	+	0.923880i
0.382683	−	0.923880i
−0.923880	−	0.382683i
0.923880	+	0.382683i

−0.707107

−

0.707107i

−0.923880

0.382683i

1.00000i

−0.382683

−

0.923880i

0.923880

0.382683i

−0.414214

1.00000i

0.707107

−

0.707107i

0.707107

−

0.707107i

−0.382683

0.923880i

121.2

−0.707107

−

0.707107i

0.923880

−

0.382683i

1.00000i

0.382683

0.923880i

−0.923880

−

0.382683i

−0.414214

1.00000i

0.707107

−

0.707107i

0.707107

−

0.707107i

0.382683

−

0.923880i

151.1

0.707107

0.707107i

−0.382683

−

0.923880i

1.00000i

0.923880

−

0.382683i

0.382683

−

0.923880i

2.41421

1.00000i

−0.707107

0.707107i

−0.707107

0.707107i

0.923880

0.382683i

151.2

0.707107

0.707107i

0.382683

0.923880i

1.00000i

−0.923880

0.382683i

−0.382683

0.923880i

2.41421

1.00000i

−0.707107

0.707107i

−0.707107

0.707107i

−0.923880

−

0.382683i

331.1

0.707107

−

0.707107i

−0.382683

0.923880i

−

1.00000i

0.923880

0.382683i

0.382683

0.923880i

2.41421

−

1.00000i

−0.707107

−

0.707107i

−0.707107

−

0.707107i

0.923880

−

0.382683i

331.2

0.707107

−

0.707107i

0.382683

−

0.923880i

−

1.00000i

−0.923880

−

0.382683i

−0.382683

−

0.923880i

2.41421

−

1.00000i

−0.707107

−

0.707107i

−0.707107

−

0.707107i

−0.923880

0.382683i

451.1

−0.707107

0.707107i

−0.923880

−

0.382683i

−

1.00000i

−0.382683

0.923880i

0.923880

−

0.382683i

−0.414214

−

1.00000i

0.707107

0.707107i

0.707107

0.707107i

−0.382683

−

0.923880i

451.2

−0.707107

0.707107i

0.923880

0.382683i

−

1.00000i

0.382683

−

0.923880i

−0.923880

0.382683i

−0.414214

−

1.00000i

0.707107

0.707107i

0.707107

0.707107i

0.382683

0.923880i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
17.d	even	8	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	510.2.u.b	✓	8
17.d	even	8	1	inner	510.2.u.b	✓	8
17.e	odd	16	1		8670.2.a.bu		4
17.e	odd	16	1		8670.2.a.bv		4

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
510.2.u.b	✓	8	1.a	even	1	1	trivial
510.2.u.b	✓	8	17.d	even	8	1	inner
8670.2.a.bu		4	17.e	odd	16	1
8670.2.a.bv		4	17.e	odd	16	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{7}^{4} - 4T_{7}^{3} + 4T_{7}^{2} + 8 $ T7^4 - 4*T7^3 + 4*T7^2 + 8 acting on $S_{2}^{\mathrm{new}}(510, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ (T^{4} + 1)^{2} $ (T^4 + 1)^2
$3$	$ T^{8} + 1 $ T^8 + 1
$5$	$ T^{8} + 1 $ T^8 + 1
$7$	$ (T^{4} - 4 T^{3} + 4 T^{2} + 8)^{2} $ (T^4 - 4T^3 + 4T^2 + 8)^2
$11$	$ T^{8} + 16 $ T^8 + 16
$13$	$ T^{8} + 24 T^{6} + \cdots + 4 $ T^8 + 24T^6 + 148T^4 + 176*T^2 + 4
$17$	$ T^{8} - 16 T^{6} + \cdots + 83521 $ T^8 - 16T^6 + 32T^5 + 290T^4 + 544T^3 - 4624*T^2 + 83521
$19$	$ (T^{2} - 4 T + 8)^{4} $ (T^2 - 4*T + 8)^4
$23$	$ T^{8} + 8 T^{7} + \cdots + 4 $ T^8 + 8T^7 - 4T^6 - 136T^5 + 200T^4 + 128T^3 + 1000T^2 - 48*T + 4
$29$	$ T^{8} - 16 T^{7} + \cdots + 4624 $ T^8 - 16T^7 + 176T^6 - 1056T^5 + 3712T^4 - 8256T^3 + 12096T^2 - 10880*T + 4624
$31$	$ T^{8} + 8 T^{7} + \cdots + 64516 $ T^8 + 8T^7 - 36T^6 - 152T^5 + 2760T^4 - 15392T^3 + 43944T^2 - 62992*T + 64516
$37$	$ (T^{4} + 16 T^{3} + \cdots + 512)^{2} $ (T^4 + 16T^3 + 96T^2 + 256*T + 512)^2
$41$	$ T^{8} - 24 T^{7} + \cdots + 141376 $ T^8 - 24T^7 + 360T^6 - 3360T^5 + 21024T^4 - 88512T^3 + 230976T^2 - 288768*T + 141376
$43$	$ T^{8} + 32 T^{7} + \cdots + 51076 $ T^8 + 32T^7 + 512T^6 + 4656T^5 + 26444T^4 + 89856T^3 + 175232T^2 + 133792*T + 51076
$47$	$ T^{8} + 144 T^{6} + \cdots + 73984 $ T^8 + 144T^6 + 5728T^4 + 71936*T^2 + 73984
$53$	$ T^{8} + 16 T^{7} + \cdots + 73984 $ T^8 + 16T^7 + 128T^6 + 512T^5 + 1120T^4 + 1280T^3 + 8192T^2 + 34816*T + 73984
$59$	$ T^{8} - 32 T^{7} + \cdots + 454276 $ T^8 - 32T^7 + 512T^6 - 3264T^5 + 4428T^4 + 84800T^3 + 346112T^2 + 560768*T + 454276
$61$	$ T^{8} - 16 T^{7} + \cdots + 1024 $ T^8 - 16T^7 + 112T^6 - 448T^5 + 1152T^4 + 10752T^2 + 6144T + 1024
$67$	$ (T^{4} + 8 T^{3} + \cdots - 2686)^{2} $ (T^4 + 8T^3 - 108T^2 - 1152*T - 2686)^2
$71$	$ (T^{4} + 12 T^{3} + \cdots + 648)^{2} $ (T^4 + 12T^3 + 108T^2 + 432*T + 648)^2
$73$	$ T^{8} + 40 T^{7} + \cdots + 4700224 $ T^8 + 40T^7 + 824T^6 + 11008T^5 + 101408T^4 + 634048T^3 + 2669248T^2 + 5619456*T + 4700224
$79$	$ T^{8} - 32 T^{7} + \cdots + 148996 $ T^8 - 32T^7 + 420T^6 - 3368T^5 + 28040T^4 - 98704T^3 + 1309912T^2 + 824496*T + 148996
$83$	$ T^{8} + 16 T^{7} + \cdots + 1024 $ T^8 + 16T^7 + 128T^6 - 1152T^5 + 4160T^4 - 7680T^3 + 8192T^2 - 4096*T + 1024
$89$	$ T^{8} + 592 T^{6} + \cdots + 51609856 $ T^8 + 592T^6 + 100896T^4 + 4317952*T^2 + 51609856
$97$	$ T^{8} - 8 T^{7} + \cdots + 64 $ T^8 - 8T^7 - 8T^6 + 128T^5 + 544T^4 + 1088T^3 + 1216T^2 + 256*T + 64
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Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 510 = 2 \cdot 3 \cdot 5 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	510.u (of order \(8\), degree \(4\), minimal)

\(f(q)\)	\(=\)	\( q + \zeta_{16}^{6} q^{2} + \zeta_{16} q^{3} - \zeta_{16}^{4} q^{4} - \zeta_{16}^{5} q^{5} + \zeta_{16}^{7} q^{6} + (\zeta_{16}^{6} - \zeta_{16}^{4} + \cdots + 1) q^{7} + \cdots + \zeta_{16}^{2} q^{9} +O(q^{10}) \) q + z^6 * q^2 + z * q^3 - z^4 * q^4 - z^5 * q^5 + z^7 * q^6 + (z^6 - z^4 - z^2 + 1) * q^7 + z^2 * q^8 + z^2 * q^9	\( q + \zeta_{16}^{6} q^{2} + \zeta_{16} q^{3} - \zeta_{16}^{4} q^{4} - \zeta_{16}^{5} q^{5} + \zeta_{16}^{7} q^{6} + (\zeta_{16}^{6} - \zeta_{16}^{4} + \cdots + 1) q^{7} + \cdots + ( - \zeta_{16}^{7} + \zeta_{16}^{3}) q^{99} +O(q^{100}) \) q + z^6 * q^2 + z * q^3 - z^4 * q^4 - z^5 * q^5 + z^7 * q^6 + (z^6 - z^4 - z^2 + 1) * q^7 + z^2 * q^8 + z^2 * q^9 + z^3 * q^10 + (-z^5 + z) * q^11 - z^5 * q^12 + (-z^5 - 2z^4 - z^3) q^13 + (z^6 - z^4 + z^2 + 1) * q^14 - z^6 * q^15 - q^16 + (-z^7 - z^5 + 2z^4 + 3z^3 - z^2 + z) * q^17 - q^18 + (-2z^4 + 2) q^19 - z * q^20 + (z^7 - z^5 - z^3 + z) * q^21 + (z^7 + z^3) * q^22 + (-2z^7 + z^6 - 2z^5 + 2z - 1) q^23 + z^3 * q^24 - z^2 * q^25 + (z^3 + 2z^2 + z) q^26 + z^3 * q^27 + (z^6 - z^4 + z^2 - 1) * q^28 + (-z^7 - 2z^6 - 2z^5 - 2z^4 - z^3 - 2z^2 + 2) * q^29 + z^4 * q^30 + (-2z^7 + 2z^6 - 2z^4 + 2z^3 - z^2 + 4z - 1) q^31 - z^6 * q^32 + (-z^6 + z^2) * q^33 + (z^7 + z^5 + z^3 - 2z^2 - 3z + 1) * q^34 + (z^7 - z^5 + z^3 - z) * q^35 - z^6 * q^36 + (-4z^2 - 4) q^37 + (2z^6 + 2z^2) * q^38 + (-z^6 - 2z^5 - z^4) q^39 - z^7 * q^40 + (3z^6 + 3z^4 + 2z^3 + 3z^2 + 3) * q^41 + (z^7 - z^5 + z^3 + z) * q^42 + (-2z^7 + 2z^5 - 4z^4 + z^3 - 2z^2 + z - 4) * q^43 + (-z^5 - z) * q^44 - z^7 * q^45 + (2z^7 - z^6 + 2z^5 - z^4 + 2z^3) q^46 + (4z^5 + 2z^4 + 4z^3) q^47 - z * q^48 + (-3z^6 - 2z^4 + 2) * q^49 + q^50 + (-z^6 + 2z^5 + 3z^4 - z^3 + z^2 + 1) * q^51 + (z^7 - z - 2) * q^52 + (2z^6 + 2z^4 - 2z^3 + 2z - 2) * q^53 - z * q^54 + (-z^6 - z^2) * q^55 + (-z^6 - z^4 + z^2 - 1) * q^56 + (-2z^5 + 2z) * q^57 + (2z^6 + z^5 + 2z^4 + 2z^3 + 2z^2 + z + 2) * q^58 + (-5z^7 + 5z^5 + 4z^4 + 2z^3 + 2z + 4) q^59 - z^2 * q^60 + (2z^6 - 4z^3 + 2) * q^61 + (4z^7 - z^6 + 2z^5 - 2z^4 + 2z^2 - 2z + 1) q^62 + (-z^6 - z^4 + z^2 - 1) * q^63 + z^4 * q^64 + (-z^2 - 2z - 1) q^65 + (z^4 - 1) * q^66 + (-2z^7 + 2z^6 + 5z^5 - 5z^3 - 2z^2 + 2z - 2) * q^67 + (-3z^7 + z^6 - z^5 - z^3 - z + 2) q^68 + (z^7 - 2z^6 + 2z^2 - z + 2) * q^69 + (-z^7 - z^5 + z^3 - z) * q^70 + (-3z^6 + 3z^4 - 3z^2 - 3) q^71 + z^4 * q^72 + (-2z^7 + 3z^6 + 2z^5 + 3z^4 - 2z^3 + 5z^2 - 5) * q^73 + (-4z^6 + 4) q^74 - z^3 * q^75 + (-2z^4 - 2) q^76 + (2z^7 - 2z^5) * q^77 + (z^4 + 2z^3 + z^2) q^78 + (-4z^6 + 6z^5 - z^4 + z^2 - 6z + 4) q^79 + z^5 * q^80 + z^4 * q^81 + (3z^6 - 3z^4 - 3z^2 - 2z - 3) * q^82 + (2z^7 - 4z^6 + 2z^5 + 2z^4 - 4z^3 + 4z - 2) * q^83 + (z^7 - z^5 + z^3 - z) * q^84 + (z^7 - z^6 - z^4 - z^2 + 2z + 3) q^85 + (z^7 - 4z^6 + 2z^5 - 2z^3 + 4z^2 - z + 2) * q^86 + (-2z^7 - 2z^6 - 2z^5 - z^4 - 2z^3 + 2z + 1) q^87 + (-z^7 + z^3) * q^88 + (-6z^6 - 6z^5 + 2z^4 - 6z^3 - 6z^2) q^89 + z^5 * q^90 + (2z^7 + 2z^6 - 2z^4 + 2z^2 - 2) * q^91 + (-2z^5 + z^4 - 2z^3 + z^2 - 2z) q^92 + (2z^7 - 2z^5 + 2z^4 - z^3 + 4z^2 - z + 2) * q^93 + (-4z^3 - 2z^2 - 4z) q^94 + (-2z^5 - 2z) * q^95 - z^7 * q^96 + (-2z^7 + z^6 + 2z^5 + z^4 - 2z^3 - z^2 + 1) q^97 + (2z^6 + 3z^4 + 2z^2) q^98 + (-z^7 + z^3) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{7}+O(q^{10}) \) 8 * q + 8 * q^7	\( 8 q + 8 q^{7} + 8 q^{14} - 8 q^{16} - 8 q^{18} + 16 q^{19} - 8 q^{23} - 8 q^{28} + 16 q^{29} - 8 q^{31} + 8 q^{34} - 32 q^{37} + 24 q^{41} - 32 q^{43} + 16 q^{49} + 8 q^{50} + 8 q^{51} - 16 q^{52} - 16 q^{53} - 8 q^{56} + 16 q^{58} + 32 q^{59} + 16 q^{61} + 8 q^{62} - 8 q^{63} - 8 q^{65} - 8 q^{66} - 16 q^{67} + 16 q^{68} + 16 q^{69} - 24 q^{71} - 40 q^{73} + 32 q^{74} - 16 q^{76} + 32 q^{79} - 24 q^{82} - 16 q^{83} + 24 q^{85} + 16 q^{86} + 8 q^{87} - 16 q^{91} + 16 q^{93} + 8 q^{97}+O(q^{100}) \) 8 * q + 8 * q^7 + 8 * q^14 - 8 * q^16 - 8 * q^18 + 16 * q^19 - 8 * q^23 - 8 * q^28 + 16 * q^29 - 8 * q^31 + 8 * q^34 - 32 * q^37 + 24 * q^41 - 32 * q^43 + 16 * q^49 + 8 * q^50 + 8 * q^51 - 16 * q^52 - 16 * q^53 - 8 * q^56 + 16 * q^58 + 32 * q^59 + 16 * q^61 + 8 * q^62 - 8 * q^63 - 8 * q^65 - 8 * q^66 - 16 * q^67 + 16 * q^68 + 16 * q^69 - 24 * q^71 - 40 * q^73 + 32 * q^74 - 16 * q^76 + 32 * q^79 - 24 * q^82 - 16 * q^83 + 24 * q^85 + 16 * q^86 + 8 * q^87 - 16 * q^91 + 16 * q^93 + 8 * q^97

Introduction

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Modular forms

Varieties

Fields

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Database

Properties

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Newspace parameters

Newform invariants

$q$-expansion

Character values

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Hecke kernels

Hecke characteristic polynomials

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	510.2.u.b

Level	$510$
Weight	$2$
Character orbit	510.u
Analytic conductor	$4.072$
Analytic rank	$0$
Dimension	$8$
CM	no
Inner twists	$2$

Self dual:	no
Analytic conductor:	\(4.07237050309\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Relative dimension:	\(2\) over \(\Q(\zeta_{8})\)
Coefficient field:	\(\Q(\zeta_{16})\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{8} + 1 \) x^8 + 1
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{8}]$

\(n\)	\(241\)	\(307\)	\(341\)
\(\chi(n)\)	\(\zeta_{16}^{2}\)	\(1\)	\(1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 121.2
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( (T^{4} + 1)^{2} \) (T^4 + 1)^2
$3$	\( T^{8} + 1 \) T^8 + 1
$5$	\( T^{8} + 1 \) T^8 + 1
$7$	\( (T^{4} - 4 T^{3} + 4 T^{2} + 8)^{2} \) (T^4 - 4T^3 + 4T^2 + 8)^2
$11$	\( T^{8} + 16 \) T^8 + 16
$13$	\( T^{8} + 24 T^{6} + \cdots + 4 \) T^8 + 24T^6 + 148T^4 + 176*T^2 + 4
$17$	\( T^{8} - 16 T^{6} + \cdots + 83521 \) T^8 - 16T^6 + 32T^5 + 290T^4 + 544T^3 - 4624*T^2 + 83521
$19$	\( (T^{2} - 4 T + 8)^{4} \) (T^2 - 4*T + 8)^4
$23$	\( T^{8} + 8 T^{7} + \cdots + 4 \) T^8 + 8T^7 - 4T^6 - 136T^5 + 200T^4 + 128T^3 + 1000T^2 - 48*T + 4
$29$	\( T^{8} - 16 T^{7} + \cdots + 4624 \) T^8 - 16T^7 + 176T^6 - 1056T^5 + 3712T^4 - 8256T^3 + 12096T^2 - 10880*T + 4624
$31$	\( T^{8} + 8 T^{7} + \cdots + 64516 \) T^8 + 8T^7 - 36T^6 - 152T^5 + 2760T^4 - 15392T^3 + 43944T^2 - 62992*T + 64516
$37$	\( (T^{4} + 16 T^{3} + \cdots + 512)^{2} \) (T^4 + 16T^3 + 96T^2 + 256*T + 512)^2
$41$	\( T^{8} - 24 T^{7} + \cdots + 141376 \) T^8 - 24T^7 + 360T^6 - 3360T^5 + 21024T^4 - 88512T^3 + 230976T^2 - 288768*T + 141376
$43$	\( T^{8} + 32 T^{7} + \cdots + 51076 \) T^8 + 32T^7 + 512T^6 + 4656T^5 + 26444T^4 + 89856T^3 + 175232T^2 + 133792*T + 51076
$47$	\( T^{8} + 144 T^{6} + \cdots + 73984 \) T^8 + 144T^6 + 5728T^4 + 71936*T^2 + 73984
$53$	\( T^{8} + 16 T^{7} + \cdots + 73984 \) T^8 + 16T^7 + 128T^6 + 512T^5 + 1120T^4 + 1280T^3 + 8192T^2 + 34816*T + 73984
$59$	\( T^{8} - 32 T^{7} + \cdots + 454276 \) T^8 - 32T^7 + 512T^6 - 3264T^5 + 4428T^4 + 84800T^3 + 346112T^2 + 560768*T + 454276
$61$	\( T^{8} - 16 T^{7} + \cdots + 1024 \) T^8 - 16T^7 + 112T^6 - 448T^5 + 1152T^4 + 10752T^2 + 6144T + 1024
$67$	\( (T^{4} + 8 T^{3} + \cdots - 2686)^{2} \) (T^4 + 8T^3 - 108T^2 - 1152*T - 2686)^2
$71$	\( (T^{4} + 12 T^{3} + \cdots + 648)^{2} \) (T^4 + 12T^3 + 108T^2 + 432*T + 648)^2
$73$	\( T^{8} + 40 T^{7} + \cdots + 4700224 \) T^8 + 40T^7 + 824T^6 + 11008T^5 + 101408T^4 + 634048T^3 + 2669248T^2 + 5619456*T + 4700224
$79$	\( T^{8} - 32 T^{7} + \cdots + 148996 \) T^8 - 32T^7 + 420T^6 - 3368T^5 + 28040T^4 - 98704T^3 + 1309912T^2 + 824496*T + 148996
$83$	\( T^{8} + 16 T^{7} + \cdots + 1024 \) T^8 + 16T^7 + 128T^6 - 1152T^5 + 4160T^4 - 7680T^3 + 8192T^2 - 4096*T + 1024
$89$	\( T^{8} + 592 T^{6} + \cdots + 51609856 \) T^8 + 592T^6 + 100896T^4 + 4317952*T^2 + 51609856
$97$	\( T^{8} - 8 T^{7} + \cdots + 64 \) T^8 - 8T^7 - 8T^6 + 128T^5 + 544T^4 + 1088T^3 + 1216T^2 + 256*T + 64
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