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Properties

Label 5148.2.e.c
Level $5148$
Weight $2$
Character orbit 5148.e
Analytic conductor $41.107$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [5148,2,Mod(1585,5148)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5148, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("5148.1585");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5148 = 2^{2} \cdot 3^{2} \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5148.e (of order \(2\), degree \(1\), 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: \(41.1069869606\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 21x^{6} + 136x^{4} + 309x^{2} + 225 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 572)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{7} - \beta_{6} - \beta_{5} - \beta_1) q^{5} + (\beta_{4} + \beta_1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{7} - \beta_{6} - \beta_{5} - \beta_1) q^{5} + (\beta_{4} + \beta_1) q^{7} - \beta_{4} q^{11} + ( - \beta_{6} - \beta_{3} - \beta_1 - 1) q^{13} + ( - \beta_{6} + \beta_{5} + 2 \beta_{2} - 2) q^{17} + ( - \beta_{7} - 2 \beta_1) q^{19} + (\beta_{6} - \beta_{5} + \beta_{3} + \cdots - 1) q^{23}+ \cdots + (\beta_{7} - \beta_{6} - \beta_{5} + \cdots - \beta_1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{13} - 16 q^{17} - 10 q^{23} - 52 q^{25} + 4 q^{29} + 8 q^{35} - 20 q^{43} + 2 q^{49} - 38 q^{53} - 36 q^{61} - 36 q^{65} + 10 q^{77} + 40 q^{79} + 22 q^{91} + 12 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 21x^{6} + 136x^{4} + 309x^{2} + 225 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{6} - 18\nu^{4} - 88\nu^{2} - 105 ) / 6 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{6} - 20\nu^{4} - 112\nu^{2} - 149 ) / 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -7\nu^{7} - 132\nu^{5} - 682\nu^{3} - 843\nu ) / 90 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -7\nu^{7} - 15\nu^{6} - 132\nu^{5} - 300\nu^{4} - 652\nu^{3} - 1620\nu^{2} - 663\nu - 1935 ) / 120 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -7\nu^{7} + 15\nu^{6} - 132\nu^{5} + 300\nu^{4} - 652\nu^{3} + 1620\nu^{2} - 663\nu + 1935 ) / 120 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 19\nu^{7} + 384\nu^{5} + 2224\nu^{3} + 3291\nu ) / 180 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{6} + \beta_{5} - \beta_{3} - 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{6} + 2\beta_{5} - 3\beta_{4} - 6\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 12\beta_{6} - 12\beta_{5} + 10\beta_{3} + 3\beta_{2} + 38 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 7\beta_{7} - 29\beta_{6} - 29\beta_{5} + 53\beta_{4} + 48\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -128\beta_{6} + 128\beta_{5} - 92\beta_{3} - 60\beta_{2} - 349 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -132\beta_{7} + 352\beta_{6} + 352\beta_{5} - 720\beta_{4} - 441\beta_1 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5148\mathbb{Z}\right)^\times\).

\(n\) \(937\) \(1145\) \(2575\) \(4357\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1585.1
1.43491i
2.51091i
− 3.32727i
− 1.25126i
1.25126i
3.32727i
− 2.51091i
− 1.43491i
0 0 0 − 4.37595i 0 2.43491i 0 0 0
1585.2 0 0 0 − 3.81560i 0 1.51091i 0 0 0
1585.3 0 0 0 − 2.74343i 0 − 4.32727i 0 0 0
1585.4 0 0 0 − 2.18309i 0 − 0.251260i 0 0 0
1585.5 0 0 0 2.18309i 0 0.251260i 0 0 0
1585.6 0 0 0 2.74343i 0 4.32727i 0 0 0
1585.7 0 0 0 3.81560i 0 − 1.51091i 0 0 0
1585.8 0 0 0 4.37595i 0 − 2.43491i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1585.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 5148.2.e.c 8
3.b odd 2 1 572.2.f.c ✓ 8
12.b even 2 1 2288.2.j.i 8
13.b even 2 1 inner 5148.2.e.c 8
39.d odd 2 1 572.2.f.c ✓ 8
39.f even 4 1 7436.2.a.o 4
39.f even 4 1 7436.2.a.p 4
156.h even 2 1 2288.2.j.i 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
572.2.f.c ✓ 8 3.b odd 2 1
572.2.f.c ✓ 8 39.d odd 2 1
2288.2.j.i 8 12.b even 2 1
2288.2.j.i 8 156.h even 2 1
5148.2.e.c 8 1.a even 1 1 trivial
5148.2.e.c 8 13.b even 2 1 inner
7436.2.a.o 4 39.f even 4 1
7436.2.a.p 4 39.f even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(5148, [\chi])\):

\( T_{5}^{8} + 46T_{5}^{6} + 729T_{5}^{4} + 4636T_{5}^{2} + 10000 \) Copy content Toggle raw display
\( T_{7}^{8} + 27T_{7}^{6} + 169T_{7}^{4} + 264T_{7}^{2} + 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} + 46 T^{6} + \cdots + 10000 \) Copy content Toggle raw display
$7$ \( T^{8} + 27 T^{6} + \cdots + 16 \) Copy content Toggle raw display
$11$ \( (T^{2} + 1)^{4} \) Copy content Toggle raw display
$13$ \( T^{8} + 8 T^{7} + \cdots + 28561 \) Copy content Toggle raw display
$17$ \( (T^{4} + 8 T^{3} + \cdots - 352)^{2} \) Copy content Toggle raw display
$19$ \( T^{8} + 107 T^{6} + \cdots + 106276 \) Copy content Toggle raw display
$23$ \( (T^{4} + 5 T^{3} + \cdots - 111)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} - 2 T^{3} + \cdots - 344)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + 190 T^{6} + \cdots + 283024 \) Copy content Toggle raw display
$37$ \( T^{8} + 122 T^{6} + \cdots + 309136 \) Copy content Toggle raw display
$41$ \( T^{8} + 275 T^{6} + \cdots + 9326916 \) Copy content Toggle raw display
$43$ \( (T^{4} + 10 T^{3} + \cdots - 1272)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + 132 T^{6} + \cdots + 30976 \) Copy content Toggle raw display
$53$ \( (T^{4} + 19 T^{3} + \cdots + 216)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + 266 T^{6} + \cdots + 156816 \) Copy content Toggle raw display
$61$ \( (T^{4} + 18 T^{3} + \cdots - 2440)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + 222 T^{6} + \cdots + 5776 \) Copy content Toggle raw display
$71$ \( T^{8} + 318 T^{6} + \cdots + 17424 \) Copy content Toggle raw display
$73$ \( T^{8} + 319 T^{6} + \cdots + 22429696 \) Copy content Toggle raw display
$79$ \( (T^{4} - 20 T^{3} + \cdots + 2456)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + 207 T^{6} + \cdots + 1119364 \) Copy content Toggle raw display
$89$ \( T^{8} + 446 T^{6} + \cdots + 23931664 \) Copy content Toggle raw display
$97$ \( T^{8} + 302 T^{6} + \cdots + 7683984 \) Copy content Toggle raw display
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