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Properties

Label 5355.2.a.bx
Level $5355$
Weight $2$
Character orbit 5355.a
Self dual yes
Analytic conductor $42.760$
Analytic rank $0$
Dimension $7$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [5355,2,Mod(1,5355)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5355, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("5355.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5355 = 3^{2} \cdot 5 \cdot 7 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5355.a (trivial)

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: yes
Analytic conductor: \(42.7598902824\)
Analytic rank: \(0\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{7} - x^{6} - 12x^{5} + 8x^{4} + 43x^{3} - 15x^{2} - 44x + 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(1,\beta_1,\ldots,\beta_{6}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + (\beta_{2} + 2) q^{4} - q^{5} + q^{7} + ( - \beta_{3} - \beta_{2} - \beta_1 - 2) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + (\beta_{2} + 2) q^{4} - q^{5} + q^{7} + ( - \beta_{3} - \beta_{2} - \beta_1 - 2) q^{8} + \beta_1 q^{10} + (\beta_{3} + \beta_{2}) q^{11} + ( - \beta_{6} + \beta_1 + 2) q^{13} - \beta_1 q^{14} + (\beta_{4} + 2 \beta_{2} + \beta_1 + 2) q^{16} - q^{17} + ( - \beta_{6} - \beta_{5} - \beta_{2}) q^{19} + ( - \beta_{2} - 2) q^{20} + ( - \beta_{4} - 3 \beta_{2} + \beta_1 - 2) q^{22} + ( - \beta_{5} - \beta_{4} - 2) q^{23} + q^{25} + ( - \beta_{6} + \beta_{5} + \beta_{4} + \cdots - 2) q^{26}+ \cdots - \beta_1 q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q - q^{2} + 11 q^{4} - 7 q^{5} + 7 q^{7} - 9 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 7 q - q^{2} + 11 q^{4} - 7 q^{5} + 7 q^{7} - 9 q^{8} + q^{10} - 6 q^{11} + 16 q^{13} - q^{14} + 11 q^{16} - 7 q^{17} + 6 q^{19} - 11 q^{20} - 6 q^{22} - 14 q^{23} + 7 q^{25} - 16 q^{26} + 11 q^{28} - 12 q^{29} + 12 q^{31} - 29 q^{32} + q^{34} - 7 q^{35} + 4 q^{37} + 30 q^{38} + 9 q^{40} + 36 q^{41} + 6 q^{43} + 18 q^{44} + 14 q^{46} + 8 q^{47} + 7 q^{49} - q^{50} + 44 q^{52} + 18 q^{53} + 6 q^{55} - 9 q^{56} + 48 q^{58} - 12 q^{59} + 38 q^{61} + 24 q^{62} + 31 q^{64} - 16 q^{65} + 26 q^{67} - 11 q^{68} + q^{70} - 6 q^{71} - 8 q^{73} - 22 q^{74} - 30 q^{76} - 6 q^{77} + 6 q^{79} - 11 q^{80} - 18 q^{82} + 7 q^{85} + 78 q^{86} - 102 q^{88} + 14 q^{89} + 16 q^{91} - 10 q^{92} - 14 q^{94} - 6 q^{95} + 20 q^{97} - q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{7} - x^{6} - 12x^{5} + 8x^{4} + 43x^{3} - 15x^{2} - 44x + 8 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - \nu^{2} - 5\nu + 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{4} - 8\nu^{2} - \nu + 10 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{6} - \nu^{5} - 10\nu^{4} + 6\nu^{3} + 25\nu^{2} - 3\nu - 10 ) / 2 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{6} + 3\nu^{5} + 8\nu^{4} - 24\nu^{3} - 13\nu^{2} + 39\nu - 2 ) / 2 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + \beta_{2} + 5\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{4} + 8\beta_{2} + \beta _1 + 22 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{6} + \beta_{5} + \beta_{4} + 9\beta_{3} + 11\beta_{2} + 28\beta _1 + 22 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{6} + 3\beta_{5} + 11\beta_{4} + 3\beta_{3} + 60\beta_{2} + 11\beta _1 + 140 \) Copy content Toggle raw display

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
2.77731
2.22224
1.30741
0.176696
−1.36925
−1.69076
−2.42366
−2.77731 0 5.71348 −1.00000 0 1.00000 −10.3135 0 2.77731
1.2 −2.22224 0 2.93835 −1.00000 0 1.00000 −2.08524 0 2.22224
1.3 −1.30741 0 −0.290667 −1.00000 0 1.00000 2.99485 0 1.30741
1.4 −0.176696 0 −1.96878 −1.00000 0 1.00000 0.701265 0 0.176696
1.5 1.36925 0 −0.125160 −1.00000 0 1.00000 −2.90987 0 −1.36925
1.6 1.69076 0 0.858669 −1.00000 0 1.00000 −1.92972 0 −1.69076
1.7 2.42366 0 3.87411 −1.00000 0 1.00000 4.54220 0 −2.42366
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.7
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \( +1 \)
\(5\) \( +1 \)
\(7\) \( -1 \)
\(17\) \( +1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 5355.2.a.bx ✓ 7
3.b odd 2 1 5355.2.a.by yes 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5355.2.a.bx ✓ 7 1.a even 1 1 trivial
5355.2.a.by yes 7 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(5355))\):

\( T_{2}^{7} + T_{2}^{6} - 12T_{2}^{5} - 8T_{2}^{4} + 43T_{2}^{3} + 15T_{2}^{2} - 44T_{2} - 8 \) Copy content Toggle raw display
\( T_{11}^{7} + 6T_{11}^{6} - 36T_{11}^{5} - 232T_{11}^{4} + 144T_{11}^{3} + 1680T_{11}^{2} + 64T_{11} - 3264 \) Copy content Toggle raw display
\( T_{13}^{7} - 16T_{13}^{6} + 57T_{13}^{5} + 284T_{13}^{4} - 2396T_{13}^{3} + 5496T_{13}^{2} - 4112T_{13} - 16 \) Copy content Toggle raw display
\( T_{19}^{7} - 6T_{19}^{6} - 84T_{19}^{5} + 536T_{19}^{4} + 1584T_{19}^{3} - 12384T_{19}^{2} + 6976T_{19} + 23424 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{7} + T^{6} - 12 T^{5} + \cdots - 8 \) Copy content Toggle raw display
$3$ \( T^{7} \) Copy content Toggle raw display
$5$ \( (T + 1)^{7} \) Copy content Toggle raw display
$7$ \( (T - 1)^{7} \) Copy content Toggle raw display
$11$ \( T^{7} + 6 T^{6} + \cdots - 3264 \) Copy content Toggle raw display
$13$ \( T^{7} - 16 T^{6} + \cdots - 16 \) Copy content Toggle raw display
$17$ \( (T + 1)^{7} \) Copy content Toggle raw display
$19$ \( T^{7} - 6 T^{6} + \cdots + 23424 \) Copy content Toggle raw display
$23$ \( T^{7} + 14 T^{6} + \cdots + 70688 \) Copy content Toggle raw display
$29$ \( T^{7} + 12 T^{6} + \cdots - 233472 \) Copy content Toggle raw display
$31$ \( T^{7} - 12 T^{6} + \cdots + 1968 \) Copy content Toggle raw display
$37$ \( T^{7} - 4 T^{6} + \cdots + 45152 \) Copy content Toggle raw display
$41$ \( T^{7} - 36 T^{6} + \cdots + 397728 \) Copy content Toggle raw display
$43$ \( T^{7} - 6 T^{6} + \cdots + 1541376 \) Copy content Toggle raw display
$47$ \( T^{7} - 8 T^{6} + \cdots + 481696 \) Copy content Toggle raw display
$53$ \( T^{7} - 18 T^{6} + \cdots - 41472 \) Copy content Toggle raw display
$59$ \( T^{7} + 12 T^{6} + \cdots - 165888 \) Copy content Toggle raw display
$61$ \( T^{7} - 38 T^{6} + \cdots + 15088 \) Copy content Toggle raw display
$67$ \( T^{7} - 26 T^{6} + \cdots - 1536 \) Copy content Toggle raw display
$71$ \( T^{7} + 6 T^{6} + \cdots - 3264 \) Copy content Toggle raw display
$73$ \( T^{7} + 8 T^{6} + \cdots + 275456 \) Copy content Toggle raw display
$79$ \( T^{7} - 6 T^{6} + \cdots + 3294336 \) Copy content Toggle raw display
$83$ \( T^{7} - 255 T^{5} + \cdots + 291168 \) Copy content Toggle raw display
$89$ \( T^{7} - 14 T^{6} + \cdots + 5248 \) Copy content Toggle raw display
$97$ \( T^{7} - 20 T^{6} + \cdots - 2227712 \) Copy content Toggle raw display
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