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Properties

Label 8415.2.a.bp
Level $8415$
Weight $2$
Character orbit 8415.a
Self dual yes
Analytic conductor $67.194$
Analytic rank $0$
Dimension $8$
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] = [8415,2,Mod(1,8415)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8415, 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("8415.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8415 = 3^{2} \cdot 5 \cdot 11 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8415.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: \(67.1941133009\)
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: \( x^{8} - 3x^{7} - 9x^{6} + 29x^{5} + 18x^{4} - 73x^{3} + 5x^{2} + 27x - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 2805)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + (\beta_{2} + 1) q^{4} - q^{5} + ( - \beta_{4} + 1) q^{7} + ( - \beta_{3} - \beta_{2} - \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + (\beta_{2} + 1) q^{4} - q^{5} + ( - \beta_{4} + 1) q^{7} + ( - \beta_{3} - \beta_{2} - \beta_1) q^{8} + \beta_1 q^{10} - q^{11} + (\beta_{7} + 1) q^{13} + ( - \beta_{7} + 2 \beta_{5} + \beta_{2} + \cdots + 1) q^{14}+ \cdots + ( - 3 \beta_{7} + 5 \beta_{5} + \cdots + 9) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 3 q^{2} + 11 q^{4} - 8 q^{5} + 9 q^{7} - 9 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 3 q^{2} + 11 q^{4} - 8 q^{5} + 9 q^{7} - 9 q^{8} + 3 q^{10} - 8 q^{11} + 10 q^{13} + q^{14} + 17 q^{16} - 8 q^{17} + 7 q^{19} - 11 q^{20} + 3 q^{22} - 6 q^{23} + 8 q^{25} + 4 q^{26} + 22 q^{28} - 6 q^{29} + 9 q^{31} - 26 q^{32} + 3 q^{34} - 9 q^{35} + 3 q^{37} + 3 q^{38} + 9 q^{40} + 16 q^{41} + 30 q^{43} - 11 q^{44} - 19 q^{46} - 9 q^{47} + 11 q^{49} - 3 q^{50} + 14 q^{52} + 2 q^{53} + 8 q^{55} - 6 q^{56} + 13 q^{58} + 6 q^{59} + q^{61} + q^{62} + 5 q^{64} - 10 q^{65} + 34 q^{67} - 11 q^{68} - q^{70} - 19 q^{71} + 20 q^{73} + 42 q^{74} - 26 q^{76} - 9 q^{77} - 6 q^{79} - 17 q^{80} + 25 q^{82} - 23 q^{83} + 8 q^{85} - 20 q^{86} + 9 q^{88} + 16 q^{89} + 5 q^{91} + 42 q^{92} - 24 q^{94} - 7 q^{95} + 9 q^{97} + 63 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 3x^{7} - 9x^{6} + 29x^{5} + 18x^{4} - 73x^{3} + 5x^{2} + 27x - 3 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - \nu^{2} - 5\nu + 3 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{7} - \nu^{6} - 10\nu^{5} + 6\nu^{4} + 24\nu^{3} - 5\nu^{2} - \nu ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{6} - \nu^{5} - 10\nu^{4} + 6\nu^{3} + 24\nu^{2} - 5\nu - 3 ) / 2 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( -\nu^{7} + \nu^{6} + 11\nu^{5} - 7\nu^{4} - 33\nu^{3} + 10\nu^{2} + 19\nu - 2 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -2\nu^{7} + 3\nu^{6} + 21\nu^{5} - 26\nu^{4} - 56\nu^{3} + 56\nu^{2} + 15\nu - 13 ) / 2 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + \beta_{2} + 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{7} + \beta_{6} + \beta_{5} + 2\beta_{3} + 8\beta_{2} + \beta _1 + 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -\beta_{7} + 2\beta_{6} + \beta_{5} + 2\beta_{4} + 11\beta_{3} + 12\beta_{2} + 28\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -11\beta_{7} + 12\beta_{6} + 13\beta_{5} + 2\beta_{4} + 25\beta_{3} + 62\beta_{2} + 13\beta _1 + 83 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -15\beta_{7} + 26\beta_{6} + 17\beta_{5} + 24\beta_{4} + 99\beta_{3} + 115\beta_{2} + 168\beta _1 + 28 \) 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
2.80084
2.16282
2.07244
0.747869
0.112491
−0.634036
−1.88191
−2.38053
−2.80084 0 5.84473 −1.00000 0 0.856821 −10.7685 0 2.80084
1.2 −2.16282 0 2.67780 −1.00000 0 3.84329 −1.46595 0 2.16282
1.3 −2.07244 0 2.29503 −1.00000 0 −0.713527 −0.611429 0 2.07244
1.4 −0.747869 0 −1.44069 −1.00000 0 −1.99392 2.57319 0 0.747869
1.5 −0.112491 0 −1.98735 −1.00000 0 1.07041 0.448541 0 0.112491
1.6 0.634036 0 −1.59800 −1.00000 0 3.80254 −2.28126 0 −0.634036
1.7 1.88191 0 1.54158 −1.00000 0 −2.74904 −0.862712 0 −1.88191
1.8 2.38053 0 3.66691 −1.00000 0 4.88343 3.96812 0 −2.38053
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(5\) \(1\)
\(11\) \(1\)
\(17\) \(1\)

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 8415.2.a.bp 8
3.b odd 2 1 2805.2.a.u ✓ 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2805.2.a.u ✓ 8 3.b odd 2 1
8415.2.a.bp 8 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8415))\):

\( T_{2}^{8} + 3T_{2}^{7} - 9T_{2}^{6} - 29T_{2}^{5} + 18T_{2}^{4} + 73T_{2}^{3} + 5T_{2}^{2} - 27T_{2} - 3 \) Copy content Toggle raw display
\( T_{7}^{8} - 9T_{7}^{7} + 7T_{7}^{6} + 113T_{7}^{5} - 187T_{7}^{4} - 376T_{7}^{3} + 569T_{7}^{2} + 144T_{7} - 256 \) Copy content Toggle raw display
\( T_{13}^{8} - 10T_{13}^{7} - T_{13}^{6} + 261T_{13}^{5} - 543T_{13}^{4} - 1353T_{13}^{3} + 4149T_{13}^{2} - 24T_{13} - 4156 \) Copy content Toggle raw display
\( T_{19}^{8} - 7T_{19}^{7} - 63T_{19}^{6} + 509T_{19}^{5} + 789T_{19}^{4} - 10120T_{19}^{3} + 3763T_{19}^{2} + 60032T_{19} - 70768 \) Copy content Toggle raw display
\( T_{23}^{8} + 6T_{23}^{7} - 82T_{23}^{6} - 458T_{23}^{5} + 2254T_{23}^{4} + 11034T_{23}^{3} - 24559T_{23}^{2} - 84768T_{23} + 89088 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 3 T^{7} + \cdots - 3 \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( (T + 1)^{8} \) Copy content Toggle raw display
$7$ \( T^{8} - 9 T^{7} + \cdots - 256 \) Copy content Toggle raw display
$11$ \( (T + 1)^{8} \) Copy content Toggle raw display
$13$ \( T^{8} - 10 T^{7} + \cdots - 4156 \) Copy content Toggle raw display
$17$ \( (T + 1)^{8} \) Copy content Toggle raw display
$19$ \( T^{8} - 7 T^{7} + \cdots - 70768 \) Copy content Toggle raw display
$23$ \( T^{8} + 6 T^{7} + \cdots + 89088 \) Copy content Toggle raw display
$29$ \( T^{8} + 6 T^{7} + \cdots + 2224 \) Copy content Toggle raw display
$31$ \( T^{8} - 9 T^{7} + \cdots - 256 \) Copy content Toggle raw display
$37$ \( T^{8} - 3 T^{7} + \cdots - 15924 \) Copy content Toggle raw display
$41$ \( T^{8} - 16 T^{7} + \cdots - 787936 \) Copy content Toggle raw display
$43$ \( T^{8} - 30 T^{7} + \cdots + 1408896 \) Copy content Toggle raw display
$47$ \( T^{8} + 9 T^{7} + \cdots - 1882112 \) Copy content Toggle raw display
$53$ \( T^{8} - 2 T^{7} + \cdots + 6286368 \) Copy content Toggle raw display
$59$ \( T^{8} - 6 T^{7} + \cdots + 5567168 \) Copy content Toggle raw display
$61$ \( T^{8} - T^{7} + \cdots + 2683428 \) Copy content Toggle raw display
$67$ \( T^{8} - 34 T^{7} + \cdots - 4837072 \) Copy content Toggle raw display
$71$ \( T^{8} + 19 T^{7} + \cdots - 81152 \) Copy content Toggle raw display
$73$ \( T^{8} - 20 T^{7} + \cdots - 20016 \) Copy content Toggle raw display
$79$ \( T^{8} + 6 T^{7} + \cdots + 121718784 \) Copy content Toggle raw display
$83$ \( T^{8} + 23 T^{7} + \cdots + 275984 \) Copy content Toggle raw display
$89$ \( T^{8} - 16 T^{7} + \cdots - 31951344 \) Copy content Toggle raw display
$97$ \( T^{8} - 9 T^{7} + \cdots + 1160708 \) Copy content Toggle raw display
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