↑

Properties

Label 4335.2.a.bd
Level $4335$
Weight $2$
Character orbit 4335.a
Self dual yes
Analytic conductor $34.615$
Analytic rank $1$
Dimension $8$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [4335,2,Mod(1,4335)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4335.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4335, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 4335 = 3 \cdot 5 \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4335.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,-4,-8,4,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(34.6151492762\)
Analytic rank: \(1\)
Dimension: \(8\)
Coefficient field: 8.8.1413480448.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} - 2x^{6} + 16x^{5} - x^{4} - 16x^{3} + 2x^{2} + 4x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 255)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content 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.

\(f(q)\) \(=\) \( q + \beta_{5} q^{2} - q^{3} + ( - \beta_{7} + \beta_{6} + \beta_1) q^{4} + q^{5} - \beta_{5} q^{6} + (\beta_{7} + \beta_{4} - \beta_{3}) q^{7} + (\beta_{7} - \beta_{6} + \beta_{4} + \cdots - \beta_1) q^{8}+ \cdots + (\beta_{6} - \beta_{4} - \beta_1 + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{2} - 8 q^{3} + 4 q^{4} + 8 q^{5} + 4 q^{6} + 8 q^{9} - 4 q^{10} + 8 q^{11} - 4 q^{12} - 8 q^{13} - 12 q^{14} - 8 q^{15} - 4 q^{16} - 4 q^{18} - 16 q^{19} + 4 q^{20} + 4 q^{22} + 16 q^{23} + 8 q^{25}+ \cdots + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} - 2x^{6} + 16x^{5} - x^{4} - 16x^{3} + 2x^{2} + 4x - 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{7} - 4\nu^{6} - 2\nu^{5} + 16\nu^{4} - \nu^{3} - 16\nu^{2} + 2\nu + 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -2\nu^{7} + 8\nu^{6} + 3\nu^{5} - 28\nu^{4} + 3\nu^{3} + 20\nu^{2} - 2\nu - 3 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -2\nu^{7} + 9\nu^{6} - \nu^{5} - 29\nu^{4} + 16\nu^{3} + 15\nu^{2} - 8\nu + 1 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -3\nu^{7} + 12\nu^{6} + 5\nu^{5} - 44\nu^{4} + 4\nu^{3} + 35\nu^{2} - 2\nu - 5 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( 2\nu^{7} - 7\nu^{6} - 8\nu^{5} + 30\nu^{4} + 14\nu^{3} - 33\nu^{2} - 11\nu + 8 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( -2\nu^{7} + 6\nu^{6} + 12\nu^{5} - 29\nu^{4} - 26\nu^{3} + 35\nu^{2} + 16\nu - 9 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{5} + \beta_{3} - \beta_{2} + 2\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} + \beta_{6} - 3\beta_{5} + \beta_{4} + 2\beta_{3} - 3\beta_{2} + 7\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 3\beta_{7} + 2\beta_{6} - 13\beta_{5} + 4\beta_{4} + 9\beta_{3} - 11\beta_{2} + 20\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 13\beta_{7} + 9\beta_{6} - 43\beta_{5} + 17\beta_{4} + 25\beta_{3} - 37\beta_{2} + 65\beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 42\beta_{7} + 25\beta_{6} - 151\beta_{5} + 60\beta_{4} + 87\beta_{3} - 125\beta_{2} + 205\beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 147\beta_{7} + 87\beta_{6} - 501\beta_{5} + 211\beta_{4} + 272\beta_{3} - 416\beta_{2} + 667\beta _1 - 25 \) 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.

Copy content comment:embeddings in the coefficient field
 
Copy content 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
0.319701
3.28192
1.13761
−0.867710
0.426500
−1.55182
1.98771
−0.733914
−2.36256 −1.00000 3.58167 1.00000 2.36256 0.558590 −3.73679 1.00000 −2.36256
1.2 −2.15246 −1.00000 2.63308 1.00000 2.15246 1.98932 −1.36267 1.00000 −2.15246
1.3 −1.64440 −1.00000 0.704063 1.00000 1.64440 1.29993 2.13104 1.00000 −1.64440
1.4 −0.695301 −1.00000 −1.51656 1.00000 0.695301 0.190263 2.44506 1.00000 −0.695301
1.5 −0.496910 −1.00000 −1.75308 1.00000 0.496910 2.87074 1.86494 1.00000 −0.496910
1.6 −0.120963 −1.00000 −1.98537 1.00000 0.120963 −4.56190 0.482083 1.00000 −0.120963
1.7 1.34467 −1.00000 −0.191866 1.00000 −1.34467 −2.22190 −2.94733 1.00000 1.34467
1.8 2.12792 −1.00000 2.52806 1.00000 −2.12792 −0.125045 1.12367 1.00000 2.12792
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
Significant digits:
Format:

Atkin-Lehner signs

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

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 4335.2.a.bd 8
17.b even 2 1 4335.2.a.be 8
17.e odd 16 2 255.2.w.a ✓ 16
51.i even 16 2 765.2.be.a 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
255.2.w.a ✓ 16 17.e odd 16 2
765.2.be.a 16 51.i even 16 2
4335.2.a.bd 8 1.a even 1 1 trivial
4335.2.a.be 8 17.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4335))\):

\( T_{2}^{8} + 4T_{2}^{7} - 2T_{2}^{6} - 24T_{2}^{5} - 19T_{2}^{4} + 24T_{2}^{3} + 34T_{2}^{2} + 12T_{2} + 1 \) Copy content Toggle raw display
\( T_{7}^{8} - 20T_{7}^{6} + 24T_{7}^{5} + 64T_{7}^{4} - 120T_{7}^{3} + 48T_{7}^{2} - 1 \) Copy content Toggle raw display
\( T_{11}^{8} - 8T_{11}^{7} + 8T_{11}^{6} + 56T_{11}^{5} - 112T_{11}^{4} - 48T_{11}^{3} + 180T_{11}^{2} - 81 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 4 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( (T + 1)^{8} \) Copy content Toggle raw display
$5$ \( (T - 1)^{8} \) Copy content Toggle raw display
$7$ \( T^{8} - 20 T^{6} + \cdots - 1 \) Copy content Toggle raw display
$11$ \( T^{8} - 8 T^{7} + \cdots - 81 \) Copy content Toggle raw display
$13$ \( T^{8} + 8 T^{7} + \cdots - 4 \) Copy content Toggle raw display
$17$ \( T^{8} \) Copy content Toggle raw display
$19$ \( T^{8} + 16 T^{7} + \cdots + 1649 \) Copy content Toggle raw display
$23$ \( T^{8} - 16 T^{7} + \cdots - 1724 \) Copy content Toggle raw display
$29$ \( T^{8} + 8 T^{7} + \cdots - 30353 \) Copy content Toggle raw display
$31$ \( T^{8} + 16 T^{7} + \cdots - 124092 \) Copy content Toggle raw display
$37$ \( T^{8} - 96 T^{6} + \cdots - 6001 \) Copy content Toggle raw display
$41$ \( T^{8} + 8 T^{7} + \cdots + 27967 \) Copy content Toggle raw display
$43$ \( T^{8} - 120 T^{6} + \cdots + 3008 \) Copy content Toggle raw display
$47$ \( T^{8} + 32 T^{7} + \cdots + 4337 \) Copy content Toggle raw display
$53$ \( T^{8} + 24 T^{7} + \cdots + 818513 \) Copy content Toggle raw display
$59$ \( T^{8} + 24 T^{7} + \cdots + 875068 \) Copy content Toggle raw display
$61$ \( T^{8} + 32 T^{7} + \cdots - 497212 \) Copy content Toggle raw display
$67$ \( T^{8} + 8 T^{7} + \cdots - 29327812 \) Copy content Toggle raw display
$71$ \( T^{8} + 8 T^{7} + \cdots + 2944768 \) Copy content Toggle raw display
$73$ \( T^{8} - 228 T^{6} + \cdots + 27871 \) Copy content Toggle raw display
$79$ \( T^{8} - 456 T^{6} + \cdots + 52606912 \) Copy content Toggle raw display
$83$ \( T^{8} - 24 T^{7} + \cdots + 272 \) Copy content Toggle raw display
$89$ \( T^{8} + 56 T^{7} + \cdots + 323068 \) Copy content Toggle raw display
$97$ \( T^{8} - 16 T^{7} + \cdots - 10348048 \) Copy content Toggle raw display
show more
show less