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Properties

Label 3825.2.a.bt
Level $3825$
Weight $2$
Character orbit 3825.a
Self dual yes
Analytic conductor $30.543$
Analytic rank $0$
Dimension $8$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [3825,2,Mod(1,3825)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3825, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3825.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 3825 = 3^{2} \cdot 5^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3825.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,4,0,8,0,0,0,18,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(30.5427787731\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.8.17830397164.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 3x^{7} - 8x^{6} + 24x^{5} + 16x^{4} - 51x^{3} - 11x^{2} + 30x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{29}]\)
Coefficient ring index: \( 2^{5} \)
Twist minimal: no (minimal twist has level 765)
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.

\(f(q)\) \(=\) \( q + ( - \beta_{6} + 1) q^{2} + ( - \beta_{6} - \beta_{2} + 1) q^{4} - \beta_{4} q^{7} + ( - \beta_{6} + \beta_{5} - \beta_{2} + 2) q^{8} + \beta_{3} q^{11} + (\beta_{4} + \beta_{3}) q^{13} + ( - 2 \beta_{4} - \beta_{3} - \beta_1) q^{14}+ \cdots + ( - 7 \beta_{6} + \beta_{5} + \cdots + 14) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{2} + 8 q^{4} + 18 q^{8} + 16 q^{16} - 8 q^{17} + 2 q^{19} + 38 q^{23} - 12 q^{31} + 40 q^{32} - 4 q^{34} + 22 q^{38} + 40 q^{46} + 36 q^{47} + 28 q^{49} + 28 q^{53} - 12 q^{61} - 4 q^{62} + 54 q^{64}+ \cdots + 102 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 3x^{7} - 8x^{6} + 24x^{5} + 16x^{4} - 51x^{3} - 11x^{2} + 30x + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -6\nu^{7} + 21\nu^{6} + 59\nu^{5} - 195\nu^{4} - 192\nu^{3} + 402\nu^{2} + 252\nu - 48 ) / 43 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 7\nu^{7} - 3\nu^{6} - 76\nu^{5} - 9\nu^{4} + 181\nu^{3} + 90\nu^{2} - 36\nu - 116 ) / 43 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -12\nu^{7} + 42\nu^{6} + 75\nu^{5} - 304\nu^{4} - 40\nu^{3} + 460\nu^{2} - 184\nu - 96 ) / 43 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 15\nu^{7} - 31\nu^{6} - 126\nu^{5} + 165\nu^{4} + 265\nu^{3} - 16\nu^{2} - 200\nu - 224 ) / 43 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -16\nu^{7} + 13\nu^{6} + 186\nu^{5} - 47\nu^{4} - 598\nu^{3} - 46\nu^{2} + 500\nu + 173 ) / 43 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -19\nu^{7} + 45\nu^{6} + 151\nu^{5} - 295\nu^{4} - 264\nu^{3} + 413\nu^{2} + 110\nu - 66 ) / 43 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -29\nu^{7} + 37\nu^{6} + 321\nu^{5} - 233\nu^{4} - 971\nu^{3} + 352\nu^{2} + 788\nu - 60 ) / 43 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{7} - 2\beta_{5} - \beta_{4} - \beta_{3} + 2 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} - \beta_{5} + \beta_{2} - \beta _1 + 7 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 4\beta_{7} - 2\beta_{6} - 7\beta_{5} - 3\beta_{4} - \beta_{3} - \beta_{2} - \beta _1 + 9 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 10\beta_{7} - 4\beta_{6} - 11\beta_{5} - 4\beta_{4} + \beta_{3} + 8\beta_{2} - 9\beta _1 + 45 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 18\beta_{7} - 12\beta_{6} - 27\beta_{5} - 12\beta_{4} - 7\beta _1 + 45 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 98\beta_{7} - 56\beta_{6} - 115\beta_{5} - 52\beta_{4} + 15\beta_{3} + 64\beta_{2} - 75\beta _1 + 365 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 332\beta_{7} - 238\beta_{6} - 461\beta_{5} - 213\beta_{4} + 31\beta_{3} + 63\beta_{2} - 157\beta _1 + 907 ) / 2 \) 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
1.87858
−0.927574
−2.27502
1.10068
1.63205
3.10385
−1.38157
−0.130988
−1.69353 0 0.868028 0 0 −1.94614 1.91702 0 0
1.2 −1.69353 0 0.868028 0 0 1.94614 1.91702 0 0
1.3 −0.329727 0 −1.89128 0 0 −2.26264 1.28306 0 0
1.4 −0.329727 0 −1.89128 0 0 2.26264 1.28306 0 0
1.5 1.32973 0 −0.231826 0 0 −3.42890 −2.96772 0 0
1.6 1.32973 0 −0.231826 0 0 3.42890 −2.96772 0 0
1.7 2.69353 0 5.25508 0 0 −4.61905 8.76763 0 0
1.8 2.69353 0 5.25508 0 0 4.61905 8.76763 0 0
\(n\): 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

Char Parity Ord Mult Type
1.a even 1 1 trivial
15.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3825.2.a.bt 8
3.b odd 2 1 3825.2.a.bs 8
5.b even 2 1 3825.2.a.bs 8
5.c odd 4 2 765.2.b.e ✓ 16
15.d odd 2 1 inner 3825.2.a.bt 8
15.e even 4 2 765.2.b.e ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
765.2.b.e ✓ 16 5.c odd 4 2
765.2.b.e ✓ 16 15.e even 4 2
3825.2.a.bs 8 3.b odd 2 1
3825.2.a.bs 8 5.b even 2 1
3825.2.a.bt 8 1.a even 1 1 trivial
3825.2.a.bt 8 15.d odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3825))\):

\( T_{2}^{4} - 2T_{2}^{3} - 4T_{2}^{2} + 5T_{2} + 2 \) Copy content Toggle raw display
\( T_{7}^{8} - 42T_{7}^{6} + 565T_{7}^{4} - 2876T_{7}^{2} + 4864 \) Copy content Toggle raw display
\( T_{11}^{8} - 65T_{11}^{6} + 1159T_{11}^{4} - 4455T_{11}^{2} + 4864 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - 2 T^{3} - 4 T^{2} + \cdots + 2)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} - 42 T^{6} + \cdots + 4864 \) Copy content Toggle raw display
$11$ \( T^{8} - 65 T^{6} + \cdots + 4864 \) Copy content Toggle raw display
$13$ \( T^{8} - 71 T^{6} + \cdots + 19456 \) Copy content Toggle raw display
$17$ \( (T + 1)^{8} \) Copy content Toggle raw display
$19$ \( (T^{4} - T^{3} - 17 T^{2} + \cdots + 4)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} - 19 T^{3} + \cdots - 16)^{2} \) Copy content Toggle raw display
$29$ \( T^{8} - 206 T^{6} + \cdots + 311296 \) Copy content Toggle raw display
$31$ \( (T^{4} + 6 T^{3} + \cdots + 128)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} - 82 T^{6} + \cdots + 1216 \) Copy content Toggle raw display
$41$ \( T^{8} - 81 T^{6} + \cdots + 304 \) Copy content Toggle raw display
$43$ \( T^{8} - 267 T^{6} + \cdots + 1245184 \) Copy content Toggle raw display
$47$ \( (T^{2} - 9 T + 16)^{4} \) Copy content Toggle raw display
$53$ \( (T^{4} - 14 T^{3} + 43 T^{2} + \cdots - 8)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} - 232 T^{6} + \cdots + 311296 \) Copy content Toggle raw display
$61$ \( (T^{4} + 6 T^{3} + \cdots - 688)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} - 392 T^{6} + \cdots + 4980736 \) Copy content Toggle raw display
$71$ \( T^{8} - 172 T^{6} + \cdots + 1245184 \) Copy content Toggle raw display
$73$ \( T^{8} - 374 T^{6} + \cdots + 71213824 \) Copy content Toggle raw display
$79$ \( (T^{4} + 18 T^{3} + \cdots - 6016)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} - 16 T^{3} + \cdots - 8192)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} - 376 T^{6} + \cdots + 77824 \) Copy content Toggle raw display
$97$ \( T^{8} - 376 T^{6} + \cdots + 77824 \) Copy content Toggle raw display
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