↑

Properties

Label 8512.2.a.cl
Level $8512$
Weight $2$
Character orbit 8512.a
Self dual yes
Analytic conductor $67.969$
Analytic rank $0$
Dimension $10$
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] = [8512,2,Mod(1,8512)] 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("8512.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8512, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 8512 = 2^{6} \cdot 7 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8512.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [10,0,0,0,0,0,10] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(67.9686622005\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 24x^{8} - 4x^{7} + 191x^{6} + 56x^{5} - 554x^{4} - 156x^{3} + 403x^{2} - 64x - 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 4256)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} + \beta_{5} q^{5} + q^{7} + (\beta_{2} + 2) q^{9} - \beta_{7} q^{11} + \beta_{3} q^{13} + (\beta_{8} - \beta_{4}) q^{15} + (\beta_{8} + \beta_{7} + \beta_{2} + \cdots + 2) q^{17} + q^{19}+ \cdots + ( - 2 \beta_{8} - \beta_{6} - 2 \beta_{4} + \cdots + 5) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 10 q^{7} + 18 q^{9} + 2 q^{11} - 4 q^{13} - 2 q^{15} + 18 q^{17} + 10 q^{19} + 4 q^{23} + 28 q^{25} - 12 q^{27} + 4 q^{29} + 2 q^{31} + 16 q^{33} + 2 q^{37} - 2 q^{39} + 18 q^{41} - 8 q^{43} - 2 q^{45}+ \cdots + 32 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 24x^{8} - 4x^{7} + 191x^{6} + 56x^{5} - 554x^{4} - 156x^{3} + 403x^{2} - 64x - 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 5 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 13 \nu^{9} + 5 \nu^{8} - 273 \nu^{7} - 157 \nu^{6} + 1644 \nu^{5} + 1453 \nu^{4} - 1749 \nu^{3} + \cdots + 1312 ) / 482 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 49 \nu^{9} + 93 \nu^{8} - 1270 \nu^{7} - 2149 \nu^{6} + 10720 \nu^{5} + 15747 \nu^{4} - 30507 \nu^{3} + \cdots + 5316 ) / 964 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 57 \nu^{9} - 59 \nu^{8} + 1438 \nu^{7} + 1467 \nu^{6} - 11880 \nu^{5} - 11265 \nu^{4} + 34327 \nu^{3} + \cdots + 328 ) / 964 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 73 \nu^{9} + 9 \nu^{8} + 1774 \nu^{7} + 103 \nu^{6} - 14200 \nu^{5} - 3265 \nu^{4} + 40039 \nu^{3} + \cdots + 48 ) / 964 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 100 \nu^{9} - 57 \nu^{8} + 2341 \nu^{7} + 1838 \nu^{6} - 17633 \nu^{5} - 17480 \nu^{4} + \cdots - 8016 ) / 964 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 108 \nu^{9} + 23 \nu^{8} - 2509 \nu^{7} - 1156 \nu^{6} + 18793 \nu^{5} + 12998 \nu^{4} - 46991 \nu^{3} + \cdots + 6228 ) / 964 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 215 \nu^{9} - 10 \nu^{8} - 4997 \nu^{7} - 891 \nu^{6} + 37923 \nu^{5} + 14687 \nu^{4} - 100614 \nu^{3} + \cdots + 2196 ) / 964 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{8} + \beta_{7} + \beta_{5} + \beta_{4} + \beta_{2} + 7\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -2\beta_{8} - 2\beta_{7} - \beta_{6} + \beta_{5} + 9\beta_{2} + 3\beta _1 + 41 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2 \beta_{9} + 11 \beta_{8} + 13 \beta_{7} + 2 \beta_{6} + 15 \beta_{5} + 15 \beta_{4} - 2 \beta_{3} + \cdots + 15 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 2 \beta_{9} - 34 \beta_{8} - 28 \beta_{7} - 15 \beta_{6} + 15 \beta_{5} + 2 \beta_{4} + 4 \beta_{3} + \cdots + 376 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 40 \beta_{9} + 102 \beta_{8} + 142 \beta_{7} + 42 \beta_{6} + 184 \beta_{5} + 182 \beta_{4} - 30 \beta_{3} + \cdots + 174 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 50 \beta_{9} - 454 \beta_{8} - 320 \beta_{7} - 162 \beta_{6} + 182 \beta_{5} + 54 \beta_{4} + \cdots + 3633 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 592 \beta_{9} + 873 \beta_{8} + 1481 \beta_{7} + 622 \beta_{6} + 2101 \beta_{5} + 2063 \beta_{4} + \cdots + 1911 \) 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
3.29848
3.00424
2.12344
0.578492
0.399218
−0.133956
−1.43531
−2.16938
−2.47210
−3.19312
0 −3.29848 0 3.48958 0 1.00000 0 7.87996 0
1.2 0 −3.00424 0 −1.53199 0 1.00000 0 6.02544 0
1.3 0 −2.12344 0 −3.57553 0 1.00000 0 1.50900 0
1.4 0 −0.578492 0 1.32386 0 1.00000 0 −2.66535 0
1.5 0 −0.399218 0 −2.37664 0 1.00000 0 −2.84063 0
1.6 0 0.133956 0 3.66471 0 1.00000 0 −2.98206 0
1.7 0 1.43531 0 1.27648 0 1.00000 0 −0.939898 0
1.8 0 2.16938 0 −4.17511 0 1.00000 0 1.70622 0
1.9 0 2.47210 0 3.07194 0 1.00000 0 3.11129 0
1.10 0 3.19312 0 −1.16730 0 1.00000 0 7.19601 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(7\) \( -1 \)
\(19\) \( -1 \)

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 8512.2.a.cl 10
4.b odd 2 1 8512.2.a.ck 10
8.b even 2 1 4256.2.a.t yes 10
8.d odd 2 1 4256.2.a.s ✓ 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4256.2.a.s ✓ 10 8.d odd 2 1
4256.2.a.t yes 10 8.b even 2 1
8512.2.a.ck 10 4.b odd 2 1
8512.2.a.cl 10 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8512))\):

\( T_{3}^{10} - 24T_{3}^{8} + 4T_{3}^{7} + 191T_{3}^{6} - 56T_{3}^{5} - 554T_{3}^{4} + 156T_{3}^{3} + 403T_{3}^{2} + 64T_{3} - 16 \) Copy content Toggle raw display
\( T_{5}^{10} - 39T_{5}^{8} + 4T_{5}^{7} + 528T_{5}^{6} - 60T_{5}^{5} - 2921T_{5}^{4} + 120T_{5}^{3} + 6147T_{5}^{2} - 108T_{5} - 4212 \) Copy content Toggle raw display
\( T_{11}^{10} - 2 T_{11}^{9} - 70 T_{11}^{8} + 120 T_{11}^{7} + 1559 T_{11}^{6} - 1864 T_{11}^{5} + \cdots + 832 \) Copy content Toggle raw display
\( T_{23}^{10} - 4 T_{23}^{9} - 150 T_{23}^{8} + 524 T_{23}^{7} + 7501 T_{23}^{6} - 19168 T_{23}^{5} + \cdots - 2027008 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \) Copy content Toggle raw display
$3$ \( T^{10} - 24 T^{8} + \cdots - 16 \) Copy content Toggle raw display
$5$ \( T^{10} - 39 T^{8} + \cdots - 4212 \) Copy content Toggle raw display
$7$ \( (T - 1)^{10} \) Copy content Toggle raw display
$11$ \( T^{10} - 2 T^{9} + \cdots + 832 \) Copy content Toggle raw display
$13$ \( T^{10} + 4 T^{9} + \cdots - 1039168 \) Copy content Toggle raw display
$17$ \( T^{10} - 18 T^{9} + \cdots - 4096 \) Copy content Toggle raw display
$19$ \( (T - 1)^{10} \) Copy content Toggle raw display
$23$ \( T^{10} - 4 T^{9} + \cdots - 2027008 \) Copy content Toggle raw display
$29$ \( T^{10} - 4 T^{9} + \cdots + 317876 \) Copy content Toggle raw display
$31$ \( T^{10} - 2 T^{9} + \cdots - 322048 \) Copy content Toggle raw display
$37$ \( T^{10} - 2 T^{9} + \cdots - 2190572 \) Copy content Toggle raw display
$41$ \( T^{10} - 18 T^{9} + \cdots - 1192388 \) Copy content Toggle raw display
$43$ \( T^{10} + 8 T^{9} + \cdots - 2854400 \) Copy content Toggle raw display
$47$ \( T^{10} - 22 T^{9} + \cdots - 1589600 \) Copy content Toggle raw display
$53$ \( T^{10} - 304 T^{8} + \cdots - 11113308 \) Copy content Toggle raw display
$59$ \( T^{10} + \cdots + 105108208 \) Copy content Toggle raw display
$61$ \( T^{10} + \cdots - 327937300 \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots - 131042944 \) Copy content Toggle raw display
$71$ \( T^{10} + 20 T^{9} + \cdots - 4917224 \) Copy content Toggle raw display
$73$ \( T^{10} - 30 T^{9} + \cdots - 62673728 \) Copy content Toggle raw display
$79$ \( T^{10} + 12 T^{9} + \cdots + 569728 \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots - 370454528 \) Copy content Toggle raw display
$89$ \( T^{10} - 24 T^{9} + \cdots - 88147712 \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots - 114782500 \) Copy content Toggle raw display
show more
show less