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Properties

Label 8512.2.a.cj
Level $8512$
Weight $2$
Character orbit 8512.a
Self dual yes
Analytic conductor $67.969$
Analytic rank $1$
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] = [8512,2,Mod(1,8512)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
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")
 
//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);
 
Level: \( N \) \(=\) \( 8512 = 2^{6} \cdot 7 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8512.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: \(67.9686622005\)
Analytic rank: \(1\)
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: \( x^{7} - 13x^{5} - 9x^{4} + 37x^{3} + 41x^{2} - 5 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 4256)
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.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 1) q^{3} + ( - \beta_{3} - 1) q^{5} - q^{7} + (\beta_{2} - \beta_1 + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 1) q^{3} + ( - \beta_{3} - 1) q^{5} - q^{7} + (\beta_{2} - \beta_1 + 2) q^{9} + (\beta_{4} - \beta_{2} - 1) q^{11} + (\beta_{6} + \beta_{4} - 2) q^{13} + ( - \beta_{6} + \beta_{5} - 2 \beta_{4} + \cdots - 1) q^{15}+ \cdots + ( - \beta_{6} + 2 \beta_{4} + \beta_{3} + \cdots - 9) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q + 7 q^{3} - 5 q^{5} - 7 q^{7} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 7 q + 7 q^{3} - 5 q^{5} - 7 q^{7} + 12 q^{9} - 3 q^{11} - 12 q^{13} - 6 q^{15} - 16 q^{17} + 7 q^{19} - 7 q^{21} - 10 q^{23} + 28 q^{25} + 16 q^{27} - 7 q^{29} - 2 q^{31} + 5 q^{33} + 5 q^{35} - 7 q^{37} + 3 q^{41} - 15 q^{43} - 32 q^{45} + 5 q^{47} + 7 q^{49} - 13 q^{53} + 14 q^{55} + 7 q^{57} + 23 q^{59} - 33 q^{61} - 12 q^{63} - 4 q^{65} - 10 q^{67} - 10 q^{69} - 13 q^{71} + 18 q^{73} + 28 q^{75} + 3 q^{77} - 21 q^{79} + 11 q^{81} + 2 q^{83} - 14 q^{85} - 64 q^{87} - 17 q^{89} + 12 q^{91} - 26 q^{93} - 5 q^{95} + q^{97} - 55 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{7} - 13x^{5} - 9x^{4} + 37x^{3} + 41x^{2} - 5 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{6} - \nu^{5} - 11\nu^{4} + \nu^{3} + 27\nu^{2} + 15\nu - 2 ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{6} - 2\nu^{5} - 10\nu^{4} + 12\nu^{3} + 24\nu^{2} - 10\nu - 5 ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{6} + 14\nu^{4} + 8\nu^{3} - 46\nu^{2} - 38\nu + 11 ) / 2 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( -\nu^{6} + \nu^{5} + 12\nu^{4} - 3\nu^{3} - 34\nu^{2} - 8\nu + 8 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{6} + \beta_{5} - \beta_{4} + \beta_{2} + 7\beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{6} + 2\beta_{5} - 2\beta_{4} + 2\beta_{3} + 9\beta_{2} + 14\beta _1 + 30 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -12\beta_{6} + 13\beta_{5} - 15\beta_{4} + 4\beta_{3} + 17\beta_{2} + 63\beta _1 + 59 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -22\beta_{6} + 34\beta_{5} - 36\beta_{4} + 28\beta_{3} + 88\beta_{2} + 168\beta _1 + 279 \) 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
3.36915
2.07750
0.312247
−0.464836
−1.22130
−1.69433
−2.37843
0 −2.36915 0 −3.19936 0 −1.00000 0 2.61287 0
1.2 0 −1.07750 0 3.27350 0 −1.00000 0 −1.83900 0
1.3 0 0.687753 0 −3.62000 0 −1.00000 0 −2.52700 0
1.4 0 1.46484 0 0.860397 0 −1.00000 0 −0.854257 0
1.5 0 2.22130 0 −0.847099 0 −1.00000 0 1.93417 0
1.6 0 2.69433 0 2.90010 0 −1.00000 0 4.25941 0
1.7 0 3.37843 0 −4.36754 0 −1.00000 0 8.41379 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.7
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.cj 7
4.b odd 2 1 8512.2.a.cg 7
8.b even 2 1 4256.2.a.o ✓ 7
8.d odd 2 1 4256.2.a.r yes 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4256.2.a.o ✓ 7 8.b even 2 1
4256.2.a.r yes 7 8.d odd 2 1
8512.2.a.cg 7 4.b odd 2 1
8512.2.a.cj 7 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8512))\):

\( T_{3}^{7} - 7T_{3}^{6} + 8T_{3}^{5} + 39T_{3}^{4} - 94T_{3}^{3} + 11T_{3}^{2} + 99T_{3} - 52 \) Copy content Toggle raw display
\( T_{5}^{7} + 5T_{5}^{6} - 19T_{5}^{5} - 102T_{5}^{4} + 95T_{5}^{3} + 551T_{5}^{2} - 65T_{5} - 350 \) Copy content Toggle raw display
\( T_{11}^{7} + 3T_{11}^{6} - 48T_{11}^{5} - 71T_{11}^{4} + 716T_{11}^{3} - 17T_{11}^{2} - 1939T_{11} + 824 \) Copy content Toggle raw display
\( T_{23}^{7} + 10T_{23}^{6} - 50T_{23}^{5} - 528T_{23}^{4} + 873T_{23}^{3} + 6254T_{23}^{2} - 9380T_{23} + 3328 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{7} \) Copy content Toggle raw display
$3$ \( T^{7} - 7 T^{6} + \cdots - 52 \) Copy content Toggle raw display
$5$ \( T^{7} + 5 T^{6} + \cdots - 350 \) Copy content Toggle raw display
$7$ \( (T + 1)^{7} \) Copy content Toggle raw display
$11$ \( T^{7} + 3 T^{6} + \cdots + 824 \) Copy content Toggle raw display
$13$ \( T^{7} + 12 T^{6} + \cdots + 7624 \) Copy content Toggle raw display
$17$ \( T^{7} + 16 T^{6} + \cdots + 8352 \) Copy content Toggle raw display
$19$ \( (T - 1)^{7} \) Copy content Toggle raw display
$23$ \( T^{7} + 10 T^{6} + \cdots + 3328 \) Copy content Toggle raw display
$29$ \( T^{7} + 7 T^{6} + \cdots - 55094 \) Copy content Toggle raw display
$31$ \( T^{7} + 2 T^{6} + \cdots - 689632 \) Copy content Toggle raw display
$37$ \( T^{7} + 7 T^{6} + \cdots - 131030 \) Copy content Toggle raw display
$41$ \( T^{7} - 3 T^{6} + \cdots - 338 \) Copy content Toggle raw display
$43$ \( T^{7} + 15 T^{6} + \cdots - 416384 \) Copy content Toggle raw display
$47$ \( T^{7} - 5 T^{6} + \cdots + 781904 \) Copy content Toggle raw display
$53$ \( T^{7} + 13 T^{6} + \cdots + 48058 \) Copy content Toggle raw display
$59$ \( T^{7} - 23 T^{6} + \cdots + 4348 \) Copy content Toggle raw display
$61$ \( T^{7} + 33 T^{6} + \cdots - 304790 \) Copy content Toggle raw display
$67$ \( T^{7} + 10 T^{6} + \cdots + 142240 \) Copy content Toggle raw display
$71$ \( T^{7} + 13 T^{6} + \cdots + 274600 \) Copy content Toggle raw display
$73$ \( T^{7} - 18 T^{6} + \cdots + 519608 \) Copy content Toggle raw display
$79$ \( T^{7} + 21 T^{6} + \cdots + 6272 \) Copy content Toggle raw display
$83$ \( T^{7} - 2 T^{6} + \cdots - 3328 \) Copy content Toggle raw display
$89$ \( T^{7} + 17 T^{6} + \cdots + 67712 \) Copy content Toggle raw display
$97$ \( T^{7} - T^{6} + \cdots + 2854 \) Copy content Toggle raw display
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