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Properties

Label 8512.2.a.ci
Level $8512$
Weight $2$
Character orbit 8512.a
Self dual yes
Analytic conductor $67.969$
Analytic rank $0$
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: \(0\)
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: \( x^{7} - 3x^{6} - 10x^{5} + 31x^{4} + 12x^{3} - 45x^{2} - 15x + 8 \) 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.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + ( - \beta_{3} + 1) q^{5} + q^{7} + (\beta_{6} + \beta_{5} - \beta_{2} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + ( - \beta_{3} + 1) q^{5} + q^{7} + (\beta_{6} + \beta_{5} - \beta_{2} + 1) q^{9} + (\beta_{6} + \beta_{5} + \beta_{4}) q^{11} + ( - \beta_{4} - \beta_{3} + 3) q^{13} + ( - \beta_{5} + \beta_{4} - 2 \beta_{3} + \cdots - 2) q^{15}+ \cdots + ( - \beta_{6} + \beta_{5} - 2 \beta_{2} + \cdots + 8) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q + 3 q^{3} + 5 q^{5} + 7 q^{7} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 7 q + 3 q^{3} + 5 q^{5} + 7 q^{7} + 8 q^{9} + 3 q^{11} + 16 q^{13} - 8 q^{15} + 8 q^{17} - 7 q^{19} + 3 q^{21} - 10 q^{23} + 8 q^{25} + 6 q^{27} + 11 q^{29} - 14 q^{31} + 3 q^{33} + 5 q^{35} + 13 q^{37} + 6 q^{39} + 11 q^{41} + 11 q^{43} - 4 q^{45} - 7 q^{47} + 7 q^{49} + 24 q^{51} + 9 q^{53} + 8 q^{55} - 3 q^{57} + 23 q^{59} + 23 q^{61} + 8 q^{63} + 40 q^{65} + 16 q^{67} + 10 q^{69} - 3 q^{71} + 4 q^{73} + 48 q^{75} + 3 q^{77} - 29 q^{79} + 23 q^{81} + 6 q^{83} + 6 q^{85} + 6 q^{87} + 15 q^{89} + 16 q^{91} - 42 q^{93} - 5 q^{95} - 13 q^{97} + 47 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{7} - 3x^{6} - 10x^{5} + 31x^{4} + 12x^{3} - 45x^{2} - 15x + 8 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{6} - 3\nu^{5} - 9\nu^{4} + 29\nu^{3} + \nu^{2} - 27\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 3\nu^{6} - 11\nu^{5} - 21\nu^{4} + 105\nu^{3} - 51\nu^{2} - 83\nu + 30 ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 4\nu^{6} - 15\nu^{5} - 28\nu^{4} + 143\nu^{3} - 68\nu^{2} - 113\nu + 40 ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -2\nu^{6} + 8\nu^{5} + 13\nu^{4} - 77\nu^{3} + 43\nu^{2} + 65\nu - 24 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 5\nu^{6} - 19\nu^{5} - 35\nu^{4} + 183\nu^{3} - 83\nu^{2} - 157\nu + 40 ) / 2 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} + \beta_{5} - \beta_{2} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{5} - 2\beta_{4} + \beta_{3} + \beta_{2} + 7\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 9\beta_{6} + 9\beta_{5} - 2\beta_{4} + 3\beta_{3} - 10\beta_{2} - 2\beta _1 + 31 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -9\beta_{5} - 24\beta_{4} + 17\beta_{3} + 9\beta_{2} + 56\beta _1 + 9 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 80\beta_{6} + 82\beta_{5} - 32\beta_{4} + 49\beta_{3} - 89\beta_{2} - 26\beta _1 + 273 \) 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
−2.95576
−0.838827
−0.811529
0.299967
1.77715
2.33548
3.19352
0 −2.95576 0 2.19253 0 1.00000 0 5.73652 0
1.2 0 −0.838827 0 2.50991 0 1.00000 0 −2.29637 0
1.3 0 −0.811529 0 −0.636007 0 1.00000 0 −2.34142 0
1.4 0 0.299967 0 −0.576611 0 1.00000 0 −2.91002 0
1.5 0 1.77715 0 0.595310 0 1.00000 0 0.158273 0
1.6 0 2.33548 0 4.35508 0 1.00000 0 2.45447 0
1.7 0 3.19352 0 −3.44021 0 1.00000 0 7.19855 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.ci 7
4.b odd 2 1 8512.2.a.ch 7
8.b even 2 1 4256.2.a.p ✓ 7
8.d odd 2 1 4256.2.a.q yes 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4256.2.a.p ✓ 7 8.b even 2 1
4256.2.a.q yes 7 8.d odd 2 1
8512.2.a.ch 7 4.b odd 2 1
8512.2.a.ci 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8512))\):

\( T_{3}^{7} - 3T_{3}^{6} - 10T_{3}^{5} + 31T_{3}^{4} + 12T_{3}^{3} - 45T_{3}^{2} - 15T_{3} + 8 \) Copy content Toggle raw display
\( T_{5}^{7} - 5T_{5}^{6} - 9T_{5}^{5} + 64T_{5}^{4} - 39T_{5}^{3} - 73T_{5}^{2} + 15T_{5} + 18 \) Copy content Toggle raw display
\( T_{11}^{7} - 3T_{11}^{6} - 46T_{11}^{5} + 123T_{11}^{4} + 362T_{11}^{3} - 1007T_{11}^{2} + 639T_{11} - 96 \) Copy content Toggle raw display
\( T_{23}^{7} + 10T_{23}^{6} - 58T_{23}^{5} - 520T_{23}^{4} + 1505T_{23}^{3} + 5646T_{23}^{2} - 11172T_{23} - 9648 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{7} \) Copy content Toggle raw display
$3$ \( T^{7} - 3 T^{6} + \cdots + 8 \) Copy content Toggle raw display
$5$ \( T^{7} - 5 T^{6} + \cdots + 18 \) Copy content Toggle raw display
$7$ \( (T - 1)^{7} \) Copy content Toggle raw display
$11$ \( T^{7} - 3 T^{6} + \cdots - 96 \) Copy content Toggle raw display
$13$ \( T^{7} - 16 T^{6} + \cdots + 192 \) Copy content Toggle raw display
$17$ \( T^{7} - 8 T^{6} + \cdots - 96 \) Copy content Toggle raw display
$19$ \( (T + 1)^{7} \) Copy content Toggle raw display
$23$ \( T^{7} + 10 T^{6} + \cdots - 9648 \) Copy content Toggle raw display
$29$ \( T^{7} - 11 T^{6} + \cdots - 7758 \) Copy content Toggle raw display
$31$ \( T^{7} + 14 T^{6} + \cdots - 27024 \) Copy content Toggle raw display
$37$ \( T^{7} - 13 T^{6} + \cdots + 5562 \) Copy content Toggle raw display
$41$ \( T^{7} - 11 T^{6} + \cdots - 5988 \) Copy content Toggle raw display
$43$ \( T^{7} - 11 T^{6} + \cdots - 192 \) Copy content Toggle raw display
$47$ \( T^{7} + 7 T^{6} + \cdots - 508032 \) Copy content Toggle raw display
$53$ \( T^{7} - 9 T^{6} + \cdots + 1443206 \) Copy content Toggle raw display
$59$ \( T^{7} - 23 T^{6} + \cdots - 35964 \) Copy content Toggle raw display
$61$ \( T^{7} - 23 T^{6} + \cdots - 1799166 \) Copy content Toggle raw display
$67$ \( T^{7} - 16 T^{6} + \cdots + 1405416 \) Copy content Toggle raw display
$71$ \( T^{7} + 3 T^{6} + \cdots - 2902 \) Copy content Toggle raw display
$73$ \( T^{7} - 4 T^{6} + \cdots - 248 \) Copy content Toggle raw display
$79$ \( T^{7} + 29 T^{6} + \cdots - 15363936 \) Copy content Toggle raw display
$83$ \( T^{7} - 6 T^{6} + \cdots - 118848 \) Copy content Toggle raw display
$89$ \( T^{7} - 15 T^{6} + \cdots + 1293568 \) Copy content Toggle raw display
$97$ \( T^{7} + 13 T^{6} + \cdots + 561644 \) Copy content Toggle raw display
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