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Properties

Label 8624.2.a.df
Level $8624$
Weight $2$
Character orbit 8624.a
Self dual yes
Analytic conductor $68.863$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [8624,2,Mod(1,8624)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8624, 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("8624.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8624 = 2^{4} \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8624.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: \(68.8629867032\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 26x^{8} + 245x^{6} - 1038x^{4} + 1884x^{2} - 968 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 539)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + \beta_{6} q^{5} + (\beta_{2} + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + \beta_{6} q^{5} + (\beta_{2} + 2) q^{9} - q^{11} + ( - \beta_{6} + \beta_{4} + \beta_{3}) q^{13} + ( - \beta_{9} - \beta_{8} + \beta_{7} - 1) q^{15} + (\beta_{6} - \beta_{4} + \beta_{3}) q^{17} + (\beta_{6} - \beta_{5} + \beta_{4}) q^{19} + (\beta_{9} + \beta_{8}) q^{23} + (\beta_{9} - 2 \beta_{8} + \beta_{7} + \cdots + 2) q^{25}+ \cdots + ( - \beta_{2} - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 22 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 22 q^{9} - 10 q^{11} - 8 q^{15} - 4 q^{23} + 18 q^{25} + 12 q^{29} + 40 q^{37} + 16 q^{39} + 8 q^{43} + 16 q^{53} - 8 q^{57} - 32 q^{65} + 4 q^{67} - 36 q^{71} - 8 q^{79} - 6 q^{81} + 88 q^{85} + 44 q^{93} + 64 q^{95} - 22 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 26x^{8} + 245x^{6} - 1038x^{4} + 1884x^{2} - 968 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 5 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 15\nu^{9} - 324\nu^{7} + 2267\nu^{5} - 5912\nu^{3} + 3972\nu ) / 88 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 25\nu^{9} - 540\nu^{7} + 3749\nu^{5} - 9472\nu^{3} + 5652\nu ) / 88 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 15\nu^{9} - 324\nu^{7} + 2245\nu^{5} - 5604\nu^{3} + 3092\nu ) / 22 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 71\nu^{9} - 1516\nu^{7} + 10355\nu^{5} - 25672\nu^{3} + 14876\nu ) / 88 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( \nu^{8} - 21\nu^{6} + 140\nu^{4} - 338\nu^{2} + 193 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -5\nu^{8} + 108\nu^{6} - 749\nu^{4} + 1880\nu^{2} - 1080 ) / 4 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -3\nu^{8} + 64\nu^{6} - 437\nu^{4} + 1086\nu^{2} - 638 ) / 2 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{5} - 3\beta_{4} + \beta_{3} + 7\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -3\beta_{9} + 2\beta_{8} - 2\beta_{7} + 13\beta_{2} + 34 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 13\beta_{5} - 42\beta_{4} + 18\beta_{3} + 58\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -49\beta_{9} + 34\beta_{8} - 31\beta_{7} + 149\beta_{2} + 277 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 5\beta_{6} + 146\beta_{5} - 502\beta_{4} + 229\beta_{3} + 541\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -609\beta_{9} + 434\beta_{8} - 370\beta_{7} + 1647\beta_{2} + 2554 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 108\beta_{6} + 1583\beta_{5} - 5678\beta_{4} + 2626\beta_{3} + 5414\beta_1 \) 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
−3.27614
−2.32267
−2.15293
−2.10267
−0.903205
0.903205
2.10267
2.15293
2.32267
3.27614
0 −3.27614 0 −0.246676 0 0 0 7.73309 0
1.2 0 −2.32267 0 3.58219 0 0 0 2.39479 0
1.3 0 −2.15293 0 −3.87589 0 0 0 1.63513 0
1.4 0 −2.10267 0 2.44342 0 0 0 1.42122 0
1.5 0 −0.903205 0 −0.337987 0 0 0 −2.18422 0
1.6 0 0.903205 0 0.337987 0 0 0 −2.18422 0
1.7 0 2.10267 0 −2.44342 0 0 0 1.42122 0
1.8 0 2.15293 0 3.87589 0 0 0 1.63513 0
1.9 0 2.32267 0 −3.58219 0 0 0 2.39479 0
1.10 0 3.27614 0 0.246676 0 0 0 7.73309 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
\(2\) \(-1\)
\(7\) \(1\)
\(11\) \(1\)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 8624.2.a.df 10
4.b odd 2 1 539.2.a.l ✓ 10
7.b odd 2 1 inner 8624.2.a.df 10
12.b even 2 1 4851.2.a.cg 10
28.d even 2 1 539.2.a.l ✓ 10
28.f even 6 2 539.2.e.o 20
28.g odd 6 2 539.2.e.o 20
44.c even 2 1 5929.2.a.bv 10
84.h odd 2 1 4851.2.a.cg 10
308.g odd 2 1 5929.2.a.bv 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
539.2.a.l ✓ 10 4.b odd 2 1
539.2.a.l ✓ 10 28.d even 2 1
539.2.e.o 20 28.f even 6 2
539.2.e.o 20 28.g odd 6 2
4851.2.a.cg 10 12.b even 2 1
4851.2.a.cg 10 84.h odd 2 1
5929.2.a.bv 10 44.c even 2 1
5929.2.a.bv 10 308.g odd 2 1
8624.2.a.df 10 1.a even 1 1 trivial
8624.2.a.df 10 7.b 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8624))\):

\( T_{3}^{10} - 26T_{3}^{8} + 245T_{3}^{6} - 1038T_{3}^{4} + 1884T_{3}^{2} - 968 \) Copy content Toggle raw display
\( T_{5}^{10} - 34T_{5}^{8} + 365T_{5}^{6} - 1214T_{5}^{4} + 204T_{5}^{2} - 8 \) Copy content Toggle raw display
\( T_{13}^{10} - 72T_{13}^{8} + 1696T_{13}^{6} - 16672T_{13}^{4} + 69120T_{13}^{2} - 100352 \) Copy content Toggle raw display
\( T_{17}^{10} - 136T_{17}^{8} + 6192T_{17}^{6} - 110752T_{17}^{4} + 705536T_{17}^{2} - 430592 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \) Copy content Toggle raw display
$3$ \( T^{10} - 26 T^{8} + \cdots - 968 \) Copy content Toggle raw display
$5$ \( T^{10} - 34 T^{8} + \cdots - 8 \) Copy content Toggle raw display
$7$ \( T^{10} \) Copy content Toggle raw display
$11$ \( (T + 1)^{10} \) Copy content Toggle raw display
$13$ \( T^{10} - 72 T^{8} + \cdots - 100352 \) Copy content Toggle raw display
$17$ \( T^{10} - 136 T^{8} + \cdots - 430592 \) Copy content Toggle raw display
$19$ \( T^{10} - 144 T^{8} + \cdots - 6422528 \) Copy content Toggle raw display
$23$ \( (T^{5} + 2 T^{4} + \cdots + 232)^{2} \) Copy content Toggle raw display
$29$ \( (T^{5} - 6 T^{4} + \cdots + 224)^{2} \) Copy content Toggle raw display
$31$ \( T^{10} - 154 T^{8} + \cdots - 3998792 \) Copy content Toggle raw display
$37$ \( (T^{5} - 20 T^{4} + \cdots - 472)^{2} \) Copy content Toggle raw display
$41$ \( T^{10} - 168 T^{8} + \cdots - 25088 \) Copy content Toggle raw display
$43$ \( (T^{5} - 4 T^{4} + \cdots - 256)^{2} \) Copy content Toggle raw display
$47$ \( T^{10} + \cdots - 1158537248 \) Copy content Toggle raw display
$53$ \( (T^{5} - 8 T^{4} + \cdots - 11264)^{2} \) Copy content Toggle raw display
$59$ \( T^{10} + \cdots - 108162632 \) Copy content Toggle raw display
$61$ \( T^{10} - 144 T^{8} + \cdots - 3527168 \) Copy content Toggle raw display
$67$ \( (T^{5} - 2 T^{4} + \cdots - 4648)^{2} \) Copy content Toggle raw display
$71$ \( (T^{5} + 18 T^{4} + \cdots - 88)^{2} \) Copy content Toggle raw display
$73$ \( T^{10} - 400 T^{8} + \cdots - 65987072 \) Copy content Toggle raw display
$79$ \( (T^{5} + 4 T^{4} + \cdots + 704)^{2} \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots - 102760448 \) Copy content Toggle raw display
$89$ \( T^{10} - 34 T^{8} + \cdots - 8 \) Copy content Toggle raw display
$97$ \( T^{10} - 226 T^{8} + \cdots - 19208 \) Copy content Toggle raw display
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