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Properties

Label 6096.2.a.bk
Level $6096$
Weight $2$
Character orbit 6096.a
Self dual yes
Analytic conductor $48.677$
Analytic rank $1$
Dimension $9$
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] = [6096,2,Mod(1,6096)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6096, 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("6096.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6096 = 2^{4} \cdot 3 \cdot 127 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6096.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: \(48.6768050722\)
Analytic rank: \(1\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{9} - 2x^{8} - 14x^{7} + 26x^{6} + 59x^{5} - 99x^{4} - 66x^{3} + 102x^{2} - 24x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 381)
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 \(1,\beta_1,\ldots,\beta_{8}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{3} - \beta_{4} q^{5} + (\beta_{8} - 1) q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{3} - \beta_{4} q^{5} + (\beta_{8} - 1) q^{7} + q^{9} + (\beta_{6} - 1) q^{11} + ( - \beta_{5} + 2) q^{13} + \beta_{4} q^{15} + (\beta_{8} + \beta_{3} - 1) q^{17} + ( - \beta_{8} - \beta_{3} - 1) q^{19} + ( - \beta_{8} + 1) q^{21} + ( - \beta_{7} - 2 \beta_{6} + \beta_{5} + \cdots + 1) q^{23}+ \cdots + (\beta_{6} - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q - 9 q^{3} - 4 q^{5} - 10 q^{7} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 9 q - 9 q^{3} - 4 q^{5} - 10 q^{7} + 9 q^{9} - 8 q^{11} + 14 q^{13} + 4 q^{15} - 6 q^{17} - 12 q^{19} + 10 q^{21} + 4 q^{23} + 21 q^{25} - 9 q^{27} - 8 q^{29} - 4 q^{31} + 8 q^{33} - 6 q^{35} + 22 q^{37} - 14 q^{39} - 2 q^{41} - 6 q^{43} - 4 q^{45} + 2 q^{47} + 23 q^{49} + 6 q^{51} - 12 q^{53} + 22 q^{55} + 12 q^{57} + 6 q^{59} + 2 q^{61} - 10 q^{63} + 4 q^{65} - 18 q^{67} - 4 q^{69} - 24 q^{71} + 14 q^{73} - 21 q^{75} - 18 q^{77} - 12 q^{79} + 9 q^{81} + 20 q^{83} - 24 q^{85} + 8 q^{87} - 30 q^{89} - 14 q^{91} + 4 q^{93} + 32 q^{95} + 12 q^{97} - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{9} - 2x^{8} - 14x^{7} + 26x^{6} + 59x^{5} - 99x^{4} - 66x^{3} + 102x^{2} - 24x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{3} - \nu^{2} - 6\nu + 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{8} + 3\nu^{7} + 17\nu^{6} - 43\nu^{5} - 94\nu^{4} + 181\nu^{3} + 161\nu^{2} - 197\nu + 29 ) / 12 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{8} + 3\nu^{7} + 14\nu^{6} - 40\nu^{5} - 58\nu^{4} + 157\nu^{3} + 59\nu^{2} - 170\nu + 32 ) / 6 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -2\nu^{8} + 3\nu^{7} + 31\nu^{6} - 38\nu^{5} - 152\nu^{4} + 134\nu^{3} + 235\nu^{2} - 109\nu - 14 ) / 6 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -5\nu^{8} + 9\nu^{7} + 73\nu^{6} - 113\nu^{5} - 338\nu^{4} + 401\nu^{3} + 499\nu^{2} - 349\nu - 5 ) / 12 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -5\nu^{8} + 9\nu^{7} + 73\nu^{6} - 113\nu^{5} - 326\nu^{4} + 401\nu^{3} + 403\nu^{2} - 361\nu + 91 ) / 12 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 3\nu^{8} - 5\nu^{7} - 43\nu^{6} + 65\nu^{5} + 190\nu^{4} - 247\nu^{3} - 247\nu^{2} + 251\nu - 27 ) / 4 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{7} + \beta_{5} + \beta_{4} - \beta_{3} + 7 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{7} + \beta_{5} + \beta_{4} - \beta_{3} + 2\beta_{2} + 6\beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -6\beta_{7} - 2\beta_{6} + 8\beta_{5} + 8\beta_{4} - 8\beta_{3} + \beta _1 + 40 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{8} - 4\beta_{7} + \beta_{6} + 5\beta_{5} + 5\beta_{4} - 6\beta_{3} + 10\beta_{2} + 20\beta _1 + 7 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 2\beta_{8} - 38\beta_{7} - 22\beta_{6} + 64\beta_{5} + 60\beta_{4} - 58\beta_{3} + 4\beta_{2} + 13\beta _1 + 250 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 30 \beta_{8} - 57 \beta_{7} + 30 \beta_{6} + 83 \beta_{5} + 91 \beta_{4} - 109 \beta_{3} + 164 \beta_{2} + \cdots + 127 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 38 \beta_{8} - 251 \beta_{7} - 182 \beta_{6} + 497 \beta_{5} + 453 \beta_{4} - 411 \beta_{3} + \cdots + 1635 ) / 2 \) 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.

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
0.0532857
2.75841
−2.63041
1.92789
−2.14900
0.247918
−1.44437
2.55353
0.682747
0 −1.00000 0 −3.85537 0 −4.59060 0 1.00000 0
1.2 0 −1.00000 0 −3.68284 0 3.98729 0 1.00000 0
1.3 0 −1.00000 0 −3.66812 0 −1.33794 0 1.00000 0
1.4 0 −1.00000 0 −1.15841 0 −2.35752 0 1.00000 0
1.5 0 −1.00000 0 0.492551 0 0.251160 0 1.00000 0
1.6 0 −1.00000 0 0.730098 0 3.26267 0 1.00000 0
1.7 0 −1.00000 0 0.766977 0 −3.56571 0 1.00000 0
1.8 0 −1.00000 0 2.44899 0 −3.89705 0 1.00000 0
1.9 0 −1.00000 0 3.92611 0 −1.75230 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.9
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(127\) \(-1\)

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 6096.2.a.bk 9
4.b odd 2 1 381.2.a.e ✓ 9
12.b even 2 1 1143.2.a.j 9
20.d odd 2 1 9525.2.a.p 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
381.2.a.e ✓ 9 4.b odd 2 1
1143.2.a.j 9 12.b even 2 1
6096.2.a.bk 9 1.a even 1 1 trivial
9525.2.a.p 9 20.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6096))\):

\( T_{5}^{9} + 4T_{5}^{8} - 25T_{5}^{7} - 94T_{5}^{6} + 185T_{5}^{5} + 524T_{5}^{4} - 612T_{5}^{3} - 384T_{5}^{2} + 592T_{5} - 160 \) Copy content Toggle raw display
\( T_{7}^{9} + 10T_{7}^{8} + 7T_{7}^{7} - 222T_{7}^{6} - 707T_{7}^{5} + 532T_{7}^{4} + 5304T_{7}^{3} + 7544T_{7}^{2} + 2352T_{7} - 1152 \) Copy content Toggle raw display
\( T_{11}^{9} + 8 T_{11}^{8} - 18 T_{11}^{7} - 238 T_{11}^{6} - 29 T_{11}^{5} + 2258 T_{11}^{4} + \cdots + 5644 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{9} \) Copy content Toggle raw display
$3$ \( (T + 1)^{9} \) Copy content Toggle raw display
$5$ \( T^{9} + 4 T^{8} + \cdots - 160 \) Copy content Toggle raw display
$7$ \( T^{9} + 10 T^{8} + \cdots - 1152 \) Copy content Toggle raw display
$11$ \( T^{9} + 8 T^{8} + \cdots + 5644 \) Copy content Toggle raw display
$13$ \( T^{9} - 14 T^{8} + \cdots + 418 \) Copy content Toggle raw display
$17$ \( T^{9} + 6 T^{8} + \cdots + 4768 \) Copy content Toggle raw display
$19$ \( T^{9} + 12 T^{8} + \cdots - 2816 \) Copy content Toggle raw display
$23$ \( T^{9} - 4 T^{8} + \cdots + 169984 \) Copy content Toggle raw display
$29$ \( T^{9} + 8 T^{8} + \cdots + 288 \) Copy content Toggle raw display
$31$ \( T^{9} + 4 T^{8} + \cdots - 123392 \) Copy content Toggle raw display
$37$ \( T^{9} - 22 T^{8} + \cdots + 136082 \) Copy content Toggle raw display
$41$ \( T^{9} + 2 T^{8} + \cdots - 30880 \) Copy content Toggle raw display
$43$ \( T^{9} + 6 T^{8} + \cdots - 292288 \) Copy content Toggle raw display
$47$ \( T^{9} - 2 T^{8} + \cdots - 1356304 \) Copy content Toggle raw display
$53$ \( T^{9} + 12 T^{8} + \cdots + 633760 \) Copy content Toggle raw display
$59$ \( T^{9} - 6 T^{8} + \cdots - 3599360 \) Copy content Toggle raw display
$61$ \( T^{9} - 2 T^{8} + \cdots - 6116062 \) Copy content Toggle raw display
$67$ \( T^{9} + 18 T^{8} + \cdots + 1696960 \) Copy content Toggle raw display
$71$ \( T^{9} + 24 T^{8} + \cdots + 165399656 \) Copy content Toggle raw display
$73$ \( T^{9} - 14 T^{8} + \cdots + 271190 \) Copy content Toggle raw display
$79$ \( T^{9} + 12 T^{8} + \cdots + 104192 \) Copy content Toggle raw display
$83$ \( T^{9} - 20 T^{8} + \cdots - 226643968 \) Copy content Toggle raw display
$89$ \( T^{9} + 30 T^{8} + \cdots - 825248 \) Copy content Toggle raw display
$97$ \( T^{9} - 12 T^{8} + \cdots - 13712896 \) Copy content Toggle raw display
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