Properties

Label 7600.2.a.ch
Level $7600$
Weight $2$
Character orbit 7600.a
Self dual yes
Analytic conductor $60.686$
Analytic rank $1$
Dimension $6$
CM no
Inner twists $1$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [7600,2,Mod(1,7600)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7600, 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("7600.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7600 = 2^{4} \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7600.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: \(60.6863055362\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} - 10x^{4} + 16x^{3} + 15x^{2} - 14x - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 3800)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} + ( - \beta_{4} - \beta_1) q^{7} + (\beta_{2} + \beta_1 + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} + ( - \beta_{4} - \beta_1) q^{7} + (\beta_{2} + \beta_1 + 1) q^{9} + ( - \beta_{5} + \beta_{4} - \beta_{3} + \cdots - 1) q^{11}+ \cdots + (\beta_{5} - \beta_{4} + 2 \beta_{3} + \cdots + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 2 q^{3} - 2 q^{7} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 2 q^{3} - 2 q^{7} + 6 q^{9} - 3 q^{11} - q^{13} - 14 q^{17} + 6 q^{19} + 15 q^{21} - 12 q^{23} - 8 q^{27} + 9 q^{29} - 5 q^{31} + 2 q^{33} - 8 q^{37} - 12 q^{39} + 3 q^{41} - 15 q^{43} - 4 q^{47} + 22 q^{49} - 33 q^{51} + 13 q^{53} - 2 q^{57} + 9 q^{61} - 21 q^{63} + 3 q^{67} - 11 q^{69} - 19 q^{71} + 3 q^{73} - 36 q^{77} + 16 q^{79} + 26 q^{81} - 31 q^{83} + 25 q^{87} + 14 q^{89} - 42 q^{91} + 39 q^{93} - 11 q^{97} + 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 2x^{5} - 10x^{4} + 16x^{3} + 15x^{2} - 14x - 3 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{4} - \nu^{3} - 9\nu^{2} + 6\nu + 5 ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{5} - 2\nu^{4} - 9\nu^{3} + 15\nu^{2} + 6\nu - 8 ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{5} - 2\nu^{4} - 10\nu^{3} + 15\nu^{2} + 15\nu - 7 ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -2\beta_{5} + 2\beta_{4} + 9\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -2\beta_{5} + 2\beta_{4} + 2\beta_{3} + 9\beta_{2} + 12\beta _1 + 32 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -22\beta_{5} + 24\beta_{4} + 4\beta_{3} + 3\beta_{2} + 84\beta _1 + 21 \) Copy content Toggle raw display

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
3.26143
1.93590
0.848258
−0.185519
−1.08999
−2.77008
0 −3.26143 0 0 0 −4.07225 0 7.63693 0
1.2 0 −1.93590 0 0 0 1.24708 0 0.747704 0
1.3 0 −0.848258 0 0 0 −1.74484 0 −2.28046 0
1.4 0 0.185519 0 0 0 4.45651 0 −2.96558 0
1.5 0 1.08999 0 0 0 −4.19727 0 −1.81192 0
1.6 0 2.77008 0 0 0 2.31077 0 4.67334 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( +1 \)
\(5\) \( -1 \)
\(19\) \( -1 \)

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 7600.2.a.ch 6
4.b odd 2 1 3800.2.a.bc yes 6
5.b even 2 1 7600.2.a.cl 6
20.d odd 2 1 3800.2.a.ba 6
20.e even 4 2 3800.2.d.q 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3800.2.a.ba 6 20.d odd 2 1
3800.2.a.bc yes 6 4.b odd 2 1
3800.2.d.q 12 20.e even 4 2
7600.2.a.ch 6 1.a even 1 1 trivial
7600.2.a.cl 6 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(7600))\):

\( T_{3}^{6} + 2T_{3}^{5} - 10T_{3}^{4} - 16T_{3}^{3} + 15T_{3}^{2} + 14T_{3} - 3 \) Copy content Toggle raw display
\( T_{7}^{6} + 2T_{7}^{5} - 30T_{7}^{4} - 48T_{7}^{3} + 223T_{7}^{2} + 154T_{7} - 383 \) Copy content Toggle raw display
\( T_{11}^{6} + 3T_{11}^{5} - 40T_{11}^{4} - 139T_{11}^{3} + 10T_{11}^{2} + 208T_{11} - 88 \) Copy content Toggle raw display
\( T_{13}^{6} + T_{13}^{5} - 65T_{13}^{4} - 15T_{13}^{3} + 883T_{13}^{2} - 1621T_{13} + 825 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} + 2 T^{5} + \cdots - 3 \) Copy content Toggle raw display
$5$ \( T^{6} \) Copy content Toggle raw display
$7$ \( T^{6} + 2 T^{5} + \cdots - 383 \) Copy content Toggle raw display
$11$ \( T^{6} + 3 T^{5} + \cdots - 88 \) Copy content Toggle raw display
$13$ \( T^{6} + T^{5} + \cdots + 825 \) Copy content Toggle raw display
$17$ \( T^{6} + 14 T^{5} + \cdots + 3147 \) Copy content Toggle raw display
$19$ \( (T - 1)^{6} \) Copy content Toggle raw display
$23$ \( T^{6} + 12 T^{5} + \cdots - 2487 \) Copy content Toggle raw display
$29$ \( T^{6} - 9 T^{5} + \cdots + 21951 \) Copy content Toggle raw display
$31$ \( T^{6} + 5 T^{5} + \cdots + 11000 \) Copy content Toggle raw display
$37$ \( T^{6} + 8 T^{5} + \cdots - 7365 \) Copy content Toggle raw display
$41$ \( T^{6} - 3 T^{5} + \cdots - 113472 \) Copy content Toggle raw display
$43$ \( T^{6} + 15 T^{5} + \cdots + 6120 \) Copy content Toggle raw display
$47$ \( T^{6} + 4 T^{5} + \cdots + 1791 \) Copy content Toggle raw display
$53$ \( T^{6} - 13 T^{5} + \cdots + 9285 \) Copy content Toggle raw display
$59$ \( T^{6} - 193 T^{4} + \cdots - 32328 \) Copy content Toggle raw display
$61$ \( T^{6} - 9 T^{5} + \cdots + 12424 \) Copy content Toggle raw display
$67$ \( T^{6} - 3 T^{5} + \cdots - 136033 \) Copy content Toggle raw display
$71$ \( T^{6} + 19 T^{5} + \cdots - 27576 \) Copy content Toggle raw display
$73$ \( T^{6} - 3 T^{5} + \cdots - 741033 \) Copy content Toggle raw display
$79$ \( T^{6} - 16 T^{5} + \cdots - 553536 \) Copy content Toggle raw display
$83$ \( T^{6} + 31 T^{5} + \cdots - 71160 \) Copy content Toggle raw display
$89$ \( T^{6} - 14 T^{5} + \cdots + 3240 \) Copy content Toggle raw display
$97$ \( T^{6} + 11 T^{5} + \cdots - 288792 \) Copy content Toggle raw display
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