Properties

Label 7600.2.a.cn
Level $7600$
Weight $2$
Character orbit 7600.a
Self dual yes
Analytic conductor $60.686$
Analytic rank $0$
Dimension $6$
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] = [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: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.253565184.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} - 11x^{4} + 20x^{3} + 22x^{2} - 32x + 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 760)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + (\beta_{3} + \beta_1 + 1) q^{7} + (\beta_{3} + \beta_{2} + \beta_1 + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + (\beta_{3} + \beta_1 + 1) q^{7} + (\beta_{3} + \beta_{2} + \beta_1 + 1) q^{9} + ( - \beta_{4} - \beta_{3} + \beta_{2}) q^{11} + ( - \beta_{5} - \beta_{4} - \beta_{2} + \cdots - 2) q^{13}+ \cdots + ( - 3 \beta_{5} - 2 \beta_{4} + \cdots + 6) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 2 q^{3} + 6 q^{7} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 2 q^{3} + 6 q^{7} + 8 q^{9} + 2 q^{11} - 14 q^{13} - 10 q^{17} - 6 q^{19} + 18 q^{21} + 2 q^{23} + 2 q^{27} - 2 q^{29} - 8 q^{31} - 8 q^{33} - 4 q^{37} + 18 q^{39} + 4 q^{41} + 4 q^{43} + 4 q^{47} + 2 q^{49} + 34 q^{51} + 14 q^{53} - 2 q^{57} + 2 q^{59} + 10 q^{61} + 44 q^{63} + 42 q^{67} + 18 q^{69} + 8 q^{71} + 2 q^{73} - 24 q^{77} + 20 q^{79} + 6 q^{81} + 16 q^{83} + 30 q^{87} + 32 q^{89} - 10 q^{91} - 12 q^{93} - 40 q^{97} + 22 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 2x^{5} - 11x^{4} + 20x^{3} + 22x^{2} - 32x + 8 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{5} + 2\nu^{4} + 11\nu^{3} - 16\nu^{2} - 26\nu + 12 ) / 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{5} - 2\nu^{4} - 11\nu^{3} + 20\nu^{2} + 22\nu - 28 ) / 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{5} - \nu^{4} - 11\nu^{3} + 9\nu^{2} + 24\nu - 10 ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -3\nu^{5} + 4\nu^{4} + 37\nu^{3} - 38\nu^{2} - 98\nu + 48 ) / 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + \beta_{2} + \beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{5} + \beta_{4} + \beta_{3} + 7\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{4} + 7\beta_{3} + 11\beta_{2} + 9\beta _1 + 26 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 11\beta_{5} + 15\beta_{4} + 9\beta_{3} + 2\beta_{2} + 53\beta_1 \) 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
−2.84742
−1.59277
0.366738
0.664406
2.43031
2.97875
0 −2.84742 0 0 0 −0.145034 0 5.10782 0
1.2 0 −1.59277 0 0 0 1.66290 0 −0.463073 0
1.3 0 0.366738 0 0 0 −3.08675 0 −2.86550 0
1.4 0 0.664406 0 0 0 −0.345799 0 −2.55856 0
1.5 0 2.43031 0 0 0 3.60737 0 2.90640 0
1.6 0 2.97875 0 0 0 4.30732 0 5.87292 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.cn 6
4.b odd 2 1 3800.2.a.z 6
5.b even 2 1 7600.2.a.cg 6
5.c odd 4 2 1520.2.d.k 12
20.d odd 2 1 3800.2.a.be 6
20.e even 4 2 760.2.d.e 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
760.2.d.e 12 20.e even 4 2
1520.2.d.k 12 5.c odd 4 2
3800.2.a.z 6 4.b odd 2 1
3800.2.a.be 6 20.d odd 2 1
7600.2.a.cg 6 5.b even 2 1
7600.2.a.cn 6 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7600))\):

\( T_{3}^{6} - 2T_{3}^{5} - 11T_{3}^{4} + 20T_{3}^{3} + 22T_{3}^{2} - 32T_{3} + 8 \) Copy content Toggle raw display
\( T_{7}^{6} - 6T_{7}^{5} - 4T_{7}^{4} + 62T_{7}^{3} - 49T_{7}^{2} - 36T_{7} - 4 \) Copy content Toggle raw display
\( T_{11}^{6} - 2T_{11}^{5} - 53T_{11}^{4} + 76T_{11}^{3} + 762T_{11}^{2} - 456T_{11} - 2584 \) Copy content Toggle raw display
\( T_{13}^{6} + 14T_{13}^{5} + 27T_{13}^{4} - 352T_{13}^{3} - 1270T_{13}^{2} + 2080T_{13} + 9376 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} - 2 T^{5} + \cdots + 8 \) Copy content Toggle raw display
$5$ \( T^{6} \) Copy content Toggle raw display
$7$ \( T^{6} - 6 T^{5} + \cdots - 4 \) Copy content Toggle raw display
$11$ \( T^{6} - 2 T^{5} + \cdots - 2584 \) Copy content Toggle raw display
$13$ \( T^{6} + 14 T^{5} + \cdots + 9376 \) Copy content Toggle raw display
$17$ \( T^{6} + 10 T^{5} + \cdots + 15864 \) Copy content Toggle raw display
$19$ \( (T + 1)^{6} \) Copy content Toggle raw display
$23$ \( T^{6} - 2 T^{5} + \cdots + 272 \) Copy content Toggle raw display
$29$ \( T^{6} + 2 T^{5} + \cdots + 10064 \) Copy content Toggle raw display
$31$ \( T^{6} + 8 T^{5} + \cdots - 2304 \) Copy content Toggle raw display
$37$ \( T^{6} + 4 T^{5} + \cdots - 12352 \) Copy content Toggle raw display
$41$ \( T^{6} - 4 T^{5} + \cdots - 4832 \) Copy content Toggle raw display
$43$ \( T^{6} - 4 T^{5} + \cdots - 4632 \) Copy content Toggle raw display
$47$ \( T^{6} - 4 T^{5} + \cdots - 1152 \) Copy content Toggle raw display
$53$ \( T^{6} - 14 T^{5} + \cdots + 2752 \) Copy content Toggle raw display
$59$ \( T^{6} - 2 T^{5} + \cdots - 374496 \) Copy content Toggle raw display
$61$ \( T^{6} - 10 T^{5} + \cdots + 40256 \) Copy content Toggle raw display
$67$ \( T^{6} - 42 T^{5} + \cdots + 2752 \) Copy content Toggle raw display
$71$ \( T^{6} - 8 T^{5} + \cdots - 64 \) Copy content Toggle raw display
$73$ \( T^{6} - 2 T^{5} + \cdots + 3208 \) Copy content Toggle raw display
$79$ \( T^{6} - 20 T^{5} + \cdots + 1024 \) Copy content Toggle raw display
$83$ \( T^{6} - 16 T^{5} + \cdots + 6208 \) Copy content Toggle raw display
$89$ \( T^{6} - 32 T^{5} + \cdots + 69504 \) Copy content Toggle raw display
$97$ \( T^{6} + 40 T^{5} + \cdots - 196992 \) Copy content Toggle raw display
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