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Properties

Label 9200.2.a.cy
Level $9200$
Weight $2$
Character orbit 9200.a
Self dual yes
Analytic conductor $73.462$
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] = [9200,2,Mod(1,9200)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9200, 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("9200.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9200 = 2^{4} \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9200.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: \(73.4623698596\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.143376304.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 12x^{4} + 22x^{2} - 6x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 460)
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.

\(f(q)\) \(=\) \( q + (\beta_{2} + 1) q^{3} + ( - \beta_{3} + 2) q^{7} + (\beta_{2} + \beta_1 + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{2} + 1) q^{3} + ( - \beta_{3} + 2) q^{7} + (\beta_{2} + \beta_1 + 2) q^{9} + (\beta_{5} - \beta_{4} + \beta_{2} - 1) q^{11} + ( - \beta_{4} - \beta_{3} + \beta_{2} + \cdots - 1) q^{13}+ \cdots + ( - 2 \beta_{4} - 2 \beta_{3} + \cdots + 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 4 q^{3} + 9 q^{7} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 4 q^{3} + 9 q^{7} + 10 q^{9} - 2 q^{11} - 8 q^{13} - 5 q^{17} - 4 q^{19} + 6 q^{23} + 22 q^{27} + 5 q^{29} - 9 q^{31} + 10 q^{33} - 21 q^{37} + 8 q^{39} - q^{41} + 16 q^{43} + 16 q^{47} + 19 q^{49} + 12 q^{51} + q^{53} - 12 q^{57} + 11 q^{59} - 4 q^{61} + 19 q^{63} + 25 q^{67} + 4 q^{69} + 17 q^{71} + 14 q^{73} - 20 q^{77} - 10 q^{79} + 14 q^{81} + 21 q^{83} + 64 q^{87} - 24 q^{89} + 4 q^{91} - 4 q^{97} + 16 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 12x^{4} + 22x^{2} - 6x - 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{5} - 3\nu^{4} + 11\nu^{3} + 33\nu^{2} - 3\nu - 27 ) / 8 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 3\nu^{5} + \nu^{4} - 33\nu^{3} - 11\nu^{2} + 41\nu - 7 ) / 8 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -7\nu^{5} + 3\nu^{4} + 85\nu^{3} - 25\nu^{2} - 157\nu + 43 ) / 16 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 9\nu^{5} + 3\nu^{4} - 107\nu^{3} - 41\nu^{2} + 195\nu + 11 ) / 16 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -9\nu^{5} - 3\nu^{4} + 107\nu^{3} + 25\nu^{2} - 179\nu + 53 ) / 16 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{4} + \beta_{3} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -2\beta_{5} - \beta_{4} + \beta_{3} + \beta _1 + 8 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{5} + 3\beta_{4} + 4\beta_{3} + 3\beta_{2} + 4\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -22\beta_{5} - 7\beta_{4} + 15\beta_{3} - 2\beta_{2} + 9\beta _1 + 66 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 22\beta_{5} + 51\beta_{4} + 73\beta_{3} + 72\beta_{2} + 75\beta _1 + 12 ) / 2 \) 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
1.26443
−3.08006
−0.116918
0.420790
−1.65047
3.16223
0 −2.40050 0 0 0 4.41307 0 2.76241 0
1.2 0 −0.873449 0 0 0 0.992530 0 −2.23709 0
1.3 0 −0.486391 0 0 0 −1.80495 0 −2.76342 0
1.4 0 1.73961 0 0 0 3.32224 0 0.0262434 0
1.5 0 2.80150 0 0 0 4.50896 0 4.84843 0
1.6 0 3.21923 0 0 0 −2.43185 0 7.36343 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
\(2\) \(-1\)
\(5\) \(-1\)
\(23\) \(-1\)

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 9200.2.a.cy 6
4.b odd 2 1 2300.2.a.n 6
5.b even 2 1 9200.2.a.cx 6
5.c odd 4 2 1840.2.e.f 12
20.d odd 2 1 2300.2.a.o 6
20.e even 4 2 460.2.c.a ✓ 12
60.l odd 4 2 4140.2.f.b 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
460.2.c.a ✓ 12 20.e even 4 2
1840.2.e.f 12 5.c odd 4 2
2300.2.a.n 6 4.b odd 2 1
2300.2.a.o 6 20.d odd 2 1
4140.2.f.b 12 60.l odd 4 2
9200.2.a.cx 6 5.b even 2 1
9200.2.a.cy 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9200))\):

\( T_{3}^{6} - 4T_{3}^{5} - 6T_{3}^{4} + 30T_{3}^{3} + 5T_{3}^{2} - 38T_{3} - 16 \) Copy content Toggle raw display
\( T_{7}^{6} - 9T_{7}^{5} + 10T_{7}^{4} + 88T_{7}^{3} - 152T_{7}^{2} - 228T_{7} + 288 \) Copy content Toggle raw display
\( T_{11}^{6} + 2T_{11}^{5} - 32T_{11}^{4} - 28T_{11}^{3} + 212T_{11}^{2} + 144T_{11} - 256 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} - 4 T^{5} + \cdots - 16 \) Copy content Toggle raw display
$5$ \( T^{6} \) Copy content Toggle raw display
$7$ \( T^{6} - 9 T^{5} + \cdots + 288 \) Copy content Toggle raw display
$11$ \( T^{6} + 2 T^{5} + \cdots - 256 \) Copy content Toggle raw display
$13$ \( T^{6} + 8 T^{5} + \cdots + 1184 \) Copy content Toggle raw display
$17$ \( T^{6} + 5 T^{5} + \cdots + 16 \) Copy content Toggle raw display
$19$ \( T^{6} + 4 T^{5} + \cdots + 256 \) Copy content Toggle raw display
$23$ \( (T - 1)^{6} \) Copy content Toggle raw display
$29$ \( T^{6} - 5 T^{5} + \cdots + 11862 \) Copy content Toggle raw display
$31$ \( T^{6} + 9 T^{5} + \cdots + 916 \) Copy content Toggle raw display
$37$ \( T^{6} + 21 T^{5} + \cdots + 63216 \) Copy content Toggle raw display
$41$ \( T^{6} + T^{5} - 64 T^{4} + \cdots - 2 \) Copy content Toggle raw display
$43$ \( T^{6} - 16 T^{5} + \cdots + 91648 \) Copy content Toggle raw display
$47$ \( T^{6} - 16 T^{5} + \cdots - 464 \) Copy content Toggle raw display
$53$ \( T^{6} - T^{5} + \cdots + 12224 \) Copy content Toggle raw display
$59$ \( T^{6} - 11 T^{5} + \cdots - 360576 \) Copy content Toggle raw display
$61$ \( T^{6} + 4 T^{5} + \cdots - 171088 \) Copy content Toggle raw display
$67$ \( T^{6} - 25 T^{5} + \cdots + 7424 \) Copy content Toggle raw display
$71$ \( T^{6} - 17 T^{5} + \cdots - 16108 \) Copy content Toggle raw display
$73$ \( T^{6} - 14 T^{5} + \cdots + 5504 \) Copy content Toggle raw display
$79$ \( T^{6} + 10 T^{5} + \cdots - 128 \) Copy content Toggle raw display
$83$ \( T^{6} - 21 T^{5} + \cdots - 633344 \) Copy content Toggle raw display
$89$ \( T^{6} + 24 T^{5} + \cdots + 180432 \) Copy content Toggle raw display
$97$ \( T^{6} + 4 T^{5} + \cdots + 45728 \) Copy content Toggle raw display
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