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Properties

Label 9200.2.a.de
Level $9200$
Weight $2$
Character orbit 9200.a
Self dual yes
Analytic conductor $73.462$
Analytic rank $0$
Dimension $8$
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: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 3x^{7} - 13x^{6} + 38x^{5} + 41x^{4} - 123x^{3} + 15x^{2} + 32x - 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 920)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + (\beta_{7} + 1) q^{7} + (\beta_{7} + \beta_{5} + \beta_{3} + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + (\beta_{7} + 1) q^{7} + (\beta_{7} + \beta_{5} + \beta_{3} + 2) q^{9} + ( - \beta_{3} - 1) q^{11} + (\beta_{2} + 1) q^{13} + (\beta_{6} + \beta_{4} - 1) q^{17} + (\beta_{7} + \beta_{4} - 1) q^{19} + (\beta_{6} - \beta_{5} - \beta_{4} + \cdots - 2) q^{21}+ \cdots + ( - \beta_{6} + \beta_{5} + \beta_{4} + \cdots - 5) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 3 q^{3} + 7 q^{7} + 11 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 3 q^{3} + 7 q^{7} + 11 q^{9} - 7 q^{11} + 11 q^{13} - 7 q^{17} - 11 q^{19} - 8 q^{23} + 12 q^{27} + 22 q^{29} - 9 q^{31} - 9 q^{33} + 4 q^{37} + 7 q^{41} + 22 q^{43} + 4 q^{47} + 39 q^{49} + 19 q^{51} + 4 q^{53} - 32 q^{59} + 17 q^{61} + 44 q^{63} - 4 q^{67} - 3 q^{69} - 15 q^{71} + 6 q^{73} + 18 q^{77} + 2 q^{79} + 24 q^{81} + 36 q^{83} - 4 q^{87} + 46 q^{89} + 35 q^{91} + 20 q^{93} + 3 q^{97} - 61 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 3x^{7} - 13x^{6} + 38x^{5} + 41x^{4} - 123x^{3} + 15x^{2} + 32x - 8 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{4} - 2\nu^{3} - 7\nu^{2} + 11\nu + 1 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{6} - 4\nu^{5} - 5\nu^{4} + 29\nu^{3} - 8\nu^{2} - 21\nu - 2 ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{7} - 3\nu^{6} - 11\nu^{5} + 30\nu^{4} + 33\nu^{3} - 69\nu^{2} - 17\nu + 12 ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{7} - 3\nu^{6} - 13\nu^{5} + 36\nu^{4} + 45\nu^{3} - 107\nu^{2} - 9\nu + 18 ) / 2 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{7} - 3\nu^{6} - 13\nu^{5} + 36\nu^{4} + 47\nu^{3} - 109\nu^{2} - 23\nu + 26 ) / 2 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -\nu^{7} + 2\nu^{6} + 17\nu^{5} - 31\nu^{4} - 74\nu^{3} + 117\nu^{2} + 30\nu - 26 ) / 2 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{7} + \beta_{5} + \beta_{3} + 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} + \beta_{6} + \beta_{3} + 7\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 9\beta_{7} + 2\beta_{6} + 7\beta_{5} + 9\beta_{3} + \beta_{2} + 3\beta _1 + 36 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 14\beta_{7} + 12\beta_{6} + \beta_{5} + \beta_{4} + 14\beta_{3} + 3\beta_{2} + 55\beta _1 + 22 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 80\beta_{7} + 29\beta_{6} + 47\beta_{5} + 4\beta_{4} + 82\beta_{3} + 17\beta_{2} + 53\beta _1 + 281 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 160\beta_{7} + 126\beta_{6} + 11\beta_{5} + 25\beta_{4} + 166\beta_{3} + 54\beta_{2} + 460\beta _1 + 305 \) 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
−2.54092
−2.51561
−0.540724
0.296848
0.493532
1.69755
2.89996
3.20935
0 −2.54092 0 0 0 −0.780573 0 3.45626 0
1.2 0 −2.51561 0 0 0 4.64022 0 3.32827 0
1.3 0 −0.540724 0 0 0 1.15693 0 −2.70762 0
1.4 0 0.296848 0 0 0 −3.46037 0 −2.91188 0
1.5 0 0.493532 0 0 0 4.54439 0 −2.75643 0
1.6 0 1.69755 0 0 0 −4.22860 0 −0.118308 0
1.7 0 2.89996 0 0 0 0.580879 0 5.40975 0
1.8 0 3.20935 0 0 0 4.54713 0 7.29996 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
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.de 8
4.b odd 2 1 4600.2.a.bj 8
5.b even 2 1 9200.2.a.dd 8
5.c odd 4 2 1840.2.e.h 16
20.d odd 2 1 4600.2.a.bk 8
20.e even 4 2 920.2.e.c ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
920.2.e.c ✓ 16 20.e even 4 2
1840.2.e.h 16 5.c odd 4 2
4600.2.a.bj 8 4.b odd 2 1
4600.2.a.bk 8 20.d odd 2 1
9200.2.a.dd 8 5.b even 2 1
9200.2.a.de 8 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9200))\):

\( T_{3}^{8} - 3T_{3}^{7} - 13T_{3}^{6} + 38T_{3}^{5} + 41T_{3}^{4} - 123T_{3}^{3} + 15T_{3}^{2} + 32T_{3} - 8 \) Copy content Toggle raw display
\( T_{7}^{8} - 7T_{7}^{7} - 23T_{7}^{6} + 218T_{7}^{5} + 20T_{7}^{4} - 1720T_{7}^{3} + 1316T_{7}^{2} + 1056T_{7} - 736 \) Copy content Toggle raw display
\( T_{11}^{8} + 7T_{11}^{7} - 29T_{11}^{6} - 248T_{11}^{5} + 162T_{11}^{4} + 2364T_{11}^{3} + 28T_{11}^{2} - 5456T_{11} + 440 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} - 3 T^{7} + \cdots - 8 \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} - 7 T^{7} + \cdots - 736 \) Copy content Toggle raw display
$11$ \( T^{8} + 7 T^{7} + \cdots + 440 \) Copy content Toggle raw display
$13$ \( T^{8} - 11 T^{7} + \cdots - 14020 \) Copy content Toggle raw display
$17$ \( T^{8} + 7 T^{7} + \cdots + 176 \) Copy content Toggle raw display
$19$ \( T^{8} + 11 T^{7} + \cdots + 11192 \) Copy content Toggle raw display
$23$ \( (T + 1)^{8} \) Copy content Toggle raw display
$29$ \( T^{8} - 22 T^{7} + \cdots - 400 \) Copy content Toggle raw display
$31$ \( T^{8} + 9 T^{7} + \cdots - 287276 \) Copy content Toggle raw display
$37$ \( T^{8} - 4 T^{7} + \cdots + 16096 \) Copy content Toggle raw display
$41$ \( T^{8} - 7 T^{7} + \cdots + 1197584 \) Copy content Toggle raw display
$43$ \( T^{8} - 22 T^{7} + \cdots - 244448 \) Copy content Toggle raw display
$47$ \( T^{8} - 4 T^{7} + \cdots - 73520 \) Copy content Toggle raw display
$53$ \( T^{8} - 4 T^{7} + \cdots + 202880 \) Copy content Toggle raw display
$59$ \( T^{8} + 32 T^{7} + \cdots + 29696 \) Copy content Toggle raw display
$61$ \( T^{8} - 17 T^{7} + \cdots + 200 \) Copy content Toggle raw display
$67$ \( T^{8} + 4 T^{7} + \cdots + 680608 \) Copy content Toggle raw display
$71$ \( T^{8} + 15 T^{7} + \cdots - 8900000 \) Copy content Toggle raw display
$73$ \( T^{8} - 6 T^{7} + \cdots - 557680 \) Copy content Toggle raw display
$79$ \( T^{8} - 2 T^{7} + \cdots - 5248 \) Copy content Toggle raw display
$83$ \( T^{8} - 36 T^{7} + \cdots - 4183520 \) Copy content Toggle raw display
$89$ \( T^{8} - 46 T^{7} + \cdots + 12804160 \) Copy content Toggle raw display
$97$ \( T^{8} - 3 T^{7} + \cdots + 2960 \) Copy content Toggle raw display
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