Properties

Label 9800.2.a.cz
Level $9800$
Weight $2$
Character orbit 9800.a
Self dual yes
Analytic conductor $78.253$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [9800,2,Mod(1,9800)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9800, 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("9800.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9800 = 2^{3} \cdot 5^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9800.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: \(78.2533939809\)
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: \( x^{8} - 18x^{6} + 85x^{4} - 38x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + (\beta_{2} + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + (\beta_{2} + 2) q^{9} + (\beta_{6} + \beta_{3} + 1) q^{11} - \beta_{5} q^{13} + ( - \beta_{5} - \beta_{4} + \beta_1) q^{17} + (\beta_{7} - \beta_{5} - \beta_{4} - \beta_1) q^{19} + ( - \beta_{3} + \beta_{2}) q^{23} + (\beta_{5} - \beta_{4} + 3 \beta_1) q^{27} + (\beta_{6} + \beta_{3} + \beta_{2} + 2) q^{29} + ( - \beta_{5} - 2 \beta_{4} - 2 \beta_1) q^{31} + (\beta_{7} - 3 \beta_{4} + \beta_1) q^{33} + (\beta_{6} - \beta_{3} - \beta_{2}) q^{37} + 2 q^{39} + (2 \beta_{5} + \beta_1) q^{41} + ( - \beta_{6} + \beta_{3} + \beta_{2} + 2) q^{43} + ( - \beta_{7} - 2 \beta_{5} + 4 \beta_{4}) q^{47} + (\beta_{3} + \beta_{2} + 7) q^{51} + ( - \beta_{6} - 2 \beta_{3} - 4) q^{53} + (\beta_{6} + 5 \beta_{3} - \beta_{2} - 1) q^{57} + ( - \beta_{7} - \beta_{5} + \beta_{4} + 2 \beta_1) q^{59} + (\beta_{7} - \beta_{5} + 2 \beta_{4} - 2 \beta_1) q^{61} + ( - 2 \beta_{3} + 5) q^{67} + ( - \beta_{7} + \beta_{5} + 4 \beta_{4} + 4 \beta_1) q^{69} + (\beta_{6} - \beta_{3} + \beta_{2} + 2) q^{71} + ( - \beta_{7} + 5 \beta_{4} + \beta_1) q^{73} + ( - 3 \beta_{3} - \beta_{2} + 2) q^{79} + (\beta_{3} + 7) q^{81} + ( - \beta_{7} - \beta_1) q^{83} + (\beta_{7} + \beta_{5} - 4 \beta_{4} + 6 \beta_1) q^{87} + (2 \beta_{5} - \beta_{4} + 5 \beta_1) q^{89} + (2 \beta_{3} - 2 \beta_{2} - 8) q^{93} + (\beta_{7} - \beta_{5} + 5 \beta_{4}) q^{97} + ( - 2 \beta_{6} + 4 \beta_{3} + \beta_{2} + 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 12 q^{9} + 4 q^{11} - 4 q^{23} + 8 q^{29} + 16 q^{39} + 16 q^{43} + 52 q^{51} - 28 q^{53} - 8 q^{57} + 40 q^{67} + 8 q^{71} + 20 q^{79} + 56 q^{81} - 56 q^{93} + 36 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 18x^{6} + 85x^{4} - 38x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 5 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{4} - 9\nu^{2} + 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{7} - 18\nu^{5} + 83\nu^{3} - 20\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{7} - 18\nu^{5} + 85\nu^{3} - 38\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( \nu^{6} - 18\nu^{4} + 83\nu^{2} - 20 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 5\nu^{7} - 88\nu^{5} + 397\nu^{3} - 96\nu ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{5} - \beta_{4} + 9\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{3} + 9\beta_{2} + 43 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{7} + 9\beta_{5} - 14\beta_{4} + 79\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{6} + 18\beta_{3} + 79\beta_{2} + 379 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 18\beta_{7} + 79\beta_{5} - 167\beta_{4} + 695\beta_1 \) 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
−3.04118
−2.87509
−0.568463
−0.402377
0.402377
0.568463
2.87509
3.04118
0 −3.04118 0 0 0 0 0 6.24878 0
1.2 0 −2.87509 0 0 0 0 0 5.26617 0
1.3 0 −0.568463 0 0 0 0 0 −2.67685 0
1.4 0 −0.402377 0 0 0 0 0 −2.83809 0
1.5 0 0.402377 0 0 0 0 0 −2.83809 0
1.6 0 0.568463 0 0 0 0 0 −2.67685 0
1.7 0 2.87509 0 0 0 0 0 5.26617 0
1.8 0 3.04118 0 0 0 0 0 6.24878 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
\(2\) \(-1\)
\(5\) \(1\)
\(7\) \(1\)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 9800.2.a.cz 8
5.b even 2 1 9800.2.a.da yes 8
7.b odd 2 1 inner 9800.2.a.cz 8
35.c odd 2 1 9800.2.a.da yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9800.2.a.cz 8 1.a even 1 1 trivial
9800.2.a.cz 8 7.b odd 2 1 inner
9800.2.a.da yes 8 5.b even 2 1
9800.2.a.da yes 8 35.c odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(9800))\):

\( T_{3}^{8} - 18T_{3}^{6} + 85T_{3}^{4} - 38T_{3}^{2} + 4 \) Copy content Toggle raw display
\( T_{11}^{4} - 2T_{11}^{3} - 38T_{11}^{2} + 62T_{11} + 257 \) Copy content Toggle raw display
\( T_{13}^{8} - 38T_{13}^{6} + 340T_{13}^{4} - 288T_{13}^{2} + 64 \) Copy content Toggle raw display
\( T_{19}^{8} - 160T_{19}^{6} + 7997T_{19}^{4} - 119588T_{19}^{2} + 84100 \) Copy content Toggle raw display
\( T_{23}^{4} + 2T_{23}^{3} - 47T_{23}^{2} - 156T_{23} - 8 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} - 18 T^{6} + 85 T^{4} - 38 T^{2} + \cdots + 4 \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( (T^{4} - 2 T^{3} - 38 T^{2} + 62 T + 257)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} - 38 T^{6} + 340 T^{4} + \cdots + 64 \) Copy content Toggle raw display
$17$ \( T^{8} - 84 T^{6} + 2089 T^{4} + \cdots + 40000 \) Copy content Toggle raw display
$19$ \( T^{8} - 160 T^{6} + 7997 T^{4} + \cdots + 84100 \) Copy content Toggle raw display
$23$ \( (T^{4} + 2 T^{3} - 47 T^{2} - 156 T - 8)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} - 4 T^{3} - 85 T^{2} + 328 T - 292)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} - 118 T^{6} + 3812 T^{4} + \cdots + 4096 \) Copy content Toggle raw display
$37$ \( (T^{4} - 105 T^{2} - 92 T + 488)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} - 138 T^{6} + 4501 T^{4} + \cdots + 24964 \) Copy content Toggle raw display
$43$ \( (T^{4} - 8 T^{3} - 81 T^{2} + 480 T - 100)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} - 404 T^{6} + \cdots + 66716224 \) Copy content Toggle raw display
$53$ \( (T^{4} + 14 T^{3} - 4 T^{2} - 320 T + 512)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} - 282 T^{6} + 19760 T^{4} + \cdots + 2704 \) Copy content Toggle raw display
$61$ \( T^{8} - 222 T^{6} + 13300 T^{4} + \cdots + 1000000 \) Copy content Toggle raw display
$67$ \( (T^{4} - 20 T^{3} + 78 T^{2} + 252 T - 1207)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} - 4 T^{3} - 105 T^{2} + 444 T - 340)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} - 314 T^{6} + 21717 T^{4} + \cdots + 676 \) Copy content Toggle raw display
$79$ \( (T^{4} - 10 T^{3} - 179 T^{2} + 2552 T - 7724)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} - 178 T^{6} + 10293 T^{4} + \cdots + 532900 \) Copy content Toggle raw display
$89$ \( T^{8} - 442 T^{6} + \cdots + 41860900 \) Copy content Toggle raw display
$97$ \( T^{8} - 386 T^{6} + 27456 T^{4} + \cdots + 38416 \) Copy content Toggle raw display
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