Properties

Label 9196.2.a.m
Level $9196$
Weight $2$
Character orbit 9196.a
Self dual yes
Analytic conductor $73.430$
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] = [9196,2,Mod(1,9196)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9196, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("9196.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9196 = 2^{2} \cdot 11^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9196.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: \(73.4304296988\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.114134848.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 10x^{4} - 6x^{3} + 20x^{2} + 14x - 5 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} + (\beta_{5} - 1) q^{5} + (\beta_{5} + \beta_{4} + \beta_1) q^{7} + (\beta_{2} + \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} + (\beta_{5} - 1) q^{5} + (\beta_{5} + \beta_{4} + \beta_1) q^{7} + (\beta_{2} + \beta_1) q^{9} + ( - \beta_{3} + \beta_{2} + 1) q^{13} + (\beta_{5} - \beta_{3} + \beta_1) q^{15} + ( - \beta_{4} - 2 \beta_{2} - \beta_1 + 2) q^{17} - q^{19} + (\beta_{5} + \beta_{4} - \beta_{3} - \beta_1 - 4) q^{21} + (\beta_{5} - \beta_{2} + \beta_1 + 1) q^{23} + ( - 2 \beta_{5} - 2 \beta_{4} - \beta_{2} - \beta_1 + 2) q^{25} + (\beta_{4} - \beta_{3} - \beta_{2} - 3) q^{27} + (\beta_{5} - 2 \beta_{3} + \beta_{2} + 2 \beta_1 - 1) q^{29} + (\beta_{5} + 2 \beta_{4} - \beta_{3} - \beta_1 - 2) q^{31} + ( - 3 \beta_{5} - \beta_{4} + 2 \beta_{3} - \beta_{2} - 2 \beta_1 + 4) q^{35} + (\beta_{5} + 3 \beta_{4} + \beta_{2} - 1) q^{37} + (\beta_{5} + \beta_{4} + \beta_{2} - 2 \beta_1) q^{39} + (2 \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} - \beta_1 + 1) q^{41} + (2 \beta_{5} - 2 \beta_{4} + \beta_{2} + \beta_1 - 2) q^{43} - \beta_{5} q^{45} + ( - 2 \beta_{3} + 2 \beta_{2}) q^{47} + ( - 2 \beta_{5} + 4 \beta_{3} - 3 \beta_{2} - \beta_1 + 5) q^{49} + ( - 3 \beta_{4} + 2 \beta_{3} + 3 \beta_1 + 4) q^{51} + (2 \beta_{5} + 2 \beta_{4} - 2 \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 2) q^{53} + \beta_1 q^{57} + (\beta_{5} + \beta_{4} - \beta_{2} + 3) q^{59} + (4 \beta_{5} + \beta_{4} - 2 \beta_{3} - \beta_1 - 2) q^{61} + ( - \beta_{5} - 2 \beta_{4} + 3 \beta_{2} + 3 \beta_1 + 2) q^{63} + (\beta_{5} - \beta_{4} + \beta_{3} - \beta_1 - 2) q^{65} + (3 \beta_{5} + 2 \beta_{4} - \beta_{3} + \beta_1) q^{67} + (\beta_{5} - \beta_{4} - \beta_{2} - 3) q^{69} + ( - 2 \beta_{4} - 2 \beta_{3} + 2 \beta_{2} + \beta_1) q^{71} + ( - 2 \beta_{5} + 2 \beta_1 + 6) q^{73} + ( - 2 \beta_{5} - 3 \beta_{4} + 3 \beta_{3} - \beta_{2} + \beta_1 + 5) q^{75} + (\beta_{4} - 2 \beta_{3} + 2 \beta_{2} - 3 \beta_1) q^{79} + (\beta_{5} + 2 \beta_{3} - \beta_{2} + 3 \beta_1 - 1) q^{81} + (\beta_{4} + 2 \beta_{3} - \beta_{2} - 6 \beta_1 + 6) q^{83} + (4 \beta_{5} - \beta_{4} - \beta_1) q^{85} + (3 \beta_{5} + \beta_{4} - \beta_1 - 6) q^{87} + (\beta_{5} + \beta_{4} + 2 \beta_{3} + \beta_{2} + 2 \beta_1 - 5) q^{89} + ( - \beta_{5} - \beta_{4} - 2 \beta_{3} + 3 \beta_{2} + \beta_1 - 3) q^{91} + (2 \beta_{5} + 2 \beta_{4} + 4 \beta_{2} + 4 \beta_1 + 1) q^{93} + ( - \beta_{5} + 1) q^{95} + ( - \beta_{5} - 3 \beta_{4} + 2 \beta_{3} - 3 \beta_{2} + 7) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 4 q^{5} + 4 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 4 q^{5} + 4 q^{7} + 2 q^{9} + 8 q^{13} + 2 q^{15} + 6 q^{17} - 6 q^{19} - 20 q^{21} + 6 q^{23} + 2 q^{25} - 18 q^{27} - 2 q^{29} - 6 q^{31} + 14 q^{35} + 4 q^{37} + 6 q^{39} + 14 q^{41} - 10 q^{43} - 2 q^{45} + 4 q^{47} + 20 q^{49} + 18 q^{51} + 20 q^{59} - 2 q^{61} + 12 q^{63} - 12 q^{65} + 10 q^{67} - 20 q^{69} + 32 q^{73} + 18 q^{75} + 6 q^{79} - 6 q^{81} + 36 q^{83} + 6 q^{85} - 28 q^{87} - 24 q^{89} - 16 q^{91} + 22 q^{93} + 4 q^{95} + 28 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 10x^{4} - 6x^{3} + 20x^{2} + 14x - 5 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{5} - \nu^{4} - 8\nu^{3} + 2\nu^{2} + 11\nu - 1 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{5} - \nu^{4} - 9\nu^{3} + 3\nu^{2} + 16\nu - 1 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -2\nu^{5} + 3\nu^{4} + 16\nu^{3} - 12\nu^{2} - 26\nu + 9 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{4} + \beta_{3} + \beta_{2} + 6\beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{5} + 2\beta_{3} + 8\beta_{2} + 12\beta _1 + 17 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{5} - 8\beta_{4} + 11\beta_{3} + 14\beta_{2} + 47\beta _1 + 36 \) 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
3.06597
1.52628
0.267039
−1.28435
−1.50521
−2.06973
0 −3.06597 0 −0.135132 0 3.35676 0 6.40017 0
1.2 0 −1.52628 0 −3.03462 0 0.757059 0 −0.670469 0
1.3 0 −0.267039 0 0.518487 0 5.09697 0 −2.92869 0
1.4 0 1.28435 0 2.85345 0 −1.18021 0 −1.35045 0
1.5 0 1.50521 0 −3.76432 0 −4.72308 0 −0.734348 0
1.6 0 2.06973 0 −0.437868 0 0.692506 0 1.28379 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
\(2\) \(-1\)
\(11\) \(1\)
\(19\) \(1\)

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 9196.2.a.m yes 6
11.b odd 2 1 9196.2.a.l 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9196.2.a.l 6 11.b odd 2 1
9196.2.a.m yes 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9196))\):

\( T_{3}^{6} - 10T_{3}^{4} + 6T_{3}^{3} + 20T_{3}^{2} - 14T_{3} - 5 \) Copy content Toggle raw display
\( T_{5}^{6} + 4T_{5}^{5} - 8T_{5}^{4} - 34T_{5}^{3} + 8T_{5} + 1 \) Copy content Toggle raw display
\( T_{7}^{6} - 4T_{7}^{5} - 23T_{7}^{4} + 92T_{7}^{3} + 3T_{7}^{2} - 110T_{7} + 50 \) Copy content Toggle raw display
\( T_{13}^{6} - 8T_{13}^{5} + T_{13}^{4} + 122T_{13}^{3} - 303T_{13}^{2} + 230T_{13} - 50 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} - 10 T^{4} + 6 T^{3} + 20 T^{2} + \cdots - 5 \) Copy content Toggle raw display
$5$ \( T^{6} + 4 T^{5} - 8 T^{4} - 34 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$7$ \( T^{6} - 4 T^{5} - 23 T^{4} + 92 T^{3} + \cdots + 50 \) Copy content Toggle raw display
$11$ \( T^{6} \) Copy content Toggle raw display
$13$ \( T^{6} - 8 T^{5} + T^{4} + 122 T^{3} + \cdots - 50 \) Copy content Toggle raw display
$17$ \( T^{6} - 6 T^{5} - 66 T^{4} + \cdots + 10400 \) Copy content Toggle raw display
$19$ \( (T + 1)^{6} \) Copy content Toggle raw display
$23$ \( T^{6} - 6 T^{5} - 27 T^{4} + \cdots + 1456 \) Copy content Toggle raw display
$29$ \( T^{6} + 2 T^{5} - 93 T^{4} - 76 T^{3} + \cdots - 50 \) Copy content Toggle raw display
$31$ \( T^{6} + 6 T^{5} - 68 T^{4} + \cdots - 7561 \) Copy content Toggle raw display
$37$ \( T^{6} - 4 T^{5} - 111 T^{4} + \cdots - 5392 \) Copy content Toggle raw display
$41$ \( T^{6} - 14 T^{5} - 49 T^{4} + \cdots + 22400 \) Copy content Toggle raw display
$43$ \( T^{6} + 10 T^{5} - 161 T^{4} + \cdots + 200000 \) Copy content Toggle raw display
$47$ \( T^{6} - 4 T^{5} - 96 T^{4} + 528 T^{3} + \cdots - 448 \) Copy content Toggle raw display
$53$ \( T^{6} - 136 T^{4} + 480 T^{3} + \cdots - 10240 \) Copy content Toggle raw display
$59$ \( T^{6} - 20 T^{5} + 121 T^{4} + \cdots + 16 \) Copy content Toggle raw display
$61$ \( T^{6} + 2 T^{5} - 258 T^{4} + \cdots - 4000 \) Copy content Toggle raw display
$67$ \( T^{6} - 10 T^{5} - 92 T^{4} + \cdots - 6349 \) Copy content Toggle raw display
$71$ \( T^{6} - 194 T^{4} + 1278 T^{3} + \cdots + 6871 \) Copy content Toggle raw display
$73$ \( T^{6} - 32 T^{5} + 320 T^{4} + \cdots - 12800 \) Copy content Toggle raw display
$79$ \( T^{6} - 6 T^{5} - 206 T^{4} + \cdots + 20000 \) Copy content Toggle raw display
$83$ \( T^{6} - 36 T^{5} + 149 T^{4} + \cdots + 2849050 \) Copy content Toggle raw display
$89$ \( T^{6} + 24 T^{5} + 17 T^{4} + \cdots + 57328 \) Copy content Toggle raw display
$97$ \( T^{6} - 28 T^{5} + 129 T^{4} + \cdots - 111808 \) Copy content Toggle raw display
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