Properties

Label 3192.2.a.bb
Level $3192$
Weight $2$
Character orbit 3192.a
Self dual yes
Analytic conductor $25.488$
Analytic rank $0$
Dimension $5$
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] = [3192,2,Mod(1,3192)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3192, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("3192.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3192 = 2^{3} \cdot 3 \cdot 7 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3192.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: \(25.4882483252\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.401584.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 2x^{4} - 7x^{3} + 8x^{2} + 12x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{3} \)
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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{3} - \beta_1 q^{5} + q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{3} - \beta_1 q^{5} + q^{7} + q^{9} + (\beta_{3} - \beta_{2} + 1) q^{11} + ( - \beta_{4} + \beta_{3}) q^{13} - \beta_1 q^{15} + ( - \beta_{3} + 1) q^{17} + q^{19} + q^{21} + (\beta_{2} - \beta_1 + 2) q^{23} + (\beta_{4} + \beta_{3} - \beta_{2} - \beta_1 + 1) q^{25} + q^{27} + (\beta_{2} + 2 \beta_1 + 2) q^{29} + ( - \beta_{4} - 2 \beta_{3} + \beta_{2} - 1) q^{31} + (\beta_{3} - \beta_{2} + 1) q^{33} - \beta_1 q^{35} + (2 \beta_{4} - \beta_{3} - \beta_1 + 1) q^{37} + ( - \beta_{4} + \beta_{3}) q^{39} + ( - 2 \beta_{2} + 2 \beta_1 + 2) q^{41} + (\beta_{4} - \beta_{3} - \beta_{2} - \beta_1) q^{43} - \beta_1 q^{45} + ( - 2 \beta_{4} + \beta_{3} + \beta_{2} + \beta_1 + 1) q^{47} + q^{49} + ( - \beta_{3} + 1) q^{51} + (\beta_{2} + 2 \beta_1 + 2) q^{53} + (2 \beta_{4} - 2 \beta_{3} - 4 \beta_1) q^{55} + q^{57} + (2 \beta_{2} + 2 \beta_1 + 4) q^{59} - 2 q^{61} + q^{63} + ( - 2 \beta_{3} + 2 \beta_{2} + 2) q^{65} + (\beta_{3} + \beta_{2} + 1) q^{67} + (\beta_{2} - \beta_1 + 2) q^{69} + (2 \beta_{4} - \beta_{2} + 6) q^{71} + (2 \beta_1 + 2) q^{73} + (\beta_{4} + \beta_{3} - \beta_{2} - \beta_1 + 1) q^{75} + (\beta_{3} - \beta_{2} + 1) q^{77} + (\beta_{2} - \beta_1 + 2) q^{79} + q^{81} + ( - 2 \beta_{3} - \beta_{2} + 2) q^{83} + ( - \beta_{4} + \beta_{3} - \beta_{2} + \beta_1) q^{85} + (\beta_{2} + 2 \beta_1 + 2) q^{87} + (2 \beta_{4} - 2 \beta_{3} + 2 \beta_{2} - 2 \beta_1 + 2) q^{89} + ( - \beta_{4} + \beta_{3}) q^{91} + ( - \beta_{4} - 2 \beta_{3} + \beta_{2} - 1) q^{93} - \beta_1 q^{95} + ( - 2 \beta_{4} - \beta_{3} + \beta_{2} + 2 \beta_1 - 1) q^{97} + (\beta_{3} - \beta_{2} + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 5 q^{3} + 2 q^{5} + 5 q^{7} + 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q + 5 q^{3} + 2 q^{5} + 5 q^{7} + 5 q^{9} + 6 q^{11} + 2 q^{15} + 6 q^{17} + 5 q^{19} + 5 q^{21} + 10 q^{23} + 7 q^{25} + 5 q^{27} + 4 q^{29} - 4 q^{31} + 6 q^{33} + 2 q^{35} + 6 q^{37} + 10 q^{41} + 4 q^{43} + 2 q^{45} + 2 q^{47} + 5 q^{49} + 6 q^{51} + 4 q^{53} + 8 q^{55} + 5 q^{57} + 12 q^{59} - 10 q^{61} + 5 q^{63} + 8 q^{65} + 2 q^{67} + 10 q^{69} + 30 q^{71} + 6 q^{73} + 7 q^{75} + 6 q^{77} + 10 q^{79} + 5 q^{81} + 14 q^{83} + 4 q^{87} + 10 q^{89} - 4 q^{93} + 2 q^{95} - 8 q^{97} + 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{5} - 2x^{4} - 7x^{3} + 8x^{2} + 12x - 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{4} - 3\nu^{3} - 2\nu^{2} + 8\nu - 4 ) / 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{4} + \nu^{3} + 8\nu^{2} - 12 ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{4} + \nu^{3} + 8\nu^{2} - 4\nu - 10 ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{4} + 3\nu^{3} + 6\nu^{2} - 12\nu - 10 ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{3} + \beta_{2} + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{4} - \beta_{3} + \beta_{2} + \beta _1 + 8 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 3\beta_{4} - 7\beta_{3} + 5\beta_{2} + \beta _1 + 12 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 11\beta_{4} - 15\beta_{3} + 9\beta_{2} + 9\beta _1 + 52 ) / 2 \) 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
−1.86937
3.06003
0.292040
1.82751
−1.31020
0 1.00000 0 −2.93275 0 1.00000 0 1.00000 0
1.2 0 1.00000 0 −1.73631 0 1.00000 0 1.00000 0
1.3 0 1.00000 0 0.950852 0 1.00000 0 1.00000 0
1.4 0 1.00000 0 1.60789 0 1.00000 0 1.00000 0
1.5 0 1.00000 0 4.11032 0 1.00000 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)
\(7\) \(-1\)
\(19\) \(-1\)

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 3192.2.a.bb 5
3.b odd 2 1 9576.2.a.cn 5
4.b odd 2 1 6384.2.a.cd 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3192.2.a.bb 5 1.a even 1 1 trivial
6384.2.a.cd 5 4.b odd 2 1
9576.2.a.cn 5 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3192))\):

\( T_{5}^{5} - 2T_{5}^{4} - 14T_{5}^{3} + 16T_{5}^{2} + 32T_{5} - 32 \) Copy content Toggle raw display
\( T_{11}^{5} - 6T_{11}^{4} - 20T_{11}^{3} + 120T_{11}^{2} + 64T_{11} - 256 \) Copy content Toggle raw display
\( T_{17}^{5} - 6T_{17}^{4} - 14T_{17}^{3} + 56T_{17}^{2} + 128T_{17} + 64 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{5} \) Copy content Toggle raw display
$3$ \( (T - 1)^{5} \) Copy content Toggle raw display
$5$ \( T^{5} - 2 T^{4} - 14 T^{3} + 16 T^{2} + \cdots - 32 \) Copy content Toggle raw display
$7$ \( (T - 1)^{5} \) Copy content Toggle raw display
$11$ \( T^{5} - 6 T^{4} - 20 T^{3} + 120 T^{2} + \cdots - 256 \) Copy content Toggle raw display
$13$ \( T^{5} - 40 T^{3} + 80 T^{2} + \cdots - 128 \) Copy content Toggle raw display
$17$ \( T^{5} - 6 T^{4} - 14 T^{3} + 56 T^{2} + \cdots + 64 \) Copy content Toggle raw display
$19$ \( (T - 1)^{5} \) Copy content Toggle raw display
$23$ \( T^{5} - 10 T^{4} - 4 T^{3} + 152 T^{2} + \cdots - 512 \) Copy content Toggle raw display
$29$ \( T^{5} - 4 T^{4} - 94 T^{3} + \cdots - 7664 \) Copy content Toggle raw display
$31$ \( T^{5} + 4 T^{4} - 148 T^{3} + \cdots - 6208 \) Copy content Toggle raw display
$37$ \( T^{5} - 6 T^{4} - 112 T^{3} + \cdots + 8864 \) Copy content Toggle raw display
$41$ \( T^{5} - 10 T^{4} - 136 T^{3} + \cdots - 5792 \) Copy content Toggle raw display
$43$ \( T^{5} - 4 T^{4} - 76 T^{3} + \cdots - 4096 \) Copy content Toggle raw display
$47$ \( T^{5} - 2 T^{4} - 134 T^{3} + \cdots - 5792 \) Copy content Toggle raw display
$53$ \( T^{5} - 4 T^{4} - 94 T^{3} + \cdots - 7664 \) Copy content Toggle raw display
$59$ \( T^{5} - 12 T^{4} - 144 T^{3} + \cdots - 59392 \) Copy content Toggle raw display
$61$ \( (T + 2)^{5} \) Copy content Toggle raw display
$67$ \( T^{5} - 2 T^{4} - 84 T^{3} + 312 T^{2} + \cdots - 64 \) Copy content Toggle raw display
$71$ \( T^{5} - 30 T^{4} + 234 T^{3} + \cdots - 6784 \) Copy content Toggle raw display
$73$ \( T^{5} - 6 T^{4} - 48 T^{3} + 224 T^{2} + \cdots - 32 \) Copy content Toggle raw display
$79$ \( T^{5} - 10 T^{4} - 4 T^{3} + 152 T^{2} + \cdots - 512 \) Copy content Toggle raw display
$83$ \( T^{5} - 14 T^{4} - 118 T^{3} + \cdots + 128 \) Copy content Toggle raw display
$89$ \( T^{5} - 10 T^{4} - 264 T^{3} + \cdots - 103072 \) Copy content Toggle raw display
$97$ \( T^{5} + 8 T^{4} - 204 T^{3} + \cdots + 128 \) Copy content Toggle raw display
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