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Properties

Label 3808.2.a.o
Level $3808$
Weight $2$
Character orbit 3808.a
Self dual yes
Analytic conductor $30.407$
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] = [3808,2,Mod(1,3808)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3808, 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("3808.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3808 = 2^{5} \cdot 7 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3808.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: \(30.4070330897\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.147697840.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} - 10x^{4} + 10x^{3} + 18x^{2} - 16x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 1) q^{3} + ( - \beta_{5} - 1) q^{5} - q^{7} + (\beta_{5} + \beta_{4} - \beta_1 + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 1) q^{3} + ( - \beta_{5} - 1) q^{5} - q^{7} + (\beta_{5} + \beta_{4} - \beta_1 + 2) q^{9} + ( - \beta_{3} + \beta_{2} + \beta_1 + 1) q^{11} + (\beta_{2} + \beta_1 - 1) q^{13} + ( - \beta_{5} + \beta_{2} + 2 \beta_1 - 1) q^{15} + q^{17} + (\beta_{4} + \beta_{3} - \beta_{2} + 3) q^{19} + (\beta_1 - 1) q^{21} + (\beta_{4} - \beta_{3} - \beta_{2} + 1) q^{23} + ( - \beta_{4} - 2 \beta_{3} + 2) q^{25} + (\beta_{5} + \beta_{4} + \beta_{3} + \cdots + 3) q^{27}+ \cdots + (2 \beta_{4} + 3 \beta_{3} + 2 \beta_{2} + \cdots + 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 4 q^{3} - 4 q^{5} - 6 q^{7} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 4 q^{3} - 4 q^{5} - 6 q^{7} + 8 q^{9} + 8 q^{11} - 4 q^{13} + 6 q^{17} + 18 q^{19} - 4 q^{21} + 6 q^{23} + 12 q^{25} + 10 q^{27} + 4 q^{29} - 8 q^{31} - 6 q^{33} + 4 q^{35} - 8 q^{37} - 18 q^{39} - 4 q^{41} + 20 q^{43} - 24 q^{45} + 6 q^{47} + 6 q^{49} + 4 q^{51} + 14 q^{53} + 4 q^{55} - 4 q^{57} + 20 q^{59} + 2 q^{61} - 8 q^{63} + 6 q^{65} + 28 q^{67} - 4 q^{69} + 18 q^{71} + 4 q^{73} + 20 q^{75} - 8 q^{77} - 2 q^{79} + 34 q^{81} + 20 q^{83} - 4 q^{85} + 22 q^{87} - 6 q^{89} + 4 q^{91} + 40 q^{93} - 6 q^{95} + 30 q^{97} + 32 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 2x^{5} - 10x^{4} + 10x^{3} + 18x^{2} - 16x + 3 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{5} - 2\nu^{4} - 9\nu^{3} + 7\nu^{2} + 13\nu - 6 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{5} - 2\nu^{4} - 10\nu^{3} + 9\nu^{2} + 20\nu - 10 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -2\nu^{5} + 3\nu^{4} + 22\nu^{3} - 10\nu^{2} - 44\nu + 14 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( 2\nu^{5} - 3\nu^{4} - 22\nu^{3} + 11\nu^{2} + 43\nu - 18 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} + \beta_{4} + \beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{5} + 2\beta_{4} - \beta_{3} + \beta_{2} + 9\beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 12\beta_{5} + 11\beta_{4} - 4\beta_{3} + 2\beta_{2} + 22\beta _1 + 34 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 35\beta_{5} + 33\beta_{4} - 17\beta_{3} + 14\beta_{2} + 105\beta _1 + 82 \) 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
3.69817
1.46371
0.478948
0.301346
−1.75839
−2.18379
0 −2.69817 0 −2.06665 0 −1.00000 0 4.28014 0
1.2 0 −0.463711 0 −0.183002 0 −1.00000 0 −2.78497 0
1.3 0 0.521052 0 −3.59354 0 −1.00000 0 −2.72850 0
1.4 0 0.698654 0 3.66503 0 −1.00000 0 −2.51188 0
1.5 0 2.75839 0 1.29021 0 −1.00000 0 4.60870 0
1.6 0 3.18379 0 −3.11205 0 −1.00000 0 7.13652 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
\(2\) \( +1 \)
\(7\) \( +1 \)
\(17\) \( -1 \)

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 3808.2.a.o yes 6
4.b odd 2 1 3808.2.a.g ✓ 6
8.b even 2 1 7616.2.a.bv 6
8.d odd 2 1 7616.2.a.cd 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3808.2.a.g ✓ 6 4.b odd 2 1
3808.2.a.o yes 6 1.a even 1 1 trivial
7616.2.a.bv 6 8.b even 2 1
7616.2.a.cd 6 8.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3808))\):

\( T_{3}^{6} - 4T_{3}^{5} - 5T_{3}^{4} + 30T_{3}^{3} - 17T_{3}^{2} - 6T_{3} + 4 \) Copy content Toggle raw display
\( T_{11}^{6} - 8T_{11}^{5} - 14T_{11}^{4} + 220T_{11}^{3} - 308T_{11}^{2} - 664T_{11} + 1168 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} - 4 T^{5} + \cdots + 4 \) Copy content Toggle raw display
$5$ \( T^{6} + 4 T^{5} + \cdots + 20 \) Copy content Toggle raw display
$7$ \( (T + 1)^{6} \) Copy content Toggle raw display
$11$ \( T^{6} - 8 T^{5} + \cdots + 1168 \) Copy content Toggle raw display
$13$ \( T^{6} + 4 T^{5} + \cdots - 64 \) Copy content Toggle raw display
$17$ \( (T - 1)^{6} \) Copy content Toggle raw display
$19$ \( T^{6} - 18 T^{5} + \cdots - 2416 \) Copy content Toggle raw display
$23$ \( T^{6} - 6 T^{5} + \cdots - 18688 \) Copy content Toggle raw display
$29$ \( T^{6} - 4 T^{5} + \cdots + 592 \) Copy content Toggle raw display
$31$ \( T^{6} + 8 T^{5} + \cdots + 80 \) Copy content Toggle raw display
$37$ \( T^{6} + 8 T^{5} + \cdots + 21200 \) Copy content Toggle raw display
$41$ \( T^{6} + 4 T^{5} + \cdots - 20 \) Copy content Toggle raw display
$43$ \( T^{6} - 20 T^{5} + \cdots - 16 \) Copy content Toggle raw display
$47$ \( T^{6} - 6 T^{5} + \cdots + 4800 \) Copy content Toggle raw display
$53$ \( T^{6} - 14 T^{5} + \cdots - 14004 \) Copy content Toggle raw display
$59$ \( T^{6} - 20 T^{5} + \cdots + 67840 \) Copy content Toggle raw display
$61$ \( T^{6} - 2 T^{5} + \cdots - 2924 \) Copy content Toggle raw display
$67$ \( T^{6} - 28 T^{5} + \cdots - 2672 \) Copy content Toggle raw display
$71$ \( T^{6} - 18 T^{5} + \cdots + 68800 \) Copy content Toggle raw display
$73$ \( T^{6} - 4 T^{5} + \cdots - 708 \) Copy content Toggle raw display
$79$ \( T^{6} + 2 T^{5} + \cdots + 3264 \) Copy content Toggle raw display
$83$ \( T^{6} - 20 T^{5} + \cdots + 85648 \) Copy content Toggle raw display
$89$ \( T^{6} + 6 T^{5} + \cdots - 432 \) Copy content Toggle raw display
$97$ \( T^{6} - 30 T^{5} + \cdots + 586604 \) Copy content Toggle raw display
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