↑

Properties

Label 3447.2.a.j
Level $3447$
Weight $2$
Character orbit 3447.a
Self dual yes
Analytic conductor $27.524$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3447,2,Mod(1,3447)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3447, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("3447.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3447 = 3^{2} \cdot 383 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3447.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: \(27.5244335767\)
Analytic rank: \(0\)
Dimension: \(24\)
Twist minimal: no (minimal twist has level 383)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 5 q^{2} + 29 q^{4} - 3 q^{5} + 17 q^{7} - 15 q^{8} + q^{10} + 28 q^{13} + 8 q^{14} + 35 q^{16} - 16 q^{17} + 13 q^{19} + 4 q^{20} + 12 q^{22} - 7 q^{23} + 67 q^{25} + 14 q^{26} + 39 q^{28} + 2 q^{29}+ \cdots + 29 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1 −2.75590 0 5.59496 1.94817 0 4.21457 −9.90734 0 −5.36895
1.2 −2.58110 0 4.66207 4.25360 0 −1.55327 −6.87105 0 −10.9790
1.3 −2.56319 0 4.56992 −0.275873 0 −0.724431 −6.58718 0 0.707113
1.4 −2.44206 0 3.96367 −4.41754 0 −1.22662 −4.79540 0 10.7879
1.5 −2.31031 0 3.33755 0.0217650 0 2.45434 −3.09016 0 −0.0502841
1.6 −1.99461 0 1.97847 −1.38116 0 −4.89803 0.0429378 0 2.75487
1.7 −1.81699 0 1.30147 0.569837 0 3.69709 1.26923 0 −1.03539
1.8 −1.55772 0 0.426486 −2.21966 0 −0.910561 2.45109 0 3.45761
1.9 −1.30644 0 −0.293212 0.910566 0 2.08417 2.99595 0 −1.18960
1.10 −1.07548 0 −0.843347 −4.23654 0 2.82145 3.05796 0 4.55631
1.11 −0.539970 0 −1.70843 −2.73147 0 1.35462 2.00244 0 1.47491
1.12 −0.378030 0 −1.85709 3.79049 0 4.13017 1.45810 0 −1.43292
1.13 −0.119680 0 −1.98568 −0.761787 0 1.36599 0.477005 0 0.0911706
1.14 0.0582123 0 −1.99661 3.28367 0 −2.97653 −0.232652 0 0.191150
1.15 0.473286 0 −1.77600 −0.826944 0 −1.86880 −1.78713 0 −0.391381
1.16 0.806619 0 −1.34937 3.72243 0 4.29879 −2.70166 0 3.00258
1.17 1.04512 0 −0.907726 −3.33279 0 −3.42998 −3.03892 0 −3.48316
1.18 1.33913 0 −0.206737 3.01439 0 −4.58337 −2.95510 0 4.03666
1.19 1.50563 0 0.266907 −1.94089 0 4.87997 −2.60939 0 −2.92225
1.20 1.56131 0 0.437676 −3.86534 0 −1.33224 −2.43927 0 −6.03498
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.24
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \( -1 \)
\(383\) \( +1 \)

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 3447.2.a.j 24
3.b odd 2 1 383.2.a.c ✓ 24
12.b even 2 1 6128.2.a.p 24
15.d odd 2 1 9575.2.a.e 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
383.2.a.c ✓ 24 3.b odd 2 1
3447.2.a.j 24 1.a even 1 1 trivial
6128.2.a.p 24 12.b even 2 1
9575.2.a.e 24 15.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3447))\):

\( T_{2}^{24} + 5 T_{2}^{23} - 26 T_{2}^{22} - 160 T_{2}^{21} + 244 T_{2}^{20} + 2173 T_{2}^{19} + \cdots - 49 \) Copy content Toggle raw display
\( T_{5}^{24} + 3 T_{5}^{23} - 89 T_{5}^{22} - 259 T_{5}^{21} + 3375 T_{5}^{20} + 9557 T_{5}^{19} + \cdots - 65536 \) Copy content Toggle raw display