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Properties

Label 9904.2.a.n
Level $9904$
Weight $2$
Character orbit 9904.a
Self dual yes
Analytic conductor $79.084$
Analytic rank $0$
Dimension $30$
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] = [9904,2,Mod(1,9904)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9904, 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("9904.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9904 = 2^{4} \cdot 619 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9904.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: \(79.0838381619\)
Analytic rank: \(0\)
Dimension: \(30\)
Twist minimal: no (minimal twist has level 619)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 30 q - q^{3} + 21 q^{5} - 2 q^{7} + 43 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 30 q - q^{3} + 21 q^{5} - 2 q^{7} + 43 q^{9} - 23 q^{11} + 9 q^{13} + 2 q^{15} + 4 q^{17} + q^{19} + 30 q^{21} - 4 q^{23} + 35 q^{25} + 5 q^{27} + 90 q^{29} - 2 q^{31} - 6 q^{33} - 9 q^{35} + 19 q^{37} - 32 q^{39} + 59 q^{41} + 4 q^{43} + 30 q^{45} - 4 q^{47} + 30 q^{49} + 34 q^{53} + 17 q^{55} - 8 q^{57} - 13 q^{59} + 16 q^{61} + 40 q^{63} + 31 q^{65} + 11 q^{67} + 6 q^{69} - 42 q^{71} - 4 q^{73} + 52 q^{75} + 29 q^{77} - 3 q^{79} + 30 q^{81} + 11 q^{83} + 19 q^{85} + 20 q^{87} + 58 q^{89} + 39 q^{91} - 15 q^{93} - 23 q^{95} - 9 q^{97} + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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 0 −3.28637 0 2.23550 0 −3.15236 0 7.80023 0
1.2 0 −3.16029 0 0.563031 0 3.36630 0 6.98741 0
1.3 0 −2.74125 0 −1.58763 0 −2.21007 0 4.51445 0
1.4 0 −2.67579 0 −2.46163 0 −1.88069 0 4.15985 0
1.5 0 −2.41034 0 −2.25507 0 1.27109 0 2.80973 0
1.6 0 −2.38151 0 2.26626 0 −3.26853 0 2.67158 0
1.7 0 −2.37814 0 1.23255 0 −1.58895 0 2.65553 0
1.8 0 −2.10519 0 3.90881 0 1.82243 0 1.43181 0
1.9 0 −2.04590 0 3.68832 0 2.71706 0 1.18571 0
1.10 0 −1.69440 0 −2.73112 0 1.24727 0 −0.129016 0
1.11 0 −1.27337 0 1.92856 0 −3.31459 0 −1.37852 0
1.12 0 −1.15474 0 0.568936 0 1.24185 0 −1.66657 0
1.13 0 −0.501992 0 2.53565 0 0.903819 0 −2.74800 0
1.14 0 −0.456554 0 0.324059 0 −2.73608 0 −2.79156 0
1.15 0 −0.252107 0 3.30541 0 1.70947 0 −2.93644 0
1.16 0 0.157868 0 2.37765 0 −4.70879 0 −2.97508 0
1.17 0 0.518455 0 −1.50291 0 4.51257 0 −2.73120 0
1.18 0 0.725873 0 −0.225133 0 −2.47282 0 −2.47311 0
1.19 0 0.800074 0 −2.40890 0 −1.00913 0 −2.35988 0
1.20 0 1.01788 0 4.11968 0 −2.70884 0 −1.96391 0
See all 30 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.30
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(619\) \(1\)

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 9904.2.a.n 30
4.b odd 2 1 619.2.a.b ✓ 30
12.b even 2 1 5571.2.a.g 30
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
619.2.a.b ✓ 30 4.b odd 2 1
5571.2.a.g 30 12.b even 2 1
9904.2.a.n 30 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9904))\):

\( T_{3}^{30} + T_{3}^{29} - 66 T_{3}^{28} - 66 T_{3}^{27} + 1939 T_{3}^{26} + 1923 T_{3}^{25} + \cdots - 155648 \) Copy content Toggle raw display
\( T_{5}^{30} - 21 T_{5}^{29} + 128 T_{5}^{28} + 304 T_{5}^{27} - 6504 T_{5}^{26} + 16067 T_{5}^{25} + \cdots + 2505487 \) Copy content Toggle raw display
\( T_{11}^{30} + 23 T_{11}^{29} + 87 T_{11}^{28} - 1931 T_{11}^{27} - 18567 T_{11}^{26} + \cdots + 464774682112 \) Copy content Toggle raw display