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Properties

Label 3004.2.a.d
Level $3004$
Weight $2$
Character orbit 3004.a
Self dual yes
Analytic conductor $23.987$
Analytic rank $1$
Dimension $31$
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] = [3004,2,Mod(1,3004)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3004, 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("3004.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3004 = 2^{2} \cdot 751 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3004.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: \(23.9870607672\)
Analytic rank: \(1\)
Dimension: \(31\)
Twist minimal: yes
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.

\(\operatorname{Tr}(f)(q) = \) \( 31 q - 10 q^{3} - 11 q^{5} - 7 q^{7} + 27 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 31 q - 10 q^{3} - 11 q^{5} - 7 q^{7} + 27 q^{9} - 28 q^{11} - 9 q^{13} - 7 q^{15} + q^{17} - 22 q^{19} - 19 q^{21} - 21 q^{23} + 26 q^{25} - 40 q^{27} - 35 q^{29} - 17 q^{31} + 8 q^{33} - 42 q^{35} - 15 q^{37} - 34 q^{39} - 3 q^{41} - 18 q^{43} - 22 q^{45} - 43 q^{47} + 24 q^{49} - 39 q^{51} - 20 q^{53} - 25 q^{55} + 7 q^{57} - 57 q^{59} - 38 q^{61} - 38 q^{63} - 5 q^{65} - 25 q^{67} - 47 q^{69} - 44 q^{71} + 2 q^{73} - 78 q^{75} - 35 q^{77} - 39 q^{79} + 11 q^{81} - 59 q^{83} - 29 q^{85} - 37 q^{87} + q^{89} - 41 q^{91} - 33 q^{93} - 96 q^{95} - 9 q^{97} - 102 q^{99}+O(q^{100}) \) 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 −3.42547 0 1.38322 0 3.99405 0 8.73386 0
1.2 0 −3.33109 0 −3.76448 0 −2.87698 0 8.09614 0
1.3 0 −2.90055 0 2.48113 0 −1.55164 0 5.41317 0
1.4 0 −2.76650 0 0.569365 0 −2.42886 0 4.65354 0
1.5 0 −2.67147 0 −2.81411 0 4.57741 0 4.13673 0
1.6 0 −2.61395 0 1.23016 0 2.20463 0 3.83271 0
1.7 0 −2.22596 0 2.14418 0 −2.41774 0 1.95489 0
1.8 0 −2.14818 0 4.30117 0 −4.57167 0 1.61470 0
1.9 0 −1.81893 0 −2.10492 0 1.73186 0 0.308506 0
1.10 0 −1.66774 0 −3.24331 0 −1.97623 0 −0.218638 0
1.11 0 −1.48767 0 −4.43484 0 3.85198 0 −0.786851 0
1.12 0 −1.47491 0 −3.07719 0 −4.67179 0 −0.824646 0
1.13 0 −1.08492 0 −0.455027 0 0.781192 0 −1.82296 0
1.14 0 −0.996212 0 −1.08287 0 0.185221 0 −2.00756 0
1.15 0 −0.639532 0 3.50910 0 2.38169 0 −2.59100 0
1.16 0 −0.440958 0 2.82232 0 −1.49147 0 −2.80556 0
1.17 0 −0.242523 0 −0.452744 0 4.09022 0 −2.94118 0
1.18 0 −0.0841173 0 −3.46973 0 0.317957 0 −2.99292 0
1.19 0 0.674692 0 1.14611 0 −1.92800 0 −2.54479 0
1.20 0 0.881010 0 0.631519 0 0.412400 0 −2.22382 0
See all 31 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.31
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(751\) \(-1\)

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 3004.2.a.d ✓ 31
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3004.2.a.d ✓ 31 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{31} + 10 T_{3}^{30} - 10 T_{3}^{29} - 400 T_{3}^{28} - 653 T_{3}^{27} + 6672 T_{3}^{26} + \cdots - 32512 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3004))\). Copy content Toggle raw display