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Properties

Label 3004.2.a.c
Level $3004$
Weight $2$
Character orbit 3004.a
Self dual yes
Analytic conductor $23.987$
Analytic rank $0$
Dimension $28$
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: \(0\)
Dimension: \(28\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 28 q + 8 q^{3} + 11 q^{5} + 5 q^{7} + 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 28 q + 8 q^{3} + 11 q^{5} + 5 q^{7} + 24 q^{9} + 26 q^{11} + 10 q^{13} + 13 q^{15} - q^{17} + 19 q^{19} + 9 q^{21} + 12 q^{23} + 37 q^{25} + 20 q^{27} + 29 q^{29} + 39 q^{31} - 4 q^{33} + 46 q^{35} - 8 q^{37} + 26 q^{39} - 3 q^{41} + 22 q^{43} + 40 q^{45} + 64 q^{47} + 17 q^{49} + 25 q^{51} + 27 q^{53} + 35 q^{55} + 3 q^{57} + 56 q^{59} + q^{61} + 50 q^{63} - 15 q^{65} - 11 q^{67} - 3 q^{69} + 56 q^{71} - 10 q^{73} + 56 q^{75} + 33 q^{77} + 55 q^{79} + 20 q^{81} + 53 q^{83} + 15 q^{85} + 47 q^{87} + 36 q^{89} + 3 q^{91} - q^{93} + 66 q^{95} - 12 q^{97} + 48 q^{99}+O(q^{100}) \) 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 −3.17861 0 −0.554963 0 −1.64012 0 7.10356 0
1.2 0 −2.66100 0 −1.62505 0 1.70557 0 4.08091 0
1.3 0 −2.56422 0 3.89286 0 3.12773 0 3.57520 0
1.4 0 −2.19828 0 −2.81009 0 −1.08623 0 1.83245 0
1.5 0 −2.16936 0 1.12512 0 2.77208 0 1.70614 0
1.6 0 −1.94943 0 1.25001 0 1.26185 0 0.800288 0
1.7 0 −1.51333 0 3.43606 0 1.34965 0 −0.709840 0
1.8 0 −1.21710 0 0.757884 0 −4.90921 0 −1.51868 0
1.9 0 −1.13374 0 −0.834242 0 −0.0956714 0 −1.71464 0
1.10 0 −0.626087 0 2.31815 0 −1.93212 0 −2.60802 0
1.11 0 −0.373619 0 1.41879 0 −3.33964 0 −2.86041 0
1.12 0 −0.245237 0 −3.51519 0 2.47739 0 −2.93986 0
1.13 0 −0.0828009 0 −1.39146 0 −4.60230 0 −2.99314 0
1.14 0 0.637291 0 −3.49521 0 0.392765 0 −2.59386 0
1.15 0 0.650269 0 −3.35400 0 −1.81063 0 −2.57715 0
1.16 0 0.791720 0 3.76512 0 1.84480 0 −2.37318 0
1.17 0 0.858367 0 −1.52565 0 2.23749 0 −2.26321 0
1.18 0 0.920424 0 2.10757 0 1.84585 0 −2.15282 0
1.19 0 0.945276 0 2.97186 0 4.62925 0 −2.10645 0
1.20 0 1.87389 0 3.04511 0 −3.43522 0 0.511472 0
See all 28 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.28
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.c ✓ 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3004.2.a.c ✓ 28 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{28} - 8 T_{3}^{27} - 22 T_{3}^{26} + 324 T_{3}^{25} - 103 T_{3}^{24} - 5482 T_{3}^{23} + \cdots - 2048 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3004))\). Copy content Toggle raw display