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Properties

Label 6003.2.a.t
Level $6003$
Weight $2$
Character orbit 6003.a
Self dual yes
Analytic conductor $47.934$
Analytic rank $1$
Dimension $22$
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] = [6003,2,Mod(1,6003)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6003, 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("6003.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6003 = 3^{2} \cdot 23 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6003.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: \(47.9341963334\)
Analytic rank: \(1\)
Dimension: \(22\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 22 q - 3 q^{2} + 17 q^{4} - 6 q^{7} - 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 22 q - 3 q^{2} + 17 q^{4} - 6 q^{7} - 6 q^{8} - 12 q^{10} - 28 q^{13} - q^{14} + 3 q^{16} - 10 q^{17} - 8 q^{19} - 11 q^{22} + 22 q^{23} + 11 q^{26} - 21 q^{28} + 22 q^{29} - 18 q^{31} + 5 q^{32} - 33 q^{34} + 2 q^{35} - 28 q^{37} + 14 q^{38} - 30 q^{40} - 10 q^{41} - 14 q^{43} + 37 q^{44} - 3 q^{46} - 18 q^{47} + 2 q^{49} + 7 q^{50} - 57 q^{52} + 20 q^{53} - 42 q^{55} - 2 q^{56} - 3 q^{58} - 20 q^{59} - 38 q^{61} + 4 q^{62} - 24 q^{64} + 12 q^{65} - 50 q^{67} + 11 q^{68} - 48 q^{70} + 12 q^{71} - 46 q^{73} - 6 q^{74} - 16 q^{76} - 14 q^{77} - 20 q^{79} - 58 q^{80} - 42 q^{82} + 22 q^{83} - 66 q^{85} + 22 q^{86} - 68 q^{88} - 14 q^{89} - 16 q^{91} + 17 q^{92} - 27 q^{94} - 20 q^{95} - 48 q^{97} - 28 q^{98}+O(q^{100}) \) 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 −2.68779 0 5.22422 −0.687677 0 −3.46312 −8.66603 0 1.84833
1.2 −2.33095 0 3.43331 2.03702 0 4.28183 −3.34096 0 −4.74817
1.3 −2.24483 0 3.03928 2.70888 0 −1.81542 −2.33301 0 −6.08099
1.4 −2.18238 0 2.76278 0.541552 0 0.936842 −1.66467 0 −1.18187
1.5 −1.96070 0 1.84436 −2.84151 0 −2.21458 0.305158 0 5.57136
1.6 −1.78641 0 1.19125 −0.326188 0 1.17242 1.44475 0 0.582705
1.7 −1.35991 0 −0.150655 4.20064 0 0.870260 2.92469 0 −5.71248
1.8 −1.24737 0 −0.444068 −1.99313 0 −4.26394 3.04866 0 2.48617
1.9 −1.06922 0 −0.856774 0.420836 0 1.13415 3.05451 0 −0.449966
1.10 −0.424349 0 −1.81993 −2.43761 0 3.41293 1.62098 0 1.03440
1.11 −0.230149 0 −1.94703 2.59479 0 −4.77655 0.908406 0 −0.597188
1.12 0.144466 0 −1.97913 −4.08805 0 −0.828594 −0.574849 0 −0.590584
1.13 0.183492 0 −1.96633 0.926834 0 −0.385808 −0.727789 0 0.170066
1.14 0.289055 0 −1.91645 −0.894239 0 3.31166 −1.13207 0 −0.258484
1.15 1.05077 0 −0.895876 3.10634 0 3.07400 −3.04291 0 3.26406
1.16 1.14621 0 −0.686192 0.461250 0 −3.03232 −3.07895 0 0.528691
1.17 1.16689 0 −0.638362 −1.66410 0 −0.970636 −3.07868 0 −1.94183
1.18 1.60427 0 0.573689 −1.49095 0 2.64378 −2.28819 0 −2.39188
1.19 1.93234 0 1.73393 2.82056 0 −3.97268 −0.514145 0 5.45026
1.20 2.08661 0 2.35393 1.43041 0 −0.274571 0.738503 0 2.98470
See all 22 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.22
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(23\) \(-1\)
\(29\) \(-1\)

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 6003.2.a.t ✓ 22
3.b odd 2 1 6003.2.a.u yes 22
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6003.2.a.t ✓ 22 1.a even 1 1 trivial
6003.2.a.u yes 22 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6003))\):

\( T_{2}^{22} + 3 T_{2}^{21} - 26 T_{2}^{20} - 81 T_{2}^{19} + 279 T_{2}^{18} + 914 T_{2}^{17} - 1603 T_{2}^{16} + \cdots - 8 \) Copy content Toggle raw display
\( T_{5}^{22} - 55 T_{5}^{20} - 8 T_{5}^{19} + 1244 T_{5}^{18} + 356 T_{5}^{17} - 15211 T_{5}^{16} + \cdots - 11776 \) Copy content Toggle raw display