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Properties

Label 6003.2.a.w
Level $6003$
Weight $2$
Character orbit 6003.a
Self dual yes
Analytic conductor $47.934$
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] = [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: \(0\)
Dimension: \(30\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 30 q + q^{2} + 37 q^{4} + 10 q^{7} + 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 30 q + q^{2} + 37 q^{4} + 10 q^{7} + 6 q^{8} + 8 q^{10} + 36 q^{13} + 7 q^{14} + 47 q^{16} + 18 q^{17} + 16 q^{19} + 25 q^{22} - 30 q^{23} + 56 q^{25} + 11 q^{26} + 27 q^{28} + 30 q^{29} + 14 q^{31} - 7 q^{32} + 3 q^{34} - 22 q^{35} + 40 q^{37} + 6 q^{38} + 30 q^{40} + 14 q^{41} + 34 q^{43} + 5 q^{44} - q^{46} - 2 q^{47} + 74 q^{49} - 21 q^{50} + 71 q^{52} + 16 q^{53} + 22 q^{55} + 14 q^{56} + q^{58} - 32 q^{59} + 46 q^{61} + 20 q^{62} + 68 q^{64} + 12 q^{65} + 14 q^{67} + 27 q^{68} - 32 q^{71} + 50 q^{73} - 26 q^{74} + 56 q^{76} + 34 q^{77} + 16 q^{79} + 2 q^{80} + 38 q^{82} - 14 q^{83} + 38 q^{85} + 10 q^{86} + 40 q^{88} - 2 q^{89} + 32 q^{91} - 37 q^{92} + 29 q^{94} - 28 q^{95} + 56 q^{97} + 8 q^{98}+O(q^{100}) \) 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 −2.64113 0 4.97557 −1.42245 0 2.00224 −7.85888 0 3.75688
1.2 −2.60906 0 4.80718 3.69173 0 1.51238 −7.32410 0 −9.63193
1.3 −2.60849 0 4.80424 −1.19758 0 −4.49655 −7.31485 0 3.12388
1.4 −2.55489 0 4.52748 −3.88186 0 2.95370 −6.45744 0 9.91773
1.5 −2.23657 0 3.00224 −0.453557 0 2.23602 −2.24159 0 1.01441
1.6 −1.79065 0 1.20642 3.95479 0 −4.84424 1.42102 0 −7.08163
1.7 −1.59749 0 0.551985 2.02263 0 −1.90895 2.31319 0 −3.23113
1.8 −1.59580 0 0.546582 1.23722 0 2.46504 2.31937 0 −1.97436
1.9 −1.56095 0 0.436554 −3.73116 0 −1.31733 2.44046 0 5.82414
1.10 −1.20001 0 −0.559985 3.40098 0 4.71373 3.07200 0 −4.08120
1.11 −1.12036 0 −0.744798 −4.16877 0 2.81319 3.07516 0 4.67051
1.12 −0.619961 0 −1.61565 −0.522930 0 −1.16804 2.24156 0 0.324196
1.13 −0.534050 0 −1.71479 −2.64563 0 −5.04348 1.98388 0 1.41290
1.14 −0.480943 0 −1.76869 −1.51137 0 3.82960 1.81253 0 0.726880
1.15 −0.395940 0 −1.84323 0.579308 0 −2.59556 1.52169 0 −0.229371
1.16 0.0795257 0 −1.99368 3.50033 0 0.586662 −0.317600 0 0.278366
1.17 0.473233 0 −1.77605 −1.04483 0 3.83278 −1.78695 0 −0.494446
1.18 0.784780 0 −1.38412 2.15464 0 −2.00724 −2.65579 0 1.69092
1.19 0.809520 0 −1.34468 2.87436 0 0.883619 −2.70758 0 2.32685
1.20 0.959410 0 −1.07953 −2.76105 0 2.95320 −2.95453 0 −2.64898
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
\(3\) \(1\)
\(23\) \(1\)
\(29\) \(-1\)

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.w yes 30
3.b odd 2 1 6003.2.a.v ✓ 30
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6003.2.a.v ✓ 30 3.b odd 2 1
6003.2.a.w yes 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6003))\):

\( T_{2}^{30} - T_{2}^{29} - 48 T_{2}^{28} + 45 T_{2}^{27} + 1028 T_{2}^{26} - 893 T_{2}^{25} - 12975 T_{2}^{24} + \cdots - 2304 \) Copy content Toggle raw display
\( T_{5}^{30} - 103 T_{5}^{28} - 12 T_{5}^{27} + 4692 T_{5}^{26} + 1088 T_{5}^{25} - 124647 T_{5}^{24} + \cdots + 1081344 \) Copy content Toggle raw display