Properties

Label 6003.2.a.v
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

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 −2.78139 0 5.73612 −0.00312667 0 2.66771 −10.3916 0 0.00869647
1.2 −2.66815 0 5.11905 −1.38302 0 0.919018 −8.32211 0 3.69010
1.3 −2.53344 0 4.41832 −4.44002 0 −2.94217 −6.12666 0 11.2485
1.4 −2.40738 0 3.79549 3.50117 0 5.09845 −4.32244 0 −8.42865
1.5 −2.27733 0 3.18622 2.98462 0 −2.44710 −2.70140 0 −6.79696
1.6 −1.99322 0 1.97291 0.735105 0 −3.83374 0.0540029 0 −1.46522
1.7 −1.96558 0 1.86349 −2.31356 0 5.13967 0.268317 0 4.54748
1.8 −1.89039 0 1.57357 −2.79891 0 1.18730 0.806117 0 5.29102
1.9 −1.58386 0 0.508607 2.90489 0 0.0968084 2.36216 0 −4.60094
1.10 −1.33909 0 −0.206831 0.887651 0 −3.28671 2.95515 0 −1.18865
1.11 −0.959410 0 −1.07953 2.76105 0 2.95320 2.95453 0 −2.64898
1.12 −0.809520 0 −1.34468 −2.87436 0 0.883619 2.70758 0 2.32685
1.13 −0.784780 0 −1.38412 −2.15464 0 −2.00724 2.65579 0 1.69092
1.14 −0.473233 0 −1.77605 1.04483 0 3.83278 1.78695 0 −0.494446
1.15 −0.0795257 0 −1.99368 −3.50033 0 0.586662 0.317600 0 0.278366
1.16 0.395940 0 −1.84323 −0.579308 0 −2.59556 −1.52169 0 −0.229371
1.17 0.480943 0 −1.76869 1.51137 0 3.82960 −1.81253 0 0.726880
1.18 0.534050 0 −1.71479 2.64563 0 −5.04348 −1.98388 0 1.41290
1.19 0.619961 0 −1.61565 0.522930 0 −1.16804 −2.24156 0 0.324196
1.20 1.12036 0 −0.744798 4.16877 0 2.81319 −3.07516 0 4.67051
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.v 30
3.b odd 2 1 6003.2.a.w yes 30
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6003.2.a.v 30 1.a even 1 1 trivial
6003.2.a.w yes 30 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6003))\):

\( T_{2}^{30} + T_{2}^{29} - 48 T_{2}^{28} - 45 T_{2}^{27} + 1028 T_{2}^{26} + 893 T_{2}^{25} - 12975 T_{2}^{24} - 10288 T_{2}^{23} + 107407 T_{2}^{22} + 76251 T_{2}^{21} - 614174 T_{2}^{20} - 380781 T_{2}^{19} + 2489025 T_{2}^{18} + \cdots - 2304 \) Copy content Toggle raw display
\( T_{5}^{30} - 103 T_{5}^{28} + 12 T_{5}^{27} + 4692 T_{5}^{26} - 1088 T_{5}^{25} - 124647 T_{5}^{24} + 43372 T_{5}^{23} + 2143650 T_{5}^{22} - 1000846 T_{5}^{21} - 25041479 T_{5}^{20} + 14812592 T_{5}^{19} + \cdots + 1081344 \) Copy content Toggle raw display