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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Newspace parameters

Level: \( N \) = \( 6003 = 3^{2} \cdot 23 \cdot 29 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 6003.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(47.9341963334\)
Analytic rank: \(0\)
Dimension: \(30\)
Coefficient ring index: multiple of None
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.

\(\operatorname{Tr}(f)(q) = \) \( 30q + q^{2} + 37q^{4} + 10q^{7} + 6q^{8} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 30q + q^{2} + 37q^{4} + 10q^{7} + 6q^{8} + 8q^{10} + 36q^{13} + 7q^{14} + 47q^{16} + 18q^{17} + 16q^{19} + 25q^{22} - 30q^{23} + 56q^{25} + 11q^{26} + 27q^{28} + 30q^{29} + 14q^{31} - 7q^{32} + 3q^{34} - 22q^{35} + 40q^{37} + 6q^{38} + 30q^{40} + 14q^{41} + 34q^{43} + 5q^{44} - q^{46} - 2q^{47} + 74q^{49} - 21q^{50} + 71q^{52} + 16q^{53} + 22q^{55} + 14q^{56} + q^{58} - 32q^{59} + 46q^{61} + 20q^{62} + 68q^{64} + 12q^{65} + 14q^{67} + 27q^{68} - 32q^{71} + 50q^{73} - 26q^{74} + 56q^{76} + 34q^{77} + 16q^{79} + 2q^{80} + 38q^{82} - 14q^{83} + 38q^{85} + 10q^{86} + 40q^{88} - 2q^{89} + 32q^{91} - 37q^{92} + 29q^{94} - 28q^{95} + 56q^{97} + 8q^{98} + O(q^{100}) \)

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.

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:

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

Atkin-Lehner signs

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

Hecke kernels

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

\(T_{2}^{30} - \cdots\)
\(T_{5}^{30} - \cdots\)

Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database