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

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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: \(1\)
Dimension: \(22\)
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.

\(\operatorname{Tr}(f)(q) = \) \(22q \) \(\mathstrut -\mathstrut 3q^{2} \) \(\mathstrut +\mathstrut 17q^{4} \) \(\mathstrut -\mathstrut 6q^{7} \) \(\mathstrut -\mathstrut 6q^{8} \) \(\mathstrut +\mathstrut O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \(22q \) \(\mathstrut -\mathstrut 3q^{2} \) \(\mathstrut +\mathstrut 17q^{4} \) \(\mathstrut -\mathstrut 6q^{7} \) \(\mathstrut -\mathstrut 6q^{8} \) \(\mathstrut -\mathstrut 12q^{10} \) \(\mathstrut -\mathstrut 28q^{13} \) \(\mathstrut -\mathstrut q^{14} \) \(\mathstrut +\mathstrut 3q^{16} \) \(\mathstrut -\mathstrut 10q^{17} \) \(\mathstrut -\mathstrut 8q^{19} \) \(\mathstrut -\mathstrut 11q^{22} \) \(\mathstrut +\mathstrut 22q^{23} \) \(\mathstrut +\mathstrut 11q^{26} \) \(\mathstrut -\mathstrut 21q^{28} \) \(\mathstrut +\mathstrut 22q^{29} \) \(\mathstrut -\mathstrut 18q^{31} \) \(\mathstrut +\mathstrut 5q^{32} \) \(\mathstrut -\mathstrut 33q^{34} \) \(\mathstrut +\mathstrut 2q^{35} \) \(\mathstrut -\mathstrut 28q^{37} \) \(\mathstrut +\mathstrut 14q^{38} \) \(\mathstrut -\mathstrut 30q^{40} \) \(\mathstrut -\mathstrut 10q^{41} \) \(\mathstrut -\mathstrut 14q^{43} \) \(\mathstrut +\mathstrut 37q^{44} \) \(\mathstrut -\mathstrut 3q^{46} \) \(\mathstrut -\mathstrut 18q^{47} \) \(\mathstrut +\mathstrut 2q^{49} \) \(\mathstrut +\mathstrut 7q^{50} \) \(\mathstrut -\mathstrut 57q^{52} \) \(\mathstrut +\mathstrut 20q^{53} \) \(\mathstrut -\mathstrut 42q^{55} \) \(\mathstrut -\mathstrut 2q^{56} \) \(\mathstrut -\mathstrut 3q^{58} \) \(\mathstrut -\mathstrut 20q^{59} \) \(\mathstrut -\mathstrut 38q^{61} \) \(\mathstrut +\mathstrut 4q^{62} \) \(\mathstrut -\mathstrut 24q^{64} \) \(\mathstrut +\mathstrut 12q^{65} \) \(\mathstrut -\mathstrut 50q^{67} \) \(\mathstrut +\mathstrut 11q^{68} \) \(\mathstrut -\mathstrut 48q^{70} \) \(\mathstrut +\mathstrut 12q^{71} \) \(\mathstrut -\mathstrut 46q^{73} \) \(\mathstrut -\mathstrut 6q^{74} \) \(\mathstrut -\mathstrut 16q^{76} \) \(\mathstrut -\mathstrut 14q^{77} \) \(\mathstrut -\mathstrut 20q^{79} \) \(\mathstrut -\mathstrut 58q^{80} \) \(\mathstrut -\mathstrut 42q^{82} \) \(\mathstrut +\mathstrut 22q^{83} \) \(\mathstrut -\mathstrut 66q^{85} \) \(\mathstrut +\mathstrut 22q^{86} \) \(\mathstrut -\mathstrut 68q^{88} \) \(\mathstrut -\mathstrut 14q^{89} \) \(\mathstrut -\mathstrut 16q^{91} \) \(\mathstrut +\mathstrut 17q^{92} \) \(\mathstrut -\mathstrut 27q^{94} \) \(\mathstrut -\mathstrut 20q^{95} \) \(\mathstrut -\mathstrut 48q^{97} \) \(\mathstrut -\mathstrut 28q^{98} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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 −2.68779 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:

Inner twists

This newform does not have CM; other inner twists have not been computed.

Atkin-Lehner signs

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

Hecke kernels

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

\(T_{2}^{22} + \cdots\)
\(T_{5}^{22} - \cdots\)