Properties

Label 4006.2.a.h
Level 4006
Weight 2
Character orbit 4006.a
Self dual Yes
Analytic conductor 31.988
Analytic rank 0
Dimension 42
CM No

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Newspace parameters

Level: \( N \) = \( 4006 = 2 \cdot 2003 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4006.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(31.9880710497\)
Analytic rank: \(0\)
Dimension: \(42\)
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.

\(\operatorname{Tr}(f)(q) = \) \(42q \) \(\mathstrut -\mathstrut 42q^{2} \) \(\mathstrut +\mathstrut 42q^{4} \) \(\mathstrut +\mathstrut 27q^{5} \) \(\mathstrut -\mathstrut 10q^{7} \) \(\mathstrut -\mathstrut 42q^{8} \) \(\mathstrut +\mathstrut 56q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \(42q \) \(\mathstrut -\mathstrut 42q^{2} \) \(\mathstrut +\mathstrut 42q^{4} \) \(\mathstrut +\mathstrut 27q^{5} \) \(\mathstrut -\mathstrut 10q^{7} \) \(\mathstrut -\mathstrut 42q^{8} \) \(\mathstrut +\mathstrut 56q^{9} \) \(\mathstrut -\mathstrut 27q^{10} \) \(\mathstrut +\mathstrut 23q^{11} \) \(\mathstrut +\mathstrut 15q^{13} \) \(\mathstrut +\mathstrut 10q^{14} \) \(\mathstrut +\mathstrut 6q^{15} \) \(\mathstrut +\mathstrut 42q^{16} \) \(\mathstrut +\mathstrut 14q^{17} \) \(\mathstrut -\mathstrut 56q^{18} \) \(\mathstrut -\mathstrut 4q^{19} \) \(\mathstrut +\mathstrut 27q^{20} \) \(\mathstrut +\mathstrut 26q^{21} \) \(\mathstrut -\mathstrut 23q^{22} \) \(\mathstrut +\mathstrut 12q^{23} \) \(\mathstrut +\mathstrut 45q^{25} \) \(\mathstrut -\mathstrut 15q^{26} \) \(\mathstrut -\mathstrut 3q^{27} \) \(\mathstrut -\mathstrut 10q^{28} \) \(\mathstrut +\mathstrut 41q^{29} \) \(\mathstrut -\mathstrut 6q^{30} \) \(\mathstrut +\mathstrut 18q^{31} \) \(\mathstrut -\mathstrut 42q^{32} \) \(\mathstrut +\mathstrut 25q^{33} \) \(\mathstrut -\mathstrut 14q^{34} \) \(\mathstrut +\mathstrut 8q^{35} \) \(\mathstrut +\mathstrut 56q^{36} \) \(\mathstrut +\mathstrut 33q^{37} \) \(\mathstrut +\mathstrut 4q^{38} \) \(\mathstrut +\mathstrut 10q^{39} \) \(\mathstrut -\mathstrut 27q^{40} \) \(\mathstrut +\mathstrut 84q^{41} \) \(\mathstrut -\mathstrut 26q^{42} \) \(\mathstrut -\mathstrut 36q^{43} \) \(\mathstrut +\mathstrut 23q^{44} \) \(\mathstrut +\mathstrut 66q^{45} \) \(\mathstrut -\mathstrut 12q^{46} \) \(\mathstrut +\mathstrut 28q^{47} \) \(\mathstrut +\mathstrut 58q^{49} \) \(\mathstrut -\mathstrut 45q^{50} \) \(\mathstrut +\mathstrut 17q^{51} \) \(\mathstrut +\mathstrut 15q^{52} \) \(\mathstrut +\mathstrut 68q^{53} \) \(\mathstrut +\mathstrut 3q^{54} \) \(\mathstrut -\mathstrut 28q^{55} \) \(\mathstrut +\mathstrut 10q^{56} \) \(\mathstrut -\mathstrut 9q^{57} \) \(\mathstrut -\mathstrut 41q^{58} \) \(\mathstrut +\mathstrut 59q^{59} \) \(\mathstrut +\mathstrut 6q^{60} \) \(\mathstrut +\mathstrut 41q^{61} \) \(\mathstrut -\mathstrut 18q^{62} \) \(\mathstrut -\mathstrut 28q^{63} \) \(\mathstrut +\mathstrut 42q^{64} \) \(\mathstrut +\mathstrut 44q^{65} \) \(\mathstrut -\mathstrut 25q^{66} \) \(\mathstrut +\mathstrut 14q^{68} \) \(\mathstrut +\mathstrut 67q^{69} \) \(\mathstrut -\mathstrut 8q^{70} \) \(\mathstrut +\mathstrut 69q^{71} \) \(\mathstrut -\mathstrut 56q^{72} \) \(\mathstrut -\mathstrut 27q^{73} \) \(\mathstrut -\mathstrut 33q^{74} \) \(\mathstrut +\mathstrut 14q^{75} \) \(\mathstrut -\mathstrut 4q^{76} \) \(\mathstrut +\mathstrut 43q^{77} \) \(\mathstrut -\mathstrut 10q^{78} \) \(\mathstrut -\mathstrut 19q^{79} \) \(\mathstrut +\mathstrut 27q^{80} \) \(\mathstrut +\mathstrut 74q^{81} \) \(\mathstrut -\mathstrut 84q^{82} \) \(\mathstrut +\mathstrut 20q^{83} \) \(\mathstrut +\mathstrut 26q^{84} \) \(\mathstrut +\mathstrut 16q^{85} \) \(\mathstrut +\mathstrut 36q^{86} \) \(\mathstrut -\mathstrut 28q^{87} \) \(\mathstrut -\mathstrut 23q^{88} \) \(\mathstrut +\mathstrut 123q^{89} \) \(\mathstrut -\mathstrut 66q^{90} \) \(\mathstrut -\mathstrut 9q^{91} \) \(\mathstrut +\mathstrut 12q^{92} \) \(\mathstrut +\mathstrut 48q^{93} \) \(\mathstrut -\mathstrut 28q^{94} \) \(\mathstrut +\mathstrut 28q^{95} \) \(\mathstrut +\mathstrut 10q^{97} \) \(\mathstrut -\mathstrut 58q^{98} \) \(\mathstrut +\mathstrut 75q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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 −1.00000 −3.21265 1.00000 3.26318 3.21265 −3.74888 −1.00000 7.32111 −3.26318
1.2 −1.00000 −3.07158 1.00000 0.0807440 3.07158 −2.46915 −1.00000 6.43459 −0.0807440
1.3 −1.00000 −3.06826 1.00000 −0.106963 3.06826 0.667572 −1.00000 6.41420 0.106963
1.4 −1.00000 −3.06520 1.00000 4.19388 3.06520 2.77423 −1.00000 6.39547 −4.19388
1.5 −1.00000 −2.83249 1.00000 −3.27043 2.83249 −1.58563 −1.00000 5.02302 3.27043
1.6 −1.00000 −2.79722 1.00000 2.06703 2.79722 3.37710 −1.00000 4.82443 −2.06703
1.7 −1.00000 −2.77917 1.00000 −2.42477 2.77917 −4.56136 −1.00000 4.72377 2.42477
1.8 −1.00000 −2.67759 1.00000 2.43345 2.67759 −4.11067 −1.00000 4.16951 −2.43345
1.9 −1.00000 −2.07354 1.00000 −1.29135 2.07354 1.20599 −1.00000 1.29958 1.29135
1.10 −1.00000 −1.84484 1.00000 −3.34479 1.84484 2.16076 −1.00000 0.403448 3.34479
1.11 −1.00000 −1.79678 1.00000 −1.46439 1.79678 2.25560 −1.00000 0.228426 1.46439
1.12 −1.00000 −1.69783 1.00000 2.46315 1.69783 −3.98093 −1.00000 −0.117367 −2.46315
1.13 −1.00000 −1.67728 1.00000 1.83767 1.67728 4.34649 −1.00000 −0.186728 −1.83767
1.14 −1.00000 −1.57856 1.00000 1.66904 1.57856 0.263990 −1.00000 −0.508157 −1.66904
1.15 −1.00000 −1.01568 1.00000 −0.411222 1.01568 −2.82966 −1.00000 −1.96840 0.411222
1.16 −1.00000 −0.963522 1.00000 3.93503 0.963522 −4.13063 −1.00000 −2.07162 −3.93503
1.17 −1.00000 −0.545281 1.00000 0.119242 0.545281 0.291946 −1.00000 −2.70267 −0.119242
1.18 −1.00000 −0.388226 1.00000 3.10956 0.388226 1.22808 −1.00000 −2.84928 −3.10956
1.19 −1.00000 −0.372757 1.00000 3.67312 0.372757 −1.58044 −1.00000 −2.86105 −3.67312
1.20 −1.00000 −0.340721 1.00000 1.77081 0.340721 4.18388 −1.00000 −2.88391 −1.77081
See all 42 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.42
Significant digits:
Format:

Inner twists

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

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(2003\) \(-1\)

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{3}^{42} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4006))\).