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
Inner twists 1

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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: \(42\)
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.

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

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 admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4006.2.a.h 42
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4006.2.a.h 42 1.a even 1 1 trivial

Atkin-Lehner signs

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

Hecke kernels

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