Properties

Label 4009.2.a.c
Level 4009
Weight 2
Character orbit 4009.a
Self dual yes
Analytic conductor 32.012
Analytic rank 1
Dimension 71
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 4009 = 19 \cdot 211 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4009.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(32.0120261703\)
Analytic rank: \(1\)
Dimension: \(71\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 71q - 15q^{2} - 8q^{3} + 69q^{4} - 18q^{5} - 9q^{6} - 19q^{7} - 39q^{8} + 63q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 71q - 15q^{2} - 8q^{3} + 69q^{4} - 18q^{5} - 9q^{6} - 19q^{7} - 39q^{8} + 63q^{9} - 10q^{10} - 52q^{11} - 9q^{12} - 15q^{13} - 53q^{14} - 33q^{15} + 53q^{16} - 10q^{17} - 35q^{18} + 71q^{19} - 33q^{20} - 38q^{21} - 6q^{22} - 65q^{23} - 30q^{24} + 51q^{25} - 4q^{26} - 23q^{27} - 29q^{28} - 97q^{29} - 27q^{30} - 53q^{31} - 78q^{32} - 17q^{33} - 24q^{34} - 38q^{35} + 24q^{36} - 33q^{37} - 15q^{38} - 86q^{39} + 25q^{40} - 69q^{41} + 64q^{42} - 10q^{43} - 94q^{44} - 34q^{45} - 6q^{46} - 37q^{47} - q^{48} + 74q^{49} - 41q^{50} - 46q^{51} - 30q^{52} - 50q^{53} - 17q^{54} - 30q^{55} - 116q^{56} - 8q^{57} + 11q^{58} - 93q^{59} - 56q^{60} - 18q^{61} - q^{62} - 84q^{63} + 93q^{64} - 78q^{65} - 53q^{66} - 5q^{67} - 9q^{68} - 69q^{69} - 10q^{70} - 221q^{71} - 73q^{72} - 34q^{73} - 58q^{74} - 70q^{75} + 69q^{76} - 2q^{77} + 7q^{78} - 68q^{79} - 71q^{80} + 39q^{81} + 26q^{82} - 45q^{83} - 10q^{84} - 44q^{85} - 80q^{86} - 7q^{87} - 46q^{88} - 143q^{89} + 41q^{90} - 30q^{91} - 46q^{92} + 32q^{93} + 41q^{94} - 18q^{95} - 140q^{96} - 18q^{97} - 97q^{98} - 142q^{99} + O(q^{100}) \)

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 −2.82251 2.93474 5.96657 0.420997 −8.28333 −3.44749 −11.1957 5.61269 −1.18827
1.2 −2.70369 1.93926 5.30992 1.68888 −5.24315 4.83108 −8.94900 0.760725 −4.56619
1.3 −2.69837 1.88273 5.28118 −2.05530 −5.08030 0.784756 −8.85381 0.544674 5.54594
1.4 −2.67657 0.00971531 5.16401 0.0508514 −0.0260037 0.312050 −8.46867 −2.99991 −0.136107
1.5 −2.66661 −1.96401 5.11081 −2.76483 5.23725 2.50280 −8.29531 0.857341 7.37272
1.6 −2.61964 −2.82043 4.86252 −3.49446 7.38851 2.54132 −7.49878 4.95481 9.15424
1.7 −2.51184 −2.51408 4.30932 3.86977 6.31496 2.24136 −5.80064 3.32060 −9.72023
1.8 −2.48400 0.221625 4.17025 −2.50863 −0.550515 2.33062 −5.39089 −2.95088 6.23143
1.9 −2.35519 0.815593 3.54692 1.05865 −1.92088 −4.06376 −3.64328 −2.33481 −2.49333
1.10 −2.30773 −2.77619 3.32561 2.46750 6.40670 −0.00709873 −3.05915 4.70724 −5.69433
1.11 −2.28791 −0.433636 3.23453 1.92947 0.992121 3.12765 −2.82450 −2.81196 −4.41445
1.12 −2.25185 −1.69642 3.07082 −1.12007 3.82008 −4.02650 −2.41131 −0.122162 2.52222
1.13 −2.08913 2.53881 2.36446 0.545075 −5.30390 0.502260 −0.761409 3.44555 −1.13873
1.14 −2.08320 2.50442 2.33972 −4.26568 −5.21720 −3.89477 −0.707698 3.27211 8.88625
1.15 −1.98460 0.183292 1.93862 −1.97615 −0.363761 −5.05020 0.121816 −2.96640 3.92186
1.16 −1.93894 −1.61421 1.75950 −0.710068 3.12986 −1.51648 0.466323 −0.394323 1.37678
1.17 −1.89616 2.32800 1.59541 −3.08512 −4.41426 2.27635 0.767159 2.41961 5.84987
1.18 −1.89239 −0.431235 1.58115 3.97235 0.816065 −2.66937 0.792636 −2.81404 −7.51724
1.19 −1.74061 −3.34117 1.02972 −2.66559 5.81567 0.0455688 1.68888 8.16342 4.63975
1.20 −1.69413 −0.649307 0.870061 −0.0469641 1.10001 3.97959 1.91426 −2.57840 0.0795630
See all 71 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.71
Significant digits:
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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 4009.2.a.c 71
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4009.2.a.c 71 1.a even 1 1 trivial

Atkin-Lehner signs

\( p \) Sign
\(19\) \(-1\)
\(211\) \(-1\)

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{2}^{71} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4009))\).

Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database