Properties

Label 4009.2.a.b
Level 4009
Weight 2
Character orbit 4009.a
Self dual yes
Analytic conductor 32.012
Analytic rank 1
Dimension 3
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: \(3\)
Coefficient field: \(\Q(\zeta_{14})^+\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a root \(\beta\) of the polynomial \(x^{3} - x^{2} - 2 x + 1\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 - \beta ) q^{2} + ( -1 + \beta ) q^{3} + ( -1 - 2 \beta + \beta^{2} ) q^{4} + ( 2 + \beta - 2 \beta^{2} ) q^{5} + ( -1 + 2 \beta - \beta^{2} ) q^{6} + ( 3 - \beta - \beta^{2} ) q^{7} + ( -2 - \beta + 2 \beta^{2} ) q^{8} + ( -2 - 2 \beta + \beta^{2} ) q^{9} +O(q^{10})\) \( q + ( 1 - \beta ) q^{2} + ( -1 + \beta ) q^{3} + ( -1 - 2 \beta + \beta^{2} ) q^{4} + ( 2 + \beta - 2 \beta^{2} ) q^{5} + ( -1 + 2 \beta - \beta^{2} ) q^{6} + ( 3 - \beta - \beta^{2} ) q^{7} + ( -2 - \beta + 2 \beta^{2} ) q^{8} + ( -2 - 2 \beta + \beta^{2} ) q^{9} + ( 3 \beta - \beta^{2} ) q^{10} + ( -3 + 2 \beta^{2} ) q^{11} + ( 3 \beta - 2 \beta^{2} ) q^{12} + ( 2 - 2 \beta + \beta^{2} ) q^{14} + ( -3 \beta + \beta^{2} ) q^{15} + ( 2 + \beta - \beta^{2} ) q^{16} + ( 1 - \beta^{2} ) q^{17} + ( -1 - 2 \beta + 2 \beta^{2} ) q^{18} + q^{19} + ( -5 + 3 \beta + \beta^{2} ) q^{20} + ( -2 + 2 \beta - \beta^{2} ) q^{21} + ( -1 - \beta ) q^{22} + ( 2 + \beta - \beta^{2} ) q^{23} + ( 3 \beta - \beta^{2} ) q^{24} + ( -1 + \beta^{2} ) q^{25} + ( 4 - \beta - 2 \beta^{2} ) q^{27} + ( -3 - 4 \beta + 4 \beta^{2} ) q^{28} + ( -1 - \beta + 2 \beta^{2} ) q^{29} + ( 1 - 5 \beta + 3 \beta^{2} ) q^{30} + ( -6 - 2 \beta + 5 \beta^{2} ) q^{31} + ( 5 + 3 \beta - 5 \beta^{2} ) q^{32} + ( 1 + \beta ) q^{33} + \beta q^{34} + ( 3 + 5 \beta - 2 \beta^{2} ) q^{35} + ( 5 - \beta ) q^{36} + ( 9 + 7 \beta - 5 \beta^{2} ) q^{37} + ( 1 - \beta ) q^{38} + ( -4 - \beta^{2} ) q^{40} + ( -10 - 2 \beta + 5 \beta^{2} ) q^{41} + ( -3 + 6 \beta - 2 \beta^{2} ) q^{42} + ( -1 + 2 \beta ) q^{43} + ( 5 - 3 \beta^{2} ) q^{44} + ( -7 + 2 \beta + 3 \beta^{2} ) q^{45} + ( 1 + \beta - \beta^{2} ) q^{46} + ( 7 \beta - 2 \beta^{2} ) q^{47} + ( -1 - \beta + \beta^{2} ) q^{48} + ( -1 - \beta ) q^{49} -\beta q^{50} -\beta q^{51} + ( -12 - \beta + 2 \beta^{2} ) q^{53} + ( 2 - \beta + \beta^{2} ) q^{54} + ( -4 - 3 \beta ) q^{55} + ( -3 - 5 \beta + 2 \beta^{2} ) q^{56} + ( -1 + \beta ) q^{57} + ( 1 - 4 \beta + \beta^{2} ) q^{58} + ( 2 - 3 \beta - \beta^{2} ) q^{59} + ( 4 - 6 \beta + 3 \beta^{2} ) q^{60} + ( 2 - 2 \beta ) q^{61} + ( -1 - 6 \beta + 2 \beta^{2} ) q^{62} + ( -6 - 3 \beta + 5 \beta^{2} ) q^{63} + ( -4 + 6 \beta - \beta^{2} ) q^{64} + ( 1 - \beta^{2} ) q^{66} + ( -12 - 6 \beta + 10 \beta^{2} ) q^{67} + ( -2 + \beta + \beta^{2} ) q^{68} + ( -1 - \beta + \beta^{2} ) q^{69} + ( 1 + 6 \beta - 5 \beta^{2} ) q^{70} + ( -8 + 3 \beta + 4 \beta^{2} ) q^{71} + ( 7 - 2 \beta - 3 \beta^{2} ) q^{72} + ( 5 - 3 \beta ) q^{73} + ( 4 + 8 \beta - 7 \beta^{2} ) q^{74} + \beta q^{75} + ( -1 - 2 \beta + \beta^{2} ) q^{76} + ( -5 - 3 \beta + \beta^{2} ) q^{77} + ( -14 - \beta + 2 \beta^{2} ) q^{79} + ( 5 - 2 \beta^{2} ) q^{80} + ( 4 + 7 \beta - 4 \beta^{2} ) q^{81} + ( -5 - 2 \beta + 2 \beta^{2} ) q^{82} + ( -4 - 5 \beta + 2 \beta^{2} ) q^{83} + ( -1 + 9 \beta - 4 \beta^{2} ) q^{84} + ( 1 + \beta + \beta^{2} ) q^{85} + ( -1 + 3 \beta - 2 \beta^{2} ) q^{86} + ( -1 + 4 \beta - \beta^{2} ) q^{87} + ( 4 + 3 \beta ) q^{88} + ( 7 + 13 \beta - 6 \beta^{2} ) q^{89} + ( -4 + 3 \beta - 2 \beta^{2} ) q^{90} + ( -4 + \beta^{2} ) q^{92} + ( 1 + 6 \beta - 2 \beta^{2} ) q^{93} + ( -2 + 11 \beta - 7 \beta^{2} ) q^{94} + ( 2 + \beta - 2 \beta^{2} ) q^{95} + ( -8 \beta + 3 \beta^{2} ) q^{96} + ( -3 - 3 \beta + \beta^{2} ) q^{97} + ( -1 + \beta^{2} ) q^{98} + ( 8 - 5 \beta^{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 2q^{2} - 2q^{3} - 3q^{5} - 6q^{6} + 3q^{7} + 3q^{8} - 3q^{9} + O(q^{10}) \) \( 3q + 2q^{2} - 2q^{3} - 3q^{5} - 6q^{6} + 3q^{7} + 3q^{8} - 3q^{9} - 2q^{10} + q^{11} - 7q^{12} + 9q^{14} + 2q^{15} + 2q^{16} - 2q^{17} + 5q^{18} + 3q^{19} - 7q^{20} - 9q^{21} - 4q^{22} + 2q^{23} - 2q^{24} + 2q^{25} + q^{27} + 7q^{28} + 6q^{29} + 13q^{30} + 5q^{31} - 7q^{32} + 4q^{33} + q^{34} + 4q^{35} + 14q^{36} + 9q^{37} + 2q^{38} - 17q^{40} - 7q^{41} - 13q^{42} - q^{43} - 4q^{45} - q^{46} - 3q^{47} + q^{48} - 4q^{49} - q^{50} - q^{51} - 27q^{53} + 10q^{54} - 15q^{55} - 4q^{56} - 2q^{57} + 4q^{58} - 2q^{59} + 21q^{60} + 4q^{61} + q^{62} + 4q^{63} - 11q^{64} - 2q^{66} + 8q^{67} + q^{69} - 16q^{70} - q^{71} + 4q^{72} + 12q^{73} - 15q^{74} + q^{75} - 13q^{77} - 33q^{79} + 5q^{80} - q^{81} - 7q^{82} - 7q^{83} - 14q^{84} + 9q^{85} - 10q^{86} - 4q^{87} + 15q^{88} + 4q^{89} - 19q^{90} - 7q^{92} - q^{93} - 30q^{94} - 3q^{95} + 7q^{96} - 7q^{97} + 2q^{98} - q^{99} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
1.80194
0.445042
−1.24698
−0.801938 0.801938 −1.35690 −2.69202 −0.643104 −2.04892 2.69202 −2.35690 2.15883
1.2 0.554958 −0.554958 −1.69202 2.04892 −0.307979 2.35690 −2.04892 −2.69202 1.13706
1.3 2.24698 −2.24698 3.04892 −2.35690 −5.04892 2.69202 2.35690 2.04892 −5.29590
\(n\): e.g. 2-40 or 990-1000
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 4009.2.a.b 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4009.2.a.b 3 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 \( T_{2}^{3} - 2 T_{2}^{2} - T_{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4009))\).