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: \( 1 \)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of \(\nu = \zeta_{14} + \zeta_{14}^{-1}\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 2 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\mathstrut +\mathstrut \) \(2\)

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.

Atkin-Lehner signs

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

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{2}^{3} \) \(\mathstrut -\mathstrut 2 T_{2}^{2} \) \(\mathstrut -\mathstrut T_{2} \) \(\mathstrut +\mathstrut 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4009))\).