Properties

Label 4029.2.a.c
Level 4029
Weight 2
Character orbit 4029.a
Self dual Yes
Analytic conductor 32.172
Analytic rank 1
Dimension 2
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 4029 = 3 \cdot 17 \cdot 79 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4029.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(32.1717269744\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
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 \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−1.41421
1.41421
−1.41421 1.00000 0 −2.41421 −1.41421 1.00000 2.82843 1.00000 3.41421
1.2 1.41421 1.00000 0 0.414214 1.41421 1.00000 −2.82843 1.00000 0.585786
\(n\): 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
\(3\) \(-1\)
\(17\) \(-1\)
\(79\) \(1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4029))\):

\(T_{2}^{2} \) \(\mathstrut -\mathstrut 2 \)
\(T_{5}^{2} \) \(\mathstrut +\mathstrut 2 T_{5} \) \(\mathstrut -\mathstrut 1 \)