Properties

Label 4026.2.a.o
Level 4026
Weight 2
Character orbit 4026.a
Self dual Yes
Analytic conductor 32.148
Analytic rank 1
Dimension 3
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 4026 = 2 \cdot 3 \cdot 11 \cdot 61 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4026.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(32.1477718538\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.1129.1
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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.

\(f(q)\) \(=\) \( q\) \(+ q^{2}\) \(- q^{3}\) \(+ q^{4}\) \( + ( -1 + \beta_{1} ) q^{5} \) \(- q^{6}\) \( + ( -1 - \beta_{2} ) q^{7} \) \(+ q^{8}\) \(+ q^{9}\) \(+O(q^{10})\) \( q\) \(+ q^{2}\) \(- q^{3}\) \(+ q^{4}\) \( + ( -1 + \beta_{1} ) q^{5} \) \(- q^{6}\) \( + ( -1 - \beta_{2} ) q^{7} \) \(+ q^{8}\) \(+ q^{9}\) \( + ( -1 + \beta_{1} ) q^{10} \) \(+ q^{11}\) \(- q^{12}\) \( + ( -2 - \beta_{1} + \beta_{2} ) q^{13} \) \( + ( -1 - \beta_{2} ) q^{14} \) \( + ( 1 - \beta_{1} ) q^{15} \) \(+ q^{16}\) \(+ q^{17}\) \(+ q^{18}\) \( + ( 1 - \beta_{1} + \beta_{2} ) q^{19} \) \( + ( -1 + \beta_{1} ) q^{20} \) \( + ( 1 + \beta_{2} ) q^{21} \) \(+ q^{22}\) \( + ( -1 - \beta_{1} + \beta_{2} ) q^{23} \) \(- q^{24}\) \( + ( 1 - \beta_{1} + \beta_{2} ) q^{25} \) \( + ( -2 - \beta_{1} + \beta_{2} ) q^{26} \) \(- q^{27}\) \( + ( -1 - \beta_{2} ) q^{28} \) \( + ( -3 - 2 \beta_{1} - \beta_{2} ) q^{29} \) \( + ( 1 - \beta_{1} ) q^{30} \) \( + ( -6 - \beta_{2} ) q^{31} \) \(+ q^{32}\) \(- q^{33}\) \(+ q^{34}\) \( + ( 3 - 2 \beta_{1} + 2 \beta_{2} ) q^{35} \) \(+ q^{36}\) \( + ( -5 - \beta_{1} - \beta_{2} ) q^{37} \) \( + ( 1 - \beta_{1} + \beta_{2} ) q^{38} \) \( + ( 2 + \beta_{1} - \beta_{2} ) q^{39} \) \( + ( -1 + \beta_{1} ) q^{40} \) \( + ( -1 + 2 \beta_{1} + 2 \beta_{2} ) q^{41} \) \( + ( 1 + \beta_{2} ) q^{42} \) \( + \beta_{1} q^{43} \) \(+ q^{44}\) \( + ( -1 + \beta_{1} ) q^{45} \) \( + ( -1 - \beta_{1} + \beta_{2} ) q^{46} \) \( + ( -1 + 3 \beta_{1} + 2 \beta_{2} ) q^{47} \) \(- q^{48}\) \( + ( 3 - 3 \beta_{1} ) q^{49} \) \( + ( 1 - \beta_{1} + \beta_{2} ) q^{50} \) \(- q^{51}\) \( + ( -2 - \beta_{1} + \beta_{2} ) q^{52} \) \( + ( 2 + \beta_{1} - 2 \beta_{2} ) q^{53} \) \(- q^{54}\) \( + ( -1 + \beta_{1} ) q^{55} \) \( + ( -1 - \beta_{2} ) q^{56} \) \( + ( -1 + \beta_{1} - \beta_{2} ) q^{57} \) \( + ( -3 - 2 \beta_{1} - \beta_{2} ) q^{58} \) \( + ( 4 - 4 \beta_{1} - \beta_{2} ) q^{59} \) \( + ( 1 - \beta_{1} ) q^{60} \) \(- q^{61}\) \( + ( -6 - \beta_{2} ) q^{62} \) \( + ( -1 - \beta_{2} ) q^{63} \) \(+ q^{64}\) \( + ( -5 - \beta_{1} - 3 \beta_{2} ) q^{65} \) \(- q^{66}\) \( + ( 6 + \beta_{2} ) q^{67} \) \(+ q^{68}\) \( + ( 1 + \beta_{1} - \beta_{2} ) q^{69} \) \( + ( 3 - 2 \beta_{1} + 2 \beta_{2} ) q^{70} \) \( + ( -6 - \beta_{2} ) q^{71} \) \(+ q^{72}\) \( + ( 3 - 2 \beta_{1} - \beta_{2} ) q^{73} \) \( + ( -5 - \beta_{1} - \beta_{2} ) q^{74} \) \( + ( -1 + \beta_{1} - \beta_{2} ) q^{75} \) \( + ( 1 - \beta_{1} + \beta_{2} ) q^{76} \) \( + ( -1 - \beta_{2} ) q^{77} \) \( + ( 2 + \beta_{1} - \beta_{2} ) q^{78} \) \( + ( -2 - \beta_{1} + \beta_{2} ) q^{79} \) \( + ( -1 + \beta_{1} ) q^{80} \) \(+ q^{81}\) \( + ( -1 + 2 \beta_{1} + 2 \beta_{2} ) q^{82} \) \( + ( -5 + 4 \beta_{1} - \beta_{2} ) q^{83} \) \( + ( 1 + \beta_{2} ) q^{84} \) \( + ( -1 + \beta_{1} ) q^{85} \) \( + \beta_{1} q^{86} \) \( + ( 3 + 2 \beta_{1} + \beta_{2} ) q^{87} \) \(+ q^{88}\) \( + ( -7 + 5 \beta_{1} + 2 \beta_{2} ) q^{89} \) \( + ( -1 + \beta_{1} ) q^{90} \) \( + ( -9 + 5 \beta_{1} + 2 \beta_{2} ) q^{91} \) \( + ( -1 - \beta_{1} + \beta_{2} ) q^{92} \) \( + ( 6 + \beta_{2} ) q^{93} \) \( + ( -1 + 3 \beta_{1} + 2 \beta_{2} ) q^{94} \) \( + ( -8 + 2 \beta_{1} - 3 \beta_{2} ) q^{95} \) \(- q^{96}\) \( + ( -2 - 4 \beta_{1} + \beta_{2} ) q^{97} \) \( + ( 3 - 3 \beta_{1} ) q^{98} \) \(+ q^{99}\) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(3q \) \(\mathstrut +\mathstrut 3q^{2} \) \(\mathstrut -\mathstrut 3q^{3} \) \(\mathstrut +\mathstrut 3q^{4} \) \(\mathstrut -\mathstrut 3q^{5} \) \(\mathstrut -\mathstrut 3q^{6} \) \(\mathstrut -\mathstrut 2q^{7} \) \(\mathstrut +\mathstrut 3q^{8} \) \(\mathstrut +\mathstrut 3q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(3q \) \(\mathstrut +\mathstrut 3q^{2} \) \(\mathstrut -\mathstrut 3q^{3} \) \(\mathstrut +\mathstrut 3q^{4} \) \(\mathstrut -\mathstrut 3q^{5} \) \(\mathstrut -\mathstrut 3q^{6} \) \(\mathstrut -\mathstrut 2q^{7} \) \(\mathstrut +\mathstrut 3q^{8} \) \(\mathstrut +\mathstrut 3q^{9} \) \(\mathstrut -\mathstrut 3q^{10} \) \(\mathstrut +\mathstrut 3q^{11} \) \(\mathstrut -\mathstrut 3q^{12} \) \(\mathstrut -\mathstrut 7q^{13} \) \(\mathstrut -\mathstrut 2q^{14} \) \(\mathstrut +\mathstrut 3q^{15} \) \(\mathstrut +\mathstrut 3q^{16} \) \(\mathstrut +\mathstrut 3q^{17} \) \(\mathstrut +\mathstrut 3q^{18} \) \(\mathstrut +\mathstrut 2q^{19} \) \(\mathstrut -\mathstrut 3q^{20} \) \(\mathstrut +\mathstrut 2q^{21} \) \(\mathstrut +\mathstrut 3q^{22} \) \(\mathstrut -\mathstrut 4q^{23} \) \(\mathstrut -\mathstrut 3q^{24} \) \(\mathstrut +\mathstrut 2q^{25} \) \(\mathstrut -\mathstrut 7q^{26} \) \(\mathstrut -\mathstrut 3q^{27} \) \(\mathstrut -\mathstrut 2q^{28} \) \(\mathstrut -\mathstrut 8q^{29} \) \(\mathstrut +\mathstrut 3q^{30} \) \(\mathstrut -\mathstrut 17q^{31} \) \(\mathstrut +\mathstrut 3q^{32} \) \(\mathstrut -\mathstrut 3q^{33} \) \(\mathstrut +\mathstrut 3q^{34} \) \(\mathstrut +\mathstrut 7q^{35} \) \(\mathstrut +\mathstrut 3q^{36} \) \(\mathstrut -\mathstrut 14q^{37} \) \(\mathstrut +\mathstrut 2q^{38} \) \(\mathstrut +\mathstrut 7q^{39} \) \(\mathstrut -\mathstrut 3q^{40} \) \(\mathstrut -\mathstrut 5q^{41} \) \(\mathstrut +\mathstrut 2q^{42} \) \(\mathstrut +\mathstrut 3q^{44} \) \(\mathstrut -\mathstrut 3q^{45} \) \(\mathstrut -\mathstrut 4q^{46} \) \(\mathstrut -\mathstrut 5q^{47} \) \(\mathstrut -\mathstrut 3q^{48} \) \(\mathstrut +\mathstrut 9q^{49} \) \(\mathstrut +\mathstrut 2q^{50} \) \(\mathstrut -\mathstrut 3q^{51} \) \(\mathstrut -\mathstrut 7q^{52} \) \(\mathstrut +\mathstrut 8q^{53} \) \(\mathstrut -\mathstrut 3q^{54} \) \(\mathstrut -\mathstrut 3q^{55} \) \(\mathstrut -\mathstrut 2q^{56} \) \(\mathstrut -\mathstrut 2q^{57} \) \(\mathstrut -\mathstrut 8q^{58} \) \(\mathstrut +\mathstrut 13q^{59} \) \(\mathstrut +\mathstrut 3q^{60} \) \(\mathstrut -\mathstrut 3q^{61} \) \(\mathstrut -\mathstrut 17q^{62} \) \(\mathstrut -\mathstrut 2q^{63} \) \(\mathstrut +\mathstrut 3q^{64} \) \(\mathstrut -\mathstrut 12q^{65} \) \(\mathstrut -\mathstrut 3q^{66} \) \(\mathstrut +\mathstrut 17q^{67} \) \(\mathstrut +\mathstrut 3q^{68} \) \(\mathstrut +\mathstrut 4q^{69} \) \(\mathstrut +\mathstrut 7q^{70} \) \(\mathstrut -\mathstrut 17q^{71} \) \(\mathstrut +\mathstrut 3q^{72} \) \(\mathstrut +\mathstrut 10q^{73} \) \(\mathstrut -\mathstrut 14q^{74} \) \(\mathstrut -\mathstrut 2q^{75} \) \(\mathstrut +\mathstrut 2q^{76} \) \(\mathstrut -\mathstrut 2q^{77} \) \(\mathstrut +\mathstrut 7q^{78} \) \(\mathstrut -\mathstrut 7q^{79} \) \(\mathstrut -\mathstrut 3q^{80} \) \(\mathstrut +\mathstrut 3q^{81} \) \(\mathstrut -\mathstrut 5q^{82} \) \(\mathstrut -\mathstrut 14q^{83} \) \(\mathstrut +\mathstrut 2q^{84} \) \(\mathstrut -\mathstrut 3q^{85} \) \(\mathstrut +\mathstrut 8q^{87} \) \(\mathstrut +\mathstrut 3q^{88} \) \(\mathstrut -\mathstrut 23q^{89} \) \(\mathstrut -\mathstrut 3q^{90} \) \(\mathstrut -\mathstrut 29q^{91} \) \(\mathstrut -\mathstrut 4q^{92} \) \(\mathstrut +\mathstrut 17q^{93} \) \(\mathstrut -\mathstrut 5q^{94} \) \(\mathstrut -\mathstrut 21q^{95} \) \(\mathstrut -\mathstrut 3q^{96} \) \(\mathstrut -\mathstrut 7q^{97} \) \(\mathstrut +\mathstrut 9q^{98} \) \(\mathstrut +\mathstrut 3q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3}\mathstrut -\mathstrut \) \(7\) \(x\mathstrut -\mathstrut \) \(3\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 5 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(5\)

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
−2.39766
−0.440808
2.83847
1.00000 −1.00000 1.00000 −3.39766 −1.00000 −4.14644 1.00000 1.00000 −3.39766
1.2 1.00000 −1.00000 1.00000 −1.44081 −1.00000 3.36488 1.00000 1.00000 −1.44081
1.3 1.00000 −1.00000 1.00000 1.83847 −1.00000 −1.21844 1.00000 1.00000 1.83847
\(n\): 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
\(2\) \(-1\)
\(3\) \(1\)
\(11\) \(-1\)
\(61\) \(1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4026))\):

\(T_{5}^{3} \) \(\mathstrut +\mathstrut 3 T_{5}^{2} \) \(\mathstrut -\mathstrut 4 T_{5} \) \(\mathstrut -\mathstrut 9 \)
\(T_{7}^{3} \) \(\mathstrut +\mathstrut 2 T_{7}^{2} \) \(\mathstrut -\mathstrut 13 T_{7} \) \(\mathstrut -\mathstrut 17 \)
\(T_{13}^{3} \) \(\mathstrut +\mathstrut 7 T_{13}^{2} \) \(\mathstrut -\mathstrut 10 T_{13} \) \(\mathstrut -\mathstrut 97 \)