Properties

Label 4026.2.a.u
Level 4026
Weight 2
Character orbit 4026.a
Self dual Yes
Analytic conductor 32.148
Analytic rank 0
Dimension 5
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: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.9176805.1
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{5}\mathstrut -\mathstrut \) \(x^{4}\mathstrut -\mathstrut \) \(12\) \(x^{3}\mathstrut +\mathstrut \) \(7\) \(x^{2}\mathstrut +\mathstrut \) \(30\) \(x\mathstrut -\mathstrut \) \(20\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{4} - \nu^{3} - 8 \nu^{2} + \nu + 6 \)\()/2\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{4} - \nu^{3} - 10 \nu^{2} + 3 \nu + 14 \)\()/2\)
\(\beta_{4}\)\(=\)\( \nu^{3} - 2 \nu^{2} - 7 \nu + 8 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\)\(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(4\)
\(\nu^{3}\)\(=\)\(\beta_{4}\mathstrut -\mathstrut \) \(2\) \(\beta_{3}\mathstrut +\mathstrut \) \(2\) \(\beta_{2}\mathstrut +\mathstrut \) \(9\) \(\beta_{1}\)
\(\nu^{4}\)\(=\)\(\beta_{4}\mathstrut -\mathstrut \) \(10\) \(\beta_{3}\mathstrut +\mathstrut \) \(12\) \(\beta_{2}\mathstrut +\mathstrut \) \(16\) \(\beta_{1}\mathstrut +\mathstrut \) \(26\)

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
3.32361
1.62906
0.689091
−2.15973
−2.48204
−1.00000 1.00000 1.00000 −3.32361 −1.00000 −3.91566 −1.00000 1.00000 3.32361
1.2 −1.00000 1.00000 1.00000 −1.62906 −1.00000 −4.09482 −1.00000 1.00000 1.62906
1.3 −1.00000 1.00000 1.00000 −0.689091 −1.00000 4.91945 −1.00000 1.00000 0.689091
1.4 −1.00000 1.00000 1.00000 2.15973 −1.00000 −1.48663 −1.00000 1.00000 −2.15973
1.5 −1.00000 1.00000 1.00000 2.48204 −1.00000 1.57766 −1.00000 1.00000 −2.48204
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4026))\):

\(T_{5}^{5} \) \(\mathstrut +\mathstrut T_{5}^{4} \) \(\mathstrut -\mathstrut 12 T_{5}^{3} \) \(\mathstrut -\mathstrut 7 T_{5}^{2} \) \(\mathstrut +\mathstrut 30 T_{5} \) \(\mathstrut +\mathstrut 20 \)
\(T_{7}^{5} \) \(\mathstrut +\mathstrut 3 T_{7}^{4} \) \(\mathstrut -\mathstrut 26 T_{7}^{3} \) \(\mathstrut -\mathstrut 84 T_{7}^{2} \) \(\mathstrut +\mathstrut 62 T_{7} \) \(\mathstrut +\mathstrut 185 \)
\(T_{13}^{5} \) \(\mathstrut -\mathstrut 2 T_{13}^{4} \) \(\mathstrut -\mathstrut 34 T_{13}^{3} \) \(\mathstrut +\mathstrut 56 T_{13}^{2} \) \(\mathstrut +\mathstrut 193 T_{13} \) \(\mathstrut -\mathstrut 339 \)