Properties

Label 4026.2.a.x
Level 4026
Weight 2
Character orbit 4026.a
Self dual Yes
Analytic conductor 32.148
Analytic rank 1
Dimension 6
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: \(6\)
Coefficient field: 6.6.46101901.1
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6}\mathstrut -\mathstrut \) \(3\) \(x^{5}\mathstrut -\mathstrut \) \(5\) \(x^{4}\mathstrut +\mathstrut \) \(12\) \(x^{3}\mathstrut +\mathstrut \) \(6\) \(x^{2}\mathstrut -\mathstrut \) \(5\) \(x\mathstrut -\mathstrut \) \(2\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( -\nu^{5} + 3 \nu^{4} + 5 \nu^{3} - 12 \nu^{2} - 5 \nu + 4 \)
\(\beta_{3}\)\(=\)\( -\nu^{5} + 4 \nu^{4} + 2 \nu^{3} - 16 \nu^{2} + 4 \nu + 6 \)
\(\beta_{4}\)\(=\)\((\)\( -3 \nu^{5} + 10 \nu^{4} + 11 \nu^{3} - 37 \nu^{2} - 3 \nu + 8 \)\()/2\)
\(\beta_{5}\)\(=\)\((\)\( -3 \nu^{5} + 10 \nu^{4} + 11 \nu^{3} - 39 \nu^{2} - \nu + 14 \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\)\(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(3\)
\(\nu^{3}\)\(=\)\(-\)\(3\) \(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(2\) \(\beta_{2}\mathstrut +\mathstrut \) \(6\) \(\beta_{1}\mathstrut +\mathstrut \) \(3\)
\(\nu^{4}\)\(=\)\(-\)\(13\) \(\beta_{5}\mathstrut +\mathstrut \) \(7\) \(\beta_{4}\mathstrut +\mathstrut \) \(4\) \(\beta_{3}\mathstrut +\mathstrut \) \(5\) \(\beta_{2}\mathstrut +\mathstrut \) \(13\) \(\beta_{1}\mathstrut +\mathstrut \) \(19\)
\(\nu^{5}\)\(=\)\(-\)\(42\) \(\beta_{5}\mathstrut +\mathstrut \) \(14\) \(\beta_{4}\mathstrut +\mathstrut \) \(17\) \(\beta_{3}\mathstrut +\mathstrut \) \(24\) \(\beta_{2}\mathstrut +\mathstrut \) \(52\) \(\beta_{1}\mathstrut +\mathstrut \) \(40\)

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
−0.654438
1.90193
3.29072
−1.86911
−0.371781
0.702675
1.00000 −1.00000 1.00000 −4.17718 −1.00000 2.87039 1.00000 1.00000 −4.17718
1.2 1.00000 −1.00000 1.00000 −4.05628 −1.00000 −1.59539 1.00000 1.00000 −4.05628
1.3 1.00000 −1.00000 1.00000 −0.656263 −1.00000 4.00423 1.00000 1.00000 −0.656263
1.4 1.00000 −1.00000 1.00000 0.199215 −1.00000 −0.938542 1.00000 1.00000 0.199215
1.5 1.00000 −1.00000 1.00000 1.28209 −1.00000 −0.306395 1.00000 1.00000 1.28209
1.6 1.00000 −1.00000 1.00000 1.40842 −1.00000 −3.03430 1.00000 1.00000 1.40842
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4026))\):

\(T_{5}^{6} \) \(\mathstrut +\mathstrut 6 T_{5}^{5} \) \(\mathstrut -\mathstrut T_{5}^{4} \) \(\mathstrut -\mathstrut 33 T_{5}^{3} \) \(\mathstrut +\mathstrut 17 T_{5}^{2} \) \(\mathstrut +\mathstrut 18 T_{5} \) \(\mathstrut -\mathstrut 4 \)
\(T_{7}^{6} \) \(\mathstrut -\mathstrut T_{7}^{5} \) \(\mathstrut -\mathstrut 18 T_{7}^{4} \) \(\mathstrut +\mathstrut 76 T_{7}^{2} \) \(\mathstrut +\mathstrut 75 T_{7} \) \(\mathstrut +\mathstrut 16 \)
\(T_{13}^{6} \) \(\mathstrut -\mathstrut 2 T_{13}^{5} \) \(\mathstrut -\mathstrut 18 T_{13}^{4} \) \(\mathstrut +\mathstrut 22 T_{13}^{3} \) \(\mathstrut +\mathstrut 69 T_{13}^{2} \) \(\mathstrut +\mathstrut 35 T_{13} \) \(\mathstrut +\mathstrut 2 \)