Properties

Label 4026.2.a.ba
Level 4026
Weight 2
Character orbit 4026.a
Self dual Yes
Analytic conductor 32.148
Analytic rank 0
Dimension 7
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: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
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 \(1,\beta_1,\ldots,\beta_{6}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{7}\mathstrut -\mathstrut \) \(2\) \(x^{6}\mathstrut -\mathstrut \) \(16\) \(x^{5}\mathstrut +\mathstrut \) \(19\) \(x^{4}\mathstrut +\mathstrut \) \(85\) \(x^{3}\mathstrut -\mathstrut \) \(23\) \(x^{2}\mathstrut -\mathstrut \) \(162\) \(x\mathstrut -\mathstrut \) \(82\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{6} + 3 \nu^{5} + 11 \nu^{4} - 28 \nu^{3} - 29 \nu^{2} + 48 \nu + 28 \)\()/2\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{6} - 3 \nu^{5} - 11 \nu^{4} + 28 \nu^{3} + 31 \nu^{2} - 50 \nu - 38 \)\()/2\)
\(\beta_{4}\)\(=\)\( -\nu^{6} + 3 \nu^{5} + 12 \nu^{4} - 30 \nu^{3} - 41 \nu^{2} + 61 \nu + 57 \)
\(\beta_{5}\)\(=\)\( -2 \nu^{6} + 7 \nu^{5} + 22 \nu^{4} - 71 \nu^{3} - 70 \nu^{2} + 148 \nu + 117 \)
\(\beta_{6}\)\(=\)\( -2 \nu^{6} + 7 \nu^{5} + 22 \nu^{4} - 72 \nu^{3} - 69 \nu^{2} + 155 \nu + 117 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(5\)
\(\nu^{3}\)\(=\)\(-\)\(\beta_{6}\mathstrut +\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(8\) \(\beta_{1}\mathstrut +\mathstrut \) \(5\)
\(\nu^{4}\)\(=\)\(-\)\(2\) \(\beta_{6}\mathstrut +\mathstrut \) \(2\) \(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(14\) \(\beta_{3}\mathstrut +\mathstrut \) \(12\) \(\beta_{2}\mathstrut +\mathstrut \) \(15\) \(\beta_{1}\mathstrut +\mathstrut \) \(41\)
\(\nu^{5}\)\(=\)\(-\)\(15\) \(\beta_{6}\mathstrut +\mathstrut \) \(16\) \(\beta_{5}\mathstrut +\mathstrut \) \(27\) \(\beta_{3}\mathstrut +\mathstrut \) \(23\) \(\beta_{2}\mathstrut +\mathstrut \) \(80\) \(\beta_{1}\mathstrut +\mathstrut \) \(74\)
\(\nu^{6}\)\(=\)\(-\)\(39\) \(\beta_{6}\mathstrut +\mathstrut \) \(42\) \(\beta_{5}\mathstrut +\mathstrut \) \(11\) \(\beta_{4}\mathstrut +\mathstrut \) \(178\) \(\beta_{3}\mathstrut +\mathstrut \) \(142\) \(\beta_{2}\mathstrut +\mathstrut \) \(200\) \(\beta_{1}\mathstrut +\mathstrut \) \(416\)

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
−2.69874
−1.47556
−1.38259
−0.749404
2.21896
2.48922
3.59811
−1.00000 1.00000 1.00000 −3.69874 −1.00000 3.64759 −1.00000 1.00000 3.69874
1.2 −1.00000 1.00000 1.00000 −2.47556 −1.00000 3.41351 −1.00000 1.00000 2.47556
1.3 −1.00000 1.00000 1.00000 −2.38259 −1.00000 0.127619 −1.00000 1.00000 2.38259
1.4 −1.00000 1.00000 1.00000 −1.74940 −1.00000 −3.94521 −1.00000 1.00000 1.74940
1.5 −1.00000 1.00000 1.00000 1.21896 −1.00000 −1.75098 −1.00000 1.00000 −1.21896
1.6 −1.00000 1.00000 1.00000 1.48922 −1.00000 4.53318 −1.00000 1.00000 −1.48922
1.7 −1.00000 1.00000 1.00000 2.59811 −1.00000 2.97430 −1.00000 1.00000 −2.59811
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.7
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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4026))\):

\(T_{5}^{7} \) \(\mathstrut +\mathstrut 5 T_{5}^{6} \) \(\mathstrut -\mathstrut 7 T_{5}^{5} \) \(\mathstrut -\mathstrut 56 T_{5}^{4} \) \(\mathstrut -\mathstrut 4 T_{5}^{3} \) \(\mathstrut +\mathstrut 177 T_{5}^{2} \) \(\mathstrut +\mathstrut 38 T_{5} \) \(\mathstrut -\mathstrut 180 \)
\(T_{7}^{7} \) \(\mathstrut -\mathstrut 9 T_{7}^{6} \) \(\mathstrut +\mathstrut 4 T_{7}^{5} \) \(\mathstrut +\mathstrut 160 T_{7}^{4} \) \(\mathstrut -\mathstrut 382 T_{7}^{3} \) \(\mathstrut -\mathstrut 301 T_{7}^{2} \) \(\mathstrut +\mathstrut 1204 T_{7} \) \(\mathstrut -\mathstrut 148 \)
\(T_{13}^{7} \) \(\mathstrut -\mathstrut 4 T_{13}^{6} \) \(\mathstrut -\mathstrut 63 T_{13}^{5} \) \(\mathstrut +\mathstrut 298 T_{13}^{4} \) \(\mathstrut +\mathstrut 865 T_{13}^{3} \) \(\mathstrut -\mathstrut 5437 T_{13}^{2} \) \(\mathstrut +\mathstrut 2219 T_{13} \) \(\mathstrut +\mathstrut 10985 \)