Properties

Label 4026.2.a.z
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: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{7}\mathstrut -\mathstrut \) \(18\) \(x^{5}\mathstrut -\mathstrut \) \(10\) \(x^{4}\mathstrut +\mathstrut \) \(91\) \(x^{3}\mathstrut +\mathstrut \) \(90\) \(x^{2}\mathstrut -\mathstrut \) \(66\) \(x\mathstrut -\mathstrut \) \(56\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{3} - \nu^{2} - 8 \nu \)\()/2\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{6} - \nu^{5} - 13 \nu^{4} + 7 \nu^{3} + 40 \nu^{2} - 2 \nu - 8 \)\()/8\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{6} + 3 \nu^{5} + 13 \nu^{4} - 25 \nu^{3} - 52 \nu^{2} + 22 \nu + 16 \)\()/8\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{6} - \nu^{5} - 17 \nu^{4} + 7 \nu^{3} + 84 \nu^{2} + 6 \nu - 72 \)\()/8\)
\(\beta_{5}\)\(=\)\((\)\( \nu^{6} - 2 \nu^{5} - 13 \nu^{4} + 18 \nu^{3} + 48 \nu^{2} - 28 \nu - 28 \)\()/4\)
\(\beta_{6}\)\(=\)\((\)\( \nu^{6} - 2 \nu^{5} - 15 \nu^{4} + 18 \nu^{3} + 66 \nu^{2} - 16 \nu - 40 \)\()/4\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{6}\mathstrut -\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut -\mathstrut \) \(\beta_{1}\mathstrut -\mathstrut \) \(1\)\()/2\)
\(\nu^{2}\)\(=\)\(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut -\mathstrut \) \(\beta_{2}\mathstrut -\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(4\)
\(\nu^{3}\)\(=\)\(4\) \(\beta_{6}\mathstrut +\mathstrut \) \(\beta_{5}\mathstrut -\mathstrut \) \(4\) \(\beta_{4}\mathstrut +\mathstrut \) \(5\) \(\beta_{3}\mathstrut -\mathstrut \) \(\beta_{2}\mathstrut -\mathstrut \) \(3\) \(\beta_{1}\)
\(\nu^{4}\)\(=\)\(\beta_{6}\mathstrut +\mathstrut \) \(11\) \(\beta_{5}\mathstrut -\mathstrut \) \(3\) \(\beta_{4}\mathstrut +\mathstrut \) \(12\) \(\beta_{3}\mathstrut -\mathstrut \) \(9\) \(\beta_{2}\mathstrut -\mathstrut \) \(12\) \(\beta_{1}\mathstrut +\mathstrut \) \(27\)
\(\nu^{5}\)\(=\)\(31\) \(\beta_{6}\mathstrut +\mathstrut \) \(15\) \(\beta_{5}\mathstrut -\mathstrut \) \(31\) \(\beta_{4}\mathstrut +\mathstrut \) \(50\) \(\beta_{3}\mathstrut -\mathstrut \) \(11\) \(\beta_{2}\mathstrut -\mathstrut \) \(28\) \(\beta_{1}\mathstrut +\mathstrut \) \(25\)
\(\nu^{6}\)\(=\)\(17\) \(\beta_{6}\mathstrut +\mathstrut \) \(111\) \(\beta_{5}\mathstrut -\mathstrut \) \(43\) \(\beta_{4}\mathstrut +\mathstrut \) \(132\) \(\beta_{3}\mathstrut -\mathstrut \) \(73\) \(\beta_{2}\mathstrut -\mathstrut \) \(124\) \(\beta_{1}\mathstrut +\mathstrut \) \(223\)

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
3.27906
0.868643
−0.647395
−1.70776
−2.63895
−2.20695
3.05336
−1.00000 −1.00000 1.00000 −2.92143 1.00000 −1.10194 −1.00000 1.00000 2.92143
1.2 −1.00000 −1.00000 1.00000 −2.19576 1.00000 −0.720188 −1.00000 1.00000 2.19576
1.3 −1.00000 −1.00000 1.00000 −0.757998 1.00000 −3.37265 −1.00000 1.00000 0.757998
1.4 −1.00000 −1.00000 1.00000 −0.745947 1.00000 −1.28209 −1.00000 1.00000 0.745947
1.5 −1.00000 −1.00000 1.00000 2.19404 1.00000 1.89767 −1.00000 1.00000 −2.19404
1.6 −1.00000 −1.00000 1.00000 3.06294 1.00000 4.73255 −1.00000 1.00000 −3.06294
1.7 −1.00000 −1.00000 1.00000 3.36415 1.00000 −4.15335 −1.00000 1.00000 −3.36415
\(n\): 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4026))\):

\(T_{5}^{7} \) \(\mathstrut -\mathstrut 2 T_{5}^{6} \) \(\mathstrut -\mathstrut 18 T_{5}^{5} \) \(\mathstrut +\mathstrut 25 T_{5}^{4} \) \(\mathstrut +\mathstrut 104 T_{5}^{3} \) \(\mathstrut -\mathstrut 57 T_{5}^{2} \) \(\mathstrut -\mathstrut 195 T_{5} \) \(\mathstrut -\mathstrut 82 \)
\(T_{7}^{7} \) \(\mathstrut +\mathstrut 4 T_{7}^{6} \) \(\mathstrut -\mathstrut 21 T_{7}^{5} \) \(\mathstrut -\mathstrut 105 T_{7}^{4} \) \(\mathstrut -\mathstrut 36 T_{7}^{3} \) \(\mathstrut +\mathstrut 284 T_{7}^{2} \) \(\mathstrut +\mathstrut 368 T_{7} \) \(\mathstrut +\mathstrut 128 \)
\(T_{13}^{7} \) \(\mathstrut +\mathstrut 7 T_{13}^{6} \) \(\mathstrut -\mathstrut 27 T_{13}^{5} \) \(\mathstrut -\mathstrut 242 T_{13}^{4} \) \(\mathstrut -\mathstrut 192 T_{13}^{3} \) \(\mathstrut +\mathstrut 565 T_{13}^{2} \) \(\mathstrut -\mathstrut 52 T_{13} \) \(\mathstrut -\mathstrut 44 \)