Properties

Label 4026.2.a.bb
Level 4026
Weight 2
Character orbit 4026.a
Self dual Yes
Analytic conductor 32.148
Analytic rank 0
Dimension 8
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: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8}\mathstrut -\mathstrut \) \(3\) \(x^{7}\mathstrut -\mathstrut \) \(22\) \(x^{6}\mathstrut +\mathstrut \) \(42\) \(x^{5}\mathstrut +\mathstrut \) \(182\) \(x^{4}\mathstrut -\mathstrut \) \(111\) \(x^{3}\mathstrut -\mathstrut \) \(538\) \(x^{2}\mathstrut -\mathstrut \) \(256\) \(x\mathstrut -\mathstrut \) \(32\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( 17 \nu^{7} - 71 \nu^{6} - 282 \nu^{5} + 978 \nu^{4} + 1918 \nu^{3} - 3271 \nu^{2} - 5518 \nu - 808 \)\()/144\)
\(\beta_{3}\)\(=\)\((\)\( 17 \nu^{7} - 71 \nu^{6} - 282 \nu^{5} + 978 \nu^{4} + 1918 \nu^{3} - 3127 \nu^{2} - 5662 \nu - 1672 \)\()/144\)
\(\beta_{4}\)\(=\)\((\)\( -17 \nu^{7} + 47 \nu^{6} + 402 \nu^{5} - 738 \nu^{4} - 3310 \nu^{3} + 2455 \nu^{2} + 9334 \nu + 2440 \)\()/144\)
\(\beta_{5}\)\(=\)\((\)\( -35 \nu^{7} + 101 \nu^{6} + 798 \nu^{5} - 1494 \nu^{4} - 6730 \nu^{3} + 4885 \nu^{2} + 20170 \nu + 4888 \)\()/288\)
\(\beta_{6}\)\(=\)\((\)\( -\nu^{7} + 3 \nu^{6} + 22 \nu^{5} - 42 \nu^{4} - 182 \nu^{3} + 119 \nu^{2} + 530 \nu + 176 \)\()/8\)
\(\beta_{7}\)\(=\)\((\)\( -21 \nu^{7} + 67 \nu^{6} + 466 \nu^{5} - 1018 \nu^{4} - 3878 \nu^{3} + 3475 \nu^{2} + 11718 \nu + 2824 \)\()/96\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3}\mathstrut -\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(6\)
\(\nu^{3}\)\(=\)\(\beta_{6}\mathstrut -\mathstrut \) \(2\) \(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(2\) \(\beta_{3}\mathstrut -\mathstrut \) \(2\) \(\beta_{2}\mathstrut +\mathstrut \) \(11\) \(\beta_{1}\mathstrut +\mathstrut \) \(7\)
\(\nu^{4}\)\(=\)\(\beta_{7}\mathstrut +\mathstrut \) \(6\) \(\beta_{6}\mathstrut -\mathstrut \) \(7\) \(\beta_{5}\mathstrut +\mathstrut \) \(16\) \(\beta_{3}\mathstrut -\mathstrut \) \(15\) \(\beta_{2}\mathstrut +\mathstrut \) \(25\) \(\beta_{1}\mathstrut +\mathstrut \) \(59\)
\(\nu^{5}\)\(=\)\(8\) \(\beta_{7}\mathstrut +\mathstrut \) \(29\) \(\beta_{6}\mathstrut -\mathstrut \) \(52\) \(\beta_{5}\mathstrut +\mathstrut \) \(13\) \(\beta_{4}\mathstrut +\mathstrut \) \(53\) \(\beta_{3}\mathstrut -\mathstrut \) \(48\) \(\beta_{2}\mathstrut +\mathstrut \) \(146\) \(\beta_{1}\mathstrut +\mathstrut \) \(135\)
\(\nu^{6}\)\(=\)\(50\) \(\beta_{7}\mathstrut +\mathstrut \) \(147\) \(\beta_{6}\mathstrut -\mathstrut \) \(214\) \(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(275\) \(\beta_{3}\mathstrut -\mathstrut \) \(246\) \(\beta_{2}\mathstrut +\mathstrut \) \(467\) \(\beta_{1}\mathstrut +\mathstrut \) \(723\)
\(\nu^{7}\)\(=\)\(284\) \(\beta_{7}\mathstrut +\mathstrut \) \(637\) \(\beta_{6}\mathstrut -\mathstrut \) \(1128\) \(\beta_{5}\mathstrut +\mathstrut \) \(107\) \(\beta_{4}\mathstrut +\mathstrut \) \(1074\) \(\beta_{3}\mathstrut -\mathstrut \) \(919\) \(\beta_{2}\mathstrut +\mathstrut \) \(2210\) \(\beta_{1}\mathstrut +\mathstrut \) \(2277\)

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
−1.83085
−0.326029
4.26126
−2.29226
−2.68569
2.67396
−0.223274
3.42288
1.00000 1.00000 1.00000 −3.68259 1.00000 3.83085 1.00000 1.00000 −3.68259
1.2 1.00000 1.00000 1.00000 −2.31671 1.00000 2.32603 1.00000 1.00000 −2.31671
1.3 1.00000 1.00000 1.00000 −1.92862 1.00000 −2.26126 1.00000 1.00000 −1.92862
1.4 1.00000 1.00000 1.00000 0.723959 1.00000 4.29226 1.00000 1.00000 0.723959
1.5 1.00000 1.00000 1.00000 1.94066 1.00000 4.68569 1.00000 1.00000 1.94066
1.6 1.00000 1.00000 1.00000 2.72927 1.00000 −0.673957 1.00000 1.00000 2.72927
1.7 1.00000 1.00000 1.00000 3.42657 1.00000 2.22327 1.00000 1.00000 3.42657
1.8 1.00000 1.00000 1.00000 4.10747 1.00000 −1.42288 1.00000 1.00000 4.10747
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4026))\):

\(T_{5}^{8} - \cdots\)
\(T_{7}^{8} - \cdots\)
\(T_{13}^{8} - \cdots\)