Properties

Label 8025.2.a.ba
Level 8025
Weight 2
Character orbit 8025.a
Self dual Yes
Analytic conductor 64.080
Analytic rank 0
Dimension 6
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 8025 = 3 \cdot 5^{2} \cdot 107 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 8025.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(64.0799476221\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.13231312.1
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6}\mathstrut -\mathstrut \) \(x^{5}\mathstrut -\mathstrut \) \(8\) \(x^{4}\mathstrut +\mathstrut \) \(9\) \(x^{3}\mathstrut +\mathstrut \) \(8\) \(x^{2}\mathstrut -\mathstrut \) \(9\) \(x\mathstrut +\mathstrut \) \(1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{4} - 7 \nu^{2} + 2 \nu + 5 \)\()/2\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{5} - 7 \nu^{3} + 2 \nu^{2} + 3 \nu - 2 \)\()/2\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{5} - \nu^{4} - 9 \nu^{3} + 9 \nu^{2} + 13 \nu - 9 \)\()/2\)
\(\beta_{5}\)\(=\)\((\)\( 2 \nu^{5} - \nu^{4} - 16 \nu^{3} + 9 \nu^{2} + 18 \nu - 5 \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\)\(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(3\)
\(\nu^{3}\)\(=\)\(-\)\(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut -\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(6\) \(\beta_{1}\mathstrut -\mathstrut \) \(1\)
\(\nu^{4}\)\(=\)\(-\)\(7\) \(\beta_{5}\mathstrut +\mathstrut \) \(7\) \(\beta_{4}\mathstrut +\mathstrut \) \(7\) \(\beta_{3}\mathstrut +\mathstrut \) \(2\) \(\beta_{2}\mathstrut +\mathstrut \) \(5\) \(\beta_{1}\mathstrut +\mathstrut \) \(16\)
\(\nu^{5}\)\(=\)\(2\) \(\beta_{5}\mathstrut -\mathstrut \) \(9\) \(\beta_{4}\mathstrut +\mathstrut \) \(7\) \(\beta_{3}\mathstrut -\mathstrut \) \(7\) \(\beta_{2}\mathstrut +\mathstrut \) \(37\) \(\beta_{1}\mathstrut -\mathstrut \) \(11\)

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
0.127357
2.46251
0.759615
−2.66308
1.42204
−1.10843
−2.57072 −1.00000 4.60860 0 2.57072 4.28120 −6.70597 1.00000 0
1.2 −2.12439 −1.00000 2.51302 0 2.12439 −2.71830 −1.08985 1.00000 0
1.3 −1.40654 −1.00000 −0.0216539 0 1.40654 −1.09008 2.84353 1.00000 0
1.4 −0.163212 −1.00000 −1.97336 0 0.163212 −1.53872 0.648498 1.00000 0
1.5 1.11100 −1.00000 −0.765670 0 −1.11100 0.814220 −3.07267 1.00000 0
1.6 2.15385 −1.00000 2.63907 0 −2.15385 0.251675 1.37646 1.00000 0
\(n\): 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
\(3\) \(1\)
\(5\) \(1\)
\(107\) \(-1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8025))\):

\(T_{2}^{6} \) \(\mathstrut +\mathstrut 3 T_{2}^{5} \) \(\mathstrut -\mathstrut 5 T_{2}^{4} \) \(\mathstrut -\mathstrut 18 T_{2}^{3} \) \(\mathstrut +\mathstrut T_{2}^{2} \) \(\mathstrut +\mathstrut 19 T_{2} \) \(\mathstrut +\mathstrut 3 \)
\(T_{7}^{6} \) \(\mathstrut -\mathstrut 15 T_{7}^{4} \) \(\mathstrut -\mathstrut 18 T_{7}^{3} \) \(\mathstrut +\mathstrut 13 T_{7}^{2} \) \(\mathstrut +\mathstrut 14 T_{7} \) \(\mathstrut -\mathstrut 4 \)
\(T_{11}^{6} \) \(\mathstrut -\mathstrut 6 T_{11}^{5} \) \(\mathstrut -\mathstrut 21 T_{11}^{4} \) \(\mathstrut +\mathstrut 138 T_{11}^{3} \) \(\mathstrut -\mathstrut 7 T_{11}^{2} \) \(\mathstrut -\mathstrut 632 T_{11} \) \(\mathstrut +\mathstrut 636 \)
\(T_{13}^{6} \) \(\mathstrut -\mathstrut 8 T_{13}^{5} \) \(\mathstrut -\mathstrut 4 T_{13}^{4} \) \(\mathstrut +\mathstrut 122 T_{13}^{3} \) \(\mathstrut -\mathstrut 20 T_{13}^{2} \) \(\mathstrut -\mathstrut 500 T_{13} \) \(\mathstrut -\mathstrut 359 \)