Properties

Label 8025.2.a.bb
Level 8025
Weight 2
Character orbit 8025.a
Self dual Yes
Analytic conductor 64.080
Analytic rank 0
Dimension 7
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: \(7\)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{7}\mathstrut -\mathstrut \) \(3\) \(x^{6}\mathstrut -\mathstrut \) \(9\) \(x^{5}\mathstrut +\mathstrut \) \(24\) \(x^{4}\mathstrut +\mathstrut \) \(13\) \(x^{3}\mathstrut -\mathstrut \) \(47\) \(x^{2}\mathstrut +\mathstrut \) \(19\) \(x\mathstrut -\mathstrut \) \(2\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{5} + \nu^{4} + 12 \nu^{3} - \nu^{2} - 27 \nu \)\()/2\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{6} - 2 \nu^{5} - 11 \nu^{4} + 13 \nu^{3} + 26 \nu^{2} - 23 \nu - 2 \)\()/2\)
\(\beta_{4}\)\(=\)\( \nu^{6} - 2 \nu^{5} - 11 \nu^{4} + 14 \nu^{3} + 24 \nu^{2} - 29 \nu + 3 \)
\(\beta_{5}\)\(=\)\((\)\( 3 \nu^{6} - 8 \nu^{5} - 29 \nu^{4} + 61 \nu^{3} + 54 \nu^{2} - 119 \nu + 24 \)\()/2\)
\(\beta_{6}\)\(=\)\((\)\( -3 \nu^{6} + 9 \nu^{5} + 28 \nu^{4} - 73 \nu^{3} - 51 \nu^{2} + 144 \nu - 32 \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{6}\mathstrut +\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(4\)
\(\nu^{3}\)\(=\)\(2\) \(\beta_{6}\mathstrut +\mathstrut \) \(2\) \(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut -\mathstrut \) \(2\) \(\beta_{3}\mathstrut +\mathstrut \) \(2\) \(\beta_{2}\mathstrut +\mathstrut \) \(8\) \(\beta_{1}\mathstrut +\mathstrut \) \(3\)
\(\nu^{4}\)\(=\)\(13\) \(\beta_{6}\mathstrut +\mathstrut \) \(14\) \(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut -\mathstrut \) \(5\) \(\beta_{3}\mathstrut +\mathstrut \) \(11\) \(\beta_{2}\mathstrut +\mathstrut \) \(17\) \(\beta_{1}\mathstrut +\mathstrut \) \(32\)
\(\nu^{5}\)\(=\)\(36\) \(\beta_{6}\mathstrut +\mathstrut \) \(37\) \(\beta_{5}\mathstrut +\mathstrut \) \(13\) \(\beta_{4}\mathstrut -\mathstrut \) \(29\) \(\beta_{3}\mathstrut +\mathstrut \) \(32\) \(\beta_{2}\mathstrut +\mathstrut \) \(85\) \(\beta_{1}\mathstrut +\mathstrut \) \(64\)
\(\nu^{6}\)\(=\)\(163\) \(\beta_{6}\mathstrut +\mathstrut \) \(176\) \(\beta_{5}\mathstrut +\mathstrut \) \(24\) \(\beta_{4}\mathstrut -\mathstrut \) \(85\) \(\beta_{3}\mathstrut +\mathstrut \) \(133\) \(\beta_{2}\mathstrut +\mathstrut \) \(250\) \(\beta_{1}\mathstrut +\mathstrut \) \(339\)

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
−2.35320
0.180939
3.62664
1.40999
−1.83986
1.67795
0.297547
−2.79760 1.00000 5.82655 0 −2.79760 −3.22568 −10.7051 1.00000 0
1.2 −2.28249 1.00000 3.20977 0 −2.28249 1.42306 −2.76128 1.00000 0
1.3 −1.21213 1.00000 −0.530744 0 −1.21213 −4.47154 3.06759 1.00000 0
1.4 0.535263 1.00000 −1.71349 0 0.535263 3.02020 −1.98769 1.00000 0
1.5 0.726480 1.00000 −1.47223 0 0.726480 −3.04775 −2.52250 1.00000 0
1.6 2.39839 1.00000 3.75227 0 2.39839 −2.59863 4.20262 1.00000 0
1.7 2.63209 1.00000 4.92788 0 2.63209 2.90033 7.70643 1.00000 0
\(n\): 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
\(3\) \(-1\)
\(5\) \(1\)
\(107\) \(1\)

Hecke kernels

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

\(T_{2}^{7} \) \(\mathstrut -\mathstrut 14 T_{2}^{5} \) \(\mathstrut +\mathstrut T_{2}^{4} \) \(\mathstrut +\mathstrut 55 T_{2}^{3} \) \(\mathstrut -\mathstrut 8 T_{2}^{2} \) \(\mathstrut -\mathstrut 46 T_{2} \) \(\mathstrut +\mathstrut 19 \)
\(T_{7}^{7} \) \(\mathstrut +\mathstrut 6 T_{7}^{6} \) \(\mathstrut -\mathstrut 15 T_{7}^{5} \) \(\mathstrut -\mathstrut 124 T_{7}^{4} \) \(\mathstrut +\mathstrut 33 T_{7}^{3} \) \(\mathstrut +\mathstrut 788 T_{7}^{2} \) \(\mathstrut +\mathstrut 188 T_{7} \) \(\mathstrut -\mathstrut 1424 \)
\(T_{11}^{7} \) \(\mathstrut -\mathstrut 4 T_{11}^{6} \) \(\mathstrut -\mathstrut 33 T_{11}^{5} \) \(\mathstrut +\mathstrut 112 T_{11}^{4} \) \(\mathstrut +\mathstrut 277 T_{11}^{3} \) \(\mathstrut -\mathstrut 610 T_{11}^{2} \) \(\mathstrut -\mathstrut 556 T_{11} \) \(\mathstrut +\mathstrut 976 \)
\(T_{13}^{7} \) \(\mathstrut +\mathstrut 6 T_{13}^{6} \) \(\mathstrut -\mathstrut 20 T_{13}^{5} \) \(\mathstrut -\mathstrut 94 T_{13}^{4} \) \(\mathstrut +\mathstrut 152 T_{13}^{3} \) \(\mathstrut +\mathstrut 276 T_{13}^{2} \) \(\mathstrut -\mathstrut 351 T_{13} \) \(\mathstrut +\mathstrut 94 \)