Properties

Label 6025.2.a.e
Level 6025
Weight 2
Character orbit 6025.a
Self dual Yes
Analytic conductor 48.110
Analytic rank 0
Dimension 5
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 6025 = 5^{2} \cdot 241 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 6025.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(48.1098672178\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.38569.1
Coefficient ring: \(\Z[a_1, a_2]\)
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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{4} - 4 \nu^{2} + 1 \)
\(\beta_{3}\)\(=\)\( -\nu^{4} + 5 \nu^{2} - 3 \)
\(\beta_{4}\)\(=\)\( \nu^{4} + \nu^{3} - 5 \nu^{2} - 4 \nu + 3 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(2\)
\(\nu^{3}\)\(=\)\(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(4\) \(\beta_{1}\)
\(\nu^{4}\)\(=\)\(4\) \(\beta_{3}\mathstrut +\mathstrut \) \(5\) \(\beta_{2}\mathstrut +\mathstrut \) \(7\)

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
0.275834
−1.95408
0.790734
−1.15098
2.03850
−2.34953 1.80652 3.52030 0 −4.24447 4.19975 −3.57200 0.263500 0
1.2 −0.442330 2.59927 −1.80434 0 −1.14974 −1.77251 1.68277 3.75621 0
1.3 0.526087 2.87779 −1.72323 0 1.51397 4.16547 −1.95874 5.28166 0
1.4 0.717838 −0.928169 −1.48471 0 −0.666275 1.52425 −2.50146 −2.13850 0
1.5 2.54794 −1.35541 4.49198 0 −3.45349 1.88303 6.34942 −1.16287 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
Significant digits:
Format:

Inner twists

This newform does not admit any (nontrivial) inner twists.

Atkin-Lehner signs

\( p \) Sign
\(5\) \(1\)
\(241\) \(-1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6025))\):

\(T_{2}^{5} \) \(\mathstrut -\mathstrut T_{2}^{4} \) \(\mathstrut -\mathstrut 6 T_{2}^{3} \) \(\mathstrut +\mathstrut 5 T_{2}^{2} \) \(\mathstrut +\mathstrut T_{2} \) \(\mathstrut -\mathstrut 1 \)
\(T_{3}^{5} \) \(\mathstrut -\mathstrut 5 T_{3}^{4} \) \(\mathstrut +\mathstrut 2 T_{3}^{3} \) \(\mathstrut +\mathstrut 17 T_{3}^{2} \) \(\mathstrut -\mathstrut 9 T_{3} \) \(\mathstrut -\mathstrut 17 \)