Properties

Label 6025.2.a.f
Level $6025$
Weight $2$
Character orbit 6025.a
Self dual yes
Analytic conductor $48.110$
Analytic rank $1$
Dimension $7$
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: \(1\)
Dimension: \(7\)
Coefficient field: 7.7.31056073.1
Defining polynomial: \(x^{7} - 3 x^{6} - 3 x^{5} + 11 x^{4} + x^{3} - 9 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 241)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{7} - 3 x^{6} - 3 x^{5} + 11 x^{4} + x^{3} - 9 x^{2} + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{5} - 2 \nu^{4} - 4 \nu^{3} + 5 \nu^{2} + 2 \nu - 1 \)
\(\beta_{3}\)\(=\)\( \nu^{5} - 2 \nu^{4} - 4 \nu^{3} + 4 \nu^{2} + 3 \nu + 1 \)
\(\beta_{4}\)\(=\)\( \nu^{6} - 3 \nu^{5} - 2 \nu^{4} + 9 \nu^{3} - 3 \nu^{2} - 4 \nu + 2 \)
\(\beta_{5}\)\(=\)\( \nu^{6} - 2 \nu^{5} - 5 \nu^{4} + 6 \nu^{3} + 7 \nu^{2} - 3 \nu - 2 \)
\(\beta_{6}\)\(=\)\( -\nu^{6} + 3 \nu^{5} + 3 \nu^{4} - 11 \nu^{3} - \nu^{2} + 8 \nu \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{3} + \beta_{2} + \beta_{1} + 2\)
\(\nu^{3}\)\(=\)\(-\beta_{6} - \beta_{5} - \beta_{3} + 2 \beta_{2} + 4 \beta_{1} + 1\)
\(\nu^{4}\)\(=\)\(-\beta_{6} - 2 \beta_{5} + \beta_{4} - 6 \beta_{3} + 8 \beta_{2} + 8 \beta_{1} + 8\)
\(\nu^{5}\)\(=\)\(-6 \beta_{6} - 8 \beta_{5} + 2 \beta_{4} - 11 \beta_{3} + 20 \beta_{2} + 25 \beta_{1} + 11\)
\(\nu^{6}\)\(=\)\(-11 \beta_{6} - 19 \beta_{5} + 9 \beta_{4} - 39 \beta_{3} + 61 \beta_{2} + 62 \beta_{1} + 44\)

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
2.73684
1.48734
1.27758
0.369356
−0.356270
−0.911223
−1.60363
−1.73684 2.37146 1.01662 0 −4.11885 2.01025 1.70797 2.62382 0
1.2 −0.487343 0.815004 −1.76250 0 −0.397187 4.61392 1.83363 −2.33577 0
1.3 −0.277577 0.494846 −1.92295 0 −0.137358 −1.36627 1.08892 −2.75513 0
1.4 0.630644 −2.33806 −1.60229 0 −1.47449 3.68231 −2.27176 2.46654 0
1.5 1.35627 2.45059 −0.160532 0 3.32366 0.283608 −2.93026 3.00540 0
1.6 1.91122 0.186202 1.65278 0 0.355874 −3.52970 −0.663624 −2.96533 0
1.7 2.60363 −0.980039 4.77887 0 −2.55166 1.30586 7.23513 −2.03952 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.7
Significant digits:
Format:

Atkin-Lehner signs

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

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 6025.2.a.f 7
5.b even 2 1 241.2.a.a 7
15.d odd 2 1 2169.2.a.e 7
20.d odd 2 1 3856.2.a.j 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
241.2.a.a 7 5.b even 2 1
2169.2.a.e 7 15.d odd 2 1
3856.2.a.j 7 20.d odd 2 1
6025.2.a.f 7 1.a even 1 1 trivial

Hecke kernels

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

\( T_{2}^{7} - 4 T_{2}^{6} + 14 T_{2}^{4} - 10 T_{2}^{3} - 6 T_{2}^{2} + 3 T_{2} + 1 \)
\( T_{3}^{7} - 3 T_{3}^{6} - 5 T_{3}^{5} + 19 T_{3}^{4} - 4 T_{3}^{3} - 14 T_{3}^{2} + 8 T_{3} - 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 3 T - 6 T^{2} - 10 T^{3} + 14 T^{4} - 4 T^{6} + T^{7} \)
$3$ \( -1 + 8 T - 14 T^{2} - 4 T^{3} + 19 T^{4} - 5 T^{5} - 3 T^{6} + T^{7} \)
$5$ \( T^{7} \)
$7$ \( -61 + 260 T - 127 T^{2} - 138 T^{3} + 98 T^{4} - 3 T^{5} - 7 T^{6} + T^{7} \)
$11$ \( 1069 - 1281 T - 1559 T^{2} - 137 T^{3} + 283 T^{4} + 117 T^{5} + 18 T^{6} + T^{7} \)
$13$ \( 1 + 13 T - 860 T^{2} + 533 T^{3} + 62 T^{4} - 48 T^{5} - T^{6} + T^{7} \)
$17$ \( 1039 - 633 T - 1382 T^{2} + 967 T^{3} + 86 T^{4} - 65 T^{5} - 2 T^{6} + T^{7} \)
$19$ \( -5983 - 6849 T + 3337 T^{2} + 1067 T^{3} - 276 T^{4} - 56 T^{5} + 6 T^{6} + T^{7} \)
$23$ \( -1369 - 532 T + 1693 T^{2} - 75 T^{3} - 463 T^{4} + 168 T^{5} - 22 T^{6} + T^{7} \)
$29$ \( -10769 - 23012 T - 17808 T^{2} - 6173 T^{3} - 839 T^{4} + 25 T^{5} + 16 T^{6} + T^{7} \)
$31$ \( -617 - 3770 T - 3600 T^{2} - 1006 T^{3} + 109 T^{4} + 104 T^{5} + 18 T^{6} + T^{7} \)
$37$ \( 78167 + 88791 T + 27243 T^{2} - 828 T^{3} - 1330 T^{4} - 119 T^{5} + 8 T^{6} + T^{7} \)
$41$ \( 101009 - 157642 T + 58348 T^{2} + 5058 T^{3} - 1974 T^{4} - 122 T^{5} + 15 T^{6} + T^{7} \)
$43$ \( -296569 + 67463 T + 74048 T^{2} - 49 T^{3} - 2460 T^{4} - 141 T^{5} + 14 T^{6} + T^{7} \)
$47$ \( 7793 - 48600 T - 18600 T^{2} + 3904 T^{3} + 997 T^{4} - 116 T^{5} - 10 T^{6} + T^{7} \)
$53$ \( 230663 + 281202 T + 84407 T^{2} - 35 T^{3} - 2311 T^{4} - 123 T^{5} + 15 T^{6} + T^{7} \)
$59$ \( -2076763 - 114293 T + 152610 T^{2} + 5036 T^{3} - 3030 T^{4} - 130 T^{5} + 18 T^{6} + T^{7} \)
$61$ \( 23149 + 291498 T - 136478 T^{2} + 13144 T^{3} + 1693 T^{4} - 254 T^{5} - 4 T^{6} + T^{7} \)
$67$ \( -2288147 + 4468 T + 189536 T^{2} + 4503 T^{3} - 3579 T^{4} - 157 T^{5} + 18 T^{6} + T^{7} \)
$71$ \( -255937 - 122885 T + 37139 T^{2} + 34990 T^{3} + 8586 T^{4} + 955 T^{5} + 50 T^{6} + T^{7} \)
$73$ \( -11879 - 5297 T + 192681 T^{2} + 37009 T^{3} - 1068 T^{4} - 378 T^{5} + T^{7} \)
$79$ \( 52709 + 79692 T + 33855 T^{2} + 711 T^{3} - 1557 T^{4} - 85 T^{5} + 15 T^{6} + T^{7} \)
$83$ \( -4333 + 2523 T + 1594 T^{2} - 1238 T^{3} + 14 T^{4} + 124 T^{5} - 24 T^{6} + T^{7} \)
$89$ \( -89477 + 26169 T + 119043 T^{2} + 12156 T^{3} - 4667 T^{4} - 363 T^{5} + 13 T^{6} + T^{7} \)
$97$ \( -40121 - 136236 T + 43003 T^{2} + 15061 T^{3} - 939 T^{4} - 283 T^{5} + T^{6} + T^{7} \)
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