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
Defining polynomial: \(x^{5} - 5 x^{3} + 4 x - 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1205)
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})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5q + q^{2} + 5q^{3} + 3q^{4} - 8q^{6} + 10q^{7} + 6q^{9} + O(q^{10}) \) \( 5q + q^{2} + 5q^{3} + 3q^{4} - 8q^{6} + 10q^{7} + 6q^{9} - 3q^{11} - 8q^{12} + q^{13} - q^{14} + 15q^{16} + 5q^{17} - 4q^{18} + 3q^{19} + 11q^{21} + 13q^{22} + 8q^{23} - 14q^{24} + 14q^{26} + 14q^{27} + 17q^{28} + 9q^{29} - 16q^{31} + 16q^{32} - 23q^{33} - 10q^{34} - 17q^{36} - 7q^{37} + 22q^{38} - 19q^{39} + 9q^{41} - 17q^{42} + 32q^{43} - 8q^{44} + 5q^{46} + 7q^{47} + 6q^{48} + 9q^{49} - 8q^{51} + 10q^{52} + 32q^{53} + 32q^{54} - 18q^{56} - 3q^{57} - 11q^{58} - 8q^{59} - 12q^{61} - 17q^{62} + 11q^{63} - 16q^{64} + 15q^{66} + 5q^{67} + 2q^{68} + 7q^{69} - 11q^{71} - 7q^{72} + 29q^{73} + 10q^{74} - 8q^{76} + 5q^{77} - 2q^{78} + 16q^{79} - 15q^{81} + 2q^{82} + 10q^{83} + 5q^{84} + 14q^{86} + 37q^{87} - 10q^{88} + 9q^{89} + 12q^{91} + 25q^{92} - 15q^{93} - 11q^{94} - 3q^{96} + 43q^{97} - 30q^{98} - 30q^{99} + O(q^{100}) \)

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

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

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.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 6025.2.a.e 5
5.b even 2 1 1205.2.a.a 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1205.2.a.a 5 5.b even 2 1
6025.2.a.e 5 1.a even 1 1 trivial

Atkin-Lehner signs

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

Hecke kernels

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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T + 4 T^{2} - 3 T^{3} + 5 T^{4} - 5 T^{5} + 10 T^{6} - 12 T^{7} + 32 T^{8} - 16 T^{9} + 32 T^{10} \)
$3$ \( 1 - 5 T + 17 T^{2} - 43 T^{3} + 99 T^{4} - 185 T^{5} + 297 T^{6} - 387 T^{7} + 459 T^{8} - 405 T^{9} + 243 T^{10} \)
$5$ 1
$7$ \( 1 - 10 T + 63 T^{2} - 277 T^{3} + 980 T^{4} - 2809 T^{5} + 6860 T^{6} - 13573 T^{7} + 21609 T^{8} - 24010 T^{9} + 16807 T^{10} \)
$11$ \( 1 + 3 T + 30 T^{2} + 64 T^{3} + 402 T^{4} + 681 T^{5} + 4422 T^{6} + 7744 T^{7} + 39930 T^{8} + 43923 T^{9} + 161051 T^{10} \)
$13$ \( 1 - T + 48 T^{2} - 34 T^{3} + 1070 T^{4} - 547 T^{5} + 13910 T^{6} - 5746 T^{7} + 105456 T^{8} - 28561 T^{9} + 371293 T^{10} \)
$17$ \( 1 - 5 T + 33 T^{2} - 181 T^{3} + 789 T^{4} - 3625 T^{5} + 13413 T^{6} - 52309 T^{7} + 162129 T^{8} - 417605 T^{9} + 1419857 T^{10} \)
$19$ \( 1 - 3 T + 72 T^{2} - 170 T^{3} + 2368 T^{4} - 4385 T^{5} + 44992 T^{6} - 61370 T^{7} + 493848 T^{8} - 390963 T^{9} + 2476099 T^{10} \)
$23$ \( 1 - 8 T + 105 T^{2} - 615 T^{3} + 4696 T^{4} - 20085 T^{5} + 108008 T^{6} - 325335 T^{7} + 1277535 T^{8} - 2238728 T^{9} + 6436343 T^{10} \)
$29$ \( 1 - 9 T + 138 T^{2} - 782 T^{3} + 7068 T^{4} - 29739 T^{5} + 204972 T^{6} - 657662 T^{7} + 3365682 T^{8} - 6365529 T^{9} + 20511149 T^{10} \)
$31$ \( 1 + 16 T + 230 T^{2} + 2063 T^{3} + 16548 T^{4} + 97141 T^{5} + 512988 T^{6} + 1982543 T^{7} + 6851930 T^{8} + 14776336 T^{9} + 28629151 T^{10} \)
$37$ \( 1 + 7 T + 136 T^{2} + 703 T^{3} + 8869 T^{4} + 36613 T^{5} + 328153 T^{6} + 962407 T^{7} + 6888808 T^{8} + 13119127 T^{9} + 69343957 T^{10} \)
$41$ \( 1 - 9 T + 232 T^{2} - 1503 T^{3} + 20127 T^{4} - 92977 T^{5} + 825207 T^{6} - 2526543 T^{7} + 15989672 T^{8} - 25431849 T^{9} + 115856201 T^{10} \)
$43$ \( 1 - 32 T + 556 T^{2} - 6781 T^{3} + 63014 T^{4} - 461877 T^{5} + 2709602 T^{6} - 12538069 T^{7} + 44205892 T^{8} - 109401632 T^{9} + 147008443 T^{10} \)
$47$ \( 1 - 7 T + 210 T^{2} - 1139 T^{3} + 18579 T^{4} - 76733 T^{5} + 873213 T^{6} - 2516051 T^{7} + 21802830 T^{8} - 34157767 T^{9} + 229345007 T^{10} \)
$53$ \( 1 - 32 T + 655 T^{2} - 9013 T^{3} + 95970 T^{4} - 781289 T^{5} + 5086410 T^{6} - 25317517 T^{7} + 97514435 T^{8} - 252495392 T^{9} + 418195493 T^{10} \)
$59$ \( 1 + 8 T + 291 T^{2} + 1707 T^{3} + 33590 T^{4} + 145309 T^{5} + 1981810 T^{6} + 5942067 T^{7} + 59765289 T^{8} + 96938888 T^{9} + 714924299 T^{10} \)
$61$ \( 1 + 12 T + 226 T^{2} + 1909 T^{3} + 24382 T^{4} + 164073 T^{5} + 1487302 T^{6} + 7103389 T^{7} + 51297706 T^{8} + 166150092 T^{9} + 844596301 T^{10} \)
$67$ \( 1 - 5 T + 152 T^{2} - 454 T^{3} + 14354 T^{4} - 39039 T^{5} + 961718 T^{6} - 2038006 T^{7} + 45715976 T^{8} - 100755605 T^{9} + 1350125107 T^{10} \)
$71$ \( 1 + 11 T + 293 T^{2} + 2666 T^{3} + 38033 T^{4} + 267381 T^{5} + 2700343 T^{6} + 13439306 T^{7} + 104867923 T^{8} + 279528491 T^{9} + 1804229351 T^{10} \)
$73$ \( 1 - 29 T + 676 T^{2} - 9972 T^{3} + 124570 T^{4} - 1149081 T^{5} + 9093610 T^{6} - 53140788 T^{7} + 262975492 T^{8} - 823548989 T^{9} + 2073071593 T^{10} \)
$79$ \( 1 - 16 T + 424 T^{2} - 4911 T^{3} + 69310 T^{4} - 576225 T^{5} + 5475490 T^{6} - 30649551 T^{7} + 209048536 T^{8} - 623201296 T^{9} + 3077056399 T^{10} \)
$83$ \( 1 - 10 T + 298 T^{2} - 2663 T^{3} + 41844 T^{4} - 313927 T^{5} + 3473052 T^{6} - 18345407 T^{7} + 170392526 T^{8} - 474583210 T^{9} + 3939040643 T^{10} \)
$89$ \( 1 - 9 T + 152 T^{2} - 1354 T^{3} + 16110 T^{4} - 77787 T^{5} + 1433790 T^{6} - 10725034 T^{7} + 107155288 T^{8} - 564680169 T^{9} + 5584059449 T^{10} \)
$97$ \( 1 - 43 T + 1116 T^{2} - 20334 T^{3} + 284864 T^{4} - 3137841 T^{5} + 27631808 T^{6} - 191322606 T^{7} + 1018543068 T^{8} - 3806759083 T^{9} + 8587340257 T^{10} \)
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