Properties

Label 4025.2.a.n
Level 4025
Weight 2
Character orbit 4025.a
Self dual yes
Analytic conductor 32.140
Analytic rank 0
Dimension 4
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 4025 = 5^{2} \cdot 7 \cdot 23 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4025.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(32.1397868136\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.22545.1
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 805)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + ( -\beta_{1} - \beta_{3} ) q^{3} + ( 3 + \beta_{3} ) q^{4} + ( 1 - 3 \beta_{1} - \beta_{2} ) q^{6} + q^{7} + ( -1 + \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{8} + ( 4 + \beta_{1} + \beta_{2} ) q^{9} +O(q^{10})\) \( q + \beta_{2} q^{2} + ( -\beta_{1} - \beta_{3} ) q^{3} + ( 3 + \beta_{3} ) q^{4} + ( 1 - 3 \beta_{1} - \beta_{2} ) q^{6} + q^{7} + ( -1 + \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{8} + ( 4 + \beta_{1} + \beta_{2} ) q^{9} + ( 1 + \beta_{2} ) q^{11} + ( -5 - 4 \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{12} + ( 4 + \beta_{1} ) q^{13} + \beta_{2} q^{14} + ( 5 + \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{16} + ( 3 - 2 \beta_{1} - \beta_{2} ) q^{17} + ( 5 + 2 \beta_{1} + 4 \beta_{2} + 2 \beta_{3} ) q^{18} + ( 1 + \beta_{1} - \beta_{2} ) q^{19} + ( -\beta_{1} - \beta_{3} ) q^{21} + ( 5 + \beta_{2} + \beta_{3} ) q^{22} - q^{23} + ( 5 - 4 \beta_{1} - 5 \beta_{2} - \beta_{3} ) q^{24} + ( 2 \beta_{1} + 4 \beta_{2} + \beta_{3} ) q^{26} + ( -1 - 4 \beta_{1} - 3 \beta_{2} - 2 \beta_{3} ) q^{27} + ( 3 + \beta_{3} ) q^{28} + ( 2 + 2 \beta_{1} ) q^{29} + ( -1 + 3 \beta_{1} - \beta_{2} ) q^{31} + ( -10 + 2 \beta_{1} + 3 \beta_{2} - \beta_{3} ) q^{32} + ( 1 - 4 \beta_{1} - \beta_{2} - \beta_{3} ) q^{33} + ( -5 - 4 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} ) q^{34} + ( 10 + 4 \beta_{1} + 5 \beta_{2} + 4 \beta_{3} ) q^{36} + ( -1 - 2 \beta_{1} + \beta_{2} ) q^{37} + ( -5 + 2 \beta_{1} + \beta_{2} ) q^{38} + ( -2 - 4 \beta_{1} - 2 \beta_{2} - 5 \beta_{3} ) q^{39} + ( 3 - 3 \beta_{1} - \beta_{2} + \beta_{3} ) q^{41} + ( 1 - 3 \beta_{1} - \beta_{2} ) q^{42} + ( 2 + \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{43} + ( 2 + \beta_{1} + 4 \beta_{2} ) q^{44} -\beta_{2} q^{46} + ( 3 - \beta_{1} + \beta_{2} ) q^{47} + ( -14 - \beta_{1} + 2 \beta_{2} - 4 \beta_{3} ) q^{48} + q^{49} + ( 3 + 5 \beta_{2} - \beta_{3} ) q^{51} + ( 11 + 3 \beta_{1} + \beta_{2} + 5 \beta_{3} ) q^{52} + ( -6 - 2 \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{53} + ( -13 - 10 \beta_{1} - 3 \beta_{2} - 5 \beta_{3} ) q^{54} + ( -1 + \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{56} + ( -3 + 2 \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{57} + ( 4 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{58} + ( 6 - \beta_{1} ) q^{59} + ( -4 - 2 \beta_{2} - \beta_{3} ) q^{61} + ( -5 + 6 \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{62} + ( 4 + \beta_{1} + \beta_{2} ) q^{63} + ( 6 + \beta_{1} - 7 \beta_{2} + 2 \beta_{3} ) q^{64} + ( -4 - 9 \beta_{1} - 4 \beta_{3} ) q^{66} + ( -2 + 3 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{67} + ( 12 - 7 \beta_{1} - 6 \beta_{2} + 2 \beta_{3} ) q^{68} + ( \beta_{1} + \beta_{3} ) q^{69} + ( 5 - \beta_{1} + 3 \beta_{2} + 2 \beta_{3} ) q^{71} + ( 11 + 8 \beta_{1} + 6 \beta_{2} + \beta_{3} ) q^{72} + ( 4 + 2 \beta_{1} - 4 \beta_{2} - \beta_{3} ) q^{73} + ( 5 - 4 \beta_{1} - \beta_{2} - \beta_{3} ) q^{74} + ( 3 + 2 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} ) q^{76} + ( 1 + \beta_{2} ) q^{77} + ( -5 - 13 \beta_{1} - 7 \beta_{2} - \beta_{3} ) q^{78} + ( 3 + 2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{79} + ( 3 + 9 \beta_{1} + 6 \beta_{2} + 3 \beta_{3} ) q^{81} + ( -6 - 5 \beta_{1} + 4 \beta_{2} - 5 \beta_{3} ) q^{82} + ( 5 \beta_{1} + 2 \beta_{3} ) q^{83} + ( -5 - 4 \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{84} + ( -11 + 3 \beta_{1} + 3 \beta_{2} - 2 \beta_{3} ) q^{86} + ( -4 - 2 \beta_{1} - 4 \beta_{2} - 4 \beta_{3} ) q^{87} + ( 10 + 2 \beta_{1} + 3 \beta_{3} ) q^{88} + ( -1 - 4 \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{89} + ( 4 + \beta_{1} ) q^{91} + ( -3 - \beta_{3} ) q^{92} + ( -7 + 4 \beta_{1} - 5 \beta_{2} - 2 \beta_{3} ) q^{93} + ( 5 - 2 \beta_{1} + 3 \beta_{2} ) q^{94} + ( 4 + 2 \beta_{1} - 8 \beta_{2} + 7 \beta_{3} ) q^{96} + ( -1 + \beta_{1} - \beta_{2} - 3 \beta_{3} ) q^{97} + \beta_{2} q^{98} + ( 9 + 3 \beta_{1} + 5 \beta_{2} + 2 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + q^{2} + 11q^{4} + 4q^{7} + 18q^{9} + O(q^{10}) \) \( 4q + q^{2} + 11q^{4} + 4q^{7} + 18q^{9} + 5q^{11} - 21q^{12} + 17q^{13} + q^{14} + 17q^{16} + 9q^{17} + 24q^{18} + 4q^{19} + 20q^{22} - 4q^{23} + 12q^{24} + 5q^{26} - 9q^{27} + 11q^{28} + 10q^{29} - 2q^{31} - 34q^{32} - 18q^{34} + 45q^{36} - 5q^{37} - 17q^{38} - 9q^{39} + 7q^{41} + 6q^{43} + 13q^{44} - q^{46} + 12q^{47} - 51q^{48} + 4q^{49} + 18q^{51} + 43q^{52} - 23q^{53} - 60q^{54} - 9q^{57} + 4q^{58} + 23q^{59} - 17q^{61} - 17q^{62} + 18q^{63} + 16q^{64} - 21q^{66} - 5q^{67} + 33q^{68} + 20q^{71} + 57q^{72} + 15q^{73} + 16q^{74} + 8q^{76} + 5q^{77} - 39q^{78} + 12q^{79} + 24q^{81} - 20q^{82} + 3q^{83} - 21q^{84} - 36q^{86} - 18q^{87} + 39q^{88} - 11q^{89} + 17q^{91} - 11q^{92} - 27q^{93} + 21q^{94} + 3q^{96} - q^{97} + q^{98} + 42q^{99} + O(q^{100}) \)

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

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

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
−0.519120
1.45106
−2.27060
2.33866
−2.73051 −1.93659 5.45571 0 5.28788 1.00000 −9.43585 0.750366 0
1.2 −0.894434 2.74893 −1.19999 0 −2.45874 1.00000 2.86218 4.55662 0
1.3 2.15561 2.62393 2.64667 0 5.65618 1.00000 1.39396 3.88502 0
1.4 2.46934 −3.43628 4.09762 0 −8.48532 1.00000 5.17972 8.80800 0
\(n\): e.g. 2-40 or 990-1000
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 4025.2.a.n 4
5.b even 2 1 805.2.a.h 4
15.d odd 2 1 7245.2.a.be 4
35.c odd 2 1 5635.2.a.t 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
805.2.a.h 4 5.b even 2 1
4025.2.a.n 4 1.a even 1 1 trivial
5635.2.a.t 4 35.c odd 2 1
7245.2.a.be 4 15.d odd 2 1

Atkin-Lehner signs

\( p \) Sign
\(5\) \(1\)
\(7\) \(-1\)
\(23\) \(1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4025))\):

\( T_{2}^{4} - T_{2}^{3} - 9 T_{2}^{2} + 8 T_{2} + 13 \)
\( T_{3}^{4} - 15 T_{3}^{2} + 3 T_{3} + 48 \)
\( T_{11}^{4} - 5 T_{11}^{3} + 19 T_{11} - 2 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 - T - T^{2} + 2 T^{3} + T^{4} + 4 T^{5} - 4 T^{6} - 8 T^{7} + 16 T^{8} \)
$3$ \( 1 - 3 T^{2} + 3 T^{3} + 12 T^{4} + 9 T^{5} - 27 T^{6} + 81 T^{8} \)
$5$ \( \)
$7$ \( ( 1 - T )^{4} \)
$11$ \( 1 - 5 T + 44 T^{2} - 146 T^{3} + 724 T^{4} - 1606 T^{5} + 5324 T^{6} - 6655 T^{7} + 14641 T^{8} \)
$13$ \( 1 - 17 T + 154 T^{2} - 914 T^{3} + 3874 T^{4} - 11882 T^{5} + 26026 T^{6} - 37349 T^{7} + 28561 T^{8} \)
$17$ \( 1 - 9 T + 62 T^{2} - 294 T^{3} + 1380 T^{4} - 4998 T^{5} + 17918 T^{6} - 44217 T^{7} + 83521 T^{8} \)
$19$ \( 1 - 4 T + 67 T^{2} - 181 T^{3} + 1792 T^{4} - 3439 T^{5} + 24187 T^{6} - 27436 T^{7} + 130321 T^{8} \)
$23$ \( ( 1 + T )^{4} \)
$29$ \( 1 - 10 T + 128 T^{2} - 790 T^{3} + 5662 T^{4} - 22910 T^{5} + 107648 T^{6} - 243890 T^{7} + 707281 T^{8} \)
$31$ \( 1 + 2 T + 61 T^{2} + 347 T^{3} + 1834 T^{4} + 10757 T^{5} + 58621 T^{6} + 59582 T^{7} + 923521 T^{8} \)
$37$ \( 1 + 5 T + 124 T^{2} + 386 T^{3} + 6202 T^{4} + 14282 T^{5} + 169756 T^{6} + 253265 T^{7} + 1874161 T^{8} \)
$41$ \( 1 - 7 T + 89 T^{2} - 574 T^{3} + 5722 T^{4} - 23534 T^{5} + 149609 T^{6} - 482447 T^{7} + 2825761 T^{8} \)
$43$ \( 1 - 6 T + 133 T^{2} - 705 T^{3} + 7716 T^{4} - 30315 T^{5} + 245917 T^{6} - 477042 T^{7} + 3418801 T^{8} \)
$47$ \( 1 - 12 T + 227 T^{2} - 1731 T^{3} + 16932 T^{4} - 81357 T^{5} + 501443 T^{6} - 1245876 T^{7} + 4879681 T^{8} \)
$53$ \( 1 + 23 T + 344 T^{2} + 3536 T^{3} + 29014 T^{4} + 187408 T^{5} + 966296 T^{6} + 3424171 T^{7} + 7890481 T^{8} \)
$59$ \( 1 - 23 T + 428 T^{2} - 4760 T^{3} + 44440 T^{4} - 280840 T^{5} + 1489868 T^{6} - 4723717 T^{7} + 12117361 T^{8} \)
$61$ \( 1 + 17 T + 304 T^{2} + 2996 T^{3} + 29488 T^{4} + 182756 T^{5} + 1131184 T^{6} + 3858677 T^{7} + 13845841 T^{8} \)
$67$ \( 1 + 5 T + 154 T^{2} + 26 T^{3} + 9622 T^{4} + 1742 T^{5} + 691306 T^{6} + 1503815 T^{7} + 20151121 T^{8} \)
$71$ \( 1 - 20 T + 293 T^{2} - 2987 T^{3} + 28828 T^{4} - 212077 T^{5} + 1477013 T^{6} - 7158220 T^{7} + 25411681 T^{8} \)
$73$ \( 1 - 15 T + 190 T^{2} - 1950 T^{3} + 20478 T^{4} - 142350 T^{5} + 1012510 T^{6} - 5835255 T^{7} + 28398241 T^{8} \)
$79$ \( 1 - 12 T + 331 T^{2} - 2637 T^{3} + 39342 T^{4} - 208323 T^{5} + 2065771 T^{6} - 5916468 T^{7} + 38950081 T^{8} \)
$83$ \( 1 - 3 T + 164 T^{2} - 546 T^{3} + 13890 T^{4} - 45318 T^{5} + 1129796 T^{6} - 1715361 T^{7} + 47458321 T^{8} \)
$89$ \( 1 + 11 T + 206 T^{2} + 1952 T^{3} + 27748 T^{4} + 173728 T^{5} + 1631726 T^{6} + 7754659 T^{7} + 62742241 T^{8} \)
$97$ \( 1 + T + 253 T^{2} - 614 T^{3} + 28606 T^{4} - 59558 T^{5} + 2380477 T^{6} + 912673 T^{7} + 88529281 T^{8} \)
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