Properties

Label 4025.2.a.m
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: 4.4.7537.1
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.

\(f(q)\) \(=\) \( q -\beta_{1} q^{2} + ( -1 - \beta_{3} ) q^{3} + ( 1 + \beta_{2} ) q^{4} + ( \beta_{1} + \beta_{2} + \beta_{3} ) q^{6} - q^{7} -\beta_{3} q^{8} + ( 1 + \beta_{1} - \beta_{2} + \beta_{3} ) q^{9} +O(q^{10})\) \( q -\beta_{1} q^{2} + ( -1 - \beta_{3} ) q^{3} + ( 1 + \beta_{2} ) q^{4} + ( \beta_{1} + \beta_{2} + \beta_{3} ) q^{6} - q^{7} -\beta_{3} q^{8} + ( 1 + \beta_{1} - \beta_{2} + \beta_{3} ) q^{9} + ( 3 - \beta_{1} ) q^{11} + ( -1 - \beta_{1} - 2 \beta_{2} ) q^{12} + ( 1 + \beta_{2} + \beta_{3} ) q^{13} + \beta_{1} q^{14} + ( -2 - \beta_{2} + \beta_{3} ) q^{16} + ( -3 - \beta_{1} ) q^{17} + ( -3 - 2 \beta_{2} ) q^{18} + ( 2 + \beta_{1} + \beta_{2} - \beta_{3} ) q^{19} + ( 1 + \beta_{3} ) q^{21} + ( 3 - 3 \beta_{1} + \beta_{2} ) q^{22} + q^{23} + ( 3 + \beta_{1} - \beta_{2} ) q^{24} + ( -2 \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{26} + ( -1 - \beta_{1} + 2 \beta_{2} ) q^{27} + ( -1 - \beta_{2} ) q^{28} + ( 2 \beta_{2} - 2 \beta_{3} ) q^{29} + ( 2 - \beta_{1} - \beta_{2} + \beta_{3} ) q^{31} + ( 3 \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{32} + ( -3 + \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{33} + ( 3 + 3 \beta_{1} + \beta_{2} ) q^{34} + ( -2 + 3 \beta_{1} + 2 \beta_{2} ) q^{36} + ( 1 + 3 \beta_{1} + 2 \beta_{2} ) q^{37} + ( -3 - 3 \beta_{1} ) q^{38} + ( -4 - 2 \beta_{1} - \beta_{2} ) q^{39} + ( 6 - 3 \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{41} + ( -\beta_{1} - \beta_{2} - \beta_{3} ) q^{42} + ( 1 - 2 \beta_{1} + 4 \beta_{2} - \beta_{3} ) q^{43} + ( 3 - 2 \beta_{1} + 3 \beta_{2} - \beta_{3} ) q^{44} -\beta_{1} q^{46} + ( \beta_{1} + 3 \beta_{2} + \beta_{3} ) q^{47} + ( -1 + 3 \beta_{2} + \beta_{3} ) q^{48} + q^{49} + ( 3 + \beta_{1} + \beta_{2} + 4 \beta_{3} ) q^{51} + ( 4 + \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{52} + ( 2 \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{53} + ( 3 - \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{54} + \beta_{3} q^{56} + ( 1 - \beta_{1} - 4 \beta_{2} - 2 \beta_{3} ) q^{57} + ( -2 \beta_{1} + 2 \beta_{2} ) q^{58} + ( 3 + 2 \beta_{1} - \beta_{2} - \beta_{3} ) q^{59} + ( 2 - 2 \beta_{1} + 5 \beta_{2} ) q^{61} + ( 3 - \beta_{1} ) q^{62} + ( -1 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{63} + ( -5 + \beta_{1} - 3 \beta_{2} - 3 \beta_{3} ) q^{64} + ( -3 + 2 \beta_{1} + \beta_{2} + \beta_{3} ) q^{66} + ( 1 - \beta_{2} - \beta_{3} ) q^{67} + ( -3 - 2 \beta_{1} - 3 \beta_{2} - \beta_{3} ) q^{68} + ( -1 - \beta_{3} ) q^{69} + ( -5 \beta_{1} + 3 \beta_{2} - 5 \beta_{3} ) q^{71} + ( -3 + \beta_{2} - 2 \beta_{3} ) q^{72} + ( -2 - 5 \beta_{2} - 2 \beta_{3} ) q^{73} + ( -9 - 3 \beta_{1} - 3 \beta_{2} - 2 \beta_{3} ) q^{74} + ( 5 + \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{76} + ( -3 + \beta_{1} ) q^{77} + ( 6 + 5 \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{78} + ( -1 + 5 \beta_{1} + \beta_{2} ) q^{79} + ( -2 - 4 \beta_{1} + \beta_{3} ) q^{81} + ( 9 - 4 \beta_{1} + 4 \beta_{2} + 3 \beta_{3} ) q^{82} + ( -3 + 4 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} ) q^{83} + ( 1 + \beta_{1} + 2 \beta_{2} ) q^{84} + ( 6 - 5 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} ) q^{86} + ( 6 - 6 \beta_{2} + 2 \beta_{3} ) q^{87} + ( \beta_{2} - 2 \beta_{3} ) q^{88} + ( 9 - \beta_{1} ) q^{89} + ( -1 - \beta_{2} - \beta_{3} ) q^{91} + ( 1 + \beta_{2} ) q^{92} + ( -5 + \beta_{1} + 4 \beta_{2} - 2 \beta_{3} ) q^{93} + ( -3 - 3 \beta_{1} - 2 \beta_{2} - 4 \beta_{3} ) q^{94} + ( -6 - 4 \beta_{1} + \beta_{2} - 4 \beta_{3} ) q^{96} + ( 4 - 3 \beta_{1} + 2 \beta_{2} + 5 \beta_{3} ) q^{97} -\beta_{1} q^{98} + ( 3 \beta_{1} - 5 \beta_{2} + 3 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - q^{2} - 4q^{3} + 3q^{4} - 4q^{7} + 6q^{9} + O(q^{10}) \) \( 4q - q^{2} - 4q^{3} + 3q^{4} - 4q^{7} + 6q^{9} + 11q^{11} - 3q^{12} + 3q^{13} + q^{14} - 7q^{16} - 13q^{17} - 10q^{18} + 8q^{19} + 4q^{21} + 8q^{22} + 4q^{23} + 14q^{24} - q^{26} - 7q^{27} - 3q^{28} - 2q^{29} + 8q^{31} + 4q^{32} - 12q^{33} + 14q^{34} - 7q^{36} + 5q^{37} - 15q^{38} - 17q^{39} + 23q^{41} - 2q^{43} + 7q^{44} - q^{46} - 2q^{47} - 7q^{48} + 4q^{49} + 12q^{51} + 15q^{52} + q^{53} + 10q^{54} + 7q^{57} - 4q^{58} + 15q^{59} + q^{61} + 11q^{62} - 6q^{63} - 16q^{64} - 11q^{66} + 5q^{67} - 11q^{68} - 4q^{69} - 8q^{71} - 13q^{72} - 3q^{73} - 36q^{74} + 20q^{76} - 11q^{77} + 27q^{78} - 12q^{81} + 28q^{82} - 5q^{83} + 3q^{84} + 16q^{86} + 30q^{87} - q^{88} + 35q^{89} - 3q^{91} + 3q^{92} - 23q^{93} - 13q^{94} - 29q^{96} + 11q^{97} - q^{98} + 8q^{99} + O(q^{100}) \)

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

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

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
2.15976
1.37933
−0.491918
−2.04717
−2.15976 −2.43525 2.66454 0 5.25956 −1.00000 −1.43525 2.93047 0
1.2 −1.37933 1.89307 −0.0974383 0 −2.61117 −1.00000 2.89307 0.583705 0
1.3 0.491918 −2.84864 −1.75802 0 −1.40130 −1.00000 −1.84864 5.11474 0
1.4 2.04717 −0.609175 2.19091 0 −1.24709 −1.00000 0.390825 −2.62891 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.m 4
5.b even 2 1 805.2.a.i 4
15.d odd 2 1 7245.2.a.bd 4
35.c odd 2 1 5635.2.a.u 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
805.2.a.i 4 5.b even 2 1
4025.2.a.m 4 1.a even 1 1 trivial
5635.2.a.u 4 35.c odd 2 1
7245.2.a.bd 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4025))\):

\( T_{2}^{4} + T_{2}^{3} - 5 T_{2}^{2} - 4 T_{2} + 3 \)
\( T_{3}^{4} + 4 T_{3}^{3} - T_{3}^{2} - 15 T_{3} - 8 \)
\( T_{11}^{4} - 11 T_{11}^{3} + 40 T_{11}^{2} - 55 T_{11} + 24 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T + 3 T^{2} + 2 T^{3} + 7 T^{4} + 4 T^{5} + 12 T^{6} + 8 T^{7} + 16 T^{8} \)
$3$ \( 1 + 4 T + 11 T^{2} + 21 T^{3} + 40 T^{4} + 63 T^{5} + 99 T^{6} + 108 T^{7} + 81 T^{8} \)
$5$ 1
$7$ \( ( 1 + T )^{4} \)
$11$ \( 1 - 11 T + 84 T^{2} - 418 T^{3} + 1630 T^{4} - 4598 T^{5} + 10164 T^{6} - 14641 T^{7} + 14641 T^{8} \)
$13$ \( 1 - 3 T + 42 T^{2} - 94 T^{3} + 752 T^{4} - 1222 T^{5} + 7098 T^{6} - 6591 T^{7} + 28561 T^{8} \)
$17$ \( 1 + 13 T + 126 T^{2} + 764 T^{3} + 3760 T^{4} + 12988 T^{5} + 36414 T^{6} + 63869 T^{7} + 83521 T^{8} \)
$19$ \( 1 - 8 T + 79 T^{2} - 375 T^{3} + 2172 T^{4} - 7125 T^{5} + 28519 T^{6} - 54872 T^{7} + 130321 T^{8} \)
$23$ \( ( 1 - T )^{4} \)
$29$ \( 1 + 2 T + 64 T^{2} + 302 T^{3} + 1982 T^{4} + 8758 T^{5} + 53824 T^{6} + 48778 T^{7} + 707281 T^{8} \)
$31$ \( 1 - 8 T + 127 T^{2} - 721 T^{3} + 5960 T^{4} - 22351 T^{5} + 122047 T^{6} - 238328 T^{7} + 923521 T^{8} \)
$37$ \( 1 - 5 T + 76 T^{2} - 504 T^{3} + 3412 T^{4} - 18648 T^{5} + 104044 T^{6} - 253265 T^{7} + 1874161 T^{8} \)
$41$ \( 1 - 23 T + 277 T^{2} - 2246 T^{3} + 15230 T^{4} - 92086 T^{5} + 465637 T^{6} - 1585183 T^{7} + 2825761 T^{8} \)
$43$ \( 1 + 2 T + 55 T^{2} + 103 T^{3} + 1724 T^{4} + 4429 T^{5} + 101695 T^{6} + 159014 T^{7} + 3418801 T^{8} \)
$47$ \( 1 + 2 T + 117 T^{2} + 73 T^{3} + 6904 T^{4} + 3431 T^{5} + 258453 T^{6} + 207646 T^{7} + 4879681 T^{8} \)
$53$ \( 1 - T + 150 T^{2} - 104 T^{3} + 10648 T^{4} - 5512 T^{5} + 421350 T^{6} - 148877 T^{7} + 7890481 T^{8} \)
$59$ \( 1 - 15 T + 286 T^{2} - 2620 T^{3} + 26558 T^{4} - 154580 T^{5} + 995566 T^{6} - 3080685 T^{7} + 12117361 T^{8} \)
$61$ \( 1 - T + 76 T^{2} - 168 T^{3} + 6712 T^{4} - 10248 T^{5} + 282796 T^{6} - 226981 T^{7} + 13845841 T^{8} \)
$67$ \( 1 - 5 T + 264 T^{2} - 984 T^{3} + 26394 T^{4} - 65928 T^{5} + 1185096 T^{6} - 1503815 T^{7} + 20151121 T^{8} \)
$71$ \( 1 + 8 T - 15 T^{2} - 323 T^{3} + 964 T^{4} - 22933 T^{5} - 75615 T^{6} + 2863288 T^{7} + 25411681 T^{8} \)
$73$ \( 1 + 3 T + 108 T^{2} + 542 T^{3} + 12212 T^{4} + 39566 T^{5} + 575532 T^{6} + 1167051 T^{7} + 28398241 T^{8} \)
$79$ \( 1 + 169 T^{2} - 121 T^{3} + 17656 T^{4} - 9559 T^{5} + 1054729 T^{6} + 38950081 T^{8} \)
$83$ \( 1 + 5 T + 162 T^{2} + 1438 T^{3} + 15118 T^{4} + 119354 T^{5} + 1116018 T^{6} + 2858935 T^{7} + 47458321 T^{8} \)
$89$ \( 1 - 35 T + 810 T^{2} - 11932 T^{3} + 133804 T^{4} - 1061948 T^{5} + 6416010 T^{6} - 24673915 T^{7} + 62742241 T^{8} \)
$97$ \( 1 - 11 T + 177 T^{2} + 318 T^{3} + 2406 T^{4} + 30846 T^{5} + 1665393 T^{6} - 10039403 T^{7} + 88529281 T^{8} \)
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