Properties

Label 4025.2.a.j
Level $4025$
Weight $2$
Character orbit 4025.a
Self dual yes
Analytic conductor $32.140$
Analytic rank $0$
Dimension $3$
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: \(3\)
Coefficient field: 3.3.148.1
Defining polynomial: \( x^{3} - x^{2} - 3x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 161)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(f(q)\) \(=\) \( q + (\beta_{2} + \beta_1) q^{2} + (\beta_1 - 1) q^{3} + (2 \beta_1 + 1) q^{4} + (\beta_1 + 1) q^{6} + q^{7} + (\beta_{2} + 3 \beta_1 + 2) q^{8} + (\beta_{2} - \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{2} + \beta_1) q^{2} + (\beta_1 - 1) q^{3} + (2 \beta_1 + 1) q^{4} + (\beta_1 + 1) q^{6} + q^{7} + (\beta_{2} + 3 \beta_1 + 2) q^{8} + (\beta_{2} - \beta_1) q^{9} + (\beta_{2} + \beta_1 + 1) q^{11} + (2 \beta_{2} + \beta_1 + 3) q^{12} - 2 \beta_1 q^{13} + (\beta_{2} + \beta_1) q^{14} + (4 \beta_{2} + 4 \beta_1 + 3) q^{16} + ( - \beta_1 - 1) q^{17} + ( - 2 \beta_{2} - 2 \beta_1 + 1) q^{18} + ( - 2 \beta_{2} + 2 \beta_1 + 2) q^{19} + (\beta_1 - 1) q^{21} + (\beta_{2} + 3 \beta_1 + 3) q^{22} - q^{23} + (2 \beta_{2} + 3 \beta_1 + 3) q^{24} + ( - 2 \beta_{2} - 4 \beta_1 - 2) q^{26} + ( - 2 \beta_{2} - 2 \beta_1) q^{27} + (2 \beta_1 + 1) q^{28} + ( - \beta_{2} + \beta_1 - 1) q^{29} + (4 \beta_{2} + \beta_1 + 5) q^{31} + (\beta_{2} + 5 \beta_1 + 8) q^{32} + 2 \beta_1 q^{33} + ( - 2 \beta_{2} - 3 \beta_1 - 1) q^{34} + ( - \beta_{2} - \beta_1 - 6) q^{36} + (2 \beta_{2} + 2) q^{37} + (6 \beta_{2} + 6 \beta_1 - 2) q^{38} + ( - 2 \beta_{2} - 4) q^{39} + (4 \beta_1 - 6) q^{41} + (\beta_1 + 1) q^{42} + (4 \beta_{2} + 2 \beta_1 + 2) q^{43} + (3 \beta_{2} + 7 \beta_1 + 3) q^{44} + ( - \beta_{2} - \beta_1) q^{46} + (2 \beta_{2} - \beta_1 - 5) q^{47} + (7 \beta_1 + 1) q^{48} + q^{49} + ( - \beta_{2} - \beta_1 - 1) q^{51} + ( - 4 \beta_{2} - 6 \beta_1 - 8) q^{52} + ( - 4 \beta_{2} - 2) q^{53} + ( - 4 \beta_1 - 6) q^{54} + (\beta_{2} + 3 \beta_1 + 2) q^{56} + (4 \beta_{2} + 4) q^{57} + (\beta_{2} + \beta_1 - 1) q^{58} + ( - 6 \beta_{2} - \beta_1 - 3) q^{59} + ( - 2 \beta_{2} - 5 \beta_1 + 5) q^{61} + (2 \beta_{2} + 7 \beta_1 + 9) q^{62} + (\beta_{2} - \beta_1) q^{63} + (4 \beta_{2} + 10 \beta_1 + 1) q^{64} + (2 \beta_{2} + 4 \beta_1 + 2) q^{66} + ( - 5 \beta_{2} - \beta_1 - 5) q^{67} + ( - 2 \beta_{2} - 5 \beta_1 - 5) q^{68} + ( - \beta_1 + 1) q^{69} + (4 \beta_{2} + 4 \beta_1) q^{71} + ( - 2 \beta_{2} - 4 \beta_1 - 5) q^{72} + 6 \beta_1 q^{73} + (2 \beta_1 + 4) q^{74} + (2 \beta_{2} + 6 \beta_1 + 14) q^{76} + (\beta_{2} + \beta_1 + 1) q^{77} + ( - 2 \beta_{2} - 4 \beta_1 - 4) q^{78} + ( - \beta_{2} - 5 \beta_1 + 7) q^{79} + ( - 3 \beta_{2} + \beta_1 - 2) q^{81} + ( - 2 \beta_{2} + 2 \beta_1 + 4) q^{82} + ( - 8 \beta_{2} - 2 \beta_1 - 6) q^{83} + (2 \beta_{2} + \beta_1 + 3) q^{84} + (6 \beta_1 + 10) q^{86} + (2 \beta_{2} - 2 \beta_1 + 4) q^{87} + (5 \beta_{2} + 11 \beta_1 + 7) q^{88} + (2 \beta_{2} - 5 \beta_1 + 3) q^{89} - 2 \beta_1 q^{91} + ( - 2 \beta_1 - 1) q^{92} + ( - 3 \beta_{2} + 9 \beta_1 - 7) q^{93} + ( - 8 \beta_{2} - 7 \beta_1 + 3) q^{94} + (4 \beta_{2} + 9 \beta_1 + 1) q^{96} + ( - 2 \beta_{2} - 3 \beta_1 - 1) q^{97} + (\beta_{2} + \beta_1) q^{98} + ( - \beta_{2} - 3 \beta_1 + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + q^{2} - 2 q^{3} + 5 q^{4} + 4 q^{6} + 3 q^{7} + 9 q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + q^{2} - 2 q^{3} + 5 q^{4} + 4 q^{6} + 3 q^{7} + 9 q^{8} - q^{9} + 4 q^{11} + 10 q^{12} - 2 q^{13} + q^{14} + 13 q^{16} - 4 q^{17} + q^{18} + 8 q^{19} - 2 q^{21} + 12 q^{22} - 3 q^{23} + 12 q^{24} - 10 q^{26} - 2 q^{27} + 5 q^{28} - 2 q^{29} + 16 q^{31} + 29 q^{32} + 2 q^{33} - 6 q^{34} - 19 q^{36} + 6 q^{37} - 12 q^{39} - 14 q^{41} + 4 q^{42} + 8 q^{43} + 16 q^{44} - q^{46} - 16 q^{47} + 10 q^{48} + 3 q^{49} - 4 q^{51} - 30 q^{52} - 6 q^{53} - 22 q^{54} + 9 q^{56} + 12 q^{57} - 2 q^{58} - 10 q^{59} + 10 q^{61} + 34 q^{62} - q^{63} + 13 q^{64} + 10 q^{66} - 16 q^{67} - 20 q^{68} + 2 q^{69} + 4 q^{71} - 19 q^{72} + 6 q^{73} + 14 q^{74} + 48 q^{76} + 4 q^{77} - 16 q^{78} + 16 q^{79} - 5 q^{81} + 14 q^{82} - 20 q^{83} + 10 q^{84} + 36 q^{86} + 10 q^{87} + 32 q^{88} + 4 q^{89} - 2 q^{91} - 5 q^{92} - 12 q^{93} + 2 q^{94} + 12 q^{96} - 6 q^{97} + q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 3x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 2 \) Copy content Toggle raw display

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
0.311108
−1.48119
2.17009
−1.90321 −0.688892 1.62222 0 1.31111 1.00000 0.719004 −2.52543 0
1.2 0.193937 −2.48119 −1.96239 0 −0.481194 1.00000 −0.768452 3.15633 0
1.3 2.70928 1.17009 5.34017 0 3.17009 1.00000 9.04945 −1.63090 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 4025.2.a.j 3
5.b even 2 1 161.2.a.c 3
15.d odd 2 1 1449.2.a.m 3
20.d odd 2 1 2576.2.a.v 3
35.c odd 2 1 1127.2.a.f 3
115.c odd 2 1 3703.2.a.c 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
161.2.a.c 3 5.b even 2 1
1127.2.a.f 3 35.c odd 2 1
1449.2.a.m 3 15.d odd 2 1
2576.2.a.v 3 20.d odd 2 1
3703.2.a.c 3 115.c odd 2 1
4025.2.a.j 3 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4025))\):

\( T_{2}^{3} - T_{2}^{2} - 5T_{2} + 1 \) Copy content Toggle raw display
\( T_{3}^{3} + 2T_{3}^{2} - 2T_{3} - 2 \) Copy content Toggle raw display
\( T_{11}^{3} - 4T_{11}^{2} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} - T^{2} - 5T + 1 \) Copy content Toggle raw display
$3$ \( T^{3} + 2 T^{2} - 2 T - 2 \) Copy content Toggle raw display
$5$ \( T^{3} \) Copy content Toggle raw display
$7$ \( (T - 1)^{3} \) Copy content Toggle raw display
$11$ \( T^{3} - 4T^{2} + 4 \) Copy content Toggle raw display
$13$ \( T^{3} + 2 T^{2} - 12 T - 8 \) Copy content Toggle raw display
$17$ \( T^{3} + 4 T^{2} + 2 T - 2 \) Copy content Toggle raw display
$19$ \( T^{3} - 8 T^{2} - 16 T + 160 \) Copy content Toggle raw display
$23$ \( (T + 1)^{3} \) Copy content Toggle raw display
$29$ \( T^{3} + 2 T^{2} - 8 T + 4 \) Copy content Toggle raw display
$31$ \( T^{3} - 16 T^{2} + 26 T + 338 \) Copy content Toggle raw display
$37$ \( T^{3} - 6 T^{2} - 4 T + 40 \) Copy content Toggle raw display
$41$ \( T^{3} + 14 T^{2} + 12 T - 152 \) Copy content Toggle raw display
$43$ \( T^{3} - 8 T^{2} - 40 T + 304 \) Copy content Toggle raw display
$47$ \( T^{3} + 16 T^{2} + 62 T + 10 \) Copy content Toggle raw display
$53$ \( T^{3} + 6 T^{2} - 52 T - 248 \) Copy content Toggle raw display
$59$ \( T^{3} + 10 T^{2} - 102 T - 970 \) Copy content Toggle raw display
$61$ \( T^{3} - 10 T^{2} - 46 T + 494 \) Copy content Toggle raw display
$67$ \( T^{3} + 16 T^{2} - 8 T - 676 \) Copy content Toggle raw display
$71$ \( T^{3} - 4 T^{2} - 80 T + 64 \) Copy content Toggle raw display
$73$ \( T^{3} - 6 T^{2} - 108 T + 216 \) Copy content Toggle raw display
$79$ \( T^{3} - 16 T^{2} + 8 T + 428 \) Copy content Toggle raw display
$83$ \( T^{3} + 20 T^{2} - 104 T - 2672 \) Copy content Toggle raw display
$89$ \( T^{3} - 4 T^{2} - 114 T - 278 \) Copy content Toggle raw display
$97$ \( T^{3} + 6 T^{2} - 22 T + 2 \) Copy content Toggle raw display
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