Properties

Label 9025.2.a.bo
Level $9025$
Weight $2$
Character orbit 9025.a
Self dual yes
Analytic conductor $72.065$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 9025 = 5^{2} \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9025.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(72.0649878242\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.2225.1
Defining polynomial: \( x^{4} - x^{3} - 5x^{2} + 2x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1805)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + ( - \beta_{3} - \beta_1) q^{3} + (\beta_{3} + \beta_1 + 1) q^{4} + ( - \beta_{3} - 2 \beta_{2} - \beta_1 - 3) q^{6} + ( - \beta_{2} + \beta_1 + 2) q^{7} + (\beta_{3} + 2 \beta_{2} + 3) q^{8} + (2 \beta_{2} + \beta_1 + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + ( - \beta_{3} - \beta_1) q^{3} + (\beta_{3} + \beta_1 + 1) q^{4} + ( - \beta_{3} - 2 \beta_{2} - \beta_1 - 3) q^{6} + ( - \beta_{2} + \beta_1 + 2) q^{7} + (\beta_{3} + 2 \beta_{2} + 3) q^{8} + (2 \beta_{2} + \beta_1 + 2) q^{9} + ( - \beta_{3} - \beta_{2} - 1) q^{11} + ( - \beta_{3} - 2 \beta_{2} - 2 \beta_1 - 5) q^{12} + (2 \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 1) q^{13} + (3 \beta_1 + 2) q^{14} + (2 \beta_{2} + \beta_1) q^{16} + ( - 2 \beta_{3} + \beta_{2} + \beta_1 + 1) q^{17} + (3 \beta_{3} + 3 \beta_1 + 5) q^{18} + ( - 3 \beta_{3} - 3 \beta_{2} - 2 \beta_1 - 3) q^{21} + ( - \beta_{3} - 2 \beta_{2} - \beta_1 - 1) q^{22} + (2 \beta_{3} + 2 \beta_{2} - \beta_1 + 5) q^{23} + ( - 2 \beta_{3} + 2 \beta_{2} - 5 \beta_1 - 2) q^{24} + (4 \beta_{2} + \beta_1 + 4) q^{26} + ( - 2 \beta_1 - 3) q^{27} + (3 \beta_{3} + 2 \beta_{2} + 3 \beta_1 + 5) q^{28} + (4 \beta_{2} - \beta_1 + 6) q^{29} + ( - \beta_{3} - 2 \beta_{2} + 3 \beta_1 - 2) q^{31} + (\beta_{3} - 4 \beta_{2} + \beta_1 - 1) q^{32} + ( - \beta_{2} + 2 \beta_1 + 2) q^{33} + (2 \beta_{3} - 4 \beta_{2} + 2 \beta_1 + 4) q^{34} + (3 \beta_{3} + 2 \beta_{2} + 6 \beta_1 + 5) q^{36} + (\beta_{3} - \beta_1 - 2) q^{37} + (\beta_{3} - 6 \beta_{2} + \beta_1 - 10) q^{39} + ( - 2 \beta_{3} - 4 \beta_{2} + 2 \beta_1 - 9) q^{41} + ( - 5 \beta_{3} - 6 \beta_{2} - 5 \beta_1 - 9) q^{42} + ( - 2 \beta_{3} + \beta_{2} + 6) q^{43} + ( - \beta_{3} - 2 \beta_1 - 3) q^{44} + (\beta_{3} + 4 \beta_{2} + 4 \beta_1 - 1) q^{46} + (2 \beta_{3} - 4 \beta_{2} + 2 \beta_1 + 5) q^{47} + ( - \beta_{3} - 3 \beta_1 - 3) q^{48} + ( - \beta_{3} - 5 \beta_{2} + 5 \beta_1 - 1) q^{49} + ( - 4 \beta_{3} - \beta_{2} - 3 \beta_1 + 1) q^{51} + (\beta_{3} + 4 \beta_{2} + \beta_1 + 9) q^{52} + ( - 2 \beta_{3} + 5 \beta_{2} - 2 \beta_1 - 2) q^{53} + ( - 2 \beta_{3} - 5 \beta_1 - 6) q^{54} + (5 \beta_{3} + 6 \beta_{2} + 2 \beta_1 + 7) q^{56} + (3 \beta_{3} + 5 \beta_1 + 1) q^{58} + (\beta_{2} - 4 \beta_1 + 4) q^{59} + ( - 3 \beta_{3} - 2 \beta_{2} - 4 \beta_1 - 1) q^{61} + (\beta_{3} - 2 \beta_{2} + \beta_1 + 7) q^{62} + (2 \beta_{3} + 4 \beta_{2} + 5 \beta_1 + 6) q^{63} + ( - 3 \beta_{3} - 2 \beta_{2} - 2 \beta_1 - 1) q^{64} + (\beta_{3} + 4 \beta_1 + 5) q^{66} + ( - \beta_{3} + 7 \beta_{2} - \beta_1 + 4) q^{67} + (2 \beta_{3} + 2 \beta_{2} + 4 \beta_1) q^{68} + ( - 2 \beta_{3} + 4 \beta_{2} - 6 \beta_1 - 1) q^{69} + (2 \beta_1 - 5) q^{71} + (2 \beta_{3} + 6 \beta_{2} + 5 \beta_1 + 10) q^{72} + (2 \beta_{3} + 4 \beta_{2} + 9) q^{73} + ( - \beta_{3} + 2 \beta_{2} - 3 \beta_1 - 3) q^{74} + ( - 4 \beta_{3} - 5 \beta_{2} - 3) q^{77} + ( - 5 \beta_{3} + 2 \beta_{2} - 9 \beta_1 - 3) q^{78} + (\beta_{3} + 2 \beta_{2} - 6) q^{79} + (5 \beta_{3} - 2 \beta_{2} + 2 \beta_1) q^{81} + ( - 2 \beta_{3} - 4 \beta_{2} - 7 \beta_1 + 2) q^{82} + ( - 3 \beta_{3} - 2 \beta_1 + 2) q^{83} + ( - 5 \beta_{3} - 4 \beta_{2} - 10 \beta_1 - 15) q^{84} + (\beta_{3} - 4 \beta_{2} + 6 \beta_1 + 1) q^{86} + ( - 5 \beta_{3} + 6 \beta_{2} - 9 \beta_1 + 3) q^{87} + (2 \beta_{2} - 3 \beta_1 - 4) q^{88} + (\beta_{3} + 3 \beta_{2} - \beta_1) q^{89} + (4 \beta_{3} + \beta_{2} + 3 \beta_1 + 4) q^{91} + (4 \beta_{3} - 2 \beta_{2} + 5 \beta_1 + 6) q^{92} + ( - 2 \beta_{3} - 8 \beta_{2} + \beta_1 - 7) q^{93} + ( - 2 \beta_{3} + 4 \beta_{2} + 7 \beta_1 + 2) q^{94} + (\beta_{3} - 6 \beta_{2} + 4 \beta_1 - 5) q^{96} + (6 \beta_{2} + 3 \beta_1 + 7) q^{97} + ( - 2 \beta_{2} + 4 \beta_1 + 10) q^{98} + ( - \beta_{3} - 2 \beta_{2} - 3 \beta_1 - 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + q^{2} + q^{3} + 3 q^{4} - 7 q^{6} + 11 q^{7} + 6 q^{8} + 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + q^{2} + q^{3} + 3 q^{4} - 7 q^{6} + 11 q^{7} + 6 q^{8} + 5 q^{9} - 16 q^{12} - 2 q^{13} + 11 q^{14} - 3 q^{16} + 7 q^{17} + 17 q^{18} - 2 q^{21} + q^{22} + 11 q^{23} - 13 q^{24} + 9 q^{26} - 14 q^{27} + 13 q^{28} + 15 q^{29} + q^{31} + 3 q^{32} + 12 q^{33} + 22 q^{34} + 16 q^{36} - 11 q^{37} - 29 q^{39} - 22 q^{41} - 19 q^{42} + 26 q^{43} - 12 q^{44} - 10 q^{46} + 26 q^{47} - 13 q^{48} + 13 q^{49} + 11 q^{51} + 27 q^{52} - 16 q^{53} - 25 q^{54} + 8 q^{56} + 3 q^{58} + 10 q^{59} + 2 q^{61} + 31 q^{62} + 17 q^{63} + 4 q^{64} + 22 q^{66} + 3 q^{67} - 4 q^{68} - 14 q^{69} - 18 q^{71} + 29 q^{72} + 24 q^{73} - 17 q^{74} + 6 q^{77} - 15 q^{78} - 30 q^{79} - 4 q^{81} + 13 q^{82} + 12 q^{83} - 52 q^{84} + 16 q^{86} + q^{87} - 23 q^{88} - 9 q^{89} + 9 q^{91} + 25 q^{92} - 7 q^{93} + 11 q^{94} - 6 q^{96} + 19 q^{97} + 48 q^{98} - 9 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} - \nu^{2} - 3\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{2} - \nu - 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + \beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 2\beta_{2} + 4\beta _1 + 3 \) 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
−1.75660
−0.820249
1.13856
2.43828
−1.75660 −0.0856374 1.08564 0 0.150431 1.86144 1.60617 −2.99267 0
1.2 −0.820249 2.32719 −1.32719 0 −1.90888 0.561717 2.72913 2.41582 0
1.3 1.13856 1.70367 −0.703671 0 1.93974 4.75660 −3.07830 −0.0975037 0
1.4 2.43828 −2.94523 3.94523 0 −7.18129 3.82025 4.74301 5.67435 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(1\)
\(19\) \(-1\)

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 9025.2.a.bo 4
5.b even 2 1 1805.2.a.j 4
19.b odd 2 1 9025.2.a.bh 4
95.d odd 2 1 1805.2.a.n yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1805.2.a.j 4 5.b even 2 1
1805.2.a.n yes 4 95.d odd 2 1
9025.2.a.bh 4 19.b odd 2 1
9025.2.a.bo 4 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9025))\):

\( T_{2}^{4} - T_{2}^{3} - 5T_{2}^{2} + 2T_{2} + 4 \) Copy content Toggle raw display
\( T_{3}^{4} - T_{3}^{3} - 8T_{3}^{2} + 11T_{3} + 1 \) Copy content Toggle raw display
\( T_{7}^{4} - 11T_{7}^{3} + 40T_{7}^{2} - 53T_{7} + 19 \) Copy content Toggle raw display
\( T_{11}^{4} - 9T_{11}^{2} - 10T_{11} - 1 \) Copy content Toggle raw display
\( T_{29}^{4} - 15T_{29}^{3} + 49T_{29}^{2} + 50T_{29} - 116 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - T^{3} - 5 T^{2} + 2 T + 4 \) Copy content Toggle raw display
$3$ \( T^{4} - T^{3} - 8 T^{2} + 11 T + 1 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} - 11 T^{3} + 40 T^{2} - 53 T + 19 \) Copy content Toggle raw display
$11$ \( T^{4} - 9 T^{2} - 10 T - 1 \) Copy content Toggle raw display
$13$ \( T^{4} + 2 T^{3} - 32 T^{2} + 22 T + 71 \) Copy content Toggle raw display
$17$ \( T^{4} - 7 T^{3} - 25 T^{2} + 256 T - 436 \) Copy content Toggle raw display
$19$ \( T^{4} \) Copy content Toggle raw display
$23$ \( T^{4} - 11 T^{3} + 2 T^{2} + 281 T - 709 \) Copy content Toggle raw display
$29$ \( T^{4} - 15 T^{3} + 49 T^{2} + \cdots - 116 \) Copy content Toggle raw display
$31$ \( T^{4} - T^{3} - 60 T^{2} - 3 T + 659 \) Copy content Toggle raw display
$37$ \( T^{4} + 11 T^{3} + 30 T^{2} + \cdots - 101 \) Copy content Toggle raw display
$41$ \( T^{4} + 22 T^{3} + 100 T^{2} + \cdots - 4061 \) Copy content Toggle raw display
$43$ \( T^{4} - 26 T^{3} + 225 T^{2} + \cdots + 379 \) Copy content Toggle raw display
$47$ \( T^{4} - 26 T^{3} + 200 T^{2} + \cdots - 1681 \) Copy content Toggle raw display
$53$ \( T^{4} + 16 T^{3} + 25 T^{2} + \cdots - 1891 \) Copy content Toggle raw display
$59$ \( T^{4} - 10 T^{3} - 41 T^{2} + \cdots + 829 \) Copy content Toggle raw display
$61$ \( T^{4} - 2 T^{3} - 131 T^{2} + \cdots - 1744 \) Copy content Toggle raw display
$67$ \( T^{4} - 3 T^{3} - 110 T^{2} + \cdots + 2389 \) Copy content Toggle raw display
$71$ \( T^{4} + 18 T^{3} + 100 T^{2} + \cdots + 19 \) Copy content Toggle raw display
$73$ \( T^{4} - 24 T^{3} + 150 T^{2} + \cdots - 2071 \) Copy content Toggle raw display
$79$ \( T^{4} + 30 T^{3} + 321 T^{2} + \cdots + 2384 \) Copy content Toggle raw display
$83$ \( T^{4} - 12 T^{3} - 5 T^{2} + 296 T + 4 \) Copy content Toggle raw display
$89$ \( T^{4} + 9 T^{3} - 13 T - 1 \) Copy content Toggle raw display
$97$ \( T^{4} - 19 T^{3} - 48 T^{2} + \cdots - 839 \) Copy content Toggle raw display
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