Properties

Label 9025.2.a.ba
Level $9025$
Weight $2$
Character orbit 9025.a
Self dual yes
Analytic conductor $72.065$
Analytic rank $0$
Dimension $3$
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: \(3\)
Coefficient field: 3.3.361.1
Defining polynomial: \( x^{3} - x^{2} - 6x + 7 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 95)
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.

\(f(q)\) \(=\) \( q + ( - \beta_{2} - \beta_1 + 1) q^{2} + \beta_{2} q^{3} + (\beta_1 + 2) q^{4} + (\beta_{2} - \beta_1 - 2) q^{6} + (\beta_{2} - 1) q^{7} + ( - 2 \beta_{2} - \beta_1 - 1) q^{8} + ( - \beta_{2} - \beta_1 + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} - \beta_1 + 1) q^{2} + \beta_{2} q^{3} + (\beta_1 + 2) q^{4} + (\beta_{2} - \beta_1 - 2) q^{6} + (\beta_{2} - 1) q^{7} + ( - 2 \beta_{2} - \beta_1 - 1) q^{8} + ( - \beta_{2} - \beta_1 + 2) q^{9} + (\beta_{2} - 2) q^{11} + (3 \beta_{2} + 2 \beta_1 - 3) q^{12} - 5 q^{13} + (2 \beta_{2} - 3) q^{14} + (\beta_{2} + 2 \beta_1) q^{16} + ( - \beta_{2} - 3 \beta_1 + 1) q^{17} + ( - \beta_{2} + 5) q^{18} + ( - 2 \beta_{2} - \beta_1 + 5) q^{21} + (3 \beta_{2} + \beta_1 - 4) q^{22} + ( - \beta_{2} - 3 \beta_1) q^{23} - 7 q^{24} + (5 \beta_{2} + 5 \beta_1 - 5) q^{26} + ( - \beta_{2} - \beta_1 - 2) q^{27} + (3 \beta_{2} + \beta_1 - 5) q^{28} + ( - \beta_{2} + 1) q^{29} + ( - \beta_{2} - \beta_1 + 1) q^{31} + (\beta_{2} - \beta_1 - 2) q^{32} + ( - 3 \beta_{2} - \beta_1 + 5) q^{33} + (4 \beta_{2} + 3 \beta_1 + 6) q^{34} + ( - 4 \beta_{2} - 2 \beta_1 + 3) q^{36} + (3 \beta_{2} - 2 \beta_1 - 1) q^{37} - 5 \beta_{2} q^{39} + (3 \beta_{2} + 2 \beta_1 - 1) q^{41} + ( - 5 \beta_{2} - 2 \beta_1 + 10) q^{42} + ( - 2 \beta_{2} - 3 \beta_1 + 2) q^{43} + (3 \beta_{2} - 7) q^{44} + (5 \beta_{2} + 4 \beta_1 + 5) q^{46} + (\beta_{2} + 2 \beta_1 - 3) q^{47} + (\beta_{2} + 3 \beta_1 - 1) q^{48} + ( - 3 \beta_{2} - \beta_1 - 1) q^{49} + ( - \beta_{2} - 5 \beta_1 + 4) q^{51} + ( - 5 \beta_1 - 10) q^{52} + (3 \beta_{2} - \beta_1 + 3) q^{53} + (3 \beta_{2} + 4 \beta_1 + 1) q^{54} + (2 \beta_{2} + \beta_1 - 6) q^{56} + ( - 2 \beta_{2} + 3) q^{58} + (\beta_{2} + 2 \beta_1 - 3) q^{59} + ( - 2 \beta_{2} - 4 \beta_1 - 1) q^{61} + (\beta_1 + 4) q^{62} + (3 \beta_{2} - 4) q^{63} + (3 \beta_{2} - 2 \beta_1 - 3) q^{64} + ( - 6 \beta_{2} - \beta_1 + 12) q^{66} + ( - 2 \beta_1 - 6) q^{67} + ( - 6 \beta_{2} - 7 \beta_1 - 7) q^{68} + ( - 2 \beta_{2} - 5 \beta_1 + 4) q^{69} + ( - 2 \beta_{2} + \beta_1 + 10) q^{71} + ( - \beta_{2} + 3 \beta_1 + 3) q^{72} + (\beta_{2} + 3 \beta_1 + 6) q^{73} + (8 \beta_{2} - 5) q^{74} + ( - 4 \beta_{2} - \beta_1 + 7) q^{77} + ( - 5 \beta_{2} + 5 \beta_1 + 10) q^{78} + ( - 2 \beta_{2} - 4 \beta_1 + 10) q^{79} + (\beta_{2} + 2 \beta_1 - 8) q^{81} + ( - 4 \beta_1 - 9) q^{82} + (3 \beta_{2} + 3 \beta_1 - 1) q^{83} + ( - 7 \beta_{2} - \beta_1 + 12) q^{84} + (2 \beta_{2} + 3 \beta_1 + 9) q^{86} + (2 \beta_{2} + \beta_1 - 5) q^{87} + (4 \beta_{2} + 2 \beta_1 - 5) q^{88} + ( - 4 \beta_1 + 6) q^{89} + ( - 5 \beta_{2} + 5) q^{91} + ( - 6 \beta_{2} - 8 \beta_1 - 9) q^{92} + (\beta_{2} - \beta_1 - 2) q^{93} - 7 q^{94} + ( - 4 \beta_{2} - 3 \beta_1 + 8) q^{96} + ( - \beta_{2} - 4 \beta_1 + 4) q^{97} + (5 \beta_1 + 6) q^{98} + (4 \beta_{2} + \beta_1 - 6) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + q^{2} + q^{3} + 7 q^{4} - 6 q^{6} - 2 q^{7} - 6 q^{8} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + q^{2} + q^{3} + 7 q^{4} - 6 q^{6} - 2 q^{7} - 6 q^{8} + 4 q^{9} - 5 q^{11} - 4 q^{12} - 15 q^{13} - 7 q^{14} + 3 q^{16} - q^{17} + 14 q^{18} + 12 q^{21} - 8 q^{22} - 4 q^{23} - 21 q^{24} - 5 q^{26} - 8 q^{27} - 11 q^{28} + 2 q^{29} + q^{31} - 6 q^{32} + 11 q^{33} + 25 q^{34} + 3 q^{36} - 2 q^{37} - 5 q^{39} + 2 q^{41} + 23 q^{42} + q^{43} - 18 q^{44} + 24 q^{46} - 6 q^{47} + q^{48} - 7 q^{49} + 6 q^{51} - 35 q^{52} + 11 q^{53} + 10 q^{54} - 15 q^{56} + 7 q^{58} - 6 q^{59} - 9 q^{61} + 13 q^{62} - 9 q^{63} - 8 q^{64} + 29 q^{66} - 20 q^{67} - 34 q^{68} + 5 q^{69} + 29 q^{71} + 11 q^{72} + 22 q^{73} - 7 q^{74} + 16 q^{77} + 30 q^{78} + 24 q^{79} - 21 q^{81} - 31 q^{82} + 3 q^{83} + 28 q^{84} + 32 q^{86} - 12 q^{87} - 9 q^{88} + 14 q^{89} + 10 q^{91} - 41 q^{92} - 6 q^{93} - 21 q^{94} + 17 q^{96} + 7 q^{97} + 23 q^{98} - 13 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 6x + 7 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 4 \) 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
2.28514
−2.50702
1.22188
−2.50702 1.22188 4.28514 0 −3.06327 0.221876 −5.72889 −1.50702 0
1.2 1.22188 2.28514 −0.507019 0 2.79216 1.28514 −3.06327 2.22188 0
1.3 2.28514 −2.50702 3.22188 0 −5.72889 −3.50702 2.79216 3.28514 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.ba 3
5.b even 2 1 1805.2.a.g 3
19.b odd 2 1 9025.2.a.z 3
19.d odd 6 2 475.2.e.d 6
95.d odd 2 1 1805.2.a.h 3
95.h odd 6 2 95.2.e.b 6
95.l even 12 4 475.2.j.b 12
285.q even 6 2 855.2.k.g 6
380.s even 6 2 1520.2.q.j 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
95.2.e.b 6 95.h odd 6 2
475.2.e.d 6 19.d odd 6 2
475.2.j.b 12 95.l even 12 4
855.2.k.g 6 285.q even 6 2
1520.2.q.j 6 380.s even 6 2
1805.2.a.g 3 5.b even 2 1
1805.2.a.h 3 95.d odd 2 1
9025.2.a.z 3 19.b odd 2 1
9025.2.a.ba 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9025))\):

\( T_{2}^{3} - T_{2}^{2} - 6T_{2} + 7 \) Copy content Toggle raw display
\( T_{3}^{3} - T_{3}^{2} - 6T_{3} + 7 \) Copy content Toggle raw display
\( T_{7}^{3} + 2T_{7}^{2} - 5T_{7} + 1 \) Copy content Toggle raw display
\( T_{11}^{3} + 5T_{11}^{2} + 2T_{11} - 1 \) Copy content Toggle raw display
\( T_{29}^{3} - 2T_{29}^{2} - 5T_{29} - 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} - T^{2} - 6T + 7 \) Copy content Toggle raw display
$3$ \( T^{3} - T^{2} - 6T + 7 \) Copy content Toggle raw display
$5$ \( T^{3} \) Copy content Toggle raw display
$7$ \( T^{3} + 2 T^{2} - 5 T + 1 \) Copy content Toggle raw display
$11$ \( T^{3} + 5 T^{2} + 2 T - 1 \) Copy content Toggle raw display
$13$ \( (T + 5)^{3} \) Copy content Toggle raw display
$17$ \( T^{3} + T^{2} - 44T - 7 \) Copy content Toggle raw display
$19$ \( T^{3} \) Copy content Toggle raw display
$23$ \( T^{3} + 4 T^{2} - 39 T - 49 \) Copy content Toggle raw display
$29$ \( T^{3} - 2 T^{2} - 5 T - 1 \) Copy content Toggle raw display
$31$ \( T^{3} - T^{2} - 6T + 7 \) Copy content Toggle raw display
$37$ \( T^{3} + 2 T^{2} - 119 T - 227 \) Copy content Toggle raw display
$41$ \( T^{3} - 2 T^{2} - 43 T + 37 \) Copy content Toggle raw display
$43$ \( T^{3} - T^{2} - 44 T + 121 \) Copy content Toggle raw display
$47$ \( T^{3} + 6 T^{2} - 7 T - 49 \) Copy content Toggle raw display
$53$ \( T^{3} - 11 T^{2} - 42 T + 311 \) Copy content Toggle raw display
$59$ \( T^{3} + 6 T^{2} - 7 T - 49 \) Copy content Toggle raw display
$61$ \( T^{3} + 9 T^{2} - 49 T - 49 \) Copy content Toggle raw display
$67$ \( T^{3} + 20 T^{2} + 108 T + 88 \) Copy content Toggle raw display
$71$ \( T^{3} - 29 T^{2} + 236 T - 467 \) Copy content Toggle raw display
$73$ \( T^{3} - 22 T^{2} + 117 T - 77 \) Copy content Toggle raw display
$79$ \( T^{3} - 24 T^{2} + 116 T + 248 \) Copy content Toggle raw display
$83$ \( T^{3} - 3 T^{2} - 54 T - 77 \) Copy content Toggle raw display
$89$ \( T^{3} - 14 T^{2} - 36 T + 56 \) Copy content Toggle raw display
$97$ \( T^{3} - 7 T^{2} - 66 T + 121 \) Copy content Toggle raw display
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