Properties

Label 9025.2.a.bt
Level $9025$
Weight $2$
Character orbit 9025.a
Self dual yes
Analytic conductor $72.065$
Analytic rank $1$
Dimension $6$
CM no
Inner twists $2$

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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: \(1\)
Dimension: \(6\)
Coefficient field: 6.6.4227136.1
Defining polynomial: \( x^{6} - 6x^{4} + 7x^{2} - 1 \) 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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{5} + \beta_1) q^{2} + (\beta_{5} - \beta_{4} + \beta_1) q^{3} + (\beta_{3} - \beta_{2}) q^{4} + ( - \beta_{2} + 2) q^{6} + ( - 2 \beta_{5} - \beta_1) q^{7} + (2 \beta_{5} + \beta_{4}) q^{8} + ( - 2 \beta_{3} - \beta_{2} + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{5} + \beta_1) q^{2} + (\beta_{5} - \beta_{4} + \beta_1) q^{3} + (\beta_{3} - \beta_{2}) q^{4} + ( - \beta_{2} + 2) q^{6} + ( - 2 \beta_{5} - \beta_1) q^{7} + (2 \beta_{5} + \beta_{4}) q^{8} + ( - 2 \beta_{3} - \beta_{2} + 2) q^{9} + \beta_{3} q^{11} + (2 \beta_{5} + 2 \beta_{4} + \beta_1) q^{12} + ( - \beta_{5} - \beta_{4} - 2 \beta_1) q^{13} + ( - 2 \beta_{3} + 2 \beta_{2} - 3) q^{14} + (\beta_{3} + 2) q^{16} + ( - \beta_{5} + 2 \beta_{4} - \beta_1) q^{17} + ( - 2 \beta_{4} + \beta_1) q^{18} + ( - \beta_{3} + \beta_{2} - 3) q^{21} + (2 \beta_{5} + \beta_{4} + \beta_1) q^{22} + (\beta_{4} - \beta_1) q^{23} + (4 \beta_{3} - 1) q^{24} + ( - 2 \beta_{3} + \beta_{2} - 3) q^{26} + ( - 2 \beta_{5} - 3 \beta_{4} + 3 \beta_1) q^{27} + ( - 7 \beta_{5} - 2 \beta_{4} - 5 \beta_1) q^{28} + ( - 3 \beta_{3} + 3) q^{29} + (\beta_{2} - 5) q^{31} + ( - \beta_{4} + 3 \beta_1) q^{32} + (\beta_{5} + 2 \beta_{4} - \beta_1) q^{33} + (\beta_{3} + \beta_{2} - 2) q^{34} + (2 \beta_{3} + 2 \beta_{2} - 3) q^{36} + ( - 2 \beta_{4} - 3 \beta_1) q^{37} + ( - \beta_{3} + 2 \beta_{2}) q^{39} + (3 \beta_{3} + 2 \beta_{2} - 3) q^{41} + ( - 7 \beta_{5} - \beta_{4} - 5 \beta_1) q^{42} + ( - \beta_{5} + 4 \beta_1) q^{43} + (\beta_{3} - 2 \beta_{2} + 3) q^{44} + (\beta_{3} - 1) q^{46} + ( - 4 \beta_{5} + 4 \beta_{4} - \beta_1) q^{47} + (3 \beta_{5} + \beta_1) q^{48} + (4 \beta_{3} - 3 \beta_{2} - 1) q^{49} + (4 \beta_{3} + \beta_{2} - 8) q^{51} + ( - 7 \beta_{5} - 2 \beta_1) q^{52} + (\beta_{5} + 2 \beta_{4} - 3 \beta_1) q^{53} + ( - 5 \beta_{3} + 2 \beta_{2} + 1) q^{54} + ( - 5 \beta_{3} + 3 \beta_{2} - 6) q^{56} + ( - 3 \beta_{5} - 3 \beta_{4}) q^{58} + ( - \beta_{3} + 4 \beta_{2} - 3) q^{59} + (2 \beta_{3} - 2 \beta_{2} - 1) q^{61} + ( - 7 \beta_{5} - 6 \beta_1) q^{62} + (\beta_{5} + \beta_{4} - \beta_1) q^{63} + ( - 3 \beta_{3} - 1) q^{64} + (3 \beta_{3} - \beta_{2}) q^{66} + (2 \beta_{5} + 4 \beta_1) q^{67} - 3 \beta_{4} q^{68} + (3 \beta_{3} + \beta_{2} - 4) q^{69} + (\beta_{3} + 5 \beta_{2}) q^{71} + ( - 3 \beta_{5} + 6 \beta_{4} - 5 \beta_1) q^{72} + (2 \beta_{5} + \beta_{4} - 5 \beta_1) q^{73} + ( - 2 \beta_{3} - 3) q^{74} + ( - 4 \beta_{5} - \beta_{4} - 2 \beta_1) q^{77} + ( - 6 \beta_{5} - \beta_{4} - 3 \beta_1) q^{78} + (4 \beta_{2} - 4) q^{79} + ( - 5 \beta_{3} + 4) q^{81} + ( - \beta_{5} + 3 \beta_{4} - 2 \beta_1) q^{82} + (5 \beta_{5} + 2 \beta_{4} + 9 \beta_1) q^{83} + ( - 6 \beta_{3} + 5 \beta_{2} - 6) q^{84} + ( - \beta_{3} + \beta_{2} + 3) q^{86} + ( - 9 \beta_{4} + 6 \beta_1) q^{87} + (5 \beta_{5} - \beta_{4} + 4 \beta_1) q^{88} + ( - 4 \beta_{2} - 6) q^{89} + (3 \beta_{3} - 4 \beta_{2} + 3) q^{91} + (\beta_{5} - \beta_{4} + 2 \beta_1) q^{92} + ( - 6 \beta_{5} + 5 \beta_{4} - 7 \beta_1) q^{93} + (4 \beta_{2} - 5) q^{94} + ( - 5 \beta_{3} - 3 \beta_{2} + 6) q^{96} + (5 \beta_{5} - 7 \beta_{4} + \beta_1) q^{97} + (13 \beta_{5} + 4 \beta_{4} + 6 \beta_1) q^{98} + (3 \beta_{3} + \beta_{2} - 6) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 2 q^{4} + 12 q^{6} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 2 q^{4} + 12 q^{6} + 8 q^{9} + 2 q^{11} - 22 q^{14} + 14 q^{16} - 20 q^{21} + 2 q^{24} - 22 q^{26} + 12 q^{29} - 30 q^{31} - 10 q^{34} - 14 q^{36} - 2 q^{39} - 12 q^{41} + 20 q^{44} - 4 q^{46} + 2 q^{49} - 40 q^{51} - 4 q^{54} - 46 q^{56} - 20 q^{59} - 2 q^{61} - 12 q^{64} + 6 q^{66} - 18 q^{69} + 2 q^{71} - 22 q^{74} - 24 q^{79} + 14 q^{81} - 48 q^{84} + 16 q^{86} - 36 q^{89} + 24 q^{91} - 30 q^{94} + 26 q^{96} - 30 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 6x^{4} + 7x^{2} - 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{4} - 5\nu^{2} + 3 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{5} - 5\nu^{3} + 3\nu \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \nu^{5} - 6\nu^{3} + 6\nu \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{5} + \beta_{4} + 3\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{3} + 5\beta_{2} + 7 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -5\beta_{5} + 6\beta_{4} + 12\beta_1 \) 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
−0.407132
−1.15904
−2.11917
2.11917
1.15904
0.407132
−2.45620 −1.56104 4.03293 0 3.83424 4.50527 −4.99330 −0.563139 0
1.2 −0.862781 −3.07914 −1.25561 0 2.65662 0.566520 2.80888 6.48108 0
1.3 −0.471884 1.04022 −1.77733 0 −0.490864 −1.17540 1.78246 −1.91794 0
1.4 0.471884 −1.04022 −1.77733 0 −0.490864 1.17540 −1.78246 −1.91794 0
1.5 0.862781 3.07914 −1.25561 0 2.65662 −0.566520 −2.80888 6.48108 0
1.6 2.45620 1.56104 4.03293 0 3.83424 −4.50527 4.99330 −0.563139 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
Significant digits:
Format:

Atkin-Lehner signs

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

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 9025.2.a.bt 6
5.b even 2 1 inner 9025.2.a.bt 6
5.c odd 4 2 1805.2.b.g 6
19.b odd 2 1 9025.2.a.bu 6
19.d odd 6 2 475.2.e.g 12
95.d odd 2 1 9025.2.a.bu 6
95.g even 4 2 1805.2.b.f 6
95.h odd 6 2 475.2.e.g 12
95.l even 12 4 95.2.i.b 12
285.w odd 12 4 855.2.be.d 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
95.2.i.b 12 95.l even 12 4
475.2.e.g 12 19.d odd 6 2
475.2.e.g 12 95.h odd 6 2
855.2.be.d 12 285.w odd 12 4
1805.2.b.f 6 95.g even 4 2
1805.2.b.g 6 5.c odd 4 2
9025.2.a.bt 6 1.a even 1 1 trivial
9025.2.a.bt 6 5.b even 2 1 inner
9025.2.a.bu 6 19.b odd 2 1
9025.2.a.bu 6 95.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(9025))\):

\( T_{2}^{6} - 7T_{2}^{4} + 6T_{2}^{2} - 1 \) Copy content Toggle raw display
\( T_{3}^{6} - 13T_{3}^{4} + 36T_{3}^{2} - 25 \) Copy content Toggle raw display
\( T_{7}^{6} - 22T_{7}^{4} + 35T_{7}^{2} - 9 \) Copy content Toggle raw display
\( T_{11}^{3} - T_{11}^{2} - 4T_{11} + 3 \) Copy content Toggle raw display
\( T_{29}^{3} - 6T_{29}^{2} - 27T_{29} + 27 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} - 7 T^{4} + 6 T^{2} - 1 \) Copy content Toggle raw display
$3$ \( T^{6} - 13 T^{4} + 36 T^{2} - 25 \) Copy content Toggle raw display
$5$ \( T^{6} \) Copy content Toggle raw display
$7$ \( T^{6} - 22 T^{4} + 35 T^{2} - 9 \) Copy content Toggle raw display
$11$ \( (T^{3} - T^{2} - 4 T + 3)^{2} \) Copy content Toggle raw display
$13$ \( T^{6} - 31 T^{4} + 239 T^{2} - 9 \) Copy content Toggle raw display
$17$ \( T^{6} - 35 T^{4} + 198 T^{2} + \cdots - 81 \) Copy content Toggle raw display
$19$ \( T^{6} \) Copy content Toggle raw display
$23$ \( T^{6} - 12 T^{4} + 7 T^{2} - 1 \) Copy content Toggle raw display
$29$ \( (T^{3} - 6 T^{2} - 27 T + 27)^{2} \) Copy content Toggle raw display
$31$ \( (T^{3} + 15 T^{2} + 70 T + 97)^{2} \) Copy content Toggle raw display
$37$ \( T^{6} - 98 T^{4} + 887 T^{2} + \cdots - 729 \) Copy content Toggle raw display
$41$ \( (T^{3} + 6 T^{2} - 41 T - 3)^{2} \) Copy content Toggle raw display
$43$ \( T^{6} - 127 T^{4} + 2516 T^{2} + \cdots - 441 \) Copy content Toggle raw display
$47$ \( T^{6} - 214 T^{4} + 13407 T^{2} + \cdots - 218089 \) Copy content Toggle raw display
$53$ \( T^{6} - 99 T^{4} + 2302 T^{2} + \cdots - 11449 \) Copy content Toggle raw display
$59$ \( (T^{3} + 10 T^{2} - 55 T - 291)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} + T^{2} - 41 T - 113)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} - 76 T^{4} + 1856 T^{2} + \cdots - 14400 \) Copy content Toggle raw display
$71$ \( (T^{3} - T^{2} - 124 T - 477)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} - 236 T^{4} + 13331 T^{2} + \cdots - 99225 \) Copy content Toggle raw display
$79$ \( (T^{3} + 12 T^{2} - 32 T - 448)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} - 459 T^{4} + 56302 T^{2} + \cdots - 966289 \) Copy content Toggle raw display
$89$ \( (T^{3} + 18 T^{2} + 28 T - 72)^{2} \) Copy content Toggle raw display
$97$ \( T^{6} - 529 T^{4} + 74192 T^{2} + \cdots - 1238769 \) Copy content Toggle raw display
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