Properties

Label 5625.2.a.ba
Level $5625$
Weight $2$
Character orbit 5625.a
Self dual yes
Analytic conductor $44.916$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 5625 = 3^{2} \cdot 5^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5625.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(44.9158511370\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.8.46980000000.2
Defining polynomial: \( x^{8} - 15x^{6} + 80x^{4} - 180x^{2} + 145 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{2} + 2) q^{4} + ( - 2 \beta_{5} + \beta_{3}) q^{7} + (\beta_{6} + \beta_{4} + \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{2} + 2) q^{4} + ( - 2 \beta_{5} + \beta_{3}) q^{7} + (\beta_{6} + \beta_{4} + \beta_1) q^{8} + ( - \beta_{7} + \beta_{6} + \beta_{4} + \beta_1) q^{11} + (2 \beta_{5} + \beta_{3} + 2) q^{13} + (\beta_{7} - \beta_{6} + \beta_1) q^{14} + (\beta_{3} + 2 \beta_{2} - 2) q^{16} + (\beta_{7} + 2 \beta_1) q^{17} + ( - \beta_{3} - 2 \beta_{2}) q^{19} + (2 \beta_{5} - 2 \beta_{3} + 4 \beta_{2} + 5) q^{22} + ( - \beta_{7} + \beta_{4} - \beta_1) q^{23} + (\beta_{7} + 3 \beta_{6} + 3 \beta_1) q^{26} + ( - \beta_{5} + \beta_{2} + 1) q^{28} + (\beta_{7} + 2 \beta_{4} + 3 \beta_1) q^{29} + (3 \beta_{3} - 3 \beta_{2} - 1) q^{31} + (\beta_{7} + \beta_{6} - \beta_1) q^{32} + ( - 2 \beta_{5} + 3 \beta_{3} + 2 \beta_{2} + 5) q^{34} + ( - 3 \beta_{3} - 3 \beta_{2}) q^{37} + ( - \beta_{7} - 3 \beta_{6} - 2 \beta_{4} - 3 \beta_1) q^{38} + ( - \beta_{7} - \beta_{6} - \beta_{4} + \beta_1) q^{41} + ( - 2 \beta_{3} + \beta_{2} + 7) q^{43} + (2 \beta_{6} + 2 \beta_{4} + 5 \beta_1) q^{44} + ( - \beta_{5} - 3 \beta_{3} + 2 \beta_{2} - 3) q^{46} + ( - 3 \beta_{7} - 2 \beta_{6} + \beta_{4} - 2 \beta_1) q^{47} + ( - 5 \beta_{5} + 4 \beta_{3} - 3 \beta_{2} - 5) q^{49} + (3 \beta_{5} + 4 \beta_{3} + 3 \beta_{2} + 5) q^{52} + ( - 2 \beta_{7} + \beta_{6} - \beta_{4}) q^{53} + ( - 2 \beta_{7} + 2 \beta_{6} + \beta_{4}) q^{56} + ( - 8 \beta_{5} + 3 \beta_{3} + 9 \beta_{2} + 5) q^{58} + ( - \beta_{7} - 2 \beta_{6} - 2 \beta_{4} + \beta_1) q^{59} + ( - 3 \beta_{5} + 3 \beta_{2} + 5) q^{61} + (3 \beta_{7} - 3 \beta_{4} - \beta_1) q^{62} + (\beta_{5} + 2 \beta_{3} - 5 \beta_{2} - 3) q^{64} + (\beta_{5} - 2 \beta_{3} - 4 \beta_{2} + 5) q^{67} + (\beta_{7} + 3 \beta_{6} + 2 \beta_{4} + 6 \beta_1) q^{68} + (4 \beta_{7} + 2 \beta_{6} - 2 \beta_{4} + 3 \beta_1) q^{71} + (2 \beta_{5} + 2 \beta_{3} - 3 \beta_{2} + 6) q^{73} + ( - 3 \beta_{7} - 6 \beta_{6} - 3 \beta_{4} - 6 \beta_1) q^{74} + ( - \beta_{5} - 4 \beta_{3} - 5 \beta_{2} - 5) q^{76} + ( - 4 \beta_{7} + 3 \beta_{6} + 3 \beta_{4} - \beta_1) q^{77} + (\beta_{5} + 4 \beta_{2}) q^{79} + (2 \beta_{5} - 4 \beta_{3} - 2 \beta_{2} + 9) q^{82} + (4 \beta_{7} + 2 \beta_{4} + 3 \beta_1) q^{83} + ( - 2 \beta_{7} - \beta_{6} + \beta_{4} + 6 \beta_1) q^{86} + ( - 4 \beta_{5} + 6 \beta_{3} + 3 \beta_{2} + 6) q^{88} + ( - \beta_{7} - 3 \beta_{6} + 2 \beta_{4} - 4 \beta_1) q^{89} + ( - \beta_{5} + 2 \beta_{3} + \beta_{2} - 2) q^{91} + ( - \beta_{7} - 2 \beta_{6} - 2 \beta_1) q^{92} + ( - 3 \beta_{5} - 11 \beta_{3} + \beta_{2} - 1) q^{94} + (4 \beta_{5} - 5 \beta_{3} + 2 \beta_{2} + 10) q^{97} + (4 \beta_{7} - 4 \beta_{6} - 3 \beta_{4} - 4 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 14 q^{4} + 10 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 14 q^{4} + 10 q^{7} + 10 q^{13} - 18 q^{16} + 2 q^{19} + 20 q^{22} + 10 q^{28} + 4 q^{31} + 50 q^{34} + 50 q^{43} - 30 q^{46} - 6 q^{49} + 30 q^{52} + 60 q^{58} + 46 q^{61} - 14 q^{64} + 40 q^{67} + 50 q^{73} - 34 q^{76} - 12 q^{79} + 60 q^{82} + 70 q^{88} - 10 q^{91} - 20 q^{94} + 50 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 15x^{6} + 80x^{4} - 180x^{2} + 145 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{4} - 8\nu^{2} + 14 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -\nu^{7} + 12\nu^{5} - 44\nu^{3} + 48\nu \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \nu^{6} - 12\nu^{4} + 45\nu^{2} - 53 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( \nu^{7} - 12\nu^{5} + 45\nu^{3} - 53\nu \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( -\nu^{7} + 13\nu^{5} - 53\nu^{3} + 66\nu \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{6} + \beta_{4} + 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{3} + 8\beta_{2} + 18 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{7} + 9\beta_{6} + 8\beta_{4} + 27\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{5} + 12\beta_{3} + 51\beta_{2} + 89 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 12\beta_{7} + 64\beta_{6} + 51\beta_{4} + 152\beta_1 \) 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
−2.44055
−2.05160
−1.63148
−1.47408
1.47408
1.63148
2.05160
2.44055
−2.44055 0 3.95630 0 0 0.591023 −4.77444 0 0
1.2 −2.05160 0 2.20906 0 0 1.27977 −0.428901 0 0
1.3 −1.63148 0 0.661739 0 0 −1.44512 2.18335 0 0
1.4 −1.47408 0 0.172909 0 0 4.57433 2.69328 0 0
1.5 1.47408 0 0.172909 0 0 4.57433 −2.69328 0 0
1.6 1.63148 0 0.661739 0 0 −1.44512 −2.18335 0 0
1.7 2.05160 0 2.20906 0 0 1.27977 0.428901 0 0
1.8 2.44055 0 3.95630 0 0 0.591023 4.77444 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(5\) \(-1\)

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 5625.2.a.ba yes 8
3.b odd 2 1 inner 5625.2.a.ba yes 8
5.b even 2 1 5625.2.a.y 8
15.d odd 2 1 5625.2.a.y 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5625.2.a.y 8 5.b even 2 1
5625.2.a.y 8 15.d odd 2 1
5625.2.a.ba yes 8 1.a even 1 1 trivial
5625.2.a.ba yes 8 3.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(5625))\):

\( T_{2}^{8} - 15T_{2}^{6} + 80T_{2}^{4} - 180T_{2}^{2} + 145 \) Copy content Toggle raw display
\( T_{7}^{4} - 5T_{7}^{3} + 10T_{7} - 5 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - 15 T^{6} + 80 T^{4} + \cdots + 145 \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( (T^{4} - 5 T^{3} + 10 T - 5)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} - 60 T^{6} + 1100 T^{4} + \cdots + 3625 \) Copy content Toggle raw display
$13$ \( (T^{4} - 5 T^{3} - 10 T^{2} + 50 T - 5)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} - 65 T^{6} + 1110 T^{4} + \cdots + 145 \) Copy content Toggle raw display
$19$ \( (T^{4} - T^{3} - 24 T^{2} + 74 T - 59)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} - 60 T^{6} + 990 T^{4} + \cdots + 11745 \) Copy content Toggle raw display
$29$ \( T^{8} - 210 T^{6} + 15050 T^{4} + \cdots + 3483625 \) Copy content Toggle raw display
$31$ \( (T^{4} - 2 T^{3} - 66 T^{2} + 67 T + 211)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} - 90 T^{2} + 405 T - 405)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} - 120 T^{6} + 3800 T^{4} + \cdots + 3625 \) Copy content Toggle raw display
$43$ \( (T^{4} - 25 T^{3} + 215 T^{2} - 755 T + 895)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} - 305 T^{6} + 28935 T^{4} + \cdots + 1148545 \) Copy content Toggle raw display
$53$ \( T^{8} - 180 T^{6} + 5705 T^{4} + \cdots + 145 \) Copy content Toggle raw display
$59$ \( T^{8} - 270 T^{6} + 21950 T^{4} + \cdots + 3483625 \) Copy content Toggle raw display
$61$ \( (T^{4} - 23 T^{3} + 159 T^{2} - 257 T - 359)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} - 20 T^{3} + 65 T^{2} + 680 T - 3455)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} - 575 T^{6} + \cdots + 118758625 \) Copy content Toggle raw display
$73$ \( (T^{4} - 25 T^{3} + 180 T^{2} - 190 T - 1055)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 6 T^{3} - 69 T^{2} - 414 T - 279)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} - 555 T^{6} + \cdots + 10648945 \) Copy content Toggle raw display
$89$ \( T^{8} - 525 T^{6} + \cdots + 76215625 \) Copy content Toggle raw display
$97$ \( (T^{4} - 25 T^{3} + 110 T^{2} + 1150 T - 7355)^{2} \) Copy content Toggle raw display
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