Properties

Label 5625.2.a.bb
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: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \( x^{8} - 15x^{6} + 70x^{4} - 105x^{2} + 45 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{2} + 2) q^{4} + ( - \beta_{4} + \beta_{2} + 1) q^{7} + (\beta_{3} + 2 \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{2} + 2) q^{4} + ( - \beta_{4} + \beta_{2} + 1) q^{7} + (\beta_{3} + 2 \beta_1) q^{8} + \beta_{7} q^{11} + (\beta_{4} + \beta_{2} + 2) q^{13} + ( - \beta_{5} + \beta_{3} + 3 \beta_1) q^{14} + (\beta_{6} + 2 \beta_{2} + 3) q^{16} + (\beta_{7} - 2 \beta_1) q^{17} + ( - \beta_{4} + \beta_{2}) q^{19} + (2 \beta_{6} + \beta_{2} + 1) q^{22} + ( - 2 \beta_{5} - \beta_{3} - \beta_1) q^{23} + (\beta_{5} + \beta_{3} + 4 \beta_1) q^{26} + ( - \beta_{4} + 3 \beta_{2} + 9) q^{28} + ( - \beta_{7} - \beta_{3} - \beta_1) q^{29} + ( - \beta_{6} + \beta_{4}) q^{31} + (\beta_{7} + 2 \beta_{5} + 3 \beta_1) q^{32} + (2 \beta_{6} - \beta_{2} - 7) q^{34} + ( - \beta_{6} + \beta_{4} + 7) q^{37} + ( - \beta_{5} + \beta_{3} + 2 \beta_1) q^{38} + ( - 4 \beta_{5} + \beta_{3} - \beta_1) q^{41} + ( - 2 \beta_{6} + \beta_{4} + 1) q^{43} + (4 \beta_{5} + \beta_{3} + 3 \beta_1) q^{44} + ( - 3 \beta_{6} - 6 \beta_{4} - 3 \beta_{2} - 3) q^{46} + ( - 2 \beta_{5} - 2 \beta_1) q^{47} + ( - \beta_{6} - \beta_{4} + 2 \beta_{2} + 2) q^{49} + (2 \beta_{6} + \beta_{4} + 4 \beta_{2} + 11) q^{52} + (2 \beta_{5} - 2 \beta_1) q^{53} + (\beta_{5} + \beta_{3} + 9 \beta_1) q^{56} + ( - 3 \beta_{6} - 4 \beta_{2} - 4) q^{58} + ( - \beta_{7} + 4 \beta_{5} + 2 \beta_1) q^{59} + ( - \beta_{6} + \beta_{4} - 6) q^{61} + ( - \beta_{7} - \beta_{5}) q^{62} + (2 \beta_{6} + 6 \beta_{4} + 7) q^{64} + (3 \beta_{6} - \beta_{4}) q^{67} + (4 \beta_{5} - \beta_{3} - 5 \beta_1) q^{68} + (\beta_{7} - 4 \beta_{5} + \beta_{3} - \beta_1) q^{71} + ( - 2 \beta_{6} - \beta_{4} - 3 \beta_{2} + 3) q^{73} + ( - \beta_{7} - \beta_{5} + 7 \beta_1) q^{74} + ( - \beta_{4} + 2 \beta_{2} + 7) q^{76} + (4 \beta_{5} + 2 \beta_1) q^{77} + ( - 2 \beta_{6} + \beta_{4} - 2 \beta_{2} + 4) q^{79} + ( - 3 \beta_{6} - 12 \beta_{4} + \beta_{2} - 5) q^{82} + (\beta_{7} - 2 \beta_{3} - 4 \beta_1) q^{83} + ( - 2 \beta_{7} - 3 \beta_{5} + \beta_1) q^{86} + (\beta_{6} + 12 \beta_{4} + 3 \beta_{2} + 9) q^{88} + (\beta_{7} + 2 \beta_{5} - 2 \beta_{3} + 4 \beta_1) q^{89} + (\beta_{6} + 3 \beta_{2} + 8) q^{91} + ( - 3 \beta_{7} - 8 \beta_{5} - \beta_{3} - 7 \beta_1) q^{92} + ( - 2 \beta_{6} - 6 \beta_{4} - 2 \beta_{2} - 8) q^{94} + (\beta_{6} - \beta_{4} - \beta_{2} - 1) q^{97} + ( - \beta_{7} - 3 \beta_{5} + 2 \beta_{3} + 6 \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 \( 8 q + 14 q^{4} + 10 q^{7} + 10 q^{13} + 22 q^{16} + 2 q^{19} + 10 q^{22} + 70 q^{28} - 6 q^{31} - 50 q^{34} + 50 q^{37} + 14 q^{49} + 80 q^{52} - 30 q^{58} - 54 q^{61} + 36 q^{64} + 10 q^{67} + 30 q^{73} + 56 q^{76} + 28 q^{79} + 20 q^{88} + 60 q^{91} - 40 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 15x^{6} + 70x^{4} - 105x^{2} + 45 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 6\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{6} - 12\nu^{4} + 40\nu^{2} - 33 ) / 6 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{7} - 12\nu^{5} + 40\nu^{3} - 33\nu ) / 6 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( \nu^{4} - 8\nu^{2} + 9 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -\nu^{7} + 15\nu^{5} - 64\nu^{3} + 60\nu ) / 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 6\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{6} + 8\beta_{2} + 23 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{7} + 2\beta_{5} + 8\beta_{3} + 39\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 12\beta_{6} + 6\beta_{4} + 56\beta_{2} + 149 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 12\beta_{7} + 30\beta_{5} + 56\beta_{3} + 261\beta_1 \) 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
−2.66202
−2.37653
−1.23762
−0.856773
0.856773
1.23762
2.37653
2.66202
−2.66202 0 5.08634 0 0 3.46831 −8.21589 0 0
1.2 −2.37653 0 3.64791 0 0 4.26594 −3.91630 0 0
1.3 −1.23762 0 −0.468306 0 0 −2.08634 3.05482 0 0
1.4 −0.856773 0 −1.26594 0 0 −0.647907 2.79817 0 0
1.5 0.856773 0 −1.26594 0 0 −0.647907 −2.79817 0 0
1.6 1.23762 0 −0.468306 0 0 −2.08634 −3.05482 0 0
1.7 2.37653 0 3.64791 0 0 4.26594 3.91630 0 0
1.8 2.66202 0 5.08634 0 0 3.46831 8.21589 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.bb yes 8
3.b odd 2 1 inner 5625.2.a.bb yes 8
5.b even 2 1 5625.2.a.z 8
15.d odd 2 1 5625.2.a.z 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5625.2.a.z 8 5.b even 2 1
5625.2.a.z 8 15.d odd 2 1
5625.2.a.bb yes 8 1.a even 1 1 trivial
5625.2.a.bb 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(5625))\):

\( T_{2}^{8} - 15T_{2}^{6} + 70T_{2}^{4} - 105T_{2}^{2} + 45 \) Copy content Toggle raw display
\( T_{7}^{4} - 5T_{7}^{3} - 5T_{7}^{2} + 30T_{7} + 20 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - 15 T^{6} + 70 T^{4} - 105 T^{2} + \cdots + 45 \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( (T^{4} - 5 T^{3} - 5 T^{2} + 30 T + 20)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} - 60 T^{6} + 1075 T^{4} + \cdots + 18000 \) Copy content Toggle raw display
$13$ \( (T^{4} - 5 T^{3} - 10 T^{2} + 35 T - 5)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} - 100 T^{6} + 3115 T^{4} + \cdots + 87120 \) Copy content Toggle raw display
$19$ \( (T^{4} - T^{3} - 14 T^{2} + 9 T + 41)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} - 165 T^{6} + 8955 T^{4} + \cdots + 933120 \) Copy content Toggle raw display
$29$ \( T^{8} - 135 T^{6} + 4675 T^{4} + \cdots + 18000 \) Copy content Toggle raw display
$31$ \( (T^{4} + 3 T^{3} - 16 T^{2} - 13 T + 41)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} - 25 T^{3} + 215 T^{2} - 720 T + 720)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} - 285 T^{6} + 24175 T^{4} + \cdots + 18000 \) Copy content Toggle raw display
$43$ \( (T^{4} - 75 T^{2} + 745)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} - 100 T^{6} + 2080 T^{4} + \cdots + 11520 \) Copy content Toggle raw display
$53$ \( T^{8} - 180 T^{6} + 4000 T^{4} + \cdots + 11520 \) Copy content Toggle raw display
$59$ \( T^{8} - 300 T^{6} + \cdots + 30258000 \) Copy content Toggle raw display
$61$ \( (T^{4} + 27 T^{3} + 254 T^{2} + 983 T + 1331)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} - 5 T^{3} - 160 T^{2} + 545 T + 3295)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} - 295 T^{6} + 26275 T^{4} + \cdots + 4608000 \) Copy content Toggle raw display
$73$ \( (T^{4} - 15 T^{3} - 85 T^{2} + 2140 T - 8080)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} - 14 T^{3} - 19 T^{2} + 636 T - 909)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} - 420 T^{6} + \cdots + 28512720 \) Copy content Toggle raw display
$89$ \( T^{8} - 600 T^{6} + 116875 T^{4} + \cdots + 7200000 \) Copy content Toggle raw display
$97$ \( (T^{4} - 45 T^{2} - 90 T - 5)^{2} \) Copy content Toggle raw display
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