Properties

Label 5625.2.a.o
Level $5625$
Weight $2$
Character orbit 5625.a
Self dual yes
Analytic conductor $44.916$
Analytic rank $1$
Dimension $6$
CM no
Inner twists $1$

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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: \(1\)
Dimension: \(6\)
Coefficient field: 6.6.46840000.1
Defining polynomial: \( x^{6} - x^{5} - 11x^{4} + 8x^{3} + 31x^{2} - 15x - 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1875)
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.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + (\beta_{2} + 2) q^{4} + (\beta_{3} - \beta_{2}) q^{7} + ( - \beta_{5} - \beta_{3} - \beta_{2} - \beta_1 - 1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + (\beta_{2} + 2) q^{4} + (\beta_{3} - \beta_{2}) q^{7} + ( - \beta_{5} - \beta_{3} - \beta_{2} - \beta_1 - 1) q^{8} + ( - \beta_{5} - \beta_{3}) q^{11} + ( - \beta_{3} - \beta_{2} + 2 \beta_1 - 1) q^{13} + (\beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} + \beta_1 + 1) q^{14} + (\beta_{5} + 2 \beta_{4} + \beta_{3} + \beta_{2} + 2 \beta_1 + 2) q^{16} + (\beta_{5} + \beta_{3} - 2 \beta_1) q^{17} + ( - 2 \beta_{4} + \beta_{3} + \beta_{2} + 1) q^{19} + (2 \beta_{4} + \beta_{2} + 1) q^{22} + ( - \beta_{5} + 2 \beta_{4} - \beta_{3} - \beta_{2} + 2 \beta_1 - 1) q^{23} + (\beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} + 2 \beta_1 - 7) q^{26} + ( - 2 \beta_{4} + \beta_{3} - \beta_{2} - 2 \beta_1 - 6) q^{28} + (\beta_{2} - 5) q^{29} + (\beta_{5} - 2 \beta_{4} + \beta_{2}) q^{31} + ( - \beta_{5} - 2 \beta_{4} - 7 \beta_{3} - 2 \beta_{2} - \beta_1 - 8) q^{32} + ( - 2 \beta_{4} + \beta_{2} + 7) q^{34} + (\beta_{5} + 2 \beta_{4} + \beta_{2} + 2 \beta_1 - 5) q^{37} + (\beta_{5} - \beta_{4} + 7 \beta_{3} - \beta_{2} - 2 \beta_1 - 1) q^{38} + (\beta_{5} + \beta_{3} + \beta_{2} - 2 \beta_1 - 5) q^{41} + ( - 2 \beta_{4} - \beta_{3} - 2 \beta_1) q^{43} + ( - \beta_{5} - 7 \beta_{3} - \beta_{2} - 2 \beta_1 - 1) q^{44} + ( - \beta_{5} + 2 \beta_{4} - 7 \beta_{3} + 2 \beta_1 - 6) q^{46} + (2 \beta_{5} - 2 \beta_{4} + 2 \beta_{3} + 2 \beta_{2} + 2) q^{47} + ( - \beta_{5} + 2 \beta_{4} - \beta_{2} + 2 \beta_1) q^{49} + ( - 2 \beta_{4} - \beta_{3} + 4 \beta_1 - 6) q^{52} + (2 \beta_{5} + 2 \beta_{4} - 4 \beta_{3} - 6) q^{53} + (\beta_{5} + \beta_{4} + 7 \beta_{3} + \beta_{2} + 5 \beta_1 + 7) q^{56} + ( - \beta_{5} - \beta_{3} - \beta_{2} + 4 \beta_1 - 1) q^{58} + (\beta_{5} + \beta_{3} + 2 \beta_{2} - 2 \beta_1 + 2) q^{59} + ( - \beta_{5} + 2 \beta_{4} - 2 \beta_{3} - \beta_{2} + 2 \beta_1 + 4) q^{61} + (\beta_{5} - \beta_{4} + 7 \beta_{3} - 2 \beta_{2} - \beta_1 - 2) q^{62} + (2 \beta_{5} + 4 \beta_{4} + 8 \beta_{3} + 2 \beta_{2} + 6 \beta_1 + 3) q^{64} + (\beta_{5} + 2 \beta_{4} + 2 \beta_{3} - \beta_{2}) q^{67} + ( - \beta_{5} + 5 \beta_{3} - \beta_{2} - 4 \beta_1 - 1) q^{68} + ( - 2 \beta_{5} - 2 \beta_{3} + \beta_{2} - 2 \beta_1 + 1) q^{71} + ( - 2 \beta_{5} - \beta_{3} - \beta_{2} + 2 \beta_1 - 6) q^{73} + ( - 3 \beta_{5} - \beta_{4} - 9 \beta_{3} - 4 \beta_{2} + 4 \beta_1 - 10) q^{74} + (2 \beta_{5} - 4 \beta_{4} + 3 \beta_{3} + 2 \beta_1 + 6) q^{76} + (6 \beta_{3} + 2 \beta_1) q^{77} + ( - \beta_{3} - 2 \beta_{2} + 2 \beta_1 + 3) q^{79} + ( - \beta_{5} - 2 \beta_{4} - \beta_{3} + 4 \beta_1 + 6) q^{82} + ( - \beta_{5} - \beta_{3} + 2 \beta_{2} + 2 \beta_1 + 2) q^{83} + (2 \beta_{5} + \beta_{4} + 8 \beta_{3} + 2 \beta_{2} + 8) q^{86} + (\beta_{5} + 4 \beta_{4} + \beta_{3} + 2 \beta_{2} + 2 \beta_1 + 8) q^{88} + ( - \beta_{5} + 2 \beta_{4} - \beta_{3} + 2 \beta_{2} + 4 \beta_1 - 4) q^{89} + ( - \beta_{5} + 4 \beta_{4} - \beta_{3} - 2 \beta_{2} + 3) q^{91} + (4 \beta_{4} - 6 \beta_{3} + \beta_{2} + 2 \beta_1 - 5) q^{92} + ( - 4 \beta_{4} + 6 \beta_{3} - 4 \beta_{2} - 4 \beta_1 - 4) q^{94} + (3 \beta_{5} + 2 \beta_{4} + 4 \beta_{3} + 1) q^{97} + ( - \beta_{5} + \beta_{4} - 7 \beta_{3} + \beta_1 - 6) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - q^{2} + 11 q^{4} - 2 q^{7} - 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - q^{2} + 11 q^{4} - 2 q^{7} - 6 q^{8} + 4 q^{14} + 17 q^{16} - 2 q^{17} - 2 q^{19} + 9 q^{22} + q^{23} - 37 q^{26} - 44 q^{28} - 31 q^{29} - 2 q^{31} - 33 q^{32} + 37 q^{34} - 22 q^{37} - 27 q^{38} - 33 q^{41} - 3 q^{43} + 11 q^{44} - 12 q^{46} + 6 q^{47} + 4 q^{49} - 33 q^{52} - 14 q^{53} + 30 q^{56} - q^{58} + 8 q^{59} + 34 q^{61} - 31 q^{62} + 12 q^{64} + 2 q^{67} - 27 q^{68} + 3 q^{71} - 36 q^{73} - 36 q^{74} + 27 q^{76} - 16 q^{77} + 25 q^{79} + 36 q^{82} + 12 q^{83} + 30 q^{86} + 56 q^{88} - 18 q^{89} + 28 q^{91} - 3 q^{92} - 50 q^{94} + 7 q^{97} - 15 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - x^{5} - 11x^{4} + 8x^{3} + 31x^{2} - 15x - 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{5} - \nu^{4} - 8\nu^{3} + 5\nu^{2} + 13\nu - 6 ) / 6 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{4} - \nu^{3} - 6\nu^{2} + 3\nu + 3 ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{5} + \nu^{4} + 14\nu^{3} - 11\nu^{2} - 43\nu + 24 ) / 6 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{5} + \beta_{3} + \beta_{2} + 5\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{5} + 2\beta_{4} + \beta_{3} + 7\beta_{2} + 2\beta _1 + 22 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 9\beta_{5} + 2\beta_{4} + 15\beta_{3} + 10\beta_{2} + 29\beta _1 + 16 \) 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
2.78712
2.13324
0.858825
−0.364088
−2.02791
−2.38719
−2.78712 0 5.76803 0 0 −3.15000 −10.5020 0 0
1.2 −2.13324 0 2.55073 0 0 −2.16876 −1.17484 0 0
1.3 −0.858825 0 −1.26242 0 0 3.88045 2.80185 0 0
1.4 0.364088 0 −1.86744 0 0 2.24941 −1.40809 0 0
1.5 2.02791 0 2.11242 0 0 0.505614 0.227977 0 0
1.6 2.38719 0 3.69868 0 0 −3.31671 4.05506 0 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
\(3\) \(-1\)
\(5\) \(-1\)

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 5625.2.a.o 6
3.b odd 2 1 1875.2.a.l yes 6
5.b even 2 1 5625.2.a.r 6
15.d odd 2 1 1875.2.a.i 6
15.e even 4 2 1875.2.b.e 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1875.2.a.i 6 15.d odd 2 1
1875.2.a.l yes 6 3.b odd 2 1
1875.2.b.e 12 15.e even 4 2
5625.2.a.o 6 1.a even 1 1 trivial
5625.2.a.r 6 5.b even 2 1

Hecke kernels

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

\( T_{2}^{6} + T_{2}^{5} - 11T_{2}^{4} - 8T_{2}^{3} + 31T_{2}^{2} + 15T_{2} - 9 \) Copy content Toggle raw display
\( T_{7}^{6} + 2T_{7}^{5} - 21T_{7}^{4} - 42T_{7}^{3} + 101T_{7}^{2} + 160T_{7} - 100 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} + T^{5} - 11 T^{4} - 8 T^{3} + \cdots - 9 \) Copy content Toggle raw display
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( T^{6} \) Copy content Toggle raw display
$7$ \( T^{6} + 2 T^{5} - 21 T^{4} - 42 T^{3} + \cdots - 100 \) Copy content Toggle raw display
$11$ \( T^{6} - 29 T^{4} + 8 T^{3} + 184 T^{2} + \cdots - 144 \) Copy content Toggle raw display
$13$ \( T^{6} - 56 T^{4} + 74 T^{3} + \cdots + 349 \) Copy content Toggle raw display
$17$ \( T^{6} + 2 T^{5} - 55 T^{4} - 80 T^{3} + \cdots - 576 \) Copy content Toggle raw display
$19$ \( T^{6} + 2 T^{5} - 74 T^{4} + \cdots - 5725 \) Copy content Toggle raw display
$23$ \( T^{6} - T^{5} - 69 T^{4} + 144 T^{3} + \cdots - 720 \) Copy content Toggle raw display
$29$ \( T^{6} + 31 T^{5} + 379 T^{4} + \cdots + 6480 \) Copy content Toggle raw display
$31$ \( T^{6} + 2 T^{5} - 86 T^{4} + \cdots + 3155 \) Copy content Toggle raw display
$37$ \( T^{6} + 22 T^{5} + 59 T^{4} + \cdots - 46100 \) Copy content Toggle raw display
$41$ \( T^{6} + 33 T^{5} + 399 T^{4} + \cdots + 720 \) Copy content Toggle raw display
$43$ \( T^{6} + 3 T^{5} - 76 T^{4} + \cdots - 1289 \) Copy content Toggle raw display
$47$ \( T^{6} - 6 T^{5} - 184 T^{4} + \cdots + 80064 \) Copy content Toggle raw display
$53$ \( T^{6} + 14 T^{5} - 164 T^{4} + \cdots + 14400 \) Copy content Toggle raw display
$59$ \( T^{6} - 8 T^{5} - 69 T^{4} + \cdots - 2880 \) Copy content Toggle raw display
$61$ \( T^{6} - 34 T^{5} + 406 T^{4} + \cdots - 72001 \) Copy content Toggle raw display
$67$ \( T^{6} - 2 T^{5} - 110 T^{4} + 540 T^{3} + \cdots + 59 \) Copy content Toggle raw display
$71$ \( T^{6} - 3 T^{5} - 225 T^{4} + \cdots - 12816 \) Copy content Toggle raw display
$73$ \( T^{6} + 36 T^{5} + 431 T^{4} + \cdots + 20380 \) Copy content Toggle raw display
$79$ \( T^{6} - 25 T^{5} + 150 T^{4} + \cdots + 2725 \) Copy content Toggle raw display
$83$ \( T^{6} - 12 T^{5} - 129 T^{4} + \cdots - 23616 \) Copy content Toggle raw display
$89$ \( T^{6} + 18 T^{5} - 219 T^{4} + \cdots - 42480 \) Copy content Toggle raw display
$97$ \( T^{6} - 7 T^{5} - 310 T^{4} + \cdots - 32291 \) Copy content Toggle raw display
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