Properties

Label 5625.2.a.x
Level $5625$
Weight $2$
Character orbit 5625.a
Self dual yes
Analytic conductor $44.916$
Analytic rank $1$
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: \(1\)
Dimension: \(8\)
Coefficient field: 8.8.14884000000.2
Defining polynomial: \(x^{8} - 11 x^{6} + 36 x^{4} - 31 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 25)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 11 x^{6} + 36 x^{4} - 31 x^{2} + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(\beta_{3}\)\(=\)\( \nu^{4} - 5 \nu^{2} + 1 \)
\(\beta_{4}\)\(=\)\((\)\( \nu^{6} - 8 \nu^{4} + 16 \nu^{2} - 7 \)\()/4\)
\(\beta_{5}\)\(=\)\((\)\( \nu^{7} - 8 \nu^{5} + 16 \nu^{3} - 7 \nu \)\()/4\)
\(\beta_{6}\)\(=\)\((\)\( -\nu^{7} + 12 \nu^{5} - 40 \nu^{3} + 27 \nu \)\()/4\)
\(\beta_{7}\)\(=\)\((\)\( \nu^{7} - 12 \nu^{5} + 44 \nu^{3} - 43 \nu \)\()/4\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 3\)
\(\nu^{3}\)\(=\)\(\beta_{7} + \beta_{6} + 4 \beta_{1}\)
\(\nu^{4}\)\(=\)\(\beta_{3} + 5 \beta_{2} + 14\)
\(\nu^{5}\)\(=\)\(6 \beta_{7} + 7 \beta_{6} + \beta_{5} + 19 \beta_{1}\)
\(\nu^{6}\)\(=\)\(4 \beta_{4} + 8 \beta_{3} + 24 \beta_{2} + 71\)
\(\nu^{7}\)\(=\)\(32 \beta_{7} + 40 \beta_{6} + 12 \beta_{5} + 95 \beta_{1}\)

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
−2.30927
−2.08529
−1.13370
−0.183172
0.183172
1.13370
2.08529
2.30927
−2.30927 0 3.33275 0 0 3.03582 −3.07768 0 0
1.2 −2.08529 0 2.34841 0 0 −0.992398 −0.726543 0 0
1.3 −1.13370 0 −0.714715 0 0 0.407162 3.07768 0 0
1.4 −0.183172 0 −1.96645 0 0 3.26086 0.726543 0 0
1.5 0.183172 0 −1.96645 0 0 −3.26086 −0.726543 0 0
1.6 1.13370 0 −0.714715 0 0 −0.407162 −3.07768 0 0
1.7 2.08529 0 2.34841 0 0 0.992398 0.726543 0 0
1.8 2.30927 0 3.33275 0 0 −3.03582 3.07768 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
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 5625.2.a.x 8
3.b odd 2 1 625.2.a.f 8
5.b even 2 1 inner 5625.2.a.x 8
12.b even 2 1 10000.2.a.bj 8
15.d odd 2 1 625.2.a.f 8
15.e even 4 2 625.2.b.c 8
25.f odd 20 2 225.2.m.a 8
60.h even 2 1 10000.2.a.bj 8
75.h odd 10 2 125.2.d.b 16
75.h odd 10 2 625.2.d.o 16
75.j odd 10 2 125.2.d.b 16
75.j odd 10 2 625.2.d.o 16
75.l even 20 2 25.2.e.a 8
75.l even 20 2 125.2.e.b 8
75.l even 20 2 625.2.e.a 8
75.l even 20 2 625.2.e.i 8
300.u odd 20 2 400.2.y.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
25.2.e.a 8 75.l even 20 2
125.2.d.b 16 75.h odd 10 2
125.2.d.b 16 75.j odd 10 2
125.2.e.b 8 75.l even 20 2
225.2.m.a 8 25.f odd 20 2
400.2.y.c 8 300.u odd 20 2
625.2.a.f 8 3.b odd 2 1
625.2.a.f 8 15.d odd 2 1
625.2.b.c 8 15.e even 4 2
625.2.d.o 16 75.h odd 10 2
625.2.d.o 16 75.j odd 10 2
625.2.e.a 8 75.l even 20 2
625.2.e.i 8 75.l even 20 2
5625.2.a.x 8 1.a even 1 1 trivial
5625.2.a.x 8 5.b even 2 1 inner
10000.2.a.bj 8 12.b even 2 1
10000.2.a.bj 8 60.h even 2 1

Hecke kernels

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

\( T_{2}^{8} - 11 T_{2}^{6} + 36 T_{2}^{4} - 31 T_{2}^{2} + 1 \)
\( T_{7}^{8} - 21 T_{7}^{6} + 121 T_{7}^{4} - 116 T_{7}^{2} + 16 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 31 T^{2} + 36 T^{4} - 11 T^{6} + T^{8} \)
$3$ \( T^{8} \)
$5$ \( T^{8} \)
$7$ \( 16 - 116 T^{2} + 121 T^{4} - 21 T^{6} + T^{8} \)
$11$ \( ( 2 + T )^{8} \)
$13$ \( 1 - 14 T^{2} + 31 T^{4} - 14 T^{6} + T^{8} \)
$17$ \( 1936 - 1636 T^{2} + 441 T^{4} - 41 T^{6} + T^{8} \)
$19$ \( ( -20 + 30 T - 5 T^{2} - 5 T^{3} + T^{4} )^{2} \)
$23$ \( 256 - 544 T^{2} + 301 T^{4} - 34 T^{6} + T^{8} \)
$29$ \( ( -695 - 290 T - 5 T^{2} + 10 T^{3} + T^{4} )^{2} \)
$31$ \( ( -44 + 328 T - 41 T^{2} - 8 T^{3} + T^{4} )^{2} \)
$37$ \( 116281 - 39331 T^{2} + 3556 T^{4} - 111 T^{6} + T^{8} \)
$41$ \( ( 116 - 148 T + 19 T^{2} + 13 T^{3} + T^{4} )^{2} \)
$43$ \( 246016 - 56784 T^{2} + 4421 T^{4} - 129 T^{6} + T^{8} \)
$47$ \( 65536 - 47616 T^{2} + 4661 T^{4} - 141 T^{6} + T^{8} \)
$53$ \( 8755681 - 722619 T^{2} + 20356 T^{4} - 239 T^{6} + T^{8} \)
$59$ \( ( -2020 - 630 T + 5 T^{2} + 15 T^{3} + T^{4} )^{2} \)
$61$ \( ( 341 - 237 T - 146 T^{2} - 3 T^{3} + T^{4} )^{2} \)
$67$ \( 246016 - 84736 T^{2} + 7776 T^{4} - 176 T^{6} + T^{8} \)
$71$ \( ( -4924 - 798 T + 99 T^{2} + 23 T^{3} + T^{4} )^{2} \)
$73$ \( 1 - 19 T^{2} + 76 T^{4} - 79 T^{6} + T^{8} \)
$79$ \( ( 5780 - 195 T^{2} - 5 T^{3} + T^{4} )^{2} \)
$83$ \( 99856 - 138164 T^{2} + 35061 T^{4} - 374 T^{6} + T^{8} \)
$89$ \( ( 1180 - 530 T - 35 T^{2} + 15 T^{3} + T^{4} )^{2} \)
$97$ \( 301334881 - 12038986 T^{2} + 146971 T^{4} - 666 T^{6} + T^{8} \)
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