Properties

Label 625.2.a.f
Level $625$
Weight $2$
Character orbit 625.a
Self dual yes
Analytic conductor $4.991$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 625 = 5^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 625.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(4.99065012633\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.8.14884000000.2
Defining polynomial: \( x^{8} - 11x^{6} + 36x^{4} - 31x^{2} + 1 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{6} - \beta_{5}) q^{3} + (\beta_{2} + 1) q^{4} + ( - 2 \beta_{4} + \beta_{2}) q^{6} + ( - \beta_{7} + \beta_{6} + \beta_1) q^{7} + (\beta_{7} + \beta_{6}) q^{8} - \beta_{3} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{6} - \beta_{5}) q^{3} + (\beta_{2} + 1) q^{4} + ( - 2 \beta_{4} + \beta_{2}) q^{6} + ( - \beta_{7} + \beta_{6} + \beta_1) q^{7} + (\beta_{7} + \beta_{6}) q^{8} - \beta_{3} q^{9} + 2 q^{11} + (\beta_{7} - \beta_{6} + \beta_1) q^{12} - \beta_{5} q^{13} + (2 \beta_{4} + \beta_{3} + 3) q^{14} + (\beta_{3} - \beta_{2}) q^{16} + ( - 2 \beta_{7} + \beta_{6} + \beta_1) q^{17} + ( - \beta_{7} - 2 \beta_{6} - \beta_{5}) q^{18} + (2 \beta_{4} - \beta_{2} + 2) q^{19} + (\beta_{3} - \beta_{2} + 1) q^{21} + 2 \beta_1 q^{22} + ( - \beta_{6} - \beta_{5}) q^{23} + (2 \beta_{4} - \beta_{3} + 3) q^{24} + ( - 3 \beta_{4} - \beta_{3} + \beta_{2} - 1) q^{26} + (2 \beta_{7} - \beta_{6} + \beta_{5} - 2 \beta_1) q^{27} + (3 \beta_{7} + 3 \beta_{5} + \beta_1) q^{28} + (3 \beta_{4} + \beta_{3} - 2 \beta_{2} + 4) q^{29} + ( - \beta_{3} - 2 \beta_{2} + 1) q^{31} + ( - 2 \beta_{7} - \beta_{6} + \beta_{5} - \beta_1) q^{32} + (2 \beta_{6} - 2 \beta_{5}) q^{33} + (3 \beta_{4} + \beta_{3} - \beta_{2} + 2) q^{34} + ( - 4 \beta_{4} - \beta_{3} - 4) q^{36} + (4 \beta_{7} + \beta_{5} - \beta_1) q^{37} + ( - \beta_{7} - \beta_{6} + 2 \beta_{5} + \beta_1) q^{38} + ( - 2 \beta_{4} - \beta_{3} + \beta_{2} + 1) q^{39} + ( - \beta_{4} - 2 \beta_{3} + \beta_{2} + 2) q^{41} + (\beta_{6} + \beta_{5}) q^{42} + ( - 3 \beta_{7} - 2 \beta_{6} + \beta_{5} + \beta_1) q^{43} + (2 \beta_{2} + 2) q^{44} + ( - 4 \beta_{4} - 2 \beta_{3} + \beta_{2} - 2) q^{46} + (\beta_{7} - \beta_{6} + 2 \beta_{5} - 3 \beta_1) q^{47} + ( - 3 \beta_{7} + \beta_{5} + \beta_1) q^{48} + (2 \beta_{3} - \beta_{2} - 1) q^{49} + (2 \beta_{3} - 2 \beta_{2}) q^{51} + ( - \beta_{6} - 2 \beta_{5}) q^{52} + ( - 3 \beta_{7} + 2 \beta_{6} - 2 \beta_{5} - 3 \beta_1) q^{53} + ( - \beta_{2} - 4) q^{54} + (2 \beta_{4} + \beta_{3} + \beta_{2} + 3) q^{56} + (\beta_{7} + 4 \beta_{6} - \beta_{5} - 3 \beta_1) q^{57} + ( - \beta_{7} + 4 \beta_{5} + 2 \beta_1) q^{58} + ( - 2 \beta_{4} + 2 \beta_{3} + \beta_{2} + 4) q^{59} + (\beta_{4} + 3 \beta_{3} + \beta_{2} + 3) q^{61} + ( - 3 \beta_{7} - 4 \beta_{6} - \beta_{5} - \beta_1) q^{62} + ( - 2 \beta_{6} - 2 \beta_1) q^{63} + (4 \beta_{4} - 2 \beta_{3} - 2 \beta_{2} - 5) q^{64} + ( - 4 \beta_{4} + 2 \beta_{2}) q^{66} + (2 \beta_{7} - 2 \beta_{6} - 4 \beta_1) q^{67} + (4 \beta_{7} - \beta_{6} + 4 \beta_{5} - \beta_1) q^{68} + ( - 4 \beta_{4} - \beta_{3} + 2 \beta_{2} - 1) q^{69} + (2 \beta_{4} - \beta_{3} + 3 \beta_{2} + 7) q^{71} + (\beta_{7} + 2 \beta_{6} - 3 \beta_{5} - 4 \beta_1) q^{72} + (3 \beta_{7} - 2 \beta_{6} + 2 \beta_{5} - \beta_1) q^{73} + ( - \beta_{4} + \beta_{3} + 2 \beta_{2} + 2) q^{74} + (2 \beta_{4} + \beta_{3} - 1) q^{76} + ( - 2 \beta_{7} + 2 \beta_{6} + 2 \beta_1) q^{77} + ( - \beta_{6} - 3 \beta_{5} + 2 \beta_1) q^{78} + ( - 4 \beta_{4} - 4 \beta_{3} + 3 \beta_{2} - 2) q^{79} + (4 \beta_{4} + 2 \beta_{3} - 1) q^{81} + ( - \beta_{7} - 3 \beta_{6} - 3 \beta_{5} + 3 \beta_1) q^{82} + (4 \beta_{7} - 5 \beta_{6} + \beta_{5} - 4 \beta_1) q^{83} + (4 \beta_{4} + \beta_{2}) q^{84} + (4 \beta_{4} - \beta_{3} - 3 \beta_{2} - 1) q^{86} + ( - \beta_{7} + 6 \beta_{6} - \beta_{5} - 3 \beta_1) q^{87} + (2 \beta_{7} + 2 \beta_{6}) q^{88} + ( - 7 \beta_{4} - \beta_{3} + \beta_{2}) q^{89} + ( - \beta_{2} - 2) q^{91} + ( - \beta_{7} - \beta_{6} - 4 \beta_{5} - \beta_1) q^{92} + (7 \beta_{6} - 5 \beta_{5} - 4 \beta_1) q^{93} + (4 \beta_{4} + \beta_{3} - 4 \beta_{2} - 7) q^{94} + (2 \beta_{4} + 3 \beta_{3} - 3 \beta_{2} - 5) q^{96} + ( - 5 \beta_{7} - 4 \beta_{6} - 4 \beta_{5} + 4 \beta_1) q^{97} + (\beta_{7} + 3 \beta_{6} + 2 \beta_{5} - 2 \beta_1) q^{98} - 2 \beta_{3} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 6 q^{4} + 6 q^{6} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 6 q^{4} + 6 q^{6} + 4 q^{9} + 16 q^{11} + 12 q^{14} - 2 q^{16} + 10 q^{19} + 6 q^{21} + 20 q^{24} + 6 q^{26} + 20 q^{29} + 16 q^{31} + 2 q^{34} - 12 q^{36} + 18 q^{39} + 26 q^{41} + 12 q^{44} + 6 q^{46} - 14 q^{49} - 4 q^{51} - 30 q^{54} + 10 q^{56} + 30 q^{59} + 6 q^{61} - 44 q^{64} + 12 q^{66} + 8 q^{69} + 46 q^{71} + 12 q^{74} - 20 q^{76} + 10 q^{79} - 32 q^{81} - 18 q^{84} - 14 q^{86} + 30 q^{89} - 14 q^{91} - 68 q^{94} - 54 q^{96} + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{4} - 5\nu^{2} + 1 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{6} - 8\nu^{4} + 16\nu^{2} - 7 ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{7} - 8\nu^{5} + 16\nu^{3} - 7\nu ) / 4 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{7} + 12\nu^{5} - 40\nu^{3} + 27\nu ) / 4 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{7} - 12\nu^{5} + 44\nu^{3} - 43\nu ) / 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} + \beta_{6} + 4\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{3} + 5\beta_{2} + 14 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 6\beta_{7} + 7\beta_{6} + \beta_{5} + 19\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 4\beta_{4} + 8\beta_{3} + 24\beta_{2} + 71 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 32\beta_{7} + 40\beta_{6} + 12\beta_{5} + 95\beta_1 \) 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
−2.30927
−2.08529
−1.13370
−0.183172
0.183172
1.13370
2.08529
2.30927
−2.30927 −0.474903 3.33275 0 1.09668 −3.03582 −3.07768 −2.77447 0
1.2 −2.08529 −2.19849 2.34841 0 4.58448 0.992398 −0.726543 1.83337 0
1.3 −1.13370 2.60278 −0.714715 0 −2.95078 −0.407162 3.07768 3.77447 0
1.4 −0.183172 −1.47195 −1.96645 0 0.269620 −3.26086 0.726543 −0.833366 0
1.5 0.183172 1.47195 −1.96645 0 0.269620 3.26086 −0.726543 −0.833366 0
1.6 1.13370 −2.60278 −0.714715 0 −2.95078 0.407162 −3.07768 3.77447 0
1.7 2.08529 2.19849 2.34841 0 4.58448 −0.992398 0.726543 1.83337 0
1.8 2.30927 0.474903 3.33275 0 1.09668 3.03582 3.07768 −2.77447 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
\(5\) \(-1\)

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 625.2.a.f 8
3.b odd 2 1 5625.2.a.x 8
4.b odd 2 1 10000.2.a.bj 8
5.b even 2 1 inner 625.2.a.f 8
5.c odd 4 2 625.2.b.c 8
15.d odd 2 1 5625.2.a.x 8
20.d odd 2 1 10000.2.a.bj 8
25.d even 5 2 125.2.d.b 16
25.d even 5 2 625.2.d.o 16
25.e even 10 2 125.2.d.b 16
25.e even 10 2 625.2.d.o 16
25.f odd 20 2 25.2.e.a 8
25.f odd 20 2 125.2.e.b 8
25.f odd 20 2 625.2.e.a 8
25.f odd 20 2 625.2.e.i 8
75.l even 20 2 225.2.m.a 8
100.l even 20 2 400.2.y.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
25.2.e.a 8 25.f odd 20 2
125.2.d.b 16 25.d even 5 2
125.2.d.b 16 25.e even 10 2
125.2.e.b 8 25.f odd 20 2
225.2.m.a 8 75.l even 20 2
400.2.y.c 8 100.l even 20 2
625.2.a.f 8 1.a even 1 1 trivial
625.2.a.f 8 5.b even 2 1 inner
625.2.b.c 8 5.c odd 4 2
625.2.d.o 16 25.d even 5 2
625.2.d.o 16 25.e even 10 2
625.2.e.a 8 25.f odd 20 2
625.2.e.i 8 25.f odd 20 2
5625.2.a.x 8 3.b odd 2 1
5625.2.a.x 8 15.d odd 2 1
10000.2.a.bj 8 4.b odd 2 1
10000.2.a.bj 8 20.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(625))\):

\( T_{2}^{8} - 11T_{2}^{6} + 36T_{2}^{4} - 31T_{2}^{2} + 1 \) Copy content Toggle raw display
\( T_{3}^{8} - 14T_{3}^{6} + 61T_{3}^{4} - 84T_{3}^{2} + 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - 11 T^{6} + 36 T^{4} - 31 T^{2} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{8} - 14 T^{6} + 61 T^{4} - 84 T^{2} + \cdots + 16 \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} - 21 T^{6} + 121 T^{4} + \cdots + 16 \) Copy content Toggle raw display
$11$ \( (T - 2)^{8} \) Copy content Toggle raw display
$13$ \( T^{8} - 14 T^{6} + 31 T^{4} - 14 T^{2} + \cdots + 1 \) Copy content Toggle raw display
$17$ \( T^{8} - 41 T^{6} + 441 T^{4} + \cdots + 1936 \) Copy content Toggle raw display
$19$ \( (T^{4} - 5 T^{3} - 5 T^{2} + 30 T - 20)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} - 34 T^{6} + 301 T^{4} + \cdots + 256 \) Copy content Toggle raw display
$29$ \( (T^{4} - 10 T^{3} - 5 T^{2} + 290 T - 695)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} - 8 T^{3} - 41 T^{2} + 328 T - 44)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} - 111 T^{6} + 3556 T^{4} + \cdots + 116281 \) Copy content Toggle raw display
$41$ \( (T^{4} - 13 T^{3} + 19 T^{2} + 148 T + 116)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} - 129 T^{6} + 4421 T^{4} + \cdots + 246016 \) Copy content Toggle raw display
$47$ \( T^{8} - 141 T^{6} + 4661 T^{4} + \cdots + 65536 \) Copy content Toggle raw display
$53$ \( T^{8} - 239 T^{6} + 20356 T^{4} + \cdots + 8755681 \) Copy content Toggle raw display
$59$ \( (T^{4} - 15 T^{3} + 5 T^{2} + 630 T - 2020)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} - 3 T^{3} - 146 T^{2} - 237 T + 341)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} - 176 T^{6} + 7776 T^{4} + \cdots + 246016 \) Copy content Toggle raw display
$71$ \( (T^{4} - 23 T^{3} + 99 T^{2} + 798 T - 4924)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} - 79 T^{6} + 76 T^{4} - 19 T^{2} + \cdots + 1 \) Copy content Toggle raw display
$79$ \( (T^{4} - 5 T^{3} - 195 T^{2} + 5780)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} - 374 T^{6} + 35061 T^{4} + \cdots + 99856 \) Copy content Toggle raw display
$89$ \( (T^{4} - 15 T^{3} - 35 T^{2} + 530 T + 1180)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} - 666 T^{6} + \cdots + 301334881 \) Copy content Toggle raw display
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