Properties

Label 625.2.e.c
Level $625$
Weight $2$
Character orbit 625.e
Analytic conductor $4.991$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 625 = 5^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 625.e (of order \(10\), degree \(4\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.99065012633\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{10})\)
Coefficient field: \(\Q(\zeta_{20})\)
Defining polynomial: \(x^{8} - x^{6} + x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 25)
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{20}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( \zeta_{20} + \zeta_{20}^{5} ) q^{2} -\zeta_{20} q^{3} + ( -1 + \zeta_{20}^{2} ) q^{4} + ( -\zeta_{20}^{2} - \zeta_{20}^{6} ) q^{6} + ( \zeta_{20}^{3} - \zeta_{20}^{5} + \zeta_{20}^{7} ) q^{7} + ( \zeta_{20} + \zeta_{20}^{3} - \zeta_{20}^{5} - \zeta_{20}^{7} ) q^{8} -2 \zeta_{20}^{2} q^{9} +O(q^{10})\) \( q + ( \zeta_{20} + \zeta_{20}^{5} ) q^{2} -\zeta_{20} q^{3} + ( -1 + \zeta_{20}^{2} ) q^{4} + ( -\zeta_{20}^{2} - \zeta_{20}^{6} ) q^{6} + ( \zeta_{20}^{3} - \zeta_{20}^{5} + \zeta_{20}^{7} ) q^{7} + ( \zeta_{20} + \zeta_{20}^{3} - \zeta_{20}^{5} - \zeta_{20}^{7} ) q^{8} -2 \zeta_{20}^{2} q^{9} + ( 4 - 2 \zeta_{20}^{2} + 2 \zeta_{20}^{4} - 4 \zeta_{20}^{6} ) q^{11} + ( \zeta_{20} - \zeta_{20}^{3} ) q^{12} + ( -3 \zeta_{20} + 3 \zeta_{20}^{3} ) q^{13} + ( -1 + \zeta_{20}^{2} - \zeta_{20}^{4} + \zeta_{20}^{6} ) q^{14} + ( 3 + 3 \zeta_{20}^{4} ) q^{16} + ( 4 \zeta_{20} - 2 \zeta_{20}^{3} + 2 \zeta_{20}^{5} - 4 \zeta_{20}^{7} ) q^{17} + ( -2 \zeta_{20}^{3} - 2 \zeta_{20}^{7} ) q^{18} + ( -3 \zeta_{20}^{2} + 4 \zeta_{20}^{4} - 3 \zeta_{20}^{6} ) q^{19} + ( 1 - \zeta_{20}^{2} ) q^{21} + ( 6 \zeta_{20} + 4 \zeta_{20}^{5} - 4 \zeta_{20}^{7} ) q^{22} + ( -2 \zeta_{20} + 7 \zeta_{20}^{3} - 2 \zeta_{20}^{5} ) q^{23} + ( -1 - 2 \zeta_{20}^{4} + 2 \zeta_{20}^{6} ) q^{24} -3 q^{26} + 5 \zeta_{20}^{3} q^{27} + ( -\zeta_{20} + \zeta_{20}^{5} - \zeta_{20}^{7} ) q^{28} + ( 1 - \zeta_{20}^{2} - 3 \zeta_{20}^{6} ) q^{29} -3 \zeta_{20}^{4} q^{31} + ( -\zeta_{20}^{3} + 5 \zeta_{20}^{5} - \zeta_{20}^{7} ) q^{32} + ( -4 \zeta_{20} + 2 \zeta_{20}^{3} - 2 \zeta_{20}^{5} + 4 \zeta_{20}^{7} ) q^{33} + ( 4 + 2 \zeta_{20}^{2} + 4 \zeta_{20}^{4} ) q^{34} + ( 2 \zeta_{20}^{2} - 2 \zeta_{20}^{4} ) q^{36} + ( -2 \zeta_{20} + 2 \zeta_{20}^{3} - \zeta_{20}^{7} ) q^{37} + ( -\zeta_{20} + \zeta_{20}^{3} - 2 \zeta_{20}^{7} ) q^{38} + ( 3 \zeta_{20}^{2} - 3 \zeta_{20}^{4} ) q^{39} + ( -2 + 4 \zeta_{20}^{2} - 2 \zeta_{20}^{4} ) q^{41} + ( \zeta_{20} - \zeta_{20}^{3} + \zeta_{20}^{5} - \zeta_{20}^{7} ) q^{42} + ( 3 \zeta_{20}^{3} + 3 \zeta_{20}^{7} ) q^{43} + ( 2 \zeta_{20}^{2} + 2 \zeta_{20}^{6} ) q^{44} + ( -5 + 5 \zeta_{20}^{2} + 3 \zeta_{20}^{6} ) q^{46} + ( \zeta_{20}^{5} - \zeta_{20}^{7} ) q^{47} + ( -3 \zeta_{20} - 3 \zeta_{20}^{5} ) q^{48} + ( 6 + \zeta_{20}^{4} - \zeta_{20}^{6} ) q^{49} + ( -4 - 2 \zeta_{20}^{4} + 2 \zeta_{20}^{6} ) q^{51} + ( 3 \zeta_{20} - 6 \zeta_{20}^{3} + 3 \zeta_{20}^{5} ) q^{52} + ( \zeta_{20} + 4 \zeta_{20}^{5} - 4 \zeta_{20}^{7} ) q^{53} + ( -5 + 5 \zeta_{20}^{2} + 5 \zeta_{20}^{6} ) q^{54} + ( -\zeta_{20}^{2} + 3 \zeta_{20}^{4} - \zeta_{20}^{6} ) q^{56} + ( 3 \zeta_{20}^{3} - 4 \zeta_{20}^{5} + 3 \zeta_{20}^{7} ) q^{57} + ( 4 \zeta_{20} - \zeta_{20}^{3} + \zeta_{20}^{5} - 4 \zeta_{20}^{7} ) q^{58} + ( -3 - 6 \zeta_{20}^{2} - 3 \zeta_{20}^{4} ) q^{59} + ( -5 - \zeta_{20}^{2} + \zeta_{20}^{4} + 5 \zeta_{20}^{6} ) q^{61} + ( 3 \zeta_{20} - 3 \zeta_{20}^{3} - 3 \zeta_{20}^{7} ) q^{62} + ( 2 \zeta_{20} - 2 \zeta_{20}^{3} ) q^{63} + ( 3 - \zeta_{20}^{2} + \zeta_{20}^{4} - 3 \zeta_{20}^{6} ) q^{64} + ( -4 - 2 \zeta_{20}^{2} - 4 \zeta_{20}^{4} ) q^{66} + ( -6 \zeta_{20} + 8 \zeta_{20}^{3} - 8 \zeta_{20}^{5} + 6 \zeta_{20}^{7} ) q^{67} + ( 2 \zeta_{20}^{3} + 2 \zeta_{20}^{7} ) q^{68} + ( 2 \zeta_{20}^{2} - 7 \zeta_{20}^{4} + 2 \zeta_{20}^{6} ) q^{69} + ( -1 + \zeta_{20}^{2} + 6 \zeta_{20}^{6} ) q^{71} + ( -2 \zeta_{20} - 4 \zeta_{20}^{5} + 4 \zeta_{20}^{7} ) q^{72} -9 \zeta_{20}^{3} q^{73} + ( -1 + \zeta_{20}^{4} - \zeta_{20}^{6} ) q^{74} + ( 3 - 4 \zeta_{20}^{4} + 4 \zeta_{20}^{6} ) q^{76} + ( 2 \zeta_{20} + 2 \zeta_{20}^{5} ) q^{77} + 3 \zeta_{20} q^{78} + ( 5 - 5 \zeta_{20}^{2} - 5 \zeta_{20}^{6} ) q^{79} + \zeta_{20}^{4} q^{81} + ( 2 \zeta_{20}^{3} - 2 \zeta_{20}^{5} + 2 \zeta_{20}^{7} ) q^{82} + ( -5 \zeta_{20} + 3 \zeta_{20}^{3} - 3 \zeta_{20}^{5} + 5 \zeta_{20}^{7} ) q^{83} + ( -1 + 2 \zeta_{20}^{2} - \zeta_{20}^{4} ) q^{84} + ( -6 + 3 \zeta_{20}^{2} - 3 \zeta_{20}^{4} + 6 \zeta_{20}^{6} ) q^{86} + ( -\zeta_{20} + \zeta_{20}^{3} + 3 \zeta_{20}^{7} ) q^{87} + ( 6 \zeta_{20} - 6 \zeta_{20}^{3} - 8 \zeta_{20}^{7} ) q^{88} + ( -4 - 4 \zeta_{20}^{2} + 4 \zeta_{20}^{4} + 4 \zeta_{20}^{6} ) q^{89} + ( 3 - 6 \zeta_{20}^{2} + 3 \zeta_{20}^{4} ) q^{91} + ( 2 \zeta_{20} - 9 \zeta_{20}^{3} + 9 \zeta_{20}^{5} - 2 \zeta_{20}^{7} ) q^{92} + 3 \zeta_{20}^{5} q^{93} + \zeta_{20}^{4} q^{94} + ( -1 + \zeta_{20}^{2} - 4 \zeta_{20}^{6} ) q^{96} + ( -2 \zeta_{20} - 3 \zeta_{20}^{5} + 3 \zeta_{20}^{7} ) q^{97} + ( 6 \zeta_{20} + \zeta_{20}^{3} + 6 \zeta_{20}^{5} ) q^{98} + ( -8 - 4 \zeta_{20}^{4} + 4 \zeta_{20}^{6} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 6q^{4} - 4q^{6} - 4q^{9} + O(q^{10}) \) \( 8q - 6q^{4} - 4q^{6} - 4q^{9} + 16q^{11} - 2q^{14} + 18q^{16} - 20q^{19} + 6q^{21} - 24q^{26} + 6q^{31} + 28q^{34} + 8q^{36} + 12q^{39} - 4q^{41} + 8q^{44} - 24q^{46} + 44q^{49} - 24q^{51} - 20q^{54} - 10q^{56} - 30q^{59} - 34q^{61} + 14q^{64} - 28q^{66} + 22q^{69} + 6q^{71} - 12q^{74} + 40q^{76} + 20q^{79} - 2q^{81} - 2q^{84} - 24q^{86} - 40q^{89} + 6q^{91} - 2q^{94} - 14q^{96} - 48q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/625\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(\zeta_{20}^{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
124.1
−0.587785 + 0.809017i
0.587785 0.809017i
−0.951057 + 0.309017i
0.951057 0.309017i
−0.951057 0.309017i
0.951057 + 0.309017i
−0.587785 0.809017i
0.587785 + 0.809017i
−0.587785 0.190983i 0.587785 0.809017i −1.30902 0.951057i 0 −0.500000 + 0.363271i 1.61803i 1.31433 + 1.80902i 0.618034 + 1.90211i 0
124.2 0.587785 + 0.190983i −0.587785 + 0.809017i −1.30902 0.951057i 0 −0.500000 + 0.363271i 1.61803i −1.31433 1.80902i 0.618034 + 1.90211i 0
249.1 −0.951057 + 1.30902i 0.951057 0.309017i −0.190983 0.587785i 0 −0.500000 + 1.53884i 0.618034i −2.12663 0.690983i −1.61803 + 1.17557i 0
249.2 0.951057 1.30902i −0.951057 + 0.309017i −0.190983 0.587785i 0 −0.500000 + 1.53884i 0.618034i 2.12663 + 0.690983i −1.61803 + 1.17557i 0
374.1 −0.951057 1.30902i 0.951057 + 0.309017i −0.190983 + 0.587785i 0 −0.500000 1.53884i 0.618034i −2.12663 + 0.690983i −1.61803 1.17557i 0
374.2 0.951057 + 1.30902i −0.951057 0.309017i −0.190983 + 0.587785i 0 −0.500000 1.53884i 0.618034i 2.12663 0.690983i −1.61803 1.17557i 0
499.1 −0.587785 + 0.190983i 0.587785 + 0.809017i −1.30902 + 0.951057i 0 −0.500000 0.363271i 1.61803i 1.31433 1.80902i 0.618034 1.90211i 0
499.2 0.587785 0.190983i −0.587785 0.809017i −1.30902 + 0.951057i 0 −0.500000 0.363271i 1.61803i −1.31433 + 1.80902i 0.618034 1.90211i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 499.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
25.d even 5 1 inner
25.e even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 625.2.e.c 8
5.b even 2 1 inner 625.2.e.c 8
5.c odd 4 1 625.2.d.b 4
5.c odd 4 1 625.2.d.h 4
25.d even 5 2 125.2.e.a 8
25.d even 5 1 625.2.b.a 4
25.d even 5 1 inner 625.2.e.c 8
25.e even 10 2 125.2.e.a 8
25.e even 10 1 625.2.b.a 4
25.e even 10 1 inner 625.2.e.c 8
25.f odd 20 2 25.2.d.a 4
25.f odd 20 2 125.2.d.a 4
25.f odd 20 1 625.2.a.b 2
25.f odd 20 1 625.2.a.c 2
25.f odd 20 1 625.2.d.b 4
25.f odd 20 1 625.2.d.h 4
75.l even 20 2 225.2.h.b 4
75.l even 20 1 5625.2.a.d 2
75.l even 20 1 5625.2.a.f 2
100.l even 20 2 400.2.u.b 4
100.l even 20 1 10000.2.a.c 2
100.l even 20 1 10000.2.a.l 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
25.2.d.a 4 25.f odd 20 2
125.2.d.a 4 25.f odd 20 2
125.2.e.a 8 25.d even 5 2
125.2.e.a 8 25.e even 10 2
225.2.h.b 4 75.l even 20 2
400.2.u.b 4 100.l even 20 2
625.2.a.b 2 25.f odd 20 1
625.2.a.c 2 25.f odd 20 1
625.2.b.a 4 25.d even 5 1
625.2.b.a 4 25.e even 10 1
625.2.d.b 4 5.c odd 4 1
625.2.d.b 4 25.f odd 20 1
625.2.d.h 4 5.c odd 4 1
625.2.d.h 4 25.f odd 20 1
625.2.e.c 8 1.a even 1 1 trivial
625.2.e.c 8 5.b even 2 1 inner
625.2.e.c 8 25.d even 5 1 inner
625.2.e.c 8 25.e even 10 1 inner
5625.2.a.d 2 75.l even 20 1
5625.2.a.f 2 75.l even 20 1
10000.2.a.c 2 100.l even 20 1
10000.2.a.l 2 100.l even 20 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(625, [\chi])\):

\( T_{2}^{8} + T_{2}^{6} + 6 T_{2}^{4} - 4 T_{2}^{2} + 1 \)
\( T_{3}^{8} - T_{3}^{6} + T_{3}^{4} - T_{3}^{2} + 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 4 T^{2} + 6 T^{4} + T^{6} + T^{8} \)
$3$ \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \)
$5$ \( T^{8} \)
$7$ \( ( 1 + 3 T^{2} + T^{4} )^{2} \)
$11$ \( ( 16 + 8 T + 24 T^{2} - 8 T^{3} + T^{4} )^{2} \)
$13$ \( 6561 + 729 T^{2} + 486 T^{4} - 36 T^{6} + T^{8} \)
$17$ \( 256 + 256 T^{2} + 736 T^{4} - 44 T^{6} + T^{8} \)
$19$ \( ( 25 + 25 T + 40 T^{2} + 10 T^{3} + T^{4} )^{2} \)
$23$ \( 923521 + 18259 T^{2} + 3841 T^{4} - 101 T^{6} + T^{8} \)
$29$ \( ( 25 + 25 T + 10 T^{2} + T^{4} )^{2} \)
$31$ \( ( 81 - 27 T + 9 T^{2} - 3 T^{3} + T^{4} )^{2} \)
$37$ \( 1 + 11 T^{2} + 321 T^{4} - 29 T^{6} + T^{8} \)
$41$ \( ( 16 - 32 T + 24 T^{2} + 2 T^{3} + T^{4} )^{2} \)
$43$ \( ( 81 + 27 T^{2} + T^{4} )^{2} \)
$47$ \( 1 - 4 T^{2} + 6 T^{4} + T^{6} + T^{8} \)
$53$ \( 130321 - 14801 T^{2} + 681 T^{4} - T^{6} + T^{8} \)
$59$ \( ( 2025 + 90 T^{2} + 15 T^{3} + T^{4} )^{2} \)
$61$ \( ( 1681 + 533 T + 139 T^{2} + 17 T^{3} + T^{4} )^{2} \)
$67$ \( 3748096 - 240064 T^{2} + 5856 T^{4} + 16 T^{6} + T^{8} \)
$71$ \( ( 841 - 232 T + 34 T^{2} - 3 T^{3} + T^{4} )^{2} \)
$73$ \( 43046721 - 531441 T^{2} + 6561 T^{4} - 81 T^{6} + T^{8} \)
$79$ \( ( 625 - 375 T + 100 T^{2} - 10 T^{3} + T^{4} )^{2} \)
$83$ \( 14641 + 2299 T^{2} + 1401 T^{4} - 61 T^{6} + T^{8} \)
$89$ \( ( 6400 + 1600 T + 240 T^{2} + 20 T^{3} + T^{4} )^{2} \)
$97$ \( 14641 - 484 T^{2} + 166 T^{4} - 19 T^{6} + T^{8} \)
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