Properties

Label 125.2.e.a
Level $125$
Weight $2$
Character orbit 125.e
Analytic conductor $0.998$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 125 = 5^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 125.e (of order \(10\), degree \(4\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.998130025266\)
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.

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/125\mathbb{Z}\right)^\times\).

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
24.1
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
−1.53884 0.500000i −0.587785 + 0.809017i 0.500000 + 0.363271i 0 1.30902 0.951057i 0.618034i 1.31433 + 1.80902i 0.618034 + 1.90211i 0
24.2 1.53884 + 0.500000i 0.587785 0.809017i 0.500000 + 0.363271i 0 1.30902 0.951057i 0.618034i −1.31433 1.80902i 0.618034 + 1.90211i 0
49.1 −0.363271 + 0.500000i −0.951057 + 0.309017i 0.500000 + 1.53884i 0 0.190983 0.587785i 1.61803i −2.12663 0.690983i −1.61803 + 1.17557i 0
49.2 0.363271 0.500000i 0.951057 0.309017i 0.500000 + 1.53884i 0 0.190983 0.587785i 1.61803i 2.12663 + 0.690983i −1.61803 + 1.17557i 0
74.1 −0.363271 0.500000i −0.951057 0.309017i 0.500000 1.53884i 0 0.190983 + 0.587785i 1.61803i −2.12663 + 0.690983i −1.61803 1.17557i 0
74.2 0.363271 + 0.500000i 0.951057 + 0.309017i 0.500000 1.53884i 0 0.190983 + 0.587785i 1.61803i 2.12663 0.690983i −1.61803 1.17557i 0
99.1 −1.53884 + 0.500000i −0.587785 0.809017i 0.500000 0.363271i 0 1.30902 + 0.951057i 0.618034i 1.31433 1.80902i 0.618034 1.90211i 0
99.2 1.53884 0.500000i 0.587785 + 0.809017i 0.500000 0.363271i 0 1.30902 + 0.951057i 0.618034i −1.31433 + 1.80902i 0.618034 1.90211i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 99.2
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 125.2.e.a 8
5.b even 2 1 inner 125.2.e.a 8
5.c odd 4 1 25.2.d.a 4
5.c odd 4 1 125.2.d.a 4
15.e even 4 1 225.2.h.b 4
20.e even 4 1 400.2.u.b 4
25.d even 5 1 inner 125.2.e.a 8
25.d even 5 1 625.2.b.a 4
25.d even 5 2 625.2.e.c 8
25.e even 10 1 inner 125.2.e.a 8
25.e even 10 1 625.2.b.a 4
25.e even 10 2 625.2.e.c 8
25.f odd 20 1 25.2.d.a 4
25.f odd 20 1 125.2.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 2 625.2.d.b 4
25.f odd 20 2 625.2.d.h 4
75.l even 20 1 225.2.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 1 400.2.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 5.c odd 4 1
25.2.d.a 4 25.f odd 20 1
125.2.d.a 4 5.c odd 4 1
125.2.d.a 4 25.f odd 20 1
125.2.e.a 8 1.a even 1 1 trivial
125.2.e.a 8 5.b even 2 1 inner
125.2.e.a 8 25.d even 5 1 inner
125.2.e.a 8 25.e even 10 1 inner
225.2.h.b 4 15.e even 4 1
225.2.h.b 4 75.l even 20 1
400.2.u.b 4 20.e even 4 1
400.2.u.b 4 100.l even 20 1
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 25.f odd 20 2
625.2.d.h 4 25.f odd 20 2
625.2.e.c 8 25.d even 5 2
625.2.e.c 8 25.e even 10 2
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 kernel of the linear operator \( T_{2}^{8} - 4 T_{2}^{6} + 6 T_{2}^{4} + T_{2}^{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(125, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T^{2} + 6 T^{4} - 4 T^{6} + T^{8} \)
$3$ \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \)
$5$ \( T^{8} \)
$7$ \( ( 1 + 3 T^{2} + T^{4} )^{2} \)
$11$ \( ( 16 - 32 T + 24 T^{2} + 2 T^{3} + T^{4} )^{2} \)
$13$ \( 6561 - 2916 T^{2} + 486 T^{4} + 9 T^{6} + T^{8} \)
$17$ \( 256 - 704 T^{2} + 736 T^{4} + 16 T^{6} + T^{8} \)
$19$ \( ( 25 - 50 T + 40 T^{2} - 5 T^{3} + T^{4} )^{2} \)
$23$ \( 923521 - 97061 T^{2} + 3841 T^{4} + 19 T^{6} + T^{8} \)
$29$ \( ( 25 + 10 T^{2} + 5 T^{3} + T^{4} )^{2} \)
$31$ \( ( 81 - 27 T + 9 T^{2} - 3 T^{3} + T^{4} )^{2} \)
$37$ \( 1 - 29 T^{2} + 321 T^{4} + 11 T^{6} + T^{8} \)
$41$ \( ( 16 + 8 T + 24 T^{2} - 8 T^{3} + T^{4} )^{2} \)
$43$ \( ( 81 + 27 T^{2} + T^{4} )^{2} \)
$47$ \( 1 + T^{2} + 6 T^{4} - 4 T^{6} + T^{8} \)
$53$ \( 130321 - 361 T^{2} + 681 T^{4} - 41 T^{6} + T^{8} \)
$59$ \( ( 2025 + 675 T + 90 T^{2} + T^{4} )^{2} \)
$61$ \( ( 1681 - 697 T + 139 T^{2} - 13 T^{3} + T^{4} )^{2} \)
$67$ \( 3748096 + 30976 T^{2} + 5856 T^{4} - 124 T^{6} + T^{8} \)
$71$ \( ( 841 - 87 T + 34 T^{2} - 8 T^{3} + T^{4} )^{2} \)
$73$ \( 43046721 - 531441 T^{2} + 6561 T^{4} - 81 T^{6} + T^{8} \)
$79$ \( ( 625 + 250 T + 100 T^{2} + 15 T^{3} + T^{4} )^{2} \)
$83$ \( 14641 - 7381 T^{2} + 1401 T^{4} + 19 T^{6} + T^{8} \)
$89$ \( ( 6400 - 1600 T + 240 T^{2} - 20 T^{3} + T^{4} )^{2} \)
$97$ \( 14641 - 2299 T^{2} + 166 T^{4} - 4 T^{6} + T^{8} \)
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