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: \( x^{8} - x^{6} + x^{4} - x^{2} + 1 \) Copy content Toggle raw display
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.

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

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 \( T_{2}^{8} - 4T_{2}^{6} + 6T_{2}^{4} + T_{2}^{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(125, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

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