Properties

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

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(0.998130025266\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{10})\)
Defining polynomial: \(x^{4} - x^{3} + x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 25)
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{10}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( \zeta_{10} - \zeta_{10}^{2} ) q^{2} + \zeta_{10}^{3} q^{3} + ( -1 + \zeta_{10} + \zeta_{10}^{3} ) q^{4} + ( \zeta_{10} - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{6} + ( 1 + \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{7} + ( 2 \zeta_{10} - \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{8} + 2 \zeta_{10} q^{9} +O(q^{10})\) \( q + ( \zeta_{10} - \zeta_{10}^{2} ) q^{2} + \zeta_{10}^{3} q^{3} + ( -1 + \zeta_{10} + \zeta_{10}^{3} ) q^{4} + ( \zeta_{10} - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{6} + ( 1 + \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{7} + ( 2 \zeta_{10} - \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{8} + 2 \zeta_{10} q^{9} + ( 2 - 4 \zeta_{10} + 4 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{11} + ( -1 - \zeta_{10}^{2} ) q^{12} + ( -3 - 3 \zeta_{10}^{2} ) q^{13} + ( 1 - \zeta_{10} + \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{14} + ( -3 + 3 \zeta_{10} - 3 \zeta_{10}^{2} ) q^{16} + ( -2 \zeta_{10} + 4 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{17} + ( 2 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{18} + ( -3 \zeta_{10} - \zeta_{10}^{2} - 3 \zeta_{10}^{3} ) q^{19} + ( -1 + \zeta_{10} + \zeta_{10}^{3} ) q^{21} + ( 4 - 4 \zeta_{10} + 2 \zeta_{10}^{3} ) q^{22} + ( 7 - 5 \zeta_{10} + 5 \zeta_{10}^{2} - 7 \zeta_{10}^{3} ) q^{23} + ( -1 - 2 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{24} -3 q^{26} + ( -5 + 5 \zeta_{10} - 5 \zeta_{10}^{2} + 5 \zeta_{10}^{3} ) q^{27} + ( -1 + \zeta_{10} + 2 \zeta_{10}^{3} ) q^{28} + ( 1 - \zeta_{10} + 2 \zeta_{10}^{3} ) q^{29} -3 \zeta_{10}^{2} q^{31} + ( -5 - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{32} + ( -2 \zeta_{10} + 4 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{33} + ( 4 - 6 \zeta_{10} + 4 \zeta_{10}^{2} ) q^{34} + ( -2 + 2 \zeta_{10}^{3} ) q^{36} + ( 2 + \zeta_{10} + 2 \zeta_{10}^{2} ) q^{37} + ( -1 - 2 \zeta_{10} - \zeta_{10}^{2} ) q^{38} + ( 3 - 3 \zeta_{10}^{3} ) q^{39} + ( 2 + 2 \zeta_{10} + 2 \zeta_{10}^{2} ) q^{41} + \zeta_{10}^{2} q^{42} + ( -3 \zeta_{10}^{2} + 3 \zeta_{10}^{3} ) q^{43} + ( 2 \zeta_{10} - 2 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{44} + ( 5 - 5 \zeta_{10} - 2 \zeta_{10}^{3} ) q^{46} + ( 1 - \zeta_{10} - \zeta_{10}^{3} ) q^{47} + ( 3 \zeta_{10} - 3 \zeta_{10}^{2} ) q^{48} + ( -5 + \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{49} + ( -2 + 2 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{51} + ( 6 - 3 \zeta_{10} + 3 \zeta_{10}^{2} - 6 \zeta_{10}^{3} ) q^{52} + ( -4 + 4 \zeta_{10} + 3 \zeta_{10}^{3} ) q^{53} + ( -5 + 5 \zeta_{10} ) q^{54} + ( \zeta_{10} + 2 \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{56} + ( 4 + 3 \zeta_{10}^{2} - 3 \zeta_{10}^{3} ) q^{57} + ( 3 \zeta_{10} - 4 \zeta_{10}^{2} + 3 \zeta_{10}^{3} ) q^{58} + ( -3 + 9 \zeta_{10} - 3 \zeta_{10}^{2} ) q^{59} + ( 1 + 5 \zeta_{10} - 5 \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{61} + ( -3 + 3 \zeta_{10} - 3 \zeta_{10}^{2} ) q^{62} + ( 2 + 2 \zeta_{10}^{2} ) q^{63} + ( -1 + 3 \zeta_{10} - 3 \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{64} + ( 4 - 6 \zeta_{10} + 4 \zeta_{10}^{2} ) q^{66} + ( -2 \zeta_{10} - 6 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{67} + ( -2 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{68} + ( 2 \zeta_{10} + 5 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{69} + ( 1 - \zeta_{10} + 5 \zeta_{10}^{3} ) q^{71} + ( -4 + 4 \zeta_{10} + 2 \zeta_{10}^{3} ) q^{72} + ( -9 + 9 \zeta_{10} - 9 \zeta_{10}^{2} + 9 \zeta_{10}^{3} ) q^{73} + ( 2 + \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{74} + ( 7 + 4 \zeta_{10}^{2} - 4 \zeta_{10}^{3} ) q^{76} + ( 2 \zeta_{10} - 2 \zeta_{10}^{2} ) q^{77} -3 \zeta_{10}^{3} q^{78} + ( 5 - 5 \zeta_{10} ) q^{79} + \zeta_{10}^{2} q^{81} + ( 2 + 2 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{82} + ( -2 \zeta_{10} + 5 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{83} + ( -1 - \zeta_{10} - \zeta_{10}^{2} ) q^{84} + ( -3 + 6 \zeta_{10} - 6 \zeta_{10}^{2} + 3 \zeta_{10}^{3} ) q^{86} + ( 1 - 3 \zeta_{10} + \zeta_{10}^{2} ) q^{87} + ( 6 - 8 \zeta_{10} + 6 \zeta_{10}^{2} ) q^{88} + ( -4 - 4 \zeta_{10} + 4 \zeta_{10}^{2} + 4 \zeta_{10}^{3} ) q^{89} + ( -3 - 3 \zeta_{10} - 3 \zeta_{10}^{2} ) q^{91} + ( 7 \zeta_{10} + 2 \zeta_{10}^{2} + 7 \zeta_{10}^{3} ) q^{92} + 3 q^{93} -\zeta_{10}^{2} q^{94} + ( 1 - \zeta_{10} - 5 \zeta_{10}^{3} ) q^{96} + ( -3 + 3 \zeta_{10} + \zeta_{10}^{3} ) q^{97} + ( 1 - 7 \zeta_{10} + 7 \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{98} + ( 4 - 4 \zeta_{10}^{2} + 4 \zeta_{10}^{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{2} + q^{3} - 2 q^{4} + 3 q^{6} + 2 q^{7} + 5 q^{8} + 2 q^{9} + O(q^{10}) \) \( 4 q + 2 q^{2} + q^{3} - 2 q^{4} + 3 q^{6} + 2 q^{7} + 5 q^{8} + 2 q^{9} - 2 q^{11} - 3 q^{12} - 9 q^{13} + q^{14} - 6 q^{16} - 8 q^{17} - 4 q^{18} - 5 q^{19} - 2 q^{21} + 14 q^{22} + 11 q^{23} - 12 q^{26} - 5 q^{27} - q^{28} + 5 q^{29} + 3 q^{31} - 18 q^{32} - 8 q^{33} + 6 q^{34} - 6 q^{36} + 7 q^{37} - 5 q^{38} + 9 q^{39} + 8 q^{41} - q^{42} + 6 q^{43} + 6 q^{44} + 13 q^{46} + 2 q^{47} + 6 q^{48} - 22 q^{49} - 12 q^{51} + 12 q^{52} - 9 q^{53} - 15 q^{54} + 10 q^{57} + 10 q^{58} + 13 q^{61} - 6 q^{62} + 6 q^{63} + 3 q^{64} + 6 q^{66} + 2 q^{67} + 4 q^{68} - q^{69} + 8 q^{71} - 10 q^{72} - 9 q^{73} + 6 q^{74} + 20 q^{76} + 4 q^{77} - 3 q^{78} + 15 q^{79} - q^{81} + 4 q^{82} - 9 q^{83} - 4 q^{84} + 3 q^{86} + 10 q^{88} - 20 q^{89} - 12 q^{91} + 12 q^{92} + 12 q^{93} + q^{94} - 2 q^{96} - 8 q^{97} - 11 q^{98} + 24 q^{99} + O(q^{100}) \)

Character values

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

\(n\) \(2\)
\(\chi(n)\) \(-\zeta_{10}^{3}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
26.1
−0.309017 + 0.951057i
0.809017 + 0.587785i
0.809017 0.587785i
−0.309017 0.951057i
0.500000 + 1.53884i 0.809017 0.587785i −0.500000 + 0.363271i 0 1.30902 + 0.951057i −0.618034 1.80902 + 1.31433i −0.618034 + 1.90211i 0
51.1 0.500000 0.363271i −0.309017 + 0.951057i −0.500000 + 1.53884i 0 0.190983 + 0.587785i 1.61803 0.690983 + 2.12663i 1.61803 + 1.17557i 0
76.1 0.500000 + 0.363271i −0.309017 0.951057i −0.500000 1.53884i 0 0.190983 0.587785i 1.61803 0.690983 2.12663i 1.61803 1.17557i 0
101.1 0.500000 1.53884i 0.809017 + 0.587785i −0.500000 0.363271i 0 1.30902 0.951057i −0.618034 1.80902 1.31433i −0.618034 1.90211i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
25.d even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 125.2.d.a 4
5.b even 2 1 25.2.d.a 4
5.c odd 4 2 125.2.e.a 8
15.d odd 2 1 225.2.h.b 4
20.d odd 2 1 400.2.u.b 4
25.d even 5 1 inner 125.2.d.a 4
25.d even 5 1 625.2.a.c 2
25.d even 5 2 625.2.d.b 4
25.e even 10 1 25.2.d.a 4
25.e even 10 1 625.2.a.b 2
25.e even 10 2 625.2.d.h 4
25.f odd 20 2 125.2.e.a 8
25.f odd 20 2 625.2.b.a 4
25.f odd 20 4 625.2.e.c 8
75.h odd 10 1 225.2.h.b 4
75.h odd 10 1 5625.2.a.f 2
75.j odd 10 1 5625.2.a.d 2
100.h odd 10 1 400.2.u.b 4
100.h odd 10 1 10000.2.a.c 2
100.j odd 10 1 10000.2.a.l 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
25.2.d.a 4 5.b even 2 1
25.2.d.a 4 25.e even 10 1
125.2.d.a 4 1.a even 1 1 trivial
125.2.d.a 4 25.d even 5 1 inner
125.2.e.a 8 5.c odd 4 2
125.2.e.a 8 25.f odd 20 2
225.2.h.b 4 15.d odd 2 1
225.2.h.b 4 75.h odd 10 1
400.2.u.b 4 20.d odd 2 1
400.2.u.b 4 100.h odd 10 1
625.2.a.b 2 25.e even 10 1
625.2.a.c 2 25.d even 5 1
625.2.b.a 4 25.f odd 20 2
625.2.d.b 4 25.d even 5 2
625.2.d.h 4 25.e even 10 2
625.2.e.c 8 25.f odd 20 4
5625.2.a.d 2 75.j odd 10 1
5625.2.a.f 2 75.h odd 10 1
10000.2.a.c 2 100.h odd 10 1
10000.2.a.l 2 100.j odd 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} - 2 T_{2}^{3} + 4 T_{2}^{2} - 3 T_{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(125, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 3 T + 4 T^{2} - 2 T^{3} + T^{4} \)
$3$ \( 1 - T + T^{2} - T^{3} + T^{4} \)
$5$ \( T^{4} \)
$7$ \( ( -1 - T + T^{2} )^{2} \)
$11$ \( 16 - 32 T + 24 T^{2} + 2 T^{3} + T^{4} \)
$13$ \( 81 + 54 T + 36 T^{2} + 9 T^{3} + T^{4} \)
$17$ \( 16 - 8 T + 24 T^{2} + 8 T^{3} + T^{4} \)
$19$ \( 25 + 50 T + 40 T^{2} + 5 T^{3} + T^{4} \)
$23$ \( 961 - 31 T + 51 T^{2} - 11 T^{3} + T^{4} \)
$29$ \( 25 + 10 T^{2} - 5 T^{3} + T^{4} \)
$31$ \( 81 - 27 T + 9 T^{2} - 3 T^{3} + T^{4} \)
$37$ \( 1 - 3 T + 19 T^{2} - 7 T^{3} + T^{4} \)
$41$ \( 16 + 8 T + 24 T^{2} - 8 T^{3} + T^{4} \)
$43$ \( ( -9 - 3 T + T^{2} )^{2} \)
$47$ \( 1 - 3 T + 4 T^{2} - 2 T^{3} + T^{4} \)
$53$ \( 361 + 209 T + 61 T^{2} + 9 T^{3} + T^{4} \)
$59$ \( 2025 - 675 T + 90 T^{2} + T^{4} \)
$61$ \( 1681 - 697 T + 139 T^{2} - 13 T^{3} + T^{4} \)
$67$ \( 1936 - 528 T + 64 T^{2} - 2 T^{3} + T^{4} \)
$71$ \( 841 - 87 T + 34 T^{2} - 8 T^{3} + T^{4} \)
$73$ \( 6561 + 729 T + 81 T^{2} + 9 T^{3} + T^{4} \)
$79$ \( 625 - 250 T + 100 T^{2} - 15 T^{3} + T^{4} \)
$83$ \( 121 - 11 T + 31 T^{2} + 9 T^{3} + T^{4} \)
$89$ \( 6400 + 1600 T + 240 T^{2} + 20 T^{3} + T^{4} \)
$97$ \( 121 + 77 T + 34 T^{2} + 8 T^{3} + T^{4} \)
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