Properties

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

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(4.99065012633\)
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.

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

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
126.1
−0.309017 + 0.951057i
0.809017 + 0.587785i
0.809017 0.587785i
−0.309017 0.951057i
−0.190983 0.587785i 0.809017 0.587785i 1.30902 0.951057i 0 −0.500000 0.363271i 1.61803 −1.80902 1.31433i −0.618034 + 1.90211i 0
251.1 −1.30902 + 0.951057i −0.309017 + 0.951057i 0.190983 0.587785i 0 −0.500000 1.53884i −0.618034 −0.690983 2.12663i 1.61803 + 1.17557i 0
376.1 −1.30902 0.951057i −0.309017 0.951057i 0.190983 + 0.587785i 0 −0.500000 + 1.53884i −0.618034 −0.690983 + 2.12663i 1.61803 1.17557i 0
501.1 −0.190983 + 0.587785i 0.809017 + 0.587785i 1.30902 + 0.951057i 0 −0.500000 + 0.363271i 1.61803 −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 625.2.d.b 4
5.b even 2 1 625.2.d.h 4
5.c odd 4 2 625.2.e.c 8
25.d even 5 2 125.2.d.a 4
25.d even 5 1 625.2.a.c 2
25.d even 5 1 inner 625.2.d.b 4
25.e even 10 2 25.2.d.a 4
25.e even 10 1 625.2.a.b 2
25.e even 10 1 625.2.d.h 4
25.f odd 20 4 125.2.e.a 8
25.f odd 20 2 625.2.b.a 4
25.f odd 20 2 625.2.e.c 8
75.h odd 10 2 225.2.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 2 400.2.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 25.e even 10 2
125.2.d.a 4 25.d even 5 2
125.2.e.a 8 25.f odd 20 4
225.2.h.b 4 75.h odd 10 2
400.2.u.b 4 100.h odd 10 2
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 1.a even 1 1 trivial
625.2.d.b 4 25.d even 5 1 inner
625.2.d.h 4 5.b even 2 1
625.2.d.h 4 25.e even 10 1
625.2.e.c 8 5.c odd 4 2
625.2.e.c 8 25.f odd 20 2
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 intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(625, [\chi])\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 2 T + 4 T^{2} + 3 T^{3} + T^{4} \)
$3$ \( 1 - T + T^{2} - T^{3} + T^{4} \)
$5$ \( T^{4} \)
$7$ \( ( -1 - T + T^{2} )^{2} \)
$11$ \( 16 + 8 T + 24 T^{2} - 8 T^{3} + T^{4} \)
$13$ \( 81 - 81 T + 36 T^{2} - 6 T^{3} + T^{4} \)
$17$ \( 16 + 32 T + 24 T^{2} - 2 T^{3} + T^{4} \)
$19$ \( 25 - 25 T + 40 T^{2} - 10 T^{3} + T^{4} \)
$23$ \( 961 - 341 T + 51 T^{2} - T^{3} + T^{4} \)
$29$ \( 25 - 25 T + 10 T^{2} + T^{4} \)
$31$ \( 81 - 27 T + 9 T^{2} - 3 T^{3} + T^{4} \)
$37$ \( 1 + 7 T + 19 T^{2} + 3 T^{3} + T^{4} \)
$41$ \( 16 - 32 T + 24 T^{2} + 2 T^{3} + T^{4} \)
$43$ \( ( -9 - 3 T + T^{2} )^{2} \)
$47$ \( 1 + 2 T + 4 T^{2} + 3 T^{3} + T^{4} \)
$53$ \( 361 - 171 T + 61 T^{2} - 11 T^{3} + T^{4} \)
$59$ \( 2025 + 90 T^{2} - 15 T^{3} + T^{4} \)
$61$ \( 1681 + 533 T + 139 T^{2} + 17 T^{3} + T^{4} \)
$67$ \( 1936 - 88 T + 64 T^{2} - 12 T^{3} + T^{4} \)
$71$ \( 841 - 232 T + 34 T^{2} - 3 T^{3} + T^{4} \)
$73$ \( 6561 + 729 T + 81 T^{2} + 9 T^{3} + T^{4} \)
$79$ \( 625 + 375 T + 100 T^{2} + 10 T^{3} + T^{4} \)
$83$ \( 121 + 99 T + 31 T^{2} - T^{3} + T^{4} \)
$89$ \( 6400 - 1600 T + 240 T^{2} - 20 T^{3} + T^{4} \)
$97$ \( 121 - 88 T + 34 T^{2} - 7 T^{3} + T^{4} \)
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