Properties

Label 625.2.d.h
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: \( x^{4} - x^{3} + x^{2} - x + 1 \) Copy content Toggle raw display
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.

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

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

Hecke kernels

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

\( T_{2}^{4} - 3T_{2}^{3} + 4T_{2}^{2} - 2T_{2} + 1 \) Copy content Toggle raw display
\( T_{3}^{4} + T_{3}^{3} + T_{3}^{2} + T_{3} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

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