Properties

Label 25.2.d.a
Level $25$
Weight $2$
Character orbit 25.d
Analytic conductor $0.200$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 25 = 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 25.d (of order \(5\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.199626005053\)
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: yes
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.

\(f(q)\) \(=\) \( q + (\zeta_{10}^{2} - \zeta_{10}) q^{2} - \zeta_{10}^{3} q^{3} + (\zeta_{10}^{3} + \zeta_{10} - 1) q^{4} + ( - 2 \zeta_{10}^{2} + \zeta_{10} - 2) q^{5} + (\zeta_{10}^{3} - \zeta_{10}^{2} + \zeta_{10}) q^{6} + (\zeta_{10}^{3} - \zeta_{10}^{2} - 1) q^{7} + ( - 2 \zeta_{10}^{3} + \zeta_{10}^{2} - 2 \zeta_{10}) q^{8} + 2 \zeta_{10} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{10}^{2} - \zeta_{10}) q^{2} - \zeta_{10}^{3} q^{3} + (\zeta_{10}^{3} + \zeta_{10} - 1) q^{4} + ( - 2 \zeta_{10}^{2} + \zeta_{10} - 2) q^{5} + (\zeta_{10}^{3} - \zeta_{10}^{2} + \zeta_{10}) q^{6} + (\zeta_{10}^{3} - \zeta_{10}^{2} - 1) q^{7} + ( - 2 \zeta_{10}^{3} + \zeta_{10}^{2} - 2 \zeta_{10}) q^{8} + 2 \zeta_{10} q^{9} + (\zeta_{10}^{3} - \zeta_{10}^{2} + 2) q^{10} + ( - 2 \zeta_{10}^{3} + 4 \zeta_{10}^{2} - 4 \zeta_{10} + 2) q^{11} + (\zeta_{10}^{2} + 1) q^{12} + (3 \zeta_{10}^{2} + 3) q^{13} + ( - \zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10} + 1) q^{14} + (\zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10} - 1) q^{15} + ( - 3 \zeta_{10}^{2} + 3 \zeta_{10} - 3) q^{16} + (2 \zeta_{10}^{3} - 4 \zeta_{10}^{2} + 2 \zeta_{10}) q^{17} + (2 \zeta_{10}^{3} - 2 \zeta_{10}^{2}) q^{18} + ( - 3 \zeta_{10}^{3} - \zeta_{10}^{2} - 3 \zeta_{10}) q^{19} + ( - 3 \zeta_{10}^{3} + 2 \zeta_{10}^{2} - 2 \zeta_{10} + 3) q^{20} + (\zeta_{10}^{3} + \zeta_{10} - 1) q^{21} + ( - 2 \zeta_{10}^{3} + 4 \zeta_{10} - 4) q^{22} + (7 \zeta_{10}^{3} - 5 \zeta_{10}^{2} + 5 \zeta_{10} - 7) q^{23} + (2 \zeta_{10}^{3} - 2 \zeta_{10}^{2} - 1) q^{24} + 5 \zeta_{10}^{2} q^{25} - 3 q^{26} + ( - 5 \zeta_{10}^{3} + 5 \zeta_{10}^{2} - 5 \zeta_{10} + 5) q^{27} + ( - 2 \zeta_{10}^{3} - \zeta_{10} + 1) q^{28} + (2 \zeta_{10}^{3} - \zeta_{10} + 1) q^{29} + ( - 2 \zeta_{10}^{3} + \zeta_{10} - 1) q^{30} - 3 \zeta_{10}^{2} q^{31} + ( - \zeta_{10}^{3} + \zeta_{10}^{2} + 5) q^{32} + (2 \zeta_{10}^{3} - 4 \zeta_{10}^{2} + 2 \zeta_{10}) q^{33} + (4 \zeta_{10}^{2} - 6 \zeta_{10} + 4) q^{34} + (\zeta_{10}^{2} + 2 \zeta_{10} + 1) q^{35} + (2 \zeta_{10}^{3} - 2) q^{36} + ( - 2 \zeta_{10}^{2} - \zeta_{10} - 2) q^{37} + (\zeta_{10}^{2} + 2 \zeta_{10} + 1) q^{38} + ( - 3 \zeta_{10}^{3} + 3) q^{39} + 5 \zeta_{10}^{3} q^{40} + (2 \zeta_{10}^{2} + 2 \zeta_{10} + 2) q^{41} - \zeta_{10}^{2} q^{42} + ( - 3 \zeta_{10}^{3} + 3 \zeta_{10}^{2}) q^{43} + (2 \zeta_{10}^{3} - 2 \zeta_{10}^{2} + 2 \zeta_{10}) q^{44} + ( - 4 \zeta_{10}^{3} + 2 \zeta_{10}^{2} - 4 \zeta_{10}) q^{45} + ( - 2 \zeta_{10}^{3} - 5 \zeta_{10} + 5) q^{46} + (\zeta_{10}^{3} + \zeta_{10} - 1) q^{47} + (3 \zeta_{10}^{2} - 3 \zeta_{10}) q^{48} + ( - \zeta_{10}^{3} + \zeta_{10}^{2} - 5) q^{49} + ( - 5 \zeta_{10}^{2} + 5 \zeta_{10} - 5) q^{50} + ( - 2 \zeta_{10}^{3} + 2 \zeta_{10}^{2} - 2) q^{51} + (6 \zeta_{10}^{3} - 3 \zeta_{10}^{2} + 3 \zeta_{10} - 6) q^{52} + ( - 3 \zeta_{10}^{3} - 4 \zeta_{10} + 4) q^{53} + (5 \zeta_{10} - 5) q^{54} + (6 \zeta_{10}^{3} - 6 \zeta_{10}^{2} + 2) q^{55} + (\zeta_{10}^{3} + 2 \zeta_{10}^{2} + \zeta_{10}) q^{56} + (3 \zeta_{10}^{3} - 3 \zeta_{10}^{2} - 4) q^{57} + ( - 3 \zeta_{10}^{3} + 4 \zeta_{10}^{2} - 3 \zeta_{10}) q^{58} + ( - 3 \zeta_{10}^{2} + 9 \zeta_{10} - 3) q^{59} + ( - \zeta_{10}^{3} - 2 \zeta_{10}^{2} - \zeta_{10}) q^{60} + ( - \zeta_{10}^{3} - 5 \zeta_{10}^{2} + 5 \zeta_{10} + 1) q^{61} + (3 \zeta_{10}^{2} - 3 \zeta_{10} + 3) q^{62} + ( - 2 \zeta_{10}^{2} - 2) q^{63} + (\zeta_{10}^{3} - 3 \zeta_{10}^{2} + 3 \zeta_{10} - 1) q^{64} + ( - 3 \zeta_{10}^{3} - 6 \zeta_{10}^{2} - 3 \zeta_{10}) q^{65} + (4 \zeta_{10}^{2} - 6 \zeta_{10} + 4) q^{66} + (2 \zeta_{10}^{3} + 6 \zeta_{10}^{2} + 2 \zeta_{10}) q^{67} + ( - 2 \zeta_{10}^{3} + 2 \zeta_{10}^{2}) q^{68} + (2 \zeta_{10}^{3} + 5 \zeta_{10}^{2} + 2 \zeta_{10}) q^{69} + (2 \zeta_{10}^{3} - 2 \zeta_{10}^{2} - 1) q^{70} + (5 \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} + 2) q^{74} + 5 q^{75} + ( - 4 \zeta_{10}^{3} + 4 \zeta_{10}^{2} + 7) q^{76} + (2 \zeta_{10}^{2} - 2 \zeta_{10}) q^{77} + 3 \zeta_{10}^{3} q^{78} + ( - 5 \zeta_{10} + 5) q^{79} + ( - 3 \zeta_{10}^{3} + 9 \zeta_{10}^{2} - 3 \zeta_{10}) q^{80} + \zeta_{10}^{2} q^{81} + (2 \zeta_{10}^{3} - 2 \zeta_{10}^{2} - 2) q^{82} + (2 \zeta_{10}^{3} - 5 \zeta_{10}^{2} + 2 \zeta_{10}) q^{83} + ( - \zeta_{10}^{2} - \zeta_{10} - 1) q^{84} + ( - 2 \zeta_{10}^{3} + 6 \zeta_{10} - 6) q^{85} + (3 \zeta_{10}^{3} - 6 \zeta_{10}^{2} + 6 \zeta_{10} - 3) q^{86} + ( - \zeta_{10}^{2} + 3 \zeta_{10} - 1) q^{87} + ( - 6 \zeta_{10}^{2} + 8 \zeta_{10} - 6) q^{88} + (4 \zeta_{10}^{3} + 4 \zeta_{10}^{2} - 4 \zeta_{10} - 4) q^{89} + ( - 2 \zeta_{10}^{2} + 6 \zeta_{10} - 2) q^{90} + ( - 3 \zeta_{10}^{2} - 3 \zeta_{10} - 3) q^{91} + ( - 7 \zeta_{10}^{3} - 2 \zeta_{10}^{2} - 7 \zeta_{10}) q^{92} - 3 q^{93} - \zeta_{10}^{2} q^{94} + (10 \zeta_{10}^{3} + 5 \zeta_{10} - 5) q^{95} + ( - 5 \zeta_{10}^{3} - \zeta_{10} + 1) q^{96} + ( - \zeta_{10}^{3} - 3 \zeta_{10} + 3) q^{97} + (\zeta_{10}^{3} - 7 \zeta_{10}^{2} + 7 \zeta_{10} - 1) q^{98} + (4 \zeta_{10}^{3} - 4 \zeta_{10}^{2} + 4) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{2} - q^{3} - 2 q^{4} - 5 q^{5} + 3 q^{6} - 2 q^{7} - 5 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{2} - q^{3} - 2 q^{4} - 5 q^{5} + 3 q^{6} - 2 q^{7} - 5 q^{8} + 2 q^{9} + 10 q^{10} - 2 q^{11} + 3 q^{12} + 9 q^{13} + q^{14} - 5 q^{15} - 6 q^{16} + 8 q^{17} + 4 q^{18} - 5 q^{19} + 5 q^{20} - 2 q^{21} - 14 q^{22} - 11 q^{23} - 5 q^{25} - 12 q^{26} + 5 q^{27} + q^{28} + 5 q^{29} - 5 q^{30} + 3 q^{31} + 18 q^{32} + 8 q^{33} + 6 q^{34} + 5 q^{35} - 6 q^{36} - 7 q^{37} + 5 q^{38} + 9 q^{39} + 5 q^{40} + 8 q^{41} + q^{42} - 6 q^{43} + 6 q^{44} - 10 q^{45} + 13 q^{46} - 2 q^{47} - 6 q^{48} - 22 q^{49} - 10 q^{50} - 12 q^{51} - 12 q^{52} + 9 q^{53} - 15 q^{54} + 20 q^{55} - 10 q^{57} - 10 q^{58} + 13 q^{61} + 6 q^{62} - 6 q^{63} + 3 q^{64} + 6 q^{66} - 2 q^{67} - 4 q^{68} - q^{69} + 8 q^{71} + 10 q^{72} + 9 q^{73} + 6 q^{74} + 20 q^{75} + 20 q^{76} - 4 q^{77} + 3 q^{78} + 15 q^{79} - 15 q^{80} - q^{81} - 4 q^{82} + 9 q^{83} - 4 q^{84} - 20 q^{85} + 3 q^{86} - 10 q^{88} - 20 q^{89} - 12 q^{91} - 12 q^{92} - 12 q^{93} + q^{94} - 5 q^{95} - 2 q^{96} + 8 q^{97} + 11 q^{98} + 24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/25\mathbb{Z}\right)^\times\).

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
6.1
−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 −0.690983 + 2.12663i 1.30902 + 0.951057i 0.618034 −1.80902 1.31433i −0.618034 + 1.90211i 3.61803
11.1 −0.500000 + 0.363271i 0.309017 0.951057i −0.500000 + 1.53884i −1.80902 1.31433i 0.190983 + 0.587785i −1.61803 −0.690983 2.12663i 1.61803 + 1.17557i 1.38197
16.1 −0.500000 0.363271i 0.309017 + 0.951057i −0.500000 1.53884i −1.80902 + 1.31433i 0.190983 0.587785i −1.61803 −0.690983 + 2.12663i 1.61803 1.17557i 1.38197
21.1 −0.500000 + 1.53884i −0.809017 0.587785i −0.500000 0.363271i −0.690983 2.12663i 1.30902 0.951057i 0.618034 −1.80902 + 1.31433i −0.618034 1.90211i 3.61803
\(n\): 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 25.2.d.a 4
3.b odd 2 1 225.2.h.b 4
4.b odd 2 1 400.2.u.b 4
5.b even 2 1 125.2.d.a 4
5.c odd 4 2 125.2.e.a 8
25.d even 5 1 inner 25.2.d.a 4
25.d even 5 1 625.2.a.b 2
25.d even 5 2 625.2.d.h 4
25.e even 10 1 125.2.d.a 4
25.e even 10 1 625.2.a.c 2
25.e even 10 2 625.2.d.b 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 5625.2.a.d 2
75.j odd 10 1 225.2.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 1 400.2.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 1.a even 1 1 trivial
25.2.d.a 4 25.d even 5 1 inner
125.2.d.a 4 5.b even 2 1
125.2.d.a 4 25.e even 10 1
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 3.b odd 2 1
225.2.h.b 4 75.j odd 10 1
400.2.u.b 4 4.b odd 2 1
400.2.u.b 4 100.j odd 10 1
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 25.e even 10 2
625.2.d.h 4 25.d even 5 2
625.2.e.c 8 25.f odd 20 4
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 is the entire newspace \(S_{2}^{\mathrm{new}}(25, [\chi])\).

Hecke characteristic polynomials

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