Properties

Label 5.4.a.a
Level $5$
Weight $4$
Character orbit 5.a
Self dual yes
Analytic conductor $0.295$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 5.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(0.295009550029\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 4 q^{2} + 2 q^{3} + 8 q^{4} - 5 q^{5} - 8 q^{6} + 6 q^{7} - 23 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 4 q^{2} + 2 q^{3} + 8 q^{4} - 5 q^{5} - 8 q^{6} + 6 q^{7} - 23 q^{9} + 20 q^{10} + 32 q^{11} + 16 q^{12} - 38 q^{13} - 24 q^{14} - 10 q^{15} - 64 q^{16} + 26 q^{17} + 92 q^{18} + 100 q^{19} - 40 q^{20} + 12 q^{21} - 128 q^{22} - 78 q^{23} + 25 q^{25} + 152 q^{26} - 100 q^{27} + 48 q^{28} - 50 q^{29} + 40 q^{30} - 108 q^{31} + 256 q^{32} + 64 q^{33} - 104 q^{34} - 30 q^{35} - 184 q^{36} + 266 q^{37} - 400 q^{38} - 76 q^{39} + 22 q^{41} - 48 q^{42} + 442 q^{43} + 256 q^{44} + 115 q^{45} + 312 q^{46} - 514 q^{47} - 128 q^{48} - 307 q^{49} - 100 q^{50} + 52 q^{51} - 304 q^{52} + 2 q^{53} + 400 q^{54} - 160 q^{55} + 200 q^{57} + 200 q^{58} + 500 q^{59} - 80 q^{60} - 518 q^{61} + 432 q^{62} - 138 q^{63} - 512 q^{64} + 190 q^{65} - 256 q^{66} + 126 q^{67} + 208 q^{68} - 156 q^{69} + 120 q^{70} + 412 q^{71} - 878 q^{73} - 1064 q^{74} + 50 q^{75} + 800 q^{76} + 192 q^{77} + 304 q^{78} + 600 q^{79} + 320 q^{80} + 421 q^{81} - 88 q^{82} + 282 q^{83} + 96 q^{84} - 130 q^{85} - 1768 q^{86} - 100 q^{87} - 150 q^{89} - 460 q^{90} - 228 q^{91} - 624 q^{92} - 216 q^{93} + 2056 q^{94} - 500 q^{95} + 512 q^{96} + 386 q^{97} + 1228 q^{98} - 736 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Expression as an eta quotient

\(f(z) = \eta(z)^{4}\eta(5z)^{4}=q\prod_{n=1}^\infty(1 - q^{n})^{4}(1 - q^{5n})^{4}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
−4.00000 2.00000 8.00000 −5.00000 −8.00000 6.00000 0 −23.0000 20.0000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 5.4.a.a 1
3.b odd 2 1 45.4.a.d 1
4.b odd 2 1 80.4.a.d 1
5.b even 2 1 25.4.a.c 1
5.c odd 4 2 25.4.b.a 2
7.b odd 2 1 245.4.a.a 1
7.c even 3 2 245.4.e.f 2
7.d odd 6 2 245.4.e.g 2
8.b even 2 1 320.4.a.g 1
8.d odd 2 1 320.4.a.h 1
9.c even 3 2 405.4.e.l 2
9.d odd 6 2 405.4.e.c 2
11.b odd 2 1 605.4.a.d 1
12.b even 2 1 720.4.a.u 1
13.b even 2 1 845.4.a.b 1
15.d odd 2 1 225.4.a.b 1
15.e even 4 2 225.4.b.c 2
16.e even 4 2 1280.4.d.e 2
16.f odd 4 2 1280.4.d.l 2
17.b even 2 1 1445.4.a.a 1
19.b odd 2 1 1805.4.a.h 1
20.d odd 2 1 400.4.a.m 1
20.e even 4 2 400.4.c.k 2
21.c even 2 1 2205.4.a.q 1
35.c odd 2 1 1225.4.a.k 1
40.e odd 2 1 1600.4.a.s 1
40.f even 2 1 1600.4.a.bi 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.4.a.a 1 1.a even 1 1 trivial
25.4.a.c 1 5.b even 2 1
25.4.b.a 2 5.c odd 4 2
45.4.a.d 1 3.b odd 2 1
80.4.a.d 1 4.b odd 2 1
225.4.a.b 1 15.d odd 2 1
225.4.b.c 2 15.e even 4 2
245.4.a.a 1 7.b odd 2 1
245.4.e.f 2 7.c even 3 2
245.4.e.g 2 7.d odd 6 2
320.4.a.g 1 8.b even 2 1
320.4.a.h 1 8.d odd 2 1
400.4.a.m 1 20.d odd 2 1
400.4.c.k 2 20.e even 4 2
405.4.e.c 2 9.d odd 6 2
405.4.e.l 2 9.c even 3 2
605.4.a.d 1 11.b odd 2 1
720.4.a.u 1 12.b even 2 1
845.4.a.b 1 13.b even 2 1
1225.4.a.k 1 35.c odd 2 1
1280.4.d.e 2 16.e even 4 2
1280.4.d.l 2 16.f odd 4 2
1445.4.a.a 1 17.b even 2 1
1600.4.a.s 1 40.e odd 2 1
1600.4.a.bi 1 40.f even 2 1
1805.4.a.h 1 19.b odd 2 1
2205.4.a.q 1 21.c even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(\Gamma_0(5))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 4 \) Copy content Toggle raw display
$3$ \( T - 2 \) Copy content Toggle raw display
$5$ \( T + 5 \) Copy content Toggle raw display
$7$ \( T - 6 \) Copy content Toggle raw display
$11$ \( T - 32 \) Copy content Toggle raw display
$13$ \( T + 38 \) Copy content Toggle raw display
$17$ \( T - 26 \) Copy content Toggle raw display
$19$ \( T - 100 \) Copy content Toggle raw display
$23$ \( T + 78 \) Copy content Toggle raw display
$29$ \( T + 50 \) Copy content Toggle raw display
$31$ \( T + 108 \) Copy content Toggle raw display
$37$ \( T - 266 \) Copy content Toggle raw display
$41$ \( T - 22 \) Copy content Toggle raw display
$43$ \( T - 442 \) Copy content Toggle raw display
$47$ \( T + 514 \) Copy content Toggle raw display
$53$ \( T - 2 \) Copy content Toggle raw display
$59$ \( T - 500 \) Copy content Toggle raw display
$61$ \( T + 518 \) Copy content Toggle raw display
$67$ \( T - 126 \) Copy content Toggle raw display
$71$ \( T - 412 \) Copy content Toggle raw display
$73$ \( T + 878 \) Copy content Toggle raw display
$79$ \( T - 600 \) Copy content Toggle raw display
$83$ \( T - 282 \) Copy content Toggle raw display
$89$ \( T + 150 \) Copy content Toggle raw display
$97$ \( T - 386 \) Copy content Toggle raw display
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