Properties

Label 10.4.a.a
Level $10$
Weight $4$
Character orbit 10.a
Self dual yes
Analytic conductor $0.590$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 10 = 2 \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 10.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(0.590019100057\)
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

\(f(q)\) \(=\) \( q + 2 q^{2} - 8 q^{3} + 4 q^{4} + 5 q^{5} - 16 q^{6} - 4 q^{7} + 8 q^{8} + 37 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 q^{2} - 8 q^{3} + 4 q^{4} + 5 q^{5} - 16 q^{6} - 4 q^{7} + 8 q^{8} + 37 q^{9} + 10 q^{10} + 12 q^{11} - 32 q^{12} - 58 q^{13} - 8 q^{14} - 40 q^{15} + 16 q^{16} + 66 q^{17} + 74 q^{18} - 100 q^{19} + 20 q^{20} + 32 q^{21} + 24 q^{22} + 132 q^{23} - 64 q^{24} + 25 q^{25} - 116 q^{26} - 80 q^{27} - 16 q^{28} - 90 q^{29} - 80 q^{30} + 152 q^{31} + 32 q^{32} - 96 q^{33} + 132 q^{34} - 20 q^{35} + 148 q^{36} - 34 q^{37} - 200 q^{38} + 464 q^{39} + 40 q^{40} - 438 q^{41} + 64 q^{42} + 32 q^{43} + 48 q^{44} + 185 q^{45} + 264 q^{46} - 204 q^{47} - 128 q^{48} - 327 q^{49} + 50 q^{50} - 528 q^{51} - 232 q^{52} + 222 q^{53} - 160 q^{54} + 60 q^{55} - 32 q^{56} + 800 q^{57} - 180 q^{58} + 420 q^{59} - 160 q^{60} + 902 q^{61} + 304 q^{62} - 148 q^{63} + 64 q^{64} - 290 q^{65} - 192 q^{66} - 1024 q^{67} + 264 q^{68} - 1056 q^{69} - 40 q^{70} + 432 q^{71} + 296 q^{72} + 362 q^{73} - 68 q^{74} - 200 q^{75} - 400 q^{76} - 48 q^{77} + 928 q^{78} - 160 q^{79} + 80 q^{80} - 359 q^{81} - 876 q^{82} + 72 q^{83} + 128 q^{84} + 330 q^{85} + 64 q^{86} + 720 q^{87} + 96 q^{88} + 810 q^{89} + 370 q^{90} + 232 q^{91} + 528 q^{92} - 1216 q^{93} - 408 q^{94} - 500 q^{95} - 256 q^{96} + 1106 q^{97} - 654 q^{98} + 444 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
2.00000 −8.00000 4.00000 5.00000 −16.0000 −4.00000 8.00000 37.0000 10.0000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(-1\)

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 10.4.a.a 1
3.b odd 2 1 90.4.a.a 1
4.b odd 2 1 80.4.a.f 1
5.b even 2 1 50.4.a.c 1
5.c odd 4 2 50.4.b.a 2
7.b odd 2 1 490.4.a.o 1
7.c even 3 2 490.4.e.i 2
7.d odd 6 2 490.4.e.a 2
8.b even 2 1 320.4.a.m 1
8.d odd 2 1 320.4.a.b 1
9.c even 3 2 810.4.e.c 2
9.d odd 6 2 810.4.e.w 2
11.b odd 2 1 1210.4.a.b 1
12.b even 2 1 720.4.a.j 1
13.b even 2 1 1690.4.a.a 1
15.d odd 2 1 450.4.a.q 1
15.e even 4 2 450.4.c.d 2
16.e even 4 2 1280.4.d.j 2
16.f odd 4 2 1280.4.d.g 2
20.d odd 2 1 400.4.a.b 1
20.e even 4 2 400.4.c.c 2
35.c odd 2 1 2450.4.a.b 1
40.e odd 2 1 1600.4.a.bx 1
40.f even 2 1 1600.4.a.d 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.4.a.a 1 1.a even 1 1 trivial
50.4.a.c 1 5.b even 2 1
50.4.b.a 2 5.c odd 4 2
80.4.a.f 1 4.b odd 2 1
90.4.a.a 1 3.b odd 2 1
320.4.a.b 1 8.d odd 2 1
320.4.a.m 1 8.b even 2 1
400.4.a.b 1 20.d odd 2 1
400.4.c.c 2 20.e even 4 2
450.4.a.q 1 15.d odd 2 1
450.4.c.d 2 15.e even 4 2
490.4.a.o 1 7.b odd 2 1
490.4.e.a 2 7.d odd 6 2
490.4.e.i 2 7.c even 3 2
720.4.a.j 1 12.b even 2 1
810.4.e.c 2 9.c even 3 2
810.4.e.w 2 9.d odd 6 2
1210.4.a.b 1 11.b odd 2 1
1280.4.d.g 2 16.f odd 4 2
1280.4.d.j 2 16.e even 4 2
1600.4.a.d 1 40.f even 2 1
1600.4.a.bx 1 40.e odd 2 1
1690.4.a.a 1 13.b even 2 1
2450.4.a.b 1 35.c odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(\Gamma_0(10))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 2 \) Copy content Toggle raw display
$3$ \( T + 8 \) Copy content Toggle raw display
$5$ \( T - 5 \) Copy content Toggle raw display
$7$ \( T + 4 \) Copy content Toggle raw display
$11$ \( T - 12 \) Copy content Toggle raw display
$13$ \( T + 58 \) Copy content Toggle raw display
$17$ \( T - 66 \) Copy content Toggle raw display
$19$ \( T + 100 \) Copy content Toggle raw display
$23$ \( T - 132 \) Copy content Toggle raw display
$29$ \( T + 90 \) Copy content Toggle raw display
$31$ \( T - 152 \) Copy content Toggle raw display
$37$ \( T + 34 \) Copy content Toggle raw display
$41$ \( T + 438 \) Copy content Toggle raw display
$43$ \( T - 32 \) Copy content Toggle raw display
$47$ \( T + 204 \) Copy content Toggle raw display
$53$ \( T - 222 \) Copy content Toggle raw display
$59$ \( T - 420 \) Copy content Toggle raw display
$61$ \( T - 902 \) Copy content Toggle raw display
$67$ \( T + 1024 \) Copy content Toggle raw display
$71$ \( T - 432 \) Copy content Toggle raw display
$73$ \( T - 362 \) Copy content Toggle raw display
$79$ \( T + 160 \) Copy content Toggle raw display
$83$ \( T - 72 \) Copy content Toggle raw display
$89$ \( T - 810 \) Copy content Toggle raw display
$97$ \( T - 1106 \) Copy content Toggle raw display
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