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

Related objects

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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

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

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:

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

Atkin-Lehner signs

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

Hecke kernels

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

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 - 2 T \)
$3$ \( 1 + 8 T + 27 T^{2} \)
$5$ \( 1 - 5 T \)
$7$ \( 1 + 4 T + 343 T^{2} \)
$11$ \( 1 - 12 T + 1331 T^{2} \)
$13$ \( 1 + 58 T + 2197 T^{2} \)
$17$ \( 1 - 66 T + 4913 T^{2} \)
$19$ \( 1 + 100 T + 6859 T^{2} \)
$23$ \( 1 - 132 T + 12167 T^{2} \)
$29$ \( 1 + 90 T + 24389 T^{2} \)
$31$ \( 1 - 152 T + 29791 T^{2} \)
$37$ \( 1 + 34 T + 50653 T^{2} \)
$41$ \( 1 + 438 T + 68921 T^{2} \)
$43$ \( 1 - 32 T + 79507 T^{2} \)
$47$ \( 1 + 204 T + 103823 T^{2} \)
$53$ \( 1 - 222 T + 148877 T^{2} \)
$59$ \( 1 - 420 T + 205379 T^{2} \)
$61$ \( 1 - 902 T + 226981 T^{2} \)
$67$ \( 1 + 1024 T + 300763 T^{2} \)
$71$ \( 1 - 432 T + 357911 T^{2} \)
$73$ \( 1 - 362 T + 389017 T^{2} \)
$79$ \( 1 + 160 T + 493039 T^{2} \)
$83$ \( 1 - 72 T + 571787 T^{2} \)
$89$ \( 1 - 810 T + 704969 T^{2} \)
$97$ \( 1 - 1106 T + 912673 T^{2} \)
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