Properties

Label 6.4.a.a
Level 6
Weight 4
Character orbit 6.a
Self dual yes
Analytic conductor 0.354
Analytic rank 0
Dimension 1
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 6 = 2 \cdot 3 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 6.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(0.354011460034\)
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

\(f(q)\) \(=\) \( q - 2q^{2} - 3q^{3} + 4q^{4} + 6q^{5} + 6q^{6} - 16q^{7} - 8q^{8} + 9q^{9} + O(q^{10}) \) \( q - 2q^{2} - 3q^{3} + 4q^{4} + 6q^{5} + 6q^{6} - 16q^{7} - 8q^{8} + 9q^{9} - 12q^{10} + 12q^{11} - 12q^{12} + 38q^{13} + 32q^{14} - 18q^{15} + 16q^{16} - 126q^{17} - 18q^{18} + 20q^{19} + 24q^{20} + 48q^{21} - 24q^{22} + 168q^{23} + 24q^{24} - 89q^{25} - 76q^{26} - 27q^{27} - 64q^{28} + 30q^{29} + 36q^{30} - 88q^{31} - 32q^{32} - 36q^{33} + 252q^{34} - 96q^{35} + 36q^{36} + 254q^{37} - 40q^{38} - 114q^{39} - 48q^{40} + 42q^{41} - 96q^{42} - 52q^{43} + 48q^{44} + 54q^{45} - 336q^{46} - 96q^{47} - 48q^{48} - 87q^{49} + 178q^{50} + 378q^{51} + 152q^{52} + 198q^{53} + 54q^{54} + 72q^{55} + 128q^{56} - 60q^{57} - 60q^{58} - 660q^{59} - 72q^{60} - 538q^{61} + 176q^{62} - 144q^{63} + 64q^{64} + 228q^{65} + 72q^{66} + 884q^{67} - 504q^{68} - 504q^{69} + 192q^{70} + 792q^{71} - 72q^{72} + 218q^{73} - 508q^{74} + 267q^{75} + 80q^{76} - 192q^{77} + 228q^{78} - 520q^{79} + 96q^{80} + 81q^{81} - 84q^{82} - 492q^{83} + 192q^{84} - 756q^{85} + 104q^{86} - 90q^{87} - 96q^{88} + 810q^{89} - 108q^{90} - 608q^{91} + 672q^{92} + 264q^{93} + 192q^{94} + 120q^{95} + 96q^{96} + 1154q^{97} + 174q^{98} + 108q^{99} + O(q^{100}) \)

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 −3.00000 4.00000 6.00000 6.00000 −16.0000 −8.00000 9.00000 −12.0000
\(n\): 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 6.4.a.a 1
3.b odd 2 1 18.4.a.a 1
4.b odd 2 1 48.4.a.c 1
5.b even 2 1 150.4.a.i 1
5.c odd 4 2 150.4.c.d 2
7.b odd 2 1 294.4.a.e 1
7.c even 3 2 294.4.e.h 2
7.d odd 6 2 294.4.e.g 2
8.b even 2 1 192.4.a.i 1
8.d odd 2 1 192.4.a.c 1
9.c even 3 2 162.4.c.f 2
9.d odd 6 2 162.4.c.c 2
11.b odd 2 1 726.4.a.f 1
12.b even 2 1 144.4.a.c 1
13.b even 2 1 1014.4.a.g 1
13.d odd 4 2 1014.4.b.d 2
15.d odd 2 1 450.4.a.h 1
15.e even 4 2 450.4.c.e 2
16.e even 4 2 768.4.d.n 2
16.f odd 4 2 768.4.d.c 2
17.b even 2 1 1734.4.a.d 1
19.b odd 2 1 2166.4.a.i 1
20.d odd 2 1 1200.4.a.b 1
20.e even 4 2 1200.4.f.j 2
21.c even 2 1 882.4.a.n 1
21.g even 6 2 882.4.g.f 2
21.h odd 6 2 882.4.g.i 2
24.f even 2 1 576.4.a.r 1
24.h odd 2 1 576.4.a.q 1
28.d even 2 1 2352.4.a.e 1
33.d even 2 1 2178.4.a.e 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6.4.a.a 1 1.a even 1 1 trivial
18.4.a.a 1 3.b odd 2 1
48.4.a.c 1 4.b odd 2 1
144.4.a.c 1 12.b even 2 1
150.4.a.i 1 5.b even 2 1
150.4.c.d 2 5.c odd 4 2
162.4.c.c 2 9.d odd 6 2
162.4.c.f 2 9.c even 3 2
192.4.a.c 1 8.d odd 2 1
192.4.a.i 1 8.b even 2 1
294.4.a.e 1 7.b odd 2 1
294.4.e.g 2 7.d odd 6 2
294.4.e.h 2 7.c even 3 2
450.4.a.h 1 15.d odd 2 1
450.4.c.e 2 15.e even 4 2
576.4.a.q 1 24.h odd 2 1
576.4.a.r 1 24.f even 2 1
726.4.a.f 1 11.b odd 2 1
768.4.d.c 2 16.f odd 4 2
768.4.d.n 2 16.e even 4 2
882.4.a.n 1 21.c even 2 1
882.4.g.f 2 21.g even 6 2
882.4.g.i 2 21.h odd 6 2
1014.4.a.g 1 13.b even 2 1
1014.4.b.d 2 13.d odd 4 2
1200.4.a.b 1 20.d odd 2 1
1200.4.f.j 2 20.e even 4 2
1734.4.a.d 1 17.b even 2 1
2166.4.a.i 1 19.b odd 2 1
2178.4.a.e 1 33.d even 2 1
2352.4.a.e 1 28.d even 2 1

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(1\)

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(\Gamma_0(6))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 2 T \)
$3$ \( 1 + 3 T \)
$5$ \( 1 - 6 T + 125 T^{2} \)
$7$ \( 1 + 16 T + 343 T^{2} \)
$11$ \( 1 - 12 T + 1331 T^{2} \)
$13$ \( 1 - 38 T + 2197 T^{2} \)
$17$ \( 1 + 126 T + 4913 T^{2} \)
$19$ \( 1 - 20 T + 6859 T^{2} \)
$23$ \( 1 - 168 T + 12167 T^{2} \)
$29$ \( 1 - 30 T + 24389 T^{2} \)
$31$ \( 1 + 88 T + 29791 T^{2} \)
$37$ \( 1 - 254 T + 50653 T^{2} \)
$41$ \( 1 - 42 T + 68921 T^{2} \)
$43$ \( 1 + 52 T + 79507 T^{2} \)
$47$ \( 1 + 96 T + 103823 T^{2} \)
$53$ \( 1 - 198 T + 148877 T^{2} \)
$59$ \( 1 + 660 T + 205379 T^{2} \)
$61$ \( 1 + 538 T + 226981 T^{2} \)
$67$ \( 1 - 884 T + 300763 T^{2} \)
$71$ \( 1 - 792 T + 357911 T^{2} \)
$73$ \( 1 - 218 T + 389017 T^{2} \)
$79$ \( 1 + 520 T + 493039 T^{2} \)
$83$ \( 1 + 492 T + 571787 T^{2} \)
$89$ \( 1 - 810 T + 704969 T^{2} \)
$97$ \( 1 - 1154 T + 912673 T^{2} \)
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Additional information

eta-product

This cusp form has an eta product $\eta(z)^2\eta(2z)^2\eta(3z)^2\eta(6z)^2=q\prod_{n=1}^\infty (1-q^n)^2(1-q^{2n})^2 (1-q^{3n})^2(1-q^{6n})^2$ where $q=\exp(2\pi i z)$.