Properties

Label 6.6.a.a
Level 6
Weight 6
Character orbit 6.a
Self dual yes
Analytic conductor 0.962
Analytic rank 0
Dimension 1
CM no
Inner twists 1

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Newspace parameters

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

Newform invariants

Self dual: yes
Analytic conductor: \(0.962302918878\)
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

\(f(q)\) \(=\) \( q + 4q^{2} - 9q^{3} + 16q^{4} - 66q^{5} - 36q^{6} + 176q^{7} + 64q^{8} + 81q^{9} + O(q^{10}) \) \( q + 4q^{2} - 9q^{3} + 16q^{4} - 66q^{5} - 36q^{6} + 176q^{7} + 64q^{8} + 81q^{9} - 264q^{10} - 60q^{11} - 144q^{12} - 658q^{13} + 704q^{14} + 594q^{15} + 256q^{16} - 414q^{17} + 324q^{18} + 956q^{19} - 1056q^{20} - 1584q^{21} - 240q^{22} + 600q^{23} - 576q^{24} + 1231q^{25} - 2632q^{26} - 729q^{27} + 2816q^{28} + 5574q^{29} + 2376q^{30} - 3592q^{31} + 1024q^{32} + 540q^{33} - 1656q^{34} - 11616q^{35} + 1296q^{36} - 8458q^{37} + 3824q^{38} + 5922q^{39} - 4224q^{40} + 19194q^{41} - 6336q^{42} + 13316q^{43} - 960q^{44} - 5346q^{45} + 2400q^{46} - 19680q^{47} - 2304q^{48} + 14169q^{49} + 4924q^{50} + 3726q^{51} - 10528q^{52} - 31266q^{53} - 2916q^{54} + 3960q^{55} + 11264q^{56} - 8604q^{57} + 22296q^{58} + 26340q^{59} + 9504q^{60} - 31090q^{61} - 14368q^{62} + 14256q^{63} + 4096q^{64} + 43428q^{65} + 2160q^{66} - 16804q^{67} - 6624q^{68} - 5400q^{69} - 46464q^{70} + 6120q^{71} + 5184q^{72} - 25558q^{73} - 33832q^{74} - 11079q^{75} + 15296q^{76} - 10560q^{77} + 23688q^{78} + 74408q^{79} - 16896q^{80} + 6561q^{81} + 76776q^{82} - 6468q^{83} - 25344q^{84} + 27324q^{85} + 53264q^{86} - 50166q^{87} - 3840q^{88} - 32742q^{89} - 21384q^{90} - 115808q^{91} + 9600q^{92} + 32328q^{93} - 78720q^{94} - 63096q^{95} - 9216q^{96} + 166082q^{97} + 56676q^{98} - 4860q^{99} + O(q^{100}) \)

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
4.00000 −9.00000 16.0000 −66.0000 −36.0000 176.000 64.0000 81.0000 −264.000
\(n\): 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 6.6.a.a 1
3.b odd 2 1 18.6.a.b 1
4.b odd 2 1 48.6.a.c 1
5.b even 2 1 150.6.a.d 1
5.c odd 4 2 150.6.c.b 2
7.b odd 2 1 294.6.a.m 1
7.c even 3 2 294.6.e.g 2
7.d odd 6 2 294.6.e.a 2
8.b even 2 1 192.6.a.o 1
8.d odd 2 1 192.6.a.g 1
9.c even 3 2 162.6.c.e 2
9.d odd 6 2 162.6.c.h 2
11.b odd 2 1 726.6.a.a 1
12.b even 2 1 144.6.a.j 1
13.b even 2 1 1014.6.a.c 1
15.d odd 2 1 450.6.a.m 1
15.e even 4 2 450.6.c.j 2
16.e even 4 2 768.6.d.c 2
16.f odd 4 2 768.6.d.p 2
21.c even 2 1 882.6.a.a 1
24.f even 2 1 576.6.a.i 1
24.h odd 2 1 576.6.a.j 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6.6.a.a 1 1.a even 1 1 trivial
18.6.a.b 1 3.b odd 2 1
48.6.a.c 1 4.b odd 2 1
144.6.a.j 1 12.b even 2 1
150.6.a.d 1 5.b even 2 1
150.6.c.b 2 5.c odd 4 2
162.6.c.e 2 9.c even 3 2
162.6.c.h 2 9.d odd 6 2
192.6.a.g 1 8.d odd 2 1
192.6.a.o 1 8.b even 2 1
294.6.a.m 1 7.b odd 2 1
294.6.e.a 2 7.d odd 6 2
294.6.e.g 2 7.c even 3 2
450.6.a.m 1 15.d odd 2 1
450.6.c.j 2 15.e even 4 2
576.6.a.i 1 24.f even 2 1
576.6.a.j 1 24.h odd 2 1
726.6.a.a 1 11.b odd 2 1
768.6.d.c 2 16.e even 4 2
768.6.d.p 2 16.f odd 4 2
882.6.a.a 1 21.c even 2 1
1014.6.a.c 1 13.b even 2 1

Atkin-Lehner signs

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

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 4 T \)
$3$ \( 1 + 9 T \)
$5$ \( 1 + 66 T + 3125 T^{2} \)
$7$ \( 1 - 176 T + 16807 T^{2} \)
$11$ \( 1 + 60 T + 161051 T^{2} \)
$13$ \( 1 + 658 T + 371293 T^{2} \)
$17$ \( 1 + 414 T + 1419857 T^{2} \)
$19$ \( 1 - 956 T + 2476099 T^{2} \)
$23$ \( 1 - 600 T + 6436343 T^{2} \)
$29$ \( 1 - 5574 T + 20511149 T^{2} \)
$31$ \( 1 + 3592 T + 28629151 T^{2} \)
$37$ \( 1 + 8458 T + 69343957 T^{2} \)
$41$ \( 1 - 19194 T + 115856201 T^{2} \)
$43$ \( 1 - 13316 T + 147008443 T^{2} \)
$47$ \( 1 + 19680 T + 229345007 T^{2} \)
$53$ \( 1 + 31266 T + 418195493 T^{2} \)
$59$ \( 1 - 26340 T + 714924299 T^{2} \)
$61$ \( 1 + 31090 T + 844596301 T^{2} \)
$67$ \( 1 + 16804 T + 1350125107 T^{2} \)
$71$ \( 1 - 6120 T + 1804229351 T^{2} \)
$73$ \( 1 + 25558 T + 2073071593 T^{2} \)
$79$ \( 1 - 74408 T + 3077056399 T^{2} \)
$83$ \( 1 + 6468 T + 3939040643 T^{2} \)
$89$ \( 1 + 32742 T + 5584059449 T^{2} \)
$97$ \( 1 - 166082 T + 8587340257 T^{2} \)
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