Properties

Label 6.12.a.b
Level $6$
Weight $12$
Character orbit 6.a
Self dual yes
Analytic conductor $4.610$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 6 = 2 \cdot 3 \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 6.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(4.61005908336\)
Analytic rank: \(1\)
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

\(f(q)\) \(=\) \( q - 32q^{2} + 243q^{3} + 1024q^{4} - 11730q^{5} - 7776q^{6} - 50008q^{7} - 32768q^{8} + 59049q^{9} + O(q^{10}) \) \( q - 32q^{2} + 243q^{3} + 1024q^{4} - 11730q^{5} - 7776q^{6} - 50008q^{7} - 32768q^{8} + 59049q^{9} + 375360q^{10} - 531420q^{11} + 248832q^{12} + 1332566q^{13} + 1600256q^{14} - 2850390q^{15} + 1048576q^{16} - 5109678q^{17} - 1889568q^{18} + 2901404q^{19} - 12011520q^{20} - 12151944q^{21} + 17005440q^{22} + 30597000q^{23} - 7962624q^{24} + 88764775q^{25} - 42642112q^{26} + 14348907q^{27} - 51208192q^{28} - 77006634q^{29} + 91212480q^{30} - 239418352q^{31} - 33554432q^{32} - 129135060q^{33} + 163509696q^{34} + 586593840q^{35} + 60466176q^{36} - 785041666q^{37} - 92844928q^{38} + 323813538q^{39} + 384368640q^{40} + 411252954q^{41} + 388862208q^{42} + 351233348q^{43} - 544174080q^{44} - 692644770q^{45} - 979104000q^{46} + 95821680q^{47} + 254803968q^{48} + 523473321q^{49} - 2840472800q^{50} - 1241651754q^{51} + 1364547584q^{52} - 1465857378q^{53} - 459165024q^{54} + 6233556600q^{55} + 1638662144q^{56} + 705041172q^{57} + 2464212288q^{58} + 5621152020q^{59} - 2918799360q^{60} - 10473587770q^{61} + 7661387264q^{62} - 2952922392q^{63} + 1073741824q^{64} - 15630999180q^{65} + 4132321920q^{66} + 4515307532q^{67} - 5232310272q^{68} + 7435071000q^{69} - 18771002880q^{70} - 8509579560q^{71} - 1934917632q^{72} + 2012496986q^{73} + 25121333312q^{74} + 21569840325q^{75} + 2971037696q^{76} + 26575251360q^{77} - 10362033216q^{78} - 22238409568q^{79} - 12299796480q^{80} + 3486784401q^{81} - 13160094528q^{82} + 6328647516q^{83} - 12443590656q^{84} + 59936522940q^{85} - 11239467136q^{86} - 18712612062q^{87} + 17413570560q^{88} - 50123706678q^{89} + 22164632640q^{90} - 66638960528q^{91} + 31331328000q^{92} - 58178659536q^{93} - 3066293760q^{94} - 34033468920q^{95} - 8153726976q^{96} + 94805961314q^{97} - 16751146272q^{98} - 31379819580q^{99} + O(q^{100}) \)

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
−32.0000 243.000 1024.00 −11730.0 −7776.00 −50008.0 −32768.0 59049.0 375360.
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 6.12.a.b 1
3.b odd 2 1 18.12.a.e 1
4.b odd 2 1 48.12.a.a 1
5.b even 2 1 150.12.a.f 1
5.c odd 4 2 150.12.c.b 2
8.b even 2 1 192.12.a.j 1
8.d odd 2 1 192.12.a.t 1
9.c even 3 2 162.12.c.j 2
9.d odd 6 2 162.12.c.a 2
12.b even 2 1 144.12.a.o 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6.12.a.b 1 1.a even 1 1 trivial
18.12.a.e 1 3.b odd 2 1
48.12.a.a 1 4.b odd 2 1
144.12.a.o 1 12.b even 2 1
150.12.a.f 1 5.b even 2 1
150.12.c.b 2 5.c odd 4 2
162.12.c.a 2 9.d odd 6 2
162.12.c.j 2 9.c even 3 2
192.12.a.j 1 8.b even 2 1
192.12.a.t 1 8.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} + 11730 \) acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(6))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 32 + T \)
$3$ \( -243 + T \)
$5$ \( 11730 + T \)
$7$ \( 50008 + T \)
$11$ \( 531420 + T \)
$13$ \( -1332566 + T \)
$17$ \( 5109678 + T \)
$19$ \( -2901404 + T \)
$23$ \( -30597000 + T \)
$29$ \( 77006634 + T \)
$31$ \( 239418352 + T \)
$37$ \( 785041666 + T \)
$41$ \( -411252954 + T \)
$43$ \( -351233348 + T \)
$47$ \( -95821680 + T \)
$53$ \( 1465857378 + T \)
$59$ \( -5621152020 + T \)
$61$ \( 10473587770 + T \)
$67$ \( -4515307532 + T \)
$71$ \( 8509579560 + T \)
$73$ \( -2012496986 + T \)
$79$ \( 22238409568 + T \)
$83$ \( -6328647516 + T \)
$89$ \( 50123706678 + T \)
$97$ \( -94805961314 + T \)
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