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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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [6,12,Mod(1,6)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 12, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("6.1");
 
S:= CuspForms(chi, 12);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6 = 2 \cdot 3 \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 6.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
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

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

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
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 \(S_{12}^{\mathrm{new}}(\Gamma_0(6))\). Copy content Toggle raw display

Hecke characteristic polynomials

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