Properties

Label 144.2.x.a
Level $144$
Weight $2$
Character orbit 144.x
Analytic conductor $1.150$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [144,2,Mod(13,144)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(144, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([0, 9, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("144.13");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 144 = 2^{4} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 144.x (of order \(12\), degree \(4\), minimal)

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: no
Analytic conductor: \(1.14984578911\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \zeta_{12}^{3} - 1) q^{2} + ( - \zeta_{12}^{3} + 2 \zeta_{12}) q^{3} + 2 \zeta_{12}^{3} q^{4} + (\zeta_{12}^{3} - \zeta_{12} - 1) q^{5} + (\zeta_{12}^{3} - 2 \zeta_{12}^{2} - 2 \zeta_{12} + 1) q^{6} + ( - \zeta_{12}^{3} - 2 \zeta_{12}^{2} + \zeta_{12} + 4) q^{7} + ( - 2 \zeta_{12}^{3} + 2) q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{12}^{3} - 1) q^{2} + ( - \zeta_{12}^{3} + 2 \zeta_{12}) q^{3} + 2 \zeta_{12}^{3} q^{4} + (\zeta_{12}^{3} - \zeta_{12} - 1) q^{5} + (\zeta_{12}^{3} - 2 \zeta_{12}^{2} - 2 \zeta_{12} + 1) q^{6} + ( - \zeta_{12}^{3} - 2 \zeta_{12}^{2} + \zeta_{12} + 4) q^{7} + ( - 2 \zeta_{12}^{3} + 2) q^{8} + 3 q^{9} + (\zeta_{12}^{2} + \zeta_{12} + 1) q^{10} + (\zeta_{12}^{3} + \zeta_{12}^{2} - 1) q^{11} + (4 \zeta_{12}^{2} - 2) q^{12} + (\zeta_{12}^{3} - 2 \zeta_{12}^{2} - 3 \zeta_{12} + 3) q^{13} + ( - \zeta_{12}^{3} + \zeta_{12}^{2} - 3 \zeta_{12} - 4) q^{14} + (\zeta_{12}^{3} + \zeta_{12}^{2} - 2 \zeta_{12} - 2) q^{15} - 4 q^{16} + 4 q^{17} + ( - 3 \zeta_{12}^{3} - 3) q^{18} + (3 \zeta_{12}^{3} - 3) q^{19} + ( - 2 \zeta_{12}^{3} - 2 \zeta_{12}^{2}) q^{20} + ( - 6 \zeta_{12}^{3} - \zeta_{12}^{2} + 6 \zeta_{12} + 2) q^{21} + ( - \zeta_{12}^{3} - \zeta_{12}^{2} + \zeta_{12} + 2) q^{22} + ( - 2 \zeta_{12}^{2} - 3 \zeta_{12} - 2) q^{23} + ( - 2 \zeta_{12}^{3} - 4 \zeta_{12}^{2} + 4 \zeta_{12} + 2) q^{24} + (3 \zeta_{12}^{3} - \zeta_{12}^{2} - 3 \zeta_{12} + 2) q^{25} + ( - 2 \zeta_{12}^{3} + 5 \zeta_{12}^{2} + \zeta_{12} - 5) q^{26} + ( - 3 \zeta_{12}^{3} + 6 \zeta_{12}) q^{27} + (4 \zeta_{12}^{3} + 2 \zeta_{12}^{2} + 4 \zeta_{12}) q^{28} + (2 \zeta_{12}^{3} - 2 \zeta_{12}^{2} - \zeta_{12} + 1) q^{29} + (\zeta_{12}^{2} + 3 \zeta_{12} + 1) q^{30} + (6 \zeta_{12}^{3} + 4 \zeta_{12}^{2} - 3 \zeta_{12} - 4) q^{31} + (4 \zeta_{12}^{3} + 4) q^{32} + (2 \zeta_{12}^{3} + 2 \zeta_{12}^{2} - \zeta_{12} - 1) q^{33} + ( - 4 \zeta_{12}^{3} - 4) q^{34} + (5 \zeta_{12}^{3} + 3 \zeta_{12}^{2} - 3 \zeta_{12} - 5) q^{35} + 6 \zeta_{12}^{3} q^{36} + ( - 7 \zeta_{12}^{3} + 2 \zeta_{12}^{2} + 2 \zeta_{12} - 7) q^{37} + 6 q^{38} + ( - 5 \zeta_{12}^{3} - \zeta_{12}^{2} + 4 \zeta_{12} - 4) q^{39} + (4 \zeta_{12}^{3} + 2 \zeta_{12}^{2} - 2 \zeta_{12} - 2) q^{40} + ( - 3 \zeta_{12}^{2} + 6 \zeta_{12} - 3) q^{41} + (5 \zeta_{12}^{3} - 5 \zeta_{12}^{2} - 7 \zeta_{12} - 2) q^{42} + (5 \zeta_{12}^{3} + 5 \zeta_{12}^{2} - 6 \zeta_{12} + 1) q^{43} + ( - 2 \zeta_{12} - 2) q^{44} + (3 \zeta_{12}^{3} - 3 \zeta_{12} - 3) q^{45} + (4 \zeta_{12}^{3} + 5 \zeta_{12}^{2} + \zeta_{12} - 1) q^{46} + ( - 3 \zeta_{12}^{3} + 4 \zeta_{12}^{2} - 3 \zeta_{12}) q^{47} + (4 \zeta_{12}^{3} - 8 \zeta_{12}) q^{48} + ( - 8 \zeta_{12}^{3} - 6 \zeta_{12}^{2} + 4 \zeta_{12} + 6) q^{49} + ( - 4 \zeta_{12}^{3} + 4 \zeta_{12}^{2} + 2 \zeta_{12} - 2) q^{50} + ( - 4 \zeta_{12}^{3} + 8 \zeta_{12}) q^{51} + (2 \zeta_{12}^{3} - 6 \zeta_{12}^{2} + 4 \zeta_{12} + 4) q^{52} + (3 \zeta_{12}^{3} + 2 \zeta_{12}^{2} + 2 \zeta_{12} + 3) q^{53} + (3 \zeta_{12}^{3} - 6 \zeta_{12}^{2} - 6 \zeta_{12} + 3) q^{54} + ( - 2 \zeta_{12}^{3} - 2 \zeta_{12}^{2} + 1) q^{55} + ( - 6 \zeta_{12}^{3} - 6 \zeta_{12}^{2} - 2 \zeta_{12} + 8) q^{56} + (3 \zeta_{12}^{3} + 6 \zeta_{12}^{2} - 6 \zeta_{12} - 3) q^{57} + ( - \zeta_{12}^{3} + 3 \zeta_{12}^{2} - \zeta_{12}) q^{58} + (3 \zeta_{12}^{3} - 3 \zeta_{12}^{2}) q^{59} + ( - 2 \zeta_{12}^{3} - 4 \zeta_{12}^{2} - 2 \zeta_{12} + 2) q^{60} + ( - 8 \zeta_{12}^{3} + 8 \zeta_{12}^{2} + \zeta_{12} - 7) q^{61} + ( - 6 \zeta_{12}^{3} - \zeta_{12}^{2} + 7 \zeta_{12} + 7) q^{62} + ( - 3 \zeta_{12}^{3} - 6 \zeta_{12}^{2} + 3 \zeta_{12} + 12) q^{63} - 8 \zeta_{12}^{3} q^{64} + (2 \zeta_{12}^{3} + \zeta_{12}^{2} + 2 \zeta_{12}) q^{65} + ( - 3 \zeta_{12}^{3} - \zeta_{12}^{2} + 3 \zeta_{12} + 2) q^{66} + ( - 3 \zeta_{12}^{3} - 5 \zeta_{12}^{2} - 2 \zeta_{12} + 2) q^{67} + 8 \zeta_{12}^{3} q^{68} + ( - 3 \zeta_{12}^{2} - 6 \zeta_{12} - 3) q^{69} + ( - 3 \zeta_{12}^{3} + 6 \zeta_{12} + 7) q^{70} + ( - 4 \zeta_{12}^{3} + 8 \zeta_{12}^{2} - 4) q^{71} + ( - 6 \zeta_{12}^{3} + 6) q^{72} + ( - 4 \zeta_{12}^{3} - 4 \zeta_{12}^{2} + 2) q^{73} + (12 \zeta_{12}^{3} - 4 \zeta_{12}^{2} + 2) q^{74} + ( - 3 \zeta_{12}^{3} + 3 \zeta_{12}^{2} + 3 \zeta_{12} - 6) q^{75} + ( - 6 \zeta_{12}^{3} - 6) q^{76} + (3 \zeta_{12}^{3} + 5 \zeta_{12}^{2} + 2 \zeta_{12} - 2) q^{77} + (10 \zeta_{12}^{3} - 3 \zeta_{12}^{2} - 5 \zeta_{12} + 3) q^{78} + ( - \zeta_{12}^{3} - \zeta_{12}) q^{79} + ( - 4 \zeta_{12}^{3} + 4 \zeta_{12} + 4) q^{80} + 9 q^{81} + (6 \zeta_{12}^{3} - 3 \zeta_{12}^{2} - 9 \zeta_{12} + 9) q^{82} + (9 \zeta_{12}^{3} - 9 \zeta_{12}^{2} - 10 \zeta_{12} - 1) q^{83} + (2 \zeta_{12}^{3} + 12 \zeta_{12}^{2} + 2 \zeta_{12}) q^{84} + (4 \zeta_{12}^{3} - 4 \zeta_{12} - 4) q^{85} + ( - 11 \zeta_{12}^{3} + \zeta_{12}^{2} + 11 \zeta_{12} - 2) q^{86} + ( - 3 \zeta_{12}^{3} + 3 \zeta_{12}^{2} - 3) q^{87} + (2 \zeta_{12}^{3} + 2 \zeta_{12}^{2} + 2 \zeta_{12}) q^{88} + ( - 2 \zeta_{12}^{3} - 16 \zeta_{12}^{2} + 8) q^{89} + (3 \zeta_{12}^{2} + 3 \zeta_{12} + 3) q^{90} + (5 \zeta_{12}^{3} - 9 \zeta_{12}^{2} - 9 \zeta_{12} + 5) q^{91} + ( - 8 \zeta_{12}^{3} - 6 \zeta_{12}^{2} + 4 \zeta_{12} + 6) q^{92} + (8 \zeta_{12}^{3} + 9 \zeta_{12}^{2} - 4 \zeta_{12} - 9) q^{93} + ( - \zeta_{12}^{3} - \zeta_{12}^{2} + 7 \zeta_{12} - 6) q^{94} + ( - 6 \zeta_{12}^{3} - 3 \zeta_{12}^{2} + 3 \zeta_{12} + 3) q^{95} + ( - 4 \zeta_{12}^{3} + 8 \zeta_{12}^{2} + 8 \zeta_{12} - 4) q^{96} - \zeta_{12}^{2} q^{97} + (8 \zeta_{12}^{3} + 2 \zeta_{12}^{2} - 10 \zeta_{12} - 10) q^{98} + (3 \zeta_{12}^{3} + 3 \zeta_{12}^{2} - 3) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{2} - 4 q^{5} + 12 q^{7} + 8 q^{8} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{2} - 4 q^{5} + 12 q^{7} + 8 q^{8} + 12 q^{9} + 6 q^{10} - 2 q^{11} + 8 q^{13} - 14 q^{14} - 6 q^{15} - 16 q^{16} + 16 q^{17} - 12 q^{18} - 12 q^{19} - 4 q^{20} + 6 q^{21} + 6 q^{22} - 12 q^{23} + 6 q^{25} - 10 q^{26} + 4 q^{28} + 6 q^{30} - 8 q^{31} + 16 q^{32} - 16 q^{34} - 14 q^{35} - 24 q^{37} + 24 q^{38} - 18 q^{39} - 4 q^{40} - 18 q^{41} - 18 q^{42} + 14 q^{43} - 8 q^{44} - 12 q^{45} + 6 q^{46} + 8 q^{47} + 12 q^{49} + 4 q^{52} + 16 q^{53} + 20 q^{56} + 6 q^{58} - 6 q^{59} - 12 q^{61} + 26 q^{62} + 36 q^{63} + 2 q^{65} + 6 q^{66} - 2 q^{67} - 18 q^{69} + 28 q^{70} + 24 q^{72} - 18 q^{75} - 24 q^{76} + 2 q^{77} + 6 q^{78} + 16 q^{80} + 36 q^{81} + 30 q^{82} - 22 q^{83} + 24 q^{84} - 16 q^{85} - 6 q^{86} - 6 q^{87} + 4 q^{88} + 18 q^{90} + 2 q^{91} + 12 q^{92} - 18 q^{93} - 26 q^{94} + 6 q^{95} - 2 q^{97} - 36 q^{98} - 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/144\mathbb{Z}\right)^\times\).

\(n\) \(37\) \(65\) \(127\)
\(\chi(n)\) \(\zeta_{12}^{3}\) \(-1 + \zeta_{12}^{2}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
13.1
−0.866025 0.500000i
0.866025 0.500000i
0.866025 + 0.500000i
−0.866025 + 0.500000i
−1.00000 + 1.00000i −1.73205 2.00000i −0.133975 0.500000i 1.73205 1.73205i 2.13397 1.23205i 2.00000 + 2.00000i 3.00000 0.633975 + 0.366025i
61.1 −1.00000 + 1.00000i 1.73205 2.00000i −1.86603 0.500000i −1.73205 + 1.73205i 3.86603 + 2.23205i 2.00000 + 2.00000i 3.00000 2.36603 1.36603i
85.1 −1.00000 1.00000i 1.73205 2.00000i −1.86603 + 0.500000i −1.73205 1.73205i 3.86603 2.23205i 2.00000 2.00000i 3.00000 2.36603 + 1.36603i
133.1 −1.00000 1.00000i −1.73205 2.00000i −0.133975 + 0.500000i 1.73205 + 1.73205i 2.13397 + 1.23205i 2.00000 2.00000i 3.00000 0.633975 0.366025i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
144.x even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 144.2.x.a 4
3.b odd 2 1 432.2.y.d 4
4.b odd 2 1 576.2.bb.a 4
9.c even 3 1 144.2.x.d yes 4
9.d odd 6 1 432.2.y.a 4
12.b even 2 1 1728.2.bc.c 4
16.e even 4 1 144.2.x.d yes 4
16.f odd 4 1 576.2.bb.b 4
36.f odd 6 1 576.2.bb.b 4
36.h even 6 1 1728.2.bc.b 4
48.i odd 4 1 432.2.y.a 4
48.k even 4 1 1728.2.bc.b 4
144.u even 12 1 1728.2.bc.c 4
144.v odd 12 1 576.2.bb.a 4
144.w odd 12 1 432.2.y.d 4
144.x even 12 1 inner 144.2.x.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
144.2.x.a 4 1.a even 1 1 trivial
144.2.x.a 4 144.x even 12 1 inner
144.2.x.d yes 4 9.c even 3 1
144.2.x.d yes 4 16.e even 4 1
432.2.y.a 4 9.d odd 6 1
432.2.y.a 4 48.i odd 4 1
432.2.y.d 4 3.b odd 2 1
432.2.y.d 4 144.w odd 12 1
576.2.bb.a 4 4.b odd 2 1
576.2.bb.a 4 144.v odd 12 1
576.2.bb.b 4 16.f odd 4 1
576.2.bb.b 4 36.f odd 6 1
1728.2.bc.b 4 36.h even 6 1
1728.2.bc.b 4 48.k even 4 1
1728.2.bc.c 4 12.b even 2 1
1728.2.bc.c 4 144.u even 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} + 4T_{5}^{3} + 5T_{5}^{2} + 2T_{5} + 1 \) acting on \(S_{2}^{\mathrm{new}}(144, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 2 T + 2)^{2} \) Copy content Toggle raw display
$3$ \( (T^{2} - 3)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} + 4 T^{3} + 5 T^{2} + 2 T + 1 \) Copy content Toggle raw display
$7$ \( T^{4} - 12 T^{3} + 59 T^{2} + \cdots + 121 \) Copy content Toggle raw display
$11$ \( T^{4} + 2 T^{3} + 5 T^{2} + 4 T + 1 \) Copy content Toggle raw display
$13$ \( T^{4} - 8 T^{3} + 17 T^{2} - 22 T + 121 \) Copy content Toggle raw display
$17$ \( (T - 4)^{4} \) Copy content Toggle raw display
$19$ \( (T^{2} + 6 T + 18)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 12 T^{3} + 51 T^{2} + 36 T + 9 \) Copy content Toggle raw display
$29$ \( T^{4} + 9 T^{2} + 18 T + 9 \) Copy content Toggle raw display
$31$ \( T^{4} + 8 T^{3} + 75 T^{2} - 88 T + 121 \) Copy content Toggle raw display
$37$ \( T^{4} + 24 T^{3} + 288 T^{2} + \cdots + 4356 \) Copy content Toggle raw display
$41$ \( T^{4} + 18 T^{3} + 99 T^{2} - 162 T + 81 \) Copy content Toggle raw display
$43$ \( T^{4} - 14 T^{3} + 65 T^{2} + \cdots + 3481 \) Copy content Toggle raw display
$47$ \( T^{4} - 8 T^{3} + 75 T^{2} + 88 T + 121 \) Copy content Toggle raw display
$53$ \( T^{4} - 16 T^{3} + 128 T^{2} + \cdots + 676 \) Copy content Toggle raw display
$59$ \( T^{4} + 6 T^{3} + 45 T^{2} + 108 T + 81 \) Copy content Toggle raw display
$61$ \( T^{4} + 12 T^{3} + 261 T^{2} + \cdots + 1089 \) Copy content Toggle raw display
$67$ \( T^{4} + 2 T^{3} + 65 T^{2} - 176 T + 121 \) Copy content Toggle raw display
$71$ \( T^{4} + 128T^{2} + 1024 \) Copy content Toggle raw display
$73$ \( T^{4} + 56T^{2} + 16 \) Copy content Toggle raw display
$79$ \( T^{4} + 3T^{2} + 9 \) Copy content Toggle raw display
$83$ \( T^{4} + 22 T^{3} + 185 T^{2} + \cdots + 32041 \) Copy content Toggle raw display
$89$ \( T^{4} + 392 T^{2} + 35344 \) Copy content Toggle raw display
$97$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
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