Properties

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

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Newspace parameters

Level: \( N \) \(=\) \( 144 = 2^{4} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 144.i (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.14984578911\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 36)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - 2 \zeta_{6} + 1) q^{3} - 3 \zeta_{6} q^{5} + (\zeta_{6} - 1) q^{7} - 3 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + ( - 2 \zeta_{6} + 1) q^{3} - 3 \zeta_{6} q^{5} + (\zeta_{6} - 1) q^{7} - 3 q^{9} + ( - 3 \zeta_{6} + 3) q^{11} + \zeta_{6} q^{13} + (3 \zeta_{6} - 6) q^{15} + 6 q^{17} + 4 q^{19} + (\zeta_{6} + 1) q^{21} - 3 \zeta_{6} q^{23} + (4 \zeta_{6} - 4) q^{25} + (6 \zeta_{6} - 3) q^{27} + (3 \zeta_{6} - 3) q^{29} + 5 \zeta_{6} q^{31} + ( - 3 \zeta_{6} - 3) q^{33} + 3 q^{35} + 2 q^{37} + ( - \zeta_{6} + 2) q^{39} - 3 \zeta_{6} q^{41} + (\zeta_{6} - 1) q^{43} + 9 \zeta_{6} q^{45} + (9 \zeta_{6} - 9) q^{47} + 6 \zeta_{6} q^{49} + ( - 12 \zeta_{6} + 6) q^{51} - 6 q^{53} - 9 q^{55} + ( - 8 \zeta_{6} + 4) q^{57} - 3 \zeta_{6} q^{59} + ( - 13 \zeta_{6} + 13) q^{61} + ( - 3 \zeta_{6} + 3) q^{63} + ( - 3 \zeta_{6} + 3) q^{65} - 7 \zeta_{6} q^{67} + (3 \zeta_{6} - 6) q^{69} + 12 q^{71} - 10 q^{73} + (4 \zeta_{6} + 4) q^{75} + 3 \zeta_{6} q^{77} + ( - 11 \zeta_{6} + 11) q^{79} + 9 q^{81} + (9 \zeta_{6} - 9) q^{83} - 18 \zeta_{6} q^{85} + (3 \zeta_{6} + 3) q^{87} + 6 q^{89} - q^{91} + ( - 5 \zeta_{6} + 10) q^{93} - 12 \zeta_{6} q^{95} + (11 \zeta_{6} - 11) q^{97} + (9 \zeta_{6} - 9) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 3 q^{5} - q^{7} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 3 q^{5} - q^{7} - 6 q^{9} + 3 q^{11} + q^{13} - 9 q^{15} + 12 q^{17} + 8 q^{19} + 3 q^{21} - 3 q^{23} - 4 q^{25} - 3 q^{29} + 5 q^{31} - 9 q^{33} + 6 q^{35} + 4 q^{37} + 3 q^{39} - 3 q^{41} - q^{43} + 9 q^{45} - 9 q^{47} + 6 q^{49} - 12 q^{53} - 18 q^{55} - 3 q^{59} + 13 q^{61} + 3 q^{63} + 3 q^{65} - 7 q^{67} - 9 q^{69} + 24 q^{71} - 20 q^{73} + 12 q^{75} + 3 q^{77} + 11 q^{79} + 18 q^{81} - 9 q^{83} - 18 q^{85} + 9 q^{87} + 12 q^{89} - 2 q^{91} + 15 q^{93} - 12 q^{95} - 11 q^{97} - 9 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(37\) \(65\) \(127\)
\(\chi(n)\) \(1\) \(-\zeta_{6}\) \(1\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
0.500000 0.866025i
0.500000 + 0.866025i
0 1.73205i 0 −1.50000 + 2.59808i 0 −0.500000 0.866025i 0 −3.00000 0
97.1 0 1.73205i 0 −1.50000 2.59808i 0 −0.500000 + 0.866025i 0 −3.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 144.2.i.a 2
3.b odd 2 1 432.2.i.c 2
4.b odd 2 1 36.2.e.a 2
8.b even 2 1 576.2.i.e 2
8.d odd 2 1 576.2.i.f 2
9.c even 3 1 inner 144.2.i.a 2
9.c even 3 1 1296.2.a.k 1
9.d odd 6 1 432.2.i.c 2
9.d odd 6 1 1296.2.a.b 1
12.b even 2 1 108.2.e.a 2
20.d odd 2 1 900.2.i.b 2
20.e even 4 2 900.2.s.b 4
24.f even 2 1 1728.2.i.d 2
24.h odd 2 1 1728.2.i.c 2
28.d even 2 1 1764.2.j.b 2
28.f even 6 1 1764.2.i.c 2
28.f even 6 1 1764.2.l.a 2
28.g odd 6 1 1764.2.i.a 2
28.g odd 6 1 1764.2.l.c 2
36.f odd 6 1 36.2.e.a 2
36.f odd 6 1 324.2.a.c 1
36.h even 6 1 108.2.e.a 2
36.h even 6 1 324.2.a.a 1
60.h even 2 1 2700.2.i.b 2
60.l odd 4 2 2700.2.s.b 4
72.j odd 6 1 1728.2.i.c 2
72.j odd 6 1 5184.2.a.bb 1
72.l even 6 1 1728.2.i.d 2
72.l even 6 1 5184.2.a.ba 1
72.n even 6 1 576.2.i.e 2
72.n even 6 1 5184.2.a.f 1
72.p odd 6 1 576.2.i.f 2
72.p odd 6 1 5184.2.a.e 1
84.h odd 2 1 5292.2.j.a 2
84.j odd 6 1 5292.2.i.a 2
84.j odd 6 1 5292.2.l.c 2
84.n even 6 1 5292.2.i.c 2
84.n even 6 1 5292.2.l.a 2
180.n even 6 1 2700.2.i.b 2
180.n even 6 1 8100.2.a.g 1
180.p odd 6 1 900.2.i.b 2
180.p odd 6 1 8100.2.a.j 1
180.v odd 12 2 2700.2.s.b 4
180.v odd 12 2 8100.2.d.c 2
180.x even 12 2 900.2.s.b 4
180.x even 12 2 8100.2.d.h 2
252.n even 6 1 1764.2.i.c 2
252.o even 6 1 5292.2.i.c 2
252.r odd 6 1 5292.2.l.c 2
252.s odd 6 1 5292.2.j.a 2
252.u odd 6 1 1764.2.l.c 2
252.bb even 6 1 5292.2.l.a 2
252.bi even 6 1 1764.2.j.b 2
252.bj even 6 1 1764.2.l.a 2
252.bl odd 6 1 1764.2.i.a 2
252.bn odd 6 1 5292.2.i.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
36.2.e.a 2 4.b odd 2 1
36.2.e.a 2 36.f odd 6 1
108.2.e.a 2 12.b even 2 1
108.2.e.a 2 36.h even 6 1
144.2.i.a 2 1.a even 1 1 trivial
144.2.i.a 2 9.c even 3 1 inner
324.2.a.a 1 36.h even 6 1
324.2.a.c 1 36.f odd 6 1
432.2.i.c 2 3.b odd 2 1
432.2.i.c 2 9.d odd 6 1
576.2.i.e 2 8.b even 2 1
576.2.i.e 2 72.n even 6 1
576.2.i.f 2 8.d odd 2 1
576.2.i.f 2 72.p odd 6 1
900.2.i.b 2 20.d odd 2 1
900.2.i.b 2 180.p odd 6 1
900.2.s.b 4 20.e even 4 2
900.2.s.b 4 180.x even 12 2
1296.2.a.b 1 9.d odd 6 1
1296.2.a.k 1 9.c even 3 1
1728.2.i.c 2 24.h odd 2 1
1728.2.i.c 2 72.j odd 6 1
1728.2.i.d 2 24.f even 2 1
1728.2.i.d 2 72.l even 6 1
1764.2.i.a 2 28.g odd 6 1
1764.2.i.a 2 252.bl odd 6 1
1764.2.i.c 2 28.f even 6 1
1764.2.i.c 2 252.n even 6 1
1764.2.j.b 2 28.d even 2 1
1764.2.j.b 2 252.bi even 6 1
1764.2.l.a 2 28.f even 6 1
1764.2.l.a 2 252.bj even 6 1
1764.2.l.c 2 28.g odd 6 1
1764.2.l.c 2 252.u odd 6 1
2700.2.i.b 2 60.h even 2 1
2700.2.i.b 2 180.n even 6 1
2700.2.s.b 4 60.l odd 4 2
2700.2.s.b 4 180.v odd 12 2
5184.2.a.e 1 72.p odd 6 1
5184.2.a.f 1 72.n even 6 1
5184.2.a.ba 1 72.l even 6 1
5184.2.a.bb 1 72.j odd 6 1
5292.2.i.a 2 84.j odd 6 1
5292.2.i.a 2 252.bn odd 6 1
5292.2.i.c 2 84.n even 6 1
5292.2.i.c 2 252.o even 6 1
5292.2.j.a 2 84.h odd 2 1
5292.2.j.a 2 252.s odd 6 1
5292.2.l.a 2 84.n even 6 1
5292.2.l.a 2 252.bb even 6 1
5292.2.l.c 2 84.j odd 6 1
5292.2.l.c 2 252.r odd 6 1
8100.2.a.g 1 180.n even 6 1
8100.2.a.j 1 180.p odd 6 1
8100.2.d.c 2 180.v odd 12 2
8100.2.d.h 2 180.x even 12 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 3T_{5} + 9 \) acting on \(S_{2}^{\mathrm{new}}(144, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 3 \) Copy content Toggle raw display
$5$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$7$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$11$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$13$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$17$ \( (T - 6)^{2} \) Copy content Toggle raw display
$19$ \( (T - 4)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$29$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$31$ \( T^{2} - 5T + 25 \) Copy content Toggle raw display
$37$ \( (T - 2)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$43$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$47$ \( T^{2} + 9T + 81 \) Copy content Toggle raw display
$53$ \( (T + 6)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$61$ \( T^{2} - 13T + 169 \) Copy content Toggle raw display
$67$ \( T^{2} + 7T + 49 \) Copy content Toggle raw display
$71$ \( (T - 12)^{2} \) Copy content Toggle raw display
$73$ \( (T + 10)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} - 11T + 121 \) Copy content Toggle raw display
$83$ \( T^{2} + 9T + 81 \) Copy content Toggle raw display
$89$ \( (T - 6)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 11T + 121 \) Copy content Toggle raw display
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