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: \(x^{2} - x + 1\)
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.

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

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 \( T_{5}^{2} + 3 T_{5} + 9 \) acting on \(S_{2}^{\mathrm{new}}(144, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 + 3 T^{2} \)
$5$ \( 1 + 3 T + 4 T^{2} + 15 T^{3} + 25 T^{4} \)
$7$ \( ( 1 - 4 T + 7 T^{2} )( 1 + 5 T + 7 T^{2} ) \)
$11$ \( 1 - 3 T - 2 T^{2} - 33 T^{3} + 121 T^{4} \)
$13$ \( 1 - T - 12 T^{2} - 13 T^{3} + 169 T^{4} \)
$17$ \( ( 1 - 6 T + 17 T^{2} )^{2} \)
$19$ \( ( 1 - 4 T + 19 T^{2} )^{2} \)
$23$ \( 1 + 3 T - 14 T^{2} + 69 T^{3} + 529 T^{4} \)
$29$ \( 1 + 3 T - 20 T^{2} + 87 T^{3} + 841 T^{4} \)
$31$ \( 1 - 5 T - 6 T^{2} - 155 T^{3} + 961 T^{4} \)
$37$ \( ( 1 - 2 T + 37 T^{2} )^{2} \)
$41$ \( 1 + 3 T - 32 T^{2} + 123 T^{3} + 1681 T^{4} \)
$43$ \( 1 + T - 42 T^{2} + 43 T^{3} + 1849 T^{4} \)
$47$ \( 1 + 9 T + 34 T^{2} + 423 T^{3} + 2209 T^{4} \)
$53$ \( ( 1 + 6 T + 53 T^{2} )^{2} \)
$59$ \( 1 + 3 T - 50 T^{2} + 177 T^{3} + 3481 T^{4} \)
$61$ \( ( 1 - 14 T + 61 T^{2} )( 1 + T + 61 T^{2} ) \)
$67$ \( 1 + 7 T - 18 T^{2} + 469 T^{3} + 4489 T^{4} \)
$71$ \( ( 1 - 12 T + 71 T^{2} )^{2} \)
$73$ \( ( 1 + 10 T + 73 T^{2} )^{2} \)
$79$ \( 1 - 11 T + 42 T^{2} - 869 T^{3} + 6241 T^{4} \)
$83$ \( 1 + 9 T - 2 T^{2} + 747 T^{3} + 6889 T^{4} \)
$89$ \( ( 1 - 6 T + 89 T^{2} )^{2} \)
$97$ \( 1 + 11 T + 24 T^{2} + 1067 T^{3} + 9409 T^{4} \)
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