Properties

Label 144.2.i.c
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 18)
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.

\(f(q)\) \(=\) \( q + (\zeta_{6} + 1) q^{3} + ( - 2 \zeta_{6} + 2) q^{7} + 3 \zeta_{6} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{6} + 1) q^{3} + ( - 2 \zeta_{6} + 2) q^{7} + 3 \zeta_{6} q^{9} + (3 \zeta_{6} - 3) q^{11} - 2 \zeta_{6} q^{13} - 3 q^{17} + q^{19} + ( - 2 \zeta_{6} + 4) q^{21} - 6 \zeta_{6} q^{23} + ( - 5 \zeta_{6} + 5) q^{25} + (6 \zeta_{6} - 3) q^{27} + (6 \zeta_{6} - 6) q^{29} - 4 \zeta_{6} q^{31} + (3 \zeta_{6} - 6) q^{33} - 4 q^{37} + ( - 4 \zeta_{6} + 2) q^{39} - 9 \zeta_{6} q^{41} + (\zeta_{6} - 1) q^{43} + (6 \zeta_{6} - 6) q^{47} + 3 \zeta_{6} q^{49} + ( - 3 \zeta_{6} - 3) q^{51} + 12 q^{53} + (\zeta_{6} + 1) q^{57} + 3 \zeta_{6} q^{59} + (8 \zeta_{6} - 8) q^{61} + 6 q^{63} + 5 \zeta_{6} q^{67} + ( - 12 \zeta_{6} + 6) q^{69} + 12 q^{71} + 11 q^{73} + ( - 5 \zeta_{6} + 10) q^{75} + 6 \zeta_{6} q^{77} + (4 \zeta_{6} - 4) q^{79} + (9 \zeta_{6} - 9) q^{81} + ( - 12 \zeta_{6} + 12) q^{83} + (6 \zeta_{6} - 12) q^{87} + 6 q^{89} - 4 q^{91} + ( - 8 \zeta_{6} + 4) q^{93} + (5 \zeta_{6} - 5) q^{97} - 9 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 3 q^{3} + 2 q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 3 q^{3} + 2 q^{7} + 3 q^{9} - 3 q^{11} - 2 q^{13} - 6 q^{17} + 2 q^{19} + 6 q^{21} - 6 q^{23} + 5 q^{25} - 6 q^{29} - 4 q^{31} - 9 q^{33} - 8 q^{37} - 9 q^{41} - q^{43} - 6 q^{47} + 3 q^{49} - 9 q^{51} + 24 q^{53} + 3 q^{57} + 3 q^{59} - 8 q^{61} + 12 q^{63} + 5 q^{67} + 24 q^{71} + 22 q^{73} + 15 q^{75} + 6 q^{77} - 4 q^{79} - 9 q^{81} + 12 q^{83} - 18 q^{87} + 12 q^{89} - 8 q^{91} - 5 q^{97} - 18 q^{99}+O(q^{100}) \) 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 1.50000 0.866025i 0 0 0 1.00000 + 1.73205i 0 1.50000 2.59808i 0
97.1 0 1.50000 + 0.866025i 0 0 0 1.00000 1.73205i 0 1.50000 + 2.59808i 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.c 2
3.b odd 2 1 432.2.i.b 2
4.b odd 2 1 18.2.c.a 2
8.b even 2 1 576.2.i.a 2
8.d odd 2 1 576.2.i.g 2
9.c even 3 1 inner 144.2.i.c 2
9.c even 3 1 1296.2.a.g 1
9.d odd 6 1 432.2.i.b 2
9.d odd 6 1 1296.2.a.f 1
12.b even 2 1 54.2.c.a 2
20.d odd 2 1 450.2.e.i 2
20.e even 4 2 450.2.j.e 4
24.f even 2 1 1728.2.i.e 2
24.h odd 2 1 1728.2.i.f 2
28.d even 2 1 882.2.f.d 2
28.f even 6 1 882.2.e.g 2
28.f even 6 1 882.2.h.b 2
28.g odd 6 1 882.2.e.i 2
28.g odd 6 1 882.2.h.c 2
36.f odd 6 1 18.2.c.a 2
36.f odd 6 1 162.2.a.c 1
36.h even 6 1 54.2.c.a 2
36.h even 6 1 162.2.a.b 1
60.h even 2 1 1350.2.e.c 2
60.l odd 4 2 1350.2.j.a 4
72.j odd 6 1 1728.2.i.f 2
72.j odd 6 1 5184.2.a.p 1
72.l even 6 1 1728.2.i.e 2
72.l even 6 1 5184.2.a.q 1
72.n even 6 1 576.2.i.a 2
72.n even 6 1 5184.2.a.o 1
72.p odd 6 1 576.2.i.g 2
72.p odd 6 1 5184.2.a.r 1
84.h odd 2 1 2646.2.f.g 2
84.j odd 6 1 2646.2.e.c 2
84.j odd 6 1 2646.2.h.i 2
84.n even 6 1 2646.2.e.b 2
84.n even 6 1 2646.2.h.h 2
180.n even 6 1 1350.2.e.c 2
180.n even 6 1 4050.2.a.v 1
180.p odd 6 1 450.2.e.i 2
180.p odd 6 1 4050.2.a.c 1
180.v odd 12 2 1350.2.j.a 4
180.v odd 12 2 4050.2.c.r 2
180.x even 12 2 450.2.j.e 4
180.x even 12 2 4050.2.c.c 2
252.n even 6 1 882.2.e.g 2
252.o even 6 1 2646.2.e.b 2
252.r odd 6 1 2646.2.h.i 2
252.s odd 6 1 2646.2.f.g 2
252.s odd 6 1 7938.2.a.i 1
252.u odd 6 1 882.2.h.c 2
252.bb even 6 1 2646.2.h.h 2
252.bi even 6 1 882.2.f.d 2
252.bi even 6 1 7938.2.a.x 1
252.bj even 6 1 882.2.h.b 2
252.bl odd 6 1 882.2.e.i 2
252.bn odd 6 1 2646.2.e.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
18.2.c.a 2 4.b odd 2 1
18.2.c.a 2 36.f odd 6 1
54.2.c.a 2 12.b even 2 1
54.2.c.a 2 36.h even 6 1
144.2.i.c 2 1.a even 1 1 trivial
144.2.i.c 2 9.c even 3 1 inner
162.2.a.b 1 36.h even 6 1
162.2.a.c 1 36.f odd 6 1
432.2.i.b 2 3.b odd 2 1
432.2.i.b 2 9.d odd 6 1
450.2.e.i 2 20.d odd 2 1
450.2.e.i 2 180.p odd 6 1
450.2.j.e 4 20.e even 4 2
450.2.j.e 4 180.x even 12 2
576.2.i.a 2 8.b even 2 1
576.2.i.a 2 72.n even 6 1
576.2.i.g 2 8.d odd 2 1
576.2.i.g 2 72.p odd 6 1
882.2.e.g 2 28.f even 6 1
882.2.e.g 2 252.n even 6 1
882.2.e.i 2 28.g odd 6 1
882.2.e.i 2 252.bl odd 6 1
882.2.f.d 2 28.d even 2 1
882.2.f.d 2 252.bi even 6 1
882.2.h.b 2 28.f even 6 1
882.2.h.b 2 252.bj even 6 1
882.2.h.c 2 28.g odd 6 1
882.2.h.c 2 252.u odd 6 1
1296.2.a.f 1 9.d odd 6 1
1296.2.a.g 1 9.c even 3 1
1350.2.e.c 2 60.h even 2 1
1350.2.e.c 2 180.n even 6 1
1350.2.j.a 4 60.l odd 4 2
1350.2.j.a 4 180.v odd 12 2
1728.2.i.e 2 24.f even 2 1
1728.2.i.e 2 72.l even 6 1
1728.2.i.f 2 24.h odd 2 1
1728.2.i.f 2 72.j odd 6 1
2646.2.e.b 2 84.n even 6 1
2646.2.e.b 2 252.o even 6 1
2646.2.e.c 2 84.j odd 6 1
2646.2.e.c 2 252.bn odd 6 1
2646.2.f.g 2 84.h odd 2 1
2646.2.f.g 2 252.s odd 6 1
2646.2.h.h 2 84.n even 6 1
2646.2.h.h 2 252.bb even 6 1
2646.2.h.i 2 84.j odd 6 1
2646.2.h.i 2 252.r odd 6 1
4050.2.a.c 1 180.p odd 6 1
4050.2.a.v 1 180.n even 6 1
4050.2.c.c 2 180.x even 12 2
4050.2.c.r 2 180.v odd 12 2
5184.2.a.o 1 72.n even 6 1
5184.2.a.p 1 72.j odd 6 1
5184.2.a.q 1 72.l even 6 1
5184.2.a.r 1 72.p odd 6 1
7938.2.a.i 1 252.s odd 6 1
7938.2.a.x 1 252.bi even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} \) 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
$3$ \( T^{2} - 3T + 3 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$11$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$13$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$17$ \( (T + 3)^{2} \) Copy content Toggle raw display
$19$ \( (T - 1)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$29$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$31$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$37$ \( (T + 4)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 9T + 81 \) Copy content Toggle raw display
$43$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$47$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$53$ \( (T - 12)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$61$ \( T^{2} + 8T + 64 \) Copy content Toggle raw display
$67$ \( T^{2} - 5T + 25 \) Copy content Toggle raw display
$71$ \( (T - 12)^{2} \) Copy content Toggle raw display
$73$ \( (T - 11)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$83$ \( T^{2} - 12T + 144 \) Copy content Toggle raw display
$89$ \( (T - 6)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 5T + 25 \) Copy content Toggle raw display
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