Properties

Label 144.3.g.c
Level $144$
Weight $3$
Character orbit 144.g
Analytic conductor $3.924$
Analytic rank $0$
Dimension $2$
CM discriminant -3
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 144 = 2^{4} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 144.g (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.92371580679\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( 8 - 16 \zeta_{6} ) q^{7} +O(q^{10})\) \( q + ( 8 - 16 \zeta_{6} ) q^{7} + 22 q^{13} + ( 16 - 32 \zeta_{6} ) q^{19} -25 q^{25} + ( -24 + 48 \zeta_{6} ) q^{31} + 26 q^{37} + ( -48 + 96 \zeta_{6} ) q^{43} -143 q^{49} + 74 q^{61} + ( -32 + 64 \zeta_{6} ) q^{67} + 46 q^{73} + ( 40 - 80 \zeta_{6} ) q^{79} + ( 176 - 352 \zeta_{6} ) q^{91} -2 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + O(q^{10}) \) \( 2q + 44q^{13} - 50q^{25} + 52q^{37} - 286q^{49} + 148q^{61} + 92q^{73} - 4q^{97} + O(q^{100}) \)

Character values

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

\(n\) \(37\) \(65\) \(127\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
127.1
0.500000 + 0.866025i
0.500000 0.866025i
0 0 0 0 0 13.8564i 0 0 0
127.2 0 0 0 0 0 13.8564i 0 0 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
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
4.b odd 2 1 inner
12.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 144.3.g.c 2
3.b odd 2 1 CM 144.3.g.c 2
4.b odd 2 1 inner 144.3.g.c 2
5.b even 2 1 3600.3.e.h 2
5.c odd 4 2 3600.3.j.e 4
8.b even 2 1 576.3.g.f 2
8.d odd 2 1 576.3.g.f 2
9.c even 3 1 1296.3.o.f 2
9.c even 3 1 1296.3.o.k 2
9.d odd 6 1 1296.3.o.f 2
9.d odd 6 1 1296.3.o.k 2
12.b even 2 1 inner 144.3.g.c 2
15.d odd 2 1 3600.3.e.h 2
15.e even 4 2 3600.3.j.e 4
16.e even 4 2 2304.3.b.m 4
16.f odd 4 2 2304.3.b.m 4
20.d odd 2 1 3600.3.e.h 2
20.e even 4 2 3600.3.j.e 4
24.f even 2 1 576.3.g.f 2
24.h odd 2 1 576.3.g.f 2
36.f odd 6 1 1296.3.o.f 2
36.f odd 6 1 1296.3.o.k 2
36.h even 6 1 1296.3.o.f 2
36.h even 6 1 1296.3.o.k 2
48.i odd 4 2 2304.3.b.m 4
48.k even 4 2 2304.3.b.m 4
60.h even 2 1 3600.3.e.h 2
60.l odd 4 2 3600.3.j.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
144.3.g.c 2 1.a even 1 1 trivial
144.3.g.c 2 3.b odd 2 1 CM
144.3.g.c 2 4.b odd 2 1 inner
144.3.g.c 2 12.b even 2 1 inner
576.3.g.f 2 8.b even 2 1
576.3.g.f 2 8.d odd 2 1
576.3.g.f 2 24.f even 2 1
576.3.g.f 2 24.h odd 2 1
1296.3.o.f 2 9.c even 3 1
1296.3.o.f 2 9.d odd 6 1
1296.3.o.f 2 36.f odd 6 1
1296.3.o.f 2 36.h even 6 1
1296.3.o.k 2 9.c even 3 1
1296.3.o.k 2 9.d odd 6 1
1296.3.o.k 2 36.f odd 6 1
1296.3.o.k 2 36.h even 6 1
2304.3.b.m 4 16.e even 4 2
2304.3.b.m 4 16.f odd 4 2
2304.3.b.m 4 48.i odd 4 2
2304.3.b.m 4 48.k even 4 2
3600.3.e.h 2 5.b even 2 1
3600.3.e.h 2 15.d odd 2 1
3600.3.e.h 2 20.d odd 2 1
3600.3.e.h 2 60.h even 2 1
3600.3.j.e 4 5.c odd 4 2
3600.3.j.e 4 15.e even 4 2
3600.3.j.e 4 20.e even 4 2
3600.3.j.e 4 60.l odd 4 2

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( T^{2} \)
$7$ \( 192 + T^{2} \)
$11$ \( T^{2} \)
$13$ \( ( -22 + T )^{2} \)
$17$ \( T^{2} \)
$19$ \( 768 + T^{2} \)
$23$ \( T^{2} \)
$29$ \( T^{2} \)
$31$ \( 1728 + T^{2} \)
$37$ \( ( -26 + T )^{2} \)
$41$ \( T^{2} \)
$43$ \( 6912 + T^{2} \)
$47$ \( T^{2} \)
$53$ \( T^{2} \)
$59$ \( T^{2} \)
$61$ \( ( -74 + T )^{2} \)
$67$ \( 3072 + T^{2} \)
$71$ \( T^{2} \)
$73$ \( ( -46 + T )^{2} \)
$79$ \( 4800 + T^{2} \)
$83$ \( T^{2} \)
$89$ \( T^{2} \)
$97$ \( ( 2 + T )^{2} \)
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