Properties

Label 144.3.g.a
Level $144$
Weight $3$
Character orbit 144.g
Self dual yes
Analytic conductor $3.924$
Analytic rank $0$
Dimension $1$
CM discriminant -4
Inner twists $2$

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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: yes
Analytic conductor: \(3.92371580679\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 16)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

\(f(q)\) \(=\) \( q + 6q^{5} + O(q^{10}) \) \( q + 6q^{5} + 10q^{13} + 30q^{17} + 11q^{25} - 42q^{29} - 70q^{37} - 18q^{41} + 49q^{49} - 90q^{53} - 22q^{61} + 60q^{65} - 110q^{73} + 180q^{85} + 78q^{89} + 130q^{97} + O(q^{100}) \)

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
0 0 0 6.00000 0 0 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
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 144.3.g.a 1
3.b odd 2 1 16.3.c.a 1
4.b odd 2 1 CM 144.3.g.a 1
5.b even 2 1 3600.3.e.c 1
5.c odd 4 2 3600.3.j.a 2
8.b even 2 1 576.3.g.b 1
8.d odd 2 1 576.3.g.b 1
9.c even 3 2 1296.3.o.b 2
9.d odd 6 2 1296.3.o.o 2
12.b even 2 1 16.3.c.a 1
15.d odd 2 1 400.3.b.a 1
15.e even 4 2 400.3.h.a 2
16.e even 4 2 2304.3.b.f 2
16.f odd 4 2 2304.3.b.f 2
20.d odd 2 1 3600.3.e.c 1
20.e even 4 2 3600.3.j.a 2
21.c even 2 1 784.3.d.b 1
21.g even 6 2 784.3.r.d 2
21.h odd 6 2 784.3.r.e 2
24.f even 2 1 64.3.c.a 1
24.h odd 2 1 64.3.c.a 1
36.f odd 6 2 1296.3.o.b 2
36.h even 6 2 1296.3.o.o 2
48.i odd 4 2 256.3.d.b 2
48.k even 4 2 256.3.d.b 2
60.h even 2 1 400.3.b.a 1
60.l odd 4 2 400.3.h.a 2
84.h odd 2 1 784.3.d.b 1
84.j odd 6 2 784.3.r.d 2
84.n even 6 2 784.3.r.e 2
120.i odd 2 1 1600.3.b.b 1
120.m even 2 1 1600.3.b.b 1
120.q odd 4 2 1600.3.h.b 2
120.w even 4 2 1600.3.h.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
16.3.c.a 1 3.b odd 2 1
16.3.c.a 1 12.b even 2 1
64.3.c.a 1 24.f even 2 1
64.3.c.a 1 24.h odd 2 1
144.3.g.a 1 1.a even 1 1 trivial
144.3.g.a 1 4.b odd 2 1 CM
256.3.d.b 2 48.i odd 4 2
256.3.d.b 2 48.k even 4 2
400.3.b.a 1 15.d odd 2 1
400.3.b.a 1 60.h even 2 1
400.3.h.a 2 15.e even 4 2
400.3.h.a 2 60.l odd 4 2
576.3.g.b 1 8.b even 2 1
576.3.g.b 1 8.d odd 2 1
784.3.d.b 1 21.c even 2 1
784.3.d.b 1 84.h odd 2 1
784.3.r.d 2 21.g even 6 2
784.3.r.d 2 84.j odd 6 2
784.3.r.e 2 21.h odd 6 2
784.3.r.e 2 84.n even 6 2
1296.3.o.b 2 9.c even 3 2
1296.3.o.b 2 36.f odd 6 2
1296.3.o.o 2 9.d odd 6 2
1296.3.o.o 2 36.h even 6 2
1600.3.b.b 1 120.i odd 2 1
1600.3.b.b 1 120.m even 2 1
1600.3.h.b 2 120.q odd 4 2
1600.3.h.b 2 120.w even 4 2
2304.3.b.f 2 16.e even 4 2
2304.3.b.f 2 16.f odd 4 2
3600.3.e.c 1 5.b even 2 1
3600.3.e.c 1 20.d odd 2 1
3600.3.j.a 2 5.c odd 4 2
3600.3.j.a 2 20.e even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} - 6 \) acting on \(S_{3}^{\mathrm{new}}(144, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( T \)
$5$ \( -6 + T \)
$7$ \( T \)
$11$ \( T \)
$13$ \( -10 + T \)
$17$ \( -30 + T \)
$19$ \( T \)
$23$ \( T \)
$29$ \( 42 + T \)
$31$ \( T \)
$37$ \( 70 + T \)
$41$ \( 18 + T \)
$43$ \( T \)
$47$ \( T \)
$53$ \( 90 + T \)
$59$ \( T \)
$61$ \( 22 + T \)
$67$ \( T \)
$71$ \( T \)
$73$ \( 110 + T \)
$79$ \( T \)
$83$ \( T \)
$89$ \( -78 + T \)
$97$ \( -130 + T \)
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