Properties

Label 144.2.c.a
Level 144
Weight 2
Character orbit 144.c
Analytic conductor 1.150
Analytic rank 0
Dimension 2
CM discriminant -4
Inner twists 4

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(1.14984578911\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 3\sqrt{-2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{5} +O(q^{10})\) \( q + \beta q^{5} + 4 q^{13} -\beta q^{17} -13 q^{25} -\beta q^{29} + 2 q^{37} -3 \beta q^{41} + 7 q^{49} + 3 \beta q^{53} -10 q^{61} + 4 \beta q^{65} + 16 q^{73} + 18 q^{85} + \beta q^{89} -8 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + O(q^{10}) \) \( 2q + 8q^{13} - 26q^{25} + 4q^{37} + 14q^{49} - 20q^{61} + 32q^{73} + 36q^{85} - 16q^{97} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
143.1
1.41421i
1.41421i
0 0 0 4.24264i 0 0 0 0 0
143.2 0 0 0 4.24264i 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 \(\Q(\sqrt{-1}) \)
3.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 144.2.c.a 2
3.b odd 2 1 inner 144.2.c.a 2
4.b odd 2 1 CM 144.2.c.a 2
5.b even 2 1 3600.2.h.b 2
5.c odd 4 2 3600.2.o.a 4
7.b odd 2 1 7056.2.h.b 2
8.b even 2 1 576.2.c.a 2
8.d odd 2 1 576.2.c.a 2
9.c even 3 2 1296.2.s.h 4
9.d odd 6 2 1296.2.s.h 4
12.b even 2 1 inner 144.2.c.a 2
15.d odd 2 1 3600.2.h.b 2
15.e even 4 2 3600.2.o.a 4
16.e even 4 2 2304.2.f.f 4
16.f odd 4 2 2304.2.f.f 4
20.d odd 2 1 3600.2.h.b 2
20.e even 4 2 3600.2.o.a 4
21.c even 2 1 7056.2.h.b 2
24.f even 2 1 576.2.c.a 2
24.h odd 2 1 576.2.c.a 2
28.d even 2 1 7056.2.h.b 2
36.f odd 6 2 1296.2.s.h 4
36.h even 6 2 1296.2.s.h 4
48.i odd 4 2 2304.2.f.f 4
48.k even 4 2 2304.2.f.f 4
60.h even 2 1 3600.2.h.b 2
60.l odd 4 2 3600.2.o.a 4
84.h odd 2 1 7056.2.h.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
144.2.c.a 2 1.a even 1 1 trivial
144.2.c.a 2 3.b odd 2 1 inner
144.2.c.a 2 4.b odd 2 1 CM
144.2.c.a 2 12.b even 2 1 inner
576.2.c.a 2 8.b even 2 1
576.2.c.a 2 8.d odd 2 1
576.2.c.a 2 24.f even 2 1
576.2.c.a 2 24.h odd 2 1
1296.2.s.h 4 9.c even 3 2
1296.2.s.h 4 9.d odd 6 2
1296.2.s.h 4 36.f odd 6 2
1296.2.s.h 4 36.h even 6 2
2304.2.f.f 4 16.e even 4 2
2304.2.f.f 4 16.f odd 4 2
2304.2.f.f 4 48.i odd 4 2
2304.2.f.f 4 48.k even 4 2
3600.2.h.b 2 5.b even 2 1
3600.2.h.b 2 15.d odd 2 1
3600.2.h.b 2 20.d odd 2 1
3600.2.h.b 2 60.h even 2 1
3600.2.o.a 4 5.c odd 4 2
3600.2.o.a 4 15.e even 4 2
3600.2.o.a 4 20.e even 4 2
3600.2.o.a 4 60.l odd 4 2
7056.2.h.b 2 7.b odd 2 1
7056.2.h.b 2 21.c even 2 1
7056.2.h.b 2 28.d even 2 1
7056.2.h.b 2 84.h odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(144, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ \( 1 + 8 T^{2} + 25 T^{4} \)
$7$ \( ( 1 - 7 T^{2} )^{2} \)
$11$ \( ( 1 + 11 T^{2} )^{2} \)
$13$ \( ( 1 - 4 T + 13 T^{2} )^{2} \)
$17$ \( 1 - 16 T^{2} + 289 T^{4} \)
$19$ \( ( 1 - 19 T^{2} )^{2} \)
$23$ \( ( 1 + 23 T^{2} )^{2} \)
$29$ \( 1 - 40 T^{2} + 841 T^{4} \)
$31$ \( ( 1 - 31 T^{2} )^{2} \)
$37$ \( ( 1 - 2 T + 37 T^{2} )^{2} \)
$41$ \( 1 + 80 T^{2} + 1681 T^{4} \)
$43$ \( ( 1 - 43 T^{2} )^{2} \)
$47$ \( ( 1 + 47 T^{2} )^{2} \)
$53$ \( 1 + 56 T^{2} + 2809 T^{4} \)
$59$ \( ( 1 + 59 T^{2} )^{2} \)
$61$ \( ( 1 + 10 T + 61 T^{2} )^{2} \)
$67$ \( ( 1 - 67 T^{2} )^{2} \)
$71$ \( ( 1 + 71 T^{2} )^{2} \)
$73$ \( ( 1 - 16 T + 73 T^{2} )^{2} \)
$79$ \( ( 1 - 79 T^{2} )^{2} \)
$83$ \( ( 1 + 83 T^{2} )^{2} \)
$89$ \( 1 - 160 T^{2} + 7921 T^{4} \)
$97$ \( ( 1 + 8 T + 97 T^{2} )^{2} \)
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