Properties

Label 144.3.e.a
Level $144$
Weight $3$
Character orbit 144.e
Analytic conductor $3.924$
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: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 144.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.92371580679\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
Defining polynomial: \(x^{2} + 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 72)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(f(q)\) \(=\) \( q + 5 \beta q^{5} -12 q^{7} +O(q^{10})\) \( q + 5 \beta q^{5} -12 q^{7} + 4 \beta q^{11} -8 q^{13} + 7 \beta q^{17} + 16 q^{19} + 28 \beta q^{23} -25 q^{25} -21 \beta q^{29} + 4 q^{31} -60 \beta q^{35} + 30 q^{37} -15 \beta q^{41} + 8 q^{43} + 12 \beta q^{47} + 95 q^{49} + 35 \beta q^{53} -40 q^{55} -56 \beta q^{59} -14 q^{61} -40 \beta q^{65} + 88 q^{67} + 20 \beta q^{71} -80 q^{73} -48 \beta q^{77} -100 q^{79} + 92 \beta q^{83} -70 q^{85} + 105 \beta q^{89} + 96 q^{91} + 80 \beta q^{95} -112 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 24q^{7} + O(q^{10}) \) \( 2q - 24q^{7} - 16q^{13} + 32q^{19} - 50q^{25} + 8q^{31} + 60q^{37} + 16q^{43} + 190q^{49} - 80q^{55} - 28q^{61} + 176q^{67} - 160q^{73} - 200q^{79} - 140q^{85} + 192q^{91} - 224q^{97} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1
1.41421i
1.41421i
0 0 0 7.07107i 0 −12.0000 0 0 0
17.2 0 0 0 7.07107i 0 −12.0000 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 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 144.3.e.a 2
3.b odd 2 1 inner 144.3.e.a 2
4.b odd 2 1 72.3.e.a 2
5.b even 2 1 3600.3.l.l 2
5.c odd 4 2 3600.3.c.c 4
8.b even 2 1 576.3.e.a 2
8.d odd 2 1 576.3.e.h 2
9.c even 3 2 1296.3.q.k 4
9.d odd 6 2 1296.3.q.k 4
12.b even 2 1 72.3.e.a 2
15.d odd 2 1 3600.3.l.l 2
15.e even 4 2 3600.3.c.c 4
16.e even 4 2 2304.3.h.h 4
16.f odd 4 2 2304.3.h.a 4
20.d odd 2 1 1800.3.l.a 2
20.e even 4 2 1800.3.c.a 4
24.f even 2 1 576.3.e.h 2
24.h odd 2 1 576.3.e.a 2
28.d even 2 1 3528.3.d.a 2
36.f odd 6 2 648.3.m.a 4
36.h even 6 2 648.3.m.a 4
48.i odd 4 2 2304.3.h.h 4
48.k even 4 2 2304.3.h.a 4
60.h even 2 1 1800.3.l.a 2
60.l odd 4 2 1800.3.c.a 4
84.h odd 2 1 3528.3.d.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.3.e.a 2 4.b odd 2 1
72.3.e.a 2 12.b even 2 1
144.3.e.a 2 1.a even 1 1 trivial
144.3.e.a 2 3.b odd 2 1 inner
576.3.e.a 2 8.b even 2 1
576.3.e.a 2 24.h odd 2 1
576.3.e.h 2 8.d odd 2 1
576.3.e.h 2 24.f even 2 1
648.3.m.a 4 36.f odd 6 2
648.3.m.a 4 36.h even 6 2
1296.3.q.k 4 9.c even 3 2
1296.3.q.k 4 9.d odd 6 2
1800.3.c.a 4 20.e even 4 2
1800.3.c.a 4 60.l odd 4 2
1800.3.l.a 2 20.d odd 2 1
1800.3.l.a 2 60.h even 2 1
2304.3.h.a 4 16.f odd 4 2
2304.3.h.a 4 48.k even 4 2
2304.3.h.h 4 16.e even 4 2
2304.3.h.h 4 48.i odd 4 2
3528.3.d.a 2 28.d even 2 1
3528.3.d.a 2 84.h odd 2 1
3600.3.c.c 4 5.c odd 4 2
3600.3.c.c 4 15.e even 4 2
3600.3.l.l 2 5.b even 2 1
3600.3.l.l 2 15.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 50 \) acting on \(S_{3}^{\mathrm{new}}(144, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( 50 + T^{2} \)
$7$ \( ( 12 + T )^{2} \)
$11$ \( 32 + T^{2} \)
$13$ \( ( 8 + T )^{2} \)
$17$ \( 98 + T^{2} \)
$19$ \( ( -16 + T )^{2} \)
$23$ \( 1568 + T^{2} \)
$29$ \( 882 + T^{2} \)
$31$ \( ( -4 + T )^{2} \)
$37$ \( ( -30 + T )^{2} \)
$41$ \( 450 + T^{2} \)
$43$ \( ( -8 + T )^{2} \)
$47$ \( 288 + T^{2} \)
$53$ \( 2450 + T^{2} \)
$59$ \( 6272 + T^{2} \)
$61$ \( ( 14 + T )^{2} \)
$67$ \( ( -88 + T )^{2} \)
$71$ \( 800 + T^{2} \)
$73$ \( ( 80 + T )^{2} \)
$79$ \( ( 100 + T )^{2} \)
$83$ \( 16928 + T^{2} \)
$89$ \( 22050 + T^{2} \)
$97$ \( ( 112 + T )^{2} \)
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