Properties

Label 144.4.c.a
Level $144$
Weight $4$
Character orbit 144.c
Analytic conductor $8.496$
Analytic rank $0$
Dimension $2$
CM discriminant -4
Inner twists $4$

Related objects

Downloads

Learn more about

Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(8.49627504083\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
Defining polynomial: \(x^{2} + 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 9\sqrt{-2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{5} +O(q^{10})\) \( q + \beta q^{5} -92 q^{13} + 11 \beta q^{17} -37 q^{25} + 23 \beta q^{29} -214 q^{37} -39 \beta q^{41} + 343 q^{49} + 3 \beta q^{53} + 830 q^{61} -92 \beta q^{65} + 592 q^{73} -1782 q^{85} -83 \beta q^{89} + 1816 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + O(q^{10}) \) \( 2q - 184q^{13} - 74q^{25} - 428q^{37} + 686q^{49} + 1660q^{61} + 1184q^{73} - 3564q^{85} + 3632q^{97} + O(q^{100}) \)

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 12.7279i 0 0 0 0 0
143.2 0 0 0 12.7279i 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 144.4.c.a 2
3.b odd 2 1 inner 144.4.c.a 2
4.b odd 2 1 CM 144.4.c.a 2
8.b even 2 1 576.4.c.b 2
8.d odd 2 1 576.4.c.b 2
12.b even 2 1 inner 144.4.c.a 2
16.e even 4 2 2304.4.f.b 4
16.f odd 4 2 2304.4.f.b 4
24.f even 2 1 576.4.c.b 2
24.h odd 2 1 576.4.c.b 2
48.i odd 4 2 2304.4.f.b 4
48.k even 4 2 2304.4.f.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
144.4.c.a 2 1.a even 1 1 trivial
144.4.c.a 2 3.b odd 2 1 inner
144.4.c.a 2 4.b odd 2 1 CM
144.4.c.a 2 12.b even 2 1 inner
576.4.c.b 2 8.b even 2 1
576.4.c.b 2 8.d odd 2 1
576.4.c.b 2 24.f even 2 1
576.4.c.b 2 24.h odd 2 1
2304.4.f.b 4 16.e even 4 2
2304.4.f.b 4 16.f odd 4 2
2304.4.f.b 4 48.i odd 4 2
2304.4.f.b 4 48.k even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 162 \) acting on \(S_{4}^{\mathrm{new}}(144, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( 162 + T^{2} \)
$7$ \( T^{2} \)
$11$ \( T^{2} \)
$13$ \( ( 92 + T )^{2} \)
$17$ \( 19602 + T^{2} \)
$19$ \( T^{2} \)
$23$ \( T^{2} \)
$29$ \( 85698 + T^{2} \)
$31$ \( T^{2} \)
$37$ \( ( 214 + T )^{2} \)
$41$ \( 246402 + T^{2} \)
$43$ \( T^{2} \)
$47$ \( T^{2} \)
$53$ \( 1458 + T^{2} \)
$59$ \( T^{2} \)
$61$ \( ( -830 + T )^{2} \)
$67$ \( T^{2} \)
$71$ \( T^{2} \)
$73$ \( ( -592 + T )^{2} \)
$79$ \( T^{2} \)
$83$ \( T^{2} \)
$89$ \( 1116018 + T^{2} \)
$97$ \( ( -1816 + T )^{2} \)
show more
show less