Properties

Label 144.2.s.d
Level 144
Weight 2
Character orbit 144.s
Analytic conductor 1.150
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: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 144.s (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.14984578911\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( 1 + \zeta_{6} ) q^{3} + ( -4 + 2 \zeta_{6} ) q^{5} + ( 2 + 2 \zeta_{6} ) q^{7} + 3 \zeta_{6} q^{9} +O(q^{10})\) \( q + ( 1 + \zeta_{6} ) q^{3} + ( -4 + 2 \zeta_{6} ) q^{5} + ( 2 + 2 \zeta_{6} ) q^{7} + 3 \zeta_{6} q^{9} + ( 3 - 3 \zeta_{6} ) q^{11} -4 \zeta_{6} q^{13} -6 q^{15} + ( 1 - 2 \zeta_{6} ) q^{17} + ( 1 - 2 \zeta_{6} ) q^{19} + 6 \zeta_{6} q^{21} + ( 7 - 7 \zeta_{6} ) q^{25} + ( -3 + 6 \zeta_{6} ) q^{27} + ( -2 - 2 \zeta_{6} ) q^{29} + ( 6 - 3 \zeta_{6} ) q^{33} -12 q^{35} + 2 q^{37} + ( 4 - 8 \zeta_{6} ) q^{39} + ( -6 + 3 \zeta_{6} ) q^{41} + ( -3 - 3 \zeta_{6} ) q^{43} + ( -6 - 6 \zeta_{6} ) q^{45} + ( 12 - 12 \zeta_{6} ) q^{47} + 5 \zeta_{6} q^{49} + ( 3 - 3 \zeta_{6} ) q^{51} + ( -6 + 12 \zeta_{6} ) q^{55} + ( 3 - 3 \zeta_{6} ) q^{57} + 15 \zeta_{6} q^{59} + ( -8 + 8 \zeta_{6} ) q^{61} + ( -6 + 12 \zeta_{6} ) q^{63} + ( 8 + 8 \zeta_{6} ) q^{65} + ( 10 - 5 \zeta_{6} ) q^{67} + 6 q^{71} -11 q^{73} + ( 14 - 7 \zeta_{6} ) q^{75} + ( 12 - 6 \zeta_{6} ) q^{77} + ( -2 - 2 \zeta_{6} ) q^{79} + ( -9 + 9 \zeta_{6} ) q^{81} + ( -12 + 12 \zeta_{6} ) q^{83} + 6 \zeta_{6} q^{85} -6 \zeta_{6} q^{87} + ( 8 - 16 \zeta_{6} ) q^{89} + ( 8 - 16 \zeta_{6} ) q^{91} + 6 \zeta_{6} q^{95} + ( -13 + 13 \zeta_{6} ) q^{97} + 9 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 3q^{3} - 6q^{5} + 6q^{7} + 3q^{9} + O(q^{10}) \) \( 2q + 3q^{3} - 6q^{5} + 6q^{7} + 3q^{9} + 3q^{11} - 4q^{13} - 12q^{15} + 6q^{21} + 7q^{25} - 6q^{29} + 9q^{33} - 24q^{35} + 4q^{37} - 9q^{41} - 9q^{43} - 18q^{45} + 12q^{47} + 5q^{49} + 3q^{51} + 3q^{57} + 15q^{59} - 8q^{61} + 24q^{65} + 15q^{67} + 12q^{71} - 22q^{73} + 21q^{75} + 18q^{77} - 6q^{79} - 9q^{81} - 12q^{83} + 6q^{85} - 6q^{87} + 6q^{95} - 13q^{97} + 18q^{99} + O(q^{100}) \)

Character values

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

\(n\) \(37\) \(65\) \(127\)
\(\chi(n)\) \(1\) \(\zeta_{6}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
47.1
0.500000 + 0.866025i
0.500000 0.866025i
0 1.50000 + 0.866025i 0 −3.00000 + 1.73205i 0 3.00000 + 1.73205i 0 1.50000 + 2.59808i 0
95.1 0 1.50000 0.866025i 0 −3.00000 1.73205i 0 3.00000 1.73205i 0 1.50000 2.59808i 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
36.h even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 144.2.s.d yes 2
3.b odd 2 1 432.2.s.d 2
4.b odd 2 1 144.2.s.a 2
8.b even 2 1 576.2.s.a 2
8.d odd 2 1 576.2.s.d 2
9.c even 3 1 432.2.s.c 2
9.c even 3 1 1296.2.c.b 2
9.d odd 6 1 144.2.s.a 2
9.d odd 6 1 1296.2.c.d 2
12.b even 2 1 432.2.s.c 2
24.f even 2 1 1728.2.s.a 2
24.h odd 2 1 1728.2.s.b 2
36.f odd 6 1 432.2.s.d 2
36.f odd 6 1 1296.2.c.d 2
36.h even 6 1 inner 144.2.s.d yes 2
36.h even 6 1 1296.2.c.b 2
72.j odd 6 1 576.2.s.d 2
72.j odd 6 1 5184.2.c.a 2
72.l even 6 1 576.2.s.a 2
72.l even 6 1 5184.2.c.c 2
72.n even 6 1 1728.2.s.a 2
72.n even 6 1 5184.2.c.c 2
72.p odd 6 1 1728.2.s.b 2
72.p odd 6 1 5184.2.c.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
144.2.s.a 2 4.b odd 2 1
144.2.s.a 2 9.d odd 6 1
144.2.s.d yes 2 1.a even 1 1 trivial
144.2.s.d yes 2 36.h even 6 1 inner
432.2.s.c 2 9.c even 3 1
432.2.s.c 2 12.b even 2 1
432.2.s.d 2 3.b odd 2 1
432.2.s.d 2 36.f odd 6 1
576.2.s.a 2 8.b even 2 1
576.2.s.a 2 72.l even 6 1
576.2.s.d 2 8.d odd 2 1
576.2.s.d 2 72.j odd 6 1
1296.2.c.b 2 9.c even 3 1
1296.2.c.b 2 36.h even 6 1
1296.2.c.d 2 9.d odd 6 1
1296.2.c.d 2 36.f odd 6 1
1728.2.s.a 2 24.f even 2 1
1728.2.s.a 2 72.n even 6 1
1728.2.s.b 2 24.h odd 2 1
1728.2.s.b 2 72.p odd 6 1
5184.2.c.a 2 72.j odd 6 1
5184.2.c.a 2 72.p odd 6 1
5184.2.c.c 2 72.l even 6 1
5184.2.c.c 2 72.n even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(144, [\chi])\):

\( T_{5}^{2} + 6 T_{5} + 12 \)
\( T_{7}^{2} - 6 T_{7} + 12 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 - 3 T + 3 T^{2} \)
$5$ \( 1 + 6 T + 17 T^{2} + 30 T^{3} + 25 T^{4} \)
$7$ \( ( 1 - 5 T + 7 T^{2} )( 1 - T + 7 T^{2} ) \)
$11$ \( 1 - 3 T - 2 T^{2} - 33 T^{3} + 121 T^{4} \)
$13$ \( 1 + 4 T + 3 T^{2} + 52 T^{3} + 169 T^{4} \)
$17$ \( 1 - 31 T^{2} + 289 T^{4} \)
$19$ \( 1 - 35 T^{2} + 361 T^{4} \)
$23$ \( 1 - 23 T^{2} + 529 T^{4} \)
$29$ \( 1 + 6 T + 41 T^{2} + 174 T^{3} + 841 T^{4} \)
$31$ \( 1 + 31 T^{2} + 961 T^{4} \)
$37$ \( ( 1 - 2 T + 37 T^{2} )^{2} \)
$41$ \( 1 + 9 T + 68 T^{2} + 369 T^{3} + 1681 T^{4} \)
$43$ \( 1 + 9 T + 70 T^{2} + 387 T^{3} + 1849 T^{4} \)
$47$ \( 1 - 12 T + 97 T^{2} - 564 T^{3} + 2209 T^{4} \)
$53$ \( ( 1 - 53 T^{2} )^{2} \)
$59$ \( 1 - 15 T + 166 T^{2} - 885 T^{3} + 3481 T^{4} \)
$61$ \( 1 + 8 T + 3 T^{2} + 488 T^{3} + 3721 T^{4} \)
$67$ \( 1 - 15 T + 142 T^{2} - 1005 T^{3} + 4489 T^{4} \)
$71$ \( ( 1 - 6 T + 71 T^{2} )^{2} \)
$73$ \( ( 1 + 11 T + 73 T^{2} )^{2} \)
$79$ \( 1 + 6 T + 91 T^{2} + 474 T^{3} + 6241 T^{4} \)
$83$ \( 1 + 12 T + 61 T^{2} + 996 T^{3} + 6889 T^{4} \)
$89$ \( 1 + 14 T^{2} + 7921 T^{4} \)
$97$ \( 1 + 13 T + 72 T^{2} + 1261 T^{3} + 9409 T^{4} \)
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