Properties

Label 144.2.s.e
Level 144
Weight 2
Character orbit 144.s
Analytic conductor 1.150
Analytic rank 0
Dimension 4
CM no
Inner twists 4

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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: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3 \)
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 \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -2 \zeta_{12} + \zeta_{12}^{3} ) q^{3} + ( 1 + \zeta_{12}^{2} ) q^{5} + ( 3 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{7} + 3 q^{9} +O(q^{10})\) \( q + ( -2 \zeta_{12} + \zeta_{12}^{3} ) q^{3} + ( 1 + \zeta_{12}^{2} ) q^{5} + ( 3 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{7} + 3 q^{9} + ( 3 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{11} + ( -1 + \zeta_{12}^{2} ) q^{13} -3 \zeta_{12} q^{15} + ( 2 - 4 \zeta_{12}^{2} ) q^{17} -6 \zeta_{12}^{3} q^{19} + ( -6 + 3 \zeta_{12}^{2} ) q^{21} + ( -3 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{23} -2 \zeta_{12}^{2} q^{25} + ( -6 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{27} + ( -10 + 5 \zeta_{12}^{2} ) q^{29} + 3 \zeta_{12} q^{31} -9 \zeta_{12}^{2} q^{33} + ( 6 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{35} -4 q^{37} + ( \zeta_{12} - 2 \zeta_{12}^{3} ) q^{39} + ( 3 + 3 \zeta_{12}^{2} ) q^{41} + ( 3 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{43} + ( 3 + 3 \zeta_{12}^{2} ) q^{45} + ( -3 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{47} + ( 2 - 2 \zeta_{12}^{2} ) q^{49} + 6 \zeta_{12}^{3} q^{51} + ( 6 - 12 \zeta_{12}^{2} ) q^{53} + 9 \zeta_{12}^{3} q^{55} + ( -6 + 12 \zeta_{12}^{2} ) q^{57} + ( 3 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{59} + 7 \zeta_{12}^{2} q^{61} + ( 9 \zeta_{12} - 9 \zeta_{12}^{3} ) q^{63} + ( -2 + \zeta_{12}^{2} ) q^{65} -9 \zeta_{12} q^{67} + ( 9 - 9 \zeta_{12}^{2} ) q^{69} + ( -12 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{71} + 4 q^{73} + ( 2 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{75} + ( 9 + 9 \zeta_{12}^{2} ) q^{77} + ( -15 \zeta_{12} + 15 \zeta_{12}^{3} ) q^{79} + 9 q^{81} + ( -3 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{83} + ( 6 - 6 \zeta_{12}^{2} ) q^{85} + ( 15 \zeta_{12} - 15 \zeta_{12}^{3} ) q^{87} + ( -2 + 4 \zeta_{12}^{2} ) q^{89} + 3 \zeta_{12}^{3} q^{91} + ( -3 - 3 \zeta_{12}^{2} ) q^{93} + ( 6 \zeta_{12} - 12 \zeta_{12}^{3} ) q^{95} -\zeta_{12}^{2} q^{97} + ( 9 \zeta_{12} + 9 \zeta_{12}^{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 6q^{5} + 12q^{9} + O(q^{10}) \) \( 4q + 6q^{5} + 12q^{9} - 2q^{13} - 18q^{21} - 4q^{25} - 30q^{29} - 18q^{33} - 16q^{37} + 18q^{41} + 18q^{45} + 4q^{49} + 14q^{61} - 6q^{65} + 18q^{69} + 16q^{73} + 54q^{77} + 36q^{81} + 12q^{85} - 18q^{93} - 2q^{97} + O(q^{100}) \)

Character values

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

\(n\) \(37\) \(65\) \(127\)
\(\chi(n)\) \(1\) \(1 - \zeta_{12}\) \(-1\)

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
0.866025 0.500000i
−0.866025 + 0.500000i
0.866025 + 0.500000i
−0.866025 0.500000i
0 −1.73205 0 1.50000 0.866025i 0 2.59808 + 1.50000i 0 3.00000 0
47.2 0 1.73205 0 1.50000 0.866025i 0 −2.59808 1.50000i 0 3.00000 0
95.1 0 −1.73205 0 1.50000 + 0.866025i 0 2.59808 1.50000i 0 3.00000 0
95.2 0 1.73205 0 1.50000 + 0.866025i 0 −2.59808 + 1.50000i 0 3.00000 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 inner
9.d odd 6 1 inner
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.e 4
3.b odd 2 1 432.2.s.e 4
4.b odd 2 1 inner 144.2.s.e 4
8.b even 2 1 576.2.s.e 4
8.d odd 2 1 576.2.s.e 4
9.c even 3 1 432.2.s.e 4
9.c even 3 1 1296.2.c.f 4
9.d odd 6 1 inner 144.2.s.e 4
9.d odd 6 1 1296.2.c.f 4
12.b even 2 1 432.2.s.e 4
24.f even 2 1 1728.2.s.e 4
24.h odd 2 1 1728.2.s.e 4
36.f odd 6 1 432.2.s.e 4
36.f odd 6 1 1296.2.c.f 4
36.h even 6 1 inner 144.2.s.e 4
36.h even 6 1 1296.2.c.f 4
72.j odd 6 1 576.2.s.e 4
72.j odd 6 1 5184.2.c.f 4
72.l even 6 1 576.2.s.e 4
72.l even 6 1 5184.2.c.f 4
72.n even 6 1 1728.2.s.e 4
72.n even 6 1 5184.2.c.f 4
72.p odd 6 1 1728.2.s.e 4
72.p odd 6 1 5184.2.c.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
144.2.s.e 4 1.a even 1 1 trivial
144.2.s.e 4 4.b odd 2 1 inner
144.2.s.e 4 9.d odd 6 1 inner
144.2.s.e 4 36.h even 6 1 inner
432.2.s.e 4 3.b odd 2 1
432.2.s.e 4 9.c even 3 1
432.2.s.e 4 12.b even 2 1
432.2.s.e 4 36.f odd 6 1
576.2.s.e 4 8.b even 2 1
576.2.s.e 4 8.d odd 2 1
576.2.s.e 4 72.j odd 6 1
576.2.s.e 4 72.l even 6 1
1296.2.c.f 4 9.c even 3 1
1296.2.c.f 4 9.d odd 6 1
1296.2.c.f 4 36.f odd 6 1
1296.2.c.f 4 36.h even 6 1
1728.2.s.e 4 24.f even 2 1
1728.2.s.e 4 24.h odd 2 1
1728.2.s.e 4 72.n even 6 1
1728.2.s.e 4 72.p odd 6 1
5184.2.c.f 4 72.j odd 6 1
5184.2.c.f 4 72.l even 6 1
5184.2.c.f 4 72.n even 6 1
5184.2.c.f 4 72.p odd 6 1

Hecke kernels

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

\( T_{5}^{2} - 3 T_{5} + 3 \)
\( T_{7}^{4} - 9 T_{7}^{2} + 81 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( ( 1 - 3 T^{2} )^{2} \)
$5$ \( ( 1 - 3 T + 8 T^{2} - 15 T^{3} + 25 T^{4} )^{2} \)
$7$ \( 1 + 5 T^{2} - 24 T^{4} + 245 T^{6} + 2401 T^{8} \)
$11$ \( 1 + 5 T^{2} - 96 T^{4} + 605 T^{6} + 14641 T^{8} \)
$13$ \( ( 1 + T - 12 T^{2} + 13 T^{3} + 169 T^{4} )^{2} \)
$17$ \( ( 1 - 22 T^{2} + 289 T^{4} )^{2} \)
$19$ \( ( 1 - 2 T^{2} + 361 T^{4} )^{2} \)
$23$ \( 1 - 19 T^{2} - 168 T^{4} - 10051 T^{6} + 279841 T^{8} \)
$29$ \( ( 1 + 15 T + 104 T^{2} + 435 T^{3} + 841 T^{4} )^{2} \)
$31$ \( 1 + 53 T^{2} + 1848 T^{4} + 50933 T^{6} + 923521 T^{8} \)
$37$ \( ( 1 + 4 T + 37 T^{2} )^{4} \)
$41$ \( ( 1 - 9 T + 68 T^{2} - 369 T^{3} + 1681 T^{4} )^{2} \)
$43$ \( 1 + 77 T^{2} + 4080 T^{4} + 142373 T^{6} + 3418801 T^{8} \)
$47$ \( 1 - 67 T^{2} + 2280 T^{4} - 148003 T^{6} + 4879681 T^{8} \)
$53$ \( ( 1 + 2 T^{2} + 2809 T^{4} )^{2} \)
$59$ \( 1 - 91 T^{2} + 4800 T^{4} - 316771 T^{6} + 12117361 T^{8} \)
$61$ \( ( 1 - 7 T - 12 T^{2} - 427 T^{3} + 3721 T^{4} )^{2} \)
$67$ \( 1 + 53 T^{2} - 1680 T^{4} + 237917 T^{6} + 20151121 T^{8} \)
$71$ \( ( 1 + 34 T^{2} + 5041 T^{4} )^{2} \)
$73$ \( ( 1 - 4 T + 73 T^{2} )^{4} \)
$79$ \( 1 - 67 T^{2} - 1752 T^{4} - 418147 T^{6} + 38950081 T^{8} \)
$83$ \( 1 - 139 T^{2} + 12432 T^{4} - 957571 T^{6} + 47458321 T^{8} \)
$89$ \( ( 1 - 166 T^{2} + 7921 T^{4} )^{2} \)
$97$ \( ( 1 + T - 96 T^{2} + 97 T^{3} + 9409 T^{4} )^{2} \)
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