Properties

Label 144.2.k.c
Level 144
Weight 2
Character orbit 144.k
Analytic conductor 1.150
Analytic rank 0
Dimension 8
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.k (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.14984578911\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: 8.0.629407744.1
Defining polynomial: \(x^{8} - 2 x^{6} + 2 x^{4} - 8 x^{2} + 16\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + \beta_{2} q^{4} + ( \beta_{4} - \beta_{6} ) q^{5} + ( -1 + \beta_{2} - \beta_{3} - \beta_{7} ) q^{7} + ( -\beta_{4} + \beta_{5} + \beta_{6} ) q^{8} +O(q^{10})\) \( q + \beta_{1} q^{2} + \beta_{2} q^{4} + ( \beta_{4} - \beta_{6} ) q^{5} + ( -1 + \beta_{2} - \beta_{3} - \beta_{7} ) q^{7} + ( -\beta_{4} + \beta_{5} + \beta_{6} ) q^{8} + ( 2 - \beta_{2} - 2 \beta_{3} - \beta_{7} ) q^{10} + ( -\beta_{1} - \beta_{5} - \beta_{6} ) q^{11} + ( 1 + \beta_{3} + 2 \beta_{7} ) q^{13} + ( -\beta_{1} - \beta_{4} + 2 \beta_{6} ) q^{14} + ( \beta_{2} + 4 \beta_{3} + \beta_{7} ) q^{16} + ( -3 \beta_{1} - \beta_{5} + \beta_{6} ) q^{17} + ( -2 - 2 \beta_{7} ) q^{19} + ( 2 \beta_{1} - 2 \beta_{5} ) q^{20} + ( -2 \beta_{2} - 4 \beta_{3} ) q^{22} + ( -2 \beta_{1} + 2 \beta_{4} + 2 \beta_{5} ) q^{23} + ( 2 - 2 \beta_{2} + \beta_{3} + 2 \beta_{7} ) q^{25} + ( \beta_{1} - \beta_{4} + 2 \beta_{5} - 2 \beta_{6} ) q^{26} + ( -4 + \beta_{2} + 4 \beta_{3} + \beta_{7} ) q^{28} + ( 3 \beta_{1} - 3 \beta_{4} + \beta_{5} - 2 \beta_{6} ) q^{29} + ( -3 - \beta_{2} - \beta_{3} - \beta_{7} ) q^{31} + ( 2 \beta_{4} + 2 \beta_{5} ) q^{32} + ( -4 - 2 \beta_{2} ) q^{34} + ( 5 \beta_{1} + 2 \beta_{4} - 3 \beta_{5} - \beta_{6} ) q^{35} + ( 3 - 2 \beta_{2} + \beta_{3} ) q^{37} + ( -2 \beta_{1} + 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} ) q^{38} + ( -4 + 2 \beta_{2} - 4 \beta_{3} ) q^{40} + ( \beta_{1} + 2 \beta_{4} - \beta_{5} + 3 \beta_{6} ) q^{41} + ( -4 + 2 \beta_{2} - 2 \beta_{3} ) q^{43} + ( -2 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} ) q^{44} + ( 4 - 2 \beta_{2} + 4 \beta_{3} - 2 \beta_{7} ) q^{46} + ( -6 \beta_{4} + 4 \beta_{5} + 2 \beta_{6} ) q^{47} + ( -1 + 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{7} ) q^{49} + ( 2 \beta_{1} + \beta_{4} - 4 \beta_{6} ) q^{50} + ( 8 - \beta_{2} + \beta_{7} ) q^{52} + ( 2 \beta_{1} + 3 \beta_{4} + 2 \beta_{5} - \beta_{6} ) q^{53} + 4 \beta_{3} q^{55} + ( -4 \beta_{1} + 2 \beta_{4} + 2 \beta_{5} ) q^{56} + ( 6 + \beta_{2} - 2 \beta_{3} + 3 \beta_{7} ) q^{58} + ( -2 \beta_{1} - 4 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} ) q^{59} + ( 1 - 3 \beta_{3} - 2 \beta_{7} ) q^{61} + ( -3 \beta_{1} + \beta_{4} - 2 \beta_{5} ) q^{62} + ( 4 + 4 \beta_{3} - 2 \beta_{7} ) q^{64} + ( -5 \beta_{1} - 4 \beta_{4} + \beta_{5} + 3 \beta_{6} ) q^{65} + ( 4 - 4 \beta_{3} ) q^{67} + ( -4 \beta_{1} + 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} ) q^{68} + ( -4 + 4 \beta_{2} - 8 \beta_{3} - 2 \beta_{7} ) q^{70} + ( 4 \beta_{4} + 4 \beta_{6} ) q^{71} + ( -2 + 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{7} ) q^{73} + ( 3 \beta_{1} + 3 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} ) q^{74} + ( -8 - 2 \beta_{7} ) q^{76} + ( 4 \beta_{1} - 2 \beta_{4} - 2 \beta_{6} ) q^{77} + ( 7 + \beta_{2} + \beta_{3} + \beta_{7} ) q^{79} + ( -4 \beta_{1} - 6 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} ) q^{80} + ( -8 + 4 \beta_{2} + 4 \beta_{3} - 2 \beta_{7} ) q^{82} + ( \beta_{1} - 2 \beta_{4} + \beta_{5} - \beta_{6} ) q^{83} + ( -4 + 4 \beta_{2} ) q^{85} + ( -4 \beta_{1} - 4 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} ) q^{86} + ( -2 \beta_{2} - 8 \beta_{3} + 2 \beta_{7} ) q^{88} + ( 2 \beta_{1} - 2 \beta_{5} + 2 \beta_{6} ) q^{89} + ( 8 - 2 \beta_{2} + 6 \beta_{3} ) q^{91} + ( 4 \beta_{1} + 8 \beta_{4} - 4 \beta_{5} ) q^{92} + ( 4 + 2 \beta_{2} + 12 \beta_{3} + 6 \beta_{7} ) q^{94} + ( 4 \beta_{1} + 2 \beta_{4} - 2 \beta_{6} ) q^{95} + ( -4 \beta_{2} - 4 \beta_{3} - 4 \beta_{7} ) q^{97} + ( -\beta_{1} - 2 \beta_{4} + 4 \beta_{5} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 4q^{4} + O(q^{10}) \) \( 8q + 4q^{4} + 16q^{10} - 8q^{19} - 8q^{22} - 32q^{28} - 24q^{31} - 40q^{34} + 16q^{37} - 24q^{40} - 24q^{43} + 32q^{46} - 8q^{49} + 56q^{52} + 40q^{58} + 16q^{61} + 40q^{64} + 32q^{67} - 8q^{70} - 56q^{76} + 56q^{79} - 40q^{82} - 16q^{85} - 16q^{88} + 56q^{91} + 16q^{94} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 2 x^{6} + 2 x^{4} - 8 x^{2} + 16\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)
\(\beta_{3}\)\(=\)\((\)\( \nu^{6} + 2 \nu^{4} - 2 \nu^{2} - 4 \)\()/12\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{7} + 2 \nu^{5} - 2 \nu^{3} - 4 \nu \)\()/12\)
\(\beta_{5}\)\(=\)\((\)\( -\nu^{7} + 4 \nu^{5} + 2 \nu^{3} + 4 \nu \)\()/12\)
\(\beta_{6}\)\(=\)\((\)\( \nu^{7} - \nu^{5} + 4 \nu^{3} - 4 \nu \)\()/6\)
\(\beta_{7}\)\(=\)\((\)\( -\nu^{6} + \nu^{4} - \nu^{2} + 4 \)\()/3\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\)
\(\nu^{3}\)\(=\)\(\beta_{6} + \beta_{5} - \beta_{4}\)
\(\nu^{4}\)\(=\)\(\beta_{7} + 4 \beta_{3} + \beta_{2}\)
\(\nu^{5}\)\(=\)\(2 \beta_{5} + 2 \beta_{4}\)
\(\nu^{6}\)\(=\)\(-2 \beta_{7} + 4 \beta_{3} + 4\)
\(\nu^{7}\)\(=\)\(2 \beta_{6} - 2 \beta_{5} + 6 \beta_{4} + 4 \beta_{1}\)

Character values

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

\(n\) \(37\) \(65\) \(127\)
\(\chi(n)\) \(\beta_{3}\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1
−1.38255 0.297594i
−0.767178 + 1.18804i
0.767178 1.18804i
1.38255 + 0.297594i
−1.38255 + 0.297594i
−0.767178 1.18804i
0.767178 + 1.18804i
1.38255 0.297594i
−1.38255 0.297594i 0 1.82288 + 0.822876i 0.595188 + 0.595188i 0 1.64575i −2.27533 1.68014i 0 −0.645751 1.00000i
37.2 −0.767178 + 1.18804i 0 −0.822876 1.82288i −2.37608 2.37608i 0 3.64575i 2.79694 + 0.420861i 0 4.64575 1.00000i
37.3 0.767178 1.18804i 0 −0.822876 1.82288i 2.37608 + 2.37608i 0 3.64575i −2.79694 0.420861i 0 4.64575 1.00000i
37.4 1.38255 + 0.297594i 0 1.82288 + 0.822876i −0.595188 0.595188i 0 1.64575i 2.27533 + 1.68014i 0 −0.645751 1.00000i
109.1 −1.38255 + 0.297594i 0 1.82288 0.822876i 0.595188 0.595188i 0 1.64575i −2.27533 + 1.68014i 0 −0.645751 + 1.00000i
109.2 −0.767178 1.18804i 0 −0.822876 + 1.82288i −2.37608 + 2.37608i 0 3.64575i 2.79694 0.420861i 0 4.64575 + 1.00000i
109.3 0.767178 + 1.18804i 0 −0.822876 + 1.82288i 2.37608 2.37608i 0 3.64575i −2.79694 + 0.420861i 0 4.64575 + 1.00000i
109.4 1.38255 0.297594i 0 1.82288 0.822876i −0.595188 + 0.595188i 0 1.64575i 2.27533 1.68014i 0 −0.645751 + 1.00000i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 109.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
16.e even 4 1 inner
48.i odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 144.2.k.c 8
3.b odd 2 1 inner 144.2.k.c 8
4.b odd 2 1 576.2.k.c 8
8.b even 2 1 1152.2.k.e 8
8.d odd 2 1 1152.2.k.d 8
12.b even 2 1 576.2.k.c 8
16.e even 4 1 inner 144.2.k.c 8
16.e even 4 1 1152.2.k.e 8
16.f odd 4 1 576.2.k.c 8
16.f odd 4 1 1152.2.k.d 8
24.f even 2 1 1152.2.k.d 8
24.h odd 2 1 1152.2.k.e 8
32.g even 8 2 9216.2.a.bq 8
32.h odd 8 2 9216.2.a.bt 8
48.i odd 4 1 inner 144.2.k.c 8
48.i odd 4 1 1152.2.k.e 8
48.k even 4 1 576.2.k.c 8
48.k even 4 1 1152.2.k.d 8
96.o even 8 2 9216.2.a.bt 8
96.p odd 8 2 9216.2.a.bq 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
144.2.k.c 8 1.a even 1 1 trivial
144.2.k.c 8 3.b odd 2 1 inner
144.2.k.c 8 16.e even 4 1 inner
144.2.k.c 8 48.i odd 4 1 inner
576.2.k.c 8 4.b odd 2 1
576.2.k.c 8 12.b even 2 1
576.2.k.c 8 16.f odd 4 1
576.2.k.c 8 48.k even 4 1
1152.2.k.d 8 8.d odd 2 1
1152.2.k.d 8 16.f odd 4 1
1152.2.k.d 8 24.f even 2 1
1152.2.k.d 8 48.k even 4 1
1152.2.k.e 8 8.b even 2 1
1152.2.k.e 8 16.e even 4 1
1152.2.k.e 8 24.h odd 2 1
1152.2.k.e 8 48.i odd 4 1
9216.2.a.bq 8 32.g even 8 2
9216.2.a.bq 8 96.p odd 8 2
9216.2.a.bt 8 32.h odd 8 2
9216.2.a.bt 8 96.o even 8 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} + 128 T_{5}^{4} + 64 \) acting on \(S_{2}^{\mathrm{new}}(144, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 2 T^{2} + 2 T^{4} - 8 T^{6} + 16 T^{8} \)
$3$ 1
$5$ \( 1 - 12 T^{4} - 506 T^{8} - 7500 T^{12} + 390625 T^{16} \)
$7$ \( ( 1 - 12 T^{2} + 106 T^{4} - 588 T^{6} + 2401 T^{8} )^{2} \)
$11$ \( 1 - 60 T^{4} - 14618 T^{8} - 878460 T^{12} + 214358881 T^{16} \)
$13$ \( ( 1 - 194 T^{4} + 28561 T^{8} )^{2} \)
$17$ \( ( 1 + 28 T^{2} + 662 T^{4} + 8092 T^{6} + 83521 T^{8} )^{2} \)
$19$ \( ( 1 + 4 T + 8 T^{2} + 28 T^{3} - 46 T^{4} + 532 T^{5} + 2888 T^{6} + 27436 T^{7} + 130321 T^{8} )^{2} \)
$23$ \( ( 1 - 12 T^{2} + 646 T^{4} - 6348 T^{6} + 279841 T^{8} )^{2} \)
$29$ \( 1 + 180 T^{4} - 772538 T^{8} + 127310580 T^{12} + 500246412961 T^{16} \)
$31$ \( ( 1 + 6 T + 64 T^{2} + 186 T^{3} + 961 T^{4} )^{4} \)
$37$ \( ( 1 - 8 T + 32 T^{2} - 248 T^{3} + 1886 T^{4} - 9176 T^{5} + 43808 T^{6} - 405224 T^{7} + 1874161 T^{8} )^{2} \)
$41$ \( ( 1 - 60 T^{2} + 4150 T^{4} - 100860 T^{6} + 2825761 T^{8} )^{2} \)
$43$ \( ( 1 + 12 T + 72 T^{2} + 564 T^{3} + 4402 T^{4} + 24252 T^{5} + 133128 T^{6} + 954084 T^{7} + 3418801 T^{8} )^{2} \)
$47$ \( ( 1 - 20 T^{2} + 4070 T^{4} - 44180 T^{6} + 4879681 T^{8} )^{2} \)
$53$ \( 1 - 5772 T^{4} + 23807110 T^{8} - 45543856332 T^{12} + 62259690411361 T^{16} \)
$59$ \( 1 - 7452 T^{4} + 27767206 T^{8} - 90298574172 T^{12} + 146830437604321 T^{16} \)
$61$ \( ( 1 - 8 T + 32 T^{2} - 440 T^{3} + 6014 T^{4} - 26840 T^{5} + 119072 T^{6} - 1815848 T^{7} + 13845841 T^{8} )^{2} \)
$67$ \( ( 1 - 8 T + 32 T^{2} - 536 T^{3} + 4489 T^{4} )^{4} \)
$71$ \( ( 1 - 92 T^{2} + 5030 T^{4} - 463772 T^{6} + 25411681 T^{8} )^{2} \)
$73$ \( ( 1 - 228 T^{2} + 23206 T^{4} - 1215012 T^{6} + 28398241 T^{8} )^{2} \)
$79$ \( ( 1 - 14 T + 200 T^{2} - 1106 T^{3} + 6241 T^{4} )^{4} \)
$83$ \( 1 + 20100 T^{4} + 185294374 T^{8} + 953912252100 T^{12} + 2252292232139041 T^{16} \)
$89$ \( ( 1 - 260 T^{2} + 30950 T^{4} - 2059460 T^{6} + 62742241 T^{8} )^{2} \)
$97$ \( ( 1 + 82 T^{2} + 9409 T^{4} )^{4} \)
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