Properties

Label 144.3.q.e
Level $144$
Weight $3$
Character orbit 144.q
Analytic conductor $3.924$
Analytic rank $0$
Dimension $8$
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.q (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.92371580679\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.19269881856.9
Defining polynomial: \( x^{8} - 2x^{7} + 15x^{6} - 2x^{5} + 133x^{4} - 84x^{3} + 276x^{2} + 144x + 144 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2}\cdot 3 \)
Twist minimal: no (minimal twist has level 72)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_{3} + \beta_{2} - 1) q^{3} + ( - \beta_{7} + \beta_{6} + \beta_{3} - \beta_{2} + \beta_1 - 1) q^{5} + ( - \beta_{7} - \beta_{5} + \beta_{3} + \beta_{2} + \beta_1 - 1) q^{7} + ( - \beta_{7} - 2 \beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} - 3 \beta_{2} - 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{3} + \beta_{2} - 1) q^{3} + ( - \beta_{7} + \beta_{6} + \beta_{3} - \beta_{2} + \beta_1 - 1) q^{5} + ( - \beta_{7} - \beta_{5} + \beta_{3} + \beta_{2} + \beta_1 - 1) q^{7} + ( - \beta_{7} - 2 \beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} - 3 \beta_{2} - 3) q^{9} + ( - 2 \beta_{7} + 4 \beta_{5} + 2 \beta_{4} - 2 \beta_{3} - 2 \beta_{2} + \beta_1 - 5) q^{11} + ( - \beta_{7} + 3 \beta_{6} + 2 \beta_{5} + 6 \beta_{4} + 3 \beta_{3} + \beta_{2} + \beta_1 - 1) q^{13} + (2 \beta_{7} + 2 \beta_{6} - 2 \beta_{5} + 4 \beta_{4} + 7 \beta_{3} + \beta_{2} + \beta_1 - 4) q^{15} + ( - 2 \beta_{7} + \beta_{6} - 3 \beta_{5} + \beta_{4} - \beta_{3} + 4 \beta_{2} + 5 \beta_1 + 1) q^{17} + ( - 6 \beta_{7} - 3 \beta_{5} + \beta_{3} - 2 \beta_{2} + \beta_1 + 1) q^{19} + (\beta_{7} - \beta_{5} + 3 \beta_{4} - 2 \beta_{3} - \beta_{2} - 2 \beta_1 - 5) q^{21} + (2 \beta_{7} - 4 \beta_{6} + 6 \beta_{5} - 5 \beta_{3} - 7 \beta_{2} + \beta_1 + 14) q^{23} + (\beta_{7} - 6 \beta_{6} + \beta_{5} - 3 \beta_{4} - 7 \beta_{3} + 9 \beta_{2} - 6 \beta_1 + 2) q^{25} + (\beta_{7} + 10 \beta_{6} - 2 \beta_{5} + 2 \beta_{4} + 3 \beta_{3} - \beta_1 + 2) q^{27} + (\beta_{7} + 3 \beta_{5} - 3 \beta_{4} - 10 \beta_{3} + \beta_{2} + 2 \beta_1 - 11) q^{29} + ( - 6 \beta_{6} - 12 \beta_{4} + 13 \beta_{3} - \beta_{2} - \beta_1) q^{31} + (4 \beta_{7} + \beta_{6} - 3 \beta_{5} - 7 \beta_{4} + 20 \beta_{3} - 6 \beta_{2} + \cdots - 20) q^{33}+ \cdots + ( - 9 \beta_{7} + 14 \beta_{6} - 16 \beta_{5} + 22 \beta_{4} + 10 \beta_{3} + \cdots + 80) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 10 q^{3} - 6 q^{5} - 6 q^{7} - 22 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 10 q^{3} - 6 q^{5} - 6 q^{7} - 22 q^{9} - 36 q^{11} + 14 q^{13} - 10 q^{15} - 4 q^{19} - 54 q^{21} + 102 q^{23} + 10 q^{25} + 20 q^{27} - 114 q^{29} + 50 q^{31} - 104 q^{33} + 120 q^{37} - 82 q^{39} + 264 q^{41} + 28 q^{43} + 206 q^{45} - 150 q^{47} + 94 q^{49} - 170 q^{51} + 244 q^{55} - 178 q^{57} + 108 q^{59} + 14 q^{61} + 210 q^{63} - 198 q^{65} + 20 q^{67} - 14 q^{69} - 76 q^{73} - 326 q^{75} + 66 q^{77} - 26 q^{79} + 194 q^{81} - 246 q^{83} - 224 q^{85} + 18 q^{87} - 108 q^{91} - 130 q^{93} + 456 q^{95} - 236 q^{97} + 634 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} + 15x^{6} - 2x^{5} + 133x^{4} - 84x^{3} + 276x^{2} + 144x + 144 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -6\nu^{7} - 169\nu^{6} + 466\nu^{5} - 2647\nu^{4} + 3180\nu^{3} - 20091\nu^{2} + 64128\nu - 19236 ) / 17700 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 331\nu^{7} - 756\nu^{6} + 8709\nu^{5} - 4178\nu^{4} + 73845\nu^{3} + 16116\nu^{2} + 432972\nu + 72936 ) / 159300 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 677 \nu^{7} + 1827 \nu^{6} - 10353 \nu^{5} + 6901 \nu^{4} - 82215 \nu^{3} + 132153 \nu^{2} - 156924 \nu + 80388 ) / 159300 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 1013 \nu^{7} - 2688 \nu^{6} + 16707 \nu^{5} - 19444 \nu^{4} + 143085 \nu^{3} - 232782 \nu^{2} + 247356 \nu - 401472 ) / 159300 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -602\nu^{7} + 252\nu^{6} - 5853\nu^{5} - 12374\nu^{4} - 64440\nu^{3} - 39297\nu^{2} + 41526\nu - 95112 ) / 79650 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 644\nu^{7} - 2019\nu^{6} + 9966\nu^{5} - 7447\nu^{4} + 68730\nu^{3} - 107691\nu^{2} + 129078\nu + 194364 ) / 79650 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 1004 \nu^{7} + 729 \nu^{6} - 12981 \nu^{5} - 14198 \nu^{4} - 139005 \nu^{3} - 62319 \nu^{2} - 104598 \nu - 259974 ) / 79650 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{7} - 2\beta_{5} + \beta_{3} + \beta_{2} + \beta _1 + 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{7} + 6\beta_{6} - \beta_{5} + 3\beta_{4} + 20\beta_{3} - \beta_{2} + 2\beta _1 - 19 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -20\beta_{7} + 3\beta_{6} + 19\beta_{5} - 3\beta_{4} + 13\beta_{3} - 26\beta_{2} + 13\beta _1 - 38 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -25\beta_{7} - 39\beta_{6} + 50\beta_{5} - 78\beta_{4} - 211\beta_{3} - 13\beta_{2} - 13\beta _1 - 25 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 79\beta_{7} - 150\beta_{6} + 79\beta_{5} - 75\beta_{4} - 626\beta_{3} + 259\beta_{2} - 248\beta _1 + 457 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 650\beta_{7} - 507\beta_{6} - 685\beta_{5} + 507\beta_{4} - 445\beta_{3} + 890\beta_{2} - 445\beta _1 + 3152 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 2293 \beta_{7} + 1335 \beta_{6} - 4586 \beta_{5} + 2670 \beta_{4} + 7609 \beta_{3} + 1039 \beta_{2} + 1039 \beta _1 + 2293 ) / 3 \) Copy content Toggle raw display

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
65.1
−1.41950 2.45865i
−0.331167 0.573598i
0.831167 + 1.43962i
1.91950 + 3.32468i
−1.41950 + 2.45865i
−0.331167 + 0.573598i
0.831167 1.43962i
1.91950 3.32468i
0 −2.91950 + 0.690286i 0 1.80902 1.04444i 0 0.781452 1.35351i 0 8.04701 4.03058i 0
65.2 0 −1.83117 2.37631i 0 3.44299 1.98781i 0 1.80469 3.12582i 0 −2.29365 + 8.70282i 0
65.3 0 −0.668833 + 2.92449i 0 −0.0440114 + 0.0254100i 0 −4.52944 + 7.84521i 0 −8.10532 3.91200i 0
65.4 0 0.419504 2.97052i 0 −8.20800 + 4.73889i 0 −1.05671 + 1.83027i 0 −8.64803 2.49230i 0
113.1 0 −2.91950 0.690286i 0 1.80902 + 1.04444i 0 0.781452 + 1.35351i 0 8.04701 + 4.03058i 0
113.2 0 −1.83117 + 2.37631i 0 3.44299 + 1.98781i 0 1.80469 + 3.12582i 0 −2.29365 8.70282i 0
113.3 0 −0.668833 2.92449i 0 −0.0440114 0.0254100i 0 −4.52944 7.84521i 0 −8.10532 + 3.91200i 0
113.4 0 0.419504 + 2.97052i 0 −8.20800 4.73889i 0 −1.05671 1.83027i 0 −8.64803 + 2.49230i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 113.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 144.3.q.e 8
3.b odd 2 1 432.3.q.e 8
4.b odd 2 1 72.3.m.b 8
8.b even 2 1 576.3.q.j 8
8.d odd 2 1 576.3.q.i 8
9.c even 3 1 432.3.q.e 8
9.c even 3 1 1296.3.e.i 8
9.d odd 6 1 inner 144.3.q.e 8
9.d odd 6 1 1296.3.e.i 8
12.b even 2 1 216.3.m.b 8
24.f even 2 1 1728.3.q.j 8
24.h odd 2 1 1728.3.q.i 8
36.f odd 6 1 216.3.m.b 8
36.f odd 6 1 648.3.e.c 8
36.h even 6 1 72.3.m.b 8
36.h even 6 1 648.3.e.c 8
72.j odd 6 1 576.3.q.j 8
72.l even 6 1 576.3.q.i 8
72.n even 6 1 1728.3.q.i 8
72.p odd 6 1 1728.3.q.j 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.3.m.b 8 4.b odd 2 1
72.3.m.b 8 36.h even 6 1
144.3.q.e 8 1.a even 1 1 trivial
144.3.q.e 8 9.d odd 6 1 inner
216.3.m.b 8 12.b even 2 1
216.3.m.b 8 36.f odd 6 1
432.3.q.e 8 3.b odd 2 1
432.3.q.e 8 9.c even 3 1
576.3.q.i 8 8.d odd 2 1
576.3.q.i 8 72.l even 6 1
576.3.q.j 8 8.b even 2 1
576.3.q.j 8 72.j odd 6 1
648.3.e.c 8 36.f odd 6 1
648.3.e.c 8 36.h even 6 1
1296.3.e.i 8 9.c even 3 1
1296.3.e.i 8 9.d odd 6 1
1728.3.q.i 8 24.h odd 2 1
1728.3.q.i 8 72.n even 6 1
1728.3.q.j 8 24.f even 2 1
1728.3.q.j 8 72.p odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} + 6T_{5}^{7} - 37T_{5}^{6} - 294T_{5}^{5} + 2661T_{5}^{4} - 6468T_{5}^{3} + 5612T_{5}^{2} + 528T_{5} + 16 \) acting on \(S_{3}^{\mathrm{new}}(144, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} + 10 T^{7} + 61 T^{6} + \cdots + 6561 \) Copy content Toggle raw display
$5$ \( T^{8} + 6 T^{7} - 37 T^{6} - 294 T^{5} + \cdots + 16 \) Copy content Toggle raw display
$7$ \( T^{8} + 6 T^{7} + 69 T^{6} + \cdots + 11664 \) Copy content Toggle raw display
$11$ \( T^{8} + 36 T^{7} + \cdots + 105616729 \) Copy content Toggle raw display
$13$ \( T^{8} - 14 T^{7} + 547 T^{6} + \cdots + 2611456 \) Copy content Toggle raw display
$17$ \( T^{8} + 1454 T^{6} + \cdots + 7020428944 \) Copy content Toggle raw display
$19$ \( (T^{4} + 2 T^{3} - 1179 T^{2} + \cdots + 226348)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} - 102 T^{7} + \cdots + 11198718976 \) Copy content Toggle raw display
$29$ \( T^{8} + 114 T^{7} + \cdots + 106450807824 \) Copy content Toggle raw display
$31$ \( T^{8} - 50 T^{7} + \cdots + 152712134656 \) Copy content Toggle raw display
$37$ \( (T^{4} - 60 T^{3} - 276 T^{2} + \cdots + 206496)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} - 264 T^{7} + \cdots + 1919025613521 \) Copy content Toggle raw display
$43$ \( T^{8} - 28 T^{7} + \cdots + 1352729498761 \) Copy content Toggle raw display
$47$ \( T^{8} + 150 T^{7} + \cdots + 4615347568896 \) Copy content Toggle raw display
$53$ \( T^{8} + 7016 T^{6} + \cdots + 78435844096 \) Copy content Toggle raw display
$59$ \( T^{8} - 108 T^{7} + \cdots + 127589696809 \) Copy content Toggle raw display
$61$ \( T^{8} - 14 T^{7} + \cdots + 133593174016 \) Copy content Toggle raw display
$67$ \( T^{8} - 20 T^{7} + \cdots + 17391015625 \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 114698616545536 \) Copy content Toggle raw display
$73$ \( (T^{4} + 38 T^{3} - 9831 T^{2} + \cdots + 2961976)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 103529078405776 \) Copy content Toggle raw display
$83$ \( T^{8} + 246 T^{7} + \cdots + 1085363908864 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 309931236458496 \) Copy content Toggle raw display
$97$ \( T^{8} + 236 T^{7} + \cdots + 3435006304129 \) Copy content Toggle raw display
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