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Properties

Label 216.3.m.b
Level $216$
Weight $3$
Character orbit 216.m
Analytic conductor $5.886$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 216 = 2^{3} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 216.m (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(5.88557371018\)
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: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{2} \)
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.

\(f(q)\) \(=\) \( q + (\beta_{4} - \beta_{2} - \beta_1 + 1) q^{5} + (\beta_{6} - 2 \beta_{2} - \beta_1 + 1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{4} - \beta_{2} - \beta_1 + 1) q^{5} + (\beta_{6} - 2 \beta_{2} - \beta_1 + 1) q^{7} + (2 \beta_{7} - \beta_{6} - 2 \beta_{5} + 3 \beta_{3} - 4 \beta_{2} - \beta_1 - 2) q^{11} + (2 \beta_{7} - 2 \beta_{6} - 6 \beta_{5} - 3 \beta_{4} + 2 \beta_{3} + 3 \beta_{2} + \cdots + 1) q^{13}+ \cdots + (2 \beta_{6} - 6 \beta_{3} + 67 \beta_{2} + 10 \beta_1 - 63) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 6 q^{5} + 6 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 6 q^{5} + 6 q^{7} - 36 q^{11} + 14 q^{13} + 4 q^{19} + 102 q^{23} + 10 q^{25} + 114 q^{29} - 50 q^{31} + 120 q^{37} - 264 q^{41} - 28 q^{43} - 150 q^{47} + 94 q^{49} - 244 q^{55} + 108 q^{59} + 14 q^{61} + 198 q^{65} - 20 q^{67} - 76 q^{73} - 66 q^{77} + 26 q^{79} - 246 q^{83} - 224 q^{85} + 108 q^{91} + 456 q^{95} - 236 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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

\(\beta_{1}\)\(=\) \( ( -473\nu^{7} + 198\nu^{6} - 5547\nu^{5} - 7826\nu^{4} - 75285\nu^{3} - 29928\nu^{2} + 140724\nu - 256788 ) / 159300 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 677\nu^{7} - 1827\nu^{6} + 10353\nu^{5} - 6901\nu^{4} + 82215\nu^{3} - 132153\nu^{2} + 156924\nu + 78912 ) / 159300 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -473\nu^{7} + 198\nu^{6} - 5547\nu^{5} - 7826\nu^{4} - 75285\nu^{3} - 29928\nu^{2} - 98226\nu - 177138 ) / 79650 \) 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
\(\beta_{5}\)\(=\) \( ( 644\nu^{7} - 2019\nu^{6} + 9966\nu^{5} - 7447\nu^{4} + 68730\nu^{3} - 107691\nu^{2} + 129078\nu + 194364 ) / 79650 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 2407 \nu^{7} + 7182 \nu^{6} - 45123 \nu^{5} + 47066 \nu^{4} - 389565 \nu^{3} + 475248 \nu^{2} - 1245834 \nu + 209808 ) / 159300 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 5753 \nu^{7} - 4878 \nu^{6} + 67467 \nu^{5} + 95186 \nu^{4} + 657585 \nu^{3} + 364008 \nu^{2} + 544536 \nu + 2015568 ) / 318600 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( -\beta_{3} + 2\beta _1 + 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 6\beta_{5} + 3\beta_{4} - 2\beta_{3} - 18\beta_{2} + \beta _1 - 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -6\beta_{7} + 6\beta_{5} - 6\beta_{4} - 13\beta_{3} - 7\beta _1 - 32 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -12\beta_{6} - 45\beta_{5} - 90\beta_{4} + 25\beta_{3} + 186\beta_{2} - 38\beta _1 - 199 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 90\beta_{7} - 90\beta_{6} - 240\beta_{5} - 120\beta_{4} + 338\beta_{3} + 288\beta_{2} - 169\beta _1 + 169 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 240\beta_{7} - 627\beta_{5} + 627\beta_{4} + 445\beta_{3} + 205\beta _1 + 2912 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 1254\beta_{6} + 1962\beta_{5} + 3924\beta_{4} - 2293\beta_{3} - 5316\beta_{2} + 3332\beta _1 + 6355 ) / 3 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/216\mathbb{Z}\right)^\times\).

\(n\) \(55\) \(109\) \(137\)
\(\chi(n)\) \(1\) \(1\) \(\beta_{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1
0.831167 + 1.43962i
1.91950 + 3.32468i
−0.331167 − 0.573598i
−1.41950 − 2.45865i
0.831167 − 1.43962i
1.91950 − 3.32468i
−0.331167 + 0.573598i
−1.41950 + 2.45865i
0 0 0 −3.44299 − 1.98781i 0 −1.80469 − 3.12582i 0 0 0
17.2 0 0 0 −1.80902 − 1.04444i 0 −0.781452 − 1.35351i 0 0 0
17.3 0 0 0 0.0440114 + 0.0254100i 0 4.52944 + 7.84521i 0 0 0
17.4 0 0 0 8.20800 + 4.73889i 0 1.05671 + 1.83027i 0 0 0
89.1 0 0 0 −3.44299 + 1.98781i 0 −1.80469 + 3.12582i 0 0 0
89.2 0 0 0 −1.80902 + 1.04444i 0 −0.781452 + 1.35351i 0 0 0
89.3 0 0 0 0.0440114 − 0.0254100i 0 4.52944 − 7.84521i 0 0 0
89.4 0 0 0 8.20800 − 4.73889i 0 1.05671 − 1.83027i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 89.4
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 216.3.m.b 8
3.b odd 2 1 72.3.m.b ✓ 8
4.b odd 2 1 432.3.q.e 8
8.b even 2 1 1728.3.q.j 8
8.d odd 2 1 1728.3.q.i 8
9.c even 3 1 72.3.m.b ✓ 8
9.c even 3 1 648.3.e.c 8
9.d odd 6 1 inner 216.3.m.b 8
9.d odd 6 1 648.3.e.c 8
12.b even 2 1 144.3.q.e 8
24.f even 2 1 576.3.q.j 8
24.h odd 2 1 576.3.q.i 8
36.f odd 6 1 144.3.q.e 8
36.f odd 6 1 1296.3.e.i 8
36.h even 6 1 432.3.q.e 8
36.h even 6 1 1296.3.e.i 8
72.j odd 6 1 1728.3.q.j 8
72.l even 6 1 1728.3.q.i 8
72.n even 6 1 576.3.q.i 8
72.p odd 6 1 576.3.q.j 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.3.m.b ✓ 8 3.b odd 2 1
72.3.m.b ✓ 8 9.c even 3 1
144.3.q.e 8 12.b even 2 1
144.3.q.e 8 36.f odd 6 1
216.3.m.b 8 1.a even 1 1 trivial
216.3.m.b 8 9.d odd 6 1 inner
432.3.q.e 8 4.b odd 2 1
432.3.q.e 8 36.h even 6 1
576.3.q.i 8 24.h odd 2 1
576.3.q.i 8 72.n even 6 1
576.3.q.j 8 24.f even 2 1
576.3.q.j 8 72.p odd 6 1
648.3.e.c 8 9.c even 3 1
648.3.e.c 8 9.d odd 6 1
1296.3.e.i 8 36.f odd 6 1
1296.3.e.i 8 36.h even 6 1
1728.3.q.i 8 8.d odd 2 1
1728.3.q.i 8 72.l even 6 1
1728.3.q.j 8 8.b even 2 1
1728.3.q.j 8 72.j 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 \(S_{3}^{\mathrm{new}}(216, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} \) 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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