Properties

Label 864.2.bh.a
Level 864
Weight 2
Character orbit 864.bh
Analytic conductor 6.899
Analytic rank 0
Dimension 12
CM discriminant -8
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 864 = 2^{5} \cdot 3^{3} \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 864.bh (of order \(18\), degree \(6\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.89907473464\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(2\) over \(\Q(\zeta_{18})\)
Coefficient field: 12.0.101559956668416.1
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 216)
Sato-Tate group: $\mathrm{U}(1)[D_{18}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -\beta_{4} - \beta_{7} + \beta_{10} ) q^{3} + ( -\beta_{2} + 2 \beta_{5} + \beta_{8} ) q^{9} +O(q^{10})\) \( q + ( -\beta_{4} - \beta_{7} + \beta_{10} ) q^{3} + ( -\beta_{2} + 2 \beta_{5} + \beta_{8} ) q^{9} + ( -\beta_{3} + \beta_{5} - 3 \beta_{6} - 3 \beta_{8} + \beta_{9} - \beta_{11} ) q^{11} + ( -2 \beta_{1} + 3 \beta_{2} - 3 \beta_{10} + 2 \beta_{11} ) q^{17} + ( -3 \beta_{1} - \beta_{2} + 3 \beta_{5} + \beta_{8} + \beta_{10} ) q^{19} -5 \beta_{4} q^{25} + ( 5 - \beta_{3} - 5 \beta_{6} ) q^{27} + ( -2 \beta_{1} - 4 \beta_{3} + 5 \beta_{4} + \beta_{6} + 2 \beta_{7} + 4 \beta_{9} ) q^{33} + ( -3 - 4 \beta_{3} + 3 \beta_{4} + 3 \beta_{6} - 4 \beta_{7} - 3 \beta_{10} ) q^{41} + ( -5 - 5 \beta_{2} - 3 \beta_{3} + 5 \beta_{6} + 3 \beta_{11} ) q^{43} -7 \beta_{2} q^{49} + ( 1 - \beta_{5} + 7 \beta_{8} - 5 \beta_{9} + \beta_{11} ) q^{51} + ( 7 - 2 \beta_{3} + 4 \beta_{5} - 7 \beta_{6} + 5 \beta_{8} - 4 \beta_{11} ) q^{57} + ( -3 - 5 \beta_{1} + 3 \beta_{4} + 5 \beta_{7} + 5 \beta_{9} ) q^{59} + ( -3 \beta_{1} - 3 \beta_{3} - 7 \beta_{4} + 7 \beta_{6} + 3 \beta_{7} + 3 \beta_{9} ) q^{67} + ( -\beta_{4} - 6 \beta_{5} + 6 \beta_{7} + \beta_{8} + \beta_{10} + 6 \beta_{11} ) q^{73} + ( 5 \beta_{2} + 5 \beta_{11} ) q^{75} + ( -4 \beta_{1} + 7 \beta_{10} ) q^{81} + ( -\beta_{1} + 9 \beta_{4} + \beta_{7} - 18 \beta_{10} ) q^{83} + ( -9 - 2 \beta_{3} - 9 \beta_{6} ) q^{89} + ( 6 \beta_{3} + 6 \beta_{5} + 5 \beta_{6} - 5 \beta_{8} - 6 \beta_{9} - 6 \beta_{11} ) q^{97} + ( 5 \beta_{1} - \beta_{2} + 7 \beta_{4} - 5 \beta_{7} - 7 \beta_{11} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q + O(q^{10}) \) \( 12q - 18q^{11} + 30q^{27} + 6q^{33} - 18q^{41} - 30q^{43} + 12q^{51} + 42q^{57} - 36q^{59} + 42q^{67} - 162q^{89} + 30q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} - 8 x^{6} + 64\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)\(/2\)
\(\beta_{3}\)\(=\)\( \nu^{3} \)\(/2\)
\(\beta_{4}\)\(=\)\( \nu^{4} \)\(/4\)
\(\beta_{5}\)\(=\)\( \nu^{5} \)\(/4\)
\(\beta_{6}\)\(=\)\( \nu^{6} \)\(/8\)
\(\beta_{7}\)\(=\)\( \nu^{7} \)\(/8\)
\(\beta_{8}\)\(=\)\( \nu^{8} \)\(/16\)
\(\beta_{9}\)\(=\)\( \nu^{9} \)\(/16\)
\(\beta_{10}\)\(=\)\( \nu^{10} \)\(/32\)
\(\beta_{11}\)\(=\)\( \nu^{11} \)\(/32\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(2 \beta_{2}\)
\(\nu^{3}\)\(=\)\(2 \beta_{3}\)
\(\nu^{4}\)\(=\)\(4 \beta_{4}\)
\(\nu^{5}\)\(=\)\(4 \beta_{5}\)
\(\nu^{6}\)\(=\)\(8 \beta_{6}\)
\(\nu^{7}\)\(=\)\(8 \beta_{7}\)
\(\nu^{8}\)\(=\)\(16 \beta_{8}\)
\(\nu^{9}\)\(=\)\(16 \beta_{9}\)
\(\nu^{10}\)\(=\)\(32 \beta_{10}\)
\(\nu^{11}\)\(=\)\(32 \beta_{11}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/864\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(353\) \(703\)
\(\chi(n)\) \(-1\) \(-\beta_{8}\) \(-1\)

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
0.909039 1.08335i
−0.909039 + 1.08335i
0.909039 + 1.08335i
−0.909039 1.08335i
1.39273 + 0.245576i
−1.39273 0.245576i
−0.483690 + 1.32893i
0.483690 1.32893i
−0.483690 1.32893i
0.483690 + 1.32893i
1.39273 0.245576i
−1.39273 + 0.245576i
0 −1.21908 1.23038i 0 0 0 0 0 −0.0276864 + 2.99987i 0
47.2 0 1.56638 0.739232i 0 0 0 0 0 1.90707 2.31583i 0
239.1 0 −1.21908 + 1.23038i 0 0 0 0 0 −0.0276864 2.99987i 0
239.2 0 1.56638 + 0.739232i 0 0 0 0 0 1.90707 + 2.31583i 0
335.1 0 −1.42338 0.986906i 0 0 0 0 0 1.05203 + 2.80949i 0
335.2 0 −0.456003 + 1.67095i 0 0 0 0 0 −2.58412 1.52391i 0
527.1 0 −0.142995 1.72614i 0 0 0 0 0 −2.95911 + 0.493657i 0
527.2 0 1.67508 + 0.440563i 0 0 0 0 0 2.61181 + 1.47596i 0
623.1 0 −0.142995 + 1.72614i 0 0 0 0 0 −2.95911 0.493657i 0
623.2 0 1.67508 0.440563i 0 0 0 0 0 2.61181 1.47596i 0
815.1 0 −1.42338 + 0.986906i 0 0 0 0 0 1.05203 2.80949i 0
815.2 0 −0.456003 1.67095i 0 0 0 0 0 −2.58412 + 1.52391i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 815.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 CM by \(\Q(\sqrt{-2}) \)
27.f odd 18 1 inner
216.v even 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 864.2.bh.a 12
4.b odd 2 1 216.2.v.a 12
8.b even 2 1 216.2.v.a 12
8.d odd 2 1 CM 864.2.bh.a 12
12.b even 2 1 648.2.v.a 12
24.h odd 2 1 648.2.v.a 12
27.f odd 18 1 inner 864.2.bh.a 12
108.j odd 18 1 648.2.v.a 12
108.l even 18 1 216.2.v.a 12
216.t even 18 1 648.2.v.a 12
216.v even 18 1 inner 864.2.bh.a 12
216.x odd 18 1 216.2.v.a 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
216.2.v.a 12 4.b odd 2 1
216.2.v.a 12 8.b even 2 1
216.2.v.a 12 108.l even 18 1
216.2.v.a 12 216.x odd 18 1
648.2.v.a 12 12.b even 2 1
648.2.v.a 12 24.h odd 2 1
648.2.v.a 12 108.j odd 18 1
648.2.v.a 12 216.t even 18 1
864.2.bh.a 12 1.a even 1 1 trivial
864.2.bh.a 12 8.d odd 2 1 CM
864.2.bh.a 12 27.f odd 18 1 inner
864.2.bh.a 12 216.v even 18 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} \) acting on \(S_{2}^{\mathrm{new}}(864, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( \)
$3$ \( 1 - 10 T^{3} + 73 T^{6} - 270 T^{9} + 729 T^{12} \)
$5$ \( ( 1 - 125 T^{6} + 15625 T^{12} )^{2} \)
$7$ \( ( 1 + 343 T^{6} + 117649 T^{12} )^{2} \)
$11$ \( ( 1 + 6 T + 25 T^{2} + 66 T^{3} + 121 T^{4} )^{3}( 1 - 18 T^{3} - 1007 T^{6} - 23958 T^{9} + 1771561 T^{12} ) \)
$13$ \( ( 1 + 2197 T^{6} + 4826809 T^{12} )^{2} \)
$17$ \( ( 1 - 90 T^{3} + 3187 T^{6} - 442170 T^{9} + 24137569 T^{12} )( 1 + 90 T^{3} + 3187 T^{6} + 442170 T^{9} + 24137569 T^{12} ) \)
$19$ \( ( 1 + 106 T^{3} + 4377 T^{6} + 727054 T^{9} + 47045881 T^{12} )^{2} \)
$23$ \( ( 1 - 12167 T^{6} + 148035889 T^{12} )^{2} \)
$29$ \( ( 1 - 24389 T^{6} + 594823321 T^{12} )^{2} \)
$31$ \( ( 1 + 29791 T^{6} + 887503681 T^{12} )^{2} \)
$37$ \( ( 1 + 37 T^{2} + 1369 T^{4} )^{6} \)
$41$ \( ( 1 + 6 T - 5 T^{2} + 246 T^{3} + 1681 T^{4} )^{3}( 1 + 522 T^{3} + 203563 T^{6} + 35976762 T^{9} + 4750104241 T^{12} ) \)
$43$ \( ( 1 + 10 T + 57 T^{2} + 430 T^{3} + 1849 T^{4} )^{3}( 1 - 290 T^{3} + 4593 T^{6} - 23057030 T^{9} + 6321363049 T^{12} ) \)
$47$ \( ( 1 - 103823 T^{6} + 10779215329 T^{12} )^{2} \)
$53$ \( ( 1 + 53 T^{2} )^{12} \)
$59$ \( ( 1 + 6 T + 59 T^{2} )^{6}( 1 - 846 T^{3} + 510337 T^{6} - 173750634 T^{9} + 42180533641 T^{12} ) \)
$61$ \( ( 1 + 226981 T^{6} + 51520374361 T^{12} )^{2} \)
$67$ \( ( 1 - 14 T + 129 T^{2} - 938 T^{3} + 4489 T^{4} )^{3}( 1 + 70 T^{3} - 295863 T^{6} + 21053410 T^{9} + 90458382169 T^{12} ) \)
$71$ \( ( 1 - 71 T^{2} + 5041 T^{4} )^{6} \)
$73$ \( ( 1 - 430 T^{3} - 204117 T^{6} - 167277310 T^{9} + 151334226289 T^{12} )^{2} \)
$79$ \( ( 1 + 493039 T^{6} + 243087455521 T^{12} )^{2} \)
$83$ \( ( 1 - 1350 T^{3} + 1250713 T^{6} - 771912450 T^{9} + 326940373369 T^{12} )( 1 + 1350 T^{3} + 1250713 T^{6} + 771912450 T^{9} + 326940373369 T^{12} ) \)
$89$ \( ( 1 + 18 T + 89 T^{2} )^{6}( 1 + 18 T + 235 T^{2} + 1602 T^{3} + 7921 T^{4} )^{3} \)
$97$ \( ( 1 - 10 T + 3 T^{2} - 970 T^{3} + 9409 T^{4} )^{3}( 1 + 1910 T^{3} + 2735427 T^{6} + 1743205430 T^{9} + 832972004929 T^{12} ) \)
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