Properties

Label 912.2.bo.d
Level $912$
Weight $2$
Character orbit 912.bo
Analytic conductor $7.282$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 912 = 2^{4} \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 912.bo (of order \(9\), degree \(6\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(7.28235666434\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\Q(\zeta_{18})\)
Defining polynomial: \(x^{6} - x^{3} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 114)
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{18}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\zeta_{18} q^{3} + ( 1 - \zeta_{18}^{2} - \zeta_{18}^{3} - 2 \zeta_{18}^{4} ) q^{5} + ( 1 - \zeta_{18} + 3 \zeta_{18}^{2} - \zeta_{18}^{3} - 2 \zeta_{18}^{4} - 2 \zeta_{18}^{5} ) q^{7} + \zeta_{18}^{2} q^{9} +O(q^{10})\) \( q -\zeta_{18} q^{3} + ( 1 - \zeta_{18}^{2} - \zeta_{18}^{3} - 2 \zeta_{18}^{4} ) q^{5} + ( 1 - \zeta_{18} + 3 \zeta_{18}^{2} - \zeta_{18}^{3} - 2 \zeta_{18}^{4} - 2 \zeta_{18}^{5} ) q^{7} + \zeta_{18}^{2} q^{9} + ( -\zeta_{18} - 2 \zeta_{18}^{2} - 2 \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{11} + ( 1 + \zeta_{18} - 2 \zeta_{18}^{2} + \zeta_{18}^{3} - 2 \zeta_{18}^{4} + 2 \zeta_{18}^{5} ) q^{13} + ( -\zeta_{18} + \zeta_{18}^{3} + \zeta_{18}^{4} + 2 \zeta_{18}^{5} ) q^{15} + ( -2 \zeta_{18} + \zeta_{18}^{2} - \zeta_{18}^{3} + 2 \zeta_{18}^{4} ) q^{17} + ( 5 \zeta_{18} - 3 \zeta_{18}^{4} ) q^{19} + ( -2 - \zeta_{18} + \zeta_{18}^{2} - \zeta_{18}^{3} + \zeta_{18}^{4} + 2 \zeta_{18}^{5} ) q^{21} + ( -6 + 3 \zeta_{18} + 3 \zeta_{18}^{3} + 3 \zeta_{18}^{4} + 2 \zeta_{18}^{5} ) q^{23} + ( -4 - 4 \zeta_{18} - \zeta_{18}^{2} + 3 \zeta_{18}^{3} + \zeta_{18}^{4} + \zeta_{18}^{5} ) q^{25} -\zeta_{18}^{3} q^{27} + ( -\zeta_{18} + 2 \zeta_{18}^{2} - \zeta_{18}^{3} ) q^{29} + ( 5 - \zeta_{18} - 5 \zeta_{18}^{3} + \zeta_{18}^{4} + \zeta_{18}^{5} ) q^{31} + ( -1 + \zeta_{18}^{2} + 3 \zeta_{18}^{3} + 2 \zeta_{18}^{5} ) q^{33} + ( -7 \zeta_{18} - 4 \zeta_{18}^{2} - 4 \zeta_{18}^{3} + 4 \zeta_{18}^{5} ) q^{35} + ( -1 + 4 \zeta_{18} + 4 \zeta_{18}^{2} - 4 \zeta_{18}^{4} ) q^{37} + ( 2 - \zeta_{18} - \zeta_{18}^{2} - \zeta_{18}^{4} + 2 \zeta_{18}^{5} ) q^{39} + ( -3 - 3 \zeta_{18} - 2 \zeta_{18}^{2} + \zeta_{18}^{3} - \zeta_{18}^{5} ) q^{41} + ( -3 + 3 \zeta_{18}^{2} + 5 \zeta_{18}^{3} - 3 \zeta_{18}^{4} + 2 \zeta_{18}^{5} ) q^{43} + ( 2 + \zeta_{18}^{2} - 2 \zeta_{18}^{3} - \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{45} + ( -4 + 3 \zeta_{18} + 3 \zeta_{18}^{2} + 3 \zeta_{18}^{3} - 4 \zeta_{18}^{4} ) q^{47} + ( 2 \zeta_{18} - \zeta_{18}^{2} - 8 \zeta_{18}^{3} - \zeta_{18}^{4} + 2 \zeta_{18}^{5} ) q^{49} + ( 2 \zeta_{18}^{2} - \zeta_{18}^{3} + \zeta_{18}^{4} - 2 \zeta_{18}^{5} ) q^{51} + ( -2 \zeta_{18} + 2 \zeta_{18}^{3} + 2 \zeta_{18}^{4} - 9 \zeta_{18}^{5} ) q^{53} + ( -8 - 4 \zeta_{18} - 7 \zeta_{18}^{2} + 7 \zeta_{18}^{3} + 4 \zeta_{18}^{4} + 8 \zeta_{18}^{5} ) q^{55} + ( -5 \zeta_{18}^{2} + 3 \zeta_{18}^{5} ) q^{57} + ( 4 - 6 \zeta_{18} - \zeta_{18}^{2} + \zeta_{18}^{3} + 6 \zeta_{18}^{4} - 4 \zeta_{18}^{5} ) q^{59} + ( -3 + 2 \zeta_{18} + \zeta_{18}^{3} + \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{61} + ( 2 + 2 \zeta_{18} + \zeta_{18}^{2} - 3 \zeta_{18}^{3} + \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{63} + ( 3 \zeta_{18} - 5 \zeta_{18}^{2} + 4 \zeta_{18}^{3} - 5 \zeta_{18}^{4} + 3 \zeta_{18}^{5} ) q^{65} + ( 6 - 11 \zeta_{18} - 4 \zeta_{18}^{2} - 11 \zeta_{18}^{3} + 6 \zeta_{18}^{4} ) q^{67} + ( 2 + 6 \zeta_{18} - 3 \zeta_{18}^{2} - 2 \zeta_{18}^{3} - 3 \zeta_{18}^{4} - 3 \zeta_{18}^{5} ) q^{69} + ( -3 + 3 \zeta_{18}^{2} + 5 \zeta_{18}^{3} - 7 \zeta_{18}^{4} + 2 \zeta_{18}^{5} ) q^{71} + ( 3 - 5 \zeta_{18} + \zeta_{18}^{2} - 2 \zeta_{18}^{3} + 2 \zeta_{18}^{5} ) q^{73} + ( 1 + 4 \zeta_{18} + 4 \zeta_{18}^{2} - 3 \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{75} + ( -7 - 6 \zeta_{18} - 6 \zeta_{18}^{2} - 4 \zeta_{18}^{4} + 10 \zeta_{18}^{5} ) q^{77} + ( -5 + 5 \zeta_{18} - 4 \zeta_{18}^{2} + \zeta_{18}^{3} - \zeta_{18}^{5} ) q^{79} + \zeta_{18}^{4} q^{81} + ( -1 + 4 \zeta_{18} - 3 \zeta_{18}^{2} + \zeta_{18}^{3} - \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{83} + ( 3 - 2 \zeta_{18} + 5 \zeta_{18}^{2} - 2 \zeta_{18}^{3} + 3 \zeta_{18}^{4} ) q^{85} + ( \zeta_{18}^{2} - 2 \zeta_{18}^{3} + \zeta_{18}^{4} ) q^{87} + ( 7 + 7 \zeta_{18} - \zeta_{18}^{2} - 7 \zeta_{18}^{4} + \zeta_{18}^{5} ) q^{89} + ( 8 - 6 \zeta_{18} - 2 \zeta_{18}^{3} - 2 \zeta_{18}^{4} + 5 \zeta_{18}^{5} ) q^{91} + ( 1 - 5 \zeta_{18} + \zeta_{18}^{2} - \zeta_{18}^{3} + 5 \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{93} + ( -3 + 2 \zeta_{18} - 6 \zeta_{18}^{2} - 2 \zeta_{18}^{3} - 5 \zeta_{18}^{4} - 4 \zeta_{18}^{5} ) q^{95} + ( 7 + 4 \zeta_{18} + 8 \zeta_{18}^{2} - 8 \zeta_{18}^{3} - 4 \zeta_{18}^{4} - 7 \zeta_{18}^{5} ) q^{97} + ( 2 + \zeta_{18} - 3 \zeta_{18}^{3} - 3 \zeta_{18}^{4} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q + 3q^{5} + 3q^{7} + O(q^{10}) \) \( 6q + 3q^{5} + 3q^{7} + 9q^{13} + 3q^{15} - 3q^{17} - 15q^{21} - 27q^{23} - 15q^{25} - 3q^{27} - 3q^{29} + 15q^{31} + 3q^{33} - 12q^{35} - 6q^{37} + 12q^{39} - 15q^{41} - 3q^{43} + 6q^{45} - 15q^{47} - 24q^{49} - 3q^{51} + 6q^{53} - 27q^{55} + 27q^{59} - 15q^{61} + 3q^{63} + 12q^{65} + 3q^{67} + 6q^{69} - 3q^{71} + 12q^{73} + 6q^{75} - 42q^{77} - 27q^{79} - 3q^{83} + 12q^{85} - 6q^{87} + 42q^{89} + 42q^{91} + 3q^{93} - 24q^{95} + 18q^{97} + 3q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/912\mathbb{Z}\right)^\times\).

\(n\) \(97\) \(229\) \(305\) \(799\)
\(\chi(n)\) \(-\zeta_{18}^{5}\) \(1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
289.1
−0.173648 0.984808i
−0.173648 + 0.984808i
0.939693 + 0.342020i
0.939693 0.342020i
−0.766044 0.642788i
−0.766044 + 0.642788i
0 0.173648 + 0.984808i 0 −0.0923963 + 0.0775297i 0 −2.14543 + 3.71599i 0 −0.939693 + 0.342020i 0
385.1 0 0.173648 0.984808i 0 −0.0923963 0.0775297i 0 −2.14543 3.71599i 0 −0.939693 0.342020i 0
481.1 0 −0.939693 0.342020i 0 −0.613341 3.47843i 0 1.85844 3.21891i 0 0.766044 + 0.642788i 0
529.1 0 −0.939693 + 0.342020i 0 −0.613341 + 3.47843i 0 1.85844 + 3.21891i 0 0.766044 0.642788i 0
625.1 0 0.766044 + 0.642788i 0 2.20574 0.802823i 0 1.78699 + 3.09516i 0 0.173648 + 0.984808i 0
769.1 0 0.766044 0.642788i 0 2.20574 + 0.802823i 0 1.78699 3.09516i 0 0.173648 0.984808i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 769.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.e even 9 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 912.2.bo.d 6
4.b odd 2 1 114.2.i.c 6
12.b even 2 1 342.2.u.b 6
19.e even 9 1 inner 912.2.bo.d 6
76.k even 18 1 2166.2.a.p 3
76.l odd 18 1 114.2.i.c 6
76.l odd 18 1 2166.2.a.r 3
228.u odd 18 1 6498.2.a.bu 3
228.v even 18 1 342.2.u.b 6
228.v even 18 1 6498.2.a.bp 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
114.2.i.c 6 4.b odd 2 1
114.2.i.c 6 76.l odd 18 1
342.2.u.b 6 12.b even 2 1
342.2.u.b 6 228.v even 18 1
912.2.bo.d 6 1.a even 1 1 trivial
912.2.bo.d 6 19.e even 9 1 inner
2166.2.a.p 3 76.k even 18 1
2166.2.a.r 3 76.l odd 18 1
6498.2.a.bp 3 228.v even 18 1
6498.2.a.bu 3 228.u odd 18 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{6} - 3 T_{5}^{5} + 12 T_{5}^{4} - 46 T_{5}^{3} + 60 T_{5}^{2} + 12 T_{5} + 1 \) acting on \(S_{2}^{\mathrm{new}}(912, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \)
$3$ \( 1 + T^{3} + T^{6} \)
$5$ \( 1 + 12 T + 60 T^{2} - 46 T^{3} + 12 T^{4} - 3 T^{5} + T^{6} \)
$7$ \( 3249 - 1026 T + 495 T^{2} - 60 T^{3} + 27 T^{4} - 3 T^{5} + T^{6} \)
$11$ \( 1369 - 777 T + 441 T^{2} - 74 T^{3} + 21 T^{4} + T^{6} \)
$13$ \( 1 - 9 T + 27 T^{2} - 28 T^{3} + 36 T^{4} - 9 T^{5} + T^{6} \)
$17$ \( 9 - 27 T + 9 T^{2} + 24 T^{3} + 18 T^{4} + 3 T^{5} + T^{6} \)
$19$ \( 6859 - 107 T^{3} + T^{6} \)
$23$ \( 157609 + 85752 T + 22626 T^{2} + 3556 T^{3} + 378 T^{4} + 27 T^{5} + T^{6} \)
$29$ \( 1 - 12 T + 60 T^{2} + 46 T^{3} + 12 T^{4} + 3 T^{5} + T^{6} \)
$31$ \( 11881 - 7848 T + 3549 T^{2} - 862 T^{3} + 153 T^{4} - 15 T^{5} + T^{6} \)
$37$ \( ( 17 - 45 T + 3 T^{2} + T^{3} )^{2} \)
$41$ \( 2809 + 2703 T + 1308 T^{2} + 316 T^{3} + 87 T^{4} + 15 T^{5} + T^{6} \)
$43$ \( 63001 - 17319 T + 2406 T^{2} - 280 T^{3} - 3 T^{4} + 3 T^{5} + T^{6} \)
$47$ \( 5041 - 1491 T + 570 T^{2} - 80 T^{3} + 51 T^{4} + 15 T^{5} + T^{6} \)
$53$ \( 395641 + 18870 T + 6186 T^{2} + 1205 T^{3} - 30 T^{4} - 6 T^{5} + T^{6} \)
$59$ \( 81 + 567 T + 1701 T^{2} + 72 T^{3} + 198 T^{4} - 27 T^{5} + T^{6} \)
$61$ \( 289 + 408 T + 375 T^{2} + 215 T^{3} + 84 T^{4} + 15 T^{5} + T^{6} \)
$67$ \( 7017201 - 619866 T + 10890 T^{2} - 834 T^{3} + 72 T^{4} - 3 T^{5} + T^{6} \)
$71$ \( 26569 - 9291 T + 14946 T^{2} - 1304 T^{3} - 87 T^{4} + 3 T^{5} + T^{6} \)
$73$ \( 3249 + 1539 T + 522 T^{2} + 300 T^{3} + 54 T^{4} - 12 T^{5} + T^{6} \)
$79$ \( 32761 + 27693 T + 9918 T^{2} + 2152 T^{3} + 351 T^{4} + 27 T^{5} + T^{6} \)
$83$ \( 2601 - 1836 T + 1143 T^{2} - 210 T^{3} + 45 T^{4} + 3 T^{5} + T^{6} \)
$89$ \( 239121 - 92421 T + 35406 T^{2} - 7104 T^{3} + 756 T^{4} - 42 T^{5} + T^{6} \)
$97$ \( 218089 + 121887 T + 19332 T^{2} + 451 T^{3} + 171 T^{4} - 18 T^{5} + T^{6} \)
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