Properties

Label 912.2.bo.c
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.

\(f(q)\) \(=\) \( q + \zeta_{18} q^{3} + ( -1 + \zeta_{18}^{2} + \zeta_{18}^{3} - 2 \zeta_{18}^{4} ) q^{5} + ( -1 - \zeta_{18} - \zeta_{18}^{2} + \zeta_{18}^{3} + 2 \zeta_{18}^{4} + 2 \zeta_{18}^{5} ) q^{7} + \zeta_{18}^{2} q^{9} +O(q^{10})\) \( q + \zeta_{18} q^{3} + ( -1 + \zeta_{18}^{2} + \zeta_{18}^{3} - 2 \zeta_{18}^{4} ) q^{5} + ( -1 - \zeta_{18} - \zeta_{18}^{2} + \zeta_{18}^{3} + 2 \zeta_{18}^{4} + 2 \zeta_{18}^{5} ) q^{7} + \zeta_{18}^{2} q^{9} + ( -\zeta_{18} - 4 \zeta_{18}^{3} - \zeta_{18}^{5} ) q^{11} + ( -3 - 3 \zeta_{18} - \zeta_{18}^{3} + 4 \zeta_{18}^{4} ) q^{13} + ( -\zeta_{18} + \zeta_{18}^{3} + \zeta_{18}^{4} - 2 \zeta_{18}^{5} ) q^{15} + ( -4 \zeta_{18} - \zeta_{18}^{2} + \zeta_{18}^{3} + 4 \zeta_{18}^{4} ) q^{17} + ( -\zeta_{18} - 2 \zeta_{18}^{2} - 2 \zeta_{18}^{3} - \zeta_{18}^{4} + 4 \zeta_{18}^{5} ) q^{19} + ( -2 - \zeta_{18} - \zeta_{18}^{2} + \zeta_{18}^{3} + \zeta_{18}^{4} + 2 \zeta_{18}^{5} ) q^{21} + ( 2 - 3 \zeta_{18} + \zeta_{18}^{3} + \zeta_{18}^{4} - 4 \zeta_{18}^{5} ) q^{23} + ( 4 + 4 \zeta_{18} - \zeta_{18}^{2} - 5 \zeta_{18}^{3} + \zeta_{18}^{4} + \zeta_{18}^{5} ) q^{25} + \zeta_{18}^{3} q^{27} + ( 4 - 3 \zeta_{18} - 2 \zeta_{18}^{2} - 3 \zeta_{18}^{3} + 4 \zeta_{18}^{4} ) q^{29} + ( -1 + \zeta_{18} - 2 \zeta_{18}^{2} + \zeta_{18}^{3} + \zeta_{18}^{4} + \zeta_{18}^{5} ) q^{31} + ( 1 - \zeta_{18}^{2} - \zeta_{18}^{3} - 4 \zeta_{18}^{4} ) q^{33} + ( -\zeta_{18} + 2 \zeta_{18}^{2} + 2 \zeta_{18}^{3} - 2 \zeta_{18}^{5} ) q^{35} + ( 1 - 2 \zeta_{18} - 2 \zeta_{18}^{2} + 2 \zeta_{18}^{5} ) q^{37} + ( -3 \zeta_{18} - 3 \zeta_{18}^{2} - \zeta_{18}^{4} + 4 \zeta_{18}^{5} ) q^{39} + ( -3 + 3 \zeta_{18} + 3 \zeta_{18}^{3} - 3 \zeta_{18}^{5} ) q^{41} + ( 3 - 3 \zeta_{18}^{2} - 3 \zeta_{18}^{3} + \zeta_{18}^{4} ) q^{43} + ( 2 - \zeta_{18}^{2} - 2 \zeta_{18}^{3} + \zeta_{18}^{4} + \zeta_{18}^{5} ) q^{45} + ( 4 - \zeta_{18} + 5 \zeta_{18}^{2} - \zeta_{18}^{3} + 4 \zeta_{18}^{4} ) q^{47} + ( -2 \zeta_{18} - 5 \zeta_{18}^{2} - 5 \zeta_{18}^{4} - 2 \zeta_{18}^{5} ) q^{49} + ( -4 \zeta_{18}^{2} - \zeta_{18}^{3} + \zeta_{18}^{4} + 4 \zeta_{18}^{5} ) q^{51} + ( 4 - 6 \zeta_{18} + 2 \zeta_{18}^{3} + 2 \zeta_{18}^{4} + \zeta_{18}^{5} ) q^{53} + ( 2 - 6 \zeta_{18} + \zeta_{18}^{2} - \zeta_{18}^{3} + 6 \zeta_{18}^{4} - 2 \zeta_{18}^{5} ) q^{55} + ( -4 - \zeta_{18}^{2} + 2 \zeta_{18}^{3} - 2 \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{57} + ( -2 + 5 \zeta_{18}^{2} - 5 \zeta_{18}^{3} + 2 \zeta_{18}^{5} ) q^{59} + ( -3 + 3 \zeta_{18}^{3} + 3 \zeta_{18}^{4} + \zeta_{18}^{5} ) q^{61} + ( -2 - 2 \zeta_{18} - \zeta_{18}^{2} + \zeta_{18}^{3} + \zeta_{18}^{4} + \zeta_{18}^{5} ) q^{63} + ( -3 \zeta_{18} + 5 \zeta_{18}^{2} - 2 \zeta_{18}^{3} + 5 \zeta_{18}^{4} - 3 \zeta_{18}^{5} ) q^{65} + ( 4 - 3 \zeta_{18} + 2 \zeta_{18}^{2} - 3 \zeta_{18}^{3} + 4 \zeta_{18}^{4} ) q^{67} + ( 4 + 2 \zeta_{18} - 3 \zeta_{18}^{2} - 4 \zeta_{18}^{3} + \zeta_{18}^{4} + \zeta_{18}^{5} ) q^{69} + ( -3 + 3 \zeta_{18}^{2} + 3 \zeta_{18}^{3} + 3 \zeta_{18}^{4} ) q^{71} + ( 7 - \zeta_{18} - 3 \zeta_{18}^{2} - 10 \zeta_{18}^{3} + 10 \zeta_{18}^{5} ) q^{73} + ( -1 + 4 \zeta_{18} + 4 \zeta_{18}^{2} - 5 \zeta_{18}^{4} + \zeta_{18}^{5} ) q^{75} + ( 7 + 10 \zeta_{18} + 10 \zeta_{18}^{2} - 4 \zeta_{18}^{4} - 6 \zeta_{18}^{5} ) q^{77} + ( -5 + 3 \zeta_{18} + 5 \zeta_{18}^{3} - 5 \zeta_{18}^{5} ) q^{79} + \zeta_{18}^{4} q^{81} + ( 1 - 6 \zeta_{18} - 5 \zeta_{18}^{2} - \zeta_{18}^{3} + 11 \zeta_{18}^{4} + 11 \zeta_{18}^{5} ) q^{83} + ( -7 + 2 \zeta_{18} + 9 \zeta_{18}^{2} + 2 \zeta_{18}^{3} - 7 \zeta_{18}^{4} ) q^{85} + ( 4 \zeta_{18} - 3 \zeta_{18}^{2} - 2 \zeta_{18}^{3} - 3 \zeta_{18}^{4} + 4 \zeta_{18}^{5} ) q^{87} + ( -5 - 5 \zeta_{18} - \zeta_{18}^{2} - 6 \zeta_{18}^{3} + 11 \zeta_{18}^{4} + \zeta_{18}^{5} ) q^{89} + ( 6 + 4 \zeta_{18} - 10 \zeta_{18}^{3} - 10 \zeta_{18}^{4} - 9 \zeta_{18}^{5} ) q^{91} + ( -1 - \zeta_{18} + \zeta_{18}^{2} - \zeta_{18}^{3} + \zeta_{18}^{4} + \zeta_{18}^{5} ) q^{93} + ( 7 - 6 \zeta_{18} - 4 \zeta_{18}^{2} + 2 \zeta_{18}^{3} + 5 \zeta_{18}^{4} ) q^{95} + ( -1 - 4 \zeta_{18} - 8 \zeta_{18}^{2} + 8 \zeta_{18}^{3} + 4 \zeta_{18}^{4} + \zeta_{18}^{5} ) q^{97} + ( \zeta_{18} - \zeta_{18}^{3} - \zeta_{18}^{4} - 4 \zeta_{18}^{5} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - 3q^{5} - 3q^{7} + O(q^{10}) \) \( 6q - 3q^{5} - 3q^{7} - 12q^{11} - 21q^{13} + 3q^{15} + 3q^{17} - 6q^{19} - 9q^{21} + 15q^{23} + 9q^{25} + 3q^{27} + 15q^{29} - 3q^{31} + 3q^{33} + 6q^{35} + 6q^{37} - 9q^{41} + 9q^{43} + 6q^{45} + 21q^{47} - 3q^{51} + 30q^{53} + 9q^{55} - 18q^{57} - 27q^{59} - 9q^{61} - 9q^{63} - 6q^{65} + 15q^{67} + 12q^{69} - 9q^{71} + 12q^{73} - 6q^{75} + 42q^{77} - 15q^{79} + 3q^{83} - 36q^{85} - 6q^{87} - 48q^{89} + 6q^{91} - 9q^{93} + 48q^{95} + 18q^{97} - 3q^{99} + O(q^{100}) \)

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 −2.97178 + 2.49362i 0 0.613341 1.06234i 0 −0.939693 + 0.342020i 0
385.1 0 −0.173648 + 0.984808i 0 −2.97178 2.49362i 0 0.613341 + 1.06234i 0 −0.939693 0.342020i 0
481.1 0 0.939693 + 0.342020i 0 −0.0812519 0.460802i 0 −2.20574 + 3.82045i 0 0.766044 + 0.642788i 0
529.1 0 0.939693 0.342020i 0 −0.0812519 + 0.460802i 0 −2.20574 3.82045i 0 0.766044 0.642788i 0
625.1 0 −0.766044 0.642788i 0 1.55303 0.565258i 0 0.0923963 + 0.160035i 0 0.173648 + 0.984808i 0
769.1 0 −0.766044 + 0.642788i 0 1.55303 + 0.565258i 0 0.0923963 0.160035i 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.c 6
4.b odd 2 1 114.2.i.b 6
12.b even 2 1 342.2.u.d 6
19.e even 9 1 inner 912.2.bo.c 6
76.k even 18 1 2166.2.a.n 3
76.l odd 18 1 114.2.i.b 6
76.l odd 18 1 2166.2.a.t 3
228.u odd 18 1 6498.2.a.bt 3
228.v even 18 1 342.2.u.d 6
228.v even 18 1 6498.2.a.bo 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
114.2.i.b 6 4.b odd 2 1
114.2.i.b 6 76.l odd 18 1
342.2.u.d 6 12.b even 2 1
342.2.u.d 6 228.v even 18 1
912.2.bo.c 6 1.a even 1 1 trivial
912.2.bo.c 6 19.e even 9 1 inner
2166.2.a.n 3 76.k even 18 1
2166.2.a.t 3 76.l odd 18 1
6498.2.a.bo 3 228.v even 18 1
6498.2.a.bt 3 228.u odd 18 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{6} + 3 T_{5}^{5} - 30 T_{5}^{3} + 36 T_{5}^{2} + 9 \) acting on \(S_{2}^{\mathrm{new}}(912, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \)
$3$ \( 1 - T^{3} + T^{6} \)
$5$ \( 9 + 36 T^{2} - 30 T^{3} + 3 T^{5} + T^{6} \)
$7$ \( 1 - 6 T + 33 T^{2} - 20 T^{3} + 15 T^{4} + 3 T^{5} + T^{6} \)
$11$ \( 2601 + 2295 T + 1413 T^{2} + 438 T^{3} + 99 T^{4} + 12 T^{5} + T^{6} \)
$13$ \( 361 - 57 T + 1407 T^{2} + 800 T^{3} + 186 T^{4} + 21 T^{5} + T^{6} \)
$17$ \( 2601 + 459 T + 495 T^{2} - 24 T^{3} - 18 T^{4} - 3 T^{5} + T^{6} \)
$19$ \( 6859 + 2166 T - 228 T^{2} - 169 T^{3} - 12 T^{4} + 6 T^{5} + T^{6} \)
$23$ \( 9 - 162 T + 846 T^{2} - 246 T^{3} + 108 T^{4} - 15 T^{5} + T^{6} \)
$29$ \( 2601 + 1836 T + 576 T^{2} - 138 T^{3} + 72 T^{4} - 15 T^{5} + T^{6} \)
$31$ \( 289 + 102 T + 87 T^{2} + 16 T^{3} + 15 T^{4} + 3 T^{5} + T^{6} \)
$37$ \( ( 19 - 9 T - 3 T^{2} + T^{3} )^{2} \)
$41$ \( 729 - 729 T + 486 T^{2} - 216 T^{3} + 27 T^{4} + 9 T^{5} + T^{6} \)
$43$ \( 1 - 45 T + 576 T^{2} - 80 T^{3} + 45 T^{4} - 9 T^{5} + T^{6} \)
$47$ \( 25281 - 27189 T + 10836 T^{2} - 1968 T^{3} + 261 T^{4} - 21 T^{5} + T^{6} \)
$53$ \( 47961 - 11826 T + 7218 T^{2} - 2373 T^{3} + 378 T^{4} - 30 T^{5} + T^{6} \)
$59$ \( 23409 + 31671 T + 19197 T^{2} + 3312 T^{3} + 360 T^{4} + 27 T^{5} + T^{6} \)
$61$ \( 1 - 36 T + 549 T^{2} + 323 T^{3} + 72 T^{4} + 9 T^{5} + T^{6} \)
$67$ \( 7921 - 8544 T + 4200 T^{2} - 1070 T^{3} + 156 T^{4} - 15 T^{5} + T^{6} \)
$71$ \( 6561 + 6561 T + 2916 T^{2} + 648 T^{3} + 81 T^{4} + 9 T^{5} + T^{6} \)
$73$ \( 546121 + 281559 T + 32586 T^{2} - 2404 T^{3} + 246 T^{4} - 12 T^{5} + T^{6} \)
$79$ \( 5329 - 7665 T + 3180 T^{2} - 152 T^{3} + 105 T^{4} + 15 T^{5} + T^{6} \)
$83$ \( 3583449 - 511110 T + 78579 T^{2} - 2976 T^{3} + 279 T^{4} - 3 T^{5} + T^{6} \)
$89$ \( 3583449 + 698517 T + 95616 T^{2} + 12174 T^{3} + 1044 T^{4} + 48 T^{5} + T^{6} \)
$97$ \( 1630729 + 34479 T + 3636 T^{2} - 613 T^{3} + 99 T^{4} - 18 T^{5} + T^{6} \)
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