Properties

Label 912.2.bo.f
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: \( x^{6} - x^{3} + 1 \) Copy content Toggle raw display
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.

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

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 0.907604 0.761570i 0 0.266044 0.460802i 0 −0.939693 + 0.342020i 0
385.1 0 0.173648 0.984808i 0 0.907604 + 0.761570i 0 0.266044 + 0.460802i 0 −0.939693 0.342020i 0
481.1 0 −0.939693 0.342020i 0 0.386659 + 2.19285i 0 −0.326352 + 0.565258i 0 0.766044 + 0.642788i 0
529.1 0 −0.939693 + 0.342020i 0 0.386659 2.19285i 0 −0.326352 0.565258i 0 0.766044 0.642788i 0
625.1 0 0.766044 + 0.642788i 0 3.20574 1.16679i 0 −1.43969 2.49362i 0 0.173648 + 0.984808i 0
769.1 0 0.766044 0.642788i 0 3.20574 + 1.16679i 0 −1.43969 + 2.49362i 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.f 6
4.b odd 2 1 114.2.i.d 6
12.b even 2 1 342.2.u.a 6
19.e even 9 1 inner 912.2.bo.f 6
76.k even 18 1 2166.2.a.u 3
76.l odd 18 1 114.2.i.d 6
76.l odd 18 1 2166.2.a.o 3
228.u odd 18 1 6498.2.a.bn 3
228.v even 18 1 342.2.u.a 6
228.v even 18 1 6498.2.a.bs 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
114.2.i.d 6 4.b odd 2 1
114.2.i.d 6 76.l odd 18 1
342.2.u.a 6 12.b even 2 1
342.2.u.a 6 228.v even 18 1
912.2.bo.f 6 1.a even 1 1 trivial
912.2.bo.f 6 19.e even 9 1 inner
2166.2.a.o 3 76.l odd 18 1
2166.2.a.u 3 76.k even 18 1
6498.2.a.bn 3 228.u odd 18 1
6498.2.a.bs 3 228.v even 18 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{6} - 9T_{5}^{5} + 36T_{5}^{4} - 90T_{5}^{3} + 162T_{5}^{2} - 162T_{5} + 81 \) acting on \(S_{2}^{\mathrm{new}}(912, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} + T^{3} + 1 \) Copy content Toggle raw display
$5$ \( T^{6} - 9 T^{5} + 36 T^{4} - 90 T^{3} + \cdots + 81 \) Copy content Toggle raw display
$7$ \( T^{6} + 3 T^{5} + 9 T^{4} + 2 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( T^{6} + 3 T^{4} + 2 T^{3} + 9 T^{2} + \cdots + 1 \) Copy content Toggle raw display
$13$ \( T^{6} + 3 T^{5} + 18 T^{4} + 24 T^{3} + \cdots + 9 \) Copy content Toggle raw display
$17$ \( T^{6} + 3 T^{5} + 48 T^{4} + 244 T^{3} + \cdots + 289 \) Copy content Toggle raw display
$19$ \( T^{6} - 18 T^{5} + 162 T^{4} + \cdots + 6859 \) Copy content Toggle raw display
$23$ \( T^{6} - 21 T^{5} + 210 T^{4} + \cdots + 72361 \) Copy content Toggle raw display
$29$ \( T^{6} + 3 T^{5} + 18 T^{4} + \cdots + 45369 \) Copy content Toggle raw display
$31$ \( T^{6} + 9 T^{5} + 81 T^{4} + 54 T^{3} + \cdots + 729 \) Copy content Toggle raw display
$37$ \( (T^{3} + 9 T^{2} - 57 T - 361)^{2} \) Copy content Toggle raw display
$41$ \( T^{6} + 15 T^{5} + 177 T^{4} + \cdots + 11881 \) Copy content Toggle raw display
$43$ \( T^{6} + 3 T^{5} + 99 T^{4} + \cdots + 3249 \) Copy content Toggle raw display
$47$ \( T^{6} - 9 T^{5} + 63 T^{4} + \cdots + 2809 \) Copy content Toggle raw display
$53$ \( T^{6} - 12 T^{5} + 174 T^{4} + \cdots + 94249 \) Copy content Toggle raw display
$59$ \( T^{6} + 27 T^{5} + 324 T^{4} + \cdots + 289 \) Copy content Toggle raw display
$61$ \( T^{6} - 3 T^{5} - 60 T^{4} + \cdots + 1682209 \) Copy content Toggle raw display
$67$ \( T^{6} + 21 T^{5} + 126 T^{4} + 24 T^{3} + \cdots + 9 \) Copy content Toggle raw display
$71$ \( T^{6} + 39 T^{5} + 561 T^{4} + \cdots + 201601 \) Copy content Toggle raw display
$73$ \( T^{6} - 36 T^{5} + 558 T^{4} + \cdots + 1369 \) Copy content Toggle raw display
$79$ \( T^{6} - 45 T^{5} + 1125 T^{4} + \cdots + 4515625 \) Copy content Toggle raw display
$83$ \( T^{6} - 27 T^{5} + 507 T^{4} + \cdots + 253009 \) Copy content Toggle raw display
$89$ \( T^{6} + 30 T^{5} + 246 T^{4} + \cdots + 11449 \) Copy content Toggle raw display
$97$ \( T^{6} + 6 T^{5} + 3 T^{4} + 35 T^{3} + \cdots + 2809 \) Copy content Toggle raw display
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