Properties

Label 912.2.ch.a
Level $912$
Weight $2$
Character orbit 912.ch
Analytic conductor $7.282$
Analytic rank $0$
Dimension $6$
CM discriminant -3
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 912 = 2^{4} \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 912.ch (of order \(18\), degree \(6\), 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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{18}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( 2 \zeta_{18} - \zeta_{18}^{4} ) q^{3} + ( 3 \zeta_{18} + \zeta_{18}^{2} - 2 \zeta_{18}^{4} + 2 \zeta_{18}^{5} ) q^{7} + ( 3 \zeta_{18}^{2} - 3 \zeta_{18}^{5} ) q^{9} +O(q^{10})\) \( q + ( 2 \zeta_{18} - \zeta_{18}^{4} ) q^{3} + ( 3 \zeta_{18} + \zeta_{18}^{2} - 2 \zeta_{18}^{4} + 2 \zeta_{18}^{5} ) q^{7} + ( 3 \zeta_{18}^{2} - 3 \zeta_{18}^{5} ) q^{9} + ( -4 + \zeta_{18} + 3 \zeta_{18}^{3} + 3 \zeta_{18}^{4} ) q^{13} + ( -3 + 5 \zeta_{18}^{3} ) q^{19} + ( -1 + 4 \zeta_{18}^{2} + 5 \zeta_{18}^{3} - 5 \zeta_{18}^{5} ) q^{21} -5 \zeta_{18}^{5} q^{25} + ( 6 - 3 \zeta_{18}^{3} ) q^{27} + ( -5 \zeta_{18} - 6 \zeta_{18}^{2} - \zeta_{18}^{4} + \zeta_{18}^{5} ) q^{31} + ( 7 \zeta_{18} + 7 \zeta_{18}^{2} - 4 \zeta_{18}^{4} - 3 \zeta_{18}^{5} ) q^{37} + ( -5 \zeta_{18} + 5 \zeta_{18}^{2} + 7 \zeta_{18}^{4} + 2 \zeta_{18}^{5} ) q^{39} + ( -7 + \zeta_{18}^{2} + \zeta_{18}^{3} - 7 \zeta_{18}^{5} ) q^{43} + ( -8 \zeta_{18} + 5 \zeta_{18}^{2} + 7 \zeta_{18}^{3} + 5 \zeta_{18}^{4} - 8 \zeta_{18}^{5} ) q^{49} + ( -\zeta_{18} + 8 \zeta_{18}^{4} ) q^{57} + ( 5 + 4 \zeta_{18} + 4 \zeta_{18}^{3} + 5 \zeta_{18}^{4} ) q^{61} + ( 9 + 3 \zeta_{18} - 6 \zeta_{18}^{3} + 6 \zeta_{18}^{4} ) q^{63} + ( -2 + 2 \zeta_{18} - 7 \zeta_{18}^{3} - 9 \zeta_{18}^{4} ) q^{67} + ( -9 + 9 \zeta_{18}^{2} + \zeta_{18}^{3} - 8 \zeta_{18}^{5} ) q^{73} + ( 5 - 10 \zeta_{18}^{3} ) q^{75} + ( -3 - 3 \zeta_{18}^{2} + 10 \zeta_{18}^{3} - 7 \zeta_{18}^{5} ) q^{79} + ( 9 \zeta_{18} - 9 \zeta_{18}^{4} ) q^{81} + ( -11 - 6 \zeta_{18} - \zeta_{18}^{2} + 6 \zeta_{18}^{3} + 11 \zeta_{18}^{4} + 2 \zeta_{18}^{5} ) q^{91} + ( -7 - 11 \zeta_{18}^{2} - 4 \zeta_{18}^{3} + 4 \zeta_{18}^{5} ) q^{93} -19 \zeta_{18} q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q + O(q^{10}) \) \( 6q - 15q^{13} - 3q^{19} + 9q^{21} + 27q^{27} - 39q^{43} + 21q^{49} + 42q^{61} + 36q^{63} - 33q^{67} - 51q^{73} + 12q^{79} - 48q^{91} - 54q^{93} + O(q^{100}) \)

Character values

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

\(n\) \(97\) \(229\) \(305\) \(799\)
\(\chi(n)\) \(\zeta_{18}^{2}\) \(1\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
47.1
−0.173648 0.984808i
0.939693 + 0.342020i
0.939693 0.342020i
−0.766044 + 0.642788i
−0.766044 0.642788i
−0.173648 + 0.984808i
0 −1.11334 1.32683i 0 0 0 −4.52481 2.61240i 0 −0.520945 + 2.95442i 0
479.1 0 1.70574 0.300767i 0 0 0 2.89053 + 1.66885i 0 2.81908 1.02606i 0
575.1 0 1.70574 + 0.300767i 0 0 0 2.89053 1.66885i 0 2.81908 + 1.02606i 0
671.1 0 −0.592396 + 1.62760i 0 0 0 1.63429 + 0.943555i 0 −2.29813 1.92836i 0
719.1 0 −0.592396 1.62760i 0 0 0 1.63429 0.943555i 0 −2.29813 + 1.92836i 0
815.1 0 −1.11334 + 1.32683i 0 0 0 −4.52481 + 2.61240i 0 −0.520945 2.95442i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 815.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
76.l odd 18 1 inner
228.v even 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 912.2.ch.a 6
3.b odd 2 1 CM 912.2.ch.a 6
4.b odd 2 1 912.2.ch.b yes 6
12.b even 2 1 912.2.ch.b yes 6
19.e even 9 1 912.2.ch.b yes 6
57.l odd 18 1 912.2.ch.b yes 6
76.l odd 18 1 inner 912.2.ch.a 6
228.v even 18 1 inner 912.2.ch.a 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
912.2.ch.a 6 1.a even 1 1 trivial
912.2.ch.a 6 3.b odd 2 1 CM
912.2.ch.a 6 76.l odd 18 1 inner
912.2.ch.a 6 228.v even 18 1 inner
912.2.ch.b yes 6 4.b odd 2 1
912.2.ch.b yes 6 12.b even 2 1
912.2.ch.b yes 6 19.e even 9 1
912.2.ch.b yes 6 57.l odd 18 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(912, [\chi])\):

\( T_{5} \)
\( T_{7}^{6} - 21 T_{7}^{4} + 441 T_{7}^{2} - 1197 T_{7} + 1083 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \)
$3$ \( 27 - 9 T^{3} + T^{6} \)
$5$ \( T^{6} \)
$7$ \( 1083 - 1197 T + 441 T^{2} - 21 T^{4} + T^{6} \)
$11$ \( T^{6} \)
$13$ \( 361 + 1254 T + 1920 T^{2} + 604 T^{3} + 114 T^{4} + 15 T^{5} + T^{6} \)
$17$ \( T^{6} \)
$19$ \( ( 19 + T + T^{2} )^{3} \)
$23$ \( T^{6} \)
$29$ \( T^{6} \)
$31$ \( 118803 - 55521 T + 8649 T^{2} - 93 T^{4} + T^{6} \)
$37$ \( ( -323 - 111 T + T^{3} )^{2} \)
$41$ \( T^{6} \)
$43$ \( 312987 + 139536 T + 36990 T^{2} + 6000 T^{3} + 636 T^{4} + 39 T^{5} + T^{6} \)
$47$ \( T^{6} \)
$53$ \( T^{6} \)
$59$ \( T^{6} \)
$61$ \( 516961 - 101379 T + 31332 T^{2} - 6967 T^{3} + 771 T^{4} - 42 T^{5} + T^{6} \)
$67$ \( 189003 - 72288 T + 20070 T^{2} + 5880 T^{3} + 564 T^{4} + 33 T^{5} + T^{6} \)
$71$ \( T^{6} \)
$73$ \( 73441 + 37398 T + 53652 T^{2} + 11440 T^{3} + 1086 T^{4} + 51 T^{5} + T^{6} \)
$79$ \( 48387 - 104013 T + 59598 T^{2} - 2463 T^{3} + 285 T^{4} - 12 T^{5} + T^{6} \)
$83$ \( T^{6} \)
$89$ \( T^{6} \)
$97$ \( 47045881 + 6859 T^{3} + T^{6} \)
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