Properties

Label 912.2.ci.b
Level $912$
Weight $2$
Character orbit 912.ci
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.ci (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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

$q$-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.

\(f(q)\) \(=\) \( q -\zeta_{18}^{2} q^{3} + ( -1 - \zeta_{18} + \zeta_{18}^{2} + \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{5} + ( 2 - \zeta_{18}^{2} - \zeta_{18}^{3} + \zeta_{18}^{4} ) q^{7} + \zeta_{18}^{4} q^{9} +O(q^{10})\) \( q -\zeta_{18}^{2} q^{3} + ( -1 - \zeta_{18} + \zeta_{18}^{2} + \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{5} + ( 2 - \zeta_{18}^{2} - \zeta_{18}^{3} + \zeta_{18}^{4} ) q^{7} + \zeta_{18}^{4} q^{9} + ( 1 + 2 \zeta_{18} + \zeta_{18}^{2} + \zeta_{18}^{3} - \zeta_{18}^{4} + \zeta_{18}^{5} ) q^{11} + ( -\zeta_{18} - 2 \zeta_{18}^{2} - 2 \zeta_{18}^{3} - \zeta_{18}^{4} ) q^{13} + ( 1 - \zeta_{18} + \zeta_{18}^{2} ) q^{15} + ( -3 + \zeta_{18} + 2 \zeta_{18}^{3} + 2 \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{17} + ( 4 - \zeta_{18} + 2 \zeta_{18}^{2} - 2 \zeta_{18}^{3} + 2 \zeta_{18}^{4} - 2 \zeta_{18}^{5} ) q^{19} + ( 1 - 2 \zeta_{18}^{2} - \zeta_{18}^{3} + \zeta_{18}^{4} + \zeta_{18}^{5} ) q^{21} + ( 2 + \zeta_{18}^{2} - 3 \zeta_{18}^{3} - 3 \zeta_{18}^{5} ) q^{23} + ( -1 - 2 \zeta_{18} - 2 \zeta_{18}^{2} + 2 \zeta_{18}^{3} + 2 \zeta_{18}^{4} + \zeta_{18}^{5} ) q^{25} + ( 1 - \zeta_{18}^{3} ) q^{27} + ( 2 + 6 \zeta_{18} + 2 \zeta_{18}^{2} - 4 \zeta_{18}^{3} - 3 \zeta_{18}^{4} + 2 \zeta_{18}^{5} ) q^{29} + ( -3 \zeta_{18} + 5 \zeta_{18}^{2} + 2 \zeta_{18}^{3} + 5 \zeta_{18}^{4} - 3 \zeta_{18}^{5} ) q^{31} + ( -1 + \zeta_{18} - \zeta_{18}^{2} - \zeta_{18}^{3} - 2 \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{33} + ( -1 - 2 \zeta_{18} + \zeta_{18}^{2} + 2 \zeta_{18}^{3} + \zeta_{18}^{4} - 2 \zeta_{18}^{5} ) q^{35} + ( 4 - \zeta_{18} + \zeta_{18}^{2} - 8 \zeta_{18}^{3} - \zeta_{18}^{4} - 2 \zeta_{18}^{5} ) q^{37} + ( -1 + 2 \zeta_{18}^{3} + 2 \zeta_{18}^{4} + 2 \zeta_{18}^{5} ) q^{39} + ( -1 + 2 \zeta_{18} + 4 \zeta_{18}^{2} - 2 \zeta_{18}^{3} + \zeta_{18}^{4} - 8 \zeta_{18}^{5} ) q^{41} + ( 2 - 2 \zeta_{18} + \zeta_{18}^{2} - 4 \zeta_{18}^{3} - 2 \zeta_{18}^{4} + \zeta_{18}^{5} ) q^{43} + ( -\zeta_{18}^{2} + \zeta_{18}^{3} - \zeta_{18}^{4} ) q^{45} + ( 7 - 2 \zeta_{18} + 7 \zeta_{18}^{2} - \zeta_{18}^{3} + \zeta_{18}^{4} - 6 \zeta_{18}^{5} ) q^{47} + ( -2 + 2 \zeta_{18} - 5 \zeta_{18}^{2} + 2 \zeta_{18}^{3} + 3 \zeta_{18}^{4} + 3 \zeta_{18}^{5} ) q^{49} + ( 2 - \zeta_{18} + 3 \zeta_{18}^{2} - 3 \zeta_{18}^{3} + \zeta_{18}^{4} - 2 \zeta_{18}^{5} ) q^{51} + ( -1 + 2 \zeta_{18} + 3 \zeta_{18}^{2} - 2 \zeta_{18}^{3} - 4 \zeta_{18}^{4} - 2 \zeta_{18}^{5} ) q^{53} + ( -3 \zeta_{18} - 3 \zeta_{18}^{3} + 3 \zeta_{18}^{4} ) q^{55} + ( 2 - 2 \zeta_{18} - 4 \zeta_{18}^{2} - \zeta_{18}^{3} + 2 \zeta_{18}^{5} ) q^{57} + ( 3 - \zeta_{18} - 2 \zeta_{18}^{3} - 2 \zeta_{18}^{4} - 11 \zeta_{18}^{5} ) q^{59} + ( 4 - \zeta_{18} + 5 \zeta_{18}^{2} + \zeta_{18}^{3} - \zeta_{18}^{5} ) q^{61} + ( 1 + \zeta_{18} - \zeta_{18}^{2} - \zeta_{18}^{3} + \zeta_{18}^{4} + \zeta_{18}^{5} ) q^{63} + ( 1 + \zeta_{18} + 2 \zeta_{18}^{2} + \zeta_{18}^{3} + \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{65} + ( 6 - 6 \zeta_{18}^{2} + 2 \zeta_{18}^{4} + 6 \zeta_{18}^{5} ) q^{67} + ( -3 \zeta_{18} - 2 \zeta_{18}^{2} + 2 \zeta_{18}^{4} + 3 \zeta_{18}^{5} ) q^{69} + ( 4 + 4 \zeta_{18} - 3 \zeta_{18}^{2} - 2 \zeta_{18}^{3} - 2 \zeta_{18}^{4} + 3 \zeta_{18}^{5} ) q^{71} + ( -3 \zeta_{18} - 8 \zeta_{18}^{2} - 3 \zeta_{18}^{3} ) q^{73} + ( 2 + \zeta_{18} + \zeta_{18}^{2} + \zeta_{18}^{4} - 2 \zeta_{18}^{5} ) q^{75} + ( 3 \zeta_{18} + 3 \zeta_{18}^{2} - 3 \zeta_{18}^{4} ) q^{77} + ( -4 + 2 \zeta_{18} + 7 \zeta_{18}^{2} + 2 \zeta_{18}^{3} - 4 \zeta_{18}^{4} ) q^{79} + ( -\zeta_{18}^{2} + \zeta_{18}^{5} ) q^{81} + ( -2 - 3 \zeta_{18} - 2 \zeta_{18}^{2} + \zeta_{18}^{3} + 2 \zeta_{18}^{4} + 3 \zeta_{18}^{5} ) q^{83} + ( 4 - 4 \zeta_{18}^{2} + \zeta_{18}^{3} - 6 \zeta_{18}^{4} + 5 \zeta_{18}^{5} ) q^{85} + ( -3 + 2 \zeta_{18} - 2 \zeta_{18}^{2} - 3 \zeta_{18}^{3} - 4 \zeta_{18}^{4} + 4 \zeta_{18}^{5} ) q^{87} + ( 4 \zeta_{18} + \zeta_{18}^{2} + \zeta_{18}^{3} + 4 \zeta_{18}^{4} ) q^{89} + ( -1 - \zeta_{18} - 3 \zeta_{18}^{2} - 2 \zeta_{18}^{3} + 2 \zeta_{18}^{5} ) q^{91} + ( 5 - 3 \zeta_{18} - 2 \zeta_{18}^{3} - 2 \zeta_{18}^{4} - 2 \zeta_{18}^{5} ) q^{93} + ( -7 + \zeta_{18} - \zeta_{18}^{2} + 6 \zeta_{18}^{3} - 3 \zeta_{18}^{5} ) q^{95} + ( -2 + \zeta_{18} + 4 \zeta_{18}^{2} + 3 \zeta_{18}^{3} - 3 \zeta_{18}^{4} - 2 \zeta_{18}^{5} ) q^{97} + ( -2 - \zeta_{18} + \zeta_{18}^{2} + \zeta_{18}^{3} + 2 \zeta_{18}^{4} + \zeta_{18}^{5} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 6 q^{5} + 9 q^{7} + O(q^{10}) \) \( 6 q - 6 q^{5} + 9 q^{7} + 9 q^{11} - 6 q^{13} + 6 q^{15} - 12 q^{17} + 18 q^{19} + 3 q^{21} + 3 q^{23} + 3 q^{27} + 6 q^{31} - 9 q^{33} - 12 q^{41} + 3 q^{45} + 39 q^{47} - 6 q^{49} + 3 q^{51} - 12 q^{53} - 9 q^{55} + 9 q^{57} + 12 q^{59} + 27 q^{61} + 3 q^{63} + 9 q^{65} + 36 q^{67} + 18 q^{71} - 9 q^{73} + 12 q^{75} - 18 q^{79} - 9 q^{83} + 27 q^{85} - 27 q^{87} + 3 q^{89} - 12 q^{91} + 24 q^{93} - 24 q^{95} - 3 q^{97} - 9 q^{99} + O(q^{100}) \)

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
79.1
−0.173648 0.984808i
−0.173648 + 0.984808i
−0.766044 + 0.642788i
−0.766044 0.642788i
0.939693 0.342020i
0.939693 + 0.342020i
0 0.939693 0.342020i 0 −0.233956 + 1.32683i 0 3.20574 1.85083i 0 0.766044 0.642788i 0
127.1 0 0.939693 + 0.342020i 0 −0.233956 1.32683i 0 3.20574 + 1.85083i 0 0.766044 + 0.642788i 0
223.1 0 −0.173648 + 0.984808i 0 −1.93969 1.62760i 0 0.386659 0.223238i 0 −0.939693 0.342020i 0
319.1 0 −0.173648 0.984808i 0 −1.93969 + 1.62760i 0 0.386659 + 0.223238i 0 −0.939693 + 0.342020i 0
751.1 0 −0.766044 + 0.642788i 0 −0.826352 0.300767i 0 0.907604 + 0.524005i 0 0.173648 0.984808i 0
895.1 0 −0.766044 0.642788i 0 −0.826352 + 0.300767i 0 0.907604 0.524005i 0 0.173648 + 0.984808i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 895.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
76.k 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.ci.b yes 6
4.b odd 2 1 912.2.ci.a 6
19.f odd 18 1 912.2.ci.a 6
76.k even 18 1 inner 912.2.ci.b yes 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
912.2.ci.a 6 4.b odd 2 1
912.2.ci.a 6 19.f odd 18 1
912.2.ci.b yes 6 1.a even 1 1 trivial
912.2.ci.b yes 6 76.k even 18 1 inner

Hecke kernels

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

\( T_{5}^{6} + 6 T_{5}^{5} + 18 T_{5}^{4} + 30 T_{5}^{3} + 36 T_{5}^{2} + 27 T_{5} + 9 \)
\( T_{7}^{6} - 9 T_{7}^{5} + 33 T_{7}^{4} - 54 T_{7}^{3} + 45 T_{7}^{2} - 18 T_{7} + 3 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \)
$3$ \( 1 - T^{3} + T^{6} \)
$5$ \( 9 + 27 T + 36 T^{2} + 30 T^{3} + 18 T^{4} + 6 T^{5} + T^{6} \)
$7$ \( 3 - 18 T + 45 T^{2} - 54 T^{3} + 33 T^{4} - 9 T^{5} + T^{6} \)
$11$ \( 243 - 81 T^{2} + 27 T^{4} - 9 T^{5} + T^{6} \)
$13$ \( 3 - 18 T + 36 T^{2} - 33 T^{3} + 24 T^{4} + 6 T^{5} + T^{6} \)
$17$ \( 9 - 81 T + 306 T^{2} + 132 T^{3} + 54 T^{4} + 12 T^{5} + T^{6} \)
$19$ \( 6859 - 6498 T + 2736 T^{2} - 737 T^{3} + 144 T^{4} - 18 T^{5} + T^{6} \)
$23$ \( 1083 - 855 T + 99 T^{2} - 6 T^{3} + 24 T^{4} - 3 T^{5} + T^{6} \)
$29$ \( 7803 - 5508 T + 5238 T^{2} - 819 T^{3} - 18 T^{4} + T^{6} \)
$31$ \( 72361 - 12105 T + 3639 T^{2} - 268 T^{3} + 81 T^{4} - 6 T^{5} + T^{6} \)
$37$ \( 70227 + 7209 T^{2} + 162 T^{4} + T^{6} \)
$41$ \( 34347 - 10593 T + 6462 T^{2} + 33 T^{3} - 39 T^{4} + 12 T^{5} + T^{6} \)
$43$ \( 9747 + 6156 T + 1242 T^{2} + 153 T^{3} + 18 T^{4} + T^{6} \)
$47$ \( 604803 - 246501 T + 54810 T^{2} - 7674 T^{3} + 699 T^{4} - 39 T^{5} + T^{6} \)
$53$ \( 4107 + 3663 T + 1890 T^{2} + 303 T^{3} + 51 T^{4} + 12 T^{5} + T^{6} \)
$59$ \( 1172889 + 185193 T - 6498 T^{2} - 456 T^{3} + 234 T^{4} - 12 T^{5} + T^{6} \)
$61$ \( 54289 - 29358 T + 9891 T^{2} - 2231 T^{3} + 324 T^{4} - 27 T^{5} + T^{6} \)
$67$ \( 179776 - 76320 T + 24480 T^{2} - 4960 T^{3} + 576 T^{4} - 36 T^{5} + T^{6} \)
$71$ \( 23409 - 19278 T + 6966 T^{2} - 1449 T^{3} + 198 T^{4} - 18 T^{5} + T^{6} \)
$73$ \( 104329 - 5814 T + 6876 T^{2} - 1376 T^{3} - 18 T^{4} + 9 T^{5} + T^{6} \)
$79$ \( 26569 - 8802 T + 4986 T^{2} - 701 T^{3} + 18 T^{4} + 18 T^{5} + T^{6} \)
$83$ \( 3 - 36 T + 153 T^{2} - 108 T^{3} + 15 T^{4} + 9 T^{5} + T^{6} \)
$89$ \( 98283 + 4887 T + 333 T^{2} + 642 T^{3} + 6 T^{4} - 3 T^{5} + T^{6} \)
$97$ \( 867 - 918 T + 990 T^{2} - 318 T^{3} + 6 T^{4} + 3 T^{5} + T^{6} \)
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