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 $2$

Related objects

Downloads

Learn more

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

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

Character values

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

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

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 \(S_{2}^{\mathrm{new}}(912, [\chi])\):

\( T_{5}^{6} + 6T_{5}^{5} + 18T_{5}^{4} + 30T_{5}^{3} + 36T_{5}^{2} + 27T_{5} + 9 \) Copy content Toggle raw display
\( T_{7}^{6} - 9T_{7}^{5} + 33T_{7}^{4} - 54T_{7}^{3} + 45T_{7}^{2} - 18T_{7} + 3 \) 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
$5$ \( T^{6} + 6 T^{5} + 18 T^{4} + 30 T^{3} + \cdots + 9 \) Copy content Toggle raw display
$7$ \( T^{6} - 9 T^{5} + 33 T^{4} - 54 T^{3} + \cdots + 3 \) Copy content Toggle raw display
$11$ \( T^{6} - 9 T^{5} + 27 T^{4} - 81 T^{2} + \cdots + 243 \) Copy content Toggle raw display
$13$ \( T^{6} + 6 T^{5} + 24 T^{4} - 33 T^{3} + \cdots + 3 \) Copy content Toggle raw display
$17$ \( T^{6} + 12 T^{5} + 54 T^{4} + 132 T^{3} + \cdots + 9 \) Copy content Toggle raw display
$19$ \( T^{6} - 18 T^{5} + 144 T^{4} + \cdots + 6859 \) Copy content Toggle raw display
$23$ \( T^{6} - 3 T^{5} + 24 T^{4} + \cdots + 1083 \) Copy content Toggle raw display
$29$ \( T^{6} - 18 T^{4} - 819 T^{3} + \cdots + 7803 \) Copy content Toggle raw display
$31$ \( T^{6} - 6 T^{5} + 81 T^{4} + \cdots + 72361 \) Copy content Toggle raw display
$37$ \( T^{6} + 162 T^{4} + 7209 T^{2} + \cdots + 70227 \) Copy content Toggle raw display
$41$ \( T^{6} + 12 T^{5} - 39 T^{4} + \cdots + 34347 \) Copy content Toggle raw display
$43$ \( T^{6} + 18 T^{4} + 153 T^{3} + \cdots + 9747 \) Copy content Toggle raw display
$47$ \( T^{6} - 39 T^{5} + 699 T^{4} + \cdots + 604803 \) Copy content Toggle raw display
$53$ \( T^{6} + 12 T^{5} + 51 T^{4} + \cdots + 4107 \) Copy content Toggle raw display
$59$ \( T^{6} - 12 T^{5} + 234 T^{4} + \cdots + 1172889 \) Copy content Toggle raw display
$61$ \( T^{6} - 27 T^{5} + 324 T^{4} + \cdots + 54289 \) Copy content Toggle raw display
$67$ \( T^{6} - 36 T^{5} + 576 T^{4} + \cdots + 179776 \) Copy content Toggle raw display
$71$ \( T^{6} - 18 T^{5} + 198 T^{4} + \cdots + 23409 \) Copy content Toggle raw display
$73$ \( T^{6} + 9 T^{5} - 18 T^{4} + \cdots + 104329 \) Copy content Toggle raw display
$79$ \( T^{6} + 18 T^{5} + 18 T^{4} + \cdots + 26569 \) Copy content Toggle raw display
$83$ \( T^{6} + 9 T^{5} + 15 T^{4} - 108 T^{3} + \cdots + 3 \) Copy content Toggle raw display
$89$ \( T^{6} - 3 T^{5} + 6 T^{4} + \cdots + 98283 \) Copy content Toggle raw display
$97$ \( T^{6} + 3 T^{5} + 6 T^{4} - 318 T^{3} + \cdots + 867 \) Copy content Toggle raw display
show more
show less