Properties

Label 912.2.ci.c
Level $912$
Weight $2$
Character orbit 912.ci
Analytic conductor $7.282$
Analytic rank $0$
Dimension $12$
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: \(12\)
Relative dimension: \(2\) over \(\Q(\zeta_{18})\)
Coefficient field: 12.0.101559956668416.1
Defining polynomial: \( x^{12} - 8x^{6} + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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 basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{8} q^{3} + ( - \beta_{10} - \beta_{9} - \beta_{6} + \beta_{4} + \beta_1 + 1) q^{5} + (\beta_{11} - \beta_{10} + \beta_{8} - \beta_{7} - \beta_{6} - \beta_{5} + \beta_{4} - 1) q^{7} + (\beta_{10} - \beta_{4}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{8} q^{3} + ( - \beta_{10} - \beta_{9} - \beta_{6} + \beta_{4} + \beta_1 + 1) q^{5} + (\beta_{11} - \beta_{10} + \beta_{8} - \beta_{7} - \beta_{6} - \beta_{5} + \beta_{4} - 1) q^{7} + (\beta_{10} - \beta_{4}) q^{9} + ( - \beta_{11} + 2 \beta_{9} - \beta_{8} - \beta_{7} + \beta_{5} + \beta_{4} - 2 \beta_{3} + \beta_1) q^{11} + ( - \beta_{11} - \beta_{8} + \beta_{6} + \beta_{5} + \beta_{3} + 2 \beta_{2} + 1) q^{13} + (\beta_{11} - \beta_{9} - \beta_{6} - \beta_{5} - \beta_{2}) q^{15} + (\beta_{11} - \beta_{10} + \beta_{6} - 2 \beta_{5} + \beta_{4} - \beta_{3} - \beta_1) q^{17} + ( - \beta_{9} - 2 \beta_{8} - \beta_{7} - 2 \beta_{6} + 2 \beta_{5} + \beta_{2} + \beta_1) q^{19} + ( - \beta_{10} + \beta_{9} + 2 \beta_{8} + \beta_{7} - \beta_{6} + \beta_{4} - \beta_{3} - \beta_{2}) q^{21} + (2 \beta_{11} + 2 \beta_{9} - 2 \beta_{8} - 2 \beta_{7} - \beta_{6} - 2 \beta_{5} + \beta_{2} + \cdots + 2) q^{23}+ \cdots + (\beta_{11} + \beta_{9} - 2 \beta_{7} + \beta_{6} - \beta_{3} - \beta_{2} + 2 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 6 q^{5} - 18 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 6 q^{5} - 18 q^{7} + 18 q^{13} - 6 q^{15} + 6 q^{17} - 12 q^{19} - 6 q^{21} + 18 q^{23} - 30 q^{25} + 6 q^{27} - 6 q^{29} - 18 q^{31} + 6 q^{33} - 36 q^{35} + 6 q^{41} + 6 q^{43} - 6 q^{47} + 12 q^{49} - 6 q^{51} - 36 q^{53} + 42 q^{55} + 6 q^{59} - 6 q^{61} - 6 q^{63} + 72 q^{65} - 6 q^{67} - 54 q^{71} - 12 q^{73} - 12 q^{75} - 36 q^{77} + 6 q^{79} - 18 q^{83} - 36 q^{85} + 36 q^{87} + 24 q^{89} - 12 q^{91} - 18 q^{93} - 24 q^{95} + 24 q^{97} + 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 8x^{6} + 64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{4} ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{5} ) / 4 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{6} ) / 8 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{7} ) / 8 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( \nu^{8} ) / 16 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( \nu^{9} ) / 16 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( \nu^{10} ) / 32 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( \nu^{11} ) / 32 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{3} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 4\beta_{4} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 4\beta_{5} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 8\beta_{6} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 8\beta_{7} \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 16\beta_{8} \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 16\beta_{9} \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 32\beta_{10} \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 32\beta_{11} \) Copy content Toggle raw display

Character values

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

\(n\) \(97\) \(229\) \(305\) \(799\)
\(\chi(n)\) \(-\beta_{4}\) \(1\) \(1\) \(-1\)

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
−0.483690 + 1.32893i
0.483690 1.32893i
−0.483690 1.32893i
0.483690 + 1.32893i
−1.39273 + 0.245576i
1.39273 0.245576i
−1.39273 0.245576i
1.39273 + 0.245576i
−0.909039 + 1.08335i
0.909039 1.08335i
−0.909039 1.08335i
0.909039 + 1.08335i
0 0.939693 0.342020i 0 −0.749734 + 4.25195i 0 −3.63109 + 2.09641i 0 0.766044 0.642788i 0
79.2 0 0.939693 0.342020i 0 0.217645 1.23433i 0 −2.78039 + 1.60526i 0 0.766044 0.642788i 0
127.1 0 0.939693 + 0.342020i 0 −0.749734 4.25195i 0 −3.63109 2.09641i 0 0.766044 + 0.642788i 0
127.2 0 0.939693 + 0.342020i 0 0.217645 + 1.23433i 0 −2.78039 1.60526i 0 0.766044 + 0.642788i 0
223.1 0 −0.173648 + 0.984808i 0 0.0469641 + 0.0394076i 0 1.48976 0.860113i 0 −0.939693 0.342020i 0
223.2 0 −0.173648 + 0.984808i 0 2.83242 + 2.37668i 0 −2.26308 + 1.30659i 0 −0.939693 0.342020i 0
319.1 0 −0.173648 0.984808i 0 0.0469641 0.0394076i 0 1.48976 + 0.860113i 0 −0.939693 + 0.342020i 0
319.2 0 −0.173648 0.984808i 0 2.83242 2.37668i 0 −2.26308 1.30659i 0 −0.939693 + 0.342020i 0
751.1 0 −0.766044 + 0.642788i 0 −0.582687 0.212081i 0 1.39416 + 0.804921i 0 0.173648 0.984808i 0
751.2 0 −0.766044 + 0.642788i 0 1.23539 + 0.449645i 0 −3.20937 1.85293i 0 0.173648 0.984808i 0
895.1 0 −0.766044 0.642788i 0 −0.582687 + 0.212081i 0 1.39416 0.804921i 0 0.173648 + 0.984808i 0
895.2 0 −0.766044 0.642788i 0 1.23539 0.449645i 0 −3.20937 + 1.85293i 0 0.173648 + 0.984808i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 895.2
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.c 12
4.b odd 2 1 912.2.ci.d yes 12
19.f odd 18 1 912.2.ci.d yes 12
76.k even 18 1 inner 912.2.ci.c 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
912.2.ci.c 12 1.a even 1 1 trivial
912.2.ci.c 12 76.k even 18 1 inner
912.2.ci.d yes 12 4.b odd 2 1
912.2.ci.d yes 12 19.f odd 18 1

Hecke kernels

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

\( T_{5}^{12} - 6 T_{5}^{11} + 33 T_{5}^{10} - 136 T_{5}^{9} + 450 T_{5}^{8} - 612 T_{5}^{7} + 432 T_{5}^{6} - 90 T_{5}^{5} - 342 T_{5}^{4} + 296 T_{5}^{3} + 240 T_{5}^{2} - 24 T_{5} + 1 \) Copy content Toggle raw display
\( T_{7}^{12} + 18 T_{7}^{11} + 135 T_{7}^{10} + 486 T_{7}^{9} + 531 T_{7}^{8} - 1782 T_{7}^{7} - 4616 T_{7}^{6} + 6570 T_{7}^{5} + 29457 T_{7}^{4} - 71478 T_{7}^{2} + 130321 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( (T^{6} - T^{3} + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{12} - 6 T^{11} + 33 T^{10} - 136 T^{9} + \cdots + 1 \) Copy content Toggle raw display
$7$ \( T^{12} + 18 T^{11} + 135 T^{10} + \cdots + 130321 \) Copy content Toggle raw display
$11$ \( T^{12} - 42 T^{10} + 1515 T^{8} + \cdots + 5329 \) Copy content Toggle raw display
$13$ \( T^{12} - 18 T^{11} + 147 T^{10} - 756 T^{9} + \cdots + 1 \) Copy content Toggle raw display
$17$ \( T^{12} - 6 T^{11} - 21 T^{10} + \cdots + 26569 \) Copy content Toggle raw display
$19$ \( T^{12} + 12 T^{11} + 48 T^{10} + \cdots + 47045881 \) Copy content Toggle raw display
$23$ \( T^{12} - 18 T^{11} + 249 T^{10} + \cdots + 3988009 \) Copy content Toggle raw display
$29$ \( T^{12} + 6 T^{11} + 39 T^{10} + \cdots + 151117849 \) Copy content Toggle raw display
$31$ \( T^{12} + 18 T^{11} + \cdots + 254625849 \) Copy content Toggle raw display
$37$ \( T^{12} + 174 T^{10} + \cdots + 166126321 \) Copy content Toggle raw display
$41$ \( T^{12} - 6 T^{11} + 93 T^{10} + \cdots + 4422609 \) Copy content Toggle raw display
$43$ \( T^{12} - 6 T^{11} - 51 T^{10} + \cdots + 71014329 \) Copy content Toggle raw display
$47$ \( T^{12} + 6 T^{11} - 21 T^{10} + \cdots + 3052009 \) Copy content Toggle raw display
$53$ \( T^{12} + 36 T^{11} + \cdots + 4583154601 \) Copy content Toggle raw display
$59$ \( T^{12} - 6 T^{11} + 15 T^{10} - 28 T^{9} + \cdots + 361 \) Copy content Toggle raw display
$61$ \( T^{12} + 6 T^{11} + \cdots + 129665528281 \) Copy content Toggle raw display
$67$ \( T^{12} + 6 T^{11} - 45 T^{10} + \cdots + 34023889 \) Copy content Toggle raw display
$71$ \( (T^{6} + 27 T^{5} + 369 T^{4} + \cdots + 104329)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} + 12 T^{11} + \cdots + 749390625 \) Copy content Toggle raw display
$79$ \( T^{12} - 6 T^{11} + \cdots + 4267094329 \) Copy content Toggle raw display
$83$ \( T^{12} + 18 T^{11} + \cdots + 5079555441 \) Copy content Toggle raw display
$89$ \( T^{12} - 24 T^{11} + \cdots + 21849865489 \) Copy content Toggle raw display
$97$ \( T^{12} - 24 T^{11} + \cdots + 1138995001 \) Copy content Toggle raw display
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