Properties

Label 912.3.be.g
Level $912$
Weight $3$
Character orbit 912.be
Analytic conductor $24.850$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 912 = 2^{4} \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 912.be (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(24.8502001097\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.520060207104.10
Defining polynomial: \( x^{8} - 44x^{6} + 664x^{4} - 3528x^{2} + 8100 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 3^{3} \)
Twist minimal: no (minimal twist has level 114)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 - 1) q^{3} + ( - \beta_{4} - \beta_1 + 1) q^{5} + ( - \beta_{6} + \beta_{2} + 3) q^{7} + 3 \beta_1 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 - 1) q^{3} + ( - \beta_{4} - \beta_1 + 1) q^{5} + ( - \beta_{6} + \beta_{2} + 3) q^{7} + 3 \beta_1 q^{9} + (\beta_{7} - \beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} + 1) q^{11} + (\beta_{7} + \beta_{6} - 2 \beta_{5} + 2 \beta_{4} - \beta_{3} + \beta_{2} - 2 \beta_1 + 4) q^{13} + (\beta_{7} + 2 \beta_{4} + \beta_1 - 2) q^{15} + (4 \beta_{6} - \beta_{5} + \beta_{4} - 3 \beta_{3} + \beta_{2} + 5 \beta_1 - 5) q^{17} + (3 \beta_{6} - 5 \beta_{5} + 8 \beta_1 - 7) q^{19} + (2 \beta_{6} - \beta_{5} + \beta_{3} - \beta_{2} - 3 \beta_1 - 3) q^{21} + (3 \beta_{6} - 2 \beta_{5} + 3 \beta_{3} + \beta_{2} - 10 \beta_1) q^{23} + (3 \beta_{7} - \beta_{6} - 2 \beta_{5} - \beta_{3} - 3 \beta_{2} - 22 \beta_1) q^{25} + ( - 6 \beta_1 + 3) q^{27} + ( - 2 \beta_{6} + 2 \beta_{3} + 2 \beta_{2} + 4 \beta_1 - 8) q^{29} + (6 \beta_{6} - 3 \beta_{5} + 3 \beta_{3} + 6 \beta_{2} + 14 \beta_1 - 7) q^{31} + ( - 2 \beta_{7} + \beta_{6} - 2 \beta_{5} - \beta_{4} - \beta_{3} - 2 \beta_{2} - \beta_1 - 1) q^{33} + (\beta_{6} + 5 \beta_{5} - 2 \beta_{4} - 6 \beta_{3} - 5 \beta_{2} + 8 \beta_1 - 8) q^{35} + ( - 2 \beta_{7} - 4 \beta_{6} + 4 \beta_{5} + 2 \beta_{4} - 4 \beta_{3} - 4 \beta_{2} + \cdots + 5) q^{37}+ \cdots + (3 \beta_{7} + 3 \beta_{5} + 3 \beta_{2} + 3 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 12 q^{3} + 4 q^{5} + 24 q^{7} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 12 q^{3} + 4 q^{5} + 24 q^{7} + 12 q^{9} + 8 q^{11} + 24 q^{13} - 12 q^{15} - 20 q^{17} - 24 q^{19} - 36 q^{21} - 40 q^{23} - 88 q^{25} - 48 q^{29} - 12 q^{33} - 32 q^{35} - 48 q^{39} + 60 q^{41} + 116 q^{43} + 24 q^{45} + 68 q^{47} - 120 q^{49} + 60 q^{51} - 168 q^{53} - 232 q^{55} + 84 q^{57} + 156 q^{59} + 72 q^{61} + 36 q^{63} + 108 q^{67} - 444 q^{71} - 68 q^{73} + 296 q^{77} - 420 q^{79} - 36 q^{81} - 424 q^{83} + 40 q^{85} + 96 q^{87} - 420 q^{89} + 228 q^{91} + 84 q^{93} + 272 q^{95} + 156 q^{97} + 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 44x^{6} + 664x^{4} - 3528x^{2} + 8100 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{7} + 44\nu^{5} - 574\nu^{3} + 1548\nu + 1080 ) / 2160 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -7\nu^{7} - 15\nu^{6} + 263\nu^{5} - 105\nu^{4} - 3208\nu^{3} + 9840\nu^{2} + 1566\nu - 61830 ) / 14040 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -7\nu^{7} + 15\nu^{6} + 263\nu^{5} + 105\nu^{4} - 3208\nu^{3} - 9840\nu^{2} + 1566\nu + 61830 ) / 14040 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -11\nu^{7} + 270\nu^{6} + 664\nu^{5} - 9810\nu^{4} - 14234\nu^{3} + 122400\nu^{2} + 86868\nu - 403380 ) / 28080 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 14\nu^{7} + 30\nu^{6} - 526\nu^{5} - 1545\nu^{4} + 6416\nu^{3} + 18930\nu^{2} - 24192\nu - 34290 ) / 14040 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -14\nu^{7} + 30\nu^{6} + 526\nu^{5} - 1545\nu^{4} - 6416\nu^{3} + 18930\nu^{2} + 24192\nu - 34290 ) / 14040 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 31\nu^{7} - 300\nu^{6} - 914\nu^{5} + 10770\nu^{4} + 5014\nu^{3} - 128460\nu^{2} + 35352\nu + 385020 ) / 28080 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{6} - \beta_{5} - 2\beta_{3} - 2\beta_{2} ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{7} - 4\beta_{6} - 3\beta_{5} + \beta_{4} - 3\beta_{3} + 2\beta_{2} + 33 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -9\beta_{7} + \beta_{6} - 4\beta_{5} - 9\beta_{4} - 38\beta_{3} - 35\beta_{2} + 54\beta _1 - 27 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -22\beta_{7} - 100\beta_{6} - 78\beta_{5} + 22\beta_{4} - 42\beta_{3} + 20\beta_{2} + 456 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -162\beta_{7} - 188\beta_{6} + 134\beta_{5} - 162\beta_{4} - 740\beta_{3} - 686\beta_{2} + 1980\beta _1 - 990 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -502\beta_{7} - 1924\beta_{6} - 1422\beta_{5} + 502\beta_{4} - 270\beta_{3} - 232\beta_{2} + 6090 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 1962 \beta_{7} - 7298 \beta_{6} + 6644 \beta_{5} - 1962 \beta_{4} - 13844 \beta_{3} - 13190 \beta_{2} + 49644 \beta _1 - 24822 ) / 3 \) Copy content Toggle raw display

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
145.1
−4.34148 0.707107i
2.03753 + 0.707107i
4.34148 0.707107i
−2.03753 + 0.707107i
−4.34148 + 0.707107i
2.03753 0.707107i
4.34148 + 0.707107i
−2.03753 0.707107i
0 −1.50000 + 0.866025i 0 −3.25884 5.64448i 0 11.0157 0 1.50000 2.59808i 0
145.2 0 −1.50000 + 0.866025i 0 −2.40185 4.16012i 0 −2.71177 0 1.50000 2.59808i 0
145.3 0 −1.50000 + 0.866025i 0 3.03409 + 5.25521i 0 2.33275 0 1.50000 2.59808i 0
145.4 0 −1.50000 + 0.866025i 0 4.62659 + 8.01349i 0 1.36330 0 1.50000 2.59808i 0
673.1 0 −1.50000 0.866025i 0 −3.25884 + 5.64448i 0 11.0157 0 1.50000 + 2.59808i 0
673.2 0 −1.50000 0.866025i 0 −2.40185 + 4.16012i 0 −2.71177 0 1.50000 + 2.59808i 0
673.3 0 −1.50000 0.866025i 0 3.03409 5.25521i 0 2.33275 0 1.50000 + 2.59808i 0
673.4 0 −1.50000 0.866025i 0 4.62659 8.01349i 0 1.36330 0 1.50000 + 2.59808i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 673.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 912.3.be.g 8
4.b odd 2 1 114.3.f.b 8
12.b even 2 1 342.3.m.b 8
19.d odd 6 1 inner 912.3.be.g 8
76.f even 6 1 114.3.f.b 8
228.n odd 6 1 342.3.m.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
114.3.f.b 8 4.b odd 2 1
114.3.f.b 8 76.f even 6 1
342.3.m.b 8 12.b even 2 1
342.3.m.b 8 228.n odd 6 1
912.3.be.g 8 1.a even 1 1 trivial
912.3.be.g 8 19.d odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(912, [\chi])\):

\( T_{5}^{8} - 4T_{5}^{7} + 102T_{5}^{6} + 32T_{5}^{5} + 6262T_{5}^{4} + 648T_{5}^{3} + 175524T_{5}^{2} + 274248T_{5} + 3090564 \) Copy content Toggle raw display
\( T_{7}^{4} - 12T_{7}^{3} + 4T_{7}^{2} + 84T_{7} - 95 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{2} + 3 T + 3)^{4} \) Copy content Toggle raw display
$5$ \( T^{8} - 4 T^{7} + 102 T^{6} + \cdots + 3090564 \) Copy content Toggle raw display
$7$ \( (T^{4} - 12 T^{3} + 4 T^{2} + 84 T - 95)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} - 4 T^{3} - 230 T^{2} - 60 T + 1902)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} - 24 T^{7} + \cdots + 845588241 \) Copy content Toggle raw display
$17$ \( T^{8} + 20 T^{7} + \cdots + 42571243584 \) Copy content Toggle raw display
$19$ \( T^{8} + 24 T^{7} + \cdots + 16983563041 \) Copy content Toggle raw display
$23$ \( T^{8} + 40 T^{7} + 1266 T^{6} + \cdots + 1726596 \) Copy content Toggle raw display
$29$ \( T^{8} + 48 T^{7} + \cdots + 269485056 \) Copy content Toggle raw display
$31$ \( T^{8} + 3288 T^{6} + \cdots + 157608206001 \) Copy content Toggle raw display
$37$ \( T^{8} + 3084 T^{6} + \cdots + 3853057329 \) Copy content Toggle raw display
$41$ \( T^{8} - 60 T^{7} + \cdots + 4588472453184 \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 196975358413681 \) Copy content Toggle raw display
$47$ \( T^{8} - 68 T^{7} + \cdots + 791744040000 \) Copy content Toggle raw display
$53$ \( T^{8} + 168 T^{7} + \cdots + 984988761156 \) Copy content Toggle raw display
$59$ \( T^{8} - 156 T^{7} + \cdots + 76622956930116 \) Copy content Toggle raw display
$61$ \( T^{8} - 72 T^{7} + \cdots + 2354429392225 \) Copy content Toggle raw display
$67$ \( T^{8} - 108 T^{7} + \cdots + 421690488129 \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 158714643240000 \) Copy content Toggle raw display
$73$ \( T^{8} + 68 T^{7} + \cdots + 14\!\cdots\!21 \) Copy content Toggle raw display
$79$ \( T^{8} + 420 T^{7} + \cdots + 36966995842401 \) Copy content Toggle raw display
$83$ \( (T^{4} + 212 T^{3} + 13252 T^{2} + \cdots - 740232)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 341891268051204 \) Copy content Toggle raw display
$97$ \( T^{8} - 156 T^{7} + \cdots + 312852286224 \) Copy content Toggle raw display
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