Properties

Label 912.3.be.d
Level $912$
Weight $3$
Character orbit 912.be
Analytic conductor $24.850$
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: \( k \) \(=\) \( 3 \)
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: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{6})\)
Coefficient field: 6.0.6967728.1
Defining polynomial: \(x^{6} - x^{5} + 8 x^{4} + 5 x^{3} + 50 x^{2} - 7 x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 57)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -2 - \beta_{3} ) q^{3} + ( -\beta_{2} + \beta_{3} ) q^{5} + ( -4 - \beta_{1} + \beta_{4} + \beta_{5} ) q^{7} + ( 3 + 3 \beta_{3} ) q^{9} +O(q^{10})\) \( q + ( -2 - \beta_{3} ) q^{3} + ( -\beta_{2} + \beta_{3} ) q^{5} + ( -4 - \beta_{1} + \beta_{4} + \beta_{5} ) q^{7} + ( 3 + 3 \beta_{3} ) q^{9} + ( -1 + 3 \beta_{1} ) q^{11} + ( -2 + 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} - \beta_{4} ) q^{13} + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{15} + ( -2 + 4 \beta_{2} + 4 \beta_{3} + 4 \beta_{4} - 2 \beta_{5} ) q^{17} + ( -2 + 3 \beta_{1} + 2 \beta_{2} - 16 \beta_{3} - 2 \beta_{5} ) q^{19} + ( 7 + 2 \beta_{1} + \beta_{2} + 5 \beta_{3} - 3 \beta_{5} ) q^{21} + ( 11 - 3 \beta_{1} - 3 \beta_{2} + 7 \beta_{3} - 2 \beta_{4} + 4 \beta_{5} ) q^{23} + ( 7 + 3 \beta_{3} - 2 \beta_{4} + 4 \beta_{5} ) q^{25} + ( -3 - 6 \beta_{3} ) q^{27} + ( 8 + 2 \beta_{1} - 2 \beta_{2} - 8 \beta_{3} - 4 \beta_{4} ) q^{29} + ( -2 - 3 \beta_{1} - 6 \beta_{2} - 2 \beta_{3} + \beta_{4} - \beta_{5} ) q^{31} + ( 2 - 6 \beta_{1} - 3 \beta_{2} + \beta_{3} ) q^{33} + ( 9 \beta_{2} - 5 \beta_{3} ) q^{35} + ( -3 - 24 \beta_{3} - 9 \beta_{4} + 9 \beta_{5} ) q^{37} + ( 7 - 6 \beta_{1} + \beta_{4} + \beta_{5} ) q^{39} + ( 12 - 12 \beta_{1} - 6 \beta_{2} + 2 \beta_{3} + 8 \beta_{5} ) q^{41} + ( -\beta_{2} + 4 \beta_{3} ) q^{43} + ( -3 + 3 \beta_{1} ) q^{45} + ( 4 - 4 \beta_{1} - 4 \beta_{2} + 16 \beta_{3} + 6 \beta_{4} - 12 \beta_{5} ) q^{47} + ( 14 + 6 \beta_{1} - 4 \beta_{4} - 4 \beta_{5} ) q^{49} + ( 4 + 4 \beta_{1} - 4 \beta_{2} - 4 \beta_{3} - 6 \beta_{4} ) q^{51} + ( 17 - 3 \beta_{1} + 3 \beta_{2} - 17 \beta_{3} + 12 \beta_{4} ) q^{53} + ( 6 - 2 \beta_{2} - 64 \beta_{3} - 12 \beta_{4} + 6 \beta_{5} ) q^{55} + ( -12 - 4 \beta_{1} - 5 \beta_{2} + 16 \beta_{3} - 2 \beta_{4} + 4 \beta_{5} ) q^{57} + ( -28 - 10 \beta_{1} - 5 \beta_{2} - 21 \beta_{3} + 14 \beta_{5} ) q^{59} + ( -48 - 12 \beta_{1} - 12 \beta_{2} - 38 \beta_{3} + 5 \beta_{4} - 10 \beta_{5} ) q^{61} + ( -9 - 3 \beta_{1} - 3 \beta_{2} - 15 \beta_{3} - 3 \beta_{4} + 6 \beta_{5} ) q^{63} + ( -27 - \beta_{1} - 2 \beta_{2} - 74 \beta_{3} - 10 \beta_{4} + 10 \beta_{5} ) q^{65} + ( 4 + \beta_{1} - \beta_{2} - 4 \beta_{3} + 8 \beta_{4} ) q^{67} + ( -13 + 3 \beta_{1} + 6 \beta_{2} - 14 \beta_{3} + 6 \beta_{4} - 6 \beta_{5} ) q^{69} + ( 86 - 4 \beta_{1} - 2 \beta_{2} + 40 \beta_{3} + 6 \beta_{5} ) q^{71} + ( -10 + 20 \beta_{2} - 31 \beta_{3} + 20 \beta_{4} - 10 \beta_{5} ) q^{73} + ( -9 - 6 \beta_{3} + 6 \beta_{4} - 6 \beta_{5} ) q^{75} + ( 7 - 29 \beta_{1} + 2 \beta_{4} + 2 \beta_{5} ) q^{77} + ( 29 + 2 \beta_{1} + \beta_{2} + 11 \beta_{3} + 7 \beta_{5} ) q^{79} + 9 \beta_{3} q^{81} + ( 42 - 2 \beta_{4} - 2 \beta_{5} ) q^{83} + ( 30 + 14 \beta_{1} + 14 \beta_{2} + 38 \beta_{3} + 4 \beta_{4} - 8 \beta_{5} ) q^{85} + ( -20 - 6 \beta_{1} + 4 \beta_{4} + 4 \beta_{5} ) q^{87} + ( 71 + \beta_{1} - \beta_{2} - 71 \beta_{3} ) q^{89} + ( -4 - 15 \beta_{1} + 15 \beta_{2} + 4 \beta_{3} - 2 \beta_{4} ) q^{91} + ( 1 + 9 \beta_{2} + 3 \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{93} + ( 42 - 12 \beta_{1} - 19 \beta_{2} + 9 \beta_{3} - 4 \beta_{4} - 2 \beta_{5} ) q^{95} + ( 14 - 16 \beta_{1} - 8 \beta_{2} - 10 \beta_{3} + 34 \beta_{5} ) q^{97} + ( -3 + 9 \beta_{1} + 9 \beta_{2} - 3 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 9 q^{3} - 2 q^{5} - 26 q^{7} + 9 q^{9} + O(q^{10}) \) \( 6 q - 9 q^{3} - 2 q^{5} - 26 q^{7} + 9 q^{9} - 15 q^{13} + 6 q^{15} - 10 q^{17} + 46 q^{19} + 39 q^{21} + 24 q^{23} + 15 q^{25} + 66 q^{29} + 6 q^{35} + 30 q^{39} + 24 q^{41} - 11 q^{43} - 12 q^{45} + 26 q^{47} + 96 q^{49} + 30 q^{51} + 180 q^{53} + 176 q^{55} - 141 q^{57} - 162 q^{59} - 141 q^{61} - 39 q^{63} + 63 q^{67} + 372 q^{71} + 103 q^{73} - 16 q^{77} + 123 q^{79} - 27 q^{81} + 252 q^{83} + 116 q^{85} - 132 q^{87} + 642 q^{89} - 87 q^{91} - 21 q^{93} + 214 q^{95} - 12 q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - x^{5} + 8 x^{4} + 5 x^{3} + 50 x^{2} - 7 x + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 2 \nu^{5} - 16 \nu^{4} + 128 \nu^{3} - 100 \nu^{2} + 14 \nu + 281 \)\()/393\)
\(\beta_{2}\)\(=\)\((\)\( -56 \nu^{5} + 55 \nu^{4} - 440 \nu^{3} - 344 \nu^{2} - 1964 \nu - 8 \)\()/393\)
\(\beta_{3}\)\(=\)\((\)\( -56 \nu^{5} + 55 \nu^{4} - 440 \nu^{3} - 344 \nu^{2} - 2750 \nu - 8 \)\()/393\)
\(\beta_{4}\)\(=\)\((\)\( 196 \nu^{5} - 127 \nu^{4} + 1540 \nu^{3} + 1466 \nu^{2} + 10018 \nu + 1469 \)\()/393\)
\(\beta_{5}\)\(=\)\((\)\( -210 \nu^{5} + 239 \nu^{4} - 1650 \nu^{3} - 766 \nu^{2} - 10116 \nu + 2459 \)\()/393\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{3} + \beta_{2}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(2 \beta_{5} - \beta_{4} - 11 \beta_{3} - 9\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{5} + \beta_{4} + 7 \beta_{1} - 15\)\()/2\)
\(\nu^{4}\)\(=\)\(-4 \beta_{5} + 8 \beta_{4} + 46 \beta_{3} - 3 \beta_{2} - 4\)
\(\nu^{5}\)\(=\)\((\)\(-28 \beta_{5} + 14 \beta_{4} + 193 \beta_{3} - 55 \beta_{2} - 55 \beta_{1} + 165\)\()/2\)

Character values

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

\(n\) \(97\) \(229\) \(305\) \(799\)
\(\chi(n)\) \(1 + \beta_{3}\) \(1\) \(1\) \(1\)

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
1.56632 + 2.71294i
0.0702177 + 0.121621i
−1.13654 1.96854i
1.56632 2.71294i
0.0702177 0.121621i
−1.13654 + 1.96854i
0 −1.50000 + 0.866025i 0 −3.13264 5.42589i 0 −8.36156 0 1.50000 2.59808i 0
145.2 0 −1.50000 + 0.866025i 0 −0.140435 0.243241i 0 5.24143 0 1.50000 2.59808i 0
145.3 0 −1.50000 + 0.866025i 0 2.27307 + 3.93708i 0 −9.87987 0 1.50000 2.59808i 0
673.1 0 −1.50000 0.866025i 0 −3.13264 + 5.42589i 0 −8.36156 0 1.50000 + 2.59808i 0
673.2 0 −1.50000 0.866025i 0 −0.140435 + 0.243241i 0 5.24143 0 1.50000 + 2.59808i 0
673.3 0 −1.50000 0.866025i 0 2.27307 3.93708i 0 −9.87987 0 1.50000 + 2.59808i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 673.3
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.d 6
4.b odd 2 1 57.3.g.a 6
12.b even 2 1 171.3.p.e 6
19.d odd 6 1 inner 912.3.be.d 6
76.f even 6 1 57.3.g.a 6
228.n odd 6 1 171.3.p.e 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
57.3.g.a 6 4.b odd 2 1
57.3.g.a 6 76.f even 6 1
171.3.p.e 6 12.b even 2 1
171.3.p.e 6 228.n odd 6 1
912.3.be.d 6 1.a even 1 1 trivial
912.3.be.d 6 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 \(S_{3}^{\mathrm{new}}(912, [\chi])\):

\( T_{5}^{6} + 2 T_{5}^{5} + 32 T_{5}^{4} - 40 T_{5}^{3} + 800 T_{5}^{2} + 224 T_{5} + 64 \)
\( T_{7}^{3} + 13 T_{7}^{2} - 13 T_{7} - 433 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \)
$3$ \( ( 3 + 3 T + T^{2} )^{3} \)
$5$ \( 64 + 224 T + 800 T^{2} - 40 T^{3} + 32 T^{4} + 2 T^{5} + T^{6} \)
$7$ \( ( -433 - 13 T + 13 T^{2} + T^{3} )^{2} \)
$11$ \( ( 304 - 264 T + T^{3} )^{2} \)
$13$ \( 3518667 + 802503 T + 44764 T^{2} - 3705 T^{3} - 172 T^{4} + 15 T^{5} + T^{6} \)
$17$ \( 5184 + 26784 T + 139104 T^{2} - 3576 T^{3} + 472 T^{4} + 10 T^{5} + T^{6} \)
$19$ \( 47045881 - 5994766 T + 499263 T^{2} - 31844 T^{3} + 1383 T^{4} - 46 T^{5} + T^{6} \)
$23$ \( 1517824 - 147840 T + 43968 T^{2} + 416 T^{3} + 696 T^{4} - 24 T^{5} + T^{6} \)
$29$ \( 19293888 + 517344 T - 162752 T^{2} - 4488 T^{3} + 1520 T^{4} - 66 T^{5} + T^{6} \)
$31$ \( 124768803 + 1160107 T^{2} + 2033 T^{4} + T^{6} \)
$37$ \( 1689765867 + 7315515 T^{2} + 5913 T^{4} + T^{6} \)
$41$ \( 1907539968 - 186396672 T + 5466112 T^{2} + 59136 T^{3} - 2272 T^{4} - 24 T^{5} + T^{6} \)
$43$ \( 2209 - 517 T + 638 T^{2} + 215 T^{3} + 110 T^{4} + 11 T^{5} + T^{6} \)
$47$ \( 2266521664 + 240134752 T + 24204128 T^{2} + 226360 T^{3} + 5720 T^{4} - 26 T^{5} + T^{6} \)
$53$ \( 2495232 + 590976 T - 117504 T^{2} - 38880 T^{3} + 11016 T^{4} - 180 T^{5} + T^{6} \)
$59$ \( 48350430912 + 623080416 T - 17889728 T^{2} - 265032 T^{3} + 7112 T^{4} + 162 T^{5} + T^{6} \)
$61$ \( 558907255201 + 2509696557 T + 116681190 T^{2} + 1021865 T^{3} + 23238 T^{4} + 141 T^{5} + T^{6} \)
$67$ \( 2349816507 - 179928423 T + 2829268 T^{2} + 135009 T^{3} - 820 T^{4} - 63 T^{5} + T^{6} \)
$71$ \( 100019515392 - 7966854144 T + 279452160 T^{2} - 5410368 T^{3} + 60672 T^{4} - 372 T^{5} + T^{6} \)
$73$ \( 214090364601 - 3052425303 T + 91178406 T^{2} - 245907 T^{3} + 17206 T^{4} - 103 T^{5} + T^{6} \)
$79$ \( 2549342403 - 32095251 T - 3450884 T^{2} + 45141 T^{3} + 4676 T^{4} - 123 T^{5} + T^{6} \)
$83$ \( ( -57704 + 4908 T - 126 T^{2} + T^{3} )^{2} \)
$89$ \( 3516172210368 - 148452636384 T + 2784260736 T^{2} - 29344536 T^{3} + 183096 T^{4} - 642 T^{5} + T^{6} \)
$97$ \( 3000768049152 + 82138512384 T + 737443840 T^{2} - 328512 T^{3} - 27328 T^{4} + 12 T^{5} + T^{6} \)
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