Properties

Label 336.6.q.e
Level $336$
Weight $6$
Character orbit 336.q
Analytic conductor $53.889$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 336.q (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(53.8889634572\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-83})\)
Defining polynomial: \( x^{4} - x^{3} - 20x^{2} - 21x + 441 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 21)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 9 \beta_1 q^{3} + (7 \beta_{3} + 20 \beta_1 + 20) q^{5} + (14 \beta_{3} - 7 \beta_{2} + 7 \beta_1 + 91) q^{7} + ( - 81 \beta_1 - 81) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 9 \beta_1 q^{3} + (7 \beta_{3} + 20 \beta_1 + 20) q^{5} + (14 \beta_{3} - 7 \beta_{2} + 7 \beta_1 + 91) q^{7} + ( - 81 \beta_1 - 81) q^{9} + ( - \beta_{3} + \beta_{2} + 568 \beta_1) q^{11} + (9 \beta_{2} + 467) q^{13} + ( - 63 \beta_{2} - 180) q^{15} + (148 \beta_{3} - 148 \beta_{2} - 88 \beta_1) q^{17} + (27 \beta_{3} + 1169 \beta_1 + 1169) q^{19} + ( - 63 \beta_{3} - 63 \beta_{2} + 756 \beta_1 - 63) q^{21} + (308 \beta_{3} + 952 \beta_1 + 952) q^{23} + (231 \beta_{3} - 231 \beta_{2} + 313 \beta_1) q^{25} + 729 q^{27} + (45 \beta_{2} - 1086) q^{29} + ( - 768 \beta_{3} + 768 \beta_{2} - 2531 \beta_1) q^{31} + (9 \beta_{3} - 5112 \beta_1 - 5112) q^{33} + (728 \beta_{3} - 231 \beta_{2} + 4858 \beta_1 - 1358) q^{35} + ( - 855 \beta_{3} + 9127 \beta_1 + 9127) q^{37} + (81 \beta_{3} - 81 \beta_{2} + 4203 \beta_1) q^{39} + (846 \beta_{2} - 6006) q^{41} + (2043 \beta_{2} + 2407) q^{43} + ( - 567 \beta_{3} + 567 \beta_{2} - 1620 \beta_1) q^{45} + ( - 604 \beta_{3} - 11882 \beta_1 - 11882) q^{47} + (2450 \beta_{3} - 1225 \beta_{2} + 1225 \beta_1 - 882) q^{49} + ( - 1332 \beta_{3} + 792 \beta_1 + 792) q^{51} + ( - 1751 \beta_{3} + 1751 \beta_{2} - 16702 \beta_1) q^{53} + ( - 3963 \beta_{2} - 10926) q^{55} + ( - 243 \beta_{2} - 10521) q^{57} + (3917 \beta_{3} - 3917 \beta_{2} - 18590 \beta_1) q^{59} + (2544 \beta_{3} - 19754 \beta_1 - 19754) q^{61} + ( - 567 \beta_{3} + 1134 \beta_{2} - 7371 \beta_1 - 6804) q^{63} + (3386 \beta_{3} + 13246 \beta_1 + 13246) q^{65} + (4461 \beta_{3} - 4461 \beta_{2} - 13151 \beta_1) q^{67} + ( - 2772 \beta_{2} - 8568) q^{69} + ( - 1404 \beta_{2} - 51750) q^{71} + (5247 \beta_{3} - 5247 \beta_{2} - 11665 \beta_1) q^{73} + ( - 2079 \beta_{3} - 2817 \beta_1 - 2817) q^{75} + ( - 4067 \beta_{3} - 3892 \beta_{2} + 48146 \beta_1 - 3108) q^{77} + (6834 \beta_{3} - 5815 \beta_1 - 5815) q^{79} + 6561 \beta_1 q^{81} + (1899 \beta_{2} - 29640) q^{83} + ( - 1308 \beta_{2} - 62472) q^{85} + (405 \beta_{3} - 405 \beta_{2} - 9774 \beta_1) q^{87} + (130 \beta_{3} - 14596 \beta_1 - 14596) q^{89} + (6475 \beta_{3} - 2450 \beta_{2} + 11081 \beta_1 + 46403) q^{91} + (6912 \beta_{3} + 22779 \beta_1 + 22779) q^{93} + (8534 \beta_{3} - 8534 \beta_{2} + 35098 \beta_1) q^{95} + ( - 1017 \beta_{2} - 5404) q^{97} + ( - 81 \beta_{2} + 46008) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 18 q^{3} + 33 q^{5} + 350 q^{7} - 162 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 18 q^{3} + 33 q^{5} + 350 q^{7} - 162 q^{9} - 1137 q^{11} + 1850 q^{13} - 594 q^{15} + 324 q^{17} + 2311 q^{19} - 1575 q^{21} + 1596 q^{23} - 395 q^{25} + 2916 q^{27} - 4434 q^{29} + 4294 q^{31} - 10233 q^{33} - 15414 q^{35} + 19109 q^{37} - 8325 q^{39} - 25716 q^{41} + 5542 q^{43} + 2673 q^{45} - 23160 q^{47} - 5978 q^{49} + 2916 q^{51} + 31653 q^{53} - 35778 q^{55} - 41598 q^{57} + 41097 q^{59} - 42052 q^{61} - 14175 q^{63} + 23106 q^{65} + 30763 q^{67} - 28728 q^{69} - 204192 q^{71} + 28577 q^{73} - 3555 q^{75} - 96873 q^{77} - 18464 q^{79} - 13122 q^{81} - 122358 q^{83} - 247272 q^{85} + 19953 q^{87} - 29322 q^{89} + 161875 q^{91} + 38646 q^{93} - 61662 q^{95} - 19582 q^{97} + 184194 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 20x^{2} - 21x + 441 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{3} + 20\nu^{2} - 20\nu - 441 ) / 420 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{3} + \nu^{2} + 41\nu ) / 21 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} + 20\nu - 41 ) / 20 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} - \beta _1 + 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{3} + 2\beta_{2} + 61\beta _1 + 62 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 40\beta_{3} - 20\beta_{2} + 20\beta _1 + 103 ) / 3 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/336\mathbb{Z}\right)^\times\).

\(n\) \(85\) \(113\) \(127\) \(241\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1 - \beta_{1}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
193.1
−3.69493 2.71062i
4.19493 + 1.84460i
−3.69493 + 2.71062i
4.19493 1.84460i
0 −4.50000 + 7.79423i 0 −19.3645 33.5404i 0 87.5000 95.6596i 0 −40.5000 70.1481i 0
193.2 0 −4.50000 + 7.79423i 0 35.8645 + 62.1192i 0 87.5000 + 95.6596i 0 −40.5000 70.1481i 0
289.1 0 −4.50000 7.79423i 0 −19.3645 + 33.5404i 0 87.5000 + 95.6596i 0 −40.5000 + 70.1481i 0
289.2 0 −4.50000 7.79423i 0 35.8645 62.1192i 0 87.5000 95.6596i 0 −40.5000 + 70.1481i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 336.6.q.e 4
4.b odd 2 1 21.6.e.b 4
7.c even 3 1 inner 336.6.q.e 4
12.b even 2 1 63.6.e.c 4
28.d even 2 1 147.6.e.l 4
28.f even 6 1 147.6.a.k 2
28.f even 6 1 147.6.e.l 4
28.g odd 6 1 21.6.e.b 4
28.g odd 6 1 147.6.a.i 2
84.j odd 6 1 441.6.a.s 2
84.n even 6 1 63.6.e.c 4
84.n even 6 1 441.6.a.t 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.6.e.b 4 4.b odd 2 1
21.6.e.b 4 28.g odd 6 1
63.6.e.c 4 12.b even 2 1
63.6.e.c 4 84.n even 6 1
147.6.a.i 2 28.g odd 6 1
147.6.a.k 2 28.f even 6 1
147.6.e.l 4 28.d even 2 1
147.6.e.l 4 28.f even 6 1
336.6.q.e 4 1.a even 1 1 trivial
336.6.q.e 4 7.c even 3 1 inner
441.6.a.s 2 84.j odd 6 1
441.6.a.t 2 84.n even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} - 33T_{5}^{3} + 3867T_{5}^{2} + 91674T_{5} + 7717284 \) acting on \(S_{6}^{\mathrm{new}}(336, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} + 9 T + 81)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} - 33 T^{3} + 3867 T^{2} + \cdots + 7717284 \) Copy content Toggle raw display
$7$ \( (T^{2} - 175 T + 16807)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} + 1137 T^{3} + \cdots + 104412996900 \) Copy content Toggle raw display
$13$ \( (T^{2} - 925 T + 208864)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} - 324 T^{3} + \cdots + 1788317798400 \) Copy content Toggle raw display
$19$ \( T^{4} - 2311 T^{3} + \cdots + 1663584040000 \) Copy content Toggle raw display
$23$ \( T^{4} - 1596 T^{3} + \cdots + 27756881510400 \) Copy content Toggle raw display
$29$ \( (T^{2} + 2217 T + 1102716)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} - 4294 T^{3} + \cdots + 10\!\cdots\!25 \) Copy content Toggle raw display
$37$ \( T^{4} - 19109 T^{3} + \cdots + 20\!\cdots\!96 \) Copy content Toggle raw display
$41$ \( (T^{2} + 12858 T - 3221280)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} - 2771 T - 257902490)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 23160 T^{3} + \cdots + 12\!\cdots\!16 \) Copy content Toggle raw display
$53$ \( T^{4} - 31653 T^{3} + \cdots + 35\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{4} - 41097 T^{3} + \cdots + 28\!\cdots\!44 \) Copy content Toggle raw display
$61$ \( T^{4} + 42052 T^{3} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{4} - 30763 T^{3} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( (T^{2} + 102096 T + 2483190108)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} - 28577 T^{3} + \cdots + 22\!\cdots\!84 \) Copy content Toggle raw display
$79$ \( T^{4} + 18464 T^{3} + \cdots + 79\!\cdots\!69 \) Copy content Toggle raw display
$83$ \( (T^{2} + 61179 T + 711231498)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} + 29322 T^{3} + \cdots + 45\!\cdots\!16 \) Copy content Toggle raw display
$97$ \( (T^{2} + 9791 T - 40418570)^{2} \) Copy content Toggle raw display
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