Properties

Label 336.6.q.k
Level $336$
Weight $6$
Character orbit 336.q
Analytic conductor $53.889$
Analytic rank $0$
Dimension $8$
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: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \( x^{8} - x^{7} + 118x^{6} + 555x^{5} + 12174x^{4} + 28215x^{3} + 199593x^{2} - 283824x + 1679616 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6}\cdot 3\cdot 7^{2} \)
Twist minimal: no (minimal twist has level 168)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

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

\(f(q)\) \(=\) \( q + ( - 9 \beta_1 + 9) q^{3} + (\beta_{2} + 16 \beta_1) q^{5} + ( - \beta_{6} + 2 \beta_{5} - 3 \beta_{4} + 3 \beta_{3} + 31 \beta_1 - 22) q^{7} - 81 \beta_1 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - 9 \beta_1 + 9) q^{3} + (\beta_{2} + 16 \beta_1) q^{5} + ( - \beta_{6} + 2 \beta_{5} - 3 \beta_{4} + 3 \beta_{3} + 31 \beta_1 - 22) q^{7} - 81 \beta_1 q^{9} + ( - \beta_{7} - 6 \beta_{4} + 5 \beta_{3} + \beta_{2} + 18 \beta_1 - 18) q^{11} + (9 \beta_{7} + 2 \beta_{6} + 11 \beta_{5} + 2 \beta_{4} + 165) q^{13} + (9 \beta_{7} + 144) q^{15} + (9 \beta_{7} + 7 \beta_{4} - 11 \beta_{3} - 9 \beta_{2} - 234 \beta_1 + 234) q^{17} + (11 \beta_{6} - 18 \beta_{5} - 18 \beta_{3} - 4 \beta_{2} - 137 \beta_1) q^{19} + (18 \beta_{6} + 27 \beta_{5} - 9 \beta_{4} + 9 \beta_{3} + 198 \beta_1 + 81) q^{21} + (21 \beta_{6} + 5 \beta_{5} + 5 \beta_{3} + 27 \beta_{2} - 154 \beta_1) q^{23} + ( - 41 \beta_{7} - 24 \beta_{4} + 77 \beta_{3} + 41 \beta_{2} + 619 \beta_1 - 619) q^{25} - 729 q^{27} + ( - 32 \beta_{7} + 55 \beta_{6} - 45 \beta_{5} + 55 \beta_{4} + \cdots - 1040) q^{29}+ \cdots + (81 \beta_{7} + 486 \beta_{6} + 405 \beta_{5} + 486 \beta_{4} + \cdots + 1458) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 36 q^{3} + 64 q^{5} - 42 q^{7} - 324 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 36 q^{3} + 64 q^{5} - 42 q^{7} - 324 q^{9} - 70 q^{11} + 1356 q^{13} + 1152 q^{15} + 944 q^{17} - 606 q^{19} + 1512 q^{21} - 648 q^{23} - 2582 q^{25} - 5832 q^{27} - 8720 q^{29} - 4354 q^{31} + 630 q^{33} - 5824 q^{35} + 19302 q^{37} + 6102 q^{39} + 1832 q^{41} - 16788 q^{43} + 5184 q^{45} - 5104 q^{47} - 15484 q^{49} - 8496 q^{51} - 13244 q^{53} - 35744 q^{55} - 10908 q^{57} - 30742 q^{59} - 2428 q^{61} + 17010 q^{63} + 126004 q^{65} + 8258 q^{67} - 11664 q^{69} - 29664 q^{71} - 12758 q^{73} + 23238 q^{75} - 91672 q^{77} - 21382 q^{79} - 26244 q^{81} + 110500 q^{83} + 275960 q^{85} - 39240 q^{87} - 49072 q^{89} + 35658 q^{91} + 39186 q^{93} + 25292 q^{95} - 135228 q^{97} + 11340 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - x^{7} + 118x^{6} + 555x^{5} + 12174x^{4} + 28215x^{3} + 199593x^{2} - 283824x + 1679616 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 39176801 \nu^{7} - 175281391 \nu^{6} - 3994853318 \nu^{5} - 45075076059 \nu^{4} - 550970443038 \nu^{3} + \cdots + 8560863130032 ) / 40543625593008 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 17578891 \nu^{7} + 3969000703 \nu^{6} - 21431359054 \nu^{5} + 420138144939 \nu^{4} + 1150045654170 \nu^{3} + \cdots + 52129165465296 ) / 4504847288112 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 100298645 \nu^{7} + 799989497 \nu^{6} + 18232629706 \nu^{5} - 95051876259 \nu^{4} + 2514645536946 \nu^{3} + \cdots + 145970281896192 ) / 20271812796504 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 222542333 \nu^{7} + 2750531273 \nu^{6} + 62687595754 \nu^{5} - 195005476659 \nu^{4} + 8645877496914 \nu^{3} + \cdots + 501876370614528 ) / 40543625593008 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 16098693 \nu^{7} - 56973323 \nu^{6} + 1751455241 \nu^{5} + 5557351381 \nu^{4} + 149522168073 \nu^{3} - 11589820599 \nu^{2} + \cdots - 6067526406732 ) / 187701970338 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 3492226855 \nu^{7} + 11312760979 \nu^{6} - 466835433310 \nu^{5} - 1087351897137 \nu^{4} - 39814681247862 \nu^{3} + \cdots + 10\!\cdots\!44 ) / 40543625593008 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 39077805 \nu^{7} + 222571003 \nu^{6} - 4251464785 \nu^{5} - 13489858685 \nu^{4} - 244207300653 \nu^{3} + \cdots - 15394821073668 ) / 375403940676 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -2\beta_{4} + 3\beta_{3} - 4\beta _1 + 4 ) / 14 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 17\beta_{6} + 8\beta_{5} + 8\beta_{3} - 7\beta_{2} - 818\beta_1 ) / 14 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -28\beta_{7} + 278\beta_{6} + 263\beta_{5} + 278\beta_{4} - 3524 ) / 14 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -847\beta_{7} + 2705\beta_{4} - 1856\beta_{3} + 847\beta_{2} + 81962\beta _1 - 81962 ) / 14 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -36362\beta_{6} - 30491\beta_{5} - 30491\beta_{3} + 5656\beta_{2} + 668228\beta_1 ) / 14 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 101815\beta_{7} - 391697\beta_{6} - 294872\beta_{5} - 391697\beta_{4} + 9969410 ) / 14 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 912772\beta_{7} - 4878158\beta_{4} + 3927935\beta_{3} - 912772\beta_{2} - 101534660\beta _1 + 101534660 ) / 14 \) Copy content Toggle raw display

Character values

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

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

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
−2.82121 + 4.88647i
1.27660 2.21113i
5.87364 10.1734i
−3.82903 + 6.63207i
−2.82121 4.88647i
1.27660 + 2.21113i
5.87364 + 10.1734i
−3.82903 6.63207i
0 4.50000 7.79423i 0 −30.1755 52.2655i 0 84.0301 + 98.7215i 0 −40.5000 70.1481i 0
193.2 0 4.50000 7.79423i 0 −7.86730 13.6266i 0 −129.640 0.644974i 0 −40.5000 70.1481i 0
193.3 0 4.50000 7.79423i 0 21.8866 + 37.9086i 0 65.0901 112.117i 0 −40.5000 70.1481i 0
193.4 0 4.50000 7.79423i 0 48.1562 + 83.4090i 0 −40.4800 + 123.160i 0 −40.5000 70.1481i 0
289.1 0 4.50000 + 7.79423i 0 −30.1755 + 52.2655i 0 84.0301 98.7215i 0 −40.5000 + 70.1481i 0
289.2 0 4.50000 + 7.79423i 0 −7.86730 + 13.6266i 0 −129.640 + 0.644974i 0 −40.5000 + 70.1481i 0
289.3 0 4.50000 + 7.79423i 0 21.8866 37.9086i 0 65.0901 + 112.117i 0 −40.5000 + 70.1481i 0
289.4 0 4.50000 + 7.79423i 0 48.1562 83.4090i 0 −40.4800 123.160i 0 −40.5000 + 70.1481i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 289.4
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.k 8
4.b odd 2 1 168.6.q.a 8
7.c even 3 1 inner 336.6.q.k 8
12.b even 2 1 504.6.s.a 8
28.g odd 6 1 168.6.q.a 8
84.n even 6 1 504.6.s.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.6.q.a 8 4.b odd 2 1
168.6.q.a 8 28.g odd 6 1
336.6.q.k 8 1.a even 1 1 trivial
336.6.q.k 8 7.c even 3 1 inner
504.6.s.a 8 12.b even 2 1
504.6.s.a 8 84.n even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} - 64 T_{5}^{7} + 9589 T_{5}^{6} - 23936 T_{5}^{5} + 38185253 T_{5}^{4} - 518841056 T_{5}^{3} + 57238551652 T_{5}^{2} + 751616582528 T_{5} + 16027307641744 \) acting on \(S_{6}^{\mathrm{new}}(336, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{2} - 9 T + 81)^{4} \) Copy content Toggle raw display
$5$ \( T^{8} - 64 T^{7} + \cdots + 16027307641744 \) Copy content Toggle raw display
$7$ \( T^{8} + 42 T^{7} + \cdots + 79\!\cdots\!01 \) Copy content Toggle raw display
$11$ \( T^{8} + 70 T^{7} + \cdots + 36\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( (T^{4} - 678 T^{3} - 591559 T^{2} + \cdots + 1300587648)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} - 944 T^{7} + \cdots + 97\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( T^{8} + 606 T^{7} + \cdots + 50\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{8} + 648 T^{7} + \cdots + 24\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( (T^{4} + 4360 T^{3} + \cdots - 130624378916400)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + 4354 T^{7} + \cdots + 63\!\cdots\!61 \) Copy content Toggle raw display
$37$ \( T^{8} - 19302 T^{7} + \cdots + 31\!\cdots\!56 \) Copy content Toggle raw display
$41$ \( (T^{4} - 916 T^{3} + \cdots + 79\!\cdots\!28)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 8394 T^{3} + \cdots - 27\!\cdots\!56)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + 5104 T^{7} + \cdots + 19\!\cdots\!04 \) Copy content Toggle raw display
$53$ \( T^{8} + 13244 T^{7} + \cdots + 25\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{8} + 30742 T^{7} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{8} + 2428 T^{7} + \cdots + 23\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{8} - 8258 T^{7} + \cdots + 96\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( (T^{4} + 14832 T^{3} + \cdots - 23\!\cdots\!40)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + 12758 T^{7} + \cdots + 78\!\cdots\!24 \) Copy content Toggle raw display
$79$ \( T^{8} + 21382 T^{7} + \cdots + 58\!\cdots\!41 \) Copy content Toggle raw display
$83$ \( (T^{4} - 55250 T^{3} + \cdots + 48\!\cdots\!16)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} + 49072 T^{7} + \cdots + 49\!\cdots\!36 \) Copy content Toggle raw display
$97$ \( (T^{4} + 67614 T^{3} + \cdots + 28\!\cdots\!08)^{2} \) Copy content Toggle raw display
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