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Properties

Label 336.6.q.j
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: \( x^{8} - x^{7} + 98x^{6} + 83x^{5} + 9122x^{4} - 91x^{3} + 28567x^{2} + 2058x + 86436 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{8}\cdot 3\cdot 7^{2} \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - 9 \beta_1 + 9) q^{3} + ( - \beta_{6} + \beta_{4}) q^{5} + (\beta_{7} - \beta_{4} + \beta_{3} - \beta_{2} - 45 \beta_1 - 9) q^{7} - 81 \beta_1 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - 9 \beta_1 + 9) q^{3} + ( - \beta_{6} + \beta_{4}) q^{5} + (\beta_{7} - \beta_{4} + \beta_{3} - \beta_{2} - 45 \beta_1 - 9) q^{7} - 81 \beta_1 q^{9} + (6 \beta_{4} + 3 \beta_{3} - \beta_{2} - 102 \beta_1 + 102) q^{11} + (5 \beta_{7} + 4 \beta_{6} - 13 \beta_{5} + 13 \beta_{2} + 118) q^{13} - 9 \beta_{6} q^{15} + (2 \beta_{4} + 6 \beta_{3} + 11 \beta_{2} + 66 \beta_1 - 66) q^{17} + ( - 7 \beta_{7} - 8 \beta_{6} - 22 \beta_{5} + 8 \beta_{4} - 7 \beta_{3} + 131 \beta_1 - 7) q^{19} + (9 \beta_{7} + 9 \beta_{6} - 9 \beta_{5} - 9 \beta_{4} + 90 \beta_1 - 486) q^{21} + (6 \beta_{7} + 14 \beta_{6} + 31 \beta_{5} - 14 \beta_{4} + 6 \beta_{3} + 1722 \beta_1 + 6) q^{23} + (70 \beta_{4} + 25 \beta_{3} - 5 \beta_{2} + 691 \beta_1 - 691) q^{25} - 729 q^{27} + (54 \beta_{7} - 49 \beta_{6} + 17 \beta_{5} - 17 \beta_{2} + 162) q^{29} + (59 \beta_{4} + 35 \beta_{3} - 19 \beta_{2} + 1585 \beta_1 - 1585) q^{31} + (27 \beta_{7} - 54 \beta_{6} - 9 \beta_{5} + 54 \beta_{4} + 27 \beta_{3} + \cdots + 27) q^{33}+ \cdots + (243 \beta_{7} - 486 \beta_{6} - 81 \beta_{5} + 81 \beta_{2} - 8019) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 36 q^{3} - 258 q^{7} - 324 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 36 q^{3} - 258 q^{7} - 324 q^{9} + 402 q^{11} + 924 q^{13} - 276 q^{17} + 510 q^{19} - 3564 q^{21} + 6900 q^{23} - 2814 q^{25} - 5832 q^{27} + 1080 q^{29} - 6410 q^{31} - 3618 q^{33} + 33108 q^{35} - 15250 q^{37} + 4158 q^{39} + 8616 q^{41} - 58396 q^{43} - 15060 q^{47} - 64252 q^{49} + 2484 q^{51} - 13692 q^{53} - 146248 q^{55} + 9180 q^{57} + 34830 q^{59} + 5364 q^{61} - 11178 q^{63} - 66864 q^{65} - 5994 q^{67} + 124200 q^{69} - 178536 q^{71} - 59638 q^{73} + 25326 q^{75} - 75660 q^{77} - 44062 q^{79} - 26244 q^{81} + 416892 q^{83} + 72648 q^{85} + 4860 q^{87} + 77520 q^{89} - 104722 q^{91} + 57690 q^{93} - 221376 q^{95} - 377260 q^{97} - 65124 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - x^{7} + 98x^{6} + 83x^{5} + 9122x^{4} - 91x^{3} + 28567x^{2} + 2058x + 86436 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 8905 \nu^{7} - 366059 \nu^{6} + 1191820 \nu^{5} - 33153695 \nu^{4} + 47979268 \nu^{3} - 3262309295 \nu^{2} + 141627885 \nu - 307508418 ) / 9888988410 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 8905 \nu^{7} - 366059 \nu^{6} + 1191820 \nu^{5} - 33153695 \nu^{4} + 47979268 \nu^{3} - 3262309295 \nu^{2} + 39697581525 \nu - 10196496828 ) / 4944494205 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 65603 \nu^{7} - 25662916 \nu^{6} + 83553680 \nu^{5} - 2627557456 \nu^{4} + 3363632432 \nu^{3} - 228707310580 \nu^{2} + \cdots - 714835153872 ) / 9888988410 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 731039 \nu^{7} - 7994302 \nu^{6} + 26027960 \nu^{5} - 628584958 \nu^{4} + 1047811304 \nu^{3} - 71245033510 \nu^{2} - 451136527737 \nu - 222679608984 ) / 9888988410 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 1437521 \nu^{7} - 1642579 \nu^{6} + 136763060 \nu^{5} + 99854873 \nu^{4} + 13093975028 \nu^{3} - 2811264295 \nu^{2} + 41000921157 \nu + 2771341902 ) / 4944494205 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 484709 \nu^{7} - 433105 \nu^{6} + 45997385 \nu^{5} + 45127907 \nu^{4} + 4285048727 \nu^{3} + 153033125 \nu^{2} + 442204518 \nu - 5171325642 ) / 282542526 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 2434409 \nu^{7} - 2482177 \nu^{6} + 231017885 \nu^{5} + 226651007 \nu^{4} + 21466747139 \nu^{3} + 768595625 \nu^{2} + 2220933918 \nu + 102356221716 ) / 1412712630 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_{2} - 2\beta _1 + 2 ) / 8 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} - \beta_{6} + \beta_{4} + \beta_{3} - 98\beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -16\beta_{6} + 95\beta_{5} - 95\beta_{2} - 542 ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -101\beta_{4} - 97\beta_{3} - 22\beta_{2} + 9050\beta _1 - 9050 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -360\beta_{7} + 1928\beta_{6} - 9009\beta_{5} - 1928\beta_{4} - 360\beta_{3} + 85442\beta _1 - 360 ) / 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -9205\beta_{7} + 9957\beta_{6} - 4220\beta_{5} + 4220\beta_{2} + 860245 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 219312\beta_{4} + 69024\beta_{3} + 862207\beta_{2} - 11352926\beta _1 + 11352926 ) / 8 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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
−0.874091 + 1.51397i
−4.61193 + 7.98809i
5.09061 − 8.81720i
0.895402 − 1.55088i
−0.874091 − 1.51397i
−4.61193 − 7.98809i
5.09061 + 8.81720i
0.895402 + 1.55088i
0 4.50000 − 7.79423i 0 −29.1836 − 50.5475i 0 21.4366 + 127.857i 0 −40.5000 − 70.1481i 0
193.2 0 4.50000 − 7.79423i 0 −11.8764 − 20.5705i 0 −30.6840 − 125.958i 0 −40.5000 − 70.1481i 0
193.3 0 4.50000 − 7.79423i 0 −11.0358 − 19.1146i 0 −126.882 − 26.6059i 0 −40.5000 − 70.1481i 0
193.4 0 4.50000 − 7.79423i 0 52.0958 + 90.2327i 0 7.12980 − 129.446i 0 −40.5000 − 70.1481i 0
289.1 0 4.50000 + 7.79423i 0 −29.1836 + 50.5475i 0 21.4366 − 127.857i 0 −40.5000 + 70.1481i 0
289.2 0 4.50000 + 7.79423i 0 −11.8764 + 20.5705i 0 −30.6840 + 125.958i 0 −40.5000 + 70.1481i 0
289.3 0 4.50000 + 7.79423i 0 −11.0358 + 19.1146i 0 −126.882 + 26.6059i 0 −40.5000 + 70.1481i 0
289.4 0 4.50000 + 7.79423i 0 52.0958 − 90.2327i 0 7.12980 + 129.446i 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.j 8
4.b odd 2 1 21.6.e.c ✓ 8
7.c even 3 1 inner 336.6.q.j 8
12.b even 2 1 63.6.e.e 8
28.d even 2 1 147.6.e.o 8
28.f even 6 1 147.6.a.l 4
28.f even 6 1 147.6.e.o 8
28.g odd 6 1 21.6.e.c ✓ 8
28.g odd 6 1 147.6.a.m 4
84.j odd 6 1 441.6.a.v 4
84.n even 6 1 63.6.e.e 8
84.n even 6 1 441.6.a.w 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.6.e.c ✓ 8 4.b odd 2 1
21.6.e.c ✓ 8 28.g odd 6 1
63.6.e.e 8 12.b even 2 1
63.6.e.e 8 84.n even 6 1
147.6.a.l 4 28.f even 6 1
147.6.a.m 4 28.g odd 6 1
147.6.e.o 8 28.d even 2 1
147.6.e.o 8 28.f even 6 1
336.6.q.j 8 1.a even 1 1 trivial
336.6.q.j 8 7.c even 3 1 inner
441.6.a.v 4 84.j odd 6 1
441.6.a.w 4 84.n even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} + 7657 T_{5}^{6} + 605400 T_{5}^{5} + 61817893 T_{5}^{4} + 2317773900 T_{5}^{3} + 67214905692 T_{5}^{2} + 965081458800 T_{5} + 10164899803536 \) 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
$5$ \( T^{8} + 7657 T^{6} + \cdots + 10164899803536 \) Copy content Toggle raw display
$7$ \( T^{8} + 258 T^{7} + \cdots + 79\!\cdots\!01 \) Copy content Toggle raw display
$11$ \( T^{8} - 402 T^{7} + \cdots + 28\!\cdots\!76 \) Copy content Toggle raw display
$13$ \( (T^{4} - 462 T^{3} + \cdots + 149501563456)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} + 276 T^{7} + \cdots + 25\!\cdots\!24 \) Copy content Toggle raw display
$19$ \( T^{8} - 510 T^{7} + \cdots + 54\!\cdots\!76 \) Copy content Toggle raw display
$23$ \( T^{8} - 6900 T^{7} + \cdots + 90\!\cdots\!76 \) Copy content Toggle raw display
$29$ \( (T^{4} - 540 T^{3} + \cdots + 408027025117872)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + 6410 T^{7} + \cdots + 75\!\cdots\!09 \) Copy content Toggle raw display
$37$ \( T^{8} + 15250 T^{7} + \cdots + 25\!\cdots\!36 \) Copy content Toggle raw display
$41$ \( (T^{4} - 4308 T^{3} + \cdots - 18\!\cdots\!52)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 29198 T^{3} + \cdots - 991662745581932)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + 15060 T^{7} + \cdots + 73\!\cdots\!36 \) Copy content Toggle raw display
$53$ \( T^{8} + 13692 T^{7} + \cdots + 72\!\cdots\!84 \) Copy content Toggle raw display
$59$ \( T^{8} - 34830 T^{7} + \cdots + 66\!\cdots\!96 \) Copy content Toggle raw display
$61$ \( T^{8} - 5364 T^{7} + \cdots + 32\!\cdots\!76 \) Copy content Toggle raw display
$67$ \( T^{8} + 5994 T^{7} + \cdots + 30\!\cdots\!56 \) Copy content Toggle raw display
$71$ \( (T^{4} + 89268 T^{3} + \cdots + 21\!\cdots\!68)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + 59638 T^{7} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{8} + 44062 T^{7} + \cdots + 27\!\cdots\!81 \) Copy content Toggle raw display
$83$ \( (T^{4} - 208446 T^{3} + \cdots - 41\!\cdots\!32)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} - 77520 T^{7} + \cdots + 22\!\cdots\!24 \) Copy content Toggle raw display
$97$ \( (T^{4} + 188630 T^{3} + \cdots - 11\!\cdots\!44)^{2} \) Copy content Toggle raw display
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