Properties

Label 336.6.q.h
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: \(\Q(\sqrt{-3}, \sqrt{505})\)
Defining polynomial: \( x^{4} - x^{3} + 127x^{2} + 126x + 15876 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 7 \)
Twist minimal: no (minimal twist has level 42)
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.

\(f(q)\) \(=\) \( q + 9 \beta_1 q^{3} + ( - 3 \beta_{3} + 2 \beta_{2} + 7 \beta_1 - 8) q^{5} + ( - 2 \beta_{3} + 3 \beta_{2} - 18 \beta_1 + 110) q^{7} + (81 \beta_1 - 81) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 9 \beta_1 q^{3} + ( - 3 \beta_{3} + 2 \beta_{2} + 7 \beta_1 - 8) q^{5} + ( - 2 \beta_{3} + 3 \beta_{2} - 18 \beta_1 + 110) q^{7} + (81 \beta_1 - 81) q^{9} + (7 \beta_{3} - 21 \beta_{2} + 76 \beta_1 + 7) q^{11} + ( - 2 \beta_{3} - \beta_{2} - \beta_1 - 356) q^{13} + ( - 18 \beta_{3} - 9 \beta_{2} - 9 \beta_1 - 63) q^{15} + (20 \beta_{3} - 60 \beta_{2} - 676 \beta_1 + 20) q^{17} + (81 \beta_{3} - 54 \beta_{2} - 500 \beta_1 + 527) q^{19} + ( - 27 \beta_{3} + 9 \beta_{2} + 828 \beta_1 + 162) q^{21} + ( - 12 \beta_{3} + 8 \beta_{2} + 2248 \beta_1 - 2252) q^{23} + (17 \beta_{3} - 51 \beta_{2} - 3125 \beta_1 + 17) q^{25} - 729 q^{27} + ( - 118 \beta_{3} - 59 \beta_{2} - 59 \beta_1 + 4021) q^{29} + ( - 14 \beta_{3} + 42 \beta_{2} + 4401 \beta_1 - 14) q^{31} + (189 \beta_{3} - 126 \beta_{2} + 747 \beta_1 - 684) q^{33} + ( - 306 \beta_{3} + 221 \beta_{2} - 1865 \beta_1 - 5171) q^{35} + ( - 243 \beta_{3} + 162 \beta_{2} - 7408 \beta_1 + 7327) q^{37} + (9 \beta_{3} - 27 \beta_{2} - 3213 \beta_1 + 9) q^{39} + (132 \beta_{3} + 66 \beta_{2} + 66 \beta_1 + 3576) q^{41} + (126 \beta_{3} + 63 \beta_{2} + 63 \beta_1 + 2866) q^{43} + (81 \beta_{3} - 243 \beta_{2} - 648 \beta_1 + 81) q^{45} + ( - 108 \beta_{3} + 72 \beta_{2} - 22458 \beta_1 + 22422) q^{47} + ( - 340 \beta_{3} + 629 \beta_{2} - 1618 \beta_1 + 4833) q^{49} + (540 \beta_{3} - 360 \beta_{2} - 5904 \beta_1 + 6084) q^{51} + (45 \beta_{3} - 135 \beta_{2} - 4686 \beta_1 + 45) q^{53} + ( - 26 \beta_{3} - 13 \beta_{2} - 13 \beta_1 + 42707) q^{55} + (486 \beta_{3} + 243 \beta_{2} + 243 \beta_1 + 4500) q^{57} + (377 \beta_{3} - 1131 \beta_{2} - 2350 \beta_1 + 377) q^{59} + ( - 276 \beta_{3} + 184 \beta_{2} + 21046 \beta_1 - 21138) q^{61} + ( - 81 \beta_{3} - 162 \beta_{2} + 8910 \beta_1 - 7452) q^{63} + (1098 \beta_{3} - 732 \beta_{2} - 8676 \beta_1 + 9042) q^{65} + (317 \beta_{3} - 951 \beta_{2} + 15409 \beta_1 + 317) q^{67} + ( - 72 \beta_{3} - 36 \beta_{2} - 36 \beta_1 - 20232) q^{69} + (440 \beta_{3} + 220 \beta_{2} + 220 \beta_1 - 46202) q^{71} + (505 \beta_{3} - 1515 \beta_{2} + 43085 \beta_1 + 505) q^{73} + (459 \beta_{3} - 306 \beta_{2} - 27972 \beta_1 + 28125) q^{75} + (199 \beta_{3} - 2010 \beta_{2} - 24053 \beta_1 + 51628) q^{77} + ( - 2064 \beta_{3} + 1376 \beta_{2} - 48355 \beta_1 + 47667) q^{79} - 6561 \beta_1 q^{81} + (62 \beta_{3} + 31 \beta_{2} + 31 \beta_1 - 16967) q^{83} + (1712 \beta_{3} + 856 \beta_{2} + 856 \beta_1 + 128272) q^{85} + (531 \beta_{3} - 1593 \beta_{2} + 35658 \beta_1 + 531) q^{87} + ( - 522 \beta_{3} + 348 \beta_{2} + 13518 \beta_1 - 13692) q^{89} + (477 \beta_{3} - 1132 \beta_{2} - 754 \beta_1 - 36525) q^{91} + ( - 378 \beta_{3} + 252 \beta_{2} + 39483 \beta_1 - 39609) q^{93} + ( - 770 \beta_{3} + 2310 \beta_{2} + 171238 \beta_1 - 770) q^{95} + ( - 1726 \beta_{3} - 863 \beta_{2} - 863 \beta_1 - 22041) q^{97} + (1134 \beta_{3} + 567 \beta_{2} + 567 \beta_1 - 6723) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 18 q^{3} - 17 q^{5} + 408 q^{7} - 162 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 18 q^{3} - 17 q^{5} + 408 q^{7} - 162 q^{9} + 145 q^{11} - 1430 q^{13} - 306 q^{15} - 1372 q^{17} + 1081 q^{19} + 2295 q^{21} - 4508 q^{23} - 6267 q^{25} - 2916 q^{27} + 15730 q^{29} + 8816 q^{31} - 1305 q^{33} - 24278 q^{35} + 14573 q^{37} - 6435 q^{39} + 14700 q^{41} + 11842 q^{43} - 1377 q^{45} + 44808 q^{47} + 17014 q^{49} + 12348 q^{51} - 9417 q^{53} + 170750 q^{55} + 19458 q^{57} - 5077 q^{59} - 42368 q^{61} - 12393 q^{63} + 18450 q^{65} + 30501 q^{67} - 81144 q^{69} - 183488 q^{71} + 85665 q^{73} + 56403 q^{75} + 154585 q^{77} + 94646 q^{79} - 13122 q^{81} - 67682 q^{83} + 518224 q^{85} + 70785 q^{87} - 27558 q^{89} - 149395 q^{91} - 79344 q^{93} + 343246 q^{95} - 93342 q^{97} - 23490 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} + 127x^{2} + 126x + 15876 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{3} + 127\nu^{2} - 127\nu + 15876 ) / 16002 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 125\nu^{3} + 127\nu^{2} - 32131\nu + 47754 ) / 16002 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 379\nu^{3} - 127\nu^{2} + 16129\nu + 63756 ) / 16002 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} - 3\beta_{2} + 4\beta _1 + 1 ) / 7 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 3\beta_{3} - 2\beta_{2} + 887\beta _1 - 886 ) / 7 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 254\beta_{3} + 127\beta_{2} + 127\beta _1 - 1517 ) / 7 \) 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
−5.36805 + 9.29774i
5.86805 10.1638i
−5.36805 9.29774i
5.86805 + 10.1638i
0 4.50000 7.79423i 0 −43.5764 75.4765i 0 113.236 63.1236i 0 −40.5000 70.1481i 0
193.2 0 4.50000 7.79423i 0 35.0764 + 60.7540i 0 90.7639 + 92.5684i 0 −40.5000 70.1481i 0
289.1 0 4.50000 + 7.79423i 0 −43.5764 + 75.4765i 0 113.236 + 63.1236i 0 −40.5000 + 70.1481i 0
289.2 0 4.50000 + 7.79423i 0 35.0764 60.7540i 0 90.7639 92.5684i 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.h 4
4.b odd 2 1 42.6.e.d 4
7.c even 3 1 inner 336.6.q.h 4
12.b even 2 1 126.6.g.g 4
28.d even 2 1 294.6.e.y 4
28.f even 6 1 294.6.a.o 2
28.f even 6 1 294.6.e.y 4
28.g odd 6 1 42.6.e.d 4
28.g odd 6 1 294.6.a.p 2
84.j odd 6 1 882.6.a.bs 2
84.n even 6 1 126.6.g.g 4
84.n even 6 1 882.6.a.bm 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.6.e.d 4 4.b odd 2 1
42.6.e.d 4 28.g odd 6 1
126.6.g.g 4 12.b even 2 1
126.6.g.g 4 84.n even 6 1
294.6.a.o 2 28.f even 6 1
294.6.a.p 2 28.g odd 6 1
294.6.e.y 4 28.d even 2 1
294.6.e.y 4 28.f even 6 1
336.6.q.h 4 1.a even 1 1 trivial
336.6.q.h 4 7.c even 3 1 inner
882.6.a.bm 2 84.n even 6 1
882.6.a.bs 2 84.j odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} + 17T_{5}^{3} + 6403T_{5}^{2} - 103938T_{5} + 37380996 \) 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
$5$ \( T^{4} + 17 T^{3} + 6403 T^{2} + \cdots + 37380996 \) Copy content Toggle raw display
$7$ \( T^{4} - 408 T^{3} + \cdots + 282475249 \) Copy content Toggle raw display
$11$ \( T^{4} - 145 T^{3} + \cdots + 88726536900 \) Copy content Toggle raw display
$13$ \( (T^{2} + 715 T + 121620)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + 1372 T^{3} + \cdots + 4015631241216 \) Copy content Toggle raw display
$19$ \( T^{4} - 1081 T^{3} + \cdots + 17788453428496 \) Copy content Toggle raw display
$23$ \( T^{4} + 4508 T^{3} + \cdots + 24815700919296 \) Copy content Toggle raw display
$29$ \( (T^{2} - 7865 T - 6069780)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + \cdots + 331894030125681 \) Copy content Toggle raw display
$37$ \( T^{4} + \cdots + 156377425969216 \) Copy content Toggle raw display
$41$ \( (T^{2} - 7350 T - 13441680)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} - 5921 T - 15788666)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} - 44808 T^{3} + \cdots + 24\!\cdots\!96 \) Copy content Toggle raw display
$53$ \( T^{4} + 9417 T^{3} + \cdots + 92983900409856 \) Copy content Toggle raw display
$59$ \( T^{4} + 5077 T^{3} + \cdots + 76\!\cdots\!36 \) Copy content Toggle raw display
$61$ \( T^{4} + 42368 T^{3} + \cdots + 15\!\cdots\!96 \) Copy content Toggle raw display
$67$ \( T^{4} - 30501 T^{3} + \cdots + 15\!\cdots\!76 \) Copy content Toggle raw display
$71$ \( (T^{2} + 91744 T + 1804825884)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} - 85665 T^{3} + \cdots + 66\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{4} - 94646 T^{3} + \cdots + 47\!\cdots\!81 \) Copy content Toggle raw display
$83$ \( (T^{2} + 33841 T + 280358334)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} + 27558 T^{3} + \cdots + 6584027556096 \) Copy content Toggle raw display
$97$ \( (T^{2} + 46671 T - 4062781666)^{2} \) Copy content Toggle raw display
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