Properties

Label 336.4.q.k
Level $336$
Weight $4$
Character orbit 336.q
Analytic conductor $19.825$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(19.8246417619\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.9924270768.1
Defining polynomial: \( x^{6} - x^{5} + 25x^{4} + 12x^{3} + 582x^{2} - 144x + 36 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{4}\cdot 3 \)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 3 \beta_{3} q^{3} + (\beta_{4} - 4 \beta_{3} + \beta_{2} - 4) q^{5} + (\beta_{5} + \beta_{4} - 2 \beta_{3} + \beta_{2} + \beta_1 + 1) q^{7} + ( - 9 \beta_{3} - 9) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 3 \beta_{3} q^{3} + (\beta_{4} - 4 \beta_{3} + \beta_{2} - 4) q^{5} + (\beta_{5} + \beta_{4} - 2 \beta_{3} + \beta_{2} + \beta_1 + 1) q^{7} + ( - 9 \beta_{3} - 9) q^{9} + (\beta_{5} - 2 \beta_{4} - 11 \beta_{3} + \beta_1) q^{11} + (2 \beta_{2} - \beta_1 + 20) q^{13} + ( - 3 \beta_{2} + 12) q^{15} + (\beta_{5} - 3 \beta_{4} + 17 \beta_{3} + \beta_1) q^{17} + ( - 2 \beta_{5} - \beta_{4} - 67 \beta_{3} - \beta_{2} - 67) q^{19} + ( - 3 \beta_{5} + 9 \beta_{3} - 3 \beta_{2} + 6) q^{21} + ( - 7 \beta_{5} - 3 \beta_{4} + 73 \beta_{3} - 3 \beta_{2} + 73) q^{23} + ( - 7 \beta_{5} - 8 \beta_{4} + 46 \beta_{3} - 7 \beta_1) q^{25} + 27 q^{27} + ( - 10 \beta_{2} + 5 \beta_1 + 21) q^{29} + ( - 5 \beta_{5} - 7 \beta_{4} + 34 \beta_{3} - 5 \beta_1) q^{31} + ( - 3 \beta_{5} + 6 \beta_{4} + 33 \beta_{3} + 6 \beta_{2} + 33) q^{33} + ( - 7 \beta_{5} - 3 \beta_{4} + 151 \beta_{3} + 14 \beta_{2} - 10 \beta_1 + 61) q^{35} + ( - 5 \beta_{5} - 4 \beta_{4} - 86 \beta_{3} - 4 \beta_{2} - 86) q^{37} + (3 \beta_{5} + 6 \beta_{4} + 60 \beta_{3} + 3 \beta_1) q^{39} + (10 \beta_{2} + 4 \beta_1 + 78) q^{41} + (15 \beta_{2} - 18 \beta_1 - 125) q^{43} + ( - 9 \beta_{4} + 36 \beta_{3}) q^{45} + ( - 25 \beta_{5} + 3 \beta_{4} - 71 \beta_{3} + 3 \beta_{2} - 71) q^{47} + ( - 12 \beta_{5} - 10 \beta_{4} - 107 \beta_{3} + 16 \beta_{2} + \cdots - 111) q^{49}+ \cdots + ( - 18 \beta_{2} - 9 \beta_1 - 99) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 9 q^{3} - 11 q^{5} + 13 q^{7} - 27 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 9 q^{3} - 11 q^{5} + 13 q^{7} - 27 q^{9} + 35 q^{11} + 124 q^{13} + 66 q^{15} - 48 q^{17} - 202 q^{19} + 3 q^{21} + 216 q^{23} - 130 q^{25} + 162 q^{27} + 106 q^{29} - 95 q^{31} + 105 q^{33} - 56 q^{35} - 262 q^{37} - 186 q^{39} + 488 q^{41} - 720 q^{43} - 99 q^{45} - 210 q^{47} - 303 q^{49} - 144 q^{51} - 393 q^{53} + 2062 q^{55} + 1212 q^{57} + 1143 q^{59} + 70 q^{61} - 126 q^{63} + 472 q^{65} - 628 q^{67} - 1296 q^{69} - 636 q^{71} - 988 q^{73} - 390 q^{75} + 1073 q^{77} + 861 q^{79} - 243 q^{81} - 1038 q^{83} + 3600 q^{85} - 159 q^{87} - 1766 q^{89} + 654 q^{91} - 285 q^{93} - 736 q^{95} + 38 q^{97} - 630 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - x^{5} + 25x^{4} + 12x^{3} + 582x^{2} - 144x + 36 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -11\nu^{5} + 275\nu^{4} + 328\nu^{3} + 6402\nu^{2} - 1584\nu + 118833 ) / 7203 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -13\nu^{5} + 325\nu^{4} - 922\nu^{3} + 7566\nu^{2} - 1872\nu + 118830 ) / 7203 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 100\nu^{5} - 99\nu^{4} + 2475\nu^{3} + 1825\nu^{2} + 57618\nu - 14256 ) / 14406 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 801\nu^{5} - 817\nu^{4} + 20425\nu^{3} + 6815\nu^{2} + 475494\nu - 117648 ) / 7203 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -1676\nu^{5} + 1083\nu^{4} - 41481\nu^{3} - 30587\nu^{2} - 947238\nu - 2514 ) / 14406 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} + 1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{5} - \beta_{4} + 33\beta_{3} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -11\beta_{2} + 13\beta _1 - 33 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -17\beta_{5} + 8\beta_{4} - 411\beta_{3} + 8\beta_{2} - 411 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -170\beta_{5} - 127\beta_{4} - 708\beta_{3} - 170\beta_1 \) Copy content Toggle raw display

Character values

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

\(n\) \(85\) \(113\) \(127\) \(241\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1 - \beta_{3}\)

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
−2.27818 3.94593i
2.65415 + 4.59712i
0.124036 + 0.214837i
−2.27818 + 3.94593i
2.65415 4.59712i
0.124036 0.214837i
0 −1.50000 + 2.59808i 0 −8.93660 15.4786i 0 −2.26047 18.3818i 0 −4.50000 7.79423i 0
193.2 0 −1.50000 + 2.59808i 0 −2.78070 4.81631i 0 −9.67799 + 15.7904i 0 −4.50000 7.79423i 0
193.3 0 −1.50000 + 2.59808i 0 6.21730 + 10.7687i 0 18.4385 1.73873i 0 −4.50000 7.79423i 0
289.1 0 −1.50000 2.59808i 0 −8.93660 + 15.4786i 0 −2.26047 + 18.3818i 0 −4.50000 + 7.79423i 0
289.2 0 −1.50000 2.59808i 0 −2.78070 + 4.81631i 0 −9.67799 15.7904i 0 −4.50000 + 7.79423i 0
289.3 0 −1.50000 2.59808i 0 6.21730 10.7687i 0 18.4385 + 1.73873i 0 −4.50000 + 7.79423i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 289.3
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 336.4.q.k 6
4.b odd 2 1 21.4.e.b 6
7.c even 3 1 inner 336.4.q.k 6
7.c even 3 1 2352.4.a.ci 3
7.d odd 6 1 2352.4.a.cg 3
12.b even 2 1 63.4.e.c 6
28.d even 2 1 147.4.e.n 6
28.f even 6 1 147.4.a.m 3
28.f even 6 1 147.4.e.n 6
28.g odd 6 1 21.4.e.b 6
28.g odd 6 1 147.4.a.l 3
84.h odd 2 1 441.4.e.w 6
84.j odd 6 1 441.4.a.t 3
84.j odd 6 1 441.4.e.w 6
84.n even 6 1 63.4.e.c 6
84.n even 6 1 441.4.a.s 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.4.e.b 6 4.b odd 2 1
21.4.e.b 6 28.g odd 6 1
63.4.e.c 6 12.b even 2 1
63.4.e.c 6 84.n even 6 1
147.4.a.l 3 28.g odd 6 1
147.4.a.m 3 28.f even 6 1
147.4.e.n 6 28.d even 2 1
147.4.e.n 6 28.f even 6 1
336.4.q.k 6 1.a even 1 1 trivial
336.4.q.k 6 7.c even 3 1 inner
441.4.a.s 3 84.n even 6 1
441.4.a.t 3 84.j odd 6 1
441.4.e.w 6 84.h odd 2 1
441.4.e.w 6 84.j odd 6 1
2352.4.a.cg 3 7.d odd 6 1
2352.4.a.ci 3 7.c even 3 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{6} + 11T_{5}^{5} + 313T_{5}^{4} + 360T_{5}^{3} + 50460T_{5}^{2} + 237312T_{5} + 1527696 \) acting on \(S_{4}^{\mathrm{new}}(336, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( (T^{2} + 3 T + 9)^{3} \) Copy content Toggle raw display
$5$ \( T^{6} + 11 T^{5} + 313 T^{4} + \cdots + 1527696 \) Copy content Toggle raw display
$7$ \( T^{6} - 13 T^{5} + 236 T^{4} + \cdots + 40353607 \) Copy content Toggle raw display
$11$ \( T^{6} - 35 T^{5} + 2593 T^{4} + \cdots + 91470096 \) Copy content Toggle raw display
$13$ \( (T^{3} - 62 T^{2} + 425 T + 18452)^{2} \) Copy content Toggle raw display
$17$ \( T^{6} + 48 T^{5} + \cdots + 12745506816 \) Copy content Toggle raw display
$19$ \( T^{6} + 202 T^{5} + \cdots + 54664310416 \) Copy content Toggle raw display
$23$ \( T^{6} - 216 T^{5} + \cdots + 2498119335936 \) Copy content Toggle raw display
$29$ \( (T^{3} - 53 T^{2} - 20472 T - 824976)^{2} \) Copy content Toggle raw display
$31$ \( T^{6} + 95 T^{5} + \cdots + 139783329 \) Copy content Toggle raw display
$37$ \( T^{6} + 262 T^{5} + \cdots + 2415919104 \) Copy content Toggle raw display
$41$ \( (T^{3} - 244 T^{2} - 18780 T - 300384)^{2} \) Copy content Toggle raw display
$43$ \( (T^{3} + 360 T^{2} - 72363 T - 18269746)^{2} \) Copy content Toggle raw display
$47$ \( T^{6} + 210 T^{5} + \cdots + 26205471480384 \) Copy content Toggle raw display
$53$ \( T^{6} + 393 T^{5} + \cdots + 11\!\cdots\!64 \) Copy content Toggle raw display
$59$ \( T^{6} - 1143 T^{5} + \cdots + 10\!\cdots\!36 \) Copy content Toggle raw display
$61$ \( T^{6} - 70 T^{5} + \cdots + 71\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{6} + \cdots + 783608160972004 \) Copy content Toggle raw display
$71$ \( (T^{3} + 318 T^{2} - 330804 T + 28535976)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} + 988 T^{5} + \cdots + 20\!\cdots\!24 \) Copy content Toggle raw display
$79$ \( T^{6} - 861 T^{5} + \cdots + 37\!\cdots\!69 \) Copy content Toggle raw display
$83$ \( (T^{3} + 519 T^{2} - 131616 T - 47916036)^{2} \) Copy content Toggle raw display
$89$ \( T^{6} + \cdots + 169118164647936 \) Copy content Toggle raw display
$97$ \( (T^{3} - 19 T^{2} - 569600 T + 44776452)^{2} \) Copy content Toggle raw display
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