Properties

Label 336.6.q.m
Level $336$
Weight $6$
Character orbit 336.q
Analytic conductor $53.889$
Analytic rank $0$
Dimension $10$
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: \(10\)
Relative dimension: \(5\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Defining polynomial: \( x^{10} - x^{9} + 1094 x^{8} - 12883 x^{7} + 1063781 x^{6} - 7555708 x^{5} + 199315216 x^{4} + 540268032 x^{3} + 20356706304 x^{2} - 26680098816 x + 37456183296 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{12}\cdot 7^{2} \)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (9 \beta_1 + 9) q^{3} + ( - \beta_{4} + \beta_{2} + 15 \beta_1) q^{5} + (\beta_{5} + 3 \beta_1 + 13) q^{7} + 81 \beta_1 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (9 \beta_1 + 9) q^{3} + ( - \beta_{4} + \beta_{2} + 15 \beta_1) q^{5} + (\beta_{5} + 3 \beta_1 + 13) q^{7} + 81 \beta_1 q^{9} + (\beta_{9} - \beta_{8} - \beta_{6} - 4 \beta_{4} - 17 \beta_1 - 17) q^{11} + (3 \beta_{8} - 3 \beta_{7} + 3 \beta_{6} - 3 \beta_{5} - \beta_{3} + 2 \beta_{2} + 56) q^{13} + (9 \beta_{2} - 135) q^{15} + ( - \beta_{9} - 5 \beta_{8} - 5 \beta_{7} + 5 \beta_{5} - 16 \beta_{4} - 224 \beta_1 - 224) q^{17} + ( - 4 \beta_{9} - \beta_{8} - 2 \beta_{7} + \beta_{6} - \beta_{5} - 4 \beta_{4} - 4 \beta_{3} + \cdots + 56 \beta_1) q^{19}+ \cdots + (81 \beta_{8} - 81 \beta_{7} + 81 \beta_{6} - 81 \beta_{5} + 81 \beta_{3} + \cdots + 1377) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 45 q^{3} - 75 q^{5} + 113 q^{7} - 405 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 45 q^{3} - 75 q^{5} + 113 q^{7} - 405 q^{9} - 91 q^{11} + 580 q^{13} - 1350 q^{15} - 1128 q^{17} - 282 q^{19} + 279 q^{21} + 1808 q^{23} - 202 q^{25} - 7290 q^{27} + 5026 q^{29} + 5069 q^{31} + 819 q^{33} + 884 q^{35} - 5010 q^{37} + 2610 q^{39} + 12872 q^{41} - 17328 q^{43} - 6075 q^{45} + 50 q^{47} + 29135 q^{49} + 10152 q^{51} + 1167 q^{53} - 97410 q^{55} - 5076 q^{57} + 42797 q^{59} - 26546 q^{61} - 6642 q^{63} - 2216 q^{65} + 13440 q^{67} + 32544 q^{69} + 39356 q^{71} - 27768 q^{73} + 1818 q^{75} + 125797 q^{77} + 123369 q^{79} - 32805 q^{81} - 334250 q^{83} - 324936 q^{85} + 22617 q^{87} - 59350 q^{89} - 113850 q^{91} - 45621 q^{93} - 41864 q^{95} + 525282 q^{97} + 14742 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - x^{9} + 1094 x^{8} - 12883 x^{7} + 1063781 x^{6} - 7555708 x^{5} + 199315216 x^{4} + 540268032 x^{3} + 20356706304 x^{2} - 26680098816 x + 37456183296 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 11\!\cdots\!03 \nu^{9} + \cdots - 28\!\cdots\!00 ) / 30\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 13\!\cdots\!15 \nu^{9} + \cdots - 30\!\cdots\!16 ) / 58\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 77\!\cdots\!61 \nu^{9} + \cdots - 36\!\cdots\!24 ) / 64\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 91\!\cdots\!39 \nu^{9} + \cdots - 17\!\cdots\!60 ) / 61\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 48\!\cdots\!29 \nu^{9} + \cdots + 23\!\cdots\!12 ) / 30\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 23\!\cdots\!23 \nu^{9} + \cdots - 16\!\cdots\!12 ) / 12\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 27\!\cdots\!49 \nu^{9} + \cdots - 86\!\cdots\!36 ) / 12\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 36\!\cdots\!33 \nu^{9} + \cdots + 16\!\cdots\!28 ) / 12\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 12\!\cdots\!17 \nu^{9} + \cdots + 23\!\cdots\!16 ) / 61\!\cdots\!60 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{8} - 2\beta_{7} + 3\beta_{6} + 2\beta_{5} + 7\beta_{4} + 4\beta _1 + 4 ) / 28 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 14 \beta_{9} + 36 \beta_{8} + 45 \beta_{7} - 36 \beta_{6} + 9 \beta_{5} - 63 \beta_{4} - 14 \beta_{3} + 63 \beta_{2} + 12220 \beta_1 ) / 28 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -983\beta_{8} + 983\beta_{7} - 267\beta_{6} + 267\beta_{5} + 168\beta_{3} - 3605\beta_{2} + 49948 ) / 14 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 12950 \beta_{9} - 40235 \beta_{8} - 45460 \beta_{7} + 5225 \beta_{6} + 45460 \beta_{5} + 127365 \beta_{4} - 10140868 \beta _1 - 10140868 ) / 28 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 452130 \beta_{9} + 1495572 \beta_{8} - 171581 \beta_{7} - 1495572 \beta_{6} - 1667153 \beta_{5} - 7228417 \beta_{4} - 452130 \beta_{3} + 7228417 \beta_{2} + 179571236 \beta_1 ) / 28 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 4610569 \beta_{8} + 4610569 \beta_{7} + 8601715 \beta_{6} - 8601715 \beta_{5} + 3530051 \beta_{3} - 43081332 \beta_{2} + 2479661140 ) / 7 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 521523366 \beta_{9} - 124123271 \beta_{8} - 1695108452 \beta_{7} + 1570985181 \beta_{6} + 1695108452 \beta_{5} + 7612226545 \beta_{4} - 241080829076 \beta _1 - 241080829076 ) / 28 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 16221080750 \beta_{9} + 59975977020 \beta_{8} + 30840278745 \beta_{7} - 59975977020 \beta_{6} - 29135698275 \beta_{5} - 212298110955 \beta_{4} + \cdots + 10437528500908 \beta_1 ) / 28 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 792993729839 \beta_{8} + 792993729839 \beta_{7} + 180564071757 \beta_{6} - 180564071757 \beta_{5} + 293820409680 \beta_{3} + \cdots + 147828855262588 ) / 14 \) 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
−16.8838 + 29.2436i
0.666317 1.15409i
8.89029 15.3984i
12.6192 21.8570i
−4.79197 + 8.29994i
−16.8838 29.2436i
0.666317 + 1.15409i
8.89029 + 15.3984i
12.6192 + 21.8570i
−4.79197 8.29994i
0 4.50000 7.79423i 0 −46.5372 80.6048i 0 −119.738 + 49.6971i 0 −40.5000 70.1481i 0
193.2 0 4.50000 7.79423i 0 −33.1247 57.3737i 0 115.972 57.9444i 0 −40.5000 70.1481i 0
193.3 0 4.50000 7.79423i 0 9.56607 + 16.5689i 0 126.222 29.5814i 0 −40.5000 70.1481i 0
193.4 0 4.50000 7.79423i 0 10.3382 + 17.9063i 0 −74.2549 106.270i 0 −40.5000 70.1481i 0
193.5 0 4.50000 7.79423i 0 22.2576 + 38.5514i 0 8.29943 + 129.376i 0 −40.5000 70.1481i 0
289.1 0 4.50000 + 7.79423i 0 −46.5372 + 80.6048i 0 −119.738 49.6971i 0 −40.5000 + 70.1481i 0
289.2 0 4.50000 + 7.79423i 0 −33.1247 + 57.3737i 0 115.972 + 57.9444i 0 −40.5000 + 70.1481i 0
289.3 0 4.50000 + 7.79423i 0 9.56607 16.5689i 0 126.222 + 29.5814i 0 −40.5000 + 70.1481i 0
289.4 0 4.50000 + 7.79423i 0 10.3382 17.9063i 0 −74.2549 + 106.270i 0 −40.5000 + 70.1481i 0
289.5 0 4.50000 + 7.79423i 0 22.2576 38.5514i 0 8.29943 129.376i 0 −40.5000 + 70.1481i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 289.5
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.m 10
4.b odd 2 1 168.6.q.b 10
7.c even 3 1 inner 336.6.q.m 10
12.b even 2 1 504.6.s.e 10
28.g odd 6 1 168.6.q.b 10
84.n even 6 1 504.6.s.e 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.6.q.b 10 4.b odd 2 1
168.6.q.b 10 28.g odd 6 1
336.6.q.m 10 1.a even 1 1 trivial
336.6.q.m 10 7.c even 3 1 inner
504.6.s.e 10 12.b even 2 1
504.6.s.e 10 84.n even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{10} + 75 T_{5}^{9} + 10726 T_{5}^{8} + 1823 T_{5}^{7} + 29874586 T_{5}^{6} - 712256737 T_{5}^{5} + 98953482841 T_{5}^{4} - 3137487320332 T_{5}^{3} + 90655492595236 T_{5}^{2} + \cdots + 11\!\cdots\!96 \) acting on \(S_{6}^{\mathrm{new}}(336, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \) Copy content Toggle raw display
$3$ \( (T^{2} - 9 T + 81)^{5} \) Copy content Toggle raw display
$5$ \( T^{10} + 75 T^{9} + \cdots + 11\!\cdots\!96 \) Copy content Toggle raw display
$7$ \( T^{10} - 113 T^{9} + \cdots + 13\!\cdots\!07 \) Copy content Toggle raw display
$11$ \( T^{10} + 91 T^{9} + \cdots + 22\!\cdots\!04 \) Copy content Toggle raw display
$13$ \( (T^{5} - 290 T^{4} + \cdots + 20518255200000)^{2} \) Copy content Toggle raw display
$17$ \( T^{10} + 1128 T^{9} + \cdots + 30\!\cdots\!16 \) Copy content Toggle raw display
$19$ \( T^{10} + 282 T^{9} + \cdots + 26\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{10} - 1808 T^{9} + \cdots + 34\!\cdots\!56 \) Copy content Toggle raw display
$29$ \( (T^{5} - 2513 T^{4} + \cdots + 21\!\cdots\!12)^{2} \) Copy content Toggle raw display
$31$ \( T^{10} - 5069 T^{9} + \cdots + 53\!\cdots\!41 \) Copy content Toggle raw display
$37$ \( T^{10} + 5010 T^{9} + \cdots + 51\!\cdots\!24 \) Copy content Toggle raw display
$41$ \( (T^{5} - 6436 T^{4} + \cdots - 16\!\cdots\!08)^{2} \) Copy content Toggle raw display
$43$ \( (T^{5} + 8664 T^{4} + \cdots + 36\!\cdots\!32)^{2} \) Copy content Toggle raw display
$47$ \( T^{10} - 50 T^{9} + \cdots + 19\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{10} - 1167 T^{9} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{10} - 42797 T^{9} + \cdots + 82\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{10} + 26546 T^{9} + \cdots + 58\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{10} - 13440 T^{9} + \cdots + 20\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( (T^{5} - 19678 T^{4} + \cdots + 45\!\cdots\!04)^{2} \) Copy content Toggle raw display
$73$ \( T^{10} + 27768 T^{9} + \cdots + 34\!\cdots\!64 \) Copy content Toggle raw display
$79$ \( T^{10} - 123369 T^{9} + \cdots + 11\!\cdots\!21 \) Copy content Toggle raw display
$83$ \( (T^{5} + 167125 T^{4} + \cdots + 27\!\cdots\!68)^{2} \) Copy content Toggle raw display
$89$ \( T^{10} + 59350 T^{9} + \cdots + 71\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( (T^{5} - 262641 T^{4} + \cdots - 39\!\cdots\!96)^{2} \) Copy content Toggle raw display
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