Properties

Label 336.2.q.f
Level 336
Weight 2
Character orbit 336.q
Analytic conductor 2.683
Analytic rank 0
Dimension 2
CM no
Inner twists 2

Related objects

Downloads

Learn more about

Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(2.68297350792\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
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 primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 - \zeta_{6} ) q^{3} + 2 \zeta_{6} q^{5} + ( 2 + \zeta_{6} ) q^{7} -\zeta_{6} q^{9} +O(q^{10})\) \( q + ( 1 - \zeta_{6} ) q^{3} + 2 \zeta_{6} q^{5} + ( 2 + \zeta_{6} ) q^{7} -\zeta_{6} q^{9} + ( -2 + 2 \zeta_{6} ) q^{11} + q^{13} + 2 q^{15} + \zeta_{6} q^{19} + ( 3 - 2 \zeta_{6} ) q^{21} + ( 1 - \zeta_{6} ) q^{25} - q^{27} + 4 q^{29} + ( 9 - 9 \zeta_{6} ) q^{31} + 2 \zeta_{6} q^{33} + ( -2 + 6 \zeta_{6} ) q^{35} -3 \zeta_{6} q^{37} + ( 1 - \zeta_{6} ) q^{39} -10 q^{41} -5 q^{43} + ( 2 - 2 \zeta_{6} ) q^{45} -6 \zeta_{6} q^{47} + ( 3 + 5 \zeta_{6} ) q^{49} + ( -12 + 12 \zeta_{6} ) q^{53} -4 q^{55} + q^{57} + ( -12 + 12 \zeta_{6} ) q^{59} -10 \zeta_{6} q^{61} + ( 1 - 3 \zeta_{6} ) q^{63} + 2 \zeta_{6} q^{65} + ( -5 + 5 \zeta_{6} ) q^{67} + 6 q^{71} + ( 3 - 3 \zeta_{6} ) q^{73} -\zeta_{6} q^{75} + ( -6 + 4 \zeta_{6} ) q^{77} -\zeta_{6} q^{79} + ( -1 + \zeta_{6} ) q^{81} -6 q^{83} + ( 4 - 4 \zeta_{6} ) q^{87} -16 \zeta_{6} q^{89} + ( 2 + \zeta_{6} ) q^{91} -9 \zeta_{6} q^{93} + ( -2 + 2 \zeta_{6} ) q^{95} -6 q^{97} + 2 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{3} + 2q^{5} + 5q^{7} - q^{9} + O(q^{10}) \) \( 2q + q^{3} + 2q^{5} + 5q^{7} - q^{9} - 2q^{11} + 2q^{13} + 4q^{15} + q^{19} + 4q^{21} + q^{25} - 2q^{27} + 8q^{29} + 9q^{31} + 2q^{33} + 2q^{35} - 3q^{37} + q^{39} - 20q^{41} - 10q^{43} + 2q^{45} - 6q^{47} + 11q^{49} - 12q^{53} - 8q^{55} + 2q^{57} - 12q^{59} - 10q^{61} - q^{63} + 2q^{65} - 5q^{67} + 12q^{71} + 3q^{73} - q^{75} - 8q^{77} - q^{79} - q^{81} - 12q^{83} + 4q^{87} - 16q^{89} + 5q^{91} - 9q^{93} - 2q^{95} - 12q^{97} + 4q^{99} + O(q^{100}) \)

Character values

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

\(n\) \(85\) \(113\) \(127\) \(241\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-\zeta_{6}\)

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
0.500000 + 0.866025i
0.500000 0.866025i
0 0.500000 0.866025i 0 1.00000 + 1.73205i 0 2.50000 + 0.866025i 0 −0.500000 0.866025i 0
289.1 0 0.500000 + 0.866025i 0 1.00000 1.73205i 0 2.50000 0.866025i 0 −0.500000 + 0.866025i 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 336.2.q.f 2
3.b odd 2 1 1008.2.s.d 2
4.b odd 2 1 21.2.e.a 2
7.b odd 2 1 2352.2.q.c 2
7.c even 3 1 inner 336.2.q.f 2
7.c even 3 1 2352.2.a.d 1
7.d odd 6 1 2352.2.a.w 1
7.d odd 6 1 2352.2.q.c 2
8.b even 2 1 1344.2.q.c 2
8.d odd 2 1 1344.2.q.m 2
12.b even 2 1 63.2.e.b 2
20.d odd 2 1 525.2.i.e 2
20.e even 4 2 525.2.r.e 4
21.g even 6 1 7056.2.a.m 1
21.h odd 6 1 1008.2.s.d 2
21.h odd 6 1 7056.2.a.bp 1
28.d even 2 1 147.2.e.a 2
28.f even 6 1 147.2.a.b 1
28.f even 6 1 147.2.e.a 2
28.g odd 6 1 21.2.e.a 2
28.g odd 6 1 147.2.a.c 1
36.f odd 6 1 567.2.g.a 2
36.f odd 6 1 567.2.h.f 2
36.h even 6 1 567.2.g.f 2
36.h even 6 1 567.2.h.a 2
56.j odd 6 1 9408.2.a.k 1
56.k odd 6 1 1344.2.q.m 2
56.k odd 6 1 9408.2.a.bg 1
56.m even 6 1 9408.2.a.bz 1
56.p even 6 1 1344.2.q.c 2
56.p even 6 1 9408.2.a.cv 1
84.h odd 2 1 441.2.e.e 2
84.j odd 6 1 441.2.a.a 1
84.j odd 6 1 441.2.e.e 2
84.n even 6 1 63.2.e.b 2
84.n even 6 1 441.2.a.b 1
140.p odd 6 1 525.2.i.e 2
140.p odd 6 1 3675.2.a.a 1
140.s even 6 1 3675.2.a.c 1
140.w even 12 2 525.2.r.e 4
252.o even 6 1 567.2.h.a 2
252.u odd 6 1 567.2.g.a 2
252.bb even 6 1 567.2.g.f 2
252.bl odd 6 1 567.2.h.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.2.e.a 2 4.b odd 2 1
21.2.e.a 2 28.g odd 6 1
63.2.e.b 2 12.b even 2 1
63.2.e.b 2 84.n even 6 1
147.2.a.b 1 28.f even 6 1
147.2.a.c 1 28.g odd 6 1
147.2.e.a 2 28.d even 2 1
147.2.e.a 2 28.f even 6 1
336.2.q.f 2 1.a even 1 1 trivial
336.2.q.f 2 7.c even 3 1 inner
441.2.a.a 1 84.j odd 6 1
441.2.a.b 1 84.n even 6 1
441.2.e.e 2 84.h odd 2 1
441.2.e.e 2 84.j odd 6 1
525.2.i.e 2 20.d odd 2 1
525.2.i.e 2 140.p odd 6 1
525.2.r.e 4 20.e even 4 2
525.2.r.e 4 140.w even 12 2
567.2.g.a 2 36.f odd 6 1
567.2.g.a 2 252.u odd 6 1
567.2.g.f 2 36.h even 6 1
567.2.g.f 2 252.bb even 6 1
567.2.h.a 2 36.h even 6 1
567.2.h.a 2 252.o even 6 1
567.2.h.f 2 36.f odd 6 1
567.2.h.f 2 252.bl odd 6 1
1008.2.s.d 2 3.b odd 2 1
1008.2.s.d 2 21.h odd 6 1
1344.2.q.c 2 8.b even 2 1
1344.2.q.c 2 56.p even 6 1
1344.2.q.m 2 8.d odd 2 1
1344.2.q.m 2 56.k odd 6 1
2352.2.a.d 1 7.c even 3 1
2352.2.a.w 1 7.d odd 6 1
2352.2.q.c 2 7.b odd 2 1
2352.2.q.c 2 7.d odd 6 1
3675.2.a.a 1 140.p odd 6 1
3675.2.a.c 1 140.s even 6 1
7056.2.a.m 1 21.g even 6 1
7056.2.a.bp 1 21.h odd 6 1
9408.2.a.k 1 56.j odd 6 1
9408.2.a.bg 1 56.k odd 6 1
9408.2.a.bz 1 56.m even 6 1
9408.2.a.cv 1 56.p even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(336, [\chi])\):

\( T_{5}^{2} - 2 T_{5} + 4 \)
\( T_{11}^{2} + 2 T_{11} + 4 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( \)
$3$ \( 1 - T + T^{2} \)
$5$ \( 1 - 2 T - T^{2} - 10 T^{3} + 25 T^{4} \)
$7$ \( 1 - 5 T + 7 T^{2} \)
$11$ \( 1 + 2 T - 7 T^{2} + 22 T^{3} + 121 T^{4} \)
$13$ \( ( 1 - T + 13 T^{2} )^{2} \)
$17$ \( 1 - 17 T^{2} + 289 T^{4} \)
$19$ \( ( 1 - 8 T + 19 T^{2} )( 1 + 7 T + 19 T^{2} ) \)
$23$ \( 1 - 23 T^{2} + 529 T^{4} \)
$29$ \( ( 1 - 4 T + 29 T^{2} )^{2} \)
$31$ \( 1 - 9 T + 50 T^{2} - 279 T^{3} + 961 T^{4} \)
$37$ \( 1 + 3 T - 28 T^{2} + 111 T^{3} + 1369 T^{4} \)
$41$ \( ( 1 + 10 T + 41 T^{2} )^{2} \)
$43$ \( ( 1 + 5 T + 43 T^{2} )^{2} \)
$47$ \( 1 + 6 T - 11 T^{2} + 282 T^{3} + 2209 T^{4} \)
$53$ \( 1 + 12 T + 91 T^{2} + 636 T^{3} + 2809 T^{4} \)
$59$ \( 1 + 12 T + 85 T^{2} + 708 T^{3} + 3481 T^{4} \)
$61$ \( 1 + 10 T + 39 T^{2} + 610 T^{3} + 3721 T^{4} \)
$67$ \( ( 1 - 11 T + 67 T^{2} )( 1 + 16 T + 67 T^{2} ) \)
$71$ \( ( 1 - 6 T + 71 T^{2} )^{2} \)
$73$ \( 1 - 3 T - 64 T^{2} - 219 T^{3} + 5329 T^{4} \)
$79$ \( 1 + T - 78 T^{2} + 79 T^{3} + 6241 T^{4} \)
$83$ \( ( 1 + 6 T + 83 T^{2} )^{2} \)
$89$ \( 1 + 16 T + 167 T^{2} + 1424 T^{3} + 7921 T^{4} \)
$97$ \( ( 1 + 6 T + 97 T^{2} )^{2} \)
show more
show less