Properties

Label 336.2.k.a
Level 336
Weight 2
Character orbit 336.k
Analytic conductor 2.683
Analytic rank 0
Dimension 2
CM discriminant -3
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 336.k (of order \(2\), degree \(1\), 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: \( 2 \)
Twist minimal: no (minimal twist has level 84)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( 1 - 2 \zeta_{6} ) q^{3} + ( -1 - 2 \zeta_{6} ) q^{7} -3 q^{9} +O(q^{10})\) \( q + ( 1 - 2 \zeta_{6} ) q^{3} + ( -1 - 2 \zeta_{6} ) q^{7} -3 q^{9} + ( 4 - 8 \zeta_{6} ) q^{13} + ( -2 + 4 \zeta_{6} ) q^{19} + ( -5 + 4 \zeta_{6} ) q^{21} -5 q^{25} + ( -3 + 6 \zeta_{6} ) q^{27} + ( 6 - 12 \zeta_{6} ) q^{31} + 10 q^{37} -12 q^{39} + 8 q^{43} + ( -3 + 8 \zeta_{6} ) q^{49} + 6 q^{57} + ( -4 + 8 \zeta_{6} ) q^{61} + ( 3 + 6 \zeta_{6} ) q^{63} + 16 q^{67} + ( 8 - 16 \zeta_{6} ) q^{73} + ( -5 + 10 \zeta_{6} ) q^{75} + 4 q^{79} + 9 q^{81} + ( -20 + 16 \zeta_{6} ) q^{91} -18 q^{93} + ( -8 + 16 \zeta_{6} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 4q^{7} - 6q^{9} + O(q^{10}) \) \( 2q - 4q^{7} - 6q^{9} - 6q^{21} - 10q^{25} + 20q^{37} - 24q^{39} + 16q^{43} + 2q^{49} + 12q^{57} + 12q^{63} + 32q^{67} + 8q^{79} + 18q^{81} - 24q^{91} - 36q^{93} + O(q^{100}) \)

Character values

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

\(n\) \(85\) \(113\) \(127\) \(241\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
209.1
0.500000 + 0.866025i
0.500000 0.866025i
0 1.73205i 0 0 0 −2.00000 1.73205i 0 −3.00000 0
209.2 0 1.73205i 0 0 0 −2.00000 + 1.73205i 0 −3.00000 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
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
7.b odd 2 1 inner
21.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 336.2.k.a 2
3.b odd 2 1 CM 336.2.k.a 2
4.b odd 2 1 84.2.f.a 2
7.b odd 2 1 inner 336.2.k.a 2
8.b even 2 1 1344.2.k.a 2
8.d odd 2 1 1344.2.k.b 2
12.b even 2 1 84.2.f.a 2
20.d odd 2 1 2100.2.d.b 2
20.e even 4 2 2100.2.f.e 4
21.c even 2 1 inner 336.2.k.a 2
24.f even 2 1 1344.2.k.b 2
24.h odd 2 1 1344.2.k.a 2
28.d even 2 1 84.2.f.a 2
28.f even 6 1 588.2.k.a 2
28.f even 6 1 588.2.k.e 2
28.g odd 6 1 588.2.k.a 2
28.g odd 6 1 588.2.k.e 2
36.f odd 6 1 2268.2.x.c 2
36.f odd 6 1 2268.2.x.e 2
36.h even 6 1 2268.2.x.c 2
36.h even 6 1 2268.2.x.e 2
56.e even 2 1 1344.2.k.b 2
56.h odd 2 1 1344.2.k.a 2
60.h even 2 1 2100.2.d.b 2
60.l odd 4 2 2100.2.f.e 4
84.h odd 2 1 84.2.f.a 2
84.j odd 6 1 588.2.k.a 2
84.j odd 6 1 588.2.k.e 2
84.n even 6 1 588.2.k.a 2
84.n even 6 1 588.2.k.e 2
140.c even 2 1 2100.2.d.b 2
140.j odd 4 2 2100.2.f.e 4
168.e odd 2 1 1344.2.k.b 2
168.i even 2 1 1344.2.k.a 2
252.s odd 6 1 2268.2.x.c 2
252.s odd 6 1 2268.2.x.e 2
252.bi even 6 1 2268.2.x.c 2
252.bi even 6 1 2268.2.x.e 2
420.o odd 2 1 2100.2.d.b 2
420.w even 4 2 2100.2.f.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.2.f.a 2 4.b odd 2 1
84.2.f.a 2 12.b even 2 1
84.2.f.a 2 28.d even 2 1
84.2.f.a 2 84.h odd 2 1
336.2.k.a 2 1.a even 1 1 trivial
336.2.k.a 2 3.b odd 2 1 CM
336.2.k.a 2 7.b odd 2 1 inner
336.2.k.a 2 21.c even 2 1 inner
588.2.k.a 2 28.f even 6 1
588.2.k.a 2 28.g odd 6 1
588.2.k.a 2 84.j odd 6 1
588.2.k.a 2 84.n even 6 1
588.2.k.e 2 28.f even 6 1
588.2.k.e 2 28.g odd 6 1
588.2.k.e 2 84.j odd 6 1
588.2.k.e 2 84.n even 6 1
1344.2.k.a 2 8.b even 2 1
1344.2.k.a 2 24.h odd 2 1
1344.2.k.a 2 56.h odd 2 1
1344.2.k.a 2 168.i even 2 1
1344.2.k.b 2 8.d odd 2 1
1344.2.k.b 2 24.f even 2 1
1344.2.k.b 2 56.e even 2 1
1344.2.k.b 2 168.e odd 2 1
2100.2.d.b 2 20.d odd 2 1
2100.2.d.b 2 60.h even 2 1
2100.2.d.b 2 140.c even 2 1
2100.2.d.b 2 420.o odd 2 1
2100.2.f.e 4 20.e even 4 2
2100.2.f.e 4 60.l odd 4 2
2100.2.f.e 4 140.j odd 4 2
2100.2.f.e 4 420.w even 4 2
2268.2.x.c 2 36.f odd 6 1
2268.2.x.c 2 36.h even 6 1
2268.2.x.c 2 252.s odd 6 1
2268.2.x.c 2 252.bi even 6 1
2268.2.x.e 2 36.f odd 6 1
2268.2.x.e 2 36.h even 6 1
2268.2.x.e 2 252.s odd 6 1
2268.2.x.e 2 252.bi even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} \) acting on \(S_{2}^{\mathrm{new}}(336, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( \)
$3$ \( 1 + 3 T^{2} \)
$5$ \( ( 1 + 5 T^{2} )^{2} \)
$7$ \( 1 + 4 T + 7 T^{2} \)
$11$ \( ( 1 - 11 T^{2} )^{2} \)
$13$ \( ( 1 - 2 T + 13 T^{2} )( 1 + 2 T + 13 T^{2} ) \)
$17$ \( ( 1 + 17 T^{2} )^{2} \)
$19$ \( ( 1 - 8 T + 19 T^{2} )( 1 + 8 T + 19 T^{2} ) \)
$23$ \( ( 1 - 23 T^{2} )^{2} \)
$29$ \( ( 1 - 29 T^{2} )^{2} \)
$31$ \( ( 1 - 4 T + 31 T^{2} )( 1 + 4 T + 31 T^{2} ) \)
$37$ \( ( 1 - 10 T + 37 T^{2} )^{2} \)
$41$ \( ( 1 + 41 T^{2} )^{2} \)
$43$ \( ( 1 - 8 T + 43 T^{2} )^{2} \)
$47$ \( ( 1 + 47 T^{2} )^{2} \)
$53$ \( ( 1 - 53 T^{2} )^{2} \)
$59$ \( ( 1 + 59 T^{2} )^{2} \)
$61$ \( ( 1 - 14 T + 61 T^{2} )( 1 + 14 T + 61 T^{2} ) \)
$67$ \( ( 1 - 16 T + 67 T^{2} )^{2} \)
$71$ \( ( 1 - 71 T^{2} )^{2} \)
$73$ \( ( 1 - 10 T + 73 T^{2} )( 1 + 10 T + 73 T^{2} ) \)
$79$ \( ( 1 - 4 T + 79 T^{2} )^{2} \)
$83$ \( ( 1 + 83 T^{2} )^{2} \)
$89$ \( ( 1 + 89 T^{2} )^{2} \)
$97$ \( ( 1 - 14 T + 97 T^{2} )( 1 + 14 T + 97 T^{2} ) \)
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