Properties

Label 336.2.k.b
Level 336
Weight 2
Character orbit 336.k
Analytic conductor 2.683
Analytic rank 0
Dimension 4
CM no
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: \(4\)
Coefficient field: \(\Q(i, \sqrt{6})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 42)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{3} + ( \beta_{1} - \beta_{3} ) q^{5} + ( 1 - \beta_{1} - \beta_{3} ) q^{7} + \beta_{2} q^{9} +O(q^{10})\) \( q + \beta_{1} q^{3} + ( \beta_{1} - \beta_{3} ) q^{5} + ( 1 - \beta_{1} - \beta_{3} ) q^{7} + \beta_{2} q^{9} + ( -\beta_{1} - \beta_{3} ) q^{13} + ( 3 + \beta_{2} ) q^{15} + ( -2 \beta_{1} + 2 \beta_{3} ) q^{17} + ( -\beta_{1} - \beta_{3} ) q^{19} + ( 3 + \beta_{1} - \beta_{2} ) q^{21} + 2 \beta_{2} q^{23} + q^{25} + 3 \beta_{3} q^{27} + 2 \beta_{2} q^{29} + ( \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{35} -2 q^{37} + ( 3 - \beta_{2} ) q^{39} + ( 2 \beta_{1} - 2 \beta_{3} ) q^{41} -4 q^{43} + ( 3 \beta_{1} + 3 \beta_{3} ) q^{45} + ( 2 \beta_{1} - 2 \beta_{3} ) q^{47} + ( -5 - 2 \beta_{1} - 2 \beta_{3} ) q^{49} + ( -6 - 2 \beta_{2} ) q^{51} -2 \beta_{2} q^{53} + ( 3 - \beta_{2} ) q^{57} + ( -5 \beta_{1} + 5 \beta_{3} ) q^{59} + ( -5 \beta_{1} - 5 \beta_{3} ) q^{61} + ( 3 \beta_{1} + \beta_{2} - 3 \beta_{3} ) q^{63} -2 \beta_{2} q^{65} -8 q^{67} + 6 \beta_{3} q^{69} + ( 4 \beta_{1} + 4 \beta_{3} ) q^{73} + \beta_{1} q^{75} + 10 q^{79} -9 q^{81} + ( \beta_{1} - \beta_{3} ) q^{83} -12 q^{85} + 6 \beta_{3} q^{87} + ( -6 - \beta_{1} - \beta_{3} ) q^{91} -2 \beta_{2} q^{95} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{7} + O(q^{10}) \) \( 4q + 4q^{7} + 12q^{15} + 12q^{21} + 4q^{25} - 8q^{37} + 12q^{39} - 16q^{43} - 20q^{49} - 24q^{51} + 12q^{57} - 32q^{67} + 40q^{79} - 36q^{81} - 48q^{85} - 24q^{91} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 9\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)
\(\beta_{3}\)\(=\)\( \nu^{3} \)\(/3\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\)
\(\nu^{3}\)\(=\)\(3 \beta_{3}\)

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
−1.22474 1.22474i
−1.22474 + 1.22474i
1.22474 1.22474i
1.22474 + 1.22474i
0 −1.22474 1.22474i 0 −2.44949 0 1.00000 + 2.44949i 0 3.00000i 0
209.2 0 −1.22474 + 1.22474i 0 −2.44949 0 1.00000 2.44949i 0 3.00000i 0
209.3 0 1.22474 1.22474i 0 2.44949 0 1.00000 + 2.44949i 0 3.00000i 0
209.4 0 1.22474 + 1.22474i 0 2.44949 0 1.00000 2.44949i 0 3.00000i 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 inner
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.b 4
3.b odd 2 1 inner 336.2.k.b 4
4.b odd 2 1 42.2.d.a 4
7.b odd 2 1 inner 336.2.k.b 4
8.b even 2 1 1344.2.k.d 4
8.d odd 2 1 1344.2.k.c 4
12.b even 2 1 42.2.d.a 4
20.d odd 2 1 1050.2.b.b 4
20.e even 4 1 1050.2.d.b 4
20.e even 4 1 1050.2.d.e 4
21.c even 2 1 inner 336.2.k.b 4
24.f even 2 1 1344.2.k.c 4
24.h odd 2 1 1344.2.k.d 4
28.d even 2 1 42.2.d.a 4
28.f even 6 2 294.2.f.b 8
28.g odd 6 2 294.2.f.b 8
36.f odd 6 2 1134.2.m.g 8
36.h even 6 2 1134.2.m.g 8
56.e even 2 1 1344.2.k.c 4
56.h odd 2 1 1344.2.k.d 4
60.h even 2 1 1050.2.b.b 4
60.l odd 4 1 1050.2.d.b 4
60.l odd 4 1 1050.2.d.e 4
84.h odd 2 1 42.2.d.a 4
84.j odd 6 2 294.2.f.b 8
84.n even 6 2 294.2.f.b 8
140.c even 2 1 1050.2.b.b 4
140.j odd 4 1 1050.2.d.b 4
140.j odd 4 1 1050.2.d.e 4
168.e odd 2 1 1344.2.k.c 4
168.i even 2 1 1344.2.k.d 4
252.s odd 6 2 1134.2.m.g 8
252.bi even 6 2 1134.2.m.g 8
420.o odd 2 1 1050.2.b.b 4
420.w even 4 1 1050.2.d.b 4
420.w even 4 1 1050.2.d.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.2.d.a 4 4.b odd 2 1
42.2.d.a 4 12.b even 2 1
42.2.d.a 4 28.d even 2 1
42.2.d.a 4 84.h odd 2 1
294.2.f.b 8 28.f even 6 2
294.2.f.b 8 28.g odd 6 2
294.2.f.b 8 84.j odd 6 2
294.2.f.b 8 84.n even 6 2
336.2.k.b 4 1.a even 1 1 trivial
336.2.k.b 4 3.b odd 2 1 inner
336.2.k.b 4 7.b odd 2 1 inner
336.2.k.b 4 21.c even 2 1 inner
1050.2.b.b 4 20.d odd 2 1
1050.2.b.b 4 60.h even 2 1
1050.2.b.b 4 140.c even 2 1
1050.2.b.b 4 420.o odd 2 1
1050.2.d.b 4 20.e even 4 1
1050.2.d.b 4 60.l odd 4 1
1050.2.d.b 4 140.j odd 4 1
1050.2.d.b 4 420.w even 4 1
1050.2.d.e 4 20.e even 4 1
1050.2.d.e 4 60.l odd 4 1
1050.2.d.e 4 140.j odd 4 1
1050.2.d.e 4 420.w even 4 1
1134.2.m.g 8 36.f odd 6 2
1134.2.m.g 8 36.h even 6 2
1134.2.m.g 8 252.s odd 6 2
1134.2.m.g 8 252.bi even 6 2
1344.2.k.c 4 8.d odd 2 1
1344.2.k.c 4 24.f even 2 1
1344.2.k.c 4 56.e even 2 1
1344.2.k.c 4 168.e odd 2 1
1344.2.k.d 4 8.b even 2 1
1344.2.k.d 4 24.h odd 2 1
1344.2.k.d 4 56.h odd 2 1
1344.2.k.d 4 168.i even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} - 6 \) acting on \(S_{2}^{\mathrm{new}}(336, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( \)
$3$ \( 1 + 9 T^{4} \)
$5$ \( ( 1 + 4 T^{2} + 25 T^{4} )^{2} \)
$7$ \( ( 1 - 2 T + 7 T^{2} )^{2} \)
$11$ \( ( 1 - 11 T^{2} )^{4} \)
$13$ \( ( 1 - 20 T^{2} + 169 T^{4} )^{2} \)
$17$ \( ( 1 + 10 T^{2} + 289 T^{4} )^{2} \)
$19$ \( ( 1 - 32 T^{2} + 361 T^{4} )^{2} \)
$23$ \( ( 1 - 10 T^{2} + 529 T^{4} )^{2} \)
$29$ \( ( 1 - 22 T^{2} + 841 T^{4} )^{2} \)
$31$ \( ( 1 - 31 T^{2} )^{4} \)
$37$ \( ( 1 + 2 T + 37 T^{2} )^{4} \)
$41$ \( ( 1 + 58 T^{2} + 1681 T^{4} )^{2} \)
$43$ \( ( 1 + 4 T + 43 T^{2} )^{4} \)
$47$ \( ( 1 + 70 T^{2} + 2209 T^{4} )^{2} \)
$53$ \( ( 1 - 70 T^{2} + 2809 T^{4} )^{2} \)
$59$ \( ( 1 - 32 T^{2} + 3481 T^{4} )^{2} \)
$61$ \( ( 1 + 28 T^{2} + 3721 T^{4} )^{2} \)
$67$ \( ( 1 + 8 T + 67 T^{2} )^{4} \)
$71$ \( ( 1 - 71 T^{2} )^{4} \)
$73$ \( ( 1 - 14 T + 73 T^{2} )^{2}( 1 + 14 T + 73 T^{2} )^{2} \)
$79$ \( ( 1 - 10 T + 79 T^{2} )^{4} \)
$83$ \( ( 1 + 160 T^{2} + 6889 T^{4} )^{2} \)
$89$ \( ( 1 + 89 T^{2} )^{4} \)
$97$ \( ( 1 - 170 T^{2} + 9409 T^{4} )^{2} \)
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