Properties

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

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Newspace parameters

Level: \( N \) = \( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 336.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.68297350792\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-6}) \)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{-6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{3} + \beta q^{5} + ( 1 - \beta ) q^{7} + q^{9} +O(q^{10})\) \( q - q^{3} + \beta q^{5} + ( 1 - \beta ) q^{7} + q^{9} + \beta q^{11} + 2 \beta q^{13} -\beta q^{15} + \beta q^{17} + 2 q^{19} + ( -1 + \beta ) q^{21} + 3 \beta q^{23} - q^{25} - q^{27} + 6 q^{29} -8 q^{31} -\beta q^{33} + ( 6 + \beta ) q^{35} + 4 q^{37} -2 \beta q^{39} -3 \beta q^{41} -2 \beta q^{43} + \beta q^{45} + 12 q^{47} + ( -5 - 2 \beta ) q^{49} -\beta q^{51} -6 q^{53} -6 q^{55} -2 q^{57} -12 q^{59} + ( 1 - \beta ) q^{63} -12 q^{65} -3 \beta q^{69} + 5 \beta q^{71} -6 \beta q^{73} + q^{75} + ( 6 + \beta ) q^{77} -4 \beta q^{79} + q^{81} -6 q^{85} -6 q^{87} -5 \beta q^{89} + ( 12 + 2 \beta ) q^{91} + 8 q^{93} + 2 \beta q^{95} + 2 \beta q^{97} + \beta q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{3} + 2q^{7} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{3} + 2q^{7} + 2q^{9} + 4q^{19} - 2q^{21} - 2q^{25} - 2q^{27} + 12q^{29} - 16q^{31} + 12q^{35} + 8q^{37} + 24q^{47} - 10q^{49} - 12q^{53} - 12q^{55} - 4q^{57} - 24q^{59} + 2q^{63} - 24q^{65} + 2q^{75} + 12q^{77} + 2q^{81} - 12q^{85} - 12q^{87} + 24q^{91} + 16q^{93} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
223.1
2.44949i
2.44949i
0 −1.00000 0 2.44949i 0 1.00000 + 2.44949i 0 1.00000 0
223.2 0 −1.00000 0 2.44949i 0 1.00000 2.44949i 0 1.00000 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
28.d 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.b.b 2
3.b odd 2 1 1008.2.b.e 2
4.b odd 2 1 336.2.b.c yes 2
7.b odd 2 1 336.2.b.c yes 2
7.c even 3 2 2352.2.bl.n 4
7.d odd 6 2 2352.2.bl.m 4
8.b even 2 1 1344.2.b.d 2
8.d odd 2 1 1344.2.b.a 2
12.b even 2 1 1008.2.b.d 2
21.c even 2 1 1008.2.b.d 2
24.f even 2 1 4032.2.b.d 2
24.h odd 2 1 4032.2.b.f 2
28.d even 2 1 inner 336.2.b.b 2
28.f even 6 2 2352.2.bl.n 4
28.g odd 6 2 2352.2.bl.m 4
56.e even 2 1 1344.2.b.d 2
56.h odd 2 1 1344.2.b.a 2
84.h odd 2 1 1008.2.b.e 2
168.e odd 2 1 4032.2.b.f 2
168.i even 2 1 4032.2.b.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
336.2.b.b 2 1.a even 1 1 trivial
336.2.b.b 2 28.d even 2 1 inner
336.2.b.c yes 2 4.b odd 2 1
336.2.b.c yes 2 7.b odd 2 1
1008.2.b.d 2 12.b even 2 1
1008.2.b.d 2 21.c even 2 1
1008.2.b.e 2 3.b odd 2 1
1008.2.b.e 2 84.h odd 2 1
1344.2.b.a 2 8.d odd 2 1
1344.2.b.a 2 56.h odd 2 1
1344.2.b.d 2 8.b even 2 1
1344.2.b.d 2 56.e even 2 1
2352.2.bl.m 4 7.d odd 6 2
2352.2.bl.m 4 28.g odd 6 2
2352.2.bl.n 4 7.c even 3 2
2352.2.bl.n 4 28.f even 6 2
4032.2.b.d 2 24.f even 2 1
4032.2.b.d 2 168.i even 2 1
4032.2.b.f 2 24.h odd 2 1
4032.2.b.f 2 168.e odd 2 1

Hecke kernels

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

\( T_{5}^{2} + 6 \)
\( T_{19} - 2 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( \)
$3$ \( ( 1 + T )^{2} \)
$5$ \( 1 - 4 T^{2} + 25 T^{4} \)
$7$ \( 1 - 2 T + 7 T^{2} \)
$11$ \( 1 - 16 T^{2} + 121 T^{4} \)
$13$ \( 1 - 2 T^{2} + 169 T^{4} \)
$17$ \( 1 - 28 T^{2} + 289 T^{4} \)
$19$ \( ( 1 - 2 T + 19 T^{2} )^{2} \)
$23$ \( 1 + 8 T^{2} + 529 T^{4} \)
$29$ \( ( 1 - 6 T + 29 T^{2} )^{2} \)
$31$ \( ( 1 + 8 T + 31 T^{2} )^{2} \)
$37$ \( ( 1 - 4 T + 37 T^{2} )^{2} \)
$41$ \( 1 - 28 T^{2} + 1681 T^{4} \)
$43$ \( 1 - 62 T^{2} + 1849 T^{4} \)
$47$ \( ( 1 - 12 T + 47 T^{2} )^{2} \)
$53$ \( ( 1 + 6 T + 53 T^{2} )^{2} \)
$59$ \( ( 1 + 12 T + 59 T^{2} )^{2} \)
$61$ \( ( 1 - 61 T^{2} )^{2} \)
$67$ \( ( 1 - 67 T^{2} )^{2} \)
$71$ \( 1 + 8 T^{2} + 5041 T^{4} \)
$73$ \( 1 + 70 T^{2} + 5329 T^{4} \)
$79$ \( 1 - 62 T^{2} + 6241 T^{4} \)
$83$ \( ( 1 + 83 T^{2} )^{2} \)
$89$ \( 1 - 28 T^{2} + 7921 T^{4} \)
$97$ \( 1 - 170 T^{2} + 9409 T^{4} \)
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