Properties

Label 336.2.bj.f
Level $336$
Weight $2$
Character orbit 336.bj
Analytic conductor $2.683$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 336.bj (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.68297350792\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.303595776.1
Defining polynomial: \( x^{8} + 5x^{6} + 16x^{4} + 45x^{2} + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + (2 \beta_{6} + \beta_{4} - 2 \beta_{2} + 1) q^{5} + (3 \beta_{5} + \beta_{3}) q^{7} + (\beta_{6} - 2 \beta_{4} - \beta_{2} - 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + (2 \beta_{6} + \beta_{4} - 2 \beta_{2} + 1) q^{5} + (3 \beta_{5} + \beta_{3}) q^{7} + (\beta_{6} - 2 \beta_{4} - \beta_{2} - 2) q^{9} + (\beta_{5} - 2 \beta_1) q^{11} - 4 q^{13} + ( - \beta_{7} - 6 \beta_{5} - 6 \beta_{3} + \beta_1) q^{15} + (\beta_{4} - 2 \beta_{2}) q^{17} + 7 \beta_{3} q^{19} + (2 \beta_{6} + 3 \beta_{4} - 3 \beta_{2} + 2) q^{21} + (2 \beta_{7} + \beta_{3}) q^{23} - 6 \beta_{4} q^{25} + (2 \beta_{7} - 3 \beta_{5} - 3 \beta_{3} - 2 \beta_1) q^{27} + (4 \beta_{6} + 2) q^{29} + 3 \beta_{5} q^{31} + ( - \beta_{6} + 5 \beta_{4} + \beta_{2} + 5) q^{33} + ( - 6 \beta_{7} - 2 \beta_{5} - 3 \beta_{3} + 4 \beta_1) q^{35} + (\beta_{4} + 1) q^{37} - 4 \beta_1 q^{39} + ( - 4 \beta_{6} - 2) q^{41} + (2 \beta_{5} + 2 \beta_{3}) q^{43} + ( - 8 \beta_{4} + 5 \beta_{2}) q^{45} + (6 \beta_{7} + 3 \beta_{3}) q^{47} + (8 \beta_{4} + 3) q^{49} + ( - \beta_{7} - 6 \beta_{3}) q^{51} + ( - \beta_{4} + 2 \beta_{2}) q^{53} + (11 \beta_{5} + 11 \beta_{3}) q^{55} + ( - 7 \beta_{6} - 7) q^{57} + ( - \beta_{5} + 2 \beta_1) q^{59} + ( - 3 \beta_{4} - 3) q^{61} + ( - 3 \beta_{7} - 6 \beta_{5} - 9 \beta_{3} + 2 \beta_1) q^{63} + ( - 8 \beta_{6} - 4 \beta_{4} + 8 \beta_{2} - 4) q^{65} + 9 \beta_{5} q^{67} + (\beta_{6} - 5) q^{69} + ( - 8 \beta_{7} - 4 \beta_{5} - 4 \beta_{3} + 8 \beta_1) q^{71} + 7 \beta_{4} q^{73} + 6 \beta_{7} q^{75} + ( - 4 \beta_{6} - 3 \beta_{4} + 6 \beta_{2} - 2) q^{77} + 9 \beta_{3} q^{79} + (\beta_{4} + 5 \beta_{2}) q^{81} + (8 \beta_{7} + 4 \beta_{5} + 4 \beta_{3} - 8 \beta_1) q^{83} + 11 q^{85} + ( - 12 \beta_{5} + 2 \beta_1) q^{87} + ( - 2 \beta_{6} - \beta_{4} + 2 \beta_{2} - 1) q^{89} + ( - 12 \beta_{5} - 4 \beta_{3}) q^{91} + (3 \beta_{6} + 3 \beta_{4} - 3 \beta_{2} + 3) q^{93} + (7 \beta_{5} - 14 \beta_1) q^{95} + 8 q^{97} + ( - 5 \beta_{7} + 3 \beta_{5} + 3 \beta_{3} + 5 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 10 q^{9} - 32 q^{13} + 2 q^{21} + 24 q^{25} + 22 q^{33} + 4 q^{37} + 22 q^{45} - 8 q^{49} - 28 q^{57} - 12 q^{61} - 44 q^{69} - 28 q^{73} - 14 q^{81} + 88 q^{85} + 6 q^{93} + 64 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 5x^{6} + 16x^{4} + 45x^{2} + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{6} + 32\nu^{4} + 16\nu^{2} + 45 ) / 144 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{7} + 32\nu^{5} + 16\nu^{3} + 45\nu ) / 432 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -5\nu^{6} - 16\nu^{4} - 80\nu^{2} - 225 ) / 144 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{7} + 13\nu ) / 48 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{6} - 13 ) / 16 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 5\nu^{7} + 16\nu^{5} + 80\nu^{3} + 225\nu ) / 144 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} - 2\beta_{4} - \beta_{2} - 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{7} - 3\beta_{5} - 3\beta_{3} - 2\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{4} + 5\beta_{2} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -\beta_{7} + 15\beta_{3} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -16\beta_{6} - 13 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 48\beta_{5} - 13\beta_1 \) Copy content Toggle raw display

Character values

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

\(n\) \(85\) \(113\) \(127\) \(241\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(-1 - \beta_{4}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
95.1
−1.26217 + 1.18614i
−0.396143 + 1.68614i
0.396143 1.68614i
1.26217 1.18614i
−1.26217 1.18614i
−0.396143 1.68614i
0.396143 + 1.68614i
1.26217 + 1.18614i
0 −1.26217 + 1.18614i 0 2.87228 1.65831i 0 −1.73205 2.00000i 0 0.186141 2.99422i 0
95.2 0 −0.396143 + 1.68614i 0 −2.87228 + 1.65831i 0 1.73205 + 2.00000i 0 −2.68614 1.33591i 0
95.3 0 0.396143 1.68614i 0 −2.87228 + 1.65831i 0 −1.73205 2.00000i 0 −2.68614 1.33591i 0
95.4 0 1.26217 1.18614i 0 2.87228 1.65831i 0 1.73205 + 2.00000i 0 0.186141 2.99422i 0
191.1 0 −1.26217 1.18614i 0 2.87228 + 1.65831i 0 −1.73205 + 2.00000i 0 0.186141 + 2.99422i 0
191.2 0 −0.396143 1.68614i 0 −2.87228 1.65831i 0 1.73205 2.00000i 0 −2.68614 + 1.33591i 0
191.3 0 0.396143 + 1.68614i 0 −2.87228 1.65831i 0 −1.73205 + 2.00000i 0 −2.68614 + 1.33591i 0
191.4 0 1.26217 + 1.18614i 0 2.87228 + 1.65831i 0 1.73205 2.00000i 0 0.186141 + 2.99422i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 191.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
7.c even 3 1 inner
12.b even 2 1 inner
21.h odd 6 1 inner
28.g odd 6 1 inner
84.n even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 336.2.bj.f 8
3.b odd 2 1 inner 336.2.bj.f 8
4.b odd 2 1 inner 336.2.bj.f 8
7.c even 3 1 inner 336.2.bj.f 8
7.c even 3 1 2352.2.h.i 4
7.d odd 6 1 2352.2.h.j 4
12.b even 2 1 inner 336.2.bj.f 8
21.g even 6 1 2352.2.h.j 4
21.h odd 6 1 inner 336.2.bj.f 8
21.h odd 6 1 2352.2.h.i 4
28.f even 6 1 2352.2.h.j 4
28.g odd 6 1 inner 336.2.bj.f 8
28.g odd 6 1 2352.2.h.i 4
84.j odd 6 1 2352.2.h.j 4
84.n even 6 1 inner 336.2.bj.f 8
84.n even 6 1 2352.2.h.i 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
336.2.bj.f 8 1.a even 1 1 trivial
336.2.bj.f 8 3.b odd 2 1 inner
336.2.bj.f 8 4.b odd 2 1 inner
336.2.bj.f 8 7.c even 3 1 inner
336.2.bj.f 8 12.b even 2 1 inner
336.2.bj.f 8 21.h odd 6 1 inner
336.2.bj.f 8 28.g odd 6 1 inner
336.2.bj.f 8 84.n even 6 1 inner
2352.2.h.i 4 7.c even 3 1
2352.2.h.i 4 21.h odd 6 1
2352.2.h.i 4 28.g odd 6 1
2352.2.h.i 4 84.n even 6 1
2352.2.h.j 4 7.d odd 6 1
2352.2.h.j 4 21.g even 6 1
2352.2.h.j 4 28.f even 6 1
2352.2.h.j 4 84.j odd 6 1

Hecke kernels

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

\( T_{5}^{4} - 11T_{5}^{2} + 121 \) Copy content Toggle raw display
\( T_{13} + 4 \) Copy content Toggle raw display
\( T_{19}^{4} - 49T_{19}^{2} + 2401 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} + 5 T^{6} + 16 T^{4} + 45 T^{2} + \cdots + 81 \) Copy content Toggle raw display
$5$ \( (T^{4} - 11 T^{2} + 121)^{2} \) Copy content Toggle raw display
$7$ \( (T^{4} + 2 T^{2} + 49)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} + 11 T^{2} + 121)^{2} \) Copy content Toggle raw display
$13$ \( (T + 4)^{8} \) Copy content Toggle raw display
$17$ \( (T^{4} - 11 T^{2} + 121)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} - 49 T^{2} + 2401)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + 11 T^{2} + 121)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 44)^{4} \) Copy content Toggle raw display
$31$ \( (T^{4} - 9 T^{2} + 81)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} - T + 1)^{4} \) Copy content Toggle raw display
$41$ \( (T^{2} + 44)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} + 4)^{4} \) Copy content Toggle raw display
$47$ \( (T^{4} + 99 T^{2} + 9801)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} - 11 T^{2} + 121)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + 11 T^{2} + 121)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 3 T + 9)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} - 81 T^{2} + 6561)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} - 176)^{4} \) Copy content Toggle raw display
$73$ \( (T^{2} + 7 T + 49)^{4} \) Copy content Toggle raw display
$79$ \( (T^{4} - 81 T^{2} + 6561)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} - 176)^{4} \) Copy content Toggle raw display
$89$ \( (T^{4} - 11 T^{2} + 121)^{2} \) Copy content Toggle raw display
$97$ \( (T - 8)^{8} \) Copy content Toggle raw display
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