Properties

Label 336.4.bc.b
Level $336$
Weight $4$
Character orbit 336.bc
Analytic conductor $19.825$
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: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 336.bc (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(19.8246417619\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 84)
Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + (3 \zeta_{6} + 3) q^{3} + ( - 19 \zeta_{6} + 18) q^{7} + 27 \zeta_{6} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (3 \zeta_{6} + 3) q^{3} + ( - 19 \zeta_{6} + 18) q^{7} + 27 \zeta_{6} q^{9} + (106 \zeta_{6} - 53) q^{13} + ( - 73 \zeta_{6} + 146) q^{19} + ( - 60 \zeta_{6} + 111) q^{21} + ( - 125 \zeta_{6} + 125) q^{25} + (162 \zeta_{6} - 81) q^{27} + (109 \zeta_{6} + 109) q^{31} + 323 \zeta_{6} q^{37} + (477 \zeta_{6} - 477) q^{39} + 71 q^{43} + ( - 323 \zeta_{6} - 37) q^{49} + 657 q^{57} + (540 \zeta_{6} - 1080) q^{61} + ( - 27 \zeta_{6} + 513) q^{63} + (127 \zeta_{6} - 127) q^{67} + (703 \zeta_{6} + 703) q^{73} + ( - 375 \zeta_{6} + 750) q^{75} - 1387 \zeta_{6} q^{79} + (729 \zeta_{6} - 729) q^{81} + (901 \zeta_{6} + 1060) q^{91} + 981 \zeta_{6} q^{93} + ( - 1584 \zeta_{6} + 792) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 9 q^{3} + 17 q^{7} + 27 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 9 q^{3} + 17 q^{7} + 27 q^{9} + 219 q^{19} + 162 q^{21} + 125 q^{25} + 327 q^{31} + 323 q^{37} - 477 q^{39} + 142 q^{43} - 397 q^{49} + 1314 q^{57} - 1620 q^{61} + 999 q^{63} - 127 q^{67} + 2109 q^{73} + 1125 q^{75} - 1387 q^{79} - 729 q^{81} + 3021 q^{91} + 981 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1
0.500000 + 0.866025i
0.500000 0.866025i
0 4.50000 + 2.59808i 0 0 0 8.50000 16.4545i 0 13.5000 + 23.3827i 0
257.1 0 4.50000 2.59808i 0 0 0 8.50000 + 16.4545i 0 13.5000 23.3827i 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.d odd 6 1 inner
21.g even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 336.4.bc.b 2
3.b odd 2 1 CM 336.4.bc.b 2
4.b odd 2 1 84.4.k.a 2
7.d odd 6 1 inner 336.4.bc.b 2
12.b even 2 1 84.4.k.a 2
21.g even 6 1 inner 336.4.bc.b 2
28.d even 2 1 588.4.k.b 2
28.f even 6 1 84.4.k.a 2
28.f even 6 1 588.4.f.a 2
28.g odd 6 1 588.4.f.a 2
28.g odd 6 1 588.4.k.b 2
84.h odd 2 1 588.4.k.b 2
84.j odd 6 1 84.4.k.a 2
84.j odd 6 1 588.4.f.a 2
84.n even 6 1 588.4.f.a 2
84.n even 6 1 588.4.k.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.4.k.a 2 4.b odd 2 1
84.4.k.a 2 12.b even 2 1
84.4.k.a 2 28.f even 6 1
84.4.k.a 2 84.j odd 6 1
336.4.bc.b 2 1.a even 1 1 trivial
336.4.bc.b 2 3.b odd 2 1 CM
336.4.bc.b 2 7.d odd 6 1 inner
336.4.bc.b 2 21.g even 6 1 inner
588.4.f.a 2 28.f even 6 1
588.4.f.a 2 28.g odd 6 1
588.4.f.a 2 84.j odd 6 1
588.4.f.a 2 84.n even 6 1
588.4.k.b 2 28.d even 2 1
588.4.k.b 2 28.g odd 6 1
588.4.k.b 2 84.h odd 2 1
588.4.k.b 2 84.n even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(336, [\chi])\):

\( T_{5} \) Copy content Toggle raw display
\( T_{13}^{2} + 8427 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 9T + 27 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 17T + 343 \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 8427 \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( T^{2} - 219T + 15987 \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 327T + 35643 \) Copy content Toggle raw display
$37$ \( T^{2} - 323T + 104329 \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( (T - 71)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( T^{2} + 1620 T + 874800 \) Copy content Toggle raw display
$67$ \( T^{2} + 127T + 16129 \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 2109 T + 1482627 \) Copy content Toggle raw display
$79$ \( T^{2} + 1387 T + 1923769 \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 1881792 \) Copy content Toggle raw display
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