Properties

Label 336.4.q.e
Level $336$
Weight $4$
Character orbit 336.q
Analytic conductor $19.825$
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: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 336.q (of order \(3\), 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: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 21)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( 3 - 3 \zeta_{6} ) q^{3} + 3 \zeta_{6} q^{5} + ( -7 + 21 \zeta_{6} ) q^{7} -9 \zeta_{6} q^{9} +O(q^{10})\) \( q + ( 3 - 3 \zeta_{6} ) q^{3} + 3 \zeta_{6} q^{5} + ( -7 + 21 \zeta_{6} ) q^{7} -9 \zeta_{6} q^{9} + ( -15 + 15 \zeta_{6} ) q^{11} -64 q^{13} + 9 q^{15} + ( -84 + 84 \zeta_{6} ) q^{17} -16 \zeta_{6} q^{19} + ( 42 + 21 \zeta_{6} ) q^{21} -84 \zeta_{6} q^{23} + ( 116 - 116 \zeta_{6} ) q^{25} -27 q^{27} -297 q^{29} + ( -253 + 253 \zeta_{6} ) q^{31} + 45 \zeta_{6} q^{33} + ( -63 + 42 \zeta_{6} ) q^{35} + 316 \zeta_{6} q^{37} + ( -192 + 192 \zeta_{6} ) q^{39} + 360 q^{41} -26 q^{43} + ( 27 - 27 \zeta_{6} ) q^{45} -30 \zeta_{6} q^{47} + ( -392 + 147 \zeta_{6} ) q^{49} + 252 \zeta_{6} q^{51} + ( -363 + 363 \zeta_{6} ) q^{53} -45 q^{55} -48 q^{57} + ( -15 + 15 \zeta_{6} ) q^{59} + 118 \zeta_{6} q^{61} + ( 189 - 126 \zeta_{6} ) q^{63} -192 \zeta_{6} q^{65} + ( -370 + 370 \zeta_{6} ) q^{67} -252 q^{69} + 342 q^{71} + ( -362 + 362 \zeta_{6} ) q^{73} -348 \zeta_{6} q^{75} + ( -210 - 105 \zeta_{6} ) q^{77} + 467 \zeta_{6} q^{79} + ( -81 + 81 \zeta_{6} ) q^{81} -477 q^{83} -252 q^{85} + ( -891 + 891 \zeta_{6} ) q^{87} -906 \zeta_{6} q^{89} + ( 448 - 1344 \zeta_{6} ) q^{91} + 759 \zeta_{6} q^{93} + ( 48 - 48 \zeta_{6} ) q^{95} + 503 q^{97} + 135 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 3q^{3} + 3q^{5} + 7q^{7} - 9q^{9} + O(q^{10}) \) \( 2q + 3q^{3} + 3q^{5} + 7q^{7} - 9q^{9} - 15q^{11} - 128q^{13} + 18q^{15} - 84q^{17} - 16q^{19} + 105q^{21} - 84q^{23} + 116q^{25} - 54q^{27} - 594q^{29} - 253q^{31} + 45q^{33} - 84q^{35} + 316q^{37} - 192q^{39} + 720q^{41} - 52q^{43} + 27q^{45} - 30q^{47} - 637q^{49} + 252q^{51} - 363q^{53} - 90q^{55} - 96q^{57} - 15q^{59} + 118q^{61} + 252q^{63} - 192q^{65} - 370q^{67} - 504q^{69} + 684q^{71} - 362q^{73} - 348q^{75} - 525q^{77} + 467q^{79} - 81q^{81} - 954q^{83} - 504q^{85} - 891q^{87} - 906q^{89} - 448q^{91} + 759q^{93} + 48q^{95} + 1006q^{97} + 270q^{99} + O(q^{100}) \)

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
193.1
0.500000 + 0.866025i
0.500000 0.866025i
0 1.50000 2.59808i 0 1.50000 + 2.59808i 0 3.50000 + 18.1865i 0 −4.50000 7.79423i 0
289.1 0 1.50000 + 2.59808i 0 1.50000 2.59808i 0 3.50000 18.1865i 0 −4.50000 + 7.79423i 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
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 336.4.q.e 2
4.b odd 2 1 21.4.e.a 2
7.c even 3 1 inner 336.4.q.e 2
7.c even 3 1 2352.4.a.i 1
7.d odd 6 1 2352.4.a.bd 1
12.b even 2 1 63.4.e.a 2
28.d even 2 1 147.4.e.h 2
28.f even 6 1 147.4.a.a 1
28.f even 6 1 147.4.e.h 2
28.g odd 6 1 21.4.e.a 2
28.g odd 6 1 147.4.a.b 1
84.h odd 2 1 441.4.e.c 2
84.j odd 6 1 441.4.a.k 1
84.j odd 6 1 441.4.e.c 2
84.n even 6 1 63.4.e.a 2
84.n even 6 1 441.4.a.l 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.4.e.a 2 4.b odd 2 1
21.4.e.a 2 28.g odd 6 1
63.4.e.a 2 12.b even 2 1
63.4.e.a 2 84.n even 6 1
147.4.a.a 1 28.f even 6 1
147.4.a.b 1 28.g odd 6 1
147.4.e.h 2 28.d even 2 1
147.4.e.h 2 28.f even 6 1
336.4.q.e 2 1.a even 1 1 trivial
336.4.q.e 2 7.c even 3 1 inner
441.4.a.k 1 84.j odd 6 1
441.4.a.l 1 84.n even 6 1
441.4.e.c 2 84.h odd 2 1
441.4.e.c 2 84.j odd 6 1
2352.4.a.i 1 7.c even 3 1
2352.4.a.bd 1 7.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} - 3 T_{5} + 9 \) acting on \(S_{4}^{\mathrm{new}}(336, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 9 - 3 T + T^{2} \)
$5$ \( 9 - 3 T + T^{2} \)
$7$ \( 343 - 7 T + T^{2} \)
$11$ \( 225 + 15 T + T^{2} \)
$13$ \( ( 64 + T )^{2} \)
$17$ \( 7056 + 84 T + T^{2} \)
$19$ \( 256 + 16 T + T^{2} \)
$23$ \( 7056 + 84 T + T^{2} \)
$29$ \( ( 297 + T )^{2} \)
$31$ \( 64009 + 253 T + T^{2} \)
$37$ \( 99856 - 316 T + T^{2} \)
$41$ \( ( -360 + T )^{2} \)
$43$ \( ( 26 + T )^{2} \)
$47$ \( 900 + 30 T + T^{2} \)
$53$ \( 131769 + 363 T + T^{2} \)
$59$ \( 225 + 15 T + T^{2} \)
$61$ \( 13924 - 118 T + T^{2} \)
$67$ \( 136900 + 370 T + T^{2} \)
$71$ \( ( -342 + T )^{2} \)
$73$ \( 131044 + 362 T + T^{2} \)
$79$ \( 218089 - 467 T + T^{2} \)
$83$ \( ( 477 + T )^{2} \)
$89$ \( 820836 + 906 T + T^{2} \)
$97$ \( ( -503 + T )^{2} \)
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