Properties

Label 336.6.q.c
Level $336$
Weight $6$
Character orbit 336.q
Analytic conductor $53.889$
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: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 336.q (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(53.8889634572\)
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 42)
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.

\(f(q)\) \(=\) \( q + ( - 9 \zeta_{6} + 9) q^{3} + 6 \zeta_{6} q^{5} + (133 \zeta_{6} - 126) q^{7} - 81 \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + ( - 9 \zeta_{6} + 9) q^{3} + 6 \zeta_{6} q^{5} + (133 \zeta_{6} - 126) q^{7} - 81 \zeta_{6} q^{9} + (666 \zeta_{6} - 666) q^{11} - 559 q^{13} + 54 q^{15} + ( - 1740 \zeta_{6} + 1740) q^{17} + 1157 \zeta_{6} q^{19} + (1134 \zeta_{6} + 63) q^{21} - 3468 \zeta_{6} q^{23} + ( - 3089 \zeta_{6} + 3089) q^{25} - 729 q^{27} + 3372 q^{29} + ( - 6293 \zeta_{6} + 6293) q^{31} + 5994 \zeta_{6} q^{33} + (42 \zeta_{6} - 798) q^{35} - 3131 \zeta_{6} q^{37} + (5031 \zeta_{6} - 5031) q^{39} - 4866 q^{41} + 11407 q^{43} + ( - 486 \zeta_{6} + 486) q^{45} + 2310 \zeta_{6} q^{47} + ( - 15827 \zeta_{6} - 1813) q^{49} - 15660 \zeta_{6} q^{51} + ( - 28296 \zeta_{6} + 28296) q^{53} - 3996 q^{55} + 10413 q^{57} + ( - 20544 \zeta_{6} + 20544) q^{59} + 4630 \zeta_{6} q^{61} + ( - 567 \zeta_{6} + 10773) q^{63} - 3354 \zeta_{6} q^{65} + (18745 \zeta_{6} - 18745) q^{67} - 31212 q^{69} + 38226 q^{71} + (70589 \zeta_{6} - 70589) q^{73} - 27801 \zeta_{6} q^{75} + ( - 83916 \zeta_{6} - 4662) q^{77} - 62293 \zeta_{6} q^{79} + (6561 \zeta_{6} - 6561) q^{81} - 79818 q^{83} + 10440 q^{85} + ( - 30348 \zeta_{6} + 30348) q^{87} + 18120 \zeta_{6} q^{89} + ( - 74347 \zeta_{6} + 70434) q^{91} - 56637 \zeta_{6} q^{93} + (6942 \zeta_{6} - 6942) q^{95} + 124754 q^{97} + 53946 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 9 q^{3} + 6 q^{5} - 119 q^{7} - 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 9 q^{3} + 6 q^{5} - 119 q^{7} - 81 q^{9} - 666 q^{11} - 1118 q^{13} + 108 q^{15} + 1740 q^{17} + 1157 q^{19} + 1260 q^{21} - 3468 q^{23} + 3089 q^{25} - 1458 q^{27} + 6744 q^{29} + 6293 q^{31} + 5994 q^{33} - 1554 q^{35} - 3131 q^{37} - 5031 q^{39} - 9732 q^{41} + 22814 q^{43} + 486 q^{45} + 2310 q^{47} - 19453 q^{49} - 15660 q^{51} + 28296 q^{53} - 7992 q^{55} + 20826 q^{57} + 20544 q^{59} + 4630 q^{61} + 20979 q^{63} - 3354 q^{65} - 18745 q^{67} - 62424 q^{69} + 76452 q^{71} - 70589 q^{73} - 27801 q^{75} - 93240 q^{77} - 62293 q^{79} - 6561 q^{81} - 159636 q^{83} + 20880 q^{85} + 30348 q^{87} + 18120 q^{89} + 66521 q^{91} - 56637 q^{93} - 6942 q^{95} + 249508 q^{97} + 107892 q^{99}+O(q^{100}) \) 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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
193.1
0.500000 + 0.866025i
0.500000 0.866025i
0 4.50000 7.79423i 0 3.00000 + 5.19615i 0 −59.5000 + 115.181i 0 −40.5000 70.1481i 0
289.1 0 4.50000 + 7.79423i 0 3.00000 5.19615i 0 −59.5000 115.181i 0 −40.5000 + 70.1481i 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 336.6.q.c 2
4.b odd 2 1 42.6.e.a 2
7.c even 3 1 inner 336.6.q.c 2
12.b even 2 1 126.6.g.c 2
28.d even 2 1 294.6.e.e 2
28.f even 6 1 294.6.a.j 1
28.f even 6 1 294.6.e.e 2
28.g odd 6 1 42.6.e.a 2
28.g odd 6 1 294.6.a.l 1
84.j odd 6 1 882.6.a.e 1
84.n even 6 1 126.6.g.c 2
84.n even 6 1 882.6.a.f 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.6.e.a 2 4.b odd 2 1
42.6.e.a 2 28.g odd 6 1
126.6.g.c 2 12.b even 2 1
126.6.g.c 2 84.n even 6 1
294.6.a.j 1 28.f even 6 1
294.6.a.l 1 28.g odd 6 1
294.6.e.e 2 28.d even 2 1
294.6.e.e 2 28.f even 6 1
336.6.q.c 2 1.a even 1 1 trivial
336.6.q.c 2 7.c even 3 1 inner
882.6.a.e 1 84.j odd 6 1
882.6.a.f 1 84.n even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} - 6T_{5} + 36 \) acting on \(S_{6}^{\mathrm{new}}(336, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 9T + 81 \) Copy content Toggle raw display
$5$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$7$ \( T^{2} + 119T + 16807 \) Copy content Toggle raw display
$11$ \( T^{2} + 666T + 443556 \) Copy content Toggle raw display
$13$ \( (T + 559)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 1740 T + 3027600 \) Copy content Toggle raw display
$19$ \( T^{2} - 1157 T + 1338649 \) Copy content Toggle raw display
$23$ \( T^{2} + 3468 T + 12027024 \) Copy content Toggle raw display
$29$ \( (T - 3372)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 6293 T + 39601849 \) Copy content Toggle raw display
$37$ \( T^{2} + 3131 T + 9803161 \) Copy content Toggle raw display
$41$ \( (T + 4866)^{2} \) Copy content Toggle raw display
$43$ \( (T - 11407)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 2310 T + 5336100 \) Copy content Toggle raw display
$53$ \( T^{2} - 28296 T + 800663616 \) Copy content Toggle raw display
$59$ \( T^{2} - 20544 T + 422055936 \) Copy content Toggle raw display
$61$ \( T^{2} - 4630 T + 21436900 \) Copy content Toggle raw display
$67$ \( T^{2} + 18745 T + 351375025 \) Copy content Toggle raw display
$71$ \( (T - 38226)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 70589 T + 4982806921 \) Copy content Toggle raw display
$79$ \( T^{2} + 62293 T + 3880417849 \) Copy content Toggle raw display
$83$ \( (T + 79818)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} - 18120 T + 328334400 \) Copy content Toggle raw display
$97$ \( (T - 124754)^{2} \) Copy content Toggle raw display
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