Properties

Label 336.6.q.a
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: \(\Z[a_1, \ldots, a_{5}]\)
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.

\(f(q)\) \(=\) \( q + (9 \zeta_{6} - 9) q^{3} - 86 \zeta_{6} q^{5} + (147 \zeta_{6} - 98) q^{7} - 81 \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + (9 \zeta_{6} - 9) q^{3} - 86 \zeta_{6} q^{5} + (147 \zeta_{6} - 98) q^{7} - 81 \zeta_{6} q^{9} + ( - 34 \zeta_{6} + 34) q^{11} - 3 q^{13} + 774 q^{15} + ( - 1904 \zeta_{6} + 1904) q^{17} - 1489 \zeta_{6} q^{19} + ( - 882 \zeta_{6} - 441) q^{21} - 224 \zeta_{6} q^{23} + (4271 \zeta_{6} - 4271) q^{25} + 729 q^{27} - 6508 q^{29} + ( - 1731 \zeta_{6} + 1731) q^{31} + 306 \zeta_{6} q^{33} + ( - 4214 \zeta_{6} + 12642) q^{35} + 7633 \zeta_{6} q^{37} + ( - 27 \zeta_{6} + 27) q^{39} + 15414 q^{41} - 18491 q^{43} + (6966 \zeta_{6} - 6966) q^{45} + 18462 \zeta_{6} q^{47} + ( - 7203 \zeta_{6} - 12005) q^{49} + 17136 \zeta_{6} q^{51} + ( - 19956 \zeta_{6} + 19956) q^{53} - 2924 q^{55} + 13401 q^{57} + (31828 \zeta_{6} - 31828) q^{59} + 57654 \zeta_{6} q^{61} + ( - 3969 \zeta_{6} + 11907) q^{63} + 258 \zeta_{6} q^{65} + (60563 \zeta_{6} - 60563) q^{67} + 2016 q^{69} + 44834 q^{71} + (20821 \zeta_{6} - 20821) q^{73} - 38439 \zeta_{6} q^{75} + (3332 \zeta_{6} + 1666) q^{77} - 30531 \zeta_{6} q^{79} + (6561 \zeta_{6} - 6561) q^{81} - 110602 q^{83} - 163744 q^{85} + ( - 58572 \zeta_{6} + 58572) q^{87} + 58992 \zeta_{6} q^{89} + ( - 441 \zeta_{6} + 294) q^{91} + 15579 \zeta_{6} q^{93} + (128054 \zeta_{6} - 128054) q^{95} - 119846 q^{97} - 2754 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 9 q^{3} - 86 q^{5} - 49 q^{7} - 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 9 q^{3} - 86 q^{5} - 49 q^{7} - 81 q^{9} + 34 q^{11} - 6 q^{13} + 1548 q^{15} + 1904 q^{17} - 1489 q^{19} - 1764 q^{21} - 224 q^{23} - 4271 q^{25} + 1458 q^{27} - 13016 q^{29} + 1731 q^{31} + 306 q^{33} + 21070 q^{35} + 7633 q^{37} + 27 q^{39} + 30828 q^{41} - 36982 q^{43} - 6966 q^{45} + 18462 q^{47} - 31213 q^{49} + 17136 q^{51} + 19956 q^{53} - 5848 q^{55} + 26802 q^{57} - 31828 q^{59} + 57654 q^{61} + 19845 q^{63} + 258 q^{65} - 60563 q^{67} + 4032 q^{69} + 89668 q^{71} - 20821 q^{73} - 38439 q^{75} + 6664 q^{77} - 30531 q^{79} - 6561 q^{81} - 221204 q^{83} - 327488 q^{85} + 58572 q^{87} + 58992 q^{89} + 147 q^{91} + 15579 q^{93} - 128054 q^{95} - 239692 q^{97} - 5508 q^{99}+O(q^{100}) \) 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 −43.0000 74.4782i 0 −24.5000 + 127.306i 0 −40.5000 70.1481i 0
289.1 0 −4.50000 7.79423i 0 −43.0000 + 74.4782i 0 −24.5000 127.306i 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.a 2
4.b odd 2 1 42.6.e.b 2
7.c even 3 1 inner 336.6.q.a 2
12.b even 2 1 126.6.g.b 2
28.d even 2 1 294.6.e.m 2
28.f even 6 1 294.6.a.e 1
28.f even 6 1 294.6.e.m 2
28.g odd 6 1 42.6.e.b 2
28.g odd 6 1 294.6.a.d 1
84.j odd 6 1 882.6.a.y 1
84.n even 6 1 126.6.g.b 2
84.n even 6 1 882.6.a.m 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.6.e.b 2 4.b odd 2 1
42.6.e.b 2 28.g odd 6 1
126.6.g.b 2 12.b even 2 1
126.6.g.b 2 84.n even 6 1
294.6.a.d 1 28.g odd 6 1
294.6.a.e 1 28.f even 6 1
294.6.e.m 2 28.d even 2 1
294.6.e.m 2 28.f even 6 1
336.6.q.a 2 1.a even 1 1 trivial
336.6.q.a 2 7.c even 3 1 inner
882.6.a.m 1 84.n even 6 1
882.6.a.y 1 84.j odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 86T_{5} + 7396 \) 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
$5$ \( T^{2} + 86T + 7396 \) Copy content Toggle raw display
$7$ \( T^{2} + 49T + 16807 \) Copy content Toggle raw display
$11$ \( T^{2} - 34T + 1156 \) Copy content Toggle raw display
$13$ \( (T + 3)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 1904 T + 3625216 \) Copy content Toggle raw display
$19$ \( T^{2} + 1489 T + 2217121 \) Copy content Toggle raw display
$23$ \( T^{2} + 224T + 50176 \) Copy content Toggle raw display
$29$ \( (T + 6508)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 1731 T + 2996361 \) Copy content Toggle raw display
$37$ \( T^{2} - 7633 T + 58262689 \) Copy content Toggle raw display
$41$ \( (T - 15414)^{2} \) Copy content Toggle raw display
$43$ \( (T + 18491)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 18462 T + 340845444 \) Copy content Toggle raw display
$53$ \( T^{2} - 19956 T + 398241936 \) Copy content Toggle raw display
$59$ \( T^{2} + 31828 T + 1013021584 \) Copy content Toggle raw display
$61$ \( T^{2} - 57654 T + 3323983716 \) Copy content Toggle raw display
$67$ \( T^{2} + 60563 T + 3667876969 \) Copy content Toggle raw display
$71$ \( (T - 44834)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 20821 T + 433514041 \) Copy content Toggle raw display
$79$ \( T^{2} + 30531 T + 932141961 \) Copy content Toggle raw display
$83$ \( (T + 110602)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} - 58992 T + 3480056064 \) Copy content Toggle raw display
$97$ \( (T + 119846)^{2} \) Copy content Toggle raw display
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