Properties

Label 336.4.q.d
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: \(\Z[a_1, \ldots, a_{7}]\)
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.

\(f(q)\) \(=\) \( q + ( -3 + 3 \zeta_{6} ) q^{3} + 15 \zeta_{6} q^{5} + ( -21 + 7 \zeta_{6} ) q^{7} -9 \zeta_{6} q^{9} +O(q^{10})\) \( q + ( -3 + 3 \zeta_{6} ) q^{3} + 15 \zeta_{6} q^{5} + ( -21 + 7 \zeta_{6} ) q^{7} -9 \zeta_{6} q^{9} + ( -9 + 9 \zeta_{6} ) q^{11} -88 q^{13} -45 q^{15} + ( 84 - 84 \zeta_{6} ) q^{17} + 104 \zeta_{6} q^{19} + ( 42 - 63 \zeta_{6} ) q^{21} -84 \zeta_{6} q^{23} + ( -100 + 100 \zeta_{6} ) q^{25} + 27 q^{27} + 51 q^{29} + ( 185 - 185 \zeta_{6} ) q^{31} -27 \zeta_{6} q^{33} + ( -105 - 210 \zeta_{6} ) q^{35} -44 \zeta_{6} q^{37} + ( 264 - 264 \zeta_{6} ) q^{39} -168 q^{41} -326 q^{43} + ( 135 - 135 \zeta_{6} ) q^{45} -138 \zeta_{6} q^{47} + ( 392 - 245 \zeta_{6} ) q^{49} + 252 \zeta_{6} q^{51} + ( -639 + 639 \zeta_{6} ) q^{53} -135 q^{55} -312 q^{57} + ( 159 - 159 \zeta_{6} ) q^{59} -722 \zeta_{6} q^{61} + ( 63 + 126 \zeta_{6} ) q^{63} -1320 \zeta_{6} q^{65} + ( -166 + 166 \zeta_{6} ) q^{67} + 252 q^{69} -1086 q^{71} + ( -218 + 218 \zeta_{6} ) q^{73} -300 \zeta_{6} q^{75} + ( 126 - 189 \zeta_{6} ) q^{77} -583 \zeta_{6} q^{79} + ( -81 + 81 \zeta_{6} ) q^{81} + 597 q^{83} + 1260 q^{85} + ( -153 + 153 \zeta_{6} ) q^{87} + 1038 \zeta_{6} q^{89} + ( 1848 - 616 \zeta_{6} ) q^{91} + 555 \zeta_{6} q^{93} + ( -1560 + 1560 \zeta_{6} ) q^{95} -169 q^{97} + 81 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 3q^{3} + 15q^{5} - 35q^{7} - 9q^{9} + O(q^{10}) \) \( 2q - 3q^{3} + 15q^{5} - 35q^{7} - 9q^{9} - 9q^{11} - 176q^{13} - 90q^{15} + 84q^{17} + 104q^{19} + 21q^{21} - 84q^{23} - 100q^{25} + 54q^{27} + 102q^{29} + 185q^{31} - 27q^{33} - 420q^{35} - 44q^{37} + 264q^{39} - 336q^{41} - 652q^{43} + 135q^{45} - 138q^{47} + 539q^{49} + 252q^{51} - 639q^{53} - 270q^{55} - 624q^{57} + 159q^{59} - 722q^{61} + 252q^{63} - 1320q^{65} - 166q^{67} + 504q^{69} - 2172q^{71} - 218q^{73} - 300q^{75} + 63q^{77} - 583q^{79} - 81q^{81} + 1194q^{83} + 2520q^{85} - 153q^{87} + 1038q^{89} + 3080q^{91} + 555q^{93} - 1560q^{95} - 338q^{97} + 162q^{99} + O(q^{100}) \)

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 7.50000 + 12.9904i 0 −17.5000 + 6.06218i 0 −4.50000 7.79423i 0
289.1 0 −1.50000 2.59808i 0 7.50000 12.9904i 0 −17.5000 6.06218i 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.d 2
4.b odd 2 1 42.4.e.b 2
7.c even 3 1 inner 336.4.q.d 2
7.c even 3 1 2352.4.a.u 1
7.d odd 6 1 2352.4.a.q 1
12.b even 2 1 126.4.g.a 2
28.d even 2 1 294.4.e.e 2
28.f even 6 1 294.4.a.g 1
28.f even 6 1 294.4.e.e 2
28.g odd 6 1 42.4.e.b 2
28.g odd 6 1 294.4.a.a 1
84.h odd 2 1 882.4.g.l 2
84.j odd 6 1 882.4.a.h 1
84.j odd 6 1 882.4.g.l 2
84.n even 6 1 126.4.g.a 2
84.n even 6 1 882.4.a.r 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.4.e.b 2 4.b odd 2 1
42.4.e.b 2 28.g odd 6 1
126.4.g.a 2 12.b even 2 1
126.4.g.a 2 84.n even 6 1
294.4.a.a 1 28.g odd 6 1
294.4.a.g 1 28.f even 6 1
294.4.e.e 2 28.d even 2 1
294.4.e.e 2 28.f even 6 1
336.4.q.d 2 1.a even 1 1 trivial
336.4.q.d 2 7.c even 3 1 inner
882.4.a.h 1 84.j odd 6 1
882.4.a.r 1 84.n even 6 1
882.4.g.l 2 84.h odd 2 1
882.4.g.l 2 84.j odd 6 1
2352.4.a.q 1 7.d odd 6 1
2352.4.a.u 1 7.c even 3 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} - 15 T_{5} + 225 \) acting on \(S_{4}^{\mathrm{new}}(336, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 9 + 3 T + T^{2} \)
$5$ \( 225 - 15 T + T^{2} \)
$7$ \( 343 + 35 T + T^{2} \)
$11$ \( 81 + 9 T + T^{2} \)
$13$ \( ( 88 + T )^{2} \)
$17$ \( 7056 - 84 T + T^{2} \)
$19$ \( 10816 - 104 T + T^{2} \)
$23$ \( 7056 + 84 T + T^{2} \)
$29$ \( ( -51 + T )^{2} \)
$31$ \( 34225 - 185 T + T^{2} \)
$37$ \( 1936 + 44 T + T^{2} \)
$41$ \( ( 168 + T )^{2} \)
$43$ \( ( 326 + T )^{2} \)
$47$ \( 19044 + 138 T + T^{2} \)
$53$ \( 408321 + 639 T + T^{2} \)
$59$ \( 25281 - 159 T + T^{2} \)
$61$ \( 521284 + 722 T + T^{2} \)
$67$ \( 27556 + 166 T + T^{2} \)
$71$ \( ( 1086 + T )^{2} \)
$73$ \( 47524 + 218 T + T^{2} \)
$79$ \( 339889 + 583 T + T^{2} \)
$83$ \( ( -597 + T )^{2} \)
$89$ \( 1077444 - 1038 T + T^{2} \)
$97$ \( ( 169 + T )^{2} \)
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