Properties

Label 336.4.a.m
Level $336$
Weight $4$
Character orbit 336.a
Self dual yes
Analytic conductor $19.825$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 336.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(19.8246417619\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{57}) \)
Defining polynomial: \(x^{2} - x - 14\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 21)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{57}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -3 q^{3} + ( 3 + \beta ) q^{5} -7 q^{7} + 9 q^{9} +O(q^{10})\) \( q -3 q^{3} + ( 3 + \beta ) q^{5} -7 q^{7} + 9 q^{9} + ( 3 - 5 \beta ) q^{11} + ( 8 - 6 \beta ) q^{13} + ( -9 - 3 \beta ) q^{15} + ( -3 - \beta ) q^{17} + ( -32 + 12 \beta ) q^{19} + 21 q^{21} + ( -3 + 17 \beta ) q^{23} + ( -59 + 6 \beta ) q^{25} -27 q^{27} + ( -126 + 12 \beta ) q^{29} + ( -20 - 36 \beta ) q^{31} + ( -9 + 15 \beta ) q^{33} + ( -21 - 7 \beta ) q^{35} + ( -124 - 18 \beta ) q^{37} + ( -24 + 18 \beta ) q^{39} + ( -225 - 15 \beta ) q^{41} + ( -188 + 24 \beta ) q^{43} + ( 27 + 9 \beta ) q^{45} + ( 6 - 34 \beta ) q^{47} + 49 q^{49} + ( 9 + 3 \beta ) q^{51} + ( -552 + 2 \beta ) q^{53} + ( -276 - 12 \beta ) q^{55} + ( 96 - 36 \beta ) q^{57} + ( -402 + 58 \beta ) q^{59} + ( -214 + 36 \beta ) q^{61} -63 q^{63} + ( -318 - 10 \beta ) q^{65} + ( -74 - 54 \beta ) q^{67} + ( 9 - 51 \beta ) q^{69} + ( -477 + 15 \beta ) q^{71} + ( 536 + 6 \beta ) q^{73} + ( 177 - 18 \beta ) q^{75} + ( -21 + 35 \beta ) q^{77} + ( 286 + 54 \beta ) q^{79} + 81 q^{81} + ( -972 - 48 \beta ) q^{83} + ( -66 - 6 \beta ) q^{85} + ( 378 - 36 \beta ) q^{87} + ( 183 - 71 \beta ) q^{89} + ( -56 + 42 \beta ) q^{91} + ( 60 + 108 \beta ) q^{93} + ( 588 + 4 \beta ) q^{95} + ( 404 + 138 \beta ) q^{97} + ( 27 - 45 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 6q^{3} + 6q^{5} - 14q^{7} + 18q^{9} + O(q^{10}) \) \( 2q - 6q^{3} + 6q^{5} - 14q^{7} + 18q^{9} + 6q^{11} + 16q^{13} - 18q^{15} - 6q^{17} - 64q^{19} + 42q^{21} - 6q^{23} - 118q^{25} - 54q^{27} - 252q^{29} - 40q^{31} - 18q^{33} - 42q^{35} - 248q^{37} - 48q^{39} - 450q^{41} - 376q^{43} + 54q^{45} + 12q^{47} + 98q^{49} + 18q^{51} - 1104q^{53} - 552q^{55} + 192q^{57} - 804q^{59} - 428q^{61} - 126q^{63} - 636q^{65} - 148q^{67} + 18q^{69} - 954q^{71} + 1072q^{73} + 354q^{75} - 42q^{77} + 572q^{79} + 162q^{81} - 1944q^{83} - 132q^{85} + 756q^{87} + 366q^{89} - 112q^{91} + 120q^{93} + 1176q^{95} + 808q^{97} + 54q^{99} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−3.27492
4.27492
0 −3.00000 0 −4.54983 0 −7.00000 0 9.00000 0
1.2 0 −3.00000 0 10.5498 0 −7.00000 0 9.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(7\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 336.4.a.m 2
3.b odd 2 1 1008.4.a.ba 2
4.b odd 2 1 21.4.a.c 2
7.b odd 2 1 2352.4.a.bz 2
8.b even 2 1 1344.4.a.bo 2
8.d odd 2 1 1344.4.a.bg 2
12.b even 2 1 63.4.a.e 2
20.d odd 2 1 525.4.a.n 2
20.e even 4 2 525.4.d.g 4
28.d even 2 1 147.4.a.i 2
28.f even 6 2 147.4.e.m 4
28.g odd 6 2 147.4.e.l 4
60.h even 2 1 1575.4.a.p 2
84.h odd 2 1 441.4.a.r 2
84.j odd 6 2 441.4.e.p 4
84.n even 6 2 441.4.e.q 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.4.a.c 2 4.b odd 2 1
63.4.a.e 2 12.b even 2 1
147.4.a.i 2 28.d even 2 1
147.4.e.l 4 28.g odd 6 2
147.4.e.m 4 28.f even 6 2
336.4.a.m 2 1.a even 1 1 trivial
441.4.a.r 2 84.h odd 2 1
441.4.e.p 4 84.j odd 6 2
441.4.e.q 4 84.n even 6 2
525.4.a.n 2 20.d odd 2 1
525.4.d.g 4 20.e even 4 2
1008.4.a.ba 2 3.b odd 2 1
1344.4.a.bg 2 8.d odd 2 1
1344.4.a.bo 2 8.b even 2 1
1575.4.a.p 2 60.h even 2 1
2352.4.a.bz 2 7.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(336))\):

\( T_{5}^{2} - 6 T_{5} - 48 \)
\( T_{11}^{2} - 6 T_{11} - 1416 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( ( 3 + T )^{2} \)
$5$ \( -48 - 6 T + T^{2} \)
$7$ \( ( 7 + T )^{2} \)
$11$ \( -1416 - 6 T + T^{2} \)
$13$ \( -1988 - 16 T + T^{2} \)
$17$ \( -48 + 6 T + T^{2} \)
$19$ \( -7184 + 64 T + T^{2} \)
$23$ \( -16464 + 6 T + T^{2} \)
$29$ \( 7668 + 252 T + T^{2} \)
$31$ \( -73472 + 40 T + T^{2} \)
$37$ \( -3092 + 248 T + T^{2} \)
$41$ \( 37800 + 450 T + T^{2} \)
$43$ \( 2512 + 376 T + T^{2} \)
$47$ \( -65856 - 12 T + T^{2} \)
$53$ \( 304476 + 1104 T + T^{2} \)
$59$ \( -30144 + 804 T + T^{2} \)
$61$ \( -28076 + 428 T + T^{2} \)
$67$ \( -160736 + 148 T + T^{2} \)
$71$ \( 214704 + 954 T + T^{2} \)
$73$ \( 285244 - 1072 T + T^{2} \)
$79$ \( -84416 - 572 T + T^{2} \)
$83$ \( 813456 + 1944 T + T^{2} \)
$89$ \( -253848 - 366 T + T^{2} \)
$97$ \( -922292 - 808 T + T^{2} \)
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