Properties

Label 336.3.f.c
Level $336$
Weight $3$
Character orbit 336.f
Analytic conductor $9.155$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 336.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(9.15533688251\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
Defining polynomial: \(x^{4} + 2 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2}\cdot 3 \)
Twist minimal: no (minimal twist has level 42)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{2} q^{3} + ( -\beta_{1} + 2 \beta_{2} ) q^{5} + ( -2 + 2 \beta_{1} + \beta_{2} + \beta_{3} ) q^{7} -3 q^{9} +O(q^{10})\) \( q -\beta_{2} q^{3} + ( -\beta_{1} + 2 \beta_{2} ) q^{5} + ( -2 + 2 \beta_{1} + \beta_{2} + \beta_{3} ) q^{7} -3 q^{9} + ( 6 + \beta_{3} ) q^{11} + ( 2 \beta_{1} - 8 \beta_{2} ) q^{13} + ( 6 - \beta_{3} ) q^{15} + ( 11 \beta_{1} + 2 \beta_{2} ) q^{17} + ( 8 \beta_{1} - 2 \beta_{2} ) q^{19} + ( 3 - 3 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{21} + ( -6 + 3 \beta_{3} ) q^{23} + ( 7 + 4 \beta_{3} ) q^{25} + 3 \beta_{2} q^{27} + 30 q^{29} + ( -12 \beta_{1} - 12 \beta_{2} ) q^{31} + ( -3 \beta_{1} - 6 \beta_{2} ) q^{33} + ( 6 + 8 \beta_{1} - 10 \beta_{2} - 3 \beta_{3} ) q^{35} + ( -20 + 12 \beta_{3} ) q^{37} + ( -24 + 2 \beta_{3} ) q^{39} + ( 7 \beta_{1} - 14 \beta_{2} ) q^{41} + ( 32 + 10 \beta_{3} ) q^{43} + ( 3 \beta_{1} - 6 \beta_{2} ) q^{45} + ( 4 \beta_{1} + 28 \beta_{2} ) q^{47} + ( -5 - 2 \beta_{1} + 20 \beta_{2} - 8 \beta_{3} ) q^{49} + ( 6 + 11 \beta_{3} ) q^{51} + ( -54 - 4 \beta_{3} ) q^{53} + 6 \beta_{2} q^{55} + ( -6 + 8 \beta_{3} ) q^{57} + ( 20 \beta_{1} - 28 \beta_{2} ) q^{59} + ( 4 \beta_{1} - 4 \beta_{2} ) q^{61} + ( 6 - 6 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} ) q^{63} + ( 60 - 12 \beta_{3} ) q^{65} + ( -44 + 4 \beta_{3} ) q^{67} + ( -9 \beta_{1} + 6 \beta_{2} ) q^{69} + ( 30 - 19 \beta_{3} ) q^{71} + ( -26 \beta_{1} - 4 \beta_{2} ) q^{73} + ( -12 \beta_{1} - 7 \beta_{2} ) q^{75} + ( 6 + 15 \beta_{1} + 18 \beta_{2} + 4 \beta_{3} ) q^{77} + ( -32 - 24 \beta_{3} ) q^{79} + 9 q^{81} + ( 20 \beta_{1} + 32 \beta_{2} ) q^{83} + ( 54 - 20 \beta_{3} ) q^{85} -30 \beta_{2} q^{87} + ( 21 \beta_{1} + 54 \beta_{2} ) q^{89} + ( -28 \beta_{1} + 28 \beta_{2} + 14 \beta_{3} ) q^{91} + ( -36 - 12 \beta_{3} ) q^{93} + ( 60 - 18 \beta_{3} ) q^{95} + ( -10 \beta_{1} - 44 \beta_{2} ) q^{97} + ( -18 - 3 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 8q^{7} - 12q^{9} + O(q^{10}) \) \( 4q - 8q^{7} - 12q^{9} + 24q^{11} + 24q^{15} + 12q^{21} - 24q^{23} + 28q^{25} + 120q^{29} + 24q^{35} - 80q^{37} - 96q^{39} + 128q^{43} - 20q^{49} + 24q^{51} - 216q^{53} - 24q^{57} + 24q^{63} + 240q^{65} - 176q^{67} + 120q^{71} + 24q^{77} - 128q^{79} + 36q^{81} + 216q^{85} - 144q^{93} + 240q^{95} - 72q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 2 x^{2} + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{3} + 4 \nu \)\()/2\)
\(\beta_{2}\)\(=\)\( \nu^{2} + 1 \)
\(\beta_{3}\)\(=\)\((\)\( -3 \nu^{3} \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + 3 \beta_{1}\)\()/6\)
\(\nu^{2}\)\(=\)\(\beta_{2} - 1\)
\(\nu^{3}\)\(=\)\((\)\(-2 \beta_{3}\)\()/3\)

Character values

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

\(n\) \(85\) \(113\) \(127\) \(241\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
97.1
0.707107 + 1.22474i
−0.707107 1.22474i
−0.707107 + 1.22474i
0.707107 1.22474i
0 1.73205i 0 1.01461i 0 2.24264 + 6.63103i 0 −3.00000 0
97.2 0 1.73205i 0 5.91359i 0 −6.24264 3.16693i 0 −3.00000 0
97.3 0 1.73205i 0 5.91359i 0 −6.24264 + 3.16693i 0 −3.00000 0
97.4 0 1.73205i 0 1.01461i 0 2.24264 6.63103i 0 −3.00000 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.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 336.3.f.c 4
3.b odd 2 1 1008.3.f.g 4
4.b odd 2 1 42.3.c.a 4
7.b odd 2 1 inner 336.3.f.c 4
8.b even 2 1 1344.3.f.e 4
8.d odd 2 1 1344.3.f.f 4
12.b even 2 1 126.3.c.b 4
20.d odd 2 1 1050.3.f.a 4
20.e even 4 2 1050.3.h.a 8
21.c even 2 1 1008.3.f.g 4
28.d even 2 1 42.3.c.a 4
28.f even 6 1 294.3.g.b 4
28.f even 6 1 294.3.g.c 4
28.g odd 6 1 294.3.g.b 4
28.g odd 6 1 294.3.g.c 4
56.e even 2 1 1344.3.f.f 4
56.h odd 2 1 1344.3.f.e 4
84.h odd 2 1 126.3.c.b 4
84.j odd 6 1 882.3.n.a 4
84.j odd 6 1 882.3.n.d 4
84.n even 6 1 882.3.n.a 4
84.n even 6 1 882.3.n.d 4
140.c even 2 1 1050.3.f.a 4
140.j odd 4 2 1050.3.h.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.3.c.a 4 4.b odd 2 1
42.3.c.a 4 28.d even 2 1
126.3.c.b 4 12.b even 2 1
126.3.c.b 4 84.h odd 2 1
294.3.g.b 4 28.f even 6 1
294.3.g.b 4 28.g odd 6 1
294.3.g.c 4 28.f even 6 1
294.3.g.c 4 28.g odd 6 1
336.3.f.c 4 1.a even 1 1 trivial
336.3.f.c 4 7.b odd 2 1 inner
882.3.n.a 4 84.j odd 6 1
882.3.n.a 4 84.n even 6 1
882.3.n.d 4 84.j odd 6 1
882.3.n.d 4 84.n even 6 1
1008.3.f.g 4 3.b odd 2 1
1008.3.f.g 4 21.c even 2 1
1050.3.f.a 4 20.d odd 2 1
1050.3.f.a 4 140.c even 2 1
1050.3.h.a 8 20.e even 4 2
1050.3.h.a 8 140.j odd 4 2
1344.3.f.e 4 8.b even 2 1
1344.3.f.e 4 56.h odd 2 1
1344.3.f.f 4 8.d odd 2 1
1344.3.f.f 4 56.e even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(336, [\chi])\):

\( T_{5}^{4} + 36 T_{5}^{2} + 36 \)
\( T_{11}^{2} - 12 T_{11} + 18 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( ( 3 + T^{2} )^{2} \)
$5$ \( 36 + 36 T^{2} + T^{4} \)
$7$ \( 2401 + 392 T + 42 T^{2} + 8 T^{3} + T^{4} \)
$11$ \( ( 18 - 12 T + T^{2} )^{2} \)
$13$ \( 28224 + 432 T^{2} + T^{4} \)
$17$ \( 509796 + 1476 T^{2} + T^{4} \)
$19$ \( 138384 + 792 T^{2} + T^{4} \)
$23$ \( ( -126 + 12 T + T^{2} )^{2} \)
$29$ \( ( -30 + T )^{4} \)
$31$ \( 186624 + 2592 T^{2} + T^{4} \)
$37$ \( ( -2192 + 40 T + T^{2} )^{2} \)
$41$ \( 86436 + 1764 T^{2} + T^{4} \)
$43$ \( ( -776 - 64 T + T^{2} )^{2} \)
$47$ \( 5089536 + 4896 T^{2} + T^{4} \)
$53$ \( ( 2628 + 108 T + T^{2} )^{2} \)
$59$ \( 2304 + 9504 T^{2} + T^{4} \)
$61$ \( 2304 + 288 T^{2} + T^{4} \)
$67$ \( ( 1648 + 88 T + T^{2} )^{2} \)
$71$ \( ( -5598 - 60 T + T^{2} )^{2} \)
$73$ \( 16064064 + 8208 T^{2} + T^{4} \)
$79$ \( ( -9344 + 64 T + T^{2} )^{2} \)
$83$ \( 451584 + 10944 T^{2} + T^{4} \)
$89$ \( 37234404 + 22788 T^{2} + T^{4} \)
$97$ \( 27123264 + 12816 T^{2} + T^{4} \)
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