# 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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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