# Properties

 Label 336.4.a.j Level $336$ Weight $4$ Character orbit 336.a Self dual yes Analytic conductor $19.825$ Analytic rank $0$ Dimension $1$ 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: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 168) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + 3q^{3} + 4q^{5} + 7q^{7} + 9q^{9} + O(q^{10})$$ $$q + 3q^{3} + 4q^{5} + 7q^{7} + 9q^{9} + 26q^{11} + 2q^{13} + 12q^{15} - 36q^{17} + 76q^{19} + 21q^{21} + 114q^{23} - 109q^{25} + 27q^{27} + 6q^{29} + 256q^{31} + 78q^{33} + 28q^{35} - 86q^{37} + 6q^{39} + 160q^{41} + 220q^{43} + 36q^{45} - 308q^{47} + 49q^{49} - 108q^{51} + 258q^{53} + 104q^{55} + 228q^{57} - 264q^{59} + 606q^{61} + 63q^{63} + 8q^{65} + 520q^{67} + 342q^{69} + 286q^{71} - 530q^{73} - 327q^{75} + 182q^{77} + 44q^{79} + 81q^{81} - 1012q^{83} - 144q^{85} + 18q^{87} + 768q^{89} + 14q^{91} + 768q^{93} + 304q^{95} + 222q^{97} + 234q^{99} + O(q^{100})$$

## 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
 0
0 3.00000 0 4.00000 0 7.00000 0 9.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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.j 1
3.b odd 2 1 1008.4.a.i 1
4.b odd 2 1 168.4.a.c 1
7.b odd 2 1 2352.4.a.h 1
8.b even 2 1 1344.4.a.e 1
8.d odd 2 1 1344.4.a.s 1
12.b even 2 1 504.4.a.b 1
28.d even 2 1 1176.4.a.j 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.4.a.c 1 4.b odd 2 1
336.4.a.j 1 1.a even 1 1 trivial
504.4.a.b 1 12.b even 2 1
1008.4.a.i 1 3.b odd 2 1
1176.4.a.j 1 28.d even 2 1
1344.4.a.e 1 8.b even 2 1
1344.4.a.s 1 8.d odd 2 1
2352.4.a.h 1 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 $$S_{4}^{\mathrm{new}}(\Gamma_0(336))$$:

 $$T_{5} - 4$$ $$T_{11} - 26$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$-3 + T$$
$5$ $$-4 + T$$
$7$ $$-7 + T$$
$11$ $$-26 + T$$
$13$ $$-2 + T$$
$17$ $$36 + T$$
$19$ $$-76 + T$$
$23$ $$-114 + T$$
$29$ $$-6 + T$$
$31$ $$-256 + T$$
$37$ $$86 + T$$
$41$ $$-160 + T$$
$43$ $$-220 + T$$
$47$ $$308 + T$$
$53$ $$-258 + T$$
$59$ $$264 + T$$
$61$ $$-606 + T$$
$67$ $$-520 + T$$
$71$ $$-286 + T$$
$73$ $$530 + T$$
$79$ $$-44 + T$$
$83$ $$1012 + T$$
$89$ $$-768 + T$$
$97$ $$-222 + T$$
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