# Properties

 Label 336.4.a.f Level $336$ Weight $4$ Character orbit 336.a Self dual yes Analytic conductor $19.825$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## 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 21) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + 3q^{3} - 18q^{5} - 7q^{7} + 9q^{9} + O(q^{10})$$ $$q + 3q^{3} - 18q^{5} - 7q^{7} + 9q^{9} + 36q^{11} - 34q^{13} - 54q^{15} + 42q^{17} + 124q^{19} - 21q^{21} + 199q^{25} + 27q^{27} + 102q^{29} + 160q^{31} + 108q^{33} + 126q^{35} + 398q^{37} - 102q^{39} - 318q^{41} + 268q^{43} - 162q^{45} - 240q^{47} + 49q^{49} + 126q^{51} - 498q^{53} - 648q^{55} + 372q^{57} + 132q^{59} + 398q^{61} - 63q^{63} + 612q^{65} - 92q^{67} + 720q^{71} - 502q^{73} + 597q^{75} - 252q^{77} + 1024q^{79} + 81q^{81} + 204q^{83} - 756q^{85} + 306q^{87} + 354q^{89} + 238q^{91} + 480q^{93} - 2232q^{95} - 286q^{97} + 324q^{99} + O(q^{100})$$

## 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 −18.0000 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.f 1
3.b odd 2 1 1008.4.a.v 1
4.b odd 2 1 21.4.a.a 1
7.b odd 2 1 2352.4.a.r 1
8.b even 2 1 1344.4.a.n 1
8.d odd 2 1 1344.4.a.ba 1
12.b even 2 1 63.4.a.c 1
20.d odd 2 1 525.4.a.g 1
20.e even 4 2 525.4.d.c 2
28.d even 2 1 147.4.a.c 1
28.f even 6 2 147.4.e.g 2
28.g odd 6 2 147.4.e.i 2
60.h even 2 1 1575.4.a.b 1
84.h odd 2 1 441.4.a.j 1
84.j odd 6 2 441.4.e.d 2
84.n even 6 2 441.4.e.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.4.a.a 1 4.b odd 2 1
63.4.a.c 1 12.b even 2 1
147.4.a.c 1 28.d even 2 1
147.4.e.g 2 28.f even 6 2
147.4.e.i 2 28.g odd 6 2
336.4.a.f 1 1.a even 1 1 trivial
441.4.a.j 1 84.h odd 2 1
441.4.e.b 2 84.n even 6 2
441.4.e.d 2 84.j odd 6 2
525.4.a.g 1 20.d odd 2 1
525.4.d.c 2 20.e even 4 2
1008.4.a.v 1 3.b odd 2 1
1344.4.a.n 1 8.b even 2 1
1344.4.a.ba 1 8.d odd 2 1
1575.4.a.b 1 60.h even 2 1
2352.4.a.r 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 $$S_{4}^{\mathrm{new}}(\Gamma_0(336))$$:

 $$T_{5} + 18$$ $$T_{11} - 36$$