# Properties

 Label 336.4.a.l 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 42) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + 3q^{3} + 18q^{5} - 7q^{7} + 9q^{9} + O(q^{10})$$ $$q + 3q^{3} + 18q^{5} - 7q^{7} + 9q^{9} + 72q^{11} - 34q^{13} + 54q^{15} + 6q^{17} - 92q^{19} - 21q^{21} + 180q^{23} + 199q^{25} + 27q^{27} - 114q^{29} - 56q^{31} + 216q^{33} - 126q^{35} - 34q^{37} - 102q^{39} + 6q^{41} - 164q^{43} + 162q^{45} - 168q^{47} + 49q^{49} + 18q^{51} + 654q^{53} + 1296q^{55} - 276q^{57} + 492q^{59} - 250q^{61} - 63q^{63} - 612q^{65} + 124q^{67} + 540q^{69} - 36q^{71} + 1010q^{73} + 597q^{75} - 504q^{77} - 56q^{79} + 81q^{81} - 228q^{83} + 108q^{85} - 342q^{87} + 390q^{89} + 238q^{91} - 168q^{93} - 1656q^{95} - 70q^{97} + 648q^{99} + O(q^{100})$$

## 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.l 1
3.b odd 2 1 1008.4.a.b 1
4.b odd 2 1 42.4.a.a 1
7.b odd 2 1 2352.4.a.a 1
8.b even 2 1 1344.4.a.a 1
8.d odd 2 1 1344.4.a.o 1
12.b even 2 1 126.4.a.a 1
20.d odd 2 1 1050.4.a.g 1
20.e even 4 2 1050.4.g.a 2
28.d even 2 1 294.4.a.i 1
28.f even 6 2 294.4.e.b 2
28.g odd 6 2 294.4.e.c 2
84.h odd 2 1 882.4.a.g 1
84.j odd 6 2 882.4.g.o 2
84.n even 6 2 882.4.g.w 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.4.a.a 1 4.b odd 2 1
126.4.a.a 1 12.b even 2 1
294.4.a.i 1 28.d even 2 1
294.4.e.b 2 28.f even 6 2
294.4.e.c 2 28.g odd 6 2
336.4.a.l 1 1.a even 1 1 trivial
882.4.a.g 1 84.h odd 2 1
882.4.g.o 2 84.j odd 6 2
882.4.g.w 2 84.n even 6 2
1008.4.a.b 1 3.b odd 2 1
1050.4.a.g 1 20.d odd 2 1
1050.4.g.a 2 20.e even 4 2
1344.4.a.a 1 8.b even 2 1
1344.4.a.o 1 8.d odd 2 1
2352.4.a.a 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 $$S_{4}^{\mathrm{new}}(\Gamma_0(336))$$:

 $$T_{5} - 18$$ $$T_{11} - 72$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$-3 + T$$
$5$ $$-18 + T$$
$7$ $$7 + T$$
$11$ $$-72 + T$$
$13$ $$34 + T$$
$17$ $$-6 + T$$
$19$ $$92 + T$$
$23$ $$-180 + T$$
$29$ $$114 + T$$
$31$ $$56 + T$$
$37$ $$34 + T$$
$41$ $$-6 + T$$
$43$ $$164 + T$$
$47$ $$168 + T$$
$53$ $$-654 + T$$
$59$ $$-492 + T$$
$61$ $$250 + T$$
$67$ $$-124 + T$$
$71$ $$36 + T$$
$73$ $$-1010 + T$$
$79$ $$56 + T$$
$83$ $$228 + T$$
$89$ $$-390 + T$$
$97$ $$70 + T$$