# Properties

 Label 336.4.a.d 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

 $$f(q)$$ $$=$$ $$q - 3q^{3} + 2q^{5} + 7q^{7} + 9q^{9} + O(q^{10})$$ $$q - 3q^{3} + 2q^{5} + 7q^{7} + 9q^{9} + 8q^{11} - 42q^{13} - 6q^{15} - 2q^{17} + 124q^{19} - 21q^{21} - 76q^{23} - 121q^{25} - 27q^{27} + 254q^{29} + 72q^{31} - 24q^{33} + 14q^{35} + 398q^{37} + 126q^{39} + 462q^{41} - 212q^{43} + 18q^{45} + 264q^{47} + 49q^{49} + 6q^{51} - 162q^{53} + 16q^{55} - 372q^{57} + 772q^{59} + 30q^{61} + 63q^{63} - 84q^{65} + 764q^{67} + 228q^{69} + 236q^{71} + 418q^{73} + 363q^{75} + 56q^{77} - 552q^{79} + 81q^{81} - 1036q^{83} - 4q^{85} - 762q^{87} + 30q^{89} - 294q^{91} - 216q^{93} + 248q^{95} - 1190q^{97} + 72q^{99} + O(q^{100})$$

## 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 2.00000 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.d 1
3.b odd 2 1 1008.4.a.j 1
4.b odd 2 1 42.4.a.b 1
7.b odd 2 1 2352.4.a.ba 1
8.b even 2 1 1344.4.a.t 1
8.d odd 2 1 1344.4.a.f 1
12.b even 2 1 126.4.a.c 1
20.d odd 2 1 1050.4.a.d 1
20.e even 4 2 1050.4.g.n 2
28.d even 2 1 294.4.a.h 1
28.f even 6 2 294.4.e.d 2
28.g odd 6 2 294.4.e.a 2
84.h odd 2 1 882.4.a.d 1
84.j odd 6 2 882.4.g.r 2
84.n even 6 2 882.4.g.s 2

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

 $$T_{5} - 2$$ $$T_{11} - 8$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$3 + T$$
$5$ $$-2 + T$$
$7$ $$-7 + T$$
$11$ $$-8 + T$$
$13$ $$42 + T$$
$17$ $$2 + T$$
$19$ $$-124 + T$$
$23$ $$76 + T$$
$29$ $$-254 + T$$
$31$ $$-72 + T$$
$37$ $$-398 + T$$
$41$ $$-462 + T$$
$43$ $$212 + T$$
$47$ $$-264 + T$$
$53$ $$162 + T$$
$59$ $$-772 + T$$
$61$ $$-30 + T$$
$67$ $$-764 + T$$
$71$ $$-236 + T$$
$73$ $$-418 + T$$
$79$ $$552 + T$$
$83$ $$1036 + T$$
$89$ $$-30 + T$$
$97$ $$1190 + T$$