# Properties

 Label 336.4.a.h Level $336$ Weight $4$ Character orbit 336.a Self dual yes Analytic conductor $19.825$ Analytic rank $1$ 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: $$1$$ 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

 $$f(q)$$ $$=$$ $$q + 3q^{3} - 4q^{5} + 7q^{7} + 9q^{9} + O(q^{10})$$ $$q + 3q^{3} - 4q^{5} + 7q^{7} + 9q^{9} - 62q^{11} - 62q^{13} - 12q^{15} + 84q^{17} - 100q^{19} + 21q^{21} + 42q^{23} - 109q^{25} + 27q^{27} - 10q^{29} + 48q^{31} - 186q^{33} - 28q^{35} - 246q^{37} - 186q^{39} - 248q^{41} - 68q^{43} - 36q^{45} - 324q^{47} + 49q^{49} + 252q^{51} + 258q^{53} + 248q^{55} - 300q^{57} - 120q^{59} + 622q^{61} + 63q^{63} + 248q^{65} - 904q^{67} + 126q^{69} + 678q^{71} - 642q^{73} - 327q^{75} - 434q^{77} - 740q^{79} + 81q^{81} - 468q^{83} - 336q^{85} - 30q^{87} + 200q^{89} - 434q^{91} + 144q^{93} + 400q^{95} - 1266q^{97} - 558q^{99} + O(q^{100})$$

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

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

 $$T_{5} + 4$$ $$T_{11} + 62$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$-3 + T$$
$5$ $$4 + T$$
$7$ $$-7 + T$$
$11$ $$62 + T$$
$13$ $$62 + T$$
$17$ $$-84 + T$$
$19$ $$100 + T$$
$23$ $$-42 + T$$
$29$ $$10 + T$$
$31$ $$-48 + T$$
$37$ $$246 + T$$
$41$ $$248 + T$$
$43$ $$68 + T$$
$47$ $$324 + T$$
$53$ $$-258 + T$$
$59$ $$120 + T$$
$61$ $$-622 + T$$
$67$ $$904 + T$$
$71$ $$-678 + T$$
$73$ $$642 + T$$
$79$ $$740 + T$$
$83$ $$468 + T$$
$89$ $$-200 + T$$
$97$ $$1266 + T$$