Properties

 Label 1216.4.a.f Level $1216$ Weight $4$ Character orbit 1216.a Self dual yes Analytic conductor $71.746$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$1216 = 2^{6} \cdot 19$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1216.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$71.7463225670$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 19) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

 $$f(q)$$ $$=$$ $$q + 5q^{3} + 12q^{5} + 11q^{7} - 2q^{9} + O(q^{10})$$ $$q + 5q^{3} + 12q^{5} + 11q^{7} - 2q^{9} + 54q^{11} - 11q^{13} + 60q^{15} - 93q^{17} - 19q^{19} + 55q^{21} + 183q^{23} + 19q^{25} - 145q^{27} + 249q^{29} + 56q^{31} + 270q^{33} + 132q^{35} + 250q^{37} - 55q^{39} + 240q^{41} + 196q^{43} - 24q^{45} - 168q^{47} - 222q^{49} - 465q^{51} - 435q^{53} + 648q^{55} - 95q^{57} - 195q^{59} + 358q^{61} - 22q^{63} - 132q^{65} + 961q^{67} + 915q^{69} - 246q^{71} + 353q^{73} + 95q^{75} + 594q^{77} - 34q^{79} - 671q^{81} - 234q^{83} - 1116q^{85} + 1245q^{87} - 168q^{89} - 121q^{91} + 280q^{93} - 228q^{95} + 758q^{97} - 108q^{99} + O(q^{100})$$

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 5.00000 0 12.0000 0 11.0000 0 −2.00000 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
$$2$$ $$1$$
$$19$$ $$1$$

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 1216.4.a.f 1
4.b odd 2 1 1216.4.a.a 1
8.b even 2 1 19.4.a.a 1
8.d odd 2 1 304.4.a.b 1
24.h odd 2 1 171.4.a.d 1
40.f even 2 1 475.4.a.e 1
40.i odd 4 2 475.4.b.c 2
56.h odd 2 1 931.4.a.a 1
88.b odd 2 1 2299.4.a.b 1
152.g odd 2 1 361.4.a.b 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
19.4.a.a 1 8.b even 2 1
171.4.a.d 1 24.h odd 2 1
304.4.a.b 1 8.d odd 2 1
361.4.a.b 1 152.g odd 2 1
475.4.a.e 1 40.f even 2 1
475.4.b.c 2 40.i odd 4 2
931.4.a.a 1 56.h odd 2 1
1216.4.a.a 1 4.b odd 2 1
1216.4.a.f 1 1.a even 1 1 trivial
2299.4.a.b 1 88.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 $$S_{4}^{\mathrm{new}}(\Gamma_0(1216))$$:

 $$T_{3} - 5$$ $$T_{5} - 12$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$-5 + T$$
$5$ $$-12 + T$$
$7$ $$-11 + T$$
$11$ $$-54 + T$$
$13$ $$11 + T$$
$17$ $$93 + T$$
$19$ $$19 + T$$
$23$ $$-183 + T$$
$29$ $$-249 + T$$
$31$ $$-56 + T$$
$37$ $$-250 + T$$
$41$ $$-240 + T$$
$43$ $$-196 + T$$
$47$ $$168 + T$$
$53$ $$435 + T$$
$59$ $$195 + T$$
$61$ $$-358 + T$$
$67$ $$-961 + T$$
$71$ $$246 + T$$
$73$ $$-353 + T$$
$79$ $$34 + T$$
$83$ $$234 + T$$
$89$ $$168 + T$$
$97$ $$-758 + T$$