# 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

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1216,4,Mod(1,1216)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1216, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1216.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

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

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 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

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

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

## 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

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 $$S_{4}^{\mathrm{new}}(\Gamma_0(1216))$$:

 $$T_{3} - 5$$ T3 - 5 $$T_{5} - 12$$ T5 - 12

## Hecke characteristic polynomials

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