# Properties

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

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