# Properties

 Label 1216.2.a.u Level $1216$ Weight $2$ Character orbit 1216.a Self dual yes Analytic conductor $9.710$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1216,2,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, 2, names="a")

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

chi := DirichletCharacter("1216.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1216 = 2^{6} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ 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: $$9.70980888579$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.961.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 10x + 8$$ x^3 - x^2 - 10*x + 8 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 152) 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)

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_1 q^{3} + \beta_{2} q^{5} + (\beta_{2} - \beta_1 + 2) q^{7} + (2 \beta_{2} - \beta_1 + 5) q^{9}+O(q^{10})$$ q - b1 * q^3 + b2 * q^5 + (b2 - b1 + 2) * q^7 + (2*b2 - b1 + 5) * q^9 $$q - \beta_1 q^{3} + \beta_{2} q^{5} + (\beta_{2} - \beta_1 + 2) q^{7} + (2 \beta_{2} - \beta_1 + 5) q^{9} + (\beta_{2} + 2) q^{11} + (\beta_1 - 2) q^{13} + ( - 2 \beta_{2} - 4) q^{15} + ( - \beta_{2} + \beta_1) q^{17} + q^{19} + ( - 3 \beta_1 + 4) q^{21} + (2 \beta_{2} - 3 \beta_1) q^{23} + ( - \beta_{2} + 2 \beta_1 + 1) q^{25} + ( - 2 \beta_{2} - 3 \beta_1) q^{27} + ( - 2 \beta_{2} + \beta_1 + 2) q^{29} + ( - 2 \beta_{2} - 2 \beta_1 - 4) q^{33} + ( - \beta_{2} + 2 \beta_1 + 2) q^{35} + 2 q^{37} + ( - 2 \beta_{2} + 3 \beta_1 - 8) q^{39} + (2 \beta_1 + 2) q^{41} + (\beta_{2} + 2 \beta_1 - 6) q^{43} + (\beta_{2} + 4 \beta_1 + 8) q^{45} + ( - 3 \beta_{2} + 2 \beta_1 - 2) q^{47} + (\beta_{2} - 3 \beta_1 + 3) q^{49} + (\beta_1 - 4) q^{51} + ( - 4 \beta_{2} + \beta_1 - 2) q^{53} + (\beta_{2} + 2 \beta_1 + 6) q^{55} - \beta_1 q^{57} + ( - \beta_1 + 8) q^{59} + (\beta_{2} - 2 \beta_1) q^{61} + (3 \beta_{2} - 4 \beta_1 + 18) q^{63} + 4 q^{65} + (2 \beta_{2} - \beta_1 - 4) q^{67} + (2 \beta_{2} - 3 \beta_1 + 16) q^{69} + (2 \beta_{2} + 2 \beta_1 - 4) q^{71} + ( - \beta_{2} + 3 \beta_1) q^{73} + ( - 2 \beta_{2} + \beta_1 - 12) q^{75} + (\beta_{2} + 6) q^{77} + (2 \beta_1 + 8) q^{79} + (4 \beta_{2} + 17) q^{81} + (4 \beta_{2} - 2 \beta_1 + 4) q^{83} + (3 \beta_{2} - 2 \beta_1 - 2) q^{85} + (2 \beta_{2} - \beta_1) q^{87} + ( - 2 \beta_{2} - 2 \beta_1 + 6) q^{89} + ( - 2 \beta_{2} + 5 \beta_1 - 8) q^{91} + \beta_{2} q^{95} + (2 \beta_{2} - 2) q^{97} + (5 \beta_{2} + 2 \beta_1 + 18) q^{99}+O(q^{100})$$ q - b1 * q^3 + b2 * q^5 + (b2 - b1 + 2) * q^7 + (2*b2 - b1 + 5) * q^9 + (b2 + 2) * q^11 + (b1 - 2) * q^13 + (-2*b2 - 4) * q^15 + (-b2 + b1) * q^17 + q^19 + (-3*b1 + 4) * q^21 + (2*b2 - 3*b1) * q^23 + (-b2 + 2*b1 + 1) * q^25 + (-2*b2 - 3*b1) * q^27 + (-2*b2 + b1 + 2) * q^29 + (-2*b2 - 2*b1 - 4) * q^33 + (-b2 + 2*b1 + 2) * q^35 + 2 * q^37 + (-2*b2 + 3*b1 - 8) * q^39 + (2*b1 + 2) * q^41 + (b2 + 2*b1 - 6) * q^43 + (b2 + 4*b1 + 8) * q^45 + (-3*b2 + 2*b1 - 2) * q^47 + (b2 - 3*b1 + 3) * q^49 + (b1 - 4) * q^51 + (-4*b2 + b1 - 2) * q^53 + (b2 + 2*b1 + 6) * q^55 - b1 * q^57 + (-b1 + 8) * q^59 + (b2 - 2*b1) * q^61 + (3*b2 - 4*b1 + 18) * q^63 + 4 * q^65 + (2*b2 - b1 - 4) * q^67 + (2*b2 - 3*b1 + 16) * q^69 + (2*b2 + 2*b1 - 4) * q^71 + (-b2 + 3*b1) * q^73 + (-2*b2 + b1 - 12) * q^75 + (b2 + 6) * q^77 + (2*b1 + 8) * q^79 + (4*b2 + 17) * q^81 + (4*b2 - 2*b1 + 4) * q^83 + (3*b2 - 2*b1 - 2) * q^85 + (2*b2 - b1) * q^87 + (-2*b2 - 2*b1 + 6) * q^89 + (-2*b2 + 5*b1 - 8) * q^91 + b2 * q^95 + (2*b2 - 2) * q^97 + (5*b2 + 2*b1 + 18) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - q^{3} - q^{5} + 4 q^{7} + 12 q^{9}+O(q^{10})$$ 3 * q - q^3 - q^5 + 4 * q^7 + 12 * q^9 $$3 q - q^{3} - q^{5} + 4 q^{7} + 12 q^{9} + 5 q^{11} - 5 q^{13} - 10 q^{15} + 2 q^{17} + 3 q^{19} + 9 q^{21} - 5 q^{23} + 6 q^{25} - q^{27} + 9 q^{29} - 12 q^{33} + 9 q^{35} + 6 q^{37} - 19 q^{39} + 8 q^{41} - 17 q^{43} + 27 q^{45} - q^{47} + 5 q^{49} - 11 q^{51} - q^{53} + 19 q^{55} - q^{57} + 23 q^{59} - 3 q^{61} + 47 q^{63} + 12 q^{65} - 15 q^{67} + 43 q^{69} - 12 q^{71} + 4 q^{73} - 33 q^{75} + 17 q^{77} + 26 q^{79} + 47 q^{81} + 6 q^{83} - 11 q^{85} - 3 q^{87} + 18 q^{89} - 17 q^{91} - q^{95} - 8 q^{97} + 51 q^{99}+O(q^{100})$$ 3 * q - q^3 - q^5 + 4 * q^7 + 12 * q^9 + 5 * q^11 - 5 * q^13 - 10 * q^15 + 2 * q^17 + 3 * q^19 + 9 * q^21 - 5 * q^23 + 6 * q^25 - q^27 + 9 * q^29 - 12 * q^33 + 9 * q^35 + 6 * q^37 - 19 * q^39 + 8 * q^41 - 17 * q^43 + 27 * q^45 - q^47 + 5 * q^49 - 11 * q^51 - q^53 + 19 * q^55 - q^57 + 23 * q^59 - 3 * q^61 + 47 * q^63 + 12 * q^65 - 15 * q^67 + 43 * q^69 - 12 * q^71 + 4 * q^73 - 33 * q^75 + 17 * q^77 + 26 * q^79 + 47 * q^81 + 6 * q^83 - 11 * q^85 - 3 * q^87 + 18 * q^89 - 17 * q^91 - q^95 - 8 * q^97 + 51 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 10x + 8$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{2} + \nu - 8 ) / 2$$ (v^2 + v - 8) / 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$2\beta_{2} - \beta _1 + 8$$ 2*b2 - b1 + 8

## 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
 3.29707 0.786802 −3.08387
0 −3.29707 0 3.08387 0 1.78680 0 7.87067 0
1.2 0 −0.786802 0 −3.29707 0 −2.08387 0 −2.38094 0
1.3 0 3.08387 0 −0.786802 0 4.29707 0 6.51027 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 1216.2.a.u 3
4.b odd 2 1 1216.2.a.v 3
8.b even 2 1 152.2.a.c 3
8.d odd 2 1 304.2.a.g 3
24.f even 2 1 2736.2.a.bd 3
24.h odd 2 1 1368.2.a.n 3
40.e odd 2 1 7600.2.a.bv 3
40.f even 2 1 3800.2.a.r 3
40.i odd 4 2 3800.2.d.j 6
56.h odd 2 1 7448.2.a.bf 3
152.b even 2 1 5776.2.a.bp 3
152.g odd 2 1 2888.2.a.o 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
152.2.a.c 3 8.b even 2 1
304.2.a.g 3 8.d odd 2 1
1216.2.a.u 3 1.a even 1 1 trivial
1216.2.a.v 3 4.b odd 2 1
1368.2.a.n 3 24.h odd 2 1
2736.2.a.bd 3 24.f even 2 1
2888.2.a.o 3 152.g odd 2 1
3800.2.a.r 3 40.f even 2 1
3800.2.d.j 6 40.i odd 4 2
5776.2.a.bp 3 152.b even 2 1
7448.2.a.bf 3 56.h odd 2 1
7600.2.a.bv 3 40.e odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1216))$$:

 $$T_{3}^{3} + T_{3}^{2} - 10T_{3} - 8$$ T3^3 + T3^2 - 10*T3 - 8 $$T_{5}^{3} + T_{5}^{2} - 10T_{5} - 8$$ T5^3 + T5^2 - 10*T5 - 8 $$T_{7}^{3} - 4T_{7}^{2} - 5T_{7} + 16$$ T7^3 - 4*T7^2 - 5*T7 + 16

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$T^{3} + T^{2} - 10T - 8$$
$5$ $$T^{3} + T^{2} - 10T - 8$$
$7$ $$T^{3} - 4 T^{2} - 5 T + 16$$
$11$ $$T^{3} - 5 T^{2} - 2 T + 8$$
$13$ $$T^{3} + 5 T^{2} - 2 T - 8$$
$17$ $$T^{3} - 2 T^{2} - 9 T + 2$$
$19$ $$(T - 1)^{3}$$
$23$ $$T^{3} + 5 T^{2} - 64 T - 256$$
$29$ $$T^{3} - 9 T^{2} - 4 T + 4$$
$31$ $$T^{3}$$
$37$ $$(T - 2)^{3}$$
$41$ $$T^{3} - 8 T^{2} - 20 T + 128$$
$43$ $$T^{3} + 17 T^{2} + 24 T - 368$$
$47$ $$T^{3} + T^{2} - 72 T - 256$$
$53$ $$T^{3} + T^{2} - 134 T - 256$$
$59$ $$T^{3} - 23 T^{2} + 166 T - 376$$
$61$ $$T^{3} + 3 T^{2} - 28 T - 92$$
$67$ $$T^{3} + 15 T^{2} + 44 T + 32$$
$71$ $$T^{3} + 12 T^{2} - 76 T - 928$$
$73$ $$T^{3} - 4 T^{2} - 67 T + 326$$
$79$ $$T^{3} - 26 T^{2} + 184 T - 256$$
$83$ $$T^{3} - 6 T^{2} - 112 T + 736$$
$89$ $$T^{3} - 18 T^{2} - 16 T + 1024$$
$97$ $$T^{3} + 8 T^{2} - 20 T - 128$$