# Properties

 Label 1216.2.a.t Level $1216$ Weight $2$ Character orbit 1216.a Self dual yes Analytic conductor $9.710$ Analytic rank $1$ Dimension $2$ 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: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{17})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 4$$ x^2 - x - 4 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 608) 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 $$\beta = \frac{1}{2}(1 + \sqrt{17})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{3} + ( - \beta + 1) q^{5} - 3 q^{7} + (\beta + 1) q^{9}+O(q^{10})$$ q + b * q^3 + (-b + 1) * q^5 - 3 * q^7 + (b + 1) * q^9 $$q + \beta q^{3} + ( - \beta + 1) q^{5} - 3 q^{7} + (\beta + 1) q^{9} + ( - \beta - 1) q^{11} - \beta q^{13} - 4 q^{15} + ( - 2 \beta - 3) q^{17} + q^{19} - 3 \beta q^{21} + (\beta - 4) q^{23} - \beta q^{25} + ( - \beta + 4) q^{27} + 3 \beta q^{29} + (2 \beta - 6) q^{31} + ( - 2 \beta - 4) q^{33} + (3 \beta - 3) q^{35} + (2 \beta - 4) q^{37} + ( - \beta - 4) q^{39} + 4 q^{41} + (\beta + 7) q^{43} + ( - \beta - 3) q^{45} + ( - 3 \beta - 1) q^{47} + 2 q^{49} + ( - 5 \beta - 8) q^{51} + (\beta + 6) q^{53} + (\beta + 3) q^{55} + \beta q^{57} + ( - \beta - 6) q^{59} + (5 \beta - 7) q^{61} + ( - 3 \beta - 3) q^{63} + 4 q^{65} + ( - \beta - 2) q^{67} + ( - 3 \beta + 4) q^{69} + ( - 4 \beta - 2) q^{71} + (4 \beta - 3) q^{73} + ( - \beta - 4) q^{75} + (3 \beta + 3) q^{77} - 10 q^{79} - 7 q^{81} + (6 \beta - 8) q^{83} + (3 \beta + 5) q^{85} + (3 \beta + 12) q^{87} + (6 \beta - 6) q^{89} + 3 \beta q^{91} + ( - 4 \beta + 8) q^{93} + ( - \beta + 1) q^{95} + ( - 2 \beta + 4) q^{97} + ( - 3 \beta - 5) q^{99} +O(q^{100})$$ q + b * q^3 + (-b + 1) * q^5 - 3 * q^7 + (b + 1) * q^9 + (-b - 1) * q^11 - b * q^13 - 4 * q^15 + (-2*b - 3) * q^17 + q^19 - 3*b * q^21 + (b - 4) * q^23 - b * q^25 + (-b + 4) * q^27 + 3*b * q^29 + (2*b - 6) * q^31 + (-2*b - 4) * q^33 + (3*b - 3) * q^35 + (2*b - 4) * q^37 + (-b - 4) * q^39 + 4 * q^41 + (b + 7) * q^43 + (-b - 3) * q^45 + (-3*b - 1) * q^47 + 2 * q^49 + (-5*b - 8) * q^51 + (b + 6) * q^53 + (b + 3) * q^55 + b * q^57 + (-b - 6) * q^59 + (5*b - 7) * q^61 + (-3*b - 3) * q^63 + 4 * q^65 + (-b - 2) * q^67 + (-3*b + 4) * q^69 + (-4*b - 2) * q^71 + (4*b - 3) * q^73 + (-b - 4) * q^75 + (3*b + 3) * q^77 - 10 * q^79 - 7 * q^81 + (6*b - 8) * q^83 + (3*b + 5) * q^85 + (3*b + 12) * q^87 + (6*b - 6) * q^89 + 3*b * q^91 + (-4*b + 8) * q^93 + (-b + 1) * q^95 + (-2*b + 4) * q^97 + (-3*b - 5) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{3} + q^{5} - 6 q^{7} + 3 q^{9}+O(q^{10})$$ 2 * q + q^3 + q^5 - 6 * q^7 + 3 * q^9 $$2 q + q^{3} + q^{5} - 6 q^{7} + 3 q^{9} - 3 q^{11} - q^{13} - 8 q^{15} - 8 q^{17} + 2 q^{19} - 3 q^{21} - 7 q^{23} - q^{25} + 7 q^{27} + 3 q^{29} - 10 q^{31} - 10 q^{33} - 3 q^{35} - 6 q^{37} - 9 q^{39} + 8 q^{41} + 15 q^{43} - 7 q^{45} - 5 q^{47} + 4 q^{49} - 21 q^{51} + 13 q^{53} + 7 q^{55} + q^{57} - 13 q^{59} - 9 q^{61} - 9 q^{63} + 8 q^{65} - 5 q^{67} + 5 q^{69} - 8 q^{71} - 2 q^{73} - 9 q^{75} + 9 q^{77} - 20 q^{79} - 14 q^{81} - 10 q^{83} + 13 q^{85} + 27 q^{87} - 6 q^{89} + 3 q^{91} + 12 q^{93} + q^{95} + 6 q^{97} - 13 q^{99}+O(q^{100})$$ 2 * q + q^3 + q^5 - 6 * q^7 + 3 * q^9 - 3 * q^11 - q^13 - 8 * q^15 - 8 * q^17 + 2 * q^19 - 3 * q^21 - 7 * q^23 - q^25 + 7 * q^27 + 3 * q^29 - 10 * q^31 - 10 * q^33 - 3 * q^35 - 6 * q^37 - 9 * q^39 + 8 * q^41 + 15 * q^43 - 7 * q^45 - 5 * q^47 + 4 * q^49 - 21 * q^51 + 13 * q^53 + 7 * q^55 + q^57 - 13 * q^59 - 9 * q^61 - 9 * q^63 + 8 * q^65 - 5 * q^67 + 5 * q^69 - 8 * q^71 - 2 * q^73 - 9 * q^75 + 9 * q^77 - 20 * q^79 - 14 * q^81 - 10 * q^83 + 13 * q^85 + 27 * q^87 - 6 * q^89 + 3 * q^91 + 12 * q^93 + q^95 + 6 * q^97 - 13 * 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
 −1.56155 2.56155
0 −1.56155 0 2.56155 0 −3.00000 0 −0.561553 0
1.2 0 2.56155 0 −1.56155 0 −3.00000 0 3.56155 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.t 2
4.b odd 2 1 1216.2.a.s 2
8.b even 2 1 608.2.a.g 2
8.d odd 2 1 608.2.a.h yes 2
24.f even 2 1 5472.2.a.bf 2
24.h odd 2 1 5472.2.a.bc 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
608.2.a.g 2 8.b even 2 1
608.2.a.h yes 2 8.d odd 2 1
1216.2.a.s 2 4.b odd 2 1
1216.2.a.t 2 1.a even 1 1 trivial
5472.2.a.bc 2 24.h odd 2 1
5472.2.a.bf 2 24.f even 2 1

## Hecke kernels

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

 $$T_{3}^{2} - T_{3} - 4$$ T3^2 - T3 - 4 $$T_{5}^{2} - T_{5} - 4$$ T5^2 - T5 - 4 $$T_{7} + 3$$ T7 + 3

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - T - 4$$
$5$ $$T^{2} - T - 4$$
$7$ $$(T + 3)^{2}$$
$11$ $$T^{2} + 3T - 2$$
$13$ $$T^{2} + T - 4$$
$17$ $$T^{2} + 8T - 1$$
$19$ $$(T - 1)^{2}$$
$23$ $$T^{2} + 7T + 8$$
$29$ $$T^{2} - 3T - 36$$
$31$ $$T^{2} + 10T + 8$$
$37$ $$T^{2} + 6T - 8$$
$41$ $$(T - 4)^{2}$$
$43$ $$T^{2} - 15T + 52$$
$47$ $$T^{2} + 5T - 32$$
$53$ $$T^{2} - 13T + 38$$
$59$ $$T^{2} + 13T + 38$$
$61$ $$T^{2} + 9T - 86$$
$67$ $$T^{2} + 5T + 2$$
$71$ $$T^{2} + 8T - 52$$
$73$ $$T^{2} + 2T - 67$$
$79$ $$(T + 10)^{2}$$
$83$ $$T^{2} + 10T - 128$$
$89$ $$T^{2} + 6T - 144$$
$97$ $$T^{2} - 6T - 8$$