# Properties

 Label 1216.2.h.b Level $1216$ Weight $2$ Character orbit 1216.h Analytic conductor $9.710$ Analytic rank $0$ Dimension $4$ CM discriminant -19 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1216,2,Mod(1215,1216)]

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([1, 0, 1]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("1216.1215");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1216 = 2^{6} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1216.h (of order $$2$$, degree $$1$$, not minimal)

## 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: no Analytic conductor: $$9.70980888579$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{-19})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{3} - 4x^{2} - 5x + 25$$ x^4 - x^3 - 4*x^2 - 5*x + 25 Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 304) Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-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 $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_{2} - 1) q^{5} + (\beta_{3} + 2 \beta_1) q^{7} - 3 q^{9}+O(q^{10})$$ q + (b2 - 1) * q^5 + (b3 + 2*b1) * q^7 - 3 * q^9 $$q + (\beta_{2} - 1) q^{5} + (\beta_{3} + 2 \beta_1) q^{7} - 3 q^{9} + ( - \beta_{3} + 2 \beta_1) q^{11} + ( - \beta_{2} - 3) q^{17} + ( - 2 \beta_{3} - \beta_1) q^{19} + ( - 4 \beta_{3} - 2 \beta_1) q^{23} + ( - \beta_{2} + 10) q^{25} + ( - 5 \beta_{3} - 8 \beta_1) q^{35} + (3 \beta_{3} + 2 \beta_1) q^{43} + ( - 3 \beta_{2} + 3) q^{45} + (\beta_{3} - 6 \beta_1) q^{47} + (3 \beta_{2} - 6) q^{49} - 7 \beta_{3} q^{55} + (\beta_{2} + 7) q^{61} + ( - 3 \beta_{3} - 6 \beta_1) q^{63} + (3 \beta_{2} - 7) q^{73} + (\beta_{2} - 7) q^{77} + 9 q^{81} + ( - 4 \beta_{3} - 2 \beta_1) q^{83} + ( - 3 \beta_{2} - 11) q^{85} + (\beta_{3} + 10 \beta_1) q^{95} + (3 \beta_{3} - 6 \beta_1) q^{99}+O(q^{100})$$ q + (b2 - 1) * q^5 + (b3 + 2*b1) * q^7 - 3 * q^9 + (-b3 + 2*b1) * q^11 + (-b2 - 3) * q^17 + (-2*b3 - b1) * q^19 + (-4*b3 - 2*b1) * q^23 + (-b2 + 10) * q^25 + (-5*b3 - 8*b1) * q^35 + (3*b3 + 2*b1) * q^43 + (-3*b2 + 3) * q^45 + (b3 - 6*b1) * q^47 + (3*b2 - 6) * q^49 - 7*b3 * q^55 + (b2 + 7) * q^61 + (-3*b3 - 6*b1) * q^63 + (3*b2 - 7) * q^73 + (b2 - 7) * q^77 + 9 * q^81 + (-4*b3 - 2*b1) * q^83 + (-3*b2 - 11) * q^85 + (b3 + 10*b1) * q^95 + (3*b3 - 6*b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 2 q^{5} - 12 q^{9}+O(q^{10})$$ 4 * q - 2 * q^5 - 12 * q^9 $$4 q - 2 q^{5} - 12 q^{9} - 14 q^{17} + 38 q^{25} + 6 q^{45} - 18 q^{49} + 30 q^{61} - 22 q^{73} - 26 q^{77} + 36 q^{81} - 50 q^{85}+O(q^{100})$$ 4 * q - 2 * q^5 - 12 * q^9 - 14 * q^17 + 38 * q^25 + 6 * q^45 - 18 * q^49 + 30 * q^61 - 22 * q^73 - 26 * q^77 + 36 * q^81 - 50 * q^85

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 4x^{2} - 5x + 25$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{3} + 4\nu^{2} - 4\nu - 15 ) / 10$$ (v^3 + 4*v^2 - 4*v - 15) / 10 $$\beta_{2}$$ $$=$$ $$( -\nu^{3} + \nu^{2} + 9\nu + 5 ) / 5$$ (-v^3 + v^2 + 9*v + 5) / 5 $$\beta_{3}$$ $$=$$ $$( -3\nu^{3} - 2\nu^{2} + 2\nu + 25 ) / 10$$ (-3*v^3 - 2*v^2 + 2*v + 25) / 10
 $$\nu$$ $$=$$ $$( -\beta_{3} + \beta_{2} - \beta_1 ) / 2$$ (-b3 + b2 - b1) / 2 $$\nu^{2}$$ $$=$$ $$( \beta_{3} + \beta_{2} + 5\beta _1 + 4 ) / 2$$ (b3 + b2 + 5*b1 + 4) / 2 $$\nu^{3}$$ $$=$$ $$-4\beta_{3} - 2\beta _1 + 7$$ -4*b3 - 2*b1 + 7

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1216\mathbb{Z}\right)^\times$$.

 $$n$$ $$191$$ $$705$$ $$837$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1215.1
 −1.63746 + 1.52274i −1.63746 − 1.52274i 2.13746 − 0.656712i 2.13746 + 0.656712i
0 0 0 −4.27492 0 4.77753i 0 −3.00000 0
1215.2 0 0 0 −4.27492 0 4.77753i 0 −3.00000 0
1215.3 0 0 0 3.27492 0 0.418627i 0 −3.00000 0
1215.4 0 0 0 3.27492 0 0.418627i 0 −3.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.b odd 2 1 CM by $$\Q(\sqrt{-19})$$
4.b odd 2 1 inner
76.d even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1216.2.h.b 4
4.b odd 2 1 inner 1216.2.h.b 4
8.b even 2 1 304.2.h.c 4
8.d odd 2 1 304.2.h.c 4
19.b odd 2 1 CM 1216.2.h.b 4
24.f even 2 1 2736.2.k.j 4
24.h odd 2 1 2736.2.k.j 4
76.d even 2 1 inner 1216.2.h.b 4
152.b even 2 1 304.2.h.c 4
152.g odd 2 1 304.2.h.c 4
456.l odd 2 1 2736.2.k.j 4
456.p even 2 1 2736.2.k.j 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
304.2.h.c 4 8.b even 2 1
304.2.h.c 4 8.d odd 2 1
304.2.h.c 4 152.b even 2 1
304.2.h.c 4 152.g odd 2 1
1216.2.h.b 4 1.a even 1 1 trivial
1216.2.h.b 4 4.b odd 2 1 inner
1216.2.h.b 4 19.b odd 2 1 CM
1216.2.h.b 4 76.d even 2 1 inner
2736.2.k.j 4 24.f even 2 1
2736.2.k.j 4 24.h odd 2 1
2736.2.k.j 4 456.l odd 2 1
2736.2.k.j 4 456.p even 2 1

## Hecke kernels

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

 $$T_{3}$$ T3 $$T_{5}^{2} + T_{5} - 14$$ T5^2 + T5 - 14

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$(T^{2} + T - 14)^{2}$$
$7$ $$T^{4} + 23T^{2} + 4$$
$11$ $$T^{4} + 47T^{2} + 196$$
$13$ $$T^{4}$$
$17$ $$(T^{2} + 7 T - 2)^{2}$$
$19$ $$(T^{2} + 19)^{2}$$
$23$ $$(T^{2} + 76)^{2}$$
$29$ $$T^{4}$$
$31$ $$T^{4}$$
$37$ $$T^{4}$$
$41$ $$T^{4}$$
$43$ $$T^{4} + 87T^{2} + 1764$$
$47$ $$T^{4} + 263 T^{2} + 14884$$
$53$ $$T^{4}$$
$59$ $$T^{4}$$
$61$ $$(T^{2} - 15 T + 42)^{2}$$
$67$ $$T^{4}$$
$71$ $$T^{4}$$
$73$ $$(T^{2} + 11 T - 98)^{2}$$
$79$ $$T^{4}$$
$83$ $$(T^{2} + 76)^{2}$$
$89$ $$T^{4}$$
$97$ $$T^{4}$$