# Properties

 Label 1216.2.c.i Level $1216$ Weight $2$ Character orbit 1216.c Analytic conductor $9.710$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1216,2,Mod(609,1216)]

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, 1, 0]))

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

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

chi := DirichletCharacter("1216.609");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1216 = 2^{6} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1216.c (of order $$2$$, degree $$1$$, 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: $$8$$ Coefficient field: 8.0.2702336256.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} + 9x^{6} + 56x^{4} + 225x^{2} + 625$$ x^8 + 9*x^6 + 56*x^4 + 225*x^2 + 625 Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$2^{6}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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 $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_{3} q^{5} + \beta_{5} q^{7} + 3 q^{9}+O(q^{10})$$ q - b3 * q^5 + b5 * q^7 + 3 * q^9 $$q - \beta_{3} q^{5} + \beta_{5} q^{7} + 3 q^{9} + ( - 2 \beta_{4} - \beta_{2}) q^{11} + (\beta_{3} - \beta_1) q^{13} - \beta_{6} q^{17} - \beta_{4} q^{19} + ( - \beta_{7} + \beta_{5}) q^{23} + (\beta_{6} - 1) q^{25} + (\beta_{3} - \beta_1) q^{29} - 2 \beta_{7} q^{31} + ( - 2 \beta_{4} + \beta_{2}) q^{35} + ( - 3 \beta_{3} - \beta_1) q^{37} + ( - 2 \beta_{6} - 2) q^{41} + ( - 6 \beta_{4} + \beta_{2}) q^{43} - 3 \beta_{3} q^{45} - \beta_{5} q^{47} + (3 \beta_{6} + 3) q^{49} + ( - 3 \beta_{3} - 3 \beta_1) q^{53} + ( - 5 \beta_{7} + 2 \beta_{5}) q^{55} + 2 \beta_{2} q^{59} - \beta_{3} q^{61} + 3 \beta_{5} q^{63} + 8 q^{65} + 2 \beta_{2} q^{67} + (4 \beta_{7} + 2 \beta_{5}) q^{71} + \beta_{6} q^{73} + (\beta_{3} - 2 \beta_1) q^{77} + ( - 4 \beta_{7} + 2 \beta_{5}) q^{79} + 9 q^{81} + ( - 4 \beta_{4} - 4 \beta_{2}) q^{83} + ( - 3 \beta_{3} - 2 \beta_1) q^{85} - 10 q^{89} + (12 \beta_{4} - 4 \beta_{2}) q^{91} - \beta_{7} q^{95} + ( - 2 \beta_{6} + 10) q^{97} + ( - 6 \beta_{4} - 3 \beta_{2}) q^{99}+O(q^{100})$$ q - b3 * q^5 + b5 * q^7 + 3 * q^9 + (-2*b4 - b2) * q^11 + (b3 - b1) * q^13 - b6 * q^17 - b4 * q^19 + (-b7 + b5) * q^23 + (b6 - 1) * q^25 + (b3 - b1) * q^29 - 2*b7 * q^31 + (-2*b4 + b2) * q^35 + (-3*b3 - b1) * q^37 + (-2*b6 - 2) * q^41 + (-6*b4 + b2) * q^43 - 3*b3 * q^45 - b5 * q^47 + (3*b6 + 3) * q^49 + (-3*b3 - 3*b1) * q^53 + (-5*b7 + 2*b5) * q^55 + 2*b2 * q^59 - b3 * q^61 + 3*b5 * q^63 + 8 * q^65 + 2*b2 * q^67 + (4*b7 + 2*b5) * q^71 + b6 * q^73 + (b3 - 2*b1) * q^77 + (-4*b7 + 2*b5) * q^79 + 9 * q^81 + (-4*b4 - 4*b2) * q^83 + (-3*b3 - 2*b1) * q^85 - 10 * q^89 + (12*b4 - 4*b2) * q^91 - b7 * q^95 + (-2*b6 + 10) * q^97 + (-6*b4 - 3*b2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 24 q^{9}+O(q^{10})$$ 8 * q + 24 * q^9 $$8 q + 24 q^{9} - 4 q^{17} - 4 q^{25} - 24 q^{41} + 36 q^{49} + 64 q^{65} + 4 q^{73} + 72 q^{81} - 80 q^{89} + 72 q^{97}+O(q^{100})$$ 8 * q + 24 * q^9 - 4 * q^17 - 4 * q^25 - 24 * q^41 + 36 * q^49 + 64 * q^65 + 4 * q^73 + 72 * q^81 - 80 * q^89 + 72 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 9x^{6} + 56x^{4} + 225x^{2} + 625$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{6} + 84\nu^{4} + 756\nu^{2} + 2325 ) / 700$$ (v^6 + 84*v^4 + 756*v^2 + 2325) / 700 $$\beta_{2}$$ $$=$$ $$( -\nu^{7} - 14\nu^{5} - 126\nu^{3} - 855\nu ) / 350$$ (-v^7 - 14*v^5 - 126*v^3 - 855*v) / 350 $$\beta_{3}$$ $$=$$ $$( 17\nu^{6} + 28\nu^{4} + 252\nu^{2} + 325 ) / 700$$ (17*v^6 + 28*v^4 + 252*v^2 + 325) / 700 $$\beta_{4}$$ $$=$$ $$( -9\nu^{7} - 56\nu^{5} - 154\nu^{3} - 625\nu ) / 1750$$ (-9*v^7 - 56*v^5 - 154*v^3 - 625*v) / 1750 $$\beta_{5}$$ $$=$$ $$( 3\nu^{7} + 52\nu^{5} + 268\nu^{3} + 575\nu ) / 500$$ (3*v^7 + 52*v^5 + 268*v^3 + 575*v) / 500 $$\beta_{6}$$ $$=$$ $$( -\nu^{6} - 9\nu^{4} - 31\nu^{2} - 100 ) / 25$$ (-v^6 - 9*v^4 - 31*v^2 - 100) / 25 $$\beta_{7}$$ $$=$$ $$( \nu^{7} + 4\nu^{5} + 36\nu^{3} + 45\nu ) / 100$$ (v^7 + 4*v^5 + 36*v^3 + 45*v) / 100
 $$\nu$$ $$=$$ $$( -\beta_{7} - \beta_{5} - 2\beta_{4} - 2\beta_{2} ) / 4$$ (-b7 - b5 - 2*b4 - 2*b2) / 4 $$\nu^{2}$$ $$=$$ $$( 2\beta_{6} + 3\beta_{3} + 5\beta _1 - 10 ) / 4$$ (2*b6 + 3*b3 + 5*b1 - 10) / 4 $$\nu^{3}$$ $$=$$ $$3\beta_{7} + \beta_{5} + 7\beta_{4}$$ 3*b7 + b5 + 7*b4 $$\nu^{4}$$ $$=$$ $$( -18\beta_{6} - 29\beta_{3} - 11\beta _1 - 22 ) / 4$$ (-18*b6 - 29*b3 - 11*b1 - 22) / 4 $$\nu^{5}$$ $$=$$ $$( -67\beta_{7} + 45\beta_{5} - 90\beta_{4} + 22\beta_{2} ) / 4$$ (-67*b7 + 45*b5 - 90*b4 + 22*b2) / 4 $$\nu^{6}$$ $$=$$ $$42\beta_{3} - 14\beta _1 + 27$$ 42*b3 - 14*b1 + 27 $$\nu^{7}$$ $$=$$ $$( 281\beta_{7} - 279\beta_{5} - 558\beta_{4} + 2\beta_{2} ) / 4$$ (281*b7 - 279*b5 - 558*b4 + 2*b2) / 4

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
609.1
 −0.656712 − 2.13746i 0.656712 + 2.13746i 1.52274 − 1.63746i −1.52274 + 1.63746i 1.52274 + 1.63746i −1.52274 − 1.63746i −0.656712 + 2.13746i 0.656712 − 2.13746i
0 0 0 3.04547i 0 −0.418627 0 3.00000 0
609.2 0 0 0 3.04547i 0 0.418627 0 3.00000 0
609.3 0 0 0 1.31342i 0 −4.77753 0 3.00000 0
609.4 0 0 0 1.31342i 0 4.77753 0 3.00000 0
609.5 0 0 0 1.31342i 0 −4.77753 0 3.00000 0
609.6 0 0 0 1.31342i 0 4.77753 0 3.00000 0
609.7 0 0 0 3.04547i 0 −0.418627 0 3.00000 0
609.8 0 0 0 3.04547i 0 0.418627 0 3.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 609.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1216.2.c.i 8
4.b odd 2 1 inner 1216.2.c.i 8
8.b even 2 1 inner 1216.2.c.i 8
8.d odd 2 1 inner 1216.2.c.i 8
16.e even 4 1 4864.2.a.bj 4
16.e even 4 1 4864.2.a.bk 4
16.f odd 4 1 4864.2.a.bj 4
16.f odd 4 1 4864.2.a.bk 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1216.2.c.i 8 1.a even 1 1 trivial
1216.2.c.i 8 4.b odd 2 1 inner
1216.2.c.i 8 8.b even 2 1 inner
1216.2.c.i 8 8.d odd 2 1 inner
4864.2.a.bj 4 16.e even 4 1
4864.2.a.bj 4 16.f odd 4 1
4864.2.a.bk 4 16.e even 4 1
4864.2.a.bk 4 16.f odd 4 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 $$T_{5}^{4} + 11T_{5}^{2} + 16$$ T5^4 + 11*T5^2 + 16 $$T_{7}^{4} - 23T_{7}^{2} + 4$$ T7^4 - 23*T7^2 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8}$$
$5$ $$(T^{4} + 11 T^{2} + 16)^{2}$$
$7$ $$(T^{4} - 23 T^{2} + 4)^{2}$$
$11$ $$(T^{4} + 33 T^{2} + 144)^{2}$$
$13$ $$(T^{4} + 44 T^{2} + 256)^{2}$$
$17$ $$(T^{2} + T - 14)^{4}$$
$19$ $$(T^{2} + 1)^{4}$$
$23$ $$(T^{2} - 12)^{4}$$
$29$ $$(T^{4} + 44 T^{2} + 256)^{2}$$
$31$ $$(T^{4} - 44 T^{2} + 256)^{2}$$
$37$ $$(T^{4} + 92 T^{2} + 64)^{2}$$
$41$ $$(T^{2} + 6 T - 48)^{4}$$
$43$ $$(T^{4} + 113 T^{2} + 784)^{2}$$
$47$ $$(T^{4} - 23 T^{2} + 4)^{2}$$
$53$ $$(T^{2} + 108)^{4}$$
$59$ $$(T^{4} + 116 T^{2} + 3136)^{2}$$
$61$ $$(T^{4} + 11 T^{2} + 16)^{2}$$
$67$ $$(T^{4} + 116 T^{2} + 3136)^{2}$$
$71$ $$(T^{4} - 348 T^{2} + 28224)^{2}$$
$73$ $$(T^{2} - T - 14)^{4}$$
$79$ $$(T^{4} - 188 T^{2} + 3136)^{2}$$
$83$ $$(T^{4} + 464 T^{2} + 50176)^{2}$$
$89$ $$(T + 10)^{8}$$
$97$ $$(T^{2} - 18 T + 24)^{4}$$