Properties

 Label 416.2.w.a Level $416$ Weight $2$ Character orbit 416.w Analytic conductor $3.322$ Analytic rank $0$ Dimension $4$ Inner twists $4$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [416,2,Mod(225,416)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(416, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 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("416.225");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$416 = 2^{5} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 416.w (of order $$6$$, degree $$2$$, 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: $$3.32177672409$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 primitive root of unity $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \zeta_{12}^{3} - \zeta_{12}) q^{3} + (4 \zeta_{12}^{2} - 2) q^{5} - 3 \zeta_{12} q^{7} +O(q^{10})$$ q + (-z^3 - z) * q^3 + (4*z^2 - 2) * q^5 - 3*z * q^7 $$q + ( - \zeta_{12}^{3} - \zeta_{12}) q^{3} + (4 \zeta_{12}^{2} - 2) q^{5} - 3 \zeta_{12} q^{7} + ( - 5 \zeta_{12}^{3} + 5 \zeta_{12}) q^{11} + ( - 4 \zeta_{12}^{2} + 1) q^{13} + ( - 6 \zeta_{12}^{3} + 6 \zeta_{12}) q^{15} + ( - 7 \zeta_{12}^{2} + 7) q^{17} + 5 \zeta_{12} q^{19} + (6 \zeta_{12}^{2} - 3) q^{21} + ( - 3 \zeta_{12}^{3} - 3 \zeta_{12}) q^{23} - 7 q^{25} + (3 \zeta_{12}^{3} - 6 \zeta_{12}) q^{27} + 5 \zeta_{12}^{2} q^{29} - 2 \zeta_{12}^{3} q^{31} + ( - 5 \zeta_{12}^{2} - 5) q^{33} + ( - 12 \zeta_{12}^{3} + 6 \zeta_{12}) q^{35} + ( - 3 \zeta_{12}^{2} + 6) q^{37} + (7 \zeta_{12}^{3} - 5 \zeta_{12}) q^{39} + (\zeta_{12}^{2} - 2) q^{41} + (6 \zeta_{12}^{3} - 3 \zeta_{12}) q^{43} - 4 \zeta_{12}^{3} q^{47} + 2 \zeta_{12}^{2} q^{49} + (7 \zeta_{12}^{3} - 14 \zeta_{12}) q^{51} + 4 q^{53} + (10 \zeta_{12}^{3} + 10 \zeta_{12}) q^{55} + ( - 10 \zeta_{12}^{2} + 5) q^{57} - 7 \zeta_{12} q^{59} + (3 \zeta_{12}^{2} - 3) q^{61} + ( - 4 \zeta_{12}^{2} + 14) q^{65} + ( - 3 \zeta_{12}^{3} + 3 \zeta_{12}) q^{67} + (9 \zeta_{12}^{2} - 9) q^{69} + 7 \zeta_{12} q^{71} + (4 \zeta_{12}^{2} - 2) q^{73} + (7 \zeta_{12}^{3} + 7 \zeta_{12}) q^{75} - 15 q^{77} + ( - 2 \zeta_{12}^{3} + 4 \zeta_{12}) q^{79} + 9 \zeta_{12}^{2} q^{81} + 14 \zeta_{12}^{3} q^{83} + (14 \zeta_{12}^{2} + 14) q^{85} + ( - 10 \zeta_{12}^{3} + 5 \zeta_{12}) q^{87} + ( - \zeta_{12}^{2} + 2) q^{89} + (12 \zeta_{12}^{3} - 3 \zeta_{12}) q^{91} + (2 \zeta_{12}^{2} - 4) q^{93} + (20 \zeta_{12}^{3} - 10 \zeta_{12}) q^{95} + (5 \zeta_{12}^{2} + 5) q^{97} +O(q^{100})$$ q + (-z^3 - z) * q^3 + (4*z^2 - 2) * q^5 - 3*z * q^7 + (-5*z^3 + 5*z) * q^11 + (-4*z^2 + 1) * q^13 + (-6*z^3 + 6*z) * q^15 + (-7*z^2 + 7) * q^17 + 5*z * q^19 + (6*z^2 - 3) * q^21 + (-3*z^3 - 3*z) * q^23 - 7 * q^25 + (3*z^3 - 6*z) * q^27 + 5*z^2 * q^29 - 2*z^3 * q^31 + (-5*z^2 - 5) * q^33 + (-12*z^3 + 6*z) * q^35 + (-3*z^2 + 6) * q^37 + (7*z^3 - 5*z) * q^39 + (z^2 - 2) * q^41 + (6*z^3 - 3*z) * q^43 - 4*z^3 * q^47 + 2*z^2 * q^49 + (7*z^3 - 14*z) * q^51 + 4 * q^53 + (10*z^3 + 10*z) * q^55 + (-10*z^2 + 5) * q^57 - 7*z * q^59 + (3*z^2 - 3) * q^61 + (-4*z^2 + 14) * q^65 + (-3*z^3 + 3*z) * q^67 + (9*z^2 - 9) * q^69 + 7*z * q^71 + (4*z^2 - 2) * q^73 + (7*z^3 + 7*z) * q^75 - 15 * q^77 + (-2*z^3 + 4*z) * q^79 + 9*z^2 * q^81 + 14*z^3 * q^83 + (14*z^2 + 14) * q^85 + (-10*z^3 + 5*z) * q^87 + (-z^2 + 2) * q^89 + (12*z^3 - 3*z) * q^91 + (2*z^2 - 4) * q^93 + (20*z^3 - 10*z) * q^95 + (5*z^2 + 5) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q+O(q^{10})$$ 4 * q $$4 q - 4 q^{13} + 14 q^{17} - 28 q^{25} + 10 q^{29} - 30 q^{33} + 18 q^{37} - 6 q^{41} + 4 q^{49} + 16 q^{53} - 6 q^{61} + 48 q^{65} - 18 q^{69} - 60 q^{77} + 18 q^{81} + 84 q^{85} + 6 q^{89} - 12 q^{93} + 30 q^{97}+O(q^{100})$$ 4 * q - 4 * q^13 + 14 * q^17 - 28 * q^25 + 10 * q^29 - 30 * q^33 + 18 * q^37 - 6 * q^41 + 4 * q^49 + 16 * q^53 - 6 * q^61 + 48 * q^65 - 18 * q^69 - 60 * q^77 + 18 * q^81 + 84 * q^85 + 6 * q^89 - 12 * q^93 + 30 * q^97

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/416\mathbb{Z}\right)^\times$$.

 $$n$$ $$261$$ $$287$$ $$353$$ $$\chi(n)$$ $$1$$ $$1$$ $$1 - \zeta_{12}^{2}$$

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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
225.1
 0.866025 − 0.500000i −0.866025 + 0.500000i 0.866025 + 0.500000i −0.866025 − 0.500000i
0 −0.866025 + 1.50000i 0 3.46410i 0 −2.59808 + 1.50000i 0 0 0
225.2 0 0.866025 1.50000i 0 3.46410i 0 2.59808 1.50000i 0 0 0
257.1 0 −0.866025 1.50000i 0 3.46410i 0 −2.59808 1.50000i 0 0 0
257.2 0 0.866025 + 1.50000i 0 3.46410i 0 2.59808 + 1.50000i 0 0 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
4.b odd 2 1 inner
13.e even 6 1 inner
52.i odd 6 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 416.2.w.a 4
4.b odd 2 1 inner 416.2.w.a 4
8.b even 2 1 832.2.w.e 4
8.d odd 2 1 832.2.w.e 4
13.e even 6 1 inner 416.2.w.a 4
13.f odd 12 1 5408.2.a.u 2
13.f odd 12 1 5408.2.a.z 2
52.i odd 6 1 inner 416.2.w.a 4
52.l even 12 1 5408.2.a.u 2
52.l even 12 1 5408.2.a.z 2
104.p odd 6 1 832.2.w.e 4
104.s even 6 1 832.2.w.e 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
416.2.w.a 4 1.a even 1 1 trivial
416.2.w.a 4 4.b odd 2 1 inner
416.2.w.a 4 13.e even 6 1 inner
416.2.w.a 4 52.i odd 6 1 inner
832.2.w.e 4 8.b even 2 1
832.2.w.e 4 8.d odd 2 1
832.2.w.e 4 104.p odd 6 1
832.2.w.e 4 104.s even 6 1
5408.2.a.u 2 13.f odd 12 1
5408.2.a.u 2 52.l even 12 1
5408.2.a.z 2 13.f odd 12 1
5408.2.a.z 2 52.l even 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{4} + 3T_{3}^{2} + 9$$ acting on $$S_{2}^{\mathrm{new}}(416, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} + 3T^{2} + 9$$
$5$ $$(T^{2} + 12)^{2}$$
$7$ $$T^{4} - 9T^{2} + 81$$
$11$ $$T^{4} - 25T^{2} + 625$$
$13$ $$(T^{2} + 2 T + 13)^{2}$$
$17$ $$(T^{2} - 7 T + 49)^{2}$$
$19$ $$T^{4} - 25T^{2} + 625$$
$23$ $$T^{4} + 27T^{2} + 729$$
$29$ $$(T^{2} - 5 T + 25)^{2}$$
$31$ $$(T^{2} + 4)^{2}$$
$37$ $$(T^{2} - 9 T + 27)^{2}$$
$41$ $$(T^{2} + 3 T + 3)^{2}$$
$43$ $$T^{4} + 27T^{2} + 729$$
$47$ $$(T^{2} + 16)^{2}$$
$53$ $$(T - 4)^{4}$$
$59$ $$T^{4} - 49T^{2} + 2401$$
$61$ $$(T^{2} + 3 T + 9)^{2}$$
$67$ $$T^{4} - 9T^{2} + 81$$
$71$ $$T^{4} - 49T^{2} + 2401$$
$73$ $$(T^{2} + 12)^{2}$$
$79$ $$(T^{2} - 12)^{2}$$
$83$ $$(T^{2} + 196)^{2}$$
$89$ $$(T^{2} - 3 T + 3)^{2}$$
$97$ $$(T^{2} - 15 T + 75)^{2}$$