# Properties

 Label 416.2.w.b Level $416$ Weight $2$ Character orbit 416.w Analytic conductor $3.322$ Analytic rank $0$ Dimension $4$ CM discriminant -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: $$\Z[a_1, \ldots, a_{9}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

## $q$-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 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.

 $$f(q)$$ $$=$$ $$q + (\beta_{3} + 2 \beta_{2} - 1) q^{5} + ( - 3 \beta_{2} + 3) q^{9}+O(q^{10})$$ q + (b3 + 2*b2 - 1) * q^5 + (-3*b2 + 3) * q^9 $$q + (\beta_{3} + 2 \beta_{2} - 1) q^{5} + ( - 3 \beta_{2} + 3) q^{9} + (\beta_{3} + 3 \beta_{2} - \beta_1) q^{13} + (4 \beta_{3} + \beta_{2} - 2 \beta_1 - 1) q^{17} + (2 \beta_{3} - 4 \beta_1 - 2) q^{25} + (\beta_{3} + 5 \beta_{2} + \beta_1) q^{29} + ( - 3 \beta_{3} - \beta_{2} + 3 \beta_1 + 2) q^{37} + ( - 2 \beta_{3} - 5 \beta_{2} + \cdots + 10) q^{41}+ \cdots + 4 \beta_1 q^{97}+O(q^{100})$$ q + (b3 + 2*b2 - 1) * q^5 + (-3*b2 + 3) * q^9 + (b3 + 3*b2 - b1) * q^13 + (4*b3 + b2 - 2*b1 - 1) * q^17 + (2*b3 - 4*b1 - 2) * q^25 + (b3 + 5*b2 + b1) * q^29 + (-3*b3 - b2 + 3*b1 + 2) * q^37 + (-2*b3 - 5*b2 + 2*b1 + 10) * q^41 + (3*b2 + 3*b1 + 3) * q^45 - 7*b2 * q^49 + (b3 - 2*b1 - 7) * q^53 + (-6*b3 + 5*b2 + 3*b1 - 5) * q^61 + (2*b3 - b2 - 4*b1 - 6) * q^65 + (-4*b3 + 6*b2 - 3) * q^73 - 9*b2 * q^81 + (-9*b2 - 7*b1 - 9) * q^85 + (-8*b3 + 8*b1) * q^89 + 4*b1 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 6 q^{9}+O(q^{10})$$ 4 * q + 6 * q^9 $$4 q + 6 q^{9} + 6 q^{13} - 2 q^{17} - 8 q^{25} + 10 q^{29} + 6 q^{37} + 30 q^{41} + 18 q^{45} - 14 q^{49} - 28 q^{53} - 10 q^{61} - 26 q^{65} - 18 q^{81} - 54 q^{85}+O(q^{100})$$ 4 * q + 6 * q^9 + 6 * q^13 - 2 * q^17 - 8 * q^25 + 10 * q^29 + 6 * q^37 + 30 * q^41 + 18 * q^45 - 14 * q^49 - 28 * q^53 - 10 * q^61 - 26 * q^65 - 18 * q^81 - 54 * q^85

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$2\zeta_{12}$$ 2*v $$\beta_{2}$$ $$=$$ $$\zeta_{12}^{2}$$ v^2 $$\beta_{3}$$ $$=$$ $$2\zeta_{12}^{3}$$ 2*v^3
 $$\zeta_{12}$$ $$=$$ $$( \beta_1 ) / 2$$ (b1) / 2 $$\zeta_{12}^{2}$$ $$=$$ $$\beta_{2}$$ b2 $$\zeta_{12}^{3}$$ $$=$$ $$( \beta_{3} ) / 2$$ (b3) / 2

## Character values

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

 $$n$$ $$261$$ $$287$$ $$353$$ $$\chi(n)$$ $$1$$ $$1$$ $$1 - \beta_{2}$$

## 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 0 3.73205i 0 0 0 1.50000 + 2.59808i 0
225.2 0 0 0 0.267949i 0 0 0 1.50000 + 2.59808i 0
257.1 0 0 0 0.267949i 0 0 0 1.50000 2.59808i 0
257.2 0 0 0 3.73205i 0 0 0 1.50000 2.59808i 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 CM by $$\Q(\sqrt{-1})$$
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.b 4
4.b odd 2 1 CM 416.2.w.b 4
8.b even 2 1 832.2.w.f 4
8.d odd 2 1 832.2.w.f 4
13.e even 6 1 inner 416.2.w.b 4
13.f odd 12 1 5408.2.a.r 2
13.f odd 12 1 5408.2.a.bc 2
52.i odd 6 1 inner 416.2.w.b 4
52.l even 12 1 5408.2.a.r 2
52.l even 12 1 5408.2.a.bc 2
104.p odd 6 1 832.2.w.f 4
104.s even 6 1 832.2.w.f 4

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

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$T^{4} + 14T^{2} + 1$$
$7$ $$T^{4}$$
$11$ $$T^{4}$$
$13$ $$T^{4} - 6 T^{3} + \cdots + 169$$
$17$ $$T^{4} + 2 T^{3} + \cdots + 2209$$
$19$ $$T^{4}$$
$23$ $$T^{4}$$
$29$ $$T^{4} - 10 T^{3} + \cdots + 169$$
$31$ $$T^{4}$$
$37$ $$T^{4} - 6 T^{3} + \cdots + 1089$$
$41$ $$T^{4} - 30 T^{3} + \cdots + 3481$$
$43$ $$T^{4}$$
$47$ $$T^{4}$$
$53$ $$(T^{2} + 14 T + 37)^{2}$$
$59$ $$T^{4}$$
$61$ $$T^{4} + 10 T^{3} + \cdots + 6889$$
$67$ $$T^{4}$$
$71$ $$T^{4}$$
$73$ $$T^{4} + 182T^{2} + 1369$$
$79$ $$T^{4}$$
$83$ $$T^{4}$$
$89$ $$T^{4} - 256 T^{2} + 65536$$
$97$ $$T^{4} - 64T^{2} + 4096$$