# Properties

 Label 832.2.bu.d Level $832$ Weight $2$ Character orbit 832.bu Analytic conductor $6.644$ Analytic rank $1$ 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] = [832,2,Mod(63,832)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(832, base_ring=CyclotomicField(12))

chi = DirichletCharacter(H, H._module([6, 0, 7]))

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

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

chi := DirichletCharacter("832.63");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$832 = 2^{6} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 832.bu (of order $$12$$, degree $$4$$, 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: $$6.64355344817$$ Analytic rank: $$1$$ Dimension: $$4$$ 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: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 52) Sato-Tate group: $\mathrm{U}(1)[D_{12}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + (2 \zeta_{12}^{3} + \zeta_{12}^{2} - \zeta_{12} - 2) q^{5} + (3 \zeta_{12}^{2} - 3) q^{9}+O(q^{10})$$ q + (2*z^3 + z^2 - z - 2) * q^5 + (3*z^2 - 3) * q^9 $$q + (2 \zeta_{12}^{3} + \zeta_{12}^{2} - \zeta_{12} - 2) q^{5} + (3 \zeta_{12}^{2} - 3) q^{9} + ( - 3 \zeta_{12}^{3} - 2 \zeta_{12}^{2} + 3 \zeta_{12}) q^{13} + ( - 4 \zeta_{12}^{2} + \zeta_{12} - 4) q^{17} + ( - \zeta_{12}^{3} - 6 \zeta_{12}^{2} + 3) q^{25} + ( - 5 \zeta_{12}^{3} - 2 \zeta_{12}^{2} - 5 \zeta_{12}) q^{29} + (\zeta_{12}^{3} + \zeta_{12}^{2} + 6 \zeta_{12} - 7) q^{37} + ( - 4 \zeta_{12}^{3} + 4 \zeta_{12}^{2} - 5 \zeta_{12} - 9) q^{41} + ( - 3 \zeta_{12}^{3} - 6 \zeta_{12}^{2} - 3 \zeta_{12} + 3) q^{45} + ( - 7 \zeta_{12}^{3} + 7 \zeta_{12}) q^{49} + ( - 2 \zeta_{12}^{3} + 4 \zeta_{12} - 7) q^{53} + (12 \zeta_{12}^{3} + 5 \zeta_{12}^{2} - 6 \zeta_{12} - 5) q^{61} + (4 \zeta_{12}^{3} + 8 \zeta_{12}^{2} + \zeta_{12} - 1) q^{65} + (8 \zeta_{12}^{3} - 5 \zeta_{12}^{2} - 5 \zeta_{12} + 8) q^{73} - 9 \zeta_{12}^{2} q^{81} + ( - 11 \zeta_{12}^{3} + \zeta_{12}^{2} + 10 \zeta_{12} + 10) q^{85} + ( - 3 \zeta_{12}^{3} - 3 \zeta_{12}^{2} + 3 \zeta_{12}) q^{89} + (5 \zeta_{12}^{2} + 5 \zeta_{12} - 5) q^{97} +O(q^{100})$$ q + (2*z^3 + z^2 - z - 2) * q^5 + (3*z^2 - 3) * q^9 + (-3*z^3 - 2*z^2 + 3*z) * q^13 + (-4*z^2 + z - 4) * q^17 + (-z^3 - 6*z^2 + 3) * q^25 + (-5*z^3 - 2*z^2 - 5*z) * q^29 + (z^3 + z^2 + 6*z - 7) * q^37 + (-4*z^3 + 4*z^2 - 5*z - 9) * q^41 + (-3*z^3 - 6*z^2 - 3*z + 3) * q^45 + (-7*z^3 + 7*z) * q^49 + (-2*z^3 + 4*z - 7) * q^53 + (12*z^3 + 5*z^2 - 6*z - 5) * q^61 + (4*z^3 + 8*z^2 + z - 1) * q^65 + (8*z^3 - 5*z^2 - 5*z + 8) * q^73 - 9*z^2 * q^81 + (-11*z^3 + z^2 + 10*z + 10) * q^85 + (-3*z^3 - 3*z^2 + 3*z) * q^89 + (5*z^2 + 5*z - 5) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 6 q^{5} - 6 q^{9}+O(q^{10})$$ 4 * q - 6 * q^5 - 6 * q^9 $$4 q - 6 q^{5} - 6 q^{9} - 4 q^{13} - 24 q^{17} - 4 q^{29} - 26 q^{37} - 28 q^{41} - 28 q^{53} - 10 q^{61} + 12 q^{65} + 22 q^{73} - 18 q^{81} + 42 q^{85} - 6 q^{89} - 10 q^{97}+O(q^{100})$$ 4 * q - 6 * q^5 - 6 * q^9 - 4 * q^13 - 24 * q^17 - 4 * q^29 - 26 * q^37 - 28 * q^41 - 28 * q^53 - 10 * q^61 + 12 * q^65 + 22 * q^73 - 18 * q^81 + 42 * q^85 - 6 * q^89 - 10 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/832\mathbb{Z}\right)^\times$$.

 $$n$$ $$261$$ $$703$$ $$769$$ $$\chi(n)$$ $$1$$ $$-1$$ $$\zeta_{12}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
63.1
 −0.866025 − 0.500000i 0.866025 − 0.500000i −0.866025 + 0.500000i 0.866025 + 0.500000i
0 0 0 −0.633975 0.633975i 0 0 0 −1.50000 + 2.59808i 0
319.1 0 0 0 −2.36603 2.36603i 0 0 0 −1.50000 2.59808i 0
383.1 0 0 0 −0.633975 + 0.633975i 0 0 0 −1.50000 2.59808i 0
639.1 0 0 0 −2.36603 + 2.36603i 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.f odd 12 1 inner
52.l even 12 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 832.2.bu.d 4
4.b odd 2 1 CM 832.2.bu.d 4
8.b even 2 1 52.2.l.a 4
8.d odd 2 1 52.2.l.a 4
13.f odd 12 1 inner 832.2.bu.d 4
24.f even 2 1 468.2.cb.d 4
24.h odd 2 1 468.2.cb.d 4
52.l even 12 1 inner 832.2.bu.d 4
104.e even 2 1 676.2.l.d 4
104.h odd 2 1 676.2.l.d 4
104.j odd 4 1 676.2.l.c 4
104.j odd 4 1 676.2.l.e 4
104.m even 4 1 676.2.l.c 4
104.m even 4 1 676.2.l.e 4
104.n odd 6 1 676.2.f.e 4
104.n odd 6 1 676.2.l.c 4
104.p odd 6 1 676.2.f.d 4
104.p odd 6 1 676.2.l.e 4
104.r even 6 1 676.2.f.e 4
104.r even 6 1 676.2.l.c 4
104.s even 6 1 676.2.f.d 4
104.s even 6 1 676.2.l.e 4
104.u even 12 1 52.2.l.a 4
104.u even 12 1 676.2.f.d 4
104.u even 12 1 676.2.f.e 4
104.u even 12 1 676.2.l.d 4
104.x odd 12 1 52.2.l.a 4
104.x odd 12 1 676.2.f.d 4
104.x odd 12 1 676.2.f.e 4
104.x odd 12 1 676.2.l.d 4
312.bo even 12 1 468.2.cb.d 4
312.bq odd 12 1 468.2.cb.d 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
52.2.l.a 4 8.b even 2 1
52.2.l.a 4 8.d odd 2 1
52.2.l.a 4 104.u even 12 1
52.2.l.a 4 104.x odd 12 1
468.2.cb.d 4 24.f even 2 1
468.2.cb.d 4 24.h odd 2 1
468.2.cb.d 4 312.bo even 12 1
468.2.cb.d 4 312.bq odd 12 1
676.2.f.d 4 104.p odd 6 1
676.2.f.d 4 104.s even 6 1
676.2.f.d 4 104.u even 12 1
676.2.f.d 4 104.x odd 12 1
676.2.f.e 4 104.n odd 6 1
676.2.f.e 4 104.r even 6 1
676.2.f.e 4 104.u even 12 1
676.2.f.e 4 104.x odd 12 1
676.2.l.c 4 104.j odd 4 1
676.2.l.c 4 104.m even 4 1
676.2.l.c 4 104.n odd 6 1
676.2.l.c 4 104.r even 6 1
676.2.l.d 4 104.e even 2 1
676.2.l.d 4 104.h odd 2 1
676.2.l.d 4 104.u even 12 1
676.2.l.d 4 104.x odd 12 1
676.2.l.e 4 104.j odd 4 1
676.2.l.e 4 104.m even 4 1
676.2.l.e 4 104.p odd 6 1
676.2.l.e 4 104.s even 6 1
832.2.bu.d 4 1.a even 1 1 trivial
832.2.bu.d 4 4.b odd 2 1 CM
832.2.bu.d 4 13.f odd 12 1 inner
832.2.bu.d 4 52.l even 12 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(832, [\chi])$$:

 $$T_{3}$$ T3 $$T_{5}^{4} + 6T_{5}^{3} + 18T_{5}^{2} + 18T_{5} + 9$$ T5^4 + 6*T5^3 + 18*T5^2 + 18*T5 + 9

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$T^{4} + 6 T^{3} + 18 T^{2} + 18 T + 9$$
$7$ $$T^{4}$$
$11$ $$T^{4}$$
$13$ $$T^{4} + 4 T^{3} + 3 T^{2} + 52 T + 169$$
$17$ $$T^{4} + 24 T^{3} + 239 T^{2} + \cdots + 2209$$
$19$ $$T^{4}$$
$23$ $$T^{4}$$
$29$ $$T^{4} + 4 T^{3} + 87 T^{2} + \cdots + 5041$$
$31$ $$T^{4}$$
$37$ $$T^{4} + 26 T^{3} + 233 T^{2} + \cdots + 3721$$
$41$ $$T^{4} + 28 T^{3} + 365 T^{2} + \cdots + 14641$$
$43$ $$T^{4}$$
$47$ $$T^{4}$$
$53$ $$(T^{2} + 14 T + 37)^{2}$$
$59$ $$T^{4}$$
$61$ $$T^{4} + 10 T^{3} + 183 T^{2} + \cdots + 6889$$
$67$ $$T^{4}$$
$71$ $$T^{4}$$
$73$ $$T^{4} - 22 T^{3} + 242 T^{2} + \cdots + 529$$
$79$ $$T^{4}$$
$83$ $$T^{4}$$
$89$ $$T^{4} + 6 T^{3} + 18 T^{2} + 108 T + 324$$
$97$ $$T^{4} + 10 T^{3} + 50 T^{2} + \cdots + 2500$$