# Properties

 Label 3267.1.l.a Level $3267$ Weight $1$ Character orbit 3267.l Analytic conductor $1.630$ Analytic rank $0$ Dimension $16$ Projective image $D_{12}$ CM discriminant -3 Inner twists $16$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3267,1,Mod(838,3267)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3267, base_ring=CyclotomicField(10))

chi = DirichletCharacter(H, H._module([0, 1]))

N = Newforms(chi, 1, names="a")

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

chi := DirichletCharacter("3267.838");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3267 = 3^{3} \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 3267.l (of order $$10$$, 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: $$1.63044539627$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$4$$ over $$\Q(\zeta_{10})$$ Coefficient field: 16.0.6879707136000000000000.7 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{16} - 4x^{14} + 15x^{12} - 56x^{10} + 209x^{8} - 56x^{6} + 15x^{4} - 4x^{2} + 1$$ x^16 - 4*x^14 + 15*x^12 - 56*x^10 + 209*x^8 - 56*x^6 + 15*x^4 - 4*x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{12}$$ Projective field: Galois closure of $$\mathbb{Q}[x]/(x^{12} + \cdots)$$

## $q$-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 $$1,\beta_1,\ldots,\beta_{15}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_{4} q^{4} - \beta_{12} q^{7}+O(q^{10})$$ q - b4 * q^4 - b12 * q^7 $$q - \beta_{4} q^{4} - \beta_{12} q^{7} + \beta_{3} q^{13} - \beta_{8} q^{16} + (\beta_{7} + \beta_{5}) q^{19} - \beta_{13} q^{25} + (\beta_{12} + \beta_{9} + \beta_{5} - \beta_1) q^{28} + \beta_{14} q^{31} + \beta_{11} q^{43} + ( - \beta_{10} + \beta_{8}) q^{49} + \beta_{7} q^{52} + ( - \beta_{15} - \beta_{11} + \beta_{7} + \beta_{5} + \beta_{3} - \beta_1) q^{61} + ( - \beta_{13} + \beta_{8} + \beta_{4} - 1) q^{64} + \beta_{2} q^{67} + ( - \beta_{15} - \beta_{12}) q^{73} - \beta_{11} q^{76} - \beta_{3} q^{79} - \beta_{13} q^{91} + ( - \beta_{13} + \beta_{8} + \beta_{4} - 1) q^{97}+O(q^{100})$$ q - b4 * q^4 - b12 * q^7 + b3 * q^13 - b8 * q^16 + (b7 + b5) * q^19 - b13 * q^25 + (b12 + b9 + b5 - b1) * q^28 + b14 * q^31 + b11 * q^43 + (-b10 + b8) * q^49 + b7 * q^52 + (-b15 - b11 + b7 + b5 + b3 - b1) * q^61 + (-b13 + b8 + b4 - 1) * q^64 + b2 * q^67 + (-b15 - b12) * q^73 - b11 * q^76 - b3 * q^79 - b13 * q^91 + (-b13 + b8 + b4 - 1) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16 q - 4 q^{4}+O(q^{10})$$ 16 * q - 4 * q^4 $$16 q - 4 q^{4} - 4 q^{16} + 4 q^{25} + 4 q^{49} - 4 q^{64} + 4 q^{91} - 4 q^{97}+O(q^{100})$$ 16 * q - 4 * q^4 - 4 * q^16 + 4 * q^25 + 4 * q^49 - 4 * q^64 + 4 * q^91 - 4 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 4x^{14} + 15x^{12} - 56x^{10} + 209x^{8} - 56x^{6} + 15x^{4} - 4x^{2} + 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{10} + 362 ) / 209$$ (v^10 + 362) / 209 $$\beta_{3}$$ $$=$$ $$( \nu^{11} + 780\nu ) / 209$$ (v^11 + 780*v) / 209 $$\beta_{4}$$ $$=$$ $$( \nu^{12} + 780\nu^{2} ) / 209$$ (v^12 + 780*v^2) / 209 $$\beta_{5}$$ $$=$$ $$( \nu^{13} + 780\nu^{3} ) / 209$$ (v^13 + 780*v^3) / 209 $$\beta_{6}$$ $$=$$ $$( -2\nu^{12} - 1351\nu^{2} ) / 209$$ (-2*v^12 - 1351*v^2) / 209 $$\beta_{7}$$ $$=$$ $$( -4\nu^{13} - 2911\nu^{3} ) / 209$$ (-4*v^13 - 2911*v^3) / 209 $$\beta_{8}$$ $$=$$ $$( -4\nu^{14} - 2911\nu^{4} ) / 209$$ (-4*v^14 - 2911*v^4) / 209 $$\beta_{9}$$ $$=$$ $$( -4\nu^{15} - 2911\nu^{5} ) / 209$$ (-4*v^15 - 2911*v^5) / 209 $$\beta_{10}$$ $$=$$ $$( -7\nu^{14} - 5042\nu^{4} ) / 209$$ (-7*v^14 - 5042*v^4) / 209 $$\beta_{11}$$ $$=$$ $$( -\nu^{15} - 723\nu^{5} ) / 19$$ (-v^15 - 723*v^5) / 19 $$\beta_{12}$$ $$=$$ $$( 60\nu^{15} - 225\nu^{13} + 840\nu^{11} - 3135\nu^{9} + 11704\nu^{7} - 225\nu^{5} + 60\nu^{3} - 15\nu ) / 209$$ (60*v^15 - 225*v^13 + 840*v^11 - 3135*v^9 + 11704*v^7 - 225*v^5 + 60*v^3 - 15*v) / 209 $$\beta_{13}$$ $$=$$ $$( 56\nu^{14} - 224\nu^{12} + 840\nu^{10} - 3135\nu^{8} + 11704\nu^{6} - 3136\nu^{4} + 840\nu^{2} - 224 ) / 209$$ (56*v^14 - 224*v^12 + 840*v^10 - 3135*v^8 + 11704*v^6 - 3136*v^4 + 840*v^2 - 224) / 209 $$\beta_{14}$$ $$=$$ $$( 104\nu^{14} - 390\nu^{12} + 1456\nu^{10} - 5434\nu^{8} + 20273\nu^{6} - 390\nu^{4} + 104\nu^{2} - 26 ) / 209$$ (104*v^14 - 390*v^12 + 1456*v^10 - 5434*v^8 + 20273*v^6 - 390*v^4 + 104*v^2 - 26) / 209 $$\beta_{15}$$ $$=$$ $$( 164\nu^{15} - 615\nu^{13} + 2296\nu^{11} - 8569\nu^{9} + 31977\nu^{7} - 615\nu^{5} + 164\nu^{3} - 41\nu ) / 209$$ (164*v^15 - 615*v^13 + 2296*v^11 - 8569*v^9 + 31977*v^7 - 615*v^5 + 164*v^3 - 41*v) / 209
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{6} + 2\beta_{4}$$ b6 + 2*b4 $$\nu^{3}$$ $$=$$ $$\beta_{7} + 4\beta_{5}$$ b7 + 4*b5 $$\nu^{4}$$ $$=$$ $$4\beta_{10} - 7\beta_{8}$$ 4*b10 - 7*b8 $$\nu^{5}$$ $$=$$ $$4\beta_{11} - 11\beta_{9}$$ 4*b11 - 11*b9 $$\nu^{6}$$ $$=$$ $$-15\beta_{14} + 26\beta_{13} - 26\beta_{8} - 26\beta_{4} + 26$$ -15*b14 + 26*b13 - 26*b8 - 26*b4 + 26 $$\nu^{7}$$ $$=$$ $$-15\beta_{15} + 41\beta_{12}$$ -15*b15 + 41*b12 $$\nu^{8}$$ $$=$$ $$-56\beta_{14} + 97\beta_{13} - 56\beta_{10} + 56\beta_{6} + 56\beta_{2}$$ -56*b14 + 97*b13 - 56*b10 + 56*b6 + 56*b2 $$\nu^{9}$$ $$=$$ $$-56\beta_{15} + 153\beta_{12} - 56\beta_{11} + 153\beta_{9} + 56\beta_{7} + 209\beta_{5} + 56\beta_{3} - 209\beta_1$$ -56*b15 + 153*b12 - 56*b11 + 153*b9 + 56*b7 + 209*b5 + 56*b3 - 209*b1 $$\nu^{10}$$ $$=$$ $$209\beta_{2} - 362$$ 209*b2 - 362 $$\nu^{11}$$ $$=$$ $$209\beta_{3} - 780\beta_1$$ 209*b3 - 780*b1 $$\nu^{12}$$ $$=$$ $$-780\beta_{6} - 1351\beta_{4}$$ -780*b6 - 1351*b4 $$\nu^{13}$$ $$=$$ $$-780\beta_{7} - 2911\beta_{5}$$ -780*b7 - 2911*b5 $$\nu^{14}$$ $$=$$ $$-2911\beta_{10} + 5042\beta_{8}$$ -2911*b10 + 5042*b8 $$\nu^{15}$$ $$=$$ $$-2911\beta_{11} + 7953\beta_{9}$$ -2911*b11 + 7953*b9

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3267\mathbb{Z}\right)^\times$$.

 $$n$$ $$244$$ $$3026$$ $$\chi(n)$$ $$\beta_{4}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
838.1
 −1.83730 − 0.596975i −0.492303 − 0.159959i 0.492303 + 0.159959i 1.83730 + 0.596975i 1.13551 + 1.56290i 0.304260 + 0.418778i −0.304260 − 0.418778i −1.13551 − 1.56290i 1.13551 − 1.56290i 0.304260 − 0.418778i −0.304260 + 0.418778i −1.13551 + 1.56290i −1.83730 + 0.596975i −0.492303 + 0.159959i 0.492303 − 0.159959i 1.83730 − 0.596975i
0 0 −0.809017 0.587785i 0 0 −1.13551 + 1.56290i 0 0 0
838.2 0 0 −0.809017 0.587785i 0 0 −0.304260 + 0.418778i 0 0 0
838.3 0 0 −0.809017 0.587785i 0 0 0.304260 0.418778i 0 0 0
838.4 0 0 −0.809017 0.587785i 0 0 1.13551 1.56290i 0 0 0
2296.1 0 0 0.309017 0.951057i 0 0 −1.83730 0.596975i 0 0 0
2296.2 0 0 0.309017 0.951057i 0 0 −0.492303 0.159959i 0 0 0
2296.3 0 0 0.309017 0.951057i 0 0 0.492303 + 0.159959i 0 0 0
2296.4 0 0 0.309017 0.951057i 0 0 1.83730 + 0.596975i 0 0 0
2944.1 0 0 0.309017 + 0.951057i 0 0 −1.83730 + 0.596975i 0 0 0
2944.2 0 0 0.309017 + 0.951057i 0 0 −0.492303 + 0.159959i 0 0 0
2944.3 0 0 0.309017 + 0.951057i 0 0 0.492303 0.159959i 0 0 0
2944.4 0 0 0.309017 + 0.951057i 0 0 1.83730 0.596975i 0 0 0
2998.1 0 0 −0.809017 + 0.587785i 0 0 −1.13551 1.56290i 0 0 0
2998.2 0 0 −0.809017 + 0.587785i 0 0 −0.304260 0.418778i 0 0 0
2998.3 0 0 −0.809017 + 0.587785i 0 0 0.304260 + 0.418778i 0 0 0
2998.4 0 0 −0.809017 + 0.587785i 0 0 1.13551 + 1.56290i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 2998.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by $$\Q(\sqrt{-3})$$
11.b odd 2 1 inner
11.c even 5 3 inner
11.d odd 10 3 inner
33.d even 2 1 inner
33.f even 10 3 inner
33.h odd 10 3 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3267.1.l.a 16
3.b odd 2 1 CM 3267.1.l.a 16
11.b odd 2 1 inner 3267.1.l.a 16
11.c even 5 1 3267.1.c.a 4
11.c even 5 3 inner 3267.1.l.a 16
11.d odd 10 1 3267.1.c.a 4
11.d odd 10 3 inner 3267.1.l.a 16
33.d even 2 1 inner 3267.1.l.a 16
33.f even 10 1 3267.1.c.a 4
33.f even 10 3 inner 3267.1.l.a 16
33.h odd 10 1 3267.1.c.a 4
33.h odd 10 3 inner 3267.1.l.a 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3267.1.c.a 4 11.c even 5 1
3267.1.c.a 4 11.d odd 10 1
3267.1.c.a 4 33.f even 10 1
3267.1.c.a 4 33.h odd 10 1
3267.1.l.a 16 1.a even 1 1 trivial
3267.1.l.a 16 3.b odd 2 1 CM
3267.1.l.a 16 11.b odd 2 1 inner
3267.1.l.a 16 11.c even 5 3 inner
3267.1.l.a 16 11.d odd 10 3 inner
3267.1.l.a 16 33.d even 2 1 inner
3267.1.l.a 16 33.f even 10 3 inner
3267.1.l.a 16 33.h odd 10 3 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(3267, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{16}$$
$3$ $$T^{16}$$
$5$ $$T^{16}$$
$7$ $$T^{16} - 4 T^{14} + 15 T^{12} - 56 T^{10} + \cdots + 1$$
$11$ $$T^{16}$$
$13$ $$T^{16} - 4 T^{14} + 15 T^{12} - 56 T^{10} + \cdots + 1$$
$17$ $$T^{16}$$
$19$ $$(T^{8} - 2 T^{6} + 4 T^{4} - 8 T^{2} + 16)^{2}$$
$23$ $$T^{16}$$
$29$ $$T^{16}$$
$31$ $$(T^{8} + 3 T^{6} + 9 T^{4} + 27 T^{2} + \cdots + 81)^{2}$$
$37$ $$T^{16}$$
$41$ $$T^{16}$$
$43$ $$(T^{2} + 2)^{8}$$
$47$ $$T^{16}$$
$53$ $$T^{16}$$
$59$ $$T^{16}$$
$61$ $$(T^{8} - 2 T^{6} + 4 T^{4} - 8 T^{2} + 16)^{2}$$
$67$ $$(T^{2} - 3)^{8}$$
$71$ $$T^{16}$$
$73$ $$T^{16} - 4 T^{14} + 15 T^{12} - 56 T^{10} + \cdots + 1$$
$79$ $$T^{16} - 4 T^{14} + 15 T^{12} - 56 T^{10} + \cdots + 1$$
$83$ $$T^{16}$$
$89$ $$T^{16}$$
$97$ $$(T^{4} + T^{3} + T^{2} + T + 1)^{4}$$