# Properties

 Label 11.7.b.a Level $11$ Weight $7$ Character orbit 11.b Self dual yes Analytic conductor $2.531$ Analytic rank $0$ Dimension $1$ CM discriminant -11 Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [11,7,Mod(10,11)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(11, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 7, names="a")

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

chi := DirichletCharacter("11.10");

S:= CuspForms(chi, 7);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$11$$ Weight: $$k$$ $$=$$ $$7$$ Character orbit: $$[\chi]$$ $$=$$ 11.b (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: yes Analytic conductor: $$2.53059491982$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q + 10 q^{3} + 64 q^{4} + 74 q^{5} - 629 q^{9}+O(q^{10})$$ q + 10 * q^3 + 64 * q^4 + 74 * q^5 - 629 * q^9 $$q + 10 q^{3} + 64 q^{4} + 74 q^{5} - 629 q^{9} - 1331 q^{11} + 640 q^{12} + 740 q^{15} + 4096 q^{16} + 4736 q^{20} - 12670 q^{23} - 10149 q^{25} - 13580 q^{27} + 56018 q^{31} - 13310 q^{33} - 40256 q^{36} + 87050 q^{37} - 85184 q^{44} - 46546 q^{45} - 206350 q^{47} + 40960 q^{48} + 117649 q^{49} + 246890 q^{53} - 98494 q^{55} + 107642 q^{59} + 47360 q^{60} + 262144 q^{64} - 428470 q^{67} - 126700 q^{69} - 341278 q^{71} - 101490 q^{75} + 303104 q^{80} + 322741 q^{81} + 1392338 q^{89} - 810880 q^{92} + 560180 q^{93} - 1824190 q^{97} + 837199 q^{99}+O(q^{100})$$ q + 10 * q^3 + 64 * q^4 + 74 * q^5 - 629 * q^9 - 1331 * q^11 + 640 * q^12 + 740 * q^15 + 4096 * q^16 + 4736 * q^20 - 12670 * q^23 - 10149 * q^25 - 13580 * q^27 + 56018 * q^31 - 13310 * q^33 - 40256 * q^36 + 87050 * q^37 - 85184 * q^44 - 46546 * q^45 - 206350 * q^47 + 40960 * q^48 + 117649 * q^49 + 246890 * q^53 - 98494 * q^55 + 107642 * q^59 + 47360 * q^60 + 262144 * q^64 - 428470 * q^67 - 126700 * q^69 - 341278 * q^71 - 101490 * q^75 + 303104 * q^80 + 322741 * q^81 + 1392338 * q^89 - 810880 * q^92 + 560180 * q^93 - 1824190 * q^97 + 837199 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/11\mathbb{Z}\right)^\times$$.

 $$n$$ $$2$$ $$\chi(n)$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
10.1
 0
0 10.0000 64.0000 74.0000 0 0 0 −629.000 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
11.b odd 2 1 CM by $$\Q(\sqrt{-11})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 11.7.b.a 1
3.b odd 2 1 99.7.c.a 1
4.b odd 2 1 176.7.h.a 1
11.b odd 2 1 CM 11.7.b.a 1
33.d even 2 1 99.7.c.a 1
44.c even 2 1 176.7.h.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
11.7.b.a 1 1.a even 1 1 trivial
11.7.b.a 1 11.b odd 2 1 CM
99.7.c.a 1 3.b odd 2 1
99.7.c.a 1 33.d even 2 1
176.7.h.a 1 4.b odd 2 1
176.7.h.a 1 44.c even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}$$ acting on $$S_{7}^{\mathrm{new}}(11, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 10$$
$5$ $$T - 74$$
$7$ $$T$$
$11$ $$T + 1331$$
$13$ $$T$$
$17$ $$T$$
$19$ $$T$$
$23$ $$T + 12670$$
$29$ $$T$$
$31$ $$T - 56018$$
$37$ $$T - 87050$$
$41$ $$T$$
$43$ $$T$$
$47$ $$T + 206350$$
$53$ $$T - 246890$$
$59$ $$T - 107642$$
$61$ $$T$$
$67$ $$T + 428470$$
$71$ $$T + 341278$$
$73$ $$T$$
$79$ $$T$$
$83$ $$T$$
$89$ $$T - 1392338$$
$97$ $$T + 1824190$$