# Properties

 Label 11.15.b.a Level $11$ Weight $15$ Character orbit 11.b Self dual yes Analytic conductor $13.676$ 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] = [11,15,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, 15, names="a")

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

chi := DirichletCharacter("11.10");

S:= CuspForms(chi, 15);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$11$$ Weight: $$k$$ $$=$$ $$15$$ 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: $$13.6761864967$$ 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)

 $$f(q)$$ $$=$$ $$q + 2515 q^{3} + 16384 q^{4} + 100799 q^{5} + 1542256 q^{9}+O(q^{10})$$ q + 2515 * q^3 + 16384 * q^4 + 100799 * q^5 + 1542256 * q^9 $$q + 2515 q^{3} + 16384 q^{4} + 100799 q^{5} + 1542256 q^{9} - 19487171 q^{11} + 41205760 q^{12} + 253509485 q^{15} + 268435456 q^{16} + 1651490816 q^{20} + 1556561195 q^{23} + 4056922776 q^{25} - 8150393195 q^{27} - 53499153997 q^{31} - 49010235065 q^{33} + 25268322304 q^{36} + 126509871575 q^{37} - 319277809664 q^{44} + 155457862544 q^{45} + 715778884850 q^{47} + 675115171840 q^{48} + 678223072849 q^{49} - 2217378708790 q^{53} - 1964287349629 q^{55} - 4954816467613 q^{59} + 4153499402240 q^{60} + 4398046511104 q^{64} - 11652648832405 q^{67} + 3914751405425 q^{69} + 14633687116307 q^{71} + 10203160781640 q^{75} + 27058025529344 q^{80} - 27874801523489 q^{81} - 68889823168417 q^{89} + 25502698618880 q^{92} - 134550372302455 q^{93} + 68911099629215 q^{97} - 30054206397776 q^{99}+O(q^{100})$$ q + 2515 * q^3 + 16384 * q^4 + 100799 * q^5 + 1542256 * q^9 - 19487171 * q^11 + 41205760 * q^12 + 253509485 * q^15 + 268435456 * q^16 + 1651490816 * q^20 + 1556561195 * q^23 + 4056922776 * q^25 - 8150393195 * q^27 - 53499153997 * q^31 - 49010235065 * q^33 + 25268322304 * q^36 + 126509871575 * q^37 - 319277809664 * q^44 + 155457862544 * q^45 + 715778884850 * q^47 + 675115171840 * q^48 + 678223072849 * q^49 - 2217378708790 * q^53 - 1964287349629 * q^55 - 4954816467613 * q^59 + 4153499402240 * q^60 + 4398046511104 * q^64 - 11652648832405 * q^67 + 3914751405425 * q^69 + 14633687116307 * q^71 + 10203160781640 * q^75 + 27058025529344 * q^80 - 27874801523489 * q^81 - 68889823168417 * q^89 + 25502698618880 * q^92 - 134550372302455 * q^93 + 68911099629215 * q^97 - 30054206397776 * 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 2515.00 16384.0 100799. 0 0 0 1.54226e6 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 11.15.b.a 1
3.b odd 2 1 99.15.c.a 1
4.b odd 2 1 176.15.h.a 1
11.b odd 2 1 CM 11.15.b.a 1
33.d even 2 1 99.15.c.a 1
44.c even 2 1 176.15.h.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
11.15.b.a 1 1.a even 1 1 trivial
11.15.b.a 1 11.b odd 2 1 CM
99.15.c.a 1 3.b odd 2 1
99.15.c.a 1 33.d even 2 1
176.15.h.a 1 4.b odd 2 1
176.15.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 $$S_{15}^{\mathrm{new}}(11, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 2515$$
$5$ $$T - 100799$$
$7$ $$T$$
$11$ $$T + 19487171$$
$13$ $$T$$
$17$ $$T$$
$19$ $$T$$
$23$ $$T - 1556561195$$
$29$ $$T$$
$31$ $$T + 53499153997$$
$37$ $$T - 126509871575$$
$41$ $$T$$
$43$ $$T$$
$47$ $$T - 715778884850$$
$53$ $$T + 2217378708790$$
$59$ $$T + 4954816467613$$
$61$ $$T$$
$67$ $$T + 11652648832405$$
$71$ $$T - 14633687116307$$
$73$ $$T$$
$79$ $$T$$
$83$ $$T$$
$89$ $$T + 68889823168417$$
$97$ $$T - 68911099629215$$