# Properties

 Label 1911.1.cc.a Level $1911$ Weight $1$ Character orbit 1911.cc Analytic conductor $0.954$ Analytic rank $0$ Dimension $4$ Projective image $D_{4}$ CM discriminant -3 Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1911 = 3 \cdot 7^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1911.cc (of order $$12$$, degree $$4$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.953713239142$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 273) Projective image: $$D_{4}$$ Projective field: Galois closure of 4.0.968877.1 Artin image: $C_6\times C_4\wr C_2$ Artin field: Galois closure of $$\mathbb{Q}[x]/(x^{48} - \cdots)$$

## $q$-expansion

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q -\zeta_{12} q^{3} + \zeta_{12} q^{4} + \zeta_{12}^{2} q^{9} +O(q^{10})$$ $$q -\zeta_{12} q^{3} + \zeta_{12} q^{4} + \zeta_{12}^{2} q^{9} -\zeta_{12}^{2} q^{12} + \zeta_{12}^{3} q^{13} + \zeta_{12}^{2} q^{16} + ( \zeta_{12}^{2} + \zeta_{12}^{5} ) q^{19} -\zeta_{12} q^{25} -\zeta_{12}^{3} q^{27} + ( \zeta_{12} - \zeta_{12}^{4} ) q^{31} + \zeta_{12}^{3} q^{36} + ( \zeta_{12}^{2} + \zeta_{12}^{5} ) q^{37} -\zeta_{12}^{4} q^{39} -\zeta_{12}^{3} q^{48} + \zeta_{12}^{4} q^{52} + ( 1 - \zeta_{12}^{3} ) q^{57} -2 \zeta_{12}^{5} q^{61} + \zeta_{12}^{3} q^{64} + ( -\zeta_{12} - \zeta_{12}^{4} ) q^{67} + ( \zeta_{12} + \zeta_{12}^{4} ) q^{73} + \zeta_{12}^{2} q^{75} + ( -1 + \zeta_{12}^{3} ) q^{76} + \zeta_{12}^{4} q^{81} + ( -\zeta_{12}^{2} + \zeta_{12}^{5} ) q^{93} + ( 1 + \zeta_{12}^{3} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{9} + O(q^{10})$$ $$4 q + 2 q^{9} - 2 q^{12} + 2 q^{16} + 2 q^{19} + 2 q^{31} + 2 q^{37} + 2 q^{39} - 2 q^{52} + 4 q^{57} + 2 q^{67} - 2 q^{73} + 2 q^{75} - 4 q^{76} - 2 q^{81} - 2 q^{93} + 4 q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1911\mathbb{Z}\right)^\times$$.

 $$n$$ $$638$$ $$1471$$ $$1522$$ $$\chi(n)$$ $$-1$$ $$-\zeta_{12}^{3}$$ $$\zeta_{12}^{2}$$

## 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.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
668.1
 0.866025 + 0.500000i −0.866025 + 0.500000i −0.866025 − 0.500000i 0.866025 − 0.500000i
0 −0.866025 0.500000i 0.866025 + 0.500000i 0 0 0 0 0.500000 + 0.866025i 0
1097.1 0 0.866025 0.500000i −0.866025 + 0.500000i 0 0 0 0 0.500000 0.866025i 0
1256.1 0 0.866025 + 0.500000i −0.866025 0.500000i 0 0 0 0 0.500000 + 0.866025i 0
1685.1 0 −0.866025 + 0.500000i 0.866025 0.500000i 0 0 0 0 0.500000 0.866025i 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
3.b odd 2 1 CM by $$\Q(\sqrt{-3})$$
7.c even 3 1 inner
21.h odd 6 1 inner
91.i even 4 1 inner
91.bb even 12 1 inner
273.o odd 4 1 inner
273.cb odd 12 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1911.1.cc.a 4
3.b odd 2 1 CM 1911.1.cc.a 4
7.b odd 2 1 1911.1.cc.b 4
7.c even 3 1 273.1.o.b yes 2
7.c even 3 1 inner 1911.1.cc.a 4
7.d odd 6 1 273.1.o.a 2
7.d odd 6 1 1911.1.cc.b 4
13.d odd 4 1 1911.1.cc.b 4
21.c even 2 1 1911.1.cc.b 4
21.g even 6 1 273.1.o.a 2
21.g even 6 1 1911.1.cc.b 4
21.h odd 6 1 273.1.o.b yes 2
21.h odd 6 1 inner 1911.1.cc.a 4
39.f even 4 1 1911.1.cc.b 4
91.g even 3 1 3549.1.ca.a 4
91.h even 3 1 3549.1.ca.a 4
91.i even 4 1 inner 1911.1.cc.a 4
91.k even 6 1 3549.1.ca.d 4
91.l odd 6 1 3549.1.ca.c 4
91.m odd 6 1 3549.1.ca.b 4
91.p odd 6 1 3549.1.ca.c 4
91.r even 6 1 3549.1.o.a 2
91.s odd 6 1 3549.1.o.b 2
91.u even 6 1 3549.1.ca.d 4
91.v odd 6 1 3549.1.ca.b 4
91.w even 12 1 3549.1.ca.a 4
91.w even 12 1 3549.1.ca.d 4
91.x odd 12 1 3549.1.ca.b 4
91.x odd 12 1 3549.1.ca.c 4
91.z odd 12 1 273.1.o.a 2
91.z odd 12 1 1911.1.cc.b 4
91.z odd 12 1 3549.1.o.b 2
91.ba even 12 1 3549.1.ca.a 4
91.ba even 12 1 3549.1.ca.d 4
91.bb even 12 1 273.1.o.b yes 2
91.bb even 12 1 inner 1911.1.cc.a 4
91.bb even 12 1 3549.1.o.a 2
91.bd odd 12 1 3549.1.ca.b 4
91.bd odd 12 1 3549.1.ca.c 4
273.o odd 4 1 inner 1911.1.cc.a 4
273.r even 6 1 3549.1.ca.b 4
273.s odd 6 1 3549.1.ca.a 4
273.w odd 6 1 3549.1.o.a 2
273.x odd 6 1 3549.1.ca.d 4
273.y even 6 1 3549.1.ca.c 4
273.ba even 6 1 3549.1.o.b 2
273.bf even 6 1 3549.1.ca.b 4
273.bm odd 6 1 3549.1.ca.a 4
273.bp odd 6 1 3549.1.ca.d 4
273.br even 6 1 3549.1.ca.c 4
273.bs odd 12 1 3549.1.ca.a 4
273.bs odd 12 1 3549.1.ca.d 4
273.bv even 12 1 3549.1.ca.b 4
273.bv even 12 1 3549.1.ca.c 4
273.bw even 12 1 3549.1.ca.b 4
273.bw even 12 1 3549.1.ca.c 4
273.cb odd 12 1 273.1.o.b yes 2
273.cb odd 12 1 inner 1911.1.cc.a 4
273.cb odd 12 1 3549.1.o.a 2
273.cd even 12 1 273.1.o.a 2
273.cd even 12 1 1911.1.cc.b 4
273.cd even 12 1 3549.1.o.b 2
273.ch odd 12 1 3549.1.ca.a 4
273.ch odd 12 1 3549.1.ca.d 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.1.o.a 2 7.d odd 6 1
273.1.o.a 2 21.g even 6 1
273.1.o.a 2 91.z odd 12 1
273.1.o.a 2 273.cd even 12 1
273.1.o.b yes 2 7.c even 3 1
273.1.o.b yes 2 21.h odd 6 1
273.1.o.b yes 2 91.bb even 12 1
273.1.o.b yes 2 273.cb odd 12 1
1911.1.cc.a 4 1.a even 1 1 trivial
1911.1.cc.a 4 3.b odd 2 1 CM
1911.1.cc.a 4 7.c even 3 1 inner
1911.1.cc.a 4 21.h odd 6 1 inner
1911.1.cc.a 4 91.i even 4 1 inner
1911.1.cc.a 4 91.bb even 12 1 inner
1911.1.cc.a 4 273.o odd 4 1 inner
1911.1.cc.a 4 273.cb odd 12 1 inner
1911.1.cc.b 4 7.b odd 2 1
1911.1.cc.b 4 7.d odd 6 1
1911.1.cc.b 4 13.d odd 4 1
1911.1.cc.b 4 21.c even 2 1
1911.1.cc.b 4 21.g even 6 1
1911.1.cc.b 4 39.f even 4 1
1911.1.cc.b 4 91.z odd 12 1
1911.1.cc.b 4 273.cd even 12 1
3549.1.o.a 2 91.r even 6 1
3549.1.o.a 2 91.bb even 12 1
3549.1.o.a 2 273.w odd 6 1
3549.1.o.a 2 273.cb odd 12 1
3549.1.o.b 2 91.s odd 6 1
3549.1.o.b 2 91.z odd 12 1
3549.1.o.b 2 273.ba even 6 1
3549.1.o.b 2 273.cd even 12 1
3549.1.ca.a 4 91.g even 3 1
3549.1.ca.a 4 91.h even 3 1
3549.1.ca.a 4 91.w even 12 1
3549.1.ca.a 4 91.ba even 12 1
3549.1.ca.a 4 273.s odd 6 1
3549.1.ca.a 4 273.bm odd 6 1
3549.1.ca.a 4 273.bs odd 12 1
3549.1.ca.a 4 273.ch odd 12 1
3549.1.ca.b 4 91.m odd 6 1
3549.1.ca.b 4 91.v odd 6 1
3549.1.ca.b 4 91.x odd 12 1
3549.1.ca.b 4 91.bd odd 12 1
3549.1.ca.b 4 273.r even 6 1
3549.1.ca.b 4 273.bf even 6 1
3549.1.ca.b 4 273.bv even 12 1
3549.1.ca.b 4 273.bw even 12 1
3549.1.ca.c 4 91.l odd 6 1
3549.1.ca.c 4 91.p odd 6 1
3549.1.ca.c 4 91.x odd 12 1
3549.1.ca.c 4 91.bd odd 12 1
3549.1.ca.c 4 273.y even 6 1
3549.1.ca.c 4 273.br even 6 1
3549.1.ca.c 4 273.bv even 12 1
3549.1.ca.c 4 273.bw even 12 1
3549.1.ca.d 4 91.k even 6 1
3549.1.ca.d 4 91.u even 6 1
3549.1.ca.d 4 91.w even 12 1
3549.1.ca.d 4 91.ba even 12 1
3549.1.ca.d 4 273.x odd 6 1
3549.1.ca.d 4 273.bp odd 6 1
3549.1.ca.d 4 273.bs odd 12 1
3549.1.ca.d 4 273.ch odd 12 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{19}^{4} - 2 T_{19}^{3} + 2 T_{19}^{2} - 4 T_{19} + 4$$ acting on $$S_{1}^{\mathrm{new}}(1911, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$1 - T^{2} + T^{4}$$
$5$ $$T^{4}$$
$7$ $$T^{4}$$
$11$ $$T^{4}$$
$13$ $$( 1 + T^{2} )^{2}$$
$17$ $$T^{4}$$
$19$ $$4 - 4 T + 2 T^{2} - 2 T^{3} + T^{4}$$
$23$ $$T^{4}$$
$29$ $$T^{4}$$
$31$ $$4 - 4 T + 2 T^{2} - 2 T^{3} + T^{4}$$
$37$ $$4 - 4 T + 2 T^{2} - 2 T^{3} + T^{4}$$
$41$ $$T^{4}$$
$43$ $$T^{4}$$
$47$ $$T^{4}$$
$53$ $$T^{4}$$
$59$ $$T^{4}$$
$61$ $$16 - 4 T^{2} + T^{4}$$
$67$ $$4 - 4 T + 2 T^{2} - 2 T^{3} + T^{4}$$
$71$ $$T^{4}$$
$73$ $$4 + 4 T + 2 T^{2} + 2 T^{3} + T^{4}$$
$79$ $$T^{4}$$
$83$ $$T^{4}$$
$89$ $$T^{4}$$
$97$ $$( 2 - 2 T + T^{2} )^{2}$$