# Properties

 Label 1911.1.be.c Level $1911$ Weight $1$ Character orbit 1911.be Analytic conductor $0.954$ Analytic rank $0$ Dimension $4$ Projective image $A_{4}$ CM/RM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1911 = 3 \cdot 7^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1911.be (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.953713239142$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 273) Projective image: $$A_{4}$$ Projective field: Galois closure of 4.0.74529.1

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + \zeta_{12}^{5} q^{2} -\zeta_{12}^{5} q^{3} -\zeta_{12}^{3} q^{5} + \zeta_{12}^{4} q^{6} -\zeta_{12}^{3} q^{8} -\zeta_{12}^{4} q^{9} +O(q^{10})$$ $$q + \zeta_{12}^{5} q^{2} -\zeta_{12}^{5} q^{3} -\zeta_{12}^{3} q^{5} + \zeta_{12}^{4} q^{6} -\zeta_{12}^{3} q^{8} -\zeta_{12}^{4} q^{9} + \zeta_{12}^{2} q^{10} - q^{13} -\zeta_{12}^{2} q^{15} + \zeta_{12}^{2} q^{16} -\zeta_{12} q^{17} + \zeta_{12}^{3} q^{18} -\zeta_{12}^{5} q^{23} -\zeta_{12}^{2} q^{24} -\zeta_{12}^{5} q^{26} -\zeta_{12}^{3} q^{27} -\zeta_{12}^{5} q^{29} + \zeta_{12} q^{30} - q^{31} + q^{34} -\zeta_{12}^{2} q^{37} + \zeta_{12}^{5} q^{39} - q^{40} + \zeta_{12}^{5} q^{41} + \zeta_{12}^{4} q^{43} -\zeta_{12} q^{45} + \zeta_{12}^{4} q^{46} -\zeta_{12}^{3} q^{47} + \zeta_{12} q^{48} - q^{51} -\zeta_{12}^{3} q^{53} + \zeta_{12}^{2} q^{54} + \zeta_{12}^{4} q^{58} + \zeta_{12} q^{59} -\zeta_{12}^{5} q^{62} - q^{64} + \zeta_{12}^{3} q^{65} -\zeta_{12}^{4} q^{69} -\zeta_{12} q^{71} -\zeta_{12} q^{72} + q^{73} + \zeta_{12} q^{74} -\zeta_{12}^{4} q^{78} + q^{79} -\zeta_{12}^{5} q^{80} -\zeta_{12}^{2} q^{81} -\zeta_{12}^{4} q^{82} + \zeta_{12}^{4} q^{85} -\zeta_{12}^{3} q^{86} -\zeta_{12}^{4} q^{87} + \zeta_{12}^{5} q^{89} + q^{90} + \zeta_{12}^{5} q^{93} + \zeta_{12}^{2} q^{94} -\zeta_{12}^{4} q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 2 q^{6} + 2 q^{9} + O(q^{10})$$ $$4 q - 2 q^{6} + 2 q^{9} + 2 q^{10} - 4 q^{13} - 2 q^{15} + 2 q^{16} - 2 q^{24} - 4 q^{31} + 4 q^{34} - 2 q^{37} - 4 q^{40} - 2 q^{43} - 2 q^{46} - 4 q^{51} + 2 q^{54} - 2 q^{58} - 4 q^{64} + 2 q^{69} + 4 q^{73} + 2 q^{78} + 4 q^{79} - 2 q^{81} + 2 q^{82} - 2 q^{85} + 2 q^{87} + 4 q^{90} + 2 q^{94} + 2 q^{97} + O(q^{100})$$

## Character values

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

 $$n$$ $$638$$ $$1471$$ $$1522$$ $$\chi(n)$$ $$-1$$ $$\zeta_{12}^{4}$$ $$1$$

## 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.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
932.1
 0.866025 − 0.500000i −0.866025 + 0.500000i 0.866025 + 0.500000i −0.866025 − 0.500000i
−0.866025 0.500000i 0.866025 + 0.500000i 0 1.00000i −0.500000 0.866025i 0 1.00000i 0.500000 + 0.866025i 0.500000 0.866025i
932.2 0.866025 + 0.500000i −0.866025 0.500000i 0 1.00000i −0.500000 0.866025i 0 1.00000i 0.500000 + 0.866025i 0.500000 0.866025i
1667.1 −0.866025 + 0.500000i 0.866025 0.500000i 0 1.00000i −0.500000 + 0.866025i 0 1.00000i 0.500000 0.866025i 0.500000 + 0.866025i
1667.2 0.866025 0.500000i −0.866025 + 0.500000i 0 1.00000i −0.500000 + 0.866025i 0 1.00000i 0.500000 0.866025i 0.500000 + 0.866025i
 $$n$$: 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 inner
13.c even 3 1 inner
39.i odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1911.1.be.c 4
3.b odd 2 1 inner 1911.1.be.c 4
7.b odd 2 1 1911.1.be.d 4
7.c even 3 1 273.1.s.b 4
7.c even 3 1 273.1.bm.b yes 4
7.d odd 6 1 1911.1.s.b 4
7.d odd 6 1 1911.1.bm.b 4
13.c even 3 1 inner 1911.1.be.c 4
21.c even 2 1 1911.1.be.d 4
21.g even 6 1 1911.1.s.b 4
21.g even 6 1 1911.1.bm.b 4
21.h odd 6 1 273.1.s.b 4
21.h odd 6 1 273.1.bm.b yes 4
39.i odd 6 1 inner 1911.1.be.c 4
91.g even 3 1 273.1.s.b 4
91.g even 3 1 3549.1.bk.d 4
91.h even 3 1 273.1.bm.b yes 4
91.h even 3 1 3549.1.bk.d 4
91.k even 6 1 3549.1.bk.c 4
91.k even 6 1 3549.1.bm.c 4
91.m odd 6 1 1911.1.s.b 4
91.n odd 6 1 1911.1.be.d 4
91.r even 6 1 3549.1.s.b 4
91.r even 6 1 3549.1.bm.c 4
91.u even 6 1 3549.1.s.b 4
91.u even 6 1 3549.1.bk.c 4
91.v odd 6 1 1911.1.bm.b 4
91.x odd 12 1 3549.1.w.c 4
91.x odd 12 1 3549.1.w.e 4
91.x odd 12 1 3549.1.x.b 4
91.x odd 12 1 3549.1.x.d 4
91.z odd 12 1 3549.1.x.b 4
91.z odd 12 1 3549.1.x.d 4
91.z odd 12 1 3549.1.bp.b 4
91.z odd 12 1 3549.1.bp.d 4
91.bd odd 12 1 3549.1.w.c 4
91.bd odd 12 1 3549.1.w.e 4
91.bd odd 12 1 3549.1.bp.b 4
91.bd odd 12 1 3549.1.bp.d 4
273.r even 6 1 1911.1.bm.b 4
273.s odd 6 1 273.1.bm.b yes 4
273.s odd 6 1 3549.1.bk.d 4
273.w odd 6 1 3549.1.s.b 4
273.w odd 6 1 3549.1.bm.c 4
273.x odd 6 1 3549.1.s.b 4
273.x odd 6 1 3549.1.bk.c 4
273.bf even 6 1 1911.1.s.b 4
273.bm odd 6 1 273.1.s.b 4
273.bm odd 6 1 3549.1.bk.d 4
273.bn even 6 1 1911.1.be.d 4
273.bp odd 6 1 3549.1.bk.c 4
273.bp odd 6 1 3549.1.bm.c 4
273.bv even 12 1 3549.1.w.c 4
273.bv even 12 1 3549.1.w.e 4
273.bv even 12 1 3549.1.x.b 4
273.bv even 12 1 3549.1.x.d 4
273.bw even 12 1 3549.1.w.c 4
273.bw even 12 1 3549.1.w.e 4
273.bw even 12 1 3549.1.bp.b 4
273.bw even 12 1 3549.1.bp.d 4
273.cd even 12 1 3549.1.x.b 4
273.cd even 12 1 3549.1.x.d 4
273.cd even 12 1 3549.1.bp.b 4
273.cd even 12 1 3549.1.bp.d 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.1.s.b 4 7.c even 3 1
273.1.s.b 4 21.h odd 6 1
273.1.s.b 4 91.g even 3 1
273.1.s.b 4 273.bm odd 6 1
273.1.bm.b yes 4 7.c even 3 1
273.1.bm.b yes 4 21.h odd 6 1
273.1.bm.b yes 4 91.h even 3 1
273.1.bm.b yes 4 273.s odd 6 1
1911.1.s.b 4 7.d odd 6 1
1911.1.s.b 4 21.g even 6 1
1911.1.s.b 4 91.m odd 6 1
1911.1.s.b 4 273.bf even 6 1
1911.1.be.c 4 1.a even 1 1 trivial
1911.1.be.c 4 3.b odd 2 1 inner
1911.1.be.c 4 13.c even 3 1 inner
1911.1.be.c 4 39.i odd 6 1 inner
1911.1.be.d 4 7.b odd 2 1
1911.1.be.d 4 21.c even 2 1
1911.1.be.d 4 91.n odd 6 1
1911.1.be.d 4 273.bn even 6 1
1911.1.bm.b 4 7.d odd 6 1
1911.1.bm.b 4 21.g even 6 1
1911.1.bm.b 4 91.v odd 6 1
1911.1.bm.b 4 273.r even 6 1
3549.1.s.b 4 91.r even 6 1
3549.1.s.b 4 91.u even 6 1
3549.1.s.b 4 273.w odd 6 1
3549.1.s.b 4 273.x odd 6 1
3549.1.w.c 4 91.x odd 12 1
3549.1.w.c 4 91.bd odd 12 1
3549.1.w.c 4 273.bv even 12 1
3549.1.w.c 4 273.bw even 12 1
3549.1.w.e 4 91.x odd 12 1
3549.1.w.e 4 91.bd odd 12 1
3549.1.w.e 4 273.bv even 12 1
3549.1.w.e 4 273.bw even 12 1
3549.1.x.b 4 91.x odd 12 1
3549.1.x.b 4 91.z odd 12 1
3549.1.x.b 4 273.bv even 12 1
3549.1.x.b 4 273.cd even 12 1
3549.1.x.d 4 91.x odd 12 1
3549.1.x.d 4 91.z odd 12 1
3549.1.x.d 4 273.bv even 12 1
3549.1.x.d 4 273.cd even 12 1
3549.1.bk.c 4 91.k even 6 1
3549.1.bk.c 4 91.u even 6 1
3549.1.bk.c 4 273.x odd 6 1
3549.1.bk.c 4 273.bp odd 6 1
3549.1.bk.d 4 91.g even 3 1
3549.1.bk.d 4 91.h even 3 1
3549.1.bk.d 4 273.s odd 6 1
3549.1.bk.d 4 273.bm odd 6 1
3549.1.bm.c 4 91.k even 6 1
3549.1.bm.c 4 91.r even 6 1
3549.1.bm.c 4 273.w odd 6 1
3549.1.bm.c 4 273.bp odd 6 1
3549.1.bp.b 4 91.z odd 12 1
3549.1.bp.b 4 91.bd odd 12 1
3549.1.bp.b 4 273.bw even 12 1
3549.1.bp.b 4 273.cd even 12 1
3549.1.bp.d 4 91.z odd 12 1
3549.1.bp.d 4 91.bd odd 12 1
3549.1.bp.d 4 273.bw even 12 1
3549.1.bp.d 4 273.cd even 12 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{1}^{\mathrm{new}}(1911, [\chi])$$:

 $$T_{2}^{4} - T_{2}^{2} + 1$$ $$T_{19}$$ $$T_{31} + 1$$

## Hecke characteristic polynomials

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