# Properties

 Label 1911.1.w.f Level $1911$ Weight $1$ Character orbit 1911.w Analytic conductor $0.954$ Analytic rank $0$ Dimension $8$ Projective image $D_{8}$ CM discriminant -39 Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1911 = 3 \cdot 7^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1911.w (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.953713239142$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{6})$$ Coefficient field: 8.0.339738624.2 Defining polynomial: $$x^{8} + 4 x^{6} + 14 x^{4} + 8 x^{2} + 4$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{8}$$ Projective field: Galois closure of 8.2.90724673403.2

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{3} q^{2} -\beta_{5} q^{3} + ( \beta_{5} - \beta_{6} ) q^{4} -\beta_{1} q^{5} -\beta_{7} q^{6} + ( \beta_{1} - \beta_{4} + \beta_{7} ) q^{8} + ( -1 - \beta_{5} ) q^{9} +O(q^{10})$$ $$q + \beta_{3} q^{2} -\beta_{5} q^{3} + ( \beta_{5} - \beta_{6} ) q^{4} -\beta_{1} q^{5} -\beta_{7} q^{6} + ( \beta_{1} - \beta_{4} + \beta_{7} ) q^{8} + ( -1 - \beta_{5} ) q^{9} -\beta_{6} q^{10} + ( \beta_{3} + \beta_{7} ) q^{11} + ( 1 - \beta_{2} + \beta_{5} - \beta_{6} ) q^{12} + q^{13} + ( -\beta_{1} + \beta_{4} ) q^{15} + ( -1 + \beta_{2} - \beta_{5} + \beta_{6} ) q^{16} + ( -\beta_{3} - \beta_{7} ) q^{18} + \beta_{7} q^{20} + ( -2 + \beta_{2} ) q^{22} + ( \beta_{3} - \beta_{4} + \beta_{7} ) q^{24} + ( \beta_{5} + \beta_{6} ) q^{25} + \beta_{3} q^{26} - q^{27} + ( -\beta_{2} - \beta_{6} ) q^{30} + ( -\beta_{3} - \beta_{7} ) q^{32} + \beta_{3} q^{33} + ( 1 - \beta_{2} ) q^{36} -\beta_{5} q^{39} + ( -2 - 2 \beta_{5} ) q^{40} + ( -\beta_{1} + \beta_{4} ) q^{41} + ( \beta_{1} - 2 \beta_{3} ) q^{44} + \beta_{4} q^{45} + \beta_{1} q^{47} + ( -1 + \beta_{2} ) q^{48} + ( -\beta_{1} + \beta_{4} ) q^{50} + ( \beta_{5} - \beta_{6} ) q^{52} -\beta_{3} q^{54} + \beta_{2} q^{55} + ( -\beta_{3} - \beta_{7} ) q^{59} + ( \beta_{3} + \beta_{7} ) q^{60} + q^{64} -\beta_{1} q^{65} + ( 2 \beta_{5} - \beta_{6} ) q^{66} + ( \beta_{1} - \beta_{4} ) q^{71} + ( -\beta_{1} + \beta_{3} ) q^{72} + ( 1 + \beta_{2} + \beta_{5} + \beta_{6} ) q^{75} -\beta_{7} q^{78} + ( -\beta_{2} - \beta_{6} ) q^{79} + ( -\beta_{3} - \beta_{7} ) q^{80} + \beta_{5} q^{81} + ( -\beta_{2} - \beta_{6} ) q^{82} -\beta_{7} q^{83} + ( -2 \beta_{5} + 2 \beta_{6} ) q^{88} -\beta_{3} q^{89} -\beta_{2} q^{90} + \beta_{6} q^{94} -\beta_{3} q^{96} -\beta_{7} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 4 q^{3} - 4 q^{4} - 4 q^{9} + O(q^{10})$$ $$8 q + 4 q^{3} - 4 q^{4} - 4 q^{9} + 4 q^{12} + 8 q^{13} - 4 q^{16} - 16 q^{22} - 4 q^{25} - 8 q^{27} + 8 q^{36} + 4 q^{39} - 8 q^{40} - 8 q^{48} - 4 q^{52} + 8 q^{64} - 8 q^{66} + 4 q^{75} - 4 q^{81} + 8 q^{88} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 4 x^{6} + 14 x^{4} + 8 x^{2} + 4$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{6} - 20$$$$)/14$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{7} - 34 \nu$$$$)/14$$ $$\beta_{4}$$ $$=$$ $$($$$$2 \nu^{7} + 7 \nu^{5} + 28 \nu^{3} + 16 \nu$$$$)/14$$ $$\beta_{5}$$ $$=$$ $$($$$$2 \nu^{6} + 7 \nu^{4} + 28 \nu^{2} + 2$$$$)/14$$ $$\beta_{6}$$ $$=$$ $$($$$$-2 \nu^{6} - 7 \nu^{4} - 21 \nu^{2} - 2$$$$)/7$$ $$\beta_{7}$$ $$=$$ $$($$$$-6 \nu^{7} - 21 \nu^{5} - 70 \nu^{3} - 6 \nu$$$$)/14$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{6} + 2 \beta_{5}$$ $$\nu^{3}$$ $$=$$ $$\beta_{7} + 3 \beta_{4} - 3 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$-4 \beta_{6} - 6 \beta_{5} - 4 \beta_{2} - 6$$ $$\nu^{5}$$ $$=$$ $$-4 \beta_{7} - 10 \beta_{4} - 4 \beta_{3}$$ $$\nu^{6}$$ $$=$$ $$14 \beta_{2} + 20$$ $$\nu^{7}$$ $$=$$ $$14 \beta_{3} + 34 \beta_{1}$$

## Character values

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

 $$n$$ $$638$$ $$1471$$ $$1522$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$-1 - \beta_{5}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
116.1
 0.382683 + 0.662827i −0.923880 − 1.60021i 0.923880 + 1.60021i −0.382683 − 0.662827i 0.382683 − 0.662827i −0.923880 + 1.60021i 0.923880 − 1.60021i −0.382683 + 0.662827i
−0.923880 1.60021i 0.500000 0.866025i −1.20711 + 2.09077i −0.382683 0.662827i −1.84776 0 2.61313 −0.500000 0.866025i −0.707107 + 1.22474i
116.2 −0.382683 0.662827i 0.500000 0.866025i 0.207107 0.358719i 0.923880 + 1.60021i −0.765367 0 −1.08239 −0.500000 0.866025i 0.707107 1.22474i
116.3 0.382683 + 0.662827i 0.500000 0.866025i 0.207107 0.358719i −0.923880 1.60021i 0.765367 0 1.08239 −0.500000 0.866025i 0.707107 1.22474i
116.4 0.923880 + 1.60021i 0.500000 0.866025i −1.20711 + 2.09077i 0.382683 + 0.662827i 1.84776 0 −2.61313 −0.500000 0.866025i −0.707107 + 1.22474i
1598.1 −0.923880 + 1.60021i 0.500000 + 0.866025i −1.20711 2.09077i −0.382683 + 0.662827i −1.84776 0 2.61313 −0.500000 + 0.866025i −0.707107 1.22474i
1598.2 −0.382683 + 0.662827i 0.500000 + 0.866025i 0.207107 + 0.358719i 0.923880 1.60021i −0.765367 0 −1.08239 −0.500000 + 0.866025i 0.707107 + 1.22474i
1598.3 0.382683 0.662827i 0.500000 + 0.866025i 0.207107 + 0.358719i −0.923880 + 1.60021i 0.765367 0 1.08239 −0.500000 + 0.866025i 0.707107 + 1.22474i
1598.4 0.923880 1.60021i 0.500000 + 0.866025i −1.20711 2.09077i 0.382683 0.662827i 1.84776 0 −2.61313 −0.500000 + 0.866025i −0.707107 1.22474i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1598.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
39.d odd 2 1 CM by $$\Q(\sqrt{-39})$$
3.b odd 2 1 inner
7.c even 3 1 inner
13.b even 2 1 inner
21.h odd 6 1 inner
91.r even 6 1 inner
273.w 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.w.f 8
3.b odd 2 1 inner 1911.1.w.f 8
7.b odd 2 1 1911.1.w.e 8
7.c even 3 1 1911.1.h.d 4
7.c even 3 1 inner 1911.1.w.f 8
7.d odd 6 1 1911.1.h.e yes 4
7.d odd 6 1 1911.1.w.e 8
13.b even 2 1 inner 1911.1.w.f 8
21.c even 2 1 1911.1.w.e 8
21.g even 6 1 1911.1.h.e yes 4
21.g even 6 1 1911.1.w.e 8
21.h odd 6 1 1911.1.h.d 4
21.h odd 6 1 inner 1911.1.w.f 8
39.d odd 2 1 CM 1911.1.w.f 8
91.b odd 2 1 1911.1.w.e 8
91.r even 6 1 1911.1.h.d 4
91.r even 6 1 inner 1911.1.w.f 8
91.s odd 6 1 1911.1.h.e yes 4
91.s odd 6 1 1911.1.w.e 8
273.g even 2 1 1911.1.w.e 8
273.w odd 6 1 1911.1.h.d 4
273.w odd 6 1 inner 1911.1.w.f 8
273.ba even 6 1 1911.1.h.e yes 4
273.ba even 6 1 1911.1.w.e 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1911.1.h.d 4 7.c even 3 1
1911.1.h.d 4 21.h odd 6 1
1911.1.h.d 4 91.r even 6 1
1911.1.h.d 4 273.w odd 6 1
1911.1.h.e yes 4 7.d odd 6 1
1911.1.h.e yes 4 21.g even 6 1
1911.1.h.e yes 4 91.s odd 6 1
1911.1.h.e yes 4 273.ba even 6 1
1911.1.w.e 8 7.b odd 2 1
1911.1.w.e 8 7.d odd 6 1
1911.1.w.e 8 21.c even 2 1
1911.1.w.e 8 21.g even 6 1
1911.1.w.e 8 91.b odd 2 1
1911.1.w.e 8 91.s odd 6 1
1911.1.w.e 8 273.g even 2 1
1911.1.w.e 8 273.ba even 6 1
1911.1.w.f 8 1.a even 1 1 trivial
1911.1.w.f 8 3.b odd 2 1 inner
1911.1.w.f 8 7.c even 3 1 inner
1911.1.w.f 8 13.b even 2 1 inner
1911.1.w.f 8 21.h odd 6 1 inner
1911.1.w.f 8 39.d odd 2 1 CM
1911.1.w.f 8 91.r even 6 1 inner
1911.1.w.f 8 273.w odd 6 1 inner

## Hecke kernels

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

 $$T_{2}^{8} + 4 T_{2}^{6} + 14 T_{2}^{4} + 8 T_{2}^{2} + 4$$ $$T_{61}$$ $$T_{199}^{2} + 2 T_{199} + 4$$

## Hecke characteristic polynomials

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