# Properties

 Label 1911.1.w.e 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: $$x^{8} + 4x^{6} + 14x^{4} + 8x^{2} + 4$$ 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.

 $$f(q)$$ $$=$$ $$q + \beta_{3} q^{2} + \beta_{5} q^{3} + ( - \beta_{6} + \beta_{5}) q^{4} + \beta_1 q^{5} + \beta_{7} q^{6} + (\beta_{7} - \beta_{4} + \beta_1) q^{8} + ( - \beta_{5} - 1) q^{9}+O(q^{10})$$ q + b3 * q^2 + b5 * q^3 + (-b6 + b5) * q^4 + b1 * q^5 + b7 * q^6 + (b7 - b4 + b1) * q^8 + (-b5 - 1) * q^9 $$q + \beta_{3} q^{2} + \beta_{5} q^{3} + ( - \beta_{6} + \beta_{5}) q^{4} + \beta_1 q^{5} + \beta_{7} q^{6} + (\beta_{7} - \beta_{4} + \beta_1) q^{8} + ( - \beta_{5} - 1) q^{9} + \beta_{6} q^{10} + (\beta_{7} + \beta_{3}) q^{11} + (\beta_{6} - \beta_{5} + \beta_{2} - 1) q^{12} - q^{13} + (\beta_{4} - \beta_1) q^{15} + (\beta_{6} - \beta_{5} + \beta_{2} - 1) q^{16} + ( - \beta_{7} - \beta_{3}) q^{18} - \beta_{7} q^{20} + (\beta_{2} - 2) q^{22} + ( - \beta_{7} + \beta_{4} - \beta_{3}) q^{24} + (\beta_{6} + \beta_{5}) q^{25} - \beta_{3} q^{26} + q^{27} + ( - \beta_{6} - \beta_{2}) q^{30} + ( - \beta_{7} - \beta_{3}) q^{32} - \beta_{3} q^{33} + ( - \beta_{2} + 1) q^{36} - \beta_{5} q^{39} + (2 \beta_{5} + 2) q^{40} + ( - \beta_{4} + \beta_1) q^{41} + ( - 2 \beta_{3} + \beta_1) q^{44} - \beta_{4} q^{45} - \beta_1 q^{47} + ( - \beta_{2} + 1) q^{48} + (\beta_{4} - \beta_1) q^{50} + (\beta_{6} - \beta_{5}) q^{52} + \beta_{3} q^{54} - \beta_{2} q^{55} + (\beta_{7} + \beta_{3}) q^{59} + (\beta_{7} + \beta_{3}) q^{60} + q^{64} - \beta_1 q^{65} + (\beta_{6} - 2 \beta_{5}) q^{66} + ( - \beta_{4} + \beta_1) q^{71} + (\beta_{3} - \beta_1) q^{72} + ( - \beta_{6} - \beta_{5} - \beta_{2} - 1) q^{75} - \beta_{7} q^{78} + ( - \beta_{6} - \beta_{2}) q^{79} + (\beta_{7} + \beta_{3}) q^{80} + \beta_{5} q^{81} + (\beta_{6} + \beta_{2}) q^{82} + \beta_{7} q^{83} + (2 \beta_{6} - 2 \beta_{5}) q^{88} + \beta_{3} q^{89} + \beta_{2} q^{90} - \beta_{6} q^{94} + \beta_{3} q^{96} - \beta_{7} q^{99}+O(q^{100})$$ q + b3 * q^2 + b5 * q^3 + (-b6 + b5) * q^4 + b1 * q^5 + b7 * q^6 + (b7 - b4 + b1) * q^8 + (-b5 - 1) * q^9 + b6 * q^10 + (b7 + b3) * q^11 + (b6 - b5 + b2 - 1) * q^12 - q^13 + (b4 - b1) * q^15 + (b6 - b5 + b2 - 1) * q^16 + (-b7 - b3) * q^18 - b7 * q^20 + (b2 - 2) * q^22 + (-b7 + b4 - b3) * q^24 + (b6 + b5) * q^25 - b3 * q^26 + q^27 + (-b6 - b2) * q^30 + (-b7 - b3) * q^32 - b3 * q^33 + (-b2 + 1) * q^36 - b5 * q^39 + (2*b5 + 2) * q^40 + (-b4 + b1) * q^41 + (-2*b3 + b1) * q^44 - b4 * q^45 - b1 * q^47 + (-b2 + 1) * q^48 + (b4 - b1) * q^50 + (b6 - b5) * q^52 + b3 * q^54 - b2 * q^55 + (b7 + b3) * q^59 + (b7 + b3) * q^60 + q^64 - b1 * q^65 + (b6 - 2*b5) * q^66 + (-b4 + b1) * q^71 + (b3 - b1) * q^72 + (-b6 - b5 - b2 - 1) * q^75 - b7 * q^78 + (-b6 - b2) * q^79 + (b7 + b3) * q^80 + b5 * q^81 + (b6 + b2) * q^82 + b7 * q^83 + (2*b6 - 2*b5) * q^88 + b3 * q^89 + b2 * q^90 - b6 * q^94 + b3 * q^96 - b7 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 4 q^{3} - 4 q^{4} - 4 q^{9}+O(q^{10})$$ 8 * q - 4 * q^3 - 4 * q^4 - 4 * q^9 $$8 q - 4 q^{3} - 4 q^{4} - 4 q^{9} - 4 q^{12} - 8 q^{13} - 4 q^{16} - 16 q^{22} - 4 q^{25} + 8 q^{27} + 8 q^{36} + 4 q^{39} + 8 q^{40} + 8 q^{48} + 4 q^{52} + 8 q^{64} + 8 q^{66} - 4 q^{75} - 4 q^{81} + 8 q^{88}+O(q^{100})$$ 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

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

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{6} - 20 ) / 14$$ (v^6 - 20) / 14 $$\beta_{3}$$ $$=$$ $$( \nu^{7} - 34\nu ) / 14$$ (v^7 - 34*v) / 14 $$\beta_{4}$$ $$=$$ $$( 2\nu^{7} + 7\nu^{5} + 28\nu^{3} + 16\nu ) / 14$$ (2*v^7 + 7*v^5 + 28*v^3 + 16*v) / 14 $$\beta_{5}$$ $$=$$ $$( 2\nu^{6} + 7\nu^{4} + 28\nu^{2} + 2 ) / 14$$ (2*v^6 + 7*v^4 + 28*v^2 + 2) / 14 $$\beta_{6}$$ $$=$$ $$( -2\nu^{6} - 7\nu^{4} - 21\nu^{2} - 2 ) / 7$$ (-2*v^6 - 7*v^4 - 21*v^2 - 2) / 7 $$\beta_{7}$$ $$=$$ $$( -6\nu^{7} - 21\nu^{5} - 70\nu^{3} - 6\nu ) / 14$$ (-6*v^7 - 21*v^5 - 70*v^3 - 6*v) / 14
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{6} + 2\beta_{5}$$ b6 + 2*b5 $$\nu^{3}$$ $$=$$ $$\beta_{7} + 3\beta_{4} - 3\beta_1$$ b7 + 3*b4 - 3*b1 $$\nu^{4}$$ $$=$$ $$-4\beta_{6} - 6\beta_{5} - 4\beta_{2} - 6$$ -4*b6 - 6*b5 - 4*b2 - 6 $$\nu^{5}$$ $$=$$ $$-4\beta_{7} - 10\beta_{4} - 4\beta_{3}$$ -4*b7 - 10*b4 - 4*b3 $$\nu^{6}$$ $$=$$ $$14\beta_{2} + 20$$ 14*b2 + 20 $$\nu^{7}$$ $$=$$ $$14\beta_{3} + 34\beta_1$$ 14*b3 + 34*b1

## 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.e 8
3.b odd 2 1 inner 1911.1.w.e 8
7.b odd 2 1 1911.1.w.f 8
7.c even 3 1 1911.1.h.e yes 4
7.c even 3 1 inner 1911.1.w.e 8
7.d odd 6 1 1911.1.h.d 4
7.d odd 6 1 1911.1.w.f 8
13.b even 2 1 inner 1911.1.w.e 8
21.c even 2 1 1911.1.w.f 8
21.g even 6 1 1911.1.h.d 4
21.g even 6 1 1911.1.w.f 8
21.h odd 6 1 1911.1.h.e yes 4
21.h odd 6 1 inner 1911.1.w.e 8
39.d odd 2 1 CM 1911.1.w.e 8
91.b odd 2 1 1911.1.w.f 8
91.r even 6 1 1911.1.h.e yes 4
91.r even 6 1 inner 1911.1.w.e 8
91.s odd 6 1 1911.1.h.d 4
91.s odd 6 1 1911.1.w.f 8
273.g even 2 1 1911.1.w.f 8
273.w odd 6 1 1911.1.h.e yes 4
273.w odd 6 1 inner 1911.1.w.e 8
273.ba even 6 1 1911.1.h.d 4
273.ba even 6 1 1911.1.w.f 8

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

## Hecke kernels

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

 $$T_{2}^{8} + 4T_{2}^{6} + 14T_{2}^{4} + 8T_{2}^{2} + 4$$ T2^8 + 4*T2^6 + 14*T2^4 + 8*T2^2 + 4 $$T_{61}$$ T61 $$T_{199}^{2} - 2T_{199} + 4$$ T199^2 - 2*T199 + 4

## Hecke characteristic polynomials

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