# Properties

 Label 1911.1.h.d Level $1911$ Weight $1$ Character orbit 1911.h Self dual yes Analytic conductor $0.954$ Analytic rank $0$ Dimension $4$ Projective image $D_{8}$ CM discriminant -39 Inner twists $4$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: yes Analytic conductor: $$0.953713239142$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{16})^+$$ Defining polynomial: $$x^{4} - 4 x^{2} + 2$$ Coefficient ring: $$\Z[a_1, a_2]$$ 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 $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

## Character values

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

 $$n$$ $$638$$ $$1471$$ $$1522$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1520.1
 1.84776 0.765367 −0.765367 −1.84776
−1.84776 −1.00000 2.41421 −0.765367 1.84776 0 −2.61313 1.00000 1.41421
1520.2 −0.765367 −1.00000 −0.414214 1.84776 0.765367 0 1.08239 1.00000 −1.41421
1520.3 0.765367 −1.00000 −0.414214 −1.84776 −0.765367 0 −1.08239 1.00000 −1.41421
1520.4 1.84776 −1.00000 2.41421 0.765367 −1.84776 0 2.61313 1.00000 1.41421
 $$n$$: 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
39.d odd 2 1 CM by $$\Q(\sqrt{-39})$$
3.b odd 2 1 inner
13.b even 2 1 inner

## Twists

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

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

## Hecke kernels

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

 $$T_{2}^{4} - 4 T_{2}^{2} + 2$$ $$T_{61}$$ $$T_{199} - 2$$

## Hecke characteristic polynomials

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