# Properties

 Label 1900.1.j.b Level $1900$ Weight $1$ Character orbit 1900.j Analytic conductor $0.948$ Analytic rank $0$ Dimension $8$ Projective image $D_{8}$ CM discriminant -95 Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1900 = 2^{2} \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1900.j (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.948223524003$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(i)$$ Coefficient field: $$\Q(\zeta_{16})$$ Defining polynomial: $$x^{8} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{8}$$ Projective field: Galois closure of 8.0.4170272000.1

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + \zeta_{16}^{3} q^{2} + ( \zeta_{16} + \zeta_{16}^{3} ) q^{3} + \zeta_{16}^{6} q^{4} + ( \zeta_{16}^{4} + \zeta_{16}^{6} ) q^{6} -\zeta_{16} q^{8} + ( \zeta_{16}^{2} + \zeta_{16}^{4} + \zeta_{16}^{6} ) q^{9} +O(q^{10})$$ $$q + \zeta_{16}^{3} q^{2} + ( \zeta_{16} + \zeta_{16}^{3} ) q^{3} + \zeta_{16}^{6} q^{4} + ( \zeta_{16}^{4} + \zeta_{16}^{6} ) q^{6} -\zeta_{16} q^{8} + ( \zeta_{16}^{2} + \zeta_{16}^{4} + \zeta_{16}^{6} ) q^{9} + ( -\zeta_{16}^{2} - \zeta_{16}^{6} ) q^{11} + ( -\zeta_{16} + \zeta_{16}^{7} ) q^{12} + ( \zeta_{16}^{5} + \zeta_{16}^{7} ) q^{13} -\zeta_{16}^{4} q^{16} + ( -\zeta_{16} + \zeta_{16}^{5} + \zeta_{16}^{7} ) q^{18} + q^{19} + ( \zeta_{16} - \zeta_{16}^{5} ) q^{22} + ( -\zeta_{16}^{2} - \zeta_{16}^{4} ) q^{24} + ( -1 - \zeta_{16}^{2} ) q^{26} + ( -\zeta_{16} + \zeta_{16}^{3} + \zeta_{16}^{5} + \zeta_{16}^{7} ) q^{27} -\zeta_{16}^{7} q^{32} + ( \zeta_{16} - \zeta_{16}^{3} - \zeta_{16}^{5} - \zeta_{16}^{7} ) q^{33} + ( -1 - \zeta_{16}^{2} - \zeta_{16}^{4} ) q^{36} + ( \zeta_{16}^{5} - \zeta_{16}^{7} ) q^{37} + \zeta_{16}^{3} q^{38} + ( -2 - \zeta_{16}^{2} + \zeta_{16}^{6} ) q^{39} + ( 1 + \zeta_{16}^{4} ) q^{44} + ( -\zeta_{16}^{5} - \zeta_{16}^{7} ) q^{48} -\zeta_{16}^{4} q^{49} + ( -\zeta_{16}^{3} - \zeta_{16}^{5} ) q^{52} + ( \zeta_{16} - \zeta_{16}^{3} ) q^{53} + ( -1 - \zeta_{16}^{2} - \zeta_{16}^{4} + \zeta_{16}^{6} ) q^{54} + ( \zeta_{16} + \zeta_{16}^{3} ) q^{57} + ( \zeta_{16}^{2} - \zeta_{16}^{6} ) q^{61} + \zeta_{16}^{2} q^{64} + ( 1 + \zeta_{16}^{2} + \zeta_{16}^{4} - \zeta_{16}^{6} ) q^{66} + ( \zeta_{16} - \zeta_{16}^{3} ) q^{67} + ( -\zeta_{16}^{3} - \zeta_{16}^{5} - \zeta_{16}^{7} ) q^{72} + ( -1 + \zeta_{16}^{2} ) q^{74} + \zeta_{16}^{6} q^{76} + ( -\zeta_{16} - 2 \zeta_{16}^{3} - \zeta_{16}^{5} ) q^{78} + ( -1 - \zeta_{16}^{2} + \zeta_{16}^{6} ) q^{81} + ( \zeta_{16}^{3} + \zeta_{16}^{7} ) q^{88} + ( 1 + \zeta_{16}^{2} ) q^{96} + ( \zeta_{16}^{5} - \zeta_{16}^{7} ) q^{97} -\zeta_{16}^{7} q^{98} + ( 2 + \zeta_{16}^{2} - \zeta_{16}^{6} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + O(q^{10})$$ $$8q + 8q^{19} - 8q^{26} - 8q^{36} - 16q^{39} + 8q^{44} - 8q^{54} + 8q^{66} - 8q^{74} - 8q^{81} + 8q^{96} + 16q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1900\mathbb{Z}\right)^\times$$.

 $$n$$ $$77$$ $$401$$ $$951$$ $$\chi(n)$$ $$-\zeta_{16}^{4}$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
607.1
 0.382683 + 0.923880i −0.923880 + 0.382683i 0.923880 − 0.382683i −0.382683 − 0.923880i 0.382683 − 0.923880i −0.923880 − 0.382683i 0.923880 + 0.382683i −0.382683 + 0.923880i
−0.923880 0.382683i −0.541196 + 0.541196i 0.707107 + 0.707107i 0 0.707107 0.292893i 0 −0.382683 0.923880i 0.414214i 0
607.2 −0.382683 + 0.923880i −1.30656 + 1.30656i −0.707107 0.707107i 0 −0.707107 1.70711i 0 0.923880 0.382683i 2.41421i 0
607.3 0.382683 0.923880i 1.30656 1.30656i −0.707107 0.707107i 0 −0.707107 1.70711i 0 −0.923880 + 0.382683i 2.41421i 0
607.4 0.923880 + 0.382683i 0.541196 0.541196i 0.707107 + 0.707107i 0 0.707107 0.292893i 0 0.382683 + 0.923880i 0.414214i 0
1443.1 −0.923880 + 0.382683i −0.541196 0.541196i 0.707107 0.707107i 0 0.707107 + 0.292893i 0 −0.382683 + 0.923880i 0.414214i 0
1443.2 −0.382683 0.923880i −1.30656 1.30656i −0.707107 + 0.707107i 0 −0.707107 + 1.70711i 0 0.923880 + 0.382683i 2.41421i 0
1443.3 0.382683 + 0.923880i 1.30656 + 1.30656i −0.707107 + 0.707107i 0 −0.707107 + 1.70711i 0 −0.923880 0.382683i 2.41421i 0
1443.4 0.923880 0.382683i 0.541196 + 0.541196i 0.707107 0.707107i 0 0.707107 + 0.292893i 0 0.382683 0.923880i 0.414214i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1443.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
95.d odd 2 1 CM by $$\Q(\sqrt{-95})$$
5.b even 2 1 inner
19.b odd 2 1 inner
20.e even 4 2 inner
380.j odd 4 2 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1900.1.j.b yes 8
4.b odd 2 1 1900.1.j.a 8
5.b even 2 1 inner 1900.1.j.b yes 8
5.c odd 4 2 1900.1.j.a 8
19.b odd 2 1 inner 1900.1.j.b yes 8
20.d odd 2 1 1900.1.j.a 8
20.e even 4 2 inner 1900.1.j.b yes 8
76.d even 2 1 1900.1.j.a 8
95.d odd 2 1 CM 1900.1.j.b yes 8
95.g even 4 2 1900.1.j.a 8
380.d even 2 1 1900.1.j.a 8
380.j odd 4 2 inner 1900.1.j.b yes 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1900.1.j.a 8 4.b odd 2 1
1900.1.j.a 8 5.c odd 4 2
1900.1.j.a 8 20.d odd 2 1
1900.1.j.a 8 76.d even 2 1
1900.1.j.a 8 95.g even 4 2
1900.1.j.a 8 380.d even 2 1
1900.1.j.b yes 8 1.a even 1 1 trivial
1900.1.j.b yes 8 5.b even 2 1 inner
1900.1.j.b yes 8 19.b odd 2 1 inner
1900.1.j.b yes 8 20.e even 4 2 inner
1900.1.j.b yes 8 95.d odd 2 1 CM
1900.1.j.b yes 8 380.j odd 4 2 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{199} + 2$$ acting on $$S_{1}^{\mathrm{new}}(1900, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T^{8}$$
$3$ $$4 + 12 T^{4} + T^{8}$$
$5$ $$T^{8}$$
$7$ $$T^{8}$$
$11$ $$( 2 + T^{2} )^{4}$$
$13$ $$4 + 12 T^{4} + T^{8}$$
$17$ $$T^{8}$$
$19$ $$( -1 + T )^{8}$$
$23$ $$T^{8}$$
$29$ $$T^{8}$$
$31$ $$T^{8}$$
$37$ $$4 + 12 T^{4} + T^{8}$$
$41$ $$T^{8}$$
$43$ $$T^{8}$$
$47$ $$T^{8}$$
$53$ $$4 + 12 T^{4} + T^{8}$$
$59$ $$T^{8}$$
$61$ $$( -2 + T^{2} )^{4}$$
$67$ $$4 + 12 T^{4} + T^{8}$$
$71$ $$T^{8}$$
$73$ $$T^{8}$$
$79$ $$T^{8}$$
$83$ $$T^{8}$$
$89$ $$T^{8}$$
$97$ $$4 + 12 T^{4} + T^{8}$$