# Properties

 Label 1900.1.bb.a Level $1900$ Weight $1$ Character orbit 1900.bb Analytic conductor $0.948$ Analytic rank $0$ Dimension $8$ Projective image $D_{15}$ CM discriminant -19 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1900 = 2^{2} \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1900.bb (of order $$10$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.948223524003$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$2$$ over $$\Q(\zeta_{10})$$ Coefficient field: $$\Q(\zeta_{15})$$ Defining polynomial: $$x^{8} - x^{7} + x^{5} - x^{4} + x^{3} - x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{15}$$ Projective field: Galois closure of $$\mathbb{Q}[x]/(x^{15} + \cdots)$$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q -\zeta_{30}^{7} q^{5} + ( \zeta_{30}^{2} - \zeta_{30}^{13} ) q^{7} + \zeta_{30}^{12} q^{9} +O(q^{10})$$ $$q -\zeta_{30}^{7} q^{5} + ( \zeta_{30}^{2} - \zeta_{30}^{13} ) q^{7} + \zeta_{30}^{12} q^{9} + ( -\zeta_{30} - \zeta_{30}^{5} ) q^{11} + ( -\zeta_{30}^{5} - \zeta_{30}^{13} ) q^{17} -\zeta_{30}^{9} q^{19} + ( 1 + \zeta_{30}^{6} ) q^{23} + \zeta_{30}^{14} q^{25} + ( -\zeta_{30}^{5} - \zeta_{30}^{9} ) q^{35} + ( -\zeta_{30} + \zeta_{30}^{14} ) q^{43} + \zeta_{30}^{4} q^{45} -\zeta_{30}^{6} q^{47} + ( 1 + \zeta_{30}^{4} - \zeta_{30}^{11} ) q^{49} + ( \zeta_{30}^{8} + \zeta_{30}^{12} ) q^{55} + ( \zeta_{30}^{2} + \zeta_{30}^{4} ) q^{61} + ( \zeta_{30}^{10} + \zeta_{30}^{14} ) q^{63} + \zeta_{30}^{3} q^{73} + ( -2 \zeta_{30}^{3} - \zeta_{30}^{7} + \zeta_{30}^{14} ) q^{77} -\zeta_{30}^{9} q^{81} + ( 1 - \zeta_{30}^{3} ) q^{83} + ( -\zeta_{30}^{5} + \zeta_{30}^{12} ) q^{85} -\zeta_{30} q^{95} + ( \zeta_{30}^{2} - \zeta_{30}^{13} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + q^{5} + 2q^{7} - 2q^{9} + O(q^{10})$$ $$8q + q^{5} + 2q^{7} - 2q^{9} - 3q^{11} - 3q^{17} - 2q^{19} + 6q^{23} + q^{25} - 6q^{35} + 2q^{43} + q^{45} + 2q^{47} + 10q^{49} - q^{55} + 2q^{61} - 3q^{63} + 2q^{73} - 2q^{77} - 2q^{81} + 6q^{83} - 6q^{85} + q^{95} + 2q^{99} + O(q^{100})$$

## Character values

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

 $$n$$ $$77$$ $$401$$ $$951$$ $$\chi(n)$$ $$\zeta_{30}^{6}$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
341.1
 −0.978148 − 0.207912i 0.669131 − 0.743145i 0.913545 − 0.406737i −0.104528 + 0.994522i 0.913545 + 0.406737i −0.104528 − 0.994522i −0.978148 + 0.207912i 0.669131 + 0.743145i
0 0 0 −0.104528 0.994522i 0 1.82709 0 −0.809017 + 0.587785i 0
341.2 0 0 0 0.913545 + 0.406737i 0 −0.209057 0 −0.809017 + 0.587785i 0
721.1 0 0 0 −0.978148 0.207912i 0 1.33826 0 0.309017 + 0.951057i 0
721.2 0 0 0 0.669131 0.743145i 0 −1.95630 0 0.309017 + 0.951057i 0
1481.1 0 0 0 −0.978148 + 0.207912i 0 1.33826 0 0.309017 0.951057i 0
1481.2 0 0 0 0.669131 + 0.743145i 0 −1.95630 0 0.309017 0.951057i 0
1861.1 0 0 0 −0.104528 + 0.994522i 0 1.82709 0 −0.809017 0.587785i 0
1861.2 0 0 0 0.913545 0.406737i 0 −0.209057 0 −0.809017 0.587785i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1861.2 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.b odd 2 1 CM by $$\Q(\sqrt{-19})$$
25.d even 5 1 inner
475.o odd 10 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1900.1.bb.a 8
19.b odd 2 1 CM 1900.1.bb.a 8
25.d even 5 1 inner 1900.1.bb.a 8
475.o odd 10 1 inner 1900.1.bb.a 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1900.1.bb.a 8 1.a even 1 1 trivial
1900.1.bb.a 8 19.b odd 2 1 CM
1900.1.bb.a 8 25.d even 5 1 inner
1900.1.bb.a 8 475.o odd 10 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(1900, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8}$$
$5$ $$1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8}$$
$7$ $$( 1 + 4 T - 4 T^{2} - T^{3} + T^{4} )^{2}$$
$11$ $$1 - 7 T + 18 T^{2} + 11 T^{3} + 15 T^{4} + 11 T^{5} + 8 T^{6} + 3 T^{7} + T^{8}$$
$13$ $$T^{8}$$
$17$ $$1 - 7 T + 18 T^{2} + 11 T^{3} + 15 T^{4} + 11 T^{5} + 8 T^{6} + 3 T^{7} + T^{8}$$
$19$ $$( 1 + T + T^{2} + T^{3} + T^{4} )^{2}$$
$23$ $$( 1 - 2 T + 4 T^{2} - 3 T^{3} + T^{4} )^{2}$$
$29$ $$T^{8}$$
$31$ $$T^{8}$$
$37$ $$T^{8}$$
$41$ $$T^{8}$$
$43$ $$( 1 + 4 T - 4 T^{2} - T^{3} + T^{4} )^{2}$$
$47$ $$( 1 - T + T^{2} - T^{3} + T^{4} )^{2}$$
$53$ $$T^{8}$$
$59$ $$T^{8}$$
$61$ $$1 + 3 T + 23 T^{2} + T^{3} + T^{5} + 3 T^{6} - 2 T^{7} + T^{8}$$
$67$ $$T^{8}$$
$71$ $$T^{8}$$
$73$ $$( 1 - T + T^{2} - T^{3} + T^{4} )^{2}$$
$79$ $$T^{8}$$
$83$ $$( 1 - 2 T + 4 T^{2} - 3 T^{3} + T^{4} )^{2}$$
$89$ $$T^{8}$$
$97$ $$T^{8}$$