# Properties

 Label 1805.2.b.b Level $1805$ Weight $2$ Character orbit 1805.b Analytic conductor $14.413$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1805 = 5 \cdot 19^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1805.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$14.4129975648$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 95) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 2 i q^{2} -2 q^{4} + ( -1 - 2 i ) q^{5} -4 i q^{7} + 3 q^{9} +O(q^{10})$$ $$q + 2 i q^{2} -2 q^{4} + ( -1 - 2 i ) q^{5} -4 i q^{7} + 3 q^{9} + ( 4 - 2 i ) q^{10} - q^{11} + 2 i q^{13} + 8 q^{14} -4 q^{16} -2 i q^{17} + 6 i q^{18} + ( 2 + 4 i ) q^{20} -2 i q^{22} -6 i q^{23} + ( -3 + 4 i ) q^{25} -4 q^{26} + 8 i q^{28} -9 q^{29} -7 q^{31} -8 i q^{32} + 4 q^{34} + ( -8 + 4 i ) q^{35} -6 q^{36} + 2 i q^{37} + 2 q^{41} -2 i q^{43} + 2 q^{44} + ( -3 - 6 i ) q^{45} + 12 q^{46} + 6 i q^{47} -9 q^{49} + ( -8 - 6 i ) q^{50} -4 i q^{52} -4 i q^{53} + ( 1 + 2 i ) q^{55} -18 i q^{58} -9 q^{59} -7 q^{61} -14 i q^{62} -12 i q^{63} + 8 q^{64} + ( 4 - 2 i ) q^{65} -10 i q^{67} + 4 i q^{68} + ( -8 - 16 i ) q^{70} + q^{71} -10 i q^{73} -4 q^{74} + 4 i q^{77} - q^{79} + ( 4 + 8 i ) q^{80} + 9 q^{81} + 4 i q^{82} + 6 i q^{83} + ( -4 + 2 i ) q^{85} + 4 q^{86} + 11 q^{89} + ( 12 - 6 i ) q^{90} + 8 q^{91} + 12 i q^{92} -12 q^{94} -6 i q^{97} -18 i q^{98} -3 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 4 q^{4} - 2 q^{5} + 6 q^{9} + O(q^{10})$$ $$2 q - 4 q^{4} - 2 q^{5} + 6 q^{9} + 8 q^{10} - 2 q^{11} + 16 q^{14} - 8 q^{16} + 4 q^{20} - 6 q^{25} - 8 q^{26} - 18 q^{29} - 14 q^{31} + 8 q^{34} - 16 q^{35} - 12 q^{36} + 4 q^{41} + 4 q^{44} - 6 q^{45} + 24 q^{46} - 18 q^{49} - 16 q^{50} + 2 q^{55} - 18 q^{59} - 14 q^{61} + 16 q^{64} + 8 q^{65} - 16 q^{70} + 2 q^{71} - 8 q^{74} - 2 q^{79} + 8 q^{80} + 18 q^{81} - 8 q^{85} + 8 q^{86} + 22 q^{89} + 24 q^{90} + 16 q^{91} - 24 q^{94} - 6 q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1805\mathbb{Z}\right)^\times$$.

 $$n$$ $$362$$ $$1446$$ $$\chi(n)$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1084.1
 − 1.00000i 1.00000i
2.00000i 0 −2.00000 −1.00000 + 2.00000i 0 4.00000i 0 3.00000 4.00000 + 2.00000i
1084.2 2.00000i 0 −2.00000 −1.00000 2.00000i 0 4.00000i 0 3.00000 4.00000 2.00000i
 $$n$$: 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
5.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1805.2.b.b 2
5.b even 2 1 inner 1805.2.b.b 2
5.c odd 4 1 9025.2.a.b 1
5.c odd 4 1 9025.2.a.i 1
19.b odd 2 1 1805.2.b.a 2
19.c even 3 2 95.2.i.a 4
57.h odd 6 2 855.2.be.a 4
95.d odd 2 1 1805.2.b.a 2
95.g even 4 1 9025.2.a.a 1
95.g even 4 1 9025.2.a.j 1
95.i even 6 2 95.2.i.a 4
95.m odd 12 2 475.2.e.a 2
95.m odd 12 2 475.2.e.c 2
285.n odd 6 2 855.2.be.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
95.2.i.a 4 19.c even 3 2
95.2.i.a 4 95.i even 6 2
475.2.e.a 2 95.m odd 12 2
475.2.e.c 2 95.m odd 12 2
855.2.be.a 4 57.h odd 6 2
855.2.be.a 4 285.n odd 6 2
1805.2.b.a 2 19.b odd 2 1
1805.2.b.a 2 95.d odd 2 1
1805.2.b.b 2 1.a even 1 1 trivial
1805.2.b.b 2 5.b even 2 1 inner
9025.2.a.a 1 95.g even 4 1
9025.2.a.b 1 5.c odd 4 1
9025.2.a.i 1 5.c odd 4 1
9025.2.a.j 1 95.g even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1805, [\chi])$$:

 $$T_{2}^{2} + 4$$ $$T_{29} + 9$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$4 + T^{2}$$
$3$ $$T^{2}$$
$5$ $$5 + 2 T + T^{2}$$
$7$ $$16 + T^{2}$$
$11$ $$( 1 + T )^{2}$$
$13$ $$4 + T^{2}$$
$17$ $$4 + T^{2}$$
$19$ $$T^{2}$$
$23$ $$36 + T^{2}$$
$29$ $$( 9 + T )^{2}$$
$31$ $$( 7 + T )^{2}$$
$37$ $$4 + T^{2}$$
$41$ $$( -2 + T )^{2}$$
$43$ $$4 + T^{2}$$
$47$ $$36 + T^{2}$$
$53$ $$16 + T^{2}$$
$59$ $$( 9 + T )^{2}$$
$61$ $$( 7 + T )^{2}$$
$67$ $$100 + T^{2}$$
$71$ $$( -1 + T )^{2}$$
$73$ $$100 + T^{2}$$
$79$ $$( 1 + T )^{2}$$
$83$ $$36 + T^{2}$$
$89$ $$( -11 + T )^{2}$$
$97$ $$36 + T^{2}$$