# Properties

 Label 800.1.p.a Level 800 Weight 1 Character orbit 800.p Analytic conductor 0.399 Analytic rank 0 Dimension 2 Projective image $$D_{4}$$ CM disc. -20 Inner twists 4

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## Newspace parameters

 Level: $$N$$ = $$800 = 2^{5} \cdot 5^{2}$$ Weight: $$k$$ = $$1$$ Character orbit: $$[\chi]$$ = 800.p (of order $$4$$ and degree $$2$$)

## Newform invariants

 Self dual: No Analytic conductor: $$0.399252010106$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Projective image $$D_{4}$$ Projective field Galois closure of 4.0.320.1

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q$$ $$+ ( -1 + i ) q^{3}$$ $$+ ( 1 + i ) q^{7}$$ $$-i q^{9}$$ $$+O(q^{10})$$ $$q$$ $$+ ( -1 + i ) q^{3}$$ $$+ ( 1 + i ) q^{7}$$ $$-i q^{9}$$ $$-2 q^{21}$$ $$+ ( -1 + i ) q^{23}$$ $$+ 2 i q^{29}$$ $$+ ( 1 - i ) q^{43}$$ $$+ ( -1 - i ) q^{47}$$ $$+ i q^{49}$$ $$+ ( 1 - i ) q^{63}$$ $$+ ( 1 + i ) q^{67}$$ $$-2 i q^{69}$$ $$+ q^{81}$$ $$+ ( 1 - i ) q^{83}$$ $$+ ( -2 - 2 i ) q^{87}$$ $$-2 i q^{89}$$ $$+O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q$$ $$\mathstrut -\mathstrut 2q^{3}$$ $$\mathstrut +\mathstrut 2q^{7}$$ $$\mathstrut +\mathstrut O(q^{10})$$ $$2q$$ $$\mathstrut -\mathstrut 2q^{3}$$ $$\mathstrut +\mathstrut 2q^{7}$$ $$\mathstrut -\mathstrut 4q^{21}$$ $$\mathstrut -\mathstrut 2q^{23}$$ $$\mathstrut +\mathstrut 2q^{43}$$ $$\mathstrut -\mathstrut 2q^{47}$$ $$\mathstrut +\mathstrut 2q^{63}$$ $$\mathstrut +\mathstrut 2q^{67}$$ $$\mathstrut +\mathstrut 2q^{81}$$ $$\mathstrut +\mathstrut 2q^{83}$$ $$\mathstrut -\mathstrut 4q^{87}$$ $$\mathstrut +\mathstrut O(q^{100})$$

## Character Values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/800\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$351$$ $$577$$ $$\chi(n)$$ $$1$$ $$1$$ $$i$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
193.1
 − 1.00000i 1.00000i
0 −1.00000 1.00000i 0 0 0 1.00000 1.00000i 0 1.00000i 0
257.1 0 −1.00000 + 1.00000i 0 0 0 1.00000 + 1.00000i 0 1.00000i 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
20.d Odd 1 CM by $$\Q(\sqrt{-5})$$ yes
5.c Odd 1 yes
20.e Even 1 yes

## Hecke kernels

This newform can be constructed as the kernel of the linear operator $$T_{3}^{2}$$ $$\mathstrut +\mathstrut 2 T_{3}$$ $$\mathstrut +\mathstrut 2$$ acting on $$S_{1}^{\mathrm{new}}(800, [\chi])$$.