# Properties

 Label 2001.1.i.b Level 2001 Weight 1 Character orbit 2001.i Analytic conductor 0.999 Analytic rank 0 Dimension 2 Projective image $$D_{4}$$ CM disc. -23 Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ = $$2001 = 3 \cdot 23 \cdot 29$$ Weight: $$k$$ = $$1$$ Character orbit: $$[\chi]$$ = 2001.i (of order $$4$$ and degree $$2$$)

## Newform invariants

 Self dual: No Analytic conductor: $$0.998629090279$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Projective image $$D_{4}$$ Projective field Galois closure of 4.0.116116029.1

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q$$ $$+ ( 1 - i ) q^{2}$$ $$+ i q^{3}$$ $$-i q^{4}$$ $$+ ( 1 + i ) q^{6}$$ $$- q^{9}$$ $$+O(q^{10})$$ $$q$$ $$+ ( 1 - i ) q^{2}$$ $$+ i q^{3}$$ $$-i q^{4}$$ $$+ ( 1 + i ) q^{6}$$ $$- q^{9}$$ $$+ q^{12}$$ $$+ 2 i q^{13}$$ $$+ q^{16}$$ $$+ ( -1 + i ) q^{18}$$ $$+ i q^{23}$$ $$+ q^{25}$$ $$+ ( 2 + 2 i ) q^{26}$$ $$-i q^{27}$$ $$-i q^{29}$$ $$+ ( -1 - i ) q^{31}$$ $$+ ( 1 - i ) q^{32}$$ $$+ i q^{36}$$ $$-2 q^{39}$$ $$+ ( -1 - i ) q^{41}$$ $$+ ( 1 + i ) q^{46}$$ $$+ ( 1 + i ) q^{47}$$ $$+ i q^{48}$$ $$+ q^{49}$$ $$+ ( 1 - i ) q^{50}$$ $$+ 2 q^{52}$$ $$+ ( -1 - i ) q^{54}$$ $$+ ( -1 - i ) q^{58}$$ $$-2 i q^{59}$$ $$-2 q^{62}$$ $$-i q^{64}$$ $$- q^{69}$$ $$+ ( -1 + i ) q^{73}$$ $$+ i q^{75}$$ $$+ ( -2 + 2 i ) q^{78}$$ $$+ q^{81}$$ $$-2 q^{82}$$ $$+ q^{87}$$ $$+ q^{92}$$ $$+ ( 1 - i ) q^{93}$$ $$+ 2 q^{94}$$ $$+ ( 1 + i ) q^{96}$$ $$+ ( 1 - i ) q^{98}$$ $$+O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q$$ $$\mathstrut +\mathstrut 2q^{2}$$ $$\mathstrut +\mathstrut 2q^{6}$$ $$\mathstrut -\mathstrut 2q^{9}$$ $$\mathstrut +\mathstrut O(q^{10})$$ $$2q$$ $$\mathstrut +\mathstrut 2q^{2}$$ $$\mathstrut +\mathstrut 2q^{6}$$ $$\mathstrut -\mathstrut 2q^{9}$$ $$\mathstrut +\mathstrut 2q^{12}$$ $$\mathstrut +\mathstrut 2q^{16}$$ $$\mathstrut -\mathstrut 2q^{18}$$ $$\mathstrut +\mathstrut 2q^{25}$$ $$\mathstrut +\mathstrut 4q^{26}$$ $$\mathstrut -\mathstrut 2q^{31}$$ $$\mathstrut +\mathstrut 2q^{32}$$ $$\mathstrut -\mathstrut 4q^{39}$$ $$\mathstrut -\mathstrut 2q^{41}$$ $$\mathstrut +\mathstrut 2q^{46}$$ $$\mathstrut +\mathstrut 2q^{47}$$ $$\mathstrut +\mathstrut 2q^{49}$$ $$\mathstrut +\mathstrut 2q^{50}$$ $$\mathstrut +\mathstrut 4q^{52}$$ $$\mathstrut -\mathstrut 2q^{54}$$ $$\mathstrut -\mathstrut 2q^{58}$$ $$\mathstrut -\mathstrut 4q^{62}$$ $$\mathstrut -\mathstrut 2q^{69}$$ $$\mathstrut -\mathstrut 2q^{73}$$ $$\mathstrut -\mathstrut 4q^{78}$$ $$\mathstrut +\mathstrut 2q^{81}$$ $$\mathstrut -\mathstrut 4q^{82}$$ $$\mathstrut +\mathstrut 2q^{87}$$ $$\mathstrut +\mathstrut 2q^{92}$$ $$\mathstrut +\mathstrut 2q^{93}$$ $$\mathstrut +\mathstrut 4q^{94}$$ $$\mathstrut +\mathstrut 2q^{96}$$ $$\mathstrut +\mathstrut 2q^{98}$$ $$\mathstrut +\mathstrut O(q^{100})$$

## Character Values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2001\mathbb{Z}\right)^\times$$.

 $$n$$ $$553$$ $$668$$ $$1132$$ $$\chi(n)$$ $$i$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1172.1
 1.00000i − 1.00000i
1.00000 1.00000i 1.00000i 1.00000i 0 1.00000 + 1.00000i 0 0 −1.00000 0
1931.1 1.00000 + 1.00000i 1.00000i 1.00000i 0 1.00000 1.00000i 0 0 −1.00000 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
23.b Odd 1 CM by $$\Q(\sqrt{-23})$$ yes
87.f Even 1 yes
2001.i Odd 1 yes

## Hecke kernels

This newform can be constructed as the kernel of the linear operator $$T_{2}^{2}$$ $$\mathstrut -\mathstrut 2 T_{2}$$ $$\mathstrut +\mathstrut 2$$ acting on $$S_{1}^{\mathrm{new}}(2001, [\chi])$$.