Properties

 Label 4000.1.p.a Level 4000 Weight 1 Character orbit 4000.p Analytic conductor 1.996 Analytic rank 0 Dimension 8 Projective image $$D_{20}$$ CM disc. -20 Inner twists 4

Related objects

Newspace parameters

 Level: $$N$$ = $$4000 = 2^{5} \cdot 5^{3}$$ Weight: $$k$$ = $$1$$ Character orbit: $$[\chi]$$ = 4000.p (of order $$4$$ and degree $$2$$)

Newform invariants

 Self dual: No Analytic conductor: $$1.99626005053$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(i)$$ Coefficient field: $$\Q(\zeta_{20})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ Projective image $$D_{20}$$ Projective field Galois closure of 20.0.320000000000000000000000000.1

$q$-expansion

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

 $$f(q)$$ $$=$$ $$q$$ $$+ ( -\zeta_{20} + \zeta_{20}^{4} ) q^{3}$$ $$+ ( \zeta_{20}^{7} - \zeta_{20}^{8} ) q^{7}$$ $$+ ( \zeta_{20}^{2} - \zeta_{20}^{5} + \zeta_{20}^{8} ) q^{9}$$ $$+O(q^{10})$$ $$q$$ $$+ ( -\zeta_{20} + \zeta_{20}^{4} ) q^{3}$$ $$+ ( \zeta_{20}^{7} - \zeta_{20}^{8} ) q^{7}$$ $$+ ( \zeta_{20}^{2} - \zeta_{20}^{5} + \zeta_{20}^{8} ) q^{9}$$ $$+ ( -\zeta_{20} + \zeta_{20}^{2} - \zeta_{20}^{8} + \zeta_{20}^{9} ) q^{21}$$ $$+ ( \zeta_{20}^{7} + \zeta_{20}^{8} ) q^{23}$$ $$+ ( -\zeta_{20}^{2} - \zeta_{20}^{3} + \zeta_{20}^{6} - \zeta_{20}^{9} ) q^{27}$$ $$+ ( \zeta_{20} + \zeta_{20}^{9} ) q^{29}$$ $$+ ( \zeta_{20} - \zeta_{20}^{9} ) q^{41}$$ $$+ ( \zeta_{20}^{6} + \zeta_{20}^{9} ) q^{43}$$ $$+ ( \zeta_{20} + \zeta_{20}^{4} ) q^{47}$$ $$+ ( -\zeta_{20}^{4} + \zeta_{20}^{5} - \zeta_{20}^{6} ) q^{49}$$ $$+ ( -\zeta_{20}^{3} + \zeta_{20}^{7} ) q^{61}$$ $$+ ( 1 + \zeta_{20}^{2} - \zeta_{20}^{3} - \zeta_{20}^{5} + \zeta_{20}^{6} + \zeta_{20}^{9} ) q^{63}$$ $$+ ( -1 - \zeta_{20}^{5} ) q^{67}$$ $$+ ( -\zeta_{20} - \zeta_{20}^{2} - \zeta_{20}^{8} - \zeta_{20}^{9} ) q^{69}$$ $$+ ( -1 + \zeta_{20}^{3} + \zeta_{20}^{4} - \zeta_{20}^{6} - \zeta_{20}^{7} ) q^{81}$$ $$+ ( \zeta_{20}^{2} - \zeta_{20}^{3} ) q^{83}$$ $$+ ( 1 - \zeta_{20}^{2} - \zeta_{20}^{3} + \zeta_{20}^{5} ) q^{87}$$ $$+ ( \zeta_{20}^{3} + \zeta_{20}^{7} ) q^{89}$$ $$+O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q$$ $$\mathstrut -\mathstrut 2q^{3}$$ $$\mathstrut +\mathstrut 2q^{7}$$ $$\mathstrut +\mathstrut O(q^{10})$$ $$8q$$ $$\mathstrut -\mathstrut 2q^{3}$$ $$\mathstrut +\mathstrut 2q^{7}$$ $$\mathstrut +\mathstrut 4q^{21}$$ $$\mathstrut -\mathstrut 2q^{23}$$ $$\mathstrut +\mathstrut 2q^{43}$$ $$\mathstrut -\mathstrut 2q^{47}$$ $$\mathstrut +\mathstrut 12q^{63}$$ $$\mathstrut -\mathstrut 8q^{67}$$ $$\mathstrut -\mathstrut 12q^{81}$$ $$\mathstrut +\mathstrut 2q^{83}$$ $$\mathstrut +\mathstrut 6q^{87}$$ $$\mathstrut +\mathstrut O(q^{100})$$

Character Values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/4000\mathbb{Z}\right)^\times$$.

 $$n$$ $$1377$$ $$2501$$ $$2751$$ $$\chi(n)$$ $$\zeta_{20}^{5}$$ $$1$$ $$1$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
193.1
 0.587785 + 0.809017i 0.951057 − 0.309017i −0.587785 + 0.809017i −0.951057 − 0.309017i 0.587785 − 0.809017i 0.951057 + 0.309017i −0.587785 − 0.809017i −0.951057 + 0.309017i
0 −1.39680 1.39680i 0 0 0 0.642040 0.642040i 0 2.90211i 0
193.2 0 −0.642040 0.642040i 0 0 0 0.221232 0.221232i 0 0.175571i 0
193.3 0 −0.221232 0.221232i 0 0 0 −1.26007 + 1.26007i 0 0.902113i 0
193.4 0 1.26007 + 1.26007i 0 0 0 1.39680 1.39680i 0 2.17557i 0
1057.1 0 −1.39680 + 1.39680i 0 0 0 0.642040 + 0.642040i 0 2.90211i 0
1057.2 0 −0.642040 + 0.642040i 0 0 0 0.221232 + 0.221232i 0 0.175571i 0
1057.3 0 −0.221232 + 0.221232i 0 0 0 −1.26007 1.26007i 0 0.902113i 0
1057.4 0 1.26007 1.26007i 0 0 0 1.39680 + 1.39680i 0 2.17557i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1057.4 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 $$T_{3}^{8}$$ $$\mathstrut +\mathstrut 2 T_{3}^{7}$$ $$\mathstrut +\mathstrut 2 T_{3}^{6}$$ $$\mathstrut +\mathstrut 11 T_{3}^{4}$$ $$\mathstrut +\mathstrut 20 T_{3}^{3}$$ $$\mathstrut +\mathstrut 18 T_{3}^{2}$$ $$\mathstrut +\mathstrut 6 T_{3}$$ $$\mathstrut +\mathstrut 1$$ acting on $$S_{1}^{\mathrm{new}}(4000, [\chi])$$.