Properties

 Label 600.1.q.a Level 600 Weight 1 Character orbit 600.q Analytic conductor 0.299 Analytic rank 0 Dimension 4 Projective image $$D_{2}$$ CM/RM disc. -8, -15, 120 Inner twists 16

Related objects

Newspace parameters

 Level: $$N$$ = $$600 = 2^{3} \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ = $$1$$ Character orbit: $$[\chi]$$ = 600.q (of order $$4$$ and degree $$2$$)

Newform invariants

 Self dual: No Analytic conductor: $$0.29943900758$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(i)$$ Coefficient field: $$\Q(\zeta_{8})$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Projective image $$D_{2}$$ Projective field Galois closure of $$\Q(\sqrt{-2}, \sqrt{-15})$$ Artin image size $$32$$ Artin image $OD_{16}:C_2$ Artin field Galois closure of 16.0.164025000000000000.1

$q$-expansion

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

 $$f(q)$$ $$=$$ $$q$$ $$+ \zeta_{8}^{3} q^{2}$$ $$-\zeta_{8} q^{3}$$ $$-\zeta_{8}^{2} q^{4}$$ $$+ q^{6}$$ $$+ \zeta_{8} q^{8}$$ $$+ \zeta_{8}^{2} q^{9}$$ $$+O(q^{10})$$ $$q$$ $$+ \zeta_{8}^{3} q^{2}$$ $$-\zeta_{8} q^{3}$$ $$-\zeta_{8}^{2} q^{4}$$ $$+ q^{6}$$ $$+ \zeta_{8} q^{8}$$ $$+ \zeta_{8}^{2} q^{9}$$ $$+ \zeta_{8}^{3} q^{12}$$ $$- q^{16}$$ $$-2 \zeta_{8}^{3} q^{17}$$ $$-\zeta_{8} q^{18}$$ $$-2 \zeta_{8}^{2} q^{19}$$ $$-\zeta_{8}^{2} q^{24}$$ $$-\zeta_{8}^{3} q^{27}$$ $$-\zeta_{8}^{3} q^{32}$$ $$+ 2 \zeta_{8}^{2} q^{34}$$ $$+ q^{36}$$ $$+ 2 \zeta_{8} q^{38}$$ $$+ \zeta_{8} q^{48}$$ $$+ \zeta_{8}^{2} q^{49}$$ $$-2 q^{51}$$ $$+ \zeta_{8}^{2} q^{54}$$ $$+ 2 \zeta_{8}^{3} q^{57}$$ $$+ \zeta_{8}^{2} q^{64}$$ $$-2 \zeta_{8} q^{68}$$ $$+ \zeta_{8}^{3} q^{72}$$ $$-2 q^{76}$$ $$- q^{81}$$ $$+ 2 \zeta_{8} q^{83}$$ $$- q^{96}$$ $$-\zeta_{8} q^{98}$$ $$+O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q$$ $$\mathstrut +\mathstrut 4q^{6}$$ $$\mathstrut +\mathstrut O(q^{10})$$ $$4q$$ $$\mathstrut +\mathstrut 4q^{6}$$ $$\mathstrut -\mathstrut 4q^{16}$$ $$\mathstrut +\mathstrut 4q^{36}$$ $$\mathstrut -\mathstrut 8q^{51}$$ $$\mathstrut -\mathstrut 8q^{76}$$ $$\mathstrut -\mathstrut 4q^{81}$$ $$\mathstrut -\mathstrut 4q^{96}$$ $$\mathstrut +\mathstrut O(q^{100})$$

Character Values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/600\mathbb{Z}\right)^\times$$.

 $$n$$ $$151$$ $$301$$ $$401$$ $$577$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$-1$$ $$\zeta_{8}^{2}$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
107.1
 0.707107 + 0.707107i −0.707107 − 0.707107i 0.707107 − 0.707107i −0.707107 + 0.707107i
−0.707107 + 0.707107i −0.707107 0.707107i 1.00000i 0 1.00000 0 0.707107 + 0.707107i 1.00000i 0
107.2 0.707107 0.707107i 0.707107 + 0.707107i 1.00000i 0 1.00000 0 −0.707107 0.707107i 1.00000i 0
443.1 −0.707107 0.707107i −0.707107 + 0.707107i 1.00000i 0 1.00000 0 0.707107 0.707107i 1.00000i 0
443.2 0.707107 + 0.707107i 0.707107 0.707107i 1.00000i 0 1.00000 0 −0.707107 + 0.707107i 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
8.d Odd 1 CM by $$\Q(\sqrt{-2})$$ yes
15.d Odd 1 CM by $$\Q(\sqrt{-15})$$ yes
120.m Even 1 RM by $$\Q(\sqrt{30})$$ yes
3.b Odd 1 yes
5.b Even 1 yes
5.c Odd 2 yes
15.e Even 2 yes
24.f Even 1 yes
40.e Odd 1 yes
40.k Even 2 yes
120.q Odd 2 yes