# Properties

 Label 600.1.bj.b Level 600 Weight 1 Character orbit 600.bj Analytic conductor 0.299 Analytic rank 0 Dimension 4 Projective image $$D_{5}$$ CM disc. -24 Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ = $$600 = 2^{3} \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ = $$1$$ Character orbit: $$[\chi]$$ = 600.bj (of order $$10$$ and degree $$4$$)

## Newform invariants

 Self dual: No Analytic conductor: $$0.29943900758$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{10})$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Projective image $$D_{5}$$ Projective field Galois closure of 5.1.225000000.2

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q$$ $$+ \zeta_{10} q^{2}$$ $$-\zeta_{10}^{2} q^{3}$$ $$+ \zeta_{10}^{2} q^{4}$$ $$+ \zeta_{10} q^{5}$$ $$-\zeta_{10}^{3} q^{6}$$ $$+ ( \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{7}$$ $$+ \zeta_{10}^{3} q^{8}$$ $$+ \zeta_{10}^{4} q^{9}$$ $$+O(q^{10})$$ $$q$$ $$+ \zeta_{10} q^{2}$$ $$-\zeta_{10}^{2} q^{3}$$ $$+ \zeta_{10}^{2} q^{4}$$ $$+ \zeta_{10} q^{5}$$ $$-\zeta_{10}^{3} q^{6}$$ $$+ ( \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{7}$$ $$+ \zeta_{10}^{3} q^{8}$$ $$+ \zeta_{10}^{4} q^{9}$$ $$+ \zeta_{10}^{2} q^{10}$$ $$+ ( -1 - \zeta_{10}^{2} ) q^{11}$$ $$-\zeta_{10}^{4} q^{12}$$ $$+ ( \zeta_{10}^{3} - \zeta_{10}^{4} ) q^{14}$$ $$-\zeta_{10}^{3} q^{15}$$ $$+ \zeta_{10}^{4} q^{16}$$ $$- q^{18}$$ $$+ \zeta_{10}^{3} q^{20}$$ $$+ ( -1 - \zeta_{10}^{4} ) q^{21}$$ $$+ ( -\zeta_{10} - \zeta_{10}^{3} ) q^{22}$$ $$+ q^{24}$$ $$+ \zeta_{10}^{2} q^{25}$$ $$+ \zeta_{10} q^{27}$$ $$+ ( 1 + \zeta_{10}^{4} ) q^{28}$$ $$-2 \zeta_{10}^{2} q^{29}$$ $$-\zeta_{10}^{4} q^{30}$$ $$+ ( 1 - \zeta_{10} ) q^{31}$$ $$- q^{32}$$ $$+ ( \zeta_{10}^{2} + \zeta_{10}^{4} ) q^{33}$$ $$+ ( \zeta_{10}^{3} - \zeta_{10}^{4} ) q^{35}$$ $$-\zeta_{10} q^{36}$$ $$+ \zeta_{10}^{4} q^{40}$$ $$+ ( 1 - \zeta_{10} ) q^{42}$$ $$+ ( -\zeta_{10}^{2} - \zeta_{10}^{4} ) q^{44}$$ $$- q^{45}$$ $$+ \zeta_{10} q^{48}$$ $$+ ( 1 - \zeta_{10} + \zeta_{10}^{4} ) q^{49}$$ $$+ \zeta_{10}^{3} q^{50}$$ $$+ ( \zeta_{10} + \zeta_{10}^{3} ) q^{53}$$ $$+ \zeta_{10}^{2} q^{54}$$ $$+ ( -\zeta_{10} - \zeta_{10}^{3} ) q^{55}$$ $$+ ( -1 + \zeta_{10} ) q^{56}$$ $$-2 \zeta_{10}^{3} q^{58}$$ $$+ ( -1 + \zeta_{10}^{3} ) q^{59}$$ $$+ q^{60}$$ $$+ ( \zeta_{10} - \zeta_{10}^{2} ) q^{62}$$ $$+ ( -\zeta_{10} + \zeta_{10}^{2} ) q^{63}$$ $$-\zeta_{10} q^{64}$$ $$+ ( -1 + \zeta_{10}^{3} ) q^{66}$$ $$+ ( 1 + \zeta_{10}^{4} ) q^{70}$$ $$-\zeta_{10}^{2} q^{72}$$ $$-2 \zeta_{10} q^{73}$$ $$-\zeta_{10}^{4} q^{75}$$ $$+ ( -1 - \zeta_{10}^{2} + \zeta_{10}^{3} - \zeta_{10}^{4} ) q^{77}$$ $$+ ( -\zeta_{10} - \zeta_{10}^{3} ) q^{79}$$ $$- q^{80}$$ $$-\zeta_{10}^{3} q^{81}$$ $$+ ( -1 + \zeta_{10} ) q^{83}$$ $$+ ( \zeta_{10} - \zeta_{10}^{2} ) q^{84}$$ $$+ 2 \zeta_{10}^{4} q^{87}$$ $$+ ( 1 - \zeta_{10}^{3} ) q^{88}$$ $$-\zeta_{10} q^{90}$$ $$+ ( -\zeta_{10}^{2} + \zeta_{10}^{3} ) q^{93}$$ $$+ \zeta_{10}^{2} q^{96}$$ $$+ ( 1 + \zeta_{10}^{4} ) q^{97}$$ $$+ ( -1 + \zeta_{10} - \zeta_{10}^{2} ) q^{98}$$ $$+ ( \zeta_{10} - \zeta_{10}^{4} ) q^{99}$$ $$+O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q$$ $$\mathstrut +\mathstrut q^{2}$$ $$\mathstrut +\mathstrut q^{3}$$ $$\mathstrut -\mathstrut q^{4}$$ $$\mathstrut +\mathstrut q^{5}$$ $$\mathstrut -\mathstrut q^{6}$$ $$\mathstrut -\mathstrut 2q^{7}$$ $$\mathstrut +\mathstrut q^{8}$$ $$\mathstrut -\mathstrut q^{9}$$ $$\mathstrut +\mathstrut O(q^{10})$$ $$4q$$ $$\mathstrut +\mathstrut q^{2}$$ $$\mathstrut +\mathstrut q^{3}$$ $$\mathstrut -\mathstrut q^{4}$$ $$\mathstrut +\mathstrut q^{5}$$ $$\mathstrut -\mathstrut q^{6}$$ $$\mathstrut -\mathstrut 2q^{7}$$ $$\mathstrut +\mathstrut q^{8}$$ $$\mathstrut -\mathstrut q^{9}$$ $$\mathstrut -\mathstrut q^{10}$$ $$\mathstrut -\mathstrut 3q^{11}$$ $$\mathstrut +\mathstrut q^{12}$$ $$\mathstrut +\mathstrut 2q^{14}$$ $$\mathstrut -\mathstrut q^{15}$$ $$\mathstrut -\mathstrut q^{16}$$ $$\mathstrut -\mathstrut 4q^{18}$$ $$\mathstrut +\mathstrut q^{20}$$ $$\mathstrut -\mathstrut 3q^{21}$$ $$\mathstrut -\mathstrut 2q^{22}$$ $$\mathstrut +\mathstrut 4q^{24}$$ $$\mathstrut -\mathstrut q^{25}$$ $$\mathstrut +\mathstrut q^{27}$$ $$\mathstrut +\mathstrut 3q^{28}$$ $$\mathstrut +\mathstrut 2q^{29}$$ $$\mathstrut +\mathstrut q^{30}$$ $$\mathstrut +\mathstrut 3q^{31}$$ $$\mathstrut -\mathstrut 4q^{32}$$ $$\mathstrut -\mathstrut 2q^{33}$$ $$\mathstrut +\mathstrut 2q^{35}$$ $$\mathstrut -\mathstrut q^{36}$$ $$\mathstrut -\mathstrut q^{40}$$ $$\mathstrut +\mathstrut 3q^{42}$$ $$\mathstrut +\mathstrut 2q^{44}$$ $$\mathstrut -\mathstrut 4q^{45}$$ $$\mathstrut +\mathstrut q^{48}$$ $$\mathstrut +\mathstrut 2q^{49}$$ $$\mathstrut +\mathstrut q^{50}$$ $$\mathstrut +\mathstrut 2q^{53}$$ $$\mathstrut -\mathstrut q^{54}$$ $$\mathstrut -\mathstrut 2q^{55}$$ $$\mathstrut -\mathstrut 3q^{56}$$ $$\mathstrut -\mathstrut 2q^{58}$$ $$\mathstrut -\mathstrut 3q^{59}$$ $$\mathstrut +\mathstrut 4q^{60}$$ $$\mathstrut +\mathstrut 2q^{62}$$ $$\mathstrut -\mathstrut 2q^{63}$$ $$\mathstrut -\mathstrut q^{64}$$ $$\mathstrut -\mathstrut 3q^{66}$$ $$\mathstrut +\mathstrut 3q^{70}$$ $$\mathstrut +\mathstrut q^{72}$$ $$\mathstrut -\mathstrut 2q^{73}$$ $$\mathstrut +\mathstrut q^{75}$$ $$\mathstrut -\mathstrut q^{77}$$ $$\mathstrut -\mathstrut 2q^{79}$$ $$\mathstrut -\mathstrut 4q^{80}$$ $$\mathstrut -\mathstrut q^{81}$$ $$\mathstrut -\mathstrut 3q^{83}$$ $$\mathstrut +\mathstrut 2q^{84}$$ $$\mathstrut -\mathstrut 2q^{87}$$ $$\mathstrut +\mathstrut 3q^{88}$$ $$\mathstrut -\mathstrut q^{90}$$ $$\mathstrut +\mathstrut 2q^{93}$$ $$\mathstrut -\mathstrut q^{96}$$ $$\mathstrut +\mathstrut 3q^{97}$$ $$\mathstrut -\mathstrut 2q^{98}$$ $$\mathstrut +\mathstrut 2q^{99}$$ $$\mathstrut +\mathstrut O(q^{100})$$

## Character Values

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

 $$n$$ $$151$$ $$301$$ $$401$$ $$577$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-1$$ $$\zeta_{10}^{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
221.1
 −0.309017 + 0.951057i 0.809017 + 0.587785i 0.809017 − 0.587785i −0.309017 − 0.951057i
−0.309017 + 0.951057i 0.809017 + 0.587785i −0.809017 0.587785i −0.309017 + 0.951057i −0.809017 + 0.587785i −1.61803 0.809017 0.587785i 0.309017 + 0.951057i −0.809017 0.587785i
341.1 0.809017 + 0.587785i −0.309017 0.951057i 0.309017 + 0.951057i 0.809017 + 0.587785i 0.309017 0.951057i 0.618034 −0.309017 + 0.951057i −0.809017 + 0.587785i 0.309017 + 0.951057i
461.1 0.809017 0.587785i −0.309017 + 0.951057i 0.309017 0.951057i 0.809017 0.587785i 0.309017 + 0.951057i 0.618034 −0.309017 0.951057i −0.809017 0.587785i 0.309017 0.951057i
581.1 −0.309017 0.951057i 0.809017 0.587785i −0.809017 + 0.587785i −0.309017 0.951057i −0.809017 0.587785i −1.61803 0.809017 + 0.587785i 0.309017 0.951057i −0.809017 + 0.587785i
 $$n$$: 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
24.h Odd 1 CM by $$\Q(\sqrt{-6})$$ yes
25.d Even 1 yes
600.bj Odd 1 yes

## Hecke kernels

This newform can be constructed as the kernel of the linear operator $$T_{11}^{4}$$ $$\mathstrut +\mathstrut 3 T_{11}^{3}$$ $$\mathstrut +\mathstrut 4 T_{11}^{2}$$ $$\mathstrut +\mathstrut 2 T_{11}$$ $$\mathstrut +\mathstrut 1$$ acting on $$S_{1}^{\mathrm{new}}(600, [\chi])$$.