# 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 discriminant -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 $$10$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.299439007580$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{10})$$ Defining polynomial: $$x^{4} - x^{3} + x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image $$D_{5}$$ Projective field Galois closure of 5.1.225000000.2

## $q$-expansion

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

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

## 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 Parity Ord Mult Type
1.a even 1 1 trivial
24.h odd 2 1 CM by $$\Q(\sqrt{-6})$$
25.d even 5 1 inner
600.bj odd 10 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 600.1.bj.b yes 4
3.b odd 2 1 600.1.bj.a 4
4.b odd 2 1 2400.1.cp.a 4
5.b even 2 1 3000.1.bj.a 4
5.c odd 4 2 3000.1.z.a 8
8.b even 2 1 600.1.bj.a 4
8.d odd 2 1 2400.1.cp.b 4
12.b even 2 1 2400.1.cp.b 4
15.d odd 2 1 3000.1.bj.b 4
15.e even 4 2 3000.1.z.b 8
24.f even 2 1 2400.1.cp.a 4
24.h odd 2 1 CM 600.1.bj.b yes 4
25.d even 5 1 inner 600.1.bj.b yes 4
25.e even 10 1 3000.1.bj.a 4
25.f odd 20 2 3000.1.z.a 8
40.f even 2 1 3000.1.bj.b 4
40.i odd 4 2 3000.1.z.b 8
75.h odd 10 1 3000.1.bj.b 4
75.j odd 10 1 600.1.bj.a 4
75.l even 20 2 3000.1.z.b 8
100.j odd 10 1 2400.1.cp.a 4
120.i odd 2 1 3000.1.bj.a 4
120.w even 4 2 3000.1.z.a 8
200.n odd 10 1 2400.1.cp.b 4
200.o even 10 1 3000.1.bj.b 4
200.t even 10 1 600.1.bj.a 4
200.x odd 20 2 3000.1.z.b 8
300.n even 10 1 2400.1.cp.b 4
600.z odd 10 1 3000.1.bj.a 4
600.bg even 10 1 2400.1.cp.a 4
600.bj odd 10 1 inner 600.1.bj.b yes 4
600.bp even 20 2 3000.1.z.a 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
600.1.bj.a 4 3.b odd 2 1
600.1.bj.a 4 8.b even 2 1
600.1.bj.a 4 75.j odd 10 1
600.1.bj.a 4 200.t even 10 1
600.1.bj.b yes 4 1.a even 1 1 trivial
600.1.bj.b yes 4 24.h odd 2 1 CM
600.1.bj.b yes 4 25.d even 5 1 inner
600.1.bj.b yes 4 600.bj odd 10 1 inner
2400.1.cp.a 4 4.b odd 2 1
2400.1.cp.a 4 24.f even 2 1
2400.1.cp.a 4 100.j odd 10 1
2400.1.cp.a 4 600.bg even 10 1
2400.1.cp.b 4 8.d odd 2 1
2400.1.cp.b 4 12.b even 2 1
2400.1.cp.b 4 200.n odd 10 1
2400.1.cp.b 4 300.n even 10 1
3000.1.z.a 8 5.c odd 4 2
3000.1.z.a 8 25.f odd 20 2
3000.1.z.a 8 120.w even 4 2
3000.1.z.a 8 600.bp even 20 2
3000.1.z.b 8 15.e even 4 2
3000.1.z.b 8 40.i odd 4 2
3000.1.z.b 8 75.l even 20 2
3000.1.z.b 8 200.x odd 20 2
3000.1.bj.a 4 5.b even 2 1
3000.1.bj.a 4 25.e even 10 1
3000.1.bj.a 4 120.i odd 2 1
3000.1.bj.a 4 600.z odd 10 1
3000.1.bj.b 4 15.d odd 2 1
3000.1.bj.b 4 40.f even 2 1
3000.1.bj.b 4 75.h odd 10 1
3000.1.bj.b 4 200.o even 10 1

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T + T^{2} - T^{3} + T^{4}$$
$3$ $$1 - T + T^{2} - T^{3} + T^{4}$$
$5$ $$1 - T + T^{2} - T^{3} + T^{4}$$
$7$ $$( 1 + T + T^{2} + T^{3} + T^{4} )^{2}$$
$11$ $$( 1 + T )^{4}( 1 - T + T^{2} - T^{3} + T^{4} )$$
$13$ $$( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} )$$
$17$ $$( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} )$$
$19$ $$( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} )$$
$23$ $$( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} )$$
$29$ $$( 1 - T + T^{2} - T^{3} + T^{4} )^{2}$$
$31$ $$( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} )$$
$37$ $$( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} )$$
$41$ $$( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} )$$
$43$ $$( 1 - T )^{4}( 1 + T )^{4}$$
$47$ $$( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} )$$
$53$ $$( 1 - T + T^{2} - T^{3} + T^{4} )^{2}$$
$59$ $$( 1 + T )^{4}( 1 - T + T^{2} - T^{3} + T^{4} )$$
$61$ $$( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} )$$
$67$ $$( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} )$$
$71$ $$( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} )$$
$73$ $$( 1 + T + T^{2} + T^{3} + T^{4} )^{2}$$
$79$ $$( 1 + T + T^{2} + T^{3} + T^{4} )^{2}$$
$83$ $$( 1 + T )^{4}( 1 - T + T^{2} - T^{3} + T^{4} )$$
$89$ $$( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} )$$
$97$ $$( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} )$$