# Properties

 Label 600.2.b.c Level 600 Weight 2 Character orbit 600.b Analytic conductor 4.791 Analytic rank 0 Dimension 4 CM discriminant -15 Inner twists 8

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$4.79102412128$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{5})$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 120) Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{2} q^{2} + ( \beta_{1} - \beta_{2} ) q^{3} + \beta_{3} q^{4} + ( -2 + \beta_{3} ) q^{6} + ( 2 \beta_{1} - \beta_{2} ) q^{8} -3 q^{9} +O(q^{10})$$ $$q -\beta_{2} q^{2} + ( \beta_{1} - \beta_{2} ) q^{3} + \beta_{3} q^{4} + ( -2 + \beta_{3} ) q^{6} + ( 2 \beta_{1} - \beta_{2} ) q^{8} -3 q^{9} + ( 2 \beta_{1} + \beta_{2} ) q^{12} + ( -4 + \beta_{3} ) q^{16} + ( 4 \beta_{1} - 4 \beta_{2} ) q^{17} + 3 \beta_{2} q^{18} -4 q^{19} + ( -4 \beta_{1} - 4 \beta_{2} ) q^{23} + ( -4 - \beta_{3} ) q^{24} + ( -3 \beta_{1} + 3 \beta_{2} ) q^{27} + ( 2 - 4 \beta_{3} ) q^{31} + ( 2 \beta_{1} + 3 \beta_{2} ) q^{32} + ( -8 + 4 \beta_{3} ) q^{34} -3 \beta_{3} q^{36} + 4 \beta_{2} q^{38} + ( 8 + 4 \beta_{3} ) q^{46} + ( -4 \beta_{1} - 4 \beta_{2} ) q^{47} + ( -2 \beta_{1} + 5 \beta_{2} ) q^{48} + 7 q^{49} -12 q^{51} + ( -2 \beta_{1} - 2 \beta_{2} ) q^{53} + ( 6 - 3 \beta_{3} ) q^{54} + ( -4 \beta_{1} + 4 \beta_{2} ) q^{57} + ( -4 + 8 \beta_{3} ) q^{61} + ( -8 \beta_{1} + 2 \beta_{2} ) q^{62} + ( -4 - 3 \beta_{3} ) q^{64} + ( 8 \beta_{1} + 4 \beta_{2} ) q^{68} + ( -4 + 8 \beta_{3} ) q^{69} + ( -6 \beta_{1} + 3 \beta_{2} ) q^{72} -4 \beta_{3} q^{76} + ( 2 - 4 \beta_{3} ) q^{79} + 9 q^{81} + ( -2 \beta_{1} + 2 \beta_{2} ) q^{83} + ( 8 \beta_{1} - 12 \beta_{2} ) q^{92} + ( -6 \beta_{1} - 6 \beta_{2} ) q^{93} + ( 8 + 4 \beta_{3} ) q^{94} + ( 4 - 5 \beta_{3} ) q^{96} -7 \beta_{2} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 2q^{4} - 6q^{6} - 12q^{9} + O(q^{10})$$ $$4q + 2q^{4} - 6q^{6} - 12q^{9} - 14q^{16} - 16q^{19} - 18q^{24} - 24q^{34} - 6q^{36} + 40q^{46} + 28q^{49} - 48q^{51} + 18q^{54} - 22q^{64} - 8q^{76} + 36q^{81} + 40q^{94} + 6q^{96} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} + 2 x^{2} + x + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{2} - \nu + 1$$ $$\beta_{2}$$ $$=$$ $$\nu^{3} - \nu^{2} + \nu + 1$$ $$\beta_{3}$$ $$=$$ $$\nu^{3} - \nu^{2} + 3 \nu + 1$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} - \beta_{2}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} - \beta_{2} + 2 \beta_{1} - 2$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$\beta_{2} + \beta_{1} - 2$$

## Character values

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

 $$n$$ $$151$$ $$301$$ $$401$$ $$577$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
251.1
 −0.309017 + 0.535233i −0.309017 − 0.535233i 0.809017 − 1.40126i 0.809017 + 1.40126i
−1.11803 0.866025i 1.73205i 0.500000 + 1.93649i 0 −1.50000 + 1.93649i 0 1.11803 2.59808i −3.00000 0
251.2 −1.11803 + 0.866025i 1.73205i 0.500000 1.93649i 0 −1.50000 1.93649i 0 1.11803 + 2.59808i −3.00000 0
251.3 1.11803 0.866025i 1.73205i 0.500000 1.93649i 0 −1.50000 1.93649i 0 −1.11803 2.59808i −3.00000 0
251.4 1.11803 + 0.866025i 1.73205i 0.500000 + 1.93649i 0 −1.50000 + 1.93649i 0 −1.11803 + 2.59808i −3.00000 0
 $$n$$: 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
15.d odd 2 1 CM by $$\Q(\sqrt{-15})$$
3.b odd 2 1 inner
5.b even 2 1 inner
8.d odd 2 1 inner
24.f even 2 1 inner
40.e odd 2 1 inner
120.m even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 600.2.b.c 4
3.b odd 2 1 inner 600.2.b.c 4
4.b odd 2 1 2400.2.b.c 4
5.b even 2 1 inner 600.2.b.c 4
5.c odd 4 2 120.2.m.a 4
8.b even 2 1 2400.2.b.c 4
8.d odd 2 1 inner 600.2.b.c 4
12.b even 2 1 2400.2.b.c 4
15.d odd 2 1 CM 600.2.b.c 4
15.e even 4 2 120.2.m.a 4
20.d odd 2 1 2400.2.b.c 4
20.e even 4 2 480.2.m.a 4
24.f even 2 1 inner 600.2.b.c 4
24.h odd 2 1 2400.2.b.c 4
40.e odd 2 1 inner 600.2.b.c 4
40.f even 2 1 2400.2.b.c 4
40.i odd 4 2 480.2.m.a 4
40.k even 4 2 120.2.m.a 4
60.h even 2 1 2400.2.b.c 4
60.l odd 4 2 480.2.m.a 4
120.i odd 2 1 2400.2.b.c 4
120.m even 2 1 inner 600.2.b.c 4
120.q odd 4 2 120.2.m.a 4
120.w even 4 2 480.2.m.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
120.2.m.a 4 5.c odd 4 2
120.2.m.a 4 15.e even 4 2
120.2.m.a 4 40.k even 4 2
120.2.m.a 4 120.q odd 4 2
480.2.m.a 4 20.e even 4 2
480.2.m.a 4 40.i odd 4 2
480.2.m.a 4 60.l odd 4 2
480.2.m.a 4 120.w even 4 2
600.2.b.c 4 1.a even 1 1 trivial
600.2.b.c 4 3.b odd 2 1 inner
600.2.b.c 4 5.b even 2 1 inner
600.2.b.c 4 8.d odd 2 1 inner
600.2.b.c 4 15.d odd 2 1 CM
600.2.b.c 4 24.f even 2 1 inner
600.2.b.c 4 40.e odd 2 1 inner
600.2.b.c 4 120.m even 2 1 inner
2400.2.b.c 4 4.b odd 2 1
2400.2.b.c 4 8.b even 2 1
2400.2.b.c 4 12.b even 2 1
2400.2.b.c 4 20.d odd 2 1
2400.2.b.c 4 24.h odd 2 1
2400.2.b.c 4 40.f even 2 1
2400.2.b.c 4 60.h even 2 1
2400.2.b.c 4 120.i odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(600, [\chi])$$:

 $$T_{7}$$ $$T_{11}$$ $$T_{23}^{2} - 80$$ $$T_{43}$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 - T^{2} + 4 T^{4}$$
$3$ $$( 1 + 3 T^{2} )^{2}$$
$5$ 1
$7$ $$( 1 - 7 T^{2} )^{4}$$
$11$ $$( 1 - 11 T^{2} )^{4}$$
$13$ $$( 1 - 13 T^{2} )^{4}$$
$17$ $$( 1 + 14 T^{2} + 289 T^{4} )^{2}$$
$19$ $$( 1 + 4 T + 19 T^{2} )^{4}$$
$23$ $$( 1 - 34 T^{2} + 529 T^{4} )^{2}$$
$29$ $$( 1 + 29 T^{2} )^{4}$$
$31$ $$( 1 - 8 T + 31 T^{2} )^{2}( 1 + 8 T + 31 T^{2} )^{2}$$
$37$ $$( 1 - 37 T^{2} )^{4}$$
$41$ $$( 1 - 41 T^{2} )^{4}$$
$43$ $$( 1 + 43 T^{2} )^{4}$$
$47$ $$( 1 + 14 T^{2} + 2209 T^{4} )^{2}$$
$53$ $$( 1 + 86 T^{2} + 2809 T^{4} )^{2}$$
$59$ $$( 1 - 59 T^{2} )^{4}$$
$61$ $$( 1 - 2 T + 61 T^{2} )^{2}( 1 + 2 T + 61 T^{2} )^{2}$$
$67$ $$( 1 + 67 T^{2} )^{4}$$
$71$ $$( 1 + 71 T^{2} )^{4}$$
$73$ $$( 1 + 73 T^{2} )^{4}$$
$79$ $$( 1 - 16 T + 79 T^{2} )^{2}( 1 + 16 T + 79 T^{2} )^{2}$$
$83$ $$( 1 - 154 T^{2} + 6889 T^{4} )^{2}$$
$89$ $$( 1 - 89 T^{2} )^{4}$$
$97$ $$( 1 + 97 T^{2} )^{4}$$