# Properties

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

# 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: $$\Q(\sqrt{-2}, \sqrt{-3})$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$3$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + ( 1 + \beta_{2} ) q^{3} -2 q^{4} + ( -1 + \beta_{1} + \beta_{3} ) q^{6} -2 \beta_{1} q^{8} + ( 1 - \beta_{1} + \beta_{2} + \beta_{3} ) q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{2} + ( 1 + \beta_{2} ) q^{3} -2 q^{4} + ( -1 + \beta_{1} + \beta_{3} ) q^{6} -2 \beta_{1} q^{8} + ( 1 - \beta_{1} + \beta_{2} + \beta_{3} ) q^{9} + ( -1 + \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{11} + ( -2 - 2 \beta_{2} ) q^{12} + 4 q^{16} + ( -1 + 4 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{17} + ( 1 + 2 \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{18} + ( 1 - \beta_{1} + 4 \beta_{2} + 2 \beta_{3} ) q^{19} + ( \beta_{1} - 4 \beta_{2} - 2 \beta_{3} ) q^{22} + ( 2 - 2 \beta_{1} - 2 \beta_{3} ) q^{24} + ( 5 - \beta_{1} ) q^{27} + 4 \beta_{1} q^{32} + ( 4 + 5 \beta_{1} - 3 \beta_{2} - \beta_{3} ) q^{33} + ( -6 + \beta_{1} - 4 \beta_{2} - 2 \beta_{3} ) q^{34} + ( -2 + 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{36} + ( -2 + 3 \beta_{1} - 4 \beta_{2} + 4 \beta_{3} ) q^{38} + ( 1 - 6 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{41} -10 q^{43} + ( 2 - 2 \beta_{1} + 4 \beta_{2} - 4 \beta_{3} ) q^{44} + ( 4 + 4 \beta_{2} ) q^{48} + 7 q^{49} + ( 1 + 8 \beta_{1} - 3 \beta_{2} + 2 \beta_{3} ) q^{51} + ( 2 + 5 \beta_{1} ) q^{54} + ( 8 - 3 \beta_{1} - \beta_{2} + 3 \beta_{3} ) q^{57} -10 \beta_{1} q^{59} -8 q^{64} + ( -7 + 3 \beta_{1} + 2 \beta_{2} - 3 \beta_{3} ) q^{66} + ( -5 - \beta_{1} + 4 \beta_{2} + 2 \beta_{3} ) q^{67} + ( 2 - 8 \beta_{1} + 4 \beta_{2} - 4 \beta_{3} ) q^{68} + ( -2 - 4 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} ) q^{72} + ( -5 + 2 \beta_{1} - 8 \beta_{2} - 4 \beta_{3} ) q^{73} + ( -2 + 2 \beta_{1} - 8 \beta_{2} - 4 \beta_{3} ) q^{76} + ( 6 - \beta_{1} + 5 \beta_{2} - \beta_{3} ) q^{81} + ( 10 - \beta_{1} + 4 \beta_{2} + 2 \beta_{3} ) q^{82} + ( 3 - 7 \beta_{1} + 6 \beta_{2} - 6 \beta_{3} ) q^{83} -10 \beta_{1} q^{86} + ( -2 \beta_{1} + 8 \beta_{2} + 4 \beta_{3} ) q^{88} + ( -3 + 8 \beta_{1} - 6 \beta_{2} + 6 \beta_{3} ) q^{89} + ( -4 + 4 \beta_{1} + 4 \beta_{3} ) q^{96} -10 q^{97} + 7 \beta_{1} q^{98} + ( -4 + 7 \beta_{1} + 5 \beta_{2} + 2 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 2q^{3} - 8q^{4} - 4q^{6} + 2q^{9} + O(q^{10})$$ $$4q + 2q^{3} - 8q^{4} - 4q^{6} + 2q^{9} - 4q^{12} + 16q^{16} + 8q^{18} - 4q^{19} + 8q^{22} + 8q^{24} + 20q^{27} + 22q^{33} - 16q^{34} - 4q^{36} - 40q^{43} + 8q^{48} + 28q^{49} + 10q^{51} + 8q^{54} + 34q^{57} - 32q^{64} - 32q^{66} - 28q^{67} - 16q^{72} - 4q^{73} + 8q^{76} + 14q^{81} + 32q^{82} - 16q^{88} - 16q^{96} - 40q^{97} - 26q^{99} + O(q^{100})$$

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

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

## 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
 −1.22474 − 0.707107i 1.22474 − 0.707107i −1.22474 + 0.707107i 1.22474 + 0.707107i
1.41421i −0.724745 1.57313i −2.00000 0 −2.22474 + 1.02494i 0 2.82843i −1.94949 + 2.28024i 0
251.2 1.41421i 1.72474 + 0.158919i −2.00000 0 0.224745 2.43916i 0 2.82843i 2.94949 + 0.548188i 0
251.3 1.41421i −0.724745 + 1.57313i −2.00000 0 −2.22474 1.02494i 0 2.82843i −1.94949 2.28024i 0
251.4 1.41421i 1.72474 0.158919i −2.00000 0 0.224745 + 2.43916i 0 2.82843i 2.94949 0.548188i 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
8.d odd 2 1 CM by $$\Q(\sqrt{-2})$$
3.b odd 2 1 inner
24.f 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.d yes 4
3.b odd 2 1 inner 600.2.b.d yes 4
4.b odd 2 1 2400.2.b.b 4
5.b even 2 1 600.2.b.b 4
5.c odd 4 2 600.2.m.b 8
8.b even 2 1 2400.2.b.b 4
8.d odd 2 1 CM 600.2.b.d yes 4
12.b even 2 1 2400.2.b.b 4
15.d odd 2 1 600.2.b.b 4
15.e even 4 2 600.2.m.b 8
20.d odd 2 1 2400.2.b.d 4
20.e even 4 2 2400.2.m.b 8
24.f even 2 1 inner 600.2.b.d yes 4
24.h odd 2 1 2400.2.b.b 4
40.e odd 2 1 600.2.b.b 4
40.f even 2 1 2400.2.b.d 4
40.i odd 4 2 2400.2.m.b 8
40.k even 4 2 600.2.m.b 8
60.h even 2 1 2400.2.b.d 4
60.l odd 4 2 2400.2.m.b 8
120.i odd 2 1 2400.2.b.d 4
120.m even 2 1 600.2.b.b 4
120.q odd 4 2 600.2.m.b 8
120.w even 4 2 2400.2.m.b 8

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

## Hecke kernels

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

 $$T_{7}$$ $$T_{11}^{4} + 58 T_{11}^{2} + 625$$ $$T_{23}$$ $$T_{43} + 10$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$( 1 + 2 T^{2} )^{2}$$
$3$ $$1 - 2 T + T^{2} - 6 T^{3} + 9 T^{4}$$
$5$ 1
$7$ $$( 1 - 7 T^{2} )^{4}$$
$11$ $$( 1 - 6 T + 25 T^{2} - 66 T^{3} + 121 T^{4} )( 1 + 6 T + 25 T^{2} + 66 T^{3} + 121 T^{4} )$$
$13$ $$( 1 - 13 T^{2} )^{4}$$
$17$ $$( 1 - 6 T + 19 T^{2} - 102 T^{3} + 289 T^{4} )( 1 + 6 T + 19 T^{2} + 102 T^{3} + 289 T^{4} )$$
$19$ $$( 1 + 2 T - 15 T^{2} + 38 T^{3} + 361 T^{4} )^{2}$$
$23$ $$( 1 + 23 T^{2} )^{4}$$
$29$ $$( 1 + 29 T^{2} )^{4}$$
$31$ $$( 1 - 31 T^{2} )^{4}$$
$37$ $$( 1 - 37 T^{2} )^{4}$$
$41$ $$( 1 - 6 T - 5 T^{2} - 246 T^{3} + 1681 T^{4} )( 1 + 6 T - 5 T^{2} + 246 T^{3} + 1681 T^{4} )$$
$43$ $$( 1 + 10 T + 43 T^{2} )^{4}$$
$47$ $$( 1 + 47 T^{2} )^{4}$$
$53$ $$( 1 + 53 T^{2} )^{4}$$
$59$ $$( 1 - 6 T + 59 T^{2} )^{2}( 1 + 6 T + 59 T^{2} )^{2}$$
$61$ $$( 1 - 61 T^{2} )^{4}$$
$67$ $$( 1 + 14 T + 129 T^{2} + 938 T^{3} + 4489 T^{4} )^{2}$$
$71$ $$( 1 + 71 T^{2} )^{4}$$
$73$ $$( 1 + 2 T - 69 T^{2} + 146 T^{3} + 5329 T^{4} )^{2}$$
$79$ $$( 1 - 79 T^{2} )^{4}$$
$83$ $$( 1 - 18 T + 241 T^{2} - 1494 T^{3} + 6889 T^{4} )( 1 + 18 T + 241 T^{2} + 1494 T^{3} + 6889 T^{4} )$$
$89$ $$( 1 - 18 T + 235 T^{2} - 1602 T^{3} + 7921 T^{4} )( 1 + 18 T + 235 T^{2} + 1602 T^{3} + 7921 T^{4} )$$
$97$ $$( 1 + 10 T + 97 T^{2} )^{4}$$