# Properties

 Label 600.2.r.a Level $600$ Weight $2$ Character orbit 600.r Analytic conductor $4.791$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$4.79102412128$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(i)$$ Coefficient field: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ x^4 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{8}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \zeta_{8}^{3} - \zeta_{8}^{2} - 1) q^{3} + (2 \zeta_{8}^{2} - 2) q^{7} + (2 \zeta_{8}^{3} + \zeta_{8}^{2} - 2 \zeta_{8}) q^{9}+O(q^{10})$$ q + (-z^3 - z^2 - 1) * q^3 + (2*z^2 - 2) * q^7 + (2*z^3 + z^2 - 2*z) * q^9 $$q + ( - \zeta_{8}^{3} - \zeta_{8}^{2} - 1) q^{3} + (2 \zeta_{8}^{2} - 2) q^{7} + (2 \zeta_{8}^{3} + \zeta_{8}^{2} - 2 \zeta_{8}) q^{9} + (4 \zeta_{8}^{3} + 4 \zeta_{8}) q^{11} - 4 \zeta_{8} q^{17} + 4 \zeta_{8}^{2} q^{19} + (2 \zeta_{8}^{3} + 2 \zeta_{8} + 4) q^{21} + 6 \zeta_{8}^{3} q^{23} + (\zeta_{8}^{2} + 5 \zeta_{8} - 1) q^{27} + (4 \zeta_{8}^{3} - 4 \zeta_{8}) q^{29} + 8 q^{31} + ( - 8 \zeta_{8}^{3} + 4 \zeta_{8}^{2} + 4) q^{33} + ( - 8 \zeta_{8}^{2} + 8) q^{37} + (4 \zeta_{8}^{3} + 4 \zeta_{8}) q^{41} + ( - 2 \zeta_{8}^{2} - 2) q^{43} + 2 \zeta_{8} q^{47} - \zeta_{8}^{2} q^{49} + (4 \zeta_{8}^{3} + 4 \zeta_{8} - 4) q^{51} + 8 \zeta_{8}^{3} q^{53} + ( - 4 \zeta_{8}^{2} + 4 \zeta_{8} + 4) q^{57} + (4 \zeta_{8}^{3} - 4 \zeta_{8}) q^{59} - 6 q^{61} + ( - 8 \zeta_{8}^{3} - 2 \zeta_{8}^{2} - 2) q^{63} + (6 \zeta_{8}^{2} - 6) q^{67} + ( - 6 \zeta_{8}^{3} + 6 \zeta_{8}^{2} + 6 \zeta_{8}) q^{69} + ( - 8 \zeta_{8}^{3} - 8 \zeta_{8}) q^{71} + (8 \zeta_{8}^{2} + 8) q^{73} - 16 \zeta_{8} q^{77} + ( - 4 \zeta_{8}^{3} - 4 \zeta_{8} + 7) q^{81} + 14 \zeta_{8}^{3} q^{83} + (4 \zeta_{8}^{2} + 8 \zeta_{8} - 4) q^{87} + ( - 8 \zeta_{8}^{3} + 8 \zeta_{8}) q^{89} + ( - 8 \zeta_{8}^{3} - 8 \zeta_{8}^{2} - 8) q^{93} + (8 \zeta_{8}^{2} - 8) q^{97} + (4 \zeta_{8}^{3} - 16 \zeta_{8}^{2} - 4 \zeta_{8}) q^{99} +O(q^{100})$$ q + (-z^3 - z^2 - 1) * q^3 + (2*z^2 - 2) * q^7 + (2*z^3 + z^2 - 2*z) * q^9 + (4*z^3 + 4*z) * q^11 - 4*z * q^17 + 4*z^2 * q^19 + (2*z^3 + 2*z + 4) * q^21 + 6*z^3 * q^23 + (z^2 + 5*z - 1) * q^27 + (4*z^3 - 4*z) * q^29 + 8 * q^31 + (-8*z^3 + 4*z^2 + 4) * q^33 + (-8*z^2 + 8) * q^37 + (4*z^3 + 4*z) * q^41 + (-2*z^2 - 2) * q^43 + 2*z * q^47 - z^2 * q^49 + (4*z^3 + 4*z - 4) * q^51 + 8*z^3 * q^53 + (-4*z^2 + 4*z + 4) * q^57 + (4*z^3 - 4*z) * q^59 - 6 * q^61 + (-8*z^3 - 2*z^2 - 2) * q^63 + (6*z^2 - 6) * q^67 + (-6*z^3 + 6*z^2 + 6*z) * q^69 + (-8*z^3 - 8*z) * q^71 + (8*z^2 + 8) * q^73 - 16*z * q^77 + (-4*z^3 - 4*z + 7) * q^81 + 14*z^3 * q^83 + (4*z^2 + 8*z - 4) * q^87 + (-8*z^3 + 8*z) * q^89 + (-8*z^3 - 8*z^2 - 8) * q^93 + (8*z^2 - 8) * q^97 + (4*z^3 - 16*z^2 - 4*z) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{3} - 8 q^{7}+O(q^{10})$$ 4 * q - 4 * q^3 - 8 * q^7 $$4 q - 4 q^{3} - 8 q^{7} + 16 q^{21} - 4 q^{27} + 32 q^{31} + 16 q^{33} + 32 q^{37} - 8 q^{43} - 16 q^{51} + 16 q^{57} - 24 q^{61} - 8 q^{63} - 24 q^{67} + 32 q^{73} + 28 q^{81} - 16 q^{87} - 32 q^{93} - 32 q^{97}+O(q^{100})$$ 4 * q - 4 * q^3 - 8 * q^7 + 16 * q^21 - 4 * q^27 + 32 * q^31 + 16 * q^33 + 32 * q^37 - 8 * q^43 - 16 * q^51 + 16 * q^57 - 24 * q^61 - 8 * q^63 - 24 * q^67 + 32 * q^73 + 28 * q^81 - 16 * q^87 - 32 * q^93 - 32 * q^97

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
257.1
 −0.707107 + 0.707107i 0.707107 − 0.707107i −0.707107 − 0.707107i 0.707107 + 0.707107i
0 −1.70711 + 0.292893i 0 0 0 −2.00000 2.00000i 0 2.82843 1.00000i 0
257.2 0 −0.292893 + 1.70711i 0 0 0 −2.00000 2.00000i 0 −2.82843 1.00000i 0
593.1 0 −1.70711 0.292893i 0 0 0 −2.00000 + 2.00000i 0 2.82843 + 1.00000i 0
593.2 0 −0.292893 1.70711i 0 0 0 −2.00000 + 2.00000i 0 −2.82843 + 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 Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.c odd 4 1 inner
15.e even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 600.2.r.a 4
3.b odd 2 1 inner 600.2.r.a 4
4.b odd 2 1 1200.2.v.k 4
5.b even 2 1 600.2.r.e yes 4
5.c odd 4 1 inner 600.2.r.a 4
5.c odd 4 1 600.2.r.e yes 4
12.b even 2 1 1200.2.v.k 4
15.d odd 2 1 600.2.r.e yes 4
15.e even 4 1 inner 600.2.r.a 4
15.e even 4 1 600.2.r.e yes 4
20.d odd 2 1 1200.2.v.a 4
20.e even 4 1 1200.2.v.a 4
20.e even 4 1 1200.2.v.k 4
60.h even 2 1 1200.2.v.a 4
60.l odd 4 1 1200.2.v.a 4
60.l odd 4 1 1200.2.v.k 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
600.2.r.a 4 1.a even 1 1 trivial
600.2.r.a 4 3.b odd 2 1 inner
600.2.r.a 4 5.c odd 4 1 inner
600.2.r.a 4 15.e even 4 1 inner
600.2.r.e yes 4 5.b even 2 1
600.2.r.e yes 4 5.c odd 4 1
600.2.r.e yes 4 15.d odd 2 1
600.2.r.e yes 4 15.e even 4 1
1200.2.v.a 4 20.d odd 2 1
1200.2.v.a 4 20.e even 4 1
1200.2.v.a 4 60.h even 2 1
1200.2.v.a 4 60.l odd 4 1
1200.2.v.k 4 4.b odd 2 1
1200.2.v.k 4 12.b even 2 1
1200.2.v.k 4 20.e even 4 1
1200.2.v.k 4 60.l odd 4 1

## Hecke kernels

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

 $$T_{7}^{2} + 4T_{7} + 8$$ T7^2 + 4*T7 + 8 $$T_{17}^{4} + 256$$ T17^4 + 256

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} + 4 T^{3} + 8 T^{2} + 12 T + 9$$
$5$ $$T^{4}$$
$7$ $$(T^{2} + 4 T + 8)^{2}$$
$11$ $$(T^{2} + 32)^{2}$$
$13$ $$T^{4}$$
$17$ $$T^{4} + 256$$
$19$ $$(T^{2} + 16)^{2}$$
$23$ $$T^{4} + 1296$$
$29$ $$(T^{2} - 32)^{2}$$
$31$ $$(T - 8)^{4}$$
$37$ $$(T^{2} - 16 T + 128)^{2}$$
$41$ $$(T^{2} + 32)^{2}$$
$43$ $$(T^{2} + 4 T + 8)^{2}$$
$47$ $$T^{4} + 16$$
$53$ $$T^{4} + 4096$$
$59$ $$(T^{2} - 32)^{2}$$
$61$ $$(T + 6)^{4}$$
$67$ $$(T^{2} + 12 T + 72)^{2}$$
$71$ $$(T^{2} + 128)^{2}$$
$73$ $$(T^{2} - 16 T + 128)^{2}$$
$79$ $$T^{4}$$
$83$ $$T^{4} + 38416$$
$89$ $$(T^{2} - 128)^{2}$$
$97$ $$(T^{2} + 16 T + 128)^{2}$$