# Properties

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

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## Newspace parameters

 Level: $$N$$ $$=$$ $$600 = 2^{3} \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 600.f (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.79102412128$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + i q^{3} -3 i q^{7} - q^{9} +O(q^{10})$$ $$q + i q^{3} -3 i q^{7} - q^{9} + 2 q^{11} -3 i q^{13} -6 i q^{17} + 7 q^{19} + 3 q^{21} + 6 i q^{23} -i q^{27} + 2 q^{29} -5 q^{31} + 2 i q^{33} -10 i q^{37} + 3 q^{39} + 12 q^{41} + 3 i q^{43} + 10 i q^{47} -2 q^{49} + 6 q^{51} + 7 i q^{57} + 6 q^{59} -13 q^{61} + 3 i q^{63} -7 i q^{67} -6 q^{69} -4 q^{71} -6 i q^{73} -6 i q^{77} + 8 q^{79} + q^{81} -6 i q^{83} + 2 i q^{87} -16 q^{89} -9 q^{91} -5 i q^{93} + 7 i q^{97} -2 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{9} + O(q^{10})$$ $$2q - 2q^{9} + 4q^{11} + 14q^{19} + 6q^{21} + 4q^{29} - 10q^{31} + 6q^{39} + 24q^{41} - 4q^{49} + 12q^{51} + 12q^{59} - 26q^{61} - 12q^{69} - 8q^{71} + 16q^{79} + 2q^{81} - 32q^{89} - 18q^{91} - 4q^{99} + O(q^{100})$$

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
49.1
 − 1.00000i 1.00000i
0 1.00000i 0 0 0 3.00000i 0 −1.00000 0
49.2 0 1.00000i 0 0 0 3.00000i 0 −1.00000 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
5.b 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.f.d 2
3.b odd 2 1 1800.2.f.e 2
4.b odd 2 1 1200.2.f.c 2
5.b even 2 1 inner 600.2.f.d 2
5.c odd 4 1 600.2.a.b 1
5.c odd 4 1 600.2.a.i yes 1
8.b even 2 1 4800.2.f.k 2
8.d odd 2 1 4800.2.f.z 2
12.b even 2 1 3600.2.f.o 2
15.d odd 2 1 1800.2.f.e 2
15.e even 4 1 1800.2.a.e 1
15.e even 4 1 1800.2.a.t 1
20.d odd 2 1 1200.2.f.c 2
20.e even 4 1 1200.2.a.b 1
20.e even 4 1 1200.2.a.q 1
40.e odd 2 1 4800.2.f.z 2
40.f even 2 1 4800.2.f.k 2
40.i odd 4 1 4800.2.a.bc 1
40.i odd 4 1 4800.2.a.bp 1
40.k even 4 1 4800.2.a.bd 1
40.k even 4 1 4800.2.a.bs 1
60.h even 2 1 3600.2.f.o 2
60.l odd 4 1 3600.2.a.i 1
60.l odd 4 1 3600.2.a.bl 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
600.2.a.b 1 5.c odd 4 1
600.2.a.i yes 1 5.c odd 4 1
600.2.f.d 2 1.a even 1 1 trivial
600.2.f.d 2 5.b even 2 1 inner
1200.2.a.b 1 20.e even 4 1
1200.2.a.q 1 20.e even 4 1
1200.2.f.c 2 4.b odd 2 1
1200.2.f.c 2 20.d odd 2 1
1800.2.a.e 1 15.e even 4 1
1800.2.a.t 1 15.e even 4 1
1800.2.f.e 2 3.b odd 2 1
1800.2.f.e 2 15.d odd 2 1
3600.2.a.i 1 60.l odd 4 1
3600.2.a.bl 1 60.l odd 4 1
3600.2.f.o 2 12.b even 2 1
3600.2.f.o 2 60.h even 2 1
4800.2.a.bc 1 40.i odd 4 1
4800.2.a.bd 1 40.k even 4 1
4800.2.a.bp 1 40.i odd 4 1
4800.2.a.bs 1 40.k even 4 1
4800.2.f.k 2 8.b even 2 1
4800.2.f.k 2 40.f even 2 1
4800.2.f.z 2 8.d odd 2 1
4800.2.f.z 2 40.e odd 2 1

## Hecke kernels

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

 $$T_{7}^{2} + 9$$ $$T_{11} - 2$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$1 + T^{2}$$
$5$ $$T^{2}$$
$7$ $$9 + T^{2}$$
$11$ $$( -2 + T )^{2}$$
$13$ $$9 + T^{2}$$
$17$ $$36 + T^{2}$$
$19$ $$( -7 + T )^{2}$$
$23$ $$36 + T^{2}$$
$29$ $$( -2 + T )^{2}$$
$31$ $$( 5 + T )^{2}$$
$37$ $$100 + T^{2}$$
$41$ $$( -12 + T )^{2}$$
$43$ $$9 + T^{2}$$
$47$ $$100 + T^{2}$$
$53$ $$T^{2}$$
$59$ $$( -6 + T )^{2}$$
$61$ $$( 13 + T )^{2}$$
$67$ $$49 + T^{2}$$
$71$ $$( 4 + T )^{2}$$
$73$ $$36 + T^{2}$$
$79$ $$( -8 + T )^{2}$$
$83$ $$36 + T^{2}$$
$89$ $$( 16 + T )^{2}$$
$97$ $$49 + T^{2}$$
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