# Properties

 Label 600.4.a.q Level $600$ Weight $4$ Character orbit 600.a Self dual yes Analytic conductor $35.401$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$600 = 2^{3} \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 600.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$35.4011460034$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 120) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + 3 q^{3} + 16 q^{7} + 9 q^{9}+O(q^{10})$$ q + 3 * q^3 + 16 * q^7 + 9 * q^9 $$q + 3 q^{3} + 16 q^{7} + 9 q^{9} - 28 q^{11} + 26 q^{13} + 62 q^{17} - 68 q^{19} + 48 q^{21} + 208 q^{23} + 27 q^{27} - 58 q^{29} + 160 q^{31} - 84 q^{33} - 270 q^{37} + 78 q^{39} + 282 q^{41} - 76 q^{43} + 280 q^{47} - 87 q^{49} + 186 q^{51} + 210 q^{53} - 204 q^{57} + 196 q^{59} + 742 q^{61} + 144 q^{63} - 836 q^{67} + 624 q^{69} - 504 q^{71} + 1062 q^{73} - 448 q^{77} + 768 q^{79} + 81 q^{81} + 1052 q^{83} - 174 q^{87} - 726 q^{89} + 416 q^{91} + 480 q^{93} + 1406 q^{97} - 252 q^{99}+O(q^{100})$$ q + 3 * q^3 + 16 * q^7 + 9 * q^9 - 28 * q^11 + 26 * q^13 + 62 * q^17 - 68 * q^19 + 48 * q^21 + 208 * q^23 + 27 * q^27 - 58 * q^29 + 160 * q^31 - 84 * q^33 - 270 * q^37 + 78 * q^39 + 282 * q^41 - 76 * q^43 + 280 * q^47 - 87 * q^49 + 186 * q^51 + 210 * q^53 - 204 * q^57 + 196 * q^59 + 742 * q^61 + 144 * q^63 - 836 * q^67 + 624 * q^69 - 504 * q^71 + 1062 * q^73 - 448 * q^77 + 768 * q^79 + 81 * q^81 + 1052 * q^83 - 174 * q^87 - 726 * q^89 + 416 * q^91 + 480 * q^93 + 1406 * q^97 - 252 * q^99

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
0 3.00000 0 0 0 16.0000 0 9.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$-1$$
$$5$$ $$1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 600.4.a.q 1
3.b odd 2 1 1800.4.a.bb 1
4.b odd 2 1 1200.4.a.c 1
5.b even 2 1 120.4.a.c 1
5.c odd 4 2 600.4.f.c 2
15.d odd 2 1 360.4.a.b 1
15.e even 4 2 1800.4.f.r 2
20.d odd 2 1 240.4.a.l 1
20.e even 4 2 1200.4.f.o 2
40.e odd 2 1 960.4.a.h 1
40.f even 2 1 960.4.a.u 1
60.h even 2 1 720.4.a.l 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
120.4.a.c 1 5.b even 2 1
240.4.a.l 1 20.d odd 2 1
360.4.a.b 1 15.d odd 2 1
600.4.a.q 1 1.a even 1 1 trivial
600.4.f.c 2 5.c odd 4 2
720.4.a.l 1 60.h even 2 1
960.4.a.h 1 40.e odd 2 1
960.4.a.u 1 40.f even 2 1
1200.4.a.c 1 4.b odd 2 1
1200.4.f.o 2 20.e even 4 2
1800.4.a.bb 1 3.b odd 2 1
1800.4.f.r 2 15.e even 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(600))$$:

 $$T_{7} - 16$$ T7 - 16 $$T_{11} + 28$$ T11 + 28

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 3$$
$5$ $$T$$
$7$ $$T - 16$$
$11$ $$T + 28$$
$13$ $$T - 26$$
$17$ $$T - 62$$
$19$ $$T + 68$$
$23$ $$T - 208$$
$29$ $$T + 58$$
$31$ $$T - 160$$
$37$ $$T + 270$$
$41$ $$T - 282$$
$43$ $$T + 76$$
$47$ $$T - 280$$
$53$ $$T - 210$$
$59$ $$T - 196$$
$61$ $$T - 742$$
$67$ $$T + 836$$
$71$ $$T + 504$$
$73$ $$T - 1062$$
$79$ $$T - 768$$
$83$ $$T - 1052$$
$89$ $$T + 726$$
$97$ $$T - 1406$$