# Properties

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

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$4.79102412128$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 120) 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.

 $$f(q)$$ $$=$$ $$q + (i - 1) q^{2} - i q^{3} - 2 i q^{4} + (i + 1) q^{6} - 2 q^{7} + (2 i + 2) q^{8} - q^{9} +O(q^{10})$$ q + (i - 1) * q^2 - i * q^3 - 2*i * q^4 + (i + 1) * q^6 - 2 * q^7 + (2*i + 2) * q^8 - q^9 $$q + (i - 1) q^{2} - i q^{3} - 2 i q^{4} + (i + 1) q^{6} - 2 q^{7} + (2 i + 2) q^{8} - q^{9} + 4 i q^{11} - 2 q^{12} + ( - 2 i + 2) q^{14} - 4 q^{16} + 6 q^{17} + ( - i + 1) q^{18} + 4 i q^{19} + 2 i q^{21} + ( - 4 i - 4) q^{22} + 4 q^{23} + ( - 2 i + 2) q^{24} + i q^{27} + 4 i q^{28} - 6 i q^{29} + 10 q^{31} + ( - 4 i + 4) q^{32} + 4 q^{33} + (6 i - 6) q^{34} + 2 i q^{36} - 4 i q^{37} + ( - 4 i - 4) q^{38} + 10 q^{41} + ( - 2 i - 2) q^{42} + 4 i q^{43} + 8 q^{44} + (4 i - 4) q^{46} + 4 q^{47} + 4 i q^{48} - 3 q^{49} - 6 i q^{51} + 10 i q^{53} + ( - i - 1) q^{54} + ( - 4 i - 4) q^{56} + 4 q^{57} + (6 i + 6) q^{58} + 8 i q^{59} - 8 i q^{61} + (10 i - 10) q^{62} + 2 q^{63} + 8 i q^{64} + (4 i - 4) q^{66} + 12 i q^{67} - 12 i q^{68} - 4 i q^{69} - 4 q^{71} + ( - 2 i - 2) q^{72} - 10 q^{73} + (4 i + 4) q^{74} + 8 q^{76} - 8 i q^{77} - 14 q^{79} + q^{81} + (10 i - 10) q^{82} + 4 q^{84} + ( - 4 i - 4) q^{86} - 6 q^{87} + (8 i - 8) q^{88} + 14 q^{89} - 8 i q^{92} - 10 i q^{93} + (4 i - 4) q^{94} + ( - 4 i - 4) q^{96} + 10 q^{97} + ( - 3 i + 3) q^{98} - 4 i q^{99} +O(q^{100})$$ q + (i - 1) * q^2 - i * q^3 - 2*i * q^4 + (i + 1) * q^6 - 2 * q^7 + (2*i + 2) * q^8 - q^9 + 4*i * q^11 - 2 * q^12 + (-2*i + 2) * q^14 - 4 * q^16 + 6 * q^17 + (-i + 1) * q^18 + 4*i * q^19 + 2*i * q^21 + (-4*i - 4) * q^22 + 4 * q^23 + (-2*i + 2) * q^24 + i * q^27 + 4*i * q^28 - 6*i * q^29 + 10 * q^31 + (-4*i + 4) * q^32 + 4 * q^33 + (6*i - 6) * q^34 + 2*i * q^36 - 4*i * q^37 + (-4*i - 4) * q^38 + 10 * q^41 + (-2*i - 2) * q^42 + 4*i * q^43 + 8 * q^44 + (4*i - 4) * q^46 + 4 * q^47 + 4*i * q^48 - 3 * q^49 - 6*i * q^51 + 10*i * q^53 + (-i - 1) * q^54 + (-4*i - 4) * q^56 + 4 * q^57 + (6*i + 6) * q^58 + 8*i * q^59 - 8*i * q^61 + (10*i - 10) * q^62 + 2 * q^63 + 8*i * q^64 + (4*i - 4) * q^66 + 12*i * q^67 - 12*i * q^68 - 4*i * q^69 - 4 * q^71 + (-2*i - 2) * q^72 - 10 * q^73 + (4*i + 4) * q^74 + 8 * q^76 - 8*i * q^77 - 14 * q^79 + q^81 + (10*i - 10) * q^82 + 4 * q^84 + (-4*i - 4) * q^86 - 6 * q^87 + (8*i - 8) * q^88 + 14 * q^89 - 8*i * q^92 - 10*i * q^93 + (4*i - 4) * q^94 + (-4*i - 4) * q^96 + 10 * q^97 + (-3*i + 3) * q^98 - 4*i * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} + 2 q^{6} - 4 q^{7} + 4 q^{8} - 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^2 + 2 * q^6 - 4 * q^7 + 4 * q^8 - 2 * q^9 $$2 q - 2 q^{2} + 2 q^{6} - 4 q^{7} + 4 q^{8} - 2 q^{9} - 4 q^{12} + 4 q^{14} - 8 q^{16} + 12 q^{17} + 2 q^{18} - 8 q^{22} + 8 q^{23} + 4 q^{24} + 20 q^{31} + 8 q^{32} + 8 q^{33} - 12 q^{34} - 8 q^{38} + 20 q^{41} - 4 q^{42} + 16 q^{44} - 8 q^{46} + 8 q^{47} - 6 q^{49} - 2 q^{54} - 8 q^{56} + 8 q^{57} + 12 q^{58} - 20 q^{62} + 4 q^{63} - 8 q^{66} - 8 q^{71} - 4 q^{72} - 20 q^{73} + 8 q^{74} + 16 q^{76} - 28 q^{79} + 2 q^{81} - 20 q^{82} + 8 q^{84} - 8 q^{86} - 12 q^{87} - 16 q^{88} + 28 q^{89} - 8 q^{94} - 8 q^{96} + 20 q^{97} + 6 q^{98}+O(q^{100})$$ 2 * q - 2 * q^2 + 2 * q^6 - 4 * q^7 + 4 * q^8 - 2 * q^9 - 4 * q^12 + 4 * q^14 - 8 * q^16 + 12 * q^17 + 2 * q^18 - 8 * q^22 + 8 * q^23 + 4 * q^24 + 20 * q^31 + 8 * q^32 + 8 * q^33 - 12 * q^34 - 8 * q^38 + 20 * q^41 - 4 * q^42 + 16 * q^44 - 8 * q^46 + 8 * q^47 - 6 * q^49 - 2 * q^54 - 8 * q^56 + 8 * q^57 + 12 * q^58 - 20 * q^62 + 4 * q^63 - 8 * q^66 - 8 * q^71 - 4 * q^72 - 20 * q^73 + 8 * q^74 + 16 * q^76 - 28 * q^79 + 2 * q^81 - 20 * q^82 + 8 * q^84 - 8 * q^86 - 12 * q^87 - 16 * q^88 + 28 * q^89 - 8 * q^94 - 8 * q^96 + 20 * q^97 + 6 * q^98

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
301.1
 − 1.00000i 1.00000i
−1.00000 1.00000i 1.00000i 2.00000i 0 1.00000 1.00000i −2.00000 2.00000 2.00000i −1.00000 0
301.2 −1.00000 + 1.00000i 1.00000i 2.00000i 0 1.00000 + 1.00000i −2.00000 2.00000 + 2.00000i −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
8.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.k.a 2
3.b odd 2 1 1800.2.k.g 2
4.b odd 2 1 2400.2.k.b 2
5.b even 2 1 120.2.k.a 2
5.c odd 4 1 600.2.d.a 2
5.c odd 4 1 600.2.d.d 2
8.b even 2 1 inner 600.2.k.a 2
8.d odd 2 1 2400.2.k.b 2
12.b even 2 1 7200.2.k.f 2
15.d odd 2 1 360.2.k.b 2
15.e even 4 1 1800.2.d.c 2
15.e even 4 1 1800.2.d.h 2
20.d odd 2 1 480.2.k.a 2
20.e even 4 1 2400.2.d.a 2
20.e even 4 1 2400.2.d.d 2
24.f even 2 1 7200.2.k.f 2
24.h odd 2 1 1800.2.k.g 2
40.e odd 2 1 480.2.k.a 2
40.f even 2 1 120.2.k.a 2
40.i odd 4 1 600.2.d.a 2
40.i odd 4 1 600.2.d.d 2
40.k even 4 1 2400.2.d.a 2
40.k even 4 1 2400.2.d.d 2
60.h even 2 1 1440.2.k.a 2
60.l odd 4 1 7200.2.d.e 2
60.l odd 4 1 7200.2.d.f 2
80.k odd 4 1 3840.2.a.m 1
80.k odd 4 1 3840.2.a.r 1
80.q even 4 1 3840.2.a.d 1
80.q even 4 1 3840.2.a.w 1
120.i odd 2 1 360.2.k.b 2
120.m even 2 1 1440.2.k.a 2
120.q odd 4 1 7200.2.d.e 2
120.q odd 4 1 7200.2.d.f 2
120.w even 4 1 1800.2.d.c 2
120.w even 4 1 1800.2.d.h 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
120.2.k.a 2 5.b even 2 1
120.2.k.a 2 40.f even 2 1
360.2.k.b 2 15.d odd 2 1
360.2.k.b 2 120.i odd 2 1
480.2.k.a 2 20.d odd 2 1
480.2.k.a 2 40.e odd 2 1
600.2.d.a 2 5.c odd 4 1
600.2.d.a 2 40.i odd 4 1
600.2.d.d 2 5.c odd 4 1
600.2.d.d 2 40.i odd 4 1
600.2.k.a 2 1.a even 1 1 trivial
600.2.k.a 2 8.b even 2 1 inner
1440.2.k.a 2 60.h even 2 1
1440.2.k.a 2 120.m even 2 1
1800.2.d.c 2 15.e even 4 1
1800.2.d.c 2 120.w even 4 1
1800.2.d.h 2 15.e even 4 1
1800.2.d.h 2 120.w even 4 1
1800.2.k.g 2 3.b odd 2 1
1800.2.k.g 2 24.h odd 2 1
2400.2.d.a 2 20.e even 4 1
2400.2.d.a 2 40.k even 4 1
2400.2.d.d 2 20.e even 4 1
2400.2.d.d 2 40.k even 4 1
2400.2.k.b 2 4.b odd 2 1
2400.2.k.b 2 8.d odd 2 1
3840.2.a.d 1 80.q even 4 1
3840.2.a.m 1 80.k odd 4 1
3840.2.a.r 1 80.k odd 4 1
3840.2.a.w 1 80.q even 4 1
7200.2.d.e 2 60.l odd 4 1
7200.2.d.e 2 120.q odd 4 1
7200.2.d.f 2 60.l odd 4 1
7200.2.d.f 2 120.q odd 4 1
7200.2.k.f 2 12.b even 2 1
7200.2.k.f 2 24.f even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7} + 2$$ acting on $$S_{2}^{\mathrm{new}}(600, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 2T + 2$$
$3$ $$T^{2} + 1$$
$5$ $$T^{2}$$
$7$ $$(T + 2)^{2}$$
$11$ $$T^{2} + 16$$
$13$ $$T^{2}$$
$17$ $$(T - 6)^{2}$$
$19$ $$T^{2} + 16$$
$23$ $$(T - 4)^{2}$$
$29$ $$T^{2} + 36$$
$31$ $$(T - 10)^{2}$$
$37$ $$T^{2} + 16$$
$41$ $$(T - 10)^{2}$$
$43$ $$T^{2} + 16$$
$47$ $$(T - 4)^{2}$$
$53$ $$T^{2} + 100$$
$59$ $$T^{2} + 64$$
$61$ $$T^{2} + 64$$
$67$ $$T^{2} + 144$$
$71$ $$(T + 4)^{2}$$
$73$ $$(T + 10)^{2}$$
$79$ $$(T + 14)^{2}$$
$83$ $$T^{2}$$
$89$ $$(T - 14)^{2}$$
$97$ $$(T - 10)^{2}$$