# Properties

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

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [600,2,Mod(251,600)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(600, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1, 1, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("600.251");

S:= CuspForms(chi, 2);

N := Newforms(S);

 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

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: no Analytic conductor: $$4.79102412128$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-2})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 2$$ x^2 + 2 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 24) Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{-2}$$. We also show the integral $$q$$-expansion of the trace form.

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

## 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
251.1
 − 1.41421i 1.41421i
1.41421i 1.00000 + 1.41421i −2.00000 0 2.00000 1.41421i 0 2.82843i −1.00000 + 2.82843i 0
251.2 1.41421i 1.00000 1.41421i −2.00000 0 2.00000 + 1.41421i 0 2.82843i −1.00000 2.82843i 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.a 2
3.b odd 2 1 inner 600.2.b.a 2
4.b odd 2 1 2400.2.b.a 2
5.b even 2 1 24.2.f.a 2
5.c odd 4 2 600.2.m.a 4
8.b even 2 1 2400.2.b.a 2
8.d odd 2 1 CM 600.2.b.a 2
12.b even 2 1 2400.2.b.a 2
15.d odd 2 1 24.2.f.a 2
15.e even 4 2 600.2.m.a 4
20.d odd 2 1 96.2.f.a 2
20.e even 4 2 2400.2.m.a 4
24.f even 2 1 inner 600.2.b.a 2
24.h odd 2 1 2400.2.b.a 2
40.e odd 2 1 24.2.f.a 2
40.f even 2 1 96.2.f.a 2
40.i odd 4 2 2400.2.m.a 4
40.k even 4 2 600.2.m.a 4
45.h odd 6 2 648.2.l.b 4
45.j even 6 2 648.2.l.b 4
60.h even 2 1 96.2.f.a 2
60.l odd 4 2 2400.2.m.a 4
80.k odd 4 2 768.2.c.h 4
80.q even 4 2 768.2.c.h 4
120.i odd 2 1 96.2.f.a 2
120.m even 2 1 24.2.f.a 2
120.q odd 4 2 600.2.m.a 4
120.w even 4 2 2400.2.m.a 4
180.n even 6 2 2592.2.p.b 4
180.p odd 6 2 2592.2.p.b 4
240.t even 4 2 768.2.c.h 4
240.bm odd 4 2 768.2.c.h 4
360.z odd 6 2 648.2.l.b 4
360.bd even 6 2 648.2.l.b 4
360.bh odd 6 2 2592.2.p.b 4
360.bk even 6 2 2592.2.p.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.2.f.a 2 5.b even 2 1
24.2.f.a 2 15.d odd 2 1
24.2.f.a 2 40.e odd 2 1
24.2.f.a 2 120.m even 2 1
96.2.f.a 2 20.d odd 2 1
96.2.f.a 2 40.f even 2 1
96.2.f.a 2 60.h even 2 1
96.2.f.a 2 120.i odd 2 1
600.2.b.a 2 1.a even 1 1 trivial
600.2.b.a 2 3.b odd 2 1 inner
600.2.b.a 2 8.d odd 2 1 CM
600.2.b.a 2 24.f even 2 1 inner
600.2.m.a 4 5.c odd 4 2
600.2.m.a 4 15.e even 4 2
600.2.m.a 4 40.k even 4 2
600.2.m.a 4 120.q odd 4 2
648.2.l.b 4 45.h odd 6 2
648.2.l.b 4 45.j even 6 2
648.2.l.b 4 360.z odd 6 2
648.2.l.b 4 360.bd even 6 2
768.2.c.h 4 80.k odd 4 2
768.2.c.h 4 80.q even 4 2
768.2.c.h 4 240.t even 4 2
768.2.c.h 4 240.bm odd 4 2
2400.2.b.a 2 4.b odd 2 1
2400.2.b.a 2 8.b even 2 1
2400.2.b.a 2 12.b even 2 1
2400.2.b.a 2 24.h odd 2 1
2400.2.m.a 4 20.e even 4 2
2400.2.m.a 4 40.i odd 4 2
2400.2.m.a 4 60.l odd 4 2
2400.2.m.a 4 120.w even 4 2
2592.2.p.b 4 180.n even 6 2
2592.2.p.b 4 180.p odd 6 2
2592.2.p.b 4 360.bh odd 6 2
2592.2.p.b 4 360.bk even 6 2

## Hecke kernels

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

 $$T_{7}$$ T7 $$T_{11}^{2} + 8$$ T11^2 + 8 $$T_{23}$$ T23 $$T_{43} - 10$$ T43 - 10

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 2$$
$3$ $$T^{2} - 2T + 3$$
$5$ $$T^{2}$$
$7$ $$T^{2}$$
$11$ $$T^{2} + 8$$
$13$ $$T^{2}$$
$17$ $$T^{2} + 32$$
$19$ $$(T - 2)^{2}$$
$23$ $$T^{2}$$
$29$ $$T^{2}$$
$31$ $$T^{2}$$
$37$ $$T^{2}$$
$41$ $$T^{2} + 128$$
$43$ $$(T - 10)^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2}$$
$59$ $$T^{2} + 200$$
$61$ $$T^{2}$$
$67$ $$(T + 14)^{2}$$
$71$ $$T^{2}$$
$73$ $$(T + 2)^{2}$$
$79$ $$T^{2}$$
$83$ $$T^{2} + 8$$
$89$ $$T^{2} + 32$$
$97$ $$(T - 10)^{2}$$