# Properties

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

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [600,2,Mod(299,600)]

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, 1]))

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

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

chi := DirichletCharacter("600.299");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$600 = 2^{3} \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 600.m (of order $$2$$, degree $$1$$, not 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: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 1$$ x^4 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ 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 a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_{3} q^{2} + ( - \beta_{3} - \beta_1) q^{3} + 2 q^{4} + (\beta_{2} + 2) q^{6} - 2 \beta_{3} q^{8} + (2 \beta_{2} + 1) q^{9}+O(q^{10})$$ q - b3 * q^2 + (-b3 - b1) * q^3 + 2 * q^4 + (b2 + 2) * q^6 - 2*b3 * q^8 + (2*b2 + 1) * q^9 $$q - \beta_{3} q^{2} + ( - \beta_{3} - \beta_1) q^{3} + 2 q^{4} + (\beta_{2} + 2) q^{6} - 2 \beta_{3} q^{8} + (2 \beta_{2} + 1) q^{9} - 2 \beta_{2} q^{11} + ( - 2 \beta_{3} - 2 \beta_1) q^{12} + 4 q^{16} + 4 \beta_{3} q^{17} + ( - \beta_{3} - 4 \beta_1) q^{18} - 2 q^{19} + 4 \beta_1 q^{22} + (2 \beta_{2} + 4) q^{24} + (\beta_{3} - 5 \beta_1) q^{27} - 4 \beta_{3} q^{32} + ( - 2 \beta_{3} + 4 \beta_1) q^{33} - 8 q^{34} + (4 \beta_{2} + 2) q^{36} + 2 \beta_{3} q^{38} - 8 \beta_{2} q^{41} - 10 \beta_1 q^{43} - 4 \beta_{2} q^{44} + ( - 4 \beta_{3} - 4 \beta_1) q^{48} - 7 q^{49} + ( - 4 \beta_{2} - 8) q^{51} + (5 \beta_{2} - 2) q^{54} + (2 \beta_{3} + 2 \beta_1) q^{57} - 10 \beta_{2} q^{59} + 8 q^{64} + ( - 4 \beta_{2} + 4) q^{66} - 14 \beta_1 q^{67} + 8 \beta_{3} q^{68} + ( - 2 \beta_{3} - 8 \beta_1) q^{72} + 2 \beta_1 q^{73} - 4 q^{76} + (4 \beta_{2} - 7) q^{81} + 16 \beta_1 q^{82} + 2 \beta_{3} q^{83} + 10 \beta_{2} q^{86} + 8 \beta_1 q^{88} - 4 \beta_{2} q^{89} + (4 \beta_{2} + 8) q^{96} + 10 \beta_1 q^{97} + 7 \beta_{3} q^{98} + ( - 2 \beta_{2} + 8) q^{99}+O(q^{100})$$ q - b3 * q^2 + (-b3 - b1) * q^3 + 2 * q^4 + (b2 + 2) * q^6 - 2*b3 * q^8 + (2*b2 + 1) * q^9 - 2*b2 * q^11 + (-2*b3 - 2*b1) * q^12 + 4 * q^16 + 4*b3 * q^17 + (-b3 - 4*b1) * q^18 - 2 * q^19 + 4*b1 * q^22 + (2*b2 + 4) * q^24 + (b3 - 5*b1) * q^27 - 4*b3 * q^32 + (-2*b3 + 4*b1) * q^33 - 8 * q^34 + (4*b2 + 2) * q^36 + 2*b3 * q^38 - 8*b2 * q^41 - 10*b1 * q^43 - 4*b2 * q^44 + (-4*b3 - 4*b1) * q^48 - 7 * q^49 + (-4*b2 - 8) * q^51 + (5*b2 - 2) * q^54 + (2*b3 + 2*b1) * q^57 - 10*b2 * q^59 + 8 * q^64 + (-4*b2 + 4) * q^66 - 14*b1 * q^67 + 8*b3 * q^68 + (-2*b3 - 8*b1) * q^72 + 2*b1 * q^73 - 4 * q^76 + (4*b2 - 7) * q^81 + 16*b1 * q^82 + 2*b3 * q^83 + 10*b2 * q^86 + 8*b1 * q^88 - 4*b2 * q^89 + (4*b2 + 8) * q^96 + 10*b1 * q^97 + 7*b3 * q^98 + (-2*b2 + 8) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 8 q^{4} + 8 q^{6} + 4 q^{9}+O(q^{10})$$ 4 * q + 8 * q^4 + 8 * q^6 + 4 * q^9 $$4 q + 8 q^{4} + 8 q^{6} + 4 q^{9} + 16 q^{16} - 8 q^{19} + 16 q^{24} - 32 q^{34} + 8 q^{36} - 28 q^{49} - 32 q^{51} - 8 q^{54} + 32 q^{64} + 16 q^{66} - 16 q^{76} - 28 q^{81} + 32 q^{96} + 32 q^{99}+O(q^{100})$$ 4 * q + 8 * q^4 + 8 * q^6 + 4 * q^9 + 16 * q^16 - 8 * q^19 + 16 * q^24 - 32 * q^34 + 8 * q^36 - 28 * q^49 - 32 * q^51 - 8 * q^54 + 32 * q^64 + 16 * q^66 - 16 * q^76 - 28 * q^81 + 32 * q^96 + 32 * q^99

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$\zeta_{8}^{2}$$ v^2 $$\beta_{2}$$ $$=$$ $$\zeta_{8}^{3} + \zeta_{8}$$ v^3 + v $$\beta_{3}$$ $$=$$ $$-\zeta_{8}^{3} + \zeta_{8}$$ -v^3 + v
 $$\zeta_{8}$$ $$=$$ $$( \beta_{3} + \beta_{2} ) / 2$$ (b3 + b2) / 2 $$\zeta_{8}^{2}$$ $$=$$ $$\beta_1$$ b1 $$\zeta_{8}^{3}$$ $$=$$ $$( -\beta_{3} + \beta_{2} ) / 2$$ (-b3 + b2) / 2

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
299.1
 0.707107 + 0.707107i 0.707107 − 0.707107i −0.707107 − 0.707107i −0.707107 + 0.707107i
−1.41421 −1.41421 1.00000i 2.00000 0 2.00000 + 1.41421i 0 −2.82843 1.00000 + 2.82843i 0
299.2 −1.41421 −1.41421 + 1.00000i 2.00000 0 2.00000 1.41421i 0 −2.82843 1.00000 2.82843i 0
299.3 1.41421 1.41421 1.00000i 2.00000 0 2.00000 1.41421i 0 2.82843 1.00000 2.82843i 0
299.4 1.41421 1.41421 + 1.00000i 2.00000 0 2.00000 + 1.41421i 0 2.82843 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
5.b even 2 1 inner
15.d odd 2 1 inner
24.f even 2 1 inner
40.e odd 2 1 inner
120.m 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.m.a 4
3.b odd 2 1 inner 600.2.m.a 4
4.b odd 2 1 2400.2.m.a 4
5.b even 2 1 inner 600.2.m.a 4
5.c odd 4 1 24.2.f.a 2
5.c odd 4 1 600.2.b.a 2
8.b even 2 1 2400.2.m.a 4
8.d odd 2 1 CM 600.2.m.a 4
12.b even 2 1 2400.2.m.a 4
15.d odd 2 1 inner 600.2.m.a 4
15.e even 4 1 24.2.f.a 2
15.e even 4 1 600.2.b.a 2
20.d odd 2 1 2400.2.m.a 4
20.e even 4 1 96.2.f.a 2
20.e even 4 1 2400.2.b.a 2
24.f even 2 1 inner 600.2.m.a 4
24.h odd 2 1 2400.2.m.a 4
40.e odd 2 1 inner 600.2.m.a 4
40.f even 2 1 2400.2.m.a 4
40.i odd 4 1 96.2.f.a 2
40.i odd 4 1 2400.2.b.a 2
40.k even 4 1 24.2.f.a 2
40.k even 4 1 600.2.b.a 2
45.k odd 12 2 648.2.l.b 4
45.l even 12 2 648.2.l.b 4
60.h even 2 1 2400.2.m.a 4
60.l odd 4 1 96.2.f.a 2
60.l odd 4 1 2400.2.b.a 2
80.i odd 4 1 768.2.c.h 4
80.j even 4 1 768.2.c.h 4
80.s even 4 1 768.2.c.h 4
80.t odd 4 1 768.2.c.h 4
120.i odd 2 1 2400.2.m.a 4
120.m even 2 1 inner 600.2.m.a 4
120.q odd 4 1 24.2.f.a 2
120.q odd 4 1 600.2.b.a 2
120.w even 4 1 96.2.f.a 2
120.w even 4 1 2400.2.b.a 2
180.v odd 12 2 2592.2.p.b 4
180.x even 12 2 2592.2.p.b 4
240.z odd 4 1 768.2.c.h 4
240.bb even 4 1 768.2.c.h 4
240.bd odd 4 1 768.2.c.h 4
240.bf even 4 1 768.2.c.h 4
360.bo even 12 2 648.2.l.b 4
360.br even 12 2 2592.2.p.b 4
360.bt odd 12 2 648.2.l.b 4
360.bu odd 12 2 2592.2.p.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.2.f.a 2 5.c odd 4 1
24.2.f.a 2 15.e even 4 1
24.2.f.a 2 40.k even 4 1
24.2.f.a 2 120.q odd 4 1
96.2.f.a 2 20.e even 4 1
96.2.f.a 2 40.i odd 4 1
96.2.f.a 2 60.l odd 4 1
96.2.f.a 2 120.w even 4 1
600.2.b.a 2 5.c odd 4 1
600.2.b.a 2 15.e even 4 1
600.2.b.a 2 40.k even 4 1
600.2.b.a 2 120.q odd 4 1
600.2.m.a 4 1.a even 1 1 trivial
600.2.m.a 4 3.b odd 2 1 inner
600.2.m.a 4 5.b even 2 1 inner
600.2.m.a 4 8.d odd 2 1 CM
600.2.m.a 4 15.d odd 2 1 inner
600.2.m.a 4 24.f even 2 1 inner
600.2.m.a 4 40.e odd 2 1 inner
600.2.m.a 4 120.m even 2 1 inner
648.2.l.b 4 45.k odd 12 2
648.2.l.b 4 45.l even 12 2
648.2.l.b 4 360.bo even 12 2
648.2.l.b 4 360.bt odd 12 2
768.2.c.h 4 80.i odd 4 1
768.2.c.h 4 80.j even 4 1
768.2.c.h 4 80.s even 4 1
768.2.c.h 4 80.t odd 4 1
768.2.c.h 4 240.z odd 4 1
768.2.c.h 4 240.bb even 4 1
768.2.c.h 4 240.bd odd 4 1
768.2.c.h 4 240.bf even 4 1
2400.2.b.a 2 20.e even 4 1
2400.2.b.a 2 40.i odd 4 1
2400.2.b.a 2 60.l odd 4 1
2400.2.b.a 2 120.w even 4 1
2400.2.m.a 4 4.b odd 2 1
2400.2.m.a 4 8.b even 2 1
2400.2.m.a 4 12.b even 2 1
2400.2.m.a 4 20.d odd 2 1
2400.2.m.a 4 24.h odd 2 1
2400.2.m.a 4 40.f even 2 1
2400.2.m.a 4 60.h even 2 1
2400.2.m.a 4 120.i odd 2 1
2592.2.p.b 4 180.v odd 12 2
2592.2.p.b 4 180.x even 12 2
2592.2.p.b 4 360.br even 12 2
2592.2.p.b 4 360.bu odd 12 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 $$T_{29}$$ T29

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} - 2)^{2}$$
$3$ $$T^{4} - 2T^{2} + 9$$
$5$ $$T^{4}$$
$7$ $$T^{4}$$
$11$ $$(T^{2} + 8)^{2}$$
$13$ $$T^{4}$$
$17$ $$(T^{2} - 32)^{2}$$
$19$ $$(T + 2)^{4}$$
$23$ $$T^{4}$$
$29$ $$T^{4}$$
$31$ $$T^{4}$$
$37$ $$T^{4}$$
$41$ $$(T^{2} + 128)^{2}$$
$43$ $$(T^{2} + 100)^{2}$$
$47$ $$T^{4}$$
$53$ $$T^{4}$$
$59$ $$(T^{2} + 200)^{2}$$
$61$ $$T^{4}$$
$67$ $$(T^{2} + 196)^{2}$$
$71$ $$T^{4}$$
$73$ $$(T^{2} + 4)^{2}$$
$79$ $$T^{4}$$
$83$ $$(T^{2} - 8)^{2}$$
$89$ $$(T^{2} + 32)^{2}$$
$97$ $$(T^{2} + 100)^{2}$$