# Properties

 Label 1200.3.l.r Level $1200$ Weight $3$ Character orbit 1200.l Analytic conductor $32.698$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1200,3,Mod(401,1200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1200, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("1200.401");

S:= CuspForms(chi, 3);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1200 = 2^{4} \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 1200.l (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: $$32.6976317232$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-5})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 5$$ x^2 + 5 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 60) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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 $$\beta = \sqrt{-5}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta + 2) q^{3} + 2 q^{7} + (4 \beta - 1) q^{9}+O(q^{10})$$ q + (b + 2) * q^3 + 2 * q^7 + (4*b - 1) * q^9 $$q + (\beta + 2) q^{3} + 2 q^{7} + (4 \beta - 1) q^{9} + 6 \beta q^{11} - 8 q^{13} + 6 \beta q^{17} + 34 q^{19} + (2 \beta + 4) q^{21} - 18 \beta q^{23} + (7 \beta - 22) q^{27} + 18 \beta q^{29} - 14 q^{31} + (12 \beta - 30) q^{33} - 56 q^{37} + ( - 8 \beta - 16) q^{39} + 12 \beta q^{41} + 8 q^{43} + 18 \beta q^{47} - 45 q^{49} + (12 \beta - 30) q^{51} - 18 \beta q^{53} + (34 \beta + 68) q^{57} + 6 \beta q^{59} - 46 q^{61} + (8 \beta - 2) q^{63} + 32 q^{67} + ( - 36 \beta + 90) q^{69} + 24 \beta q^{71} + 106 q^{73} + 12 \beta q^{77} + 22 q^{79} + ( - 8 \beta - 79) q^{81} + 54 \beta q^{83} + (36 \beta - 90) q^{87} - 48 \beta q^{89} - 16 q^{91} + ( - 14 \beta - 28) q^{93} - 122 q^{97} + ( - 6 \beta - 120) q^{99} +O(q^{100})$$ q + (b + 2) * q^3 + 2 * q^7 + (4*b - 1) * q^9 + 6*b * q^11 - 8 * q^13 + 6*b * q^17 + 34 * q^19 + (2*b + 4) * q^21 - 18*b * q^23 + (7*b - 22) * q^27 + 18*b * q^29 - 14 * q^31 + (12*b - 30) * q^33 - 56 * q^37 + (-8*b - 16) * q^39 + 12*b * q^41 + 8 * q^43 + 18*b * q^47 - 45 * q^49 + (12*b - 30) * q^51 - 18*b * q^53 + (34*b + 68) * q^57 + 6*b * q^59 - 46 * q^61 + (8*b - 2) * q^63 + 32 * q^67 + (-36*b + 90) * q^69 + 24*b * q^71 + 106 * q^73 + 12*b * q^77 + 22 * q^79 + (-8*b - 79) * q^81 + 54*b * q^83 + (36*b - 90) * q^87 - 48*b * q^89 - 16 * q^91 + (-14*b - 28) * q^93 - 122 * q^97 + (-6*b - 120) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 4 q^{3} + 4 q^{7} - 2 q^{9}+O(q^{10})$$ 2 * q + 4 * q^3 + 4 * q^7 - 2 * q^9 $$2 q + 4 q^{3} + 4 q^{7} - 2 q^{9} - 16 q^{13} + 68 q^{19} + 8 q^{21} - 44 q^{27} - 28 q^{31} - 60 q^{33} - 112 q^{37} - 32 q^{39} + 16 q^{43} - 90 q^{49} - 60 q^{51} + 136 q^{57} - 92 q^{61} - 4 q^{63} + 64 q^{67} + 180 q^{69} + 212 q^{73} + 44 q^{79} - 158 q^{81} - 180 q^{87} - 32 q^{91} - 56 q^{93} - 244 q^{97} - 240 q^{99}+O(q^{100})$$ 2 * q + 4 * q^3 + 4 * q^7 - 2 * q^9 - 16 * q^13 + 68 * q^19 + 8 * q^21 - 44 * q^27 - 28 * q^31 - 60 * q^33 - 112 * q^37 - 32 * q^39 + 16 * q^43 - 90 * q^49 - 60 * q^51 + 136 * q^57 - 92 * q^61 - 4 * q^63 + 64 * q^67 + 180 * q^69 + 212 * q^73 + 44 * q^79 - 158 * q^81 - 180 * q^87 - 32 * q^91 - 56 * q^93 - 244 * q^97 - 240 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1200\mathbb{Z}\right)^\times$$.

 $$n$$ $$401$$ $$577$$ $$751$$ $$901$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
401.1
 − 2.23607i 2.23607i
0 2.00000 2.23607i 0 0 0 2.00000 0 −1.00000 8.94427i 0
401.2 0 2.00000 + 2.23607i 0 0 0 2.00000 0 −1.00000 + 8.94427i 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
3.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1200.3.l.r 2
3.b odd 2 1 inner 1200.3.l.r 2
4.b odd 2 1 300.3.g.d 2
5.b even 2 1 240.3.l.a 2
5.c odd 4 2 1200.3.c.e 4
12.b even 2 1 300.3.g.d 2
15.d odd 2 1 240.3.l.a 2
15.e even 4 2 1200.3.c.e 4
20.d odd 2 1 60.3.g.a 2
20.e even 4 2 300.3.b.c 4
40.e odd 2 1 960.3.l.a 2
40.f even 2 1 960.3.l.d 2
60.h even 2 1 60.3.g.a 2
60.l odd 4 2 300.3.b.c 4
120.i odd 2 1 960.3.l.d 2
120.m even 2 1 960.3.l.a 2
180.n even 6 2 1620.3.o.b 4
180.p odd 6 2 1620.3.o.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.3.g.a 2 20.d odd 2 1
60.3.g.a 2 60.h even 2 1
240.3.l.a 2 5.b even 2 1
240.3.l.a 2 15.d odd 2 1
300.3.b.c 4 20.e even 4 2
300.3.b.c 4 60.l odd 4 2
300.3.g.d 2 4.b odd 2 1
300.3.g.d 2 12.b even 2 1
960.3.l.a 2 40.e odd 2 1
960.3.l.a 2 120.m even 2 1
960.3.l.d 2 40.f even 2 1
960.3.l.d 2 120.i odd 2 1
1200.3.c.e 4 5.c odd 4 2
1200.3.c.e 4 15.e even 4 2
1200.3.l.r 2 1.a even 1 1 trivial
1200.3.l.r 2 3.b odd 2 1 inner
1620.3.o.b 4 180.n even 6 2
1620.3.o.b 4 180.p odd 6 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(1200, [\chi])$$:

 $$T_{7} - 2$$ T7 - 2 $$T_{11}^{2} + 180$$ T11^2 + 180 $$T_{13} + 8$$ T13 + 8

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - 4T + 9$$
$5$ $$T^{2}$$
$7$ $$(T - 2)^{2}$$
$11$ $$T^{2} + 180$$
$13$ $$(T + 8)^{2}$$
$17$ $$T^{2} + 180$$
$19$ $$(T - 34)^{2}$$
$23$ $$T^{2} + 1620$$
$29$ $$T^{2} + 1620$$
$31$ $$(T + 14)^{2}$$
$37$ $$(T + 56)^{2}$$
$41$ $$T^{2} + 720$$
$43$ $$(T - 8)^{2}$$
$47$ $$T^{2} + 1620$$
$53$ $$T^{2} + 1620$$
$59$ $$T^{2} + 180$$
$61$ $$(T + 46)^{2}$$
$67$ $$(T - 32)^{2}$$
$71$ $$T^{2} + 2880$$
$73$ $$(T - 106)^{2}$$
$79$ $$(T - 22)^{2}$$
$83$ $$T^{2} + 14580$$
$89$ $$T^{2} + 11520$$
$97$ $$(T + 122)^{2}$$