# Properties

 Label 1200.3.l.k 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: $$\Q(\sqrt{-35})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x + 9$$ x^2 - x + 9 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 300) 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 $$\beta = \frac{1}{2}(1 + \sqrt{-35})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta q^{3} - 8 q^{7} + (\beta - 9) q^{9} +O(q^{10})$$ q - b * q^3 - 8 * q^7 + (b - 9) * q^9 $$q - \beta q^{3} - 8 q^{7} + (\beta - 9) q^{9} + ( - 6 \beta + 3) q^{11} + 2 q^{13} + ( - 6 \beta + 3) q^{17} - 11 q^{19} + 8 \beta q^{21} + ( - 12 \beta + 6) q^{23} + (8 \beta + 9) q^{27} + (12 \beta - 6) q^{29} + 46 q^{31} + (3 \beta - 54) q^{33} - 16 q^{37} - 2 \beta q^{39} + (18 \beta - 9) q^{41} - 62 q^{43} + (12 \beta - 6) q^{47} + 15 q^{49} + (3 \beta - 54) q^{51} + ( - 12 \beta + 6) q^{53} + 11 \beta q^{57} + (24 \beta - 12) q^{59} - 16 q^{61} + ( - 8 \beta + 72) q^{63} - 113 q^{67} + (6 \beta - 108) q^{69} + (36 \beta - 18) q^{71} + 101 q^{73} + (48 \beta - 24) q^{77} - 68 q^{79} + ( - 17 \beta + 72) q^{81} + (6 \beta - 3) q^{83} + ( - 6 \beta + 108) q^{87} + (18 \beta - 9) q^{89} - 16 q^{91} - 46 \beta q^{93} - 22 q^{97} + (51 \beta + 27) q^{99} +O(q^{100})$$ q - b * q^3 - 8 * q^7 + (b - 9) * q^9 + (-6*b + 3) * q^11 + 2 * q^13 + (-6*b + 3) * q^17 - 11 * q^19 + 8*b * q^21 + (-12*b + 6) * q^23 + (8*b + 9) * q^27 + (12*b - 6) * q^29 + 46 * q^31 + (3*b - 54) * q^33 - 16 * q^37 - 2*b * q^39 + (18*b - 9) * q^41 - 62 * q^43 + (12*b - 6) * q^47 + 15 * q^49 + (3*b - 54) * q^51 + (-12*b + 6) * q^53 + 11*b * q^57 + (24*b - 12) * q^59 - 16 * q^61 + (-8*b + 72) * q^63 - 113 * q^67 + (6*b - 108) * q^69 + (36*b - 18) * q^71 + 101 * q^73 + (48*b - 24) * q^77 - 68 * q^79 + (-17*b + 72) * q^81 + (6*b - 3) * q^83 + (-6*b + 108) * q^87 + (18*b - 9) * q^89 - 16 * q^91 - 46*b * q^93 - 22 * q^97 + (51*b + 27) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{3} - 16 q^{7} - 17 q^{9}+O(q^{10})$$ 2 * q - q^3 - 16 * q^7 - 17 * q^9 $$2 q - q^{3} - 16 q^{7} - 17 q^{9} + 4 q^{13} - 22 q^{19} + 8 q^{21} + 26 q^{27} + 92 q^{31} - 105 q^{33} - 32 q^{37} - 2 q^{39} - 124 q^{43} + 30 q^{49} - 105 q^{51} + 11 q^{57} - 32 q^{61} + 136 q^{63} - 226 q^{67} - 210 q^{69} + 202 q^{73} - 136 q^{79} + 127 q^{81} + 210 q^{87} - 32 q^{91} - 46 q^{93} - 44 q^{97} + 105 q^{99}+O(q^{100})$$ 2 * q - q^3 - 16 * q^7 - 17 * q^9 + 4 * q^13 - 22 * q^19 + 8 * q^21 + 26 * q^27 + 92 * q^31 - 105 * q^33 - 32 * q^37 - 2 * q^39 - 124 * q^43 + 30 * q^49 - 105 * q^51 + 11 * q^57 - 32 * q^61 + 136 * q^63 - 226 * q^67 - 210 * q^69 + 202 * q^73 - 136 * q^79 + 127 * q^81 + 210 * q^87 - 32 * q^91 - 46 * q^93 - 44 * q^97 + 105 * 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
 0.5 + 2.95804i 0.5 − 2.95804i
0 −0.500000 2.95804i 0 0 0 −8.00000 0 −8.50000 + 2.95804i 0
401.2 0 −0.500000 + 2.95804i 0 0 0 −8.00000 0 −8.50000 2.95804i 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.k 2
3.b odd 2 1 inner 1200.3.l.k 2
4.b odd 2 1 300.3.g.g yes 2
5.b even 2 1 1200.3.l.m 2
5.c odd 4 2 1200.3.c.j 4
12.b even 2 1 300.3.g.g yes 2
15.d odd 2 1 1200.3.l.m 2
15.e even 4 2 1200.3.c.j 4
20.d odd 2 1 300.3.g.f 2
20.e even 4 2 300.3.b.d 4
60.h even 2 1 300.3.g.f 2
60.l odd 4 2 300.3.b.d 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
300.3.b.d 4 20.e even 4 2
300.3.b.d 4 60.l odd 4 2
300.3.g.f 2 20.d odd 2 1
300.3.g.f 2 60.h even 2 1
300.3.g.g yes 2 4.b odd 2 1
300.3.g.g yes 2 12.b even 2 1
1200.3.c.j 4 5.c odd 4 2
1200.3.c.j 4 15.e even 4 2
1200.3.l.k 2 1.a even 1 1 trivial
1200.3.l.k 2 3.b odd 2 1 inner
1200.3.l.m 2 5.b even 2 1
1200.3.l.m 2 15.d odd 2 1

## Hecke kernels

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

 $$T_{7} + 8$$ T7 + 8 $$T_{11}^{2} + 315$$ T11^2 + 315 $$T_{13} - 2$$ T13 - 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + T + 9$$
$5$ $$T^{2}$$
$7$ $$(T + 8)^{2}$$
$11$ $$T^{2} + 315$$
$13$ $$(T - 2)^{2}$$
$17$ $$T^{2} + 315$$
$19$ $$(T + 11)^{2}$$
$23$ $$T^{2} + 1260$$
$29$ $$T^{2} + 1260$$
$31$ $$(T - 46)^{2}$$
$37$ $$(T + 16)^{2}$$
$41$ $$T^{2} + 2835$$
$43$ $$(T + 62)^{2}$$
$47$ $$T^{2} + 1260$$
$53$ $$T^{2} + 1260$$
$59$ $$T^{2} + 5040$$
$61$ $$(T + 16)^{2}$$
$67$ $$(T + 113)^{2}$$
$71$ $$T^{2} + 11340$$
$73$ $$(T - 101)^{2}$$
$79$ $$(T + 68)^{2}$$
$83$ $$T^{2} + 315$$
$89$ $$T^{2} + 2835$$
$97$ $$(T + 22)^{2}$$