# Properties

 Label 1200.4.f.a Level $1200$ Weight $4$ Character orbit 1200.f Analytic conductor $70.802$ Analytic rank $0$ Dimension $2$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1200,4,Mod(49,1200)]

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, 0, 1]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1200.49");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1200 = 2^{4} \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1200.f (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: $$70.8022920069$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 120) 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 $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 3 i q^{3} + 4 i q^{7} - 9 q^{9}+O(q^{10})$$ q + 3*i * q^3 + 4*i * q^7 - 9 * q^9 $$q + 3 i q^{3} + 4 i q^{7} - 9 q^{9} - 72 q^{11} - 6 i q^{13} - 38 i q^{17} + 52 q^{19} - 12 q^{21} - 152 i q^{23} - 27 i q^{27} + 78 q^{29} - 120 q^{31} - 216 i q^{33} + 150 i q^{37} + 18 q^{39} + 362 q^{41} + 484 i q^{43} + 280 i q^{47} + 327 q^{49} + 114 q^{51} - 670 i q^{53} + 156 i q^{57} + 696 q^{59} + 222 q^{61} - 36 i q^{63} - 4 i q^{67} + 456 q^{69} - 96 q^{71} + 178 i q^{73} - 288 i q^{77} - 632 q^{79} + 81 q^{81} + 612 i q^{83} + 234 i q^{87} - 994 q^{89} + 24 q^{91} - 360 i q^{93} - 1634 i q^{97} + 648 q^{99} +O(q^{100})$$ q + 3*i * q^3 + 4*i * q^7 - 9 * q^9 - 72 * q^11 - 6*i * q^13 - 38*i * q^17 + 52 * q^19 - 12 * q^21 - 152*i * q^23 - 27*i * q^27 + 78 * q^29 - 120 * q^31 - 216*i * q^33 + 150*i * q^37 + 18 * q^39 + 362 * q^41 + 484*i * q^43 + 280*i * q^47 + 327 * q^49 + 114 * q^51 - 670*i * q^53 + 156*i * q^57 + 696 * q^59 + 222 * q^61 - 36*i * q^63 - 4*i * q^67 + 456 * q^69 - 96 * q^71 + 178*i * q^73 - 288*i * q^77 - 632 * q^79 + 81 * q^81 + 612*i * q^83 + 234*i * q^87 - 994 * q^89 + 24 * q^91 - 360*i * q^93 - 1634*i * q^97 + 648 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 18 q^{9}+O(q^{10})$$ 2 * q - 18 * q^9 $$2 q - 18 q^{9} - 144 q^{11} + 104 q^{19} - 24 q^{21} + 156 q^{29} - 240 q^{31} + 36 q^{39} + 724 q^{41} + 654 q^{49} + 228 q^{51} + 1392 q^{59} + 444 q^{61} + 912 q^{69} - 192 q^{71} - 1264 q^{79} + 162 q^{81} - 1988 q^{89} + 48 q^{91} + 1296 q^{99}+O(q^{100})$$ 2 * q - 18 * q^9 - 144 * q^11 + 104 * q^19 - 24 * q^21 + 156 * q^29 - 240 * q^31 + 36 * q^39 + 724 * q^41 + 654 * q^49 + 228 * q^51 + 1392 * q^59 + 444 * q^61 + 912 * q^69 - 192 * q^71 - 1264 * q^79 + 162 * q^81 - 1988 * q^89 + 48 * q^91 + 1296 * 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
49.1
 − 1.00000i 1.00000i
0 3.00000i 0 0 0 4.00000i 0 −9.00000 0
49.2 0 3.00000i 0 0 0 4.00000i 0 −9.00000 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
5.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1200.4.f.a 2
4.b odd 2 1 600.4.f.i 2
5.b even 2 1 inner 1200.4.f.a 2
5.c odd 4 1 240.4.a.h 1
5.c odd 4 1 1200.4.a.k 1
12.b even 2 1 1800.4.f.a 2
15.e even 4 1 720.4.a.v 1
20.d odd 2 1 600.4.f.i 2
20.e even 4 1 120.4.a.a 1
20.e even 4 1 600.4.a.l 1
40.i odd 4 1 960.4.a.o 1
40.k even 4 1 960.4.a.bf 1
60.h even 2 1 1800.4.f.a 2
60.l odd 4 1 360.4.a.l 1
60.l odd 4 1 1800.4.a.n 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
120.4.a.a 1 20.e even 4 1
240.4.a.h 1 5.c odd 4 1
360.4.a.l 1 60.l odd 4 1
600.4.a.l 1 20.e even 4 1
600.4.f.i 2 4.b odd 2 1
600.4.f.i 2 20.d odd 2 1
720.4.a.v 1 15.e even 4 1
960.4.a.o 1 40.i odd 4 1
960.4.a.bf 1 40.k even 4 1
1200.4.a.k 1 5.c odd 4 1
1200.4.f.a 2 1.a even 1 1 trivial
1200.4.f.a 2 5.b even 2 1 inner
1800.4.a.n 1 60.l odd 4 1
1800.4.f.a 2 12.b even 2 1
1800.4.f.a 2 60.h even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(1200, [\chi])$$:

 $$T_{7}^{2} + 16$$ T7^2 + 16 $$T_{11} + 72$$ T11 + 72

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 9$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 16$$
$11$ $$(T + 72)^{2}$$
$13$ $$T^{2} + 36$$
$17$ $$T^{2} + 1444$$
$19$ $$(T - 52)^{2}$$
$23$ $$T^{2} + 23104$$
$29$ $$(T - 78)^{2}$$
$31$ $$(T + 120)^{2}$$
$37$ $$T^{2} + 22500$$
$41$ $$(T - 362)^{2}$$
$43$ $$T^{2} + 234256$$
$47$ $$T^{2} + 78400$$
$53$ $$T^{2} + 448900$$
$59$ $$(T - 696)^{2}$$
$61$ $$(T - 222)^{2}$$
$67$ $$T^{2} + 16$$
$71$ $$(T + 96)^{2}$$
$73$ $$T^{2} + 31684$$
$79$ $$(T + 632)^{2}$$
$83$ $$T^{2} + 374544$$
$89$ $$(T + 994)^{2}$$
$97$ $$T^{2} + 2669956$$