# Properties

 Label 1200.4.f.e Level $1200$ Weight $4$ Character orbit 1200.f Analytic conductor $70.802$ 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] = [1200,4,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 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 $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 3 i q^{3} + 32 i q^{7} - 9 q^{9}+O(q^{10})$$ q + 3*i * q^3 + 32*i * q^7 - 9 * q^9 $$q + 3 i q^{3} + 32 i q^{7} - 9 q^{9} - 36 q^{11} - 10 i q^{13} + 78 i q^{17} + 140 q^{19} - 96 q^{21} + 192 i q^{23} - 27 i q^{27} - 6 q^{29} + 16 q^{31} - 108 i q^{33} + 34 i q^{37} + 30 q^{39} - 390 q^{41} + 52 i q^{43} + 408 i q^{47} - 681 q^{49} - 234 q^{51} - 114 i q^{53} + 420 i q^{57} + 516 q^{59} - 58 q^{61} - 288 i q^{63} - 892 i q^{67} - 576 q^{69} + 120 q^{71} - 646 i q^{73} - 1152 i q^{77} - 1168 q^{79} + 81 q^{81} + 732 i q^{83} - 18 i q^{87} + 1590 q^{89} + 320 q^{91} + 48 i q^{93} - 194 i q^{97} + 324 q^{99} +O(q^{100})$$ q + 3*i * q^3 + 32*i * q^7 - 9 * q^9 - 36 * q^11 - 10*i * q^13 + 78*i * q^17 + 140 * q^19 - 96 * q^21 + 192*i * q^23 - 27*i * q^27 - 6 * q^29 + 16 * q^31 - 108*i * q^33 + 34*i * q^37 + 30 * q^39 - 390 * q^41 + 52*i * q^43 + 408*i * q^47 - 681 * q^49 - 234 * q^51 - 114*i * q^53 + 420*i * q^57 + 516 * q^59 - 58 * q^61 - 288*i * q^63 - 892*i * q^67 - 576 * q^69 + 120 * q^71 - 646*i * q^73 - 1152*i * q^77 - 1168 * q^79 + 81 * q^81 + 732*i * q^83 - 18*i * q^87 + 1590 * q^89 + 320 * q^91 + 48*i * q^93 - 194*i * q^97 + 324 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 18 q^{9}+O(q^{10})$$ 2 * q - 18 * q^9 $$2 q - 18 q^{9} - 72 q^{11} + 280 q^{19} - 192 q^{21} - 12 q^{29} + 32 q^{31} + 60 q^{39} - 780 q^{41} - 1362 q^{49} - 468 q^{51} + 1032 q^{59} - 116 q^{61} - 1152 q^{69} + 240 q^{71} - 2336 q^{79} + 162 q^{81} + 3180 q^{89} + 640 q^{91} + 648 q^{99}+O(q^{100})$$ 2 * q - 18 * q^9 - 72 * q^11 + 280 * q^19 - 192 * q^21 - 12 * q^29 + 32 * q^31 + 60 * q^39 - 780 * q^41 - 1362 * q^49 - 468 * q^51 + 1032 * q^59 - 116 * q^61 - 1152 * q^69 + 240 * q^71 - 2336 * q^79 + 162 * q^81 + 3180 * q^89 + 640 * q^91 + 648 * 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 32.0000i 0 −9.00000 0
49.2 0 3.00000i 0 0 0 32.0000i 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.e 2
4.b odd 2 1 300.4.d.d 2
5.b even 2 1 inner 1200.4.f.e 2
5.c odd 4 1 240.4.a.j 1
5.c odd 4 1 1200.4.a.s 1
12.b even 2 1 900.4.d.b 2
15.e even 4 1 720.4.a.c 1
20.d odd 2 1 300.4.d.d 2
20.e even 4 1 60.4.a.b 1
20.e even 4 1 300.4.a.e 1
40.i odd 4 1 960.4.a.a 1
40.k even 4 1 960.4.a.bb 1
60.h even 2 1 900.4.d.b 2
60.l odd 4 1 180.4.a.c 1
60.l odd 4 1 900.4.a.b 1
180.v odd 12 2 1620.4.i.g 2
180.x even 12 2 1620.4.i.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.4.a.b 1 20.e even 4 1
180.4.a.c 1 60.l odd 4 1
240.4.a.j 1 5.c odd 4 1
300.4.a.e 1 20.e even 4 1
300.4.d.d 2 4.b odd 2 1
300.4.d.d 2 20.d odd 2 1
720.4.a.c 1 15.e even 4 1
900.4.a.b 1 60.l odd 4 1
900.4.d.b 2 12.b even 2 1
900.4.d.b 2 60.h even 2 1
960.4.a.a 1 40.i odd 4 1
960.4.a.bb 1 40.k even 4 1
1200.4.a.s 1 5.c odd 4 1
1200.4.f.e 2 1.a even 1 1 trivial
1200.4.f.e 2 5.b even 2 1 inner
1620.4.i.a 2 180.x even 12 2
1620.4.i.g 2 180.v odd 12 2

## Hecke kernels

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

 $$T_{7}^{2} + 1024$$ T7^2 + 1024 $$T_{11} + 36$$ T11 + 36

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 9$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 1024$$
$11$ $$(T + 36)^{2}$$
$13$ $$T^{2} + 100$$
$17$ $$T^{2} + 6084$$
$19$ $$(T - 140)^{2}$$
$23$ $$T^{2} + 36864$$
$29$ $$(T + 6)^{2}$$
$31$ $$(T - 16)^{2}$$
$37$ $$T^{2} + 1156$$
$41$ $$(T + 390)^{2}$$
$43$ $$T^{2} + 2704$$
$47$ $$T^{2} + 166464$$
$53$ $$T^{2} + 12996$$
$59$ $$(T - 516)^{2}$$
$61$ $$(T + 58)^{2}$$
$67$ $$T^{2} + 795664$$
$71$ $$(T - 120)^{2}$$
$73$ $$T^{2} + 417316$$
$79$ $$(T + 1168)^{2}$$
$83$ $$T^{2} + 535824$$
$89$ $$(T - 1590)^{2}$$
$97$ $$T^{2} + 37636$$