# Properties

 Label 1200.4.f.j Level $1200$ Weight $4$ Character orbit 1200.f Analytic conductor $70.802$ Analytic rank $1$ 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: $$1$$ 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 6) 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.

 $$f(q)$$ $$=$$ $$q - 3 i q^{3} + 16 i q^{7} - 9 q^{9} +O(q^{10})$$ q - 3*i * q^3 + 16*i * q^7 - 9 * q^9 $$q - 3 i q^{3} + 16 i q^{7} - 9 q^{9} - 12 q^{11} - 38 i q^{13} - 126 i q^{17} + 20 q^{19} + 48 q^{21} + 168 i q^{23} + 27 i q^{27} - 30 q^{29} + 88 q^{31} + 36 i q^{33} + 254 i q^{37} - 114 q^{39} + 42 q^{41} - 52 i q^{43} + 96 i q^{47} + 87 q^{49} - 378 q^{51} - 198 i q^{53} - 60 i q^{57} - 660 q^{59} - 538 q^{61} - 144 i q^{63} - 884 i q^{67} + 504 q^{69} - 792 q^{71} - 218 i q^{73} - 192 i q^{77} - 520 q^{79} + 81 q^{81} - 492 i q^{83} + 90 i q^{87} - 810 q^{89} + 608 q^{91} - 264 i q^{93} + 1154 i q^{97} + 108 q^{99} +O(q^{100})$$ q - 3*i * q^3 + 16*i * q^7 - 9 * q^9 - 12 * q^11 - 38*i * q^13 - 126*i * q^17 + 20 * q^19 + 48 * q^21 + 168*i * q^23 + 27*i * q^27 - 30 * q^29 + 88 * q^31 + 36*i * q^33 + 254*i * q^37 - 114 * q^39 + 42 * q^41 - 52*i * q^43 + 96*i * q^47 + 87 * q^49 - 378 * q^51 - 198*i * q^53 - 60*i * q^57 - 660 * q^59 - 538 * q^61 - 144*i * q^63 - 884*i * q^67 + 504 * q^69 - 792 * q^71 - 218*i * q^73 - 192*i * q^77 - 520 * q^79 + 81 * q^81 - 492*i * q^83 + 90*i * q^87 - 810 * q^89 + 608 * q^91 - 264*i * q^93 + 1154*i * q^97 + 108 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 18 q^{9}+O(q^{10})$$ 2 * q - 18 * q^9 $$2 q - 18 q^{9} - 24 q^{11} + 40 q^{19} + 96 q^{21} - 60 q^{29} + 176 q^{31} - 228 q^{39} + 84 q^{41} + 174 q^{49} - 756 q^{51} - 1320 q^{59} - 1076 q^{61} + 1008 q^{69} - 1584 q^{71} - 1040 q^{79} + 162 q^{81} - 1620 q^{89} + 1216 q^{91} + 216 q^{99}+O(q^{100})$$ 2 * q - 18 * q^9 - 24 * q^11 + 40 * q^19 + 96 * q^21 - 60 * q^29 + 176 * q^31 - 228 * q^39 + 84 * q^41 + 174 * q^49 - 756 * q^51 - 1320 * q^59 - 1076 * q^61 + 1008 * q^69 - 1584 * q^71 - 1040 * q^79 + 162 * q^81 - 1620 * q^89 + 1216 * q^91 + 216 * 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 16.0000i 0 −9.00000 0
49.2 0 3.00000i 0 0 0 16.0000i 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.j 2
4.b odd 2 1 150.4.c.d 2
5.b even 2 1 inner 1200.4.f.j 2
5.c odd 4 1 48.4.a.c 1
5.c odd 4 1 1200.4.a.b 1
12.b even 2 1 450.4.c.e 2
15.e even 4 1 144.4.a.c 1
20.d odd 2 1 150.4.c.d 2
20.e even 4 1 6.4.a.a 1
20.e even 4 1 150.4.a.i 1
35.f even 4 1 2352.4.a.e 1
40.i odd 4 1 192.4.a.c 1
40.k even 4 1 192.4.a.i 1
60.h even 2 1 450.4.c.e 2
60.l odd 4 1 18.4.a.a 1
60.l odd 4 1 450.4.a.h 1
80.i odd 4 1 768.4.d.c 2
80.j even 4 1 768.4.d.n 2
80.s even 4 1 768.4.d.n 2
80.t odd 4 1 768.4.d.c 2
120.q odd 4 1 576.4.a.q 1
120.w even 4 1 576.4.a.r 1
140.j odd 4 1 294.4.a.e 1
140.w even 12 2 294.4.e.h 2
140.x odd 12 2 294.4.e.g 2
180.v odd 12 2 162.4.c.c 2
180.x even 12 2 162.4.c.f 2
220.i odd 4 1 726.4.a.f 1
260.l odd 4 1 1014.4.b.d 2
260.p even 4 1 1014.4.a.g 1
260.s odd 4 1 1014.4.b.d 2
340.r even 4 1 1734.4.a.d 1
380.j odd 4 1 2166.4.a.i 1
420.w even 4 1 882.4.a.n 1
420.bp odd 12 2 882.4.g.i 2
420.br even 12 2 882.4.g.f 2
660.q even 4 1 2178.4.a.e 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6.4.a.a 1 20.e even 4 1
18.4.a.a 1 60.l odd 4 1
48.4.a.c 1 5.c odd 4 1
144.4.a.c 1 15.e even 4 1
150.4.a.i 1 20.e even 4 1
150.4.c.d 2 4.b odd 2 1
150.4.c.d 2 20.d odd 2 1
162.4.c.c 2 180.v odd 12 2
162.4.c.f 2 180.x even 12 2
192.4.a.c 1 40.i odd 4 1
192.4.a.i 1 40.k even 4 1
294.4.a.e 1 140.j odd 4 1
294.4.e.g 2 140.x odd 12 2
294.4.e.h 2 140.w even 12 2
450.4.a.h 1 60.l odd 4 1
450.4.c.e 2 12.b even 2 1
450.4.c.e 2 60.h even 2 1
576.4.a.q 1 120.q odd 4 1
576.4.a.r 1 120.w even 4 1
726.4.a.f 1 220.i odd 4 1
768.4.d.c 2 80.i odd 4 1
768.4.d.c 2 80.t odd 4 1
768.4.d.n 2 80.j even 4 1
768.4.d.n 2 80.s even 4 1
882.4.a.n 1 420.w even 4 1
882.4.g.f 2 420.br even 12 2
882.4.g.i 2 420.bp odd 12 2
1014.4.a.g 1 260.p even 4 1
1014.4.b.d 2 260.l odd 4 1
1014.4.b.d 2 260.s odd 4 1
1200.4.a.b 1 5.c odd 4 1
1200.4.f.j 2 1.a even 1 1 trivial
1200.4.f.j 2 5.b even 2 1 inner
1734.4.a.d 1 340.r even 4 1
2166.4.a.i 1 380.j odd 4 1
2178.4.a.e 1 660.q even 4 1
2352.4.a.e 1 35.f even 4 1

## Hecke kernels

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

 $$T_{7}^{2} + 256$$ T7^2 + 256 $$T_{11} + 12$$ T11 + 12

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 9$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 256$$
$11$ $$(T + 12)^{2}$$
$13$ $$T^{2} + 1444$$
$17$ $$T^{2} + 15876$$
$19$ $$(T - 20)^{2}$$
$23$ $$T^{2} + 28224$$
$29$ $$(T + 30)^{2}$$
$31$ $$(T - 88)^{2}$$
$37$ $$T^{2} + 64516$$
$41$ $$(T - 42)^{2}$$
$43$ $$T^{2} + 2704$$
$47$ $$T^{2} + 9216$$
$53$ $$T^{2} + 39204$$
$59$ $$(T + 660)^{2}$$
$61$ $$(T + 538)^{2}$$
$67$ $$T^{2} + 781456$$
$71$ $$(T + 792)^{2}$$
$73$ $$T^{2} + 47524$$
$79$ $$(T + 520)^{2}$$
$83$ $$T^{2} + 242064$$
$89$ $$(T + 810)^{2}$$
$97$ $$T^{2} + 1331716$$