# Properties

 Label 1200.4.f.b 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: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 15) 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.

 $$f(q)$$ $$=$$ $$q + 3 i q^{3} + 24 i q^{7} - 9 q^{9}+O(q^{10})$$ q + 3*i * q^3 + 24*i * q^7 - 9 * q^9 $$q + 3 i q^{3} + 24 i q^{7} - 9 q^{9} - 52 q^{11} - 22 i q^{13} - 14 i q^{17} - 20 q^{19} - 72 q^{21} - 168 i q^{23} - 27 i q^{27} - 230 q^{29} + 288 q^{31} - 156 i q^{33} - 34 i q^{37} + 66 q^{39} + 122 q^{41} - 188 i q^{43} - 256 i q^{47} - 233 q^{49} + 42 q^{51} + 338 i q^{53} - 60 i q^{57} + 100 q^{59} + 742 q^{61} - 216 i q^{63} + 84 i q^{67} + 504 q^{69} + 328 q^{71} + 38 i q^{73} - 1248 i q^{77} - 240 q^{79} + 81 q^{81} + 1212 i q^{83} - 690 i q^{87} - 330 q^{89} + 528 q^{91} + 864 i q^{93} + 866 i q^{97} + 468 q^{99} +O(q^{100})$$ q + 3*i * q^3 + 24*i * q^7 - 9 * q^9 - 52 * q^11 - 22*i * q^13 - 14*i * q^17 - 20 * q^19 - 72 * q^21 - 168*i * q^23 - 27*i * q^27 - 230 * q^29 + 288 * q^31 - 156*i * q^33 - 34*i * q^37 + 66 * q^39 + 122 * q^41 - 188*i * q^43 - 256*i * q^47 - 233 * q^49 + 42 * q^51 + 338*i * q^53 - 60*i * q^57 + 100 * q^59 + 742 * q^61 - 216*i * q^63 + 84*i * q^67 + 504 * q^69 + 328 * q^71 + 38*i * q^73 - 1248*i * q^77 - 240 * q^79 + 81 * q^81 + 1212*i * q^83 - 690*i * q^87 - 330 * q^89 + 528 * q^91 + 864*i * q^93 + 866*i * q^97 + 468 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 18 q^{9}+O(q^{10})$$ 2 * q - 18 * q^9 $$2 q - 18 q^{9} - 104 q^{11} - 40 q^{19} - 144 q^{21} - 460 q^{29} + 576 q^{31} + 132 q^{39} + 244 q^{41} - 466 q^{49} + 84 q^{51} + 200 q^{59} + 1484 q^{61} + 1008 q^{69} + 656 q^{71} - 480 q^{79} + 162 q^{81} - 660 q^{89} + 1056 q^{91} + 936 q^{99}+O(q^{100})$$ 2 * q - 18 * q^9 - 104 * q^11 - 40 * q^19 - 144 * q^21 - 460 * q^29 + 576 * q^31 + 132 * q^39 + 244 * q^41 - 466 * q^49 + 84 * q^51 + 200 * q^59 + 1484 * q^61 + 1008 * q^69 + 656 * q^71 - 480 * q^79 + 162 * q^81 - 660 * q^89 + 1056 * q^91 + 936 * 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 24.0000i 0 −9.00000 0
49.2 0 3.00000i 0 0 0 24.0000i 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.b 2
4.b odd 2 1 75.4.b.b 2
5.b even 2 1 inner 1200.4.f.b 2
5.c odd 4 1 240.4.a.e 1
5.c odd 4 1 1200.4.a.t 1
12.b even 2 1 225.4.b.e 2
15.e even 4 1 720.4.a.n 1
20.d odd 2 1 75.4.b.b 2
20.e even 4 1 15.4.a.a 1
20.e even 4 1 75.4.a.b 1
40.i odd 4 1 960.4.a.ba 1
40.k even 4 1 960.4.a.b 1
60.h even 2 1 225.4.b.e 2
60.l odd 4 1 45.4.a.c 1
60.l odd 4 1 225.4.a.f 1
140.j odd 4 1 735.4.a.e 1
180.v odd 12 2 405.4.e.i 2
180.x even 12 2 405.4.e.g 2
220.i odd 4 1 1815.4.a.e 1
420.w even 4 1 2205.4.a.l 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.4.a.a 1 20.e even 4 1
45.4.a.c 1 60.l odd 4 1
75.4.a.b 1 20.e even 4 1
75.4.b.b 2 4.b odd 2 1
75.4.b.b 2 20.d odd 2 1
225.4.a.f 1 60.l odd 4 1
225.4.b.e 2 12.b even 2 1
225.4.b.e 2 60.h even 2 1
240.4.a.e 1 5.c odd 4 1
405.4.e.g 2 180.x even 12 2
405.4.e.i 2 180.v odd 12 2
720.4.a.n 1 15.e even 4 1
735.4.a.e 1 140.j odd 4 1
960.4.a.b 1 40.k even 4 1
960.4.a.ba 1 40.i odd 4 1
1200.4.a.t 1 5.c odd 4 1
1200.4.f.b 2 1.a even 1 1 trivial
1200.4.f.b 2 5.b even 2 1 inner
1815.4.a.e 1 220.i odd 4 1
2205.4.a.l 1 420.w even 4 1

## Hecke kernels

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

 $$T_{7}^{2} + 576$$ T7^2 + 576 $$T_{11} + 52$$ T11 + 52

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 9$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 576$$
$11$ $$(T + 52)^{2}$$
$13$ $$T^{2} + 484$$
$17$ $$T^{2} + 196$$
$19$ $$(T + 20)^{2}$$
$23$ $$T^{2} + 28224$$
$29$ $$(T + 230)^{2}$$
$31$ $$(T - 288)^{2}$$
$37$ $$T^{2} + 1156$$
$41$ $$(T - 122)^{2}$$
$43$ $$T^{2} + 35344$$
$47$ $$T^{2} + 65536$$
$53$ $$T^{2} + 114244$$
$59$ $$(T - 100)^{2}$$
$61$ $$(T - 742)^{2}$$
$67$ $$T^{2} + 7056$$
$71$ $$(T - 328)^{2}$$
$73$ $$T^{2} + 1444$$
$79$ $$(T + 240)^{2}$$
$83$ $$T^{2} + 1468944$$
$89$ $$(T + 330)^{2}$$
$97$ $$T^{2} + 749956$$