# Properties

 Label 1200.4.f.m Level $1200$ Weight $4$ Character orbit 1200.f Analytic conductor $70.802$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

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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 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.

 $$f(q)$$ $$=$$ $$q + 3 i q^{3} + 20 i q^{7} - 9 q^{9}+O(q^{10})$$ q + 3*i * q^3 + 20*i * q^7 - 9 * q^9 $$q + 3 i q^{3} + 20 i q^{7} - 9 q^{9} + 24 q^{11} + 74 i q^{13} - 54 i q^{17} - 124 q^{19} - 60 q^{21} + 120 i q^{23} - 27 i q^{27} + 78 q^{29} - 200 q^{31} + 72 i q^{33} + 70 i q^{37} - 222 q^{39} + 330 q^{41} - 92 i q^{43} - 24 i q^{47} - 57 q^{49} + 162 q^{51} + 450 i q^{53} - 372 i q^{57} + 24 q^{59} - 322 q^{61} - 180 i q^{63} - 196 i q^{67} - 360 q^{69} + 288 q^{71} - 430 i q^{73} + 480 i q^{77} - 520 q^{79} + 81 q^{81} - 156 i q^{83} + 234 i q^{87} - 1026 q^{89} - 1480 q^{91} - 600 i q^{93} + 286 i q^{97} - 216 q^{99} +O(q^{100})$$ q + 3*i * q^3 + 20*i * q^7 - 9 * q^9 + 24 * q^11 + 74*i * q^13 - 54*i * q^17 - 124 * q^19 - 60 * q^21 + 120*i * q^23 - 27*i * q^27 + 78 * q^29 - 200 * q^31 + 72*i * q^33 + 70*i * q^37 - 222 * q^39 + 330 * q^41 - 92*i * q^43 - 24*i * q^47 - 57 * q^49 + 162 * q^51 + 450*i * q^53 - 372*i * q^57 + 24 * q^59 - 322 * q^61 - 180*i * q^63 - 196*i * q^67 - 360 * q^69 + 288 * q^71 - 430*i * q^73 + 480*i * q^77 - 520 * q^79 + 81 * q^81 - 156*i * q^83 + 234*i * q^87 - 1026 * q^89 - 1480 * q^91 - 600*i * q^93 + 286*i * q^97 - 216 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 18 q^{9}+O(q^{10})$$ 2 * q - 18 * q^9 $$2 q - 18 q^{9} + 48 q^{11} - 248 q^{19} - 120 q^{21} + 156 q^{29} - 400 q^{31} - 444 q^{39} + 660 q^{41} - 114 q^{49} + 324 q^{51} + 48 q^{59} - 644 q^{61} - 720 q^{69} + 576 q^{71} - 1040 q^{79} + 162 q^{81} - 2052 q^{89} - 2960 q^{91} - 432 q^{99}+O(q^{100})$$ 2 * q - 18 * q^9 + 48 * q^11 - 248 * q^19 - 120 * q^21 + 156 * q^29 - 400 * q^31 - 444 * q^39 + 660 * q^41 - 114 * q^49 + 324 * q^51 + 48 * q^59 - 644 * q^61 - 720 * q^69 + 576 * q^71 - 1040 * q^79 + 162 * q^81 - 2052 * q^89 - 2960 * q^91 - 432 * 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 20.0000i 0 −9.00000 0
49.2 0 3.00000i 0 0 0 20.0000i 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.m 2
4.b odd 2 1 75.4.b.a 2
5.b even 2 1 inner 1200.4.f.m 2
5.c odd 4 1 240.4.a.f 1
5.c odd 4 1 1200.4.a.o 1
12.b even 2 1 225.4.b.d 2
15.e even 4 1 720.4.a.r 1
20.d odd 2 1 75.4.b.a 2
20.e even 4 1 15.4.a.b 1
20.e even 4 1 75.4.a.a 1
40.i odd 4 1 960.4.a.l 1
40.k even 4 1 960.4.a.bi 1
60.h even 2 1 225.4.b.d 2
60.l odd 4 1 45.4.a.b 1
60.l odd 4 1 225.4.a.g 1
140.j odd 4 1 735.4.a.i 1
180.v odd 12 2 405.4.e.k 2
180.x even 12 2 405.4.e.d 2
220.i odd 4 1 1815.4.a.a 1
420.w even 4 1 2205.4.a.c 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.4.a.b 1 20.e even 4 1
45.4.a.b 1 60.l odd 4 1
75.4.a.a 1 20.e even 4 1
75.4.b.a 2 4.b odd 2 1
75.4.b.a 2 20.d odd 2 1
225.4.a.g 1 60.l odd 4 1
225.4.b.d 2 12.b even 2 1
225.4.b.d 2 60.h even 2 1
240.4.a.f 1 5.c odd 4 1
405.4.e.d 2 180.x even 12 2
405.4.e.k 2 180.v odd 12 2
720.4.a.r 1 15.e even 4 1
735.4.a.i 1 140.j odd 4 1
960.4.a.l 1 40.i odd 4 1
960.4.a.bi 1 40.k even 4 1
1200.4.a.o 1 5.c odd 4 1
1200.4.f.m 2 1.a even 1 1 trivial
1200.4.f.m 2 5.b even 2 1 inner
1815.4.a.a 1 220.i odd 4 1
2205.4.a.c 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 $$S_{4}^{\mathrm{new}}(1200, [\chi])$$:

 $$T_{7}^{2} + 400$$ T7^2 + 400 $$T_{11} - 24$$ T11 - 24

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 9$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 400$$
$11$ $$(T - 24)^{2}$$
$13$ $$T^{2} + 5476$$
$17$ $$T^{2} + 2916$$
$19$ $$(T + 124)^{2}$$
$23$ $$T^{2} + 14400$$
$29$ $$(T - 78)^{2}$$
$31$ $$(T + 200)^{2}$$
$37$ $$T^{2} + 4900$$
$41$ $$(T - 330)^{2}$$
$43$ $$T^{2} + 8464$$
$47$ $$T^{2} + 576$$
$53$ $$T^{2} + 202500$$
$59$ $$(T - 24)^{2}$$
$61$ $$(T + 322)^{2}$$
$67$ $$T^{2} + 38416$$
$71$ $$(T - 288)^{2}$$
$73$ $$T^{2} + 184900$$
$79$ $$(T + 520)^{2}$$
$83$ $$T^{2} + 24336$$
$89$ $$(T + 1026)^{2}$$
$97$ $$T^{2} + 81796$$
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