# Properties

 Label 1200.4.a.b Level $1200$ Weight $4$ Character orbit 1200.a Self dual yes Analytic conductor $70.802$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1200,4,Mod(1,1200)]

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, 0]))

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

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

chi := DirichletCharacter("1200.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1200 = 2^{4} \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1200.a (trivial)

## 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: yes Analytic conductor: $$70.8022920069$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 6) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q - 3 q^{3} - 16 q^{7} + 9 q^{9}+O(q^{10})$$ q - 3 * q^3 - 16 * q^7 + 9 * q^9 $$q - 3 q^{3} - 16 q^{7} + 9 q^{9} - 12 q^{11} - 38 q^{13} + 126 q^{17} - 20 q^{19} + 48 q^{21} + 168 q^{23} - 27 q^{27} + 30 q^{29} + 88 q^{31} + 36 q^{33} - 254 q^{37} + 114 q^{39} + 42 q^{41} - 52 q^{43} - 96 q^{47} - 87 q^{49} - 378 q^{51} - 198 q^{53} + 60 q^{57} + 660 q^{59} - 538 q^{61} - 144 q^{63} + 884 q^{67} - 504 q^{69} - 792 q^{71} - 218 q^{73} + 192 q^{77} + 520 q^{79} + 81 q^{81} - 492 q^{83} - 90 q^{87} + 810 q^{89} + 608 q^{91} - 264 q^{93} - 1154 q^{97} - 108 q^{99}+O(q^{100})$$ q - 3 * q^3 - 16 * q^7 + 9 * q^9 - 12 * q^11 - 38 * q^13 + 126 * q^17 - 20 * q^19 + 48 * q^21 + 168 * q^23 - 27 * q^27 + 30 * q^29 + 88 * q^31 + 36 * q^33 - 254 * q^37 + 114 * q^39 + 42 * q^41 - 52 * q^43 - 96 * q^47 - 87 * q^49 - 378 * q^51 - 198 * q^53 + 60 * q^57 + 660 * q^59 - 538 * q^61 - 144 * q^63 + 884 * q^67 - 504 * q^69 - 792 * q^71 - 218 * q^73 + 192 * q^77 + 520 * q^79 + 81 * q^81 - 492 * q^83 - 90 * q^87 + 810 * q^89 + 608 * q^91 - 264 * q^93 - 1154 * q^97 - 108 * q^99

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
0 −3.00000 0 0 0 −16.0000 0 9.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$1$$
$$5$$ $$1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1200.4.a.b 1
4.b odd 2 1 150.4.a.i 1
5.b even 2 1 48.4.a.c 1
5.c odd 4 2 1200.4.f.j 2
12.b even 2 1 450.4.a.h 1
15.d odd 2 1 144.4.a.c 1
20.d odd 2 1 6.4.a.a 1
20.e even 4 2 150.4.c.d 2
35.c odd 2 1 2352.4.a.e 1
40.e odd 2 1 192.4.a.i 1
40.f even 2 1 192.4.a.c 1
60.h even 2 1 18.4.a.a 1
60.l odd 4 2 450.4.c.e 2
80.k odd 4 2 768.4.d.n 2
80.q even 4 2 768.4.d.c 2
120.i odd 2 1 576.4.a.r 1
120.m even 2 1 576.4.a.q 1
140.c even 2 1 294.4.a.e 1
140.p odd 6 2 294.4.e.h 2
140.s even 6 2 294.4.e.g 2
180.n even 6 2 162.4.c.c 2
180.p odd 6 2 162.4.c.f 2
220.g even 2 1 726.4.a.f 1
260.g odd 2 1 1014.4.a.g 1
260.u even 4 2 1014.4.b.d 2
340.d odd 2 1 1734.4.a.d 1
380.d even 2 1 2166.4.a.i 1
420.o odd 2 1 882.4.a.n 1
420.ba even 6 2 882.4.g.i 2
420.be odd 6 2 882.4.g.f 2
660.g odd 2 1 2178.4.a.e 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6.4.a.a 1 20.d odd 2 1
18.4.a.a 1 60.h even 2 1
48.4.a.c 1 5.b even 2 1
144.4.a.c 1 15.d odd 2 1
150.4.a.i 1 4.b odd 2 1
150.4.c.d 2 20.e even 4 2
162.4.c.c 2 180.n even 6 2
162.4.c.f 2 180.p odd 6 2
192.4.a.c 1 40.f even 2 1
192.4.a.i 1 40.e odd 2 1
294.4.a.e 1 140.c even 2 1
294.4.e.g 2 140.s even 6 2
294.4.e.h 2 140.p odd 6 2
450.4.a.h 1 12.b even 2 1
450.4.c.e 2 60.l odd 4 2
576.4.a.q 1 120.m even 2 1
576.4.a.r 1 120.i odd 2 1
726.4.a.f 1 220.g even 2 1
768.4.d.c 2 80.q even 4 2
768.4.d.n 2 80.k odd 4 2
882.4.a.n 1 420.o odd 2 1
882.4.g.f 2 420.be odd 6 2
882.4.g.i 2 420.ba even 6 2
1014.4.a.g 1 260.g odd 2 1
1014.4.b.d 2 260.u even 4 2
1200.4.a.b 1 1.a even 1 1 trivial
1200.4.f.j 2 5.c odd 4 2
1734.4.a.d 1 340.d odd 2 1
2166.4.a.i 1 380.d even 2 1
2178.4.a.e 1 660.g odd 2 1
2352.4.a.e 1 35.c odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(1200))$$:

 $$T_{7} + 16$$ T7 + 16 $$T_{11} + 12$$ T11 + 12

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T + 3$$
$5$ $$T$$
$7$ $$T + 16$$
$11$ $$T + 12$$
$13$ $$T + 38$$
$17$ $$T - 126$$
$19$ $$T + 20$$
$23$ $$T - 168$$
$29$ $$T - 30$$
$31$ $$T - 88$$
$37$ $$T + 254$$
$41$ $$T - 42$$
$43$ $$T + 52$$
$47$ $$T + 96$$
$53$ $$T + 198$$
$59$ $$T - 660$$
$61$ $$T + 538$$
$67$ $$T - 884$$
$71$ $$T + 792$$
$73$ $$T + 218$$
$79$ $$T - 520$$
$83$ $$T + 492$$
$89$ $$T - 810$$
$97$ $$T + 1154$$