# Properties

 Label 1200.4.a.bt Level $1200$ Weight $4$ Character orbit 1200.a Self dual yes Analytic conductor $70.802$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Learn more

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: $$2$$ Coefficient field: $$\Q(\sqrt{41})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 10$$ x^2 - x - 10 Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 15) 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)

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{41}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 3 q^{3} + (3 \beta + 3) q^{7} + 9 q^{9}+O(q^{10})$$ q + 3 * q^3 + (3*b + 3) * q^7 + 9 * q^9 $$q + 3 q^{3} + (3 \beta + 3) q^{7} + 9 q^{9} + ( - 3 \beta + 21) q^{11} + ( - 3 \beta - 39) q^{13} + (5 \beta - 51) q^{17} + ( - 12 \beta - 28) q^{19} + (9 \beta + 9) q^{21} + ( - 4 \beta - 24) q^{23} + 27 q^{27} + ( - 21 \beta - 159) q^{29} + ( - 6 \beta - 26) q^{31} + ( - 9 \beta + 63) q^{33} + (27 \beta - 153) q^{37} + ( - 9 \beta - 117) q^{39} + (6 \beta - 204) q^{41} + ( - 48 \beta + 60) q^{43} + (46 \beta + 90) q^{47} + (18 \beta + 35) q^{49} + (15 \beta - 153) q^{51} + ( - 41 \beta - 201) q^{53} + ( - 36 \beta - 84) q^{57} + ( - 3 \beta + 93) q^{59} + ( - 48 \beta + 170) q^{61} + (27 \beta + 27) q^{63} + ( - 30 \beta + 366) q^{67} + ( - 12 \beta - 72) q^{69} + (90 \beta + 18) q^{71} + (54 \beta - 666) q^{73} + (54 \beta - 306) q^{77} + (150 \beta - 190) q^{79} + 81 q^{81} + (104 \beta - 492) q^{83} + ( - 63 \beta - 477) q^{87} + (72 \beta + 558) q^{89} + ( - 126 \beta - 486) q^{91} + ( - 18 \beta - 78) q^{93} + ( - 120 \beta + 384) q^{97} + ( - 27 \beta + 189) q^{99}+O(q^{100})$$ q + 3 * q^3 + (3*b + 3) * q^7 + 9 * q^9 + (-3*b + 21) * q^11 + (-3*b - 39) * q^13 + (5*b - 51) * q^17 + (-12*b - 28) * q^19 + (9*b + 9) * q^21 + (-4*b - 24) * q^23 + 27 * q^27 + (-21*b - 159) * q^29 + (-6*b - 26) * q^31 + (-9*b + 63) * q^33 + (27*b - 153) * q^37 + (-9*b - 117) * q^39 + (6*b - 204) * q^41 + (-48*b + 60) * q^43 + (46*b + 90) * q^47 + (18*b + 35) * q^49 + (15*b - 153) * q^51 + (-41*b - 201) * q^53 + (-36*b - 84) * q^57 + (-3*b + 93) * q^59 + (-48*b + 170) * q^61 + (27*b + 27) * q^63 + (-30*b + 366) * q^67 + (-12*b - 72) * q^69 + (90*b + 18) * q^71 + (54*b - 666) * q^73 + (54*b - 306) * q^77 + (150*b - 190) * q^79 + 81 * q^81 + (104*b - 492) * q^83 + (-63*b - 477) * q^87 + (72*b + 558) * q^89 + (-126*b - 486) * q^91 + (-18*b - 78) * q^93 + (-120*b + 384) * q^97 + (-27*b + 189) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 6 q^{3} + 6 q^{7} + 18 q^{9}+O(q^{10})$$ 2 * q + 6 * q^3 + 6 * q^7 + 18 * q^9 $$2 q + 6 q^{3} + 6 q^{7} + 18 q^{9} + 42 q^{11} - 78 q^{13} - 102 q^{17} - 56 q^{19} + 18 q^{21} - 48 q^{23} + 54 q^{27} - 318 q^{29} - 52 q^{31} + 126 q^{33} - 306 q^{37} - 234 q^{39} - 408 q^{41} + 120 q^{43} + 180 q^{47} + 70 q^{49} - 306 q^{51} - 402 q^{53} - 168 q^{57} + 186 q^{59} + 340 q^{61} + 54 q^{63} + 732 q^{67} - 144 q^{69} + 36 q^{71} - 1332 q^{73} - 612 q^{77} - 380 q^{79} + 162 q^{81} - 984 q^{83} - 954 q^{87} + 1116 q^{89} - 972 q^{91} - 156 q^{93} + 768 q^{97} + 378 q^{99}+O(q^{100})$$ 2 * q + 6 * q^3 + 6 * q^7 + 18 * q^9 + 42 * q^11 - 78 * q^13 - 102 * q^17 - 56 * q^19 + 18 * q^21 - 48 * q^23 + 54 * q^27 - 318 * q^29 - 52 * q^31 + 126 * q^33 - 306 * q^37 - 234 * q^39 - 408 * q^41 + 120 * q^43 + 180 * q^47 + 70 * q^49 - 306 * q^51 - 402 * q^53 - 168 * q^57 + 186 * q^59 + 340 * q^61 + 54 * q^63 + 732 * q^67 - 144 * q^69 + 36 * q^71 - 1332 * q^73 - 612 * q^77 - 380 * q^79 + 162 * q^81 - 984 * q^83 - 954 * q^87 + 1116 * q^89 - 972 * q^91 - 156 * q^93 + 768 * q^97 + 378 * 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
 −2.70156 3.70156
0 3.00000 0 0 0 −16.2094 0 9.00000 0
1.2 0 3.00000 0 0 0 22.2094 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.bt 2
4.b odd 2 1 75.4.a.c 2
5.b even 2 1 1200.4.a.bn 2
5.c odd 4 2 240.4.f.f 4
12.b even 2 1 225.4.a.o 2
15.e even 4 2 720.4.f.j 4
20.d odd 2 1 75.4.a.f 2
20.e even 4 2 15.4.b.a 4
40.i odd 4 2 960.4.f.p 4
40.k even 4 2 960.4.f.q 4
60.h even 2 1 225.4.a.i 2
60.l odd 4 2 45.4.b.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.4.b.a 4 20.e even 4 2
45.4.b.b 4 60.l odd 4 2
75.4.a.c 2 4.b odd 2 1
75.4.a.f 2 20.d odd 2 1
225.4.a.i 2 60.h even 2 1
225.4.a.o 2 12.b even 2 1
240.4.f.f 4 5.c odd 4 2
720.4.f.j 4 15.e even 4 2
960.4.f.p 4 40.i odd 4 2
960.4.f.q 4 40.k even 4 2
1200.4.a.bn 2 5.b even 2 1
1200.4.a.bt 2 1.a even 1 1 trivial

## Hecke kernels

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

 $$T_{7}^{2} - 6T_{7} - 360$$ T7^2 - 6*T7 - 360 $$T_{11}^{2} - 42T_{11} + 72$$ T11^2 - 42*T11 + 72

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$(T - 3)^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} - 6T - 360$$
$11$ $$T^{2} - 42T + 72$$
$13$ $$T^{2} + 78T + 1152$$
$17$ $$T^{2} + 102T + 1576$$
$19$ $$T^{2} + 56T - 5120$$
$23$ $$T^{2} + 48T - 80$$
$29$ $$T^{2} + 318T + 7200$$
$31$ $$T^{2} + 52T - 800$$
$37$ $$T^{2} + 306T - 6480$$
$41$ $$T^{2} + 408T + 40140$$
$43$ $$T^{2} - 120T - 90864$$
$47$ $$T^{2} - 180T - 78656$$
$53$ $$T^{2} + 402T - 28520$$
$59$ $$T^{2} - 186T + 8280$$
$61$ $$T^{2} - 340T - 65564$$
$67$ $$T^{2} - 732T + 97056$$
$71$ $$T^{2} - 36T - 331776$$
$73$ $$T^{2} + 1332 T + 324000$$
$79$ $$T^{2} + 380T - 886400$$
$83$ $$T^{2} + 984T - 201392$$
$89$ $$T^{2} - 1116T + 98820$$
$97$ $$T^{2} - 768T - 442944$$
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