# Properties

 Label 400.3.p.g Level $400$ Weight $3$ Character orbit 400.p Analytic conductor $10.899$ 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] = [400,3,Mod(193,400)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(400, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 0, 3]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("400.193");

S:= CuspForms(chi, 3);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$400 = 2^{4} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 400.p (of order $$4$$, degree $$2$$, 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: $$10.8992105744$$ 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: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 50) Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( - 3 i + 3) q^{3} + ( - 3 i - 3) q^{7} - 9 i q^{9} +O(q^{10})$$ q + (-3*i + 3) * q^3 + (-3*i - 3) * q^7 - 9*i * q^9 $$q + ( - 3 i + 3) q^{3} + ( - 3 i - 3) q^{7} - 9 i q^{9} - 12 q^{11} + ( - 12 i + 12) q^{13} + ( - 12 i - 12) q^{17} - 20 i q^{19} - 18 q^{21} + ( - 3 i + 3) q^{23} - 30 i q^{29} + 8 q^{31} + (36 i - 36) q^{33} + (48 i + 48) q^{37} - 72 i q^{39} - 48 q^{41} + (27 i - 27) q^{43} + (27 i + 27) q^{47} - 31 i q^{49} - 72 q^{51} + ( - 12 i + 12) q^{53} + ( - 60 i - 60) q^{57} + 60 i q^{59} + 32 q^{61} + (27 i - 27) q^{63} + ( - 3 i - 3) q^{67} - 18 i q^{69} + 48 q^{71} + ( - 12 i + 12) q^{73} + (36 i + 36) q^{77} - 40 i q^{79} + 81 q^{81} + ( - 93 i + 93) q^{83} + ( - 90 i - 90) q^{87} + 30 i q^{89} - 72 q^{91} + ( - 24 i + 24) q^{93} + ( - 12 i - 12) q^{97} + 108 i q^{99} +O(q^{100})$$ q + (-3*i + 3) * q^3 + (-3*i - 3) * q^7 - 9*i * q^9 - 12 * q^11 + (-12*i + 12) * q^13 + (-12*i - 12) * q^17 - 20*i * q^19 - 18 * q^21 + (-3*i + 3) * q^23 - 30*i * q^29 + 8 * q^31 + (36*i - 36) * q^33 + (48*i + 48) * q^37 - 72*i * q^39 - 48 * q^41 + (27*i - 27) * q^43 + (27*i + 27) * q^47 - 31*i * q^49 - 72 * q^51 + (-12*i + 12) * q^53 + (-60*i - 60) * q^57 + 60*i * q^59 + 32 * q^61 + (27*i - 27) * q^63 + (-3*i - 3) * q^67 - 18*i * q^69 + 48 * q^71 + (-12*i + 12) * q^73 + (36*i + 36) * q^77 - 40*i * q^79 + 81 * q^81 + (-93*i + 93) * q^83 + (-90*i - 90) * q^87 + 30*i * q^89 - 72 * q^91 + (-24*i + 24) * q^93 + (-12*i - 12) * q^97 + 108*i * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 6 q^{3} - 6 q^{7}+O(q^{10})$$ 2 * q + 6 * q^3 - 6 * q^7 $$2 q + 6 q^{3} - 6 q^{7} - 24 q^{11} + 24 q^{13} - 24 q^{17} - 36 q^{21} + 6 q^{23} + 16 q^{31} - 72 q^{33} + 96 q^{37} - 96 q^{41} - 54 q^{43} + 54 q^{47} - 144 q^{51} + 24 q^{53} - 120 q^{57} + 64 q^{61} - 54 q^{63} - 6 q^{67} + 96 q^{71} + 24 q^{73} + 72 q^{77} + 162 q^{81} + 186 q^{83} - 180 q^{87} - 144 q^{91} + 48 q^{93} - 24 q^{97}+O(q^{100})$$ 2 * q + 6 * q^3 - 6 * q^7 - 24 * q^11 + 24 * q^13 - 24 * q^17 - 36 * q^21 + 6 * q^23 + 16 * q^31 - 72 * q^33 + 96 * q^37 - 96 * q^41 - 54 * q^43 + 54 * q^47 - 144 * q^51 + 24 * q^53 - 120 * q^57 + 64 * q^61 - 54 * q^63 - 6 * q^67 + 96 * q^71 + 24 * q^73 + 72 * q^77 + 162 * q^81 + 186 * q^83 - 180 * q^87 - 144 * q^91 + 48 * q^93 - 24 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/400\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$177$$ $$351$$ $$\chi(n)$$ $$1$$ $$i$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
193.1
 − 1.00000i 1.00000i
0 3.00000 + 3.00000i 0 0 0 −3.00000 + 3.00000i 0 9.00000i 0
257.1 0 3.00000 3.00000i 0 0 0 −3.00000 3.00000i 0 9.00000i 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.c odd 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 400.3.p.g 2
4.b odd 2 1 50.3.c.b yes 2
5.b even 2 1 400.3.p.a 2
5.c odd 4 1 400.3.p.a 2
5.c odd 4 1 inner 400.3.p.g 2
12.b even 2 1 450.3.g.c 2
20.d odd 2 1 50.3.c.a 2
20.e even 4 1 50.3.c.a 2
20.e even 4 1 50.3.c.b yes 2
60.h even 2 1 450.3.g.e 2
60.l odd 4 1 450.3.g.c 2
60.l odd 4 1 450.3.g.e 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
50.3.c.a 2 20.d odd 2 1
50.3.c.a 2 20.e even 4 1
50.3.c.b yes 2 4.b odd 2 1
50.3.c.b yes 2 20.e even 4 1
400.3.p.a 2 5.b even 2 1
400.3.p.a 2 5.c odd 4 1
400.3.p.g 2 1.a even 1 1 trivial
400.3.p.g 2 5.c odd 4 1 inner
450.3.g.c 2 12.b even 2 1
450.3.g.c 2 60.l odd 4 1
450.3.g.e 2 60.h even 2 1
450.3.g.e 2 60.l odd 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(400, [\chi])$$:

 $$T_{3}^{2} - 6T_{3} + 18$$ T3^2 - 6*T3 + 18 $$T_{7}^{2} + 6T_{7} + 18$$ T7^2 + 6*T7 + 18

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - 6T + 18$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 6T + 18$$
$11$ $$(T + 12)^{2}$$
$13$ $$T^{2} - 24T + 288$$
$17$ $$T^{2} + 24T + 288$$
$19$ $$T^{2} + 400$$
$23$ $$T^{2} - 6T + 18$$
$29$ $$T^{2} + 900$$
$31$ $$(T - 8)^{2}$$
$37$ $$T^{2} - 96T + 4608$$
$41$ $$(T + 48)^{2}$$
$43$ $$T^{2} + 54T + 1458$$
$47$ $$T^{2} - 54T + 1458$$
$53$ $$T^{2} - 24T + 288$$
$59$ $$T^{2} + 3600$$
$61$ $$(T - 32)^{2}$$
$67$ $$T^{2} + 6T + 18$$
$71$ $$(T - 48)^{2}$$
$73$ $$T^{2} - 24T + 288$$
$79$ $$T^{2} + 1600$$
$83$ $$T^{2} - 186T + 17298$$
$89$ $$T^{2} + 900$$
$97$ $$T^{2} + 24T + 288$$