# Properties

 Label 400.3.p.f 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: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 200) 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.

 $$f(q)$$ $$=$$ $$q + ( - i + 1) q^{3} + (7 i + 7) q^{7} + 7 i q^{9}+O(q^{10})$$ q + (-i + 1) * q^3 + (7*i + 7) * q^7 + 7*i * q^9 $$q + ( - i + 1) q^{3} + (7 i + 7) q^{7} + 7 i q^{9} + 4 q^{11} + (12 i - 12) q^{13} + ( - 20 i - 20) q^{17} + 28 i q^{19} + 14 q^{21} + ( - 9 i + 9) q^{23} + (16 i + 16) q^{27} + 34 i q^{29} + 40 q^{31} + ( - 4 i + 4) q^{33} + (16 i + 16) q^{37} + 24 i q^{39} + 32 q^{41} + ( - 7 i + 7) q^{43} + ( - 31 i - 31) q^{47} + 49 i q^{49} - 40 q^{51} + ( - 52 i + 52) q^{53} + (28 i + 28) q^{57} + 44 i q^{59} + (49 i - 49) q^{63} + ( - 81 i - 81) q^{67} - 18 i q^{69} + 112 q^{71} + (44 i - 44) q^{73} + (28 i + 28) q^{77} - 72 i q^{79} - 31 q^{81} + (49 i - 49) q^{83} + (34 i + 34) q^{87} - 98 i q^{89} - 168 q^{91} + ( - 40 i + 40) q^{93} + (44 i + 44) q^{97} + 28 i q^{99} +O(q^{100})$$ q + (-i + 1) * q^3 + (7*i + 7) * q^7 + 7*i * q^9 + 4 * q^11 + (12*i - 12) * q^13 + (-20*i - 20) * q^17 + 28*i * q^19 + 14 * q^21 + (-9*i + 9) * q^23 + (16*i + 16) * q^27 + 34*i * q^29 + 40 * q^31 + (-4*i + 4) * q^33 + (16*i + 16) * q^37 + 24*i * q^39 + 32 * q^41 + (-7*i + 7) * q^43 + (-31*i - 31) * q^47 + 49*i * q^49 - 40 * q^51 + (-52*i + 52) * q^53 + (28*i + 28) * q^57 + 44*i * q^59 + (49*i - 49) * q^63 + (-81*i - 81) * q^67 - 18*i * q^69 + 112 * q^71 + (44*i - 44) * q^73 + (28*i + 28) * q^77 - 72*i * q^79 - 31 * q^81 + (49*i - 49) * q^83 + (34*i + 34) * q^87 - 98*i * q^89 - 168 * q^91 + (-40*i + 40) * q^93 + (44*i + 44) * q^97 + 28*i * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{3} + 14 q^{7}+O(q^{10})$$ 2 * q + 2 * q^3 + 14 * q^7 $$2 q + 2 q^{3} + 14 q^{7} + 8 q^{11} - 24 q^{13} - 40 q^{17} + 28 q^{21} + 18 q^{23} + 32 q^{27} + 80 q^{31} + 8 q^{33} + 32 q^{37} + 64 q^{41} + 14 q^{43} - 62 q^{47} - 80 q^{51} + 104 q^{53} + 56 q^{57} - 98 q^{63} - 162 q^{67} + 224 q^{71} - 88 q^{73} + 56 q^{77} - 62 q^{81} - 98 q^{83} + 68 q^{87} - 336 q^{91} + 80 q^{93} + 88 q^{97}+O(q^{100})$$ 2 * q + 2 * q^3 + 14 * q^7 + 8 * q^11 - 24 * q^13 - 40 * q^17 + 28 * q^21 + 18 * q^23 + 32 * q^27 + 80 * q^31 + 8 * q^33 + 32 * q^37 + 64 * q^41 + 14 * q^43 - 62 * q^47 - 80 * q^51 + 104 * q^53 + 56 * q^57 - 98 * q^63 - 162 * q^67 + 224 * q^71 - 88 * q^73 + 56 * q^77 - 62 * q^81 - 98 * q^83 + 68 * q^87 - 336 * q^91 + 80 * q^93 + 88 * 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 1.00000 + 1.00000i 0 0 0 7.00000 7.00000i 0 7.00000i 0
257.1 0 1.00000 1.00000i 0 0 0 7.00000 + 7.00000i 0 7.00000i 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.f 2
4.b odd 2 1 200.3.l.a 2
5.b even 2 1 400.3.p.c 2
5.c odd 4 1 400.3.p.c 2
5.c odd 4 1 inner 400.3.p.f 2
12.b even 2 1 1800.3.v.a 2
20.d odd 2 1 200.3.l.c yes 2
20.e even 4 1 200.3.l.a 2
20.e even 4 1 200.3.l.c yes 2
60.h even 2 1 1800.3.v.g 2
60.l odd 4 1 1800.3.v.a 2
60.l odd 4 1 1800.3.v.g 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
200.3.l.a 2 4.b odd 2 1
200.3.l.a 2 20.e even 4 1
200.3.l.c yes 2 20.d odd 2 1
200.3.l.c yes 2 20.e even 4 1
400.3.p.c 2 5.b even 2 1
400.3.p.c 2 5.c odd 4 1
400.3.p.f 2 1.a even 1 1 trivial
400.3.p.f 2 5.c odd 4 1 inner
1800.3.v.a 2 12.b even 2 1
1800.3.v.a 2 60.l odd 4 1
1800.3.v.g 2 60.h even 2 1
1800.3.v.g 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 $$S_{3}^{\mathrm{new}}(400, [\chi])$$:

 $$T_{3}^{2} - 2T_{3} + 2$$ T3^2 - 2*T3 + 2 $$T_{7}^{2} - 14T_{7} + 98$$ T7^2 - 14*T7 + 98

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - 2T + 2$$
$5$ $$T^{2}$$
$7$ $$T^{2} - 14T + 98$$
$11$ $$(T - 4)^{2}$$
$13$ $$T^{2} + 24T + 288$$
$17$ $$T^{2} + 40T + 800$$
$19$ $$T^{2} + 784$$
$23$ $$T^{2} - 18T + 162$$
$29$ $$T^{2} + 1156$$
$31$ $$(T - 40)^{2}$$
$37$ $$T^{2} - 32T + 512$$
$41$ $$(T - 32)^{2}$$
$43$ $$T^{2} - 14T + 98$$
$47$ $$T^{2} + 62T + 1922$$
$53$ $$T^{2} - 104T + 5408$$
$59$ $$T^{2} + 1936$$
$61$ $$T^{2}$$
$67$ $$T^{2} + 162T + 13122$$
$71$ $$(T - 112)^{2}$$
$73$ $$T^{2} + 88T + 3872$$
$79$ $$T^{2} + 5184$$
$83$ $$T^{2} + 98T + 4802$$
$89$ $$T^{2} + 9604$$
$97$ $$T^{2} - 88T + 3872$$