# Properties

 Label 400.3.p.l Level $400$ Weight $3$ Character orbit 400.p Analytic conductor $10.899$ Analytic rank $0$ Dimension $4$ 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: $$4$$ Relative dimension: $$2$$ over $$\Q(i)$$ Coefficient field: $$\Q(i, \sqrt{6})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 9$$ x^4 + 9 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ 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 a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_{3} - 2 \beta_{2} + 2) q^{3} + ( - 4 \beta_{2} + 4 \beta_1 - 4) q^{7} + (4 \beta_{3} - 2 \beta_{2} + 4 \beta_1) q^{9}+O(q^{10})$$ q + (b3 - 2*b2 + 2) * q^3 + (-4*b2 + 4*b1 - 4) * q^7 + (4*b3 - 2*b2 + 4*b1) * q^9 $$q + (\beta_{3} - 2 \beta_{2} + 2) q^{3} + ( - 4 \beta_{2} + 4 \beta_1 - 4) q^{7} + (4 \beta_{3} - 2 \beta_{2} + 4 \beta_1) q^{9} + (6 \beta_{3} - 6 \beta_1 - 5) q^{11} + (2 \beta_{3} + 12 \beta_{2} - 12) q^{13} + ( - 4 \beta_{2} + 13 \beta_1 - 4) q^{17} + ( - 2 \beta_{3} - 5 \beta_{2} - 2 \beta_1) q^{19} + ( - 12 \beta_{3} + 12 \beta_1 - 28) q^{21} + (2 \beta_{3} - 12 \beta_{2} + 12) q^{23} + (2 \beta_{2} + 9 \beta_1 + 2) q^{27} + (4 \beta_{3} + 10 \beta_{2} + 4 \beta_1) q^{29} + (4 \beta_{3} - 4 \beta_1 - 26) q^{31} + (19 \beta_{3} - 8 \beta_{2} + 8) q^{33} + (20 \beta_{2} + 8 \beta_1 + 20) q^{37} + ( - 8 \beta_{3} + 42 \beta_{2} - 8 \beta_1) q^{39} + (16 \beta_{3} - 16 \beta_1 - 7) q^{41} + (36 \beta_{3} + 4 \beta_{2} - 4) q^{43} + (16 \beta_{2} + 30 \beta_1 + 16) q^{47} + ( - 32 \beta_{3} + 31 \beta_{2} - 32 \beta_1) q^{49} + ( - 30 \beta_{3} + 30 \beta_1 - 55) q^{51} + ( - 10 \beta_{3} - 8 \beta_{2} + 8) q^{53} + ( - 4 \beta_{2} - 3 \beta_1 - 4) q^{57} + (8 \beta_{3} - 34 \beta_{2} + 8 \beta_1) q^{59} + (4 \beta_{3} - 4 \beta_1 - 36) q^{61} + ( - 40 \beta_{3} + 56 \beta_{2} - 56) q^{63} + ( - 6 \beta_{2} - 31 \beta_1 - 6) q^{67} + (16 \beta_{3} - 54 \beta_{2} + 16 \beta_1) q^{69} + ( - 16 \beta_{3} + 16 \beta_1 + 46) q^{71} + (13 \beta_{3} - 68 \beta_{2} + 68) q^{73} + ( - 52 \beta_{2} + 28 \beta_1 - 52) q^{77} + (44 \beta_{3} + 24 \beta_{2} + 44 \beta_1) q^{79} + (20 \beta_{3} - 20 \beta_1 - 1) q^{81} + (19 \beta_{3} + 14 \beta_{2} - 14) q^{83} + (8 \beta_{2} + 6 \beta_1 + 8) q^{87} + (12 \beta_{3} + 25 \beta_{2} + 12 \beta_1) q^{89} + (40 \beta_{3} - 40 \beta_1 + 72) q^{91} + ( - 10 \beta_{3} + 40 \beta_{2} - 40) q^{93} + (16 \beta_{2} - 20 \beta_1 + 16) q^{97} + ( - 8 \beta_{3} - 134 \beta_{2} - 8 \beta_1) q^{99}+O(q^{100})$$ q + (b3 - 2*b2 + 2) * q^3 + (-4*b2 + 4*b1 - 4) * q^7 + (4*b3 - 2*b2 + 4*b1) * q^9 + (6*b3 - 6*b1 - 5) * q^11 + (2*b3 + 12*b2 - 12) * q^13 + (-4*b2 + 13*b1 - 4) * q^17 + (-2*b3 - 5*b2 - 2*b1) * q^19 + (-12*b3 + 12*b1 - 28) * q^21 + (2*b3 - 12*b2 + 12) * q^23 + (2*b2 + 9*b1 + 2) * q^27 + (4*b3 + 10*b2 + 4*b1) * q^29 + (4*b3 - 4*b1 - 26) * q^31 + (19*b3 - 8*b2 + 8) * q^33 + (20*b2 + 8*b1 + 20) * q^37 + (-8*b3 + 42*b2 - 8*b1) * q^39 + (16*b3 - 16*b1 - 7) * q^41 + (36*b3 + 4*b2 - 4) * q^43 + (16*b2 + 30*b1 + 16) * q^47 + (-32*b3 + 31*b2 - 32*b1) * q^49 + (-30*b3 + 30*b1 - 55) * q^51 + (-10*b3 - 8*b2 + 8) * q^53 + (-4*b2 - 3*b1 - 4) * q^57 + (8*b3 - 34*b2 + 8*b1) * q^59 + (4*b3 - 4*b1 - 36) * q^61 + (-40*b3 + 56*b2 - 56) * q^63 + (-6*b2 - 31*b1 - 6) * q^67 + (16*b3 - 54*b2 + 16*b1) * q^69 + (-16*b3 + 16*b1 + 46) * q^71 + (13*b3 - 68*b2 + 68) * q^73 + (-52*b2 + 28*b1 - 52) * q^77 + (44*b3 + 24*b2 + 44*b1) * q^79 + (20*b3 - 20*b1 - 1) * q^81 + (19*b3 + 14*b2 - 14) * q^83 + (8*b2 + 6*b1 + 8) * q^87 + (12*b3 + 25*b2 + 12*b1) * q^89 + (40*b3 - 40*b1 + 72) * q^91 + (-10*b3 + 40*b2 - 40) * q^93 + (16*b2 - 20*b1 + 16) * q^97 + (-8*b3 - 134*b2 - 8*b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 8 q^{3} - 16 q^{7}+O(q^{10})$$ 4 * q + 8 * q^3 - 16 * q^7 $$4 q + 8 q^{3} - 16 q^{7} - 20 q^{11} - 48 q^{13} - 16 q^{17} - 112 q^{21} + 48 q^{23} + 8 q^{27} - 104 q^{31} + 32 q^{33} + 80 q^{37} - 28 q^{41} - 16 q^{43} + 64 q^{47} - 220 q^{51} + 32 q^{53} - 16 q^{57} - 144 q^{61} - 224 q^{63} - 24 q^{67} + 184 q^{71} + 272 q^{73} - 208 q^{77} - 4 q^{81} - 56 q^{83} + 32 q^{87} + 288 q^{91} - 160 q^{93} + 64 q^{97}+O(q^{100})$$ 4 * q + 8 * q^3 - 16 * q^7 - 20 * q^11 - 48 * q^13 - 16 * q^17 - 112 * q^21 + 48 * q^23 + 8 * q^27 - 104 * q^31 + 32 * q^33 + 80 * q^37 - 28 * q^41 - 16 * q^43 + 64 * q^47 - 220 * q^51 + 32 * q^53 - 16 * q^57 - 144 * q^61 - 224 * q^63 - 24 * q^67 + 184 * q^71 + 272 * q^73 - 208 * q^77 - 4 * q^81 - 56 * q^83 + 32 * q^87 + 288 * q^91 - 160 * q^93 + 64 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 9$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{2} ) / 3$$ (v^2) / 3 $$\beta_{3}$$ $$=$$ $$( \nu^{3} ) / 3$$ (v^3) / 3
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$3\beta_{2}$$ 3*b2 $$\nu^{3}$$ $$=$$ $$3\beta_{3}$$ 3*b3

## Character values

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

 $$n$$ $$101$$ $$177$$ $$351$$ $$\chi(n)$$ $$1$$ $$\beta_{2}$$ $$1$$

## 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
 1.22474 − 1.22474i −1.22474 + 1.22474i 1.22474 + 1.22474i −1.22474 − 1.22474i
0 0.775255 + 0.775255i 0 0 0 0.898979 0.898979i 0 7.79796i 0
193.2 0 3.22474 + 3.22474i 0 0 0 −8.89898 + 8.89898i 0 11.7980i 0
257.1 0 0.775255 0.775255i 0 0 0 0.898979 + 0.898979i 0 7.79796i 0
257.2 0 3.22474 3.22474i 0 0 0 −8.89898 8.89898i 0 11.7980i 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.l 4
4.b odd 2 1 200.3.l.d 4
5.b even 2 1 400.3.p.h 4
5.c odd 4 1 400.3.p.h 4
5.c odd 4 1 inner 400.3.p.l 4
12.b even 2 1 1800.3.v.o 4
20.d odd 2 1 200.3.l.f yes 4
20.e even 4 1 200.3.l.d 4
20.e even 4 1 200.3.l.f yes 4
60.h even 2 1 1800.3.v.h 4
60.l odd 4 1 1800.3.v.h 4
60.l odd 4 1 1800.3.v.o 4

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

 $$T_{3}^{4} - 8T_{3}^{3} + 32T_{3}^{2} - 40T_{3} + 25$$ T3^4 - 8*T3^3 + 32*T3^2 - 40*T3 + 25 $$T_{7}^{4} + 16T_{7}^{3} + 128T_{7}^{2} - 256T_{7} + 256$$ T7^4 + 16*T7^3 + 128*T7^2 - 256*T7 + 256

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} - 8 T^{3} + 32 T^{2} - 40 T + 25$$
$5$ $$T^{4}$$
$7$ $$T^{4} + 16 T^{3} + 128 T^{2} + \cdots + 256$$
$11$ $$(T^{2} + 10 T - 191)^{2}$$
$13$ $$T^{4} + 48 T^{3} + 1152 T^{2} + \cdots + 76176$$
$17$ $$T^{4} + 16 T^{3} + 128 T^{2} + \cdots + 225625$$
$19$ $$T^{4} + 98T^{2} + 1$$
$23$ $$T^{4} - 48 T^{3} + 1152 T^{2} + \cdots + 76176$$
$29$ $$T^{4} + 392T^{2} + 16$$
$31$ $$(T^{2} + 52 T + 580)^{2}$$
$37$ $$T^{4} - 80 T^{3} + 3200 T^{2} + \cdots + 369664$$
$41$ $$(T^{2} + 14 T - 1487)^{2}$$
$43$ $$T^{4} + 16 T^{3} + 128 T^{2} + \cdots + 14868736$$
$47$ $$T^{4} - 64 T^{3} + 2048 T^{2} + \cdots + 4787344$$
$53$ $$T^{4} - 32 T^{3} + 512 T^{2} + \cdots + 29584$$
$59$ $$T^{4} + 3080 T^{2} + 595984$$
$61$ $$(T^{2} + 72 T + 1200)^{2}$$
$67$ $$T^{4} + 24 T^{3} + 288 T^{2} + \cdots + 7901721$$
$71$ $$(T^{2} - 92 T + 580)^{2}$$
$73$ $$T^{4} - 272 T^{3} + \cdots + 76405081$$
$79$ $$T^{4} + 24384 T^{2} + \cdots + 121881600$$
$83$ $$T^{4} + 56 T^{3} + 1568 T^{2} + \cdots + 477481$$
$89$ $$T^{4} + 2978 T^{2} + 57121$$
$97$ $$T^{4} - 64 T^{3} + 2048 T^{2} + \cdots + 473344$$