# Properties

 Label 400.3.p.k Level $400$ Weight $3$ Character orbit 400.p Analytic conductor $10.899$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# 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: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 100) 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.

 $$f(q)$$ $$=$$ $$q + 3 \beta_1 q^{3} - 4 \beta_{3} q^{7} + 18 \beta_{2} q^{9}+O(q^{10})$$ q + 3*b1 * q^3 - 4*b3 * q^7 + 18*b2 * q^9 $$q + 3 \beta_1 q^{3} - 4 \beta_{3} q^{7} + 18 \beta_{2} q^{9} + 15 q^{11} - 2 \beta_1 q^{13} + 3 \beta_{3} q^{17} + 17 \beta_{2} q^{19} + 36 q^{21} + 6 \beta_1 q^{23} + 27 \beta_{3} q^{27} - 42 \beta_{2} q^{29} + 14 q^{31} + 45 \beta_1 q^{33} + 24 \beta_{3} q^{37} - 18 \beta_{2} q^{39} - 39 q^{41} - 20 \beta_1 q^{43} + 42 \beta_{3} q^{47} + \beta_{2} q^{49} - 27 q^{51} - 54 \beta_1 q^{53} + 51 \beta_{3} q^{57} - 6 \beta_{2} q^{59} + 92 q^{61} + 72 \beta_1 q^{63} - 29 \beta_{3} q^{67} + 54 \beta_{2} q^{69} + 102 q^{71} + 3 \beta_1 q^{73} - 60 \beta_{3} q^{77} - 104 \beta_{2} q^{79} - 81 q^{81} - 39 \beta_1 q^{83} - 126 \beta_{3} q^{87} + 87 \beta_{2} q^{89} - 24 q^{91} + 42 \beta_1 q^{93} - 76 \beta_{3} q^{97} + 270 \beta_{2} q^{99}+O(q^{100})$$ q + 3*b1 * q^3 - 4*b3 * q^7 + 18*b2 * q^9 + 15 * q^11 - 2*b1 * q^13 + 3*b3 * q^17 + 17*b2 * q^19 + 36 * q^21 + 6*b1 * q^23 + 27*b3 * q^27 - 42*b2 * q^29 + 14 * q^31 + 45*b1 * q^33 + 24*b3 * q^37 - 18*b2 * q^39 - 39 * q^41 - 20*b1 * q^43 + 42*b3 * q^47 + b2 * q^49 - 27 * q^51 - 54*b1 * q^53 + 51*b3 * q^57 - 6*b2 * q^59 + 92 * q^61 + 72*b1 * q^63 - 29*b3 * q^67 + 54*b2 * q^69 + 102 * q^71 + 3*b1 * q^73 - 60*b3 * q^77 - 104*b2 * q^79 - 81 * q^81 - 39*b1 * q^83 - 126*b3 * q^87 + 87*b2 * q^89 - 24 * q^91 + 42*b1 * q^93 - 76*b3 * q^97 + 270*b2 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q+O(q^{10})$$ 4 * q $$4 q + 60 q^{11} + 144 q^{21} + 56 q^{31} - 156 q^{41} - 108 q^{51} + 368 q^{61} + 408 q^{71} - 324 q^{81} - 96 q^{91}+O(q^{100})$$ 4 * q + 60 * q^11 + 144 * q^21 + 56 * q^31 - 156 * q^41 - 108 * q^51 + 368 * q^61 + 408 * q^71 - 324 * q^81 - 96 * q^91

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 −3.67423 3.67423i 0 0 0 −4.89898 + 4.89898i 0 18.0000i 0
193.2 0 3.67423 + 3.67423i 0 0 0 4.89898 4.89898i 0 18.0000i 0
257.1 0 −3.67423 + 3.67423i 0 0 0 −4.89898 4.89898i 0 18.0000i 0
257.2 0 3.67423 3.67423i 0 0 0 4.89898 + 4.89898i 0 18.0000i 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.b even 2 1 inner
5.c odd 4 2 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 400.3.p.k 4
4.b odd 2 1 100.3.f.b 4
5.b even 2 1 inner 400.3.p.k 4
5.c odd 4 2 inner 400.3.p.k 4
12.b even 2 1 900.3.l.g 4
20.d odd 2 1 100.3.f.b 4
20.e even 4 2 100.3.f.b 4
60.h even 2 1 900.3.l.g 4
60.l odd 4 2 900.3.l.g 4

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

## Hecke kernels

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

 $$T_{3}^{4} + 729$$ T3^4 + 729 $$T_{7}^{4} + 2304$$ T7^4 + 2304

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} + 729$$
$5$ $$T^{4}$$
$7$ $$T^{4} + 2304$$
$11$ $$(T - 15)^{4}$$
$13$ $$T^{4} + 144$$
$17$ $$T^{4} + 729$$
$19$ $$(T^{2} + 289)^{2}$$
$23$ $$T^{4} + 11664$$
$29$ $$(T^{2} + 1764)^{2}$$
$31$ $$(T - 14)^{4}$$
$37$ $$T^{4} + 2985984$$
$41$ $$(T + 39)^{4}$$
$43$ $$T^{4} + 1440000$$
$47$ $$T^{4} + 28005264$$
$53$ $$T^{4} + 76527504$$
$59$ $$(T^{2} + 36)^{2}$$
$61$ $$(T - 92)^{4}$$
$67$ $$T^{4} + 6365529$$
$71$ $$(T - 102)^{4}$$
$73$ $$T^{4} + 729$$
$79$ $$(T^{2} + 10816)^{2}$$
$83$ $$T^{4} + 20820969$$
$89$ $$(T^{2} + 7569)^{2}$$
$97$ $$T^{4} + 300259584$$