# Properties

 Label 400.3.p.d 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 20) 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 $$q + ( - i + 1) q^{3} + ( - 7 i - 7) q^{7} + 7 i q^{9} - 10 q^{11} + (9 i - 9) q^{13} + ( - i - 1) q^{17} + 8 i q^{19} - 14 q^{21} + (23 i - 23) q^{23} + (16 i + 16) q^{27} - 8 i q^{29} + 14 q^{31} + (10 i - 10) q^{33} + ( - 33 i - 33) q^{37} + 18 i q^{39} - 14 q^{41} + (15 i - 15) q^{43} + ( - 39 i - 39) q^{47} + 49 i q^{49} - 2 q^{51} + ( - 7 i + 7) q^{53} + (8 i + 8) q^{57} + 56 i q^{59} + 42 q^{61} + ( - 49 i + 49) q^{63} + ( - 7 i - 7) q^{67} + 46 i q^{69} - 98 q^{71} + (49 i - 49) q^{73} + (70 i + 70) q^{77} - 96 i q^{79} - 31 q^{81} + (63 i - 63) q^{83} + ( - 8 i - 8) q^{87} - 112 i q^{89} + 126 q^{91} + ( - 14 i + 14) q^{93} + ( - 33 i - 33) q^{97} - 70 i q^{99} +O(q^{100})$$ q + (-i + 1) * q^3 + (-7*i - 7) * q^7 + 7*i * q^9 - 10 * q^11 + (9*i - 9) * q^13 + (-i - 1) * q^17 + 8*i * q^19 - 14 * q^21 + (23*i - 23) * q^23 + (16*i + 16) * q^27 - 8*i * q^29 + 14 * q^31 + (10*i - 10) * q^33 + (-33*i - 33) * q^37 + 18*i * q^39 - 14 * q^41 + (15*i - 15) * q^43 + (-39*i - 39) * q^47 + 49*i * q^49 - 2 * q^51 + (-7*i + 7) * q^53 + (8*i + 8) * q^57 + 56*i * q^59 + 42 * q^61 + (-49*i + 49) * q^63 + (-7*i - 7) * q^67 + 46*i * q^69 - 98 * q^71 + (49*i - 49) * q^73 + (70*i + 70) * q^77 - 96*i * q^79 - 31 * q^81 + (63*i - 63) * q^83 + (-8*i - 8) * q^87 - 112*i * q^89 + 126 * q^91 + (-14*i + 14) * q^93 + (-33*i - 33) * q^97 - 70*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 $$2 q + 2 q^{3} - 14 q^{7} - 20 q^{11} - 18 q^{13} - 2 q^{17} - 28 q^{21} - 46 q^{23} + 32 q^{27} + 28 q^{31} - 20 q^{33} - 66 q^{37} - 28 q^{41} - 30 q^{43} - 78 q^{47} - 4 q^{51} + 14 q^{53} + 16 q^{57} + 84 q^{61} + 98 q^{63} - 14 q^{67} - 196 q^{71} - 98 q^{73} + 140 q^{77} - 62 q^{81} - 126 q^{83} - 16 q^{87} + 252 q^{91} + 28 q^{93} - 66 q^{97}+O(q^{100})$$ 2 * q + 2 * q^3 - 14 * q^7 - 20 * q^11 - 18 * q^13 - 2 * q^17 - 28 * q^21 - 46 * q^23 + 32 * q^27 + 28 * q^31 - 20 * q^33 - 66 * q^37 - 28 * q^41 - 30 * q^43 - 78 * q^47 - 4 * q^51 + 14 * q^53 + 16 * q^57 + 84 * q^61 + 98 * q^63 - 14 * q^67 - 196 * q^71 - 98 * q^73 + 140 * q^77 - 62 * q^81 - 126 * q^83 - 16 * q^87 + 252 * q^91 + 28 * q^93 - 66 * 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.d 2
4.b odd 2 1 100.3.f.a 2
5.b even 2 1 80.3.p.a 2
5.c odd 4 1 80.3.p.a 2
5.c odd 4 1 inner 400.3.p.d 2
12.b even 2 1 900.3.l.a 2
15.d odd 2 1 720.3.bh.e 2
15.e even 4 1 720.3.bh.e 2
20.d odd 2 1 20.3.f.a 2
20.e even 4 1 20.3.f.a 2
20.e even 4 1 100.3.f.a 2
40.e odd 2 1 320.3.p.c 2
40.f even 2 1 320.3.p.g 2
40.i odd 4 1 320.3.p.g 2
40.k even 4 1 320.3.p.c 2
60.h even 2 1 180.3.l.a 2
60.l odd 4 1 180.3.l.a 2
60.l odd 4 1 900.3.l.a 2
140.c even 2 1 980.3.l.a 2
140.j odd 4 1 980.3.l.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.3.f.a 2 20.d odd 2 1
20.3.f.a 2 20.e even 4 1
80.3.p.a 2 5.b even 2 1
80.3.p.a 2 5.c odd 4 1
100.3.f.a 2 4.b odd 2 1
100.3.f.a 2 20.e even 4 1
180.3.l.a 2 60.h even 2 1
180.3.l.a 2 60.l odd 4 1
320.3.p.c 2 40.e odd 2 1
320.3.p.c 2 40.k even 4 1
320.3.p.g 2 40.f even 2 1
320.3.p.g 2 40.i odd 4 1
400.3.p.d 2 1.a even 1 1 trivial
400.3.p.d 2 5.c odd 4 1 inner
720.3.bh.e 2 15.d odd 2 1
720.3.bh.e 2 15.e even 4 1
900.3.l.a 2 12.b even 2 1
900.3.l.a 2 60.l odd 4 1
980.3.l.a 2 140.c even 2 1
980.3.l.a 2 140.j 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

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - 2T + 2$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 14T + 98$$
$11$ $$(T + 10)^{2}$$
$13$ $$T^{2} + 18T + 162$$
$17$ $$T^{2} + 2T + 2$$
$19$ $$T^{2} + 64$$
$23$ $$T^{2} + 46T + 1058$$
$29$ $$T^{2} + 64$$
$31$ $$(T - 14)^{2}$$
$37$ $$T^{2} + 66T + 2178$$
$41$ $$(T + 14)^{2}$$
$43$ $$T^{2} + 30T + 450$$
$47$ $$T^{2} + 78T + 3042$$
$53$ $$T^{2} - 14T + 98$$
$59$ $$T^{2} + 3136$$
$61$ $$(T - 42)^{2}$$
$67$ $$T^{2} + 14T + 98$$
$71$ $$(T + 98)^{2}$$
$73$ $$T^{2} + 98T + 4802$$
$79$ $$T^{2} + 9216$$
$83$ $$T^{2} + 126T + 7938$$
$89$ $$T^{2} + 12544$$
$97$ $$T^{2} + 66T + 2178$$