# Properties

 Label 400.3.p.e 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 40) 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.

 $$f(q)$$ $$=$$ $$q + ( - i + 1) q^{3} + ( - 3 i - 3) q^{7} + 7 i q^{9}+O(q^{10})$$ q + (-i + 1) * q^3 + (-3*i - 3) * q^7 + 7*i * q^9 $$q + ( - i + 1) q^{3} + ( - 3 i - 3) q^{7} + 7 i q^{9} + 14 q^{11} + ( - 3 i + 3) q^{13} + (15 i + 15) q^{17} - 32 i q^{19} - 6 q^{21} + ( - 29 i + 29) q^{23} + (16 i + 16) q^{27} - 16 i q^{29} - 10 q^{31} + ( - 14 i + 14) q^{33} + (11 i + 11) q^{37} - 6 i q^{39} + 2 q^{41} + (23 i - 23) q^{43} + ( - 11 i - 11) q^{47} - 31 i q^{49} + 30 q^{51} + ( - 27 i + 27) q^{53} + ( - 32 i - 32) q^{57} + 64 i q^{59} + 90 q^{61} + ( - 21 i + 21) q^{63} + (49 i + 49) q^{67} - 58 i q^{69} - 58 q^{71} + (9 i - 9) q^{73} + ( - 42 i - 42) q^{77} - 32 i q^{79} - 31 q^{81} + ( - i + 1) q^{83} + ( - 16 i - 16) q^{87} + 32 i q^{89} - 18 q^{91} + (10 i - 10) q^{93} + (79 i + 79) q^{97} + 98 i q^{99} +O(q^{100})$$ q + (-i + 1) * q^3 + (-3*i - 3) * q^7 + 7*i * q^9 + 14 * q^11 + (-3*i + 3) * q^13 + (15*i + 15) * q^17 - 32*i * q^19 - 6 * q^21 + (-29*i + 29) * q^23 + (16*i + 16) * q^27 - 16*i * q^29 - 10 * q^31 + (-14*i + 14) * q^33 + (11*i + 11) * q^37 - 6*i * q^39 + 2 * q^41 + (23*i - 23) * q^43 + (-11*i - 11) * q^47 - 31*i * q^49 + 30 * q^51 + (-27*i + 27) * q^53 + (-32*i - 32) * q^57 + 64*i * q^59 + 90 * q^61 + (-21*i + 21) * q^63 + (49*i + 49) * q^67 - 58*i * q^69 - 58 * q^71 + (9*i - 9) * q^73 + (-42*i - 42) * q^77 - 32*i * q^79 - 31 * q^81 + (-i + 1) * q^83 + (-16*i - 16) * q^87 + 32*i * q^89 - 18 * q^91 + (10*i - 10) * q^93 + (79*i + 79) * q^97 + 98*i * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{3} - 6 q^{7}+O(q^{10})$$ 2 * q + 2 * q^3 - 6 * q^7 $$2 q + 2 q^{3} - 6 q^{7} + 28 q^{11} + 6 q^{13} + 30 q^{17} - 12 q^{21} + 58 q^{23} + 32 q^{27} - 20 q^{31} + 28 q^{33} + 22 q^{37} + 4 q^{41} - 46 q^{43} - 22 q^{47} + 60 q^{51} + 54 q^{53} - 64 q^{57} + 180 q^{61} + 42 q^{63} + 98 q^{67} - 116 q^{71} - 18 q^{73} - 84 q^{77} - 62 q^{81} + 2 q^{83} - 32 q^{87} - 36 q^{91} - 20 q^{93} + 158 q^{97}+O(q^{100})$$ 2 * q + 2 * q^3 - 6 * q^7 + 28 * q^11 + 6 * q^13 + 30 * q^17 - 12 * q^21 + 58 * q^23 + 32 * q^27 - 20 * q^31 + 28 * q^33 + 22 * q^37 + 4 * q^41 - 46 * q^43 - 22 * q^47 + 60 * q^51 + 54 * q^53 - 64 * q^57 + 180 * q^61 + 42 * q^63 + 98 * q^67 - 116 * q^71 - 18 * q^73 - 84 * q^77 - 62 * q^81 + 2 * q^83 - 32 * q^87 - 36 * q^91 - 20 * q^93 + 158 * 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 −3.00000 + 3.00000i 0 7.00000i 0
257.1 0 1.00000 1.00000i 0 0 0 −3.00000 3.00000i 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.e 2
4.b odd 2 1 200.3.l.b 2
5.b even 2 1 80.3.p.b 2
5.c odd 4 1 80.3.p.b 2
5.c odd 4 1 inner 400.3.p.e 2
12.b even 2 1 1800.3.v.d 2
15.d odd 2 1 720.3.bh.a 2
15.e even 4 1 720.3.bh.a 2
20.d odd 2 1 40.3.l.a 2
20.e even 4 1 40.3.l.a 2
20.e even 4 1 200.3.l.b 2
40.e odd 2 1 320.3.p.b 2
40.f even 2 1 320.3.p.f 2
40.i odd 4 1 320.3.p.f 2
40.k even 4 1 320.3.p.b 2
60.h even 2 1 360.3.v.a 2
60.l odd 4 1 360.3.v.a 2
60.l odd 4 1 1800.3.v.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
40.3.l.a 2 20.d odd 2 1
40.3.l.a 2 20.e even 4 1
80.3.p.b 2 5.b even 2 1
80.3.p.b 2 5.c odd 4 1
200.3.l.b 2 4.b odd 2 1
200.3.l.b 2 20.e even 4 1
320.3.p.b 2 40.e odd 2 1
320.3.p.b 2 40.k even 4 1
320.3.p.f 2 40.f even 2 1
320.3.p.f 2 40.i odd 4 1
360.3.v.a 2 60.h even 2 1
360.3.v.a 2 60.l odd 4 1
400.3.p.e 2 1.a even 1 1 trivial
400.3.p.e 2 5.c odd 4 1 inner
720.3.bh.a 2 15.d odd 2 1
720.3.bh.a 2 15.e even 4 1
1800.3.v.d 2 12.b even 2 1
1800.3.v.d 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 $$T_{7}^{2} + 6T_{7} + 18$$ T7^2 + 6*T7 + 18

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - 2T + 2$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 6T + 18$$
$11$ $$(T - 14)^{2}$$
$13$ $$T^{2} - 6T + 18$$
$17$ $$T^{2} - 30T + 450$$
$19$ $$T^{2} + 1024$$
$23$ $$T^{2} - 58T + 1682$$
$29$ $$T^{2} + 256$$
$31$ $$(T + 10)^{2}$$
$37$ $$T^{2} - 22T + 242$$
$41$ $$(T - 2)^{2}$$
$43$ $$T^{2} + 46T + 1058$$
$47$ $$T^{2} + 22T + 242$$
$53$ $$T^{2} - 54T + 1458$$
$59$ $$T^{2} + 4096$$
$61$ $$(T - 90)^{2}$$
$67$ $$T^{2} - 98T + 4802$$
$71$ $$(T + 58)^{2}$$
$73$ $$T^{2} + 18T + 162$$
$79$ $$T^{2} + 1024$$
$83$ $$T^{2} - 2T + 2$$
$89$ $$T^{2} + 1024$$
$97$ $$T^{2} - 158T + 12482$$