# Properties

 Label 400.3.p.b 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: $$\Z[a_1, \ldots, a_{9}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 10) 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.

 $$f(q)$$ $$=$$ $$q + (2 i - 2) q^{3} + (2 i + 2) q^{7} + i q^{9}+O(q^{10})$$ q + (2*i - 2) * q^3 + (2*i + 2) * q^7 + i * q^9 $$q + (2 i - 2) q^{3} + (2 i + 2) q^{7} + i q^{9} + 8 q^{11} + (3 i - 3) q^{13} + ( - 7 i - 7) q^{17} + 20 i q^{19} - 8 q^{21} + (2 i - 2) q^{23} + ( - 20 i - 20) q^{27} + 40 i q^{29} - 52 q^{31} + (16 i - 16) q^{33} + (3 i + 3) q^{37} - 12 i q^{39} - 8 q^{41} + (42 i - 42) q^{43} + ( - 18 i - 18) q^{47} - 41 i q^{49} + 28 q^{51} + (53 i - 53) q^{53} + ( - 40 i - 40) q^{57} + 20 i q^{59} - 48 q^{61} + (2 i - 2) q^{63} + (62 i + 62) q^{67} - 8 i q^{69} + 28 q^{71} + ( - 47 i + 47) q^{73} + (16 i + 16) q^{77} + 71 q^{81} + ( - 18 i + 18) q^{83} + ( - 80 i - 80) q^{87} + 80 i q^{89} - 12 q^{91} + ( - 104 i + 104) q^{93} + (63 i + 63) q^{97} + 8 i q^{99} +O(q^{100})$$ q + (2*i - 2) * q^3 + (2*i + 2) * q^7 + i * q^9 + 8 * q^11 + (3*i - 3) * q^13 + (-7*i - 7) * q^17 + 20*i * q^19 - 8 * q^21 + (2*i - 2) * q^23 + (-20*i - 20) * q^27 + 40*i * q^29 - 52 * q^31 + (16*i - 16) * q^33 + (3*i + 3) * q^37 - 12*i * q^39 - 8 * q^41 + (42*i - 42) * q^43 + (-18*i - 18) * q^47 - 41*i * q^49 + 28 * q^51 + (53*i - 53) * q^53 + (-40*i - 40) * q^57 + 20*i * q^59 - 48 * q^61 + (2*i - 2) * q^63 + (62*i + 62) * q^67 - 8*i * q^69 + 28 * q^71 + (-47*i + 47) * q^73 + (16*i + 16) * q^77 + 71 * q^81 + (-18*i + 18) * q^83 + (-80*i - 80) * q^87 + 80*i * q^89 - 12 * q^91 + (-104*i + 104) * q^93 + (63*i + 63) * q^97 + 8*i * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 4 q^{3} + 4 q^{7}+O(q^{10})$$ 2 * q - 4 * q^3 + 4 * q^7 $$2 q - 4 q^{3} + 4 q^{7} + 16 q^{11} - 6 q^{13} - 14 q^{17} - 16 q^{21} - 4 q^{23} - 40 q^{27} - 104 q^{31} - 32 q^{33} + 6 q^{37} - 16 q^{41} - 84 q^{43} - 36 q^{47} + 56 q^{51} - 106 q^{53} - 80 q^{57} - 96 q^{61} - 4 q^{63} + 124 q^{67} + 56 q^{71} + 94 q^{73} + 32 q^{77} + 142 q^{81} + 36 q^{83} - 160 q^{87} - 24 q^{91} + 208 q^{93} + 126 q^{97}+O(q^{100})$$ 2 * q - 4 * q^3 + 4 * q^7 + 16 * q^11 - 6 * q^13 - 14 * q^17 - 16 * q^21 - 4 * q^23 - 40 * q^27 - 104 * q^31 - 32 * q^33 + 6 * q^37 - 16 * q^41 - 84 * q^43 - 36 * q^47 + 56 * q^51 - 106 * q^53 - 80 * q^57 - 96 * q^61 - 4 * q^63 + 124 * q^67 + 56 * q^71 + 94 * q^73 + 32 * q^77 + 142 * q^81 + 36 * q^83 - 160 * q^87 - 24 * q^91 + 208 * q^93 + 126 * 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 −2.00000 2.00000i 0 0 0 2.00000 2.00000i 0 1.00000i 0
257.1 0 −2.00000 + 2.00000i 0 0 0 2.00000 + 2.00000i 0 1.00000i 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.b 2
4.b odd 2 1 50.3.c.c 2
5.b even 2 1 80.3.p.c 2
5.c odd 4 1 80.3.p.c 2
5.c odd 4 1 inner 400.3.p.b 2
12.b even 2 1 450.3.g.b 2
15.d odd 2 1 720.3.bh.c 2
15.e even 4 1 720.3.bh.c 2
20.d odd 2 1 10.3.c.a 2
20.e even 4 1 10.3.c.a 2
20.e even 4 1 50.3.c.c 2
40.e odd 2 1 320.3.p.h 2
40.f even 2 1 320.3.p.a 2
40.i odd 4 1 320.3.p.a 2
40.k even 4 1 320.3.p.h 2
60.h even 2 1 90.3.g.b 2
60.l odd 4 1 90.3.g.b 2
60.l odd 4 1 450.3.g.b 2
140.c even 2 1 490.3.f.b 2
140.j odd 4 1 490.3.f.b 2

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

## Hecke kernels

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

 $$T_{3}^{2} + 4T_{3} + 8$$ T3^2 + 4*T3 + 8 $$T_{7}^{2} - 4T_{7} + 8$$ T7^2 - 4*T7 + 8

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 4T + 8$$
$5$ $$T^{2}$$
$7$ $$T^{2} - 4T + 8$$
$11$ $$(T - 8)^{2}$$
$13$ $$T^{2} + 6T + 18$$
$17$ $$T^{2} + 14T + 98$$
$19$ $$T^{2} + 400$$
$23$ $$T^{2} + 4T + 8$$
$29$ $$T^{2} + 1600$$
$31$ $$(T + 52)^{2}$$
$37$ $$T^{2} - 6T + 18$$
$41$ $$(T + 8)^{2}$$
$43$ $$T^{2} + 84T + 3528$$
$47$ $$T^{2} + 36T + 648$$
$53$ $$T^{2} + 106T + 5618$$
$59$ $$T^{2} + 400$$
$61$ $$(T + 48)^{2}$$
$67$ $$T^{2} - 124T + 7688$$
$71$ $$(T - 28)^{2}$$
$73$ $$T^{2} - 94T + 4418$$
$79$ $$T^{2}$$
$83$ $$T^{2} - 36T + 648$$
$89$ $$T^{2} + 6400$$
$97$ $$T^{2} - 126T + 7938$$