# Properties

 Label 400.3.p.i 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: $$\Q(i, \sqrt{41})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 21x^{2} + 100$$ x^4 + 21*x^2 + 100 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2$$ 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 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.

 $$f(q)$$ $$=$$ $$q + (\beta_{2} - 1) q^{3} + ( - \beta_{3} - 3 \beta_1 + 3) q^{7} + ( - \beta_{3} - \beta_{2} + 12 \beta_1) q^{9}+O(q^{10})$$ q + (b2 - 1) * q^3 + (-b3 - 3*b1 + 3) * q^7 + (-b3 - b2 + 12*b1) * q^9 $$q + (\beta_{2} - 1) q^{3} + ( - \beta_{3} - 3 \beta_1 + 3) q^{7} + ( - \beta_{3} - \beta_{2} + 12 \beta_1) q^{9} + ( - \beta_{3} + \beta_{2} + \beta_1 - 6) q^{11} + (2 \beta_{2} + 9 \beta_1 + 7) q^{13} + ( - 2 \beta_{3} + 5 \beta_1 - 5) q^{17} + ( - 2 \beta_{3} - 2 \beta_{2} - 10 \beta_1) q^{19} + ( - 3 \beta_{3} + 3 \beta_{2} + 3 \beta_1 + 14) q^{21} + ( - 3 \beta_{2} + 8 \beta_1 + 11) q^{23} + (4 \beta_{3} - 24 \beta_1 + 24) q^{27} + (4 \beta_{3} + 4 \beta_{2} + 20 \beta_1) q^{29} + (\beta_{3} - \beta_{2} - \beta_1 + 10) q^{31} + ( - 6 \beta_{2} + 20 \beta_1 + 26) q^{33} + (8 \beta_{3} - 19 \beta_1 + 19) q^{37} + (7 \beta_{3} + 7 \beta_{2} + 33 \beta_1) q^{39} + (9 \beta_{3} - 9 \beta_{2} - 9 \beta_1 + 22) q^{41} + (\beta_{2} + 24 \beta_1 + 23) q^{43} + ( - 5 \beta_{3} + 29 \beta_1 - 29) q^{47} + ( - 7 \beta_{3} - 7 \beta_{2} + 4 \beta_1) q^{49} + (5 \beta_{3} - 5 \beta_{2} - 5 \beta_1 + 50) q^{51} + (3 \beta_1 + 3) q^{53} + ( - 8 \beta_{3} - 32 \beta_1 + 32) q^{57} + ( - 2 \beta_{3} - 2 \beta_{2} + 54 \beta_1) q^{59} + (\beta_{3} - \beta_{2} - \beta_1 - 50) q^{61} + (5 \beta_{2} + 24 \beta_1 + 19) q^{63} + ( - \beta_{3} - 31 \beta_1 + 31) q^{67} + (11 \beta_{3} + 11 \beta_{2} - 71 \beta_1) q^{69} + (\beta_{3} - \beta_{2} - \beta_1 - 38) q^{71} + (8 \beta_{2} - 33 \beta_1 - 41) q^{73} + ( - 2 \beta_{3} - 2 \beta_1 + 2) q^{77} + ( - 16 \beta_{3} - 16 \beta_{2} + 16 \beta_1) q^{79} + ( - 15 \beta_{3} + 15 \beta_{2} + 15 \beta_1 - 11) q^{81} + ( - 11 \beta_{2} - 52 \beta_1 - 41) q^{83} + (16 \beta_{3} + 64 \beta_1 - 64) q^{87} + ( - 8 \beta_{3} - 8 \beta_{2} - 40 \beta_1) q^{89} + ( - 15 \beta_{3} + 15 \beta_{2} + 15 \beta_1 + 82) q^{91} + (10 \beta_{2} - 20 \beta_1 - 30) q^{93} + (49 \beta_1 - 49) q^{97} + (17 \beta_{3} + 17 \beta_{2} - 101 \beta_1) q^{99}+O(q^{100})$$ q + (b2 - 1) * q^3 + (-b3 - 3*b1 + 3) * q^7 + (-b3 - b2 + 12*b1) * q^9 + (-b3 + b2 + b1 - 6) * q^11 + (2*b2 + 9*b1 + 7) * q^13 + (-2*b3 + 5*b1 - 5) * q^17 + (-2*b3 - 2*b2 - 10*b1) * q^19 + (-3*b3 + 3*b2 + 3*b1 + 14) * q^21 + (-3*b2 + 8*b1 + 11) * q^23 + (4*b3 - 24*b1 + 24) * q^27 + (4*b3 + 4*b2 + 20*b1) * q^29 + (b3 - b2 - b1 + 10) * q^31 + (-6*b2 + 20*b1 + 26) * q^33 + (8*b3 - 19*b1 + 19) * q^37 + (7*b3 + 7*b2 + 33*b1) * q^39 + (9*b3 - 9*b2 - 9*b1 + 22) * q^41 + (b2 + 24*b1 + 23) * q^43 + (-5*b3 + 29*b1 - 29) * q^47 + (-7*b3 - 7*b2 + 4*b1) * q^49 + (5*b3 - 5*b2 - 5*b1 + 50) * q^51 + (3*b1 + 3) * q^53 + (-8*b3 - 32*b1 + 32) * q^57 + (-2*b3 - 2*b2 + 54*b1) * q^59 + (b3 - b2 - b1 - 50) * q^61 + (5*b2 + 24*b1 + 19) * q^63 + (-b3 - 31*b1 + 31) * q^67 + (11*b3 + 11*b2 - 71*b1) * q^69 + (b3 - b2 - b1 - 38) * q^71 + (8*b2 - 33*b1 - 41) * q^73 + (-2*b3 - 2*b1 + 2) * q^77 + (-16*b3 - 16*b2 + 16*b1) * q^79 + (-15*b3 + 15*b2 + 15*b1 - 11) * q^81 + (-11*b2 - 52*b1 - 41) * q^83 + (16*b3 + 64*b1 - 64) * q^87 + (-8*b3 - 8*b2 - 40*b1) * q^89 + (-15*b3 + 15*b2 + 15*b1 + 82) * q^91 + (10*b2 - 20*b1 - 30) * q^93 + (49*b1 - 49) * q^97 + (17*b3 + 17*b2 - 101*b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 2 q^{3} + 14 q^{7}+O(q^{10})$$ 4 * q - 2 * q^3 + 14 * q^7 $$4 q - 2 q^{3} + 14 q^{7} - 20 q^{11} + 32 q^{13} - 16 q^{17} + 68 q^{21} + 38 q^{23} + 88 q^{27} + 36 q^{31} + 92 q^{33} + 60 q^{37} + 52 q^{41} + 94 q^{43} - 106 q^{47} + 180 q^{51} + 12 q^{53} + 144 q^{57} - 204 q^{61} + 86 q^{63} + 126 q^{67} - 156 q^{71} - 148 q^{73} + 12 q^{77} + 16 q^{81} - 186 q^{83} - 288 q^{87} + 388 q^{91} - 100 q^{93} - 196 q^{97}+O(q^{100})$$ 4 * q - 2 * q^3 + 14 * q^7 - 20 * q^11 + 32 * q^13 - 16 * q^17 + 68 * q^21 + 38 * q^23 + 88 * q^27 + 36 * q^31 + 92 * q^33 + 60 * q^37 + 52 * q^41 + 94 * q^43 - 106 * q^47 + 180 * q^51 + 12 * q^53 + 144 * q^57 - 204 * q^61 + 86 * q^63 + 126 * q^67 - 156 * q^71 - 148 * q^73 + 12 * q^77 + 16 * q^81 - 186 * q^83 - 288 * q^87 + 388 * q^91 - 100 * q^93 - 196 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 21x^{2} + 100$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{3} + 11\nu ) / 10$$ (v^3 + 11*v) / 10 $$\beta_{2}$$ $$=$$ $$\nu^{2} + \nu + 11$$ v^2 + v + 11 $$\beta_{3}$$ $$=$$ $$( \nu^{3} - 10\nu^{2} + 21\nu - 110 ) / 10$$ (v^3 - 10*v^2 + 21*v - 110) / 10
 $$\nu$$ $$=$$ $$( \beta_{3} + \beta_{2} - \beta_1 ) / 2$$ (b3 + b2 - b1) / 2 $$\nu^{2}$$ $$=$$ $$( -\beta_{3} + \beta_{2} + \beta _1 - 22 ) / 2$$ (-b3 + b2 + b1 - 22) / 2 $$\nu^{3}$$ $$=$$ $$( -11\beta_{3} - 11\beta_{2} + 31\beta_1 ) / 2$$ (-11*b3 - 11*b2 + 31*b1) / 2

## Character values

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

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

## 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
 − 3.70156i 2.70156i 3.70156i − 2.70156i
0 −3.70156 3.70156i 0 0 0 0.298438 0.298438i 0 18.4031i 0
193.2 0 2.70156 + 2.70156i 0 0 0 6.70156 6.70156i 0 5.59688i 0
257.1 0 −3.70156 + 3.70156i 0 0 0 0.298438 + 0.298438i 0 18.4031i 0
257.2 0 2.70156 2.70156i 0 0 0 6.70156 + 6.70156i 0 5.59688i 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.i 4
4.b odd 2 1 200.3.l.e 4
5.b even 2 1 80.3.p.d 4
5.c odd 4 1 80.3.p.d 4
5.c odd 4 1 inner 400.3.p.i 4
12.b even 2 1 1800.3.v.k 4
15.d odd 2 1 720.3.bh.l 4
15.e even 4 1 720.3.bh.l 4
20.d odd 2 1 40.3.l.b 4
20.e even 4 1 40.3.l.b 4
20.e even 4 1 200.3.l.e 4
40.e odd 2 1 320.3.p.l 4
40.f even 2 1 320.3.p.i 4
40.i odd 4 1 320.3.p.i 4
40.k even 4 1 320.3.p.l 4
60.h even 2 1 360.3.v.c 4
60.l odd 4 1 360.3.v.c 4
60.l odd 4 1 1800.3.v.k 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
40.3.l.b 4 20.d odd 2 1
40.3.l.b 4 20.e even 4 1
80.3.p.d 4 5.b even 2 1
80.3.p.d 4 5.c odd 4 1
200.3.l.e 4 4.b odd 2 1
200.3.l.e 4 20.e even 4 1
320.3.p.i 4 40.f even 2 1
320.3.p.i 4 40.i odd 4 1
320.3.p.l 4 40.e odd 2 1
320.3.p.l 4 40.k even 4 1
360.3.v.c 4 60.h even 2 1
360.3.v.c 4 60.l odd 4 1
400.3.p.i 4 1.a even 1 1 trivial
400.3.p.i 4 5.c odd 4 1 inner
720.3.bh.l 4 15.d odd 2 1
720.3.bh.l 4 15.e even 4 1
1800.3.v.k 4 12.b even 2 1
1800.3.v.k 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 $$S_{3}^{\mathrm{new}}(400, [\chi])$$:

 $$T_{3}^{4} + 2T_{3}^{3} + 2T_{3}^{2} - 40T_{3} + 400$$ T3^4 + 2*T3^3 + 2*T3^2 - 40*T3 + 400 $$T_{7}^{4} - 14T_{7}^{3} + 98T_{7}^{2} - 56T_{7} + 16$$ T7^4 - 14*T7^3 + 98*T7^2 - 56*T7 + 16

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} + 2 T^{3} + 2 T^{2} - 40 T + 400$$
$5$ $$T^{4}$$
$7$ $$T^{4} - 14 T^{3} + 98 T^{2} - 56 T + 16$$
$11$ $$(T^{2} + 10 T - 16)^{2}$$
$13$ $$T^{4} - 32 T^{3} + 512 T^{2} + \cdots + 2116$$
$17$ $$T^{4} + 16 T^{3} + 128 T^{2} + \cdots + 2500$$
$19$ $$T^{4} + 528T^{2} + 4096$$
$23$ $$T^{4} - 38 T^{3} + 722 T^{2} + \cdots + 16$$
$29$ $$T^{4} + 2112 T^{2} + 65536$$
$31$ $$(T^{2} - 18 T + 40)^{2}$$
$37$ $$T^{4} - 60 T^{3} + 1800 T^{2} + \cdots + 743044$$
$41$ $$(T^{2} - 26 T - 3152)^{2}$$
$43$ $$T^{4} - 94 T^{3} + 4418 T^{2} + \cdots + 1175056$$
$47$ $$T^{4} + 106 T^{3} + 5618 T^{2} + \cdots + 795664$$
$53$ $$(T^{2} - 6 T + 18)^{2}$$
$59$ $$T^{4} + 6160 T^{2} + \cdots + 7573504$$
$61$ $$(T^{2} + 102 T + 2560)^{2}$$
$67$ $$T^{4} - 126 T^{3} + 7938 T^{2} + \cdots + 3857296$$
$71$ $$(T^{2} + 78 T + 1480)^{2}$$
$73$ $$T^{4} + 148 T^{3} + 10952 T^{2} + \cdots + 2033476$$
$79$ $$T^{4} + 21504 T^{2} + \cdots + 104857600$$
$83$ $$T^{4} + 186 T^{3} + 17298 T^{2} + \cdots + 3400336$$
$89$ $$T^{4} + 8448 T^{2} + \cdots + 1048576$$
$97$ $$(T^{2} + 98 T + 4802)^{2}$$