# Properties

 Label 400.4.c.m Level $400$ Weight $4$ Character orbit 400.c Analytic conductor $23.601$ Analytic rank $0$ Dimension $2$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [400,4,Mod(49,400)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(400, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("400.49");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$400 = 2^{4} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 400.c (of order $$2$$, degree $$1$$, 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: $$23.6007640023$$ 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 200) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + i q^{3} + 6 i q^{7} + 26 q^{9}+O(q^{10})$$ q + i * q^3 + 6*i * q^7 + 26 * q^9 $$q + i q^{3} + 6 i q^{7} + 26 q^{9} + 19 q^{11} - 12 i q^{13} - 75 i q^{17} - 91 q^{19} - 6 q^{21} + 174 i q^{23} + 53 i q^{27} + 272 q^{29} + 230 q^{31} + 19 i q^{33} - 182 i q^{37} + 12 q^{39} + 117 q^{41} + 372 i q^{43} + 52 i q^{47} + 307 q^{49} + 75 q^{51} + 402 i q^{53} - 91 i q^{57} + 312 q^{59} + 170 q^{61} + 156 i q^{63} - 763 i q^{67} - 174 q^{69} + 52 q^{71} + 981 i q^{73} + 114 i q^{77} + 1054 q^{79} + 649 q^{81} + 351 i q^{83} + 272 i q^{87} - 799 q^{89} + 72 q^{91} + 230 i q^{93} + 962 i q^{97} + 494 q^{99} +O(q^{100})$$ q + i * q^3 + 6*i * q^7 + 26 * q^9 + 19 * q^11 - 12*i * q^13 - 75*i * q^17 - 91 * q^19 - 6 * q^21 + 174*i * q^23 + 53*i * q^27 + 272 * q^29 + 230 * q^31 + 19*i * q^33 - 182*i * q^37 + 12 * q^39 + 117 * q^41 + 372*i * q^43 + 52*i * q^47 + 307 * q^49 + 75 * q^51 + 402*i * q^53 - 91*i * q^57 + 312 * q^59 + 170 * q^61 + 156*i * q^63 - 763*i * q^67 - 174 * q^69 + 52 * q^71 + 981*i * q^73 + 114*i * q^77 + 1054 * q^79 + 649 * q^81 + 351*i * q^83 + 272*i * q^87 - 799 * q^89 + 72 * q^91 + 230*i * q^93 + 962*i * q^97 + 494 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 52 q^{9}+O(q^{10})$$ 2 * q + 52 * q^9 $$2 q + 52 q^{9} + 38 q^{11} - 182 q^{19} - 12 q^{21} + 544 q^{29} + 460 q^{31} + 24 q^{39} + 234 q^{41} + 614 q^{49} + 150 q^{51} + 624 q^{59} + 340 q^{61} - 348 q^{69} + 104 q^{71} + 2108 q^{79} + 1298 q^{81} - 1598 q^{89} + 144 q^{91} + 988 q^{99}+O(q^{100})$$ 2 * q + 52 * q^9 + 38 * q^11 - 182 * q^19 - 12 * q^21 + 544 * q^29 + 460 * q^31 + 24 * q^39 + 234 * q^41 + 614 * q^49 + 150 * q^51 + 624 * q^59 + 340 * q^61 - 348 * q^69 + 104 * q^71 + 2108 * q^79 + 1298 * q^81 - 1598 * q^89 + 144 * q^91 + 988 * q^99

## Character values

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

 $$n$$ $$101$$ $$177$$ $$351$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
49.1
 − 1.00000i 1.00000i
0 1.00000i 0 0 0 6.00000i 0 26.0000 0
49.2 0 1.00000i 0 0 0 6.00000i 0 26.0000 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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 400.4.c.m 2
4.b odd 2 1 200.4.c.g 2
5.b even 2 1 inner 400.4.c.m 2
5.c odd 4 1 400.4.a.j 1
5.c odd 4 1 400.4.a.k 1
12.b even 2 1 1800.4.f.p 2
20.d odd 2 1 200.4.c.g 2
20.e even 4 1 200.4.a.e 1
20.e even 4 1 200.4.a.f yes 1
40.i odd 4 1 1600.4.a.v 1
40.i odd 4 1 1600.4.a.be 1
40.k even 4 1 1600.4.a.w 1
40.k even 4 1 1600.4.a.bf 1
60.h even 2 1 1800.4.f.p 2
60.l odd 4 1 1800.4.a.l 1
60.l odd 4 1 1800.4.a.w 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
200.4.a.e 1 20.e even 4 1
200.4.a.f yes 1 20.e even 4 1
200.4.c.g 2 4.b odd 2 1
200.4.c.g 2 20.d odd 2 1
400.4.a.j 1 5.c odd 4 1
400.4.a.k 1 5.c odd 4 1
400.4.c.m 2 1.a even 1 1 trivial
400.4.c.m 2 5.b even 2 1 inner
1600.4.a.v 1 40.i odd 4 1
1600.4.a.w 1 40.k even 4 1
1600.4.a.be 1 40.i odd 4 1
1600.4.a.bf 1 40.k even 4 1
1800.4.a.l 1 60.l odd 4 1
1800.4.a.w 1 60.l odd 4 1
1800.4.f.p 2 12.b even 2 1
1800.4.f.p 2 60.h even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(400, [\chi])$$:

 $$T_{3}^{2} + 1$$ T3^2 + 1 $$T_{7}^{2} + 36$$ T7^2 + 36 $$T_{11} - 19$$ T11 - 19

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 1$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 36$$
$11$ $$(T - 19)^{2}$$
$13$ $$T^{2} + 144$$
$17$ $$T^{2} + 5625$$
$19$ $$(T + 91)^{2}$$
$23$ $$T^{2} + 30276$$
$29$ $$(T - 272)^{2}$$
$31$ $$(T - 230)^{2}$$
$37$ $$T^{2} + 33124$$
$41$ $$(T - 117)^{2}$$
$43$ $$T^{2} + 138384$$
$47$ $$T^{2} + 2704$$
$53$ $$T^{2} + 161604$$
$59$ $$(T - 312)^{2}$$
$61$ $$(T - 170)^{2}$$
$67$ $$T^{2} + 582169$$
$71$ $$(T - 52)^{2}$$
$73$ $$T^{2} + 962361$$
$79$ $$(T - 1054)^{2}$$
$83$ $$T^{2} + 123201$$
$89$ $$(T + 799)^{2}$$
$97$ $$T^{2} + 925444$$