# Properties

 Label 400.4.c.g 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: $$\Z[a_1, \ldots, a_{7}]$$ 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.

 $$f(q)$$ $$=$$ $$q + 5 i q^{3} - 2 i q^{7} + 2 q^{9} +O(q^{10})$$ q + 5*i * q^3 - 2*i * q^7 + 2 * q^9 $$q + 5 i q^{3} - 2 i q^{7} + 2 q^{9} - 39 q^{11} - 84 i q^{13} - 61 i q^{17} + 151 q^{19} + 10 q^{21} - 58 i q^{23} + 145 i q^{27} - 192 q^{29} + 18 q^{31} - 195 i q^{33} - 138 i q^{37} + 420 q^{39} + 229 q^{41} - 164 i q^{43} + 212 i q^{47} + 339 q^{49} + 305 q^{51} - 578 i q^{53} + 755 i q^{57} - 336 q^{59} + 858 q^{61} - 4 i q^{63} + 209 i q^{67} + 290 q^{69} + 780 q^{71} + 403 i q^{73} + 78 i q^{77} - 230 q^{79} - 671 q^{81} - 1293 i q^{83} - 960 i q^{87} + 1369 q^{89} - 168 q^{91} + 90 i q^{93} + 382 i q^{97} - 78 q^{99} +O(q^{100})$$ q + 5*i * q^3 - 2*i * q^7 + 2 * q^9 - 39 * q^11 - 84*i * q^13 - 61*i * q^17 + 151 * q^19 + 10 * q^21 - 58*i * q^23 + 145*i * q^27 - 192 * q^29 + 18 * q^31 - 195*i * q^33 - 138*i * q^37 + 420 * q^39 + 229 * q^41 - 164*i * q^43 + 212*i * q^47 + 339 * q^49 + 305 * q^51 - 578*i * q^53 + 755*i * q^57 - 336 * q^59 + 858 * q^61 - 4*i * q^63 + 209*i * q^67 + 290 * q^69 + 780 * q^71 + 403*i * q^73 + 78*i * q^77 - 230 * q^79 - 671 * q^81 - 1293*i * q^83 - 960*i * q^87 + 1369 * q^89 - 168 * q^91 + 90*i * q^93 + 382*i * q^97 - 78 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 4 q^{9}+O(q^{10})$$ 2 * q + 4 * q^9 $$2 q + 4 q^{9} - 78 q^{11} + 302 q^{19} + 20 q^{21} - 384 q^{29} + 36 q^{31} + 840 q^{39} + 458 q^{41} + 678 q^{49} + 610 q^{51} - 672 q^{59} + 1716 q^{61} + 580 q^{69} + 1560 q^{71} - 460 q^{79} - 1342 q^{81} + 2738 q^{89} - 336 q^{91} - 156 q^{99}+O(q^{100})$$ 2 * q + 4 * q^9 - 78 * q^11 + 302 * q^19 + 20 * q^21 - 384 * q^29 + 36 * q^31 + 840 * q^39 + 458 * q^41 + 678 * q^49 + 610 * q^51 - 672 * q^59 + 1716 * q^61 + 580 * q^69 + 1560 * q^71 - 460 * q^79 - 1342 * q^81 + 2738 * q^89 - 336 * q^91 - 156 * 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 5.00000i 0 0 0 2.00000i 0 2.00000 0
49.2 0 5.00000i 0 0 0 2.00000i 0 2.00000 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.g 2
4.b odd 2 1 200.4.c.d 2
5.b even 2 1 inner 400.4.c.g 2
5.c odd 4 1 400.4.a.f 1
5.c odd 4 1 400.4.a.q 1
12.b even 2 1 1800.4.f.c 2
20.d odd 2 1 200.4.c.d 2
20.e even 4 1 200.4.a.c 1
20.e even 4 1 200.4.a.h yes 1
40.i odd 4 1 1600.4.a.n 1
40.i odd 4 1 1600.4.a.bo 1
40.k even 4 1 1600.4.a.m 1
40.k even 4 1 1600.4.a.bn 1
60.h even 2 1 1800.4.f.c 2
60.l odd 4 1 1800.4.a.p 1
60.l odd 4 1 1800.4.a.t 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
200.4.a.c 1 20.e even 4 1
200.4.a.h yes 1 20.e even 4 1
200.4.c.d 2 4.b odd 2 1
200.4.c.d 2 20.d odd 2 1
400.4.a.f 1 5.c odd 4 1
400.4.a.q 1 5.c odd 4 1
400.4.c.g 2 1.a even 1 1 trivial
400.4.c.g 2 5.b even 2 1 inner
1600.4.a.m 1 40.k even 4 1
1600.4.a.n 1 40.i odd 4 1
1600.4.a.bn 1 40.k even 4 1
1600.4.a.bo 1 40.i odd 4 1
1800.4.a.p 1 60.l odd 4 1
1800.4.a.t 1 60.l odd 4 1
1800.4.f.c 2 12.b even 2 1
1800.4.f.c 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 $$S_{4}^{\mathrm{new}}(400, [\chi])$$:

 $$T_{3}^{2} + 25$$ T3^2 + 25 $$T_{7}^{2} + 4$$ T7^2 + 4 $$T_{11} + 39$$ T11 + 39

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 25$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 4$$
$11$ $$(T + 39)^{2}$$
$13$ $$T^{2} + 7056$$
$17$ $$T^{2} + 3721$$
$19$ $$(T - 151)^{2}$$
$23$ $$T^{2} + 3364$$
$29$ $$(T + 192)^{2}$$
$31$ $$(T - 18)^{2}$$
$37$ $$T^{2} + 19044$$
$41$ $$(T - 229)^{2}$$
$43$ $$T^{2} + 26896$$
$47$ $$T^{2} + 44944$$
$53$ $$T^{2} + 334084$$
$59$ $$(T + 336)^{2}$$
$61$ $$(T - 858)^{2}$$
$67$ $$T^{2} + 43681$$
$71$ $$(T - 780)^{2}$$
$73$ $$T^{2} + 162409$$
$79$ $$(T + 230)^{2}$$
$83$ $$T^{2} + 1671849$$
$89$ $$(T - 1369)^{2}$$
$97$ $$T^{2} + 145924$$