# Properties

 Label 400.4.c.b Level $400$ Weight $4$ Character orbit 400.c Analytic conductor $23.601$ 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] = [400,4,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.

 $$f(q)$$ $$=$$ $$q + 9 i q^{3} - 26 i q^{7} - 54 q^{9} +O(q^{10})$$ q + 9*i * q^3 - 26*i * q^7 - 54 * q^9 $$q + 9 i q^{3} - 26 i q^{7} - 54 q^{9} + 59 q^{11} - 28 i q^{13} + 5 i q^{17} + 109 q^{19} + 234 q^{21} - 194 i q^{23} - 243 i q^{27} + 32 q^{29} - 10 q^{31} + 531 i q^{33} - 198 i q^{37} + 252 q^{39} + 117 q^{41} + 388 i q^{43} + 68 i q^{47} - 333 q^{49} - 45 q^{51} + 18 i q^{53} + 981 i q^{57} + 392 q^{59} - 710 q^{61} + 1404 i q^{63} + 253 i q^{67} + 1746 q^{69} + 612 q^{71} + 549 i q^{73} - 1534 i q^{77} + 414 q^{79} + 729 q^{81} - 121 i q^{83} + 288 i q^{87} + 81 q^{89} - 728 q^{91} - 90 i q^{93} - 1502 i q^{97} - 3186 q^{99} +O(q^{100})$$ q + 9*i * q^3 - 26*i * q^7 - 54 * q^9 + 59 * q^11 - 28*i * q^13 + 5*i * q^17 + 109 * q^19 + 234 * q^21 - 194*i * q^23 - 243*i * q^27 + 32 * q^29 - 10 * q^31 + 531*i * q^33 - 198*i * q^37 + 252 * q^39 + 117 * q^41 + 388*i * q^43 + 68*i * q^47 - 333 * q^49 - 45 * q^51 + 18*i * q^53 + 981*i * q^57 + 392 * q^59 - 710 * q^61 + 1404*i * q^63 + 253*i * q^67 + 1746 * q^69 + 612 * q^71 + 549*i * q^73 - 1534*i * q^77 + 414 * q^79 + 729 * q^81 - 121*i * q^83 + 288*i * q^87 + 81 * q^89 - 728 * q^91 - 90*i * q^93 - 1502*i * q^97 - 3186 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 108 q^{9}+O(q^{10})$$ 2 * q - 108 * q^9 $$2 q - 108 q^{9} + 118 q^{11} + 218 q^{19} + 468 q^{21} + 64 q^{29} - 20 q^{31} + 504 q^{39} + 234 q^{41} - 666 q^{49} - 90 q^{51} + 784 q^{59} - 1420 q^{61} + 3492 q^{69} + 1224 q^{71} + 828 q^{79} + 1458 q^{81} + 162 q^{89} - 1456 q^{91} - 6372 q^{99}+O(q^{100})$$ 2 * q - 108 * q^9 + 118 * q^11 + 218 * q^19 + 468 * q^21 + 64 * q^29 - 20 * q^31 + 504 * q^39 + 234 * q^41 - 666 * q^49 - 90 * q^51 + 784 * q^59 - 1420 * q^61 + 3492 * q^69 + 1224 * q^71 + 828 * q^79 + 1458 * q^81 + 162 * q^89 - 1456 * q^91 - 6372 * 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 9.00000i 0 0 0 26.0000i 0 −54.0000 0
49.2 0 9.00000i 0 0 0 26.0000i 0 −54.0000 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.b 2
4.b odd 2 1 200.4.c.b 2
5.b even 2 1 inner 400.4.c.b 2
5.c odd 4 1 400.4.a.a 1
5.c odd 4 1 400.4.a.t 1
12.b even 2 1 1800.4.f.w 2
20.d odd 2 1 200.4.c.b 2
20.e even 4 1 200.4.a.b 1
20.e even 4 1 200.4.a.j yes 1
40.i odd 4 1 1600.4.a.b 1
40.i odd 4 1 1600.4.a.by 1
40.k even 4 1 1600.4.a.c 1
40.k even 4 1 1600.4.a.bz 1
60.h even 2 1 1800.4.f.w 2
60.l odd 4 1 1800.4.a.c 1
60.l odd 4 1 1800.4.a.bh 1

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

 $$T_{3}^{2} + 81$$ T3^2 + 81 $$T_{7}^{2} + 676$$ T7^2 + 676 $$T_{11} - 59$$ T11 - 59

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 81$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 676$$
$11$ $$(T - 59)^{2}$$
$13$ $$T^{2} + 784$$
$17$ $$T^{2} + 25$$
$19$ $$(T - 109)^{2}$$
$23$ $$T^{2} + 37636$$
$29$ $$(T - 32)^{2}$$
$31$ $$(T + 10)^{2}$$
$37$ $$T^{2} + 39204$$
$41$ $$(T - 117)^{2}$$
$43$ $$T^{2} + 150544$$
$47$ $$T^{2} + 4624$$
$53$ $$T^{2} + 324$$
$59$ $$(T - 392)^{2}$$
$61$ $$(T + 710)^{2}$$
$67$ $$T^{2} + 64009$$
$71$ $$(T - 612)^{2}$$
$73$ $$T^{2} + 301401$$
$79$ $$(T - 414)^{2}$$
$83$ $$T^{2} + 14641$$
$89$ $$(T - 81)^{2}$$
$97$ $$T^{2} + 2256004$$