# Properties

 Label 400.4.c.e 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 25) 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.

 $$f(q)$$ $$=$$ $$q + 7 i q^{3} - 6 i q^{7} - 22 q^{9} +O(q^{10})$$ q + 7*i * q^3 - 6*i * q^7 - 22 * q^9 $$q + 7 i q^{3} - 6 i q^{7} - 22 q^{9} + 43 q^{11} + 28 i q^{13} + 91 i q^{17} - 35 q^{19} + 42 q^{21} + 162 i q^{23} + 35 i q^{27} - 160 q^{29} - 42 q^{31} + 301 i q^{33} - 314 i q^{37} - 196 q^{39} - 203 q^{41} + 92 i q^{43} - 196 i q^{47} + 307 q^{49} - 637 q^{51} - 82 i q^{53} - 245 i q^{57} - 280 q^{59} - 518 q^{61} + 132 i q^{63} - 141 i q^{67} - 1134 q^{69} - 412 q^{71} + 763 i q^{73} - 258 i q^{77} + 510 q^{79} - 839 q^{81} + 777 i q^{83} - 1120 i q^{87} + 945 q^{89} + 168 q^{91} - 294 i q^{93} + 1246 i q^{97} - 946 q^{99} +O(q^{100})$$ q + 7*i * q^3 - 6*i * q^7 - 22 * q^9 + 43 * q^11 + 28*i * q^13 + 91*i * q^17 - 35 * q^19 + 42 * q^21 + 162*i * q^23 + 35*i * q^27 - 160 * q^29 - 42 * q^31 + 301*i * q^33 - 314*i * q^37 - 196 * q^39 - 203 * q^41 + 92*i * q^43 - 196*i * q^47 + 307 * q^49 - 637 * q^51 - 82*i * q^53 - 245*i * q^57 - 280 * q^59 - 518 * q^61 + 132*i * q^63 - 141*i * q^67 - 1134 * q^69 - 412 * q^71 + 763*i * q^73 - 258*i * q^77 + 510 * q^79 - 839 * q^81 + 777*i * q^83 - 1120*i * q^87 + 945 * q^89 + 168 * q^91 - 294*i * q^93 + 1246*i * q^97 - 946 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 44 q^{9}+O(q^{10})$$ 2 * q - 44 * q^9 $$2 q - 44 q^{9} + 86 q^{11} - 70 q^{19} + 84 q^{21} - 320 q^{29} - 84 q^{31} - 392 q^{39} - 406 q^{41} + 614 q^{49} - 1274 q^{51} - 560 q^{59} - 1036 q^{61} - 2268 q^{69} - 824 q^{71} + 1020 q^{79} - 1678 q^{81} + 1890 q^{89} + 336 q^{91} - 1892 q^{99}+O(q^{100})$$ 2 * q - 44 * q^9 + 86 * q^11 - 70 * q^19 + 84 * q^21 - 320 * q^29 - 84 * q^31 - 392 * q^39 - 406 * q^41 + 614 * q^49 - 1274 * q^51 - 560 * q^59 - 1036 * q^61 - 2268 * q^69 - 824 * q^71 + 1020 * q^79 - 1678 * q^81 + 1890 * q^89 + 336 * q^91 - 1892 * 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 7.00000i 0 0 0 6.00000i 0 −22.0000 0
49.2 0 7.00000i 0 0 0 6.00000i 0 −22.0000 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.e 2
4.b odd 2 1 25.4.b.b 2
5.b even 2 1 inner 400.4.c.e 2
5.c odd 4 1 400.4.a.c 1
5.c odd 4 1 400.4.a.s 1
12.b even 2 1 225.4.b.f 2
20.d odd 2 1 25.4.b.b 2
20.e even 4 1 25.4.a.a 1
20.e even 4 1 25.4.a.b yes 1
40.i odd 4 1 1600.4.a.h 1
40.i odd 4 1 1600.4.a.bs 1
40.k even 4 1 1600.4.a.i 1
40.k even 4 1 1600.4.a.bt 1
60.h even 2 1 225.4.b.f 2
60.l odd 4 1 225.4.a.c 1
60.l odd 4 1 225.4.a.e 1
140.j odd 4 1 1225.4.a.h 1
140.j odd 4 1 1225.4.a.i 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
25.4.a.a 1 20.e even 4 1
25.4.a.b yes 1 20.e even 4 1
25.4.b.b 2 4.b odd 2 1
25.4.b.b 2 20.d odd 2 1
225.4.a.c 1 60.l odd 4 1
225.4.a.e 1 60.l odd 4 1
225.4.b.f 2 12.b even 2 1
225.4.b.f 2 60.h even 2 1
400.4.a.c 1 5.c odd 4 1
400.4.a.s 1 5.c odd 4 1
400.4.c.e 2 1.a even 1 1 trivial
400.4.c.e 2 5.b even 2 1 inner
1225.4.a.h 1 140.j odd 4 1
1225.4.a.i 1 140.j odd 4 1
1600.4.a.h 1 40.i odd 4 1
1600.4.a.i 1 40.k even 4 1
1600.4.a.bs 1 40.i odd 4 1
1600.4.a.bt 1 40.k even 4 1

## Hecke kernels

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

 $$T_{3}^{2} + 49$$ T3^2 + 49 $$T_{7}^{2} + 36$$ T7^2 + 36 $$T_{11} - 43$$ T11 - 43

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 49$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 36$$
$11$ $$(T - 43)^{2}$$
$13$ $$T^{2} + 784$$
$17$ $$T^{2} + 8281$$
$19$ $$(T + 35)^{2}$$
$23$ $$T^{2} + 26244$$
$29$ $$(T + 160)^{2}$$
$31$ $$(T + 42)^{2}$$
$37$ $$T^{2} + 98596$$
$41$ $$(T + 203)^{2}$$
$43$ $$T^{2} + 8464$$
$47$ $$T^{2} + 38416$$
$53$ $$T^{2} + 6724$$
$59$ $$(T + 280)^{2}$$
$61$ $$(T + 518)^{2}$$
$67$ $$T^{2} + 19881$$
$71$ $$(T + 412)^{2}$$
$73$ $$T^{2} + 582169$$
$79$ $$(T - 510)^{2}$$
$83$ $$T^{2} + 603729$$
$89$ $$(T - 945)^{2}$$
$97$ $$T^{2} + 1552516$$