# Properties

 Label 400.4.c.c 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: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 10) 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 $$\beta = 2i$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 4 \beta q^{3} - 2 \beta q^{7} - 37 q^{9} +O(q^{10})$$ q + 4*b * q^3 - 2*b * q^7 - 37 * q^9 $$q + 4 \beta q^{3} - 2 \beta q^{7} - 37 q^{9} - 12 q^{11} - 29 \beta q^{13} - 33 \beta q^{17} - 100 q^{19} + 32 q^{21} - 66 \beta q^{23} - 40 \beta q^{27} + 90 q^{29} - 152 q^{31} - 48 \beta q^{33} + 17 \beta q^{37} + 464 q^{39} - 438 q^{41} - 16 \beta q^{43} - 102 \beta q^{47} + 327 q^{49} + 528 q^{51} + 111 \beta q^{53} - 400 \beta q^{57} + 420 q^{59} + 902 q^{61} + 74 \beta q^{63} - 512 \beta q^{67} + 1056 q^{69} - 432 q^{71} + 181 \beta q^{73} + 24 \beta q^{77} - 160 q^{79} - 359 q^{81} - 36 \beta q^{83} + 360 \beta q^{87} - 810 q^{89} - 232 q^{91} - 608 \beta q^{93} - 553 \beta q^{97} + 444 q^{99} +O(q^{100})$$ q + 4*b * q^3 - 2*b * q^7 - 37 * q^9 - 12 * q^11 - 29*b * q^13 - 33*b * q^17 - 100 * q^19 + 32 * q^21 - 66*b * q^23 - 40*b * q^27 + 90 * q^29 - 152 * q^31 - 48*b * q^33 + 17*b * q^37 + 464 * q^39 - 438 * q^41 - 16*b * q^43 - 102*b * q^47 + 327 * q^49 + 528 * q^51 + 111*b * q^53 - 400*b * q^57 + 420 * q^59 + 902 * q^61 + 74*b * q^63 - 512*b * q^67 + 1056 * q^69 - 432 * q^71 + 181*b * q^73 + 24*b * q^77 - 160 * q^79 - 359 * q^81 - 36*b * q^83 + 360*b * q^87 - 810 * q^89 - 232 * q^91 - 608*b * q^93 - 553*b * q^97 + 444 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 74 q^{9}+O(q^{10})$$ 2 * q - 74 * q^9 $$2 q - 74 q^{9} - 24 q^{11} - 200 q^{19} + 64 q^{21} + 180 q^{29} - 304 q^{31} + 928 q^{39} - 876 q^{41} + 654 q^{49} + 1056 q^{51} + 840 q^{59} + 1804 q^{61} + 2112 q^{69} - 864 q^{71} - 320 q^{79} - 718 q^{81} - 1620 q^{89} - 464 q^{91} + 888 q^{99}+O(q^{100})$$ 2 * q - 74 * q^9 - 24 * q^11 - 200 * q^19 + 64 * q^21 + 180 * q^29 - 304 * q^31 + 928 * q^39 - 876 * q^41 + 654 * q^49 + 1056 * q^51 + 840 * q^59 + 1804 * q^61 + 2112 * q^69 - 864 * q^71 - 320 * q^79 - 718 * q^81 - 1620 * q^89 - 464 * q^91 + 888 * 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 8.00000i 0 0 0 4.00000i 0 −37.0000 0
49.2 0 8.00000i 0 0 0 4.00000i 0 −37.0000 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.c 2
4.b odd 2 1 50.4.b.a 2
5.b even 2 1 inner 400.4.c.c 2
5.c odd 4 1 80.4.a.f 1
5.c odd 4 1 400.4.a.b 1
12.b even 2 1 450.4.c.d 2
15.e even 4 1 720.4.a.j 1
20.d odd 2 1 50.4.b.a 2
20.e even 4 1 10.4.a.a 1
20.e even 4 1 50.4.a.c 1
40.i odd 4 1 320.4.a.b 1
40.i odd 4 1 1600.4.a.bx 1
40.k even 4 1 320.4.a.m 1
40.k even 4 1 1600.4.a.d 1
60.h even 2 1 450.4.c.d 2
60.l odd 4 1 90.4.a.a 1
60.l odd 4 1 450.4.a.q 1
80.i odd 4 1 1280.4.d.g 2
80.j even 4 1 1280.4.d.j 2
80.s even 4 1 1280.4.d.j 2
80.t odd 4 1 1280.4.d.g 2
140.j odd 4 1 490.4.a.o 1
140.j odd 4 1 2450.4.a.b 1
140.w even 12 2 490.4.e.i 2
140.x odd 12 2 490.4.e.a 2
180.v odd 12 2 810.4.e.w 2
180.x even 12 2 810.4.e.c 2
220.i odd 4 1 1210.4.a.b 1
260.p even 4 1 1690.4.a.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.4.a.a 1 20.e even 4 1
50.4.a.c 1 20.e even 4 1
50.4.b.a 2 4.b odd 2 1
50.4.b.a 2 20.d odd 2 1
80.4.a.f 1 5.c odd 4 1
90.4.a.a 1 60.l odd 4 1
320.4.a.b 1 40.i odd 4 1
320.4.a.m 1 40.k even 4 1
400.4.a.b 1 5.c odd 4 1
400.4.c.c 2 1.a even 1 1 trivial
400.4.c.c 2 5.b even 2 1 inner
450.4.a.q 1 60.l odd 4 1
450.4.c.d 2 12.b even 2 1
450.4.c.d 2 60.h even 2 1
490.4.a.o 1 140.j odd 4 1
490.4.e.a 2 140.x odd 12 2
490.4.e.i 2 140.w even 12 2
720.4.a.j 1 15.e even 4 1
810.4.e.c 2 180.x even 12 2
810.4.e.w 2 180.v odd 12 2
1210.4.a.b 1 220.i odd 4 1
1280.4.d.g 2 80.i odd 4 1
1280.4.d.g 2 80.t odd 4 1
1280.4.d.j 2 80.j even 4 1
1280.4.d.j 2 80.s even 4 1
1600.4.a.d 1 40.k even 4 1
1600.4.a.bx 1 40.i odd 4 1
1690.4.a.a 1 260.p even 4 1
2450.4.a.b 1 140.j odd 4 1

## Hecke kernels

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

 $$T_{3}^{2} + 64$$ T3^2 + 64 $$T_{7}^{2} + 16$$ T7^2 + 16 $$T_{11} + 12$$ T11 + 12

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 64$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 16$$
$11$ $$(T + 12)^{2}$$
$13$ $$T^{2} + 3364$$
$17$ $$T^{2} + 4356$$
$19$ $$(T + 100)^{2}$$
$23$ $$T^{2} + 17424$$
$29$ $$(T - 90)^{2}$$
$31$ $$(T + 152)^{2}$$
$37$ $$T^{2} + 1156$$
$41$ $$(T + 438)^{2}$$
$43$ $$T^{2} + 1024$$
$47$ $$T^{2} + 41616$$
$53$ $$T^{2} + 49284$$
$59$ $$(T - 420)^{2}$$
$61$ $$(T - 902)^{2}$$
$67$ $$T^{2} + 1048576$$
$71$ $$(T + 432)^{2}$$
$73$ $$T^{2} + 131044$$
$79$ $$(T + 160)^{2}$$
$83$ $$T^{2} + 5184$$
$89$ $$(T + 810)^{2}$$
$97$ $$T^{2} + 1223236$$