# Properties

 Label 400.4.c.h 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 40) 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.

 $$f(q)$$ $$=$$ $$q + 2 \beta q^{3} - 8 \beta q^{7} + 11 q^{9} +O(q^{10})$$ q + 2*b * q^3 - 8*b * q^7 + 11 * q^9 $$q + 2 \beta q^{3} - 8 \beta q^{7} + 11 q^{9} - 36 q^{11} + 21 \beta q^{13} - 55 \beta q^{17} - 116 q^{19} + 64 q^{21} + 8 \beta q^{23} + 76 \beta q^{27} - 198 q^{29} - 240 q^{31} - 72 \beta q^{33} - 129 \beta q^{37} - 168 q^{39} + 442 q^{41} - 146 \beta q^{43} - 196 \beta q^{47} + 87 q^{49} + 440 q^{51} - 71 \beta q^{53} - 232 \beta q^{57} - 348 q^{59} - 570 q^{61} - 88 \beta q^{63} - 346 \beta q^{67} - 64 q^{69} - 168 q^{71} + 67 \beta q^{73} + 288 \beta q^{77} + 784 q^{79} - 311 q^{81} + 282 \beta q^{83} - 396 \beta q^{87} - 1034 q^{89} + 672 q^{91} - 480 \beta q^{93} - 191 \beta q^{97} - 396 q^{99} +O(q^{100})$$ q + 2*b * q^3 - 8*b * q^7 + 11 * q^9 - 36 * q^11 + 21*b * q^13 - 55*b * q^17 - 116 * q^19 + 64 * q^21 + 8*b * q^23 + 76*b * q^27 - 198 * q^29 - 240 * q^31 - 72*b * q^33 - 129*b * q^37 - 168 * q^39 + 442 * q^41 - 146*b * q^43 - 196*b * q^47 + 87 * q^49 + 440 * q^51 - 71*b * q^53 - 232*b * q^57 - 348 * q^59 - 570 * q^61 - 88*b * q^63 - 346*b * q^67 - 64 * q^69 - 168 * q^71 + 67*b * q^73 + 288*b * q^77 + 784 * q^79 - 311 * q^81 + 282*b * q^83 - 396*b * q^87 - 1034 * q^89 + 672 * q^91 - 480*b * q^93 - 191*b * q^97 - 396 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 22 q^{9}+O(q^{10})$$ 2 * q + 22 * q^9 $$2 q + 22 q^{9} - 72 q^{11} - 232 q^{19} + 128 q^{21} - 396 q^{29} - 480 q^{31} - 336 q^{39} + 884 q^{41} + 174 q^{49} + 880 q^{51} - 696 q^{59} - 1140 q^{61} - 128 q^{69} - 336 q^{71} + 1568 q^{79} - 622 q^{81} - 2068 q^{89} + 1344 q^{91} - 792 q^{99}+O(q^{100})$$ 2 * q + 22 * q^9 - 72 * q^11 - 232 * q^19 + 128 * q^21 - 396 * q^29 - 480 * q^31 - 336 * q^39 + 884 * q^41 + 174 * q^49 + 880 * q^51 - 696 * q^59 - 1140 * q^61 - 128 * q^69 - 336 * q^71 + 1568 * q^79 - 622 * q^81 - 2068 * q^89 + 1344 * q^91 - 792 * 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 4.00000i 0 0 0 16.0000i 0 11.0000 0
49.2 0 4.00000i 0 0 0 16.0000i 0 11.0000 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.h 2
4.b odd 2 1 200.4.c.f 2
5.b even 2 1 inner 400.4.c.h 2
5.c odd 4 1 80.4.a.b 1
5.c odd 4 1 400.4.a.p 1
12.b even 2 1 1800.4.f.d 2
15.e even 4 1 720.4.a.d 1
20.d odd 2 1 200.4.c.f 2
20.e even 4 1 40.4.a.b 1
20.e even 4 1 200.4.a.d 1
40.i odd 4 1 320.4.a.j 1
40.i odd 4 1 1600.4.a.q 1
40.k even 4 1 320.4.a.e 1
40.k even 4 1 1600.4.a.bk 1
60.h even 2 1 1800.4.f.d 2
60.l odd 4 1 360.4.a.f 1
60.l odd 4 1 1800.4.a.h 1
80.i odd 4 1 1280.4.d.m 2
80.j even 4 1 1280.4.d.d 2
80.s even 4 1 1280.4.d.d 2
80.t odd 4 1 1280.4.d.m 2
140.j odd 4 1 1960.4.a.e 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
40.4.a.b 1 20.e even 4 1
80.4.a.b 1 5.c odd 4 1
200.4.a.d 1 20.e even 4 1
200.4.c.f 2 4.b odd 2 1
200.4.c.f 2 20.d odd 2 1
320.4.a.e 1 40.k even 4 1
320.4.a.j 1 40.i odd 4 1
360.4.a.f 1 60.l odd 4 1
400.4.a.p 1 5.c odd 4 1
400.4.c.h 2 1.a even 1 1 trivial
400.4.c.h 2 5.b even 2 1 inner
720.4.a.d 1 15.e even 4 1
1280.4.d.d 2 80.j even 4 1
1280.4.d.d 2 80.s even 4 1
1280.4.d.m 2 80.i odd 4 1
1280.4.d.m 2 80.t odd 4 1
1600.4.a.q 1 40.i odd 4 1
1600.4.a.bk 1 40.k even 4 1
1800.4.a.h 1 60.l odd 4 1
1800.4.f.d 2 12.b even 2 1
1800.4.f.d 2 60.h even 2 1
1960.4.a.e 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 $$S_{4}^{\mathrm{new}}(400, [\chi])$$:

 $$T_{3}^{2} + 16$$ T3^2 + 16 $$T_{7}^{2} + 256$$ T7^2 + 256 $$T_{11} + 36$$ T11 + 36

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 16$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 256$$
$11$ $$(T + 36)^{2}$$
$13$ $$T^{2} + 1764$$
$17$ $$T^{2} + 12100$$
$19$ $$(T + 116)^{2}$$
$23$ $$T^{2} + 256$$
$29$ $$(T + 198)^{2}$$
$31$ $$(T + 240)^{2}$$
$37$ $$T^{2} + 66564$$
$41$ $$(T - 442)^{2}$$
$43$ $$T^{2} + 85264$$
$47$ $$T^{2} + 153664$$
$53$ $$T^{2} + 20164$$
$59$ $$(T + 348)^{2}$$
$61$ $$(T + 570)^{2}$$
$67$ $$T^{2} + 478864$$
$71$ $$(T + 168)^{2}$$
$73$ $$T^{2} + 17956$$
$79$ $$(T - 784)^{2}$$
$83$ $$T^{2} + 318096$$
$89$ $$(T + 1034)^{2}$$
$97$ $$T^{2} + 145924$$