# Properties

 Label 1100.4.b.c Level $1100$ Weight $4$ Character orbit 1100.b Analytic conductor $64.902$ 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] = [1100,4,Mod(749,1100)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1100, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1100.749");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1100 = 2^{2} \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1100.b (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: $$64.9021010063$$ 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 44) 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.

 $$f(q)$$ $$=$$ $$q + 5 i q^{3} - 26 i q^{7} + 2 q^{9} +O(q^{10})$$ q + 5*i * q^3 - 26*i * q^7 + 2 * q^9 $$q + 5 i q^{3} - 26 i q^{7} + 2 q^{9} - 11 q^{11} - 52 i q^{13} + 46 i q^{17} + 96 q^{19} + 130 q^{21} - 27 i q^{23} + 145 i q^{27} - 16 q^{29} - 293 q^{31} - 55 i q^{33} - 29 i q^{37} + 260 q^{39} - 472 q^{41} + 110 i q^{43} - 224 i q^{47} - 333 q^{49} - 230 q^{51} - 754 i q^{53} + 480 i q^{57} - 825 q^{59} - 548 q^{61} - 52 i q^{63} - 123 i q^{67} + 135 q^{69} + 1001 q^{71} + 1020 i q^{73} + 286 i q^{77} - 526 q^{79} - 671 q^{81} + 158 i q^{83} - 80 i q^{87} + 1217 q^{89} - 1352 q^{91} - 1465 i q^{93} - 263 i q^{97} - 22 q^{99} +O(q^{100})$$ q + 5*i * q^3 - 26*i * q^7 + 2 * q^9 - 11 * q^11 - 52*i * q^13 + 46*i * q^17 + 96 * q^19 + 130 * q^21 - 27*i * q^23 + 145*i * q^27 - 16 * q^29 - 293 * q^31 - 55*i * q^33 - 29*i * q^37 + 260 * q^39 - 472 * q^41 + 110*i * q^43 - 224*i * q^47 - 333 * q^49 - 230 * q^51 - 754*i * q^53 + 480*i * q^57 - 825 * q^59 - 548 * q^61 - 52*i * q^63 - 123*i * q^67 + 135 * q^69 + 1001 * q^71 + 1020*i * q^73 + 286*i * q^77 - 526 * q^79 - 671 * q^81 + 158*i * q^83 - 80*i * q^87 + 1217 * q^89 - 1352 * q^91 - 1465*i * q^93 - 263*i * q^97 - 22 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 4 q^{9}+O(q^{10})$$ 2 * q + 4 * q^9 $$2 q + 4 q^{9} - 22 q^{11} + 192 q^{19} + 260 q^{21} - 32 q^{29} - 586 q^{31} + 520 q^{39} - 944 q^{41} - 666 q^{49} - 460 q^{51} - 1650 q^{59} - 1096 q^{61} + 270 q^{69} + 2002 q^{71} - 1052 q^{79} - 1342 q^{81} + 2434 q^{89} - 2704 q^{91} - 44 q^{99}+O(q^{100})$$ 2 * q + 4 * q^9 - 22 * q^11 + 192 * q^19 + 260 * q^21 - 32 * q^29 - 586 * q^31 + 520 * q^39 - 944 * q^41 - 666 * q^49 - 460 * q^51 - 1650 * q^59 - 1096 * q^61 + 270 * q^69 + 2002 * q^71 - 1052 * q^79 - 1342 * q^81 + 2434 * q^89 - 2704 * q^91 - 44 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1100\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$177$$ $$551$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
749.1
 − 1.00000i 1.00000i
0 5.00000i 0 0 0 26.0000i 0 2.00000 0
749.2 0 5.00000i 0 0 0 26.0000i 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 1100.4.b.c 2
5.b even 2 1 inner 1100.4.b.c 2
5.c odd 4 1 44.4.a.a 1
5.c odd 4 1 1100.4.a.d 1
15.e even 4 1 396.4.a.e 1
20.e even 4 1 176.4.a.e 1
35.f even 4 1 2156.4.a.b 1
40.i odd 4 1 704.4.a.j 1
40.k even 4 1 704.4.a.c 1
55.e even 4 1 484.4.a.a 1
60.l odd 4 1 1584.4.a.p 1
220.i odd 4 1 1936.4.a.m 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
44.4.a.a 1 5.c odd 4 1
176.4.a.e 1 20.e even 4 1
396.4.a.e 1 15.e even 4 1
484.4.a.a 1 55.e even 4 1
704.4.a.c 1 40.k even 4 1
704.4.a.j 1 40.i odd 4 1
1100.4.a.d 1 5.c odd 4 1
1100.4.b.c 2 1.a even 1 1 trivial
1100.4.b.c 2 5.b even 2 1 inner
1584.4.a.p 1 60.l odd 4 1
1936.4.a.m 1 220.i odd 4 1
2156.4.a.b 1 35.f even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(1100, [\chi])$$:

 $$T_{3}^{2} + 25$$ T3^2 + 25 $$T_{7}^{2} + 676$$ T7^2 + 676

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 25$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 676$$
$11$ $$(T + 11)^{2}$$
$13$ $$T^{2} + 2704$$
$17$ $$T^{2} + 2116$$
$19$ $$(T - 96)^{2}$$
$23$ $$T^{2} + 729$$
$29$ $$(T + 16)^{2}$$
$31$ $$(T + 293)^{2}$$
$37$ $$T^{2} + 841$$
$41$ $$(T + 472)^{2}$$
$43$ $$T^{2} + 12100$$
$47$ $$T^{2} + 50176$$
$53$ $$T^{2} + 568516$$
$59$ $$(T + 825)^{2}$$
$61$ $$(T + 548)^{2}$$
$67$ $$T^{2} + 15129$$
$71$ $$(T - 1001)^{2}$$
$73$ $$T^{2} + 1040400$$
$79$ $$(T + 526)^{2}$$
$83$ $$T^{2} + 24964$$
$89$ $$(T - 1217)^{2}$$
$97$ $$T^{2} + 69169$$