LMFDB - Newform orbit 1100.6.b.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1100,6,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, 6, names="a")

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

chi := DirichletCharacter("1100.749");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$N$	$=$	$1100 = 2^{2} \cdot 5^{2} \cdot 11$
Weight:	$k$	$=$	$6$
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:	$176.422201794$
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.

$f(q)$	$=$	$q + 7 i q^{3} + 50 i q^{7} + 194 q^{9} + 121 q^{11} - 380 i q^{13} + 1154 i q^{17} + 1824 q^{19} - 350 q^{21} + 3591 i q^{23} + 3059 i q^{27} - 8032 q^{29} - 2945 q^{31} + 847 i q^{33} - 6979 i q^{37} + \cdots + 23474 q^{99} +O(q^{100})$ q + 7i q^3 + 50i q^7 + 194 * q^9 + 121 * q^11 - 380i q^13 + 1154i q^17 + 1824 * q^19 - 350 * q^21 + 3591i q^23 + 3059i q^27 - 8032 * q^29 - 2945 * q^31 + 847i q^33 - 6979i q^37 + 2660 * q^39 - 520 * q^41 - 2486i q^43 + 6920i q^47 + 14307 * q^49 - 8078 * q^51 - 13718i q^53 + 12768i q^57 + 31779 * q^59 + 34156 * q^61 + 9700i q^63 + 61503i q^67 - 25137 * q^69 - 14971 * q^71 - 36444i q^73 + 6050i q^77 + 28538 * q^79 + 25729 * q^81 + 77482i q^83 - 56224i q^87 - 36271 * q^89 + 19000 * q^91 - 20615i q^93 + 49799i q^97 + 23474 * q^99
$\operatorname{Tr}(f)(q)$	$=$	$2 q + 388 q^{9} + 242 q^{11} + 3648 q^{19} - 700 q^{21} - 16064 q^{29} - 5890 q^{31} + 5320 q^{39} - 1040 q^{41} + 28614 q^{49} - 16156 q^{51} + 63558 q^{59} + 68312 q^{61} - 50274 q^{69} - 29942 q^{71}+ \cdots + 46948 q^{99}+O(q^{100})$ 2 * q + 388 * q^9 + 242 * q^11 + 3648 * q^19 - 700 * q^21 - 16064 * q^29 - 5890 * q^31 + 5320 * q^39 - 1040 * q^41 + 28614 * q^49 - 16156 * q^51 + 63558 * q^59 + 68312 * q^61 - 50274 * q^69 - 29942 * q^71 + 57076 * q^79 + 51458 * q^81 - 72542 * q^89 + 38000 * q^91 + 46948 * 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

−

7.00000i

−

50.0000i

194.000

749.2

7.00000i

50.0000i

194.000

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	1100.6.b.a		2
5.b	even	2	1	inner	1100.6.b.a		2
5.c	odd	4	1		44.6.a.a	✓	1
5.c	odd	4	1		1100.6.a.a		1
15.e	even	4	1		396.6.a.e		1
20.e	even	4	1		176.6.a.a		1
40.i	odd	4	1		704.6.a.d		1
40.k	even	4	1		704.6.a.g		1
55.e	even	4	1		484.6.a.b		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
44.6.a.a	✓	1	5.c	odd	4	1
176.6.a.a		1	20.e	even	4	1
396.6.a.e		1	15.e	even	4	1
484.6.a.b		1	55.e	even	4	1
704.6.a.d		1	40.i	odd	4	1
704.6.a.g		1	40.k	even	4	1
1100.6.a.a		1	5.c	odd	4	1
1100.6.b.a		2	1.a	even	1	1	trivial
1100.6.b.a		2	5.b	even	2	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{3}^{2} + 49$ T3^2 + 49 acting on $S_{6}^{\mathrm{new}}(1100, [\chi])$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2}$ T^2
$3$	$T^{2} + 49$ T^2 + 49
$5$	$T^{2}$ T^2
$7$	$T^{2} + 2500$ T^2 + 2500
$11$	$(T - 121)^{2}$ (T - 121)^2
$13$	$T^{2} + 144400$ T^2 + 144400
$17$	$T^{2} + 1331716$ T^2 + 1331716
$19$	$(T - 1824)^{2}$ (T - 1824)^2
$23$	$T^{2} + 12895281$ T^2 + 12895281
$29$	$(T + 8032)^{2}$ (T + 8032)^2
$31$	$(T + 2945)^{2}$ (T + 2945)^2
$37$	$T^{2} + 48706441$ T^2 + 48706441
$41$	$(T + 520)^{2}$ (T + 520)^2
$43$	$T^{2} + 6180196$ T^2 + 6180196
$47$	$T^{2} + 47886400$ T^2 + 47886400
$53$	$T^{2} + 188183524$ T^2 + 188183524
$59$	$(T - 31779)^{2}$ (T - 31779)^2
$61$	$(T - 34156)^{2}$ (T - 34156)^2
$67$	$T^{2} + 3782619009$ T^2 + 3782619009
$71$	$(T + 14971)^{2}$ (T + 14971)^2
$73$	$T^{2} + 1328165136$ T^2 + 1328165136
$79$	$(T - 28538)^{2}$ (T - 28538)^2
$83$	$T^{2} + 6003460324$ T^2 + 6003460324
$89$	$(T + 36271)^{2}$ (T + 36271)^2
$97$	$T^{2} + 2479940401$ T^2 + 2479940401
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