Properties

 Label 1156.2.e.a Level $1156$ Weight $2$ Character orbit 1156.e Analytic conductor $9.231$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1156,2,Mod(829,1156)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1156, base_ring=CyclotomicField(4))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("1156.829");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1156 = 2^{2} \cdot 17^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1156.e (of order $$4$$, degree $$2$$, 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: $$9.23070647366$$ 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: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 68) Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

 $$f(q)$$ $$=$$ $$q + ( - i - 1) q^{3} + (2 i + 2) q^{5} + ( - 3 i + 3) q^{7} - i q^{9} +O(q^{10})$$ q + (-i - 1) * q^3 + (2*i + 2) * q^5 + (-3*i + 3) * q^7 - i * q^9 $$q + ( - i - 1) q^{3} + (2 i + 2) q^{5} + ( - 3 i + 3) q^{7} - i q^{9} + (i - 1) q^{11} + 4 q^{13} - 4 i q^{15} - 4 i q^{19} - 6 q^{21} + (i - 1) q^{23} + 3 i q^{25} + (4 i - 4) q^{27} + (2 i + 2) q^{29} + ( - 3 i - 3) q^{31} + 2 q^{33} + 12 q^{35} + (6 i + 6) q^{37} + ( - 4 i - 4) q^{39} + (8 i - 8) q^{41} - 8 i q^{43} + ( - 2 i + 2) q^{45} + 12 q^{47} - 11 i q^{49} - 6 i q^{53} - 4 q^{55} + (4 i - 4) q^{57} + ( - 6 i + 6) q^{61} + ( - 3 i - 3) q^{63} + (8 i + 8) q^{65} - 4 q^{67} + 2 q^{69} + ( - 5 i - 5) q^{71} + ( - 3 i + 3) q^{75} + 6 i q^{77} + ( - 3 i + 3) q^{79} + 5 q^{81} - 4 i q^{87} - 12 q^{89} + ( - 12 i + 12) q^{91} + 6 i q^{93} + ( - 8 i + 8) q^{95} + (i + 1) q^{99} +O(q^{100})$$ q + (-i - 1) * q^3 + (2*i + 2) * q^5 + (-3*i + 3) * q^7 - i * q^9 + (i - 1) * q^11 + 4 * q^13 - 4*i * q^15 - 4*i * q^19 - 6 * q^21 + (i - 1) * q^23 + 3*i * q^25 + (4*i - 4) * q^27 + (2*i + 2) * q^29 + (-3*i - 3) * q^31 + 2 * q^33 + 12 * q^35 + (6*i + 6) * q^37 + (-4*i - 4) * q^39 + (8*i - 8) * q^41 - 8*i * q^43 + (-2*i + 2) * q^45 + 12 * q^47 - 11*i * q^49 - 6*i * q^53 - 4 * q^55 + (4*i - 4) * q^57 + (-6*i + 6) * q^61 + (-3*i - 3) * q^63 + (8*i + 8) * q^65 - 4 * q^67 + 2 * q^69 + (-5*i - 5) * q^71 + (-3*i + 3) * q^75 + 6*i * q^77 + (-3*i + 3) * q^79 + 5 * q^81 - 4*i * q^87 - 12 * q^89 + (-12*i + 12) * q^91 + 6*i * q^93 + (-8*i + 8) * q^95 + (i + 1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{3} + 4 q^{5} + 6 q^{7}+O(q^{10})$$ 2 * q - 2 * q^3 + 4 * q^5 + 6 * q^7 $$2 q - 2 q^{3} + 4 q^{5} + 6 q^{7} - 2 q^{11} + 8 q^{13} - 12 q^{21} - 2 q^{23} - 8 q^{27} + 4 q^{29} - 6 q^{31} + 4 q^{33} + 24 q^{35} + 12 q^{37} - 8 q^{39} - 16 q^{41} + 4 q^{45} + 24 q^{47} - 8 q^{55} - 8 q^{57} + 12 q^{61} - 6 q^{63} + 16 q^{65} - 8 q^{67} + 4 q^{69} - 10 q^{71} + 6 q^{75} + 6 q^{79} + 10 q^{81} - 24 q^{89} + 24 q^{91} + 16 q^{95} + 2 q^{99}+O(q^{100})$$ 2 * q - 2 * q^3 + 4 * q^5 + 6 * q^7 - 2 * q^11 + 8 * q^13 - 12 * q^21 - 2 * q^23 - 8 * q^27 + 4 * q^29 - 6 * q^31 + 4 * q^33 + 24 * q^35 + 12 * q^37 - 8 * q^39 - 16 * q^41 + 4 * q^45 + 24 * q^47 - 8 * q^55 - 8 * q^57 + 12 * q^61 - 6 * q^63 + 16 * q^65 - 8 * q^67 + 4 * q^69 - 10 * q^71 + 6 * q^75 + 6 * q^79 + 10 * q^81 - 24 * q^89 + 24 * q^91 + 16 * q^95 + 2 * q^99

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1156\mathbb{Z}\right)^\times$$.

 $$n$$ $$579$$ $$581$$ $$\chi(n)$$ $$1$$ $$i$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
829.1
 1.00000i − 1.00000i
0 −1.00000 1.00000i 0 2.00000 + 2.00000i 0 3.00000 3.00000i 0 1.00000i 0
905.1 0 −1.00000 + 1.00000i 0 2.00000 2.00000i 0 3.00000 + 3.00000i 0 1.00000i 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
17.c even 4 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1156.2.e.a 2
17.b even 2 1 1156.2.e.b 2
17.c even 4 1 inner 1156.2.e.a 2
17.c even 4 1 1156.2.e.b 2
17.d even 8 2 68.2.b.a 2
17.d even 8 2 1156.2.a.c 2
17.e odd 16 8 1156.2.h.d 8
51.g odd 8 2 612.2.b.a 2
68.g odd 8 2 272.2.b.c 2
68.g odd 8 2 4624.2.a.n 2
85.k odd 8 2 1700.2.g.a 4
85.m even 8 2 1700.2.c.a 2
85.n odd 8 2 1700.2.g.a 4
119.l odd 8 2 3332.2.b.a 2
136.o even 8 2 1088.2.b.e 2
136.p odd 8 2 1088.2.b.f 2
204.p even 8 2 2448.2.c.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
68.2.b.a 2 17.d even 8 2
272.2.b.c 2 68.g odd 8 2
612.2.b.a 2 51.g odd 8 2
1088.2.b.e 2 136.o even 8 2
1088.2.b.f 2 136.p odd 8 2
1156.2.a.c 2 17.d even 8 2
1156.2.e.a 2 1.a even 1 1 trivial
1156.2.e.a 2 17.c even 4 1 inner
1156.2.e.b 2 17.b even 2 1
1156.2.e.b 2 17.c even 4 1
1156.2.h.d 8 17.e odd 16 8
1700.2.c.a 2 85.m even 8 2
1700.2.g.a 4 85.k odd 8 2
1700.2.g.a 4 85.n odd 8 2
2448.2.c.d 2 204.p even 8 2
3332.2.b.a 2 119.l odd 8 2
4624.2.a.n 2 68.g odd 8 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} + 2T_{3} + 2$$ acting on $$S_{2}^{\mathrm{new}}(1156, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 2T + 2$$
$5$ $$T^{2} - 4T + 8$$
$7$ $$T^{2} - 6T + 18$$
$11$ $$T^{2} + 2T + 2$$
$13$ $$(T - 4)^{2}$$
$17$ $$T^{2}$$
$19$ $$T^{2} + 16$$
$23$ $$T^{2} + 2T + 2$$
$29$ $$T^{2} - 4T + 8$$
$31$ $$T^{2} + 6T + 18$$
$37$ $$T^{2} - 12T + 72$$
$41$ $$T^{2} + 16T + 128$$
$43$ $$T^{2} + 64$$
$47$ $$(T - 12)^{2}$$
$53$ $$T^{2} + 36$$
$59$ $$T^{2}$$
$61$ $$T^{2} - 12T + 72$$
$67$ $$(T + 4)^{2}$$
$71$ $$T^{2} + 10T + 50$$
$73$ $$T^{2}$$
$79$ $$T^{2} - 6T + 18$$
$83$ $$T^{2}$$
$89$ $$(T + 12)^{2}$$
$97$ $$T^{2}$$