# Properties

 Label 968.2.i.i Level $968$ Weight $2$ Character orbit 968.i Analytic conductor $7.730$ Analytic rank $0$ Dimension $4$ Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [968,2,Mod(9,968)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(968, base_ring=CyclotomicField(10))

chi = DirichletCharacter(H, H._module([0, 0, 6]))

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

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

chi := DirichletCharacter("968.9");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$968 = 2^{3} \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 968.i (of order $$5$$, degree $$4$$, 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: $$7.72951891566$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{10})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{3} + x^{2} - x + 1$$ x^4 - x^3 + x^2 - x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 88) Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

## $q$-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 a primitive root of unity $$\zeta_{10}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - 3 \zeta_{10}^{2} q^{3} + 3 \zeta_{10} q^{5} - 2 \zeta_{10}^{3} q^{7} + (6 \zeta_{10}^{3} - 6 \zeta_{10}^{2} + \cdots - 6) q^{9} +O(q^{10})$$ q - 3*z^2 * q^3 + 3*z * q^5 - 2*z^3 * q^7 + (6*z^3 - 6*z^2 + 6*z - 6) * q^9 $$q - 3 \zeta_{10}^{2} q^{3} + 3 \zeta_{10} q^{5} - 2 \zeta_{10}^{3} q^{7} + (6 \zeta_{10}^{3} - 6 \zeta_{10}^{2} + \cdots - 6) q^{9} + \cdots + ( - 7 \zeta_{10}^{3} + 7 \zeta_{10}^{2} + \cdots + 7) q^{97} +O(q^{100})$$ q - 3*z^2 * q^3 + 3*z * q^5 - 2*z^3 * q^7 + (6*z^3 - 6*z^2 + 6*z - 6) * q^9 - 9*z^3 * q^15 - 6*z * q^17 - 4*z^2 * q^19 - 6 * q^21 + q^23 + 4*z^2 * q^25 + 9*z * q^27 - 8*z^3 * q^29 + (-7*z^3 + 7*z^2 - 7*z + 7) * q^31 + (-6*z^3 + 6*z^2 - 6*z + 6) * q^35 + z^3 * q^37 - 4*z^2 * q^41 - 6 * q^43 - 18 * q^45 - 8*z^2 * q^47 + 3*z * q^49 + 18*z^3 * q^51 + (2*z^3 - 2*z^2 + 2*z - 2) * q^53 + (12*z^3 - 12*z^2 + 12*z - 12) * q^57 + z^3 * q^59 + 4*z * q^61 + 12*z^2 * q^63 - 5 * q^67 - 3*z^2 * q^69 - 3*z * q^71 + 16*z^3 * q^73 + (-12*z^3 + 12*z^2 - 12*z + 12) * q^75 + (-2*z^3 + 2*z^2 - 2*z + 2) * q^79 - 9*z^3 * q^81 - 2*z * q^83 - 18*z^2 * q^85 - 24 * q^87 + 15 * q^89 - 21*z * q^93 - 12*z^3 * q^95 + (-7*z^3 + 7*z^2 - 7*z + 7) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 3 q^{3} + 3 q^{5} - 2 q^{7} - 6 q^{9}+O(q^{10})$$ 4 * q + 3 * q^3 + 3 * q^5 - 2 * q^7 - 6 * q^9 $$4 q + 3 q^{3} + 3 q^{5} - 2 q^{7} - 6 q^{9} - 9 q^{15} - 6 q^{17} + 4 q^{19} - 24 q^{21} + 4 q^{23} - 4 q^{25} + 9 q^{27} - 8 q^{29} + 7 q^{31} + 6 q^{35} + q^{37} + 4 q^{41} - 24 q^{43} - 72 q^{45} + 8 q^{47} + 3 q^{49} + 18 q^{51} - 2 q^{53} - 12 q^{57} + q^{59} + 4 q^{61} - 12 q^{63} - 20 q^{67} + 3 q^{69} - 3 q^{71} + 16 q^{73} + 12 q^{75} + 2 q^{79} - 9 q^{81} - 2 q^{83} + 18 q^{85} - 96 q^{87} + 60 q^{89} - 21 q^{93} - 12 q^{95} + 7 q^{97}+O(q^{100})$$ 4 * q + 3 * q^3 + 3 * q^5 - 2 * q^7 - 6 * q^9 - 9 * q^15 - 6 * q^17 + 4 * q^19 - 24 * q^21 + 4 * q^23 - 4 * q^25 + 9 * q^27 - 8 * q^29 + 7 * q^31 + 6 * q^35 + q^37 + 4 * q^41 - 24 * q^43 - 72 * q^45 + 8 * q^47 + 3 * q^49 + 18 * q^51 - 2 * q^53 - 12 * q^57 + q^59 + 4 * q^61 - 12 * q^63 - 20 * q^67 + 3 * q^69 - 3 * q^71 + 16 * q^73 + 12 * q^75 + 2 * q^79 - 9 * q^81 - 2 * q^83 + 18 * q^85 - 96 * q^87 + 60 * q^89 - 21 * q^93 - 12 * q^95 + 7 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/968\mathbb{Z}\right)^\times$$.

 $$n$$ $$485$$ $$727$$ $$849$$ $$\chi(n)$$ $$1$$ $$1$$ $$-\zeta_{10}^{3}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
9.1
 −0.309017 − 0.951057i 0.809017 − 0.587785i 0.809017 + 0.587785i −0.309017 + 0.951057i
0 2.42705 1.76336i 0 −0.927051 2.85317i 0 −1.61803 1.17557i 0 1.85410 5.70634i 0
81.1 0 −0.927051 + 2.85317i 0 2.42705 1.76336i 0 0.618034 + 1.90211i 0 −4.85410 3.52671i 0
729.1 0 −0.927051 2.85317i 0 2.42705 + 1.76336i 0 0.618034 1.90211i 0 −4.85410 + 3.52671i 0
753.1 0 2.42705 + 1.76336i 0 −0.927051 + 2.85317i 0 −1.61803 + 1.17557i 0 1.85410 + 5.70634i 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
11.c even 5 3 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 968.2.i.i 4
11.b odd 2 1 968.2.i.j 4
11.c even 5 1 968.2.a.a 1
11.c even 5 3 inner 968.2.i.i 4
11.d odd 10 1 88.2.a.a 1
11.d odd 10 3 968.2.i.j 4
33.f even 10 1 792.2.a.g 1
33.h odd 10 1 8712.2.a.x 1
44.g even 10 1 176.2.a.c 1
44.h odd 10 1 1936.2.a.l 1
55.h odd 10 1 2200.2.a.k 1
55.l even 20 2 2200.2.b.a 2
77.l even 10 1 4312.2.a.l 1
88.k even 10 1 704.2.a.b 1
88.l odd 10 1 7744.2.a.b 1
88.o even 10 1 7744.2.a.bk 1
88.p odd 10 1 704.2.a.l 1
132.n odd 10 1 1584.2.a.q 1
176.u odd 20 2 2816.2.c.i 2
176.x even 20 2 2816.2.c.d 2
220.o even 10 1 4400.2.a.a 1
220.w odd 20 2 4400.2.b.b 2
264.r odd 10 1 6336.2.a.k 1
264.u even 10 1 6336.2.a.h 1
308.s odd 10 1 8624.2.a.c 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
88.2.a.a 1 11.d odd 10 1
176.2.a.c 1 44.g even 10 1
704.2.a.b 1 88.k even 10 1
704.2.a.l 1 88.p odd 10 1
792.2.a.g 1 33.f even 10 1
968.2.a.a 1 11.c even 5 1
968.2.i.i 4 1.a even 1 1 trivial
968.2.i.i 4 11.c even 5 3 inner
968.2.i.j 4 11.b odd 2 1
968.2.i.j 4 11.d odd 10 3
1584.2.a.q 1 132.n odd 10 1
1936.2.a.l 1 44.h odd 10 1
2200.2.a.k 1 55.h odd 10 1
2200.2.b.a 2 55.l even 20 2
2816.2.c.d 2 176.x even 20 2
2816.2.c.i 2 176.u odd 20 2
4312.2.a.l 1 77.l even 10 1
4400.2.a.a 1 220.o even 10 1
4400.2.b.b 2 220.w odd 20 2
6336.2.a.h 1 264.u even 10 1
6336.2.a.k 1 264.r odd 10 1
7744.2.a.b 1 88.l odd 10 1
7744.2.a.bk 1 88.o even 10 1
8624.2.a.c 1 308.s odd 10 1
8712.2.a.x 1 33.h odd 10 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(968, [\chi])$$:

 $$T_{3}^{4} - 3T_{3}^{3} + 9T_{3}^{2} - 27T_{3} + 81$$ T3^4 - 3*T3^3 + 9*T3^2 - 27*T3 + 81 $$T_{7}^{4} + 2T_{7}^{3} + 4T_{7}^{2} + 8T_{7} + 16$$ T7^4 + 2*T7^3 + 4*T7^2 + 8*T7 + 16

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} - 3 T^{3} + \cdots + 81$$
$5$ $$T^{4} - 3 T^{3} + \cdots + 81$$
$7$ $$T^{4} + 2 T^{3} + \cdots + 16$$
$11$ $$T^{4}$$
$13$ $$T^{4}$$
$17$ $$T^{4} + 6 T^{3} + \cdots + 1296$$
$19$ $$T^{4} - 4 T^{3} + \cdots + 256$$
$23$ $$(T - 1)^{4}$$
$29$ $$T^{4} + 8 T^{3} + \cdots + 4096$$
$31$ $$T^{4} - 7 T^{3} + \cdots + 2401$$
$37$ $$T^{4} - T^{3} + T^{2} + \cdots + 1$$
$41$ $$T^{4} - 4 T^{3} + \cdots + 256$$
$43$ $$(T + 6)^{4}$$
$47$ $$T^{4} - 8 T^{3} + \cdots + 4096$$
$53$ $$T^{4} + 2 T^{3} + \cdots + 16$$
$59$ $$T^{4} - T^{3} + T^{2} + \cdots + 1$$
$61$ $$T^{4} - 4 T^{3} + \cdots + 256$$
$67$ $$(T + 5)^{4}$$
$71$ $$T^{4} + 3 T^{3} + \cdots + 81$$
$73$ $$T^{4} - 16 T^{3} + \cdots + 65536$$
$79$ $$T^{4} - 2 T^{3} + \cdots + 16$$
$83$ $$T^{4} + 2 T^{3} + \cdots + 16$$
$89$ $$(T - 15)^{4}$$
$97$ $$T^{4} - 7 T^{3} + \cdots + 2401$$