# Properties

 Label 88.1.l.a Level $88$ Weight $1$ Character orbit 88.l Analytic conductor $0.044$ Analytic rank $0$ Dimension $4$ Projective image $D_{5}$ CM discriminant -8 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$88 = 2^{3} \cdot 11$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 88.l (of order $$10$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.0439177211117$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{10})$$ Defining polynomial: $$x^{4} - x^{3} + x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{5}$$ Projective field: Galois closure of 5.1.937024.1

## $q$-expansion

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q -\zeta_{10} q^{2} + ( \zeta_{10}^{2} + \zeta_{10}^{4} ) q^{3} + \zeta_{10}^{2} q^{4} + ( 1 - \zeta_{10}^{3} ) q^{6} -\zeta_{10}^{3} q^{8} + ( -\zeta_{10} - \zeta_{10}^{3} + \zeta_{10}^{4} ) q^{9} +O(q^{10})$$ $$q -\zeta_{10} q^{2} + ( \zeta_{10}^{2} + \zeta_{10}^{4} ) q^{3} + \zeta_{10}^{2} q^{4} + ( 1 - \zeta_{10}^{3} ) q^{6} -\zeta_{10}^{3} q^{8} + ( -\zeta_{10} - \zeta_{10}^{3} + \zeta_{10}^{4} ) q^{9} -\zeta_{10}^{3} q^{11} + ( -\zeta_{10} + \zeta_{10}^{4} ) q^{12} + \zeta_{10}^{4} q^{16} + ( -\zeta_{10} + \zeta_{10}^{2} ) q^{17} + ( 1 + \zeta_{10}^{2} + \zeta_{10}^{4} ) q^{18} + ( 1 - \zeta_{10} ) q^{19} + \zeta_{10}^{4} q^{22} + ( 1 + \zeta_{10}^{2} ) q^{24} -\zeta_{10}^{3} q^{25} + ( 1 - \zeta_{10} + \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{27} + q^{32} + ( 1 + \zeta_{10}^{2} ) q^{33} + ( \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{34} + ( 1 - \zeta_{10} - \zeta_{10}^{3} ) q^{36} + ( -\zeta_{10} + \zeta_{10}^{2} ) q^{38} + ( \zeta_{10}^{2} + \zeta_{10}^{4} ) q^{41} + ( -\zeta_{10} + \zeta_{10}^{4} ) q^{43} + q^{44} + ( -\zeta_{10} - \zeta_{10}^{3} ) q^{48} + \zeta_{10}^{4} q^{49} + \zeta_{10}^{4} q^{50} + ( 1 - \zeta_{10} - \zeta_{10}^{3} + \zeta_{10}^{4} ) q^{51} + ( -\zeta_{10} + \zeta_{10}^{2} - \zeta_{10}^{3} + \zeta_{10}^{4} ) q^{54} + ( 1 + \zeta_{10}^{2} - \zeta_{10}^{3} + \zeta_{10}^{4} ) q^{57} + ( 1 + \zeta_{10}^{4} ) q^{59} -\zeta_{10} q^{64} + ( -\zeta_{10} - \zeta_{10}^{3} ) q^{66} + ( \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{67} + ( -\zeta_{10}^{3} + \zeta_{10}^{4} ) q^{68} + ( -\zeta_{10} + \zeta_{10}^{2} + \zeta_{10}^{4} ) q^{72} + ( -\zeta_{10} - \zeta_{10}^{3} ) q^{73} + ( 1 + \zeta_{10}^{2} ) q^{75} + ( \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{76} + ( 1 - \zeta_{10} + \zeta_{10}^{2} - \zeta_{10}^{3} + \zeta_{10}^{4} ) q^{81} + ( 1 - \zeta_{10}^{3} ) q^{82} + ( 1 - \zeta_{10}^{3} ) q^{83} + ( 1 + \zeta_{10}^{2} ) q^{86} -\zeta_{10} q^{88} + ( -\zeta_{10} + \zeta_{10}^{4} ) q^{89} + ( \zeta_{10}^{2} + \zeta_{10}^{4} ) q^{96} + ( 1 + \zeta_{10}^{2} ) q^{97} + q^{98} + ( -\zeta_{10} + \zeta_{10}^{2} + \zeta_{10}^{4} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - q^{2} - 2q^{3} - q^{4} + 3q^{6} - q^{8} - 3q^{9} + O(q^{10})$$ $$4q - q^{2} - 2q^{3} - q^{4} + 3q^{6} - q^{8} - 3q^{9} - q^{11} - 2q^{12} - q^{16} - 2q^{17} + 2q^{18} + 3q^{19} - q^{22} + 3q^{24} - q^{25} + q^{27} + 4q^{32} + 3q^{33} - 2q^{34} + 2q^{36} - 2q^{38} - 2q^{41} - 2q^{43} + 4q^{44} - 2q^{48} - q^{49} - q^{50} + q^{51} - 4q^{54} + q^{57} + 3q^{59} - q^{64} - 2q^{66} - 2q^{67} - 2q^{68} - 3q^{72} - 2q^{73} + 3q^{75} - 2q^{76} + 3q^{82} + 3q^{83} + 3q^{86} - q^{88} - 2q^{89} - 2q^{96} + 3q^{97} + 4q^{98} - 3q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/88\mathbb{Z}\right)^\times$$.

 $$n$$ $$23$$ $$45$$ $$57$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$\zeta_{10}^{2}$$

## 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.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
3.1
 0.809017 − 0.587785i −0.309017 − 0.951057i 0.809017 + 0.587785i −0.309017 + 0.951057i
−0.809017 + 0.587785i −0.500000 1.53884i 0.309017 0.951057i 0 1.30902 + 0.951057i 0 0.309017 + 0.951057i −1.30902 + 0.951057i 0
27.1 0.309017 + 0.951057i −0.500000 0.363271i −0.809017 + 0.587785i 0 0.190983 0.587785i 0 −0.809017 0.587785i −0.190983 0.587785i 0
59.1 −0.809017 0.587785i −0.500000 + 1.53884i 0.309017 + 0.951057i 0 1.30902 0.951057i 0 0.309017 0.951057i −1.30902 0.951057i 0
75.1 0.309017 0.951057i −0.500000 + 0.363271i −0.809017 0.587785i 0 0.190983 + 0.587785i 0 −0.809017 + 0.587785i −0.190983 + 0.587785i 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
8.d odd 2 1 CM by $$\Q(\sqrt{-2})$$
11.c even 5 1 inner
88.l odd 10 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 88.1.l.a 4
3.b odd 2 1 792.1.bu.a 4
4.b odd 2 1 352.1.t.a 4
5.b even 2 1 2200.1.cl.a 4
5.c odd 4 2 2200.1.dd.a 8
8.b even 2 1 352.1.t.a 4
8.d odd 2 1 CM 88.1.l.a 4
11.b odd 2 1 968.1.l.b 4
11.c even 5 1 inner 88.1.l.a 4
11.c even 5 1 968.1.f.b 2
11.c even 5 2 968.1.l.a 4
11.d odd 10 1 968.1.f.a 2
11.d odd 10 1 968.1.l.b 4
11.d odd 10 2 968.1.l.c 4
12.b even 2 1 3168.1.ck.a 4
16.e even 4 2 2816.1.v.c 8
16.f odd 4 2 2816.1.v.c 8
24.f even 2 1 792.1.bu.a 4
24.h odd 2 1 3168.1.ck.a 4
33.h odd 10 1 792.1.bu.a 4
40.e odd 2 1 2200.1.cl.a 4
40.k even 4 2 2200.1.dd.a 8
44.c even 2 1 3872.1.t.c 4
44.g even 10 1 3872.1.f.b 2
44.g even 10 2 3872.1.t.a 4
44.g even 10 1 3872.1.t.c 4
44.h odd 10 1 352.1.t.a 4
44.h odd 10 1 3872.1.f.a 2
44.h odd 10 2 3872.1.t.b 4
55.j even 10 1 2200.1.cl.a 4
55.k odd 20 2 2200.1.dd.a 8
88.b odd 2 1 3872.1.t.c 4
88.g even 2 1 968.1.l.b 4
88.k even 10 1 968.1.f.a 2
88.k even 10 1 968.1.l.b 4
88.k even 10 2 968.1.l.c 4
88.l odd 10 1 inner 88.1.l.a 4
88.l odd 10 1 968.1.f.b 2
88.l odd 10 2 968.1.l.a 4
88.o even 10 1 352.1.t.a 4
88.o even 10 1 3872.1.f.a 2
88.o even 10 2 3872.1.t.b 4
88.p odd 10 1 3872.1.f.b 2
88.p odd 10 2 3872.1.t.a 4
88.p odd 10 1 3872.1.t.c 4
132.o even 10 1 3168.1.ck.a 4
176.v odd 20 2 2816.1.v.c 8
176.w even 20 2 2816.1.v.c 8
264.t odd 10 1 3168.1.ck.a 4
264.w even 10 1 792.1.bu.a 4
440.bh odd 10 1 2200.1.cl.a 4
440.bs even 20 2 2200.1.dd.a 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
88.1.l.a 4 1.a even 1 1 trivial
88.1.l.a 4 8.d odd 2 1 CM
88.1.l.a 4 11.c even 5 1 inner
88.1.l.a 4 88.l odd 10 1 inner
352.1.t.a 4 4.b odd 2 1
352.1.t.a 4 8.b even 2 1
352.1.t.a 4 44.h odd 10 1
352.1.t.a 4 88.o even 10 1
792.1.bu.a 4 3.b odd 2 1
792.1.bu.a 4 24.f even 2 1
792.1.bu.a 4 33.h odd 10 1
792.1.bu.a 4 264.w even 10 1
968.1.f.a 2 11.d odd 10 1
968.1.f.a 2 88.k even 10 1
968.1.f.b 2 11.c even 5 1
968.1.f.b 2 88.l odd 10 1
968.1.l.a 4 11.c even 5 2
968.1.l.a 4 88.l odd 10 2
968.1.l.b 4 11.b odd 2 1
968.1.l.b 4 11.d odd 10 1
968.1.l.b 4 88.g even 2 1
968.1.l.b 4 88.k even 10 1
968.1.l.c 4 11.d odd 10 2
968.1.l.c 4 88.k even 10 2
2200.1.cl.a 4 5.b even 2 1
2200.1.cl.a 4 40.e odd 2 1
2200.1.cl.a 4 55.j even 10 1
2200.1.cl.a 4 440.bh odd 10 1
2200.1.dd.a 8 5.c odd 4 2
2200.1.dd.a 8 40.k even 4 2
2200.1.dd.a 8 55.k odd 20 2
2200.1.dd.a 8 440.bs even 20 2
2816.1.v.c 8 16.e even 4 2
2816.1.v.c 8 16.f odd 4 2
2816.1.v.c 8 176.v odd 20 2
2816.1.v.c 8 176.w even 20 2
3168.1.ck.a 4 12.b even 2 1
3168.1.ck.a 4 24.h odd 2 1
3168.1.ck.a 4 132.o even 10 1
3168.1.ck.a 4 264.t odd 10 1
3872.1.f.a 2 44.h odd 10 1
3872.1.f.a 2 88.o even 10 1
3872.1.f.b 2 44.g even 10 1
3872.1.f.b 2 88.p odd 10 1
3872.1.t.a 4 44.g even 10 2
3872.1.t.a 4 88.p odd 10 2
3872.1.t.b 4 44.h odd 10 2
3872.1.t.b 4 88.o even 10 2
3872.1.t.c 4 44.c even 2 1
3872.1.t.c 4 44.g even 10 1
3872.1.t.c 4 88.b odd 2 1
3872.1.t.c 4 88.p odd 10 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(88, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T + T^{2} + T^{3} + T^{4}$$
$3$ $$1 + 3 T + 4 T^{2} + 2 T^{3} + T^{4}$$
$5$ $$T^{4}$$
$7$ $$T^{4}$$
$11$ $$1 + T + T^{2} + T^{3} + T^{4}$$
$13$ $$T^{4}$$
$17$ $$1 + 3 T + 4 T^{2} + 2 T^{3} + T^{4}$$
$19$ $$1 - 2 T + 4 T^{2} - 3 T^{3} + T^{4}$$
$23$ $$T^{4}$$
$29$ $$T^{4}$$
$31$ $$T^{4}$$
$37$ $$T^{4}$$
$41$ $$1 + 3 T + 4 T^{2} + 2 T^{3} + T^{4}$$
$43$ $$( -1 + T + T^{2} )^{2}$$
$47$ $$T^{4}$$
$53$ $$T^{4}$$
$59$ $$1 - 2 T + 4 T^{2} - 3 T^{3} + T^{4}$$
$61$ $$T^{4}$$
$67$ $$( -1 + T + T^{2} )^{2}$$
$71$ $$T^{4}$$
$73$ $$1 + 3 T + 4 T^{2} + 2 T^{3} + T^{4}$$
$79$ $$T^{4}$$
$83$ $$1 - 2 T + 4 T^{2} - 3 T^{3} + T^{4}$$
$89$ $$( -1 + T + T^{2} )^{2}$$
$97$ $$1 - 2 T + 4 T^{2} - 3 T^{3} + T^{4}$$