Properties

 Label 136.1.p.a Level $136$ Weight $1$ Character orbit 136.p Analytic conductor $0.068$ Analytic rank $0$ Dimension $4$ Projective image $D_{8}$ CM discriminant -8 Inner twists $4$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$136 = 2^{3} \cdot 17$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 136.p (of order $$8$$, degree $$4$$, minimal)

Newform invariants

 Self dual: no Analytic conductor: $$0.0678728417181$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{8}$$ Projective field: Galois closure of 8.0.1680747204608.3

$q$-expansion

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

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

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/136\mathbb{Z}\right)^\times$$.

 $$n$$ $$69$$ $$103$$ $$105$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$\zeta_{8}$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
19.1
 0.707107 − 0.707107i 0.707107 + 0.707107i −0.707107 − 0.707107i −0.707107 + 0.707107i
−0.707107 0.707107i 0.707107 0.292893i 1.00000i 0 −0.707107 0.292893i 0 0.707107 0.707107i −0.292893 + 0.292893i 0
43.1 −0.707107 + 0.707107i 0.707107 + 0.292893i 1.00000i 0 −0.707107 + 0.292893i 0 0.707107 + 0.707107i −0.292893 0.292893i 0
59.1 0.707107 0.707107i −0.707107 + 1.70711i 1.00000i 0 0.707107 + 1.70711i 0 −0.707107 0.707107i −1.70711 1.70711i 0
83.1 0.707107 + 0.707107i −0.707107 1.70711i 1.00000i 0 0.707107 1.70711i 0 −0.707107 + 0.707107i −1.70711 + 1.70711i 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 $$\Q(\sqrt{-2})$$
17.d even 8 1 inner
136.p odd 8 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 136.1.p.a 4
3.b odd 2 1 1224.1.bv.a 4
4.b odd 2 1 544.1.bl.a 4
5.b even 2 1 3400.1.ce.a 4
5.c odd 4 1 3400.1.br.a 4
5.c odd 4 1 3400.1.br.b 4
8.b even 2 1 544.1.bl.a 4
8.d odd 2 1 CM 136.1.p.a 4
17.b even 2 1 2312.1.p.b 4
17.c even 4 1 2312.1.p.a 4
17.c even 4 1 2312.1.p.d 4
17.d even 8 1 inner 136.1.p.a 4
17.d even 8 1 2312.1.p.a 4
17.d even 8 1 2312.1.p.b 4
17.d even 8 1 2312.1.p.d 4
17.e odd 16 2 2312.1.e.b 4
17.e odd 16 2 2312.1.f.c 4
17.e odd 16 4 2312.1.j.c 8
24.f even 2 1 1224.1.bv.a 4
40.e odd 2 1 3400.1.ce.a 4
40.k even 4 1 3400.1.br.a 4
40.k even 4 1 3400.1.br.b 4
51.g odd 8 1 1224.1.bv.a 4
68.g odd 8 1 544.1.bl.a 4
85.k odd 8 1 3400.1.br.a 4
85.m even 8 1 3400.1.ce.a 4
85.n odd 8 1 3400.1.br.b 4
136.e odd 2 1 2312.1.p.b 4
136.j odd 4 1 2312.1.p.a 4
136.j odd 4 1 2312.1.p.d 4
136.o even 8 1 544.1.bl.a 4
136.p odd 8 1 inner 136.1.p.a 4
136.p odd 8 1 2312.1.p.a 4
136.p odd 8 1 2312.1.p.b 4
136.p odd 8 1 2312.1.p.d 4
136.s even 16 2 2312.1.e.b 4
136.s even 16 2 2312.1.f.c 4
136.s even 16 4 2312.1.j.c 8
408.bd even 8 1 1224.1.bv.a 4
680.bq odd 8 1 3400.1.ce.a 4
680.bw even 8 1 3400.1.br.a 4
680.bz even 8 1 3400.1.br.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
136.1.p.a 4 1.a even 1 1 trivial
136.1.p.a 4 8.d odd 2 1 CM
136.1.p.a 4 17.d even 8 1 inner
136.1.p.a 4 136.p odd 8 1 inner
544.1.bl.a 4 4.b odd 2 1
544.1.bl.a 4 8.b even 2 1
544.1.bl.a 4 68.g odd 8 1
544.1.bl.a 4 136.o even 8 1
1224.1.bv.a 4 3.b odd 2 1
1224.1.bv.a 4 24.f even 2 1
1224.1.bv.a 4 51.g odd 8 1
1224.1.bv.a 4 408.bd even 8 1
2312.1.e.b 4 17.e odd 16 2
2312.1.e.b 4 136.s even 16 2
2312.1.f.c 4 17.e odd 16 2
2312.1.f.c 4 136.s even 16 2
2312.1.j.c 8 17.e odd 16 4
2312.1.j.c 8 136.s even 16 4
2312.1.p.a 4 17.c even 4 1
2312.1.p.a 4 17.d even 8 1
2312.1.p.a 4 136.j odd 4 1
2312.1.p.a 4 136.p odd 8 1
2312.1.p.b 4 17.b even 2 1
2312.1.p.b 4 17.d even 8 1
2312.1.p.b 4 136.e odd 2 1
2312.1.p.b 4 136.p odd 8 1
2312.1.p.d 4 17.c even 4 1
2312.1.p.d 4 17.d even 8 1
2312.1.p.d 4 136.j odd 4 1
2312.1.p.d 4 136.p odd 8 1
3400.1.br.a 4 5.c odd 4 1
3400.1.br.a 4 40.k even 4 1
3400.1.br.a 4 85.k odd 8 1
3400.1.br.a 4 680.bw even 8 1
3400.1.br.b 4 5.c odd 4 1
3400.1.br.b 4 40.k even 4 1
3400.1.br.b 4 85.n odd 8 1
3400.1.br.b 4 680.bz even 8 1
3400.1.ce.a 4 5.b even 2 1
3400.1.ce.a 4 40.e odd 2 1
3400.1.ce.a 4 85.m even 8 1
3400.1.ce.a 4 680.bq odd 8 1

Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(136, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T^{4}$$
$3$ $$2 - 4 T + 2 T^{2} + T^{4}$$
$5$ $$T^{4}$$
$7$ $$T^{4}$$
$11$ $$2 + 4 T + 6 T^{2} + 4 T^{3} + T^{4}$$
$13$ $$T^{4}$$
$17$ $$1 + T^{4}$$
$19$ $$T^{4}$$
$23$ $$T^{4}$$
$29$ $$T^{4}$$
$31$ $$T^{4}$$
$37$ $$T^{4}$$
$41$ $$2 + 4 T + 2 T^{2} + T^{4}$$
$43$ $$( 2 - 2 T + T^{2} )^{2}$$
$47$ $$T^{4}$$
$53$ $$T^{4}$$
$59$ $$( 2 + 2 T + T^{2} )^{2}$$
$61$ $$T^{4}$$
$67$ $$( -2 + T^{2} )^{2}$$
$71$ $$T^{4}$$
$73$ $$2 - 4 T + 2 T^{2} + T^{4}$$
$79$ $$T^{4}$$
$83$ $$( 2 - 2 T + T^{2} )^{2}$$
$89$ $$( 2 + T^{2} )^{2}$$
$97$ $$2 + 4 T + 6 T^{2} + 4 T^{3} + T^{4}$$