# Properties

 Label 136.2.k.b Level $136$ Weight $2$ Character orbit 136.k Analytic conductor $1.086$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$1.08596546749$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -1 - i ) q^{3} + ( 2 + 2 i ) q^{5} + ( 2 - 2 i ) q^{7} -i q^{9} +O(q^{10})$$ $$q + ( -1 - i ) q^{3} + ( 2 + 2 i ) q^{5} + ( 2 - 2 i ) q^{7} -i q^{9} + ( 1 - i ) q^{11} + 2 q^{13} -4 i q^{15} + ( -4 + i ) q^{17} + 4 i q^{19} -4 q^{21} + ( -4 + 4 i ) q^{23} + 3 i q^{25} + ( -4 + 4 i ) q^{27} + ( 6 + 6 i ) q^{29} + ( -6 - 6 i ) q^{31} -2 q^{33} + 8 q^{35} + ( -8 - 8 i ) q^{37} + ( -2 - 2 i ) q^{39} + ( 1 - i ) q^{41} + 2 i q^{43} + ( 2 - 2 i ) q^{45} -i q^{49} + ( 5 + 3 i ) q^{51} -6 i q^{53} + 4 q^{55} + ( 4 - 4 i ) q^{57} + 14 i q^{59} + ( -4 + 4 i ) q^{61} + ( -2 - 2 i ) q^{63} + ( 4 + 4 i ) q^{65} + 6 q^{67} + 8 q^{69} + ( -9 - 9 i ) q^{73} + ( 3 - 3 i ) q^{75} -4 i q^{77} + ( 4 - 4 i ) q^{79} + 5 q^{81} + 6 i q^{83} + ( -10 - 6 i ) q^{85} -12 i q^{87} + 16 q^{89} + ( 4 - 4 i ) q^{91} + 12 i q^{93} + ( -8 + 8 i ) q^{95} + ( 3 + 3 i ) q^{97} + ( -1 - i ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{3} + 4 q^{5} + 4 q^{7} + O(q^{10})$$ $$2 q - 2 q^{3} + 4 q^{5} + 4 q^{7} + 2 q^{11} + 4 q^{13} - 8 q^{17} - 8 q^{21} - 8 q^{23} - 8 q^{27} + 12 q^{29} - 12 q^{31} - 4 q^{33} + 16 q^{35} - 16 q^{37} - 4 q^{39} + 2 q^{41} + 4 q^{45} + 10 q^{51} + 8 q^{55} + 8 q^{57} - 8 q^{61} - 4 q^{63} + 8 q^{65} + 12 q^{67} + 16 q^{69} - 18 q^{73} + 6 q^{75} + 8 q^{79} + 10 q^{81} - 20 q^{85} + 32 q^{89} + 8 q^{91} - 16 q^{95} + 6 q^{97} - 2 q^{99} + O(q^{100})$$

## Character values

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

 $$n$$ $$69$$ $$103$$ $$105$$ $$\chi(n)$$ $$1$$ $$1$$ $$i$$

## 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.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
81.1
 1.00000i − 1.00000i
0 −1.00000 1.00000i 0 2.00000 + 2.00000i 0 2.00000 2.00000i 0 1.00000i 0
89.1 0 −1.00000 + 1.00000i 0 2.00000 2.00000i 0 2.00000 + 2.00000i 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 136.2.k.b 2
3.b odd 2 1 1224.2.w.a 2
4.b odd 2 1 272.2.o.e 2
8.b even 2 1 1088.2.o.m 2
8.d odd 2 1 1088.2.o.c 2
12.b even 2 1 2448.2.be.a 2
17.c even 4 1 inner 136.2.k.b 2
17.d even 8 2 2312.2.a.j 2
17.d even 8 2 2312.2.b.d 2
51.f odd 4 1 1224.2.w.a 2
68.f odd 4 1 272.2.o.e 2
68.g odd 8 2 4624.2.a.o 2
136.i even 4 1 1088.2.o.m 2
136.j odd 4 1 1088.2.o.c 2
204.l even 4 1 2448.2.be.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
136.2.k.b 2 1.a even 1 1 trivial
136.2.k.b 2 17.c even 4 1 inner
272.2.o.e 2 4.b odd 2 1
272.2.o.e 2 68.f odd 4 1
1088.2.o.c 2 8.d odd 2 1
1088.2.o.c 2 136.j odd 4 1
1088.2.o.m 2 8.b even 2 1
1088.2.o.m 2 136.i even 4 1
1224.2.w.a 2 3.b odd 2 1
1224.2.w.a 2 51.f odd 4 1
2312.2.a.j 2 17.d even 8 2
2312.2.b.d 2 17.d even 8 2
2448.2.be.a 2 12.b even 2 1
2448.2.be.a 2 204.l even 4 1
4624.2.a.o 2 68.g odd 8 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(136, [\chi])$$:

 $$T_{3}^{2} + 2 T_{3} + 2$$ $$T_{5}^{2} - 4 T_{5} + 8$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$2 + 2 T + T^{2}$$
$5$ $$8 - 4 T + T^{2}$$
$7$ $$8 - 4 T + T^{2}$$
$11$ $$2 - 2 T + T^{2}$$
$13$ $$( -2 + T )^{2}$$
$17$ $$17 + 8 T + T^{2}$$
$19$ $$16 + T^{2}$$
$23$ $$32 + 8 T + T^{2}$$
$29$ $$72 - 12 T + T^{2}$$
$31$ $$72 + 12 T + T^{2}$$
$37$ $$128 + 16 T + T^{2}$$
$41$ $$2 - 2 T + T^{2}$$
$43$ $$4 + T^{2}$$
$47$ $$T^{2}$$
$53$ $$36 + T^{2}$$
$59$ $$196 + T^{2}$$
$61$ $$32 + 8 T + T^{2}$$
$67$ $$( -6 + T )^{2}$$
$71$ $$T^{2}$$
$73$ $$162 + 18 T + T^{2}$$
$79$ $$32 - 8 T + T^{2}$$
$83$ $$36 + T^{2}$$
$89$ $$( -16 + T )^{2}$$
$97$ $$18 - 6 T + T^{2}$$