# Properties

 Label 136.2.b.a Level $136$ Weight $2$ Character orbit 136.b 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.b (of order $$2$$, degree $$1$$, 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: $$2$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + 2 i q^{3} + 2 i q^{7} - q^{9} +O(q^{10})$$ $$q + 2 i q^{3} + 2 i q^{7} - q^{9} + 2 i q^{11} -2 q^{13} + ( 1 - 4 i ) q^{17} + 4 q^{19} -4 q^{21} -6 i q^{23} + 5 q^{25} + 4 i q^{27} -8 i q^{29} -6 i q^{31} -4 q^{33} + 8 i q^{37} -4 i q^{39} -12 q^{43} -8 q^{47} + 3 q^{49} + ( 8 + 2 i ) q^{51} + 6 q^{53} + 8 i q^{57} + 4 q^{59} + 8 i q^{61} -2 i q^{63} + 4 q^{67} + 12 q^{69} -6 i q^{71} + 8 i q^{73} + 10 i q^{75} -4 q^{77} + 2 i q^{79} -11 q^{81} -4 q^{83} + 16 q^{87} -14 q^{89} -4 i q^{91} + 12 q^{93} + 8 i q^{97} -2 i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{9} + O(q^{10})$$ $$2 q - 2 q^{9} - 4 q^{13} + 2 q^{17} + 8 q^{19} - 8 q^{21} + 10 q^{25} - 8 q^{33} - 24 q^{43} - 16 q^{47} + 6 q^{49} + 16 q^{51} + 12 q^{53} + 8 q^{59} + 8 q^{67} + 24 q^{69} - 8 q^{77} - 22 q^{81} - 8 q^{83} + 32 q^{87} - 28 q^{89} + 24 q^{93} + O(q^{100})$$

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
33.1
 − 1.00000i 1.00000i
0 2.00000i 0 0 0 2.00000i 0 −1.00000 0
33.2 0 2.00000i 0 0 0 2.00000i 0 −1.00000 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.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 136.2.b.a 2
3.b odd 2 1 1224.2.c.c 2
4.b odd 2 1 272.2.b.b 2
5.b even 2 1 3400.2.c.c 2
5.c odd 4 1 3400.2.o.a 2
5.c odd 4 1 3400.2.o.d 2
8.b even 2 1 1088.2.b.c 2
8.d odd 2 1 1088.2.b.d 2
12.b even 2 1 2448.2.c.e 2
17.b even 2 1 inner 136.2.b.a 2
17.c even 4 1 2312.2.a.b 1
17.c even 4 1 2312.2.a.c 1
51.c odd 2 1 1224.2.c.c 2
68.d odd 2 1 272.2.b.b 2
68.f odd 4 1 4624.2.a.b 1
68.f odd 4 1 4624.2.a.e 1
85.c even 2 1 3400.2.c.c 2
85.g odd 4 1 3400.2.o.a 2
85.g odd 4 1 3400.2.o.d 2
136.e odd 2 1 1088.2.b.d 2
136.h even 2 1 1088.2.b.c 2
204.h even 2 1 2448.2.c.e 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
136.2.b.a 2 1.a even 1 1 trivial
136.2.b.a 2 17.b even 2 1 inner
272.2.b.b 2 4.b odd 2 1
272.2.b.b 2 68.d odd 2 1
1088.2.b.c 2 8.b even 2 1
1088.2.b.c 2 136.h even 2 1
1088.2.b.d 2 8.d odd 2 1
1088.2.b.d 2 136.e odd 2 1
1224.2.c.c 2 3.b odd 2 1
1224.2.c.c 2 51.c odd 2 1
2312.2.a.b 1 17.c even 4 1
2312.2.a.c 1 17.c even 4 1
2448.2.c.e 2 12.b even 2 1
2448.2.c.e 2 204.h even 2 1
3400.2.c.c 2 5.b even 2 1
3400.2.c.c 2 85.c even 2 1
3400.2.o.a 2 5.c odd 4 1
3400.2.o.a 2 85.g odd 4 1
3400.2.o.d 2 5.c odd 4 1
3400.2.o.d 2 85.g odd 4 1
4624.2.a.b 1 68.f odd 4 1
4624.2.a.e 1 68.f odd 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} + 4$$ acting on $$S_{2}^{\mathrm{new}}(136, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$4 + T^{2}$$
$5$ $$T^{2}$$
$7$ $$4 + T^{2}$$
$11$ $$4 + T^{2}$$
$13$ $$( 2 + T )^{2}$$
$17$ $$17 - 2 T + T^{2}$$
$19$ $$( -4 + T )^{2}$$
$23$ $$36 + T^{2}$$
$29$ $$64 + T^{2}$$
$31$ $$36 + T^{2}$$
$37$ $$64 + T^{2}$$
$41$ $$T^{2}$$
$43$ $$( 12 + T )^{2}$$
$47$ $$( 8 + T )^{2}$$
$53$ $$( -6 + T )^{2}$$
$59$ $$( -4 + T )^{2}$$
$61$ $$64 + T^{2}$$
$67$ $$( -4 + T )^{2}$$
$71$ $$36 + T^{2}$$
$73$ $$64 + T^{2}$$
$79$ $$4 + T^{2}$$
$83$ $$( 4 + T )^{2}$$
$89$ $$( 14 + T )^{2}$$
$97$ $$64 + T^{2}$$