# Properties

 Label 136.1.j.a Level $136$ Weight $1$ Character orbit 136.j Analytic conductor $0.068$ Analytic rank $0$ Dimension $2$ Projective image $D_{4}$ CM discriminant -8 Inner twists $4$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$0.0678728417181$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{4}$$ Projective field: Galois closure of 4.0.314432.1 Artin image: $C_4\wr C_2$ Artin field: Galois closure of 8.0.20123648.1

## $q$-expansion

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
115.1
 1.00000i − 1.00000i
1.00000i −1.00000 1.00000i −1.00000 0 −1.00000 + 1.00000i 0 1.00000i 1.00000i 0
123.1 1.00000i −1.00000 + 1.00000i −1.00000 0 −1.00000 1.00000i 0 1.00000i 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
8.d odd 2 1 CM by $$\Q(\sqrt{-2})$$
17.c even 4 1 inner
136.j odd 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 136.1.j.a 2
3.b odd 2 1 1224.1.s.a 2
4.b odd 2 1 544.1.n.a 2
5.b even 2 1 3400.1.y.a 2
5.c odd 4 1 3400.1.bc.a 2
5.c odd 4 1 3400.1.bc.b 2
8.b even 2 1 544.1.n.a 2
8.d odd 2 1 CM 136.1.j.a 2
17.b even 2 1 2312.1.j.b 2
17.c even 4 1 inner 136.1.j.a 2
17.c even 4 1 2312.1.j.b 2
17.d even 8 2 2312.1.e.a 2
17.d even 8 2 2312.1.f.b 2
17.e odd 16 8 2312.1.p.e 8
24.f even 2 1 1224.1.s.a 2
40.e odd 2 1 3400.1.y.a 2
40.k even 4 1 3400.1.bc.a 2
40.k even 4 1 3400.1.bc.b 2
51.f odd 4 1 1224.1.s.a 2
68.f odd 4 1 544.1.n.a 2
85.f odd 4 1 3400.1.bc.a 2
85.i odd 4 1 3400.1.bc.b 2
85.j even 4 1 3400.1.y.a 2
136.e odd 2 1 2312.1.j.b 2
136.i even 4 1 544.1.n.a 2
136.j odd 4 1 inner 136.1.j.a 2
136.j odd 4 1 2312.1.j.b 2
136.p odd 8 2 2312.1.e.a 2
136.p odd 8 2 2312.1.f.b 2
136.s even 16 8 2312.1.p.e 8
408.q even 4 1 1224.1.s.a 2
680.t even 4 1 3400.1.bc.a 2
680.bc odd 4 1 3400.1.y.a 2
680.bl even 4 1 3400.1.bc.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
136.1.j.a 2 1.a even 1 1 trivial
136.1.j.a 2 8.d odd 2 1 CM
136.1.j.a 2 17.c even 4 1 inner
136.1.j.a 2 136.j odd 4 1 inner
544.1.n.a 2 4.b odd 2 1
544.1.n.a 2 8.b even 2 1
544.1.n.a 2 68.f odd 4 1
544.1.n.a 2 136.i even 4 1
1224.1.s.a 2 3.b odd 2 1
1224.1.s.a 2 24.f even 2 1
1224.1.s.a 2 51.f odd 4 1
1224.1.s.a 2 408.q even 4 1
2312.1.e.a 2 17.d even 8 2
2312.1.e.a 2 136.p odd 8 2
2312.1.f.b 2 17.d even 8 2
2312.1.f.b 2 136.p odd 8 2
2312.1.j.b 2 17.b even 2 1
2312.1.j.b 2 17.c even 4 1
2312.1.j.b 2 136.e odd 2 1
2312.1.j.b 2 136.j odd 4 1
2312.1.p.e 8 17.e odd 16 8
2312.1.p.e 8 136.s even 16 8
3400.1.y.a 2 5.b even 2 1
3400.1.y.a 2 40.e odd 2 1
3400.1.y.a 2 85.j even 4 1
3400.1.y.a 2 680.bc odd 4 1
3400.1.bc.a 2 5.c odd 4 1
3400.1.bc.a 2 40.k even 4 1
3400.1.bc.a 2 85.f odd 4 1
3400.1.bc.a 2 680.t even 4 1
3400.1.bc.b 2 5.c odd 4 1
3400.1.bc.b 2 40.k even 4 1
3400.1.bc.b 2 85.i odd 4 1
3400.1.bc.b 2 680.bl even 4 1

## Hecke kernels

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

## Hecke characteristic polynomials

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