# Properties

 Label 136.1.e.a Level 136 Weight 1 Character orbit 136.e Self dual yes Analytic conductor 0.068 Analytic rank 0 Dimension 1 Projective image $$D_{2}$$ CM/RM discs -8, -136, 17 Inner twists 4

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: yes Analytic conductor: $$0.0678728417181$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image $$D_{2}$$ Projective field Galois closure of $$\Q(\sqrt{-2}, \sqrt{17})$$ Artin image $D_4$ Artin field Galois closure of 4.0.1088.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{4} - q^{8} + q^{9} + O(q^{10})$$ $$q - q^{2} + q^{4} - q^{8} + q^{9} + q^{16} - q^{17} - q^{18} - 2q^{19} - q^{25} - q^{32} + q^{34} + q^{36} + 2q^{38} + 2q^{43} - q^{49} + q^{50} + 2q^{59} + q^{64} - 2q^{67} - q^{68} - q^{72} - 2q^{76} + q^{81} + 2q^{83} - 2q^{86} - 2q^{89} + q^{98} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
67.1
 0
−1.00000 0 1.00000 0 0 0 −1.00000 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
8.d odd 2 1 CM by $$\Q(\sqrt{-2})$$
17.b even 2 1 RM by $$\Q(\sqrt{17})$$
136.e odd 2 1 CM by $$\Q(\sqrt{-34})$$

## Twists

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

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

## Hecke kernels

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

## Hecke characteristic polynomials

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