# Properties

 Label 136.3.e.a Level $136$ Weight $3$ Character orbit 136.e Self dual yes Analytic conductor $3.706$ Analytic rank $0$ Dimension $2$ CM discriminant -136 Inner twists $4$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: yes Analytic conductor: $$3.70573159530$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{17})$$ Defining polynomial: $$x^{2} - x - 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 2\sqrt{17}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -2 q^{2} + 4 q^{4} + \beta q^{5} -\beta q^{7} -8 q^{8} + 9 q^{9} +O(q^{10})$$ $$q -2 q^{2} + 4 q^{4} + \beta q^{5} -\beta q^{7} -8 q^{8} + 9 q^{9} -2 \beta q^{10} + 2 \beta q^{14} + 16 q^{16} + 17 q^{17} -18 q^{18} + 30 q^{19} + 4 \beta q^{20} -\beta q^{23} + 43 q^{25} -4 \beta q^{28} -7 \beta q^{29} + 7 \beta q^{31} -32 q^{32} -34 q^{34} -68 q^{35} + 36 q^{36} + \beta q^{37} -60 q^{38} -8 \beta q^{40} -50 q^{43} + 9 \beta q^{45} + 2 \beta q^{46} + 19 q^{49} -86 q^{50} + 8 \beta q^{56} + 14 \beta q^{58} -18 q^{59} -7 \beta q^{61} -14 \beta q^{62} -9 \beta q^{63} + 64 q^{64} -66 q^{67} + 68 q^{68} + 136 q^{70} -17 \beta q^{71} -72 q^{72} -2 \beta q^{74} + 120 q^{76} + 7 \beta q^{79} + 16 \beta q^{80} + 81 q^{81} + 30 q^{83} + 17 \beta q^{85} + 100 q^{86} -110 q^{89} -18 \beta q^{90} -4 \beta q^{92} + 30 \beta q^{95} -38 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 4 q^{2} + 8 q^{4} - 16 q^{8} + 18 q^{9} + O(q^{10})$$ $$2 q - 4 q^{2} + 8 q^{4} - 16 q^{8} + 18 q^{9} + 32 q^{16} + 34 q^{17} - 36 q^{18} + 60 q^{19} + 86 q^{25} - 64 q^{32} - 68 q^{34} - 136 q^{35} + 72 q^{36} - 120 q^{38} - 100 q^{43} + 38 q^{49} - 172 q^{50} - 36 q^{59} + 128 q^{64} - 132 q^{67} + 136 q^{68} + 272 q^{70} - 144 q^{72} + 240 q^{76} + 162 q^{81} + 60 q^{83} + 200 q^{86} - 220 q^{89} - 76 q^{98} + O(q^{100})$$

## 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
 −1.56155 2.56155
−2.00000 0 4.00000 −8.24621 0 8.24621 −8.00000 9.00000 16.4924
67.2 −2.00000 0 4.00000 8.24621 0 −8.24621 −8.00000 9.00000 −16.4924
 $$n$$: 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
136.e odd 2 1 CM by $$\Q(\sqrt{-34})$$
8.d odd 2 1 inner
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 136.3.e.a 2
4.b odd 2 1 544.3.e.c 2
8.b even 2 1 544.3.e.c 2
8.d odd 2 1 inner 136.3.e.a 2
17.b even 2 1 inner 136.3.e.a 2
68.d odd 2 1 544.3.e.c 2
136.e odd 2 1 CM 136.3.e.a 2
136.h even 2 1 544.3.e.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
136.3.e.a 2 1.a even 1 1 trivial
136.3.e.a 2 8.d odd 2 1 inner
136.3.e.a 2 17.b even 2 1 inner
136.3.e.a 2 136.e odd 2 1 CM
544.3.e.c 2 4.b odd 2 1
544.3.e.c 2 8.b even 2 1
544.3.e.c 2 68.d odd 2 1
544.3.e.c 2 136.h even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(136, [\chi])$$:

 $$T_{3}$$ $$T_{5}^{2} - 68$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 2 + T )^{2}$$
$3$ $$T^{2}$$
$5$ $$-68 + T^{2}$$
$7$ $$-68 + T^{2}$$
$11$ $$T^{2}$$
$13$ $$T^{2}$$
$17$ $$( -17 + T )^{2}$$
$19$ $$( -30 + T )^{2}$$
$23$ $$-68 + T^{2}$$
$29$ $$-3332 + T^{2}$$
$31$ $$-3332 + T^{2}$$
$37$ $$-68 + T^{2}$$
$41$ $$T^{2}$$
$43$ $$( 50 + T )^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2}$$
$59$ $$( 18 + T )^{2}$$
$61$ $$-3332 + T^{2}$$
$67$ $$( 66 + T )^{2}$$
$71$ $$-19652 + T^{2}$$
$73$ $$T^{2}$$
$79$ $$-3332 + T^{2}$$
$83$ $$( -30 + T )^{2}$$
$89$ $$( 110 + T )^{2}$$
$97$ $$T^{2}$$