# Properties

 Label 136.3.e.b Level $136$ Weight $3$ Character orbit 136.e Analytic conductor $3.706$ Analytic rank $0$ Dimension $2$ CM discriminant -8 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: no Analytic conductor: $$3.70573159530$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-2})$$ Defining polynomial: $$x^{2} + 2$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ 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 $$\beta = 4\sqrt{-2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 2 q^{2} + \beta q^{3} + 4 q^{4} + 2 \beta q^{6} + 8 q^{8} -23 q^{9} +O(q^{10})$$ $$q + 2 q^{2} + \beta q^{3} + 4 q^{4} + 2 \beta q^{6} + 8 q^{8} -23 q^{9} -3 \beta q^{11} + 4 \beta q^{12} + 16 q^{16} + ( -1 + 3 \beta ) q^{17} -46 q^{18} + 34 q^{19} -6 \beta q^{22} + 8 \beta q^{24} -25 q^{25} -14 \beta q^{27} + 32 q^{32} + 96 q^{33} + ( -2 + 6 \beta ) q^{34} -92 q^{36} + 68 q^{38} -12 \beta q^{41} + 14 q^{43} -12 \beta q^{44} + 16 \beta q^{48} -49 q^{49} -50 q^{50} + ( -96 - \beta ) q^{51} -28 \beta q^{54} + 34 \beta q^{57} -82 q^{59} + 64 q^{64} + 192 q^{66} -62 q^{67} + ( -4 + 12 \beta ) q^{68} -184 q^{72} + 6 \beta q^{73} -25 \beta q^{75} + 136 q^{76} + 241 q^{81} -24 \beta q^{82} + 158 q^{83} + 28 q^{86} -24 \beta q^{88} -146 q^{89} + 32 \beta q^{96} + 30 \beta q^{97} -98 q^{98} + 69 \beta q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 4 q^{2} + 8 q^{4} + 16 q^{8} - 46 q^{9} + O(q^{10})$$ $$2 q + 4 q^{2} + 8 q^{4} + 16 q^{8} - 46 q^{9} + 32 q^{16} - 2 q^{17} - 92 q^{18} + 68 q^{19} - 50 q^{25} + 64 q^{32} + 192 q^{33} - 4 q^{34} - 184 q^{36} + 136 q^{38} + 28 q^{43} - 98 q^{49} - 100 q^{50} - 192 q^{51} - 164 q^{59} + 128 q^{64} + 384 q^{66} - 124 q^{67} - 8 q^{68} - 368 q^{72} + 272 q^{76} + 482 q^{81} + 316 q^{83} + 56 q^{86} - 292 q^{89} - 196 q^{98} + O(q^{100})$$

## 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
 − 1.41421i 1.41421i
2.00000 5.65685i 4.00000 0 11.3137i 0 8.00000 −23.0000 0
67.2 2.00000 5.65685i 4.00000 0 11.3137i 0 8.00000 −23.0000 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 inner
136.e odd 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.b 2
4.b odd 2 1 544.3.e.a 2
8.b even 2 1 544.3.e.a 2
8.d odd 2 1 CM 136.3.e.b 2
17.b even 2 1 inner 136.3.e.b 2
68.d odd 2 1 544.3.e.a 2
136.e odd 2 1 inner 136.3.e.b 2
136.h even 2 1 544.3.e.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
136.3.e.b 2 1.a even 1 1 trivial
136.3.e.b 2 8.d odd 2 1 CM
136.3.e.b 2 17.b even 2 1 inner
136.3.e.b 2 136.e odd 2 1 inner
544.3.e.a 2 4.b odd 2 1
544.3.e.a 2 8.b even 2 1
544.3.e.a 2 68.d odd 2 1
544.3.e.a 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 $$S_{3}^{\mathrm{new}}(136, [\chi])$$:

 $$T_{3}^{2} + 32$$ $$T_{5}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -2 + T )^{2}$$
$3$ $$32 + T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2}$$
$11$ $$288 + T^{2}$$
$13$ $$T^{2}$$
$17$ $$289 + 2 T + T^{2}$$
$19$ $$( -34 + T )^{2}$$
$23$ $$T^{2}$$
$29$ $$T^{2}$$
$31$ $$T^{2}$$
$37$ $$T^{2}$$
$41$ $$4608 + T^{2}$$
$43$ $$( -14 + T )^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2}$$
$59$ $$( 82 + T )^{2}$$
$61$ $$T^{2}$$
$67$ $$( 62 + T )^{2}$$
$71$ $$T^{2}$$
$73$ $$1152 + T^{2}$$
$79$ $$T^{2}$$
$83$ $$( -158 + T )^{2}$$
$89$ $$( 146 + T )^{2}$$
$97$ $$28800 + T^{2}$$