# Properties

 Label 136.3.p.a Level $136$ Weight $3$ Character orbit 136.p Analytic conductor $3.706$ Analytic rank $0$ Dimension $4$ CM discriminant -8 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$136 = 2^{3} \cdot 17$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 136.p (of order $$8$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.70573159530$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{8}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{8}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -2 \zeta_{8}^{3} q^{2} + ( -2 - 2 \zeta_{8} - 3 \zeta_{8}^{2} + 3 \zeta_{8}^{3} ) q^{3} -4 \zeta_{8}^{2} q^{4} + ( -4 - 6 \zeta_{8} + 6 \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{6} -8 \zeta_{8} q^{8} + ( 7 + 17 \zeta_{8} + 7 \zeta_{8}^{2} ) q^{9} +O(q^{10})$$ $$q -2 \zeta_{8}^{3} q^{2} + ( -2 - 2 \zeta_{8} - 3 \zeta_{8}^{2} + 3 \zeta_{8}^{3} ) q^{3} -4 \zeta_{8}^{2} q^{4} + ( -4 - 6 \zeta_{8} + 6 \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{6} -8 \zeta_{8} q^{8} + ( 7 + 17 \zeta_{8} + 7 \zeta_{8}^{2} ) q^{9} + ( -13 + 6 \zeta_{8} - 6 \zeta_{8}^{2} + 13 \zeta_{8}^{3} ) q^{11} + ( -12 + 12 \zeta_{8} + 8 \zeta_{8}^{2} + 8 \zeta_{8}^{3} ) q^{12} -16 q^{16} + ( -12 - 12 \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{17} + ( 34 + 14 \zeta_{8} - 14 \zeta_{8}^{3} ) q^{18} + ( 12 - 12 \zeta_{8}^{2} ) q^{19} + ( 12 - 12 \zeta_{8} + 26 \zeta_{8}^{2} + 26 \zeta_{8}^{3} ) q^{22} + ( 24 + 16 \zeta_{8} + 16 \zeta_{8}^{2} + 24 \zeta_{8}^{3} ) q^{24} -25 \zeta_{8} q^{25} + ( -17 - 51 \zeta_{8} - 51 \zeta_{8}^{2} - 17 \zeta_{8}^{3} ) q^{27} + 32 \zeta_{8}^{3} q^{32} + ( 16 + 71 \zeta_{8} - 71 \zeta_{8}^{3} ) q^{33} + ( -24 \zeta_{8} - 2 \zeta_{8}^{2} + 24 \zeta_{8}^{3} ) q^{34} + ( 28 - 28 \zeta_{8}^{2} - 68 \zeta_{8}^{3} ) q^{36} + ( -24 \zeta_{8} - 24 \zeta_{8}^{3} ) q^{38} + ( -24 + \zeta_{8} + \zeta_{8}^{2} - 24 \zeta_{8}^{3} ) q^{41} + ( 7 - 60 \zeta_{8} + 7 \zeta_{8}^{2} ) q^{43} + ( -24 + 52 \zeta_{8} + 52 \zeta_{8}^{2} - 24 \zeta_{8}^{3} ) q^{44} + ( 32 + 32 \zeta_{8} + 48 \zeta_{8}^{2} - 48 \zeta_{8}^{3} ) q^{48} + 49 \zeta_{8}^{3} q^{49} -50 q^{50} + ( -14 + 57 \zeta_{8} + 63 \zeta_{8}^{2} - 10 \zeta_{8}^{3} ) q^{51} + ( -102 - 102 \zeta_{8} - 34 \zeta_{8}^{2} + 34 \zeta_{8}^{3} ) q^{54} + ( -60 + 12 \zeta_{8} - 12 \zeta_{8}^{2} + 60 \zeta_{8}^{3} ) q^{57} + ( 41 - 60 \zeta_{8} + 41 \zeta_{8}^{2} ) q^{59} + 64 \zeta_{8}^{2} q^{64} + ( 142 - 142 \zeta_{8}^{2} - 32 \zeta_{8}^{3} ) q^{66} + ( 84 + 31 \zeta_{8} - 31 \zeta_{8}^{3} ) q^{67} + ( -48 - 4 \zeta_{8} + 48 \zeta_{8}^{2} ) q^{68} + ( -56 \zeta_{8} - 136 \zeta_{8}^{2} - 56 \zeta_{8}^{3} ) q^{72} + ( 12 - 12 \zeta_{8} - 59 \zeta_{8}^{2} - 59 \zeta_{8}^{3} ) q^{73} + ( 75 + 50 \zeta_{8} + 50 \zeta_{8}^{2} + 75 \zeta_{8}^{3} ) q^{75} + ( -48 - 48 \zeta_{8}^{2} ) q^{76} + ( 175 \zeta_{8} + 153 \zeta_{8}^{2} + 175 \zeta_{8}^{3} ) q^{81} + ( 2 + 2 \zeta_{8} - 48 \zeta_{8}^{2} + 48 \zeta_{8}^{3} ) q^{82} + ( 79 - 79 \zeta_{8}^{2} + 36 \zeta_{8}^{3} ) q^{83} + ( -120 + 14 \zeta_{8} - 14 \zeta_{8}^{3} ) q^{86} + ( 104 + 104 \zeta_{8} - 48 \zeta_{8}^{2} + 48 \zeta_{8}^{3} ) q^{88} + ( -73 \zeta_{8} + 72 \zeta_{8}^{2} - 73 \zeta_{8}^{3} ) q^{89} + ( 64 + 96 \zeta_{8} - 96 \zeta_{8}^{2} - 64 \zeta_{8}^{3} ) q^{96} + ( 107 - 107 \zeta_{8} - 60 \zeta_{8}^{2} - 60 \zeta_{8}^{3} ) q^{97} + 98 \zeta_{8}^{2} q^{98} + ( -270 - 270 \zeta_{8} - 31 \zeta_{8}^{2} + 31 \zeta_{8}^{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 8 q^{3} - 16 q^{6} + 28 q^{9} + O(q^{10})$$ $$4 q - 8 q^{3} - 16 q^{6} + 28 q^{9} - 52 q^{11} - 48 q^{12} - 64 q^{16} - 48 q^{17} + 136 q^{18} + 48 q^{19} + 48 q^{22} + 96 q^{24} - 68 q^{27} + 64 q^{33} + 112 q^{36} - 96 q^{41} + 28 q^{43} - 96 q^{44} + 128 q^{48} - 200 q^{50} - 56 q^{51} - 408 q^{54} - 240 q^{57} + 164 q^{59} + 568 q^{66} + 336 q^{67} - 192 q^{68} + 48 q^{73} + 300 q^{75} - 192 q^{76} + 8 q^{82} + 316 q^{83} - 480 q^{86} + 416 q^{88} + 256 q^{96} + 428 q^{97} - 1080 q^{99} + O(q^{100})$$

## Character values

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

 $$n$$ $$69$$ $$103$$ $$105$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$\zeta_{8}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
19.1
 0.707107 − 0.707107i 0.707107 + 0.707107i −0.707107 − 0.707107i −0.707107 + 0.707107i
1.41421 + 1.41421i −5.53553 + 2.29289i 4.00000i 0 −11.0711 4.58579i 0 −5.65685 + 5.65685i 19.0208 19.0208i 0
43.1 1.41421 1.41421i −5.53553 2.29289i 4.00000i 0 −11.0711 + 4.58579i 0 −5.65685 5.65685i 19.0208 + 19.0208i 0
59.1 −1.41421 + 1.41421i 1.53553 3.70711i 4.00000i 0 3.07107 + 7.41421i 0 5.65685 + 5.65685i −5.02082 5.02082i 0
83.1 −1.41421 1.41421i 1.53553 + 3.70711i 4.00000i 0 3.07107 7.41421i 0 5.65685 5.65685i −5.02082 + 5.02082i 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.d even 8 1 inner
136.p odd 8 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 136.3.p.a 4
8.d odd 2 1 CM 136.3.p.a 4
17.d even 8 1 inner 136.3.p.a 4
136.p odd 8 1 inner 136.3.p.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
136.3.p.a 4 1.a even 1 1 trivial
136.3.p.a 4 8.d odd 2 1 CM
136.3.p.a 4 17.d even 8 1 inner
136.3.p.a 4 136.p odd 8 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{4} + 8 T_{3}^{3} + 18 T_{3}^{2} + 68 T_{3} + 578$$ acting on $$S_{3}^{\mathrm{new}}(136, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$16 + T^{4}$$
$3$ $$578 + 68 T + 18 T^{2} + 8 T^{3} + T^{4}$$
$5$ $$T^{4}$$
$7$ $$T^{4}$$
$11$ $$167042 + 21964 T + 1398 T^{2} + 52 T^{3} + T^{4}$$
$13$ $$T^{4}$$
$17$ $$83521 + 13872 T + 1152 T^{2} + 48 T^{3} + T^{4}$$
$19$ $$( 288 - 24 T + T^{2} )^{2}$$
$23$ $$T^{4}$$
$29$ $$T^{4}$$
$31$ $$T^{4}$$
$37$ $$T^{4}$$
$41$ $$555458 + 48484 T + 3362 T^{2} + 96 T^{3} + T^{4}$$
$43$ $$12264004 + 98056 T + 392 T^{2} - 28 T^{3} + T^{4}$$
$47$ $$T^{4}$$
$53$ $$T^{4}$$
$59$ $$56644 + 39032 T + 13448 T^{2} - 164 T^{3} + T^{4}$$
$61$ $$T^{4}$$
$67$ $$( 5134 - 168 T + T^{2} )^{2}$$
$71$ $$T^{4}$$
$73$ $$7380482 + 545564 T + 10658 T^{2} - 48 T^{3} + T^{4}$$
$79$ $$T^{4}$$
$83$ $$125126596 - 3534776 T + 49928 T^{2} - 316 T^{3} + T^{4}$$
$89$ $$29964676 + 31684 T^{2} + T^{4}$$
$97$ $$856069442 - 13820252 T + 101574 T^{2} - 428 T^{3} + T^{4}$$