Properties

 Label 136.2.n.a Level $136$ Weight $2$ Character orbit 136.n Analytic conductor $1.086$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

Related objects

Newspace parameters

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

Newform invariants

 Self dual: no Analytic conductor: $$1.08596546749$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ x^4 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

$q$-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.

 $$f(q)$$ $$=$$ $$q + ( - \zeta_{8} - 1) q^{3} + ( - \zeta_{8} + 1) q^{5} + ( - \zeta_{8}^{3} - 2 \zeta_{8}^{2} - 2 \zeta_{8} - 1) q^{7} + (\zeta_{8}^{2} - \zeta_{8} + 1) q^{9}+O(q^{10})$$ q + (-z - 1) * q^3 + (-z + 1) * q^5 + (-z^3 - 2*z^2 - 2*z - 1) * q^7 + (z^2 - z + 1) * q^9 $$q + ( - \zeta_{8} - 1) q^{3} + ( - \zeta_{8} + 1) q^{5} + ( - \zeta_{8}^{3} - 2 \zeta_{8}^{2} - 2 \zeta_{8} - 1) q^{7} + (\zeta_{8}^{2} - \zeta_{8} + 1) q^{9} + ( - \zeta_{8}^{3} + 1) q^{11} + (\zeta_{8}^{2} - 1) q^{15} + ( - 4 \zeta_{8}^{2} + 1) q^{17} + (2 \zeta_{8}^{3} + 3 \zeta_{8}^{2} - 3) q^{19} + (3 \zeta_{8}^{3} + 4 \zeta_{8}^{2} + 3 \zeta_{8}) q^{21} + ( - \zeta_{8}^{3} - 4 \zeta_{8}^{2} + 4 \zeta_{8} + 1) q^{23} + (\zeta_{8}^{2} + 3 \zeta_{8} + 1) q^{25} + ( - \zeta_{8}^{3} + 3 \zeta_{8}^{2} + 3 \zeta_{8} - 1) q^{27} + (3 \zeta_{8} - 3) q^{29} + (3 \zeta_{8} + 3) q^{31} + (\zeta_{8}^{3} - \zeta_{8} - 2) q^{33} + (\zeta_{8}^{3} - \zeta_{8} - 2) q^{35} + (\zeta_{8} + 1) q^{37} + ( - 3 \zeta_{8}^{3} - 4 \zeta_{8}^{2} - 4 \zeta_{8} - 3) q^{41} + (3 \zeta_{8}^{2} - 6 \zeta_{8} + 3) q^{43} + ( - \zeta_{8}^{3} + 2 \zeta_{8}^{2} - 2 \zeta_{8} + 1) q^{45} + (2 \zeta_{8}^{3} - 10 \zeta_{8}^{2} + 2 \zeta_{8}) q^{47} + (3 \zeta_{8}^{3} + 7 \zeta_{8}^{2} - 7) q^{49} + (4 \zeta_{8}^{3} + 4 \zeta_{8}^{2} - \zeta_{8} - 1) q^{51} + ( - 4 \zeta_{8}^{3} - 5 \zeta_{8}^{2} + 5) q^{53} + ( - \zeta_{8}^{3} - \zeta_{8}) q^{55} + ( - 5 \zeta_{8}^{3} - 3 \zeta_{8}^{2} + 3 \zeta_{8} + 5) q^{57} + (7 \zeta_{8}^{2} + 2 \zeta_{8} + 7) q^{59} + ( - 7 \zeta_{8}^{3} - 4 \zeta_{8}^{2} - 4 \zeta_{8} - 7) q^{61} + ( - \zeta_{8}^{3} - \zeta_{8}^{2}) q^{63} + (4 \zeta_{8}^{3} - 4 \zeta_{8} + 4) q^{67} + (5 \zeta_{8}^{3} - 5 \zeta_{8} - 2) q^{69} + (4 \zeta_{8}^{3} - 4 \zeta_{8}^{2} + 3 \zeta_{8} + 3) q^{71} + (8 \zeta_{8}^{3} + 8 \zeta_{8}^{2} - \zeta_{8} + 1) q^{73} + ( - \zeta_{8}^{3} - 4 \zeta_{8}^{2} - 4 \zeta_{8} - 1) q^{75} + ( - 3 \zeta_{8}^{2} - 4 \zeta_{8} - 3) q^{77} + ( - 9 \zeta_{8}^{3} + 4 \zeta_{8}^{2} - 4 \zeta_{8} + 9) q^{79} + ( - 5 \zeta_{8}^{3} - 3 \zeta_{8}^{2} - 5 \zeta_{8}) q^{81} + ( - 6 \zeta_{8}^{3} + 3 \zeta_{8}^{2} - 3) q^{83} + (4 \zeta_{8}^{3} - 4 \zeta_{8}^{2} - \zeta_{8} + 1) q^{85} + ( - 3 \zeta_{8}^{2} + 3) q^{87} + (4 \zeta_{8}^{3} + 4 \zeta_{8}^{2} + 4 \zeta_{8}) q^{89} + ( - 3 \zeta_{8}^{2} - 6 \zeta_{8} - 3) q^{93} + ( - \zeta_{8}^{3} + 3 \zeta_{8}^{2} + 3 \zeta_{8} - 1) q^{95} + ( - 4 \zeta_{8}^{3} - 4 \zeta_{8}^{2} + 7 \zeta_{8} - 7) q^{97} + ( - \zeta_{8}^{3} + \zeta_{8}^{2}) q^{99}+O(q^{100})$$ q + (-z - 1) * q^3 + (-z + 1) * q^5 + (-z^3 - 2*z^2 - 2*z - 1) * q^7 + (z^2 - z + 1) * q^9 + (-z^3 + 1) * q^11 + (z^2 - 1) * q^15 + (-4*z^2 + 1) * q^17 + (2*z^3 + 3*z^2 - 3) * q^19 + (3*z^3 + 4*z^2 + 3*z) * q^21 + (-z^3 - 4*z^2 + 4*z + 1) * q^23 + (z^2 + 3*z + 1) * q^25 + (-z^3 + 3*z^2 + 3*z - 1) * q^27 + (3*z - 3) * q^29 + (3*z + 3) * q^31 + (z^3 - z - 2) * q^33 + (z^3 - z - 2) * q^35 + (z + 1) * q^37 + (-3*z^3 - 4*z^2 - 4*z - 3) * q^41 + (3*z^2 - 6*z + 3) * q^43 + (-z^3 + 2*z^2 - 2*z + 1) * q^45 + (2*z^3 - 10*z^2 + 2*z) * q^47 + (3*z^3 + 7*z^2 - 7) * q^49 + (4*z^3 + 4*z^2 - z - 1) * q^51 + (-4*z^3 - 5*z^2 + 5) * q^53 + (-z^3 - z) * q^55 + (-5*z^3 - 3*z^2 + 3*z + 5) * q^57 + (7*z^2 + 2*z + 7) * q^59 + (-7*z^3 - 4*z^2 - 4*z - 7) * q^61 + (-z^3 - z^2) * q^63 + (4*z^3 - 4*z + 4) * q^67 + (5*z^3 - 5*z - 2) * q^69 + (4*z^3 - 4*z^2 + 3*z + 3) * q^71 + (8*z^3 + 8*z^2 - z + 1) * q^73 + (-z^3 - 4*z^2 - 4*z - 1) * q^75 + (-3*z^2 - 4*z - 3) * q^77 + (-9*z^3 + 4*z^2 - 4*z + 9) * q^79 + (-5*z^3 - 3*z^2 - 5*z) * q^81 + (-6*z^3 + 3*z^2 - 3) * q^83 + (4*z^3 - 4*z^2 - z + 1) * q^85 + (-3*z^2 + 3) * q^87 + (4*z^3 + 4*z^2 + 4*z) * q^89 + (-3*z^2 - 6*z - 3) * q^93 + (-z^3 + 3*z^2 + 3*z - 1) * q^95 + (-4*z^3 - 4*z^2 + 7*z - 7) * q^97 + (-z^3 + z^2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{3} + 4 q^{5} - 4 q^{7} + 4 q^{9}+O(q^{10})$$ 4 * q - 4 * q^3 + 4 * q^5 - 4 * q^7 + 4 * q^9 $$4 q - 4 q^{3} + 4 q^{5} - 4 q^{7} + 4 q^{9} + 4 q^{11} - 4 q^{15} + 4 q^{17} - 12 q^{19} + 4 q^{23} + 4 q^{25} - 4 q^{27} - 12 q^{29} + 12 q^{31} - 8 q^{33} - 8 q^{35} + 4 q^{37} - 12 q^{41} + 12 q^{43} + 4 q^{45} - 28 q^{49} - 4 q^{51} + 20 q^{53} + 20 q^{57} + 28 q^{59} - 28 q^{61} + 16 q^{67} - 8 q^{69} + 12 q^{71} + 4 q^{73} - 4 q^{75} - 12 q^{77} + 36 q^{79} - 12 q^{83} + 4 q^{85} + 12 q^{87} - 12 q^{93} - 4 q^{95} - 28 q^{97}+O(q^{100})$$ 4 * q - 4 * q^3 + 4 * q^5 - 4 * q^7 + 4 * q^9 + 4 * q^11 - 4 * q^15 + 4 * q^17 - 12 * q^19 + 4 * q^23 + 4 * q^25 - 4 * q^27 - 12 * q^29 + 12 * q^31 - 8 * q^33 - 8 * q^35 + 4 * q^37 - 12 * q^41 + 12 * q^43 + 4 * q^45 - 28 * q^49 - 4 * q^51 + 20 * q^53 + 20 * q^57 + 28 * q^59 - 28 * q^61 + 16 * q^67 - 8 * q^69 + 12 * q^71 + 4 * q^73 - 4 * q^75 - 12 * q^77 + 36 * q^79 - 12 * q^83 + 4 * q^85 + 12 * q^87 - 12 * q^93 - 4 * q^95 - 28 * q^97

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
9.1
 0.707107 + 0.707107i −0.707107 − 0.707107i −0.707107 + 0.707107i 0.707107 − 0.707107i
0 −1.70711 0.707107i 0 0.292893 0.707107i 0 −1.70711 4.12132i 0 0.292893 + 0.292893i 0
25.1 0 −0.292893 + 0.707107i 0 1.70711 + 0.707107i 0 −0.292893 + 0.121320i 0 1.70711 + 1.70711i 0
49.1 0 −0.292893 0.707107i 0 1.70711 0.707107i 0 −0.292893 0.121320i 0 1.70711 1.70711i 0
121.1 0 −1.70711 + 0.707107i 0 0.292893 + 0.707107i 0 −1.70711 + 4.12132i 0 0.292893 0.292893i 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
17.d even 8 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 136.2.n.a 4
3.b odd 2 1 1224.2.bq.a 4
4.b odd 2 1 272.2.v.e 4
17.d even 8 1 inner 136.2.n.a 4
17.e odd 16 2 2312.2.a.s 4
17.e odd 16 2 2312.2.b.j 4
51.g odd 8 1 1224.2.bq.a 4
68.g odd 8 1 272.2.v.e 4
68.i even 16 2 4624.2.a.bm 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
136.2.n.a 4 1.a even 1 1 trivial
136.2.n.a 4 17.d even 8 1 inner
272.2.v.e 4 4.b odd 2 1
272.2.v.e 4 68.g odd 8 1
1224.2.bq.a 4 3.b odd 2 1
1224.2.bq.a 4 51.g odd 8 1
2312.2.a.s 4 17.e odd 16 2
2312.2.b.j 4 17.e odd 16 2
4624.2.a.bm 4 68.i even 16 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{4} + 4T_{3}^{3} + 6T_{3}^{2} + 4T_{3} + 2$$ acting on $$S_{2}^{\mathrm{new}}(136, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} + 4 T^{3} + 6 T^{2} + 4 T + 2$$
$5$ $$T^{4} - 4 T^{3} + 6 T^{2} - 4 T + 2$$
$7$ $$T^{4} + 4 T^{3} + 22 T^{2} + 12 T + 2$$
$11$ $$T^{4} - 4 T^{3} + 6 T^{2} - 4 T + 2$$
$13$ $$T^{4}$$
$17$ $$(T^{2} - 2 T + 17)^{2}$$
$19$ $$T^{4} + 12 T^{3} + 72 T^{2} + \cdots + 196$$
$23$ $$T^{4} - 4 T^{3} + 22 T^{2} + \cdots + 1058$$
$29$ $$T^{4} + 12 T^{3} + 54 T^{2} + \cdots + 162$$
$31$ $$T^{4} - 12 T^{3} + 54 T^{2} + \cdots + 162$$
$37$ $$T^{4} - 4 T^{3} + 6 T^{2} - 4 T + 2$$
$41$ $$T^{4} + 12 T^{3} + 134 T^{2} + \cdots + 578$$
$43$ $$T^{4} - 12 T^{3} + 72 T^{2} + \cdots + 324$$
$47$ $$T^{4} + 216T^{2} + 8464$$
$53$ $$T^{4} - 20 T^{3} + 200 T^{2} + \cdots + 1156$$
$59$ $$T^{4} - 28 T^{3} + 392 T^{2} + \cdots + 8836$$
$61$ $$T^{4} + 28 T^{3} + 438 T^{2} + \cdots + 15842$$
$67$ $$(T^{2} - 8 T - 16)^{2}$$
$71$ $$T^{4} - 12 T^{3} + 134 T^{2} + \cdots + 578$$
$73$ $$T^{4} - 4 T^{3} + 102 T^{2} + \cdots + 12482$$
$79$ $$T^{4} - 36 T^{3} + 662 T^{2} + \cdots + 37538$$
$83$ $$T^{4} + 12 T^{3} + 72 T^{2} + \cdots + 324$$
$89$ $$T^{4} + 96T^{2} + 256$$
$97$ $$T^{4} + 28 T^{3} + 214 T^{2} + \cdots + 1058$$