Properties

 Label 136.2.n.b 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: $$\Z[a_1, \ldots, a_{5}]$$ 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.

 $$f(q)$$ $$=$$ $$q + (\zeta_{8}^{3} - \zeta_{8}^{2} + \zeta_{8} + 1) q^{3} + (\zeta_{8}^{3} + \zeta_{8}^{2} + 2 \zeta_{8} - 2) q^{5} + (\zeta_{8}^{3} + \zeta_{8}^{2} + \zeta_{8} + 1) q^{7} + ( - 2 \zeta_{8}^{2} + \zeta_{8} - 2) q^{9}+O(q^{10})$$ q + (z^3 - z^2 + z + 1) * q^3 + (z^3 + z^2 + 2*z - 2) * q^5 + (z^3 + z^2 + z + 1) * q^7 + (-2*z^2 + z - 2) * q^9 $$q + (\zeta_{8}^{3} - \zeta_{8}^{2} + \zeta_{8} + 1) q^{3} + (\zeta_{8}^{3} + \zeta_{8}^{2} + 2 \zeta_{8} - 2) q^{5} + (\zeta_{8}^{3} + \zeta_{8}^{2} + \zeta_{8} + 1) q^{7} + ( - 2 \zeta_{8}^{2} + \zeta_{8} - 2) q^{9} + ( - 3 \zeta_{8}^{3} + \zeta_{8}^{2} - \zeta_{8} + 3) q^{11} + ( - \zeta_{8}^{3} - 4 \zeta_{8}^{2} - \zeta_{8}) q^{13} + ( - 2 \zeta_{8}^{3} + 4 \zeta_{8}^{2} - 4) q^{15} + (2 \zeta_{8}^{3} + 3 \zeta_{8}^{2} - 2 \zeta_{8}) q^{17} + (2 \zeta_{8}^{3} - 4 \zeta_{8}^{2} + 4) q^{19} + (2 \zeta_{8}^{3} + 2 \zeta_{8}) q^{21} + ( - \zeta_{8}^{3} + 3 \zeta_{8}^{2} - 3 \zeta_{8} + 1) q^{23} + ( - \zeta_{8}^{2} - 5 \zeta_{8} - 1) q^{25} + ( - 2 \zeta_{8}^{3} - 2 \zeta_{8}^{2} - 2 \zeta_{8} - 2) q^{27} + (3 \zeta_{8} - 3) q^{29} + ( - \zeta_{8}^{3} + \zeta_{8}^{2} + \zeta_{8} + 1) q^{31} + (2 \zeta_{8}^{3} - 2 \zeta_{8} + 8) q^{33} + (2 \zeta_{8}^{3} - 2 \zeta_{8} - 6) q^{35} + ( - 5 \zeta_{8}^{3} + 5 \zeta_{8}^{2} - 2 \zeta_{8} - 2) q^{37} + ( - 4 \zeta_{8}^{3} - 4 \zeta_{8}^{2} + 2 \zeta_{8} - 2) q^{39} + ( - 5 \zeta_{8}^{3} - 5) q^{41} + ( - 4 \zeta_{8}^{2} - 6 \zeta_{8} - 4) q^{43} + ( - 5 \zeta_{8}^{3} + 4 \zeta_{8}^{2} - 4 \zeta_{8} + 5) q^{45} + (2 \zeta_{8}^{3} + 4 \zeta_{8}^{2} + 2 \zeta_{8}) q^{47} + ( - 3 \zeta_{8}^{3} + 2 \zeta_{8}^{2} - 2) q^{49} + (7 \zeta_{8}^{3} - \zeta_{8}^{2} - 3 \zeta_{8} + 3) q^{51} + (4 \zeta_{8}^{3} - 3 \zeta_{8}^{2} + 3) q^{53} + (10 \zeta_{8}^{3} + 2 \zeta_{8}^{2} + 10 \zeta_{8}) q^{55} + (2 \zeta_{8}^{3} - 10 \zeta_{8}^{2} + 10 \zeta_{8} - 2) q^{57} + (2 \zeta_{8}^{2} + 10 \zeta_{8} + 2) q^{59} + (2 \zeta_{8}^{3} - \zeta_{8}^{2} - \zeta_{8} + 2) q^{61} + ( - 3 \zeta_{8}^{3} - 3 \zeta_{8}^{2} + \zeta_{8} - 1) q^{63} + ( - 7 \zeta_{8}^{3} + 7 \zeta_{8}^{2} + 7 \zeta_{8} + 7) q^{65} + ( - 2 \zeta_{8}^{3} + 2 \zeta_{8}) q^{67} + (6 \zeta_{8}^{3} - 6 \zeta_{8} + 8) q^{69} + (7 \zeta_{8}^{3} - 7 \zeta_{8}^{2} - 5 \zeta_{8} - 5) q^{71} + (8 \zeta_{8}^{3} + 8 \zeta_{8}^{2} - \zeta_{8} + 1) q^{73} + (3 \zeta_{8}^{3} - 5 \zeta_{8}^{2} - 5 \zeta_{8} + 3) q^{75} + (6 \zeta_{8}^{2} + 4 \zeta_{8} + 6) q^{77} + ( - 3 \zeta_{8}^{3} + \zeta_{8}^{2} - \zeta_{8} + 3) q^{79} + (2 \zeta_{8}^{3} - 3 \zeta_{8}^{2} + 2 \zeta_{8}) q^{81} + ( - 10 \zeta_{8}^{3} + 2 \zeta_{8}^{2} - 2) q^{83} + ( - 12 \zeta_{8}^{2} - \zeta_{8} - 5) q^{85} + ( - 6 \zeta_{8}^{3} + 6 \zeta_{8}^{2} - 6) q^{87} + (3 \zeta_{8}^{3} + 3 \zeta_{8}) q^{89} + ( - 6 \zeta_{8}^{3} - 4 \zeta_{8}^{2} + 4 \zeta_{8} + 6) q^{91} + (2 \zeta_{8}^{2} + 2) q^{93} + ( - 8 \zeta_{8}^{3} + 10 \zeta_{8}^{2} + 10 \zeta_{8} - 8) q^{95} + ( - \zeta_{8}^{3} - \zeta_{8}^{2} + 2 \zeta_{8} - 2) q^{97} + (9 \zeta_{8}^{3} - 9 \zeta_{8}^{2} - \zeta_{8} - 1) q^{99}+O(q^{100})$$ q + (z^3 - z^2 + z + 1) * q^3 + (z^3 + z^2 + 2*z - 2) * q^5 + (z^3 + z^2 + z + 1) * q^7 + (-2*z^2 + z - 2) * q^9 + (-3*z^3 + z^2 - z + 3) * q^11 + (-z^3 - 4*z^2 - z) * q^13 + (-2*z^3 + 4*z^2 - 4) * q^15 + (2*z^3 + 3*z^2 - 2*z) * q^17 + (2*z^3 - 4*z^2 + 4) * q^19 + (2*z^3 + 2*z) * q^21 + (-z^3 + 3*z^2 - 3*z + 1) * q^23 + (-z^2 - 5*z - 1) * q^25 + (-2*z^3 - 2*z^2 - 2*z - 2) * q^27 + (3*z - 3) * q^29 + (-z^3 + z^2 + z + 1) * q^31 + (2*z^3 - 2*z + 8) * q^33 + (2*z^3 - 2*z - 6) * q^35 + (-5*z^3 + 5*z^2 - 2*z - 2) * q^37 + (-4*z^3 - 4*z^2 + 2*z - 2) * q^39 + (-5*z^3 - 5) * q^41 + (-4*z^2 - 6*z - 4) * q^43 + (-5*z^3 + 4*z^2 - 4*z + 5) * q^45 + (2*z^3 + 4*z^2 + 2*z) * q^47 + (-3*z^3 + 2*z^2 - 2) * q^49 + (7*z^3 - z^2 - 3*z + 3) * q^51 + (4*z^3 - 3*z^2 + 3) * q^53 + (10*z^3 + 2*z^2 + 10*z) * q^55 + (2*z^3 - 10*z^2 + 10*z - 2) * q^57 + (2*z^2 + 10*z + 2) * q^59 + (2*z^3 - z^2 - z + 2) * q^61 + (-3*z^3 - 3*z^2 + z - 1) * q^63 + (-7*z^3 + 7*z^2 + 7*z + 7) * q^65 + (-2*z^3 + 2*z) * q^67 + (6*z^3 - 6*z + 8) * q^69 + (7*z^3 - 7*z^2 - 5*z - 5) * q^71 + (8*z^3 + 8*z^2 - z + 1) * q^73 + (3*z^3 - 5*z^2 - 5*z + 3) * q^75 + (6*z^2 + 4*z + 6) * q^77 + (-3*z^3 + z^2 - z + 3) * q^79 + (2*z^3 - 3*z^2 + 2*z) * q^81 + (-10*z^3 + 2*z^2 - 2) * q^83 + (-12*z^2 - z - 5) * q^85 + (-6*z^3 + 6*z^2 - 6) * q^87 + (3*z^3 + 3*z) * q^89 + (-6*z^3 - 4*z^2 + 4*z + 6) * q^91 + (2*z^2 + 2) * q^93 + (-8*z^3 + 10*z^2 + 10*z - 8) * q^95 + (-z^3 - z^2 + 2*z - 2) * q^97 + (9*z^3 - 9*z^2 - z - 1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 4 q^{3} - 8 q^{5} + 4 q^{7} - 8 q^{9}+O(q^{10})$$ 4 * q + 4 * q^3 - 8 * q^5 + 4 * q^7 - 8 * q^9 $$4 q + 4 q^{3} - 8 q^{5} + 4 q^{7} - 8 q^{9} + 12 q^{11} - 16 q^{15} + 16 q^{19} + 4 q^{23} - 4 q^{25} - 8 q^{27} - 12 q^{29} + 4 q^{31} + 32 q^{33} - 24 q^{35} - 8 q^{37} - 8 q^{39} - 20 q^{41} - 16 q^{43} + 20 q^{45} - 8 q^{49} + 12 q^{51} + 12 q^{53} - 8 q^{57} + 8 q^{59} + 8 q^{61} - 4 q^{63} + 28 q^{65} + 32 q^{69} - 20 q^{71} + 4 q^{73} + 12 q^{75} + 24 q^{77} + 12 q^{79} - 8 q^{83} - 20 q^{85} - 24 q^{87} + 24 q^{91} + 8 q^{93} - 32 q^{95} - 8 q^{97} - 4 q^{99}+O(q^{100})$$ 4 * q + 4 * q^3 - 8 * q^5 + 4 * q^7 - 8 * q^9 + 12 * q^11 - 16 * q^15 + 16 * q^19 + 4 * q^23 - 4 * q^25 - 8 * q^27 - 12 * q^29 + 4 * q^31 + 32 * q^33 - 24 * q^35 - 8 * q^37 - 8 * q^39 - 20 * q^41 - 16 * q^43 + 20 * q^45 - 8 * q^49 + 12 * q^51 + 12 * q^53 - 8 * q^57 + 8 * q^59 + 8 * q^61 - 4 * q^63 + 28 * q^65 + 32 * q^69 - 20 * q^71 + 4 * q^73 + 12 * q^75 + 24 * q^77 + 12 * q^79 - 8 * q^83 - 20 * q^85 - 24 * q^87 + 24 * q^91 + 8 * q^93 - 32 * q^95 - 8 * q^97 - 4 * q^99

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 1.00000 + 0.414214i 0 −1.29289 + 3.12132i 0 1.00000 + 2.41421i 0 −1.29289 1.29289i 0
25.1 0 1.00000 2.41421i 0 −2.70711 1.12132i 0 1.00000 0.414214i 0 −2.70711 2.70711i 0
49.1 0 1.00000 + 2.41421i 0 −2.70711 + 1.12132i 0 1.00000 + 0.414214i 0 −2.70711 + 2.70711i 0
121.1 0 1.00000 0.414214i 0 −1.29289 3.12132i 0 1.00000 2.41421i 0 −1.29289 + 1.29289i 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.b 4
3.b odd 2 1 1224.2.bq.b 4
4.b odd 2 1 272.2.v.a 4
17.d even 8 1 inner 136.2.n.b 4
17.e odd 16 2 2312.2.a.t 4
17.e odd 16 2 2312.2.b.i 4
51.g odd 8 1 1224.2.bq.b 4
68.g odd 8 1 272.2.v.a 4
68.i even 16 2 4624.2.a.bo 4

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{4} - 4T_{3}^{3} + 12T_{3}^{2} - 16T_{3} + 8$$ acting on $$S_{2}^{\mathrm{new}}(136, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} - 4 T^{3} + 12 T^{2} - 16 T + 8$$
$5$ $$T^{4} + 8 T^{3} + 34 T^{2} + 84 T + 98$$
$7$ $$T^{4} - 4 T^{3} + 12 T^{2} - 16 T + 8$$
$11$ $$T^{4} - 12 T^{3} + 68 T^{2} + \cdots + 392$$
$13$ $$T^{4} + 36T^{2} + 196$$
$17$ $$T^{4} + 2T^{2} + 289$$
$19$ $$T^{4} - 16 T^{3} + 128 T^{2} + \cdots + 784$$
$23$ $$T^{4} - 4 T^{3} + 36 T^{2} + 32 T + 8$$
$29$ $$T^{4} + 12 T^{3} + 54 T^{2} + \cdots + 162$$
$31$ $$T^{4} - 4 T^{3} + 4 T^{2} + 8$$
$37$ $$T^{4} + 8 T^{3} + 114 T^{2} - 28 T + 2$$
$41$ $$T^{4} + 20 T^{3} + 150 T^{2} + \cdots + 1250$$
$43$ $$T^{4} + 16 T^{3} + 128 T^{2} + \cdots + 16$$
$47$ $$T^{4} + 48T^{2} + 64$$
$53$ $$T^{4} - 12 T^{3} + 72 T^{2} - 24 T + 4$$
$59$ $$T^{4} - 8 T^{3} + 32 T^{2} + \cdots + 8464$$
$61$ $$T^{4} - 8 T^{3} + 18 T^{2} + 4 T + 2$$
$67$ $$(T^{2} - 8)^{2}$$
$71$ $$T^{4} + 20 T^{3} + 108 T^{2} + \cdots + 17672$$
$73$ $$T^{4} - 4 T^{3} + 102 T^{2} + \cdots + 12482$$
$79$ $$T^{4} - 12 T^{3} + 68 T^{2} + \cdots + 392$$
$83$ $$T^{4} + 8 T^{3} + 32 T^{2} + \cdots + 8464$$
$89$ $$(T^{2} + 18)^{2}$$
$97$ $$T^{4} + 8 T^{3} + 18 T^{2} - 4 T + 2$$