# Properties

 Label 136.4.s.a Level $136$ Weight $4$ Character orbit 136.s Analytic conductor $8.024$ Analytic rank $0$ Dimension $8$ CM discriminant -8 Inner twists $4$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$136 = 2^{3} \cdot 17$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 136.s (of order $$16$$, degree $$8$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$8.02425976078$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: $$\Q(\zeta_{16})$$ Defining polynomial: $$x^{8} + 1$$ x^8 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{16}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{16}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - 2 \zeta_{16}^{5} + 2 \zeta_{16}) q^{2} + ( - 5 \zeta_{16}^{7} - \zeta_{16}^{5} + \zeta_{16}^{4} + 5 \zeta_{16}^{2} - \zeta_{16} - 1) q^{3} - 8 \zeta_{16}^{6} q^{4} + ( - 10 \zeta_{16}^{7} + 4 \zeta_{16}^{5} - 10 \zeta_{16}^{4} + 10 \zeta_{16}^{3} - 4 \zeta_{16}^{2} + \cdots + 10) q^{6} + \cdots + ( - 13 \zeta_{16}^{6} + 13 \zeta_{16}^{4} - 10 \zeta_{16}^{2} + 27 \zeta_{16} - 10) q^{9} +O(q^{10})$$ q + (-2*z^5 + 2*z) * q^2 + (-5*z^7 - z^5 + z^4 + 5*z^2 - z - 1) * q^3 - 8*z^6 * q^4 + (-10*z^7 + 4*z^5 - 10*z^4 + 10*z^3 - 4*z^2 + 10) * q^6 + (-16*z^7 - 16*z^3) * q^8 + (-13*z^6 + 13*z^4 - 10*z^2 + 27*z - 10) * q^9 $$q + ( - 2 \zeta_{16}^{5} + 2 \zeta_{16}) q^{2} + ( - 5 \zeta_{16}^{7} - \zeta_{16}^{5} + \zeta_{16}^{4} + 5 \zeta_{16}^{2} - \zeta_{16} - 1) q^{3} - 8 \zeta_{16}^{6} q^{4} + ( - 10 \zeta_{16}^{7} + 4 \zeta_{16}^{5} - 10 \zeta_{16}^{4} + 10 \zeta_{16}^{3} - 4 \zeta_{16}^{2} + \cdots + 10) q^{6} + \cdots + (717 \zeta_{16}^{7} - 1340 \zeta_{16}^{6} + 728 \zeta_{16}^{5} + 728 \zeta_{16}^{4} + \cdots - 192) q^{99} +O(q^{100})$$ q + (-2*z^5 + 2*z) * q^2 + (-5*z^7 - z^5 + z^4 + 5*z^2 - z - 1) * q^3 - 8*z^6 * q^4 + (-10*z^7 + 4*z^5 - 10*z^4 + 10*z^3 - 4*z^2 + 10) * q^6 + (-16*z^7 - 16*z^3) * q^8 + (-13*z^6 + 13*z^4 - 10*z^2 + 27*z - 10) * q^9 + (25*z^6 - 25*z^5 + 9*z^4 + 9*z^3 - 25*z^2 + 25*z) * q^11 + (8*z^7 + 8*z^6 - 40*z^5 - 8*z^3 + 8*z^2 + 40) * q^12 - 64*z^4 * q^16 + (45*z^7 + 38*z^5 + 38*z) * q^17 + (-6*z^7 - 54*z^6 + 46*z^5 - 46*z^3 + 54*z^2 + 6*z) * q^18 + (90*z^7 - 53*z^5 + 53*z) * q^19 + (100*z^7 - 100*z^6 + 18*z^5 + 18*z^4 + 18*z + 18) * q^22 + (-80*z^6 - 80*z^5 + 32*z^3 - 80*z^2 + 80*z - 32) * q^24 + 125*z^5 * q^25 + (73*z^7 + 68*z^6 - 68*z^5 - 73*z^4 + 62*z^3 - 27*z^2 - 27*z + 62) * q^27 + (-128*z^5 - 128*z) * q^32 + (-116*z^7 - 5*z^6 + 211*z^5 - 268*z^4 + 211*z^3 - 5*z^2 - 116*z) * q^33 + (90*z^4 + 152*z^2 - 90) * q^34 + (-216*z^7 + 80*z^6 - 104*z^4 + 104*z^2 - 80) * q^36 + (-212*z^6 + 180*z^4 - 180) * q^38 + (-20*z^7 - 20*z^4 - 20*z^3 - 261*z^2 + 261*z + 20) * q^41 + (171*z^6 - 171*z^4 + 26*z^2 + 26) * q^43 + (-200*z^7 + 200*z^4 - 200*z^3 + 72*z^2 + 72*z - 200) * q^44 + (-320*z^6 + 64*z^5 + 64*z^4 - 320*z^3 - 64*z + 64) * q^48 - 343*z^3 * q^49 + (250*z^6 + 250*z^2) * q^50 + (145*z^7 + 149*z^6 + 235*z^4 + 145*z^3 - 301*z + 235) * q^51 + (190*z^7 - 82*z^6 - 270*z^5 + 270*z^4 + 82*z^3 - 190*z^2 - 22*z - 22) * q^54 + (-355*z^7 + 450*z^6 + 106*z^5 - 175*z^4 + 175*z^3 - 106*z^2 - 450*z + 355) * q^57 + (308*z^6 + 308*z^4 + 115*z^2 - 115) * q^59 - 512*z^2 * q^64 + (654*z^6 - 536*z^5 + 190*z^4 - 20*z^3 + 190*z^2 - 536*z + 654) * q^66 + (352*z^7 + 387*z^5 + 387*z^3 + 352*z) * q^67 + (-304*z^7 + 360*z^5 + 304*z^3) * q^68 + (-48*z^7 - 48*z^5 - 432*z^4 + 368*z^3 - 368*z + 432) * q^72 + (-414*z^5 + 414*z^4 - 215*z^3 - 215*z^2 + 414*z - 414) * q^73 + (625*z^7 - 125*z^6 - 125*z^5 + 625*z^4 + 125*z^2 - 125*z) * q^75 + (-424*z^7 + 720*z^5 - 424*z^3) * q^76 + (-351*z^7 + 351*z^5 - 329*z^4 - 270*z^3 - 191*z^2 - 270*z - 329) * q^81 + (522*z^7 - 522*z^6 - 80*z^5 - 80*z^4 - 522*z^3 + 522*z^2) * q^82 + (-434*z^6 + 241*z^4 + 241*z^2 - 434) * q^83 + (290*z^7 - 394*z^5 + 394*z^3 - 290*z) * q^86 + (-144*z^7 - 144*z^6 + 800*z^5 - 800*z^4 + 144*z^3 + 144*z^2) * q^88 + (-983*z^7 - 983*z^5 + 470*z^3 - 470*z) * q^89 + (-640*z^7 + 256*z^6 - 640*z^4 - 640*z^3 + 256*z - 640) * q^96 + (-18*z^7 - 18*z^6 - 955*z^4 - 18*z^3 + 18*z^2 + 955*z) * q^97 + (-686*z^4 - 686) * q^98 + (717*z^7 - 1340*z^6 + 728*z^5 + 728*z^4 - 1340*z^3 + 717*z^2 + 192*z - 192) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 8 q^{3} + 80 q^{6} - 80 q^{9}+O(q^{10})$$ 8 * q - 8 * q^3 + 80 * q^6 - 80 * q^9 $$8 q - 8 q^{3} + 80 q^{6} - 80 q^{9} + 320 q^{12} + 144 q^{22} - 256 q^{24} + 496 q^{27} - 720 q^{34} - 640 q^{36} - 1440 q^{38} + 160 q^{41} + 208 q^{43} - 1600 q^{44} + 512 q^{48} + 1880 q^{51} - 176 q^{54} + 2840 q^{57} - 920 q^{59} + 5232 q^{66} + 3456 q^{72} - 3312 q^{73} - 2632 q^{81} - 3472 q^{83} - 5120 q^{96} - 5488 q^{98} - 1536 q^{99}+O(q^{100})$$ 8 * q - 8 * q^3 + 80 * q^6 - 80 * q^9 + 320 * q^12 + 144 * q^22 - 256 * q^24 + 496 * q^27 - 720 * q^34 - 640 * q^36 - 1440 * q^38 + 160 * q^41 + 208 * q^43 - 1600 * q^44 + 512 * q^48 + 1880 * q^51 - 176 * q^54 + 2840 * q^57 - 920 * q^59 + 5232 * q^66 + 3456 * q^72 - 3312 * q^73 - 2632 * q^81 - 3472 * q^83 - 5120 * q^96 - 5488 * q^98 - 1536 * q^99

## Character values

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

 $$n$$ $$69$$ $$103$$ $$105$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$\zeta_{16}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
3.1
 0.923880 + 0.382683i −0.923880 + 0.382683i 0.382683 + 0.923880i −0.382683 − 0.923880i 0.923880 − 0.382683i −0.923880 − 0.382683i −0.382683 + 0.923880i 0.382683 − 0.923880i
2.61313 1.08239i 6.61374 + 1.31555i 5.65685 5.65685i 0 18.7065 3.72095i 0 8.65914 20.9050i 17.0661 + 7.06900i 0
11.1 −2.61313 1.08239i −1.54267 7.75551i 5.65685 + 5.65685i 0 −4.36332 + 21.9359i 0 −8.65914 20.9050i −32.8234 + 13.5959i 0
27.1 −1.08239 + 2.61313i −3.92868 2.62506i −5.65685 5.65685i 0 11.1120 7.42479i 0 20.9050 8.65914i −1.78887 4.31871i 0
75.1 1.08239 2.61313i −5.14239 + 7.69613i −5.65685 5.65685i 0 14.5449 + 21.7679i 0 −20.9050 + 8.65914i −22.4538 54.2082i 0
91.1 2.61313 + 1.08239i 6.61374 1.31555i 5.65685 + 5.65685i 0 18.7065 + 3.72095i 0 8.65914 + 20.9050i 17.0661 7.06900i 0
99.1 −2.61313 + 1.08239i −1.54267 + 7.75551i 5.65685 5.65685i 0 −4.36332 21.9359i 0 −8.65914 + 20.9050i −32.8234 13.5959i 0
107.1 1.08239 + 2.61313i −5.14239 7.69613i −5.65685 + 5.65685i 0 14.5449 21.7679i 0 −20.9050 8.65914i −22.4538 + 54.2082i 0
131.1 −1.08239 2.61313i −3.92868 + 2.62506i −5.65685 + 5.65685i 0 11.1120 + 7.42479i 0 20.9050 + 8.65914i −1.78887 + 4.31871i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 131.1 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.e odd 16 1 inner
136.s even 16 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 136.4.s.a 8
8.d odd 2 1 CM 136.4.s.a 8
17.e odd 16 1 inner 136.4.s.a 8
136.s even 16 1 inner 136.4.s.a 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
136.4.s.a 8 1.a even 1 1 trivial
136.4.s.a 8 8.d odd 2 1 CM
136.4.s.a 8 17.e odd 16 1 inner
136.4.s.a 8 136.s even 16 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{8} + 8T_{3}^{7} + 72T_{3}^{6} - 480T_{3}^{5} - 4174T_{3}^{4} - 36832T_{3}^{3} + 45332T_{3}^{2} + 1253240T_{3} + 5438402$$ acting on $$S_{4}^{\mathrm{new}}(136, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8} + 4096$$
$3$ $$T^{8} + 8 T^{7} + 72 T^{6} + \cdots + 5438402$$
$5$ $$T^{8}$$
$7$ $$T^{8}$$
$11$ $$T^{8} - 6476 T^{6} + \cdots + 221297408642$$
$13$ $$T^{8}$$
$17$ $$T^{8} + \cdots + 582622237229761$$
$19$ $$T^{8} + 38160 T^{6} + \cdots + 37949591784976$$
$23$ $$T^{8}$$
$29$ $$T^{8}$$
$31$ $$T^{8}$$
$37$ $$T^{8}$$
$41$ $$T^{8} - 160 T^{7} + \cdots + 11\!\cdots\!02$$
$43$ $$(T^{4} - 104 T^{3} + 80322 T^{2} + \cdots + 774053858)^{2}$$
$47$ $$T^{8}$$
$53$ $$T^{8}$$
$59$ $$(T^{4} + 460 T^{3} + 410758 T^{2} + \cdots + 233236802)^{2}$$
$61$ $$T^{8}$$
$67$ $$(T^{4} + 1094692 T^{2} + \cdots + 121606351778)^{2}$$
$71$ $$T^{8}$$
$73$ $$T^{8} + 3312 T^{7} + \cdots + 71\!\cdots\!82$$
$79$ $$T^{8}$$
$83$ $$(T^{4} + 1736 T^{3} + \cdots + 12454523138)^{2}$$
$89$ $$T^{8} + 11402977910412 T^{4} + \cdots + 31\!\cdots\!44$$
$97$ $$T^{8} + 3507988 T^{6} + \cdots + 18\!\cdots\!22$$
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