# Properties

 Label 136.4.b.a Level $136$ Weight $4$ Character orbit 136.b Analytic conductor $8.024$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$136 = 2^{3} \cdot 17$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 136.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$8.02425976078$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: $$\mathbb{Q}[x]/(x^{6} + \cdots)$$ Defining polynomial: $$x^{6} + 81x^{4} + 222x^{2} + 64$$ x^6 + 81*x^4 + 222*x^2 + 64 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{8}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_1 q^{3} - \beta_{4} q^{5} + (\beta_{4} + \beta_{2} - \beta_1) q^{7} + (\beta_{3} - 7) q^{9}+O(q^{10})$$ q - b1 * q^3 - b4 * q^5 + (b4 + b2 - b1) * q^7 + (b3 - 7) * q^9 $$q - \beta_1 q^{3} - \beta_{4} q^{5} + (\beta_{4} + \beta_{2} - \beta_1) q^{7} + (\beta_{3} - 7) q^{9} + (\beta_{2} - \beta_1) q^{11} + (\beta_{5} - 12) q^{13} + (\beta_{5} - \beta_{3} - 4) q^{15} + (\beta_{5} + \beta_{4} + 2 \beta_{3} - \beta_{2} + 2 \beta_1 - 21) q^{17} + ( - \beta_{5} + 3 \beta_{3} - 4) q^{19} + (5 \beta_{3} - 34) q^{21} + ( - 7 \beta_{4} - 2 \beta_{2} + 3 \beta_1) q^{23} + (3 \beta_{5} - \beta_{3} - 19) q^{25} + ( - 10 \beta_{4} - 7 \beta_{2} - 2 \beta_1) q^{27} + ( - 9 \beta_{4} + 6 \beta_{2} + 4 \beta_1) q^{29} + (15 \beta_{4} + 6 \beta_{2} + 3 \beta_1) q^{31} + (\beta_{5} + 4 \beta_{3} - 38) q^{33} + ( - 2 \beta_{5} - 4 \beta_{3} + 68) q^{35} + (9 \beta_{4} + 12 \beta_{2} - 28 \beta_1) q^{37} + (18 \beta_{4} - \beta_{2} + 2 \beta_1) q^{39} + ( - 2 \beta_{4} - 2 \beta_{2} - 24 \beta_1) q^{41} + (\beta_{5} - 13 \beta_{3} + 32) q^{43} + (\beta_{4} + 6 \beta_{2} - 24 \beta_1) q^{45} + ( - 5 \beta_{5} + \beta_{3} - 12) q^{47} + ( - 3 \beta_{5} + 6 \beta_{3} - 3) q^{49} + ( - 2 \beta_{5} - 2 \beta_{4} - 4 \beta_{3} - 15 \beta_{2} + 47 \beta_1 + 76) q^{51} + ( - \beta_{5} - 7 \beta_{3} + 154) q^{53} + (\beta_{5} - 5 \beta_{3} - 76) q^{55} + ( - 48 \beta_{4} - 20 \beta_{2} + 68 \beta_1) q^{57} + ( - 9 \beta_{5} - 11 \beta_{3} + 280) q^{59} + ( - 5 \beta_{4} - 14 \beta_{2} - 52 \beta_1) q^{61} + ( - 23 \beta_{4} - 8 \beta_{2} + 97 \beta_1) q^{63} + (66 \beta_{4} + 10 \beta_{2} + 72 \beta_1) q^{65} + ( - 7 \beta_{5} + 5 \beta_{3} - 52) q^{67} + (5 \beta_{5} - 16 \beta_{3} + 82) q^{69} + (7 \beta_{4} - 19 \beta_{2} - 117 \beta_1) q^{71} + ( - 8 \beta_{4} - 22 \beta_{2} + 44 \beta_1) q^{73} + (64 \beta_{4} + 4 \beta_{2} - 29 \beta_1) q^{75} + ( - 5 \beta_{5} + 2 \beta_{3} - 278) q^{77} + ( - 27 \beta_{4} + 34 \beta_{2} - 25 \beta_1) q^{79} + (3 \beta_{5} - 2 \beta_{3} - 269) q^{81} + ( - \beta_{5} - 23 \beta_{3} + 216) q^{83} + ( - 5 \beta_{5} + 63 \beta_{4} + 7 \beta_{3} + 22 \beta_{2} + 24 \beta_1 + 224) q^{85} + (15 \beta_{5} + 5 \beta_{3} + 76) q^{87} + ( - 3 \beta_{5} - 14 \beta_{3} + 4) q^{89} + ( - 20 \beta_{4} + 21 \beta_{2} - 22 \beta_1) q^{91} + ( - 9 \beta_{5} + 30 \beta_{3} + 138) q^{93} + ( - 68 \beta_{4} + 8 \beta_{2} - 144 \beta_1) q^{95} + ( - 70 \beta_{4} + 8 \beta_{2} + 116 \beta_1) q^{97} + ( - 22 \beta_{4} - 2 \beta_{2} + 73 \beta_1) q^{99}+O(q^{100})$$ q - b1 * q^3 - b4 * q^5 + (b4 + b2 - b1) * q^7 + (b3 - 7) * q^9 + (b2 - b1) * q^11 + (b5 - 12) * q^13 + (b5 - b3 - 4) * q^15 + (b5 + b4 + 2*b3 - b2 + 2*b1 - 21) * q^17 + (-b5 + 3*b3 - 4) * q^19 + (5*b3 - 34) * q^21 + (-7*b4 - 2*b2 + 3*b1) * q^23 + (3*b5 - b3 - 19) * q^25 + (-10*b4 - 7*b2 - 2*b1) * q^27 + (-9*b4 + 6*b2 + 4*b1) * q^29 + (15*b4 + 6*b2 + 3*b1) * q^31 + (b5 + 4*b3 - 38) * q^33 + (-2*b5 - 4*b3 + 68) * q^35 + (9*b4 + 12*b2 - 28*b1) * q^37 + (18*b4 - b2 + 2*b1) * q^39 + (-2*b4 - 2*b2 - 24*b1) * q^41 + (b5 - 13*b3 + 32) * q^43 + (b4 + 6*b2 - 24*b1) * q^45 + (-5*b5 + b3 - 12) * q^47 + (-3*b5 + 6*b3 - 3) * q^49 + (-2*b5 - 2*b4 - 4*b3 - 15*b2 + 47*b1 + 76) * q^51 + (-b5 - 7*b3 + 154) * q^53 + (b5 - 5*b3 - 76) * q^55 + (-48*b4 - 20*b2 + 68*b1) * q^57 + (-9*b5 - 11*b3 + 280) * q^59 + (-5*b4 - 14*b2 - 52*b1) * q^61 + (-23*b4 - 8*b2 + 97*b1) * q^63 + (66*b4 + 10*b2 + 72*b1) * q^65 + (-7*b5 + 5*b3 - 52) * q^67 + (5*b5 - 16*b3 + 82) * q^69 + (7*b4 - 19*b2 - 117*b1) * q^71 + (-8*b4 - 22*b2 + 44*b1) * q^73 + (64*b4 + 4*b2 - 29*b1) * q^75 + (-5*b5 + 2*b3 - 278) * q^77 + (-27*b4 + 34*b2 - 25*b1) * q^79 + (3*b5 - 2*b3 - 269) * q^81 + (-b5 - 23*b3 + 216) * q^83 + (-5*b5 + 63*b4 + 7*b3 + 22*b2 + 24*b1 + 224) * q^85 + (15*b5 + 5*b3 + 76) * q^87 + (-3*b5 - 14*b3 + 4) * q^89 + (-20*b4 + 21*b2 - 22*b1) * q^91 + (-9*b5 + 30*b3 + 138) * q^93 + (-68*b4 + 8*b2 - 144*b1) * q^95 + (-70*b4 + 8*b2 + 116*b1) * q^97 + (-22*b4 - 2*b2 + 73*b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 42 q^{9}+O(q^{10})$$ 6 * q - 42 * q^9 $$6 q - 42 q^{9} - 72 q^{13} - 24 q^{15} - 126 q^{17} - 24 q^{19} - 204 q^{21} - 114 q^{25} - 228 q^{33} + 408 q^{35} + 192 q^{43} - 72 q^{47} - 18 q^{49} + 456 q^{51} + 924 q^{53} - 456 q^{55} + 1680 q^{59} - 312 q^{67} + 492 q^{69} - 1668 q^{77} - 1614 q^{81} + 1296 q^{83} + 1344 q^{85} + 456 q^{87} + 24 q^{89} + 828 q^{93}+O(q^{100})$$ 6 * q - 42 * q^9 - 72 * q^13 - 24 * q^15 - 126 * q^17 - 24 * q^19 - 204 * q^21 - 114 * q^25 - 228 * q^33 + 408 * q^35 + 192 * q^43 - 72 * q^47 - 18 * q^49 + 456 * q^51 + 924 * q^53 - 456 * q^55 + 1680 * q^59 - 312 * q^67 + 492 * q^69 - 1668 * q^77 - 1614 * q^81 + 1296 * q^83 + 1344 * q^85 + 456 * q^87 + 24 * q^89 + 828 * q^93

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} + 81x^{4} + 222x^{2} + 64$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{5} + 85\nu^{3} + 562\nu ) / 68$$ (v^5 + 85*v^3 + 562*v) / 68 $$\beta_{2}$$ $$=$$ $$( -\nu^{5} - 85\nu^{3} - 426\nu ) / 34$$ (-v^5 - 85*v^3 - 426*v) / 34 $$\beta_{3}$$ $$=$$ $$( 5\nu^{4} + 391\nu^{2} + 362 ) / 17$$ (5*v^4 + 391*v^2 + 362) / 17 $$\beta_{4}$$ $$=$$ $$( 15\nu^{5} + 1207\nu^{3} + 2650\nu ) / 68$$ (15*v^5 + 1207*v^3 + 2650*v) / 68 $$\beta_{5}$$ $$=$$ $$( -9\nu^{4} - 731\nu^{2} - 1386 ) / 17$$ (-9*v^4 - 731*v^2 - 1386) / 17
 $$\nu$$ $$=$$ $$( \beta_{2} + 2\beta_1 ) / 4$$ (b2 + 2*b1) / 4 $$\nu^{2}$$ $$=$$ $$( -5\beta_{5} - 9\beta_{3} - 216 ) / 8$$ (-5*b5 - 9*b3 - 216) / 8 $$\nu^{3}$$ $$=$$ $$( -4\beta_{4} - 85\beta_{2} - 110\beta_1 ) / 4$$ (-4*b4 - 85*b2 - 110*b1) / 4 $$\nu^{4}$$ $$=$$ $$( 391\beta_{5} + 731\beta_{3} + 16312 ) / 8$$ (391*b5 + 731*b3 + 16312) / 8 $$\nu^{5}$$ $$=$$ $$( 340\beta_{4} + 6663\beta_{2} + 8498\beta_1 ) / 4$$ (340*b4 + 6663*b2 + 8498*b1) / 4

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
33.1
 1.58186i 0.572006i 8.84141i − 8.84141i − 0.572006i − 1.58186i
0 8.27144i 0 6.42821i 0 24.9151i 0 −41.4167 0
33.2 0 4.49442i 0 18.9829i 0 7.78768i 0 6.80022 0
33.3 0 3.65834i 0 5.50702i 0 18.8836i 0 13.6165 0
33.4 0 3.65834i 0 5.50702i 0 18.8836i 0 13.6165 0
33.5 0 4.49442i 0 18.9829i 0 7.78768i 0 6.80022 0
33.6 0 8.27144i 0 6.42821i 0 24.9151i 0 −41.4167 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 33.6 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.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 136.4.b.a 6
3.b odd 2 1 1224.4.c.c 6
4.b odd 2 1 272.4.b.e 6
17.b even 2 1 inner 136.4.b.a 6
17.c even 4 2 2312.4.a.h 6
51.c odd 2 1 1224.4.c.c 6
68.d odd 2 1 272.4.b.e 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
136.4.b.a 6 1.a even 1 1 trivial
136.4.b.a 6 17.b even 2 1 inner
272.4.b.e 6 4.b odd 2 1
272.4.b.e 6 68.d odd 2 1
1224.4.c.c 6 3.b odd 2 1
1224.4.c.c 6 51.c odd 2 1
2312.4.a.h 6 17.c even 4 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{6} + 102T_{3}^{4} + 2568T_{3}^{2} + 18496$$ acting on $$S_{4}^{\mathrm{new}}(136, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6}$$
$3$ $$T^{6} + 102 T^{4} + 2568 T^{2} + \cdots + 18496$$
$5$ $$T^{6} + 432 T^{4} + 27072 T^{2} + \cdots + 451584$$
$7$ $$T^{6} + 1038 T^{4} + \cdots + 13424896$$
$11$ $$T^{6} + 1062 T^{4} + \cdots + 25482304$$
$13$ $$(T^{3} + 36 T^{2} - 3108 T + 28016)^{2}$$
$17$ $$T^{6} + 126 T^{5} + \cdots + 118587876497$$
$19$ $$(T^{3} + 12 T^{2} - 13968 T + 178496)^{2}$$
$23$ $$T^{6} + 19422 T^{4} + \cdots + 3953894400$$
$29$ $$T^{6} + 92208 T^{4} + \cdots + 1770986054656$$
$31$ $$T^{6} + 93582 T^{4} + \cdots + 10987474749696$$
$37$ $$T^{6} + 205104 T^{4} + \cdots + 7252378264576$$
$41$ $$T^{6} + 62496 T^{4} + \cdots + 8454416891904$$
$43$ $$(T^{3} - 96 T^{2} - 162864 T - 20641152)^{2}$$
$47$ $$(T^{3} + 36 T^{2} - 92928 T - 11699200)^{2}$$
$53$ $$(T^{3} - 462 T^{2} + 29052 T + 1630776)^{2}$$
$59$ $$(T^{3} - 840 T^{2} - 82032 T + 116758016)^{2}$$
$61$ $$T^{6} + 428592 T^{4} + \cdots + 11\!\cdots\!76$$
$67$ $$(T^{3} + 156 T^{2} - 215568 T - 49226176)^{2}$$
$71$ $$T^{6} + 1739790 T^{4} + \cdots + 47\!\cdots\!76$$
$73$ $$T^{6} + 616896 T^{4} + \cdots + 25\!\cdots\!64$$
$79$ $$T^{6} + 1893870 T^{4} + \cdots + 10\!\cdots\!44$$
$83$ $$(T^{3} - 648 T^{2} - 321456 T - 10110464)^{2}$$
$89$ $$(T^{3} - 12 T^{2} - 174948 T + 2334896)^{2}$$
$97$ $$T^{6} + 3573984 T^{4} + \cdots + 13\!\cdots\!16$$