# Properties

 Label 256.8.a.q Level $256$ Weight $8$ Character orbit 256.a Self dual yes Analytic conductor $79.971$ Analytic rank $1$ Dimension $6$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$256 = 2^{8}$$ Weight: $$k$$ $$=$$ $$8$$ Character orbit: $$[\chi]$$ $$=$$ 256.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$79.9705665239$$ Analytic rank: $$1$$ Dimension: $$6$$ Coefficient field: $$\mathbb{Q}[x]/(x^{6} - \cdots)$$ Defining polynomial: $$x^{6} - 163x^{4} + 4820x^{2} - 15296$$ x^6 - 163*x^4 + 4820*x^2 - 15296 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{27}$$ Twist minimal: no (minimal twist has level 8) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-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.

 $$f(q)$$ $$=$$ $$q - \beta_1 q^{3} + (\beta_{2} + \beta_1) q^{5} + (\beta_{4} - 115) q^{7} + ( - \beta_{5} - \beta_{4} + 487) q^{9}+O(q^{10})$$ q - b1 * q^3 + (b2 + b1) * q^5 + (b4 - 115) * q^7 + (-b5 - b4 + 487) * q^9 $$q - \beta_1 q^{3} + (\beta_{2} + \beta_1) q^{5} + (\beta_{4} - 115) q^{7} + ( - \beta_{5} - \beta_{4} + 487) q^{9} + ( - \beta_{3} - 7 \beta_{2} - 3 \beta_1) q^{11} + (2 \beta_{3} + 3 \beta_{2} + 21 \beta_1) q^{13} + (8 \beta_{5} - \beta_{4} - 2981) q^{15} + (7 \beta_{5} - 25 \beta_{4} + 248) q^{17} + (5 \beta_{3} - 93 \beta_{2} - 105 \beta_1) q^{19} + ( - 6 \beta_{3} - 6 \beta_{2} + 452 \beta_1) q^{21} + ( - 8 \beta_{5} + 11 \beta_{4} - 217) q^{23} + ( - 6 \beta_{5} - 38 \beta_{4} + 6567) q^{25} + (3 \beta_{3} - 363 \beta_{2} - 310 \beta_1) q^{27} + ( - 16 \beta_{3} - 79 \beta_{2} + 2337 \beta_1) q^{29} + ( - 72 \beta_{5} - 12 \beta_{4} + 14908) q^{31} + ( - 63 \beta_{5} + 129 \beta_{4} + 8958) q^{33} + ( - 50 \beta_{3} - 478 \beta_{2} - 368 \beta_1) q^{35} + (58 \beta_{3} - 33 \beta_{2} + 6633 \beta_1) q^{37} + (64 \beta_{5} - 221 \beta_{4} - 54649) q^{39} + (142 \beta_{5} + 302 \beta_{4} - 87022) q^{41} + (38 \beta_{3} + 522 \beta_{2} - 579 \beta_1) q^{43} + (30 \beta_{3} + 771 \beta_{2} + 13841 \beta_1) q^{45} + (168 \beta_{5} + 210 \beta_{4} - 261198) q^{47} + (168 \beta_{5} + 168 \beta_{4} - 85287) q^{49} + (171 \beta_{3} + 2733 \beta_{2} + 3038 \beta_1) q^{51} + ( - 170 \beta_{3} + 535 \beta_{2} + 17949 \beta_1) q^{53} + ( - 96 \beta_{5} + 1237 \beta_{4} - 545423) q^{55} + ( - 701 \beta_{5} - 509 \beta_{4} + 315386) q^{57} + ( - 224 \beta_{3} + 3680 \beta_{2} + 10933 \beta_1) q^{59} + (66 \beta_{3} - 3993 \beta_{2} + 7353 \beta_1) q^{61} + (344 \beta_{5} - 1015 \beta_{4} - 962579) q^{63} + (50 \beta_{5} - 2030 \beta_{4} + 236740) q^{65} + ( - 123 \beta_{3} + 1443 \beta_{2} - 4017 \beta_1) q^{67} + ( - 90 \beta_{3} - 3018 \beta_{2} - 9460 \beta_1) q^{69} + ( - 504 \beta_{5} - 3279 \beta_{4} - 1264923) q^{71} + (1239 \beta_{5} - 809 \beta_{4} - 348404) q^{73} + (210 \beta_{3} - 1986 \beta_{2} - 29411 \beta_1) q^{75} + ( - 218 \beta_{3} + 12598 \beta_{2} - 31860 \beta_1) q^{77} + ( - 2016 \beta_{5} + 2054 \beta_{4} - 2669330) q^{79} + ( - 631 \beta_{5} + 2249 \beta_{4} - 121049) q^{81} + ( - 568 \beta_{3} - 9992 \beta_{2} + 13319 \beta_1) q^{83} + (1390 \beta_{3} + 9876 \beta_{2} - 85134 \beta_1) q^{85} + (1608 \beta_{5} + 4383 \beta_{4} - 6244293) q^{87} + (439 \beta_{5} + 1079 \beta_{4} - 362020) q^{89} + (1142 \beta_{3} - 19782 \beta_{2} + 56016 \beta_1) q^{91} + ( - 144 \beta_{3} - 26496 \beta_{2} - 139408 \beta_1) q^{93} + (1184 \beta_{5} - 1313 \beta_{4} - 8089613) q^{95} + ( - 849 \beta_{5} + 7183 \beta_{4} - 183496) q^{97} + (1224 \beta_{3} - 8712 \beta_{2} - 64323 \beta_1) q^{99}+O(q^{100})$$ q - b1 * q^3 + (b2 + b1) * q^5 + (b4 - 115) * q^7 + (-b5 - b4 + 487) * q^9 + (-b3 - 7*b2 - 3*b1) * q^11 + (2*b3 + 3*b2 + 21*b1) * q^13 + (8*b5 - b4 - 2981) * q^15 + (7*b5 - 25*b4 + 248) * q^17 + (5*b3 - 93*b2 - 105*b1) * q^19 + (-6*b3 - 6*b2 + 452*b1) * q^21 + (-8*b5 + 11*b4 - 217) * q^23 + (-6*b5 - 38*b4 + 6567) * q^25 + (3*b3 - 363*b2 - 310*b1) * q^27 + (-16*b3 - 79*b2 + 2337*b1) * q^29 + (-72*b5 - 12*b4 + 14908) * q^31 + (-63*b5 + 129*b4 + 8958) * q^33 + (-50*b3 - 478*b2 - 368*b1) * q^35 + (58*b3 - 33*b2 + 6633*b1) * q^37 + (64*b5 - 221*b4 - 54649) * q^39 + (142*b5 + 302*b4 - 87022) * q^41 + (38*b3 + 522*b2 - 579*b1) * q^43 + (30*b3 + 771*b2 + 13841*b1) * q^45 + (168*b5 + 210*b4 - 261198) * q^47 + (168*b5 + 168*b4 - 85287) * q^49 + (171*b3 + 2733*b2 + 3038*b1) * q^51 + (-170*b3 + 535*b2 + 17949*b1) * q^53 + (-96*b5 + 1237*b4 - 545423) * q^55 + (-701*b5 - 509*b4 + 315386) * q^57 + (-224*b3 + 3680*b2 + 10933*b1) * q^59 + (66*b3 - 3993*b2 + 7353*b1) * q^61 + (344*b5 - 1015*b4 - 962579) * q^63 + (50*b5 - 2030*b4 + 236740) * q^65 + (-123*b3 + 1443*b2 - 4017*b1) * q^67 + (-90*b3 - 3018*b2 - 9460*b1) * q^69 + (-504*b5 - 3279*b4 - 1264923) * q^71 + (1239*b5 - 809*b4 - 348404) * q^73 + (210*b3 - 1986*b2 - 29411*b1) * q^75 + (-218*b3 + 12598*b2 - 31860*b1) * q^77 + (-2016*b5 + 2054*b4 - 2669330) * q^79 + (-631*b5 + 2249*b4 - 121049) * q^81 + (-568*b3 - 9992*b2 + 13319*b1) * q^83 + (1390*b3 + 9876*b2 - 85134*b1) * q^85 + (1608*b5 + 4383*b4 - 6244293) * q^87 + (439*b5 + 1079*b4 - 362020) * q^89 + (1142*b3 - 19782*b2 + 56016*b1) * q^91 + (-144*b3 - 26496*b2 - 139408*b1) * q^93 + (1184*b5 - 1313*b4 - 8089613) * q^95 + (-849*b5 + 7183*b4 - 183496) * q^97 + (1224*b3 - 8712*b2 - 64323*b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 688 q^{7} + 2918 q^{9}+O(q^{10})$$ 6 * q - 688 * q^7 + 2918 * q^9 $$6 q - 688 q^{7} + 2918 q^{9} - 17872 q^{15} + 1452 q^{17} - 1296 q^{23} + 39314 q^{25} + 89280 q^{31} + 53880 q^{33} - 328208 q^{39} - 521244 q^{41} - 1566432 q^{47} - 511050 q^{49} - 3270256 q^{55} + 1889896 q^{57} - 5776816 q^{63} + 1416480 q^{65} - 7597104 q^{71} - 2089564 q^{73} - 16015904 q^{79} - 723058 q^{81} - 37453776 q^{87} - 2169084 q^{89} - 48537936 q^{95} - 1088308 q^{97}+O(q^{100})$$ 6 * q - 688 * q^7 + 2918 * q^9 - 17872 * q^15 + 1452 * q^17 - 1296 * q^23 + 39314 * q^25 + 89280 * q^31 + 53880 * q^33 - 328208 * q^39 - 521244 * q^41 - 1566432 * q^47 - 511050 * q^49 - 3270256 * q^55 + 1889896 * q^57 - 5776816 * q^63 + 1416480 * q^65 - 7597104 * q^71 - 2089564 * q^73 - 16015904 * q^79 - 723058 * q^81 - 37453776 * q^87 - 2169084 * q^89 - 48537936 * q^95 - 1088308 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 163x^{4} + 4820x^{2} - 15296$$ :

 $$\beta_{1}$$ $$=$$ $$( -\nu^{5} + 171\nu^{3} - 5468\nu ) / 120$$ (-v^5 + 171*v^3 - 5468*v) / 120 $$\beta_{2}$$ $$=$$ $$( -37\nu^{5} + 5367\nu^{3} - 101516\nu ) / 600$$ (-37*v^5 + 5367*v^3 - 101516*v) / 600 $$\beta_{3}$$ $$=$$ $$( -77\nu^{5} + 12207\nu^{3} - 13036\nu ) / 600$$ (-77*v^5 + 12207*v^3 - 13036*v) / 600 $$\beta_{4}$$ $$=$$ $$( 8\nu^{4} - 728\nu^{2} - 5581 ) / 25$$ (8*v^4 - 728*v^2 - 5581) / 25 $$\beta_{5}$$ $$=$$ $$( -104\nu^{4} + 15864\nu^{2} - 275047 ) / 75$$ (-104*v^4 + 15864*v^2 - 275047) / 75
 $$\nu$$ $$=$$ $$( \beta_{3} - \beta_{2} - 8\beta_1 ) / 512$$ (b3 - b2 - 8*b1) / 512 $$\nu^{2}$$ $$=$$ $$( 3\beta_{5} + 13\beta_{4} + 13904 ) / 256$$ (3*b5 + 13*b4 + 13904) / 256 $$\nu^{3}$$ $$=$$ $$( 105\beta_{3} - 425\beta_{2} + 1528\beta_1 ) / 512$$ (105*b3 - 425*b2 + 1528*b1) / 512 $$\nu^{4}$$ $$=$$ $$( 273\beta_{5} + 1983\beta_{4} + 1443856 ) / 256$$ (273*b5 + 1983*b4 + 1443856) / 256 $$\nu^{5}$$ $$=$$ $$( 12487\beta_{3} - 67207\beta_{2} + 243592\beta_1 ) / 512$$ (12487*b3 - 67207*b2 + 243592*b1) / 512

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −1.89807 −5.81430 11.2068 −11.2068 5.81430 1.89807
0 −76.9497 0 338.443 0 −438.996 0 3734.25 0
1.2 0 −40.2163 0 −324.492 0 −956.960 0 −569.651 0
1.3 0 −21.9408 0 −184.916 0 1051.96 0 −1705.60 0
1.4 0 21.9408 0 184.916 0 1051.96 0 −1705.60 0
1.5 0 40.2163 0 324.492 0 −956.960 0 −569.651 0
1.6 0 76.9497 0 −338.443 0 −438.996 0 3734.25 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 256.8.a.q 6
4.b odd 2 1 256.8.a.r 6
8.b even 2 1 inner 256.8.a.q 6
8.d odd 2 1 256.8.a.r 6
16.e even 4 2 32.8.b.a 6
16.f odd 4 2 8.8.b.a 6
48.i odd 4 2 288.8.d.b 6
48.k even 4 2 72.8.d.b 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.8.b.a 6 16.f odd 4 2
32.8.b.a 6 16.e even 4 2
72.8.d.b 6 48.k even 4 2
256.8.a.q 6 1.a even 1 1 trivial
256.8.a.q 6 8.b even 2 1 inner
256.8.a.r 6 4.b odd 2 1
256.8.a.r 6 8.d odd 2 1
288.8.d.b 6 48.i odd 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{8}^{\mathrm{new}}(\Gamma_0(256))$$:

 $$T_{3}^{6} - 8020T_{3}^{4} + 13205808T_{3}^{2} - 4610229696$$ T3^6 - 8020*T3^4 + 13205808*T3^2 - 4610229696 $$T_{5}^{6} - 254032T_{5}^{4} + 19577926400T_{5}^{2} - 412405245440000$$ T5^6 - 254032*T5^4 + 19577926400*T5^2 - 412405245440000 $$T_{7}^{3} + 344T_{7}^{2} - 1048384T_{7} - 441929216$$ T7^3 + 344*T7^2 - 1048384*T7 - 441929216

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6}$$
$3$ $$T^{6} - 8020 T^{4} + \cdots - 4610229696$$
$5$ $$T^{6} + \cdots - 412405245440000$$
$7$ $$(T^{3} + 344 T^{2} - 1048384 T - 441929216)^{2}$$
$11$ $$T^{6} - 52294004 T^{4} + \cdots - 21\!\cdots\!00$$
$13$ $$T^{6} - 171080144 T^{4} + \cdots - 24\!\cdots\!00$$
$17$ $$(T^{3} - 726 T^{2} + \cdots - 9112197964104)^{2}$$
$19$ $$T^{6} - 3360814100 T^{4} + \cdots - 47\!\cdots\!04$$
$23$ $$(T^{3} + 648 T^{2} + \cdots + 2134822184448)^{2}$$
$29$ $$T^{6} - 55662621776 T^{4} + \cdots - 42\!\cdots\!00$$
$31$ $$(T^{3} - 44640 T^{2} + \cdots - 18\!\cdots\!28)^{2}$$
$37$ $$T^{6} - 490654094672 T^{4} + \cdots - 61\!\cdots\!64$$
$41$ $$(T^{3} + 260622 T^{2} + \cdots - 17\!\cdots\!00)^{2}$$
$43$ $$T^{6} - 124911737588 T^{4} + \cdots - 77\!\cdots\!56$$
$47$ $$(T^{3} + 783216 T^{2} + \cdots - 15\!\cdots\!96)^{2}$$
$53$ $$T^{6} - 3916631783120 T^{4} + \cdots - 12\!\cdots\!36$$
$59$ $$T^{6} - 6619585104052 T^{4} + \cdots - 55\!\cdots\!04$$
$61$ $$T^{6} - 4505952081744 T^{4} + \cdots - 10\!\cdots\!00$$
$67$ $$T^{6} - 1291377394260 T^{4} + \cdots - 75\!\cdots\!24$$
$71$ $$(T^{3} + 3798552 T^{2} + \cdots + 38\!\cdots\!92)^{2}$$
$73$ $$(T^{3} + 1044782 T^{2} + \cdots - 21\!\cdots\!72)^{2}$$
$79$ $$(T^{3} + 8007952 T^{2} + \cdots - 49\!\cdots\!40)^{2}$$
$83$ $$T^{6} - 37884069033748 T^{4} + \cdots - 63\!\cdots\!16$$
$89$ $$(T^{3} + 1084542 T^{2} + \cdots - 11\!\cdots\!20)^{2}$$
$97$ $$(T^{3} + 544154 T^{2} + \cdots - 16\!\cdots\!24)^{2}$$