# Properties

 Label 32.8.b.a Level $32$ Weight $8$ Character orbit 32.b Analytic conductor $9.996$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$32 = 2^{5}$$ Weight: $$k$$ $$=$$ $$8$$ Character orbit: $$[\chi]$$ $$=$$ 32.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$9.99632081549$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: $$\mathbb{Q}[x]/(x^{6} - \cdots)$$ Defining polynomial: $$x^{6} - 3x^{5} - 10x^{4} - 24x^{3} - 320x^{2} - 3072x + 32768$$ x^6 - 3*x^5 - 10*x^4 - 24*x^3 - 320*x^2 - 3072*x + 32768 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{30}$$ Twist minimal: no (minimal twist has level 8) 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.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{3} + (\beta_{3} + \beta_{2}) q^{5} + (\beta_1 + 115) q^{7} + ( - \beta_{4} - \beta_1 - 487) q^{9}+O(q^{10})$$ q + b2 * q^3 + (b3 + b2) * q^5 + (b1 + 115) * q^7 + (-b4 - b1 - 487) * q^9 $$q + \beta_{2} q^{3} + (\beta_{3} + \beta_{2}) q^{5} + (\beta_1 + 115) q^{7} + ( - \beta_{4} - \beta_1 - 487) q^{9} + ( - \beta_{5} - 7 \beta_{3} - 3 \beta_{2}) q^{11} + ( - 2 \beta_{5} - 3 \beta_{3} - 21 \beta_{2}) q^{13} + ( - 8 \beta_{4} + \beta_1 - 2981) q^{15} + ( - 7 \beta_{4} + 25 \beta_1 + 248) q^{17} + ( - 5 \beta_{5} + 93 \beta_{3} + 105 \beta_{2}) q^{19} + ( - 6 \beta_{5} - 6 \beta_{3} + 452 \beta_{2}) q^{21} + ( - 8 \beta_{4} + 11 \beta_1 + 217) q^{23} + ( - 6 \beta_{4} - 38 \beta_1 - 6567) q^{25} + (3 \beta_{5} - 363 \beta_{3} - 310 \beta_{2}) q^{27} + (16 \beta_{5} + 79 \beta_{3} - 2337 \beta_{2}) q^{29} + (72 \beta_{4} + 12 \beta_1 + 14908) q^{31} + (63 \beta_{4} - 129 \beta_1 + 8958) q^{33} + (50 \beta_{5} + 478 \beta_{3} + 368 \beta_{2}) q^{35} + (58 \beta_{5} - 33 \beta_{3} + 6633 \beta_{2}) q^{37} + (64 \beta_{4} - 221 \beta_1 + 54649) q^{39} + (142 \beta_{4} + 302 \beta_1 + 87022) q^{41} + (38 \beta_{5} + 522 \beta_{3} - 579 \beta_{2}) q^{43} + ( - 30 \beta_{5} - 771 \beta_{3} - 13841 \beta_{2}) q^{45} + ( - 168 \beta_{4} - 210 \beta_1 - 261198) q^{47} + ( - 168 \beta_{4} - 168 \beta_1 - 85287) q^{49} + ( - 171 \beta_{5} - 2733 \beta_{3} - 3038 \beta_{2}) q^{51} + ( - 170 \beta_{5} + 535 \beta_{3} + 17949 \beta_{2}) q^{53} + ( - 96 \beta_{4} + 1237 \beta_1 + 545423) q^{55} + ( - 701 \beta_{4} - 509 \beta_1 - 315386) q^{57} + ( - 224 \beta_{5} + 3680 \beta_{3} + 10933 \beta_{2}) q^{59} + ( - 66 \beta_{5} + 3993 \beta_{3} - 7353 \beta_{2}) q^{61} + ( - 344 \beta_{4} + 1015 \beta_1 - 962579) q^{63} + ( - 50 \beta_{4} + 2030 \beta_1 + 236740) q^{65} + (123 \beta_{5} - 1443 \beta_{3} + 4017 \beta_{2}) q^{67} + ( - 90 \beta_{5} - 3018 \beta_{3} - 9460 \beta_{2}) q^{69} + ( - 504 \beta_{4} - 3279 \beta_1 + 1264923) q^{71} + (1239 \beta_{4} - 809 \beta_1 + 348404) q^{73} + (210 \beta_{5} - 1986 \beta_{3} - 29411 \beta_{2}) q^{75} + (218 \beta_{5} - 12598 \beta_{3} + 31860 \beta_{2}) q^{77} + (2016 \beta_{4} - 2054 \beta_1 - 2669330) q^{79} + (631 \beta_{4} - 2249 \beta_1 - 121049) q^{81} + (568 \beta_{5} + 9992 \beta_{3} - 13319 \beta_{2}) q^{83} + (1390 \beta_{5} + 9876 \beta_{3} - 85134 \beta_{2}) q^{85} + (1608 \beta_{4} + 4383 \beta_1 + 6244293) q^{87} + (439 \beta_{4} + 1079 \beta_1 + 362020) q^{89} + (1142 \beta_{5} - 19782 \beta_{3} + 56016 \beta_{2}) q^{91} + (144 \beta_{5} + 26496 \beta_{3} + 139408 \beta_{2}) q^{93} + ( - 1184 \beta_{4} + 1313 \beta_1 - 8089613) q^{95} + (849 \beta_{4} - 7183 \beta_1 - 183496) q^{97} + ( - 1224 \beta_{5} + 8712 \beta_{3} + 64323 \beta_{2}) q^{99}+O(q^{100})$$ q + b2 * q^3 + (b3 + b2) * q^5 + (b1 + 115) * q^7 + (-b4 - b1 - 487) * q^9 + (-b5 - 7*b3 - 3*b2) * q^11 + (-2*b5 - 3*b3 - 21*b2) * q^13 + (-8*b4 + b1 - 2981) * q^15 + (-7*b4 + 25*b1 + 248) * q^17 + (-5*b5 + 93*b3 + 105*b2) * q^19 + (-6*b5 - 6*b3 + 452*b2) * q^21 + (-8*b4 + 11*b1 + 217) * q^23 + (-6*b4 - 38*b1 - 6567) * q^25 + (3*b5 - 363*b3 - 310*b2) * q^27 + (16*b5 + 79*b3 - 2337*b2) * q^29 + (72*b4 + 12*b1 + 14908) * q^31 + (63*b4 - 129*b1 + 8958) * q^33 + (50*b5 + 478*b3 + 368*b2) * q^35 + (58*b5 - 33*b3 + 6633*b2) * q^37 + (64*b4 - 221*b1 + 54649) * q^39 + (142*b4 + 302*b1 + 87022) * q^41 + (38*b5 + 522*b3 - 579*b2) * q^43 + (-30*b5 - 771*b3 - 13841*b2) * q^45 + (-168*b4 - 210*b1 - 261198) * q^47 + (-168*b4 - 168*b1 - 85287) * q^49 + (-171*b5 - 2733*b3 - 3038*b2) * q^51 + (-170*b5 + 535*b3 + 17949*b2) * q^53 + (-96*b4 + 1237*b1 + 545423) * q^55 + (-701*b4 - 509*b1 - 315386) * q^57 + (-224*b5 + 3680*b3 + 10933*b2) * q^59 + (-66*b5 + 3993*b3 - 7353*b2) * q^61 + (-344*b4 + 1015*b1 - 962579) * q^63 + (-50*b4 + 2030*b1 + 236740) * q^65 + (123*b5 - 1443*b3 + 4017*b2) * q^67 + (-90*b5 - 3018*b3 - 9460*b2) * q^69 + (-504*b4 - 3279*b1 + 1264923) * q^71 + (1239*b4 - 809*b1 + 348404) * q^73 + (210*b5 - 1986*b3 - 29411*b2) * q^75 + (218*b5 - 12598*b3 + 31860*b2) * q^77 + (2016*b4 - 2054*b1 - 2669330) * q^79 + (631*b4 - 2249*b1 - 121049) * q^81 + (568*b5 + 9992*b3 - 13319*b2) * q^83 + (1390*b5 + 9876*b3 - 85134*b2) * q^85 + (1608*b4 + 4383*b1 + 6244293) * q^87 + (439*b4 + 1079*b1 + 362020) * q^89 + (1142*b5 - 19782*b3 + 56016*b2) * q^91 + (144*b5 + 26496*b3 + 139408*b2) * q^93 + (-1184*b4 + 1313*b1 - 8089613) * q^95 + (849*b4 - 7183*b1 - 183496) * q^97 + (-1224*b5 + 8712*b3 + 64323*b2) * 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 $$x^{6} - 3x^{5} - 10x^{4} - 24x^{3} - 320x^{2} - 3072x + 32768$$ :

 $$\beta_{1}$$ $$=$$ $$( -\nu^{4} + 3\nu^{3} + 42\nu^{2} - 72\nu + 106 ) / 2$$ (-v^4 + 3*v^3 + 42*v^2 - 72*v + 106) / 2 $$\beta_{2}$$ $$=$$ $$( -3\nu^{5} - 7\nu^{4} - 50\nu^{3} + 104\nu^{2} + 1088\nu + 15872 ) / 256$$ (-3*v^5 - 7*v^4 - 50*v^3 + 104*v^2 + 1088*v + 15872) / 256 $$\beta_{3}$$ $$=$$ $$( -11\nu^{5} - 47\nu^{4} + 222\nu^{3} - 1112\nu^{2} + 3136\nu + 60928 ) / 256$$ (-11*v^5 - 47*v^4 + 222*v^3 - 1112*v^2 + 3136*v + 60928) / 256 $$\beta_{4}$$ $$=$$ $$( -\nu^{5} + \nu^{4} + 16\nu^{3} + 108\nu^{2} + 1200\nu + 2260 ) / 4$$ (-v^5 + v^4 + 16*v^3 + 108*v^2 + 1200*v + 2260) / 4 $$\beta_{5}$$ $$=$$ $$( 93\nu^{5} - 487\nu^{4} - 1458\nu^{3} - 3352\nu^{2} + 101952\nu - 205312 ) / 256$$ (93*v^5 - 487*v^4 - 1458*v^3 - 3352*v^2 + 101952*v - 205312) / 256
 $$\nu$$ $$=$$ $$( \beta_{5} + 2\beta_{4} - \beta_{3} - 8\beta_{2} - 2\beta _1 + 512 ) / 1024$$ (b5 + 2*b4 - b3 - 8*b2 - 2*b1 + 512) / 1024 $$\nu^{2}$$ $$=$$ $$( -\beta_{5} + 6\beta_{4} - 63\beta_{3} + 72\beta_{2} + 26\beta _1 + 4960 ) / 1024$$ (-b5 + 6*b4 - 63*b3 + 72*b2 + 26*b1 + 4960) / 1024 $$\nu^{3}$$ $$=$$ $$( -9\beta_{5} + 38\beta_{4} + 329\beta_{3} - 2296\beta_{2} + 58\beta _1 + 32288 ) / 1024$$ (-9*b5 + 38*b4 + 329*b3 - 2296*b2 + 58*b1 + 32288) / 1024 $$\nu^{4}$$ $$=$$ $$( -141\beta_{5} + 222\beta_{4} - 1587\beta_{3} - 3288\beta_{2} - 638\beta _1 + 376864 ) / 1024$$ (-141*b5 + 222*b4 - 1587*b3 - 3288*b2 - 638*b1 + 376864) / 1024 $$\nu^{5}$$ $$=$$ $$( 807\beta_{5} - 218\beta_{4} - 4327\beta_{3} - 41848\beta_{2} + 698\beta _1 + 4357792 ) / 1024$$ (807*b5 - 218*b4 - 4327*b3 - 41848*b2 + 698*b1 + 4357792) / 1024

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/32\mathbb{Z}\right)^\times$$.

 $$n$$ $$5$$ $$31$$ $$\chi(n)$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
17.1
 5.57668 + 0.949035i −4.85268 + 2.90715i 0.776001 − 5.60338i 0.776001 + 5.60338i −4.85268 − 2.90715i 5.57668 − 0.949035i
0 76.9497i 0 338.443i 0 438.996 0 −3734.25 0
17.2 0 40.2163i 0 324.492i 0 956.960 0 569.651 0
17.3 0 21.9408i 0 184.916i 0 −1051.96 0 1705.60 0
17.4 0 21.9408i 0 184.916i 0 −1051.96 0 1705.60 0
17.5 0 40.2163i 0 324.492i 0 956.960 0 569.651 0
17.6 0 76.9497i 0 338.443i 0 438.996 0 −3734.25 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 17.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
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 32.8.b.a 6
3.b odd 2 1 288.8.d.b 6
4.b odd 2 1 8.8.b.a 6
8.b even 2 1 inner 32.8.b.a 6
8.d odd 2 1 8.8.b.a 6
12.b even 2 1 72.8.d.b 6
16.e even 4 2 256.8.a.q 6
16.f odd 4 2 256.8.a.r 6
24.f even 2 1 72.8.d.b 6
24.h odd 2 1 288.8.d.b 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.8.b.a 6 4.b odd 2 1
8.8.b.a 6 8.d odd 2 1
32.8.b.a 6 1.a even 1 1 trivial
32.8.b.a 6 8.b even 2 1 inner
72.8.d.b 6 12.b even 2 1
72.8.d.b 6 24.f even 2 1
256.8.a.q 6 16.e even 4 2
256.8.a.r 6 16.f odd 4 2
288.8.d.b 6 3.b odd 2 1
288.8.d.b 6 24.h odd 2 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{8}^{\mathrm{new}}(32, [\chi])$$.

## Hecke characteristic polynomials

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