# Properties

 Label 72.8.d.b Level $72$ Weight $8$ Character orbit 72.d Analytic conductor $22.492$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$72 = 2^{3} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$8$$ Character orbit: $$[\chi]$$ $$=$$ 72.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$22.4917218349$$ 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: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{15}$$ 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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_1 - 1) q^{2} + (\beta_{3} + \beta_1 + 19) q^{4} + (\beta_{5} + \beta_{4} - \beta_{3} - 2 \beta_1) q^{5} + ( - 2 \beta_{5} + 2 \beta_{4} - 6 \beta_{3} + 2 \beta_{2} + 16 \beta_1 - 112) q^{7} + ( - 4 \beta_{5} + 20 \beta_{4} - 2 \beta_{3} - 2 \beta_{2} - 10 \beta_1 - 250) q^{8}+O(q^{10})$$ q + (-b1 - 1) * q^2 + (b3 + b1 + 19) * q^4 + (b5 + b4 - b3 - 2*b1) * q^5 + (-2*b5 + 2*b4 - 6*b3 + 2*b2 + 16*b1 - 112) * q^7 + (-4*b5 + 20*b4 - 2*b3 - 2*b2 - 10*b1 - 250) * q^8 $$q + ( - \beta_1 - 1) q^{2} + (\beta_{3} + \beta_1 + 19) q^{4} + (\beta_{5} + \beta_{4} - \beta_{3} - 2 \beta_1) q^{5} + ( - 2 \beta_{5} + 2 \beta_{4} - 6 \beta_{3} + 2 \beta_{2} + 16 \beta_1 - 112) q^{7} + ( - 4 \beta_{5} + 20 \beta_{4} - 2 \beta_{3} - 2 \beta_{2} - 10 \beta_1 - 250) q^{8} + (4 \beta_{5} - 52 \beta_{4} + 8 \beta_{3} - 10 \beta_{2} - 12 \beta_1 - 280) q^{10} + (8 \beta_{5} + 11 \beta_{4} - 8 \beta_{3} - 16 \beta_{2} + 208 \beta_1) q^{11} + (5 \beta_{5} + 37 \beta_{4} - 5 \beta_{3} - 32 \beta_{2} + 438 \beta_1) q^{13} + (16 \beta_{5} - 80 \beta_{4} - 16 \beta_{3} + 40 \beta_{2} + 88 \beta_1 - 2008) q^{14} + ( - 40 \beta_{5} - 56 \beta_{4} - 12 \beta_{3} + 76 \beta_{2} + \cdots + 5908) q^{16}+ \cdots + (5376 \beta_{5} - 26880 \beta_{4} + 16128 \beta_{3} + \cdots + 2490039) q^{98}+O(q^{100})$$ q + (-b1 - 1) * q^2 + (b3 + b1 + 19) * q^4 + (b5 + b4 - b3 - 2*b1) * q^5 + (-2*b5 + 2*b4 - 6*b3 + 2*b2 + 16*b1 - 112) * q^7 + (-4*b5 + 20*b4 - 2*b3 - 2*b2 - 10*b1 - 250) * q^8 + (4*b5 - 52*b4 + 8*b3 - 10*b2 - 12*b1 - 280) * q^10 + (8*b5 + 11*b4 - 8*b3 - 16*b2 + 208*b1) * q^11 + (5*b5 + 37*b4 - 5*b3 - 32*b2 + 438*b1) * q^13 + (16*b5 - 80*b4 - 16*b3 + 40*b2 + 88*b1 - 2008) * q^14 + (-40*b5 - 56*b4 - 12*b3 + 76*b2 + 164*b1 + 5908) * q^16 + (-36*b5 + 36*b4 - 108*b3 + 92*b2 + 1296*b1 - 194) * q^17 + (88*b5 + 65*b4 - 88*b3 + 80*b2 - 1296*b1) * q^19 + (80*b5 + 368*b4 + 4*b3 - 120*b2 + 292*b1 + 19100) * q^20 + (26*b5 - 434*b4 - 192*b3 - 77*b2 - 358*b1 + 25532) * q^22 + (6*b5 - 6*b4 + 18*b3 - 70*b2 - 1200*b1 + 208) * q^23 + (-88*b5 + 88*b4 - 264*b3 + 40*b2 - 160*b1 - 6435) * q^25 + (-44*b5 - 452*b4 - 472*b3 - 18*b2 - 636*b1 + 53000) * q^26 + (256*b5 - 256*b4 + 88*b3 - 256*b2 + 2648*b1 - 80248) * q^28 + (95*b5 - 2209*b4 - 95*b3 - 256*b2 + 3394*b1) * q^29 + (-168*b5 + 168*b4 - 504*b3 - 408*b2 - 9024*b1 - 14656) * q^31 + (80*b5 + 1136*b4 - 200*b3 + 488*b2 - 4648*b1 - 136136) * q^32 + (288*b5 - 1440*b4 - 1184*b3 + 720*b2 + 658*b1 - 167886) * q^34 + (-528*b5 - 768*b4 + 528*b3 + 800*b2 - 10144*b1) * q^35 + (-25*b5 - 7097*b4 + 25*b3 + 928*b2 - 12942*b1) * q^37 + (398*b5 - 4438*b4 + 1984*b3 - 903*b2 + 270*b1 - 163148) * q^38 + (-592*b5 - 4208*b4 - 136*b3 - 680*b2 - 21800*b1 + 159320) * q^40 + (-888*b5 + 888*b4 - 2664*b3 - 248*b2 - 13344*b1 - 85690) * q^41 + (560*b5 - 275*b4 - 560*b3 - 608*b2 + 7392*b1) * q^43 + (1688*b5 - 1912*b4 + 526*b3 - 388*b2 - 27650*b1 + 181986) * q^44 + (-48*b5 + 240*b4 + 1072*b3 - 120*b2 - 1160*b1 + 154632) * q^46 + (-756*b5 + 756*b4 - 2268*b3 - 588*b2 - 18144*b1 - 260064) * q^47 + (-672*b5 + 672*b4 - 2016*b3 - 672*b2 - 18816*b1 - 84279) * q^49 + (704*b5 - 3520*b4 + 64*b3 + 1760*b2 + 4611*b1 + 24515) * q^50 + (2704*b5 - 5584*b4 + 852*b3 + 1064*b2 - 56588*b1 + 136140) * q^52 + (365*b5 + 16589*b4 - 365*b3 + 2720*b2 - 38810*b1) * q^53 + (-2282*b5 + 2282*b4 - 6846*b3 + 3050*b2 + 32080*b1 - 542000) * q^55 + (672*b5 - 3360*b4 - 1968*b3 - 3760*b2 + 77200*b1 + 411920) * q^56 + (4988*b5 + 8884*b4 - 3336*b3 - 3254*b2 - 628*b1 + 521240) * q^58 + (-3456*b5 - 9141*b4 + 3456*b3 - 3584*b2 + 57088*b1) * q^59 + (-3927*b5 + 7881*b4 + 3927*b3 - 1056*b2 + 22638*b1) * q^61 + (1344*b5 - 6720*b4 + 7872*b3 + 3360*b2 + 3424*b1 + 1172896) * q^62 + (-1312*b5 - 12896*b4 + 6224*b3 + 496*b2 + 139792*b1 - 800432) * q^64 + (-3960*b5 + 3960*b4 - 11880*b3 + 4360*b2 + 38880*b1 - 230800) * q^65 + (-1320*b5 + 5001*b4 + 1320*b3 - 1968*b2 + 30192*b1) * q^67 + (8192*b5 - 22528*b4 + 3406*b3 - 2816*b2 + 171342*b1 - 1105206) * q^68 + (-1632*b5 + 28896*b4 + 8576*b3 + 5040*b2 + 19616*b1 - 1236800) * q^70 + (-7566*b5 + 7566*b4 - 22698*b3 + 3534*b2 - 12048*b1 + 1276272) * q^71 + (860*b5 - 860*b4 + 2580*b3 + 9052*b2 + 171536*b1 + 347114) * q^73 + (14044*b5 + 43732*b4 + 14648*b3 - 6822*b2 + 29292*b1 - 1300264) * q^74 + (1736*b5 + 55960*b4 - 2070*b3 - 13580*b2 + 175034*b1 + 1540326) * q^76 + (-12380*b5 + 33604*b4 + 12380*b3 - 3488*b2 + 73592*b1) * q^77 + (76*b5 - 76*b4 + 228*b3 - 16204*b2 - 290912*b1 + 2669216) * q^79 + (7776*b5 + 37920*b4 + 17616*b3 + 3760*b2 - 162416*b1 + 2091600) * q^80 + (7104*b5 - 35520*b4 + 11072*b3 + 17760*b2 + 56858*b1 + 1779930) * q^82 + (-10560*b5 + 8775*b4 + 10560*b3 + 9088*b2 - 106112*b1) * q^83 + (-11266*b5 + 74014*b4 + 11266*b3 + 22240*b2 - 288828*b1) * q^85 + (3910*b5 - 24110*b4 - 5248*b3 - 6435*b2 - 14778*b1 + 940292) * q^86 + (5096*b5 - 21896*b4 + 34788*b3 - 24908*b2 - 181324*b1 + 243572) * q^88 + (-3036*b5 + 3036*b4 - 9108*b3 - 476*b2 - 38928*b1 - 357466) * q^89 + (-18640*b5 + 65152*b4 + 18640*b3 - 18272*b2 + 293088*b1) * q^91 + (-4864*b5 + 21248*b4 - 392*b3 - 1280*b2 - 147336*b1 - 109848) * q^92 + (6048*b5 - 30240*b4 + 15456*b3 + 15120*b2 + 229488*b1 + 2576784) * q^94 + (258*b5 - 258*b4 + 774*b3 - 9730*b2 - 172560*b1 - 8090000) * q^95 + (-12668*b5 + 12668*b4 - 38004*b3 + 19460*b2 + 223600*b1 - 164494) * q^97 + (5376*b5 - 26880*b4 + 16128*b3 + 13440*b2 + 54711*b1 + 2490039) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 6 q^{2} + 116 q^{4} - 688 q^{7} - 1512 q^{8}+O(q^{10})$$ 6 * q - 6 * q^2 + 116 * q^4 - 688 * q^7 - 1512 * q^8 $$6 q - 6 q^{2} + 116 q^{4} - 688 q^{7} - 1512 q^{8} - 1656 q^{10} - 12048 q^{14} + 35344 q^{16} - 1452 q^{17} + 114768 q^{20} + 152860 q^{22} + 1296 q^{23} - 39314 q^{25} + 316968 q^{26} - 480800 q^{28} - 89280 q^{31} - 817056 q^{32} - 1009108 q^{34} - 974124 q^{38} + 954464 q^{40} - 521244 q^{41} + 1096344 q^{44} + 929840 q^{46} - 1566432 q^{47} - 511050 q^{49} + 148626 q^{50} + 823952 q^{52} - 3270256 q^{55} + 2468928 q^{56} + 3130744 q^{58} + 7055808 q^{62} - 4792768 q^{64} - 1416480 q^{65} - 6608040 q^{68} - 7406912 q^{70} + 7597104 q^{71} + 2089564 q^{73} - 7744200 q^{74} + 9241288 q^{76} + 16015904 q^{79} + 12600384 q^{80} + 10715932 q^{82} + 5639076 q^{86} + 1541200 q^{88} - 2169084 q^{89} - 669600 q^{92} + 15503712 q^{94} - 48537936 q^{95} - 1088308 q^{97} + 14983242 q^{98}+O(q^{100})$$ 6 * q - 6 * q^2 + 116 * q^4 - 688 * q^7 - 1512 * q^8 - 1656 * q^10 - 12048 * q^14 + 35344 * q^16 - 1452 * q^17 + 114768 * q^20 + 152860 * q^22 + 1296 * q^23 - 39314 * q^25 + 316968 * q^26 - 480800 * q^28 - 89280 * q^31 - 817056 * q^32 - 1009108 * q^34 - 974124 * q^38 + 954464 * q^40 - 521244 * q^41 + 1096344 * q^44 + 929840 * q^46 - 1566432 * q^47 - 511050 * q^49 + 148626 * q^50 + 823952 * q^52 - 3270256 * q^55 + 2468928 * q^56 + 3130744 * q^58 + 7055808 * q^62 - 4792768 * q^64 - 1416480 * q^65 - 6608040 * q^68 - 7406912 * q^70 + 7597104 * q^71 + 2089564 * q^73 - 7744200 * q^74 + 9241288 * q^76 + 16015904 * q^79 + 12600384 * q^80 + 10715932 * q^82 + 5639076 * q^86 + 1541200 * q^88 - 2169084 * q^89 - 669600 * q^92 + 15503712 * q^94 - 48537936 * q^95 - 1088308 * q^97 + 14983242 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 3x^{5} - 10x^{4} - 24x^{3} - 320x^{2} - 3072x + 32768$$ :

 $$\beta_{1}$$ $$=$$ $$( -\nu^{5} + 3\nu^{4} + 10\nu^{3} + 24\nu^{2} + 320\nu + 2560 ) / 512$$ (-v^5 + 3*v^4 + 10*v^3 + 24*v^2 + 320*v + 2560) / 512 $$\beta_{2}$$ $$=$$ $$( \nu^{5} - 3\nu^{4} - 10\nu^{3} - 24\nu^{2} + 7872\nu - 6656 ) / 256$$ (v^5 - 3*v^4 - 10*v^3 - 24*v^2 + 7872*v - 6656) / 256 $$\beta_{3}$$ $$=$$ $$( -5\nu^{5} - 49\nu^{4} + 242\nu^{3} + 760\nu^{2} + 3136\nu + 26624 ) / 512$$ (-5*v^5 - 49*v^4 + 242*v^3 + 760*v^2 + 3136*v + 26624) / 512 $$\beta_{4}$$ $$=$$ $$( 3\nu^{5} + 7\nu^{4} + 50\nu^{3} - 104\nu^{2} - 1088\nu - 15872 ) / 256$$ (3*v^5 + 7*v^4 + 50*v^3 - 104*v^2 - 1088*v - 15872) / 256 $$\beta_{5}$$ $$=$$ $$( 15\nu^{5} + 51\nu^{4} - 182\nu^{3} + 3032\nu^{2} - 2496\nu - 90112 ) / 512$$ (15*v^5 + 51*v^4 - 182*v^3 + 3032*v^2 - 2496*v - 90112) / 512
 $$\nu$$ $$=$$ $$( \beta_{2} + 2\beta _1 + 16 ) / 32$$ (b2 + 2*b1 + 16) / 32 $$\nu^{2}$$ $$=$$ $$( 4\beta_{5} - 4\beta_{4} + 4\beta_{3} - \beta_{2} + 14\beta _1 + 152 ) / 32$$ (4*b5 - 4*b4 + 4*b3 - b2 + 14*b1 + 152) / 32 $$\nu^{3}$$ $$=$$ $$( -4\beta_{5} + 68\beta_{4} + 28\beta_{3} - \beta_{2} + 206\beta _1 + 1000 ) / 32$$ (-4*b5 + 68*b4 + 28*b3 - b2 + 206*b1 + 1000) / 32 $$\nu^{4}$$ $$=$$ $$( 28\beta_{5} + 164\beta_{4} - 132\beta_{3} + 11\beta_{2} + 2086\beta _1 + 11816 ) / 32$$ (28*b5 + 164*b4 - 132*b3 + 11*b2 + 2086*b1 + 11816) / 32 $$\nu^{5}$$ $$=$$ $$( 140\beta_{5} + 1076\beta_{4} - 20\beta_{3} + 319\beta_{2} - 7090\beta _1 + 136136 ) / 32$$ (140*b5 + 1076*b4 - 20*b3 + 319*b2 - 7090*b1 + 136136) / 32

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/72\mathbb{Z}\right)^\times$$.

 $$n$$ $$37$$ $$55$$ $$65$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
37.1
 5.57668 − 0.949035i 5.57668 + 0.949035i 0.776001 − 5.60338i 0.776001 + 5.60338i −4.85268 − 2.90715i −4.85268 + 2.90715i
−11.1534 1.89807i 0 120.795 + 42.3397i 338.443i 0 −438.996 −1266.90 701.506i 0 −642.387 + 3774.77i
37.2 −11.1534 + 1.89807i 0 120.795 42.3397i 338.443i 0 −438.996 −1266.90 + 701.506i 0 −642.387 3774.77i
37.3 −1.55200 11.2068i 0 −123.183 + 34.7858i 184.916i 0 1051.96 581.015 + 1326.49i 0 −2072.30 + 286.989i
37.4 −1.55200 + 11.2068i 0 −123.183 34.7858i 184.916i 0 1051.96 581.015 1326.49i 0 −2072.30 286.989i
37.5 9.70536 5.81430i 0 60.3879 112.860i 324.492i 0 −956.960 −70.1132 1446.46i 0 1886.69 + 3149.31i
37.6 9.70536 + 5.81430i 0 60.3879 + 112.860i 324.492i 0 −956.960 −70.1132 + 1446.46i 0 1886.69 3149.31i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 37.6 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 72.8.d.b 6
3.b odd 2 1 8.8.b.a 6
4.b odd 2 1 288.8.d.b 6
8.b even 2 1 inner 72.8.d.b 6
8.d odd 2 1 288.8.d.b 6
12.b even 2 1 32.8.b.a 6
24.f even 2 1 32.8.b.a 6
24.h odd 2 1 8.8.b.a 6
48.i odd 4 2 256.8.a.r 6
48.k even 4 2 256.8.a.q 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.8.b.a 6 3.b odd 2 1
8.8.b.a 6 24.h odd 2 1
32.8.b.a 6 12.b even 2 1
32.8.b.a 6 24.f even 2 1
72.8.d.b 6 1.a even 1 1 trivial
72.8.d.b 6 8.b even 2 1 inner
256.8.a.q 6 48.k even 4 2
256.8.a.r 6 48.i odd 4 2
288.8.d.b 6 4.b odd 2 1
288.8.d.b 6 8.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{6} + 254032T_{5}^{4} + 19577926400T_{5}^{2} + 412405245440000$$ acting on $$S_{8}^{\mathrm{new}}(72, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6} + 6 T^{5} - 40 T^{4} + \cdots + 2097152$$
$3$ $$T^{6}$$
$5$ $$T^{6} + \cdots + 412405245440000$$
$7$ $$(T^{3} + 344 T^{2} - 1048384 T - 441929216)^{2}$$
$11$ $$T^{6} + 52294004 T^{4} + \cdots + 21\!\cdots\!00$$
$13$ $$T^{6} + 171080144 T^{4} + \cdots + 24\!\cdots\!00$$
$17$ $$(T^{3} + 726 T^{2} + \cdots + 9112197964104)^{2}$$
$19$ $$T^{6} + 3360814100 T^{4} + \cdots + 47\!\cdots\!04$$
$23$ $$(T^{3} - 648 T^{2} + \cdots - 2134822184448)^{2}$$
$29$ $$T^{6} + 55662621776 T^{4} + \cdots + 42\!\cdots\!00$$
$31$ $$(T^{3} + 44640 T^{2} + \cdots + 18\!\cdots\!28)^{2}$$
$37$ $$T^{6} + 490654094672 T^{4} + \cdots + 61\!\cdots\!64$$
$41$ $$(T^{3} + 260622 T^{2} + \cdots - 17\!\cdots\!00)^{2}$$
$43$ $$T^{6} + 124911737588 T^{4} + \cdots + 77\!\cdots\!56$$
$47$ $$(T^{3} + 783216 T^{2} + \cdots - 15\!\cdots\!96)^{2}$$
$53$ $$T^{6} + 3916631783120 T^{4} + \cdots + 12\!\cdots\!36$$
$59$ $$T^{6} + 6619585104052 T^{4} + \cdots + 55\!\cdots\!04$$
$61$ $$T^{6} + 4505952081744 T^{4} + \cdots + 10\!\cdots\!00$$
$67$ $$T^{6} + 1291377394260 T^{4} + \cdots + 75\!\cdots\!24$$
$71$ $$(T^{3} - 3798552 T^{2} + \cdots - 38\!\cdots\!92)^{2}$$
$73$ $$(T^{3} - 1044782 T^{2} + \cdots + 21\!\cdots\!72)^{2}$$
$79$ $$(T^{3} - 8007952 T^{2} + \cdots + 49\!\cdots\!40)^{2}$$
$83$ $$T^{6} + 37884069033748 T^{4} + \cdots + 63\!\cdots\!16$$
$89$ $$(T^{3} + 1084542 T^{2} + \cdots - 11\!\cdots\!20)^{2}$$
$97$ $$(T^{3} + 544154 T^{2} + \cdots - 16\!\cdots\!24)^{2}$$