# Properties

 Label 63.10.a.e Level $63$ Weight $10$ Character orbit 63.a Self dual yes Analytic conductor $32.447$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$63 = 3^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$10$$ Character orbit: $$[\chi]$$ $$=$$ 63.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$32.4472576783$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: $$\mathbb{Q}[x]/(x^{3} - \cdots)$$ Defining polynomial: $$x^{3} - x^{2} - 426x + 2016$$ x^3 - x^2 - 426*x + 2016 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2\cdot 3$$ Twist minimal: no (minimal twist has level 7) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_{2} - 7) q^{2} + ( - 8 \beta_{2} + 7 \beta_1 + 519) q^{4} + ( - 43 \beta_{2} + 13 \beta_1 - 518) q^{5} + 2401 q^{7} + (470 \beta_{2} - 147 \beta_1 - 4685) q^{8}+O(q^{10})$$ q + (b2 - 7) * q^2 + (-8*b2 + 7*b1 + 519) * q^4 + (-43*b2 + 13*b1 - 518) * q^5 + 2401 * q^7 + (470*b2 - 147*b1 - 4685) * q^8 $$q + (\beta_{2} - 7) q^{2} + ( - 8 \beta_{2} + 7 \beta_1 + 519) q^{4} + ( - 43 \beta_{2} + 13 \beta_1 - 518) q^{5} + 2401 q^{7} + (470 \beta_{2} - 147 \beta_1 - 4685) q^{8} + (370 \beta_{2} - 470 \beta_1 - 32620) q^{10} + ( - 650 \beta_{2} - 658 \beta_1 + 1148) q^{11} + (3017 \beta_{2} - 175 \beta_1 - 6594) q^{13} + (2401 \beta_{2} - 16807) q^{14} + ( - 10614 \beta_{2} + 1617 \beta_1 + 160987) q^{16} + ( - 3030 \beta_{2} - 1574 \beta_1 - 338898) q^{17} + ( - 15371 \beta_{2} + 2437 \beta_1 + 74284) q^{19} + ( - 41524 \beta_{2} + 2044 \beta_1 + 640696) q^{20} + ( - 40972 \beta_{2} + 4004 \beta_1 - 949016) q^{22} + (24200 \beta_{2} + 3808 \beta_1 - 628544) q^{23} + ( - 15338 \beta_{2} - 4802 \beta_1 + 1024407) q^{25} + ( - 20986 \beta_{2} + 23394 \beta_1 + 2928352) q^{26} + ( - 19208 \beta_{2} + 16807 \beta_1 + 1246119) q^{28} + ( - 54866 \beta_{2} - 18914 \beta_1 - 1360606) q^{29} + (55698 \beta_{2} - 70302 \beta_1 + 956480) q^{31} + (36066 \beta_{2} - 20055 \beta_1 - 8407317) q^{32} + ( - 438178 \beta_{2} - 748 \beta_1 - 1327214) q^{34} + ( - 103243 \beta_{2} + 31213 \beta_1 - 1243718) q^{35} + (209418 \beta_{2} + 60522 \beta_1 + 465206) q^{37} + (248060 \beta_{2} - 139278 \beta_1 - 14493290) q^{38} + (625640 \beta_{2} - 76600 \beta_1 - 27619760) q^{40} + (163478 \beta_{2} + 131894 \beta_1 + 4806886) q^{41} + (121982 \beta_{2} + 65366 \beta_1 - 20543724) q^{43} + ( - 314984 \beta_{2} - 1960 \beta_1 - 32337328) q^{44} + ( - 405224 \beta_{2} + 119896 \beta_1 + 29915888) q^{46} + (534778 \beta_{2} - 83238 \beta_1 + 3456320) q^{47} + 5764801 q^{49} + (727615 \beta_{2} - 44940 \beta_1 - 24441685) q^{50} + (2925244 \beta_{2} - 361424 \beta_1 - 26969348) q^{52} + ( - 1553376 \beta_{2} + 450352 \beta_1 - 22500870) q^{53} + (1659356 \beta_{2} + 988364 \beta_1 - 35274344) q^{55} + (1128470 \beta_{2} - 352947 \beta_1 - 11248685) q^{56} + ( - 2535150 \beta_{2} - 138180 \beta_1 - 53054610) q^{58} + (2231195 \beta_{2} + 49659 \beta_1 + 14196700) q^{59} + (589107 \beta_{2} - 844773 \beta_1 + 63915614) q^{61} + ( - 3668848 \beta_{2} + 1303812 \beta_1 + 15661156) q^{62} + ( - 4312590 \beta_{2} - 314727 \beta_1 + 2617387) q^{64} + (958090 \beta_{2} - 1035790 \beta_1 - 121427740) q^{65} + (3939816 \beta_{2} + 47712 \beta_1 - 85058596) q^{67} + (613704 \beta_{2} - 2251634 \beta_1 - 247828602) q^{68} + (888370 \beta_{2} - 1128470 \beta_1 - 78320620) q^{70} + ( - 3499356 \beta_{2} + 526260 \beta_1 - 98838168) q^{71} + ( - 9544844 \beta_{2} + 118516 \beta_1 + 114737770) q^{73} + (4189718 \beta_{2} + 679140 \beta_1 + 230232154) q^{74} + ( - 15924468 \beta_{2} + 2299290 \beta_1 + 242946662) q^{76} + ( - 1560650 \beta_{2} - 1579858 \beta_1 + 2756348) q^{77} + ( - 7475532 \beta_{2} + 1679412 \beta_1 - 320137552) q^{79} + ( - 11964112 \beta_{2} + 4328752 \beta_1 + 444444448) q^{80} + (13216518 \beta_{2} - 570276 \beta_1 + 187558434) q^{82} + (4479559 \beta_{2} + 1977367 \beta_1 + 366839060) q^{83} + (18558254 \beta_{2} - 1518034 \beta_1 + 146059804) q^{85} + ( - 16416916 \beta_{2} + 4116 \beta_1 + 293660752) q^{86} + ( - 11172080 \beta_{2} - 4229456 \beta_1 + 402041600) q^{88} + (10612104 \beta_{2} - 1815976 \beta_1 - 168938826) q^{89} + (7243817 \beta_{2} - 420175 \beta_1 - 15832194) q^{91} + (25723952 \beta_{2} - 6344912 \beta_1 - 230374496) q^{92} + ( - 2488928 \beta_{2} + 4825540 \beta_1 + 462668276) q^{94} + ( - 10749416 \beta_{2} + 4098416 \beta_1 + 734357024) q^{95} + (33276782 \beta_{2} - 864850 \beta_1 - 215832750) q^{97} + (5764801 \beta_{2} - 40353607) q^{98}+O(q^{100})$$ q + (b2 - 7) * q^2 + (-8*b2 + 7*b1 + 519) * q^4 + (-43*b2 + 13*b1 - 518) * q^5 + 2401 * q^7 + (470*b2 - 147*b1 - 4685) * q^8 + (370*b2 - 470*b1 - 32620) * q^10 + (-650*b2 - 658*b1 + 1148) * q^11 + (3017*b2 - 175*b1 - 6594) * q^13 + (2401*b2 - 16807) * q^14 + (-10614*b2 + 1617*b1 + 160987) * q^16 + (-3030*b2 - 1574*b1 - 338898) * q^17 + (-15371*b2 + 2437*b1 + 74284) * q^19 + (-41524*b2 + 2044*b1 + 640696) * q^20 + (-40972*b2 + 4004*b1 - 949016) * q^22 + (24200*b2 + 3808*b1 - 628544) * q^23 + (-15338*b2 - 4802*b1 + 1024407) * q^25 + (-20986*b2 + 23394*b1 + 2928352) * q^26 + (-19208*b2 + 16807*b1 + 1246119) * q^28 + (-54866*b2 - 18914*b1 - 1360606) * q^29 + (55698*b2 - 70302*b1 + 956480) * q^31 + (36066*b2 - 20055*b1 - 8407317) * q^32 + (-438178*b2 - 748*b1 - 1327214) * q^34 + (-103243*b2 + 31213*b1 - 1243718) * q^35 + (209418*b2 + 60522*b1 + 465206) * q^37 + (248060*b2 - 139278*b1 - 14493290) * q^38 + (625640*b2 - 76600*b1 - 27619760) * q^40 + (163478*b2 + 131894*b1 + 4806886) * q^41 + (121982*b2 + 65366*b1 - 20543724) * q^43 + (-314984*b2 - 1960*b1 - 32337328) * q^44 + (-405224*b2 + 119896*b1 + 29915888) * q^46 + (534778*b2 - 83238*b1 + 3456320) * q^47 + 5764801 * q^49 + (727615*b2 - 44940*b1 - 24441685) * q^50 + (2925244*b2 - 361424*b1 - 26969348) * q^52 + (-1553376*b2 + 450352*b1 - 22500870) * q^53 + (1659356*b2 + 988364*b1 - 35274344) * q^55 + (1128470*b2 - 352947*b1 - 11248685) * q^56 + (-2535150*b2 - 138180*b1 - 53054610) * q^58 + (2231195*b2 + 49659*b1 + 14196700) * q^59 + (589107*b2 - 844773*b1 + 63915614) * q^61 + (-3668848*b2 + 1303812*b1 + 15661156) * q^62 + (-4312590*b2 - 314727*b1 + 2617387) * q^64 + (958090*b2 - 1035790*b1 - 121427740) * q^65 + (3939816*b2 + 47712*b1 - 85058596) * q^67 + (613704*b2 - 2251634*b1 - 247828602) * q^68 + (888370*b2 - 1128470*b1 - 78320620) * q^70 + (-3499356*b2 + 526260*b1 - 98838168) * q^71 + (-9544844*b2 + 118516*b1 + 114737770) * q^73 + (4189718*b2 + 679140*b1 + 230232154) * q^74 + (-15924468*b2 + 2299290*b1 + 242946662) * q^76 + (-1560650*b2 - 1579858*b1 + 2756348) * q^77 + (-7475532*b2 + 1679412*b1 - 320137552) * q^79 + (-11964112*b2 + 4328752*b1 + 444444448) * q^80 + (13216518*b2 - 570276*b1 + 187558434) * q^82 + (4479559*b2 + 1977367*b1 + 366839060) * q^83 + (18558254*b2 - 1518034*b1 + 146059804) * q^85 + (-16416916*b2 + 4116*b1 + 293660752) * q^86 + (-11172080*b2 - 4229456*b1 + 402041600) * q^88 + (10612104*b2 - 1815976*b1 - 168938826) * q^89 + (7243817*b2 - 420175*b1 - 15832194) * q^91 + (25723952*b2 - 6344912*b1 - 230374496) * q^92 + (-2488928*b2 + 4825540*b1 + 462668276) * q^94 + (-10749416*b2 + 4098416*b1 + 734357024) * q^95 + (33276782*b2 - 864850*b1 - 215832750) * q^97 + (5764801*b2 - 40353607) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 21 q^{2} + 1557 q^{4} - 1554 q^{5} + 7203 q^{7} - 14055 q^{8}+O(q^{10})$$ 3 * q - 21 * q^2 + 1557 * q^4 - 1554 * q^5 + 7203 * q^7 - 14055 * q^8 $$3 q - 21 q^{2} + 1557 q^{4} - 1554 q^{5} + 7203 q^{7} - 14055 q^{8} - 97860 q^{10} + 3444 q^{11} - 19782 q^{13} - 50421 q^{14} + 482961 q^{16} - 1016694 q^{17} + 222852 q^{19} + 1922088 q^{20} - 2847048 q^{22} - 1885632 q^{23} + 3073221 q^{25} + 8785056 q^{26} + 3738357 q^{28} - 4081818 q^{29} + 2869440 q^{31} - 25221951 q^{32} - 3981642 q^{34} - 3731154 q^{35} + 1395618 q^{37} - 43479870 q^{38} - 82859280 q^{40} + 14420658 q^{41} - 61631172 q^{43} - 97011984 q^{44} + 89747664 q^{46} + 10368960 q^{47} + 17294403 q^{49} - 73325055 q^{50} - 80908044 q^{52} - 67502610 q^{53} - 105823032 q^{55} - 33746055 q^{56} - 159163830 q^{58} + 42590100 q^{59} + 191746842 q^{61} + 46983468 q^{62} + 7852161 q^{64} - 364283220 q^{65} - 255175788 q^{67} - 743485806 q^{68} - 234961860 q^{70} - 296514504 q^{71} + 344213310 q^{73} + 690696462 q^{74} + 728839986 q^{76} + 8269044 q^{77} - 960412656 q^{79} + 1333333344 q^{80} + 562675302 q^{82} + 1100517180 q^{83} + 438179412 q^{85} + 880982256 q^{86} + 1206124800 q^{88} - 506816478 q^{89} - 47496582 q^{91} - 691123488 q^{92} + 1388004828 q^{94} + 2203071072 q^{95} - 647498250 q^{97} - 121060821 q^{98}+O(q^{100})$$ 3 * q - 21 * q^2 + 1557 * q^4 - 1554 * q^5 + 7203 * q^7 - 14055 * q^8 - 97860 * q^10 + 3444 * q^11 - 19782 * q^13 - 50421 * q^14 + 482961 * q^16 - 1016694 * q^17 + 222852 * q^19 + 1922088 * q^20 - 2847048 * q^22 - 1885632 * q^23 + 3073221 * q^25 + 8785056 * q^26 + 3738357 * q^28 - 4081818 * q^29 + 2869440 * q^31 - 25221951 * q^32 - 3981642 * q^34 - 3731154 * q^35 + 1395618 * q^37 - 43479870 * q^38 - 82859280 * q^40 + 14420658 * q^41 - 61631172 * q^43 - 97011984 * q^44 + 89747664 * q^46 + 10368960 * q^47 + 17294403 * q^49 - 73325055 * q^50 - 80908044 * q^52 - 67502610 * q^53 - 105823032 * q^55 - 33746055 * q^56 - 159163830 * q^58 + 42590100 * q^59 + 191746842 * q^61 + 46983468 * q^62 + 7852161 * q^64 - 364283220 * q^65 - 255175788 * q^67 - 743485806 * q^68 - 234961860 * q^70 - 296514504 * q^71 + 344213310 * q^73 + 690696462 * q^74 + 728839986 * q^76 + 8269044 * q^77 - 960412656 * q^79 + 1333333344 * q^80 + 562675302 * q^82 + 1100517180 * q^83 + 438179412 * q^85 + 880982256 * q^86 + 1206124800 * q^88 - 506816478 * q^89 - 47496582 * q^91 - 691123488 * q^92 + 1388004828 * q^94 + 2203071072 * q^95 - 647498250 * q^97 - 121060821 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 426x + 2016$$ :

 $$\beta_{1}$$ $$=$$ $$( -\nu^{2} + 25\nu + 276 ) / 6$$ (-v^2 + 25*v + 276) / 6 $$\beta_{2}$$ $$=$$ $$( \nu^{2} + 11\nu - 288 ) / 6$$ (v^2 + 11*v - 288) / 6
 $$\nu$$ $$=$$ $$( \beta_{2} + \beta _1 + 2 ) / 6$$ (b2 + b1 + 2) / 6 $$\nu^{2}$$ $$=$$ $$( 25\beta_{2} - 11\beta _1 + 1706 ) / 6$$ (25*b2 - 11*b1 + 1706) / 6

## 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
 4.96128 −22.2358 18.2745
−41.8019 0 1235.40 1791.89 0 2401.00 −30239.6 0 −74904.4
1.2 −13.3607 0 −333.491 −1922.19 0 2401.00 11296.4 0 25681.8
1.3 34.1627 0 655.088 −1423.70 0 2401.00 4888.28 0 −48637.4
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$7$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 63.10.a.e 3
3.b odd 2 1 7.10.a.b 3
12.b even 2 1 112.10.a.h 3
15.d odd 2 1 175.10.a.d 3
15.e even 4 2 175.10.b.d 6
21.c even 2 1 49.10.a.c 3
21.g even 6 2 49.10.c.e 6
21.h odd 6 2 49.10.c.d 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.10.a.b 3 3.b odd 2 1
49.10.a.c 3 21.c even 2 1
49.10.c.d 6 21.h odd 6 2
49.10.c.e 6 21.g even 6 2
63.10.a.e 3 1.a even 1 1 trivial
112.10.a.h 3 12.b even 2 1
175.10.a.d 3 15.d odd 2 1
175.10.b.d 6 15.e even 4 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{3} + 21T_{2}^{2} - 1326T_{2} - 19080$$ acting on $$S_{10}^{\mathrm{new}}(\Gamma_0(63))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3} + 21 T^{2} - 1326 T - 19080$$
$3$ $$T^{3}$$
$5$ $$T^{3} + 1554 T^{2} + \cdots - 4903718400$$
$7$ $$(T - 2401)^{3}$$
$11$ $$T^{3} + \cdots - 108859759460352$$
$13$ $$T^{3} + 19782 T^{2} + \cdots - 41548412541440$$
$17$ $$T^{3} + 1016694 T^{2} + \cdots + 21\!\cdots\!32$$
$19$ $$T^{3} - 222852 T^{2} + \cdots - 43\!\cdots\!60$$
$23$ $$T^{3} + 1885632 T^{2} + \cdots - 97\!\cdots\!36$$
$29$ $$T^{3} + 4081818 T^{2} + \cdots - 44\!\cdots\!00$$
$31$ $$T^{3} - 2869440 T^{2} + \cdots - 74\!\cdots\!84$$
$37$ $$T^{3} - 1395618 T^{2} + \cdots - 34\!\cdots\!28$$
$41$ $$T^{3} - 14420658 T^{2} + \cdots + 19\!\cdots\!12$$
$43$ $$T^{3} + 61631172 T^{2} + \cdots + 68\!\cdots\!80$$
$47$ $$T^{3} - 10368960 T^{2} + \cdots + 43\!\cdots\!16$$
$53$ $$T^{3} + 67502610 T^{2} + \cdots - 23\!\cdots\!28$$
$59$ $$T^{3} - 42590100 T^{2} + \cdots - 42\!\cdots\!00$$
$61$ $$T^{3} - 191746842 T^{2} + \cdots + 51\!\cdots\!08$$
$67$ $$T^{3} + 255175788 T^{2} + \cdots - 20\!\cdots\!64$$
$71$ $$T^{3} + 296514504 T^{2} + \cdots - 16\!\cdots\!80$$
$73$ $$T^{3} - 344213310 T^{2} + \cdots + 19\!\cdots\!48$$
$79$ $$T^{3} + 960412656 T^{2} + \cdots - 11\!\cdots\!00$$
$83$ $$T^{3} - 1100517180 T^{2} + \cdots - 18\!\cdots\!48$$
$89$ $$T^{3} + 506816478 T^{2} + \cdots - 19\!\cdots\!40$$
$97$ $$T^{3} + 647498250 T^{2} + \cdots - 49\!\cdots\!16$$