# Properties

 Label 63.10.a.d Level $63$ Weight $10$ Character orbit 63.a Self dual yes Analytic conductor $32.447$ Analytic rank $0$ Dimension $2$ 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: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{193})$$ Defining polynomial: $$x^{2} - x - 48$$ x^2 - x - 48 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2$$ 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 $$\beta = \sqrt{193}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta + 3) q^{2} + (6 \beta - 310) q^{4} + (95 \beta + 1119) q^{5} - 2401 q^{7} + ( - 804 \beta - 1308) q^{8}+O(q^{10})$$ q + (b + 3) * q^2 + (6*b - 310) * q^4 + (95*b + 1119) * q^5 - 2401 * q^7 + (-804*b - 1308) * q^8 $$q + (\beta + 3) q^{2} + (6 \beta - 310) q^{4} + (95 \beta + 1119) q^{5} - 2401 q^{7} + ( - 804 \beta - 1308) q^{8} + (1404 \beta + 21692) q^{10} + (3326 \beta - 17658) q^{11} + (10899 \beta - 13265) q^{13} + ( - 2401 \beta - 7203) q^{14} + ( - 6792 \beta - 376) q^{16} + ( - 9426 \beta + 231960) q^{17} + ( - 1887 \beta - 462713) q^{19} + ( - 22736 \beta - 236880) q^{20} + ( - 7680 \beta + 588944) q^{22} + (38088 \beta - 389064) q^{23} + (212610 \beta + 1040861) q^{25} + (19432 \beta + 2063712) q^{26} + ( - 14406 \beta + 744310) q^{28} + (94682 \beta + 5001792) q^{29} + ( - 161430 \beta + 1233630) q^{31} + (390896 \beta - 642288) q^{32} + (203682 \beta - 1123338) q^{34} + ( - 228095 \beta - 2686719) q^{35} + ( - 248130 \beta + 15367776) q^{37} + ( - 468374 \beta - 1752330) q^{38} + ( - 1023936 \beta - 16204992) q^{40} + (860818 \beta + 9551724) q^{41} + (1048278 \beta + 2032550) q^{43} + ( - 1137008 \beta + 9325488) q^{44} + ( - 274800 \beta + 6183792) q^{46} + ( - 1033182 \beta + 41097510) q^{47} + 5764801 q^{49} + (1678691 \beta + 44156313) q^{50} + ( - 3458280 \beta + 16733192) q^{52} + ( - 4685568 \beta + 27594906) q^{53} + (2044284 \beta + 41222908) q^{55} + (1930404 \beta + 3140508) q^{56} + (5285838 \beta + 33279002) q^{58} + ( - 1563825 \beta + 3534609) q^{59} + ( - 3395319 \beta + 22158193) q^{61} + (749340 \beta - 27455100) q^{62} + (4007904 \beta + 73708576) q^{64} + (10935806 \beta + 184989630) q^{65} + ( - 7026216 \beta - 120960668) q^{67} + (4313820 \beta - 82822908) q^{68} + ( - 3371004 \beta - 52082492) q^{70} + ( - 15075900 \beta - 103246908) q^{71} + ( - 2840484 \beta - 249576594) q^{73} + (14623386 \beta - 1785762) q^{74} + ( - 2191308 \beta + 141255884) q^{76} + ( - 7985726 \beta + 42396858) q^{77} + (16873716 \beta + 234267548) q^{79} + ( - 7635968 \beta - 124952064) q^{80} + (12134178 \beta + 194793046) q^{82} + (21562275 \beta - 222011979) q^{83} + (11488506 \beta + 86737530) q^{85} + (5177384 \beta + 208415304) q^{86} + (9846624 \beta - 493005408) q^{88} + ( - 1406968 \beta - 318133698) q^{89} + ( - 26168499 \beta + 31849265) q^{91} + ( - 14141664 \beta + 164715744) q^{92} + (37997964 \beta - 76111596) q^{94} + ( - 46069288 \beta - 552373992) q^{95} + (5731530 \beta - 816358032) q^{97} + (5764801 \beta + 17294403) q^{98}+O(q^{100})$$ q + (b + 3) * q^2 + (6*b - 310) * q^4 + (95*b + 1119) * q^5 - 2401 * q^7 + (-804*b - 1308) * q^8 + (1404*b + 21692) * q^10 + (3326*b - 17658) * q^11 + (10899*b - 13265) * q^13 + (-2401*b - 7203) * q^14 + (-6792*b - 376) * q^16 + (-9426*b + 231960) * q^17 + (-1887*b - 462713) * q^19 + (-22736*b - 236880) * q^20 + (-7680*b + 588944) * q^22 + (38088*b - 389064) * q^23 + (212610*b + 1040861) * q^25 + (19432*b + 2063712) * q^26 + (-14406*b + 744310) * q^28 + (94682*b + 5001792) * q^29 + (-161430*b + 1233630) * q^31 + (390896*b - 642288) * q^32 + (203682*b - 1123338) * q^34 + (-228095*b - 2686719) * q^35 + (-248130*b + 15367776) * q^37 + (-468374*b - 1752330) * q^38 + (-1023936*b - 16204992) * q^40 + (860818*b + 9551724) * q^41 + (1048278*b + 2032550) * q^43 + (-1137008*b + 9325488) * q^44 + (-274800*b + 6183792) * q^46 + (-1033182*b + 41097510) * q^47 + 5764801 * q^49 + (1678691*b + 44156313) * q^50 + (-3458280*b + 16733192) * q^52 + (-4685568*b + 27594906) * q^53 + (2044284*b + 41222908) * q^55 + (1930404*b + 3140508) * q^56 + (5285838*b + 33279002) * q^58 + (-1563825*b + 3534609) * q^59 + (-3395319*b + 22158193) * q^61 + (749340*b - 27455100) * q^62 + (4007904*b + 73708576) * q^64 + (10935806*b + 184989630) * q^65 + (-7026216*b - 120960668) * q^67 + (4313820*b - 82822908) * q^68 + (-3371004*b - 52082492) * q^70 + (-15075900*b - 103246908) * q^71 + (-2840484*b - 249576594) * q^73 + (14623386*b - 1785762) * q^74 + (-2191308*b + 141255884) * q^76 + (-7985726*b + 42396858) * q^77 + (16873716*b + 234267548) * q^79 + (-7635968*b - 124952064) * q^80 + (12134178*b + 194793046) * q^82 + (21562275*b - 222011979) * q^83 + (11488506*b + 86737530) * q^85 + (5177384*b + 208415304) * q^86 + (9846624*b - 493005408) * q^88 + (-1406968*b - 318133698) * q^89 + (-26168499*b + 31849265) * q^91 + (-14141664*b + 164715744) * q^92 + (37997964*b - 76111596) * q^94 + (-46069288*b - 552373992) * q^95 + (5731530*b - 816358032) * q^97 + (5764801*b + 17294403) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 6 q^{2} - 620 q^{4} + 2238 q^{5} - 4802 q^{7} - 2616 q^{8}+O(q^{10})$$ 2 * q + 6 * q^2 - 620 * q^4 + 2238 * q^5 - 4802 * q^7 - 2616 * q^8 $$2 q + 6 q^{2} - 620 q^{4} + 2238 q^{5} - 4802 q^{7} - 2616 q^{8} + 43384 q^{10} - 35316 q^{11} - 26530 q^{13} - 14406 q^{14} - 752 q^{16} + 463920 q^{17} - 925426 q^{19} - 473760 q^{20} + 1177888 q^{22} - 778128 q^{23} + 2081722 q^{25} + 4127424 q^{26} + 1488620 q^{28} + 10003584 q^{29} + 2467260 q^{31} - 1284576 q^{32} - 2246676 q^{34} - 5373438 q^{35} + 30735552 q^{37} - 3504660 q^{38} - 32409984 q^{40} + 19103448 q^{41} + 4065100 q^{43} + 18650976 q^{44} + 12367584 q^{46} + 82195020 q^{47} + 11529602 q^{49} + 88312626 q^{50} + 33466384 q^{52} + 55189812 q^{53} + 82445816 q^{55} + 6281016 q^{56} + 66558004 q^{58} + 7069218 q^{59} + 44316386 q^{61} - 54910200 q^{62} + 147417152 q^{64} + 369979260 q^{65} - 241921336 q^{67} - 165645816 q^{68} - 104164984 q^{70} - 206493816 q^{71} - 499153188 q^{73} - 3571524 q^{74} + 282511768 q^{76} + 84793716 q^{77} + 468535096 q^{79} - 249904128 q^{80} + 389586092 q^{82} - 444023958 q^{83} + 173475060 q^{85} + 416830608 q^{86} - 986010816 q^{88} - 636267396 q^{89} + 63698530 q^{91} + 329431488 q^{92} - 152223192 q^{94} - 1104747984 q^{95} - 1632716064 q^{97} + 34588806 q^{98}+O(q^{100})$$ 2 * q + 6 * q^2 - 620 * q^4 + 2238 * q^5 - 4802 * q^7 - 2616 * q^8 + 43384 * q^10 - 35316 * q^11 - 26530 * q^13 - 14406 * q^14 - 752 * q^16 + 463920 * q^17 - 925426 * q^19 - 473760 * q^20 + 1177888 * q^22 - 778128 * q^23 + 2081722 * q^25 + 4127424 * q^26 + 1488620 * q^28 + 10003584 * q^29 + 2467260 * q^31 - 1284576 * q^32 - 2246676 * q^34 - 5373438 * q^35 + 30735552 * q^37 - 3504660 * q^38 - 32409984 * q^40 + 19103448 * q^41 + 4065100 * q^43 + 18650976 * q^44 + 12367584 * q^46 + 82195020 * q^47 + 11529602 * q^49 + 88312626 * q^50 + 33466384 * q^52 + 55189812 * q^53 + 82445816 * q^55 + 6281016 * q^56 + 66558004 * q^58 + 7069218 * q^59 + 44316386 * q^61 - 54910200 * q^62 + 147417152 * q^64 + 369979260 * q^65 - 241921336 * q^67 - 165645816 * q^68 - 104164984 * q^70 - 206493816 * q^71 - 499153188 * q^73 - 3571524 * q^74 + 282511768 * q^76 + 84793716 * q^77 + 468535096 * q^79 - 249904128 * q^80 + 389586092 * q^82 - 444023958 * q^83 + 173475060 * q^85 + 416830608 * q^86 - 986010816 * q^88 - 636267396 * q^89 + 63698530 * q^91 + 329431488 * q^92 - 152223192 * q^94 - 1104747984 * q^95 - 1632716064 * q^97 + 34588806 * q^98

## 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
 −6.44622 7.44622
−10.8924 0 −393.355 −200.782 0 −2401.00 9861.52 0 2187.01
1.2 16.8924 0 −226.645 2438.78 0 −2401.00 −12477.5 0 41197.0
 $$n$$: 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.d 2
3.b odd 2 1 7.10.a.a 2
12.b even 2 1 112.10.a.e 2
15.d odd 2 1 175.10.a.b 2
15.e even 4 2 175.10.b.b 4
21.c even 2 1 49.10.a.b 2
21.g even 6 2 49.10.c.b 4
21.h odd 6 2 49.10.c.c 4

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} - 6T_{2} - 184$$ acting on $$S_{10}^{\mathrm{new}}(\Gamma_0(63))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 6T - 184$$
$3$ $$T^{2}$$
$5$ $$T^{2} - 2238 T - 489664$$
$7$ $$(T + 2401)^{2}$$
$11$ $$T^{2} + 35316 T - 1823214304$$
$13$ $$T^{2} + 26530 T - 22750162568$$
$17$ $$T^{2} - 463920 T + 36657492732$$
$19$ $$T^{2} + 925426 T + 213416091952$$
$23$ $$T^{2} + 778128 T - 128613482496$$
$29$ $$T^{2} - 10003584 T + 23287739754332$$
$31$ $$T^{2} - 2467260 T - 3507668488800$$
$37$ $$T^{2} + \cdots + 224285819284476$$
$41$ $$T^{2} - 19103448 T - 51779041048756$$
$43$ $$T^{2} + \cdots - 207953886197312$$
$47$ $$T^{2} - 82195020 T + 14\!\cdots\!68$$
$53$ $$T^{2} - 55189812 T - 34\!\cdots\!96$$
$59$ $$T^{2} + \cdots - 459497424927744$$
$61$ $$T^{2} - 44316386 T - 17\!\cdots\!24$$
$67$ $$T^{2} + 241921336 T + 51\!\cdots\!16$$
$71$ $$T^{2} + 206493816 T - 33\!\cdots\!36$$
$73$ $$T^{2} + 499153188 T + 60\!\cdots\!28$$
$79$ $$T^{2} - 468535096 T - 70118242258304$$
$83$ $$T^{2} + 444023958 T - 40\!\cdots\!84$$
$89$ $$T^{2} + 636267396 T + 10\!\cdots\!72$$
$97$ $$T^{2} + 1632716064 T + 66\!\cdots\!24$$