# Properties

 Label 63.6.a.f Level 63 Weight 6 Character orbit 63.a Self dual yes Analytic conductor 10.104 Analytic rank 1 Dimension 2 CM no Inner twists 1

# Learn more about

## Newspace parameters

 Level: $$N$$ $$=$$ $$63 = 3^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 63.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$10.1041806482$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{57})$$ Defining polynomial: $$x^{2} - x - 14$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ 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 $$\beta = \frac{1}{2}(1 + \sqrt{57})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -4 - \beta ) q^{2} + ( -2 + 9 \beta ) q^{4} + ( 14 - 10 \beta ) q^{5} + 49 q^{7} + ( 10 - 11 \beta ) q^{8} +O(q^{10})$$ $$q + ( -4 - \beta ) q^{2} + ( -2 + 9 \beta ) q^{4} + ( 14 - 10 \beta ) q^{5} + 49 q^{7} + ( 10 - 11 \beta ) q^{8} + ( 84 + 36 \beta ) q^{10} + ( -260 + 124 \beta ) q^{11} + ( -238 + 126 \beta ) q^{13} + ( -196 - 49 \beta ) q^{14} + ( 178 - 243 \beta ) q^{16} + ( -938 + 76 \beta ) q^{17} + ( -1624 - 18 \beta ) q^{19} + ( -1288 + 56 \beta ) q^{20} + ( -696 - 360 \beta ) q^{22} + ( -760 - 568 \beta ) q^{23} + ( -1529 - 180 \beta ) q^{25} + ( -812 - 392 \beta ) q^{26} + ( -98 + 441 \beta ) q^{28} + ( -3222 - 252 \beta ) q^{29} + ( -280 + 540 \beta ) q^{31} + ( 2370 + 1389 \beta ) q^{32} + ( 2688 + 558 \beta ) q^{34} + ( 686 - 490 \beta ) q^{35} + ( 2846 + 540 \beta ) q^{37} + ( 6748 + 1714 \beta ) q^{38} + ( 1680 - 144 \beta ) q^{40} + ( 2478 + 1092 \beta ) q^{41} + ( 884 - 4788 \beta ) q^{43} + ( 16144 - 1472 \beta ) q^{44} + ( 10992 + 3600 \beta ) q^{46} + ( -3976 - 3748 \beta ) q^{47} + 2401 q^{49} + ( 8636 + 2429 \beta ) q^{50} + ( 16352 - 1260 \beta ) q^{52} + ( -4838 + 208 \beta ) q^{53} + ( -21000 + 3096 \beta ) q^{55} + ( 490 - 539 \beta ) q^{56} + ( 16416 + 4482 \beta ) q^{58} + ( 20944 + 2050 \beta ) q^{59} + ( -29974 - 4806 \beta ) q^{61} + ( -6440 - 2420 \beta ) q^{62} + ( -34622 - 1539 \beta ) q^{64} + ( -20972 + 2884 \beta ) q^{65} + ( 13364 - 1944 \beta ) q^{67} + ( 11452 - 7910 \beta ) q^{68} + ( 4116 + 1764 \beta ) q^{70} + ( -50808 + 4200 \beta ) q^{71} + ( 11354 - 5256 \beta ) q^{73} + ( -18944 - 5546 \beta ) q^{74} + ( 980 - 14742 \beta ) q^{76} + ( -12740 + 6076 \beta ) q^{77} + ( 18176 + 14904 \beta ) q^{79} + ( 36512 - 2752 \beta ) q^{80} + ( -25200 - 7938 \beta ) q^{82} + ( -50904 - 15750 \beta ) q^{83} + ( -23772 + 9684 \beta ) q^{85} + ( 63496 + 23056 \beta ) q^{86} + ( -21696 + 2736 \beta ) q^{88} + ( -53242 + 22208 \beta ) q^{89} + ( -11662 + 6174 \beta ) q^{91} + ( -70048 - 10816 \beta ) q^{92} + ( 68376 + 22716 \beta ) q^{94} + ( -20216 + 16168 \beta ) q^{95} + ( 5978 + 8820 \beta ) q^{97} + ( -9604 - 2401 \beta ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 9q^{2} + 5q^{4} + 18q^{5} + 98q^{7} + 9q^{8} + O(q^{10})$$ $$2q - 9q^{2} + 5q^{4} + 18q^{5} + 98q^{7} + 9q^{8} + 204q^{10} - 396q^{11} - 350q^{13} - 441q^{14} + 113q^{16} - 1800q^{17} - 3266q^{19} - 2520q^{20} - 1752q^{22} - 2088q^{23} - 3238q^{25} - 2016q^{26} + 245q^{28} - 6696q^{29} - 20q^{31} + 6129q^{32} + 5934q^{34} + 882q^{35} + 6232q^{37} + 15210q^{38} + 3216q^{40} + 6048q^{41} - 3020q^{43} + 30816q^{44} + 25584q^{46} - 11700q^{47} + 4802q^{49} + 19701q^{50} + 31444q^{52} - 9468q^{53} - 38904q^{55} + 441q^{56} + 37314q^{58} + 43938q^{59} - 64754q^{61} - 15300q^{62} - 70783q^{64} - 39060q^{65} + 24784q^{67} + 14994q^{68} + 9996q^{70} - 97416q^{71} + 17452q^{73} - 43434q^{74} - 12782q^{76} - 19404q^{77} + 51256q^{79} + 70272q^{80} - 58338q^{82} - 117558q^{83} - 37860q^{85} + 150048q^{86} - 40656q^{88} - 84276q^{89} - 17150q^{91} - 150912q^{92} + 159468q^{94} - 24264q^{95} + 20776q^{97} - 21609q^{98} + O(q^{100})$$

## 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
 4.27492 −3.27492
−8.27492 0 36.4743 −28.7492 0 49.0000 −37.0241 0 237.897
1.2 −0.725083 0 −31.4743 46.7492 0 49.0000 46.0241 0 −33.8970
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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 63.6.a.f 2
3.b odd 2 1 7.6.a.b 2
4.b odd 2 1 1008.6.a.bq 2
7.b odd 2 1 441.6.a.l 2
12.b even 2 1 112.6.a.h 2
15.d odd 2 1 175.6.a.c 2
15.e even 4 2 175.6.b.c 4
21.c even 2 1 49.6.a.f 2
21.g even 6 2 49.6.c.d 4
21.h odd 6 2 49.6.c.e 4
24.f even 2 1 448.6.a.u 2
24.h odd 2 1 448.6.a.w 2
33.d even 2 1 847.6.a.c 2
84.h odd 2 1 784.6.a.v 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.6.a.b 2 3.b odd 2 1
49.6.a.f 2 21.c even 2 1
49.6.c.d 4 21.g even 6 2
49.6.c.e 4 21.h odd 6 2
63.6.a.f 2 1.a even 1 1 trivial
112.6.a.h 2 12.b even 2 1
175.6.a.c 2 15.d odd 2 1
175.6.b.c 4 15.e even 4 2
441.6.a.l 2 7.b odd 2 1
448.6.a.u 2 24.f even 2 1
448.6.a.w 2 24.h odd 2 1
784.6.a.v 2 84.h odd 2 1
847.6.a.c 2 33.d even 2 1
1008.6.a.bq 2 4.b odd 2 1

## Atkin-Lehner signs

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + 9 T_{2} + 6$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(63))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 9 T + 70 T^{2} + 288 T^{3} + 1024 T^{4}$$
$3$ 1
$5$ $$1 - 18 T + 4906 T^{2} - 56250 T^{3} + 9765625 T^{4}$$
$7$ $$( 1 - 49 T )^{2}$$
$11$ $$1 + 396 T + 142198 T^{2} + 63776196 T^{3} + 25937424601 T^{4}$$
$13$ $$1 + 350 T + 546978 T^{2} + 129952550 T^{3} + 137858491849 T^{4}$$
$17$ $$1 + 1800 T + 3567406 T^{2} + 2555742600 T^{3} + 2015993900449 T^{4}$$
$19$ $$1 + 3266 T + 7614270 T^{2} + 8086939334 T^{3} + 6131066257801 T^{4}$$
$23$ $$1 + 2088 T + 9365230 T^{2} + 13439084184 T^{3} + 41426511213649 T^{4}$$
$29$ $$1 + 6696 T + 51326470 T^{2} + 137342653704 T^{3} + 420707233300201 T^{4}$$
$31$ $$1 + 20 T + 53103102 T^{2} + 572583020 T^{3} + 819628286980801 T^{4}$$
$37$ $$1 - 6232 T + 144242070 T^{2} - 432151540024 T^{3} + 4808584372417849 T^{4}$$
$41$ $$1 - 6048 T + 223864366 T^{2} - 700698303648 T^{3} + 13422659310152401 T^{4}$$
$43$ $$1 + 3020 T - 30383466 T^{2} + 443965497860 T^{3} + 21611482313284249 T^{4}$$
$47$ $$1 + 11700 T + 292735582 T^{2} + 2683336581900 T^{3} + 52599132235830049 T^{4}$$
$53$ $$1 + 9468 T + 858185230 T^{2} + 3959474927724 T^{3} + 174887470365513049 T^{4}$$
$59$ $$1 - 43938 T + 1852599934 T^{2} - 31412343849462 T^{3} + 511116753300641401 T^{4}$$
$61$ $$1 + 64754 T + 2408321418 T^{2} + 54690988874954 T^{3} + 713342911662882601 T^{4}$$
$67$ $$1 - 24784 T + 2799959190 T^{2} - 33461500651888 T^{3} + 1822837804551761449 T^{4}$$
$71$ $$1 + 97416 T + 5729557966 T^{2} + 175760806457016 T^{3} + 3255243551009881201 T^{4}$$
$73$ $$1 - 17452 T + 3828622374 T^{2} - 36179245441036 T^{3} + 4297625829703557649 T^{4}$$
$79$ $$1 - 51256 T + 3645565854 T^{2} - 157717602787144 T^{3} + 9468276082626847201 T^{4}$$
$83$ $$1 + 117558 T + 7798161502 T^{2} + 463065739909794 T^{3} + 15516041187205853449 T^{4}$$
$89$ $$1 + 84276 T + 5915697430 T^{2} + 470602194123924 T^{3} + 31181719929966183601 T^{4}$$
$97$ $$1 - 20776 T + 16174049358 T^{2} - 178410581179432 T^{3} + 73742412689492826049 T^{4}$$
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