# Properties

 Label 63.4.f.a Level $63$ Weight $4$ Character orbit 63.f Analytic conductor $3.717$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$63 = 3^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 63.f (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.71712033036$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \zeta_{6} q^{2} + ( -6 + 3 \zeta_{6} ) q^{3} + ( 7 - 7 \zeta_{6} ) q^{4} + ( 14 - 14 \zeta_{6} ) q^{5} + ( -3 - 3 \zeta_{6} ) q^{6} + 7 \zeta_{6} q^{7} + 15 q^{8} + ( 27 - 27 \zeta_{6} ) q^{9} +O(q^{10})$$ $$q + \zeta_{6} q^{2} + ( -6 + 3 \zeta_{6} ) q^{3} + ( 7 - 7 \zeta_{6} ) q^{4} + ( 14 - 14 \zeta_{6} ) q^{5} + ( -3 - 3 \zeta_{6} ) q^{6} + 7 \zeta_{6} q^{7} + 15 q^{8} + ( 27 - 27 \zeta_{6} ) q^{9} + 14 q^{10} + 47 \zeta_{6} q^{11} + ( -21 + 42 \zeta_{6} ) q^{12} + ( 86 - 86 \zeta_{6} ) q^{13} + ( -7 + 7 \zeta_{6} ) q^{14} + ( -42 + 84 \zeta_{6} ) q^{15} -41 \zeta_{6} q^{16} -9 q^{17} + 27 q^{18} -131 q^{19} -98 \zeta_{6} q^{20} + ( -21 - 21 \zeta_{6} ) q^{21} + ( -47 + 47 \zeta_{6} ) q^{22} + ( 12 - 12 \zeta_{6} ) q^{23} + ( -90 + 45 \zeta_{6} ) q^{24} -71 \zeta_{6} q^{25} + 86 q^{26} + ( -81 + 162 \zeta_{6} ) q^{27} + 49 q^{28} + 260 \zeta_{6} q^{29} + ( -84 + 42 \zeta_{6} ) q^{30} + ( 54 - 54 \zeta_{6} ) q^{31} + ( 161 - 161 \zeta_{6} ) q^{32} + ( -141 - 141 \zeta_{6} ) q^{33} -9 \zeta_{6} q^{34} + 98 q^{35} -189 \zeta_{6} q^{36} -246 q^{37} -131 \zeta_{6} q^{38} + ( -258 + 516 \zeta_{6} ) q^{39} + ( 210 - 210 \zeta_{6} ) q^{40} + ( -383 + 383 \zeta_{6} ) q^{41} + ( 21 - 42 \zeta_{6} ) q^{42} + 169 \zeta_{6} q^{43} + 329 q^{44} -378 \zeta_{6} q^{45} + 12 q^{46} -96 \zeta_{6} q^{47} + ( 123 + 123 \zeta_{6} ) q^{48} + ( -49 + 49 \zeta_{6} ) q^{49} + ( 71 - 71 \zeta_{6} ) q^{50} + ( 54 - 27 \zeta_{6} ) q^{51} -602 \zeta_{6} q^{52} + 300 q^{53} + ( -162 + 81 \zeta_{6} ) q^{54} + 658 q^{55} + 105 \zeta_{6} q^{56} + ( 786 - 393 \zeta_{6} ) q^{57} + ( -260 + 260 \zeta_{6} ) q^{58} + ( -429 + 429 \zeta_{6} ) q^{59} + ( 294 + 294 \zeta_{6} ) q^{60} + 380 \zeta_{6} q^{61} + 54 q^{62} + 189 q^{63} -167 q^{64} -1204 \zeta_{6} q^{65} + ( 141 - 282 \zeta_{6} ) q^{66} + ( 155 - 155 \zeta_{6} ) q^{67} + ( -63 + 63 \zeta_{6} ) q^{68} + ( -36 + 72 \zeta_{6} ) q^{69} + 98 \zeta_{6} q^{70} + 72 q^{71} + ( 405 - 405 \zeta_{6} ) q^{72} + 117 q^{73} -246 \zeta_{6} q^{74} + ( 213 + 213 \zeta_{6} ) q^{75} + ( -917 + 917 \zeta_{6} ) q^{76} + ( -329 + 329 \zeta_{6} ) q^{77} + ( -516 + 258 \zeta_{6} ) q^{78} + 526 \zeta_{6} q^{79} -574 q^{80} -729 \zeta_{6} q^{81} -383 q^{82} -576 \zeta_{6} q^{83} + ( -294 + 147 \zeta_{6} ) q^{84} + ( -126 + 126 \zeta_{6} ) q^{85} + ( -169 + 169 \zeta_{6} ) q^{86} + ( -780 - 780 \zeta_{6} ) q^{87} + 705 \zeta_{6} q^{88} -278 q^{89} + ( 378 - 378 \zeta_{6} ) q^{90} + 602 q^{91} -84 \zeta_{6} q^{92} + ( -162 + 324 \zeta_{6} ) q^{93} + ( 96 - 96 \zeta_{6} ) q^{94} + ( -1834 + 1834 \zeta_{6} ) q^{95} + ( -483 + 966 \zeta_{6} ) q^{96} + 201 \zeta_{6} q^{97} -49 q^{98} + 1269 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + q^{2} - 9q^{3} + 7q^{4} + 14q^{5} - 9q^{6} + 7q^{7} + 30q^{8} + 27q^{9} + O(q^{10})$$ $$2q + q^{2} - 9q^{3} + 7q^{4} + 14q^{5} - 9q^{6} + 7q^{7} + 30q^{8} + 27q^{9} + 28q^{10} + 47q^{11} + 86q^{13} - 7q^{14} - 41q^{16} - 18q^{17} + 54q^{18} - 262q^{19} - 98q^{20} - 63q^{21} - 47q^{22} + 12q^{23} - 135q^{24} - 71q^{25} + 172q^{26} + 98q^{28} + 260q^{29} - 126q^{30} + 54q^{31} + 161q^{32} - 423q^{33} - 9q^{34} + 196q^{35} - 189q^{36} - 492q^{37} - 131q^{38} + 210q^{40} - 383q^{41} + 169q^{43} + 658q^{44} - 378q^{45} + 24q^{46} - 96q^{47} + 369q^{48} - 49q^{49} + 71q^{50} + 81q^{51} - 602q^{52} + 600q^{53} - 243q^{54} + 1316q^{55} + 105q^{56} + 1179q^{57} - 260q^{58} - 429q^{59} + 882q^{60} + 380q^{61} + 108q^{62} + 378q^{63} - 334q^{64} - 1204q^{65} + 155q^{67} - 63q^{68} + 98q^{70} + 144q^{71} + 405q^{72} + 234q^{73} - 246q^{74} + 639q^{75} - 917q^{76} - 329q^{77} - 774q^{78} + 526q^{79} - 1148q^{80} - 729q^{81} - 766q^{82} - 576q^{83} - 441q^{84} - 126q^{85} - 169q^{86} - 2340q^{87} + 705q^{88} - 556q^{89} + 378q^{90} + 1204q^{91} - 84q^{92} + 96q^{94} - 1834q^{95} + 201q^{97} - 98q^{98} + 2538q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/63\mathbb{Z}\right)^\times$$.

 $$n$$ $$10$$ $$29$$ $$\chi(n)$$ $$1$$ $$-1 + \zeta_{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
22.1
 0.5 + 0.866025i 0.5 − 0.866025i
0.500000 + 0.866025i −4.50000 + 2.59808i 3.50000 6.06218i 7.00000 12.1244i −4.50000 2.59808i 3.50000 + 6.06218i 15.0000 13.5000 23.3827i 14.0000
43.1 0.500000 0.866025i −4.50000 2.59808i 3.50000 + 6.06218i 7.00000 + 12.1244i −4.50000 + 2.59808i 3.50000 6.06218i 15.0000 13.5000 + 23.3827i 14.0000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 63.4.f.a 2
3.b odd 2 1 189.4.f.a 2
9.c even 3 1 inner 63.4.f.a 2
9.c even 3 1 567.4.a.a 1
9.d odd 6 1 189.4.f.a 2
9.d odd 6 1 567.4.a.b 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.4.f.a 2 1.a even 1 1 trivial
63.4.f.a 2 9.c even 3 1 inner
189.4.f.a 2 3.b odd 2 1
189.4.f.a 2 9.d odd 6 1
567.4.a.a 1 9.c even 3 1
567.4.a.b 1 9.d odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} - T_{2} + 1$$ acting on $$S_{4}^{\mathrm{new}}(63, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T + T^{2}$$
$3$ $$27 + 9 T + T^{2}$$
$5$ $$196 - 14 T + T^{2}$$
$7$ $$49 - 7 T + T^{2}$$
$11$ $$2209 - 47 T + T^{2}$$
$13$ $$7396 - 86 T + T^{2}$$
$17$ $$( 9 + T )^{2}$$
$19$ $$( 131 + T )^{2}$$
$23$ $$144 - 12 T + T^{2}$$
$29$ $$67600 - 260 T + T^{2}$$
$31$ $$2916 - 54 T + T^{2}$$
$37$ $$( 246 + T )^{2}$$
$41$ $$146689 + 383 T + T^{2}$$
$43$ $$28561 - 169 T + T^{2}$$
$47$ $$9216 + 96 T + T^{2}$$
$53$ $$( -300 + T )^{2}$$
$59$ $$184041 + 429 T + T^{2}$$
$61$ $$144400 - 380 T + T^{2}$$
$67$ $$24025 - 155 T + T^{2}$$
$71$ $$( -72 + T )^{2}$$
$73$ $$( -117 + T )^{2}$$
$79$ $$276676 - 526 T + T^{2}$$
$83$ $$331776 + 576 T + T^{2}$$
$89$ $$( 278 + T )^{2}$$
$97$ $$40401 - 201 T + T^{2}$$