# Properties

 Label 63.4.e.b Level $63$ Weight $4$ Character orbit 63.e 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.e (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: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 7) 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.

 $$f(q)$$ $$=$$ $$q + ( 2 - 2 \zeta_{6} ) q^{2} + 4 \zeta_{6} q^{4} + ( 7 - 7 \zeta_{6} ) q^{5} + ( 21 - 14 \zeta_{6} ) q^{7} + 24 q^{8} +O(q^{10})$$ $$q + ( 2 - 2 \zeta_{6} ) q^{2} + 4 \zeta_{6} q^{4} + ( 7 - 7 \zeta_{6} ) q^{5} + ( 21 - 14 \zeta_{6} ) q^{7} + 24 q^{8} -14 \zeta_{6} q^{10} -5 \zeta_{6} q^{11} -14 q^{13} + ( 14 - 42 \zeta_{6} ) q^{14} + ( 16 - 16 \zeta_{6} ) q^{16} -21 \zeta_{6} q^{17} + ( -49 + 49 \zeta_{6} ) q^{19} + 28 q^{20} -10 q^{22} + ( -159 + 159 \zeta_{6} ) q^{23} + 76 \zeta_{6} q^{25} + ( -28 + 28 \zeta_{6} ) q^{26} + ( 56 + 28 \zeta_{6} ) q^{28} -58 q^{29} -147 \zeta_{6} q^{31} + 160 \zeta_{6} q^{32} -42 q^{34} + ( 49 - 147 \zeta_{6} ) q^{35} + ( -219 + 219 \zeta_{6} ) q^{37} + 98 \zeta_{6} q^{38} + ( 168 - 168 \zeta_{6} ) q^{40} -350 q^{41} -124 q^{43} + ( 20 - 20 \zeta_{6} ) q^{44} + 318 \zeta_{6} q^{46} + ( 525 - 525 \zeta_{6} ) q^{47} + ( 245 - 392 \zeta_{6} ) q^{49} + 152 q^{50} -56 \zeta_{6} q^{52} + 303 \zeta_{6} q^{53} -35 q^{55} + ( 504 - 336 \zeta_{6} ) q^{56} + ( -116 + 116 \zeta_{6} ) q^{58} -105 \zeta_{6} q^{59} + ( 413 - 413 \zeta_{6} ) q^{61} -294 q^{62} + 448 q^{64} + ( -98 + 98 \zeta_{6} ) q^{65} -415 \zeta_{6} q^{67} + ( 84 - 84 \zeta_{6} ) q^{68} + ( -196 - 98 \zeta_{6} ) q^{70} + 432 q^{71} + 1113 \zeta_{6} q^{73} + 438 \zeta_{6} q^{74} -196 q^{76} + ( -70 - 35 \zeta_{6} ) q^{77} + ( 103 - 103 \zeta_{6} ) q^{79} -112 \zeta_{6} q^{80} + ( -700 + 700 \zeta_{6} ) q^{82} -1092 q^{83} -147 q^{85} + ( -248 + 248 \zeta_{6} ) q^{86} -120 \zeta_{6} q^{88} + ( -329 + 329 \zeta_{6} ) q^{89} + ( -294 + 196 \zeta_{6} ) q^{91} -636 q^{92} -1050 \zeta_{6} q^{94} + 343 \zeta_{6} q^{95} -882 q^{97} + ( -294 - 490 \zeta_{6} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{2} + 4q^{4} + 7q^{5} + 28q^{7} + 48q^{8} + O(q^{10})$$ $$2q + 2q^{2} + 4q^{4} + 7q^{5} + 28q^{7} + 48q^{8} - 14q^{10} - 5q^{11} - 28q^{13} - 14q^{14} + 16q^{16} - 21q^{17} - 49q^{19} + 56q^{20} - 20q^{22} - 159q^{23} + 76q^{25} - 28q^{26} + 140q^{28} - 116q^{29} - 147q^{31} + 160q^{32} - 84q^{34} - 49q^{35} - 219q^{37} + 98q^{38} + 168q^{40} - 700q^{41} - 248q^{43} + 20q^{44} + 318q^{46} + 525q^{47} + 98q^{49} + 304q^{50} - 56q^{52} + 303q^{53} - 70q^{55} + 672q^{56} - 116q^{58} - 105q^{59} + 413q^{61} - 588q^{62} + 896q^{64} - 98q^{65} - 415q^{67} + 84q^{68} - 490q^{70} + 864q^{71} + 1113q^{73} + 438q^{74} - 392q^{76} - 175q^{77} + 103q^{79} - 112q^{80} - 700q^{82} - 2184q^{83} - 294q^{85} - 248q^{86} - 120q^{88} - 329q^{89} - 392q^{91} - 1272q^{92} - 1050q^{94} + 343q^{95} - 1764q^{97} - 1078q^{98} + O(q^{100})$$

## Character values

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

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

## 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
 0.5 + 0.866025i 0.5 − 0.866025i
1.00000 1.73205i 0 2.00000 + 3.46410i 3.50000 6.06218i 0 14.0000 12.1244i 24.0000 0 −7.00000 12.1244i
46.1 1.00000 + 1.73205i 0 2.00000 3.46410i 3.50000 + 6.06218i 0 14.0000 + 12.1244i 24.0000 0 −7.00000 + 12.1244i
 $$n$$: 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
7.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.e.b 2
3.b odd 2 1 7.4.c.a 2
7.b odd 2 1 441.4.e.k 2
7.c even 3 1 inner 63.4.e.b 2
7.c even 3 1 441.4.a.d 1
7.d odd 6 1 441.4.a.e 1
7.d odd 6 1 441.4.e.k 2
12.b even 2 1 112.4.i.c 2
15.d odd 2 1 175.4.e.a 2
15.e even 4 2 175.4.k.a 4
21.c even 2 1 49.4.c.a 2
21.g even 6 1 49.4.a.c 1
21.g even 6 1 49.4.c.a 2
21.h odd 6 1 7.4.c.a 2
21.h odd 6 1 49.4.a.d 1
24.f even 2 1 448.4.i.a 2
24.h odd 2 1 448.4.i.f 2
84.j odd 6 1 784.4.a.r 1
84.n even 6 1 112.4.i.c 2
84.n even 6 1 784.4.a.b 1
105.o odd 6 1 175.4.e.a 2
105.o odd 6 1 1225.4.a.c 1
105.p even 6 1 1225.4.a.d 1
105.x even 12 2 175.4.k.a 4
168.s odd 6 1 448.4.i.f 2
168.v even 6 1 448.4.i.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.4.c.a 2 3.b odd 2 1
7.4.c.a 2 21.h odd 6 1
49.4.a.c 1 21.g even 6 1
49.4.a.d 1 21.h odd 6 1
49.4.c.a 2 21.c even 2 1
49.4.c.a 2 21.g even 6 1
63.4.e.b 2 1.a even 1 1 trivial
63.4.e.b 2 7.c even 3 1 inner
112.4.i.c 2 12.b even 2 1
112.4.i.c 2 84.n even 6 1
175.4.e.a 2 15.d odd 2 1
175.4.e.a 2 105.o odd 6 1
175.4.k.a 4 15.e even 4 2
175.4.k.a 4 105.x even 12 2
441.4.a.d 1 7.c even 3 1
441.4.a.e 1 7.d odd 6 1
441.4.e.k 2 7.b odd 2 1
441.4.e.k 2 7.d odd 6 1
448.4.i.a 2 24.f even 2 1
448.4.i.a 2 168.v even 6 1
448.4.i.f 2 24.h odd 2 1
448.4.i.f 2 168.s odd 6 1
784.4.a.b 1 84.n even 6 1
784.4.a.r 1 84.j odd 6 1
1225.4.a.c 1 105.o odd 6 1
1225.4.a.d 1 105.p even 6 1

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$4 - 2 T + T^{2}$$
$3$ $$T^{2}$$
$5$ $$49 - 7 T + T^{2}$$
$7$ $$343 - 28 T + T^{2}$$
$11$ $$25 + 5 T + T^{2}$$
$13$ $$( 14 + T )^{2}$$
$17$ $$441 + 21 T + T^{2}$$
$19$ $$2401 + 49 T + T^{2}$$
$23$ $$25281 + 159 T + T^{2}$$
$29$ $$( 58 + T )^{2}$$
$31$ $$21609 + 147 T + T^{2}$$
$37$ $$47961 + 219 T + T^{2}$$
$41$ $$( 350 + T )^{2}$$
$43$ $$( 124 + T )^{2}$$
$47$ $$275625 - 525 T + T^{2}$$
$53$ $$91809 - 303 T + T^{2}$$
$59$ $$11025 + 105 T + T^{2}$$
$61$ $$170569 - 413 T + T^{2}$$
$67$ $$172225 + 415 T + T^{2}$$
$71$ $$( -432 + T )^{2}$$
$73$ $$1238769 - 1113 T + T^{2}$$
$79$ $$10609 - 103 T + T^{2}$$
$83$ $$( 1092 + T )^{2}$$
$89$ $$108241 + 329 T + T^{2}$$
$97$ $$( 882 + T )^{2}$$