# Properties

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

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$63 = 3^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 63.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.71712033036$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{111})$$ Defining polynomial: $$x^{4} + 112 x^{2} + 3025$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 2 \beta_{1} q^{2} + \beta_{3} q^{5} + ( -11 - \beta_{2} ) q^{7} + 16 \beta_{1} q^{8} +O(q^{10})$$ $$q + 2 \beta_{1} q^{2} + \beta_{3} q^{5} + ( -11 - \beta_{2} ) q^{7} + 16 \beta_{1} q^{8} + 4 \beta_{2} q^{10} + 11 \beta_{1} q^{11} + 2 \beta_{2} q^{13} + ( -22 \beta_{1} + 2 \beta_{3} ) q^{14} -64 q^{16} -3 \beta_{3} q^{17} -6 \beta_{2} q^{19} -44 q^{22} -55 \beta_{1} q^{23} + 319 q^{25} -4 \beta_{3} q^{26} + 89 \beta_{1} q^{29} -16 \beta_{2} q^{31} -12 \beta_{2} q^{34} + ( -222 \beta_{1} - 11 \beta_{3} ) q^{35} -184 q^{37} + 12 \beta_{3} q^{38} + 32 \beta_{2} q^{40} -5 \beta_{3} q^{41} -190 q^{43} + 220 q^{46} + 2 \beta_{3} q^{47} + ( -101 + 22 \beta_{2} ) q^{49} + 638 \beta_{1} q^{50} -253 \beta_{1} q^{53} + 22 \beta_{2} q^{55} + ( -176 \beta_{1} + 16 \beta_{3} ) q^{56} -356 q^{58} + 4 \beta_{3} q^{59} -44 \beta_{2} q^{61} + 32 \beta_{3} q^{62} -512 q^{64} + 444 \beta_{1} q^{65} + 296 q^{67} + ( 888 - 44 \beta_{2} ) q^{70} + 233 \beta_{1} q^{71} + 54 \beta_{2} q^{73} -368 \beta_{1} q^{74} + ( -121 \beta_{1} + 11 \beta_{3} ) q^{77} + 836 q^{79} -64 \beta_{3} q^{80} -20 \beta_{2} q^{82} -58 \beta_{3} q^{83} -1332 q^{85} -380 \beta_{1} q^{86} -352 q^{88} + 33 \beta_{3} q^{89} + ( 444 - 22 \beta_{2} ) q^{91} + 8 \beta_{2} q^{94} -1332 \beta_{1} q^{95} -38 \beta_{2} q^{97} + ( -202 \beta_{1} - 44 \beta_{3} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 44q^{7} + O(q^{10})$$ $$4q - 44q^{7} - 256q^{16} - 176q^{22} + 1276q^{25} - 736q^{37} - 760q^{43} + 880q^{46} - 404q^{49} - 1424q^{58} - 2048q^{64} + 1184q^{67} + 3552q^{70} + 3344q^{79} - 5328q^{85} - 1408q^{88} + 1776q^{91} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 112 x^{2} + 3025$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{3} + 57 \nu$$$$)/55$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} + 167 \nu$$$$)/55$$ $$\beta_{3}$$ $$=$$ $$2 \nu^{2} + 112$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} - \beta_{1}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} - 112$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$-57 \beta_{2} + 167 \beta_{1}$$$$)/2$$

## Character values

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

 $$n$$ $$10$$ $$29$$ $$\chi(n)$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
62.1
 8.15694i − 6.74273i − 8.15694i 6.74273i
2.82843i 0 0 −21.0713 0 −11.0000 14.8997i 22.6274i 0 59.5987i
62.2 2.82843i 0 0 21.0713 0 −11.0000 + 14.8997i 22.6274i 0 59.5987i
62.3 2.82843i 0 0 −21.0713 0 −11.0000 + 14.8997i 22.6274i 0 59.5987i
62.4 2.82843i 0 0 21.0713 0 −11.0000 14.8997i 22.6274i 0 59.5987i
 $$n$$: 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
3.b odd 2 1 inner
7.b odd 2 1 inner
21.c even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 63.4.c.b 4
3.b odd 2 1 inner 63.4.c.b 4
4.b odd 2 1 1008.4.k.b 4
7.b odd 2 1 inner 63.4.c.b 4
7.c even 3 2 441.4.p.a 8
7.d odd 6 2 441.4.p.a 8
12.b even 2 1 1008.4.k.b 4
21.c even 2 1 inner 63.4.c.b 4
21.g even 6 2 441.4.p.a 8
21.h odd 6 2 441.4.p.a 8
28.d even 2 1 1008.4.k.b 4
84.h odd 2 1 1008.4.k.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.4.c.b 4 1.a even 1 1 trivial
63.4.c.b 4 3.b odd 2 1 inner
63.4.c.b 4 7.b odd 2 1 inner
63.4.c.b 4 21.c even 2 1 inner
441.4.p.a 8 7.c even 3 2
441.4.p.a 8 7.d odd 6 2
441.4.p.a 8 21.g even 6 2
441.4.p.a 8 21.h odd 6 2
1008.4.k.b 4 4.b odd 2 1
1008.4.k.b 4 12.b even 2 1
1008.4.k.b 4 28.d even 2 1
1008.4.k.b 4 84.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + 8$$ acting on $$S_{4}^{\mathrm{new}}(63, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 8 + T^{2} )^{2}$$
$3$ $$T^{4}$$
$5$ $$( -444 + T^{2} )^{2}$$
$7$ $$( 343 + 22 T + T^{2} )^{2}$$
$11$ $$( 242 + T^{2} )^{2}$$
$13$ $$( 888 + T^{2} )^{2}$$
$17$ $$( -3996 + T^{2} )^{2}$$
$19$ $$( 7992 + T^{2} )^{2}$$
$23$ $$( 6050 + T^{2} )^{2}$$
$29$ $$( 15842 + T^{2} )^{2}$$
$31$ $$( 56832 + T^{2} )^{2}$$
$37$ $$( 184 + T )^{4}$$
$41$ $$( -11100 + T^{2} )^{2}$$
$43$ $$( 190 + T )^{4}$$
$47$ $$( -1776 + T^{2} )^{2}$$
$53$ $$( 128018 + T^{2} )^{2}$$
$59$ $$( -7104 + T^{2} )^{2}$$
$61$ $$( 429792 + T^{2} )^{2}$$
$67$ $$( -296 + T )^{4}$$
$71$ $$( 108578 + T^{2} )^{2}$$
$73$ $$( 647352 + T^{2} )^{2}$$
$79$ $$( -836 + T )^{4}$$
$83$ $$( -1493616 + T^{2} )^{2}$$
$89$ $$( -483516 + T^{2} )^{2}$$
$97$ $$( 320568 + T^{2} )^{2}$$