# Properties

 Label 63.4.c.a Level $63$ Weight $4$ Character orbit 63.c Analytic conductor $3.717$ Analytic rank $0$ Dimension $4$ CM discriminant -7 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: $$\Q(\sqrt{-2}, \sqrt{7})$$ Defining polynomial: $$x^{4} + 8 x^{2} + 9$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$3^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{2} + ( -8 - 5 \beta_{3} ) q^{4} -7 \beta_{3} q^{7} + ( -5 \beta_{1} - 10 \beta_{2} ) q^{8} +O(q^{10})$$ $$q + \beta_{2} q^{2} + ( -8 - 5 \beta_{3} ) q^{4} -7 \beta_{3} q^{7} + ( -5 \beta_{1} - 10 \beta_{2} ) q^{8} + ( -\beta_{1} + 13 \beta_{2} ) q^{11} + ( -7 \beta_{1} - 14 \beta_{2} ) q^{14} + ( 111 + 40 \beta_{3} ) q^{16} + ( -205 - 59 \beta_{3} ) q^{22} + ( 25 \beta_{1} + 7 \beta_{2} ) q^{23} -125 q^{25} + ( 245 + 56 \beta_{3} ) q^{28} + ( 37 \beta_{1} - 11 \beta_{2} ) q^{29} + 111 \beta_{2} q^{32} + 4 \beta_{3} q^{37} + 202 \beta_{3} q^{43} + ( -67 \beta_{1} - 219 \beta_{2} ) q^{44} + ( -187 - 185 \beta_{3} ) q^{46} + 343 q^{49} -125 \beta_{2} q^{50} + ( -85 \beta_{1} + 67 \beta_{2} ) q^{53} + 245 \beta_{2} q^{56} + ( 65 - 167 \beta_{3} ) q^{58} + ( -888 - 235 \beta_{3} ) q^{64} + 740 q^{67} + ( 141 \beta_{1} - 53 \beta_{2} ) q^{71} + ( 4 \beta_{1} + 8 \beta_{2} ) q^{74} + ( -105 \beta_{1} - 161 \beta_{2} ) q^{77} -1384 q^{79} + ( 202 \beta_{1} + 404 \beta_{2} ) q^{86} + ( 2065 + 1025 \beta_{3} ) q^{88} + ( 15 \beta_{1} - 501 \beta_{2} ) q^{92} + 343 \beta_{2} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 32q^{4} + O(q^{10})$$ $$4q - 32q^{4} + 444q^{16} - 820q^{22} - 500q^{25} + 980q^{28} - 748q^{46} + 1372q^{49} + 260q^{58} - 3552q^{64} + 2960q^{67} - 5536q^{79} + 8260q^{88} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 8 x^{2} + 9$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$3 \nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{3} + 6 \nu$$ $$\beta_{3}$$ $$=$$ $$\nu^{2} + 4$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$$$/3$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} - 4$$ $$\nu^{3}$$ $$=$$ $$\beta_{2} - 2 \beta_{1}$$

## 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
 − 1.16372i 2.57794i − 2.57794i 1.16372i
5.40636i 0 −21.2288 0 0 −18.5203 71.5195i 0 0
62.2 1.66471i 0 5.22876 0 0 18.5203 22.0220i 0 0
62.3 1.66471i 0 5.22876 0 0 18.5203 22.0220i 0 0
62.4 5.40636i 0 −21.2288 0 0 −18.5203 71.5195i 0 0
 $$n$$: 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.b odd 2 1 CM by $$\Q(\sqrt{-7})$$
3.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.a 4
3.b odd 2 1 inner 63.4.c.a 4
4.b odd 2 1 1008.4.k.a 4
7.b odd 2 1 CM 63.4.c.a 4
7.c even 3 2 441.4.p.b 8
7.d odd 6 2 441.4.p.b 8
12.b even 2 1 1008.4.k.a 4
21.c even 2 1 inner 63.4.c.a 4
21.g even 6 2 441.4.p.b 8
21.h odd 6 2 441.4.p.b 8
28.d even 2 1 1008.4.k.a 4
84.h odd 2 1 1008.4.k.a 4

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} + 32 T_{2}^{2} + 81$$ acting on $$S_{4}^{\mathrm{new}}(63, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$81 + 32 T^{2} + T^{4}$$
$3$ $$T^{4}$$
$5$ $$T^{4}$$
$7$ $$( -343 + T^{2} )^{2}$$
$11$ $$3849444 + 5324 T^{2} + T^{4}$$
$13$ $$T^{4}$$
$17$ $$T^{4}$$
$19$ $$T^{4}$$
$23$ $$516834756 + 48668 T^{2} + T^{4}$$
$29$ $$450373284 + 97556 T^{2} + T^{4}$$
$31$ $$T^{4}$$
$37$ $$( -112 + T^{2} )^{2}$$
$41$ $$T^{4}$$
$43$ $$( -285628 + T^{2} )^{2}$$
$47$ $$T^{4}$$
$53$ $$2534719716 + 595508 T^{2} + T^{4}$$
$59$ $$T^{4}$$
$61$ $$T^{4}$$
$67$ $$( -740 + T )^{4}$$
$71$ $$58795580484 + 1431644 T^{2} + T^{4}$$
$73$ $$T^{4}$$
$79$ $$( 1384 + T )^{4}$$
$83$ $$T^{4}$$
$89$ $$T^{4}$$
$97$ $$T^{4}$$