# Properties

 Label 722.3.b.b Level $722$ Weight $3$ Character orbit 722.b Analytic conductor $19.673$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$722 = 2 \cdot 19^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 722.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$19.6730750868$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{-3})$$ Defining polynomial: $$x^{4} - 2 x^{2} + 4$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 38) 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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + ( -\beta_{1} + \beta_{2} ) q^{3} -2 q^{4} - q^{5} + ( 2 - \beta_{3} ) q^{6} + ( 2 - 2 \beta_{3} ) q^{7} -2 \beta_{1} q^{8} + ( 4 + 2 \beta_{3} ) q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{2} + ( -\beta_{1} + \beta_{2} ) q^{3} -2 q^{4} - q^{5} + ( 2 - \beta_{3} ) q^{6} + ( 2 - 2 \beta_{3} ) q^{7} -2 \beta_{1} q^{8} + ( 4 + 2 \beta_{3} ) q^{9} -\beta_{1} q^{10} + ( -10 + 2 \beta_{3} ) q^{11} + ( 2 \beta_{1} - 2 \beta_{2} ) q^{12} + ( -6 \beta_{1} + 5 \beta_{2} ) q^{13} + ( 2 \beta_{1} - 4 \beta_{2} ) q^{14} + ( \beta_{1} - \beta_{2} ) q^{15} + 4 q^{16} + ( -7 - 2 \beta_{3} ) q^{17} + ( 4 \beta_{1} + 4 \beta_{2} ) q^{18} + 2 q^{20} + ( -8 \beta_{1} + 6 \beta_{2} ) q^{21} + ( -10 \beta_{1} + 4 \beta_{2} ) q^{22} + ( 5 - 9 \beta_{3} ) q^{23} + ( -4 + 2 \beta_{3} ) q^{24} -24 q^{25} + ( 12 - 5 \beta_{3} ) q^{26} + ( -7 \beta_{1} + 9 \beta_{2} ) q^{27} + ( -4 + 4 \beta_{3} ) q^{28} + ( -18 \beta_{1} - 11 \beta_{2} ) q^{29} + ( -2 + \beta_{3} ) q^{30} -18 \beta_{2} q^{31} + 4 \beta_{1} q^{32} + ( 16 \beta_{1} - 14 \beta_{2} ) q^{33} + ( -7 \beta_{1} - 4 \beta_{2} ) q^{34} + ( -2 + 2 \beta_{3} ) q^{35} + ( -8 - 4 \beta_{3} ) q^{36} + ( 18 \beta_{1} - 2 \beta_{2} ) q^{37} + ( -27 + 11 \beta_{3} ) q^{39} + 2 \beta_{1} q^{40} + ( -42 \beta_{1} + 3 \beta_{2} ) q^{41} + ( 16 - 6 \beta_{3} ) q^{42} + ( -19 + 23 \beta_{3} ) q^{43} + ( 20 - 4 \beta_{3} ) q^{44} + ( -4 - 2 \beta_{3} ) q^{45} + ( 5 \beta_{1} - 18 \beta_{2} ) q^{46} + ( 35 + 19 \beta_{3} ) q^{47} + ( -4 \beta_{1} + 4 \beta_{2} ) q^{48} + ( -21 - 8 \beta_{3} ) q^{49} -24 \beta_{1} q^{50} + ( \beta_{1} - 3 \beta_{2} ) q^{51} + ( 12 \beta_{1} - 10 \beta_{2} ) q^{52} + ( -48 \beta_{1} + 7 \beta_{2} ) q^{53} + ( 14 - 9 \beta_{3} ) q^{54} + ( 10 - 2 \beta_{3} ) q^{55} + ( -4 \beta_{1} + 8 \beta_{2} ) q^{56} + ( 36 + 11 \beta_{3} ) q^{58} + ( -33 \beta_{1} + 7 \beta_{2} ) q^{59} + ( -2 \beta_{1} + 2 \beta_{2} ) q^{60} + ( -37 - 16 \beta_{3} ) q^{61} + 18 \beta_{3} q^{62} + ( -16 - 4 \beta_{3} ) q^{63} -8 q^{64} + ( 6 \beta_{1} - 5 \beta_{2} ) q^{65} + ( -32 + 14 \beta_{3} ) q^{66} + ( -63 \beta_{1} - 17 \beta_{2} ) q^{67} + ( 14 + 4 \beta_{3} ) q^{68} + ( -32 \beta_{1} + 23 \beta_{2} ) q^{69} + ( -2 \beta_{1} + 4 \beta_{2} ) q^{70} + ( -9 \beta_{1} + 51 \beta_{2} ) q^{71} + ( -8 \beta_{1} - 8 \beta_{2} ) q^{72} + ( -49 + 32 \beta_{3} ) q^{73} + ( -36 + 2 \beta_{3} ) q^{74} + ( 24 \beta_{1} - 24 \beta_{2} ) q^{75} + ( -44 + 24 \beta_{3} ) q^{77} + ( -27 \beta_{1} + 22 \beta_{2} ) q^{78} + ( -21 \beta_{1} - 21 \beta_{2} ) q^{79} -4 q^{80} + ( -5 + 34 \beta_{3} ) q^{81} + ( 84 - 3 \beta_{3} ) q^{82} + ( -16 - 6 \beta_{3} ) q^{83} + ( 16 \beta_{1} - 12 \beta_{2} ) q^{84} + ( 7 + 2 \beta_{3} ) q^{85} + ( -19 \beta_{1} + 46 \beta_{2} ) q^{86} + ( -3 + 7 \beta_{3} ) q^{87} + ( 20 \beta_{1} - 8 \beta_{2} ) q^{88} + ( 6 \beta_{1} + \beta_{2} ) q^{89} + ( -4 \beta_{1} - 4 \beta_{2} ) q^{90} + ( -42 \beta_{1} + 34 \beta_{2} ) q^{91} + ( -10 + 18 \beta_{3} ) q^{92} + ( 54 - 18 \beta_{3} ) q^{93} + ( 35 \beta_{1} + 38 \beta_{2} ) q^{94} + ( 8 - 4 \beta_{3} ) q^{96} + ( -6 \beta_{1} - 81 \beta_{2} ) q^{97} + ( -21 \beta_{1} - 16 \beta_{2} ) q^{98} + ( -16 - 12 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 8q^{4} - 4q^{5} + 8q^{6} + 8q^{7} + 16q^{9} + O(q^{10})$$ $$4q - 8q^{4} - 4q^{5} + 8q^{6} + 8q^{7} + 16q^{9} - 40q^{11} + 16q^{16} - 28q^{17} + 8q^{20} + 20q^{23} - 16q^{24} - 96q^{25} + 48q^{26} - 16q^{28} - 8q^{30} - 8q^{35} - 32q^{36} - 108q^{39} + 64q^{42} - 76q^{43} + 80q^{44} - 16q^{45} + 140q^{47} - 84q^{49} + 56q^{54} + 40q^{55} + 144q^{58} - 148q^{61} - 64q^{63} - 32q^{64} - 128q^{66} + 56q^{68} - 196q^{73} - 144q^{74} - 176q^{77} - 16q^{80} - 20q^{81} + 336q^{82} - 64q^{83} + 28q^{85} - 12q^{87} - 40q^{92} + 216q^{93} + 32q^{96} - 64q^{99} + O(q^{100})$$

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

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

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/722\mathbb{Z}\right)^\times$$.

 $$n$$ $$363$$ $$\chi(n)$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
721.1
 1.22474 − 0.707107i −1.22474 − 0.707107i −1.22474 + 0.707107i 1.22474 + 0.707107i
1.41421i 0.317837i −2.00000 −1.00000 −0.449490 −2.89898 2.82843i 8.89898 1.41421i
721.2 1.41421i 3.14626i −2.00000 −1.00000 4.44949 6.89898 2.82843i −0.898979 1.41421i
721.3 1.41421i 3.14626i −2.00000 −1.00000 4.44949 6.89898 2.82843i −0.898979 1.41421i
721.4 1.41421i 0.317837i −2.00000 −1.00000 −0.449490 −2.89898 2.82843i 8.89898 1.41421i
 $$n$$: 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
19.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 722.3.b.b 4
19.b odd 2 1 inner 722.3.b.b 4
19.c even 3 1 38.3.d.a 4
19.d odd 6 1 38.3.d.a 4
57.f even 6 1 342.3.m.a 4
57.h odd 6 1 342.3.m.a 4
76.f even 6 1 304.3.r.a 4
76.g odd 6 1 304.3.r.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.3.d.a 4 19.c even 3 1
38.3.d.a 4 19.d odd 6 1
304.3.r.a 4 76.f even 6 1
304.3.r.a 4 76.g odd 6 1
342.3.m.a 4 57.f even 6 1
342.3.m.a 4 57.h odd 6 1
722.3.b.b 4 1.a even 1 1 trivial
722.3.b.b 4 19.b odd 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{4} + 10 T_{3}^{2} + 1$$ acting on $$S_{3}^{\mathrm{new}}(722, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 2 + T^{2} )^{2}$$
$3$ $$1 + 10 T^{2} + T^{4}$$
$5$ $$( 1 + T )^{4}$$
$7$ $$( -20 - 4 T + T^{2} )^{2}$$
$11$ $$( 76 + 20 T + T^{2} )^{2}$$
$13$ $$9 + 294 T^{2} + T^{4}$$
$17$ $$( 25 + 14 T + T^{2} )^{2}$$
$19$ $$T^{4}$$
$23$ $$( -461 - 10 T + T^{2} )^{2}$$
$29$ $$81225 + 2022 T^{2} + T^{4}$$
$31$ $$( 972 + T^{2} )^{2}$$
$37$ $$404496 + 1320 T^{2} + T^{4}$$
$41$ $$12257001 + 7110 T^{2} + T^{4}$$
$43$ $$( -2813 + 38 T + T^{2} )^{2}$$
$47$ $$( -941 - 70 T + T^{2} )^{2}$$
$53$ $$19900521 + 9510 T^{2} + T^{4}$$
$59$ $$4124961 + 4650 T^{2} + T^{4}$$
$61$ $$( -167 + 74 T + T^{2} )^{2}$$
$67$ $$49999041 + 17610 T^{2} + T^{4}$$
$71$ $$58384881 + 15930 T^{2} + T^{4}$$
$73$ $$( -3743 + 98 T + T^{2} )^{2}$$
$79$ $$194481 + 4410 T^{2} + T^{4}$$
$83$ $$( 40 + 32 T + T^{2} )^{2}$$
$89$ $$4761 + 150 T^{2} + T^{4}$$
$97$ $$384591321 + 39510 T^{2} + T^{4}$$