# Properties

 Label 483.2.h.a Level $483$ Weight $2$ Character orbit 483.h Analytic conductor $3.857$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$483 = 3 \cdot 7 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 483.h (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.85677441763$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.342102016.5 Defining polynomial: $$x^{8} + x^{6} + 4 x^{4} + 4 x^{2} + 16$$ Coefficient ring: $$\Z[a_1, \ldots, a_{23}]$$ Coefficient ring index: $$2^{3}$$ 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 $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{2} -\beta_{1} q^{3} - q^{4} + ( \beta_{4} - \beta_{5} ) q^{5} + \beta_{1} q^{6} + ( \beta_{2} - \beta_{5} - \beta_{7} ) q^{7} + 3 q^{8} - q^{9} +O(q^{10})$$ $$q - q^{2} -\beta_{1} q^{3} - q^{4} + ( \beta_{4} - \beta_{5} ) q^{5} + \beta_{1} q^{6} + ( \beta_{2} - \beta_{5} - \beta_{7} ) q^{7} + 3 q^{8} - q^{9} + ( -\beta_{4} + \beta_{5} ) q^{10} + ( \beta_{2} + \beta_{4} + \beta_{5} + \beta_{7} ) q^{11} + \beta_{1} q^{12} + ( -\beta_{2} - 2 \beta_{3} + \beta_{4} + \beta_{5} + \beta_{7} ) q^{13} + ( -\beta_{2} + \beta_{5} + \beta_{7} ) q^{14} + ( -\beta_{2} - \beta_{7} ) q^{15} - q^{16} + ( -\beta_{4} + \beta_{5} ) q^{17} + q^{18} + ( -2 \beta_{2} + \beta_{4} - \beta_{5} + 2 \beta_{7} ) q^{19} + ( -\beta_{4} + \beta_{5} ) q^{20} + ( -\beta_{2} - \beta_{4} - \beta_{5} ) q^{21} + ( -\beta_{2} - \beta_{4} - \beta_{5} - \beta_{7} ) q^{22} + ( -1 - 2 \beta_{2} - \beta_{6} - 2 \beta_{7} ) q^{23} -3 \beta_{1} q^{24} + ( 1 - \beta_{2} - \beta_{4} - \beta_{5} - 2 \beta_{6} - \beta_{7} ) q^{25} + ( \beta_{2} + 2 \beta_{3} - \beta_{4} - \beta_{5} - \beta_{7} ) q^{26} + \beta_{1} q^{27} + ( -\beta_{2} + \beta_{5} + \beta_{7} ) q^{28} + ( -2 - 2 \beta_{2} - 2 \beta_{4} - 2 \beta_{5} - 4 \beta_{6} - 2 \beta_{7} ) q^{29} + ( \beta_{2} + \beta_{7} ) q^{30} + ( 4 \beta_{1} - 2 \beta_{2} - 4 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{7} ) q^{31} -5 q^{32} + ( \beta_{2} + \beta_{4} - \beta_{5} - \beta_{7} ) q^{33} + ( \beta_{4} - \beta_{5} ) q^{34} + ( 1 - \beta_{2} + \beta_{3} - 2 \beta_{4} - 2 \beta_{5} - 3 \beta_{6} - 2 \beta_{7} ) q^{35} + q^{36} + ( 2 \beta_{4} + 2 \beta_{5} ) q^{37} + ( 2 \beta_{2} - \beta_{4} + \beta_{5} - 2 \beta_{7} ) q^{38} + ( -2 - \beta_{2} - \beta_{4} - \beta_{5} - 2 \beta_{6} - \beta_{7} ) q^{39} + ( 3 \beta_{4} - 3 \beta_{5} ) q^{40} + ( -8 \beta_{1} + \beta_{2} + 2 \beta_{3} - \beta_{4} - \beta_{5} - \beta_{7} ) q^{41} + ( \beta_{2} + \beta_{4} + \beta_{5} ) q^{42} + ( \beta_{4} + \beta_{5} ) q^{43} + ( -\beta_{2} - \beta_{4} - \beta_{5} - \beta_{7} ) q^{44} + ( -\beta_{4} + \beta_{5} ) q^{45} + ( 1 + 2 \beta_{2} + \beta_{6} + 2 \beta_{7} ) q^{46} + ( -4 \beta_{1} + \beta_{2} + 2 \beta_{3} - \beta_{4} - \beta_{5} - \beta_{7} ) q^{47} + \beta_{1} q^{48} + ( 4 - 4 \beta_{1} - \beta_{2} - \beta_{3} - \beta_{6} ) q^{49} + ( -1 + \beta_{2} + \beta_{4} + \beta_{5} + 2 \beta_{6} + \beta_{7} ) q^{50} + ( \beta_{2} + \beta_{7} ) q^{51} + ( \beta_{2} + 2 \beta_{3} - \beta_{4} - \beta_{5} - \beta_{7} ) q^{52} + ( -3 \beta_{2} - 3 \beta_{7} ) q^{53} -\beta_{1} q^{54} + ( 8 \beta_{1} - 2 \beta_{2} - 4 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{7} ) q^{55} + ( 3 \beta_{2} - 3 \beta_{5} - 3 \beta_{7} ) q^{56} + ( -\beta_{2} + 2 \beta_{4} + 2 \beta_{5} - \beta_{7} ) q^{57} + ( 2 + 2 \beta_{2} + 2 \beta_{4} + 2 \beta_{5} + 4 \beta_{6} + 2 \beta_{7} ) q^{58} + ( 4 \beta_{1} - \beta_{2} - 2 \beta_{3} + \beta_{4} + \beta_{5} + \beta_{7} ) q^{59} + ( \beta_{2} + \beta_{7} ) q^{60} + ( -2 \beta_{2} + 2 \beta_{7} ) q^{61} + ( -4 \beta_{1} + 2 \beta_{2} + 4 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{7} ) q^{62} + ( -\beta_{2} + \beta_{5} + \beta_{7} ) q^{63} + 7 q^{64} + ( 2 \beta_{2} + 4 \beta_{4} + 4 \beta_{5} + 2 \beta_{7} ) q^{65} + ( -\beta_{2} - \beta_{4} + \beta_{5} + \beta_{7} ) q^{66} + ( -5 \beta_{4} - 5 \beta_{5} ) q^{67} + ( \beta_{4} - \beta_{5} ) q^{68} + ( \beta_{2} + \beta_{3} - 2 \beta_{4} + \beta_{5} - \beta_{7} ) q^{69} + ( -1 + \beta_{2} - \beta_{3} + 2 \beta_{4} + 2 \beta_{5} + 3 \beta_{6} + 2 \beta_{7} ) q^{70} + ( -6 + \beta_{2} + \beta_{4} + \beta_{5} + 2 \beta_{6} + \beta_{7} ) q^{71} -3 q^{72} + ( -2 \beta_{2} - 4 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{7} ) q^{73} + ( -2 \beta_{4} - 2 \beta_{5} ) q^{74} + ( -3 \beta_{1} + \beta_{2} + 2 \beta_{3} - \beta_{4} - \beta_{5} - \beta_{7} ) q^{75} + ( 2 \beta_{2} - \beta_{4} + \beta_{5} - 2 \beta_{7} ) q^{76} + ( 2 + 8 \beta_{1} - \beta_{2} - 2 \beta_{3} + \beta_{4} + \beta_{5} + \beta_{7} ) q^{77} + ( 2 + \beta_{2} + \beta_{4} + \beta_{5} + 2 \beta_{6} + \beta_{7} ) q^{78} + ( -4 \beta_{2} - 3 \beta_{4} - 3 \beta_{5} - 4 \beta_{7} ) q^{79} + ( -\beta_{4} + \beta_{5} ) q^{80} + q^{81} + ( 8 \beta_{1} - \beta_{2} - 2 \beta_{3} + \beta_{4} + \beta_{5} + \beta_{7} ) q^{82} + ( -4 \beta_{2} - 4 \beta_{4} + 4 \beta_{5} + 4 \beta_{7} ) q^{83} + ( \beta_{2} + \beta_{4} + \beta_{5} ) q^{84} + ( -6 + \beta_{2} + \beta_{4} + \beta_{5} + 2 \beta_{6} + \beta_{7} ) q^{85} + ( -\beta_{4} - \beta_{5} ) q^{86} + ( -2 \beta_{1} + 2 \beta_{2} + 4 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{7} ) q^{87} + ( 3 \beta_{2} + 3 \beta_{4} + 3 \beta_{5} + 3 \beta_{7} ) q^{88} + ( 3 \beta_{4} - 3 \beta_{5} ) q^{89} + ( \beta_{4} - \beta_{5} ) q^{90} + ( \beta_{2} - \beta_{4} - 3 \beta_{5} + 5 \beta_{7} ) q^{91} + ( 1 + 2 \beta_{2} + \beta_{6} + 2 \beta_{7} ) q^{92} + ( -2 \beta_{2} - 2 \beta_{4} - 2 \beta_{5} - 4 \beta_{6} - 2 \beta_{7} ) q^{93} + ( 4 \beta_{1} - \beta_{2} - 2 \beta_{3} + \beta_{4} + \beta_{5} + \beta_{7} ) q^{94} + ( 10 + \beta_{2} + \beta_{4} + \beta_{5} + 2 \beta_{6} + \beta_{7} ) q^{95} + 5 \beta_{1} q^{96} + ( 2 \beta_{2} - 2 \beta_{7} ) q^{97} + ( -4 + 4 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{6} ) q^{98} + ( -\beta_{2} - \beta_{4} - \beta_{5} - \beta_{7} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 8q^{2} - 8q^{4} + 24q^{8} - 8q^{9} + O(q^{10})$$ $$8q - 8q^{2} - 8q^{4} + 24q^{8} - 8q^{9} - 8q^{16} + 8q^{18} - 4q^{23} + 16q^{25} - 40q^{32} + 20q^{35} + 8q^{36} - 8q^{39} + 4q^{46} + 36q^{49} - 16q^{50} + 56q^{64} - 20q^{70} - 56q^{71} - 24q^{72} + 16q^{77} + 8q^{78} + 8q^{81} - 56q^{85} + 4q^{92} + 16q^{93} + 72q^{95} - 36q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + x^{6} + 4 x^{4} + 4 x^{2} + 16$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-\nu^{7} + \nu^{5} + 2 \nu^{3}$$$$)/16$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{7} - \nu^{6} + 3 \nu^{5} + 5 \nu^{4} + 2 \nu^{3} - 2 \nu^{2} + 8$$$$)/16$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{6} - 3 \nu^{4} + 8 \nu^{3} - 10 \nu^{2} + 24 \nu - 8$$$$)/16$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{7} - \nu^{6} + \nu^{5} - 3 \nu^{4} - 6 \nu^{3} - 10 \nu^{2} + 8 \nu - 8$$$$)/16$$ $$\beta_{5}$$ $$=$$ $$($$$$\nu^{7} - \nu^{6} - \nu^{5} - 3 \nu^{4} + 6 \nu^{3} - 10 \nu^{2} - 8 \nu - 8$$$$)/16$$ $$\beta_{6}$$ $$=$$ $$($$$$-\nu^{6} - 3 \nu^{4} + 6 \nu^{2} - 8$$$$)/8$$ $$\beta_{7}$$ $$=$$ $$($$$$-\nu^{7} - \nu^{6} - 3 \nu^{5} + 5 \nu^{4} - 2 \nu^{3} - 2 \nu^{2} + 8$$$$)/16$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{5} + \beta_{3} - \beta_{1}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{6} - \beta_{5} - \beta_{4}$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$\beta_{5} - 2 \beta_{4} + \beta_{3} + 3 \beta_{1}$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$2 \beta_{7} - \beta_{6} - \beta_{5} - \beta_{4} + 2 \beta_{2} - 4$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$-4 \beta_{7} - \beta_{5} + 2 \beta_{4} - \beta_{3} + 4 \beta_{2} + 5 \beta_{1}$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$($$$$-6 \beta_{7} - 7 \beta_{6} - 3 \beta_{5} - 3 \beta_{4} - 6 \beta_{2} - 4$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$-4 \beta_{7} + \beta_{5} - 2 \beta_{4} + \beta_{3} + 4 \beta_{2} - 21 \beta_{1}$$$$)/2$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/483\mathbb{Z}\right)^\times$$.

 $$n$$ $$323$$ $$346$$ $$442$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
160.1
 −0.599676 − 1.28078i −1.17915 + 0.780776i 1.17915 + 0.780776i 0.599676 − 1.28078i −0.599676 + 1.28078i −1.17915 − 0.780776i 1.17915 − 0.780776i 0.599676 + 1.28078i
−1.00000 1.00000i −1.00000 −3.33513 1.00000i −2.60399 + 0.468213i 3.00000 −1.00000 3.33513
160.2 −1.00000 1.00000i −1.00000 −1.69614 1.00000i 2.17238 1.51022i 3.00000 −1.00000 1.69614
160.3 −1.00000 1.00000i −1.00000 1.69614 1.00000i −2.17238 + 1.51022i 3.00000 −1.00000 −1.69614
160.4 −1.00000 1.00000i −1.00000 3.33513 1.00000i 2.60399 0.468213i 3.00000 −1.00000 −3.33513
160.5 −1.00000 1.00000i −1.00000 −3.33513 1.00000i −2.60399 0.468213i 3.00000 −1.00000 3.33513
160.6 −1.00000 1.00000i −1.00000 −1.69614 1.00000i 2.17238 + 1.51022i 3.00000 −1.00000 1.69614
160.7 −1.00000 1.00000i −1.00000 1.69614 1.00000i −2.17238 1.51022i 3.00000 −1.00000 −1.69614
160.8 −1.00000 1.00000i −1.00000 3.33513 1.00000i 2.60399 + 0.468213i 3.00000 −1.00000 −3.33513
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 160.8 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 inner
23.b odd 2 1 inner
161.c even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 483.2.h.a 8
3.b odd 2 1 1449.2.h.d 8
7.b odd 2 1 inner 483.2.h.a 8
21.c even 2 1 1449.2.h.d 8
23.b odd 2 1 inner 483.2.h.a 8
69.c even 2 1 1449.2.h.d 8
161.c even 2 1 inner 483.2.h.a 8
483.c odd 2 1 1449.2.h.d 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
483.2.h.a 8 1.a even 1 1 trivial
483.2.h.a 8 7.b odd 2 1 inner
483.2.h.a 8 23.b odd 2 1 inner
483.2.h.a 8 161.c even 2 1 inner
1449.2.h.d 8 3.b odd 2 1
1449.2.h.d 8 21.c even 2 1
1449.2.h.d 8 69.c even 2 1
1449.2.h.d 8 483.c odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(483, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{8}$$
$3$ $$( 1 + T^{2} )^{4}$$
$5$ $$( 32 - 14 T^{2} + T^{4} )^{2}$$
$7$ $$2401 - 882 T^{2} + 162 T^{4} - 18 T^{6} + T^{8}$$
$11$ $$( 32 + 20 T^{2} + T^{4} )^{2}$$
$13$ $$( 256 + 36 T^{2} + T^{4} )^{2}$$
$17$ $$( 32 - 14 T^{2} + T^{4} )^{2}$$
$19$ $$( 128 - 62 T^{2} + T^{4} )^{2}$$
$23$ $$( 529 + 46 T + 30 T^{2} + 2 T^{3} + T^{4} )^{2}$$
$29$ $$( -68 + T^{2} )^{4}$$
$31$ $$( 4096 + 144 T^{2} + T^{4} )^{2}$$
$37$ $$( 128 + 40 T^{2} + T^{4} )^{2}$$
$41$ $$( 1024 + 132 T^{2} + T^{4} )^{2}$$
$43$ $$( 8 + 10 T^{2} + T^{4} )^{2}$$
$47$ $$( 64 + 52 T^{2} + T^{4} )^{2}$$
$53$ $$( 2592 + 126 T^{2} + T^{4} )^{2}$$
$59$ $$( 64 + 52 T^{2} + T^{4} )^{2}$$
$61$ $$( 128 - 40 T^{2} + T^{4} )^{2}$$
$67$ $$( 5000 + 250 T^{2} + T^{4} )^{2}$$
$71$ $$( 32 + 14 T + T^{2} )^{4}$$
$73$ $$( 4096 + 144 T^{2} + T^{4} )^{2}$$
$79$ $$( 1352 + 266 T^{2} + T^{4} )^{2}$$
$83$ $$( 8192 - 320 T^{2} + T^{4} )^{2}$$
$89$ $$( 2592 - 126 T^{2} + T^{4} )^{2}$$
$97$ $$( 128 - 40 T^{2} + T^{4} )^{2}$$