# Properties

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

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$3.85677441763$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{17})$$ Defining polynomial: $$x^{4} - x^{3} + 5x^{2} + 4x + 16$$ x^4 - x^3 + 5*x^2 + 4*x + 16 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + (\beta_{2} - 1) q^{3} + (\beta_{3} + 2 \beta_{2} + \beta_1 - 2) q^{4} + \beta_1 q^{5} + \beta_{3} q^{6} + (\beta_{2} + 2) q^{7} + (\beta_{3} - 4) q^{8} - \beta_{2} q^{9}+O(q^{10})$$ q + b1 * q^2 + (b2 - 1) * q^3 + (b3 + 2*b2 + b1 - 2) * q^4 + b1 * q^5 + b3 * q^6 + (b2 + 2) * q^7 + (b3 - 4) * q^8 - b2 * q^9 $$q + \beta_1 q^{2} + (\beta_{2} - 1) q^{3} + (\beta_{3} + 2 \beta_{2} + \beta_1 - 2) q^{4} + \beta_1 q^{5} + \beta_{3} q^{6} + (\beta_{2} + 2) q^{7} + (\beta_{3} - 4) q^{8} - \beta_{2} q^{9} + (\beta_{3} + 4 \beta_{2} + \beta_1 - 4) q^{10} + ( - 2 \beta_{2} + 2) q^{11} + ( - 2 \beta_{2} - \beta_1) q^{12} + ( - 2 \beta_{3} - 3) q^{13} + (\beta_{3} + 3 \beta_1) q^{14} + \beta_{3} q^{15} - 3 \beta_1 q^{16} + (\beta_{3} - 6 \beta_{2} + \beta_1 + 6) q^{17} + ( - \beta_{3} - \beta_1) q^{18} + ( - 3 \beta_{2} - \beta_1) q^{19} + (3 \beta_{3} - 4) q^{20} + (2 \beta_{2} - 3) q^{21} - 2 \beta_{3} q^{22} - \beta_{2} q^{23} + ( - \beta_{3} - 4 \beta_{2} - \beta_1 + 4) q^{24} + (\beta_{3} - \beta_{2} + \beta_1 + 1) q^{25} + (8 \beta_{2} - \beta_1) q^{26} + q^{27} + (3 \beta_{3} + 4 \beta_{2} + 2 \beta_1 - 6) q^{28} + ( - 2 \beta_{3} - 6) q^{29} + ( - 4 \beta_{2} - \beta_1) q^{30} + ( - 3 \beta_{3} + \beta_{2} - 3 \beta_1 - 1) q^{31} + ( - \beta_{3} - 4 \beta_{2} - \beta_1 + 4) q^{32} + 2 \beta_{2} q^{33} + ( - 5 \beta_{3} - 4) q^{34} + (\beta_{3} + 3 \beta_1) q^{35} + ( - \beta_{3} + 2) q^{36} + ( - 5 \beta_{2} - \beta_1) q^{37} + ( - 4 \beta_{3} - 4 \beta_{2} - 4 \beta_1 + 4) q^{38} + (2 \beta_{3} - 3 \beta_{2} + 2 \beta_1 + 3) q^{39} + ( - 4 \beta_{2} - 5 \beta_1) q^{40} + ( - 4 \beta_{3} - 4) q^{41} + (2 \beta_{3} - \beta_1) q^{42} + (3 \beta_{3} - 5) q^{43} + (4 \beta_{2} + 2 \beta_1) q^{44} + ( - \beta_{3} - \beta_1) q^{45} + ( - \beta_{3} - \beta_1) q^{46} + ( - 2 \beta_{2} + 3 \beta_1) q^{47} - 3 \beta_{3} q^{48} + (5 \beta_{2} + 3) q^{49} - 4 q^{50} + (6 \beta_{2} - \beta_1) q^{51} + (3 \beta_{3} + 2 \beta_{2} + 3 \beta_1 - 2) q^{52} + (5 \beta_{3} + 5 \beta_1) q^{53} + \beta_1 q^{54} - 2 \beta_{3} q^{55} + (2 \beta_{3} - 4 \beta_{2} - \beta_1 - 8) q^{56} + ( - \beta_{3} + 3) q^{57} + (8 \beta_{2} - 4 \beta_1) q^{58} + ( - 2 \beta_{3} + 6 \beta_{2} - 2 \beta_1 - 6) q^{59} + ( - 3 \beta_{3} - 4 \beta_{2} - 3 \beta_1 + 4) q^{60} - 6 \beta_{2} q^{61} + ( - 2 \beta_{3} + 12) q^{62} + ( - 3 \beta_{2} + 1) q^{63} + (\beta_{3} + 4) q^{64} + (8 \beta_{2} - \beta_1) q^{65} + (2 \beta_{3} + 2 \beta_1) q^{66} + (2 \beta_{3} - 11 \beta_{2} + 2 \beta_1 + 11) q^{67} + (8 \beta_{2} + 3 \beta_1) q^{68} + q^{69} + (3 \beta_{3} + 8 \beta_{2} + 2 \beta_1 - 12) q^{70} + (5 \beta_{3} + 6) q^{71} + (4 \beta_{2} + \beta_1) q^{72} + (2 \beta_{3} + 9 \beta_{2} + 2 \beta_1 - 9) q^{73} + ( - 6 \beta_{3} - 4 \beta_{2} - 6 \beta_1 + 4) q^{74} + (\beta_{2} - \beta_1) q^{75} + ( - 6 \beta_{3} + 10) q^{76} + ( - 4 \beta_{2} + 6) q^{77} + ( - \beta_{3} - 8) q^{78} + (7 \beta_{2} + \beta_1) q^{79} + ( - 3 \beta_{3} - 12 \beta_{2} - 3 \beta_1 + 12) q^{80} + (\beta_{2} - 1) q^{81} + 16 \beta_{2} q^{82} + ( - 2 \beta_{3} + 8) q^{83} + ( - \beta_{3} - 6 \beta_{2} - 3 \beta_1 + 2) q^{84} + ( - 5 \beta_{3} - 4) q^{85} + ( - 12 \beta_{2} - 8 \beta_1) q^{86} + (2 \beta_{3} - 6 \beta_{2} + 2 \beta_1 + 6) q^{87} + (2 \beta_{3} + 8 \beta_{2} + 2 \beta_1 - 8) q^{88} + 10 \beta_{2} q^{89} + ( - \beta_{3} + 4) q^{90} + ( - 4 \beta_{3} - 3 \beta_{2} + 2 \beta_1 - 6) q^{91} + ( - \beta_{3} + 2) q^{92} + ( - \beta_{2} + 3 \beta_1) q^{93} + (\beta_{3} + 12 \beta_{2} + \beta_1 - 12) q^{94} + ( - 4 \beta_{3} - 4 \beta_{2} - 4 \beta_1 + 4) q^{95} + (4 \beta_{2} + \beta_1) q^{96} + (6 \beta_{3} + 6) q^{97} + (5 \beta_{3} + 8 \beta_1) q^{98} - 2 q^{99}+O(q^{100})$$ q + b1 * q^2 + (b2 - 1) * q^3 + (b3 + 2*b2 + b1 - 2) * q^4 + b1 * q^5 + b3 * q^6 + (b2 + 2) * q^7 + (b3 - 4) * q^8 - b2 * q^9 + (b3 + 4*b2 + b1 - 4) * q^10 + (-2*b2 + 2) * q^11 + (-2*b2 - b1) * q^12 + (-2*b3 - 3) * q^13 + (b3 + 3*b1) * q^14 + b3 * q^15 - 3*b1 * q^16 + (b3 - 6*b2 + b1 + 6) * q^17 + (-b3 - b1) * q^18 + (-3*b2 - b1) * q^19 + (3*b3 - 4) * q^20 + (2*b2 - 3) * q^21 - 2*b3 * q^22 - b2 * q^23 + (-b3 - 4*b2 - b1 + 4) * q^24 + (b3 - b2 + b1 + 1) * q^25 + (8*b2 - b1) * q^26 + q^27 + (3*b3 + 4*b2 + 2*b1 - 6) * q^28 + (-2*b3 - 6) * q^29 + (-4*b2 - b1) * q^30 + (-3*b3 + b2 - 3*b1 - 1) * q^31 + (-b3 - 4*b2 - b1 + 4) * q^32 + 2*b2 * q^33 + (-5*b3 - 4) * q^34 + (b3 + 3*b1) * q^35 + (-b3 + 2) * q^36 + (-5*b2 - b1) * q^37 + (-4*b3 - 4*b2 - 4*b1 + 4) * q^38 + (2*b3 - 3*b2 + 2*b1 + 3) * q^39 + (-4*b2 - 5*b1) * q^40 + (-4*b3 - 4) * q^41 + (2*b3 - b1) * q^42 + (3*b3 - 5) * q^43 + (4*b2 + 2*b1) * q^44 + (-b3 - b1) * q^45 + (-b3 - b1) * q^46 + (-2*b2 + 3*b1) * q^47 - 3*b3 * q^48 + (5*b2 + 3) * q^49 - 4 * q^50 + (6*b2 - b1) * q^51 + (3*b3 + 2*b2 + 3*b1 - 2) * q^52 + (5*b3 + 5*b1) * q^53 + b1 * q^54 - 2*b3 * q^55 + (2*b3 - 4*b2 - b1 - 8) * q^56 + (-b3 + 3) * q^57 + (8*b2 - 4*b1) * q^58 + (-2*b3 + 6*b2 - 2*b1 - 6) * q^59 + (-3*b3 - 4*b2 - 3*b1 + 4) * q^60 - 6*b2 * q^61 + (-2*b3 + 12) * q^62 + (-3*b2 + 1) * q^63 + (b3 + 4) * q^64 + (8*b2 - b1) * q^65 + (2*b3 + 2*b1) * q^66 + (2*b3 - 11*b2 + 2*b1 + 11) * q^67 + (8*b2 + 3*b1) * q^68 + q^69 + (3*b3 + 8*b2 + 2*b1 - 12) * q^70 + (5*b3 + 6) * q^71 + (4*b2 + b1) * q^72 + (2*b3 + 9*b2 + 2*b1 - 9) * q^73 + (-6*b3 - 4*b2 - 6*b1 + 4) * q^74 + (b2 - b1) * q^75 + (-6*b3 + 10) * q^76 + (-4*b2 + 6) * q^77 + (-b3 - 8) * q^78 + (7*b2 + b1) * q^79 + (-3*b3 - 12*b2 - 3*b1 + 12) * q^80 + (b2 - 1) * q^81 + 16*b2 * q^82 + (-2*b3 + 8) * q^83 + (-b3 - 6*b2 - 3*b1 + 2) * q^84 + (-5*b3 - 4) * q^85 + (-12*b2 - 8*b1) * q^86 + (2*b3 - 6*b2 + 2*b1 + 6) * q^87 + (2*b3 + 8*b2 + 2*b1 - 8) * q^88 + 10*b2 * q^89 + (-b3 + 4) * q^90 + (-4*b3 - 3*b2 + 2*b1 - 6) * q^91 + (-b3 + 2) * q^92 + (-b2 + 3*b1) * q^93 + (b3 + 12*b2 + b1 - 12) * q^94 + (-4*b3 - 4*b2 - 4*b1 + 4) * q^95 + (4*b2 + b1) * q^96 + (6*b3 + 6) * q^97 + (5*b3 + 8*b1) * q^98 - 2 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + q^{2} - 2 q^{3} - 5 q^{4} + q^{5} - 2 q^{6} + 10 q^{7} - 18 q^{8} - 2 q^{9}+O(q^{10})$$ 4 * q + q^2 - 2 * q^3 - 5 * q^4 + q^5 - 2 * q^6 + 10 * q^7 - 18 * q^8 - 2 * q^9 $$4 q + q^{2} - 2 q^{3} - 5 q^{4} + q^{5} - 2 q^{6} + 10 q^{7} - 18 q^{8} - 2 q^{9} - 9 q^{10} + 4 q^{11} - 5 q^{12} - 8 q^{13} + q^{14} - 2 q^{15} - 3 q^{16} + 11 q^{17} + q^{18} - 7 q^{19} - 22 q^{20} - 8 q^{21} + 4 q^{22} - 2 q^{23} + 9 q^{24} + q^{25} + 15 q^{26} + 4 q^{27} - 20 q^{28} - 20 q^{29} - 9 q^{30} + q^{31} + 9 q^{32} + 4 q^{33} - 6 q^{34} + q^{35} + 10 q^{36} - 11 q^{37} + 12 q^{38} + 4 q^{39} - 13 q^{40} - 8 q^{41} - 5 q^{42} - 26 q^{43} + 10 q^{44} + q^{45} + q^{46} - q^{47} + 6 q^{48} + 22 q^{49} - 16 q^{50} + 11 q^{51} - 7 q^{52} - 5 q^{53} + q^{54} + 4 q^{55} - 45 q^{56} + 14 q^{57} + 12 q^{58} - 10 q^{59} + 11 q^{60} - 12 q^{61} + 52 q^{62} - 2 q^{63} + 14 q^{64} + 15 q^{65} - 2 q^{66} + 20 q^{67} + 19 q^{68} + 4 q^{69} - 36 q^{70} + 14 q^{71} + 9 q^{72} - 20 q^{73} + 14 q^{74} + q^{75} + 52 q^{76} + 16 q^{77} - 30 q^{78} + 15 q^{79} + 27 q^{80} - 2 q^{81} + 32 q^{82} + 36 q^{83} - 5 q^{84} - 6 q^{85} - 32 q^{86} + 10 q^{87} - 18 q^{88} + 20 q^{89} + 18 q^{90} - 20 q^{91} + 10 q^{92} + q^{93} - 25 q^{94} + 12 q^{95} + 9 q^{96} + 12 q^{97} - 2 q^{98} - 8 q^{99}+O(q^{100})$$ 4 * q + q^2 - 2 * q^3 - 5 * q^4 + q^5 - 2 * q^6 + 10 * q^7 - 18 * q^8 - 2 * q^9 - 9 * q^10 + 4 * q^11 - 5 * q^12 - 8 * q^13 + q^14 - 2 * q^15 - 3 * q^16 + 11 * q^17 + q^18 - 7 * q^19 - 22 * q^20 - 8 * q^21 + 4 * q^22 - 2 * q^23 + 9 * q^24 + q^25 + 15 * q^26 + 4 * q^27 - 20 * q^28 - 20 * q^29 - 9 * q^30 + q^31 + 9 * q^32 + 4 * q^33 - 6 * q^34 + q^35 + 10 * q^36 - 11 * q^37 + 12 * q^38 + 4 * q^39 - 13 * q^40 - 8 * q^41 - 5 * q^42 - 26 * q^43 + 10 * q^44 + q^45 + q^46 - q^47 + 6 * q^48 + 22 * q^49 - 16 * q^50 + 11 * q^51 - 7 * q^52 - 5 * q^53 + q^54 + 4 * q^55 - 45 * q^56 + 14 * q^57 + 12 * q^58 - 10 * q^59 + 11 * q^60 - 12 * q^61 + 52 * q^62 - 2 * q^63 + 14 * q^64 + 15 * q^65 - 2 * q^66 + 20 * q^67 + 19 * q^68 + 4 * q^69 - 36 * q^70 + 14 * q^71 + 9 * q^72 - 20 * q^73 + 14 * q^74 + q^75 + 52 * q^76 + 16 * q^77 - 30 * q^78 + 15 * q^79 + 27 * q^80 - 2 * q^81 + 32 * q^82 + 36 * q^83 - 5 * q^84 - 6 * q^85 - 32 * q^86 + 10 * q^87 - 18 * q^88 + 20 * q^89 + 18 * q^90 - 20 * q^91 + 10 * q^92 + q^93 - 25 * q^94 + 12 * q^95 + 9 * q^96 + 12 * q^97 - 2 * q^98 - 8 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} + 5x^{2} + 4x + 16$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( -\nu^{3} + 5\nu^{2} - 5\nu + 16 ) / 20$$ (-v^3 + 5*v^2 - 5*v + 16) / 20 $$\beta_{3}$$ $$=$$ $$( \nu^{3} + 4 ) / 5$$ (v^3 + 4) / 5
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{3} + 4\beta_{2} + \beta _1 - 4$$ b3 + 4*b2 + b1 - 4 $$\nu^{3}$$ $$=$$ $$5\beta_{3} - 4$$ 5*b3 - 4

## Character values

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

 $$n$$ $$323$$ $$346$$ $$442$$ $$\chi(n)$$ $$1$$ $$-\beta_{2}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
277.1
 −0.780776 − 1.35234i 1.28078 + 2.21837i −0.780776 + 1.35234i 1.28078 − 2.21837i
−0.780776 1.35234i −0.500000 + 0.866025i −0.219224 + 0.379706i −0.780776 1.35234i 1.56155 2.50000 + 0.866025i −2.43845 −0.500000 0.866025i −1.21922 + 2.11176i
277.2 1.28078 + 2.21837i −0.500000 + 0.866025i −2.28078 + 3.95042i 1.28078 + 2.21837i −2.56155 2.50000 + 0.866025i −6.56155 −0.500000 0.866025i −3.28078 + 5.68247i
415.1 −0.780776 + 1.35234i −0.500000 0.866025i −0.219224 0.379706i −0.780776 + 1.35234i 1.56155 2.50000 0.866025i −2.43845 −0.500000 + 0.866025i −1.21922 2.11176i
415.2 1.28078 2.21837i −0.500000 0.866025i −2.28078 3.95042i 1.28078 2.21837i −2.56155 2.50000 0.866025i −6.56155 −0.500000 + 0.866025i −3.28078 5.68247i
 $$n$$: 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 483.2.i.e 4
7.c even 3 1 inner 483.2.i.e 4
7.c even 3 1 3381.2.a.s 2
7.d odd 6 1 3381.2.a.q 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
483.2.i.e 4 1.a even 1 1 trivial
483.2.i.e 4 7.c even 3 1 inner
3381.2.a.q 2 7.d odd 6 1
3381.2.a.s 2 7.c even 3 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(483, [\chi])$$:

 $$T_{2}^{4} - T_{2}^{3} + 5T_{2}^{2} + 4T_{2} + 16$$ T2^4 - T2^3 + 5*T2^2 + 4*T2 + 16 $$T_{5}^{4} - T_{5}^{3} + 5T_{5}^{2} + 4T_{5} + 16$$ T5^4 - T5^3 + 5*T5^2 + 4*T5 + 16 $$T_{11}^{2} - 2T_{11} + 4$$ T11^2 - 2*T11 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - T^{3} + 5 T^{2} + 4 T + 16$$
$3$ $$(T^{2} + T + 1)^{2}$$
$5$ $$T^{4} - T^{3} + 5 T^{2} + 4 T + 16$$
$7$ $$(T^{2} - 5 T + 7)^{2}$$
$11$ $$(T^{2} - 2 T + 4)^{2}$$
$13$ $$(T^{2} + 4 T - 13)^{2}$$
$17$ $$T^{4} - 11 T^{3} + 95 T^{2} + \cdots + 676$$
$19$ $$T^{4} + 7 T^{3} + 41 T^{2} + 56 T + 64$$
$23$ $$(T^{2} + T + 1)^{2}$$
$29$ $$(T^{2} + 10 T + 8)^{2}$$
$31$ $$T^{4} - T^{3} + 39 T^{2} + 38 T + 1444$$
$37$ $$T^{4} + 11 T^{3} + 95 T^{2} + \cdots + 676$$
$41$ $$(T^{2} + 4 T - 64)^{2}$$
$43$ $$(T^{2} + 13 T + 4)^{2}$$
$47$ $$T^{4} + T^{3} + 39 T^{2} - 38 T + 1444$$
$53$ $$T^{4} + 5 T^{3} + 125 T^{2} + \cdots + 10000$$
$59$ $$T^{4} + 10 T^{3} + 92 T^{2} + 80 T + 64$$
$61$ $$(T^{2} + 6 T + 36)^{2}$$
$67$ $$T^{4} - 20 T^{3} + 317 T^{2} + \cdots + 6889$$
$71$ $$(T^{2} - 7 T - 94)^{2}$$
$73$ $$T^{4} + 20 T^{3} + 317 T^{2} + \cdots + 6889$$
$79$ $$T^{4} - 15 T^{3} + 173 T^{2} + \cdots + 2704$$
$83$ $$(T^{2} - 18 T + 64)^{2}$$
$89$ $$(T^{2} - 10 T + 100)^{2}$$
$97$ $$(T^{2} - 6 T - 144)^{2}$$