# Properties

 Label 1083.2.d.b Level $1083$ Weight $2$ Character orbit 1083.d Analytic conductor $8.648$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$1083 = 3 \cdot 19^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1083.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$8.64779853890$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: $$\Q(\zeta_{24})$$ Defining polynomial: $$x^{8} - x^{4} + 1$$ x^8 - x^4 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: no (minimal twist has level 57) 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.

 $$f(q)$$ $$=$$ $$q - \beta_{6} q^{2} + (\beta_{6} + \beta_{5} - \beta_1) q^{3} + \beta_{3} q^{4} + (\beta_{7} + \beta_{4}) q^{5} + (\beta_{7} - \beta_{3} - 1) q^{6} + ( - \beta_{3} - 2) q^{7} - \beta_{5} q^{8} + ( - 2 \beta_{7} - 2 \beta_{4} + 1) q^{9}+O(q^{10})$$ q - b6 * q^2 + (b6 + b5 - b1) * q^3 + b3 * q^4 + (b7 + b4) * q^5 + (b7 - b3 - 1) * q^6 + (-b3 - 2) * q^7 - b5 * q^8 + (-2*b7 - 2*b4 + 1) * q^9 $$q - \beta_{6} q^{2} + (\beta_{6} + \beta_{5} - \beta_1) q^{3} + \beta_{3} q^{4} + (\beta_{7} + \beta_{4}) q^{5} + (\beta_{7} - \beta_{3} - 1) q^{6} + ( - \beta_{3} - 2) q^{7} - \beta_{5} q^{8} + ( - 2 \beta_{7} - 2 \beta_{4} + 1) q^{9} + ( - \beta_{2} - \beta_1) q^{10} + (\beta_{7} + 3 \beta_{4}) q^{11} + (\beta_{6} - \beta_{5} - \beta_{2}) q^{12} + (\beta_{2} + 2 \beta_1) q^{13} + (4 \beta_{6} + \beta_{5}) q^{14} + (\beta_{6} + \beta_{5} + 2 \beta_1) q^{15} + ( - 2 \beta_{3} - 1) q^{16} + ( - 2 \beta_{7} + 2 \beta_{4}) q^{17} + ( - \beta_{6} + 2 \beta_{2} + 2 \beta_1) q^{18} + (\beta_{7} - \beta_{4}) q^{20} + ( - 3 \beta_{6} - \beta_{5} + \beta_{2} + 2 \beta_1) q^{21} + ( - \beta_{2} + \beta_1) q^{22} + (\beta_{7} + 3 \beta_{4}) q^{23} + (\beta_{4} + \beta_{3} - 1) q^{24} + 3 q^{25} + ( - 4 \beta_{7} - \beta_{4}) q^{26} + ( - \beta_{6} - \beta_{5} - 5 \beta_1) q^{27} + ( - 2 \beta_{3} - 3) q^{28} + 4 \beta_{6} q^{29} + ( - 2 \beta_{7} - \beta_{3} - 1) q^{30} + (2 \beta_{2} + \beta_1) q^{31} + (5 \beta_{6} + 4 \beta_{5}) q^{32} + (\beta_{6} + 3 \beta_{5} - 2 \beta_{2} + 4 \beta_1) q^{33} + (2 \beta_{2} + 6 \beta_1) q^{34} + ( - 3 \beta_{7} - \beta_{4}) q^{35} + ( - 2 \beta_{7} + 2 \beta_{4} + \beta_{3}) q^{36} + ( - \beta_{2} + 6 \beta_1) q^{37} + (3 \beta_{7} + \beta_{4} + \beta_{3} + 2) q^{39} + (\beta_{2} - \beta_1) q^{40} + (4 \beta_{6} + 4 \beta_{5}) q^{41} + ( - 4 \beta_{7} - \beta_{4} + 3 \beta_{3} + 5) q^{42} + ( - \beta_{3} + 4) q^{43} + ( - \beta_{7} - 5 \beta_{4}) q^{44} + (\beta_{7} + \beta_{4} + 4) q^{45} + ( - \beta_{2} + \beta_1) q^{46} + (6 \beta_{7} + 2 \beta_{4}) q^{47} + ( - 3 \beta_{6} + \beta_{5} + 2 \beta_{2} + \beta_1) q^{48} + 4 \beta_{3} q^{49} - 3 \beta_{6} q^{50} + ( - 2 \beta_{6} + 2 \beta_{5} - 4 \beta_{2}) q^{51} + (2 \beta_{2} + 3 \beta_1) q^{52} + (5 \beta_{6} + 7 \beta_{5}) q^{53} + (5 \beta_{7} + \beta_{3} + 1) q^{54} + (2 \beta_{3} - 4) q^{55} - \beta_{6} q^{56} + ( - 4 \beta_{3} - 8) q^{58} + (3 \beta_{6} - 5 \beta_{5}) q^{59} + (\beta_{6} - \beta_{5} + 2 \beta_{2}) q^{60} + (2 \beta_{3} - 7) q^{61} + ( - 5 \beta_{7} - 2 \beta_{4}) q^{62} + (6 \beta_{7} + 2 \beta_{4} - \beta_{3} - 2) q^{63} + ( - \beta_{3} - 4) q^{64} + ( - 3 \beta_{6} - \beta_{5}) q^{65} + (2 \beta_{4} - \beta_{3} + 1) q^{66} + \beta_1 q^{67} + ( - 6 \beta_{7} - 6 \beta_{4}) q^{68} + (\beta_{6} + 3 \beta_{5} - 2 \beta_{2} + 4 \beta_1) q^{69} + (3 \beta_{2} + 5 \beta_1) q^{70} + (4 \beta_{6} + 8 \beta_{5}) q^{71} + ( - \beta_{5} - 2 \beta_{2} + 2 \beta_1) q^{72} - 3 q^{73} + ( - 4 \beta_{7} + \beta_{4}) q^{74} + (3 \beta_{6} + 3 \beta_{5} - 3 \beta_1) q^{75} + ( - \beta_{7} - \beta_{4}) q^{77} + ( - 4 \beta_{6} - \beta_{5} - 3 \beta_{2} - 5 \beta_1) q^{78} + ( - 2 \beta_{2} + 7 \beta_1) q^{79} + ( - 3 \beta_{7} + \beta_{4}) q^{80} + ( - 4 \beta_{7} - 4 \beta_{4} - 7) q^{81} + ( - 4 \beta_{3} - 4) q^{82} + 4 \beta_{7} q^{83} + ( - 5 \beta_{6} - \beta_{5} + 2 \beta_{2} + 3 \beta_1) q^{84} + 4 \beta_{3} q^{85} + ( - 2 \beta_{6} + \beta_{5}) q^{86} + ( - 4 \beta_{7} + 4 \beta_{3} + 4) q^{87} + (3 \beta_{2} - 5 \beta_1) q^{88} + (3 \beta_{6} - 3 \beta_{5}) q^{89} + ( - 4 \beta_{6} - \beta_{2} - \beta_1) q^{90} + ( - 4 \beta_{2} - 7 \beta_1) q^{91} + ( - \beta_{7} - 5 \beta_{4}) q^{92} + (3 \beta_{7} - \beta_{4} + 2 \beta_{3} + 1) q^{93} + ( - 6 \beta_{2} - 10 \beta_1) q^{94} + ( - 5 \beta_{7} - 4 \beta_{4} + \beta_{3} + 9) q^{96} + ( - 2 \beta_{2} - 4 \beta_1) q^{97} + ( - 8 \beta_{6} - 4 \beta_{5}) q^{98} + (\beta_{7} + 3 \beta_{4} - 4 \beta_{3} + 8) q^{99}+O(q^{100})$$ q - b6 * q^2 + (b6 + b5 - b1) * q^3 + b3 * q^4 + (b7 + b4) * q^5 + (b7 - b3 - 1) * q^6 + (-b3 - 2) * q^7 - b5 * q^8 + (-2*b7 - 2*b4 + 1) * q^9 + (-b2 - b1) * q^10 + (b7 + 3*b4) * q^11 + (b6 - b5 - b2) * q^12 + (b2 + 2*b1) * q^13 + (4*b6 + b5) * q^14 + (b6 + b5 + 2*b1) * q^15 + (-2*b3 - 1) * q^16 + (-2*b7 + 2*b4) * q^17 + (-b6 + 2*b2 + 2*b1) * q^18 + (b7 - b4) * q^20 + (-3*b6 - b5 + b2 + 2*b1) * q^21 + (-b2 + b1) * q^22 + (b7 + 3*b4) * q^23 + (b4 + b3 - 1) * q^24 + 3 * q^25 + (-4*b7 - b4) * q^26 + (-b6 - b5 - 5*b1) * q^27 + (-2*b3 - 3) * q^28 + 4*b6 * q^29 + (-2*b7 - b3 - 1) * q^30 + (2*b2 + b1) * q^31 + (5*b6 + 4*b5) * q^32 + (b6 + 3*b5 - 2*b2 + 4*b1) * q^33 + (2*b2 + 6*b1) * q^34 + (-3*b7 - b4) * q^35 + (-2*b7 + 2*b4 + b3) * q^36 + (-b2 + 6*b1) * q^37 + (3*b7 + b4 + b3 + 2) * q^39 + (b2 - b1) * q^40 + (4*b6 + 4*b5) * q^41 + (-4*b7 - b4 + 3*b3 + 5) * q^42 + (-b3 + 4) * q^43 + (-b7 - 5*b4) * q^44 + (b7 + b4 + 4) * q^45 + (-b2 + b1) * q^46 + (6*b7 + 2*b4) * q^47 + (-3*b6 + b5 + 2*b2 + b1) * q^48 + 4*b3 * q^49 - 3*b6 * q^50 + (-2*b6 + 2*b5 - 4*b2) * q^51 + (2*b2 + 3*b1) * q^52 + (5*b6 + 7*b5) * q^53 + (5*b7 + b3 + 1) * q^54 + (2*b3 - 4) * q^55 - b6 * q^56 + (-4*b3 - 8) * q^58 + (3*b6 - 5*b5) * q^59 + (b6 - b5 + 2*b2) * q^60 + (2*b3 - 7) * q^61 + (-5*b7 - 2*b4) * q^62 + (6*b7 + 2*b4 - b3 - 2) * q^63 + (-b3 - 4) * q^64 + (-3*b6 - b5) * q^65 + (2*b4 - b3 + 1) * q^66 + b1 * q^67 + (-6*b7 - 6*b4) * q^68 + (b6 + 3*b5 - 2*b2 + 4*b1) * q^69 + (3*b2 + 5*b1) * q^70 + (4*b6 + 8*b5) * q^71 + (-b5 - 2*b2 + 2*b1) * q^72 - 3 * q^73 + (-4*b7 + b4) * q^74 + (3*b6 + 3*b5 - 3*b1) * q^75 + (-b7 - b4) * q^77 + (-4*b6 - b5 - 3*b2 - 5*b1) * q^78 + (-2*b2 + 7*b1) * q^79 + (-3*b7 + b4) * q^80 + (-4*b7 - 4*b4 - 7) * q^81 + (-4*b3 - 4) * q^82 + 4*b7 * q^83 + (-5*b6 - b5 + 2*b2 + 3*b1) * q^84 + 4*b3 * q^85 + (-2*b6 + b5) * q^86 + (-4*b7 + 4*b3 + 4) * q^87 + (3*b2 - 5*b1) * q^88 + (3*b6 - 3*b5) * q^89 + (-4*b6 - b2 - b1) * q^90 + (-4*b2 - 7*b1) * q^91 + (-b7 - 5*b4) * q^92 + (3*b7 - b4 + 2*b3 + 1) * q^93 + (-6*b2 - 10*b1) * q^94 + (-5*b7 - 4*b4 + b3 + 9) * q^96 + (-2*b2 - 4*b1) * q^97 + (-8*b6 - 4*b5) * q^98 + (b7 + 3*b4 - 4*b3 + 8) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 8 q^{6} - 16 q^{7} + 8 q^{9}+O(q^{10})$$ 8 * q - 8 * q^6 - 16 * q^7 + 8 * q^9 $$8 q - 8 q^{6} - 16 q^{7} + 8 q^{9} - 8 q^{16} - 8 q^{24} + 24 q^{25} - 24 q^{28} - 8 q^{30} + 16 q^{39} + 40 q^{42} + 32 q^{43} + 32 q^{45} + 8 q^{54} - 32 q^{55} - 64 q^{58} - 56 q^{61} - 16 q^{63} - 32 q^{64} + 8 q^{66} - 24 q^{73} - 56 q^{81} - 32 q^{82} + 32 q^{87} + 8 q^{93} + 72 q^{96} + 64 q^{99}+O(q^{100})$$ 8 * q - 8 * q^6 - 16 * q^7 + 8 * q^9 - 8 * q^16 - 8 * q^24 + 24 * q^25 - 24 * q^28 - 8 * q^30 + 16 * q^39 + 40 * q^42 + 32 * q^43 + 32 * q^45 + 8 * q^54 - 32 * q^55 - 64 * q^58 - 56 * q^61 - 16 * q^63 - 32 * q^64 + 8 * q^66 - 24 * q^73 - 56 * q^81 - 32 * q^82 + 32 * q^87 + 8 * q^93 + 72 * q^96 + 64 * q^99

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$\zeta_{24}^{6}$$ v^6 $$\beta_{2}$$ $$=$$ $$2\zeta_{24}^{4} - 1$$ 2*v^4 - 1 $$\beta_{3}$$ $$=$$ $$-\zeta_{24}^{6} + 2\zeta_{24}^{2}$$ -v^6 + 2*v^2 $$\beta_{4}$$ $$=$$ $$-\zeta_{24}^{7} + \zeta_{24}^{3} - \zeta_{24}$$ -v^7 + v^3 - v $$\beta_{5}$$ $$=$$ $$\zeta_{24}^{7} - \zeta_{24}^{5}$$ v^7 - v^5 $$\beta_{6}$$ $$=$$ $$-\zeta_{24}^{7} + \zeta_{24}^{3} + \zeta_{24}$$ -v^7 + v^3 + v $$\beta_{7}$$ $$=$$ $$\zeta_{24}^{7} + \zeta_{24}^{5}$$ v^7 + v^5
 $$\zeta_{24}$$ $$=$$ $$( \beta_{6} - \beta_{4} ) / 2$$ (b6 - b4) / 2 $$\zeta_{24}^{2}$$ $$=$$ $$( \beta_{3} + \beta_1 ) / 2$$ (b3 + b1) / 2 $$\zeta_{24}^{3}$$ $$=$$ $$( \beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} ) / 2$$ (b7 + b6 + b5 + b4) / 2 $$\zeta_{24}^{4}$$ $$=$$ $$( \beta_{2} + 1 ) / 2$$ (b2 + 1) / 2 $$\zeta_{24}^{5}$$ $$=$$ $$( \beta_{7} - \beta_{5} ) / 2$$ (b7 - b5) / 2 $$\zeta_{24}^{6}$$ $$=$$ $$\beta_1$$ b1 $$\zeta_{24}^{7}$$ $$=$$ $$( \beta_{7} + \beta_{5} ) / 2$$ (b7 + b5) / 2

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1083\mathbb{Z}\right)^\times$$.

 $$n$$ $$362$$ $$724$$ $$\chi(n)$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1082.1
 0.965926 + 0.258819i 0.965926 − 0.258819i 0.258819 + 0.965926i 0.258819 − 0.965926i −0.258819 − 0.965926i −0.258819 + 0.965926i −0.965926 − 0.258819i −0.965926 + 0.258819i
−1.93185 1.41421 1.00000i 1.73205 1.41421i −2.73205 + 1.93185i −3.73205 0.517638 1.00000 2.82843i 2.73205i
1082.2 −1.93185 1.41421 + 1.00000i 1.73205 1.41421i −2.73205 1.93185i −3.73205 0.517638 1.00000 + 2.82843i 2.73205i
1082.3 −0.517638 −1.41421 1.00000i −1.73205 1.41421i 0.732051 + 0.517638i −0.267949 1.93185 1.00000 + 2.82843i 0.732051i
1082.4 −0.517638 −1.41421 + 1.00000i −1.73205 1.41421i 0.732051 0.517638i −0.267949 1.93185 1.00000 2.82843i 0.732051i
1082.5 0.517638 1.41421 1.00000i −1.73205 1.41421i 0.732051 0.517638i −0.267949 −1.93185 1.00000 2.82843i 0.732051i
1082.6 0.517638 1.41421 + 1.00000i −1.73205 1.41421i 0.732051 + 0.517638i −0.267949 −1.93185 1.00000 + 2.82843i 0.732051i
1082.7 1.93185 −1.41421 1.00000i 1.73205 1.41421i −2.73205 1.93185i −3.73205 −0.517638 1.00000 + 2.82843i 2.73205i
1082.8 1.93185 −1.41421 + 1.00000i 1.73205 1.41421i −2.73205 + 1.93185i −3.73205 −0.517638 1.00000 2.82843i 2.73205i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1082.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
19.b odd 2 1 inner
57.d even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1083.2.d.b 8
3.b odd 2 1 inner 1083.2.d.b 8
19.b odd 2 1 inner 1083.2.d.b 8
19.c even 3 1 57.2.f.a 8
19.d odd 6 1 57.2.f.a 8
57.d even 2 1 inner 1083.2.d.b 8
57.f even 6 1 57.2.f.a 8
57.h odd 6 1 57.2.f.a 8
76.f even 6 1 912.2.bn.m 8
76.g odd 6 1 912.2.bn.m 8
228.m even 6 1 912.2.bn.m 8
228.n odd 6 1 912.2.bn.m 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
57.2.f.a 8 19.c even 3 1
57.2.f.a 8 19.d odd 6 1
57.2.f.a 8 57.f even 6 1
57.2.f.a 8 57.h odd 6 1
912.2.bn.m 8 76.f even 6 1
912.2.bn.m 8 76.g odd 6 1
912.2.bn.m 8 228.m even 6 1
912.2.bn.m 8 228.n odd 6 1
1083.2.d.b 8 1.a even 1 1 trivial
1083.2.d.b 8 3.b odd 2 1 inner
1083.2.d.b 8 19.b odd 2 1 inner
1083.2.d.b 8 57.d even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} - 4T_{2}^{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(1083, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{4} - 4 T^{2} + 1)^{2}$$
$3$ $$(T^{4} - 2 T^{2} + 9)^{2}$$
$5$ $$(T^{2} + 2)^{4}$$
$7$ $$(T^{2} + 4 T + 1)^{4}$$
$11$ $$(T^{4} + 28 T^{2} + 4)^{2}$$
$13$ $$(T^{4} + 14 T^{2} + 1)^{2}$$
$17$ $$(T^{2} + 24)^{4}$$
$19$ $$T^{8}$$
$23$ $$(T^{4} + 28 T^{2} + 4)^{2}$$
$29$ $$(T^{4} - 64 T^{2} + 256)^{2}$$
$31$ $$(T^{4} + 26 T^{2} + 121)^{2}$$
$37$ $$(T^{4} + 78 T^{2} + 1089)^{2}$$
$41$ $$(T^{2} - 32)^{4}$$
$43$ $$(T^{2} - 8 T + 13)^{4}$$
$47$ $$(T^{4} + 112 T^{2} + 64)^{2}$$
$53$ $$(T^{4} - 156 T^{2} + 4356)^{2}$$
$59$ $$(T^{4} - 196 T^{2} + 8836)^{2}$$
$61$ $$(T^{2} + 14 T + 37)^{4}$$
$67$ $$(T^{2} + 1)^{4}$$
$71$ $$(T^{4} - 192 T^{2} + 2304)^{2}$$
$73$ $$(T + 3)^{8}$$
$79$ $$(T^{4} + 122 T^{2} + 1369)^{2}$$
$83$ $$(T^{4} + 64 T^{2} + 256)^{2}$$
$89$ $$(T^{2} - 54)^{4}$$
$97$ $$(T^{4} + 56 T^{2} + 16)^{2}$$
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