# Properties

 Label 6027.2.a.bc Level 6027 Weight 2 Character orbit 6027.a Self dual yes Analytic conductor 48.126 Analytic rank 0 Dimension 8 CM no Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ = $$6027 = 3 \cdot 7^{2} \cdot 41$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 6027.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$48.1258372982$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.8.7457527933.1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 861) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 3 x^{7} - 6 x^{6} + 23 x^{5} - 4 x^{4} - 27 x^{3} + 8 x^{2} + 8 x + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{7} - 2 \nu^{6} - 7 \nu^{5} + 14 \nu^{4} + 3 \nu^{3} - 10 \nu^{2} + \nu$$ $$\beta_{3}$$ $$=$$ $$-\nu^{6} + 2 \nu^{5} + 8 \nu^{4} - 15 \nu^{3} - 10 \nu^{2} + 16 \nu + 3$$ $$\beta_{4}$$ $$=$$ $$\nu^{6} - 2 \nu^{5} - 8 \nu^{4} + 15 \nu^{3} + 11 \nu^{2} - 16 \nu - 6$$ $$\beta_{5}$$ $$=$$ $$-\nu^{7} + 2 \nu^{6} + 8 \nu^{5} - 15 \nu^{4} - 11 \nu^{3} + 16 \nu^{2} + 7 \nu - 1$$ $$\beta_{6}$$ $$=$$ $$2 \nu^{6} - 4 \nu^{5} - 15 \nu^{4} + 29 \nu^{3} + 14 \nu^{2} - 26 \nu - 6$$ $$\beta_{7}$$ $$=$$ $$-3 \nu^{7} + 6 \nu^{6} + 23 \nu^{5} - 44 \nu^{4} - 24 \nu^{3} + 42 \nu^{2} + 8 \nu$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{4} + \beta_{3} + 3$$ $$\nu^{3}$$ $$=$$ $$\beta_{7} - 2 \beta_{5} + \beta_{2} + 5 \beta_{1} - 2$$ $$\nu^{4}$$ $$=$$ $$\beta_{7} + \beta_{6} - 2 \beta_{5} + 6 \beta_{4} + 8 \beta_{3} + \beta_{2} - \beta_{1} + 16$$ $$\nu^{5}$$ $$=$$ $$9 \beta_{7} + \beta_{6} - 17 \beta_{5} + 2 \beta_{3} + 10 \beta_{2} + 31 \beta_{1} - 17$$ $$\nu^{6}$$ $$=$$ $$11 \beta_{7} + 10 \beta_{6} - 20 \beta_{5} + 38 \beta_{4} + 57 \beta_{3} + 13 \beta_{2} - 5 \beta_{1} + 97$$ $$\nu^{7}$$ $$=$$ $$68 \beta_{7} + 13 \beta_{6} - 125 \beta_{5} + 2 \beta_{4} + 26 \beta_{3} + 80 \beta_{2} + 205 \beta_{1} - 113$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 2.70360 1.35554 1.21768 −0.978012 −0.169079 1.70821 −2.52358 −0.314356
−2.23115 1.00000 2.97801 −0.893036 −2.23115 0 −2.18209 1.00000 1.99249
1.2 −2.04183 1.00000 2.16908 3.68950 −2.04183 0 −0.345232 1.00000 −7.53333
1.3 −1.51387 1.00000 0.291794 −3.80505 −1.51387 0 2.58600 1.00000 5.76034
1.4 −1.13860 1.00000 −0.703600 2.27123 −1.13860 0 3.07831 1.00000 −2.58601
1.5 1.62618 1.00000 0.644462 1.67363 1.62618 0 −2.20435 1.00000 2.72162
1.6 1.66803 1.00000 0.782321 4.32429 1.66803 0 −2.03112 1.00000 7.21304
1.7 2.07710 1.00000 2.31436 −2.30039 2.07710 0 0.652949 1.00000 −4.77814
1.8 2.55413 1.00000 4.52358 2.03983 2.55413 0 6.44554 1.00000 5.20999
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 6027.2.a.bc 8
7.b odd 2 1 6027.2.a.bb 8
7.d odd 6 2 861.2.i.d 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
861.2.i.d 16 7.d odd 6 2
6027.2.a.bb 8 7.b odd 2 1
6027.2.a.bc 8 1.a even 1 1 trivial

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$7$$ $$-1$$
$$41$$ $$-1$$

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(6027))$$:

 $$T_{2}^{8} - \cdots$$ $$T_{5}^{8} - \cdots$$ $$T_{13}^{4} - 5 T_{13}^{3} - 10 T_{13}^{2} + 44 T_{13} + 49$$