# Properties

 Label 1127.2.a.d Level 1127 Weight 2 Character orbit 1127.a Self dual yes Analytic conductor 8.999 Analytic rank 0 Dimension 2 CM no Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1127 = 7^{2} \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1127.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$8.99914030780$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{5})$$ Defining polynomial: $$x^{2} - x - 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 161) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{5})$$. We also show the integral $$q$$-expansion of the trace form.

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

## 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
 1.61803 −0.618034
−1.61803 1.00000 0.618034 −1.23607 −1.61803 0 2.23607 −2.00000 2.00000
1.2 0.618034 1.00000 −1.61803 3.23607 0.618034 0 −2.23607 −2.00000 2.00000
 $$n$$: e.g. 2-40 or 990-1000 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 1127.2.a.d 2
7.b odd 2 1 161.2.a.b 2
21.c even 2 1 1449.2.a.i 2
28.d even 2 1 2576.2.a.s 2
35.c odd 2 1 4025.2.a.i 2
161.c even 2 1 3703.2.a.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
161.2.a.b 2 7.b odd 2 1
1127.2.a.d 2 1.a even 1 1 trivial
1449.2.a.i 2 21.c even 2 1
2576.2.a.s 2 28.d even 2 1
3703.2.a.b 2 161.c even 2 1
4025.2.a.i 2 35.c odd 2 1

## Atkin-Lehner signs

$$p$$ Sign
$$7$$ $$-1$$
$$23$$ $$1$$

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1127))$$:

 $$T_{2}^{2} + T_{2} - 1$$ $$T_{3} - 1$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T + 3 T^{2} + 2 T^{3} + 4 T^{4}$$
$3$ $$( 1 - T + 3 T^{2} )^{2}$$
$5$ $$1 - 2 T + 6 T^{2} - 10 T^{3} + 25 T^{4}$$
$7$ 1
$11$ $$1 + 2 T^{2} + 121 T^{4}$$
$13$ $$1 - 4 T + 25 T^{2} - 52 T^{3} + 169 T^{4}$$
$17$ $$( 1 + 17 T^{2} )^{2}$$
$19$ $$1 - 10 T + 58 T^{2} - 190 T^{3} + 361 T^{4}$$
$23$ $$( 1 + T )^{2}$$
$29$ $$1 - 6 T + 47 T^{2} - 174 T^{3} + 841 T^{4}$$
$31$ $$( 1 - 9 T + 31 T^{2} )^{2}$$
$37$ $$1 - 2 T + 30 T^{2} - 74 T^{3} + 1369 T^{4}$$
$41$ $$1 + 77 T^{2} + 1681 T^{4}$$
$43$ $$1 + 4 T + 70 T^{2} + 172 T^{3} + 1849 T^{4}$$
$47$ $$1 + 2 T + 75 T^{2} + 94 T^{3} + 2209 T^{4}$$
$53$ $$1 - 18 T + 182 T^{2} - 954 T^{3} + 2809 T^{4}$$
$59$ $$1 - 12 T + 134 T^{2} - 708 T^{3} + 3481 T^{4}$$
$61$ $$1 - 58 T^{2} + 3721 T^{4}$$
$67$ $$1 + 2 T + 10 T^{2} + 134 T^{3} + 4489 T^{4}$$
$71$ $$1 + 16 T + 201 T^{2} + 1136 T^{3} + 5041 T^{4}$$
$73$ $$1 + 101 T^{2} + 5329 T^{4}$$
$79$ $$1 + 10 T + 178 T^{2} + 790 T^{3} + 6241 T^{4}$$
$83$ $$1 + 4 T + 150 T^{2} + 332 T^{3} + 6889 T^{4}$$
$89$ $$1 + 98 T^{2} + 7921 T^{4}$$
$97$ $$1 - 6 T + 158 T^{2} - 582 T^{3} + 9409 T^{4}$$