# Properties

 Label 945.2.a.e Level 945 Weight 2 Character orbit 945.a Self dual yes Analytic conductor 7.546 Analytic rank 0 Dimension 2 CM no Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$945 = 3^{3} \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 945.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$7.54586299101$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{5})$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q -\beta q^{2} + ( -1 + \beta ) q^{4} + q^{5} - q^{7} + ( -1 + 2 \beta ) q^{8} +O(q^{10})$$ $$q -\beta q^{2} + ( -1 + \beta ) q^{4} + q^{5} - q^{7} + ( -1 + 2 \beta ) q^{8} -\beta q^{10} + ( 1 + 2 \beta ) q^{11} + ( 3 - \beta ) q^{13} + \beta q^{14} -3 \beta q^{16} + \beta q^{17} + ( 1 - 5 \beta ) q^{19} + ( -1 + \beta ) q^{20} + ( -2 - 3 \beta ) q^{22} + ( 7 - \beta ) q^{23} + q^{25} + ( 1 - 2 \beta ) q^{26} + ( 1 - \beta ) q^{28} + ( 5 - 9 \beta ) q^{29} + ( -3 + 6 \beta ) q^{31} + ( 5 - \beta ) q^{32} + ( -1 - \beta ) q^{34} - q^{35} + ( -3 + 6 \beta ) q^{37} + ( 5 + 4 \beta ) q^{38} + ( -1 + 2 \beta ) q^{40} + 5 \beta q^{41} + ( 3 - 8 \beta ) q^{43} + ( 1 + \beta ) q^{44} + ( 1 - 6 \beta ) q^{46} + 11 q^{47} + q^{49} -\beta q^{50} + ( -4 + 3 \beta ) q^{52} + ( -6 + \beta ) q^{53} + ( 1 + 2 \beta ) q^{55} + ( 1 - 2 \beta ) q^{56} + ( 9 + 4 \beta ) q^{58} + ( 11 - 6 \beta ) q^{59} + ( -9 - 3 \beta ) q^{61} + ( -6 - 3 \beta ) q^{62} + ( 1 + 2 \beta ) q^{64} + ( 3 - \beta ) q^{65} + 7 \beta q^{67} + q^{68} + \beta q^{70} + ( -1 + 3 \beta ) q^{71} + ( 1 + 8 \beta ) q^{73} + ( -6 - 3 \beta ) q^{74} + ( -6 + \beta ) q^{76} + ( -1 - 2 \beta ) q^{77} + ( 4 + \beta ) q^{79} -3 \beta q^{80} + ( -5 - 5 \beta ) q^{82} + ( 1 + 6 \beta ) q^{83} + \beta q^{85} + ( 8 + 5 \beta ) q^{86} + ( 3 + 4 \beta ) q^{88} + ( -5 + 4 \beta ) q^{89} + ( -3 + \beta ) q^{91} + ( -8 + 7 \beta ) q^{92} -11 \beta q^{94} + ( 1 - 5 \beta ) q^{95} + ( 8 - 7 \beta ) q^{97} -\beta q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - q^{2} - q^{4} + 2q^{5} - 2q^{7} + O(q^{10})$$ $$2q - q^{2} - q^{4} + 2q^{5} - 2q^{7} - q^{10} + 4q^{11} + 5q^{13} + q^{14} - 3q^{16} + q^{17} - 3q^{19} - q^{20} - 7q^{22} + 13q^{23} + 2q^{25} + q^{28} + q^{29} + 9q^{32} - 3q^{34} - 2q^{35} + 14q^{38} + 5q^{41} - 2q^{43} + 3q^{44} - 4q^{46} + 22q^{47} + 2q^{49} - q^{50} - 5q^{52} - 11q^{53} + 4q^{55} + 22q^{58} + 16q^{59} - 21q^{61} - 15q^{62} + 4q^{64} + 5q^{65} + 7q^{67} + 2q^{68} + q^{70} + q^{71} + 10q^{73} - 15q^{74} - 11q^{76} - 4q^{77} + 9q^{79} - 3q^{80} - 15q^{82} + 8q^{83} + q^{85} + 21q^{86} + 10q^{88} - 6q^{89} - 5q^{91} - 9q^{92} - 11q^{94} - 3q^{95} + 9q^{97} - q^{98} + O(q^{100})$$

## 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 0 0.618034 1.00000 0 −1.00000 2.23607 0 −1.61803
1.2 0.618034 0 −1.61803 1.00000 0 −1.00000 −2.23607 0 0.618034
 $$n$$: 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 945.2.a.e 2
3.b odd 2 1 945.2.a.g yes 2
5.b even 2 1 4725.2.a.be 2
7.b odd 2 1 6615.2.a.o 2
15.d odd 2 1 4725.2.a.w 2
21.c even 2 1 6615.2.a.r 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
945.2.a.e 2 1.a even 1 1 trivial
945.2.a.g yes 2 3.b odd 2 1
4725.2.a.w 2 15.d odd 2 1
4725.2.a.be 2 5.b even 2 1
6615.2.a.o 2 7.b odd 2 1
6615.2.a.r 2 21.c even 2 1

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$1$$
$$5$$ $$-1$$
$$7$$ $$1$$

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(945))$$:

 $$T_{2}^{2} + T_{2} - 1$$ $$T_{11}^{2} - 4 T_{11} - 1$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T + 3 T^{2} + 2 T^{3} + 4 T^{4}$$
$3$ 1
$5$ $$( 1 - T )^{2}$$
$7$ $$( 1 + T )^{2}$$
$11$ $$1 - 4 T + 21 T^{2} - 44 T^{3} + 121 T^{4}$$
$13$ $$1 - 5 T + 31 T^{2} - 65 T^{3} + 169 T^{4}$$
$17$ $$1 - T + 33 T^{2} - 17 T^{3} + 289 T^{4}$$
$19$ $$1 + 3 T + 9 T^{2} + 57 T^{3} + 361 T^{4}$$
$23$ $$1 - 13 T + 87 T^{2} - 299 T^{3} + 529 T^{4}$$
$29$ $$1 - T - 43 T^{2} - 29 T^{3} + 841 T^{4}$$
$31$ $$1 + 17 T^{2} + 961 T^{4}$$
$37$ $$1 + 29 T^{2} + 1369 T^{4}$$
$41$ $$1 - 5 T + 57 T^{2} - 205 T^{3} + 1681 T^{4}$$
$43$ $$1 + 2 T + 7 T^{2} + 86 T^{3} + 1849 T^{4}$$
$47$ $$( 1 - 11 T + 47 T^{2} )^{2}$$
$53$ $$1 + 11 T + 135 T^{2} + 583 T^{3} + 2809 T^{4}$$
$59$ $$1 - 16 T + 137 T^{2} - 944 T^{3} + 3481 T^{4}$$
$61$ $$1 + 21 T + 221 T^{2} + 1281 T^{3} + 3721 T^{4}$$
$67$ $$1 - 7 T + 85 T^{2} - 469 T^{3} + 4489 T^{4}$$
$71$ $$1 - T + 131 T^{2} - 71 T^{3} + 5041 T^{4}$$
$73$ $$1 - 10 T + 91 T^{2} - 730 T^{3} + 5329 T^{4}$$
$79$ $$1 - 9 T + 177 T^{2} - 711 T^{3} + 6241 T^{4}$$
$83$ $$1 - 8 T + 137 T^{2} - 664 T^{3} + 6889 T^{4}$$
$89$ $$1 + 6 T + 167 T^{2} + 534 T^{3} + 7921 T^{4}$$
$97$ $$1 - 9 T + 153 T^{2} - 873 T^{3} + 9409 T^{4}$$