# Properties

 Label 945.2.a.a Level 945 Weight 2 Character orbit 945.a Self dual Yes Analytic conductor 7.546 Analytic rank 1 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: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{5})$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ 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.

 $$f(q)$$ $$=$$ $$q$$ $$+ ( -1 - \beta ) q^{2}$$ $$+ 3 \beta q^{4}$$ $$+ q^{5}$$ $$+ q^{7}$$ $$+ ( -1 - 4 \beta ) q^{8}$$ $$+O(q^{10})$$ $$q$$ $$+ ( -1 - \beta ) q^{2}$$ $$+ 3 \beta q^{4}$$ $$+ q^{5}$$ $$+ q^{7}$$ $$+ ( -1 - 4 \beta ) q^{8}$$ $$+ ( -1 - \beta ) q^{10}$$ $$+ ( -3 + 2 \beta ) q^{11}$$ $$+ ( -2 + \beta ) q^{13}$$ $$+ ( -1 - \beta ) q^{14}$$ $$+ ( 5 + 3 \beta ) q^{16}$$ $$+ ( -1 - 3 \beta ) q^{17}$$ $$+ ( -2 - \beta ) q^{19}$$ $$+ 3 \beta q^{20}$$ $$+ ( 1 - \beta ) q^{22}$$ $$+ ( -4 - 3 \beta ) q^{23}$$ $$+ q^{25}$$ $$+ q^{26}$$ $$+ 3 \beta q^{28}$$ $$+ ( -4 + 3 \beta ) q^{29}$$ $$+ ( -1 - 4 \beta ) q^{31}$$ $$+ ( -6 - 3 \beta ) q^{32}$$ $$+ ( 4 + 7 \beta ) q^{34}$$ $$+ q^{35}$$ $$+ ( 3 - 8 \beta ) q^{37}$$ $$+ ( 3 + 4 \beta ) q^{38}$$ $$+ ( -1 - 4 \beta ) q^{40}$$ $$+ ( -7 + 9 \beta ) q^{41}$$ $$+ ( -1 + 2 \beta ) q^{43}$$ $$+ ( 6 - 3 \beta ) q^{44}$$ $$+ ( 7 + 10 \beta ) q^{46}$$ $$+ ( -3 + 4 \beta ) q^{47}$$ $$+ q^{49}$$ $$+ ( -1 - \beta ) q^{50}$$ $$+ ( 3 - 3 \beta ) q^{52}$$ $$+ ( -5 + 5 \beta ) q^{53}$$ $$+ ( -3 + 2 \beta ) q^{55}$$ $$+ ( -1 - 4 \beta ) q^{56}$$ $$+ ( 1 - 2 \beta ) q^{58}$$ $$+ ( -1 + 8 \beta ) q^{59}$$ $$+ ( -2 - 3 \beta ) q^{61}$$ $$+ ( 5 + 9 \beta ) q^{62}$$ $$+ ( -1 + 6 \beta ) q^{64}$$ $$+ ( -2 + \beta ) q^{65}$$ $$+ ( -7 - 3 \beta ) q^{67}$$ $$+ ( -9 - 12 \beta ) q^{68}$$ $$+ ( -1 - \beta ) q^{70}$$ $$+ ( -6 - 5 \beta ) q^{71}$$ $$+ ( -3 + 8 \beta ) q^{73}$$ $$+ ( 5 + 13 \beta ) q^{74}$$ $$+ ( -3 - 9 \beta ) q^{76}$$ $$+ ( -3 + 2 \beta ) q^{77}$$ $$+ ( -3 + 11 \beta ) q^{79}$$ $$+ ( 5 + 3 \beta ) q^{80}$$ $$+ ( -2 - 11 \beta ) q^{82}$$ $$+ ( -11 + 2 \beta ) q^{83}$$ $$+ ( -1 - 3 \beta ) q^{85}$$ $$+ ( -1 - 3 \beta ) q^{86}$$ $$+ ( -5 + 2 \beta ) q^{88}$$ $$+ ( 5 - 6 \beta ) q^{89}$$ $$+ ( -2 + \beta ) q^{91}$$ $$+ ( -9 - 21 \beta ) q^{92}$$ $$+ ( -1 - 5 \beta ) q^{94}$$ $$+ ( -2 - \beta ) q^{95}$$ $$+ ( 5 - 5 \beta ) q^{97}$$ $$+ ( -1 - \beta ) q^{98}$$ $$+O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q$$ $$\mathstrut -\mathstrut 3q^{2}$$ $$\mathstrut +\mathstrut 3q^{4}$$ $$\mathstrut +\mathstrut 2q^{5}$$ $$\mathstrut +\mathstrut 2q^{7}$$ $$\mathstrut -\mathstrut 6q^{8}$$ $$\mathstrut +\mathstrut O(q^{10})$$ $$2q$$ $$\mathstrut -\mathstrut 3q^{2}$$ $$\mathstrut +\mathstrut 3q^{4}$$ $$\mathstrut +\mathstrut 2q^{5}$$ $$\mathstrut +\mathstrut 2q^{7}$$ $$\mathstrut -\mathstrut 6q^{8}$$ $$\mathstrut -\mathstrut 3q^{10}$$ $$\mathstrut -\mathstrut 4q^{11}$$ $$\mathstrut -\mathstrut 3q^{13}$$ $$\mathstrut -\mathstrut 3q^{14}$$ $$\mathstrut +\mathstrut 13q^{16}$$ $$\mathstrut -\mathstrut 5q^{17}$$ $$\mathstrut -\mathstrut 5q^{19}$$ $$\mathstrut +\mathstrut 3q^{20}$$ $$\mathstrut +\mathstrut q^{22}$$ $$\mathstrut -\mathstrut 11q^{23}$$ $$\mathstrut +\mathstrut 2q^{25}$$ $$\mathstrut +\mathstrut 2q^{26}$$ $$\mathstrut +\mathstrut 3q^{28}$$ $$\mathstrut -\mathstrut 5q^{29}$$ $$\mathstrut -\mathstrut 6q^{31}$$ $$\mathstrut -\mathstrut 15q^{32}$$ $$\mathstrut +\mathstrut 15q^{34}$$ $$\mathstrut +\mathstrut 2q^{35}$$ $$\mathstrut -\mathstrut 2q^{37}$$ $$\mathstrut +\mathstrut 10q^{38}$$ $$\mathstrut -\mathstrut 6q^{40}$$ $$\mathstrut -\mathstrut 5q^{41}$$ $$\mathstrut +\mathstrut 9q^{44}$$ $$\mathstrut +\mathstrut 24q^{46}$$ $$\mathstrut -\mathstrut 2q^{47}$$ $$\mathstrut +\mathstrut 2q^{49}$$ $$\mathstrut -\mathstrut 3q^{50}$$ $$\mathstrut +\mathstrut 3q^{52}$$ $$\mathstrut -\mathstrut 5q^{53}$$ $$\mathstrut -\mathstrut 4q^{55}$$ $$\mathstrut -\mathstrut 6q^{56}$$ $$\mathstrut +\mathstrut 6q^{59}$$ $$\mathstrut -\mathstrut 7q^{61}$$ $$\mathstrut +\mathstrut 19q^{62}$$ $$\mathstrut +\mathstrut 4q^{64}$$ $$\mathstrut -\mathstrut 3q^{65}$$ $$\mathstrut -\mathstrut 17q^{67}$$ $$\mathstrut -\mathstrut 30q^{68}$$ $$\mathstrut -\mathstrut 3q^{70}$$ $$\mathstrut -\mathstrut 17q^{71}$$ $$\mathstrut +\mathstrut 2q^{73}$$ $$\mathstrut +\mathstrut 23q^{74}$$ $$\mathstrut -\mathstrut 15q^{76}$$ $$\mathstrut -\mathstrut 4q^{77}$$ $$\mathstrut +\mathstrut 5q^{79}$$ $$\mathstrut +\mathstrut 13q^{80}$$ $$\mathstrut -\mathstrut 15q^{82}$$ $$\mathstrut -\mathstrut 20q^{83}$$ $$\mathstrut -\mathstrut 5q^{85}$$ $$\mathstrut -\mathstrut 5q^{86}$$ $$\mathstrut -\mathstrut 8q^{88}$$ $$\mathstrut +\mathstrut 4q^{89}$$ $$\mathstrut -\mathstrut 3q^{91}$$ $$\mathstrut -\mathstrut 39q^{92}$$ $$\mathstrut -\mathstrut 7q^{94}$$ $$\mathstrut -\mathstrut 5q^{95}$$ $$\mathstrut +\mathstrut 5q^{97}$$ $$\mathstrut -\mathstrut 3q^{98}$$ $$\mathstrut +\mathstrut O(q^{100})$$

## 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
−2.61803 0 4.85410 1.00000 0 1.00000 −7.47214 0 −2.61803
1.2 −0.381966 0 −1.85410 1.00000 0 1.00000 1.47214 0 −0.381966
 $$n$$: 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.

## Atkin-Lehner signs

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

## Hecke kernels

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

 $$T_{2}^{2}$$ $$\mathstrut +\mathstrut 3 T_{2}$$ $$\mathstrut +\mathstrut 1$$ $$T_{11}^{2}$$ $$\mathstrut +\mathstrut 4 T_{11}$$ $$\mathstrut -\mathstrut 1$$