# Properties

 Label 945.2.a.m Level 945 Weight 2 Character orbit 945.a Self dual yes Analytic conductor 7.546 Analytic rank 0 Dimension 4 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: $$4$$ Coefficient field: 4.4.144344.1 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 a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{2} + ( 2 + \beta_{2} ) q^{4} - q^{5} + q^{7} + ( -1 - 2 \beta_{1} - \beta_{3} ) q^{8} +O(q^{10})$$ $$q -\beta_{1} q^{2} + ( 2 + \beta_{2} ) q^{4} - q^{5} + q^{7} + ( -1 - 2 \beta_{1} - \beta_{3} ) q^{8} + \beta_{1} q^{10} + ( 1 - \beta_{1} + \beta_{2} ) q^{11} + ( 1 - \beta_{1} - \beta_{2} ) q^{13} -\beta_{1} q^{14} + ( 4 + \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{16} + ( \beta_{1} - \beta_{2} + \beta_{3} ) q^{17} + ( 1 - \beta_{3} ) q^{19} + ( -2 - \beta_{2} ) q^{20} + ( 3 - 3 \beta_{1} + \beta_{2} - \beta_{3} ) q^{22} + ( \beta_{1} - \beta_{2} + \beta_{3} ) q^{23} + q^{25} + ( 5 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{26} + ( 2 + \beta_{2} ) q^{28} + ( 2 + 2 \beta_{2} ) q^{29} + ( 2 - \beta_{1} - \beta_{2} + \beta_{3} ) q^{31} + ( -4 - 4 \beta_{1} - 3 \beta_{2} - \beta_{3} ) q^{32} + ( -3 + 2 \beta_{1} - 3 \beta_{2} ) q^{34} - q^{35} + ( 2 + 2 \beta_{1} ) q^{37} + ( -\beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{38} + ( 1 + 2 \beta_{1} + \beta_{3} ) q^{40} + ( 1 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{41} + ( 5 - \beta_{1} - \beta_{2} + \beta_{3} ) q^{43} + ( 9 - 3 \beta_{1} + 3 \beta_{2} ) q^{44} + ( -3 + 2 \beta_{1} - 3 \beta_{2} ) q^{46} + ( -5 + 3 \beta_{1} - \beta_{2} - \beta_{3} ) q^{47} + q^{49} -\beta_{1} q^{50} + ( -7 - 5 \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{52} + ( 1 - 2 \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{53} + ( -1 + \beta_{1} - \beta_{2} ) q^{55} + ( -1 - 2 \beta_{1} - \beta_{3} ) q^{56} + ( -2 - 6 \beta_{1} - 2 \beta_{3} ) q^{58} + 2 \beta_{1} q^{59} + ( 2 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{61} + ( 5 - \beta_{2} ) q^{62} + ( 11 + 8 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{64} + ( -1 + \beta_{1} + \beta_{2} ) q^{65} + ( 3 + \beta_{1} - 3 \beta_{2} + \beta_{3} ) q^{67} + ( -5 + 7 \beta_{1} + \beta_{3} ) q^{68} + \beta_{1} q^{70} + ( 2 + 4 \beta_{1} - 4 \beta_{2} ) q^{71} + ( -3 + \beta_{1} + \beta_{2} ) q^{73} + ( -8 - 2 \beta_{1} - 2 \beta_{2} ) q^{74} + ( -4 \beta_{1} - \beta_{2} - \beta_{3} ) q^{76} + ( 1 - \beta_{1} + \beta_{2} ) q^{77} + ( 2 + 3 \beta_{1} + \beta_{2} - \beta_{3} ) q^{79} + ( -4 - \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{80} + ( 5 + \beta_{1} + 3 \beta_{2} + 2 \beta_{3} ) q^{82} + ( -7 + 2 \beta_{1} + 2 \beta_{2} ) q^{83} + ( -\beta_{1} + \beta_{2} - \beta_{3} ) q^{85} + ( 5 - 3 \beta_{1} - \beta_{2} ) q^{86} + ( 3 - 9 \beta_{1} + \beta_{2} - \beta_{3} ) q^{88} + ( 1 + 5 \beta_{1} - \beta_{2} - \beta_{3} ) q^{89} + ( 1 - \beta_{1} - \beta_{2} ) q^{91} + ( -5 + 7 \beta_{1} + \beta_{3} ) q^{92} + ( -11 + 7 \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{94} + ( -1 + \beta_{3} ) q^{95} + ( 2 + 4 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{97} -\beta_{1} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - q^{2} + 9q^{4} - 4q^{5} + 4q^{7} - 6q^{8} + O(q^{10})$$ $$4q - q^{2} + 9q^{4} - 4q^{5} + 4q^{7} - 6q^{8} + q^{10} + 4q^{11} + 2q^{13} - q^{14} + 19q^{16} + 4q^{19} - 9q^{20} + 10q^{22} + 4q^{25} + 22q^{26} + 9q^{28} + 10q^{29} + 6q^{31} - 23q^{32} - 13q^{34} - 4q^{35} + 10q^{37} + q^{38} + 6q^{40} + 2q^{41} + 18q^{43} + 36q^{44} - 13q^{46} - 18q^{47} + 4q^{49} - q^{50} - 34q^{52} + 4q^{53} - 4q^{55} - 6q^{56} - 14q^{58} + 2q^{59} + 8q^{61} + 19q^{62} + 54q^{64} - 2q^{65} + 10q^{67} - 13q^{68} + q^{70} + 8q^{71} - 10q^{73} - 36q^{74} - 5q^{76} + 4q^{77} + 12q^{79} - 19q^{80} + 24q^{82} - 24q^{83} + 16q^{86} + 4q^{88} + 8q^{89} + 2q^{91} - 13q^{92} - 38q^{94} - 4q^{95} + 10q^{97} - q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 8 x^{2} + 5 x + 9$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 4$$ $$\beta_{3}$$ $$=$$ $$\nu^{3} - 6 \nu - 1$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 4$$ $$\nu^{3}$$ $$=$$ $$\beta_{3} + 6 \beta_{1} + 1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 2.80834 1.51533 −0.857589 −2.46608
−2.80834 0 5.88678 −1.00000 0 1.00000 −10.9154 0 2.80834
1.2 −1.51533 0 0.296215 −1.00000 0 1.00000 2.58179 0 1.51533
1.3 0.857589 0 −1.26454 −1.00000 0 1.00000 −2.79964 0 −0.857589
1.4 2.46608 0 4.08154 −1.00000 0 1.00000 5.13325 0 −2.46608
 $$n$$: 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.m 4
3.b odd 2 1 945.2.a.n yes 4
5.b even 2 1 4725.2.a.bx 4
7.b odd 2 1 6615.2.a.be 4
15.d odd 2 1 4725.2.a.bo 4
21.c even 2 1 6615.2.a.bh 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
945.2.a.m 4 1.a even 1 1 trivial
945.2.a.n yes 4 3.b odd 2 1
4725.2.a.bo 4 15.d odd 2 1
4725.2.a.bx 4 5.b even 2 1
6615.2.a.be 4 7.b odd 2 1
6615.2.a.bh 4 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(945))$$:

 $$T_{2}^{4} + T_{2}^{3} - 8 T_{2}^{2} - 5 T_{2} + 9$$ $$T_{11}^{4} - 4 T_{11}^{3} - 13 T_{11}^{2} + 18 T_{11} + 36$$