# Properties

 Label 2695.2.a.g Level $2695$ Weight $2$ Character orbit 2695.a Self dual yes Analytic conductor $21.520$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2695 = 5 \cdot 7^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2695.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$21.5196833447$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.148.1 Defining polynomial: $$x^{3} - x^{2} - 3 x + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 385) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -1 - \beta_{2} ) q^{2} + ( 1 - \beta_{1} ) q^{3} + ( 2 - \beta_{1} + \beta_{2} ) q^{4} + q^{5} + ( -2 + 2 \beta_{1} - \beta_{2} ) q^{6} + ( -4 + 3 \beta_{1} ) q^{8} + ( -\beta_{1} + \beta_{2} ) q^{9} +O(q^{10})$$ $$q + ( -1 - \beta_{2} ) q^{2} + ( 1 - \beta_{1} ) q^{3} + ( 2 - \beta_{1} + \beta_{2} ) q^{4} + q^{5} + ( -2 + 2 \beta_{1} - \beta_{2} ) q^{6} + ( -4 + 3 \beta_{1} ) q^{8} + ( -\beta_{1} + \beta_{2} ) q^{9} + ( -1 - \beta_{2} ) q^{10} - q^{11} + ( 5 - 3 \beta_{1} + 2 \beta_{2} ) q^{12} + ( -1 + \beta_{1} - 2 \beta_{2} ) q^{13} + ( 1 - \beta_{1} ) q^{15} + ( 3 - 4 \beta_{1} + 2 \beta_{2} ) q^{16} + ( -1 + 3 \beta_{1} ) q^{17} + ( -4 + 3 \beta_{1} ) q^{18} + ( -2 + 2 \beta_{2} ) q^{19} + ( 2 - \beta_{1} + \beta_{2} ) q^{20} + ( 1 + \beta_{2} ) q^{22} + ( -3 - \beta_{1} - \beta_{2} ) q^{23} + ( -10 + 4 \beta_{1} - 3 \beta_{2} ) q^{24} + q^{25} + ( 8 - 4 \beta_{1} + \beta_{2} ) q^{26} + ( 2 \beta_{1} + 2 \beta_{2} ) q^{27} + ( -4 + 2 \beta_{1} + 2 \beta_{2} ) q^{29} + ( -2 + 2 \beta_{1} - \beta_{2} ) q^{30} + ( 4 - 2 \beta_{1} - \beta_{2} ) q^{31} + ( -5 + 4 \beta_{1} - 3 \beta_{2} ) q^{32} + ( -1 + \beta_{1} ) q^{33} + ( 4 - 6 \beta_{1} + \beta_{2} ) q^{34} + ( 7 - 4 \beta_{1} + 2 \beta_{2} ) q^{36} + ( -5 - \beta_{1} - 3 \beta_{2} ) q^{37} + ( -4 + 2 \beta_{1} + 2 \beta_{2} ) q^{38} + ( -5 + 3 \beta_{1} - 3 \beta_{2} ) q^{39} + ( -4 + 3 \beta_{1} ) q^{40} + 3 \beta_{2} q^{41} + ( -1 + \beta_{1} + 5 \beta_{2} ) q^{43} + ( -2 + \beta_{1} - \beta_{2} ) q^{44} + ( -\beta_{1} + \beta_{2} ) q^{45} + ( 5 + \beta_{1} + 3 \beta_{2} ) q^{46} + ( 7 - \beta_{1} + 2 \beta_{2} ) q^{47} + ( 13 - 5 \beta_{1} + 6 \beta_{2} ) q^{48} + ( -1 - \beta_{2} ) q^{50} + ( -7 + \beta_{1} - 3 \beta_{2} ) q^{51} + ( -13 + 7 \beta_{1} - 4 \beta_{2} ) q^{52} + ( -3 - 3 \beta_{1} - \beta_{2} ) q^{53} + ( -4 - 2 \beta_{1} ) q^{54} - q^{55} + 2 \beta_{2} q^{57} + ( -2 \beta_{1} + 4 \beta_{2} ) q^{58} + ( -6 + 4 \beta_{1} - \beta_{2} ) q^{59} + ( 5 - 3 \beta_{1} + 2 \beta_{2} ) q^{60} + ( -4 + 2 \beta_{1} + 5 \beta_{2} ) q^{61} + ( -3 + 3 \beta_{1} - 4 \beta_{2} ) q^{62} + ( 12 - 3 \beta_{1} + \beta_{2} ) q^{64} + ( -1 + \beta_{1} - 2 \beta_{2} ) q^{65} + ( 2 - 2 \beta_{1} + \beta_{2} ) q^{66} + ( -1 + \beta_{1} - 5 \beta_{2} ) q^{67} + ( -11 + 7 \beta_{1} - 4 \beta_{2} ) q^{68} + ( -2 + 4 \beta_{1} ) q^{69} + ( -10 + 6 \beta_{1} + 2 \beta_{2} ) q^{71} + ( -9 + 4 \beta_{1} - 7 \beta_{2} ) q^{72} + ( 1 - 7 \beta_{1} ) q^{73} + ( 13 - \beta_{1} + 5 \beta_{2} ) q^{74} + ( 1 - \beta_{1} ) q^{75} + ( 4 - 2 \beta_{1} ) q^{76} + ( 17 - 9 \beta_{1} + 5 \beta_{2} ) q^{78} + ( 1 + 5 \beta_{1} + 5 \beta_{2} ) q^{79} + ( 3 - 4 \beta_{1} + 2 \beta_{2} ) q^{80} + ( -2 + \beta_{1} - 3 \beta_{2} ) q^{81} + ( -9 + 3 \beta_{1} ) q^{82} + ( 2 + 4 \beta_{1} - 4 \beta_{2} ) q^{83} + ( -1 + 3 \beta_{1} ) q^{85} + ( -13 + 3 \beta_{1} + \beta_{2} ) q^{86} + ( -6 + 2 \beta_{1} ) q^{87} + ( 4 - 3 \beta_{1} ) q^{88} + ( -8 + 4 \beta_{1} + 4 \beta_{2} ) q^{89} + ( -4 + 3 \beta_{1} ) q^{90} + ( -7 + 3 \beta_{1} - 3 \beta_{2} ) q^{92} + ( 7 - 3 \beta_{1} + \beta_{2} ) q^{93} + ( -14 + 4 \beta_{1} - 7 \beta_{2} ) q^{94} + ( -2 + 2 \beta_{2} ) q^{95} + ( -16 + 8 \beta_{1} - 7 \beta_{2} ) q^{96} + ( 2 - 6 \beta_{1} - 6 \beta_{2} ) q^{97} + ( \beta_{1} - \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q - 3q^{2} + 2q^{3} + 5q^{4} + 3q^{5} - 4q^{6} - 9q^{8} - q^{9} + O(q^{10})$$ $$3q - 3q^{2} + 2q^{3} + 5q^{4} + 3q^{5} - 4q^{6} - 9q^{8} - q^{9} - 3q^{10} - 3q^{11} + 12q^{12} - 2q^{13} + 2q^{15} + 5q^{16} - 9q^{18} - 6q^{19} + 5q^{20} + 3q^{22} - 10q^{23} - 26q^{24} + 3q^{25} + 20q^{26} + 2q^{27} - 10q^{29} - 4q^{30} + 10q^{31} - 11q^{32} - 2q^{33} + 6q^{34} + 17q^{36} - 16q^{37} - 10q^{38} - 12q^{39} - 9q^{40} - 2q^{43} - 5q^{44} - q^{45} + 16q^{46} + 20q^{47} + 34q^{48} - 3q^{50} - 20q^{51} - 32q^{52} - 12q^{53} - 14q^{54} - 3q^{55} - 2q^{58} - 14q^{59} + 12q^{60} - 10q^{61} - 6q^{62} + 33q^{64} - 2q^{65} + 4q^{66} - 2q^{67} - 26q^{68} - 2q^{69} - 24q^{71} - 23q^{72} - 4q^{73} + 38q^{74} + 2q^{75} + 10q^{76} + 42q^{78} + 8q^{79} + 5q^{80} - 5q^{81} - 24q^{82} + 10q^{83} - 36q^{86} - 16q^{87} + 9q^{88} - 20q^{89} - 9q^{90} - 18q^{92} + 18q^{93} - 38q^{94} - 6q^{95} - 40q^{96} + q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 3 x + 1$$:

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

## 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
 −1.48119 2.17009 0.311108
−2.67513 2.48119 5.15633 1.00000 −6.63752 0 −8.44358 3.15633 −2.67513
1.2 −1.53919 −1.17009 0.369102 1.00000 1.80098 0 2.51026 −1.63090 −1.53919
1.3 1.21432 0.688892 −0.525428 1.00000 0.836535 0 −3.06668 −2.52543 1.21432
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$5$$ $$-1$$
$$7$$ $$-1$$
$$11$$ $$1$$

## 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 2695.2.a.g 3
7.b odd 2 1 385.2.a.f 3
21.c even 2 1 3465.2.a.bh 3
28.d even 2 1 6160.2.a.bn 3
35.c odd 2 1 1925.2.a.v 3
35.f even 4 2 1925.2.b.n 6
77.b even 2 1 4235.2.a.q 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
385.2.a.f 3 7.b odd 2 1
1925.2.a.v 3 35.c odd 2 1
1925.2.b.n 6 35.f even 4 2
2695.2.a.g 3 1.a even 1 1 trivial
3465.2.a.bh 3 21.c even 2 1
4235.2.a.q 3 77.b even 2 1
6160.2.a.bn 3 28.d even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(2695))$$:

 $$T_{2}^{3} + 3 T_{2}^{2} - T_{2} - 5$$ $$T_{3}^{3} - 2 T_{3}^{2} - 2 T_{3} + 2$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-5 - T + 3 T^{2} + T^{3}$$
$3$ $$2 - 2 T - 2 T^{2} + T^{3}$$
$5$ $$( -1 + T )^{3}$$
$7$ $$T^{3}$$
$11$ $$( 1 + T )^{3}$$
$13$ $$2 - 22 T + 2 T^{2} + T^{3}$$
$17$ $$-2 - 30 T + T^{3}$$
$19$ $$-8 - 4 T + 6 T^{2} + T^{3}$$
$23$ $$20 + 28 T + 10 T^{2} + T^{3}$$
$29$ $$-40 + 12 T + 10 T^{2} + T^{3}$$
$31$ $$26 + 20 T - 10 T^{2} + T^{3}$$
$37$ $$-100 + 52 T + 16 T^{2} + T^{3}$$
$41$ $$54 - 36 T + T^{3}$$
$43$ $$268 - 92 T + 2 T^{2} + T^{3}$$
$47$ $$-158 + 110 T - 20 T^{2} + T^{3}$$
$53$ $$4 + 20 T + 12 T^{2} + T^{3}$$
$59$ $$-74 + 14 T^{2} + T^{3}$$
$61$ $$62 - 60 T + 10 T^{2} + T^{3}$$
$67$ $$-172 - 112 T + 2 T^{2} + T^{3}$$
$71$ $$-800 + 80 T + 24 T^{2} + T^{3}$$
$73$ $$-190 - 158 T + 4 T^{2} + T^{3}$$
$79$ $$244 - 112 T - 8 T^{2} + T^{3}$$
$83$ $$1096 - 116 T - 10 T^{2} + T^{3}$$
$89$ $$-320 + 48 T + 20 T^{2} + T^{3}$$
$97$ $$160 - 192 T + T^{3}$$