# Properties

 Label 2695.2.a.h Level $2695$ Weight $2$ Character orbit 2695.a Self dual yes Analytic conductor $21.520$ Analytic rank $0$ 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: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.148.1 Defining polynomial: $$x^{3} - x^{2} - 3x + 1$$ 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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{2} - 1) q^{2} + (\beta_1 + 1) q^{3} + (\beta_{2} - \beta_1 + 2) q^{4} + q^{5} + ( - \beta_{2} - 2 \beta_1) q^{6} + (3 \beta_1 - 4) q^{8} + (\beta_{2} + 3 \beta_1) q^{9}+O(q^{10})$$ q + (-b2 - 1) * q^2 + (b1 + 1) * q^3 + (b2 - b1 + 2) * q^4 + q^5 + (-b2 - 2*b1) * q^6 + (3*b1 - 4) * q^8 + (b2 + 3*b1) * q^9 $$q + ( - \beta_{2} - 1) q^{2} + (\beta_1 + 1) q^{3} + (\beta_{2} - \beta_1 + 2) q^{4} + q^{5} + ( - \beta_{2} - 2 \beta_1) q^{6} + (3 \beta_1 - 4) q^{8} + (\beta_{2} + 3 \beta_1) q^{9} + ( - \beta_{2} - 1) q^{10} + q^{11} + (\beta_1 - 1) q^{12} + ( - \beta_1 + 3) q^{13} + (\beta_1 + 1) q^{15} + (2 \beta_{2} - 4 \beta_1 + 3) q^{16} + ( - \beta_1 + 5) q^{17} - 5 \beta_1 q^{18} + ( - 4 \beta_{2} - 2 \beta_1) q^{19} + (\beta_{2} - \beta_1 + 2) q^{20} + ( - \beta_{2} - 1) q^{22} + (\beta_{2} - 3 \beta_1 - 1) q^{23} + (3 \beta_{2} + 2 \beta_1 + 2) q^{24} + q^{25} + ( - 3 \beta_{2} + 2 \beta_1 - 4) q^{26} + (4 \beta_{2} + 4 \beta_1 + 2) q^{27} + ( - 2 \beta_{2} + 2 \beta_1) q^{29} + ( - \beta_{2} - 2 \beta_1) q^{30} + (3 \beta_{2} + 4 \beta_1) q^{31} + ( - 3 \beta_{2} + 4 \beta_1 - 5) q^{32} + (\beta_1 + 1) q^{33} + ( - 5 \beta_{2} + 2 \beta_1 - 6) q^{34} + ( - 2 \beta_{2} + 4 \beta_1 - 5) q^{36} + ( - \beta_{2} + 5 \beta_1 + 1) q^{37} + 10 q^{38} + ( - \beta_{2} + \beta_1 + 1) q^{39} + (3 \beta_1 - 4) q^{40} + (\beta_{2} - 2 \beta_1 + 4) q^{41} + ( - 3 \beta_{2} + \beta_1 - 1) q^{43} + (\beta_{2} - \beta_1 + 2) q^{44} + (\beta_{2} + 3 \beta_1) q^{45} + (\beta_{2} + 7 \beta_1 - 5) q^{46} + ( - \beta_1 + 9) q^{47} + ( - 2 \beta_{2} - 3 \beta_1 - 7) q^{48} + ( - \beta_{2} - 1) q^{50} + ( - \beta_{2} + 3 \beta_1 + 3) q^{51} + (4 \beta_{2} - 5 \beta_1 + 9) q^{52} + (\beta_{2} - 5 \beta_1 + 3) q^{53} + ( - 2 \beta_{2} - 4 \beta_1 - 10) q^{54} + q^{55} + ( - 6 \beta_{2} - 8 \beta_1) q^{57} + ( - 6 \beta_1 + 8) q^{58} + (3 \beta_{2} + 2 \beta_1 + 2) q^{59} + (\beta_1 - 1) q^{60} + (\beta_{2} + 2 \beta_1 + 2) q^{61} + ( - 5 \beta_1 - 5) q^{62} + (\beta_{2} - 3 \beta_1 + 12) q^{64} + ( - \beta_1 + 3) q^{65} + ( - \beta_{2} - 2 \beta_1) q^{66} + (5 \beta_{2} + 3 \beta_1 - 3) q^{67} + (6 \beta_{2} - 7 \beta_1 + 13) q^{68} + ( - 2 \beta_{2} - 6 \beta_1 - 8) q^{69} + ( - 2 \beta_{2} - 2 \beta_1 - 6) q^{71} + (5 \beta_{2} + 15) q^{72} + ( - 4 \beta_{2} - 3 \beta_1 + 3) q^{73} + ( - \beta_{2} - 11 \beta_1 + 7) q^{74} + (\beta_1 + 1) q^{75} + ( - 2 \beta_{2} + 4 \beta_1 - 10) q^{76} + ( - \beta_{2} - 3 \beta_1 + 3) q^{78} + ( - 5 \beta_{2} - 5 \beta_1 - 1) q^{79} + (2 \beta_{2} - 4 \beta_1 + 3) q^{80} + (5 \beta_{2} + 5 \beta_1 + 6) q^{81} + ( - 4 \beta_{2} + 5 \beta_1 - 9) q^{82} + ( - 6 \beta_{2} + 2 \beta_1 - 4) q^{83} + ( - \beta_1 + 5) q^{85} + (\beta_{2} - 5 \beta_1 + 11) q^{86} + (2 \beta_1 + 6) q^{87} + (3 \beta_1 - 4) q^{88} + ( - 2 \beta_{2} - 2 \beta_1 + 6) q^{89} - 5 \beta_1 q^{90} + (3 \beta_{2} - 7 \beta_1 + 11) q^{92} + (7 \beta_{2} + 11 \beta_1 + 5) q^{93} + ( - 9 \beta_{2} + 2 \beta_1 - 10) q^{94} + ( - 4 \beta_{2} - 2 \beta_1) q^{95} + (\beta_{2} + 6) q^{96} + 4 \beta_{2} q^{97} + (\beta_{2} + 3 \beta_1) q^{99}+O(q^{100})$$ q + (-b2 - 1) * q^2 + (b1 + 1) * q^3 + (b2 - b1 + 2) * q^4 + q^5 + (-b2 - 2*b1) * q^6 + (3*b1 - 4) * q^8 + (b2 + 3*b1) * q^9 + (-b2 - 1) * q^10 + q^11 + (b1 - 1) * q^12 + (-b1 + 3) * q^13 + (b1 + 1) * q^15 + (2*b2 - 4*b1 + 3) * q^16 + (-b1 + 5) * q^17 - 5*b1 * q^18 + (-4*b2 - 2*b1) * q^19 + (b2 - b1 + 2) * q^20 + (-b2 - 1) * q^22 + (b2 - 3*b1 - 1) * q^23 + (3*b2 + 2*b1 + 2) * q^24 + q^25 + (-3*b2 + 2*b1 - 4) * q^26 + (4*b2 + 4*b1 + 2) * q^27 + (-2*b2 + 2*b1) * q^29 + (-b2 - 2*b1) * q^30 + (3*b2 + 4*b1) * q^31 + (-3*b2 + 4*b1 - 5) * q^32 + (b1 + 1) * q^33 + (-5*b2 + 2*b1 - 6) * q^34 + (-2*b2 + 4*b1 - 5) * q^36 + (-b2 + 5*b1 + 1) * q^37 + 10 * q^38 + (-b2 + b1 + 1) * q^39 + (3*b1 - 4) * q^40 + (b2 - 2*b1 + 4) * q^41 + (-3*b2 + b1 - 1) * q^43 + (b2 - b1 + 2) * q^44 + (b2 + 3*b1) * q^45 + (b2 + 7*b1 - 5) * q^46 + (-b1 + 9) * q^47 + (-2*b2 - 3*b1 - 7) * q^48 + (-b2 - 1) * q^50 + (-b2 + 3*b1 + 3) * q^51 + (4*b2 - 5*b1 + 9) * q^52 + (b2 - 5*b1 + 3) * q^53 + (-2*b2 - 4*b1 - 10) * q^54 + q^55 + (-6*b2 - 8*b1) * q^57 + (-6*b1 + 8) * q^58 + (3*b2 + 2*b1 + 2) * q^59 + (b1 - 1) * q^60 + (b2 + 2*b1 + 2) * q^61 + (-5*b1 - 5) * q^62 + (b2 - 3*b1 + 12) * q^64 + (-b1 + 3) * q^65 + (-b2 - 2*b1) * q^66 + (5*b2 + 3*b1 - 3) * q^67 + (6*b2 - 7*b1 + 13) * q^68 + (-2*b2 - 6*b1 - 8) * q^69 + (-2*b2 - 2*b1 - 6) * q^71 + (5*b2 + 15) * q^72 + (-4*b2 - 3*b1 + 3) * q^73 + (-b2 - 11*b1 + 7) * q^74 + (b1 + 1) * q^75 + (-2*b2 + 4*b1 - 10) * q^76 + (-b2 - 3*b1 + 3) * q^78 + (-5*b2 - 5*b1 - 1) * q^79 + (2*b2 - 4*b1 + 3) * q^80 + (5*b2 + 5*b1 + 6) * q^81 + (-4*b2 + 5*b1 - 9) * q^82 + (-6*b2 + 2*b1 - 4) * q^83 + (-b1 + 5) * q^85 + (b2 - 5*b1 + 11) * q^86 + (2*b1 + 6) * q^87 + (3*b1 - 4) * q^88 + (-2*b2 - 2*b1 + 6) * q^89 - 5*b1 * q^90 + (3*b2 - 7*b1 + 11) * q^92 + (7*b2 + 11*b1 + 5) * q^93 + (-9*b2 + 2*b1 - 10) * q^94 + (-4*b2 - 2*b1) * q^95 + (b2 + 6) * q^96 + 4*b2 * q^97 + (b2 + 3*b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 3 q^{2} + 4 q^{3} + 5 q^{4} + 3 q^{5} - 2 q^{6} - 9 q^{8} + 3 q^{9}+O(q^{10})$$ 3 * q - 3 * q^2 + 4 * q^3 + 5 * q^4 + 3 * q^5 - 2 * q^6 - 9 * q^8 + 3 * q^9 $$3 q - 3 q^{2} + 4 q^{3} + 5 q^{4} + 3 q^{5} - 2 q^{6} - 9 q^{8} + 3 q^{9} - 3 q^{10} + 3 q^{11} - 2 q^{12} + 8 q^{13} + 4 q^{15} + 5 q^{16} + 14 q^{17} - 5 q^{18} - 2 q^{19} + 5 q^{20} - 3 q^{22} - 6 q^{23} + 8 q^{24} + 3 q^{25} - 10 q^{26} + 10 q^{27} + 2 q^{29} - 2 q^{30} + 4 q^{31} - 11 q^{32} + 4 q^{33} - 16 q^{34} - 11 q^{36} + 8 q^{37} + 30 q^{38} + 4 q^{39} - 9 q^{40} + 10 q^{41} - 2 q^{43} + 5 q^{44} + 3 q^{45} - 8 q^{46} + 26 q^{47} - 24 q^{48} - 3 q^{50} + 12 q^{51} + 22 q^{52} + 4 q^{53} - 34 q^{54} + 3 q^{55} - 8 q^{57} + 18 q^{58} + 8 q^{59} - 2 q^{60} + 8 q^{61} - 20 q^{62} + 33 q^{64} + 8 q^{65} - 2 q^{66} - 6 q^{67} + 32 q^{68} - 30 q^{69} - 20 q^{71} + 45 q^{72} + 6 q^{73} + 10 q^{74} + 4 q^{75} - 26 q^{76} + 6 q^{78} - 8 q^{79} + 5 q^{80} + 23 q^{81} - 22 q^{82} - 10 q^{83} + 14 q^{85} + 28 q^{86} + 20 q^{87} - 9 q^{88} + 16 q^{89} - 5 q^{90} + 26 q^{92} + 26 q^{93} - 28 q^{94} - 2 q^{95} + 18 q^{96} + 3 q^{99}+O(q^{100})$$ 3 * q - 3 * q^2 + 4 * q^3 + 5 * q^4 + 3 * q^5 - 2 * q^6 - 9 * q^8 + 3 * q^9 - 3 * q^10 + 3 * q^11 - 2 * q^12 + 8 * q^13 + 4 * q^15 + 5 * q^16 + 14 * q^17 - 5 * q^18 - 2 * q^19 + 5 * q^20 - 3 * q^22 - 6 * q^23 + 8 * q^24 + 3 * q^25 - 10 * q^26 + 10 * q^27 + 2 * q^29 - 2 * q^30 + 4 * q^31 - 11 * q^32 + 4 * q^33 - 16 * q^34 - 11 * q^36 + 8 * q^37 + 30 * q^38 + 4 * q^39 - 9 * q^40 + 10 * q^41 - 2 * q^43 + 5 * q^44 + 3 * q^45 - 8 * q^46 + 26 * q^47 - 24 * q^48 - 3 * q^50 + 12 * q^51 + 22 * q^52 + 4 * q^53 - 34 * q^54 + 3 * q^55 - 8 * q^57 + 18 * q^58 + 8 * q^59 - 2 * q^60 + 8 * q^61 - 20 * q^62 + 33 * q^64 + 8 * q^65 - 2 * q^66 - 6 * q^67 + 32 * q^68 - 30 * q^69 - 20 * q^71 + 45 * q^72 + 6 * q^73 + 10 * q^74 + 4 * q^75 - 26 * q^76 + 6 * q^78 - 8 * q^79 + 5 * q^80 + 23 * q^81 - 22 * q^82 - 10 * q^83 + 14 * q^85 + 28 * q^86 + 20 * q^87 - 9 * q^88 + 16 * q^89 - 5 * q^90 + 26 * q^92 + 26 * q^93 - 28 * q^94 - 2 * q^95 + 18 * q^96 + 3 * q^99

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

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 2$$ v^2 - v - 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta _1 + 2$$ b2 + b1 + 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 −0.481194 5.15633 1.00000 1.28726 0 −8.44358 −2.76845 −2.67513
1.2 −1.53919 3.17009 0.369102 1.00000 −4.87936 0 2.51026 7.04945 −1.53919
1.3 1.21432 1.31111 −0.525428 1.00000 1.59210 0 −3.06668 −1.28100 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.h 3
7.b odd 2 1 385.2.a.e 3
21.c even 2 1 3465.2.a.bi 3
28.d even 2 1 6160.2.a.bo 3
35.c odd 2 1 1925.2.a.w 3
35.f even 4 2 1925.2.b.m 6
77.b even 2 1 4235.2.a.p 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
385.2.a.e 3 7.b odd 2 1
1925.2.a.w 3 35.c odd 2 1
1925.2.b.m 6 35.f even 4 2
2695.2.a.h 3 1.a even 1 1 trivial
3465.2.a.bi 3 21.c even 2 1
4235.2.a.p 3 77.b even 2 1
6160.2.a.bo 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(2695))$$:

 $$T_{2}^{3} + 3T_{2}^{2} - T_{2} - 5$$ T2^3 + 3*T2^2 - T2 - 5 $$T_{3}^{3} - 4T_{3}^{2} + 2T_{3} + 2$$ T3^3 - 4*T3^2 + 2*T3 + 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3} + 3T^{2} - T - 5$$
$3$ $$T^{3} - 4 T^{2} + 2 T + 2$$
$5$ $$(T - 1)^{3}$$
$7$ $$T^{3}$$
$11$ $$(T - 1)^{3}$$
$13$ $$T^{3} - 8 T^{2} + 18 T - 10$$
$17$ $$T^{3} - 14 T^{2} + 62 T - 86$$
$19$ $$T^{3} + 2 T^{2} - 60 T - 200$$
$23$ $$T^{3} + 6 T^{2} - 28 T - 148$$
$29$ $$T^{3} - 2 T^{2} - 36 T + 104$$
$31$ $$T^{3} - 4 T^{2} - 60 T - 50$$
$37$ $$T^{3} - 8 T^{2} - 76 T + 436$$
$41$ $$T^{3} - 10 T^{2} + 12 T - 2$$
$43$ $$T^{3} + 2 T^{2} - 44 T - 20$$
$47$ $$T^{3} - 26 T^{2} + 222 T - 622$$
$53$ $$T^{3} - 4 T^{2} - 92 T - 68$$
$59$ $$T^{3} - 8 T^{2} - 16 T + 130$$
$61$ $$T^{3} - 8 T^{2} + 8 T - 2$$
$67$ $$T^{3} + 6 T^{2} - 88 T + 76$$
$71$ $$T^{3} + 20 T^{2} + 112 T + 160$$
$73$ $$T^{3} - 6 T^{2} - 58 T + 46$$
$79$ $$T^{3} + 8 T^{2} - 112 T - 244$$
$83$ $$T^{3} + 10 T^{2} - 148 T - 488$$
$89$ $$T^{3} - 16 T^{2} + 64 T - 32$$
$97$ $$T^{3} - 64T + 128$$