# Properties

 Label 2695.2.a.l Level $2695$ Weight $2$ Character orbit 2695.a Self dual yes Analytic conductor $21.520$ Analytic rank $0$ Dimension $4$ 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: $$4$$ Coefficient field: 4.4.11348.1 Defining polynomial: $$x^{4} - x^{3} - 5x^{2} + x + 2$$ x^4 - x^3 - 5*x^2 + x + 2 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2$$ 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 $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_{2} + 1) q^{2} + ( - \beta_{3} - 1) q^{3} + ( - \beta_{3} + \beta_{2} + 2) q^{4} - q^{5} + (\beta_{2} - \beta_1) q^{6} + ( - \beta_{3} + 2 \beta_{2} - \beta_1 + 4) q^{8} + (\beta_{3} - \beta_{2} + 2) q^{9}+O(q^{10})$$ q + (b2 + 1) * q^2 + (-b3 - 1) * q^3 + (-b3 + b2 + 2) * q^4 - q^5 + (b2 - b1) * q^6 + (-b3 + 2*b2 - b1 + 4) * q^8 + (b3 - b2 + 2) * q^9 $$q + (\beta_{2} + 1) q^{2} + ( - \beta_{3} - 1) q^{3} + ( - \beta_{3} + \beta_{2} + 2) q^{4} - q^{5} + (\beta_{2} - \beta_1) q^{6} + ( - \beta_{3} + 2 \beta_{2} - \beta_1 + 4) q^{8} + (\beta_{3} - \beta_{2} + 2) q^{9} + ( - \beta_{2} - 1) q^{10} - q^{11} + ( - \beta_{3} - \beta_1 + 3) q^{12} + ( - \beta_{3} + \beta_1 + 1) q^{13} + (\beta_{3} + 1) q^{15} + ( - 2 \beta_{3} + 4 \beta_{2} - 2 \beta_1 + 5) q^{16} + ( - \beta_{3} + 1) q^{17} + (\beta_{3} + \beta_1 - 2) q^{18} + (2 \beta_{2} - \beta_1) q^{19} + (\beta_{3} - \beta_{2} - 2) q^{20} + ( - \beta_{2} - 1) q^{22} + ( - \beta_{3} - \beta_{2} - \beta_1 + 3) q^{23} + ( - 2 \beta_{3} + 3 \beta_{2} + 2) q^{24} + q^{25} + (2 \beta_{3} + 3 \beta_{2} + 4) q^{26} + (\beta_1 - 4) q^{27} + (2 \beta_1 - 2) q^{29} + ( - \beta_{2} + \beta_1) q^{30} + ( - 3 \beta_{2} - 4) q^{31} + ( - 6 \beta_{3} + 5 \beta_{2} - 2 \beta_1 + 7) q^{32} + (\beta_{3} + 1) q^{33} + (3 \beta_{2} - \beta_1 + 2) q^{34} + ( - 2 \beta_{2} + 2 \beta_1 - 5) q^{36} + (\beta_{3} - \beta_{2} + \beta_1 - 1) q^{37} + ( - 4 \beta_{3} - \beta_1 + 4) q^{38} + ( - \beta_{3} - 3 \beta_{2} - 2 \beta_1 + 3) q^{39} + (\beta_{3} - 2 \beta_{2} + \beta_1 - 4) q^{40} + (2 \beta_{3} + \beta_{2} + 4) q^{41} + ( - \beta_{3} - \beta_{2} + \beta_1 - 5) q^{43} + (\beta_{3} - \beta_{2} - 2) q^{44} + ( - \beta_{3} + \beta_{2} - 2) q^{45} + ( - \beta_{3} + 5 \beta_{2} - 2 \beta_1 - 1) q^{46} + ( - 3 \beta_{3} + 4 \beta_{2} - \beta_1 - 3) q^{47} + ( - \beta_{3} + 6 \beta_{2} + 7) q^{48} + (\beta_{2} + 1) q^{50} + ( - \beta_{3} - \beta_{2} + 3) q^{51} + ( - \beta_{3} + 9) q^{52} + ( - \beta_{3} - 3 \beta_{2} + 3 \beta_1 - 3) q^{53} + (2 \beta_{3} - 4 \beta_{2} + \beta_1 - 2) q^{54} + q^{55} + (2 \beta_{3} + 4 \beta_{2} + 2) q^{57} + (4 \beta_{3} - 2 \beta_{2} + 2 \beta_1 + 2) q^{58} + (2 \beta_{3} + \beta_{2} + 2 \beta_1 - 6) q^{59} + (\beta_{3} + \beta_1 - 3) q^{60} + ( - \beta_{2} + \beta_1 + 2) q^{61} + (3 \beta_{3} - 4 \beta_{2} - 13) q^{62} + ( - 5 \beta_{3} + 11 \beta_{2} - 4 \beta_1 + 14) q^{64} + (\beta_{3} - \beta_1 - 1) q^{65} + ( - \beta_{2} + \beta_1) q^{66} + ( - \beta_{3} + 5 \beta_{2} + \beta_1 - 1) q^{67} + ( - 3 \beta_{3} + 2 \beta_{2} - \beta_1 + 7) q^{68} + ( - 4 \beta_{3} + 3 \beta_1) q^{69} + 8 q^{71} + (4 \beta_{3} - 5 \beta_{2} - 3) q^{72} + (\beta_{3} + 2 \beta_1 + 3) q^{73} + (3 \beta_{3} - 3 \beta_{2} + 2 \beta_1 - 3) q^{74} + ( - \beta_{3} - 1) q^{75} + ( - 2 \beta_{3} + 8 \beta_{2} - 3 \beta_1 + 6) q^{76} + ( - \beta_{3} + 5 \beta_{2} - 3 \beta_1 - 9) q^{78} + ( - \beta_{3} - \beta_{2} + 2 \beta_1 + 3) q^{79} + (2 \beta_{3} - 4 \beta_{2} + 2 \beta_1 - 5) q^{80} + (\beta_{3} + \beta_{2} - 2 \beta_1 - 2) q^{81} + ( - \beta_{3} + 2 \beta_1 + 5) q^{82} + (2 \beta_{3} - 2 \beta_{2} + \beta_1 + 2) q^{83} + (\beta_{3} - 1) q^{85} + (3 \beta_{3} - 3 \beta_{2} - 5) q^{86} + (2 \beta_{3} - 4 \beta_{2} - 4 \beta_1 + 2) q^{87} + (\beta_{3} - 2 \beta_{2} + \beta_1 - 4) q^{88} + (\beta_1 + 2) q^{89} + ( - \beta_{3} - \beta_1 + 2) q^{90} + ( - 7 \beta_{3} + 3 \beta_{2} - \beta_1 + 5) q^{92} + (\beta_{3} - 3 \beta_{2} + 3 \beta_1 + 1) q^{93} + ( - 6 \beta_{3} + 3 \beta_{2} - 4 \beta_1 + 10) q^{94} + ( - 2 \beta_{2} + \beta_1) q^{95} + ( - 2 \beta_{3} + 3 \beta_{2} - \beta_1 + 22) q^{96} + ( - 3 \beta_1 + 2) q^{97} + ( - \beta_{3} + \beta_{2} - 2) q^{99}+O(q^{100})$$ q + (b2 + 1) * q^2 + (-b3 - 1) * q^3 + (-b3 + b2 + 2) * q^4 - q^5 + (b2 - b1) * q^6 + (-b3 + 2*b2 - b1 + 4) * q^8 + (b3 - b2 + 2) * q^9 + (-b2 - 1) * q^10 - q^11 + (-b3 - b1 + 3) * q^12 + (-b3 + b1 + 1) * q^13 + (b3 + 1) * q^15 + (-2*b3 + 4*b2 - 2*b1 + 5) * q^16 + (-b3 + 1) * q^17 + (b3 + b1 - 2) * q^18 + (2*b2 - b1) * q^19 + (b3 - b2 - 2) * q^20 + (-b2 - 1) * q^22 + (-b3 - b2 - b1 + 3) * q^23 + (-2*b3 + 3*b2 + 2) * q^24 + q^25 + (2*b3 + 3*b2 + 4) * q^26 + (b1 - 4) * q^27 + (2*b1 - 2) * q^29 + (-b2 + b1) * q^30 + (-3*b2 - 4) * q^31 + (-6*b3 + 5*b2 - 2*b1 + 7) * q^32 + (b3 + 1) * q^33 + (3*b2 - b1 + 2) * q^34 + (-2*b2 + 2*b1 - 5) * q^36 + (b3 - b2 + b1 - 1) * q^37 + (-4*b3 - b1 + 4) * q^38 + (-b3 - 3*b2 - 2*b1 + 3) * q^39 + (b3 - 2*b2 + b1 - 4) * q^40 + (2*b3 + b2 + 4) * q^41 + (-b3 - b2 + b1 - 5) * q^43 + (b3 - b2 - 2) * q^44 + (-b3 + b2 - 2) * q^45 + (-b3 + 5*b2 - 2*b1 - 1) * q^46 + (-3*b3 + 4*b2 - b1 - 3) * q^47 + (-b3 + 6*b2 + 7) * q^48 + (b2 + 1) * q^50 + (-b3 - b2 + 3) * q^51 + (-b3 + 9) * q^52 + (-b3 - 3*b2 + 3*b1 - 3) * q^53 + (2*b3 - 4*b2 + b1 - 2) * q^54 + q^55 + (2*b3 + 4*b2 + 2) * q^57 + (4*b3 - 2*b2 + 2*b1 + 2) * q^58 + (2*b3 + b2 + 2*b1 - 6) * q^59 + (b3 + b1 - 3) * q^60 + (-b2 + b1 + 2) * q^61 + (3*b3 - 4*b2 - 13) * q^62 + (-5*b3 + 11*b2 - 4*b1 + 14) * q^64 + (b3 - b1 - 1) * q^65 + (-b2 + b1) * q^66 + (-b3 + 5*b2 + b1 - 1) * q^67 + (-3*b3 + 2*b2 - b1 + 7) * q^68 + (-4*b3 + 3*b1) * q^69 + 8 * q^71 + (4*b3 - 5*b2 - 3) * q^72 + (b3 + 2*b1 + 3) * q^73 + (3*b3 - 3*b2 + 2*b1 - 3) * q^74 + (-b3 - 1) * q^75 + (-2*b3 + 8*b2 - 3*b1 + 6) * q^76 + (-b3 + 5*b2 - 3*b1 - 9) * q^78 + (-b3 - b2 + 2*b1 + 3) * q^79 + (2*b3 - 4*b2 + 2*b1 - 5) * q^80 + (b3 + b2 - 2*b1 - 2) * q^81 + (-b3 + 2*b1 + 5) * q^82 + (2*b3 - 2*b2 + b1 + 2) * q^83 + (b3 - 1) * q^85 + (3*b3 - 3*b2 - 5) * q^86 + (2*b3 - 4*b2 - 4*b1 + 2) * q^87 + (b3 - 2*b2 + b1 - 4) * q^88 + (b1 + 2) * q^89 + (-b3 - b1 + 2) * q^90 + (-7*b3 + 3*b2 - b1 + 5) * q^92 + (b3 - 3*b2 + 3*b1 + 1) * q^93 + (-6*b3 + 3*b2 - 4*b1 + 10) * q^94 + (-2*b2 + b1) * q^95 + (-2*b3 + 3*b2 - b1 + 22) * q^96 + (-3*b1 + 2) * q^97 + (-b3 + b2 - 2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{2} - 2 q^{3} + 8 q^{4} - 4 q^{5} - 4 q^{6} + 12 q^{8} + 8 q^{9}+O(q^{10})$$ 4 * q + 2 * q^2 - 2 * q^3 + 8 * q^4 - 4 * q^5 - 4 * q^6 + 12 * q^8 + 8 * q^9 $$4 q + 2 q^{2} - 2 q^{3} + 8 q^{4} - 4 q^{5} - 4 q^{6} + 12 q^{8} + 8 q^{9} - 2 q^{10} - 4 q^{11} + 12 q^{12} + 8 q^{13} + 2 q^{15} + 12 q^{16} + 6 q^{17} - 8 q^{18} - 6 q^{19} - 8 q^{20} - 2 q^{22} + 14 q^{23} + 6 q^{24} + 4 q^{25} + 6 q^{26} - 14 q^{27} - 4 q^{29} + 4 q^{30} - 10 q^{31} + 26 q^{32} + 2 q^{33} - 12 q^{36} - 2 q^{37} + 22 q^{38} + 16 q^{39} - 12 q^{40} + 10 q^{41} - 14 q^{43} - 8 q^{44} - 8 q^{45} - 16 q^{46} - 16 q^{47} + 18 q^{48} + 2 q^{50} + 16 q^{51} + 38 q^{52} + 2 q^{53} - 2 q^{54} + 4 q^{55} - 4 q^{57} + 8 q^{58} - 26 q^{59} - 12 q^{60} + 12 q^{61} - 50 q^{62} + 36 q^{64} - 8 q^{65} + 4 q^{66} - 10 q^{67} + 28 q^{68} + 14 q^{69} + 32 q^{71} - 10 q^{72} + 14 q^{73} - 8 q^{74} - 2 q^{75} + 6 q^{76} - 50 q^{78} + 20 q^{79} - 12 q^{80} - 16 q^{81} + 26 q^{82} + 10 q^{83} - 6 q^{85} - 20 q^{86} + 4 q^{87} - 12 q^{88} + 10 q^{89} + 8 q^{90} + 26 q^{92} + 14 q^{93} + 38 q^{94} + 6 q^{95} + 84 q^{96} + 2 q^{97} - 8 q^{99}+O(q^{100})$$ 4 * q + 2 * q^2 - 2 * q^3 + 8 * q^4 - 4 * q^5 - 4 * q^6 + 12 * q^8 + 8 * q^9 - 2 * q^10 - 4 * q^11 + 12 * q^12 + 8 * q^13 + 2 * q^15 + 12 * q^16 + 6 * q^17 - 8 * q^18 - 6 * q^19 - 8 * q^20 - 2 * q^22 + 14 * q^23 + 6 * q^24 + 4 * q^25 + 6 * q^26 - 14 * q^27 - 4 * q^29 + 4 * q^30 - 10 * q^31 + 26 * q^32 + 2 * q^33 - 12 * q^36 - 2 * q^37 + 22 * q^38 + 16 * q^39 - 12 * q^40 + 10 * q^41 - 14 * q^43 - 8 * q^44 - 8 * q^45 - 16 * q^46 - 16 * q^47 + 18 * q^48 + 2 * q^50 + 16 * q^51 + 38 * q^52 + 2 * q^53 - 2 * q^54 + 4 * q^55 - 4 * q^57 + 8 * q^58 - 26 * q^59 - 12 * q^60 + 12 * q^61 - 50 * q^62 + 36 * q^64 - 8 * q^65 + 4 * q^66 - 10 * q^67 + 28 * q^68 + 14 * q^69 + 32 * q^71 - 10 * q^72 + 14 * q^73 - 8 * q^74 - 2 * q^75 + 6 * q^76 - 50 * q^78 + 20 * q^79 - 12 * q^80 - 16 * q^81 + 26 * q^82 + 10 * q^83 - 6 * q^85 - 20 * q^86 + 4 * q^87 - 12 * q^88 + 10 * q^89 + 8 * q^90 + 26 * q^92 + 14 * q^93 + 38 * q^94 + 6 * q^95 + 84 * q^96 + 2 * q^97 - 8 * q^99

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

 $$\beta_{1}$$ $$=$$ $$2\nu$$ 2*v $$\beta_{2}$$ $$=$$ $$\nu^{3} - \nu^{2} - 4\nu$$ v^3 - v^2 - 4*v $$\beta_{3}$$ $$=$$ $$\nu^{2} - \nu - 3$$ v^2 - v - 3
 $$\nu$$ $$=$$ $$( \beta_1 ) / 2$$ (b1) / 2 $$\nu^{2}$$ $$=$$ $$( 2\beta_{3} + \beta _1 + 6 ) / 2$$ (2*b3 + b1 + 6) / 2 $$\nu^{3}$$ $$=$$ $$( 2\beta_{3} + 2\beta_{2} + 5\beta _1 + 6 ) / 2$$ (2*b3 + 2*b2 + 5*b1 + 6) / 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
 0.723742 −1.77571 2.64119 −0.589216
−2.03967 2.19994 2.16027 −1.00000 −4.48716 0 −0.326891 1.83973 2.03967
1.2 −0.649405 −2.92887 −1.57827 −1.00000 1.90202 0 2.32375 5.57827 0.649405
1.3 1.88395 −2.33468 1.54927 −1.00000 −4.39842 0 −0.849150 2.45073 −1.88395
1.4 2.80513 1.06361 5.86874 −1.00000 2.98356 0 10.8523 −1.86874 −2.80513
 $$n$$: 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.l 4
7.b odd 2 1 385.2.a.h 4
21.c even 2 1 3465.2.a.bk 4
28.d even 2 1 6160.2.a.br 4
35.c odd 2 1 1925.2.a.x 4
35.f even 4 2 1925.2.b.p 8
77.b even 2 1 4235.2.a.r 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
385.2.a.h 4 7.b odd 2 1
1925.2.a.x 4 35.c odd 2 1
1925.2.b.p 8 35.f even 4 2
2695.2.a.l 4 1.a even 1 1 trivial
3465.2.a.bk 4 21.c even 2 1
4235.2.a.r 4 77.b even 2 1
6160.2.a.br 4 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(2695))$$:

 $$T_{2}^{4} - 2T_{2}^{3} - 6T_{2}^{2} + 8T_{2} + 7$$ T2^4 - 2*T2^3 - 6*T2^2 + 8*T2 + 7 $$T_{3}^{4} + 2T_{3}^{3} - 8T_{3}^{2} - 10T_{3} + 16$$ T3^4 + 2*T3^3 - 8*T3^2 - 10*T3 + 16

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - 2 T^{3} - 6 T^{2} + 8 T + 7$$
$3$ $$T^{4} + 2 T^{3} - 8 T^{2} - 10 T + 16$$
$5$ $$(T + 1)^{4}$$
$7$ $$T^{4}$$
$11$ $$(T + 1)^{4}$$
$13$ $$T^{4} - 8 T^{3} - 8 T^{2} + 162 T - 236$$
$17$ $$T^{4} - 6 T^{3} + 4 T^{2} + 14 T + 4$$
$19$ $$T^{4} + 6 T^{3} - 28 T^{2} - 120 T + 32$$
$23$ $$T^{4} - 14 T^{3} + 28 T^{2} + \cdots - 976$$
$29$ $$T^{4} + 4 T^{3} - 80 T^{2} - 272 T + 304$$
$31$ $$T^{4} + 10 T^{3} - 30 T^{2} + \cdots + 304$$
$37$ $$T^{4} + 2 T^{3} - 28 T^{2} - 20 T + 8$$
$41$ $$T^{4} - 10 T^{3} - 14 T^{2} + \cdots - 428$$
$43$ $$T^{4} + 14 T^{3} + 36 T^{2} + \cdots - 272$$
$47$ $$T^{4} + 16 T^{3} - 72 T^{2} + \cdots - 8408$$
$53$ $$T^{4} - 2 T^{3} - 236 T^{2} + \cdots + 12728$$
$59$ $$T^{4} + 26 T^{3} + 110 T^{2} + \cdots - 11848$$
$61$ $$T^{4} - 12 T^{3} + 30 T^{2} + 38 T - 4$$
$67$ $$T^{4} + 10 T^{3} - 192 T^{2} + \cdots + 11168$$
$71$ $$(T - 8)^{4}$$
$73$ $$T^{4} - 14 T^{3} - 20 T^{2} + \cdots + 124$$
$79$ $$T^{4} - 20 T^{3} + 52 T^{2} + \cdots - 544$$
$83$ $$T^{4} - 10 T^{3} - 28 T^{2} + \cdots - 992$$
$89$ $$T^{4} - 10 T^{3} + 16 T^{2} + 32 T - 32$$
$97$ $$T^{4} - 2 T^{3} - 192 T^{2} + \cdots + 2272$$