# Properties

 Label 2695.2.a.k Level $2695$ Weight $2$ Character orbit 2695.a Self dual yes Analytic conductor $21.520$ Analytic rank $1$ 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: $$1$$ Dimension: $$4$$ Coefficient field: 4.4.1957.1 Defining polynomial: $$x^{4} - 4x^{2} - x + 1$$ x^4 - 4*x^2 - 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 $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 4x^{2} - x + 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{3} - 4\nu - 1$$ v^3 - 4*v - 1 $$\beta_{3}$$ $$=$$ $$-\nu^{3} + \nu^{2} + 3\nu - 1$$ -v^3 + v^2 + 3*v - 1
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{3} + \beta_{2} + \beta _1 + 2$$ b3 + b2 + b1 + 2 $$\nu^{3}$$ $$=$$ $$\beta_{2} + 4\beta _1 + 1$$ b2 + 4*b1 + 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
 2.06150 −0.693822 −1.76401 0.396339
−2.57641 0.514916 4.63791 −1.00000 −1.32664 0 −6.79636 −2.73486 2.57641
1.2 −1.74747 2.44129 1.05365 −1.00000 −4.26608 0 1.65372 2.95990 1.74747
1.3 0.197126 1.56689 −1.96114 −1.00000 0.308875 0 −0.780845 −0.544860 −0.197126
1.4 1.12676 −1.52310 −0.730419 −1.00000 −1.71616 0 −3.07652 −0.680180 −1.12676
 $$n$$: 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.k 4
7.b odd 2 1 2695.2.a.j 4
7.d odd 6 2 385.2.i.a 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
385.2.i.a 8 7.d odd 6 2
2695.2.a.j 4 7.b odd 2 1
2695.2.a.k 4 1.a even 1 1 trivial

## Hecke kernels

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

 $$T_{2}^{4} + 3T_{2}^{3} - T_{2}^{2} - 5T_{2} + 1$$ T2^4 + 3*T2^3 - T2^2 - 5*T2 + 1 $$T_{3}^{4} - 3T_{3}^{3} - T_{3}^{2} + 7T_{3} - 3$$ T3^4 - 3*T3^3 - T3^2 + 7*T3 - 3

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 3 T^{3} - T^{2} - 5 T + 1$$
$3$ $$T^{4} - 3 T^{3} - T^{2} + 7 T - 3$$
$5$ $$(T + 1)^{4}$$
$7$ $$T^{4}$$
$11$ $$(T - 1)^{4}$$
$13$ $$T^{4} - 6 T^{3} - 4 T^{2} + 22 T + 3$$
$17$ $$T^{4} - 3 T^{3} - 7 T^{2} + 29 T - 21$$
$19$ $$T^{4} - 3 T^{3} - 40 T^{2} + 56 T + 373$$
$23$ $$T^{4} + 6 T^{3} - 14 T^{2} - 109 T - 127$$
$29$ $$T^{4} + 8 T^{3} - 9 T^{2} - 42 T + 43$$
$31$ $$T^{4} + 10 T^{3} - 14 T^{2} - 113 T - 43$$
$37$ $$T^{4} + 9 T^{3} - 22 T^{2} - 334 T - 621$$
$41$ $$T^{4} + 15 T^{3} + 21 T^{2} + \cdots - 557$$
$43$ $$T^{4} + 2 T^{3} - 127 T^{2} + \cdots + 2851$$
$47$ $$T^{4} - 15 T^{3} + 61 T^{2} + \cdots - 301$$
$53$ $$T^{4} + 30 T^{3} + 303 T^{2} + \cdots + 1293$$
$59$ $$T^{4} + 17 T^{3} - 21 T^{2} + \cdots - 1647$$
$61$ $$T^{4} - 46 T^{2} - 19 T + 57$$
$67$ $$T^{4} + 25 T^{3} - 58 T^{2} + \cdots - 33217$$
$71$ $$T^{4} + 13 T^{3} - 29 T^{2} + \cdots + 717$$
$73$ $$T^{4} + 3 T^{3} - 76 T^{2} + 22 T + 3$$
$79$ $$T^{4} + 4 T^{3} - 132 T^{2} + \cdots + 157$$
$83$ $$T^{4} + 18 T^{3} - 18 T^{2} + \cdots - 7629$$
$89$ $$T^{4} + 25 T^{3} - 39 T^{2} + \cdots - 29273$$
$97$ $$T^{4} - 23 T^{3} + 150 T^{2} + \cdots - 881$$