# Properties

 Label 2695.2.a.e Level $2695$ Weight $2$ Character orbit 2695.a Self dual yes Analytic conductor $21.520$ Analytic rank $0$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ Defining polynomial: $$x^{2} - 2$$ x^2 - 2 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 $$\beta = \sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

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

## 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
 −1.41421 1.41421
−0.414214 1.41421 −1.82843 1.00000 −0.585786 0 1.58579 −1.00000 −0.414214
1.2 2.41421 −1.41421 3.82843 1.00000 −3.41421 0 4.41421 −1.00000 2.41421
 $$n$$: 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.e 2
7.b odd 2 1 385.2.a.d 2
21.c even 2 1 3465.2.a.u 2
28.d even 2 1 6160.2.a.y 2
35.c odd 2 1 1925.2.a.n 2
35.f even 4 2 1925.2.b.j 4
77.b even 2 1 4235.2.a.h 2

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

 $$T_{2}^{2} - 2T_{2} - 1$$ T2^2 - 2*T2 - 1 $$T_{3}^{2} - 2$$ T3^2 - 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 2T - 1$$
$3$ $$T^{2} - 2$$
$5$ $$(T - 1)^{2}$$
$7$ $$T^{2}$$
$11$ $$(T - 1)^{2}$$
$13$ $$T^{2} + 4T + 2$$
$17$ $$T^{2} - 4T + 2$$
$19$ $$T^{2}$$
$23$ $$T^{2} - 12T + 28$$
$29$ $$T^{2} - 4T - 4$$
$31$ $$T^{2} - 12T + 18$$
$37$ $$T^{2} - 8T + 8$$
$41$ $$T^{2} - 18$$
$43$ $$(T - 6)^{2}$$
$47$ $$T^{2} - 2$$
$53$ $$T^{2} - 8$$
$59$ $$T^{2} + 4T - 94$$
$61$ $$T^{2} - 8T - 82$$
$67$ $$T^{2} + 4T - 68$$
$71$ $$T^{2} - 24T + 136$$
$73$ $$T^{2} + 4T - 46$$
$79$ $$T^{2} + 4T - 68$$
$83$ $$(T + 12)^{2}$$
$89$ $$T^{2} - 4T - 4$$
$97$ $$T^{2} - 4T - 124$$