Properties

 Label 8281.2.a.x Level $8281$ Weight $2$ Character orbit 8281.a Self dual yes Analytic conductor $66.124$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$8281 = 7^{2} \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 8281.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$66.1241179138$$ 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 637) 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.

 $$f(q)$$ $$=$$ $$q + (\beta + 1) q^{2} - \beta q^{3} + (2 \beta + 1) q^{4} + ( - 2 \beta - 1) q^{5} + ( - \beta - 2) q^{6} + (\beta + 3) q^{8} - q^{9} +O(q^{10})$$ q + (b + 1) * q^2 - b * q^3 + (2*b + 1) * q^4 + (-2*b - 1) * q^5 + (-b - 2) * q^6 + (b + 3) * q^8 - q^9 $$q + (\beta + 1) q^{2} - \beta q^{3} + (2 \beta + 1) q^{4} + ( - 2 \beta - 1) q^{5} + ( - \beta - 2) q^{6} + (\beta + 3) q^{8} - q^{9} + ( - 3 \beta - 5) q^{10} + ( - \beta - 2) q^{11} + ( - \beta - 4) q^{12} + (\beta + 4) q^{15} + 3 q^{16} + (2 \beta - 3) q^{17} + ( - \beta - 1) q^{18} - 6 q^{19} + ( - 4 \beta - 9) q^{20} + ( - 3 \beta - 4) q^{22} + \beta q^{23} + ( - 3 \beta - 2) q^{24} + (4 \beta + 4) q^{25} + 4 \beta q^{27} + (2 \beta + 7) q^{29} + (5 \beta + 6) q^{30} + ( - \beta - 4) q^{31} + (\beta - 3) q^{32} + (2 \beta + 2) q^{33} + ( - \beta + 1) q^{34} + ( - 2 \beta - 1) q^{36} + ( - 6 \beta + 1) q^{37} + ( - 6 \beta - 6) q^{38} + ( - 7 \beta - 7) q^{40} + ( - 2 \beta - 3) q^{41} + ( - \beta + 2) q^{43} + ( - 5 \beta - 6) q^{44} + (2 \beta + 1) q^{45} + (\beta + 2) q^{46} + ( - 4 \beta - 2) q^{47} - 3 \beta q^{48} + (8 \beta + 12) q^{50} + (3 \beta - 4) q^{51} - 3 q^{53} + (4 \beta + 8) q^{54} + (5 \beta + 6) q^{55} + 6 \beta q^{57} + (9 \beta + 11) q^{58} + ( - 3 \beta + 6) q^{59} + (9 \beta + 8) q^{60} + (2 \beta + 7) q^{61} + ( - 5 \beta - 6) q^{62} + ( - 2 \beta - 7) q^{64} + (4 \beta + 6) q^{66} + 3 \beta q^{67} + ( - 4 \beta + 5) q^{68} - 2 q^{69} + ( - 4 \beta + 6) q^{71} + ( - \beta - 3) q^{72} + ( - 4 \beta + 5) q^{73} + ( - 5 \beta - 11) q^{74} + ( - 4 \beta - 8) q^{75} + ( - 12 \beta - 6) q^{76} + (3 \beta + 6) q^{79} + ( - 6 \beta - 3) q^{80} - 5 q^{81} + ( - 5 \beta - 7) q^{82} + (5 \beta + 6) q^{83} + (4 \beta - 5) q^{85} + \beta q^{86} + ( - 7 \beta - 4) q^{87} + ( - 5 \beta - 8) q^{88} + (8 \beta - 4) q^{89} + (3 \beta + 5) q^{90} + (\beta + 4) q^{92} + (4 \beta + 2) q^{93} + ( - 6 \beta - 10) q^{94} + (12 \beta + 6) q^{95} + (3 \beta - 2) q^{96} + ( - 2 \beta + 8) q^{97} + (\beta + 2) q^{99} +O(q^{100})$$ q + (b + 1) * q^2 - b * q^3 + (2*b + 1) * q^4 + (-2*b - 1) * q^5 + (-b - 2) * q^6 + (b + 3) * q^8 - q^9 + (-3*b - 5) * q^10 + (-b - 2) * q^11 + (-b - 4) * q^12 + (b + 4) * q^15 + 3 * q^16 + (2*b - 3) * q^17 + (-b - 1) * q^18 - 6 * q^19 + (-4*b - 9) * q^20 + (-3*b - 4) * q^22 + b * q^23 + (-3*b - 2) * q^24 + (4*b + 4) * q^25 + 4*b * q^27 + (2*b + 7) * q^29 + (5*b + 6) * q^30 + (-b - 4) * q^31 + (b - 3) * q^32 + (2*b + 2) * q^33 + (-b + 1) * q^34 + (-2*b - 1) * q^36 + (-6*b + 1) * q^37 + (-6*b - 6) * q^38 + (-7*b - 7) * q^40 + (-2*b - 3) * q^41 + (-b + 2) * q^43 + (-5*b - 6) * q^44 + (2*b + 1) * q^45 + (b + 2) * q^46 + (-4*b - 2) * q^47 - 3*b * q^48 + (8*b + 12) * q^50 + (3*b - 4) * q^51 - 3 * q^53 + (4*b + 8) * q^54 + (5*b + 6) * q^55 + 6*b * q^57 + (9*b + 11) * q^58 + (-3*b + 6) * q^59 + (9*b + 8) * q^60 + (2*b + 7) * q^61 + (-5*b - 6) * q^62 + (-2*b - 7) * q^64 + (4*b + 6) * q^66 + 3*b * q^67 + (-4*b + 5) * q^68 - 2 * q^69 + (-4*b + 6) * q^71 + (-b - 3) * q^72 + (-4*b + 5) * q^73 + (-5*b - 11) * q^74 + (-4*b - 8) * q^75 + (-12*b - 6) * q^76 + (3*b + 6) * q^79 + (-6*b - 3) * q^80 - 5 * q^81 + (-5*b - 7) * q^82 + (5*b + 6) * q^83 + (4*b - 5) * q^85 + b * q^86 + (-7*b - 4) * q^87 + (-5*b - 8) * q^88 + (8*b - 4) * q^89 + (3*b + 5) * q^90 + (b + 4) * q^92 + (4*b + 2) * q^93 + (-6*b - 10) * q^94 + (12*b + 6) * q^95 + (3*b - 2) * q^96 + (-2*b + 8) * q^97 + (b + 2) * 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 $$2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} - 4 q^{6} + 6 q^{8} - 2 q^{9} - 10 q^{10} - 4 q^{11} - 8 q^{12} + 8 q^{15} + 6 q^{16} - 6 q^{17} - 2 q^{18} - 12 q^{19} - 18 q^{20} - 8 q^{22} - 4 q^{24} + 8 q^{25} + 14 q^{29} + 12 q^{30} - 8 q^{31} - 6 q^{32} + 4 q^{33} + 2 q^{34} - 2 q^{36} + 2 q^{37} - 12 q^{38} - 14 q^{40} - 6 q^{41} + 4 q^{43} - 12 q^{44} + 2 q^{45} + 4 q^{46} - 4 q^{47} + 24 q^{50} - 8 q^{51} - 6 q^{53} + 16 q^{54} + 12 q^{55} + 22 q^{58} + 12 q^{59} + 16 q^{60} + 14 q^{61} - 12 q^{62} - 14 q^{64} + 12 q^{66} + 10 q^{68} - 4 q^{69} + 12 q^{71} - 6 q^{72} + 10 q^{73} - 22 q^{74} - 16 q^{75} - 12 q^{76} + 12 q^{79} - 6 q^{80} - 10 q^{81} - 14 q^{82} + 12 q^{83} - 10 q^{85} - 8 q^{87} - 16 q^{88} - 8 q^{89} + 10 q^{90} + 8 q^{92} + 4 q^{93} - 20 q^{94} + 12 q^{95} - 4 q^{96} + 16 q^{97} + 4 q^{99}+O(q^{100})$$ 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^5 - 4 * q^6 + 6 * q^8 - 2 * q^9 - 10 * q^10 - 4 * q^11 - 8 * q^12 + 8 * q^15 + 6 * q^16 - 6 * q^17 - 2 * q^18 - 12 * q^19 - 18 * q^20 - 8 * q^22 - 4 * q^24 + 8 * q^25 + 14 * q^29 + 12 * q^30 - 8 * q^31 - 6 * q^32 + 4 * q^33 + 2 * q^34 - 2 * q^36 + 2 * q^37 - 12 * q^38 - 14 * q^40 - 6 * q^41 + 4 * q^43 - 12 * q^44 + 2 * q^45 + 4 * q^46 - 4 * q^47 + 24 * q^50 - 8 * q^51 - 6 * q^53 + 16 * q^54 + 12 * q^55 + 22 * q^58 + 12 * q^59 + 16 * q^60 + 14 * q^61 - 12 * q^62 - 14 * q^64 + 12 * q^66 + 10 * q^68 - 4 * q^69 + 12 * q^71 - 6 * q^72 + 10 * q^73 - 22 * q^74 - 16 * q^75 - 12 * q^76 + 12 * q^79 - 6 * q^80 - 10 * q^81 - 14 * q^82 + 12 * q^83 - 10 * q^85 - 8 * q^87 - 16 * q^88 - 8 * q^89 + 10 * q^90 + 8 * q^92 + 4 * q^93 - 20 * q^94 + 12 * q^95 - 4 * q^96 + 16 * q^97 + 4 * 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 1.82843 −0.585786 0 1.58579 −1.00000 −0.757359
1.2 2.41421 −1.41421 3.82843 −3.82843 −3.41421 0 4.41421 −1.00000 −9.24264
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Atkin-Lehner signs

$$p$$ Sign
$$7$$ $$-1$$
$$13$$ $$1$$

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 8281.2.a.x 2
7.b odd 2 1 8281.2.a.y 2
13.b even 2 1 8281.2.a.p 2
13.e even 6 2 637.2.f.f yes 4
91.b odd 2 1 8281.2.a.o 2
91.k even 6 2 637.2.h.b 4
91.l odd 6 2 637.2.h.c 4
91.p odd 6 2 637.2.g.g 4
91.t odd 6 2 637.2.f.e 4
91.u even 6 2 637.2.g.f 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
637.2.f.e 4 91.t odd 6 2
637.2.f.f yes 4 13.e even 6 2
637.2.g.f 4 91.u even 6 2
637.2.g.g 4 91.p odd 6 2
637.2.h.b 4 91.k even 6 2
637.2.h.c 4 91.l odd 6 2
8281.2.a.o 2 91.b odd 2 1
8281.2.a.p 2 13.b even 2 1
8281.2.a.x 2 1.a even 1 1 trivial
8281.2.a.y 2 7.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(8281))$$:

 $$T_{2}^{2} - 2T_{2} - 1$$ T2^2 - 2*T2 - 1 $$T_{3}^{2} - 2$$ T3^2 - 2 $$T_{5}^{2} + 2T_{5} - 7$$ T5^2 + 2*T5 - 7 $$T_{11}^{2} + 4T_{11} + 2$$ T11^2 + 4*T11 + 2 $$T_{17}^{2} + 6T_{17} + 1$$ T17^2 + 6*T17 + 1

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 2T - 1$$
$3$ $$T^{2} - 2$$
$5$ $$T^{2} + 2T - 7$$
$7$ $$T^{2}$$
$11$ $$T^{2} + 4T + 2$$
$13$ $$T^{2}$$
$17$ $$T^{2} + 6T + 1$$
$19$ $$(T + 6)^{2}$$
$23$ $$T^{2} - 2$$
$29$ $$T^{2} - 14T + 41$$
$31$ $$T^{2} + 8T + 14$$
$37$ $$T^{2} - 2T - 71$$
$41$ $$T^{2} + 6T + 1$$
$43$ $$T^{2} - 4T + 2$$
$47$ $$T^{2} + 4T - 28$$
$53$ $$(T + 3)^{2}$$
$59$ $$T^{2} - 12T + 18$$
$61$ $$T^{2} - 14T + 41$$
$67$ $$T^{2} - 18$$
$71$ $$T^{2} - 12T + 4$$
$73$ $$T^{2} - 10T - 7$$
$79$ $$T^{2} - 12T + 18$$
$83$ $$T^{2} - 12T - 14$$
$89$ $$T^{2} + 8T - 112$$
$97$ $$T^{2} - 16T + 56$$