Properties

 Label 825.2.a.n Level $825$ Weight $2$ Character orbit 825.a Self dual yes Analytic conductor $6.588$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$825 = 3 \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 825.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$6.58765816676$$ 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 165) 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.

 $$f(q)$$ $$=$$ $$q + (\beta_{2} + 1) q^{2} - q^{3} + (\beta_{2} - \beta_1 + 2) q^{4} + ( - \beta_{2} - 1) q^{6} + ( - \beta_{2} - \beta_1 + 3) q^{7} + ( - 3 \beta_1 + 4) q^{8} + q^{9}+O(q^{10})$$ q + (b2 + 1) * q^2 - q^3 + (b2 - b1 + 2) * q^4 + (-b2 - 1) * q^6 + (-b2 - b1 + 3) * q^7 + (-3*b1 + 4) * q^8 + q^9 $$q + (\beta_{2} + 1) q^{2} - q^{3} + (\beta_{2} - \beta_1 + 2) q^{4} + ( - \beta_{2} - 1) q^{6} + ( - \beta_{2} - \beta_1 + 3) q^{7} + ( - 3 \beta_1 + 4) q^{8} + q^{9} - q^{11} + ( - \beta_{2} + \beta_1 - 2) q^{12} + ( - \beta_{2} + 3 \beta_1 + 1) q^{13} + (3 \beta_{2} - \beta_1 + 1) q^{14} + (2 \beta_{2} - 4 \beta_1 + 3) q^{16} + (3 \beta_{2} + \beta_1 + 1) q^{17} + (\beta_{2} + 1) q^{18} + (2 \beta_{2} + 4 \beta_1 - 2) q^{19} + (\beta_{2} + \beta_1 - 3) q^{21} + ( - \beta_{2} - 1) q^{22} + 4 q^{23} + (3 \beta_1 - 4) q^{24} + (\beta_{2} + 7 \beta_1 - 5) q^{26} - q^{27} + (3 \beta_{2} - 3 \beta_1 + 5) q^{28} + ( - 2 \beta_{2} - 2 \beta_1 - 2) q^{29} + (2 \beta_1 + 2) q^{31} + (3 \beta_{2} - 4 \beta_1 + 5) q^{32} + q^{33} + (\beta_{2} - \beta_1 + 9) q^{34} + (\beta_{2} - \beta_1 + 2) q^{36} + ( - 2 \beta_{2} - 2 \beta_1 + 2) q^{37} + ( - 2 \beta_{2} + 6 \beta_1) q^{38} + (\beta_{2} - 3 \beta_1 - 1) q^{39} + ( - 2 \beta_{2} - 2 \beta_1 - 2) q^{41} + ( - 3 \beta_{2} + \beta_1 - 1) q^{42} + ( - \beta_{2} - \beta_1 + 3) q^{43} + ( - \beta_{2} + \beta_1 - 2) q^{44} + (4 \beta_{2} + 4) q^{46} + ( - 2 \beta_{2} + 2 \beta_1 + 2) q^{47} + ( - 2 \beta_{2} + 4 \beta_1 - 3) q^{48} + ( - 6 \beta_{2} - 4 \beta_1 + 5) q^{49} + ( - 3 \beta_{2} - \beta_1 - 1) q^{51} + ( - 3 \beta_{2} + 7 \beta_1 - 11) q^{52} + (2 \beta_{2} + 4 \beta_1 - 4) q^{53} + ( - \beta_{2} - 1) q^{54} + ( - \beta_{2} - 7 \beta_1 + 15) q^{56} + ( - 2 \beta_{2} - 4 \beta_1 + 2) q^{57} + ( - 2 \beta_{2} - 2 \beta_1 - 6) q^{58} + ( - 2 \beta_{2} + 4 \beta_1 - 4) q^{59} + ( - 4 \beta_1 + 2) q^{61} + (2 \beta_{2} + 4 \beta_1) q^{62} + ( - \beta_{2} - \beta_1 + 3) q^{63} + (\beta_{2} - 3 \beta_1 + 12) q^{64} + (\beta_{2} + 1) q^{66} + (4 \beta_{2} + 4) q^{67} + (3 \beta_{2} - 5 \beta_1 + 11) q^{68} - 4 q^{69} + ( - 2 \beta_{2} - 4) q^{71} + ( - 3 \beta_1 + 4) q^{72} + ( - 3 \beta_{2} - 3 \beta_1 + 7) q^{73} + (2 \beta_{2} - 2 \beta_1 - 2) q^{74} + ( - 4 \beta_{2} + 6 \beta_1 - 8) q^{76} + (\beta_{2} + \beta_1 - 3) q^{77} + ( - \beta_{2} - 7 \beta_1 + 5) q^{78} + (2 \beta_{2} - 2) q^{79} + q^{81} + ( - 2 \beta_{2} - 2 \beta_1 - 6) q^{82} + (\beta_{2} + \beta_1 - 1) q^{83} + ( - 3 \beta_{2} + 3 \beta_1 - 5) q^{84} + (3 \beta_{2} - \beta_1 + 1) q^{86} + (2 \beta_{2} + 2 \beta_1 + 2) q^{87} + (3 \beta_1 - 4) q^{88} + 2 \beta_1 q^{89} + ( - 8 \beta_{2} + 2 \beta_1 + 2) q^{91} + (4 \beta_{2} - 4 \beta_1 + 8) q^{92} + ( - 2 \beta_1 - 2) q^{93} + (2 \beta_{2} + 6 \beta_1 - 6) q^{94} + ( - 3 \beta_{2} + 4 \beta_1 - 5) q^{96} + (4 \beta_1 - 4) q^{97} + (5 \beta_{2} - 2 \beta_1 - 9) q^{98} - q^{99}+O(q^{100})$$ q + (b2 + 1) * q^2 - q^3 + (b2 - b1 + 2) * q^4 + (-b2 - 1) * q^6 + (-b2 - b1 + 3) * q^7 + (-3*b1 + 4) * q^8 + q^9 - q^11 + (-b2 + b1 - 2) * q^12 + (-b2 + 3*b1 + 1) * q^13 + (3*b2 - b1 + 1) * q^14 + (2*b2 - 4*b1 + 3) * q^16 + (3*b2 + b1 + 1) * q^17 + (b2 + 1) * q^18 + (2*b2 + 4*b1 - 2) * q^19 + (b2 + b1 - 3) * q^21 + (-b2 - 1) * q^22 + 4 * q^23 + (3*b1 - 4) * q^24 + (b2 + 7*b1 - 5) * q^26 - q^27 + (3*b2 - 3*b1 + 5) * q^28 + (-2*b2 - 2*b1 - 2) * q^29 + (2*b1 + 2) * q^31 + (3*b2 - 4*b1 + 5) * q^32 + q^33 + (b2 - b1 + 9) * q^34 + (b2 - b1 + 2) * q^36 + (-2*b2 - 2*b1 + 2) * q^37 + (-2*b2 + 6*b1) * q^38 + (b2 - 3*b1 - 1) * q^39 + (-2*b2 - 2*b1 - 2) * q^41 + (-3*b2 + b1 - 1) * q^42 + (-b2 - b1 + 3) * q^43 + (-b2 + b1 - 2) * q^44 + (4*b2 + 4) * q^46 + (-2*b2 + 2*b1 + 2) * q^47 + (-2*b2 + 4*b1 - 3) * q^48 + (-6*b2 - 4*b1 + 5) * q^49 + (-3*b2 - b1 - 1) * q^51 + (-3*b2 + 7*b1 - 11) * q^52 + (2*b2 + 4*b1 - 4) * q^53 + (-b2 - 1) * q^54 + (-b2 - 7*b1 + 15) * q^56 + (-2*b2 - 4*b1 + 2) * q^57 + (-2*b2 - 2*b1 - 6) * q^58 + (-2*b2 + 4*b1 - 4) * q^59 + (-4*b1 + 2) * q^61 + (2*b2 + 4*b1) * q^62 + (-b2 - b1 + 3) * q^63 + (b2 - 3*b1 + 12) * q^64 + (b2 + 1) * q^66 + (4*b2 + 4) * q^67 + (3*b2 - 5*b1 + 11) * q^68 - 4 * q^69 + (-2*b2 - 4) * q^71 + (-3*b1 + 4) * q^72 + (-3*b2 - 3*b1 + 7) * q^73 + (2*b2 - 2*b1 - 2) * q^74 + (-4*b2 + 6*b1 - 8) * q^76 + (b2 + b1 - 3) * q^77 + (-b2 - 7*b1 + 5) * q^78 + (2*b2 - 2) * q^79 + q^81 + (-2*b2 - 2*b1 - 6) * q^82 + (b2 + b1 - 1) * q^83 + (-3*b2 + 3*b1 - 5) * q^84 + (3*b2 - b1 + 1) * q^86 + (2*b2 + 2*b1 + 2) * q^87 + (3*b1 - 4) * q^88 + 2*b1 * q^89 + (-8*b2 + 2*b1 + 2) * q^91 + (4*b2 - 4*b1 + 8) * q^92 + (-2*b1 - 2) * q^93 + (2*b2 + 6*b1 - 6) * q^94 + (-3*b2 + 4*b1 - 5) * q^96 + (4*b1 - 4) * q^97 + (5*b2 - 2*b1 - 9) * q^98 - q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 3 q^{2} - 3 q^{3} + 5 q^{4} - 3 q^{6} + 8 q^{7} + 9 q^{8} + 3 q^{9}+O(q^{10})$$ 3 * q + 3 * q^2 - 3 * q^3 + 5 * q^4 - 3 * q^6 + 8 * q^7 + 9 * q^8 + 3 * q^9 $$3 q + 3 q^{2} - 3 q^{3} + 5 q^{4} - 3 q^{6} + 8 q^{7} + 9 q^{8} + 3 q^{9} - 3 q^{11} - 5 q^{12} + 6 q^{13} + 2 q^{14} + 5 q^{16} + 4 q^{17} + 3 q^{18} - 2 q^{19} - 8 q^{21} - 3 q^{22} + 12 q^{23} - 9 q^{24} - 8 q^{26} - 3 q^{27} + 12 q^{28} - 8 q^{29} + 8 q^{31} + 11 q^{32} + 3 q^{33} + 26 q^{34} + 5 q^{36} + 4 q^{37} + 6 q^{38} - 6 q^{39} - 8 q^{41} - 2 q^{42} + 8 q^{43} - 5 q^{44} + 12 q^{46} + 8 q^{47} - 5 q^{48} + 11 q^{49} - 4 q^{51} - 26 q^{52} - 8 q^{53} - 3 q^{54} + 38 q^{56} + 2 q^{57} - 20 q^{58} - 8 q^{59} + 2 q^{61} + 4 q^{62} + 8 q^{63} + 33 q^{64} + 3 q^{66} + 12 q^{67} + 28 q^{68} - 12 q^{69} - 12 q^{71} + 9 q^{72} + 18 q^{73} - 8 q^{74} - 18 q^{76} - 8 q^{77} + 8 q^{78} - 6 q^{79} + 3 q^{81} - 20 q^{82} - 2 q^{83} - 12 q^{84} + 2 q^{86} + 8 q^{87} - 9 q^{88} + 2 q^{89} + 8 q^{91} + 20 q^{92} - 8 q^{93} - 12 q^{94} - 11 q^{96} - 8 q^{97} - 29 q^{98} - 3 q^{99}+O(q^{100})$$ 3 * q + 3 * q^2 - 3 * q^3 + 5 * q^4 - 3 * q^6 + 8 * q^7 + 9 * q^8 + 3 * q^9 - 3 * q^11 - 5 * q^12 + 6 * q^13 + 2 * q^14 + 5 * q^16 + 4 * q^17 + 3 * q^18 - 2 * q^19 - 8 * q^21 - 3 * q^22 + 12 * q^23 - 9 * q^24 - 8 * q^26 - 3 * q^27 + 12 * q^28 - 8 * q^29 + 8 * q^31 + 11 * q^32 + 3 * q^33 + 26 * q^34 + 5 * q^36 + 4 * q^37 + 6 * q^38 - 6 * q^39 - 8 * q^41 - 2 * q^42 + 8 * q^43 - 5 * q^44 + 12 * q^46 + 8 * q^47 - 5 * q^48 + 11 * q^49 - 4 * q^51 - 26 * q^52 - 8 * q^53 - 3 * q^54 + 38 * q^56 + 2 * q^57 - 20 * q^58 - 8 * q^59 + 2 * q^61 + 4 * q^62 + 8 * q^63 + 33 * q^64 + 3 * q^66 + 12 * q^67 + 28 * q^68 - 12 * q^69 - 12 * q^71 + 9 * q^72 + 18 * q^73 - 8 * q^74 - 18 * q^76 - 8 * q^77 + 8 * q^78 - 6 * q^79 + 3 * q^81 - 20 * q^82 - 2 * q^83 - 12 * q^84 + 2 * q^86 + 8 * q^87 - 9 * q^88 + 2 * q^89 + 8 * q^91 + 20 * q^92 - 8 * q^93 - 12 * q^94 - 11 * q^96 - 8 * q^97 - 29 * q^98 - 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
 0.311108 2.17009 −1.48119
−1.21432 −1.00000 −0.525428 0 1.21432 4.90321 3.06668 1.00000 0
1.2 1.53919 −1.00000 0.369102 0 −1.53919 0.290725 −2.51026 1.00000 0
1.3 2.67513 −1.00000 5.15633 0 −2.67513 2.80606 8.44358 1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$1$$
$$5$$ $$-1$$
$$11$$ $$1$$

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 825.2.a.n 3
3.b odd 2 1 2475.2.a.y 3
5.b even 2 1 825.2.a.h 3
5.c odd 4 2 165.2.c.a 6
11.b odd 2 1 9075.2.a.cc 3
15.d odd 2 1 2475.2.a.be 3
15.e even 4 2 495.2.c.d 6
20.e even 4 2 2640.2.d.i 6
55.d odd 2 1 9075.2.a.ck 3
55.e even 4 2 1815.2.c.d 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.2.c.a 6 5.c odd 4 2
495.2.c.d 6 15.e even 4 2
825.2.a.h 3 5.b even 2 1
825.2.a.n 3 1.a even 1 1 trivial
1815.2.c.d 6 55.e even 4 2
2475.2.a.y 3 3.b odd 2 1
2475.2.a.be 3 15.d odd 2 1
2640.2.d.i 6 20.e even 4 2
9075.2.a.cc 3 11.b odd 2 1
9075.2.a.ck 3 55.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(825))$$:

 $$T_{2}^{3} - 3T_{2}^{2} - T_{2} + 5$$ T2^3 - 3*T2^2 - T2 + 5 $$T_{7}^{3} - 8T_{7}^{2} + 16T_{7} - 4$$ T7^3 - 8*T7^2 + 16*T7 - 4

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3} - 3T^{2} - T + 5$$
$3$ $$(T + 1)^{3}$$
$5$ $$T^{3}$$
$7$ $$T^{3} - 8 T^{2} + 16 T - 4$$
$11$ $$(T + 1)^{3}$$
$13$ $$T^{3} - 6 T^{2} - 28 T + 148$$
$17$ $$T^{3} - 4 T^{2} - 28 T + 116$$
$19$ $$T^{3} + 2 T^{2} - 52 T - 184$$
$23$ $$(T - 4)^{3}$$
$29$ $$T^{3} + 8T^{2} - 32$$
$31$ $$T^{3} - 8 T^{2} + 8 T + 16$$
$37$ $$T^{3} - 4 T^{2} - 16 T + 32$$
$41$ $$T^{3} + 8T^{2} - 32$$
$43$ $$T^{3} - 8 T^{2} + 16 T - 4$$
$47$ $$T^{3} - 8 T^{2} - 16 T + 160$$
$53$ $$T^{3} + 8 T^{2} - 32 T - 272$$
$59$ $$T^{3} + 8 T^{2} - 64 T + 80$$
$61$ $$T^{3} - 2 T^{2} - 52 T + 40$$
$67$ $$T^{3} - 12 T^{2} - 16 T + 320$$
$71$ $$T^{3} + 12 T^{2} + 32 T - 16$$
$73$ $$T^{3} - 18 T^{2} + 60 T + 92$$
$79$ $$T^{3} + 6 T^{2} - 4 T - 8$$
$83$ $$T^{3} + 2 T^{2} - 4 T - 4$$
$89$ $$T^{3} - 2 T^{2} - 12 T + 8$$
$97$ $$T^{3} + 8 T^{2} - 32 T - 128$$