# Properties

 Label 825.2.a.j Level $825$ Weight $2$ Character orbit 825.a Self dual yes Analytic conductor $6.588$ Analytic rank $1$ 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: $$1$$ 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.

 $$f(q)$$ $$=$$ $$q - \beta_1 q^{2} - q^{3} + (\beta_{2} + \beta_1) q^{4} + \beta_1 q^{6} + ( - \beta_{2} - \beta_1 - 1) q^{7} + ( - \beta_{2} - 1) q^{8} + q^{9}+O(q^{10})$$ q - b1 * q^2 - q^3 + (b2 + b1) * q^4 + b1 * q^6 + (-b2 - b1 - 1) * q^7 + (-b2 - 1) * q^8 + q^9 $$q - \beta_1 q^{2} - q^{3} + (\beta_{2} + \beta_1) q^{4} + \beta_1 q^{6} + ( - \beta_{2} - \beta_1 - 1) q^{7} + ( - \beta_{2} - 1) q^{8} + q^{9} + q^{11} + ( - \beta_{2} - \beta_1) q^{12} + (\beta_{2} + \beta_1 - 1) q^{13} + (\beta_{2} + 3 \beta_1 + 1) q^{14} + ( - 2 \beta_{2} - 1) q^{16} + (\beta_{2} + 3 \beta_1 - 1) q^{17} - \beta_1 q^{18} + ( - 2 \beta_{2} - 2) q^{19} + (\beta_{2} + \beta_1 + 1) q^{21} - \beta_1 q^{22} - 4 q^{23} + (\beta_{2} + 1) q^{24} + ( - \beta_{2} - \beta_1 - 1) q^{26} - q^{27} + ( - \beta_{2} - 3 \beta_1 - 3) q^{28} + ( - 2 \beta_{2} + 2 \beta_1 + 2) q^{29} + ( - 2 \beta_1 - 2) q^{31} + (2 \beta_{2} + 3 \beta_1) q^{32} - q^{33} + ( - 3 \beta_{2} - 3 \beta_1 - 5) q^{34} + (\beta_{2} + \beta_1) q^{36} + (2 \beta_{2} + 2 \beta_1 - 2) q^{37} + (4 \beta_1 - 2) q^{38} + ( - \beta_{2} - \beta_1 + 1) q^{39} + (6 \beta_{2} + 2 \beta_1 + 2) q^{41} + ( - \beta_{2} - 3 \beta_1 - 1) q^{42} + (3 \beta_{2} + 3 \beta_1 - 5) q^{43} + (\beta_{2} + \beta_1) q^{44} + 4 \beta_1 q^{46} + ( - 2 \beta_{2} + 2 \beta_1 - 6) q^{47} + (2 \beta_{2} + 1) q^{48} + (2 \beta_{2} + 4 \beta_1 - 3) q^{49} + ( - \beta_{2} - 3 \beta_1 + 1) q^{51} + ( - \beta_{2} + \beta_1 + 3) q^{52} + ( - 2 \beta_{2} - 4 \beta_1 - 4) q^{53} + \beta_1 q^{54} + (\beta_{2} + \beta_1 + 3) q^{56} + (2 \beta_{2} + 2) q^{57} + ( - 2 \beta_{2} - 2 \beta_1 - 6) q^{58} + (2 \beta_{2} - 4 \beta_1 + 4) q^{59} + ( - 4 \beta_1 - 6) q^{61} + (2 \beta_{2} + 4 \beta_1 + 4) q^{62} + ( - \beta_{2} - \beta_1 - 1) q^{63} + (\beta_{2} - 5 \beta_1 - 2) q^{64} + \beta_1 q^{66} + (4 \beta_{2} - 4) q^{67} + (\beta_{2} + 5 \beta_1 + 5) q^{68} + 4 q^{69} + ( - 6 \beta_{2} - 4) q^{71} + ( - \beta_{2} - 1) q^{72} + ( - \beta_{2} - 5 \beta_1 + 5) q^{73} + ( - 2 \beta_{2} - 2 \beta_1 - 2) q^{74} + ( - 2 \beta_1 - 4) q^{76} + ( - \beta_{2} - \beta_1 - 1) q^{77} + (\beta_{2} + \beta_1 + 1) q^{78} + (6 \beta_{2} - 4 \beta_1 - 2) q^{79} + q^{81} + ( - 2 \beta_{2} - 10 \beta_1 + 2) q^{82} + (5 \beta_{2} + \beta_1 + 3) q^{83} + (\beta_{2} + 3 \beta_1 + 3) q^{84} + ( - 3 \beta_{2} - \beta_1 - 3) q^{86} + (2 \beta_{2} - 2 \beta_1 - 2) q^{87} + ( - \beta_{2} - 1) q^{88} + ( - 6 \beta_1 + 8) q^{89} + ( - 2 \beta_1 - 2) q^{91} + ( - 4 \beta_{2} - 4 \beta_1) q^{92} + (2 \beta_1 + 2) q^{93} + ( - 2 \beta_{2} + 6 \beta_1 - 6) q^{94} + ( - 2 \beta_{2} - 3 \beta_1) q^{96} + ( - 8 \beta_{2} - 4 \beta_1 - 4) q^{97} + ( - 4 \beta_{2} - 3 \beta_1 - 6) q^{98} + q^{99}+O(q^{100})$$ q - b1 * q^2 - q^3 + (b2 + b1) * q^4 + b1 * q^6 + (-b2 - b1 - 1) * q^7 + (-b2 - 1) * q^8 + q^9 + q^11 + (-b2 - b1) * q^12 + (b2 + b1 - 1) * q^13 + (b2 + 3*b1 + 1) * q^14 + (-2*b2 - 1) * q^16 + (b2 + 3*b1 - 1) * q^17 - b1 * q^18 + (-2*b2 - 2) * q^19 + (b2 + b1 + 1) * q^21 - b1 * q^22 - 4 * q^23 + (b2 + 1) * q^24 + (-b2 - b1 - 1) * q^26 - q^27 + (-b2 - 3*b1 - 3) * q^28 + (-2*b2 + 2*b1 + 2) * q^29 + (-2*b1 - 2) * q^31 + (2*b2 + 3*b1) * q^32 - q^33 + (-3*b2 - 3*b1 - 5) * q^34 + (b2 + b1) * q^36 + (2*b2 + 2*b1 - 2) * q^37 + (4*b1 - 2) * q^38 + (-b2 - b1 + 1) * q^39 + (6*b2 + 2*b1 + 2) * q^41 + (-b2 - 3*b1 - 1) * q^42 + (3*b2 + 3*b1 - 5) * q^43 + (b2 + b1) * q^44 + 4*b1 * q^46 + (-2*b2 + 2*b1 - 6) * q^47 + (2*b2 + 1) * q^48 + (2*b2 + 4*b1 - 3) * q^49 + (-b2 - 3*b1 + 1) * q^51 + (-b2 + b1 + 3) * q^52 + (-2*b2 - 4*b1 - 4) * q^53 + b1 * q^54 + (b2 + b1 + 3) * q^56 + (2*b2 + 2) * q^57 + (-2*b2 - 2*b1 - 6) * q^58 + (2*b2 - 4*b1 + 4) * q^59 + (-4*b1 - 6) * q^61 + (2*b2 + 4*b1 + 4) * q^62 + (-b2 - b1 - 1) * q^63 + (b2 - 5*b1 - 2) * q^64 + b1 * q^66 + (4*b2 - 4) * q^67 + (b2 + 5*b1 + 5) * q^68 + 4 * q^69 + (-6*b2 - 4) * q^71 + (-b2 - 1) * q^72 + (-b2 - 5*b1 + 5) * q^73 + (-2*b2 - 2*b1 - 2) * q^74 + (-2*b1 - 4) * q^76 + (-b2 - b1 - 1) * q^77 + (b2 + b1 + 1) * q^78 + (6*b2 - 4*b1 - 2) * q^79 + q^81 + (-2*b2 - 10*b1 + 2) * q^82 + (5*b2 + b1 + 3) * q^83 + (b2 + 3*b1 + 3) * q^84 + (-3*b2 - b1 - 3) * q^86 + (2*b2 - 2*b1 - 2) * q^87 + (-b2 - 1) * q^88 + (-6*b1 + 8) * q^89 + (-2*b1 - 2) * q^91 + (-4*b2 - 4*b1) * q^92 + (2*b1 + 2) * q^93 + (-2*b2 + 6*b1 - 6) * q^94 + (-2*b2 - 3*b1) * q^96 + (-8*b2 - 4*b1 - 4) * q^97 + (-4*b2 - 3*b1 - 6) * q^98 + q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - q^{2} - 3 q^{3} + q^{4} + q^{6} - 4 q^{7} - 3 q^{8} + 3 q^{9}+O(q^{10})$$ 3 * q - q^2 - 3 * q^3 + q^4 + q^6 - 4 * q^7 - 3 * q^8 + 3 * q^9 $$3 q - q^{2} - 3 q^{3} + q^{4} + q^{6} - 4 q^{7} - 3 q^{8} + 3 q^{9} + 3 q^{11} - q^{12} - 2 q^{13} + 6 q^{14} - 3 q^{16} - q^{18} - 6 q^{19} + 4 q^{21} - q^{22} - 12 q^{23} + 3 q^{24} - 4 q^{26} - 3 q^{27} - 12 q^{28} + 8 q^{29} - 8 q^{31} + 3 q^{32} - 3 q^{33} - 18 q^{34} + q^{36} - 4 q^{37} - 2 q^{38} + 2 q^{39} + 8 q^{41} - 6 q^{42} - 12 q^{43} + q^{44} + 4 q^{46} - 16 q^{47} + 3 q^{48} - 5 q^{49} + 10 q^{52} - 16 q^{53} + q^{54} + 10 q^{56} + 6 q^{57} - 20 q^{58} + 8 q^{59} - 22 q^{61} + 16 q^{62} - 4 q^{63} - 11 q^{64} + q^{66} - 12 q^{67} + 20 q^{68} + 12 q^{69} - 12 q^{71} - 3 q^{72} + 10 q^{73} - 8 q^{74} - 14 q^{76} - 4 q^{77} + 4 q^{78} - 10 q^{79} + 3 q^{81} - 4 q^{82} + 10 q^{83} + 12 q^{84} - 10 q^{86} - 8 q^{87} - 3 q^{88} + 18 q^{89} - 8 q^{91} - 4 q^{92} + 8 q^{93} - 12 q^{94} - 3 q^{96} - 16 q^{97} - 21 q^{98} + 3 q^{99}+O(q^{100})$$ 3 * q - q^2 - 3 * q^3 + q^4 + q^6 - 4 * q^7 - 3 * q^8 + 3 * q^9 + 3 * q^11 - q^12 - 2 * q^13 + 6 * q^14 - 3 * q^16 - q^18 - 6 * q^19 + 4 * q^21 - q^22 - 12 * q^23 + 3 * q^24 - 4 * q^26 - 3 * q^27 - 12 * q^28 + 8 * q^29 - 8 * q^31 + 3 * q^32 - 3 * q^33 - 18 * q^34 + q^36 - 4 * q^37 - 2 * q^38 + 2 * q^39 + 8 * q^41 - 6 * q^42 - 12 * q^43 + q^44 + 4 * q^46 - 16 * q^47 + 3 * q^48 - 5 * q^49 + 10 * q^52 - 16 * q^53 + q^54 + 10 * q^56 + 6 * q^57 - 20 * q^58 + 8 * q^59 - 22 * q^61 + 16 * q^62 - 4 * q^63 - 11 * q^64 + q^66 - 12 * q^67 + 20 * q^68 + 12 * q^69 - 12 * q^71 - 3 * q^72 + 10 * q^73 - 8 * q^74 - 14 * q^76 - 4 * q^77 + 4 * q^78 - 10 * q^79 + 3 * q^81 - 4 * q^82 + 10 * q^83 + 12 * q^84 - 10 * q^86 - 8 * q^87 - 3 * q^88 + 18 * q^89 - 8 * q^91 - 4 * q^92 + 8 * q^93 - 12 * q^94 - 3 * q^96 - 16 * q^97 - 21 * 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
 2.17009 0.311108 −1.48119
−2.17009 −1.00000 2.70928 0 2.17009 −3.70928 −1.53919 1.00000 0
1.2 −0.311108 −1.00000 −1.90321 0 0.311108 0.903212 1.21432 1.00000 0
1.3 1.48119 −1.00000 0.193937 0 −1.48119 −1.19394 −2.67513 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.j 3
3.b odd 2 1 2475.2.a.bc 3
5.b even 2 1 825.2.a.l 3
5.c odd 4 2 165.2.c.b 6
11.b odd 2 1 9075.2.a.ch 3
15.d odd 2 1 2475.2.a.ba 3
15.e even 4 2 495.2.c.e 6
20.e even 4 2 2640.2.d.h 6
55.d odd 2 1 9075.2.a.cg 3
55.e even 4 2 1815.2.c.e 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.2.c.b 6 5.c odd 4 2
495.2.c.e 6 15.e even 4 2
825.2.a.j 3 1.a even 1 1 trivial
825.2.a.l 3 5.b even 2 1
1815.2.c.e 6 55.e even 4 2
2475.2.a.ba 3 15.d odd 2 1
2475.2.a.bc 3 3.b odd 2 1
2640.2.d.h 6 20.e even 4 2
9075.2.a.cg 3 55.d odd 2 1
9075.2.a.ch 3 11.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(825))$$:

 $$T_{2}^{3} + T_{2}^{2} - 3T_{2} - 1$$ T2^3 + T2^2 - 3*T2 - 1 $$T_{7}^{3} + 4T_{7}^{2} - 4$$ T7^3 + 4*T7^2 - 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3} + T^{2} - 3T - 1$$
$3$ $$(T + 1)^{3}$$
$5$ $$T^{3}$$
$7$ $$T^{3} + 4T^{2} - 4$$
$11$ $$(T - 1)^{3}$$
$13$ $$T^{3} + 2 T^{2} - 4 T - 4$$
$17$ $$T^{3} - 28T - 52$$
$19$ $$T^{3} + 6 T^{2} - 4 T - 40$$
$23$ $$(T + 4)^{3}$$
$29$ $$T^{3} - 8 T^{2} - 16 T + 160$$
$31$ $$T^{3} + 8 T^{2} + 8 T - 16$$
$37$ $$T^{3} + 4 T^{2} - 16 T - 32$$
$41$ $$T^{3} - 8 T^{2} - 112 T + 928$$
$43$ $$T^{3} + 12T^{2} - 148$$
$47$ $$T^{3} + 16 T^{2} + 48 T + 32$$
$53$ $$T^{3} + 16 T^{2} + 32 T + 16$$
$59$ $$T^{3} - 8 T^{2} - 64 T - 80$$
$61$ $$T^{3} + 22 T^{2} + 108 T + 8$$
$67$ $$T^{3} + 12 T^{2} - 16 T - 64$$
$71$ $$T^{3} + 12 T^{2} - 96 T - 944$$
$73$ $$T^{3} - 10 T^{2} - 44 T + 388$$
$79$ $$T^{3} + 10 T^{2} - 212 T - 1720$$
$83$ $$T^{3} - 10 T^{2} - 60 T + 604$$
$89$ $$T^{3} - 18 T^{2} - 12 T + 520$$
$97$ $$T^{3} + 16 T^{2} - 160 T - 2432$$