Properties

 Label 825.2.a.a Level $825$ Weight $2$ Character orbit 825.a Self dual yes Analytic conductor $6.588$ Analytic rank $1$ Dimension $1$ 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: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 33) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{3} - q^{4} - q^{6} - 4 q^{7} + 3 q^{8} + q^{9}+O(q^{10})$$ q - q^2 + q^3 - q^4 - q^6 - 4 * q^7 + 3 * q^8 + q^9 $$q - q^{2} + q^{3} - q^{4} - q^{6} - 4 q^{7} + 3 q^{8} + q^{9} + q^{11} - q^{12} + 2 q^{13} + 4 q^{14} - q^{16} + 2 q^{17} - q^{18} - 4 q^{21} - q^{22} - 8 q^{23} + 3 q^{24} - 2 q^{26} + q^{27} + 4 q^{28} - 6 q^{29} - 8 q^{31} - 5 q^{32} + q^{33} - 2 q^{34} - q^{36} - 6 q^{37} + 2 q^{39} - 2 q^{41} + 4 q^{42} - q^{44} + 8 q^{46} - 8 q^{47} - q^{48} + 9 q^{49} + 2 q^{51} - 2 q^{52} - 6 q^{53} - q^{54} - 12 q^{56} + 6 q^{58} - 4 q^{59} + 6 q^{61} + 8 q^{62} - 4 q^{63} + 7 q^{64} - q^{66} + 4 q^{67} - 2 q^{68} - 8 q^{69} + 3 q^{72} + 14 q^{73} + 6 q^{74} - 4 q^{77} - 2 q^{78} - 4 q^{79} + q^{81} + 2 q^{82} - 12 q^{83} + 4 q^{84} - 6 q^{87} + 3 q^{88} - 6 q^{89} - 8 q^{91} + 8 q^{92} - 8 q^{93} + 8 q^{94} - 5 q^{96} - 2 q^{97} - 9 q^{98} + q^{99}+O(q^{100})$$ q - q^2 + q^3 - q^4 - q^6 - 4 * q^7 + 3 * q^8 + q^9 + q^11 - q^12 + 2 * q^13 + 4 * q^14 - q^16 + 2 * q^17 - q^18 - 4 * q^21 - q^22 - 8 * q^23 + 3 * q^24 - 2 * q^26 + q^27 + 4 * q^28 - 6 * q^29 - 8 * q^31 - 5 * q^32 + q^33 - 2 * q^34 - q^36 - 6 * q^37 + 2 * q^39 - 2 * q^41 + 4 * q^42 - q^44 + 8 * q^46 - 8 * q^47 - q^48 + 9 * q^49 + 2 * q^51 - 2 * q^52 - 6 * q^53 - q^54 - 12 * q^56 + 6 * q^58 - 4 * q^59 + 6 * q^61 + 8 * q^62 - 4 * q^63 + 7 * q^64 - q^66 + 4 * q^67 - 2 * q^68 - 8 * q^69 + 3 * q^72 + 14 * q^73 + 6 * q^74 - 4 * q^77 - 2 * q^78 - 4 * q^79 + q^81 + 2 * q^82 - 12 * q^83 + 4 * q^84 - 6 * q^87 + 3 * q^88 - 6 * q^89 - 8 * q^91 + 8 * q^92 - 8 * q^93 + 8 * q^94 - 5 * q^96 - 2 * q^97 - 9 * q^98 + 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
 0
−1.00000 1.00000 −1.00000 0 −1.00000 −4.00000 3.00000 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.a 1
3.b odd 2 1 2475.2.a.g 1
5.b even 2 1 33.2.a.a 1
5.c odd 4 2 825.2.c.a 2
11.b odd 2 1 9075.2.a.q 1
15.d odd 2 1 99.2.a.b 1
15.e even 4 2 2475.2.c.d 2
20.d odd 2 1 528.2.a.g 1
35.c odd 2 1 1617.2.a.j 1
40.e odd 2 1 2112.2.a.j 1
40.f even 2 1 2112.2.a.bb 1
45.h odd 6 2 891.2.e.g 2
45.j even 6 2 891.2.e.e 2
55.d odd 2 1 363.2.a.b 1
55.h odd 10 4 363.2.e.g 4
55.j even 10 4 363.2.e.e 4
60.h even 2 1 1584.2.a.o 1
65.d even 2 1 5577.2.a.a 1
85.c even 2 1 9537.2.a.m 1
105.g even 2 1 4851.2.a.b 1
120.i odd 2 1 6336.2.a.x 1
120.m even 2 1 6336.2.a.n 1
165.d even 2 1 1089.2.a.j 1
220.g even 2 1 5808.2.a.t 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.2.a.a 1 5.b even 2 1
99.2.a.b 1 15.d odd 2 1
363.2.a.b 1 55.d odd 2 1
363.2.e.e 4 55.j even 10 4
363.2.e.g 4 55.h odd 10 4
528.2.a.g 1 20.d odd 2 1
825.2.a.a 1 1.a even 1 1 trivial
825.2.c.a 2 5.c odd 4 2
891.2.e.e 2 45.j even 6 2
891.2.e.g 2 45.h odd 6 2
1089.2.a.j 1 165.d even 2 1
1584.2.a.o 1 60.h even 2 1
1617.2.a.j 1 35.c odd 2 1
2112.2.a.j 1 40.e odd 2 1
2112.2.a.bb 1 40.f even 2 1
2475.2.a.g 1 3.b odd 2 1
2475.2.c.d 2 15.e even 4 2
4851.2.a.b 1 105.g even 2 1
5577.2.a.a 1 65.d even 2 1
5808.2.a.t 1 220.g even 2 1
6336.2.a.n 1 120.m even 2 1
6336.2.a.x 1 120.i odd 2 1
9075.2.a.q 1 11.b odd 2 1
9537.2.a.m 1 85.c even 2 1

Hecke kernels

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

 $$T_{2} + 1$$ T2 + 1 $$T_{7} + 4$$ T7 + 4

Hecke characteristic polynomials

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