# Properties

 Label 825.6.a.a Level $825$ Weight $6$ Character orbit 825.a Self dual yes Analytic conductor $132.317$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$825 = 3 \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 825.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$132.316651346$$ Analytic rank: $$0$$ 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

 $$f(q)$$ $$=$$ $$q - q^{2} - 9q^{3} - 31q^{4} + 9q^{6} + 26q^{7} + 63q^{8} + 81q^{9} + O(q^{10})$$ $$q - q^{2} - 9q^{3} - 31q^{4} + 9q^{6} + 26q^{7} + 63q^{8} + 81q^{9} + 121q^{11} + 279q^{12} + 692q^{13} - 26q^{14} + 929q^{16} + 1442q^{17} - 81q^{18} + 2160q^{19} - 234q^{21} - 121q^{22} + 1582q^{23} - 567q^{24} - 692q^{26} - 729q^{27} - 806q^{28} - 5526q^{29} + 4792q^{31} - 2945q^{32} - 1089q^{33} - 1442q^{34} - 2511q^{36} + 10194q^{37} - 2160q^{38} - 6228q^{39} - 10622q^{41} + 234q^{42} - 8580q^{43} - 3751q^{44} - 1582q^{46} + 2362q^{47} - 8361q^{48} - 16131q^{49} - 12978q^{51} - 21452q^{52} + 30804q^{53} + 729q^{54} + 1638q^{56} - 19440q^{57} + 5526q^{58} + 6416q^{59} + 42096q^{61} - 4792q^{62} + 2106q^{63} - 26783q^{64} + 1089q^{66} + 28444q^{67} - 44702q^{68} - 14238q^{69} + 45690q^{71} + 5103q^{72} + 18374q^{73} - 10194q^{74} - 66960q^{76} + 3146q^{77} + 6228q^{78} - 105214q^{79} + 6561q^{81} + 10622q^{82} - 62292q^{83} + 7254q^{84} + 8580q^{86} + 49734q^{87} + 7623q^{88} - 72246q^{89} + 17992q^{91} - 49042q^{92} - 43128q^{93} - 2362q^{94} + 26505q^{96} - 79262q^{97} + 16131q^{98} + 9801q^{99} + O(q^{100})$$

## 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 −9.00000 −31.0000 0 9.00000 26.0000 63.0000 81.0000 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 825.6.a.a 1
5.b even 2 1 33.6.a.b 1
15.d odd 2 1 99.6.a.a 1
20.d odd 2 1 528.6.a.a 1
55.d odd 2 1 363.6.a.b 1
165.d even 2 1 1089.6.a.h 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.6.a.b 1 5.b even 2 1
99.6.a.a 1 15.d odd 2 1
363.6.a.b 1 55.d odd 2 1
528.6.a.a 1 20.d odd 2 1
825.6.a.a 1 1.a even 1 1 trivial
1089.6.a.h 1 165.d even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2} + 1$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(825))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T$$
$3$ $$9 + T$$
$5$ $$T$$
$7$ $$-26 + T$$
$11$ $$-121 + T$$
$13$ $$-692 + T$$
$17$ $$-1442 + T$$
$19$ $$-2160 + T$$
$23$ $$-1582 + T$$
$29$ $$5526 + T$$
$31$ $$-4792 + T$$
$37$ $$-10194 + T$$
$41$ $$10622 + T$$
$43$ $$8580 + T$$
$47$ $$-2362 + T$$
$53$ $$-30804 + T$$
$59$ $$-6416 + T$$
$61$ $$-42096 + T$$
$67$ $$-28444 + T$$
$71$ $$-45690 + T$$
$73$ $$-18374 + T$$
$79$ $$105214 + T$$
$83$ $$62292 + T$$
$89$ $$72246 + T$$
$97$ $$79262 + T$$