# Properties

 Label 105.2.a.a Level $105$ Weight $2$ Character orbit 105.a Self dual yes Analytic conductor $0.838$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$105 = 3 \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 105.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$0.838429221223$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{3} - q^{4} + q^{5} + q^{6} + q^{7} - 3 q^{8} + q^{9}+O(q^{10})$$ q + q^2 + q^3 - q^4 + q^5 + q^6 + q^7 - 3 * q^8 + q^9 $$q + q^{2} + q^{3} - q^{4} + q^{5} + q^{6} + q^{7} - 3 q^{8} + q^{9} + q^{10} - q^{12} - 6 q^{13} + q^{14} + q^{15} - q^{16} + 2 q^{17} + q^{18} - 8 q^{19} - q^{20} + q^{21} + 8 q^{23} - 3 q^{24} + q^{25} - 6 q^{26} + q^{27} - q^{28} - 2 q^{29} + q^{30} + 4 q^{31} + 5 q^{32} + 2 q^{34} + q^{35} - q^{36} - 2 q^{37} - 8 q^{38} - 6 q^{39} - 3 q^{40} - 6 q^{41} + q^{42} + 4 q^{43} + q^{45} + 8 q^{46} + 8 q^{47} - q^{48} + q^{49} + q^{50} + 2 q^{51} + 6 q^{52} + 10 q^{53} + q^{54} - 3 q^{56} - 8 q^{57} - 2 q^{58} + 4 q^{59} - q^{60} - 2 q^{61} + 4 q^{62} + q^{63} + 7 q^{64} - 6 q^{65} + 4 q^{67} - 2 q^{68} + 8 q^{69} + q^{70} - 12 q^{71} - 3 q^{72} - 2 q^{73} - 2 q^{74} + q^{75} + 8 q^{76} - 6 q^{78} + 8 q^{79} - q^{80} + q^{81} - 6 q^{82} - 4 q^{83} - q^{84} + 2 q^{85} + 4 q^{86} - 2 q^{87} - 6 q^{89} + q^{90} - 6 q^{91} - 8 q^{92} + 4 q^{93} + 8 q^{94} - 8 q^{95} + 5 q^{96} - 18 q^{97} + q^{98}+O(q^{100})$$ q + q^2 + q^3 - q^4 + q^5 + q^6 + q^7 - 3 * q^8 + q^9 + q^10 - q^12 - 6 * q^13 + q^14 + q^15 - q^16 + 2 * q^17 + q^18 - 8 * q^19 - q^20 + q^21 + 8 * q^23 - 3 * q^24 + q^25 - 6 * q^26 + q^27 - q^28 - 2 * q^29 + q^30 + 4 * q^31 + 5 * q^32 + 2 * q^34 + q^35 - q^36 - 2 * q^37 - 8 * q^38 - 6 * q^39 - 3 * q^40 - 6 * q^41 + q^42 + 4 * q^43 + q^45 + 8 * q^46 + 8 * q^47 - q^48 + q^49 + q^50 + 2 * q^51 + 6 * q^52 + 10 * q^53 + q^54 - 3 * q^56 - 8 * q^57 - 2 * q^58 + 4 * q^59 - q^60 - 2 * q^61 + 4 * q^62 + q^63 + 7 * q^64 - 6 * q^65 + 4 * q^67 - 2 * q^68 + 8 * q^69 + q^70 - 12 * q^71 - 3 * q^72 - 2 * q^73 - 2 * q^74 + q^75 + 8 * q^76 - 6 * q^78 + 8 * q^79 - q^80 + q^81 - 6 * q^82 - 4 * q^83 - q^84 + 2 * q^85 + 4 * q^86 - 2 * q^87 - 6 * q^89 + q^90 - 6 * q^91 - 8 * q^92 + 4 * q^93 + 8 * q^94 - 8 * q^95 + 5 * q^96 - 18 * q^97 + q^98

## 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 1.00000 1.00000 1.00000 −3.00000 1.00000 1.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$5$$ $$-1$$
$$7$$ $$-1$$

## 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 105.2.a.a 1
3.b odd 2 1 315.2.a.a 1
4.b odd 2 1 1680.2.a.f 1
5.b even 2 1 525.2.a.a 1
5.c odd 4 2 525.2.d.b 2
7.b odd 2 1 735.2.a.f 1
7.c even 3 2 735.2.i.a 2
7.d odd 6 2 735.2.i.b 2
8.b even 2 1 6720.2.a.p 1
8.d odd 2 1 6720.2.a.bk 1
12.b even 2 1 5040.2.a.d 1
15.d odd 2 1 1575.2.a.h 1
15.e even 4 2 1575.2.d.b 2
20.d odd 2 1 8400.2.a.co 1
21.c even 2 1 2205.2.a.b 1
35.c odd 2 1 3675.2.a.f 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.2.a.a 1 1.a even 1 1 trivial
315.2.a.a 1 3.b odd 2 1
525.2.a.a 1 5.b even 2 1
525.2.d.b 2 5.c odd 4 2
735.2.a.f 1 7.b odd 2 1
735.2.i.a 2 7.c even 3 2
735.2.i.b 2 7.d odd 6 2
1575.2.a.h 1 15.d odd 2 1
1575.2.d.b 2 15.e even 4 2
1680.2.a.f 1 4.b odd 2 1
2205.2.a.b 1 21.c even 2 1
3675.2.a.f 1 35.c odd 2 1
5040.2.a.d 1 12.b even 2 1
6720.2.a.p 1 8.b even 2 1
6720.2.a.bk 1 8.d odd 2 1
8400.2.a.co 1 20.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2} - 1$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(105))$$.

## Hecke characteristic polynomials

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