# Properties

 Label 105.4.a.a Level $105$ Weight $4$ Character orbit 105.a Self dual yes Analytic conductor $6.195$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Learn more about

## Newspace parameters

 Level: $$N$$ $$=$$ $$105 = 3 \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 105.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$6.19520055060$$ 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

 $$f(q)$$ $$=$$ $$q - 3q^{3} - 8q^{4} + 5q^{5} + 7q^{7} + 9q^{9} + O(q^{10})$$ $$q - 3q^{3} - 8q^{4} + 5q^{5} + 7q^{7} + 9q^{9} + 42q^{11} + 24q^{12} + 20q^{13} - 15q^{15} + 64q^{16} + 66q^{17} + 38q^{19} - 40q^{20} - 21q^{21} + 12q^{23} + 25q^{25} - 27q^{27} - 56q^{28} - 258q^{29} + 146q^{31} - 126q^{33} + 35q^{35} - 72q^{36} + 434q^{37} - 60q^{39} - 282q^{41} + 20q^{43} - 336q^{44} + 45q^{45} - 72q^{47} - 192q^{48} + 49q^{49} - 198q^{51} - 160q^{52} + 336q^{53} + 210q^{55} - 114q^{57} - 360q^{59} + 120q^{60} - 682q^{61} + 63q^{63} - 512q^{64} + 100q^{65} + 812q^{67} - 528q^{68} - 36q^{69} + 810q^{71} - 124q^{73} - 75q^{75} - 304q^{76} + 294q^{77} + 1136q^{79} + 320q^{80} + 81q^{81} + 156q^{83} + 168q^{84} + 330q^{85} + 774q^{87} - 1038q^{89} + 140q^{91} - 96q^{92} - 438q^{93} + 190q^{95} + 1208q^{97} + 378q^{99} + O(q^{100})$$

## 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
0 −3.00000 −8.00000 5.00000 0 7.00000 0 9.00000 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
$$3$$ $$1$$
$$5$$ $$-1$$
$$7$$ $$-1$$

## 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 105.4.a.a 1
3.b odd 2 1 315.4.a.d 1
4.b odd 2 1 1680.4.a.s 1
5.b even 2 1 525.4.a.e 1
5.c odd 4 2 525.4.d.f 2
7.b odd 2 1 735.4.a.c 1
15.d odd 2 1 1575.4.a.f 1
21.c even 2 1 2205.4.a.o 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.4.a.a 1 1.a even 1 1 trivial
315.4.a.d 1 3.b odd 2 1
525.4.a.e 1 5.b even 2 1
525.4.d.f 2 5.c odd 4 2
735.4.a.c 1 7.b odd 2 1
1575.4.a.f 1 15.d odd 2 1
1680.4.a.s 1 4.b odd 2 1
2205.4.a.o 1 21.c even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(105))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$3 + T$$
$5$ $$-5 + T$$
$7$ $$-7 + T$$
$11$ $$-42 + T$$
$13$ $$-20 + T$$
$17$ $$-66 + T$$
$19$ $$-38 + T$$
$23$ $$-12 + T$$
$29$ $$258 + T$$
$31$ $$-146 + T$$
$37$ $$-434 + T$$
$41$ $$282 + T$$
$43$ $$-20 + T$$
$47$ $$72 + T$$
$53$ $$-336 + T$$
$59$ $$360 + T$$
$61$ $$682 + T$$
$67$ $$-812 + T$$
$71$ $$-810 + T$$
$73$ $$124 + T$$
$79$ $$-1136 + T$$
$83$ $$-156 + T$$
$89$ $$1038 + T$$
$97$ $$-1208 + T$$
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