# 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 $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [105,4,Mod(1,105)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(105, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("105.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

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

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 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

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

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

comment: embeddings in the coefficient field

gp: mfembed(f)

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

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