# Properties

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

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$105 = 3 \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 105.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.838429221223$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 1 - 2 \zeta_{6} ) q^{2} + ( 1 - 2 \zeta_{6} ) q^{3} - q^{4} - q^{5} -3 q^{6} + ( 1 + 2 \zeta_{6} ) q^{7} + ( 1 - 2 \zeta_{6} ) q^{8} -3 q^{9} +O(q^{10})$$ $$q + ( 1 - 2 \zeta_{6} ) q^{2} + ( 1 - 2 \zeta_{6} ) q^{3} - q^{4} - q^{5} -3 q^{6} + ( 1 + 2 \zeta_{6} ) q^{7} + ( 1 - 2 \zeta_{6} ) q^{8} -3 q^{9} + ( -1 + 2 \zeta_{6} ) q^{10} + ( -2 + 4 \zeta_{6} ) q^{11} + ( -1 + 2 \zeta_{6} ) q^{12} + ( 5 - 4 \zeta_{6} ) q^{14} + ( -1 + 2 \zeta_{6} ) q^{15} -5 q^{16} + 6 q^{17} + ( -3 + 6 \zeta_{6} ) q^{18} + ( 2 - 4 \zeta_{6} ) q^{19} + q^{20} + ( 5 - 4 \zeta_{6} ) q^{21} + 6 q^{22} + ( 2 - 4 \zeta_{6} ) q^{23} -3 q^{24} + q^{25} + ( -3 + 6 \zeta_{6} ) q^{27} + ( -1 - 2 \zeta_{6} ) q^{28} + ( -4 + 8 \zeta_{6} ) q^{29} + 3 q^{30} + ( -2 + 4 \zeta_{6} ) q^{31} + ( -3 + 6 \zeta_{6} ) q^{32} + 6 q^{33} + ( 6 - 12 \zeta_{6} ) q^{34} + ( -1 - 2 \zeta_{6} ) q^{35} + 3 q^{36} -2 q^{37} -6 q^{38} + ( -1 + 2 \zeta_{6} ) q^{40} + 6 q^{41} + ( -3 - 6 \zeta_{6} ) q^{42} -8 q^{43} + ( 2 - 4 \zeta_{6} ) q^{44} + 3 q^{45} -6 q^{46} -12 q^{47} + ( -5 + 10 \zeta_{6} ) q^{48} + ( -3 + 8 \zeta_{6} ) q^{49} + ( 1 - 2 \zeta_{6} ) q^{50} + ( 6 - 12 \zeta_{6} ) q^{51} + 9 q^{54} + ( 2 - 4 \zeta_{6} ) q^{55} + ( 5 - 4 \zeta_{6} ) q^{56} -6 q^{57} + 12 q^{58} -12 q^{59} + ( 1 - 2 \zeta_{6} ) q^{60} + ( -4 + 8 \zeta_{6} ) q^{61} + 6 q^{62} + ( -3 - 6 \zeta_{6} ) q^{63} - q^{64} + ( 6 - 12 \zeta_{6} ) q^{66} + 8 q^{67} -6 q^{68} -6 q^{69} + ( -5 + 4 \zeta_{6} ) q^{70} + ( 2 - 4 \zeta_{6} ) q^{71} + ( -3 + 6 \zeta_{6} ) q^{72} + ( 4 - 8 \zeta_{6} ) q^{73} + ( -2 + 4 \zeta_{6} ) q^{74} + ( 1 - 2 \zeta_{6} ) q^{75} + ( -2 + 4 \zeta_{6} ) q^{76} + ( -10 + 8 \zeta_{6} ) q^{77} + 8 q^{79} + 5 q^{80} + 9 q^{81} + ( 6 - 12 \zeta_{6} ) q^{82} + ( -5 + 4 \zeta_{6} ) q^{84} -6 q^{85} + ( -8 + 16 \zeta_{6} ) q^{86} + 12 q^{87} + 6 q^{88} + 6 q^{89} + ( 3 - 6 \zeta_{6} ) q^{90} + ( -2 + 4 \zeta_{6} ) q^{92} + 6 q^{93} + ( -12 + 24 \zeta_{6} ) q^{94} + ( -2 + 4 \zeta_{6} ) q^{95} + 9 q^{96} + ( 4 - 8 \zeta_{6} ) q^{97} + ( 13 - 2 \zeta_{6} ) q^{98} + ( 6 - 12 \zeta_{6} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{4} - 2q^{5} - 6q^{6} + 4q^{7} - 6q^{9} + O(q^{10})$$ $$2q - 2q^{4} - 2q^{5} - 6q^{6} + 4q^{7} - 6q^{9} + 6q^{14} - 10q^{16} + 12q^{17} + 2q^{20} + 6q^{21} + 12q^{22} - 6q^{24} + 2q^{25} - 4q^{28} + 6q^{30} + 12q^{33} - 4q^{35} + 6q^{36} - 4q^{37} - 12q^{38} + 12q^{41} - 12q^{42} - 16q^{43} + 6q^{45} - 12q^{46} - 24q^{47} + 2q^{49} + 18q^{54} + 6q^{56} - 12q^{57} + 24q^{58} - 24q^{59} + 12q^{62} - 12q^{63} - 2q^{64} + 16q^{67} - 12q^{68} - 12q^{69} - 6q^{70} - 12q^{77} + 16q^{79} + 10q^{80} + 18q^{81} - 6q^{84} - 12q^{85} + 24q^{87} + 12q^{88} + 12q^{89} + 12q^{93} + 18q^{96} + 24q^{98} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/105\mathbb{Z}\right)^\times$$.

 $$n$$ $$22$$ $$31$$ $$71$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
41.1
 0.5 + 0.866025i 0.5 − 0.866025i
1.73205i 1.73205i −1.00000 −1.00000 −3.00000 2.00000 + 1.73205i 1.73205i −3.00000 1.73205i
41.2 1.73205i 1.73205i −1.00000 −1.00000 −3.00000 2.00000 1.73205i 1.73205i −3.00000 1.73205i
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
21.c even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 105.2.b.a 2
3.b odd 2 1 105.2.b.b yes 2
4.b odd 2 1 1680.2.f.b 2
5.b even 2 1 525.2.b.a 2
5.c odd 4 2 525.2.g.b 4
7.b odd 2 1 105.2.b.b yes 2
7.c even 3 1 735.2.s.b 2
7.c even 3 1 735.2.s.d 2
7.d odd 6 1 735.2.s.a 2
7.d odd 6 1 735.2.s.f 2
12.b even 2 1 1680.2.f.c 2
15.d odd 2 1 525.2.b.b 2
15.e even 4 2 525.2.g.c 4
21.c even 2 1 inner 105.2.b.a 2
21.g even 6 1 735.2.s.b 2
21.g even 6 1 735.2.s.d 2
21.h odd 6 1 735.2.s.a 2
21.h odd 6 1 735.2.s.f 2
28.d even 2 1 1680.2.f.c 2
35.c odd 2 1 525.2.b.b 2
35.f even 4 2 525.2.g.c 4
84.h odd 2 1 1680.2.f.b 2
105.g even 2 1 525.2.b.a 2
105.k odd 4 2 525.2.g.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.2.b.a 2 1.a even 1 1 trivial
105.2.b.a 2 21.c even 2 1 inner
105.2.b.b yes 2 3.b odd 2 1
105.2.b.b yes 2 7.b odd 2 1
525.2.b.a 2 5.b even 2 1
525.2.b.a 2 105.g even 2 1
525.2.b.b 2 15.d odd 2 1
525.2.b.b 2 35.c odd 2 1
525.2.g.b 4 5.c odd 4 2
525.2.g.b 4 105.k odd 4 2
525.2.g.c 4 15.e even 4 2
525.2.g.c 4 35.f even 4 2
735.2.s.a 2 7.d odd 6 1
735.2.s.a 2 21.h odd 6 1
735.2.s.b 2 7.c even 3 1
735.2.s.b 2 21.g even 6 1
735.2.s.d 2 7.c even 3 1
735.2.s.d 2 21.g even 6 1
735.2.s.f 2 7.d odd 6 1
735.2.s.f 2 21.h odd 6 1
1680.2.f.b 2 4.b odd 2 1
1680.2.f.b 2 84.h odd 2 1
1680.2.f.c 2 12.b even 2 1
1680.2.f.c 2 28.d even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(105, [\chi])$$:

 $$T_{2}^{2} + 3$$ $$T_{17} - 6$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$3 + T^{2}$$
$3$ $$3 + T^{2}$$
$5$ $$( 1 + T )^{2}$$
$7$ $$7 - 4 T + T^{2}$$
$11$ $$12 + T^{2}$$
$13$ $$T^{2}$$
$17$ $$( -6 + T )^{2}$$
$19$ $$12 + T^{2}$$
$23$ $$12 + T^{2}$$
$29$ $$48 + T^{2}$$
$31$ $$12 + T^{2}$$
$37$ $$( 2 + T )^{2}$$
$41$ $$( -6 + T )^{2}$$
$43$ $$( 8 + T )^{2}$$
$47$ $$( 12 + T )^{2}$$
$53$ $$T^{2}$$
$59$ $$( 12 + T )^{2}$$
$61$ $$48 + T^{2}$$
$67$ $$( -8 + T )^{2}$$
$71$ $$12 + T^{2}$$
$73$ $$48 + T^{2}$$
$79$ $$( -8 + T )^{2}$$
$83$ $$T^{2}$$
$89$ $$( -6 + T )^{2}$$
$97$ $$48 + T^{2}$$