# Properties

 Label 105.2.g.a Level 105 Weight 2 Character orbit 105.g Analytic conductor 0.838 Analytic rank 0 Dimension 4 CM no Inner twists 4

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$0.838429221223$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{3})$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ 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 basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{2} q^{2} + ( -1 - \beta_{1} ) q^{3} + q^{4} + ( \beta_{1} - \beta_{2} ) q^{5} + ( \beta_{2} + \beta_{3} ) q^{6} + ( 1 + \beta_{3} ) q^{7} + \beta_{2} q^{8} + ( -1 + 2 \beta_{1} ) q^{9} +O(q^{10})$$ $$q -\beta_{2} q^{2} + ( -1 - \beta_{1} ) q^{3} + q^{4} + ( \beta_{1} - \beta_{2} ) q^{5} + ( \beta_{2} + \beta_{3} ) q^{6} + ( 1 + \beta_{3} ) q^{7} + \beta_{2} q^{8} + ( -1 + 2 \beta_{1} ) q^{9} + ( 3 - \beta_{3} ) q^{10} + 2 \beta_{1} q^{11} + ( -1 - \beta_{1} ) q^{12} -4 q^{13} + ( -3 \beta_{1} - \beta_{2} ) q^{14} + ( 2 - \beta_{1} + \beta_{2} + \beta_{3} ) q^{15} -5 q^{16} + 2 \beta_{1} q^{17} + ( \beta_{2} - 2 \beta_{3} ) q^{18} + ( \beta_{1} - \beta_{2} ) q^{20} + ( -1 - \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{21} -2 \beta_{3} q^{22} -2 \beta_{2} q^{23} + ( -\beta_{2} - \beta_{3} ) q^{24} + ( 1 - 2 \beta_{3} ) q^{25} + 4 \beta_{2} q^{26} + ( 5 - \beta_{1} ) q^{27} + ( 1 + \beta_{3} ) q^{28} -4 \beta_{1} q^{29} + ( -3 - 3 \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{30} + 4 \beta_{3} q^{31} + 3 \beta_{2} q^{32} + ( 4 - 2 \beta_{1} ) q^{33} -2 \beta_{3} q^{34} + ( -2 \beta_{1} - 3 \beta_{2} ) q^{35} + ( -1 + 2 \beta_{1} ) q^{36} + ( 4 + 4 \beta_{1} ) q^{39} + ( -3 + \beta_{3} ) q^{40} -2 \beta_{2} q^{41} + ( -6 + 3 \beta_{1} + \beta_{2} + \beta_{3} ) q^{42} + 2 \beta_{3} q^{43} + 2 \beta_{1} q^{44} + ( -4 - \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{45} + 6 q^{46} + 2 \beta_{1} q^{47} + ( 5 + 5 \beta_{1} ) q^{48} + ( -5 + 2 \beta_{3} ) q^{49} + ( 6 \beta_{1} - \beta_{2} ) q^{50} + ( 4 - 2 \beta_{1} ) q^{51} -4 q^{52} + ( -5 \beta_{2} + \beta_{3} ) q^{54} + ( -4 - 2 \beta_{3} ) q^{55} + ( 3 \beta_{1} + \beta_{2} ) q^{56} + 4 \beta_{3} q^{58} + 4 \beta_{2} q^{59} + ( 2 - \beta_{1} + \beta_{2} + \beta_{3} ) q^{60} -4 \beta_{3} q^{61} -12 \beta_{1} q^{62} + ( -1 + 2 \beta_{1} - 4 \beta_{2} - \beta_{3} ) q^{63} + q^{64} + ( -4 \beta_{1} + 4 \beta_{2} ) q^{65} + ( -4 \beta_{2} + 2 \beta_{3} ) q^{66} -2 \beta_{3} q^{67} + 2 \beta_{1} q^{68} + ( 2 \beta_{2} + 2 \beta_{3} ) q^{69} + ( 9 + 2 \beta_{3} ) q^{70} + 2 \beta_{1} q^{71} + ( -\beta_{2} + 2 \beta_{3} ) q^{72} + 8 q^{73} + ( -1 - \beta_{1} - 4 \beta_{2} + 2 \beta_{3} ) q^{75} + ( 2 \beta_{1} - 4 \beta_{2} ) q^{77} + ( -4 \beta_{2} - 4 \beta_{3} ) q^{78} + 8 q^{79} + ( -5 \beta_{1} + 5 \beta_{2} ) q^{80} + ( -7 - 4 \beta_{1} ) q^{81} + 6 q^{82} + 2 \beta_{1} q^{83} + ( -1 - \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{84} + ( -4 - 2 \beta_{3} ) q^{85} -6 \beta_{1} q^{86} + ( -8 + 4 \beta_{1} ) q^{87} + 2 \beta_{3} q^{88} + 6 \beta_{2} q^{89} + ( -3 + 6 \beta_{1} + 4 \beta_{2} + \beta_{3} ) q^{90} + ( -4 - 4 \beta_{3} ) q^{91} -2 \beta_{2} q^{92} + ( 8 \beta_{2} - 4 \beta_{3} ) q^{93} -2 \beta_{3} q^{94} + ( -3 \beta_{2} - 3 \beta_{3} ) q^{96} + 8 q^{97} + ( -6 \beta_{1} + 5 \beta_{2} ) q^{98} + ( -8 - 2 \beta_{1} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 4q^{3} + 4q^{4} + 4q^{7} - 4q^{9} + O(q^{10})$$ $$4q - 4q^{3} + 4q^{4} + 4q^{7} - 4q^{9} + 12q^{10} - 4q^{12} - 16q^{13} + 8q^{15} - 20q^{16} - 4q^{21} + 4q^{25} + 20q^{27} + 4q^{28} - 12q^{30} + 16q^{33} - 4q^{36} + 16q^{39} - 12q^{40} - 24q^{42} - 16q^{45} + 24q^{46} + 20q^{48} - 20q^{49} + 16q^{51} - 16q^{52} - 16q^{55} + 8q^{60} - 4q^{63} + 4q^{64} + 36q^{70} + 32q^{73} - 4q^{75} + 32q^{79} - 28q^{81} + 24q^{82} - 4q^{84} - 16q^{85} - 32q^{87} - 12q^{90} - 16q^{91} + 32q^{97} - 32q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 4 x^{2} + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{3} + 3 \nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} + 2$$ $$\beta_{3}$$ $$=$$ $$\nu^{3} + 5 \nu$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} - \beta_{1}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} - 2$$ $$\nu^{3}$$ $$=$$ $$($$$$-3 \beta_{3} + 5 \beta_{1}$$$$)/2$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
104.1
 0.517638i − 0.517638i − 1.93185i 1.93185i
−1.73205 −1.00000 1.41421i 1.00000 −1.73205 + 1.41421i 1.73205 + 2.44949i 1.00000 + 2.44949i 1.73205 −1.00000 + 2.82843i 3.00000 2.44949i
104.2 −1.73205 −1.00000 + 1.41421i 1.00000 −1.73205 1.41421i 1.73205 2.44949i 1.00000 2.44949i 1.73205 −1.00000 2.82843i 3.00000 + 2.44949i
104.3 1.73205 −1.00000 1.41421i 1.00000 1.73205 + 1.41421i −1.73205 2.44949i 1.00000 2.44949i −1.73205 −1.00000 + 2.82843i 3.00000 + 2.44949i
104.4 1.73205 −1.00000 + 1.41421i 1.00000 1.73205 1.41421i −1.73205 + 2.44949i 1.00000 + 2.44949i −1.73205 −1.00000 2.82843i 3.00000 2.44949i
 $$n$$: 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
3.b odd 2 1 inner
35.c odd 2 1 inner
105.g 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.g.a 4
3.b odd 2 1 inner 105.2.g.a 4
4.b odd 2 1 1680.2.k.c 4
5.b even 2 1 105.2.g.c yes 4
5.c odd 4 2 525.2.b.j 8
7.b odd 2 1 105.2.g.c yes 4
7.c even 3 2 735.2.p.c 8
7.d odd 6 2 735.2.p.a 8
12.b even 2 1 1680.2.k.c 4
15.d odd 2 1 105.2.g.c yes 4
15.e even 4 2 525.2.b.j 8
20.d odd 2 1 1680.2.k.a 4
21.c even 2 1 105.2.g.c yes 4
21.g even 6 2 735.2.p.a 8
21.h odd 6 2 735.2.p.c 8
28.d even 2 1 1680.2.k.a 4
35.c odd 2 1 inner 105.2.g.a 4
35.f even 4 2 525.2.b.j 8
35.i odd 6 2 735.2.p.c 8
35.j even 6 2 735.2.p.a 8
60.h even 2 1 1680.2.k.a 4
84.h odd 2 1 1680.2.k.a 4
105.g even 2 1 inner 105.2.g.a 4
105.k odd 4 2 525.2.b.j 8
105.o odd 6 2 735.2.p.a 8
105.p even 6 2 735.2.p.c 8
140.c even 2 1 1680.2.k.c 4
420.o odd 2 1 1680.2.k.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.2.g.a 4 1.a even 1 1 trivial
105.2.g.a 4 3.b odd 2 1 inner
105.2.g.a 4 35.c odd 2 1 inner
105.2.g.a 4 105.g even 2 1 inner
105.2.g.c yes 4 5.b even 2 1
105.2.g.c yes 4 7.b odd 2 1
105.2.g.c yes 4 15.d odd 2 1
105.2.g.c yes 4 21.c even 2 1
525.2.b.j 8 5.c odd 4 2
525.2.b.j 8 15.e even 4 2
525.2.b.j 8 35.f even 4 2
525.2.b.j 8 105.k odd 4 2
735.2.p.a 8 7.d odd 6 2
735.2.p.a 8 21.g even 6 2
735.2.p.a 8 35.j even 6 2
735.2.p.a 8 105.o odd 6 2
735.2.p.c 8 7.c even 3 2
735.2.p.c 8 21.h odd 6 2
735.2.p.c 8 35.i odd 6 2
735.2.p.c 8 105.p even 6 2
1680.2.k.a 4 20.d odd 2 1
1680.2.k.a 4 28.d even 2 1
1680.2.k.a 4 60.h even 2 1
1680.2.k.a 4 84.h odd 2 1
1680.2.k.c 4 4.b odd 2 1
1680.2.k.c 4 12.b even 2 1
1680.2.k.c 4 140.c even 2 1
1680.2.k.c 4 420.o odd 2 1

## Hecke kernels

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

 $$T_{2}^{2} - 3$$ $$T_{13} + 4$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T^{2} + 4 T^{4} )^{2}$$
$3$ $$( 1 + 2 T + 3 T^{2} )^{2}$$
$5$ $$1 - 2 T^{2} + 25 T^{4}$$
$7$ $$( 1 - 2 T + 7 T^{2} )^{2}$$
$11$ $$( 1 - 6 T + 11 T^{2} )^{2}( 1 + 6 T + 11 T^{2} )^{2}$$
$13$ $$( 1 + 4 T + 13 T^{2} )^{4}$$
$17$ $$( 1 - 26 T^{2} + 289 T^{4} )^{2}$$
$19$ $$( 1 - 19 T^{2} )^{4}$$
$23$ $$( 1 + 34 T^{2} + 529 T^{4} )^{2}$$
$29$ $$( 1 - 26 T^{2} + 841 T^{4} )^{2}$$
$31$ $$( 1 + 34 T^{2} + 961 T^{4} )^{2}$$
$37$ $$( 1 - 37 T^{2} )^{4}$$
$41$ $$( 1 + 70 T^{2} + 1681 T^{4} )^{2}$$
$43$ $$( 1 - 62 T^{2} + 1849 T^{4} )^{2}$$
$47$ $$( 1 - 86 T^{2} + 2209 T^{4} )^{2}$$
$53$ $$( 1 + 53 T^{2} )^{4}$$
$59$ $$( 1 + 70 T^{2} + 3481 T^{4} )^{2}$$
$61$ $$( 1 - 26 T^{2} + 3721 T^{4} )^{2}$$
$67$ $$( 1 - 110 T^{2} + 4489 T^{4} )^{2}$$
$71$ $$( 1 - 134 T^{2} + 5041 T^{4} )^{2}$$
$73$ $$( 1 - 8 T + 73 T^{2} )^{4}$$
$79$ $$( 1 - 8 T + 79 T^{2} )^{4}$$
$83$ $$( 1 - 18 T + 83 T^{2} )^{2}( 1 + 18 T + 83 T^{2} )^{2}$$
$89$ $$( 1 + 70 T^{2} + 7921 T^{4} )^{2}$$
$97$ $$( 1 - 8 T + 97 T^{2} )^{4}$$