# Properties

 Label 105.2.g.b Level 105 Weight 2 Character orbit 105.g Analytic conductor 0.838 Analytic rank 0 Dimension 4 CM discriminant -35 Inner twists 8

# 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: $$\Q(\sqrt{-5}, \sqrt{7})$$ Defining polynomial: $$x^{4} - x^{2} + 9$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{3} q^{3} -2 q^{4} + ( \beta_{1} + \beta_{3} ) q^{5} + ( -\beta_{1} + \beta_{3} ) q^{7} + ( 1 - \beta_{2} ) q^{9} +O(q^{10})$$ $$q + \beta_{3} q^{3} -2 q^{4} + ( \beta_{1} + \beta_{3} ) q^{5} + ( -\beta_{1} + \beta_{3} ) q^{7} + ( 1 - \beta_{2} ) q^{9} + ( -1 + 2 \beta_{2} ) q^{11} -2 \beta_{3} q^{12} + ( \beta_{1} - \beta_{3} ) q^{13} + ( -2 - \beta_{2} ) q^{15} + 4 q^{16} + ( -\beta_{1} - \beta_{3} ) q^{17} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{20} + ( 4 - \beta_{2} ) q^{21} -5 q^{25} + ( 3 \beta_{1} + \beta_{3} ) q^{27} + ( 2 \beta_{1} - 2 \beta_{3} ) q^{28} + ( -1 + 2 \beta_{2} ) q^{29} + ( -6 \beta_{1} - \beta_{3} ) q^{33} + ( 1 - 2 \beta_{2} ) q^{35} + ( -2 + 2 \beta_{2} ) q^{36} + ( -4 + \beta_{2} ) q^{39} + ( 2 - 4 \beta_{2} ) q^{44} + ( 3 \beta_{1} - 2 \beta_{3} ) q^{45} + ( 5 \beta_{1} + 5 \beta_{3} ) q^{47} + 4 \beta_{3} q^{48} + 7 q^{49} + ( 2 + \beta_{2} ) q^{51} + ( -2 \beta_{1} + 2 \beta_{3} ) q^{52} + ( -5 \beta_{1} + 5 \beta_{3} ) q^{55} + ( 4 + 2 \beta_{2} ) q^{60} + ( 3 \beta_{1} + 4 \beta_{3} ) q^{63} -8 q^{64} + ( -1 + 2 \beta_{2} ) q^{65} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{68} + ( 2 - 4 \beta_{2} ) q^{71} + ( 4 \beta_{1} - 4 \beta_{3} ) q^{73} -5 \beta_{3} q^{75} + ( -7 \beta_{1} - 7 \beta_{3} ) q^{77} - q^{79} + ( 4 \beta_{1} + 4 \beta_{3} ) q^{80} + ( -8 - \beta_{2} ) q^{81} + ( -4 \beta_{1} - 4 \beta_{3} ) q^{83} + ( -8 + 2 \beta_{2} ) q^{84} + 5 q^{85} + ( -6 \beta_{1} - \beta_{3} ) q^{87} -7 q^{91} + ( 7 \beta_{1} - 7 \beta_{3} ) q^{97} + ( 17 + \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 8q^{4} + 2q^{9} + O(q^{10})$$ $$4q - 8q^{4} + 2q^{9} - 10q^{15} + 16q^{16} + 14q^{21} - 20q^{25} - 4q^{36} - 14q^{39} + 28q^{49} + 10q^{51} + 20q^{60} - 32q^{64} - 4q^{79} - 34q^{81} - 28q^{84} + 20q^{85} - 28q^{91} + 70q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{2} + 9$$:

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

## 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
 1.32288 − 1.11803i 1.32288 + 1.11803i −1.32288 − 1.11803i −1.32288 + 1.11803i
0 −1.32288 1.11803i −2.00000 2.23607i 0 −2.64575 0 0.500000 + 2.95804i 0
104.2 0 −1.32288 + 1.11803i −2.00000 2.23607i 0 −2.64575 0 0.500000 2.95804i 0
104.3 0 1.32288 1.11803i −2.00000 2.23607i 0 2.64575 0 0.500000 2.95804i 0
104.4 0 1.32288 + 1.11803i −2.00000 2.23607i 0 2.64575 0 0.500000 + 2.95804i 0
 $$n$$: 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
35.c odd 2 1 CM by $$\Q(\sqrt{-35})$$
3.b odd 2 1 inner
5.b even 2 1 inner
7.b odd 2 1 inner
15.d odd 2 1 inner
21.c even 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.b 4
3.b odd 2 1 inner 105.2.g.b 4
4.b odd 2 1 1680.2.k.b 4
5.b even 2 1 inner 105.2.g.b 4
5.c odd 4 2 525.2.b.f 4
7.b odd 2 1 inner 105.2.g.b 4
7.c even 3 2 735.2.p.b 8
7.d odd 6 2 735.2.p.b 8
12.b even 2 1 1680.2.k.b 4
15.d odd 2 1 inner 105.2.g.b 4
15.e even 4 2 525.2.b.f 4
20.d odd 2 1 1680.2.k.b 4
21.c even 2 1 inner 105.2.g.b 4
21.g even 6 2 735.2.p.b 8
21.h odd 6 2 735.2.p.b 8
28.d even 2 1 1680.2.k.b 4
35.c odd 2 1 CM 105.2.g.b 4
35.f even 4 2 525.2.b.f 4
35.i odd 6 2 735.2.p.b 8
35.j even 6 2 735.2.p.b 8
60.h even 2 1 1680.2.k.b 4
84.h odd 2 1 1680.2.k.b 4
105.g even 2 1 inner 105.2.g.b 4
105.k odd 4 2 525.2.b.f 4
105.o odd 6 2 735.2.p.b 8
105.p even 6 2 735.2.p.b 8
140.c even 2 1 1680.2.k.b 4
420.o odd 2 1 1680.2.k.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.2.g.b 4 1.a even 1 1 trivial
105.2.g.b 4 3.b odd 2 1 inner
105.2.g.b 4 5.b even 2 1 inner
105.2.g.b 4 7.b odd 2 1 inner
105.2.g.b 4 15.d odd 2 1 inner
105.2.g.b 4 21.c even 2 1 inner
105.2.g.b 4 35.c odd 2 1 CM
105.2.g.b 4 105.g even 2 1 inner
525.2.b.f 4 5.c odd 4 2
525.2.b.f 4 15.e even 4 2
525.2.b.f 4 35.f even 4 2
525.2.b.f 4 105.k odd 4 2
735.2.p.b 8 7.c even 3 2
735.2.p.b 8 7.d odd 6 2
735.2.p.b 8 21.g even 6 2
735.2.p.b 8 21.h odd 6 2
735.2.p.b 8 35.i odd 6 2
735.2.p.b 8 35.j even 6 2
735.2.p.b 8 105.o odd 6 2
735.2.p.b 8 105.p even 6 2
1680.2.k.b 4 4.b odd 2 1
1680.2.k.b 4 12.b even 2 1
1680.2.k.b 4 20.d odd 2 1
1680.2.k.b 4 28.d even 2 1
1680.2.k.b 4 60.h even 2 1
1680.2.k.b 4 84.h odd 2 1
1680.2.k.b 4 140.c even 2 1
1680.2.k.b 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 $$S_{2}^{\mathrm{new}}(105, [\chi])$$:

 $$T_{2}$$ $$T_{13}^{2} - 7$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + 2 T^{2} )^{4}$$
$3$ $$1 - T^{2} + 9 T^{4}$$
$5$ $$( 1 + 5 T^{2} )^{2}$$
$7$ $$( 1 - 7 T^{2} )^{2}$$
$11$ $$( 1 - 3 T + 11 T^{2} )^{2}( 1 + 3 T + 11 T^{2} )^{2}$$
$13$ $$( 1 + 19 T^{2} + 169 T^{4} )^{2}$$
$17$ $$( 1 - 29 T^{2} + 289 T^{4} )^{2}$$
$19$ $$( 1 - 19 T^{2} )^{4}$$
$23$ $$( 1 + 23 T^{2} )^{4}$$
$29$ $$( 1 - 9 T + 29 T^{2} )^{2}( 1 + 9 T + 29 T^{2} )^{2}$$
$31$ $$( 1 - 31 T^{2} )^{4}$$
$37$ $$( 1 - 37 T^{2} )^{4}$$
$41$ $$( 1 + 41 T^{2} )^{4}$$
$43$ $$( 1 - 43 T^{2} )^{4}$$
$47$ $$( 1 + 31 T^{2} + 2209 T^{4} )^{2}$$
$53$ $$( 1 + 53 T^{2} )^{4}$$
$59$ $$( 1 + 59 T^{2} )^{4}$$
$61$ $$( 1 - 61 T^{2} )^{4}$$
$67$ $$( 1 - 67 T^{2} )^{4}$$
$71$ $$( 1 - 12 T + 71 T^{2} )^{2}( 1 + 12 T + 71 T^{2} )^{2}$$
$73$ $$( 1 + 34 T^{2} + 5329 T^{4} )^{2}$$
$79$ $$( 1 + T + 79 T^{2} )^{4}$$
$83$ $$( 1 - 86 T^{2} + 6889 T^{4} )^{2}$$
$89$ $$( 1 + 89 T^{2} )^{4}$$
$97$ $$( 1 - 149 T^{2} + 9409 T^{4} )^{2}$$