# Properties

 Label 105.2.p.a Level 105 Weight 2 Character orbit 105.p Analytic conductor 0.838 Analytic rank 0 Dimension 24 CM no Inner twists 8

# Related objects

## Newspace parameters

 Level: $$N$$ = $$105 = 3 \cdot 5 \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 105.p (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.838429221223$$ Analytic rank: $$0$$ Dimension: $$24$$ Relative dimension: $$12$$ over $$\Q(\zeta_{6})$$ Coefficient ring index: multiple of None Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

The dimension is sufficiently large that we do not compute an algebraic $$q$$-expansion, but we have computed the trace expansion.

 $$\operatorname{Tr}(f)(q) =$$ $$24q - 12q^{4} - 6q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$24q - 12q^{4} - 6q^{9} - 24q^{15} - 12q^{16} - 6q^{21} + 18q^{24} - 12q^{25} + 18q^{30} + 84q^{36} - 12q^{39} - 72q^{40} - 18q^{45} + 36q^{46} - 12q^{49} - 12q^{51} - 36q^{54} + 12q^{60} + 36q^{61} + 24q^{64} - 72q^{66} + 108q^{70} + 72q^{75} + 48q^{79} - 6q^{81} + 48q^{84} + 48q^{85} - 96q^{91} - 72q^{94} - 90q^{96} - 48q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
59.1 −1.25399 2.17197i 0.0401158 + 1.73159i −2.14497 + 3.71520i 0.00874528 + 2.23605i 3.71065 2.25852i 1.65993 + 2.06025i 5.74313 −2.99678 + 0.138928i 4.84567 2.82298i
59.2 −1.25399 2.17197i 1.51966 0.831052i −2.14497 + 3.71520i −1.93210 1.12560i −3.71065 2.25852i −1.65993 2.06025i 5.74313 1.61871 2.52582i −0.0219329 + 5.60796i
59.3 −0.757344 1.31176i −1.24551 + 1.20362i −0.147140 + 0.254854i 1.42830 1.72046i 2.52214 + 0.722254i −0.0753638 2.64468i −2.58363 0.102593 2.99825i −3.33853 0.570605i
59.4 −0.757344 1.31176i 0.419611 1.68045i −0.147140 + 0.254854i 2.20411 0.376714i −2.52214 + 0.722254i 0.0753638 + 2.64468i −2.58363 −2.64785 1.41027i −2.16343 2.60595i
59.5 −0.322403 0.558418i 1.15761 + 1.28838i 0.792113 1.37198i −1.60193 1.56008i 0.346239 1.06181i 2.64366 + 0.105130i −2.31113 −0.319861 + 2.98290i −0.354709 + 1.39752i
59.6 −0.322403 0.558418i 1.69458 + 0.358331i 0.792113 1.37198i 0.550103 + 2.16735i −0.346239 1.06181i −2.64366 0.105130i −2.31113 2.74320 + 1.21444i 1.03293 1.00595i
59.7 0.322403 + 0.558418i −1.69458 0.358331i 0.792113 1.37198i 1.60193 + 1.56008i −0.346239 1.06181i 2.64366 + 0.105130i 2.31113 2.74320 + 1.21444i −0.354709 + 1.39752i
59.8 0.322403 + 0.558418i −1.15761 1.28838i 0.792113 1.37198i −0.550103 2.16735i 0.346239 1.06181i −2.64366 0.105130i 2.31113 −0.319861 + 2.98290i 1.03293 1.00595i
59.9 0.757344 + 1.31176i −0.419611 + 1.68045i −0.147140 + 0.254854i −1.42830 + 1.72046i −2.52214 + 0.722254i −0.0753638 2.64468i 2.58363 −2.64785 1.41027i −3.33853 0.570605i
59.10 0.757344 + 1.31176i 1.24551 1.20362i −0.147140 + 0.254854i −2.20411 + 0.376714i 2.52214 + 0.722254i 0.0753638 + 2.64468i 2.58363 0.102593 2.99825i −2.16343 2.60595i
59.11 1.25399 + 2.17197i −1.51966 + 0.831052i −2.14497 + 3.71520i −0.00874528 2.23605i −3.71065 2.25852i 1.65993 + 2.06025i −5.74313 1.61871 2.52582i 4.84567 2.82298i
59.12 1.25399 + 2.17197i −0.0401158 1.73159i −2.14497 + 3.71520i 1.93210 + 1.12560i 3.71065 2.25852i −1.65993 2.06025i −5.74313 −2.99678 + 0.138928i −0.0219329 + 5.60796i
89.1 −1.25399 + 2.17197i 0.0401158 1.73159i −2.14497 3.71520i 0.00874528 2.23605i 3.71065 + 2.25852i 1.65993 2.06025i 5.74313 −2.99678 0.138928i 4.84567 + 2.82298i
89.2 −1.25399 + 2.17197i 1.51966 + 0.831052i −2.14497 3.71520i −1.93210 + 1.12560i −3.71065 + 2.25852i −1.65993 + 2.06025i 5.74313 1.61871 + 2.52582i −0.0219329 5.60796i
89.3 −0.757344 + 1.31176i −1.24551 1.20362i −0.147140 0.254854i 1.42830 + 1.72046i 2.52214 0.722254i −0.0753638 + 2.64468i −2.58363 0.102593 + 2.99825i −3.33853 + 0.570605i
89.4 −0.757344 + 1.31176i 0.419611 + 1.68045i −0.147140 0.254854i 2.20411 + 0.376714i −2.52214 0.722254i 0.0753638 2.64468i −2.58363 −2.64785 + 1.41027i −2.16343 + 2.60595i
89.5 −0.322403 + 0.558418i 1.15761 1.28838i 0.792113 + 1.37198i −1.60193 + 1.56008i 0.346239 + 1.06181i 2.64366 0.105130i −2.31113 −0.319861 2.98290i −0.354709 1.39752i
89.6 −0.322403 + 0.558418i 1.69458 0.358331i 0.792113 + 1.37198i 0.550103 2.16735i −0.346239 + 1.06181i −2.64366 + 0.105130i −2.31113 2.74320 1.21444i 1.03293 + 1.00595i
89.7 0.322403 0.558418i −1.69458 + 0.358331i 0.792113 + 1.37198i 1.60193 1.56008i −0.346239 + 1.06181i 2.64366 0.105130i 2.31113 2.74320 1.21444i −0.354709 1.39752i
89.8 0.322403 0.558418i −1.15761 + 1.28838i 0.792113 + 1.37198i −0.550103 + 2.16735i 0.346239 + 1.06181i −2.64366 + 0.105130i 2.31113 −0.319861 2.98290i 1.03293 + 1.00595i
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 89.12 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
5.b even 2 1 inner
7.d odd 6 1 inner
15.d odd 2 1 inner
21.g even 6 1 inner
35.i odd 6 1 inner
105.p even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 105.2.p.a 24
3.b odd 2 1 inner 105.2.p.a 24
5.b even 2 1 inner 105.2.p.a 24
5.c odd 4 2 525.2.t.j 24
7.b odd 2 1 735.2.p.f 24
7.c even 3 1 735.2.g.b 24
7.c even 3 1 735.2.p.f 24
7.d odd 6 1 inner 105.2.p.a 24
7.d odd 6 1 735.2.g.b 24
15.d odd 2 1 inner 105.2.p.a 24
15.e even 4 2 525.2.t.j 24
21.c even 2 1 735.2.p.f 24
21.g even 6 1 inner 105.2.p.a 24
21.g even 6 1 735.2.g.b 24
21.h odd 6 1 735.2.g.b 24
21.h odd 6 1 735.2.p.f 24
35.c odd 2 1 735.2.p.f 24
35.i odd 6 1 inner 105.2.p.a 24
35.i odd 6 1 735.2.g.b 24
35.j even 6 1 735.2.g.b 24
35.j even 6 1 735.2.p.f 24
35.k even 12 2 525.2.t.j 24
105.g even 2 1 735.2.p.f 24
105.o odd 6 1 735.2.g.b 24
105.o odd 6 1 735.2.p.f 24
105.p even 6 1 inner 105.2.p.a 24
105.p even 6 1 735.2.g.b 24
105.w odd 12 2 525.2.t.j 24

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.2.p.a 24 1.a even 1 1 trivial
105.2.p.a 24 3.b odd 2 1 inner
105.2.p.a 24 5.b even 2 1 inner
105.2.p.a 24 7.d odd 6 1 inner
105.2.p.a 24 15.d odd 2 1 inner
105.2.p.a 24 21.g even 6 1 inner
105.2.p.a 24 35.i odd 6 1 inner
105.2.p.a 24 105.p even 6 1 inner
525.2.t.j 24 5.c odd 4 2
525.2.t.j 24 15.e even 4 2
525.2.t.j 24 35.k even 12 2
525.2.t.j 24 105.w odd 12 2
735.2.g.b 24 7.c even 3 1
735.2.g.b 24 7.d odd 6 1
735.2.g.b 24 21.g even 6 1
735.2.g.b 24 21.h odd 6 1
735.2.g.b 24 35.i odd 6 1
735.2.g.b 24 35.j even 6 1
735.2.g.b 24 105.o odd 6 1
735.2.g.b 24 105.p even 6 1
735.2.p.f 24 7.b odd 2 1
735.2.p.f 24 7.c even 3 1
735.2.p.f 24 21.c even 2 1
735.2.p.f 24 21.h odd 6 1
735.2.p.f 24 35.c odd 2 1
735.2.p.f 24 35.j even 6 1
735.2.p.f 24 105.g even 2 1
735.2.p.f 24 105.o odd 6 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(105, [\chi])$$.

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database