# Properties

 Label 105.2.x.a Level 105 Weight 2 Character orbit 105.x Analytic conductor 0.838 Analytic rank 0 Dimension 48 CM no Inner twists 8

# Related objects

## Newspace parameters

 Level: $$N$$ = $$105 = 3 \cdot 5 \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 105.x (of order $$12$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.838429221223$$ Analytic rank: $$0$$ Dimension: $$48$$ Relative dimension: $$12$$ over $$\Q(\zeta_{12})$$ Coefficient ring index: multiple of None Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$48q - 2q^{3} - 24q^{6} - 12q^{7} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$48q - 2q^{3} - 24q^{6} - 12q^{7} - 8q^{10} - 10q^{12} - 16q^{13} + 4q^{15} - 8q^{16} + 14q^{18} - 28q^{21} - 8q^{22} + 4q^{25} + 40q^{27} - 60q^{28} + 40q^{30} - 24q^{31} - 4q^{33} + 8q^{36} + 4q^{37} - 16q^{40} + 14q^{42} + 16q^{43} + 40q^{45} - 32q^{46} + 44q^{48} + 8q^{51} + 36q^{52} - 40q^{55} - 88q^{57} + 56q^{58} - 50q^{60} - 8q^{61} + 44q^{63} + 76q^{66} + 12q^{67} + 140q^{70} - 34q^{72} + 52q^{73} + 6q^{75} + 64q^{76} - 120q^{78} + 20q^{81} + 104q^{82} - 24q^{85} - 46q^{87} - 84q^{90} + 72q^{91} - 44q^{93} + 12q^{96} - 120q^{97} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2.1 −0.631395 2.35640i 0.102851 1.72899i −3.42191 + 1.97564i −1.90893 + 1.16447i −4.13914 + 0.849321i 1.82148 1.91891i 3.36596 + 3.36596i −2.97884 0.355658i 3.94925 + 3.76295i
2.2 −0.582118 2.17249i 1.60750 + 0.644934i −2.64881 + 1.52929i 2.21039 + 0.337883i 0.465359 3.86771i −1.15099 2.38227i 1.68355 + 1.68355i 2.16812 + 2.07346i −0.552660 4.99875i
2.3 −0.391246 1.46015i −0.879005 1.49243i −0.246919 + 0.142558i 1.82416 1.29322i −1.83527 + 1.86739i −1.17707 + 2.36949i −1.83305 1.83305i −1.45470 + 2.62371i −2.60200 2.15759i
2.4 −0.340162 1.26950i −0.664627 + 1.59946i 0.236127 0.136328i 1.25032 + 1.85383i 2.25660 + 0.299670i 2.32676 + 1.25943i −2.11207 2.11207i −2.11654 2.12609i 1.92813 2.21789i
2.5 −0.243110 0.907300i −1.70312 + 0.315275i 0.967960 0.558852i −1.66520 1.49235i 0.700094 + 1.46859i 0.0144144 2.64571i −2.07075 2.07075i 2.80120 1.07390i −0.949181 + 1.87364i
2.6 −0.0799329 0.298314i 1.29105 1.15464i 1.64945 0.952310i −0.596180 + 2.15513i −0.447643 0.292843i −2.46856 + 0.951942i −0.852694 0.852694i 0.333606 2.98139i 0.690558 + 0.00558322i
2.7 0.0799329 + 0.298314i 0.540759 + 1.64547i 1.64945 0.952310i 0.596180 2.15513i −0.447643 + 0.292843i −2.46856 + 0.951942i 0.852694 + 0.852694i −2.41516 + 1.77961i 0.690558 + 0.00558322i
2.8 0.243110 + 0.907300i −1.31730 1.12459i 0.967960 0.558852i 1.66520 + 1.49235i 0.700094 1.46859i 0.0144144 2.64571i 2.07075 + 2.07075i 0.470578 + 2.96286i −0.949181 + 1.87364i
2.9 0.340162 + 1.26950i 0.224146 1.71749i 0.236127 0.136328i −1.25032 1.85383i 2.25660 0.299670i 2.32676 + 1.25943i 2.11207 + 2.11207i −2.89952 0.769934i 1.92813 2.21789i
2.10 0.391246 + 1.46015i −1.50746 + 0.852980i −0.246919 + 0.142558i −1.82416 + 1.29322i −1.83527 1.86739i −1.17707 + 2.36949i 1.83305 + 1.83305i 1.54485 2.57166i −2.60200 2.15759i
2.11 0.582118 + 2.17249i 1.71460 + 0.245221i −2.64881 + 1.52929i −2.21039 0.337883i 0.465359 + 3.86771i −1.15099 2.38227i −1.68355 1.68355i 2.87973 + 0.840915i −0.552660 4.99875i
2.12 0.631395 + 2.35640i −0.775426 + 1.54878i −3.42191 + 1.97564i 1.90893 1.16447i −4.13914 0.849321i 1.82148 1.91891i −3.36596 3.36596i −1.79743 2.40193i 3.94925 + 3.76295i
23.1 −2.35640 + 0.631395i 1.72899 + 0.102851i 3.42191 1.97564i 0.0540016 2.23542i −4.13914 + 0.849321i −1.91891 1.82148i −3.36596 + 3.36596i 2.97884 + 0.355658i 1.28418 + 5.30163i
23.2 −2.17249 + 0.582118i −0.644934 + 1.60750i 2.64881 1.52929i 1.39781 + 1.74531i 0.465359 3.86771i −2.38227 + 1.15099i −1.68355 + 1.68355i −2.16812 2.07346i −4.05271 2.97799i
23.3 −1.46015 + 0.391246i 1.49243 0.879005i 0.246919 0.142558i −0.207883 + 2.22638i −1.83527 + 1.86739i 2.36949 + 1.17707i 1.83305 1.83305i 1.45470 2.62371i −0.567525 3.33219i
23.4 −1.26950 + 0.340162i −1.59946 0.664627i −0.236127 + 0.136328i 2.23063 + 0.155895i 2.25660 + 0.299670i 1.25943 2.32676i 2.11207 2.11207i 2.11654 + 2.12609i −2.88481 + 0.560865i
23.5 −0.907300 + 0.243110i −0.315275 1.70312i −0.967960 + 0.558852i −2.12501 0.695932i 0.700094 + 1.46859i −2.64571 0.0144144i 2.07075 2.07075i −2.80120 + 1.07390i 2.09721 + 0.114806i
23.6 −0.298314 + 0.0799329i 1.15464 + 1.29105i −1.64945 + 0.952310i 1.56830 1.59387i −0.447643 0.292843i 0.951942 + 2.46856i 0.852694 0.852694i −0.333606 + 2.98139i −0.340444 + 0.600832i
23.7 0.298314 0.0799329i −1.64547 + 0.540759i −1.64945 + 0.952310i −1.56830 + 1.59387i −0.447643 + 0.292843i 0.951942 + 2.46856i −0.852694 + 0.852694i 2.41516 1.77961i −0.340444 + 0.600832i
23.8 0.907300 0.243110i 1.12459 1.31730i −0.967960 + 0.558852i 2.12501 + 0.695932i 0.700094 1.46859i −2.64571 0.0144144i −2.07075 + 2.07075i −0.470578 2.96286i 2.09721 + 0.114806i
See all 48 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 53.12 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.c odd 4 1 inner
7.c even 3 1 inner
15.e even 4 1 inner
21.h odd 6 1 inner
35.l odd 12 1 inner
105.x even 12 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 105.2.x.a 48
3.b odd 2 1 inner 105.2.x.a 48
5.b even 2 1 525.2.bf.f 48
5.c odd 4 1 inner 105.2.x.a 48
5.c odd 4 1 525.2.bf.f 48
7.b odd 2 1 735.2.y.i 48
7.c even 3 1 inner 105.2.x.a 48
7.c even 3 1 735.2.j.g 24
7.d odd 6 1 735.2.j.e 24
7.d odd 6 1 735.2.y.i 48
15.d odd 2 1 525.2.bf.f 48
15.e even 4 1 inner 105.2.x.a 48
15.e even 4 1 525.2.bf.f 48
21.c even 2 1 735.2.y.i 48
21.g even 6 1 735.2.j.e 24
21.g even 6 1 735.2.y.i 48
21.h odd 6 1 inner 105.2.x.a 48
21.h odd 6 1 735.2.j.g 24
35.f even 4 1 735.2.y.i 48
35.j even 6 1 525.2.bf.f 48
35.k even 12 1 735.2.j.e 24
35.k even 12 1 735.2.y.i 48
35.l odd 12 1 inner 105.2.x.a 48
35.l odd 12 1 525.2.bf.f 48
35.l odd 12 1 735.2.j.g 24
105.k odd 4 1 735.2.y.i 48
105.o odd 6 1 525.2.bf.f 48
105.w odd 12 1 735.2.j.e 24
105.w odd 12 1 735.2.y.i 48
105.x even 12 1 inner 105.2.x.a 48
105.x even 12 1 525.2.bf.f 48
105.x even 12 1 735.2.j.g 24

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.2.x.a 48 1.a even 1 1 trivial
105.2.x.a 48 3.b odd 2 1 inner
105.2.x.a 48 5.c odd 4 1 inner
105.2.x.a 48 7.c even 3 1 inner
105.2.x.a 48 15.e even 4 1 inner
105.2.x.a 48 21.h odd 6 1 inner
105.2.x.a 48 35.l odd 12 1 inner
105.2.x.a 48 105.x even 12 1 inner
525.2.bf.f 48 5.b even 2 1
525.2.bf.f 48 5.c odd 4 1
525.2.bf.f 48 15.d odd 2 1
525.2.bf.f 48 15.e even 4 1
525.2.bf.f 48 35.j even 6 1
525.2.bf.f 48 35.l odd 12 1
525.2.bf.f 48 105.o odd 6 1
525.2.bf.f 48 105.x even 12 1
735.2.j.e 24 7.d odd 6 1
735.2.j.e 24 21.g even 6 1
735.2.j.e 24 35.k even 12 1
735.2.j.e 24 105.w odd 12 1
735.2.j.g 24 7.c even 3 1
735.2.j.g 24 21.h odd 6 1
735.2.j.g 24 35.l odd 12 1
735.2.j.g 24 105.x even 12 1
735.2.y.i 48 7.b odd 2 1
735.2.y.i 48 7.d odd 6 1
735.2.y.i 48 21.c even 2 1
735.2.y.i 48 21.g even 6 1
735.2.y.i 48 35.f even 4 1
735.2.y.i 48 35.k even 12 1
735.2.y.i 48 105.k odd 4 1
735.2.y.i 48 105.w odd 12 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