# Properties

 Label 105.3.v.a Level 105 Weight 3 Character orbit 105.v Analytic conductor 2.861 Analytic rank 0 Dimension 64 CM no Inner twists 4

# Learn more about

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$2.86104277578$$ Analytic rank: $$0$$ Dimension: $$64$$ Relative dimension: $$16$$ over $$\Q(\zeta_{12})$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$64q + 4q^{5} - 4q^{7} + 24q^{8} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$64q + 4q^{5} - 4q^{7} + 24q^{8} - 16q^{10} + 16q^{11} - 48q^{15} + 80q^{16} + 56q^{17} + 24q^{21} - 96q^{22} + 72q^{23} - 4q^{25} - 288q^{26} - 380q^{28} - 48q^{30} - 136q^{31} - 48q^{32} - 72q^{33} + 76q^{35} + 384q^{36} - 28q^{37} - 68q^{38} + 164q^{40} + 128q^{41} - 12q^{42} + 344q^{43} + 240q^{46} + 412q^{47} - 288q^{48} - 72q^{50} - 24q^{51} + 388q^{52} - 40q^{53} - 8q^{55} - 864q^{56} - 192q^{57} + 56q^{58} - 180q^{60} - 216q^{61} - 912q^{62} - 84q^{63} + 20q^{65} - 72q^{66} - 368q^{67} - 492q^{68} + 416q^{70} + 784q^{71} + 36q^{72} - 316q^{73} + 96q^{75} - 32q^{76} + 844q^{77} + 624q^{78} + 908q^{80} + 288q^{81} + 556q^{82} + 1408q^{83} - 536q^{85} + 1024q^{86} + 108q^{87} + 372q^{88} + 216q^{90} - 1064q^{91} - 1704q^{92} + 144q^{93} + 260q^{95} + 352q^{97} + 272q^{98} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
37.1 −3.56483 + 0.955194i −1.67303 0.448288i 8.33154 4.81021i −1.05941 + 4.88648i 6.39228 −6.28878 + 3.07429i −14.6673 + 14.6673i 2.59808 + 1.50000i −0.890898 18.4314i
37.2 −3.18498 + 0.853412i 1.67303 + 0.448288i 5.95167 3.43620i −4.91224 + 0.932701i −5.71115 5.00478 + 4.89410i −6.69718 + 6.69718i 2.59808 + 1.50000i 14.8494 7.16279i
37.3 −2.87375 + 0.770020i 1.67303 + 0.448288i 4.20143 2.42570i 3.78688 + 3.26489i −5.15308 −1.65502 6.80154i −1.79111 + 1.79111i 2.59808 + 1.50000i −13.3966 6.46653i
37.4 −2.38023 + 0.637781i −1.67303 0.448288i 1.79464 1.03614i 2.91424 4.06291i 4.26812 −4.25379 + 5.55925i 3.35897 3.35897i 2.59808 + 1.50000i −4.34532 + 11.5293i
37.5 −1.91023 + 0.511845i −1.67303 0.448288i −0.0771041 + 0.0445161i −4.99333 0.258093i 3.42533 6.99587 + 0.240410i 5.71805 5.71805i 2.59808 + 1.50000i 9.67053 2.06280i
37.6 −1.84280 + 0.493776i 1.67303 + 0.448288i −0.312015 + 0.180142i −1.82066 4.65674i −3.30441 −6.63712 2.22456i 5.88211 5.88211i 2.59808 + 1.50000i 5.65448 + 7.68243i
37.7 −0.445275 + 0.119311i −1.67303 0.448288i −3.28007 + 1.89375i 4.59550 1.97013i 0.798445 −0.171680 6.99789i 2.53844 2.53844i 2.59808 + 1.50000i −1.81120 + 1.42554i
37.8 −0.435396 + 0.116664i 1.67303 + 0.448288i −3.28814 + 1.89841i 3.62365 3.44517i −0.780730 6.87993 + 1.29096i 2.48509 2.48509i 2.59808 + 1.50000i −1.17579 + 1.92276i
37.9 0.232416 0.0622758i 1.67303 + 0.448288i −3.41396 + 1.97105i −2.65242 + 4.23847i 0.416757 −4.06422 + 5.69931i −1.35127 + 1.35127i 2.59808 + 1.50000i −0.352512 + 1.15027i
37.10 0.901161 0.241465i −1.67303 0.448288i −2.71032 + 1.56480i 2.44472 + 4.36158i −1.61592 5.41162 + 4.44009i −4.70337 + 4.70337i 2.59808 + 1.50000i 3.25625 + 3.34017i
37.11 0.984292 0.263740i −1.67303 0.448288i −2.56483 + 1.48081i −4.04958 2.93273i −1.76498 −5.81621 + 3.89508i −5.01620 + 5.01620i 2.59808 + 1.50000i −4.75945 1.81862i
37.12 2.13226 0.571337i 1.67303 + 0.448288i 0.756006 0.436480i 2.78563 + 4.15214i 3.82346 2.73668 6.44287i −4.88107 + 4.88107i 2.59808 + 1.50000i 8.31195 + 7.26192i
37.13 2.56059 0.686107i 1.67303 + 0.448288i 2.62176 1.51367i 1.32982 4.81992i 4.59152 0.158915 + 6.99820i −1.82323 + 1.82323i 2.59808 + 1.50000i 0.0981374 13.2542i
37.14 2.88660 0.773463i −1.67303 0.448288i 4.27013 2.46536i 0.160937 4.99741i −5.17611 5.19830 4.68804i 1.96674 1.96674i 2.59808 + 1.50000i −3.40075 14.5500i
37.15 3.41166 0.914152i 1.67303 + 0.448288i 7.33966 4.23756i −4.98672 + 0.364163i 6.11763 −6.35750 2.92953i 11.1766 11.1766i 2.59808 + 1.50000i −16.6801 + 5.80102i
37.16 3.52852 0.945463i −1.67303 0.448288i 8.09242 4.67216i 3.83301 + 3.21062i −6.32716 −6.80203 + 1.65300i 13.8047 13.8047i 2.59808 + 1.50000i 16.5603 + 7.70474i
58.1 −0.945463 3.52852i 0.448288 1.67303i −8.09242 + 4.67216i −4.69698 1.71417i −6.32716 1.65300 + 6.80203i 13.8047 + 13.8047i −2.59808 1.50000i −1.60767 + 18.1940i
58.2 −0.914152 3.41166i −0.448288 + 1.67303i −7.33966 + 4.23756i 2.17799 + 4.50071i 6.11763 −2.92953 + 6.35750i 11.1766 + 11.1766i −2.59808 1.50000i 13.3639 11.5449i
58.3 −0.773463 2.88660i 0.448288 1.67303i −4.27013 + 2.46536i 4.24741 2.63808i −5.17611 −4.68804 5.19830i 1.96674 + 1.96674i −2.59808 1.50000i −10.9003 10.2201i
58.4 −0.686107 2.56059i −0.448288 + 1.67303i −2.62176 + 1.51367i 3.50926 3.56161i 4.59152 6.99820 0.158915i −1.82323 1.82323i −2.59808 1.50000i −11.5275 6.54211i
See all 64 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 88.16 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
7.c even 3 1 inner
35.l odd 12 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 105.3.v.a 64
3.b odd 2 1 315.3.ca.b 64
5.c odd 4 1 inner 105.3.v.a 64
7.c even 3 1 inner 105.3.v.a 64
15.e even 4 1 315.3.ca.b 64
21.h odd 6 1 315.3.ca.b 64
35.l odd 12 1 inner 105.3.v.a 64
105.x even 12 1 315.3.ca.b 64

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.3.v.a 64 1.a even 1 1 trivial
105.3.v.a 64 5.c odd 4 1 inner
105.3.v.a 64 7.c even 3 1 inner
105.3.v.a 64 35.l odd 12 1 inner
315.3.ca.b 64 3.b odd 2 1
315.3.ca.b 64 15.e even 4 1
315.3.ca.b 64 21.h odd 6 1
315.3.ca.b 64 105.x even 12 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{3}^{\mathrm{new}}(105, [\chi])$$.

## Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database