# Properties

 Label 105.2.u.a Level 105 Weight 2 Character orbit 105.u Analytic conductor 0.838 Analytic rank 0 Dimension 32 CM no Inner twists 4

# Learn more about

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$0.838429221223$$ Analytic rank: $$0$$ Dimension: $$32$$ Relative dimension: $$8$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$32q - 12q^{5} + 8q^{7} - 24q^{8} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$32q - 12q^{5} + 8q^{7} - 24q^{8} - 12q^{10} - 8q^{11} - 8q^{15} - 8q^{21} - 8q^{22} - 8q^{23} + 12q^{25} + 24q^{26} - 24q^{28} + 8q^{30} + 24q^{31} + 24q^{32} - 36q^{33} + 44q^{35} - 32q^{36} + 4q^{37} + 12q^{38} + 12q^{40} + 16q^{42} + 40q^{43} - 40q^{46} - 60q^{47} + 72q^{50} - 8q^{51} - 108q^{52} - 24q^{53} - 48q^{56} + 16q^{57} + 4q^{58} + 20q^{60} - 24q^{61} + 4q^{63} - 4q^{65} + 72q^{66} + 8q^{67} + 132q^{68} + 4q^{70} - 16q^{71} + 12q^{72} + 36q^{73} + 48q^{75} + 60q^{77} + 80q^{78} - 12q^{80} + 16q^{81} + 12q^{82} - 72q^{85} - 16q^{86} - 24q^{87} - 32q^{88} - 24q^{91} - 56q^{92} - 24q^{93} - 12q^{95} - 72q^{98} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
52.1 −0.582519 + 2.17399i −0.965926 + 0.258819i −2.65485 1.53278i −0.0591736 + 2.23528i 2.25068i −2.56205 0.660211i 1.69581 1.69581i 0.866025 0.500000i −4.82502 1.43074i
52.2 −0.461174 + 1.72112i 0.965926 0.258819i −1.01754 0.587476i −1.92237 + 1.14215i 1.78184i 2.04329 + 1.68076i −1.03952 + 1.03952i 0.866025 0.500000i −1.07923 3.83536i
52.3 −0.401003 + 1.49657i 0.965926 0.258819i −0.346853 0.200256i 2.14688 0.625221i 1.54936i −1.01885 2.44171i −1.75234 + 1.75234i 0.866025 0.500000i 0.0747765 + 3.46366i
52.4 −0.173702 + 0.648264i −0.965926 + 0.258819i 1.34198 + 0.774791i 1.64858 1.51070i 0.671132i −0.588837 + 2.57939i −1.68450 + 1.68450i 0.866025 0.500000i 0.692969 + 1.33112i
52.5 0.105703 0.394487i −0.965926 + 0.258819i 1.58760 + 0.916603i −0.699113 + 2.12397i 0.408404i 2.57548 0.605712i 1.10697 1.10697i 0.866025 0.500000i 0.763981 + 0.500300i
52.6 0.259789 0.969545i 0.965926 0.258819i 0.859523 + 0.496246i −1.40510 1.73945i 1.00375i −1.06195 + 2.42328i 2.12394 2.12394i 0.866025 0.500000i −2.05151 + 0.910421i
52.7 0.602389 2.24814i 0.965926 0.258819i −2.95923 1.70851i −1.28534 + 1.82973i 2.32745i 0.519864 2.59417i −2.33208 + 2.33208i 0.866025 0.500000i 3.33923 + 3.99183i
52.8 0.650518 2.42777i −0.965926 + 0.258819i −3.73883 2.15861i −1.42436 1.72371i 2.51341i 2.09305 + 1.61838i −4.11829 + 4.11829i 0.866025 0.500000i −5.11135 + 2.33671i
73.1 −2.42777 0.650518i −0.258819 0.965926i 3.73883 + 2.15861i 0.780598 + 2.09539i 2.51341i 1.61838 2.09305i −4.11829 4.11829i −0.866025 + 0.500000i −0.532020 5.59492i
73.2 −2.24814 0.602389i 0.258819 + 0.965926i 2.95923 + 1.70851i −2.22726 + 0.198269i 2.32745i −2.59417 0.519864i −2.33208 2.33208i −0.866025 + 0.500000i 5.12664 + 0.895939i
73.3 −0.969545 0.259789i 0.258819 + 0.965926i −0.859523 0.496246i 0.803857 + 2.08658i 1.00375i 2.42328 + 1.06195i 2.12394 + 2.12394i −0.866025 + 0.500000i −0.237305 2.23187i
73.4 −0.394487 0.105703i −0.258819 0.965926i −1.58760 0.916603i −2.18897 0.456535i 0.408404i −0.605712 2.57548i 1.10697 + 1.10697i −0.866025 + 0.500000i 0.815263 + 0.411477i
73.5 0.648264 + 0.173702i −0.258819 0.965926i −1.34198 0.774791i 2.13259 0.672361i 0.671132i 2.57939 + 0.588837i −1.68450 1.68450i −0.866025 + 0.500000i 1.49927 0.0654328i
73.6 1.49657 + 0.401003i 0.258819 + 0.965926i 0.346853 + 0.200256i 1.61490 1.54664i 1.54936i −2.44171 + 1.01885i −1.75234 1.75234i −0.866025 + 0.500000i 3.03701 1.66707i
73.7 1.72112 + 0.461174i 0.258819 + 0.965926i 1.01754 + 0.587476i −1.95031 + 1.09375i 1.78184i 1.68076 2.04329i −1.03952 1.03952i −0.866025 + 0.500000i −3.86114 + 0.983040i
73.8 2.17399 + 0.582519i −0.258819 0.965926i 2.65485 + 1.53278i −1.96540 1.06640i 2.25068i −0.660211 + 2.56205i 1.69581 + 1.69581i −0.866025 + 0.500000i −3.65156 3.46322i
82.1 −2.42777 + 0.650518i −0.258819 + 0.965926i 3.73883 2.15861i 0.780598 2.09539i 2.51341i 1.61838 + 2.09305i −4.11829 + 4.11829i −0.866025 0.500000i −0.532020 + 5.59492i
82.2 −2.24814 + 0.602389i 0.258819 0.965926i 2.95923 1.70851i −2.22726 0.198269i 2.32745i −2.59417 + 0.519864i −2.33208 + 2.33208i −0.866025 0.500000i 5.12664 0.895939i
82.3 −0.969545 + 0.259789i 0.258819 0.965926i −0.859523 + 0.496246i 0.803857 2.08658i 1.00375i 2.42328 1.06195i 2.12394 2.12394i −0.866025 0.500000i −0.237305 + 2.23187i
82.4 −0.394487 + 0.105703i −0.258819 + 0.965926i −1.58760 + 0.916603i −2.18897 + 0.456535i 0.408404i −0.605712 + 2.57548i 1.10697 1.10697i −0.866025 0.500000i 0.815263 0.411477i
See all 32 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 103.8 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.d odd 6 1 inner
35.k 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.u.a 32
3.b odd 2 1 315.2.bz.d 32
5.b even 2 1 525.2.bc.e 32
5.c odd 4 1 inner 105.2.u.a 32
5.c odd 4 1 525.2.bc.e 32
7.b odd 2 1 735.2.v.b 32
7.c even 3 1 735.2.m.c 32
7.c even 3 1 735.2.v.b 32
7.d odd 6 1 inner 105.2.u.a 32
7.d odd 6 1 735.2.m.c 32
15.e even 4 1 315.2.bz.d 32
21.g even 6 1 315.2.bz.d 32
35.f even 4 1 735.2.v.b 32
35.i odd 6 1 525.2.bc.e 32
35.k even 12 1 inner 105.2.u.a 32
35.k even 12 1 525.2.bc.e 32
35.k even 12 1 735.2.m.c 32
35.l odd 12 1 735.2.m.c 32
35.l odd 12 1 735.2.v.b 32
105.w odd 12 1 315.2.bz.d 32

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.2.u.a 32 1.a even 1 1 trivial
105.2.u.a 32 5.c odd 4 1 inner
105.2.u.a 32 7.d odd 6 1 inner
105.2.u.a 32 35.k even 12 1 inner
315.2.bz.d 32 3.b odd 2 1
315.2.bz.d 32 15.e even 4 1
315.2.bz.d 32 21.g even 6 1
315.2.bz.d 32 105.w odd 12 1
525.2.bc.e 32 5.b even 2 1
525.2.bc.e 32 5.c odd 4 1
525.2.bc.e 32 35.i odd 6 1
525.2.bc.e 32 35.k even 12 1
735.2.m.c 32 7.c even 3 1
735.2.m.c 32 7.d odd 6 1
735.2.m.c 32 35.k even 12 1
735.2.m.c 32 35.l odd 12 1
735.2.v.b 32 7.b odd 2 1
735.2.v.b 32 7.c even 3 1
735.2.v.b 32 35.f even 4 1
735.2.v.b 32 35.l 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