# Properties

 Label 210.3.v.a Level 210 Weight 3 Character orbit 210.v Analytic conductor 5.722 Analytic rank 0 Dimension 32 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$210 = 2 \cdot 3 \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 210.v (of order $$12$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$5.72208555157$$ Analytic rank: $$0$$ Dimension: $$32$$ Relative dimension: $$8$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$32q - 16q^{2} - 8q^{7} - 64q^{8} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$32q - 16q^{2} - 8q^{7} - 64q^{8} + 4q^{10} - 32q^{11} - 32q^{13} + 64q^{16} - 56q^{17} - 48q^{18} - 16q^{20} - 48q^{21} + 64q^{22} - 48q^{23} + 68q^{25} + 32q^{26} + 40q^{28} + 12q^{30} + 160q^{31} + 64q^{32} + 12q^{33} + 152q^{35} + 192q^{36} + 44q^{37} - 64q^{38} + 8q^{40} - 80q^{41} - 48q^{42} - 184q^{43} - 12q^{45} - 96q^{46} - 228q^{47} - 96q^{50} + 192q^{51} + 32q^{52} + 48q^{53} + 104q^{55} + 32q^{56} + 144q^{57} - 112q^{58} + 24q^{60} + 216q^{61} - 320q^{62} + 84q^{63} - 384q^{65} + 24q^{66} + 112q^{68} - 24q^{70} + 368q^{71} - 96q^{72} + 52q^{73} + 48q^{75} + 256q^{76} - 836q^{77} - 240q^{78} + 144q^{81} + 40q^{82} - 736q^{83} - 72q^{85} + 184q^{86} - 72q^{87} + 64q^{88} + 24q^{90} + 216q^{91} + 192q^{92} - 216q^{93} + 272q^{95} - 408q^{97} + 200q^{98} + O(q^{100})$$

## 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 −1.36603 + 0.366025i −1.67303 0.448288i 1.73205 1.00000i −4.60436 1.94932i 2.44949 −4.28537 5.53495i −2.00000 + 2.00000i 2.59808 + 1.50000i 7.00318 + 0.977504i
37.2 −1.36603 + 0.366025i −1.67303 0.448288i 1.73205 1.00000i −1.25847 4.83903i 2.44949 5.72194 + 4.03230i −2.00000 + 2.00000i 2.59808 + 1.50000i 3.49032 + 6.14961i
37.3 −1.36603 + 0.366025i −1.67303 0.448288i 1.73205 1.00000i −0.438103 + 4.98077i 2.44949 6.08450 3.46105i −2.00000 + 2.00000i 2.59808 + 1.50000i −1.22463 6.96421i
37.4 −1.36603 + 0.366025i −1.67303 0.448288i 1.73205 1.00000i 4.98663 0.365451i 2.44949 −3.26382 6.19254i −2.00000 + 2.00000i 2.59808 + 1.50000i −6.67809 + 2.32445i
37.5 −1.36603 + 0.366025i 1.67303 + 0.448288i 1.73205 1.00000i −4.60928 + 1.93765i −2.44949 −1.45688 6.84672i −2.00000 + 2.00000i 2.59808 + 1.50000i 5.58717 4.33400i
37.6 −1.36603 + 0.366025i 1.67303 + 0.448288i 1.73205 1.00000i −2.69832 4.20940i −2.44949 1.39786 + 6.85901i −2.00000 + 2.00000i 2.59808 + 1.50000i 5.22672 + 4.76250i
37.7 −1.36603 + 0.366025i 1.67303 + 0.448288i 1.73205 1.00000i 2.01909 + 4.57420i −2.44949 −6.19056 + 3.26756i −2.00000 + 2.00000i 2.59808 + 1.50000i −4.43240 5.50943i
37.8 −1.36603 + 0.366025i 1.67303 + 0.448288i 1.73205 1.00000i 4.87077 1.12941i −2.44949 6.92053 1.05182i −2.00000 + 2.00000i 2.59808 + 1.50000i −6.24021 + 3.32563i
67.1 0.366025 1.36603i −0.448288 1.67303i −1.73205 1.00000i −4.97092 0.538514i −2.44949 3.26756 6.19056i −2.00000 + 2.00000i −2.59808 + 1.50000i −2.55510 + 6.59329i
67.2 0.366025 1.36603i −0.448288 1.67303i −1.73205 1.00000i −1.45729 + 4.78292i −2.44949 −1.05182 + 6.92053i −2.00000 + 2.00000i −2.59808 + 1.50000i 6.00018 + 3.74136i
67.3 0.366025 1.36603i −0.448288 1.67303i −1.73205 1.00000i 0.626586 4.96058i −2.44949 −6.84672 1.45688i −2.00000 + 2.00000i −2.59808 + 1.50000i −6.54694 2.67163i
67.4 0.366025 1.36603i −0.448288 1.67303i −1.73205 1.00000i 4.99461 0.232109i −2.44949 6.85901 + 1.39786i −2.00000 + 2.00000i −2.59808 + 1.50000i 1.51109 6.90772i
67.5 0.366025 1.36603i 0.448288 + 1.67303i −1.73205 1.00000i −4.09442 2.86979i 2.44949 −3.46105 + 6.08450i −2.00000 + 2.00000i −2.59808 + 1.50000i −5.41887 + 4.54267i
67.6 0.366025 1.36603i 0.448288 + 1.67303i −1.73205 1.00000i −2.17682 + 4.50127i 2.44949 −6.19254 3.26382i −2.00000 + 2.00000i −2.59808 + 1.50000i 5.35208 + 4.62118i
67.7 0.366025 1.36603i 0.448288 + 1.67303i −1.73205 1.00000i 3.99034 3.01284i 2.44949 −5.53495 4.28537i −2.00000 + 2.00000i −2.59808 + 1.50000i −2.65505 6.55368i
67.8 0.366025 1.36603i 0.448288 + 1.67303i −1.73205 1.00000i 4.81996 + 1.32965i 2.44949 4.03230 + 5.72194i −2.00000 + 2.00000i −2.59808 + 1.50000i 3.58056 6.09751i
163.1 0.366025 + 1.36603i −0.448288 + 1.67303i −1.73205 + 1.00000i −4.97092 + 0.538514i −2.44949 3.26756 + 6.19056i −2.00000 2.00000i −2.59808 1.50000i −2.55510 6.59329i
163.2 0.366025 + 1.36603i −0.448288 + 1.67303i −1.73205 + 1.00000i −1.45729 4.78292i −2.44949 −1.05182 6.92053i −2.00000 2.00000i −2.59808 1.50000i 6.00018 3.74136i
163.3 0.366025 + 1.36603i −0.448288 + 1.67303i −1.73205 + 1.00000i 0.626586 + 4.96058i −2.44949 −6.84672 + 1.45688i −2.00000 2.00000i −2.59808 1.50000i −6.54694 + 2.67163i
163.4 0.366025 + 1.36603i −0.448288 + 1.67303i −1.73205 + 1.00000i 4.99461 + 0.232109i −2.44949 6.85901 1.39786i −2.00000 2.00000i −2.59808 1.50000i 1.51109 + 6.90772i
See all 32 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 193.8 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.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 210.3.v.a 32
5.c odd 4 1 inner 210.3.v.a 32
7.c even 3 1 inner 210.3.v.a 32
35.l odd 12 1 inner 210.3.v.a 32

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.3.v.a 32 1.a even 1 1 trivial
210.3.v.a 32 5.c odd 4 1 inner
210.3.v.a 32 7.c even 3 1 inner
210.3.v.a 32 35.l odd 12 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{11}^{16} + \cdots$$ acting on $$S_{3}^{\mathrm{new}}(210, [\chi])$$.

## Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database