# Properties

 Label 210.3.v.b 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.

 $$\operatorname{Tr}(f)(q) =$$ $$32q + 16q^{2} - 8q^{5} + 24q^{7} + 64q^{8} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$32q + 16q^{2} - 8q^{5} + 24q^{7} + 64q^{8} + 12q^{10} + 16q^{11} + 32q^{13} + 48q^{15} + 64q^{16} - 56q^{17} + 48q^{18} + 16q^{20} + 32q^{22} - 28q^{25} + 32q^{26} + 72q^{28} + 36q^{30} + 112q^{31} - 64q^{32} + 12q^{33} - 112q^{35} + 192q^{36} - 52q^{37} - 8q^{40} - 336q^{41} - 312q^{43} + 12q^{45} - 212q^{47} + 96q^{50} - 144q^{51} - 32q^{52} - 96q^{53} - 312q^{55} + 96q^{56} + 48q^{57} - 96q^{58} - 24q^{60} + 216q^{61} + 224q^{62} + 36q^{63} + 248q^{65} - 24q^{66} + 128q^{67} + 112q^{68} - 264q^{70} - 848q^{71} + 96q^{72} + 84q^{73} - 144q^{75} - 324q^{77} + 48q^{78} + 32q^{80} + 144q^{81} - 168q^{82} - 416q^{83} + 536q^{85} - 312q^{86} - 72q^{87} + 32q^{88} - 24q^{90} + 504q^{91} + 168q^{93} + 168q^{95} + 488q^{97} - 328q^{98} + O(q^{100})$$

## 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 −4.95862 + 0.641960i −2.44949 2.95853 6.34406i 2.00000 2.00000i 2.59808 + 1.50000i −6.53862 + 2.69191i
37.2 1.36603 0.366025i −1.67303 0.448288i 1.73205 1.00000i −4.19534 + 2.72013i −2.44949 −1.84650 + 6.75207i 2.00000 2.00000i 2.59808 + 1.50000i −4.73531 + 5.25136i
37.3 1.36603 0.366025i −1.67303 0.448288i 1.73205 1.00000i 2.17130 + 4.50394i −2.44949 1.63812 6.80563i 2.00000 2.00000i 2.59808 + 1.50000i 4.61461 + 5.35774i
37.4 1.36603 0.366025i −1.67303 0.448288i 1.73205 1.00000i 3.95091 3.06436i −2.44949 3.71395 + 5.93351i 2.00000 2.00000i 2.59808 + 1.50000i 4.27541 5.63213i
37.5 1.36603 0.366025i 1.67303 + 0.448288i 1.73205 1.00000i −3.99375 3.00831i 2.44949 6.54111 2.49277i 2.00000 2.00000i 2.59808 + 1.50000i −6.55669 2.64762i
37.6 1.36603 0.366025i 1.67303 + 0.448288i 1.73205 1.00000i −1.22909 + 4.84658i 2.44949 6.32429 + 3.00056i 2.00000 2.00000i 2.59808 + 1.50000i 0.0950035 + 7.07043i
37.7 1.36603 0.366025i 1.67303 + 0.448288i 1.73205 1.00000i 3.25594 3.79459i 2.44949 −2.26902 6.62205i 2.00000 2.00000i 2.59808 + 1.50000i 3.05878 6.37526i
37.8 1.36603 0.366025i 1.67303 + 0.448288i 1.73205 1.00000i 4.73071 + 1.61876i 2.44949 −4.13228 + 5.65016i 2.00000 2.00000i 2.59808 + 1.50000i 7.05478 + 0.479709i
67.1 −0.366025 + 1.36603i −0.448288 1.67303i −1.73205 1.00000i −3.76724 + 3.28753i 2.44949 5.65016 4.13228i 2.00000 2.00000i −2.59808 + 1.50000i −3.11195 6.34947i
67.2 −0.366025 + 1.36603i −0.448288 1.67303i −1.73205 1.00000i −3.58272 3.48771i 2.44949 3.00056 + 6.32429i 2.00000 2.00000i −2.59808 + 1.50000i 6.07567 3.61749i
67.3 −0.366025 + 1.36603i −0.448288 1.67303i −1.73205 1.00000i 1.65824 + 4.71702i 2.44949 −6.62205 2.26902i 2.00000 2.00000i −2.59808 + 1.50000i −7.05052 + 0.538651i
67.4 −0.366025 + 1.36603i −0.448288 1.67303i −1.73205 1.00000i 4.60215 1.95454i 2.44949 −2.49277 + 6.54111i 2.00000 2.00000i −2.59808 + 1.50000i 0.985440 + 7.00206i
67.5 −0.366025 + 1.36603i 0.448288 + 1.67303i −1.73205 1.00000i −4.98617 0.371566i −2.44949 −6.80563 + 1.63812i 2.00000 2.00000i −2.59808 + 1.50000i 2.33264 6.67524i
67.6 −0.366025 + 1.36603i 0.448288 + 1.67303i −1.73205 1.00000i −0.258026 4.99334i −2.44949 6.75207 1.84650i 2.00000 2.00000i −2.59808 + 1.50000i 6.91547 + 1.47522i
67.7 −0.366025 + 1.36603i 0.448288 + 1.67303i −1.73205 1.00000i 0.678362 + 4.95377i −2.44949 5.93351 + 3.71395i 2.00000 2.00000i −2.59808 + 1.50000i −7.01527 0.886545i
67.8 −0.366025 + 1.36603i 0.448288 + 1.67303i −1.73205 1.00000i 1.92336 4.61527i −2.44949 −6.34406 + 2.95853i 2.00000 2.00000i −2.59808 + 1.50000i 5.60058 + 4.31666i
163.1 −0.366025 1.36603i −0.448288 + 1.67303i −1.73205 + 1.00000i −3.76724 3.28753i 2.44949 5.65016 + 4.13228i 2.00000 + 2.00000i −2.59808 1.50000i −3.11195 + 6.34947i
163.2 −0.366025 1.36603i −0.448288 + 1.67303i −1.73205 + 1.00000i −3.58272 + 3.48771i 2.44949 3.00056 6.32429i 2.00000 + 2.00000i −2.59808 1.50000i 6.07567 + 3.61749i
163.3 −0.366025 1.36603i −0.448288 + 1.67303i −1.73205 + 1.00000i 1.65824 4.71702i 2.44949 −6.62205 + 2.26902i 2.00000 + 2.00000i −2.59808 1.50000i −7.05052 0.538651i
163.4 −0.366025 1.36603i −0.448288 + 1.67303i −1.73205 + 1.00000i 4.60215 + 1.95454i 2.44949 −2.49277 6.54111i 2.00000 + 2.00000i −2.59808 1.50000i 0.985440 7.00206i
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.b 32
5.c odd 4 1 inner 210.3.v.b 32
7.c even 3 1 inner 210.3.v.b 32
35.l odd 12 1 inner 210.3.v.b 32

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