# Properties

 Label 210.3.p.a Level $210$ Weight $3$ Character orbit 210.p 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.p (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$5.72208555157$$ Analytic rank: $$0$$ Dimension: $$32$$ Relative dimension: $$16$$ over $$\Q(\zeta_{6})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$32q + 32q^{4} + 12q^{5} - 48q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$32q + 32q^{4} + 12q^{5} - 48q^{9} - 24q^{10} + 48q^{11} - 16q^{14} + 24q^{15} - 64q^{16} + 48q^{19} - 24q^{21} + 72q^{25} + 96q^{26} + 176q^{29} - 24q^{30} - 48q^{31} + 68q^{35} - 192q^{36} - 72q^{39} - 48q^{40} - 96q^{44} - 36q^{45} + 32q^{46} - 272q^{49} + 192q^{50} - 24q^{51} - 64q^{56} + 744q^{59} + 24q^{60} - 672q^{61} - 256q^{64} + 172q^{65} + 320q^{70} - 144q^{71} - 416q^{74} - 144q^{75} + 128q^{79} - 48q^{80} - 144q^{81} - 96q^{84} - 736q^{85} + 304q^{86} - 48q^{89} + 976q^{91} + 528q^{94} + 236q^{95} - 288q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
19.1 −1.22474 0.707107i −0.866025 1.50000i 1.00000 + 1.73205i −3.49700 + 3.57365i 2.44949i 0.180689 6.99767i 2.82843i −1.50000 + 2.59808i 6.80989 1.90405i
19.2 −1.22474 0.707107i −0.866025 1.50000i 1.00000 + 1.73205i −0.214021 4.99542i 2.44949i 5.88548 3.78961i 2.82843i −1.50000 + 2.59808i −3.27017 + 6.26945i
19.3 −1.22474 0.707107i −0.866025 1.50000i 1.00000 + 1.73205i 2.84271 + 4.11327i 2.44949i −6.72066 1.95772i 2.82843i −1.50000 + 2.59808i −0.573066 7.04781i
19.4 −1.22474 0.707107i −0.866025 1.50000i 1.00000 + 1.73205i 4.84836 1.22205i 2.44949i −1.07756 + 6.91657i 2.82843i −1.50000 + 2.59808i −6.80213 1.93160i
19.5 −1.22474 0.707107i 0.866025 + 1.50000i 1.00000 + 1.73205i −3.86586 3.17098i 2.44949i 6.18245 + 3.28288i 2.82843i −1.50000 + 2.59808i 2.49248 + 6.61722i
19.6 −1.22474 0.707107i 0.866025 + 1.50000i 1.00000 + 1.73205i −2.13749 4.52008i 2.44949i −6.07868 3.47127i 2.82843i −1.50000 + 2.59808i −0.578308 + 7.04738i
19.7 −1.22474 0.707107i 0.866025 + 1.50000i 1.00000 + 1.73205i 2.51052 + 4.32404i 2.44949i −0.686090 + 6.96630i 2.82843i −1.50000 + 2.59808i −0.0171898 7.07105i
19.8 −1.22474 0.707107i 0.866025 + 1.50000i 1.00000 + 1.73205i 4.96228 0.613015i 2.44949i 2.31437 6.60634i 2.82843i −1.50000 + 2.59808i −6.51099 2.75807i
19.9 1.22474 + 0.707107i −0.866025 1.50000i 1.00000 + 1.73205i −4.98325 + 0.408925i 2.44949i 6.07868 + 3.47127i 2.82843i −1.50000 + 2.59808i −6.39236 3.02286i
19.10 1.22474 + 0.707107i −0.866025 1.50000i 1.00000 + 1.73205i −4.67908 1.76245i 2.44949i −6.18245 3.28288i 2.82843i −1.50000 + 2.59808i −4.48444 5.46716i
19.11 1.22474 + 0.707107i −0.866025 1.50000i 1.00000 + 1.73205i 1.95025 + 4.60397i 2.44949i −2.31437 + 6.60634i 2.82843i −1.50000 + 2.59808i −0.866933 + 7.01772i
19.12 1.22474 + 0.707107i −0.866025 1.50000i 1.00000 + 1.73205i 4.99999 + 0.0121550i 2.44949i 0.686090 6.96630i 2.82843i −1.50000 + 2.59808i 6.11511 + 3.55041i
19.13 1.22474 + 0.707107i 0.866025 + 1.50000i 1.00000 + 1.73205i −4.43317 + 2.31236i 2.44949i −5.88548 + 3.78961i 2.82843i −1.50000 + 2.59808i −7.06459 0.302672i
19.14 1.22474 + 0.707107i 0.866025 + 1.50000i 1.00000 + 1.73205i 1.34637 4.81532i 2.44949i −0.180689 + 6.99767i 2.82843i −1.50000 + 2.59808i 5.05390 4.94551i
19.15 1.22474 + 0.707107i 0.866025 + 1.50000i 1.00000 + 1.73205i 1.36585 + 4.80983i 2.44949i 1.07756 6.91657i 2.82843i −1.50000 + 2.59808i −1.72825 + 6.85661i
19.16 1.22474 + 0.707107i 0.866025 + 1.50000i 1.00000 + 1.73205i 4.98355 + 0.405219i 2.44949i 6.72066 + 1.95772i 2.82843i −1.50000 + 2.59808i 5.81705 + 4.02019i
199.1 −1.22474 + 0.707107i −0.866025 + 1.50000i 1.00000 1.73205i −3.49700 3.57365i 2.44949i 0.180689 + 6.99767i 2.82843i −1.50000 2.59808i 6.80989 + 1.90405i
199.2 −1.22474 + 0.707107i −0.866025 + 1.50000i 1.00000 1.73205i −0.214021 + 4.99542i 2.44949i 5.88548 + 3.78961i 2.82843i −1.50000 2.59808i −3.27017 6.26945i
199.3 −1.22474 + 0.707107i −0.866025 + 1.50000i 1.00000 1.73205i 2.84271 4.11327i 2.44949i −6.72066 + 1.95772i 2.82843i −1.50000 2.59808i −0.573066 + 7.04781i
199.4 −1.22474 + 0.707107i −0.866025 + 1.50000i 1.00000 1.73205i 4.84836 + 1.22205i 2.44949i −1.07756 6.91657i 2.82843i −1.50000 2.59808i −6.80213 + 1.93160i
See all 32 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 199.16 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.b even 2 1 inner
7.d odd 6 1 inner
35.i odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 210.3.p.a 32
3.b odd 2 1 630.3.bc.b 32
5.b even 2 1 inner 210.3.p.a 32
5.c odd 4 1 1050.3.p.g 16
5.c odd 4 1 1050.3.p.h 16
7.c even 3 1 1470.3.h.a 32
7.d odd 6 1 inner 210.3.p.a 32
7.d odd 6 1 1470.3.h.a 32
15.d odd 2 1 630.3.bc.b 32
21.g even 6 1 630.3.bc.b 32
35.i odd 6 1 inner 210.3.p.a 32
35.i odd 6 1 1470.3.h.a 32
35.j even 6 1 1470.3.h.a 32
35.k even 12 1 1050.3.p.g 16
35.k even 12 1 1050.3.p.h 16
105.p even 6 1 630.3.bc.b 32

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.3.p.a 32 1.a even 1 1 trivial
210.3.p.a 32 5.b even 2 1 inner
210.3.p.a 32 7.d odd 6 1 inner
210.3.p.a 32 35.i odd 6 1 inner
630.3.bc.b 32 3.b odd 2 1
630.3.bc.b 32 15.d odd 2 1
630.3.bc.b 32 21.g even 6 1
630.3.bc.b 32 105.p even 6 1
1050.3.p.g 16 5.c odd 4 1
1050.3.p.g 16 35.k even 12 1
1050.3.p.h 16 5.c odd 4 1
1050.3.p.h 16 35.k even 12 1
1470.3.h.a 32 7.c even 3 1
1470.3.h.a 32 7.d odd 6 1
1470.3.h.a 32 35.i odd 6 1
1470.3.h.a 32 35.j even 6 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{3}^{\mathrm{new}}(210, [\chi])$$.

## Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database