# Properties

 Label 210.3.s.a Level $210$ Weight $3$ Character orbit 210.s Analytic conductor $5.722$ Analytic rank $0$ Dimension $40$ 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.s (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$5.72208555157$$ Analytic rank: $$0$$ Dimension: $$40$$ Relative dimension: $$20$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$40q + 40q^{4} - 8q^{6} + 20q^{7} - 4q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$40q + 40q^{4} - 8q^{6} + 20q^{7} - 4q^{9} + 136q^{13} + 40q^{15} - 80q^{16} + 16q^{18} - 140q^{19} + 36q^{21} - 8q^{24} + 100q^{25} - 120q^{27} - 16q^{28} - 20q^{30} + 4q^{31} + 232q^{33} + 32q^{34} - 16q^{36} - 76q^{37} - 4q^{39} + 128q^{42} - 104q^{43} - 20q^{45} - 56q^{46} + 100q^{49} + 168q^{51} + 136q^{52} + 40q^{54} + 80q^{55} + 200q^{57} + 144q^{58} + 40q^{60} - 120q^{61} - 324q^{63} - 320q^{64} - 288q^{66} - 20q^{67} - 416q^{69} - 120q^{70} - 32q^{72} - 476q^{73} - 560q^{76} - 192q^{78} - 508q^{79} - 304q^{81} + 224q^{82} + 144q^{84} - 240q^{85} - 324q^{87} + 468q^{91} + 204q^{93} + 400q^{94} + 16q^{96} - 512q^{97} + 208q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
11.1 −1.22474 + 0.707107i −2.97598 0.378837i 1.00000 1.73205i 1.93649 1.11803i 3.91270 1.64036i −2.08959 + 6.68084i 2.82843i 8.71297 + 2.25482i −1.58114 + 2.73861i
11.2 −1.22474 + 0.707107i −2.87585 0.854111i 1.00000 1.73205i −1.93649 + 1.11803i 4.12613 0.987462i 6.85537 1.41561i 2.82843i 7.54099 + 4.91259i 1.58114 2.73861i
11.3 −1.22474 + 0.707107i −1.55909 + 2.56305i 1.00000 1.73205i −1.93649 + 1.11803i 0.0971326 4.24153i 1.98659 6.71219i 2.82843i −4.13849 7.99205i 1.58114 2.73861i
11.4 −1.22474 + 0.707107i −0.690423 2.91947i 1.00000 1.73205i −1.93649 + 1.11803i 2.90997 + 3.08741i 0.399356 + 6.98860i 2.82843i −8.04663 + 4.03134i 1.58114 2.73861i
11.5 −1.22474 + 0.707107i −0.328980 + 2.98191i 1.00000 1.73205i 1.93649 1.11803i −1.70561 3.88470i −5.45202 4.39039i 2.82843i −8.78354 1.96197i −1.58114 + 2.73861i
11.6 −1.22474 + 0.707107i 1.03277 + 2.81663i 1.00000 1.73205i 1.93649 1.11803i −3.25654 2.71937i 5.52236 + 4.30158i 2.82843i −6.86677 + 5.81786i −1.58114 + 2.73861i
11.7 −1.22474 + 0.707107i 1.03973 2.81407i 1.00000 1.73205i 1.93649 1.11803i 0.716446 + 4.18171i 6.69226 2.05274i 2.82843i −6.83794 5.85172i −1.58114 + 2.73861i
11.8 −1.22474 + 0.707107i 1.92630 2.29987i 1.00000 1.73205i −1.93649 + 1.11803i −0.732971 + 4.17885i −5.85650 3.83424i 2.82843i −1.57876 8.86045i 1.58114 2.73861i
11.9 −1.22474 + 0.707107i 2.66551 + 1.37661i 1.00000 1.73205i −1.93649 + 1.11803i −4.23798 + 0.198807i 3.85860 5.84048i 2.82843i 5.20990 + 7.33873i 1.58114 2.73861i
11.10 −1.22474 + 0.707107i 2.99076 + 0.235263i 1.00000 1.73205i 1.93649 1.11803i −3.82928 + 1.82665i −6.91642 + 1.07847i 2.82843i 8.88930 + 1.40723i −1.58114 + 2.73861i
11.11 1.22474 0.707107i −2.95566 + 0.513907i 1.00000 1.73205i −1.93649 + 1.11803i −3.25654 + 2.71937i 5.52236 + 4.30158i 2.82843i 8.47180 3.03786i −1.58114 + 2.73861i
11.12 1.22474 0.707107i −2.52493 1.62010i 1.00000 1.73205i 1.93649 1.11803i −4.23798 0.198807i 3.85860 5.84048i 2.82843i 3.75058 + 8.18127i 1.58114 2.73861i
11.13 1.22474 0.707107i −2.41792 + 1.77586i 1.00000 1.73205i −1.93649 + 1.11803i −1.70561 + 3.88470i −5.45202 4.39039i 2.82843i 2.69265 8.58776i −1.58114 + 2.73861i
11.14 1.22474 0.707107i −1.69912 2.47244i 1.00000 1.73205i −1.93649 + 1.11803i −3.82928 1.82665i −6.91642 + 1.07847i 2.82843i −3.22595 + 8.40198i −1.58114 + 2.73861i
11.15 1.22474 0.707107i −1.44013 + 2.63174i 1.00000 1.73205i 1.93649 1.11803i 0.0971326 + 4.24153i 1.98659 6.71219i 2.82843i −4.85208 7.58006i 1.58114 2.73861i
11.16 1.22474 0.707107i 1.02859 2.81815i 1.00000 1.73205i 1.93649 1.11803i −0.732971 4.17885i −5.85650 3.83424i 2.82843i −6.88399 5.79747i 1.58114 2.73861i
11.17 1.22474 0.707107i 1.81607 + 2.38786i 1.00000 1.73205i −1.93649 + 1.11803i 3.91270 + 1.64036i −2.08959 + 6.68084i 2.82843i −2.40375 + 8.67306i −1.58114 + 2.73861i
11.18 1.22474 0.707107i 1.91719 2.30746i 1.00000 1.73205i −1.93649 + 1.11803i 0.716446 4.18171i 6.69226 2.05274i 2.82843i −1.64877 8.84769i −1.58114 + 2.73861i
11.19 1.22474 0.707107i 2.17761 + 2.06350i 1.00000 1.73205i 1.93649 1.11803i 4.12613 + 0.987462i 6.85537 1.41561i 2.82843i 0.483930 + 8.98698i 1.58114 2.73861i
11.20 1.22474 0.707107i 2.87355 0.861812i 1.00000 1.73205i 1.93649 1.11803i 2.90997 3.08741i 0.399356 + 6.98860i 2.82843i 7.51456 4.95292i 1.58114 2.73861i
See all 40 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 191.20 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.c even 3 1 inner
21.h 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.s.a 40
3.b odd 2 1 inner 210.3.s.a 40
7.c even 3 1 inner 210.3.s.a 40
21.h odd 6 1 inner 210.3.s.a 40

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.3.s.a 40 1.a even 1 1 trivial
210.3.s.a 40 3.b odd 2 1 inner
210.3.s.a 40 7.c even 3 1 inner
210.3.s.a 40 21.h odd 6 1 inner

## 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