Properties

 Label 210.3.w.a Level 210 Weight 3 Character orbit 210.w Analytic conductor 5.722 Analytic rank 0 Dimension 64 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.w (of order $$12$$, degree $$4$$, minimal)

Newform invariants

 Self dual: no Analytic conductor: $$5.72208555157$$ Analytic rank: $$0$$ Dimension: $$64$$ Relative dimension: $$16$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$64q - 32q^{2} - 6q^{3} - 12q^{5} + 4q^{7} - 128q^{8} - 16q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$64q - 32q^{2} - 6q^{3} - 12q^{5} + 4q^{7} - 128q^{8} - 16q^{9} + 24q^{10} + 12q^{12} - 16q^{14} - 44q^{15} + 128q^{16} - 20q^{18} + 36q^{21} + 16q^{22} - 12q^{23} - 16q^{25} + 8q^{28} - 112q^{29} + 26q^{30} + 128q^{32} + 30q^{33} + 16q^{36} - 32q^{37} + 24q^{38} + 64q^{39} - 136q^{42} + 32q^{43} - 16q^{44} - 114q^{45} - 24q^{46} - 96q^{47} + 40q^{50} - 84q^{51} + 56q^{53} - 72q^{54} - 316q^{57} + 56q^{58} + 672q^{59} + 8q^{60} + 600q^{61} - 210q^{63} + 28q^{65} + 16q^{67} + 24q^{72} - 624q^{73} - 64q^{74} + 48q^{75} + 208q^{77} - 8q^{78} - 48q^{80} - 64q^{81} - 192q^{82} + 160q^{84} - 152q^{85} + 60q^{87} - 16q^{88} + 144q^{89} - 232q^{91} + 48q^{92} - 170q^{93} + 136q^{95} - 48q^{96} + 128q^{98} + 160q^{99} + O(q^{100})$$

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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
17.1 −1.36603 0.366025i −2.99585 0.157742i 1.73205 + 1.00000i 2.05899 + 4.55638i 4.03467 + 1.31204i −6.80611 1.63612i −2.00000 2.00000i 8.95023 + 0.945145i −1.14488 6.97777i
17.2 −1.36603 0.366025i −2.72119 1.26298i 1.73205 + 1.00000i −2.34013 4.41857i 3.25493 + 2.72129i −6.07917 + 3.47039i −2.00000 2.00000i 5.80976 + 6.87362i 1.57936 + 6.89243i
17.3 −1.36603 0.366025i −2.49614 + 1.66411i 1.73205 + 1.00000i −4.84166 1.24831i 4.01890 1.35957i −0.400482 6.98853i −2.00000 2.00000i 3.46146 8.30773i 6.15692 + 3.47740i
17.4 −1.36603 0.366025i −2.44741 + 1.73499i 1.73205 + 1.00000i 4.38913 + 2.39490i 3.97827 1.47423i 5.69479 + 4.07055i −2.00000 2.00000i 2.97959 8.49247i −5.11906 4.87803i
17.5 −1.36603 0.366025i −2.25410 1.97966i 1.73205 + 1.00000i −3.83488 + 3.20838i 2.35455 + 3.52932i 6.92770 1.00349i −2.00000 2.00000i 1.16192 + 8.92468i 6.41289 2.97907i
17.6 −1.36603 0.366025i −1.37229 2.66774i 1.73205 + 1.00000i 4.92543 0.860317i 0.898127 + 4.14649i 1.68496 + 6.79418i −2.00000 2.00000i −5.23363 + 7.32183i −7.04316 0.627617i
17.7 −1.36603 0.366025i −0.897104 + 2.86273i 1.73205 + 1.00000i 4.28576 2.57532i 2.27330 3.58219i −1.16075 6.90309i −2.00000 2.00000i −7.39041 5.13633i −6.79709 + 1.94926i
17.8 −1.36603 0.366025i −0.197383 + 2.99350i 1.73205 + 1.00000i −1.09236 4.87922i 1.36533 4.01695i −1.72310 + 6.78461i −2.00000 2.00000i −8.92208 1.18173i −0.293729 + 7.06496i
17.9 −1.36603 0.366025i 0.587638 2.94188i 1.73205 + 1.00000i −3.75994 3.29588i −1.87953 + 3.80360i 5.65896 4.12022i −2.00000 2.00000i −8.30936 3.45752i 3.92980 + 5.87849i
17.10 −1.36603 0.366025i 0.941591 2.84840i 1.73205 + 1.00000i 2.72423 4.19268i −2.32883 + 3.54635i −5.35730 4.50548i −2.00000 2.00000i −7.22681 5.36406i −5.25600 + 4.73017i
17.11 −1.36603 0.366025i 1.05588 + 2.80804i 1.73205 + 1.00000i −1.47500 + 4.77749i −0.414547 4.22234i 6.99874 + 0.132847i −2.00000 2.00000i −6.77022 + 5.92993i 3.76357 5.98628i
17.12 −1.36603 0.366025i 1.06031 2.80638i 1.73205 + 1.00000i −0.695196 + 4.95143i −2.47561 + 3.44548i −2.48720 + 6.54323i −2.00000 2.00000i −6.75151 5.95123i 2.76201 6.50933i
17.13 −1.36603 0.366025i 2.42903 + 1.76063i 1.73205 + 1.00000i −4.98956 + 0.322873i −2.67368 3.29415i −4.85698 5.04081i −2.00000 2.00000i 2.80035 + 8.55325i 6.93405 + 1.38525i
17.14 −1.36603 0.366025i 2.82505 1.00950i 1.73205 + 1.00000i 2.52260 + 4.31700i −4.22859 + 0.344962i 0.692615 6.96565i −2.00000 2.00000i 6.96182 5.70377i −1.86581 6.82047i
17.15 −1.36603 0.366025i 2.86958 0.874940i 1.73205 + 1.00000i −4.99957 + 0.0656422i −4.24017 + 0.144852i −2.70652 + 6.45560i −2.00000 2.00000i 7.46896 5.02142i 6.85357 + 1.74030i
17.16 −1.36603 0.366025i 2.97842 + 0.359186i 1.73205 + 1.00000i 2.39011 4.39174i −3.93713 1.58084i 6.65191 + 2.17994i −2.00000 2.00000i 8.74197 + 2.13962i −4.87243 + 5.12439i
47.1 0.366025 + 1.36603i −2.99647 0.145428i −1.73205 + 1.00000i 3.20777 + 3.83539i −0.898127 4.14649i 6.79418 + 1.68496i −2.00000 2.00000i 8.95770 + 0.871542i −4.06511 + 5.78575i
47.2 0.366025 + 1.36603i −2.84148 + 0.962278i −1.73205 + 1.00000i −4.69598 1.71691i −2.35455 3.52932i −1.00349 + 6.92770i −2.00000 2.00000i 7.14804 5.46859i 0.626491 7.04326i
47.3 0.366025 + 1.36603i −2.45437 + 1.72513i −1.73205 + 1.00000i 2.65653 4.23590i −3.25493 2.72129i 3.47039 6.07917i −2.00000 2.00000i 3.04785 8.46821i 6.75870 + 2.07845i
47.4 0.366025 + 1.36603i −2.25393 1.97985i −1.73205 + 1.00000i 0.974348 4.90415i 1.87953 3.80360i −4.12022 + 5.65896i −2.00000 2.00000i 1.16038 + 8.92488i 7.05582 0.464058i
See all 64 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 173.16 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner
15.e even 4 1 inner
105.w 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.w.a 64
3.b odd 2 1 210.3.w.b yes 64
5.c odd 4 1 210.3.w.b yes 64
7.d odd 6 1 inner 210.3.w.a 64
15.e even 4 1 inner 210.3.w.a 64
21.g even 6 1 210.3.w.b yes 64
35.k even 12 1 210.3.w.b yes 64
105.w odd 12 1 inner 210.3.w.a 64

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.3.w.a 64 1.a even 1 1 trivial
210.3.w.a 64 7.d odd 6 1 inner
210.3.w.a 64 15.e even 4 1 inner
210.3.w.a 64 105.w odd 12 1 inner
210.3.w.b yes 64 3.b odd 2 1
210.3.w.b yes 64 5.c odd 4 1
210.3.w.b yes 64 21.g even 6 1
210.3.w.b yes 64 35.k even 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{17}^{64} - \cdots$$ acting on $$S_{3}^{\mathrm{new}}(210, [\chi])$$.

Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database