# Properties

 Label 210.3.w.b 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.

 $$\operatorname{Tr}(f)(q) =$$ $$64q + 32q^{2} + 6q^{3} + 12q^{5} + 4q^{7} + 128q^{8} + 16q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$64q + 32q^{2} + 6q^{3} + 12q^{5} + 4q^{7} + 128q^{8} + 16q^{9} + 24q^{10} - 12q^{12} + 16q^{14} + 68q^{15} + 128q^{16} - 12q^{18} + 36q^{21} + 16q^{22} + 12q^{23} - 16q^{25} + 8q^{28} + 112q^{29} + 22q^{30} - 128q^{32} + 30q^{33} + 16q^{36} - 32q^{37} - 24q^{38} - 64q^{39} - 88q^{42} + 32q^{43} + 16q^{44} - 474q^{45} - 24q^{46} + 96q^{47} - 40q^{50} - 84q^{51} - 56q^{53} + 72q^{54} - 220q^{57} + 56q^{58} - 672q^{59} + 24q^{60} + 600q^{61} - 114q^{63} - 28q^{65} + 16q^{67} + 40q^{72} - 624q^{73} + 64q^{74} - 144q^{75} - 208q^{77} - 248q^{78} + 48q^{80} - 64q^{81} - 192q^{82} - 160q^{84} - 152q^{85} - 672q^{87} - 16q^{88} - 144q^{89} - 232q^{91} - 48q^{92} - 202q^{93} - 136q^{95} - 48q^{96} - 128q^{98} - 160q^{99} + O(q^{100})$$

## 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 −2.99378 0.193092i 1.73205 + 1.00000i 4.84166 + 1.24831i −4.01890 1.35957i −0.400482 6.98853i 2.00000 + 2.00000i 8.92543 + 1.15615i 6.15692 + 3.47740i
17.2 1.36603 + 0.366025i −2.98701 0.278846i 1.73205 + 1.00000i −4.38913 2.39490i −3.97827 1.47423i 5.69479 + 4.07055i 2.00000 + 2.00000i 8.84449 + 1.66583i −5.11906 4.87803i
17.3 1.36603 + 0.366025i −2.51561 + 1.63453i 1.73205 + 1.00000i −2.05899 4.55638i −4.03467 + 1.31204i −6.80611 1.63612i 2.00000 + 2.00000i 3.65660 8.22370i −1.14488 6.97777i
17.4 1.36603 + 0.366025i −2.20828 2.03064i 1.73205 + 1.00000i −4.28576 + 2.57532i −2.27330 3.58219i −1.16075 6.90309i 2.00000 + 2.00000i 0.752986 + 8.96845i −6.79709 + 1.94926i
17.5 1.36603 + 0.366025i −1.72513 + 2.45437i 1.73205 + 1.00000i 2.34013 + 4.41857i −3.25493 + 2.72129i −6.07917 + 3.47039i 2.00000 + 2.00000i −3.04785 8.46821i 1.57936 + 6.89243i
17.6 1.36603 + 0.366025i −1.66769 2.49375i 1.73205 + 1.00000i 1.09236 + 4.87922i −1.36533 4.01695i −1.72310 + 6.78461i 2.00000 + 2.00000i −3.43763 + 8.31761i −0.293729 + 7.06496i
17.7 1.36603 + 0.366025i −0.962278 + 2.84148i 1.73205 + 1.00000i 3.83488 3.20838i −2.35455 + 3.52932i 6.92770 1.00349i 2.00000 + 2.00000i −7.14804 5.46859i 6.41289 2.97907i
17.8 1.36603 + 0.366025i −0.489601 2.95978i 1.73205 + 1.00000i 1.47500 4.77749i 0.414547 4.22234i 6.99874 + 0.132847i 2.00000 + 2.00000i −8.52058 + 2.89822i 3.76357 5.98628i
17.9 1.36603 + 0.366025i 0.145428 + 2.99647i 1.73205 + 1.00000i −4.92543 + 0.860317i −0.898127 + 4.14649i 1.68496 + 6.79418i 2.00000 + 2.00000i −8.95770 + 0.871542i −7.04316 0.627617i
17.10 1.36603 + 0.366025i 1.22328 2.73927i 1.73205 + 1.00000i 4.98956 0.322873i 2.67368 3.29415i −4.85698 5.04081i 2.00000 + 2.00000i −6.00715 6.70180i 6.93405 + 1.38525i
17.11 1.36603 + 0.366025i 1.97985 + 2.25393i 1.73205 + 1.00000i 3.75994 + 3.29588i 1.87953 + 3.80360i 5.65896 4.12022i 2.00000 + 2.00000i −1.16038 + 8.92488i 3.92980 + 5.87849i
17.12 1.36603 + 0.366025i 2.23964 + 1.99599i 1.73205 + 1.00000i −2.72423 + 4.19268i 2.32883 + 3.54635i −5.35730 4.50548i 2.00000 + 2.00000i 1.03201 + 8.94064i −5.25600 + 4.73017i
17.13 1.36603 + 0.366025i 2.32144 + 1.90024i 1.73205 + 1.00000i 0.695196 4.95143i 2.47561 + 3.44548i −2.48720 + 6.54323i 2.00000 + 2.00000i 1.77817 + 8.82259i 2.76201 6.50933i
17.14 1.36603 + 0.366025i 2.39979 1.80027i 1.73205 + 1.00000i −2.39011 + 4.39174i 3.93713 1.58084i 6.65191 + 2.17994i 2.00000 + 2.00000i 2.51802 8.64058i −4.87243 + 5.12439i
17.15 1.36603 + 0.366025i 2.92260 0.677069i 1.73205 + 1.00000i 4.99957 0.0656422i 4.24017 + 0.144852i −2.70652 + 6.45560i 2.00000 + 2.00000i 8.08316 3.95760i 6.85357 + 1.74030i
17.16 1.36603 + 0.366025i 2.95132 0.538273i 1.73205 + 1.00000i −2.52260 4.31700i 4.22859 + 0.344962i 0.692615 6.96565i 2.00000 + 2.00000i 8.42052 3.17723i −1.86581 6.82047i
47.1 −0.366025 1.36603i −2.99350 + 0.197383i −1.73205 + 1.00000i −3.67935 + 3.38562i 1.36533 + 4.01695i 6.78461 1.72310i 2.00000 + 2.00000i 8.92208 1.18173i 5.97157 + 3.78686i
47.2 −0.366025 1.36603i −2.86273 + 0.897104i −1.73205 + 1.00000i −4.37317 2.42391i 2.27330 + 3.58219i −6.90309 1.16075i 2.00000 + 2.00000i 7.39041 5.13633i −1.71043 + 6.86108i
47.3 −0.366025 1.36603i −2.80804 1.05588i −1.73205 + 1.00000i 4.87492 1.11135i −0.414547 + 4.22234i 0.132847 + 6.99874i 2.00000 + 2.00000i 6.77022 + 5.92993i −3.30248 6.25249i
47.4 −0.366025 1.36603i −1.76063 2.42903i −1.73205 + 1.00000i 2.77440 + 4.15965i −2.67368 + 3.29415i −5.04081 4.85698i 2.00000 + 2.00000i −2.80035 + 8.55325i 4.66669 5.31244i
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.b yes 64
3.b odd 2 1 210.3.w.a 64
5.c odd 4 1 210.3.w.a 64
7.d odd 6 1 inner 210.3.w.b yes 64
15.e even 4 1 inner 210.3.w.b yes 64
21.g even 6 1 210.3.w.a 64
35.k even 12 1 210.3.w.a 64
105.w odd 12 1 inner 210.3.w.b yes 64

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

## 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