# Properties

 Label 210.3.k.b Level 210 Weight 3 Character orbit 210.k 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.k (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$5.72208555157$$ Analytic rank: $$0$$ Dimension: $$32$$ Relative dimension: $$16$$ over $$\Q(i)$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$32q + 32q^{2} - 4q^{7} - 64q^{8} + 16q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$32q + 32q^{2} - 4q^{7} - 64q^{8} + 16q^{9} - 8q^{14} - 4q^{15} - 128q^{16} - 4q^{18} + 12q^{21} - 40q^{22} - 24q^{23} + 16q^{25} - 8q^{28} + 112q^{29} + 28q^{30} - 128q^{32} + 48q^{35} - 40q^{36} + 32q^{37} - 64q^{39} - 20q^{42} - 32q^{43} - 80q^{44} - 48q^{46} + 8q^{50} + 84q^{51} + 136q^{53} + 340q^{57} + 112q^{58} + 64q^{60} + 168q^{63} + 200q^{65} + 32q^{67} - 72q^{72} + 64q^{74} - 88q^{77} - 4q^{78} + 76q^{81} - 64q^{84} - 40q^{85} - 80q^{88} - 272q^{91} - 48q^{92} - 388q^{93} - 544q^{95} - 128q^{98} - 160q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
83.1 1.00000 + 1.00000i −2.99999 0.00829838i 2.00000i −3.67015 3.39558i −2.99169 3.00829i 6.84640 1.45834i −2.00000 + 2.00000i 8.99986 + 0.0497901i −0.274569 7.06574i
83.2 1.00000 + 1.00000i −2.98664 + 0.282815i 2.00000i 3.28357 3.77070i −3.26945 2.70382i −3.67639 + 5.95686i −2.00000 + 2.00000i 8.84003 1.68933i 7.05427 0.487124i
83.3 1.00000 + 1.00000i −2.74188 + 1.21742i 2.00000i −3.32079 + 3.73796i −3.95930 1.52446i −6.61294 2.29543i −2.00000 + 2.00000i 6.03579 6.67602i −7.05875 + 0.417165i
83.4 1.00000 + 1.00000i −2.09499 2.14733i 2.00000i −1.13661 + 4.86910i 0.0523328 4.24232i 2.31291 6.60685i −2.00000 + 2.00000i −0.222012 + 8.99726i −6.00571 + 3.73249i
83.5 1.00000 + 1.00000i −1.94039 2.28799i 2.00000i 4.58449 1.99562i 0.347606 4.22838i −4.23091 5.57668i −2.00000 + 2.00000i −1.46981 + 8.87917i 6.58010 + 2.58887i
83.6 1.00000 + 1.00000i −1.86291 + 2.35150i 2.00000i 1.91622 + 4.61824i −4.21441 + 0.488596i 6.26242 + 3.12763i −2.00000 + 2.00000i −2.05914 8.76127i −2.70202 + 6.53446i
83.7 1.00000 + 1.00000i −0.947561 2.84642i 2.00000i −4.64638 1.84693i 1.89886 3.79398i −3.13705 + 6.25771i −2.00000 + 2.00000i −7.20426 + 5.39432i −2.79945 6.49331i
83.8 1.00000 + 1.00000i −0.199427 + 2.99336i 2.00000i −4.37611 2.41861i −3.19279 + 2.79394i −5.12625 4.76671i −2.00000 + 2.00000i −8.92046 1.19391i −1.95750 6.79472i
83.9 1.00000 + 1.00000i 0.199427 2.99336i 2.00000i 4.37611 + 2.41861i 3.19279 2.79394i 4.76671 + 5.12625i −2.00000 + 2.00000i −8.92046 1.19391i 1.95750 + 6.79472i
83.10 1.00000 + 1.00000i 0.947561 + 2.84642i 2.00000i 4.64638 + 1.84693i −1.89886 + 3.79398i −6.25771 + 3.13705i −2.00000 + 2.00000i −7.20426 + 5.39432i 2.79945 + 6.49331i
83.11 1.00000 + 1.00000i 1.86291 2.35150i 2.00000i −1.91622 4.61824i 4.21441 0.488596i −3.12763 6.26242i −2.00000 + 2.00000i −2.05914 8.76127i 2.70202 6.53446i
83.12 1.00000 + 1.00000i 1.94039 + 2.28799i 2.00000i −4.58449 + 1.99562i −0.347606 + 4.22838i 5.57668 + 4.23091i −2.00000 + 2.00000i −1.46981 + 8.87917i −6.58010 2.58887i
83.13 1.00000 + 1.00000i 2.09499 + 2.14733i 2.00000i 1.13661 4.86910i −0.0523328 + 4.24232i 6.60685 2.31291i −2.00000 + 2.00000i −0.222012 + 8.99726i 6.00571 3.73249i
83.14 1.00000 + 1.00000i 2.74188 1.21742i 2.00000i 3.32079 3.73796i 3.95930 + 1.52446i 2.29543 + 6.61294i −2.00000 + 2.00000i 6.03579 6.67602i 7.05875 0.417165i
83.15 1.00000 + 1.00000i 2.98664 0.282815i 2.00000i −3.28357 + 3.77070i 3.26945 + 2.70382i −5.95686 + 3.67639i −2.00000 + 2.00000i 8.84003 1.68933i −7.05427 + 0.487124i
83.16 1.00000 + 1.00000i 2.99999 + 0.00829838i 2.00000i 3.67015 + 3.39558i 2.99169 + 3.00829i 1.45834 6.84640i −2.00000 + 2.00000i 8.99986 + 0.0497901i 0.274569 + 7.06574i
167.1 1.00000 1.00000i −2.99999 + 0.00829838i 2.00000i −3.67015 + 3.39558i −2.99169 + 3.00829i 6.84640 + 1.45834i −2.00000 2.00000i 8.99986 0.0497901i −0.274569 + 7.06574i
167.2 1.00000 1.00000i −2.98664 0.282815i 2.00000i 3.28357 + 3.77070i −3.26945 + 2.70382i −3.67639 5.95686i −2.00000 2.00000i 8.84003 + 1.68933i 7.05427 + 0.487124i
167.3 1.00000 1.00000i −2.74188 1.21742i 2.00000i −3.32079 3.73796i −3.95930 + 1.52446i −6.61294 + 2.29543i −2.00000 2.00000i 6.03579 + 6.67602i −7.05875 0.417165i
167.4 1.00000 1.00000i −2.09499 + 2.14733i 2.00000i −1.13661 4.86910i 0.0523328 + 4.24232i 2.31291 + 6.60685i −2.00000 2.00000i −0.222012 8.99726i −6.00571 3.73249i
See all 32 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 167.16 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.b odd 2 1 inner
15.e even 4 1 inner
105.k odd 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 210.3.k.b yes 32
3.b odd 2 1 210.3.k.a 32
5.c odd 4 1 210.3.k.a 32
7.b odd 2 1 inner 210.3.k.b yes 32
15.e even 4 1 inner 210.3.k.b yes 32
21.c even 2 1 210.3.k.a 32
35.f even 4 1 210.3.k.a 32
105.k odd 4 1 inner 210.3.k.b yes 32

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.3.k.a 32 3.b odd 2 1
210.3.k.a 32 5.c odd 4 1
210.3.k.a 32 21.c even 2 1
210.3.k.a 32 35.f even 4 1
210.3.k.b yes 32 1.a even 1 1 trivial
210.3.k.b yes 32 7.b odd 2 1 inner
210.3.k.b yes 32 15.e even 4 1 inner
210.3.k.b yes 32 105.k odd 4 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{23}^{16} + \cdots$$ acting on $$S_{3}^{\mathrm{new}}(210, [\chi])$$.

## Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database