# Properties

 Label 210.3.q.a Level $210$ Weight $3$ Character orbit 210.q Analytic conductor $5.722$ Analytic rank $0$ Dimension $64$ CM no Inner twists $8$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$210 = 2 \cdot 3 \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 210.q (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$5.72208555157$$ Analytic rank: $$0$$ Dimension: $$64$$ Relative dimension: $$32$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$64q - 64q^{4} + 8q^{6} - 4q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$64q - 64q^{4} + 8q^{6} - 4q^{9} - 8q^{10} + 4q^{15} - 128q^{16} + 8q^{19} - 88q^{21} - 8q^{24} + 12q^{25} - 8q^{30} + 152q^{31} + 16q^{36} - 208q^{39} - 16q^{40} + 106q^{45} - 56q^{46} - 64q^{49} - 140q^{51} - 56q^{54} + 616q^{55} - 4q^{60} + 104q^{61} + 512q^{64} - 160q^{66} + 456q^{69} - 144q^{70} + 298q^{75} - 32q^{76} - 360q^{79} + 304q^{81} - 80q^{84} - 408q^{85} - 688q^{90} - 288q^{91} + 240q^{94} - 16q^{96} - 568q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
149.1 −0.707107 + 1.22474i −2.99996 + 0.0151730i −1.00000 1.73205i −3.19601 3.84519i 2.10271 3.68492i −6.79135 + 1.69633i 2.82843 8.99954 0.0910371i 6.96930 1.19534i
149.2 −0.707107 + 1.22474i −2.97672 + 0.373050i −1.00000 1.73205i −2.48660 + 4.33784i 1.64796 3.90950i 1.32164 + 6.87410i 2.82843 8.72167 2.22093i −3.55445 6.11276i
149.3 −0.707107 + 1.22474i −2.59977 1.49707i −1.00000 1.73205i 2.43105 + 4.36921i 3.67184 2.12546i −1.98724 6.71199i 2.82843 4.51756 + 7.78406i −7.07018 0.112086i
149.4 −0.707107 + 1.22474i −2.31822 + 1.90417i −1.00000 1.73205i 4.61270 + 1.92950i −0.692887 4.18568i 6.90570 + 1.14510i 2.82843 1.74830 8.82856i −5.62482 + 4.28502i
149.5 −0.707107 + 1.22474i −1.64845 2.50651i −1.00000 1.73205i 4.68918 1.73539i 4.23547 0.246562i −3.35661 + 6.14273i 2.82843 −3.56521 + 8.26373i −1.19034 + 6.97016i
149.6 −0.707107 + 1.22474i −1.34648 2.68086i −1.00000 1.73205i −0.841696 4.92865i 4.23547 + 0.246562i 3.35661 6.14273i 2.82843 −5.37400 + 7.21943i 6.63150 + 2.45422i
149.7 −0.707107 + 1.22474i −1.10765 + 2.78803i −1.00000 1.73205i −4.37658 2.41777i −2.63140 3.32803i 5.67547 4.09745i 2.82843 −6.54621 6.17633i 6.05585 3.65057i
149.8 −0.707107 + 1.22474i −0.355071 + 2.97891i −1.00000 1.73205i −1.67341 + 4.71166i −3.39734 2.54128i −7.00000 + 0.00780758i 2.82843 −8.74785 2.11545i −4.58730 5.38114i
149.9 −0.707107 + 1.22474i 0.00338259 3.00000i −1.00000 1.73205i −4.99937 + 0.0792566i 3.67184 + 2.12546i 1.98724 + 6.71199i 2.82843 −8.99998 0.0202955i 3.43802 6.17900i
149.10 −0.707107 + 1.22474i 0.422498 + 2.97010i −1.00000 1.73205i 1.86933 4.63741i −3.93637 1.58273i −0.123332 + 6.99891i 2.82843 −8.64299 + 2.50972i 4.35784 + 5.56860i
149.11 −0.707107 + 1.22474i 1.51312 2.59046i −1.00000 1.73205i 4.92804 + 0.845231i 2.10271 + 3.68492i 6.79135 1.69633i 2.82843 −4.42093 7.83935i −4.51984 + 5.43792i
149.12 −0.707107 + 1.22474i 1.81143 2.39139i −1.00000 1.73205i −2.51338 + 4.32238i 1.64796 + 3.90950i −1.32164 6.87410i 2.82843 −2.43745 8.66365i −3.51658 6.13463i
149.13 −0.707107 + 1.22474i 2.36093 + 1.85094i −1.00000 1.73205i 3.08146 3.93759i −3.93637 + 1.58273i 0.123332 6.99891i 2.82843 2.14801 + 8.73991i 2.64363 + 6.55830i
149.14 −0.707107 + 1.22474i 2.75735 + 1.18196i −1.00000 1.73205i −3.24371 + 3.80504i −3.39734 + 2.54128i 7.00000 0.00780758i 2.82843 6.20596 + 6.51813i −2.36656 6.66329i
149.15 −0.707107 + 1.22474i 2.80817 1.05556i −1.00000 1.73205i −3.97735 3.02996i −0.692887 + 4.18568i −6.90570 1.14510i 2.82843 6.77160 5.92835i 6.52334 2.72873i
149.16 −0.707107 + 1.22474i 2.96833 + 0.434759i −1.00000 1.73205i 4.28213 + 2.58134i −2.63140 + 3.32803i −5.67547 + 4.09745i 2.82843 8.62197 + 2.58102i −6.18941 + 3.41924i
149.17 0.707107 1.22474i −2.96833 0.434759i −1.00000 1.73205i 4.37658 + 2.41777i −2.63140 + 3.32803i 5.67547 4.09745i −2.82843 8.62197 + 2.58102i 6.05585 3.65057i
149.18 0.707107 1.22474i −2.80817 + 1.05556i −1.00000 1.73205i −4.61270 1.92950i −0.692887 + 4.18568i 6.90570 + 1.14510i −2.82843 6.77160 5.92835i −5.62482 + 4.28502i
149.19 0.707107 1.22474i −2.75735 1.18196i −1.00000 1.73205i 1.67341 4.71166i −3.39734 + 2.54128i −7.00000 + 0.00780758i −2.82843 6.20596 + 6.51813i −4.58730 5.38114i
149.20 0.707107 1.22474i −2.36093 1.85094i −1.00000 1.73205i −1.86933 + 4.63741i −3.93637 + 1.58273i −0.123332 + 6.99891i −2.82843 2.14801 + 8.73991i 4.35784 + 5.56860i
See all 64 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 179.32 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
5.b even 2 1 inner
7.c even 3 1 inner
15.d odd 2 1 inner
21.h odd 6 1 inner
35.j even 6 1 inner
105.o 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.q.a 64
3.b odd 2 1 inner 210.3.q.a 64
5.b even 2 1 inner 210.3.q.a 64
7.c even 3 1 inner 210.3.q.a 64
15.d odd 2 1 inner 210.3.q.a 64
21.h odd 6 1 inner 210.3.q.a 64
35.j even 6 1 inner 210.3.q.a 64
105.o odd 6 1 inner 210.3.q.a 64

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.3.q.a 64 1.a even 1 1 trivial
210.3.q.a 64 3.b odd 2 1 inner
210.3.q.a 64 5.b even 2 1 inner
210.3.q.a 64 7.c even 3 1 inner
210.3.q.a 64 15.d odd 2 1 inner
210.3.q.a 64 21.h odd 6 1 inner
210.3.q.a 64 35.j even 6 1 inner
210.3.q.a 64 105.o 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