# Properties

 Label 210.2.x.a Level 210 Weight 2 Character orbit 210.x Analytic conductor 1.677 Analytic rank 0 Dimension 64 CM no Inner twists 8

# Related objects

## Newspace parameters

 Level: $$N$$ = $$210 = 2 \cdot 3 \cdot 5 \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 210.x (of order $$12$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.67685844245$$ Analytic rank: $$0$$ Dimension: $$64$$ Relative dimension: $$16$$ over $$\Q(\zeta_{12})$$ Coefficient ring index: multiple of None 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.

 $$\operatorname{Tr}(f)(q) =$$ $$64q + 8q^{6} + 4q^{7} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$64q + 8q^{6} + 4q^{7} + 4q^{10} - 24q^{15} + 32q^{16} - 8q^{18} + 12q^{21} - 24q^{22} - 8q^{25} - 72q^{27} + 4q^{28} - 12q^{30} + 32q^{31} - 20q^{33} - 24q^{36} - 8q^{37} - 40q^{42} - 64q^{43} - 28q^{45} + 24q^{46} - 16q^{55} + 24q^{57} - 28q^{58} + 8q^{60} + 24q^{61} - 88q^{63} - 16q^{67} - 12q^{70} + 8q^{72} - 56q^{73} + 8q^{75} - 32q^{76} - 16q^{78} + 4q^{81} + 16q^{82} + 32q^{85} + 76q^{87} + 12q^{88} + 40q^{90} - 48q^{91} + 84q^{93} + 4q^{96} + 104q^{97} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
23.1 −0.965926 + 0.258819i −1.64884 0.530415i 0.866025 0.500000i 0.680454 2.13002i 1.72993 + 0.0855915i −2.36464 + 1.18680i −0.707107 + 0.707107i 2.43732 + 1.74913i −0.105978 + 2.23356i
23.2 −0.965926 + 0.258819i −1.45883 + 0.933717i 0.866025 0.500000i −2.00119 + 0.997619i 1.16745 1.27947i −1.20893 2.35340i −0.707107 + 0.707107i 1.25635 2.72426i 1.67480 1.48157i
23.3 −0.965926 + 0.258819i −0.806628 1.53276i 0.866025 0.500000i 0.480420 + 2.18385i 1.17585 + 1.27176i −0.998710 + 2.45002i −0.707107 + 0.707107i −1.69870 + 2.47273i −1.02927 1.98509i
23.4 −0.965926 + 0.258819i −0.755113 + 1.55878i 0.866025 0.500000i 1.36035 1.77467i 0.325941 1.70111i 2.62857 + 0.301012i −0.707107 + 0.707107i −1.85961 2.35411i −0.854674 + 2.06628i
23.5 −0.965926 + 0.258819i 0.378582 1.69017i 0.866025 0.500000i 2.16375 0.564071i 0.0717662 + 1.73056i 0.113038 2.64334i −0.707107 + 0.707107i −2.71335 1.27974i −1.94403 + 1.10487i
23.6 −0.965926 + 0.258819i 0.929705 1.46139i 0.866025 0.500000i −1.68046 1.47514i −0.519792 + 1.65222i 2.42513 + 1.05770i −0.707107 + 0.707107i −1.27130 2.71732i 2.00500 + 0.989939i
23.7 −0.965926 + 0.258819i 1.03943 + 1.38549i 0.866025 0.500000i −2.22506 0.221605i −1.36260 1.06926i −0.736347 + 2.54122i −0.707107 + 0.707107i −0.839170 + 2.88024i 2.20660 0.361834i
23.8 −0.965926 + 0.258819i 1.35576 + 1.07792i 0.866025 0.500000i 0.962923 + 2.01811i −1.58855 0.690295i 1.50791 2.17399i −0.707107 + 0.707107i 0.676173 + 2.92281i −1.45244 1.70012i
23.9 0.965926 0.258819i −1.71308 + 0.255627i 0.866025 0.500000i −0.962923 2.01811i −1.58855 + 0.690295i 1.50791 2.17399i 0.707107 0.707107i 2.86931 0.875819i −1.45244 1.70012i
23.10 0.965926 0.258819i −1.59292 + 0.680155i 0.866025 0.500000i 2.22506 + 0.221605i −1.36260 + 1.06926i −0.736347 + 2.54122i 0.707107 0.707107i 2.07478 2.16686i 2.20660 0.361834i
23.11 0.965926 0.258819i −0.125444 + 1.72750i 0.866025 0.500000i −1.36035 + 1.77467i 0.325941 + 1.70111i 2.62857 + 0.301012i 0.707107 0.707107i −2.96853 0.433411i −0.854674 + 2.06628i
23.12 0.965926 0.258819i −0.0744552 1.73045i 0.866025 0.500000i 1.68046 + 1.47514i −0.519792 1.65222i 2.42513 + 1.05770i 0.707107 0.707107i −2.98891 + 0.257682i 2.00500 + 0.989939i
23.13 0.965926 0.258819i 0.517224 1.65302i 0.866025 0.500000i −2.16375 + 0.564071i 0.0717662 1.73056i 0.113038 2.64334i 0.707107 0.707107i −2.46496 1.70996i −1.94403 + 1.10487i
23.14 0.965926 0.258819i 0.796522 + 1.53804i 0.866025 0.500000i 2.00119 0.997619i 1.16745 + 1.27947i −1.20893 2.35340i 0.707107 0.707107i −1.73111 + 2.45016i 1.67480 1.48157i
23.15 0.965926 0.258819i 1.46494 0.924094i 0.866025 0.500000i −0.480420 2.18385i 1.17585 1.27176i −0.998710 + 2.45002i 0.707107 0.707107i 1.29210 2.70749i −1.02927 1.98509i
23.16 0.965926 0.258819i 1.69314 + 0.365065i 0.866025 0.500000i −0.680454 + 2.13002i 1.72993 0.0855915i −2.36464 + 1.18680i 0.707107 0.707107i 2.73346 + 1.23621i −0.105978 + 2.23356i
53.1 −0.258819 + 0.965926i −1.69017 + 0.378582i −0.866025 0.500000i 0.593376 2.15590i 0.0717662 1.73056i 2.64334 0.113038i 0.707107 0.707107i 2.71335 1.27974i 1.92886 + 1.13115i
53.2 −0.258819 + 0.965926i −1.53276 0.806628i −0.866025 0.500000i 2.13148 + 0.675868i 1.17585 1.27176i −2.45002 + 0.998710i 0.707107 0.707107i 1.69870 + 2.47273i −1.20451 + 1.88392i
53.3 −0.258819 + 0.965926i −1.46139 + 0.929705i −0.866025 0.500000i −2.11774 + 0.717755i −0.519792 1.65222i −1.05770 2.42513i 0.707107 0.707107i 1.27130 2.71732i −0.145187 2.23135i
53.4 −0.258819 + 0.965926i −0.530415 1.64884i −0.866025 0.500000i −1.50442 1.65430i 1.72993 0.0855915i −1.18680 + 2.36464i 0.707107 0.707107i −2.43732 + 1.74913i 1.98730 1.02500i
See all 64 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 137.16 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.c odd 4 1 inner
7.c even 3 1 inner
15.e even 4 1 inner
21.h odd 6 1 inner
35.l odd 12 1 inner
105.x even 12 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 210.2.x.a 64
3.b odd 2 1 inner 210.2.x.a 64
5.c odd 4 1 inner 210.2.x.a 64
7.c even 3 1 inner 210.2.x.a 64
15.e even 4 1 inner 210.2.x.a 64
21.h odd 6 1 inner 210.2.x.a 64
35.l odd 12 1 inner 210.2.x.a 64
105.x even 12 1 inner 210.2.x.a 64

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.2.x.a 64 1.a even 1 1 trivial
210.2.x.a 64 3.b odd 2 1 inner
210.2.x.a 64 5.c odd 4 1 inner
210.2.x.a 64 7.c even 3 1 inner
210.2.x.a 64 15.e even 4 1 inner
210.2.x.a 64 21.h odd 6 1 inner
210.2.x.a 64 35.l odd 12 1 inner
210.2.x.a 64 105.x even 12 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(210, [\chi])$$.

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database