# Properties

 Label 84.2.n.a Level $84$ Weight $2$ Character orbit 84.n Analytic conductor $0.671$ Analytic rank $0$ Dimension $24$ CM no Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$84 = 2^{2} \cdot 3 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 84.n (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.670743376979$$ Analytic rank: $$0$$ Dimension: $$24$$ Relative dimension: $$12$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$24q - 2q^{4} - 2q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$24q - 2q^{4} - 2q^{9} - 10q^{10} - 12q^{12} - 24q^{13} - 10q^{16} - 10q^{18} - 6q^{21} + 28q^{22} - 14q^{24} - 12q^{25} + 10q^{28} - 14q^{30} + 10q^{33} - 8q^{34} + 44q^{36} - 8q^{37} + 34q^{40} + 38q^{42} - 18q^{45} + 24q^{46} + 8q^{48} + 16q^{52} + 38q^{54} - 4q^{57} + 14q^{58} + 14q^{60} + 4q^{61} - 68q^{64} + 30q^{66} + 36q^{69} - 90q^{70} + 20q^{72} - 24q^{76} - 104q^{78} + 26q^{81} - 68q^{82} - 76q^{84} - 40q^{85} - 34q^{88} - 40q^{90} - 6q^{93} - 24q^{94} - 62q^{96} + 72q^{97} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
11.1 −1.32175 + 0.502962i −1.72058 0.199011i 1.49406 1.32958i 1.79791 1.03802i 2.37428 0.602344i 2.61347 + 0.412056i −1.30604 + 2.50883i 2.92079 + 0.684828i −1.85431 + 2.27629i
11.2 −1.29776 0.561968i 0.373317 1.69134i 1.36838 + 1.45860i 0.432549 0.249732i −1.43496 + 1.98517i −0.261928 2.63275i −0.956154 2.66191i −2.72127 1.26281i −0.701688 + 0.0810151i
11.3 −1.09645 + 0.893190i 1.72058 + 0.199011i 0.404424 1.95868i 1.79791 1.03802i −2.06429 + 1.31860i −2.61347 0.412056i 1.30604 + 2.50883i 2.92079 + 0.684828i −1.04417 + 2.74402i
11.4 −0.951145 1.04658i −1.66539 + 0.475894i −0.190646 + 1.99089i −2.36229 + 1.36387i 2.08209 + 1.29031i −1.89768 + 1.84359i 2.26495 1.69410i 2.54705 1.58510i 3.67427 + 1.17508i
11.5 −0.430790 1.34700i 0.420559 + 1.68022i −1.62884 + 1.16055i 2.36229 1.36387i 2.08209 1.29031i 1.89768 1.84359i 2.26495 + 1.69410i −2.64626 + 1.41326i −2.85479 2.59447i
11.6 −0.162204 + 1.40488i −0.373317 + 1.69134i −1.94738 0.455755i 0.432549 0.249732i −2.31558 0.798808i 0.261928 + 2.63275i 0.956154 2.66191i −2.72127 1.26281i 0.280683 + 0.648187i
11.7 0.162204 1.40488i 1.27809 1.16897i −1.94738 0.455755i −0.432549 + 0.249732i −1.43496 1.98517i 0.261928 + 2.63275i −0.956154 + 2.66191i 0.267006 2.98809i 0.280683 + 0.648187i
11.8 0.430790 + 1.34700i 1.66539 0.475894i −1.62884 + 1.16055i −2.36229 + 1.36387i 1.35846 + 2.03828i 1.89768 1.84359i −2.26495 1.69410i 2.54705 1.58510i −2.85479 2.59447i
11.9 0.951145 + 1.04658i −0.420559 1.68022i −0.190646 + 1.99089i 2.36229 1.36387i 1.35846 2.03828i −1.89768 + 1.84359i −2.26495 + 1.69410i −2.64626 + 1.41326i 3.67427 + 1.17508i
11.10 1.09645 0.893190i 1.03264 + 1.39056i 0.404424 1.95868i −1.79791 + 1.03802i 2.37428 + 0.602344i −2.61347 0.412056i −1.30604 2.50883i −0.867316 + 2.87189i −1.04417 + 2.74402i
11.11 1.29776 + 0.561968i −1.27809 + 1.16897i 1.36838 + 1.45860i −0.432549 + 0.249732i −2.31558 + 0.798808i −0.261928 2.63275i 0.956154 + 2.66191i 0.267006 2.98809i −0.701688 + 0.0810151i
11.12 1.32175 0.502962i −1.03264 1.39056i 1.49406 1.32958i −1.79791 + 1.03802i −2.06429 1.31860i 2.61347 + 0.412056i 1.30604 2.50883i −0.867316 + 2.87189i −1.85431 + 2.27629i
23.1 −1.32175 0.502962i −1.72058 + 0.199011i 1.49406 + 1.32958i 1.79791 + 1.03802i 2.37428 + 0.602344i 2.61347 0.412056i −1.30604 2.50883i 2.92079 0.684828i −1.85431 2.27629i
23.2 −1.29776 + 0.561968i 0.373317 + 1.69134i 1.36838 1.45860i 0.432549 + 0.249732i −1.43496 1.98517i −0.261928 + 2.63275i −0.956154 + 2.66191i −2.72127 + 1.26281i −0.701688 0.0810151i
23.3 −1.09645 0.893190i 1.72058 0.199011i 0.404424 + 1.95868i 1.79791 + 1.03802i −2.06429 1.31860i −2.61347 + 0.412056i 1.30604 2.50883i 2.92079 0.684828i −1.04417 2.74402i
23.4 −0.951145 + 1.04658i −1.66539 0.475894i −0.190646 1.99089i −2.36229 1.36387i 2.08209 1.29031i −1.89768 1.84359i 2.26495 + 1.69410i 2.54705 + 1.58510i 3.67427 1.17508i
23.5 −0.430790 + 1.34700i 0.420559 1.68022i −1.62884 1.16055i 2.36229 + 1.36387i 2.08209 + 1.29031i 1.89768 + 1.84359i 2.26495 1.69410i −2.64626 1.41326i −2.85479 + 2.59447i
23.6 −0.162204 1.40488i −0.373317 1.69134i −1.94738 + 0.455755i 0.432549 + 0.249732i −2.31558 + 0.798808i 0.261928 2.63275i 0.956154 + 2.66191i −2.72127 + 1.26281i 0.280683 0.648187i
23.7 0.162204 + 1.40488i 1.27809 + 1.16897i −1.94738 + 0.455755i −0.432549 0.249732i −1.43496 + 1.98517i 0.261928 2.63275i −0.956154 2.66191i 0.267006 + 2.98809i 0.280683 0.648187i
23.8 0.430790 1.34700i 1.66539 + 0.475894i −1.62884 1.16055i −2.36229 1.36387i 1.35846 2.03828i 1.89768 + 1.84359i −2.26495 + 1.69410i 2.54705 + 1.58510i −2.85479 + 2.59447i
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 23.12 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
4.b odd 2 1 inner
7.c even 3 1 inner
12.b even 2 1 inner
21.h odd 6 1 inner
28.g odd 6 1 inner
84.n even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 84.2.n.a 24
3.b odd 2 1 inner 84.2.n.a 24
4.b odd 2 1 inner 84.2.n.a 24
7.b odd 2 1 588.2.n.e 24
7.c even 3 1 inner 84.2.n.a 24
7.c even 3 1 588.2.e.e 12
7.d odd 6 1 588.2.e.d 12
7.d odd 6 1 588.2.n.e 24
12.b even 2 1 inner 84.2.n.a 24
21.c even 2 1 588.2.n.e 24
21.g even 6 1 588.2.e.d 12
21.g even 6 1 588.2.n.e 24
21.h odd 6 1 inner 84.2.n.a 24
21.h odd 6 1 588.2.e.e 12
28.d even 2 1 588.2.n.e 24
28.f even 6 1 588.2.e.d 12
28.f even 6 1 588.2.n.e 24
28.g odd 6 1 inner 84.2.n.a 24
28.g odd 6 1 588.2.e.e 12
84.h odd 2 1 588.2.n.e 24
84.j odd 6 1 588.2.e.d 12
84.j odd 6 1 588.2.n.e 24
84.n even 6 1 inner 84.2.n.a 24
84.n even 6 1 588.2.e.e 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.2.n.a 24 1.a even 1 1 trivial
84.2.n.a 24 3.b odd 2 1 inner
84.2.n.a 24 4.b odd 2 1 inner
84.2.n.a 24 7.c even 3 1 inner
84.2.n.a 24 12.b even 2 1 inner
84.2.n.a 24 21.h odd 6 1 inner
84.2.n.a 24 28.g odd 6 1 inner
84.2.n.a 24 84.n even 6 1 inner
588.2.e.d 12 7.d odd 6 1
588.2.e.d 12 21.g even 6 1
588.2.e.d 12 28.f even 6 1
588.2.e.d 12 84.j odd 6 1
588.2.e.e 12 7.c even 3 1
588.2.e.e 12 21.h odd 6 1
588.2.e.e 12 28.g odd 6 1
588.2.e.e 12 84.n even 6 1
588.2.n.e 24 7.b odd 2 1
588.2.n.e 24 7.d odd 6 1
588.2.n.e 24 21.c even 2 1
588.2.n.e 24 21.g even 6 1
588.2.n.e 24 28.d even 2 1
588.2.n.e 24 28.f even 6 1
588.2.n.e 24 84.h odd 2 1
588.2.n.e 24 84.j odd 6 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(84, [\chi])$$.