# Properties

 Label 108.6.h.a Level 108 Weight 6 Character orbit 108.h Analytic conductor 17.321 Analytic rank 0 Dimension 56 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$108 = 2^{2} \cdot 3^{3}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 108.h (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$17.3214525398$$ Analytic rank: $$0$$ Dimension: $$56$$ Relative dimension: $$28$$ over $$\Q(\zeta_{6})$$ Twist minimal: no (minimal twist has level 36) 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.

 $$\operatorname{Tr}(f)(q) =$$ $$56q + 3q^{2} - q^{4} + 6q^{5} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$56q + 3q^{2} - q^{4} + 6q^{5} - 68q^{10} - 2q^{13} + 1518q^{14} - q^{16} + 1242q^{20} + 63q^{22} + 12498q^{25} - 2052q^{28} + 11946q^{29} + 7233q^{32} + 6361q^{34} - 8q^{37} + 14877q^{38} - 1526q^{40} + 43536q^{41} - 26880q^{46} + 38414q^{49} - 38631q^{50} + 24988q^{52} - 21186q^{56} - 3314q^{58} - 2q^{61} - 106342q^{64} - 35970q^{65} - 31413q^{68} + 10524q^{70} + 53620q^{73} + 20406q^{74} + 26193q^{76} - 26178q^{77} - 151286q^{82} + 6248q^{85} - 279237q^{86} - 122541q^{88} - 435804q^{92} + 63480q^{94} - 58148q^{97} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
35.1 −5.65362 + 0.191212i 0 31.9269 2.16208i −3.68052 + 2.12495i 0 −13.9322 8.04377i −180.089 + 18.3284i 0 20.4019 12.7174i
35.2 −5.52836 + 1.19885i 0 29.1255 13.2554i 20.1454 11.6310i 0 156.832 + 90.5473i −145.125 + 108.198i 0 −97.4272 + 88.4515i
35.3 −5.48009 1.40307i 0 28.0628 + 15.3779i 77.8058 44.9212i 0 −124.034 71.6112i −132.210 123.647i 0 −489.410 + 137.005i
35.4 −5.12358 2.39768i 0 20.5022 + 24.5695i −61.0162 + 35.2277i 0 −181.157 104.591i −46.1351 175.042i 0 397.087 34.1946i
35.5 −4.87383 + 2.87155i 0 15.5084 27.9909i −70.1957 + 40.5275i 0 −89.9985 51.9607i 4.79147 + 180.956i 0 225.745 399.094i
35.6 −4.63825 3.23831i 0 11.0267 + 30.0402i −61.0162 + 35.2277i 0 181.157 + 104.591i 46.1351 175.042i 0 397.087 + 34.1946i
35.7 −3.95514 4.04436i 0 −0.713689 + 31.9920i 77.8058 44.9212i 0 124.034 + 71.6112i 132.210 123.647i 0 −489.410 137.005i
35.8 −3.75843 + 4.22779i 0 −3.74838 31.7797i −12.8631 + 7.42651i 0 15.2989 + 8.83282i 148.446 + 103.595i 0 16.9473 82.2944i
35.9 −3.65969 + 4.31354i 0 −5.21330 31.5725i 51.8636 29.9435i 0 33.8827 + 19.5622i 155.268 + 93.0578i 0 −60.6425 + 333.300i
35.10 −2.66122 4.99179i 0 −17.8359 + 26.5684i −3.68052 + 2.12495i 0 13.9322 + 8.04377i 180.089 + 18.3284i 0 20.4019 + 12.7174i
35.11 −1.78768 + 5.36696i 0 −25.6084 19.1888i 52.1793 30.1257i 0 −185.316 106.992i 148.765 103.136i 0 68.4038 + 333.899i
35.12 −1.72594 5.38713i 0 −26.0423 + 18.5957i 20.1454 11.6310i 0 −156.832 90.5473i 145.125 + 108.198i 0 −97.4272 88.4515i
35.13 −0.253656 + 5.65116i 0 −31.8713 2.86690i −84.3255 + 48.6853i 0 −87.9661 50.7872i 24.2857 179.383i 0 −253.739 488.886i
35.14 −0.216257 + 5.65272i 0 −31.9065 2.44488i 0.143132 0.0826372i 0 192.634 + 111.217i 20.7202 179.830i 0 0.436172 + 0.826955i
35.15 0.0499164 5.65663i 0 −31.9950 0.564717i −70.1957 + 40.5275i 0 89.9985 + 51.9607i −4.79147 + 180.956i 0 225.745 + 399.094i
35.16 1.64008 + 5.41389i 0 −26.6203 + 17.7584i −38.3603 + 22.1474i 0 102.940 + 59.4323i −139.801 114.994i 0 −182.817 171.355i
35.17 1.78216 5.36879i 0 −25.6478 19.1360i −12.8631 + 7.42651i 0 −15.2989 8.83282i −148.446 + 103.595i 0 16.9473 + 82.2944i
35.18 1.90579 5.32616i 0 −24.7359 20.3011i 51.8636 29.9435i 0 −33.8827 19.5622i −155.268 + 93.0578i 0 −60.6425 333.300i
35.19 2.01287 + 5.28662i 0 −23.8967 + 21.2825i 78.5398 45.3450i 0 −69.6792 40.2293i −160.614 83.4942i 0 397.812 + 323.937i
35.20 3.75408 4.23165i 0 −3.81374 31.7719i 52.1793 30.1257i 0 185.316 + 106.992i −148.765 103.136i 0 68.4038 333.899i
See all 56 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 71.28 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
9.d odd 6 1 inner
36.h even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 108.6.h.a 56
3.b odd 2 1 36.6.h.a 56
4.b odd 2 1 inner 108.6.h.a 56
9.c even 3 1 36.6.h.a 56
9.d odd 6 1 inner 108.6.h.a 56
12.b even 2 1 36.6.h.a 56
36.f odd 6 1 36.6.h.a 56
36.h even 6 1 inner 108.6.h.a 56

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
36.6.h.a 56 3.b odd 2 1
36.6.h.a 56 9.c even 3 1
36.6.h.a 56 12.b even 2 1
36.6.h.a 56 36.f odd 6 1
108.6.h.a 56 1.a even 1 1 trivial
108.6.h.a 56 4.b odd 2 1 inner
108.6.h.a 56 9.d odd 6 1 inner
108.6.h.a 56 36.h even 6 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{6}^{\mathrm{new}}(108, [\chi])$$.

## Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database