# Properties

 Label 76.6.f.a Level $76$ Weight $6$ Character orbit 76.f Analytic conductor $12.189$ Analytic rank $0$ Dimension $96$ CM no Inner twists $4$

# Learn more

## Newspace parameters

 Level: $$N$$ $$=$$ $$76 = 2^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 76.f (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$12.1891703058$$ Analytic rank: $$0$$ Dimension: $$96$$ Relative dimension: $$48$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$96q - 3q^{2} - 11q^{4} - 2q^{5} - 27q^{6} - 3588q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$96q - 3q^{2} - 11q^{4} - 2q^{5} - 27q^{6} - 3588q^{9} + 648q^{10} - 354q^{13} + 2154q^{14} - 611q^{16} - 1006q^{17} + 10524q^{20} + 7824q^{21} + 93q^{22} - 5311q^{24} - 25002q^{25} - 13428q^{26} + 1556q^{28} - 6q^{29} - 12656q^{30} - 30453q^{32} + 8586q^{33} + 30102q^{34} - 2306q^{36} + 29846q^{38} - 1392q^{40} + 29064q^{41} + 18616q^{42} + 9155q^{44} + 13464q^{45} + 38217q^{48} - 116224q^{49} + 36858q^{52} + 26262q^{53} - 26355q^{54} + 53974q^{57} - 112740q^{58} - 108810q^{60} - 77370q^{61} - 106848q^{62} + 337966q^{64} + 166855q^{66} + 175796q^{68} + 148122q^{70} - 259842q^{72} - 52040q^{73} + 103690q^{74} - 194709q^{76} + 197584q^{77} + 253206q^{78} - 171934q^{80} - 313100q^{81} + 63919q^{82} - 161762q^{85} - 38442q^{86} + 10482q^{89} - 372450q^{90} + 64544q^{92} + 94012q^{93} + 537614q^{96} + 177180q^{97} - 509397q^{98} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
27.1 −5.65655 + 0.0584141i 6.88277 11.9213i 31.9932 0.660845i 12.1360 21.0202i −38.2364 + 67.8356i 229.350i −180.932 + 5.60696i 26.7549 + 46.3408i −67.4201 + 119.611i
27.2 −5.64509 + 0.364630i 2.01363 3.48771i 31.7341 4.11673i 54.6845 94.7163i −10.0954 + 20.4227i 177.092i −177.641 + 34.8105i 113.391 + 196.398i −274.162 + 554.621i
27.3 −5.64436 0.375742i −5.80366 + 10.0522i 31.7176 + 4.24165i −48.4181 + 83.8626i 36.5350 54.5577i 77.0285i −177.432 35.8590i 54.1352 + 93.7648i 304.800 455.158i
27.4 −5.63922 + 0.446278i 12.1499 21.0442i 31.6017 5.03332i −36.1399 + 62.5962i −59.1243 + 124.095i 149.741i −175.963 + 42.4871i −173.739 300.925i 175.866 369.123i
27.5 −5.57270 + 0.972140i −9.46379 + 16.3918i 30.1099 10.8349i −13.0553 + 22.6124i 36.8037 100.546i 118.461i −157.260 + 89.6506i −57.6266 99.8122i 50.7707 138.704i
27.6 −5.16544 2.30614i −14.1896 + 24.5772i 21.3634 + 23.8244i 24.5038 42.4418i 129.974 94.2285i 6.61611i −55.4091 172.331i −281.192 487.038i −224.449 + 162.721i
27.7 −5.03826 2.57214i 3.22504 5.58593i 18.7682 + 25.9182i −2.36309 + 4.09299i −30.6164 + 19.8482i 80.2152i −27.8940 178.857i 100.698 + 174.414i 22.4336 14.5434i
27.8 −5.00676 + 2.63294i −8.66479 + 15.0079i 18.1352 26.3650i 25.4990 44.1655i 3.86765 97.9546i 143.799i −21.3810 + 179.752i −28.6572 49.6358i −11.3818 + 288.263i
27.9 −4.74667 3.07720i −3.22504 + 5.58593i 13.0617 + 29.2129i −2.36309 + 4.09299i 32.4972 16.5905i 80.2152i 27.8940 178.857i 100.698 + 174.414i 23.8117 12.1564i
27.10 −4.70334 + 3.14303i 7.75035 13.4240i 12.2427 29.5654i −8.78700 + 15.2195i 5.73951 + 87.4972i 77.0445i 35.3433 + 177.535i 1.36406 + 2.36263i −6.50720 99.2004i
27.11 −4.57989 3.32033i 14.1896 24.5772i 9.95083 + 30.4135i 24.5038 42.4418i −146.591 + 65.4465i 6.61611i 55.4091 172.331i −281.192 487.038i −253.145 + 113.018i
27.12 −4.29234 + 3.68454i −1.79179 + 3.10347i 4.84840 31.6306i −10.1109 + 17.5126i −3.74388 19.9231i 149.904i 95.7329 + 153.633i 115.079 + 199.323i −21.1264 112.424i
27.13 −3.30431 + 4.59146i 10.9615 18.9858i −10.1630 30.3432i 32.1573 55.6980i 50.9526 + 113.064i 65.5487i 172.902 + 53.6003i −118.808 205.781i 149.478 + 331.693i
27.14 −3.14758 4.70029i 5.80366 10.0522i −12.1854 + 29.5891i −48.4181 + 83.8626i −65.5159 + 4.36135i 77.0285i 177.432 35.8590i 54.1352 + 93.7648i 546.579 36.3854i
27.15 −2.77769 4.92793i −6.88277 + 11.9213i −16.5689 + 27.3765i 12.1360 21.0202i 77.8655 + 0.804103i 229.350i 180.932 + 5.60696i 26.7549 + 46.3408i −137.296 1.41783i
27.16 −2.77038 + 4.93204i −1.48160 + 2.56620i −16.6500 27.3272i −49.5981 + 85.9065i −8.55202 14.4167i 147.623i 180.906 6.41167i 117.110 + 202.840i −286.288 482.613i
27.17 −2.64697 + 4.99936i −15.2473 + 26.4090i −17.9871 26.4663i −23.0916 + 39.9958i −91.6692 146.130i 131.131i 179.926 19.8689i −343.458 594.887i −138.831 221.311i
27.18 −2.50677 5.07111i −2.01363 + 3.48771i −19.4322 + 25.4242i 54.6845 94.7163i 22.7343 + 1.46846i 177.092i 177.641 + 34.8105i 113.391 + 196.398i −617.397 39.8792i
27.19 −2.43312 5.10685i −12.1499 + 21.0442i −20.1598 + 24.8512i −36.1399 + 62.5962i 137.032 + 10.8444i 149.741i 175.963 + 42.4871i −173.739 300.925i 407.602 + 32.2569i
27.20 −2.12496 + 5.24257i −8.46878 + 14.6684i −22.9691 22.2805i 29.9102 51.8059i −58.9040 75.5679i 59.1673i 165.616 73.0716i −21.9405 38.0021i 208.038 + 266.892i
See all 96 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 31.48 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
19.d odd 6 1 inner
76.f even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 76.6.f.a 96
4.b odd 2 1 inner 76.6.f.a 96
19.d odd 6 1 inner 76.6.f.a 96
76.f even 6 1 inner 76.6.f.a 96

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
76.6.f.a 96 1.a even 1 1 trivial
76.6.f.a 96 4.b odd 2 1 inner
76.6.f.a 96 19.d odd 6 1 inner
76.6.f.a 96 76.f even 6 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{6}^{\mathrm{new}}(76, [\chi])$$.