Properties

 Label 81.9.h.a Level $81$ Weight $9$ Character orbit 81.h Analytic conductor $32.998$ Analytic rank $0$ Dimension $1278$ CM no Inner twists $2$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [81,9,Mod(2,81)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(81, base_ring=CyclotomicField(54))

chi = DirichletCharacter(H, H._module([1]))

N = Newforms(chi, 9, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("81.2");

S:= CuspForms(chi, 9);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$81 = 3^{4}$$ Weight: $$k$$ $$=$$ $$9$$ Character orbit: $$[\chi]$$ $$=$$ 81.h (of order $$54$$, degree $$18$$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: no Analytic conductor: $$32.9976674150$$ Analytic rank: $$0$$ Dimension: $$1278$$ Relative dimension: $$71$$ over $$\Q(\zeta_{54})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{54}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic $$q$$-expansion, but we have computed the trace expansion.

 $$\operatorname{Tr}(f)(q) =$$ $$1278 q - 18 q^{2} - 18 q^{3} - 18 q^{4} - 18 q^{5} - 18 q^{6} - 18 q^{7} - 18 q^{8} - 18 q^{9}+O(q^{10})$$ 1278 * q - 18 * q^2 - 18 * q^3 - 18 * q^4 - 18 * q^5 - 18 * q^6 - 18 * q^7 - 18 * q^8 - 18 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$1278 q - 18 q^{2} - 18 q^{3} - 18 q^{4} - 18 q^{5} - 18 q^{6} - 18 q^{7} - 18 q^{8} - 18 q^{9} - 18 q^{10} - 18 q^{11} - 18 q^{12} - 18 q^{13} - 18 q^{14} - 18 q^{15} - 18 q^{16} - 18 q^{17} - 274194 q^{18} - 18 q^{19} - 677394 q^{20} - 1360953 q^{21} - 18 q^{22} + 932049 q^{23} + 3725550 q^{24} - 18 q^{25} - 27 q^{26} - 1933299 q^{27} - 9 q^{28} - 2853729 q^{29} - 1292562 q^{30} - 18 q^{31} + 11466990 q^{32} + 6069339 q^{33} - 18 q^{34} - 6225435 q^{35} - 18224658 q^{36} - 18 q^{37} + 12749166 q^{38} - 18 q^{39} - 18 q^{40} - 6454962 q^{41} + 39542067 q^{42} - 18 q^{43} - 17341146 q^{44} - 37896786 q^{45} - 18 q^{46} - 19806786 q^{47} + 15217767 q^{48} - 18 q^{49} + 104798403 q^{50} + 54355878 q^{51} - 2322 q^{52} - 27 q^{53} - 75728412 q^{54} - 9 q^{55} - 168854472 q^{56} - 65747826 q^{57} - 18 q^{58} + 7377030 q^{59} + 125543673 q^{60} - 18 q^{61} + 223709589 q^{62} + 101958462 q^{63} - 18 q^{64} - 45382806 q^{65} - 552090186 q^{66} + 95245587 q^{67} - 190809207 q^{68} + 273910662 q^{69} - 177120018 q^{70} + 251437590 q^{71} + 548536302 q^{72} - 18 q^{73} - 154054674 q^{74} - 246093768 q^{75} + 305980398 q^{76} - 574220898 q^{77} - 478793403 q^{78} + 103151808 q^{79} + 148365990 q^{81} - 36 q^{82} + 368657334 q^{83} + 1377777303 q^{84} - 257411268 q^{85} + 435234798 q^{86} + 61361982 q^{87} + 213663726 q^{88} - 893853972 q^{89} - 624764277 q^{90} - 18 q^{91} - 745285608 q^{92} - 53952426 q^{93} - 106493202 q^{94} - 313936362 q^{95} + 626058639 q^{96} - 427023783 q^{97} - 590433561 q^{98} - 967332078 q^{99}+O(q^{100})$$ 1278 * q - 18 * q^2 - 18 * q^3 - 18 * q^4 - 18 * q^5 - 18 * q^6 - 18 * q^7 - 18 * q^8 - 18 * q^9 - 18 * q^10 - 18 * q^11 - 18 * q^12 - 18 * q^13 - 18 * q^14 - 18 * q^15 - 18 * q^16 - 18 * q^17 - 274194 * q^18 - 18 * q^19 - 677394 * q^20 - 1360953 * q^21 - 18 * q^22 + 932049 * q^23 + 3725550 * q^24 - 18 * q^25 - 27 * q^26 - 1933299 * q^27 - 9 * q^28 - 2853729 * q^29 - 1292562 * q^30 - 18 * q^31 + 11466990 * q^32 + 6069339 * q^33 - 18 * q^34 - 6225435 * q^35 - 18224658 * q^36 - 18 * q^37 + 12749166 * q^38 - 18 * q^39 - 18 * q^40 - 6454962 * q^41 + 39542067 * q^42 - 18 * q^43 - 17341146 * q^44 - 37896786 * q^45 - 18 * q^46 - 19806786 * q^47 + 15217767 * q^48 - 18 * q^49 + 104798403 * q^50 + 54355878 * q^51 - 2322 * q^52 - 27 * q^53 - 75728412 * q^54 - 9 * q^55 - 168854472 * q^56 - 65747826 * q^57 - 18 * q^58 + 7377030 * q^59 + 125543673 * q^60 - 18 * q^61 + 223709589 * q^62 + 101958462 * q^63 - 18 * q^64 - 45382806 * q^65 - 552090186 * q^66 + 95245587 * q^67 - 190809207 * q^68 + 273910662 * q^69 - 177120018 * q^70 + 251437590 * q^71 + 548536302 * q^72 - 18 * q^73 - 154054674 * q^74 - 246093768 * q^75 + 305980398 * q^76 - 574220898 * q^77 - 478793403 * q^78 + 103151808 * q^79 + 148365990 * q^81 - 36 * q^82 + 368657334 * q^83 + 1377777303 * q^84 - 257411268 * q^85 + 435234798 * q^86 + 61361982 * q^87 + 213663726 * q^88 - 893853972 * q^89 - 624764277 * q^90 - 18 * q^91 - 745285608 * q^92 - 53952426 * q^93 - 106493202 * q^94 - 313936362 * q^95 + 626058639 * q^96 - 427023783 * q^97 - 590433561 * q^98 - 967332078 * q^99

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.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label   $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2.1 −31.4247 1.83028i −49.2077 64.3398i 729.893 + 85.3122i 55.5600 + 234.426i 1428.58 + 2111.92i 565.067 + 759.017i −14844.6 2617.50i −1718.21 + 6332.02i −1316.89 7468.45i
2.2 −29.6556 1.72724i 79.6167 + 14.9058i 622.200 + 72.7247i 40.0859 + 169.136i −2335.33 579.556i −896.647 1204.41i −10836.9 1910.84i 6116.64 + 2373.49i −896.631 5085.05i
2.3 −29.5522 1.72122i −2.99020 + 80.9448i 616.102 + 72.0120i −151.007 637.148i 227.691 2386.95i 2120.94 + 2848.92i −10620.2 1872.62i −6543.12 484.082i 3365.91 + 19089.0i
2.4 −29.1725 1.69911i −71.7738 + 37.5435i 593.879 + 69.4145i −2.78495 11.7506i 2157.61 973.288i −1224.85 1645.25i −9839.83 1735.03i 3741.97 5389.29i 61.2785 + 347.528i
2.5 −28.2144 1.64330i 18.1456 78.9414i 539.083 + 63.0097i −257.498 1086.47i −641.692 + 2197.46i −2491.18 3346.23i −7981.14 1407.29i −5902.47 2864.88i 5479.75 + 31077.2i
2.6 −27.5640 1.60542i 64.9324 48.4230i 502.927 + 58.7837i −129.334 545.702i −1867.53 + 1230.49i 2195.49 + 2949.06i −6807.35 1200.32i 1871.42 6288.44i 2688.87 + 15249.3i
2.7 −27.0854 1.57755i 43.2512 68.4860i 476.862 + 55.7372i 261.554 + 1103.58i −1279.52 + 1786.74i −405.970 545.312i −5987.98 1055.84i −2819.67 5924.20i −5343.34 30303.6i
2.8 −26.4374 1.53980i −51.8941 + 62.1933i 442.295 + 51.6969i 262.586 + 1107.94i 1467.71 1564.32i 923.730 + 1240.79i −4937.07 870.539i −1175.00 6454.93i −5236.09 29695.3i
2.9 −26.2584 1.52938i 7.96860 + 80.6071i 432.896 + 50.5983i −93.6433 395.112i −85.9642 2128.80i −2506.37 3366.65i −4658.54 821.427i −6434.00 + 1284.65i 1854.65 + 10518.2i
2.10 −25.1840 1.46680i −79.9667 12.8969i 377.813 + 44.1600i 70.3664 + 296.899i 1994.96 + 442.090i 323.959 + 435.153i −3090.15 544.877i 6228.34 + 2062.64i −1336.62 7580.33i
2.11 −25.0803 1.46076i 48.9585 + 64.5296i 372.621 + 43.5531i 140.643 + 593.421i −1133.63 1689.94i 969.865 + 1302.75i −2948.10 519.829i −1767.13 + 6318.54i −2660.53 15088.6i
2.12 −24.5500 1.42988i −78.3366 20.6003i 346.389 + 40.4871i −273.470 1153.86i 1893.71 + 617.750i 749.183 + 1006.33i −2246.16 396.059i 5712.25 + 3227.52i 5063.82 + 28718.4i
2.13 −22.2975 1.29868i −34.3614 73.3505i 241.222 + 28.1948i 3.79280 + 16.0031i 670.914 + 1680.15i −996.330 1338.30i 288.943 + 50.9484i −4199.59 + 5040.85i −63.7870 361.754i
2.14 −21.7243 1.26530i −14.9748 79.6037i 216.076 + 25.2557i 6.64933 + 28.0557i 224.595 + 1748.29i 2479.42 + 3330.44i 824.060 + 145.304i −6112.51 + 2384.10i −108.953 617.906i
2.15 −20.4271 1.18974i 77.1800 24.5813i 161.582 + 18.8862i −79.3009 334.597i −1605.81 + 410.300i −712.264 956.736i 1880.44 + 331.572i 5352.52 3794.37i 1221.80 + 6929.19i
2.16 −18.9741 1.10511i 32.1670 74.3390i 104.526 + 12.2173i 65.9484 + 278.258i −692.492 + 1374.97i −746.281 1002.43i 2821.90 + 497.577i −4491.57 4782.52i −943.803 5352.57i
2.17 −18.7332 1.09109i −57.9381 + 56.6055i 95.4740 + 11.1593i −119.240 503.113i 1147.13 997.187i −78.6555 105.653i 2954.49 + 520.956i 152.639 6559.22i 1684.81 + 9555.04i
2.18 −18.4528 1.07475i 67.5589 + 44.6855i 85.0808 + 9.94452i −199.438 841.495i −1198.62 897.181i 110.076 + 147.857i 3100.74 + 546.744i 2567.41 + 6037.81i 2775.79 + 15742.3i
2.19 −18.4357 1.07376i −75.3885 29.6239i 84.4526 + 9.87110i 216.946 + 915.368i 1358.03 + 627.086i −2638.29 3543.84i 3109.37 + 548.266i 4805.85 + 4466.60i −3016.67 17108.4i
2.20 −17.8510 1.03970i 17.7984 + 79.0204i 63.3069 + 7.39952i 179.679 + 758.126i −235.561 1429.09i −2206.31 2963.59i 3385.65 + 596.982i −5927.43 + 2812.87i −2419.22 13720.1i
See next 80 embeddings (of 1278 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 2.71 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
81.h odd 54 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 81.9.h.a 1278
81.h odd 54 1 inner 81.9.h.a 1278

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
81.9.h.a 1278 1.a even 1 1 trivial
81.9.h.a 1278 81.h odd 54 1 inner

Hecke kernels

This newform subspace is the entire newspace $$S_{9}^{\mathrm{new}}(81, [\chi])$$.