# Properties

 Label 64.8.i.a Level $64$ Weight $8$ Character orbit 64.i Analytic conductor $19.993$ Analytic rank $0$ Dimension $440$ CM no Inner twists $2$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [64,8,Mod(5,64)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(64, base_ring=CyclotomicField(16))

chi = DirichletCharacter(H, H._module([0, 1]))

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("64.5");

S:= CuspForms(chi, 8);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$64 = 2^{6}$$ Weight: $$k$$ $$=$$ $$8$$ Character orbit: $$[\chi]$$ $$=$$ 64.i (of order $$16$$, degree $$8$$, 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: $$19.9926416310$$ Analytic rank: $$0$$ Dimension: $$440$$ Relative dimension: $$55$$ over $$\Q(\zeta_{16})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{16}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$440 q - 8 q^{2} - 8 q^{3} - 8 q^{4} - 8 q^{5} - 8 q^{6} - 8 q^{7} - 8 q^{8} - 8 q^{9}+O(q^{10})$$ 440 * q - 8 * q^2 - 8 * q^3 - 8 * q^4 - 8 * q^5 - 8 * q^6 - 8 * q^7 - 8 * q^8 - 8 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$440 q - 8 q^{2} - 8 q^{3} - 8 q^{4} - 8 q^{5} - 8 q^{6} - 8 q^{7} - 8 q^{8} - 8 q^{9} - 8 q^{10} - 8 q^{11} - 8 q^{12} - 8 q^{13} - 8 q^{14} - 8 q^{15} - 8 q^{16} - 8 q^{17} - 8 q^{18} - 8 q^{19} - 8 q^{20} - 8 q^{21} + 274448 q^{22} - 8 q^{23} - 598048 q^{24} - 8 q^{25} + 727952 q^{26} - 8 q^{27} + 390912 q^{28} - 8 q^{29} - 1739928 q^{30} - 1071328 q^{32} + 403152 q^{34} - 8 q^{35} + 4069512 q^{36} - 8 q^{37} + 1244912 q^{38} - 8 q^{39} - 2686368 q^{40} - 8 q^{41} - 3858928 q^{42} - 8 q^{43} + 3370464 q^{44} - 8 q^{45} - 8 q^{46} - 8 q^{47} - 8 q^{48} - 8 q^{49} - 2317424 q^{50} - 5986664 q^{51} + 10204936 q^{52} - 8 q^{53} - 1259720 q^{54} + 8382008 q^{55} - 10737280 q^{56} - 8 q^{57} - 10358144 q^{58} + 3671864 q^{59} + 4336408 q^{60} - 8 q^{61} + 10348520 q^{62} - 20003776 q^{63} + 22837144 q^{64} - 16 q^{65} + 7515224 q^{66} + 3881352 q^{67} - 8750072 q^{68} - 8 q^{69} - 35990312 q^{70} + 24697400 q^{71} - 22727312 q^{72} - 8 q^{73} + 4214384 q^{74} - 22521032 q^{75} + 38942904 q^{76} - 8 q^{77} + 68832880 q^{78} - 523752 q^{79} - 78362464 q^{80} - 8 q^{81} - 49314648 q^{82} - 8 q^{83} + 17056808 q^{84} - 8 q^{85} + 69535536 q^{86} - 8 q^{87} + 52517832 q^{88} - 8 q^{89} + 31733992 q^{90} - 8 q^{91} - 54440784 q^{92} + 17488 q^{93} - 74691560 q^{94} - 112868992 q^{96} - 62769040 q^{98} + 17488 q^{99}+O(q^{100})$$ 440 * q - 8 * q^2 - 8 * q^3 - 8 * q^4 - 8 * q^5 - 8 * q^6 - 8 * q^7 - 8 * q^8 - 8 * q^9 - 8 * q^10 - 8 * q^11 - 8 * q^12 - 8 * q^13 - 8 * q^14 - 8 * q^15 - 8 * q^16 - 8 * q^17 - 8 * q^18 - 8 * q^19 - 8 * q^20 - 8 * q^21 + 274448 * q^22 - 8 * q^23 - 598048 * q^24 - 8 * q^25 + 727952 * q^26 - 8 * q^27 + 390912 * q^28 - 8 * q^29 - 1739928 * q^30 - 1071328 * q^32 + 403152 * q^34 - 8 * q^35 + 4069512 * q^36 - 8 * q^37 + 1244912 * q^38 - 8 * q^39 - 2686368 * q^40 - 8 * q^41 - 3858928 * q^42 - 8 * q^43 + 3370464 * q^44 - 8 * q^45 - 8 * q^46 - 8 * q^47 - 8 * q^48 - 8 * q^49 - 2317424 * q^50 - 5986664 * q^51 + 10204936 * q^52 - 8 * q^53 - 1259720 * q^54 + 8382008 * q^55 - 10737280 * q^56 - 8 * q^57 - 10358144 * q^58 + 3671864 * q^59 + 4336408 * q^60 - 8 * q^61 + 10348520 * q^62 - 20003776 * q^63 + 22837144 * q^64 - 16 * q^65 + 7515224 * q^66 + 3881352 * q^67 - 8750072 * q^68 - 8 * q^69 - 35990312 * q^70 + 24697400 * q^71 - 22727312 * q^72 - 8 * q^73 + 4214384 * q^74 - 22521032 * q^75 + 38942904 * q^76 - 8 * q^77 + 68832880 * q^78 - 523752 * q^79 - 78362464 * q^80 - 8 * q^81 - 49314648 * q^82 - 8 * q^83 + 17056808 * q^84 - 8 * q^85 + 69535536 * q^86 - 8 * q^87 + 52517832 * q^88 - 8 * q^89 + 31733992 * q^90 - 8 * q^91 - 54440784 * q^92 + 17488 * q^93 - 74691560 * q^94 - 112868992 * q^96 - 62769040 * q^98 + 17488 * 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   $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
5.1 −11.3083 0.351269i −58.8200 39.3023i 127.753 + 7.94447i 330.459 + 65.7324i 651.346 + 465.102i 49.4248 119.322i −1441.88 134.714i 1078.20 + 2602.99i −3713.82 859.398i
5.2 −11.2858 + 0.794822i 62.7283 + 41.9137i 126.737 17.9403i −472.644 94.0147i −741.251 423.170i 447.256 1079.77i −1416.06 + 303.203i 1341.16 + 3237.83i 5408.87 + 685.359i
5.3 −11.1803 + 1.73216i −30.0164 20.0563i 121.999 38.7321i −371.645 73.9247i 370.334 + 172.243i 215.597 520.498i −1296.90 + 644.360i −338.200 816.486i 4283.16 + 182.756i
5.4 −11.1256 + 2.05426i 75.7717 + 50.6290i 119.560 45.7099i 402.001 + 79.9629i −947.014 407.626i −355.392 + 857.993i −1236.28 + 754.159i 2341.12 + 5651.97i −4636.78 63.8267i
5.5 −11.0200 + 2.56116i 4.98311 + 3.32961i 114.881 56.4480i 93.6874 + 18.6356i −63.4416 23.9297i −322.998 + 779.786i −1121.41 + 916.286i −823.184 1987.34i −1080.16 + 34.5846i
5.6 −10.8718 3.13113i −16.4647 11.0014i 108.392 + 68.0820i −349.067 69.4337i 144.555 + 171.158i −339.076 + 818.601i −965.243 1079.56i −686.871 1658.25i 3577.58 + 1847.84i
5.7 −10.7958 3.38396i 13.8786 + 9.27336i 105.098 + 73.0650i 413.266 + 82.2037i −118.449 147.078i 570.140 1376.44i −887.360 1144.44i −730.309 1763.12i −4183.35 2285.93i
5.8 −10.5533 4.07776i 35.5330 + 23.7424i 94.7438 + 86.0675i 19.2350 + 3.82607i −278.174 395.455i −17.6231 + 42.5460i −648.897 1294.64i −138.036 333.248i −187.390 118.813i
5.9 −9.78313 5.68246i −65.2874 43.6236i 63.4192 + 111.185i −221.237 44.0068i 390.825 + 797.769i 188.633 455.401i 11.3635 1448.11i 1522.49 + 3675.62i 1914.32 + 1687.69i
5.10 −9.53136 + 6.09534i 32.1674 + 21.4936i 53.6936 116.194i 75.4308 + 15.0041i −437.610 8.79155i 415.616 1003.38i 196.468 + 1434.77i −264.161 637.740i −810.413 + 316.767i
5.11 −9.48528 + 6.16681i −42.5883 28.4566i 51.9410 116.988i −5.90191 1.17396i 579.448 + 7.28486i 468.708 1131.56i 228.766 + 1429.97i 167.055 + 403.307i 63.2208 25.2606i
5.12 −8.62267 + 7.32458i −70.8076 47.3121i 20.7010 126.315i −255.007 50.7241i 957.092 110.679i −594.970 + 1436.38i 746.707 + 1240.80i 1938.35 + 4679.58i 2570.38 1430.45i
5.13 −8.56889 7.38743i −33.1663 22.1610i 18.8519 + 126.604i 252.282 + 50.1820i 120.486 + 434.908i −626.314 + 1512.05i 773.739 1224.12i −228.037 550.531i −1791.06 2293.72i
5.14 −8.04024 + 7.95955i 39.3577 + 26.2980i 1.29101 127.993i −306.788 61.0240i −525.766 + 101.828i −379.700 + 916.677i 1008.39 + 1039.37i 20.5165 + 49.5312i 2952.37 1951.25i
5.15 −7.91328 8.08579i 55.9515 + 37.3856i −2.76005 + 127.970i −163.225 32.4674i −140.468 748.255i −215.563 + 520.415i 1056.58 990.347i 895.959 + 2163.04i 1029.12 + 1576.72i
5.16 −7.31338 + 8.63218i −22.4221 14.9820i −21.0290 126.261i 466.664 + 92.8252i 293.309 83.9828i −211.546 + 510.718i 1243.70 + 741.867i −558.637 1348.67i −4214.17 + 3349.46i
5.17 −6.93324 8.94037i −29.5675 19.7564i −31.8604 + 123.971i 244.950 + 48.7235i 28.3693 + 401.320i 123.109 297.212i 1329.25 574.680i −353.006 852.232i −1262.69 2527.75i
5.18 −6.81413 9.03148i −5.99631 4.00661i −35.1353 + 123.083i −290.762 57.8362i 4.67403 + 81.4571i 560.385 1352.89i 1351.04 521.381i −817.026 1972.47i 1458.95 + 3020.12i
5.19 −4.89711 + 10.1989i 54.4484 + 36.3812i −80.0365 99.8907i 229.290 + 45.6085i −637.690 + 377.152i 341.434 824.294i 1410.73 327.111i 804.103 + 1941.28i −1588.02 + 2115.16i
5.20 −4.58707 10.3421i 47.8706 + 31.9861i −85.9175 + 94.8798i 406.433 + 80.8446i 111.217 641.805i −41.5209 + 100.240i 1375.37 + 453.346i 431.557 + 1041.87i −1028.24 4574.21i
See next 80 embeddings (of 440 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 5.55 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
64.i even 16 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 64.8.i.a 440
64.i even 16 1 inner 64.8.i.a 440

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
64.8.i.a 440 1.a even 1 1 trivial
64.8.i.a 440 64.i even 16 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{8}^{\mathrm{new}}(64, [\chi])$$.