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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 $2$

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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.

\(\operatorname{Tr}(f)(q) = \) \( 440 q - 8 q^{2} - 8 q^{3} - 8 q^{4} - 8 q^{5} - 8 q^{6} - 8 q^{7} - 8 q^{8} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 440 q - 8 q^{2} - 8 q^{3} - 8 q^{4} - 8 q^{5} - 8 q^{6} - 8 q^{7} - 8 q^{8} - 8 q^{9} - 8 q^{10} - 8 q^{11} - 8 q^{12} - 8 q^{13} - 8 q^{14} - 8 q^{15} - 8 q^{16} - 8 q^{17} - 8 q^{18} - 8 q^{19} - 8 q^{20} - 8 q^{21} + 274448 q^{22} - 8 q^{23} - 598048 q^{24} - 8 q^{25} + 727952 q^{26} - 8 q^{27} + 390912 q^{28} - 8 q^{29} - 1739928 q^{30} - 1071328 q^{32} + 403152 q^{34} - 8 q^{35} + 4069512 q^{36} - 8 q^{37} + 1244912 q^{38} - 8 q^{39} - 2686368 q^{40} - 8 q^{41} - 3858928 q^{42} - 8 q^{43} + 3370464 q^{44} - 8 q^{45} - 8 q^{46} - 8 q^{47} - 8 q^{48} - 8 q^{49} - 2317424 q^{50} - 5986664 q^{51} + 10204936 q^{52} - 8 q^{53} - 1259720 q^{54} + 8382008 q^{55} - 10737280 q^{56} - 8 q^{57} - 10358144 q^{58} + 3671864 q^{59} + 4336408 q^{60} - 8 q^{61} + 10348520 q^{62} - 20003776 q^{63} + 22837144 q^{64} - 16 q^{65} + 7515224 q^{66} + 3881352 q^{67} - 8750072 q^{68} - 8 q^{69} - 35990312 q^{70} + 24697400 q^{71} - 22727312 q^{72} - 8 q^{73} + 4214384 q^{74} - 22521032 q^{75} + 38942904 q^{76} - 8 q^{77} + 68832880 q^{78} - 523752 q^{79} - 78362464 q^{80} - 8 q^{81} - 49314648 q^{82} - 8 q^{83} + 17056808 q^{84} - 8 q^{85} + 69535536 q^{86} - 8 q^{87} + 52517832 q^{88} - 8 q^{89} + 31733992 q^{90} - 8 q^{91} - 54440784 q^{92} + 17488 q^{93} - 74691560 q^{94} - 112868992 q^{96} - 62769040 q^{98} + 17488 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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:

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 \(S_{8}^{\mathrm{new}}(64, [\chi])\).