Properties

Label 16.9.f.a
Level 16
Weight 9
Character orbit 16.f
Analytic conductor 6.518
Analytic rank 0
Dimension 30
CM no
Inner twists 2

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 16 = 2^{4} \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 16.f (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.51805776098\)
Analytic rank: \(0\)
Dimension: \(30\)
Relative dimension: \(15\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 30q - 2q^{2} - 2q^{3} + 184q^{4} - 2q^{5} + 3232q^{6} - 4q^{7} - 8732q^{8} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 30q - 2q^{2} - 2q^{3} + 184q^{4} - 2q^{5} + 3232q^{6} - 4q^{7} - 8732q^{8} - 1860q^{10} - 19778q^{11} + 14068q^{12} - 2q^{13} - 58900q^{14} + 245336q^{16} - 4q^{17} - 223730q^{18} + 167550q^{19} - 135380q^{20} - 13124q^{21} + 985700q^{22} - 845572q^{23} - 800592q^{24} - 184760q^{26} + 38656q^{27} + 94136q^{28} + 1066174q^{29} + 1881700q^{30} - 3395192q^{32} - 4q^{33} + 1634900q^{34} + 426620q^{35} + 1877324q^{36} + 2360254q^{37} - 3010352q^{38} - 7650052q^{39} + 3256936q^{40} - 3731304q^{42} + 6314814q^{43} - 10360940q^{44} + 794370q^{45} + 1510780q^{46} + 4591848q^{48} + 14823770q^{49} + 14752710q^{50} - 28969860q^{51} + 4335652q^{52} + 2679358q^{53} + 18667040q^{54} + 46326780q^{55} - 36346408q^{56} - 1034696q^{58} - 46004162q^{59} - 20687200q^{60} - 24476034q^{61} - 64043472q^{62} + 50827456q^{64} - 14970820q^{65} + 65035780q^{66} + 59474558q^{67} + 74339312q^{68} + 8623420q^{69} + 39082496q^{70} - 79832068q^{71} + 929812q^{72} - 168043460q^{74} + 88519490q^{75} - 166219388q^{76} + 35952572q^{77} + 5059788q^{78} - 12904856q^{80} - 66961570q^{81} + 261305296q^{82} - 34471682q^{83} + 456781480q^{84} - 107741252q^{85} + 9841444q^{86} + 149712636q^{87} - 173775752q^{88} - 650366536q^{90} - 76429060q^{91} - 449169832q^{92} + 138269056q^{93} - 239278032q^{94} + 378359728q^{96} - 4q^{97} + 630040554q^{98} + 184838270q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1 −15.9265 1.53176i 110.050 110.050i 251.307 + 48.7911i −128.245 + 128.245i −1921.29 + 1584.14i 3057.02 −3927.71 1162.01i 17661.1i 2238.94 1846.06i
3.2 −15.9118 1.67767i −93.3844 + 93.3844i 250.371 + 53.3896i −695.406 + 695.406i 1642.58 1329.25i 805.482 −3894.28 1269.56i 10880.3i 12231.8 9898.49i
3.3 −14.5765 6.59745i −7.14041 + 7.14041i 168.947 + 192.335i 610.669 610.669i 151.190 56.9735i −3410.30 −1193.74 3918.19i 6459.03i −12930.3 + 4872.54i
3.4 −13.4426 + 8.67733i −4.15109 + 4.15109i 105.408 233.292i 178.569 178.569i 19.7811 91.8219i 678.560 607.393 + 4050.71i 6526.54i −850.931 + 3949.93i
3.5 −8.02121 13.8441i 17.4297 17.4297i −127.320 + 222.093i −396.274 + 396.274i −381.106 101.492i 432.698 4095.96 18.8169i 5953.41i 8664.66 + 2307.47i
3.6 −5.55129 + 15.0061i 65.0768 65.0768i −194.366 166.606i −526.870 + 526.870i 615.290 + 1337.81i −3638.93 3579.10 1991.80i 1908.99i −4981.46 10831.1i
3.7 −2.48211 15.8063i −100.879 + 100.879i −243.678 + 78.4659i 623.715 623.715i 1844.91 + 1344.13i 1797.37 1845.09 + 3656.89i 13792.1i −11406.8 8310.50i
3.8 −1.81651 + 15.8965i −71.2555 + 71.2555i −249.401 57.7524i −8.38358 + 8.38358i −1003.28 1262.15i 1087.59 1371.10 3859.70i 3593.69i −118.041 148.499i
3.9 4.89750 15.2320i 60.9999 60.9999i −208.029 149.198i 283.594 283.594i −630.404 1227.90i 502.933 −3291.41 + 2438.00i 880.969i −2930.81 5708.61i
3.10 5.41380 + 15.0563i 63.2572 63.2572i −197.381 + 163.023i 825.937 825.937i 1294.88 + 609.955i 711.293 −3523.10 2089.25i 1441.95i 16907.0 + 7964.05i
3.11 10.2604 12.2770i −49.7393 + 49.7393i −45.4477 251.934i −502.907 + 502.907i 100.302 + 1120.99i −2327.86 −3559.29 2026.98i 1613.00i 1014.13 + 11334.2i
3.12 11.2357 + 11.3912i 23.8972 23.8972i −3.51862 + 255.976i −695.590 + 695.590i 540.720 + 3.71617i 3167.79 −2955.40 + 2835.98i 5418.84i −15739.0 108.169i
3.13 13.3691 + 8.79011i −67.4818 + 67.4818i 101.468 + 235.033i 191.722 191.722i −1495.35 + 309.001i −3935.28 −709.426 + 4034.10i 2546.60i 4248.42 877.901i
3.14 15.6071 3.52385i −28.7960 + 28.7960i 231.165 109.995i 301.267 301.267i −347.951 + 550.897i 3827.65 3220.22 2531.29i 4902.58i 3640.29 5763.53i
3.15 15.9448 1.32739i 81.1164 81.1164i 252.476 42.3300i −62.7969 + 62.7969i 1185.72 1401.06i −2758.02 3969.50 1010.08i 6598.75i −917.931 + 1084.64i
11.1 −15.9265 + 1.53176i 110.050 + 110.050i 251.307 48.7911i −128.245 128.245i −1921.29 1584.14i 3057.02 −3927.71 + 1162.01i 17661.1i 2238.94 + 1846.06i
11.2 −15.9118 + 1.67767i −93.3844 93.3844i 250.371 53.3896i −695.406 695.406i 1642.58 + 1329.25i 805.482 −3894.28 + 1269.56i 10880.3i 12231.8 + 9898.49i
11.3 −14.5765 + 6.59745i −7.14041 7.14041i 168.947 192.335i 610.669 + 610.669i 151.190 + 56.9735i −3410.30 −1193.74 + 3918.19i 6459.03i −12930.3 4872.54i
11.4 −13.4426 8.67733i −4.15109 4.15109i 105.408 + 233.292i 178.569 + 178.569i 19.7811 + 91.8219i 678.560 607.393 4050.71i 6526.54i −850.931 3949.93i
11.5 −8.02121 + 13.8441i 17.4297 + 17.4297i −127.320 222.093i −396.274 396.274i −381.106 + 101.492i 432.698 4095.96 + 18.8169i 5953.41i 8664.66 2307.47i
See all 30 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 11.15
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
16.f odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 16.9.f.a 30
4.b odd 2 1 64.9.f.a 30
8.b even 2 1 128.9.f.b 30
8.d odd 2 1 128.9.f.a 30
16.e even 4 1 64.9.f.a 30
16.e even 4 1 128.9.f.a 30
16.f odd 4 1 inner 16.9.f.a 30
16.f odd 4 1 128.9.f.b 30
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
16.9.f.a 30 1.a even 1 1 trivial
16.9.f.a 30 16.f odd 4 1 inner
64.9.f.a 30 4.b odd 2 1
64.9.f.a 30 16.e even 4 1
128.9.f.a 30 8.d odd 2 1
128.9.f.a 30 16.e even 4 1
128.9.f.b 30 8.b even 2 1
128.9.f.b 30 16.f odd 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{9}^{\mathrm{new}}(16, [\chi])\).

Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database