Properties

Label 16.10.e.a
Level 16
Weight 10
Character orbit 16.e
Analytic conductor 8.241
Analytic rank 0
Dimension 34
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 16 = 2^{4} \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 16.e (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(8.24057337862\)
Analytic rank: \(0\)
Dimension: \(34\)
Relative dimension: \(17\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 34q - 2q^{2} - 2q^{3} + 168q^{4} - 2q^{5} - 2192q^{6} - 716q^{8} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 34q - 2q^{2} - 2q^{3} + 168q^{4} - 2q^{5} - 2192q^{6} - 716q^{8} + 4684q^{10} + 65858q^{11} + 217156q^{12} - 2q^{13} - 133540q^{14} + 404996q^{15} - 817256q^{16} - 4q^{17} + 2179902q^{18} - 480890q^{19} - 3277988q^{20} + 39364q^{21} - 34300q^{22} + 1863856q^{24} - 6347768q^{26} + 6364408q^{27} + 3629528q^{28} - 633402q^{29} - 4677804q^{30} - 11082256q^{31} - 1655992q^{32} - 4q^{33} - 4528204q^{34} + 23026556q^{35} + 23564140q^{36} - 1214778q^{37} - 14996352q^{38} - 11751288q^{40} + 105399160q^{42} - 34881078q^{43} - 94223356q^{44} + 3866882q^{45} + 15432396q^{46} + 97593616q^{47} + 7831528q^{48} - 126825626q^{49} - 6289530q^{50} - 27382140q^{51} - 28284428q^{52} - 74907850q^{53} + 105311008q^{54} + 26511256q^{56} - 178271784q^{58} + 94503354q^{59} + 229034112q^{60} - 90121106q^{61} - 136788944q^{62} - 153644796q^{63} - 349292928q^{64} + 62806084q^{65} + 192273844q^{66} + 406167670q^{67} - 285757488q^{68} + 382399540q^{69} + 606051200q^{70} + 882075684q^{72} + 123910860q^{74} - 1038797114q^{75} - 453269100q^{76} + 286751428q^{77} - 1735024804q^{78} + 1913565952q^{79} - 700272856q^{80} - 774840982q^{81} - 674287584q^{82} - 1637053442q^{83} + 1281919912q^{84} + 425303748q^{85} + 243980676q^{86} + 1781147128q^{88} + 3084087192q^{90} + 2019528196q^{91} - 3131634504q^{92} + 1071927184q^{93} - 2068460272q^{94} - 5087413260q^{95} - 3896483152q^{96} - 4q^{97} - 3100632102q^{98} + 3713751422q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1 −22.5065 + 2.33610i −82.6799 + 82.6799i 501.085 105.155i 1214.44 + 1214.44i 1667.69 2053.98i 16.4286i −11032.0 + 3537.26i 6011.08i −30169.9 24495.7i
5.2 −21.3919 7.37468i 10.9414 10.9414i 403.228 + 315.517i −1361.43 1361.43i −314.745 + 153.367i 1171.64i −6298.98 9723.20i 19443.6i 19083.5 + 39163.8i
5.3 −20.7667 + 8.98584i 155.975 155.975i 350.509 373.212i −404.055 404.055i −1837.51 + 4640.65i 637.759i −3925.29 + 10900.0i 28973.4i 12021.7 + 4760.11i
5.4 −16.2245 15.7723i 111.916 111.916i 14.4713 + 511.795i 1353.12 + 1353.12i −3580.96 + 50.6167i 6223.61i 7837.38 8531.89i 5367.52i −611.981 43295.6i
5.5 −15.3981 + 16.5801i −149.427 + 149.427i −37.7985 510.603i −1560.48 1560.48i −176.625 4778.42i 10771.6i 9047.86 + 7235.60i 24974.1i 49901.2 1844.50i
5.6 −12.8269 + 18.6406i −4.54374 + 4.54374i −182.942 478.201i 335.590 + 335.590i −26.4158 142.980i 11538.0i 11260.5 + 2723.70i 19641.7i −10560.2 + 1951.01i
5.7 −11.9169 19.2351i −128.561 + 128.561i −227.974 + 458.445i −28.4518 28.4518i 4004.93 + 940.828i 2525.84i 11535.0 1078.14i 13372.9i −208.215 + 886.330i
5.8 −3.01687 + 22.4254i 80.3683 80.3683i −493.797 135.309i 652.415 + 652.415i 1559.83 + 2044.75i 9250.15i 4524.08 10665.4i 6764.87i −16598.9 + 12662.4i
5.9 −0.735037 22.6155i 100.059 100.059i −510.919 + 33.2464i −601.526 601.526i −2336.42 2189.32i 5438.71i 1127.43 + 11530.2i 340.415i −13161.7 + 14046.0i
5.10 7.39503 + 21.3849i −180.934 + 180.934i −402.627 + 316.284i 1444.92 + 1444.92i −5207.26 2531.24i 567.986i −9741.14 6271.20i 45790.9i −20214.2 + 41584.6i
5.11 8.52251 20.9611i −77.8288 + 77.8288i −366.734 357.282i 206.519 + 206.519i 968.080 + 2294.67i 6914.62i −10614.5 + 4642.20i 7568.35i 6088.93 2568.81i
5.12 8.74072 + 20.8710i −8.26401 + 8.26401i −359.200 + 364.856i −1313.02 1313.02i −244.712 100.245i 3840.19i −10754.6 4307.77i 19546.4i 15927.3 38880.8i
5.13 17.2877 + 14.5992i 185.041 185.041i 85.7268 + 504.772i 356.050 + 356.050i 5900.38 497.477i 6406.17i −5887.25 + 9977.87i 48797.5i 957.228 + 11353.3i
5.14 18.5834 12.9096i 62.9496 62.9496i 178.687 479.807i 1829.14 + 1829.14i 357.167 1982.47i 7941.79i −2873.49 11223.2i 11757.7i 57605.1 + 10378.3i
5.15 20.5458 9.47996i 117.096 117.096i 332.261 389.547i −1418.63 1418.63i 1295.77 3515.91i 7313.79i 3133.68 11153.4i 7740.11i −42595.4 15698.3i
5.16 21.2709 7.71668i −159.536 + 159.536i 392.906 328.282i −1021.24 1021.24i −2162.40 + 4624.58i 10191.8i 5824.23 10014.8i 31220.8i −29603.4 13842.2i
5.17 21.4373 + 7.24163i −33.5715 + 33.5715i 407.118 + 310.482i 315.639 + 315.639i −962.796 + 476.571i 3725.04i 6479.11 + 9604.11i 17428.9i 4480.71 + 9052.18i
13.1 −22.5065 2.33610i −82.6799 82.6799i 501.085 + 105.155i 1214.44 1214.44i 1667.69 + 2053.98i 16.4286i −11032.0 3537.26i 6011.08i −30169.9 + 24495.7i
13.2 −21.3919 + 7.37468i 10.9414 + 10.9414i 403.228 315.517i −1361.43 + 1361.43i −314.745 153.367i 1171.64i −6298.98 + 9723.20i 19443.6i 19083.5 39163.8i
13.3 −20.7667 8.98584i 155.975 + 155.975i 350.509 + 373.212i −404.055 + 404.055i −1837.51 4640.65i 637.759i −3925.29 10900.0i 28973.4i 12021.7 4760.11i
See all 34 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 13.17
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 16.10.e.a 34
4.b odd 2 1 64.10.e.a 34
8.b even 2 1 128.10.e.b 34
8.d odd 2 1 128.10.e.a 34
16.e even 4 1 inner 16.10.e.a 34
16.e even 4 1 128.10.e.b 34
16.f odd 4 1 64.10.e.a 34
16.f odd 4 1 128.10.e.a 34
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
16.10.e.a 34 1.a even 1 1 trivial
16.10.e.a 34 16.e even 4 1 inner
64.10.e.a 34 4.b odd 2 1
64.10.e.a 34 16.f odd 4 1
128.10.e.a 34 8.d odd 2 1
128.10.e.a 34 16.f odd 4 1
128.10.e.b 34 8.b even 2 1
128.10.e.b 34 16.e even 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{10}^{\mathrm{new}}(16, [\chi])\).

Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database