Properties

Label 16.13.f.a
Level 16
Weight 13
Character orbit 16.f
Analytic conductor 14.624
Analytic rank 0
Dimension 46
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 16 = 2^{4} \)
Weight: \( k \) \(=\) \( 13 \)
Character orbit: \([\chi]\) \(=\) 16.f (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(14.6239010764\)
Analytic rank: \(0\)
Dimension: \(46\)
Relative dimension: \(23\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 46q - 2q^{2} - 2q^{3} - 4232q^{4} - 2q^{5} + 72928q^{6} - 4q^{7} - 478172q^{8} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 46q - 2q^{2} - 2q^{3} - 4232q^{4} - 2q^{5} + 72928q^{6} - 4q^{7} - 478172q^{8} + 2806780q^{10} + 2668318q^{11} + 7885108q^{12} - 2q^{13} - 26829652q^{14} + 49704280q^{16} - 4q^{17} + 15306190q^{18} + 51868606q^{19} - 72195860q^{20} - 1062884q^{21} + 47303908q^{22} + 298270076q^{23} - 131729232q^{24} + 526315336q^{26} + 970053760q^{27} + 711713336q^{28} + 704570398q^{29} - 6112100060q^{30} + 7182362248q^{32} - 4q^{33} - 6346386988q^{34} - 3815032900q^{35} - 11830365748q^{36} + 364298398q^{37} + 14395577488q^{38} + 15553507196q^{39} - 19774396184q^{40} + 1899889176q^{42} + 363863518q^{43} - 12173301164q^{44} + 489344130q^{45} - 10837581892q^{46} + 7988573928q^{48} + 67229109258q^{49} + 1530058950q^{50} + 33806024892q^{51} - 9789537692q^{52} - 11168756642q^{53} - 21820351456q^{54} - 74491808260q^{55} + 61228226264q^{56} - 33805733192q^{58} + 104334793054q^{59} - 76265037280q^{60} - 106371743810q^{61} + 280885397040q^{62} - 234402872384q^{64} - 75186419620q^{65} + 148565995972q^{66} - 43778233922q^{67} + 232865553776q^{68} - 214340079908q^{69} - 411016751104q^{70} - 188251854340q^{71} + 632336686420q^{72} - 207892484228q^{74} + 308961520610q^{75} + 21519708868q^{76} - 341607754084q^{77} - 115706090484q^{78} + 477844371304q^{80} - 941431788274q^{81} - 114119320560q^{82} - 1025936323202q^{83} - 718734617432q^{84} + 436332718748q^{85} + 2196044404900q^{86} + 2368412421756q^{87} - 2058716398472q^{88} + 1566205087544q^{90} - 2028231531652q^{91} + 151792482776q^{92} + 1534541270080q^{93} - 1756088243280q^{94} + 1830456305584q^{96} - 4q^{97} + 707523027114q^{98} + 4950023059646q^{99} + O(q^{100}) \)

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 −63.9395 + 2.78179i 398.082 398.082i 4080.52 355.733i 14857.0 14857.0i −24345.8 + 26560.6i 113515. −259917. + 34096.6i 214502.i −908623. + 991281.i
3.2 −62.9799 + 11.3812i −429.462 + 429.462i 3836.94 1433.57i −2473.08 + 2473.08i 22159.7 31935.2i −198740. −225334. + 133955.i 162567.i 127608. 183901.i
3.3 −59.3506 23.9480i −234.930 + 234.930i 2948.99 + 2842.66i −18167.9 + 18167.9i 19569.4 8317.14i 112043. −106948. 239336.i 421057.i 1.51336e6 643191.i
3.4 −55.0768 + 32.5967i 769.939 769.939i 1970.91 3590.64i −17644.5 + 17644.5i −17308.3 + 67503.3i 15384.5 8491.78 + 262006.i 654171.i 396649. 1.54695e6i
3.5 −50.3472 39.5115i 739.301 739.301i 973.684 + 3978.59i −805.082 + 805.082i −66432.6 + 8010.85i −97582.7 108178. 238782.i 561690.i 72343.6 8723.64i
3.6 −48.8546 + 41.3429i −913.246 + 913.246i 677.536 4039.57i 70.8144 70.8144i 6860.04 82372.5i 225982. 133907. + 225363.i 1.13660e6i −531.937 + 6387.27i
3.7 −48.1592 42.1508i −640.742 + 640.742i 542.622 + 4059.90i 10737.8 10737.8i 57865.4 3849.85i 25233.9 144996. 218394.i 289658.i −969728. + 64517.2i
3.8 −34.8144 + 53.7025i 78.2777 78.2777i −1671.91 3739.24i 5406.83 5406.83i 1478.51 + 6928.90i −30061.3 259013. + 40393.8i 519186.i 102124. + 478596.i
3.9 −18.3780 61.3046i 37.8347 37.8347i −3420.50 + 2253.31i 5270.38 5270.38i −3014.77 1624.12i −115235. 201000. + 168281.i 528578.i −419958. 226239.i
3.10 −11.7829 62.9060i 252.451 252.451i −3818.33 + 1482.43i −9569.73 + 9569.73i −18855.3 12906.1i 168743. 138244. + 222729.i 403978.i 714752. + 489234.i
3.11 2.53819 + 63.9496i −322.288 + 322.288i −4083.12 + 324.633i −16463.9 + 16463.9i −21428.3 19792.2i 21051.9 −31123.9 260290.i 323701.i −1.09465e6 1.01107e6i
3.12 3.36222 + 63.9116i 837.232 837.232i −4073.39 + 429.769i 5671.68 5671.68i 56323.8 + 50693.9i −4585.41 −41162.9 258892.i 870475.i 381556. + 343417.i
3.13 4.15344 63.8651i −980.484 + 980.484i −4061.50 530.520i −15901.7 + 15901.7i 58546.3 + 66691.1i −94838.7 −50750.9 + 257184.i 1.39126e6i 949514. + 1.08161e6i
3.14 14.0860 + 62.4306i −832.058 + 832.058i −3699.17 + 1758.79i 17612.4 17612.4i −63666.3 40225.6i −164049. −161909. 206167.i 853201.i 1.34764e6 + 851466.i
3.15 24.3843 59.1727i 949.435 949.435i −2906.81 2885.77i 10755.1 10755.1i −33029.3 79332.0i 35819.9 −241639. + 101636.i 1.27141e6i −374153. 898665.i
3.16 26.6058 58.2077i −435.597 + 435.597i −2680.26 3097.32i 17337.6 17337.6i 13765.7 + 36944.5i 102408. −251598. + 73605.0i 151952.i −547901. 1.47046e6i
3.17 40.2856 + 49.7300i 87.0633 87.0633i −850.147 + 4006.80i 8119.76 8119.76i 7837.05 + 822.264i 149756. −233507. + 119138.i 516281.i 730905. + 76686.6i
3.18 41.7334 48.5213i 193.300 193.300i −612.641 4049.92i −10355.8 + 10355.8i −1312.10 17446.2i −108466. −222075. 139291.i 456712.i 70294.5 + 934665.i
3.19 49.5161 + 40.5482i 543.347 543.347i 807.695 + 4015.58i −12801.7 + 12801.7i 48936.2 4872.73i −188146. −122830. + 231586.i 59011.2i −1.15298e6 + 114806.i
3.20 58.9783 24.8508i −513.579 + 513.579i 2860.88 2931.31i 208.882 208.882i −17527.2 + 43052.8i 42912.4 95884.1 243979.i 3914.51i 7128.62 17510.4i
See all 46 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 11.23
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 16.13.f.a 46
4.b odd 2 1 64.13.f.a 46
8.b even 2 1 128.13.f.b 46
8.d odd 2 1 128.13.f.a 46
16.e even 4 1 64.13.f.a 46
16.e even 4 1 128.13.f.a 46
16.f odd 4 1 inner 16.13.f.a 46
16.f odd 4 1 128.13.f.b 46
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
16.13.f.a 46 1.a even 1 1 trivial
16.13.f.a 46 16.f odd 4 1 inner
64.13.f.a 46 4.b odd 2 1
64.13.f.a 46 16.e even 4 1
128.13.f.a 46 8.d odd 2 1
128.13.f.a 46 16.e even 4 1
128.13.f.b 46 8.b even 2 1
128.13.f.b 46 16.f odd 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{13}^{\mathrm{new}}(16, [\chi])\).

Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database