Properties

Label 19.7.f.a
Level $19$
Weight $7$
Character orbit 19.f
Analytic conductor $4.371$
Analytic rank $0$
Dimension $54$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 19 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 19.f (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.37102758878\)
Analytic rank: \(0\)
Dimension: \(54\)
Relative dimension: \(9\) over \(\Q(\zeta_{18})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 54q - 6q^{2} - 36q^{3} + 192q^{4} - 6q^{5} + 720q^{6} - 219q^{7} - 9q^{8} - 2076q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 54q - 6q^{2} - 36q^{3} + 192q^{4} - 6q^{5} + 720q^{6} - 219q^{7} - 9q^{8} - 2076q^{9} - 1689q^{10} + 1677q^{11} - 9q^{12} - 10506q^{13} + 6105q^{14} + 24138q^{15} + 23760q^{16} - 3438q^{17} - 30918q^{19} - 76650q^{20} - 59055q^{21} + 20490q^{22} + 42498q^{23} + 31848q^{24} + 7914q^{25} + 109989q^{26} + 124821q^{27} - 109476q^{28} - 27726q^{29} - 153516q^{30} - 30789q^{31} - 114375q^{32} + 126039q^{33} + 359796q^{34} + 160161q^{35} + 184995q^{36} - 245244q^{38} - 390564q^{39} - 372678q^{40} - 24708q^{41} - 244893q^{42} - 30726q^{43} - 301419q^{44} + 293037q^{45} + 69246q^{46} - 71454q^{47} + 1024041q^{48} - 350346q^{49} - 104148q^{50} - 180672q^{51} + 921801q^{52} + 822558q^{53} + 308469q^{54} + 623517q^{55} - 17862q^{57} - 1150644q^{58} - 463812q^{59} - 908130q^{60} - 583698q^{61} - 429732q^{62} - 2576043q^{63} - 328293q^{64} - 1056789q^{65} - 1038756q^{66} + 1120464q^{67} + 717270q^{68} + 3771567q^{69} + 4845345q^{70} + 94494q^{71} + 2351634q^{72} + 950850q^{73} + 55149q^{74} - 218814q^{76} - 343086q^{77} + 1833471q^{78} - 4936494q^{79} - 9642039q^{80} - 769551q^{81} - 8627793q^{82} - 412131q^{83} - 3125205q^{84} - 896910q^{85} + 8878236q^{86} + 610737q^{87} + 13604751q^{88} + 6653022q^{89} + 3142734q^{90} + 3409821q^{91} - 3043740q^{92} - 816393q^{93} + 4839186q^{95} - 5667138q^{96} - 853656q^{97} - 17679315q^{98} - 13991937q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1 −13.8473 2.44165i 19.5791 23.3335i 125.645 + 45.7311i 43.8845 15.9727i −328.090 + 275.300i −286.306 495.896i −848.852 490.085i −34.5199 195.772i −646.681 + 114.027i
2.2 −13.2866 2.34278i −30.7619 + 36.6606i 110.904 + 40.3656i −169.577 + 61.7209i 494.607 415.025i −29.1620 50.5101i −631.184 364.414i −271.115 1537.57i 2397.69 422.777i
2.3 −8.05321 1.42000i 3.00004 3.57530i 2.69755 + 0.981826i −7.95224 + 2.89438i −29.2369 + 24.5326i 213.780 + 370.279i 432.910 + 249.941i 122.807 + 696.473i 68.1511 12.0169i
2.4 −4.80629 0.847479i −21.7309 + 25.8979i −37.7581 13.7428i 225.514 82.0804i 126.393 106.056i −232.429 402.578i 440.331 + 254.225i −71.8793 407.648i −1153.45 + 203.384i
2.5 −0.612104 0.107930i 29.3902 35.0259i −59.7773 21.7572i −60.2750 + 21.9383i −21.7702 + 18.2674i −70.8586 122.731i 68.6913 + 39.6589i −236.440 1340.92i 39.2624 6.92302i
2.6 3.23760 + 0.570877i −11.8162 + 14.0820i −49.9842 18.1927i −103.262 + 37.5842i −46.2952 + 38.8462i −64.9288 112.460i −333.657 192.637i 67.9097 + 385.135i −355.777 + 62.7330i
2.7 8.59619 + 1.51574i 11.8818 14.1601i 11.4566 + 4.16987i 148.419 54.0202i 123.601 103.714i 77.3177 + 133.918i −391.636 226.111i 67.2563 + 381.430i 1357.72 239.403i
2.8 12.3683 + 2.18087i −27.8366 + 33.1744i 88.0791 + 32.0582i 29.3643 10.6877i −416.642 + 349.604i 202.665 + 351.026i 323.379 + 186.703i −199.074 1129.00i 386.496 68.1497i
2.9 14.4636 + 2.55033i 14.8077 17.6471i 142.552 + 51.8848i −160.505 + 58.4191i 259.179 217.477i −201.619 349.214i 1115.48 + 644.022i 34.4366 + 195.300i −2470.47 + 435.611i
3.1 −8.55141 + 10.1912i −4.72892 12.9926i −19.6199 111.270i 12.1561 68.9408i 172.849 + 62.9118i 144.225 + 249.805i 564.387 + 325.849i 412.002 345.710i 598.636 + 713.426i
3.2 −6.22147 + 7.41446i 3.87015 + 10.6332i −5.15404 29.2300i −26.3442 + 149.405i −102.917 37.4588i −266.785 462.086i −287.668 166.085i 460.361 386.288i −943.861 1124.85i
3.3 −4.60931 + 5.49316i 16.2253 + 44.5787i 2.18441 + 12.3884i 17.5596 99.5852i −319.665 116.349i 137.474 + 238.111i −475.567 274.569i −1165.55 + 978.015i 466.100 + 555.476i
3.4 −2.65924 + 3.16916i −16.9615 46.6013i 8.14146 + 46.1725i −28.7281 + 162.925i 192.792 + 70.1705i 152.866 + 264.772i −397.277 229.368i −1325.54 + 1112.26i −439.942 524.303i
3.5 −0.521164 + 0.621099i −6.41153 17.6155i 10.9993 + 62.3803i 35.4222 200.889i 14.2825 + 5.19839i −216.833 375.565i −89.4152 51.6239i 289.247 242.707i 106.311 + 126.697i
3.6 1.25256 1.49274i 1.82612 + 5.01723i 10.4541 + 59.2882i −2.44738 + 13.8798i 9.77677 + 3.55845i 212.620 + 368.269i 209.601 + 121.013i 536.609 450.268i 17.6535 + 21.0386i
3.7 4.29634 5.12017i 10.1613 + 27.9178i 3.35580 + 19.0317i −23.2992 + 132.137i 186.600 + 67.9170i −124.815 216.186i 482.323 + 278.470i −117.708 + 98.7685i 576.461 + 686.999i
3.8 7.32193 8.72593i −9.97412 27.4037i −11.4178 64.7536i −5.58040 + 31.6480i −312.152 113.614i −45.3132 78.4847i −17.2882 9.98136i −93.0319 + 78.0631i 235.299 + 280.419i
3.9 8.86541 10.5654i 10.1562 + 27.9038i −21.9184 124.305i 30.9255 175.387i 384.853 + 140.075i 131.445 + 227.670i −743.213 429.094i −117.030 + 98.1996i −1578.87 1881.62i
10.1 −13.8473 + 2.44165i 19.5791 + 23.3335i 125.645 45.7311i 43.8845 + 15.9727i −328.090 275.300i −286.306 + 495.896i −848.852 + 490.085i −34.5199 + 195.772i −646.681 114.027i
10.2 −13.2866 + 2.34278i −30.7619 36.6606i 110.904 40.3656i −169.577 61.7209i 494.607 + 415.025i −29.1620 + 50.5101i −631.184 + 364.414i −271.115 + 1537.57i 2397.69 + 422.777i
See all 54 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 15.9
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.f odd 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 19.7.f.a 54
19.f odd 18 1 inner 19.7.f.a 54
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
19.7.f.a 54 1.a even 1 1 trivial
19.7.f.a 54 19.f odd 18 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{7}^{\mathrm{new}}(19, [\chi])\).