↑

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 $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [19,7,Mod(2,19)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(19, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("19.2");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 19 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 19.f (of order \(18\), degree \(6\), 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: \(4.37102758878\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 54 q - 6 q^{2} - 36 q^{3} + 192 q^{4} - 6 q^{5} + 720 q^{6} - 219 q^{7} - 9 q^{8} - 2076 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 54 q - 6 q^{2} - 36 q^{3} + 192 q^{4} - 6 q^{5} + 720 q^{6} - 219 q^{7} - 9 q^{8} - 2076 q^{9} - 1689 q^{10} + 1677 q^{11} - 9 q^{12} - 10506 q^{13} + 6105 q^{14} + 24138 q^{15} + 23760 q^{16} - 3438 q^{17} - 30918 q^{19} - 76650 q^{20} - 59055 q^{21} + 20490 q^{22} + 42498 q^{23} + 31848 q^{24} + 7914 q^{25} + 109989 q^{26} + 124821 q^{27} - 109476 q^{28} - 27726 q^{29} - 153516 q^{30} - 30789 q^{31} - 114375 q^{32} + 126039 q^{33} + 359796 q^{34} + 160161 q^{35} + 184995 q^{36} - 245244 q^{38} - 390564 q^{39} - 372678 q^{40} - 24708 q^{41} - 244893 q^{42} - 30726 q^{43} - 301419 q^{44} + 293037 q^{45} + 69246 q^{46} - 71454 q^{47} + 1024041 q^{48} - 350346 q^{49} - 104148 q^{50} - 180672 q^{51} + 921801 q^{52} + 822558 q^{53} + 308469 q^{54} + 623517 q^{55} - 17862 q^{57} - 1150644 q^{58} - 463812 q^{59} - 908130 q^{60} - 583698 q^{61} - 429732 q^{62} - 2576043 q^{63} - 328293 q^{64} - 1056789 q^{65} - 1038756 q^{66} + 1120464 q^{67} + 717270 q^{68} + 3771567 q^{69} + 4845345 q^{70} + 94494 q^{71} + 2351634 q^{72} + 950850 q^{73} + 55149 q^{74} - 218814 q^{76} - 343086 q^{77} + 1833471 q^{78} - 4936494 q^{79} - 9642039 q^{80} - 769551 q^{81} - 8627793 q^{82} - 412131 q^{83} - 3125205 q^{84} - 896910 q^{85} + 8878236 q^{86} + 610737 q^{87} + 13604751 q^{88} + 6653022 q^{89} + 3142734 q^{90} + 3409821 q^{91} - 3043740 q^{92} - 816393 q^{93} + 4839186 q^{95} - 5667138 q^{96} - 853656 q^{97} - 17679315 q^{98} - 13991937 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
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 2.9
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 \(S_{7}^{\mathrm{new}}(19, [\chi])\).