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Properties

Label 19.4.e.a.5.1
Level $19$
Weight $4$
Character 19.5
Analytic conductor $1.121$
Analytic rank $0$
Dimension $24$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [19,4,Mod(4,19)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("19.4"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(19, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([2])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 19.e (of order \(9\), degree \(6\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.12103629011\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(4\) over \(\Q(\zeta_{9})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label 5.1
Character \(\chi\) \(=\) 19.5
Dual form 19.4.e.a.4.1

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-5.03941 + 1.83419i) q^{2} +(0.944789 - 5.35816i) q^{3} +(15.9030 - 13.3442i) q^{4} +(-7.64679 - 6.41642i) q^{5} +(5.06673 + 28.7349i) q^{6} +(-9.43070 - 16.3344i) q^{7} +(-34.2145 + 59.2612i) q^{8} +(-2.44557 - 0.890116i) q^{9} +(50.3042 + 18.3092i) q^{10} +(11.4425 - 19.8190i) q^{11} +(-56.4754 - 97.8182i) q^{12} +(3.39564 + 19.2576i) q^{13} +(77.4857 + 65.0182i) q^{14} +(-41.6048 + 34.9106i) q^{15} +(34.8848 - 197.841i) q^{16} +(-6.05373 + 2.20338i) q^{17} +13.9569 q^{18} +(68.3688 + 46.7409i) q^{19} -207.229 q^{20} +(-96.4326 + 35.0986i) q^{21} +(-21.3115 + 120.864i) q^{22} +(77.7328 - 65.2256i) q^{23} +(285.206 + 239.316i) q^{24} +(-4.40306 - 24.9710i) q^{25} +(-52.4343 - 90.8188i) q^{26} +(66.3711 - 114.958i) q^{27} +(-367.946 - 133.921i) q^{28} +(96.3201 + 35.0576i) q^{29} +(145.631 - 252.240i) q^{30} +(43.6655 + 75.6309i) q^{31} +(92.0203 + 521.873i) q^{32} +(-95.3826 - 80.0355i) q^{33} +(26.4658 - 22.2074i) q^{34} +(-32.6941 + 185.417i) q^{35} +(-50.7698 + 18.4787i) q^{36} -389.438 q^{37} +(-430.270 - 110.145i) q^{38} +106.394 q^{39} +(641.875 - 233.624i) q^{40} +(35.0252 - 198.638i) q^{41} +(421.586 - 353.752i) q^{42} +(289.606 + 243.009i) q^{43} +(-82.4984 - 467.872i) q^{44} +(12.9894 + 22.4984i) q^{45} +(-272.091 + 471.275i) q^{46} +(120.013 + 43.6812i) q^{47} +(-1027.11 - 373.836i) q^{48} +(-6.37616 + 11.0438i) q^{49} +(67.9904 + 117.763i) q^{50} +(6.08655 + 34.5186i) q^{51} +(310.979 + 260.942i) q^{52} +(67.0183 - 56.2350i) q^{53} +(-123.616 + 701.058i) q^{54} +(-214.665 + 78.1317i) q^{55} +1290.67 q^{56} +(315.039 - 322.171i) q^{57} -549.698 q^{58} +(-243.431 + 88.6015i) q^{59} +(-195.787 + 1110.36i) q^{60} +(-385.587 + 323.546i) q^{61} +(-358.770 - 301.044i) q^{62} +(8.52391 + 48.3415i) q^{63} +(-617.371 - 1069.32i) q^{64} +(97.5993 - 169.047i) q^{65} +(627.472 + 228.381i) q^{66} +(601.364 + 218.878i) q^{67} +(-66.8700 + 115.822i) q^{68} +(-276.048 - 478.129i) q^{69} +(-175.333 - 994.361i) q^{70} +(-532.440 - 446.770i) q^{71} +(136.423 - 114.473i) q^{72} +(-41.0761 + 232.954i) q^{73} +(1962.54 - 714.305i) q^{74} -137.959 q^{75} +(1710.99 - 169.006i) q^{76} -431.643 q^{77} +(-536.161 + 195.147i) q^{78} +(199.021 - 1128.70i) q^{79} +(-1536.19 + 1289.02i) q^{80} +(-607.086 - 509.406i) q^{81} +(187.834 + 1065.26i) q^{82} +(359.782 + 623.160i) q^{83} +(-1065.20 + 1844.99i) q^{84} +(60.4293 + 21.9945i) q^{85} +(-1905.17 - 693.425i) q^{86} +(278.847 - 482.977i) q^{87} +(782.998 + 1356.19i) q^{88} +(237.214 + 1345.31i) q^{89} +(-106.725 - 89.5532i) q^{90} +(282.540 - 237.079i) q^{91} +(365.801 - 2074.56i) q^{92} +(446.498 - 162.512i) q^{93} -684.915 q^{94} +(-222.892 - 796.100i) q^{95} +2883.22 q^{96} +(-277.899 + 101.147i) q^{97} +(11.8755 - 67.3495i) q^{98} +(-45.6247 + 38.2836i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q - 6 q^{2} - 3 q^{3} - 24 q^{4} - 6 q^{5} + 42 q^{6} + 3 q^{7} - 75 q^{8} - 51 q^{9} + 75 q^{10} + 39 q^{11} - 219 q^{12} - 156 q^{13} + 93 q^{14} - 192 q^{15} + 504 q^{16} + 12 q^{17} + 264 q^{18}+ \cdots + 492 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/19\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(e\left(\frac{8}{9}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −5.03941 + 1.83419i −1.78170 + 0.648485i −0.782017 + 0.623257i \(0.785809\pi\)
−0.999682 + 0.0252285i \(0.991969\pi\)
\(3\) 0.944789 − 5.35816i 0.181825 − 1.03118i −0.748143 − 0.663537i \(-0.769054\pi\)
0.929968 − 0.367641i \(-0.119835\pi\)
\(4\) 15.9030 − 13.3442i 1.98787 − 1.66802i
\(5\) −7.64679 − 6.41642i −0.683950 − 0.573902i 0.233208 − 0.972427i \(-0.425078\pi\)
−0.917157 + 0.398525i \(0.869522\pi\)
\(6\) 5.06673 + 28.7349i 0.344748 + 1.95516i
\(7\) −9.43070 − 16.3344i −0.509210 − 0.881977i −0.999943 − 0.0106675i \(-0.996604\pi\)
0.490733 − 0.871310i \(-0.336729\pi\)
\(8\) −34.2145 + 59.2612i −1.51208 + 2.61900i
\(9\) −2.44557 − 0.890116i −0.0905768 − 0.0329673i
\(10\) 50.3042 + 18.3092i 1.59076 + 0.578989i
\(11\) 11.4425 − 19.8190i 0.313640 − 0.543241i −0.665507 − 0.746391i \(-0.731785\pi\)
0.979147 + 0.203151i \(0.0651181\pi\)
\(12\) −56.4754 − 97.8182i −1.35859 − 2.35314i
\(13\) 3.39564 + 19.2576i 0.0724447 + 0.410855i 0.999366 + 0.0355991i \(0.0113339\pi\)
−0.926921 + 0.375256i \(0.877555\pi\)
\(14\) 77.4857 + 65.0182i 1.47921 + 1.24120i
\(15\) −41.6048 + 34.9106i −0.716154 + 0.600925i
\(16\) 34.8848 − 197.841i 0.545074 − 3.09127i
\(17\) −6.05373 + 2.20338i −0.0863673 + 0.0314351i −0.384842 − 0.922982i \(-0.625744\pi\)
0.298475 + 0.954417i \(0.403522\pi\)
\(18\) 13.9569 0.182759
\(19\) 68.3688 + 46.7409i 0.825519 + 0.564374i
\(20\) −207.229 −2.31689
\(21\) −96.4326 + 35.0986i −1.00206 + 0.364721i
\(22\) −21.3115 + 120.864i −0.206529 + 1.17128i
\(23\) 77.7328 − 65.2256i 0.704713 − 0.591325i −0.218397 − 0.975860i \(-0.570083\pi\)
0.923110 + 0.384535i \(0.125638\pi\)
\(24\) 285.206 + 239.316i 2.42572 + 2.03542i
\(25\) −4.40306 − 24.9710i −0.0352245 − 0.199768i
\(26\) −52.4343 − 90.8188i −0.395508 − 0.685040i
\(27\) 66.3711 − 114.958i 0.473079 − 0.819397i
\(28\) −367.946 − 133.921i −2.48340 − 0.903885i
\(29\) 96.3201 + 35.0576i 0.616765 + 0.224484i 0.631461 − 0.775408i \(-0.282456\pi\)
−0.0146956 + 0.999892i \(0.504678\pi\)
\(30\) 145.631 − 252.240i 0.886280 − 1.53508i
\(31\) 43.6655 + 75.6309i 0.252986 + 0.438184i 0.964347 − 0.264643i \(-0.0852540\pi\)
−0.711361 + 0.702827i \(0.751921\pi\)
\(32\) 92.0203 + 521.873i 0.508345 + 2.88297i
\(33\) −95.3826 − 80.0355i −0.503151 − 0.422194i
\(34\) 26.4658 − 22.2074i 0.133495 − 0.112016i
\(35\) −32.6941 + 185.417i −0.157895 + 0.895465i
\(36\) −50.7698 + 18.4787i −0.235045 + 0.0855495i
\(37\) −389.438 −1.73036 −0.865178 − 0.501464i \(-0.832795\pi\)
−0.865178 + 0.501464i \(0.832795\pi\)
\(38\) −430.270 − 110.145i −1.83682 − 0.470207i
\(39\) 106.394 0.436837
\(40\) 641.875 − 233.624i 2.53724 − 0.923478i
\(41\) 35.0252 − 198.638i 0.133415 − 0.756635i −0.842535 − 0.538641i \(-0.818938\pi\)
0.975950 − 0.217993i \(-0.0699511\pi\)
\(42\) 421.586 − 353.752i 1.54886 − 1.29965i
\(43\) 289.606 + 243.009i 1.02708 + 0.861825i 0.990501 − 0.137506i \(-0.0439086\pi\)
0.0365818 + 0.999331i \(0.488353\pi\)
\(44\) −82.4984 − 467.872i −0.282661 − 1.60305i
\(45\) 12.9894 + 22.4984i 0.0430300 + 0.0745301i
\(46\) −272.091 + 471.275i −0.872122 + 1.51056i
\(47\) 120.013 + 43.6812i 0.372462 + 0.135565i 0.521467 − 0.853271i \(-0.325385\pi\)
−0.149005 + 0.988836i \(0.547607\pi\)
\(48\) −1027.11 − 373.836i −3.08854 − 1.12414i
\(49\) −6.37616 + 11.0438i −0.0185894 + 0.0321978i
\(50\) 67.9904 + 117.763i 0.192306 + 0.333084i
\(51\) 6.08655 + 34.5186i 0.0167115 + 0.0947758i
\(52\) 310.979 + 260.942i 0.829326 + 0.695887i
\(53\) 67.0183 − 56.2350i 0.173692 − 0.145745i −0.551798 − 0.833978i \(-0.686058\pi\)
0.725489 + 0.688233i \(0.241613\pi\)
\(54\) −123.616 + 701.058i −0.311517 + 1.76670i
\(55\) −214.665 + 78.1317i −0.526281 + 0.191551i
\(56\) 1290.67 3.07986
\(57\) 315.039 − 322.171i 0.732070 − 0.748641i
\(58\) −549.698 −1.24446
\(59\) −243.431 + 88.6015i −0.537152 + 0.195507i −0.596329 − 0.802740i \(-0.703375\pi\)
0.0591768 + 0.998248i \(0.481152\pi\)
\(60\) −195.787 + 1110.36i −0.421267 + 2.38912i
\(61\) −385.587 + 323.546i −0.809334 + 0.679112i −0.950449 − 0.310882i \(-0.899376\pi\)
0.141115 + 0.989993i \(0.454931\pi\)
\(62\) −358.770 − 301.044i −0.734901 − 0.616655i
\(63\) 8.52391 + 48.3415i 0.0170462 + 0.0966739i
\(64\) −617.371 − 1069.32i −1.20580 − 2.08851i
\(65\) 97.5993 − 169.047i 0.186242 − 0.322580i
\(66\) 627.472 + 228.381i 1.17025 + 0.425936i
\(67\) 601.364 + 218.878i 1.09654 + 0.399108i 0.826040 − 0.563611i \(-0.190588\pi\)
0.270501 + 0.962720i \(0.412811\pi\)
\(68\) −66.8700 + 115.822i −0.119253 + 0.206552i
\(69\) −276.048 − 478.129i −0.481627 − 0.834203i
\(70\) −175.333 − 994.361i −0.299375 − 1.69784i
\(71\) −532.440 − 446.770i −0.889985 − 0.746786i 0.0782217 − 0.996936i \(-0.475076\pi\)
−0.968207 + 0.250149i \(0.919520\pi\)
\(72\) 136.423 − 114.473i 0.223301 − 0.187371i
\(73\) −41.0761 + 232.954i −0.0658574 + 0.373496i 0.934011 + 0.357245i \(0.116284\pi\)
−0.999868 + 0.0162504i \(0.994827\pi\)
\(74\) 1962.54 − 714.305i 3.08297 − 1.12211i
\(75\) −137.959 −0.212401
\(76\) 1710.99 − 169.006i 2.58242 − 0.255083i
\(77\) −431.643 −0.638835
\(78\) −536.161 + 195.147i −0.778312 + 0.283282i
\(79\) 199.021 − 1128.70i 0.283438 − 1.60745i −0.427376 − 0.904074i \(-0.640562\pi\)
0.710813 − 0.703381i \(-0.248327\pi\)
\(80\) −1536.19 + 1289.02i −2.14689 + 1.80145i
\(81\) −607.086 − 509.406i −0.832766 − 0.698774i
\(82\) 187.834 + 1065.26i 0.252961 + 1.43461i
\(83\) 359.782 + 623.160i 0.475797 + 0.824105i 0.999616 − 0.0277253i \(-0.00882635\pi\)
−0.523819 + 0.851830i \(0.675493\pi\)
\(84\) −1065.20 + 1844.99i −1.38361 + 2.39648i
\(85\) 60.4293 + 21.9945i 0.0771116 + 0.0280663i
\(86\) −1905.17 − 693.425i −2.38883 − 0.869464i
\(87\) 278.847 − 482.977i 0.343626 − 0.595178i
\(88\) 782.998 + 1356.19i 0.948498 + 1.64285i
\(89\) 237.214 + 1345.31i 0.282524 + 1.60227i 0.713999 + 0.700147i \(0.246882\pi\)
−0.431475 + 0.902125i \(0.642007\pi\)
\(90\) −106.725 − 89.5532i −0.124998 − 0.104886i
\(91\) 282.540 − 237.079i 0.325475 − 0.273106i
\(92\) 365.801 − 2074.56i 0.414537 − 2.35096i
\(93\) 446.498 − 162.512i 0.497846 − 0.181201i
\(94\) −684.915 −0.751527
\(95\) −222.892 − 796.100i −0.240719 − 0.859770i
\(96\) 2883.22 3.06529
\(97\) −277.899 + 101.147i −0.290891 + 0.105876i −0.483344 − 0.875431i \(-0.660578\pi\)
0.192453 + 0.981306i \(0.438356\pi\)
\(98\) 11.8755 − 67.3495i 0.0122409 − 0.0694217i
\(99\) −45.6247 + 38.2836i −0.0463177 + 0.0388651i
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 19.4.e.a.5.1 yes 24
3.2 odd 2 171.4.u.b.100.4 24
19.2 odd 18 361.4.a.m.1.1 12
19.4 even 9 inner 19.4.e.a.4.1 ✓ 24
19.17 even 9 361.4.a.n.1.12 12
57.23 odd 18 171.4.u.b.118.4 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
19.4.e.a.4.1 ✓ 24 19.4 even 9 inner
19.4.e.a.5.1 yes 24 1.1 even 1 trivial
171.4.u.b.100.4 24 3.2 odd 2
171.4.u.b.118.4 24 57.23 odd 18
361.4.a.m.1.1 12 19.2 odd 18
361.4.a.n.1.12 12 19.17 even 9