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Properties

Label 19.2.e.a.4.1
Level $19$
Weight $2$
Character 19.4
Analytic conductor $0.152$
Analytic rank $0$
Dimension $6$
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] = [19,2,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, 2); 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, 2, names="a")
 
Level: \( N \) \(=\) \( 19 \)
Weight: \( k \) \(=\) \( 2 \)
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: \(0.151715763840\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\Q(\zeta_{18})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{3} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label 4.1
Root \(0.939693 + 0.342020i\) of defining polynomial
Character \(\chi\) \(=\) 19.4
Dual form 19.2.e.a.5.1

$q$-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+(-0.826352 - 0.300767i) q^{2} +(0.0923963 + 0.524005i) q^{3} +(-0.939693 - 0.788496i) q^{4} +(-1.93969 + 1.62760i) q^{5} +(0.0812519 - 0.460802i) q^{6} +(0.939693 - 1.62760i) q^{7} +(1.41875 + 2.45734i) q^{8} +(2.55303 - 0.929228i) q^{9} +(2.09240 - 0.761570i) q^{10} +(-1.70574 - 2.95442i) q^{11} +(0.326352 - 0.565258i) q^{12} +(-0.918748 + 5.21048i) q^{13} +(-1.26604 + 1.06234i) q^{14} +(-1.03209 - 0.866025i) q^{15} +(-0.00727396 - 0.0412527i) q^{16} +(-1.55303 - 0.565258i) q^{17} -2.38919 q^{18} +(-2.52094 - 3.55596i) q^{19} +3.10607 q^{20} +(0.939693 + 0.342020i) q^{21} +(0.520945 + 2.95442i) q^{22} +(1.34730 + 1.13052i) q^{23} +(-1.15657 + 0.970481i) q^{24} +(0.245100 - 1.39003i) q^{25} +(2.32635 - 4.02936i) q^{26} +(1.52094 + 2.63435i) q^{27} +(-2.16637 + 0.788496i) q^{28} +(3.25877 - 1.18610i) q^{29} +(0.592396 + 1.02606i) q^{30} +(-0.971782 + 1.68317i) q^{31} +(0.979055 - 5.55250i) q^{32} +(1.39053 - 1.16679i) q^{33} +(1.11334 + 0.934204i) q^{34} +(0.826352 + 4.68647i) q^{35} +(-3.13176 - 1.13987i) q^{36} -0.837496 q^{37} +(1.01367 + 3.69669i) q^{38} -2.81521 q^{39} +(-6.75150 - 2.45734i) q^{40} +(-0.779715 - 4.42198i) q^{41} +(-0.673648 - 0.565258i) q^{42} +(3.67752 - 3.08580i) q^{43} +(-0.726682 + 4.12122i) q^{44} +(-3.43969 + 5.95772i) q^{45} +(-0.773318 - 1.33943i) q^{46} +(-0.673648 + 0.245188i) q^{47} +(0.0209445 - 0.00762319i) q^{48} +(1.73396 + 3.00330i) q^{49} +(-0.620615 + 1.07494i) q^{50} +(0.152704 - 0.866025i) q^{51} +(4.97178 - 4.17182i) q^{52} +(-4.67752 - 3.92490i) q^{53} +(-0.464508 - 2.63435i) q^{54} +(8.11721 + 2.95442i) q^{55} +5.33275 q^{56} +(1.63041 - 1.64955i) q^{57} -3.04963 q^{58} +(10.1099 + 3.67972i) q^{59} +(0.286989 + 1.62760i) q^{60} +(3.36231 + 2.82131i) q^{61} +(1.30928 - 1.09861i) q^{62} +(0.886659 - 5.02849i) q^{63} +(-2.52094 + 4.36640i) q^{64} +(-6.69846 - 11.6021i) q^{65} +(-1.50000 + 0.545955i) q^{66} +(-13.3550 + 4.86084i) q^{67} +(1.01367 + 1.75573i) q^{68} +(-0.467911 + 0.810446i) q^{69} +(0.726682 - 4.12122i) q^{70} +(-10.5398 + 8.84397i) q^{71} +(5.90554 + 4.95534i) q^{72} +(-1.30541 - 7.40333i) q^{73} +(0.692066 + 0.251892i) q^{74} +0.751030 q^{75} +(-0.434945 + 5.32926i) q^{76} -6.41147 q^{77} +(2.32635 + 0.846723i) q^{78} +(-1.20914 - 6.85738i) q^{79} +(0.0812519 + 0.0681784i) q^{80} +(5.00387 - 4.19875i) q^{81} +(-0.685670 + 3.88863i) q^{82} +(-1.25624 + 2.17588i) q^{83} +(-0.613341 - 1.06234i) q^{84} +(3.93242 - 1.43128i) q^{85} +(-3.96703 + 1.44388i) q^{86} +(0.922618 + 1.59802i) q^{87} +(4.84002 - 8.38316i) q^{88} +(-0.396459 + 2.24843i) q^{89} +(4.63429 - 3.88863i) q^{90} +(7.61721 + 6.39160i) q^{91} +(-0.374638 - 2.12467i) q^{92} +(-0.971782 - 0.353700i) q^{93} +0.630415 q^{94} +(10.6775 + 2.79439i) q^{95} +3.00000 q^{96} +(1.71301 + 0.623485i) q^{97} +(-0.529563 - 3.00330i) q^{98} +(-7.10014 - 5.95772i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 6 q^{2} - 3 q^{3} - 6 q^{5} + 3 q^{6} + 6 q^{8} + 3 q^{9} + 9 q^{10} + 3 q^{12} - 3 q^{13} - 3 q^{14} + 3 q^{15} - 18 q^{16} + 3 q^{17} - 6 q^{18} - 12 q^{19} - 6 q^{20} + 6 q^{23} + 15 q^{24} + 15 q^{26}+ \cdots + 9 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(2\)
\(\chi(n)\) \(e\left(\frac{1}{9}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.826352 − 0.300767i −0.584319 − 0.212675i 0.0329100 − 0.999458i \(-0.489523\pi\)
−0.617229 + 0.786784i \(0.711745\pi\)
\(3\) 0.0923963 + 0.524005i 0.0533450 + 0.302535i 0.999794 − 0.0203202i \(-0.00646857\pi\)
−0.946449 + 0.322855i \(0.895357\pi\)
\(4\) −0.939693 − 0.788496i −0.469846 − 0.394248i
\(5\) −1.93969 + 1.62760i −0.867457 + 0.727883i −0.963561 − 0.267489i \(-0.913806\pi\)
0.0961041 + 0.995371i \(0.469362\pi\)
\(6\) 0.0812519 − 0.460802i 0.0331710 − 0.188122i
\(7\) 0.939693 − 1.62760i 0.355170 − 0.615173i −0.631977 − 0.774987i \(-0.717756\pi\)
0.987147 + 0.159814i \(0.0510895\pi\)
\(8\) 1.41875 + 2.45734i 0.501603 + 0.868802i
\(9\) 2.55303 − 0.929228i 0.851011 − 0.309743i
\(10\) 2.09240 − 0.761570i 0.661674 − 0.240830i
\(11\) −1.70574 − 2.95442i −0.514299 − 0.890792i −0.999862 − 0.0165906i \(-0.994719\pi\)
0.485563 − 0.874202i \(-0.338615\pi\)
\(12\) 0.326352 − 0.565258i 0.0942097 − 0.163176i
\(13\) −0.918748 + 5.21048i −0.254815 + 1.44513i 0.541733 + 0.840551i \(0.317769\pi\)
−0.796547 + 0.604576i \(0.793343\pi\)
\(14\) −1.26604 + 1.06234i −0.338365 + 0.283922i
\(15\) −1.03209 − 0.866025i −0.266484 − 0.223607i
\(16\) −0.00727396 − 0.0412527i −0.00181849 − 0.0103132i
\(17\) −1.55303 − 0.565258i −0.376666 − 0.137095i 0.146748 − 0.989174i \(-0.453119\pi\)
−0.523414 + 0.852079i \(0.675342\pi\)
\(18\) −2.38919 −0.563136
\(19\) −2.52094 − 3.55596i −0.578344 − 0.815793i
\(20\) 3.10607 0.694538
\(21\) 0.939693 + 0.342020i 0.205058 + 0.0746349i
\(22\) 0.520945 + 2.95442i 0.111066 + 0.629885i
\(23\) 1.34730 + 1.13052i 0.280931 + 0.235729i 0.772354 − 0.635192i \(-0.219079\pi\)
−0.491424 + 0.870921i \(0.663523\pi\)
\(24\) −1.15657 + 0.970481i −0.236085 + 0.198099i
\(25\) 0.245100 − 1.39003i 0.0490200 − 0.278006i
\(26\) 2.32635 − 4.02936i 0.456235 − 0.790222i
\(27\) 1.52094 + 2.63435i 0.292706 + 0.506982i
\(28\) −2.16637 + 0.788496i −0.409406 + 0.149012i
\(29\) 3.25877 − 1.18610i 0.605138 − 0.220252i −0.0212363 − 0.999774i \(-0.506760\pi\)
0.626375 + 0.779522i \(0.284538\pi\)
\(30\) 0.592396 + 1.02606i 0.108156 + 0.187332i
\(31\) −0.971782 + 1.68317i −0.174537 + 0.302307i −0.940001 − 0.341172i \(-0.889176\pi\)
0.765464 + 0.643479i \(0.222510\pi\)
\(32\) 0.979055 − 5.55250i 0.173074 − 0.981553i
\(33\) 1.39053 − 1.16679i 0.242060 − 0.203113i
\(34\) 1.11334 + 0.934204i 0.190936 + 0.160215i
\(35\) 0.826352 + 4.68647i 0.139679 + 0.792159i
\(36\) −3.13176 − 1.13987i −0.521960 − 0.189978i
\(37\) −0.837496 −0.137684 −0.0688418 − 0.997628i \(-0.521930\pi\)
−0.0688418 + 0.997628i \(0.521930\pi\)
\(38\) 1.01367 + 3.69669i 0.164439 + 0.599682i
\(39\) −2.81521 −0.450794
\(40\) −6.75150 − 2.45734i −1.06751 − 0.388540i
\(41\) −0.779715 − 4.42198i −0.121771 − 0.690598i −0.983173 − 0.182675i \(-0.941524\pi\)
0.861402 − 0.507923i \(-0.169587\pi\)
\(42\) −0.673648 − 0.565258i −0.103946 − 0.0872212i
\(43\) 3.67752 − 3.08580i 0.560816 − 0.470581i −0.317768 − 0.948169i \(-0.602933\pi\)
0.878584 + 0.477588i \(0.158489\pi\)
\(44\) −0.726682 + 4.12122i −0.109551 + 0.621297i
\(45\) −3.43969 + 5.95772i −0.512759 + 0.888125i
\(46\) −0.773318 − 1.33943i −0.114020 − 0.197488i
\(47\) −0.673648 + 0.245188i −0.0982617 + 0.0357643i −0.390683 − 0.920525i \(-0.627761\pi\)
0.292422 + 0.956290i \(0.405539\pi\)
\(48\) 0.0209445 − 0.00762319i 0.00302308 − 0.00110031i
\(49\) 1.73396 + 3.00330i 0.247708 + 0.429043i
\(50\) −0.620615 + 1.07494i −0.0877682 + 0.152019i
\(51\) 0.152704 − 0.866025i 0.0213828 − 0.121268i
\(52\) 4.97178 − 4.17182i 0.689462 − 0.578527i
\(53\) −4.67752 − 3.92490i −0.642507 − 0.539127i 0.262280 − 0.964992i \(-0.415526\pi\)
−0.904787 + 0.425865i \(0.859970\pi\)
\(54\) −0.464508 − 2.63435i −0.0632115 − 0.358490i
\(55\) 8.11721 + 2.95442i 1.09452 + 0.398374i
\(56\) 5.33275 0.712618
\(57\) 1.63041 − 1.64955i 0.215954 − 0.218488i
\(58\) −3.04963 −0.400436
\(59\) 10.1099 + 3.67972i 1.31620 + 0.479058i 0.902239 − 0.431236i \(-0.141922\pi\)
0.413962 + 0.910294i \(0.364144\pi\)
\(60\) 0.286989 + 1.62760i 0.0370501 + 0.210122i
\(61\) 3.36231 + 2.82131i 0.430500 + 0.361232i 0.832140 − 0.554565i \(-0.187115\pi\)
−0.401640 + 0.915797i \(0.631560\pi\)
\(62\) 1.30928 − 1.09861i 0.166278 − 0.139524i
\(63\) 0.886659 − 5.02849i 0.111709 − 0.633531i
\(64\) −2.52094 + 4.36640i −0.315118 + 0.545801i
\(65\) −6.69846 − 11.6021i −0.830842 − 1.43906i
\(66\) −1.50000 + 0.545955i −0.184637 + 0.0672025i
\(67\) −13.3550 + 4.86084i −1.63158 + 0.593846i −0.985537 − 0.169458i \(-0.945798\pi\)
−0.646040 + 0.763304i \(0.723576\pi\)
\(68\) 1.01367 + 1.75573i 0.122926 + 0.212913i
\(69\) −0.467911 + 0.810446i −0.0563299 + 0.0975662i
\(70\) 0.726682 − 4.12122i 0.0868551 − 0.492580i
\(71\) −10.5398 + 8.84397i −1.25085 + 1.04959i −0.254252 + 0.967138i \(0.581829\pi\)
−0.996595 + 0.0824479i \(0.973726\pi\)
\(72\) 5.90554 + 4.95534i 0.695975 + 0.583992i
\(73\) −1.30541 − 7.40333i −0.152786 − 0.866495i −0.960782 − 0.277306i \(-0.910559\pi\)
0.807995 − 0.589189i \(-0.200553\pi\)
\(74\) 0.692066 + 0.251892i 0.0804511 + 0.0292818i
\(75\) 0.751030 0.0867214
\(76\) −0.434945 + 5.32926i −0.0498916 + 0.611308i
\(77\) −6.41147 −0.730655
\(78\) 2.32635 + 0.846723i 0.263407 + 0.0958725i
\(79\) −1.20914 − 6.85738i −0.136039 − 0.771515i −0.974131 − 0.225986i \(-0.927440\pi\)
0.838092 − 0.545529i \(-0.183671\pi\)
\(80\) 0.0812519 + 0.0681784i 0.00908424 + 0.00762258i
\(81\) 5.00387 − 4.19875i 0.555986 − 0.466527i
\(82\) −0.685670 + 3.88863i −0.0757196 + 0.429427i
\(83\) −1.25624 + 2.17588i −0.137891 + 0.238834i −0.926698 − 0.375807i \(-0.877366\pi\)
0.788807 + 0.614641i \(0.210699\pi\)
\(84\) −0.613341 − 1.06234i −0.0669210 − 0.115911i
\(85\) 3.93242 − 1.43128i 0.426531 − 0.155244i
\(86\) −3.96703 + 1.44388i −0.427776 + 0.155698i
\(87\) 0.922618 + 1.59802i 0.0989151 + 0.171326i
\(88\) 4.84002 − 8.38316i 0.515948 − 0.893648i
\(89\) −0.396459 + 2.24843i −0.0420246 + 0.238333i −0.998584 − 0.0532055i \(-0.983056\pi\)
0.956559 + 0.291539i \(0.0941673\pi\)
\(90\) 4.63429 − 3.88863i 0.488497 − 0.409897i
\(91\) 7.61721 + 6.39160i 0.798501 + 0.670022i
\(92\) −0.374638 − 2.12467i −0.0390587 − 0.221513i
\(93\) −0.971782 − 0.353700i −0.100769 − 0.0366769i
\(94\) 0.630415 0.0650223
\(95\) 10.6775 + 2.79439i 1.09549 + 0.286698i
\(96\) 3.00000 0.306186
\(97\) 1.71301 + 0.623485i 0.173930 + 0.0633053i 0.427517 − 0.904007i \(-0.359388\pi\)
−0.253587 + 0.967312i \(0.581611\pi\)
\(98\) −0.529563 − 3.00330i −0.0534939 − 0.303379i
\(99\) −7.10014 − 5.95772i −0.713591 − 0.598774i
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 19.2.e.a.4.1 ✓ 6
3.2 odd 2 171.2.u.c.118.1 6
4.3 odd 2 304.2.u.b.289.1 6
5.2 odd 4 475.2.u.a.99.2 12
5.3 odd 4 475.2.u.a.99.1 12
5.4 even 2 475.2.l.a.251.1 6
7.2 even 3 931.2.x.a.802.1 6
7.3 odd 6 931.2.v.a.422.1 6
7.4 even 3 931.2.v.b.422.1 6
7.5 odd 6 931.2.x.b.802.1 6
7.6 odd 2 931.2.w.a.99.1 6
19.2 odd 18 361.2.e.b.54.1 6
19.3 odd 18 361.2.e.a.245.1 6
19.4 even 9 361.2.c.i.292.1 6
19.5 even 9 inner 19.2.e.a.5.1 yes 6
19.6 even 9 361.2.c.i.68.1 6
19.7 even 3 361.2.e.f.234.1 6
19.8 odd 6 361.2.e.a.28.1 6
19.9 even 9 361.2.a.g.1.3 3
19.10 odd 18 361.2.a.h.1.1 3
19.11 even 3 361.2.e.g.28.1 6
19.12 odd 6 361.2.e.b.234.1 6
19.13 odd 18 361.2.c.h.68.3 6
19.14 odd 18 361.2.e.h.62.1 6
19.15 odd 18 361.2.c.h.292.3 6
19.16 even 9 361.2.e.g.245.1 6
19.17 even 9 361.2.e.f.54.1 6
19.18 odd 2 361.2.e.h.99.1 6
57.5 odd 18 171.2.u.c.100.1 6
57.29 even 18 3249.2.a.s.1.3 3
57.47 odd 18 3249.2.a.z.1.1 3
76.43 odd 18 304.2.u.b.81.1 6
76.47 odd 18 5776.2.a.br.1.1 3
76.67 even 18 5776.2.a.bi.1.3 3
95.9 even 18 9025.2.a.bd.1.1 3
95.24 even 18 475.2.l.a.176.1 6
95.29 odd 18 9025.2.a.x.1.3 3
95.43 odd 36 475.2.u.a.24.2 12
95.62 odd 36 475.2.u.a.24.1 12
133.5 odd 18 931.2.v.a.214.1 6
133.24 odd 18 931.2.x.b.765.1 6
133.62 odd 18 931.2.w.a.442.1 6
133.81 even 9 931.2.x.a.765.1 6
133.100 even 9 931.2.v.b.214.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
19.2.e.a.4.1 ✓ 6 1.1 even 1 trivial
19.2.e.a.5.1 yes 6 19.5 even 9 inner
171.2.u.c.100.1 6 57.5 odd 18
171.2.u.c.118.1 6 3.2 odd 2
304.2.u.b.81.1 6 76.43 odd 18
304.2.u.b.289.1 6 4.3 odd 2
361.2.a.g.1.3 3 19.9 even 9
361.2.a.h.1.1 3 19.10 odd 18
361.2.c.h.68.3 6 19.13 odd 18
361.2.c.h.292.3 6 19.15 odd 18
361.2.c.i.68.1 6 19.6 even 9
361.2.c.i.292.1 6 19.4 even 9
361.2.e.a.28.1 6 19.8 odd 6
361.2.e.a.245.1 6 19.3 odd 18
361.2.e.b.54.1 6 19.2 odd 18
361.2.e.b.234.1 6 19.12 odd 6
361.2.e.f.54.1 6 19.17 even 9
361.2.e.f.234.1 6 19.7 even 3
361.2.e.g.28.1 6 19.11 even 3
361.2.e.g.245.1 6 19.16 even 9
361.2.e.h.62.1 6 19.14 odd 18
361.2.e.h.99.1 6 19.18 odd 2
475.2.l.a.176.1 6 95.24 even 18
475.2.l.a.251.1 6 5.4 even 2
475.2.u.a.24.1 12 95.62 odd 36
475.2.u.a.24.2 12 95.43 odd 36
475.2.u.a.99.1 12 5.3 odd 4
475.2.u.a.99.2 12 5.2 odd 4
931.2.v.a.214.1 6 133.5 odd 18
931.2.v.a.422.1 6 7.3 odd 6
931.2.v.b.214.1 6 133.100 even 9
931.2.v.b.422.1 6 7.4 even 3
931.2.w.a.99.1 6 7.6 odd 2
931.2.w.a.442.1 6 133.62 odd 18
931.2.x.a.765.1 6 133.81 even 9
931.2.x.a.802.1 6 7.2 even 3
931.2.x.b.765.1 6 133.24 odd 18
931.2.x.b.802.1 6 7.5 odd 6
3249.2.a.s.1.3 3 57.29 even 18
3249.2.a.z.1.1 3 57.47 odd 18
5776.2.a.bi.1.3 3 76.67 even 18
5776.2.a.br.1.1 3 76.47 odd 18
9025.2.a.x.1.3 3 95.29 odd 18
9025.2.a.bd.1.1 3 95.9 even 18