Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [639,2,Mod(10,639)] 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("639.10"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(639, base_ring=CyclotomicField(70)) chi = DirichletCharacter(H, H._module([0, 34])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 639 = 3^{2} \cdot 71 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 639.v (of order \(35\), degree \(24\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [120,22,0,-18] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(5.10244068916\)
Analytic rank: \(0\)
Dimension: \(120\)
Relative dimension: \(5\) over \(\Q(\zeta_{35})\)
Twist minimal: no (minimal twist has level 71)
Sato-Tate group: $\mathrm{SU}(2)[C_{35}]$

Embedding invariants

Embedding label 19.1
Character \(\chi\) \(=\) 639.19
Dual form 639.2.v.a.370.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.871177 - 2.03822i) q^{2} +(-2.01326 + 2.10571i) q^{4} +(0.768083 + 2.36392i) q^{5} +(-1.65970 - 1.45003i) q^{7} +(1.89530 + 0.711319i) q^{8} +(4.14905 - 3.62491i) q^{10} +(-0.236549 - 0.0652833i) q^{11} +(-0.742793 + 0.204998i) q^{13} +(-1.50960 + 4.64606i) q^{14} +(0.0600888 + 1.33798i) q^{16} +(-6.29398 - 4.57284i) q^{17} +(-0.932339 + 1.73258i) q^{19} +(-6.52408 - 3.14183i) q^{20} +(0.0730141 + 0.539011i) q^{22} +(1.49584 + 6.55371i) q^{23} +(-0.953069 + 0.692445i) q^{25} +(1.06493 + 1.33539i) q^{26} +(6.39476 - 0.575539i) q^{28} +(-1.11272 + 8.21446i) q^{29} +(-0.336431 + 7.49122i) q^{31} +(6.32257 - 3.04479i) q^{32} +(-3.83729 + 16.8123i) q^{34} +(2.15298 - 5.03714i) q^{35} +(-0.840104 + 3.68073i) q^{37} +(4.34360 + 0.390931i) q^{38} +(-0.225749 + 5.02669i) q^{40} +(-2.72472 + 3.41669i) q^{41} +(-10.1766 + 0.915913i) q^{43} +(0.613703 - 0.366670i) q^{44} +(12.0548 - 8.75829i) q^{46} +(-2.78273 - 0.504992i) q^{47} +(-0.287636 - 2.12341i) q^{49} +(2.24165 + 1.33932i) q^{50} +(1.06377 - 1.97682i) q^{52} +(-2.93485 - 3.06961i) q^{53} +(-0.0273648 - 0.609325i) q^{55} +(-2.11419 - 3.92883i) q^{56} +(17.7123 - 4.88827i) q^{58} +(-6.29307 + 9.53360i) q^{59} +(8.64549 - 7.55334i) q^{61} +(15.5618 - 5.84046i) q^{62} +(-9.69681 - 8.47185i) q^{64} +(-1.05512 - 1.59845i) q^{65} +(-1.97431 + 2.06496i) q^{67} +(22.3005 - 4.04695i) q^{68} -12.1424 q^{70} +(7.80170 - 3.18332i) q^{71} +(0.820375 + 1.91936i) q^{73} +(8.23402 - 1.49426i) q^{74} +(-1.77126 - 5.45137i) q^{76} +(0.297936 + 0.451354i) q^{77} +(2.55518 + 0.958974i) q^{79} +(-3.11672 + 1.16973i) q^{80} +(9.33767 + 2.57703i) q^{82} +(4.34310 - 6.57952i) q^{83} +(5.97552 - 18.3908i) q^{85} +(10.7325 + 19.9443i) q^{86} +(-0.401894 - 0.291993i) q^{88} +(-8.90745 - 9.31646i) q^{89} +(1.53007 + 0.736841i) q^{91} +(-16.8117 - 10.0445i) q^{92} +(1.39497 + 6.11176i) q^{94} +(-4.81178 - 0.873210i) q^{95} +(0.923002 + 1.15741i) q^{97} +(-4.07740 + 2.43613i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 120 q + 22 q^{2} - 18 q^{4} + 20 q^{5} - 27 q^{7} + 27 q^{8} - 8 q^{10} + 27 q^{11} - 31 q^{13} - 2 q^{14} + 30 q^{16} - 9 q^{17} - 31 q^{19} - 72 q^{20} - 24 q^{22} + 6 q^{23} - 42 q^{25} - 77 q^{26} - 18 q^{28}+ \cdots - 128 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/639\mathbb{Z}\right)^\times\).

\(n\) \(433\) \(569\)
\(\chi(n)\) \(e\left(\frac{8}{35}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.871177 − 2.03822i −0.616015 − 1.44124i −0.879181 − 0.476488i \(-0.841909\pi\)
0.263166 − 0.964751i \(-0.415233\pi\)
\(3\) 0 0
\(4\) −2.01326 + 2.10571i −1.00663 + 1.05285i
\(5\) 0.768083 + 2.36392i 0.343497 + 1.05718i 0.962383 + 0.271695i \(0.0875843\pi\)
−0.618886 + 0.785481i \(0.712416\pi\)
\(6\) 0 0
\(7\) −1.65970 − 1.45003i −0.627307 − 0.548061i 0.284503 − 0.958675i \(-0.408172\pi\)
−0.911809 + 0.410614i \(0.865314\pi\)
\(8\) 1.89530 + 0.711319i 0.670091 + 0.251489i
\(9\) 0 0
\(10\) 4.14905 − 3.62491i 1.31204 − 1.14630i
\(11\) −0.236549 − 0.0652833i −0.0713221 − 0.0196837i 0.230194 − 0.973145i \(-0.426064\pi\)
−0.301516 + 0.953461i \(0.597493\pi\)
\(12\) 0 0
\(13\) −0.742793 + 0.204998i −0.206014 + 0.0568561i −0.367520 − 0.930015i \(-0.619793\pi\)
0.161507 + 0.986872i \(0.448365\pi\)
\(14\) −1.50960 + 4.64606i −0.403457 + 1.24171i
\(15\) 0 0
\(16\) 0.0600888 + 1.33798i 0.0150222 + 0.334495i
\(17\) −6.29398 − 4.57284i −1.52651 − 1.10908i −0.958137 − 0.286309i \(-0.907572\pi\)
−0.568376 − 0.822769i \(-0.692428\pi\)
\(18\) 0 0
\(19\) −0.932339 + 1.73258i −0.213893 + 0.397480i −0.964822 − 0.262903i \(-0.915320\pi\)
0.750929 + 0.660383i \(0.229606\pi\)
\(20\) −6.52408 − 3.14183i −1.45883 − 0.702535i
\(21\) 0 0
\(22\) 0.0730141 + 0.539011i 0.0155667 + 0.114918i
\(23\) 1.49584 + 6.55371i 0.311905 + 1.36654i 0.851383 + 0.524544i \(0.175764\pi\)
−0.539479 + 0.841999i \(0.681379\pi\)
\(24\) 0 0
\(25\) −0.953069 + 0.692445i −0.190614 + 0.138489i
\(26\) 1.06493 + 1.33539i 0.208851 + 0.261891i
\(27\) 0 0
\(28\) 6.39476 − 0.575539i 1.20850 − 0.108767i
\(29\) −1.11272 + 8.21446i −0.206628 + 1.52539i 0.529363 + 0.848395i \(0.322431\pi\)
−0.735991 + 0.676992i \(0.763283\pi\)
\(30\) 0 0
\(31\) −0.336431 + 7.49122i −0.0604248 + 1.34546i 0.708313 + 0.705899i \(0.249457\pi\)
−0.768738 + 0.639564i \(0.779115\pi\)
\(32\) 6.32257 − 3.04479i 1.11768 − 0.538248i
\(33\) 0 0
\(34\) −3.83729 + 16.8123i −0.658090 + 2.88328i
\(35\) 2.15298 − 5.03714i 0.363919 − 0.851431i
\(36\) 0 0
\(37\) −0.840104 + 3.68073i −0.138112 + 0.605109i 0.857737 + 0.514089i \(0.171870\pi\)
−0.995849 + 0.0910201i \(0.970987\pi\)
\(38\) 4.34360 + 0.390931i 0.704625 + 0.0634174i
\(39\) 0 0
\(40\) −0.225749 + 5.02669i −0.0356941 + 0.794790i
\(41\) −2.72472 + 3.41669i −0.425529 + 0.533597i −0.947665 − 0.319265i \(-0.896564\pi\)
0.522136 + 0.852862i \(0.325135\pi\)
\(42\) 0 0
\(43\) −10.1766 + 0.915913i −1.55192 + 0.139675i −0.832156 − 0.554542i \(-0.812894\pi\)
−0.719766 + 0.694217i \(0.755751\pi\)
\(44\) 0.613703 − 0.366670i 0.0925192 − 0.0552776i
\(45\) 0 0
\(46\) 12.0548 − 8.75829i 1.77738 − 1.29134i
\(47\) −2.78273 − 0.504992i −0.405904 − 0.0736606i −0.0282346 − 0.999601i \(-0.508989\pi\)
−0.377669 + 0.925941i \(0.623274\pi\)
\(48\) 0 0
\(49\) −0.287636 − 2.12341i −0.0410908 − 0.303344i
\(50\) 2.24165 + 1.33932i 0.317017 + 0.189409i
\(51\) 0 0
\(52\) 1.06377 − 1.97682i 0.147519 − 0.274136i
\(53\) −2.93485 − 3.06961i −0.403133 − 0.421644i 0.489799 − 0.871835i \(-0.337070\pi\)
−0.892932 + 0.450191i \(0.851356\pi\)
\(54\) 0 0
\(55\) −0.0273648 − 0.609325i −0.00368987 − 0.0821613i
\(56\) −2.11419 − 3.92883i −0.282521 − 0.525012i
\(57\) 0 0
\(58\) 17.7123 − 4.88827i 2.32573 − 0.641861i
\(59\) −6.29307 + 9.53360i −0.819288 + 1.24117i 0.148377 + 0.988931i \(0.452595\pi\)
−0.967665 + 0.252238i \(0.918833\pi\)
\(60\) 0 0
\(61\) 8.64549 − 7.55334i 1.10694 − 0.967106i 0.107309 − 0.994226i \(-0.465777\pi\)
0.999632 + 0.0271201i \(0.00863366\pi\)
\(62\) 15.5618 − 5.84046i 1.97636 − 0.741739i
\(63\) 0 0
\(64\) −9.69681 − 8.47185i −1.21210 − 1.05898i
\(65\) −1.05512 − 1.59845i −0.130872 − 0.198263i
\(66\) 0 0
\(67\) −1.97431 + 2.06496i −0.241200 + 0.252275i −0.832303 − 0.554320i \(-0.812978\pi\)
0.591103 + 0.806596i \(0.298692\pi\)
\(68\) 22.3005 − 4.04695i 2.70434 − 0.490765i
\(69\) 0 0
\(70\) −12.1424 −1.45130
\(71\) 7.80170 − 3.18332i 0.925891 − 0.377791i
\(72\) 0 0
\(73\) 0.820375 + 1.91936i 0.0960177 + 0.224644i 0.960651 − 0.277760i \(-0.0895919\pi\)
−0.864633 + 0.502404i \(0.832449\pi\)
\(74\) 8.23402 − 1.49426i 0.957186 − 0.173704i
\(75\) 0 0
\(76\) −1.77126 − 5.45137i −0.203177 − 0.625315i
\(77\) 0.297936 + 0.451354i 0.0339530 + 0.0514366i
\(78\) 0 0
\(79\) 2.55518 + 0.958974i 0.287480 + 0.107893i 0.490949 − 0.871188i \(-0.336650\pi\)
−0.203469 + 0.979081i \(0.565222\pi\)
\(80\) −3.11672 + 1.16973i −0.348460 + 0.130779i
\(81\) 0 0
\(82\) 9.33767 + 2.57703i 1.03117 + 0.284586i
\(83\) 4.34310 − 6.57952i 0.476718 − 0.722196i −0.514051 − 0.857760i \(-0.671856\pi\)
0.990769 + 0.135563i \(0.0432844\pi\)
\(84\) 0 0
\(85\) 5.97552 − 18.3908i 0.648136 − 1.99476i
\(86\) 10.7325 + 19.9443i 1.15731 + 2.15065i
\(87\) 0 0
\(88\) −0.401894 − 0.291993i −0.0428421 − 0.0311266i
\(89\) −8.90745 − 9.31646i −0.944188 − 0.987543i 0.0557529 − 0.998445i \(-0.482244\pi\)
−0.999941 + 0.0109018i \(0.996530\pi\)
\(90\) 0 0
\(91\) 1.53007 + 0.736841i 0.160394 + 0.0772419i
\(92\) −16.8117 − 10.0445i −1.75274 − 1.04722i
\(93\) 0 0
\(94\) 1.39497 + 6.11176i 0.143880 + 0.630380i
\(95\) −4.81178 − 0.873210i −0.493678 − 0.0895894i
\(96\) 0 0
\(97\) 0.923002 + 1.15741i 0.0937166 + 0.117517i 0.826479 − 0.562968i \(-0.190341\pi\)
−0.732762 + 0.680485i \(0.761769\pi\)
\(98\) −4.07740 + 2.43613i −0.411879 + 0.246086i
\(99\) 0 0
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 639.2.v.a.19.1 120
3.2 odd 2 71.2.g.a.19.5 yes 120
71.15 even 35 inner 639.2.v.a.370.1 120
213.50 odd 70 5041.2.a.s.1.5 60
213.86 odd 70 71.2.g.a.15.5 ✓ 120
213.92 even 70 5041.2.a.t.1.5 60
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
71.2.g.a.15.5 ✓ 120 213.86 odd 70
71.2.g.a.19.5 yes 120 3.2 odd 2
639.2.v.a.19.1 120 1.1 even 1 trivial
639.2.v.a.370.1 120 71.15 even 35 inner
5041.2.a.s.1.5 60 213.50 odd 70
5041.2.a.t.1.5 60 213.92 even 70