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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [380,3,Mod(11,380)] 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("380.11"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(380, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 0, 4])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 380 = 2^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 380.q (of order \(6\), degree \(2\), 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: \(10.3542500457\)
Analytic rank: \(0\)
Dimension: \(160\)
Relative dimension: \(80\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 11.2
Character \(\chi\) \(=\) 380.11
Dual form 380.3.q.a.311.2

$q$-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+(-1.99382 + 0.157147i) q^{2} +(3.38036 + 1.95165i) q^{3} +(3.95061 - 0.626645i) q^{4} +(1.11803 - 1.93649i) q^{5} +(-7.04652 - 3.36002i) q^{6} -4.79106i q^{7} +(-7.77832 + 1.87024i) q^{8} +(3.11789 + 5.40035i) q^{9} +(-1.92484 + 4.03670i) q^{10} -10.8826i q^{11} +(14.5775 + 5.59193i) q^{12} +(-9.01390 - 15.6125i) q^{13} +(0.752901 + 9.55250i) q^{14} +(7.55872 - 4.36403i) q^{15} +(15.2146 - 4.95126i) q^{16} +(12.2766 - 21.2637i) q^{17} +(-7.06516 - 10.2773i) q^{18} +(-17.2968 - 7.86254i) q^{19} +(3.20342 - 8.35093i) q^{20} +(9.35049 - 16.1955i) q^{21} +(1.71017 + 21.6979i) q^{22} +(-23.2887 + 13.4457i) q^{23} +(-29.9436 - 8.85848i) q^{24} +(-2.50000 - 4.33013i) q^{25} +(20.4255 + 29.7120i) q^{26} -10.7896i q^{27} +(-3.00229 - 18.9276i) q^{28} +(-10.5279 - 18.2348i) q^{29} +(-14.3849 + 9.88890i) q^{30} +57.2377i q^{31} +(-29.5571 + 12.2628i) q^{32} +(21.2391 - 36.7871i) q^{33} +(-21.1357 + 44.3251i) q^{34} +(-9.27786 - 5.35657i) q^{35} +(15.7017 + 19.3809i) q^{36} +70.4471 q^{37} +(35.7223 + 12.9583i) q^{38} -70.3680i q^{39} +(-5.07472 + 17.1536i) q^{40} +(-26.5857 + 46.0478i) q^{41} +(-16.0981 + 33.7603i) q^{42} +(24.1071 + 13.9182i) q^{43} +(-6.81952 - 42.9929i) q^{44} +13.9436 q^{45} +(44.3205 - 30.4681i) q^{46} +(-46.7446 + 26.9880i) q^{47} +(61.0941 + 12.9566i) q^{48} +26.0457 q^{49} +(5.66501 + 8.24061i) q^{50} +(82.9986 - 47.9193i) q^{51} +(-45.3939 - 56.0305i) q^{52} +(-15.9654 - 27.6529i) q^{53} +(1.69555 + 21.5124i) q^{54} +(-21.0741 - 12.1671i) q^{55} +(8.96044 + 37.2664i) q^{56} +(-43.1246 - 60.3356i) q^{57} +(23.8562 + 34.7025i) q^{58} +(-8.18096 - 4.72328i) q^{59} +(27.1268 - 21.9772i) q^{60} +(4.88891 + 8.46783i) q^{61} +(-8.99473 - 114.121i) q^{62} +(25.8734 - 14.9380i) q^{63} +(57.0044 - 29.0946i) q^{64} -40.3114 q^{65} +(-36.5658 + 76.6844i) q^{66} +(96.6689 - 55.8118i) q^{67} +(35.1752 - 91.6975i) q^{68} -104.966 q^{69} +(19.3401 + 9.22204i) q^{70} +(99.6655 + 57.5419i) q^{71} +(-34.3519 - 36.1744i) q^{72} +(60.1309 - 104.150i) q^{73} +(-140.459 + 11.0705i) q^{74} -19.5165i q^{75} +(-73.2601 - 20.2229i) q^{76} -52.1392 q^{77} +(11.0581 + 140.301i) q^{78} +(86.5330 + 49.9598i) q^{79} +(7.42241 - 34.9987i) q^{80} +(49.1185 - 85.0758i) q^{81} +(45.7708 - 95.9887i) q^{82} -22.2344i q^{83} +(26.7913 - 69.8416i) q^{84} +(-27.4513 - 47.5470i) q^{85} +(-50.2523 - 23.9621i) q^{86} -82.1871i q^{87} +(20.3531 + 84.6483i) q^{88} +(-8.17721 - 14.1634i) q^{89} +(-27.8011 + 2.19120i) q^{90} +(-74.8007 + 43.1862i) q^{91} +(-83.5789 + 67.7126i) q^{92} +(-111.708 + 193.484i) q^{93} +(88.9591 - 61.1549i) q^{94} +(-34.5642 + 24.7046i) q^{95} +(-123.846 - 16.2324i) q^{96} +(-33.5594 + 58.1266i) q^{97} +(-51.9304 + 4.09300i) q^{98} +(58.7699 - 33.9308i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 160 q + 2 q^{4} + 6 q^{6} + 248 q^{9} - 10 q^{10} - 16 q^{13} - 14 q^{16} + 48 q^{17} + 48 q^{21} - 44 q^{24} - 400 q^{25} + 68 q^{26} + 60 q^{28} - 80 q^{30} + 30 q^{32} - 40 q^{33} - 22 q^{34} + 52 q^{36}+ \cdots - 226 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/380\mathbb{Z}\right)^\times\).

\(n\) \(21\) \(77\) \(191\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.99382 + 0.157147i −0.996908 + 0.0785735i
\(3\) 3.38036 + 1.95165i 1.12679 + 0.650551i 0.943125 − 0.332439i \(-0.107871\pi\)
0.183662 + 0.982989i \(0.441205\pi\)
\(4\) 3.95061 − 0.626645i 0.987652 − 0.156661i
\(5\) 1.11803 − 1.93649i 0.223607 − 0.387298i
\(6\) −7.04652 − 3.36002i −1.17442 − 0.560004i
\(7\) − 4.79106i − 0.684438i −0.939620 − 0.342219i \(-0.888822\pi\)
0.939620 − 0.342219i \(-0.111178\pi\)
\(8\) −7.77832 + 1.87024i −0.972290 + 0.233780i
\(9\) 3.11789 + 5.40035i 0.346433 + 0.600039i
\(10\) −1.92484 + 4.03670i −0.192484 + 0.403670i
\(11\) − 10.8826i − 0.989327i −0.869084 − 0.494664i \(-0.835291\pi\)
0.869084 − 0.494664i \(-0.164709\pi\)
\(12\) 14.5775 + 5.59193i 1.21479 + 0.465994i
\(13\) −9.01390 − 15.6125i −0.693377 − 1.20096i −0.970725 − 0.240195i \(-0.922789\pi\)
0.277348 − 0.960770i \(-0.410545\pi\)
\(14\) 0.752901 + 9.55250i 0.0537787 + 0.682322i
\(15\) 7.55872 − 4.36403i 0.503914 − 0.290935i
\(16\) 15.2146 − 4.95126i 0.950915 − 0.309453i
\(17\) 12.2766 − 21.2637i 0.722152 − 1.25080i −0.237983 − 0.971269i \(-0.576486\pi\)
0.960135 − 0.279535i \(-0.0901803\pi\)
\(18\) −7.06516 − 10.2773i −0.392509 − 0.570963i
\(19\) −17.2968 − 7.86254i −0.910360 − 0.413818i
\(20\) 3.20342 − 8.35093i 0.160171 − 0.417547i
\(21\) 9.35049 − 16.1955i 0.445262 − 0.771216i
\(22\) 1.71017 + 21.6979i 0.0777349 + 0.986269i
\(23\) −23.2887 + 13.4457i −1.01255 + 0.584598i −0.911938 − 0.410328i \(-0.865414\pi\)
−0.100615 + 0.994925i \(0.532081\pi\)
\(24\) −29.9436 − 8.85848i −1.24765 − 0.369103i
\(25\) −2.50000 − 4.33013i −0.100000 − 0.173205i
\(26\) 20.4255 + 29.7120i 0.785597 + 1.14277i
\(27\) − 10.7896i − 0.399614i
\(28\) −3.00229 − 18.9276i −0.107225 − 0.675987i
\(29\) −10.5279 − 18.2348i −0.363030 − 0.628787i 0.625428 − 0.780282i \(-0.284925\pi\)
−0.988458 + 0.151495i \(0.951591\pi\)
\(30\) −14.3849 + 9.88890i −0.479497 + 0.329630i
\(31\) 57.2377i 1.84638i 0.384348 + 0.923188i \(0.374426\pi\)
−0.384348 + 0.923188i \(0.625574\pi\)
\(32\) −29.5571 + 12.2628i −0.923660 + 0.383213i
\(33\) 21.2391 − 36.7871i 0.643608 − 1.11476i
\(34\) −21.1357 + 44.3251i −0.621640 + 1.30368i
\(35\) −9.27786 − 5.35657i −0.265082 − 0.153045i
\(36\) 15.7017 + 19.3809i 0.436158 + 0.538357i
\(37\) 70.4471 1.90398 0.951988 − 0.306136i \(-0.0990363\pi\)
0.951988 + 0.306136i \(0.0990363\pi\)
\(38\) 35.7223 + 12.9583i 0.940060 + 0.341008i
\(39\) − 70.3680i − 1.80431i
\(40\) −5.07472 + 17.1536i −0.126868 + 0.428841i
\(41\) −26.5857 + 46.0478i −0.648432 + 1.12312i 0.335066 + 0.942195i \(0.391242\pi\)
−0.983497 + 0.180922i \(0.942092\pi\)
\(42\) −16.0981 + 33.7603i −0.383288 + 0.803817i
\(43\) 24.1071 + 13.9182i 0.560630 + 0.323680i 0.753398 − 0.657564i \(-0.228413\pi\)
−0.192768 + 0.981244i \(0.561747\pi\)
\(44\) −6.81952 − 42.9929i −0.154989 − 0.977112i
\(45\) 13.9436 0.309859
\(46\) 44.3205 − 30.4681i 0.963488 − 0.662350i
\(47\) −46.7446 + 26.9880i −0.994566 + 0.574213i −0.906636 − 0.421914i \(-0.861358\pi\)
−0.0879299 + 0.996127i \(0.528025\pi\)
\(48\) 61.0941 + 12.9566i 1.27279 + 0.269930i
\(49\) 26.0457 0.531545
\(50\) 5.66501 + 8.24061i 0.113300 + 0.164812i
\(51\) 82.9986 − 47.9193i 1.62742 − 0.939593i
\(52\) −45.3939 − 56.0305i −0.872960 − 1.07751i
\(53\) −15.9654 − 27.6529i −0.301234 − 0.521753i 0.675182 − 0.737652i \(-0.264065\pi\)
−0.976416 + 0.215899i \(0.930732\pi\)
\(54\) 1.69555 + 21.5124i 0.0313990 + 0.398378i
\(55\) −21.0741 − 12.1671i −0.383165 − 0.221220i
\(56\) 8.96044 + 37.2664i 0.160008 + 0.665472i
\(57\) −43.1246 − 60.3356i −0.756572 − 1.05852i
\(58\) 23.8562 + 34.7025i 0.411314 + 0.598318i
\(59\) −8.18096 − 4.72328i −0.138660 − 0.0800556i 0.429065 − 0.903274i \(-0.358843\pi\)
−0.567725 + 0.823218i \(0.692176\pi\)
\(60\) 27.1268 − 21.9772i 0.452114 − 0.366287i
\(61\) 4.88891 + 8.46783i 0.0801460 + 0.138817i 0.903313 − 0.428983i \(-0.141128\pi\)
−0.823167 + 0.567800i \(0.807795\pi\)
\(62\) −8.99473 − 114.121i −0.145076 − 1.84067i
\(63\) 25.8734 − 14.9380i 0.410689 − 0.237112i
\(64\) 57.0044 − 29.0946i 0.890694 − 0.454604i
\(65\) −40.3114 −0.620175
\(66\) −36.5658 + 76.6844i −0.554027 + 1.16189i
\(67\) 96.6689 − 55.8118i 1.44282 − 0.833012i 0.444782 − 0.895639i \(-0.353281\pi\)
0.998037 + 0.0626268i \(0.0199478\pi\)
\(68\) 35.1752 − 91.6975i 0.517283 − 1.34849i
\(69\) −104.966 −1.52124
\(70\) 19.3401 + 9.22204i 0.276287 + 0.131743i
\(71\) 99.6655 + 57.5419i 1.40374 + 0.810449i 0.994774 − 0.102101i \(-0.0325565\pi\)
0.408965 + 0.912550i \(0.365890\pi\)
\(72\) −34.3519 − 36.1744i −0.477110 − 0.502422i
\(73\) 60.1309 − 104.150i 0.823711 − 1.42671i −0.0791895 − 0.996860i \(-0.525233\pi\)
0.902900 − 0.429850i \(-0.141433\pi\)
\(74\) −140.459 + 11.0705i −1.89809 + 0.149602i
\(75\) − 19.5165i − 0.260220i
\(76\) −73.2601 − 20.2229i −0.963948 − 0.266090i
\(77\) −52.1392 −0.677133
\(78\) 11.0581 + 140.301i 0.141771 + 1.79873i
\(79\) 86.5330 + 49.9598i 1.09535 + 0.632403i 0.934997 − 0.354656i \(-0.115402\pi\)
0.160357 + 0.987059i \(0.448735\pi\)
\(80\) 7.42241 − 34.9987i 0.0927801 − 0.437484i
\(81\) 49.1185 − 85.0758i 0.606402 − 1.05032i
\(82\) 45.7708 − 95.9887i 0.558180 − 1.17059i
\(83\) − 22.2344i − 0.267885i −0.990989 − 0.133942i \(-0.957236\pi\)
0.990989 − 0.133942i \(-0.0427637\pi\)
\(84\) 26.7913 − 69.8416i 0.318944 − 0.831448i
\(85\) −27.4513 − 47.5470i −0.322956 − 0.559377i
\(86\) −50.2523 − 23.9621i −0.584330 − 0.278629i
\(87\) − 82.1871i − 0.944679i
\(88\) 20.3531 + 84.6483i 0.231285 + 0.961913i
\(89\) −8.17721 − 14.1634i −0.0918788 − 0.159139i 0.816423 − 0.577454i \(-0.195954\pi\)
−0.908302 + 0.418316i \(0.862621\pi\)
\(90\) −27.8011 + 2.19120i −0.308901 + 0.0243467i
\(91\) −74.8007 + 43.1862i −0.821985 + 0.474573i
\(92\) −83.5789 + 67.7126i −0.908466 + 0.736007i
\(93\) −111.708 + 193.484i −1.20116 + 2.08047i
\(94\) 88.9591 − 61.1549i 0.946373 − 0.650584i
\(95\) −34.5642 + 24.7046i −0.363834 + 0.260048i
\(96\) −123.846 − 16.2324i −1.29007 − 0.169088i
\(97\) −33.5594 + 58.1266i −0.345973 + 0.599243i −0.985530 − 0.169501i \(-0.945785\pi\)
0.639557 + 0.768744i \(0.279118\pi\)
\(98\) −51.9304 + 4.09300i −0.529902 + 0.0417653i
\(99\) 58.7699 − 33.9308i 0.593635 − 0.342735i
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 380.3.q.a.11.2 ✓ 160
4.3 odd 2 inner 380.3.q.a.11.52 yes 160
19.7 even 3 inner 380.3.q.a.311.52 yes 160
76.7 odd 6 inner 380.3.q.a.311.2 yes 160
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.3.q.a.11.2 ✓ 160 1.1 even 1 trivial
380.3.q.a.11.52 yes 160 4.3 odd 2 inner
380.3.q.a.311.2 yes 160 76.7 odd 6 inner
380.3.q.a.311.52 yes 160 19.7 even 3 inner