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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [380,2,Mod(179,380)] 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("380.179"); S:= CuspForms(chi, 2); 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, 3, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 380 = 2^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 380.s (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: \(3.03431527681\)
Analytic rank: \(0\)
Dimension: \(112\)
Relative dimension: \(56\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 179.12
Character \(\chi\) \(=\) 380.179
Dual form 380.2.s.a.259.12

$q$-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+(-1.08964 + 0.901488i) q^{2} +(2.51144 + 1.44998i) q^{3} +(0.374639 - 1.96460i) q^{4} +(-0.944748 + 2.02668i) q^{5} +(-4.04370 + 0.684072i) q^{6} +2.35031 q^{7} +(1.36284 + 2.47844i) q^{8} +(2.70487 + 4.68497i) q^{9} +(-0.797594 - 3.06004i) q^{10} +0.0923348i q^{11} +(3.78950 - 4.39074i) q^{12} +(-2.35791 - 4.08402i) q^{13} +(-2.56100 + 2.11878i) q^{14} +(-5.31132 + 3.72002i) q^{15} +(-3.71929 - 1.47203i) q^{16} +(5.63305 + 3.25224i) q^{17} +(-7.17079 - 2.66654i) q^{18} +(-2.81184 + 3.33070i) q^{19} +(3.62768 + 2.61533i) q^{20} +(5.90265 + 3.40790i) q^{21} +(-0.0832388 - 0.100612i) q^{22} +(-1.83997 - 3.18692i) q^{23} +(-0.171001 + 8.20053i) q^{24} +(-3.21490 - 3.82941i) q^{25} +(6.25098 + 2.32449i) q^{26} +6.98814i q^{27} +(0.880518 - 4.61742i) q^{28} +(3.83165 - 2.21221i) q^{29} +(2.43388 - 8.84159i) q^{30} -5.93969 q^{31} +(5.37971 - 1.74891i) q^{32} +(-0.133883 + 0.231893i) q^{33} +(-9.06986 + 1.53435i) q^{34} +(-2.22045 + 4.76334i) q^{35} +(10.2174 - 3.55881i) q^{36} +6.02238 q^{37} +(0.0613109 - 6.16411i) q^{38} -13.6757i q^{39} +(-6.31056 + 0.420543i) q^{40} +(-4.92432 - 2.84306i) q^{41} +(-9.50396 + 1.60778i) q^{42} +(-0.432234 + 0.748652i) q^{43} +(0.181401 + 0.0345922i) q^{44} +(-12.0504 + 1.05580i) q^{45} +(4.87788 + 1.81389i) q^{46} +(2.36202 + 4.09114i) q^{47} +(-7.20635 - 9.08980i) q^{48} -1.47604 q^{49} +(6.95526 + 1.27450i) q^{50} +(9.43135 + 16.3356i) q^{51} +(-8.90683 + 3.10231i) q^{52} +(-1.54252 - 2.67172i) q^{53} +(-6.29973 - 7.61457i) q^{54} +(-0.187134 - 0.0872332i) q^{55} +(3.20310 + 5.82511i) q^{56} +(-11.8912 + 4.28773i) q^{57} +(-2.18085 + 5.86470i) q^{58} +(-2.32266 + 4.02297i) q^{59} +(5.31852 + 11.8283i) q^{60} +(-2.86919 - 4.96959i) q^{61} +(6.47213 - 5.35456i) q^{62} +(6.35729 + 11.0111i) q^{63} +(-4.28534 + 6.75543i) q^{64} +(10.5047 - 0.920369i) q^{65} +(-0.0631637 - 0.373375i) q^{66} +(7.96171 - 4.59669i) q^{67} +(8.49970 - 9.84825i) q^{68} -10.6717i q^{69} +(-1.87459 - 7.19204i) q^{70} +(-1.23099 + 2.13214i) q^{71} +(-7.92513 + 13.0887i) q^{72} +(8.66211 + 5.00107i) q^{73} +(-6.56223 + 5.42910i) q^{74} +(-2.52145 - 14.2789i) q^{75} +(5.49006 + 6.77194i) q^{76} +0.217016i q^{77} +(12.3285 + 14.9016i) q^{78} +(7.95547 - 13.7793i) q^{79} +(6.49713 - 6.14713i) q^{80} +(-2.01804 + 3.49535i) q^{81} +(7.92873 - 1.34130i) q^{82} -17.9514 q^{83} +(8.90651 - 10.3196i) q^{84} +(-11.9131 + 8.34386i) q^{85} +(-0.203920 - 1.20542i) q^{86} +12.8306 q^{87} +(-0.228846 + 0.125838i) q^{88} +(11.2790 - 6.51193i) q^{89} +(12.1788 - 12.0137i) q^{90} +(-5.54182 - 9.59872i) q^{91} +(-6.95035 + 2.42086i) q^{92} +(-14.9171 - 8.61241i) q^{93} +(-6.26187 - 2.32854i) q^{94} +(-4.09380 - 8.84538i) q^{95} +(16.0467 + 3.40819i) q^{96} +(7.79770 - 13.5060i) q^{97} +(1.60835 - 1.33063i) q^{98} +(-0.432586 + 0.249754i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 112 q - 2 q^{5} - 8 q^{6} + 44 q^{9} + 6 q^{10} - 36 q^{14} - 4 q^{16} + 44 q^{20} - 48 q^{21} + 2 q^{24} - 2 q^{25} - 36 q^{26} - 12 q^{29} - 32 q^{30} - 30 q^{34} + 20 q^{36} - 24 q^{40} - 24 q^{41} - 14 q^{44}+ \cdots - 84 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(21\) \(77\) \(191\)
\(\chi(n)\) \(e\left(\frac{1}{6}\right)\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.08964 + 0.901488i −0.770493 + 0.637448i
\(3\) 2.51144 + 1.44998i 1.44998 + 0.837145i 0.998479 − 0.0551269i \(-0.0175563\pi\)
0.451498 + 0.892272i \(0.350890\pi\)
\(4\) 0.374639 − 1.96460i 0.187319 − 0.982299i
\(5\) −0.944748 + 2.02668i −0.422504 + 0.906361i
\(6\) −4.04370 + 0.684072i −1.65083 + 0.279271i
\(7\) 2.35031 0.888334 0.444167 − 0.895944i \(-0.353500\pi\)
0.444167 + 0.895944i \(0.353500\pi\)
\(8\) 1.36284 + 2.47844i 0.481836 + 0.876261i
\(9\) 2.70487 + 4.68497i 0.901624 + 1.56166i
\(10\) −0.797594 − 3.06004i −0.252221 − 0.967670i
\(11\) 0.0923348i 0.0278400i 0.999903 + 0.0139200i \(0.00443102\pi\)
−0.999903 + 0.0139200i \(0.995569\pi\)
\(12\) 3.78950 − 4.39074i 1.09394 − 1.26750i
\(13\) −2.35791 − 4.08402i −0.653967 − 1.13270i −0.982152 − 0.188090i \(-0.939770\pi\)
0.328185 − 0.944614i \(-0.393563\pi\)
\(14\) −2.56100 + 2.11878i −0.684455 + 0.566267i
\(15\) −5.31132 + 3.72002i −1.37138 + 0.960506i
\(16\) −3.71929 − 1.47203i −0.929823 − 0.368007i
\(17\) 5.63305 + 3.25224i 1.36621 + 0.788784i 0.990442 − 0.137928i \(-0.0440443\pi\)
0.375772 + 0.926712i \(0.377378\pi\)
\(18\) −7.17079 − 2.66654i −1.69017 − 0.628508i
\(19\) −2.81184 + 3.33070i −0.645080 + 0.764115i
\(20\) 3.62768 + 2.61533i 0.811174 + 0.584805i
\(21\) 5.90265 + 3.40790i 1.28806 + 0.743664i
\(22\) −0.0832388 − 0.100612i −0.0177466 − 0.0214505i
\(23\) −1.83997 − 3.18692i −0.383661 − 0.664520i 0.607922 − 0.793997i \(-0.292003\pi\)
−0.991582 + 0.129477i \(0.958670\pi\)
\(24\) −0.171001 + 8.20053i −0.0349055 + 1.67393i
\(25\) −3.21490 − 3.82941i −0.642980 − 0.765883i
\(26\) 6.25098 + 2.32449i 1.22592 + 0.455871i
\(27\) 6.98814i 1.34487i
\(28\) 0.880518 − 4.61742i 0.166402 − 0.872610i
\(29\) 3.83165 − 2.21221i 0.711520 − 0.410796i −0.100103 − 0.994977i \(-0.531917\pi\)
0.811624 + 0.584181i \(0.198584\pi\)
\(30\) 2.43388 − 8.84159i 0.444364 − 1.61425i
\(31\) −5.93969 −1.06680 −0.533399 − 0.845863i \(-0.679086\pi\)
−0.533399 + 0.845863i \(0.679086\pi\)
\(32\) 5.37971 − 1.74891i 0.951008 − 0.309167i
\(33\) −0.133883 + 0.231893i −0.0233061 + 0.0403674i
\(34\) −9.06986 + 1.53435i −1.55547 + 0.263138i
\(35\) −2.22045 + 4.76334i −0.375325 + 0.805151i
\(36\) 10.2174 − 3.55881i 1.70291 − 0.593135i
\(37\) 6.02238 0.990073 0.495036 − 0.868872i \(-0.335155\pi\)
0.495036 + 0.868872i \(0.335155\pi\)
\(38\) 0.0613109 − 6.16411i 0.00994595 − 0.999951i
\(39\) − 13.6757i − 2.18986i
\(40\) −6.31056 + 0.420543i −0.997787 + 0.0664936i
\(41\) −4.92432 − 2.84306i −0.769050 − 0.444011i 0.0634858 − 0.997983i \(-0.479778\pi\)
−0.832535 + 0.553972i \(0.813112\pi\)
\(42\) −9.50396 + 1.60778i −1.46649 + 0.248086i
\(43\) −0.432234 + 0.748652i −0.0659151 + 0.114168i −0.897100 − 0.441828i \(-0.854330\pi\)
0.831184 + 0.555997i \(0.187663\pi\)
\(44\) 0.181401 + 0.0345922i 0.0273472 + 0.00521497i
\(45\) −12.0504 + 1.05580i −1.79637 + 0.157389i
\(46\) 4.87788 + 1.81389i 0.719205 + 0.267444i
\(47\) 2.36202 + 4.09114i 0.344536 + 0.596754i 0.985269 − 0.171010i \(-0.0547030\pi\)
−0.640733 + 0.767763i \(0.721370\pi\)
\(48\) −7.20635 − 9.08980i −1.04015 − 1.31200i
\(49\) −1.47604 −0.210863
\(50\) 6.95526 + 1.27450i 0.983623 + 0.180241i
\(51\) 9.43135 + 16.3356i 1.32065 + 2.28744i
\(52\) −8.90683 + 3.10231i −1.23515 + 0.430214i
\(53\) −1.54252 − 2.67172i −0.211881 − 0.366989i 0.740422 − 0.672142i \(-0.234626\pi\)
−0.952303 + 0.305153i \(0.901292\pi\)
\(54\) −6.29973 − 7.61457i −0.857284 − 1.03621i
\(55\) −0.187134 − 0.0872332i −0.0252331 − 0.0117625i
\(56\) 3.20310 + 5.82511i 0.428032 + 0.778413i
\(57\) −11.8912 + 4.28773i −1.57503 + 0.567924i
\(58\) −2.18085 + 5.86470i −0.286360 + 0.770073i
\(59\) −2.32266 + 4.02297i −0.302385 + 0.523746i −0.976676 − 0.214720i \(-0.931116\pi\)
0.674291 + 0.738466i \(0.264450\pi\)
\(60\) 5.31852 + 11.8283i 0.686618 + 1.52702i
\(61\) −2.86919 − 4.96959i −0.367362 − 0.636290i 0.621790 − 0.783184i \(-0.286406\pi\)
−0.989152 + 0.146894i \(0.953072\pi\)
\(62\) 6.47213 − 5.35456i 0.821961 − 0.680029i
\(63\) 6.35729 + 11.0111i 0.800943 + 1.38727i
\(64\) −4.28534 + 6.75543i −0.535667 + 0.844429i
\(65\) 10.5047 − 0.920369i 1.30294 − 0.114158i
\(66\) −0.0631637 − 0.373375i −0.00777491 − 0.0459592i
\(67\) 7.96171 − 4.59669i 0.972677 − 0.561576i 0.0726260 − 0.997359i \(-0.476862\pi\)
0.900051 + 0.435784i \(0.143529\pi\)
\(68\) 8.49970 − 9.84825i 1.03074 − 1.19428i
\(69\) − 10.6717i − 1.28472i
\(70\) −1.87459 − 7.19204i −0.224057 − 0.859614i
\(71\) −1.23099 + 2.13214i −0.146092 + 0.253039i −0.929780 − 0.368116i \(-0.880003\pi\)
0.783688 + 0.621155i \(0.213336\pi\)
\(72\) −7.92513 + 13.0887i −0.933985 + 1.54252i
\(73\) 8.66211 + 5.00107i 1.01382 + 0.585331i 0.912309 − 0.409503i \(-0.134298\pi\)
0.101515 + 0.994834i \(0.467631\pi\)
\(74\) −6.56223 + 5.42910i −0.762844 + 0.631120i
\(75\) −2.52145 − 14.2789i −0.291152 − 1.64878i
\(76\) 5.49006 + 6.77194i 0.629753 + 0.776795i
\(77\) 0.217016i 0.0247312i
\(78\) 12.3285 + 14.9016i 1.39592 + 1.68727i
\(79\) 7.95547 − 13.7793i 0.895060 − 1.55029i 0.0613305 − 0.998118i \(-0.480466\pi\)
0.833730 − 0.552173i \(-0.186201\pi\)
\(80\) 6.49713 − 6.14713i 0.726402 − 0.687270i
\(81\) −2.01804 + 3.49535i −0.224227 + 0.388372i
\(82\) 7.92873 − 1.34130i 0.875582 − 0.148122i
\(83\) −17.9514 −1.97042 −0.985209 − 0.171358i \(-0.945185\pi\)
−0.985209 + 0.171358i \(0.945185\pi\)
\(84\) 8.90651 − 10.3196i 0.971780 − 1.12596i
\(85\) −11.9131 + 8.34386i −1.29215 + 0.905018i
\(86\) −0.203920 − 1.20542i −0.0219893 − 0.129983i
\(87\) 12.8306 1.37558
\(88\) −0.228846 + 0.125838i −0.0243951 + 0.0134143i
\(89\) 11.2790 − 6.51193i 1.19557 − 0.690264i 0.236007 − 0.971751i \(-0.424161\pi\)
0.959565 + 0.281488i \(0.0908279\pi\)
\(90\) 12.1788 − 12.0137i 1.28376 − 1.26636i
\(91\) −5.54182 − 9.59872i −0.580941 − 1.00622i
\(92\) −6.95035 + 2.42086i −0.724624 + 0.252392i
\(93\) −14.9171 − 8.61241i −1.54683 − 0.893065i
\(94\) −6.26187 − 2.32854i −0.645862 − 0.240171i
\(95\) −4.09380 − 8.84538i −0.420015 − 0.907517i
\(96\) 16.0467 + 3.40819i 1.63776 + 0.347847i
\(97\) 7.79770 − 13.5060i 0.791737 − 1.37133i −0.133154 − 0.991095i \(-0.542510\pi\)
0.924891 − 0.380233i \(-0.124156\pi\)
\(98\) 1.60835 − 1.33063i 0.162468 − 0.134414i
\(99\) −0.432586 + 0.249754i −0.0434766 + 0.0251012i
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 380.2.s.a.179.12 ✓ 112
4.3 odd 2 inner 380.2.s.a.179.26 yes 112
5.4 even 2 inner 380.2.s.a.179.45 yes 112
19.12 odd 6 inner 380.2.s.a.259.31 yes 112
20.19 odd 2 inner 380.2.s.a.179.31 yes 112
76.31 even 6 inner 380.2.s.a.259.45 yes 112
95.69 odd 6 inner 380.2.s.a.259.26 yes 112
380.259 even 6 inner 380.2.s.a.259.12 yes 112
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.2.s.a.179.12 ✓ 112 1.1 even 1 trivial
380.2.s.a.179.26 yes 112 4.3 odd 2 inner
380.2.s.a.179.31 yes 112 20.19 odd 2 inner
380.2.s.a.179.45 yes 112 5.4 even 2 inner
380.2.s.a.259.12 yes 112 380.259 even 6 inner
380.2.s.a.259.26 yes 112 95.69 odd 6 inner
380.2.s.a.259.31 yes 112 19.12 odd 6 inner
380.2.s.a.259.45 yes 112 76.31 even 6 inner