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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [380,2,Mod(7,380)] 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("380.7"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(380, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([6, 3, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 380 = 2^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 380.v (of order \(12\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [216,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.03431527681\)
Analytic rank: \(0\)
Dimension: \(216\)
Relative dimension: \(54\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label 163.48
Character \(\chi\) \(=\) 380.163
Dual form 380.2.v.c.7.48

$q$-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+(1.32283 - 0.500120i) q^{2} +(0.00674551 + 0.0251746i) q^{3} +(1.49976 - 1.32315i) q^{4} +(-2.17162 - 0.532967i) q^{5} +(0.0215135 + 0.0299281i) q^{6} +(3.02167 + 3.02167i) q^{7} +(1.32219 - 2.50036i) q^{8} +(2.59749 - 1.49966i) q^{9} +(-3.13924 + 0.381048i) q^{10} -5.59169i q^{11} +(0.0434264 + 0.0288305i) q^{12} +(-1.20316 + 4.49027i) q^{13} +(5.50835 + 2.48596i) q^{14} +(-0.00123148 - 0.0582648i) q^{15} +(0.498557 - 3.96881i) q^{16} +(0.369102 + 1.37751i) q^{17} +(2.68602 - 3.28285i) q^{18} +(1.63762 - 4.03958i) q^{19} +(-3.96211 + 2.07406i) q^{20} +(-0.0556865 + 0.0964519i) q^{21} +(-2.79652 - 7.39686i) q^{22} +(-3.40135 - 0.911389i) q^{23} +(0.0718644 + 0.0164195i) q^{24} +(4.43189 + 2.31481i) q^{25} +(0.654093 + 6.54159i) q^{26} +(0.110562 + 0.110562i) q^{27} +(8.52989 + 0.533660i) q^{28} +(-6.80095 + 3.92653i) q^{29} +(-0.0307685 - 0.0764586i) q^{30} +4.44156i q^{31} +(-1.32538 - 5.49940i) q^{32} +(0.140768 - 0.0377188i) q^{33} +(1.17718 + 1.63761i) q^{34} +(-4.95147 - 8.17237i) q^{35} +(1.91133 - 5.68599i) q^{36} +(0.0225777 - 0.0225777i) q^{37} +(0.146011 - 6.16268i) q^{38} -0.121157 q^{39} +(-4.20392 + 4.72516i) q^{40} +(-4.62151 + 8.00469i) q^{41} +(-0.0254263 + 0.155439i) q^{42} +(1.31048 + 4.89080i) q^{43} +(-7.39864 - 8.38619i) q^{44} +(-6.44003 + 1.87232i) q^{45} +(-4.95521 + 0.495471i) q^{46} +(0.998551 - 3.72664i) q^{47} +(0.103276 - 0.0142207i) q^{48} +11.2609i q^{49} +(7.02032 + 0.845617i) q^{50} +(-0.0321884 + 0.0185840i) q^{51} +(4.13684 + 8.32629i) q^{52} +(-1.82051 + 6.79424i) q^{53} +(0.201549 + 0.0909604i) q^{54} +(-2.98019 + 12.1430i) q^{55} +(11.5505 - 3.56003i) q^{56} +(0.112741 + 0.0139773i) q^{57} +(-7.03276 + 8.59543i) q^{58} +(-2.46104 + 4.26265i) q^{59} +(-0.0789400 - 0.0857538i) q^{60} +(-3.05728 - 5.29537i) q^{61} +(2.22132 + 5.87543i) q^{62} +(12.3802 + 3.31727i) q^{63} +(-4.50361 - 6.61192i) q^{64} +(5.00599 - 9.10993i) q^{65} +(0.167349 - 0.120297i) q^{66} +(1.31656 - 4.91348i) q^{67} +(2.37621 + 1.57755i) q^{68} -0.0917753i q^{69} +(-10.6371 - 8.33433i) q^{70} +(1.45766 + 0.841583i) q^{71} +(-0.315310 - 8.47750i) q^{72} +(-8.63579 + 2.31395i) q^{73} +(0.0185749 - 0.0411581i) q^{74} +(-0.0283789 + 0.127186i) q^{75} +(-2.88894 - 8.22521i) q^{76} +(16.8962 - 16.8962i) q^{77} +(-0.160270 + 0.0605929i) q^{78} +(-0.107623 + 0.186408i) q^{79} +(-3.19792 + 8.35304i) q^{80} +(4.49694 - 7.78893i) q^{81} +(-2.11016 + 12.9002i) q^{82} +(-1.16162 + 1.16162i) q^{83} +(0.0441038 + 0.218336i) q^{84} +(-0.0673843 - 3.18815i) q^{85} +(4.17953 + 5.81429i) q^{86} +(-0.144725 - 0.144725i) q^{87} +(-13.9812 - 7.39330i) q^{88} +(4.77522 - 2.75698i) q^{89} +(-7.58269 + 5.69755i) q^{90} +(-17.2037 + 9.93254i) q^{91} +(-6.30711 + 3.13363i) q^{92} +(-0.111815 + 0.0299606i) q^{93} +(-0.542857 - 5.42911i) q^{94} +(-5.70925 + 7.89965i) q^{95} +(0.129505 - 0.0704620i) q^{96} +(-0.845041 - 3.15374i) q^{97} +(5.63183 + 14.8963i) q^{98} +(-8.38564 - 14.5243i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 216 q + 12 q^{6} - 36 q^{8} + 4 q^{10} - 20 q^{12} - 8 q^{13} + 8 q^{16} - 16 q^{17} + 12 q^{18} - 48 q^{20} + 40 q^{21} + 16 q^{22} - 16 q^{25} + 24 q^{26} + 8 q^{28} - 12 q^{30} - 20 q^{32} + 20 q^{33}+ \cdots + 34 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(21\) \(77\) \(191\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{3}{4}\right)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.32283 − 0.500120i 0.935382 − 0.353638i
\(3\) 0.00674551 + 0.0251746i 0.00389452 + 0.0145346i 0.967846 − 0.251544i \(-0.0809385\pi\)
−0.963951 + 0.266079i \(0.914272\pi\)
\(4\) 1.49976 − 1.32315i 0.749880 − 0.661574i
\(5\) −2.17162 − 0.532967i −0.971179 − 0.238350i
\(6\) 0.0215135 + 0.0299281i 0.00878284 + 0.0122181i
\(7\) 3.02167 + 3.02167i 1.14208 + 1.14208i 0.988069 + 0.154014i \(0.0492201\pi\)
0.154014 + 0.988069i \(0.450780\pi\)
\(8\) 1.32219 − 2.50036i 0.467466 − 0.884011i
\(9\) 2.59749 − 1.49966i 0.865829 − 0.499887i
\(10\) −3.13924 + 0.381048i −0.992714 + 0.120498i
\(11\) − 5.59169i − 1.68596i −0.537946 − 0.842979i \(-0.680800\pi\)
0.537946 − 0.842979i \(-0.319200\pi\)
\(12\) 0.0434264 + 0.0288305i 0.0125361 + 0.00832265i
\(13\) −1.20316 + 4.49027i −0.333698 + 1.24538i 0.571576 + 0.820549i \(0.306332\pi\)
−0.905274 + 0.424828i \(0.860335\pi\)
\(14\) 5.50835 + 2.48596i 1.47217 + 0.664400i
\(15\) −0.00123148 − 0.0582648i −0.000317966 − 0.0150439i
\(16\) 0.498557 − 3.96881i 0.124639 − 0.992202i
\(17\) 0.369102 + 1.37751i 0.0895204 + 0.334095i 0.996132 − 0.0878719i \(-0.0280066\pi\)
−0.906611 + 0.421967i \(0.861340\pi\)
\(18\) 2.68602 − 3.28285i 0.633102 − 0.773776i
\(19\) 1.63762 − 4.03958i 0.375695 − 0.926743i
\(20\) −3.96211 + 2.07406i −0.885954 + 0.463773i
\(21\) −0.0556865 + 0.0964519i −0.0121518 + 0.0210475i
\(22\) −2.79652 − 7.39686i −0.596220 − 1.57702i
\(23\) −3.40135 − 0.911389i −0.709230 − 0.190038i −0.113869 − 0.993496i \(-0.536324\pi\)
−0.595361 + 0.803458i \(0.702991\pi\)
\(24\) 0.0718644 + 0.0164195i 0.0146693 + 0.00335161i
\(25\) 4.43189 + 2.31481i 0.886378 + 0.462961i
\(26\) 0.654093 + 6.54159i 0.128278 + 1.28291i
\(27\) 0.110562 + 0.110562i 0.0212777 + 0.0212777i
\(28\) 8.52989 + 0.533660i 1.61200 + 0.100852i
\(29\) −6.80095 + 3.92653i −1.26290 + 0.729138i −0.973636 − 0.228109i \(-0.926746\pi\)
−0.289269 + 0.957248i \(0.593412\pi\)
\(30\) −0.0307685 − 0.0764586i −0.00561753 − 0.0139594i
\(31\) 4.44156i 0.797728i 0.917010 + 0.398864i \(0.130595\pi\)
−0.917010 + 0.398864i \(0.869405\pi\)
\(32\) −1.32538 − 5.49940i −0.234296 − 0.972165i
\(33\) 0.140768 − 0.0377188i 0.0245046 − 0.00656600i
\(34\) 1.17718 + 1.63761i 0.201885 + 0.280848i
\(35\) −4.95147 − 8.17237i −0.836952 − 1.38138i
\(36\) 1.91133 − 5.68599i 0.318556 − 0.947665i
\(37\) 0.0225777 − 0.0225777i 0.00371176 − 0.00371176i −0.705248 − 0.708960i \(-0.749165\pi\)
0.708960 + 0.705248i \(0.249165\pi\)
\(38\) 0.146011 − 6.16268i 0.0236861 − 0.999719i
\(39\) −0.121157 −0.0194006
\(40\) −4.20392 + 4.72516i −0.664697 + 0.747113i
\(41\) −4.62151 + 8.00469i −0.721758 + 1.25012i 0.238536 + 0.971134i \(0.423332\pi\)
−0.960294 + 0.278989i \(0.910001\pi\)
\(42\) −0.0254263 + 0.155439i −0.00392336 + 0.0239848i
\(43\) 1.31048 + 4.89080i 0.199847 + 0.745839i 0.990959 + 0.134168i \(0.0428360\pi\)
−0.791112 + 0.611672i \(0.790497\pi\)
\(44\) −7.39864 − 8.38619i −1.11539 − 1.26427i
\(45\) −6.44003 + 1.87232i −0.960024 + 0.279109i
\(46\) −4.95521 + 0.495471i −0.730606 + 0.0730532i
\(47\) 0.998551 − 3.72664i 0.145654 − 0.543587i −0.854072 − 0.520155i \(-0.825874\pi\)
0.999725 − 0.0234317i \(-0.00745923\pi\)
\(48\) 0.103276 − 0.0142207i 0.0149066 − 0.00205258i
\(49\) 11.2609i 1.60871i
\(50\) 7.02032 + 0.845617i 0.992824 + 0.119588i
\(51\) −0.0321884 + 0.0185840i −0.00450728 + 0.00260228i
\(52\) 4.13684 + 8.32629i 0.573676 + 1.15465i
\(53\) −1.82051 + 6.79424i −0.250066 + 0.933261i 0.720702 + 0.693245i \(0.243819\pi\)
−0.970769 + 0.240016i \(0.922847\pi\)
\(54\) 0.201549 + 0.0909604i 0.0274273 + 0.0123781i
\(55\) −2.98019 + 12.1430i −0.401848 + 1.63737i
\(56\) 11.5505 − 3.56003i 1.54350 − 0.475729i
\(57\) 0.112741 + 0.0139773i 0.0149330 + 0.00185133i
\(58\) −7.03276 + 8.59543i −0.923447 + 1.12863i
\(59\) −2.46104 + 4.26265i −0.320400 + 0.554949i −0.980571 − 0.196167i \(-0.937151\pi\)
0.660171 + 0.751116i \(0.270484\pi\)
\(60\) −0.0789400 − 0.0857538i −0.0101911 − 0.0110708i
\(61\) −3.05728 − 5.29537i −0.391445 − 0.678003i 0.601195 − 0.799102i \(-0.294691\pi\)
−0.992640 + 0.121099i \(0.961358\pi\)
\(62\) 2.22132 + 5.87543i 0.282107 + 0.746181i
\(63\) 12.3802 + 3.31727i 1.55976 + 0.417937i
\(64\) −4.50361 − 6.61192i −0.562951 − 0.826490i
\(65\) 5.00599 − 9.10993i 0.620916 − 1.12995i
\(66\) 0.167349 − 0.120297i 0.0205992 − 0.0148075i
\(67\) 1.31656 − 4.91348i 0.160844 − 0.600277i −0.837690 − 0.546146i \(-0.816094\pi\)
0.998534 − 0.0541310i \(-0.0172389\pi\)
\(68\) 2.37621 + 1.57755i 0.288158 + 0.191306i
\(69\) − 0.0917753i − 0.0110485i
\(70\) −10.6371 − 8.33433i −1.27138 − 0.996143i
\(71\) 1.45766 + 0.841583i 0.172993 + 0.0998776i 0.583996 − 0.811756i \(-0.301488\pi\)
−0.411003 + 0.911634i \(0.634822\pi\)
\(72\) −0.315310 − 8.47750i −0.0371596 − 0.999083i
\(73\) −8.63579 + 2.31395i −1.01074 + 0.270828i −0.725940 − 0.687758i \(-0.758595\pi\)
−0.284803 + 0.958586i \(0.591928\pi\)
\(74\) 0.0185749 − 0.0411581i 0.00215929 − 0.00478453i
\(75\) −0.0283789 + 0.127186i −0.00327692 + 0.0146861i
\(76\) −2.88894 − 8.22521i −0.331384 − 0.943496i
\(77\) 16.8962 − 16.8962i 1.92550 − 1.92550i
\(78\) −0.160270 + 0.0605929i −0.0181470 + 0.00686080i
\(79\) −0.107623 + 0.186408i −0.0121085 + 0.0209725i −0.872016 − 0.489477i \(-0.837188\pi\)
0.859908 + 0.510450i \(0.170521\pi\)
\(80\) −3.19792 + 8.35304i −0.357538 + 0.933898i
\(81\) 4.49694 − 7.78893i 0.499660 − 0.865437i
\(82\) −2.11016 + 12.9002i −0.233029 + 1.42458i
\(83\) −1.16162 + 1.16162i −0.127505 + 0.127505i −0.767979 − 0.640475i \(-0.778738\pi\)
0.640475 + 0.767979i \(0.278738\pi\)
\(84\) 0.0441038 + 0.218336i 0.00481212 + 0.0238224i
\(85\) −0.0673843 − 3.18815i −0.00730885 − 0.345803i
\(86\) 4.17953 + 5.81429i 0.450691 + 0.626971i
\(87\) −0.144725 − 0.144725i −0.0155161 − 0.0155161i
\(88\) −13.9812 − 7.39330i −1.49041 − 0.788128i
\(89\) 4.77522 − 2.75698i 0.506173 − 0.292239i −0.225086 − 0.974339i \(-0.572266\pi\)
0.731259 + 0.682100i \(0.238933\pi\)
\(90\) −7.58269 + 5.69755i −0.799285 + 0.600575i
\(91\) −17.2037 + 9.93254i −1.80343 + 1.04121i
\(92\) −6.30711 + 3.13363i −0.657561 + 0.326703i
\(93\) −0.111815 + 0.0299606i −0.0115946 + 0.00310677i
\(94\) −0.542857 − 5.42911i −0.0559914 − 0.559970i
\(95\) −5.70925 + 7.89965i −0.585756 + 0.810487i
\(96\) 0.129505 − 0.0704620i 0.0132175 − 0.00719150i
\(97\) −0.845041 − 3.15374i −0.0858009 − 0.320213i 0.909664 − 0.415345i \(-0.136339\pi\)
−0.995465 + 0.0951321i \(0.969673\pi\)
\(98\) 5.63183 + 14.8963i 0.568900 + 1.50475i
\(99\) −8.38564 − 14.5243i −0.842788 − 1.45975i
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.v.c.163.48 yes 216
4.3 odd 2 inner 380.2.v.c.163.3 yes 216
5.2 odd 4 inner 380.2.v.c.87.34 yes 216
19.7 even 3 inner 380.2.v.c.83.25 yes 216
20.7 even 4 inner 380.2.v.c.87.25 yes 216
76.7 odd 6 inner 380.2.v.c.83.34 yes 216
95.7 odd 12 inner 380.2.v.c.7.3 ✓ 216
380.7 even 12 inner 380.2.v.c.7.48 yes 216
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.2.v.c.7.3 ✓ 216 95.7 odd 12 inner
380.2.v.c.7.48 yes 216 380.7 even 12 inner
380.2.v.c.83.25 yes 216 19.7 even 3 inner
380.2.v.c.83.34 yes 216 76.7 odd 6 inner
380.2.v.c.87.25 yes 216 20.7 even 4 inner
380.2.v.c.87.34 yes 216 5.2 odd 4 inner
380.2.v.c.163.3 yes 216 4.3 odd 2 inner
380.2.v.c.163.48 yes 216 1.1 even 1 trivial