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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [380,2,Mod(13,380)] 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("380.13"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(380, base_ring=CyclotomicField(36)) chi = DirichletCharacter(H, H._module([0, 27, 10])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 380 = 2^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 380.bh (of order \(36\), degree \(12\), 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: \(120\)
Relative dimension: \(10\) over \(\Q(\zeta_{36})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{36}]$

Embedding invariants

Embedding label 53.3
Character \(\chi\) \(=\) 380.53
Dual form 380.2.bh.a.337.3

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.46147 - 1.02333i) q^{3} +(2.21130 - 0.331920i) q^{5} +(0.711084 + 2.65380i) q^{7} +(0.0626300 + 0.172074i) q^{9} +(2.84045 + 4.91980i) q^{11} +(-1.31501 - 1.87803i) q^{13} +(-3.57141 - 1.77780i) q^{15} +(3.38777 - 1.57974i) q^{17} +(3.00247 - 3.15994i) q^{19} +(1.67650 - 4.60614i) q^{21} +(-0.294424 - 3.36528i) q^{23} +(4.77966 - 1.46795i) q^{25} +(-1.30074 + 4.85444i) q^{27} +(1.09210 - 0.397492i) q^{29} +(7.63721 + 4.40934i) q^{31} +(0.883363 - 10.0969i) q^{33} +(2.45327 + 5.63232i) q^{35} +(-3.85089 - 3.85089i) q^{37} +4.09038i q^{39} +(-2.15157 + 0.379380i) q^{41} +(-1.44809 - 0.126691i) q^{43} +(0.195608 + 0.359719i) q^{45} +(0.0130497 - 0.0279851i) q^{47} +(-0.474851 + 0.274156i) q^{49} +(-6.56773 - 1.15807i) q^{51} +(-1.23364 + 0.107930i) q^{53} +(7.91405 + 9.93633i) q^{55} +(-7.62170 + 1.54564i) q^{57} +(-6.00753 - 2.18656i) q^{59} +(8.16469 + 6.85099i) q^{61} +(-0.412116 + 0.288567i) q^{63} +(-3.53122 - 3.71639i) q^{65} +(-12.5590 - 5.85638i) q^{67} +(-3.01352 + 5.21957i) q^{69} +(-7.74802 - 9.23373i) q^{71} +(-7.08552 + 10.1192i) q^{73} +(-8.48754 - 2.74583i) q^{75} +(-11.0364 + 11.0364i) q^{77} +(2.15191 + 12.2041i) q^{79} +(7.28954 - 6.11665i) q^{81} +(-11.2432 + 3.01260i) q^{83} +(6.96701 - 4.61774i) q^{85} +(-2.00284 - 0.536661i) q^{87} +(-1.92434 + 10.9135i) q^{89} +(4.04883 - 4.82520i) q^{91} +(-6.64934 - 14.2596i) q^{93} +(5.59049 - 7.98413i) q^{95} +(4.90928 + 10.5280i) q^{97} +(-0.668674 + 0.796895i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 120 q + 6 q^{7} + 18 q^{15} - 18 q^{17} + 48 q^{21} - 36 q^{23} - 24 q^{25} - 60 q^{33} - 18 q^{35} - 12 q^{41} - 36 q^{43} + 18 q^{45} - 24 q^{47} + 96 q^{51} - 18 q^{53} + 72 q^{55} - 6 q^{57} - 24 q^{61}+ \cdots - 60 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(21\) \(77\) \(191\)
\(\chi(n)\) \(e\left(\frac{11}{18}\right)\) \(e\left(\frac{3}{4}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.46147 − 1.02333i −0.843782 − 0.590823i 0.0697615 − 0.997564i \(-0.477776\pi\)
−0.913543 + 0.406741i \(0.866665\pi\)
\(4\) 0 0
\(5\) 2.21130 − 0.331920i 0.988922 − 0.148439i
\(6\) 0 0
\(7\) 0.711084 + 2.65380i 0.268765 + 1.00304i 0.959906 + 0.280323i \(0.0904417\pi\)
−0.691141 + 0.722720i \(0.742892\pi\)
\(8\) 0 0
\(9\) 0.0626300 + 0.172074i 0.0208767 + 0.0573581i
\(10\) 0 0
\(11\) 2.84045 + 4.91980i 0.856427 + 1.48337i 0.875315 + 0.483553i \(0.160654\pi\)
−0.0188880 + 0.999822i \(0.506013\pi\)
\(12\) 0 0
\(13\) −1.31501 − 1.87803i −0.364717 − 0.520870i 0.594290 − 0.804251i \(-0.297433\pi\)
−0.959008 + 0.283380i \(0.908544\pi\)
\(14\) 0 0
\(15\) −3.57141 − 1.77780i −0.922135 − 0.459027i
\(16\) 0 0
\(17\) 3.38777 − 1.57974i 0.821654 − 0.383144i 0.0341223 − 0.999418i \(-0.489136\pi\)
0.787532 + 0.616274i \(0.211359\pi\)
\(18\) 0 0
\(19\) 3.00247 − 3.15994i 0.688813 − 0.724939i
\(20\) 0 0
\(21\) 1.67650 − 4.60614i 0.365842 − 1.00514i
\(22\) 0 0
\(23\) −0.294424 − 3.36528i −0.0613917 − 0.701710i −0.963061 − 0.269284i \(-0.913213\pi\)
0.901669 − 0.432426i \(-0.142342\pi\)
\(24\) 0 0
\(25\) 4.77966 − 1.46795i 0.955932 − 0.293589i
\(26\) 0 0
\(27\) −1.30074 + 4.85444i −0.250328 + 0.934237i
\(28\) 0 0
\(29\) 1.09210 − 0.397492i 0.202798 − 0.0738125i −0.238624 − 0.971112i \(-0.576696\pi\)
0.441422 + 0.897300i \(0.354474\pi\)
\(30\) 0 0
\(31\) 7.63721 + 4.40934i 1.37168 + 0.791942i 0.991140 − 0.132822i \(-0.0424037\pi\)
0.380543 + 0.924763i \(0.375737\pi\)
\(32\) 0 0
\(33\) 0.883363 − 10.0969i 0.153774 − 1.75764i
\(34\) 0 0
\(35\) 2.45327 + 5.63232i 0.414678 + 0.952036i
\(36\) 0 0
\(37\) −3.85089 − 3.85089i −0.633083 − 0.633083i 0.315757 − 0.948840i \(-0.397742\pi\)
−0.948840 + 0.315757i \(0.897742\pi\)
\(38\) 0 0
\(39\) 4.09038i 0.654984i
\(40\) 0 0
\(41\) −2.15157 + 0.379380i −0.336019 + 0.0592491i −0.339112 − 0.940746i \(-0.610127\pi\)
0.00309324 + 0.999995i \(0.499015\pi\)
\(42\) 0 0
\(43\) −1.44809 − 0.126691i −0.220832 − 0.0193203i −0.0237970 − 0.999717i \(-0.507576\pi\)
−0.197035 + 0.980397i \(0.563131\pi\)
\(44\) 0 0
\(45\) 0.195608 + 0.359719i 0.0291596 + 0.0536238i
\(46\) 0 0
\(47\) 0.0130497 − 0.0279851i 0.00190349 − 0.00408204i −0.905354 − 0.424658i \(-0.860394\pi\)
0.907257 + 0.420576i \(0.138172\pi\)
\(48\) 0 0
\(49\) −0.474851 + 0.274156i −0.0678359 + 0.0391651i
\(50\) 0 0
\(51\) −6.56773 − 1.15807i −0.919667 − 0.162162i
\(52\) 0 0
\(53\) −1.23364 + 0.107930i −0.169454 + 0.0148253i −0.171567 − 0.985172i \(-0.554883\pi\)
0.00211296 + 0.999998i \(0.499327\pi\)
\(54\) 0 0
\(55\) 7.91405 + 9.93633i 1.06713 + 1.33981i
\(56\) 0 0
\(57\) −7.62170 + 1.54564i −1.00952 + 0.204724i
\(58\) 0 0
\(59\) −6.00753 − 2.18656i −0.782114 − 0.284666i −0.0800600 − 0.996790i \(-0.525511\pi\)
−0.702054 + 0.712124i \(0.747733\pi\)
\(60\) 0 0
\(61\) 8.16469 + 6.85099i 1.04538 + 0.877179i 0.992600 − 0.121427i \(-0.0387472\pi\)
0.0527806 + 0.998606i \(0.483192\pi\)
\(62\) 0 0
\(63\) −0.412116 + 0.288567i −0.0519218 + 0.0363560i
\(64\) 0 0
\(65\) −3.53122 − 3.71639i −0.437994 − 0.460962i
\(66\) 0 0
\(67\) −12.5590 − 5.85638i −1.53433 − 0.715471i −0.542151 − 0.840281i \(-0.682390\pi\)
−0.992181 + 0.124810i \(0.960168\pi\)
\(68\) 0 0
\(69\) −3.01352 + 5.21957i −0.362785 + 0.628362i
\(70\) 0 0
\(71\) −7.74802 − 9.23373i −0.919521 − 1.09584i −0.995117 − 0.0987040i \(-0.968530\pi\)
0.0755958 − 0.997139i \(-0.475914\pi\)
\(72\) 0 0
\(73\) −7.08552 + 10.1192i −0.829298 + 1.18436i 0.151331 + 0.988483i \(0.451644\pi\)
−0.980629 + 0.195877i \(0.937245\pi\)
\(74\) 0 0
\(75\) −8.48754 − 2.74583i −0.980057 − 0.317061i
\(76\) 0 0
\(77\) −11.0364 + 11.0364i −1.25771 + 1.25771i
\(78\) 0 0
\(79\) 2.15191 + 12.2041i 0.242109 + 1.37307i 0.827111 + 0.562038i \(0.189983\pi\)
−0.585002 + 0.811032i \(0.698906\pi\)
\(80\) 0 0
\(81\) 7.28954 − 6.11665i 0.809949 − 0.679628i
\(82\) 0 0
\(83\) −11.2432 + 3.01260i −1.23410 + 0.330676i −0.816174 − 0.577806i \(-0.803909\pi\)
−0.417925 + 0.908482i \(0.637242\pi\)
\(84\) 0 0
\(85\) 6.96701 − 4.61774i 0.755678 − 0.500864i
\(86\) 0 0
\(87\) −2.00284 − 0.536661i −0.214727 − 0.0575361i
\(88\) 0 0
\(89\) −1.92434 + 10.9135i −0.203980 + 1.15683i 0.695056 + 0.718955i \(0.255379\pi\)
−0.899037 + 0.437874i \(0.855732\pi\)
\(90\) 0 0
\(91\) 4.04883 − 4.82520i 0.424432 − 0.505819i
\(92\) 0 0
\(93\) −6.64934 − 14.2596i −0.689505 − 1.47865i
\(94\) 0 0
\(95\) 5.59049 − 7.98413i 0.573573 − 0.819155i
\(96\) 0 0
\(97\) 4.90928 + 10.5280i 0.498462 + 1.06896i 0.981165 + 0.193171i \(0.0618772\pi\)
−0.482703 + 0.875784i \(0.660345\pi\)
\(98\) 0 0
\(99\) −0.668674 + 0.796895i −0.0672043 + 0.0800910i
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.bh.a.53.3 yes 120
5.2 odd 4 inner 380.2.bh.a.357.3 yes 120
19.14 odd 18 inner 380.2.bh.a.33.3 ✓ 120
95.52 even 36 inner 380.2.bh.a.337.3 yes 120
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.2.bh.a.33.3 ✓ 120 19.14 odd 18 inner
380.2.bh.a.53.3 yes 120 1.1 even 1 trivial
380.2.bh.a.337.3 yes 120 95.52 even 36 inner
380.2.bh.a.357.3 yes 120 5.2 odd 4 inner