Show commands: Magma / Pari/GP / SageMath

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 337.3
Character \(\chi\) \(=\) 380.337
Dual form 380.2.bh.a.53.3

$q$-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 \(\left(\mathbb{Z}/380\mathbb{Z}\right)^\times\).

\(n\) \(21\) \(77\) \(191\)
\(\chi(n)\) \(e\left(\frac{7}{18}\right)\) \(e\left(\frac{1}{4}\right)\) \(1\)

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 −0.913543 − 0.406741i \(-0.866665\pi\)
0.0697615 + 0.997564i \(0.477776\pi\)
\(4\) 0 0
\(5\) 2.21130 + 0.331920i 0.988922 + 0.148439i
\(6\) 0 0
\(7\) 0.711084 − 2.65380i 0.268765 − 1.00304i −0.691141 − 0.722720i \(-0.742892\pi\)
0.959906 − 0.280323i \(-0.0904417\pi\)
\(8\) 0 0
\(9\) 0.0626300 − 0.172074i 0.0208767 − 0.0573581i
\(10\) 0 0
\(11\) 2.84045 − 4.91980i 0.856427 − 1.48337i −0.0188880 − 0.999822i \(-0.506013\pi\)
0.875315 − 0.483553i \(-0.160654\pi\)
\(12\) 0 0
\(13\) −1.31501 + 1.87803i −0.364717 + 0.520870i −0.959008 − 0.283380i \(-0.908544\pi\)
0.594290 + 0.804251i \(0.297433\pi\)
\(14\) 0 0
\(15\) −3.57141 + 1.77780i −0.922135 + 0.459027i
\(16\) 0 0
\(17\) 3.38777 + 1.57974i 0.821654 + 0.383144i 0.787532 − 0.616274i \(-0.211359\pi\)
0.0341223 + 0.999418i \(0.489136\pi\)
\(18\) 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 0.901669 + 0.432426i \(0.142342\pi\)
−0.963061 + 0.269284i \(0.913213\pi\)
\(24\) 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 0.441422 − 0.897300i \(-0.354474\pi\)
−0.238624 + 0.971112i \(0.576696\pi\)
\(30\) 0 0
\(31\) 7.63721 − 4.40934i 1.37168 − 0.791942i 0.380543 − 0.924763i \(-0.375737\pi\)
0.991140 + 0.132822i \(0.0424037\pi\)
\(32\) 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 −0.948840 − 0.315757i \(-0.897742\pi\)
0.315757 + 0.948840i \(0.397742\pi\)
\(38\) 0 0
\(39\) − 4.09038i − 0.654984i
\(40\) 0 0
\(41\) −2.15157 − 0.379380i −0.336019 − 0.0592491i 0.00309324 − 0.999995i \(-0.499015\pi\)
−0.339112 + 0.940746i \(0.610127\pi\)
\(42\) 0 0
\(43\) −1.44809 + 0.126691i −0.220832 + 0.0193203i −0.197035 − 0.980397i \(-0.563131\pi\)
−0.0237970 + 0.999717i \(0.507576\pi\)
\(44\) 0 0
\(45\) 0.195608 − 0.359719i 0.0291596 − 0.0536238i
\(46\) 0 0
\(47\) 0.0130497 + 0.0279851i 0.00190349 + 0.00408204i 0.907257 − 0.420576i \(-0.138172\pi\)
−0.905354 + 0.424658i \(0.860394\pi\)
\(48\) 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 0.00211296 − 0.999998i \(-0.499327\pi\)
−0.171567 + 0.985172i \(0.554883\pi\)
\(54\) 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 −0.702054 − 0.712124i \(-0.747733\pi\)
−0.0800600 + 0.996790i \(0.525511\pi\)
\(60\) 0 0
\(61\) 8.16469 − 6.85099i 1.04538 − 0.877179i 0.0527806 − 0.998606i \(-0.483192\pi\)
0.992600 + 0.121427i \(0.0387472\pi\)
\(62\) 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 −0.992181 − 0.124810i \(-0.960168\pi\)
−0.542151 + 0.840281i \(0.682390\pi\)
\(68\) 0 0
\(69\) −3.01352 − 5.21957i −0.362785 − 0.628362i
\(70\) 0 0
\(71\) −7.74802 + 9.23373i −0.919521 + 1.09584i 0.0755958 + 0.997139i \(0.475914\pi\)
−0.995117 + 0.0987040i \(0.968530\pi\)
\(72\) 0 0
\(73\) −7.08552 − 10.1192i −0.829298 − 1.18436i −0.980629 − 0.195877i \(-0.937245\pi\)
0.151331 − 0.988483i \(-0.451644\pi\)
\(74\) 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 −0.585002 − 0.811032i \(-0.698906\pi\)
0.827111 − 0.562038i \(-0.189983\pi\)
\(80\) 0 0
\(81\) 7.28954 + 6.11665i 0.809949 + 0.679628i
\(82\) 0 0
\(83\) −11.2432 − 3.01260i −1.23410 − 0.330676i −0.417925 − 0.908482i \(-0.637242\pi\)
−0.816174 + 0.577806i \(0.803909\pi\)
\(84\) 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 −0.899037 − 0.437874i \(-0.855732\pi\)
0.695056 − 0.718955i \(-0.255379\pi\)
\(90\) 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 −0.482703 − 0.875784i \(-0.660345\pi\)
0.981165 − 0.193171i \(-0.0618772\pi\)
\(98\) 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.337.3 yes 120
5.3 odd 4 inner 380.2.bh.a.33.3 ✓ 120
19.15 odd 18 inner 380.2.bh.a.357.3 yes 120
95.53 even 36 inner 380.2.bh.a.53.3 yes 120
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.2.bh.a.33.3 ✓ 120 5.3 odd 4 inner
380.2.bh.a.53.3 yes 120 95.53 even 36 inner
380.2.bh.a.337.3 yes 120 1.1 even 1 trivial
380.2.bh.a.357.3 yes 120 19.15 odd 18 inner