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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 357.3
Character \(\chi\) \(=\) 380.357
Dual form 380.2.bh.a.33.3

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.02333 + 1.46147i) q^{3} +(0.710865 - 2.12006i) q^{5} +(-2.65380 + 0.711084i) q^{7} +(-0.0626300 - 0.172074i) q^{9} +(2.84045 + 4.91980i) q^{11} +(-1.87803 + 1.31501i) q^{13} +(2.37096 + 3.20844i) q^{15} +(1.57974 + 3.38777i) q^{17} +(-3.00247 + 3.15994i) q^{19} +(1.67650 - 4.60614i) q^{21} +(-3.36528 + 0.294424i) q^{23} +(-3.98934 - 3.01416i) q^{25} +(-4.85444 - 1.30074i) q^{27} +(-1.09210 + 0.397492i) q^{29} +(7.63721 + 4.40934i) q^{31} +(-10.0969 - 0.883363i) q^{33} +(-0.378951 + 6.13172i) q^{35} +(3.85089 - 3.85089i) q^{37} -4.09038i q^{39} +(-2.15157 + 0.379380i) q^{41} +(-0.126691 + 1.44809i) q^{43} +(-0.409330 + 0.0104579i) q^{45} +(0.0279851 + 0.0130497i) q^{47} +(0.474851 - 0.274156i) q^{49} +(-6.56773 - 1.15807i) q^{51} +(0.107930 + 1.23364i) q^{53} +(12.4495 - 2.52462i) q^{55} +(-1.54564 - 7.62170i) q^{57} +(6.00753 + 2.18656i) q^{59} +(8.16469 + 6.85099i) q^{61} +(0.288567 + 0.412116i) q^{63} +(1.45288 + 4.91633i) q^{65} +(5.85638 - 12.5590i) q^{67} +(3.01352 - 5.21957i) q^{69} +(-7.74802 - 9.23373i) q^{71} +(10.1192 + 7.08552i) q^{73} +(8.48754 - 2.74583i) q^{75} +(-11.0364 - 11.0364i) q^{77} +(-2.15191 - 12.2041i) q^{79} +(7.28954 - 6.11665i) q^{81} +(3.01260 + 11.2432i) q^{83} +(8.30526 - 0.940909i) q^{85} +(0.536661 - 2.00284i) q^{87} +(1.92434 - 10.9135i) q^{89} +(4.04883 - 4.82520i) q^{91} +(-14.2596 + 6.64934i) q^{93} +(4.56492 + 8.61171i) q^{95} +(-10.5280 + 4.90928i) q^{97} +(0.668674 - 0.796895i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 120 q + 6 q^{7} + 18 q^{15} - 18 q^{17} + 48 q^{21} - 36 q^{23} - 24 q^{25} - 60 q^{33} - 18 q^{35} - 12 q^{41} - 36 q^{43} + 18 q^{45} - 24 q^{47} + 96 q^{51} - 18 q^{53} + 72 q^{55} - 6 q^{57} - 24 q^{61}+ \cdots - 60 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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
\(3\) −1.02333 + 1.46147i −0.590823 + 0.843782i −0.997564 − 0.0697615i \(-0.977776\pi\)
0.406741 + 0.913543i \(0.366665\pi\)
\(4\) 0 0
\(5\) 0.710865 − 2.12006i 0.317908 − 0.948121i
\(6\) 0 0
\(7\) −2.65380 + 0.711084i −1.00304 + 0.268765i −0.722720 − 0.691141i \(-0.757108\pi\)
−0.280323 + 0.959906i \(0.590442\pi\)
\(8\) 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
\(13\) −1.87803 + 1.31501i −0.520870 + 0.364717i −0.804251 − 0.594290i \(-0.797433\pi\)
0.283380 + 0.959008i \(0.408544\pi\)
\(14\) 0 0
\(15\) 2.37096 + 3.20844i 0.612180 + 0.828417i
\(16\) 0 0
\(17\) 1.57974 + 3.38777i 0.383144 + 0.821654i 0.999418 + 0.0341223i \(0.0108636\pi\)
−0.616274 + 0.787532i \(0.711359\pi\)
\(18\) 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
\(23\) −3.36528 + 0.294424i −0.701710 + 0.0613917i −0.432426 − 0.901669i \(-0.642342\pi\)
−0.269284 + 0.963061i \(0.586787\pi\)
\(24\) 0 0
\(25\) −3.98934 − 3.01416i −0.797869 − 0.602832i
\(26\) 0 0
\(27\) −4.85444 − 1.30074i −0.934237 − 0.250328i
\(28\) 0 0
\(29\) −1.09210 + 0.397492i −0.202798 + 0.0738125i −0.441422 − 0.897300i \(-0.645526\pi\)
0.238624 + 0.971112i \(0.423304\pi\)
\(30\) 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
\(33\) −10.0969 − 0.883363i −1.75764 − 0.153774i
\(34\) 0 0
\(35\) −0.378951 + 6.13172i −0.0640544 + 1.03645i
\(36\) 0 0
\(37\) 3.85089 − 3.85089i 0.633083 − 0.633083i −0.315757 − 0.948840i \(-0.602258\pi\)
0.948840 + 0.315757i \(0.102258\pi\)
\(38\) 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
\(43\) −0.126691 + 1.44809i −0.0193203 + 0.220832i 0.980397 + 0.197035i \(0.0631311\pi\)
−0.999717 + 0.0237970i \(0.992424\pi\)
\(44\) 0 0
\(45\) −0.409330 + 0.0104579i −0.0610193 + 0.00155897i
\(46\) 0 0
\(47\) 0.0279851 + 0.0130497i 0.00408204 + 0.00190349i 0.424658 − 0.905354i \(-0.360394\pi\)
−0.420576 + 0.907257i \(0.638172\pi\)
\(48\) 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
\(53\) 0.107930 + 1.23364i 0.0148253 + 0.169454i 0.999998 − 0.00211296i \(-0.000672577\pi\)
−0.985172 + 0.171567i \(0.945117\pi\)
\(54\) 0 0
\(55\) 12.4495 − 2.52462i 1.67868 − 0.340419i
\(56\) 0 0
\(57\) −1.54564 − 7.62170i −0.204724 − 1.00952i
\(58\) 0 0
\(59\) 6.00753 + 2.18656i 0.782114 + 0.284666i 0.702054 − 0.712124i \(-0.252267\pi\)
0.0800600 + 0.996790i \(0.474489\pi\)
\(60\) 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
\(63\) 0.288567 + 0.412116i 0.0363560 + 0.0519218i
\(64\) 0 0
\(65\) 1.45288 + 4.91633i 0.180207 + 0.609795i
\(66\) 0 0
\(67\) 5.85638 − 12.5590i 0.715471 − 1.53433i −0.124810 − 0.992181i \(-0.539832\pi\)
0.840281 − 0.542151i \(-0.182390\pi\)
\(68\) 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
\(73\) 10.1192 + 7.08552i 1.18436 + 0.829298i 0.988483 − 0.151331i \(-0.0483559\pi\)
0.195877 + 0.980629i \(0.437245\pi\)
\(74\) 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 \(-0.810017\pi\)
0.585002 − 0.811032i \(-0.301094\pi\)
\(80\) 0 0
\(81\) 7.28954 − 6.11665i 0.809949 − 0.679628i
\(82\) 0 0
\(83\) 3.01260 + 11.2432i 0.330676 + 1.23410i 0.908482 + 0.417925i \(0.137242\pi\)
−0.577806 + 0.816174i \(0.696091\pi\)
\(84\) 0 0
\(85\) 8.30526 − 0.940909i 0.900832 − 0.102056i
\(86\) 0 0
\(87\) 0.536661 − 2.00284i 0.0575361 − 0.214727i
\(88\) 0 0
\(89\) 1.92434 − 10.9135i 0.203980 − 1.15683i −0.695056 − 0.718955i \(-0.744621\pi\)
0.899037 − 0.437874i \(-0.144268\pi\)
\(90\) 0 0
\(91\) 4.04883 − 4.82520i 0.424432 − 0.505819i
\(92\) 0 0
\(93\) −14.2596 + 6.64934i −1.47865 + 0.689505i
\(94\) 0 0
\(95\) 4.56492 + 8.61171i 0.468351 + 0.883543i
\(96\) 0 0
\(97\) −10.5280 + 4.90928i −1.06896 + 0.498462i −0.875784 − 0.482703i \(-0.839655\pi\)
−0.193171 + 0.981165i \(0.561877\pi\)
\(98\) 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.357.3 yes 120
5.3 odd 4 inner 380.2.bh.a.53.3 yes 120
19.14 odd 18 inner 380.2.bh.a.337.3 yes 120
95.33 even 36 inner 380.2.bh.a.33.3 ✓ 120
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.2.bh.a.33.3 ✓ 120 95.33 even 36 inner
380.2.bh.a.53.3 yes 120 5.3 odd 4 inner
380.2.bh.a.337.3 yes 120 19.14 odd 18 inner
380.2.bh.a.357.3 yes 120 1.1 even 1 trivial