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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 13.9
Character \(\chi\) \(=\) 380.13
Dual form 380.2.bh.a.117.9

$q$-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)
 
\(f(q)\) \(=\) \(q+(1.84872 - 0.862074i) q^{3} +(2.22179 + 0.252247i) q^{5} +(-1.05366 - 3.93231i) q^{7} +(0.746246 - 0.889341i) q^{9} +(-2.62986 - 4.55506i) q^{11} +(-0.168100 + 0.360492i) q^{13} +(4.32494 - 1.44902i) q^{15} +(-0.612760 + 7.00388i) q^{17} +(3.17822 + 2.98311i) q^{19} +(-5.33787 - 6.36142i) q^{21} +(-2.31544 + 1.62129i) q^{23} +(4.87274 + 1.12088i) q^{25} +(-0.970925 + 3.62354i) q^{27} +(2.22257 + 1.86496i) q^{29} +(3.19971 + 1.84735i) q^{31} +(-8.78869 - 6.15390i) q^{33} +(-1.34910 - 9.00257i) q^{35} +(0.837316 + 0.837316i) q^{37} +0.811364i q^{39} +(2.89465 - 7.95299i) q^{41} +(-1.46246 + 2.08860i) q^{43} +(1.88234 - 1.78769i) q^{45} +(3.24199 - 0.283638i) q^{47} +(-8.29069 + 4.78663i) q^{49} +(4.90504 + 13.4765i) q^{51} +(3.42197 + 4.88708i) q^{53} +(-4.69402 - 10.7838i) q^{55} +(8.44731 + 2.77509i) q^{57} +(-9.57764 + 8.03660i) q^{59} +(0.260608 - 1.47798i) q^{61} +(-4.28345 - 1.99741i) q^{63} +(-0.464416 + 0.758536i) q^{65} +(-0.959802 - 10.9706i) q^{67} +(-2.88294 + 4.99339i) q^{69} +(9.90257 - 1.74609i) q^{71} +(-4.60382 - 9.87292i) q^{73} +(9.97464 - 2.12847i) q^{75} +(-15.1409 + 15.1409i) q^{77} +(11.6936 + 4.25614i) q^{79} +(1.93358 + 10.9659i) q^{81} +(-9.88658 + 2.64910i) q^{83} +(-3.12813 + 15.4066i) q^{85} +(5.71664 + 1.53177i) q^{87} +(-5.85780 + 2.13206i) q^{89} +(1.59468 + 0.281186i) q^{91} +(7.50794 + 0.656859i) q^{93} +(6.30887 + 7.42955i) q^{95} +(-6.89617 - 0.603337i) q^{97} +(-6.01352 - 1.06035i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 120 q + 6 q^{7} + 18 q^{15} - 18 q^{17} + 48 q^{21} - 36 q^{23} - 24 q^{25} - 60 q^{33} - 18 q^{35} - 12 q^{41} - 36 q^{43} + 18 q^{45} - 24 q^{47} + 96 q^{51} - 18 q^{53} + 72 q^{55} - 6 q^{57} - 24 q^{61}+ \cdots - 60 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(21\) \(77\) \(191\)
\(\chi(n)\) \(e\left(\frac{5}{18}\right)\) \(e\left(\frac{3}{4}\right)\) \(1\)

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
\(3\) 1.84872 − 0.862074i 1.06736 − 0.497719i 0.192103 − 0.981375i \(-0.438469\pi\)
0.875258 + 0.483656i \(0.160691\pi\)
\(4\) 0 0
\(5\) 2.22179 + 0.252247i 0.993617 + 0.112808i
\(6\) 0 0
\(7\) −1.05366 − 3.93231i −0.398246 − 1.48627i −0.816181 − 0.577797i \(-0.803913\pi\)
0.417935 − 0.908477i \(-0.362754\pi\)
\(8\) 0 0
\(9\) 0.746246 − 0.889341i 0.248749 − 0.296447i
\(10\) 0 0
\(11\) −2.62986 − 4.55506i −0.792934 − 1.37340i −0.924143 − 0.382047i \(-0.875219\pi\)
0.131210 − 0.991355i \(-0.458114\pi\)
\(12\) 0 0
\(13\) −0.168100 + 0.360492i −0.0466225 + 0.0999824i −0.928239 − 0.371984i \(-0.878678\pi\)
0.881617 + 0.471966i \(0.156456\pi\)
\(14\) 0 0
\(15\) 4.32494 − 1.44902i 1.11669 − 0.374135i
\(16\) 0 0
\(17\) −0.612760 + 7.00388i −0.148616 + 1.69869i 0.446285 + 0.894891i \(0.352747\pi\)
−0.594901 + 0.803799i \(0.702809\pi\)
\(18\) 0 0
\(19\) 3.17822 + 2.98311i 0.729133 + 0.684372i
\(20\) 0 0
\(21\) −5.33787 − 6.36142i −1.16482 − 1.38818i
\(22\) 0 0
\(23\) −2.31544 + 1.62129i −0.482802 + 0.338062i −0.789506 − 0.613743i \(-0.789663\pi\)
0.306703 + 0.951805i \(0.400774\pi\)
\(24\) 0 0
\(25\) 4.87274 + 1.12088i 0.974549 + 0.224176i
\(26\) 0 0
\(27\) −0.970925 + 3.62354i −0.186855 + 0.697351i
\(28\) 0 0
\(29\) 2.22257 + 1.86496i 0.412720 + 0.346313i 0.825386 − 0.564569i \(-0.190958\pi\)
−0.412665 + 0.910883i \(0.635402\pi\)
\(30\) 0 0
\(31\) 3.19971 + 1.84735i 0.574685 + 0.331794i 0.759018 − 0.651069i \(-0.225679\pi\)
−0.184333 + 0.982864i \(0.559013\pi\)
\(32\) 0 0
\(33\) −8.78869 − 6.15390i −1.52991 − 1.07126i
\(34\) 0 0
\(35\) −1.34910 − 9.00257i −0.228040 − 1.52171i
\(36\) 0 0
\(37\) 0.837316 + 0.837316i 0.137654 + 0.137654i 0.772576 − 0.634922i \(-0.218968\pi\)
−0.634922 + 0.772576i \(0.718968\pi\)
\(38\) 0 0
\(39\) 0.811364i 0.129922i
\(40\) 0 0
\(41\) 2.89465 − 7.95299i 0.452069 − 1.24205i −0.479197 − 0.877707i \(-0.659072\pi\)
0.931266 − 0.364341i \(-0.118706\pi\)
\(42\) 0 0
\(43\) −1.46246 + 2.08860i −0.223023 + 0.318509i −0.915034 − 0.403378i \(-0.867836\pi\)
0.692011 + 0.721887i \(0.256725\pi\)
\(44\) 0 0
\(45\) 1.88234 − 1.78769i 0.280602 − 0.266494i
\(46\) 0 0
\(47\) 3.24199 − 0.283638i 0.472894 − 0.0413728i 0.151782 − 0.988414i \(-0.451499\pi\)
0.321112 + 0.947041i \(0.395943\pi\)
\(48\) 0 0
\(49\) −8.29069 + 4.78663i −1.18438 + 0.683805i
\(50\) 0 0
\(51\) 4.90504 + 13.4765i 0.686843 + 1.88709i
\(52\) 0 0
\(53\) 3.42197 + 4.88708i 0.470044 + 0.671292i 0.982251 − 0.187570i \(-0.0600611\pi\)
−0.512208 + 0.858862i \(0.671172\pi\)
\(54\) 0 0
\(55\) −4.69402 − 10.7838i −0.632941 − 1.45408i
\(56\) 0 0
\(57\) 8.44731 + 2.77509i 1.11887 + 0.367569i
\(58\) 0 0
\(59\) −9.57764 + 8.03660i −1.24690 + 1.04628i −0.249951 + 0.968259i \(0.580414\pi\)
−0.996952 + 0.0780171i \(0.975141\pi\)
\(60\) 0 0
\(61\) 0.260608 − 1.47798i 0.0333674 − 0.189236i −0.963568 − 0.267463i \(-0.913815\pi\)
0.996936 + 0.0782267i \(0.0249258\pi\)
\(62\) 0 0
\(63\) −4.28345 − 1.99741i −0.539664 − 0.251650i
\(64\) 0 0
\(65\) −0.464416 + 0.758536i −0.0576038 + 0.0940848i
\(66\) 0 0
\(67\) −0.959802 − 10.9706i −0.117258 − 1.34027i −0.795733 − 0.605647i \(-0.792914\pi\)
0.678475 − 0.734624i \(-0.262641\pi\)
\(68\) 0 0
\(69\) −2.88294 + 4.99339i −0.347065 + 0.601134i
\(70\) 0 0
\(71\) 9.90257 − 1.74609i 1.17522 − 0.207223i 0.448260 − 0.893903i \(-0.352044\pi\)
0.726959 + 0.686681i \(0.240933\pi\)
\(72\) 0 0
\(73\) −4.60382 − 9.87292i −0.538836 − 1.15554i −0.967685 − 0.252162i \(-0.918859\pi\)
0.428849 − 0.903376i \(-0.358919\pi\)
\(74\) 0 0
\(75\) 9.97464 − 2.12847i 1.15177 − 0.245774i
\(76\) 0 0
\(77\) −15.1409 + 15.1409i −1.72547 + 1.72547i
\(78\) 0 0
\(79\) 11.6936 + 4.25614i 1.31564 + 0.478853i 0.902058 − 0.431615i \(-0.142056\pi\)
0.413580 + 0.910468i \(0.364278\pi\)
\(80\) 0 0
\(81\) 1.93358 + 10.9659i 0.214842 + 1.21843i
\(82\) 0 0
\(83\) −9.88658 + 2.64910i −1.08519 + 0.290777i −0.756722 − 0.653737i \(-0.773200\pi\)
−0.328472 + 0.944514i \(0.606534\pi\)
\(84\) 0 0
\(85\) −3.12813 + 15.4066i −0.339294 + 1.67108i
\(86\) 0 0
\(87\) 5.71664 + 1.53177i 0.612888 + 0.164223i
\(88\) 0 0
\(89\) −5.85780 + 2.13206i −0.620925 + 0.225998i −0.633277 − 0.773925i \(-0.718291\pi\)
0.0123514 + 0.999924i \(0.496068\pi\)
\(90\) 0 0
\(91\) 1.59468 + 0.281186i 0.167168 + 0.0294763i
\(92\) 0 0
\(93\) 7.50794 + 0.656859i 0.778537 + 0.0681131i
\(94\) 0 0
\(95\) 6.30887 + 7.42955i 0.647276 + 0.762256i
\(96\) 0 0
\(97\) −6.89617 − 0.603337i −0.700200 − 0.0612596i −0.268504 − 0.963279i \(-0.586529\pi\)
−0.431696 + 0.902019i \(0.642085\pi\)
\(98\) 0 0
\(99\) −6.01352 − 1.06035i −0.604382 − 0.106569i
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.13.9 ✓ 120
5.2 odd 4 inner 380.2.bh.a.317.2 yes 120
19.3 odd 18 inner 380.2.bh.a.193.2 yes 120
95.22 even 36 inner 380.2.bh.a.117.9 yes 120
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.2.bh.a.13.9 ✓ 120 1.1 even 1 trivial
380.2.bh.a.117.9 yes 120 95.22 even 36 inner
380.2.bh.a.193.2 yes 120 19.3 odd 18 inner
380.2.bh.a.317.2 yes 120 5.2 odd 4 inner