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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 317.8
Character \(\chi\) \(=\) 380.317
Dual form 380.2.bh.a.193.8

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.660272 + 1.41596i) q^{3} +(1.58495 + 1.57732i) q^{5} +(4.18524 - 1.12143i) q^{7} +(0.359387 - 0.428300i) q^{9} +(-0.627484 - 1.08683i) q^{11} +(-4.12588 - 1.92393i) q^{13} +(-1.18692 + 3.28568i) q^{15} +(0.831080 + 0.0727101i) q^{17} +(-3.83747 + 2.06733i) q^{19} +(4.35129 + 5.18567i) q^{21} +(-0.411516 - 0.587706i) q^{23} +(0.0241238 + 4.99994i) q^{25} +(5.37105 + 1.43917i) q^{27} +(3.22929 + 2.70969i) q^{29} +(-6.47928 - 3.74082i) q^{31} +(1.12460 - 1.60610i) q^{33} +(8.40224 + 4.82405i) q^{35} +(-1.31792 + 1.31792i) q^{37} -7.11239i q^{39} +(-2.83535 + 7.79006i) q^{41} +(-9.60211 - 6.72347i) q^{43} +(1.24518 - 0.111966i) q^{45} +(-0.832071 - 9.51062i) q^{47} +(10.1964 - 5.88691i) q^{49} +(0.445784 + 1.22478i) q^{51} +(-7.84194 + 5.49098i) q^{53} +(0.719755 - 2.71232i) q^{55} +(-5.46102 - 4.06869i) q^{57} +(2.93676 - 2.46424i) q^{59} +(-1.31237 + 7.44285i) q^{61} +(1.02381 - 2.19557i) q^{63} +(-3.50466 - 9.55716i) q^{65} +(-7.04871 + 0.616682i) q^{67} +(0.560454 - 0.970734i) q^{69} +(11.6384 - 2.05217i) q^{71} +(2.56798 - 1.19747i) q^{73} +(-7.06378 + 3.33548i) q^{75} +(-3.84498 - 3.84498i) q^{77} +(5.16648 + 1.88045i) q^{79} +(1.21729 + 6.90358i) q^{81} +(-1.93497 - 7.22139i) q^{83} +(1.20253 + 1.42612i) q^{85} +(-1.70460 + 6.36167i) q^{87} +(-3.31376 + 1.20611i) q^{89} +(-19.4253 - 3.42521i) q^{91} +(1.01875 - 11.6443i) q^{93} +(-9.34302 - 2.77631i) q^{95} +(0.476660 - 5.44825i) q^{97} +(-0.691001 - 0.121842i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 120 q + 6 q^{7} + 18 q^{15} - 18 q^{17} + 48 q^{21} - 36 q^{23} - 24 q^{25} - 60 q^{33} - 18 q^{35} - 12 q^{41} - 36 q^{43} + 18 q^{45} - 24 q^{47} + 96 q^{51} - 18 q^{53} + 72 q^{55} - 6 q^{57} - 24 q^{61}+ \cdots - 60 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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
\(3\) 0.660272 + 1.41596i 0.381208 + 0.817503i 0.999502 + 0.0315563i \(0.0100464\pi\)
−0.618294 + 0.785947i \(0.712176\pi\)
\(4\) 0 0
\(5\) 1.58495 + 1.57732i 0.708811 + 0.705399i
\(6\) 0 0
\(7\) 4.18524 − 1.12143i 1.58187 − 0.423861i 0.642367 − 0.766397i \(-0.277953\pi\)
0.939505 + 0.342536i \(0.111286\pi\)
\(8\) 0 0
\(9\) 0.359387 − 0.428300i 0.119796 − 0.142767i
\(10\) 0 0
\(11\) −0.627484 − 1.08683i −0.189194 − 0.327693i 0.755788 − 0.654816i \(-0.227254\pi\)
−0.944982 + 0.327123i \(0.893921\pi\)
\(12\) 0 0
\(13\) −4.12588 − 1.92393i −1.14431 − 0.533602i −0.244402 − 0.969674i \(-0.578591\pi\)
−0.899912 + 0.436072i \(0.856369\pi\)
\(14\) 0 0
\(15\) −1.18692 + 3.28568i −0.306462 + 0.848359i
\(16\) 0 0
\(17\) 0.831080 + 0.0727101i 0.201567 + 0.0176348i 0.187492 − 0.982266i \(-0.439964\pi\)
0.0140747 + 0.999901i \(0.495520\pi\)
\(18\) 0 0
\(19\) −3.83747 + 2.06733i −0.880376 + 0.474277i
\(20\) 0 0
\(21\) 4.35129 + 5.18567i 0.949530 + 1.13161i
\(22\) 0 0
\(23\) −0.411516 − 0.587706i −0.0858070 − 0.122545i 0.773954 − 0.633242i \(-0.218276\pi\)
−0.859761 + 0.510697i \(0.829387\pi\)
\(24\) 0 0
\(25\) 0.0241238 + 4.99994i 0.00482476 + 0.999988i
\(26\) 0 0
\(27\) 5.37105 + 1.43917i 1.03366 + 0.276968i
\(28\) 0 0
\(29\) 3.22929 + 2.70969i 0.599664 + 0.503177i 0.891337 − 0.453340i \(-0.149768\pi\)
−0.291674 + 0.956518i \(0.594212\pi\)
\(30\) 0 0
\(31\) −6.47928 − 3.74082i −1.16371 − 0.671870i −0.211522 − 0.977373i \(-0.567842\pi\)
−0.952191 + 0.305503i \(0.901175\pi\)
\(32\) 0 0
\(33\) 1.12460 − 1.60610i 0.195768 − 0.279586i
\(34\) 0 0
\(35\) 8.40224 + 4.82405i 1.42024 + 0.815413i
\(36\) 0 0
\(37\) −1.31792 + 1.31792i −0.216665 + 0.216665i −0.807091 − 0.590427i \(-0.798960\pi\)
0.590427 + 0.807091i \(0.298960\pi\)
\(38\) 0 0
\(39\) − 7.11239i − 1.13889i
\(40\) 0 0
\(41\) −2.83535 + 7.79006i −0.442807 + 1.21660i 0.494831 + 0.868989i \(0.335230\pi\)
−0.937638 + 0.347613i \(0.886992\pi\)
\(42\) 0 0
\(43\) −9.60211 − 6.72347i −1.46431 − 1.02532i −0.989389 − 0.145290i \(-0.953589\pi\)
−0.474919 − 0.880030i \(-0.657523\pi\)
\(44\) 0 0
\(45\) 1.24518 − 0.111966i 0.185620 − 0.0166909i
\(46\) 0 0
\(47\) −0.832071 − 9.51062i −0.121370 − 1.38727i −0.775721 − 0.631076i \(-0.782614\pi\)
0.654351 − 0.756191i \(-0.272942\pi\)
\(48\) 0 0
\(49\) 10.1964 − 5.88691i 1.45663 − 0.840988i
\(50\) 0 0
\(51\) 0.445784 + 1.22478i 0.0624223 + 0.171504i
\(52\) 0 0
\(53\) −7.84194 + 5.49098i −1.07717 + 0.754245i −0.970754 − 0.240077i \(-0.922827\pi\)
−0.106419 + 0.994321i \(0.533938\pi\)
\(54\) 0 0
\(55\) 0.719755 − 2.71232i 0.0970518 − 0.365729i
\(56\) 0 0
\(57\) −5.46102 − 4.06869i −0.723329 − 0.538912i
\(58\) 0 0
\(59\) 2.93676 − 2.46424i 0.382334 − 0.320816i −0.431284 − 0.902216i \(-0.641939\pi\)
0.813618 + 0.581400i \(0.197495\pi\)
\(60\) 0 0
\(61\) −1.31237 + 7.44285i −0.168032 + 0.952959i 0.777850 + 0.628450i \(0.216310\pi\)
−0.945883 + 0.324509i \(0.894801\pi\)
\(62\) 0 0
\(63\) 1.02381 − 2.19557i 0.128988 − 0.276615i
\(64\) 0 0
\(65\) −3.50466 − 9.55716i −0.434699 − 1.18542i
\(66\) 0 0
\(67\) −7.04871 + 0.616682i −0.861137 + 0.0753398i −0.509163 − 0.860670i \(-0.670045\pi\)
−0.351974 + 0.936010i \(0.614489\pi\)
\(68\) 0 0
\(69\) 0.560454 − 0.970734i 0.0674707 − 0.116863i
\(70\) 0 0
\(71\) 11.6384 − 2.05217i 1.38123 − 0.243548i 0.566821 − 0.823841i \(-0.308173\pi\)
0.814408 + 0.580293i \(0.197062\pi\)
\(72\) 0 0
\(73\) 2.56798 − 1.19747i 0.300560 − 0.140153i −0.266488 − 0.963838i \(-0.585863\pi\)
0.567048 + 0.823685i \(0.308085\pi\)
\(74\) 0 0
\(75\) −7.06378 + 3.33548i −0.815655 + 0.385148i
\(76\) 0 0
\(77\) −3.84498 − 3.84498i −0.438176 − 0.438176i
\(78\) 0 0
\(79\) 5.16648 + 1.88045i 0.581275 + 0.211567i 0.615888 − 0.787834i \(-0.288798\pi\)
−0.0346128 + 0.999401i \(0.511020\pi\)
\(80\) 0 0
\(81\) 1.21729 + 6.90358i 0.135254 + 0.767065i
\(82\) 0 0
\(83\) −1.93497 − 7.22139i −0.212390 − 0.792651i −0.987069 − 0.160296i \(-0.948755\pi\)
0.774679 − 0.632355i \(-0.217912\pi\)
\(84\) 0 0
\(85\) 1.20253 + 1.42612i 0.130433 + 0.154685i
\(86\) 0 0
\(87\) −1.70460 + 6.36167i −0.182753 + 0.682042i
\(88\) 0 0
\(89\) −3.31376 + 1.20611i −0.351258 + 0.127847i −0.511622 − 0.859210i \(-0.670955\pi\)
0.160364 + 0.987058i \(0.448733\pi\)
\(90\) 0 0
\(91\) −19.4253 − 3.42521i −2.03633 − 0.359060i
\(92\) 0 0
\(93\) 1.01875 − 11.6443i 0.105639 − 1.20746i
\(94\) 0 0
\(95\) −9.34302 − 2.77631i −0.958574 − 0.284843i
\(96\) 0 0
\(97\) 0.476660 − 5.44825i 0.0483975 − 0.553186i −0.932828 − 0.360322i \(-0.882667\pi\)
0.981225 − 0.192864i \(-0.0617776\pi\)
\(98\) 0 0
\(99\) −0.691001 − 0.121842i −0.0694482 − 0.0122456i
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.317.8 yes 120
5.3 odd 4 inner 380.2.bh.a.13.3 ✓ 120
19.3 odd 18 inner 380.2.bh.a.117.3 yes 120
95.3 even 36 inner 380.2.bh.a.193.8 yes 120
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.2.bh.a.13.3 ✓ 120 5.3 odd 4 inner
380.2.bh.a.117.3 yes 120 19.3 odd 18 inner
380.2.bh.a.193.8 yes 120 95.3 even 36 inner
380.2.bh.a.317.8 yes 120 1.1 even 1 trivial