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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 53.10
Character \(\chi\) \(=\) 380.53
Dual form 380.2.bh.a.337.10

$q$-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)
 
\(f(q)\) \(=\) \(q+(2.49753 + 1.74879i) q^{3} +(1.88119 + 1.20877i) q^{5} +(-0.789905 - 2.94797i) q^{7} +(2.15333 + 5.91624i) q^{9} +(1.47232 + 2.55014i) q^{11} +(-3.96647 - 5.66470i) q^{13} +(2.58446 + 6.30874i) q^{15} +(1.46384 - 0.682601i) q^{17} +(-3.77813 - 2.17387i) q^{19} +(3.18256 - 8.74402i) q^{21} +(0.0211134 + 0.241328i) q^{23} +(2.07777 + 4.54784i) q^{25} +(-2.60088 + 9.70662i) q^{27} +(-9.03438 + 3.28825i) q^{29} +(-6.03549 - 3.48459i) q^{31} +(-0.782485 + 8.94384i) q^{33} +(2.07744 - 6.50050i) q^{35} +(3.20416 + 3.20416i) q^{37} -21.0843i q^{39} +(4.03303 - 0.711131i) q^{41} +(0.239871 + 0.0209860i) q^{43} +(-3.10051 + 13.7325i) q^{45} +(-0.704303 + 1.51038i) q^{47} +(-2.00438 + 1.15723i) q^{49} +(4.84972 + 0.855137i) q^{51} +(11.2003 - 0.979898i) q^{53} +(-0.312796 + 6.57700i) q^{55} +(-5.63436 - 12.0365i) q^{57} +(-5.39264 - 1.96276i) q^{59} +(6.17924 + 5.18500i) q^{61} +(15.7399 - 11.0212i) q^{63} +(-0.614392 - 15.4509i) q^{65} +(5.55998 + 2.59266i) q^{67} +(-0.369300 + 0.639646i) q^{69} +(-6.21776 - 7.41004i) q^{71} +(-1.98316 + 2.83225i) q^{73} +(-2.76392 + 14.9920i) q^{75} +(6.35473 - 6.35473i) q^{77} +(-0.417461 - 2.36754i) q^{79} +(-9.00174 + 7.55335i) q^{81} +(-4.13615 + 1.10828i) q^{83} +(3.57888 + 0.485339i) q^{85} +(-28.3141 - 7.58674i) q^{87} +(0.238274 - 1.35132i) q^{89} +(-13.5662 + 16.1676i) q^{91} +(-8.98000 - 19.2577i) q^{93} +(-4.47970 - 8.65634i) q^{95} +(-2.50121 - 5.36386i) q^{97} +(-11.9168 + 14.2019i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 120 q + 6 q^{7} + 18 q^{15} - 18 q^{17} + 48 q^{21} - 36 q^{23} - 24 q^{25} - 60 q^{33} - 18 q^{35} - 12 q^{41} - 36 q^{43} + 18 q^{45} - 24 q^{47} + 96 q^{51} - 18 q^{53} + 72 q^{55} - 6 q^{57} - 24 q^{61}+ \cdots - 60 q^{97}+O(q^{100}) \) 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
\(3\) 2.49753 + 1.74879i 1.44195 + 1.00966i 0.993251 + 0.115981i \(0.0370013\pi\)
0.448699 + 0.893683i \(0.351888\pi\)
\(4\) 0 0
\(5\) 1.88119 + 1.20877i 0.841295 + 0.540576i
\(6\) 0 0
\(7\) −0.789905 − 2.94797i −0.298556 − 1.11423i −0.938352 − 0.345682i \(-0.887648\pi\)
0.639796 − 0.768545i \(-0.279019\pi\)
\(8\) 0 0
\(9\) 2.15333 + 5.91624i 0.717778 + 1.97208i
\(10\) 0 0
\(11\) 1.47232 + 2.55014i 0.443923 + 0.768896i 0.997976 − 0.0635844i \(-0.0202532\pi\)
−0.554054 + 0.832481i \(0.686920\pi\)
\(12\) 0 0
\(13\) −3.96647 − 5.66470i −1.10010 − 1.57111i −0.782251 − 0.622964i \(-0.785928\pi\)
−0.317849 − 0.948141i \(-0.602961\pi\)
\(14\) 0 0
\(15\) 2.58446 + 6.30874i 0.667305 + 1.62891i
\(16\) 0 0
\(17\) 1.46384 − 0.682601i 0.355034 − 0.165555i −0.236915 − 0.971530i \(-0.576136\pi\)
0.591949 + 0.805975i \(0.298359\pi\)
\(18\) 0 0
\(19\) −3.77813 − 2.17387i −0.866763 − 0.498720i
\(20\) 0 0
\(21\) 3.18256 − 8.74402i 0.694492 − 1.90810i
\(22\) 0 0
\(23\) 0.0211134 + 0.241328i 0.00440245 + 0.0503203i 0.998078 − 0.0619678i \(-0.0197376\pi\)
−0.993676 + 0.112288i \(0.964182\pi\)
\(24\) 0 0
\(25\) 2.07777 + 4.54784i 0.415554 + 0.909568i
\(26\) 0 0
\(27\) −2.60088 + 9.70662i −0.500540 + 1.86804i
\(28\) 0 0
\(29\) −9.03438 + 3.28825i −1.67764 + 0.610612i −0.992984 − 0.118252i \(-0.962271\pi\)
−0.684658 + 0.728864i \(0.740049\pi\)
\(30\) 0 0
\(31\) −6.03549 − 3.48459i −1.08401 − 0.625851i −0.152031 − 0.988376i \(-0.548581\pi\)
−0.931974 + 0.362525i \(0.881915\pi\)
\(32\) 0 0
\(33\) −0.782485 + 8.94384i −0.136213 + 1.55692i
\(34\) 0 0
\(35\) 2.07744 − 6.50050i 0.351151 − 1.09879i
\(36\) 0 0
\(37\) 3.20416 + 3.20416i 0.526760 + 0.526760i 0.919605 − 0.392845i \(-0.128509\pi\)
−0.392845 + 0.919605i \(0.628509\pi\)
\(38\) 0 0
\(39\) − 21.0843i − 3.37619i
\(40\) 0 0
\(41\) 4.03303 − 0.711131i 0.629853 − 0.111060i 0.150396 − 0.988626i \(-0.451945\pi\)
0.479456 + 0.877566i \(0.340834\pi\)
\(42\) 0 0
\(43\) 0.239871 + 0.0209860i 0.0365799 + 0.00320033i 0.105431 − 0.994427i \(-0.466378\pi\)
−0.0688511 + 0.997627i \(0.521933\pi\)
\(44\) 0 0
\(45\) −3.10051 + 13.7325i −0.462196 + 2.04711i
\(46\) 0 0
\(47\) −0.704303 + 1.51038i −0.102733 + 0.220312i −0.950955 − 0.309329i \(-0.899896\pi\)
0.848222 + 0.529641i \(0.177673\pi\)
\(48\) 0 0
\(49\) −2.00438 + 1.15723i −0.286340 + 0.165318i
\(50\) 0 0
\(51\) 4.84972 + 0.855137i 0.679097 + 0.119743i
\(52\) 0 0
\(53\) 11.2003 − 0.979898i 1.53848 − 0.134599i 0.713860 − 0.700289i \(-0.246945\pi\)
0.824619 + 0.565689i \(0.191390\pi\)
\(54\) 0 0
\(55\) −0.312796 + 6.57700i −0.0421774 + 0.886843i
\(56\) 0 0
\(57\) −5.63436 − 12.0365i −0.746290 − 1.59427i
\(58\) 0 0
\(59\) −5.39264 − 1.96276i −0.702061 − 0.255529i −0.0337707 − 0.999430i \(-0.510752\pi\)
−0.668291 + 0.743900i \(0.732974\pi\)
\(60\) 0 0
\(61\) 6.17924 + 5.18500i 0.791171 + 0.663871i 0.946035 − 0.324065i \(-0.105050\pi\)
−0.154864 + 0.987936i \(0.549494\pi\)
\(62\) 0 0
\(63\) 15.7399 − 11.0212i 1.98305 − 1.38854i
\(64\) 0 0
\(65\) −0.614392 − 15.4509i −0.0762060 − 1.91645i
\(66\) 0 0
\(67\) 5.55998 + 2.59266i 0.679259 + 0.316744i 0.731452 − 0.681893i \(-0.238843\pi\)
−0.0521924 + 0.998637i \(0.516621\pi\)
\(68\) 0 0
\(69\) −0.369300 + 0.639646i −0.0444585 + 0.0770043i
\(70\) 0 0
\(71\) −6.21776 − 7.41004i −0.737913 − 0.879410i 0.258326 − 0.966058i \(-0.416829\pi\)
−0.996239 + 0.0866474i \(0.972385\pi\)
\(72\) 0 0
\(73\) −1.98316 + 2.83225i −0.232111 + 0.331489i −0.918264 − 0.395968i \(-0.870409\pi\)
0.686153 + 0.727457i \(0.259298\pi\)
\(74\) 0 0
\(75\) −2.76392 + 14.9920i −0.319150 + 1.73112i
\(76\) 0 0
\(77\) 6.35473 − 6.35473i 0.724189 − 0.724189i
\(78\) 0 0
\(79\) −0.417461 − 2.36754i −0.0469680 − 0.266369i 0.952276 − 0.305237i \(-0.0987358\pi\)
−0.999244 + 0.0388684i \(0.987625\pi\)
\(80\) 0 0
\(81\) −9.00174 + 7.55335i −1.00019 + 0.839261i
\(82\) 0 0
\(83\) −4.13615 + 1.10828i −0.454001 + 0.121649i −0.478571 − 0.878049i \(-0.658845\pi\)
0.0245698 + 0.999698i \(0.492178\pi\)
\(84\) 0 0
\(85\) 3.57888 + 0.485339i 0.388184 + 0.0526424i
\(86\) 0 0
\(87\) −28.3141 − 7.58674i −3.03559 − 0.813384i
\(88\) 0 0
\(89\) 0.238274 − 1.35132i 0.0252570 − 0.143240i −0.969572 − 0.244808i \(-0.921275\pi\)
0.994829 + 0.101568i \(0.0323861\pi\)
\(90\) 0 0
\(91\) −13.5662 + 16.1676i −1.42213 + 1.69482i
\(92\) 0 0
\(93\) −8.98000 − 19.2577i −0.931182 − 1.99693i
\(94\) 0 0
\(95\) −4.47970 − 8.65634i −0.459607 − 0.888122i
\(96\) 0 0
\(97\) −2.50121 − 5.36386i −0.253959 − 0.544617i 0.737452 − 0.675400i \(-0.236029\pi\)
−0.991411 + 0.130782i \(0.958251\pi\)
\(98\) 0 0
\(99\) −11.9168 + 14.2019i −1.19769 + 1.42735i
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.53.10 yes 120
5.2 odd 4 inner 380.2.bh.a.357.10 yes 120
19.14 odd 18 inner 380.2.bh.a.33.10 ✓ 120
95.52 even 36 inner 380.2.bh.a.337.10 yes 120
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.2.bh.a.33.10 ✓ 120 19.14 odd 18 inner
380.2.bh.a.53.10 yes 120 1.1 even 1 trivial
380.2.bh.a.337.10 yes 120 95.52 even 36 inner
380.2.bh.a.357.10 yes 120 5.2 odd 4 inner