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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 357.10
Character \(\chi\) \(=\) 380.357
Dual form 380.2.bh.a.33.10

$q$-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)
 
\(f(q)\) \(=\) \(q+(1.74879 - 2.49753i) q^{3} +(-0.863736 - 2.06251i) q^{5} +(2.94797 - 0.789905i) q^{7} +(-2.15333 - 5.91624i) q^{9} +(1.47232 + 2.55014i) q^{11} +(-5.66470 + 3.96647i) q^{13} +(-6.66168 - 1.44970i) q^{15} +(0.682601 + 1.46384i) q^{17} +(3.77813 + 2.17387i) q^{19} +(3.18256 - 8.74402i) q^{21} +(0.241328 - 0.0211134i) q^{23} +(-3.50792 + 3.56293i) q^{25} +(-9.70662 - 2.60088i) q^{27} +(9.03438 - 3.28825i) q^{29} +(-6.03549 - 3.48459i) q^{31} +(8.94384 + 0.782485i) q^{33} +(-4.17545 - 5.39795i) q^{35} +(-3.20416 + 3.20416i) q^{37} +21.0843i q^{39} +(4.03303 - 0.711131i) q^{41} +(0.0209860 - 0.239871i) q^{43} +(-10.3424 + 9.55135i) q^{45} +(-1.51038 - 0.704303i) q^{47} +(2.00438 - 1.15723i) q^{49} +(4.84972 + 0.855137i) q^{51} +(-0.979898 - 11.2003i) q^{53} +(3.98800 - 5.23934i) q^{55} +(12.0365 - 5.63436i) q^{57} +(5.39264 + 1.96276i) q^{59} +(6.17924 + 5.18500i) q^{61} +(-11.0212 - 15.7399i) q^{63} +(13.0737 + 8.25754i) q^{65} +(-2.59266 + 5.55998i) q^{67} +(0.369300 - 0.639646i) q^{69} +(-6.21776 - 7.41004i) q^{71} +(2.83225 + 1.98316i) q^{73} +(2.76392 + 14.9920i) q^{75} +(6.35473 + 6.35473i) q^{77} +(0.417461 + 2.36754i) q^{79} +(-9.00174 + 7.55335i) q^{81} +(1.10828 + 4.13615i) q^{83} +(2.42961 - 2.67225i) q^{85} +(7.58674 - 28.3141i) q^{87} +(-0.238274 + 1.35132i) q^{89} +(-13.5662 + 16.1676i) q^{91} +(-19.2577 + 8.98000i) q^{93} +(1.22033 - 9.67010i) q^{95} +(5.36386 - 2.50121i) q^{97} +(11.9168 - 14.2019i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 120 q + 6 q^{7} + 18 q^{15} - 18 q^{17} + 48 q^{21} - 36 q^{23} - 24 q^{25} - 60 q^{33} - 18 q^{35} - 12 q^{41} - 36 q^{43} + 18 q^{45} - 24 q^{47} + 96 q^{51} - 18 q^{53} + 72 q^{55} - 6 q^{57} - 24 q^{61}+ \cdots - 60 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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
\(3\) 1.74879 − 2.49753i 1.00966 − 1.44195i 0.115981 − 0.993251i \(-0.462999\pi\)
0.893683 − 0.448699i \(-0.148112\pi\)
\(4\) 0 0
\(5\) −0.863736 − 2.06251i −0.386274 − 0.922384i
\(6\) 0 0
\(7\) 2.94797 − 0.789905i 1.11423 − 0.298556i 0.345682 − 0.938352i \(-0.387648\pi\)
0.768545 + 0.639796i \(0.220981\pi\)
\(8\) 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
\(13\) −5.66470 + 3.96647i −1.57111 + 1.10010i −0.622964 + 0.782251i \(0.714072\pi\)
−0.948141 + 0.317849i \(0.897039\pi\)
\(14\) 0 0
\(15\) −6.66168 − 1.44970i −1.72004 − 0.374310i
\(16\) 0 0
\(17\) 0.682601 + 1.46384i 0.165555 + 0.355034i 0.971530 − 0.236915i \(-0.0761363\pi\)
−0.805975 + 0.591949i \(0.798359\pi\)
\(18\) 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
\(23\) 0.241328 − 0.0211134i 0.0503203 − 0.00440245i −0.0619678 − 0.998078i \(-0.519738\pi\)
0.112288 + 0.993676i \(0.464182\pi\)
\(24\) 0 0
\(25\) −3.50792 + 3.56293i −0.701584 + 0.712587i
\(26\) 0 0
\(27\) −9.70662 − 2.60088i −1.86804 − 0.500540i
\(28\) 0 0
\(29\) 9.03438 − 3.28825i 1.67764 − 0.610612i 0.684658 − 0.728864i \(-0.259951\pi\)
0.992984 + 0.118252i \(0.0377291\pi\)
\(30\) 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
\(33\) 8.94384 + 0.782485i 1.55692 + 0.136213i
\(34\) 0 0
\(35\) −4.17545 − 5.39795i −0.705781 − 0.912420i
\(36\) 0 0
\(37\) −3.20416 + 3.20416i −0.526760 + 0.526760i −0.919605 − 0.392845i \(-0.871491\pi\)
0.392845 + 0.919605i \(0.371491\pi\)
\(38\) 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
\(43\) 0.0209860 − 0.239871i 0.00320033 − 0.0365799i −0.994427 − 0.105431i \(-0.966378\pi\)
0.997627 + 0.0688511i \(0.0219333\pi\)
\(44\) 0 0
\(45\) −10.3424 + 9.55135i −1.54175 + 1.42383i
\(46\) 0 0
\(47\) −1.51038 − 0.704303i −0.220312 − 0.102733i 0.309329 − 0.950955i \(-0.399896\pi\)
−0.529641 + 0.848222i \(0.677673\pi\)
\(48\) 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
\(53\) −0.979898 − 11.2003i −0.134599 − 1.53848i −0.700289 − 0.713860i \(-0.746945\pi\)
0.565689 − 0.824619i \(-0.308610\pi\)
\(54\) 0 0
\(55\) 3.98800 − 5.23934i 0.537742 − 0.706472i
\(56\) 0 0
\(57\) 12.0365 − 5.63436i 1.59427 − 0.746290i
\(58\) 0 0
\(59\) 5.39264 + 1.96276i 0.702061 + 0.255529i 0.668291 − 0.743900i \(-0.267026\pi\)
0.0337707 + 0.999430i \(0.489248\pi\)
\(60\) 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
\(63\) −11.0212 − 15.7399i −1.38854 − 1.98305i
\(64\) 0 0
\(65\) 13.0737 + 8.25754i 1.62159 + 1.02422i
\(66\) 0 0
\(67\) −2.59266 + 5.55998i −0.316744 + 0.679259i −0.998637 − 0.0521924i \(-0.983379\pi\)
0.681893 + 0.731452i \(0.261157\pi\)
\(68\) 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
\(73\) 2.83225 + 1.98316i 0.331489 + 0.232111i 0.727457 − 0.686153i \(-0.240702\pi\)
−0.395968 + 0.918264i \(0.629591\pi\)
\(74\) 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 0.999244 − 0.0388684i \(-0.0123753\pi\)
−0.952276 + 0.305237i \(0.901264\pi\)
\(80\) 0 0
\(81\) −9.00174 + 7.55335i −1.00019 + 0.839261i
\(82\) 0 0
\(83\) 1.10828 + 4.13615i 0.121649 + 0.454001i 0.999698 − 0.0245698i \(-0.00782161\pi\)
−0.878049 + 0.478571i \(0.841155\pi\)
\(84\) 0 0
\(85\) 2.42961 − 2.67225i 0.263528 − 0.289846i
\(86\) 0 0
\(87\) 7.58674 − 28.3141i 0.813384 − 3.03559i
\(88\) 0 0
\(89\) −0.238274 + 1.35132i −0.0252570 + 0.143240i −0.994829 − 0.101568i \(-0.967614\pi\)
0.969572 + 0.244808i \(0.0787250\pi\)
\(90\) 0 0
\(91\) −13.5662 + 16.1676i −1.42213 + 1.69482i
\(92\) 0 0
\(93\) −19.2577 + 8.98000i −1.99693 + 0.931182i
\(94\) 0 0
\(95\) 1.22033 − 9.67010i 0.125203 − 0.992131i
\(96\) 0 0
\(97\) 5.36386 − 2.50121i 0.544617 − 0.253959i −0.130782 − 0.991411i \(-0.541749\pi\)
0.675400 + 0.737452i \(0.263971\pi\)
\(98\) 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.357.10 yes 120
5.3 odd 4 inner 380.2.bh.a.53.10 yes 120
19.14 odd 18 inner 380.2.bh.a.337.10 yes 120
95.33 even 36 inner 380.2.bh.a.33.10 ✓ 120
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.2.bh.a.33.10 ✓ 120 95.33 even 36 inner
380.2.bh.a.53.10 yes 120 5.3 odd 4 inner
380.2.bh.a.337.10 yes 120 19.14 odd 18 inner
380.2.bh.a.357.10 yes 120 1.1 even 1 trivial