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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 117.4
Character \(\chi\) \(=\) 380.117
Dual form 380.2.bh.a.13.4

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.927188 - 0.432355i) q^{3} +(0.140904 + 2.23162i) q^{5} +(-0.231730 + 0.864827i) q^{7} +(-1.25562 - 1.49639i) q^{9} +(-3.29119 + 5.70051i) q^{11} +(-2.04641 - 4.38853i) q^{13} +(0.834209 - 2.13006i) q^{15} +(0.154168 + 1.76215i) q^{17} +(-3.61204 + 2.43990i) q^{19} +(0.588769 - 0.701667i) q^{21} +(5.10986 + 3.57796i) q^{23} +(-4.96029 + 0.628891i) q^{25} +(1.31157 + 4.89484i) q^{27} +(-3.21899 + 2.70106i) q^{29} +(-2.00481 + 1.15748i) q^{31} +(5.51619 - 3.86248i) q^{33} +(-1.96262 - 0.395276i) q^{35} +(-3.30447 + 3.30447i) q^{37} +4.95377i q^{39} +(-2.25386 - 6.19243i) q^{41} +(1.82673 + 2.60884i) q^{43} +(3.16245 - 3.01291i) q^{45} +(5.57863 + 0.488067i) q^{47} +(5.36795 + 3.09919i) q^{49} +(0.618932 - 1.70050i) q^{51} +(6.39127 - 9.12769i) q^{53} +(-13.1851 - 6.54147i) q^{55} +(4.40395 - 0.700564i) q^{57} +(1.36717 + 1.14719i) q^{59} +(1.13891 + 6.45906i) q^{61} +(1.58508 - 0.739134i) q^{63} +(9.50521 - 5.18517i) q^{65} +(0.953238 - 10.8956i) q^{67} +(-3.19085 - 5.52671i) q^{69} +(-10.2750 - 1.81176i) q^{71} +(3.06340 - 6.56949i) q^{73} +(4.87103 + 1.56151i) q^{75} +(-4.16729 - 4.16729i) q^{77} +(-1.84631 + 0.672002i) q^{79} +(-0.117372 + 0.665647i) q^{81} +(-15.1740 - 4.06586i) q^{83} +(-3.91074 + 0.592340i) q^{85} +(4.15243 - 1.11264i) q^{87} +(13.1850 + 4.79895i) q^{89} +(4.26953 - 0.752834i) q^{91} +(2.35928 - 0.206410i) q^{93} +(-5.95390 - 7.71693i) q^{95} +(12.2510 - 1.07182i) q^{97} +(12.6626 - 2.23276i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 120 q + 6 q^{7} + 18 q^{15} - 18 q^{17} + 48 q^{21} - 36 q^{23} - 24 q^{25} - 60 q^{33} - 18 q^{35} - 12 q^{41} - 36 q^{43} + 18 q^{45} - 24 q^{47} + 96 q^{51} - 18 q^{53} + 72 q^{55} - 6 q^{57} - 24 q^{61}+ \cdots - 60 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(21\) \(77\) \(191\)
\(\chi(n)\) \(e\left(\frac{13}{18}\right)\) \(e\left(\frac{1}{4}\right)\) \(1\)

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
\(3\) −0.927188 − 0.432355i −0.535312 − 0.249620i 0.136112 − 0.990693i \(-0.456539\pi\)
−0.671424 + 0.741073i \(0.734317\pi\)
\(4\) 0 0
\(5\) 0.140904 + 2.23162i 0.0630143 + 0.998013i
\(6\) 0 0
\(7\) −0.231730 + 0.864827i −0.0875856 + 0.326874i −0.995791 − 0.0916503i \(-0.970786\pi\)
0.908206 + 0.418524i \(0.137452\pi\)
\(8\) 0 0
\(9\) −1.25562 − 1.49639i −0.418539 − 0.498795i
\(10\) 0 0
\(11\) −3.29119 + 5.70051i −0.992331 + 1.71877i −0.389118 + 0.921188i \(0.627220\pi\)
−0.603213 + 0.797580i \(0.706113\pi\)
\(12\) 0 0
\(13\) −2.04641 − 4.38853i −0.567571 − 1.21716i −0.955369 − 0.295416i \(-0.904542\pi\)
0.387798 − 0.921744i \(-0.373236\pi\)
\(14\) 0 0
\(15\) 0.834209 − 2.13006i 0.215392 − 0.549978i
\(16\) 0 0
\(17\) 0.154168 + 1.76215i 0.0373913 + 0.427384i 0.991701 + 0.128565i \(0.0410372\pi\)
−0.954310 + 0.298819i \(0.903407\pi\)
\(18\) 0 0
\(19\) −3.61204 + 2.43990i −0.828660 + 0.559752i
\(20\) 0 0
\(21\) 0.588769 − 0.701667i 0.128480 − 0.153116i
\(22\) 0 0
\(23\) 5.10986 + 3.57796i 1.06548 + 0.746056i 0.968446 − 0.249225i \(-0.0801758\pi\)
0.0970331 + 0.995281i \(0.469065\pi\)
\(24\) 0 0
\(25\) −4.96029 + 0.628891i −0.992058 + 0.125778i
\(26\) 0 0
\(27\) 1.31157 + 4.89484i 0.252411 + 0.942012i
\(28\) 0 0
\(29\) −3.21899 + 2.70106i −0.597752 + 0.501574i −0.890722 − 0.454548i \(-0.849801\pi\)
0.292970 + 0.956122i \(0.405356\pi\)
\(30\) 0 0
\(31\) −2.00481 + 1.15748i −0.360075 + 0.207889i −0.669113 − 0.743160i \(-0.733326\pi\)
0.309039 + 0.951049i \(0.399993\pi\)
\(32\) 0 0
\(33\) 5.51619 − 3.86248i 0.960246 − 0.672372i
\(34\) 0 0
\(35\) −1.96262 − 0.395276i −0.331743 − 0.0668138i
\(36\) 0 0
\(37\) −3.30447 + 3.30447i −0.543251 + 0.543251i −0.924481 − 0.381229i \(-0.875501\pi\)
0.381229 + 0.924481i \(0.375501\pi\)
\(38\) 0 0
\(39\) 4.95377i 0.793238i
\(40\) 0 0
\(41\) −2.25386 − 6.19243i −0.351994 − 0.967096i −0.981729 − 0.190285i \(-0.939059\pi\)
0.629735 − 0.776810i \(-0.283164\pi\)
\(42\) 0 0
\(43\) 1.82673 + 2.60884i 0.278574 + 0.397845i 0.933919 − 0.357485i \(-0.116366\pi\)
−0.655345 + 0.755330i \(0.727477\pi\)
\(44\) 0 0
\(45\) 3.16245 − 3.01291i 0.471430 − 0.449138i
\(46\) 0 0
\(47\) 5.57863 + 0.488067i 0.813727 + 0.0711919i 0.486419 − 0.873726i \(-0.338303\pi\)
0.327308 + 0.944918i \(0.393858\pi\)
\(48\) 0 0
\(49\) 5.36795 + 3.09919i 0.766850 + 0.442741i
\(50\) 0 0
\(51\) 0.618932 − 1.70050i 0.0866678 − 0.238118i
\(52\) 0 0
\(53\) 6.39127 − 9.12769i 0.877909 − 1.25378i −0.0883182 − 0.996092i \(-0.528149\pi\)
0.966227 − 0.257692i \(-0.0829619\pi\)
\(54\) 0 0
\(55\) −13.1851 − 6.54147i −1.77788 − 0.882052i
\(56\) 0 0
\(57\) 4.40395 − 0.700564i 0.583317 − 0.0927920i
\(58\) 0 0
\(59\) 1.36717 + 1.14719i 0.177991 + 0.149352i 0.727430 − 0.686181i \(-0.240714\pi\)
−0.549440 + 0.835533i \(0.685159\pi\)
\(60\) 0 0
\(61\) 1.13891 + 6.45906i 0.145822 + 0.826998i 0.966703 + 0.255900i \(0.0823716\pi\)
−0.820881 + 0.571099i \(0.806517\pi\)
\(62\) 0 0
\(63\) 1.58508 − 0.739134i 0.199701 − 0.0931221i
\(64\) 0 0
\(65\) 9.50521 − 5.18517i 1.17898 − 0.643142i
\(66\) 0 0
\(67\) 0.953238 − 10.8956i 0.116457 − 1.33110i −0.683017 − 0.730403i \(-0.739332\pi\)
0.799473 − 0.600702i \(-0.205112\pi\)
\(68\) 0 0
\(69\) −3.19085 − 5.52671i −0.384133 − 0.665338i
\(70\) 0 0
\(71\) −10.2750 − 1.81176i −1.21942 − 0.215016i −0.473343 − 0.880878i \(-0.656953\pi\)
−0.746075 + 0.665862i \(0.768064\pi\)
\(72\) 0 0
\(73\) 3.06340 − 6.56949i 0.358544 − 0.768901i −0.641454 − 0.767161i \(-0.721669\pi\)
0.999998 − 0.00173942i \(-0.000553676\pi\)
\(74\) 0 0
\(75\) 4.87103 + 1.56151i 0.562458 + 0.180307i
\(76\) 0 0
\(77\) −4.16729 − 4.16729i −0.474906 − 0.474906i
\(78\) 0 0
\(79\) −1.84631 + 0.672002i −0.207726 + 0.0756061i −0.443788 − 0.896132i \(-0.646366\pi\)
0.236062 + 0.971738i \(0.424143\pi\)
\(80\) 0 0
\(81\) −0.117372 + 0.665647i −0.0130413 + 0.0739608i
\(82\) 0 0
\(83\) −15.1740 − 4.06586i −1.66556 − 0.446287i −0.701655 − 0.712517i \(-0.747555\pi\)
−0.963909 + 0.266231i \(0.914222\pi\)
\(84\) 0 0
\(85\) −3.91074 + 0.592340i −0.424179 + 0.0642483i
\(86\) 0 0
\(87\) 4.15243 − 1.11264i 0.445187 − 0.119287i
\(88\) 0 0
\(89\) 13.1850 + 4.79895i 1.39761 + 0.508688i 0.927467 − 0.373904i \(-0.121981\pi\)
0.470140 + 0.882592i \(0.344203\pi\)
\(90\) 0 0
\(91\) 4.26953 − 0.752834i 0.447569 − 0.0789184i
\(92\) 0 0
\(93\) 2.35928 − 0.206410i 0.244646 − 0.0214037i
\(94\) 0 0
\(95\) −5.95390 − 7.71693i −0.610857 − 0.791741i
\(96\) 0 0
\(97\) 12.2510 − 1.07182i 1.24390 − 0.108827i 0.553865 − 0.832606i \(-0.313152\pi\)
0.690031 + 0.723779i \(0.257597\pi\)
\(98\) 0 0
\(99\) 12.6626 − 2.23276i 1.27264 − 0.224401i
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.117.4 yes 120
5.3 odd 4 inner 380.2.bh.a.193.7 yes 120
19.13 odd 18 inner 380.2.bh.a.317.7 yes 120
95.13 even 36 inner 380.2.bh.a.13.4 ✓ 120
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.2.bh.a.13.4 ✓ 120 95.13 even 36 inner
380.2.bh.a.117.4 yes 120 1.1 even 1 trivial
380.2.bh.a.193.7 yes 120 5.3 odd 4 inner
380.2.bh.a.317.7 yes 120 19.13 odd 18 inner