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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 193.7
Character \(\chi\) \(=\) 380.193
Dual form 380.2.bh.a.317.7

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.432355 - 0.927188i) q^{3} +(-0.895667 + 2.04885i) q^{5} +(0.864827 + 0.231730i) q^{7} +(1.25562 + 1.49639i) q^{9} +(-3.29119 + 5.70051i) q^{11} +(4.38853 - 2.04641i) q^{13} +(1.51242 + 1.71628i) q^{15} +(1.76215 - 0.154168i) q^{17} +(3.61204 - 2.43990i) q^{19} +(0.588769 - 0.701667i) q^{21} +(-3.57796 + 5.10986i) q^{23} +(-3.39556 - 3.67017i) q^{25} +(4.89484 - 1.31157i) q^{27} +(3.21899 - 2.70106i) q^{29} +(-2.00481 + 1.15748i) q^{31} +(3.86248 + 5.51619i) q^{33} +(-1.24938 + 1.56435i) q^{35} +(3.30447 + 3.30447i) q^{37} -4.95377i q^{39} +(-2.25386 - 6.19243i) q^{41} +(-2.60884 + 1.82673i) q^{43} +(-4.19048 + 1.23230i) q^{45} +(0.488067 - 5.57863i) q^{47} +(-5.36795 - 3.09919i) q^{49} +(0.618932 - 1.70050i) q^{51} +(9.12769 + 6.39127i) q^{53} +(-8.73167 - 11.8489i) q^{55} +(-0.700564 - 4.40395i) q^{57} +(-1.36717 - 1.14719i) q^{59} +(1.13891 + 6.45906i) q^{61} +(0.739134 + 1.58508i) q^{63} +(0.262113 + 10.8243i) q^{65} +(-10.8956 - 0.953238i) q^{67} +(3.19085 + 5.52671i) q^{69} +(-10.2750 - 1.81176i) q^{71} +(6.56949 + 3.06340i) q^{73} +(-4.87103 + 1.56151i) q^{75} +(-4.16729 + 4.16729i) q^{77} +(1.84631 - 0.672002i) q^{79} +(-0.117372 + 0.665647i) q^{81} +(4.06586 - 15.1740i) q^{83} +(-1.26243 + 3.74846i) q^{85} +(-1.11264 - 4.15243i) q^{87} +(-13.1850 - 4.79895i) q^{89} +(4.26953 - 0.752834i) q^{91} +(0.206410 + 2.35928i) q^{93} +(1.76380 + 9.58587i) q^{95} +(-1.07182 - 12.2510i) q^{97} +(-12.6626 + 2.23276i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 120 q + 6 q^{7} + 18 q^{15} - 18 q^{17} + 48 q^{21} - 36 q^{23} - 24 q^{25} - 60 q^{33} - 18 q^{35} - 12 q^{41} - 36 q^{43} + 18 q^{45} - 24 q^{47} + 96 q^{51} - 18 q^{53} + 72 q^{55} - 6 q^{57} - 24 q^{61}+ \cdots - 60 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(21\) \(77\) \(191\)
\(\chi(n)\) \(e\left(\frac{13}{18}\right)\) \(e\left(\frac{3}{4}\right)\) \(1\)

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
\(3\) 0.432355 − 0.927188i 0.249620 − 0.535312i −0.741073 − 0.671424i \(-0.765683\pi\)
0.990693 + 0.136112i \(0.0434607\pi\)
\(4\) 0 0
\(5\) −0.895667 + 2.04885i −0.400554 + 0.916273i
\(6\) 0 0
\(7\) 0.864827 + 0.231730i 0.326874 + 0.0875856i 0.418524 − 0.908206i \(-0.362548\pi\)
−0.0916503 + 0.995791i \(0.529214\pi\)
\(8\) 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
\(13\) 4.38853 − 2.04641i 1.21716 − 0.567571i 0.295416 − 0.955369i \(-0.404542\pi\)
0.921744 + 0.387798i \(0.126764\pi\)
\(14\) 0 0
\(15\) 1.51242 + 1.71628i 0.390506 + 0.443142i
\(16\) 0 0
\(17\) 1.76215 − 0.154168i 0.427384 − 0.0373913i 0.128565 − 0.991701i \(-0.458963\pi\)
0.298819 + 0.954310i \(0.403407\pi\)
\(18\) 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
\(23\) −3.57796 + 5.10986i −0.746056 + 1.06548i 0.249225 + 0.968446i \(0.419824\pi\)
−0.995281 + 0.0970331i \(0.969065\pi\)
\(24\) 0 0
\(25\) −3.39556 − 3.67017i −0.679112 − 0.734034i
\(26\) 0 0
\(27\) 4.89484 − 1.31157i 0.942012 − 0.252411i
\(28\) 0 0
\(29\) 3.21899 − 2.70106i 0.597752 − 0.501574i −0.292970 − 0.956122i \(-0.594644\pi\)
0.890722 + 0.454548i \(0.150199\pi\)
\(30\) 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
\(33\) 3.86248 + 5.51619i 0.672372 + 0.960246i
\(34\) 0 0
\(35\) −1.24938 + 1.56435i −0.211183 + 0.264423i
\(36\) 0 0
\(37\) 3.30447 + 3.30447i 0.543251 + 0.543251i 0.924481 − 0.381229i \(-0.124499\pi\)
−0.381229 + 0.924481i \(0.624499\pi\)
\(38\) 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
\(43\) −2.60884 + 1.82673i −0.397845 + 0.278574i −0.755330 − 0.655345i \(-0.772523\pi\)
0.357485 + 0.933919i \(0.383634\pi\)
\(44\) 0 0
\(45\) −4.19048 + 1.23230i −0.624680 + 0.183701i
\(46\) 0 0
\(47\) 0.488067 − 5.57863i 0.0711919 − 0.813727i −0.873726 − 0.486419i \(-0.838303\pi\)
0.944918 − 0.327308i \(-0.106142\pi\)
\(48\) 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
\(53\) 9.12769 + 6.39127i 1.25378 + 0.877909i 0.996092 − 0.0883182i \(-0.0281492\pi\)
0.257692 + 0.966227i \(0.417038\pi\)
\(54\) 0 0
\(55\) −8.73167 − 11.8489i −1.17738 − 1.59771i
\(56\) 0 0
\(57\) −0.700564 − 4.40395i −0.0927920 − 0.583317i
\(58\) 0 0
\(59\) −1.36717 − 1.14719i −0.177991 − 0.149352i 0.549440 − 0.835533i \(-0.314841\pi\)
−0.727430 + 0.686181i \(0.759286\pi\)
\(60\) 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
\(63\) 0.739134 + 1.58508i 0.0931221 + 0.199701i
\(64\) 0 0
\(65\) 0.262113 + 10.8243i 0.0325111 + 1.34259i
\(66\) 0 0
\(67\) −10.8956 − 0.953238i −1.33110 − 0.116457i −0.600702 − 0.799473i \(-0.705112\pi\)
−0.730403 + 0.683017i \(0.760668\pi\)
\(68\) 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
\(73\) 6.56949 + 3.06340i 0.768901 + 0.358544i 0.767161 − 0.641454i \(-0.221669\pi\)
0.00173942 + 0.999998i \(0.499446\pi\)
\(74\) 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 −0.236062 − 0.971738i \(-0.575857\pi\)
0.443788 + 0.896132i \(0.353634\pi\)
\(80\) 0 0
\(81\) −0.117372 + 0.665647i −0.0130413 + 0.0739608i
\(82\) 0 0
\(83\) 4.06586 − 15.1740i 0.446287 − 1.66556i −0.266231 − 0.963909i \(-0.585778\pi\)
0.712517 − 0.701655i \(-0.247555\pi\)
\(84\) 0 0
\(85\) −1.26243 + 3.74846i −0.136930 + 0.406578i
\(86\) 0 0
\(87\) −1.11264 − 4.15243i −0.119287 − 0.445187i
\(88\) 0 0
\(89\) −13.1850 − 4.79895i −1.39761 − 0.508688i −0.470140 − 0.882592i \(-0.655797\pi\)
−0.927467 + 0.373904i \(0.878019\pi\)
\(90\) 0 0
\(91\) 4.26953 − 0.752834i 0.447569 − 0.0789184i
\(92\) 0 0
\(93\) 0.206410 + 2.35928i 0.0214037 + 0.244646i
\(94\) 0 0
\(95\) 1.76380 + 9.58587i 0.180962 + 0.983490i
\(96\) 0 0
\(97\) −1.07182 − 12.2510i −0.108827 − 1.24390i −0.832606 − 0.553865i \(-0.813152\pi\)
0.723779 − 0.690031i \(-0.242403\pi\)
\(98\) 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.193.7 yes 120
5.2 odd 4 inner 380.2.bh.a.117.4 yes 120
19.13 odd 18 inner 380.2.bh.a.13.4 ✓ 120
95.32 even 36 inner 380.2.bh.a.317.7 yes 120
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.2.bh.a.13.4 ✓ 120 19.13 odd 18 inner
380.2.bh.a.117.4 yes 120 5.2 odd 4 inner
380.2.bh.a.193.7 yes 120 1.1 even 1 trivial
380.2.bh.a.317.7 yes 120 95.32 even 36 inner