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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 317.7
Character \(\chi\) \(=\) 380.317
Dual form 380.2.bh.a.193.7

$q$-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 \(\left(\mathbb{Z}/380\mathbb{Z}\right)^\times\).

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

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
\(3\) 0.432355 + 0.927188i 0.249620 + 0.535312i 0.990693 − 0.136112i \(-0.0434607\pi\)
−0.741073 + 0.671424i \(0.765683\pi\)
\(4\) 0 0
\(5\) −0.895667 − 2.04885i −0.400554 − 0.916273i
\(6\) 0 0
\(7\) 0.864827 − 0.231730i 0.326874 − 0.0875856i −0.0916503 − 0.995791i \(-0.529214\pi\)
0.418524 + 0.908206i \(0.362548\pi\)
\(8\) 0 0
\(9\) 1.25562 − 1.49639i 0.418539 − 0.498795i
\(10\) 0 0
\(11\) −3.29119 − 5.70051i −0.992331 − 1.71877i −0.603213 − 0.797580i \(-0.706113\pi\)
−0.389118 − 0.921188i \(-0.627220\pi\)
\(12\) 0 0
\(13\) 4.38853 + 2.04641i 1.21716 + 0.567571i 0.921744 − 0.387798i \(-0.126764\pi\)
0.295416 + 0.955369i \(0.404542\pi\)
\(14\) 0 0
\(15\) 1.51242 − 1.71628i 0.390506 − 0.443142i
\(16\) 0 0
\(17\) 1.76215 + 0.154168i 0.427384 + 0.0373913i 0.298819 − 0.954310i \(-0.403407\pi\)
0.128565 + 0.991701i \(0.458963\pi\)
\(18\) 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 −0.995281 − 0.0970331i \(-0.969065\pi\)
0.249225 − 0.968446i \(-0.419824\pi\)
\(24\) 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 0.890722 − 0.454548i \(-0.150199\pi\)
−0.292970 + 0.956122i \(0.594644\pi\)
\(30\) 0 0
\(31\) −2.00481 − 1.15748i −0.360075 − 0.207889i 0.309039 − 0.951049i \(-0.399993\pi\)
−0.669113 + 0.743160i \(0.733326\pi\)
\(32\) 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 −0.381229 − 0.924481i \(-0.624499\pi\)
0.924481 + 0.381229i \(0.124499\pi\)
\(38\) 0 0
\(39\) 4.95377i 0.793238i
\(40\) 0 0
\(41\) −2.25386 + 6.19243i −0.351994 + 0.967096i 0.629735 + 0.776810i \(0.283164\pi\)
−0.981729 + 0.190285i \(0.939059\pi\)
\(42\) 0 0
\(43\) −2.60884 − 1.82673i −0.397845 − 0.278574i 0.357485 − 0.933919i \(-0.383634\pi\)
−0.755330 + 0.655345i \(0.772523\pi\)
\(44\) 0 0
\(45\) −4.19048 − 1.23230i −0.624680 − 0.183701i
\(46\) 0 0
\(47\) 0.488067 + 5.57863i 0.0711919 + 0.813727i 0.944918 + 0.327308i \(0.106142\pi\)
−0.873726 + 0.486419i \(0.838303\pi\)
\(48\) 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 0.257692 − 0.966227i \(-0.417038\pi\)
0.996092 + 0.0883182i \(0.0281492\pi\)
\(54\) 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 −0.727430 − 0.686181i \(-0.759286\pi\)
0.549440 + 0.835533i \(0.314841\pi\)
\(60\) 0 0
\(61\) 1.13891 − 6.45906i 0.145822 − 0.826998i −0.820881 − 0.571099i \(-0.806517\pi\)
0.966703 − 0.255900i \(-0.0823716\pi\)
\(62\) 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 −0.730403 − 0.683017i \(-0.760668\pi\)
−0.600702 + 0.799473i \(0.705112\pi\)
\(68\) 0 0
\(69\) 3.19085 − 5.52671i 0.384133 − 0.665338i
\(70\) 0 0
\(71\) −10.2750 + 1.81176i −1.21942 + 0.215016i −0.746075 − 0.665862i \(-0.768064\pi\)
−0.473343 + 0.880878i \(0.656953\pi\)
\(72\) 0 0
\(73\) 6.56949 − 3.06340i 0.768901 − 0.358544i 0.00173942 − 0.999998i \(-0.499446\pi\)
0.767161 + 0.641454i \(0.221669\pi\)
\(74\) 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 \(-0.353634\pi\)
−0.236062 + 0.971738i \(0.575857\pi\)
\(80\) 0 0
\(81\) −0.117372 − 0.665647i −0.0130413 − 0.0739608i
\(82\) 0 0
\(83\) 4.06586 + 15.1740i 0.446287 + 1.66556i 0.712517 + 0.701655i \(0.247555\pi\)
−0.266231 + 0.963909i \(0.585778\pi\)
\(84\) 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 −0.927467 − 0.373904i \(-0.878019\pi\)
−0.470140 + 0.882592i \(0.655797\pi\)
\(90\) 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 0.723779 + 0.690031i \(0.242403\pi\)
−0.832606 + 0.553865i \(0.813152\pi\)
\(98\) 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.317.7 yes 120
5.3 odd 4 inner 380.2.bh.a.13.4 ✓ 120
19.3 odd 18 inner 380.2.bh.a.117.4 yes 120
95.3 even 36 inner 380.2.bh.a.193.7 yes 120
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.2.bh.a.13.4 ✓ 120 5.3 odd 4 inner
380.2.bh.a.117.4 yes 120 19.3 odd 18 inner
380.2.bh.a.193.7 yes 120 95.3 even 36 inner
380.2.bh.a.317.7 yes 120 1.1 even 1 trivial