Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [342,2,Mod(55,342)] 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("342.55"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(342, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([0, 10])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 342 = 2 \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 342.u (of order \(9\), degree \(6\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,0,0,3,0,3,3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(8)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.73088374913\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\Q(\zeta_{18})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{3} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 114)
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label 289.1
Root \(0.939693 + 0.342020i\) of defining polynomial
Character \(\chi\) \(=\) 342.289
Dual form 342.2.u.d.271.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.939693 + 0.342020i) q^{2} +(0.766044 + 0.642788i) q^{4} +(2.97178 - 2.49362i) q^{5} +(-0.613341 + 1.06234i) q^{7} +(0.500000 + 0.866025i) q^{8} +(3.64543 - 1.32683i) q^{10} +(-1.06031 - 1.83651i) q^{11} +(0.0851223 - 0.482753i) q^{13} +(-0.939693 + 0.788496i) q^{14} +(0.173648 + 0.984808i) q^{16} +(-5.19846 - 1.89209i) q^{17} +(2.77719 + 3.35965i) q^{19} +3.87939 q^{20} +(-0.368241 - 2.08840i) q^{22} +(6.85117 + 5.74881i) q^{23} +(1.74510 - 9.89695i) q^{25} +(0.245100 - 0.424525i) q^{26} +(-1.15270 + 0.419550i) q^{28} +(-7.96451 + 2.89884i) q^{29} +(-1.20574 + 2.08840i) q^{31} +(-0.173648 + 0.984808i) q^{32} +(-4.23783 - 3.55596i) q^{34} +(0.826352 + 4.68647i) q^{35} +1.69459 q^{37} +(1.46064 + 4.10689i) q^{38} +(3.64543 + 1.32683i) q^{40} +(-0.277189 - 1.57202i) q^{41} +(-5.08512 + 4.26692i) q^{43} +(0.368241 - 2.08840i) q^{44} +(4.47178 + 7.74535i) q^{46} +(2.03936 - 0.742267i) q^{47} +(2.74763 + 4.75903i) q^{49} +(5.02481 - 8.70323i) q^{50} +(0.375515 - 0.315094i) q^{52} +(-6.80793 - 5.71253i) q^{53} +(-7.73055 - 2.81369i) q^{55} -1.22668 q^{56} -8.47565 q^{58} +(-10.7306 - 3.90560i) q^{59} +(0.0320889 + 0.0269258i) q^{61} +(-1.84730 + 1.55007i) q^{62} +(-0.500000 + 0.866025i) q^{64} +(-0.950837 - 1.64690i) q^{65} +(-4.20574 + 1.53076i) q^{67} +(-2.76604 - 4.79093i) q^{68} +(-0.826352 + 4.68647i) q^{70} +(-2.02094 + 1.69577i) q^{71} +(-2.66772 - 15.1294i) q^{73} +(1.59240 + 0.579585i) q^{74} +(-0.0320889 + 4.35878i) q^{76} +2.60132 q^{77} +(-0.809278 - 4.58964i) q^{79} +(2.97178 + 2.49362i) q^{80} +(0.277189 - 1.57202i) q^{82} +(6.24035 - 10.8086i) q^{83} +(-20.1668 + 7.34013i) q^{85} +(-6.23783 + 2.27038i) q^{86} +(1.06031 - 1.83651i) q^{88} +(-1.46838 + 8.32759i) q^{89} +(0.460637 + 0.386520i) q^{91} +(1.55303 + 8.80769i) q^{92} +2.17024 q^{94} +(16.6309 + 3.05888i) q^{95} +(13.5103 + 4.91734i) q^{97} +(0.954241 + 5.41177i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 3 q^{5} + 3 q^{7} + 3 q^{8} + 6 q^{10} - 12 q^{11} - 21 q^{13} - 3 q^{17} + 6 q^{19} + 12 q^{20} + 3 q^{22} + 15 q^{23} + 9 q^{25} - 9 q^{28} - 15 q^{29} + 3 q^{31} - 6 q^{34} + 6 q^{35} + 6 q^{37}+ \cdots + 36 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/342\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(325\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{9}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.939693 + 0.342020i 0.664463 + 0.241845i
\(3\) 0 0
\(4\) 0.766044 + 0.642788i 0.383022 + 0.321394i
\(5\) 2.97178 − 2.49362i 1.32902 − 1.11518i 0.344716 − 0.938707i \(-0.387975\pi\)
0.984305 − 0.176474i \(-0.0564692\pi\)
\(6\) 0 0
\(7\) −0.613341 + 1.06234i −0.231821 + 0.401526i −0.958344 − 0.285616i \(-0.907802\pi\)
0.726523 + 0.687142i \(0.241135\pi\)
\(8\) 0.500000 + 0.866025i 0.176777 + 0.306186i
\(9\) 0 0
\(10\) 3.64543 − 1.32683i 1.15279 − 0.419580i
\(11\) −1.06031 − 1.83651i −0.319695 − 0.553727i 0.660730 − 0.750624i \(-0.270247\pi\)
−0.980424 + 0.196897i \(0.936914\pi\)
\(12\) 0 0
\(13\) 0.0851223 − 0.482753i 0.0236087 − 0.133891i −0.970725 − 0.240192i \(-0.922790\pi\)
0.994334 + 0.106301i \(0.0339006\pi\)
\(14\) −0.939693 + 0.788496i −0.251143 + 0.210734i
\(15\) 0 0
\(16\) 0.173648 + 0.984808i 0.0434120 + 0.246202i
\(17\) −5.19846 − 1.89209i −1.26081 − 0.458898i −0.376771 − 0.926306i \(-0.622966\pi\)
−0.884042 + 0.467408i \(0.845188\pi\)
\(18\) 0 0
\(19\) 2.77719 + 3.35965i 0.637131 + 0.770756i
\(20\) 3.87939 0.867457
\(21\) 0 0
\(22\) −0.368241 − 2.08840i −0.0785092 − 0.445248i
\(23\) 6.85117 + 5.74881i 1.42857 + 1.19871i 0.946554 + 0.322546i \(0.104539\pi\)
0.482013 + 0.876164i \(0.339906\pi\)
\(24\) 0 0
\(25\) 1.74510 − 9.89695i 0.349020 − 1.97939i
\(26\) 0.245100 − 0.424525i 0.0480680 − 0.0832563i
\(27\) 0 0
\(28\) −1.15270 + 0.419550i −0.217841 + 0.0792875i
\(29\) −7.96451 + 2.89884i −1.47897 + 0.538302i −0.950521 − 0.310662i \(-0.899449\pi\)
−0.528451 + 0.848964i \(0.677227\pi\)
\(30\) 0 0
\(31\) −1.20574 + 2.08840i −0.216557 + 0.375087i −0.953753 − 0.300591i \(-0.902816\pi\)
0.737196 + 0.675679i \(0.236149\pi\)
\(32\) −0.173648 + 0.984808i −0.0306970 + 0.174091i
\(33\) 0 0
\(34\) −4.23783 − 3.55596i −0.726781 − 0.609842i
\(35\) 0.826352 + 4.68647i 0.139679 + 0.792159i
\(36\) 0 0
\(37\) 1.69459 0.278589 0.139295 − 0.990251i \(-0.455516\pi\)
0.139295 + 0.990251i \(0.455516\pi\)
\(38\) 1.46064 + 4.10689i 0.236947 + 0.666225i
\(39\) 0 0
\(40\) 3.64543 + 1.32683i 0.576393 + 0.209790i
\(41\) −0.277189 − 1.57202i −0.0432896 − 0.245508i 0.955483 − 0.295048i \(-0.0953355\pi\)
−0.998772 + 0.0495401i \(0.984224\pi\)
\(42\) 0 0
\(43\) −5.08512 + 4.26692i −0.775474 + 0.650700i −0.942104 − 0.335320i \(-0.891156\pi\)
0.166631 + 0.986019i \(0.446711\pi\)
\(44\) 0.368241 − 2.08840i 0.0555144 − 0.314838i
\(45\) 0 0
\(46\) 4.47178 + 7.74535i 0.659328 + 1.14199i
\(47\) 2.03936 − 0.742267i 0.297472 − 0.108271i −0.188973 − 0.981982i \(-0.560516\pi\)
0.486444 + 0.873711i \(0.338294\pi\)
\(48\) 0 0
\(49\) 2.74763 + 4.75903i 0.392518 + 0.679861i
\(50\) 5.02481 − 8.70323i 0.710616 − 1.23082i
\(51\) 0 0
\(52\) 0.375515 − 0.315094i 0.0520745 − 0.0436957i
\(53\) −6.80793 − 5.71253i −0.935142 − 0.784677i 0.0415917 − 0.999135i \(-0.486757\pi\)
−0.976733 + 0.214458i \(0.931202\pi\)
\(54\) 0 0
\(55\) −7.73055 − 2.81369i −1.04239 − 0.379398i
\(56\) −1.22668 −0.163922
\(57\) 0 0
\(58\) −8.47565 −1.11291
\(59\) −10.7306 − 3.90560i −1.39700 − 0.508466i −0.469713 − 0.882819i \(-0.655643\pi\)
−0.927286 + 0.374353i \(0.877865\pi\)
\(60\) 0 0
\(61\) 0.0320889 + 0.0269258i 0.00410856 + 0.00344749i 0.644840 − 0.764318i \(-0.276924\pi\)
−0.640731 + 0.767765i \(0.721369\pi\)
\(62\) −1.84730 + 1.55007i −0.234607 + 0.196859i
\(63\) 0 0
\(64\) −0.500000 + 0.866025i −0.0625000 + 0.108253i
\(65\) −0.950837 − 1.64690i −0.117937 − 0.204273i
\(66\) 0 0
\(67\) −4.20574 + 1.53076i −0.513813 + 0.187012i −0.585896 − 0.810386i \(-0.699257\pi\)
0.0720836 + 0.997399i \(0.477035\pi\)
\(68\) −2.76604 − 4.79093i −0.335432 − 0.580986i
\(69\) 0 0
\(70\) −0.826352 + 4.68647i −0.0987679 + 0.560141i
\(71\) −2.02094 + 1.69577i −0.239842 + 0.201251i −0.754783 − 0.655974i \(-0.772258\pi\)
0.514941 + 0.857225i \(0.327814\pi\)
\(72\) 0 0
\(73\) −2.66772 − 15.1294i −0.312233 − 1.77076i −0.587334 − 0.809345i \(-0.699822\pi\)
0.275101 − 0.961415i \(-0.411289\pi\)
\(74\) 1.59240 + 0.579585i 0.185112 + 0.0673754i
\(75\) 0 0
\(76\) −0.0320889 + 4.35878i −0.00368085 + 0.499986i
\(77\) 2.60132 0.296448
\(78\) 0 0
\(79\) −0.809278 − 4.58964i −0.0910509 − 0.516375i −0.995886 − 0.0906133i \(-0.971117\pi\)
0.904835 − 0.425762i \(-0.139994\pi\)
\(80\) 2.97178 + 2.49362i 0.332255 + 0.278795i
\(81\) 0 0
\(82\) 0.277189 − 1.57202i 0.0306104 − 0.173600i
\(83\) 6.24035 − 10.8086i 0.684968 − 1.18640i −0.288479 − 0.957486i \(-0.593150\pi\)
0.973447 − 0.228913i \(-0.0735170\pi\)
\(84\) 0 0
\(85\) −20.1668 + 7.34013i −2.18740 + 0.796149i
\(86\) −6.23783 + 2.27038i −0.672642 + 0.244822i
\(87\) 0 0
\(88\) 1.06031 − 1.83651i 0.113029 − 0.195772i
\(89\) −1.46838 + 8.32759i −0.155648 + 0.882722i 0.802543 + 0.596594i \(0.203480\pi\)
−0.958191 + 0.286129i \(0.907632\pi\)
\(90\) 0 0
\(91\) 0.460637 + 0.386520i 0.0482879 + 0.0405184i
\(92\) 1.55303 + 8.80769i 0.161915 + 0.918265i
\(93\) 0 0
\(94\) 2.17024 0.223844
\(95\) 16.6309 + 3.05888i 1.70629 + 0.313834i
\(96\) 0 0
\(97\) 13.5103 + 4.91734i 1.37176 + 0.499280i 0.919670 − 0.392693i \(-0.128456\pi\)
0.452090 + 0.891972i \(0.350679\pi\)
\(98\) 0.954241 + 5.41177i 0.0963929 + 0.546671i
\(99\) 0 0
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 342.2.u.d.289.1 6
3.2 odd 2 114.2.i.b.61.1 yes 6
12.11 even 2 912.2.bo.c.289.1 6
19.5 even 9 inner 342.2.u.d.271.1 6
19.9 even 9 6498.2.a.bo.1.3 3
19.10 odd 18 6498.2.a.bt.1.3 3
57.5 odd 18 114.2.i.b.43.1 ✓ 6
57.29 even 18 2166.2.a.n.1.1 3
57.47 odd 18 2166.2.a.t.1.1 3
228.119 even 18 912.2.bo.c.385.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
114.2.i.b.43.1 ✓ 6 57.5 odd 18
114.2.i.b.61.1 yes 6 3.2 odd 2
342.2.u.d.271.1 6 19.5 even 9 inner
342.2.u.d.289.1 6 1.1 even 1 trivial
912.2.bo.c.289.1 6 12.11 even 2
912.2.bo.c.385.1 6 228.119 even 18
2166.2.a.n.1.1 3 57.29 even 18
2166.2.a.t.1.1 3 57.47 odd 18
6498.2.a.bo.1.3 3 19.9 even 9
6498.2.a.bt.1.3 3 19.10 odd 18