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(293,342)] 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("342.293"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(342, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([5, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 342 = 2 \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 342.n (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [18,-18,-1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.73088374913\)
Analytic rank: \(0\)
Dimension: \(18\)
Relative dimension: \(9\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} - 5 x^{17} + 13 x^{16} - 30 x^{15} + 54 x^{14} - 69 x^{13} + 66 x^{12} + 36 x^{11} - 243 x^{10} + \cdots + 19683 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3^{3} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 293.6
Root \(-1.68196 - 0.413538i\) of defining polynomial
Character \(\chi\) \(=\) 342.293
Dual form 342.2.n.f.335.6

$q$-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-1.00000 q^{2} +(0.482845 - 1.66339i) q^{3} +1.00000 q^{4} +(1.62258 - 0.936797i) q^{5} +(-0.482845 + 1.66339i) q^{6} +(-0.813726 - 1.40941i) q^{7} -1.00000 q^{8} +(-2.53372 - 1.60632i) q^{9} +(-1.62258 + 0.936797i) q^{10} +(4.29611 - 2.48036i) q^{11} +(0.482845 - 1.66339i) q^{12} +2.89351i q^{13} +(0.813726 + 1.40941i) q^{14} +(-0.774802 - 3.15131i) q^{15} +1.00000 q^{16} +(-5.64383 - 3.25847i) q^{17} +(2.53372 + 1.60632i) q^{18} +(2.60278 + 3.49650i) q^{19} +(1.62258 - 0.936797i) q^{20} +(-2.73731 + 0.673013i) q^{21} +(-4.29611 + 2.48036i) q^{22} -0.380616i q^{23} +(-0.482845 + 1.66339i) q^{24} +(-0.744823 + 1.29007i) q^{25} -2.89351i q^{26} +(-3.89533 + 3.43896i) q^{27} +(-0.813726 - 1.40941i) q^{28} +(4.38107 - 7.58824i) q^{29} +(0.774802 + 3.15131i) q^{30} +(-2.03149 - 1.17288i) q^{31} -1.00000 q^{32} +(-2.05145 - 8.34374i) q^{33} +(5.64383 + 3.25847i) q^{34} +(-2.64067 - 1.52459i) q^{35} +(-2.53372 - 1.60632i) q^{36} +0.616895i q^{37} +(-2.60278 - 3.49650i) q^{38} +(4.81303 + 1.39712i) q^{39} +(-1.62258 + 0.936797i) q^{40} +(-4.25502 - 7.36991i) q^{41} +(2.73731 - 0.673013i) q^{42} +0.834414 q^{43} +(4.29611 - 2.48036i) q^{44} +(-5.61596 - 0.232797i) q^{45} +0.380616i q^{46} +(2.32983 + 1.34513i) q^{47} +(0.482845 - 1.66339i) q^{48} +(2.17570 - 3.76842i) q^{49} +(0.744823 - 1.29007i) q^{50} +(-8.14519 + 7.81454i) q^{51} +2.89351i q^{52} +(5.03924 + 8.72823i) q^{53} +(3.89533 - 3.43896i) q^{54} +(4.64719 - 8.04917i) q^{55} +(0.813726 + 1.40941i) q^{56} +(7.07278 - 2.64117i) q^{57} +(-4.38107 + 7.58824i) q^{58} +(6.26424 + 10.8500i) q^{59} +(-0.774802 - 3.15131i) q^{60} +(3.43040 - 5.94162i) q^{61} +(2.03149 + 1.17288i) q^{62} +(-0.202214 + 4.87817i) q^{63} +1.00000 q^{64} +(2.71063 + 4.69495i) q^{65} +(2.05145 + 8.34374i) q^{66} +13.9657i q^{67} +(-5.64383 - 3.25847i) q^{68} +(-0.633113 - 0.183779i) q^{69} +(2.64067 + 1.52459i) q^{70} +(-3.68375 + 6.38043i) q^{71} +(2.53372 + 1.60632i) q^{72} +(-4.23388 + 7.33329i) q^{73} -0.616895i q^{74} +(1.78626 + 1.86183i) q^{75} +(2.60278 + 3.49650i) q^{76} +(-6.99172 - 4.03667i) q^{77} +(-4.81303 - 1.39712i) q^{78} -7.11620i q^{79} +(1.62258 - 0.936797i) q^{80} +(3.83948 + 8.13992i) q^{81} +(4.25502 + 7.36991i) q^{82} +(9.46181 - 5.46278i) q^{83} +(-2.73731 + 0.673013i) q^{84} -12.2101 q^{85} -0.834414 q^{86} +(-10.5068 - 10.9514i) q^{87} +(-4.29611 + 2.48036i) q^{88} +(1.28517 + 2.22598i) q^{89} +(5.61596 + 0.232797i) q^{90} +(4.07815 - 2.35452i) q^{91} -0.380616i q^{92} +(-2.93186 + 2.81284i) q^{93} +(-2.32983 - 1.34513i) q^{94} +(7.49873 + 3.23508i) q^{95} +(-0.482845 + 1.66339i) q^{96} +4.39065i q^{97} +(-2.17570 + 3.76842i) q^{98} +(-14.8694 - 0.616379i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q - 18 q^{2} - q^{3} + 18 q^{4} + 3 q^{5} + q^{6} - 18 q^{8} + 5 q^{9} - 3 q^{10} + 15 q^{11} - q^{12} - 17 q^{15} + 18 q^{16} - 15 q^{17} - 5 q^{18} + 6 q^{19} + 3 q^{20} + q^{21} - 15 q^{22} + q^{24}+ \cdots - 56 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(191\) \(325\)
\(\chi(n)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{1}{6}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 0.482845 − 1.66339i 0.278771 − 0.960358i
\(4\) 1.00000 0.500000
\(5\) 1.62258 − 0.936797i 0.725640 − 0.418948i −0.0911851 − 0.995834i \(-0.529066\pi\)
0.816825 + 0.576886i \(0.195732\pi\)
\(6\) −0.482845 + 1.66339i −0.197121 + 0.679075i
\(7\) −0.813726 − 1.40941i −0.307560 − 0.532709i 0.670268 − 0.742119i \(-0.266179\pi\)
−0.977828 + 0.209410i \(0.932846\pi\)
\(8\) −1.00000 −0.353553
\(9\) −2.53372 − 1.60632i −0.844574 − 0.535439i
\(10\) −1.62258 + 0.936797i −0.513105 + 0.296241i
\(11\) 4.29611 − 2.48036i 1.29533 − 0.747858i 0.315734 − 0.948848i \(-0.397749\pi\)
0.979593 + 0.200990i \(0.0644159\pi\)
\(12\) 0.482845 − 1.66339i 0.139385 − 0.480179i
\(13\) 2.89351i 0.802515i 0.915965 + 0.401257i \(0.131427\pi\)
−0.915965 + 0.401257i \(0.868573\pi\)
\(14\) 0.813726 + 1.40941i 0.217477 + 0.376682i
\(15\) −0.774802 − 3.15131i −0.200053 − 0.813664i
\(16\) 1.00000 0.250000
\(17\) −5.64383 − 3.25847i −1.36883 − 0.790294i −0.378051 − 0.925785i \(-0.623406\pi\)
−0.990779 + 0.135491i \(0.956739\pi\)
\(18\) 2.53372 + 1.60632i 0.597204 + 0.378613i
\(19\) 2.60278 + 3.49650i 0.597119 + 0.802153i
\(20\) 1.62258 − 0.936797i 0.362820 − 0.209474i
\(21\) −2.73731 + 0.673013i −0.597330 + 0.146864i
\(22\) −4.29611 + 2.48036i −0.915935 + 0.528815i
\(23\) − 0.380616i − 0.0793640i −0.999212 − 0.0396820i \(-0.987366\pi\)
0.999212 − 0.0396820i \(-0.0126345\pi\)
\(24\) −0.482845 + 1.66339i −0.0985604 + 0.339538i
\(25\) −0.744823 + 1.29007i −0.148965 + 0.258014i
\(26\) − 2.89351i − 0.567463i
\(27\) −3.89533 + 3.43896i −0.749656 + 0.661828i
\(28\) −0.813726 − 1.40941i −0.153780 − 0.266354i
\(29\) 4.38107 − 7.58824i 0.813545 − 1.40910i −0.0968233 − 0.995302i \(-0.530868\pi\)
0.910368 − 0.413799i \(-0.135799\pi\)
\(30\) 0.774802 + 3.15131i 0.141459 + 0.575348i
\(31\) −2.03149 − 1.17288i −0.364867 − 0.210656i 0.306347 − 0.951920i \(-0.400893\pi\)
−0.671214 + 0.741264i \(0.734227\pi\)
\(32\) −1.00000 −0.176777
\(33\) −2.05145 − 8.34374i −0.357111 − 1.45246i
\(34\) 5.64383 + 3.25847i 0.967909 + 0.558822i
\(35\) −2.64067 − 1.52459i −0.446355 − 0.257703i
\(36\) −2.53372 − 1.60632i −0.422287 − 0.267720i
\(37\) 0.616895i 0.101417i 0.998713 + 0.0507085i \(0.0161479\pi\)
−0.998713 + 0.0507085i \(0.983852\pi\)
\(38\) −2.60278 − 3.49650i −0.422227 − 0.567208i
\(39\) 4.81303 + 1.39712i 0.770701 + 0.223718i
\(40\) −1.62258 + 0.936797i −0.256552 + 0.148121i
\(41\) −4.25502 − 7.36991i −0.664522 − 1.15099i −0.979415 − 0.201859i \(-0.935302\pi\)
0.314892 − 0.949127i \(-0.398032\pi\)
\(42\) 2.73731 − 0.673013i 0.422376 − 0.103848i
\(43\) 0.834414 0.127247 0.0636235 − 0.997974i \(-0.479734\pi\)
0.0636235 + 0.997974i \(0.479734\pi\)
\(44\) 4.29611 − 2.48036i 0.647664 − 0.373929i
\(45\) −5.61596 − 0.232797i −0.837178 − 0.0347034i
\(46\) 0.380616i 0.0561188i
\(47\) 2.32983 + 1.34513i 0.339841 + 0.196207i 0.660202 − 0.751088i \(-0.270471\pi\)
−0.320361 + 0.947296i \(0.603804\pi\)
\(48\) 0.482845 − 1.66339i 0.0696927 − 0.240089i
\(49\) 2.17570 − 3.76842i 0.310814 − 0.538346i
\(50\) 0.744823 − 1.29007i 0.105334 − 0.182444i
\(51\) −8.14519 + 7.81454i −1.14055 + 1.09426i
\(52\) 2.89351i 0.401257i
\(53\) 5.03924 + 8.72823i 0.692193 + 1.19891i 0.971118 + 0.238601i \(0.0766889\pi\)
−0.278924 + 0.960313i \(0.589978\pi\)
\(54\) 3.89533 − 3.43896i 0.530087 − 0.467983i
\(55\) 4.64719 − 8.04917i 0.626627 − 1.08535i
\(56\) 0.813726 + 1.40941i 0.108739 + 0.188341i
\(57\) 7.07278 − 2.64117i 0.936813 − 0.349831i
\(58\) −4.38107 + 7.58824i −0.575263 + 0.996385i
\(59\) 6.26424 + 10.8500i 0.815535 + 1.41255i 0.908943 + 0.416920i \(0.136891\pi\)
−0.0934080 + 0.995628i \(0.529776\pi\)
\(60\) −0.774802 − 3.15131i −0.100027 − 0.406832i
\(61\) 3.43040 − 5.94162i 0.439217 − 0.760747i −0.558412 − 0.829564i \(-0.688589\pi\)
0.997629 + 0.0688171i \(0.0219225\pi\)
\(62\) 2.03149 + 1.17288i 0.258000 + 0.148956i
\(63\) −0.202214 + 4.87817i −0.0254765 + 0.614591i
\(64\) 1.00000 0.125000
\(65\) 2.71063 + 4.69495i 0.336212 + 0.582336i
\(66\) 2.05145 + 8.34374i 0.252516 + 1.02704i
\(67\) 13.9657i 1.70618i 0.521764 + 0.853090i \(0.325274\pi\)
−0.521764 + 0.853090i \(0.674726\pi\)
\(68\) −5.64383 − 3.25847i −0.684415 − 0.395147i
\(69\) −0.633113 − 0.183779i −0.0762178 − 0.0221244i
\(70\) 2.64067 + 1.52459i 0.315621 + 0.182224i
\(71\) −3.68375 + 6.38043i −0.437180 + 0.757218i −0.997471 − 0.0710779i \(-0.977356\pi\)
0.560291 + 0.828296i \(0.310689\pi\)
\(72\) 2.53372 + 1.60632i 0.298602 + 0.189306i
\(73\) −4.23388 + 7.33329i −0.495538 + 0.858297i −0.999987 − 0.00514450i \(-0.998362\pi\)
0.504449 + 0.863442i \(0.331696\pi\)
\(74\) − 0.616895i − 0.0717126i
\(75\) 1.78626 + 1.86183i 0.206259 + 0.214986i
\(76\) 2.60278 + 3.49650i 0.298559 + 0.401076i
\(77\) −6.99172 − 4.03667i −0.796780 − 0.460021i
\(78\) −4.81303 − 1.39712i −0.544968 − 0.158192i
\(79\) − 7.11620i − 0.800635i −0.916376 − 0.400318i \(-0.868900\pi\)
0.916376 − 0.400318i \(-0.131100\pi\)
\(80\) 1.62258 − 0.936797i 0.181410 − 0.104737i
\(81\) 3.83948 + 8.13992i 0.426609 + 0.904436i
\(82\) 4.25502 + 7.36991i 0.469888 + 0.813870i
\(83\) 9.46181 − 5.46278i 1.03857 − 0.599618i 0.119141 − 0.992877i \(-0.461986\pi\)
0.919428 + 0.393259i \(0.128653\pi\)
\(84\) −2.73731 + 0.673013i −0.298665 + 0.0734318i
\(85\) −12.2101 −1.32437
\(86\) −0.834414 −0.0899772
\(87\) −10.5068 − 10.9514i −1.12645 − 1.17411i
\(88\) −4.29611 + 2.48036i −0.457967 + 0.264408i
\(89\) 1.28517 + 2.22598i 0.136228 + 0.235954i 0.926066 − 0.377362i \(-0.123169\pi\)
−0.789838 + 0.613316i \(0.789835\pi\)
\(90\) 5.61596 + 0.232797i 0.591974 + 0.0245390i
\(91\) 4.07815 − 2.35452i 0.427506 − 0.246821i
\(92\) − 0.380616i − 0.0396820i
\(93\) −2.93186 + 2.81284i −0.304019 + 0.291678i
\(94\) −2.32983 − 1.34513i −0.240304 − 0.138740i
\(95\) 7.49873 + 3.23508i 0.769354 + 0.331912i
\(96\) −0.482845 + 1.66339i −0.0492802 + 0.169769i
\(97\) 4.39065i 0.445803i 0.974841 + 0.222901i \(0.0715528\pi\)
−0.974841 + 0.222901i \(0.928447\pi\)
\(98\) −2.17570 + 3.76842i −0.219779 + 0.380668i
\(99\) −14.8694 − 0.616379i −1.49443 − 0.0619484i
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.n.f.293.6 yes 18
3.2 odd 2 1026.2.n.f.179.3 18
9.2 odd 6 342.2.j.f.65.2 ✓ 18
9.7 even 3 1026.2.j.f.521.7 18
19.12 odd 6 342.2.j.f.221.2 yes 18
57.50 even 6 1026.2.j.f.449.3 18
171.88 odd 6 1026.2.n.f.791.3 18
171.164 even 6 inner 342.2.n.f.335.6 yes 18
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
342.2.j.f.65.2 ✓ 18 9.2 odd 6
342.2.j.f.221.2 yes 18 19.12 odd 6
342.2.n.f.293.6 yes 18 1.1 even 1 trivial
342.2.n.f.335.6 yes 18 171.164 even 6 inner
1026.2.j.f.449.3 18 57.50 even 6
1026.2.j.f.521.7 18 9.7 even 3
1026.2.n.f.179.3 18 3.2 odd 2
1026.2.n.f.791.3 18 171.88 odd 6