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 335.7
Root \(0.539325 - 1.64594i\) of defining polynomial
Character \(\chi\) \(=\) 342.335
Dual form 342.2.n.f.293.7

$q$-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-1.00000 q^{2} +(1.15577 - 1.29004i) q^{3} +1.00000 q^{4} +(0.400415 + 0.231180i) q^{5} +(-1.15577 + 1.29004i) q^{6} +(1.36198 - 2.35901i) q^{7} -1.00000 q^{8} +(-0.328410 - 2.98197i) q^{9} +(-0.400415 - 0.231180i) q^{10} +(-1.33471 - 0.770594i) q^{11} +(1.15577 - 1.29004i) q^{12} +0.534112i q^{13} +(-1.36198 + 2.35901i) q^{14} +(0.761017 - 0.249362i) q^{15} +1.00000 q^{16} +(-0.233094 + 0.134577i) q^{17} +(0.328410 + 2.98197i) q^{18} +(4.33785 + 0.427839i) q^{19} +(0.400415 + 0.231180i) q^{20} +(-1.46910 - 4.48347i) q^{21} +(1.33471 + 0.770594i) q^{22} +3.53215i q^{23} +(-1.15577 + 1.29004i) q^{24} +(-2.39311 - 4.14499i) q^{25} -0.534112i q^{26} +(-4.22643 - 3.02280i) q^{27} +(1.36198 - 2.35901i) q^{28} +(-0.734416 - 1.27205i) q^{29} +(-0.761017 + 0.249362i) q^{30} +(2.29038 - 1.32235i) q^{31} -1.00000 q^{32} +(-2.53671 + 0.831202i) q^{33} +(0.233094 - 0.134577i) q^{34} +(1.09071 - 0.629722i) q^{35} +(-0.328410 - 2.98197i) q^{36} -3.86471i q^{37} +(-4.33785 - 0.427839i) q^{38} +(0.689026 + 0.617308i) q^{39} +(-0.400415 - 0.231180i) q^{40} +(-3.11772 + 5.40004i) q^{41} +(1.46910 + 4.48347i) q^{42} +3.84975 q^{43} +(-1.33471 - 0.770594i) q^{44} +(0.557871 - 1.26995i) q^{45} -3.53215i q^{46} +(3.63534 - 2.09887i) q^{47} +(1.15577 - 1.29004i) q^{48} +(-0.209958 - 0.363659i) q^{49} +(2.39311 + 4.14499i) q^{50} +(-0.0957925 + 0.456241i) q^{51} +0.534112i q^{52} +(2.81077 - 4.86839i) q^{53} +(4.22643 + 3.02280i) q^{54} +(-0.356291 - 0.617115i) q^{55} +(-1.36198 + 2.35901i) q^{56} +(5.56547 - 5.10152i) q^{57} +(0.734416 + 1.27205i) q^{58} +(-2.39166 + 4.14247i) q^{59} +(0.761017 - 0.249362i) q^{60} +(5.30609 + 9.19042i) q^{61} +(-2.29038 + 1.32235i) q^{62} +(-7.48179 - 3.28665i) q^{63} +1.00000 q^{64} +(-0.123476 + 0.213866i) q^{65} +(2.53671 - 0.831202i) q^{66} +11.4457i q^{67} +(-0.233094 + 0.134577i) q^{68} +(4.55661 + 4.08234i) q^{69} +(-1.09071 + 0.629722i) q^{70} +(2.20447 + 3.81825i) q^{71} +(0.328410 + 2.98197i) q^{72} +(-4.06718 - 7.04456i) q^{73} +3.86471i q^{74} +(-8.11309 - 1.70343i) q^{75} +(4.33785 + 0.427839i) q^{76} +(-3.63568 + 2.09906i) q^{77} +(-0.689026 - 0.617308i) q^{78} +14.6123i q^{79} +(0.400415 + 0.231180i) q^{80} +(-8.78429 + 1.95862i) q^{81} +(3.11772 - 5.40004i) q^{82} +(1.64361 + 0.948939i) q^{83} +(-1.46910 - 4.48347i) q^{84} -0.124446 q^{85} -3.84975 q^{86} +(-2.48980 - 0.522761i) q^{87} +(1.33471 + 0.770594i) q^{88} +(-6.10595 + 10.5758i) q^{89} +(-0.557871 + 1.26995i) q^{90} +(1.25998 + 0.727448i) q^{91} +3.53215i q^{92} +(0.941258 - 4.48302i) q^{93} +(-3.63534 + 2.09887i) q^{94} +(1.63803 + 1.17414i) q^{95} +(-1.15577 + 1.29004i) q^{96} +13.9173i q^{97} +(0.209958 + 0.363659i) q^{98} +(-1.85956 + 4.23313i) 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 \(\left(\mathbb{Z}/342\mathbb{Z}\right)^\times\).

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

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
\(3\) 1.15577 − 1.29004i 0.667282 − 0.744805i
\(4\) 1.00000 0.500000
\(5\) 0.400415 + 0.231180i 0.179071 + 0.103387i 0.586856 − 0.809691i \(-0.300365\pi\)
−0.407785 + 0.913078i \(0.633699\pi\)
\(6\) −1.15577 + 1.29004i −0.471839 + 0.526657i
\(7\) 1.36198 − 2.35901i 0.514779 − 0.891623i −0.485074 − 0.874473i \(-0.661207\pi\)
0.999853 − 0.0171498i \(-0.00545922\pi\)
\(8\) −1.00000 −0.353553
\(9\) −0.328410 − 2.98197i −0.109470 − 0.993990i
\(10\) −0.400415 − 0.231180i −0.126622 − 0.0731054i
\(11\) −1.33471 − 0.770594i −0.402430 − 0.232343i 0.285102 − 0.958497i \(-0.407972\pi\)
−0.687532 + 0.726154i \(0.741306\pi\)
\(12\) 1.15577 − 1.29004i 0.333641 − 0.372403i
\(13\) 0.534112i 0.148136i 0.997253 + 0.0740680i \(0.0235982\pi\)
−0.997253 + 0.0740680i \(0.976402\pi\)
\(14\) −1.36198 + 2.35901i −0.364003 + 0.630472i
\(15\) 0.761017 − 0.249362i 0.196494 − 0.0643850i
\(16\) 1.00000 0.250000
\(17\) −0.233094 + 0.134577i −0.0565337 + 0.0326397i −0.528000 − 0.849244i \(-0.677058\pi\)
0.471467 + 0.881884i \(0.343725\pi\)
\(18\) 0.328410 + 2.98197i 0.0774070 + 0.702857i
\(19\) 4.33785 + 0.427839i 0.995171 + 0.0981530i
\(20\) 0.400415 + 0.231180i 0.0895355 + 0.0516933i
\(21\) −1.46910 − 4.48347i −0.320583 − 0.978373i
\(22\) 1.33471 + 0.770594i 0.284561 + 0.164291i
\(23\) 3.53215i 0.736504i 0.929726 + 0.368252i \(0.120044\pi\)
−0.929726 + 0.368252i \(0.879956\pi\)
\(24\) −1.15577 + 1.29004i −0.235920 + 0.263328i
\(25\) −2.39311 − 4.14499i −0.478622 − 0.828998i
\(26\) − 0.534112i − 0.104748i
\(27\) −4.22643 − 3.02280i −0.813377 − 0.581738i
\(28\) 1.36198 − 2.35901i 0.257389 − 0.445811i
\(29\) −0.734416 − 1.27205i −0.136378 − 0.236213i 0.789745 − 0.613435i \(-0.210213\pi\)
−0.926123 + 0.377222i \(0.876879\pi\)
\(30\) −0.761017 + 0.249362i −0.138942 + 0.0455271i
\(31\) 2.29038 − 1.32235i 0.411365 − 0.237502i −0.280011 − 0.959997i \(-0.590338\pi\)
0.691376 + 0.722495i \(0.257005\pi\)
\(32\) −1.00000 −0.176777
\(33\) −2.53671 + 0.831202i −0.441584 + 0.144694i
\(34\) 0.233094 − 0.134577i 0.0399753 − 0.0230798i
\(35\) 1.09071 − 0.629722i 0.184364 − 0.106443i
\(36\) −0.328410 − 2.98197i −0.0547350 − 0.496995i
\(37\) − 3.86471i − 0.635355i −0.948199 − 0.317677i \(-0.897097\pi\)
0.948199 − 0.317677i \(-0.102903\pi\)
\(38\) −4.33785 − 0.427839i −0.703692 − 0.0694046i
\(39\) 0.689026 + 0.617308i 0.110332 + 0.0988485i
\(40\) −0.400415 − 0.231180i −0.0633112 − 0.0365527i
\(41\) −3.11772 + 5.40004i −0.486905 + 0.843345i −0.999887 − 0.0150550i \(-0.995208\pi\)
0.512981 + 0.858400i \(0.328541\pi\)
\(42\) 1.46910 + 4.48347i 0.226686 + 0.691815i
\(43\) 3.84975 0.587081 0.293540 − 0.955947i \(-0.405167\pi\)
0.293540 + 0.955947i \(0.405167\pi\)
\(44\) −1.33471 − 0.770594i −0.201215 − 0.116171i
\(45\) 0.557871 − 1.26995i 0.0831624 − 0.189313i
\(46\) − 3.53215i − 0.520787i
\(47\) 3.63534 − 2.09887i 0.530269 − 0.306151i −0.210857 − 0.977517i \(-0.567625\pi\)
0.741126 + 0.671366i \(0.234292\pi\)
\(48\) 1.15577 − 1.29004i 0.166820 − 0.186201i
\(49\) −0.209958 − 0.363659i −0.0299941 − 0.0519512i
\(50\) 2.39311 + 4.14499i 0.338437 + 0.586190i
\(51\) −0.0957925 + 0.456241i −0.0134136 + 0.0638865i
\(52\) 0.534112i 0.0740680i
\(53\) 2.81077 − 4.86839i 0.386088 − 0.668725i −0.605831 − 0.795593i \(-0.707159\pi\)
0.991920 + 0.126869i \(0.0404927\pi\)
\(54\) 4.22643 + 3.02280i 0.575144 + 0.411351i
\(55\) −0.356291 − 0.617115i −0.0480423 − 0.0832118i
\(56\) −1.36198 + 2.35901i −0.182002 + 0.315236i
\(57\) 5.56547 − 5.10152i 0.737165 − 0.675713i
\(58\) 0.734416 + 1.27205i 0.0964336 + 0.167028i
\(59\) −2.39166 + 4.14247i −0.311367 + 0.539304i −0.978659 − 0.205493i \(-0.934120\pi\)
0.667291 + 0.744797i \(0.267454\pi\)
\(60\) 0.761017 − 0.249362i 0.0982469 − 0.0321925i
\(61\) 5.30609 + 9.19042i 0.679375 + 1.17671i 0.975169 + 0.221461i \(0.0710825\pi\)
−0.295794 + 0.955252i \(0.595584\pi\)
\(62\) −2.29038 + 1.32235i −0.290879 + 0.167939i
\(63\) −7.48179 − 3.28665i −0.942617 − 0.414079i
\(64\) 1.00000 0.125000
\(65\) −0.123476 + 0.213866i −0.0153153 + 0.0265269i
\(66\) 2.53671 − 0.831202i 0.312247 − 0.102314i
\(67\) 11.4457i 1.39831i 0.714970 + 0.699156i \(0.246441\pi\)
−0.714970 + 0.699156i \(0.753559\pi\)
\(68\) −0.233094 + 0.134577i −0.0282668 + 0.0163199i
\(69\) 4.55661 + 4.08234i 0.548552 + 0.491455i
\(70\) −1.09071 + 0.629722i −0.130365 + 0.0752662i
\(71\) 2.20447 + 3.81825i 0.261622 + 0.453143i 0.966673 − 0.256014i \(-0.0824092\pi\)
−0.705051 + 0.709157i \(0.749076\pi\)
\(72\) 0.328410 + 2.98197i 0.0387035 + 0.351429i
\(73\) −4.06718 − 7.04456i −0.476027 − 0.824503i 0.523596 − 0.851967i \(-0.324590\pi\)
−0.999623 + 0.0274637i \(0.991257\pi\)
\(74\) 3.86471i 0.449264i
\(75\) −8.11309 − 1.70343i −0.936818 − 0.196695i
\(76\) 4.33785 + 0.427839i 0.497586 + 0.0490765i
\(77\) −3.63568 + 2.09906i −0.414324 + 0.239210i
\(78\) −0.689026 − 0.617308i −0.0780169 − 0.0698964i
\(79\) 14.6123i 1.64401i 0.569480 + 0.822005i \(0.307145\pi\)
−0.569480 + 0.822005i \(0.692855\pi\)
\(80\) 0.400415 + 0.231180i 0.0447678 + 0.0258467i
\(81\) −8.78429 + 1.95862i −0.976033 + 0.217624i
\(82\) 3.11772 − 5.40004i 0.344294 − 0.596335i
\(83\) 1.64361 + 0.948939i 0.180410 + 0.104160i 0.587485 − 0.809235i \(-0.300118\pi\)
−0.407075 + 0.913395i \(0.633451\pi\)
\(84\) −1.46910 − 4.48347i −0.160292 − 0.489187i
\(85\) −0.124446 −0.0134981
\(86\) −3.84975 −0.415129
\(87\) −2.48980 − 0.522761i −0.266935 − 0.0560458i
\(88\) 1.33471 + 0.770594i 0.142280 + 0.0821456i
\(89\) −6.10595 + 10.5758i −0.647229 + 1.12103i 0.336553 + 0.941665i \(0.390739\pi\)
−0.983782 + 0.179369i \(0.942594\pi\)
\(90\) −0.557871 + 1.26995i −0.0588047 + 0.133864i
\(91\) 1.25998 + 0.727448i 0.132081 + 0.0762573i
\(92\) 3.53215i 0.368252i
\(93\) 0.941258 − 4.48302i 0.0976039 − 0.464868i
\(94\) −3.63534 + 2.09887i −0.374957 + 0.216481i
\(95\) 1.63803 + 1.17414i 0.168059 + 0.120464i
\(96\) −1.15577 + 1.29004i −0.117960 + 0.131664i
\(97\) 13.9173i 1.41308i 0.707671 + 0.706542i \(0.249746\pi\)
−0.707671 + 0.706542i \(0.750254\pi\)
\(98\) 0.209958 + 0.363659i 0.0212090 + 0.0367351i
\(99\) −1.85956 + 4.23313i −0.186893 + 0.425446i
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.335.7 yes 18
3.2 odd 2 1026.2.n.f.791.6 18
9.4 even 3 1026.2.j.f.449.6 18
9.5 odd 6 342.2.j.f.221.9 yes 18
19.8 odd 6 342.2.j.f.65.9 ✓ 18
57.8 even 6 1026.2.j.f.521.4 18
171.103 odd 6 1026.2.n.f.179.6 18
171.122 even 6 inner 342.2.n.f.293.7 yes 18
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
342.2.j.f.65.9 ✓ 18 19.8 odd 6
342.2.j.f.221.9 yes 18 9.5 odd 6
342.2.n.f.293.7 yes 18 171.122 even 6 inner
342.2.n.f.335.7 yes 18 1.1 even 1 trivial
1026.2.j.f.449.6 18 9.4 even 3
1026.2.j.f.521.4 18 57.8 even 6
1026.2.n.f.179.6 18 171.103 odd 6
1026.2.n.f.791.6 18 3.2 odd 2