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 335.8
Root \(-0.599249 - 1.62508i\) of defining polynomial
Character \(\chi\) \(=\) 342.335
Dual form 342.2.n.f.293.8

$q$-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} +(1.70699 - 0.293577i) q^{3} +1.00000 q^{4} +(1.10624 + 0.638690i) q^{5} +(-1.70699 + 0.293577i) q^{6} +(-1.74174 + 3.01679i) q^{7} -1.00000 q^{8} +(2.82762 - 1.00227i) q^{9} +(-1.10624 - 0.638690i) q^{10} +(1.65904 + 0.957850i) q^{11} +(1.70699 - 0.293577i) q^{12} +5.37510i q^{13} +(1.74174 - 3.01679i) q^{14} +(2.07585 + 0.765469i) q^{15} +1.00000 q^{16} +(1.94083 - 1.12054i) q^{17} +(-2.82762 + 1.00227i) q^{18} +(-4.35810 + 0.0834976i) q^{19} +(1.10624 + 0.638690i) q^{20} +(-2.08748 + 5.66097i) q^{21} +(-1.65904 - 0.957850i) q^{22} -7.32914i q^{23} +(-1.70699 + 0.293577i) q^{24} +(-1.68415 - 2.91703i) q^{25} -5.37510i q^{26} +(4.53248 - 2.54098i) q^{27} +(-1.74174 + 3.01679i) q^{28} +(0.991510 + 1.71734i) q^{29} +(-2.07585 - 0.765469i) q^{30} +(7.98945 - 4.61271i) q^{31} -1.00000 q^{32} +(3.11317 + 1.14798i) q^{33} +(-1.94083 + 1.12054i) q^{34} +(-3.85359 + 2.22487i) q^{35} +(2.82762 - 1.00227i) q^{36} +10.4153i q^{37} +(4.35810 - 0.0834976i) q^{38} +(1.57801 + 9.17523i) q^{39} +(-1.10624 - 0.638690i) q^{40} +(3.85318 - 6.67390i) q^{41} +(2.08748 - 5.66097i) q^{42} -2.90523 q^{43} +(1.65904 + 0.957850i) q^{44} +(3.76818 + 0.697226i) q^{45} +7.32914i q^{46} +(-8.13361 + 4.69594i) q^{47} +(1.70699 - 0.293577i) q^{48} +(-2.56735 - 4.44678i) q^{49} +(1.68415 + 2.91703i) q^{50} +(2.98402 - 2.48254i) q^{51} +5.37510i q^{52} +(5.35373 - 9.27293i) q^{53} +(-4.53248 + 2.54098i) q^{54} +(1.22354 + 2.11923i) q^{55} +(1.74174 - 3.01679i) q^{56} +(-7.41472 + 1.42197i) q^{57} +(-0.991510 - 1.71734i) q^{58} +(0.785560 - 1.36063i) q^{59} +(2.07585 + 0.765469i) q^{60} +(-0.721334 - 1.24939i) q^{61} +(-7.98945 + 4.61271i) q^{62} +(-1.90137 + 10.2760i) q^{63} +1.00000 q^{64} +(-3.43302 + 5.94617i) q^{65} +(-3.11317 - 1.14798i) q^{66} -2.02926i q^{67} +(1.94083 - 1.12054i) q^{68} +(-2.15167 - 12.5108i) q^{69} +(3.85359 - 2.22487i) q^{70} +(0.756995 + 1.31115i) q^{71} +(-2.82762 + 1.00227i) q^{72} +(-3.96278 - 6.86374i) q^{73} -10.4153i q^{74} +(-3.73120 - 4.48492i) q^{75} +(-4.35810 + 0.0834976i) q^{76} +(-5.77926 + 3.33666i) q^{77} +(-1.57801 - 9.17523i) q^{78} -11.2082i q^{79} +(1.10624 + 0.638690i) q^{80} +(6.99093 - 5.66806i) q^{81} +(-3.85318 + 6.67390i) q^{82} +(-12.8857 - 7.43955i) q^{83} +(-2.08748 + 5.66097i) q^{84} +2.86271 q^{85} +2.90523 q^{86} +(2.19667 + 2.64040i) q^{87} +(-1.65904 - 0.957850i) q^{88} +(-2.25713 + 3.90947i) q^{89} +(-3.76818 - 0.697226i) q^{90} +(-16.2155 - 9.36205i) q^{91} -7.32914i q^{92} +(12.2837 - 10.2194i) q^{93} +(8.13361 - 4.69594i) q^{94} +(-4.87445 - 2.69111i) q^{95} +(-1.70699 + 0.293577i) q^{96} +7.95995i q^{97} +(2.56735 + 4.44678i) q^{98} +(5.65118 + 1.04564i) 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
\(3\) 1.70699 − 0.293577i 0.985531 − 0.169497i
\(4\) 1.00000 0.500000
\(5\) 1.10624 + 0.638690i 0.494727 + 0.285631i 0.726533 − 0.687131i \(-0.241130\pi\)
−0.231806 + 0.972762i \(0.574464\pi\)
\(6\) −1.70699 + 0.293577i −0.696875 + 0.119852i
\(7\) −1.74174 + 3.01679i −0.658318 + 1.14024i 0.322733 + 0.946490i \(0.395398\pi\)
−0.981051 + 0.193750i \(0.937935\pi\)
\(8\) −1.00000 −0.353553
\(9\) 2.82762 − 1.00227i 0.942542 − 0.334089i
\(10\) −1.10624 − 0.638690i −0.349825 − 0.201972i
\(11\) 1.65904 + 0.957850i 0.500221 + 0.288803i 0.728805 − 0.684722i \(-0.240076\pi\)
−0.228584 + 0.973524i \(0.573410\pi\)
\(12\) 1.70699 − 0.293577i 0.492765 − 0.0847484i
\(13\) 5.37510i 1.49078i 0.666626 + 0.745392i \(0.267738\pi\)
−0.666626 + 0.745392i \(0.732262\pi\)
\(14\) 1.74174 − 3.01679i 0.465501 − 0.806271i
\(15\) 2.07585 + 0.765469i 0.535982 + 0.197643i
\(16\) 1.00000 0.250000
\(17\) 1.94083 − 1.12054i 0.470721 − 0.271771i −0.245820 − 0.969315i \(-0.579057\pi\)
0.716542 + 0.697544i \(0.245724\pi\)
\(18\) −2.82762 + 1.00227i −0.666478 + 0.236236i
\(19\) −4.35810 + 0.0834976i −0.999817 + 0.0191557i
\(20\) 1.10624 + 0.638690i 0.247364 + 0.142815i
\(21\) −2.08748 + 5.66097i −0.455525 + 1.23532i
\(22\) −1.65904 − 0.957850i −0.353710 − 0.204214i
\(23\) − 7.32914i − 1.52823i −0.645079 − 0.764116i \(-0.723176\pi\)
0.645079 − 0.764116i \(-0.276824\pi\)
\(24\) −1.70699 + 0.293577i −0.348438 + 0.0599262i
\(25\) −1.68415 − 2.91703i −0.336830 − 0.583407i
\(26\) − 5.37510i − 1.05414i
\(27\) 4.53248 − 2.54098i 0.872277 − 0.489012i
\(28\) −1.74174 + 3.01679i −0.329159 + 0.570120i
\(29\) 0.991510 + 1.71734i 0.184119 + 0.318903i 0.943279 − 0.332001i \(-0.107724\pi\)
−0.759161 + 0.650903i \(0.774390\pi\)
\(30\) −2.07585 − 0.765469i −0.378997 − 0.139755i
\(31\) 7.98945 − 4.61271i 1.43495 − 0.828467i 0.437455 − 0.899240i \(-0.355880\pi\)
0.997492 + 0.0707732i \(0.0225467\pi\)
\(32\) −1.00000 −0.176777
\(33\) 3.11317 + 1.14798i 0.541934 + 0.199838i
\(34\) −1.94083 + 1.12054i −0.332850 + 0.192171i
\(35\) −3.85359 + 2.22487i −0.651375 + 0.376072i
\(36\) 2.82762 − 1.00227i 0.471271 − 0.167044i
\(37\) 10.4153i 1.71227i 0.516754 + 0.856134i \(0.327140\pi\)
−0.516754 + 0.856134i \(0.672860\pi\)
\(38\) 4.35810 − 0.0834976i 0.706977 − 0.0135451i
\(39\) 1.57801 + 9.17523i 0.252683 + 1.46921i
\(40\) −1.10624 − 0.638690i −0.174913 − 0.100986i
\(41\) 3.85318 − 6.67390i 0.601765 − 1.04229i −0.390788 − 0.920481i \(-0.627798\pi\)
0.992554 − 0.121807i \(-0.0388691\pi\)
\(42\) 2.08748 − 5.66097i 0.322105 − 0.873506i
\(43\) −2.90523 −0.443044 −0.221522 − 0.975155i \(-0.571102\pi\)
−0.221522 + 0.975155i \(0.571102\pi\)
\(44\) 1.65904 + 0.957850i 0.250110 + 0.144401i
\(45\) 3.76818 + 0.697226i 0.561727 + 0.103936i
\(46\) 7.32914i 1.08062i
\(47\) −8.13361 + 4.69594i −1.18641 + 0.684973i −0.957488 − 0.288472i \(-0.906853\pi\)
−0.228920 + 0.973445i \(0.573520\pi\)
\(48\) 1.70699 − 0.293577i 0.246383 − 0.0423742i
\(49\) −2.56735 − 4.44678i −0.366764 − 0.635254i
\(50\) 1.68415 + 2.91703i 0.238175 + 0.412531i
\(51\) 2.98402 − 2.48254i 0.417846 − 0.347625i
\(52\) 5.37510i 0.745392i
\(53\) 5.35373 − 9.27293i 0.735391 − 1.27373i −0.219161 − 0.975689i \(-0.570332\pi\)
0.954552 − 0.298046i \(-0.0963348\pi\)
\(54\) −4.53248 + 2.54098i −0.616793 + 0.345784i
\(55\) 1.22354 + 2.11923i 0.164982 + 0.285757i
\(56\) 1.74174 − 3.01679i 0.232750 − 0.403136i
\(57\) −7.41472 + 1.42197i −0.982103 + 0.188344i
\(58\) −0.991510 − 1.71734i −0.130192 − 0.225498i
\(59\) 0.785560 − 1.36063i 0.102271 − 0.177139i −0.810349 − 0.585948i \(-0.800722\pi\)
0.912620 + 0.408809i \(0.134056\pi\)
\(60\) 2.07585 + 0.765469i 0.267991 + 0.0988217i
\(61\) −0.721334 − 1.24939i −0.0923574 − 0.159968i 0.816145 − 0.577847i \(-0.196107\pi\)
−0.908503 + 0.417879i \(0.862774\pi\)
\(62\) −7.98945 + 4.61271i −1.01466 + 0.585815i
\(63\) −1.90137 + 10.2760i −0.239551 + 1.29466i
\(64\) 1.00000 0.125000
\(65\) −3.43302 + 5.94617i −0.425814 + 0.737531i
\(66\) −3.11317 − 1.14798i −0.383205 − 0.141307i
\(67\) − 2.02926i − 0.247914i −0.992288 − 0.123957i \(-0.960442\pi\)
0.992288 − 0.123957i \(-0.0395585\pi\)
\(68\) 1.94083 − 1.12054i 0.235361 − 0.135886i
\(69\) −2.15167 − 12.5108i −0.259030 − 1.50612i
\(70\) 3.85359 − 2.22487i 0.460592 − 0.265923i
\(71\) 0.756995 + 1.31115i 0.0898388 + 0.155605i 0.907443 − 0.420175i \(-0.138031\pi\)
−0.817604 + 0.575781i \(0.804698\pi\)
\(72\) −2.82762 + 1.00227i −0.333239 + 0.118118i
\(73\) −3.96278 − 6.86374i −0.463808 − 0.803340i 0.535338 − 0.844638i \(-0.320184\pi\)
−0.999147 + 0.0412979i \(0.986851\pi\)
\(74\) − 10.4153i − 1.21076i
\(75\) −3.73120 − 4.48492i −0.430842 − 0.517874i
\(76\) −4.35810 + 0.0834976i −0.499908 + 0.00957783i
\(77\) −5.77926 + 3.33666i −0.658608 + 0.380248i
\(78\) −1.57801 − 9.17523i −0.178674 − 1.03889i
\(79\) − 11.2082i − 1.26102i −0.776179 − 0.630512i \(-0.782845\pi\)
0.776179 − 0.630512i \(-0.217155\pi\)
\(80\) 1.10624 + 0.638690i 0.123682 + 0.0714077i
\(81\) 6.99093 − 5.66806i 0.776770 − 0.629785i
\(82\) −3.85318 + 6.67390i −0.425512 + 0.737009i
\(83\) −12.8857 − 7.43955i −1.41439 − 0.816596i −0.418589 − 0.908176i \(-0.637475\pi\)
−0.995798 + 0.0915797i \(0.970808\pi\)
\(84\) −2.08748 + 5.66097i −0.227763 + 0.617662i
\(85\) 2.86271 0.310505
\(86\) 2.90523 0.313279
\(87\) 2.19667 + 2.64040i 0.235508 + 0.283081i
\(88\) −1.65904 − 0.957850i −0.176855 − 0.102107i
\(89\) −2.25713 + 3.90947i −0.239256 + 0.414403i −0.960501 − 0.278277i \(-0.910237\pi\)
0.721245 + 0.692680i \(0.243570\pi\)
\(90\) −3.76818 − 0.697226i −0.397201 − 0.0734940i
\(91\) −16.2155 − 9.36205i −1.69985 − 0.981409i
\(92\) − 7.32914i − 0.764116i
\(93\) 12.2837 − 10.2194i 1.27376 − 1.05970i
\(94\) 8.13361 − 4.69594i 0.838918 − 0.484349i
\(95\) −4.87445 − 2.69111i −0.500108 − 0.276102i
\(96\) −1.70699 + 0.293577i −0.174219 + 0.0299631i
\(97\) 7.95995i 0.808210i 0.914713 + 0.404105i \(0.132417\pi\)
−0.914713 + 0.404105i \(0.867583\pi\)
\(98\) 2.56735 + 4.44678i 0.259341 + 0.449193i
\(99\) 5.65118 + 1.04564i 0.567965 + 0.105090i
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.335.8 yes 18
3.2 odd 2 1026.2.n.f.791.4 18
9.4 even 3 1026.2.j.f.449.4 18
9.5 odd 6 342.2.j.f.221.7 yes 18
19.8 odd 6 342.2.j.f.65.7 ✓ 18
57.8 even 6 1026.2.j.f.521.6 18
171.103 odd 6 1026.2.n.f.179.4 18
171.122 even 6 inner 342.2.n.f.293.8 yes 18
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
342.2.j.f.65.7 ✓ 18 19.8 odd 6
342.2.j.f.221.7 yes 18 9.5 odd 6
342.2.n.f.293.8 yes 18 171.122 even 6 inner
342.2.n.f.335.8 yes 18 1.1 even 1 trivial
1026.2.j.f.449.4 18 9.4 even 3
1026.2.j.f.521.6 18 57.8 even 6
1026.2.n.f.179.4 18 171.103 odd 6
1026.2.n.f.791.4 18 3.2 odd 2