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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [342,2,Mod(29,342)] 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("342.29"); 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([3, 17])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 342 = 2 \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 342.x (of order \(18\), degree \(6\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [48] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.73088374913\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(8\) over \(\Q(\zeta_{18})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label 281.6
Character \(\chi\) \(=\) 342.281
Dual form 342.2.x.b.185.6

$q$-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+(-0.766044 + 0.642788i) q^{2} +(1.16912 + 1.27795i) q^{3} +(0.173648 - 0.984808i) q^{4} +(-0.846718 + 2.32634i) q^{5} +(-1.71705 - 0.227474i) q^{6} +(1.68533 + 2.91907i) q^{7} +(0.500000 + 0.866025i) q^{8} +(-0.266325 + 2.98816i) q^{9} +(-0.846718 - 2.32634i) q^{10} -1.87152i q^{11} +(1.46155 - 0.929443i) q^{12} +(-2.43087 - 6.67877i) q^{13} +(-3.16738 - 1.15283i) q^{14} +(-3.96287 + 1.63770i) q^{15} +(-0.939693 - 0.342020i) q^{16} +(-1.76521 + 4.84988i) q^{17} +(-1.71673 - 2.46025i) q^{18} +(4.24229 + 1.00150i) q^{19} +(2.14397 + 1.23782i) q^{20} +(-1.76009 + 5.56650i) q^{21} +(1.20299 + 1.43367i) q^{22} +(-1.38671 - 0.244515i) q^{23} +(-0.522180 + 1.65146i) q^{24} +(-0.864701 - 0.725570i) q^{25} +(6.15519 + 3.55370i) q^{26} +(-4.13009 + 3.15316i) q^{27} +(3.16738 - 1.15283i) q^{28} +(0.0802322 - 0.455020i) q^{29} +(1.98304 - 3.80183i) q^{30} -0.245984i q^{31} +(0.939693 - 0.342020i) q^{32} +(2.39171 - 2.18803i) q^{33} +(-1.76521 - 4.84988i) q^{34} +(-8.21774 + 1.44901i) q^{35} +(2.89651 + 0.781167i) q^{36} -2.64332i q^{37} +(-3.89353 + 1.95970i) q^{38} +(5.69317 - 10.9148i) q^{39} +(-2.43803 + 0.429890i) q^{40} +(-2.57077 + 2.15713i) q^{41} +(-2.22977 - 5.39555i) q^{42} +(0.540887 + 3.06752i) q^{43} +(-1.84309 - 0.324986i) q^{44} +(-6.72596 - 3.14969i) q^{45} +(1.21945 - 0.704052i) q^{46} +(6.44758 + 1.13688i) q^{47} +(-0.661526 - 1.60074i) q^{48} +(-2.18065 + 3.77699i) q^{49} +1.12879 q^{50} +(-8.26165 + 3.41422i) q^{51} +(-6.99942 + 1.23419i) q^{52} +(7.46859 + 6.26689i) q^{53} +(1.13702 - 5.07023i) q^{54} +(4.35379 + 1.58465i) q^{55} +(-1.68533 + 2.91907i) q^{56} +(3.67987 + 6.59231i) q^{57} +(0.231019 + 0.400137i) q^{58} +(-1.41017 - 7.99747i) q^{59} +(0.924675 + 4.18704i) q^{60} +(9.05773 - 3.29674i) q^{61} +(0.158116 + 0.188435i) q^{62} +(-9.17148 + 4.25859i) q^{63} +(-0.500000 + 0.866025i) q^{64} +17.5953 q^{65} +(-0.425721 + 3.21349i) q^{66} +(-3.40868 + 4.06231i) q^{67} +(4.46967 + 2.58056i) q^{68} +(-1.30875 - 2.05802i) q^{69} +(5.36375 - 6.39227i) q^{70} +(10.9539 - 9.19138i) q^{71} +(-2.72098 + 1.26343i) q^{72} +(-1.08991 - 6.18119i) q^{73} +(1.69910 + 2.02490i) q^{74} +(-0.0836933 - 1.95332i) q^{75} +(1.72295 - 4.00393i) q^{76} +(5.46309 - 3.15412i) q^{77} +(2.65468 + 12.0207i) q^{78} +(1.87290 - 5.14576i) q^{79} +(1.59131 - 1.89645i) q^{80} +(-8.85814 - 1.59164i) q^{81} +(0.582747 - 3.30492i) q^{82} +(-10.3388 + 5.96913i) q^{83} +(5.17630 + 2.69996i) q^{84} +(-9.78782 - 8.21296i) q^{85} +(-2.38611 - 2.00218i) q^{86} +(0.675294 - 0.429439i) q^{87} +(1.62078 - 0.935759i) q^{88} +(-0.229972 + 1.30423i) q^{89} +(7.17697 - 1.91056i) q^{90} +(15.3990 - 18.3518i) q^{91} +(-0.481600 + 1.32319i) q^{92} +(0.314356 - 0.287585i) q^{93} +(-5.66991 + 3.27352i) q^{94} +(-5.92184 + 9.02102i) q^{95} +(1.53570 + 0.801021i) q^{96} +(11.0702 + 13.1930i) q^{97} +(-0.757331 - 4.29504i) q^{98} +(5.59239 + 0.498433i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 48 q - 6 q^{3} - 9 q^{5} - 9 q^{6} + 9 q^{7} + 24 q^{8} - 9 q^{10} - 6 q^{12} + 15 q^{13} - 6 q^{14} + 12 q^{15} - 27 q^{17} - 9 q^{18} - 12 q^{19} - 9 q^{20} - 15 q^{21} + 18 q^{22} - 3 q^{24} - 9 q^{25}+ \cdots + 15 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(191\) \(325\)
\(\chi(n)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{11}{18}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.766044 + 0.642788i −0.541675 + 0.454519i
\(3\) 1.16912 + 1.27795i 0.674991 + 0.737826i
\(4\) 0.173648 − 0.984808i 0.0868241 − 0.492404i
\(5\) −0.846718 + 2.32634i −0.378664 + 1.04037i 0.593247 + 0.805021i \(0.297846\pi\)
−0.971911 + 0.235350i \(0.924376\pi\)
\(6\) −1.71705 − 0.227474i −0.700982 − 0.0928658i
\(7\) 1.68533 + 2.91907i 0.636993 + 1.10330i 0.986089 + 0.166218i \(0.0531555\pi\)
−0.349096 + 0.937087i \(0.613511\pi\)
\(8\) 0.500000 + 0.866025i 0.176777 + 0.306186i
\(9\) −0.266325 + 2.98816i −0.0887751 + 0.996052i
\(10\) −0.846718 − 2.32634i −0.267756 − 0.735653i
\(11\) − 1.87152i − 0.564284i −0.959373 − 0.282142i \(-0.908955\pi\)
0.959373 − 0.282142i \(-0.0910450\pi\)
\(12\) 1.46155 − 0.929443i 0.421914 − 0.268307i
\(13\) −2.43087 − 6.67877i −0.674203 − 1.85236i −0.495896 − 0.868382i \(-0.665160\pi\)
−0.178307 − 0.983975i \(-0.557062\pi\)
\(14\) −3.16738 − 1.15283i −0.846517 − 0.308107i
\(15\) −3.96287 + 1.63770i −1.02321 + 0.422852i
\(16\) −0.939693 − 0.342020i −0.234923 − 0.0855050i
\(17\) −1.76521 + 4.84988i −0.428126 + 1.17627i 0.518822 + 0.854882i \(0.326371\pi\)
−0.946948 + 0.321385i \(0.895852\pi\)
\(18\) −1.71673 − 2.46025i −0.404638 − 0.579887i
\(19\) 4.24229 + 1.00150i 0.973248 + 0.229759i
\(20\) 2.14397 + 1.23782i 0.479405 + 0.276785i
\(21\) −1.76009 + 5.56650i −0.384083 + 1.21471i
\(22\) 1.20299 + 1.43367i 0.256478 + 0.305659i
\(23\) −1.38671 − 0.244515i −0.289150 − 0.0509849i 0.0271920 − 0.999630i \(-0.491343\pi\)
−0.316342 + 0.948645i \(0.602455\pi\)
\(24\) −0.522180 + 1.65146i −0.106590 + 0.337103i
\(25\) −0.864701 − 0.725570i −0.172940 − 0.145114i
\(26\) 6.15519 + 3.55370i 1.20713 + 0.696938i
\(27\) −4.13009 + 3.15316i −0.794835 + 0.606825i
\(28\) 3.16738 − 1.15283i 0.598578 − 0.217865i
\(29\) 0.0802322 − 0.455020i 0.0148987 − 0.0844950i −0.976452 − 0.215736i \(-0.930785\pi\)
0.991351 + 0.131241i \(0.0418961\pi\)
\(30\) 1.98304 − 3.80183i 0.362051 − 0.694116i
\(31\) − 0.245984i − 0.0441801i −0.999756 − 0.0220900i \(-0.992968\pi\)
0.999756 − 0.0220900i \(-0.00703205\pi\)
\(32\) 0.939693 − 0.342020i 0.166116 − 0.0604612i
\(33\) 2.39171 − 2.18803i 0.416344 − 0.380887i
\(34\) −1.76521 − 4.84988i −0.302731 − 0.831747i
\(35\) −8.21774 + 1.44901i −1.38905 + 0.244927i
\(36\) 2.89651 + 0.781167i 0.482752 + 0.130194i
\(37\) − 2.64332i − 0.434560i −0.976109 − 0.217280i \(-0.930282\pi\)
0.976109 − 0.217280i \(-0.0697184\pi\)
\(38\) −3.89353 + 1.95970i −0.631614 + 0.317905i
\(39\) 5.69317 − 10.9148i 0.911637 − 1.74777i
\(40\) −2.43803 + 0.429890i −0.385486 + 0.0679716i
\(41\) −2.57077 + 2.15713i −0.401487 + 0.336888i −0.821068 − 0.570830i \(-0.806621\pi\)
0.419581 + 0.907718i \(0.362177\pi\)
\(42\) −2.22977 − 5.39555i −0.344062 − 0.832552i
\(43\) 0.540887 + 3.06752i 0.0824845 + 0.467793i 0.997871 + 0.0652168i \(0.0207739\pi\)
−0.915387 + 0.402576i \(0.868115\pi\)
\(44\) −1.84309 − 0.324986i −0.277856 − 0.0489935i
\(45\) −6.72596 − 3.14969i −1.00265 − 0.469528i
\(46\) 1.21945 − 0.704052i 0.179799 − 0.103807i
\(47\) 6.44758 + 1.13688i 0.940476 + 0.165831i 0.622812 − 0.782372i \(-0.285990\pi\)
0.317665 + 0.948203i \(0.397101\pi\)
\(48\) −0.661526 − 1.60074i −0.0954831 − 0.231048i
\(49\) −2.18065 + 3.77699i −0.311521 + 0.539570i
\(50\) 1.12879 0.159635
\(51\) −8.26165 + 3.41422i −1.15686 + 0.478087i
\(52\) −6.99942 + 1.23419i −0.970645 + 0.171151i
\(53\) 7.46859 + 6.26689i 1.02589 + 0.860823i 0.990356 − 0.138545i \(-0.0442426\pi\)
0.0355331 + 0.999368i \(0.488687\pi\)
\(54\) 1.13702 − 5.07023i 0.154729 − 0.689970i
\(55\) 4.35379 + 1.58465i 0.587065 + 0.213674i
\(56\) −1.68533 + 2.91907i −0.225211 + 0.390077i
\(57\) 3.67987 + 6.59231i 0.487411 + 0.873173i
\(58\) 0.231019 + 0.400137i 0.0303343 + 0.0525406i
\(59\) −1.41017 − 7.99747i −0.183588 − 1.04118i −0.927756 − 0.373188i \(-0.878265\pi\)
0.744167 − 0.667993i \(-0.232846\pi\)
\(60\) 0.924675 + 4.18704i 0.119375 + 0.540545i
\(61\) 9.05773 − 3.29674i 1.15972 − 0.422105i 0.310725 − 0.950500i \(-0.399428\pi\)
0.848999 + 0.528395i \(0.177206\pi\)
\(62\) 0.158116 + 0.188435i 0.0200807 + 0.0239312i
\(63\) −9.17148 + 4.25859i −1.15550 + 0.536532i
\(64\) −0.500000 + 0.866025i −0.0625000 + 0.108253i
\(65\) 17.5953 2.18243
\(66\) −0.425721 + 3.21349i −0.0524027 + 0.395553i
\(67\) −3.40868 + 4.06231i −0.416437 + 0.496290i −0.932958 − 0.359984i \(-0.882782\pi\)
0.516522 + 0.856274i \(0.327227\pi\)
\(68\) 4.46967 + 2.58056i 0.542027 + 0.312939i
\(69\) −1.30875 − 2.05802i −0.157555 − 0.247756i
\(70\) 5.36375 − 6.39227i 0.641091 − 0.764022i
\(71\) 10.9539 − 9.19138i 1.29998 − 1.09082i 0.309837 − 0.950790i \(-0.399726\pi\)
0.990148 − 0.140027i \(-0.0447188\pi\)
\(72\) −2.72098 + 1.26343i −0.320671 + 0.148897i
\(73\) −1.08991 − 6.18119i −0.127564 − 0.723453i −0.979752 − 0.200216i \(-0.935835\pi\)
0.852187 − 0.523237i \(-0.175276\pi\)
\(74\) 1.69910 + 2.02490i 0.197516 + 0.235390i
\(75\) −0.0836933 − 1.95332i −0.00966407 − 0.225550i
\(76\) 1.72295 − 4.00393i 0.197636 − 0.459282i
\(77\) 5.46309 − 3.15412i 0.622577 − 0.359445i
\(78\) 2.65468 + 12.0207i 0.300584 + 1.36108i
\(79\) 1.87290 − 5.14576i 0.210718 − 0.578943i −0.788637 − 0.614859i \(-0.789213\pi\)
0.999355 + 0.0359163i \(0.0114350\pi\)
\(80\) 1.59131 − 1.89645i 0.177914 − 0.212029i
\(81\) −8.85814 − 1.59164i −0.984238 − 0.176849i
\(82\) 0.582747 − 3.30492i 0.0643536 − 0.364967i
\(83\) −10.3388 + 5.96913i −1.13483 + 0.655197i −0.945146 − 0.326647i \(-0.894081\pi\)
−0.189688 + 0.981844i \(0.560748\pi\)
\(84\) 5.17630 + 2.69996i 0.564781 + 0.294590i
\(85\) −9.78782 − 8.21296i −1.06164 − 0.890820i
\(86\) −2.38611 − 2.00218i −0.257301 − 0.215901i
\(87\) 0.675294 − 0.429439i 0.0723991 − 0.0460407i
\(88\) 1.62078 − 0.935759i 0.172776 − 0.0997523i
\(89\) −0.229972 + 1.30423i −0.0243769 + 0.138248i −0.994567 − 0.104096i \(-0.966805\pi\)
0.970190 + 0.242344i \(0.0779163\pi\)
\(90\) 7.17697 − 1.91056i 0.756519 − 0.201391i
\(91\) 15.3990 − 18.3518i 1.61425 − 1.92379i
\(92\) −0.481600 + 1.32319i −0.0502103 + 0.137952i
\(93\) 0.314356 − 0.287585i 0.0325972 − 0.0298211i
\(94\) −5.66991 + 3.27352i −0.584806 + 0.337638i
\(95\) −5.92184 + 9.02102i −0.607568 + 0.925537i
\(96\) 1.53570 + 0.801021i 0.156736 + 0.0817538i
\(97\) 11.0702 + 13.1930i 1.12401 + 1.33955i 0.933797 + 0.357803i \(0.116474\pi\)
0.190215 + 0.981742i \(0.439081\pi\)
\(98\) −0.757331 − 4.29504i −0.0765019 − 0.433864i
\(99\) 5.59239 + 0.498433i 0.562056 + 0.0500944i
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.x.b.281.6 yes 48
9.5 odd 6 342.2.bf.b.167.6 yes 48
19.14 odd 18 342.2.bf.b.299.6 yes 48
171.14 even 18 inner 342.2.x.b.185.6 ✓ 48
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
342.2.x.b.185.6 ✓ 48 171.14 even 18 inner
342.2.x.b.281.6 yes 48 1.1 even 1 trivial
342.2.bf.b.167.6 yes 48 9.5 odd 6
342.2.bf.b.299.6 yes 48 19.14 odd 18