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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [162,2,Mod(7,162)] 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("162.7"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(162, base_ring=CyclotomicField(54)) chi = DirichletCharacter(H, H._module([16])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 162 = 2 \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 162.g (of order \(27\), degree \(18\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [90] 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: \(1.29357651274\)
Analytic rank: \(0\)
Dimension: \(90\)
Relative dimension: \(5\) over \(\Q(\zeta_{27})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{27}]$

Embedding invariants

Embedding label 13.3
Character \(\chi\) \(=\) 162.13
Dual form 162.2.g.b.25.3

$q$-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+(0.893633 + 0.448799i) q^{2} +(-0.530234 + 1.64889i) q^{3} +(0.597159 + 0.802123i) q^{4} +(-0.536865 - 1.79325i) q^{5} +(-1.21386 + 1.23554i) q^{6} +(1.99514 + 4.62524i) q^{7} +(0.173648 + 0.984808i) q^{8} +(-2.43770 - 1.74860i) q^{9} +(0.325051 - 1.84346i) q^{10} +(1.04359 - 0.247336i) q^{11} +(-1.63925 + 0.559338i) q^{12} +(-2.13082 - 1.40147i) q^{13} +(-0.292887 + 5.02868i) q^{14} +(3.24155 + 0.0656115i) q^{15} +(-0.286803 + 0.957990i) q^{16} +(5.49113 - 1.99861i) q^{17} +(-1.39364 - 2.65665i) q^{18} +(-5.64228 - 2.05362i) q^{19} +(1.11782 - 1.50149i) q^{20} +(-8.68443 + 0.837303i) q^{21} +(1.04359 + 0.247336i) q^{22} +(1.77254 - 4.10922i) q^{23} +(-1.71592 - 0.235852i) q^{24} +(1.24990 - 0.822072i) q^{25} +(-1.27520 - 2.20871i) q^{26} +(4.17581 - 3.09235i) q^{27} +(-2.51860 + 4.36235i) q^{28} +(-0.0699925 - 1.20173i) q^{29} +(2.86731 + 1.51344i) q^{30} +(6.69052 - 0.782010i) q^{31} +(-0.686242 + 0.727374i) q^{32} +(-0.145518 + 1.85192i) q^{33} +(5.80403 + 0.678393i) q^{34} +(7.22312 - 6.06092i) q^{35} +(-0.0531021 - 2.99953i) q^{36} +(1.39018 + 1.16650i) q^{37} +(-4.12046 - 4.36743i) q^{38} +(3.44070 - 2.77040i) q^{39} +(1.67279 - 0.840105i) q^{40} +(-5.78256 + 2.90411i) q^{41} +(-8.13647 - 3.14932i) q^{42} +(-1.11720 - 1.18417i) q^{43} +(0.821586 + 0.689392i) q^{44} +(-1.82697 + 5.31019i) q^{45} +(3.42822 - 2.87662i) q^{46} +(-11.9271 - 1.39408i) q^{47} +(-1.42755 - 0.980867i) q^{48} +(-12.6086 + 13.3644i) q^{49} +(1.48590 - 0.173676i) q^{50} +(0.383906 + 10.1140i) q^{51} +(-0.148292 - 2.54608i) q^{52} +(2.50648 - 4.34135i) q^{53} +(5.11948 - 0.889320i) q^{54} +(-1.00381 - 1.73864i) q^{55} +(-4.20852 + 2.76799i) q^{56} +(6.37794 - 8.21462i) q^{57} +(0.476786 - 1.10531i) q^{58} +(-2.74121 - 0.649679i) q^{59} +(1.88309 + 2.63930i) q^{60} +(-2.62080 + 3.52034i) q^{61} +(6.32983 + 2.30387i) q^{62} +(3.22416 - 14.7637i) q^{63} +(-0.939693 + 0.342020i) q^{64} +(-1.36922 + 4.57351i) q^{65} +(-0.961181 + 1.58963i) q^{66} +(-0.636693 + 10.9316i) q^{67} +(4.88220 + 3.21108i) q^{68} +(5.83580 + 5.10159i) q^{69} +(9.17495 - 2.17450i) q^{70} +(-1.51839 + 8.61121i) q^{71} +(1.29873 - 2.70431i) q^{72} +(-1.22998 - 6.97557i) q^{73} +(0.718785 + 1.66633i) q^{74} +(0.692770 + 2.49684i) q^{75} +(-1.72208 - 5.75214i) q^{76} +(3.22610 + 4.33341i) q^{77} +(4.31808 - 0.931533i) q^{78} +(5.34642 + 2.68507i) q^{79} +1.87189 q^{80} +(2.88479 + 8.52514i) q^{81} -6.47084 q^{82} +(-8.01224 - 4.02390i) q^{83} +(-5.85760 - 6.46598i) q^{84} +(-6.53201 - 8.77401i) q^{85} +(-0.466916 - 1.55961i) q^{86} +(2.01863 + 0.521786i) q^{87} +(0.424797 + 0.984790i) q^{88} +(-1.24608 - 7.06686i) q^{89} +(-4.01585 + 3.92541i) q^{90} +(2.23084 - 12.6517i) q^{91} +(4.35459 - 1.03206i) q^{92} +(-2.25809 + 11.4466i) q^{93} +(-10.0328 - 6.59869i) q^{94} +(-0.653523 + 11.2206i) q^{95} +(-0.835493 - 1.51722i) q^{96} +(-2.10743 + 7.03931i) q^{97} +(-17.2654 + 6.28409i) q^{98} +(-2.97647 - 1.22190i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 90 q - 9 q^{6} - 18 q^{13} - 9 q^{18} - 9 q^{20} - 54 q^{21} + 27 q^{23} - 18 q^{25} - 27 q^{26} - 27 q^{27} - 18 q^{28} - 27 q^{29} + 9 q^{30} + 54 q^{31} - 63 q^{33} - 27 q^{35} - 9 q^{36} - 18 q^{38}+ \cdots - 81 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/162\mathbb{Z}\right)^\times\).

\(n\) \(83\)
\(\chi(n)\) \(e\left(\frac{4}{27}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.893633 + 0.448799i 0.631894 + 0.317349i
\(3\) −0.530234 + 1.64889i −0.306131 + 0.951989i
\(4\) 0.597159 + 0.802123i 0.298579 + 0.401062i
\(5\) −0.536865 − 1.79325i −0.240093 − 0.801968i −0.990079 − 0.140514i \(-0.955125\pi\)
0.749985 − 0.661454i \(-0.230061\pi\)
\(6\) −1.21386 + 1.23554i −0.495555 + 0.504406i
\(7\) 1.99514 + 4.62524i 0.754090 + 1.74818i 0.658274 + 0.752779i \(0.271287\pi\)
0.0958168 + 0.995399i \(0.469454\pi\)
\(8\) 0.173648 + 0.984808i 0.0613939 + 0.348182i
\(9\) −2.43770 − 1.74860i −0.812568 − 0.582867i
\(10\) 0.325051 − 1.84346i 0.102790 − 0.582952i
\(11\) 1.04359 − 0.247336i 0.314656 − 0.0745747i −0.0702537 − 0.997529i \(-0.522381\pi\)
0.384909 + 0.922954i \(0.374233\pi\)
\(12\) −1.63925 + 0.559338i −0.473211 + 0.161467i
\(13\) −2.13082 − 1.40147i −0.590984 − 0.388697i 0.218501 − 0.975837i \(-0.429883\pi\)
−0.809485 + 0.587140i \(0.800254\pi\)
\(14\) −0.292887 + 5.02868i −0.0782775 + 1.34397i
\(15\) 3.24155 + 0.0656115i 0.836965 + 0.0169408i
\(16\) −0.286803 + 0.957990i −0.0717008 + 0.239497i
\(17\) 5.49113 − 1.99861i 1.33179 − 0.484734i 0.424576 − 0.905392i \(-0.360423\pi\)
0.907219 + 0.420659i \(0.138201\pi\)
\(18\) −1.39364 − 2.65665i −0.328484 − 0.626177i
\(19\) −5.64228 − 2.05362i −1.29443 − 0.471133i −0.399250 − 0.916842i \(-0.630729\pi\)
−0.895178 + 0.445709i \(0.852952\pi\)
\(20\) 1.11782 − 1.50149i 0.249952 − 0.335743i
\(21\) −8.68443 + 0.837303i −1.89510 + 0.182715i
\(22\) 1.04359 + 0.247336i 0.222495 + 0.0527323i
\(23\) 1.77254 − 4.10922i 0.369601 − 0.856831i −0.627345 − 0.778741i \(-0.715859\pi\)
0.996946 − 0.0780902i \(-0.0248822\pi\)
\(24\) −1.71592 − 0.235852i −0.350260 − 0.0481430i
\(25\) 1.24990 − 0.822072i 0.249980 − 0.164414i
\(26\) −1.27520 − 2.20871i −0.250087 − 0.433163i
\(27\) 4.17581 − 3.09235i 0.803635 − 0.595122i
\(28\) −2.51860 + 4.36235i −0.475971 + 0.824406i
\(29\) −0.0699925 − 1.20173i −0.0129973 − 0.223155i −0.998552 − 0.0537969i \(-0.982868\pi\)
0.985555 − 0.169358i \(-0.0541694\pi\)
\(30\) 2.86731 + 1.51344i 0.523497 + 0.276315i
\(31\) 6.69052 − 0.782010i 1.20165 − 0.140453i 0.508383 − 0.861131i \(-0.330243\pi\)
0.693270 + 0.720678i \(0.256169\pi\)
\(32\) −0.686242 + 0.727374i −0.121312 + 0.128583i
\(33\) −0.145518 + 1.85192i −0.0253314 + 0.322378i
\(34\) 5.80403 + 0.678393i 0.995382 + 0.116343i
\(35\) 7.22312 − 6.06092i 1.22093 − 1.02448i
\(36\) −0.0531021 − 2.99953i −0.00885035 − 0.499922i
\(37\) 1.39018 + 1.16650i 0.228544 + 0.191771i 0.749868 − 0.661588i \(-0.230117\pi\)
−0.521324 + 0.853359i \(0.674562\pi\)
\(38\) −4.12046 − 4.36743i −0.668427 − 0.708491i
\(39\) 3.44070 − 2.77040i 0.550954 − 0.443619i
\(40\) 1.67279 − 0.840105i 0.264491 − 0.132832i
\(41\) −5.78256 + 2.90411i −0.903084 + 0.453546i −0.838833 − 0.544388i \(-0.816762\pi\)
−0.0642502 + 0.997934i \(0.520466\pi\)
\(42\) −8.13647 − 3.14932i −1.25548 − 0.485951i
\(43\) −1.11720 − 1.18417i −0.170372 − 0.180584i 0.636562 − 0.771226i \(-0.280356\pi\)
−0.806934 + 0.590642i \(0.798875\pi\)
\(44\) 0.821586 + 0.689392i 0.123859 + 0.103930i
\(45\) −1.82697 + 5.31019i −0.272348 + 0.791596i
\(46\) 3.42822 − 2.87662i 0.505463 − 0.424134i
\(47\) −11.9271 − 1.39408i −1.73975 − 0.203348i −0.813840 − 0.581088i \(-0.802627\pi\)
−0.925911 + 0.377741i \(0.876701\pi\)
\(48\) −1.42755 − 0.980867i −0.206049 − 0.141576i
\(49\) −12.6086 + 13.3644i −1.80123 + 1.90919i
\(50\) 1.48590 − 0.173676i 0.210138 − 0.0245616i
\(51\) 0.383906 + 10.1140i 0.0537577 + 1.41625i
\(52\) −0.148292 − 2.54608i −0.0205644 − 0.353078i
\(53\) 2.50648 − 4.34135i 0.344291 − 0.596330i −0.640933 − 0.767597i \(-0.721452\pi\)
0.985225 + 0.171266i \(0.0547858\pi\)
\(54\) 5.11948 − 0.889320i 0.696673 − 0.121021i
\(55\) −1.00381 − 1.73864i −0.135353 − 0.234439i
\(56\) −4.20852 + 2.76799i −0.562388 + 0.369888i
\(57\) 6.37794 − 8.21462i 0.844778 − 1.08805i
\(58\) 0.476786 − 1.10531i 0.0626050 − 0.145135i
\(59\) −2.74121 − 0.649679i −0.356875 − 0.0845810i 0.0482675 − 0.998834i \(-0.484630\pi\)
−0.405143 + 0.914253i \(0.632778\pi\)
\(60\) 1.88309 + 2.63930i 0.243106 + 0.340733i
\(61\) −2.62080 + 3.52034i −0.335559 + 0.450734i −0.937592 − 0.347737i \(-0.886950\pi\)
0.602033 + 0.798471i \(0.294358\pi\)
\(62\) 6.32983 + 2.30387i 0.803890 + 0.292592i
\(63\) 3.22416 − 14.7637i 0.406206 − 1.86005i
\(64\) −0.939693 + 0.342020i −0.117462 + 0.0427525i
\(65\) −1.36922 + 4.57351i −0.169831 + 0.567274i
\(66\) −0.961181 + 1.58963i −0.118313 + 0.195670i
\(67\) −0.636693 + 10.9316i −0.0777844 + 1.33551i 0.703216 + 0.710976i \(0.251747\pi\)
−0.781001 + 0.624530i \(0.785290\pi\)
\(68\) 4.88220 + 3.21108i 0.592054 + 0.389400i
\(69\) 5.83580 + 5.10159i 0.702548 + 0.614159i
\(70\) 9.17495 − 2.17450i 1.09662 − 0.259903i
\(71\) −1.51839 + 8.61121i −0.180200 + 1.02196i 0.751770 + 0.659426i \(0.229200\pi\)
−0.931969 + 0.362537i \(0.881911\pi\)
\(72\) 1.29873 − 2.70431i 0.153057 − 0.318706i
\(73\) −1.22998 − 6.97557i −0.143958 − 0.816429i −0.968198 − 0.250187i \(-0.919508\pi\)
0.824239 − 0.566242i \(-0.191603\pi\)
\(74\) 0.718785 + 1.66633i 0.0835570 + 0.193707i
\(75\) 0.692770 + 2.49684i 0.0799942 + 0.288311i
\(76\) −1.72208 − 5.75214i −0.197536 − 0.659816i
\(77\) 3.22610 + 4.33341i 0.367649 + 0.493838i
\(78\) 4.31808 − 0.931533i 0.488926 − 0.105475i
\(79\) 5.34642 + 2.68507i 0.601519 + 0.302094i 0.723377 − 0.690453i \(-0.242589\pi\)
−0.121858 + 0.992548i \(0.538885\pi\)
\(80\) 1.87189 0.209284
\(81\) 2.88479 + 8.52514i 0.320532 + 0.947238i
\(82\) −6.47084 −0.714585
\(83\) −8.01224 − 4.02390i −0.879457 − 0.441680i −0.0490238 − 0.998798i \(-0.515611\pi\)
−0.830434 + 0.557117i \(0.811907\pi\)
\(84\) −5.85760 − 6.46598i −0.639117 − 0.705496i
\(85\) −6.53201 − 8.77401i −0.708496 − 0.951675i
\(86\) −0.466916 − 1.55961i −0.0503489 − 0.168177i
\(87\) 2.01863 + 0.521786i 0.216420 + 0.0559413i
\(88\) 0.424797 + 0.984790i 0.0452835 + 0.104979i
\(89\) −1.24608 − 7.06686i −0.132084 − 0.749086i −0.976846 − 0.213944i \(-0.931369\pi\)
0.844762 − 0.535142i \(-0.179742\pi\)
\(90\) −4.01585 + 3.92541i −0.423307 + 0.413775i
\(91\) 2.23084 − 12.6517i 0.233855 − 1.32626i
\(92\) 4.35459 − 1.03206i 0.453997 − 0.107599i
\(93\) −2.25809 + 11.4466i −0.234153 + 1.18696i
\(94\) −10.0328 − 6.59869i −1.03481 − 0.680603i
\(95\) −0.653523 + 11.2206i −0.0670501 + 1.15121i
\(96\) −0.835493 − 1.51722i −0.0852722 − 0.154850i
\(97\) −2.10743 + 7.03931i −0.213977 + 0.714733i 0.781849 + 0.623467i \(0.214277\pi\)
−0.995827 + 0.0912660i \(0.970909\pi\)
\(98\) −17.2654 + 6.28409i −1.74407 + 0.634789i
\(99\) −2.97647 − 1.22190i −0.299146 − 0.122805i
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 162.2.g.b.13.3 ✓ 90
3.2 odd 2 486.2.g.b.253.4 90
81.25 even 27 inner 162.2.g.b.25.3 yes 90
81.56 odd 54 486.2.g.b.73.4 90
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
162.2.g.b.13.3 ✓ 90 1.1 even 1 trivial
162.2.g.b.25.3 yes 90 81.25 even 27 inner
486.2.g.b.73.4 90 81.56 odd 54
486.2.g.b.253.4 90 3.2 odd 2