Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [114,2,Mod(29,114)] 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("114.29"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(114, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([9, 17])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 114 = 2 \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 114.l (of order \(18\), degree \(6\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [18,0,0] 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: \(0.910294583043\)
Analytic rank: \(0\)
Dimension: \(18\)
Relative dimension: \(3\) over \(\Q(\zeta_{18})\)
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: \( x^{18} - x^{15} - 18 x^{14} + 36 x^{13} + 10 x^{12} + 18 x^{11} + 90 x^{10} - 567 x^{9} + 270 x^{8} + \cdots + 19683 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label 71.1
Root \(-0.363139 + 1.69356i\) of defining polynomial
Character \(\chi\) \(=\) 114.71
Dual form 114.2.l.a.53.1

$q$-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+(0.939693 - 0.342020i) q^{2} +(-1.64823 + 0.532290i) q^{3} +(0.766044 - 0.642788i) q^{4} +(2.20556 - 2.62849i) q^{5} +(-1.36678 + 1.06392i) q^{6} +(1.68651 + 2.92113i) q^{7} +(0.500000 - 0.866025i) q^{8} +(2.43333 - 1.75467i) q^{9} +(1.17355 - 3.22432i) q^{10} +(-2.33635 - 1.34889i) q^{11} +(-0.920469 + 1.46722i) q^{12} +(-5.05419 + 0.891189i) q^{13} +(2.58389 + 2.16814i) q^{14} +(-2.23616 + 5.50635i) q^{15} +(0.173648 - 0.984808i) q^{16} +(1.44531 + 3.97095i) q^{17} +(1.68645 - 2.48110i) q^{18} +(-2.73048 + 3.39772i) q^{19} -3.43124i q^{20} +(-4.33465 - 3.91698i) q^{21} +(-2.65680 - 0.468466i) q^{22} +(-1.69398 - 2.01881i) q^{23} +(-0.363139 + 1.69356i) q^{24} +(-1.17620 - 6.67054i) q^{25} +(-4.44458 + 2.56608i) q^{26} +(-3.07670 + 4.18735i) q^{27} +(3.16961 + 1.15364i) q^{28} +(-3.54249 - 1.28936i) q^{29} +(-0.218019 + 5.93909i) q^{30} +(4.78254 - 2.76120i) q^{31} +(-0.173648 - 0.984808i) q^{32} +(4.56886 + 0.979673i) q^{33} +(2.71629 + 3.23715i) q^{34} +(11.3979 + 2.00975i) q^{35} +(0.736160 - 2.90828i) q^{36} +5.17636i q^{37} +(-1.40372 + 4.12669i) q^{38} +(7.85610 - 4.15918i) q^{39} +(-1.17355 - 3.22432i) q^{40} +(-0.289735 + 1.64317i) q^{41} +(-5.41293 - 2.19822i) q^{42} +(1.85806 + 1.55910i) q^{43} +(-2.65680 + 0.468466i) q^{44} +(0.754733 - 10.2660i) q^{45} +(-2.28230 - 1.31768i) q^{46} +(-0.0440069 + 0.120908i) q^{47} +(0.237991 + 1.71562i) q^{48} +(-2.18866 + 3.79087i) q^{49} +(-3.38672 - 5.86597i) q^{50} +(-4.49590 - 5.77572i) q^{51} +(-3.29889 + 3.93146i) q^{52} +(6.53342 - 5.48219i) q^{53} +(-1.45900 + 4.98712i) q^{54} +(-8.69853 + 3.16600i) q^{55} +3.37303 q^{56} +(2.69188 - 7.05363i) q^{57} -3.76983 q^{58} +(-3.87665 + 1.41099i) q^{59} +(1.82642 + 5.65549i) q^{60} +(3.53369 - 2.96512i) q^{61} +(3.54973 - 4.23041i) q^{62} +(9.22948 + 4.14880i) q^{63} +(-0.500000 - 0.866025i) q^{64} +(-8.80484 + 15.2504i) q^{65} +(4.62839 - 0.642049i) q^{66} +(3.81629 - 10.4852i) q^{67} +(3.65965 + 2.11290i) q^{68} +(3.86667 + 2.42578i) q^{69} +(11.3979 - 2.00975i) q^{70} +(-9.91131 - 8.31658i) q^{71} +(-0.302925 - 2.98467i) q^{72} +(-0.414656 + 2.35163i) q^{73} +(1.77042 + 4.86419i) q^{74} +(5.48930 + 10.3685i) q^{75} +(0.0923461 + 4.35792i) q^{76} -9.09972i q^{77} +(5.95979 - 6.59529i) q^{78} +(2.22246 + 0.391880i) q^{79} +(-2.20556 - 2.62849i) q^{80} +(2.84224 - 8.53942i) q^{81} +(0.289735 + 1.64317i) q^{82} +(-6.27861 + 3.62496i) q^{83} +(-5.83832 - 0.214320i) q^{84} +(13.6253 + 4.95920i) q^{85} +(2.27924 + 0.829577i) q^{86} +(6.52515 + 0.239533i) q^{87} +(-2.33635 + 1.34889i) q^{88} +(-0.209662 - 1.18905i) q^{89} +(-2.80197 - 9.90505i) q^{90} +(-11.1272 - 13.2609i) q^{91} +(-2.59533 - 0.457627i) q^{92} +(-6.41298 + 7.09680i) q^{93} +0.128667i q^{94} +(2.90863 + 14.6709i) q^{95} +(0.810416 + 1.53076i) q^{96} +(3.13271 + 8.60706i) q^{97} +(-0.760113 + 4.31081i) q^{98} +(-8.05200 + 0.817228i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q + 3 q^{6} + 9 q^{8} - 12 q^{13} + 12 q^{14} - 24 q^{15} - 6 q^{17} - 6 q^{19} - 24 q^{22} - 18 q^{25} - 18 q^{26} - 3 q^{27} + 6 q^{28} + 6 q^{29} - 27 q^{33} - 6 q^{34} + 24 q^{35} - 3 q^{36} - 3 q^{38}+ \cdots - 30 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/114\mathbb{Z}\right)^\times\).

\(n\) \(77\) \(97\)
\(\chi(n)\) \(-1\) \(e\left(\frac{7}{18}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.939693 − 0.342020i 0.664463 − 0.241845i
\(3\) −1.64823 + 0.532290i −0.951607 + 0.307318i
\(4\) 0.766044 − 0.642788i 0.383022 − 0.321394i
\(5\) 2.20556 − 2.62849i 0.986357 − 1.17549i 0.00187711 − 0.999998i \(-0.499402\pi\)
0.984480 − 0.175496i \(-0.0561531\pi\)
\(6\) −1.36678 + 1.06392i −0.557984 + 0.434342i
\(7\) 1.68651 + 2.92113i 0.637442 + 1.10408i 0.985992 + 0.166792i \(0.0533409\pi\)
−0.348550 + 0.937290i \(0.613326\pi\)
\(8\) 0.500000 − 0.866025i 0.176777 − 0.306186i
\(9\) 2.43333 − 1.75467i 0.811112 − 0.584891i
\(10\) 1.17355 − 3.22432i 0.371111 − 1.01962i
\(11\) −2.33635 − 1.34889i −0.704437 − 0.406707i 0.104561 − 0.994519i \(-0.466656\pi\)
−0.808998 + 0.587811i \(0.799990\pi\)
\(12\) −0.920469 + 1.46722i −0.265717 + 0.423550i
\(13\) −5.05419 + 0.891189i −1.40178 + 0.247171i −0.822874 − 0.568224i \(-0.807631\pi\)
−0.578905 + 0.815395i \(0.696520\pi\)
\(14\) 2.58389 + 2.16814i 0.690573 + 0.579460i
\(15\) −2.23616 + 5.50635i −0.577374 + 1.42173i
\(16\) 0.173648 − 0.984808i 0.0434120 − 0.246202i
\(17\) 1.44531 + 3.97095i 0.350538 + 0.963096i 0.982198 + 0.187850i \(0.0601520\pi\)
−0.631659 + 0.775246i \(0.717626\pi\)
\(18\) 1.68645 − 2.48110i 0.397501 − 0.584802i
\(19\) −2.73048 + 3.39772i −0.626414 + 0.779490i
\(20\) − 3.43124i − 0.767250i
\(21\) −4.33465 − 3.91698i −0.945899 − 0.854755i
\(22\) −2.65680 − 0.468466i −0.566433 − 0.0998773i
\(23\) −1.69398 − 2.01881i −0.353220 − 0.420951i 0.559952 − 0.828525i \(-0.310819\pi\)
−0.913172 + 0.407574i \(0.866375\pi\)
\(24\) −0.363139 + 1.69356i −0.0741255 + 0.345696i
\(25\) −1.17620 − 6.67054i −0.235239 − 1.33411i
\(26\) −4.44458 + 2.56608i −0.871653 + 0.503249i
\(27\) −3.07670 + 4.18735i −0.592112 + 0.805856i
\(28\) 3.16961 + 1.15364i 0.599000 + 0.218018i
\(29\) −3.54249 − 1.28936i −0.657823 − 0.239428i −0.00852691 − 0.999964i \(-0.502714\pi\)
−0.649296 + 0.760536i \(0.724936\pi\)
\(30\) −0.218019 + 5.93909i −0.0398047 + 1.08432i
\(31\) 4.78254 − 2.76120i 0.858970 − 0.495927i −0.00469717 − 0.999989i \(-0.501495\pi\)
0.863667 + 0.504062i \(0.168162\pi\)
\(32\) −0.173648 − 0.984808i −0.0306970 − 0.174091i
\(33\) 4.56886 + 0.979673i 0.795336 + 0.170539i
\(34\) 2.71629 + 3.23715i 0.465840 + 0.555166i
\(35\) 11.3979 + 2.00975i 1.92659 + 0.339710i
\(36\) 0.736160 − 2.90828i 0.122693 − 0.484713i
\(37\) 5.17636i 0.850989i 0.904961 + 0.425494i \(0.139900\pi\)
−0.904961 + 0.425494i \(0.860100\pi\)
\(38\) −1.40372 + 4.12669i −0.227713 + 0.669438i
\(39\) 7.85610 − 4.15918i 1.25798 − 0.666002i
\(40\) −1.17355 − 3.22432i −0.185555 − 0.509809i
\(41\) −0.289735 + 1.64317i −0.0452490 + 0.256620i −0.999038 − 0.0438581i \(-0.986035\pi\)
0.953789 + 0.300478i \(0.0971462\pi\)
\(42\) −5.41293 − 2.19822i −0.835233 − 0.339193i
\(43\) 1.85806 + 1.55910i 0.283351 + 0.237760i 0.773374 − 0.633950i \(-0.218567\pi\)
−0.490023 + 0.871709i \(0.663012\pi\)
\(44\) −2.65680 + 0.468466i −0.400528 + 0.0706240i
\(45\) 0.754733 − 10.2660i 0.112509 − 1.53037i
\(46\) −2.28230 − 1.31768i −0.336506 − 0.194282i
\(47\) −0.0440069 + 0.120908i −0.00641906 + 0.0176362i −0.942861 − 0.333187i \(-0.891876\pi\)
0.936442 + 0.350823i \(0.114098\pi\)
\(48\) 0.237991 + 1.71562i 0.0343510 + 0.247629i
\(49\) −2.18866 + 3.79087i −0.312665 + 0.541552i
\(50\) −3.38672 − 5.86597i −0.478955 − 0.829574i
\(51\) −4.49590 − 5.77572i −0.629551 − 0.808763i
\(52\) −3.29889 + 3.93146i −0.457473 + 0.545195i
\(53\) 6.53342 − 5.48219i 0.897434 − 0.753036i −0.0722533 − 0.997386i \(-0.523019\pi\)
0.969687 + 0.244350i \(0.0785746\pi\)
\(54\) −1.45900 + 4.98712i −0.198544 + 0.678661i
\(55\) −8.69853 + 3.16600i −1.17291 + 0.426904i
\(56\) 3.37303 0.450740
\(57\) 2.69188 − 7.05363i 0.356549 − 0.934277i
\(58\) −3.76983 −0.495004
\(59\) −3.87665 + 1.41099i −0.504697 + 0.183695i −0.581805 − 0.813328i \(-0.697653\pi\)
0.0771085 + 0.997023i \(0.475431\pi\)
\(60\) 1.82642 + 5.65549i 0.235789 + 0.730120i
\(61\) 3.53369 − 2.96512i 0.452443 − 0.379645i −0.387898 − 0.921702i \(-0.626799\pi\)
0.840342 + 0.542057i \(0.182354\pi\)
\(62\) 3.54973 − 4.23041i 0.450817 − 0.537262i
\(63\) 9.22948 + 4.14880i 1.16281 + 0.522700i
\(64\) −0.500000 − 0.866025i −0.0625000 − 0.108253i
\(65\) −8.80484 + 15.2504i −1.09211 + 1.89158i
\(66\) 4.62839 − 0.642049i 0.569715 − 0.0790308i
\(67\) 3.81629 − 10.4852i 0.466234 − 1.28097i −0.454490 − 0.890752i \(-0.650179\pi\)
0.920724 − 0.390215i \(-0.127599\pi\)
\(68\) 3.65965 + 2.11290i 0.443797 + 0.256226i
\(69\) 3.86667 + 2.42578i 0.465492 + 0.292029i
\(70\) 11.3979 − 2.00975i 1.36230 − 0.240211i
\(71\) −9.91131 − 8.31658i −1.17626 − 0.986996i −0.999996 − 0.00265261i \(-0.999156\pi\)
−0.176260 − 0.984344i \(-0.556400\pi\)
\(72\) −0.302925 − 2.98467i −0.0357001 − 0.351746i
\(73\) −0.414656 + 2.35163i −0.0485318 + 0.275238i −0.999411 − 0.0343255i \(-0.989072\pi\)
0.950879 + 0.309563i \(0.100183\pi\)
\(74\) 1.77042 + 4.86419i 0.205807 + 0.565450i
\(75\) 5.48930 + 10.3685i 0.633850 + 1.19725i
\(76\) 0.0923461 + 4.35792i 0.0105928 + 0.499888i
\(77\) − 9.09972i − 1.03701i
\(78\) 5.95979 − 6.59529i 0.674814 − 0.746770i
\(79\) 2.22246 + 0.391880i 0.250046 + 0.0440899i 0.297266 − 0.954795i \(-0.403925\pi\)
−0.0472200 + 0.998885i \(0.515036\pi\)
\(80\) −2.20556 − 2.62849i −0.246589 − 0.293874i
\(81\) 2.84224 − 8.53942i 0.315804 − 0.948824i
\(82\) 0.289735 + 1.64317i 0.0319959 + 0.181458i
\(83\) −6.27861 + 3.62496i −0.689167 + 0.397891i −0.803300 − 0.595575i \(-0.796924\pi\)
0.114133 + 0.993465i \(0.463591\pi\)
\(84\) −5.83832 − 0.214320i −0.637013 − 0.0233843i
\(85\) 13.6253 + 4.95920i 1.47787 + 0.537901i
\(86\) 2.27924 + 0.829577i 0.245777 + 0.0894556i
\(87\) 6.52515 + 0.239533i 0.699570 + 0.0256807i
\(88\) −2.33635 + 1.34889i −0.249056 + 0.143793i
\(89\) −0.209662 − 1.18905i −0.0222241 − 0.126039i 0.971677 − 0.236312i \(-0.0759387\pi\)
−0.993901 + 0.110273i \(0.964828\pi\)
\(90\) −2.80197 − 9.90505i −0.295354 − 1.04408i
\(91\) −11.1272 − 13.2609i −1.16645 − 1.39012i
\(92\) −2.59533 − 0.457627i −0.270582 − 0.0477109i
\(93\) −6.41298 + 7.09680i −0.664995 + 0.735904i
\(94\) 0.128667i 0.0132710i
\(95\) 2.90863 + 14.6709i 0.298419 + 1.50520i
\(96\) 0.810416 + 1.53076i 0.0827127 + 0.156233i
\(97\) 3.13271 + 8.60706i 0.318079 + 0.873914i 0.990959 + 0.134163i \(0.0428346\pi\)
−0.672880 + 0.739751i \(0.734943\pi\)
\(98\) −0.760113 + 4.31081i −0.0767830 + 0.435458i
\(99\) −8.05200 + 0.817228i −0.809257 + 0.0821345i
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 114.2.l.a.71.1 yes 18
3.2 odd 2 114.2.l.b.71.1 yes 18
4.3 odd 2 912.2.cc.d.641.3 18
12.11 even 2 912.2.cc.c.641.3 18
19.15 odd 18 114.2.l.b.53.1 yes 18
57.53 even 18 inner 114.2.l.a.53.1 ✓ 18
76.15 even 18 912.2.cc.c.737.3 18
228.167 odd 18 912.2.cc.d.737.3 18
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
114.2.l.a.53.1 ✓ 18 57.53 even 18 inner
114.2.l.a.71.1 yes 18 1.1 even 1 trivial
114.2.l.b.53.1 yes 18 19.15 odd 18
114.2.l.b.71.1 yes 18 3.2 odd 2
912.2.cc.c.641.3 18 12.11 even 2
912.2.cc.c.737.3 18 76.15 even 18
912.2.cc.d.641.3 18 4.3 odd 2
912.2.cc.d.737.3 18 228.167 odd 18