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 = [18,0,3] 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 59.3
Root \(-1.72388 + 0.168030i\) of defining polynomial
Character \(\chi\) \(=\) 114.59
Dual form 114.2.l.b.29.3

$q$-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+(0.173648 - 0.984808i) q^{2} +(1.67739 + 0.431705i) q^{3} +(-0.939693 - 0.342020i) q^{4} +(1.14133 + 3.13578i) q^{5} +(0.716422 - 1.57694i) q^{6} +(-1.07356 - 1.85947i) q^{7} +(-0.500000 + 0.866025i) q^{8} +(2.62726 + 1.44827i) q^{9} +(3.28633 - 0.579469i) q^{10} +(-5.41799 - 3.12808i) q^{11} +(-1.42858 - 0.979371i) q^{12} +(-2.56208 - 3.05336i) q^{13} +(-2.01764 + 0.734361i) q^{14} +(0.560722 + 5.75264i) q^{15} +(0.766044 + 0.642788i) q^{16} +(-0.403611 - 0.0711674i) q^{17} +(1.88249 - 2.33586i) q^{18} +(4.34640 - 0.329887i) q^{19} -3.33703i q^{20} +(-0.998042 - 3.58251i) q^{21} +(-4.02138 + 4.79249i) q^{22} +(-0.280411 + 0.770422i) q^{23} +(-1.21256 + 1.23681i) q^{24} +(-4.70025 + 3.94398i) q^{25} +(-3.45187 + 1.99294i) q^{26} +(3.78171 + 3.56352i) q^{27} +(0.372845 + 2.11451i) q^{28} +(0.805141 + 4.56618i) q^{29} +(5.76261 + 0.446732i) q^{30} +(-2.02597 + 1.16970i) q^{31} +(0.766044 - 0.642788i) q^{32} +(-7.73767 - 7.58597i) q^{33} +(-0.140172 + 0.385121i) q^{34} +(4.60559 - 5.48872i) q^{35} +(-1.97348 - 2.25951i) q^{36} +6.01346i q^{37} +(0.429869 - 4.33765i) q^{38} +(-2.97944 - 6.22774i) q^{39} +(-3.28633 - 0.579469i) q^{40} +(0.926617 + 0.777524i) q^{41} +(-3.70139 + 0.360783i) q^{42} +(5.87377 - 2.13788i) q^{43} +(4.02138 + 4.79249i) q^{44} +(-1.54289 + 9.89147i) q^{45} +(0.710025 + 0.409933i) q^{46} +(-7.59919 + 1.33994i) q^{47} +(1.00746 + 1.40891i) q^{48} +(1.19492 - 2.06967i) q^{49} +(3.06787 + 5.31371i) q^{50} +(-0.646288 - 0.293616i) q^{51} +(1.36325 + 3.74550i) q^{52} +(-0.220516 - 0.0802612i) q^{53} +(4.16607 - 3.10546i) q^{54} +(3.62525 - 20.5598i) q^{55} +2.14713 q^{56} +(7.43301 + 1.32301i) q^{57} +4.63662 q^{58} +(0.930375 - 5.27642i) q^{59} +(1.44061 - 5.59749i) q^{60} +(7.30705 + 2.65955i) q^{61} +(0.800119 + 2.19831i) q^{62} +(-0.127515 - 6.44012i) q^{63} +(-0.500000 - 0.866025i) q^{64} +(6.65050 - 11.5190i) q^{65} +(-8.81436 + 6.30282i) q^{66} +(3.48689 - 0.614832i) q^{67} +(0.354929 + 0.204918i) q^{68} +(-0.802953 + 1.17124i) q^{69} +(-4.60559 - 5.48872i) q^{70} +(4.19799 - 1.52794i) q^{71} +(-2.56787 + 1.55114i) q^{72} +(-4.33185 - 3.63485i) q^{73} +(5.92210 + 1.04423i) q^{74} +(-9.58679 + 4.58646i) q^{75} +(-4.19711 - 1.17656i) q^{76} +13.4328i q^{77} +(-6.65050 + 1.85274i) q^{78} +(-8.05412 + 9.59853i) q^{79} +(-1.14133 + 3.13578i) q^{80} +(4.80501 + 7.60999i) q^{81} +(0.926617 - 0.777524i) q^{82} +(8.01579 - 4.62792i) q^{83} +(-0.287438 + 3.70781i) q^{84} +(-0.237488 - 1.34686i) q^{85} +(-1.08543 - 6.15577i) q^{86} +(-0.620710 + 8.00685i) q^{87} +(5.41799 - 3.12808i) q^{88} +(-5.61888 + 4.71480i) q^{89} +(9.47328 + 3.23709i) q^{90} +(-2.92708 + 8.04207i) q^{91} +(0.527000 - 0.628054i) q^{92} +(-3.90331 + 1.08741i) q^{93} +7.71642i q^{94} +(5.99513 + 13.2528i) q^{95} +(1.56245 - 0.747499i) q^{96} +(-16.0734 - 2.83418i) q^{97} +(-1.83073 - 1.53616i) q^{98} +(-9.70416 - 16.0650i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q + 3 q^{3} - 9 q^{8} - 3 q^{9} - 12 q^{13} - 12 q^{14} + 18 q^{15} + 6 q^{17} - 6 q^{19} - 24 q^{22} + 3 q^{24} - 18 q^{25} + 18 q^{26} - 6 q^{27} + 6 q^{28} - 6 q^{29} - 24 q^{33} - 6 q^{34} - 24 q^{35}+ \cdots - 102 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.173648 − 0.984808i 0.122788 − 0.696364i
\(3\) 1.67739 + 0.431705i 0.968440 + 0.249245i
\(4\) −0.939693 − 0.342020i −0.469846 − 0.171010i
\(5\) 1.14133 + 3.13578i 0.510418 + 1.40236i 0.880802 + 0.473484i \(0.157004\pi\)
−0.370384 + 0.928879i \(0.620774\pi\)
\(6\) 0.716422 − 1.57694i 0.292478 − 0.643783i
\(7\) −1.07356 − 1.85947i −0.405769 − 0.702812i 0.588642 − 0.808394i \(-0.299663\pi\)
−0.994411 + 0.105582i \(0.966330\pi\)
\(8\) −0.500000 + 0.866025i −0.176777 + 0.306186i
\(9\) 2.62726 + 1.44827i 0.875754 + 0.482758i
\(10\) 3.28633 − 0.579469i 1.03923 − 0.183244i
\(11\) −5.41799 − 3.12808i −1.63359 − 0.943151i −0.982975 − 0.183740i \(-0.941180\pi\)
−0.650611 − 0.759412i \(-0.725487\pi\)
\(12\) −1.42858 − 0.979371i −0.412395 − 0.282720i
\(13\) −2.56208 − 3.05336i −0.710592 − 0.846850i 0.283089 − 0.959094i \(-0.408641\pi\)
−0.993681 + 0.112243i \(0.964196\pi\)
\(14\) −2.01764 + 0.734361i −0.539237 + 0.196266i
\(15\) 0.560722 + 5.75264i 0.144778 + 1.48532i
\(16\) 0.766044 + 0.642788i 0.191511 + 0.160697i
\(17\) −0.403611 − 0.0711674i −0.0978899 − 0.0172606i 0.124489 − 0.992221i \(-0.460271\pi\)
−0.222379 + 0.974960i \(0.571382\pi\)
\(18\) 1.88249 − 2.33586i 0.443707 − 0.550567i
\(19\) 4.34640 − 0.329887i 0.997132 − 0.0756812i
\(20\) − 3.33703i − 0.746182i
\(21\) −0.998042 − 3.58251i −0.217791 − 0.781768i
\(22\) −4.02138 + 4.79249i −0.857361 + 1.02176i
\(23\) −0.280411 + 0.770422i −0.0584697 + 0.160644i −0.965488 − 0.260446i \(-0.916130\pi\)
0.907019 + 0.421090i \(0.138353\pi\)
\(24\) −1.21256 + 1.23681i −0.247513 + 0.252462i
\(25\) −4.70025 + 3.94398i −0.940051 + 0.788796i
\(26\) −3.45187 + 1.99294i −0.676968 + 0.390848i
\(27\) 3.78171 + 3.56352i 0.727790 + 0.685800i
\(28\) 0.372845 + 2.11451i 0.0704610 + 0.399604i
\(29\) 0.805141 + 4.56618i 0.149511 + 0.847919i 0.963634 + 0.267226i \(0.0861071\pi\)
−0.814123 + 0.580693i \(0.802782\pi\)
\(30\) 5.76261 + 0.446732i 1.05210 + 0.0815616i
\(31\) −2.02597 + 1.16970i −0.363875 + 0.210084i −0.670779 − 0.741657i \(-0.734040\pi\)
0.306904 + 0.951740i \(0.400707\pi\)
\(32\) 0.766044 − 0.642788i 0.135419 − 0.113630i
\(33\) −7.73767 − 7.58597i −1.34695 − 1.32055i
\(34\) −0.140172 + 0.385121i −0.0240394 + 0.0660477i
\(35\) 4.60559 − 5.48872i 0.778486 − 0.927764i
\(36\) −1.97348 − 2.25951i −0.328913 − 0.376585i
\(37\) 6.01346i 0.988607i 0.869289 + 0.494303i \(0.164577\pi\)
−0.869289 + 0.494303i \(0.835423\pi\)
\(38\) 0.429869 − 4.33765i 0.0697340 − 0.703660i
\(39\) −2.97944 − 6.22774i −0.477093 − 0.997236i
\(40\) −3.28633 − 0.579469i −0.519614 − 0.0916220i
\(41\) 0.926617 + 0.777524i 0.144713 + 0.121429i 0.712270 − 0.701905i \(-0.247667\pi\)
−0.567557 + 0.823334i \(0.692111\pi\)
\(42\) −3.70139 + 0.360783i −0.571137 + 0.0556700i
\(43\) 5.87377 − 2.13788i 0.895741 − 0.326023i 0.147196 − 0.989107i \(-0.452975\pi\)
0.748545 + 0.663084i \(0.230753\pi\)
\(44\) 4.02138 + 4.79249i 0.606246 + 0.722496i
\(45\) −1.54289 + 9.89147i −0.230001 + 1.47453i
\(46\) 0.710025 + 0.409933i 0.104687 + 0.0604413i
\(47\) −7.59919 + 1.33994i −1.10846 + 0.195451i −0.697767 − 0.716325i \(-0.745823\pi\)
−0.410689 + 0.911776i \(0.634712\pi\)
\(48\) 1.00746 + 1.40891i 0.145414 + 0.203359i
\(49\) 1.19492 − 2.06967i 0.170703 − 0.295666i
\(50\) 3.06787 + 5.31371i 0.433863 + 0.751472i
\(51\) −0.646288 − 0.293616i −0.0904984 − 0.0411145i
\(52\) 1.36325 + 3.74550i 0.189049 + 0.519408i
\(53\) −0.220516 − 0.0802612i −0.0302902 − 0.0110247i 0.326831 − 0.945083i \(-0.394019\pi\)
−0.357121 + 0.934058i \(0.616242\pi\)
\(54\) 4.16607 − 3.10546i 0.566930 − 0.422599i
\(55\) 3.62525 − 20.5598i 0.488828 − 2.77228i
\(56\) 2.14713 0.286922
\(57\) 7.43301 + 1.32301i 0.984526 + 0.175237i
\(58\) 4.63662 0.608819
\(59\) 0.930375 − 5.27642i 0.121124 − 0.686931i −0.862410 − 0.506210i \(-0.831046\pi\)
0.983534 − 0.180721i \(-0.0578430\pi\)
\(60\) 1.44061 − 5.59749i 0.185982 − 0.722633i
\(61\) 7.30705 + 2.65955i 0.935572 + 0.340520i 0.764416 − 0.644723i \(-0.223027\pi\)
0.171156 + 0.985244i \(0.445250\pi\)
\(62\) 0.800119 + 2.19831i 0.101615 + 0.279186i
\(63\) −0.127515 − 6.44012i −0.0160654 − 0.811379i
\(64\) −0.500000 − 0.866025i −0.0625000 − 0.108253i
\(65\) 6.65050 − 11.5190i 0.824893 − 1.42876i
\(66\) −8.81436 + 6.30282i −1.08497 + 0.775824i
\(67\) 3.48689 − 0.614832i 0.425991 − 0.0751137i 0.0434574 − 0.999055i \(-0.486163\pi\)
0.382534 + 0.923942i \(0.375052\pi\)
\(68\) 0.354929 + 0.204918i 0.0430415 + 0.0248500i
\(69\) −0.802953 + 1.17124i −0.0966642 + 0.141001i
\(70\) −4.60559 − 5.48872i −0.550473 − 0.656028i
\(71\) 4.19799 − 1.52794i 0.498210 − 0.181334i −0.0806788 − 0.996740i \(-0.525709\pi\)
0.578889 + 0.815407i \(0.303487\pi\)
\(72\) −2.56787 + 1.55114i −0.302627 + 0.182803i
\(73\) −4.33185 − 3.63485i −0.507005 − 0.425427i 0.353069 − 0.935597i \(-0.385138\pi\)
−0.860074 + 0.510170i \(0.829583\pi\)
\(74\) 5.92210 + 1.04423i 0.688430 + 0.121389i
\(75\) −9.58679 + 4.58646i −1.10699 + 0.529599i
\(76\) −4.19711 − 1.17656i −0.481441 − 0.134961i
\(77\) 13.4328i 1.53081i
\(78\) −6.65050 + 1.85274i −0.753020 + 0.209782i
\(79\) −8.05412 + 9.59853i −0.906159 + 1.07992i 0.0903059 + 0.995914i \(0.471216\pi\)
−0.996465 + 0.0840047i \(0.973229\pi\)
\(80\) −1.14133 + 3.13578i −0.127605 + 0.350591i
\(81\) 4.80501 + 7.60999i 0.533889 + 0.845554i
\(82\) 0.926617 − 0.777524i 0.102328 − 0.0858631i
\(83\) 8.01579 − 4.62792i 0.879848 − 0.507980i 0.00923947 − 0.999957i \(-0.497059\pi\)
0.870608 + 0.491977i \(0.163726\pi\)
\(84\) −0.287438 + 3.70781i −0.0313621 + 0.404555i
\(85\) −0.237488 − 1.34686i −0.0257591 − 0.146087i
\(86\) −1.08543 − 6.15577i −0.117045 − 0.663794i
\(87\) −0.620710 + 8.00685i −0.0665471 + 0.858424i
\(88\) 5.41799 − 3.12808i 0.577560 − 0.333454i
\(89\) −5.61888 + 4.71480i −0.595600 + 0.499767i −0.890028 − 0.455906i \(-0.849315\pi\)
0.294428 + 0.955674i \(0.404871\pi\)
\(90\) 9.47328 + 3.23709i 0.998571 + 0.341219i
\(91\) −2.92708 + 8.04207i −0.306841 + 0.843038i
\(92\) 0.527000 − 0.628054i 0.0549435 − 0.0654792i
\(93\) −3.90331 + 1.08741i −0.404754 + 0.112759i
\(94\) 7.71642i 0.795888i
\(95\) 5.99513 + 13.2528i 0.615087 + 1.35971i
\(96\) 1.56245 − 0.747499i 0.159467 − 0.0762913i
\(97\) −16.0734 − 2.83418i −1.63201 − 0.287767i −0.718786 − 0.695231i \(-0.755302\pi\)
−0.913221 + 0.407464i \(0.866413\pi\)
\(98\) −1.83073 − 1.53616i −0.184931 − 0.155176i
\(99\) −9.70416 − 16.0650i −0.975305 − 1.61459i
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.b.59.3 yes 18
3.2 odd 2 114.2.l.a.59.1 yes 18
4.3 odd 2 912.2.cc.c.401.1 18
12.11 even 2 912.2.cc.d.401.3 18
19.10 odd 18 114.2.l.a.29.1 ✓ 18
57.29 even 18 inner 114.2.l.b.29.3 yes 18
76.67 even 18 912.2.cc.d.257.3 18
228.143 odd 18 912.2.cc.c.257.1 18
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
114.2.l.a.29.1 ✓ 18 19.10 odd 18
114.2.l.a.59.1 yes 18 3.2 odd 2
114.2.l.b.29.3 yes 18 57.29 even 18 inner
114.2.l.b.59.3 yes 18 1.1 even 1 trivial
912.2.cc.c.257.1 18 228.143 odd 18
912.2.cc.c.401.1 18 4.3 odd 2
912.2.cc.d.257.3 18 76.67 even 18
912.2.cc.d.401.3 18 12.11 even 2