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(25,114)] 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("114.25"); 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([0, 14])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 114 = 2 \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 114.i (of order \(9\), degree \(6\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,0,0,3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.910294583043\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\Q(\zeta_{18})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{3} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label 73.1
Root \(-0.766044 - 0.642788i\) of defining polynomial
Character \(\chi\) \(=\) 114.73
Dual form 114.2.i.c.25.1

$q$-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+(0.766044 + 0.642788i) q^{2} +(0.939693 - 0.342020i) q^{3} +(0.173648 + 0.984808i) q^{4} +(-0.613341 + 3.47843i) q^{5} +(0.939693 + 0.342020i) q^{6} +(-1.85844 - 3.21891i) q^{7} +(-0.500000 + 0.866025i) q^{8} +(0.766044 - 0.642788i) q^{9} +(-2.70574 + 2.27038i) q^{10} +(2.64543 - 4.58202i) q^{11} +(0.500000 + 0.866025i) q^{12} +(0.213011 + 0.0775297i) q^{13} +(0.645430 - 3.66041i) q^{14} +(0.613341 + 3.47843i) q^{15} +(-0.939693 + 0.342020i) q^{16} +(-1.26604 - 1.06234i) q^{17} +1.00000 q^{18} +(-4.17752 - 1.24432i) q^{19} -3.53209 q^{20} +(-2.84730 - 2.38917i) q^{21} +(4.97178 - 1.80958i) q^{22} +(1.50727 + 8.54818i) q^{23} +(-0.173648 + 0.984808i) q^{24} +(-7.02481 - 2.55682i) q^{25} +(0.113341 + 0.196312i) q^{26} +(0.500000 - 0.866025i) q^{27} +(2.84730 - 2.38917i) q^{28} +(0.0923963 - 0.0775297i) q^{29} +(-1.76604 + 3.05888i) q^{30} +(-1.56031 - 2.70253i) q^{31} +(-0.939693 - 0.342020i) q^{32} +(0.918748 - 5.21048i) q^{33} +(-0.286989 - 1.62760i) q^{34} +(12.3366 - 4.49016i) q^{35} +(0.766044 + 0.642788i) q^{36} +5.12836 q^{37} +(-2.40033 - 3.63846i) q^{38} +0.226682 q^{39} +(-2.70574 - 2.27038i) q^{40} +(-6.67752 + 2.43042i) q^{41} +(-0.645430 - 3.66041i) q^{42} +(-0.929892 + 5.27368i) q^{43} +(4.97178 + 1.80958i) q^{44} +(1.76604 + 3.05888i) q^{45} +(-4.34002 + 7.51714i) q^{46} +(-1.92262 + 1.61327i) q^{47} +(-0.766044 + 0.642788i) q^{48} +(-3.40760 + 5.90214i) q^{49} +(-3.73783 - 6.47410i) q^{50} +(-1.55303 - 0.565258i) q^{51} +(-0.0393628 + 0.223238i) q^{52} +(1.03074 + 5.84564i) q^{53} +(0.939693 - 0.342020i) q^{54} +(14.3157 + 12.0123i) q^{55} +3.71688 q^{56} +(-4.35117 + 0.259515i) q^{57} +0.120615 q^{58} +(0.167718 + 0.140732i) q^{59} +(-3.31908 + 1.20805i) q^{60} +(-0.273318 - 1.55007i) q^{61} +(0.541889 - 3.07321i) q^{62} +(-3.49273 - 1.27125i) q^{63} +(-0.500000 - 0.866025i) q^{64} +(-0.400330 + 0.693392i) q^{65} +(4.05303 - 3.40090i) q^{66} +(11.8589 - 9.95080i) q^{67} +(0.826352 - 1.43128i) q^{68} +(4.34002 + 7.51714i) q^{69} +(12.3366 + 4.49016i) q^{70} +(-0.235300 + 1.33445i) q^{71} +(0.173648 + 0.984808i) q^{72} +(-2.27972 + 0.829748i) q^{73} +(3.92855 + 3.29644i) q^{74} -7.47565 q^{75} +(0.500000 - 4.33013i) q^{76} -19.6655 q^{77} +(0.173648 + 0.145708i) q^{78} +(2.69207 - 0.979832i) q^{79} +(-0.613341 - 3.47843i) q^{80} +(0.173648 - 0.984808i) q^{81} +(-6.67752 - 2.43042i) q^{82} +(-0.960637 - 1.66387i) q^{83} +(1.85844 - 3.21891i) q^{84} +(4.47178 - 3.75227i) q^{85} +(-4.10220 + 3.44215i) q^{86} +(0.0603074 - 0.104455i) q^{87} +(2.64543 + 4.58202i) q^{88} +(11.4226 + 4.15749i) q^{89} +(-0.613341 + 3.47843i) q^{90} +(-0.146307 - 0.829748i) q^{91} +(-8.15657 + 2.96875i) q^{92} +(-2.39053 - 2.00589i) q^{93} -2.50980 q^{94} +(6.89053 - 13.7680i) q^{95} -1.00000 q^{96} +(13.4081 + 11.2507i) q^{97} +(-6.40420 + 2.33094i) q^{98} +(-0.918748 - 5.21048i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 3 q^{5} - 3 q^{7} - 3 q^{8} - 6 q^{10} + 3 q^{12} + 9 q^{13} - 12 q^{14} - 3 q^{15} - 3 q^{17} + 6 q^{18} - 12 q^{20} - 15 q^{21} + 15 q^{22} + 27 q^{23} - 15 q^{25} - 6 q^{26} + 3 q^{27} + 15 q^{28}+ \cdots - 3 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(77\) \(97\)
\(\chi(n)\) \(1\) \(e\left(\frac{2}{9}\right)\)

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
\(3\) 0.939693 − 0.342020i 0.542532 − 0.197465i
\(4\) 0.173648 + 0.984808i 0.0868241 + 0.492404i
\(5\) −0.613341 + 3.47843i −0.274294 + 1.55560i 0.466900 + 0.884310i \(0.345371\pi\)
−0.741194 + 0.671290i \(0.765740\pi\)
\(6\) 0.939693 + 0.342020i 0.383628 + 0.139629i
\(7\) −1.85844 − 3.21891i −0.702425 − 1.21664i −0.967613 − 0.252438i \(-0.918767\pi\)
0.265188 − 0.964197i \(-0.414566\pi\)
\(8\) −0.500000 + 0.866025i −0.176777 + 0.306186i
\(9\) 0.766044 − 0.642788i 0.255348 − 0.214263i
\(10\) −2.70574 + 2.27038i −0.855629 + 0.717958i
\(11\) 2.64543 − 4.58202i 0.797627 − 1.38153i −0.123531 − 0.992341i \(-0.539422\pi\)
0.921158 − 0.389190i \(-0.127245\pi\)
\(12\) 0.500000 + 0.866025i 0.144338 + 0.250000i
\(13\) 0.213011 + 0.0775297i 0.0590786 + 0.0215029i 0.371390 − 0.928477i \(-0.378881\pi\)
−0.312312 + 0.949980i \(0.601103\pi\)
\(14\) 0.645430 − 3.66041i 0.172498 − 0.978287i
\(15\) 0.613341 + 3.47843i 0.158364 + 0.898126i
\(16\) −0.939693 + 0.342020i −0.234923 + 0.0855050i
\(17\) −1.26604 − 1.06234i −0.307061 − 0.257655i 0.476215 − 0.879329i \(-0.342008\pi\)
−0.783276 + 0.621674i \(0.786453\pi\)
\(18\) 1.00000 0.235702
\(19\) −4.17752 − 1.24432i −0.958388 − 0.285467i
\(20\) −3.53209 −0.789799
\(21\) −2.84730 − 2.38917i −0.621331 − 0.521359i
\(22\) 4.97178 − 1.80958i 1.05999 − 0.385804i
\(23\) 1.50727 + 8.54818i 0.314288 + 1.78242i 0.576182 + 0.817321i \(0.304542\pi\)
−0.261894 + 0.965097i \(0.584347\pi\)
\(24\) −0.173648 + 0.984808i −0.0354458 + 0.201023i
\(25\) −7.02481 − 2.55682i −1.40496 − 0.511365i
\(26\) 0.113341 + 0.196312i 0.0222280 + 0.0385000i
\(27\) 0.500000 − 0.866025i 0.0962250 − 0.166667i
\(28\) 2.84730 − 2.38917i 0.538088 − 0.451510i
\(29\) 0.0923963 − 0.0775297i 0.0171576 − 0.0143969i −0.634169 − 0.773195i \(-0.718657\pi\)
0.651326 + 0.758798i \(0.274213\pi\)
\(30\) −1.76604 + 3.05888i −0.322434 + 0.558472i
\(31\) −1.56031 − 2.70253i −0.280239 − 0.485389i 0.691204 − 0.722660i \(-0.257081\pi\)
−0.971444 + 0.237271i \(0.923747\pi\)
\(32\) −0.939693 − 0.342020i −0.166116 − 0.0604612i
\(33\) 0.918748 − 5.21048i 0.159934 − 0.907028i
\(34\) −0.286989 − 1.62760i −0.0492182 − 0.279130i
\(35\) 12.3366 − 4.49016i 2.08527 − 0.758976i
\(36\) 0.766044 + 0.642788i 0.127674 + 0.107131i
\(37\) 5.12836 0.843096 0.421548 − 0.906806i \(-0.361487\pi\)
0.421548 + 0.906806i \(0.361487\pi\)
\(38\) −2.40033 − 3.63846i −0.389385 − 0.590237i
\(39\) 0.226682 0.0362981
\(40\) −2.70574 − 2.27038i −0.427815 − 0.358979i
\(41\) −6.67752 + 2.43042i −1.04285 + 0.379568i −0.805961 − 0.591968i \(-0.798351\pi\)
−0.236892 + 0.971536i \(0.576129\pi\)
\(42\) −0.645430 − 3.66041i −0.0995920 − 0.564814i
\(43\) −0.929892 + 5.27368i −0.141807 + 0.804229i 0.828068 + 0.560627i \(0.189440\pi\)
−0.969875 + 0.243602i \(0.921671\pi\)
\(44\) 4.97178 + 1.80958i 0.749524 + 0.272805i
\(45\) 1.76604 + 3.05888i 0.263266 + 0.455991i
\(46\) −4.34002 + 7.51714i −0.639901 + 1.10834i
\(47\) −1.92262 + 1.61327i −0.280443 + 0.235319i −0.772149 − 0.635442i \(-0.780818\pi\)
0.491706 + 0.870761i \(0.336374\pi\)
\(48\) −0.766044 + 0.642788i −0.110569 + 0.0927784i
\(49\) −3.40760 + 5.90214i −0.486801 + 0.843163i
\(50\) −3.73783 − 6.47410i −0.528608 − 0.915577i
\(51\) −1.55303 − 0.565258i −0.217468 − 0.0791519i
\(52\) −0.0393628 + 0.223238i −0.00545864 + 0.0309575i
\(53\) 1.03074 + 5.84564i 0.141584 + 0.802961i 0.970047 + 0.242918i \(0.0781044\pi\)
−0.828463 + 0.560043i \(0.810784\pi\)
\(54\) 0.939693 − 0.342020i 0.127876 − 0.0465430i
\(55\) 14.3157 + 12.0123i 1.93033 + 1.61974i
\(56\) 3.71688 0.496689
\(57\) −4.35117 + 0.259515i −0.576326 + 0.0343736i
\(58\) 0.120615 0.0158375
\(59\) 0.167718 + 0.140732i 0.0218351 + 0.0183218i 0.653640 − 0.756806i \(-0.273241\pi\)
−0.631805 + 0.775128i \(0.717686\pi\)
\(60\) −3.31908 + 1.20805i −0.428491 + 0.155958i
\(61\) −0.273318 − 1.55007i −0.0349948 − 0.198466i 0.962298 − 0.271997i \(-0.0876841\pi\)
−0.997293 + 0.0735316i \(0.976573\pi\)
\(62\) 0.541889 − 3.07321i 0.0688200 − 0.390298i
\(63\) −3.49273 − 1.27125i −0.440042 − 0.160162i
\(64\) −0.500000 − 0.866025i −0.0625000 − 0.108253i
\(65\) −0.400330 + 0.693392i −0.0496548 + 0.0860046i
\(66\) 4.05303 − 3.40090i 0.498894 − 0.418622i
\(67\) 11.8589 − 9.95080i 1.44880 − 1.21568i 0.515343 − 0.856984i \(-0.327664\pi\)
0.933453 − 0.358701i \(-0.116780\pi\)
\(68\) 0.826352 − 1.43128i 0.100210 − 0.173569i
\(69\) 4.34002 + 7.51714i 0.522477 + 0.904957i
\(70\) 12.3366 + 4.49016i 1.47451 + 0.536677i
\(71\) −0.235300 + 1.33445i −0.0279249 + 0.158370i −0.995582 − 0.0939008i \(-0.970066\pi\)
0.967657 + 0.252271i \(0.0811775\pi\)
\(72\) 0.173648 + 0.984808i 0.0204646 + 0.116061i
\(73\) −2.27972 + 0.829748i −0.266820 + 0.0971147i −0.471966 − 0.881617i \(-0.656455\pi\)
0.205145 + 0.978731i \(0.434233\pi\)
\(74\) 3.92855 + 3.29644i 0.456684 + 0.383204i
\(75\) −7.47565 −0.863214
\(76\) 0.500000 − 4.33013i 0.0573539 − 0.496700i
\(77\) −19.6655 −2.24109
\(78\) 0.173648 + 0.145708i 0.0196618 + 0.0164982i
\(79\) 2.69207 − 0.979832i 0.302881 − 0.110240i −0.186109 − 0.982529i \(-0.559588\pi\)
0.488990 + 0.872289i \(0.337365\pi\)
\(80\) −0.613341 − 3.47843i −0.0685736 − 0.388900i
\(81\) 0.173648 − 0.984808i 0.0192942 − 0.109423i
\(82\) −6.67752 − 2.43042i −0.737409 − 0.268395i
\(83\) −0.960637 − 1.66387i −0.105444 − 0.182634i 0.808476 − 0.588530i \(-0.200293\pi\)
−0.913919 + 0.405896i \(0.866960\pi\)
\(84\) 1.85844 − 3.21891i 0.202773 − 0.351212i
\(85\) 4.47178 − 3.75227i 0.485033 − 0.406991i
\(86\) −4.10220 + 3.44215i −0.442351 + 0.371177i
\(87\) 0.0603074 − 0.104455i 0.00646563 − 0.0111988i
\(88\) 2.64543 + 4.58202i 0.282004 + 0.488445i
\(89\) 11.4226 + 4.15749i 1.21080 + 0.440693i 0.866980 − 0.498344i \(-0.166058\pi\)
0.343816 + 0.939037i \(0.388280\pi\)
\(90\) −0.613341 + 3.47843i −0.0646518 + 0.366659i
\(91\) −0.146307 − 0.829748i −0.0153371 − 0.0869813i
\(92\) −8.15657 + 2.96875i −0.850382 + 0.309514i
\(93\) −2.39053 − 2.00589i −0.247886 − 0.208001i
\(94\) −2.50980 −0.258866
\(95\) 6.89053 − 13.7680i 0.706953 − 1.41257i
\(96\) −1.00000 −0.102062
\(97\) 13.4081 + 11.2507i 1.36138 + 1.14234i 0.975552 + 0.219767i \(0.0705296\pi\)
0.385831 + 0.922570i \(0.373915\pi\)
\(98\) −6.40420 + 2.33094i −0.646922 + 0.235460i
\(99\) −0.918748 − 5.21048i −0.0923377 − 0.523673i
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.i.c.73.1 yes 6
3.2 odd 2 342.2.u.b.73.1 6
4.3 odd 2 912.2.bo.d.529.1 6
19.5 even 9 2166.2.a.r.1.1 3
19.6 even 9 inner 114.2.i.c.25.1 ✓ 6
19.14 odd 18 2166.2.a.p.1.1 3
57.5 odd 18 6498.2.a.bp.1.3 3
57.14 even 18 6498.2.a.bu.1.3 3
57.44 odd 18 342.2.u.b.253.1 6
76.63 odd 18 912.2.bo.d.481.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
114.2.i.c.25.1 ✓ 6 19.6 even 9 inner
114.2.i.c.73.1 yes 6 1.1 even 1 trivial
342.2.u.b.73.1 6 3.2 odd 2
342.2.u.b.253.1 6 57.44 odd 18
912.2.bo.d.481.1 6 76.63 odd 18
912.2.bo.d.529.1 6 4.3 odd 2
2166.2.a.p.1.1 3 19.14 odd 18
2166.2.a.r.1.1 3 19.5 even 9
6498.2.a.bp.1.3 3 57.5 odd 18
6498.2.a.bu.1.3 3 57.14 even 18