Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [342,2,Mod(55,342)] 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("342.55"); 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([0, 10])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 342 = 2 \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 342.u (of order \(9\), degree \(6\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,0,0,-3,0,-3,3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(8)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.73088374913\)
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: no (minimal twist has level 114)
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label 73.1
Root \(-0.766044 - 0.642788i\) of defining polynomial
Character \(\chi\) \(=\) 342.73
Dual form 342.2.u.b.253.1

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

Character values

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

\(n\) \(191\) \(325\)
\(\chi(n)\) \(1\) \(e\left(\frac{2}{9}\right)\)

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 0
\(4\) 0.173648 + 0.984808i 0.0868241 + 0.492404i
\(5\) 0.613341 − 3.47843i 0.274294 − 1.55560i −0.466900 − 0.884310i \(-0.654629\pi\)
0.741194 − 0.671290i \(-0.234260\pi\)
\(6\) 0 0
\(7\) −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 0
\(10\) −2.70574 + 2.27038i −0.855629 + 0.717958i
\(11\) −2.64543 + 4.58202i −0.797627 + 1.38153i 0.123531 + 0.992341i \(0.460578\pi\)
−0.921158 + 0.389190i \(0.872755\pi\)
\(12\) 0 0
\(13\) 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 0
\(16\) −0.939693 + 0.342020i −0.234923 + 0.0855050i
\(17\) 1.26604 + 1.06234i 0.307061 + 0.257655i 0.783276 − 0.621674i \(-0.213547\pi\)
−0.476215 + 0.879329i \(0.657992\pi\)
\(18\) 0 0
\(19\) −4.17752 − 1.24432i −0.958388 − 0.285467i
\(20\) 3.53209 0.789799
\(21\) 0 0
\(22\) 4.97178 − 1.80958i 1.05999 − 0.385804i
\(23\) −1.50727 − 8.54818i −0.314288 − 1.78242i −0.576182 − 0.817321i \(-0.695458\pi\)
0.261894 − 0.965097i \(-0.415653\pi\)
\(24\) 0 0
\(25\) −7.02481 − 2.55682i −1.40496 − 0.511365i
\(26\) −0.113341 − 0.196312i −0.0222280 − 0.0385000i
\(27\) 0 0
\(28\) 2.84730 − 2.38917i 0.538088 − 0.451510i
\(29\) −0.0923963 + 0.0775297i −0.0171576 + 0.0143969i −0.651326 − 0.758798i \(-0.725787\pi\)
0.634169 + 0.773195i \(0.281343\pi\)
\(30\) 0 0
\(31\) −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 0
\(34\) −0.286989 − 1.62760i −0.0492182 − 0.279130i
\(35\) −12.3366 + 4.49016i −2.08527 + 0.758976i
\(36\) 0 0
\(37\) 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 0
\(40\) −2.70574 − 2.27038i −0.427815 − 0.358979i
\(41\) 6.67752 − 2.43042i 1.04285 − 0.379568i 0.236892 − 0.971536i \(-0.423871\pi\)
0.805961 + 0.591968i \(0.201649\pi\)
\(42\) 0 0
\(43\) −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
\(45\) 0 0
\(46\) −4.34002 + 7.51714i −0.639901 + 1.10834i
\(47\) 1.92262 − 1.61327i 0.280443 − 0.235319i −0.491706 − 0.870761i \(-0.663626\pi\)
0.772149 + 0.635442i \(0.219182\pi\)
\(48\) 0 0
\(49\) −3.40760 + 5.90214i −0.486801 + 0.843163i
\(50\) 3.73783 + 6.47410i 0.528608 + 0.915577i
\(51\) 0 0
\(52\) −0.0393628 + 0.223238i −0.00545864 + 0.0309575i
\(53\) −1.03074 − 5.84564i −0.141584 − 0.802961i −0.970047 − 0.242918i \(-0.921896\pi\)
0.828463 − 0.560043i \(-0.189216\pi\)
\(54\) 0 0
\(55\) 14.3157 + 12.0123i 1.93033 + 1.61974i
\(56\) −3.71688 −0.496689
\(57\) 0 0
\(58\) 0.120615 0.0158375
\(59\) −0.167718 − 0.140732i −0.0218351 − 0.0183218i 0.631805 − 0.775128i \(-0.282314\pi\)
−0.653640 + 0.756806i \(0.726759\pi\)
\(60\) 0 0
\(61\) −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
\(63\) 0 0
\(64\) −0.500000 − 0.866025i −0.0625000 − 0.108253i
\(65\) 0.400330 − 0.693392i 0.0496548 − 0.0860046i
\(66\) 0 0
\(67\) 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
\(69\) 0 0
\(70\) 12.3366 + 4.49016i 1.47451 + 0.536677i
\(71\) 0.235300 − 1.33445i 0.0279249 − 0.158370i −0.967657 − 0.252271i \(-0.918823\pi\)
0.995582 + 0.0939008i \(0.0299337\pi\)
\(72\) 0 0
\(73\) −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
\(75\) 0 0
\(76\) 0.500000 − 4.33013i 0.0573539 − 0.496700i
\(77\) 19.6655 2.24109
\(78\) 0 0
\(79\) 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 0
\(82\) −6.67752 − 2.43042i −0.737409 − 0.268395i
\(83\) 0.960637 + 1.66387i 0.105444 + 0.182634i 0.913919 − 0.405896i \(-0.133040\pi\)
−0.808476 + 0.588530i \(0.799707\pi\)
\(84\) 0 0
\(85\) 4.47178 − 3.75227i 0.485033 − 0.406991i
\(86\) 4.10220 − 3.44215i 0.442351 − 0.371177i
\(87\) 0 0
\(88\) 2.64543 + 4.58202i 0.282004 + 0.488445i
\(89\) −11.4226 − 4.15749i −1.21080 − 0.440693i −0.343816 − 0.939037i \(-0.611720\pi\)
−0.866980 + 0.498344i \(0.833942\pi\)
\(90\) 0 0
\(91\) −0.146307 − 0.829748i −0.0153371 − 0.0869813i
\(92\) 8.15657 − 2.96875i 0.850382 − 0.309514i
\(93\) 0 0
\(94\) −2.50980 −0.258866
\(95\) −6.89053 + 13.7680i −0.706953 + 1.41257i
\(96\) 0 0
\(97\) 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 0
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.u.b.73.1 6
3.2 odd 2 114.2.i.c.73.1 yes 6
12.11 even 2 912.2.bo.d.529.1 6
19.5 even 9 6498.2.a.bp.1.3 3
19.6 even 9 inner 342.2.u.b.253.1 6
19.14 odd 18 6498.2.a.bu.1.3 3
57.5 odd 18 2166.2.a.r.1.1 3
57.14 even 18 2166.2.a.p.1.1 3
57.44 odd 18 114.2.i.c.25.1 ✓ 6
228.215 even 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 57.44 odd 18
114.2.i.c.73.1 yes 6 3.2 odd 2
342.2.u.b.73.1 6 1.1 even 1 trivial
342.2.u.b.253.1 6 19.6 even 9 inner
912.2.bo.d.481.1 6 228.215 even 18
912.2.bo.d.529.1 6 12.11 even 2
2166.2.a.p.1.1 3 57.14 even 18
2166.2.a.r.1.1 3 57.5 odd 18
6498.2.a.bp.1.3 3 19.5 even 9
6498.2.a.bu.1.3 3 19.14 odd 18