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 253.1
Root \(-0.766044 + 0.642788i\) of defining polynomial
Character \(\chi\) \(=\) 342.253
Dual form 342.2.u.b.73.1

$q$-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 \(\left(\mathbb{Z}/342\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(325\)
\(\chi(n)\) \(1\) \(e\left(\frac{7}{9}\right)\)

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 0.741194 + 0.671290i \(0.234260\pi\)
−0.466900 + 0.884310i \(0.654629\pi\)
\(6\) 0 0
\(7\) −1.85844 + 3.21891i −0.702425 + 1.21664i 0.265188 + 0.964197i \(0.414566\pi\)
−0.967613 + 0.252438i \(0.918767\pi\)
\(8\) 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 −0.921158 − 0.389190i \(-0.872755\pi\)
0.123531 − 0.992341i \(-0.460578\pi\)
\(12\) 0 0
\(13\) 0.213011 − 0.0775297i 0.0590786 − 0.0215029i −0.312312 − 0.949980i \(-0.601103\pi\)
0.371390 + 0.928477i \(0.378881\pi\)
\(14\) −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 −0.476215 − 0.879329i \(-0.657992\pi\)
0.783276 + 0.621674i \(0.213547\pi\)
\(18\) 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 0.261894 + 0.965097i \(0.415653\pi\)
−0.576182 + 0.817321i \(0.695458\pi\)
\(24\) 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 0.634169 − 0.773195i \(-0.281343\pi\)
−0.651326 + 0.758798i \(0.725787\pi\)
\(30\) 0 0
\(31\) −1.56031 + 2.70253i −0.280239 + 0.485389i −0.971444 − 0.237271i \(-0.923747\pi\)
0.691204 + 0.722660i \(0.257081\pi\)
\(32\) 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 0.805961 − 0.591968i \(-0.201649\pi\)
0.236892 + 0.971536i \(0.423871\pi\)
\(42\) 0 0
\(43\) −0.929892 − 5.27368i −0.141807 − 0.804229i −0.969875 − 0.243602i \(-0.921671\pi\)
0.828068 − 0.560627i \(-0.189440\pi\)
\(44\) −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 0.772149 − 0.635442i \(-0.219182\pi\)
−0.491706 + 0.870761i \(0.663626\pi\)
\(48\) 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 0.828463 + 0.560043i \(0.189216\pi\)
−0.970047 + 0.242918i \(0.921896\pi\)
\(54\) 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 −0.653640 − 0.756806i \(-0.726759\pi\)
0.631805 + 0.775128i \(0.282314\pi\)
\(60\) 0 0
\(61\) −0.273318 + 1.55007i −0.0349948 + 0.198466i −0.997293 − 0.0735316i \(-0.976573\pi\)
0.962298 + 0.271997i \(0.0876841\pi\)
\(62\) −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 0.933453 + 0.358701i \(0.116780\pi\)
0.515343 + 0.856984i \(0.327664\pi\)
\(68\) −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 0.995582 − 0.0939008i \(-0.0299337\pi\)
−0.967657 + 0.252271i \(0.918823\pi\)
\(72\) 0 0
\(73\) −2.27972 − 0.829748i −0.266820 − 0.0971147i 0.205145 − 0.978731i \(-0.434233\pi\)
−0.471966 + 0.881617i \(0.656455\pi\)
\(74\) −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 0.488990 − 0.872289i \(-0.337365\pi\)
−0.186109 + 0.982529i \(0.559588\pi\)
\(80\) 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 −0.808476 − 0.588530i \(-0.799707\pi\)
0.913919 + 0.405896i \(0.133040\pi\)
\(84\) 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 −0.866980 − 0.498344i \(-0.833942\pi\)
−0.343816 + 0.939037i \(0.611720\pi\)
\(90\) 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 0.385831 − 0.922570i \(-0.373915\pi\)
0.975552 − 0.219767i \(-0.0705296\pi\)
\(98\) 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.253.1 6
3.2 odd 2 114.2.i.c.25.1 ✓ 6
12.11 even 2 912.2.bo.d.481.1 6
19.4 even 9 6498.2.a.bp.1.3 3
19.15 odd 18 6498.2.a.bu.1.3 3
19.16 even 9 inner 342.2.u.b.73.1 6
57.23 odd 18 2166.2.a.r.1.1 3
57.35 odd 18 114.2.i.c.73.1 yes 6
57.53 even 18 2166.2.a.p.1.1 3
228.35 even 18 912.2.bo.d.529.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
114.2.i.c.25.1 ✓ 6 3.2 odd 2
114.2.i.c.73.1 yes 6 57.35 odd 18
342.2.u.b.73.1 6 19.16 even 9 inner
342.2.u.b.253.1 6 1.1 even 1 trivial
912.2.bo.d.481.1 6 12.11 even 2
912.2.bo.d.529.1 6 228.35 even 18
2166.2.a.p.1.1 3 57.53 even 18
2166.2.a.r.1.1 3 57.23 odd 18
6498.2.a.bp.1.3 3 19.4 even 9
6498.2.a.bu.1.3 3 19.15 odd 18