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,-9,0,3,-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 55.1
Root \(-0.173648 + 0.984808i\) of defining polynomial
Character \(\chi\) \(=\) 342.55
Dual form 342.2.u.a.199.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.173648 - 0.984808i) q^{2} +(-0.939693 - 0.342020i) q^{4} +(-3.20574 + 1.16679i) q^{5} +(1.43969 + 2.49362i) q^{7} +(-0.500000 + 0.866025i) q^{8} +(0.592396 + 3.35965i) q^{10} +(-0.173648 + 0.300767i) q^{11} +(-1.26604 + 1.06234i) q^{13} +(2.70574 - 0.984808i) q^{14} +(0.766044 + 0.642788i) q^{16} +(-1.20574 + 6.83807i) q^{17} +(-2.82635 + 3.31839i) q^{19} +3.41147 q^{20} +(0.266044 + 0.223238i) q^{22} +(6.39053 + 2.32596i) q^{23} +(5.08512 - 4.26692i) q^{25} +(0.826352 + 1.43128i) q^{26} +(-0.500000 - 2.83564i) q^{28} +(-1.10354 - 6.25849i) q^{29} +(-0.798133 - 1.38241i) q^{31} +(0.766044 - 0.642788i) q^{32} +(6.52481 + 2.37484i) q^{34} +(-7.52481 - 6.31407i) q^{35} -11.2121 q^{37} +(2.77719 + 3.35965i) q^{38} +(0.592396 - 3.35965i) q^{40} +(2.67365 + 2.24346i) q^{41} +(-2.14543 + 0.780873i) q^{43} +(0.266044 - 0.223238i) q^{44} +(3.40033 - 5.88954i) q^{46} +(-0.971782 - 5.51125i) q^{47} +(-0.645430 + 1.11792i) q^{49} +(-3.31908 - 5.74881i) q^{50} +(1.55303 - 0.565258i) q^{52} +(1.86097 + 0.677337i) q^{53} +(0.205737 - 1.16679i) q^{55} -2.87939 q^{56} -6.35504 q^{58} +(-0.0773815 + 0.438852i) q^{59} +(11.7763 + 4.28623i) q^{61} +(-1.50000 + 0.545955i) q^{62} +(-0.500000 - 0.866025i) q^{64} +(2.81908 - 4.88279i) q^{65} +(-0.187319 - 1.06234i) q^{67} +(3.47178 - 6.01330i) q^{68} +(-7.52481 + 6.31407i) q^{70} +(-15.6211 + 5.68561i) q^{71} +(9.51367 + 7.98292i) q^{73} +(-1.94697 + 11.0418i) q^{74} +(3.79086 - 2.15160i) q^{76} -1.00000 q^{77} +(-8.36824 - 7.02179i) q^{79} +(-3.20574 - 1.16679i) q^{80} +(2.67365 - 2.24346i) q^{82} +(5.85844 + 10.1471i) q^{83} +(-4.11334 - 23.3279i) q^{85} +(0.396459 + 2.24843i) q^{86} +(-0.173648 - 0.300767i) q^{88} +(-1.37346 + 1.15247i) q^{89} +(-4.47178 - 1.62760i) q^{91} +(-5.20961 - 4.37138i) q^{92} -5.59627 q^{94} +(5.18866 - 13.9357i) q^{95} +(0.634285 - 3.59721i) q^{97} +(0.988856 + 0.829748i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 9 q^{5} + 3 q^{7} - 3 q^{8} - 3 q^{13} + 6 q^{14} + 3 q^{17} - 18 q^{19} - 3 q^{22} + 21 q^{23} + 9 q^{25} + 6 q^{26} - 3 q^{28} + 3 q^{29} + 9 q^{31} + 12 q^{34} - 18 q^{35} - 18 q^{37} + 6 q^{38}+ \cdots + 12 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(191\) \(325\)
\(\chi(n)\) \(1\) \(e\left(\frac{5}{9}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.173648 − 0.984808i 0.122788 − 0.696364i
\(3\) 0 0
\(4\) −0.939693 − 0.342020i −0.469846 − 0.171010i
\(5\) −3.20574 + 1.16679i −1.43365 + 0.521806i −0.937975 − 0.346703i \(-0.887301\pi\)
−0.495674 + 0.868509i \(0.665079\pi\)
\(6\) 0 0
\(7\) 1.43969 + 2.49362i 0.544153 + 0.942500i 0.998660 + 0.0517569i \(0.0164821\pi\)
−0.454507 + 0.890743i \(0.650185\pi\)
\(8\) −0.500000 + 0.866025i −0.176777 + 0.306186i
\(9\) 0 0
\(10\) 0.592396 + 3.35965i 0.187332 + 1.06241i
\(11\) −0.173648 + 0.300767i −0.0523569 + 0.0906848i −0.891016 − 0.453972i \(-0.850007\pi\)
0.838659 + 0.544657i \(0.183340\pi\)
\(12\) 0 0
\(13\) −1.26604 + 1.06234i −0.351138 + 0.294639i −0.801247 − 0.598334i \(-0.795830\pi\)
0.450109 + 0.892974i \(0.351385\pi\)
\(14\) 2.70574 − 0.984808i 0.723139 − 0.263201i
\(15\) 0 0
\(16\) 0.766044 + 0.642788i 0.191511 + 0.160697i
\(17\) −1.20574 + 6.83807i −0.292434 + 1.65848i 0.385017 + 0.922909i \(0.374195\pi\)
−0.677452 + 0.735567i \(0.736916\pi\)
\(18\) 0 0
\(19\) −2.82635 + 3.31839i −0.648410 + 0.761292i
\(20\) 3.41147 0.762829
\(21\) 0 0
\(22\) 0.266044 + 0.223238i 0.0567209 + 0.0475945i
\(23\) 6.39053 + 2.32596i 1.33252 + 0.484997i 0.907448 − 0.420164i \(-0.138027\pi\)
0.425069 + 0.905161i \(0.360250\pi\)
\(24\) 0 0
\(25\) 5.08512 − 4.26692i 1.01702 − 0.853385i
\(26\) 0.826352 + 1.43128i 0.162061 + 0.280698i
\(27\) 0 0
\(28\) −0.500000 − 2.83564i −0.0944911 − 0.535886i
\(29\) −1.10354 − 6.25849i −0.204922 − 1.16217i −0.897562 − 0.440888i \(-0.854663\pi\)
0.692640 − 0.721284i \(-0.256448\pi\)
\(30\) 0 0
\(31\) −0.798133 − 1.38241i −0.143349 − 0.248288i 0.785407 − 0.618980i \(-0.212454\pi\)
−0.928756 + 0.370692i \(0.879120\pi\)
\(32\) 0.766044 − 0.642788i 0.135419 − 0.113630i
\(33\) 0 0
\(34\) 6.52481 + 2.37484i 1.11900 + 0.407281i
\(35\) −7.52481 − 6.31407i −1.27193 − 1.06727i
\(36\) 0 0
\(37\) −11.2121 −1.84326 −0.921632 − 0.388066i \(-0.873143\pi\)
−0.921632 + 0.388066i \(0.873143\pi\)
\(38\) 2.77719 + 3.35965i 0.450520 + 0.545007i
\(39\) 0 0
\(40\) 0.592396 − 3.35965i 0.0936661 − 0.531207i
\(41\) 2.67365 + 2.24346i 0.417554 + 0.350369i 0.827232 − 0.561861i \(-0.189914\pi\)
−0.409678 + 0.912230i \(0.634359\pi\)
\(42\) 0 0
\(43\) −2.14543 + 0.780873i −0.327175 + 0.119082i −0.500385 − 0.865803i \(-0.666808\pi\)
0.173210 + 0.984885i \(0.444586\pi\)
\(44\) 0.266044 − 0.223238i 0.0401077 − 0.0336544i
\(45\) 0 0
\(46\) 3.40033 − 5.88954i 0.501351 − 0.868366i
\(47\) −0.971782 − 5.51125i −0.141749 − 0.803898i −0.969920 − 0.243423i \(-0.921730\pi\)
0.828171 − 0.560475i \(-0.189381\pi\)
\(48\) 0 0
\(49\) −0.645430 + 1.11792i −0.0922042 + 0.159702i
\(50\) −3.31908 − 5.74881i −0.469388 − 0.813005i
\(51\) 0 0
\(52\) 1.55303 − 0.565258i 0.215367 − 0.0783872i
\(53\) 1.86097 + 0.677337i 0.255623 + 0.0930393i 0.466653 − 0.884440i \(-0.345460\pi\)
−0.211030 + 0.977480i \(0.567682\pi\)
\(54\) 0 0
\(55\) 0.205737 − 1.16679i 0.0277416 − 0.157330i
\(56\) −2.87939 −0.384774
\(57\) 0 0
\(58\) −6.35504 −0.834457
\(59\) −0.0773815 + 0.438852i −0.0100742 + 0.0571337i −0.989430 − 0.145008i \(-0.953679\pi\)
0.979356 + 0.202142i \(0.0647902\pi\)
\(60\) 0 0
\(61\) 11.7763 + 4.28623i 1.50780 + 0.548795i 0.958067 − 0.286544i \(-0.0925064\pi\)
0.549735 + 0.835339i \(0.314729\pi\)
\(62\) −1.50000 + 0.545955i −0.190500 + 0.0693364i
\(63\) 0 0
\(64\) −0.500000 − 0.866025i −0.0625000 − 0.108253i
\(65\) 2.81908 − 4.88279i 0.349664 − 0.605635i
\(66\) 0 0
\(67\) −0.187319 − 1.06234i −0.0228846 − 0.129785i 0.971225 − 0.238163i \(-0.0765454\pi\)
−0.994110 + 0.108378i \(0.965434\pi\)
\(68\) 3.47178 − 6.01330i 0.421015 − 0.729220i
\(69\) 0 0
\(70\) −7.52481 + 6.31407i −0.899387 + 0.754676i
\(71\) −15.6211 + 5.68561i −1.85388 + 0.674758i −0.870786 + 0.491662i \(0.836390\pi\)
−0.983095 + 0.183096i \(0.941388\pi\)
\(72\) 0 0
\(73\) 9.51367 + 7.98292i 1.11349 + 0.934330i 0.998257 − 0.0590086i \(-0.0187939\pi\)
0.115233 + 0.993338i \(0.463238\pi\)
\(74\) −1.94697 + 11.0418i −0.226330 + 1.28358i
\(75\) 0 0
\(76\) 3.79086 − 2.15160i 0.434841 − 0.246806i
\(77\) −1.00000 −0.113961
\(78\) 0 0
\(79\) −8.36824 − 7.02179i −0.941501 − 0.790013i 0.0363452 − 0.999339i \(-0.488428\pi\)
−0.977846 + 0.209326i \(0.932873\pi\)
\(80\) −3.20574 − 1.16679i −0.358412 − 0.130451i
\(81\) 0 0
\(82\) 2.67365 − 2.24346i 0.295255 − 0.247748i
\(83\) 5.85844 + 10.1471i 0.643047 + 1.11379i 0.984749 + 0.173982i \(0.0556635\pi\)
−0.341701 + 0.939809i \(0.611003\pi\)
\(84\) 0 0
\(85\) −4.11334 − 23.3279i −0.446154 − 2.53027i
\(86\) 0.396459 + 2.24843i 0.0427513 + 0.242455i
\(87\) 0 0
\(88\) −0.173648 − 0.300767i −0.0185110 − 0.0320619i
\(89\) −1.37346 + 1.15247i −0.145586 + 0.122161i −0.712672 − 0.701497i \(-0.752515\pi\)
0.567086 + 0.823659i \(0.308071\pi\)
\(90\) 0 0
\(91\) −4.47178 − 1.62760i −0.468770 − 0.170618i
\(92\) −5.20961 − 4.37138i −0.543139 − 0.455748i
\(93\) 0 0
\(94\) −5.59627 −0.577211
\(95\) 5.18866 − 13.9357i 0.532346 − 1.42977i
\(96\) 0 0
\(97\) 0.634285 − 3.59721i 0.0644019 − 0.365241i −0.935526 − 0.353257i \(-0.885074\pi\)
0.999928 − 0.0119843i \(-0.00381481\pi\)
\(98\) 0.988856 + 0.829748i 0.0998895 + 0.0838172i
\(99\) 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.a.55.1 6
3.2 odd 2 114.2.i.d.55.1 ✓ 6
12.11 even 2 912.2.bo.f.625.1 6
19.3 odd 18 6498.2.a.bn.1.3 3
19.9 even 9 inner 342.2.u.a.199.1 6
19.16 even 9 6498.2.a.bs.1.3 3
57.35 odd 18 2166.2.a.o.1.1 3
57.41 even 18 2166.2.a.u.1.1 3
57.47 odd 18 114.2.i.d.85.1 yes 6
228.47 even 18 912.2.bo.f.769.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
114.2.i.d.55.1 ✓ 6 3.2 odd 2
114.2.i.d.85.1 yes 6 57.47 odd 18
342.2.u.a.55.1 6 1.1 even 1 trivial
342.2.u.a.199.1 6 19.9 even 9 inner
912.2.bo.f.625.1 6 12.11 even 2
912.2.bo.f.769.1 6 228.47 even 18
2166.2.a.o.1.1 3 57.35 odd 18
2166.2.a.u.1.1 3 57.41 even 18
6498.2.a.bn.1.3 3 19.3 odd 18
6498.2.a.bs.1.3 3 19.16 even 9