Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [58,2,Mod(5,58)] 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("58.5"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(58, base_ring=CyclotomicField(14)) chi = DirichletCharacter(H, H._module([11])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 58 = 2 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 58.e (of order \(14\), degree \(6\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.463132331723\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(2\) over \(\Q(\zeta_{14})\)
Coefficient field: \(\Q(\zeta_{28})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - x^{10} + x^{8} - x^{6} + x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{14}]$

Embedding invariants

Embedding label 33.2
Root \(-0.781831 + 0.623490i\) of defining polynomial
Character \(\chi\) \(=\) 58.33
Dual form 58.2.e.a.51.2

$q$-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+(0.974928 + 0.222521i) q^{2} +(-0.626980 - 1.30194i) q^{3} +(0.900969 + 0.433884i) q^{4} +(-0.136585 + 0.598418i) q^{5} +(-0.321552 - 1.40881i) q^{6} +(-2.59161 + 1.24805i) q^{7} +(0.781831 + 0.623490i) q^{8} +(0.568532 - 0.712916i) q^{9} +(-0.266321 + 0.553021i) q^{10} +(-3.39687 + 2.70891i) q^{11} -1.44504i q^{12} +(-0.298199 - 0.373929i) q^{13} +(-2.80435 + 0.640075i) q^{14} +(0.864739 - 0.197371i) q^{15} +(0.623490 + 0.781831i) q^{16} -0.259558i q^{17} +(0.712916 - 0.568532i) q^{18} +(3.65470 - 7.58906i) q^{19} +(-0.382702 + 0.479894i) q^{20} +(3.24978 + 2.59161i) q^{21} +(-3.91449 + 1.88512i) q^{22} +(0.0317259 + 0.139000i) q^{23} +(0.321552 - 1.40881i) q^{24} +(4.16540 + 2.00595i) q^{25} +(-0.207515 - 0.430910i) q^{26} +(-5.51107 - 1.25786i) q^{27} -2.87647 q^{28} +(1.07561 + 5.27665i) q^{29} +0.886977 q^{30} +(6.46089 + 1.47465i) q^{31} +(0.433884 + 0.900969i) q^{32} +(5.65660 + 2.72407i) q^{33} +(0.0577572 - 0.253051i) q^{34} +(-0.392883 - 1.72133i) q^{35} +(0.821552 - 0.395639i) q^{36} +(-7.50895 - 5.98819i) q^{37} +(5.25179 - 6.58554i) q^{38} +(-0.299868 + 0.622683i) q^{39} +(-0.479894 + 0.382702i) q^{40} -4.28236i q^{41} +(2.59161 + 3.24978i) q^{42} +(3.17741 - 0.725223i) q^{43} +(-4.23582 + 0.966799i) q^{44} +(0.348969 + 0.437593i) q^{45} +0.142575i q^{46} +(-3.97456 + 3.16960i) q^{47} +(0.626980 - 1.30194i) q^{48} +(0.794384 - 0.996126i) q^{49} +(3.61460 + 2.88254i) q^{50} +(-0.337929 + 0.162738i) q^{51} +(-0.106426 - 0.466282i) q^{52} +(-2.06111 + 9.03032i) q^{53} +(-5.09299 - 2.45265i) q^{54} +(-1.15710 - 2.40274i) q^{55} +(-2.80435 - 0.640075i) q^{56} -12.1719 q^{57} +(-0.125524 + 5.38370i) q^{58} -10.2463 q^{59} +(0.864739 + 0.197371i) q^{60} +(-4.31279 - 8.95559i) q^{61} +(5.97076 + 2.87536i) q^{62} +(-0.583655 + 2.55716i) q^{63} +(0.222521 + 0.974928i) q^{64} +(0.264495 - 0.127374i) q^{65} +(4.90861 + 3.91449i) q^{66} +(-1.16176 + 1.45680i) q^{67} +(0.112618 - 0.233854i) q^{68} +(0.161078 - 0.128456i) q^{69} -1.76560i q^{70} +(5.97581 + 7.49342i) q^{71} +(0.888992 - 0.202907i) q^{72} +(2.90704 - 0.663513i) q^{73} +(-5.98819 - 7.50895i) q^{74} -6.68078i q^{75} +(6.58554 - 5.25179i) q^{76} +(5.42249 - 11.2599i) q^{77} +(-0.430910 + 0.540344i) q^{78} +(10.5977 + 8.45137i) q^{79} +(-0.553021 + 0.266321i) q^{80} +(1.20895 + 5.29674i) q^{81} +(0.952915 - 4.17499i) q^{82} +(-0.950401 - 0.457689i) q^{83} +(1.80349 + 3.74499i) q^{84} +(0.155324 + 0.0354518i) q^{85} +3.25912 q^{86} +(6.19549 - 4.70873i) q^{87} -4.34475 q^{88} +(2.51138 + 0.573205i) q^{89} +(0.242846 + 0.504275i) q^{90} +(1.23950 + 0.596912i) q^{91} +(-0.0317259 + 0.139000i) q^{92} +(-2.13094 - 9.33625i) q^{93} +(-4.58021 + 2.20571i) q^{94} +(4.04225 + 3.22359i) q^{95} +(0.900969 - 1.12978i) q^{96} +(-7.22194 + 14.9965i) q^{97} +(0.996126 - 0.794384i) q^{98} +3.96178i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 2 q^{4} - 2 q^{5} - 12 q^{6} + 4 q^{7} - 4 q^{9} - 26 q^{13} - 14 q^{15} - 2 q^{16} + 2 q^{20} + 14 q^{21} + 4 q^{22} - 16 q^{23} + 12 q^{24} + 22 q^{25} + 14 q^{26} - 4 q^{28} + 18 q^{29} + 16 q^{30}+ \cdots - 28 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/58\mathbb{Z}\right)^\times\).

\(n\) \(31\)
\(\chi(n)\) \(e\left(\frac{1}{14}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.974928 + 0.222521i 0.689378 + 0.157346i
\(3\) −0.626980 − 1.30194i −0.361987 − 0.751674i 0.637842 − 0.770167i \(-0.279827\pi\)
−0.999829 + 0.0184933i \(0.994113\pi\)
\(4\) 0.900969 + 0.433884i 0.450484 + 0.216942i
\(5\) −0.136585 + 0.598418i −0.0610826 + 0.267621i −0.996243 − 0.0866048i \(-0.972398\pi\)
0.935160 + 0.354225i \(0.115255\pi\)
\(6\) −0.321552 − 1.40881i −0.131273 − 0.575145i
\(7\) −2.59161 + 1.24805i −0.979537 + 0.471720i −0.853946 − 0.520362i \(-0.825797\pi\)
−0.125591 + 0.992082i \(0.540083\pi\)
\(8\) 0.781831 + 0.623490i 0.276419 + 0.220437i
\(9\) 0.568532 − 0.712916i 0.189511 − 0.237639i
\(10\) −0.266321 + 0.553021i −0.0842181 + 0.174881i
\(11\) −3.39687 + 2.70891i −1.02419 + 0.816767i −0.983226 − 0.182394i \(-0.941616\pi\)
−0.0409677 + 0.999160i \(0.513044\pi\)
\(12\) − 1.44504i − 0.417148i
\(13\) −0.298199 − 0.373929i −0.0827054 − 0.103709i 0.738757 − 0.673972i \(-0.235413\pi\)
−0.821463 + 0.570262i \(0.806842\pi\)
\(14\) −2.80435 + 0.640075i −0.749495 + 0.171067i
\(15\) 0.864739 − 0.197371i 0.223275 − 0.0509610i
\(16\) 0.623490 + 0.781831i 0.155872 + 0.195458i
\(17\) − 0.259558i − 0.0629522i −0.999505 − 0.0314761i \(-0.989979\pi\)
0.999505 − 0.0314761i \(-0.0100208\pi\)
\(18\) 0.712916 − 0.568532i 0.168036 − 0.134004i
\(19\) 3.65470 − 7.58906i 0.838446 − 1.74105i 0.186992 − 0.982361i \(-0.440126\pi\)
0.651453 − 0.758689i \(-0.274160\pi\)
\(20\) −0.382702 + 0.479894i −0.0855749 + 0.107308i
\(21\) 3.24978 + 2.59161i 0.709160 + 0.565536i
\(22\) −3.91449 + 1.88512i −0.834572 + 0.401908i
\(23\) 0.0317259 + 0.139000i 0.00661531 + 0.0289836i 0.978128 − 0.208005i \(-0.0666969\pi\)
−0.971513 + 0.236988i \(0.923840\pi\)
\(24\) 0.321552 − 1.40881i 0.0656365 − 0.287572i
\(25\) 4.16540 + 2.00595i 0.833079 + 0.401190i
\(26\) −0.207515 − 0.430910i −0.0406971 − 0.0845083i
\(27\) −5.51107 − 1.25786i −1.06061 − 0.242076i
\(28\) −2.87647 −0.543602
\(29\) 1.07561 + 5.27665i 0.199736 + 0.979850i
\(30\) 0.886977 0.161939
\(31\) 6.46089 + 1.47465i 1.16041 + 0.264856i 0.759030 − 0.651056i \(-0.225674\pi\)
0.401380 + 0.915912i \(0.368531\pi\)
\(32\) 0.433884 + 0.900969i 0.0767005 + 0.159270i
\(33\) 5.65660 + 2.72407i 0.984687 + 0.474200i
\(34\) 0.0577572 − 0.253051i 0.00990528 − 0.0433979i
\(35\) −0.392883 − 1.72133i −0.0664093 − 0.290958i
\(36\) 0.821552 − 0.395639i 0.136925 − 0.0659398i
\(37\) −7.50895 − 5.98819i −1.23446 − 0.984452i −0.999924 − 0.0123461i \(-0.996070\pi\)
−0.234541 − 0.972106i \(-0.575359\pi\)
\(38\) 5.25179 − 6.58554i 0.851953 − 1.06832i
\(39\) −0.299868 + 0.622683i −0.0480173 + 0.0997090i
\(40\) −0.479894 + 0.382702i −0.0758779 + 0.0605106i
\(41\) − 4.28236i − 0.668792i −0.942433 − 0.334396i \(-0.891468\pi\)
0.942433 − 0.334396i \(-0.108532\pi\)
\(42\) 2.59161 + 3.24978i 0.399894 + 0.501452i
\(43\) 3.17741 − 0.725223i 0.484550 − 0.110595i 0.0267346 − 0.999643i \(-0.491489\pi\)
0.457816 + 0.889047i \(0.348632\pi\)
\(44\) −4.23582 + 0.966799i −0.638574 + 0.145750i
\(45\) 0.348969 + 0.437593i 0.0520212 + 0.0652325i
\(46\) 0.142575i 0.0210215i
\(47\) −3.97456 + 3.16960i −0.579749 + 0.462334i −0.868926 − 0.494941i \(-0.835190\pi\)
0.289178 + 0.957275i \(0.406618\pi\)
\(48\) 0.626980 − 1.30194i 0.0904968 − 0.187919i
\(49\) 0.794384 − 0.996126i 0.113483 − 0.142304i
\(50\) 3.61460 + 2.88254i 0.511181 + 0.407653i
\(51\) −0.337929 + 0.162738i −0.0473195 + 0.0227879i
\(52\) −0.106426 − 0.466282i −0.0147586 − 0.0646617i
\(53\) −2.06111 + 9.03032i −0.283116 + 1.24041i 0.610659 + 0.791894i \(0.290905\pi\)
−0.893774 + 0.448517i \(0.851952\pi\)
\(54\) −5.09299 − 2.45265i −0.693068 − 0.333764i
\(55\) −1.15710 − 2.40274i −0.156023 − 0.323985i
\(56\) −2.80435 − 0.640075i −0.374747 − 0.0855337i
\(57\) −12.1719 −1.61221
\(58\) −0.125524 + 5.38370i −0.0164821 + 0.706915i
\(59\) −10.2463 −1.33395 −0.666977 − 0.745078i \(-0.732412\pi\)
−0.666977 + 0.745078i \(0.732412\pi\)
\(60\) 0.864739 + 0.197371i 0.111637 + 0.0254805i
\(61\) −4.31279 − 8.95559i −0.552196 − 1.14665i −0.971113 − 0.238622i \(-0.923304\pi\)
0.418917 − 0.908025i \(-0.362410\pi\)
\(62\) 5.97076 + 2.87536i 0.758287 + 0.365172i
\(63\) −0.583655 + 2.55716i −0.0735336 + 0.322172i
\(64\) 0.222521 + 0.974928i 0.0278151 + 0.121866i
\(65\) 0.264495 − 0.127374i 0.0328066 − 0.0157988i
\(66\) 4.90861 + 3.91449i 0.604208 + 0.481840i
\(67\) −1.16176 + 1.45680i −0.141931 + 0.177976i −0.847716 − 0.530450i \(-0.822023\pi\)
0.705785 + 0.708426i \(0.250594\pi\)
\(68\) 0.112618 − 0.233854i 0.0136570 − 0.0283590i
\(69\) 0.161078 − 0.128456i 0.0193915 − 0.0154642i
\(70\) − 1.76560i − 0.211029i
\(71\) 5.97581 + 7.49342i 0.709198 + 0.889306i 0.997673 − 0.0681816i \(-0.0217197\pi\)
−0.288475 + 0.957487i \(0.593148\pi\)
\(72\) 0.888992 − 0.202907i 0.104769 − 0.0239128i
\(73\) 2.90704 − 0.663513i 0.340243 − 0.0776583i −0.0489853 − 0.998799i \(-0.515599\pi\)
0.389229 + 0.921141i \(0.372742\pi\)
\(74\) −5.98819 − 7.50895i −0.696113 − 0.872898i
\(75\) − 6.68078i − 0.771430i
\(76\) 6.58554 − 5.25179i 0.755413 − 0.602422i
\(77\) 5.42249 − 11.2599i 0.617950 − 1.28319i
\(78\) −0.430910 + 0.540344i −0.0487909 + 0.0611819i
\(79\) 10.5977 + 8.45137i 1.19233 + 0.950853i 0.999538 − 0.0303860i \(-0.00967364\pi\)
0.192794 + 0.981239i \(0.438245\pi\)
\(80\) −0.553021 + 0.266321i −0.0618296 + 0.0297756i
\(81\) 1.20895 + 5.29674i 0.134327 + 0.588527i
\(82\) 0.952915 − 4.17499i 0.105232 − 0.461051i
\(83\) −0.950401 − 0.457689i −0.104320 − 0.0502379i 0.380996 − 0.924577i \(-0.375581\pi\)
−0.485316 + 0.874339i \(0.661295\pi\)
\(84\) 1.80349 + 3.74499i 0.196777 + 0.408612i
\(85\) 0.155324 + 0.0354518i 0.0168473 + 0.00384528i
\(86\) 3.25912 0.351440
\(87\) 6.19549 − 4.70873i 0.664226 − 0.504829i
\(88\) −4.34475 −0.463152
\(89\) 2.51138 + 0.573205i 0.266205 + 0.0607596i 0.353540 − 0.935419i \(-0.384978\pi\)
−0.0873347 + 0.996179i \(0.527835\pi\)
\(90\) 0.242846 + 0.504275i 0.0255982 + 0.0531552i
\(91\) 1.23950 + 0.596912i 0.129935 + 0.0625733i
\(92\) −0.0317259 + 0.139000i −0.00330766 + 0.0144918i
\(93\) −2.13094 − 9.33625i −0.220968 − 0.968124i
\(94\) −4.58021 + 2.20571i −0.472413 + 0.227502i
\(95\) 4.04225 + 3.22359i 0.414726 + 0.330733i
\(96\) 0.900969 − 1.12978i 0.0919548 − 0.115308i
\(97\) −7.22194 + 14.9965i −0.733277 + 1.52267i 0.115147 + 0.993348i \(0.463266\pi\)
−0.848424 + 0.529317i \(0.822448\pi\)
\(98\) 0.996126 − 0.794384i 0.100624 − 0.0802449i
\(99\) 3.96178i 0.398174i
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 58.2.e.a.33.2 ✓ 12
3.2 odd 2 522.2.n.a.91.1 12
4.3 odd 2 464.2.y.c.33.2 12
29.6 even 14 1682.2.b.j.1681.9 12
29.14 odd 28 1682.2.a.r.1.4 6
29.15 odd 28 1682.2.a.s.1.4 6
29.22 even 14 inner 58.2.e.a.51.2 yes 12
29.23 even 7 1682.2.b.j.1681.3 12
87.80 odd 14 522.2.n.a.109.1 12
116.51 odd 14 464.2.y.c.225.2 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
58.2.e.a.33.2 ✓ 12 1.1 even 1 trivial
58.2.e.a.51.2 yes 12 29.22 even 14 inner
464.2.y.c.33.2 12 4.3 odd 2
464.2.y.c.225.2 12 116.51 odd 14
522.2.n.a.91.1 12 3.2 odd 2
522.2.n.a.109.1 12 87.80 odd 14
1682.2.a.r.1.4 6 29.14 odd 28
1682.2.a.s.1.4 6 29.15 odd 28
1682.2.b.j.1681.3 12 29.23 even 7
1682.2.b.j.1681.9 12 29.6 even 14