Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [74,2,Mod(7,74)] 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("74.7"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(74, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([16])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 74 = 2 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 74.f (of order \(9\), degree \(6\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.590892974957\)
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 7.1
Root \(-0.173648 + 0.984808i\) of defining polynomial
Character \(\chi\) \(=\) 74.7
Dual form 74.2.f.a.53.1

$q$-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+(0.939693 - 0.342020i) q^{2} +(-0.326352 - 0.118782i) q^{3} +(0.766044 - 0.642788i) q^{4} +(-0.0209445 - 0.118782i) q^{5} -0.347296 q^{6} +(0.233956 + 1.32683i) q^{7} +(0.500000 - 0.866025i) q^{8} +(-2.20574 - 1.85083i) q^{9} +(-0.0603074 - 0.104455i) q^{10} +(-2.26604 + 3.92490i) q^{11} +(-0.326352 + 0.118782i) q^{12} +(-0.592396 + 0.497079i) q^{13} +(0.673648 + 1.16679i) q^{14} +(-0.00727396 + 0.0412527i) q^{15} +(0.173648 - 0.984808i) q^{16} +(-2.29813 - 1.92836i) q^{17} +(-2.70574 - 0.984808i) q^{18} +(1.91875 + 0.698367i) q^{19} +(-0.0923963 - 0.0775297i) q^{20} +(0.0812519 - 0.460802i) q^{21} +(-0.786989 + 4.46324i) q^{22} +(-0.0282185 - 0.0488759i) q^{23} +(-0.266044 + 0.223238i) q^{24} +(4.68479 - 1.70513i) q^{25} +(-0.386659 + 0.669713i) q^{26} +(1.02094 + 1.76833i) q^{27} +(1.03209 + 0.866025i) q^{28} +(2.89053 - 5.00654i) q^{29} +(0.00727396 + 0.0412527i) q^{30} -3.34730 q^{31} +(-0.173648 - 0.984808i) q^{32} +(1.20574 - 1.01173i) q^{33} +(-2.81908 - 1.02606i) q^{34} +(0.152704 - 0.0555796i) q^{35} -2.87939 q^{36} +(5.60607 - 2.36051i) q^{37} +2.04189 q^{38} +(0.252374 - 0.0918566i) q^{39} +(-0.113341 - 0.0412527i) q^{40} +(-5.47565 + 4.59462i) q^{41} +(-0.0812519 - 0.460802i) q^{42} +9.31315 q^{43} +(0.786989 + 4.46324i) q^{44} +(-0.173648 + 0.300767i) q^{45} +(-0.0432332 - 0.0362770i) q^{46} +(-4.25877 - 7.37641i) q^{47} +(-0.173648 + 0.300767i) q^{48} +(4.87211 - 1.77330i) q^{49} +(3.81908 - 3.20459i) q^{50} +(0.520945 + 0.902302i) q^{51} +(-0.134285 + 0.761570i) q^{52} +(0.482926 - 2.73881i) q^{53} +(1.56418 + 1.31250i) q^{54} +(0.513671 + 0.186961i) q^{55} +(1.26604 + 0.460802i) q^{56} +(-0.543233 - 0.455827i) q^{57} +(1.00387 - 5.69323i) q^{58} +(-2.25624 + 12.7958i) q^{59} +(0.0209445 + 0.0362770i) q^{60} +(-8.82295 + 7.40333i) q^{61} +(-3.14543 + 1.14484i) q^{62} +(1.93969 - 3.35965i) q^{63} +(-0.500000 - 0.866025i) q^{64} +(0.0714517 + 0.0599551i) q^{65} +(0.786989 - 1.36310i) q^{66} +(0.889185 + 5.04282i) q^{67} -3.00000 q^{68} +(0.00340357 + 0.0193026i) q^{69} +(0.124485 - 0.104455i) q^{70} +(-12.6236 - 4.59462i) q^{71} +(-2.70574 + 0.984808i) q^{72} -8.71688 q^{73} +(4.46064 - 4.13554i) q^{74} -1.73143 q^{75} +(1.91875 - 0.698367i) q^{76} +(-5.73783 - 2.08840i) q^{77} +(0.205737 - 0.172634i) q^{78} +(0.720285 + 4.08494i) q^{79} -0.120615 q^{80} +(1.37686 + 7.80856i) q^{81} +(-3.57398 + 6.19031i) q^{82} +(-4.53596 - 3.80612i) q^{83} +(-0.233956 - 0.405223i) q^{84} +(-0.180922 + 0.313366i) q^{85} +(8.75150 - 3.18528i) q^{86} +(-1.53802 + 1.29055i) q^{87} +(2.26604 + 3.92490i) q^{88} +(1.32295 - 7.50281i) q^{89} +(-0.0603074 + 0.342020i) q^{90} +(-0.798133 - 0.669713i) q^{91} +(-0.0530334 - 0.0193026i) q^{92} +(1.09240 + 0.397600i) q^{93} +(-6.52481 - 5.47497i) q^{94} +(0.0427664 - 0.242540i) q^{95} +(-0.0603074 + 0.342020i) q^{96} +(5.12061 + 8.86916i) q^{97} +(3.97178 - 3.33272i) q^{98} +(12.2626 - 4.46324i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 3 q^{3} + 3 q^{5} + 6 q^{7} + 3 q^{8} - 3 q^{9} - 6 q^{10} - 9 q^{11} - 3 q^{12} + 3 q^{14} - 18 q^{15} - 6 q^{18} + 9 q^{19} + 3 q^{20} + 3 q^{21} + 3 q^{22} - 15 q^{23} + 3 q^{24} + 21 q^{25} - 9 q^{26}+ \cdots + 27 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/74\mathbb{Z}\right)^\times\).

\(n\) \(39\)
\(\chi(n)\) \(e\left(\frac{8}{9}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.939693 − 0.342020i 0.664463 − 0.241845i
\(3\) −0.326352 − 0.118782i −0.188419 − 0.0685790i 0.246087 − 0.969248i \(-0.420855\pi\)
−0.434507 + 0.900669i \(0.643077\pi\)
\(4\) 0.766044 − 0.642788i 0.383022 − 0.321394i
\(5\) −0.0209445 − 0.118782i −0.00936668 − 0.0531211i 0.979766 − 0.200146i \(-0.0641415\pi\)
−0.989133 + 0.147024i \(0.953030\pi\)
\(6\) −0.347296 −0.141783
\(7\) 0.233956 + 1.32683i 0.0884269 + 0.501494i 0.996564 + 0.0828217i \(0.0263932\pi\)
−0.908137 + 0.418672i \(0.862496\pi\)
\(8\) 0.500000 − 0.866025i 0.176777 − 0.306186i
\(9\) −2.20574 − 1.85083i −0.735246 − 0.616944i
\(10\) −0.0603074 − 0.104455i −0.0190709 − 0.0330317i
\(11\) −2.26604 + 3.92490i −0.683238 + 1.18340i 0.290749 + 0.956799i \(0.406096\pi\)
−0.973987 + 0.226604i \(0.927238\pi\)
\(12\) −0.326352 + 0.118782i −0.0942097 + 0.0342895i
\(13\) −0.592396 + 0.497079i −0.164301 + 0.137865i −0.721231 − 0.692694i \(-0.756424\pi\)
0.556930 + 0.830559i \(0.311979\pi\)
\(14\) 0.673648 + 1.16679i 0.180040 + 0.311839i
\(15\) −0.00727396 + 0.0412527i −0.00187813 + 0.0106514i
\(16\) 0.173648 − 0.984808i 0.0434120 − 0.246202i
\(17\) −2.29813 − 1.92836i −0.557379 − 0.467697i 0.320051 − 0.947400i \(-0.396300\pi\)
−0.877431 + 0.479703i \(0.840744\pi\)
\(18\) −2.70574 − 0.984808i −0.637748 − 0.232121i
\(19\) 1.91875 + 0.698367i 0.440191 + 0.160216i 0.552602 − 0.833445i \(-0.313635\pi\)
−0.112411 + 0.993662i \(0.535857\pi\)
\(20\) −0.0923963 − 0.0775297i −0.0206604 − 0.0173362i
\(21\) 0.0812519 − 0.460802i 0.0177306 − 0.100555i
\(22\) −0.786989 + 4.46324i −0.167787 + 0.951565i
\(23\) −0.0282185 − 0.0488759i −0.00588396 − 0.0101913i 0.863068 − 0.505087i \(-0.168540\pi\)
−0.868952 + 0.494896i \(0.835206\pi\)
\(24\) −0.266044 + 0.223238i −0.0543061 + 0.0455682i
\(25\) 4.68479 − 1.70513i 0.936959 − 0.341025i
\(26\) −0.386659 + 0.669713i −0.0758301 + 0.131342i
\(27\) 1.02094 + 1.76833i 0.196481 + 0.340315i
\(28\) 1.03209 + 0.866025i 0.195046 + 0.163663i
\(29\) 2.89053 − 5.00654i 0.536758 − 0.929692i −0.462318 − 0.886714i \(-0.652982\pi\)
0.999076 − 0.0429778i \(-0.0136845\pi\)
\(30\) 0.00727396 + 0.0412527i 0.00132804 + 0.00753167i
\(31\) −3.34730 −0.601192 −0.300596 − 0.953752i \(-0.597186\pi\)
−0.300596 + 0.953752i \(0.597186\pi\)
\(32\) −0.173648 − 0.984808i −0.0306970 − 0.174091i
\(33\) 1.20574 − 1.01173i 0.209892 − 0.176120i
\(34\) −2.81908 − 1.02606i −0.483468 − 0.175968i
\(35\) 0.152704 − 0.0555796i 0.0258116 − 0.00939466i
\(36\) −2.87939 −0.479898
\(37\) 5.60607 − 2.36051i 0.921632 − 0.388066i
\(38\) 2.04189 0.331238
\(39\) 0.252374 − 0.0918566i 0.0404122 − 0.0147088i
\(40\) −0.113341 − 0.0412527i −0.0179208 − 0.00652262i
\(41\) −5.47565 + 4.59462i −0.855153 + 0.717559i −0.960918 − 0.276832i \(-0.910715\pi\)
0.105765 + 0.994391i \(0.466271\pi\)
\(42\) −0.0812519 − 0.460802i −0.0125374 − 0.0711034i
\(43\) 9.31315 1.42024 0.710121 − 0.704080i \(-0.248640\pi\)
0.710121 + 0.704080i \(0.248640\pi\)
\(44\) 0.786989 + 4.46324i 0.118643 + 0.672858i
\(45\) −0.173648 + 0.300767i −0.0258859 + 0.0448358i
\(46\) −0.0432332 − 0.0362770i −0.00637439 − 0.00534875i
\(47\) −4.25877 − 7.37641i −0.621206 − 1.07596i −0.989262 − 0.146156i \(-0.953310\pi\)
0.368056 − 0.929804i \(-0.380023\pi\)
\(48\) −0.173648 + 0.300767i −0.0250640 + 0.0434120i
\(49\) 4.87211 − 1.77330i 0.696016 − 0.253329i
\(50\) 3.81908 − 3.20459i 0.540099 − 0.453197i
\(51\) 0.520945 + 0.902302i 0.0729468 + 0.126348i
\(52\) −0.134285 + 0.761570i −0.0186220 + 0.105611i
\(53\) 0.482926 − 2.73881i 0.0663350 − 0.376204i −0.933509 − 0.358553i \(-0.883270\pi\)
0.999844 − 0.0176510i \(-0.00561877\pi\)
\(54\) 1.56418 + 1.31250i 0.212858 + 0.178609i
\(55\) 0.513671 + 0.186961i 0.0692633 + 0.0252098i
\(56\) 1.26604 + 0.460802i 0.169182 + 0.0615773i
\(57\) −0.543233 − 0.455827i −0.0719530 − 0.0603757i
\(58\) 1.00387 − 5.69323i 0.131815 − 0.747558i
\(59\) −2.25624 + 12.7958i −0.293738 + 1.66587i 0.378550 + 0.925581i \(0.376423\pi\)
−0.672288 + 0.740290i \(0.734688\pi\)
\(60\) 0.0209445 + 0.0362770i 0.00270393 + 0.00468334i
\(61\) −8.82295 + 7.40333i −1.12966 + 0.947900i −0.999052 − 0.0435341i \(-0.986138\pi\)
−0.130611 + 0.991434i \(0.541694\pi\)
\(62\) −3.14543 + 1.14484i −0.399470 + 0.145395i
\(63\) 1.93969 − 3.35965i 0.244378 − 0.423276i
\(64\) −0.500000 − 0.866025i −0.0625000 − 0.108253i
\(65\) 0.0714517 + 0.0599551i 0.00886250 + 0.00743652i
\(66\) 0.786989 − 1.36310i 0.0968716 − 0.167787i
\(67\) 0.889185 + 5.04282i 0.108631 + 0.616079i 0.989708 + 0.143104i \(0.0457085\pi\)
−0.881076 + 0.472974i \(0.843180\pi\)
\(68\) −3.00000 −0.363803
\(69\) 0.00340357 + 0.0193026i 0.000409741 + 0.00232376i
\(70\) 0.124485 − 0.104455i 0.0148788 − 0.0124848i
\(71\) −12.6236 − 4.59462i −1.49815 − 0.545281i −0.542567 − 0.840013i \(-0.682547\pi\)
−0.955580 + 0.294732i \(0.904770\pi\)
\(72\) −2.70574 + 0.984808i −0.318874 + 0.116061i
\(73\) −8.71688 −1.02023 −0.510117 − 0.860105i \(-0.670398\pi\)
−0.510117 + 0.860105i \(0.670398\pi\)
\(74\) 4.46064 − 4.13554i 0.518539 − 0.480747i
\(75\) −1.73143 −0.199928
\(76\) 1.91875 − 0.698367i 0.220096 − 0.0801082i
\(77\) −5.73783 − 2.08840i −0.653886 − 0.237995i
\(78\) 0.205737 − 0.172634i 0.0232951 − 0.0195469i
\(79\) 0.720285 + 4.08494i 0.0810384 + 0.459592i 0.998141 + 0.0609412i \(0.0194102\pi\)
−0.917103 + 0.398650i \(0.869479\pi\)
\(80\) −0.120615 −0.0134851
\(81\) 1.37686 + 7.80856i 0.152984 + 0.867617i
\(82\) −3.57398 + 6.19031i −0.394680 + 0.683606i
\(83\) −4.53596 − 3.80612i −0.497886 − 0.417776i 0.358956 − 0.933354i \(-0.383133\pi\)
−0.856843 + 0.515578i \(0.827577\pi\)
\(84\) −0.233956 − 0.405223i −0.0255266 − 0.0442134i
\(85\) −0.180922 + 0.313366i −0.0196238 + 0.0339894i
\(86\) 8.75150 − 3.18528i 0.943698 − 0.343478i
\(87\) −1.53802 + 1.29055i −0.164893 + 0.138362i
\(88\) 2.26604 + 3.92490i 0.241561 + 0.418396i
\(89\) 1.32295 − 7.50281i 0.140232 − 0.795297i −0.830840 − 0.556511i \(-0.812140\pi\)
0.971072 − 0.238785i \(-0.0767492\pi\)
\(90\) −0.0603074 + 0.342020i −0.00635696 + 0.0360521i
\(91\) −0.798133 − 0.669713i −0.0836671 − 0.0702050i
\(92\) −0.0530334 − 0.0193026i −0.00552912 − 0.00201243i
\(93\) 1.09240 + 0.397600i 0.113276 + 0.0412292i
\(94\) −6.52481 − 5.47497i −0.672983 − 0.564700i
\(95\) 0.0427664 − 0.242540i 0.00438774 − 0.0248841i
\(96\) −0.0603074 + 0.342020i −0.00615510 + 0.0349073i
\(97\) 5.12061 + 8.86916i 0.519920 + 0.900527i 0.999732 + 0.0231560i \(0.00737146\pi\)
−0.479812 + 0.877371i \(0.659295\pi\)
\(98\) 3.97178 − 3.33272i 0.401211 − 0.336656i
\(99\) 12.2626 − 4.46324i 1.23244 − 0.448572i
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 74.2.f.a.7.1 ✓ 6
3.2 odd 2 666.2.x.c.451.1 6
4.3 odd 2 592.2.bc.b.81.1 6
37.4 even 18 2738.2.a.p.1.2 3
37.16 even 9 inner 74.2.f.a.53.1 yes 6
37.33 even 9 2738.2.a.m.1.2 3
111.53 odd 18 666.2.x.c.127.1 6
148.127 odd 18 592.2.bc.b.497.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
74.2.f.a.7.1 ✓ 6 1.1 even 1 trivial
74.2.f.a.53.1 yes 6 37.16 even 9 inner
592.2.bc.b.81.1 6 4.3 odd 2
592.2.bc.b.497.1 6 148.127 odd 18
666.2.x.c.127.1 6 111.53 odd 18
666.2.x.c.451.1 6 3.2 odd 2
2738.2.a.m.1.2 3 37.33 even 9
2738.2.a.p.1.2 3 37.4 even 18