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 = [12] 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: \(12\)
Relative dimension: \(2\) over \(\Q(\zeta_{9})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 24x^{10} + 264x^{8} - 1687x^{6} + 6600x^{4} - 15000x^{2} + 15625 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label 49.2
Root \(2.20976 + 0.342020i\) of defining polynomial
Character \(\chi\) \(=\) 74.49
Dual form 74.2.f.b.71.2

$q$-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+(0.766044 - 0.642788i) q^{2} +(0.972925 + 0.816381i) q^{3} +(0.173648 - 0.984808i) q^{4} +(-1.43969 + 0.524005i) q^{5} +1.27006 q^{6} +(-1.82850 + 0.665520i) q^{7} +(-0.500000 - 0.866025i) q^{8} +(-0.240839 - 1.36587i) q^{9} +(-0.766044 + 1.32683i) q^{10} +(-0.220544 - 0.381994i) q^{11} +(0.972925 - 0.816381i) q^{12} +(-0.367592 + 2.08472i) q^{13} +(-0.972925 + 1.68516i) q^{14} +(-1.82850 - 0.665520i) q^{15} +(-0.939693 - 0.342020i) q^{16} +(-0.664245 - 3.76712i) q^{17} +(-1.06246 - 0.891507i) q^{18} +(4.55479 + 3.82192i) q^{19} +(0.266044 + 1.50881i) q^{20} +(-2.32231 - 0.845253i) q^{21} +(-0.414488 - 0.150861i) q^{22} +(-2.85558 + 4.94600i) q^{23} +(0.220544 - 1.25077i) q^{24} +(-2.03209 + 1.70513i) q^{25} +(1.05844 + 1.83327i) q^{26} +(2.78585 - 4.82523i) q^{27} +(0.337893 + 1.91629i) q^{28} +(-1.36712 - 2.36792i) q^{29} +(-1.82850 + 0.665520i) q^{30} +9.87516 q^{31} +(-0.939693 + 0.342020i) q^{32} +(0.0972795 - 0.551699i) q^{33} +(-2.93030 - 2.45881i) q^{34} +(2.28374 - 1.91629i) q^{35} -1.38694 q^{36} +(2.18831 - 5.67550i) q^{37} +5.94585 q^{38} +(-2.05956 + 1.72818i) q^{39} +(1.17365 + 0.984808i) q^{40} +(1.80615 - 10.2432i) q^{41} +(-2.32231 + 0.845253i) q^{42} +3.84681 q^{43} +(-0.414488 + 0.150861i) q^{44} +(1.06246 + 1.84023i) q^{45} +(0.991731 + 5.62439i) q^{46} +(-3.84316 + 6.65655i) q^{47} +(-0.635032 - 1.09991i) q^{48} +(-2.46181 + 2.06571i) q^{49} +(-0.460637 + 2.61240i) q^{50} +(2.42915 - 4.20740i) q^{51} +(1.98921 + 0.724014i) q^{52} +(-1.62561 - 0.591673i) q^{53} +(-0.967514 - 5.48704i) q^{54} +(0.517683 + 0.434387i) q^{55} +(1.49061 + 1.25077i) q^{56} +(1.31132 + 7.43688i) q^{57} +(-2.56934 - 0.935163i) q^{58} +(-1.93027 - 0.702561i) q^{59} +(-0.972925 + 1.68516i) q^{60} +(-1.89894 + 10.7694i) q^{61} +(7.56481 - 6.34763i) q^{62} +(1.34939 + 2.33721i) q^{63} +(-0.500000 + 0.866025i) q^{64} +(-0.563183 - 3.19397i) q^{65} +(-0.280105 - 0.485156i) q^{66} +(1.57261 - 0.572384i) q^{67} -3.82524 q^{68} +(-6.81608 + 2.48085i) q^{69} +(0.517683 - 2.93592i) q^{70} +(-7.70462 - 6.46494i) q^{71} +(-1.06246 + 0.891507i) q^{72} -9.88570 q^{73} +(-1.97180 - 5.75430i) q^{74} -3.36910 q^{75} +(4.55479 - 3.82192i) q^{76} +(0.657490 + 0.551699i) q^{77} +(-0.466865 + 2.64772i) q^{78} +(-12.5395 + 4.56401i) q^{79} +1.53209 q^{80} +(2.73975 - 0.997189i) q^{81} +(-5.20061 - 9.00771i) q^{82} +(0.617999 + 3.50484i) q^{83} +(-1.23568 + 2.14025i) q^{84} +(2.93030 + 5.07543i) q^{85} +(2.94683 - 2.47268i) q^{86} +(0.603020 - 3.41989i) q^{87} +(-0.220544 + 0.381994i) q^{88} +(11.1543 + 4.05982i) q^{89} +(1.99677 + 0.726763i) q^{90} +(-0.715278 - 4.05654i) q^{91} +(4.37500 + 3.67106i) q^{92} +(9.60779 + 8.06190i) q^{93} +(1.33472 + 7.56955i) q^{94} +(-8.56020 - 3.11566i) q^{95} +(-1.19347 - 0.434387i) q^{96} +(0.296748 - 0.513983i) q^{97} +(-0.558047 + 3.16484i) q^{98} +(-0.468637 + 0.393233i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 3 q^{3} - 6 q^{5} + 6 q^{7} - 6 q^{8} - 3 q^{9} + 3 q^{11} - 3 q^{12} - 6 q^{13} + 3 q^{14} + 6 q^{15} - 3 q^{17} + 6 q^{18} - 3 q^{19} - 6 q^{20} - 33 q^{21} - 3 q^{22} - 21 q^{23} - 3 q^{24} - 6 q^{25}+ \cdots - 33 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(39\)
\(\chi(n)\) \(e\left(\frac{7}{9}\right)\)

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
\(3\) 0.972925 + 0.816381i 0.561719 + 0.471338i 0.878886 − 0.477032i \(-0.158287\pi\)
−0.317168 + 0.948370i \(0.602732\pi\)
\(4\) 0.173648 − 0.984808i 0.0868241 − 0.492404i
\(5\) −1.43969 + 0.524005i −0.643850 + 0.234342i −0.643248 − 0.765658i \(-0.722414\pi\)
−0.000601695 1.00000i \(0.500192\pi\)
\(6\) 1.27006 0.518501
\(7\) −1.82850 + 0.665520i −0.691108 + 0.251543i −0.663610 − 0.748079i \(-0.730976\pi\)
−0.0274987 + 0.999622i \(0.508754\pi\)
\(8\) −0.500000 − 0.866025i −0.176777 − 0.306186i
\(9\) −0.240839 − 1.36587i −0.0802798 − 0.455289i
\(10\) −0.766044 + 1.32683i −0.242245 + 0.419580i
\(11\) −0.220544 − 0.381994i −0.0664966 − 0.115175i 0.830860 − 0.556481i \(-0.187849\pi\)
−0.897357 + 0.441306i \(0.854515\pi\)
\(12\) 0.972925 − 0.816381i 0.280859 − 0.235669i
\(13\) −0.367592 + 2.08472i −0.101952 + 0.578196i 0.890443 + 0.455095i \(0.150395\pi\)
−0.992394 + 0.123100i \(0.960716\pi\)
\(14\) −0.972925 + 1.68516i −0.260025 + 0.450377i
\(15\) −1.82850 − 0.665520i −0.472117 − 0.171837i
\(16\) −0.939693 − 0.342020i −0.234923 − 0.0855050i
\(17\) −0.664245 − 3.76712i −0.161103 − 0.913661i −0.952992 − 0.302996i \(-0.902013\pi\)
0.791889 − 0.610665i \(-0.209098\pi\)
\(18\) −1.06246 − 0.891507i −0.250423 − 0.210130i
\(19\) 4.55479 + 3.82192i 1.04494 + 0.876808i 0.992552 − 0.121819i \(-0.0388726\pi\)
0.0523871 + 0.998627i \(0.483317\pi\)
\(20\) 0.266044 + 1.50881i 0.0594893 + 0.337381i
\(21\) −2.32231 − 0.845253i −0.506770 − 0.184449i
\(22\) −0.414488 − 0.150861i −0.0883690 − 0.0321637i
\(23\) −2.85558 + 4.94600i −0.595429 + 1.03131i 0.398057 + 0.917360i \(0.369684\pi\)
−0.993486 + 0.113952i \(0.963649\pi\)
\(24\) 0.220544 − 1.25077i 0.0450184 − 0.255312i
\(25\) −2.03209 + 1.70513i −0.406418 + 0.341025i
\(26\) 1.05844 + 1.83327i 0.207577 + 0.359533i
\(27\) 2.78585 − 4.82523i 0.536136 − 0.928615i
\(28\) 0.337893 + 1.91629i 0.0638558 + 0.362144i
\(29\) −1.36712 − 2.36792i −0.253867 − 0.439711i 0.710720 − 0.703475i \(-0.248369\pi\)
−0.964587 + 0.263764i \(0.915036\pi\)
\(30\) −1.82850 + 0.665520i −0.333837 + 0.121507i
\(31\) 9.87516 1.77363 0.886816 − 0.462123i \(-0.152912\pi\)
0.886816 + 0.462123i \(0.152912\pi\)
\(32\) −0.939693 + 0.342020i −0.166116 + 0.0604612i
\(33\) 0.0972795 − 0.551699i 0.0169342 − 0.0960386i
\(34\) −2.93030 − 2.45881i −0.502542 − 0.421683i
\(35\) 2.28374 − 1.91629i 0.386023 − 0.323912i
\(36\) −1.38694 −0.231156
\(37\) 2.18831 − 5.67550i 0.359755 − 0.933047i
\(38\) 5.94585 0.964544
\(39\) −2.05956 + 1.72818i −0.329794 + 0.276730i
\(40\) 1.17365 + 0.984808i 0.185570 + 0.155712i
\(41\) 1.80615 − 10.2432i 0.282073 − 1.59972i −0.433482 − 0.901162i \(-0.642715\pi\)
0.715556 − 0.698556i \(-0.246174\pi\)
\(42\) −2.32231 + 0.845253i −0.358341 + 0.130425i
\(43\) 3.84681 0.586633 0.293317 − 0.956015i \(-0.405241\pi\)
0.293317 + 0.956015i \(0.405241\pi\)
\(44\) −0.414488 + 0.150861i −0.0624863 + 0.0227432i
\(45\) 1.06246 + 1.84023i 0.158382 + 0.274325i
\(46\) 0.991731 + 5.62439i 0.146223 + 0.829271i
\(47\) −3.84316 + 6.65655i −0.560582 + 0.970957i 0.436863 + 0.899528i \(0.356089\pi\)
−0.997446 + 0.0714293i \(0.977244\pi\)
\(48\) −0.635032 − 1.09991i −0.0916589 − 0.158758i
\(49\) −2.46181 + 2.06571i −0.351687 + 0.295101i
\(50\) −0.460637 + 2.61240i −0.0651439 + 0.369450i
\(51\) 2.42915 − 4.20740i 0.340148 − 0.589154i
\(52\) 1.98921 + 0.724014i 0.275854 + 0.100403i
\(53\) −1.62561 − 0.591673i −0.223295 − 0.0812726i 0.227950 − 0.973673i \(-0.426798\pi\)
−0.451245 + 0.892400i \(0.649020\pi\)
\(54\) −0.967514 − 5.48704i −0.131662 − 0.746692i
\(55\) 0.517683 + 0.434387i 0.0698043 + 0.0585728i
\(56\) 1.49061 + 1.25077i 0.199191 + 0.167141i
\(57\) 1.31132 + 7.43688i 0.173689 + 0.985039i
\(58\) −2.56934 − 0.935163i −0.337371 − 0.122793i
\(59\) −1.93027 − 0.702561i −0.251300 − 0.0914656i 0.213299 − 0.976987i \(-0.431579\pi\)
−0.464599 + 0.885521i \(0.653801\pi\)
\(60\) −0.972925 + 1.68516i −0.125604 + 0.217553i
\(61\) −1.89894 + 10.7694i −0.243134 + 1.37888i 0.581652 + 0.813437i \(0.302406\pi\)
−0.824786 + 0.565444i \(0.808705\pi\)
\(62\) 7.56481 − 6.34763i 0.960732 − 0.806150i
\(63\) 1.34939 + 2.33721i 0.170007 + 0.294460i
\(64\) −0.500000 + 0.866025i −0.0625000 + 0.108253i
\(65\) −0.563183 − 3.19397i −0.0698542 − 0.396163i
\(66\) −0.280105 − 0.485156i −0.0344786 − 0.0597186i
\(67\) 1.57261 − 0.572384i 0.192125 − 0.0699279i −0.244166 − 0.969734i \(-0.578514\pi\)
0.436291 + 0.899806i \(0.356292\pi\)
\(68\) −3.82524 −0.463878
\(69\) −6.81608 + 2.48085i −0.820560 + 0.298659i
\(70\) 0.517683 − 2.93592i 0.0618749 − 0.350910i
\(71\) −7.70462 − 6.46494i −0.914370 − 0.767248i 0.0585751 − 0.998283i \(-0.481344\pi\)
−0.972945 + 0.231035i \(0.925789\pi\)
\(72\) −1.06246 + 0.891507i −0.125212 + 0.105065i
\(73\) −9.88570 −1.15703 −0.578517 − 0.815671i \(-0.696368\pi\)
−0.578517 + 0.815671i \(0.696368\pi\)
\(74\) −1.97180 − 5.75430i −0.229217 − 0.668924i
\(75\) −3.36910 −0.389030
\(76\) 4.55479 − 3.82192i 0.522470 − 0.438404i
\(77\) 0.657490 + 0.551699i 0.0749279 + 0.0628720i
\(78\) −0.466865 + 2.64772i −0.0528620 + 0.299795i
\(79\) −12.5395 + 4.56401i −1.41081 + 0.513491i −0.931366 − 0.364084i \(-0.881382\pi\)
−0.479439 + 0.877575i \(0.659160\pi\)
\(80\) 1.53209 0.171293
\(81\) 2.73975 − 0.997189i 0.304417 − 0.110799i
\(82\) −5.20061 − 9.00771i −0.574311 − 0.994735i
\(83\) 0.617999 + 3.50484i 0.0678341 + 0.384707i 0.999757 + 0.0220497i \(0.00701920\pi\)
−0.931923 + 0.362657i \(0.881870\pi\)
\(84\) −1.23568 + 2.14025i −0.134823 + 0.233521i
\(85\) 2.93030 + 5.07543i 0.317836 + 0.550508i
\(86\) 2.94683 − 2.47268i 0.317765 − 0.266636i
\(87\) 0.603020 − 3.41989i 0.0646505 − 0.366651i
\(88\) −0.220544 + 0.381994i −0.0235101 + 0.0407207i
\(89\) 11.1543 + 4.05982i 1.18235 + 0.430340i 0.857031 − 0.515264i \(-0.172306\pi\)
0.325319 + 0.945604i \(0.394528\pi\)
\(90\) 1.99677 + 0.726763i 0.210478 + 0.0766076i
\(91\) −0.715278 − 4.05654i −0.0749815 − 0.425241i
\(92\) 4.37500 + 3.67106i 0.456125 + 0.382734i
\(93\) 9.60779 + 8.06190i 0.996282 + 0.835980i
\(94\) 1.33472 + 7.56955i 0.137665 + 0.780739i
\(95\) −8.56020 − 3.11566i −0.878258 − 0.319660i
\(96\) −1.19347 − 0.434387i −0.121808 − 0.0443345i
\(97\) 0.296748 − 0.513983i 0.0301302 − 0.0521871i −0.850567 − 0.525867i \(-0.823741\pi\)
0.880697 + 0.473679i \(0.157074\pi\)
\(98\) −0.558047 + 3.16484i −0.0563713 + 0.319698i
\(99\) −0.468637 + 0.393233i −0.0470998 + 0.0395214i
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.b.49.2 ✓ 12
3.2 odd 2 666.2.x.g.271.1 12
4.3 odd 2 592.2.bc.d.49.1 12
37.16 even 9 2738.2.a.t.1.4 6
37.21 even 18 2738.2.a.q.1.4 6
37.34 even 9 inner 74.2.f.b.71.2 yes 12
111.71 odd 18 666.2.x.g.145.1 12
148.71 odd 18 592.2.bc.d.145.1 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
74.2.f.b.49.2 ✓ 12 1.1 even 1 trivial
74.2.f.b.71.2 yes 12 37.34 even 9 inner
592.2.bc.d.49.1 12 4.3 odd 2
592.2.bc.d.145.1 12 148.71 odd 18
666.2.x.g.145.1 12 111.71 odd 18
666.2.x.g.271.1 12 3.2 odd 2
2738.2.a.q.1.4 6 37.21 even 18
2738.2.a.t.1.4 6 37.16 even 9