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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [483,2,Mod(10,483)] 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("483.10"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(483, base_ring=CyclotomicField(66)) chi = DirichletCharacter(H, H._module([0, 11, 9])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 483 = 3 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 483.bf (of order \(66\), degree \(20\), 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: \(3.85677441763\)
Analytic rank: \(0\)
Dimension: \(640\)
Relative dimension: \(32\) over \(\Q(\zeta_{66})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{66}]$

Embedding invariants

Embedding label 10.4
Character \(\chi\) \(=\) 483.10
Dual form 483.2.bf.a.145.4

$q$-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+(-0.797371 + 2.30385i) q^{2} +(0.458227 + 0.888835i) q^{3} +(-3.09983 - 2.43773i) q^{4} +(1.48893 - 0.142176i) q^{5} +(-2.41312 + 0.346955i) q^{6} +(2.64327 + 0.114618i) q^{7} +(3.98605 - 2.56168i) q^{8} +(-0.580057 + 0.814576i) q^{9} +(-0.859680 + 3.54365i) q^{10} +(5.25822 - 1.81989i) q^{11} +(0.746319 - 3.87227i) q^{12} +(1.14332 + 3.89380i) q^{13} +(-2.37173 + 5.99831i) q^{14} +(0.808640 + 1.25827i) q^{15} +(0.863919 + 3.56112i) q^{16} +(2.36184 + 0.945539i) q^{17} +(-1.41414 - 1.98589i) q^{18} +(-4.40203 + 1.76231i) q^{19} +(-4.96203 - 3.18890i) q^{20} +(1.10934 + 2.40195i) q^{21} +13.5653i q^{22} +(0.501961 - 4.76949i) q^{23} +(4.10342 + 2.36911i) q^{24} +(-2.71293 + 0.522875i) q^{25} +(-9.88239 - 0.470756i) q^{26} +(-0.989821 - 0.142315i) q^{27} +(-7.91427 - 6.79888i) q^{28} +(-0.100916 - 0.701885i) q^{29} +(-3.54365 + 0.859680i) q^{30} +(1.67480 - 0.0797805i) q^{31} +(0.540375 + 0.0515995i) q^{32} +(4.02704 + 3.83977i) q^{33} +(-4.06165 + 4.68739i) q^{34} +(3.95195 - 0.205150i) q^{35} +(3.78380 - 1.11102i) q^{36} +(-0.543946 - 0.387342i) q^{37} +(-0.550043 - 11.5468i) q^{38} +(-2.93704 + 2.80047i) q^{39} +(5.57075 - 4.38089i) q^{40} +(-5.32488 - 2.43179i) q^{41} +(-6.41830 + 0.640506i) q^{42} +(-4.30916 + 6.70519i) q^{43} +(-20.7360 - 7.17680i) q^{44} +(-0.747853 + 1.29532i) q^{45} +(10.5880 + 4.95950i) q^{46} +(1.90751 - 1.10130i) q^{47} +(-2.76938 + 2.39968i) q^{48} +(6.97373 + 0.605934i) q^{49} +(0.958588 - 6.66713i) q^{50} +(0.241830 + 2.53256i) q^{51} +(5.94794 - 14.8572i) q^{52} +(-1.07652 - 1.12902i) q^{53} +(1.11713 - 2.16693i) q^{54} +(7.57040 - 3.45729i) q^{55} +(10.8298 - 6.31432i) q^{56} +(-3.58353 - 3.10514i) q^{57} +(1.69751 + 0.327168i) q^{58} +(-0.288871 - 0.0700794i) q^{59} +(0.560676 - 5.87167i) q^{60} +(-10.0666 - 5.18972i) q^{61} +(-1.15163 + 3.92210i) q^{62} +(-1.62661 + 2.08666i) q^{63} +(-3.59426 + 7.87034i) q^{64} +(2.25593 + 5.63505i) q^{65} +(-12.0573 + 6.21598i) q^{66} +(-2.58709 - 13.4231i) q^{67} +(-5.01634 - 8.68855i) q^{68} +(4.46930 - 1.73935i) q^{69} +(-2.67853 + 9.26828i) q^{70} +(-2.97261 - 3.43057i) q^{71} +(-0.225454 + 4.73286i) q^{72} +(2.80639 - 3.56861i) q^{73} +(1.32611 - 0.944317i) q^{74} +(-1.70789 - 2.17176i) q^{75} +(17.9416 + 5.26812i) q^{76} +(14.1075 - 4.20776i) q^{77} +(-4.10995 - 8.99953i) q^{78} +(-8.36812 + 8.77623i) q^{79} +(1.79262 + 5.17944i) q^{80} +(-0.327068 - 0.945001i) q^{81} +(9.84840 - 10.3287i) q^{82} +(2.45765 + 5.38150i) q^{83} +(2.41656 - 10.1499i) q^{84} +(3.65106 + 1.07205i) q^{85} +(-12.0118 - 15.2742i) q^{86} +(0.577618 - 0.411320i) q^{87} +(16.2976 - 20.7240i) q^{88} +(-0.806837 + 16.9376i) q^{89} +(-2.38791 - 2.75579i) q^{90} +(2.57581 + 10.4234i) q^{91} +(-13.1827 + 13.5610i) q^{92} +(0.838349 + 1.45206i) q^{93} +(1.01624 + 5.27277i) q^{94} +(-6.30377 + 3.24982i) q^{95} +(0.201751 + 0.503949i) q^{96} +(4.75583 - 10.4138i) q^{97} +(-6.95663 + 15.5833i) q^{98} +(-1.56763 + 5.33886i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 640 q - 4 q^{2} + 36 q^{4} + 24 q^{8} - 32 q^{9} + 4 q^{18} - 28 q^{23} + 56 q^{25} - 84 q^{26} - 176 q^{28} - 24 q^{29} + 12 q^{31} + 36 q^{32} - 76 q^{35} + 28 q^{36} + 44 q^{37} - 110 q^{42} - 88 q^{43}+ \cdots - 44 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/483\mathbb{Z}\right)^\times\).

\(n\) \(323\) \(346\) \(442\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{3}{22}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.797371 + 2.30385i −0.563827 + 1.62907i 0.197210 + 0.980361i \(0.436812\pi\)
−0.761037 + 0.648709i \(0.775309\pi\)
\(3\) 0.458227 + 0.888835i 0.264557 + 0.513169i
\(4\) −3.09983 − 2.43773i −1.54992 − 1.21887i
\(5\) 1.48893 − 0.142176i 0.665871 − 0.0635830i 0.243354 − 0.969938i \(-0.421752\pi\)
0.422518 + 0.906355i \(0.361146\pi\)
\(6\) −2.41312 + 0.346955i −0.985153 + 0.141644i
\(7\) 2.64327 + 0.114618i 0.999061 + 0.0433217i
\(8\) 3.98605 − 2.56168i 1.40928 − 0.905690i
\(9\) −0.580057 + 0.814576i −0.193352 + 0.271525i
\(10\) −0.859680 + 3.54365i −0.271855 + 1.12060i
\(11\) 5.25822 − 1.81989i 1.58541 − 0.548717i 0.614659 − 0.788793i \(-0.289293\pi\)
0.970754 + 0.240076i \(0.0771723\pi\)
\(12\) 0.746319 − 3.87227i 0.215444 − 1.11783i
\(13\) 1.14332 + 3.89380i 0.317101 + 1.07995i 0.951681 + 0.307088i \(0.0993547\pi\)
−0.634580 + 0.772857i \(0.718827\pi\)
\(14\) −2.37173 + 5.99831i −0.633871 + 1.60311i
\(15\) 0.808640 + 1.25827i 0.208790 + 0.324883i
\(16\) 0.863919 + 3.56112i 0.215980 + 0.890280i
\(17\) 2.36184 + 0.945539i 0.572831 + 0.229327i 0.639948 − 0.768418i \(-0.278956\pi\)
−0.0671173 + 0.997745i \(0.521380\pi\)
\(18\) −1.41414 − 1.98589i −0.333317 − 0.468078i
\(19\) −4.40203 + 1.76231i −1.00989 + 0.404301i −0.816826 − 0.576885i \(-0.804268\pi\)
−0.193069 + 0.981185i \(0.561844\pi\)
\(20\) −4.96203 − 3.18890i −1.10954 − 0.713060i
\(21\) 1.10934 + 2.40195i 0.242077 + 0.524149i
\(22\) 13.5653i 2.89213i
\(23\) 0.501961 − 4.76949i 0.104666 − 0.994507i
\(24\) 4.10342 + 2.36911i 0.837607 + 0.483593i
\(25\) −2.71293 + 0.522875i −0.542587 + 0.104575i
\(26\) −9.88239 − 0.470756i −1.93810 − 0.0923229i
\(27\) −0.989821 − 0.142315i −0.190491 − 0.0273885i
\(28\) −7.91427 − 6.79888i −1.49566 − 1.28487i
\(29\) −0.100916 − 0.701885i −0.0187396 − 0.130337i 0.978304 − 0.207174i \(-0.0664268\pi\)
−0.997044 + 0.0768377i \(0.975518\pi\)
\(30\) −3.54365 + 0.859680i −0.646979 + 0.156955i
\(31\) 1.67480 − 0.0797805i 0.300803 − 0.0143290i 0.103362 − 0.994644i \(-0.467040\pi\)
0.197441 + 0.980315i \(0.436737\pi\)
\(32\) 0.540375 + 0.0515995i 0.0955257 + 0.00912160i
\(33\) 4.02704 + 3.83977i 0.701017 + 0.668419i
\(34\) −4.06165 + 4.68739i −0.696567 + 0.803881i
\(35\) 3.95195 − 0.205150i 0.668001 − 0.0346767i
\(36\) 3.78380 − 1.11102i 0.630633 − 0.185170i
\(37\) −0.543946 − 0.387342i −0.0894242 − 0.0636787i 0.534470 − 0.845187i \(-0.320511\pi\)
−0.623895 + 0.781509i \(0.714451\pi\)
\(38\) −0.550043 − 11.5468i −0.0892288 − 1.87314i
\(39\) −2.93704 + 2.80047i −0.470304 + 0.448434i
\(40\) 5.57075 − 4.38089i 0.880813 − 0.692679i
\(41\) −5.32488 − 2.43179i −0.831607 − 0.379782i −0.0463296 − 0.998926i \(-0.514752\pi\)
−0.785277 + 0.619144i \(0.787480\pi\)
\(42\) −6.41830 + 0.640506i −0.990364 + 0.0988322i
\(43\) −4.30916 + 6.70519i −0.657141 + 1.02253i 0.339498 + 0.940607i \(0.389743\pi\)
−0.996639 + 0.0819244i \(0.973893\pi\)
\(44\) −20.7360 − 7.17680i −3.12607 − 1.08194i
\(45\) −0.747853 + 1.29532i −0.111483 + 0.193095i
\(46\) 10.5880 + 4.95950i 1.56111 + 0.731238i
\(47\) 1.90751 − 1.10130i 0.278239 − 0.160641i −0.354387 − 0.935099i \(-0.615310\pi\)
0.632626 + 0.774458i \(0.281977\pi\)
\(48\) −2.76938 + 2.39968i −0.399726 + 0.346364i
\(49\) 6.97373 + 0.605934i 0.996246 + 0.0865620i
\(50\) 0.958588 − 6.66713i 0.135565 − 0.942874i
\(51\) 0.241830 + 2.53256i 0.0338630 + 0.354629i
\(52\) 5.94794 − 14.8572i 0.824830 − 2.06033i
\(53\) −1.07652 − 1.12902i −0.147872 − 0.155083i 0.645632 − 0.763648i \(-0.276594\pi\)
−0.793504 + 0.608565i \(0.791745\pi\)
\(54\) 1.11713 − 2.16693i 0.152022 − 0.294881i
\(55\) 7.57040 − 3.45729i 1.02079 − 0.466180i
\(56\) 10.8298 − 6.31432i 1.44719 − 0.843787i
\(57\) −3.58353 − 3.10514i −0.474650 − 0.411286i
\(58\) 1.69751 + 0.327168i 0.222894 + 0.0429592i
\(59\) −0.288871 − 0.0700794i −0.0376078 − 0.00912356i 0.216911 − 0.976191i \(-0.430402\pi\)
−0.254519 + 0.967068i \(0.581917\pi\)
\(60\) 0.560676 − 5.87167i 0.0723830 − 0.758029i
\(61\) −10.0666 − 5.18972i −1.28890 − 0.664475i −0.328872 − 0.944374i \(-0.606669\pi\)
−0.960029 + 0.279899i \(0.909699\pi\)
\(62\) −1.15163 + 3.92210i −0.146258 + 0.498108i
\(63\) −1.62661 + 2.08666i −0.204934 + 0.262894i
\(64\) −3.59426 + 7.87034i −0.449283 + 0.983792i
\(65\) 2.25593 + 5.63505i 0.279814 + 0.698942i
\(66\) −12.0573 + 6.21598i −1.48415 + 0.765134i
\(67\) −2.58709 − 13.4231i −0.316064 − 1.63989i −0.698283 − 0.715822i \(-0.746052\pi\)
0.382219 − 0.924072i \(-0.375160\pi\)
\(68\) −5.01634 − 8.68855i −0.608320 − 1.05364i
\(69\) 4.46930 − 1.73935i 0.538041 − 0.209393i
\(70\) −2.67853 + 9.26828i −0.320146 + 1.10777i
\(71\) −2.97261 − 3.43057i −0.352784 − 0.407134i 0.551425 − 0.834224i \(-0.314084\pi\)
−0.904209 + 0.427090i \(0.859539\pi\)
\(72\) −0.225454 + 4.73286i −0.0265700 + 0.557772i
\(73\) 2.80639 − 3.56861i 0.328463 − 0.417674i −0.593572 − 0.804781i \(-0.702283\pi\)
0.922035 + 0.387106i \(0.126525\pi\)
\(74\) 1.32611 − 0.944317i 0.154157 − 0.109775i
\(75\) −1.70789 − 2.17176i −0.197210 − 0.250773i
\(76\) 17.9416 + 5.26812i 2.05804 + 0.604295i
\(77\) 14.1075 − 4.20776i 1.60770 − 0.479519i
\(78\) −4.10995 − 8.99953i −0.465360 − 1.01900i
\(79\) −8.36812 + 8.77623i −0.941487 + 0.987403i −0.999941 − 0.0108874i \(-0.996534\pi\)
0.0584541 + 0.998290i \(0.481383\pi\)
\(80\) 1.79262 + 5.17944i 0.200421 + 0.579079i
\(81\) −0.327068 − 0.945001i −0.0363409 − 0.105000i
\(82\) 9.84840 − 10.3287i 1.08757 − 1.14061i
\(83\) 2.45765 + 5.38150i 0.269762 + 0.590697i 0.995230 − 0.0975597i \(-0.0311037\pi\)
−0.725468 + 0.688256i \(0.758376\pi\)
\(84\) 2.41656 − 10.1499i 0.263668 − 1.10745i
\(85\) 3.65106 + 1.07205i 0.396013 + 0.116280i
\(86\) −12.0118 − 15.2742i −1.29526 − 1.64706i
\(87\) 0.577618 − 0.411320i 0.0619271 − 0.0440981i
\(88\) 16.2976 − 20.7240i 1.73733 − 2.20919i
\(89\) −0.806837 + 16.9376i −0.0855246 + 1.79538i 0.395287 + 0.918558i \(0.370645\pi\)
−0.480811 + 0.876824i \(0.659658\pi\)
\(90\) −2.38791 − 2.75579i −0.251708 − 0.290486i
\(91\) 2.57581 + 10.4234i 0.270018 + 1.09267i
\(92\) −13.1827 + 13.5610i −1.37440 + 1.41383i
\(93\) 0.838349 + 1.45206i 0.0869327 + 0.150572i
\(94\) 1.01624 + 5.27277i 0.104817 + 0.543845i
\(95\) −6.30377 + 3.24982i −0.646753 + 0.333424i
\(96\) 0.201751 + 0.503949i 0.0205911 + 0.0514340i
\(97\) 4.75583 − 10.4138i 0.482882 − 1.05736i −0.498779 − 0.866729i \(-0.666218\pi\)
0.981661 − 0.190635i \(-0.0610546\pi\)
\(98\) −6.95663 + 15.5833i −0.702726 + 1.57415i
\(99\) −1.56763 + 5.33886i −0.157553 + 0.536576i
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 483.2.bf.a.10.4 ✓ 640
7.5 odd 6 inner 483.2.bf.a.355.29 yes 640
23.7 odd 22 inner 483.2.bf.a.283.29 yes 640
161.145 even 66 inner 483.2.bf.a.145.4 yes 640
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
483.2.bf.a.10.4 ✓ 640 1.1 even 1 trivial
483.2.bf.a.145.4 yes 640 161.145 even 66 inner
483.2.bf.a.283.29 yes 640 23.7 odd 22 inner
483.2.bf.a.355.29 yes 640 7.5 odd 6 inner