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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [522,2,Mod(41,522)] 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("522.41"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(522, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([10, 3])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 522 = 2 \cdot 3^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 522.l (of order \(12\), degree \(4\), 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: \(4.16819098551\)
Analytic rank: \(0\)
Dimension: \(120\)
Relative dimension: \(30\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label 41.17
Character \(\chi\) \(=\) 522.41
Dual form 522.2.l.a.191.17

$q$-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+(0.965926 + 0.258819i) q^{2} +(-1.63468 + 0.572547i) q^{3} +(0.866025 + 0.500000i) q^{4} +(1.02346 - 1.77269i) q^{5} +(-1.72717 + 0.129950i) q^{6} +(-1.33363 - 2.30991i) q^{7} +(0.707107 + 0.707107i) q^{8} +(2.34438 - 1.87187i) q^{9} +(1.44739 - 1.44739i) q^{10} +(-5.33855 - 1.43046i) q^{11} +(-1.70195 - 0.321502i) q^{12} +(-1.22828 - 0.709147i) q^{13} +(-0.690336 - 2.57637i) q^{14} +(-0.658090 + 3.48377i) q^{15} +(0.500000 + 0.866025i) q^{16} +(4.70074 - 4.70074i) q^{17} +(2.74897 - 1.20131i) q^{18} +(-3.75403 - 3.75403i) q^{19} +(1.77269 - 1.02346i) q^{20} +(3.50259 + 3.01241i) q^{21} +(-4.78641 - 2.76343i) q^{22} +(4.07945 + 2.35527i) q^{23} +(-1.56075 - 0.751044i) q^{24} +(0.405049 + 0.701566i) q^{25} +(-1.00289 - 1.00289i) q^{26} +(-2.76059 + 4.40217i) q^{27} -2.66725i q^{28} +(-4.62431 - 2.75967i) q^{29} +(-1.53733 + 3.19473i) q^{30} +(3.75561 - 1.00631i) q^{31} +(0.258819 + 0.965926i) q^{32} +(9.54584 - 0.718219i) q^{33} +(5.75720 - 3.32392i) q^{34} -5.45966 q^{35} +(2.96623 - 0.448893i) q^{36} +(-2.00441 + 2.00441i) q^{37} +(-2.65450 - 4.59773i) q^{38} +(2.41387 + 0.455984i) q^{39} +(1.97718 - 0.529783i) q^{40} +(7.36863 - 1.97442i) q^{41} +(2.60357 + 3.81630i) q^{42} +(1.92694 - 7.19142i) q^{43} +(-3.90809 - 3.90809i) q^{44} +(-0.918849 - 6.07164i) q^{45} +(3.33086 + 3.33086i) q^{46} +(0.354390 - 1.32260i) q^{47} +(-1.31318 - 1.12940i) q^{48} +(-0.0571168 + 0.0989292i) q^{49} +(0.209669 + 0.782495i) q^{50} +(-4.99283 + 10.3756i) q^{51} +(-0.709147 - 1.22828i) q^{52} -0.189952i q^{53} +(-3.80589 + 3.53768i) q^{54} +(-7.99956 + 7.99956i) q^{55} +(0.690336 - 2.57637i) q^{56} +(8.28600 + 3.98729i) q^{57} +(-3.75248 - 3.86250i) q^{58} +(9.80639 + 5.66172i) q^{59} +(-2.31181 + 2.68798i) q^{60} +(-2.42793 + 9.06118i) q^{61} +3.88810 q^{62} +(-7.45036 - 2.91894i) q^{63} +1.00000i q^{64} +(-2.51419 + 1.45157i) q^{65} +(9.40646 + 1.77690i) q^{66} +(-9.42679 - 5.44256i) q^{67} +(6.42133 - 1.72059i) q^{68} +(-8.01712 - 1.51445i) q^{69} +(-5.27363 - 1.41307i) q^{70} +2.02501 q^{71} +(2.98134 + 0.334119i) q^{72} +(-9.40866 + 9.40866i) q^{73} +(-2.45489 + 1.41733i) q^{74} +(-1.06381 - 0.914929i) q^{75} +(-1.37407 - 5.12810i) q^{76} +(3.81540 + 14.2392i) q^{77} +(2.21360 + 1.06520i) q^{78} +(-1.87994 + 7.01603i) q^{79} +2.04692 q^{80} +(1.99224 - 8.77673i) q^{81} +7.62857 q^{82} +(5.86507 - 3.38620i) q^{83} +(1.52713 + 4.36011i) q^{84} +(-3.52192 - 13.1440i) q^{85} +(3.72255 - 6.44765i) q^{86} +(9.13932 + 1.86356i) q^{87} +(-2.76343 - 4.78641i) q^{88} +(-6.14677 + 6.14677i) q^{89} +(0.683916 - 6.10257i) q^{90} +3.78295i q^{91} +(2.35527 + 4.07945i) q^{92} +(-5.56308 + 3.79527i) q^{93} +(0.684629 - 1.18581i) q^{94} +(-10.4968 + 2.81262i) q^{95} +(-0.976125 - 1.43080i) q^{96} +(-8.36035 - 2.24015i) q^{97} +(-0.0807753 + 0.0807753i) q^{98} +(-15.1932 + 6.63950i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 120 q - 24 q^{11} + 12 q^{14} + 20 q^{15} + 60 q^{16} - 28 q^{21} + 8 q^{24} - 60 q^{25} - 24 q^{27} + 12 q^{29} + 12 q^{30} - 16 q^{36} + 16 q^{39} + 12 q^{41} + 104 q^{45} - 24 q^{46} + 24 q^{47} - 60 q^{49}+ \cdots - 24 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/522\mathbb{Z}\right)^\times\).

\(n\) \(379\) \(407\)
\(\chi(n)\) \(e\left(\frac{1}{4}\right)\) \(e\left(\frac{5}{6}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.965926 + 0.258819i 0.683013 + 0.183013i
\(3\) −1.63468 + 0.572547i −0.943785 + 0.330560i
\(4\) 0.866025 + 0.500000i 0.433013 + 0.250000i
\(5\) 1.02346 − 1.77269i 0.457706 − 0.792771i −0.541133 − 0.840937i \(-0.682004\pi\)
0.998839 + 0.0481664i \(0.0153378\pi\)
\(6\) −1.72717 + 0.129950i −0.705114 + 0.0530520i
\(7\) −1.33363 − 2.30991i −0.504063 − 0.873063i −0.999989 − 0.00469822i \(-0.998505\pi\)
0.495926 − 0.868365i \(-0.334829\pi\)
\(8\) 0.707107 + 0.707107i 0.250000 + 0.250000i
\(9\) 2.34438 − 1.87187i 0.781460 − 0.623955i
\(10\) 1.44739 − 1.44739i 0.457706 − 0.457706i
\(11\) −5.33855 − 1.43046i −1.60963 − 0.431300i −0.661699 − 0.749769i \(-0.730164\pi\)
−0.947933 + 0.318470i \(0.896831\pi\)
\(12\) −1.70195 − 0.321502i −0.491311 − 0.0928096i
\(13\) −1.22828 − 0.709147i −0.340663 − 0.196682i 0.319902 − 0.947451i \(-0.396350\pi\)
−0.660565 + 0.750769i \(0.729683\pi\)
\(14\) −0.690336 − 2.57637i −0.184500 − 0.688563i
\(15\) −0.658090 + 3.48377i −0.169918 + 0.899504i
\(16\) 0.500000 + 0.866025i 0.125000 + 0.216506i
\(17\) 4.70074 − 4.70074i 1.14010 − 1.14010i 0.151664 − 0.988432i \(-0.451537\pi\)
0.988432 − 0.151664i \(-0.0484632\pi\)
\(18\) 2.74897 − 1.20131i 0.647939 − 0.283152i
\(19\) −3.75403 − 3.75403i −0.861233 − 0.861233i 0.130249 − 0.991481i \(-0.458422\pi\)
−0.991481 + 0.130249i \(0.958422\pi\)
\(20\) 1.77269 − 1.02346i 0.396385 − 0.228853i
\(21\) 3.50259 + 3.01241i 0.764327 + 0.657361i
\(22\) −4.78641 − 2.76343i −1.02047 − 0.589166i
\(23\) 4.07945 + 2.35527i 0.850625 + 0.491109i 0.860862 − 0.508839i \(-0.169925\pi\)
−0.0102366 + 0.999948i \(0.503258\pi\)
\(24\) −1.56075 − 0.751044i −0.318586 − 0.153306i
\(25\) 0.405049 + 0.701566i 0.0810099 + 0.140313i
\(26\) −1.00289 − 1.00289i −0.196682 − 0.196682i
\(27\) −2.76059 + 4.40217i −0.531276 + 0.847199i
\(28\) − 2.66725i − 0.504063i
\(29\) −4.62431 − 2.75967i −0.858712 − 0.512458i
\(30\) −1.53733 + 3.19473i −0.280677 + 0.583276i
\(31\) 3.75561 − 1.00631i 0.674528 − 0.180739i 0.0947345 − 0.995503i \(-0.469800\pi\)
0.579794 + 0.814763i \(0.303133\pi\)
\(32\) 0.258819 + 0.965926i 0.0457532 + 0.170753i
\(33\) 9.54584 − 0.718219i 1.66172 − 0.125026i
\(34\) 5.75720 − 3.32392i 0.987352 − 0.570048i
\(35\) −5.45966 −0.922852
\(36\) 2.96623 − 0.448893i 0.494371 − 0.0748154i
\(37\) −2.00441 + 2.00441i −0.329522 + 0.329522i −0.852405 − 0.522883i \(-0.824857\pi\)
0.522883 + 0.852405i \(0.324857\pi\)
\(38\) −2.65450 − 4.59773i −0.430616 − 0.745850i
\(39\) 2.41387 + 0.455984i 0.386528 + 0.0730159i
\(40\) 1.97718 − 0.529783i 0.312619 − 0.0837661i
\(41\) 7.36863 − 1.97442i 1.15079 − 0.308352i 0.367506 − 0.930021i \(-0.380212\pi\)
0.783280 + 0.621669i \(0.213545\pi\)
\(42\) 2.60357 + 3.81630i 0.401740 + 0.588867i
\(43\) 1.92694 − 7.19142i 0.293855 − 1.09668i −0.648267 − 0.761413i \(-0.724506\pi\)
0.942122 − 0.335269i \(-0.108827\pi\)
\(44\) −3.90809 − 3.90809i −0.589166 − 0.589166i
\(45\) −0.918849 − 6.07164i −0.136974 − 0.905107i
\(46\) 3.33086 + 3.33086i 0.491109 + 0.491109i
\(47\) 0.354390 − 1.32260i 0.0516931 − 0.192921i −0.935251 − 0.353986i \(-0.884826\pi\)
0.986944 + 0.161065i \(0.0514928\pi\)
\(48\) −1.31318 − 1.12940i −0.189541 − 0.163015i
\(49\) −0.0571168 + 0.0989292i −0.00815954 + 0.0141327i
\(50\) 0.209669 + 0.782495i 0.0296517 + 0.110662i
\(51\) −4.99283 + 10.3756i −0.699135 + 1.45288i
\(52\) −0.709147 − 1.22828i −0.0983410 − 0.170332i
\(53\) − 0.189952i − 0.0260919i −0.999915 − 0.0130459i \(-0.995847\pi\)
0.999915 − 0.0130459i \(-0.00415277\pi\)
\(54\) −3.80589 + 3.53768i −0.517916 + 0.481417i
\(55\) −7.99956 + 7.99956i −1.07866 + 1.07866i
\(56\) 0.690336 − 2.57637i 0.0922500 − 0.344282i
\(57\) 8.28600 + 3.98729i 1.09751 + 0.528130i
\(58\) −3.75248 − 3.86250i −0.492725 − 0.507171i
\(59\) 9.80639 + 5.66172i 1.27668 + 0.737093i 0.976237 − 0.216705i \(-0.0695311\pi\)
0.300446 + 0.953799i \(0.402864\pi\)
\(60\) −2.31181 + 2.68798i −0.298453 + 0.347017i
\(61\) −2.42793 + 9.06118i −0.310865 + 1.16016i 0.616913 + 0.787032i \(0.288383\pi\)
−0.927778 + 0.373133i \(0.878283\pi\)
\(62\) 3.88810 0.493789
\(63\) −7.45036 − 2.91894i −0.938658 − 0.367751i
\(64\) 1.00000i 0.125000i
\(65\) −2.51419 + 1.45157i −0.311847 + 0.180045i
\(66\) 9.40646 + 1.77690i 1.15786 + 0.218721i
\(67\) −9.42679 − 5.44256i −1.15167 − 0.664914i −0.202373 − 0.979309i \(-0.564865\pi\)
−0.949292 + 0.314394i \(0.898199\pi\)
\(68\) 6.42133 − 1.72059i 0.778700 − 0.208652i
\(69\) −8.01712 − 1.51445i −0.965148 − 0.182318i
\(70\) −5.27363 − 1.41307i −0.630319 − 0.168894i
\(71\) 2.02501 0.240324 0.120162 − 0.992754i \(-0.461659\pi\)
0.120162 + 0.992754i \(0.461659\pi\)
\(72\) 2.98134 + 0.334119i 0.351354 + 0.0393763i
\(73\) −9.40866 + 9.40866i −1.10120 + 1.10120i −0.106934 + 0.994266i \(0.534103\pi\)
−0.994266 + 0.106934i \(0.965897\pi\)
\(74\) −2.45489 + 1.41733i −0.285375 + 0.164761i
\(75\) −1.06381 − 0.914929i −0.122838 − 0.105647i
\(76\) −1.37407 − 5.12810i −0.157617 − 0.588233i
\(77\) 3.81540 + 14.2392i 0.434805 + 1.62271i
\(78\) 2.21360 + 1.06520i 0.250641 + 0.120610i
\(79\) −1.87994 + 7.01603i −0.211510 + 0.789365i 0.775856 + 0.630910i \(0.217318\pi\)
−0.987366 + 0.158456i \(0.949349\pi\)
\(80\) 2.04692 0.228853
\(81\) 1.99224 − 8.77673i 0.221360 − 0.975192i
\(82\) 7.62857 0.842434
\(83\) 5.86507 − 3.38620i 0.643775 − 0.371684i −0.142292 − 0.989825i \(-0.545447\pi\)
0.786067 + 0.618141i \(0.212114\pi\)
\(84\) 1.52713 + 4.36011i 0.166623 + 0.475727i
\(85\) −3.52192 − 13.1440i −0.382006 − 1.42566i
\(86\) 3.72255 − 6.44765i 0.401413 − 0.695268i
\(87\) 9.13932 + 1.86356i 0.979838 + 0.199795i
\(88\) −2.76343 − 4.78641i −0.294583 − 0.510233i
\(89\) −6.14677 + 6.14677i −0.651556 + 0.651556i −0.953368 − 0.301812i \(-0.902409\pi\)
0.301812 + 0.953368i \(0.402409\pi\)
\(90\) 0.683916 − 6.10257i 0.0720911 − 0.643267i
\(91\) 3.78295i 0.396561i
\(92\) 2.35527 + 4.07945i 0.245554 + 0.425313i
\(93\) −5.56308 + 3.79527i −0.576864 + 0.393551i
\(94\) 0.684629 − 1.18581i 0.0706141 − 0.122307i
\(95\) −10.4968 + 2.81262i −1.07695 + 0.288568i
\(96\) −0.976125 − 1.43080i −0.0996253 − 0.146030i
\(97\) −8.36035 − 2.24015i −0.848864 − 0.227453i −0.191938 − 0.981407i \(-0.561477\pi\)
−0.656927 + 0.753955i \(0.728144\pi\)
\(98\) −0.0807753 + 0.0807753i −0.00815954 + 0.00815954i
\(99\) −15.1932 + 6.63950i −1.52698 + 0.667295i
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 522.2.l.a.41.17 ✓ 120
9.2 odd 6 inner 522.2.l.a.389.25 yes 120
29.17 odd 4 inner 522.2.l.a.365.25 yes 120
261.191 even 12 inner 522.2.l.a.191.17 yes 120
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
522.2.l.a.41.17 ✓ 120 1.1 even 1 trivial
522.2.l.a.191.17 yes 120 261.191 even 12 inner
522.2.l.a.365.25 yes 120 29.17 odd 4 inner
522.2.l.a.389.25 yes 120 9.2 odd 6 inner