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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [147,2,Mod(5,147)] 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("147.5"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(147, base_ring=CyclotomicField(42)) chi = DirichletCharacter(H, H._module([21, 29])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 147 = 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 147.o (of order \(42\), degree \(12\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [192] 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: \(1.17380090971\)
Analytic rank: \(0\)
Dimension: \(192\)
Relative dimension: \(16\) over \(\Q(\zeta_{42})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{42}]$

Embedding invariants

Embedding label 101.4
Character \(\chi\) \(=\) 147.101
Dual form 147.2.o.b.131.4

$q$-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+(-1.58081 - 0.620423i) q^{2} +(0.678774 - 1.59351i) q^{3} +(0.647935 + 0.601196i) q^{4} +(-0.761363 - 0.519089i) q^{5} +(-2.06166 + 2.09791i) q^{6} +(-0.654203 - 2.56359i) q^{7} +(0.822376 + 1.70768i) q^{8} +(-2.07853 - 2.16326i) q^{9} +(0.881517 + 1.29295i) q^{10} +(0.0278066 + 0.184485i) q^{11} +(1.39781 - 0.624413i) q^{12} +(-2.33987 + 1.86598i) q^{13} +(-0.556341 + 4.45844i) q^{14} +(-1.34397 + 0.860894i) q^{15} +(-0.372643 - 4.97257i) q^{16} +(-3.71746 - 1.14668i) q^{17} +(1.94363 + 4.70928i) q^{18} +(2.14851 - 1.24044i) q^{19} +(-0.181240 - 0.794064i) q^{20} +(-4.52916 - 0.697625i) q^{21} +(0.0705014 - 0.308887i) q^{22} +(-0.363328 - 1.17788i) q^{23} +(3.27941 - 0.151332i) q^{24} +(-1.51648 - 3.86394i) q^{25} +(4.85658 - 1.49806i) q^{26} +(-4.85803 + 1.84379i) q^{27} +(1.11734 - 2.05435i) q^{28} +(9.18424 - 2.09624i) q^{29} +(2.65867 - 0.527084i) q^{30} +(6.62989 + 3.82777i) q^{31} +(-1.37867 + 4.46954i) q^{32} +(0.312852 + 0.0809134i) q^{33} +(5.16517 + 4.11909i) q^{34} +(-0.832647 + 2.29142i) q^{35} +(-0.0462080 - 2.65126i) q^{36} +(2.37319 - 2.20200i) q^{37} +(-4.16598 + 0.627921i) q^{38} +(1.38521 + 4.99517i) q^{39} +(0.260311 - 1.72705i) q^{40} +(5.40314 - 2.60202i) q^{41} +(6.72693 + 3.91281i) q^{42} +(4.29773 + 2.06968i) q^{43} +(-0.0928945 + 0.136251i) q^{44} +(0.459591 + 2.72597i) q^{45} +(-0.156431 + 2.08742i) q^{46} +(3.79145 - 9.66046i) q^{47} +(-8.17677 - 2.78145i) q^{48} +(-6.14404 + 3.35422i) q^{49} +7.04902i q^{50} +(-4.35057 + 5.14546i) q^{51} +(-2.63790 - 0.197683i) q^{52} +(1.45148 - 1.56432i) q^{53} +(8.82355 + 0.0993532i) q^{54} +(0.0745929 - 0.154894i) q^{55} +(3.83980 - 3.22541i) q^{56} +(-0.518301 - 4.26564i) q^{57} +(-15.8191 - 2.38434i) q^{58} +(-8.18966 + 5.58362i) q^{59} +(-1.38837 - 0.250183i) q^{60} +(-3.96416 - 4.27234i) q^{61} +(-8.10576 - 10.1643i) q^{62} +(-4.18595 + 6.74372i) q^{63} +(-1.26567 + 1.58709i) q^{64} +(2.75010 - 0.206091i) q^{65} +(-0.444359 - 0.322009i) q^{66} +(5.08746 - 8.81174i) q^{67} +(-1.71929 - 2.97790i) q^{68} +(-2.12358 - 0.220549i) q^{69} +(2.73790 - 3.10570i) q^{70} +(0.367586 + 0.0838990i) q^{71} +(1.98483 - 5.32849i) q^{72} +(-7.84877 + 3.08042i) q^{73} +(-5.11773 + 2.00856i) q^{74} +(-7.18657 - 0.206214i) q^{75} +(2.13784 + 0.487948i) q^{76} +(0.454752 - 0.191975i) q^{77} +(0.909358 - 8.75584i) q^{78} +(7.67502 + 13.2935i) q^{79} +(-2.29749 + 3.97937i) q^{80} +(-0.359418 + 8.99282i) q^{81} +(-10.1557 + 0.761064i) q^{82} +(0.837332 - 1.04998i) q^{83} +(-2.51519 - 3.17493i) q^{84} +(2.23511 + 2.80273i) q^{85} +(-5.50983 - 5.93818i) q^{86} +(2.89365 - 16.0580i) q^{87} +(-0.292173 + 0.199201i) q^{88} +(-5.96918 - 0.899709i) q^{89} +(0.964727 - 4.59438i) q^{90} +(6.31437 + 4.77774i) q^{91} +(0.472723 - 0.981620i) q^{92} +(10.5998 - 7.96658i) q^{93} +(-11.9871 + 12.9191i) q^{94} +(-2.27969 - 0.170839i) q^{95} +(6.18644 + 5.23073i) q^{96} -5.60940i q^{97} +(11.7936 - 1.49049i) q^{98} +(0.341292 - 0.443610i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 192 q - 14 q^{3} - 44 q^{4} - 28 q^{6} - 28 q^{7} - 26 q^{9} - 28 q^{10} + 21 q^{12} - 28 q^{13} - 10 q^{15} - 20 q^{16} - 3 q^{18} - 42 q^{19} - 14 q^{21} - 34 q^{22} - 14 q^{24} - 16 q^{25} + 7 q^{27}+ \cdots - 136 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/147\mathbb{Z}\right)^\times\).

\(n\) \(50\) \(52\)
\(\chi(n)\) \(-1\) \(e\left(\frac{1}{42}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.58081 − 0.620423i −1.11780 − 0.438705i −0.266739 − 0.963769i \(-0.585946\pi\)
−0.851063 + 0.525064i \(0.824041\pi\)
\(3\) 0.678774 − 1.59351i 0.391891 − 0.920012i
\(4\) 0.647935 + 0.601196i 0.323967 + 0.300598i
\(5\) −0.761363 − 0.519089i −0.340492 − 0.232143i 0.380986 − 0.924581i \(-0.375585\pi\)
−0.721478 + 0.692437i \(0.756537\pi\)
\(6\) −2.06166 + 2.09791i −0.841670 + 0.856467i
\(7\) −0.654203 − 2.56359i −0.247265 − 0.968948i
\(8\) 0.822376 + 1.70768i 0.290754 + 0.603757i
\(9\) −2.07853 − 2.16326i −0.692844 − 0.721088i
\(10\) 0.881517 + 1.29295i 0.278760 + 0.408866i
\(11\) 0.0278066 + 0.184485i 0.00838400 + 0.0556242i 0.992632 − 0.121171i \(-0.0386650\pi\)
−0.984248 + 0.176795i \(0.943427\pi\)
\(12\) 1.39781 − 0.624413i 0.403513 − 0.180252i
\(13\) −2.33987 + 1.86598i −0.648962 + 0.517530i −0.891738 − 0.452553i \(-0.850513\pi\)
0.242776 + 0.970082i \(0.421942\pi\)
\(14\) −0.556341 + 4.45844i −0.148688 + 1.19157i
\(15\) −1.34397 + 0.860894i −0.347010 + 0.222282i
\(16\) −0.372643 − 4.97257i −0.0931607 − 1.24314i
\(17\) −3.71746 − 1.14668i −0.901616 − 0.278112i −0.190916 − 0.981606i \(-0.561146\pi\)
−0.710701 + 0.703495i \(0.751622\pi\)
\(18\) 1.94363 + 4.70928i 0.458117 + 1.10999i
\(19\) 2.14851 − 1.24044i 0.492901 − 0.284577i −0.232876 − 0.972506i \(-0.574814\pi\)
0.725777 + 0.687930i \(0.241480\pi\)
\(20\) −0.181240 − 0.794064i −0.0405265 − 0.177558i
\(21\) −4.52916 − 0.697625i −0.988344 − 0.152234i
\(22\) 0.0705014 − 0.308887i 0.0150310 − 0.0658549i
\(23\) −0.363328 − 1.17788i −0.0757590 − 0.245605i 0.909770 − 0.415113i \(-0.136258\pi\)
−0.985529 + 0.169509i \(0.945782\pi\)
\(24\) 3.27941 − 0.151332i 0.669407 − 0.0308905i
\(25\) −1.51648 − 3.86394i −0.303297 − 0.772788i
\(26\) 4.85658 − 1.49806i 0.952454 − 0.293793i
\(27\) −4.85803 + 1.84379i −0.934928 + 0.354837i
\(28\) 1.11734 − 2.05435i 0.211158 − 0.388235i
\(29\) 9.18424 − 2.09624i 1.70547 − 0.389262i 0.744873 − 0.667207i \(-0.232510\pi\)
0.960597 + 0.277945i \(0.0896531\pi\)
\(30\) 2.65867 − 0.527084i 0.485405 − 0.0962319i
\(31\) 6.62989 + 3.82777i 1.19076 + 0.687487i 0.958480 − 0.285161i \(-0.0920471\pi\)
0.232283 + 0.972648i \(0.425380\pi\)
\(32\) −1.37867 + 4.46954i −0.243717 + 0.790111i
\(33\) 0.312852 + 0.0809134i 0.0544605 + 0.0140852i
\(34\) 5.16517 + 4.11909i 0.885820 + 0.706418i
\(35\) −0.832647 + 2.29142i −0.140743 + 0.387320i
\(36\) −0.0462080 − 2.65126i −0.00770133 − 0.441876i
\(37\) 2.37319 − 2.20200i 0.390150 − 0.362006i −0.460572 − 0.887622i \(-0.652356\pi\)
0.850722 + 0.525616i \(0.176165\pi\)
\(38\) −4.16598 + 0.627921i −0.675811 + 0.101862i
\(39\) 1.38521 + 4.99517i 0.221812 + 0.799868i
\(40\) 0.260311 − 1.72705i 0.0411588 − 0.273071i
\(41\) 5.40314 − 2.60202i 0.843829 − 0.406367i 0.0385457 − 0.999257i \(-0.487727\pi\)
0.805283 + 0.592890i \(0.202013\pi\)
\(42\) 6.72693 + 3.91281i 1.03799 + 0.603759i
\(43\) 4.29773 + 2.06968i 0.655398 + 0.315623i 0.731866 − 0.681449i \(-0.238650\pi\)
−0.0764674 + 0.997072i \(0.524364\pi\)
\(44\) −0.0928945 + 0.136251i −0.0140044 + 0.0205406i
\(45\) 0.459591 + 2.72597i 0.0685118 + 0.406364i
\(46\) −0.156431 + 2.08742i −0.0230644 + 0.307773i
\(47\) 3.79145 − 9.66046i 0.553040 − 1.40912i −0.331854 − 0.943331i \(-0.607674\pi\)
0.884894 − 0.465792i \(-0.154231\pi\)
\(48\) −8.17677 − 2.78145i −1.18022 − 0.401467i
\(49\) −6.14404 + 3.35422i −0.877720 + 0.479175i
\(50\) 7.04902i 0.996882i
\(51\) −4.35057 + 5.14546i −0.609201 + 0.720508i
\(52\) −2.63790 − 0.197683i −0.365811 − 0.0274137i
\(53\) 1.45148 − 1.56432i 0.199376 − 0.214876i −0.625448 − 0.780266i \(-0.715084\pi\)
0.824824 + 0.565390i \(0.191274\pi\)
\(54\) 8.82355 + 0.0993532i 1.20073 + 0.0135203i
\(55\) 0.0745929 − 0.154894i 0.0100581 − 0.0208859i
\(56\) 3.83980 − 3.22541i 0.513115 − 0.431014i
\(57\) −0.518301 − 4.26564i −0.0686506 − 0.564998i
\(58\) −15.8191 − 2.38434i −2.07715 − 0.313080i
\(59\) −8.18966 + 5.58362i −1.06620 + 0.726925i −0.963479 − 0.267785i \(-0.913708\pi\)
−0.102724 + 0.994710i \(0.532756\pi\)
\(60\) −1.38837 − 0.250183i −0.179237 − 0.0322985i
\(61\) −3.96416 − 4.27234i −0.507558 − 0.547018i 0.426337 − 0.904564i \(-0.359804\pi\)
−0.933895 + 0.357547i \(0.883613\pi\)
\(62\) −8.10576 − 10.1643i −1.02943 − 1.29087i
\(63\) −4.18595 + 6.74372i −0.527380 + 0.849629i
\(64\) −1.26567 + 1.58709i −0.158208 + 0.198387i
\(65\) 2.75010 − 0.206091i 0.341107 − 0.0255625i
\(66\) −0.444359 − 0.322009i −0.0546968 − 0.0396366i
\(67\) 5.08746 − 8.81174i 0.621532 − 1.07653i −0.367668 − 0.929957i \(-0.619844\pi\)
0.989201 − 0.146568i \(-0.0468228\pi\)
\(68\) −1.71929 − 2.97790i −0.208495 − 0.361123i
\(69\) −2.12358 − 0.220549i −0.255649 − 0.0265510i
\(70\) 2.73790 − 3.10570i 0.327242 − 0.371202i
\(71\) 0.367586 + 0.0838990i 0.0436244 + 0.00995698i 0.244277 − 0.969705i \(-0.421449\pi\)
−0.200653 + 0.979662i \(0.564306\pi\)
\(72\) 1.98483 − 5.32849i 0.233915 − 0.627968i
\(73\) −7.84877 + 3.08042i −0.918630 + 0.360536i −0.777085 − 0.629396i \(-0.783302\pi\)
−0.141545 + 0.989932i \(0.545207\pi\)
\(74\) −5.11773 + 2.00856i −0.594925 + 0.233491i
\(75\) −7.18657 − 0.206214i −0.829833 − 0.0238115i
\(76\) 2.13784 + 0.487948i 0.245227 + 0.0559715i
\(77\) 0.454752 − 0.191975i 0.0518238 − 0.0218776i
\(78\) 0.909358 − 8.75584i 0.102965 − 0.991404i
\(79\) 7.67502 + 13.2935i 0.863507 + 1.49564i 0.868522 + 0.495650i \(0.165070\pi\)
−0.00501530 + 0.999987i \(0.501596\pi\)
\(80\) −2.29749 + 3.97937i −0.256867 + 0.444907i
\(81\) −0.359418 + 8.99282i −0.0399354 + 0.999202i
\(82\) −10.1557 + 0.761064i −1.12151 + 0.0840455i
\(83\) 0.837332 − 1.04998i 0.0919091 − 0.115250i −0.733750 − 0.679419i \(-0.762232\pi\)
0.825660 + 0.564169i \(0.190803\pi\)
\(84\) −2.51519 − 3.17493i −0.274430 − 0.346413i
\(85\) 2.23511 + 2.80273i 0.242431 + 0.303999i
\(86\) −5.50983 − 5.93818i −0.594140 − 0.640331i
\(87\) 2.89365 − 16.0580i 0.310231 − 1.72160i
\(88\) −0.292173 + 0.199201i −0.0311458 + 0.0212348i
\(89\) −5.96918 − 0.899709i −0.632732 − 0.0953690i −0.175158 − 0.984540i \(-0.556044\pi\)
−0.457574 + 0.889171i \(0.651282\pi\)
\(90\) 0.964727 − 4.59438i 0.101691 − 0.484291i
\(91\) 6.31437 + 4.77774i 0.661925 + 0.500843i
\(92\) 0.472723 − 0.981620i 0.0492848 − 0.102341i
\(93\) 10.5998 − 7.96658i 1.09914 − 0.826096i
\(94\) −11.9871 + 12.9191i −1.23638 + 1.33250i
\(95\) −2.27969 − 0.170839i −0.233892 − 0.0175278i
\(96\) 6.18644 + 5.23073i 0.631401 + 0.533859i
\(97\) − 5.60940i − 0.569548i −0.958595 − 0.284774i \(-0.908081\pi\)
0.958595 − 0.284774i \(-0.0919185\pi\)
\(98\) 11.7936 − 1.49049i 1.19133 − 0.150562i
\(99\) 0.341292 − 0.443610i 0.0343011 − 0.0445845i
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 147.2.o.b.101.4 ✓ 192
3.2 odd 2 inner 147.2.o.b.101.13 yes 192
49.33 odd 42 inner 147.2.o.b.131.13 yes 192
147.131 even 42 inner 147.2.o.b.131.4 yes 192
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
147.2.o.b.101.4 ✓ 192 1.1 even 1 trivial
147.2.o.b.101.13 yes 192 3.2 odd 2 inner
147.2.o.b.131.4 yes 192 147.131 even 42 inner
147.2.o.b.131.13 yes 192 49.33 odd 42 inner