Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [441,2,Mod(214,441)] 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("441.214"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(441, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([4, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 441.h (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,-2,2,6,5] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.52140272914\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.309123.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 3x^{5} + 10x^{4} - 15x^{3} + 19x^{2} - 12x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 63)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 373.2
Root \(0.500000 - 1.41036i\) of defining polynomial
Character \(\chi\) \(=\) 441.373
Dual form 441.2.h.c.214.2

$q$-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-0.239123 q^{2} +(-1.09097 + 1.34528i) q^{3} -1.94282 q^{4} +(-0.590972 - 1.02359i) q^{5} +(0.260877 - 0.321688i) q^{6} +0.942820 q^{8} +(-0.619562 - 2.93533i) q^{9} +(0.141315 + 0.244765i) q^{10} +(1.85185 - 3.20750i) q^{11} +(2.11956 - 2.61364i) q^{12} +(-0.500000 + 0.866025i) q^{13} +(2.02175 + 0.321688i) q^{15} +3.66019 q^{16} +(3.47141 + 6.01266i) q^{17} +(0.148152 + 0.701905i) q^{18} +(-0.971410 + 1.68253i) q^{19} +(1.14815 + 1.98866i) q^{20} +(-0.442820 + 0.766987i) q^{22} +(2.80150 + 4.85235i) q^{23} +(-1.02859 + 1.26836i) q^{24} +(1.80150 - 3.12030i) q^{25} +(0.119562 - 0.207087i) q^{26} +(4.62476 + 2.36887i) q^{27} +(-0.119562 - 0.207087i) q^{29} +(-0.483448 - 0.0769231i) q^{30} +1.66019 q^{31} -2.76088 q^{32} +(2.29467 + 5.99054i) q^{33} +(-0.830095 - 1.43777i) q^{34} +(1.20370 + 5.70281i) q^{36} +(4.77292 - 8.26693i) q^{37} +(0.232287 - 0.402332i) q^{38} +(-0.619562 - 1.61745i) q^{39} +(-0.557180 - 0.965064i) q^{40} +(5.09097 - 8.81782i) q^{41} +(-1.11273 - 1.92730i) q^{43} +(-3.59781 + 6.23159i) q^{44} +(-2.63844 + 2.36887i) q^{45} +(-0.669905 - 1.16031i) q^{46} +5.82846 q^{47} +(-3.99316 + 4.92398i) q^{48} +(-0.430782 + 0.746136i) q^{50} +(-11.8759 - 1.88962i) q^{51} +(0.971410 - 1.68253i) q^{52} +(5.80150 + 10.0485i) q^{53} +(-1.10589 - 0.566453i) q^{54} -4.37756 q^{55} +(-1.20370 - 3.14241i) q^{57} +(0.0285900 + 0.0495193i) q^{58} +2.60301 q^{59} +(-3.92790 - 0.624982i) q^{60} -7.60301 q^{61} -0.396990 q^{62} -6.66019 q^{64} +1.18194 q^{65} +(-0.548709 - 1.43248i) q^{66} +3.50808 q^{67} +(-6.74433 - 11.6815i) q^{68} +(-9.58414 - 1.52496i) q^{69} +8.60301 q^{71} +(-0.584135 - 2.76748i) q^{72} +(-7.57442 - 13.1193i) q^{73} +(-1.14132 + 1.97682i) q^{74} +(2.23229 + 5.82769i) q^{75} +(1.88727 - 3.26886i) q^{76} +(0.148152 + 0.386770i) q^{78} +7.37756 q^{79} +(-2.16307 - 3.74654i) q^{80} +(-8.23229 + 3.63723i) q^{81} +(-1.21737 + 2.10855i) q^{82} +(3.47141 + 6.01266i) q^{83} +(4.10301 - 7.10662i) q^{85} +(0.266078 + 0.460861i) q^{86} +(0.409028 + 0.0650819i) q^{87} +(1.74596 - 3.02409i) q^{88} +(-1.37360 + 2.37915i) q^{89} +(0.630912 - 0.566453i) q^{90} +(-5.44282 - 9.42724i) q^{92} +(-1.81122 + 2.23342i) q^{93} -1.39372 q^{94} +2.29630 q^{95} +(3.01204 - 3.71415i) q^{96} +(-3.58414 - 6.20790i) q^{97} +(-10.5624 - 3.44854i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 2 q^{2} + 2 q^{3} + 6 q^{4} + 5 q^{5} + q^{6} - 12 q^{8} - 4 q^{9} + 2 q^{11} + 13 q^{12} - 3 q^{13} + 11 q^{15} + 6 q^{16} + 12 q^{17} + 10 q^{18} + 3 q^{19} + 16 q^{20} + 15 q^{22} - 15 q^{24}+ \cdots - 46 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/441\mathbb{Z}\right)^\times\).

\(n\) \(199\) \(344\)
\(\chi(n)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.239123 −0.169086 −0.0845428 − 0.996420i \(-0.526943\pi\)
−0.0845428 + 0.996420i \(0.526943\pi\)
\(3\) −1.09097 + 1.34528i −0.629873 + 0.776698i
\(4\) −1.94282 −0.971410
\(5\) −0.590972 − 1.02359i −0.264291 − 0.457765i 0.703087 − 0.711104i \(-0.251804\pi\)
−0.967378 + 0.253339i \(0.918471\pi\)
\(6\) 0.260877 − 0.321688i 0.106502 − 0.131329i
\(7\) 0 0
\(8\) 0.942820 0.333337
\(9\) −0.619562 − 2.93533i −0.206521 − 0.978442i
\(10\) 0.141315 + 0.244765i 0.0446878 + 0.0774015i
\(11\) 1.85185 − 3.20750i 0.558353 − 0.967096i −0.439281 − 0.898350i \(-0.644767\pi\)
0.997634 − 0.0687465i \(-0.0219000\pi\)
\(12\) 2.11956 − 2.61364i 0.611865 − 0.754493i
\(13\) −0.500000 + 0.866025i −0.138675 + 0.240192i −0.926995 − 0.375073i \(-0.877618\pi\)
0.788320 + 0.615265i \(0.210951\pi\)
\(14\) 0 0
\(15\) 2.02175 + 0.321688i 0.522014 + 0.0830595i
\(16\) 3.66019 0.915047
\(17\) 3.47141 + 6.01266i 0.841941 + 1.45828i 0.888252 + 0.459357i \(0.151920\pi\)
−0.0463112 + 0.998927i \(0.514747\pi\)
\(18\) 0.148152 + 0.701905i 0.0349197 + 0.165441i
\(19\) −0.971410 + 1.68253i −0.222857 + 0.385999i −0.955674 − 0.294426i \(-0.904872\pi\)
0.732818 + 0.680425i \(0.238205\pi\)
\(20\) 1.14815 + 1.98866i 0.256735 + 0.444677i
\(21\) 0 0
\(22\) −0.442820 + 0.766987i −0.0944096 + 0.163522i
\(23\) 2.80150 + 4.85235i 0.584154 + 1.01178i 0.994980 + 0.100071i \(0.0319070\pi\)
−0.410826 + 0.911714i \(0.634760\pi\)
\(24\) −1.02859 + 1.26836i −0.209960 + 0.258902i
\(25\) 1.80150 − 3.12030i 0.360301 − 0.624060i
\(26\) 0.119562 − 0.207087i 0.0234480 − 0.0406131i
\(27\) 4.62476 + 2.36887i 0.890036 + 0.455890i
\(28\) 0 0
\(29\) −0.119562 − 0.207087i −0.0222020 − 0.0384551i 0.854711 − 0.519104i \(-0.173734\pi\)
−0.876913 + 0.480649i \(0.840401\pi\)
\(30\) −0.483448 − 0.0769231i −0.0882652 − 0.0140442i
\(31\) 1.66019 0.298179 0.149089 − 0.988824i \(-0.452366\pi\)
0.149089 + 0.988824i \(0.452366\pi\)
\(32\) −2.76088 −0.488059
\(33\) 2.29467 + 5.99054i 0.399451 + 1.04282i
\(34\) −0.830095 − 1.43777i −0.142360 − 0.246575i
\(35\) 0 0
\(36\) 1.20370 + 5.70281i 0.200616 + 0.950469i
\(37\) 4.77292 − 8.26693i 0.784662 − 1.35908i −0.144538 − 0.989499i \(-0.546170\pi\)
0.929201 − 0.369576i \(-0.120497\pi\)
\(38\) 0.232287 − 0.402332i 0.0376819 − 0.0652669i
\(39\) −0.619562 − 1.61745i −0.0992093 − 0.258999i
\(40\) −0.557180 − 0.965064i −0.0880979 − 0.152590i
\(41\) 5.09097 − 8.81782i 0.795076 − 1.37711i −0.127715 − 0.991811i \(-0.540764\pi\)
0.922791 − 0.385301i \(-0.125903\pi\)
\(42\) 0 0
\(43\) −1.11273 − 1.92730i −0.169689 − 0.293910i 0.768622 − 0.639704i \(-0.220943\pi\)
−0.938311 + 0.345794i \(0.887610\pi\)
\(44\) −3.59781 + 6.23159i −0.542390 + 0.939447i
\(45\) −2.63844 + 2.36887i −0.393315 + 0.353131i
\(46\) −0.669905 − 1.16031i −0.0987721 − 0.171078i
\(47\) 5.82846 0.850168 0.425084 − 0.905154i \(-0.360245\pi\)
0.425084 + 0.905154i \(0.360245\pi\)
\(48\) −3.99316 + 4.92398i −0.576364 + 0.710716i
\(49\) 0 0
\(50\) −0.430782 + 0.746136i −0.0609217 + 0.105520i
\(51\) −11.8759 − 1.88962i −1.66296 − 0.264599i
\(52\) 0.971410 − 1.68253i 0.134710 − 0.233325i
\(53\) 5.80150 + 10.0485i 0.796898 + 1.38027i 0.921627 + 0.388077i \(0.126861\pi\)
−0.124729 + 0.992191i \(0.539806\pi\)
\(54\) −1.10589 − 0.566453i −0.150492 − 0.0770845i
\(55\) −4.37756 −0.590270
\(56\) 0 0
\(57\) −1.20370 − 3.14241i −0.159434 − 0.416223i
\(58\) 0.0285900 + 0.0495193i 0.00375405 + 0.00650220i
\(59\) 2.60301 0.338883 0.169442 − 0.985540i \(-0.445804\pi\)
0.169442 + 0.985540i \(0.445804\pi\)
\(60\) −3.92790 − 0.624982i −0.507090 − 0.0806848i
\(61\) −7.60301 −0.973466 −0.486733 − 0.873551i \(-0.661811\pi\)
−0.486733 + 0.873551i \(0.661811\pi\)
\(62\) −0.396990 −0.0504178
\(63\) 0 0
\(64\) −6.66019 −0.832524
\(65\) 1.18194 0.146602
\(66\) −0.548709 − 1.43248i −0.0675414 − 0.176326i
\(67\) 3.50808 0.428580 0.214290 − 0.976770i \(-0.431256\pi\)
0.214290 + 0.976770i \(0.431256\pi\)
\(68\) −6.74433 − 11.6815i −0.817870 − 1.41659i
\(69\) −9.58414 − 1.52496i −1.15379 − 0.183584i
\(70\) 0 0
\(71\) 8.60301 1.02099 0.510495 − 0.859881i \(-0.329462\pi\)
0.510495 + 0.859881i \(0.329462\pi\)
\(72\) −0.584135 − 2.76748i −0.0688410 − 0.326151i
\(73\) −7.57442 − 13.1193i −0.886519 − 1.53550i −0.843963 − 0.536402i \(-0.819783\pi\)
−0.0425559 − 0.999094i \(-0.513550\pi\)
\(74\) −1.14132 + 1.97682i −0.132675 + 0.229800i
\(75\) 2.23229 + 5.82769i 0.257762 + 0.672923i
\(76\) 1.88727 − 3.26886i 0.216485 − 0.374963i
\(77\) 0 0
\(78\) 0.148152 + 0.386770i 0.0167749 + 0.0437931i
\(79\) 7.37756 0.830040 0.415020 − 0.909812i \(-0.363775\pi\)
0.415020 + 0.909812i \(0.363775\pi\)
\(80\) −2.16307 − 3.74654i −0.241838 − 0.418876i
\(81\) −8.23229 + 3.63723i −0.914699 + 0.404137i
\(82\) −1.21737 + 2.10855i −0.134436 + 0.232850i
\(83\) 3.47141 + 6.01266i 0.381037 + 0.659975i 0.991211 − 0.132292i \(-0.0422338\pi\)
−0.610174 + 0.792267i \(0.708900\pi\)
\(84\) 0 0
\(85\) 4.10301 − 7.10662i 0.445034 − 0.770821i
\(86\) 0.266078 + 0.460861i 0.0286920 + 0.0496960i
\(87\) 0.409028 + 0.0650819i 0.0438524 + 0.00697751i
\(88\) 1.74596 − 3.02409i 0.186120 − 0.322369i
\(89\) −1.37360 + 2.37915i −0.145602 + 0.252189i −0.929597 − 0.368577i \(-0.879845\pi\)
0.783996 + 0.620766i \(0.213178\pi\)
\(90\) 0.630912 − 0.566453i 0.0665039 − 0.0597094i
\(91\) 0 0
\(92\) −5.44282 − 9.42724i −0.567453 − 0.982858i
\(93\) −1.81122 + 2.23342i −0.187815 + 0.231595i
\(94\) −1.39372 −0.143751
\(95\) 2.29630 0.235596
\(96\) 3.01204 − 3.71415i 0.307415 − 0.379074i
\(97\) −3.58414 − 6.20790i −0.363914 − 0.630317i 0.624687 − 0.780875i \(-0.285226\pi\)
−0.988601 + 0.150558i \(0.951893\pi\)
\(98\) 0 0
\(99\) −10.5624 − 3.44854i −1.06156 − 0.346591i
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 441.2.h.c.373.2 6
3.2 odd 2 1323.2.h.d.226.2 6
7.2 even 3 63.2.f.b.22.2 ✓ 6
7.3 odd 6 441.2.g.d.67.2 6
7.4 even 3 441.2.g.e.67.2 6
7.5 odd 6 441.2.f.d.148.2 6
7.6 odd 2 441.2.h.b.373.2 6
9.2 odd 6 1323.2.g.c.667.2 6
9.7 even 3 441.2.g.e.79.2 6
21.2 odd 6 189.2.f.a.64.2 6
21.5 even 6 1323.2.f.c.442.2 6
21.11 odd 6 1323.2.g.c.361.2 6
21.17 even 6 1323.2.g.b.361.2 6
21.20 even 2 1323.2.h.e.226.2 6
28.23 odd 6 1008.2.r.k.337.2 6
63.2 odd 6 189.2.f.a.127.2 6
63.5 even 6 3969.2.a.p.1.2 3
63.11 odd 6 1323.2.h.d.802.2 6
63.16 even 3 63.2.f.b.43.2 yes 6
63.20 even 6 1323.2.g.b.667.2 6
63.23 odd 6 567.2.a.g.1.2 3
63.25 even 3 inner 441.2.h.c.214.2 6
63.34 odd 6 441.2.g.d.79.2 6
63.38 even 6 1323.2.h.e.802.2 6
63.40 odd 6 3969.2.a.m.1.2 3
63.47 even 6 1323.2.f.c.883.2 6
63.52 odd 6 441.2.h.b.214.2 6
63.58 even 3 567.2.a.d.1.2 3
63.61 odd 6 441.2.f.d.295.2 6
84.23 even 6 3024.2.r.g.1009.3 6
252.23 even 6 9072.2.a.cd.1.1 3
252.79 odd 6 1008.2.r.k.673.2 6
252.191 even 6 3024.2.r.g.2017.3 6
252.247 odd 6 9072.2.a.bq.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
63.2.f.b.22.2 ✓ 6 7.2 even 3
63.2.f.b.43.2 yes 6 63.16 even 3
189.2.f.a.64.2 6 21.2 odd 6
189.2.f.a.127.2 6 63.2 odd 6
441.2.f.d.148.2 6 7.5 odd 6
441.2.f.d.295.2 6 63.61 odd 6
441.2.g.d.67.2 6 7.3 odd 6
441.2.g.d.79.2 6 63.34 odd 6
441.2.g.e.67.2 6 7.4 even 3
441.2.g.e.79.2 6 9.7 even 3
441.2.h.b.214.2 6 63.52 odd 6
441.2.h.b.373.2 6 7.6 odd 2
441.2.h.c.214.2 6 63.25 even 3 inner
441.2.h.c.373.2 6 1.1 even 1 trivial
567.2.a.d.1.2 3 63.58 even 3
567.2.a.g.1.2 3 63.23 odd 6
1008.2.r.k.337.2 6 28.23 odd 6
1008.2.r.k.673.2 6 252.79 odd 6
1323.2.f.c.442.2 6 21.5 even 6
1323.2.f.c.883.2 6 63.47 even 6
1323.2.g.b.361.2 6 21.17 even 6
1323.2.g.b.667.2 6 63.20 even 6
1323.2.g.c.361.2 6 21.11 odd 6
1323.2.g.c.667.2 6 9.2 odd 6
1323.2.h.d.226.2 6 3.2 odd 2
1323.2.h.d.802.2 6 63.11 odd 6
1323.2.h.e.226.2 6 21.20 even 2
1323.2.h.e.802.2 6 63.38 even 6
3024.2.r.g.1009.3 6 84.23 even 6
3024.2.r.g.2017.3 6 252.191 even 6
3969.2.a.m.1.2 3 63.40 odd 6
3969.2.a.p.1.2 3 63.5 even 6
9072.2.a.bq.1.3 3 252.247 odd 6
9072.2.a.cd.1.1 3 252.23 even 6