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 214.1
Root \(0.500000 - 2.05195i\) of defining polynomial
Character \(\chi\) \(=\) 441.214
Dual form 441.2.h.b.373.1

$q$-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-2.46050 q^{2} +(-0.796790 + 1.53790i) q^{3} +4.05408 q^{4} +(-1.29679 + 2.24611i) q^{5} +(1.96050 - 3.78400i) q^{6} -5.05408 q^{8} +(-1.73025 - 2.45076i) q^{9} +(3.19076 - 5.52655i) q^{10} +(-2.25729 - 3.90975i) q^{11} +(-3.23025 + 6.23476i) q^{12} +(0.500000 + 0.866025i) q^{13} +(-2.42101 - 3.78400i) q^{15} +4.32743 q^{16} +(-0.472958 + 0.819187i) q^{17} +(4.25729 + 6.03011i) q^{18} +(-2.02704 - 3.51094i) q^{19} +(-5.25729 + 9.10590i) q^{20} +(5.55408 + 9.61996i) q^{22} +(0.136673 - 0.236725i) q^{23} +(4.02704 - 7.77266i) q^{24} +(-0.863327 - 1.49533i) q^{25} +(-1.23025 - 2.13086i) q^{26} +(5.14766 - 0.708209i) q^{27} +(-1.23025 + 2.13086i) q^{29} +(5.95691 + 9.31056i) q^{30} -2.32743 q^{31} -0.539495 q^{32} +(7.81138 - 0.356238i) q^{33} +(1.16372 - 2.01561i) q^{34} +(-7.01459 - 9.93559i) q^{36} +(-0.890369 - 1.54216i) q^{37} +(4.98755 + 8.63868i) q^{38} +(-1.73025 + 0.0789082i) q^{39} +(6.55408 - 11.3520i) q^{40} +(-3.20321 - 5.54812i) q^{41} +(5.21780 - 9.03749i) q^{43} +(-9.15126 - 15.8505i) q^{44} +(7.74844 - 0.708209i) q^{45} +(-0.336285 + 0.582462i) q^{46} +12.1623 q^{47} +(-3.44805 + 6.65514i) q^{48} +(2.12422 + 3.67926i) q^{50} +(-0.882977 - 1.38008i) q^{51} +(2.02704 + 3.51094i) q^{52} +(3.13667 - 5.43288i) q^{53} +(-12.6659 + 1.74255i) q^{54} +11.7089 q^{55} +(7.01459 - 0.319901i) q^{57} +(3.02704 - 5.24299i) q^{58} +2.72665 q^{59} +(-9.81498 - 15.3407i) q^{60} +2.27335 q^{61} +5.72665 q^{62} -7.32743 q^{64} -2.59358 q^{65} +(-19.2199 + 0.876526i) q^{66} -15.8171 q^{67} +(-1.91741 + 3.32105i) q^{68} +(0.255158 + 0.398809i) q^{69} +3.27335 q^{71} +(8.74484 + 12.3863i) q^{72} +(-0.753696 + 1.30544i) q^{73} +(2.19076 + 3.79450i) q^{74} +(2.98755 - 0.136247i) q^{75} +(-8.21780 - 14.2336i) q^{76} +(4.25729 - 0.194154i) q^{78} +14.7089 q^{79} +(-5.61177 + 9.71987i) q^{80} +(-3.01245 + 8.48087i) q^{81} +(7.88151 + 13.6512i) q^{82} +(-0.472958 + 0.819187i) q^{83} +(-1.22665 - 2.12463i) q^{85} +(-12.8384 + 22.2368i) q^{86} +(-2.29679 - 3.58985i) q^{87} +(11.4086 + 19.7602i) q^{88} +(-7.17830 - 12.4332i) q^{89} +(-19.0651 + 1.74255i) q^{90} +(0.554084 - 0.959702i) q^{92} +(1.85447 - 3.57935i) q^{93} -29.9253 q^{94} +10.5146 q^{95} +(0.429864 - 0.829688i) q^{96} +(-5.74484 + 9.95036i) q^{97} +(-5.67617 + 12.2969i) 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 \(\left(\mathbb{Z}/441\mathbb{Z}\right)^\times\).

\(n\) \(199\) \(344\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{2}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.46050 −1.73984 −0.869920 − 0.493193i \(-0.835830\pi\)
−0.869920 + 0.493193i \(0.835830\pi\)
\(3\) −0.796790 + 1.53790i −0.460027 + 0.887905i
\(4\) 4.05408 2.02704
\(5\) −1.29679 + 2.24611i −0.579942 + 1.00449i 0.415543 + 0.909573i \(0.363591\pi\)
−0.995485 + 0.0949156i \(0.969742\pi\)
\(6\) 1.96050 − 3.78400i 0.800373 − 1.54481i
\(7\) 0 0
\(8\) −5.05408 −1.78689
\(9\) −1.73025 − 2.45076i −0.576751 − 0.816920i
\(10\) 3.19076 − 5.52655i 1.00901 − 1.74765i
\(11\) −2.25729 − 3.90975i −0.680600 − 1.17883i −0.974798 − 0.223089i \(-0.928386\pi\)
0.294198 − 0.955744i \(-0.404947\pi\)
\(12\) −3.23025 + 6.23476i −0.932494 + 1.79982i
\(13\) 0.500000 + 0.866025i 0.138675 + 0.240192i 0.926995 − 0.375073i \(-0.122382\pi\)
−0.788320 + 0.615265i \(0.789049\pi\)
\(14\) 0 0
\(15\) −2.42101 − 3.78400i −0.625102 − 0.977025i
\(16\) 4.32743 1.08186
\(17\) −0.472958 + 0.819187i −0.114709 + 0.198682i −0.917663 − 0.397359i \(-0.869927\pi\)
0.802954 + 0.596041i \(0.203260\pi\)
\(18\) 4.25729 + 6.03011i 1.00345 + 1.42131i
\(19\) −2.02704 − 3.51094i −0.465035 − 0.805465i 0.534168 − 0.845378i \(-0.320625\pi\)
−0.999203 + 0.0399136i \(0.987292\pi\)
\(20\) −5.25729 + 9.10590i −1.17557 + 2.03614i
\(21\) 0 0
\(22\) 5.55408 + 9.61996i 1.18413 + 2.05098i
\(23\) 0.136673 − 0.236725i 0.0284983 − 0.0493605i −0.851425 − 0.524477i \(-0.824261\pi\)
0.879923 + 0.475117i \(0.157594\pi\)
\(24\) 4.02704 − 7.77266i 0.822017 − 1.58659i
\(25\) −0.863327 − 1.49533i −0.172665 − 0.299065i
\(26\) −1.23025 − 2.13086i −0.241272 − 0.417896i
\(27\) 5.14766 − 0.708209i 0.990668 − 0.136295i
\(28\) 0 0
\(29\) −1.23025 + 2.13086i −0.228452 + 0.395691i −0.957350 − 0.288932i \(-0.906700\pi\)
0.728897 + 0.684623i \(0.240033\pi\)
\(30\) 5.95691 + 9.31056i 1.08758 + 1.69987i
\(31\) −2.32743 −0.418019 −0.209009 − 0.977914i \(-0.567024\pi\)
−0.209009 + 0.977914i \(0.567024\pi\)
\(32\) −0.539495 −0.0953702
\(33\) 7.81138 − 0.356238i 1.35979 − 0.0620131i
\(34\) 1.16372 − 2.01561i 0.199576 − 0.345675i
\(35\) 0 0
\(36\) −7.01459 − 9.93559i −1.16910 − 1.65593i
\(37\) −0.890369 − 1.54216i −0.146376 − 0.253530i 0.783510 − 0.621380i \(-0.213428\pi\)
−0.929885 + 0.367849i \(0.880094\pi\)
\(38\) 4.98755 + 8.63868i 0.809087 + 1.40138i
\(39\) −1.73025 + 0.0789082i −0.277062 + 0.0126354i
\(40\) 6.55408 − 11.3520i 1.03629 − 1.79491i
\(41\) −3.20321 − 5.54812i −0.500257 − 0.866471i −1.00000 0.000297253i \(-0.999905\pi\)
0.499743 − 0.866174i \(-0.333428\pi\)
\(42\) 0 0
\(43\) 5.21780 − 9.03749i 0.795707 − 1.37820i −0.126682 − 0.991943i \(-0.540433\pi\)
0.922389 − 0.386262i \(-0.126234\pi\)
\(44\) −9.15126 − 15.8505i −1.37960 − 2.38955i
\(45\) 7.74844 − 0.708209i 1.15507 − 0.105574i
\(46\) −0.336285 + 0.582462i −0.0495825 + 0.0858794i
\(47\) 12.1623 1.77405 0.887023 − 0.461724i \(-0.152769\pi\)
0.887023 + 0.461724i \(0.152769\pi\)
\(48\) −3.44805 + 6.65514i −0.497683 + 0.960587i
\(49\) 0 0
\(50\) 2.12422 + 3.67926i 0.300410 + 0.520326i
\(51\) −0.882977 − 1.38008i −0.123642 − 0.193250i
\(52\) 2.02704 + 3.51094i 0.281100 + 0.486880i
\(53\) 3.13667 − 5.43288i 0.430855 − 0.746263i −0.566092 − 0.824342i \(-0.691545\pi\)
0.996947 + 0.0780790i \(0.0248786\pi\)
\(54\) −12.6659 + 1.74255i −1.72360 + 0.237131i
\(55\) 11.7089 1.57883
\(56\) 0 0
\(57\) 7.01459 − 0.319901i 0.929105 − 0.0423719i
\(58\) 3.02704 − 5.24299i 0.397470 − 0.688438i
\(59\) 2.72665 0.354980 0.177490 − 0.984123i \(-0.443202\pi\)
0.177490 + 0.984123i \(0.443202\pi\)
\(60\) −9.81498 − 15.3407i −1.26711 − 1.98047i
\(61\) 2.27335 0.291072 0.145536 − 0.989353i \(-0.453509\pi\)
0.145536 + 0.989353i \(0.453509\pi\)
\(62\) 5.72665 0.727286
\(63\) 0 0
\(64\) −7.32743 −0.915929
\(65\) −2.59358 −0.321694
\(66\) −19.2199 + 0.876526i −2.36581 + 0.107893i
\(67\) −15.8171 −1.93237 −0.966184 − 0.257854i \(-0.916985\pi\)
−0.966184 + 0.257854i \(0.916985\pi\)
\(68\) −1.91741 + 3.32105i −0.232520 + 0.402737i
\(69\) 0.255158 + 0.398809i 0.0307175 + 0.0480110i
\(70\) 0 0
\(71\) 3.27335 0.388475 0.194237 − 0.980955i \(-0.437777\pi\)
0.194237 + 0.980955i \(0.437777\pi\)
\(72\) 8.74484 + 12.3863i 1.03059 + 1.45975i
\(73\) −0.753696 + 1.30544i −0.0882134 + 0.152790i −0.906756 − 0.421656i \(-0.861449\pi\)
0.818543 + 0.574446i \(0.194782\pi\)
\(74\) 2.19076 + 3.79450i 0.254670 + 0.441102i
\(75\) 2.98755 − 0.136247i 0.344972 − 0.0157325i
\(76\) −8.21780 − 14.2336i −0.942646 − 1.63271i
\(77\) 0 0
\(78\) 4.25729 − 0.194154i 0.482044 − 0.0219836i
\(79\) 14.7089 1.65489 0.827443 − 0.561550i \(-0.189795\pi\)
0.827443 + 0.561550i \(0.189795\pi\)
\(80\) −5.61177 + 9.71987i −0.627415 + 1.08671i
\(81\) −3.01245 + 8.48087i −0.334717 + 0.942319i
\(82\) 7.88151 + 13.6512i 0.870368 + 1.50752i
\(83\) −0.472958 + 0.819187i −0.0519139 + 0.0899175i −0.890815 − 0.454367i \(-0.849865\pi\)
0.838901 + 0.544285i \(0.183199\pi\)
\(84\) 0 0
\(85\) −1.22665 − 2.12463i −0.133049 − 0.230448i
\(86\) −12.8384 + 22.2368i −1.38440 + 2.39786i
\(87\) −2.29679 − 3.58985i −0.246242 − 0.384872i
\(88\) 11.4086 + 19.7602i 1.21616 + 2.10644i
\(89\) −7.17830 − 12.4332i −0.760899 − 1.31792i −0.942388 − 0.334522i \(-0.891425\pi\)
0.181489 − 0.983393i \(-0.441908\pi\)
\(90\) −19.0651 + 1.74255i −2.00964 + 0.183681i
\(91\) 0 0
\(92\) 0.554084 − 0.959702i 0.0577673 − 0.100056i
\(93\) 1.85447 − 3.57935i 0.192300 − 0.371161i
\(94\) −29.9253 −3.08656
\(95\) 10.5146 1.07877
\(96\) 0.429864 − 0.829688i 0.0438728 − 0.0846797i
\(97\) −5.74484 + 9.95036i −0.583300 + 1.01031i 0.411785 + 0.911281i \(0.364906\pi\)
−0.995085 + 0.0990246i \(0.968428\pi\)
\(98\) 0 0
\(99\) −5.67617 + 12.2969i −0.570476 + 1.23589i
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.b.214.1 6
3.2 odd 2 1323.2.h.e.802.3 6
7.2 even 3 441.2.g.d.79.3 6
7.3 odd 6 63.2.f.b.43.3 yes 6
7.4 even 3 441.2.f.d.295.3 6
7.5 odd 6 441.2.g.e.79.3 6
7.6 odd 2 441.2.h.c.214.1 6
9.4 even 3 441.2.g.d.67.3 6
9.5 odd 6 1323.2.g.b.361.1 6
21.2 odd 6 1323.2.g.b.667.1 6
21.5 even 6 1323.2.g.c.667.1 6
21.11 odd 6 1323.2.f.c.883.1 6
21.17 even 6 189.2.f.a.127.1 6
21.20 even 2 1323.2.h.d.802.3 6
28.3 even 6 1008.2.r.k.673.3 6
63.4 even 3 441.2.f.d.148.3 6
63.5 even 6 1323.2.h.d.226.3 6
63.11 odd 6 3969.2.a.p.1.3 3
63.13 odd 6 441.2.g.e.67.3 6
63.23 odd 6 1323.2.h.e.226.3 6
63.25 even 3 3969.2.a.m.1.1 3
63.31 odd 6 63.2.f.b.22.3 ✓ 6
63.32 odd 6 1323.2.f.c.442.1 6
63.38 even 6 567.2.a.g.1.3 3
63.40 odd 6 441.2.h.c.373.1 6
63.41 even 6 1323.2.g.c.361.1 6
63.52 odd 6 567.2.a.d.1.1 3
63.58 even 3 inner 441.2.h.b.373.1 6
63.59 even 6 189.2.f.a.64.1 6
84.59 odd 6 3024.2.r.g.2017.2 6
252.31 even 6 1008.2.r.k.337.3 6
252.59 odd 6 3024.2.r.g.1009.2 6
252.115 even 6 9072.2.a.bq.1.2 3
252.227 odd 6 9072.2.a.cd.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
63.2.f.b.22.3 ✓ 6 63.31 odd 6
63.2.f.b.43.3 yes 6 7.3 odd 6
189.2.f.a.64.1 6 63.59 even 6
189.2.f.a.127.1 6 21.17 even 6
441.2.f.d.148.3 6 63.4 even 3
441.2.f.d.295.3 6 7.4 even 3
441.2.g.d.67.3 6 9.4 even 3
441.2.g.d.79.3 6 7.2 even 3
441.2.g.e.67.3 6 63.13 odd 6
441.2.g.e.79.3 6 7.5 odd 6
441.2.h.b.214.1 6 1.1 even 1 trivial
441.2.h.b.373.1 6 63.58 even 3 inner
441.2.h.c.214.1 6 7.6 odd 2
441.2.h.c.373.1 6 63.40 odd 6
567.2.a.d.1.1 3 63.52 odd 6
567.2.a.g.1.3 3 63.38 even 6
1008.2.r.k.337.3 6 252.31 even 6
1008.2.r.k.673.3 6 28.3 even 6
1323.2.f.c.442.1 6 63.32 odd 6
1323.2.f.c.883.1 6 21.11 odd 6
1323.2.g.b.361.1 6 9.5 odd 6
1323.2.g.b.667.1 6 21.2 odd 6
1323.2.g.c.361.1 6 63.41 even 6
1323.2.g.c.667.1 6 21.5 even 6
1323.2.h.d.226.3 6 63.5 even 6
1323.2.h.d.802.3 6 21.20 even 2
1323.2.h.e.226.3 6 63.23 odd 6
1323.2.h.e.802.3 6 3.2 odd 2
3024.2.r.g.1009.2 6 252.59 odd 6
3024.2.r.g.2017.2 6 84.59 odd 6
3969.2.a.m.1.1 3 63.25 even 3
3969.2.a.p.1.3 3 63.11 odd 6
9072.2.a.bq.1.2 3 252.115 even 6
9072.2.a.cd.1.2 3 252.227 odd 6