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Properties

Label 441.2.g.d.67.2
Level $441$
Weight $2$
Character 441.67
Analytic conductor $3.521$
Analytic rank $0$
Dimension $6$
Inner twists $2$

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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(67,441)] 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("441.67"); 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([2, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 441.g (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,1,-2,-3,10] 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: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 63)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 67.2
Root \(0.500000 - 1.41036i\) of defining polynomial
Character \(\chi\) \(=\) 441.67
Dual form 441.2.g.d.79.2

$q$-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+(0.119562 - 0.207087i) q^{2} +(-1.71053 - 0.272169i) q^{3} +(0.971410 + 1.68253i) q^{4} -1.18194 q^{5} +(-0.260877 + 0.321688i) q^{6} +0.942820 q^{8} +(2.85185 + 0.931107i) q^{9} +(-0.141315 + 0.244765i) q^{10} -3.70370 q^{11} +(-1.20370 - 3.14241i) q^{12} +(0.500000 - 0.866025i) q^{13} +(2.02175 + 0.321688i) q^{15} +(-1.83009 + 3.16982i) q^{16} +(-3.47141 + 6.01266i) q^{17} +(0.533792 - 0.479256i) q^{18} +(0.971410 + 1.68253i) q^{19} +(-1.14815 - 1.98866i) q^{20} +(-0.442820 + 0.766987i) q^{22} -5.60301 q^{23} +(-1.61273 - 0.256606i) q^{24} -3.60301 q^{25} +(-0.119562 - 0.207087i) q^{26} +(-4.62476 - 2.36887i) q^{27} +(-0.119562 - 0.207087i) q^{29} +(0.308342 - 0.380217i) q^{30} +(0.830095 + 1.43777i) q^{31} +(1.38044 + 2.39099i) q^{32} +(6.33530 + 1.00803i) q^{33} +(0.830095 + 1.43777i) q^{34} +(1.20370 + 5.70281i) q^{36} +(4.77292 + 8.26693i) q^{37} +0.464574 q^{38} +(-1.09097 + 1.34528i) q^{39} -1.11436 q^{40} +(-5.09097 + 8.81782i) q^{41} +(-1.11273 - 1.92730i) q^{43} +(-3.59781 - 6.23159i) q^{44} +(-3.37072 - 1.10052i) q^{45} +(-0.669905 + 1.16031i) q^{46} +(2.91423 - 5.04759i) q^{47} +(3.99316 - 4.92398i) q^{48} +(-0.430782 + 0.746136i) q^{50} +(7.57442 - 9.34004i) q^{51} +1.94282 q^{52} +(5.80150 - 10.0485i) q^{53} +(-1.04351 + 0.674501i) q^{54} +4.37756 q^{55} +(-1.20370 - 3.14241i) q^{57} -0.0571799 q^{58} +(1.30150 + 2.25427i) q^{59} +(1.42270 + 3.71415i) q^{60} +(-3.80150 + 6.58440i) q^{61} +0.396990 q^{62} -6.66019 q^{64} +(-0.590972 + 1.02359i) q^{65} +(0.966208 - 1.19143i) q^{66} +(-1.75404 - 3.03809i) q^{67} -13.4887 q^{68} +(9.58414 + 1.52496i) q^{69} +8.60301 q^{71} +(2.68878 + 0.877867i) q^{72} +(7.57442 - 13.1193i) q^{73} +2.28263 q^{74} +(6.16307 + 0.980627i) q^{75} +(-1.88727 + 3.26886i) q^{76} +(0.148152 + 0.386770i) q^{78} +(-3.68878 + 6.38915i) q^{79} +(2.16307 - 3.74654i) q^{80} +(7.26608 + 5.31075i) q^{81} +(1.21737 + 2.10855i) q^{82} +(-3.47141 - 6.01266i) q^{83} +(4.10301 - 7.10662i) q^{85} -0.532157 q^{86} +(0.148152 + 0.386770i) q^{87} -3.49192 q^{88} +(1.37360 + 2.37915i) q^{89} +(-0.630912 + 0.566453i) q^{90} +(-5.44282 - 9.42724i) q^{92} +(-1.02859 - 2.68527i) q^{93} +(-0.696860 - 1.20700i) q^{94} +(-1.14815 - 1.98866i) q^{95} +(-1.71053 - 4.46558i) q^{96} +(3.58414 + 6.20790i) q^{97} +(-10.5624 - 3.44854i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + q^{2} - 2 q^{3} - 3 q^{4} + 10 q^{5} - q^{6} - 12 q^{8} + 8 q^{9} - 4 q^{11} + 11 q^{12} + 3 q^{13} + 11 q^{15} - 3 q^{16} - 12 q^{17} - 23 q^{18} - 3 q^{19} - 16 q^{20} + 15 q^{22} + 12 q^{25}+ \cdots - 46 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(199\) \(344\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.119562 − 0.207087i 0.0845428 − 0.146433i −0.820653 − 0.571426i \(-0.806390\pi\)
0.905196 + 0.424994i \(0.139724\pi\)
\(3\) −1.71053 − 0.272169i −0.987577 − 0.157137i
\(4\) 0.971410 + 1.68253i 0.485705 + 0.841266i
\(5\) −1.18194 −0.528581 −0.264291 − 0.964443i \(-0.585138\pi\)
−0.264291 + 0.964443i \(0.585138\pi\)
\(6\) −0.260877 + 0.321688i −0.106502 + 0.131329i
\(7\) 0 0
\(8\) 0.942820 0.333337
\(9\) 2.85185 + 0.931107i 0.950616 + 0.310369i
\(10\) −0.141315 + 0.244765i −0.0446878 + 0.0774015i
\(11\) −3.70370 −1.11671 −0.558353 − 0.829603i \(-0.688567\pi\)
−0.558353 + 0.829603i \(0.688567\pi\)
\(12\) −1.20370 − 3.14241i −0.347477 − 0.907137i
\(13\) 0.500000 − 0.866025i 0.138675 − 0.240192i −0.788320 − 0.615265i \(-0.789049\pi\)
0.926995 + 0.375073i \(0.122382\pi\)
\(14\) 0 0
\(15\) 2.02175 + 0.321688i 0.522014 + 0.0830595i
\(16\) −1.83009 + 3.16982i −0.457524 + 0.792454i
\(17\) −3.47141 + 6.01266i −0.841941 + 1.45828i 0.0463112 + 0.998927i \(0.485253\pi\)
−0.888252 + 0.459357i \(0.848080\pi\)
\(18\) 0.533792 − 0.479256i 0.125816 − 0.112962i
\(19\) 0.971410 + 1.68253i 0.222857 + 0.385999i 0.955674 − 0.294426i \(-0.0951285\pi\)
−0.732818 + 0.680425i \(0.761795\pi\)
\(20\) −1.14815 − 1.98866i −0.256735 − 0.444677i
\(21\) 0 0
\(22\) −0.442820 + 0.766987i −0.0944096 + 0.163522i
\(23\) −5.60301 −1.16831 −0.584154 − 0.811643i \(-0.698574\pi\)
−0.584154 + 0.811643i \(0.698574\pi\)
\(24\) −1.61273 − 0.256606i −0.329196 − 0.0523795i
\(25\) −3.60301 −0.720602
\(26\) −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 \(0.840401\pi\)
\(30\) 0.308342 − 0.380217i 0.0562952 − 0.0694178i
\(31\) 0.830095 + 1.43777i 0.149089 + 0.258231i 0.930891 − 0.365297i \(-0.119032\pi\)
−0.781802 + 0.623527i \(0.785699\pi\)
\(32\) 1.38044 + 2.39099i 0.244029 + 0.422671i
\(33\) 6.33530 + 1.00803i 1.10283 + 0.175476i
\(34\) 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 0.929201 + 0.369576i \(0.120497\pi\)
−0.144538 + 0.989499i \(0.546170\pi\)
\(38\) 0.464574 0.0753638
\(39\) −1.09097 + 1.34528i −0.174695 + 0.215417i
\(40\) −1.11436 −0.176196
\(41\) −5.09097 + 8.81782i −0.795076 + 1.37711i 0.127715 + 0.991811i \(0.459236\pi\)
−0.922791 + 0.385301i \(0.874097\pi\)
\(42\) 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
\(45\) −3.37072 − 1.10052i −0.502478 − 0.164055i
\(46\) −0.669905 + 1.16031i −0.0987721 + 0.171078i
\(47\) 2.91423 − 5.04759i 0.425084 − 0.736267i −0.571344 − 0.820711i \(-0.693578\pi\)
0.996428 + 0.0844432i \(0.0269112\pi\)
\(48\) 3.99316 − 4.92398i 0.576364 − 0.710716i
\(49\) 0 0
\(50\) −0.430782 + 0.746136i −0.0609217 + 0.105520i
\(51\) 7.57442 − 9.34004i 1.06063 − 1.30787i
\(52\) 1.94282 0.269421
\(53\) 5.80150 − 10.0485i 0.796898 − 1.38027i −0.124729 − 0.992191i \(-0.539806\pi\)
0.921627 − 0.388077i \(-0.126861\pi\)
\(54\) −1.04351 + 0.674501i −0.142003 + 0.0917880i
\(55\) 4.37756 0.590270
\(56\) 0 0
\(57\) −1.20370 − 3.14241i −0.159434 − 0.416223i
\(58\) −0.0571799 −0.00750809
\(59\) 1.30150 + 2.25427i 0.169442 + 0.293481i 0.938224 − 0.346029i \(-0.112470\pi\)
−0.768782 + 0.639511i \(0.779137\pi\)
\(60\) 1.42270 + 3.71415i 0.183670 + 0.479495i
\(61\) −3.80150 + 6.58440i −0.486733 + 0.843046i −0.999884 − 0.0152524i \(-0.995145\pi\)
0.513151 + 0.858298i \(0.328478\pi\)
\(62\) 0.396990 0.0504178
\(63\) 0 0
\(64\) −6.66019 −0.832524
\(65\) −0.590972 + 1.02359i −0.0733010 + 0.126961i
\(66\) 0.966208 − 1.19143i 0.118932 − 0.146655i
\(67\) −1.75404 − 3.03809i −0.214290 − 0.371161i 0.738763 − 0.673966i \(-0.235410\pi\)
−0.953053 + 0.302804i \(0.902077\pi\)
\(68\) −13.4887 −1.63574
\(69\) 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 \(0.329462\pi\)
\(72\) 2.68878 + 0.877867i 0.316876 + 0.103458i
\(73\) 7.57442 − 13.1193i 0.886519 − 1.53550i 0.0425559 − 0.999094i \(-0.486450\pi\)
0.843963 − 0.536402i \(-0.180217\pi\)
\(74\) 2.28263 0.265350
\(75\) 6.16307 + 0.980627i 0.711650 + 0.113233i
\(76\) −1.88727 + 3.26886i −0.216485 + 0.374963i
\(77\) 0 0
\(78\) 0.148152 + 0.386770i 0.0167749 + 0.0437931i
\(79\) −3.68878 + 6.38915i −0.415020 + 0.718836i −0.995431 − 0.0954881i \(-0.969559\pi\)
0.580410 + 0.814324i \(0.302892\pi\)
\(80\) 2.16307 − 3.74654i 0.241838 − 0.418876i
\(81\) 7.26608 + 5.31075i 0.807342 + 0.590084i
\(82\) 1.21737 + 2.10855i 0.134436 + 0.232850i
\(83\) −3.47141 − 6.01266i −0.381037 − 0.659975i 0.610174 − 0.792267i \(-0.291100\pi\)
−0.991211 + 0.132292i \(0.957766\pi\)
\(84\) 0 0
\(85\) 4.10301 − 7.10662i 0.445034 − 0.770821i
\(86\) −0.532157 −0.0573840
\(87\) 0.148152 + 0.386770i 0.0158835 + 0.0414661i
\(88\) −3.49192 −0.372240
\(89\) 1.37360 + 2.37915i 0.145602 + 0.252189i 0.929597 − 0.368577i \(-0.120155\pi\)
−0.783996 + 0.620766i \(0.786822\pi\)
\(90\) −0.630912 + 0.566453i −0.0665039 + 0.0597094i
\(91\) 0 0
\(92\) −5.44282 − 9.42724i −0.567453 − 0.982858i
\(93\) −1.02859 − 2.68527i −0.106660 − 0.278450i
\(94\) −0.696860 − 1.20700i −0.0718756 − 0.124492i
\(95\) −1.14815 − 1.98866i −0.117798 − 0.204032i
\(96\) −1.71053 − 4.46558i −0.174581 − 0.455766i
\(97\) 3.58414 + 6.20790i 0.363914 + 0.630317i 0.988601 − 0.150558i \(-0.0481069\pi\)
−0.624687 + 0.780875i \(0.714774\pi\)
\(98\) 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.g.d.67.2 6
3.2 odd 2 1323.2.g.b.361.2 6
7.2 even 3 441.2.h.b.373.2 6
7.3 odd 6 63.2.f.b.22.2 ✓ 6
7.4 even 3 441.2.f.d.148.2 6
7.5 odd 6 441.2.h.c.373.2 6
7.6 odd 2 441.2.g.e.67.2 6
9.2 odd 6 1323.2.h.e.802.2 6
9.7 even 3 441.2.h.b.214.2 6
21.2 odd 6 1323.2.h.e.226.2 6
21.5 even 6 1323.2.h.d.226.2 6
21.11 odd 6 1323.2.f.c.442.2 6
21.17 even 6 189.2.f.a.64.2 6
21.20 even 2 1323.2.g.c.361.2 6
28.3 even 6 1008.2.r.k.337.2 6
63.2 odd 6 1323.2.g.b.667.2 6
63.4 even 3 3969.2.a.m.1.2 3
63.11 odd 6 1323.2.f.c.883.2 6
63.16 even 3 inner 441.2.g.d.79.2 6
63.20 even 6 1323.2.h.d.802.2 6
63.25 even 3 441.2.f.d.295.2 6
63.31 odd 6 567.2.a.d.1.2 3
63.32 odd 6 3969.2.a.p.1.2 3
63.34 odd 6 441.2.h.c.214.2 6
63.38 even 6 189.2.f.a.127.2 6
63.47 even 6 1323.2.g.c.667.2 6
63.52 odd 6 63.2.f.b.43.2 yes 6
63.59 even 6 567.2.a.g.1.2 3
63.61 odd 6 441.2.g.e.79.2 6
84.59 odd 6 3024.2.r.g.1009.3 6
252.31 even 6 9072.2.a.bq.1.3 3
252.59 odd 6 9072.2.a.cd.1.1 3
252.115 even 6 1008.2.r.k.673.2 6
252.227 odd 6 3024.2.r.g.2017.3 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
63.2.f.b.22.2 ✓ 6 7.3 odd 6
63.2.f.b.43.2 yes 6 63.52 odd 6
189.2.f.a.64.2 6 21.17 even 6
189.2.f.a.127.2 6 63.38 even 6
441.2.f.d.148.2 6 7.4 even 3
441.2.f.d.295.2 6 63.25 even 3
441.2.g.d.67.2 6 1.1 even 1 trivial
441.2.g.d.79.2 6 63.16 even 3 inner
441.2.g.e.67.2 6 7.6 odd 2
441.2.g.e.79.2 6 63.61 odd 6
441.2.h.b.214.2 6 9.7 even 3
441.2.h.b.373.2 6 7.2 even 3
441.2.h.c.214.2 6 63.34 odd 6
441.2.h.c.373.2 6 7.5 odd 6
567.2.a.d.1.2 3 63.31 odd 6
567.2.a.g.1.2 3 63.59 even 6
1008.2.r.k.337.2 6 28.3 even 6
1008.2.r.k.673.2 6 252.115 even 6
1323.2.f.c.442.2 6 21.11 odd 6
1323.2.f.c.883.2 6 63.11 odd 6
1323.2.g.b.361.2 6 3.2 odd 2
1323.2.g.b.667.2 6 63.2 odd 6
1323.2.g.c.361.2 6 21.20 even 2
1323.2.g.c.667.2 6 63.47 even 6
1323.2.h.d.226.2 6 21.5 even 6
1323.2.h.d.802.2 6 63.20 even 6
1323.2.h.e.226.2 6 21.2 odd 6
1323.2.h.e.802.2 6 9.2 odd 6
3024.2.r.g.1009.3 6 84.59 odd 6
3024.2.r.g.2017.3 6 252.227 odd 6
3969.2.a.m.1.2 3 63.4 even 3
3969.2.a.p.1.2 3 63.32 odd 6
9072.2.a.bq.1.3 3 252.31 even 6
9072.2.a.cd.1.1 3 252.59 odd 6