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Properties

Label 441.2.g.e.79.2
Level $441$
Weight $2$
Character 441.79
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 79.2
Root \(0.500000 + 1.41036i\) of defining polynomial
Character \(\chi\) \(=\) 441.79
Dual form 441.2.g.e.67.2

$q$-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 \(\left(\mathbb{Z}/441\mathbb{Z}\right)^\times\).

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.119562 + 0.207087i 0.0845428 + 0.146433i 0.905196 − 0.424994i \(-0.139724\pi\)
−0.820653 + 0.571426i \(0.806390\pi\)
\(3\) 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 \(-0.414862\pi\)
0.264291 + 0.964443i \(0.414862\pi\)
\(6\) 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 \(-0.210951\pi\)
−0.926995 + 0.375073i \(0.877618\pi\)
\(14\) 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 0.888252 + 0.459357i \(0.151920\pi\)
−0.0463112 + 0.998927i \(0.514747\pi\)
\(18\) 0.533792 + 0.479256i 0.125816 + 0.112962i
\(19\) −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
\(23\) −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 −0.876913 − 0.480649i \(-0.840401\pi\)
0.854711 + 0.519104i \(0.173734\pi\)
\(30\) 0.308342 + 0.380217i 0.0562952 + 0.0694178i
\(31\) −0.830095 + 1.43777i −0.149089 + 0.258231i −0.930891 − 0.365297i \(-0.880968\pi\)
0.781802 + 0.623527i \(0.214301\pi\)
\(32\) 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 −0.144538 − 0.989499i \(-0.546170\pi\)
0.929201 − 0.369576i \(-0.120497\pi\)
\(38\) −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 0.922791 + 0.385301i \(0.125903\pi\)
−0.127715 + 0.991811i \(0.540764\pi\)
\(42\) 0 0
\(43\) −1.11273 + 1.92730i −0.169689 + 0.293910i −0.938311 − 0.345794i \(-0.887610\pi\)
0.768622 + 0.639704i \(0.220943\pi\)
\(44\) −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 \(-0.306422\pi\)
−0.996428 + 0.0844432i \(0.973089\pi\)
\(48\) −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 0.921627 + 0.388077i \(0.126861\pi\)
−0.124729 + 0.992191i \(0.539806\pi\)
\(54\) 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 \(-0.887530\pi\)
0.768782 + 0.639511i \(0.220863\pi\)
\(60\) 1.42270 − 3.71415i 0.183670 − 0.479495i
\(61\) 3.80150 + 6.58440i 0.486733 + 0.843046i 0.999884 − 0.0152524i \(-0.00485519\pi\)
−0.513151 + 0.858298i \(0.671522\pi\)
\(62\) −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 −0.953053 − 0.302804i \(-0.902077\pi\)
0.738763 + 0.673966i \(0.235410\pi\)
\(68\) 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 −0.843963 − 0.536402i \(-0.819783\pi\)
−0.0425559 − 0.999094i \(-0.513550\pi\)
\(74\) 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 0.580410 − 0.814324i \(-0.302892\pi\)
−0.995431 + 0.0954881i \(0.969559\pi\)
\(80\) −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 \(-0.708900\pi\)
0.991211 + 0.132292i \(0.0422338\pi\)
\(84\) 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 \(-0.879845\pi\)
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
\(93\) −1.02859 + 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 \(-0.951893\pi\)
0.624687 + 0.780875i \(0.285226\pi\)
\(98\) 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.e.79.2 6
3.2 odd 2 1323.2.g.c.667.2 6
7.2 even 3 63.2.f.b.43.2 yes 6
7.3 odd 6 441.2.h.b.214.2 6
7.4 even 3 441.2.h.c.214.2 6
7.5 odd 6 441.2.f.d.295.2 6
7.6 odd 2 441.2.g.d.79.2 6
9.4 even 3 441.2.h.c.373.2 6
9.5 odd 6 1323.2.h.d.226.2 6
21.2 odd 6 189.2.f.a.127.2 6
21.5 even 6 1323.2.f.c.883.2 6
21.11 odd 6 1323.2.h.d.802.2 6
21.17 even 6 1323.2.h.e.802.2 6
21.20 even 2 1323.2.g.b.667.2 6
28.23 odd 6 1008.2.r.k.673.2 6
63.2 odd 6 567.2.a.g.1.2 3
63.4 even 3 inner 441.2.g.e.67.2 6
63.5 even 6 1323.2.f.c.442.2 6
63.13 odd 6 441.2.h.b.373.2 6
63.16 even 3 567.2.a.d.1.2 3
63.23 odd 6 189.2.f.a.64.2 6
63.31 odd 6 441.2.g.d.67.2 6
63.32 odd 6 1323.2.g.c.361.2 6
63.40 odd 6 441.2.f.d.148.2 6
63.41 even 6 1323.2.h.e.226.2 6
63.47 even 6 3969.2.a.p.1.2 3
63.58 even 3 63.2.f.b.22.2 ✓ 6
63.59 even 6 1323.2.g.b.361.2 6
63.61 odd 6 3969.2.a.m.1.2 3
84.23 even 6 3024.2.r.g.2017.3 6
252.23 even 6 3024.2.r.g.1009.3 6
252.79 odd 6 9072.2.a.bq.1.3 3
252.191 even 6 9072.2.a.cd.1.1 3
252.247 odd 6 1008.2.r.k.337.2 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
63.2.f.b.22.2 ✓ 6 63.58 even 3
63.2.f.b.43.2 yes 6 7.2 even 3
189.2.f.a.64.2 6 63.23 odd 6
189.2.f.a.127.2 6 21.2 odd 6
441.2.f.d.148.2 6 63.40 odd 6
441.2.f.d.295.2 6 7.5 odd 6
441.2.g.d.67.2 6 63.31 odd 6
441.2.g.d.79.2 6 7.6 odd 2
441.2.g.e.67.2 6 63.4 even 3 inner
441.2.g.e.79.2 6 1.1 even 1 trivial
441.2.h.b.214.2 6 7.3 odd 6
441.2.h.b.373.2 6 63.13 odd 6
441.2.h.c.214.2 6 7.4 even 3
441.2.h.c.373.2 6 9.4 even 3
567.2.a.d.1.2 3 63.16 even 3
567.2.a.g.1.2 3 63.2 odd 6
1008.2.r.k.337.2 6 252.247 odd 6
1008.2.r.k.673.2 6 28.23 odd 6
1323.2.f.c.442.2 6 63.5 even 6
1323.2.f.c.883.2 6 21.5 even 6
1323.2.g.b.361.2 6 63.59 even 6
1323.2.g.b.667.2 6 21.20 even 2
1323.2.g.c.361.2 6 63.32 odd 6
1323.2.g.c.667.2 6 3.2 odd 2
1323.2.h.d.226.2 6 9.5 odd 6
1323.2.h.d.802.2 6 21.11 odd 6
1323.2.h.e.226.2 6 63.41 even 6
1323.2.h.e.802.2 6 21.17 even 6
3024.2.r.g.1009.3 6 252.23 even 6
3024.2.r.g.2017.3 6 84.23 even 6
3969.2.a.m.1.2 3 63.61 odd 6
3969.2.a.p.1.2 3 63.47 even 6
9072.2.a.bq.1.3 3 252.79 odd 6
9072.2.a.cd.1.1 3 252.191 even 6