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Properties

Label 441.2.h.e.373.2
Level $441$
Weight $2$
Character 441.373
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(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 = [6,6,0,6,3] 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: \(\Q(\zeta_{18})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{3} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 63)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 373.2
Root \(-0.173648 - 0.984808i\) of defining polynomial
Character \(\chi\) \(=\) 441.373
Dual form 441.2.h.e.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+1.34730 q^{2} +(1.11334 + 1.32683i) q^{3} -0.184793 q^{4} +(1.26604 + 2.19285i) q^{5} +(1.50000 + 1.78763i) q^{6} -2.94356 q^{8} +(-0.520945 + 2.95442i) q^{9} +(1.70574 + 2.95442i) q^{10} +(-0.233956 + 0.405223i) q^{11} +(-0.205737 - 0.245188i) q^{12} +(2.91147 - 5.04282i) q^{13} +(-1.50000 + 4.12122i) q^{15} -3.59627 q^{16} +(1.93969 + 3.35965i) q^{17} +(-0.701867 + 3.98048i) q^{18} +(-1.09240 + 1.89209i) q^{19} +(-0.233956 - 0.405223i) q^{20} +(-0.315207 + 0.545955i) q^{22} +(0.0530334 + 0.0918566i) q^{23} +(-3.27719 - 3.90560i) q^{24} +(-0.705737 + 1.22237i) q^{25} +(3.92262 - 6.79417i) q^{26} +(-4.50000 + 2.59808i) q^{27} +(-4.39053 - 7.60462i) q^{29} +(-2.02094 + 5.55250i) q^{30} +7.68004 q^{31} +1.04189 q^{32} +(-0.798133 + 0.140732i) q^{33} +(2.61334 + 4.52644i) q^{34} +(0.0962667 - 0.545955i) q^{36} +(3.84002 - 6.65111i) q^{37} +(-1.47178 + 2.54920i) q^{38} +(9.93242 - 1.75135i) q^{39} +(-3.72668 - 6.45480i) q^{40} +(-1.11334 + 1.92836i) q^{41} +(-0.613341 - 1.06234i) q^{43} +(0.0432332 - 0.0748822i) q^{44} +(-7.13816 + 2.59808i) q^{45} +(0.0714517 + 0.123758i) q^{46} +5.33275 q^{47} +(-4.00387 - 4.77163i) q^{48} +(-0.950837 + 1.64690i) q^{50} +(-2.29813 + 6.31407i) q^{51} +(-0.538019 + 0.931876i) q^{52} +(0.358441 + 0.620838i) q^{53} +(-6.06283 + 3.50038i) q^{54} -1.18479 q^{55} +(-3.72668 + 0.657115i) q^{57} +(-5.91534 - 10.2457i) q^{58} -0.736482 q^{59} +(0.277189 - 0.761570i) q^{60} -0.958111 q^{61} +10.3473 q^{62} +8.59627 q^{64} +14.7442 q^{65} +(-1.07532 + 0.189608i) q^{66} -9.63816 q^{67} +(-0.358441 - 0.620838i) q^{68} +(-0.0628336 + 0.172634i) q^{69} +13.2344 q^{71} +(1.53343 - 8.69653i) q^{72} +(-5.13429 - 8.89284i) q^{73} +(5.17365 - 8.96102i) q^{74} +(-2.40760 + 0.424525i) q^{75} +(0.201867 - 0.349643i) q^{76} +(13.3819 - 2.35959i) q^{78} -12.6382 q^{79} +(-4.55303 - 7.88609i) q^{80} +(-8.45723 - 3.07818i) q^{81} +(-1.50000 + 2.59808i) q^{82} +(-1.36571 - 2.36549i) q^{83} +(-4.91147 + 8.50692i) q^{85} +(-0.826352 - 1.43128i) q^{86} +(5.20187 - 14.2920i) q^{87} +(0.688663 - 1.19280i) q^{88} +(-4.05690 + 7.02676i) q^{89} +(-9.61721 + 3.50038i) q^{90} +(-0.00980018 - 0.0169744i) q^{92} +(8.55051 + 10.1901i) q^{93} +7.18479 q^{94} -5.53209 q^{95} +(1.15998 + 1.38241i) q^{96} +(-6.80200 - 11.7814i) q^{97} +(-1.07532 - 0.902302i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 6 q^{2} + 6 q^{4} + 3 q^{5} + 9 q^{6} + 12 q^{8} - 6 q^{11} + 9 q^{12} - 3 q^{13} - 9 q^{15} + 6 q^{16} + 6 q^{17} - 18 q^{18} - 3 q^{19} - 6 q^{20} - 9 q^{22} - 12 q^{23} - 9 q^{24} + 6 q^{25} - 3 q^{26}+ \cdots + 18 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.34730 0.952682 0.476341 − 0.879261i \(-0.341963\pi\)
0.476341 + 0.879261i \(0.341963\pi\)
\(3\) 1.11334 + 1.32683i 0.642788 + 0.766044i
\(4\) −0.184793 −0.0923963
\(5\) 1.26604 + 2.19285i 0.566192 + 0.980674i 0.996938 + 0.0782003i \(0.0249174\pi\)
−0.430745 + 0.902473i \(0.641749\pi\)
\(6\) 1.50000 + 1.78763i 0.612372 + 0.729797i
\(7\) 0 0
\(8\) −2.94356 −1.04071
\(9\) −0.520945 + 2.95442i −0.173648 + 0.984808i
\(10\) 1.70574 + 2.95442i 0.539401 + 0.934271i
\(11\) −0.233956 + 0.405223i −0.0705403 + 0.122179i −0.899138 − 0.437665i \(-0.855806\pi\)
0.828598 + 0.559844i \(0.189139\pi\)
\(12\) −0.205737 − 0.245188i −0.0593912 − 0.0707796i
\(13\) 2.91147 − 5.04282i 0.807498 − 1.39863i −0.107094 − 0.994249i \(-0.534155\pi\)
0.914592 − 0.404378i \(-0.132512\pi\)
\(14\) 0 0
\(15\) −1.50000 + 4.12122i −0.387298 + 1.06409i
\(16\) −3.59627 −0.899067
\(17\) 1.93969 + 3.35965i 0.470445 + 0.814834i 0.999429 − 0.0337978i \(-0.0107602\pi\)
−0.528984 + 0.848632i \(0.677427\pi\)
\(18\) −0.701867 + 3.98048i −0.165432 + 0.938209i
\(19\) −1.09240 + 1.89209i −0.250613 + 0.434074i −0.963695 − 0.267007i \(-0.913965\pi\)
0.713082 + 0.701081i \(0.247299\pi\)
\(20\) −0.233956 − 0.405223i −0.0523141 − 0.0906106i
\(21\) 0 0
\(22\) −0.315207 + 0.545955i −0.0672025 + 0.116398i
\(23\) 0.0530334 + 0.0918566i 0.0110582 + 0.0191534i 0.871502 − 0.490393i \(-0.163147\pi\)
−0.860443 + 0.509546i \(0.829813\pi\)
\(24\) −3.27719 − 3.90560i −0.668953 − 0.797228i
\(25\) −0.705737 + 1.22237i −0.141147 + 0.244474i
\(26\) 3.92262 − 6.79417i 0.769289 − 1.33245i
\(27\) −4.50000 + 2.59808i −0.866025 + 0.500000i
\(28\) 0 0
\(29\) −4.39053 − 7.60462i −0.815301 − 1.41214i −0.909112 − 0.416552i \(-0.863238\pi\)
0.0938108 − 0.995590i \(-0.470095\pi\)
\(30\) −2.02094 + 5.55250i −0.368972 + 1.01374i
\(31\) 7.68004 1.37938 0.689688 − 0.724106i \(-0.257748\pi\)
0.689688 + 0.724106i \(0.257748\pi\)
\(32\) 1.04189 0.184182
\(33\) −0.798133 + 0.140732i −0.138937 + 0.0244984i
\(34\) 2.61334 + 4.52644i 0.448184 + 0.776278i
\(35\) 0 0
\(36\) 0.0962667 − 0.545955i 0.0160444 − 0.0909926i
\(37\) 3.84002 − 6.65111i 0.631296 − 1.09344i −0.355991 − 0.934489i \(-0.615857\pi\)
0.987287 − 0.158947i \(-0.0508099\pi\)
\(38\) −1.47178 + 2.54920i −0.238754 + 0.413535i
\(39\) 9.93242 − 1.75135i 1.59046 − 0.280441i
\(40\) −3.72668 − 6.45480i −0.589240 − 1.02059i
\(41\) −1.11334 + 1.92836i −0.173875 + 0.301160i −0.939771 − 0.341804i \(-0.888962\pi\)
0.765897 + 0.642964i \(0.222295\pi\)
\(42\) 0 0
\(43\) −0.613341 − 1.06234i −0.0935336 − 0.162005i 0.815462 − 0.578811i \(-0.196483\pi\)
−0.908996 + 0.416806i \(0.863150\pi\)
\(44\) 0.0432332 − 0.0748822i 0.00651766 − 0.0112889i
\(45\) −7.13816 + 2.59808i −1.06409 + 0.387298i
\(46\) 0.0714517 + 0.123758i 0.0105350 + 0.0182471i
\(47\) 5.33275 0.777861 0.388931 − 0.921267i \(-0.372845\pi\)
0.388931 + 0.921267i \(0.372845\pi\)
\(48\) −4.00387 − 4.77163i −0.577909 − 0.688725i
\(49\) 0 0
\(50\) −0.950837 + 1.64690i −0.134469 + 0.232907i
\(51\) −2.29813 + 6.31407i −0.321803 + 0.884147i
\(52\) −0.538019 + 0.931876i −0.0746098 + 0.129228i
\(53\) 0.358441 + 0.620838i 0.0492356 + 0.0852786i 0.889593 − 0.456754i \(-0.150988\pi\)
−0.840357 + 0.542033i \(0.817655\pi\)
\(54\) −6.06283 + 3.50038i −0.825047 + 0.476341i
\(55\) −1.18479 −0.159757
\(56\) 0 0
\(57\) −3.72668 + 0.657115i −0.493611 + 0.0870369i
\(58\) −5.91534 − 10.2457i −0.776723 − 1.34532i
\(59\) −0.736482 −0.0958818 −0.0479409 − 0.998850i \(-0.515266\pi\)
−0.0479409 + 0.998850i \(0.515266\pi\)
\(60\) 0.277189 − 0.761570i 0.0357849 − 0.0983183i
\(61\) −0.958111 −0.122674 −0.0613368 − 0.998117i \(-0.519536\pi\)
−0.0613368 + 0.998117i \(0.519536\pi\)
\(62\) 10.3473 1.31411
\(63\) 0 0
\(64\) 8.59627 1.07453
\(65\) 14.7442 1.82880
\(66\) −1.07532 + 0.189608i −0.132363 + 0.0233392i
\(67\) −9.63816 −1.17749 −0.588744 − 0.808320i \(-0.700377\pi\)
−0.588744 + 0.808320i \(0.700377\pi\)
\(68\) −0.358441 − 0.620838i −0.0434673 − 0.0752876i
\(69\) −0.0628336 + 0.172634i −0.00756428 + 0.0207827i
\(70\) 0 0
\(71\) 13.2344 1.57064 0.785318 − 0.619092i \(-0.212499\pi\)
0.785318 + 0.619092i \(0.212499\pi\)
\(72\) 1.53343 − 8.69653i 0.180717 − 1.02490i
\(73\) −5.13429 − 8.89284i −0.600923 − 1.04083i −0.992682 − 0.120761i \(-0.961467\pi\)
0.391759 − 0.920068i \(-0.371867\pi\)
\(74\) 5.17365 − 8.96102i 0.601424 − 1.04170i
\(75\) −2.40760 + 0.424525i −0.278006 + 0.0490200i
\(76\) 0.201867 − 0.349643i 0.0231557 − 0.0401068i
\(77\) 0 0
\(78\) 13.3819 − 2.35959i 1.51520 − 0.267171i
\(79\) −12.6382 −1.42190 −0.710952 − 0.703241i \(-0.751736\pi\)
−0.710952 + 0.703241i \(0.751736\pi\)
\(80\) −4.55303 − 7.88609i −0.509045 − 0.881691i
\(81\) −8.45723 − 3.07818i −0.939693 − 0.342020i
\(82\) −1.50000 + 2.59808i −0.165647 + 0.286910i
\(83\) −1.36571 − 2.36549i −0.149907 − 0.259646i 0.781286 − 0.624173i \(-0.214564\pi\)
−0.931193 + 0.364527i \(0.881231\pi\)
\(84\) 0 0
\(85\) −4.91147 + 8.50692i −0.532724 + 0.922705i
\(86\) −0.826352 − 1.43128i −0.0891078 − 0.154339i
\(87\) 5.20187 − 14.2920i 0.557699 − 1.53226i
\(88\) 0.688663 − 1.19280i 0.0734117 − 0.127153i
\(89\) −4.05690 + 7.02676i −0.430031 + 0.744835i −0.996875 − 0.0789894i \(-0.974831\pi\)
0.566845 + 0.823825i \(0.308164\pi\)
\(90\) −9.61721 + 3.50038i −1.01374 + 0.368972i
\(91\) 0 0
\(92\) −0.00980018 − 0.0169744i −0.00102174 − 0.00176970i
\(93\) 8.55051 + 10.1901i 0.886646 + 1.05666i
\(94\) 7.18479 0.741055
\(95\) −5.53209 −0.567580
\(96\) 1.15998 + 1.38241i 0.118390 + 0.141091i
\(97\) −6.80200 − 11.7814i −0.690639 − 1.19622i −0.971629 − 0.236511i \(-0.923996\pi\)
0.280990 − 0.959711i \(-0.409337\pi\)
\(98\) 0 0
\(99\) −1.07532 − 0.902302i −0.108074 − 0.0906848i
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.e.373.2 6
3.2 odd 2 1323.2.h.b.226.2 6
7.2 even 3 441.2.f.c.148.2 6
7.3 odd 6 441.2.g.c.67.2 6
7.4 even 3 441.2.g.b.67.2 6
7.5 odd 6 63.2.f.a.22.2 ✓ 6
7.6 odd 2 441.2.h.d.373.2 6
9.2 odd 6 1323.2.g.e.667.2 6
9.7 even 3 441.2.g.b.79.2 6
21.2 odd 6 1323.2.f.d.442.2 6
21.5 even 6 189.2.f.b.64.2 6
21.11 odd 6 1323.2.g.e.361.2 6
21.17 even 6 1323.2.g.d.361.2 6
21.20 even 2 1323.2.h.c.226.2 6
28.19 even 6 1008.2.r.h.337.1 6
63.2 odd 6 1323.2.f.d.883.2 6
63.5 even 6 567.2.a.c.1.2 3
63.11 odd 6 1323.2.h.b.802.2 6
63.16 even 3 441.2.f.c.295.2 6
63.20 even 6 1323.2.g.d.667.2 6
63.23 odd 6 3969.2.a.l.1.2 3
63.25 even 3 inner 441.2.h.e.214.2 6
63.34 odd 6 441.2.g.c.79.2 6
63.38 even 6 1323.2.h.c.802.2 6
63.40 odd 6 567.2.a.h.1.2 3
63.47 even 6 189.2.f.b.127.2 6
63.52 odd 6 441.2.h.d.214.2 6
63.58 even 3 3969.2.a.q.1.2 3
63.61 odd 6 63.2.f.a.43.2 yes 6
84.47 odd 6 3024.2.r.k.1009.3 6
252.47 odd 6 3024.2.r.k.2017.3 6
252.103 even 6 9072.2.a.ca.1.3 3
252.131 odd 6 9072.2.a.bs.1.1 3
252.187 even 6 1008.2.r.h.673.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
63.2.f.a.22.2 ✓ 6 7.5 odd 6
63.2.f.a.43.2 yes 6 63.61 odd 6
189.2.f.b.64.2 6 21.5 even 6
189.2.f.b.127.2 6 63.47 even 6
441.2.f.c.148.2 6 7.2 even 3
441.2.f.c.295.2 6 63.16 even 3
441.2.g.b.67.2 6 7.4 even 3
441.2.g.b.79.2 6 9.7 even 3
441.2.g.c.67.2 6 7.3 odd 6
441.2.g.c.79.2 6 63.34 odd 6
441.2.h.d.214.2 6 63.52 odd 6
441.2.h.d.373.2 6 7.6 odd 2
441.2.h.e.214.2 6 63.25 even 3 inner
441.2.h.e.373.2 6 1.1 even 1 trivial
567.2.a.c.1.2 3 63.5 even 6
567.2.a.h.1.2 3 63.40 odd 6
1008.2.r.h.337.1 6 28.19 even 6
1008.2.r.h.673.1 6 252.187 even 6
1323.2.f.d.442.2 6 21.2 odd 6
1323.2.f.d.883.2 6 63.2 odd 6
1323.2.g.d.361.2 6 21.17 even 6
1323.2.g.d.667.2 6 63.20 even 6
1323.2.g.e.361.2 6 21.11 odd 6
1323.2.g.e.667.2 6 9.2 odd 6
1323.2.h.b.226.2 6 3.2 odd 2
1323.2.h.b.802.2 6 63.11 odd 6
1323.2.h.c.226.2 6 21.20 even 2
1323.2.h.c.802.2 6 63.38 even 6
3024.2.r.k.1009.3 6 84.47 odd 6
3024.2.r.k.2017.3 6 252.47 odd 6
3969.2.a.l.1.2 3 63.23 odd 6
3969.2.a.q.1.2 3 63.58 even 3
9072.2.a.bs.1.1 3 252.131 odd 6
9072.2.a.ca.1.3 3 252.103 even 6