Properties

Label 441.6.a.b.1.1
Level $441$
Weight $6$
Character 441.1
Self dual yes
Analytic conductor $70.729$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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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] = [441,6,Mod(1,441)] 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("441.1"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(441, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 441.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,-10,0,68,-106,0,0,-360,0,1060,-92,0,-670] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(70.7292645375\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 21)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 441.1

$q$-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)
 
\(f(q)\) \(=\) \(q-10.0000 q^{2} +68.0000 q^{4} -106.000 q^{5} -360.000 q^{8} +1060.00 q^{10} -92.0000 q^{11} -670.000 q^{13} +1424.00 q^{16} -222.000 q^{17} +908.000 q^{19} -7208.00 q^{20} +920.000 q^{22} +1176.00 q^{23} +8111.00 q^{25} +6700.00 q^{26} -1118.00 q^{29} -3696.00 q^{31} -2720.00 q^{32} +2220.00 q^{34} +4182.00 q^{37} -9080.00 q^{38} +38160.0 q^{40} -6662.00 q^{41} -3700.00 q^{43} -6256.00 q^{44} -11760.0 q^{46} -7056.00 q^{47} -81110.0 q^{50} -45560.0 q^{52} +37578.0 q^{53} +9752.00 q^{55} +11180.0 q^{58} +32700.0 q^{59} +10802.0 q^{61} +36960.0 q^{62} -18368.0 q^{64} +71020.0 q^{65} +64996.0 q^{67} -15096.0 q^{68} +61320.0 q^{71} -38922.0 q^{73} -41820.0 q^{74} +61744.0 q^{76} -88096.0 q^{79} -150944. q^{80} +66620.0 q^{82} +71892.0 q^{83} +23532.0 q^{85} +37000.0 q^{86} +33120.0 q^{88} +111818. q^{89} +79968.0 q^{92} +70560.0 q^{94} -96248.0 q^{95} +150846. q^{97} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −10.0000 −1.76777 −0.883883 − 0.467707i \(-0.845080\pi\)
−0.883883 + 0.467707i \(0.845080\pi\)
\(3\) 0 0
\(4\) 68.0000 2.12500
\(5\) −106.000 −1.89619 −0.948093 − 0.317994i \(-0.896991\pi\)
−0.948093 + 0.317994i \(0.896991\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) −360.000 −1.98874
\(9\) 0 0
\(10\) 1060.00 3.35201
\(11\) −92.0000 −0.229248 −0.114624 − 0.993409i \(-0.536566\pi\)
−0.114624 + 0.993409i \(0.536566\pi\)
\(12\) 0 0
\(13\) −670.000 −1.09955 −0.549777 − 0.835312i \(-0.685287\pi\)
−0.549777 + 0.835312i \(0.685287\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1424.00 1.39062
\(17\) −222.000 −0.186308 −0.0931538 − 0.995652i \(-0.529695\pi\)
−0.0931538 + 0.995652i \(0.529695\pi\)
\(18\) 0 0
\(19\) 908.000 0.577035 0.288517 − 0.957475i \(-0.406838\pi\)
0.288517 + 0.957475i \(0.406838\pi\)
\(20\) −7208.00 −4.02939
\(21\) 0 0
\(22\) 920.000 0.405258
\(23\) 1176.00 0.463541 0.231770 − 0.972771i \(-0.425548\pi\)
0.231770 + 0.972771i \(0.425548\pi\)
\(24\) 0 0
\(25\) 8111.00 2.59552
\(26\) 6700.00 1.94375
\(27\) 0 0
\(28\) 0 0
\(29\) −1118.00 −0.246858 −0.123429 − 0.992353i \(-0.539389\pi\)
−0.123429 + 0.992353i \(0.539389\pi\)
\(30\) 0 0
\(31\) −3696.00 −0.690761 −0.345380 − 0.938463i \(-0.612250\pi\)
−0.345380 + 0.938463i \(0.612250\pi\)
\(32\) −2720.00 −0.469563
\(33\) 0 0
\(34\) 2220.00 0.329348
\(35\) 0 0
\(36\) 0 0
\(37\) 4182.00 0.502203 0.251102 − 0.967961i \(-0.419207\pi\)
0.251102 + 0.967961i \(0.419207\pi\)
\(38\) −9080.00 −1.02006
\(39\) 0 0
\(40\) 38160.0 3.77102
\(41\) −6662.00 −0.618935 −0.309467 − 0.950910i \(-0.600151\pi\)
−0.309467 + 0.950910i \(0.600151\pi\)
\(42\) 0 0
\(43\) −3700.00 −0.305162 −0.152581 − 0.988291i \(-0.548759\pi\)
−0.152581 + 0.988291i \(0.548759\pi\)
\(44\) −6256.00 −0.487153
\(45\) 0 0
\(46\) −11760.0 −0.819432
\(47\) −7056.00 −0.465923 −0.232961 − 0.972486i \(-0.574842\pi\)
−0.232961 + 0.972486i \(0.574842\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) −81110.0 −4.58827
\(51\) 0 0
\(52\) −45560.0 −2.33655
\(53\) 37578.0 1.83757 0.918785 − 0.394758i \(-0.129172\pi\)
0.918785 + 0.394758i \(0.129172\pi\)
\(54\) 0 0
\(55\) 9752.00 0.434697
\(56\) 0 0
\(57\) 0 0
\(58\) 11180.0 0.436387
\(59\) 32700.0 1.22298 0.611488 − 0.791254i \(-0.290571\pi\)
0.611488 + 0.791254i \(0.290571\pi\)
\(60\) 0 0
\(61\) 10802.0 0.371689 0.185844 − 0.982579i \(-0.440498\pi\)
0.185844 + 0.982579i \(0.440498\pi\)
\(62\) 36960.0 1.22110
\(63\) 0 0
\(64\) −18368.0 −0.560547
\(65\) 71020.0 2.08496
\(66\) 0 0
\(67\) 64996.0 1.76889 0.884443 − 0.466649i \(-0.154539\pi\)
0.884443 + 0.466649i \(0.154539\pi\)
\(68\) −15096.0 −0.395904
\(69\) 0 0
\(70\) 0 0
\(71\) 61320.0 1.44363 0.721816 − 0.692085i \(-0.243308\pi\)
0.721816 + 0.692085i \(0.243308\pi\)
\(72\) 0 0
\(73\) −38922.0 −0.854846 −0.427423 − 0.904052i \(-0.640579\pi\)
−0.427423 + 0.904052i \(0.640579\pi\)
\(74\) −41820.0 −0.887779
\(75\) 0 0
\(76\) 61744.0 1.22620
\(77\) 0 0
\(78\) 0 0
\(79\) −88096.0 −1.58814 −0.794069 − 0.607827i \(-0.792041\pi\)
−0.794069 + 0.607827i \(0.792041\pi\)
\(80\) −150944. −2.63688
\(81\) 0 0
\(82\) 66620.0 1.09413
\(83\) 71892.0 1.14547 0.572737 − 0.819739i \(-0.305882\pi\)
0.572737 + 0.819739i \(0.305882\pi\)
\(84\) 0 0
\(85\) 23532.0 0.353274
\(86\) 37000.0 0.539455
\(87\) 0 0
\(88\) 33120.0 0.455915
\(89\) 111818. 1.49636 0.748181 − 0.663495i \(-0.230927\pi\)
0.748181 + 0.663495i \(0.230927\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 79968.0 0.985024
\(93\) 0 0
\(94\) 70560.0 0.823643
\(95\) −96248.0 −1.09416
\(96\) 0 0
\(97\) 150846. 1.62781 0.813906 − 0.580996i \(-0.197337\pi\)
0.813906 + 0.580996i \(0.197337\pi\)
\(98\) 0 0
\(99\) 0 0
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 441.6.a.b.1.1 1
3.2 odd 2 147.6.a.g.1.1 1
7.6 odd 2 63.6.a.a.1.1 1
21.2 odd 6 147.6.e.b.67.1 2
21.5 even 6 147.6.e.a.67.1 2
21.11 odd 6 147.6.e.b.79.1 2
21.17 even 6 147.6.e.a.79.1 2
21.20 even 2 21.6.a.d.1.1 ✓ 1
28.27 even 2 1008.6.a.bc.1.1 1
84.83 odd 2 336.6.a.a.1.1 1
105.62 odd 4 525.6.d.a.274.2 2
105.83 odd 4 525.6.d.a.274.1 2
105.104 even 2 525.6.a.a.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
21.6.a.d.1.1 ✓ 1 21.20 even 2
63.6.a.a.1.1 1 7.6 odd 2
147.6.a.g.1.1 1 3.2 odd 2
147.6.e.a.67.1 2 21.5 even 6
147.6.e.a.79.1 2 21.17 even 6
147.6.e.b.67.1 2 21.2 odd 6
147.6.e.b.79.1 2 21.11 odd 6
336.6.a.a.1.1 1 84.83 odd 2
441.6.a.b.1.1 1 1.1 even 1 trivial
525.6.a.a.1.1 1 105.104 even 2
525.6.d.a.274.1 2 105.83 odd 4
525.6.d.a.274.2 2 105.62 odd 4
1008.6.a.bc.1.1 1 28.27 even 2