Properties

Label 21.6.a.d.1.1
Level $21$
Weight $6$
Character 21.1
Self dual yes
Analytic conductor $3.368$
Analytic rank $0$
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] = [21,6,Mod(1,21)] 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("21.1"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(21, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 21 = 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 21.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,10] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(3.36806021607\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 21.1

$q$-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)
 
\(f(q)\) \(=\) \(q+10.0000 q^{2} +9.00000 q^{3} +68.0000 q^{4} -106.000 q^{5} +90.0000 q^{6} -49.0000 q^{7} +360.000 q^{8} +81.0000 q^{9} -1060.00 q^{10} +92.0000 q^{11} +612.000 q^{12} +670.000 q^{13} -490.000 q^{14} -954.000 q^{15} +1424.00 q^{16} -222.000 q^{17} +810.000 q^{18} -908.000 q^{19} -7208.00 q^{20} -441.000 q^{21} +920.000 q^{22} -1176.00 q^{23} +3240.00 q^{24} +8111.00 q^{25} +6700.00 q^{26} +729.000 q^{27} -3332.00 q^{28} +1118.00 q^{29} -9540.00 q^{30} +3696.00 q^{31} +2720.00 q^{32} +828.000 q^{33} -2220.00 q^{34} +5194.00 q^{35} +5508.00 q^{36} +4182.00 q^{37} -9080.00 q^{38} +6030.00 q^{39} -38160.0 q^{40} -6662.00 q^{41} -4410.00 q^{42} -3700.00 q^{43} +6256.00 q^{44} -8586.00 q^{45} -11760.0 q^{46} -7056.00 q^{47} +12816.0 q^{48} +2401.00 q^{49} +81110.0 q^{50} -1998.00 q^{51} +45560.0 q^{52} -37578.0 q^{53} +7290.00 q^{54} -9752.00 q^{55} -17640.0 q^{56} -8172.00 q^{57} +11180.0 q^{58} +32700.0 q^{59} -64872.0 q^{60} -10802.0 q^{61} +36960.0 q^{62} -3969.00 q^{63} -18368.0 q^{64} -71020.0 q^{65} +8280.00 q^{66} +64996.0 q^{67} -15096.0 q^{68} -10584.0 q^{69} +51940.0 q^{70} -61320.0 q^{71} +29160.0 q^{72} +38922.0 q^{73} +41820.0 q^{74} +72999.0 q^{75} -61744.0 q^{76} -4508.00 q^{77} +60300.0 q^{78} -88096.0 q^{79} -150944. q^{80} +6561.00 q^{81} -66620.0 q^{82} +71892.0 q^{83} -29988.0 q^{84} +23532.0 q^{85} -37000.0 q^{86} +10062.0 q^{87} +33120.0 q^{88} +111818. q^{89} -85860.0 q^{90} -32830.0 q^{91} -79968.0 q^{92} +33264.0 q^{93} -70560.0 q^{94} +96248.0 q^{95} +24480.0 q^{96} -150846. q^{97} +24010.0 q^{98} +7452.00 q^{99} +O(q^{100})\)

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 \(-0.154920\pi\)
0.883883 + 0.467707i \(0.154920\pi\)
\(3\) 9.00000 0.577350
\(4\) 68.0000 2.12500
\(5\) −106.000 −1.89619 −0.948093 − 0.317994i \(-0.896991\pi\)
−0.948093 + 0.317994i \(0.896991\pi\)
\(6\) 90.0000 1.02062
\(7\) −49.0000 −0.377964
\(8\) 360.000 1.98874
\(9\) 81.0000 0.333333
\(10\) −1060.00 −3.35201
\(11\) 92.0000 0.229248 0.114624 − 0.993409i \(-0.463434\pi\)
0.114624 + 0.993409i \(0.463434\pi\)
\(12\) 612.000 1.22687
\(13\) 670.000 1.09955 0.549777 − 0.835312i \(-0.314713\pi\)
0.549777 + 0.835312i \(0.314713\pi\)
\(14\) −490.000 −0.668153
\(15\) −954.000 −1.09476
\(16\) 1424.00 1.39062
\(17\) −222.000 −0.186308 −0.0931538 − 0.995652i \(-0.529695\pi\)
−0.0931538 + 0.995652i \(0.529695\pi\)
\(18\) 810.000 0.589256
\(19\) −908.000 −0.577035 −0.288517 − 0.957475i \(-0.593162\pi\)
−0.288517 + 0.957475i \(0.593162\pi\)
\(20\) −7208.00 −4.02939
\(21\) −441.000 −0.218218
\(22\) 920.000 0.405258
\(23\) −1176.00 −0.463541 −0.231770 − 0.972771i \(-0.574452\pi\)
−0.231770 + 0.972771i \(0.574452\pi\)
\(24\) 3240.00 1.14820
\(25\) 8111.00 2.59552
\(26\) 6700.00 1.94375
\(27\) 729.000 0.192450
\(28\) −3332.00 −0.803175
\(29\) 1118.00 0.246858 0.123429 − 0.992353i \(-0.460611\pi\)
0.123429 + 0.992353i \(0.460611\pi\)
\(30\) −9540.00 −1.93529
\(31\) 3696.00 0.690761 0.345380 − 0.938463i \(-0.387750\pi\)
0.345380 + 0.938463i \(0.387750\pi\)
\(32\) 2720.00 0.469563
\(33\) 828.000 0.132357
\(34\) −2220.00 −0.329348
\(35\) 5194.00 0.716691
\(36\) 5508.00 0.708333
\(37\) 4182.00 0.502203 0.251102 − 0.967961i \(-0.419207\pi\)
0.251102 + 0.967961i \(0.419207\pi\)
\(38\) −9080.00 −1.02006
\(39\) 6030.00 0.634828
\(40\) −38160.0 −3.77102
\(41\) −6662.00 −0.618935 −0.309467 − 0.950910i \(-0.600151\pi\)
−0.309467 + 0.950910i \(0.600151\pi\)
\(42\) −4410.00 −0.385758
\(43\) −3700.00 −0.305162 −0.152581 − 0.988291i \(-0.548759\pi\)
−0.152581 + 0.988291i \(0.548759\pi\)
\(44\) 6256.00 0.487153
\(45\) −8586.00 −0.632062
\(46\) −11760.0 −0.819432
\(47\) −7056.00 −0.465923 −0.232961 − 0.972486i \(-0.574842\pi\)
−0.232961 + 0.972486i \(0.574842\pi\)
\(48\) 12816.0 0.802878
\(49\) 2401.00 0.142857
\(50\) 81110.0 4.58827
\(51\) −1998.00 −0.107565
\(52\) 45560.0 2.33655
\(53\) −37578.0 −1.83757 −0.918785 − 0.394758i \(-0.870828\pi\)
−0.918785 + 0.394758i \(0.870828\pi\)
\(54\) 7290.00 0.340207
\(55\) −9752.00 −0.434697
\(56\) −17640.0 −0.751672
\(57\) −8172.00 −0.333151
\(58\) 11180.0 0.436387
\(59\) 32700.0 1.22298 0.611488 − 0.791254i \(-0.290571\pi\)
0.611488 + 0.791254i \(0.290571\pi\)
\(60\) −64872.0 −2.32637
\(61\) −10802.0 −0.371689 −0.185844 − 0.982579i \(-0.559502\pi\)
−0.185844 + 0.982579i \(0.559502\pi\)
\(62\) 36960.0 1.22110
\(63\) −3969.00 −0.125988
\(64\) −18368.0 −0.560547
\(65\) −71020.0 −2.08496
\(66\) 8280.00 0.233976
\(67\) 64996.0 1.76889 0.884443 − 0.466649i \(-0.154539\pi\)
0.884443 + 0.466649i \(0.154539\pi\)
\(68\) −15096.0 −0.395904
\(69\) −10584.0 −0.267625
\(70\) 51940.0 1.26694
\(71\) −61320.0 −1.44363 −0.721816 − 0.692085i \(-0.756692\pi\)
−0.721816 + 0.692085i \(0.756692\pi\)
\(72\) 29160.0 0.662913
\(73\) 38922.0 0.854846 0.427423 − 0.904052i \(-0.359421\pi\)
0.427423 + 0.904052i \(0.359421\pi\)
\(74\) 41820.0 0.887779
\(75\) 72999.0 1.49852
\(76\) −61744.0 −1.22620
\(77\) −4508.00 −0.0866477
\(78\) 60300.0 1.12223
\(79\) −88096.0 −1.58814 −0.794069 − 0.607827i \(-0.792041\pi\)
−0.794069 + 0.607827i \(0.792041\pi\)
\(80\) −150944. −2.63688
\(81\) 6561.00 0.111111
\(82\) −66620.0 −1.09413
\(83\) 71892.0 1.14547 0.572737 − 0.819739i \(-0.305882\pi\)
0.572737 + 0.819739i \(0.305882\pi\)
\(84\) −29988.0 −0.463713
\(85\) 23532.0 0.353274
\(86\) −37000.0 −0.539455
\(87\) 10062.0 0.142523
\(88\) 33120.0 0.455915
\(89\) 111818. 1.49636 0.748181 − 0.663495i \(-0.230927\pi\)
0.748181 + 0.663495i \(0.230927\pi\)
\(90\) −85860.0 −1.11734
\(91\) −32830.0 −0.415592
\(92\) −79968.0 −0.985024
\(93\) 33264.0 0.398811
\(94\) −70560.0 −0.823643
\(95\) 96248.0 1.09416
\(96\) 24480.0 0.271102
\(97\) −150846. −1.62781 −0.813906 − 0.580996i \(-0.802663\pi\)
−0.813906 + 0.580996i \(0.802663\pi\)
\(98\) 24010.0 0.252538
\(99\) 7452.00 0.0764161
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 21.6.a.d.1.1 ✓ 1
3.2 odd 2 63.6.a.a.1.1 1
4.3 odd 2 336.6.a.a.1.1 1
5.2 odd 4 525.6.d.a.274.2 2
5.3 odd 4 525.6.d.a.274.1 2
5.4 even 2 525.6.a.a.1.1 1
7.2 even 3 147.6.e.a.67.1 2
7.3 odd 6 147.6.e.b.79.1 2
7.4 even 3 147.6.e.a.79.1 2
7.5 odd 6 147.6.e.b.67.1 2
7.6 odd 2 147.6.a.g.1.1 1
12.11 even 2 1008.6.a.bc.1.1 1
21.20 even 2 441.6.a.b.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
21.6.a.d.1.1 ✓ 1 1.1 even 1 trivial
63.6.a.a.1.1 1 3.2 odd 2
147.6.a.g.1.1 1 7.6 odd 2
147.6.e.a.67.1 2 7.2 even 3
147.6.e.a.79.1 2 7.4 even 3
147.6.e.b.67.1 2 7.5 odd 6
147.6.e.b.79.1 2 7.3 odd 6
336.6.a.a.1.1 1 4.3 odd 2
441.6.a.b.1.1 1 21.20 even 2
525.6.a.a.1.1 1 5.4 even 2
525.6.d.a.274.1 2 5.3 odd 4
525.6.d.a.274.2 2 5.2 odd 4
1008.6.a.bc.1.1 1 12.11 even 2