↑

Properties

Label 21.4.a.c.1.2
Level $21$
Weight $4$
Character 21.1
Self dual yes
Analytic conductor $1.239$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [21,4,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, 4); 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, 4, names="a")
 
Level: \( N \) \(=\) \( 21 = 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 21.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,-3] 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: \(1.23904011012\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{57}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 14 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-3.27492\) of defining polynomial
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)
 
\(f(q)\) \(=\) \(q+2.27492 q^{2} +3.00000 q^{3} -2.82475 q^{4} -4.54983 q^{5} +6.82475 q^{6} +7.00000 q^{7} -24.6254 q^{8} +9.00000 q^{9} -10.3505 q^{10} -40.7492 q^{11} -8.47425 q^{12} +53.2990 q^{13} +15.9244 q^{14} -13.6495 q^{15} -33.4228 q^{16} +4.54983 q^{17} +20.4743 q^{18} +122.598 q^{19} +12.8522 q^{20} +21.0000 q^{21} -92.7010 q^{22} +131.347 q^{23} -73.8762 q^{24} -104.299 q^{25} +121.251 q^{26} +27.0000 q^{27} -19.7733 q^{28} -216.598 q^{29} -31.0515 q^{30} -251.794 q^{31} +120.969 q^{32} -122.248 q^{33} +10.3505 q^{34} -31.8488 q^{35} -25.4228 q^{36} +11.8970 q^{37} +278.900 q^{38} +159.897 q^{39} +112.042 q^{40} -111.752 q^{41} +47.7733 q^{42} +369.196 q^{43} +115.106 q^{44} -40.9485 q^{45} +298.804 q^{46} -262.694 q^{47} -100.268 q^{48} +49.0000 q^{49} -237.272 q^{50} +13.6495 q^{51} -150.556 q^{52} -567.100 q^{53} +61.4228 q^{54} +185.402 q^{55} -172.378 q^{56} +367.794 q^{57} -492.743 q^{58} +839.890 q^{59} +38.5565 q^{60} -485.794 q^{61} -572.811 q^{62} +63.0000 q^{63} +542.577 q^{64} -242.502 q^{65} -278.103 q^{66} -333.691 q^{67} -12.8522 q^{68} +394.042 q^{69} -72.4535 q^{70} +590.248 q^{71} -221.629 q^{72} +490.701 q^{73} +27.0647 q^{74} -312.897 q^{75} -346.309 q^{76} -285.244 q^{77} +363.752 q^{78} +121.691 q^{79} +152.068 q^{80} +81.0000 q^{81} -254.228 q^{82} +609.608 q^{83} -59.3198 q^{84} -20.7010 q^{85} +839.890 q^{86} -649.794 q^{87} +1003.47 q^{88} +719.038 q^{89} -93.1545 q^{90} +373.093 q^{91} -371.023 q^{92} -755.382 q^{93} -597.608 q^{94} -557.801 q^{95} +362.908 q^{96} -637.877 q^{97} +111.471 q^{98} -366.743 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 3 q^{2} + 6 q^{3} + 17 q^{4} + 6 q^{5} - 9 q^{6} + 14 q^{7} - 87 q^{8} + 18 q^{9} - 66 q^{10} - 6 q^{11} + 51 q^{12} + 16 q^{13} - 21 q^{14} + 18 q^{15} + 137 q^{16} - 6 q^{17} - 27 q^{18} + 64 q^{19}+ \cdots - 54 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.27492 0.804305 0.402152 − 0.915573i \(-0.368262\pi\)
0.402152 + 0.915573i \(0.368262\pi\)
\(3\) 3.00000 0.577350
\(4\) −2.82475 −0.353094
\(5\) −4.54983 −0.406950 −0.203475 − 0.979080i \(-0.565223\pi\)
−0.203475 + 0.979080i \(0.565223\pi\)
\(6\) 6.82475 0.464366
\(7\) 7.00000 0.377964
\(8\) −24.6254 −1.08830
\(9\) 9.00000 0.333333
\(10\) −10.3505 −0.327311
\(11\) −40.7492 −1.11694 −0.558470 − 0.829525i \(-0.688611\pi\)
−0.558470 + 0.829525i \(0.688611\pi\)
\(12\) −8.47425 −0.203859
\(13\) 53.2990 1.13711 0.568557 − 0.822644i \(-0.307502\pi\)
0.568557 + 0.822644i \(0.307502\pi\)
\(14\) 15.9244 0.303999
\(15\) −13.6495 −0.234952
\(16\) −33.4228 −0.522231
\(17\) 4.54983 0.0649116 0.0324558 − 0.999473i \(-0.489667\pi\)
0.0324558 + 0.999473i \(0.489667\pi\)
\(18\) 20.4743 0.268102
\(19\) 122.598 1.48031 0.740156 − 0.672436i \(-0.234752\pi\)
0.740156 + 0.672436i \(0.234752\pi\)
\(20\) 12.8522 0.143691
\(21\) 21.0000 0.218218
\(22\) −92.7010 −0.898360
\(23\) 131.347 1.19077 0.595387 − 0.803439i \(-0.296999\pi\)
0.595387 + 0.803439i \(0.296999\pi\)
\(24\) −73.8762 −0.628330
\(25\) −104.299 −0.834392
\(26\) 121.251 0.914586
\(27\) 27.0000 0.192450
\(28\) −19.7733 −0.133457
\(29\) −216.598 −1.38694 −0.693470 − 0.720486i \(-0.743919\pi\)
−0.693470 + 0.720486i \(0.743919\pi\)
\(30\) −31.0515 −0.188973
\(31\) −251.794 −1.45882 −0.729412 − 0.684075i \(-0.760206\pi\)
−0.729412 + 0.684075i \(0.760206\pi\)
\(32\) 120.969 0.668267
\(33\) −122.248 −0.644865
\(34\) 10.3505 0.0522087
\(35\) −31.8488 −0.153812
\(36\) −25.4228 −0.117698
\(37\) 11.8970 0.0528610 0.0264305 − 0.999651i \(-0.491586\pi\)
0.0264305 + 0.999651i \(0.491586\pi\)
\(38\) 278.900 1.19062
\(39\) 159.897 0.656513
\(40\) 112.042 0.442883
\(41\) −111.752 −0.425678 −0.212839 − 0.977087i \(-0.568271\pi\)
−0.212839 + 0.977087i \(0.568271\pi\)
\(42\) 47.7733 0.175514
\(43\) 369.196 1.30935 0.654673 − 0.755912i \(-0.272806\pi\)
0.654673 + 0.755912i \(0.272806\pi\)
\(44\) 115.106 0.394385
\(45\) −40.9485 −0.135650
\(46\) 298.804 0.957744
\(47\) −262.694 −0.815275 −0.407637 − 0.913144i \(-0.633647\pi\)
−0.407637 + 0.913144i \(0.633647\pi\)
\(48\) −100.268 −0.301510
\(49\) 49.0000 0.142857
\(50\) −237.272 −0.671105
\(51\) 13.6495 0.0374767
\(52\) −150.556 −0.401508
\(53\) −567.100 −1.46976 −0.734879 − 0.678199i \(-0.762761\pi\)
−0.734879 + 0.678199i \(0.762761\pi\)
\(54\) 61.4228 0.154789
\(55\) 185.402 0.454538
\(56\) −172.378 −0.411339
\(57\) 367.794 0.854658
\(58\) −492.743 −1.11552
\(59\) 839.890 1.85330 0.926648 − 0.375931i \(-0.122677\pi\)
0.926648 + 0.375931i \(0.122677\pi\)
\(60\) 38.5565 0.0829603
\(61\) −485.794 −1.01966 −0.509832 − 0.860274i \(-0.670293\pi\)
−0.509832 + 0.860274i \(0.670293\pi\)
\(62\) −572.811 −1.17334
\(63\) 63.0000 0.125988
\(64\) 542.577 1.05972
\(65\) −242.502 −0.462748
\(66\) −278.103 −0.518668
\(67\) −333.691 −0.608460 −0.304230 − 0.952599i \(-0.598399\pi\)
−0.304230 + 0.952599i \(0.598399\pi\)
\(68\) −12.8522 −0.0229199
\(69\) 394.042 0.687493
\(70\) −72.4535 −0.123712
\(71\) 590.248 0.986613 0.493306 − 0.869856i \(-0.335788\pi\)
0.493306 + 0.869856i \(0.335788\pi\)
\(72\) −221.629 −0.362767
\(73\) 490.701 0.786743 0.393371 − 0.919380i \(-0.371309\pi\)
0.393371 + 0.919380i \(0.371309\pi\)
\(74\) 27.0647 0.0425164
\(75\) −312.897 −0.481736
\(76\) −346.309 −0.522689
\(77\) −285.244 −0.422164
\(78\) 363.752 0.528037
\(79\) 121.691 0.173308 0.0866539 − 0.996238i \(-0.472383\pi\)
0.0866539 + 0.996238i \(0.472383\pi\)
\(80\) 152.068 0.212522
\(81\) 81.0000 0.111111
\(82\) −254.228 −0.342375
\(83\) 609.608 0.806183 0.403091 − 0.915160i \(-0.367936\pi\)
0.403091 + 0.915160i \(0.367936\pi\)
\(84\) −59.3198 −0.0770514
\(85\) −20.7010 −0.0264157
\(86\) 839.890 1.05311
\(87\) −649.794 −0.800750
\(88\) 1003.47 1.21557
\(89\) 719.038 0.856381 0.428190 − 0.903689i \(-0.359151\pi\)
0.428190 + 0.903689i \(0.359151\pi\)
\(90\) −93.1545 −0.109104
\(91\) 373.093 0.429789
\(92\) −371.023 −0.420455
\(93\) −755.382 −0.842252
\(94\) −597.608 −0.655729
\(95\) −557.801 −0.602412
\(96\) 362.908 0.385824
\(97\) −637.877 −0.667697 −0.333849 − 0.942627i \(-0.608347\pi\)
−0.333849 + 0.942627i \(0.608347\pi\)
\(98\) 111.471 0.114901
\(99\) −366.743 −0.372313
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 21.4.a.c.1.2 ✓ 2
3.2 odd 2 63.4.a.e.1.1 2
4.3 odd 2 336.4.a.m.1.1 2
5.2 odd 4 525.4.d.g.274.3 4
5.3 odd 4 525.4.d.g.274.2 4
5.4 even 2 525.4.a.n.1.1 2
7.2 even 3 147.4.e.l.67.1 4
7.3 odd 6 147.4.e.m.79.1 4
7.4 even 3 147.4.e.l.79.1 4
7.5 odd 6 147.4.e.m.67.1 4
7.6 odd 2 147.4.a.i.1.2 2
8.3 odd 2 1344.4.a.bo.1.2 2
8.5 even 2 1344.4.a.bg.1.2 2
12.11 even 2 1008.4.a.ba.1.2 2
15.14 odd 2 1575.4.a.p.1.2 2
21.2 odd 6 441.4.e.q.361.2 4
21.5 even 6 441.4.e.p.361.2 4
21.11 odd 6 441.4.e.q.226.2 4
21.17 even 6 441.4.e.p.226.2 4
21.20 even 2 441.4.a.r.1.1 2
28.27 even 2 2352.4.a.bz.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
21.4.a.c.1.2 ✓ 2 1.1 even 1 trivial
63.4.a.e.1.1 2 3.2 odd 2
147.4.a.i.1.2 2 7.6 odd 2
147.4.e.l.67.1 4 7.2 even 3
147.4.e.l.79.1 4 7.4 even 3
147.4.e.m.67.1 4 7.5 odd 6
147.4.e.m.79.1 4 7.3 odd 6
336.4.a.m.1.1 2 4.3 odd 2
441.4.a.r.1.1 2 21.20 even 2
441.4.e.p.226.2 4 21.17 even 6
441.4.e.p.361.2 4 21.5 even 6
441.4.e.q.226.2 4 21.11 odd 6
441.4.e.q.361.2 4 21.2 odd 6
525.4.a.n.1.1 2 5.4 even 2
525.4.d.g.274.2 4 5.3 odd 4
525.4.d.g.274.3 4 5.2 odd 4
1008.4.a.ba.1.2 2 12.11 even 2
1344.4.a.bg.1.2 2 8.5 even 2
1344.4.a.bo.1.2 2 8.3 odd 2
1575.4.a.p.1.2 2 15.14 odd 2
2352.4.a.bz.1.2 2 28.27 even 2