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Properties

Label 21.3.f.b.10.1
Level $21$
Weight $3$
Character 21.10
Analytic conductor $0.572$
Analytic rank $0$
Dimension $2$
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] = [21,3,Mod(10,21)] 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("21.10"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(21, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 21 = 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 21.f (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,-1] 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: no
Analytic conductor: \(0.572208555157\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 10.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 21.10
Dual form 21.3.f.b.19.1

$q$-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+(-0.500000 - 0.866025i) q^{2} +(1.50000 + 0.866025i) q^{3} +(1.50000 - 2.59808i) q^{4} +(-4.50000 + 2.59808i) q^{5} -1.73205i q^{6} +(-3.50000 + 6.06218i) q^{7} -7.00000 q^{8} +(1.50000 + 2.59808i) q^{9} +(4.50000 + 2.59808i) q^{10} +(5.50000 - 9.52628i) q^{11} +(4.50000 - 2.59808i) q^{12} -6.92820i q^{13} +7.00000 q^{14} -9.00000 q^{15} +(-2.50000 - 4.33013i) q^{16} +(21.0000 + 12.1244i) q^{17} +(1.50000 - 2.59808i) q^{18} +(-3.00000 + 1.73205i) q^{19} +15.5885i q^{20} +(-10.5000 + 6.06218i) q^{21} -11.0000 q^{22} +(-14.0000 - 24.2487i) q^{23} +(-10.5000 - 6.06218i) q^{24} +(1.00000 - 1.73205i) q^{25} +(-6.00000 + 3.46410i) q^{26} +5.19615i q^{27} +(10.5000 + 18.1865i) q^{28} +25.0000 q^{29} +(4.50000 + 7.79423i) q^{30} +(-28.5000 - 16.4545i) q^{31} +(-16.5000 + 28.5788i) q^{32} +(16.5000 - 9.52628i) q^{33} -24.2487i q^{34} -36.3731i q^{35} +9.00000 q^{36} +(29.0000 + 50.2295i) q^{37} +(3.00000 + 1.73205i) q^{38} +(6.00000 - 10.3923i) q^{39} +(31.5000 - 18.1865i) q^{40} -3.46410i q^{41} +(10.5000 + 6.06218i) q^{42} +26.0000 q^{43} +(-16.5000 - 28.5788i) q^{44} +(-13.5000 - 7.79423i) q^{45} +(-14.0000 + 24.2487i) q^{46} +(-66.0000 + 38.1051i) q^{47} -8.66025i q^{48} +(-24.5000 - 42.4352i) q^{49} -2.00000 q^{50} +(21.0000 + 36.3731i) q^{51} +(-18.0000 - 10.3923i) q^{52} +(-15.5000 + 26.8468i) q^{53} +(4.50000 - 2.59808i) q^{54} +57.1577i q^{55} +(24.5000 - 42.4352i) q^{56} -6.00000 q^{57} +(-12.5000 - 21.6506i) q^{58} +(-7.50000 - 4.33013i) q^{59} +(-13.5000 + 23.3827i) q^{60} +(12.0000 - 6.92820i) q^{61} +32.9090i q^{62} -21.0000 q^{63} +13.0000 q^{64} +(18.0000 + 31.1769i) q^{65} +(-16.5000 - 9.52628i) q^{66} +(26.0000 - 45.0333i) q^{67} +(63.0000 - 36.3731i) q^{68} -48.4974i q^{69} +(-31.5000 + 18.1865i) q^{70} +64.0000 q^{71} +(-10.5000 - 18.1865i) q^{72} +(6.00000 + 3.46410i) q^{73} +(29.0000 - 50.2295i) q^{74} +(3.00000 - 1.73205i) q^{75} +10.3923i q^{76} +(38.5000 + 66.6840i) q^{77} -12.0000 q^{78} +(-8.50000 - 14.7224i) q^{79} +(22.5000 + 12.9904i) q^{80} +(-4.50000 + 7.79423i) q^{81} +(-3.00000 + 1.73205i) q^{82} -53.6936i q^{83} +36.3731i q^{84} -126.000 q^{85} +(-13.0000 - 22.5167i) q^{86} +(37.5000 + 21.6506i) q^{87} +(-38.5000 + 66.6840i) q^{88} +(-69.0000 + 39.8372i) q^{89} +15.5885i q^{90} +(42.0000 + 24.2487i) q^{91} -84.0000 q^{92} +(-28.5000 - 49.3634i) q^{93} +(66.0000 + 38.1051i) q^{94} +(9.00000 - 15.5885i) q^{95} +(-49.5000 + 28.5788i) q^{96} -91.7987i q^{97} +(-24.5000 + 42.4352i) q^{98} +33.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} + 3 q^{3} + 3 q^{4} - 9 q^{5} - 7 q^{7} - 14 q^{8} + 3 q^{9} + 9 q^{10} + 11 q^{11} + 9 q^{12} + 14 q^{14} - 18 q^{15} - 5 q^{16} + 42 q^{17} + 3 q^{18} - 6 q^{19} - 21 q^{21} - 22 q^{22} - 28 q^{23}+ \cdots + 66 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/21\mathbb{Z}\right)^\times\).

\(n\) \(8\) \(10\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{6}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.500000 − 0.866025i −0.250000 − 0.433013i 0.713525 − 0.700629i \(-0.247097\pi\)
−0.963525 + 0.267617i \(0.913764\pi\)
\(3\) 1.50000 + 0.866025i 0.500000 + 0.288675i
\(4\) 1.50000 − 2.59808i 0.375000 − 0.649519i
\(5\) −4.50000 + 2.59808i −0.900000 + 0.519615i −0.877200 − 0.480125i \(-0.840591\pi\)
−0.0227998 + 0.999740i \(0.507258\pi\)
\(6\) − 1.73205i − 0.288675i
\(7\) −3.50000 + 6.06218i −0.500000 + 0.866025i
\(8\) −7.00000 −0.875000
\(9\) 1.50000 + 2.59808i 0.166667 + 0.288675i
\(10\) 4.50000 + 2.59808i 0.450000 + 0.259808i
\(11\) 5.50000 − 9.52628i 0.500000 − 0.866025i −0.500000 − 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(12\) 4.50000 − 2.59808i 0.375000 − 0.216506i
\(13\) − 6.92820i − 0.532939i −0.963843 − 0.266469i \(-0.914143\pi\)
0.963843 − 0.266469i \(-0.0858571\pi\)
\(14\) 7.00000 0.500000
\(15\) −9.00000 −0.600000
\(16\) −2.50000 − 4.33013i −0.156250 − 0.270633i
\(17\) 21.0000 + 12.1244i 1.23529 + 0.713197i 0.968129 − 0.250453i \(-0.0805797\pi\)
0.267165 + 0.963651i \(0.413913\pi\)
\(18\) 1.50000 − 2.59808i 0.0833333 − 0.144338i
\(19\) −3.00000 + 1.73205i −0.157895 + 0.0911606i −0.576865 − 0.816839i \(-0.695724\pi\)
0.418971 + 0.908000i \(0.362391\pi\)
\(20\) 15.5885i 0.779423i
\(21\) −10.5000 + 6.06218i −0.500000 + 0.288675i
\(22\) −11.0000 −0.500000
\(23\) −14.0000 − 24.2487i −0.608696 − 1.05429i −0.991456 − 0.130444i \(-0.958360\pi\)
0.382760 − 0.923848i \(-0.374974\pi\)
\(24\) −10.5000 − 6.06218i −0.437500 − 0.252591i
\(25\) 1.00000 − 1.73205i 0.0400000 − 0.0692820i
\(26\) −6.00000 + 3.46410i −0.230769 + 0.133235i
\(27\) 5.19615i 0.192450i
\(28\) 10.5000 + 18.1865i 0.375000 + 0.649519i
\(29\) 25.0000 0.862069 0.431034 − 0.902335i \(-0.358149\pi\)
0.431034 + 0.902335i \(0.358149\pi\)
\(30\) 4.50000 + 7.79423i 0.150000 + 0.259808i
\(31\) −28.5000 − 16.4545i −0.919355 − 0.530790i −0.0359257 − 0.999354i \(-0.511438\pi\)
−0.883429 + 0.468565i \(0.844771\pi\)
\(32\) −16.5000 + 28.5788i −0.515625 + 0.893089i
\(33\) 16.5000 − 9.52628i 0.500000 − 0.288675i
\(34\) − 24.2487i − 0.713197i
\(35\) − 36.3731i − 1.03923i
\(36\) 9.00000 0.250000
\(37\) 29.0000 + 50.2295i 0.783784 + 1.35755i 0.929723 + 0.368260i \(0.120046\pi\)
−0.145939 + 0.989294i \(0.546620\pi\)
\(38\) 3.00000 + 1.73205i 0.0789474 + 0.0455803i
\(39\) 6.00000 − 10.3923i 0.153846 − 0.266469i
\(40\) 31.5000 − 18.1865i 0.787500 − 0.454663i
\(41\) − 3.46410i − 0.0844903i −0.999107 − 0.0422451i \(-0.986549\pi\)
0.999107 − 0.0422451i \(-0.0134510\pi\)
\(42\) 10.5000 + 6.06218i 0.250000 + 0.144338i
\(43\) 26.0000 0.604651 0.302326 − 0.953205i \(-0.402237\pi\)
0.302326 + 0.953205i \(0.402237\pi\)
\(44\) −16.5000 − 28.5788i −0.375000 − 0.649519i
\(45\) −13.5000 − 7.79423i −0.300000 − 0.173205i
\(46\) −14.0000 + 24.2487i −0.304348 + 0.527146i
\(47\) −66.0000 + 38.1051i −1.40426 + 0.810747i −0.994826 − 0.101595i \(-0.967606\pi\)
−0.409429 + 0.912342i \(0.634272\pi\)
\(48\) − 8.66025i − 0.180422i
\(49\) −24.5000 − 42.4352i −0.500000 − 0.866025i
\(50\) −2.00000 −0.0400000
\(51\) 21.0000 + 36.3731i 0.411765 + 0.713197i
\(52\) −18.0000 − 10.3923i −0.346154 − 0.199852i
\(53\) −15.5000 + 26.8468i −0.292453 + 0.506543i −0.974389 − 0.224868i \(-0.927805\pi\)
0.681936 + 0.731412i \(0.261138\pi\)
\(54\) 4.50000 − 2.59808i 0.0833333 − 0.0481125i
\(55\) 57.1577i 1.03923i
\(56\) 24.5000 − 42.4352i 0.437500 − 0.757772i
\(57\) −6.00000 −0.105263
\(58\) −12.5000 − 21.6506i −0.215517 − 0.373287i
\(59\) −7.50000 − 4.33013i −0.127119 − 0.0733920i 0.435092 − 0.900386i \(-0.356716\pi\)
−0.562211 + 0.826994i \(0.690049\pi\)
\(60\) −13.5000 + 23.3827i −0.225000 + 0.389711i
\(61\) 12.0000 − 6.92820i 0.196721 − 0.113577i −0.398404 − 0.917210i \(-0.630436\pi\)
0.595125 + 0.803633i \(0.297102\pi\)
\(62\) 32.9090i 0.530790i
\(63\) −21.0000 −0.333333
\(64\) 13.0000 0.203125
\(65\) 18.0000 + 31.1769i 0.276923 + 0.479645i
\(66\) −16.5000 − 9.52628i −0.250000 − 0.144338i
\(67\) 26.0000 − 45.0333i 0.388060 − 0.672139i −0.604129 − 0.796887i \(-0.706479\pi\)
0.992189 + 0.124748i \(0.0398121\pi\)
\(68\) 63.0000 − 36.3731i 0.926471 − 0.534898i
\(69\) − 48.4974i − 0.702861i
\(70\) −31.5000 + 18.1865i −0.450000 + 0.259808i
\(71\) 64.0000 0.901408 0.450704 − 0.892673i \(-0.351173\pi\)
0.450704 + 0.892673i \(0.351173\pi\)
\(72\) −10.5000 − 18.1865i −0.145833 − 0.252591i
\(73\) 6.00000 + 3.46410i 0.0821918 + 0.0474534i 0.540533 − 0.841323i \(-0.318223\pi\)
−0.458341 + 0.888777i \(0.651556\pi\)
\(74\) 29.0000 − 50.2295i 0.391892 − 0.678777i
\(75\) 3.00000 − 1.73205i 0.0400000 − 0.0230940i
\(76\) 10.3923i 0.136741i
\(77\) 38.5000 + 66.6840i 0.500000 + 0.866025i
\(78\) −12.0000 −0.153846
\(79\) −8.50000 − 14.7224i −0.107595 − 0.186360i 0.807200 − 0.590277i \(-0.200982\pi\)
−0.914795 + 0.403917i \(0.867648\pi\)
\(80\) 22.5000 + 12.9904i 0.281250 + 0.162380i
\(81\) −4.50000 + 7.79423i −0.0555556 + 0.0962250i
\(82\) −3.00000 + 1.73205i −0.0365854 + 0.0211226i
\(83\) − 53.6936i − 0.646911i −0.946243 − 0.323455i \(-0.895155\pi\)
0.946243 − 0.323455i \(-0.104845\pi\)
\(84\) 36.3731i 0.433013i
\(85\) −126.000 −1.48235
\(86\) −13.0000 − 22.5167i −0.151163 − 0.261822i
\(87\) 37.5000 + 21.6506i 0.431034 + 0.248858i
\(88\) −38.5000 + 66.6840i −0.437500 + 0.757772i
\(89\) −69.0000 + 39.8372i −0.775281 + 0.447609i −0.834755 − 0.550621i \(-0.814391\pi\)
0.0594743 + 0.998230i \(0.481058\pi\)
\(90\) 15.5885i 0.173205i
\(91\) 42.0000 + 24.2487i 0.461538 + 0.266469i
\(92\) −84.0000 −0.913043
\(93\) −28.5000 − 49.3634i −0.306452 − 0.530790i
\(94\) 66.0000 + 38.1051i 0.702128 + 0.405374i
\(95\) 9.00000 − 15.5885i 0.0947368 − 0.164089i
\(96\) −49.5000 + 28.5788i −0.515625 + 0.297696i
\(97\) − 91.7987i − 0.946378i −0.880961 − 0.473189i \(-0.843103\pi\)
0.880961 − 0.473189i \(-0.156897\pi\)
\(98\) −24.5000 + 42.4352i −0.250000 + 0.433013i
\(99\) 33.0000 0.333333
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 21.3.f.b.10.1 ✓ 2
3.2 odd 2 63.3.m.c.10.1 2
4.3 odd 2 336.3.bh.a.241.1 2
5.2 odd 4 525.3.s.c.199.2 4
5.3 odd 4 525.3.s.c.199.1 4
5.4 even 2 525.3.o.g.451.1 2
7.2 even 3 147.3.f.c.19.1 2
7.3 odd 6 147.3.d.b.97.2 2
7.4 even 3 147.3.d.b.97.1 2
7.5 odd 6 inner 21.3.f.b.19.1 yes 2
7.6 odd 2 147.3.f.c.31.1 2
12.11 even 2 1008.3.cg.g.577.1 2
21.2 odd 6 441.3.m.e.19.1 2
21.5 even 6 63.3.m.c.19.1 2
21.11 odd 6 441.3.d.b.244.2 2
21.17 even 6 441.3.d.b.244.1 2
21.20 even 2 441.3.m.e.325.1 2
28.3 even 6 2352.3.f.d.97.1 2
28.11 odd 6 2352.3.f.d.97.2 2
28.19 even 6 336.3.bh.a.145.1 2
35.12 even 12 525.3.s.c.124.1 4
35.19 odd 6 525.3.o.g.376.1 2
35.33 even 12 525.3.s.c.124.2 4
84.47 odd 6 1008.3.cg.g.145.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
21.3.f.b.10.1 ✓ 2 1.1 even 1 trivial
21.3.f.b.19.1 yes 2 7.5 odd 6 inner
63.3.m.c.10.1 2 3.2 odd 2
63.3.m.c.19.1 2 21.5 even 6
147.3.d.b.97.1 2 7.4 even 3
147.3.d.b.97.2 2 7.3 odd 6
147.3.f.c.19.1 2 7.2 even 3
147.3.f.c.31.1 2 7.6 odd 2
336.3.bh.a.145.1 2 28.19 even 6
336.3.bh.a.241.1 2 4.3 odd 2
441.3.d.b.244.1 2 21.17 even 6
441.3.d.b.244.2 2 21.11 odd 6
441.3.m.e.19.1 2 21.2 odd 6
441.3.m.e.325.1 2 21.20 even 2
525.3.o.g.376.1 2 35.19 odd 6
525.3.o.g.451.1 2 5.4 even 2
525.3.s.c.124.1 4 35.12 even 12
525.3.s.c.124.2 4 35.33 even 12
525.3.s.c.199.1 4 5.3 odd 4
525.3.s.c.199.2 4 5.2 odd 4
1008.3.cg.g.145.1 2 84.47 odd 6
1008.3.cg.g.577.1 2 12.11 even 2
2352.3.f.d.97.1 2 28.3 even 6
2352.3.f.d.97.2 2 28.11 odd 6