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Properties

Label 21.2.g.a.5.1
Level $21$
Weight $2$
Character 21.5
Analytic conductor $0.168$
Analytic rank $0$
Dimension $2$
CM discriminant -3
Inner twists $4$

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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] = [21,2,Mod(5,21)] 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("21.5"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(21, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 5])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 21 = 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 21.g (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.167685844245\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

Embedding invariants

Embedding label 5.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 21.5
Dual form 21.2.g.a.17.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.50000 + 0.866025i) q^{3} +(-1.00000 - 1.73205i) q^{4} +(0.500000 + 2.59808i) q^{7} +(1.50000 - 2.59808i) q^{9} +(3.00000 + 1.73205i) q^{12} -1.73205i q^{13} +(-2.00000 + 3.46410i) q^{16} +(-4.50000 - 2.59808i) q^{19} +(-3.00000 - 3.46410i) q^{21} +(2.50000 + 4.33013i) q^{25} +5.19615i q^{27} +(4.00000 - 3.46410i) q^{28} +(7.50000 - 4.33013i) q^{31} -6.00000 q^{36} +(-0.500000 + 0.866025i) q^{37} +(1.50000 + 2.59808i) q^{39} -5.00000 q^{43} -6.92820i q^{48} +(-6.50000 + 2.59808i) q^{49} +(-3.00000 + 1.73205i) q^{52} +9.00000 q^{57} +(6.00000 + 3.46410i) q^{61} +(7.50000 + 2.59808i) q^{63} +8.00000 q^{64} +(-5.50000 - 9.52628i) q^{67} +(-13.5000 + 7.79423i) q^{73} +(-7.50000 - 4.33013i) q^{75} +10.3923i q^{76} +(6.50000 - 11.2583i) q^{79} +(-4.50000 - 7.79423i) q^{81} +(-3.00000 + 8.66025i) q^{84} +(4.50000 - 0.866025i) q^{91} +(-7.50000 + 12.9904i) q^{93} -13.8564i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 3 q^{3} - 2 q^{4} + q^{7} + 3 q^{9} + 6 q^{12} - 4 q^{16} - 9 q^{19} - 6 q^{21} + 5 q^{25} + 8 q^{28} + 15 q^{31} - 12 q^{36} - q^{37} + 3 q^{39} - 10 q^{43} - 13 q^{49} - 6 q^{52} + 18 q^{57} + 12 q^{61}+ \cdots - 15 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(8\) \(10\)
\(\chi(n)\) \(-1\) \(e\left(\frac{5}{6}\right)\)

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 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(3\) −1.50000 + 0.866025i −0.866025 + 0.500000i
\(4\) −1.00000 − 1.73205i −0.500000 − 0.866025i
\(5\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(6\) 0 0
\(7\) 0.500000 + 2.59808i 0.188982 + 0.981981i
\(8\) 0 0
\(9\) 1.50000 − 2.59808i 0.500000 − 0.866025i
\(10\) 0 0
\(11\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(12\) 3.00000 + 1.73205i 0.866025 + 0.500000i
\(13\) − 1.73205i − 0.480384i −0.970725 − 0.240192i \(-0.922790\pi\)
0.970725 − 0.240192i \(-0.0772105\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −2.00000 + 3.46410i −0.500000 + 0.866025i
\(17\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(18\) 0 0
\(19\) −4.50000 − 2.59808i −1.03237 − 0.596040i −0.114708 − 0.993399i \(-0.536593\pi\)
−0.917663 + 0.397360i \(0.869927\pi\)
\(20\) 0 0
\(21\) −3.00000 − 3.46410i −0.654654 − 0.755929i
\(22\) 0 0
\(23\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(24\) 0 0
\(25\) 2.50000 + 4.33013i 0.500000 + 0.866025i
\(26\) 0 0
\(27\) 5.19615i 1.00000i
\(28\) 4.00000 − 3.46410i 0.755929 − 0.654654i
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) 7.50000 − 4.33013i 1.34704 − 0.777714i 0.359211 − 0.933257i \(-0.383046\pi\)
0.987829 + 0.155543i \(0.0497126\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) −6.00000 −1.00000
\(37\) −0.500000 + 0.866025i −0.0821995 + 0.142374i −0.904194 − 0.427121i \(-0.859528\pi\)
0.821995 + 0.569495i \(0.192861\pi\)
\(38\) 0 0
\(39\) 1.50000 + 2.59808i 0.240192 + 0.416025i
\(40\) 0 0
\(41\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) −5.00000 −0.762493 −0.381246 − 0.924473i \(-0.624505\pi\)
−0.381246 + 0.924473i \(0.624505\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(48\) − 6.92820i − 1.00000i
\(49\) −6.50000 + 2.59808i −0.928571 + 0.371154i
\(50\) 0 0
\(51\) 0 0
\(52\) −3.00000 + 1.73205i −0.416025 + 0.240192i
\(53\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 9.00000 1.19208
\(58\) 0 0
\(59\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(60\) 0 0
\(61\) 6.00000 + 3.46410i 0.768221 + 0.443533i 0.832240 − 0.554416i \(-0.187058\pi\)
−0.0640184 + 0.997949i \(0.520392\pi\)
\(62\) 0 0
\(63\) 7.50000 + 2.59808i 0.944911 + 0.327327i
\(64\) 8.00000 1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) −5.50000 − 9.52628i −0.671932 − 1.16382i −0.977356 − 0.211604i \(-0.932131\pi\)
0.305424 − 0.952217i \(-0.401202\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) −13.5000 + 7.79423i −1.58006 + 0.912245i −0.585206 + 0.810885i \(0.698986\pi\)
−0.994850 + 0.101361i \(0.967680\pi\)
\(74\) 0 0
\(75\) −7.50000 − 4.33013i −0.866025 − 0.500000i
\(76\) 10.3923i 1.19208i
\(77\) 0 0
\(78\) 0 0
\(79\) 6.50000 − 11.2583i 0.731307 − 1.26666i −0.225018 − 0.974355i \(-0.572244\pi\)
0.956325 − 0.292306i \(-0.0944227\pi\)
\(80\) 0 0
\(81\) −4.50000 − 7.79423i −0.500000 − 0.866025i
\(82\) 0 0
\(83\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) −3.00000 + 8.66025i −0.327327 + 0.944911i
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(90\) 0 0
\(91\) 4.50000 − 0.866025i 0.471728 − 0.0907841i
\(92\) 0 0
\(93\) −7.50000 + 12.9904i −0.777714 + 1.34704i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) − 13.8564i − 1.40690i −0.710742 − 0.703452i \(-0.751641\pi\)
0.710742 − 0.703452i \(-0.248359\pi\)
\(98\) 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 21.2.g.a.5.1 ✓ 2
3.2 odd 2 CM 21.2.g.a.5.1 ✓ 2
4.3 odd 2 336.2.bc.c.257.1 2
5.2 odd 4 525.2.q.d.299.2 4
5.3 odd 4 525.2.q.d.299.1 4
5.4 even 2 525.2.t.c.26.1 2
7.2 even 3 147.2.c.a.146.1 2
7.3 odd 6 inner 21.2.g.a.17.1 yes 2
7.4 even 3 147.2.g.a.80.1 2
7.5 odd 6 147.2.c.a.146.2 2
7.6 odd 2 147.2.g.a.68.1 2
9.2 odd 6 567.2.i.b.215.1 2
9.4 even 3 567.2.s.a.26.1 2
9.5 odd 6 567.2.s.a.26.1 2
9.7 even 3 567.2.i.b.215.1 2
12.11 even 2 336.2.bc.c.257.1 2
15.2 even 4 525.2.q.d.299.2 4
15.8 even 4 525.2.q.d.299.1 4
15.14 odd 2 525.2.t.c.26.1 2
21.2 odd 6 147.2.c.a.146.1 2
21.5 even 6 147.2.c.a.146.2 2
21.11 odd 6 147.2.g.a.80.1 2
21.17 even 6 inner 21.2.g.a.17.1 yes 2
21.20 even 2 147.2.g.a.68.1 2
28.3 even 6 336.2.bc.c.17.1 2
28.19 even 6 2352.2.k.c.881.1 2
28.23 odd 6 2352.2.k.c.881.2 2
35.3 even 12 525.2.q.d.374.2 4
35.17 even 12 525.2.q.d.374.1 4
35.24 odd 6 525.2.t.c.101.1 2
63.31 odd 6 567.2.i.b.269.1 2
63.38 even 6 567.2.s.a.458.1 2
63.52 odd 6 567.2.s.a.458.1 2
63.59 even 6 567.2.i.b.269.1 2
84.23 even 6 2352.2.k.c.881.2 2
84.47 odd 6 2352.2.k.c.881.1 2
84.59 odd 6 336.2.bc.c.17.1 2
105.17 odd 12 525.2.q.d.374.1 4
105.38 odd 12 525.2.q.d.374.2 4
105.59 even 6 525.2.t.c.101.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
21.2.g.a.5.1 ✓ 2 1.1 even 1 trivial
21.2.g.a.5.1 ✓ 2 3.2 odd 2 CM
21.2.g.a.17.1 yes 2 7.3 odd 6 inner
21.2.g.a.17.1 yes 2 21.17 even 6 inner
147.2.c.a.146.1 2 7.2 even 3
147.2.c.a.146.1 2 21.2 odd 6
147.2.c.a.146.2 2 7.5 odd 6
147.2.c.a.146.2 2 21.5 even 6
147.2.g.a.68.1 2 7.6 odd 2
147.2.g.a.68.1 2 21.20 even 2
147.2.g.a.80.1 2 7.4 even 3
147.2.g.a.80.1 2 21.11 odd 6
336.2.bc.c.17.1 2 28.3 even 6
336.2.bc.c.17.1 2 84.59 odd 6
336.2.bc.c.257.1 2 4.3 odd 2
336.2.bc.c.257.1 2 12.11 even 2
525.2.q.d.299.1 4 5.3 odd 4
525.2.q.d.299.1 4 15.8 even 4
525.2.q.d.299.2 4 5.2 odd 4
525.2.q.d.299.2 4 15.2 even 4
525.2.q.d.374.1 4 35.17 even 12
525.2.q.d.374.1 4 105.17 odd 12
525.2.q.d.374.2 4 35.3 even 12
525.2.q.d.374.2 4 105.38 odd 12
525.2.t.c.26.1 2 5.4 even 2
525.2.t.c.26.1 2 15.14 odd 2
525.2.t.c.101.1 2 35.24 odd 6
525.2.t.c.101.1 2 105.59 even 6
567.2.i.b.215.1 2 9.2 odd 6
567.2.i.b.215.1 2 9.7 even 3
567.2.i.b.269.1 2 63.31 odd 6
567.2.i.b.269.1 2 63.59 even 6
567.2.s.a.26.1 2 9.4 even 3
567.2.s.a.26.1 2 9.5 odd 6
567.2.s.a.458.1 2 63.38 even 6
567.2.s.a.458.1 2 63.52 odd 6
2352.2.k.c.881.1 2 28.19 even 6
2352.2.k.c.881.1 2 84.47 odd 6
2352.2.k.c.881.2 2 28.23 odd 6
2352.2.k.c.881.2 2 84.23 even 6