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Properties

Label 21.4.c.a.20.2
Level $21$
Weight $4$
Character 21.20
Analytic conductor $1.239$
Analytic rank $0$
Dimension $2$
CM discriminant -3
Inner twists $4$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [21,4,Mod(20,21)] 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("21.20"); 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([1, 1])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 21 = 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 21.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.23904011012\)
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: \( 2\cdot 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 20.2
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 21.20
Dual form 21.4.c.a.20.1

$q$-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)
 
\(f(q)\) \(=\) \(q+5.19615i q^{3} +8.00000 q^{4} +(-10.0000 - 15.5885i) q^{7} -27.0000 q^{9} +41.5692i q^{12} -62.3538i q^{13} +64.0000 q^{16} +155.885i q^{19} +(81.0000 - 51.9615i) q^{21} -125.000 q^{25} -140.296i q^{27} +(-80.0000 - 124.708i) q^{28} +155.885i q^{31} -216.000 q^{36} -110.000 q^{37} +324.000 q^{39} +520.000 q^{43} +332.554i q^{48} +(-143.000 + 311.769i) q^{49} -498.831i q^{52} -810.000 q^{57} -935.307i q^{61} +(270.000 + 420.888i) q^{63} +512.000 q^{64} -880.000 q^{67} +374.123i q^{73} -649.519i q^{75} +1247.08i q^{76} +884.000 q^{79} +729.000 q^{81} +(648.000 - 415.692i) q^{84} +(-972.000 + 623.538i) q^{91} -810.000 q^{93} -1371.78i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 16 q^{4} - 20 q^{7} - 54 q^{9} + 128 q^{16} + 162 q^{21} - 250 q^{25} - 160 q^{28} - 432 q^{36} - 220 q^{37} + 648 q^{39} + 1040 q^{43} - 286 q^{49} - 1620 q^{57} + 540 q^{63} + 1024 q^{64} - 1760 q^{67}+ \cdots - 1620 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(8\) \(10\)
\(\chi(n)\) \(-1\) \(-1\)

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 1.00000 \(0\)
−1.00000 \(\pi\)
\(3\) 5.19615i 1.00000i
\(4\) 8.00000 1.00000
\(5\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(6\) 0 0
\(7\) −10.0000 − 15.5885i −0.539949 − 0.841698i
\(8\) 0 0
\(9\) −27.0000 −1.00000
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 41.5692i 1.00000i
\(13\) − 62.3538i − 1.33030i −0.746712 − 0.665148i \(-0.768369\pi\)
0.746712 − 0.665148i \(-0.231631\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 64.0000 1.00000
\(17\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) 155.885i 1.88223i 0.338086 + 0.941115i \(0.390220\pi\)
−0.338086 + 0.941115i \(0.609780\pi\)
\(20\) 0 0
\(21\) 81.0000 − 51.9615i 0.841698 − 0.539949i
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) −125.000 −1.00000
\(26\) 0 0
\(27\) − 140.296i − 1.00000i
\(28\) −80.0000 − 124.708i −0.539949 − 0.841698i
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) 155.885i 0.903151i 0.892233 + 0.451576i \(0.149138\pi\)
−0.892233 + 0.451576i \(0.850862\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) −216.000 −1.00000
\(37\) −110.000 −0.488754 −0.244377 − 0.969680i \(-0.578583\pi\)
−0.244377 + 0.969680i \(0.578583\pi\)
\(38\) 0 0
\(39\) 324.000 1.33030
\(40\) 0 0
\(41\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) 520.000 1.84417 0.922084 − 0.386989i \(-0.126485\pi\)
0.922084 + 0.386989i \(0.126485\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 332.554i 1.00000i
\(49\) −143.000 + 311.769i −0.416910 + 0.908948i
\(50\) 0 0
\(51\) 0 0
\(52\) − 498.831i − 1.33030i
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −810.000 −1.88223
\(58\) 0 0
\(59\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) − 935.307i − 1.96318i −0.191006 − 0.981589i \(-0.561175\pi\)
0.191006 − 0.981589i \(-0.438825\pi\)
\(62\) 0 0
\(63\) 270.000 + 420.888i 0.539949 + 0.841698i
\(64\) 512.000 1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) −880.000 −1.60461 −0.802307 − 0.596912i \(-0.796394\pi\)
−0.802307 + 0.596912i \(0.796394\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) 374.123i 0.599833i 0.953966 + 0.299916i \(0.0969588\pi\)
−0.953966 + 0.299916i \(0.903041\pi\)
\(74\) 0 0
\(75\) − 649.519i − 1.00000i
\(76\) 1247.08i 1.88223i
\(77\) 0 0
\(78\) 0 0
\(79\) 884.000 1.25896 0.629480 − 0.777017i \(-0.283268\pi\)
0.629480 + 0.777017i \(0.283268\pi\)
\(80\) 0 0
\(81\) 729.000 1.00000
\(82\) 0 0
\(83\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 648.000 − 415.692i 0.841698 − 0.539949i
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) −972.000 + 623.538i −1.11971 + 0.718292i
\(92\) 0 0
\(93\) −810.000 −0.903151
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) − 1371.78i − 1.43591i −0.696088 − 0.717957i \(-0.745078\pi\)
0.696088 − 0.717957i \(-0.254922\pi\)
\(98\) 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 21.4.c.a.20.2 yes 2
3.2 odd 2 CM 21.4.c.a.20.2 yes 2
4.3 odd 2 336.4.k.a.209.1 2
7.2 even 3 147.4.g.a.80.1 2
7.3 odd 6 147.4.g.a.68.1 2
7.4 even 3 147.4.g.b.68.1 2
7.5 odd 6 147.4.g.b.80.1 2
7.6 odd 2 inner 21.4.c.a.20.1 ✓ 2
12.11 even 2 336.4.k.a.209.1 2
21.2 odd 6 147.4.g.a.80.1 2
21.5 even 6 147.4.g.b.80.1 2
21.11 odd 6 147.4.g.b.68.1 2
21.17 even 6 147.4.g.a.68.1 2
21.20 even 2 inner 21.4.c.a.20.1 ✓ 2
28.27 even 2 336.4.k.a.209.2 2
84.83 odd 2 336.4.k.a.209.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
21.4.c.a.20.1 ✓ 2 7.6 odd 2 inner
21.4.c.a.20.1 ✓ 2 21.20 even 2 inner
21.4.c.a.20.2 yes 2 1.1 even 1 trivial
21.4.c.a.20.2 yes 2 3.2 odd 2 CM
147.4.g.a.68.1 2 7.3 odd 6
147.4.g.a.68.1 2 21.17 even 6
147.4.g.a.80.1 2 7.2 even 3
147.4.g.a.80.1 2 21.2 odd 6
147.4.g.b.68.1 2 7.4 even 3
147.4.g.b.68.1 2 21.11 odd 6
147.4.g.b.80.1 2 7.5 odd 6
147.4.g.b.80.1 2 21.5 even 6
336.4.k.a.209.1 2 4.3 odd 2
336.4.k.a.209.1 2 12.11 even 2
336.4.k.a.209.2 2 28.27 even 2
336.4.k.a.209.2 2 84.83 odd 2