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Properties

Label 140.2.c.a.139.4
Level $140$
Weight $2$
Character 140.139
Analytic conductor $1.118$
Analytic rank $0$
Dimension $4$
CM discriminant -20
Inner twists $8$

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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] = [140,2,Mod(139,140)] 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("140.139"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(140, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 140 = 2^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 140.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4] 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: \(1.11790562830\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-5})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 4x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 139.4
Root \(-0.707107 - 1.58114i\) of defining polynomial
Character \(\chi\) \(=\) 140.139
Dual form 140.2.c.a.139.3

$q$-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)
 
\(f(q)\) \(=\) \(q+1.41421 q^{2} +3.16228i q^{3} +2.00000 q^{4} -2.23607i q^{5} +4.47214i q^{6} +(-2.12132 - 1.58114i) q^{7} +2.82843 q^{8} -7.00000 q^{9} -3.16228i q^{10} +6.32456i q^{12} +(-3.00000 - 2.23607i) q^{14} +7.07107 q^{15} +4.00000 q^{16} -9.89949 q^{18} -4.47214i q^{20} +(5.00000 - 6.70820i) q^{21} -1.41421 q^{23} +8.94427i q^{24} -5.00000 q^{25} -12.6491i q^{27} +(-4.24264 - 3.16228i) q^{28} +6.00000 q^{29} +10.0000 q^{30} +5.65685 q^{32} +(-3.53553 + 4.74342i) q^{35} -14.0000 q^{36} -6.32456i q^{40} +4.47214i q^{41} +(7.07107 - 9.48683i) q^{42} -12.7279 q^{43} +15.6525i q^{45} -2.00000 q^{46} +9.48683i q^{47} +12.6491i q^{48} +(2.00000 + 6.70820i) q^{49} -7.07107 q^{50} -17.8885i q^{54} +(-6.00000 - 4.47214i) q^{56} +8.48528 q^{58} +14.1421 q^{60} -13.4164i q^{61} +(14.8492 + 11.0680i) q^{63} +8.00000 q^{64} +4.24264 q^{67} -4.47214i q^{69} +(-5.00000 + 6.70820i) q^{70} -19.7990 q^{72} -15.8114i q^{75} -8.94427i q^{80} +19.0000 q^{81} +6.32456i q^{82} -9.48683i q^{83} +(10.0000 - 13.4164i) q^{84} -18.0000 q^{86} +18.9737i q^{87} +17.8885i q^{89} +22.1359i q^{90} -2.82843 q^{92} +13.4164i q^{94} +17.8885i q^{96} +(2.82843 + 9.48683i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 8 q^{4} - 28 q^{9} - 12 q^{14} + 16 q^{16} + 20 q^{21} - 20 q^{25} + 24 q^{29} + 40 q^{30} - 56 q^{36} - 8 q^{46} + 8 q^{49} - 24 q^{56} + 32 q^{64} - 20 q^{70} + 76 q^{81} + 40 q^{84} - 72 q^{86}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/140\mathbb{Z}\right)^\times\).

\(n\) \(57\) \(71\) \(101\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.41421 1.00000
\(3\) 3.16228i 1.82574i 0.408248 + 0.912871i \(0.366140\pi\)
−0.408248 + 0.912871i \(0.633860\pi\)
\(4\) 2.00000 1.00000
\(5\) − 2.23607i − 1.00000i
\(6\) 4.47214i 1.82574i
\(7\) −2.12132 − 1.58114i −0.801784 − 0.597614i
\(8\) 2.82843 1.00000
\(9\) −7.00000 −2.33333
\(10\) − 3.16228i − 1.00000i
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 6.32456i 1.82574i
\(13\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) −3.00000 − 2.23607i −0.801784 − 0.597614i
\(15\) 7.07107 1.82574
\(16\) 4.00000 1.00000
\(17\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) −9.89949 −2.33333
\(19\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) − 4.47214i − 1.00000i
\(21\) 5.00000 − 6.70820i 1.09109 − 1.46385i
\(22\) 0 0
\(23\) −1.41421 −0.294884 −0.147442 − 0.989071i \(-0.547104\pi\)
−0.147442 + 0.989071i \(0.547104\pi\)
\(24\) 8.94427i 1.82574i
\(25\) −5.00000 −1.00000
\(26\) 0 0
\(27\) − 12.6491i − 2.43432i
\(28\) −4.24264 − 3.16228i −0.801784 − 0.597614i
\(29\) 6.00000 1.11417 0.557086 − 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 10.0000 1.82574
\(31\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 5.65685 1.00000
\(33\) 0 0
\(34\) 0 0
\(35\) −3.53553 + 4.74342i −0.597614 + 0.801784i
\(36\) −14.0000 −2.33333
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) − 6.32456i − 1.00000i
\(41\) 4.47214i 0.698430i 0.937043 + 0.349215i \(0.113552\pi\)
−0.937043 + 0.349215i \(0.886448\pi\)
\(42\) 7.07107 − 9.48683i 1.09109 − 1.46385i
\(43\) −12.7279 −1.94099 −0.970495 − 0.241121i \(-0.922485\pi\)
−0.970495 + 0.241121i \(0.922485\pi\)
\(44\) 0 0
\(45\) 15.6525i 2.33333i
\(46\) −2.00000 −0.294884
\(47\) 9.48683i 1.38380i 0.721995 + 0.691898i \(0.243225\pi\)
−0.721995 + 0.691898i \(0.756775\pi\)
\(48\) 12.6491i 1.82574i
\(49\) 2.00000 + 6.70820i 0.285714 + 0.958315i
\(50\) −7.07107 −1.00000
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) − 17.8885i − 2.43432i
\(55\) 0 0
\(56\) −6.00000 − 4.47214i −0.801784 − 0.597614i
\(57\) 0 0
\(58\) 8.48528 1.11417
\(59\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 14.1421 1.82574
\(61\) − 13.4164i − 1.71780i −0.512148 − 0.858898i \(-0.671150\pi\)
0.512148 − 0.858898i \(-0.328850\pi\)
\(62\) 0 0
\(63\) 14.8492 + 11.0680i 1.87083 + 1.39443i
\(64\) 8.00000 1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) 4.24264 0.518321 0.259161 − 0.965834i \(-0.416554\pi\)
0.259161 + 0.965834i \(0.416554\pi\)
\(68\) 0 0
\(69\) − 4.47214i − 0.538382i
\(70\) −5.00000 + 6.70820i −0.597614 + 0.801784i
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) −19.7990 −2.33333
\(73\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(74\) 0 0
\(75\) − 15.8114i − 1.82574i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) − 8.94427i − 1.00000i
\(81\) 19.0000 2.11111
\(82\) 6.32456i 0.698430i
\(83\) − 9.48683i − 1.04132i −0.853766 − 0.520658i \(-0.825687\pi\)
0.853766 − 0.520658i \(-0.174313\pi\)
\(84\) 10.0000 − 13.4164i 1.09109 − 1.46385i
\(85\) 0 0
\(86\) −18.0000 −1.94099
\(87\) 18.9737i 2.03419i
\(88\) 0 0
\(89\) 17.8885i 1.89618i 0.317999 + 0.948091i \(0.396989\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 22.1359i 2.33333i
\(91\) 0 0
\(92\) −2.82843 −0.294884
\(93\) 0 0
\(94\) 13.4164i 1.38380i
\(95\) 0 0
\(96\) 17.8885i 1.82574i
\(97\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(98\) 2.82843 + 9.48683i 0.285714 + 0.958315i
\(99\) 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 140.2.c.a.139.4 yes 4
4.3 odd 2 inner 140.2.c.a.139.1 ✓ 4
5.2 odd 4 700.2.g.d.251.4 4
5.3 odd 4 700.2.g.d.251.1 4
5.4 even 2 inner 140.2.c.a.139.1 ✓ 4
7.2 even 3 980.2.s.b.619.1 8
7.3 odd 6 980.2.s.b.19.1 8
7.4 even 3 980.2.s.b.19.2 8
7.5 odd 6 980.2.s.b.619.2 8
7.6 odd 2 inner 140.2.c.a.139.3 yes 4
8.3 odd 2 2240.2.e.a.2239.4 4
8.5 even 2 2240.2.e.a.2239.2 4
20.3 even 4 700.2.g.d.251.4 4
20.7 even 4 700.2.g.d.251.1 4
20.19 odd 2 CM 140.2.c.a.139.4 yes 4
28.3 even 6 980.2.s.b.19.4 8
28.11 odd 6 980.2.s.b.19.3 8
28.19 even 6 980.2.s.b.619.3 8
28.23 odd 6 980.2.s.b.619.4 8
28.27 even 2 inner 140.2.c.a.139.2 yes 4
35.4 even 6 980.2.s.b.19.3 8
35.9 even 6 980.2.s.b.619.4 8
35.13 even 4 700.2.g.d.251.2 4
35.19 odd 6 980.2.s.b.619.3 8
35.24 odd 6 980.2.s.b.19.4 8
35.27 even 4 700.2.g.d.251.3 4
35.34 odd 2 inner 140.2.c.a.139.2 yes 4
40.19 odd 2 2240.2.e.a.2239.2 4
40.29 even 2 2240.2.e.a.2239.4 4
56.13 odd 2 2240.2.e.a.2239.3 4
56.27 even 2 2240.2.e.a.2239.1 4
140.19 even 6 980.2.s.b.619.2 8
140.27 odd 4 700.2.g.d.251.2 4
140.39 odd 6 980.2.s.b.19.2 8
140.59 even 6 980.2.s.b.19.1 8
140.79 odd 6 980.2.s.b.619.1 8
140.83 odd 4 700.2.g.d.251.3 4
140.139 even 2 inner 140.2.c.a.139.3 yes 4
280.69 odd 2 2240.2.e.a.2239.1 4
280.139 even 2 2240.2.e.a.2239.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
140.2.c.a.139.1 ✓ 4 4.3 odd 2 inner
140.2.c.a.139.1 ✓ 4 5.4 even 2 inner
140.2.c.a.139.2 yes 4 28.27 even 2 inner
140.2.c.a.139.2 yes 4 35.34 odd 2 inner
140.2.c.a.139.3 yes 4 7.6 odd 2 inner
140.2.c.a.139.3 yes 4 140.139 even 2 inner
140.2.c.a.139.4 yes 4 1.1 even 1 trivial
140.2.c.a.139.4 yes 4 20.19 odd 2 CM
700.2.g.d.251.1 4 5.3 odd 4
700.2.g.d.251.1 4 20.7 even 4
700.2.g.d.251.2 4 35.13 even 4
700.2.g.d.251.2 4 140.27 odd 4
700.2.g.d.251.3 4 35.27 even 4
700.2.g.d.251.3 4 140.83 odd 4
700.2.g.d.251.4 4 5.2 odd 4
700.2.g.d.251.4 4 20.3 even 4
980.2.s.b.19.1 8 7.3 odd 6
980.2.s.b.19.1 8 140.59 even 6
980.2.s.b.19.2 8 7.4 even 3
980.2.s.b.19.2 8 140.39 odd 6
980.2.s.b.19.3 8 28.11 odd 6
980.2.s.b.19.3 8 35.4 even 6
980.2.s.b.19.4 8 28.3 even 6
980.2.s.b.19.4 8 35.24 odd 6
980.2.s.b.619.1 8 7.2 even 3
980.2.s.b.619.1 8 140.79 odd 6
980.2.s.b.619.2 8 7.5 odd 6
980.2.s.b.619.2 8 140.19 even 6
980.2.s.b.619.3 8 28.19 even 6
980.2.s.b.619.3 8 35.19 odd 6
980.2.s.b.619.4 8 28.23 odd 6
980.2.s.b.619.4 8 35.9 even 6
2240.2.e.a.2239.1 4 56.27 even 2
2240.2.e.a.2239.1 4 280.69 odd 2
2240.2.e.a.2239.2 4 8.5 even 2
2240.2.e.a.2239.2 4 40.19 odd 2
2240.2.e.a.2239.3 4 56.13 odd 2
2240.2.e.a.2239.3 4 280.139 even 2
2240.2.e.a.2239.4 4 8.3 odd 2
2240.2.e.a.2239.4 4 40.29 even 2