Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [105,2,Mod(104,105)] 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("105.104"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(105, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 105 = 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 105.g (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.838429221223\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-5}, \sqrt{7})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 104.4
Root \(-1.32288 + 1.11803i\) of defining polynomial
Character \(\chi\) \(=\) 105.104
Dual form 105.2.g.b.104.3

$q$-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)
 
\(f(q)\) \(=\) \(q+(1.32288 + 1.11803i) q^{3} -2.00000 q^{4} +2.23607i q^{5} +2.64575 q^{7} +(0.500000 + 2.95804i) q^{9} -5.91608i q^{11} +(-2.64575 - 2.23607i) q^{12} -2.64575 q^{13} +(-2.50000 + 2.95804i) q^{15} +4.00000 q^{16} -2.23607i q^{17} -4.47214i q^{20} +(3.50000 + 2.95804i) q^{21} -5.00000 q^{25} +(-2.64575 + 4.47214i) q^{27} -5.29150 q^{28} -5.91608i q^{29} +(6.61438 - 7.82624i) q^{33} +5.91608i q^{35} +(-1.00000 - 5.91608i) q^{36} +(-3.50000 - 2.95804i) q^{39} +11.8322i q^{44} +(-6.61438 + 1.11803i) q^{45} +11.1803i q^{47} +(5.29150 + 4.47214i) q^{48} +7.00000 q^{49} +(2.50000 - 2.95804i) q^{51} +5.29150 q^{52} +13.2288 q^{55} +(5.00000 - 5.91608i) q^{60} +(1.32288 + 7.82624i) q^{63} -8.00000 q^{64} -5.91608i q^{65} +4.47214i q^{68} +11.8322i q^{71} -10.5830 q^{73} +(-6.61438 - 5.59017i) q^{75} -15.6525i q^{77} -1.00000 q^{79} +8.94427i q^{80} +(-8.50000 + 2.95804i) q^{81} -8.94427i q^{83} +(-7.00000 - 5.91608i) q^{84} +5.00000 q^{85} +(6.61438 - 7.82624i) q^{87} -7.00000 q^{91} -18.5203 q^{97} +(17.5000 - 2.95804i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 8 q^{4} + 2 q^{9} - 10 q^{15} + 16 q^{16} + 14 q^{21} - 20 q^{25} - 4 q^{36} - 14 q^{39} + 28 q^{49} + 10 q^{51} + 20 q^{60} - 32 q^{64} - 4 q^{79} - 34 q^{81} - 28 q^{84} + 20 q^{85} - 28 q^{91}+ \cdots + 70 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/105\mathbb{Z}\right)^\times\).

\(n\) \(22\) \(31\) \(71\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(3\) 1.32288 + 1.11803i 0.763763 + 0.645497i
\(4\) −2.00000 −1.00000
\(5\) 2.23607i 1.00000i
\(6\) 0 0
\(7\) 2.64575 1.00000
\(8\) 0 0
\(9\) 0.500000 + 2.95804i 0.166667 + 0.986013i
\(10\) 0 0
\(11\) − 5.91608i − 1.78377i −0.452267 − 0.891883i \(-0.649385\pi\)
0.452267 − 0.891883i \(-0.350615\pi\)
\(12\) −2.64575 − 2.23607i −0.763763 − 0.645497i
\(13\) −2.64575 −0.733799 −0.366900 − 0.930261i \(-0.619581\pi\)
−0.366900 + 0.930261i \(0.619581\pi\)
\(14\) 0 0
\(15\) −2.50000 + 2.95804i −0.645497 + 0.763763i
\(16\) 4.00000 1.00000
\(17\) − 2.23607i − 0.542326i −0.962533 − 0.271163i \(-0.912592\pi\)
0.962533 − 0.271163i \(-0.0874083\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) − 4.47214i − 1.00000i
\(21\) 3.50000 + 2.95804i 0.763763 + 0.645497i
\(22\) 0 0
\(23\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) −5.00000 −1.00000
\(26\) 0 0
\(27\) −2.64575 + 4.47214i −0.509175 + 0.860663i
\(28\) −5.29150 −1.00000
\(29\) − 5.91608i − 1.09859i −0.835629 − 0.549294i \(-0.814897\pi\)
0.835629 − 0.549294i \(-0.185103\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) 0 0
\(33\) 6.61438 − 7.82624i 1.15142 − 1.36237i
\(34\) 0 0
\(35\) 5.91608i 1.00000i
\(36\) −1.00000 − 5.91608i −0.166667 − 0.986013i
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) −3.50000 − 2.95804i −0.560449 − 0.473665i
\(40\) 0 0
\(41\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 11.8322i 1.78377i
\(45\) −6.61438 + 1.11803i −0.986013 + 0.166667i
\(46\) 0 0
\(47\) 11.1803i 1.63082i 0.578884 + 0.815410i \(0.303489\pi\)
−0.578884 + 0.815410i \(0.696511\pi\)
\(48\) 5.29150 + 4.47214i 0.763763 + 0.645497i
\(49\) 7.00000 1.00000
\(50\) 0 0
\(51\) 2.50000 − 2.95804i 0.350070 − 0.414208i
\(52\) 5.29150 0.733799
\(53\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 0 0
\(55\) 13.2288 1.78377
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 5.00000 − 5.91608i 0.645497 − 0.763763i
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) 1.32288 + 7.82624i 0.166667 + 0.986013i
\(64\) −8.00000 −1.00000
\(65\) − 5.91608i − 0.733799i
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) 4.47214i 0.542326i
\(69\) 0 0
\(70\) 0 0
\(71\) 11.8322i 1.40422i 0.712069 + 0.702109i \(0.247758\pi\)
−0.712069 + 0.702109i \(0.752242\pi\)
\(72\) 0 0
\(73\) −10.5830 −1.23865 −0.619324 − 0.785136i \(-0.712593\pi\)
−0.619324 + 0.785136i \(0.712593\pi\)
\(74\) 0 0
\(75\) −6.61438 − 5.59017i −0.763763 − 0.645497i
\(76\) 0 0
\(77\) − 15.6525i − 1.78377i
\(78\) 0 0
\(79\) −1.00000 −0.112509 −0.0562544 − 0.998416i \(-0.517916\pi\)
−0.0562544 + 0.998416i \(0.517916\pi\)
\(80\) 8.94427i 1.00000i
\(81\) −8.50000 + 2.95804i −0.944444 + 0.328671i
\(82\) 0 0
\(83\) − 8.94427i − 0.981761i −0.871227 − 0.490881i \(-0.836675\pi\)
0.871227 − 0.490881i \(-0.163325\pi\)
\(84\) −7.00000 − 5.91608i −0.763763 − 0.645497i
\(85\) 5.00000 0.542326
\(86\) 0 0
\(87\) 6.61438 − 7.82624i 0.709136 − 0.839061i
\(88\) 0 0
\(89\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) −7.00000 −0.733799
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −18.5203 −1.88045 −0.940224 − 0.340557i \(-0.889384\pi\)
−0.940224 + 0.340557i \(0.889384\pi\)
\(98\) 0 0
\(99\) 17.5000 − 2.95804i 1.75882 − 0.297294i
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 105.2.g.b.104.4 yes 4
3.2 odd 2 inner 105.2.g.b.104.3 yes 4
4.3 odd 2 1680.2.k.b.209.1 4
5.2 odd 4 525.2.b.f.251.3 4
5.3 odd 4 525.2.b.f.251.2 4
5.4 even 2 inner 105.2.g.b.104.1 ✓ 4
7.2 even 3 735.2.p.b.374.1 8
7.3 odd 6 735.2.p.b.509.2 8
7.4 even 3 735.2.p.b.509.3 8
7.5 odd 6 735.2.p.b.374.4 8
7.6 odd 2 inner 105.2.g.b.104.1 ✓ 4
12.11 even 2 1680.2.k.b.209.2 4
15.2 even 4 525.2.b.f.251.1 4
15.8 even 4 525.2.b.f.251.4 4
15.14 odd 2 inner 105.2.g.b.104.2 yes 4
20.19 odd 2 1680.2.k.b.209.4 4
21.2 odd 6 735.2.p.b.374.3 8
21.5 even 6 735.2.p.b.374.2 8
21.11 odd 6 735.2.p.b.509.1 8
21.17 even 6 735.2.p.b.509.4 8
21.20 even 2 inner 105.2.g.b.104.2 yes 4
28.27 even 2 1680.2.k.b.209.4 4
35.4 even 6 735.2.p.b.509.2 8
35.9 even 6 735.2.p.b.374.4 8
35.13 even 4 525.2.b.f.251.3 4
35.19 odd 6 735.2.p.b.374.1 8
35.24 odd 6 735.2.p.b.509.3 8
35.27 even 4 525.2.b.f.251.2 4
35.34 odd 2 CM 105.2.g.b.104.4 yes 4
60.59 even 2 1680.2.k.b.209.3 4
84.83 odd 2 1680.2.k.b.209.3 4
105.44 odd 6 735.2.p.b.374.2 8
105.59 even 6 735.2.p.b.509.1 8
105.62 odd 4 525.2.b.f.251.4 4
105.74 odd 6 735.2.p.b.509.4 8
105.83 odd 4 525.2.b.f.251.1 4
105.89 even 6 735.2.p.b.374.3 8
105.104 even 2 inner 105.2.g.b.104.3 yes 4
140.139 even 2 1680.2.k.b.209.1 4
420.419 odd 2 1680.2.k.b.209.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
105.2.g.b.104.1 ✓ 4 5.4 even 2 inner
105.2.g.b.104.1 ✓ 4 7.6 odd 2 inner
105.2.g.b.104.2 yes 4 15.14 odd 2 inner
105.2.g.b.104.2 yes 4 21.20 even 2 inner
105.2.g.b.104.3 yes 4 3.2 odd 2 inner
105.2.g.b.104.3 yes 4 105.104 even 2 inner
105.2.g.b.104.4 yes 4 1.1 even 1 trivial
105.2.g.b.104.4 yes 4 35.34 odd 2 CM
525.2.b.f.251.1 4 15.2 even 4
525.2.b.f.251.1 4 105.83 odd 4
525.2.b.f.251.2 4 5.3 odd 4
525.2.b.f.251.2 4 35.27 even 4
525.2.b.f.251.3 4 5.2 odd 4
525.2.b.f.251.3 4 35.13 even 4
525.2.b.f.251.4 4 15.8 even 4
525.2.b.f.251.4 4 105.62 odd 4
735.2.p.b.374.1 8 7.2 even 3
735.2.p.b.374.1 8 35.19 odd 6
735.2.p.b.374.2 8 21.5 even 6
735.2.p.b.374.2 8 105.44 odd 6
735.2.p.b.374.3 8 21.2 odd 6
735.2.p.b.374.3 8 105.89 even 6
735.2.p.b.374.4 8 7.5 odd 6
735.2.p.b.374.4 8 35.9 even 6
735.2.p.b.509.1 8 21.11 odd 6
735.2.p.b.509.1 8 105.59 even 6
735.2.p.b.509.2 8 7.3 odd 6
735.2.p.b.509.2 8 35.4 even 6
735.2.p.b.509.3 8 7.4 even 3
735.2.p.b.509.3 8 35.24 odd 6
735.2.p.b.509.4 8 21.17 even 6
735.2.p.b.509.4 8 105.74 odd 6
1680.2.k.b.209.1 4 4.3 odd 2
1680.2.k.b.209.1 4 140.139 even 2
1680.2.k.b.209.2 4 12.11 even 2
1680.2.k.b.209.2 4 420.419 odd 2
1680.2.k.b.209.3 4 60.59 even 2
1680.2.k.b.209.3 4 84.83 odd 2
1680.2.k.b.209.4 4 20.19 odd 2
1680.2.k.b.209.4 4 28.27 even 2