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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2106,2,Mod(649,2106)] 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("2106.649"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2106, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 2106 = 2 \cdot 3^{4} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2106.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,0,-6,0,0,0,0,0,8,0,0,2,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(14)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(16.8164946657\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.5089536.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 16x^{2} - 24x + 18 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 649.3
Root \(1.66044 + 1.66044i\) of defining polynomial
Character \(\chi\) \(=\) 2106.649
Dual form 2106.2.b.a.649.4

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000i q^{2} -1.00000 q^{4} +3.51414i q^{5} +2.32088i q^{7} +1.00000i q^{8} +3.51414 q^{10} -5.83502i q^{11} +(0.806748 - 3.51414i) q^{13} +2.32088 q^{14} +1.00000 q^{16} +6.34916 q^{17} -4.12763i q^{19} -3.51414i q^{20} -5.83502 q^{22} +1.80675 q^{23} -7.34916 q^{25} +(-3.51414 - 0.806748i) q^{26} -2.32088i q^{28} +1.19325 q^{29} +5.32088i q^{31} -1.00000i q^{32} -6.34916i q^{34} -8.15591 q^{35} +8.05655i q^{37} -4.12763 q^{38} -3.51414 q^{40} -7.47679i q^{41} -0.386505 q^{43} +5.83502i q^{44} -1.80675i q^{46} +1.70739i q^{47} +1.61350 q^{49} +7.34916i q^{50} +(-0.806748 + 3.51414i) q^{52} +13.5424 q^{53} +20.5051 q^{55} -2.32088 q^{56} -1.19325i q^{58} +8.57068i q^{59} +6.76940 q^{61} +5.32088 q^{62} -1.00000 q^{64} +(12.3492 + 2.83502i) q^{65} +3.61350i q^{67} -6.34916 q^{68} +8.15591i q^{70} -12.9627i q^{71} +2.38650i q^{73} +8.05655 q^{74} +4.12763i q^{76} +13.5424 q^{77} -3.80675 q^{79} +3.51414i q^{80} -7.47679 q^{82} +8.22153i q^{83} +22.3118i q^{85} +0.386505i q^{86} +5.83502 q^{88} -8.83502i q^{89} +(8.15591 + 1.87237i) q^{91} -1.80675 q^{92} +1.70739 q^{94} +14.5051 q^{95} +18.2498i q^{97} -1.61350i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 6 q^{4} + 8 q^{10} + 2 q^{13} - 2 q^{14} + 6 q^{16} - 4 q^{17} - 6 q^{22} + 8 q^{23} - 2 q^{25} - 8 q^{26} + 10 q^{29} - 4 q^{35} - 6 q^{38} - 8 q^{40} - 8 q^{43} + 4 q^{49} - 2 q^{52} + 42 q^{53}+ \cdots - 8 q^{92}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2106\mathbb{Z}\right)^\times\).

\(n\) \(1379\) \(1783\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 1.00000i − 0.707107i
\(3\) 0 0
\(4\) −1.00000 −0.500000
\(5\) 3.51414i 1.57157i 0.618500 + 0.785785i \(0.287741\pi\)
−0.618500 + 0.785785i \(0.712259\pi\)
\(6\) 0 0
\(7\) 2.32088i 0.877212i 0.898679 + 0.438606i \(0.144528\pi\)
−0.898679 + 0.438606i \(0.855472\pi\)
\(8\) 1.00000i 0.353553i
\(9\) 0 0
\(10\) 3.51414 1.11127
\(11\) − 5.83502i − 1.75933i −0.475598 − 0.879663i \(-0.657768\pi\)
0.475598 − 0.879663i \(-0.342232\pi\)
\(12\) 0 0
\(13\) 0.806748 − 3.51414i 0.223752 − 0.974646i
\(14\) 2.32088 0.620282
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 6.34916 1.53990 0.769949 − 0.638106i \(-0.220282\pi\)
0.769949 + 0.638106i \(0.220282\pi\)
\(18\) 0 0
\(19\) − 4.12763i − 0.946944i −0.880809 − 0.473472i \(-0.843001\pi\)
0.880809 − 0.473472i \(-0.156999\pi\)
\(20\) − 3.51414i − 0.785785i
\(21\) 0 0
\(22\) −5.83502 −1.24403
\(23\) 1.80675 0.376733 0.188366 − 0.982099i \(-0.439681\pi\)
0.188366 + 0.982099i \(0.439681\pi\)
\(24\) 0 0
\(25\) −7.34916 −1.46983
\(26\) −3.51414 − 0.806748i −0.689179 − 0.158216i
\(27\) 0 0
\(28\) − 2.32088i − 0.438606i
\(29\) 1.19325 0.221581 0.110791 − 0.993844i \(-0.464662\pi\)
0.110791 + 0.993844i \(0.464662\pi\)
\(30\) 0 0
\(31\) 5.32088i 0.955659i 0.878453 + 0.477830i \(0.158576\pi\)
−0.878453 + 0.477830i \(0.841424\pi\)
\(32\) − 1.00000i − 0.176777i
\(33\) 0 0
\(34\) − 6.34916i − 1.08887i
\(35\) −8.15591 −1.37860
\(36\) 0 0
\(37\) 8.05655i 1.32449i 0.749288 + 0.662244i \(0.230396\pi\)
−0.749288 + 0.662244i \(0.769604\pi\)
\(38\) −4.12763 −0.669590
\(39\) 0 0
\(40\) −3.51414 −0.555634
\(41\) − 7.47679i − 1.16768i −0.811869 − 0.583839i \(-0.801550\pi\)
0.811869 − 0.583839i \(-0.198450\pi\)
\(42\) 0 0
\(43\) −0.386505 −0.0589414 −0.0294707 − 0.999566i \(-0.509382\pi\)
−0.0294707 + 0.999566i \(0.509382\pi\)
\(44\) 5.83502i 0.879663i
\(45\) 0 0
\(46\) − 1.80675i − 0.266390i
\(47\) 1.70739i 0.249048i 0.992217 + 0.124524i \(0.0397404\pi\)
−0.992217 + 0.124524i \(0.960260\pi\)
\(48\) 0 0
\(49\) 1.61350 0.230499
\(50\) 7.34916i 1.03933i
\(51\) 0 0
\(52\) −0.806748 + 3.51414i −0.111876 + 0.487323i
\(53\) 13.5424 1.86019 0.930097 − 0.367315i \(-0.119723\pi\)
0.930097 + 0.367315i \(0.119723\pi\)
\(54\) 0 0
\(55\) 20.5051 2.76490
\(56\) −2.32088 −0.310141
\(57\) 0 0
\(58\) − 1.19325i − 0.156682i
\(59\) 8.57068i 1.11581i 0.829905 + 0.557904i \(0.188394\pi\)
−0.829905 + 0.557904i \(0.811606\pi\)
\(60\) 0 0
\(61\) 6.76940 0.866733 0.433366 − 0.901218i \(-0.357326\pi\)
0.433366 + 0.901218i \(0.357326\pi\)
\(62\) 5.32088 0.675753
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) 12.3492 + 2.83502i 1.53172 + 0.351641i
\(66\) 0 0
\(67\) 3.61350i 0.441459i 0.975335 + 0.220729i \(0.0708437\pi\)
−0.975335 + 0.220729i \(0.929156\pi\)
\(68\) −6.34916 −0.769949
\(69\) 0 0
\(70\) 8.15591i 0.974817i
\(71\) − 12.9627i − 1.53838i −0.639018 − 0.769192i \(-0.720659\pi\)
0.639018 − 0.769192i \(-0.279341\pi\)
\(72\) 0 0
\(73\) 2.38650i 0.279319i 0.990200 + 0.139660i \(0.0446008\pi\)
−0.990200 + 0.139660i \(0.955399\pi\)
\(74\) 8.05655 0.936555
\(75\) 0 0
\(76\) 4.12763i 0.473472i
\(77\) 13.5424 1.54330
\(78\) 0 0
\(79\) −3.80675 −0.428293 −0.214146 − 0.976802i \(-0.568697\pi\)
−0.214146 + 0.976802i \(0.568697\pi\)
\(80\) 3.51414i 0.392892i
\(81\) 0 0
\(82\) −7.47679 −0.825673
\(83\) 8.22153i 0.902430i 0.892415 + 0.451215i \(0.149009\pi\)
−0.892415 + 0.451215i \(0.850991\pi\)
\(84\) 0 0
\(85\) 22.3118i 2.42006i
\(86\) 0.386505i 0.0416779i
\(87\) 0 0
\(88\) 5.83502 0.622015
\(89\) − 8.83502i − 0.936510i −0.883593 − 0.468255i \(-0.844883\pi\)
0.883593 − 0.468255i \(-0.155117\pi\)
\(90\) 0 0
\(91\) 8.15591 + 1.87237i 0.854971 + 0.196277i
\(92\) −1.80675 −0.188366
\(93\) 0 0
\(94\) 1.70739 0.176104
\(95\) 14.5051 1.48819
\(96\) 0 0
\(97\) 18.2498i 1.85299i 0.376311 + 0.926493i \(0.377192\pi\)
−0.376311 + 0.926493i \(0.622808\pi\)
\(98\) − 1.61350i − 0.162988i
\(99\) 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 2106.2.b.a.649.3 ✓ 6
3.2 odd 2 2106.2.b.b.649.4 yes 6
13.12 even 2 inner 2106.2.b.a.649.4 yes 6
39.38 odd 2 2106.2.b.b.649.3 yes 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2106.2.b.a.649.3 ✓ 6 1.1 even 1 trivial
2106.2.b.a.649.4 yes 6 13.12 even 2 inner
2106.2.b.b.649.3 yes 6 39.38 odd 2
2106.2.b.b.649.4 yes 6 3.2 odd 2