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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 649.1
Root \(0.675970 + 0.675970i\) of defining polynomial
Character \(\chi\) \(=\) 2106.649
Dual form 2106.2.b.a.649.6

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000i q^{2} -1.00000 q^{4} -1.08613i q^{5} +0.351939i q^{7} +1.00000i q^{8} -1.08613 q^{10} +0.734191i q^{11} +(3.43807 + 1.08613i) q^{13} +0.351939 q^{14} +1.00000 q^{16} -4.82032 q^{17} -4.79001i q^{19} +1.08613i q^{20} +0.734191 q^{22} +4.43807 q^{23} +3.82032 q^{25} +(1.08613 - 3.43807i) q^{26} -0.351939i q^{28} -1.43807 q^{29} +3.35194i q^{31} -1.00000i q^{32} +4.82032i q^{34} +0.382252 q^{35} -10.3445i q^{37} -4.79001 q^{38} +1.08613 q^{40} +3.03031i q^{41} +4.87614 q^{43} -0.734191i q^{44} -4.43807i q^{46} -5.52420i q^{47} +6.87614 q^{49} -3.82032i q^{50} +(-3.43807 - 1.08613i) q^{52} -0.258391 q^{53} +0.797427 q^{55} -0.351939 q^{56} +1.43807i q^{58} -14.4307i q^{59} +3.49389 q^{61} +3.35194 q^{62} -1.00000 q^{64} +(1.17968 - 3.73419i) q^{65} +8.87614i q^{67} +4.82032 q^{68} -0.382252i q^{70} -7.05582i q^{71} -2.87614i q^{73} -10.3445 q^{74} +4.79001i q^{76} -0.258391 q^{77} -6.43807 q^{79} -1.08613i q^{80} +3.03031 q^{82} -3.61033i q^{83} +5.23550i q^{85} -4.87614i q^{86} -0.734191 q^{88} -2.26581i q^{89} +(-0.382252 + 1.20999i) q^{91} -4.43807 q^{92} -5.52420 q^{94} -5.20257 q^{95} -2.78259i q^{97} -6.87614i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 6 q^{4} + 8 q^{10} + 2 q^{13} - 2 q^{14} + 6 q^{16} - 4 q^{17} - 6 q^{22} + 8 q^{23} - 2 q^{25} - 8 q^{26} + 10 q^{29} - 4 q^{35} - 6 q^{38} - 8 q^{40} - 8 q^{43} + 4 q^{49} - 2 q^{52} + 42 q^{53}+ \cdots - 8 q^{92}+O(q^{100}) \) 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
\(5\) − 1.08613i − 0.485732i −0.970060 − 0.242866i \(-0.921912\pi\)
0.970060 − 0.242866i \(-0.0780876\pi\)
\(6\) 0 0
\(7\) 0.351939i 0.133021i 0.997786 + 0.0665103i \(0.0211865\pi\)
−0.997786 + 0.0665103i \(0.978813\pi\)
\(8\) 1.00000i 0.353553i
\(9\) 0 0
\(10\) −1.08613 −0.343465
\(11\) 0.734191i 0.221367i 0.993856 + 0.110683i \(0.0353040\pi\)
−0.993856 + 0.110683i \(0.964696\pi\)
\(12\) 0 0
\(13\) 3.43807 + 1.08613i 0.953549 + 0.301238i
\(14\) 0.351939 0.0940597
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −4.82032 −1.16910 −0.584550 − 0.811358i \(-0.698729\pi\)
−0.584550 + 0.811358i \(0.698729\pi\)
\(18\) 0 0
\(19\) − 4.79001i − 1.09890i −0.835526 − 0.549452i \(-0.814837\pi\)
0.835526 − 0.549452i \(-0.185163\pi\)
\(20\) 1.08613i 0.242866i
\(21\) 0 0
\(22\) 0.734191 0.156530
\(23\) 4.43807 0.925401 0.462701 − 0.886515i \(-0.346880\pi\)
0.462701 + 0.886515i \(0.346880\pi\)
\(24\) 0 0
\(25\) 3.82032 0.764064
\(26\) 1.08613 − 3.43807i 0.213008 − 0.674261i
\(27\) 0 0
\(28\) − 0.351939i − 0.0665103i
\(29\) −1.43807 −0.267043 −0.133521 − 0.991046i \(-0.542628\pi\)
−0.133521 + 0.991046i \(0.542628\pi\)
\(30\) 0 0
\(31\) 3.35194i 0.602026i 0.953620 + 0.301013i \(0.0973248\pi\)
−0.953620 + 0.301013i \(0.902675\pi\)
\(32\) − 1.00000i − 0.176777i
\(33\) 0 0
\(34\) 4.82032i 0.826678i
\(35\) 0.382252 0.0646123
\(36\) 0 0
\(37\) − 10.3445i − 1.70063i −0.526275 − 0.850314i \(-0.676412\pi\)
0.526275 − 0.850314i \(-0.323588\pi\)
\(38\) −4.79001 −0.777042
\(39\) 0 0
\(40\) 1.08613 0.171732
\(41\) 3.03031i 0.473255i 0.971600 + 0.236628i \(0.0760422\pi\)
−0.971600 + 0.236628i \(0.923958\pi\)
\(42\) 0 0
\(43\) 4.87614 0.743604 0.371802 − 0.928312i \(-0.378740\pi\)
0.371802 + 0.928312i \(0.378740\pi\)
\(44\) − 0.734191i − 0.110683i
\(45\) 0 0
\(46\) − 4.43807i − 0.654358i
\(47\) − 5.52420i − 0.805787i −0.915247 − 0.402894i \(-0.868004\pi\)
0.915247 − 0.402894i \(-0.131996\pi\)
\(48\) 0 0
\(49\) 6.87614 0.982306
\(50\) − 3.82032i − 0.540275i
\(51\) 0 0
\(52\) −3.43807 − 1.08613i −0.476774 − 0.150619i
\(53\) −0.258391 −0.0354927 −0.0177463 − 0.999843i \(-0.505649\pi\)
−0.0177463 + 0.999843i \(0.505649\pi\)
\(54\) 0 0
\(55\) 0.797427 0.107525
\(56\) −0.351939 −0.0470299
\(57\) 0 0
\(58\) 1.43807i 0.188828i
\(59\) − 14.4307i − 1.87871i −0.342945 − 0.939355i \(-0.611425\pi\)
0.342945 − 0.939355i \(-0.388575\pi\)
\(60\) 0 0
\(61\) 3.49389 0.447346 0.223673 − 0.974664i \(-0.428195\pi\)
0.223673 + 0.974664i \(0.428195\pi\)
\(62\) 3.35194 0.425697
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) 1.17968 − 3.73419i 0.146321 − 0.463169i
\(66\) 0 0
\(67\) 8.87614i 1.08439i 0.840252 + 0.542196i \(0.182407\pi\)
−0.840252 + 0.542196i \(0.817593\pi\)
\(68\) 4.82032 0.584550
\(69\) 0 0
\(70\) − 0.382252i − 0.0456878i
\(71\) − 7.05582i − 0.837371i −0.908131 − 0.418686i \(-0.862491\pi\)
0.908131 − 0.418686i \(-0.137509\pi\)
\(72\) 0 0
\(73\) − 2.87614i − 0.336627i −0.985734 − 0.168313i \(-0.946168\pi\)
0.985734 − 0.168313i \(-0.0538320\pi\)
\(74\) −10.3445 −1.20253
\(75\) 0 0
\(76\) 4.79001i 0.549452i
\(77\) −0.258391 −0.0294463
\(78\) 0 0
\(79\) −6.43807 −0.724339 −0.362170 − 0.932112i \(-0.617964\pi\)
−0.362170 + 0.932112i \(0.617964\pi\)
\(80\) − 1.08613i − 0.121433i
\(81\) 0 0
\(82\) 3.03031 0.334642
\(83\) − 3.61033i − 0.396285i −0.980173 − 0.198143i \(-0.936509\pi\)
0.980173 − 0.198143i \(-0.0634909\pi\)
\(84\) 0 0
\(85\) 5.23550i 0.567869i
\(86\) − 4.87614i − 0.525808i
\(87\) 0 0
\(88\) −0.734191 −0.0782650
\(89\) − 2.26581i − 0.240175i −0.992763 − 0.120088i \(-0.961682\pi\)
0.992763 − 0.120088i \(-0.0383176\pi\)
\(90\) 0 0
\(91\) −0.382252 + 1.20999i −0.0400709 + 0.126842i
\(92\) −4.43807 −0.462701
\(93\) 0 0
\(94\) −5.52420 −0.569778
\(95\) −5.20257 −0.533773
\(96\) 0 0
\(97\) − 2.78259i − 0.282529i −0.989972 − 0.141265i \(-0.954883\pi\)
0.989972 − 0.141265i \(-0.0451168\pi\)
\(98\) − 6.87614i − 0.694595i
\(99\) 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.649.1 ✓ 6
3.2 odd 2 2106.2.b.b.649.6 yes 6
13.12 even 2 inner 2106.2.b.a.649.6 yes 6
39.38 odd 2 2106.2.b.b.649.1 yes 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2106.2.b.a.649.1 ✓ 6 1.1 even 1 trivial
2106.2.b.a.649.6 yes 6 13.12 even 2 inner
2106.2.b.b.649.1 yes 6 39.38 odd 2
2106.2.b.b.649.6 yes 6 3.2 odd 2