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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 649.2
Root \(-1.33641 - 1.33641i\) of defining polynomial
Character \(\chi\) \(=\) 2106.649
Dual form 2106.2.b.a.649.5

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000i q^{2} -1.00000 q^{4} +1.57199i q^{5} -3.67282i q^{7} +1.00000i q^{8} +1.57199 q^{10} +2.10083i q^{11} +(-3.24482 - 1.57199i) q^{13} -3.67282 q^{14} +1.00000 q^{16} -3.52884 q^{17} +5.91764i q^{19} -1.57199i q^{20} +2.10083 q^{22} -2.24482 q^{23} +2.52884 q^{25} +(-1.57199 + 3.24482i) q^{26} +3.67282i q^{28} +5.24482 q^{29} -0.672824i q^{31} -1.00000i q^{32} +3.52884i q^{34} +5.77365 q^{35} +0.287973i q^{37} +5.91764 q^{38} -1.57199 q^{40} +12.4465i q^{41} -8.48963 q^{43} -2.10083i q^{44} +2.24482i q^{46} +3.81681i q^{47} -6.48963 q^{49} -2.52884i q^{50} +(3.24482 + 1.57199i) q^{52} +7.71598 q^{53} -3.30249 q^{55} +3.67282 q^{56} -5.24482i q^{58} -1.14003i q^{59} -15.2633 q^{61} -0.672824 q^{62} -1.00000 q^{64} +(2.47116 - 5.10083i) q^{65} -4.48963i q^{67} +3.52884 q^{68} -5.77365i q^{70} +5.01847i q^{71} +10.4896i q^{73} +0.287973 q^{74} -5.91764i q^{76} +7.71598 q^{77} +0.244817 q^{79} +1.57199i q^{80} +12.4465 q^{82} +8.38880i q^{83} -5.54731i q^{85} +8.48963i q^{86} -2.10083 q^{88} -0.899170i q^{89} +(-5.77365 + 11.9176i) q^{91} +2.24482 q^{92} +3.81681 q^{94} -9.30249 q^{95} +14.5328i q^{97} +6.48963i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 6 q^{4} + 8 q^{10} + 2 q^{13} - 2 q^{14} + 6 q^{16} - 4 q^{17} - 6 q^{22} + 8 q^{23} - 2 q^{25} - 8 q^{26} + 10 q^{29} - 4 q^{35} - 6 q^{38} - 8 q^{40} - 8 q^{43} + 4 q^{49} - 2 q^{52} + 42 q^{53}+ \cdots - 8 q^{92}+O(q^{100}) \) 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
\(5\) 1.57199i 0.703017i 0.936185 + 0.351508i \(0.114331\pi\)
−0.936185 + 0.351508i \(0.885669\pi\)
\(6\) 0 0
\(7\) − 3.67282i − 1.38820i −0.719880 − 0.694098i \(-0.755803\pi\)
0.719880 − 0.694098i \(-0.244197\pi\)
\(8\) 1.00000i 0.353553i
\(9\) 0 0
\(10\) 1.57199 0.497108
\(11\) 2.10083i 0.633424i 0.948522 + 0.316712i \(0.102579\pi\)
−0.948522 + 0.316712i \(0.897421\pi\)
\(12\) 0 0
\(13\) −3.24482 − 1.57199i −0.899950 − 0.435992i
\(14\) −3.67282 −0.981603
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −3.52884 −0.855869 −0.427934 − 0.903810i \(-0.640759\pi\)
−0.427934 + 0.903810i \(0.640759\pi\)
\(18\) 0 0
\(19\) 5.91764i 1.35760i 0.734323 + 0.678800i \(0.237500\pi\)
−0.734323 + 0.678800i \(0.762500\pi\)
\(20\) − 1.57199i − 0.351508i
\(21\) 0 0
\(22\) 2.10083 0.447899
\(23\) −2.24482 −0.468077 −0.234038 − 0.972227i \(-0.575194\pi\)
−0.234038 + 0.972227i \(0.575194\pi\)
\(24\) 0 0
\(25\) 2.52884 0.505767
\(26\) −1.57199 + 3.24482i −0.308293 + 0.636361i
\(27\) 0 0
\(28\) 3.67282i 0.694098i
\(29\) 5.24482 0.973938 0.486969 − 0.873419i \(-0.338102\pi\)
0.486969 + 0.873419i \(0.338102\pi\)
\(30\) 0 0
\(31\) − 0.672824i − 0.120843i −0.998173 − 0.0604214i \(-0.980756\pi\)
0.998173 − 0.0604214i \(-0.0192444\pi\)
\(32\) − 1.00000i − 0.176777i
\(33\) 0 0
\(34\) 3.52884i 0.605191i
\(35\) 5.77365 0.975926
\(36\) 0 0
\(37\) 0.287973i 0.0473425i 0.999720 + 0.0236712i \(0.00753549\pi\)
−0.999720 + 0.0236712i \(0.992465\pi\)
\(38\) 5.91764 0.959968
\(39\) 0 0
\(40\) −1.57199 −0.248554
\(41\) 12.4465i 1.94381i 0.235367 + 0.971907i \(0.424371\pi\)
−0.235367 + 0.971907i \(0.575629\pi\)
\(42\) 0 0
\(43\) −8.48963 −1.29466 −0.647329 − 0.762211i \(-0.724114\pi\)
−0.647329 + 0.762211i \(0.724114\pi\)
\(44\) − 2.10083i − 0.316712i
\(45\) 0 0
\(46\) 2.24482i 0.330980i
\(47\) 3.81681i 0.556739i 0.960474 + 0.278369i \(0.0897940\pi\)
−0.960474 + 0.278369i \(0.910206\pi\)
\(48\) 0 0
\(49\) −6.48963 −0.927091
\(50\) − 2.52884i − 0.357632i
\(51\) 0 0
\(52\) 3.24482 + 1.57199i 0.449975 + 0.217996i
\(53\) 7.71598 1.05987 0.529936 − 0.848038i \(-0.322216\pi\)
0.529936 + 0.848038i \(0.322216\pi\)
\(54\) 0 0
\(55\) −3.30249 −0.445308
\(56\) 3.67282 0.490802
\(57\) 0 0
\(58\) − 5.24482i − 0.688678i
\(59\) − 1.14003i − 0.148420i −0.997243 − 0.0742099i \(-0.976357\pi\)
0.997243 − 0.0742099i \(-0.0236435\pi\)
\(60\) 0 0
\(61\) −15.2633 −1.95426 −0.977132 − 0.212634i \(-0.931796\pi\)
−0.977132 + 0.212634i \(0.931796\pi\)
\(62\) −0.672824 −0.0854487
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) 2.47116 − 5.10083i 0.306510 − 0.632680i
\(66\) 0 0
\(67\) − 4.48963i − 0.548496i −0.961659 − 0.274248i \(-0.911571\pi\)
0.961659 − 0.274248i \(-0.0884289\pi\)
\(68\) 3.52884 0.427934
\(69\) 0 0
\(70\) − 5.77365i − 0.690084i
\(71\) 5.01847i 0.595583i 0.954631 + 0.297791i \(0.0962500\pi\)
−0.954631 + 0.297791i \(0.903750\pi\)
\(72\) 0 0
\(73\) 10.4896i 1.22772i 0.789415 + 0.613859i \(0.210384\pi\)
−0.789415 + 0.613859i \(0.789616\pi\)
\(74\) 0.287973 0.0334762
\(75\) 0 0
\(76\) − 5.91764i − 0.678800i
\(77\) 7.71598 0.879318
\(78\) 0 0
\(79\) 0.244817 0.0275441 0.0137720 − 0.999905i \(-0.495616\pi\)
0.0137720 + 0.999905i \(0.495616\pi\)
\(80\) 1.57199i 0.175754i
\(81\) 0 0
\(82\) 12.4465 1.37448
\(83\) 8.38880i 0.920791i 0.887714 + 0.460395i \(0.152292\pi\)
−0.887714 + 0.460395i \(0.847708\pi\)
\(84\) 0 0
\(85\) − 5.54731i − 0.601690i
\(86\) 8.48963i 0.915461i
\(87\) 0 0
\(88\) −2.10083 −0.223949
\(89\) − 0.899170i − 0.0953118i −0.998864 − 0.0476559i \(-0.984825\pi\)
0.998864 − 0.0476559i \(-0.0151751\pi\)
\(90\) 0 0
\(91\) −5.77365 + 11.9176i −0.605243 + 1.24931i
\(92\) 2.24482 0.234038
\(93\) 0 0
\(94\) 3.81681 0.393674
\(95\) −9.30249 −0.954415
\(96\) 0 0
\(97\) 14.5328i 1.47558i 0.675030 + 0.737791i \(0.264131\pi\)
−0.675030 + 0.737791i \(0.735869\pi\)
\(98\) 6.48963i 0.655552i
\(99\) 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.649.2 ✓ 6
3.2 odd 2 2106.2.b.b.649.5 yes 6
13.12 even 2 inner 2106.2.b.a.649.5 yes 6
39.38 odd 2 2106.2.b.b.649.2 yes 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2106.2.b.a.649.2 ✓ 6 1.1 even 1 trivial
2106.2.b.a.649.5 yes 6 13.12 even 2 inner
2106.2.b.b.649.2 yes 6 39.38 odd 2
2106.2.b.b.649.5 yes 6 3.2 odd 2