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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2664,2,Mod(433,2664)] 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("2664.433"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2664, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 0, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 2664 = 2^{3} \cdot 3^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2664.r (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,0,0,-1,0,2,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(21.2721470985\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.591408.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} + 4x^{4} + x^{3} + 10x^{2} - 3x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 296)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 433.3
Root \(0.155554 - 0.269427i\) of defining polynomial
Character \(\chi\) \(=\) 2664.433
Dual form 2664.2.r.i.1009.3

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.951606 - 1.64823i) q^{5} +(-0.762714 + 1.32106i) q^{7} +(1.00000 - 1.73205i) q^{13} +(3.02543 + 5.24019i) q^{17} +(-3.52543 + 6.10622i) q^{19} -4.28100 q^{23} +(0.688892 + 1.19320i) q^{25} +1.28100 q^{29} -9.80642 q^{31} +(1.45161 + 2.51426i) q^{35} +(-0.854818 + 6.02240i) q^{37} +(0.811108 - 1.40488i) q^{41} +7.76049 q^{43} +5.80642 q^{47} +(2.33654 + 4.04700i) q^{49} +(-3.21432 - 5.56737i) q^{53} +(-3.52543 - 6.10622i) q^{59} +(-5.64050 + 9.76963i) q^{61} +(-1.90321 - 3.29646i) q^{65} +(3.80642 - 6.59292i) q^{67} +(-1.38493 + 2.39877i) q^{71} +11.2859 q^{73} +(-7.95407 + 13.7768i) q^{79} +(6.78346 + 11.7493i) q^{83} +11.5161 q^{85} +(4.46989 + 7.74207i) q^{89} +(1.52543 + 2.64212i) q^{91} +(6.70964 + 11.6214i) q^{95} +11.9906 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - q^{5} + 2 q^{7} + 6 q^{13} + 5 q^{17} - 8 q^{19} - 12 q^{23} + 4 q^{25} - 6 q^{29} - 32 q^{31} + 2 q^{35} + 15 q^{37} + 5 q^{41} - 20 q^{43} + 8 q^{47} + q^{49} - 6 q^{53} - 8 q^{59} - 27 q^{61}+ \cdots + 18 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2664\mathbb{Z}\right)^\times\).

\(n\) \(1297\) \(1333\) \(1999\) \(2369\)
\(\chi(n)\) \(e\left(\frac{1}{3}\right)\) \(1\) \(1\) \(1\)

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
\(3\) 0 0
\(4\) 0 0
\(5\) 0.951606 − 1.64823i 0.425571 − 0.737111i −0.570902 − 0.821018i \(-0.693407\pi\)
0.996474 + 0.0839071i \(0.0267399\pi\)
\(6\) 0 0
\(7\) −0.762714 + 1.32106i −0.288279 + 0.499313i −0.973399 − 0.229117i \(-0.926416\pi\)
0.685120 + 0.728430i \(0.259750\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) 1.00000 − 1.73205i 0.277350 − 0.480384i −0.693375 − 0.720577i \(-0.743877\pi\)
0.970725 + 0.240192i \(0.0772105\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 3.02543 + 5.24019i 0.733774 + 1.27093i 0.955259 + 0.295770i \(0.0955762\pi\)
−0.221485 + 0.975164i \(0.571090\pi\)
\(18\) 0 0
\(19\) −3.52543 + 6.10622i −0.808789 + 1.40086i 0.104915 + 0.994481i \(0.466543\pi\)
−0.913703 + 0.406382i \(0.866790\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −4.28100 −0.892649 −0.446325 − 0.894871i \(-0.647267\pi\)
−0.446325 + 0.894871i \(0.647267\pi\)
\(24\) 0 0
\(25\) 0.688892 + 1.19320i 0.137778 + 0.238639i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 1.28100 0.237875 0.118938 − 0.992902i \(-0.462051\pi\)
0.118938 + 0.992902i \(0.462051\pi\)
\(30\) 0 0
\(31\) −9.80642 −1.76129 −0.880643 − 0.473781i \(-0.842889\pi\)
−0.880643 + 0.473781i \(0.842889\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 1.45161 + 2.51426i 0.245366 + 0.424987i
\(36\) 0 0
\(37\) −0.854818 + 6.02240i −0.140531 + 0.990076i
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0.811108 − 1.40488i 0.126674 − 0.219405i −0.795712 − 0.605675i \(-0.792903\pi\)
0.922386 + 0.386270i \(0.126237\pi\)
\(42\) 0 0
\(43\) 7.76049 1.18346 0.591732 − 0.806135i \(-0.298444\pi\)
0.591732 + 0.806135i \(0.298444\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 5.80642 0.846954 0.423477 − 0.905907i \(-0.360809\pi\)
0.423477 + 0.905907i \(0.360809\pi\)
\(48\) 0 0
\(49\) 2.33654 + 4.04700i 0.333791 + 0.578143i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −3.21432 − 5.56737i −0.441521 − 0.764736i 0.556282 − 0.830994i \(-0.312228\pi\)
−0.997803 + 0.0662573i \(0.978894\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −3.52543 − 6.10622i −0.458972 − 0.794962i 0.539935 − 0.841707i \(-0.318449\pi\)
−0.998907 + 0.0467444i \(0.985115\pi\)
\(60\) 0 0
\(61\) −5.64050 + 9.76963i −0.722192 + 1.25087i 0.237928 + 0.971283i \(0.423532\pi\)
−0.960119 + 0.279590i \(0.909801\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −1.90321 − 3.29646i −0.236064 − 0.408876i
\(66\) 0 0
\(67\) 3.80642 − 6.59292i 0.465029 − 0.805453i −0.534174 − 0.845374i \(-0.679377\pi\)
0.999203 + 0.0399211i \(0.0127107\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −1.38493 + 2.39877i −0.164361 + 0.284681i −0.936428 − 0.350859i \(-0.885889\pi\)
0.772067 + 0.635541i \(0.219223\pi\)
\(72\) 0 0
\(73\) 11.2859 1.32092 0.660458 − 0.750863i \(-0.270362\pi\)
0.660458 + 0.750863i \(0.270362\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −7.95407 + 13.7768i −0.894902 + 1.55002i −0.0609772 + 0.998139i \(0.519422\pi\)
−0.833925 + 0.551877i \(0.813912\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 6.78346 + 11.7493i 0.744581 + 1.28965i 0.950390 + 0.311060i \(0.100684\pi\)
−0.205809 + 0.978592i \(0.565983\pi\)
\(84\) 0 0
\(85\) 11.5161 1.24909
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 4.46989 + 7.74207i 0.473807 + 0.820658i 0.999550 − 0.0299852i \(-0.00954601\pi\)
−0.525743 + 0.850643i \(0.676213\pi\)
\(90\) 0 0
\(91\) 1.52543 + 2.64212i 0.159908 + 0.276969i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 6.70964 + 11.6214i 0.688394 + 1.19233i
\(96\) 0 0
\(97\) 11.9906 1.21746 0.608732 − 0.793376i \(-0.291678\pi\)
0.608732 + 0.793376i \(0.291678\pi\)
\(98\) 0 0
\(99\) 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 2664.2.r.i.433.3 6
3.2 odd 2 296.2.i.b.137.1 yes 6
12.11 even 2 592.2.i.g.433.3 6
37.10 even 3 inner 2664.2.r.i.1009.3 6
111.47 odd 6 296.2.i.b.121.1 ✓ 6
444.47 even 6 592.2.i.g.417.3 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
296.2.i.b.121.1 ✓ 6 111.47 odd 6
296.2.i.b.137.1 yes 6 3.2 odd 2
592.2.i.g.417.3 6 444.47 even 6
592.2.i.g.433.3 6 12.11 even 2
2664.2.r.i.433.3 6 1.1 even 1 trivial
2664.2.r.i.1009.3 6 37.10 even 3 inner