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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1296,2,Mod(433,1296)] 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("1296.433"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1296, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1296 = 2^{4} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1296.i (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,-3,0,-1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(10.3486121020\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 54)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 865.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 1296.865
Dual form 1296.2.i.c.433.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.50000 - 2.59808i) q^{5} +(-0.500000 + 0.866025i) q^{7} +(-1.50000 + 2.59808i) q^{11} +(2.00000 + 3.46410i) q^{13} -2.00000 q^{19} +(-3.00000 - 5.19615i) q^{23} +(-2.00000 + 3.46410i) q^{25} +(-3.00000 + 5.19615i) q^{29} +(2.50000 + 4.33013i) q^{31} +3.00000 q^{35} +2.00000 q^{37} +(3.00000 + 5.19615i) q^{41} +(-5.00000 + 8.66025i) q^{43} +(3.00000 - 5.19615i) q^{47} +(3.00000 + 5.19615i) q^{49} +9.00000 q^{53} +9.00000 q^{55} +(6.00000 + 10.3923i) q^{59} +(-4.00000 + 6.92820i) q^{61} +(6.00000 - 10.3923i) q^{65} +(7.00000 + 12.1244i) q^{67} -7.00000 q^{73} +(-1.50000 - 2.59808i) q^{77} +(4.00000 - 6.92820i) q^{79} +(-1.50000 + 2.59808i) q^{83} -18.0000 q^{89} -4.00000 q^{91} +(3.00000 + 5.19615i) q^{95} +(0.500000 - 0.866025i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 3 q^{5} - q^{7} - 3 q^{11} + 4 q^{13} - 4 q^{19} - 6 q^{23} - 4 q^{25} - 6 q^{29} + 5 q^{31} + 6 q^{35} + 4 q^{37} + 6 q^{41} - 10 q^{43} + 6 q^{47} + 6 q^{49} + 18 q^{53} + 18 q^{55} + 12 q^{59}+ \cdots + q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1296\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(1135\) \(1217\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{2}{3}\right)\)

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
\(5\) −1.50000 − 2.59808i −0.670820 − 1.16190i −0.977672 − 0.210138i \(-0.932609\pi\)
0.306851 − 0.951757i \(-0.400725\pi\)
\(6\) 0 0
\(7\) −0.500000 + 0.866025i −0.188982 + 0.327327i −0.944911 − 0.327327i \(-0.893852\pi\)
0.755929 + 0.654654i \(0.227186\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −1.50000 + 2.59808i −0.452267 + 0.783349i −0.998526 − 0.0542666i \(-0.982718\pi\)
0.546259 + 0.837616i \(0.316051\pi\)
\(12\) 0 0
\(13\) 2.00000 + 3.46410i 0.554700 + 0.960769i 0.997927 + 0.0643593i \(0.0205004\pi\)
−0.443227 + 0.896410i \(0.646166\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) −2.00000 −0.458831 −0.229416 − 0.973329i \(-0.573682\pi\)
−0.229416 + 0.973329i \(0.573682\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −3.00000 − 5.19615i −0.625543 − 1.08347i −0.988436 − 0.151642i \(-0.951544\pi\)
0.362892 − 0.931831i \(-0.381789\pi\)
\(24\) 0 0
\(25\) −2.00000 + 3.46410i −0.400000 + 0.692820i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −3.00000 + 5.19615i −0.557086 + 0.964901i 0.440652 + 0.897678i \(0.354747\pi\)
−0.997738 + 0.0672232i \(0.978586\pi\)
\(30\) 0 0
\(31\) 2.50000 + 4.33013i 0.449013 + 0.777714i 0.998322 − 0.0579057i \(-0.0184423\pi\)
−0.549309 + 0.835619i \(0.685109\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 3.00000 0.507093
\(36\) 0 0
\(37\) 2.00000 0.328798 0.164399 − 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 3.00000 + 5.19615i 0.468521 + 0.811503i 0.999353 − 0.0359748i \(-0.0114536\pi\)
−0.530831 + 0.847477i \(0.678120\pi\)
\(42\) 0 0
\(43\) −5.00000 + 8.66025i −0.762493 + 1.32068i 0.179069 + 0.983836i \(0.442691\pi\)
−0.941562 + 0.336840i \(0.890642\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 3.00000 − 5.19615i 0.437595 − 0.757937i −0.559908 − 0.828554i \(-0.689164\pi\)
0.997503 + 0.0706177i \(0.0224970\pi\)
\(48\) 0 0
\(49\) 3.00000 + 5.19615i 0.428571 + 0.742307i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 9.00000 1.23625 0.618123 − 0.786082i \(-0.287894\pi\)
0.618123 + 0.786082i \(0.287894\pi\)
\(54\) 0 0
\(55\) 9.00000 1.21356
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 6.00000 + 10.3923i 0.781133 + 1.35296i 0.931282 + 0.364299i \(0.118692\pi\)
−0.150148 + 0.988663i \(0.547975\pi\)
\(60\) 0 0
\(61\) −4.00000 + 6.92820i −0.512148 + 0.887066i 0.487753 + 0.872982i \(0.337817\pi\)
−0.999901 + 0.0140840i \(0.995517\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 6.00000 − 10.3923i 0.744208 − 1.28901i
\(66\) 0 0
\(67\) 7.00000 + 12.1244i 0.855186 + 1.48123i 0.876472 + 0.481452i \(0.159891\pi\)
−0.0212861 + 0.999773i \(0.506776\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) −7.00000 −0.819288 −0.409644 − 0.912245i \(-0.634347\pi\)
−0.409644 + 0.912245i \(0.634347\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −1.50000 − 2.59808i −0.170941 − 0.296078i
\(78\) 0 0
\(79\) 4.00000 − 6.92820i 0.450035 − 0.779484i −0.548352 − 0.836247i \(-0.684745\pi\)
0.998388 + 0.0567635i \(0.0180781\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −1.50000 + 2.59808i −0.164646 + 0.285176i −0.936530 − 0.350588i \(-0.885982\pi\)
0.771883 + 0.635764i \(0.219315\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −18.0000 −1.90800 −0.953998 − 0.299813i \(-0.903076\pi\)
−0.953998 + 0.299813i \(0.903076\pi\)
\(90\) 0 0
\(91\) −4.00000 −0.419314
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 3.00000 + 5.19615i 0.307794 + 0.533114i
\(96\) 0 0
\(97\) 0.500000 − 0.866025i 0.0507673 − 0.0879316i −0.839525 − 0.543321i \(-0.817167\pi\)
0.890292 + 0.455389i \(0.150500\pi\)
\(98\) 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 1296.2.i.c.865.1 2
3.2 odd 2 1296.2.i.o.865.1 2
4.3 odd 2 162.2.c.c.55.1 2
9.2 odd 6 432.2.a.b.1.1 1
9.4 even 3 inner 1296.2.i.c.433.1 2
9.5 odd 6 1296.2.i.o.433.1 2
9.7 even 3 432.2.a.g.1.1 1
12.11 even 2 162.2.c.b.55.1 2
36.7 odd 6 54.2.a.a.1.1 ✓ 1
36.11 even 6 54.2.a.b.1.1 yes 1
36.23 even 6 162.2.c.b.109.1 2
36.31 odd 6 162.2.c.c.109.1 2
72.11 even 6 1728.2.a.y.1.1 1
72.29 odd 6 1728.2.a.z.1.1 1
72.43 odd 6 1728.2.a.c.1.1 1
72.61 even 6 1728.2.a.d.1.1 1
180.7 even 12 1350.2.c.b.649.1 2
180.43 even 12 1350.2.c.b.649.2 2
180.47 odd 12 1350.2.c.k.649.2 2
180.79 odd 6 1350.2.a.r.1.1 1
180.83 odd 12 1350.2.c.k.649.1 2
180.119 even 6 1350.2.a.h.1.1 1
252.83 odd 6 2646.2.a.bd.1.1 1
252.223 even 6 2646.2.a.a.1.1 1
396.43 even 6 6534.2.a.bc.1.1 1
396.263 odd 6 6534.2.a.b.1.1 1
468.155 even 6 9126.2.a.r.1.1 1
468.259 odd 6 9126.2.a.u.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
54.2.a.a.1.1 ✓ 1 36.7 odd 6
54.2.a.b.1.1 yes 1 36.11 even 6
162.2.c.b.55.1 2 12.11 even 2
162.2.c.b.109.1 2 36.23 even 6
162.2.c.c.55.1 2 4.3 odd 2
162.2.c.c.109.1 2 36.31 odd 6
432.2.a.b.1.1 1 9.2 odd 6
432.2.a.g.1.1 1 9.7 even 3
1296.2.i.c.433.1 2 9.4 even 3 inner
1296.2.i.c.865.1 2 1.1 even 1 trivial
1296.2.i.o.433.1 2 9.5 odd 6
1296.2.i.o.865.1 2 3.2 odd 2
1350.2.a.h.1.1 1 180.119 even 6
1350.2.a.r.1.1 1 180.79 odd 6
1350.2.c.b.649.1 2 180.7 even 12
1350.2.c.b.649.2 2 180.43 even 12
1350.2.c.k.649.1 2 180.83 odd 12
1350.2.c.k.649.2 2 180.47 odd 12
1728.2.a.c.1.1 1 72.43 odd 6
1728.2.a.d.1.1 1 72.61 even 6
1728.2.a.y.1.1 1 72.11 even 6
1728.2.a.z.1.1 1 72.29 odd 6
2646.2.a.a.1.1 1 252.223 even 6
2646.2.a.bd.1.1 1 252.83 odd 6
6534.2.a.b.1.1 1 396.263 odd 6
6534.2.a.bc.1.1 1 396.43 even 6
9126.2.a.r.1.1 1 468.155 even 6
9126.2.a.u.1.1 1 468.259 odd 6