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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1728,2,Mod(145,1728)] 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("1728.145"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1728, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([0, 9, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1728 = 2^{6} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1728.bc (of order \(12\), degree \(4\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(13.7981494693\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 144)
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label 1009.1
Root \(0.866025 + 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 1728.1009
Dual form 1728.2.bc.a.721.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.00000 + 0.267949i) q^{5} +(-2.36603 + 1.36603i) q^{7} +(1.13397 - 4.23205i) q^{11} +(0.901924 + 3.36603i) q^{13} +5.73205 q^{17} +(2.36603 - 2.36603i) q^{19} +(-4.09808 - 2.36603i) q^{23} +(-3.40192 + 1.96410i) q^{25} +(-2.36603 - 0.633975i) q^{29} +(0.267949 - 0.464102i) q^{31} +(2.00000 - 2.00000i) q^{35} +(4.73205 + 4.73205i) q^{37} +(2.59808 + 1.50000i) q^{41} +(-2.23205 + 8.33013i) q^{43} +(3.83013 + 6.63397i) q^{47} +(0.232051 - 0.401924i) q^{49} +(7.46410 + 7.46410i) q^{53} +4.53590i q^{55} +(7.33013 - 1.96410i) q^{59} +(11.1962 + 3.00000i) q^{61} +(-1.80385 - 3.12436i) q^{65} +(1.76795 + 6.59808i) q^{67} -2.92820i q^{71} +6.26795i q^{73} +(3.09808 + 11.5622i) q^{77} +(6.00000 + 10.3923i) q^{79} +(-1.36603 - 0.366025i) q^{83} +(-5.73205 + 1.53590i) q^{85} -2.00000i q^{89} +(-6.73205 - 6.73205i) q^{91} +(-1.73205 + 3.00000i) q^{95} +(-5.86603 - 10.1603i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{5} - 6 q^{7} + 8 q^{11} + 14 q^{13} + 16 q^{17} + 6 q^{19} - 6 q^{23} - 24 q^{25} - 6 q^{29} + 8 q^{31} + 8 q^{35} + 12 q^{37} - 2 q^{43} - 2 q^{47} - 6 q^{49} + 16 q^{53} + 12 q^{59} + 24 q^{61}+ \cdots - 20 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1728\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(703\) \(1217\)
\(\chi(n)\) \(e\left(\frac{1}{4}\right)\) \(1\) \(e\left(\frac{1}{3}\right)\)

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
\(5\) −1.00000 + 0.267949i −0.447214 + 0.119831i −0.475395 − 0.879772i \(-0.657695\pi\)
0.0281817 + 0.999603i \(0.491028\pi\)
\(6\) 0 0
\(7\) −2.36603 + 1.36603i −0.894274 + 0.516309i −0.875338 − 0.483512i \(-0.839361\pi\)
−0.0189356 + 0.999821i \(0.506028\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 1.13397 − 4.23205i 0.341906 − 1.27601i −0.554279 − 0.832331i \(-0.687006\pi\)
0.896185 − 0.443680i \(-0.146327\pi\)
\(12\) 0 0
\(13\) 0.901924 + 3.36603i 0.250149 + 0.933567i 0.970725 + 0.240192i \(0.0772105\pi\)
−0.720577 + 0.693375i \(0.756123\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 5.73205 1.39023 0.695113 − 0.718900i \(-0.255354\pi\)
0.695113 + 0.718900i \(0.255354\pi\)
\(18\) 0 0
\(19\) 2.36603 − 2.36603i 0.542803 − 0.542803i −0.381546 − 0.924350i \(-0.624608\pi\)
0.924350 + 0.381546i \(0.124608\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −4.09808 − 2.36603i −0.854508 − 0.493350i 0.00766135 − 0.999971i \(-0.497561\pi\)
−0.862169 + 0.506620i \(0.830895\pi\)
\(24\) 0 0
\(25\) −3.40192 + 1.96410i −0.680385 + 0.392820i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −2.36603 − 0.633975i −0.439360 − 0.117726i 0.0323566 − 0.999476i \(-0.489699\pi\)
−0.471717 + 0.881750i \(0.656365\pi\)
\(30\) 0 0
\(31\) 0.267949 − 0.464102i 0.0481251 − 0.0833551i −0.840959 − 0.541098i \(-0.818009\pi\)
0.889085 + 0.457743i \(0.151342\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 2.00000 − 2.00000i 0.338062 − 0.338062i
\(36\) 0 0
\(37\) 4.73205 + 4.73205i 0.777944 + 0.777944i 0.979481 − 0.201537i \(-0.0645935\pi\)
−0.201537 + 0.979481i \(0.564594\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 2.59808 + 1.50000i 0.405751 + 0.234261i 0.688963 − 0.724797i \(-0.258066\pi\)
−0.283211 + 0.959058i \(0.591400\pi\)
\(42\) 0 0
\(43\) −2.23205 + 8.33013i −0.340385 + 1.27033i 0.557528 + 0.830158i \(0.311750\pi\)
−0.897912 + 0.440174i \(0.854917\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 3.83013 + 6.63397i 0.558681 + 0.967665i 0.997607 + 0.0691412i \(0.0220259\pi\)
−0.438925 + 0.898523i \(0.644641\pi\)
\(48\) 0 0
\(49\) 0.232051 − 0.401924i 0.0331501 − 0.0574177i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 7.46410 + 7.46410i 1.02527 + 1.02527i 0.999672 + 0.0256010i \(0.00814993\pi\)
0.0256010 + 0.999672i \(0.491850\pi\)
\(54\) 0 0
\(55\) 4.53590i 0.611620i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 7.33013 − 1.96410i 0.954301 − 0.255704i 0.252115 − 0.967697i \(-0.418874\pi\)
0.702186 + 0.711993i \(0.252207\pi\)
\(60\) 0 0
\(61\) 11.1962 + 3.00000i 1.43352 + 0.384111i 0.890260 − 0.455453i \(-0.150523\pi\)
0.543261 + 0.839564i \(0.317189\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −1.80385 − 3.12436i −0.223740 − 0.387529i
\(66\) 0 0
\(67\) 1.76795 + 6.59808i 0.215989 + 0.806083i 0.985816 + 0.167830i \(0.0536760\pi\)
−0.769827 + 0.638253i \(0.779657\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) − 2.92820i − 0.347514i −0.984789 − 0.173757i \(-0.944409\pi\)
0.984789 − 0.173757i \(-0.0555907\pi\)
\(72\) 0 0
\(73\) 6.26795i 0.733608i 0.930298 + 0.366804i \(0.119548\pi\)
−0.930298 + 0.366804i \(0.880452\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 3.09808 + 11.5622i 0.353059 + 1.31763i
\(78\) 0 0
\(79\) 6.00000 + 10.3923i 0.675053 + 1.16923i 0.976453 + 0.215728i \(0.0692125\pi\)
−0.301401 + 0.953498i \(0.597454\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −1.36603 − 0.366025i −0.149941 − 0.0401765i 0.183068 − 0.983100i \(-0.441397\pi\)
−0.333009 + 0.942924i \(0.608064\pi\)
\(84\) 0 0
\(85\) −5.73205 + 1.53590i −0.621728 + 0.166592i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) − 2.00000i − 0.212000i −0.994366 − 0.106000i \(-0.966196\pi\)
0.994366 − 0.106000i \(-0.0338043\pi\)
\(90\) 0 0
\(91\) −6.73205 − 6.73205i −0.705711 − 0.705711i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −1.73205 + 3.00000i −0.177705 + 0.307794i
\(96\) 0 0
\(97\) −5.86603 − 10.1603i −0.595605 − 1.03162i −0.993461 − 0.114170i \(-0.963579\pi\)
0.397857 − 0.917448i \(-0.369754\pi\)
\(98\) 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 1728.2.bc.a.1009.1 4
3.2 odd 2 576.2.bb.c.49.1 4
4.3 odd 2 432.2.y.b.37.1 4
9.2 odd 6 576.2.bb.d.241.1 4
9.7 even 3 1728.2.bc.d.1585.1 4
12.11 even 2 144.2.x.c.85.1 yes 4
16.3 odd 4 432.2.y.c.253.1 4
16.13 even 4 1728.2.bc.d.145.1 4
36.7 odd 6 432.2.y.c.181.1 4
36.11 even 6 144.2.x.b.133.1 yes 4
48.29 odd 4 576.2.bb.d.337.1 4
48.35 even 4 144.2.x.b.13.1 ✓ 4
144.29 odd 12 576.2.bb.c.529.1 4
144.61 even 12 inner 1728.2.bc.a.721.1 4
144.83 even 12 144.2.x.c.61.1 yes 4
144.115 odd 12 432.2.y.b.397.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
144.2.x.b.13.1 ✓ 4 48.35 even 4
144.2.x.b.133.1 yes 4 36.11 even 6
144.2.x.c.61.1 yes 4 144.83 even 12
144.2.x.c.85.1 yes 4 12.11 even 2
432.2.y.b.37.1 4 4.3 odd 2
432.2.y.b.397.1 4 144.115 odd 12
432.2.y.c.181.1 4 36.7 odd 6
432.2.y.c.253.1 4 16.3 odd 4
576.2.bb.c.49.1 4 3.2 odd 2
576.2.bb.c.529.1 4 144.29 odd 12
576.2.bb.d.241.1 4 9.2 odd 6
576.2.bb.d.337.1 4 48.29 odd 4
1728.2.bc.a.721.1 4 144.61 even 12 inner
1728.2.bc.a.1009.1 4 1.1 even 1 trivial
1728.2.bc.d.145.1 4 16.13 even 4
1728.2.bc.d.1585.1 4 9.7 even 3