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Properties

Label 1728.2.c.e.1727.3
Level $1728$
Weight $2$
Character 1728.1727
Analytic conductor $13.798$
Analytic rank $0$
Dimension $4$
Inner twists $4$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1728,2,Mod(1727,1728)] 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("1728.1727"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1728, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1728 = 2^{6} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1728.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,0,0,0,0,0,0,0,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == 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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 432)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1727.3
Root \(0.866025 + 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 1728.1727
Dual form 1728.2.c.e.1727.2

$q$-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)
 
\(f(q)\) \(=\) \(q+3.00000i q^{5} -1.73205i q^{7} +5.19615 q^{11} +2.00000 q^{13} -6.00000i q^{17} -6.92820i q^{19} -4.00000 q^{25} +6.00000i q^{29} -5.19615i q^{31} +5.19615 q^{35} -8.00000 q^{37} +10.3923i q^{43} +10.3923 q^{47} +4.00000 q^{49} -9.00000i q^{53} +15.5885i q^{55} +10.3923 q^{59} +4.00000 q^{61} +6.00000i q^{65} +3.46410i q^{67} +10.3923 q^{71} +1.00000 q^{73} -9.00000i q^{77} -3.46410i q^{79} -5.19615 q^{83} +18.0000 q^{85} +6.00000i q^{89} -3.46410i q^{91} +20.7846 q^{95} -5.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 8 q^{13} - 16 q^{25} - 32 q^{37} + 16 q^{49} + 16 q^{61} + 4 q^{73} + 72 q^{85} - 20 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(325\) \(703\) \(1217\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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
\(5\) 3.00000i 1.34164i 0.741620 + 0.670820i \(0.234058\pi\)
−0.741620 + 0.670820i \(0.765942\pi\)
\(6\) 0 0
\(7\) − 1.73205i − 0.654654i −0.944911 − 0.327327i \(-0.893852\pi\)
0.944911 − 0.327327i \(-0.106148\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 5.19615 1.56670 0.783349 − 0.621582i \(-0.213510\pi\)
0.783349 + 0.621582i \(0.213510\pi\)
\(12\) 0 0
\(13\) 2.00000 0.554700 0.277350 − 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 6.00000i − 1.45521i −0.685994 − 0.727607i \(-0.740633\pi\)
0.685994 − 0.727607i \(-0.259367\pi\)
\(18\) 0 0
\(19\) − 6.92820i − 1.58944i −0.606977 − 0.794719i \(-0.707618\pi\)
0.606977 − 0.794719i \(-0.292382\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) −4.00000 −0.800000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 6.00000i 1.11417i 0.830455 + 0.557086i \(0.188081\pi\)
−0.830455 + 0.557086i \(0.811919\pi\)
\(30\) 0 0
\(31\) − 5.19615i − 0.933257i −0.884454 − 0.466628i \(-0.845469\pi\)
0.884454 − 0.466628i \(-0.154531\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 5.19615 0.878310
\(36\) 0 0
\(37\) −8.00000 −1.31519 −0.657596 − 0.753371i \(-0.728427\pi\)
−0.657596 + 0.753371i \(0.728427\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) 10.3923i 1.58481i 0.609994 + 0.792406i \(0.291172\pi\)
−0.609994 + 0.792406i \(0.708828\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 10.3923 1.51587 0.757937 − 0.652328i \(-0.226208\pi\)
0.757937 + 0.652328i \(0.226208\pi\)
\(48\) 0 0
\(49\) 4.00000 0.571429
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) − 9.00000i − 1.23625i −0.786082 − 0.618123i \(-0.787894\pi\)
0.786082 − 0.618123i \(-0.212106\pi\)
\(54\) 0 0
\(55\) 15.5885i 2.10195i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 10.3923 1.35296 0.676481 − 0.736460i \(-0.263504\pi\)
0.676481 + 0.736460i \(0.263504\pi\)
\(60\) 0 0
\(61\) 4.00000 0.512148 0.256074 − 0.966657i \(-0.417571\pi\)
0.256074 + 0.966657i \(0.417571\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 6.00000i 0.744208i
\(66\) 0 0
\(67\) 3.46410i 0.423207i 0.977356 + 0.211604i \(0.0678686\pi\)
−0.977356 + 0.211604i \(0.932131\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 10.3923 1.23334 0.616670 − 0.787222i \(-0.288481\pi\)
0.616670 + 0.787222i \(0.288481\pi\)
\(72\) 0 0
\(73\) 1.00000 0.117041 0.0585206 − 0.998286i \(-0.481362\pi\)
0.0585206 + 0.998286i \(0.481362\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 9.00000i − 1.02565i
\(78\) 0 0
\(79\) − 3.46410i − 0.389742i −0.980829 − 0.194871i \(-0.937571\pi\)
0.980829 − 0.194871i \(-0.0624288\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −5.19615 −0.570352 −0.285176 − 0.958475i \(-0.592052\pi\)
−0.285176 + 0.958475i \(0.592052\pi\)
\(84\) 0 0
\(85\) 18.0000 1.95237
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 6.00000i 0.635999i 0.948091 + 0.317999i \(0.103011\pi\)
−0.948091 + 0.317999i \(0.896989\pi\)
\(90\) 0 0
\(91\) − 3.46410i − 0.363137i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 20.7846 2.13246
\(96\) 0 0
\(97\) −5.00000 −0.507673 −0.253837 − 0.967247i \(-0.581693\pi\)
−0.253837 + 0.967247i \(0.581693\pi\)
\(98\) 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.c.e.1727.3 4
3.2 odd 2 inner 1728.2.c.e.1727.1 4
4.3 odd 2 inner 1728.2.c.e.1727.4 4
8.3 odd 2 432.2.c.c.431.2 yes 4
8.5 even 2 432.2.c.c.431.1 ✓ 4
12.11 even 2 inner 1728.2.c.e.1727.2 4
24.5 odd 2 432.2.c.c.431.3 yes 4
24.11 even 2 432.2.c.c.431.4 yes 4
72.5 odd 6 1296.2.s.i.431.2 4
72.11 even 6 1296.2.s.i.863.1 4
72.13 even 6 1296.2.s.i.431.1 4
72.29 odd 6 1296.2.s.g.863.1 4
72.43 odd 6 1296.2.s.i.863.2 4
72.59 even 6 1296.2.s.g.431.2 4
72.61 even 6 1296.2.s.g.863.2 4
72.67 odd 6 1296.2.s.g.431.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
432.2.c.c.431.1 ✓ 4 8.5 even 2
432.2.c.c.431.2 yes 4 8.3 odd 2
432.2.c.c.431.3 yes 4 24.5 odd 2
432.2.c.c.431.4 yes 4 24.11 even 2
1296.2.s.g.431.1 4 72.67 odd 6
1296.2.s.g.431.2 4 72.59 even 6
1296.2.s.g.863.1 4 72.29 odd 6
1296.2.s.g.863.2 4 72.61 even 6
1296.2.s.i.431.1 4 72.13 even 6
1296.2.s.i.431.2 4 72.5 odd 6
1296.2.s.i.863.1 4 72.11 even 6
1296.2.s.i.863.2 4 72.43 odd 6
1728.2.c.e.1727.1 4 3.2 odd 2 inner
1728.2.c.e.1727.2 4 12.11 even 2 inner
1728.2.c.e.1727.3 4 1.1 even 1 trivial
1728.2.c.e.1727.4 4 4.3 odd 2 inner