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Properties

Label 1728.2.c.d.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 = [4,0,0,0,0,0,0,0,0,0,0,0,4] 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: \(\Q(\sqrt{-2}, \sqrt{-3})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 108)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1727.3
Root \(-1.22474 + 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 1728.1727
Dual form 1728.2.c.d.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)
 
\(f(q)\) \(=\) \(q+2.82843i q^{5} -1.73205i q^{7} -4.89898 q^{11} +1.00000 q^{13} -2.82843i q^{17} -5.19615i q^{19} -4.89898 q^{23} -3.00000 q^{25} -5.65685i q^{29} -3.46410i q^{31} +4.89898 q^{35} +1.00000 q^{37} +5.65685i q^{41} -3.46410i q^{43} +4.89898 q^{47} +4.00000 q^{49} -5.65685i q^{53} -13.8564i q^{55} +4.89898 q^{59} -11.0000 q^{61} +2.82843i q^{65} -12.1244i q^{67} -1.00000 q^{73} +8.48528i q^{77} -1.73205i q^{79} +9.79796 q^{83} +8.00000 q^{85} -2.82843i q^{89} -1.73205i q^{91} +14.6969 q^{95} -13.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{13} - 12 q^{25} + 4 q^{37} + 16 q^{49} - 44 q^{61} - 4 q^{73} + 32 q^{85} - 52 q^{97}+O(q^{100}) \) 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
\(5\) 2.82843i 1.26491i 0.774597 + 0.632456i \(0.217953\pi\)
−0.774597 + 0.632456i \(0.782047\pi\)
\(6\) 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
\(11\) −4.89898 −1.47710 −0.738549 − 0.674200i \(-0.764489\pi\)
−0.738549 + 0.674200i \(0.764489\pi\)
\(12\) 0 0
\(13\) 1.00000 0.277350 0.138675 − 0.990338i \(-0.455716\pi\)
0.138675 + 0.990338i \(0.455716\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 2.82843i − 0.685994i −0.939336 − 0.342997i \(-0.888558\pi\)
0.939336 − 0.342997i \(-0.111442\pi\)
\(18\) 0 0
\(19\) − 5.19615i − 1.19208i −0.802955 − 0.596040i \(-0.796740\pi\)
0.802955 − 0.596040i \(-0.203260\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −4.89898 −1.02151 −0.510754 − 0.859727i \(-0.670634\pi\)
−0.510754 + 0.859727i \(0.670634\pi\)
\(24\) 0 0
\(25\) −3.00000 −0.600000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) − 5.65685i − 1.05045i −0.850963 − 0.525226i \(-0.823981\pi\)
0.850963 − 0.525226i \(-0.176019\pi\)
\(30\) 0 0
\(31\) − 3.46410i − 0.622171i −0.950382 − 0.311086i \(-0.899307\pi\)
0.950382 − 0.311086i \(-0.100693\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 4.89898 0.828079
\(36\) 0 0
\(37\) 1.00000 0.164399 0.0821995 − 0.996616i \(-0.473806\pi\)
0.0821995 + 0.996616i \(0.473806\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 5.65685i 0.883452i 0.897150 + 0.441726i \(0.145634\pi\)
−0.897150 + 0.441726i \(0.854366\pi\)
\(42\) 0 0
\(43\) − 3.46410i − 0.528271i −0.964486 − 0.264135i \(-0.914913\pi\)
0.964486 − 0.264135i \(-0.0850865\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 4.89898 0.714590 0.357295 − 0.933992i \(-0.383699\pi\)
0.357295 + 0.933992i \(0.383699\pi\)
\(48\) 0 0
\(49\) 4.00000 0.571429
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) − 5.65685i − 0.777029i −0.921443 − 0.388514i \(-0.872988\pi\)
0.921443 − 0.388514i \(-0.127012\pi\)
\(54\) 0 0
\(55\) − 13.8564i − 1.86840i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 4.89898 0.637793 0.318896 − 0.947790i \(-0.396688\pi\)
0.318896 + 0.947790i \(0.396688\pi\)
\(60\) 0 0
\(61\) −11.0000 −1.40841 −0.704203 − 0.709999i \(-0.748695\pi\)
−0.704203 + 0.709999i \(0.748695\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 2.82843i 0.350823i
\(66\) 0 0
\(67\) − 12.1244i − 1.48123i −0.671932 − 0.740613i \(-0.734535\pi\)
0.671932 − 0.740613i \(-0.265465\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) −1.00000 −0.117041 −0.0585206 − 0.998286i \(-0.518638\pi\)
−0.0585206 + 0.998286i \(0.518638\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 8.48528i 0.966988i
\(78\) 0 0
\(79\) − 1.73205i − 0.194871i −0.995242 − 0.0974355i \(-0.968936\pi\)
0.995242 − 0.0974355i \(-0.0310640\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 9.79796 1.07547 0.537733 − 0.843115i \(-0.319281\pi\)
0.537733 + 0.843115i \(0.319281\pi\)
\(84\) 0 0
\(85\) 8.00000 0.867722
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) − 2.82843i − 0.299813i −0.988700 − 0.149906i \(-0.952103\pi\)
0.988700 − 0.149906i \(-0.0478972\pi\)
\(90\) 0 0
\(91\) − 1.73205i − 0.181568i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 14.6969 1.50787
\(96\) 0 0
\(97\) −13.0000 −1.31995 −0.659975 − 0.751288i \(-0.729433\pi\)
−0.659975 + 0.751288i \(0.729433\pi\)
\(98\) 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.d.1727.3 4
3.2 odd 2 inner 1728.2.c.d.1727.1 4
4.3 odd 2 inner 1728.2.c.d.1727.4 4
8.3 odd 2 108.2.b.b.107.4 yes 4
8.5 even 2 108.2.b.b.107.2 yes 4
12.11 even 2 inner 1728.2.c.d.1727.2 4
24.5 odd 2 108.2.b.b.107.3 yes 4
24.11 even 2 108.2.b.b.107.1 ✓ 4
72.5 odd 6 324.2.h.a.107.2 4
72.11 even 6 324.2.h.a.215.2 4
72.13 even 6 324.2.h.a.107.1 4
72.29 odd 6 324.2.h.b.215.1 4
72.43 odd 6 324.2.h.a.215.1 4
72.59 even 6 324.2.h.b.107.2 4
72.61 even 6 324.2.h.b.215.2 4
72.67 odd 6 324.2.h.b.107.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
108.2.b.b.107.1 ✓ 4 24.11 even 2
108.2.b.b.107.2 yes 4 8.5 even 2
108.2.b.b.107.3 yes 4 24.5 odd 2
108.2.b.b.107.4 yes 4 8.3 odd 2
324.2.h.a.107.1 4 72.13 even 6
324.2.h.a.107.2 4 72.5 odd 6
324.2.h.a.215.1 4 72.43 odd 6
324.2.h.a.215.2 4 72.11 even 6
324.2.h.b.107.1 4 72.67 odd 6
324.2.h.b.107.2 4 72.59 even 6
324.2.h.b.215.1 4 72.29 odd 6
324.2.h.b.215.2 4 72.61 even 6
1728.2.c.d.1727.1 4 3.2 odd 2 inner
1728.2.c.d.1727.2 4 12.11 even 2 inner
1728.2.c.d.1727.3 4 1.1 even 1 trivial
1728.2.c.d.1727.4 4 4.3 odd 2 inner