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Properties

Label 1728.2.z.a.719.7
Level $1728$
Weight $2$
Character 1728.719
Analytic conductor $13.798$
Analytic rank $0$
Dimension $88$
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(143,1728)] 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("1728.143"); 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([6, 3, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1728 = 2^{6} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1728.z (of order \(12\), degree \(4\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(13.7981494693\)
Analytic rank: \(0\)
Dimension: \(88\)
Relative dimension: \(22\) over \(\Q(\zeta_{12})\)
Twist minimal: no (minimal twist has level 144)
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label 719.7
Character \(\chi\) \(=\) 1728.719
Dual form 1728.2.z.a.1007.7

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.323102 - 1.20583i) q^{5} +(-0.140266 - 0.242948i) q^{7} +(-0.823794 + 3.07444i) q^{11} +(-0.740984 - 2.76539i) q^{13} +3.72031i q^{17} +(4.10860 + 4.10860i) q^{19} +(-1.57595 - 0.909876i) q^{23} +(2.98049 - 1.72078i) q^{25} +(1.02626 - 3.83006i) q^{29} +(8.81101 + 5.08704i) q^{31} +(-0.247635 + 0.247635i) q^{35} +(1.76964 + 1.76964i) q^{37} +(2.66819 - 4.62144i) q^{41} +(-6.89490 - 1.84748i) q^{43} +(5.48486 + 9.50006i) q^{47} +(3.46065 - 5.99402i) q^{49} +(8.58403 - 8.58403i) q^{53} +3.97344 q^{55} +(5.38532 - 1.44299i) q^{59} +(-6.23168 - 1.66977i) q^{61} +(-3.09519 + 1.78701i) q^{65} +(-3.75640 + 1.00652i) q^{67} -10.6808i q^{71} +9.30419i q^{73} +(0.862479 - 0.231101i) q^{77} +(8.70990 - 5.02866i) q^{79} +(2.19703 + 0.588691i) q^{83} +(4.48608 - 1.20204i) q^{85} -1.87637 q^{89} +(-0.567911 + 0.567911i) q^{91} +(3.62679 - 6.28179i) q^{95} +(9.19070 + 15.9188i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 88 q + 6 q^{5} + 4 q^{7} - 6 q^{11} - 2 q^{13} + 8 q^{19} - 12 q^{23} + 6 q^{29} - 8 q^{37} + 2 q^{43} - 24 q^{49} + 16 q^{55} - 42 q^{59} - 2 q^{61} + 12 q^{65} + 2 q^{67} + 6 q^{77} + 54 q^{83} + 8 q^{85}+ \cdots - 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(325\) \(703\) \(1217\)
\(\chi(n)\) \(e\left(\frac{1}{4}\right)\) \(-1\) \(e\left(\frac{5}{6}\right)\)

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
\(5\) −0.323102 − 1.20583i −0.144496 − 0.539266i −0.999777 − 0.0211020i \(-0.993283\pi\)
0.855282 − 0.518164i \(-0.173384\pi\)
\(6\) 0 0
\(7\) −0.140266 − 0.242948i −0.0530156 − 0.0918256i 0.838300 − 0.545210i \(-0.183550\pi\)
−0.891315 + 0.453384i \(0.850217\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −0.823794 + 3.07444i −0.248383 + 0.926979i 0.723269 + 0.690566i \(0.242639\pi\)
−0.971653 + 0.236413i \(0.924028\pi\)
\(12\) 0 0
\(13\) −0.740984 − 2.76539i −0.205512 − 0.766982i −0.989293 − 0.145944i \(-0.953378\pi\)
0.783781 − 0.621038i \(-0.213289\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 3.72031i 0.902309i 0.892446 + 0.451154i \(0.148988\pi\)
−0.892446 + 0.451154i \(0.851012\pi\)
\(18\) 0 0
\(19\) 4.10860 + 4.10860i 0.942577 + 0.942577i 0.998439 − 0.0558614i \(-0.0177905\pi\)
−0.0558614 + 0.998439i \(0.517791\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −1.57595 − 0.909876i −0.328609 − 0.189722i 0.326615 − 0.945158i \(-0.394092\pi\)
−0.655223 + 0.755435i \(0.727425\pi\)
\(24\) 0 0
\(25\) 2.98049 − 1.72078i 0.596097 − 0.344157i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 1.02626 − 3.83006i 0.190572 − 0.711224i −0.802797 − 0.596253i \(-0.796656\pi\)
0.993369 − 0.114972i \(-0.0366777\pi\)
\(30\) 0 0
\(31\) 8.81101 + 5.08704i 1.58250 + 0.913659i 0.994493 + 0.104807i \(0.0334225\pi\)
0.588012 + 0.808852i \(0.299911\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −0.247635 + 0.247635i −0.0418579 + 0.0418579i
\(36\) 0 0
\(37\) 1.76964 + 1.76964i 0.290928 + 0.290928i 0.837447 − 0.546519i \(-0.184047\pi\)
−0.546519 + 0.837447i \(0.684047\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 2.66819 − 4.62144i 0.416701 − 0.721747i −0.578904 − 0.815395i \(-0.696520\pi\)
0.995605 + 0.0936482i \(0.0298529\pi\)
\(42\) 0 0
\(43\) −6.89490 − 1.84748i −1.05146 − 0.281738i −0.308606 − 0.951190i \(-0.599862\pi\)
−0.742856 + 0.669452i \(0.766529\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 5.48486 + 9.50006i 0.800049 + 1.38573i 0.919583 + 0.392896i \(0.128527\pi\)
−0.119534 + 0.992830i \(0.538140\pi\)
\(48\) 0 0
\(49\) 3.46065 − 5.99402i 0.494379 − 0.856289i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 8.58403 − 8.58403i 1.17911 − 1.17911i 0.199135 − 0.979972i \(-0.436187\pi\)
0.979972 − 0.199135i \(-0.0638133\pi\)
\(54\) 0 0
\(55\) 3.97344 0.535778
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 5.38532 − 1.44299i 0.701109 − 0.187862i 0.109382 − 0.994000i \(-0.465113\pi\)
0.591727 + 0.806138i \(0.298446\pi\)
\(60\) 0 0
\(61\) −6.23168 − 1.66977i −0.797885 − 0.213793i −0.163230 − 0.986588i \(-0.552191\pi\)
−0.634655 + 0.772796i \(0.718858\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −3.09519 + 1.78701i −0.383911 + 0.221651i
\(66\) 0 0
\(67\) −3.75640 + 1.00652i −0.458918 + 0.122967i −0.480869 − 0.876792i \(-0.659679\pi\)
0.0219514 + 0.999759i \(0.493012\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) − 10.6808i − 1.26758i −0.773506 − 0.633789i \(-0.781499\pi\)
0.773506 − 0.633789i \(-0.218501\pi\)
\(72\) 0 0
\(73\) 9.30419i 1.08897i 0.838770 + 0.544487i \(0.183275\pi\)
−0.838770 + 0.544487i \(0.816725\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0.862479 − 0.231101i 0.0982886 − 0.0263364i
\(78\) 0 0
\(79\) 8.70990 − 5.02866i 0.979940 − 0.565769i 0.0776882 − 0.996978i \(-0.475246\pi\)
0.902252 + 0.431209i \(0.141913\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 2.19703 + 0.588691i 0.241155 + 0.0646173i 0.377372 − 0.926062i \(-0.376828\pi\)
−0.136217 + 0.990679i \(0.543494\pi\)
\(84\) 0 0
\(85\) 4.48608 − 1.20204i 0.486584 − 0.130380i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −1.87637 −0.198895 −0.0994475 − 0.995043i \(-0.531708\pi\)
−0.0994475 + 0.995043i \(0.531708\pi\)
\(90\) 0 0
\(91\) −0.567911 + 0.567911i −0.0595332 + 0.0595332i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 3.62679 − 6.28179i 0.372101 − 0.644498i
\(96\) 0 0
\(97\) 9.19070 + 15.9188i 0.933175 + 1.61631i 0.777857 + 0.628441i \(0.216307\pi\)
0.155317 + 0.987865i \(0.450360\pi\)
\(98\) 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.z.a.719.7 88
3.2 odd 2 576.2.y.a.527.17 88
4.3 odd 2 432.2.v.a.395.1 88
9.2 odd 6 inner 1728.2.z.a.143.7 88
9.7 even 3 576.2.y.a.335.6 88
12.11 even 2 144.2.u.a.59.22 yes 88
16.3 odd 4 inner 1728.2.z.a.1583.7 88
16.13 even 4 432.2.v.a.179.7 88
36.7 odd 6 144.2.u.a.11.16 ✓ 88
36.11 even 6 432.2.v.a.251.7 88
48.29 odd 4 144.2.u.a.131.16 yes 88
48.35 even 4 576.2.y.a.239.6 88
144.29 odd 12 432.2.v.a.35.1 88
144.61 even 12 144.2.u.a.83.22 yes 88
144.83 even 12 inner 1728.2.z.a.1007.7 88
144.115 odd 12 576.2.y.a.47.17 88
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
144.2.u.a.11.16 ✓ 88 36.7 odd 6
144.2.u.a.59.22 yes 88 12.11 even 2
144.2.u.a.83.22 yes 88 144.61 even 12
144.2.u.a.131.16 yes 88 48.29 odd 4
432.2.v.a.35.1 88 144.29 odd 12
432.2.v.a.179.7 88 16.13 even 4
432.2.v.a.251.7 88 36.11 even 6
432.2.v.a.395.1 88 4.3 odd 2
576.2.y.a.47.17 88 144.115 odd 12
576.2.y.a.239.6 88 48.35 even 4
576.2.y.a.335.6 88 9.7 even 3
576.2.y.a.527.17 88 3.2 odd 2
1728.2.z.a.143.7 88 9.2 odd 6 inner
1728.2.z.a.719.7 88 1.1 even 1 trivial
1728.2.z.a.1007.7 88 144.83 even 12 inner
1728.2.z.a.1583.7 88 16.3 odd 4 inner