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Properties

Label 1728.2.i.m.577.2
Level $1728$
Weight $2$
Character 1728.577
Analytic conductor $13.798$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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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(577,1728)] 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("1728.577"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1728, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1728 = 2^{6} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1728.i (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,2,0,2] 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: \(13.7981494693\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
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: \( 3 \)
Twist minimal: no (minimal twist has level 288)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 577.2
Root \(-1.22474 - 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 1728.577
Dual form 1728.2.i.m.1153.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.500000 - 0.866025i) q^{5} +(1.72474 + 2.98735i) q^{7} +(0.724745 + 1.25529i) q^{11} +(2.94949 - 5.10867i) q^{13} -4.89898 q^{17} +4.00000 q^{19} +(2.72474 - 4.71940i) q^{23} +(2.00000 + 3.46410i) q^{25} +(-0.0505103 - 0.0874863i) q^{29} +(1.27526 - 2.20881i) q^{31} +3.44949 q^{35} +0.898979 q^{37} +(-5.94949 + 10.3048i) q^{41} +(1.17423 + 2.03383i) q^{43} +(3.17423 + 5.49794i) q^{47} +(-2.44949 + 4.24264i) q^{49} +8.89898 q^{53} +1.44949 q^{55} +(7.17423 - 12.4261i) q^{59} +(-3.94949 - 6.84072i) q^{61} +(-2.94949 - 5.10867i) q^{65} +(-6.17423 + 10.6941i) q^{67} +7.79796 q^{71} -4.89898 q^{73} +(-2.50000 + 4.33013i) q^{77} +(6.72474 + 11.6476i) q^{79} +(-0.275255 - 0.476756i) q^{83} +(-2.44949 + 4.24264i) q^{85} +12.8990 q^{89} +20.3485 q^{91} +(2.00000 - 3.46410i) q^{95} +(1.94949 + 3.37662i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{5} + 2 q^{7} - 2 q^{11} + 2 q^{13} + 16 q^{19} + 6 q^{23} + 8 q^{25} - 10 q^{29} + 10 q^{31} + 4 q^{35} - 16 q^{37} - 14 q^{41} - 10 q^{43} - 2 q^{47} + 16 q^{53} - 4 q^{55} + 14 q^{59} - 6 q^{61}+ \cdots - 2 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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
\(5\) 0.500000 − 0.866025i 0.223607 − 0.387298i −0.732294 − 0.680989i \(-0.761550\pi\)
0.955901 + 0.293691i \(0.0948835\pi\)
\(6\) 0 0
\(7\) 1.72474 + 2.98735i 0.651892 + 1.12911i 0.982663 + 0.185399i \(0.0593579\pi\)
−0.330771 + 0.943711i \(0.607309\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0.724745 + 1.25529i 0.218519 + 0.378486i 0.954355 − 0.298674i \(-0.0965442\pi\)
−0.735837 + 0.677159i \(0.763211\pi\)
\(12\) 0 0
\(13\) 2.94949 − 5.10867i 0.818041 − 1.41689i −0.0890821 − 0.996024i \(-0.528393\pi\)
0.907123 − 0.420865i \(-0.138273\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −4.89898 −1.18818 −0.594089 − 0.804400i \(-0.702487\pi\)
−0.594089 + 0.804400i \(0.702487\pi\)
\(18\) 0 0
\(19\) 4.00000 0.917663 0.458831 − 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 2.72474 − 4.71940i 0.568149 − 0.984062i −0.428601 − 0.903494i \(-0.640993\pi\)
0.996749 − 0.0805681i \(-0.0256735\pi\)
\(24\) 0 0
\(25\) 2.00000 + 3.46410i 0.400000 + 0.692820i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −0.0505103 − 0.0874863i −0.00937952 − 0.0162458i 0.861298 − 0.508101i \(-0.169652\pi\)
−0.870677 + 0.491855i \(0.836319\pi\)
\(30\) 0 0
\(31\) 1.27526 − 2.20881i 0.229043 − 0.396713i −0.728482 − 0.685065i \(-0.759774\pi\)
0.957525 + 0.288352i \(0.0931072\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 3.44949 0.583070
\(36\) 0 0
\(37\) 0.898979 0.147791 0.0738957 − 0.997266i \(-0.476457\pi\)
0.0738957 + 0.997266i \(0.476457\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −5.94949 + 10.3048i −0.929154 + 1.60934i −0.144414 + 0.989517i \(0.546130\pi\)
−0.784740 + 0.619825i \(0.787204\pi\)
\(42\) 0 0
\(43\) 1.17423 + 2.03383i 0.179069 + 0.310157i 0.941562 − 0.336840i \(-0.109358\pi\)
−0.762493 + 0.646997i \(0.776025\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 3.17423 + 5.49794i 0.463010 + 0.801956i 0.999109 − 0.0421984i \(-0.0134362\pi\)
−0.536100 + 0.844155i \(0.680103\pi\)
\(48\) 0 0
\(49\) −2.44949 + 4.24264i −0.349927 + 0.606092i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 8.89898 1.22237 0.611184 − 0.791488i \(-0.290693\pi\)
0.611184 + 0.791488i \(0.290693\pi\)
\(54\) 0 0
\(55\) 1.44949 0.195449
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 7.17423 − 12.4261i 0.934006 − 1.61775i 0.157609 − 0.987502i \(-0.449622\pi\)
0.776397 − 0.630244i \(-0.217045\pi\)
\(60\) 0 0
\(61\) −3.94949 − 6.84072i −0.505680 − 0.875864i −0.999978 − 0.00657156i \(-0.997908\pi\)
0.494298 − 0.869292i \(-0.335425\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −2.94949 − 5.10867i −0.365839 − 0.633652i
\(66\) 0 0
\(67\) −6.17423 + 10.6941i −0.754303 + 1.30649i 0.191417 + 0.981509i \(0.438692\pi\)
−0.945720 + 0.324982i \(0.894642\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 7.79796 0.925447 0.462724 − 0.886503i \(-0.346872\pi\)
0.462724 + 0.886503i \(0.346872\pi\)
\(72\) 0 0
\(73\) −4.89898 −0.573382 −0.286691 − 0.958023i \(-0.592555\pi\)
−0.286691 + 0.958023i \(0.592555\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −2.50000 + 4.33013i −0.284901 + 0.493464i
\(78\) 0 0
\(79\) 6.72474 + 11.6476i 0.756593 + 1.31046i 0.944579 + 0.328286i \(0.106471\pi\)
−0.187986 + 0.982172i \(0.560196\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −0.275255 − 0.476756i −0.0302132 − 0.0523308i 0.850523 − 0.525937i \(-0.176285\pi\)
−0.880737 + 0.473606i \(0.842952\pi\)
\(84\) 0 0
\(85\) −2.44949 + 4.24264i −0.265684 + 0.460179i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 12.8990 1.36729 0.683645 − 0.729815i \(-0.260394\pi\)
0.683645 + 0.729815i \(0.260394\pi\)
\(90\) 0 0
\(91\) 20.3485 2.13310
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 2.00000 − 3.46410i 0.205196 − 0.355409i
\(96\) 0 0
\(97\) 1.94949 + 3.37662i 0.197941 + 0.342843i 0.947861 − 0.318685i \(-0.103241\pi\)
−0.749920 + 0.661529i \(0.769908\pi\)
\(98\) 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.i.m.577.2 4
3.2 odd 2 576.2.i.m.193.1 4
4.3 odd 2 1728.2.i.k.577.1 4
8.3 odd 2 864.2.i.c.577.1 4
8.5 even 2 864.2.i.e.577.2 4
9.2 odd 6 576.2.i.m.385.2 4
9.4 even 3 5184.2.a.bj.1.1 2
9.5 odd 6 5184.2.a.bu.1.1 2
9.7 even 3 inner 1728.2.i.m.1153.2 4
12.11 even 2 576.2.i.i.193.2 4
24.5 odd 2 288.2.i.c.193.2 yes 4
24.11 even 2 288.2.i.e.193.1 yes 4
36.7 odd 6 1728.2.i.k.1153.1 4
36.11 even 6 576.2.i.i.385.1 4
36.23 even 6 5184.2.a.by.1.2 2
36.31 odd 6 5184.2.a.bn.1.2 2
72.5 odd 6 2592.2.a.j.1.1 2
72.11 even 6 288.2.i.e.97.2 yes 4
72.13 even 6 2592.2.a.o.1.1 2
72.29 odd 6 288.2.i.c.97.1 ✓ 4
72.43 odd 6 864.2.i.c.289.1 4
72.59 even 6 2592.2.a.n.1.2 2
72.61 even 6 864.2.i.e.289.2 4
72.67 odd 6 2592.2.a.s.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
288.2.i.c.97.1 ✓ 4 72.29 odd 6
288.2.i.c.193.2 yes 4 24.5 odd 2
288.2.i.e.97.2 yes 4 72.11 even 6
288.2.i.e.193.1 yes 4 24.11 even 2
576.2.i.i.193.2 4 12.11 even 2
576.2.i.i.385.1 4 36.11 even 6
576.2.i.m.193.1 4 3.2 odd 2
576.2.i.m.385.2 4 9.2 odd 6
864.2.i.c.289.1 4 72.43 odd 6
864.2.i.c.577.1 4 8.3 odd 2
864.2.i.e.289.2 4 72.61 even 6
864.2.i.e.577.2 4 8.5 even 2
1728.2.i.k.577.1 4 4.3 odd 2
1728.2.i.k.1153.1 4 36.7 odd 6
1728.2.i.m.577.2 4 1.1 even 1 trivial
1728.2.i.m.1153.2 4 9.7 even 3 inner
2592.2.a.j.1.1 2 72.5 odd 6
2592.2.a.n.1.2 2 72.59 even 6
2592.2.a.o.1.1 2 72.13 even 6
2592.2.a.s.1.2 2 72.67 odd 6
5184.2.a.bj.1.1 2 9.4 even 3
5184.2.a.bn.1.2 2 36.31 odd 6
5184.2.a.bu.1.1 2 9.5 odd 6
5184.2.a.by.1.2 2 36.23 even 6