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Properties

Label 1728.2.s.a.1151.1
Level $1728$
Weight $2$
Character 1728.1151
Analytic conductor $13.798$
Analytic rank $0$
Dimension $2$
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(575,1728)] 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("1728.575"); 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([3, 0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1728 = 2^{6} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1728.s (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,-6,0,-6] 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: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) 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: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 1151.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 1728.1151
Dual form 1728.2.s.a.575.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-3.00000 - 1.73205i) q^{5} +(-3.00000 + 1.73205i) q^{7} +(-1.50000 - 2.59808i) q^{11} +(2.00000 - 3.46410i) q^{13} -1.73205i q^{17} +1.73205i q^{19} +(3.50000 + 6.06218i) q^{25} +(-3.00000 + 1.73205i) q^{29} +12.0000 q^{35} -2.00000 q^{37} +(4.50000 + 2.59808i) q^{41} +(-4.50000 + 2.59808i) q^{43} +(6.00000 + 10.3923i) q^{47} +(2.50000 - 4.33013i) q^{49} +10.3923i q^{55} +(-7.50000 + 12.9904i) q^{59} +(4.00000 + 6.92820i) q^{61} +(-12.0000 + 6.92820i) q^{65} +(7.50000 + 4.33013i) q^{67} +6.00000 q^{71} -11.0000 q^{73} +(9.00000 + 5.19615i) q^{77} +(3.00000 - 1.73205i) q^{79} +(6.00000 + 10.3923i) q^{83} +(-3.00000 + 5.19615i) q^{85} -13.8564i q^{89} +13.8564i q^{91} +(3.00000 - 5.19615i) q^{95} +(-6.50000 - 11.2583i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 6 q^{5} - 6 q^{7} - 3 q^{11} + 4 q^{13} + 7 q^{25} - 6 q^{29} + 24 q^{35} - 4 q^{37} + 9 q^{41} - 9 q^{43} + 12 q^{47} + 5 q^{49} - 15 q^{59} + 8 q^{61} - 24 q^{65} + 15 q^{67} + 12 q^{71} - 22 q^{73}+ \cdots - 13 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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
\(5\) −3.00000 − 1.73205i −1.34164 − 0.774597i −0.354593 − 0.935021i \(-0.615380\pi\)
−0.987048 + 0.160424i \(0.948714\pi\)
\(6\) 0 0
\(7\) −3.00000 + 1.73205i −1.13389 + 0.654654i −0.944911 − 0.327327i \(-0.893852\pi\)
−0.188982 + 0.981981i \(0.560519\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −1.50000 − 2.59808i −0.452267 − 0.783349i 0.546259 − 0.837616i \(-0.316051\pi\)
−0.998526 + 0.0542666i \(0.982718\pi\)
\(12\) 0 0
\(13\) 2.00000 − 3.46410i 0.554700 − 0.960769i −0.443227 − 0.896410i \(-0.646166\pi\)
0.997927 − 0.0643593i \(-0.0205004\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 1.73205i − 0.420084i −0.977692 − 0.210042i \(-0.932640\pi\)
0.977692 − 0.210042i \(-0.0673601\pi\)
\(18\) 0 0
\(19\) 1.73205i 0.397360i 0.980064 + 0.198680i \(0.0636654\pi\)
−0.980064 + 0.198680i \(0.936335\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(24\) 0 0
\(25\) 3.50000 + 6.06218i 0.700000 + 1.21244i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −3.00000 + 1.73205i −0.557086 + 0.321634i −0.751975 − 0.659192i \(-0.770899\pi\)
0.194889 + 0.980825i \(0.437565\pi\)
\(30\) 0 0
\(31\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 12.0000 2.02837
\(36\) 0 0
\(37\) −2.00000 −0.328798 −0.164399 − 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 4.50000 + 2.59808i 0.702782 + 0.405751i 0.808383 − 0.588657i \(-0.200343\pi\)
−0.105601 + 0.994409i \(0.533677\pi\)
\(42\) 0 0
\(43\) −4.50000 + 2.59808i −0.686244 + 0.396203i −0.802203 − 0.597051i \(-0.796339\pi\)
0.115960 + 0.993254i \(0.463006\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 6.00000 + 10.3923i 0.875190 + 1.51587i 0.856560 + 0.516047i \(0.172597\pi\)
0.0186297 + 0.999826i \(0.494070\pi\)
\(48\) 0 0
\(49\) 2.50000 − 4.33013i 0.357143 − 0.618590i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) 10.3923i 1.40130i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −7.50000 + 12.9904i −0.976417 + 1.69120i −0.301239 + 0.953549i \(0.597400\pi\)
−0.675178 + 0.737655i \(0.735933\pi\)
\(60\) 0 0
\(61\) 4.00000 + 6.92820i 0.512148 + 0.887066i 0.999901 + 0.0140840i \(0.00448323\pi\)
−0.487753 + 0.872982i \(0.662183\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −12.0000 + 6.92820i −1.48842 + 0.859338i
\(66\) 0 0
\(67\) 7.50000 + 4.33013i 0.916271 + 0.529009i 0.882443 − 0.470418i \(-0.155897\pi\)
0.0338274 + 0.999428i \(0.489230\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 6.00000 0.712069 0.356034 − 0.934473i \(-0.384129\pi\)
0.356034 + 0.934473i \(0.384129\pi\)
\(72\) 0 0
\(73\) −11.0000 −1.28745 −0.643726 − 0.765256i \(-0.722612\pi\)
−0.643726 + 0.765256i \(0.722612\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 9.00000 + 5.19615i 1.02565 + 0.592157i
\(78\) 0 0
\(79\) 3.00000 − 1.73205i 0.337526 − 0.194871i −0.321651 − 0.946858i \(-0.604238\pi\)
0.659178 + 0.751987i \(0.270905\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 6.00000 + 10.3923i 0.658586 + 1.14070i 0.980982 + 0.194099i \(0.0621783\pi\)
−0.322396 + 0.946605i \(0.604488\pi\)
\(84\) 0 0
\(85\) −3.00000 + 5.19615i −0.325396 + 0.563602i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) − 13.8564i − 1.46878i −0.678730 − 0.734388i \(-0.737469\pi\)
0.678730 − 0.734388i \(-0.262531\pi\)
\(90\) 0 0
\(91\) 13.8564i 1.45255i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 3.00000 − 5.19615i 0.307794 − 0.533114i
\(96\) 0 0
\(97\) −6.50000 − 11.2583i −0.659975 − 1.14311i −0.980622 − 0.195911i \(-0.937234\pi\)
0.320647 − 0.947199i \(-0.396100\pi\)
\(98\) 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.s.a.1151.1 2
3.2 odd 2 576.2.s.d.383.1 2
4.3 odd 2 1728.2.s.b.1151.1 2
8.3 odd 2 432.2.s.d.287.1 2
8.5 even 2 432.2.s.c.287.1 2
9.2 odd 6 1728.2.s.b.575.1 2
9.4 even 3 5184.2.c.c.5183.2 2
9.5 odd 6 5184.2.c.a.5183.1 2
9.7 even 3 576.2.s.a.191.1 2
12.11 even 2 576.2.s.a.383.1 2
24.5 odd 2 144.2.s.a.95.1 yes 2
24.11 even 2 144.2.s.d.95.1 yes 2
36.7 odd 6 576.2.s.d.191.1 2
36.11 even 6 inner 1728.2.s.a.575.1 2
36.23 even 6 5184.2.c.c.5183.1 2
36.31 odd 6 5184.2.c.a.5183.2 2
72.5 odd 6 1296.2.c.d.1295.2 2
72.11 even 6 432.2.s.c.143.1 2
72.13 even 6 1296.2.c.b.1295.1 2
72.29 odd 6 432.2.s.d.143.1 2
72.43 odd 6 144.2.s.a.47.1 ✓ 2
72.59 even 6 1296.2.c.b.1295.2 2
72.61 even 6 144.2.s.d.47.1 yes 2
72.67 odd 6 1296.2.c.d.1295.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
144.2.s.a.47.1 ✓ 2 72.43 odd 6
144.2.s.a.95.1 yes 2 24.5 odd 2
144.2.s.d.47.1 yes 2 72.61 even 6
144.2.s.d.95.1 yes 2 24.11 even 2
432.2.s.c.143.1 2 72.11 even 6
432.2.s.c.287.1 2 8.5 even 2
432.2.s.d.143.1 2 72.29 odd 6
432.2.s.d.287.1 2 8.3 odd 2
576.2.s.a.191.1 2 9.7 even 3
576.2.s.a.383.1 2 12.11 even 2
576.2.s.d.191.1 2 36.7 odd 6
576.2.s.d.383.1 2 3.2 odd 2
1296.2.c.b.1295.1 2 72.13 even 6
1296.2.c.b.1295.2 2 72.59 even 6
1296.2.c.d.1295.1 2 72.67 odd 6
1296.2.c.d.1295.2 2 72.5 odd 6
1728.2.s.a.575.1 2 36.11 even 6 inner
1728.2.s.a.1151.1 2 1.1 even 1 trivial
1728.2.s.b.575.1 2 9.2 odd 6
1728.2.s.b.1151.1 2 4.3 odd 2
5184.2.c.a.5183.1 2 9.5 odd 6
5184.2.c.a.5183.2 2 36.31 odd 6
5184.2.c.c.5183.1 2 36.23 even 6
5184.2.c.c.5183.2 2 9.4 even 3