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Properties

Label 1152.2.l.a.287.3
Level $1152$
Weight $2$
Character 1152.287
Analytic conductor $9.199$
Analytic rank $0$
Dimension $16$
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] = [1152,2,Mod(287,1152)] 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("1152.287"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1152, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 1, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1152 = 2^{7} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1152.l (of order \(4\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(9.19876631285\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 4x^{14} + 6x^{12} - 12x^{10} + 33x^{8} - 48x^{6} + 96x^{4} - 256x^{2} + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{18} \)
Twist minimal: no (minimal twist has level 144)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 287.3
Root \(1.36166 - 0.381939i\) of defining polynomial
Character \(\chi\) \(=\) 1152.287
Dual form 1152.2.l.a.863.3

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.763878 + 0.763878i) q^{5} +1.33620 q^{7} +(-1.95945 - 1.95945i) q^{11} +(-4.18757 + 4.18757i) q^{13} -4.03243i q^{17} +(4.26785 + 4.26785i) q^{19} +8.86408i q^{23} +3.83298i q^{25} +(-1.23934 - 1.23934i) q^{29} +2.87835i q^{31} +(-1.02070 + 1.02070i) q^{35} +(-0.434870 - 0.434870i) q^{37} -7.81179 q^{41} +(5.49678 - 5.49678i) q^{43} -3.20723 q^{47} -5.21456 q^{49} +(-4.06777 + 4.06777i) q^{53} +2.99355 q^{55} +(4.71811 + 4.71811i) q^{59} +(-3.26785 + 3.26785i) q^{61} -6.39758i q^{65} +(5.44348 + 5.44348i) q^{67} -3.76718i q^{71} +10.5357i q^{73} +(-2.61822 - 2.61822i) q^{77} +11.1995i q^{79} +(-9.73306 + 9.73306i) q^{83} +(3.08029 + 3.08029i) q^{85} -1.64130 q^{89} +(-5.59544 + 5.59544i) q^{91} -6.52023 q^{95} -5.70272 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 16 q^{19} + 32 q^{43} + 16 q^{49} - 64 q^{55} + 32 q^{61} + 16 q^{67} + 32 q^{85} - 48 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1152\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(641\) \(901\)
\(\chi(n)\) \(-1\) \(-1\) \(e\left(\frac{1}{4}\right)\)

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
\(5\) −0.763878 + 0.763878i −0.341617 + 0.341617i −0.856975 − 0.515358i \(-0.827659\pi\)
0.515358 + 0.856975i \(0.327659\pi\)
\(6\) 0 0
\(7\) 1.33620 0.505038 0.252519 − 0.967592i \(-0.418741\pi\)
0.252519 + 0.967592i \(0.418741\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −1.95945 − 1.95945i −0.590795 − 0.590795i 0.347051 − 0.937846i \(-0.387183\pi\)
−0.937846 + 0.347051i \(0.887183\pi\)
\(12\) 0 0
\(13\) −4.18757 + 4.18757i −1.16142 + 1.16142i −0.177257 + 0.984165i \(0.556722\pi\)
−0.984165 + 0.177257i \(0.943278\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 4.03243i − 0.978009i −0.872281 − 0.489004i \(-0.837360\pi\)
0.872281 − 0.489004i \(-0.162640\pi\)
\(18\) 0 0
\(19\) 4.26785 + 4.26785i 0.979112 + 0.979112i 0.999786 − 0.0206739i \(-0.00658119\pi\)
−0.0206739 + 0.999786i \(0.506581\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 8.86408i 1.84829i 0.382044 + 0.924144i \(0.375220\pi\)
−0.382044 + 0.924144i \(0.624780\pi\)
\(24\) 0 0
\(25\) 3.83298i 0.766596i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −1.23934 − 1.23934i −0.230140 − 0.230140i 0.582611 − 0.812751i \(-0.302031\pi\)
−0.812751 + 0.582611i \(0.802031\pi\)
\(30\) 0 0
\(31\) 2.87835i 0.516968i 0.966016 + 0.258484i \(0.0832229\pi\)
−0.966016 + 0.258484i \(0.916777\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −1.02070 + 1.02070i −0.172529 + 0.172529i
\(36\) 0 0
\(37\) −0.434870 − 0.434870i −0.0714922 − 0.0714922i 0.670457 − 0.741949i \(-0.266098\pi\)
−0.741949 + 0.670457i \(0.766098\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −7.81179 −1.22000 −0.609998 − 0.792403i \(-0.708830\pi\)
−0.609998 + 0.792403i \(0.708830\pi\)
\(42\) 0 0
\(43\) 5.49678 − 5.49678i 0.838251 − 0.838251i −0.150378 − 0.988629i \(-0.548049\pi\)
0.988629 + 0.150378i \(0.0480491\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −3.20723 −0.467822 −0.233911 − 0.972258i \(-0.575152\pi\)
−0.233911 + 0.972258i \(0.575152\pi\)
\(48\) 0 0
\(49\) −5.21456 −0.744937
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −4.06777 + 4.06777i −0.558751 + 0.558751i −0.928952 − 0.370201i \(-0.879289\pi\)
0.370201 + 0.928952i \(0.379289\pi\)
\(54\) 0 0
\(55\) 2.99355 0.403651
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 4.71811 + 4.71811i 0.614245 + 0.614245i 0.944049 − 0.329804i \(-0.106983\pi\)
−0.329804 + 0.944049i \(0.606983\pi\)
\(60\) 0 0
\(61\) −3.26785 + 3.26785i −0.418406 + 0.418406i −0.884654 − 0.466248i \(-0.845605\pi\)
0.466248 + 0.884654i \(0.345605\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) − 6.39758i − 0.793522i
\(66\) 0 0
\(67\) 5.44348 + 5.44348i 0.665027 + 0.665027i 0.956561 − 0.291533i \(-0.0941654\pi\)
−0.291533 + 0.956561i \(0.594165\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) − 3.76718i − 0.447082i −0.974695 − 0.223541i \(-0.928238\pi\)
0.974695 − 0.223541i \(-0.0717616\pi\)
\(72\) 0 0
\(73\) 10.5357i 1.23311i 0.787312 + 0.616555i \(0.211472\pi\)
−0.787312 + 0.616555i \(0.788528\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −2.61822 − 2.61822i −0.298374 − 0.298374i
\(78\) 0 0
\(79\) 11.1995i 1.26004i 0.776578 + 0.630021i \(0.216954\pi\)
−0.776578 + 0.630021i \(0.783046\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −9.73306 + 9.73306i −1.06834 + 1.06834i −0.0708558 + 0.997487i \(0.522573\pi\)
−0.997487 + 0.0708558i \(0.977427\pi\)
\(84\) 0 0
\(85\) 3.08029 + 3.08029i 0.334104 + 0.334104i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −1.64130 −0.173977 −0.0869886 − 0.996209i \(-0.527724\pi\)
−0.0869886 + 0.996209i \(0.527724\pi\)
\(90\) 0 0
\(91\) −5.59544 + 5.59544i −0.586562 + 0.586562i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −6.52023 −0.668962
\(96\) 0 0
\(97\) −5.70272 −0.579024 −0.289512 − 0.957174i \(-0.593493\pi\)
−0.289512 + 0.957174i \(0.593493\pi\)
\(98\) 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 1152.2.l.a.287.3 16
3.2 odd 2 inner 1152.2.l.a.287.6 16
4.3 odd 2 1152.2.l.b.287.3 16
8.3 odd 2 576.2.l.a.143.6 16
8.5 even 2 144.2.l.a.107.2 yes 16
12.11 even 2 1152.2.l.b.287.6 16
16.3 odd 4 inner 1152.2.l.a.863.6 16
16.5 even 4 576.2.l.a.431.3 16
16.11 odd 4 144.2.l.a.35.7 yes 16
16.13 even 4 1152.2.l.b.863.6 16
24.5 odd 2 144.2.l.a.107.7 yes 16
24.11 even 2 576.2.l.a.143.3 16
48.5 odd 4 576.2.l.a.431.6 16
48.11 even 4 144.2.l.a.35.2 ✓ 16
48.29 odd 4 1152.2.l.b.863.3 16
48.35 even 4 inner 1152.2.l.a.863.3 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
144.2.l.a.35.2 ✓ 16 48.11 even 4
144.2.l.a.35.7 yes 16 16.11 odd 4
144.2.l.a.107.2 yes 16 8.5 even 2
144.2.l.a.107.7 yes 16 24.5 odd 2
576.2.l.a.143.3 16 24.11 even 2
576.2.l.a.143.6 16 8.3 odd 2
576.2.l.a.431.3 16 16.5 even 4
576.2.l.a.431.6 16 48.5 odd 4
1152.2.l.a.287.3 16 1.1 even 1 trivial
1152.2.l.a.287.6 16 3.2 odd 2 inner
1152.2.l.a.863.3 16 48.35 even 4 inner
1152.2.l.a.863.6 16 16.3 odd 4 inner
1152.2.l.b.287.3 16 4.3 odd 2
1152.2.l.b.287.6 16 12.11 even 2
1152.2.l.b.863.3 16 48.29 odd 4
1152.2.l.b.863.6 16 16.13 even 4