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Properties

Label 1152.2.l.a.287.4
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 287.4
Root \(1.40927 - 0.118126i\) of defining polynomial
Character \(\chi\) \(=\) 1152.287
Dual form 1152.2.l.a.863.4

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.236253 + 0.236253i) q^{5} +3.27830 q^{7} +(-2.58229 - 2.58229i) q^{11} +(1.70773 - 1.70773i) q^{13} -7.05130i q^{17} +(-3.04184 - 3.04184i) q^{19} +1.47338i q^{23} +4.88837i q^{25} +(2.98575 + 2.98575i) q^{29} -8.02552i q^{31} +(-0.774506 + 0.774506i) q^{35} +(7.93021 + 7.93021i) q^{37} +2.22112 q^{41} +(4.61007 - 4.61007i) q^{43} -7.13023 q^{47} +3.74723 q^{49} +(5.81417 - 5.81417i) q^{53} +1.22015 q^{55} +(-7.46464 - 7.46464i) q^{59} +(4.04184 - 4.04184i) q^{61} +0.806909i q^{65} +(2.90468 + 2.90468i) q^{67} +1.02064i q^{71} -4.08367i q^{73} +(-8.46551 - 8.46551i) q^{77} -5.36197i q^{79} +(3.93734 - 3.93734i) q^{83} +(1.66589 + 1.66589i) q^{85} -2.35922 q^{89} +(5.59843 - 5.59843i) q^{91} +1.43728 q^{95} +9.97204 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 16 q^{19} + 32 q^{43} + 16 q^{49} - 64 q^{55} + 32 q^{61} + 16 q^{67} + 32 q^{85} - 48 q^{91}+O(q^{100}) \) 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
\(5\) −0.236253 + 0.236253i −0.105655 + 0.105655i −0.757958 − 0.652303i \(-0.773803\pi\)
0.652303 + 0.757958i \(0.273803\pi\)
\(6\) 0 0
\(7\) 3.27830 1.23908 0.619540 − 0.784965i \(-0.287319\pi\)
0.619540 + 0.784965i \(0.287319\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −2.58229 − 2.58229i −0.778590 − 0.778590i 0.201001 − 0.979591i \(-0.435580\pi\)
−0.979591 + 0.201001i \(0.935580\pi\)
\(12\) 0 0
\(13\) 1.70773 − 1.70773i 0.473638 − 0.473638i −0.429452 − 0.903090i \(-0.641293\pi\)
0.903090 + 0.429452i \(0.141293\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 7.05130i − 1.71019i −0.518470 − 0.855096i \(-0.673498\pi\)
0.518470 − 0.855096i \(-0.326502\pi\)
\(18\) 0 0
\(19\) −3.04184 − 3.04184i −0.697845 − 0.697845i 0.266100 − 0.963945i \(-0.414265\pi\)
−0.963945 + 0.266100i \(0.914265\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 1.47338i 0.307221i 0.988131 + 0.153610i \(0.0490901\pi\)
−0.988131 + 0.153610i \(0.950910\pi\)
\(24\) 0 0
\(25\) 4.88837i 0.977674i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 2.98575 + 2.98575i 0.554439 + 0.554439i 0.927719 − 0.373280i \(-0.121767\pi\)
−0.373280 + 0.927719i \(0.621767\pi\)
\(30\) 0 0
\(31\) − 8.02552i − 1.44143i −0.693233 − 0.720713i \(-0.743814\pi\)
0.693233 − 0.720713i \(-0.256186\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −0.774506 + 0.774506i −0.130915 + 0.130915i
\(36\) 0 0
\(37\) 7.93021 + 7.93021i 1.30372 + 1.30372i 0.925866 + 0.377852i \(0.123337\pi\)
0.377852 + 0.925866i \(0.376663\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 2.22112 0.346881 0.173441 − 0.984844i \(-0.444512\pi\)
0.173441 + 0.984844i \(0.444512\pi\)
\(42\) 0 0
\(43\) 4.61007 − 4.61007i 0.703030 − 0.703030i −0.262030 − 0.965060i \(-0.584392\pi\)
0.965060 + 0.262030i \(0.0843920\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −7.13023 −1.04005 −0.520026 − 0.854151i \(-0.674078\pi\)
−0.520026 + 0.854151i \(0.674078\pi\)
\(48\) 0 0
\(49\) 3.74723 0.535318
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 5.81417 − 5.81417i 0.798638 − 0.798638i −0.184243 − 0.982881i \(-0.558983\pi\)
0.982881 + 0.184243i \(0.0589833\pi\)
\(54\) 0 0
\(55\) 1.22015 0.164524
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −7.46464 − 7.46464i −0.971813 − 0.971813i 0.0278004 − 0.999613i \(-0.491150\pi\)
−0.999613 + 0.0278004i \(0.991150\pi\)
\(60\) 0 0
\(61\) 4.04184 − 4.04184i 0.517504 − 0.517504i −0.399311 − 0.916815i \(-0.630751\pi\)
0.916815 + 0.399311i \(0.130751\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0.806909i 0.100085i
\(66\) 0 0
\(67\) 2.90468 + 2.90468i 0.354863 + 0.354863i 0.861915 − 0.507052i \(-0.169265\pi\)
−0.507052 + 0.861915i \(0.669265\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 1.02064i 0.121128i 0.998164 + 0.0605640i \(0.0192899\pi\)
−0.998164 + 0.0605640i \(0.980710\pi\)
\(72\) 0 0
\(73\) − 4.08367i − 0.477958i −0.971025 − 0.238979i \(-0.923187\pi\)
0.971025 − 0.238979i \(-0.0768127\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −8.46551 − 8.46551i −0.964735 − 0.964735i
\(78\) 0 0
\(79\) − 5.36197i − 0.603269i −0.953424 − 0.301634i \(-0.902468\pi\)
0.953424 − 0.301634i \(-0.0975322\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 3.93734 − 3.93734i 0.432179 − 0.432179i −0.457190 − 0.889369i \(-0.651144\pi\)
0.889369 + 0.457190i \(0.151144\pi\)
\(84\) 0 0
\(85\) 1.66589 + 1.66589i 0.180691 + 0.180691i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −2.35922 −0.250077 −0.125039 − 0.992152i \(-0.539905\pi\)
−0.125039 + 0.992152i \(0.539905\pi\)
\(90\) 0 0
\(91\) 5.59843 − 5.59843i 0.586875 − 0.586875i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 1.43728 0.147462
\(96\) 0 0
\(97\) 9.97204 1.01251 0.506254 − 0.862385i \(-0.331030\pi\)
0.506254 + 0.862385i \(0.331030\pi\)
\(98\) 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.287.4 16
3.2 odd 2 inner 1152.2.l.a.287.5 16
4.3 odd 2 1152.2.l.b.287.4 16
8.3 odd 2 576.2.l.a.143.5 16
8.5 even 2 144.2.l.a.107.3 yes 16
12.11 even 2 1152.2.l.b.287.5 16
16.3 odd 4 inner 1152.2.l.a.863.5 16
16.5 even 4 576.2.l.a.431.4 16
16.11 odd 4 144.2.l.a.35.6 yes 16
16.13 even 4 1152.2.l.b.863.5 16
24.5 odd 2 144.2.l.a.107.6 yes 16
24.11 even 2 576.2.l.a.143.4 16
48.5 odd 4 576.2.l.a.431.5 16
48.11 even 4 144.2.l.a.35.3 ✓ 16
48.29 odd 4 1152.2.l.b.863.4 16
48.35 even 4 inner 1152.2.l.a.863.4 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
144.2.l.a.35.3 ✓ 16 48.11 even 4
144.2.l.a.35.6 yes 16 16.11 odd 4
144.2.l.a.107.3 yes 16 8.5 even 2
144.2.l.a.107.6 yes 16 24.5 odd 2
576.2.l.a.143.4 16 24.11 even 2
576.2.l.a.143.5 16 8.3 odd 2
576.2.l.a.431.4 16 16.5 even 4
576.2.l.a.431.5 16 48.5 odd 4
1152.2.l.a.287.4 16 1.1 even 1 trivial
1152.2.l.a.287.5 16 3.2 odd 2 inner
1152.2.l.a.863.4 16 48.35 even 4 inner
1152.2.l.a.863.5 16 16.3 odd 4 inner
1152.2.l.b.287.4 16 4.3 odd 2
1152.2.l.b.287.5 16 12.11 even 2
1152.2.l.b.863.4 16 48.29 odd 4
1152.2.l.b.863.5 16 16.13 even 4