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Properties

Label 1152.2.l.a.287.7
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 287.7
Root \(-0.944649 + 1.05244i\) of defining polynomial
Character \(\chi\) \(=\) 1152.287
Dual form 1152.2.l.a.863.7

$q$-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)
 
\(f(q)\) \(=\) \(q+(2.10489 - 2.10489i) q^{5} -4.40731 q^{7} +(-0.215589 - 0.215589i) q^{11} +(2.73544 - 2.73544i) q^{13} +2.36438i q^{17} +(-0.758681 - 0.758681i) q^{19} -1.75549i q^{23} -3.86110i q^{25} +(-5.54221 - 5.54221i) q^{29} -9.01709i q^{31} +(-9.27690 + 9.27690i) q^{35} +(-3.10242 - 3.10242i) q^{37} -10.1014 q^{41} +(3.54621 - 3.54621i) q^{43} -3.90136 q^{47} +12.4244 q^{49} +(-2.71378 + 2.71378i) q^{53} -0.907583 q^{55} +(3.40445 + 3.40445i) q^{59} +(1.75868 - 1.75868i) q^{61} -11.5156i q^{65} +(-9.11951 - 9.11951i) q^{67} +11.8897i q^{71} +0.482639i q^{73} +(0.950169 + 0.950169i) q^{77} +6.88995i q^{79} +(4.79951 - 4.79951i) q^{83} +(4.97676 + 4.97676i) q^{85} +7.00534 q^{89} +(-12.0559 + 12.0559i) q^{91} -3.19387 q^{95} -3.34374 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 16 q^{19} + 32 q^{43} + 16 q^{49} - 64 q^{55} + 32 q^{61} + 16 q^{67} + 32 q^{85} - 48 q^{91}+O(q^{100}) \) 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
\(5\) 2.10489 − 2.10489i 0.941334 − 0.941334i −0.0570377 − 0.998372i \(-0.518166\pi\)
0.998372 + 0.0570377i \(0.0181655\pi\)
\(6\) 0 0
\(7\) −4.40731 −1.66581 −0.832904 − 0.553418i \(-0.813323\pi\)
−0.832904 + 0.553418i \(0.813323\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −0.215589 − 0.215589i −0.0650026 − 0.0650026i 0.673858 − 0.738861i \(-0.264636\pi\)
−0.738861 + 0.673858i \(0.764636\pi\)
\(12\) 0 0
\(13\) 2.73544 − 2.73544i 0.758675 − 0.758675i −0.217406 − 0.976081i \(-0.569760\pi\)
0.976081 + 0.217406i \(0.0697597\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 2.36438i 0.573447i 0.958013 + 0.286724i \(0.0925661\pi\)
−0.958013 + 0.286724i \(0.907434\pi\)
\(18\) 0 0
\(19\) −0.758681 − 0.758681i −0.174053 − 0.174053i 0.614704 − 0.788758i \(-0.289275\pi\)
−0.788758 + 0.614704i \(0.789275\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) − 1.75549i − 0.366045i −0.983109 − 0.183023i \(-0.941412\pi\)
0.983109 − 0.183023i \(-0.0585881\pi\)
\(24\) 0 0
\(25\) − 3.86110i − 0.772221i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −5.54221 − 5.54221i −1.02916 − 1.02916i −0.999562 − 0.0296002i \(-0.990577\pi\)
−0.0296002 − 0.999562i \(-0.509423\pi\)
\(30\) 0 0
\(31\) − 9.01709i − 1.61952i −0.586763 − 0.809759i \(-0.699598\pi\)
0.586763 − 0.809759i \(-0.300402\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −9.27690 + 9.27690i −1.56808 + 1.56808i
\(36\) 0 0
\(37\) −3.10242 − 3.10242i −0.510035 − 0.510035i 0.404502 − 0.914537i \(-0.367445\pi\)
−0.914537 + 0.404502i \(0.867445\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −10.1014 −1.57757 −0.788785 − 0.614669i \(-0.789290\pi\)
−0.788785 + 0.614669i \(0.789290\pi\)
\(42\) 0 0
\(43\) 3.54621 − 3.54621i 0.540792 − 0.540792i −0.382969 − 0.923761i \(-0.625099\pi\)
0.923761 + 0.382969i \(0.125099\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −3.90136 −0.569072 −0.284536 − 0.958665i \(-0.591840\pi\)
−0.284536 + 0.958665i \(0.591840\pi\)
\(48\) 0 0
\(49\) 12.4244 1.77491
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −2.71378 + 2.71378i −0.372766 + 0.372766i −0.868484 − 0.495717i \(-0.834905\pi\)
0.495717 + 0.868484i \(0.334905\pi\)
\(54\) 0 0
\(55\) −0.907583 −0.122378
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 3.40445 + 3.40445i 0.443222 + 0.443222i 0.893093 − 0.449871i \(-0.148530\pi\)
−0.449871 + 0.893093i \(0.648530\pi\)
\(60\) 0 0
\(61\) 1.75868 − 1.75868i 0.225176 − 0.225176i −0.585498 − 0.810674i \(-0.699101\pi\)
0.810674 + 0.585498i \(0.199101\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) − 11.5156i − 1.42833i
\(66\) 0 0
\(67\) −9.11951 − 9.11951i −1.11413 − 1.11413i −0.992587 − 0.121539i \(-0.961217\pi\)
−0.121539 − 0.992587i \(-0.538783\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 11.8897i 1.41105i 0.708684 + 0.705526i \(0.249289\pi\)
−0.708684 + 0.705526i \(0.750711\pi\)
\(72\) 0 0
\(73\) 0.482639i 0.0564886i 0.999601 + 0.0282443i \(0.00899163\pi\)
−0.999601 + 0.0282443i \(0.991008\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0.950169 + 0.950169i 0.108282 + 0.108282i
\(78\) 0 0
\(79\) 6.88995i 0.775180i 0.921832 + 0.387590i \(0.126692\pi\)
−0.921832 + 0.387590i \(0.873308\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 4.79951 − 4.79951i 0.526814 − 0.526814i −0.392807 − 0.919621i \(-0.628496\pi\)
0.919621 + 0.392807i \(0.128496\pi\)
\(84\) 0 0
\(85\) 4.97676 + 4.97676i 0.539805 + 0.539805i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 7.00534 0.742564 0.371282 − 0.928520i \(-0.378918\pi\)
0.371282 + 0.928520i \(0.378918\pi\)
\(90\) 0 0
\(91\) −12.0559 + 12.0559i −1.26381 + 1.26381i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −3.19387 −0.327685
\(96\) 0 0
\(97\) −3.34374 −0.339506 −0.169753 − 0.985487i \(-0.554297\pi\)
−0.169753 + 0.985487i \(0.554297\pi\)
\(98\) 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.287.7 16
3.2 odd 2 inner 1152.2.l.a.287.2 16
4.3 odd 2 1152.2.l.b.287.7 16
8.3 odd 2 576.2.l.a.143.2 16
8.5 even 2 144.2.l.a.107.1 yes 16
12.11 even 2 1152.2.l.b.287.2 16
16.3 odd 4 inner 1152.2.l.a.863.2 16
16.5 even 4 576.2.l.a.431.7 16
16.11 odd 4 144.2.l.a.35.8 yes 16
16.13 even 4 1152.2.l.b.863.2 16
24.5 odd 2 144.2.l.a.107.8 yes 16
24.11 even 2 576.2.l.a.143.7 16
48.5 odd 4 576.2.l.a.431.2 16
48.11 even 4 144.2.l.a.35.1 ✓ 16
48.29 odd 4 1152.2.l.b.863.7 16
48.35 even 4 inner 1152.2.l.a.863.7 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
144.2.l.a.35.1 ✓ 16 48.11 even 4
144.2.l.a.35.8 yes 16 16.11 odd 4
144.2.l.a.107.1 yes 16 8.5 even 2
144.2.l.a.107.8 yes 16 24.5 odd 2
576.2.l.a.143.2 16 8.3 odd 2
576.2.l.a.143.7 16 24.11 even 2
576.2.l.a.431.2 16 48.5 odd 4
576.2.l.a.431.7 16 16.5 even 4
1152.2.l.a.287.2 16 3.2 odd 2 inner
1152.2.l.a.287.7 16 1.1 even 1 trivial
1152.2.l.a.863.2 16 16.3 odd 4 inner
1152.2.l.a.863.7 16 48.35 even 4 inner
1152.2.l.b.287.2 16 12.11 even 2
1152.2.l.b.287.7 16 4.3 odd 2
1152.2.l.b.863.2 16 16.13 even 4
1152.2.l.b.863.7 16 48.29 odd 4