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Properties

Label 1152.5.b.h.703.1
Level $1152$
Weight $5$
Character 1152.703
Analytic conductor $119.082$
Analytic rank $0$
Dimension $4$
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] = [1152,5,Mod(703,1152)] 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("1152.703"); S:= CuspForms(chi, 5); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1152, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 0])) N = Newforms(chi, 5, names="a")
 
Level: \( N \) \(=\) \( 1152 = 2^{7} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 1152.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,64] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(119.082197473\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-17})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 16x^{2} + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{7}\cdot 3^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 703.1
Root \(2.91548 - 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 1152.703
Dual form 1152.5.b.h.703.4

$q$-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)
 
\(f(q)\) \(=\) \(q-24.7386i q^{5} -50.9117i q^{7} +139.943 q^{11} +98.9545i q^{13} +16.0000 q^{17} +559.771 q^{19} +814.587i q^{23} +13.0000 q^{25} -420.557i q^{29} +661.852i q^{31} -1259.49 q^{35} +2473.86i q^{37} +464.000 q^{41} -559.771 q^{43} +2443.76i q^{47} -191.000 q^{49} +3191.28i q^{53} -3461.99i q^{55} +3638.51 q^{59} +4255.05i q^{61} +2448.00 q^{65} -5597.71 q^{67} -3258.35i q^{71} +898.000 q^{73} -7124.73i q^{77} +8604.08i q^{79} -419.829 q^{83} -395.818i q^{85} +7072.00 q^{89} +5037.94 q^{91} -13848.0i q^{95} -2366.00 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 64 q^{17} + 52 q^{25} + 1856 q^{41} - 764 q^{49} + 9792 q^{65} + 3592 q^{73} + 28288 q^{89} - 9464 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(127\) \(641\) \(901\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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
\(5\) − 24.7386i − 0.989545i −0.869022 − 0.494773i \(-0.835251\pi\)
0.869022 − 0.494773i \(-0.164749\pi\)
\(6\) 0 0
\(7\) − 50.9117i − 1.03901i −0.854466 − 0.519507i \(-0.826116\pi\)
0.854466 − 0.519507i \(-0.173884\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 139.943 1.15655 0.578276 − 0.815841i \(-0.303726\pi\)
0.578276 + 0.815841i \(0.303726\pi\)
\(12\) 0 0
\(13\) 98.9545i 0.585530i 0.956184 + 0.292765i \(0.0945753\pi\)
−0.956184 + 0.292765i \(0.905425\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 16.0000 0.0553633 0.0276817 − 0.999617i \(-0.491188\pi\)
0.0276817 + 0.999617i \(0.491188\pi\)
\(18\) 0 0
\(19\) 559.771 1.55061 0.775307 − 0.631585i \(-0.217595\pi\)
0.775307 + 0.631585i \(0.217595\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 814.587i 1.53986i 0.638127 + 0.769931i \(0.279709\pi\)
−0.638127 + 0.769931i \(0.720291\pi\)
\(24\) 0 0
\(25\) 13.0000 0.0208000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) − 420.557i − 0.500068i −0.968237 − 0.250034i \(-0.919558\pi\)
0.968237 − 0.250034i \(-0.0804417\pi\)
\(30\) 0 0
\(31\) 661.852i 0.688712i 0.938839 + 0.344356i \(0.111903\pi\)
−0.938839 + 0.344356i \(0.888097\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −1259.49 −1.02815
\(36\) 0 0
\(37\) 2473.86i 1.80706i 0.428526 + 0.903529i \(0.359033\pi\)
−0.428526 + 0.903529i \(0.640967\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 464.000 0.276026 0.138013 − 0.990430i \(-0.455928\pi\)
0.138013 + 0.990430i \(0.455928\pi\)
\(42\) 0 0
\(43\) −559.771 −0.302743 −0.151371 − 0.988477i \(-0.548369\pi\)
−0.151371 + 0.988477i \(0.548369\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 2443.76i 1.10627i 0.833090 + 0.553137i \(0.186570\pi\)
−0.833090 + 0.553137i \(0.813430\pi\)
\(48\) 0 0
\(49\) −191.000 −0.0795502
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 3191.28i 1.13609i 0.822997 + 0.568046i \(0.192301\pi\)
−0.822997 + 0.568046i \(0.807699\pi\)
\(54\) 0 0
\(55\) − 3461.99i − 1.14446i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 3638.51 1.04525 0.522625 − 0.852563i \(-0.324953\pi\)
0.522625 + 0.852563i \(0.324953\pi\)
\(60\) 0 0
\(61\) 4255.05i 1.14352i 0.820420 + 0.571761i \(0.193740\pi\)
−0.820420 + 0.571761i \(0.806260\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 2448.00 0.579408
\(66\) 0 0
\(67\) −5597.71 −1.24698 −0.623492 − 0.781829i \(-0.714287\pi\)
−0.623492 + 0.781829i \(0.714287\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) − 3258.35i − 0.646369i −0.946336 − 0.323185i \(-0.895246\pi\)
0.946336 − 0.323185i \(-0.104754\pi\)
\(72\) 0 0
\(73\) 898.000 0.168512 0.0842560 − 0.996444i \(-0.473149\pi\)
0.0842560 + 0.996444i \(0.473149\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 7124.73i − 1.20167i
\(78\) 0 0
\(79\) 8604.08i 1.37864i 0.724458 + 0.689319i \(0.242090\pi\)
−0.724458 + 0.689319i \(0.757910\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −419.829 −0.0609419 −0.0304709 − 0.999536i \(-0.509701\pi\)
−0.0304709 + 0.999536i \(0.509701\pi\)
\(84\) 0 0
\(85\) − 395.818i − 0.0547845i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 7072.00 0.892817 0.446408 − 0.894829i \(-0.352703\pi\)
0.446408 + 0.894829i \(0.352703\pi\)
\(90\) 0 0
\(91\) 5037.94 0.608374
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) − 13848.0i − 1.53440i
\(96\) 0 0
\(97\) −2366.00 −0.251461 −0.125731 − 0.992064i \(-0.540128\pi\)
−0.125731 + 0.992064i \(0.540128\pi\)
\(98\) 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 1152.5.b.h.703.1 yes 4
3.2 odd 2 1152.5.b.f.703.3 yes 4
4.3 odd 2 inner 1152.5.b.h.703.2 yes 4
8.3 odd 2 inner 1152.5.b.h.703.4 yes 4
8.5 even 2 inner 1152.5.b.h.703.3 yes 4
12.11 even 2 1152.5.b.f.703.4 yes 4
24.5 odd 2 1152.5.b.f.703.1 ✓ 4
24.11 even 2 1152.5.b.f.703.2 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1152.5.b.f.703.1 ✓ 4 24.5 odd 2
1152.5.b.f.703.2 yes 4 24.11 even 2
1152.5.b.f.703.3 yes 4 3.2 odd 2
1152.5.b.f.703.4 yes 4 12.11 even 2
1152.5.b.h.703.1 yes 4 1.1 even 1 trivial
1152.5.b.h.703.2 yes 4 4.3 odd 2 inner
1152.5.b.h.703.3 yes 4 8.5 even 2 inner
1152.5.b.h.703.4 yes 4 8.3 odd 2 inner