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Properties

Label 1152.2.f.a.575.2
Level $1152$
Weight $2$
Character 1152.575
Analytic conductor $9.199$
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] = [1152,2,Mod(575,1152)] 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("1152.575"); S:= CuspForms(chi, 2); 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, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1152 = 2^{7} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1152.f (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-16,0,-12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(25)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(9.19876631285\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 575.2
Root \(0.707107 - 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 1152.575
Dual form 1152.2.f.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)
 
\(f(q)\) \(=\) \(q-1.41421 q^{5} +2.82843i q^{7} -4.00000i q^{11} +2.00000i q^{13} +1.41421i q^{17} -5.65685 q^{19} -4.00000 q^{23} -3.00000 q^{25} +7.07107 q^{29} +8.48528i q^{31} -4.00000i q^{35} -8.00000i q^{37} -4.24264i q^{41} -11.3137 q^{43} -12.0000 q^{47} -1.00000 q^{49} -12.7279 q^{53} +5.65685i q^{55} +8.00000i q^{61} -2.82843i q^{65} -5.65685 q^{67} -4.00000 q^{71} +8.00000 q^{73} +11.3137 q^{77} +2.82843i q^{79} -12.0000i q^{83} -2.00000i q^{85} +15.5563i q^{89} -5.65685 q^{91} +8.00000 q^{95} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 16 q^{23} - 12 q^{25} - 48 q^{47} - 4 q^{49} - 16 q^{71} + 32 q^{73} + 32 q^{95}+O(q^{100}) \) 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
\(5\) −1.41421 −0.632456 −0.316228 − 0.948683i \(-0.602416\pi\)
−0.316228 + 0.948683i \(0.602416\pi\)
\(6\) 0 0
\(7\) 2.82843i 1.06904i 0.845154 + 0.534522i \(0.179509\pi\)
−0.845154 + 0.534522i \(0.820491\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) − 4.00000i − 1.20605i −0.797724 − 0.603023i \(-0.793963\pi\)
0.797724 − 0.603023i \(-0.206037\pi\)
\(12\) 0 0
\(13\) 2.00000i 0.554700i 0.960769 + 0.277350i \(0.0894562\pi\)
−0.960769 + 0.277350i \(0.910544\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 1.41421i 0.342997i 0.985184 + 0.171499i \(0.0548609\pi\)
−0.985184 + 0.171499i \(0.945139\pi\)
\(18\) 0 0
\(19\) −5.65685 −1.29777 −0.648886 − 0.760886i \(-0.724765\pi\)
−0.648886 + 0.760886i \(0.724765\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −4.00000 −0.834058 −0.417029 − 0.908893i \(-0.636929\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(24\) 0 0
\(25\) −3.00000 −0.600000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 7.07107 1.31306 0.656532 − 0.754298i \(-0.272023\pi\)
0.656532 + 0.754298i \(0.272023\pi\)
\(30\) 0 0
\(31\) 8.48528i 1.52400i 0.647576 + 0.762001i \(0.275783\pi\)
−0.647576 + 0.762001i \(0.724217\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) − 4.00000i − 0.676123i
\(36\) 0 0
\(37\) − 8.00000i − 1.31519i −0.753371 − 0.657596i \(-0.771573\pi\)
0.753371 − 0.657596i \(-0.228427\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) − 4.24264i − 0.662589i −0.943527 − 0.331295i \(-0.892515\pi\)
0.943527 − 0.331295i \(-0.107485\pi\)
\(42\) 0 0
\(43\) −11.3137 −1.72532 −0.862662 − 0.505781i \(-0.831205\pi\)
−0.862662 + 0.505781i \(0.831205\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −12.0000 −1.75038 −0.875190 − 0.483779i \(-0.839264\pi\)
−0.875190 + 0.483779i \(0.839264\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −12.7279 −1.74831 −0.874157 − 0.485643i \(-0.838586\pi\)
−0.874157 + 0.485643i \(0.838586\pi\)
\(54\) 0 0
\(55\) 5.65685i 0.762770i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 0 0
\(61\) 8.00000i 1.02430i 0.858898 + 0.512148i \(0.171150\pi\)
−0.858898 + 0.512148i \(0.828850\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) − 2.82843i − 0.350823i
\(66\) 0 0
\(67\) −5.65685 −0.691095 −0.345547 − 0.938401i \(-0.612307\pi\)
−0.345547 + 0.938401i \(0.612307\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −4.00000 −0.474713 −0.237356 − 0.971423i \(-0.576281\pi\)
−0.237356 + 0.971423i \(0.576281\pi\)
\(72\) 0 0
\(73\) 8.00000 0.936329 0.468165 − 0.883641i \(-0.344915\pi\)
0.468165 + 0.883641i \(0.344915\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 11.3137 1.28932
\(78\) 0 0
\(79\) 2.82843i 0.318223i 0.987261 + 0.159111i \(0.0508629\pi\)
−0.987261 + 0.159111i \(0.949137\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) − 12.0000i − 1.31717i −0.752506 − 0.658586i \(-0.771155\pi\)
0.752506 − 0.658586i \(-0.228845\pi\)
\(84\) 0 0
\(85\) − 2.00000i − 0.216930i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 15.5563i 1.64897i 0.565884 + 0.824485i \(0.308535\pi\)
−0.565884 + 0.824485i \(0.691465\pi\)
\(90\) 0 0
\(91\) −5.65685 −0.592999
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 8.00000 0.820783
\(96\) 0 0
\(97\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(98\) 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.f.a.575.2 yes 4
3.2 odd 2 1152.2.f.d.575.4 yes 4
4.3 odd 2 1152.2.f.d.575.1 yes 4
8.3 odd 2 1152.2.f.d.575.3 yes 4
8.5 even 2 inner 1152.2.f.a.575.4 yes 4
12.11 even 2 inner 1152.2.f.a.575.3 yes 4
16.3 odd 4 2304.2.c.g.2303.2 2
16.5 even 4 2304.2.c.h.2303.1 2
16.11 odd 4 2304.2.c.b.2303.1 2
16.13 even 4 2304.2.c.a.2303.2 2
24.5 odd 2 1152.2.f.d.575.2 yes 4
24.11 even 2 inner 1152.2.f.a.575.1 ✓ 4
48.5 odd 4 2304.2.c.b.2303.2 2
48.11 even 4 2304.2.c.h.2303.2 2
48.29 odd 4 2304.2.c.g.2303.1 2
48.35 even 4 2304.2.c.a.2303.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1152.2.f.a.575.1 ✓ 4 24.11 even 2 inner
1152.2.f.a.575.2 yes 4 1.1 even 1 trivial
1152.2.f.a.575.3 yes 4 12.11 even 2 inner
1152.2.f.a.575.4 yes 4 8.5 even 2 inner
1152.2.f.d.575.1 yes 4 4.3 odd 2
1152.2.f.d.575.2 yes 4 24.5 odd 2
1152.2.f.d.575.3 yes 4 8.3 odd 2
1152.2.f.d.575.4 yes 4 3.2 odd 2
2304.2.c.a.2303.1 2 48.35 even 4
2304.2.c.a.2303.2 2 16.13 even 4
2304.2.c.b.2303.1 2 16.11 odd 4
2304.2.c.b.2303.2 2 48.5 odd 4
2304.2.c.g.2303.1 2 48.29 odd 4
2304.2.c.g.2303.2 2 16.3 odd 4
2304.2.c.h.2303.1 2 16.5 even 4
2304.2.c.h.2303.2 2 48.11 even 4