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Properties

Label 5184.2.c.h.5183.7
Level $5184$
Weight $2$
Character 5184.5183
Analytic conductor $41.394$
Analytic rank $0$
Dimension $8$
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] = [5184,2,Mod(5183,5184)] 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("5184.5183"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5184, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 5184 = 2^{6} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5184.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,0,0,0,0,0,0,0,0,0,-8,0,0,0,0,0,0,0,0,0,0,0,-8] 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: \(41.3944484078\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{24})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{6} \)
Twist minimal: no (minimal twist has level 1296)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 5183.7
Root \(0.258819 - 0.965926i\) of defining polynomial
Character \(\chi\) \(=\) 5184.5183
Dual form 5184.2.c.h.5183.2

$q$-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)
 
\(f(q)\) \(=\) \(q+3.34607i q^{5} -4.73205i q^{7} -4.24264 q^{11} -1.00000 q^{13} +3.34607i q^{17} +1.26795i q^{19} +7.34847 q^{23} -6.19615 q^{25} +4.00240i q^{29} -6.00000i q^{31} +15.8338 q^{35} +9.19615 q^{37} +7.34847i q^{41} -2.19615i q^{43} -3.10583 q^{47} -15.3923 q^{49} -7.34847i q^{53} -14.1962i q^{55} -3.10583 q^{59} -7.19615 q^{61} -3.34607i q^{65} +7.26795i q^{67} -15.8338 q^{71} -1.19615 q^{73} +20.0764i q^{77} -1.26795i q^{79} -8.48528 q^{83} -11.1962 q^{85} +9.38186i q^{89} +4.73205i q^{91} -4.24264 q^{95} -6.39230 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{13} - 8 q^{25} + 32 q^{37} - 40 q^{49} - 16 q^{61} + 32 q^{73} - 48 q^{85} + 32 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5184\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(1217\) \(2431\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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
\(5\) 3.34607i 1.49641i 0.663470 + 0.748203i \(0.269083\pi\)
−0.663470 + 0.748203i \(0.730917\pi\)
\(6\) 0 0
\(7\) − 4.73205i − 1.78855i −0.447521 − 0.894274i \(-0.647693\pi\)
0.447521 − 0.894274i \(-0.352307\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −4.24264 −1.27920 −0.639602 − 0.768706i \(-0.720901\pi\)
−0.639602 + 0.768706i \(0.720901\pi\)
\(12\) 0 0
\(13\) −1.00000 −0.277350 −0.138675 − 0.990338i \(-0.544284\pi\)
−0.138675 + 0.990338i \(0.544284\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 3.34607i 0.811540i 0.913975 + 0.405770i \(0.132997\pi\)
−0.913975 + 0.405770i \(0.867003\pi\)
\(18\) 0 0
\(19\) 1.26795i 0.290887i 0.989367 + 0.145444i \(0.0464610\pi\)
−0.989367 + 0.145444i \(0.953539\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 7.34847 1.53226 0.766131 − 0.642685i \(-0.222179\pi\)
0.766131 + 0.642685i \(0.222179\pi\)
\(24\) 0 0
\(25\) −6.19615 −1.23923
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 4.00240i 0.743228i 0.928387 + 0.371614i \(0.121195\pi\)
−0.928387 + 0.371614i \(0.878805\pi\)
\(30\) 0 0
\(31\) − 6.00000i − 1.07763i −0.842424 − 0.538816i \(-0.818872\pi\)
0.842424 − 0.538816i \(-0.181128\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 15.8338 2.67639
\(36\) 0 0
\(37\) 9.19615 1.51184 0.755919 − 0.654665i \(-0.227190\pi\)
0.755919 + 0.654665i \(0.227190\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 7.34847i 1.14764i 0.818982 + 0.573819i \(0.194539\pi\)
−0.818982 + 0.573819i \(0.805461\pi\)
\(42\) 0 0
\(43\) − 2.19615i − 0.334910i −0.985880 − 0.167455i \(-0.946445\pi\)
0.985880 − 0.167455i \(-0.0535549\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −3.10583 −0.453032 −0.226516 − 0.974007i \(-0.572734\pi\)
−0.226516 + 0.974007i \(0.572734\pi\)
\(48\) 0 0
\(49\) −15.3923 −2.19890
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) − 7.34847i − 1.00939i −0.863298 − 0.504695i \(-0.831605\pi\)
0.863298 − 0.504695i \(-0.168395\pi\)
\(54\) 0 0
\(55\) − 14.1962i − 1.91421i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −3.10583 −0.404344 −0.202172 − 0.979350i \(-0.564800\pi\)
−0.202172 + 0.979350i \(0.564800\pi\)
\(60\) 0 0
\(61\) −7.19615 −0.921373 −0.460686 − 0.887563i \(-0.652397\pi\)
−0.460686 + 0.887563i \(0.652397\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) − 3.34607i − 0.415028i
\(66\) 0 0
\(67\) 7.26795i 0.887921i 0.896046 + 0.443961i \(0.146427\pi\)
−0.896046 + 0.443961i \(0.853573\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −15.8338 −1.87912 −0.939560 − 0.342384i \(-0.888766\pi\)
−0.939560 + 0.342384i \(0.888766\pi\)
\(72\) 0 0
\(73\) −1.19615 −0.139999 −0.0699995 − 0.997547i \(-0.522300\pi\)
−0.0699995 + 0.997547i \(0.522300\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 20.0764i 2.28792i
\(78\) 0 0
\(79\) − 1.26795i − 0.142655i −0.997453 − 0.0713277i \(-0.977276\pi\)
0.997453 − 0.0713277i \(-0.0227236\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −8.48528 −0.931381 −0.465690 − 0.884948i \(-0.654194\pi\)
−0.465690 + 0.884948i \(0.654194\pi\)
\(84\) 0 0
\(85\) −11.1962 −1.21439
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 9.38186i 0.994475i 0.867615 + 0.497237i \(0.165652\pi\)
−0.867615 + 0.497237i \(0.834348\pi\)
\(90\) 0 0
\(91\) 4.73205i 0.496054i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −4.24264 −0.435286
\(96\) 0 0
\(97\) −6.39230 −0.649040 −0.324520 − 0.945879i \(-0.605203\pi\)
−0.324520 + 0.945879i \(0.605203\pi\)
\(98\) 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 5184.2.c.h.5183.7 8
3.2 odd 2 inner 5184.2.c.h.5183.1 8
4.3 odd 2 inner 5184.2.c.h.5183.8 8
8.3 odd 2 1296.2.c.g.1295.2 yes 8
8.5 even 2 1296.2.c.g.1295.1 ✓ 8
12.11 even 2 inner 5184.2.c.h.5183.2 8
24.5 odd 2 1296.2.c.g.1295.7 yes 8
24.11 even 2 1296.2.c.g.1295.8 yes 8
72.5 odd 6 1296.2.s.l.431.4 8
72.11 even 6 1296.2.s.l.863.1 8
72.13 even 6 1296.2.s.l.431.1 8
72.29 odd 6 1296.2.s.j.863.1 8
72.43 odd 6 1296.2.s.l.863.4 8
72.59 even 6 1296.2.s.j.431.4 8
72.61 even 6 1296.2.s.j.863.4 8
72.67 odd 6 1296.2.s.j.431.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1296.2.c.g.1295.1 ✓ 8 8.5 even 2
1296.2.c.g.1295.2 yes 8 8.3 odd 2
1296.2.c.g.1295.7 yes 8 24.5 odd 2
1296.2.c.g.1295.8 yes 8 24.11 even 2
1296.2.s.j.431.1 8 72.67 odd 6
1296.2.s.j.431.4 8 72.59 even 6
1296.2.s.j.863.1 8 72.29 odd 6
1296.2.s.j.863.4 8 72.61 even 6
1296.2.s.l.431.1 8 72.13 even 6
1296.2.s.l.431.4 8 72.5 odd 6
1296.2.s.l.863.1 8 72.11 even 6
1296.2.s.l.863.4 8 72.43 odd 6
5184.2.c.h.5183.1 8 3.2 odd 2 inner
5184.2.c.h.5183.2 8 12.11 even 2 inner
5184.2.c.h.5183.7 8 1.1 even 1 trivial
5184.2.c.h.5183.8 8 4.3 odd 2 inner