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Properties

Label 5184.2.c.h.5183.5
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 5183.5
Root \(-0.965926 + 0.258819i\) of defining polynomial
Character \(\chi\) \(=\) 5184.5183
Dual form 5184.2.c.h.5183.4

$q$-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)
 
\(f(q)\) \(=\) \(q+0.896575i q^{5} -1.26795i q^{7} -4.24264 q^{11} -1.00000 q^{13} +0.896575i q^{17} +4.73205i q^{19} -7.34847 q^{23} +4.19615 q^{25} -8.24504i q^{29} -6.00000i q^{31} +1.13681 q^{35} -1.19615 q^{37} -7.34847i q^{41} +8.19615i q^{43} +11.5911 q^{47} +5.39230 q^{49} +7.34847i q^{53} -3.80385i q^{55} +11.5911 q^{59} +3.19615 q^{61} -0.896575i q^{65} +10.7321i q^{67} -1.13681 q^{71} +9.19615 q^{73} +5.37945i q^{77} -4.73205i q^{79} -8.48528 q^{83} -0.803848 q^{85} +11.8313i q^{89} +1.26795i q^{91} -4.24264 q^{95} +14.3923 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{13} - 8 q^{25} + 32 q^{37} - 40 q^{49} - 16 q^{61} + 32 q^{73} - 48 q^{85} + 32 q^{97}+O(q^{100}) \) 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
\(5\) 0.896575i 0.400961i 0.979698 + 0.200480i \(0.0642503\pi\)
−0.979698 + 0.200480i \(0.935750\pi\)
\(6\) 0 0
\(7\) − 1.26795i − 0.479240i −0.970867 − 0.239620i \(-0.922977\pi\)
0.970867 − 0.239620i \(-0.0770228\pi\)
\(8\) 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
\(17\) 0.896575i 0.217451i 0.994072 + 0.108726i \(0.0346770\pi\)
−0.994072 + 0.108726i \(0.965323\pi\)
\(18\) 0 0
\(19\) 4.73205i 1.08561i 0.839860 + 0.542803i \(0.182637\pi\)
−0.839860 + 0.542803i \(0.817363\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −7.34847 −1.53226 −0.766131 − 0.642685i \(-0.777821\pi\)
−0.766131 + 0.642685i \(0.777821\pi\)
\(24\) 0 0
\(25\) 4.19615 0.839230
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) − 8.24504i − 1.53107i −0.643396 − 0.765533i \(-0.722475\pi\)
0.643396 − 0.765533i \(-0.277525\pi\)
\(30\) 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
\(35\) 1.13681 0.192156
\(36\) 0 0
\(37\) −1.19615 −0.196646 −0.0983231 − 0.995155i \(-0.531348\pi\)
−0.0983231 + 0.995155i \(0.531348\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) − 7.34847i − 1.14764i −0.818982 − 0.573819i \(-0.805461\pi\)
0.818982 − 0.573819i \(-0.194539\pi\)
\(42\) 0 0
\(43\) 8.19615i 1.24990i 0.780664 + 0.624951i \(0.214881\pi\)
−0.780664 + 0.624951i \(0.785119\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 11.5911 1.69074 0.845369 − 0.534183i \(-0.179381\pi\)
0.845369 + 0.534183i \(0.179381\pi\)
\(48\) 0 0
\(49\) 5.39230 0.770329
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 7.34847i 1.00939i 0.863298 + 0.504695i \(0.168395\pi\)
−0.863298 + 0.504695i \(0.831605\pi\)
\(54\) 0 0
\(55\) − 3.80385i − 0.512911i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 11.5911 1.50903 0.754517 − 0.656281i \(-0.227871\pi\)
0.754517 + 0.656281i \(0.227871\pi\)
\(60\) 0 0
\(61\) 3.19615 0.409225 0.204613 − 0.978843i \(-0.434407\pi\)
0.204613 + 0.978843i \(0.434407\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) − 0.896575i − 0.111207i
\(66\) 0 0
\(67\) 10.7321i 1.31113i 0.755139 + 0.655564i \(0.227569\pi\)
−0.755139 + 0.655564i \(0.772431\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −1.13681 −0.134915 −0.0674574 − 0.997722i \(-0.521489\pi\)
−0.0674574 + 0.997722i \(0.521489\pi\)
\(72\) 0 0
\(73\) 9.19615 1.07633 0.538164 − 0.842840i \(-0.319118\pi\)
0.538164 + 0.842840i \(0.319118\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 5.37945i 0.613046i
\(78\) 0 0
\(79\) − 4.73205i − 0.532397i −0.963918 − 0.266199i \(-0.914232\pi\)
0.963918 − 0.266199i \(-0.0857677\pi\)
\(80\) 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
\(85\) −0.803848 −0.0871895
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 11.8313i 1.25412i 0.778971 + 0.627060i \(0.215742\pi\)
−0.778971 + 0.627060i \(0.784258\pi\)
\(90\) 0 0
\(91\) 1.26795i 0.132917i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −4.24264 −0.435286
\(96\) 0 0
\(97\) 14.3923 1.46132 0.730659 − 0.682743i \(-0.239213\pi\)
0.730659 + 0.682743i \(0.239213\pi\)
\(98\) 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.5183.5 8
3.2 odd 2 inner 5184.2.c.h.5183.3 8
4.3 odd 2 inner 5184.2.c.h.5183.6 8
8.3 odd 2 1296.2.c.g.1295.4 yes 8
8.5 even 2 1296.2.c.g.1295.3 ✓ 8
12.11 even 2 inner 5184.2.c.h.5183.4 8
24.5 odd 2 1296.2.c.g.1295.5 yes 8
24.11 even 2 1296.2.c.g.1295.6 yes 8
72.5 odd 6 1296.2.s.j.431.3 8
72.11 even 6 1296.2.s.j.863.2 8
72.13 even 6 1296.2.s.j.431.2 8
72.29 odd 6 1296.2.s.l.863.2 8
72.43 odd 6 1296.2.s.j.863.3 8
72.59 even 6 1296.2.s.l.431.3 8
72.61 even 6 1296.2.s.l.863.3 8
72.67 odd 6 1296.2.s.l.431.2 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1296.2.c.g.1295.3 ✓ 8 8.5 even 2
1296.2.c.g.1295.4 yes 8 8.3 odd 2
1296.2.c.g.1295.5 yes 8 24.5 odd 2
1296.2.c.g.1295.6 yes 8 24.11 even 2
1296.2.s.j.431.2 8 72.13 even 6
1296.2.s.j.431.3 8 72.5 odd 6
1296.2.s.j.863.2 8 72.11 even 6
1296.2.s.j.863.3 8 72.43 odd 6
1296.2.s.l.431.2 8 72.67 odd 6
1296.2.s.l.431.3 8 72.59 even 6
1296.2.s.l.863.2 8 72.29 odd 6
1296.2.s.l.863.3 8 72.61 even 6
5184.2.c.h.5183.3 8 3.2 odd 2 inner
5184.2.c.h.5183.4 8 12.11 even 2 inner
5184.2.c.h.5183.5 8 1.1 even 1 trivial
5184.2.c.h.5183.6 8 4.3 odd 2 inner