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Properties

Label 5184.2.c.k.5183.2
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 = [8,0,0,0,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,0,0,24] 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: 8.0.5780865024.3
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{6} + 9x^{4} - 16x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 324)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 5183.2
Root \(1.39033 - 0.258819i\) of defining polynomial
Character \(\chi\) \(=\) 5184.5183
Dual form 5184.2.c.k.5183.7

$q$-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)
 
\(f(q)\) \(=\) \(q-1.93185i q^{5} +3.93244i q^{7} +2.03558 q^{11} -2.46410 q^{13} -4.76028i q^{17} +6.81119i q^{19} -7.59689 q^{23} +1.26795 q^{25} +2.31079i q^{29} +2.87875i q^{31} +7.59689 q^{35} -3.73205 q^{37} -3.20736i q^{41} -3.93244i q^{43} -5.56131 q^{47} -8.46410 q^{49} -10.1769i q^{53} -3.93244i q^{55} -5.56131 q^{59} +5.73205 q^{61} +4.76028i q^{65} +3.93244i q^{67} +3.52573 q^{71} -12.6603 q^{73} +8.00481i q^{77} -1.05369i q^{79} -4.07116 q^{83} -9.19615 q^{85} -3.62347i q^{89} -9.68994i q^{91} +13.1582 q^{95} -7.46410 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{13} + 24 q^{25} - 16 q^{37} - 40 q^{49} + 32 q^{61} - 32 q^{73} - 32 q^{85} - 32 q^{97}+O(q^{100}) \) 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
\(5\) − 1.93185i − 0.863950i −0.901886 − 0.431975i \(-0.857817\pi\)
0.901886 − 0.431975i \(-0.142183\pi\)
\(6\) 0 0
\(7\) 3.93244i 1.48632i 0.669112 + 0.743162i \(0.266675\pi\)
−0.669112 + 0.743162i \(0.733325\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 2.03558 0.613751 0.306876 − 0.951750i \(-0.400716\pi\)
0.306876 + 0.951750i \(0.400716\pi\)
\(12\) 0 0
\(13\) −2.46410 −0.683419 −0.341709 − 0.939806i \(-0.611006\pi\)
−0.341709 + 0.939806i \(0.611006\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 4.76028i − 1.15454i −0.816554 − 0.577269i \(-0.804119\pi\)
0.816554 − 0.577269i \(-0.195881\pi\)
\(18\) 0 0
\(19\) 6.81119i 1.56259i 0.624159 + 0.781297i \(0.285442\pi\)
−0.624159 + 0.781297i \(0.714558\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −7.59689 −1.58406 −0.792031 − 0.610481i \(-0.790976\pi\)
−0.792031 + 0.610481i \(0.790976\pi\)
\(24\) 0 0
\(25\) 1.26795 0.253590
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 2.31079i 0.429103i 0.976713 + 0.214551i \(0.0688289\pi\)
−0.976713 + 0.214551i \(0.931171\pi\)
\(30\) 0 0
\(31\) 2.87875i 0.517038i 0.966006 + 0.258519i \(0.0832345\pi\)
−0.966006 + 0.258519i \(0.916765\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 7.59689 1.28411
\(36\) 0 0
\(37\) −3.73205 −0.613545 −0.306773 − 0.951783i \(-0.599249\pi\)
−0.306773 + 0.951783i \(0.599249\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) − 3.20736i − 0.500906i −0.968129 − 0.250453i \(-0.919420\pi\)
0.968129 − 0.250453i \(-0.0805796\pi\)
\(42\) 0 0
\(43\) − 3.93244i − 0.599692i −0.953988 − 0.299846i \(-0.903065\pi\)
0.953988 − 0.299846i \(-0.0969353\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −5.56131 −0.811201 −0.405600 − 0.914050i \(-0.632938\pi\)
−0.405600 + 0.914050i \(0.632938\pi\)
\(48\) 0 0
\(49\) −8.46410 −1.20916
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) − 10.1769i − 1.39790i −0.715168 − 0.698952i \(-0.753650\pi\)
0.715168 − 0.698952i \(-0.246350\pi\)
\(54\) 0 0
\(55\) − 3.93244i − 0.530250i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −5.56131 −0.724021 −0.362011 − 0.932174i \(-0.617910\pi\)
−0.362011 + 0.932174i \(0.617910\pi\)
\(60\) 0 0
\(61\) 5.73205 0.733914 0.366957 − 0.930238i \(-0.380400\pi\)
0.366957 + 0.930238i \(0.380400\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 4.76028i 0.590440i
\(66\) 0 0
\(67\) 3.93244i 0.480424i 0.970720 + 0.240212i \(0.0772170\pi\)
−0.970720 + 0.240212i \(0.922783\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 3.52573 0.418427 0.209214 − 0.977870i \(-0.432910\pi\)
0.209214 + 0.977870i \(0.432910\pi\)
\(72\) 0 0
\(73\) −12.6603 −1.48177 −0.740885 − 0.671632i \(-0.765594\pi\)
−0.740885 + 0.671632i \(0.765594\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 8.00481i 0.912233i
\(78\) 0 0
\(79\) − 1.05369i − 0.118550i −0.998242 − 0.0592750i \(-0.981121\pi\)
0.998242 − 0.0592750i \(-0.0188789\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −4.07116 −0.446868 −0.223434 − 0.974719i \(-0.571727\pi\)
−0.223434 + 0.974719i \(0.571727\pi\)
\(84\) 0 0
\(85\) −9.19615 −0.997463
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) − 3.62347i − 0.384087i −0.981386 − 0.192043i \(-0.938489\pi\)
0.981386 − 0.192043i \(-0.0615114\pi\)
\(90\) 0 0
\(91\) − 9.68994i − 1.01578i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 13.1582 1.35000
\(96\) 0 0
\(97\) −7.46410 −0.757865 −0.378932 − 0.925424i \(-0.623709\pi\)
−0.378932 + 0.925424i \(0.623709\pi\)
\(98\) 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.k.5183.2 8
3.2 odd 2 inner 5184.2.c.k.5183.8 8
4.3 odd 2 inner 5184.2.c.k.5183.1 8
8.3 odd 2 324.2.b.c.323.1 ✓ 8
8.5 even 2 324.2.b.c.323.7 yes 8
12.11 even 2 inner 5184.2.c.k.5183.7 8
24.5 odd 2 324.2.b.c.323.2 yes 8
24.11 even 2 324.2.b.c.323.8 yes 8
72.5 odd 6 324.2.h.f.107.6 16
72.11 even 6 324.2.h.f.215.3 16
72.13 even 6 324.2.h.f.107.3 16
72.29 odd 6 324.2.h.f.215.7 16
72.43 odd 6 324.2.h.f.215.6 16
72.59 even 6 324.2.h.f.107.2 16
72.61 even 6 324.2.h.f.215.2 16
72.67 odd 6 324.2.h.f.107.7 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
324.2.b.c.323.1 ✓ 8 8.3 odd 2
324.2.b.c.323.2 yes 8 24.5 odd 2
324.2.b.c.323.7 yes 8 8.5 even 2
324.2.b.c.323.8 yes 8 24.11 even 2
324.2.h.f.107.2 16 72.59 even 6
324.2.h.f.107.3 16 72.13 even 6
324.2.h.f.107.6 16 72.5 odd 6
324.2.h.f.107.7 16 72.67 odd 6
324.2.h.f.215.2 16 72.61 even 6
324.2.h.f.215.3 16 72.11 even 6
324.2.h.f.215.6 16 72.43 odd 6
324.2.h.f.215.7 16 72.29 odd 6
5184.2.c.k.5183.1 8 4.3 odd 2 inner
5184.2.c.k.5183.2 8 1.1 even 1 trivial
5184.2.c.k.5183.7 8 12.11 even 2 inner
5184.2.c.k.5183.8 8 3.2 odd 2 inner