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Properties

Label 5184.2.d.p.2593.5
Level $5184$
Weight $2$
Character 5184.2593
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(2593,5184)] 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("5184.2593"); 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([0, 1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 5184 = 2^{6} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5184.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,24,0,0,0,0,0,0,0,-32,0,0,0,0, 0,0,0,0,0,0,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(41)] == 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: \( 2^{10}\cdot 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2593.5
Root \(-0.965926 + 0.258819i\) of defining polynomial
Character \(\chi\) \(=\) 5184.2593
Dual form 5184.2.d.p.2593.3

$q$-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)
 
\(f(q)\) \(=\) \(q+0.717439i q^{5} -2.44949 q^{7} +4.24264i q^{11} -4.18154i q^{13} +3.00000 q^{17} +2.24264i q^{19} -5.91359 q^{23} +4.48528 q^{25} +0.717439i q^{29} +1.43488 q^{31} -1.75736i q^{35} -2.74666i q^{37} -8.48528 q^{41} +3.75736i q^{43} +1.43488 q^{47} -1.00000 q^{49} +11.8272i q^{53} -3.04384 q^{55} -9.08052i q^{61} +3.00000 q^{65} +6.24264i q^{67} +16.3059 q^{71} -9.48528 q^{73} -10.3923i q^{77} -1.01461 q^{79} -6.00000i q^{83} +2.15232i q^{85} +0.514719 q^{89} +10.2426i q^{91} -1.60896 q^{95} -16.4853 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 24 q^{17} - 32 q^{25} - 8 q^{49} + 24 q^{65} - 8 q^{73} + 72 q^{89} - 64 q^{97}+O(q^{100}) \) 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
\(5\) 0.717439i 0.320848i 0.987048 + 0.160424i \(0.0512862\pi\)
−0.987048 + 0.160424i \(0.948714\pi\)
\(6\) 0 0
\(7\) −2.44949 −0.925820 −0.462910 − 0.886405i \(-0.653195\pi\)
−0.462910 + 0.886405i \(0.653195\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 4.24264i 1.27920i 0.768706 + 0.639602i \(0.220901\pi\)
−0.768706 + 0.639602i \(0.779099\pi\)
\(12\) 0 0
\(13\) − 4.18154i − 1.15975i −0.814705 − 0.579875i \(-0.803101\pi\)
0.814705 − 0.579875i \(-0.196899\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 3.00000 0.727607 0.363803 − 0.931476i \(-0.381478\pi\)
0.363803 + 0.931476i \(0.381478\pi\)
\(18\) 0 0
\(19\) 2.24264i 0.514497i 0.966345 + 0.257249i \(0.0828159\pi\)
−0.966345 + 0.257249i \(0.917184\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −5.91359 −1.23307 −0.616535 − 0.787328i \(-0.711464\pi\)
−0.616535 + 0.787328i \(0.711464\pi\)
\(24\) 0 0
\(25\) 4.48528 0.897056
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 0.717439i 0.133225i 0.997779 + 0.0666125i \(0.0212191\pi\)
−0.997779 + 0.0666125i \(0.978781\pi\)
\(30\) 0 0
\(31\) 1.43488 0.257712 0.128856 − 0.991663i \(-0.458870\pi\)
0.128856 + 0.991663i \(0.458870\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) − 1.75736i − 0.297048i
\(36\) 0 0
\(37\) − 2.74666i − 0.451549i −0.974180 − 0.225774i \(-0.927509\pi\)
0.974180 − 0.225774i \(-0.0724912\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −8.48528 −1.32518 −0.662589 − 0.748983i \(-0.730542\pi\)
−0.662589 + 0.748983i \(0.730542\pi\)
\(42\) 0 0
\(43\) 3.75736i 0.572992i 0.958081 + 0.286496i \(0.0924905\pi\)
−0.958081 + 0.286496i \(0.907509\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 1.43488 0.209298 0.104649 − 0.994509i \(-0.466628\pi\)
0.104649 + 0.994509i \(0.466628\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 11.8272i 1.62459i 0.583248 + 0.812294i \(0.301782\pi\)
−0.583248 + 0.812294i \(0.698218\pi\)
\(54\) 0 0
\(55\) −3.04384 −0.410431
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 0 0
\(61\) − 9.08052i − 1.16264i −0.813675 − 0.581321i \(-0.802536\pi\)
0.813675 − 0.581321i \(-0.197464\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 3.00000 0.372104
\(66\) 0 0
\(67\) 6.24264i 0.762660i 0.924439 + 0.381330i \(0.124534\pi\)
−0.924439 + 0.381330i \(0.875466\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 16.3059 1.93515 0.967577 − 0.252577i \(-0.0812779\pi\)
0.967577 + 0.252577i \(0.0812779\pi\)
\(72\) 0 0
\(73\) −9.48528 −1.11017 −0.555084 − 0.831794i \(-0.687314\pi\)
−0.555084 + 0.831794i \(0.687314\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 10.3923i − 1.18431i
\(78\) 0 0
\(79\) −1.01461 −0.114153 −0.0570764 − 0.998370i \(-0.518178\pi\)
−0.0570764 + 0.998370i \(0.518178\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) − 6.00000i − 0.658586i −0.944228 − 0.329293i \(-0.893190\pi\)
0.944228 − 0.329293i \(-0.106810\pi\)
\(84\) 0 0
\(85\) 2.15232i 0.233452i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0.514719 0.0545601 0.0272800 − 0.999628i \(-0.491315\pi\)
0.0272800 + 0.999628i \(0.491315\pi\)
\(90\) 0 0
\(91\) 10.2426i 1.07372i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −1.60896 −0.165076
\(96\) 0 0
\(97\) −16.4853 −1.67383 −0.836913 − 0.547335i \(-0.815642\pi\)
−0.836913 + 0.547335i \(0.815642\pi\)
\(98\) 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.d.p.2593.5 yes 8
3.2 odd 2 5184.2.d.o.2593.3 ✓ 8
4.3 odd 2 inner 5184.2.d.p.2593.6 yes 8
8.3 odd 2 inner 5184.2.d.p.2593.4 yes 8
8.5 even 2 inner 5184.2.d.p.2593.3 yes 8
12.11 even 2 5184.2.d.o.2593.4 yes 8
24.5 odd 2 5184.2.d.o.2593.5 yes 8
24.11 even 2 5184.2.d.o.2593.6 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5184.2.d.o.2593.3 ✓ 8 3.2 odd 2
5184.2.d.o.2593.4 yes 8 12.11 even 2
5184.2.d.o.2593.5 yes 8 24.5 odd 2
5184.2.d.o.2593.6 yes 8 24.11 even 2
5184.2.d.p.2593.3 yes 8 8.5 even 2 inner
5184.2.d.p.2593.4 yes 8 8.3 odd 2 inner
5184.2.d.p.2593.5 yes 8 1.1 even 1 trivial
5184.2.d.p.2593.6 yes 8 4.3 odd 2 inner