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Properties

Label 5184.2.d.i.2593.2
Level $5184$
Weight $2$
Character 5184.2593
Analytic conductor $41.394$
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] = [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 = [4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,24,0,-16,0,0,0,0, 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: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2593.2
Root \(0.866025 - 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 5184.2593
Dual form 5184.2.d.i.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)
 
\(f(q)\) \(=\) \(q-3.00000i q^{5} +3.46410i q^{11} +5.19615i q^{13} -1.73205 q^{17} -4.00000i q^{19} +6.00000 q^{23} -4.00000 q^{25} -3.00000i q^{29} -10.3923 q^{31} -5.19615i q^{37} -6.92820 q^{41} -2.00000i q^{43} +6.00000 q^{47} -7.00000 q^{49} -6.00000i q^{53} +10.3923 q^{55} -6.92820i q^{59} -5.19615i q^{61} +15.5885 q^{65} -8.00000i q^{67} +12.0000 q^{71} -13.0000 q^{73} +10.3923 q^{79} +10.3923i q^{83} +5.19615i q^{85} -5.19615 q^{89} -12.0000 q^{95} -2.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 24 q^{23} - 16 q^{25} + 24 q^{47} - 28 q^{49} + 48 q^{71} - 52 q^{73} - 48 q^{95} - 8 q^{97}+O(q^{100}) \) 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
\(5\) − 3.00000i − 1.34164i −0.741620 − 0.670820i \(-0.765942\pi\)
0.741620 − 0.670820i \(-0.234058\pi\)
\(6\) 0 0
\(7\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 3.46410i 1.04447i 0.852803 + 0.522233i \(0.174901\pi\)
−0.852803 + 0.522233i \(0.825099\pi\)
\(12\) 0 0
\(13\) 5.19615i 1.44115i 0.693375 + 0.720577i \(0.256123\pi\)
−0.693375 + 0.720577i \(0.743877\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −1.73205 −0.420084 −0.210042 − 0.977692i \(-0.567360\pi\)
−0.210042 + 0.977692i \(0.567360\pi\)
\(18\) 0 0
\(19\) − 4.00000i − 0.917663i −0.888523 − 0.458831i \(-0.848268\pi\)
0.888523 − 0.458831i \(-0.151732\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 6.00000 1.25109 0.625543 − 0.780189i \(-0.284877\pi\)
0.625543 + 0.780189i \(0.284877\pi\)
\(24\) 0 0
\(25\) −4.00000 −0.800000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) − 3.00000i − 0.557086i −0.960424 − 0.278543i \(-0.910149\pi\)
0.960424 − 0.278543i \(-0.0898515\pi\)
\(30\) 0 0
\(31\) −10.3923 −1.86651 −0.933257 − 0.359211i \(-0.883046\pi\)
−0.933257 + 0.359211i \(0.883046\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) − 5.19615i − 0.854242i −0.904194 − 0.427121i \(-0.859528\pi\)
0.904194 − 0.427121i \(-0.140472\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −6.92820 −1.08200 −0.541002 − 0.841021i \(-0.681955\pi\)
−0.541002 + 0.841021i \(0.681955\pi\)
\(42\) 0 0
\(43\) − 2.00000i − 0.304997i −0.988304 − 0.152499i \(-0.951268\pi\)
0.988304 − 0.152499i \(-0.0487319\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 6.00000 0.875190 0.437595 − 0.899172i \(-0.355830\pi\)
0.437595 + 0.899172i \(0.355830\pi\)
\(48\) 0 0
\(49\) −7.00000 −1.00000
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) − 6.00000i − 0.824163i −0.911147 − 0.412082i \(-0.864802\pi\)
0.911147 − 0.412082i \(-0.135198\pi\)
\(54\) 0 0
\(55\) 10.3923 1.40130
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) − 6.92820i − 0.901975i −0.892530 − 0.450988i \(-0.851072\pi\)
0.892530 − 0.450988i \(-0.148928\pi\)
\(60\) 0 0
\(61\) − 5.19615i − 0.665299i −0.943051 − 0.332650i \(-0.892057\pi\)
0.943051 − 0.332650i \(-0.107943\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 15.5885 1.93351
\(66\) 0 0
\(67\) − 8.00000i − 0.977356i −0.872464 − 0.488678i \(-0.837479\pi\)
0.872464 − 0.488678i \(-0.162521\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 12.0000 1.42414 0.712069 − 0.702109i \(-0.247758\pi\)
0.712069 + 0.702109i \(0.247758\pi\)
\(72\) 0 0
\(73\) −13.0000 −1.52153 −0.760767 − 0.649025i \(-0.775177\pi\)
−0.760767 + 0.649025i \(0.775177\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 10.3923 1.16923 0.584613 − 0.811312i \(-0.301246\pi\)
0.584613 + 0.811312i \(0.301246\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 10.3923i 1.14070i 0.821401 + 0.570352i \(0.193193\pi\)
−0.821401 + 0.570352i \(0.806807\pi\)
\(84\) 0 0
\(85\) 5.19615i 0.563602i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −5.19615 −0.550791 −0.275396 − 0.961331i \(-0.588809\pi\)
−0.275396 + 0.961331i \(0.588809\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −12.0000 −1.23117
\(96\) 0 0
\(97\) −2.00000 −0.203069 −0.101535 − 0.994832i \(-0.532375\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) 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.i.2593.2 yes 4
3.2 odd 2 5184.2.d.f.2593.3 yes 4
4.3 odd 2 5184.2.d.f.2593.1 ✓ 4
8.3 odd 2 5184.2.d.f.2593.4 yes 4
8.5 even 2 inner 5184.2.d.i.2593.3 yes 4
12.11 even 2 inner 5184.2.d.i.2593.4 yes 4
24.5 odd 2 5184.2.d.f.2593.2 yes 4
24.11 even 2 inner 5184.2.d.i.2593.1 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5184.2.d.f.2593.1 ✓ 4 4.3 odd 2
5184.2.d.f.2593.2 yes 4 24.5 odd 2
5184.2.d.f.2593.3 yes 4 3.2 odd 2
5184.2.d.f.2593.4 yes 4 8.3 odd 2
5184.2.d.i.2593.1 yes 4 24.11 even 2 inner
5184.2.d.i.2593.2 yes 4 1.1 even 1 trivial
5184.2.d.i.2593.3 yes 4 8.5 even 2 inner
5184.2.d.i.2593.4 yes 4 12.11 even 2 inner