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Properties

Label 5184.2.d.m.2593.1
Level $5184$
Weight $2$
Character 5184.2593
Analytic conductor $41.394$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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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 = [4,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,-24,0,0,0,0,0,12,0,8,0,0,0, 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 2593.1
Root \(0.866025 + 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 5184.2593
Dual form 5184.2.d.m.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)
 
\(f(q)\) \(=\) \(q-1.73205i q^{5} +1.26795 q^{7} -1.26795i q^{11} +3.00000i q^{13} -4.26795 q^{17} +4.19615i q^{19} +1.26795 q^{23} +2.00000 q^{25} +4.26795i q^{29} -3.46410 q^{31} -2.19615i q^{35} +0.464102i q^{37} -3.46410 q^{41} +6.19615i q^{43} -12.9282 q^{47} -5.39230 q^{49} +0.928203i q^{53} -2.19615 q^{55} -9.46410i q^{59} +6.46410i q^{61} +5.19615 q^{65} -4.19615i q^{67} +4.73205 q^{71} +5.00000 q^{73} -1.60770i q^{77} -14.1962 q^{79} +10.3923i q^{83} +7.39230i q^{85} +0.803848 q^{89} +3.80385i q^{91} +7.26795 q^{95} +4.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 12 q^{7} - 24 q^{17} + 12 q^{23} + 8 q^{25} - 24 q^{47} + 20 q^{49} + 12 q^{55} + 12 q^{71} + 20 q^{73} - 36 q^{79} + 24 q^{89} + 36 q^{95} + 16 q^{97}+O(q^{100}) \) 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
\(5\) − 1.73205i − 0.774597i −0.921954 − 0.387298i \(-0.873408\pi\)
0.921954 − 0.387298i \(-0.126592\pi\)
\(6\) 0 0
\(7\) 1.26795 0.479240 0.239620 − 0.970867i \(-0.422977\pi\)
0.239620 + 0.970867i \(0.422977\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) − 1.26795i − 0.382301i −0.981561 − 0.191151i \(-0.938778\pi\)
0.981561 − 0.191151i \(-0.0612219\pi\)
\(12\) 0 0
\(13\) 3.00000i 0.832050i 0.909353 + 0.416025i \(0.136577\pi\)
−0.909353 + 0.416025i \(0.863423\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −4.26795 −1.03513 −0.517565 − 0.855644i \(-0.673161\pi\)
−0.517565 + 0.855644i \(0.673161\pi\)
\(18\) 0 0
\(19\) 4.19615i 0.962663i 0.876539 + 0.481332i \(0.159847\pi\)
−0.876539 + 0.481332i \(0.840153\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 1.26795 0.264386 0.132193 − 0.991224i \(-0.457798\pi\)
0.132193 + 0.991224i \(0.457798\pi\)
\(24\) 0 0
\(25\) 2.00000 0.400000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 4.26795i 0.792538i 0.918134 + 0.396269i \(0.129695\pi\)
−0.918134 + 0.396269i \(0.870305\pi\)
\(30\) 0 0
\(31\) −3.46410 −0.622171 −0.311086 − 0.950382i \(-0.600693\pi\)
−0.311086 + 0.950382i \(0.600693\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) − 2.19615i − 0.371218i
\(36\) 0 0
\(37\) 0.464102i 0.0762978i 0.999272 + 0.0381489i \(0.0121461\pi\)
−0.999272 + 0.0381489i \(0.987854\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −3.46410 −0.541002 −0.270501 − 0.962720i \(-0.587189\pi\)
−0.270501 + 0.962720i \(0.587189\pi\)
\(42\) 0 0
\(43\) 6.19615i 0.944904i 0.881356 + 0.472452i \(0.156631\pi\)
−0.881356 + 0.472452i \(0.843369\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −12.9282 −1.88577 −0.942886 − 0.333115i \(-0.891900\pi\)
−0.942886 + 0.333115i \(0.891900\pi\)
\(48\) 0 0
\(49\) −5.39230 −0.770329
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0.928203i 0.127499i 0.997966 + 0.0637493i \(0.0203058\pi\)
−0.997966 + 0.0637493i \(0.979694\pi\)
\(54\) 0 0
\(55\) −2.19615 −0.296129
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) − 9.46410i − 1.23212i −0.787699 − 0.616061i \(-0.788728\pi\)
0.787699 − 0.616061i \(-0.211272\pi\)
\(60\) 0 0
\(61\) 6.46410i 0.827643i 0.910358 + 0.413822i \(0.135806\pi\)
−0.910358 + 0.413822i \(0.864194\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 5.19615 0.644503
\(66\) 0 0
\(67\) − 4.19615i − 0.512642i −0.966592 − 0.256321i \(-0.917490\pi\)
0.966592 − 0.256321i \(-0.0825104\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 4.73205 0.561591 0.280796 − 0.959768i \(-0.409402\pi\)
0.280796 + 0.959768i \(0.409402\pi\)
\(72\) 0 0
\(73\) 5.00000 0.585206 0.292603 − 0.956234i \(-0.405479\pi\)
0.292603 + 0.956234i \(0.405479\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 1.60770i − 0.183214i
\(78\) 0 0
\(79\) −14.1962 −1.59719 −0.798596 − 0.601867i \(-0.794424\pi\)
−0.798596 + 0.601867i \(0.794424\pi\)
\(80\) 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
\(85\) 7.39230i 0.801808i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0.803848 0.0852077 0.0426038 − 0.999092i \(-0.486435\pi\)
0.0426038 + 0.999092i \(0.486435\pi\)
\(90\) 0 0
\(91\) 3.80385i 0.398752i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 7.26795 0.745676
\(96\) 0 0
\(97\) 4.00000 0.406138 0.203069 − 0.979164i \(-0.434908\pi\)
0.203069 + 0.979164i \(0.434908\pi\)
\(98\) 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.m.2593.1 yes 4
3.2 odd 2 5184.2.d.n.2593.3 yes 4
4.3 odd 2 5184.2.d.a.2593.2 ✓ 4
8.3 odd 2 5184.2.d.a.2593.4 yes 4
8.5 even 2 inner 5184.2.d.m.2593.3 yes 4
12.11 even 2 5184.2.d.b.2593.4 yes 4
24.5 odd 2 5184.2.d.n.2593.1 yes 4
24.11 even 2 5184.2.d.b.2593.2 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5184.2.d.a.2593.2 ✓ 4 4.3 odd 2
5184.2.d.a.2593.4 yes 4 8.3 odd 2
5184.2.d.b.2593.2 yes 4 24.11 even 2
5184.2.d.b.2593.4 yes 4 12.11 even 2
5184.2.d.m.2593.1 yes 4 1.1 even 1 trivial
5184.2.d.m.2593.3 yes 4 8.5 even 2 inner
5184.2.d.n.2593.1 yes 4 24.5 odd 2
5184.2.d.n.2593.3 yes 4 3.2 odd 2