↑

Properties

Label 5184.2.d.m.2593.4
Level $5184$
Weight $2$
Character 5184.2593
Analytic conductor $41.394$
Analytic rank $0$
Dimension $4$
Inner twists $2$

Related objects

Downloads

Learn more

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 2593.4
Root \(-0.866025 + 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 5184.2593
Dual form 5184.2.d.m.2593.2

$q$-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)
 
\(f(q)\) \(=\) \(q+1.73205i q^{5} +4.73205 q^{7} -4.73205i q^{11} +3.00000i q^{13} -7.73205 q^{17} -6.19615i q^{19} +4.73205 q^{23} +2.00000 q^{25} +7.73205i q^{29} +3.46410 q^{31} +8.19615i q^{35} -6.46410i q^{37} +3.46410 q^{41} -4.19615i q^{43} +0.928203 q^{47} +15.3923 q^{49} -12.9282i q^{53} +8.19615 q^{55} -2.53590i q^{59} -0.464102i q^{61} -5.19615 q^{65} +6.19615i q^{67} +1.26795 q^{71} +5.00000 q^{73} -22.3923i q^{77} -3.80385 q^{79} -10.3923i q^{83} -13.3923i q^{85} +11.1962 q^{89} +14.1962i q^{91} +10.7321 q^{95} +4.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 12 q^{7} - 24 q^{17} + 12 q^{23} + 8 q^{25} - 24 q^{47} + 20 q^{49} + 12 q^{55} + 12 q^{71} + 20 q^{73} - 36 q^{79} + 24 q^{89} + 36 q^{95} + 16 q^{97}+O(q^{100}) \) 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 \(0.126592\pi\)
−0.921954 + 0.387298i \(0.873408\pi\)
\(6\) 0 0
\(7\) 4.73205 1.78855 0.894274 − 0.447521i \(-0.147693\pi\)
0.894274 + 0.447521i \(0.147693\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) − 4.73205i − 1.42677i −0.700774 − 0.713384i \(-0.747162\pi\)
0.700774 − 0.713384i \(-0.252838\pi\)
\(12\) 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
\(17\) −7.73205 −1.87530 −0.937649 − 0.347584i \(-0.887002\pi\)
−0.937649 + 0.347584i \(0.887002\pi\)
\(18\) 0 0
\(19\) − 6.19615i − 1.42149i −0.703447 − 0.710747i \(-0.748357\pi\)
0.703447 − 0.710747i \(-0.251643\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 4.73205 0.986701 0.493350 − 0.869831i \(-0.335772\pi\)
0.493350 + 0.869831i \(0.335772\pi\)
\(24\) 0 0
\(25\) 2.00000 0.400000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 7.73205i 1.43581i 0.696143 + 0.717903i \(0.254898\pi\)
−0.696143 + 0.717903i \(0.745102\pi\)
\(30\) 0 0
\(31\) 3.46410 0.622171 0.311086 − 0.950382i \(-0.399307\pi\)
0.311086 + 0.950382i \(0.399307\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 8.19615i 1.38540i
\(36\) 0 0
\(37\) − 6.46410i − 1.06269i −0.847155 − 0.531346i \(-0.821686\pi\)
0.847155 − 0.531346i \(-0.178314\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 3.46410 0.541002 0.270501 − 0.962720i \(-0.412811\pi\)
0.270501 + 0.962720i \(0.412811\pi\)
\(42\) 0 0
\(43\) − 4.19615i − 0.639907i −0.947433 − 0.319954i \(-0.896333\pi\)
0.947433 − 0.319954i \(-0.103667\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0.928203 0.135392 0.0676962 − 0.997706i \(-0.478435\pi\)
0.0676962 + 0.997706i \(0.478435\pi\)
\(48\) 0 0
\(49\) 15.3923 2.19890
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) − 12.9282i − 1.77583i −0.460012 − 0.887913i \(-0.652155\pi\)
0.460012 − 0.887913i \(-0.347845\pi\)
\(54\) 0 0
\(55\) 8.19615 1.10517
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) − 2.53590i − 0.330146i −0.986281 − 0.165073i \(-0.947214\pi\)
0.986281 − 0.165073i \(-0.0527859\pi\)
\(60\) 0 0
\(61\) − 0.464102i − 0.0594221i −0.999559 − 0.0297111i \(-0.990541\pi\)
0.999559 − 0.0297111i \(-0.00945872\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −5.19615 −0.644503
\(66\) 0 0
\(67\) 6.19615i 0.756980i 0.925605 + 0.378490i \(0.123557\pi\)
−0.925605 + 0.378490i \(0.876443\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 1.26795 0.150478 0.0752389 − 0.997166i \(-0.476028\pi\)
0.0752389 + 0.997166i \(0.476028\pi\)
\(72\) 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
\(77\) − 22.3923i − 2.55184i
\(78\) 0 0
\(79\) −3.80385 −0.427966 −0.213983 − 0.976837i \(-0.568644\pi\)
−0.213983 + 0.976837i \(0.568644\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) − 10.3923i − 1.14070i −0.821401 − 0.570352i \(-0.806807\pi\)
0.821401 − 0.570352i \(-0.193193\pi\)
\(84\) 0 0
\(85\) − 13.3923i − 1.45260i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 11.1962 1.18679 0.593395 − 0.804911i \(-0.297787\pi\)
0.593395 + 0.804911i \(0.297787\pi\)
\(90\) 0 0
\(91\) 14.1962i 1.48816i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 10.7321 1.10109
\(96\) 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.2593.4 yes 4
3.2 odd 2 5184.2.d.n.2593.2 yes 4
4.3 odd 2 5184.2.d.a.2593.3 yes 4
8.3 odd 2 5184.2.d.a.2593.1 ✓ 4
8.5 even 2 inner 5184.2.d.m.2593.2 yes 4
12.11 even 2 5184.2.d.b.2593.1 yes 4
24.5 odd 2 5184.2.d.n.2593.4 yes 4
24.11 even 2 5184.2.d.b.2593.3 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5184.2.d.a.2593.1 ✓ 4 8.3 odd 2
5184.2.d.a.2593.3 yes 4 4.3 odd 2
5184.2.d.b.2593.1 yes 4 12.11 even 2
5184.2.d.b.2593.3 yes 4 24.11 even 2
5184.2.d.m.2593.2 yes 4 8.5 even 2 inner
5184.2.d.m.2593.4 yes 4 1.1 even 1 trivial
5184.2.d.n.2593.2 yes 4 3.2 odd 2
5184.2.d.n.2593.4 yes 4 24.5 odd 2