Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2592,2,Mod(865,2592)] 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("2592.865"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2592, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 2592 = 2^{5} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2592.i (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,4,0,0,0,0,0,0,0,-6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(20.6972242039\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
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: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{3}]$

Embedding invariants

Embedding label 1729.2
Root \(0.866025 + 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 2592.1729
Dual form 2592.2.i.bf.865.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(1.86603 - 3.23205i) q^{5} +(-3.23205 + 5.59808i) q^{13} -5.73205 q^{17} +(-4.46410 - 7.73205i) q^{25} +(-5.33013 - 9.23205i) q^{29} -9.39230 q^{37} +(4.00000 - 6.92820i) q^{41} +(3.50000 - 6.06218i) q^{49} -4.00000 q^{53} +(-7.69615 - 13.3301i) q^{61} +(12.0622 + 20.8923i) q^{65} -16.8564 q^{73} +(-10.6962 + 18.5263i) q^{85} +0.660254 q^{89} +(9.00000 + 15.5885i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{5} - 6 q^{13} - 16 q^{17} - 4 q^{25} - 4 q^{29} + 4 q^{37} + 16 q^{41} + 14 q^{49} - 16 q^{53} - 10 q^{61} + 24 q^{65} - 12 q^{73} - 22 q^{85} - 32 q^{89} + 36 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2592\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(1217\) \(2431\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{3}\right)\) \(1\)

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
\(5\) 1.86603 − 3.23205i 0.834512 − 1.44542i −0.0599153 − 0.998203i \(-0.519083\pi\)
0.894427 − 0.447214i \(-0.147584\pi\)
\(6\) 0 0
\(7\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(12\) 0 0
\(13\) −3.23205 + 5.59808i −0.896410 + 1.55263i −0.0643593 + 0.997927i \(0.520500\pi\)
−0.832050 + 0.554700i \(0.812833\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −5.73205 −1.39023 −0.695113 − 0.718900i \(-0.744646\pi\)
−0.695113 + 0.718900i \(0.744646\pi\)
\(18\) 0 0
\(19\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(24\) 0 0
\(25\) −4.46410 − 7.73205i −0.892820 − 1.54641i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −5.33013 − 9.23205i −0.989780 − 1.71435i −0.618389 − 0.785872i \(-0.712214\pi\)
−0.371391 − 0.928477i \(-0.621119\pi\)
\(30\) 0 0
\(31\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −9.39230 −1.54409 −0.772043 − 0.635571i \(-0.780765\pi\)
−0.772043 + 0.635571i \(0.780765\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 4.00000 − 6.92820i 0.624695 − 1.08200i −0.363905 − 0.931436i \(-0.618557\pi\)
0.988600 − 0.150567i \(-0.0481100\pi\)
\(42\) 0 0
\(43\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(48\) 0 0
\(49\) 3.50000 − 6.06218i 0.500000 − 0.866025i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −4.00000 −0.549442 −0.274721 − 0.961524i \(-0.588586\pi\)
−0.274721 + 0.961524i \(0.588586\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(60\) 0 0
\(61\) −7.69615 − 13.3301i −0.985391 − 1.70675i −0.640184 − 0.768221i \(-0.721142\pi\)
−0.345207 − 0.938527i \(-0.612191\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 12.0622 + 20.8923i 1.49613 + 2.59137i
\(66\) 0 0
\(67\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) −16.8564 −1.97289 −0.986447 − 0.164083i \(-0.947534\pi\)
−0.986447 + 0.164083i \(0.947534\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(84\) 0 0
\(85\) −10.6962 + 18.5263i −1.16016 + 2.00946i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0.660254 0.0699868 0.0349934 − 0.999388i \(-0.488859\pi\)
0.0349934 + 0.999388i \(0.488859\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 9.00000 + 15.5885i 0.913812 + 1.58277i 0.808632 + 0.588315i \(0.200208\pi\)
0.105180 + 0.994453i \(0.466458\pi\)
\(98\) 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 2592.2.i.bf.1729.2 4
3.2 odd 2 2592.2.i.y.1729.1 4
4.3 odd 2 CM 2592.2.i.bf.1729.2 4
9.2 odd 6 2592.2.i.y.865.1 4
9.4 even 3 2592.2.a.i.1.1 ✓ 2
9.5 odd 6 2592.2.a.t.1.2 yes 2
9.7 even 3 inner 2592.2.i.bf.865.2 4
12.11 even 2 2592.2.i.y.1729.1 4
36.7 odd 6 inner 2592.2.i.bf.865.2 4
36.11 even 6 2592.2.i.y.865.1 4
36.23 even 6 2592.2.a.t.1.2 yes 2
36.31 odd 6 2592.2.a.i.1.1 ✓ 2
72.5 odd 6 5184.2.a.bh.1.1 2
72.13 even 6 5184.2.a.ca.1.2 2
72.59 even 6 5184.2.a.bh.1.1 2
72.67 odd 6 5184.2.a.ca.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2592.2.a.i.1.1 ✓ 2 9.4 even 3
2592.2.a.i.1.1 ✓ 2 36.31 odd 6
2592.2.a.t.1.2 yes 2 9.5 odd 6
2592.2.a.t.1.2 yes 2 36.23 even 6
2592.2.i.y.865.1 4 9.2 odd 6
2592.2.i.y.865.1 4 36.11 even 6
2592.2.i.y.1729.1 4 3.2 odd 2
2592.2.i.y.1729.1 4 12.11 even 2
2592.2.i.bf.865.2 4 9.7 even 3 inner
2592.2.i.bf.865.2 4 36.7 odd 6 inner
2592.2.i.bf.1729.2 4 1.1 even 1 trivial
2592.2.i.bf.1729.2 4 4.3 odd 2 CM
5184.2.a.bh.1.1 2 72.5 odd 6
5184.2.a.bh.1.1 2 72.59 even 6
5184.2.a.ca.1.2 2 72.13 even 6
5184.2.a.ca.1.2 2 72.67 odd 6