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 = [2,0,0,0,-2,0,1,0,0,0,2,0,-1] 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: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 864)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 865.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 2592.865
Dual form 2592.2.i.f.1729.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.00000 - 1.73205i) q^{5} +(0.500000 - 0.866025i) q^{7} +(1.00000 - 1.73205i) q^{11} +(-0.500000 - 0.866025i) q^{13} +6.00000 q^{17} +5.00000 q^{19} +(-3.00000 - 5.19615i) q^{23} +(0.500000 - 0.866025i) q^{25} +(-4.00000 + 6.92820i) q^{29} +(4.00000 + 6.92820i) q^{31} -2.00000 q^{35} -5.00000 q^{37} +(-4.00000 - 6.92820i) q^{41} +(-2.00000 + 3.46410i) q^{43} +(5.00000 - 8.66025i) q^{47} +(3.00000 + 5.19615i) q^{49} +4.00000 q^{53} -4.00000 q^{55} +(-7.00000 - 12.1244i) q^{59} +(-1.50000 + 2.59808i) q^{61} +(-1.00000 + 1.73205i) q^{65} +(-6.50000 - 11.2583i) q^{67} -4.00000 q^{71} +9.00000 q^{73} +(-1.00000 - 1.73205i) q^{77} +(5.50000 - 9.52628i) q^{79} +(6.00000 - 10.3923i) q^{83} +(-6.00000 - 10.3923i) q^{85} -2.00000 q^{89} -1.00000 q^{91} +(-5.00000 - 8.66025i) q^{95} +(-0.500000 + 0.866025i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{5} + q^{7} + 2 q^{11} - q^{13} + 12 q^{17} + 10 q^{19} - 6 q^{23} + q^{25} - 8 q^{29} + 8 q^{31} - 4 q^{35} - 10 q^{37} - 8 q^{41} - 4 q^{43} + 10 q^{47} + 6 q^{49} + 8 q^{53} - 8 q^{55} - 14 q^{59}+ \cdots - q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(325\) \(1217\) \(2431\)
\(\chi(n)\) \(1\) \(e\left(\frac{2}{3}\right)\) \(1\)

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
\(5\) −1.00000 − 1.73205i −0.447214 − 0.774597i 0.550990 − 0.834512i \(-0.314250\pi\)
−0.998203 + 0.0599153i \(0.980917\pi\)
\(6\) 0 0
\(7\) 0.500000 − 0.866025i 0.188982 − 0.327327i −0.755929 − 0.654654i \(-0.772814\pi\)
0.944911 + 0.327327i \(0.106148\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 1.00000 − 1.73205i 0.301511 − 0.522233i −0.674967 − 0.737848i \(-0.735842\pi\)
0.976478 + 0.215615i \(0.0691756\pi\)
\(12\) 0 0
\(13\) −0.500000 − 0.866025i −0.138675 − 0.240192i 0.788320 − 0.615265i \(-0.210951\pi\)
−0.926995 + 0.375073i \(0.877618\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 6.00000 1.45521 0.727607 − 0.685994i \(-0.240633\pi\)
0.727607 + 0.685994i \(0.240633\pi\)
\(18\) 0 0
\(19\) 5.00000 1.14708 0.573539 − 0.819178i \(-0.305570\pi\)
0.573539 + 0.819178i \(0.305570\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −3.00000 − 5.19615i −0.625543 − 1.08347i −0.988436 − 0.151642i \(-0.951544\pi\)
0.362892 − 0.931831i \(-0.381789\pi\)
\(24\) 0 0
\(25\) 0.500000 − 0.866025i 0.100000 − 0.173205i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −4.00000 + 6.92820i −0.742781 + 1.28654i 0.208443 + 0.978035i \(0.433160\pi\)
−0.951224 + 0.308500i \(0.900173\pi\)
\(30\) 0 0
\(31\) 4.00000 + 6.92820i 0.718421 + 1.24434i 0.961625 + 0.274367i \(0.0884683\pi\)
−0.243204 + 0.969975i \(0.578198\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −2.00000 −0.338062
\(36\) 0 0
\(37\) −5.00000 −0.821995 −0.410997 − 0.911636i \(-0.634819\pi\)
−0.410997 + 0.911636i \(0.634819\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −4.00000 − 6.92820i −0.624695 − 1.08200i −0.988600 − 0.150567i \(-0.951890\pi\)
0.363905 − 0.931436i \(-0.381443\pi\)
\(42\) 0 0
\(43\) −2.00000 + 3.46410i −0.304997 + 0.528271i −0.977261 − 0.212041i \(-0.931989\pi\)
0.672264 + 0.740312i \(0.265322\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 5.00000 − 8.66025i 0.729325 − 1.26323i −0.227844 − 0.973698i \(-0.573168\pi\)
0.957169 − 0.289530i \(-0.0934991\pi\)
\(48\) 0 0
\(49\) 3.00000 + 5.19615i 0.428571 + 0.742307i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 4.00000 0.549442 0.274721 − 0.961524i \(-0.411414\pi\)
0.274721 + 0.961524i \(0.411414\pi\)
\(54\) 0 0
\(55\) −4.00000 −0.539360
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −7.00000 − 12.1244i −0.911322 − 1.57846i −0.812198 − 0.583382i \(-0.801729\pi\)
−0.0991242 − 0.995075i \(-0.531604\pi\)
\(60\) 0 0
\(61\) −1.50000 + 2.59808i −0.192055 + 0.332650i −0.945931 − 0.324367i \(-0.894849\pi\)
0.753876 + 0.657017i \(0.228182\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −1.00000 + 1.73205i −0.124035 + 0.214834i
\(66\) 0 0
\(67\) −6.50000 − 11.2583i −0.794101 − 1.37542i −0.923408 − 0.383819i \(-0.874609\pi\)
0.129307 − 0.991605i \(-0.458725\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −4.00000 −0.474713 −0.237356 − 0.971423i \(-0.576281\pi\)
−0.237356 + 0.971423i \(0.576281\pi\)
\(72\) 0 0
\(73\) 9.00000 1.05337 0.526685 − 0.850060i \(-0.323435\pi\)
0.526685 + 0.850060i \(0.323435\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −1.00000 − 1.73205i −0.113961 − 0.197386i
\(78\) 0 0
\(79\) 5.50000 − 9.52628i 0.618798 − 1.07179i −0.370907 − 0.928670i \(-0.620953\pi\)
0.989705 − 0.143120i \(-0.0457135\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 6.00000 − 10.3923i 0.658586 − 1.14070i −0.322396 − 0.946605i \(-0.604488\pi\)
0.980982 − 0.194099i \(-0.0621783\pi\)
\(84\) 0 0
\(85\) −6.00000 − 10.3923i −0.650791 − 1.12720i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −2.00000 −0.212000 −0.106000 − 0.994366i \(-0.533804\pi\)
−0.106000 + 0.994366i \(0.533804\pi\)
\(90\) 0 0
\(91\) −1.00000 −0.104828
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −5.00000 − 8.66025i −0.512989 − 0.888523i
\(96\) 0 0
\(97\) −0.500000 + 0.866025i −0.0507673 + 0.0879316i −0.890292 − 0.455389i \(-0.849500\pi\)
0.839525 + 0.543321i \(0.182833\pi\)
\(98\) 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.f.865.1 2
3.2 odd 2 2592.2.i.u.865.1 2
4.3 odd 2 2592.2.i.d.865.1 2
9.2 odd 6 864.2.a.b.1.1 ✓ 1
9.4 even 3 inner 2592.2.i.f.1729.1 2
9.5 odd 6 2592.2.i.u.1729.1 2
9.7 even 3 864.2.a.j.1.1 yes 1
12.11 even 2 2592.2.i.s.865.1 2
36.7 odd 6 864.2.a.k.1.1 yes 1
36.11 even 6 864.2.a.c.1.1 yes 1
36.23 even 6 2592.2.i.s.1729.1 2
36.31 odd 6 2592.2.i.d.1729.1 2
72.11 even 6 1728.2.a.w.1.1 1
72.29 odd 6 1728.2.a.v.1.1 1
72.43 odd 6 1728.2.a.g.1.1 1
72.61 even 6 1728.2.a.f.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
864.2.a.b.1.1 ✓ 1 9.2 odd 6
864.2.a.c.1.1 yes 1 36.11 even 6
864.2.a.j.1.1 yes 1 9.7 even 3
864.2.a.k.1.1 yes 1 36.7 odd 6
1728.2.a.f.1.1 1 72.61 even 6
1728.2.a.g.1.1 1 72.43 odd 6
1728.2.a.v.1.1 1 72.29 odd 6
1728.2.a.w.1.1 1 72.11 even 6
2592.2.i.d.865.1 2 4.3 odd 2
2592.2.i.d.1729.1 2 36.31 odd 6
2592.2.i.f.865.1 2 1.1 even 1 trivial
2592.2.i.f.1729.1 2 9.4 even 3 inner
2592.2.i.s.865.1 2 12.11 even 2
2592.2.i.s.1729.1 2 36.23 even 6
2592.2.i.u.865.1 2 3.2 odd 2
2592.2.i.u.1729.1 2 9.5 odd 6