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(1297,2592)] 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("2592.1297"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2592, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 2592 = 2^{5} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2592.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,0,0,-8,0,0,0,0,0,0,0,0,0,10] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == 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: \(\Q(i)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 72)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1297.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 2592.1297
Dual form 2592.2.d.b.1297.2

$q$-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)
 
\(f(q)\) \(=\) \(q-2.00000i q^{5} -4.00000 q^{7} +3.00000i q^{11} +2.00000i q^{13} +5.00000 q^{17} -1.00000i q^{19} -2.00000 q^{23} +1.00000 q^{25} +4.00000 q^{31} +8.00000i q^{35} +2.00000i q^{37} -5.00000 q^{41} -11.0000i q^{43} +6.00000 q^{47} +9.00000 q^{49} +6.00000 q^{55} +1.00000i q^{59} -12.0000i q^{61} +4.00000 q^{65} -3.00000i q^{67} +6.00000 q^{71} +9.00000 q^{73} -12.0000i q^{77} +14.0000 q^{79} +4.00000i q^{83} -10.0000i q^{85} -14.0000 q^{89} -8.00000i q^{91} -2.00000 q^{95} +1.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 8 q^{7} + 10 q^{17} - 4 q^{23} + 2 q^{25} + 8 q^{31} - 10 q^{41} + 12 q^{47} + 18 q^{49} + 12 q^{55} + 8 q^{65} + 12 q^{71} + 18 q^{73} + 28 q^{79} - 28 q^{89} - 4 q^{95} + 2 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(325\) \(1217\) \(2431\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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
\(5\) − 2.00000i − 0.894427i −0.894427 − 0.447214i \(-0.852416\pi\)
0.894427 − 0.447214i \(-0.147584\pi\)
\(6\) 0 0
\(7\) −4.00000 −1.51186 −0.755929 − 0.654654i \(-0.772814\pi\)
−0.755929 + 0.654654i \(0.772814\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 3.00000i 0.904534i 0.891883 + 0.452267i \(0.149385\pi\)
−0.891883 + 0.452267i \(0.850615\pi\)
\(12\) 0 0
\(13\) 2.00000i 0.554700i 0.960769 + 0.277350i \(0.0894562\pi\)
−0.960769 + 0.277350i \(0.910544\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 5.00000 1.21268 0.606339 − 0.795206i \(-0.292637\pi\)
0.606339 + 0.795206i \(0.292637\pi\)
\(18\) 0 0
\(19\) − 1.00000i − 0.229416i −0.993399 − 0.114708i \(-0.963407\pi\)
0.993399 − 0.114708i \(-0.0365932\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −2.00000 −0.417029 −0.208514 − 0.978019i \(-0.566863\pi\)
−0.208514 + 0.978019i \(0.566863\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) 4.00000 0.718421 0.359211 − 0.933257i \(-0.383046\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 8.00000i 1.35225i
\(36\) 0 0
\(37\) 2.00000i 0.328798i 0.986394 + 0.164399i \(0.0525685\pi\)
−0.986394 + 0.164399i \(0.947432\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −5.00000 −0.780869 −0.390434 − 0.920631i \(-0.627675\pi\)
−0.390434 + 0.920631i \(0.627675\pi\)
\(42\) 0 0
\(43\) − 11.0000i − 1.67748i −0.544529 − 0.838742i \(-0.683292\pi\)
0.544529 − 0.838742i \(-0.316708\pi\)
\(44\) 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
\(49\) 9.00000 1.28571
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) 6.00000 0.809040
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 1.00000i 0.130189i 0.997879 + 0.0650945i \(0.0207349\pi\)
−0.997879 + 0.0650945i \(0.979265\pi\)
\(60\) 0 0
\(61\) − 12.0000i − 1.53644i −0.640184 − 0.768221i \(-0.721142\pi\)
0.640184 − 0.768221i \(-0.278858\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 4.00000 0.496139
\(66\) 0 0
\(67\) − 3.00000i − 0.366508i −0.983066 − 0.183254i \(-0.941337\pi\)
0.983066 − 0.183254i \(-0.0586631\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 6.00000 0.712069 0.356034 − 0.934473i \(-0.384129\pi\)
0.356034 + 0.934473i \(0.384129\pi\)
\(72\) 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
\(77\) − 12.0000i − 1.36753i
\(78\) 0 0
\(79\) 14.0000 1.57512 0.787562 − 0.616236i \(-0.211343\pi\)
0.787562 + 0.616236i \(0.211343\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 4.00000i 0.439057i 0.975606 + 0.219529i \(0.0704519\pi\)
−0.975606 + 0.219529i \(0.929548\pi\)
\(84\) 0 0
\(85\) − 10.0000i − 1.08465i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −14.0000 −1.48400 −0.741999 − 0.670402i \(-0.766122\pi\)
−0.741999 + 0.670402i \(0.766122\pi\)
\(90\) 0 0
\(91\) − 8.00000i − 0.838628i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −2.00000 −0.205196
\(96\) 0 0
\(97\) 1.00000 0.101535 0.0507673 − 0.998711i \(-0.483833\pi\)
0.0507673 + 0.998711i \(0.483833\pi\)
\(98\) 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.d.b.1297.1 2
3.2 odd 2 2592.2.d.a.1297.2 2
4.3 odd 2 648.2.d.d.325.2 2
8.3 odd 2 648.2.d.d.325.1 2
8.5 even 2 inner 2592.2.d.b.1297.2 2
9.2 odd 6 864.2.r.a.145.1 4
9.4 even 3 288.2.r.a.241.2 4
9.5 odd 6 864.2.r.a.721.2 4
9.7 even 3 288.2.r.a.49.1 4
12.11 even 2 648.2.d.a.325.1 2
24.5 odd 2 2592.2.d.a.1297.1 2
24.11 even 2 648.2.d.a.325.2 2
36.7 odd 6 72.2.n.a.13.2 yes 4
36.11 even 6 216.2.n.a.37.1 4
36.23 even 6 216.2.n.a.181.2 4
36.31 odd 6 72.2.n.a.61.1 yes 4
72.5 odd 6 864.2.r.a.721.1 4
72.11 even 6 216.2.n.a.37.2 4
72.13 even 6 288.2.r.a.241.1 4
72.29 odd 6 864.2.r.a.145.2 4
72.43 odd 6 72.2.n.a.13.1 ✓ 4
72.59 even 6 216.2.n.a.181.1 4
72.61 even 6 288.2.r.a.49.2 4
72.67 odd 6 72.2.n.a.61.2 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
72.2.n.a.13.1 ✓ 4 72.43 odd 6
72.2.n.a.13.2 yes 4 36.7 odd 6
72.2.n.a.61.1 yes 4 36.31 odd 6
72.2.n.a.61.2 yes 4 72.67 odd 6
216.2.n.a.37.1 4 36.11 even 6
216.2.n.a.37.2 4 72.11 even 6
216.2.n.a.181.1 4 72.59 even 6
216.2.n.a.181.2 4 36.23 even 6
288.2.r.a.49.1 4 9.7 even 3
288.2.r.a.49.2 4 72.61 even 6
288.2.r.a.241.1 4 72.13 even 6
288.2.r.a.241.2 4 9.4 even 3
648.2.d.a.325.1 2 12.11 even 2
648.2.d.a.325.2 2 24.11 even 2
648.2.d.d.325.1 2 8.3 odd 2
648.2.d.d.325.2 2 4.3 odd 2
864.2.r.a.145.1 4 9.2 odd 6
864.2.r.a.145.2 4 72.29 odd 6
864.2.r.a.721.1 4 72.5 odd 6
864.2.r.a.721.2 4 9.5 odd 6
2592.2.d.a.1297.1 2 24.5 odd 2
2592.2.d.a.1297.2 2 3.2 odd 2
2592.2.d.b.1297.1 2 1.1 even 1 trivial
2592.2.d.b.1297.2 2 8.5 even 2 inner