Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [288,2,Mod(145,288)] 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("288.145"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(288, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 288 = 2^{5} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 288.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,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.29969157821\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 72)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 145.1
Root \(-1.41421i\) of defining polynomial
Character \(\chi\) \(=\) 288.145
Dual form 288.2.d.a.145.2

$q$-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)
 
\(f(q)\) \(=\) \(q-2.82843i q^{5} -2.00000 q^{7} -5.65685i q^{11} -3.00000 q^{25} -2.82843i q^{29} +10.0000 q^{31} +5.65685i q^{35} -3.00000 q^{49} +14.1421i q^{53} -16.0000 q^{55} +11.3137i q^{59} +14.0000 q^{73} +11.3137i q^{77} +10.0000 q^{79} -5.65685i q^{83} +2.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{7} - 6 q^{25} + 20 q^{31} - 6 q^{49} - 32 q^{55} + 28 q^{73} + 20 q^{79} + 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/288\mathbb{Z}\right)^\times\).

\(n\) \(37\) \(65\) \(127\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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
\(5\) − 2.82843i − 1.26491i −0.774597 − 0.632456i \(-0.782047\pi\)
0.774597 − 0.632456i \(-0.217953\pi\)
\(6\) 0 0
\(7\) −2.00000 −0.755929 −0.377964 − 0.925820i \(-0.623376\pi\)
−0.377964 + 0.925820i \(0.623376\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) − 5.65685i − 1.70561i −0.522233 − 0.852803i \(-0.674901\pi\)
0.522233 − 0.852803i \(-0.325099\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) −3.00000 −0.600000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) − 2.82843i − 0.525226i −0.964901 − 0.262613i \(-0.915416\pi\)
0.964901 − 0.262613i \(-0.0845842\pi\)
\(30\) 0 0
\(31\) 10.0000 1.79605 0.898027 − 0.439941i \(-0.145001\pi\)
0.898027 + 0.439941i \(0.145001\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 5.65685i 0.956183i
\(36\) 0 0
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) −3.00000 −0.428571
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 14.1421i 1.94257i 0.237915 + 0.971286i \(0.423536\pi\)
−0.237915 + 0.971286i \(0.576464\pi\)
\(54\) 0 0
\(55\) −16.0000 −2.15744
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 11.3137i 1.47292i 0.676481 + 0.736460i \(0.263504\pi\)
−0.676481 + 0.736460i \(0.736496\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 14.0000 1.63858 0.819288 − 0.573382i \(-0.194369\pi\)
0.819288 + 0.573382i \(0.194369\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 11.3137i 1.28932i
\(78\) 0 0
\(79\) 10.0000 1.12509 0.562544 − 0.826767i \(-0.309823\pi\)
0.562544 + 0.826767i \(0.309823\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) − 5.65685i − 0.620920i −0.950586 − 0.310460i \(-0.899517\pi\)
0.950586 − 0.310460i \(-0.100483\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 2.00000 0.203069 0.101535 − 0.994832i \(-0.467625\pi\)
0.101535 + 0.994832i \(0.467625\pi\)
\(98\) 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 288.2.d.a.145.1 2
3.2 odd 2 inner 288.2.d.a.145.2 2
4.3 odd 2 72.2.d.a.37.1 ✓ 2
5.2 odd 4 7200.2.d.p.2449.1 4
5.3 odd 4 7200.2.d.p.2449.3 4
5.4 even 2 7200.2.k.h.3601.1 2
8.3 odd 2 72.2.d.a.37.2 yes 2
8.5 even 2 inner 288.2.d.a.145.2 2
9.2 odd 6 2592.2.r.i.433.1 4
9.4 even 3 2592.2.r.i.2161.1 4
9.5 odd 6 2592.2.r.i.2161.2 4
9.7 even 3 2592.2.r.i.433.2 4
12.11 even 2 72.2.d.a.37.2 yes 2
15.2 even 4 7200.2.d.p.2449.2 4
15.8 even 4 7200.2.d.p.2449.4 4
15.14 odd 2 7200.2.k.h.3601.2 2
16.3 odd 4 2304.2.a.q.1.1 2
16.5 even 4 2304.2.a.y.1.2 2
16.11 odd 4 2304.2.a.q.1.2 2
16.13 even 4 2304.2.a.y.1.1 2
20.3 even 4 1800.2.d.n.1549.1 4
20.7 even 4 1800.2.d.n.1549.4 4
20.19 odd 2 1800.2.k.e.901.2 2
24.5 odd 2 CM 288.2.d.a.145.1 2
24.11 even 2 72.2.d.a.37.1 ✓ 2
36.7 odd 6 648.2.n.h.109.1 4
36.11 even 6 648.2.n.h.109.2 4
36.23 even 6 648.2.n.h.541.1 4
36.31 odd 6 648.2.n.h.541.2 4
40.3 even 4 1800.2.d.n.1549.3 4
40.13 odd 4 7200.2.d.p.2449.4 4
40.19 odd 2 1800.2.k.e.901.1 2
40.27 even 4 1800.2.d.n.1549.2 4
40.29 even 2 7200.2.k.h.3601.2 2
40.37 odd 4 7200.2.d.p.2449.2 4
48.5 odd 4 2304.2.a.y.1.1 2
48.11 even 4 2304.2.a.q.1.1 2
48.29 odd 4 2304.2.a.y.1.2 2
48.35 even 4 2304.2.a.q.1.2 2
60.23 odd 4 1800.2.d.n.1549.3 4
60.47 odd 4 1800.2.d.n.1549.2 4
60.59 even 2 1800.2.k.e.901.1 2
72.5 odd 6 2592.2.r.i.2161.1 4
72.11 even 6 648.2.n.h.109.1 4
72.13 even 6 2592.2.r.i.2161.2 4
72.29 odd 6 2592.2.r.i.433.2 4
72.43 odd 6 648.2.n.h.109.2 4
72.59 even 6 648.2.n.h.541.2 4
72.61 even 6 2592.2.r.i.433.1 4
72.67 odd 6 648.2.n.h.541.1 4
120.29 odd 2 7200.2.k.h.3601.1 2
120.53 even 4 7200.2.d.p.2449.3 4
120.59 even 2 1800.2.k.e.901.2 2
120.77 even 4 7200.2.d.p.2449.1 4
120.83 odd 4 1800.2.d.n.1549.1 4
120.107 odd 4 1800.2.d.n.1549.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
72.2.d.a.37.1 ✓ 2 4.3 odd 2
72.2.d.a.37.1 ✓ 2 24.11 even 2
72.2.d.a.37.2 yes 2 8.3 odd 2
72.2.d.a.37.2 yes 2 12.11 even 2
288.2.d.a.145.1 2 1.1 even 1 trivial
288.2.d.a.145.1 2 24.5 odd 2 CM
288.2.d.a.145.2 2 3.2 odd 2 inner
288.2.d.a.145.2 2 8.5 even 2 inner
648.2.n.h.109.1 4 36.7 odd 6
648.2.n.h.109.1 4 72.11 even 6
648.2.n.h.109.2 4 36.11 even 6
648.2.n.h.109.2 4 72.43 odd 6
648.2.n.h.541.1 4 36.23 even 6
648.2.n.h.541.1 4 72.67 odd 6
648.2.n.h.541.2 4 36.31 odd 6
648.2.n.h.541.2 4 72.59 even 6
1800.2.d.n.1549.1 4 20.3 even 4
1800.2.d.n.1549.1 4 120.83 odd 4
1800.2.d.n.1549.2 4 40.27 even 4
1800.2.d.n.1549.2 4 60.47 odd 4
1800.2.d.n.1549.3 4 40.3 even 4
1800.2.d.n.1549.3 4 60.23 odd 4
1800.2.d.n.1549.4 4 20.7 even 4
1800.2.d.n.1549.4 4 120.107 odd 4
1800.2.k.e.901.1 2 40.19 odd 2
1800.2.k.e.901.1 2 60.59 even 2
1800.2.k.e.901.2 2 20.19 odd 2
1800.2.k.e.901.2 2 120.59 even 2
2304.2.a.q.1.1 2 16.3 odd 4
2304.2.a.q.1.1 2 48.11 even 4
2304.2.a.q.1.2 2 16.11 odd 4
2304.2.a.q.1.2 2 48.35 even 4
2304.2.a.y.1.1 2 16.13 even 4
2304.2.a.y.1.1 2 48.5 odd 4
2304.2.a.y.1.2 2 16.5 even 4
2304.2.a.y.1.2 2 48.29 odd 4
2592.2.r.i.433.1 4 9.2 odd 6
2592.2.r.i.433.1 4 72.61 even 6
2592.2.r.i.433.2 4 9.7 even 3
2592.2.r.i.433.2 4 72.29 odd 6
2592.2.r.i.2161.1 4 9.4 even 3
2592.2.r.i.2161.1 4 72.5 odd 6
2592.2.r.i.2161.2 4 9.5 odd 6
2592.2.r.i.2161.2 4 72.13 even 6
7200.2.d.p.2449.1 4 5.2 odd 4
7200.2.d.p.2449.1 4 120.77 even 4
7200.2.d.p.2449.2 4 15.2 even 4
7200.2.d.p.2449.2 4 40.37 odd 4
7200.2.d.p.2449.3 4 5.3 odd 4
7200.2.d.p.2449.3 4 120.53 even 4
7200.2.d.p.2449.4 4 15.8 even 4
7200.2.d.p.2449.4 4 40.13 odd 4
7200.2.k.h.3601.1 2 5.4 even 2
7200.2.k.h.3601.1 2 120.29 odd 2
7200.2.k.h.3601.2 2 15.14 odd 2
7200.2.k.h.3601.2 2 40.29 even 2