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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [288,2,Mod(287,288)] 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("288.287"); 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([1, 0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 288 = 2^{5} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 288.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.29969157821\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{5} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 287.4
Root \(-0.707107 - 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 288.287
Dual form 288.2.c.a.287.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.41421i q^{5} +4.00000i q^{7} -5.65685 q^{11} +4.00000 q^{13} +4.24264i q^{17} +5.65685 q^{23} +3.00000 q^{25} -1.41421i q^{29} +4.00000i q^{31} -5.65685 q^{35} -6.00000 q^{37} -9.89949i q^{41} -8.00000i q^{43} +5.65685 q^{47} -9.00000 q^{49} +4.24264i q^{53} -8.00000i q^{55} +11.3137 q^{59} -2.00000 q^{61} +5.65685i q^{65} -8.00000i q^{67} -5.65685 q^{71} -22.6274i q^{77} -4.00000i q^{79} +5.65685 q^{83} -6.00000 q^{85} -4.24264i q^{89} +16.0000i q^{91} -8.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 16 q^{13} + 12 q^{25} - 24 q^{37} - 36 q^{49} - 8 q^{61} - 24 q^{85} - 32 q^{97}+O(q^{100}) \) 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
\(5\) 1.41421i 0.632456i 0.948683 + 0.316228i \(0.102416\pi\)
−0.948683 + 0.316228i \(0.897584\pi\)
\(6\) 0 0
\(7\) 4.00000i 1.51186i 0.654654 + 0.755929i \(0.272814\pi\)
−0.654654 + 0.755929i \(0.727186\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −5.65685 −1.70561 −0.852803 − 0.522233i \(-0.825099\pi\)
−0.852803 + 0.522233i \(0.825099\pi\)
\(12\) 0 0
\(13\) 4.00000 1.10940 0.554700 − 0.832050i \(-0.312833\pi\)
0.554700 + 0.832050i \(0.312833\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 4.24264i 1.02899i 0.857493 + 0.514496i \(0.172021\pi\)
−0.857493 + 0.514496i \(0.827979\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 5.65685 1.17954 0.589768 − 0.807573i \(-0.299219\pi\)
0.589768 + 0.807573i \(0.299219\pi\)
\(24\) 0 0
\(25\) 3.00000 0.600000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) − 1.41421i − 0.262613i −0.991342 − 0.131306i \(-0.958083\pi\)
0.991342 − 0.131306i \(-0.0419172\pi\)
\(30\) 0 0
\(31\) 4.00000i 0.718421i 0.933257 + 0.359211i \(0.116954\pi\)
−0.933257 + 0.359211i \(0.883046\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −5.65685 −0.956183
\(36\) 0 0
\(37\) −6.00000 −0.986394 −0.493197 − 0.869918i \(-0.664172\pi\)
−0.493197 + 0.869918i \(0.664172\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) − 9.89949i − 1.54604i −0.634381 − 0.773021i \(-0.718745\pi\)
0.634381 − 0.773021i \(-0.281255\pi\)
\(42\) 0 0
\(43\) − 8.00000i − 1.21999i −0.792406 − 0.609994i \(-0.791172\pi\)
0.792406 − 0.609994i \(-0.208828\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 5.65685 0.825137 0.412568 − 0.910927i \(-0.364632\pi\)
0.412568 + 0.910927i \(0.364632\pi\)
\(48\) 0 0
\(49\) −9.00000 −1.28571
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 4.24264i 0.582772i 0.956606 + 0.291386i \(0.0941163\pi\)
−0.956606 + 0.291386i \(0.905884\pi\)
\(54\) 0 0
\(55\) − 8.00000i − 1.07872i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 11.3137 1.47292 0.736460 − 0.676481i \(-0.236496\pi\)
0.736460 + 0.676481i \(0.236496\pi\)
\(60\) 0 0
\(61\) −2.00000 −0.256074 −0.128037 − 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 5.65685i 0.701646i
\(66\) 0 0
\(67\) − 8.00000i − 0.977356i −0.872464 − 0.488678i \(-0.837479\pi\)
0.872464 − 0.488678i \(-0.162521\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −5.65685 −0.671345 −0.335673 − 0.941979i \(-0.608964\pi\)
−0.335673 + 0.941979i \(0.608964\pi\)
\(72\) 0 0
\(73\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 22.6274i − 2.57863i
\(78\) 0 0
\(79\) − 4.00000i − 0.450035i −0.974355 − 0.225018i \(-0.927756\pi\)
0.974355 − 0.225018i \(-0.0722440\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 5.65685 0.620920 0.310460 − 0.950586i \(-0.399517\pi\)
0.310460 + 0.950586i \(0.399517\pi\)
\(84\) 0 0
\(85\) −6.00000 −0.650791
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) − 4.24264i − 0.449719i −0.974391 − 0.224860i \(-0.927808\pi\)
0.974391 − 0.224860i \(-0.0721923\pi\)
\(90\) 0 0
\(91\) 16.0000i 1.67726i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −8.00000 −0.812277 −0.406138 − 0.913812i \(-0.633125\pi\)
−0.406138 + 0.913812i \(0.633125\pi\)
\(98\) 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.c.a.287.4 yes 4
3.2 odd 2 inner 288.2.c.a.287.2 yes 4
4.3 odd 2 inner 288.2.c.a.287.3 yes 4
5.2 odd 4 7200.2.o.a.7199.1 4
5.3 odd 4 7200.2.o.n.7199.2 4
5.4 even 2 7200.2.h.d.1151.1 4
8.3 odd 2 576.2.c.c.575.1 4
8.5 even 2 576.2.c.c.575.2 4
9.2 odd 6 2592.2.s.d.863.4 8
9.4 even 3 2592.2.s.d.1727.3 8
9.5 odd 6 2592.2.s.d.1727.1 8
9.7 even 3 2592.2.s.d.863.2 8
12.11 even 2 inner 288.2.c.a.287.1 ✓ 4
15.2 even 4 7200.2.o.a.7199.3 4
15.8 even 4 7200.2.o.n.7199.4 4
15.14 odd 2 7200.2.h.d.1151.2 4
16.3 odd 4 2304.2.f.e.1151.4 4
16.5 even 4 2304.2.f.e.1151.1 4
16.11 odd 4 2304.2.f.c.1151.2 4
16.13 even 4 2304.2.f.c.1151.3 4
20.3 even 4 7200.2.o.a.7199.4 4
20.7 even 4 7200.2.o.n.7199.3 4
20.19 odd 2 7200.2.h.d.1151.4 4
24.5 odd 2 576.2.c.c.575.4 4
24.11 even 2 576.2.c.c.575.3 4
36.7 odd 6 2592.2.s.d.863.1 8
36.11 even 6 2592.2.s.d.863.3 8
36.23 even 6 2592.2.s.d.1727.2 8
36.31 odd 6 2592.2.s.d.1727.4 8
48.5 odd 4 2304.2.f.e.1151.3 4
48.11 even 4 2304.2.f.c.1151.4 4
48.29 odd 4 2304.2.f.c.1151.1 4
48.35 even 4 2304.2.f.e.1151.2 4
60.23 odd 4 7200.2.o.a.7199.2 4
60.47 odd 4 7200.2.o.n.7199.1 4
60.59 even 2 7200.2.h.d.1151.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
288.2.c.a.287.1 ✓ 4 12.11 even 2 inner
288.2.c.a.287.2 yes 4 3.2 odd 2 inner
288.2.c.a.287.3 yes 4 4.3 odd 2 inner
288.2.c.a.287.4 yes 4 1.1 even 1 trivial
576.2.c.c.575.1 4 8.3 odd 2
576.2.c.c.575.2 4 8.5 even 2
576.2.c.c.575.3 4 24.11 even 2
576.2.c.c.575.4 4 24.5 odd 2
2304.2.f.c.1151.1 4 48.29 odd 4
2304.2.f.c.1151.2 4 16.11 odd 4
2304.2.f.c.1151.3 4 16.13 even 4
2304.2.f.c.1151.4 4 48.11 even 4
2304.2.f.e.1151.1 4 16.5 even 4
2304.2.f.e.1151.2 4 48.35 even 4
2304.2.f.e.1151.3 4 48.5 odd 4
2304.2.f.e.1151.4 4 16.3 odd 4
2592.2.s.d.863.1 8 36.7 odd 6
2592.2.s.d.863.2 8 9.7 even 3
2592.2.s.d.863.3 8 36.11 even 6
2592.2.s.d.863.4 8 9.2 odd 6
2592.2.s.d.1727.1 8 9.5 odd 6
2592.2.s.d.1727.2 8 36.23 even 6
2592.2.s.d.1727.3 8 9.4 even 3
2592.2.s.d.1727.4 8 36.31 odd 6
7200.2.h.d.1151.1 4 5.4 even 2
7200.2.h.d.1151.2 4 15.14 odd 2
7200.2.h.d.1151.3 4 60.59 even 2
7200.2.h.d.1151.4 4 20.19 odd 2
7200.2.o.a.7199.1 4 5.2 odd 4
7200.2.o.a.7199.2 4 60.23 odd 4
7200.2.o.a.7199.3 4 15.2 even 4
7200.2.o.a.7199.4 4 20.3 even 4
7200.2.o.n.7199.1 4 60.47 odd 4
7200.2.o.n.7199.2 4 5.3 odd 4
7200.2.o.n.7199.3 4 20.7 even 4
7200.2.o.n.7199.4 4 15.8 even 4