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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [288,7,Mod(127,288)] 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("288.127"); S:= CuspForms(chi, 7); 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, 0])) N = Newforms(chi, 7, names="a")
 
Level: \( N \) \(=\) \( 288 = 2^{5} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 288.g (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,56] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(66.2555760825\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{6})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{17} \)
Twist minimal: no (minimal twist has level 32)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 127.1
Root \(-1.22474 + 1.22474i\) of defining polynomial
Character \(\chi\) \(=\) 288.127
Dual form 288.7.g.b.127.2

$q$-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)
 
\(f(q)\) \(=\) \(q-142.767 q^{5} -217.616i q^{7} +25.7388i q^{11} -1390.16 q^{13} -4004.09 q^{17} +129.021i q^{19} -3697.19i q^{23} +4757.51 q^{25} -40871.1 q^{29} -46174.6i q^{31} +31068.5i q^{35} +65959.6 q^{37} +41801.8 q^{41} +30093.2i q^{43} +79785.8i q^{47} +70292.1 q^{49} -13717.2 q^{53} -3674.66i q^{55} +300272. i q^{59} +243442. q^{61} +198470. q^{65} +406008. i q^{67} -249585. i q^{71} -366195. q^{73} +5601.18 q^{77} +452154. i q^{79} -512869. i q^{83} +571653. q^{85} +797498. q^{89} +302522. i q^{91} -18419.9i q^{95} -8788.19 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 56 q^{5} - 8696 q^{13} + 5304 q^{17} + 36588 q^{25} - 77576 q^{29} + 125256 q^{37} + 209848 q^{41} + 95556 q^{49} + 68664 q^{53} + 238216 q^{61} - 613264 q^{65} + 219528 q^{73} + 1028224 q^{77} + 3416592 q^{85}+ \cdots + 2918344 q^{97}+O(q^{100}) \) 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
\(5\) −142.767 −1.14214 −0.571069 − 0.820902i \(-0.693471\pi\)
−0.571069 + 0.820902i \(0.693471\pi\)
\(6\) 0 0
\(7\) − 217.616i − 0.634450i −0.948350 − 0.317225i \(-0.897249\pi\)
0.948350 − 0.317225i \(-0.102751\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 25.7388i 0.0193379i 0.999953 + 0.00966897i \(0.00307778\pi\)
−0.999953 + 0.00966897i \(0.996922\pi\)
\(12\) 0 0
\(13\) −1390.16 −0.632755 −0.316378 − 0.948633i \(-0.602467\pi\)
−0.316378 + 0.948633i \(0.602467\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −4004.09 −0.814999 −0.407499 − 0.913205i \(-0.633599\pi\)
−0.407499 + 0.913205i \(0.633599\pi\)
\(18\) 0 0
\(19\) 129.021i 0.0188104i 0.999956 + 0.00940520i \(0.00299381\pi\)
−0.999956 + 0.00940520i \(0.997006\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) − 3697.19i − 0.303870i −0.988390 − 0.151935i \(-0.951449\pi\)
0.988390 − 0.151935i \(-0.0485505\pi\)
\(24\) 0 0
\(25\) 4757.51 0.304481
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −40871.1 −1.67580 −0.837901 − 0.545823i \(-0.816217\pi\)
−0.837901 + 0.545823i \(0.816217\pi\)
\(30\) 0 0
\(31\) − 46174.6i − 1.54995i −0.631991 − 0.774976i \(-0.717762\pi\)
0.631991 − 0.774976i \(-0.282238\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 31068.5i 0.724630i
\(36\) 0 0
\(37\) 65959.6 1.30219 0.651093 − 0.758998i \(-0.274311\pi\)
0.651093 + 0.758998i \(0.274311\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 41801.8 0.606518 0.303259 − 0.952908i \(-0.401925\pi\)
0.303259 + 0.952908i \(0.401925\pi\)
\(42\) 0 0
\(43\) 30093.2i 0.378498i 0.981929 + 0.189249i \(0.0606053\pi\)
−0.981929 + 0.189249i \(0.939395\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 79785.8i 0.768479i 0.923234 + 0.384239i \(0.125536\pi\)
−0.923234 + 0.384239i \(0.874464\pi\)
\(48\) 0 0
\(49\) 70292.1 0.597473
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −13717.2 −0.0921376 −0.0460688 − 0.998938i \(-0.514669\pi\)
−0.0460688 + 0.998938i \(0.514669\pi\)
\(54\) 0 0
\(55\) − 3674.66i − 0.0220866i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 300272.i 1.46204i 0.682357 + 0.731019i \(0.260955\pi\)
−0.682357 + 0.731019i \(0.739045\pi\)
\(60\) 0 0
\(61\) 243442. 1.07252 0.536261 − 0.844052i \(-0.319836\pi\)
0.536261 + 0.844052i \(0.319836\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 198470. 0.722694
\(66\) 0 0
\(67\) 406008.i 1.34993i 0.737851 + 0.674964i \(0.235841\pi\)
−0.737851 + 0.674964i \(0.764159\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) − 249585.i − 0.697337i −0.937246 − 0.348669i \(-0.886634\pi\)
0.937246 − 0.348669i \(-0.113366\pi\)
\(72\) 0 0
\(73\) −366195. −0.941334 −0.470667 − 0.882311i \(-0.655987\pi\)
−0.470667 + 0.882311i \(0.655987\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 5601.18 0.0122690
\(78\) 0 0
\(79\) 452154.i 0.917076i 0.888675 + 0.458538i \(0.151627\pi\)
−0.888675 + 0.458538i \(0.848373\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) − 512869.i − 0.896958i −0.893793 − 0.448479i \(-0.851966\pi\)
0.893793 − 0.448479i \(-0.148034\pi\)
\(84\) 0 0
\(85\) 571653. 0.930842
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 797498. 1.13125 0.565626 − 0.824662i \(-0.308635\pi\)
0.565626 + 0.824662i \(0.308635\pi\)
\(90\) 0 0
\(91\) 302522.i 0.401452i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) − 18419.9i − 0.0214841i
\(96\) 0 0
\(97\) −8788.19 −0.00962907 −0.00481453 − 0.999988i \(-0.501533\pi\)
−0.00481453 + 0.999988i \(0.501533\pi\)
\(98\) 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 288.7.g.b.127.1 4
3.2 odd 2 32.7.c.b.31.1 ✓ 4
4.3 odd 2 inner 288.7.g.b.127.2 4
8.3 odd 2 576.7.g.l.127.4 4
8.5 even 2 576.7.g.l.127.3 4
12.11 even 2 32.7.c.b.31.4 yes 4
24.5 odd 2 64.7.c.e.63.4 4
24.11 even 2 64.7.c.e.63.1 4
48.5 odd 4 256.7.d.d.127.2 4
48.11 even 4 256.7.d.g.127.4 4
48.29 odd 4 256.7.d.g.127.3 4
48.35 even 4 256.7.d.d.127.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
32.7.c.b.31.1 ✓ 4 3.2 odd 2
32.7.c.b.31.4 yes 4 12.11 even 2
64.7.c.e.63.1 4 24.11 even 2
64.7.c.e.63.4 4 24.5 odd 2
256.7.d.d.127.1 4 48.35 even 4
256.7.d.d.127.2 4 48.5 odd 4
256.7.d.g.127.3 4 48.29 odd 4
256.7.d.g.127.4 4 48.11 even 4
288.7.g.b.127.1 4 1.1 even 1 trivial
288.7.g.b.127.2 4 4.3 odd 2 inner
576.7.g.l.127.3 4 8.5 even 2
576.7.g.l.127.4 4 8.3 odd 2