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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 127.4
Root \(1.22474 + 1.22474i\) of defining polynomial
Character \(\chi\) \(=\) 288.127
Dual form 288.7.g.b.127.3

$q$-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)
 
\(f(q)\) \(=\) \(q+170.767 q^{5} +374.384i q^{7} -1358.26i q^{11} -2957.84 q^{13} +6656.09 q^{17} -9926.98i q^{19} -15585.2i q^{23} +13536.5 q^{25} +2083.13 q^{29} -28446.6i q^{31} +63932.5i q^{35} -3331.58 q^{37} +63122.2 q^{41} +88501.2i q^{43} +10825.8i q^{47} -22514.1 q^{49} +48049.2 q^{53} -231947. i q^{55} -147032. i q^{59} -124334. q^{61} -505102. q^{65} -87535.6i q^{67} +225599. i q^{71} +475959. q^{73} +508511. q^{77} -276262. i q^{79} -485661. i q^{83} +1.13664e6 q^{85} +1.06400e6 q^{89} -1.10737e6i q^{91} -1.69520e6i q^{95} +1.46796e6 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 56 q^{5} - 8696 q^{13} + 5304 q^{17} + 36588 q^{25} - 77576 q^{29} + 125256 q^{37} + 209848 q^{41} + 95556 q^{49} + 68664 q^{53} + 238216 q^{61} - 613264 q^{65} + 219528 q^{73} + 1028224 q^{77} + 3416592 q^{85}+ \cdots + 2918344 q^{97}+O(q^{100}) \) 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
\(5\) 170.767 1.36614 0.683069 − 0.730353i \(-0.260645\pi\)
0.683069 + 0.730353i \(0.260645\pi\)
\(6\) 0 0
\(7\) 374.384i 1.09150i 0.837949 + 0.545749i \(0.183755\pi\)
−0.837949 + 0.545749i \(0.816245\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) − 1358.26i − 1.02048i −0.860032 − 0.510241i \(-0.829556\pi\)
0.860032 − 0.510241i \(-0.170444\pi\)
\(12\) 0 0
\(13\) −2957.84 −1.34631 −0.673154 − 0.739503i \(-0.735061\pi\)
−0.673154 + 0.739503i \(0.735061\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 6656.09 1.35479 0.677396 − 0.735619i \(-0.263109\pi\)
0.677396 + 0.735619i \(0.263109\pi\)
\(18\) 0 0
\(19\) − 9926.98i − 1.44729i −0.690171 − 0.723646i \(-0.742465\pi\)
0.690171 − 0.723646i \(-0.257535\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) − 15585.2i − 1.28094i −0.767983 − 0.640470i \(-0.778740\pi\)
0.767983 − 0.640470i \(-0.221260\pi\)
\(24\) 0 0
\(25\) 13536.5 0.866335
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 2083.13 0.0854125 0.0427063 − 0.999088i \(-0.486402\pi\)
0.0427063 + 0.999088i \(0.486402\pi\)
\(30\) 0 0
\(31\) − 28446.6i − 0.954873i −0.878666 − 0.477437i \(-0.841566\pi\)
0.878666 − 0.477437i \(-0.158434\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 63932.5i 1.49114i
\(36\) 0 0
\(37\) −3331.58 −0.0657727 −0.0328863 − 0.999459i \(-0.510470\pi\)
−0.0328863 + 0.999459i \(0.510470\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 63122.2 0.915863 0.457931 − 0.888988i \(-0.348591\pi\)
0.457931 + 0.888988i \(0.348591\pi\)
\(42\) 0 0
\(43\) 88501.2i 1.11313i 0.830806 + 0.556563i \(0.187880\pi\)
−0.830806 + 0.556563i \(0.812120\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 10825.8i 0.104271i 0.998640 + 0.0521357i \(0.0166028\pi\)
−0.998640 + 0.0521357i \(0.983397\pi\)
\(48\) 0 0
\(49\) −22514.1 −0.191367
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 48049.2 0.322744 0.161372 − 0.986894i \(-0.448408\pi\)
0.161372 + 0.986894i \(0.448408\pi\)
\(54\) 0 0
\(55\) − 231947.i − 1.39412i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) − 147032.i − 0.715906i −0.933740 − 0.357953i \(-0.883475\pi\)
0.933740 − 0.357953i \(-0.116525\pi\)
\(60\) 0 0
\(61\) −124334. −0.547773 −0.273887 − 0.961762i \(-0.588309\pi\)
−0.273887 + 0.961762i \(0.588309\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −505102. −1.83924
\(66\) 0 0
\(67\) − 87535.6i − 0.291045i −0.989355 − 0.145523i \(-0.953514\pi\)
0.989355 − 0.145523i \(-0.0464863\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 225599.i 0.630322i 0.949038 + 0.315161i \(0.102059\pi\)
−0.949038 + 0.315161i \(0.897941\pi\)
\(72\) 0 0
\(73\) 475959. 1.22349 0.611746 − 0.791054i \(-0.290468\pi\)
0.611746 + 0.791054i \(0.290468\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 508511. 1.11385
\(78\) 0 0
\(79\) − 276262.i − 0.560324i −0.959953 − 0.280162i \(-0.909612\pi\)
0.959953 − 0.280162i \(-0.0903882\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) − 485661.i − 0.849374i −0.905340 − 0.424687i \(-0.860384\pi\)
0.905340 − 0.424687i \(-0.139616\pi\)
\(84\) 0 0
\(85\) 1.13664e6 1.85083
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 1.06400e6 1.50929 0.754645 − 0.656134i \(-0.227809\pi\)
0.754645 + 0.656134i \(0.227809\pi\)
\(90\) 0 0
\(91\) − 1.10737e6i − 1.46949i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) − 1.69520e6i − 1.97720i
\(96\) 0 0
\(97\) 1.46796e6 1.60842 0.804209 − 0.594346i \(-0.202589\pi\)
0.804209 + 0.594346i \(0.202589\pi\)
\(98\) 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.127.4 4
3.2 odd 2 32.7.c.b.31.2 ✓ 4
4.3 odd 2 inner 288.7.g.b.127.3 4
8.3 odd 2 576.7.g.l.127.1 4
8.5 even 2 576.7.g.l.127.2 4
12.11 even 2 32.7.c.b.31.3 yes 4
24.5 odd 2 64.7.c.e.63.3 4
24.11 even 2 64.7.c.e.63.2 4
48.5 odd 4 256.7.d.g.127.1 4
48.11 even 4 256.7.d.d.127.3 4
48.29 odd 4 256.7.d.d.127.4 4
48.35 even 4 256.7.d.g.127.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
32.7.c.b.31.2 ✓ 4 3.2 odd 2
32.7.c.b.31.3 yes 4 12.11 even 2
64.7.c.e.63.2 4 24.11 even 2
64.7.c.e.63.3 4 24.5 odd 2
256.7.d.d.127.3 4 48.11 even 4
256.7.d.d.127.4 4 48.29 odd 4
256.7.d.g.127.1 4 48.5 odd 4
256.7.d.g.127.2 4 48.35 even 4
288.7.g.b.127.3 4 4.3 odd 2 inner
288.7.g.b.127.4 4 1.1 even 1 trivial
576.7.g.l.127.1 4 8.3 odd 2
576.7.g.l.127.2 4 8.5 even 2