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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [784,5,Mod(687,784)] 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("784.687"); S:= CuspForms(chi, 5); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(784, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 0])) N = Newforms(chi, 5, names="a")
 
Level: \( N \) \(=\) \( 784 = 2^{4} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 784.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,0,36,0,0,0,80,0,0,0,236,0,0,0,-24,0,0,0,0,0,0,0,624] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(25)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(81.0420510577\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.207528535809.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} + 21x^{6} - 2x^{5} + 265x^{4} - 66x^{3} + 1344x^{2} + 1080x + 3600 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{22}\cdot 7^{2} \)
Twist minimal: no (minimal twist has level 112)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 687.3
Root \(-0.830211 + 1.43797i\) of defining polynomial
Character \(\chi\) \(=\) 784.687
Dual form 784.5.d.j.687.6

$q$-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)
 
\(f(q)\) \(=\) \(q-3.80490i q^{3} +10.4721 q^{5} +66.5227 q^{9} +224.693i q^{11} +312.948 q^{13} -39.8451i q^{15} +321.929 q^{17} +199.468i q^{19} +450.504i q^{23} -515.336 q^{25} -561.309i q^{27} -807.360 q^{29} -141.630i q^{31} +854.934 q^{33} -667.809 q^{37} -1190.74i q^{39} -2679.08 q^{41} +3003.84i q^{43} +696.630 q^{45} +201.110i q^{47} -1224.91i q^{51} +1514.86 q^{53} +2353.00i q^{55} +758.957 q^{57} +4972.94i q^{59} -5627.76 q^{61} +3277.21 q^{65} +5773.68i q^{67} +1714.12 q^{69} -3854.50i q^{71} +2627.42 q^{73} +1960.80i q^{75} -7405.72i q^{79} +3252.62 q^{81} -7216.05i q^{83} +3371.26 q^{85} +3071.92i q^{87} +2842.38 q^{89} -538.890 q^{93} +2088.84i q^{95} -2062.71 q^{97} +14947.2i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 36 q^{5} + 80 q^{9} + 236 q^{13} - 24 q^{17} + 624 q^{25} + 3864 q^{29} + 4160 q^{33} - 2408 q^{37} + 1464 q^{41} + 116 q^{45} + 6960 q^{53} + 23480 q^{57} + 10836 q^{61} - 15240 q^{65} + 8336 q^{69}+ \cdots - 19576 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/784\mathbb{Z}\right)^\times\).

\(n\) \(197\) \(687\) \(689\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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
\(3\) − 3.80490i − 0.422766i −0.977403 − 0.211383i \(-0.932203\pi\)
0.977403 − 0.211383i \(-0.0677968\pi\)
\(4\) 0 0
\(5\) 10.4721 0.418882 0.209441 − 0.977821i \(-0.432836\pi\)
0.209441 + 0.977821i \(0.432836\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 66.5227 0.821268
\(10\) 0 0
\(11\) 224.693i 1.85697i 0.371374 + 0.928483i \(0.378887\pi\)
−0.371374 + 0.928483i \(0.621113\pi\)
\(12\) 0 0
\(13\) 312.948 1.85177 0.925883 − 0.377811i \(-0.123323\pi\)
0.925883 + 0.377811i \(0.123323\pi\)
\(14\) 0 0
\(15\) − 39.8451i − 0.177089i
\(16\) 0 0
\(17\) 321.929 1.11394 0.556971 − 0.830532i \(-0.311963\pi\)
0.556971 + 0.830532i \(0.311963\pi\)
\(18\) 0 0
\(19\) 199.468i 0.552544i 0.961079 + 0.276272i \(0.0890991\pi\)
−0.961079 + 0.276272i \(0.910901\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 450.504i 0.851615i 0.904814 + 0.425807i \(0.140010\pi\)
−0.904814 + 0.425807i \(0.859990\pi\)
\(24\) 0 0
\(25\) −515.336 −0.824538
\(26\) 0 0
\(27\) − 561.309i − 0.769971i
\(28\) 0 0
\(29\) −807.360 −0.960000 −0.480000 − 0.877269i \(-0.659363\pi\)
−0.480000 + 0.877269i \(0.659363\pi\)
\(30\) 0 0
\(31\) − 141.630i − 0.147378i −0.997281 − 0.0736891i \(-0.976523\pi\)
0.997281 − 0.0736891i \(-0.0234773\pi\)
\(32\) 0 0
\(33\) 854.934 0.785063
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −667.809 −0.487808 −0.243904 − 0.969799i \(-0.578428\pi\)
−0.243904 + 0.969799i \(0.578428\pi\)
\(38\) 0 0
\(39\) − 1190.74i − 0.782864i
\(40\) 0 0
\(41\) −2679.08 −1.59374 −0.796870 − 0.604151i \(-0.793512\pi\)
−0.796870 + 0.604151i \(0.793512\pi\)
\(42\) 0 0
\(43\) 3003.84i 1.62458i 0.583257 + 0.812288i \(0.301778\pi\)
−0.583257 + 0.812288i \(0.698222\pi\)
\(44\) 0 0
\(45\) 696.630 0.344015
\(46\) 0 0
\(47\) 201.110i 0.0910413i 0.998963 + 0.0455206i \(0.0144947\pi\)
−0.998963 + 0.0455206i \(0.985505\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) − 1224.91i − 0.470938i
\(52\) 0 0
\(53\) 1514.86 0.539289 0.269644 − 0.962960i \(-0.413094\pi\)
0.269644 + 0.962960i \(0.413094\pi\)
\(54\) 0 0
\(55\) 2353.00i 0.777850i
\(56\) 0 0
\(57\) 758.957 0.233597
\(58\) 0 0
\(59\) 4972.94i 1.42860i 0.699841 + 0.714298i \(0.253254\pi\)
−0.699841 + 0.714298i \(0.746746\pi\)
\(60\) 0 0
\(61\) −5627.76 −1.51243 −0.756217 − 0.654321i \(-0.772954\pi\)
−0.756217 + 0.654321i \(0.772954\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 3277.21 0.775671
\(66\) 0 0
\(67\) 5773.68i 1.28618i 0.765789 + 0.643092i \(0.222349\pi\)
−0.765789 + 0.643092i \(0.777651\pi\)
\(68\) 0 0
\(69\) 1714.12 0.360034
\(70\) 0 0
\(71\) − 3854.50i − 0.764631i −0.924032 − 0.382315i \(-0.875127\pi\)
0.924032 − 0.382315i \(-0.124873\pi\)
\(72\) 0 0
\(73\) 2627.42 0.493042 0.246521 − 0.969137i \(-0.420712\pi\)
0.246521 + 0.969137i \(0.420712\pi\)
\(74\) 0 0
\(75\) 1960.80i 0.348587i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) − 7405.72i − 1.18662i −0.804973 − 0.593312i \(-0.797820\pi\)
0.804973 − 0.593312i \(-0.202180\pi\)
\(80\) 0 0
\(81\) 3252.62 0.495750
\(82\) 0 0
\(83\) − 7216.05i − 1.04747i −0.851880 − 0.523737i \(-0.824537\pi\)
0.851880 − 0.523737i \(-0.175463\pi\)
\(84\) 0 0
\(85\) 3371.26 0.466611
\(86\) 0 0
\(87\) 3071.92i 0.405856i
\(88\) 0 0
\(89\) 2842.38 0.358841 0.179421 − 0.983772i \(-0.442578\pi\)
0.179421 + 0.983772i \(0.442578\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −538.890 −0.0623066
\(94\) 0 0
\(95\) 2088.84i 0.231451i
\(96\) 0 0
\(97\) −2062.71 −0.219227 −0.109614 − 0.993974i \(-0.534961\pi\)
−0.109614 + 0.993974i \(0.534961\pi\)
\(98\) 0 0
\(99\) 14947.2i 1.52507i
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 784.5.d.j.687.3 8
4.3 odd 2 inner 784.5.d.j.687.6 8
7.6 odd 2 112.5.d.b.15.6 yes 8
21.20 even 2 1008.5.m.e.127.3 8
28.27 even 2 112.5.d.b.15.3 ✓ 8
56.13 odd 2 448.5.d.d.127.3 8
56.27 even 2 448.5.d.d.127.6 8
84.83 odd 2 1008.5.m.e.127.4 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
112.5.d.b.15.3 ✓ 8 28.27 even 2
112.5.d.b.15.6 yes 8 7.6 odd 2
448.5.d.d.127.3 8 56.13 odd 2
448.5.d.d.127.6 8 56.27 even 2
784.5.d.j.687.3 8 1.1 even 1 trivial
784.5.d.j.687.6 8 4.3 odd 2 inner
1008.5.m.e.127.3 8 21.20 even 2
1008.5.m.e.127.4 8 84.83 odd 2