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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1008,5,Mod(127,1008)] 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("1008.127"); S:= CuspForms(chi, 5); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1008, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 0, 0])) N = Newforms(chi, 5, names="a")
 
Level: \( N \) \(=\) \( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 1008.m (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,0,36,0,0,0,0,0,0,0,-236] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(104.196922789\)
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: \( 2^{18}\cdot 7^{2} \)
Twist minimal: no (minimal twist has level 112)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 127.4
Root \(-0.830211 + 1.43797i\) of defining polynomial
Character \(\chi\) \(=\) 1008.127
Dual form 1008.5.m.e.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)
 
\(f(q)\) \(=\) \(q+10.4721 q^{5} +18.5203i q^{7} +224.693i q^{11} -312.948 q^{13} +321.929 q^{17} +199.468i q^{19} +450.504i q^{23} -515.336 q^{25} +807.360 q^{29} -141.630i q^{31} +193.945i q^{35} -667.809 q^{37} -2679.08 q^{41} -3003.84i q^{43} -201.110i q^{47} -343.000 q^{49} -1514.86 q^{53} +2353.00i q^{55} -4972.94i q^{59} +5627.76 q^{61} -3277.21 q^{65} -5773.68i q^{67} -3854.50i q^{71} -2627.42 q^{73} -4161.37 q^{77} +7405.72i q^{79} +7216.05i q^{83} +3371.26 q^{85} +2842.38 q^{89} -5795.88i q^{91} +2088.84i q^{95} +2062.71 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 36 q^{5} - 236 q^{13} - 24 q^{17} + 624 q^{25} - 3864 q^{29} - 2408 q^{37} + 1464 q^{41} - 2744 q^{49} - 6960 q^{53} - 10836 q^{61} + 15240 q^{65} - 12352 q^{73} - 5880 q^{77} + 27544 q^{85} + 16272 q^{89}+ \cdots + 19576 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1008\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(577\) \(757\) \(785\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\)

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
\(5\) 10.4721 0.418882 0.209441 − 0.977821i \(-0.432836\pi\)
0.209441 + 0.977821i \(0.432836\pi\)
\(6\) 0 0
\(7\) 18.5203i 0.377964i
\(8\) 0 0
\(9\) 0 0
\(10\) 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 \(-0.876677\pi\)
−0.925883 + 0.377811i \(0.876677\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 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
\(27\) 0 0
\(28\) 0 0
\(29\) 807.360 0.960000 0.480000 − 0.877269i \(-0.340637\pi\)
0.480000 + 0.877269i \(0.340637\pi\)
\(30\) 0 0
\(31\) − 141.630i − 0.147378i −0.997281 − 0.0736891i \(-0.976523\pi\)
0.997281 − 0.0736891i \(-0.0234773\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 193.945i 0.158323i
\(36\) 0 0
\(37\) −667.809 −0.487808 −0.243904 − 0.969799i \(-0.578428\pi\)
−0.243904 + 0.969799i \(0.578428\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 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 \(-0.698222\pi\)
0.583257 − 0.812288i \(-0.301778\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 201.110i − 0.0910413i −0.998963 − 0.0455206i \(-0.985505\pi\)
0.998963 − 0.0455206i \(-0.0144947\pi\)
\(48\) 0 0
\(49\) −343.000 −0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −1514.86 −0.539289 −0.269644 − 0.962960i \(-0.586906\pi\)
−0.269644 + 0.962960i \(0.586906\pi\)
\(54\) 0 0
\(55\) 2353.00i 0.777850i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) − 4972.94i − 1.42860i −0.699841 − 0.714298i \(-0.746746\pi\)
0.699841 − 0.714298i \(-0.253254\pi\)
\(60\) 0 0
\(61\) 5627.76 1.51243 0.756217 − 0.654321i \(-0.227046\pi\)
0.756217 + 0.654321i \(0.227046\pi\)
\(62\) 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 \(-0.777651\pi\)
0.765789 − 0.643092i \(-0.222349\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 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 \(-0.579288\pi\)
−0.246521 + 0.969137i \(0.579288\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −4161.37 −0.701867
\(78\) 0 0
\(79\) 7405.72i 1.18662i 0.804973 + 0.593312i \(0.202180\pi\)
−0.804973 + 0.593312i \(0.797820\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 7216.05i 1.04747i 0.851880 + 0.523737i \(0.175463\pi\)
−0.851880 + 0.523737i \(0.824537\pi\)
\(84\) 0 0
\(85\) 3371.26 0.466611
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 2842.38 0.358841 0.179421 − 0.983772i \(-0.442578\pi\)
0.179421 + 0.983772i \(0.442578\pi\)
\(90\) 0 0
\(91\) − 5795.88i − 0.699902i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 2088.84i 0.231451i
\(96\) 0 0
\(97\) 2062.71 0.219227 0.109614 − 0.993974i \(-0.465039\pi\)
0.109614 + 0.993974i \(0.465039\pi\)
\(98\) 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 1008.5.m.e.127.4 8
3.2 odd 2 112.5.d.b.15.3 ✓ 8
4.3 odd 2 inner 1008.5.m.e.127.3 8
12.11 even 2 112.5.d.b.15.6 yes 8
21.20 even 2 784.5.d.j.687.6 8
24.5 odd 2 448.5.d.d.127.6 8
24.11 even 2 448.5.d.d.127.3 8
84.83 odd 2 784.5.d.j.687.3 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
112.5.d.b.15.3 ✓ 8 3.2 odd 2
112.5.d.b.15.6 yes 8 12.11 even 2
448.5.d.d.127.3 8 24.11 even 2
448.5.d.d.127.6 8 24.5 odd 2
784.5.d.j.687.3 8 84.83 odd 2
784.5.d.j.687.6 8 21.20 even 2
1008.5.m.e.127.3 8 4.3 odd 2 inner
1008.5.m.e.127.4 8 1.1 even 1 trivial