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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 687.1
Root \(1.33021 + 2.30399i\) of defining polynomial
Character \(\chi\) \(=\) 784.687
Dual form 784.5.d.j.687.8

$q$-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)
 
\(f(q)\) \(=\) \(q-15.1109i q^{3} +12.2757 q^{5} -147.339 q^{9} +114.144i q^{11} +7.25844 q^{13} -185.497i q^{15} +194.484 q^{17} +699.118i q^{19} -275.146i q^{23} -474.308 q^{25} +1002.45i q^{27} +1030.98 q^{29} -1002.17i q^{31} +1724.81 q^{33} -859.871 q^{37} -109.682i q^{39} +3292.53 q^{41} +2647.21i q^{43} -1808.69 q^{45} -2564.73i q^{47} -2938.83i q^{51} +4844.37 q^{53} +1401.19i q^{55} +10564.3 q^{57} -2811.16i q^{59} +5582.64 q^{61} +89.1023 q^{65} -2393.56i q^{67} -4157.71 q^{69} +4154.55i q^{71} -3737.06 q^{73} +7167.22i q^{75} +10088.0i q^{79} +3213.41 q^{81} +1257.71i q^{83} +2387.42 q^{85} -15579.1i q^{87} -5153.33 q^{89} -15143.7 q^{93} +8582.15i q^{95} +8542.66 q^{97} -16817.8i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 36 q^{5} + 80 q^{9} + 236 q^{13} - 24 q^{17} + 624 q^{25} + 3864 q^{29} + 4160 q^{33} - 2408 q^{37} + 1464 q^{41} + 116 q^{45} + 6960 q^{53} + 23480 q^{57} + 10836 q^{61} - 15240 q^{65} + 8336 q^{69}+ \cdots - 19576 q^{97}+O(q^{100}) \) 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
\(3\) − 15.1109i − 1.67899i −0.543368 − 0.839495i \(-0.682851\pi\)
0.543368 − 0.839495i \(-0.317149\pi\)
\(4\) 0 0
\(5\) 12.2757 0.491027 0.245514 − 0.969393i \(-0.421043\pi\)
0.245514 + 0.969393i \(0.421043\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) −147.339 −1.81901
\(10\) 0 0
\(11\) 114.144i 0.943335i 0.881777 + 0.471667i \(0.156348\pi\)
−0.881777 + 0.471667i \(0.843652\pi\)
\(12\) 0 0
\(13\) 7.25844 0.0429494 0.0214747 − 0.999769i \(-0.493164\pi\)
0.0214747 + 0.999769i \(0.493164\pi\)
\(14\) 0 0
\(15\) − 185.497i − 0.824429i
\(16\) 0 0
\(17\) 194.484 0.672956 0.336478 − 0.941691i \(-0.390764\pi\)
0.336478 + 0.941691i \(0.390764\pi\)
\(18\) 0 0
\(19\) 699.118i 1.93662i 0.249760 + 0.968308i \(0.419648\pi\)
−0.249760 + 0.968308i \(0.580352\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) − 275.146i − 0.520125i −0.965592 − 0.260063i \(-0.916257\pi\)
0.965592 − 0.260063i \(-0.0837432\pi\)
\(24\) 0 0
\(25\) −474.308 −0.758892
\(26\) 0 0
\(27\) 1002.45i 1.37510i
\(28\) 0 0
\(29\) 1030.98 1.22590 0.612950 − 0.790121i \(-0.289983\pi\)
0.612950 + 0.790121i \(0.289983\pi\)
\(30\) 0 0
\(31\) − 1002.17i − 1.04284i −0.853299 − 0.521422i \(-0.825402\pi\)
0.853299 − 0.521422i \(-0.174598\pi\)
\(32\) 0 0
\(33\) 1724.81 1.58385
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −859.871 −0.628101 −0.314051 − 0.949406i \(-0.601686\pi\)
−0.314051 + 0.949406i \(0.601686\pi\)
\(38\) 0 0
\(39\) − 109.682i − 0.0721115i
\(40\) 0 0
\(41\) 3292.53 1.95868 0.979338 − 0.202230i \(-0.0648189\pi\)
0.979338 + 0.202230i \(0.0648189\pi\)
\(42\) 0 0
\(43\) 2647.21i 1.43170i 0.698254 + 0.715850i \(0.253960\pi\)
−0.698254 + 0.715850i \(0.746040\pi\)
\(44\) 0 0
\(45\) −1808.69 −0.893181
\(46\) 0 0
\(47\) − 2564.73i − 1.16103i −0.814248 − 0.580517i \(-0.802850\pi\)
0.814248 − 0.580517i \(-0.197150\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) − 2938.83i − 1.12989i
\(52\) 0 0
\(53\) 4844.37 1.72459 0.862295 − 0.506406i \(-0.169026\pi\)
0.862295 + 0.506406i \(0.169026\pi\)
\(54\) 0 0
\(55\) 1401.19i 0.463203i
\(56\) 0 0
\(57\) 10564.3 3.25156
\(58\) 0 0
\(59\) − 2811.16i − 0.807571i −0.914854 − 0.403786i \(-0.867694\pi\)
0.914854 − 0.403786i \(-0.132306\pi\)
\(60\) 0 0
\(61\) 5582.64 1.50031 0.750153 − 0.661265i \(-0.229980\pi\)
0.750153 + 0.661265i \(0.229980\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 89.1023 0.0210893
\(66\) 0 0
\(67\) − 2393.56i − 0.533206i −0.963806 − 0.266603i \(-0.914099\pi\)
0.963806 − 0.266603i \(-0.0859012\pi\)
\(68\) 0 0
\(69\) −4157.71 −0.873285
\(70\) 0 0
\(71\) 4154.55i 0.824151i 0.911150 + 0.412076i \(0.135196\pi\)
−0.911150 + 0.412076i \(0.864804\pi\)
\(72\) 0 0
\(73\) −3737.06 −0.701269 −0.350635 − 0.936512i \(-0.614034\pi\)
−0.350635 + 0.936512i \(0.614034\pi\)
\(74\) 0 0
\(75\) 7167.22i 1.27417i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 10088.0i 1.61640i 0.588905 + 0.808202i \(0.299559\pi\)
−0.588905 + 0.808202i \(0.700441\pi\)
\(80\) 0 0
\(81\) 3213.41 0.489774
\(82\) 0 0
\(83\) 1257.71i 0.182569i 0.995825 + 0.0912843i \(0.0290972\pi\)
−0.995825 + 0.0912843i \(0.970903\pi\)
\(84\) 0 0
\(85\) 2387.42 0.330439
\(86\) 0 0
\(87\) − 15579.1i − 2.05827i
\(88\) 0 0
\(89\) −5153.33 −0.650591 −0.325295 − 0.945612i \(-0.605464\pi\)
−0.325295 + 0.945612i \(0.605464\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −15143.7 −1.75092
\(94\) 0 0
\(95\) 8582.15i 0.950931i
\(96\) 0 0
\(97\) 8542.66 0.907925 0.453962 − 0.891021i \(-0.350010\pi\)
0.453962 + 0.891021i \(0.350010\pi\)
\(98\) 0 0
\(99\) − 16817.8i − 1.71593i
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.687.1 8
4.3 odd 2 inner 784.5.d.j.687.8 8
7.6 odd 2 112.5.d.b.15.8 yes 8
21.20 even 2 1008.5.m.e.127.6 8
28.27 even 2 112.5.d.b.15.1 ✓ 8
56.13 odd 2 448.5.d.d.127.1 8
56.27 even 2 448.5.d.d.127.8 8
84.83 odd 2 1008.5.m.e.127.5 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
112.5.d.b.15.1 ✓ 8 28.27 even 2
112.5.d.b.15.8 yes 8 7.6 odd 2
448.5.d.d.127.1 8 56.13 odd 2
448.5.d.d.127.8 8 56.27 even 2
784.5.d.j.687.1 8 1.1 even 1 trivial
784.5.d.j.687.8 8 4.3 odd 2 inner
1008.5.m.e.127.5 8 84.83 odd 2
1008.5.m.e.127.6 8 21.20 even 2