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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1248,4,Mod(337,1248)] 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("1248.337"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1248, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0, 1])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 1248 = 2^{5} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1248.m (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(73.6343836872\)
Analytic rank: \(0\)
Dimension: \(84\)
Twist minimal: no (minimal twist has level 312)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 337.60
Character \(\chi\) \(=\) 1248.337
Dual form 1248.4.m.a.337.58

$q$-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)
 
\(f(q)\) \(=\) \(q+3.00000i q^{3} +8.37088 q^{5} -16.5971i q^{7} -9.00000 q^{9} +23.2256 q^{11} +(-46.2169 - 7.80987i) q^{13} +25.1126i q^{15} -14.7565 q^{17} -124.395 q^{19} +49.7913 q^{21} +173.645 q^{23} -54.9283 q^{25} -27.0000i q^{27} +126.409i q^{29} +27.9669i q^{31} +69.6768i q^{33} -138.932i q^{35} -362.661 q^{37} +(23.4296 - 138.651i) q^{39} +309.872i q^{41} +556.370i q^{43} -75.3379 q^{45} +163.307i q^{47} +67.5362 q^{49} -44.2696i q^{51} +334.459i q^{53} +194.419 q^{55} -373.186i q^{57} +809.123 q^{59} +6.20734i q^{61} +149.374i q^{63} +(-386.877 - 65.3755i) q^{65} +252.494 q^{67} +520.936i q^{69} -822.887i q^{71} +449.440i q^{73} -164.785i q^{75} -385.478i q^{77} +547.122 q^{79} +81.0000 q^{81} +959.070 q^{83} -123.525 q^{85} -379.227 q^{87} +656.191i q^{89} +(-129.621 + 767.067i) q^{91} -83.9007 q^{93} -1041.30 q^{95} +417.791i q^{97} -209.030 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 84 q - 756 q^{9} - 104 q^{17} + 2188 q^{25} - 3396 q^{49} + 1616 q^{55} + 696 q^{65} - 3160 q^{79} + 6804 q^{81} + 2088 q^{87} - 2480 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1248\mathbb{Z}\right)^\times\).

\(n\) \(703\) \(769\) \(833\) \(1093\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(-1\)

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
\(3\) 3.00000i 0.577350i
\(4\) 0 0
\(5\) 8.37088 0.748714 0.374357 − 0.927285i \(-0.377863\pi\)
0.374357 + 0.927285i \(0.377863\pi\)
\(6\) 0 0
\(7\) − 16.5971i − 0.896159i −0.893994 − 0.448080i \(-0.852108\pi\)
0.893994 − 0.448080i \(-0.147892\pi\)
\(8\) 0 0
\(9\) −9.00000 −0.333333
\(10\) 0 0
\(11\) 23.2256 0.636616 0.318308 − 0.947987i \(-0.396885\pi\)
0.318308 + 0.947987i \(0.396885\pi\)
\(12\) 0 0
\(13\) −46.2169 − 7.80987i −0.986021 − 0.166621i
\(14\) 0 0
\(15\) 25.1126i 0.432270i
\(16\) 0 0
\(17\) −14.7565 −0.210529 −0.105264 − 0.994444i \(-0.533569\pi\)
−0.105264 + 0.994444i \(0.533569\pi\)
\(18\) 0 0
\(19\) −124.395 −1.50201 −0.751006 − 0.660295i \(-0.770431\pi\)
−0.751006 + 0.660295i \(0.770431\pi\)
\(20\) 0 0
\(21\) 49.7913 0.517398
\(22\) 0 0
\(23\) 173.645 1.57424 0.787121 − 0.616798i \(-0.211571\pi\)
0.787121 + 0.616798i \(0.211571\pi\)
\(24\) 0 0
\(25\) −54.9283 −0.439427
\(26\) 0 0
\(27\) − 27.0000i − 0.192450i
\(28\) 0 0
\(29\) 126.409i 0.809433i 0.914442 + 0.404716i \(0.132630\pi\)
−0.914442 + 0.404716i \(0.867370\pi\)
\(30\) 0 0
\(31\) 27.9669i 0.162032i 0.996713 + 0.0810162i \(0.0258165\pi\)
−0.996713 + 0.0810162i \(0.974183\pi\)
\(32\) 0 0
\(33\) 69.6768i 0.367551i
\(34\) 0 0
\(35\) − 138.932i − 0.670967i
\(36\) 0 0
\(37\) −362.661 −1.61138 −0.805692 − 0.592335i \(-0.798206\pi\)
−0.805692 + 0.592335i \(0.798206\pi\)
\(38\) 0 0
\(39\) 23.4296 − 138.651i 0.0961985 − 0.569280i
\(40\) 0 0
\(41\) 309.872i 1.18034i 0.807279 + 0.590170i \(0.200939\pi\)
−0.807279 + 0.590170i \(0.799061\pi\)
\(42\) 0 0
\(43\) 556.370i 1.97316i 0.163293 + 0.986578i \(0.447788\pi\)
−0.163293 + 0.986578i \(0.552212\pi\)
\(44\) 0 0
\(45\) −75.3379 −0.249571
\(46\) 0 0
\(47\) 163.307i 0.506825i 0.967358 + 0.253412i \(0.0815530\pi\)
−0.967358 + 0.253412i \(0.918447\pi\)
\(48\) 0 0
\(49\) 67.5362 0.196899
\(50\) 0 0
\(51\) − 44.2696i − 0.121549i
\(52\) 0 0
\(53\) 334.459i 0.866821i 0.901197 + 0.433411i \(0.142690\pi\)
−0.901197 + 0.433411i \(0.857310\pi\)
\(54\) 0 0
\(55\) 194.419 0.476644
\(56\) 0 0
\(57\) − 373.186i − 0.867187i
\(58\) 0 0
\(59\) 809.123 1.78540 0.892702 − 0.450647i \(-0.148807\pi\)
0.892702 + 0.450647i \(0.148807\pi\)
\(60\) 0 0
\(61\) 6.20734i 0.0130290i 0.999979 + 0.00651450i \(0.00207364\pi\)
−0.999979 + 0.00651450i \(0.997926\pi\)
\(62\) 0 0
\(63\) 149.374i 0.298720i
\(64\) 0 0
\(65\) −386.877 − 65.3755i −0.738248 − 0.124751i
\(66\) 0 0
\(67\) 252.494 0.460403 0.230202 − 0.973143i \(-0.426061\pi\)
0.230202 + 0.973143i \(0.426061\pi\)
\(68\) 0 0
\(69\) 520.936i 0.908889i
\(70\) 0 0
\(71\) − 822.887i − 1.37548i −0.725959 − 0.687738i \(-0.758604\pi\)
0.725959 − 0.687738i \(-0.241396\pi\)
\(72\) 0 0
\(73\) 449.440i 0.720589i 0.932839 + 0.360294i \(0.117324\pi\)
−0.932839 + 0.360294i \(0.882676\pi\)
\(74\) 0 0
\(75\) − 164.785i − 0.253703i
\(76\) 0 0
\(77\) − 385.478i − 0.570510i
\(78\) 0 0
\(79\) 547.122 0.779191 0.389595 − 0.920986i \(-0.372615\pi\)
0.389595 + 0.920986i \(0.372615\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) 959.070 1.26833 0.634166 − 0.773197i \(-0.281344\pi\)
0.634166 + 0.773197i \(0.281344\pi\)
\(84\) 0 0
\(85\) −123.525 −0.157626
\(86\) 0 0
\(87\) −379.227 −0.467326
\(88\) 0 0
\(89\) 656.191i 0.781529i 0.920491 + 0.390765i \(0.127789\pi\)
−0.920491 + 0.390765i \(0.872211\pi\)
\(90\) 0 0
\(91\) −129.621 + 767.067i −0.149319 + 0.883632i
\(92\) 0 0
\(93\) −83.9007 −0.0935494
\(94\) 0 0
\(95\) −1041.30 −1.12458
\(96\) 0 0
\(97\) 417.791i 0.437322i 0.975801 + 0.218661i \(0.0701689\pi\)
−0.975801 + 0.218661i \(0.929831\pi\)
\(98\) 0 0
\(99\) −209.030 −0.212205
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 1248.4.m.a.337.60 84
4.3 odd 2 312.4.m.a.181.12 yes 84
8.3 odd 2 312.4.m.a.181.74 yes 84
8.5 even 2 inner 1248.4.m.a.337.57 84
13.12 even 2 inner 1248.4.m.a.337.59 84
52.51 odd 2 312.4.m.a.181.73 yes 84
104.51 odd 2 312.4.m.a.181.11 ✓ 84
104.77 even 2 inner 1248.4.m.a.337.58 84
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
312.4.m.a.181.11 ✓ 84 104.51 odd 2
312.4.m.a.181.12 yes 84 4.3 odd 2
312.4.m.a.181.73 yes 84 52.51 odd 2
312.4.m.a.181.74 yes 84 8.3 odd 2
1248.4.m.a.337.57 84 8.5 even 2 inner
1248.4.m.a.337.58 84 104.77 even 2 inner
1248.4.m.a.337.59 84 13.12 even 2 inner
1248.4.m.a.337.60 84 1.1 even 1 trivial