Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [448,8,Mod(1,448)] 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("448.1"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(448, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 8, names="a")
 
Level: \( N \) \(=\) \( 448 = 2^{6} \cdot 7 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 448.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,14,0,-42,0,686] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(139.948491417\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{3529}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 882 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 28)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(30.2027\) of defining polynomial
Character \(\chi\) \(=\) 448.1

$q$-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)
 
\(f(q)\) \(=\) \(q-52.4054 q^{3} -199.216 q^{5} +343.000 q^{7} +559.325 q^{9} -1218.97 q^{11} -6449.65 q^{13} +10440.0 q^{15} +26786.9 q^{17} -14910.9 q^{19} -17975.0 q^{21} -73500.9 q^{23} -38437.9 q^{25} +85299.0 q^{27} +106453. q^{29} +66816.7 q^{31} +63880.8 q^{33} -68331.1 q^{35} +479502. q^{37} +337996. q^{39} -644539. q^{41} -145165. q^{43} -111426. q^{45} +1.01085e6 q^{47} +117649. q^{49} -1.40378e6 q^{51} -34208.5 q^{53} +242839. q^{55} +781414. q^{57} +443317. q^{59} +1.14043e6 q^{61} +191848. q^{63} +1.28487e6 q^{65} +4.31928e6 q^{67} +3.85185e6 q^{69} +2.54867e6 q^{71} -3.67723e6 q^{73} +2.01435e6 q^{75} -418108. q^{77} +8.55564e6 q^{79} -5.69337e6 q^{81} +1.79721e6 q^{83} -5.33639e6 q^{85} -5.57873e6 q^{87} +5.56317e6 q^{89} -2.21223e6 q^{91} -3.50155e6 q^{93} +2.97050e6 q^{95} -1.72057e7 q^{97} -681802. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 14 q^{3} - 42 q^{5} + 686 q^{7} + 2782 q^{9} - 7428 q^{11} - 11830 q^{13} + 20880 q^{15} + 15792 q^{17} - 26614 q^{19} + 4802 q^{21} + 32640 q^{23} - 91846 q^{25} + 87668 q^{27} + 158016 q^{29} - 180740 q^{31}+ \cdots - 14482452 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
\(3\) −52.4054 −1.12060 −0.560301 − 0.828289i \(-0.689315\pi\)
−0.560301 + 0.828289i \(0.689315\pi\)
\(4\) 0 0
\(5\) −199.216 −0.712737 −0.356369 − 0.934345i \(-0.615985\pi\)
−0.356369 + 0.934345i \(0.615985\pi\)
\(6\) 0 0
\(7\) 343.000 0.377964
\(8\) 0 0
\(9\) 559.325 0.255750
\(10\) 0 0
\(11\) −1218.97 −0.276134 −0.138067 − 0.990423i \(-0.544089\pi\)
−0.138067 + 0.990423i \(0.544089\pi\)
\(12\) 0 0
\(13\) −6449.65 −0.814206 −0.407103 − 0.913382i \(-0.633461\pi\)
−0.407103 + 0.913382i \(0.633461\pi\)
\(14\) 0 0
\(15\) 10440.0 0.798695
\(16\) 0 0
\(17\) 26786.9 1.32237 0.661183 − 0.750225i \(-0.270055\pi\)
0.661183 + 0.750225i \(0.270055\pi\)
\(18\) 0 0
\(19\) −14910.9 −0.498732 −0.249366 − 0.968409i \(-0.580222\pi\)
−0.249366 + 0.968409i \(0.580222\pi\)
\(20\) 0 0
\(21\) −17975.0 −0.423548
\(22\) 0 0
\(23\) −73500.9 −1.25964 −0.629819 − 0.776742i \(-0.716871\pi\)
−0.629819 + 0.776742i \(0.716871\pi\)
\(24\) 0 0
\(25\) −38437.9 −0.492005
\(26\) 0 0
\(27\) 85299.0 0.834009
\(28\) 0 0
\(29\) 106453. 0.810524 0.405262 − 0.914200i \(-0.367180\pi\)
0.405262 + 0.914200i \(0.367180\pi\)
\(30\) 0 0
\(31\) 66816.7 0.402827 0.201414 − 0.979506i \(-0.435446\pi\)
0.201414 + 0.979506i \(0.435446\pi\)
\(32\) 0 0
\(33\) 63880.8 0.309436
\(34\) 0 0
\(35\) −68331.1 −0.269389
\(36\) 0 0
\(37\) 479502. 1.55627 0.778134 − 0.628099i \(-0.216167\pi\)
0.778134 + 0.628099i \(0.216167\pi\)
\(38\) 0 0
\(39\) 337996. 0.912401
\(40\) 0 0
\(41\) −644539. −1.46051 −0.730257 − 0.683173i \(-0.760599\pi\)
−0.730257 + 0.683173i \(0.760599\pi\)
\(42\) 0 0
\(43\) −145165. −0.278434 −0.139217 − 0.990262i \(-0.544459\pi\)
−0.139217 + 0.990262i \(0.544459\pi\)
\(44\) 0 0
\(45\) −111426. −0.182282
\(46\) 0 0
\(47\) 1.01085e6 1.42018 0.710088 − 0.704113i \(-0.248655\pi\)
0.710088 + 0.704113i \(0.248655\pi\)
\(48\) 0 0
\(49\) 117649. 0.142857
\(50\) 0 0
\(51\) −1.40378e6 −1.48185
\(52\) 0 0
\(53\) −34208.5 −0.0315623 −0.0157812 − 0.999875i \(-0.505024\pi\)
−0.0157812 + 0.999875i \(0.505024\pi\)
\(54\) 0 0
\(55\) 242839. 0.196811
\(56\) 0 0
\(57\) 781414. 0.558881
\(58\) 0 0
\(59\) 443317. 0.281017 0.140508 − 0.990079i \(-0.455126\pi\)
0.140508 + 0.990079i \(0.455126\pi\)
\(60\) 0 0
\(61\) 1.14043e6 0.643300 0.321650 − 0.946859i \(-0.395763\pi\)
0.321650 + 0.946859i \(0.395763\pi\)
\(62\) 0 0
\(63\) 191848. 0.0966643
\(64\) 0 0
\(65\) 1.28487e6 0.580315
\(66\) 0 0
\(67\) 4.31928e6 1.75448 0.877242 − 0.480048i \(-0.159381\pi\)
0.877242 + 0.480048i \(0.159381\pi\)
\(68\) 0 0
\(69\) 3.85185e6 1.41155
\(70\) 0 0
\(71\) 2.54867e6 0.845103 0.422551 − 0.906339i \(-0.361135\pi\)
0.422551 + 0.906339i \(0.361135\pi\)
\(72\) 0 0
\(73\) −3.67723e6 −1.10635 −0.553173 − 0.833067i \(-0.686583\pi\)
−0.553173 + 0.833067i \(0.686583\pi\)
\(74\) 0 0
\(75\) 2.01435e6 0.551342
\(76\) 0 0
\(77\) −418108. −0.104369
\(78\) 0 0
\(79\) 8.55564e6 1.95235 0.976174 − 0.216987i \(-0.0696230\pi\)
0.976174 + 0.216987i \(0.0696230\pi\)
\(80\) 0 0
\(81\) −5.69337e6 −1.19034
\(82\) 0 0
\(83\) 1.79721e6 0.345006 0.172503 − 0.985009i \(-0.444815\pi\)
0.172503 + 0.985009i \(0.444815\pi\)
\(84\) 0 0
\(85\) −5.33639e6 −0.942499
\(86\) 0 0
\(87\) −5.57873e6 −0.908276
\(88\) 0 0
\(89\) 5.56317e6 0.836484 0.418242 − 0.908336i \(-0.362646\pi\)
0.418242 + 0.908336i \(0.362646\pi\)
\(90\) 0 0
\(91\) −2.21223e6 −0.307741
\(92\) 0 0
\(93\) −3.50155e6 −0.451409
\(94\) 0 0
\(95\) 2.97050e6 0.355465
\(96\) 0 0
\(97\) −1.72057e7 −1.91413 −0.957064 − 0.289877i \(-0.906386\pi\)
−0.957064 + 0.289877i \(0.906386\pi\)
\(98\) 0 0
\(99\) −681802. −0.0706212
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 448.8.a.p.1.1 2
4.3 odd 2 448.8.a.n.1.2 2
8.3 odd 2 112.8.a.i.1.1 2
8.5 even 2 28.8.a.a.1.2 ✓ 2
24.5 odd 2 252.8.a.e.1.1 2
56.5 odd 6 196.8.e.a.165.2 4
56.13 odd 2 196.8.a.b.1.1 2
56.37 even 6 196.8.e.d.165.1 4
56.45 odd 6 196.8.e.a.177.2 4
56.53 even 6 196.8.e.d.177.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
28.8.a.a.1.2 ✓ 2 8.5 even 2
112.8.a.i.1.1 2 8.3 odd 2
196.8.a.b.1.1 2 56.13 odd 2
196.8.e.a.165.2 4 56.5 odd 6
196.8.e.a.177.2 4 56.45 odd 6
196.8.e.d.165.1 4 56.37 even 6
196.8.e.d.177.1 4 56.53 even 6
252.8.a.e.1.1 2 24.5 odd 2
448.8.a.n.1.2 2 4.3 odd 2
448.8.a.p.1.1 2 1.1 even 1 trivial