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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [3328,2,Mod(1665,3328)] 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("3328.1665"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3328, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 3328 = 2^{8} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3328.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [10,0,0,0,0,0,-10,0,-16,0,0,0,0,0,18,0,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(26.5742137927\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: 10.0.89857052655616.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} + 16x^{8} + 88x^{6} + 185x^{4} + 100x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{5} \)
Twist minimal: no (minimal twist has level 1664)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1665.5
Root \(2.19441i\) of defining polynomial
Character \(\chi\) \(=\) 3328.1665
Dual form 3328.2.b.bc.1665.6

$q$-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)
 
\(f(q)\) \(=\) \(q-0.378965i q^{3} +4.27753i q^{5} -4.43100 q^{7} +2.85639 q^{9} -2.38883i q^{11} +1.00000i q^{13} +1.62103 q^{15} -3.51960 q^{17} -7.74509i q^{19} +1.67920i q^{21} -1.35626 q^{23} -13.2972 q^{25} -2.21937i q^{27} +0.643737i q^{29} +7.63090 q^{31} -0.905282 q^{33} -18.9537i q^{35} -5.09845i q^{37} +0.378965 q^{39} +5.19879 q^{41} +1.19989i q^{43} +12.2183i q^{45} -2.09422 q^{47} +12.6338 q^{49} +1.33381i q^{51} -9.19879i q^{53} +10.2183 q^{55} -2.93512 q^{57} -2.38883i q^{59} -3.68293i q^{61} -12.6567 q^{63} -4.27753 q^{65} +9.36503i q^{67} +0.513976i q^{69} -9.96659 q^{71} -6.26367 q^{73} +5.03920i q^{75} +10.5849i q^{77} -0.777651 q^{79} +7.72809 q^{81} +7.76481i q^{83} -15.0552i q^{85} +0.243954 q^{87} +7.61994 q^{89} -4.43100i q^{91} -2.89184i q^{93} +33.1298 q^{95} +8.97620 q^{97} -6.82341i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 10 q^{7} - 16 q^{9} + 18 q^{15} + 6 q^{17} + 4 q^{23} - 20 q^{25} + 44 q^{31} - 16 q^{33} + 2 q^{39} - 20 q^{41} + 10 q^{47} + 64 q^{49} + 16 q^{55} - 12 q^{57} + 88 q^{63} + 2 q^{65} + 10 q^{71}+ \cdots + 84 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3328\mathbb{Z}\right)^\times\).

\(n\) \(261\) \(769\) \(1535\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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
\(3\) − 0.378965i − 0.218796i −0.993998 − 0.109398i \(-0.965108\pi\)
0.993998 − 0.109398i \(-0.0348923\pi\)
\(4\) 0 0
\(5\) 4.27753i 1.91297i 0.291783 + 0.956484i \(0.405751\pi\)
−0.291783 + 0.956484i \(0.594249\pi\)
\(6\) 0 0
\(7\) −4.43100 −1.67476 −0.837381 − 0.546620i \(-0.815914\pi\)
−0.837381 + 0.546620i \(0.815914\pi\)
\(8\) 0 0
\(9\) 2.85639 0.952128
\(10\) 0 0
\(11\) − 2.38883i − 0.720258i −0.932903 − 0.360129i \(-0.882733\pi\)
0.932903 − 0.360129i \(-0.117267\pi\)
\(12\) 0 0
\(13\) 1.00000i 0.277350i
\(14\) 0 0
\(15\) 1.62103 0.418549
\(16\) 0 0
\(17\) −3.51960 −0.853628 −0.426814 − 0.904339i \(-0.640364\pi\)
−0.426814 + 0.904339i \(0.640364\pi\)
\(18\) 0 0
\(19\) − 7.74509i − 1.77685i −0.459027 − 0.888423i \(-0.651802\pi\)
0.459027 − 0.888423i \(-0.348198\pi\)
\(20\) 0 0
\(21\) 1.67920i 0.366431i
\(22\) 0 0
\(23\) −1.35626 −0.282800 −0.141400 − 0.989953i \(-0.545160\pi\)
−0.141400 + 0.989953i \(0.545160\pi\)
\(24\) 0 0
\(25\) −13.2972 −2.65945
\(26\) 0 0
\(27\) − 2.21937i − 0.427117i
\(28\) 0 0
\(29\) 0.643737i 0.119539i 0.998212 + 0.0597695i \(0.0190366\pi\)
−0.998212 + 0.0597695i \(0.980963\pi\)
\(30\) 0 0
\(31\) 7.63090 1.37055 0.685275 − 0.728285i \(-0.259682\pi\)
0.685275 + 0.728285i \(0.259682\pi\)
\(32\) 0 0
\(33\) −0.905282 −0.157589
\(34\) 0 0
\(35\) − 18.9537i − 3.20377i
\(36\) 0 0
\(37\) − 5.09845i − 0.838181i −0.907945 − 0.419090i \(-0.862349\pi\)
0.907945 − 0.419090i \(-0.137651\pi\)
\(38\) 0 0
\(39\) 0.378965 0.0606830
\(40\) 0 0
\(41\) 5.19879 0.811915 0.405958 − 0.913892i \(-0.366938\pi\)
0.405958 + 0.913892i \(0.366938\pi\)
\(42\) 0 0
\(43\) 1.19989i 0.182982i 0.995806 + 0.0914909i \(0.0291632\pi\)
−0.995806 + 0.0914909i \(0.970837\pi\)
\(44\) 0 0
\(45\) 12.2183i 1.82139i
\(46\) 0 0
\(47\) −2.09422 −0.305473 −0.152736 − 0.988267i \(-0.548809\pi\)
−0.152736 + 0.988267i \(0.548809\pi\)
\(48\) 0 0
\(49\) 12.6338 1.80483
\(50\) 0 0
\(51\) 1.33381i 0.186770i
\(52\) 0 0
\(53\) − 9.19879i − 1.26355i −0.775151 − 0.631776i \(-0.782326\pi\)
0.775151 − 0.631776i \(-0.217674\pi\)
\(54\) 0 0
\(55\) 10.2183 1.37783
\(56\) 0 0
\(57\) −2.93512 −0.388766
\(58\) 0 0
\(59\) − 2.38883i − 0.310999i −0.987836 − 0.155499i \(-0.950301\pi\)
0.987836 − 0.155499i \(-0.0496986\pi\)
\(60\) 0 0
\(61\) − 3.68293i − 0.471551i −0.971808 − 0.235776i \(-0.924237\pi\)
0.971808 − 0.235776i \(-0.0757630\pi\)
\(62\) 0 0
\(63\) −12.6567 −1.59459
\(64\) 0 0
\(65\) −4.27753 −0.530562
\(66\) 0 0
\(67\) 9.36503i 1.14412i 0.820212 + 0.572060i \(0.193856\pi\)
−0.820212 + 0.572060i \(0.806144\pi\)
\(68\) 0 0
\(69\) 0.513976i 0.0618755i
\(70\) 0 0
\(71\) −9.96659 −1.18282 −0.591408 − 0.806372i \(-0.701428\pi\)
−0.591408 + 0.806372i \(0.701428\pi\)
\(72\) 0 0
\(73\) −6.26367 −0.733108 −0.366554 − 0.930397i \(-0.619462\pi\)
−0.366554 + 0.930397i \(0.619462\pi\)
\(74\) 0 0
\(75\) 5.03920i 0.581876i
\(76\) 0 0
\(77\) 10.5849i 1.20626i
\(78\) 0 0
\(79\) −0.777651 −0.0874926 −0.0437463 − 0.999043i \(-0.513929\pi\)
−0.0437463 + 0.999043i \(0.513929\pi\)
\(80\) 0 0
\(81\) 7.72809 0.858677
\(82\) 0 0
\(83\) 7.76481i 0.852298i 0.904653 + 0.426149i \(0.140130\pi\)
−0.904653 + 0.426149i \(0.859870\pi\)
\(84\) 0 0
\(85\) − 15.0552i − 1.63296i
\(86\) 0 0
\(87\) 0.243954 0.0261546
\(88\) 0 0
\(89\) 7.61994 0.807712 0.403856 − 0.914823i \(-0.367670\pi\)
0.403856 + 0.914823i \(0.367670\pi\)
\(90\) 0 0
\(91\) − 4.43100i − 0.464495i
\(92\) 0 0
\(93\) − 2.89184i − 0.299870i
\(94\) 0 0
\(95\) 33.1298 3.39905
\(96\) 0 0
\(97\) 8.97620 0.911395 0.455698 − 0.890135i \(-0.349390\pi\)
0.455698 + 0.890135i \(0.349390\pi\)
\(98\) 0 0
\(99\) − 6.82341i − 0.685778i
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 3328.2.b.bc.1665.5 10
4.3 odd 2 3328.2.b.bd.1665.6 10
8.3 odd 2 3328.2.b.bd.1665.5 10
8.5 even 2 inner 3328.2.b.bc.1665.6 10
16.3 odd 4 1664.2.a.y.1.3 ✓ 5
16.5 even 4 1664.2.a.z.1.3 yes 5
16.11 odd 4 1664.2.a.bb.1.3 yes 5
16.13 even 4 1664.2.a.ba.1.3 yes 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1664.2.a.y.1.3 ✓ 5 16.3 odd 4
1664.2.a.z.1.3 yes 5 16.5 even 4
1664.2.a.ba.1.3 yes 5 16.13 even 4
1664.2.a.bb.1.3 yes 5 16.11 odd 4
3328.2.b.bc.1665.5 10 1.1 even 1 trivial
3328.2.b.bc.1665.6 10 8.5 even 2 inner
3328.2.b.bd.1665.5 10 8.3 odd 2
3328.2.b.bd.1665.6 10 4.3 odd 2