Show commands: Magma / Pari/GP / SageMath

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 = [4,0,0,0,0,0,4,0,0,0,0,0,0,0,20,0,20] 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: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 1664)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1665.3
Root \(-0.707107 + 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 3328.1665
Dual form 3328.2.b.z.1665.2

$q$-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)
 
\(f(q)\) \(=\) \(q+0.414214i q^{3} -1.82843i q^{5} +2.41421 q^{7} +2.82843 q^{9} +4.82843i q^{11} +1.00000i q^{13} +0.757359 q^{15} +5.00000 q^{17} +4.82843i q^{19} +1.00000i q^{21} +4.00000 q^{23} +1.65685 q^{25} +2.41421i q^{27} -3.65685i q^{29} -6.00000 q^{31} -2.00000 q^{33} -4.41421i q^{35} +8.65685i q^{37} -0.414214 q^{39} -8.00000 q^{41} +0.757359i q^{43} -5.17157i q^{45} -6.07107 q^{47} -1.17157 q^{49} +2.07107i q^{51} +8.00000i q^{53} +8.82843 q^{55} -2.00000 q^{57} +14.4853i q^{59} -9.65685i q^{61} +6.82843 q^{63} +1.82843 q^{65} -8.82843i q^{67} +1.65685i q^{69} -9.72792 q^{71} -13.3137 q^{73} +0.686292i q^{75} +11.6569i q^{77} +7.31371 q^{79} +7.48528 q^{81} +6.34315i q^{83} -9.14214i q^{85} +1.51472 q^{87} +7.65685 q^{89} +2.41421i q^{91} -2.48528i q^{93} +8.82843 q^{95} +14.0000 q^{97} +13.6569i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{7} + 20 q^{15} + 20 q^{17} + 16 q^{23} - 16 q^{25} - 24 q^{31} - 8 q^{33} + 4 q^{39} - 32 q^{41} + 4 q^{47} - 16 q^{49} + 24 q^{55} - 8 q^{57} + 16 q^{63} - 4 q^{65} + 12 q^{71} - 8 q^{73} - 16 q^{79}+ \cdots + 56 q^{97}+O(q^{100}) \) 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
\(3\) 0.414214i 0.239146i 0.992825 + 0.119573i \(0.0381526\pi\)
−0.992825 + 0.119573i \(0.961847\pi\)
\(4\) 0 0
\(5\) − 1.82843i − 0.817697i −0.912602 − 0.408849i \(-0.865930\pi\)
0.912602 − 0.408849i \(-0.134070\pi\)
\(6\) 0 0
\(7\) 2.41421 0.912487 0.456243 − 0.889855i \(-0.349195\pi\)
0.456243 + 0.889855i \(0.349195\pi\)
\(8\) 0 0
\(9\) 2.82843 0.942809
\(10\) 0 0
\(11\) 4.82843i 1.45583i 0.685670 + 0.727913i \(0.259509\pi\)
−0.685670 + 0.727913i \(0.740491\pi\)
\(12\) 0 0
\(13\) 1.00000i 0.277350i
\(14\) 0 0
\(15\) 0.757359 0.195549
\(16\) 0 0
\(17\) 5.00000 1.21268 0.606339 − 0.795206i \(-0.292637\pi\)
0.606339 + 0.795206i \(0.292637\pi\)
\(18\) 0 0
\(19\) 4.82843i 1.10772i 0.832611 + 0.553859i \(0.186845\pi\)
−0.832611 + 0.553859i \(0.813155\pi\)
\(20\) 0 0
\(21\) 1.00000i 0.218218i
\(22\) 0 0
\(23\) 4.00000 0.834058 0.417029 − 0.908893i \(-0.363071\pi\)
0.417029 + 0.908893i \(0.363071\pi\)
\(24\) 0 0
\(25\) 1.65685 0.331371
\(26\) 0 0
\(27\) 2.41421i 0.464616i
\(28\) 0 0
\(29\) − 3.65685i − 0.679061i −0.940595 − 0.339530i \(-0.889732\pi\)
0.940595 − 0.339530i \(-0.110268\pi\)
\(30\) 0 0
\(31\) −6.00000 −1.07763 −0.538816 − 0.842424i \(-0.681128\pi\)
−0.538816 + 0.842424i \(0.681128\pi\)
\(32\) 0 0
\(33\) −2.00000 −0.348155
\(34\) 0 0
\(35\) − 4.41421i − 0.746138i
\(36\) 0 0
\(37\) 8.65685i 1.42318i 0.702596 + 0.711589i \(0.252024\pi\)
−0.702596 + 0.711589i \(0.747976\pi\)
\(38\) 0 0
\(39\) −0.414214 −0.0663273
\(40\) 0 0
\(41\) −8.00000 −1.24939 −0.624695 − 0.780869i \(-0.714777\pi\)
−0.624695 + 0.780869i \(0.714777\pi\)
\(42\) 0 0
\(43\) 0.757359i 0.115496i 0.998331 + 0.0577481i \(0.0183920\pi\)
−0.998331 + 0.0577481i \(0.981608\pi\)
\(44\) 0 0
\(45\) − 5.17157i − 0.770933i
\(46\) 0 0
\(47\) −6.07107 −0.885556 −0.442778 − 0.896631i \(-0.646007\pi\)
−0.442778 + 0.896631i \(0.646007\pi\)
\(48\) 0 0
\(49\) −1.17157 −0.167368
\(50\) 0 0
\(51\) 2.07107i 0.290008i
\(52\) 0 0
\(53\) 8.00000i 1.09888i 0.835532 + 0.549442i \(0.185160\pi\)
−0.835532 + 0.549442i \(0.814840\pi\)
\(54\) 0 0
\(55\) 8.82843 1.19042
\(56\) 0 0
\(57\) −2.00000 −0.264906
\(58\) 0 0
\(59\) 14.4853i 1.88582i 0.333043 + 0.942912i \(0.391924\pi\)
−0.333043 + 0.942912i \(0.608076\pi\)
\(60\) 0 0
\(61\) − 9.65685i − 1.23643i −0.786008 − 0.618217i \(-0.787855\pi\)
0.786008 − 0.618217i \(-0.212145\pi\)
\(62\) 0 0
\(63\) 6.82843 0.860301
\(64\) 0 0
\(65\) 1.82843 0.226788
\(66\) 0 0
\(67\) − 8.82843i − 1.07856i −0.842125 − 0.539282i \(-0.818696\pi\)
0.842125 − 0.539282i \(-0.181304\pi\)
\(68\) 0 0
\(69\) 1.65685i 0.199462i
\(70\) 0 0
\(71\) −9.72792 −1.15449 −0.577246 − 0.816570i \(-0.695873\pi\)
−0.577246 + 0.816570i \(0.695873\pi\)
\(72\) 0 0
\(73\) −13.3137 −1.55825 −0.779126 − 0.626868i \(-0.784337\pi\)
−0.779126 + 0.626868i \(0.784337\pi\)
\(74\) 0 0
\(75\) 0.686292i 0.0792461i
\(76\) 0 0
\(77\) 11.6569i 1.32842i
\(78\) 0 0
\(79\) 7.31371 0.822856 0.411428 − 0.911442i \(-0.365030\pi\)
0.411428 + 0.911442i \(0.365030\pi\)
\(80\) 0 0
\(81\) 7.48528 0.831698
\(82\) 0 0
\(83\) 6.34315i 0.696251i 0.937448 + 0.348125i \(0.113182\pi\)
−0.937448 + 0.348125i \(0.886818\pi\)
\(84\) 0 0
\(85\) − 9.14214i − 0.991604i
\(86\) 0 0
\(87\) 1.51472 0.162395
\(88\) 0 0
\(89\) 7.65685 0.811625 0.405812 − 0.913956i \(-0.366989\pi\)
0.405812 + 0.913956i \(0.366989\pi\)
\(90\) 0 0
\(91\) 2.41421i 0.253078i
\(92\) 0 0
\(93\) − 2.48528i − 0.257712i
\(94\) 0 0
\(95\) 8.82843 0.905778
\(96\) 0 0
\(97\) 14.0000 1.42148 0.710742 − 0.703452i \(-0.248359\pi\)
0.710742 + 0.703452i \(0.248359\pi\)
\(98\) 0 0
\(99\) 13.6569i 1.37257i
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.z.1665.3 4
4.3 odd 2 3328.2.b.v.1665.2 4
8.3 odd 2 3328.2.b.v.1665.3 4
8.5 even 2 inner 3328.2.b.z.1665.2 4
16.3 odd 4 1664.2.a.v.1.2 yes 2
16.5 even 4 1664.2.a.u.1.2 ✓ 2
16.11 odd 4 1664.2.a.w.1.1 yes 2
16.13 even 4 1664.2.a.x.1.1 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1664.2.a.u.1.2 ✓ 2 16.5 even 4
1664.2.a.v.1.2 yes 2 16.3 odd 4
1664.2.a.w.1.1 yes 2 16.11 odd 4
1664.2.a.x.1.1 yes 2 16.13 even 4
3328.2.b.v.1665.2 4 4.3 odd 2
3328.2.b.v.1665.3 4 8.3 odd 2
3328.2.b.z.1665.2 4 8.5 even 2 inner
3328.2.b.z.1665.3 4 1.1 even 1 trivial