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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 1665.2
Root \(0.208364i\) of defining polynomial
Character \(\chi\) \(=\) 3328.1665
Dual form 3328.2.b.bc.1665.9

$q$-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)
 
\(f(q)\) \(=\) \(q-3.16495i q^{3} -0.368078i q^{5} -2.90060 q^{7} -7.01690 q^{9} +1.58327i q^{11} +1.00000i q^{13} -1.16495 q^{15} +6.69798 q^{17} +2.80171i q^{19} +9.18025i q^{21} +5.21843 q^{23} +4.86452 q^{25} +12.7133i q^{27} +7.21843i q^{29} -1.91317 q^{31} +5.01097 q^{33} +1.06765i q^{35} +10.3468i q^{37} +3.16495 q^{39} +2.48228 q^{41} -6.81377i q^{43} +2.58277i q^{45} -0.219525 q^{47} +1.41349 q^{49} -21.1987i q^{51} -6.48228i q^{53} +0.582768 q^{55} +8.86726 q^{57} +1.58327i q^{59} +10.1775i q^{61} +20.3532 q^{63} +0.368078 q^{65} +1.32939i q^{67} -16.5161i q^{69} -6.06396 q^{71} -15.3495 q^{73} -15.3960i q^{75} -4.59244i q^{77} +7.16654 q^{79} +19.1862 q^{81} -16.2981i q^{83} -2.46538i q^{85} +22.8460 q^{87} +10.1311 q^{89} -2.90060i q^{91} +6.05508i q^{93} +1.03125 q^{95} +4.91266 q^{97} -11.1097i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 10 q^{7} - 16 q^{9} + 18 q^{15} + 6 q^{17} + 4 q^{23} - 20 q^{25} + 44 q^{31} - 16 q^{33} + 2 q^{39} - 20 q^{41} + 10 q^{47} + 64 q^{49} + 16 q^{55} - 12 q^{57} + 88 q^{63} + 2 q^{65} + 10 q^{71}+ \cdots + 84 q^{95}+O(q^{100}) \) 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
\(3\) − 3.16495i − 1.82728i −0.406520 − 0.913642i \(-0.633258\pi\)
0.406520 − 0.913642i \(-0.366742\pi\)
\(4\) 0 0
\(5\) − 0.368078i − 0.164610i −0.996607 − 0.0823048i \(-0.973772\pi\)
0.996607 − 0.0823048i \(-0.0262281\pi\)
\(6\) 0 0
\(7\) −2.90060 −1.09632 −0.548162 − 0.836372i \(-0.684672\pi\)
−0.548162 + 0.836372i \(0.684672\pi\)
\(8\) 0 0
\(9\) −7.01690 −2.33897
\(10\) 0 0
\(11\) 1.58327i 0.477374i 0.971097 + 0.238687i \(0.0767170\pi\)
−0.971097 + 0.238687i \(0.923283\pi\)
\(12\) 0 0
\(13\) 1.00000i 0.277350i
\(14\) 0 0
\(15\) −1.16495 −0.300788
\(16\) 0 0
\(17\) 6.69798 1.62450 0.812249 − 0.583311i \(-0.198243\pi\)
0.812249 + 0.583311i \(0.198243\pi\)
\(18\) 0 0
\(19\) 2.80171i 0.642755i 0.946951 + 0.321378i \(0.104146\pi\)
−0.946951 + 0.321378i \(0.895854\pi\)
\(20\) 0 0
\(21\) 9.18025i 2.00330i
\(22\) 0 0
\(23\) 5.21843 1.08812 0.544059 − 0.839047i \(-0.316887\pi\)
0.544059 + 0.839047i \(0.316887\pi\)
\(24\) 0 0
\(25\) 4.86452 0.972904
\(26\) 0 0
\(27\) 12.7133i 2.44667i
\(28\) 0 0
\(29\) 7.21843i 1.34043i 0.742167 + 0.670215i \(0.233798\pi\)
−0.742167 + 0.670215i \(0.766202\pi\)
\(30\) 0 0
\(31\) −1.91317 −0.343615 −0.171808 − 0.985131i \(-0.554961\pi\)
−0.171808 + 0.985131i \(0.554961\pi\)
\(32\) 0 0
\(33\) 5.01097 0.872298
\(34\) 0 0
\(35\) 1.06765i 0.180465i
\(36\) 0 0
\(37\) 10.3468i 1.70100i 0.525973 + 0.850501i \(0.323701\pi\)
−0.525973 + 0.850501i \(0.676299\pi\)
\(38\) 0 0
\(39\) 3.16495 0.506797
\(40\) 0 0
\(41\) 2.48228 0.387667 0.193833 − 0.981034i \(-0.437908\pi\)
0.193833 + 0.981034i \(0.437908\pi\)
\(42\) 0 0
\(43\) − 6.81377i − 1.03909i −0.854443 − 0.519545i \(-0.826101\pi\)
0.854443 − 0.519545i \(-0.173899\pi\)
\(44\) 0 0
\(45\) 2.58277i 0.385016i
\(46\) 0 0
\(47\) −0.219525 −0.0320211 −0.0160105 − 0.999872i \(-0.505097\pi\)
−0.0160105 + 0.999872i \(0.505097\pi\)
\(48\) 0 0
\(49\) 1.41349 0.201927
\(50\) 0 0
\(51\) − 21.1987i − 2.96842i
\(52\) 0 0
\(53\) − 6.48228i − 0.890409i −0.895429 − 0.445205i \(-0.853131\pi\)
0.895429 − 0.445205i \(-0.146869\pi\)
\(54\) 0 0
\(55\) 0.582768 0.0785804
\(56\) 0 0
\(57\) 8.86726 1.17450
\(58\) 0 0
\(59\) 1.58327i 0.206124i 0.994675 + 0.103062i \(0.0328641\pi\)
−0.994675 + 0.103062i \(0.967136\pi\)
\(60\) 0 0
\(61\) 10.1775i 1.30310i 0.758607 + 0.651549i \(0.225880\pi\)
−0.758607 + 0.651549i \(0.774120\pi\)
\(62\) 0 0
\(63\) 20.3532 2.56427
\(64\) 0 0
\(65\) 0.368078 0.0456545
\(66\) 0 0
\(67\) 1.32939i 0.162411i 0.996697 + 0.0812056i \(0.0258770\pi\)
−0.996697 + 0.0812056i \(0.974123\pi\)
\(68\) 0 0
\(69\) − 16.5161i − 1.98830i
\(70\) 0 0
\(71\) −6.06396 −0.719659 −0.359830 − 0.933018i \(-0.617165\pi\)
−0.359830 + 0.933018i \(0.617165\pi\)
\(72\) 0 0
\(73\) −15.3495 −1.79653 −0.898264 − 0.439457i \(-0.855171\pi\)
−0.898264 + 0.439457i \(0.855171\pi\)
\(74\) 0 0
\(75\) − 15.3960i − 1.77777i
\(76\) 0 0
\(77\) − 4.59244i − 0.523357i
\(78\) 0 0
\(79\) 7.16654 0.806299 0.403150 − 0.915134i \(-0.367915\pi\)
0.403150 + 0.915134i \(0.367915\pi\)
\(80\) 0 0
\(81\) 19.1862 2.13180
\(82\) 0 0
\(83\) − 16.2981i − 1.78895i −0.447114 − 0.894477i \(-0.647548\pi\)
0.447114 − 0.894477i \(-0.352452\pi\)
\(84\) 0 0
\(85\) − 2.46538i − 0.267408i
\(86\) 0 0
\(87\) 22.8460 2.44935
\(88\) 0 0
\(89\) 10.1311 1.07389 0.536947 − 0.843616i \(-0.319577\pi\)
0.536947 + 0.843616i \(0.319577\pi\)
\(90\) 0 0
\(91\) − 2.90060i − 0.304066i
\(92\) 0 0
\(93\) 6.05508i 0.627883i
\(94\) 0 0
\(95\) 1.03125 0.105804
\(96\) 0 0
\(97\) 4.91266 0.498806 0.249403 − 0.968400i \(-0.419766\pi\)
0.249403 + 0.968400i \(0.419766\pi\)
\(98\) 0 0
\(99\) − 11.1097i − 1.11656i
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.1665.2 10
4.3 odd 2 3328.2.b.bd.1665.9 10
8.3 odd 2 3328.2.b.bd.1665.2 10
8.5 even 2 inner 3328.2.b.bc.1665.9 10
16.3 odd 4 1664.2.a.y.1.1 ✓ 5
16.5 even 4 1664.2.a.z.1.1 yes 5
16.11 odd 4 1664.2.a.bb.1.5 yes 5
16.13 even 4 1664.2.a.ba.1.5 yes 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1664.2.a.y.1.1 ✓ 5 16.3 odd 4
1664.2.a.z.1.1 yes 5 16.5 even 4
1664.2.a.ba.1.5 yes 5 16.13 even 4
1664.2.a.bb.1.5 yes 5 16.11 odd 4
3328.2.b.bc.1665.2 10 1.1 even 1 trivial
3328.2.b.bc.1665.9 10 8.5 even 2 inner
3328.2.b.bd.1665.2 10 8.3 odd 2
3328.2.b.bd.1665.9 10 4.3 odd 2