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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [736,2,Mod(49,736)] 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("736.49"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(736, base_ring=CyclotomicField(22)) chi = DirichletCharacter(H, H._module([0, 11, 16])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 736 = 2^{5} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 736.x (of order \(22\), degree \(10\), 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: \(5.87698958877\)
Analytic rank: \(0\)
Dimension: \(220\)
Relative dimension: \(22\) over \(\Q(\zeta_{22})\)
Twist minimal: no (minimal twist has level 184)
Sato-Tate group: $\mathrm{SU}(2)[C_{22}]$

Embedding invariants

Embedding label 49.1
Character \(\chi\) \(=\) 736.49
Dual form 736.2.x.a.721.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-3.05941 - 1.39719i) q^{3} +(-1.58431 - 1.37281i) q^{5} +(-2.86721 + 1.84264i) q^{7} +(5.44328 + 6.28187i) q^{9} +(1.04646 + 0.150458i) q^{11} +(-1.62922 + 2.53511i) q^{13} +(2.92898 + 6.41357i) q^{15} +(-5.14438 - 1.51053i) q^{17} +(0.644621 + 2.19538i) q^{19} +(11.3465 - 1.63138i) q^{21} +(3.56630 - 3.20648i) q^{23} +(-0.0861503 - 0.599189i) q^{25} +(-5.03357 - 17.1428i) q^{27} +(0.164140 - 0.559009i) q^{29} +(0.410217 + 0.898250i) q^{31} +(-2.99132 - 1.92240i) q^{33} +(7.07214 + 1.01682i) q^{35} +(-0.517929 + 0.448788i) q^{37} +(8.52646 - 5.47962i) q^{39} +(4.78863 - 5.52637i) q^{41} +(-2.44215 - 1.11529i) q^{43} -17.4250i q^{45} +6.60921 q^{47} +(1.91764 - 4.19905i) q^{49} +(13.6283 + 11.8090i) q^{51} +(2.06856 + 3.21875i) q^{53} +(-1.45136 - 1.67496i) q^{55} +(1.09519 - 7.61722i) q^{57} +(-0.859136 + 1.33684i) q^{59} +(3.33233 - 1.52182i) q^{61} +(-27.1822 - 7.98143i) q^{63} +(6.06141 - 1.77979i) q^{65} +(3.41724 - 0.491326i) q^{67} +(-15.3908 + 4.82714i) q^{69} +(-0.0686942 - 0.477779i) q^{71} +(2.14164 - 0.628843i) q^{73} +(-0.573609 + 1.95353i) q^{75} +(-3.27764 + 1.49685i) q^{77} +(10.7432 + 6.90427i) q^{79} +(-5.00306 + 34.7970i) q^{81} +(-6.55507 + 5.68000i) q^{83} +(6.07662 + 9.45541i) q^{85} +(-1.28321 + 1.48090i) q^{87} +(-2.12317 + 4.64909i) q^{89} -10.2707i q^{91} -3.32126i q^{93} +(1.99256 - 4.36310i) q^{95} +(7.30416 - 8.42945i) q^{97} +(4.75099 + 7.39268i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 220 q + 22 q^{7} + 22 q^{15} - 18 q^{17} + 16 q^{23} - 4 q^{25} + 34 q^{31} - 30 q^{33} + 18 q^{39} - 18 q^{41} + 40 q^{47} - 28 q^{49} + 38 q^{55} - 30 q^{57} - 18 q^{63} - 38 q^{65} + 26 q^{71} - 18 q^{73}+ \cdots - 18 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/736\mathbb{Z}\right)^\times\).

\(n\) \(97\) \(415\) \(645\)
\(\chi(n)\) \(e\left(\frac{8}{11}\right)\) \(1\) \(-1\)

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
\(3\) −3.05941 − 1.39719i −1.76635 − 0.806665i −0.982636 − 0.185545i \(-0.940595\pi\)
−0.783715 − 0.621121i \(-0.786678\pi\)
\(4\) 0 0
\(5\) −1.58431 − 1.37281i −0.708525 − 0.613940i 0.224194 − 0.974545i \(-0.428025\pi\)
−0.932719 + 0.360604i \(0.882570\pi\)
\(6\) 0 0
\(7\) −2.86721 + 1.84264i −1.08370 + 0.696453i −0.955410 − 0.295283i \(-0.904586\pi\)
−0.128292 + 0.991736i \(0.540950\pi\)
\(8\) 0 0
\(9\) 5.44328 + 6.28187i 1.81443 + 2.09396i
\(10\) 0 0
\(11\) 1.04646 + 0.150458i 0.315518 + 0.0453647i 0.298255 − 0.954486i \(-0.403596\pi\)
0.0172634 + 0.999851i \(0.494505\pi\)
\(12\) 0 0
\(13\) −1.62922 + 2.53511i −0.451863 + 0.703113i −0.990206 − 0.139613i \(-0.955414\pi\)
0.538343 + 0.842726i \(0.319051\pi\)
\(14\) 0 0
\(15\) 2.92898 + 6.41357i 0.756259 + 1.65598i
\(16\) 0 0
\(17\) −5.14438 − 1.51053i −1.24770 − 0.366357i −0.409794 − 0.912178i \(-0.634400\pi\)
−0.837902 + 0.545821i \(0.816218\pi\)
\(18\) 0 0
\(19\) 0.644621 + 2.19538i 0.147886 + 0.503654i 0.999799 − 0.0200288i \(-0.00637578\pi\)
−0.851913 + 0.523683i \(0.824558\pi\)
\(20\) 0 0
\(21\) 11.3465 − 1.63138i 2.47600 − 0.355995i
\(22\) 0 0
\(23\) 3.56630 − 3.20648i 0.743625 − 0.668597i
\(24\) 0 0
\(25\) −0.0861503 − 0.599189i −0.0172301 − 0.119838i
\(26\) 0 0
\(27\) −5.03357 − 17.1428i −0.968711 − 3.29913i
\(28\) 0 0
\(29\) 0.164140 − 0.559009i 0.0304800 − 0.103805i −0.942848 − 0.333223i \(-0.891864\pi\)
0.973328 + 0.229417i \(0.0736821\pi\)
\(30\) 0 0
\(31\) 0.410217 + 0.898250i 0.0736772 + 0.161330i 0.942887 − 0.333113i \(-0.108099\pi\)
−0.869210 + 0.494444i \(0.835372\pi\)
\(32\) 0 0
\(33\) −2.99132 − 1.92240i −0.520722 − 0.334647i
\(34\) 0 0
\(35\) 7.07214 + 1.01682i 1.19541 + 0.171874i
\(36\) 0 0
\(37\) −0.517929 + 0.448788i −0.0851470 + 0.0737803i −0.696394 − 0.717659i \(-0.745214\pi\)
0.611247 + 0.791440i \(0.290668\pi\)
\(38\) 0 0
\(39\) 8.52646 − 5.47962i 1.36533 − 0.877442i
\(40\) 0 0
\(41\) 4.78863 − 5.52637i 0.747858 − 0.863074i −0.246501 − 0.969143i \(-0.579281\pi\)
0.994359 + 0.106068i \(0.0338262\pi\)
\(42\) 0 0
\(43\) −2.44215 − 1.11529i −0.372425 − 0.170081i 0.220405 − 0.975408i \(-0.429262\pi\)
−0.592830 + 0.805328i \(0.701989\pi\)
\(44\) 0 0
\(45\) − 17.4250i − 2.59757i
\(46\) 0 0
\(47\) 6.60921 0.964052 0.482026 − 0.876157i \(-0.339901\pi\)
0.482026 + 0.876157i \(0.339901\pi\)
\(48\) 0 0
\(49\) 1.91764 − 4.19905i 0.273949 − 0.599864i
\(50\) 0 0
\(51\) 13.6283 + 11.8090i 1.90834 + 1.65359i
\(52\) 0 0
\(53\) 2.06856 + 3.21875i 0.284139 + 0.442129i 0.953758 − 0.300576i \(-0.0971789\pi\)
−0.669619 + 0.742705i \(0.733543\pi\)
\(54\) 0 0
\(55\) −1.45136 − 1.67496i −0.195701 − 0.225851i
\(56\) 0 0
\(57\) 1.09519 − 7.61722i 0.145062 − 1.00892i
\(58\) 0 0
\(59\) −0.859136 + 1.33684i −0.111850 + 0.174042i −0.892666 − 0.450720i \(-0.851167\pi\)
0.780816 + 0.624762i \(0.214804\pi\)
\(60\) 0 0
\(61\) 3.33233 − 1.52182i 0.426661 − 0.194850i −0.190499 − 0.981687i \(-0.561010\pi\)
0.617160 + 0.786838i \(0.288283\pi\)
\(62\) 0 0
\(63\) −27.1822 − 7.98143i −3.42464 − 1.00557i
\(64\) 0 0
\(65\) 6.06141 − 1.77979i 0.751826 − 0.220756i
\(66\) 0 0
\(67\) 3.41724 − 0.491326i 0.417483 − 0.0600250i 0.0696272 − 0.997573i \(-0.477819\pi\)
0.347856 + 0.937548i \(0.386910\pi\)
\(68\) 0 0
\(69\) −15.3908 + 4.82714i −1.85284 + 0.581120i
\(70\) 0 0
\(71\) −0.0686942 − 0.477779i −0.00815250 − 0.0567019i 0.985339 − 0.170607i \(-0.0545730\pi\)
−0.993492 + 0.113906i \(0.963664\pi\)
\(72\) 0 0
\(73\) 2.14164 − 0.628843i 0.250660 − 0.0736005i −0.153989 − 0.988073i \(-0.549212\pi\)
0.404649 + 0.914472i \(0.367394\pi\)
\(74\) 0 0
\(75\) −0.573609 + 1.95353i −0.0662346 + 0.225574i
\(76\) 0 0
\(77\) −3.27764 + 1.49685i −0.373522 + 0.170582i
\(78\) 0 0
\(79\) 10.7432 + 6.90427i 1.20871 + 0.776791i 0.980442 − 0.196808i \(-0.0630575\pi\)
0.228268 + 0.973598i \(0.426694\pi\)
\(80\) 0 0
\(81\) −5.00306 + 34.7970i −0.555895 + 3.86634i
\(82\) 0 0
\(83\) −6.55507 + 5.68000i −0.719513 + 0.623461i −0.935661 − 0.352899i \(-0.885196\pi\)
0.216149 + 0.976360i \(0.430650\pi\)
\(84\) 0 0
\(85\) 6.07662 + 9.45541i 0.659102 + 1.02558i
\(86\) 0 0
\(87\) −1.28321 + 1.48090i −0.137575 + 0.158769i
\(88\) 0 0
\(89\) −2.12317 + 4.64909i −0.225055 + 0.492803i −0.988151 − 0.153483i \(-0.950951\pi\)
0.763096 + 0.646285i \(0.223678\pi\)
\(90\) 0 0
\(91\) − 10.2707i − 1.07667i
\(92\) 0 0
\(93\) − 3.32126i − 0.344399i
\(94\) 0 0
\(95\) 1.99256 − 4.36310i 0.204433 − 0.447645i
\(96\) 0 0
\(97\) 7.30416 − 8.42945i 0.741625 − 0.855881i −0.252103 − 0.967700i \(-0.581122\pi\)
0.993729 + 0.111819i \(0.0356677\pi\)
\(98\) 0 0
\(99\) 4.75099 + 7.39268i 0.477492 + 0.742993i
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 736.2.x.a.49.1 220
4.3 odd 2 184.2.p.a.141.6 yes 220
8.3 odd 2 184.2.p.a.141.19 yes 220
8.5 even 2 inner 736.2.x.a.49.22 220
23.8 even 11 inner 736.2.x.a.721.22 220
92.31 odd 22 184.2.p.a.77.19 yes 220
184.77 even 22 inner 736.2.x.a.721.1 220
184.123 odd 22 184.2.p.a.77.6 ✓ 220
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
184.2.p.a.77.6 ✓ 220 184.123 odd 22
184.2.p.a.77.19 yes 220 92.31 odd 22
184.2.p.a.141.6 yes 220 4.3 odd 2
184.2.p.a.141.19 yes 220 8.3 odd 2
736.2.x.a.49.1 220 1.1 even 1 trivial
736.2.x.a.49.22 220 8.5 even 2 inner
736.2.x.a.721.1 220 184.77 even 22 inner
736.2.x.a.721.22 220 23.8 even 11 inner