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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 49.11
Character \(\chi\) \(=\) 736.49
Dual form 736.2.x.a.721.11

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.247795 - 0.113164i) q^{3} +(-0.131544 - 0.113984i) q^{5} +(-1.39962 + 0.899478i) q^{7} +(-1.91599 - 2.21117i) q^{9} +(2.64684 + 0.380559i) q^{11} +(-3.17249 + 4.93649i) q^{13} +(0.0196971 + 0.0431306i) q^{15} +(6.51388 + 1.91265i) q^{17} +(2.16986 + 7.38986i) q^{19} +(0.448607 - 0.0644999i) q^{21} +(4.64816 - 1.18094i) q^{23} +(-0.707263 - 4.91912i) q^{25} +(0.454790 + 1.54887i) q^{27} +(-0.640836 + 2.18249i) q^{29} +(2.23249 + 4.88848i) q^{31} +(-0.612810 - 0.393829i) q^{33} +(0.286637 + 0.0412121i) q^{35} +(-0.496922 + 0.430586i) q^{37} +(1.34476 - 0.864225i) q^{39} +(-3.27866 + 3.78378i) q^{41} +(-4.47162 - 2.04212i) q^{43} +0.509256i q^{45} +0.262765 q^{47} +(-1.75804 + 3.84958i) q^{49} +(-1.39766 - 1.21108i) q^{51} +(4.17439 + 6.49548i) q^{53} +(-0.304799 - 0.351757i) q^{55} +(0.298588 - 2.07672i) q^{57} +(-5.22872 + 8.13605i) q^{59} +(10.6297 - 4.85440i) q^{61} +(4.67054 + 1.37139i) q^{63} +(0.980000 - 0.287754i) q^{65} +(7.49178 - 1.07715i) q^{67} +(-1.28543 - 0.233373i) q^{69} +(-1.61107 - 11.2053i) q^{71} +(8.10508 - 2.37987i) q^{73} +(-0.381412 + 1.29897i) q^{75} +(-4.04687 + 1.84814i) q^{77} +(-7.70944 - 4.95456i) q^{79} +(-1.18657 + 8.25276i) q^{81} +(-3.71593 + 3.21987i) q^{83} +(-0.638851 - 0.994072i) q^{85} +(0.405775 - 0.468290i) q^{87} +(-0.981932 + 2.15013i) q^{89} -9.76276i q^{91} -1.46398i q^{93} +(0.556890 - 1.21942i) q^{95} +(1.58641 - 1.83082i) q^{97} +(-4.22984 - 6.58176i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 220 q + 22 q^{7} + 22 q^{15} - 18 q^{17} + 16 q^{23} - 4 q^{25} + 34 q^{31} - 30 q^{33} + 18 q^{39} - 18 q^{41} + 40 q^{47} - 28 q^{49} + 38 q^{55} - 30 q^{57} - 18 q^{63} - 38 q^{65} + 26 q^{71} - 18 q^{73}+ \cdots - 18 q^{97}+O(q^{100}) \) 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
\(3\) −0.247795 − 0.113164i −0.143065 − 0.0653354i 0.342596 − 0.939483i \(-0.388694\pi\)
−0.485661 + 0.874147i \(0.661421\pi\)
\(4\) 0 0
\(5\) −0.131544 − 0.113984i −0.0588283 − 0.0509750i 0.624950 − 0.780664i \(-0.285119\pi\)
−0.683779 + 0.729689i \(0.739665\pi\)
\(6\) 0 0
\(7\) −1.39962 + 0.899478i −0.529005 + 0.339971i −0.777725 − 0.628605i \(-0.783626\pi\)
0.248720 + 0.968575i \(0.419990\pi\)
\(8\) 0 0
\(9\) −1.91599 − 2.21117i −0.638662 − 0.737055i
\(10\) 0 0
\(11\) 2.64684 + 0.380559i 0.798054 + 0.114743i 0.529265 − 0.848456i \(-0.322468\pi\)
0.268788 + 0.963199i \(0.413377\pi\)
\(12\) 0 0
\(13\) −3.17249 + 4.93649i −0.879889 + 1.36913i 0.0490083 + 0.998798i \(0.484394\pi\)
−0.928898 + 0.370336i \(0.879242\pi\)
\(14\) 0 0
\(15\) 0.0196971 + 0.0431306i 0.00508577 + 0.0111363i
\(16\) 0 0
\(17\) 6.51388 + 1.91265i 1.57985 + 0.463885i 0.949848 − 0.312711i \(-0.101237\pi\)
0.629999 + 0.776596i \(0.283055\pi\)
\(18\) 0 0
\(19\) 2.16986 + 7.38986i 0.497800 + 1.69535i 0.698426 + 0.715682i \(0.253884\pi\)
−0.200626 + 0.979668i \(0.564298\pi\)
\(20\) 0 0
\(21\) 0.448607 − 0.0644999i 0.0978940 − 0.0140750i
\(22\) 0 0
\(23\) 4.64816 − 1.18094i 0.969208 − 0.246244i
\(24\) 0 0
\(25\) −0.707263 − 4.91912i −0.141453 − 0.983824i
\(26\) 0 0
\(27\) 0.454790 + 1.54887i 0.0875243 + 0.298080i
\(28\) 0 0
\(29\) −0.640836 + 2.18249i −0.119000 + 0.405278i −0.997351 − 0.0727391i \(-0.976826\pi\)
0.878351 + 0.478017i \(0.158644\pi\)
\(30\) 0 0
\(31\) 2.23249 + 4.88848i 0.400968 + 0.877997i 0.997171 + 0.0751636i \(0.0239479\pi\)
−0.596203 + 0.802833i \(0.703325\pi\)
\(32\) 0 0
\(33\) −0.612810 − 0.393829i −0.106676 − 0.0685568i
\(34\) 0 0
\(35\) 0.286637 + 0.0412121i 0.0484504 + 0.00696612i
\(36\) 0 0
\(37\) −0.496922 + 0.430586i −0.0816935 + 0.0707878i −0.694750 − 0.719251i \(-0.744485\pi\)
0.613057 + 0.790039i \(0.289940\pi\)
\(38\) 0 0
\(39\) 1.34476 − 0.864225i 0.215334 − 0.138387i
\(40\) 0 0
\(41\) −3.27866 + 3.78378i −0.512041 + 0.590927i −0.951620 − 0.307277i \(-0.900582\pi\)
0.439579 + 0.898204i \(0.355128\pi\)
\(42\) 0 0
\(43\) −4.47162 − 2.04212i −0.681915 − 0.311420i 0.0441711 − 0.999024i \(-0.485935\pi\)
−0.726086 + 0.687604i \(0.758663\pi\)
\(44\) 0 0
\(45\) 0.509256i 0.0759155i
\(46\) 0 0
\(47\) 0.262765 0.0383282 0.0191641 − 0.999816i \(-0.493899\pi\)
0.0191641 + 0.999816i \(0.493899\pi\)
\(48\) 0 0
\(49\) −1.75804 + 3.84958i −0.251149 + 0.549940i
\(50\) 0 0
\(51\) −1.39766 − 1.21108i −0.195712 − 0.169586i
\(52\) 0 0
\(53\) 4.17439 + 6.49548i 0.573397 + 0.892223i 0.999925 − 0.0122514i \(-0.00389984\pi\)
−0.426528 + 0.904474i \(0.640263\pi\)
\(54\) 0 0
\(55\) −0.304799 − 0.351757i −0.0410991 − 0.0474309i
\(56\) 0 0
\(57\) 0.298588 − 2.07672i 0.0395489 − 0.275069i
\(58\) 0 0
\(59\) −5.22872 + 8.13605i −0.680721 + 1.05922i 0.313259 + 0.949668i \(0.398579\pi\)
−0.993981 + 0.109556i \(0.965057\pi\)
\(60\) 0 0
\(61\) 10.6297 − 4.85440i 1.36099 − 0.621542i 0.404829 − 0.914392i \(-0.367331\pi\)
0.956158 + 0.292850i \(0.0946037\pi\)
\(62\) 0 0
\(63\) 4.67054 + 1.37139i 0.588432 + 0.172779i
\(64\) 0 0
\(65\) 0.980000 − 0.287754i 0.121554 − 0.0356915i
\(66\) 0 0
\(67\) 7.49178 − 1.07715i 0.915266 − 0.131595i 0.331442 − 0.943476i \(-0.392465\pi\)
0.583824 + 0.811880i \(0.301556\pi\)
\(68\) 0 0
\(69\) −1.28543 − 0.233373i −0.154748 − 0.0280948i
\(70\) 0 0
\(71\) −1.61107 − 11.2053i −0.191199 − 1.32982i −0.828840 − 0.559486i \(-0.810998\pi\)
0.637640 − 0.770334i \(-0.279911\pi\)
\(72\) 0 0
\(73\) 8.10508 − 2.37987i 0.948628 − 0.278542i 0.229412 − 0.973329i \(-0.426320\pi\)
0.719216 + 0.694787i \(0.244501\pi\)
\(74\) 0 0
\(75\) −0.381412 + 1.29897i −0.0440417 + 0.149992i
\(76\) 0 0
\(77\) −4.04687 + 1.84814i −0.461183 + 0.210615i
\(78\) 0 0
\(79\) −7.70944 − 4.95456i −0.867380 − 0.557431i 0.0295704 − 0.999563i \(-0.490586\pi\)
−0.896950 + 0.442132i \(0.854222\pi\)
\(80\) 0 0
\(81\) −1.18657 + 8.25276i −0.131841 + 0.916974i
\(82\) 0 0
\(83\) −3.71593 + 3.21987i −0.407877 + 0.353427i −0.834505 − 0.551000i \(-0.814247\pi\)
0.426629 + 0.904427i \(0.359701\pi\)
\(84\) 0 0
\(85\) −0.638851 − 0.994072i −0.0692932 − 0.107822i
\(86\) 0 0
\(87\) 0.405775 − 0.468290i 0.0435037 − 0.0502059i
\(88\) 0 0
\(89\) −0.981932 + 2.15013i −0.104085 + 0.227913i −0.954508 − 0.298186i \(-0.903619\pi\)
0.850423 + 0.526099i \(0.176346\pi\)
\(90\) 0 0
\(91\) − 9.76276i − 1.02342i
\(92\) 0 0
\(93\) − 1.46398i − 0.151808i
\(94\) 0 0
\(95\) 0.556890 − 1.21942i 0.0571358 − 0.125110i
\(96\) 0 0
\(97\) 1.58641 − 1.83082i 0.161076 − 0.185891i −0.669474 − 0.742835i \(-0.733481\pi\)
0.830550 + 0.556944i \(0.188026\pi\)
\(98\) 0 0
\(99\) −4.22984 − 6.58176i −0.425115 − 0.661491i
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.49.11 220
4.3 odd 2 184.2.p.a.141.20 yes 220
8.3 odd 2 184.2.p.a.141.4 yes 220
8.5 even 2 inner 736.2.x.a.49.12 220
23.8 even 11 inner 736.2.x.a.721.12 220
92.31 odd 22 184.2.p.a.77.4 ✓ 220
184.77 even 22 inner 736.2.x.a.721.11 220
184.123 odd 22 184.2.p.a.77.20 yes 220
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
184.2.p.a.77.4 ✓ 220 92.31 odd 22
184.2.p.a.77.20 yes 220 184.123 odd 22
184.2.p.a.141.4 yes 220 8.3 odd 2
184.2.p.a.141.20 yes 220 4.3 odd 2
736.2.x.a.49.11 220 1.1 even 1 trivial
736.2.x.a.49.12 220 8.5 even 2 inner
736.2.x.a.721.11 220 184.77 even 22 inner
736.2.x.a.721.12 220 23.8 even 11 inner