Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [91,2,Mod(22,91)] 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("91.22"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(91, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 91 = 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 91.f (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.726638658394\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: 8.0.59066497296.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} + 7x^{6} + 38x^{4} - 16x^{3} + 15x^{2} + 3x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 29.2
Root \(-0.115680 - 0.200364i\) of defining polynomial
Character \(\chi\) \(=\) 91.29
Dual form 91.2.f.c.22.2

$q$-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+(-0.115680 - 0.200364i) q^{2} +(1.66113 + 2.87716i) q^{3} +(0.973236 - 1.68569i) q^{4} -2.23136 q^{5} +(0.384320 - 0.665661i) q^{6} +(0.500000 - 0.866025i) q^{7} -0.913059 q^{8} +(-4.01868 + 6.96056i) q^{9} +(0.258125 + 0.447085i) q^{10} +(-1.66113 - 2.87716i) q^{11} +6.46667 q^{12} +(3.40300 - 1.19146i) q^{13} -0.231361 q^{14} +(-3.70657 - 6.41997i) q^{15} +(-1.84085 - 3.18844i) q^{16} +(0.687890 - 1.19146i) q^{17} +1.85953 q^{18} +(-1.61766 + 2.80186i) q^{19} +(-2.17164 + 3.76139i) q^{20} +3.32225 q^{21} +(-0.384320 + 0.665661i) q^{22} +(-0.419251 - 0.726164i) q^{23} +(-1.51671 - 2.62701i) q^{24} -0.0210289 q^{25} +(-0.632387 - 0.544012i) q^{26} -16.7354 q^{27} +(-0.973236 - 1.68569i) q^{28} +(0.303571 + 0.525800i) q^{29} +(-0.857556 + 1.48533i) q^{30} +1.71511 q^{31} +(-1.33896 + 2.31915i) q^{32} +(5.51868 - 9.55864i) q^{33} -0.318302 q^{34} +(-1.11568 + 1.93242i) q^{35} +(7.82225 + 13.5485i) q^{36} +(-0.776807 - 1.34547i) q^{37} +0.748524 q^{38} +(9.08083 + 7.81180i) q^{39} +2.03736 q^{40} +(4.58892 + 7.94824i) q^{41} +(-0.384320 - 0.665661i) q^{42} +(-0.615680 + 1.06639i) q^{43} -6.46667 q^{44} +(8.96713 - 15.5315i) q^{45} +(-0.0969983 + 0.168006i) q^{46} -1.62817 q^{47} +(6.11577 - 10.5928i) q^{48} +(-0.500000 - 0.866025i) q^{49} +(0.00243263 + 0.00421343i) q^{50} +4.57069 q^{51} +(1.30348 - 6.89599i) q^{52} +8.39607 q^{53} +(1.93596 + 3.35318i) q^{54} +(3.70657 + 6.41997i) q^{55} +(-0.456530 + 0.790732i) q^{56} -10.7485 q^{57} +(0.0702344 - 0.121650i) q^{58} +(-4.41117 + 7.64037i) q^{59} -14.4295 q^{60} +(-2.73334 + 4.73428i) q^{61} +(-0.198405 - 0.343647i) q^{62} +(4.01868 + 6.96056i) q^{63} -6.74383 q^{64} +(-7.59332 + 2.65858i) q^{65} -2.55361 q^{66} +(5.09287 + 8.82111i) q^{67} +(-1.33896 - 2.31915i) q^{68} +(1.39286 - 2.41250i) q^{69} +0.516249 q^{70} +(2.60714 - 4.51570i) q^{71} +(3.66929 - 6.35540i) q^{72} -3.96355 q^{73} +(-0.179723 + 0.311289i) q^{74} +(-0.0349316 - 0.0605033i) q^{75} +(3.14872 + 5.45375i) q^{76} -3.32225 q^{77} +(0.514731 - 2.72315i) q^{78} +6.45051 q^{79} +(4.10760 + 7.11457i) q^{80} +(-15.7436 - 27.2687i) q^{81} +(1.06170 - 1.83891i) q^{82} -4.64055 q^{83} +(3.23334 - 5.60030i) q^{84} +(-1.53493 + 2.65858i) q^{85} +0.284889 q^{86} +(-1.00854 + 1.74684i) q^{87} +(1.51671 + 2.62701i) q^{88} +(-4.56413 - 7.90530i) q^{89} -4.14929 q^{90} +(0.669665 - 3.54282i) q^{91} -1.63212 q^{92} +(2.84902 + 4.93464i) q^{93} +(0.188347 + 0.326227i) q^{94} +(3.60957 - 6.25197i) q^{95} -8.89672 q^{96} +(7.67944 - 13.3012i) q^{97} +(-0.115680 + 0.200364i) q^{98} +26.7022 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + q^{2} - q^{3} - 5 q^{4} - 14 q^{5} + 5 q^{6} + 4 q^{7} - 12 q^{8} - 7 q^{9} + 11 q^{10} + q^{11} + 24 q^{12} + 4 q^{13} + 2 q^{14} - 3 q^{15} - 19 q^{16} + 4 q^{17} - 6 q^{18} - q^{19} + 2 q^{20}+ \cdots + 46 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/91\mathbb{Z}\right)^\times\).

\(n\) \(15\) \(66\)
\(\chi(n)\) \(e\left(\frac{1}{3}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.115680 − 0.200364i −0.0817984 − 0.141679i 0.822224 − 0.569164i \(-0.192733\pi\)
−0.904022 + 0.427485i \(0.859400\pi\)
\(3\) 1.66113 + 2.87716i 0.959052 + 1.66113i 0.724811 + 0.688948i \(0.241927\pi\)
0.234241 + 0.972179i \(0.424740\pi\)
\(4\) 0.973236 − 1.68569i 0.486618 − 0.842847i
\(5\) −2.23136 −0.997895 −0.498947 − 0.866632i \(-0.666280\pi\)
−0.498947 + 0.866632i \(0.666280\pi\)
\(6\) 0.384320 − 0.665661i 0.156898 − 0.271755i
\(7\) 0.500000 − 0.866025i 0.188982 − 0.327327i
\(8\) −0.913059 −0.322815
\(9\) −4.01868 + 6.96056i −1.33956 + 2.32019i
\(10\) 0.258125 + 0.447085i 0.0816262 + 0.141381i
\(11\) −1.66113 − 2.87716i −0.500848 − 0.867495i −1.00000 0.000980003i \(-0.999688\pi\)
0.499151 − 0.866515i \(-0.333645\pi\)
\(12\) 6.46667 1.86677
\(13\) 3.40300 − 1.19146i 0.943823 − 0.330452i
\(14\) −0.231361 −0.0618338
\(15\) −3.70657 − 6.41997i −0.957033 − 1.65763i
\(16\) −1.84085 − 3.18844i −0.460212 − 0.797111i
\(17\) 0.687890 − 1.19146i 0.166838 − 0.288972i −0.770469 − 0.637478i \(-0.779978\pi\)
0.937306 + 0.348506i \(0.113311\pi\)
\(18\) 1.85953 0.438296
\(19\) −1.61766 + 2.80186i −0.371116 + 0.642791i −0.989737 − 0.142898i \(-0.954358\pi\)
0.618622 + 0.785689i \(0.287691\pi\)
\(20\) −2.17164 + 3.76139i −0.485594 + 0.841073i
\(21\) 3.32225 0.724975
\(22\) −0.384320 + 0.665661i −0.0819372 + 0.141919i
\(23\) −0.419251 − 0.726164i −0.0874199 − 0.151416i 0.819000 − 0.573794i \(-0.194529\pi\)
−0.906420 + 0.422378i \(0.861196\pi\)
\(24\) −1.51671 − 2.62701i −0.309596 − 0.536237i
\(25\) −0.0210289 −0.00420577
\(26\) −0.632387 − 0.544012i −0.124021 − 0.106689i
\(27\) −16.7354 −3.22073
\(28\) −0.973236 − 1.68569i −0.183924 − 0.318566i
\(29\) 0.303571 + 0.525800i 0.0563717 + 0.0976386i 0.892834 − 0.450386i \(-0.148713\pi\)
−0.836462 + 0.548024i \(0.815380\pi\)
\(30\) −0.857556 + 1.48533i −0.156568 + 0.271183i
\(31\) 1.71511 0.308043 0.154022 − 0.988067i \(-0.450777\pi\)
0.154022 + 0.988067i \(0.450777\pi\)
\(32\) −1.33896 + 2.31915i −0.236697 + 0.409971i
\(33\) 5.51868 − 9.55864i 0.960679 − 1.66395i
\(34\) −0.318302 −0.0545883
\(35\) −1.11568 + 1.93242i −0.188584 + 0.326638i
\(36\) 7.82225 + 13.5485i 1.30371 + 2.25809i
\(37\) −0.776807 − 1.34547i −0.127706 − 0.221194i 0.795081 − 0.606503i \(-0.207428\pi\)
−0.922788 + 0.385309i \(0.874095\pi\)
\(38\) 0.748524 0.121427
\(39\) 9.08083 + 7.81180i 1.45410 + 1.25089i
\(40\) 2.03736 0.322136
\(41\) 4.58892 + 7.94824i 0.716668 + 1.24131i 0.962313 + 0.271946i \(0.0876672\pi\)
−0.245644 + 0.969360i \(0.578999\pi\)
\(42\) −0.384320 − 0.665661i −0.0593018 − 0.102714i
\(43\) −0.615680 + 1.06639i −0.0938904 + 0.162623i −0.909145 − 0.416480i \(-0.863264\pi\)
0.815255 + 0.579103i \(0.196597\pi\)
\(44\) −6.46667 −0.974888
\(45\) 8.96713 − 15.5315i 1.33674 − 2.31530i
\(46\) −0.0969983 + 0.168006i −0.0143016 + 0.0247711i
\(47\) −1.62817 −0.237493 −0.118747 − 0.992925i \(-0.537888\pi\)
−0.118747 + 0.992925i \(0.537888\pi\)
\(48\) 6.11577 − 10.5928i 0.882735 − 1.52894i
\(49\) −0.500000 − 0.866025i −0.0714286 − 0.123718i
\(50\) 0.00243263 + 0.00421343i 0.000344025 + 0.000595870i
\(51\) 4.57069 0.640025
\(52\) 1.30348 − 6.89599i 0.180761 − 0.956302i
\(53\) 8.39607 1.15329 0.576644 − 0.816995i \(-0.304362\pi\)
0.576644 + 0.816995i \(0.304362\pi\)
\(54\) 1.93596 + 3.35318i 0.263450 + 0.456310i
\(55\) 3.70657 + 6.41997i 0.499794 + 0.865669i
\(56\) −0.456530 + 0.790732i −0.0610063 + 0.105666i
\(57\) −10.7485 −1.42368
\(58\) 0.0702344 − 0.121650i 0.00922223 − 0.0159734i
\(59\) −4.41117 + 7.64037i −0.574285 + 0.994691i 0.421834 + 0.906673i \(0.361387\pi\)
−0.996119 + 0.0880181i \(0.971947\pi\)
\(60\) −14.4295 −1.86284
\(61\) −2.73334 + 4.73428i −0.349968 + 0.606162i −0.986243 − 0.165300i \(-0.947141\pi\)
0.636276 + 0.771462i \(0.280474\pi\)
\(62\) −0.198405 − 0.343647i −0.0251974 − 0.0436432i
\(63\) 4.01868 + 6.96056i 0.506306 + 0.876948i
\(64\) −6.74383 −0.842979
\(65\) −7.59332 + 2.65858i −0.941836 + 0.329756i
\(66\) −2.55361 −0.314328
\(67\) 5.09287 + 8.82111i 0.622193 + 1.07767i 0.989077 + 0.147403i \(0.0470913\pi\)
−0.366884 + 0.930267i \(0.619575\pi\)
\(68\) −1.33896 − 2.31915i −0.162373 − 0.281238i
\(69\) 1.39286 − 2.41250i 0.167680 − 0.290431i
\(70\) 0.516249 0.0617036
\(71\) 2.60714 − 4.51570i 0.309411 − 0.535915i −0.668823 − 0.743422i \(-0.733202\pi\)
0.978234 + 0.207507i \(0.0665349\pi\)
\(72\) 3.66929 − 6.35540i 0.432430 − 0.748992i
\(73\) −3.96355 −0.463898 −0.231949 − 0.972728i \(-0.574510\pi\)
−0.231949 + 0.972728i \(0.574510\pi\)
\(74\) −0.179723 + 0.311289i −0.0208923 + 0.0361866i
\(75\) −0.0349316 − 0.0605033i −0.00403355 − 0.00698632i
\(76\) 3.14872 + 5.45375i 0.361183 + 0.625588i
\(77\) −3.32225 −0.378606
\(78\) 0.514731 − 2.72315i 0.0582818 − 0.308336i
\(79\) 6.45051 0.725739 0.362869 − 0.931840i \(-0.381797\pi\)
0.362869 + 0.931840i \(0.381797\pi\)
\(80\) 4.10760 + 7.11457i 0.459243 + 0.795433i
\(81\) −15.7436 − 27.2687i −1.74929 − 3.02985i
\(82\) 1.06170 − 1.83891i 0.117245 − 0.203074i
\(83\) −4.64055 −0.509367 −0.254684 − 0.967024i \(-0.581971\pi\)
−0.254684 + 0.967024i \(0.581971\pi\)
\(84\) 3.23334 − 5.60030i 0.352786 − 0.611043i
\(85\) −1.53493 + 2.65858i −0.166487 + 0.288363i
\(86\) 0.284889 0.0307203
\(87\) −1.00854 + 1.74684i −0.108127 + 0.187281i
\(88\) 1.51671 + 2.62701i 0.161681 + 0.280041i
\(89\) −4.56413 − 7.90530i −0.483797 − 0.837960i 0.516030 − 0.856570i \(-0.327409\pi\)
−0.999827 + 0.0186101i \(0.994076\pi\)
\(90\) −4.14929 −0.437373
\(91\) 0.669665 − 3.54282i 0.0702000 − 0.371388i
\(92\) −1.63212 −0.170160
\(93\) 2.84902 + 4.93464i 0.295429 + 0.511699i
\(94\) 0.188347 + 0.326227i 0.0194266 + 0.0336478i
\(95\) 3.60957 − 6.25197i 0.370334 − 0.641438i
\(96\) −8.89672 −0.908018
\(97\) 7.67944 − 13.3012i 0.779729 − 1.35053i −0.152369 − 0.988324i \(-0.548690\pi\)
0.932098 − 0.362206i \(-0.117976\pi\)
\(98\) −0.115680 + 0.200364i −0.0116855 + 0.0202399i
\(99\) 26.7022 2.68367
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 91.2.f.c.29.2 yes 8
3.2 odd 2 819.2.o.h.757.3 8
4.3 odd 2 1456.2.s.q.1121.1 8
7.2 even 3 637.2.g.k.263.2 8
7.3 odd 6 637.2.h.i.471.3 8
7.4 even 3 637.2.h.h.471.3 8
7.5 odd 6 637.2.g.j.263.2 8
7.6 odd 2 637.2.f.i.393.2 8
13.2 odd 12 1183.2.c.g.337.4 8
13.3 even 3 1183.2.a.k.1.3 4
13.9 even 3 inner 91.2.f.c.22.2 ✓ 8
13.10 even 6 1183.2.a.l.1.2 4
13.11 odd 12 1183.2.c.g.337.5 8
39.35 odd 6 819.2.o.h.568.3 8
52.35 odd 6 1456.2.s.q.113.1 8
91.9 even 3 637.2.h.h.165.3 8
91.48 odd 6 637.2.f.i.295.2 8
91.55 odd 6 8281.2.a.bp.1.3 4
91.61 odd 6 637.2.h.i.165.3 8
91.62 odd 6 8281.2.a.bt.1.2 4
91.74 even 3 637.2.g.k.373.2 8
91.87 odd 6 637.2.g.j.373.2 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
91.2.f.c.22.2 ✓ 8 13.9 even 3 inner
91.2.f.c.29.2 yes 8 1.1 even 1 trivial
637.2.f.i.295.2 8 91.48 odd 6
637.2.f.i.393.2 8 7.6 odd 2
637.2.g.j.263.2 8 7.5 odd 6
637.2.g.j.373.2 8 91.87 odd 6
637.2.g.k.263.2 8 7.2 even 3
637.2.g.k.373.2 8 91.74 even 3
637.2.h.h.165.3 8 91.9 even 3
637.2.h.h.471.3 8 7.4 even 3
637.2.h.i.165.3 8 91.61 odd 6
637.2.h.i.471.3 8 7.3 odd 6
819.2.o.h.568.3 8 39.35 odd 6
819.2.o.h.757.3 8 3.2 odd 2
1183.2.a.k.1.3 4 13.3 even 3
1183.2.a.l.1.2 4 13.10 even 6
1183.2.c.g.337.4 8 13.2 odd 12
1183.2.c.g.337.5 8 13.11 odd 12
1456.2.s.q.113.1 8 52.35 odd 6
1456.2.s.q.1121.1 8 4.3 odd 2
8281.2.a.bp.1.3 4 91.55 odd 6
8281.2.a.bt.1.2 4 91.62 odd 6