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 22.4
Root \(1.37054 - 2.37385i\) of defining polynomial
Character \(\chi\) \(=\) 91.22
Dual form 91.2.f.c.29.4

$q$-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+(1.37054 - 2.37385i) q^{2} +(-0.682410 + 1.18197i) q^{3} +(-2.75677 - 4.77486i) q^{4} +0.741082 q^{5} +(1.87054 + 3.23987i) q^{6} +(0.500000 + 0.866025i) q^{7} -9.63087 q^{8} +(0.568634 + 0.984903i) q^{9} +(1.01568 - 1.75921i) q^{10} +(0.682410 - 1.18197i) q^{11} +7.52497 q^{12} +(0.301907 + 3.59289i) q^{13} +2.74108 q^{14} +(-0.505722 + 0.875935i) q^{15} +(-7.68598 + 13.3125i) q^{16} +(2.07436 + 3.59289i) q^{17} +3.11734 q^{18} +(-3.63303 - 6.29259i) q^{19} +(-2.04299 - 3.53856i) q^{20} -1.36482 q^{21} +(-1.87054 - 3.23987i) q^{22} +(1.16673 - 2.02083i) q^{23} +(6.57220 - 11.3834i) q^{24} -4.45080 q^{25} +(8.94274 + 4.20752i) q^{26} -5.64662 q^{27} +(2.75677 - 4.77486i) q^{28} +(0.203815 - 0.353017i) q^{29} +(1.38622 + 2.40101i) q^{30} -2.77245 q^{31} +(11.4370 + 19.8095i) q^{32} +(0.931366 + 1.61317i) q^{33} +11.3720 q^{34} +(0.370541 + 0.641796i) q^{35} +(3.13518 - 5.43029i) q^{36} +(3.05295 - 5.28787i) q^{37} -19.9169 q^{38} +(-4.45271 - 2.09498i) q^{39} -7.13727 q^{40} +(-0.627306 + 1.08653i) q^{41} +(-1.87054 + 3.23987i) q^{42} +(0.870541 + 1.50782i) q^{43} -7.52497 q^{44} +(0.421404 + 0.729894i) q^{45} +(-3.19809 - 5.53926i) q^{46} -5.85843 q^{47} +(-10.4900 - 18.1692i) q^{48} +(-0.500000 + 0.866025i) q^{49} +(-6.10000 + 10.5655i) q^{50} -5.66224 q^{51} +(16.3232 - 11.3463i) q^{52} +4.56778 q^{53} +(-7.73893 + 13.4042i) q^{54} +(0.505722 - 0.875935i) q^{55} +(-4.81544 - 8.34058i) q^{56} +9.91685 q^{57} +(-0.558672 - 0.967649i) q^{58} +(5.49213 + 9.51264i) q^{59} +5.57662 q^{60} +(-3.26249 - 5.65079i) q^{61} +(-3.79975 + 6.58137i) q^{62} +(-0.568634 + 0.984903i) q^{63} +31.9557 q^{64} +(0.223738 + 2.66263i) q^{65} +5.10590 q^{66} +(6.87983 - 11.9162i) q^{67} +(11.4370 - 19.8095i) q^{68} +(1.59237 + 2.75807i) q^{69} +2.03137 q^{70} +(2.40763 + 4.17014i) q^{71} +(-5.47644 - 9.48548i) q^{72} +6.06987 q^{73} +(-8.36839 - 14.4945i) q^{74} +(3.03727 - 5.26070i) q^{75} +(-20.0308 + 34.6944i) q^{76} +1.36482 q^{77} +(-11.0758 + 7.69879i) q^{78} -9.12582 q^{79} +(-5.69594 + 9.86566i) q^{80} +(2.14741 - 3.71942i) q^{81} +(1.71950 + 2.97826i) q^{82} +11.7368 q^{83} +(3.76249 + 6.51682i) q^{84} +(1.53727 + 2.66263i) q^{85} +4.77245 q^{86} +(0.278170 + 0.481805i) q^{87} +(-6.57220 + 11.3834i) q^{88} +(0.880503 - 1.52508i) q^{89} +2.31021 q^{90} +(-2.96058 + 2.05790i) q^{91} -12.8656 q^{92} +(1.89195 - 3.27695i) q^{93} +(-8.02921 + 13.9070i) q^{94} +(-2.69237 - 4.66332i) q^{95} -31.2189 q^{96} +(-4.76691 - 8.25652i) q^{97} +(1.37054 + 2.37385i) q^{98} +1.55217 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 \(\left(\mathbb{Z}/91\mathbb{Z}\right)^\times\).

\(n\) \(15\) \(66\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.37054 − 2.37385i 0.969119 − 1.67856i 0.271003 − 0.962579i \(-0.412645\pi\)
0.698116 − 0.715984i \(-0.254022\pi\)
\(3\) −0.682410 + 1.18197i −0.393989 + 0.682410i −0.992972 − 0.118353i \(-0.962239\pi\)
0.598982 + 0.800762i \(0.295572\pi\)
\(4\) −2.75677 − 4.77486i −1.37838 − 2.38743i
\(5\) 0.741082 0.331422 0.165711 − 0.986174i \(-0.447008\pi\)
0.165711 + 0.986174i \(0.447008\pi\)
\(6\) 1.87054 + 3.23987i 0.763645 + 1.32267i
\(7\) 0.500000 + 0.866025i 0.188982 + 0.327327i
\(8\) −9.63087 −3.40503
\(9\) 0.568634 + 0.984903i 0.189545 + 0.328301i
\(10\) 1.01568 − 1.75921i 0.321187 − 0.556313i
\(11\) 0.682410 − 1.18197i 0.205754 − 0.356377i −0.744619 − 0.667490i \(-0.767369\pi\)
0.950373 + 0.311113i \(0.100702\pi\)
\(12\) 7.52497 2.17227
\(13\) 0.301907 + 3.59289i 0.0837339 + 0.996488i
\(14\) 2.74108 0.732585
\(15\) −0.505722 + 0.875935i −0.130577 + 0.226166i
\(16\) −7.68598 + 13.3125i −1.92149 + 3.32813i
\(17\) 2.07436 + 3.59289i 0.503105 + 0.871404i 0.999994 + 0.00358919i \(0.00114248\pi\)
−0.496888 + 0.867814i \(0.665524\pi\)
\(18\) 3.11734 0.734765
\(19\) −3.63303 − 6.29259i −0.833474 − 1.44362i −0.895267 − 0.445530i \(-0.853015\pi\)
0.0617933 − 0.998089i \(-0.480318\pi\)
\(20\) −2.04299 − 3.53856i −0.456826 − 0.791246i
\(21\) −1.36482 −0.297828
\(22\) −1.87054 − 3.23987i −0.398801 − 0.690743i
\(23\) 1.16673 − 2.02083i 0.243279 − 0.421372i −0.718367 − 0.695664i \(-0.755110\pi\)
0.961646 + 0.274292i \(0.0884435\pi\)
\(24\) 6.57220 − 11.3834i 1.34155 − 2.32362i
\(25\) −4.45080 −0.890159
\(26\) 8.94274 + 4.20752i 1.75382 + 0.825163i
\(27\) −5.64662 −1.08669
\(28\) 2.75677 − 4.77486i 0.520980 − 0.902363i
\(29\) 0.203815 − 0.353017i 0.0378474 − 0.0655536i −0.846481 − 0.532419i \(-0.821283\pi\)
0.884329 + 0.466865i \(0.154617\pi\)
\(30\) 1.38622 + 2.40101i 0.253089 + 0.438363i
\(31\) −2.77245 −0.497946 −0.248973 − 0.968510i \(-0.580093\pi\)
−0.248973 + 0.968510i \(0.580093\pi\)
\(32\) 11.4370 + 19.8095i 2.02180 + 3.50186i
\(33\) 0.931366 + 1.61317i 0.162130 + 0.280817i
\(34\) 11.3720 1.95027
\(35\) 0.370541 + 0.641796i 0.0626329 + 0.108483i
\(36\) 3.13518 − 5.43029i 0.522530 − 0.905049i
\(37\) 3.05295 − 5.28787i 0.501902 − 0.869320i −0.498096 − 0.867122i \(-0.665967\pi\)
0.999998 − 0.00219764i \(-0.000699531\pi\)
\(38\) −19.9169 −3.23094
\(39\) −4.45271 − 2.09498i −0.713003 − 0.335465i
\(40\) −7.13727 −1.12850
\(41\) −0.627306 + 1.08653i −0.0979688 + 0.169687i −0.910844 − 0.412751i \(-0.864568\pi\)
0.812875 + 0.582438i \(0.197901\pi\)
\(42\) −1.87054 + 3.23987i −0.288631 + 0.499923i
\(43\) 0.870541 + 1.50782i 0.132756 + 0.229941i 0.924738 − 0.380604i \(-0.124284\pi\)
−0.791982 + 0.610545i \(0.790951\pi\)
\(44\) −7.52497 −1.13443
\(45\) 0.421404 + 0.729894i 0.0628193 + 0.108806i
\(46\) −3.19809 − 5.53926i −0.471533 − 0.816719i
\(47\) −5.85843 −0.854539 −0.427270 − 0.904124i \(-0.640525\pi\)
−0.427270 + 0.904124i \(0.640525\pi\)
\(48\) −10.4900 − 18.1692i −1.51410 − 2.62249i
\(49\) −0.500000 + 0.866025i −0.0714286 + 0.123718i
\(50\) −6.10000 + 10.5655i −0.862670 + 1.49419i
\(51\) −5.66224 −0.792872
\(52\) 16.3232 − 11.3463i 2.26363 − 1.57345i
\(53\) 4.56778 0.627433 0.313717 − 0.949517i \(-0.398426\pi\)
0.313717 + 0.949517i \(0.398426\pi\)
\(54\) −7.73893 + 13.4042i −1.05313 + 1.82408i
\(55\) 0.505722 − 0.875935i 0.0681915 − 0.118111i
\(56\) −4.81544 − 8.34058i −0.643490 − 1.11456i
\(57\) 9.91685 1.31352
\(58\) −0.558672 − 0.967649i −0.0733573 − 0.127059i
\(59\) 5.49213 + 9.51264i 0.715014 + 1.23844i 0.962954 + 0.269665i \(0.0869130\pi\)
−0.247940 + 0.968775i \(0.579754\pi\)
\(60\) 5.57662 0.719939
\(61\) −3.26249 − 5.65079i −0.417719 − 0.723510i 0.577991 − 0.816043i \(-0.303837\pi\)
−0.995710 + 0.0925333i \(0.970504\pi\)
\(62\) −3.79975 + 6.58137i −0.482569 + 0.835834i
\(63\) −0.568634 + 0.984903i −0.0716411 + 0.124086i
\(64\) 31.9557 3.99446
\(65\) 0.223738 + 2.66263i 0.0277513 + 0.330258i
\(66\) 5.10590 0.628493
\(67\) 6.87983 − 11.9162i 0.840505 − 1.45580i −0.0489630 − 0.998801i \(-0.515592\pi\)
0.889468 − 0.456997i \(-0.151075\pi\)
\(68\) 11.4370 − 19.8095i 1.38694 − 2.40226i
\(69\) 1.59237 + 2.75807i 0.191699 + 0.332032i
\(70\) 2.03137 0.242795
\(71\) 2.40763 + 4.17014i 0.285733 + 0.494904i 0.972787 − 0.231703i \(-0.0744296\pi\)
−0.687054 + 0.726607i \(0.741096\pi\)
\(72\) −5.47644 − 9.48548i −0.645405 − 1.11787i
\(73\) 6.06987 0.710425 0.355212 − 0.934786i \(-0.384409\pi\)
0.355212 + 0.934786i \(0.384409\pi\)
\(74\) −8.36839 − 14.4945i −0.972805 − 1.68495i
\(75\) 3.03727 − 5.26070i 0.350713 − 0.607453i
\(76\) −20.0308 + 34.6944i −2.29769 + 3.97972i
\(77\) 1.36482 0.155536
\(78\) −11.0758 + 7.69879i −1.25408 + 0.871716i
\(79\) −9.12582 −1.02674 −0.513368 − 0.858169i \(-0.671602\pi\)
−0.513368 + 0.858169i \(0.671602\pi\)
\(80\) −5.69594 + 9.86566i −0.636825 + 1.10301i
\(81\) 2.14741 − 3.71942i 0.238601 − 0.413269i
\(82\) 1.71950 + 2.97826i 0.189887 + 0.328894i
\(83\) 11.7368 1.28828 0.644139 − 0.764908i \(-0.277216\pi\)
0.644139 + 0.764908i \(0.277216\pi\)
\(84\) 3.76249 + 6.51682i 0.410521 + 0.711043i
\(85\) 1.53727 + 2.66263i 0.166740 + 0.288802i
\(86\) 4.77245 0.514626
\(87\) 0.278170 + 0.481805i 0.0298230 + 0.0516549i
\(88\) −6.57220 + 11.3834i −0.700599 + 1.21347i
\(89\) 0.880503 − 1.52508i 0.0933331 − 0.161658i −0.815579 − 0.578646i \(-0.803581\pi\)
0.908912 + 0.416989i \(0.136915\pi\)
\(90\) 2.31021 0.243517
\(91\) −2.96058 + 2.05790i −0.310353 + 0.215727i
\(92\) −12.8656 −1.34133
\(93\) 1.89195 − 3.27695i 0.196186 − 0.339803i
\(94\) −8.02921 + 13.9070i −0.828150 + 1.43440i
\(95\) −2.69237 − 4.66332i −0.276231 − 0.478447i
\(96\) −31.2189 −3.18627
\(97\) −4.76691 − 8.25652i −0.484006 − 0.838323i 0.515825 − 0.856694i \(-0.327485\pi\)
−0.999831 + 0.0183708i \(0.994152\pi\)
\(98\) 1.37054 + 2.37385i 0.138446 + 0.239795i
\(99\) 1.55217 0.155998
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.22.4 ✓ 8
3.2 odd 2 819.2.o.h.568.1 8
4.3 odd 2 1456.2.s.q.113.3 8
7.2 even 3 637.2.h.h.165.1 8
7.3 odd 6 637.2.g.j.373.4 8
7.4 even 3 637.2.g.k.373.4 8
7.5 odd 6 637.2.h.i.165.1 8
7.6 odd 2 637.2.f.i.295.4 8
13.3 even 3 inner 91.2.f.c.29.4 yes 8
13.4 even 6 1183.2.a.l.1.4 4
13.6 odd 12 1183.2.c.g.337.8 8
13.7 odd 12 1183.2.c.g.337.1 8
13.9 even 3 1183.2.a.k.1.1 4
39.29 odd 6 819.2.o.h.757.1 8
52.3 odd 6 1456.2.s.q.1121.3 8
91.3 odd 6 637.2.h.i.471.1 8
91.16 even 3 637.2.g.k.263.4 8
91.48 odd 6 8281.2.a.bp.1.1 4
91.55 odd 6 637.2.f.i.393.4 8
91.68 odd 6 637.2.g.j.263.4 8
91.69 odd 6 8281.2.a.bt.1.4 4
91.81 even 3 637.2.h.h.471.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
91.2.f.c.22.4 ✓ 8 1.1 even 1 trivial
91.2.f.c.29.4 yes 8 13.3 even 3 inner
637.2.f.i.295.4 8 7.6 odd 2
637.2.f.i.393.4 8 91.55 odd 6
637.2.g.j.263.4 8 91.68 odd 6
637.2.g.j.373.4 8 7.3 odd 6
637.2.g.k.263.4 8 91.16 even 3
637.2.g.k.373.4 8 7.4 even 3
637.2.h.h.165.1 8 7.2 even 3
637.2.h.h.471.1 8 91.81 even 3
637.2.h.i.165.1 8 7.5 odd 6
637.2.h.i.471.1 8 91.3 odd 6
819.2.o.h.568.1 8 3.2 odd 2
819.2.o.h.757.1 8 39.29 odd 6
1183.2.a.k.1.1 4 13.9 even 3
1183.2.a.l.1.4 4 13.4 even 6
1183.2.c.g.337.1 8 13.7 odd 12
1183.2.c.g.337.8 8 13.6 odd 12
1456.2.s.q.113.3 8 4.3 odd 2
1456.2.s.q.1121.3 8 52.3 odd 6
8281.2.a.bp.1.1 4 91.48 odd 6
8281.2.a.bt.1.4 4 91.69 odd 6