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Properties

Label 117.2.h.a.22.7
Level $117$
Weight $2$
Character 117.22
Analytic conductor $0.934$
Analytic rank $0$
Dimension $24$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [117,2,Mod(16,117)] 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("117.16"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(117, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([4, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 117 = 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 117.h (of order \(3\), degree \(2\), 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: \(0.934249703649\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(\zeta_{3})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 22.7
Character \(\chi\) \(=\) 117.22
Dual form 117.2.h.a.16.7

$q$-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+0.216364 q^{2} +(-1.67508 + 0.440592i) q^{3} -1.95319 q^{4} +(-0.702153 + 1.21616i) q^{5} +(-0.362427 + 0.0953285i) q^{6} +(-1.65726 + 2.87046i) q^{7} -0.855329 q^{8} +(2.61176 - 1.47605i) q^{9} +(-0.151921 + 0.263135i) q^{10} -4.20649 q^{11} +(3.27174 - 0.860559i) q^{12} +(3.00054 + 1.99919i) q^{13} +(-0.358572 + 0.621065i) q^{14} +(0.640326 - 2.34653i) q^{15} +3.72131 q^{16} +(-2.86476 - 4.96191i) q^{17} +(0.565091 - 0.319365i) q^{18} +(0.190838 + 0.330541i) q^{19} +(1.37144 - 2.37540i) q^{20} +(1.51133 - 5.53841i) q^{21} -0.910136 q^{22} +(3.40591 + 5.89922i) q^{23} +(1.43274 - 0.376851i) q^{24} +(1.51396 + 2.62226i) q^{25} +(0.649210 + 0.432554i) q^{26} +(-3.72455 + 3.62322i) q^{27} +(3.23694 - 5.60654i) q^{28} -4.56664 q^{29} +(0.138544 - 0.507706i) q^{30} +(0.165976 - 0.287478i) q^{31} +2.51582 q^{32} +(7.04620 - 1.85335i) q^{33} +(-0.619832 - 1.07358i) q^{34} +(-2.32730 - 4.03100i) q^{35} +(-5.10125 + 2.88300i) q^{36} +(-4.85317 + 8.40594i) q^{37} +(0.0412905 + 0.0715172i) q^{38} +(-5.90696 - 2.02679i) q^{39} +(0.600572 - 1.04022i) q^{40} +(-0.557112 - 0.964947i) q^{41} +(0.326999 - 1.19831i) q^{42} +(1.05270 - 1.82333i) q^{43} +8.21607 q^{44} +(-0.0387317 + 4.21274i) q^{45} +(0.736918 + 1.27638i) q^{46} +(0.177929 + 0.308182i) q^{47} +(-6.23348 + 1.63958i) q^{48} +(-1.99302 - 3.45201i) q^{49} +(0.327568 + 0.567364i) q^{50} +(6.98487 + 7.04938i) q^{51} +(-5.86061 - 3.90480i) q^{52} +8.84890 q^{53} +(-0.805861 + 0.783935i) q^{54} +(2.95360 - 5.11579i) q^{55} +(1.41750 - 2.45518i) q^{56} +(-0.465301 - 0.469599i) q^{57} -0.988058 q^{58} -13.4147 q^{59} +(-1.25068 + 4.58321i) q^{60} +(-0.445428 + 0.771504i) q^{61} +(0.0359112 - 0.0622000i) q^{62} +(-0.0914166 + 9.94313i) q^{63} -6.89829 q^{64} +(-4.53818 + 2.24541i) q^{65} +(1.52455 - 0.400999i) q^{66} +(-0.390882 - 0.677027i) q^{67} +(5.59541 + 9.69153i) q^{68} +(-8.30431 - 8.38101i) q^{69} +(-0.503545 - 0.872165i) q^{70} +(5.32069 + 9.21571i) q^{71} +(-2.23391 + 1.26251i) q^{72} +0.857988 q^{73} +(-1.05005 + 1.81875i) q^{74} +(-3.69135 - 3.72544i) q^{75} +(-0.372742 - 0.645607i) q^{76} +(6.97125 - 12.0746i) q^{77} +(-1.27806 - 0.438524i) q^{78} +(-2.58775 - 4.48211i) q^{79} +(-2.61293 + 4.52573i) q^{80} +(4.64255 - 7.71017i) q^{81} +(-0.120539 - 0.208780i) q^{82} +(4.78446 + 8.28693i) q^{83} +(-2.95192 + 10.8175i) q^{84} +8.04599 q^{85} +(0.227767 - 0.394504i) q^{86} +(7.64946 - 2.01203i) q^{87} +3.59794 q^{88} +(2.55436 - 4.42428i) q^{89} +(-0.00838016 + 0.911487i) q^{90} +(-10.7113 + 5.29973i) q^{91} +(-6.65238 - 11.5223i) q^{92} +(-0.151361 + 0.554675i) q^{93} +(0.0384975 + 0.0666797i) q^{94} -0.535989 q^{95} +(-4.21418 + 1.10845i) q^{96} +(3.66171 - 6.34226i) q^{97} +(-0.431218 - 0.746891i) q^{98} +(-10.9863 + 6.20900i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q - 2 q^{2} - q^{3} + 18 q^{4} - 2 q^{5} - 12 q^{6} + 3 q^{7} - 18 q^{8} - 3 q^{9} + 6 q^{11} - 3 q^{12} + 2 q^{14} + 11 q^{15} + 6 q^{16} + 6 q^{17} - 8 q^{18} - 3 q^{19} - 11 q^{20} - 25 q^{21} - 18 q^{22}+ \cdots + 18 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/117\mathbb{Z}\right)^\times\).

\(n\) \(28\) \(92\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.216364 0.152993 0.0764964 − 0.997070i \(-0.475627\pi\)
0.0764964 + 0.997070i \(0.475627\pi\)
\(3\) −1.67508 + 0.440592i −0.967105 + 0.254376i
\(4\) −1.95319 −0.976593
\(5\) −0.702153 + 1.21616i −0.314012 + 0.543885i −0.979227 − 0.202767i \(-0.935007\pi\)
0.665215 + 0.746652i \(0.268340\pi\)
\(6\) −0.362427 + 0.0953285i −0.147960 + 0.0389177i
\(7\) −1.65726 + 2.87046i −0.626385 + 1.08493i 0.361886 + 0.932222i \(0.382133\pi\)
−0.988271 + 0.152709i \(0.951200\pi\)
\(8\) −0.855329 −0.302404
\(9\) 2.61176 − 1.47605i 0.870586 − 0.492017i
\(10\) −0.151921 + 0.263135i −0.0480416 + 0.0832105i
\(11\) −4.20649 −1.26831 −0.634153 − 0.773208i \(-0.718651\pi\)
−0.634153 + 0.773208i \(0.718651\pi\)
\(12\) 3.27174 − 0.860559i 0.944469 − 0.248422i
\(13\) 3.00054 + 1.99919i 0.832199 + 0.554476i
\(14\) −0.358572 + 0.621065i −0.0958324 + 0.165987i
\(15\) 0.640326 − 2.34653i 0.165332 − 0.605872i
\(16\) 3.72131 0.930328
\(17\) −2.86476 − 4.96191i −0.694806 − 1.20344i −0.970246 − 0.242121i \(-0.922157\pi\)
0.275440 − 0.961318i \(-0.411176\pi\)
\(18\) 0.565091 − 0.319365i 0.133193 − 0.0752750i
\(19\) 0.190838 + 0.330541i 0.0437812 + 0.0758312i 0.887086 − 0.461605i \(-0.152726\pi\)
−0.843304 + 0.537436i \(0.819393\pi\)
\(20\) 1.37144 − 2.37540i 0.306662 − 0.531155i
\(21\) 1.51133 − 5.53841i 0.329800 − 1.20858i
\(22\) −0.910136 −0.194042
\(23\) 3.40591 + 5.89922i 0.710182 + 1.23007i 0.964789 + 0.263027i \(0.0847207\pi\)
−0.254607 + 0.967045i \(0.581946\pi\)
\(24\) 1.43274 − 0.376851i 0.292457 − 0.0769245i
\(25\) 1.51396 + 2.62226i 0.302793 + 0.524452i
\(26\) 0.649210 + 0.432554i 0.127320 + 0.0848309i
\(27\) −3.72455 + 3.62322i −0.716791 + 0.697288i
\(28\) 3.23694 − 5.60654i 0.611723 − 1.05954i
\(29\) −4.56664 −0.848003 −0.424002 − 0.905661i \(-0.639375\pi\)
−0.424002 + 0.905661i \(0.639375\pi\)
\(30\) 0.138544 − 0.507706i 0.0252945 − 0.0926940i
\(31\) 0.165976 − 0.287478i 0.0298101 − 0.0516326i −0.850735 − 0.525594i \(-0.823843\pi\)
0.880546 + 0.473961i \(0.157176\pi\)
\(32\) 2.51582 0.444738
\(33\) 7.04620 − 1.85335i 1.22659 − 0.322627i
\(34\) −0.619832 − 1.07358i −0.106300 − 0.184117i
\(35\) −2.32730 − 4.03100i −0.393385 − 0.681363i
\(36\) −5.10125 + 2.88300i −0.850208 + 0.480500i
\(37\) −4.85317 + 8.40594i −0.797857 + 1.38193i 0.123152 + 0.992388i \(0.460700\pi\)
−0.921009 + 0.389541i \(0.872634\pi\)
\(38\) 0.0412905 + 0.0715172i 0.00669820 + 0.0116016i
\(39\) −5.90696 − 2.02679i −0.945870 − 0.324545i
\(40\) 0.600572 − 1.04022i 0.0949587 − 0.164473i
\(41\) −0.557112 − 0.964947i −0.0870063 − 0.150699i 0.819238 − 0.573454i \(-0.194397\pi\)
−0.906244 + 0.422754i \(0.861063\pi\)
\(42\) 0.326999 − 1.19831i 0.0504570 − 0.184904i
\(43\) 1.05270 − 1.82333i 0.160536 − 0.278056i −0.774525 − 0.632543i \(-0.782011\pi\)
0.935061 + 0.354487i \(0.115345\pi\)
\(44\) 8.21607 1.23862
\(45\) −0.0387317 + 4.21274i −0.00577378 + 0.627998i
\(46\) 0.736918 + 1.27638i 0.108653 + 0.188192i
\(47\) 0.177929 + 0.308182i 0.0259536 + 0.0449530i 0.878710 − 0.477355i \(-0.158404\pi\)
−0.852757 + 0.522308i \(0.825071\pi\)
\(48\) −6.23348 + 1.63958i −0.899725 + 0.236653i
\(49\) −1.99302 − 3.45201i −0.284717 − 0.493144i
\(50\) 0.327568 + 0.567364i 0.0463251 + 0.0802373i
\(51\) 6.98487 + 7.04938i 0.978077 + 0.987110i
\(52\) −5.86061 − 3.90480i −0.812720 − 0.541498i
\(53\) 8.84890 1.21549 0.607745 − 0.794132i \(-0.292074\pi\)
0.607745 + 0.794132i \(0.292074\pi\)
\(54\) −0.805861 + 0.783935i −0.109664 + 0.106680i
\(55\) 2.95360 − 5.11579i 0.398264 − 0.689813i
\(56\) 1.41750 − 2.45518i 0.189422 − 0.328088i
\(57\) −0.465301 − 0.469599i −0.0616307 − 0.0621999i
\(58\) −0.988058 −0.129738
\(59\) −13.4147 −1.74645 −0.873226 − 0.487316i \(-0.837976\pi\)
−0.873226 + 0.487316i \(0.837976\pi\)
\(60\) −1.25068 + 4.58321i −0.161462 + 0.591690i
\(61\) −0.445428 + 0.771504i −0.0570313 + 0.0987810i −0.893131 − 0.449796i \(-0.851497\pi\)
0.836100 + 0.548577i \(0.184830\pi\)
\(62\) 0.0359112 − 0.0622000i 0.00456073 − 0.00789941i
\(63\) −0.0914166 + 9.94313i −0.0115174 + 1.25272i
\(64\) −6.89829 −0.862286
\(65\) −4.53818 + 2.24541i −0.562892 + 0.278509i
\(66\) 1.52455 − 0.400999i 0.187659 − 0.0493595i
\(67\) −0.390882 − 0.677027i −0.0477538 − 0.0827120i 0.841160 − 0.540786i \(-0.181873\pi\)
−0.888914 + 0.458074i \(0.848540\pi\)
\(68\) 5.59541 + 9.69153i 0.678543 + 1.17527i
\(69\) −8.30431 − 8.38101i −0.999722 − 1.00896i
\(70\) −0.503545 − 0.872165i −0.0601851 − 0.104244i
\(71\) 5.32069 + 9.21571i 0.631450 + 1.09370i 0.987256 + 0.159143i \(0.0508731\pi\)
−0.355806 + 0.934560i \(0.615794\pi\)
\(72\) −2.23391 + 1.26251i −0.263269 + 0.148788i
\(73\) 0.857988 0.100420 0.0502100 − 0.998739i \(-0.484011\pi\)
0.0502100 + 0.998739i \(0.484011\pi\)
\(74\) −1.05005 + 1.81875i −0.122066 + 0.211425i
\(75\) −3.69135 − 3.72544i −0.426240 − 0.430177i
\(76\) −0.372742 − 0.645607i −0.0427564 − 0.0740562i
\(77\) 6.97125 − 12.0746i 0.794448 − 1.37602i
\(78\) −1.27806 − 0.438524i −0.144711 − 0.0496531i
\(79\) −2.58775 − 4.48211i −0.291145 − 0.504277i 0.682936 − 0.730478i \(-0.260703\pi\)
−0.974081 + 0.226201i \(0.927369\pi\)
\(80\) −2.61293 + 4.52573i −0.292134 + 0.505991i
\(81\) 4.64255 − 7.71017i 0.515839 − 0.856686i
\(82\) −0.120539 − 0.208780i −0.0133113 − 0.0230559i
\(83\) 4.78446 + 8.28693i 0.525163 + 0.909608i 0.999571 + 0.0293033i \(0.00932886\pi\)
−0.474408 + 0.880305i \(0.657338\pi\)
\(84\) −2.95192 + 10.8175i −0.322080 + 1.18029i
\(85\) 8.04599 0.872711
\(86\) 0.227767 − 0.394504i 0.0245608 − 0.0425405i
\(87\) 7.64946 − 2.01203i 0.820109 − 0.215712i
\(88\) 3.59794 0.383541
\(89\) 2.55436 − 4.42428i 0.270761 − 0.468972i −0.698296 − 0.715809i \(-0.746058\pi\)
0.969057 + 0.246837i \(0.0793913\pi\)
\(90\) −0.00838016 + 0.911487i −0.000883346 + 0.0960792i
\(91\) −10.7113 + 5.29973i −1.12285 + 0.555563i
\(92\) −6.65238 − 11.5223i −0.693559 − 1.20128i
\(93\) −0.151361 + 0.554675i −0.0156954 + 0.0575172i
\(94\) 0.0384975 + 0.0666797i 0.00397072 + 0.00687748i
\(95\) −0.535989 −0.0549913
\(96\) −4.21418 + 1.10845i −0.430108 + 0.113131i
\(97\) 3.66171 − 6.34226i 0.371790 − 0.643959i −0.618051 − 0.786138i \(-0.712077\pi\)
0.989841 + 0.142179i \(0.0454108\pi\)
\(98\) −0.431218 − 0.746891i −0.0435596 − 0.0754474i
\(99\) −10.9863 + 6.20900i −1.10417 + 0.624028i
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 117.2.h.a.22.7 yes 24
3.2 odd 2 351.2.h.a.334.6 24
9.2 odd 6 351.2.f.a.100.7 24
9.7 even 3 117.2.f.a.61.6 ✓ 24
13.3 even 3 117.2.f.a.94.6 yes 24
39.29 odd 6 351.2.f.a.172.7 24
117.16 even 3 inner 117.2.h.a.16.7 yes 24
117.29 odd 6 351.2.h.a.289.6 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
117.2.f.a.61.6 ✓ 24 9.7 even 3
117.2.f.a.94.6 yes 24 13.3 even 3
117.2.h.a.16.7 yes 24 117.16 even 3 inner
117.2.h.a.22.7 yes 24 1.1 even 1 trivial
351.2.f.a.100.7 24 9.2 odd 6
351.2.f.a.172.7 24 39.29 odd 6
351.2.h.a.289.6 24 117.29 odd 6
351.2.h.a.334.6 24 3.2 odd 2