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(25,117)] 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("117.25"); 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, 3])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 117 = 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 117.t (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [20,0,2,12,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.934249703649\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 6x^{16} + 9x^{14} + 54x^{12} + 81x^{10} + 486x^{8} + 729x^{6} - 4374x^{4} + 59049 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3^{6} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 103.9
Root \(1.23798 + 1.21137i\) of defining polynomial
Character \(\chi\) \(=\) 117.103
Dual form 117.2.t.c.25.9

$q$-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+(1.97712 - 1.14149i) q^{2} +(0.833228 + 1.51846i) q^{3} +(1.60600 - 2.78168i) q^{4} +(-2.78501 - 1.60793i) q^{5} +(3.38070 + 2.05106i) q^{6} +(-2.09815 + 1.21137i) q^{7} -2.76698i q^{8} +(-1.61146 + 2.53045i) q^{9} -7.34174 q^{10} +(1.27730 - 0.737448i) q^{11} +(5.56204 + 0.120883i) q^{12} +(3.56557 + 0.535475i) q^{13} +(-2.76553 + 4.79003i) q^{14} +(0.121028 - 5.56871i) q^{15} +(0.0535232 + 0.0927049i) q^{16} -5.12974 q^{17} +(-0.297563 + 6.84248i) q^{18} -1.13065i q^{19} +(-8.94547 + 5.16467i) q^{20} +(-3.58765 - 2.17662i) q^{21} +(1.68358 - 2.91604i) q^{22} +(4.61735 - 7.99748i) q^{23} +(4.20155 - 2.30552i) q^{24} +(2.67087 + 4.62608i) q^{25} +(7.66079 - 3.01136i) q^{26} +(-5.18512 - 0.338499i) q^{27} +7.78182i q^{28} +(-0.487293 - 0.844016i) q^{29} +(-6.11735 - 11.1482i) q^{30} +(-3.16380 - 1.82662i) q^{31} +(5.00419 + 2.88917i) q^{32} +(2.18407 + 1.32507i) q^{33} +(-10.1421 + 5.85555i) q^{34} +7.79116 q^{35} +(4.45089 + 8.54647i) q^{36} +4.22691i q^{37} +(-1.29063 - 2.23543i) q^{38} +(2.15783 + 5.86035i) q^{39} +(-4.44910 + 7.70607i) q^{40} +(3.47188 + 2.00449i) q^{41} +(-9.57780 - 0.208159i) q^{42} +(4.33040 + 7.50047i) q^{43} -4.73737i q^{44} +(8.55673 - 4.45623i) q^{45} -21.0826i q^{46} +(1.33337 - 0.769820i) q^{47} +(-0.0961719 + 0.158517i) q^{48} +(-0.565185 + 0.978929i) q^{49} +(10.5613 + 6.09755i) q^{50} +(-4.27425 - 7.78932i) q^{51} +(7.21582 - 9.05828i) q^{52} +0.739889 q^{53} +(-10.6380 + 5.24951i) q^{54} -4.74305 q^{55} +(3.35182 + 5.80553i) q^{56} +(1.71685 - 0.942089i) q^{57} +(-1.92687 - 1.11248i) q^{58} +(-6.72630 - 3.88343i) q^{59} +(-15.2960 - 9.28002i) q^{60} +(-4.06781 - 7.04566i) q^{61} -8.34028 q^{62} +(0.315778 - 7.26133i) q^{63} +12.9777 q^{64} +(-9.06915 - 7.22448i) q^{65} +(5.83071 + 0.126722i) q^{66} +(-0.669411 - 0.386485i) q^{67} +(-8.23837 + 14.2693i) q^{68} +(15.9912 + 0.347545i) q^{69} +(15.4041 - 8.89354i) q^{70} -3.01136i q^{71} +(7.00171 + 4.45888i) q^{72} +9.21010i q^{73} +(4.82498 + 8.35711i) q^{74} +(-4.79909 + 7.91020i) q^{75} +(-3.14510 - 1.81582i) q^{76} +(-1.78664 + 3.09455i) q^{77} +(10.9558 + 9.12348i) q^{78} +(-1.86858 - 3.23648i) q^{79} -0.344246i q^{80} +(-3.80639 - 8.15545i) q^{81} +9.15243 q^{82} +(-12.3640 + 7.13838i) q^{83} +(-11.8164 + 6.48403i) q^{84} +(14.2864 + 8.24826i) q^{85} +(17.1234 + 9.88621i) q^{86} +(0.875581 - 1.44319i) q^{87} +(-2.04050 - 3.53425i) q^{88} +8.21257i q^{89} +(11.8309 - 18.5779i) q^{90} +(-8.12974 + 3.19570i) q^{91} +(-14.8309 - 25.6879i) q^{92} +(0.137489 - 6.32611i) q^{93} +(1.75748 - 3.04405i) q^{94} +(-1.81800 + 3.14887i) q^{95} +(-0.217466 + 10.0060i) q^{96} +(13.1880 - 7.61407i) q^{97} +2.58061i q^{98} +(-0.192237 + 4.42051i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 2 q^{3} + 12 q^{4} - 2 q^{9} - 16 q^{10} - 2 q^{12} - 4 q^{13} - 18 q^{14} + 4 q^{16} - 12 q^{17} - 10 q^{22} + 24 q^{23} - 12 q^{25} - 12 q^{26} - 22 q^{27} + 12 q^{29} - 54 q^{30} - 12 q^{35} + 50 q^{36}+ \cdots + 24 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.97712 − 1.14149i 1.39803 − 0.807156i 0.403848 − 0.914826i \(-0.367672\pi\)
0.994187 + 0.107670i \(0.0343391\pi\)
\(3\) 0.833228 + 1.51846i 0.481065 + 0.876685i
\(4\) 1.60600 − 2.78168i 0.803001 − 1.39084i
\(5\) −2.78501 − 1.60793i −1.24550 − 0.719088i −0.275289 − 0.961362i \(-0.588773\pi\)
−0.970208 + 0.242274i \(0.922107\pi\)
\(6\) 3.38070 + 2.05106i 1.38017 + 0.837342i
\(7\) −2.09815 + 1.21137i −0.793025 + 0.457853i −0.841026 − 0.540994i \(-0.818048\pi\)
0.0480013 + 0.998847i \(0.484715\pi\)
\(8\) − 2.76698i − 0.978274i
\(9\) −1.61146 + 2.53045i −0.537154 + 0.843484i
\(10\) −7.34174 −2.32166
\(11\) 1.27730 − 0.737448i 0.385119 − 0.222349i −0.294924 − 0.955521i \(-0.595294\pi\)
0.680043 + 0.733172i \(0.261961\pi\)
\(12\) 5.56204 + 0.120883i 1.60562 + 0.0348958i
\(13\) 3.56557 + 0.535475i 0.988910 + 0.148514i
\(14\) −2.76553 + 4.79003i −0.739118 + 1.28019i
\(15\) 0.121028 − 5.56871i 0.0312492 − 1.43784i
\(16\) 0.0535232 + 0.0927049i 0.0133808 + 0.0231762i
\(17\) −5.12974 −1.24414 −0.622072 − 0.782960i \(-0.713709\pi\)
−0.622072 + 0.782960i \(0.713709\pi\)
\(18\) −0.297563 + 6.84248i −0.0701362 + 1.61279i
\(19\) − 1.13065i − 0.259389i −0.991554 − 0.129694i \(-0.958600\pi\)
0.991554 − 0.129694i \(-0.0413996\pi\)
\(20\) −8.94547 + 5.16467i −2.00027 + 1.15486i
\(21\) −3.58765 − 2.17662i −0.782890 − 0.474976i
\(22\) 1.68358 − 2.91604i 0.358940 − 0.621703i
\(23\) 4.61735 − 7.99748i 0.962784 − 1.66759i 0.247328 − 0.968932i \(-0.420447\pi\)
0.715455 − 0.698658i \(-0.246219\pi\)
\(24\) 4.20155 − 2.30552i 0.857639 − 0.470613i
\(25\) 2.67087 + 4.62608i 0.534174 + 0.925217i
\(26\) 7.66079 − 3.01136i 1.50240 − 0.590577i
\(27\) −5.18512 − 0.338499i −0.997876 − 0.0651442i
\(28\) 7.78182i 1.47063i
\(29\) −0.487293 − 0.844016i −0.0904880 − 0.156730i 0.817229 − 0.576314i \(-0.195509\pi\)
−0.907717 + 0.419584i \(0.862176\pi\)
\(30\) −6.11735 − 11.1482i −1.11687 − 2.03537i
\(31\) −3.16380 − 1.82662i −0.568235 − 0.328071i 0.188209 − 0.982129i \(-0.439732\pi\)
−0.756444 + 0.654058i \(0.773065\pi\)
\(32\) 5.00419 + 2.88917i 0.884624 + 0.510738i
\(33\) 2.18407 + 1.32507i 0.380197 + 0.230664i
\(34\) −10.1421 + 5.85555i −1.73936 + 1.00422i
\(35\) 7.79116 1.31695
\(36\) 4.45089 + 8.54647i 0.741815 + 1.42441i
\(37\) 4.22691i 0.694900i 0.937698 + 0.347450i \(0.112952\pi\)
−0.937698 + 0.347450i \(0.887048\pi\)
\(38\) −1.29063 − 2.23543i −0.209367 − 0.362634i
\(39\) 2.15783 + 5.86035i 0.345530 + 0.938408i
\(40\) −4.44910 + 7.70607i −0.703465 + 1.21844i
\(41\) 3.47188 + 2.00449i 0.542217 + 0.313049i 0.745977 − 0.665972i \(-0.231983\pi\)
−0.203760 + 0.979021i \(0.565316\pi\)
\(42\) −9.57780 − 0.208159i −1.47789 − 0.0321197i
\(43\) 4.33040 + 7.50047i 0.660379 + 1.14381i 0.980516 + 0.196439i \(0.0629377\pi\)
−0.320137 + 0.947371i \(0.603729\pi\)
\(44\) − 4.73737i − 0.714185i
\(45\) 8.55673 − 4.45623i 1.27556 − 0.664296i
\(46\) − 21.0826i − 3.10847i
\(47\) 1.33337 − 0.769820i 0.194492 − 0.112290i −0.399592 − 0.916693i \(-0.630848\pi\)
0.594084 + 0.804403i \(0.297515\pi\)
\(48\) −0.0961719 + 0.158517i −0.0138812 + 0.0228800i
\(49\) −0.565185 + 0.978929i −0.0807407 + 0.139847i
\(50\) 10.5613 + 6.09755i 1.49359 + 0.862324i
\(51\) −4.27425 − 7.78932i −0.598514 − 1.09072i
\(52\) 7.21582 − 9.05828i 1.00065 − 1.25616i
\(53\) 0.739889 0.101632 0.0508158 − 0.998708i \(-0.483818\pi\)
0.0508158 + 0.998708i \(0.483818\pi\)
\(54\) −10.6380 + 5.24951i −1.44765 + 0.714367i
\(55\) −4.74305 −0.639553
\(56\) 3.35182 + 5.80553i 0.447906 + 0.775796i
\(57\) 1.71685 − 0.942089i 0.227402 − 0.124783i
\(58\) −1.92687 − 1.11248i −0.253011 − 0.146076i
\(59\) −6.72630 − 3.88343i −0.875689 − 0.505580i −0.00645471 − 0.999979i \(-0.502055\pi\)
−0.869235 + 0.494400i \(0.835388\pi\)
\(60\) −15.2960 − 9.28002i −1.97470 − 1.19805i
\(61\) −4.06781 − 7.04566i −0.520830 − 0.902104i −0.999707 − 0.0242218i \(-0.992289\pi\)
0.478877 − 0.877882i \(-0.341044\pi\)
\(62\) −8.34028 −1.05922
\(63\) 0.315778 − 7.26133i 0.0397843 − 0.914842i
\(64\) 12.9777 1.62222
\(65\) −9.06915 − 7.22448i −1.12489 − 0.896087i
\(66\) 5.83071 + 0.126722i 0.717711 + 0.0155984i
\(67\) −0.669411 − 0.386485i −0.0817816 − 0.0472166i 0.458552 − 0.888668i \(-0.348368\pi\)
−0.540333 + 0.841451i \(0.681702\pi\)
\(68\) −8.23837 + 14.2693i −0.999049 + 1.73040i
\(69\) 15.9912 + 0.347545i 1.92511 + 0.0418395i
\(70\) 15.4041 − 8.89354i 1.84114 − 1.06298i
\(71\) − 3.01136i − 0.357383i −0.983905 − 0.178692i \(-0.942814\pi\)
0.983905 − 0.178692i \(-0.0571864\pi\)
\(72\) 7.00171 + 4.45888i 0.825159 + 0.525484i
\(73\) 9.21010i 1.07796i 0.842318 + 0.538980i \(0.181190\pi\)
−0.842318 + 0.538980i \(0.818810\pi\)
\(74\) 4.82498 + 8.35711i 0.560893 + 0.971495i
\(75\) −4.79909 + 7.91020i −0.554152 + 0.913392i
\(76\) −3.14510 − 1.81582i −0.360768 − 0.208289i
\(77\) −1.78664 + 3.09455i −0.203606 + 0.352656i
\(78\) 10.9558 + 9.12348i 1.24050 + 1.03303i
\(79\) −1.86858 − 3.23648i −0.210232 − 0.364133i 0.741555 − 0.670892i \(-0.234089\pi\)
−0.951787 + 0.306759i \(0.900755\pi\)
\(80\) − 0.344246i − 0.0384879i
\(81\) −3.80639 − 8.15545i −0.422932 − 0.906161i
\(82\) 9.15243 1.01072
\(83\) −12.3640 + 7.13838i −1.35713 + 0.783539i −0.989236 − 0.146329i \(-0.953254\pi\)
−0.367893 + 0.929868i \(0.619921\pi\)
\(84\) −11.8164 + 6.48403i −1.28928 + 0.707466i
\(85\) 14.2864 + 8.24826i 1.54958 + 0.894649i
\(86\) 17.1234 + 9.88621i 1.84647 + 1.06606i
\(87\) 0.875581 − 1.44319i 0.0938721 − 0.154727i
\(88\) −2.04050 − 3.53425i −0.217518 − 0.376752i
\(89\) 8.21257i 0.870531i 0.900302 + 0.435265i \(0.143345\pi\)
−0.900302 + 0.435265i \(0.856655\pi\)
\(90\) 11.8309 − 18.5779i 1.24709 − 1.95829i
\(91\) −8.12974 + 3.19570i −0.852228 + 0.335000i
\(92\) −14.8309 − 25.6879i −1.54623 − 2.67815i
\(93\) 0.137489 − 6.32611i 0.0142569 − 0.655987i
\(94\) 1.75748 − 3.04405i 0.181271 − 0.313970i
\(95\) −1.81800 + 3.14887i −0.186523 + 0.323068i
\(96\) −0.217466 + 10.0060i −0.0221950 + 1.02123i
\(97\) 13.1880 − 7.61407i 1.33903 − 0.773092i 0.352370 − 0.935861i \(-0.385376\pi\)
0.986664 + 0.162769i \(0.0520425\pi\)
\(98\) 2.58061i 0.260681i
\(99\) −0.192237 + 4.42051i −0.0193206 + 0.444278i
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.t.c.103.9 yes 20
3.2 odd 2 351.2.t.c.64.2 20
9.2 odd 6 351.2.t.c.181.9 20
9.4 even 3 1053.2.b.j.649.9 10
9.5 odd 6 1053.2.b.i.649.2 10
9.7 even 3 inner 117.2.t.c.25.2 ✓ 20
13.12 even 2 inner 117.2.t.c.103.2 yes 20
39.38 odd 2 351.2.t.c.64.9 20
117.25 even 6 inner 117.2.t.c.25.9 yes 20
117.38 odd 6 351.2.t.c.181.2 20
117.77 odd 6 1053.2.b.i.649.9 10
117.103 even 6 1053.2.b.j.649.2 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
117.2.t.c.25.2 ✓ 20 9.7 even 3 inner
117.2.t.c.25.9 yes 20 117.25 even 6 inner
117.2.t.c.103.2 yes 20 13.12 even 2 inner
117.2.t.c.103.9 yes 20 1.1 even 1 trivial
351.2.t.c.64.2 20 3.2 odd 2
351.2.t.c.64.9 20 39.38 odd 2
351.2.t.c.181.2 20 117.38 odd 6
351.2.t.c.181.9 20 9.2 odd 6
1053.2.b.i.649.2 10 9.5 odd 6
1053.2.b.i.649.9 10 117.77 odd 6
1053.2.b.j.649.2 10 117.103 even 6
1053.2.b.j.649.9 10 9.4 even 3