↑

Properties

Label 117.2.bc.a.50.6
Level $117$
Weight $2$
Character 117.50
Analytic conductor $0.934$
Analytic rank $0$
Dimension $48$
Inner twists $2$

Related objects

Downloads

Learn more

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(20,117)] 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("117.20"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(117, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([2, 11])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 117 = 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 117.bc (of order \(12\), degree \(4\), 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: \(48\)
Relative dimension: \(12\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label 50.6
Character \(\chi\) \(=\) 117.50
Dual form 117.2.bc.a.110.6

$q$-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+(-0.104868 - 0.391374i) q^{2} +(0.735256 - 1.56825i) q^{3} +(1.58987 - 0.917915i) q^{4} +(0.537559 + 2.00620i) q^{5} +(-0.690875 - 0.123300i) q^{6} +(-1.39480 + 1.39480i) q^{7} +(-1.09899 - 1.09899i) q^{8} +(-1.91880 - 2.30613i) q^{9} +(0.728800 - 0.420773i) q^{10} +(-2.68955 + 0.720663i) q^{11} +(-0.270552 - 3.16822i) q^{12} +(3.47606 - 0.957605i) q^{13} +(0.692156 + 0.399616i) q^{14} +(3.54146 + 0.632043i) q^{15} +(1.52096 - 2.63439i) q^{16} +(-2.88941 + 5.00460i) q^{17} +(-0.701336 + 0.992806i) q^{18} +(-4.00984 + 1.07443i) q^{19} +(2.69617 + 2.69617i) q^{20} +(1.16185 + 3.21292i) q^{21} +(0.564096 + 0.977044i) q^{22} +2.93102 q^{23} +(-2.53152 + 0.915445i) q^{24} +(0.594268 - 0.343101i) q^{25} +(-0.739309 - 1.26002i) q^{26} +(-5.02738 + 1.31356i) q^{27} +(-0.937247 + 3.49785i) q^{28} +(-0.0236075 - 0.0136298i) q^{29} +(-0.124021 - 1.45231i) q^{30} +(-4.49489 + 1.20440i) q^{31} +(-4.19301 - 1.12351i) q^{32} +(-0.847330 + 4.74775i) q^{33} +(2.26167 + 0.606014i) q^{34} +(-3.54802 - 2.04845i) q^{35} +(-5.16747 - 1.90516i) q^{36} +(6.63559 + 1.77800i) q^{37} +(0.841010 + 1.45667i) q^{38} +(1.05403 - 6.15541i) q^{39} +(1.61401 - 2.79555i) q^{40} +(6.64791 - 6.64791i) q^{41} +(1.13561 - 0.791651i) q^{42} +6.27230i q^{43} +(-3.61454 + 3.61454i) q^{44} +(3.59508 - 5.08917i) q^{45} +(-0.307370 - 1.14712i) q^{46} +(-1.14324 + 4.26664i) q^{47} +(-3.01307 - 4.32220i) q^{48} +3.10909i q^{49} +(-0.196600 - 0.196600i) q^{50} +(5.72399 + 8.21096i) q^{51} +(4.64750 - 4.71320i) q^{52} -8.65927i q^{53} +(1.04130 + 1.82983i) q^{54} +(-2.89158 - 5.00837i) q^{55} +3.06572 q^{56} +(-1.26328 + 7.07840i) q^{57} +(-0.00285867 + 0.0106687i) q^{58} +(3.56328 - 13.2984i) q^{59} +(6.21064 - 2.24589i) q^{60} -9.28187 q^{61} +(0.942743 + 1.63288i) q^{62} +(5.89290 + 0.540244i) q^{63} -4.32500i q^{64} +(3.78973 + 6.45889i) q^{65} +(1.94700 - 0.166265i) q^{66} +(-4.68587 - 4.68587i) q^{67} +10.6089i q^{68} +(2.15505 - 4.59656i) q^{69} +(-0.429635 + 1.60342i) q^{70} +(-2.46714 - 9.20749i) q^{71} +(-0.425668 + 4.64313i) q^{72} +(0.696641 - 0.696641i) q^{73} -2.78345i q^{74} +(-0.101128 - 1.18423i) q^{75} +(-5.38891 + 5.38891i) q^{76} +(2.74619 - 4.75655i) q^{77} +(-2.51960 + 0.232986i) q^{78} +(-6.46914 - 11.2049i) q^{79} +(6.10271 + 1.63522i) q^{80} +(-1.63643 + 8.84998i) q^{81} +(-3.29897 - 1.90466i) q^{82} +(3.21235 + 0.860748i) q^{83} +(4.79638 + 4.04165i) q^{84} +(-11.5934 - 3.10645i) q^{85} +(2.45481 - 0.657764i) q^{86} +(-0.0387325 + 0.0270010i) q^{87} +(3.74777 + 2.16378i) q^{88} +(0.569878 - 2.12681i) q^{89} +(-2.36877 - 0.873326i) q^{90} +(-3.51273 + 6.18406i) q^{91} +(4.65995 - 2.69042i) q^{92} +(-1.41610 + 7.93465i) q^{93} +1.78974 q^{94} +(-4.31105 - 7.46696i) q^{95} +(-4.84489 + 5.74961i) q^{96} +(8.64769 + 8.64769i) q^{97} +(1.21682 - 0.326045i) q^{98} +(6.82264 + 4.81963i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 48 q - 6 q^{2} - 2 q^{3} - 6 q^{4} - 6 q^{5} - 2 q^{6} + 2 q^{7} - 30 q^{8} - 2 q^{9} - 12 q^{10} + 6 q^{11} - 18 q^{12} - 2 q^{13} - 12 q^{14} + 4 q^{15} + 14 q^{16} - 2 q^{18} - 4 q^{19} - 6 q^{20} + 22 q^{21}+ \cdots - 32 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(28\) \(92\)
\(\chi(n)\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{5}{6}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.104868 − 0.391374i −0.0741530 − 0.276743i 0.918887 − 0.394521i \(-0.129089\pi\)
−0.993040 + 0.117778i \(0.962423\pi\)
\(3\) 0.735256 − 1.56825i 0.424500 − 0.905428i
\(4\) 1.58987 − 0.917915i 0.794937 − 0.458957i
\(5\) 0.537559 + 2.00620i 0.240404 + 0.897199i 0.975638 + 0.219386i \(0.0704054\pi\)
−0.735234 + 0.677813i \(0.762928\pi\)
\(6\) −0.690875 − 0.123300i −0.282049 − 0.0503372i
\(7\) −1.39480 + 1.39480i −0.527183 + 0.527183i −0.919731 − 0.392548i \(-0.871594\pi\)
0.392548 + 0.919731i \(0.371594\pi\)
\(8\) −1.09899 − 1.09899i −0.388550 − 0.388550i
\(9\) −1.91880 − 2.30613i −0.639599 − 0.768709i
\(10\) 0.728800 − 0.420773i 0.230467 − 0.133060i
\(11\) −2.68955 + 0.720663i −0.810930 + 0.217288i −0.640377 − 0.768061i \(-0.721222\pi\)
−0.170552 + 0.985349i \(0.554555\pi\)
\(12\) −0.270552 − 3.16822i −0.0781016 − 0.914586i
\(13\) 3.47606 − 0.957605i 0.964086 − 0.265592i
\(14\) 0.692156 + 0.399616i 0.184986 + 0.106802i
\(15\) 3.54146 + 0.632043i 0.914400 + 0.163193i
\(16\) 1.52096 − 2.63439i 0.380241 − 0.658597i
\(17\) −2.88941 + 5.00460i −0.700784 + 1.21379i 0.267408 + 0.963583i \(0.413833\pi\)
−0.968192 + 0.250210i \(0.919500\pi\)
\(18\) −0.701336 + 0.992806i −0.165306 + 0.234007i
\(19\) −4.00984 + 1.07443i −0.919920 + 0.246492i −0.687551 − 0.726136i \(-0.741314\pi\)
−0.232369 + 0.972628i \(0.574648\pi\)
\(20\) 2.69617 + 2.69617i 0.602882 + 0.602882i
\(21\) 1.16185 + 3.21292i 0.253537 + 0.701116i
\(22\) 0.564096 + 0.977044i 0.120266 + 0.208306i
\(23\) 2.93102 0.611159 0.305580 − 0.952167i \(-0.401150\pi\)
0.305580 + 0.952167i \(0.401150\pi\)
\(24\) −2.53152 + 0.915445i −0.516744 + 0.186864i
\(25\) 0.594268 − 0.343101i 0.118854 − 0.0686202i
\(26\) −0.739309 − 1.26002i −0.144991 − 0.247109i
\(27\) −5.02738 + 1.31356i −0.967520 + 0.252794i
\(28\) −0.937247 + 3.49785i −0.177123 + 0.661032i
\(29\) −0.0236075 − 0.0136298i −0.00438381 − 0.00253099i 0.497806 − 0.867288i \(-0.334139\pi\)
−0.502190 + 0.864757i \(0.667472\pi\)
\(30\) −0.124021 − 1.45231i −0.0226431 − 0.265155i
\(31\) −4.49489 + 1.20440i −0.807307 + 0.216317i −0.638790 − 0.769381i \(-0.720565\pi\)
−0.168517 + 0.985699i \(0.553898\pi\)
\(32\) −4.19301 − 1.12351i −0.741227 − 0.198611i
\(33\) −0.847330 + 4.74775i −0.147501 + 0.826477i
\(34\) 2.26167 + 0.606014i 0.387874 + 0.103930i
\(35\) −3.54802 − 2.04845i −0.599725 − 0.346251i
\(36\) −5.16747 − 1.90516i −0.861246 − 0.317527i
\(37\) 6.63559 + 1.77800i 1.09088 + 0.292302i 0.759047 − 0.651035i \(-0.225665\pi\)
0.331837 + 0.943337i \(0.392332\pi\)
\(38\) 0.841010 + 1.45667i 0.136430 + 0.236303i
\(39\) 1.05403 − 6.15541i 0.168780 − 0.985654i
\(40\) 1.61401 − 2.79555i 0.255198 − 0.442016i
\(41\) 6.64791 − 6.64791i 1.03823 − 1.03823i 0.0389900 − 0.999240i \(-0.487586\pi\)
0.999240 − 0.0389900i \(-0.0124140\pi\)
\(42\) 1.13561 − 0.791651i 0.175228 − 0.122154i
\(43\) 6.27230i 0.956516i 0.878219 + 0.478258i \(0.158732\pi\)
−0.878219 + 0.478258i \(0.841268\pi\)
\(44\) −3.61454 + 3.61454i −0.544912 + 0.544912i
\(45\) 3.59508 − 5.08917i 0.535922 − 0.758648i
\(46\) −0.307370 − 1.14712i −0.0453193 − 0.169134i
\(47\) −1.14324 + 4.26664i −0.166759 + 0.622353i 0.831050 + 0.556197i \(0.187740\pi\)
−0.997809 + 0.0661557i \(0.978927\pi\)
\(48\) −3.01307 − 4.32220i −0.434899 − 0.623855i
\(49\) 3.10909i 0.444156i
\(50\) −0.196600 − 0.196600i −0.0278035 − 0.0278035i
\(51\) 5.72399 + 8.21096i 0.801519 + 1.14976i
\(52\) 4.64750 − 4.71320i 0.644492 − 0.653603i
\(53\) − 8.65927i − 1.18944i −0.803932 − 0.594721i \(-0.797263\pi\)
0.803932 − 0.594721i \(-0.202737\pi\)
\(54\) 1.04130 + 1.82983i 0.141703 + 0.249009i
\(55\) −2.89158 − 5.00837i −0.389901 − 0.675328i
\(56\) 3.06572 0.409674
\(57\) −1.26328 + 7.07840i −0.167326 + 0.937557i
\(58\) −0.00285867 + 0.0106687i −0.000375362 + 0.00140087i
\(59\) 3.56328 − 13.2984i 0.463900 − 1.73130i −0.196610 − 0.980482i \(-0.562993\pi\)
0.660510 − 0.750817i \(-0.270340\pi\)
\(60\) 6.21064 − 2.24589i 0.801790 − 0.289943i
\(61\) −9.28187 −1.18842 −0.594211 − 0.804309i \(-0.702535\pi\)
−0.594211 + 0.804309i \(0.702535\pi\)
\(62\) 0.942743 + 1.63288i 0.119729 + 0.207376i
\(63\) 5.89290 + 0.540244i 0.742436 + 0.0680644i
\(64\) − 4.32500i − 0.540625i
\(65\) 3.78973 + 6.45889i 0.470058 + 0.801127i
\(66\) 1.94700 − 0.166265i 0.239659 − 0.0204659i
\(67\) −4.68587 − 4.68587i −0.572470 − 0.572470i 0.360348 − 0.932818i \(-0.382658\pi\)
−0.932818 + 0.360348i \(0.882658\pi\)
\(68\) 10.6089i 1.28652i
\(69\) 2.15505 − 4.59656i 0.259437 − 0.553361i
\(70\) −0.429635 + 1.60342i −0.0513512 + 0.191645i
\(71\) −2.46714 − 9.20749i −0.292795 − 1.09273i −0.942952 − 0.332928i \(-0.891964\pi\)
0.650157 − 0.759800i \(-0.274703\pi\)
\(72\) −0.425668 + 4.64313i −0.0501655 + 0.547198i
\(73\) 0.696641 − 0.696641i 0.0815356 − 0.0815356i −0.665163 − 0.746698i \(-0.731638\pi\)
0.746698 + 0.665163i \(0.231638\pi\)
\(74\) − 2.78345i − 0.323569i
\(75\) −0.101128 − 1.18423i −0.0116772 − 0.136743i
\(76\) −5.38891 + 5.38891i −0.618150 + 0.618150i
\(77\) 2.74619 − 4.75655i 0.312958 − 0.542059i
\(78\) −2.51960 + 0.232986i −0.285288 + 0.0263805i
\(79\) −6.46914 − 11.2049i −0.727835 − 1.26065i −0.957796 − 0.287447i \(-0.907193\pi\)
0.229962 − 0.973200i \(-0.426140\pi\)
\(80\) 6.10271 + 1.63522i 0.682304 + 0.182823i
\(81\) −1.63643 + 8.84998i −0.181826 + 0.983331i
\(82\) −3.29897 − 1.90466i −0.364311 − 0.210335i
\(83\) 3.21235 + 0.860748i 0.352602 + 0.0944793i 0.430772 − 0.902461i \(-0.358241\pi\)
−0.0781706 + 0.996940i \(0.524908\pi\)
\(84\) 4.79638 + 4.04165i 0.523328 + 0.440980i
\(85\) −11.5934 − 3.10645i −1.25748 − 0.336942i
\(86\) 2.45481 − 0.657764i 0.264709 − 0.0709286i
\(87\) −0.0387325 + 0.0270010i −0.00415256 + 0.00289481i
\(88\) 3.74777 + 2.16378i 0.399514 + 0.230659i
\(89\) 0.569878 − 2.12681i 0.0604069 − 0.225442i −0.929123 − 0.369771i \(-0.879436\pi\)
0.989530 + 0.144330i \(0.0461026\pi\)
\(90\) −2.36877 − 0.873326i −0.249691 − 0.0920567i
\(91\) −3.51273 + 6.18406i −0.368234 + 0.648265i
\(92\) 4.65995 − 2.69042i 0.485833 − 0.280496i
\(93\) −1.41610 + 7.93465i −0.146842 + 0.822785i
\(94\) 1.78974 0.184597
\(95\) −4.31105 − 7.46696i −0.442305 − 0.766094i
\(96\) −4.84489 + 5.74961i −0.494479 + 0.586817i
\(97\) 8.64769 + 8.64769i 0.878040 + 0.878040i 0.993332 − 0.115292i \(-0.0367803\pi\)
−0.115292 + 0.993332i \(0.536780\pi\)
\(98\) 1.21682 − 0.326045i 0.122917 − 0.0329355i
\(99\) 6.82264 + 4.81963i 0.685701 + 0.484391i
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.bc.a.50.6 yes 48
3.2 odd 2 351.2.bf.a.206.7 48
9.2 odd 6 117.2.x.a.11.7 ✓ 48
9.7 even 3 351.2.ba.a.89.6 48
13.6 odd 12 117.2.x.a.32.7 yes 48
39.32 even 12 351.2.ba.a.71.6 48
117.97 odd 12 351.2.bf.a.305.7 48
117.110 even 12 inner 117.2.bc.a.110.6 yes 48
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
117.2.x.a.11.7 ✓ 48 9.2 odd 6
117.2.x.a.32.7 yes 48 13.6 odd 12
117.2.bc.a.50.6 yes 48 1.1 even 1 trivial
117.2.bc.a.110.6 yes 48 117.110 even 12 inner
351.2.ba.a.71.6 48 39.32 even 12
351.2.ba.a.89.6 48 9.7 even 3
351.2.bf.a.206.7 48 3.2 odd 2
351.2.bf.a.305.7 48 117.97 odd 12