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Properties

Label 117.2.bc.a.110.6
Level $117$
Weight $2$
Character 117.110
Analytic conductor $0.934$
Analytic rank $0$
Dimension $48$
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(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 110.6
Character \(\chi\) \(=\) 117.110
Dual form 117.2.bc.a.50.6

$q$-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 \(\left(\mathbb{Z}/117\mathbb{Z}\right)^\times\).

\(n\) \(28\) \(92\)
\(\chi(n)\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{1}{6}\right)\)

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 −0.993040 − 0.117778i \(-0.962423\pi\)
0.918887 + 0.394521i \(0.129089\pi\)
\(3\) 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 −0.735234 − 0.677813i \(-0.762928\pi\)
0.975638 − 0.219386i \(-0.0704054\pi\)
\(6\) −0.690875 + 0.123300i −0.282049 + 0.0503372i
\(7\) −1.39480 − 1.39480i −0.527183 − 0.527183i 0.392548 − 0.919731i \(-0.371594\pi\)
−0.919731 + 0.392548i \(0.871594\pi\)
\(8\) −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 −0.170552 − 0.985349i \(-0.554555\pi\)
−0.640377 + 0.768061i \(0.721222\pi\)
\(12\) −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 −0.968192 − 0.250210i \(-0.919500\pi\)
0.267408 − 0.963583i \(-0.413833\pi\)
\(18\) −0.701336 − 0.992806i −0.165306 − 0.234007i
\(19\) −4.00984 − 1.07443i −0.919920 − 0.246492i −0.232369 − 0.972628i \(-0.574648\pi\)
−0.687551 + 0.726136i \(0.741314\pi\)
\(20\) 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 −0.502190 − 0.864757i \(-0.667472\pi\)
0.497806 + 0.867288i \(0.334139\pi\)
\(30\) −0.124021 + 1.45231i −0.0226431 + 0.265155i
\(31\) −4.49489 − 1.20440i −0.807307 − 0.216317i −0.168517 − 0.985699i \(-0.553898\pi\)
−0.638790 + 0.769381i \(0.720565\pi\)
\(32\) −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 0.331837 − 0.943337i \(-0.392332\pi\)
0.759047 + 0.651035i \(0.225665\pi\)
\(38\) 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 0.999240 + 0.0389900i \(0.0124140\pi\)
0.0389900 + 0.999240i \(0.487586\pi\)
\(42\) 1.13561 + 0.791651i 0.175228 + 0.122154i
\(43\) − 6.27230i − 0.956516i −0.878219 − 0.478258i \(-0.841268\pi\)
0.878219 − 0.478258i \(-0.158732\pi\)
\(44\) −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 −0.997809 − 0.0661557i \(-0.978927\pi\)
0.831050 − 0.556197i \(-0.187740\pi\)
\(48\) −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 \(0.202737\pi\)
−0.803932 + 0.594721i \(0.797263\pi\)
\(54\) 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 0.660510 + 0.750817i \(0.270340\pi\)
−0.196610 + 0.980482i \(0.562993\pi\)
\(60\) 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 −0.932818 − 0.360348i \(-0.882658\pi\)
0.360348 + 0.932818i \(0.382658\pi\)
\(68\) − 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 0.650157 + 0.759800i \(0.274703\pi\)
−0.942952 + 0.332928i \(0.891964\pi\)
\(72\) −0.425668 − 4.64313i −0.0501655 − 0.547198i
\(73\) 0.696641 + 0.696641i 0.0815356 + 0.0815356i 0.746698 − 0.665163i \(-0.231638\pi\)
−0.665163 + 0.746698i \(0.731638\pi\)
\(74\) 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 0.229962 + 0.973200i \(0.426140\pi\)
−0.957796 + 0.287447i \(0.907193\pi\)
\(80\) 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 −0.0781706 − 0.996940i \(-0.524908\pi\)
0.430772 + 0.902461i \(0.358241\pi\)
\(84\) 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 0.989530 − 0.144330i \(-0.0461026\pi\)
−0.929123 + 0.369771i \(0.879436\pi\)
\(90\) −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 −0.115292 − 0.993332i \(-0.536780\pi\)
0.993332 + 0.115292i \(0.0367803\pi\)
\(98\) 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.110.6 yes 48
3.2 odd 2 351.2.bf.a.305.7 48
9.4 even 3 351.2.ba.a.71.6 48
9.5 odd 6 117.2.x.a.32.7 yes 48
13.11 odd 12 117.2.x.a.11.7 ✓ 48
39.11 even 12 351.2.ba.a.89.6 48
117.50 even 12 inner 117.2.bc.a.50.6 yes 48
117.76 odd 12 351.2.bf.a.206.7 48
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
117.2.x.a.11.7 ✓ 48 13.11 odd 12
117.2.x.a.32.7 yes 48 9.5 odd 6
117.2.bc.a.50.6 yes 48 117.50 even 12 inner
117.2.bc.a.110.6 yes 48 1.1 even 1 trivial
351.2.ba.a.71.6 48 9.4 even 3
351.2.ba.a.89.6 48 39.11 even 12
351.2.bf.a.206.7 48 117.76 odd 12
351.2.bf.a.305.7 48 3.2 odd 2