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Properties

Label 117.2.bc.a.110.12
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 110.12
Character \(\chi\) \(=\) 117.110
Dual form 117.2.bc.a.50.12

$q$-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+(0.683347 - 2.55028i) q^{2} +(1.41657 - 0.996665i) q^{3} +(-4.30494 - 2.48546i) q^{4} +(-0.697740 + 2.60400i) q^{5} +(-1.57377 - 4.29371i) q^{6} +(1.60053 + 1.60053i) q^{7} +(-5.54651 + 5.54651i) q^{8} +(1.01332 - 2.82368i) q^{9} +(6.16415 + 3.55887i) q^{10} +(-1.49528 - 0.400660i) q^{11} +(-8.57539 + 0.769767i) q^{12} +(2.76885 + 2.30943i) q^{13} +(5.17552 - 2.98809i) q^{14} +(1.60692 + 4.38416i) q^{15} +(5.38407 + 9.32548i) q^{16} +(-1.06695 - 1.84800i) q^{17} +(-6.50875 - 4.51380i) q^{18} +(-3.84372 - 1.02992i) q^{19} +(9.47586 - 9.47586i) q^{20} +(3.86245 + 0.672064i) q^{21} +(-2.04359 + 3.53961i) q^{22} -7.11178 q^{23} +(-2.32898 + 13.3850i) q^{24} +(-1.96386 - 1.13384i) q^{25} +(7.78180 - 5.48320i) q^{26} +(-1.37884 - 5.00987i) q^{27} +(-2.91213 - 10.8682i) q^{28} +(-0.203285 + 0.117367i) q^{29} +(12.2789 - 1.10221i) q^{30} +(6.02345 + 1.61398i) q^{31} +(12.3085 - 3.29805i) q^{32} +(-2.51749 + 0.922735i) q^{33} +(-5.44203 + 1.45819i) q^{34} +(-5.28454 + 3.05103i) q^{35} +(-11.3804 + 9.63722i) q^{36} +(2.21625 - 0.593843i) q^{37} +(-5.25318 + 9.09878i) q^{38} +(6.22399 + 0.511852i) q^{39} +(-10.5731 - 18.3131i) q^{40} +(5.99425 + 5.99425i) q^{41} +(4.35335 - 9.39109i) q^{42} +0.761002i q^{43} +(5.44127 + 5.44127i) q^{44} +(6.64585 + 4.60888i) q^{45} +(-4.85981 + 18.1371i) q^{46} +(-0.640422 - 2.39009i) q^{47} +(16.9213 + 7.84405i) q^{48} -1.87661i q^{49} +(-4.23360 + 4.23360i) q^{50} +(-3.35324 - 1.55443i) q^{51} +(-6.17971 - 16.8238i) q^{52} +3.34230i q^{53} +(-13.7188 + 0.0929427i) q^{54} +(2.08664 - 3.61416i) q^{55} -17.7547 q^{56} +(-6.47136 + 2.37195i) q^{57} +(0.160404 + 0.598636i) q^{58} +(-3.20908 - 11.9765i) q^{59} +(3.97892 - 22.8674i) q^{60} -9.67608 q^{61} +(8.23221 - 14.2586i) q^{62} +(6.14124 - 2.89754i) q^{63} -12.1076i q^{64} +(-7.94571 + 5.59870i) q^{65} +(0.632918 + 7.05086i) q^{66} +(-5.41390 + 5.41390i) q^{67} +10.6074i q^{68} +(-10.0743 + 7.08807i) q^{69} +(4.16982 + 15.5620i) q^{70} +(-0.833742 + 3.11157i) q^{71} +(10.0412 + 21.2820i) q^{72} +(-0.706395 - 0.706395i) q^{73} -6.05787i q^{74} +(-3.91200 + 0.351159i) q^{75} +(13.9871 + 13.9871i) q^{76} +(-1.75198 - 3.03451i) q^{77} +(5.55851 - 15.5232i) q^{78} +(4.39999 - 7.62101i) q^{79} +(-28.0403 + 7.51337i) q^{80} +(-6.94637 - 5.72258i) q^{81} +(19.3832 - 11.1909i) q^{82} +(3.82540 - 1.02501i) q^{83} +(-14.9572 - 12.4931i) q^{84} +(5.55666 - 1.48890i) q^{85} +(1.94077 + 0.520028i) q^{86} +(-0.170991 + 0.368865i) q^{87} +(10.5159 - 6.07133i) q^{88} +(0.560753 + 2.09276i) q^{89} +(16.2954 - 13.7993i) q^{90} +(0.735305 + 8.12794i) q^{91} +(30.6158 + 17.6760i) q^{92} +(10.1412 - 3.71705i) q^{93} -6.53304 q^{94} +(5.36383 - 9.29043i) q^{95} +(14.1487 - 16.9393i) q^{96} +(-0.451327 + 0.451327i) q^{97} +(-4.78588 - 1.28237i) q^{98} +(-2.64653 + 3.81621i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.683347 − 2.55028i 0.483199 − 1.80332i −0.104838 − 0.994489i \(-0.533432\pi\)
0.588037 − 0.808834i \(-0.299901\pi\)
\(3\) 1.41657 − 0.996665i 0.817855 − 0.575425i
\(4\) −4.30494 − 2.48546i −2.15247 − 1.24273i
\(5\) −0.697740 + 2.60400i −0.312039 + 1.16455i 0.614676 + 0.788780i \(0.289287\pi\)
−0.926715 + 0.375766i \(0.877380\pi\)
\(6\) −1.57377 − 4.29371i −0.642490 − 1.75290i
\(7\) 1.60053 + 1.60053i 0.604944 + 0.604944i 0.941620 − 0.336677i \(-0.109303\pi\)
−0.336677 + 0.941620i \(0.609303\pi\)
\(8\) −5.54651 + 5.54651i −1.96099 + 1.96099i
\(9\) 1.01332 − 2.82368i 0.337773 − 0.941228i
\(10\) 6.16415 + 3.55887i 1.94928 + 1.12541i
\(11\) −1.49528 − 0.400660i −0.450845 − 0.120803i 0.0262499 − 0.999655i \(-0.491643\pi\)
−0.477095 + 0.878852i \(0.658310\pi\)
\(12\) −8.57539 + 0.769767i −2.47550 + 0.222213i
\(13\) 2.76885 + 2.30943i 0.767940 + 0.640522i
\(14\) 5.17552 − 2.98809i 1.38322 − 0.798601i
\(15\) 1.60692 + 4.38416i 0.414906 + 1.13198i
\(16\) 5.38407 + 9.32548i 1.34602 + 2.33137i
\(17\) −1.06695 − 1.84800i −0.258772 − 0.448207i 0.707141 − 0.707073i \(-0.249985\pi\)
−0.965913 + 0.258866i \(0.916651\pi\)
\(18\) −6.50875 − 4.51380i −1.53413 − 1.06391i
\(19\) −3.84372 − 1.02992i −0.881809 − 0.236280i −0.210622 − 0.977568i \(-0.567549\pi\)
−0.671187 + 0.741288i \(0.734215\pi\)
\(20\) 9.47586 − 9.47586i 2.11887 − 2.11887i
\(21\) 3.86245 + 0.672064i 0.842855 + 0.146656i
\(22\) −2.04359 + 3.53961i −0.435695 + 0.754647i
\(23\) −7.11178 −1.48291 −0.741455 − 0.671003i \(-0.765864\pi\)
−0.741455 + 0.671003i \(0.765864\pi\)
\(24\) −2.32898 + 13.3850i −0.475402 + 2.73220i
\(25\) −1.96386 − 1.13384i −0.392773 − 0.226767i
\(26\) 7.78180 − 5.48320i 1.52614 − 1.07534i
\(27\) −1.37884 − 5.00987i −0.265357 − 0.964150i
\(28\) −2.91213 − 10.8682i −0.550341 − 2.05390i
\(29\) −0.203285 + 0.117367i −0.0377491 + 0.0217944i −0.518756 − 0.854922i \(-0.673605\pi\)
0.481007 + 0.876717i \(0.340271\pi\)
\(30\) 12.2789 − 1.10221i 2.24182 − 0.201236i
\(31\) 6.02345 + 1.61398i 1.08184 + 0.289879i 0.755351 − 0.655321i \(-0.227466\pi\)
0.326493 + 0.945200i \(0.394133\pi\)
\(32\) 12.3085 − 3.29805i 2.17585 − 0.583018i
\(33\) −2.51749 + 0.922735i −0.438239 + 0.160628i
\(34\) −5.44203 + 1.45819i −0.933300 + 0.250077i
\(35\) −5.28454 + 3.05103i −0.893250 + 0.515718i
\(36\) −11.3804 + 9.63722i −1.89673 + 1.60620i
\(37\) 2.21625 − 0.593843i 0.364349 − 0.0976271i −0.0719995 − 0.997405i \(-0.522938\pi\)
0.436349 + 0.899778i \(0.356271\pi\)
\(38\) −5.25318 + 9.09878i −0.852178 + 1.47602i
\(39\) 6.22399 + 0.511852i 0.996635 + 0.0819619i
\(40\) −10.5731 − 18.3131i −1.67175 − 2.89556i
\(41\) 5.99425 + 5.99425i 0.936145 + 0.936145i 0.998080 − 0.0619356i \(-0.0197273\pi\)
−0.0619356 + 0.998080i \(0.519727\pi\)
\(42\) 4.35335 − 9.39109i 0.671736 − 1.44908i
\(43\) 0.761002i 0.116052i 0.998315 + 0.0580259i \(0.0184806\pi\)
−0.998315 + 0.0580259i \(0.981519\pi\)
\(44\) 5.44127 + 5.44127i 0.820303 + 0.820303i
\(45\) 6.64585 + 4.60888i 0.990704 + 0.687051i
\(46\) −4.85981 + 18.1371i −0.716540 + 2.67416i
\(47\) −0.640422 − 2.39009i −0.0934152 − 0.348630i 0.903359 − 0.428885i \(-0.141093\pi\)
−0.996774 + 0.0802545i \(0.974427\pi\)
\(48\) 16.9213 + 7.84405i 2.44238 + 1.13219i
\(49\) − 1.87661i − 0.268087i
\(50\) −4.23360 + 4.23360i −0.598722 + 0.598722i
\(51\) −3.35324 − 1.55443i −0.469547 − 0.217664i
\(52\) −6.17971 − 16.8238i −0.856972 − 2.33304i
\(53\) 3.34230i 0.459100i 0.973297 + 0.229550i \(0.0737254\pi\)
−0.973297 + 0.229550i \(0.926275\pi\)
\(54\) −13.7188 + 0.0929427i −1.86689 + 0.0126479i
\(55\) 2.08664 − 3.61416i 0.281362 − 0.487334i
\(56\) −17.7547 −2.37257
\(57\) −6.47136 + 2.37195i −0.857153 + 0.314172i
\(58\) 0.160404 + 0.598636i 0.0210621 + 0.0786048i
\(59\) −3.20908 − 11.9765i −0.417787 − 1.55920i −0.779187 − 0.626792i \(-0.784368\pi\)
0.361399 − 0.932411i \(-0.382299\pi\)
\(60\) 3.97892 − 22.8674i 0.513677 − 2.95217i
\(61\) −9.67608 −1.23889 −0.619447 − 0.785038i \(-0.712643\pi\)
−0.619447 + 0.785038i \(0.712643\pi\)
\(62\) 8.23221 − 14.2586i 1.04549 − 1.81084i
\(63\) 6.14124 − 2.89754i 0.773723 − 0.365056i
\(64\) − 12.1076i − 1.51344i
\(65\) −7.94571 + 5.59870i −0.985544 + 0.694433i
\(66\) 0.632918 + 7.05086i 0.0779068 + 0.867901i
\(67\) −5.41390 + 5.41390i −0.661414 + 0.661414i −0.955713 − 0.294300i \(-0.904914\pi\)
0.294300 + 0.955713i \(0.404914\pi\)
\(68\) 10.6074i 1.28633i
\(69\) −10.0743 + 7.08807i −1.21280 + 0.853303i
\(70\) 4.16982 + 15.5620i 0.498389 + 1.86001i
\(71\) −0.833742 + 3.11157i −0.0989469 + 0.369275i −0.997588 − 0.0694091i \(-0.977889\pi\)
0.898641 + 0.438684i \(0.144555\pi\)
\(72\) 10.0412 + 21.2820i 1.18337 + 2.50810i
\(73\) −0.706395 − 0.706395i −0.0826773 − 0.0826773i 0.664559 − 0.747236i \(-0.268620\pi\)
−0.747236 + 0.664559i \(0.768620\pi\)
\(74\) − 6.05787i − 0.704213i
\(75\) −3.91200 + 0.351159i −0.451718 + 0.0405483i
\(76\) 13.9871 + 13.9871i 1.60443 + 1.60443i
\(77\) −1.75198 − 3.03451i −0.199656 − 0.345815i
\(78\) 5.55851 − 15.5232i 0.629377 − 1.75765i
\(79\) 4.39999 − 7.62101i 0.495038 − 0.857430i −0.504946 − 0.863151i \(-0.668488\pi\)
0.999984 + 0.00572053i \(0.00182091\pi\)
\(80\) −28.0403 + 7.51337i −3.13500 + 0.840020i
\(81\) −6.94637 − 5.72258i −0.771819 − 0.635842i
\(82\) 19.3832 − 11.1909i 2.14052 − 1.23583i
\(83\) 3.82540 − 1.02501i 0.419892 − 0.112510i −0.0426854 − 0.999089i \(-0.513591\pi\)
0.462578 + 0.886579i \(0.346925\pi\)
\(84\) −14.9572 − 12.4931i −1.63197 − 1.36311i
\(85\) 5.55666 − 1.48890i 0.602704 − 0.161494i
\(86\) 1.94077 + 0.520028i 0.209279 + 0.0560761i
\(87\) −0.170991 + 0.368865i −0.0183322 + 0.0395464i
\(88\) 10.5159 − 6.07133i 1.12099 − 0.647206i
\(89\) 0.560753 + 2.09276i 0.0594397 + 0.221832i 0.989256 − 0.146191i \(-0.0467014\pi\)
−0.929817 + 0.368023i \(0.880035\pi\)
\(90\) 16.2954 − 13.7993i 1.71768 − 1.45458i
\(91\) 0.735305 + 8.12794i 0.0770809 + 0.852040i
\(92\) 30.6158 + 17.6760i 3.19192 + 1.84285i
\(93\) 10.1412 − 3.71705i 1.05159 − 0.385441i
\(94\) −6.53304 −0.673831
\(95\) 5.36383 − 9.29043i 0.550318 − 0.953178i
\(96\) 14.1487 − 16.9393i 1.44405 − 1.72886i
\(97\) −0.451327 + 0.451327i −0.0458253 + 0.0458253i −0.729648 − 0.683823i \(-0.760316\pi\)
0.683823 + 0.729648i \(0.260316\pi\)
\(98\) −4.78588 − 1.28237i −0.483447 − 0.129539i
\(99\) −2.64653 + 3.81621i −0.265987 + 0.383543i
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.110.12 yes 48
3.2 odd 2 351.2.bf.a.305.1 48
9.4 even 3 351.2.ba.a.71.12 48
9.5 odd 6 117.2.x.a.32.1 yes 48
13.11 odd 12 117.2.x.a.11.1 ✓ 48
39.11 even 12 351.2.ba.a.89.12 48
117.50 even 12 inner 117.2.bc.a.50.12 yes 48
117.76 odd 12 351.2.bf.a.206.1 48
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
117.2.x.a.11.1 ✓ 48 13.11 odd 12
117.2.x.a.32.1 yes 48 9.5 odd 6
117.2.bc.a.50.12 yes 48 117.50 even 12 inner
117.2.bc.a.110.12 yes 48 1.1 even 1 trivial
351.2.ba.a.71.12 48 9.4 even 3
351.2.ba.a.89.12 48 39.11 even 12
351.2.bf.a.206.1 48 117.76 odd 12
351.2.bf.a.305.1 48 3.2 odd 2