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Properties

Label 117.2.t.c.103.6
Level $117$
Weight $2$
Character 117.103
Analytic conductor $0.934$
Analytic rank $0$
Dimension $20$
Inner twists $4$

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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(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 103.6
Root \(-0.219737 - 1.71806i\) of defining polynomial
Character \(\chi\) \(=\) 117.103
Dual form 117.2.t.c.25.6

$q$-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+(0.784270 - 0.452798i) q^{2} +(1.66968 - 0.460628i) q^{3} +(-0.589947 + 1.02182i) q^{4} +(-1.94254 - 1.12153i) q^{5} +(1.10091 - 1.11728i) q^{6} +(2.97576 - 1.71806i) q^{7} +2.87970i q^{8} +(2.57564 - 1.53820i) q^{9} -2.03130 q^{10} +(-3.20133 + 1.84829i) q^{11} +(-0.514343 + 1.97785i) q^{12} +(-3.28340 + 1.48972i) q^{13} +(1.55587 - 2.69484i) q^{14} +(-3.76003 - 0.977800i) q^{15} +(0.124029 + 0.214825i) q^{16} -4.21120 q^{17} +(1.32350 - 2.37261i) q^{18} +4.25298i q^{19} +(2.29200 - 1.32329i) q^{20} +(4.17717 - 4.23932i) q^{21} +(-1.67380 + 2.89911i) q^{22} +(1.89162 - 3.27639i) q^{23} +(1.32647 + 4.80817i) q^{24} +(0.0156524 + 0.0271108i) q^{25} +(-1.90053 + 2.65506i) q^{26} +(3.59195 - 3.75471i) q^{27} +4.05425i q^{28} +(-1.18945 - 2.06020i) q^{29} +(-3.39162 + 0.935677i) q^{30} +(6.37163 + 3.67866i) q^{31} +(-4.79325 - 2.76738i) q^{32} +(-4.49381 + 4.56066i) q^{33} +(-3.30272 + 1.90682i) q^{34} -7.70739 q^{35} +(0.0522689 + 3.53930i) q^{36} -5.49928i q^{37} +(1.92574 + 3.33549i) q^{38} +(-4.79602 + 3.99978i) q^{39} +(3.22967 - 5.59395i) q^{40} +(6.86085 + 3.96111i) q^{41} +(1.35647 - 5.21619i) q^{42} +(-0.450266 - 0.779883i) q^{43} -4.36157i q^{44} +(-6.72844 + 0.0993666i) q^{45} -3.42609i q^{46} +(4.80060 - 2.77163i) q^{47} +(0.306043 + 0.301557i) q^{48} +(2.40343 - 4.16287i) q^{49} +(0.0245514 + 0.0141748i) q^{50} +(-7.03134 + 1.93980i) q^{51} +(0.414812 - 4.23390i) q^{52} +7.59566 q^{53} +(1.11693 - 4.57114i) q^{54} +8.29163 q^{55} +(4.94749 + 8.56930i) q^{56} +(1.95904 + 7.10111i) q^{57} +(-1.86571 - 1.07717i) q^{58} +(-4.44379 - 2.56562i) q^{59} +(3.21735 - 3.26522i) q^{60} +(6.50907 + 11.2740i) q^{61} +6.66277 q^{62} +(5.02178 - 9.00242i) q^{63} -5.50838 q^{64} +(8.04892 + 0.788585i) q^{65} +(-1.45930 + 5.61158i) q^{66} +(-11.7002 - 6.75511i) q^{67} +(2.48439 - 4.30308i) q^{68} +(1.64920 - 6.34184i) q^{69} +(-6.04468 + 3.48989i) q^{70} +2.65506i q^{71} +(4.42956 + 7.41708i) q^{72} -5.45741i q^{73} +(-2.49006 - 4.31292i) q^{74} +(0.0386224 + 0.0380563i) q^{75} +(-4.34578 - 2.50904i) q^{76} +(-6.35092 + 11.0001i) q^{77} +(-1.95028 + 5.30853i) q^{78} +(-5.46886 - 9.47234i) q^{79} -0.556410i q^{80} +(4.26787 - 7.92371i) q^{81} +7.17434 q^{82} +(-0.465547 + 0.268784i) q^{83} +(1.86750 + 6.76929i) q^{84} +(8.18044 + 4.72298i) q^{85} +(-0.706259 - 0.407759i) q^{86} +(-2.93499 - 2.89197i) q^{87} +(-5.32252 - 9.21887i) q^{88} +5.75227i q^{89} +(-5.23192 + 3.12456i) q^{90} +(-7.21120 + 10.0741i) q^{91} +(2.23192 + 3.86579i) q^{92} +(12.3331 + 3.20723i) q^{93} +(2.50998 - 4.34740i) q^{94} +(4.76984 - 8.26161i) q^{95} +(-9.27791 - 2.41273i) q^{96} +(5.87585 - 3.39243i) q^{97} -4.35308i q^{98} +(-5.40244 + 9.68481i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.784270 − 0.452798i 0.554562 − 0.320177i −0.196398 − 0.980524i \(-0.562924\pi\)
0.750960 + 0.660348i \(0.229591\pi\)
\(3\) 1.66968 − 0.460628i 0.963989 − 0.265944i
\(4\) −0.589947 + 1.02182i −0.294974 + 0.510909i
\(5\) −1.94254 − 1.12153i −0.868732 − 0.501563i −0.00180550 − 0.999998i \(-0.500575\pi\)
−0.866927 + 0.498436i \(0.833908\pi\)
\(6\) 1.10091 − 1.11728i 0.449443 − 0.456129i
\(7\) 2.97576 − 1.71806i 1.12473 − 0.649364i 0.182127 − 0.983275i \(-0.441702\pi\)
0.942605 + 0.333911i \(0.108368\pi\)
\(8\) 2.87970i 1.01813i
\(9\) 2.57564 − 1.53820i 0.858548 − 0.512734i
\(10\) −2.03130 −0.642355
\(11\) −3.20133 + 1.84829i −0.965236 + 0.557279i −0.897781 − 0.440443i \(-0.854821\pi\)
−0.0674557 + 0.997722i \(0.521488\pi\)
\(12\) −0.514343 + 1.97785i −0.148478 + 0.570957i
\(13\) −3.28340 + 1.48972i −0.910652 + 0.413174i
\(14\) 1.55587 − 2.69484i 0.415823 − 0.720226i
\(15\) −3.76003 − 0.977800i −0.970835 − 0.252467i
\(16\) 0.124029 + 0.214825i 0.0310073 + 0.0537063i
\(17\) −4.21120 −1.02137 −0.510683 − 0.859769i \(-0.670607\pi\)
−0.510683 + 0.859769i \(0.670607\pi\)
\(18\) 1.32350 − 2.37261i 0.311953 − 0.559230i
\(19\) 4.25298i 0.975701i 0.872927 + 0.487851i \(0.162219\pi\)
−0.872927 + 0.487851i \(0.837781\pi\)
\(20\) 2.29200 − 1.32329i 0.512506 − 0.295896i
\(21\) 4.17717 − 4.23932i 0.911534 − 0.925095i
\(22\) −1.67380 + 2.89911i −0.356856 + 0.618092i
\(23\) 1.89162 − 3.27639i 0.394431 − 0.683174i −0.598598 − 0.801050i \(-0.704275\pi\)
0.993028 + 0.117876i \(0.0376085\pi\)
\(24\) 1.32647 + 4.80817i 0.270765 + 0.981464i
\(25\) 0.0156524 + 0.0271108i 0.00313048 + 0.00542215i
\(26\) −1.90053 + 2.65506i −0.372725 + 0.520700i
\(27\) 3.59195 − 3.75471i 0.691272 − 0.722595i
\(28\) 4.05425i 0.766181i
\(29\) −1.18945 − 2.06020i −0.220876 − 0.382569i 0.734198 − 0.678935i \(-0.237558\pi\)
−0.955074 + 0.296367i \(0.904225\pi\)
\(30\) −3.39162 + 0.935677i −0.619223 + 0.170830i
\(31\) 6.37163 + 3.67866i 1.14438 + 0.660707i 0.947511 − 0.319723i \(-0.103590\pi\)
0.196868 + 0.980430i \(0.436923\pi\)
\(32\) −4.79325 − 2.76738i −0.847334 − 0.489209i
\(33\) −4.49381 + 4.56066i −0.782272 + 0.793910i
\(34\) −3.30272 + 1.90682i −0.566411 + 0.327018i
\(35\) −7.70739 −1.30279
\(36\) 0.0522689 + 3.53930i 0.00871149 + 0.589883i
\(37\) − 5.49928i − 0.904076i −0.891999 − 0.452038i \(-0.850697\pi\)
0.891999 − 0.452038i \(-0.149303\pi\)
\(38\) 1.92574 + 3.33549i 0.312397 + 0.541087i
\(39\) −4.79602 + 3.99978i −0.767977 + 0.640477i
\(40\) 3.22967 − 5.59395i 0.510655 − 0.884481i
\(41\) 6.86085 + 3.96111i 1.07148 + 0.618622i 0.928587 − 0.371116i \(-0.121025\pi\)
0.142898 + 0.989737i \(0.454358\pi\)
\(42\) 1.35647 − 5.21619i 0.209309 − 0.804875i
\(43\) −0.450266 − 0.779883i −0.0686649 − 0.118931i 0.829649 − 0.558285i \(-0.188541\pi\)
−0.898314 + 0.439354i \(0.855207\pi\)
\(44\) − 4.36157i − 0.657531i
\(45\) −6.72844 + 0.0993666i −1.00302 + 0.0148127i
\(46\) − 3.42609i − 0.505150i
\(47\) 4.80060 − 2.77163i 0.700239 − 0.404283i −0.107197 − 0.994238i \(-0.534188\pi\)
0.807436 + 0.589955i \(0.200854\pi\)
\(48\) 0.306043 + 0.301557i 0.0441736 + 0.0435260i
\(49\) 2.40343 − 4.16287i 0.343347 − 0.594695i
\(50\) 0.0245514 + 0.0141748i 0.00347209 + 0.00200461i
\(51\) −7.03134 + 1.93980i −0.984585 + 0.271626i
\(52\) 0.414812 − 4.23390i 0.0575241 − 0.587136i
\(53\) 7.59566 1.04334 0.521672 − 0.853146i \(-0.325308\pi\)
0.521672 + 0.853146i \(0.325308\pi\)
\(54\) 1.11693 − 4.57114i 0.151995 − 0.622053i
\(55\) 8.29163 1.11804
\(56\) 4.94749 + 8.56930i 0.661136 + 1.14512i
\(57\) 1.95904 + 7.10111i 0.259482 + 0.940565i
\(58\) −1.86571 − 1.07717i −0.244979 − 0.141439i
\(59\) −4.44379 − 2.56562i −0.578532 − 0.334016i 0.182018 − 0.983295i \(-0.441737\pi\)
−0.760550 + 0.649279i \(0.775071\pi\)
\(60\) 3.21735 − 3.26522i 0.415359 − 0.421538i
\(61\) 6.50907 + 11.2740i 0.833401 + 1.44349i 0.895326 + 0.445412i \(0.146943\pi\)
−0.0619247 + 0.998081i \(0.519724\pi\)
\(62\) 6.66277 0.846173
\(63\) 5.02178 − 9.00242i 0.632685 − 1.13420i
\(64\) −5.50838 −0.688547
\(65\) 8.04892 + 0.788585i 0.998346 + 0.0978120i
\(66\) −1.45930 + 5.61158i −0.179627 + 0.690738i
\(67\) −11.7002 − 6.75511i −1.42941 − 0.825269i −0.432333 − 0.901714i \(-0.642310\pi\)
−0.997074 + 0.0764454i \(0.975643\pi\)
\(68\) 2.48439 − 4.30308i 0.301276 − 0.521825i
\(69\) 1.64920 − 6.34184i 0.198541 − 0.763468i
\(70\) −6.04468 + 3.48989i −0.722477 + 0.417122i
\(71\) 2.65506i 0.315098i 0.987511 + 0.157549i \(0.0503592\pi\)
−0.987511 + 0.157549i \(0.949641\pi\)
\(72\) 4.42956 + 7.41708i 0.522029 + 0.874112i
\(73\) − 5.45741i − 0.638741i −0.947630 − 0.319371i \(-0.896528\pi\)
0.947630 − 0.319371i \(-0.103472\pi\)
\(74\) −2.49006 − 4.31292i −0.289464 − 0.501366i
\(75\) 0.0386224 + 0.0380563i 0.00445974 + 0.00439436i
\(76\) −4.34578 − 2.50904i −0.498495 − 0.287806i
\(77\) −6.35092 + 11.0001i −0.723755 + 1.25358i
\(78\) −1.95028 + 5.30853i −0.220825 + 0.601073i
\(79\) −5.46886 − 9.47234i −0.615294 − 1.06572i −0.990333 − 0.138712i \(-0.955704\pi\)
0.375038 − 0.927009i \(-0.377630\pi\)
\(80\) − 0.556410i − 0.0622085i
\(81\) 4.26787 − 7.92371i 0.474208 − 0.880413i
\(82\) 7.17434 0.792273
\(83\) −0.465547 + 0.268784i −0.0511004 + 0.0295029i −0.525333 − 0.850897i \(-0.676059\pi\)
0.474232 + 0.880400i \(0.342726\pi\)
\(84\) 1.86750 + 6.76929i 0.203761 + 0.738590i
\(85\) 8.18044 + 4.72298i 0.887294 + 0.512279i
\(86\) −0.706259 − 0.407759i −0.0761579 − 0.0439698i
\(87\) −2.93499 − 2.89197i −0.314664 − 0.310051i
\(88\) −5.32252 − 9.21887i −0.567382 − 0.982735i
\(89\) 5.75227i 0.609739i 0.952394 + 0.304870i \(0.0986129\pi\)
−0.952394 + 0.304870i \(0.901387\pi\)
\(90\) −5.23192 + 3.12456i −0.551492 + 0.329357i
\(91\) −7.21120 + 10.0741i −0.755939 + 1.05605i
\(92\) 2.23192 + 3.86579i 0.232693 + 0.403037i
\(93\) 12.3331 + 3.20723i 1.27888 + 0.332574i
\(94\) 2.50998 − 4.34740i 0.258884 − 0.448401i
\(95\) 4.76984 − 8.26161i 0.489375 − 0.847623i
\(96\) −9.27791 − 2.41273i −0.946922 − 0.246248i
\(97\) 5.87585 − 3.39243i 0.596603 − 0.344449i −0.171101 − 0.985253i \(-0.554733\pi\)
0.767704 + 0.640805i \(0.221399\pi\)
\(98\) − 4.35308i − 0.439727i
\(99\) −5.40244 + 9.68481i −0.542965 + 0.973360i
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.103.6 yes 20
3.2 odd 2 351.2.t.c.64.5 20
9.2 odd 6 351.2.t.c.181.6 20
9.4 even 3 1053.2.b.j.649.6 10
9.5 odd 6 1053.2.b.i.649.5 10
9.7 even 3 inner 117.2.t.c.25.5 ✓ 20
13.12 even 2 inner 117.2.t.c.103.5 yes 20
39.38 odd 2 351.2.t.c.64.6 20
117.25 even 6 inner 117.2.t.c.25.6 yes 20
117.38 odd 6 351.2.t.c.181.5 20
117.77 odd 6 1053.2.b.i.649.6 10
117.103 even 6 1053.2.b.j.649.5 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
117.2.t.c.25.5 ✓ 20 9.7 even 3 inner
117.2.t.c.25.6 yes 20 117.25 even 6 inner
117.2.t.c.103.5 yes 20 13.12 even 2 inner
117.2.t.c.103.6 yes 20 1.1 even 1 trivial
351.2.t.c.64.5 20 3.2 odd 2
351.2.t.c.64.6 20 39.38 odd 2
351.2.t.c.181.5 20 117.38 odd 6
351.2.t.c.181.6 20 9.2 odd 6
1053.2.b.i.649.5 10 9.5 odd 6
1053.2.b.i.649.6 10 117.77 odd 6
1053.2.b.j.649.5 10 117.103 even 6
1053.2.b.j.649.6 10 9.4 even 3