Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [138,2,Mod(13,138)] 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("138.13"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(138, base_ring=CyclotomicField(22)) chi = DirichletCharacter(H, H._module([0, 14])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 138 = 2 \cdot 3 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 138.e (of order \(11\), degree \(10\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [10,-1,1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.10193554789\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\Q(\zeta_{22})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - x^{9} + x^{8} - x^{7} + x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{11}]$

Embedding invariants

Embedding label 85.1
Root \(0.654861 - 0.755750i\) of defining polynomial
Character \(\chi\) \(=\) 138.85
Dual form 138.2.e.b.13.1

$q$-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+(-0.654861 + 0.755750i) q^{2} +(-0.841254 + 0.540641i) q^{3} +(-0.142315 - 0.989821i) q^{4} +(-1.61435 - 3.53494i) q^{5} +(0.142315 - 0.989821i) q^{6} +(-3.99283 + 1.17240i) q^{7} +(0.841254 + 0.540641i) q^{8} +(0.415415 - 0.909632i) q^{9} +(3.72871 + 1.09485i) q^{10} +(-2.96537 - 3.42222i) q^{11} +(0.654861 + 0.755750i) q^{12} +(3.13097 + 0.919336i) q^{13} +(1.72871 - 3.78534i) q^{14} +(3.26921 + 2.10100i) q^{15} +(-0.959493 + 0.281733i) q^{16} +(0.260239 - 1.81001i) q^{17} +(0.415415 + 0.909632i) q^{18} +(-0.194351 - 1.35174i) q^{19} +(-3.26921 + 2.10100i) q^{20} +(2.72514 - 3.14498i) q^{21} +4.52825 q^{22} +(-2.28185 + 4.21819i) q^{23} -1.00000 q^{24} +(-6.61537 + 7.63454i) q^{25} +(-2.74514 + 1.76419i) q^{26} +(0.142315 + 0.989821i) q^{27} +(1.72871 + 3.78534i) q^{28} +(-0.0798548 + 0.555402i) q^{29} +(-3.72871 + 1.09485i) q^{30} +(-3.90053 - 2.50672i) q^{31} +(0.415415 - 0.909632i) q^{32} +(4.34482 + 1.27576i) q^{33} +(1.19749 + 1.38198i) q^{34} +(10.5902 + 12.2218i) q^{35} +(-0.959493 - 0.281733i) q^{36} +(1.04639 - 2.29127i) q^{37} +(1.14885 + 0.738323i) q^{38} +(-3.13097 + 0.919336i) q^{39} +(0.553053 - 3.84657i) q^{40} +(-2.54130 - 5.56467i) q^{41} +(0.592229 + 4.11904i) q^{42} +(8.28836 - 5.32661i) q^{43} +(-2.96537 + 3.42222i) q^{44} -3.88612 q^{45} +(-1.69360 - 4.48684i) q^{46} +2.38128 q^{47} +(0.654861 - 0.755750i) q^{48} +(8.67941 - 5.57792i) q^{49} +(-1.43766 - 9.99913i) q^{50} +(0.759635 + 1.66337i) q^{51} +(0.464395 - 3.22994i) q^{52} +(-10.1804 + 2.98924i) q^{53} +(-0.841254 - 0.540641i) q^{54} +(-7.31020 + 16.0071i) q^{55} +(-3.99283 - 1.17240i) q^{56} +(0.894306 + 1.03208i) q^{57} +(-0.367451 - 0.424061i) q^{58} +(6.86289 + 2.01513i) q^{59} +(1.61435 - 3.53494i) q^{60} +(-2.73982 - 1.76078i) q^{61} +(4.44875 - 1.30627i) q^{62} +(-0.592229 + 4.11904i) q^{63} +(0.415415 + 0.909632i) q^{64} +(-1.80470 - 12.5519i) q^{65} +(-3.80941 + 2.44816i) q^{66} +(4.11751 - 4.75186i) q^{67} -1.82862 q^{68} +(-0.360914 - 4.78223i) q^{69} -16.1717 q^{70} +(4.22647 - 4.87760i) q^{71} +(0.841254 - 0.540641i) q^{72} +(-1.75805 - 12.2275i) q^{73} +(1.04639 + 2.29127i) q^{74} +(1.43766 - 9.99913i) q^{75} +(-1.31033 + 0.384746i) q^{76} +(15.8525 + 10.1878i) q^{77} +(1.35556 - 2.96827i) q^{78} +(-8.09543 - 2.37703i) q^{79} +(2.54487 + 2.93694i) q^{80} +(-0.654861 - 0.755750i) q^{81} +(5.86969 + 1.72350i) q^{82} +(-0.156624 + 0.342959i) q^{83} +(-3.50079 - 2.24982i) q^{84} +(-6.81838 + 2.00206i) q^{85} +(-1.40214 + 9.75211i) q^{86} +(-0.233095 - 0.510407i) q^{87} +(-0.644437 - 4.48216i) q^{88} +(12.7912 - 8.22041i) q^{89} +(2.54487 - 2.93694i) q^{90} -13.5793 q^{91} +(4.50000 + 1.65831i) q^{92} +4.63657 q^{93} +(-1.55940 + 1.79965i) q^{94} +(-4.46458 + 2.86921i) q^{95} +(0.142315 + 0.989821i) q^{96} +(7.84133 + 17.1701i) q^{97} +(-1.46830 + 10.2122i) q^{98} +(-4.34482 + 1.27576i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - q^{2} + q^{3} - q^{4} - 2 q^{5} + q^{6} - q^{8} - q^{9} + 9 q^{10} + 11 q^{11} + q^{12} + 13 q^{13} - 11 q^{14} + 13 q^{15} - q^{16} - 24 q^{17} - q^{18} - 14 q^{19} - 13 q^{20} + 11 q^{21}+ \cdots - 22 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/138\mathbb{Z}\right)^\times\).

\(n\) \(47\) \(97\)
\(\chi(n)\) \(1\) \(e\left(\frac{4}{11}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.654861 + 0.755750i −0.463056 + 0.534396i
\(3\) −0.841254 + 0.540641i −0.485698 + 0.312139i
\(4\) −0.142315 − 0.989821i −0.0711574 − 0.494911i
\(5\) −1.61435 − 3.53494i −0.721961 − 1.58087i −0.811136 − 0.584858i \(-0.801150\pi\)
0.0891748 − 0.996016i \(-0.471577\pi\)
\(6\) 0.142315 − 0.989821i 0.0580998 − 0.404093i
\(7\) −3.99283 + 1.17240i −1.50915 + 0.443126i −0.928595 − 0.371096i \(-0.878982\pi\)
−0.580554 + 0.814222i \(0.697164\pi\)
\(8\) 0.841254 + 0.540641i 0.297428 + 0.191145i
\(9\) 0.415415 − 0.909632i 0.138472 − 0.303211i
\(10\) 3.72871 + 1.09485i 1.17912 + 0.346221i
\(11\) −2.96537 − 3.42222i −0.894094 − 1.03184i −0.999301 − 0.0373888i \(-0.988096\pi\)
0.105207 − 0.994450i \(-0.466449\pi\)
\(12\) 0.654861 + 0.755750i 0.189042 + 0.218166i
\(13\) 3.13097 + 0.919336i 0.868376 + 0.254978i 0.685424 − 0.728144i \(-0.259617\pi\)
0.182951 + 0.983122i \(0.441435\pi\)
\(14\) 1.72871 − 3.78534i 0.462016 − 1.01167i
\(15\) 3.26921 + 2.10100i 0.844108 + 0.542475i
\(16\) −0.959493 + 0.281733i −0.239873 + 0.0704331i
\(17\) 0.260239 − 1.81001i 0.0631173 − 0.438991i −0.933619 − 0.358267i \(-0.883368\pi\)
0.996737 − 0.0807238i \(-0.0257232\pi\)
\(18\) 0.415415 + 0.909632i 0.0979143 + 0.214402i
\(19\) −0.194351 − 1.35174i −0.0445873 − 0.310111i −0.999895 − 0.0144978i \(-0.995385\pi\)
0.955308 − 0.295613i \(-0.0955240\pi\)
\(20\) −3.26921 + 2.10100i −0.731019 + 0.469797i
\(21\) 2.72514 − 3.14498i 0.594674 − 0.686290i
\(22\) 4.52825 0.965426
\(23\) −2.28185 + 4.21819i −0.475799 + 0.879554i
\(24\) −1.00000 −0.204124
\(25\) −6.61537 + 7.63454i −1.32307 + 1.52691i
\(26\) −2.74514 + 1.76419i −0.538366 + 0.345987i
\(27\) 0.142315 + 0.989821i 0.0273885 + 0.190491i
\(28\) 1.72871 + 3.78534i 0.326695 + 0.715362i
\(29\) −0.0798548 + 0.555402i −0.0148287 + 0.103136i −0.995892 − 0.0905538i \(-0.971136\pi\)
0.981063 + 0.193689i \(0.0620454\pi\)
\(30\) −3.72871 + 1.09485i −0.680766 + 0.199891i
\(31\) −3.90053 − 2.50672i −0.700556 − 0.450220i 0.141269 − 0.989971i \(-0.454882\pi\)
−0.841824 + 0.539751i \(0.818518\pi\)
\(32\) 0.415415 − 0.909632i 0.0734357 − 0.160802i
\(33\) 4.34482 + 1.27576i 0.756337 + 0.222081i
\(34\) 1.19749 + 1.38198i 0.205368 + 0.237007i
\(35\) 10.5902 + 12.2218i 1.79007 + 2.06585i
\(36\) −0.959493 − 0.281733i −0.159915 − 0.0469554i
\(37\) 1.04639 − 2.29127i 0.172025 − 0.376682i −0.803908 − 0.594754i \(-0.797249\pi\)
0.975933 + 0.218072i \(0.0699767\pi\)
\(38\) 1.14885 + 0.738323i 0.186369 + 0.119772i
\(39\) −3.13097 + 0.919336i −0.501357 + 0.147212i
\(40\) 0.553053 − 3.84657i 0.0874453 − 0.608196i
\(41\) −2.54130 − 5.56467i −0.396884 − 0.869055i −0.997577 − 0.0695749i \(-0.977836\pi\)
0.600693 − 0.799480i \(-0.294892\pi\)
\(42\) 0.592229 + 4.11904i 0.0913829 + 0.635582i
\(43\) 8.28836 − 5.32661i 1.26396 − 0.812300i 0.275142 − 0.961404i \(-0.411275\pi\)
0.988822 + 0.149104i \(0.0476388\pi\)
\(44\) −2.96537 + 3.42222i −0.447047 + 0.515920i
\(45\) −3.88612 −0.579309
\(46\) −1.69360 − 4.48684i −0.249708 − 0.661548i
\(47\) 2.38128 0.347345 0.173672 − 0.984803i \(-0.444437\pi\)
0.173672 + 0.984803i \(0.444437\pi\)
\(48\) 0.654861 − 0.755750i 0.0945210 − 0.109083i
\(49\) 8.67941 − 5.57792i 1.23992 − 0.796846i
\(50\) −1.43766 − 9.99913i −0.203315 − 1.41409i
\(51\) 0.759635 + 1.66337i 0.106370 + 0.232918i
\(52\) 0.464395 − 3.22994i 0.0644000 − 0.447912i
\(53\) −10.1804 + 2.98924i −1.39839 + 0.410604i −0.892131 − 0.451777i \(-0.850790\pi\)
−0.506259 + 0.862382i \(0.668972\pi\)
\(54\) −0.841254 − 0.540641i −0.114480 − 0.0735719i
\(55\) −7.31020 + 16.0071i −0.985707 + 2.15840i
\(56\) −3.99283 − 1.17240i −0.533565 − 0.156669i
\(57\) 0.894306 + 1.03208i 0.118454 + 0.136703i
\(58\) −0.367451 − 0.424061i −0.0482487 − 0.0556820i
\(59\) 6.86289 + 2.01513i 0.893472 + 0.262347i 0.696069 − 0.717975i \(-0.254931\pi\)
0.197404 + 0.980322i \(0.436749\pi\)
\(60\) 1.61435 − 3.53494i 0.208412 − 0.456359i
\(61\) −2.73982 − 1.76078i −0.350799 − 0.225445i 0.353360 − 0.935487i \(-0.385039\pi\)
−0.704158 + 0.710043i \(0.748676\pi\)
\(62\) 4.44875 − 1.30627i 0.564992 − 0.165897i
\(63\) −0.592229 + 4.11904i −0.0746138 + 0.518950i
\(64\) 0.415415 + 0.909632i 0.0519269 + 0.113704i
\(65\) −1.80470 − 12.5519i −0.223845 − 1.55688i
\(66\) −3.80941 + 2.44816i −0.468906 + 0.301347i
\(67\) 4.11751 − 4.75186i 0.503034 − 0.580532i −0.446267 − 0.894900i \(-0.647247\pi\)
0.949301 + 0.314367i \(0.101792\pi\)
\(68\) −1.82862 −0.221753
\(69\) −0.360914 − 4.78223i −0.0434489 − 0.575713i
\(70\) −16.1717 −1.93289
\(71\) 4.22647 − 4.87760i 0.501589 − 0.578865i −0.447336 − 0.894366i \(-0.647627\pi\)
0.948925 + 0.315501i \(0.102173\pi\)
\(72\) 0.841254 − 0.540641i 0.0991427 − 0.0637151i
\(73\) −1.75805 − 12.2275i −0.205765 − 1.43112i −0.786781 − 0.617232i \(-0.788254\pi\)
0.581017 − 0.813892i \(-0.302655\pi\)
\(74\) 1.04639 + 2.29127i 0.121640 + 0.266355i
\(75\) 1.43766 − 9.99913i 0.166006 − 1.15460i
\(76\) −1.31033 + 0.384746i −0.150305 + 0.0441334i
\(77\) 15.8525 + 10.1878i 1.80656 + 1.16100i
\(78\) 1.35556 − 2.96827i 0.153487 − 0.336090i
\(79\) −8.09543 − 2.37703i −0.910807 − 0.267437i −0.207427 − 0.978251i \(-0.566509\pi\)
−0.703381 + 0.710813i \(0.748327\pi\)
\(80\) 2.54487 + 2.93694i 0.284525 + 0.328359i
\(81\) −0.654861 − 0.755750i −0.0727623 − 0.0839722i
\(82\) 5.86969 + 1.72350i 0.648199 + 0.190328i
\(83\) −0.156624 + 0.342959i −0.0171917 + 0.0376446i −0.918033 − 0.396504i \(-0.870223\pi\)
0.900841 + 0.434148i \(0.142951\pi\)
\(84\) −3.50079 − 2.24982i −0.381968 − 0.245476i
\(85\) −6.81838 + 2.00206i −0.739557 + 0.217154i
\(86\) −1.40214 + 9.75211i −0.151197 + 1.05160i
\(87\) −0.233095 − 0.510407i −0.0249904 − 0.0547214i
\(88\) −0.644437 − 4.48216i −0.0686972 − 0.477800i
\(89\) 12.7912 − 8.22041i 1.35587 − 0.871362i 0.357816 − 0.933792i \(-0.383521\pi\)
0.998049 + 0.0624300i \(0.0198850\pi\)
\(90\) 2.54487 − 2.93694i 0.268253 − 0.309580i
\(91\) −13.5793 −1.42350
\(92\) 4.50000 + 1.65831i 0.469157 + 0.172891i
\(93\) 4.63657 0.480790
\(94\) −1.55940 + 1.79965i −0.160840 + 0.185619i
\(95\) −4.46458 + 2.86921i −0.458056 + 0.294375i
\(96\) 0.142315 + 0.989821i 0.0145249 + 0.101023i
\(97\) 7.84133 + 17.1701i 0.796166 + 1.74336i 0.658091 + 0.752939i \(0.271364\pi\)
0.138075 + 0.990422i \(0.455908\pi\)
\(98\) −1.46830 + 10.2122i −0.148320 + 1.03159i
\(99\) −4.34482 + 1.27576i −0.436671 + 0.128218i
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 138.2.e.b.85.1 yes 10
3.2 odd 2 414.2.i.e.361.1 10
23.6 even 11 3174.2.a.ba.1.1 5
23.13 even 11 inner 138.2.e.b.13.1 ✓ 10
23.17 odd 22 3174.2.a.bb.1.5 5
69.17 even 22 9522.2.a.br.1.1 5
69.29 odd 22 9522.2.a.bs.1.5 5
69.59 odd 22 414.2.i.e.289.1 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
138.2.e.b.13.1 ✓ 10 23.13 even 11 inner
138.2.e.b.85.1 yes 10 1.1 even 1 trivial
414.2.i.e.289.1 10 69.59 odd 22
414.2.i.e.361.1 10 3.2 odd 2
3174.2.a.ba.1.1 5 23.6 even 11
3174.2.a.bb.1.5 5 23.17 odd 22
9522.2.a.br.1.1 5 69.17 even 22
9522.2.a.bs.1.5 5 69.29 odd 22