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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [138,4,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, 4); 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, 4, names="a")
 
Level: \( N \) \(=\) \( 138 = 2 \cdot 3 \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 138.e (of order \(11\), degree \(10\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [30,-6,9] 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: \(8.14226358079\)
Analytic rank: \(0\)
Dimension: \(30\)
Relative dimension: \(3\) over \(\Q(\zeta_{11})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{11}]$

Embedding invariants

Embedding label 25.1
Character \(\chi\) \(=\) 138.25
Dual form 138.4.e.b.127.1

$q$-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+(1.68251 + 1.08128i) q^{2} +(0.426945 + 2.96946i) q^{3} +(1.66166 + 3.63853i) q^{4} +(-20.3614 - 5.97866i) q^{5} +(-2.49249 + 5.45779i) q^{6} +(5.13322 - 5.92406i) q^{7} +(-1.13852 + 7.91857i) q^{8} +(-8.63544 + 2.53559i) q^{9} +(-27.7936 - 32.0756i) q^{10} +(-33.5772 + 21.5787i) q^{11} +(-10.0950 + 6.48769i) q^{12} +(-36.4799 - 42.1001i) q^{13} +(15.0423 - 4.41680i) q^{14} +(9.06020 - 63.0151i) q^{15} +(-10.4778 + 12.0920i) q^{16} +(-37.7694 + 82.7034i) q^{17} +(-17.2709 - 5.07119i) q^{18} +(-47.3670 - 103.719i) q^{19} +(-12.0803 - 84.0201i) q^{20} +(19.7829 + 12.7137i) q^{21} -79.8265 q^{22} +(107.478 - 24.8105i) q^{23} -24.0000 q^{24} +(273.687 + 175.888i) q^{25} +(-15.8557 - 110.279i) q^{26} +(-11.2162 - 24.5601i) q^{27} +(30.0845 + 8.83361i) q^{28} +(-62.5068 + 136.871i) q^{29} +(83.3809 - 96.2267i) q^{30} +(0.528140 - 3.67329i) q^{31} +(-30.7038 + 9.01544i) q^{32} +(-78.4129 - 90.4933i) q^{33} +(-152.973 + 98.3097i) q^{34} +(-139.938 + 89.9325i) q^{35} +(-23.5750 - 27.2070i) q^{36} +(-210.432 + 61.7884i) q^{37} +(32.4544 - 225.725i) q^{38} +(109.440 - 126.300i) q^{39} +(70.5243 - 154.427i) q^{40} +(-136.943 - 40.2102i) q^{41} +(19.5378 + 42.7817i) q^{42} +(52.5290 + 365.347i) q^{43} +(-134.309 - 86.3149i) q^{44} +190.989 q^{45} +(207.659 + 74.4698i) q^{46} -145.843 q^{47} +(-40.3802 - 25.9508i) q^{48} +(40.0695 + 278.690i) q^{49} +(270.296 + 591.865i) q^{50} +(-261.710 - 76.8450i) q^{51} +(92.5651 - 202.689i) q^{52} +(-84.5597 + 97.5871i) q^{53} +(7.68500 - 53.4504i) q^{54} +(812.691 - 238.628i) q^{55} +(41.0658 + 47.3924i) q^{56} +(287.768 - 184.937i) q^{57} +(-253.164 + 162.699i) q^{58} +(328.232 + 378.800i) q^{59} +(244.337 - 71.7439i) q^{60} +(117.472 - 817.037i) q^{61} +(4.86046 - 5.60927i) q^{62} +(-29.3066 + 64.1726i) q^{63} +(-61.4076 - 18.0309i) q^{64} +(491.082 + 1075.32i) q^{65} +(-34.0815 - 237.042i) q^{66} +(-384.841 - 247.322i) q^{67} -363.678 q^{68} +(119.561 + 308.558i) q^{69} -332.688 q^{70} +(-109.019 - 70.0621i) q^{71} +(-10.2467 - 71.2671i) q^{72} +(102.202 + 223.790i) q^{73} +(-420.864 - 123.577i) q^{74} +(-405.444 + 887.798i) q^{75} +(298.678 - 344.692i) q^{76} +(-44.5255 + 309.681i) q^{77} +(320.699 - 94.1658i) q^{78} +(-178.961 - 206.532i) q^{79} +(285.636 - 183.567i) q^{80} +(68.1415 - 43.7919i) q^{81} +(-186.930 - 215.728i) q^{82} +(-368.913 + 108.323i) q^{83} +(-13.3867 + 93.1063i) q^{84} +(1263.49 - 1458.15i) q^{85} +(-306.663 + 671.497i) q^{86} +(-433.120 - 127.175i) q^{87} +(-132.645 - 290.451i) q^{88} +(-183.455 - 1275.96i) q^{89} +(321.341 + 206.513i) q^{90} -436.663 q^{91} +(268.865 + 349.834i) q^{92} +11.1332 q^{93} +(-245.382 - 157.697i) q^{94} +(344.358 + 2395.06i) q^{95} +(-39.8798 - 87.3247i) q^{96} +(1192.07 + 350.024i) q^{97} +(-233.925 + 512.224i) q^{98} +(235.239 - 271.480i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 30 q - 6 q^{2} + 9 q^{3} - 12 q^{4} - 6 q^{5} + 18 q^{6} + 22 q^{7} - 24 q^{8} - 27 q^{9} - 56 q^{10} - 105 q^{11} + 36 q^{12} - 21 q^{13} - 114 q^{15} - 48 q^{16} + 41 q^{17} - 54 q^{18} - 149 q^{19} + 152 q^{20}+ \cdots - 945 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.68251 + 1.08128i 0.594856 + 0.382291i
\(3\) 0.426945 + 2.96946i 0.0821655 + 0.571474i
\(4\) 1.66166 + 3.63853i 0.207708 + 0.454816i
\(5\) −20.3614 − 5.97866i −1.82118 − 0.534747i −0.821796 − 0.569782i \(-0.807028\pi\)
−0.999386 + 0.0350350i \(0.988846\pi\)
\(6\) −2.49249 + 5.45779i −0.169592 + 0.371356i
\(7\) 5.13322 − 5.92406i 0.277168 − 0.319869i −0.600049 − 0.799963i \(-0.704852\pi\)
0.877217 + 0.480094i \(0.159398\pi\)
\(8\) −1.13852 + 7.91857i −0.0503159 + 0.349955i
\(9\) −8.63544 + 2.53559i −0.319831 + 0.0939109i
\(10\) −27.7936 − 32.0756i −0.878912 − 1.01432i
\(11\) −33.5772 + 21.5787i −0.920354 + 0.591476i −0.912760 − 0.408495i \(-0.866054\pi\)
−0.00759384 + 0.999971i \(0.502417\pi\)
\(12\) −10.0950 + 6.48769i −0.242849 + 0.156070i
\(13\) −36.4799 − 42.1001i −0.778286 − 0.898189i 0.218699 − 0.975792i \(-0.429819\pi\)
−0.996984 + 0.0776030i \(0.975273\pi\)
\(14\) 15.0423 − 4.41680i 0.287158 − 0.0843172i
\(15\) 9.06020 − 63.0151i 0.155956 − 1.08470i
\(16\) −10.4778 + 12.0920i −0.163715 + 0.188937i
\(17\) −37.7694 + 82.7034i −0.538848 + 1.17991i 0.422950 + 0.906153i \(0.360995\pi\)
−0.961798 + 0.273760i \(0.911733\pi\)
\(18\) −17.2709 − 5.07119i −0.226155 − 0.0664050i
\(19\) −47.3670 − 103.719i −0.571933 − 1.25236i −0.945762 − 0.324861i \(-0.894682\pi\)
0.373828 − 0.927498i \(-0.378045\pi\)
\(20\) −12.0803 − 84.0201i −0.135062 − 0.939374i
\(21\) 19.7829 + 12.7137i 0.205570 + 0.132112i
\(22\) −79.8265 −0.773594
\(23\) 107.478 − 24.8105i 0.974375 − 0.224928i
\(24\) −24.0000 −0.204124
\(25\) 273.687 + 175.888i 2.18950 + 1.40710i
\(26\) −15.8557 − 110.279i −0.119598 − 0.831825i
\(27\) −11.2162 − 24.5601i −0.0799467 − 0.175059i
\(28\) 30.0845 + 8.83361i 0.203051 + 0.0596213i
\(29\) −62.5068 + 136.871i −0.400249 + 0.876423i 0.596996 + 0.802244i \(0.296361\pi\)
−0.997245 + 0.0741786i \(0.976367\pi\)
\(30\) 83.3809 − 96.2267i 0.507440 − 0.585617i
\(31\) 0.528140 − 3.67329i 0.00305989 − 0.0212820i −0.988234 − 0.152948i \(-0.951123\pi\)
0.991294 + 0.131666i \(0.0420325\pi\)
\(32\) −30.7038 + 9.01544i −0.169616 + 0.0498038i
\(33\) −78.4129 − 90.4933i −0.413634 − 0.477359i
\(34\) −152.973 + 98.3097i −0.771607 + 0.495882i
\(35\) −139.938 + 89.9325i −0.675822 + 0.434325i
\(36\) −23.5750 − 27.2070i −0.109143 − 0.125958i
\(37\) −210.432 + 61.7884i −0.934995 + 0.274539i −0.713527 − 0.700628i \(-0.752903\pi\)
−0.221469 + 0.975168i \(0.571085\pi\)
\(38\) 32.4544 − 225.725i 0.138547 − 0.963618i
\(39\) 109.440 − 126.300i 0.449343 − 0.518570i
\(40\) 70.5243 − 154.427i 0.278772 − 0.610425i
\(41\) −136.943 − 40.2102i −0.521633 − 0.153165i 0.0103074 − 0.999947i \(-0.496719\pi\)
−0.531941 + 0.846781i \(0.678537\pi\)
\(42\) 19.5378 + 42.7817i 0.0717795 + 0.157175i
\(43\) 52.5290 + 365.347i 0.186293 + 1.29569i 0.841505 + 0.540250i \(0.181670\pi\)
−0.655212 + 0.755445i \(0.727421\pi\)
\(44\) −134.309 − 86.3149i −0.460177 − 0.295738i
\(45\) 190.989 0.632689
\(46\) 207.659 + 74.4698i 0.665601 + 0.238695i
\(47\) −145.843 −0.452626 −0.226313 − 0.974055i \(-0.572667\pi\)
−0.226313 + 0.974055i \(0.572667\pi\)
\(48\) −40.3802 − 25.9508i −0.121424 − 0.0780348i
\(49\) 40.0695 + 278.690i 0.116821 + 0.812507i
\(50\) 270.296 + 591.865i 0.764512 + 1.67405i
\(51\) −261.710 − 76.8450i −0.718564 − 0.210989i
\(52\) 92.5651 − 202.689i 0.246855 − 0.540537i
\(53\) −84.5597 + 97.5871i −0.219154 + 0.252917i −0.854671 − 0.519169i \(-0.826241\pi\)
0.635517 + 0.772087i \(0.280787\pi\)
\(54\) 7.68500 − 53.4504i 0.0193666 − 0.134698i
\(55\) 812.691 − 238.628i 1.99242 − 0.585028i
\(56\) 41.0658 + 47.3924i 0.0979937 + 0.113091i
\(57\) 287.768 − 184.937i 0.668697 − 0.429746i
\(58\) −253.164 + 162.699i −0.573139 + 0.368334i
\(59\) 328.232 + 378.800i 0.724274 + 0.835857i 0.991814 − 0.127690i \(-0.0407562\pi\)
−0.267540 + 0.963547i \(0.586211\pi\)
\(60\) 244.337 − 71.7439i 0.525730 − 0.154368i
\(61\) 117.472 − 817.037i 0.246570 − 1.71493i −0.371183 − 0.928560i \(-0.621048\pi\)
0.617753 − 0.786372i \(-0.288043\pi\)
\(62\) 4.86046 − 5.60927i 0.00995612 − 0.0114900i
\(63\) −29.3066 + 64.1726i −0.0586078 + 0.128333i
\(64\) −61.4076 − 18.0309i −0.119937 − 0.0352166i
\(65\) 491.082 + 1075.32i 0.937095 + 2.05195i
\(66\) −34.0815 − 237.042i −0.0635627 − 0.442089i
\(67\) −384.841 − 247.322i −0.701728 − 0.450973i 0.140510 − 0.990079i \(-0.455126\pi\)
−0.842238 + 0.539106i \(0.818762\pi\)
\(68\) −363.678 −0.648566
\(69\) 119.561 + 308.558i 0.208600 + 0.538349i
\(70\) −332.688 −0.568055
\(71\) −109.019 − 70.0621i −0.182227 − 0.117110i 0.446347 − 0.894860i \(-0.352725\pi\)
−0.628574 + 0.777750i \(0.716361\pi\)
\(72\) −10.2467 − 71.2671i −0.0167720 − 0.116652i
\(73\) 102.202 + 223.790i 0.163860 + 0.358804i 0.973695 − 0.227854i \(-0.0731708\pi\)
−0.809835 + 0.586657i \(0.800444\pi\)
\(74\) −420.864 − 123.577i −0.661141 − 0.194129i
\(75\) −405.444 + 887.798i −0.624221 + 1.36685i
\(76\) 298.678 − 344.692i 0.450798 − 0.520249i
\(77\) −44.5255 + 309.681i −0.0658980 + 0.458331i
\(78\) 320.699 − 94.1658i 0.465539 − 0.136695i
\(79\) −178.961 − 206.532i −0.254870 − 0.294135i 0.613867 − 0.789409i \(-0.289613\pi\)
−0.868737 + 0.495274i \(0.835068\pi\)
\(80\) 285.636 − 183.567i 0.399189 − 0.256543i
\(81\) 68.1415 − 43.7919i 0.0934726 − 0.0600712i
\(82\) −186.930 − 215.728i −0.251743 − 0.290527i
\(83\) −368.913 + 108.323i −0.487874 + 0.143253i −0.516412 − 0.856340i \(-0.672733\pi\)
0.0285387 + 0.999593i \(0.490915\pi\)
\(84\) −13.3867 + 93.1063i −0.0173882 + 0.120937i
\(85\) 1263.49 − 1458.15i 1.61230 − 1.86069i
\(86\) −306.663 + 671.497i −0.384515 + 0.841970i
\(87\) −433.120 − 127.175i −0.533739 − 0.156720i
\(88\) −132.645 − 290.451i −0.160681 − 0.351843i
\(89\) −183.455 − 1275.96i −0.218497 − 1.51968i −0.743590 − 0.668636i \(-0.766879\pi\)
0.525093 − 0.851045i \(-0.324030\pi\)
\(90\) 321.341 + 206.513i 0.376359 + 0.241871i
\(91\) −436.663 −0.503019
\(92\) 268.865 + 349.834i 0.304686 + 0.396442i
\(93\) 11.1332 0.0124135
\(94\) −245.382 − 157.697i −0.269247 − 0.173035i
\(95\) 344.358 + 2395.06i 0.371899 + 2.58661i
\(96\) −39.8798 − 87.3247i −0.0423981 − 0.0928389i
\(97\) 1192.07 + 350.024i 1.24780 + 0.366387i 0.837940 − 0.545762i \(-0.183760\pi\)
0.409859 + 0.912149i \(0.365578\pi\)
\(98\) −233.925 + 512.224i −0.241122 + 0.527984i
\(99\) 235.239 − 271.480i 0.238812 − 0.275604i
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 138.4.e.b.25.1 ✓ 30
23.12 even 11 inner 138.4.e.b.127.1 yes 30
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
138.4.e.b.25.1 ✓ 30 1.1 even 1 trivial
138.4.e.b.127.1 yes 30 23.12 even 11 inner