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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [230,4,Mod(31,230)] 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("230.31"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(230, base_ring=CyclotomicField(22)) chi = DirichletCharacter(H, H._module([0, 6])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 230 = 2 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 230.g (of order \(11\), degree \(10\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [60,12,-3] 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: \(13.5704393013\)
Analytic rank: \(0\)
Dimension: \(60\)
Relative dimension: \(6\) over \(\Q(\zeta_{11})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{11}]$

Embedding invariants

Embedding label 71.3
Character \(\chi\) \(=\) 230.71
Dual form 230.4.g.b.81.3

$q$-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} +(-0.259790 - 1.80688i) q^{3} +(1.66166 + 3.63853i) q^{4} +(-4.79746 - 1.40866i) q^{5} +(-1.51665 + 3.32099i) q^{6} +(-3.50130 + 4.04072i) q^{7} +(1.13852 - 7.91857i) q^{8} +(22.7090 - 6.66796i) q^{9} +(6.54861 + 7.55750i) q^{10} +(-20.4137 + 13.1191i) q^{11} +(6.14270 - 3.94767i) q^{12} +(-42.9889 - 49.6119i) q^{13} +(10.2601 - 3.01264i) q^{14} +(-1.29895 + 9.03439i) q^{15} +(-10.4778 + 12.0920i) q^{16} +(-34.5899 + 75.7413i) q^{17} +(-45.4180 - 13.3359i) q^{18} +(30.0343 + 65.7660i) q^{19} +(-2.84630 - 19.7964i) q^{20} +(8.21069 + 5.27669i) q^{21} +48.5315 q^{22} +(47.3096 + 99.6434i) q^{23} -14.6037 q^{24} +(21.0313 + 13.5160i) q^{25} +(18.6848 + 129.955i) q^{26} +(-38.4225 - 84.1335i) q^{27} +(-20.5202 - 6.02528i) q^{28} +(2.18701 - 4.78889i) q^{29} +(11.9542 - 13.7959i) q^{30} +(-11.1970 + 77.8766i) q^{31} +(30.7038 - 9.01544i) q^{32} +(29.0078 + 33.4768i) q^{33} +(140.095 - 90.0338i) q^{34} +(22.4894 - 14.4530i) q^{35} +(61.9962 + 71.5474i) q^{36} +(-280.098 + 82.2443i) q^{37} +(20.5786 - 143.127i) q^{38} +(-78.4745 + 90.5645i) q^{39} +(-16.6166 + 36.3853i) q^{40} +(394.291 + 115.774i) q^{41} +(-8.10895 - 17.7561i) q^{42} +(62.0217 + 431.370i) q^{43} +(-81.6546 - 52.4763i) q^{44} -118.338 q^{45} +(28.1438 - 218.806i) q^{46} +548.118 q^{47} +(24.5708 + 15.7907i) q^{48} +(44.7457 + 311.213i) q^{49} +(-20.7708 - 45.4816i) q^{50} +(145.841 + 42.8229i) q^{51} +(109.081 - 238.855i) q^{52} +(-198.782 + 229.407i) q^{53} +(-26.3259 + 183.101i) q^{54} +(116.414 - 34.1823i) q^{55} +(28.0104 + 32.3257i) q^{56} +(111.029 - 71.3538i) q^{57} +(-8.85780 + 5.69256i) q^{58} +(190.609 + 219.975i) q^{59} +(-35.0303 + 10.2858i) q^{60} +(-117.619 + 818.061i) q^{61} +(103.045 - 118.921i) q^{62} +(-52.5677 + 115.107i) q^{63} +(-61.4076 - 18.0309i) q^{64} +(136.351 + 298.568i) q^{65} +(-12.6080 - 87.6906i) q^{66} +(-742.504 - 477.179i) q^{67} -333.063 q^{68} +(167.753 - 111.369i) q^{69} -53.4663 q^{70} +(-299.418 - 192.424i) q^{71} +(-26.9461 - 187.414i) q^{72} +(-377.642 - 826.921i) q^{73} +(560.197 + 164.489i) q^{74} +(18.9581 - 41.5124i) q^{75} +(-189.385 + 218.562i) q^{76} +(18.4640 - 128.420i) q^{77} +(229.960 - 67.5223i) q^{78} +(-353.150 - 407.557i) q^{79} +(67.3003 - 43.2513i) q^{80} +(395.547 - 254.203i) q^{81} +(-538.213 - 621.131i) q^{82} +(-150.181 + 44.0971i) q^{83} +(-5.55601 + 38.6429i) q^{84} +(272.638 - 314.641i) q^{85} +(362.081 - 792.846i) q^{86} +(-9.22111 - 2.70756i) q^{87} +(80.6429 + 176.583i) q^{88} +(84.3682 + 586.794i) q^{89} +(199.105 + 127.957i) q^{90} +350.985 q^{91} +(-283.943 + 337.711i) q^{92} +143.622 q^{93} +(-922.212 - 592.670i) q^{94} +(-51.4465 - 357.818i) q^{95} +(-24.2663 - 53.1359i) q^{96} +(-978.602 - 287.344i) q^{97} +(261.224 - 572.001i) q^{98} +(-376.096 + 434.038i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 60 q + 12 q^{2} - 3 q^{3} - 24 q^{4} - 30 q^{5} + 6 q^{6} + 100 q^{7} + 48 q^{8} + 69 q^{9} + 60 q^{10} - 51 q^{11} + 120 q^{12} + 184 q^{13} + 20 q^{14} - 15 q^{15} - 96 q^{16} - 334 q^{17} - 138 q^{18}+ \cdots - 10589 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/230\mathbb{Z}\right)^\times\).

\(n\) \(47\) \(51\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{11}\right)\)

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
\(3\) −0.259790 − 1.80688i −0.0499966 − 0.347734i −0.999430 − 0.0337649i \(-0.989250\pi\)
0.949433 − 0.313969i \(-0.101659\pi\)
\(4\) 1.66166 + 3.63853i 0.207708 + 0.454816i
\(5\) −4.79746 − 1.40866i −0.429098 − 0.125995i
\(6\) −1.51665 + 3.32099i −0.103195 + 0.225965i
\(7\) −3.50130 + 4.04072i −0.189052 + 0.218178i −0.842361 − 0.538914i \(-0.818835\pi\)
0.653309 + 0.757092i \(0.273380\pi\)
\(8\) 1.13852 − 7.91857i 0.0503159 − 0.349955i
\(9\) 22.7090 − 6.66796i 0.841074 − 0.246962i
\(10\) 6.54861 + 7.55750i 0.207085 + 0.238989i
\(11\) −20.4137 + 13.1191i −0.559541 + 0.359595i −0.789638 − 0.613572i \(-0.789732\pi\)
0.230097 + 0.973168i \(0.426095\pi\)
\(12\) 6.14270 − 3.94767i 0.147770 − 0.0949662i
\(13\) −42.9889 − 49.6119i −0.917153 − 1.05845i −0.998093 − 0.0617317i \(-0.980338\pi\)
0.0809401 − 0.996719i \(-0.474208\pi\)
\(14\) 10.2601 − 3.01264i 0.195867 − 0.0575116i
\(15\) −1.29895 + 9.03439i −0.0223592 + 0.155511i
\(16\) −10.4778 + 12.0920i −0.163715 + 0.188937i
\(17\) −34.5899 + 75.7413i −0.493487 + 1.08059i 0.485045 + 0.874489i \(0.338803\pi\)
−0.978532 + 0.206096i \(0.933924\pi\)
\(18\) −45.4180 − 13.3359i −0.594729 − 0.174628i
\(19\) 30.0343 + 65.7660i 0.362650 + 0.794093i 0.999729 + 0.0232940i \(0.00741537\pi\)
−0.637079 + 0.770799i \(0.719857\pi\)
\(20\) −2.84630 − 19.7964i −0.0318226 − 0.221331i
\(21\) 8.21069 + 5.27669i 0.0853200 + 0.0548318i
\(22\) 48.5315 0.470316
\(23\) 47.3096 + 99.6434i 0.428901 + 0.903351i
\(24\) −14.6037 −0.124207
\(25\) 21.0313 + 13.5160i 0.168251 + 0.108128i
\(26\) 18.6848 + 129.955i 0.140938 + 0.980245i
\(27\) −38.4225 − 84.1335i −0.273867 − 0.599685i
\(28\) −20.5202 − 6.02528i −0.138499 − 0.0406668i
\(29\) 2.18701 − 4.78889i 0.0140041 − 0.0306646i −0.902501 − 0.430688i \(-0.858271\pi\)
0.916505 + 0.400024i \(0.130998\pi\)
\(30\) 11.9542 − 13.7959i 0.0727510 − 0.0839592i
\(31\) −11.1970 + 77.8766i −0.0648721 + 0.451195i 0.931334 + 0.364167i \(0.118646\pi\)
−0.996206 + 0.0870284i \(0.972263\pi\)
\(32\) 30.7038 − 9.01544i 0.169616 − 0.0498038i
\(33\) 29.0078 + 33.4768i 0.153019 + 0.176593i
\(34\) 140.095 − 90.0338i 0.706652 − 0.454137i
\(35\) 22.4894 − 14.4530i 0.108611 − 0.0698003i
\(36\) 61.9962 + 71.5474i 0.287019 + 0.331238i
\(37\) −280.098 + 82.2443i −1.24454 + 0.365429i −0.836718 − 0.547633i \(-0.815529\pi\)
−0.407819 + 0.913063i \(0.633711\pi\)
\(38\) 20.5786 − 143.127i 0.0878498 − 0.611009i
\(39\) −78.4745 + 90.5645i −0.322205 + 0.371844i
\(40\) −16.6166 + 36.3853i −0.0656829 + 0.143825i
\(41\) 394.291 + 115.774i 1.50190 + 0.440998i 0.926317 − 0.376744i \(-0.122956\pi\)
0.575584 + 0.817742i \(0.304775\pi\)
\(42\) −8.10895 − 17.7561i −0.0297914 − 0.0652341i
\(43\) 62.0217 + 431.370i 0.219958 + 1.52984i 0.738183 + 0.674600i \(0.235684\pi\)
−0.518225 + 0.855245i \(0.673407\pi\)
\(44\) −81.6546 − 52.4763i −0.279770 − 0.179798i
\(45\) −118.338 −0.392019
\(46\) 28.1438 − 218.806i 0.0902083 − 0.701329i
\(47\) 548.118 1.70109 0.850545 − 0.525902i \(-0.176272\pi\)
0.850545 + 0.525902i \(0.176272\pi\)
\(48\) 24.5708 + 15.7907i 0.0738852 + 0.0474831i
\(49\) 44.7457 + 311.213i 0.130454 + 0.907327i
\(50\) −20.7708 − 45.4816i −0.0587486 − 0.128641i
\(51\) 145.841 + 42.8229i 0.400429 + 0.117577i
\(52\) 109.081 − 238.855i 0.290901 − 0.636984i
\(53\) −198.782 + 229.407i −0.515186 + 0.594556i −0.952419 − 0.304792i \(-0.901413\pi\)
0.437233 + 0.899348i \(0.355958\pi\)
\(54\) −26.3259 + 183.101i −0.0663427 + 0.461423i
\(55\) 116.414 − 34.1823i 0.285405 − 0.0838025i
\(56\) 28.0104 + 32.3257i 0.0668401 + 0.0771376i
\(57\) 111.029 − 71.3538i 0.258002 − 0.165808i
\(58\) −8.85780 + 5.69256i −0.0200532 + 0.0128874i
\(59\) 190.609 + 219.975i 0.420597 + 0.485394i 0.926019 − 0.377478i \(-0.123209\pi\)
−0.505422 + 0.862872i \(0.668663\pi\)
\(60\) −35.0303 + 10.2858i −0.0753732 + 0.0221316i
\(61\) −117.619 + 818.061i −0.246879 + 1.71708i 0.369159 + 0.929366i \(0.379646\pi\)
−0.616038 + 0.787716i \(0.711263\pi\)
\(62\) 103.045 − 118.921i 0.211077 − 0.243596i
\(63\) −52.5677 + 115.107i −0.105125 + 0.230193i
\(64\) −61.4076 − 18.0309i −0.119937 − 0.0352166i
\(65\) 136.351 + 298.568i 0.260190 + 0.569736i
\(66\) −12.6080 − 87.6906i −0.0235142 − 0.163545i
\(67\) −742.504 − 477.179i −1.35390 − 0.870099i −0.355976 − 0.934495i \(-0.615852\pi\)
−0.997924 + 0.0643961i \(0.979488\pi\)
\(68\) −333.063 −0.593969
\(69\) 167.753 − 111.369i 0.292682 − 0.194308i
\(70\) −53.4663 −0.0912922
\(71\) −299.418 − 192.424i −0.500483 − 0.321641i 0.265926 − 0.963993i \(-0.414322\pi\)
−0.766410 + 0.642352i \(0.777959\pi\)
\(72\) −26.9461 − 187.414i −0.0441060 − 0.306764i
\(73\) −377.642 − 826.921i −0.605475 − 1.32581i −0.925626 − 0.378439i \(-0.876461\pi\)
0.320151 − 0.947367i \(-0.396266\pi\)
\(74\) 560.197 + 164.489i 0.880021 + 0.258397i
\(75\) 18.9581 − 41.5124i 0.0291879 − 0.0639125i
\(76\) −189.385 + 218.562i −0.285841 + 0.329878i
\(77\) 18.4640 − 128.420i 0.0273268 − 0.190062i
\(78\) 229.960 − 67.5223i 0.333818 − 0.0980178i
\(79\) −353.150 − 407.557i −0.502943 − 0.580428i 0.446334 − 0.894866i \(-0.352729\pi\)
−0.949278 + 0.314439i \(0.898184\pi\)
\(80\) 67.3003 − 43.2513i 0.0940550 − 0.0604455i
\(81\) 395.547 − 254.203i 0.542589 − 0.348701i
\(82\) −538.213 − 621.131i −0.724826 − 0.836494i
\(83\) −150.181 + 44.0971i −0.198608 + 0.0583167i −0.379524 − 0.925182i \(-0.623912\pi\)
0.180915 + 0.983499i \(0.442094\pi\)
\(84\) −5.55601 + 38.6429i −0.00721679 + 0.0501939i
\(85\) 272.638 − 314.641i 0.347902 − 0.401501i
\(86\) 362.081 − 792.846i 0.454002 − 0.994126i
\(87\) −9.22111 − 2.70756i −0.0113633 − 0.00333656i
\(88\) 80.6429 + 176.583i 0.0976882 + 0.213907i
\(89\) 84.3682 + 586.794i 0.100483 + 0.698876i 0.976330 + 0.216287i \(0.0693946\pi\)
−0.875847 + 0.482590i \(0.839696\pi\)
\(90\) 199.105 + 127.957i 0.233195 + 0.149865i
\(91\) 350.985 0.404321
\(92\) −283.943 + 337.711i −0.321773 + 0.382704i
\(93\) 143.622 0.160139
\(94\) −922.212 − 592.670i −1.01190 − 0.650311i
\(95\) −51.4465 − 357.818i −0.0555611 − 0.386436i
\(96\) −24.2663 − 53.1359i −0.0257987 − 0.0564912i
\(97\) −978.602 − 287.344i −1.02435 − 0.300776i −0.273939 − 0.961747i \(-0.588327\pi\)
−0.750412 + 0.660971i \(0.770145\pi\)
\(98\) 261.224 − 572.001i 0.269261 − 0.589600i
\(99\) −376.096 + 434.038i −0.381809 + 0.440631i
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 230.4.g.b.71.3 ✓ 60
23.12 even 11 inner 230.4.g.b.81.3 yes 60
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.4.g.b.71.3 ✓ 60 1.1 even 1 trivial
230.4.g.b.81.3 yes 60 23.12 even 11 inner