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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [230,4,Mod(137,230)] 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("230.137"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(230, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([1, 2])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 230 = 2 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 230.e (of order \(4\), degree \(2\), 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: \(13.5704393013\)
Analytic rank: \(0\)
Dimension: \(72\)
Relative dimension: \(36\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 137.8
Character \(\chi\) \(=\) 230.137
Dual form 230.4.e.a.183.8

$q$-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+(-1.41421 - 1.41421i) q^{2} +(-2.02411 + 2.02411i) q^{3} +4.00000i q^{4} +(2.14096 - 10.9734i) q^{5} +5.72506 q^{6} +(1.46247 - 1.46247i) q^{7} +(5.65685 - 5.65685i) q^{8} +18.8059i q^{9} +(-18.5466 + 12.4910i) q^{10} -21.4009i q^{11} +(-8.09645 - 8.09645i) q^{12} +(14.5475 - 14.5475i) q^{13} -4.13650 q^{14} +(17.8779 + 26.5450i) q^{15} -16.0000 q^{16} +(-34.9835 + 34.9835i) q^{17} +(26.5956 - 26.5956i) q^{18} +102.683 q^{19} +(43.8937 + 8.56385i) q^{20} +5.92042i q^{21} +(-30.2655 + 30.2655i) q^{22} +(-102.355 - 41.1151i) q^{23} +22.9002i q^{24} +(-115.833 - 46.9874i) q^{25} -41.1464 q^{26} +(-92.7164 - 92.7164i) q^{27} +(5.84989 + 5.84989i) q^{28} -275.499i q^{29} +(12.2571 - 62.8235i) q^{30} -44.8903 q^{31} +(22.6274 + 22.6274i) q^{32} +(43.3179 + 43.3179i) q^{33} +98.9483 q^{34} +(-12.9172 - 19.1794i) q^{35} -75.2237 q^{36} +(244.472 - 244.472i) q^{37} +(-145.216 - 145.216i) q^{38} +58.8914i q^{39} +(-49.9640 - 74.1862i) q^{40} -399.252 q^{41} +(8.37274 - 8.37274i) q^{42} +(-311.304 - 311.304i) q^{43} +85.6038 q^{44} +(206.366 + 40.2628i) q^{45} +(86.6062 + 202.897i) q^{46} +(-403.349 - 403.349i) q^{47} +(32.3858 - 32.3858i) q^{48} +338.722i q^{49} +(97.3617 + 230.262i) q^{50} -141.621i q^{51} +(58.1898 + 58.1898i) q^{52} +(-40.1517 - 40.1517i) q^{53} +262.242i q^{54} +(-234.842 - 45.8186i) q^{55} -16.5460i q^{56} +(-207.843 + 207.843i) q^{57} +(-389.615 + 389.615i) q^{58} -99.7810i q^{59} +(-106.180 + 71.5117i) q^{60} +534.925i q^{61} +(63.4845 + 63.4845i) q^{62} +(27.5032 + 27.5032i) q^{63} -64.0000i q^{64} +(-128.490 - 190.781i) q^{65} -122.522i q^{66} +(83.3100 - 83.3100i) q^{67} +(-139.934 - 139.934i) q^{68} +(290.400 - 123.956i) q^{69} +(-8.85608 + 45.3916i) q^{70} -236.189 q^{71} +(106.382 + 106.382i) q^{72} +(782.038 - 782.038i) q^{73} -691.472 q^{74} +(329.566 - 139.350i) q^{75} +410.733i q^{76} +(-31.2983 - 31.2983i) q^{77} +(83.2850 - 83.2850i) q^{78} +648.939 q^{79} +(-34.2554 + 175.575i) q^{80} -132.423 q^{81} +(564.627 + 564.627i) q^{82} +(-113.220 - 113.220i) q^{83} -23.6817 q^{84} +(308.991 + 458.787i) q^{85} +880.500i q^{86} +(557.641 + 557.641i) q^{87} +(-121.062 - 121.062i) q^{88} +1256.85 q^{89} +(-234.905 - 348.785i) q^{90} -42.5505i q^{91} +(164.461 - 409.420i) q^{92} +(90.8631 - 90.8631i) q^{93} +1140.84i q^{94} +(219.841 - 1126.79i) q^{95} -91.6009 q^{96} +(-1036.42 + 1036.42i) q^{97} +(479.026 - 479.026i) q^{98} +402.465 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 72 q - 16 q^{3} - 16 q^{6} - 64 q^{12} + 192 q^{13} - 1152 q^{16} + 32 q^{18} + 276 q^{23} + 880 q^{25} + 304 q^{26} + 728 q^{27} + 608 q^{31} + 688 q^{35} + 2816 q^{36} - 2208 q^{41} - 256 q^{46} + 144 q^{47}+ \cdots - 448 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(47\) \(51\)
\(\chi(n)\) \(e\left(\frac{1}{4}\right)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.41421 − 1.41421i −0.500000 − 0.500000i
\(3\) −2.02411 + 2.02411i −0.389541 + 0.389541i −0.874524 − 0.484983i \(-0.838826\pi\)
0.484983 + 0.874524i \(0.338826\pi\)
\(4\) 4.00000i 0.500000i
\(5\) 2.14096 − 10.9734i 0.191493 − 0.981494i
\(6\) 5.72506 0.389541
\(7\) 1.46247 − 1.46247i 0.0789661 − 0.0789661i −0.666521 − 0.745487i \(-0.732217\pi\)
0.745487 + 0.666521i \(0.232217\pi\)
\(8\) 5.65685 − 5.65685i 0.250000 − 0.250000i
\(9\) 18.8059i 0.696516i
\(10\) −18.5466 + 12.4910i −0.586494 + 0.395000i
\(11\) − 21.4009i − 0.586603i −0.956020 − 0.293301i \(-0.905246\pi\)
0.956020 − 0.293301i \(-0.0947539\pi\)
\(12\) −8.09645 − 8.09645i −0.194770 − 0.194770i
\(13\) 14.5475 − 14.5475i 0.310364 − 0.310364i −0.534686 − 0.845051i \(-0.679570\pi\)
0.845051 + 0.534686i \(0.179570\pi\)
\(14\) −4.13650 −0.0789661
\(15\) 17.8779 + 26.5450i 0.307737 + 0.456926i
\(16\) −16.0000 −0.250000
\(17\) −34.9835 + 34.9835i −0.499103 + 0.499103i −0.911159 − 0.412056i \(-0.864811\pi\)
0.412056 + 0.911159i \(0.364811\pi\)
\(18\) 26.5956 − 26.5956i 0.348258 − 0.348258i
\(19\) 102.683 1.23985 0.619925 − 0.784661i \(-0.287163\pi\)
0.619925 + 0.784661i \(0.287163\pi\)
\(20\) 43.8937 + 8.56385i 0.490747 + 0.0957467i
\(21\) 5.92042i 0.0615210i
\(22\) −30.2655 + 30.2655i −0.293301 + 0.293301i
\(23\) −102.355 − 41.1151i −0.927934 − 0.372743i
\(24\) 22.9002i 0.194770i
\(25\) −115.833 − 46.9874i −0.926660 − 0.375899i
\(26\) −41.1464 −0.310364
\(27\) −92.7164 − 92.7164i −0.660862 − 0.660862i
\(28\) 5.84989 + 5.84989i 0.0394830 + 0.0394830i
\(29\) − 275.499i − 1.76410i −0.471156 − 0.882050i \(-0.656163\pi\)
0.471156 − 0.882050i \(-0.343837\pi\)
\(30\) 12.2571 − 62.8235i 0.0745945 − 0.382332i
\(31\) −44.8903 −0.260082 −0.130041 − 0.991509i \(-0.541511\pi\)
−0.130041 + 0.991509i \(0.541511\pi\)
\(32\) 22.6274 + 22.6274i 0.125000 + 0.125000i
\(33\) 43.3179 + 43.3179i 0.228506 + 0.228506i
\(34\) 98.9483 0.499103
\(35\) −12.9172 − 19.1794i −0.0623832 − 0.0926262i
\(36\) −75.2237 −0.348258
\(37\) 244.472 − 244.472i 1.08624 − 1.08624i 0.0903307 − 0.995912i \(-0.471208\pi\)
0.995912 − 0.0903307i \(-0.0287924\pi\)
\(38\) −145.216 − 145.216i −0.619925 − 0.619925i
\(39\) 58.8914i 0.241799i
\(40\) −49.9640 − 74.1862i −0.197500 − 0.293247i
\(41\) −399.252 −1.52080 −0.760398 − 0.649458i \(-0.774996\pi\)
−0.760398 + 0.649458i \(0.774996\pi\)
\(42\) 8.37274 − 8.37274i 0.0307605 − 0.0307605i
\(43\) −311.304 − 311.304i −1.10403 − 1.10403i −0.993919 − 0.110112i \(-0.964879\pi\)
−0.110112 − 0.993919i \(-0.535121\pi\)
\(44\) 85.6038 0.293301
\(45\) 206.366 + 40.2628i 0.683626 + 0.133378i
\(46\) 86.6062 + 202.897i 0.277596 + 0.650339i
\(47\) −403.349 − 403.349i −1.25180 − 1.25180i −0.954914 − 0.296884i \(-0.904053\pi\)
−0.296884 − 0.954914i \(-0.595947\pi\)
\(48\) 32.3858 − 32.3858i 0.0973852 − 0.0973852i
\(49\) 338.722i 0.987529i
\(50\) 97.3617 + 230.262i 0.275381 + 0.651280i
\(51\) − 141.621i − 0.388841i
\(52\) 58.1898 + 58.1898i 0.155182 + 0.155182i
\(53\) −40.1517 − 40.1517i −0.104062 − 0.104062i 0.653159 − 0.757221i \(-0.273443\pi\)
−0.757221 + 0.653159i \(0.773443\pi\)
\(54\) 262.242i 0.660862i
\(55\) −234.842 − 45.8186i −0.575747 − 0.112331i
\(56\) − 16.5460i − 0.0394830i
\(57\) −207.843 + 207.843i −0.482972 + 0.482972i
\(58\) −389.615 + 389.615i −0.882050 + 0.882050i
\(59\) − 99.7810i − 0.220176i −0.993922 − 0.110088i \(-0.964887\pi\)
0.993922 − 0.110088i \(-0.0351132\pi\)
\(60\) −106.180 + 71.5117i −0.228463 + 0.153869i
\(61\) 534.925i 1.12279i 0.827549 + 0.561394i \(0.189735\pi\)
−0.827549 + 0.561394i \(0.810265\pi\)
\(62\) 63.4845 + 63.4845i 0.130041 + 0.130041i
\(63\) 27.5032 + 27.5032i 0.0550012 + 0.0550012i
\(64\) − 64.0000i − 0.125000i
\(65\) −128.490 − 190.781i −0.245188 − 0.364054i
\(66\) − 122.522i − 0.228506i
\(67\) 83.3100 − 83.3100i 0.151910 − 0.151910i −0.627061 − 0.778970i \(-0.715742\pi\)
0.778970 + 0.627061i \(0.215742\pi\)
\(68\) −139.934 − 139.934i −0.249551 − 0.249551i
\(69\) 290.400 − 123.956i 0.506667 − 0.216270i
\(70\) −8.85608 + 45.3916i −0.0151215 + 0.0775047i
\(71\) −236.189 −0.394796 −0.197398 − 0.980323i \(-0.563249\pi\)
−0.197398 + 0.980323i \(0.563249\pi\)
\(72\) 106.382 + 106.382i 0.174129 + 0.174129i
\(73\) 782.038 − 782.038i 1.25384 − 1.25384i 0.299862 − 0.953983i \(-0.403059\pi\)
0.953983 − 0.299862i \(-0.0969406\pi\)
\(74\) −691.472 −1.08624
\(75\) 329.566 − 139.350i 0.507400 − 0.214544i
\(76\) 410.733i 0.619925i
\(77\) −31.2983 − 31.2983i −0.0463217 − 0.0463217i
\(78\) 83.2850 − 83.2850i 0.120900 − 0.120900i
\(79\) 648.939 0.924194 0.462097 − 0.886829i \(-0.347097\pi\)
0.462097 + 0.886829i \(0.347097\pi\)
\(80\) −34.2554 + 175.575i −0.0478734 + 0.245373i
\(81\) −132.423 −0.181651
\(82\) 564.627 + 564.627i 0.760398 + 0.760398i
\(83\) −113.220 − 113.220i −0.149729 − 0.149729i 0.628268 − 0.777997i \(-0.283764\pi\)
−0.777997 + 0.628268i \(0.783764\pi\)
\(84\) −23.6817 −0.0307605
\(85\) 308.991 + 458.787i 0.394291 + 0.585441i
\(86\) 880.500i 1.10403i
\(87\) 557.641 + 557.641i 0.687189 + 0.687189i
\(88\) −121.062 − 121.062i −0.146651 − 0.146651i
\(89\) 1256.85 1.49692 0.748461 − 0.663178i \(-0.230793\pi\)
0.748461 + 0.663178i \(0.230793\pi\)
\(90\) −234.905 − 348.785i −0.275124 − 0.408502i
\(91\) − 42.5505i − 0.0490165i
\(92\) 164.461 − 409.420i 0.186372 − 0.463967i
\(93\) 90.8631 − 90.8631i 0.101313 − 0.101313i
\(94\) 1140.84i 1.25180i
\(95\) 219.841 − 1126.79i 0.237423 − 1.21691i
\(96\) −91.6009 −0.0973852
\(97\) −1036.42 + 1036.42i −1.08487 + 1.08487i −0.0888263 + 0.996047i \(0.528312\pi\)
−0.996047 + 0.0888263i \(0.971688\pi\)
\(98\) 479.026 − 479.026i 0.493764 − 0.493764i
\(99\) 402.465 0.408578
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.e.a.137.8 yes 72
5.3 odd 4 inner 230.4.e.a.183.7 yes 72
23.22 odd 2 inner 230.4.e.a.137.7 ✓ 72
115.68 even 4 inner 230.4.e.a.183.8 yes 72
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.4.e.a.137.7 ✓ 72 23.22 odd 2 inner
230.4.e.a.137.8 yes 72 1.1 even 1 trivial
230.4.e.a.183.7 yes 72 5.3 odd 4 inner
230.4.e.a.183.8 yes 72 115.68 even 4 inner