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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [230,3,Mod(3,230)] 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("230.3"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(230, base_ring=CyclotomicField(44)) chi = DirichletCharacter(H, H._module([33, 32])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 230 = 2 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 230.k (of order \(44\), degree \(20\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [240,24] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(6.26704608029\)
Analytic rank: \(0\)
Dimension: \(240\)
Relative dimension: \(12\) over \(\Q(\zeta_{44})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{44}]$

Embedding invariants

Embedding label 13.4
Character \(\chi\) \(=\) 230.13
Dual form 230.3.k.b.177.4

$q$-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+(1.41061 + 0.100889i) q^{2} +(-0.526675 - 2.42109i) q^{3} +(1.97964 + 0.284630i) q^{4} +(1.87475 + 4.63522i) q^{5} +(-0.498673 - 3.46835i) q^{6} +(1.72156 - 0.940042i) q^{7} +(2.76379 + 0.601225i) q^{8} +(2.60241 - 1.18848i) q^{9} +(2.17690 + 6.72764i) q^{10} +(2.86593 - 3.30746i) q^{11} +(-0.353516 - 4.94280i) q^{12} +(11.4674 + 6.26168i) q^{13} +(2.52329 - 1.15235i) q^{14} +(10.2349 - 6.98020i) q^{15} +(3.83797 + 1.12693i) q^{16} +(-9.99859 - 7.48485i) q^{17} +(3.79089 - 1.41393i) q^{18} +(-0.916560 - 0.131781i) q^{19} +(2.39202 + 9.70970i) q^{20} +(-3.18262 - 3.67294i) q^{21} +(4.37640 - 4.37640i) q^{22} +(20.2116 - 10.9769i) q^{23} -7.00802i q^{24} +(-17.9706 + 17.3798i) q^{25} +(15.5443 + 9.98972i) q^{26} +(-17.6116 - 23.5263i) q^{27} +(3.67563 - 1.37094i) q^{28} +(13.0868 - 1.88160i) q^{29} +(15.1417 - 8.81375i) q^{30} +(3.80822 - 2.44739i) q^{31} +(5.30019 + 1.97687i) q^{32} +(-9.51706 - 5.19671i) q^{33} +(-13.3490 - 11.5669i) q^{34} +(7.58480 + 6.21746i) q^{35} +(5.49012 - 1.61205i) q^{36} +(-25.4199 - 9.48112i) q^{37} +(-1.27961 - 0.278363i) q^{38} +(9.12047 - 31.0615i) q^{39} +(2.39461 + 13.9379i) q^{40} +(-25.0594 + 54.8725i) q^{41} +(-4.11888 - 5.50218i) q^{42} +(-7.03854 - 32.3557i) q^{43} +(6.61492 - 5.73186i) q^{44} +(10.3878 + 9.83465i) q^{45} +(29.6181 - 13.4449i) q^{46} +(-36.2557 + 36.2557i) q^{47} +(0.707031 - 9.88559i) q^{48} +(-24.4113 + 37.9848i) q^{49} +(-27.1029 + 22.7031i) q^{50} +(-12.8555 + 28.1495i) q^{51} +(20.9191 + 15.6599i) q^{52} +(0.176320 + 0.322907i) q^{53} +(-22.4695 - 34.9633i) q^{54} +(20.7037 + 7.08356i) q^{55} +(5.32320 - 1.56303i) q^{56} +(0.163675 + 2.28848i) q^{57} +(18.6503 - 1.33389i) q^{58} +(20.7558 + 70.6878i) q^{59} +(22.2482 - 10.9051i) q^{60} +(-52.0119 + 33.4260i) q^{61} +(5.61883 - 3.06811i) q^{62} +(3.36298 - 4.49241i) q^{63} +(7.27706 + 3.32332i) q^{64} +(-7.52573 + 64.8931i) q^{65} +(-12.9006 - 8.29070i) q^{66} +(-102.030 - 7.29733i) q^{67} +(-17.6632 - 17.6632i) q^{68} +(-37.2209 - 43.1528i) q^{69} +(10.0719 + 9.53563i) q^{70} +(-9.82094 - 11.3340i) q^{71} +(7.90706 - 1.72008i) q^{72} +(-29.6164 + 22.1706i) q^{73} +(-34.9010 - 15.9388i) q^{74} +(51.5427 + 34.3549i) q^{75} +(-1.77695 - 0.521760i) q^{76} +(1.82471 - 8.38808i) q^{77} +(15.9992 - 42.8955i) q^{78} +(-8.10819 - 27.6140i) q^{79} +(1.97168 + 19.9026i) q^{80} +(-30.8219 + 35.5704i) q^{81} +(-40.8851 + 74.8755i) q^{82} +(50.8007 - 136.202i) q^{83} +(-5.25503 - 8.17699i) q^{84} +(15.9491 - 60.3779i) q^{85} +(-6.66432 - 46.3513i) q^{86} +(-11.4480 - 30.6934i) q^{87} +(9.90936 - 7.41805i) q^{88} +(44.7429 - 69.6214i) q^{89} +(13.6609 + 14.9209i) q^{90} +25.6280 q^{91} +(43.1361 - 15.9774i) q^{92} +(-7.93105 - 7.93105i) q^{93} +(-54.8004 + 47.4849i) q^{94} +(-1.10749 - 4.49552i) q^{95} +(1.99469 - 13.8734i) q^{96} +(-26.6553 - 71.4657i) q^{97} +(-38.2671 + 51.1189i) q^{98} +(3.52748 - 12.0135i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 240 q + 24 q^{2} + 8 q^{3} - 4 q^{5} - 16 q^{6} + 50 q^{7} - 48 q^{8} + 16 q^{10} - 24 q^{11} - 28 q^{12} - 8 q^{13} + 8 q^{15} + 96 q^{16} + 44 q^{17} + 200 q^{18} - 24 q^{20} + 24 q^{21} + 24 q^{22} - 40 q^{23}+ \cdots - 776 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(47\) \(51\)
\(\chi(n)\) \(e\left(\frac{3}{4}\right)\) \(e\left(\frac{7}{11}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.41061 + 0.100889i 0.705305 + 0.0504444i
\(3\) −0.526675 − 2.42109i −0.175558 − 0.807029i −0.977719 − 0.209920i \(-0.932680\pi\)
0.802160 − 0.597109i \(-0.203684\pi\)
\(4\) 1.97964 + 0.284630i 0.494911 + 0.0711574i
\(5\) 1.87475 + 4.63522i 0.374951 + 0.927045i
\(6\) −0.498673 − 3.46835i −0.0831121 − 0.578058i
\(7\) 1.72156 − 0.940042i 0.245937 − 0.134292i −0.351553 − 0.936168i \(-0.614346\pi\)
0.597490 + 0.801876i \(0.296165\pi\)
\(8\) 2.76379 + 0.601225i 0.345474 + 0.0751532i
\(9\) 2.60241 − 1.18848i 0.289157 − 0.132054i
\(10\) 2.17690 + 6.72764i 0.217690 + 0.672764i
\(11\) 2.86593 − 3.30746i 0.260539 − 0.300678i −0.610376 − 0.792112i \(-0.708982\pi\)
0.870915 + 0.491434i \(0.163527\pi\)
\(12\) −0.353516 − 4.94280i −0.0294596 − 0.411900i
\(13\) 11.4674 + 6.26168i 0.882109 + 0.481668i 0.855396 − 0.517975i \(-0.173314\pi\)
0.0267131 + 0.999643i \(0.491496\pi\)
\(14\) 2.52329 − 1.15235i 0.180235 − 0.0823105i
\(15\) 10.2349 − 6.98020i 0.682326 − 0.465347i
\(16\) 3.83797 + 1.12693i 0.239873 + 0.0704331i
\(17\) −9.99859 − 7.48485i −0.588152 − 0.440285i 0.263303 − 0.964713i \(-0.415188\pi\)
−0.851455 + 0.524428i \(0.824279\pi\)
\(18\) 3.79089 − 1.41393i 0.210605 − 0.0785517i
\(19\) −0.916560 − 0.131781i −0.0482400 − 0.00693587i 0.118153 − 0.992995i \(-0.462303\pi\)
−0.166393 + 0.986060i \(0.553212\pi\)
\(20\) 2.39202 + 9.70970i 0.119601 + 0.485485i
\(21\) −3.18262 − 3.67294i −0.151554 − 0.174902i
\(22\) 4.37640 − 4.37640i 0.198927 − 0.198927i
\(23\) 20.2116 − 10.9769i 0.878765 − 0.477255i
\(24\) − 7.00802i − 0.292001i
\(25\) −17.9706 + 17.3798i −0.718824 + 0.695192i
\(26\) 15.5443 + 9.98972i 0.597858 + 0.384220i
\(27\) −17.6116 − 23.5263i −0.652280 − 0.871345i
\(28\) 3.67563 − 1.37094i 0.131273 − 0.0489622i
\(29\) 13.0868 − 1.88160i 0.451270 − 0.0648829i 0.0870673 − 0.996202i \(-0.472250\pi\)
0.364203 + 0.931320i \(0.381341\pi\)
\(30\) 15.1417 − 8.81375i 0.504722 − 0.293792i
\(31\) 3.80822 − 2.44739i 0.122846 − 0.0789482i −0.477775 − 0.878482i \(-0.658557\pi\)
0.600621 + 0.799534i \(0.294920\pi\)
\(32\) 5.30019 + 1.97687i 0.165631 + 0.0617771i
\(33\) −9.51706 − 5.19671i −0.288396 − 0.157476i
\(34\) −13.3490 − 11.5669i −0.392617 − 0.340204i
\(35\) 7.58480 + 6.21746i 0.216709 + 0.177642i
\(36\) 5.49012 − 1.61205i 0.152503 − 0.0447790i
\(37\) −25.4199 − 9.48112i −0.687024 − 0.256247i −0.0183816 − 0.999831i \(-0.505851\pi\)
−0.668642 + 0.743584i \(0.733124\pi\)
\(38\) −1.27961 − 0.278363i −0.0336740 − 0.00732534i
\(39\) 9.12047 − 31.0615i 0.233858 − 0.796448i
\(40\) 2.39461 + 13.9379i 0.0598652 + 0.348448i
\(41\) −25.0594 + 54.8725i −0.611206 + 1.33835i 0.310540 + 0.950560i \(0.399490\pi\)
−0.921746 + 0.387794i \(0.873237\pi\)
\(42\) −4.11888 − 5.50218i −0.0980687 − 0.131004i
\(43\) −7.03854 − 32.3557i −0.163687 − 0.752457i −0.983657 − 0.180051i \(-0.942374\pi\)
0.819970 − 0.572406i \(-0.193990\pi\)
\(44\) 6.61492 − 5.73186i 0.150339 − 0.130270i
\(45\) 10.3878 + 9.83465i 0.230839 + 0.218548i
\(46\) 29.6181 − 13.4449i 0.643872 − 0.292281i
\(47\) −36.2557 + 36.2557i −0.771398 + 0.771398i −0.978351 − 0.206953i \(-0.933645\pi\)
0.206953 + 0.978351i \(0.433645\pi\)
\(48\) 0.707031 − 9.88559i 0.0147298 − 0.205950i
\(49\) −24.4113 + 37.9848i −0.498190 + 0.775199i
\(50\) −27.1029 + 22.7031i −0.542059 + 0.454062i
\(51\) −12.8555 + 28.1495i −0.252068 + 0.551952i
\(52\) 20.9191 + 15.6599i 0.402291 + 0.301151i
\(53\) 0.176320 + 0.322907i 0.00332680 + 0.00609258i 0.879337 − 0.476200i \(-0.157986\pi\)
−0.876010 + 0.482292i \(0.839804\pi\)
\(54\) −22.4695 − 34.9633i −0.416102 − 0.647468i
\(55\) 20.7037 + 7.08356i 0.376431 + 0.128792i
\(56\) 5.32320 − 1.56303i 0.0950571 − 0.0279113i
\(57\) 0.163675 + 2.28848i 0.00287149 + 0.0401487i
\(58\) 18.6503 − 1.33389i 0.321556 − 0.0229982i
\(59\) 20.7558 + 70.6878i 0.351793 + 1.19810i 0.925408 + 0.378973i \(0.123723\pi\)
−0.573614 + 0.819125i \(0.694459\pi\)
\(60\) 22.2482 − 10.9051i 0.370803 − 0.181752i
\(61\) −52.0119 + 33.4260i −0.852655 + 0.547968i −0.892402 − 0.451242i \(-0.850981\pi\)
0.0397471 + 0.999210i \(0.487345\pi\)
\(62\) 5.61883 − 3.06811i 0.0906263 − 0.0494857i
\(63\) 3.36298 − 4.49241i 0.0533806 − 0.0713082i
\(64\) 7.27706 + 3.32332i 0.113704 + 0.0519269i
\(65\) −7.52573 + 64.8931i −0.115780 + 0.998356i
\(66\) −12.9006 − 8.29070i −0.195463 − 0.125617i
\(67\) −102.030 − 7.29733i −1.52284 − 0.108915i −0.715335 − 0.698781i \(-0.753726\pi\)
−0.807501 + 0.589866i \(0.799181\pi\)
\(68\) −17.6632 − 17.6632i −0.259753 − 0.259753i
\(69\) −37.2209 − 43.1528i −0.539433 − 0.625403i
\(70\) 10.0719 + 9.53563i 0.143885 + 0.136223i
\(71\) −9.82094 − 11.3340i −0.138323 − 0.159633i 0.682361 − 0.731015i \(-0.260953\pi\)
−0.820684 + 0.571382i \(0.806408\pi\)
\(72\) 7.90706 − 1.72008i 0.109820 − 0.0238899i
\(73\) −29.6164 + 22.1706i −0.405705 + 0.303707i −0.782571 − 0.622561i \(-0.786092\pi\)
0.376867 + 0.926267i \(0.377001\pi\)
\(74\) −34.9010 − 15.9388i −0.471635 − 0.215389i
\(75\) 51.5427 + 34.3549i 0.687236 + 0.458065i
\(76\) −1.77695 − 0.521760i −0.0233810 − 0.00686527i
\(77\) 1.82471 − 8.38808i 0.0236976 − 0.108936i
\(78\) 15.9992 − 42.8955i 0.205118 − 0.549942i
\(79\) −8.10819 − 27.6140i −0.102635 − 0.349544i 0.892124 − 0.451791i \(-0.149215\pi\)
−0.994759 + 0.102247i \(0.967397\pi\)
\(80\) 1.97168 + 19.9026i 0.0246459 + 0.248782i
\(81\) −30.8219 + 35.5704i −0.380518 + 0.439141i
\(82\) −40.8851 + 74.8755i −0.498599 + 0.913116i
\(83\) 50.8007 − 136.202i 0.612056 − 1.64099i −0.146372 − 0.989230i \(-0.546760\pi\)
0.758428 − 0.651756i \(-0.225968\pi\)
\(84\) −5.25503 − 8.17699i −0.0625599 − 0.0973451i
\(85\) 15.9491 − 60.3779i 0.187636 − 0.710329i
\(86\) −6.66432 − 46.3513i −0.0774921 − 0.538969i
\(87\) −11.4480 − 30.6934i −0.131587 − 0.352798i
\(88\) 9.90936 − 7.41805i 0.112606 − 0.0842960i
\(89\) 44.7429 − 69.6214i 0.502730 − 0.782262i −0.493433 − 0.869784i \(-0.664258\pi\)
0.996163 + 0.0875213i \(0.0278946\pi\)
\(90\) 13.6609 + 14.9209i 0.151787 + 0.165787i
\(91\) 25.6280 0.281627
\(92\) 43.1361 − 15.9774i 0.468870 − 0.173668i
\(93\) −7.93105 − 7.93105i −0.0852801 − 0.0852801i
\(94\) −54.8004 + 47.4849i −0.582983 + 0.505158i
\(95\) −1.10749 − 4.49552i −0.0116578 − 0.0473213i
\(96\) 1.99469 − 13.8734i 0.0207780 − 0.144514i
\(97\) −26.6553 − 71.4657i −0.274797 − 0.736760i −0.998892 − 0.0470594i \(-0.985015\pi\)
0.724095 − 0.689700i \(-0.242258\pi\)
\(98\) −38.2671 + 51.1189i −0.390481 + 0.521621i
\(99\) 3.52748 − 12.0135i 0.0356311 − 0.121348i
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 230.3.k.b.13.4 ✓ 240
5.2 odd 4 inner 230.3.k.b.197.4 yes 240
23.16 even 11 inner 230.3.k.b.223.4 yes 240
115.62 odd 44 inner 230.3.k.b.177.4 yes 240
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.3.k.b.13.4 ✓ 240 1.1 even 1 trivial
230.3.k.b.177.4 yes 240 115.62 odd 44 inner
230.3.k.b.197.4 yes 240 5.2 odd 4 inner
230.3.k.b.223.4 yes 240 23.16 even 11 inner