↑

Properties

Label 230.4.j.a.9.7
Level $230$
Weight $4$
Character 230.9
Analytic conductor $13.570$
Analytic rank $0$
Dimension $360$
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [230,4,Mod(9,230)] 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("230.9"); 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([11, 10])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 230 = 2 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 230.j (of order \(22\), degree \(10\), 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: \(360\)
Relative dimension: \(36\) over \(\Q(\zeta_{22})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{22}]$

Embedding invariants

Embedding label 9.7
Character \(\chi\) \(=\) 230.9
Dual form 230.4.j.a.179.7

$q$-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+(-0.563465 - 1.91899i) q^{2} +(-3.14614 + 2.72615i) q^{3} +(-3.36501 + 2.16256i) q^{4} +(-3.36640 - 10.6615i) q^{5} +(7.00418 + 4.50131i) q^{6} +(17.7210 + 8.09292i) q^{7} +(6.04600 + 5.23889i) q^{8} +(-1.37617 + 9.57148i) q^{9} +(-18.5624 + 12.4675i) q^{10} +(-26.0618 - 7.65243i) q^{11} +(4.69135 - 15.9773i) q^{12} +(46.8693 - 21.4045i) q^{13} +(5.54502 - 38.5665i) q^{14} +(39.6560 + 24.3652i) q^{15} +(6.64664 - 14.5541i) q^{16} +(10.0534 - 15.6435i) q^{17} +(19.1430 - 2.75234i) q^{18} +(-40.2014 + 25.8359i) q^{19} +(34.3841 + 28.5960i) q^{20} +(-77.8154 + 22.8487i) q^{21} +54.3241i q^{22} +(32.0242 - 105.553i) q^{23} -33.3035 q^{24} +(-102.335 + 71.7818i) q^{25} +(-67.4842 - 77.8809i) q^{26} +(-82.5314 - 128.421i) q^{27} +(-77.1329 + 11.0900i) q^{28} +(-223.109 - 143.383i) q^{29} +(24.4118 - 89.8282i) q^{30} +(201.546 - 232.596i) q^{31} +(-31.6743 - 4.55407i) q^{32} +(102.856 - 46.9727i) q^{33} +(-35.6844 - 10.4779i) q^{34} +(26.6264 - 216.176i) q^{35} +(-16.0681 - 35.1842i) q^{36} +(-301.727 - 43.3818i) q^{37} +(72.2308 + 62.5883i) q^{38} +(-89.1058 + 195.114i) q^{39} +(35.5011 - 82.0955i) q^{40} +(-40.1360 - 279.152i) q^{41} +(87.6925 + 136.452i) q^{42} +(135.544 - 117.450i) q^{43} +(104.247 - 30.6097i) q^{44} +(106.679 - 17.5494i) q^{45} +(-220.599 - 1.97856i) q^{46} +325.865i q^{47} +(18.7654 + 63.9090i) q^{48} +(23.9220 + 27.6075i) q^{49} +(195.410 + 155.932i) q^{50} +(11.0168 + 76.6238i) q^{51} +(-111.427 + 173.384i) q^{52} +(-67.4741 - 30.8144i) q^{53} +(-199.935 + 230.737i) q^{54} +(6.14821 + 303.619i) q^{55} +(64.7434 + 141.768i) q^{56} +(56.0470 - 190.878i) q^{57} +(-149.436 + 508.934i) q^{58} +(-318.947 - 698.397i) q^{59} +(-186.134 + 3.76917i) q^{60} +(148.324 - 171.175i) q^{61} +(-559.913 - 255.704i) q^{62} +(-101.848 + 158.479i) q^{63} +(9.10815 + 63.3486i) q^{64} +(-385.985 - 427.641i) q^{65} +(-148.096 - 170.911i) q^{66} +(-7.27596 - 24.7797i) q^{67} +74.3817i q^{68} +(187.001 + 419.388i) q^{69} +(-429.843 + 70.7121i) q^{70} +(393.903 - 115.660i) q^{71} +(-58.4642 + 50.6595i) q^{72} +(490.327 + 762.963i) q^{73} +(86.7636 + 603.454i) q^{74} +(126.272 - 504.815i) q^{75} +(79.4066 - 173.876i) q^{76} +(-399.911 - 346.525i) q^{77} +(424.630 + 61.0525i) q^{78} +(311.024 + 681.048i) q^{79} +(-177.544 - 21.8681i) q^{80} +(359.240 + 105.482i) q^{81} +(-513.074 + 234.313i) q^{82} +(595.559 + 85.6284i) q^{83} +(212.438 - 245.167i) q^{84} +(-200.627 - 54.5225i) q^{85} +(-301.759 - 193.929i) q^{86} +(1092.81 - 157.123i) q^{87} +(-117.479 - 182.801i) q^{88} +(-412.880 - 476.489i) q^{89} +(-93.7870 - 194.827i) q^{90} +1003.80 q^{91} +(120.503 + 424.442i) q^{92} +1281.23i q^{93} +(625.331 - 183.614i) q^{94} +(410.783 + 341.633i) q^{95} +(112.067 - 72.0210i) q^{96} +(688.014 - 98.9215i) q^{97} +(39.4992 - 61.4619i) q^{98} +(109.111 - 238.919i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 360 q + 144 q^{4} + 32 q^{5} + 8 q^{6} + 224 q^{9} + 96 q^{11} + 208 q^{14} - 752 q^{15} - 576 q^{16} + 48 q^{19} + 224 q^{20} - 272 q^{21} - 32 q^{24} + 904 q^{25} - 152 q^{26} - 20 q^{29} - 1272 q^{30}+ \cdots + 23448 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(47\) \(51\)
\(\chi(n)\) \(-1\) \(e\left(\frac{5}{11}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.563465 − 1.91899i −0.199215 − 0.678464i
\(3\) −3.14614 + 2.72615i −0.605475 + 0.524647i −0.902762 − 0.430141i \(-0.858464\pi\)
0.297286 + 0.954788i \(0.403918\pi\)
\(4\) −3.36501 + 2.16256i −0.420627 + 0.270320i
\(5\) −3.36640 − 10.6615i −0.301100 − 0.953592i
\(6\) 7.00418 + 4.50131i 0.476574 + 0.306276i
\(7\) 17.7210 + 8.09292i 0.956845 + 0.436977i 0.831739 − 0.555167i \(-0.187346\pi\)
0.125106 + 0.992143i \(0.460073\pi\)
\(8\) 6.04600 + 5.23889i 0.267198 + 0.231528i
\(9\) −1.37617 + 9.57148i −0.0509693 + 0.354499i
\(10\) −18.5624 + 12.4675i −0.586994 + 0.394256i
\(11\) −26.0618 − 7.65243i −0.714357 − 0.209754i −0.0956942 − 0.995411i \(-0.530507\pi\)
−0.618663 + 0.785657i \(0.712325\pi\)
\(12\) 4.69135 − 15.9773i 0.112856 − 0.384353i
\(13\) 46.8693 − 21.4045i 0.999940 − 0.456657i 0.152931 − 0.988237i \(-0.451129\pi\)
0.847008 + 0.531580i \(0.178401\pi\)
\(14\) 5.54502 − 38.5665i 0.105855 − 0.736237i
\(15\) 39.6560 + 24.3652i 0.682609 + 0.419405i
\(16\) 6.64664 − 14.5541i 0.103854 − 0.227408i
\(17\) 10.0534 − 15.6435i 0.143431 − 0.223182i −0.762104 − 0.647454i \(-0.775834\pi\)
0.905535 + 0.424272i \(0.139470\pi\)
\(18\) 19.1430 − 2.75234i 0.250669 − 0.0360407i
\(19\) −40.2014 + 25.8359i −0.485413 + 0.311956i −0.760358 − 0.649504i \(-0.774977\pi\)
0.274946 + 0.961460i \(0.411340\pi\)
\(20\) 34.3841 + 28.5960i 0.384426 + 0.319713i
\(21\) −77.8154 + 22.8487i −0.808605 + 0.237428i
\(22\) 54.3241i 0.526452i
\(23\) 32.0242 − 105.553i 0.290327 − 0.956928i
\(24\) −33.3035 −0.283252
\(25\) −102.335 + 71.7818i −0.818677 + 0.574254i
\(26\) −67.4842 − 77.8809i −0.509028 − 0.587450i
\(27\) −82.5314 − 128.421i −0.588266 − 0.915359i
\(28\) −77.1329 + 11.0900i −0.520598 + 0.0748508i
\(29\) −223.109 − 143.383i −1.42863 − 0.918123i −0.999892 − 0.0146936i \(-0.995323\pi\)
−0.428736 − 0.903430i \(-0.641041\pi\)
\(30\) 24.4118 − 89.8282i 0.148565 − 0.546677i
\(31\) 201.546 − 232.596i 1.16770 − 1.34760i 0.241575 − 0.970382i \(-0.422336\pi\)
0.926125 − 0.377216i \(-0.123119\pi\)
\(32\) −31.6743 − 4.55407i −0.174977 − 0.0251579i
\(33\) 102.856 − 46.9727i 0.542573 − 0.247785i
\(34\) −35.6844 − 10.4779i −0.179995 − 0.0528512i
\(35\) 26.6264 − 216.176i 0.128591 − 1.04401i
\(36\) −16.0681 − 35.1842i −0.0743893 − 0.162890i
\(37\) −301.727 − 43.3818i −1.34064 − 0.192755i −0.565606 − 0.824676i \(-0.691358\pi\)
−0.775033 + 0.631921i \(0.782267\pi\)
\(38\) 72.2308 + 62.5883i 0.308352 + 0.267189i
\(39\) −89.1058 + 195.114i −0.365855 + 0.801110i
\(40\) 35.5011 − 82.0955i 0.140330 − 0.324511i
\(41\) −40.1360 − 279.152i −0.152883 − 1.06332i −0.911356 − 0.411619i \(-0.864964\pi\)
0.758473 − 0.651704i \(-0.225946\pi\)
\(42\) 87.6925 + 136.452i 0.322172 + 0.501310i
\(43\) 135.544 − 117.450i 0.480705 − 0.416533i −0.380508 − 0.924778i \(-0.624251\pi\)
0.861213 + 0.508245i \(0.169705\pi\)
\(44\) 104.247 − 30.6097i 0.357179 − 0.104877i
\(45\) 106.679 − 17.5494i 0.353395 − 0.0581359i
\(46\) −220.599 − 1.97856i −0.707078 − 0.00634181i
\(47\) 325.865i 1.01133i 0.862731 + 0.505663i \(0.168752\pi\)
−0.862731 + 0.505663i \(0.831248\pi\)
\(48\) 18.7654 + 63.9090i 0.0564281 + 0.192177i
\(49\) 23.9220 + 27.6075i 0.0697436 + 0.0804884i
\(50\) 195.410 + 155.932i 0.552703 + 0.441043i
\(51\) 11.0168 + 76.6238i 0.0302483 + 0.210382i
\(52\) −111.427 + 173.384i −0.297158 + 0.462386i
\(53\) −67.4741 − 30.8144i −0.174873 − 0.0798620i 0.326055 − 0.945351i \(-0.394281\pi\)
−0.500928 + 0.865489i \(0.667008\pi\)
\(54\) −199.935 + 230.737i −0.503847 + 0.581470i
\(55\) 6.14821 + 303.619i 0.0150732 + 0.744363i
\(56\) 64.7434 + 141.768i 0.154495 + 0.338296i
\(57\) 56.0470 − 190.878i 0.130239 − 0.443552i
\(58\) −149.436 + 508.934i −0.338310 + 1.15218i
\(59\) −318.947 − 698.397i −0.703786 − 1.54108i −0.835320 − 0.549765i \(-0.814717\pi\)
0.131534 − 0.991312i \(-0.458010\pi\)
\(60\) −186.134 + 3.76917i −0.400497 + 0.00810997i
\(61\) 148.324 − 171.175i 0.311327 − 0.359291i −0.578424 − 0.815736i \(-0.696332\pi\)
0.889751 + 0.456445i \(0.150878\pi\)
\(62\) −559.913 − 255.704i −1.14692 − 0.523781i
\(63\) −101.848 + 158.479i −0.203678 + 0.316928i
\(64\) 9.10815 + 63.3486i 0.0177894 + 0.123728i
\(65\) −385.985 − 427.641i −0.736547 − 0.816035i
\(66\) −148.096 − 170.911i −0.276202 − 0.318754i
\(67\) −7.27596 − 24.7797i −0.0132672 − 0.0451838i 0.952593 − 0.304246i \(-0.0984046\pi\)
−0.965861 + 0.259063i \(0.916586\pi\)
\(68\) 74.3817i 0.132649i
\(69\) 187.001 + 419.388i 0.326264 + 0.731715i
\(70\) −429.843 + 70.7121i −0.733943 + 0.120739i
\(71\) 393.903 − 115.660i 0.658418 − 0.193329i 0.0645787 − 0.997913i \(-0.479430\pi\)
0.593839 + 0.804584i \(0.297611\pi\)
\(72\) −58.4642 + 50.6595i −0.0956955 + 0.0829206i
\(73\) 490.327 + 762.963i 0.786143 + 1.22326i 0.970666 + 0.240433i \(0.0772896\pi\)
−0.184523 + 0.982828i \(0.559074\pi\)
\(74\) 86.7636 + 603.454i 0.136298 + 0.947975i
\(75\) 126.272 − 504.815i 0.194408 − 0.777214i
\(76\) 79.4066 − 173.876i 0.119850 − 0.262434i
\(77\) −399.911 − 346.525i −0.591871 − 0.512860i
\(78\) 424.630 + 61.0525i 0.616408 + 0.0886261i
\(79\) 311.024 + 681.048i 0.442949 + 0.969923i 0.991048 + 0.133509i \(0.0426245\pi\)
−0.548099 + 0.836414i \(0.684648\pi\)
\(80\) −177.544 − 21.8681i −0.248125 − 0.0305615i
\(81\) 359.240 + 105.482i 0.492784 + 0.144694i
\(82\) −513.074 + 234.313i −0.690970 + 0.315555i
\(83\) 595.559 + 85.6284i 0.787603 + 0.113240i 0.524368 − 0.851491i \(-0.324301\pi\)
0.263235 + 0.964732i \(0.415211\pi\)
\(84\) 212.438 − 245.167i 0.275939 − 0.318451i
\(85\) −200.627 − 54.5225i −0.256012 − 0.0695740i
\(86\) −301.759 − 193.929i −0.378366 − 0.243161i
\(87\) 1092.81 − 157.123i 1.34669 − 0.193625i
\(88\) −117.479 − 182.801i −0.142311 − 0.221440i
\(89\) −412.880 − 476.489i −0.491744 − 0.567502i 0.454587 − 0.890702i \(-0.349787\pi\)
−0.946331 + 0.323200i \(0.895241\pi\)
\(90\) −93.7870 − 194.827i −0.109845 − 0.228184i
\(91\) 1003.80 1.15634
\(92\) 120.503 + 424.442i 0.136558 + 0.480991i
\(93\) 1281.23i 1.42857i
\(94\) 625.331 − 183.614i 0.686149 − 0.201471i
\(95\) 410.783 + 341.633i 0.443636 + 0.368956i
\(96\) 112.067 − 72.0210i 0.119144 − 0.0765689i
\(97\) 688.014 − 98.9215i 0.720178 − 0.103546i 0.227525 − 0.973772i \(-0.426937\pi\)
0.492652 + 0.870226i \(0.336027\pi\)
\(98\) 39.4992 − 61.4619i 0.0407145 − 0.0633530i
\(99\) 109.111 − 238.919i 0.110768 − 0.242548i
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.j.a.9.7 ✓ 360
5.4 even 2 inner 230.4.j.a.9.30 yes 360
23.18 even 11 inner 230.4.j.a.179.30 yes 360
115.64 even 22 inner 230.4.j.a.179.7 yes 360
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.4.j.a.9.7 ✓ 360 1.1 even 1 trivial
230.4.j.a.9.30 yes 360 5.4 even 2 inner
230.4.j.a.179.7 yes 360 115.64 even 22 inner
230.4.j.a.179.30 yes 360 23.18 even 11 inner