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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 13.5
Character \(\chi\) \(=\) 230.13
Dual form 230.3.k.b.177.5

$q$-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} +(-0.110736 - 0.509045i) q^{3} +(1.97964 + 0.284630i) q^{4} +(-4.91212 + 0.933335i) q^{5} +(-0.104848 - 0.729237i) q^{6} +(5.72192 - 3.12441i) q^{7} +(2.76379 + 0.601225i) q^{8} +(7.93982 - 3.62600i) q^{9} +(-7.02324 + 0.820995i) q^{10} +(8.03552 - 9.27349i) q^{11} +(-0.0743284 - 1.03925i) q^{12} +(0.195417 + 0.106706i) q^{13} +(8.38662 - 3.83004i) q^{14} +(1.01906 + 2.39714i) q^{15} +(3.83797 + 1.12693i) q^{16} +(25.1514 + 18.8281i) q^{17} +(11.5658 - 4.31383i) q^{18} +(-0.992090 - 0.142641i) q^{19} +(-9.98989 + 0.449536i) q^{20} +(-2.22409 - 2.56673i) q^{21} +(12.2706 - 12.2706i) q^{22} +(-13.2873 - 18.7736i) q^{23} -1.47347i q^{24} +(23.2578 - 9.16930i) q^{25} +(0.264892 + 0.170236i) q^{26} +(-5.53477 - 7.39358i) q^{27} +(12.2167 - 4.55658i) q^{28} +(-6.83846 + 0.983222i) q^{29} +(1.19565 + 3.48424i) q^{30} +(-47.7322 + 30.6756i) q^{31} +(5.30019 + 1.97687i) q^{32} +(-5.61045 - 3.06354i) q^{33} +(33.5793 + 29.0966i) q^{34} +(-25.1906 + 20.6879i) q^{35} +(16.7501 - 4.91827i) q^{36} +(-58.3461 - 21.7620i) q^{37} +(-1.38506 - 0.301302i) q^{38} +(0.0326784 - 0.111293i) q^{39} +(-14.1372 - 0.373748i) q^{40} +(12.0341 - 26.3510i) q^{41} +(-2.87837 - 3.84505i) q^{42} +(11.0269 + 50.6896i) q^{43} +(18.5470 - 16.0710i) q^{44} +(-35.6171 + 25.2218i) q^{45} +(-16.8492 - 27.8227i) q^{46} +(53.9305 - 53.9305i) q^{47} +(0.148657 - 2.07849i) q^{48} +(-3.51293 + 5.46622i) q^{49} +(33.7327 - 10.5879i) q^{50} +(6.79919 - 14.8881i) q^{51} +(0.356485 + 0.266861i) q^{52} +(35.6701 + 65.3249i) q^{53} +(-7.06147 - 10.9879i) q^{54} +(-30.8161 + 53.0523i) q^{55} +(17.6927 - 5.19503i) q^{56} +(0.0372494 + 0.520814i) q^{57} +(-9.74560 + 0.697019i) q^{58} +(-14.4643 - 49.2610i) q^{59} +(1.33508 + 5.03553i) q^{60} +(-75.0546 + 48.2347i) q^{61} +(-70.4264 + 38.4557i) q^{62} +(34.1020 - 45.5549i) q^{63} +(7.27706 + 3.32332i) q^{64} +(-1.05951 - 0.341762i) q^{65} +(-7.60508 - 4.88749i) q^{66} +(27.0219 + 1.93264i) q^{67} +(44.4317 + 44.4317i) q^{68} +(-8.08520 + 8.84277i) q^{69} +(-37.6213 + 26.6411i) q^{70} +(6.14596 + 7.09282i) q^{71} +(24.1240 - 5.24786i) q^{72} +(-62.4132 + 46.7220i) q^{73} +(-80.1081 - 36.5841i) q^{74} +(-7.24306 - 10.8239i) q^{75} +(-1.92338 - 0.564756i) q^{76} +(17.0045 - 78.1684i) q^{77} +(0.0573247 - 0.153694i) q^{78} +(-4.01425 - 13.6713i) q^{79} +(-19.9044 - 1.95350i) q^{80} +(48.2934 - 55.7336i) q^{81} +(19.6340 - 35.9569i) q^{82} +(-52.4819 + 140.709i) q^{83} +(-3.67233 - 5.71426i) q^{84} +(-141.119 - 69.0111i) q^{85} +(10.4406 + 72.6158i) q^{86} +(1.25777 + 3.37221i) q^{87} +(27.7839 - 20.7988i) q^{88} +(-69.6061 + 108.309i) q^{89} +(-52.7864 + 31.9848i) q^{90} +1.45156 q^{91} +(-20.9607 - 40.9469i) q^{92} +(20.9010 + 20.9010i) q^{93} +(81.5159 - 70.6339i) q^{94} +(5.00639 - 0.225283i) q^{95} +(0.419394 - 2.91695i) q^{96} +(18.9776 + 50.8809i) q^{97} +(-5.50685 + 7.35629i) q^{98} +(30.1750 - 102.767i) 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
\(3\) −0.110736 − 0.509045i −0.0369120 − 0.169682i 0.955001 − 0.296603i \(-0.0958540\pi\)
−0.991913 + 0.126922i \(0.959490\pi\)
\(4\) 1.97964 + 0.284630i 0.494911 + 0.0711574i
\(5\) −4.91212 + 0.933335i −0.982423 + 0.186667i
\(6\) −0.104848 − 0.729237i −0.0174747 − 0.121539i
\(7\) 5.72192 − 3.12441i 0.817417 − 0.446344i −0.0154068 − 0.999881i \(-0.504904\pi\)
0.832824 + 0.553538i \(0.186723\pi\)
\(8\) 2.76379 + 0.601225i 0.345474 + 0.0751532i
\(9\) 7.93982 − 3.62600i 0.882203 − 0.402888i
\(10\) −7.02324 + 0.820995i −0.702324 + 0.0820995i
\(11\) 8.03552 − 9.27349i 0.730502 − 0.843044i −0.262026 − 0.965061i \(-0.584391\pi\)
0.992528 + 0.122017i \(0.0389361\pi\)
\(12\) −0.0743284 − 1.03925i −0.00619403 − 0.0866039i
\(13\) 0.195417 + 0.106706i 0.0150321 + 0.00820816i 0.486747 − 0.873543i \(-0.338183\pi\)
−0.471715 + 0.881751i \(0.656365\pi\)
\(14\) 8.38662 − 3.83004i 0.599044 − 0.273574i
\(15\) 1.01906 + 2.39714i 0.0679372 + 0.159809i
\(16\) 3.83797 + 1.12693i 0.239873 + 0.0704331i
\(17\) 25.1514 + 18.8281i 1.47949 + 1.10753i 0.968822 + 0.247759i \(0.0796940\pi\)
0.510671 + 0.859776i \(0.329397\pi\)
\(18\) 11.5658 − 4.31383i 0.642546 − 0.239657i
\(19\) −0.992090 − 0.142641i −0.0522153 − 0.00750742i 0.116158 − 0.993231i \(-0.462942\pi\)
−0.168373 + 0.985723i \(0.553851\pi\)
\(20\) −9.98989 + 0.449536i −0.499495 + 0.0224768i
\(21\) −2.22409 − 2.56673i −0.105909 − 0.122225i
\(22\) 12.2706 − 12.2706i 0.557754 − 0.557754i
\(23\) −13.2873 − 18.7736i −0.577711 − 0.816242i
\(24\) − 1.47347i − 0.0613946i
\(25\) 23.2578 − 9.16930i 0.930311 − 0.366772i
\(26\) 0.264892 + 0.170236i 0.0101882 + 0.00654754i
\(27\) −5.53477 − 7.39358i −0.204991 − 0.273836i
\(28\) 12.2167 − 4.55658i 0.436309 − 0.162735i
\(29\) −6.83846 + 0.983222i −0.235809 + 0.0339042i −0.259206 − 0.965822i \(-0.583461\pi\)
0.0233971 + 0.999726i \(0.492552\pi\)
\(30\) 1.19565 + 3.48424i 0.0398550 + 0.116141i
\(31\) −47.7322 + 30.6756i −1.53975 + 0.989536i −0.551937 + 0.833886i \(0.686111\pi\)
−0.987812 + 0.155650i \(0.950253\pi\)
\(32\) 5.30019 + 1.97687i 0.165631 + 0.0617771i
\(33\) −5.61045 − 3.06354i −0.170014 − 0.0928344i
\(34\) 33.5793 + 29.0966i 0.987625 + 0.855782i
\(35\) −25.1906 + 20.6879i −0.719732 + 0.591083i
\(36\) 16.7501 − 4.91827i 0.465280 − 0.136619i
\(37\) −58.3461 − 21.7620i −1.57692 − 0.588161i −0.599670 − 0.800247i \(-0.704702\pi\)
−0.977251 + 0.212086i \(0.931974\pi\)
\(38\) −1.38506 − 0.301302i −0.0364490 − 0.00792899i
\(39\) 0.0326784 − 0.111293i 0.000837909 − 0.00285366i
\(40\) −14.1372 − 0.373748i −0.353430 − 0.00934370i
\(41\) 12.0341 − 26.3510i 0.293515 − 0.642708i −0.704220 − 0.709982i \(-0.748703\pi\)
0.997735 + 0.0672743i \(0.0214302\pi\)
\(42\) −2.87837 − 3.84505i −0.0685325 − 0.0915487i
\(43\) 11.0269 + 50.6896i 0.256439 + 1.17883i 0.907582 + 0.419874i \(0.137926\pi\)
−0.651144 + 0.758954i \(0.725711\pi\)
\(44\) 18.5470 − 16.0710i 0.421522 − 0.365251i
\(45\) −35.6171 + 25.2218i −0.791490 + 0.560485i
\(46\) −16.8492 − 27.8227i −0.366288 − 0.604842i
\(47\) 53.9305 − 53.9305i 1.14746 − 1.14746i 0.160405 − 0.987051i \(-0.448720\pi\)
0.987051 − 0.160405i \(-0.0512802\pi\)
\(48\) 0.148657 − 2.07849i 0.00309702 − 0.0433019i
\(49\) −3.51293 + 5.46622i −0.0716924 + 0.111555i
\(50\) 33.7327 − 10.5879i 0.674655 − 0.211757i
\(51\) 6.79919 − 14.8881i 0.133317 − 0.291924i
\(52\) 0.356485 + 0.266861i 0.00685548 + 0.00513195i
\(53\) 35.6701 + 65.3249i 0.673021 + 1.23255i 0.961433 + 0.275040i \(0.0886909\pi\)
−0.288412 + 0.957506i \(0.593127\pi\)
\(54\) −7.06147 − 10.9879i −0.130768 − 0.203479i
\(55\) −30.8161 + 53.0523i −0.560294 + 0.964587i
\(56\) 17.6927 − 5.19503i 0.315940 − 0.0927684i
\(57\) 0.0372494 + 0.520814i 0.000653498 + 0.00913709i
\(58\) −9.74560 + 0.697019i −0.168028 + 0.0120176i
\(59\) −14.4643 − 49.2610i −0.245158 − 0.834933i −0.986494 − 0.163798i \(-0.947625\pi\)
0.741335 − 0.671135i \(-0.234193\pi\)
\(60\) 1.33508 + 5.03553i 0.0222513 + 0.0839255i
\(61\) −75.0546 + 48.2347i −1.23040 + 0.790732i −0.983954 − 0.178423i \(-0.942900\pi\)
−0.246450 + 0.969156i \(0.579264\pi\)
\(62\) −70.4264 + 38.4557i −1.13591 + 0.620253i
\(63\) 34.1020 − 45.5549i 0.541301 − 0.723093i
\(64\) 7.27706 + 3.32332i 0.113704 + 0.0519269i
\(65\) −1.05951 − 0.341762i −0.0163001 − 0.00525788i
\(66\) −7.60508 − 4.88749i −0.115228 − 0.0740528i
\(67\) 27.0219 + 1.93264i 0.403311 + 0.0288454i 0.271522 − 0.962432i \(-0.412473\pi\)
0.131790 + 0.991278i \(0.457928\pi\)
\(68\) 44.4317 + 44.4317i 0.653408 + 0.653408i
\(69\) −8.08520 + 8.84277i −0.117177 + 0.128156i
\(70\) −37.6213 + 26.6411i −0.537448 + 0.380588i
\(71\) 6.14596 + 7.09282i 0.0865629 + 0.0998989i 0.797379 − 0.603479i \(-0.206219\pi\)
−0.710816 + 0.703378i \(0.751674\pi\)
\(72\) 24.1240 − 5.24786i 0.335056 − 0.0728870i
\(73\) −62.4132 + 46.7220i −0.854976 + 0.640027i −0.934168 − 0.356833i \(-0.883857\pi\)
0.0791927 + 0.996859i \(0.474766\pi\)
\(74\) −80.1081 − 36.5841i −1.08254 − 0.494380i
\(75\) −7.24306 − 10.8239i −0.0965742 − 0.144319i
\(76\) −1.92338 − 0.564756i −0.0253077 − 0.00743100i
\(77\) 17.0045 − 78.1684i 0.220838 − 1.01517i
\(78\) 0.0573247 − 0.153694i 0.000734932 − 0.00197043i
\(79\) −4.01425 − 13.6713i −0.0508133 − 0.173054i 0.930177 − 0.367112i \(-0.119653\pi\)
−0.980990 + 0.194057i \(0.937835\pi\)
\(80\) −19.9044 − 1.95350i −0.248805 − 0.0244187i
\(81\) 48.2934 − 55.7336i 0.596215 − 0.688069i
\(82\) 19.6340 − 35.9569i 0.239439 − 0.438499i
\(83\) −52.4819 + 140.709i −0.632312 + 1.69529i 0.0830551 + 0.996545i \(0.473532\pi\)
−0.715367 + 0.698749i \(0.753740\pi\)
\(84\) −3.67233 − 5.71426i −0.0437182 − 0.0680269i
\(85\) −141.119 − 69.0111i −1.66023 − 0.811895i
\(86\) 10.4406 + 72.6158i 0.121402 + 0.844370i
\(87\) 1.25777 + 3.37221i 0.0144571 + 0.0387610i
\(88\) 27.7839 − 20.7988i 0.315727 − 0.236350i
\(89\) −69.6061 + 108.309i −0.782091 + 1.21696i 0.189868 + 0.981810i \(0.439194\pi\)
−0.971959 + 0.235148i \(0.924442\pi\)
\(90\) −52.7864 + 31.9848i −0.586516 + 0.355387i
\(91\) 1.45156 0.0159512
\(92\) −20.9607 − 40.9469i −0.227834 − 0.445075i
\(93\) 20.9010 + 20.9010i 0.224742 + 0.224742i
\(94\) 81.5159 − 70.6339i 0.867190 − 0.751424i
\(95\) 5.00639 − 0.225283i 0.0526989 − 0.00237140i
\(96\) 0.419394 − 2.91695i 0.00436868 − 0.0303849i
\(97\) 18.9776 + 50.8809i 0.195645 + 0.524545i 0.997316 − 0.0732212i \(-0.0233279\pi\)
−0.801671 + 0.597766i \(0.796055\pi\)
\(98\) −5.50685 + 7.35629i −0.0561923 + 0.0750642i
\(99\) 30.1750 − 102.767i 0.304798 − 1.03805i
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.13.5 ✓ 240
5.2 odd 4 inner 230.3.k.b.197.5 yes 240
23.16 even 11 inner 230.3.k.b.223.5 yes 240
115.62 odd 44 inner 230.3.k.b.177.5 yes 240
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.3.k.b.13.5 ✓ 240 1.1 even 1 trivial
230.3.k.b.177.5 yes 240 115.62 odd 44 inner
230.3.k.b.197.5 yes 240 5.2 odd 4 inner
230.3.k.b.223.5 yes 240 23.16 even 11 inner