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Properties

Label 230.4.j.a.9.3
Level $230$
Weight $4$
Character 230.9
Analytic conductor $13.570$
Analytic rank $0$
Dimension $360$
Inner twists $4$

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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 9.3
Character \(\chi\) \(=\) 230.9
Dual form 230.4.j.a.179.3

$q$-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+(-0.563465 - 1.91899i) q^{2} +(-6.12185 + 5.30461i) q^{3} +(-3.36501 + 2.16256i) q^{4} +(-10.8097 + 2.85485i) q^{5} +(13.6289 + 8.75878i) q^{6} +(32.7383 + 14.9511i) q^{7} +(6.04600 + 5.23889i) q^{8} +(5.49562 - 38.2229i) q^{9} +(11.5693 + 19.1351i) q^{10} +(47.2262 + 13.8669i) q^{11} +(9.12855 - 31.0890i) q^{12} +(-33.9422 + 15.5009i) q^{13} +(10.2440 - 71.2487i) q^{14} +(51.0316 - 74.8183i) q^{15} +(6.64664 - 14.5541i) q^{16} +(-44.9759 + 69.9839i) q^{17} +(-76.4458 + 10.9912i) q^{18} +(-1.92814 + 1.23914i) q^{19} +(30.2010 - 32.9833i) q^{20} +(-279.729 + 82.1357i) q^{21} -98.4399i q^{22} +(1.81307 + 110.289i) q^{23} -64.8029 q^{24} +(108.700 - 61.7202i) q^{25} +(48.8712 + 56.4004i) q^{26} +(50.8707 + 79.1564i) q^{27} +(-142.497 + 20.4880i) q^{28} +(-118.812 - 76.3559i) q^{29} +(-172.330 - 55.7713i) q^{30} +(-130.847 + 151.006i) q^{31} +(-31.6743 - 4.55407i) q^{32} +(-362.670 + 165.626i) q^{33} +(159.640 + 46.8746i) q^{34} +(-396.575 - 68.1540i) q^{35} +(64.1666 + 140.505i) q^{36} +(7.20064 + 1.03530i) q^{37} +(3.46434 + 3.00187i) q^{38} +(125.563 - 274.944i) q^{39} +(-80.3117 - 39.3704i) q^{40} +(-37.4347 - 260.364i) q^{41} +(315.235 + 490.515i) q^{42} +(197.946 - 171.521i) q^{43} +(-188.905 + 55.4675i) q^{44} +(49.7145 + 428.868i) q^{45} +(210.622 - 65.6234i) q^{46} -167.410i q^{47} +(36.5142 + 124.356i) q^{48} +(623.644 + 719.723i) q^{49} +(-179.689 - 173.816i) q^{50} +(-95.9016 - 667.010i) q^{51} +(80.6943 - 125.563i) q^{52} +(-586.436 - 267.816i) q^{53} +(123.236 - 142.222i) q^{54} +(-550.089 - 15.0731i) q^{55} +(119.609 + 261.906i) q^{56} +(5.23064 - 17.8139i) q^{57} +(-79.5794 + 271.023i) q^{58} +(178.588 + 391.054i) q^{59} +(-9.92264 + 362.124i) q^{60} +(-323.451 + 373.283i) q^{61} +(363.506 + 166.008i) q^{62} +(751.391 - 1169.19i) q^{63} +(9.10815 + 63.3486i) q^{64} +(322.653 - 264.460i) q^{65} +(522.186 + 602.634i) q^{66} +(-156.736 - 533.793i) q^{67} -332.760i q^{68} +(-596.141 - 665.556i) q^{69} +(92.6694 + 799.423i) q^{70} +(-728.021 + 213.766i) q^{71} +(233.472 - 202.305i) q^{72} +(214.714 + 334.101i) q^{73} +(-2.07059 - 14.4013i) q^{74} +(-338.041 + 954.451i) q^{75} +(3.80851 - 8.33947i) q^{76} +(1338.78 + 1160.06i) q^{77} +(-598.364 - 86.0318i) q^{78} +(461.225 + 1009.94i) q^{79} +(-30.2985 + 176.301i) q^{80} +(269.079 + 79.0089i) q^{81} +(-478.542 + 218.543i) q^{82} +(-995.172 - 143.084i) q^{83} +(763.667 - 881.319i) q^{84} +(286.383 - 884.905i) q^{85} +(-440.682 - 283.209i) q^{86} +(1132.39 - 162.813i) q^{87} +(212.883 + 331.252i) q^{88} +(-382.453 - 441.374i) q^{89} +(794.978 - 337.053i) q^{90} -1342.96 q^{91} +(-244.608 - 367.204i) q^{92} -1618.53i q^{93} +(-321.257 + 94.3296i) q^{94} +(17.3051 - 18.8993i) q^{95} +(218.063 - 140.140i) q^{96} +(982.745 - 141.297i) q^{97} +(1029.74 - 1602.30i) q^{98} +(789.569 - 1728.92i) 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
\(3\) −6.12185 + 5.30461i −1.17815 + 1.02087i −0.178835 + 0.983879i \(0.557233\pi\)
−0.999315 + 0.0369940i \(0.988222\pi\)
\(4\) −3.36501 + 2.16256i −0.420627 + 0.270320i
\(5\) −10.8097 + 2.85485i −0.966850 + 0.255345i
\(6\) 13.6289 + 8.75878i 0.927331 + 0.595959i
\(7\) 32.7383 + 14.9511i 1.76770 + 0.807282i 0.982052 + 0.188611i \(0.0603984\pi\)
0.785650 + 0.618672i \(0.212329\pi\)
\(8\) 6.04600 + 5.23889i 0.267198 + 0.231528i
\(9\) 5.49562 − 38.2229i 0.203542 − 1.41566i
\(10\) 11.5693 + 19.1351i 0.365854 + 0.605104i
\(11\) 47.2262 + 13.8669i 1.29448 + 0.380092i 0.855218 − 0.518268i \(-0.173423\pi\)
0.439258 + 0.898361i \(0.355241\pi\)
\(12\) 9.12855 − 31.0890i 0.219599 − 0.747885i
\(13\) −33.9422 + 15.5009i −0.724144 + 0.330705i −0.743160 − 0.669114i \(-0.766674\pi\)
0.0190161 + 0.999819i \(0.493947\pi\)
\(14\) 10.2440 − 71.2487i 0.195559 − 1.36014i
\(15\) 51.0316 − 74.8183i 0.878419 − 1.28787i
\(16\) 6.64664 − 14.5541i 0.103854 − 0.227408i
\(17\) −44.9759 + 69.9839i −0.641662 + 0.998446i 0.356283 + 0.934378i \(0.384044\pi\)
−0.997946 + 0.0640678i \(0.979593\pi\)
\(18\) −76.4458 + 10.9912i −1.00102 + 0.143926i
\(19\) −1.92814 + 1.23914i −0.0232814 + 0.0149621i −0.552230 − 0.833692i \(-0.686223\pi\)
0.528948 + 0.848654i \(0.322587\pi\)
\(20\) 30.2010 − 32.9833i 0.337658 − 0.368764i
\(21\) −279.729 + 82.1357i −2.90675 + 0.853499i
\(22\) − 98.4399i − 0.953976i
\(23\) 1.81307 + 110.289i 0.0164370 + 0.999865i
\(24\) −64.8029 −0.551160
\(25\) 108.700 − 61.7202i 0.869597 − 0.493761i
\(26\) 48.8712 + 56.4004i 0.368632 + 0.425424i
\(27\) 50.8707 + 79.1564i 0.362596 + 0.564210i
\(28\) −142.497 + 20.4880i −0.961768 + 0.138281i
\(29\) −118.812 − 76.3559i −0.760788 − 0.488929i 0.101819 − 0.994803i \(-0.467534\pi\)
−0.862607 + 0.505874i \(0.831170\pi\)
\(30\) −172.330 − 55.7713i −1.04877 − 0.339414i
\(31\) −130.847 + 151.006i −0.758092 + 0.874885i −0.995326 − 0.0965674i \(-0.969214\pi\)
0.237234 + 0.971452i \(0.423759\pi\)
\(32\) −31.6743 − 4.55407i −0.174977 − 0.0251579i
\(33\) −362.670 + 165.626i −1.91311 + 0.873690i
\(34\) 159.640 + 46.8746i 0.805238 + 0.236439i
\(35\) −396.575 − 68.1540i −1.91524 − 0.329146i
\(36\) 64.1666 + 140.505i 0.297067 + 0.650487i
\(37\) 7.20064 + 1.03530i 0.0319940 + 0.00460004i 0.158293 − 0.987392i \(-0.449401\pi\)
−0.126299 + 0.991992i \(0.540310\pi\)
\(38\) 3.46434 + 3.00187i 0.0147892 + 0.0128149i
\(39\) 125.563 − 274.944i 0.515542 − 1.12888i
\(40\) −80.3117 − 39.3704i −0.317460 − 0.155625i
\(41\) −37.4347 − 260.364i −0.142593 − 0.991758i −0.927947 − 0.372711i \(-0.878428\pi\)
0.785354 − 0.619047i \(-0.212481\pi\)
\(42\) 315.235 + 490.515i 1.15814 + 1.80210i
\(43\) 197.946 − 171.521i 0.702011 − 0.608296i −0.228940 − 0.973441i \(-0.573526\pi\)
0.930951 + 0.365145i \(0.118980\pi\)
\(44\) −188.905 + 55.4675i −0.647238 + 0.190046i
\(45\) 49.7145 + 428.868i 0.164689 + 1.42071i
\(46\) 210.622 − 65.6234i 0.675098 − 0.210340i
\(47\) − 167.410i − 0.519558i −0.965668 − 0.259779i \(-0.916350\pi\)
0.965668 − 0.259779i \(-0.0836498\pi\)
\(48\) 36.5142 + 124.356i 0.109799 + 0.373942i
\(49\) 623.644 + 719.723i 1.81820 + 2.09832i
\(50\) −179.689 − 173.816i −0.508236 − 0.491626i
\(51\) −95.9016 − 667.010i −0.263312 − 1.83137i
\(52\) 80.6943 − 125.563i 0.215198 − 0.334854i
\(53\) −586.436 − 267.816i −1.51987 − 0.694102i −0.531626 − 0.846979i \(-0.678419\pi\)
−0.988245 + 0.152877i \(0.951146\pi\)
\(54\) 123.236 − 142.222i 0.310561 − 0.358407i
\(55\) −550.089 − 15.0731i −1.34862 − 0.0369538i
\(56\) 119.609 + 261.906i 0.285417 + 0.624977i
\(57\) 5.23064 − 17.8139i 0.0121546 − 0.0413949i
\(58\) −79.5794 + 271.023i −0.180160 + 0.613569i
\(59\) 178.588 + 391.054i 0.394072 + 0.862897i 0.997837 + 0.0657346i \(0.0209391\pi\)
−0.603766 + 0.797162i \(0.706334\pi\)
\(60\) −9.92264 + 362.124i −0.0213501 + 0.779166i
\(61\) −323.451 + 373.283i −0.678913 + 0.783507i −0.985743 − 0.168257i \(-0.946186\pi\)
0.306830 + 0.951764i \(0.400732\pi\)
\(62\) 363.506 + 166.008i 0.744601 + 0.340048i
\(63\) 751.391 − 1169.19i 1.50264 − 2.33815i
\(64\) 9.10815 + 63.3486i 0.0177894 + 0.123728i
\(65\) 322.653 − 264.460i 0.615694 − 0.504649i
\(66\) 522.186 + 602.634i 0.973888 + 1.12393i
\(67\) −156.736 − 533.793i −0.285796 − 0.973332i −0.969810 − 0.243863i \(-0.921585\pi\)
0.684014 − 0.729469i \(-0.260233\pi\)
\(68\) − 332.760i − 0.593427i
\(69\) −596.141 − 665.556i −1.04010 − 1.16121i
\(70\) 92.6694 + 799.423i 0.158230 + 1.36499i
\(71\) −728.021 + 213.766i −1.21690 + 0.357315i −0.826293 − 0.563241i \(-0.809554\pi\)
−0.390611 + 0.920556i \(0.627736\pi\)
\(72\) 233.472 − 202.305i 0.382152 − 0.331136i
\(73\) 214.714 + 334.101i 0.344251 + 0.535666i 0.969605 − 0.244676i \(-0.0786816\pi\)
−0.625353 + 0.780342i \(0.715045\pi\)
\(74\) −2.07059 − 14.4013i −0.00325272 − 0.0226232i
\(75\) −338.041 + 954.451i −0.520449 + 1.46947i
\(76\) 3.80851 − 8.33947i 0.00574824 − 0.0125869i
\(77\) 1338.78 + 1160.06i 1.98141 + 1.71690i
\(78\) −598.364 − 86.0318i −0.868608 − 0.124887i
\(79\) 461.225 + 1009.94i 0.656859 + 1.43832i 0.885420 + 0.464791i \(0.153871\pi\)
−0.228562 + 0.973529i \(0.573402\pi\)
\(80\) −30.2985 + 176.301i −0.0423434 + 0.246388i
\(81\) 269.079 + 79.0089i 0.369108 + 0.108380i
\(82\) −478.542 + 218.543i −0.644465 + 0.294317i
\(83\) −995.172 − 143.084i −1.31608 − 0.189223i −0.551726 − 0.834025i \(-0.686031\pi\)
−0.764350 + 0.644802i \(0.776940\pi\)
\(84\) 763.667 − 881.319i 0.991939 − 1.14476i
\(85\) 286.383 − 884.905i 0.365442 − 1.12919i
\(86\) −440.682 − 283.209i −0.552558 − 0.355108i
\(87\) 1132.39 − 162.813i 1.39546 − 0.200636i
\(88\) 212.883 + 331.252i 0.257879 + 0.401268i
\(89\) −382.453 − 441.374i −0.455505 − 0.525681i 0.480818 − 0.876820i \(-0.340340\pi\)
−0.936323 + 0.351140i \(0.885794\pi\)
\(90\) 794.978 − 337.053i 0.931090 − 0.394762i
\(91\) −1342.96 −1.54704
\(92\) −244.608 − 367.204i −0.277198 − 0.416127i
\(93\) − 1618.53i − 1.80466i
\(94\) −321.257 + 94.3296i −0.352502 + 0.103504i
\(95\) 17.3051 − 18.8993i 0.0186891 − 0.0204109i
\(96\) 218.063 − 140.140i 0.231833 − 0.148990i
\(97\) 982.745 − 141.297i 1.02869 − 0.147903i 0.392751 − 0.919645i \(-0.371523\pi\)
0.635936 + 0.771742i \(0.280614\pi\)
\(98\) 1029.74 − 1602.30i 1.06142 − 1.65160i
\(99\) 789.569 − 1728.92i 0.801562 − 1.75518i
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.9.3 ✓ 360
5.4 even 2 inner 230.4.j.a.9.34 yes 360
23.18 even 11 inner 230.4.j.a.179.34 yes 360
115.64 even 22 inner 230.4.j.a.179.3 yes 360
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.4.j.a.9.3 ✓ 360 1.1 even 1 trivial
230.4.j.a.9.34 yes 360 5.4 even 2 inner
230.4.j.a.179.3 yes 360 115.64 even 22 inner
230.4.j.a.179.34 yes 360 23.18 even 11 inner