↑

Properties

Label 230.4.g.d.41.6
Level $230$
Weight $4$
Character 230.41
Analytic conductor $13.570$
Analytic rank $0$
Dimension $70$
Inner twists $2$

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(31,230)] 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("230.31"); 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([0, 6])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 230 = 2 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 230.g (of order \(11\), degree \(10\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [70,-14,3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(13.5704393013\)
Analytic rank: \(0\)
Dimension: \(70\)
Relative dimension: \(7\) over \(\Q(\zeta_{11})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{11}]$

Embedding invariants

Embedding label 41.6
Character \(\chi\) \(=\) 230.41
Dual form 230.4.g.d.101.6

$q$-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+(-1.91899 - 0.563465i) q^{2} +(3.35821 - 3.87558i) q^{3} +(3.36501 + 2.16256i) q^{4} +(-0.711574 + 4.94911i) q^{5} +(-8.62810 + 5.54495i) q^{6} +(3.71145 + 8.12693i) q^{7} +(-5.23889 - 6.04600i) q^{8} +(0.0999547 + 0.695200i) q^{9} +(4.15415 - 9.09632i) q^{10} +(-35.7109 + 10.4857i) q^{11} +(19.6816 - 5.77904i) q^{12} +(-12.2875 + 26.9058i) q^{13} +(-2.54297 - 17.6867i) q^{14} +(16.7910 + 19.3779i) q^{15} +(6.64664 + 14.5541i) q^{16} +(-82.1276 + 52.7802i) q^{17} +(0.199909 - 1.39040i) q^{18} +(-98.2121 - 63.1171i) q^{19} +(-13.0972 + 15.1150i) q^{20} +(43.9604 + 12.9079i) q^{21} +74.4370 q^{22} +(-104.868 - 34.2004i) q^{23} -41.0250 q^{24} +(-23.9873 - 7.04331i) q^{25} +(38.7400 - 44.7084i) q^{26} +(119.509 + 76.8040i) q^{27} +(-5.08594 + 35.3735i) q^{28} +(36.5214 - 23.4709i) q^{29} +(-21.3030 - 46.6471i) q^{30} +(86.9295 + 100.322i) q^{31} +(-4.55407 - 31.6743i) q^{32} +(-79.2866 + 173.613i) q^{33} +(187.341 - 55.0084i) q^{34} +(-42.8620 + 12.5854i) q^{35} +(-1.16707 + 2.55552i) q^{36} +(11.4258 + 79.4682i) q^{37} +(152.903 + 176.460i) q^{38} +(63.0117 + 137.976i) q^{39} +(33.6501 - 21.6256i) q^{40} +(-5.65753 + 39.3490i) q^{41} +(-77.0862 - 49.5403i) q^{42} +(-71.9691 + 83.0568i) q^{43} +(-142.844 - 41.9427i) q^{44} -3.51175 q^{45} +(181.970 + 124.720i) q^{46} -311.890 q^{47} +(78.7264 + 23.1161i) q^{48} +(172.345 - 198.897i) q^{49} +(42.0627 + 27.0320i) q^{50} +(-71.2477 + 495.539i) q^{51} +(-99.5331 + 63.9661i) q^{52} +(233.890 + 512.147i) q^{53} +(-186.060 - 214.725i) q^{54} +(-26.4837 - 184.198i) q^{55} +(29.6916 - 65.0155i) q^{56} +(-574.432 + 168.668i) q^{57} +(-83.3091 + 24.4617i) q^{58} +(-202.556 + 443.535i) q^{59} +(14.5962 + 101.519i) q^{60} +(344.094 + 397.106i) q^{61} +(-110.289 - 241.498i) q^{62} +(-5.27887 + 3.39252i) q^{63} +(-9.10815 + 63.3486i) q^{64} +(-124.416 - 79.9576i) q^{65} +(249.975 - 288.486i) q^{66} +(-402.672 - 118.235i) q^{67} -390.501 q^{68} +(-484.715 + 291.573i) q^{69} +89.3431 q^{70} +(485.972 + 142.694i) q^{71} +(3.67953 - 4.24640i) q^{72} +(464.645 + 298.609i) q^{73} +(22.8516 - 158.936i) q^{74} +(-107.851 + 69.3118i) q^{75} +(-193.991 - 424.780i) q^{76} +(-217.755 - 251.303i) q^{77} +(-43.1737 - 300.280i) q^{78} +(419.950 - 919.561i) q^{79} +(-76.7594 + 22.5386i) q^{80} +(680.802 - 199.901i) q^{81} +(33.0285 - 72.3224i) q^{82} +(-160.205 - 1114.25i) q^{83} +(120.013 + 138.502i) q^{84} +(-202.775 - 444.015i) q^{85} +(184.907 - 118.833i) q^{86} +(31.6832 - 220.362i) q^{87} +(250.482 + 160.975i) q^{88} +(-71.4272 + 82.4314i) q^{89} +(6.73899 + 1.97875i) q^{90} -264.266 q^{91} +(-278.923 - 341.869i) q^{92} +680.733 q^{93} +(598.513 + 175.739i) q^{94} +(382.259 - 441.150i) q^{95} +(-138.050 - 88.7191i) q^{96} +(-103.585 + 720.449i) q^{97} +(-442.799 + 284.570i) q^{98} +(-10.8591 - 23.7781i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 70 q - 14 q^{2} + 3 q^{3} - 28 q^{4} - 35 q^{5} + 6 q^{6} - 78 q^{7} - 56 q^{8} - 24 q^{9} - 70 q^{10} - 15 q^{11} - 120 q^{12} - 270 q^{13} + 64 q^{14} + 15 q^{15} - 112 q^{16} + 114 q^{17} - 48 q^{18}+ \cdots + 11285 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(47\) \(51\)
\(\chi(n)\) \(1\) \(e\left(\frac{6}{11}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.91899 − 0.563465i −0.678464 − 0.199215i
\(3\) 3.35821 − 3.87558i 0.646287 − 0.745855i −0.334186 − 0.942507i \(-0.608461\pi\)
0.980473 + 0.196652i \(0.0630069\pi\)
\(4\) 3.36501 + 2.16256i 0.420627 + 0.270320i
\(5\) −0.711574 + 4.94911i −0.0636451 + 0.442662i
\(6\) −8.62810 + 5.54495i −0.587068 + 0.377286i
\(7\) 3.71145 + 8.12693i 0.200399 + 0.438813i 0.982974 − 0.183743i \(-0.0588214\pi\)
−0.782575 + 0.622556i \(0.786094\pi\)
\(8\) −5.23889 − 6.04600i −0.231528 − 0.267198i
\(9\) 0.0999547 + 0.695200i 0.00370203 + 0.0257482i
\(10\) 4.15415 − 9.09632i 0.131366 − 0.287651i
\(11\) −35.7109 + 10.4857i −0.978840 + 0.287413i −0.731746 − 0.681578i \(-0.761294\pi\)
−0.247095 + 0.968991i \(0.579476\pi\)
\(12\) 19.6816 − 5.77904i 0.473466 − 0.139022i
\(13\) −12.2875 + 26.9058i −0.262149 + 0.574026i −0.994239 − 0.107181i \(-0.965817\pi\)
0.732091 + 0.681207i \(0.238545\pi\)
\(14\) −2.54297 − 17.6867i −0.0485455 − 0.337641i
\(15\) 16.7910 + 19.3779i 0.289028 + 0.333557i
\(16\) 6.64664 + 14.5541i 0.103854 + 0.227408i
\(17\) −82.1276 + 52.7802i −1.17170 + 0.753005i −0.973842 − 0.227227i \(-0.927034\pi\)
−0.197856 + 0.980231i \(0.563398\pi\)
\(18\) 0.199909 − 1.39040i 0.00261773 − 0.0182067i
\(19\) −98.2121 − 63.1171i −1.18586 − 0.762108i −0.209408 − 0.977828i \(-0.567154\pi\)
−0.976455 + 0.215720i \(0.930790\pi\)
\(20\) −13.0972 + 15.1150i −0.146431 + 0.168991i
\(21\) 43.9604 + 12.9079i 0.456807 + 0.134131i
\(22\) 74.4370 0.721365
\(23\) −104.868 − 34.2004i −0.950719 − 0.310055i
\(24\) −41.0250 −0.348925
\(25\) −23.9873 − 7.04331i −0.191899 − 0.0563465i
\(26\) 38.7400 − 44.7084i 0.292213 − 0.337232i
\(27\) 119.509 + 76.8040i 0.851837 + 0.547442i
\(28\) −5.08594 + 35.3735i −0.0343269 + 0.238749i
\(29\) 36.5214 − 23.4709i 0.233857 − 0.150291i −0.418465 − 0.908233i \(-0.637432\pi\)
0.652321 + 0.757942i \(0.273795\pi\)
\(30\) −21.3030 − 46.6471i −0.129646 − 0.283885i
\(31\) 86.9295 + 100.322i 0.503645 + 0.581237i 0.949460 − 0.313888i \(-0.101632\pi\)
−0.445815 + 0.895125i \(0.647086\pi\)
\(32\) −4.55407 − 31.6743i −0.0251579 − 0.174977i
\(33\) −79.2866 + 173.613i −0.418243 + 0.915825i
\(34\) 187.341 − 55.0084i 0.944965 − 0.277467i
\(35\) −42.8620 + 12.5854i −0.207000 + 0.0607807i
\(36\) −1.16707 + 2.55552i −0.00540308 + 0.0118311i
\(37\) 11.4258 + 79.4682i 0.0507673 + 0.353094i 0.999333 + 0.0365143i \(0.0116255\pi\)
−0.948566 + 0.316580i \(0.897465\pi\)
\(38\) 152.903 + 176.460i 0.652742 + 0.753305i
\(39\) 63.0117 + 137.976i 0.258717 + 0.566511i
\(40\) 33.6501 − 21.6256i 0.133014 − 0.0854828i
\(41\) −5.65753 + 39.3490i −0.0215502 + 0.149885i −0.997755 − 0.0669639i \(-0.978669\pi\)
0.976205 + 0.216849i \(0.0695779\pi\)
\(42\) −77.0862 − 49.5403i −0.283206 − 0.182005i
\(43\) −71.9691 + 83.0568i −0.255237 + 0.294559i −0.868878 − 0.495026i \(-0.835159\pi\)
0.613641 + 0.789585i \(0.289704\pi\)
\(44\) −142.844 − 41.9427i −0.489420 − 0.143707i
\(45\) −3.51175 −0.0116333
\(46\) 181.970 + 124.720i 0.583261 + 0.399759i
\(47\) −311.890 −0.967955 −0.483978 − 0.875080i \(-0.660808\pi\)
−0.483978 + 0.875080i \(0.660808\pi\)
\(48\) 78.7264 + 23.1161i 0.236733 + 0.0695110i
\(49\) 172.345 − 198.897i 0.502464 − 0.579874i
\(50\) 42.0627 + 27.0320i 0.118971 + 0.0764582i
\(51\) −71.2477 + 495.539i −0.195621 + 1.36057i
\(52\) −99.5331 + 63.9661i −0.265438 + 0.170586i
\(53\) 233.890 + 512.147i 0.606174 + 1.32734i 0.925160 + 0.379577i \(0.123930\pi\)
−0.318986 + 0.947759i \(0.603342\pi\)
\(54\) −186.060 − 214.725i −0.468882 − 0.541118i
\(55\) −26.4837 − 184.198i −0.0649285 − 0.451587i
\(56\) 29.6916 − 65.0155i 0.0708518 − 0.155144i
\(57\) −574.432 + 168.668i −1.33483 + 0.391942i
\(58\) −83.3091 + 24.4617i −0.188604 + 0.0553790i
\(59\) −202.556 + 443.535i −0.446958 + 0.978701i 0.543311 + 0.839532i \(0.317171\pi\)
−0.990269 + 0.139169i \(0.955557\pi\)
\(60\) 14.5962 + 101.519i 0.0314059 + 0.218433i
\(61\) 344.094 + 397.106i 0.722242 + 0.833511i 0.991575 − 0.129536i \(-0.0413487\pi\)
−0.269333 + 0.963047i \(0.586803\pi\)
\(62\) −110.289 − 241.498i −0.225914 − 0.494682i
\(63\) −5.27887 + 3.39252i −0.0105568 + 0.00678441i
\(64\) −9.10815 + 63.3486i −0.0177894 + 0.123728i
\(65\) −124.416 − 79.9576i −0.237415 − 0.152577i
\(66\) 249.975 − 288.486i 0.466209 − 0.538034i
\(67\) −402.672 − 118.235i −0.734241 − 0.215593i −0.106826 − 0.994278i \(-0.534069\pi\)
−0.627415 + 0.778685i \(0.715887\pi\)
\(68\) −390.501 −0.696400
\(69\) −484.715 + 291.573i −0.845693 + 0.508714i
\(70\) 89.3431 0.152551
\(71\) 485.972 + 142.694i 0.812313 + 0.238517i 0.661403 − 0.750031i \(-0.269961\pi\)
0.150910 + 0.988547i \(0.451779\pi\)
\(72\) 3.67953 − 4.24640i 0.00602273 − 0.00695060i
\(73\) 464.645 + 298.609i 0.744967 + 0.478761i 0.857241 − 0.514916i \(-0.172177\pi\)
−0.112273 + 0.993677i \(0.535813\pi\)
\(74\) 22.8516 − 158.936i 0.0358979 − 0.249675i
\(75\) −107.851 + 69.3118i −0.166048 + 0.106713i
\(76\) −193.991 − 424.780i −0.292793 − 0.641126i
\(77\) −217.755 − 251.303i −0.322280 − 0.371931i
\(78\) −43.1737 − 300.280i −0.0626726 − 0.435897i
\(79\) 419.950 − 919.561i 0.598076 − 1.30960i −0.332361 − 0.943152i \(-0.607845\pi\)
0.930437 − 0.366452i \(-0.119428\pi\)
\(80\) −76.7594 + 22.5386i −0.107275 + 0.0314987i
\(81\) 680.802 − 199.901i 0.933884 − 0.274213i
\(82\) 33.0285 − 72.3224i 0.0444804 − 0.0973984i
\(83\) −160.205 − 1114.25i −0.211865 − 1.47355i −0.766923 − 0.641739i \(-0.778213\pi\)
0.555058 − 0.831811i \(-0.312696\pi\)
\(84\) 120.013 + 138.502i 0.155887 + 0.179903i
\(85\) −202.775 − 444.015i −0.258753 − 0.566591i
\(86\) 184.907 − 118.833i 0.231850 − 0.149001i
\(87\) 31.6832 − 220.362i 0.0390436 − 0.271554i
\(88\) 250.482 + 160.975i 0.303425 + 0.195000i
\(89\) −71.4272 + 82.4314i −0.0850704 + 0.0981765i −0.796689 − 0.604390i \(-0.793417\pi\)
0.711618 + 0.702566i \(0.247962\pi\)
\(90\) 6.73899 + 1.97875i 0.00789280 + 0.00231754i
\(91\) −264.266 −0.304425
\(92\) −278.923 − 341.869i −0.316083 − 0.387416i
\(93\) 680.733 0.759018
\(94\) 598.513 + 175.739i 0.656723 + 0.192831i
\(95\) 382.259 − 441.150i 0.412831 − 0.476432i
\(96\) −138.050 − 88.7191i −0.146767 − 0.0943214i
\(97\) −103.585 + 720.449i −0.108427 + 0.754129i 0.860974 + 0.508649i \(0.169855\pi\)
−0.969402 + 0.245480i \(0.921054\pi\)
\(98\) −442.799 + 284.570i −0.456423 + 0.293325i
\(99\) −10.8591 − 23.7781i −0.0110241 − 0.0241393i
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.g.d.41.6 ✓ 70
23.9 even 11 inner 230.4.g.d.101.6 yes 70
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.4.g.d.41.6 ✓ 70 1.1 even 1 trivial
230.4.g.d.101.6 yes 70 23.9 even 11 inner