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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 71.3
Character \(\chi\) \(=\) 230.71
Dual form 230.4.g.d.81.3

$q$-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+(1.68251 + 1.08128i) q^{2} +(-0.533706 - 3.71201i) q^{3} +(1.66166 + 3.63853i) q^{4} +(-4.79746 - 1.40866i) q^{5} +(3.11576 - 6.82257i) q^{6} +(-11.4866 + 13.2562i) q^{7} +(-1.13852 + 7.91857i) q^{8} +(12.4121 - 3.64453i) q^{9} +(-6.54861 - 7.55750i) q^{10} +(36.6950 - 23.5824i) q^{11} +(12.6194 - 8.11000i) q^{12} +(46.9952 + 54.2354i) q^{13} +(-33.6600 + 9.88345i) q^{14} +(-2.66853 + 18.5600i) q^{15} +(-10.4778 + 12.0920i) q^{16} +(49.9997 - 109.484i) q^{17} +(24.8243 + 7.28907i) q^{18} +(58.3700 + 127.812i) q^{19} +(-2.84630 - 19.7964i) q^{20} +(55.3377 + 35.5634i) q^{21} +87.2389 q^{22} +(58.3067 - 93.6340i) q^{23} +30.0014 q^{24} +(21.0313 + 13.5160i) q^{25} +(20.4261 + 142.066i) q^{26} +(-62.2158 - 136.234i) q^{27} +(-67.3199 - 19.7669i) q^{28} +(51.6022 - 112.993i) q^{29} +(-24.5585 + 28.3420i) q^{30} +(-18.2819 + 127.154i) q^{31} +(-30.7038 + 9.01544i) q^{32} +(-107.123 - 123.626i) q^{33} +(202.508 - 130.144i) q^{34} +(73.7800 - 47.4155i) q^{35} +(33.8855 + 39.1059i) q^{36} +(29.0346 - 8.52534i) q^{37} +(-39.9933 + 278.160i) q^{38} +(176.241 - 203.392i) q^{39} +(16.6166 - 36.3853i) q^{40} +(-14.9436 - 4.38783i) q^{41} +(54.6520 + 119.671i) q^{42} +(28.3569 + 197.226i) q^{43} +(146.780 + 94.3298i) q^{44} -64.6807 q^{45} +(199.346 - 94.4939i) q^{46} +225.326 q^{47} +(50.4776 + 32.4400i) q^{48} +(5.02815 + 34.9716i) q^{49} +(20.7708 + 45.4816i) q^{50} +(-433.091 - 127.167i) q^{51} +(-119.247 + 261.114i) q^{52} +(-462.414 + 533.654i) q^{53} +(42.6283 - 296.487i) q^{54} +(-209.263 + 61.4451i) q^{55} +(-91.8926 - 106.050i) q^{56} +(443.289 - 284.884i) q^{57} +(208.998 - 134.315i) q^{58} +(195.833 + 226.003i) q^{59} +(-71.9654 + 21.1310i) q^{60} +(33.2526 - 231.277i) q^{61} +(-168.248 + 194.169i) q^{62} +(-94.2603 + 206.401i) q^{63} +(-61.4076 - 18.0309i) q^{64} +(-149.059 - 326.393i) q^{65} +(-46.5599 - 323.831i) q^{66} +(-440.685 - 283.211i) q^{67} +481.443 q^{68} +(-378.689 - 166.462i) q^{69} +175.405 q^{70} +(-169.263 - 108.779i) q^{71} +(14.7280 + 102.436i) q^{72} +(-98.0358 - 214.668i) q^{73} +(58.0693 + 17.0507i) q^{74} +(38.9470 - 85.2821i) q^{75} +(-368.058 + 424.762i) q^{76} +(-108.886 + 757.319i) q^{77} +(516.450 - 151.644i) q^{78} +(773.589 + 892.770i) q^{79} +(67.3003 - 43.2513i) q^{80} +(-178.666 + 114.821i) q^{81} +(-20.3982 - 23.5408i) q^{82} +(-703.471 + 206.558i) q^{83} +(-37.4459 + 260.442i) q^{84} +(-394.098 + 454.813i) q^{85} +(-165.547 + 362.496i) q^{86} +(-446.972 - 131.243i) q^{87} +(144.961 + 317.421i) q^{88} +(-104.728 - 728.397i) q^{89} +(-108.826 - 69.9381i) q^{90} -1258.77 q^{91} +(437.576 + 56.5627i) q^{92} +481.753 q^{93} +(379.112 + 243.641i) q^{94} +(-99.9834 - 695.400i) q^{95} +(49.8522 + 109.161i) q^{96} +(-1341.04 - 393.766i) q^{97} +(-29.3542 + 64.2768i) q^{98} +(369.517 - 426.445i) 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 \(\left(\mathbb{Z}/230\mathbb{Z}\right)^\times\).

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.68251 + 1.08128i 0.594856 + 0.382291i
\(3\) −0.533706 − 3.71201i −0.102712 − 0.714376i −0.974483 − 0.224463i \(-0.927937\pi\)
0.871771 − 0.489914i \(-0.162972\pi\)
\(4\) 1.66166 + 3.63853i 0.207708 + 0.454816i
\(5\) −4.79746 − 1.40866i −0.429098 − 0.125995i
\(6\) 3.11576 − 6.82257i 0.212001 − 0.464217i
\(7\) −11.4866 + 13.2562i −0.620217 + 0.715769i −0.975748 − 0.218896i \(-0.929754\pi\)
0.355531 + 0.934664i \(0.384300\pi\)
\(8\) −1.13852 + 7.91857i −0.0503159 + 0.349955i
\(9\) 12.4121 − 3.64453i 0.459709 − 0.134983i
\(10\) −6.54861 − 7.55750i −0.207085 − 0.238989i
\(11\) 36.6950 − 23.5824i 1.00581 − 0.646398i 0.0695079 − 0.997581i \(-0.477857\pi\)
0.936307 + 0.351184i \(0.114221\pi\)
\(12\) 12.6194 − 8.11000i 0.303576 − 0.195096i
\(13\) 46.9952 + 54.2354i 1.00263 + 1.15709i 0.987566 + 0.157205i \(0.0502482\pi\)
0.0150592 + 0.999887i \(0.495206\pi\)
\(14\) −33.6600 + 9.88345i −0.642572 + 0.188676i
\(15\) −2.66853 + 18.5600i −0.0459341 + 0.319479i
\(16\) −10.4778 + 12.0920i −0.163715 + 0.188937i
\(17\) 49.9997 − 109.484i 0.713336 − 1.56199i −0.109679 − 0.993967i \(-0.534982\pi\)
0.823015 − 0.568020i \(-0.192290\pi\)
\(18\) 24.8243 + 7.28907i 0.325063 + 0.0954472i
\(19\) 58.3700 + 127.812i 0.704789 + 1.54327i 0.834066 + 0.551665i \(0.186007\pi\)
−0.129277 + 0.991609i \(0.541265\pi\)
\(20\) −2.84630 − 19.7964i −0.0318226 − 0.221331i
\(21\) 55.3377 + 35.5634i 0.575032 + 0.369550i
\(22\) 87.2389 0.845427
\(23\) 58.3067 − 93.6340i 0.528600 − 0.848871i
\(24\) 30.0014 0.255167
\(25\) 21.0313 + 13.5160i 0.168251 + 0.108128i
\(26\) 20.4261 + 142.066i 0.154072 + 1.07160i
\(27\) −62.2158 − 136.234i −0.443460 − 0.971043i
\(28\) −67.3199 − 19.7669i −0.454367 − 0.133414i
\(29\) 51.6022 − 112.993i 0.330424 − 0.723527i −0.669388 − 0.742913i \(-0.733444\pi\)
0.999812 + 0.0193857i \(0.00617103\pi\)
\(30\) −24.5585 + 28.3420i −0.149458 + 0.172484i
\(31\) −18.2819 + 127.154i −0.105920 + 0.736693i 0.865771 + 0.500440i \(0.166828\pi\)
−0.971692 + 0.236253i \(0.924081\pi\)
\(32\) −30.7038 + 9.01544i −0.169616 + 0.0498038i
\(33\) −107.123 − 123.626i −0.565080 − 0.652138i
\(34\) 202.508 − 130.144i 1.02147 − 0.656456i
\(35\) 73.7800 − 47.4155i 0.356317 − 0.228991i
\(36\) 33.8855 + 39.1059i 0.156877 + 0.181046i
\(37\) 29.0346 − 8.52534i 0.129007 − 0.0378799i −0.216592 − 0.976262i \(-0.569494\pi\)
0.345599 + 0.938382i \(0.387676\pi\)
\(38\) −39.9933 + 278.160i −0.170731 + 1.18746i
\(39\) 176.241 − 203.392i 0.723617 − 0.835099i
\(40\) 16.6166 − 36.3853i 0.0656829 − 0.143825i
\(41\) −14.9436 − 4.38783i −0.0569218 − 0.0167138i 0.253148 − 0.967428i \(-0.418534\pi\)
−0.310069 + 0.950714i \(0.600352\pi\)
\(42\) 54.6520 + 119.671i 0.200785 + 0.439659i
\(43\) 28.3569 + 197.226i 0.100567 + 0.699459i 0.976262 + 0.216594i \(0.0694947\pi\)
−0.875695 + 0.482865i \(0.839596\pi\)
\(44\) 146.780 + 94.3298i 0.502907 + 0.323199i
\(45\) −64.6807 −0.214267
\(46\) 199.346 − 94.4939i 0.638956 − 0.302878i
\(47\) 225.326 0.699301 0.349651 − 0.936880i \(-0.386300\pi\)
0.349651 + 0.936880i \(0.386300\pi\)
\(48\) 50.4776 + 32.4400i 0.151788 + 0.0975482i
\(49\) 5.02815 + 34.9716i 0.0146593 + 0.101958i
\(50\) 20.7708 + 45.4816i 0.0587486 + 0.128641i
\(51\) −433.091 − 127.167i −1.18911 − 0.349156i
\(52\) −119.247 + 261.114i −0.318011 + 0.696347i
\(53\) −462.414 + 533.654i −1.19844 + 1.38308i −0.294374 + 0.955690i \(0.595111\pi\)
−0.904069 + 0.427387i \(0.859434\pi\)
\(54\) 42.6283 − 296.487i 0.107426 − 0.747162i
\(55\) −209.263 + 61.4451i −0.513036 + 0.150641i
\(56\) −91.8926 − 106.050i −0.219280 − 0.253062i
\(57\) 443.289 − 284.884i 1.03009 − 0.661997i
\(58\) 208.998 − 134.315i 0.473152 − 0.304076i
\(59\) 195.833 + 226.003i 0.432124 + 0.498697i 0.929492 − 0.368842i \(-0.120246\pi\)
−0.497368 + 0.867539i \(0.665700\pi\)
\(60\) −71.9654 + 21.1310i −0.154845 + 0.0454666i
\(61\) 33.2526 − 231.277i 0.0697960 − 0.485442i −0.924702 − 0.380691i \(-0.875686\pi\)
0.994498 − 0.104751i \(-0.0334047\pi\)
\(62\) −168.248 + 194.169i −0.344638 + 0.397734i
\(63\) −94.2603 + 206.401i −0.188503 + 0.412764i
\(64\) −61.4076 − 18.0309i −0.119937 − 0.0352166i
\(65\) −149.059 − 326.393i −0.284438 − 0.622831i
\(66\) −46.5599 − 323.831i −0.0868353 − 0.603953i
\(67\) −440.685 − 283.211i −0.803556 − 0.516414i 0.0732184 − 0.997316i \(-0.476673\pi\)
−0.876774 + 0.480902i \(0.840309\pi\)
\(68\) 481.443 0.858582
\(69\) −378.689 − 166.462i −0.660707 − 0.290430i
\(70\) 175.405 0.299499
\(71\) −169.263 − 108.779i −0.282927 − 0.181826i 0.391479 − 0.920187i \(-0.371964\pi\)
−0.674406 + 0.738361i \(0.735600\pi\)
\(72\) 14.7280 + 102.436i 0.0241072 + 0.167669i
\(73\) −98.0358 − 214.668i −0.157181 − 0.344179i 0.814615 − 0.580002i \(-0.196948\pi\)
−0.971796 + 0.235824i \(0.924221\pi\)
\(74\) 58.0693 + 17.0507i 0.0912218 + 0.0267851i
\(75\) 38.9470 − 85.2821i 0.0599629 − 0.131300i
\(76\) −368.058 + 424.762i −0.555515 + 0.641099i
\(77\) −108.886 + 757.319i −0.161152 + 1.12084i
\(78\) 516.450 − 151.644i 0.749699 − 0.220131i
\(79\) 773.589 + 892.770i 1.10172 + 1.27145i 0.959529 + 0.281609i \(0.0908681\pi\)
0.142187 + 0.989840i \(0.454586\pi\)
\(80\) 67.3003 − 43.2513i 0.0940550 − 0.0604455i
\(81\) −178.666 + 114.821i −0.245083 + 0.157505i
\(82\) −20.3982 − 23.5408i −0.0274708 − 0.0317030i
\(83\) −703.471 + 206.558i −0.930313 + 0.273165i −0.711568 − 0.702617i \(-0.752015\pi\)
−0.218745 + 0.975782i \(0.570196\pi\)
\(84\) −37.4459 + 260.442i −0.0486391 + 0.338292i
\(85\) −394.098 + 454.813i −0.502893 + 0.580369i
\(86\) −165.547 + 362.496i −0.207574 + 0.454523i
\(87\) −446.972 − 131.243i −0.550809 − 0.161732i
\(88\) 144.961 + 317.421i 0.175601 + 0.384514i
\(89\) −104.728 − 728.397i −0.124732 − 0.867527i −0.952082 − 0.305842i \(-0.901062\pi\)
0.827351 − 0.561686i \(-0.189847\pi\)
\(90\) −108.826 − 69.9381i −0.127458 − 0.0819125i
\(91\) −1258.77 −1.45005
\(92\) 437.576 + 56.5627i 0.495874 + 0.0640986i
\(93\) 481.753 0.537155
\(94\) 379.112 + 243.641i 0.415984 + 0.267336i
\(95\) −99.9834 − 695.400i −0.107980 − 0.751016i
\(96\) 49.8522 + 109.161i 0.0530002 + 0.116054i
\(97\) −1341.04 − 393.766i −1.40374 − 0.412174i −0.509771 − 0.860310i \(-0.670270\pi\)
−0.893965 + 0.448136i \(0.852088\pi\)
\(98\) −29.3542 + 64.2768i −0.0302574 + 0.0662544i
\(99\) 369.517 − 426.445i 0.375129 − 0.432922i
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.71.3 ✓ 70
23.12 even 11 inner 230.4.g.d.81.3 yes 70
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.4.g.d.71.3 ✓ 70 1.1 even 1 trivial
230.4.g.d.81.3 yes 70 23.12 even 11 inner