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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 = [60,12,-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: \(60\)
Relative dimension: \(6\) over \(\Q(\zeta_{11})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{11}]$

Embedding invariants

Embedding label 71.5
Character \(\chi\) \(=\) 230.71
Dual form 230.4.g.b.81.5

$q$-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+(-1.68251 - 1.08128i) q^{2} +(0.839469 + 5.83864i) q^{3} +(1.66166 + 3.63853i) q^{4} +(-4.79746 - 1.40866i) q^{5} +(4.90080 - 10.7312i) q^{6} +(5.10412 - 5.89047i) q^{7} +(1.13852 - 7.91857i) q^{8} +(-7.47864 + 2.19593i) q^{9} +(6.54861 + 7.55750i) q^{10} +(32.9368 - 21.1672i) q^{11} +(-19.8491 + 12.7563i) q^{12} +(4.12196 + 4.75700i) q^{13} +(-14.9570 + 4.39176i) q^{14} +(4.19734 - 29.1932i) q^{15} +(-10.4778 + 12.0920i) q^{16} +(53.3867 - 116.901i) q^{17} +(14.9573 + 4.39185i) q^{18} +(34.0453 + 74.5488i) q^{19} +(-2.84630 - 19.7964i) q^{20} +(38.6771 + 24.8562i) q^{21} -78.3040 q^{22} +(-28.9900 + 106.426i) q^{23} +47.1894 q^{24} +(21.0313 + 13.5160i) q^{25} +(-1.79158 - 12.4607i) q^{26} +(47.0614 + 103.050i) q^{27} +(29.9140 + 8.78353i) q^{28} +(-41.6960 + 91.3016i) q^{29} +(-38.6281 + 44.5792i) q^{30} +(-16.5320 + 114.983i) q^{31} +(30.7038 - 9.01544i) q^{32} +(151.237 + 174.537i) q^{33} +(-216.226 + 138.960i) q^{34} +(-32.7845 + 21.0693i) q^{35} +(-20.4169 - 23.5624i) q^{36} +(149.431 - 43.8768i) q^{37} +(23.3268 - 162.241i) q^{38} +(-24.3141 + 28.0600i) q^{39} +(-16.6166 + 36.3853i) q^{40} +(320.732 + 94.1754i) q^{41} +(-38.1978 - 83.6416i) q^{42} +(-20.3111 - 141.267i) q^{43} +(131.747 + 84.6687i) q^{44} +38.9718 q^{45} +(163.853 - 147.717i) q^{46} +51.3169 q^{47} +(-79.3965 - 51.0250i) q^{48} +(40.1684 + 279.377i) q^{49} +(-20.7708 - 45.4816i) q^{50} +(727.357 + 213.571i) q^{51} +(-10.4592 + 22.9024i) q^{52} +(156.058 - 180.101i) q^{53} +(32.2451 - 224.269i) q^{54} +(-187.830 + 55.1520i) q^{55} +(-40.8330 - 47.1238i) q^{56} +(-406.683 + 261.360i) q^{57} +(168.877 - 108.530i) q^{58} +(163.088 + 188.213i) q^{59} +(113.195 - 33.2370i) q^{60} +(129.077 - 897.750i) q^{61} +(152.144 - 175.583i) q^{62} +(-25.2368 + 55.2610i) q^{63} +(-61.4076 - 18.0309i) q^{64} +(-13.0740 - 28.6280i) q^{65} +(-65.7338 - 457.189i) q^{66} +(-252.176 - 162.064i) q^{67} +514.057 q^{68} +(-645.721 - 79.9201i) q^{69} +77.9421 q^{70} +(915.460 + 588.330i) q^{71} +(8.87403 + 61.7203i) q^{72} +(78.6079 + 172.127i) q^{73} +(-298.861 - 87.7535i) q^{74} +(-61.2600 + 134.141i) q^{75} +(-214.676 + 247.750i) q^{76} +(43.4287 - 302.053i) q^{77} +(71.2494 - 20.9207i) q^{78} +(264.691 + 305.470i) q^{79} +(67.3003 - 43.2513i) q^{80} +(-739.206 + 475.059i) q^{81} +(-437.803 - 505.252i) q^{82} +(271.864 - 79.8266i) q^{83} +(-26.1720 + 182.030i) q^{84} +(-420.795 + 485.623i) q^{85} +(-118.575 + 259.644i) q^{86} +(-568.079 - 166.803i) q^{87} +(-130.115 - 284.911i) q^{88} +(-20.2934 - 141.144i) q^{89} +(-65.5704 - 42.1395i) q^{90} +49.0600 q^{91} +(-435.407 + 71.3638i) q^{92} -685.219 q^{93} +(-86.3411 - 55.4880i) q^{94} +(-58.3170 - 405.604i) q^{95} +(78.4127 + 171.700i) q^{96} +(-1696.01 - 497.995i) q^{97} +(234.502 - 513.488i) q^{98} +(-199.841 + 230.628i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 60 q + 12 q^{2} - 3 q^{3} - 24 q^{4} - 30 q^{5} + 6 q^{6} + 100 q^{7} + 48 q^{8} + 69 q^{9} + 60 q^{10} - 51 q^{11} + 120 q^{12} + 184 q^{13} + 20 q^{14} - 15 q^{15} - 96 q^{16} - 334 q^{17} - 138 q^{18}+ \cdots - 10589 q^{99}+O(q^{100}) \) 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
\(3\) 0.839469 + 5.83864i 0.161556 + 1.12365i 0.895702 + 0.444655i \(0.146674\pi\)
−0.734146 + 0.678991i \(0.762417\pi\)
\(4\) 1.66166 + 3.63853i 0.207708 + 0.454816i
\(5\) −4.79746 − 1.40866i −0.429098 − 0.125995i
\(6\) 4.90080 − 10.7312i 0.333457 − 0.730169i
\(7\) 5.10412 − 5.89047i 0.275597 − 0.318055i −0.601030 − 0.799226i \(-0.705243\pi\)
0.876627 + 0.481171i \(0.159788\pi\)
\(8\) 1.13852 − 7.91857i 0.0503159 − 0.349955i
\(9\) −7.47864 + 2.19593i −0.276987 + 0.0813306i
\(10\) 6.54861 + 7.55750i 0.207085 + 0.238989i
\(11\) 32.9368 − 21.1672i 0.902801 − 0.580195i −0.00481864 − 0.999988i \(-0.501534\pi\)
0.907620 + 0.419793i \(0.137897\pi\)
\(12\) −19.8491 + 12.7563i −0.477496 + 0.306868i
\(13\) 4.12196 + 4.75700i 0.0879405 + 0.101489i 0.798014 − 0.602639i \(-0.205884\pi\)
−0.710074 + 0.704127i \(0.751338\pi\)
\(14\) −14.9570 + 4.39176i −0.285530 + 0.0838392i
\(15\) 4.19734 − 29.1932i 0.0722500 − 0.502510i
\(16\) −10.4778 + 12.0920i −0.163715 + 0.188937i
\(17\) 53.3867 − 116.901i 0.761658 − 1.66780i 0.0174523 − 0.999848i \(-0.494444\pi\)
0.744206 − 0.667951i \(-0.232828\pi\)
\(18\) 14.9573 + 4.39185i 0.195859 + 0.0575094i
\(19\) 34.0453 + 74.5488i 0.411081 + 0.900141i 0.996026 + 0.0890678i \(0.0283888\pi\)
−0.584945 + 0.811073i \(0.698884\pi\)
\(20\) −2.84630 − 19.7964i −0.0318226 − 0.221331i
\(21\) 38.6771 + 24.8562i 0.401906 + 0.258289i
\(22\) −78.3040 −0.758840
\(23\) −28.9900 + 106.426i −0.262818 + 0.964845i
\(24\) 47.1894 0.401354
\(25\) 21.0313 + 13.5160i 0.168251 + 0.108128i
\(26\) −1.79158 − 12.4607i −0.0135137 − 0.0939901i
\(27\) 47.0614 + 103.050i 0.335444 + 0.734519i
\(28\) 29.9140 + 8.78353i 0.201900 + 0.0592833i
\(29\) −41.6960 + 91.3016i −0.266992 + 0.584630i −0.994880 − 0.101066i \(-0.967775\pi\)
0.727888 + 0.685696i \(0.240502\pi\)
\(30\) −38.6281 + 44.5792i −0.235083 + 0.271300i
\(31\) −16.5320 + 114.983i −0.0957817 + 0.666176i 0.884203 + 0.467103i \(0.154702\pi\)
−0.979985 + 0.199073i \(0.936207\pi\)
\(32\) 30.7038 − 9.01544i 0.169616 − 0.0498038i
\(33\) 151.237 + 174.537i 0.797787 + 0.920695i
\(34\) −216.226 + 138.960i −1.09066 + 0.700925i
\(35\) −32.7845 + 21.0693i −0.158331 + 0.101753i
\(36\) −20.4169 − 23.5624i −0.0945227 − 0.109085i
\(37\) 149.431 − 43.8768i 0.663952 − 0.194954i 0.0676440 − 0.997710i \(-0.478452\pi\)
0.596308 + 0.802756i \(0.296634\pi\)
\(38\) 23.3268 − 162.241i 0.0995818 − 0.692606i
\(39\) −24.3141 + 28.0600i −0.0998301 + 0.115210i
\(40\) −16.6166 + 36.3853i −0.0656829 + 0.143825i
\(41\) 320.732 + 94.1754i 1.22170 + 0.358725i 0.828113 − 0.560561i \(-0.189414\pi\)
0.393591 + 0.919285i \(0.371232\pi\)
\(42\) −38.1978 − 83.6416i −0.140335 − 0.307290i
\(43\) −20.3111 − 141.267i −0.0720328 − 0.500999i −0.993616 − 0.112818i \(-0.964012\pi\)
0.921583 − 0.388182i \(-0.126897\pi\)
\(44\) 131.747 + 84.6687i 0.451401 + 0.290098i
\(45\) 38.9718 0.129102
\(46\) 163.853 − 147.717i 0.525191 − 0.473471i
\(47\) 51.3169 0.159263 0.0796313 − 0.996824i \(-0.474626\pi\)
0.0796313 + 0.996824i \(0.474626\pi\)
\(48\) −79.3965 − 51.0250i −0.238748 − 0.153434i
\(49\) 40.1684 + 279.377i 0.117109 + 0.814511i
\(50\) −20.7708 − 45.4816i −0.0587486 − 0.128641i
\(51\) 727.357 + 213.571i 1.99706 + 0.586391i
\(52\) −10.4592 + 22.9024i −0.0278928 + 0.0610767i
\(53\) 156.058 − 180.101i 0.404457 − 0.466768i −0.516583 − 0.856237i \(-0.672796\pi\)
0.921039 + 0.389469i \(0.127342\pi\)
\(54\) 32.2451 − 224.269i 0.0812592 − 0.565170i
\(55\) −187.830 + 55.1520i −0.460492 + 0.135213i
\(56\) −40.8330 − 47.1238i −0.0974381 − 0.112450i
\(57\) −406.683 + 261.360i −0.945027 + 0.607332i
\(58\) 168.877 − 108.530i 0.382321 − 0.245702i
\(59\) 163.088 + 188.213i 0.359868 + 0.415310i 0.906595 − 0.422001i \(-0.138672\pi\)
−0.546728 + 0.837311i \(0.684127\pi\)
\(60\) 113.195 − 33.2370i 0.243556 − 0.0715146i
\(61\) 129.077 − 897.750i 0.270928 − 1.88435i −0.167943 − 0.985797i \(-0.553712\pi\)
0.438871 − 0.898550i \(-0.355378\pi\)
\(62\) 152.144 − 175.583i 0.311649 − 0.359663i
\(63\) −25.2368 + 55.2610i −0.0504690 + 0.110512i
\(64\) −61.4076 − 18.0309i −0.119937 − 0.0352166i
\(65\) −13.0740 − 28.6280i −0.0249481 − 0.0546287i
\(66\) −65.7338 − 457.189i −0.122595 − 0.852667i
\(67\) −252.176 − 162.064i −0.459824 − 0.295511i 0.290147 − 0.956982i \(-0.406296\pi\)
−0.749970 + 0.661472i \(0.769932\pi\)
\(68\) 514.057 0.916743
\(69\) −645.721 − 79.9201i −1.12660 − 0.139438i
\(70\) 77.9421 0.133084
\(71\) 915.460 + 588.330i 1.53021 + 0.983408i 0.989872 + 0.141962i \(0.0453410\pi\)
0.540341 + 0.841446i \(0.318295\pi\)
\(72\) 8.87403 + 61.7203i 0.0145252 + 0.101025i
\(73\) 78.6079 + 172.127i 0.126032 + 0.275972i 0.962121 − 0.272621i \(-0.0878907\pi\)
−0.836089 + 0.548594i \(0.815163\pi\)
\(74\) −298.861 − 87.7535i −0.469485 − 0.137853i
\(75\) −61.2600 + 134.141i −0.0943159 + 0.206523i
\(76\) −214.676 + 247.750i −0.324014 + 0.373932i
\(77\) 43.4287 − 302.053i 0.0642747 − 0.447041i
\(78\) 71.2494 − 20.9207i 0.103428 − 0.0303693i
\(79\) 264.691 + 305.470i 0.376963 + 0.435038i 0.912251 − 0.409631i \(-0.134343\pi\)
−0.535289 + 0.844669i \(0.679797\pi\)
\(80\) 67.3003 − 43.2513i 0.0940550 − 0.0604455i
\(81\) −739.206 + 475.059i −1.01400 + 0.651658i
\(82\) −437.803 − 505.252i −0.589601 − 0.680436i
\(83\) 271.864 − 79.8266i 0.359530 − 0.105568i −0.0969787 − 0.995286i \(-0.530918\pi\)
0.456509 + 0.889719i \(0.349100\pi\)
\(84\) −26.1720 + 182.030i −0.0339952 + 0.236442i
\(85\) −420.795 + 485.623i −0.536960 + 0.619685i
\(86\) −118.575 + 259.644i −0.148678 + 0.325560i
\(87\) −568.079 − 166.803i −0.700052 − 0.205554i
\(88\) −130.115 − 284.911i −0.157617 − 0.345133i
\(89\) −20.2934 − 141.144i −0.0241696 − 0.168103i 0.974162 − 0.225853i \(-0.0725167\pi\)
−0.998331 + 0.0577491i \(0.981608\pi\)
\(90\) −65.5704 − 42.1395i −0.0767970 − 0.0493544i
\(91\) 49.0600 0.0565152
\(92\) −435.407 + 71.3638i −0.493416 + 0.0808716i
\(93\) −685.219 −0.764021
\(94\) −86.3411 − 55.4880i −0.0947383 − 0.0608846i
\(95\) −58.3170 − 405.604i −0.0629810 − 0.438043i
\(96\) 78.4127 + 171.700i 0.0833642 + 0.182542i
\(97\) −1696.01 − 497.995i −1.77530 − 0.521275i −0.780687 − 0.624922i \(-0.785131\pi\)
−0.994614 + 0.103647i \(0.966949\pi\)
\(98\) 234.502 − 513.488i 0.241717 − 0.529287i
\(99\) −199.841 + 230.628i −0.202876 + 0.234132i
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.b.71.5 ✓ 60
23.12 even 11 inner 230.4.g.b.81.5 yes 60
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.4.g.b.71.5 ✓ 60 1.1 even 1 trivial
230.4.g.b.81.5 yes 60 23.12 even 11 inner