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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [230,2,Mod(7,230)] 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("230.7"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(230, base_ring=CyclotomicField(44)) chi = DirichletCharacter(H, H._module([11, 38])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 230 = 2 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 230.l (of order \(44\), degree \(20\), 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: \(1.83655924649\)
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 63.12
Character \(\chi\) \(=\) 230.63
Dual form 230.2.l.a.157.12

$q$-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+(0.349464 + 0.936950i) q^{2} +(1.41339 + 2.58843i) q^{3} +(-0.755750 + 0.654861i) q^{4} +(0.600273 - 2.15399i) q^{5} +(-1.93130 + 2.22883i) q^{6} +(-3.44357 + 2.57783i) q^{7} +(-0.877679 - 0.479249i) q^{8} +(-3.08036 + 4.79313i) q^{9} +(2.22795 - 0.190317i) q^{10} +(2.62000 - 1.19652i) q^{11} +(-2.76322 - 1.03063i) q^{12} +(0.656485 - 0.876961i) q^{13} +(-3.61870 - 2.32560i) q^{14} +(6.42386 - 1.49066i) q^{15} +(0.142315 - 0.989821i) q^{16} +(6.87139 - 0.491452i) q^{17} +(-5.56740 - 1.21111i) q^{18} +(-0.238182 - 0.274877i) q^{19} +(0.956907 + 2.02097i) q^{20} +(-11.5396 - 5.26996i) q^{21} +(2.03667 + 2.03667i) q^{22} +(3.83522 + 2.87942i) q^{23} -2.94917i q^{24} +(-4.27934 - 2.58596i) q^{25} +(1.05109 + 0.308627i) q^{26} +(-7.93543 - 0.567553i) q^{27} +(0.914361 - 4.20325i) q^{28} +(-7.29106 - 6.31774i) q^{29} +(3.64158 + 5.49790i) q^{30} +(2.32578 - 0.682910i) q^{31} +(0.977147 - 0.212565i) q^{32} +(6.80017 + 5.09054i) q^{33} +(2.86177 + 6.26640i) q^{34} +(3.48553 + 8.96482i) q^{35} +(-0.810854 - 5.63961i) q^{36} +(-0.0397705 - 0.182822i) q^{37} +(0.174310 - 0.319224i) q^{38} +(3.19782 + 0.459776i) q^{39} +(-1.55914 + 1.60283i) q^{40} +(-1.27399 + 0.818744i) q^{41} +(0.905010 - 12.6537i) q^{42} +(7.35235 - 4.01469i) q^{43} +(-1.19652 + 2.62000i) q^{44} +(8.47530 + 9.51225i) q^{45} +(-1.35760 + 4.59967i) q^{46} +(-2.00541 + 2.00541i) q^{47} +(2.76322 - 1.03063i) q^{48} +(3.24088 - 11.0374i) q^{49} +(0.927441 - 4.91323i) q^{50} +(10.9840 + 17.0915i) q^{51} +(0.0781492 + 1.09267i) q^{52} +(-2.72639 - 3.64203i) q^{53} +(-2.24138 - 7.63344i) q^{54} +(-1.00456 - 6.36170i) q^{55} +(4.25777 - 0.612175i) q^{56} +(0.374855 - 1.00502i) q^{57} +(3.37144 - 9.03917i) q^{58} +(-7.97533 + 1.14668i) q^{59} +(-3.87865 + 5.33330i) q^{60} +(0.0664217 + 0.226212i) q^{61} +(1.45263 + 1.94048i) q^{62} +(-1.74842 - 24.4461i) q^{63} +(0.540641 + 0.841254i) q^{64} +(-1.49490 - 1.94048i) q^{65} +(-2.39317 + 8.15038i) q^{66} +(-1.69439 + 0.631976i) q^{67} +(-4.87122 + 4.87122i) q^{68} +(-2.03251 + 13.9969i) q^{69} +(-7.18152 + 6.39865i) q^{70} +(-2.82162 + 6.17848i) q^{71} +(5.00067 - 2.73057i) q^{72} +(-0.120682 + 1.68736i) q^{73} +(0.157397 - 0.101153i) q^{74} +(0.645205 - 14.7317i) q^{75} +(0.360012 + 0.0517619i) q^{76} +(-5.93776 + 10.8742i) q^{77} +(0.686735 + 3.15687i) q^{78} +(-2.16274 - 15.0422i) q^{79} +(-2.04664 - 0.900708i) q^{80} +(-2.64616 - 5.79427i) q^{81} +(-1.21234 - 0.907543i) q^{82} +(-9.93729 + 2.16172i) q^{83} +(12.1721 - 3.57407i) q^{84} +(3.06613 - 15.0959i) q^{85} +(6.33094 + 5.48579i) q^{86} +(6.04790 - 27.8018i) q^{87} +(-2.87295 - 0.205477i) q^{88} +(16.0974 + 4.72663i) q^{89} +(-5.95069 + 11.2651i) q^{90} +4.71218i q^{91} +(-4.78409 + 0.335416i) q^{92} +(5.05488 + 5.05488i) q^{93} +(-2.57979 - 1.17815i) q^{94} +(-0.735056 + 0.348041i) q^{95} +(1.93130 + 2.22883i) q^{96} +(-9.23402 - 2.00874i) q^{97} +(11.4741 - 0.820642i) q^{98} +(-2.33549 + 16.2437i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 240 q - 8 q^{3} - 8 q^{6} - 8 q^{12} + 16 q^{13} + 24 q^{16} - 72 q^{18} - 80 q^{23} - 8 q^{26} + 16 q^{27} - 44 q^{28} + 24 q^{31} - 44 q^{33} - 8 q^{35} - 32 q^{36} - 88 q^{37} - 24 q^{41} - 8 q^{46}+ \cdots + 24 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(47\) \(51\)
\(\chi(n)\) \(e\left(\frac{3}{4}\right)\) \(e\left(\frac{7}{22}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.349464 + 0.936950i 0.247108 + 0.662524i
\(3\) 1.41339 + 2.58843i 0.816019 + 1.49443i 0.868435 + 0.495803i \(0.165126\pi\)
−0.0524159 + 0.998625i \(0.516692\pi\)
\(4\) −0.755750 + 0.654861i −0.377875 + 0.327430i
\(5\) 0.600273 − 2.15399i 0.268450 − 0.963294i
\(6\) −1.93130 + 2.22883i −0.788448 + 0.909918i
\(7\) −3.44357 + 2.57783i −1.30155 + 0.974327i −0.301768 + 0.953382i \(0.597577\pi\)
−0.999781 + 0.0209453i \(0.993332\pi\)
\(8\) −0.877679 − 0.479249i −0.310306 − 0.169440i
\(9\) −3.08036 + 4.79313i −1.02679 + 1.59771i
\(10\) 2.22795 − 0.190317i 0.704541 − 0.0601834i
\(11\) 2.62000 − 1.19652i 0.789961 − 0.360763i 0.0207716 − 0.999784i \(-0.493388\pi\)
0.769189 + 0.639021i \(0.220660\pi\)
\(12\) −2.76322 − 1.03063i −0.797674 − 0.297517i
\(13\) 0.656485 − 0.876961i 0.182076 − 0.243225i −0.700212 − 0.713935i \(-0.746911\pi\)
0.882288 + 0.470710i \(0.156002\pi\)
\(14\) −3.61870 − 2.32560i −0.967138 − 0.621542i
\(15\) 6.42386 − 1.49066i 1.65863 − 0.384887i
\(16\) 0.142315 − 0.989821i 0.0355787 − 0.247455i
\(17\) 6.87139 − 0.491452i 1.66656 − 0.119195i 0.794068 − 0.607829i \(-0.207959\pi\)
0.872489 + 0.488634i \(0.162505\pi\)
\(18\) −5.56740 − 1.21111i −1.31225 − 0.285462i
\(19\) −0.238182 − 0.274877i −0.0546427 − 0.0630611i 0.727770 − 0.685821i \(-0.240557\pi\)
−0.782413 + 0.622760i \(0.786011\pi\)
\(20\) 0.956907 + 2.02097i 0.213971 + 0.451903i
\(21\) −11.5396 − 5.26996i −2.51815 − 1.15000i
\(22\) 2.03667 + 2.03667i 0.434220 + 0.434220i
\(23\) 3.83522 + 2.87942i 0.799699 + 0.600401i
\(24\) − 2.94917i − 0.601997i
\(25\) −4.27934 − 2.58596i −0.855869 − 0.517193i
\(26\) 1.05109 + 0.308627i 0.206135 + 0.0605267i
\(27\) −7.93543 − 0.567553i −1.52718 − 0.109226i
\(28\) 0.914361 − 4.20325i 0.172798 − 0.794340i
\(29\) −7.29106 − 6.31774i −1.35391 − 1.17317i −0.968090 − 0.250602i \(-0.919371\pi\)
−0.385825 − 0.922572i \(-0.626083\pi\)
\(30\) 3.64158 + 5.49790i 0.664859 + 1.00377i
\(31\) 2.32578 − 0.682910i 0.417722 − 0.122654i −0.0661158 − 0.997812i \(-0.521061\pi\)
0.483838 + 0.875158i \(0.339242\pi\)
\(32\) 0.977147 − 0.212565i 0.172737 − 0.0375766i
\(33\) 6.80017 + 5.09054i 1.18376 + 0.886150i
\(34\) 2.86177 + 6.26640i 0.490790 + 1.07468i
\(35\) 3.48553 + 8.96482i 0.589162 + 1.51533i
\(36\) −0.810854 − 5.63961i −0.135142 − 0.939936i
\(37\) −0.0397705 − 0.182822i −0.00653823 − 0.0300557i 0.973762 − 0.227568i \(-0.0730774\pi\)
−0.980300 + 0.197512i \(0.936714\pi\)
\(38\) 0.174310 − 0.319224i 0.0282768 − 0.0517850i
\(39\) 3.19782 + 0.459776i 0.512060 + 0.0736232i
\(40\) −1.55914 + 1.60283i −0.246522 + 0.253430i
\(41\) −1.27399 + 0.818744i −0.198964 + 0.127866i −0.636329 − 0.771418i \(-0.719548\pi\)
0.437365 + 0.899284i \(0.355912\pi\)
\(42\) 0.905010 − 12.6537i 0.139646 − 1.95251i
\(43\) 7.35235 − 4.01469i 1.12122 − 0.612234i 0.191868 − 0.981421i \(-0.438545\pi\)
0.929355 + 0.369187i \(0.120364\pi\)
\(44\) −1.19652 + 2.62000i −0.180381 + 0.394980i
\(45\) 8.47530 + 9.51225i 1.26342 + 1.41800i
\(46\) −1.35760 + 4.59967i −0.200167 + 0.678184i
\(47\) −2.00541 + 2.00541i −0.292520 + 0.292520i −0.838075 − 0.545555i \(-0.816319\pi\)
0.545555 + 0.838075i \(0.316319\pi\)
\(48\) 2.76322 − 1.03063i 0.398837 − 0.148759i
\(49\) 3.24088 − 11.0374i 0.462983 − 1.57677i
\(50\) 0.927441 − 4.91323i 0.131160 − 0.694836i
\(51\) 10.9840 + 17.0915i 1.53807 + 2.39328i
\(52\) 0.0781492 + 1.09267i 0.0108374 + 0.151526i
\(53\) −2.72639 − 3.64203i −0.374498 − 0.500271i 0.573211 − 0.819408i \(-0.305698\pi\)
−0.947708 + 0.319138i \(0.896607\pi\)
\(54\) −2.24138 − 7.63344i −0.305013 − 1.03878i
\(55\) −1.00456 − 6.36170i −0.135455 − 0.857811i
\(56\) 4.25777 − 0.612175i 0.568969 − 0.0818054i
\(57\) 0.374855 − 1.00502i 0.0496507 − 0.133119i
\(58\) 3.37144 − 9.03917i 0.442691 − 1.18690i
\(59\) −7.97533 + 1.14668i −1.03830 + 0.149285i −0.640319 − 0.768109i \(-0.721198\pi\)
−0.397981 + 0.917394i \(0.630289\pi\)
\(60\) −3.87865 + 5.33330i −0.500732 + 0.688526i
\(61\) 0.0664217 + 0.226212i 0.00850443 + 0.0289634i 0.963636 − 0.267219i \(-0.0861046\pi\)
−0.955131 + 0.296182i \(0.904286\pi\)
\(62\) 1.45263 + 1.94048i 0.184484 + 0.246442i
\(63\) −1.74842 − 24.4461i −0.220280 − 3.07992i
\(64\) 0.540641 + 0.841254i 0.0675801 + 0.105157i
\(65\) −1.49490 − 1.94048i −0.185419 − 0.240687i
\(66\) −2.39317 + 8.15038i −0.294579 + 1.00324i
\(67\) −1.69439 + 0.631976i −0.207003 + 0.0772082i −0.450827 − 0.892611i \(-0.648871\pi\)
0.243824 + 0.969820i \(0.421598\pi\)
\(68\) −4.87122 + 4.87122i −0.590722 + 0.590722i
\(69\) −2.03251 + 13.9969i −0.244686 + 1.68503i
\(70\) −7.18152 + 6.39865i −0.858356 + 0.764785i
\(71\) −2.82162 + 6.17848i −0.334864 + 0.733250i −0.999908 − 0.0135540i \(-0.995685\pi\)
0.665044 + 0.746804i \(0.268413\pi\)
\(72\) 5.00067 − 2.73057i 0.589335 − 0.321801i
\(73\) −0.120682 + 1.68736i −0.0141248 + 0.197490i 0.985490 + 0.169734i \(0.0542908\pi\)
−0.999615 + 0.0277565i \(0.991164\pi\)
\(74\) 0.157397 − 0.101153i 0.0182970 − 0.0117588i
\(75\) 0.645205 − 14.7317i 0.0745018 − 1.70107i
\(76\) 0.360012 + 0.0517619i 0.0412962 + 0.00593750i
\(77\) −5.93776 + 10.8742i −0.676671 + 1.23923i
\(78\) 0.686735 + 3.15687i 0.0777574 + 0.357445i
\(79\) −2.16274 − 15.0422i −0.243327 − 1.69237i −0.635194 − 0.772353i \(-0.719080\pi\)
0.391867 − 0.920022i \(-0.371829\pi\)
\(80\) −2.04664 − 0.900708i −0.228821 − 0.100702i
\(81\) −2.64616 − 5.79427i −0.294017 − 0.643808i
\(82\) −1.21234 − 0.907543i −0.133880 − 0.100221i
\(83\) −9.93729 + 2.16172i −1.09076 + 0.237280i −0.721756 − 0.692148i \(-0.756665\pi\)
−0.369003 + 0.929428i \(0.620301\pi\)
\(84\) 12.1721 − 3.57407i 1.32809 − 0.389963i
\(85\) 3.06613 − 15.0959i 0.332568 − 1.63738i
\(86\) 6.33094 + 5.48579i 0.682683 + 0.591548i
\(87\) 6.04790 − 27.8018i 0.648403 − 2.98066i
\(88\) −2.87295 − 0.205477i −0.306258 − 0.0219040i
\(89\) 16.0974 + 4.72663i 1.70633 + 0.501022i 0.982071 − 0.188511i \(-0.0603661\pi\)
0.724254 + 0.689533i \(0.242184\pi\)
\(90\) −5.95069 + 11.2651i −0.627257 + 1.18745i
\(91\) 4.71218i 0.493971i
\(92\) −4.78409 + 0.335416i −0.498776 + 0.0349695i
\(93\) 5.05488 + 5.05488i 0.524167 + 0.524167i
\(94\) −2.57979 − 1.17815i −0.266085 − 0.121517i
\(95\) −0.735056 + 0.348041i −0.0754152 + 0.0357082i
\(96\) 1.93130 + 2.22883i 0.197112 + 0.227479i
\(97\) −9.23402 − 2.00874i −0.937573 − 0.203956i −0.282280 − 0.959332i \(-0.591091\pi\)
−0.655292 + 0.755376i \(0.727454\pi\)
\(98\) 11.4741 − 0.820642i 1.15906 − 0.0828974i
\(99\) −2.33549 + 16.2437i −0.234726 + 1.63256i
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 230.2.l.a.63.12 yes 240
5.2 odd 4 inner 230.2.l.a.17.6 ✓ 240
23.19 odd 22 inner 230.2.l.a.203.6 yes 240
115.42 even 44 inner 230.2.l.a.157.12 yes 240
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.2.l.a.17.6 ✓ 240 5.2 odd 4 inner
230.2.l.a.63.12 yes 240 1.1 even 1 trivial
230.2.l.a.157.12 yes 240 115.42 even 44 inner
230.2.l.a.203.6 yes 240 23.19 odd 22 inner