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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 157.12
Character \(\chi\) \(=\) 230.157
Dual form 230.2.l.a.63.12

$q$-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 \(\left(\mathbb{Z}/230\mathbb{Z}\right)^\times\).

\(n\) \(47\) \(51\)
\(\chi(n)\) \(e\left(\frac{1}{4}\right)\) \(e\left(\frac{15}{22}\right)\)

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 −0.0524159 − 0.998625i \(-0.516692\pi\)
0.868435 − 0.495803i \(-0.165126\pi\)
\(4\) −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 −0.999781 − 0.0209453i \(-0.993332\pi\)
−0.301768 − 0.953382i \(-0.597577\pi\)
\(8\) −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 0.769189 − 0.639021i \(-0.220660\pi\)
0.0207716 + 0.999784i \(0.493388\pi\)
\(12\) −2.76322 + 1.03063i −0.797674 + 0.297517i
\(13\) 0.656485 + 0.876961i 0.182076 + 0.243225i 0.882288 − 0.470710i \(-0.156002\pi\)
−0.700212 + 0.713935i \(0.746911\pi\)
\(14\) −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 0.872489 − 0.488634i \(-0.162505\pi\)
0.794068 + 0.607829i \(0.207959\pi\)
\(18\) −5.56740 + 1.21111i −1.31225 + 0.285462i
\(19\) −0.238182 + 0.274877i −0.0546427 + 0.0630611i −0.782413 − 0.622760i \(-0.786011\pi\)
0.727770 + 0.685821i \(0.240557\pi\)
\(20\) 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 −0.385825 + 0.922572i \(0.626083\pi\)
−0.968090 + 0.250602i \(0.919371\pi\)
\(30\) 3.64158 − 5.49790i 0.664859 − 1.00377i
\(31\) 2.32578 + 0.682910i 0.417722 + 0.122654i 0.483838 − 0.875158i \(-0.339242\pi\)
−0.0661158 + 0.997812i \(0.521061\pi\)
\(32\) 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 −0.980300 − 0.197512i \(-0.936714\pi\)
0.973762 + 0.227568i \(0.0730774\pi\)
\(38\) 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 0.437365 − 0.899284i \(-0.355912\pi\)
−0.636329 + 0.771418i \(0.719548\pi\)
\(42\) 0.905010 + 12.6537i 0.139646 + 1.95251i
\(43\) 7.35235 + 4.01469i 1.12122 + 0.612234i 0.929355 − 0.369187i \(-0.120364\pi\)
0.191868 + 0.981421i \(0.438545\pi\)
\(44\) −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 0.545555 − 0.838075i \(-0.316319\pi\)
−0.838075 + 0.545555i \(0.816319\pi\)
\(48\) 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 −0.947708 − 0.319138i \(-0.896607\pi\)
0.573211 + 0.819408i \(0.305698\pi\)
\(54\) −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 −0.397981 − 0.917394i \(-0.630289\pi\)
−0.640319 + 0.768109i \(0.721198\pi\)
\(60\) −3.87865 − 5.33330i −0.500732 − 0.688526i
\(61\) 0.0664217 − 0.226212i 0.00850443 − 0.0289634i −0.955131 − 0.296182i \(-0.904286\pi\)
0.963636 + 0.267219i \(0.0861046\pi\)
\(62\) 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 0.243824 − 0.969820i \(-0.421598\pi\)
−0.450827 + 0.892611i \(0.648871\pi\)
\(68\) −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 0.665044 − 0.746804i \(-0.268413\pi\)
−0.999908 + 0.0135540i \(0.995685\pi\)
\(72\) 5.00067 + 2.73057i 0.589335 + 0.321801i
\(73\) −0.120682 − 1.68736i −0.0141248 − 0.197490i −0.999615 − 0.0277565i \(-0.991164\pi\)
0.985490 − 0.169734i \(-0.0542908\pi\)
\(74\) 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 0.391867 + 0.920022i \(0.371829\pi\)
−0.635194 + 0.772353i \(0.719080\pi\)
\(80\) −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 −0.369003 − 0.929428i \(-0.620301\pi\)
−0.721756 + 0.692148i \(0.756665\pi\)
\(84\) 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 0.724254 − 0.689533i \(-0.242184\pi\)
0.982071 + 0.188511i \(0.0603661\pi\)
\(90\) −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 −0.655292 − 0.755376i \(-0.727454\pi\)
−0.282280 + 0.959332i \(0.591091\pi\)
\(98\) 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.157.12 yes 240
5.3 odd 4 inner 230.2.l.a.203.6 yes 240
23.17 odd 22 inner 230.2.l.a.17.6 ✓ 240
115.63 even 44 inner 230.2.l.a.63.12 yes 240
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.2.l.a.17.6 ✓ 240 23.17 odd 22 inner
230.2.l.a.63.12 yes 240 115.63 even 44 inner
230.2.l.a.157.12 yes 240 1.1 even 1 trivial
230.2.l.a.203.6 yes 240 5.3 odd 4 inner