Show commands: Magma / Pari/GP / SageMath

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 203.6
Character \(\chi\) \(=\) 230.203
Dual form 230.2.l.a.17.6

$q$-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+(-0.936950 - 0.349464i) q^{2} +(2.58843 + 1.41339i) q^{3} +(0.755750 + 0.654861i) q^{4} +(-2.20870 + 0.348772i) q^{5} +(-1.93130 - 2.22883i) q^{6} +(-2.57783 + 3.44357i) q^{7} +(-0.479249 - 0.877679i) q^{8} +(3.08036 + 4.79313i) q^{9} +(2.19132 + 0.445080i) q^{10} +(2.62000 + 1.19652i) q^{11} +(1.03063 + 2.76322i) q^{12} +(-0.876961 + 0.656485i) q^{13} +(3.61870 - 2.32560i) q^{14} +(-6.21001 - 2.21898i) q^{15} +(0.142315 + 0.989821i) q^{16} +(0.491452 - 6.87139i) q^{17} +(-1.21111 - 5.56740i) q^{18} +(0.238182 - 0.274877i) q^{19} +(-1.89762 - 1.18281i) q^{20} +(-11.5396 + 5.26996i) q^{21} +(-2.03667 - 2.03667i) q^{22} +(2.87942 + 3.83522i) q^{23} -2.94917i q^{24} +(4.75672 - 1.54067i) q^{25} +(1.05109 - 0.308627i) q^{26} +(0.567553 + 7.93543i) q^{27} +(-4.20325 + 0.914361i) q^{28} +(7.29106 - 6.31774i) q^{29} +(5.04301 + 4.24925i) q^{30} +(2.32578 + 0.682910i) q^{31} +(0.212565 - 0.977147i) q^{32} +(5.09054 + 6.80017i) q^{33} +(-2.86177 + 6.26640i) q^{34} +(4.49262 - 8.50490i) q^{35} +(-0.810854 + 5.63961i) q^{36} +(0.182822 + 0.0397705i) q^{37} +(-0.319224 + 0.174310i) q^{38} +(-3.19782 + 0.459776i) q^{39} +(1.36463 + 1.77138i) q^{40} +(-1.27399 - 0.818744i) q^{41} +(12.6537 - 0.905010i) q^{42} +(-4.01469 + 7.35235i) 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} +(-1.03063 + 2.76322i) q^{48} +(-3.24088 - 11.0374i) q^{49} +(-4.99521 - 0.218775i) q^{50} +(10.9840 - 17.0915i) q^{51} +(-1.09267 - 0.0781492i) q^{52} +(-3.64203 - 2.72639i) q^{53} +(2.24138 - 7.63344i) q^{54} +(-6.20411 - 1.72896i) q^{55} +(4.25777 + 0.612175i) q^{56} +(1.00502 - 0.374855i) q^{57} +(-9.03917 + 3.37144i) q^{58} +(7.97533 + 1.14668i) q^{59} +(-3.24009 - 5.74368i) q^{60} +(0.0664217 - 0.226212i) q^{61} +(-1.94048 - 1.45263i) q^{62} +(-24.4461 - 1.74842i) q^{63} +(-0.540641 + 0.841254i) q^{64} +(1.70798 - 1.75584i) q^{65} +(-2.39317 - 8.15038i) q^{66} +(-0.631976 + 1.69439i) 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} +(2.73057 - 5.00067i) q^{72} +(1.68736 - 0.120682i) q^{73} +(-0.157397 - 0.101153i) q^{74} +(14.4900 + 2.73518i) q^{75} +(0.360012 - 0.0517619i) q^{76} +(-10.8742 + 5.93776i) q^{77} +(3.15687 + 0.686735i) q^{78} +(2.16274 - 15.0422i) q^{79} +(-0.659553 - 2.13658i) q^{80} +(-2.64616 + 5.79427i) q^{81} +(0.907543 + 1.21234i) q^{82} +(2.16172 - 9.93729i) q^{83} +(-12.1721 - 3.57407i) q^{84} +(1.31108 + 15.3482i) q^{85} +(6.33094 - 5.48579i) q^{86} +(27.8018 - 6.04790i) q^{87} +(-0.205477 - 2.87295i) q^{88} +(-16.0974 + 4.72663i) q^{89} +(4.61674 + 11.8743i) q^{90} -4.71218i q^{91} +(-0.335416 + 4.78409i) q^{92} +(5.05488 + 5.05488i) q^{93} +(2.57979 - 1.17815i) q^{94} +(-0.430204 + 0.690192i) q^{95} +(1.93130 - 2.22883i) q^{96} +(2.00874 + 9.23402i) q^{97} +(-0.820642 + 11.4741i) 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 \(\left(\mathbb{Z}/230\mathbb{Z}\right)^\times\).

\(n\) \(47\) \(51\)
\(\chi(n)\) \(e\left(\frac{3}{4}\right)\) \(e\left(\frac{15}{22}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.936950 − 0.349464i −0.662524 − 0.247108i
\(3\) 2.58843 + 1.41339i 1.49443 + 0.816019i 0.998625 − 0.0524159i \(-0.0166921\pi\)
0.495803 + 0.868435i \(0.334874\pi\)
\(4\) 0.755750 + 0.654861i 0.377875 + 0.327430i
\(5\) −2.20870 + 0.348772i −0.987761 + 0.155976i
\(6\) −1.93130 − 2.22883i −0.788448 − 0.909918i
\(7\) −2.57783 + 3.44357i −0.974327 + 1.30155i −0.0209453 + 0.999781i \(0.506668\pi\)
−0.953382 + 0.301768i \(0.902423\pi\)
\(8\) −0.479249 − 0.877679i −0.169440 − 0.310306i
\(9\) 3.08036 + 4.79313i 1.02679 + 1.59771i
\(10\) 2.19132 + 0.445080i 0.692958 + 0.140747i
\(11\) 2.62000 + 1.19652i 0.789961 + 0.360763i 0.769189 − 0.639021i \(-0.220660\pi\)
0.0207716 + 0.999784i \(0.493388\pi\)
\(12\) 1.03063 + 2.76322i 0.297517 + 0.797674i
\(13\) −0.876961 + 0.656485i −0.243225 + 0.182076i −0.713935 − 0.700212i \(-0.753089\pi\)
0.470710 + 0.882288i \(0.343998\pi\)
\(14\) 3.61870 − 2.32560i 0.967138 − 0.621542i
\(15\) −6.21001 − 2.21898i −1.60342 − 0.572938i
\(16\) 0.142315 + 0.989821i 0.0355787 + 0.247455i
\(17\) 0.491452 − 6.87139i 0.119195 − 1.66656i −0.488634 − 0.872489i \(-0.662505\pi\)
0.607829 − 0.794068i \(-0.292041\pi\)
\(18\) −1.21111 − 5.56740i −0.285462 − 1.31225i
\(19\) 0.238182 − 0.274877i 0.0546427 − 0.0630611i −0.727770 − 0.685821i \(-0.759443\pi\)
0.782413 + 0.622760i \(0.213989\pi\)
\(20\) −1.89762 − 1.18281i −0.424321 − 0.264484i
\(21\) −11.5396 + 5.26996i −2.51815 + 1.15000i
\(22\) −2.03667 − 2.03667i −0.434220 − 0.434220i
\(23\) 2.87942 + 3.83522i 0.600401 + 0.799699i
\(24\) − 2.94917i − 0.601997i
\(25\) 4.75672 − 1.54067i 0.951343 − 0.308133i
\(26\) 1.05109 − 0.308627i 0.206135 − 0.0605267i
\(27\) 0.567553 + 7.93543i 0.109226 + 1.52718i
\(28\) −4.20325 + 0.914361i −0.794340 + 0.172798i
\(29\) 7.29106 − 6.31774i 1.35391 − 1.17317i 0.385825 − 0.922572i \(-0.373917\pi\)
0.968090 − 0.250602i \(-0.0806285\pi\)
\(30\) 5.04301 + 4.24925i 0.920723 + 0.775803i
\(31\) 2.32578 + 0.682910i 0.417722 + 0.122654i 0.483838 − 0.875158i \(-0.339242\pi\)
−0.0661158 + 0.997812i \(0.521061\pi\)
\(32\) 0.212565 − 0.977147i 0.0375766 − 0.172737i
\(33\) 5.09054 + 6.80017i 0.886150 + 1.18376i
\(34\) −2.86177 + 6.26640i −0.490790 + 1.07468i
\(35\) 4.49262 − 8.50490i 0.759392 − 1.43759i
\(36\) −0.810854 + 5.63961i −0.135142 + 0.939936i
\(37\) 0.182822 + 0.0397705i 0.0300557 + 0.00653823i 0.227568 − 0.973762i \(-0.426923\pi\)
−0.197512 + 0.980300i \(0.563286\pi\)
\(38\) −0.319224 + 0.174310i −0.0517850 + 0.0282768i
\(39\) −3.19782 + 0.459776i −0.512060 + 0.0736232i
\(40\) 1.36463 + 1.77138i 0.215767 + 0.280080i
\(41\) −1.27399 − 0.818744i −0.198964 − 0.127866i 0.437365 − 0.899284i \(-0.355912\pi\)
−0.636329 + 0.771418i \(0.719548\pi\)
\(42\) 12.6537 − 0.905010i 1.95251 − 0.139646i
\(43\) −4.01469 + 7.35235i −0.612234 + 1.12122i 0.369187 + 0.929355i \(0.379636\pi\)
−0.981421 + 0.191868i \(0.938545\pi\)
\(44\) 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 \(0.316319\pi\)
\(48\) −1.03063 + 2.76322i −0.148759 + 0.398837i
\(49\) −3.24088 − 11.0374i −0.462983 − 1.57677i
\(50\) −4.99521 − 0.218775i −0.706430 − 0.0309395i
\(51\) 10.9840 − 17.0915i 1.53807 − 2.39328i
\(52\) −1.09267 − 0.0781492i −0.151526 − 0.0108374i
\(53\) −3.64203 − 2.72639i −0.500271 − 0.374498i 0.319138 − 0.947708i \(-0.396607\pi\)
−0.819408 + 0.573211i \(0.805698\pi\)
\(54\) 2.24138 − 7.63344i 0.305013 − 1.03878i
\(55\) −6.20411 − 1.72896i −0.836562 − 0.233133i
\(56\) 4.25777 + 0.612175i 0.568969 + 0.0818054i
\(57\) 1.00502 − 0.374855i 0.133119 − 0.0496507i
\(58\) −9.03917 + 3.37144i −1.18690 + 0.442691i
\(59\) 7.97533 + 1.14668i 1.03830 + 0.149285i 0.640319 − 0.768109i \(-0.278802\pi\)
0.397981 + 0.917394i \(0.369711\pi\)
\(60\) −3.24009 − 5.74368i −0.418294 − 0.741506i
\(61\) 0.0664217 − 0.226212i 0.00850443 − 0.0289634i −0.955131 − 0.296182i \(-0.904286\pi\)
0.963636 + 0.267219i \(0.0861046\pi\)
\(62\) −1.94048 − 1.45263i −0.246442 − 0.184484i
\(63\) −24.4461 − 1.74842i −3.07992 − 0.220280i
\(64\) −0.540641 + 0.841254i −0.0675801 + 0.105157i
\(65\) 1.70798 − 1.75584i 0.211849 − 0.217785i
\(66\) −2.39317 − 8.15038i −0.294579 − 1.00324i
\(67\) −0.631976 + 1.69439i −0.0772082 + 0.207003i −0.969820 − 0.243824i \(-0.921598\pi\)
0.892611 + 0.450827i \(0.148871\pi\)
\(68\) 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 \(0.995685\pi\)
\(72\) 2.73057 − 5.00067i 0.321801 − 0.589335i
\(73\) 1.68736 − 0.120682i 0.197490 − 0.0141248i 0.0277565 − 0.999615i \(-0.491164\pi\)
0.169734 + 0.985490i \(0.445709\pi\)
\(74\) −0.157397 − 0.101153i −0.0182970 − 0.0117588i
\(75\) 14.4900 + 2.73518i 1.67316 + 0.315832i
\(76\) 0.360012 − 0.0517619i 0.0412962 − 0.00593750i
\(77\) −10.8742 + 5.93776i −1.23923 + 0.676671i
\(78\) 3.15687 + 0.686735i 0.357445 + 0.0777574i
\(79\) 2.16274 − 15.0422i 0.243327 − 1.69237i −0.391867 − 0.920022i \(-0.628171\pi\)
0.635194 − 0.772353i \(-0.280920\pi\)
\(80\) −0.659553 − 2.13658i −0.0737403 − 0.238877i
\(81\) −2.64616 + 5.79427i −0.294017 + 0.643808i
\(82\) 0.907543 + 1.21234i 0.100221 + 0.133880i
\(83\) 2.16172 − 9.93729i 0.237280 − 1.09076i −0.692148 − 0.721756i \(-0.743335\pi\)
0.929428 − 0.369003i \(-0.120301\pi\)
\(84\) −12.1721 − 3.57407i −1.32809 − 0.389963i
\(85\) 1.31108 + 15.3482i 0.142207 + 1.66475i
\(86\) 6.33094 − 5.48579i 0.682683 − 0.591548i
\(87\) 27.8018 − 6.04790i 2.98066 − 0.648403i
\(88\) −0.205477 − 2.87295i −0.0219040 − 0.306258i
\(89\) −16.0974 + 4.72663i −1.70633 + 0.501022i −0.982071 − 0.188511i \(-0.939634\pi\)
−0.724254 + 0.689533i \(0.757816\pi\)
\(90\) 4.61674 + 11.8743i 0.486647 + 1.25166i
\(91\) − 4.71218i − 0.493971i
\(92\) −0.335416 + 4.78409i −0.0349695 + 0.498776i
\(93\) 5.05488 + 5.05488i 0.524167 + 0.524167i
\(94\) 2.57979 − 1.17815i 0.266085 − 0.121517i
\(95\) −0.430204 + 0.690192i −0.0441380 + 0.0708122i
\(96\) 1.93130 − 2.22883i 0.197112 − 0.227479i
\(97\) 2.00874 + 9.23402i 0.203956 + 0.937573i 0.959332 + 0.282280i \(0.0910909\pi\)
−0.755376 + 0.655292i \(0.772546\pi\)
\(98\) −0.820642 + 11.4741i −0.0828974 + 1.15906i
\(99\) 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.203.6 yes 240
5.2 odd 4 inner 230.2.l.a.157.12 yes 240
23.17 odd 22 inner 230.2.l.a.63.12 yes 240
115.17 even 44 inner 230.2.l.a.17.6 ✓ 240
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.2.l.a.17.6 ✓ 240 115.17 even 44 inner
230.2.l.a.63.12 yes 240 23.17 odd 22 inner
230.2.l.a.157.12 yes 240 5.2 odd 4 inner
230.2.l.a.203.6 yes 240 1.1 even 1 trivial