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Properties

Label 230.2.l.a.17.6
Level $230$
Weight $2$
Character 230.17
Analytic conductor $1.837$
Analytic rank $0$
Dimension $240$
Inner twists $4$

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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 17.6
Character \(\chi\) \(=\) 230.17
Dual form 230.2.l.a.203.6

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

\(n\) \(47\) \(51\)
\(\chi(n)\) \(e\left(\frac{1}{4}\right)\) \(e\left(\frac{7}{22}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.936950 + 0.349464i −0.662524 + 0.247108i
\(3\) 2.58843 − 1.41339i 1.49443 − 0.816019i 0.495803 − 0.868435i \(-0.334874\pi\)
0.998625 + 0.0524159i \(0.0166921\pi\)
\(4\) 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 −0.953382 − 0.301768i \(-0.902423\pi\)
−0.0209453 − 0.999781i \(-0.506668\pi\)
\(8\) −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 0.0207716 − 0.999784i \(-0.493388\pi\)
0.769189 + 0.639021i \(0.220660\pi\)
\(12\) 1.03063 − 2.76322i 0.297517 − 0.797674i
\(13\) −0.876961 − 0.656485i −0.243225 − 0.182076i 0.470710 − 0.882288i \(-0.343998\pi\)
−0.713935 + 0.700212i \(0.753089\pi\)
\(14\) 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 0.607829 + 0.794068i \(0.292041\pi\)
−0.488634 + 0.872489i \(0.662505\pi\)
\(18\) −1.21111 + 5.56740i −0.285462 + 1.31225i
\(19\) 0.238182 + 0.274877i 0.0546427 + 0.0630611i 0.782413 − 0.622760i \(-0.213989\pi\)
−0.727770 + 0.685821i \(0.759443\pi\)
\(20\) −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 0.968090 + 0.250602i \(0.0806285\pi\)
0.385825 + 0.922572i \(0.373917\pi\)
\(30\) 5.04301 − 4.24925i 0.920723 − 0.775803i
\(31\) 2.32578 − 0.682910i 0.417722 − 0.122654i −0.0661158 − 0.997812i \(-0.521061\pi\)
0.483838 + 0.875158i \(0.339242\pi\)
\(32\) 0.212565 + 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 −0.197512 − 0.980300i \(-0.563286\pi\)
0.227568 + 0.973762i \(0.426923\pi\)
\(38\) −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 −0.636329 − 0.771418i \(-0.719548\pi\)
0.437365 + 0.899284i \(0.355912\pi\)
\(42\) 12.6537 + 0.905010i 1.95251 + 0.139646i
\(43\) −4.01469 − 7.35235i −0.612234 − 1.12122i −0.981421 − 0.191868i \(-0.938545\pi\)
0.369187 − 0.929355i \(-0.379636\pi\)
\(44\) 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 \(0.816319\pi\)
\(48\) −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 −0.819408 − 0.573211i \(-0.805698\pi\)
0.319138 + 0.947708i \(0.396607\pi\)
\(54\) 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 0.397981 − 0.917394i \(-0.369711\pi\)
0.640319 + 0.768109i \(0.278802\pi\)
\(60\) −3.24009 + 5.74368i −0.418294 + 0.741506i
\(61\) 0.0664217 + 0.226212i 0.00850443 + 0.0289634i 0.963636 − 0.267219i \(-0.0861046\pi\)
−0.955131 + 0.296182i \(0.904286\pi\)
\(62\) −1.94048 + 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 0.892611 − 0.450827i \(-0.148871\pi\)
−0.969820 + 0.243824i \(0.921598\pi\)
\(68\) 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 \(0.268413\pi\)
\(72\) 2.73057 + 5.00067i 0.321801 + 0.589335i
\(73\) 1.68736 + 0.120682i 0.197490 + 0.0141248i 0.169734 − 0.985490i \(-0.445709\pi\)
0.0277565 + 0.999615i \(0.491164\pi\)
\(74\) −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 0.635194 + 0.772353i \(0.280920\pi\)
−0.391867 + 0.920022i \(0.628171\pi\)
\(80\) −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 0.929428 + 0.369003i \(0.120301\pi\)
−0.692148 + 0.721756i \(0.743335\pi\)
\(84\) −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 −0.724254 − 0.689533i \(-0.757816\pi\)
−0.982071 + 0.188511i \(0.939634\pi\)
\(90\) 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 −0.755376 − 0.655292i \(-0.772546\pi\)
0.959332 − 0.282280i \(-0.0910909\pi\)
\(98\) −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.17.6 ✓ 240
5.3 odd 4 inner 230.2.l.a.63.12 yes 240
23.19 odd 22 inner 230.2.l.a.157.12 yes 240
115.88 even 44 inner 230.2.l.a.203.6 yes 240
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.2.l.a.17.6 ✓ 240 1.1 even 1 trivial
230.2.l.a.63.12 yes 240 5.3 odd 4 inner
230.2.l.a.157.12 yes 240 23.19 odd 22 inner
230.2.l.a.203.6 yes 240 115.88 even 44 inner