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Properties

Label 555.2.bp.a.4.12
Level $555$
Weight $2$
Character 555.4
Analytic conductor $4.432$
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] = [555,2,Mod(4,555)] 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("555.4"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(555, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([0, 9, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 555 = 3 \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 555.bp (of order \(18\), degree \(6\), 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: \(4.43169731218\)
Analytic rank: \(0\)
Dimension: \(240\)
Relative dimension: \(40\) over \(\Q(\zeta_{18})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label 4.12
Character \(\chi\) \(=\) 555.4
Dual form 555.2.bp.a.139.12

$q$-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+(-0.208975 + 1.18516i) q^{2} +(0.984808 - 0.173648i) q^{3} +(0.518457 + 0.188703i) q^{4} +(2.23588 - 0.0286419i) q^{5} +1.20344i q^{6} +(1.21615 + 1.44935i) q^{7} +(-1.53543 + 2.65944i) q^{8} +(0.939693 - 0.342020i) q^{9} +(-0.433299 + 2.65586i) q^{10} +(-1.53393 + 2.65684i) q^{11} +(0.543348 + 0.0958070i) q^{12} +(-3.86806 - 1.40786i) q^{13} +(-1.97186 + 1.13845i) q^{14} +(2.19694 - 0.416464i) q^{15} +(-1.98569 - 1.66619i) q^{16} +(-1.72469 + 0.627737i) q^{17} +(0.208975 + 1.18516i) q^{18} +(5.06455 - 0.893017i) q^{19} +(1.16461 + 0.407068i) q^{20} +(1.44935 + 1.21615i) q^{21} +(-2.82822 - 2.37316i) q^{22} +(-4.39279 - 7.60853i) q^{23} +(-1.05029 + 2.88566i) q^{24} +(4.99836 - 0.128080i) q^{25} +(2.47686 - 4.29005i) q^{26} +(0.866025 - 0.500000i) q^{27} +(0.357025 + 0.980918i) q^{28} +(7.41057 + 4.27850i) q^{29} +(0.0344688 + 2.69075i) q^{30} -2.12029i q^{31} +(-2.31517 + 1.94265i) q^{32} +(-1.04927 + 2.88284i) q^{33} +(-0.383549 - 2.17521i) q^{34} +(2.76068 + 3.20575i) q^{35} +0.551730 q^{36} +(-3.35395 - 5.07455i) q^{37} +6.18891i q^{38} +(-4.05376 - 0.714788i) q^{39} +(-3.35687 + 5.99018i) q^{40} +(-3.78829 - 1.37882i) q^{41} +(-1.74421 + 1.46357i) q^{42} +1.42178 q^{43} +(-1.29663 + 1.08800i) q^{44} +(2.09125 - 0.791632i) q^{45} +(9.93529 - 3.61615i) q^{46} +(9.38600 - 5.41901i) q^{47} +(-2.24485 - 1.29607i) q^{48} +(0.593939 - 3.36839i) q^{49} +(-0.892739 + 5.95061i) q^{50} +(-1.58949 + 0.917690i) q^{51} +(-1.73975 - 1.45983i) q^{52} +(-6.74027 + 8.03274i) q^{53} +(0.411601 + 1.13086i) q^{54} +(-3.35358 + 5.98432i) q^{55} +(-5.72178 + 1.00890i) q^{56} +(4.83254 - 1.75890i) q^{57} +(-6.61932 + 7.88860i) q^{58} +(-7.98485 + 9.51597i) q^{59} +(1.21761 + 0.198651i) q^{60} +(4.52144 - 12.4226i) q^{61} +(2.51288 + 0.443088i) q^{62} +(1.63852 + 0.945997i) q^{63} +(-4.41067 - 7.63951i) q^{64} +(-8.68885 - 3.03702i) q^{65} +(-3.19735 - 1.84599i) q^{66} +(5.97626 + 7.12223i) q^{67} -1.01263 q^{68} +(-5.64726 - 6.73014i) q^{69} +(-4.37624 + 2.60192i) q^{70} +(-1.54417 - 8.75743i) q^{71} +(-0.533249 + 3.02420i) q^{72} -3.23072i q^{73} +(6.71503 - 2.91451i) q^{74} +(4.90018 - 0.994090i) q^{75} +(2.79427 + 0.492705i) q^{76} +(-5.71618 + 1.00792i) q^{77} +(1.69427 - 4.65498i) q^{78} +(0.439615 + 0.523913i) q^{79} +(-4.48749 - 3.66854i) q^{80} +(0.766044 - 0.642788i) q^{81} +(2.42578 - 4.20158i) q^{82} +(4.97998 + 13.6824i) q^{83} +(0.521935 + 0.904019i) q^{84} +(-3.83823 + 1.45295i) q^{85} +(-0.297117 + 1.68504i) q^{86} +(8.04094 + 2.92666i) q^{87} +(-4.71046 - 8.15876i) q^{88} +(-9.23895 + 11.0106i) q^{89} +(0.501190 + 2.64389i) q^{90} +(-2.66366 - 7.31834i) q^{91} +(-0.841719 - 4.77362i) q^{92} +(-0.368184 - 2.08808i) q^{93} +(4.46094 + 12.2563i) q^{94} +(11.2982 - 2.14174i) q^{95} +(-1.94265 + 2.31517i) q^{96} +(-3.04716 - 5.27783i) q^{97} +(3.86796 + 1.40782i) q^{98} +(-0.532727 + 3.02124i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 240 q + 12 q^{5} + 36 q^{14} + 12 q^{19} - 30 q^{20} + 12 q^{21} - 48 q^{25} - 36 q^{26} + 24 q^{30} + 48 q^{34} + 12 q^{35} - 264 q^{36} + 36 q^{39} + 66 q^{40} - 144 q^{41} + 192 q^{44} + 24 q^{46} - 12 q^{49}+ \cdots + 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/555\mathbb{Z}\right)^\times\).

\(n\) \(76\) \(112\) \(371\)
\(\chi(n)\) \(e\left(\frac{1}{18}\right)\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.208975 + 1.18516i −0.147768 + 0.838033i 0.817335 + 0.576163i \(0.195450\pi\)
−0.965103 + 0.261871i \(0.915661\pi\)
\(3\) 0.984808 − 0.173648i 0.568579 − 0.100256i
\(4\) 0.518457 + 0.188703i 0.259228 + 0.0943514i
\(5\) 2.23588 − 0.0286419i 0.999918 − 0.0128091i
\(6\) 1.20344i 0.491303i
\(7\) 1.21615 + 1.44935i 0.459662 + 0.547804i 0.945234 − 0.326393i \(-0.105833\pi\)
−0.485572 + 0.874196i \(0.661389\pi\)
\(8\) −1.53543 + 2.65944i −0.542856 + 0.940254i
\(9\) 0.939693 − 0.342020i 0.313231 − 0.114007i
\(10\) −0.433299 + 2.65586i −0.137021 + 0.839857i
\(11\) −1.53393 + 2.65684i −0.462496 + 0.801066i −0.999085 − 0.0427775i \(-0.986379\pi\)
0.536589 + 0.843844i \(0.319713\pi\)
\(12\) 0.543348 + 0.0958070i 0.156851 + 0.0276571i
\(13\) −3.86806 − 1.40786i −1.07281 − 0.390469i −0.255581 − 0.966788i \(-0.582267\pi\)
−0.817225 + 0.576318i \(0.804489\pi\)
\(14\) −1.97186 + 1.13845i −0.527001 + 0.304264i
\(15\) 2.19694 − 0.416464i 0.567248 − 0.107531i
\(16\) −1.98569 − 1.66619i −0.496422 − 0.416548i
\(17\) −1.72469 + 0.627737i −0.418300 + 0.152249i −0.542591 − 0.839997i \(-0.682557\pi\)
0.124291 + 0.992246i \(0.460334\pi\)
\(18\) 0.208975 + 1.18516i 0.0492560 + 0.279344i
\(19\) 5.06455 − 0.893017i 1.16189 − 0.204872i 0.440728 − 0.897641i \(-0.354720\pi\)
0.721160 + 0.692768i \(0.243609\pi\)
\(20\) 1.16461 + 0.407068i 0.260416 + 0.0910232i
\(21\) 1.44935 + 1.21615i 0.316275 + 0.265386i
\(22\) −2.82822 − 2.37316i −0.602978 − 0.505959i
\(23\) −4.39279 − 7.60853i −0.915959 − 1.58649i −0.805491 − 0.592608i \(-0.798098\pi\)
−0.110468 − 0.993880i \(-0.535235\pi\)
\(24\) −1.05029 + 2.88566i −0.214391 + 0.589033i
\(25\) 4.99836 − 0.128080i 0.999672 − 0.0256160i
\(26\) 2.47686 − 4.29005i 0.485753 − 0.841348i
\(27\) 0.866025 − 0.500000i 0.166667 − 0.0962250i
\(28\) 0.357025 + 0.980918i 0.0674713 + 0.185376i
\(29\) 7.41057 + 4.27850i 1.37611 + 0.794497i 0.991689 − 0.128661i \(-0.0410680\pi\)
0.384420 + 0.923158i \(0.374401\pi\)
\(30\) 0.0344688 + 2.69075i 0.00629312 + 0.491262i
\(31\) − 2.12029i − 0.380815i −0.981705 − 0.190407i \(-0.939019\pi\)
0.981705 − 0.190407i \(-0.0609809\pi\)
\(32\) −2.31517 + 1.94265i −0.409267 + 0.343416i
\(33\) −1.04927 + 2.88284i −0.182654 + 0.501837i
\(34\) −0.383549 − 2.17521i −0.0657781 − 0.373046i
\(35\) 2.76068 + 3.20575i 0.466641 + 0.541871i
\(36\) 0.551730 0.0919550
\(37\) −3.35395 − 5.07455i −0.551386 − 0.834250i
\(38\) 6.18891i 1.00397i
\(39\) −4.05376 − 0.714788i −0.649122 − 0.114458i
\(40\) −3.35687 + 5.99018i −0.530768 + 0.947130i
\(41\) −3.78829 − 1.37882i −0.591631 − 0.215336i 0.0288155 − 0.999585i \(-0.490826\pi\)
−0.620447 + 0.784249i \(0.713049\pi\)
\(42\) −1.74421 + 1.46357i −0.269137 + 0.225833i
\(43\) 1.42178 0.216820 0.108410 − 0.994106i \(-0.465424\pi\)
0.108410 + 0.994106i \(0.465424\pi\)
\(44\) −1.29663 + 1.08800i −0.195474 + 0.164022i
\(45\) 2.09125 − 0.791632i 0.311745 − 0.118010i
\(46\) 9.93529 − 3.61615i 1.46488 − 0.533172i
\(47\) 9.38600 − 5.41901i 1.36909 − 0.790444i 0.378278 − 0.925692i \(-0.376516\pi\)
0.990812 + 0.135248i \(0.0431832\pi\)
\(48\) −2.24485 − 1.29607i −0.324017 − 0.187071i
\(49\) 0.593939 − 3.36839i 0.0848484 − 0.481199i
\(50\) −0.892739 + 5.95061i −0.126252 + 0.841543i
\(51\) −1.58949 + 0.917690i −0.222573 + 0.128502i
\(52\) −1.73975 − 1.45983i −0.241260 − 0.202442i
\(53\) −6.74027 + 8.03274i −0.925847 + 1.10338i 0.0685475 + 0.997648i \(0.478164\pi\)
−0.994394 + 0.105734i \(0.966281\pi\)
\(54\) 0.411601 + 1.13086i 0.0560118 + 0.153891i
\(55\) −3.35358 + 5.98432i −0.452197 + 0.806925i
\(56\) −5.72178 + 1.00890i −0.764605 + 0.134820i
\(57\) 4.83254 − 1.75890i 0.640086 − 0.232972i
\(58\) −6.61932 + 7.88860i −0.869159 + 1.03582i
\(59\) −7.98485 + 9.51597i −1.03954 + 1.23887i −0.0690791 + 0.997611i \(0.522006\pi\)
−0.970460 + 0.241263i \(0.922438\pi\)
\(60\) 1.21761 + 0.198651i 0.157193 + 0.0256457i
\(61\) 4.52144 − 12.4226i 0.578911 − 1.59055i −0.211107 − 0.977463i \(-0.567707\pi\)
0.790019 − 0.613083i \(-0.210071\pi\)
\(62\) 2.51288 + 0.443088i 0.319136 + 0.0562722i
\(63\) 1.63852 + 0.945997i 0.206434 + 0.119184i
\(64\) −4.41067 − 7.63951i −0.551334 − 0.954939i
\(65\) −8.68885 − 3.03702i −1.07772 − 0.376696i
\(66\) −3.19735 − 1.84599i −0.393566 − 0.227225i
\(67\) 5.97626 + 7.12223i 0.730116 + 0.870119i 0.995572 − 0.0940043i \(-0.0299668\pi\)
−0.265455 + 0.964123i \(0.585522\pi\)
\(68\) −1.01263 −0.122800
\(69\) −5.64726 − 6.73014i −0.679850 − 0.810213i
\(70\) −4.37624 + 2.60192i −0.523060 + 0.310989i
\(71\) −1.54417 − 8.75743i −0.183259 − 1.03932i −0.928171 − 0.372153i \(-0.878620\pi\)
0.744912 − 0.667163i \(-0.232491\pi\)
\(72\) −0.533249 + 3.02420i −0.0628439 + 0.356406i
\(73\) − 3.23072i − 0.378128i −0.981965 − 0.189064i \(-0.939455\pi\)
0.981965 − 0.189064i \(-0.0605453\pi\)
\(74\) 6.71503 − 2.91451i 0.780606 − 0.338805i
\(75\) 4.90018 − 0.994090i 0.565824 − 0.114788i
\(76\) 2.79427 + 0.492705i 0.320524 + 0.0565171i
\(77\) −5.71618 + 1.00792i −0.651419 + 0.114863i
\(78\) 1.69427 − 4.65498i 0.191839 − 0.527072i
\(79\) 0.439615 + 0.523913i 0.0494605 + 0.0589448i 0.790208 − 0.612839i \(-0.209973\pi\)
−0.740747 + 0.671784i \(0.765528\pi\)
\(80\) −4.48749 − 3.66854i −0.501717 − 0.410155i
\(81\) 0.766044 − 0.642788i 0.0851160 − 0.0714208i
\(82\) 2.42578 − 4.20158i 0.267883 − 0.463987i
\(83\) 4.97998 + 13.6824i 0.546624 + 1.50184i 0.838240 + 0.545301i \(0.183585\pi\)
−0.291617 + 0.956535i \(0.594193\pi\)
\(84\) 0.521935 + 0.904019i 0.0569478 + 0.0986365i
\(85\) −3.83823 + 1.45295i −0.416315 + 0.157594i
\(86\) −0.297117 + 1.68504i −0.0320390 + 0.181702i
\(87\) 8.04094 + 2.92666i 0.862080 + 0.313771i
\(88\) −4.71046 − 8.15876i −0.502137 − 0.869727i
\(89\) −9.23895 + 11.0106i −0.979327 + 1.16712i 0.00660674 + 0.999978i \(0.497897\pi\)
−0.985934 + 0.167138i \(0.946547\pi\)
\(90\) 0.501190 + 2.64389i 0.0528300 + 0.278691i
\(91\) −2.66366 − 7.31834i −0.279227 − 0.767171i
\(92\) −0.841719 − 4.77362i −0.0877552 − 0.497685i
\(93\) −0.368184 − 2.08808i −0.0381789 − 0.216523i
\(94\) 4.46094 + 12.2563i 0.460111 + 1.26414i
\(95\) 11.2982 − 2.14174i 1.15917 − 0.219738i
\(96\) −1.94265 + 2.31517i −0.198271 + 0.236291i
\(97\) −3.04716 − 5.27783i −0.309392 − 0.535883i 0.668837 − 0.743409i \(-0.266792\pi\)
−0.978230 + 0.207526i \(0.933459\pi\)
\(98\) 3.86796 + 1.40782i 0.390723 + 0.142211i
\(99\) −0.532727 + 3.02124i −0.0535411 + 0.303646i
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 555.2.bp.a.4.12 ✓ 240
5.4 even 2 inner 555.2.bp.a.4.29 yes 240
37.28 even 18 inner 555.2.bp.a.139.29 yes 240
185.139 even 18 inner 555.2.bp.a.139.12 yes 240
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
555.2.bp.a.4.12 ✓ 240 1.1 even 1 trivial
555.2.bp.a.4.29 yes 240 5.4 even 2 inner
555.2.bp.a.139.12 yes 240 185.139 even 18 inner
555.2.bp.a.139.29 yes 240 37.28 even 18 inner