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Properties

Label 555.2.bp.a.4.1
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 4.1
Character \(\chi\) \(=\) 555.4
Dual form 555.2.bp.a.139.1

$q$-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+(-0.483654 + 2.74294i) q^{2} +(-0.984808 + 0.173648i) q^{3} +(-5.41042 - 1.96923i) q^{4} +(2.17371 - 0.524382i) q^{5} -2.78526i q^{6} +(2.76078 + 3.29017i) q^{7} +(5.23300 - 9.06382i) q^{8} +(0.939693 - 0.342020i) q^{9} +(0.387024 + 6.21598i) q^{10} +(-1.59569 + 2.76381i) q^{11} +(5.67017 + 0.999805i) q^{12} +(1.51655 + 0.551981i) q^{13} +(-10.3600 + 5.98135i) q^{14} +(-2.04963 + 0.893877i) q^{15} +(13.5093 + 11.3357i) q^{16} +(-1.80359 + 0.656453i) q^{17} +(0.483654 + 2.74294i) q^{18} +(-4.60729 + 0.812389i) q^{19} +(-12.7933 - 1.44341i) q^{20} +(-3.29017 - 2.76078i) q^{21} +(-6.80920 - 5.71360i) q^{22} +(3.22774 + 5.59061i) q^{23} +(-3.57958 + 9.83483i) q^{24} +(4.45005 - 2.27971i) q^{25} +(-2.24754 + 3.89285i) q^{26} +(-0.866025 + 0.500000i) q^{27} +(-8.45787 - 23.2378i) q^{28} +(3.34098 + 1.92891i) q^{29} +(-1.46054 - 6.05434i) q^{30} -3.47435i q^{31} +(-21.5921 + 18.1179i) q^{32} +(1.09151 - 2.99891i) q^{33} +(-0.928297 - 5.26463i) q^{34} +(7.72645 + 5.70418i) q^{35} -5.75765 q^{36} +(4.43439 - 4.16367i) q^{37} -13.0304i q^{38} +(-1.58937 - 0.280248i) q^{39} +(6.62213 - 22.4462i) q^{40} +(-8.49367 - 3.09144i) q^{41} +(9.16396 - 7.68948i) q^{42} +0.00771474 q^{43} +(14.0759 - 11.8111i) q^{44} +(1.86327 - 1.23621i) q^{45} +(-16.8958 + 6.14957i) q^{46} +(-5.76744 + 3.32983i) q^{47} +(-15.2725 - 8.81760i) q^{48} +(-1.98777 + 11.2732i) q^{49} +(4.10083 + 13.3088i) q^{50} +(1.66220 - 0.959670i) q^{51} +(-7.11822 - 5.97289i) q^{52} +(-3.53088 + 4.20794i) q^{53} +(-0.952613 - 2.61728i) q^{54} +(-2.01927 + 6.84447i) q^{55} +(44.2687 - 7.80576i) q^{56} +(4.39622 - 1.60009i) q^{57} +(-6.90677 + 8.23117i) q^{58} +(-5.51859 + 6.57680i) q^{59} +(12.8496 - 0.800051i) q^{60} +(1.16391 - 3.19780i) q^{61} +(9.52995 + 1.68039i) q^{62} +(3.71959 + 2.14751i) q^{63} +(-21.6181 - 37.4437i) q^{64} +(3.58600 + 0.404593i) q^{65} +(7.69791 + 4.44439i) q^{66} +(4.79764 + 5.71760i) q^{67} +11.0509 q^{68} +(-4.14950 - 4.94518i) q^{69} +(-19.3831 + 18.4343i) q^{70} +(-1.92850 - 10.9371i) q^{71} +(1.81740 - 10.3070i) q^{72} -1.09541i q^{73} +(9.27599 + 14.1771i) q^{74} +(-3.98657 + 3.01782i) q^{75} +(26.5271 + 4.67745i) q^{76} +(-13.4987 + 2.38019i) q^{77} +(1.53741 - 4.22399i) q^{78} +(-1.35224 - 1.61154i) q^{79} +(35.3097 + 17.5565i) q^{80} +(0.766044 - 0.642788i) q^{81} +(12.5876 - 21.8024i) q^{82} +(3.95447 + 10.8648i) q^{83} +(12.3646 + 21.4161i) q^{84} +(-3.57625 + 2.37271i) q^{85} +(-0.00373127 + 0.0211611i) q^{86} +(-3.62517 - 1.31945i) q^{87} +(16.7004 + 28.9260i) q^{88} +(1.26203 - 1.50403i) q^{89} +(2.48967 + 5.70874i) q^{90} +(2.37076 + 6.51362i) q^{91} +(-6.45422 - 36.6037i) q^{92} +(0.603315 + 3.42157i) q^{93} +(-6.34409 - 17.4302i) q^{94} +(-9.58892 + 4.18188i) q^{95} +(18.1179 - 21.5921i) q^{96} +(-0.773775 - 1.34022i) q^{97} +(-29.9604 - 10.9047i) q^{98} +(-0.554176 + 3.14289i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.483654 + 2.74294i −0.341995 + 1.93955i 0.000410493 1.00000i \(0.499869\pi\)
−0.342406 + 0.939552i \(0.611242\pi\)
\(3\) −0.984808 + 0.173648i −0.568579 + 0.100256i
\(4\) −5.41042 − 1.96923i −2.70521 − 0.984615i
\(5\) 2.17371 − 0.524382i 0.972114 − 0.234511i
\(6\) − 2.78526i − 1.13708i
\(7\) 2.76078 + 3.29017i 1.04348 + 1.24357i 0.969186 + 0.246330i \(0.0792248\pi\)
0.0742907 + 0.997237i \(0.476331\pi\)
\(8\) 5.23300 − 9.06382i 1.85015 − 3.20455i
\(9\) 0.939693 − 0.342020i 0.313231 − 0.114007i
\(10\) 0.387024 + 6.21598i 0.122388 + 1.96567i
\(11\) −1.59569 + 2.76381i −0.481117 + 0.833319i −0.999765 − 0.0216684i \(-0.993102\pi\)
0.518648 + 0.854988i \(0.326436\pi\)
\(12\) 5.67017 + 0.999805i 1.63684 + 0.288619i
\(13\) 1.51655 + 0.551981i 0.420617 + 0.153092i 0.543653 − 0.839310i \(-0.317041\pi\)
−0.123036 + 0.992402i \(0.539263\pi\)
\(14\) −10.3600 + 5.98135i −2.76883 + 1.59858i
\(15\) −2.04963 + 0.893877i −0.529212 + 0.230798i
\(16\) 13.5093 + 11.3357i 3.37734 + 2.83392i
\(17\) −1.80359 + 0.656453i −0.437435 + 0.159213i −0.551344 − 0.834278i \(-0.685885\pi\)
0.113909 + 0.993491i \(0.463663\pi\)
\(18\) 0.483654 + 2.74294i 0.113998 + 0.646517i
\(19\) −4.60729 + 0.812389i −1.05698 + 0.186375i −0.675018 − 0.737801i \(-0.735864\pi\)
−0.381967 + 0.924176i \(0.624753\pi\)
\(20\) −12.7933 − 1.44341i −2.86067 − 0.322757i
\(21\) −3.29017 − 2.76078i −0.717974 − 0.602452i
\(22\) −6.80920 − 5.71360i −1.45173 − 1.21814i
\(23\) 3.22774 + 5.59061i 0.673030 + 1.16572i 0.977041 + 0.213054i \(0.0683409\pi\)
−0.304011 + 0.952669i \(0.598326\pi\)
\(24\) −3.57958 + 9.83483i −0.730679 + 2.00753i
\(25\) 4.45005 − 2.27971i 0.890009 − 0.455942i
\(26\) −2.24754 + 3.89285i −0.440779 + 0.763451i
\(27\) −0.866025 + 0.500000i −0.166667 + 0.0962250i
\(28\) −8.45787 − 23.2378i −1.59839 − 4.39153i
\(29\) 3.34098 + 1.92891i 0.620404 + 0.358190i 0.777026 − 0.629468i \(-0.216727\pi\)
−0.156622 + 0.987659i \(0.550061\pi\)
\(30\) −1.46054 − 6.05434i −0.266657 − 1.10537i
\(31\) − 3.47435i − 0.624013i −0.950080 − 0.312006i \(-0.898999\pi\)
0.950080 − 0.312006i \(-0.101001\pi\)
\(32\) −21.5921 + 18.1179i −3.81698 + 3.20283i
\(33\) 1.09151 − 2.99891i 0.190008 − 0.522043i
\(34\) −0.928297 − 5.26463i −0.159202 − 0.902877i
\(35\) 7.72645 + 5.70418i 1.30601 + 0.964182i
\(36\) −5.75765 −0.959608
\(37\) 4.43439 − 4.16367i 0.729010 − 0.684503i
\(38\) − 13.0304i − 2.11382i
\(39\) −1.58937 − 0.280248i −0.254502 − 0.0448756i
\(40\) 6.62213 − 22.4462i 1.04705 − 3.54906i
\(41\) −8.49367 − 3.09144i −1.32649 − 0.482802i −0.420957 − 0.907081i \(-0.638306\pi\)
−0.905532 + 0.424278i \(0.860528\pi\)
\(42\) 9.16396 − 7.68948i 1.41403 − 1.18651i
\(43\) 0.00771474 0.00117649 0.000588244 − 1.00000i \(-0.499813\pi\)
0.000588244 1.00000i \(0.499813\pi\)
\(44\) 14.0759 − 11.8111i 2.12202 − 1.78059i
\(45\) 1.86327 − 1.23621i 0.277760 − 0.184283i
\(46\) −16.8958 + 6.14957i −2.49115 + 0.906705i
\(47\) −5.76744 + 3.32983i −0.841268 + 0.485706i −0.857695 − 0.514159i \(-0.828104\pi\)
0.0164271 + 0.999865i \(0.494771\pi\)
\(48\) −15.2725 − 8.81760i −2.20440 − 1.27271i
\(49\) −1.98777 + 11.2732i −0.283967 + 1.61046i
\(50\) 4.10083 + 13.3088i 0.579945 + 1.88215i
\(51\) 1.66220 − 0.959670i 0.232754 − 0.134381i
\(52\) −7.11822 − 5.97289i −0.987119 − 0.828291i
\(53\) −3.53088 + 4.20794i −0.485004 + 0.578005i −0.951939 − 0.306286i \(-0.900914\pi\)
0.466936 + 0.884291i \(0.345358\pi\)
\(54\) −0.952613 − 2.61728i −0.129634 − 0.356167i
\(55\) −2.01927 + 6.84447i −0.272278 + 0.922908i
\(56\) 44.2687 − 7.80576i 5.91565 − 1.04309i
\(57\) 4.39622 − 1.60009i 0.582294 − 0.211938i
\(58\) −6.90677 + 8.23117i −0.906904 + 1.08081i
\(59\) −5.51859 + 6.57680i −0.718460 + 0.856227i −0.994480 − 0.104925i \(-0.966540\pi\)
0.276020 + 0.961152i \(0.410984\pi\)
\(60\) 12.8496 − 0.800051i 1.65888 − 0.103286i
\(61\) 1.16391 − 3.19780i 0.149023 − 0.409437i −0.842611 − 0.538523i \(-0.818982\pi\)
0.991633 + 0.129087i \(0.0412046\pi\)
\(62\) 9.52995 + 1.68039i 1.21030 + 0.213409i
\(63\) 3.71959 + 2.14751i 0.468624 + 0.270560i
\(64\) −21.6181 − 37.4437i −2.70226 − 4.68046i
\(65\) 3.58600 + 0.404593i 0.444789 + 0.0501836i
\(66\) 7.69791 + 4.44439i 0.947547 + 0.547067i
\(67\) 4.79764 + 5.71760i 0.586124 + 0.698516i 0.974856 − 0.222835i \(-0.0715311\pi\)
−0.388732 + 0.921351i \(0.627087\pi\)
\(68\) 11.0509 1.34012
\(69\) −4.14950 − 4.94518i −0.499541 − 0.595330i
\(70\) −19.3831 + 18.4343i −2.31673 + 2.20332i
\(71\) −1.92850 − 10.9371i −0.228871 − 1.29799i −0.855146 − 0.518388i \(-0.826532\pi\)
0.626275 − 0.779602i \(-0.284579\pi\)
\(72\) 1.81740 − 10.3070i 0.214183 − 1.21469i
\(73\) − 1.09541i − 0.128208i −0.997943 − 0.0641038i \(-0.979581\pi\)
0.997943 − 0.0641038i \(-0.0204189\pi\)
\(74\) 9.27599 + 14.1771i 1.07831 + 1.64805i
\(75\) −3.98657 + 3.01782i −0.460330 + 0.348468i
\(76\) 26.5271 + 4.67745i 3.04287 + 0.536540i
\(77\) −13.4987 + 2.38019i −1.53832 + 0.271248i
\(78\) 1.53741 − 4.22399i 0.174077 − 0.478273i
\(79\) −1.35224 − 1.61154i −0.152139 − 0.181313i 0.684592 − 0.728927i \(-0.259981\pi\)
−0.836731 + 0.547614i \(0.815536\pi\)
\(80\) 35.3097 + 17.5565i 3.94774 + 1.96287i
\(81\) 0.766044 − 0.642788i 0.0851160 − 0.0714208i
\(82\) 12.5876 − 21.8024i 1.39007 − 2.40768i
\(83\) 3.95447 + 10.8648i 0.434060 + 1.19257i 0.943299 + 0.331944i \(0.107705\pi\)
−0.509239 + 0.860625i \(0.670073\pi\)
\(84\) 12.3646 + 21.4161i 1.34909 + 2.33669i
\(85\) −3.57625 + 2.37271i −0.387899 + 0.257356i
\(86\) −0.00373127 + 0.0211611i −0.000402353 + 0.00228186i
\(87\) −3.62517 − 1.31945i −0.388659 − 0.141460i
\(88\) 16.7004 + 28.9260i 1.78027 + 3.08352i
\(89\) 1.26203 − 1.50403i 0.133775 − 0.159427i −0.694998 − 0.719011i \(-0.744595\pi\)
0.828773 + 0.559585i \(0.189039\pi\)
\(90\) 2.48967 + 5.70874i 0.262435 + 0.601754i
\(91\) 2.37076 + 6.51362i 0.248524 + 0.682813i
\(92\) −6.45422 − 36.6037i −0.672899 − 3.81620i
\(93\) 0.603315 + 3.42157i 0.0625609 + 0.354800i
\(94\) −6.34409 − 17.4302i −0.654343 − 1.79779i
\(95\) −9.58892 + 4.18188i −0.983802 + 0.429052i
\(96\) 18.1179 − 21.5921i 1.84915 − 2.20374i
\(97\) −0.773775 − 1.34022i −0.0785649 − 0.136078i 0.824066 − 0.566494i \(-0.191700\pi\)
−0.902631 + 0.430415i \(0.858367\pi\)
\(98\) −29.9604 − 10.9047i −3.02645 − 1.10154i
\(99\) −0.554176 + 3.14289i −0.0556967 + 0.315872i
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.4.1 ✓ 240
5.4 even 2 inner 555.2.bp.a.4.40 yes 240
37.28 even 18 inner 555.2.bp.a.139.40 yes 240
185.139 even 18 inner 555.2.bp.a.139.1 yes 240
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
555.2.bp.a.4.1 ✓ 240 1.1 even 1 trivial
555.2.bp.a.4.40 yes 240 5.4 even 2 inner
555.2.bp.a.139.1 yes 240 185.139 even 18 inner
555.2.bp.a.139.40 yes 240 37.28 even 18 inner