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Properties

Label 555.2.bp.a.4.18
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 4.18
Character \(\chi\) \(=\) 555.4
Dual form 555.2.bp.a.139.18

$q$-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+(-0.0741078 + 0.420286i) q^{2} +(0.984808 - 0.173648i) q^{3} +(1.70824 + 0.621747i) q^{4} +(1.35836 + 1.77619i) q^{5} +0.426770i q^{6} +(-2.53805 - 3.02473i) q^{7} +(-0.814676 + 1.41106i) q^{8} +(0.939693 - 0.342020i) q^{9} +(-0.847173 + 0.439272i) q^{10} +(-1.47477 + 2.55439i) q^{11} +(1.79025 + 0.315669i) q^{12} +(3.84770 + 1.40045i) q^{13} +(1.45934 - 0.842551i) q^{14} +(1.64616 + 1.51333i) q^{15} +(2.25246 + 1.89004i) q^{16} +(3.24549 - 1.18126i) q^{17} +(0.0741078 + 0.420286i) q^{18} +(1.07481 - 0.189518i) q^{19} +(1.21607 + 3.87871i) q^{20} +(-3.02473 - 2.53805i) q^{21} +(-0.964281 - 0.809128i) q^{22} +(2.26665 + 3.92596i) q^{23} +(-0.557271 + 1.53109i) q^{24} +(-1.30969 + 4.82542i) q^{25} +(-0.873734 + 1.51335i) q^{26} +(0.866025 - 0.500000i) q^{27} +(-2.45497 - 6.74497i) q^{28} +(-6.44351 - 3.72016i) q^{29} +(-0.758024 + 0.579709i) q^{30} -7.15328i q^{31} +(-3.45759 + 2.90126i) q^{32} +(-1.00881 + 2.77167i) q^{33} +(0.255952 + 1.45158i) q^{34} +(1.92489 - 8.61673i) q^{35} +1.81787 q^{36} +(-4.42509 - 4.17356i) q^{37} +0.465772i q^{38} +(4.03243 + 0.711026i) q^{39} +(-3.61293 + 0.469715i) q^{40} +(8.60102 + 3.13051i) q^{41} +(1.29086 - 1.08316i) q^{42} -12.0675 q^{43} +(-4.10745 + 3.44656i) q^{44} +(1.88394 + 1.20448i) q^{45} +(-1.81800 + 0.661699i) q^{46} +(3.06310 - 1.76848i) q^{47} +(2.54644 + 1.47019i) q^{48} +(-1.49175 + 8.46014i) q^{49} +(-1.93100 - 0.908048i) q^{50} +(2.99106 - 1.72689i) q^{51} +(5.70205 + 4.78459i) q^{52} +(-4.53969 + 5.41019i) q^{53} +(0.145964 + 0.401033i) q^{54} +(-6.54035 + 0.850307i) q^{55} +(6.33575 - 1.11716i) q^{56} +(1.02557 - 0.373277i) q^{57} +(2.04105 - 2.43243i) q^{58} +(7.07351 - 8.42988i) q^{59} +(1.87112 + 3.60862i) q^{60} +(1.69125 - 4.64668i) q^{61} +(3.00643 + 0.530114i) q^{62} +(-3.41950 - 1.97425i) q^{63} +(1.97725 + 3.42470i) q^{64} +(2.73912 + 8.73656i) q^{65} +(-1.09013 - 0.629390i) q^{66} +(-2.62214 - 3.12494i) q^{67} +6.27851 q^{68} +(2.91395 + 3.47271i) q^{69} +(3.47884 + 1.44757i) q^{70} +(-0.547099 - 3.10275i) q^{71} +(-0.282934 + 1.60460i) q^{72} -5.18306i q^{73} +(2.08202 - 1.55051i) q^{74} +(-0.451870 + 4.97954i) q^{75} +(1.95386 + 0.344518i) q^{76} +(11.4694 - 2.02236i) q^{77} +(-0.597669 + 1.64208i) q^{78} +(-8.26589 - 9.85090i) q^{79} +(-0.297404 + 6.56815i) q^{80} +(0.766044 - 0.642788i) q^{81} +(-1.95312 + 3.38290i) q^{82} +(-1.20488 - 3.31038i) q^{83} +(-3.58892 - 6.21620i) q^{84} +(6.50670 + 4.16002i) q^{85} +(0.894297 - 5.07181i) q^{86} +(-6.99162 - 2.54474i) q^{87} +(-2.40293 - 4.16199i) q^{88} +(-11.3811 + 13.5635i) q^{89} +(-0.645842 + 0.702531i) q^{90} +(-5.52967 - 15.1926i) q^{91} +(1.43103 + 8.11575i) q^{92} +(-1.24215 - 7.04461i) q^{93} +(0.516270 + 1.41844i) q^{94} +(1.79660 + 1.65163i) q^{95} +(-2.90126 + 3.45759i) q^{96} +(-2.14750 - 3.71959i) q^{97} +(-3.44513 - 1.25392i) q^{98} +(-0.512184 + 2.90474i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.0741078 + 0.420286i −0.0524021 + 0.297187i −0.999734 − 0.0230664i \(-0.992657\pi\)
0.947332 + 0.320254i \(0.103768\pi\)
\(3\) 0.984808 − 0.173648i 0.568579 − 0.100256i
\(4\) 1.70824 + 0.621747i 0.854118 + 0.310874i
\(5\) 1.35836 + 1.77619i 0.607479 + 0.794336i
\(6\) 0.426770i 0.174228i
\(7\) −2.53805 − 3.02473i −0.959291 − 1.14324i −0.989621 − 0.143699i \(-0.954100\pi\)
0.0303299 − 0.999540i \(-0.490344\pi\)
\(8\) −0.814676 + 1.41106i −0.288031 + 0.498885i
\(9\) 0.939693 − 0.342020i 0.313231 − 0.114007i
\(10\) −0.847173 + 0.439272i −0.267900 + 0.138910i
\(11\) −1.47477 + 2.55439i −0.444661 + 0.770176i −0.998029 − 0.0627614i \(-0.980009\pi\)
0.553367 + 0.832937i \(0.313343\pi\)
\(12\) 1.79025 + 0.315669i 0.516801 + 0.0911259i
\(13\) 3.84770 + 1.40045i 1.06716 + 0.388414i 0.815114 − 0.579301i \(-0.196674\pi\)
0.252046 + 0.967715i \(0.418897\pi\)
\(14\) 1.45934 − 0.842551i 0.390025 − 0.225181i
\(15\) 1.64616 + 1.51333i 0.425037 + 0.390739i
\(16\) 2.25246 + 1.89004i 0.563115 + 0.472509i
\(17\) 3.24549 − 1.18126i 0.787147 − 0.286498i 0.0829973 − 0.996550i \(-0.473551\pi\)
0.704149 + 0.710052i \(0.251328\pi\)
\(18\) 0.0741078 + 0.420286i 0.0174674 + 0.0990624i
\(19\) 1.07481 − 0.189518i 0.246578 − 0.0434783i −0.0489931 − 0.998799i \(-0.515601\pi\)
0.295571 + 0.955321i \(0.404490\pi\)
\(20\) 1.21607 + 3.87871i 0.271921 + 0.867306i
\(21\) −3.02473 − 2.53805i −0.660049 − 0.553847i
\(22\) −0.964281 − 0.809128i −0.205585 − 0.172507i
\(23\) 2.26665 + 3.92596i 0.472630 + 0.818619i 0.999509 − 0.0313209i \(-0.00997139\pi\)
−0.526879 + 0.849940i \(0.676638\pi\)
\(24\) −0.557271 + 1.53109i −0.113752 + 0.312532i
\(25\) −1.30969 + 4.82542i −0.261939 + 0.965085i
\(26\) −0.873734 + 1.51335i −0.171353 + 0.296793i
\(27\) 0.866025 − 0.500000i 0.166667 − 0.0962250i
\(28\) −2.45497 − 6.74497i −0.463946 − 1.27468i
\(29\) −6.44351 − 3.72016i −1.19653 − 0.690817i −0.236750 − 0.971571i \(-0.576082\pi\)
−0.959780 + 0.280754i \(0.909416\pi\)
\(30\) −0.758024 + 0.579709i −0.138396 + 0.105840i
\(31\) − 7.15328i − 1.28477i −0.766383 − 0.642384i \(-0.777946\pi\)
0.766383 − 0.642384i \(-0.222054\pi\)
\(32\) −3.45759 + 2.90126i −0.611222 + 0.512876i
\(33\) −1.00881 + 2.77167i −0.175610 + 0.482486i
\(34\) 0.255952 + 1.45158i 0.0438954 + 0.248943i
\(35\) 1.92489 − 8.61673i 0.325366 − 1.45649i
\(36\) 1.81787 0.302978
\(37\) −4.42509 − 4.17356i −0.727480 − 0.686129i
\(38\) 0.465772i 0.0755582i
\(39\) 4.03243 + 0.711026i 0.645705 + 0.113855i
\(40\) −3.61293 + 0.469715i −0.571255 + 0.0742685i
\(41\) 8.60102 + 3.13051i 1.34325 + 0.488904i 0.910836 − 0.412769i \(-0.135438\pi\)
0.432418 + 0.901673i \(0.357661\pi\)
\(42\) 1.29086 − 1.08316i 0.199184 − 0.167136i
\(43\) −12.0675 −1.84028 −0.920139 − 0.391592i \(-0.871924\pi\)
−0.920139 + 0.391592i \(0.871924\pi\)
\(44\) −4.10745 + 3.44656i −0.619221 + 0.519588i
\(45\) 1.88394 + 1.20448i 0.280841 + 0.179554i
\(46\) −1.81800 + 0.661699i −0.268050 + 0.0975622i
\(47\) 3.06310 − 1.76848i 0.446799 − 0.257960i −0.259678 − 0.965695i \(-0.583616\pi\)
0.706478 + 0.707735i \(0.250283\pi\)
\(48\) 2.54644 + 1.47019i 0.367547 + 0.212203i
\(49\) −1.49175 + 8.46014i −0.213107 + 1.20859i
\(50\) −1.93100 − 0.908048i −0.273085 − 0.128417i
\(51\) 2.99106 − 1.72689i 0.418832 − 0.241813i
\(52\) 5.70205 + 4.78459i 0.790733 + 0.663503i
\(53\) −4.53969 + 5.41019i −0.623575 + 0.743147i −0.981681 − 0.190534i \(-0.938978\pi\)
0.358106 + 0.933681i \(0.383423\pi\)
\(54\) 0.145964 + 0.401033i 0.0198632 + 0.0545736i
\(55\) −6.54035 + 0.850307i −0.881901 + 0.114655i
\(56\) 6.33575 − 1.11716i 0.846651 − 0.149287i
\(57\) 1.02557 − 0.373277i 0.135840 − 0.0494417i
\(58\) 2.04105 − 2.43243i 0.268003 − 0.319393i
\(59\) 7.07351 − 8.42988i 0.920892 − 1.09748i −0.0740730 − 0.997253i \(-0.523600\pi\)
0.994965 − 0.100223i \(-0.0319558\pi\)
\(60\) 1.87112 + 3.60862i 0.241561 + 0.465870i
\(61\) 1.69125 − 4.64668i 0.216543 − 0.594946i −0.783094 − 0.621903i \(-0.786360\pi\)
0.999637 + 0.0269574i \(0.00858183\pi\)
\(62\) 3.00643 + 0.530114i 0.381817 + 0.0673246i
\(63\) −3.41950 − 1.97425i −0.430817 − 0.248732i
\(64\) 1.97725 + 3.42470i 0.247156 + 0.428087i
\(65\) 2.73912 + 8.73656i 0.339746 + 1.08364i
\(66\) −1.09013 − 0.629390i −0.134186 − 0.0774725i
\(67\) −2.62214 − 3.12494i −0.320345 − 0.381772i 0.581708 − 0.813398i \(-0.302385\pi\)
−0.902053 + 0.431626i \(0.857940\pi\)
\(68\) 6.27851 0.761381
\(69\) 2.91395 + 3.47271i 0.350799 + 0.418066i
\(70\) 3.47884 + 1.44757i 0.415801 + 0.173018i
\(71\) −0.547099 − 3.10275i −0.0649287 − 0.368229i −0.999908 − 0.0135281i \(-0.995694\pi\)
0.934980 − 0.354701i \(-0.115417\pi\)
\(72\) −0.282934 + 1.60460i −0.0333441 + 0.189104i
\(73\) − 5.18306i − 0.606631i −0.952890 − 0.303315i \(-0.901906\pi\)
0.952890 − 0.303315i \(-0.0980936\pi\)
\(74\) 2.08202 − 1.55051i 0.242030 − 0.180243i
\(75\) −0.451870 + 4.97954i −0.0521774 + 0.574988i
\(76\) 1.95386 + 0.344518i 0.224123 + 0.0395189i
\(77\) 11.4694 − 2.02236i 1.30706 − 0.230469i
\(78\) −0.597669 + 1.64208i −0.0676727 + 0.185929i
\(79\) −8.26589 − 9.85090i −0.929985 − 1.10831i −0.993892 − 0.110357i \(-0.964801\pi\)
0.0639072 − 0.997956i \(-0.479644\pi\)
\(80\) −0.297404 + 6.56815i −0.0332507 + 0.734342i
\(81\) 0.766044 − 0.642788i 0.0851160 − 0.0714208i
\(82\) −1.95312 + 3.38290i −0.215686 + 0.373578i
\(83\) −1.20488 − 3.31038i −0.132253 − 0.363362i 0.855836 − 0.517248i \(-0.173043\pi\)
−0.988089 + 0.153886i \(0.950821\pi\)
\(84\) −3.58892 − 6.21620i −0.391584 − 0.678243i
\(85\) 6.50670 + 4.16002i 0.705751 + 0.451217i
\(86\) 0.894297 − 5.07181i 0.0964345 − 0.546907i
\(87\) −6.99162 − 2.54474i −0.749580 − 0.272825i
\(88\) −2.40293 − 4.16199i −0.256153 − 0.443670i
\(89\) −11.3811 + 13.5635i −1.20639 + 1.43772i −0.338506 + 0.940964i \(0.609922\pi\)
−0.867888 + 0.496760i \(0.834523\pi\)
\(90\) −0.645842 + 0.702531i −0.0680778 + 0.0740533i
\(91\) −5.52967 − 15.1926i −0.579667 − 1.59262i
\(92\) 1.43103 + 8.11575i 0.149195 + 0.846126i
\(93\) −1.24215 − 7.04461i −0.128805 − 0.730492i
\(94\) 0.516270 + 1.41844i 0.0532491 + 0.146301i
\(95\) 1.79660 + 1.65163i 0.184327 + 0.169453i
\(96\) −2.90126 + 3.45759i −0.296109 + 0.352889i
\(97\) −2.14750 − 3.71959i −0.218046 − 0.377667i 0.736165 − 0.676803i \(-0.236635\pi\)
−0.954211 + 0.299136i \(0.903302\pi\)
\(98\) −3.44513 − 1.25392i −0.348011 − 0.126666i
\(99\) −0.512184 + 2.90474i −0.0514764 + 0.291937i
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.4.18 ✓ 240
5.4 even 2 inner 555.2.bp.a.4.23 yes 240
37.28 even 18 inner 555.2.bp.a.139.23 yes 240
185.139 even 18 inner 555.2.bp.a.139.18 yes 240
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
555.2.bp.a.4.18 ✓ 240 1.1 even 1 trivial
555.2.bp.a.4.23 yes 240 5.4 even 2 inner
555.2.bp.a.139.18 yes 240 185.139 even 18 inner
555.2.bp.a.139.23 yes 240 37.28 even 18 inner