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Properties

Label 555.2.bp.a.4.9
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 4.9
Character \(\chi\) \(=\) 555.4
Dual form 555.2.bp.a.139.9

$q$-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+(-0.326358 + 1.85087i) q^{2} +(0.984808 - 0.173648i) q^{3} +(-1.43983 - 0.524054i) q^{4} +(-2.21686 - 0.292450i) q^{5} +1.87942i q^{6} +(-2.07200 - 2.46931i) q^{7} +(-0.439568 + 0.761354i) q^{8} +(0.939693 - 0.342020i) q^{9} +(1.26478 - 4.00768i) q^{10} +(2.30244 - 3.98795i) q^{11} +(-1.50895 - 0.266069i) q^{12} +(-6.07926 - 2.21267i) q^{13} +(5.24658 - 3.02912i) q^{14} +(-2.23397 + 0.0969473i) q^{15} +(-3.61322 - 3.03186i) q^{16} +(3.00281 - 1.09293i) q^{17} +(0.326358 + 1.85087i) q^{18} +(-2.15623 + 0.380202i) q^{19} +(3.03864 + 1.58283i) q^{20} +(-2.46931 - 2.07200i) q^{21} +(6.62976 + 5.56303i) q^{22} +(1.04986 + 1.81841i) q^{23} +(-0.300682 + 0.826117i) q^{24} +(4.82895 + 1.29664i) q^{25} +(6.07938 - 10.5298i) q^{26} +(0.866025 - 0.500000i) q^{27} +(1.68926 + 4.64121i) q^{28} +(-4.38440 - 2.53133i) q^{29} +(0.549636 - 4.16642i) q^{30} -1.97280i q^{31} +(5.44386 - 4.56794i) q^{32} +(1.57496 - 4.32718i) q^{33} +(1.04289 + 5.91450i) q^{34} +(3.87118 + 6.08007i) q^{35} -1.53223 q^{36} +(-6.01492 - 0.905962i) q^{37} -4.11499i q^{38} +(-6.37113 - 1.12340i) q^{39} +(1.19712 - 1.55926i) q^{40} +(0.888860 + 0.323518i) q^{41} +(4.64087 - 3.89416i) q^{42} -4.16467 q^{43} +(-5.40502 + 4.53535i) q^{44} +(-2.18319 + 0.483398i) q^{45} +(-3.70828 + 1.34970i) q^{46} +(9.11856 - 5.26460i) q^{47} +(-4.08481 - 2.35837i) q^{48} +(-0.588780 + 3.33914i) q^{49} +(-3.97588 + 8.51458i) q^{50} +(2.76740 - 1.59776i) q^{51} +(7.59352 + 6.37172i) q^{52} +(-6.76718 + 8.06482i) q^{53} +(0.642801 + 1.76608i) q^{54} +(-6.27047 + 8.16738i) q^{55} +(2.79080 - 0.492093i) q^{56} +(-2.05745 + 0.748851i) q^{57} +(6.11606 - 7.28884i) q^{58} +(-5.54526 + 6.60858i) q^{59} +(3.26733 + 1.03113i) q^{60} +(2.67590 - 7.35198i) q^{61} +(3.65139 + 0.643839i) q^{62} +(-2.79159 - 1.61173i) q^{63} +(1.96129 + 3.39706i) q^{64} +(12.8298 + 6.68305i) q^{65} +(7.49505 + 4.32727i) q^{66} +(1.70753 + 2.03496i) q^{67} -4.89628 q^{68} +(1.34968 + 1.60848i) q^{69} +(-12.5168 + 5.18077i) q^{70} +(2.38075 + 13.5019i) q^{71} +(-0.152660 + 0.865780i) q^{72} -11.6066i q^{73} +(3.63984 - 10.8372i) q^{74} +(4.98074 + 0.438403i) q^{75} +(3.30385 + 0.582557i) q^{76} +(-14.6181 + 2.57757i) q^{77} +(4.15854 - 11.4255i) q^{78} +(0.378412 + 0.450973i) q^{79} +(7.12335 + 7.77789i) q^{80} +(0.766044 - 0.642788i) q^{81} +(-0.888877 + 1.53958i) q^{82} +(-4.56245 - 12.5352i) q^{83} +(2.46954 + 4.27736i) q^{84} +(-6.97644 + 1.54471i) q^{85} +(1.35918 - 7.70827i) q^{86} +(-4.75735 - 1.73154i) q^{87} +(2.02416 + 3.50595i) q^{88} +(5.57899 - 6.64878i) q^{89} +(-0.182205 - 4.19857i) q^{90} +(7.13243 + 19.5962i) q^{91} +(-0.558672 - 3.16838i) q^{92} +(-0.342573 - 1.94283i) q^{93} +(6.76818 + 18.5954i) q^{94} +(4.89126 - 0.212266i) q^{95} +(4.56794 - 5.44386i) q^{96} +(3.05850 + 5.29748i) q^{97} +(-5.98815 - 2.17951i) q^{98} +(0.799631 - 4.53493i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.326358 + 1.85087i −0.230770 + 1.30876i 0.620571 + 0.784150i \(0.286901\pi\)
−0.851341 + 0.524613i \(0.824210\pi\)
\(3\) 0.984808 − 0.173648i 0.568579 − 0.100256i
\(4\) −1.43983 − 0.524054i −0.719913 − 0.262027i
\(5\) −2.21686 − 0.292450i −0.991410 − 0.130787i
\(6\) 1.87942i 0.767271i
\(7\) −2.07200 − 2.46931i −0.783141 − 0.933311i 0.215930 − 0.976409i \(-0.430722\pi\)
−0.999071 + 0.0430981i \(0.986277\pi\)
\(8\) −0.439568 + 0.761354i −0.155411 + 0.269179i
\(9\) 0.939693 − 0.342020i 0.313231 − 0.114007i
\(10\) 1.26478 − 4.00768i 0.399958 − 1.26734i
\(11\) 2.30244 − 3.98795i 0.694213 − 1.20241i −0.276232 − 0.961091i \(-0.589086\pi\)
0.970445 − 0.241321i \(-0.0775808\pi\)
\(12\) −1.50895 − 0.266069i −0.435597 − 0.0768075i
\(13\) −6.07926 − 2.21267i −1.68608 − 0.613684i −0.691959 − 0.721937i \(-0.743252\pi\)
−0.994123 + 0.108253i \(0.965474\pi\)
\(14\) 5.24658 − 3.02912i 1.40221 − 0.809565i
\(15\) −2.23397 + 0.0969473i −0.576807 + 0.0250317i
\(16\) −3.61322 − 3.03186i −0.903306 − 0.757964i
\(17\) 3.00281 − 1.09293i 0.728288 − 0.265075i 0.0488480 − 0.998806i \(-0.484445\pi\)
0.679440 + 0.733731i \(0.262223\pi\)
\(18\) 0.326358 + 1.85087i 0.0769234 + 0.436254i
\(19\) −2.15623 + 0.380202i −0.494673 + 0.0872243i −0.415419 − 0.909630i \(-0.636365\pi\)
−0.0792544 + 0.996854i \(0.525254\pi\)
\(20\) 3.03864 + 1.58283i 0.679460 + 0.353932i
\(21\) −2.46931 − 2.07200i −0.538847 − 0.452146i
\(22\) 6.62976 + 5.56303i 1.41347 + 1.18604i
\(23\) 1.04986 + 1.81841i 0.218911 + 0.379166i 0.954475 − 0.298290i \(-0.0964161\pi\)
−0.735564 + 0.677455i \(0.763083\pi\)
\(24\) −0.300682 + 0.826117i −0.0613765 + 0.168631i
\(25\) 4.82895 + 1.29664i 0.965789 + 0.259328i
\(26\) 6.07938 − 10.5298i 1.19226 − 2.06506i
\(27\) 0.866025 − 0.500000i 0.166667 − 0.0962250i
\(28\) 1.68926 + 4.64121i 0.319241 + 0.877107i
\(29\) −4.38440 − 2.53133i −0.814163 − 0.470057i 0.0342366 − 0.999414i \(-0.489100\pi\)
−0.848399 + 0.529357i \(0.822433\pi\)
\(30\) 0.549636 − 4.16642i 0.100349 − 0.760681i
\(31\) − 1.97280i − 0.354325i −0.984182 − 0.177162i \(-0.943308\pi\)
0.984182 − 0.177162i \(-0.0566918\pi\)
\(32\) 5.44386 − 4.56794i 0.962348 − 0.807506i
\(33\) 1.57496 − 4.32718i 0.274166 − 0.753265i
\(34\) 1.04289 + 5.91450i 0.178853 + 1.01433i
\(35\) 3.87118 + 6.08007i 0.654349 + 1.02772i
\(36\) −1.53223 −0.255372
\(37\) −6.01492 − 0.905962i −0.988846 − 0.148939i
\(38\) − 4.11499i − 0.667539i
\(39\) −6.37113 − 1.12340i −1.02020 − 0.179888i
\(40\) 1.19712 − 1.55926i 0.189281 − 0.246541i
\(41\) 0.888860 + 0.323518i 0.138817 + 0.0505251i 0.410494 − 0.911863i \(-0.365356\pi\)
−0.271678 + 0.962388i \(0.587578\pi\)
\(42\) 4.64087 − 3.89416i 0.716102 − 0.600881i
\(43\) −4.16467 −0.635106 −0.317553 − 0.948240i \(-0.602861\pi\)
−0.317553 + 0.948240i \(0.602861\pi\)
\(44\) −5.40502 + 4.53535i −0.814838 + 0.683730i
\(45\) −2.18319 + 0.483398i −0.325451 + 0.0720608i
\(46\) −3.70828 + 1.34970i −0.546756 + 0.199003i
\(47\) 9.11856 − 5.26460i 1.33008 − 0.767921i 0.344767 − 0.938688i \(-0.387958\pi\)
0.985311 + 0.170767i \(0.0546246\pi\)
\(48\) −4.08481 − 2.35837i −0.589591 − 0.340401i
\(49\) −0.588780 + 3.33914i −0.0841114 + 0.477019i
\(50\) −3.97588 + 8.51458i −0.562274 + 1.20414i
\(51\) 2.76740 − 1.59776i 0.387514 − 0.223731i
\(52\) 7.59352 + 6.37172i 1.05303 + 0.883598i
\(53\) −6.76718 + 8.06482i −0.929544 + 1.10779i 0.0644025 + 0.997924i \(0.479486\pi\)
−0.993947 + 0.109864i \(0.964959\pi\)
\(54\) 0.642801 + 1.76608i 0.0874741 + 0.240333i
\(55\) −6.27047 + 8.16738i −0.845511 + 1.10129i
\(56\) 2.79080 − 0.492093i 0.372936 − 0.0657587i
\(57\) −2.05745 + 0.748851i −0.272516 + 0.0991878i
\(58\) 6.11606 − 7.28884i 0.803078 − 0.957071i
\(59\) −5.54526 + 6.60858i −0.721931 + 0.860364i −0.994817 − 0.101682i \(-0.967577\pi\)
0.272886 + 0.962046i \(0.412022\pi\)
\(60\) 3.26733 + 1.03113i 0.421810 + 0.133118i
\(61\) 2.67590 − 7.35198i 0.342614 − 0.941324i −0.642019 − 0.766689i \(-0.721903\pi\)
0.984633 − 0.174636i \(-0.0558748\pi\)
\(62\) 3.65139 + 0.643839i 0.463727 + 0.0817676i
\(63\) −2.79159 − 1.61173i −0.351708 − 0.203058i
\(64\) 1.96129 + 3.39706i 0.245161 + 0.424632i
\(65\) 12.8298 + 6.68305i 1.59134 + 0.828931i
\(66\) 7.49505 + 4.32727i 0.922576 + 0.532650i
\(67\) 1.70753 + 2.03496i 0.208608 + 0.248610i 0.860196 − 0.509964i \(-0.170341\pi\)
−0.651588 + 0.758573i \(0.725897\pi\)
\(68\) −4.89628 −0.593761
\(69\) 1.34968 + 1.60848i 0.162482 + 0.193638i
\(70\) −12.5168 + 5.18077i −1.49604 + 0.619220i
\(71\) 2.38075 + 13.5019i 0.282543 + 1.60238i 0.713934 + 0.700213i \(0.246912\pi\)
−0.431391 + 0.902165i \(0.641977\pi\)
\(72\) −0.152660 + 0.865780i −0.0179912 + 0.102033i
\(73\) − 11.6066i − 1.35845i −0.733931 − 0.679224i \(-0.762317\pi\)
0.733931 − 0.679224i \(-0.237683\pi\)
\(74\) 3.63984 − 10.8372i 0.423122 − 1.25979i
\(75\) 4.98074 + 0.438403i 0.575127 + 0.0506225i
\(76\) 3.30385 + 0.582557i 0.378977 + 0.0668239i
\(77\) −14.6181 + 2.57757i −1.66589 + 0.293742i
\(78\) 4.15854 − 11.4255i 0.470862 − 1.29368i
\(79\) 0.378412 + 0.450973i 0.0425746 + 0.0507385i 0.786912 − 0.617066i \(-0.211679\pi\)
−0.744337 + 0.667804i \(0.767234\pi\)
\(80\) 7.12335 + 7.77789i 0.796415 + 0.869594i
\(81\) 0.766044 − 0.642788i 0.0851160 − 0.0714208i
\(82\) −0.888877 + 1.53958i −0.0981601 + 0.170018i
\(83\) −4.56245 − 12.5352i −0.500794 − 1.37592i −0.890501 − 0.454981i \(-0.849646\pi\)
0.389707 − 0.920939i \(-0.372576\pi\)
\(84\) 2.46954 + 4.27736i 0.269449 + 0.466699i
\(85\) −6.97644 + 1.54471i −0.756701 + 0.167547i
\(86\) 1.35918 − 7.70827i 0.146564 − 0.831204i
\(87\) −4.75735 − 1.73154i −0.510042 − 0.185640i
\(88\) 2.02416 + 3.50595i 0.215776 + 0.373736i
\(89\) 5.57899 − 6.64878i 0.591371 − 0.704769i −0.384498 − 0.923126i \(-0.625625\pi\)
0.975869 + 0.218357i \(0.0700697\pi\)
\(90\) −0.182205 − 4.19857i −0.0192061 − 0.442568i
\(91\) 7.13243 + 19.5962i 0.747682 + 2.05424i
\(92\) −0.558672 − 3.16838i −0.0582455 − 0.330327i
\(93\) −0.342573 − 1.94283i −0.0355231 − 0.201462i
\(94\) 6.76818 + 18.5954i 0.698084 + 1.91797i
\(95\) 4.89126 − 0.212266i 0.501832 − 0.0217780i
\(96\) 4.56794 − 5.44386i 0.466214 − 0.555612i
\(97\) 3.05850 + 5.29748i 0.310544 + 0.537877i 0.978480 − 0.206341i \(-0.0661556\pi\)
−0.667936 + 0.744218i \(0.732822\pi\)
\(98\) −5.98815 − 2.17951i −0.604895 − 0.220164i
\(99\) 0.799631 − 4.53493i 0.0803659 − 0.455778i
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.4.9 ✓ 240
5.4 even 2 inner 555.2.bp.a.4.32 yes 240
37.28 even 18 inner 555.2.bp.a.139.32 yes 240
185.139 even 18 inner 555.2.bp.a.139.9 yes 240
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
555.2.bp.a.4.9 ✓ 240 1.1 even 1 trivial
555.2.bp.a.4.32 yes 240 5.4 even 2 inner
555.2.bp.a.139.9 yes 240 185.139 even 18 inner
555.2.bp.a.139.32 yes 240 37.28 even 18 inner