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Properties

Label 585.2.i.e.391.8
Level $585$
Weight $2$
Character 585.391
Analytic conductor $4.671$
Analytic rank $0$
Dimension $16$
Inner twists $2$

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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] = [585,2,Mod(196,585)] 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("585.196"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(585, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([4, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 585 = 3^{2} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 585.i (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16,-3,1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.67124851824\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 5 x^{15} + 20 x^{14} - 44 x^{13} + 96 x^{12} - 107 x^{11} + 178 x^{10} - 19 x^{9} + 231 x^{8} + \cdots + 268 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 391.8
Root \(1.79305 - 1.53983i\) of defining polynomial
Character \(\chi\) \(=\) 585.391
Dual form 585.2.i.e.196.8

$q$-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+(1.29305 + 2.23963i) q^{2} +(1.68259 - 0.410979i) q^{3} +(-2.34397 + 4.05987i) q^{4} +(0.500000 - 0.866025i) q^{5} +(3.09611 + 3.23696i) q^{6} +(2.13745 + 3.70217i) q^{7} -6.95128 q^{8} +(2.66219 - 1.38302i) q^{9} +2.58610 q^{10} +(-2.19223 - 3.79706i) q^{11} +(-2.27540 + 7.79440i) q^{12} +(0.500000 - 0.866025i) q^{13} +(-5.52766 + 9.57419i) q^{14} +(0.485374 - 1.66265i) q^{15} +(-4.30043 - 7.44856i) q^{16} +0.619151 q^{17} +(6.53980 + 4.17402i) q^{18} -2.61288 q^{19} +(2.34397 + 4.05987i) q^{20} +(5.11795 + 5.35077i) q^{21} +(5.66934 - 9.81959i) q^{22} +(-2.71443 + 4.70154i) q^{23} +(-11.6961 + 2.85683i) q^{24} +(-0.500000 - 0.866025i) q^{25} +2.58610 q^{26} +(3.91098 - 3.42115i) q^{27} -20.0404 q^{28} +(-3.47887 - 6.02558i) q^{29} +(4.35134 - 1.06284i) q^{30} +(3.44304 - 5.96353i) q^{31} +(4.17008 - 7.22278i) q^{32} +(-5.24913 - 5.48791i) q^{33} +(0.800595 + 1.38667i) q^{34} +4.27489 q^{35} +(-0.625222 + 14.0499i) q^{36} -4.02164 q^{37} +(-3.37859 - 5.85188i) q^{38} +(0.485374 - 1.66265i) q^{39} +(-3.47564 + 6.01998i) q^{40} +(2.03714 - 3.52842i) q^{41} +(-5.36597 + 18.3811i) q^{42} +(5.01845 + 8.69221i) q^{43} +20.5541 q^{44} +(0.133368 - 2.99703i) q^{45} -14.0396 q^{46} +(0.202993 + 0.351595i) q^{47} +(-10.2970 - 10.7655i) q^{48} +(-5.63736 + 9.76419i) q^{49} +(1.29305 - 2.23963i) q^{50} +(1.04178 - 0.254458i) q^{51} +(2.34397 + 4.05987i) q^{52} -8.71205 q^{53} +(12.7192 + 4.33542i) q^{54} -4.38446 q^{55} +(-14.8580 - 25.7348i) q^{56} +(-4.39639 + 1.07384i) q^{57} +(8.99672 - 15.5828i) q^{58} +(5.36073 - 9.28505i) q^{59} +(5.61245 + 5.86776i) q^{60} +(-2.53609 - 4.39263i) q^{61} +17.8081 q^{62} +(10.8105 + 6.89975i) q^{63} +4.36679 q^{64} +(-0.500000 - 0.866025i) q^{65} +(5.50350 - 18.8523i) q^{66} +(-7.27894 + 12.6075i) q^{67} +(-1.45127 + 2.51367i) q^{68} +(-2.63503 + 9.02632i) q^{69} +(5.52766 + 9.57419i) q^{70} +6.57523 q^{71} +(-18.5056 + 9.61373i) q^{72} +16.7313 q^{73} +(-5.20019 - 9.00700i) q^{74} +(-1.19721 - 1.25167i) q^{75} +(6.12450 - 10.6079i) q^{76} +(9.37156 - 16.2320i) q^{77} +(4.35134 - 1.06284i) q^{78} +(-7.27288 - 12.5970i) q^{79} -8.60086 q^{80} +(5.17453 - 7.36371i) q^{81} +10.5365 q^{82} +(1.52057 + 2.63370i) q^{83} +(-33.7197 + 8.23620i) q^{84} +(0.309576 - 0.536201i) q^{85} +(-12.9782 + 22.4790i) q^{86} +(-8.32989 - 8.70881i) q^{87} +(15.2388 + 26.3944i) q^{88} +7.27070 q^{89} +(6.88470 - 3.57663i) q^{90} +4.27489 q^{91} +(-12.7251 - 22.0405i) q^{92} +(3.34233 - 11.4492i) q^{93} +(-0.524962 + 0.909260i) q^{94} +(-1.30644 + 2.26282i) q^{95} +(4.04810 - 13.8668i) q^{96} +(4.81738 + 8.34394i) q^{97} -29.1576 q^{98} +(-11.0875 - 7.07660i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 3 q^{2} + q^{3} - 9 q^{4} + 8 q^{5} + 8 q^{6} + 11 q^{7} - 12 q^{8} + q^{9} - 6 q^{10} - 6 q^{11} + 2 q^{12} + 8 q^{13} - 10 q^{14} + 5 q^{15} - 11 q^{16} - 4 q^{17} + 5 q^{18} - 20 q^{19} + 9 q^{20}+ \cdots - 43 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/585\mathbb{Z}\right)^\times\).

\(n\) \(326\) \(352\) \(496\)
\(\chi(n)\) \(e\left(\frac{1}{3}\right)\) \(1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.29305 + 2.23963i 0.914326 + 1.58366i 0.807885 + 0.589340i \(0.200612\pi\)
0.106440 + 0.994319i \(0.466055\pi\)
\(3\) 1.68259 − 0.410979i 0.971442 − 0.237279i
\(4\) −2.34397 + 4.05987i −1.17198 + 2.02993i
\(5\) 0.500000 − 0.866025i 0.223607 − 0.387298i
\(6\) 3.09611 + 3.23696i 1.26398 + 1.32148i
\(7\) 2.13745 + 3.70217i 0.807879 + 1.39929i 0.914330 + 0.404969i \(0.132718\pi\)
−0.106451 + 0.994318i \(0.533949\pi\)
\(8\) −6.95128 −2.45765
\(9\) 2.66219 − 1.38302i 0.887397 − 0.461006i
\(10\) 2.58610 0.817798
\(11\) −2.19223 − 3.79706i −0.660983 − 1.14486i −0.980358 − 0.197227i \(-0.936806\pi\)
0.319375 − 0.947628i \(-0.396527\pi\)
\(12\) −2.27540 + 7.79440i −0.656852 + 2.25005i
\(13\) 0.500000 − 0.866025i 0.138675 − 0.240192i
\(14\) −5.52766 + 9.57419i −1.47733 + 2.55881i
\(15\) 0.485374 − 1.66265i 0.125323 − 0.429295i
\(16\) −4.30043 − 7.44856i −1.07511 − 1.86214i
\(17\) 0.619151 0.150166 0.0750831 − 0.997177i \(-0.476078\pi\)
0.0750831 + 0.997177i \(0.476078\pi\)
\(18\) 6.53980 + 4.17402i 1.54145 + 0.983825i
\(19\) −2.61288 −0.599435 −0.299718 − 0.954028i \(-0.596892\pi\)
−0.299718 + 0.954028i \(0.596892\pi\)
\(20\) 2.34397 + 4.05987i 0.524127 + 0.907814i
\(21\) 5.11795 + 5.35077i 1.11683 + 1.16763i
\(22\) 5.66934 − 9.81959i 1.20871 − 2.09354i
\(23\) −2.71443 + 4.70154i −0.565998 + 0.980338i 0.430958 + 0.902372i \(0.358176\pi\)
−0.996956 + 0.0779658i \(0.975157\pi\)
\(24\) −11.6961 + 2.85683i −2.38746 + 0.583148i
\(25\) −0.500000 − 0.866025i −0.100000 − 0.173205i
\(26\) 2.58610 0.507177
\(27\) 3.91098 − 3.42115i 0.752668 − 0.658401i
\(28\) −20.0404 −3.78728
\(29\) −3.47887 − 6.02558i −0.646010 − 1.11892i −0.984067 − 0.177796i \(-0.943103\pi\)
0.338057 − 0.941125i \(-0.390230\pi\)
\(30\) 4.35134 − 1.06284i 0.794443 − 0.194046i
\(31\) 3.44304 − 5.96353i 0.618389 − 1.07108i −0.371391 − 0.928477i \(-0.621119\pi\)
0.989780 − 0.142605i \(-0.0455477\pi\)
\(32\) 4.17008 − 7.22278i 0.737172 − 1.27682i
\(33\) −5.24913 − 5.48791i −0.913757 − 0.955323i
\(34\) 0.800595 + 1.38667i 0.137301 + 0.237812i
\(35\) 4.27489 0.722589
\(36\) −0.625222 + 14.0499i −0.104204 + 2.34165i
\(37\) −4.02164 −0.661154 −0.330577 − 0.943779i \(-0.607243\pi\)
−0.330577 + 0.943779i \(0.607243\pi\)
\(38\) −3.37859 − 5.85188i −0.548079 − 0.949301i
\(39\) 0.485374 − 1.66265i 0.0777221 − 0.266237i
\(40\) −3.47564 + 6.01998i −0.549547 + 0.951843i
\(41\) 2.03714 − 3.52842i 0.318147 − 0.551047i −0.661954 − 0.749544i \(-0.730273\pi\)
0.980101 + 0.198497i \(0.0636060\pi\)
\(42\) −5.36597 + 18.3811i −0.827987 + 2.83627i
\(43\) 5.01845 + 8.69221i 0.765307 + 1.32555i 0.940084 + 0.340942i \(0.110746\pi\)
−0.174778 + 0.984608i \(0.555921\pi\)
\(44\) 20.5541 3.09864
\(45\) 0.133368 − 2.99703i 0.0198814 − 0.446771i
\(46\) −14.0396 −2.07003
\(47\) 0.202993 + 0.351595i 0.0296096 + 0.0512854i 0.880450 − 0.474138i \(-0.157240\pi\)
−0.850841 + 0.525423i \(0.823907\pi\)
\(48\) −10.2970 − 10.7655i −1.48625 − 1.55386i
\(49\) −5.63736 + 9.76419i −0.805337 + 1.39488i
\(50\) 1.29305 − 2.23963i 0.182865 − 0.316732i
\(51\) 1.04178 − 0.254458i 0.145878 − 0.0356313i
\(52\) 2.34397 + 4.05987i 0.325050 + 0.563003i
\(53\) −8.71205 −1.19669 −0.598346 − 0.801238i \(-0.704175\pi\)
−0.598346 + 0.801238i \(0.704175\pi\)
\(54\) 12.7192 + 4.33542i 1.73087 + 0.589976i
\(55\) −4.38446 −0.591201
\(56\) −14.8580 − 25.7348i −1.98548 − 3.43895i
\(57\) −4.39639 + 1.07384i −0.582316 + 0.142233i
\(58\) 8.99672 − 15.5828i 1.18133 − 2.04612i
\(59\) 5.36073 − 9.28505i 0.697907 − 1.20881i −0.271284 − 0.962499i \(-0.587448\pi\)
0.969191 − 0.246311i \(-0.0792185\pi\)
\(60\) 5.61245 + 5.86776i 0.724564 + 0.757524i
\(61\) −2.53609 − 4.39263i −0.324713 − 0.562419i 0.656742 − 0.754116i \(-0.271934\pi\)
−0.981454 + 0.191697i \(0.938601\pi\)
\(62\) 17.8081 2.26164
\(63\) 10.8105 + 6.89975i 1.36199 + 0.869287i
\(64\) 4.36679 0.545849
\(65\) −0.500000 − 0.866025i −0.0620174 − 0.107417i
\(66\) 5.50350 − 18.8523i 0.677435 − 2.32056i
\(67\) −7.27894 + 12.6075i −0.889264 + 1.54025i −0.0485164 + 0.998822i \(0.515449\pi\)
−0.840747 + 0.541428i \(0.817884\pi\)
\(68\) −1.45127 + 2.51367i −0.175992 + 0.304828i
\(69\) −2.63503 + 9.02632i −0.317221 + 1.08664i
\(70\) 5.52766 + 9.57419i 0.660682 + 1.14433i
\(71\) 6.57523 0.780336 0.390168 − 0.920744i \(-0.372417\pi\)
0.390168 + 0.920744i \(0.372417\pi\)
\(72\) −18.5056 + 9.61373i −2.18091 + 1.13299i
\(73\) 16.7313 1.95825 0.979125 − 0.203257i \(-0.0651527\pi\)
0.979125 + 0.203257i \(0.0651527\pi\)
\(74\) −5.20019 − 9.00700i −0.604510 − 1.04704i
\(75\) −1.19721 − 1.25167i −0.138242 − 0.144531i
\(76\) 6.12450 − 10.6079i 0.702528 − 1.21681i
\(77\) 9.37156 − 16.2320i 1.06799 − 1.84981i
\(78\) 4.35134 − 1.06284i 0.492693 − 0.120342i
\(79\) −7.27288 − 12.5970i −0.818263 − 1.41727i −0.906961 − 0.421214i \(-0.861604\pi\)
0.0886982 − 0.996059i \(-0.471729\pi\)
\(80\) −8.60086 −0.961605
\(81\) 5.17453 − 7.36371i 0.574948 − 0.818190i
\(82\) 10.5365 1.16356
\(83\) 1.52057 + 2.63370i 0.166904 + 0.289086i 0.937330 − 0.348444i \(-0.113290\pi\)
−0.770426 + 0.637530i \(0.779956\pi\)
\(84\) −33.7197 + 8.23620i −3.67912 + 0.898643i
\(85\) 0.309576 − 0.536201i 0.0335782 − 0.0581591i
\(86\) −12.9782 + 22.4790i −1.39948 + 2.42397i
\(87\) −8.32989 − 8.70881i −0.893058 − 0.933682i
\(88\) 15.2388 + 26.3944i 1.62446 + 2.81365i
\(89\) 7.27070 0.770693 0.385347 − 0.922772i \(-0.374082\pi\)
0.385347 + 0.922772i \(0.374082\pi\)
\(90\) 6.88470 − 3.57663i 0.725712 − 0.377009i
\(91\) 4.27489 0.448131
\(92\) −12.7251 − 22.0405i −1.32668 − 2.29788i
\(93\) 3.34233 − 11.4492i 0.346584 − 1.18722i
\(94\) −0.524962 + 0.909260i −0.0541457 + 0.0937830i
\(95\) −1.30644 + 2.26282i −0.134038 + 0.232160i
\(96\) 4.04810 − 13.8668i 0.413157 − 1.41527i
\(97\) 4.81738 + 8.34394i 0.489130 + 0.847199i 0.999922 − 0.0125059i \(-0.00398085\pi\)
−0.510791 + 0.859705i \(0.670648\pi\)
\(98\) −29.1576 −2.94536
\(99\) −11.0875 − 7.07660i −1.11434 − 0.711225i
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 585.2.i.e.391.8 yes 16
3.2 odd 2 1755.2.i.f.1171.1 16
9.2 odd 6 1755.2.i.f.586.1 16
9.4 even 3 5265.2.a.bf.1.1 8
9.5 odd 6 5265.2.a.ba.1.8 8
9.7 even 3 inner 585.2.i.e.196.8 ✓ 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
585.2.i.e.196.8 ✓ 16 9.7 even 3 inner
585.2.i.e.391.8 yes 16 1.1 even 1 trivial
1755.2.i.f.586.1 16 9.2 odd 6
1755.2.i.f.1171.1 16 3.2 odd 2
5265.2.a.ba.1.8 8 9.5 odd 6
5265.2.a.bf.1.1 8 9.4 even 3