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Properties

Label 585.2.bt.b.571.35
Level $585$
Weight $2$
Character 585.571
Analytic conductor $4.671$
Analytic rank $0$
Dimension $108$
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] = [585,2,Mod(376,585)] 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("585.376"); 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, 3])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 585 = 3^{2} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 585.bt (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [108] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.67124851824\)
Analytic rank: \(0\)
Dimension: \(108\)
Relative dimension: \(54\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 571.35
Character \(\chi\) \(=\) 585.571
Dual form 585.2.bt.b.376.35

$q$-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+(0.774258 - 0.447018i) q^{2} +(-0.513274 + 1.65425i) q^{3} +(-0.600350 + 1.03984i) q^{4} +(0.866025 + 0.500000i) q^{5} +(0.342074 + 1.51026i) q^{6} +(-1.76954 + 1.02164i) q^{7} +2.86154i q^{8} +(-2.47310 - 1.69817i) q^{9} +0.894036 q^{10} +(-0.659790 + 0.380930i) q^{11} +(-1.41201 - 1.52685i) q^{12} +(2.22492 - 2.83720i) q^{13} +(-0.913387 + 1.58203i) q^{14} +(-1.27163 + 1.17599i) q^{15} +(0.0784618 + 0.135900i) q^{16} -2.67783 q^{17} +(-2.67393 - 0.209302i) q^{18} +7.38079i q^{19} +(-1.03984 + 0.600350i) q^{20} +(-0.781798 - 3.45165i) q^{21} +(-0.340565 + 0.589876i) q^{22} +(-1.72917 + 2.99502i) q^{23} +(-4.73371 - 1.46876i) q^{24} +(0.500000 + 0.866025i) q^{25} +(0.454381 - 3.19131i) q^{26} +(4.07858 - 3.21950i) q^{27} -2.45338i q^{28} +(-4.28619 - 7.42389i) q^{29} +(-0.458886 + 1.47896i) q^{30} +(-7.28497 - 4.20598i) q^{31} +(-4.83484 - 2.79139i) q^{32} +(-0.291501 - 1.28698i) q^{33} +(-2.07333 + 1.19704i) q^{34} -2.04329 q^{35} +(3.25054 - 1.55212i) q^{36} +3.24522i q^{37} +(3.29935 + 5.71463i) q^{38} +(3.55146 + 5.13684i) q^{39} +(-1.43077 + 2.47817i) q^{40} +(5.25284 + 3.03273i) q^{41} +(-2.14826 - 2.32299i) q^{42} +(5.42269 + 9.39238i) q^{43} -0.914765i q^{44} +(-1.29268 - 2.70721i) q^{45} +3.09189i q^{46} +(-0.719689 + 0.415513i) q^{47} +(-0.265085 + 0.0600417i) q^{48} +(-1.41248 + 2.44650i) q^{49} +(0.774258 + 0.447018i) q^{50} +(1.37446 - 4.42981i) q^{51} +(1.61450 + 4.01687i) q^{52} +12.9596 q^{53} +(1.71870 - 4.31592i) q^{54} -0.761860 q^{55} +(-2.92348 - 5.06361i) q^{56} +(-12.2097 - 3.78837i) q^{57} +(-6.63723 - 3.83201i) q^{58} +(8.57721 + 4.95206i) q^{59} +(-0.459408 - 2.02829i) q^{60} +(1.11958 + 1.93918i) q^{61} -7.52059 q^{62} +(6.11117 + 0.478351i) q^{63} -5.30506 q^{64} +(3.34544 - 1.34463i) q^{65} +(-0.801001 - 0.866149i) q^{66} +(0.874153 + 0.504693i) q^{67} +(1.60764 - 2.78451i) q^{68} +(-4.06697 - 4.39775i) q^{69} +(-1.58203 + 0.913387i) q^{70} +16.2829i q^{71} +(4.85938 - 7.07688i) q^{72} +8.99447i q^{73} +(1.45067 + 2.51264i) q^{74} +(-1.68926 + 0.382617i) q^{75} +(-7.67481 - 4.43105i) q^{76} +(0.778350 - 1.34814i) q^{77} +(5.04601 + 2.38968i) q^{78} +(-7.11738 - 12.3277i) q^{79} +0.156924i q^{80} +(3.23244 + 8.39948i) q^{81} +5.42274 q^{82} +(7.52521 - 4.34468i) q^{83} +(4.05850 + 1.25925i) q^{84} +(-2.31907 - 1.33892i) q^{85} +(8.39713 + 4.84808i) q^{86} +(14.4810 - 3.27994i) q^{87} +(-1.09005 - 1.88802i) q^{88} -6.84329i q^{89} +(-2.21104 - 1.51823i) q^{90} +(-1.03847 + 7.29363i) q^{91} +(-2.07622 - 3.59611i) q^{92} +(10.6969 - 9.89235i) q^{93} +(-0.371484 + 0.643428i) q^{94} +(-3.69039 + 6.39195i) q^{95} +(7.09926 - 6.56529i) q^{96} +(-0.0788540 + 0.0455264i) q^{97} +2.52563i q^{98} +(2.27861 + 0.178358i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 108 q - 6 q^{3} + 52 q^{4} - 14 q^{9} - 8 q^{10} - 36 q^{12} - 4 q^{13} - 8 q^{14} - 64 q^{16} + 36 q^{17} + 24 q^{22} - 22 q^{23} + 54 q^{25} + 40 q^{26} + 48 q^{27} - 16 q^{29} + 20 q^{30} - 40 q^{35}+ \cdots - 56 q^{95}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.774258 − 0.447018i 0.547483 − 0.316090i −0.200623 − 0.979669i \(-0.564297\pi\)
0.748106 + 0.663579i \(0.230963\pi\)
\(3\) −0.513274 + 1.65425i −0.296339 + 0.955083i
\(4\) −0.600350 + 1.03984i −0.300175 + 0.519918i
\(5\) 0.866025 + 0.500000i 0.387298 + 0.223607i
\(6\) 0.342074 + 1.51026i 0.139651 + 0.616561i
\(7\) −1.76954 + 1.02164i −0.668823 + 0.386145i −0.795631 − 0.605782i \(-0.792860\pi\)
0.126807 + 0.991927i \(0.459527\pi\)
\(8\) 2.86154i 1.01171i
\(9\) −2.47310 − 1.69817i −0.824366 − 0.566057i
\(10\) 0.894036 0.282719
\(11\) −0.659790 + 0.380930i −0.198934 + 0.114855i −0.596158 − 0.802867i \(-0.703307\pi\)
0.397224 + 0.917722i \(0.369974\pi\)
\(12\) −1.41201 − 1.52685i −0.407611 − 0.440764i
\(13\) 2.22492 − 2.83720i 0.617082 − 0.786899i
\(14\) −0.913387 + 1.58203i −0.244113 + 0.422816i
\(15\) −1.27163 + 1.17599i −0.328335 + 0.303639i
\(16\) 0.0784618 + 0.135900i 0.0196155 + 0.0339750i
\(17\) −2.67783 −0.649470 −0.324735 − 0.945805i \(-0.605275\pi\)
−0.324735 + 0.945805i \(0.605275\pi\)
\(18\) −2.67393 − 0.209302i −0.630251 − 0.0493328i
\(19\) 7.38079i 1.69327i 0.532175 + 0.846634i \(0.321375\pi\)
−0.532175 + 0.846634i \(0.678625\pi\)
\(20\) −1.03984 + 0.600350i −0.232514 + 0.134242i
\(21\) −0.781798 − 3.45165i −0.170602 − 0.753212i
\(22\) −0.340565 + 0.589876i −0.0726088 + 0.125762i
\(23\) −1.72917 + 2.99502i −0.360558 + 0.624504i −0.988053 − 0.154116i \(-0.950747\pi\)
0.627495 + 0.778620i \(0.284080\pi\)
\(24\) −4.73371 − 1.46876i −0.966265 − 0.299808i
\(25\) 0.500000 + 0.866025i 0.100000 + 0.173205i
\(26\) 0.454381 − 3.19131i 0.0891114 − 0.625867i
\(27\) 4.07858 − 3.21950i 0.784923 − 0.619594i
\(28\) − 2.45338i − 0.463644i
\(29\) −4.28619 − 7.42389i −0.795925 − 1.37858i −0.922250 − 0.386593i \(-0.873652\pi\)
0.126326 − 0.991989i \(-0.459682\pi\)
\(30\) −0.458886 + 1.47896i −0.0837807 + 0.270020i
\(31\) −7.28497 − 4.20598i −1.30842 − 0.755416i −0.326586 − 0.945167i \(-0.605898\pi\)
−0.981832 + 0.189751i \(0.939232\pi\)
\(32\) −4.83484 − 2.79139i −0.854686 − 0.493453i
\(33\) −0.291501 − 1.28698i −0.0507438 − 0.224035i
\(34\) −2.07333 + 1.19704i −0.355574 + 0.205291i
\(35\) −2.04329 −0.345379
\(36\) 3.25054 − 1.55212i 0.541757 − 0.258687i
\(37\) 3.24522i 0.533511i 0.963764 + 0.266756i \(0.0859517\pi\)
−0.963764 + 0.266756i \(0.914048\pi\)
\(38\) 3.29935 + 5.71463i 0.535224 + 0.927036i
\(39\) 3.55146 + 5.13684i 0.568688 + 0.822553i
\(40\) −1.43077 + 2.47817i −0.226225 + 0.391833i
\(41\) 5.25284 + 3.03273i 0.820356 + 0.473633i 0.850539 − 0.525911i \(-0.176276\pi\)
−0.0301829 + 0.999544i \(0.509609\pi\)
\(42\) −2.14826 − 2.32299i −0.331484 − 0.358445i
\(43\) 5.42269 + 9.39238i 0.826953 + 1.43232i 0.900418 + 0.435026i \(0.143261\pi\)
−0.0734653 + 0.997298i \(0.523406\pi\)
\(44\) − 0.914765i − 0.137906i
\(45\) −1.29268 − 2.70721i −0.192702 − 0.403567i
\(46\) 3.09189i 0.455874i
\(47\) −0.719689 + 0.415513i −0.104977 + 0.0606088i −0.551570 − 0.834129i \(-0.685971\pi\)
0.446592 + 0.894738i \(0.352638\pi\)
\(48\) −0.265085 + 0.0600417i −0.0382617 + 0.00866628i
\(49\) −1.41248 + 2.44650i −0.201784 + 0.349499i
\(50\) 0.774258 + 0.447018i 0.109497 + 0.0632179i
\(51\) 1.37446 − 4.42981i 0.192463 − 0.620297i
\(52\) 1.61450 + 4.01687i 0.223891 + 0.557039i
\(53\) 12.9596 1.78014 0.890072 − 0.455819i \(-0.150654\pi\)
0.890072 + 0.455819i \(0.150654\pi\)
\(54\) 1.71870 − 4.31592i 0.233885 − 0.587323i
\(55\) −0.761860 −0.102729
\(56\) −2.92348 − 5.06361i −0.390666 − 0.676654i
\(57\) −12.2097 − 3.78837i −1.61721 − 0.501781i
\(58\) −6.63723 − 3.83201i −0.871511 − 0.503167i
\(59\) 8.57721 + 4.95206i 1.11666 + 0.644703i 0.940546 − 0.339667i \(-0.110314\pi\)
0.176112 + 0.984370i \(0.443648\pi\)
\(60\) −0.459408 − 2.02829i −0.0593094 − 0.261852i
\(61\) 1.11958 + 1.93918i 0.143348 + 0.248286i 0.928755 − 0.370693i \(-0.120880\pi\)
−0.785407 + 0.618979i \(0.787546\pi\)
\(62\) −7.52059 −0.955116
\(63\) 6.11117 + 0.478351i 0.769936 + 0.0602666i
\(64\) −5.30506 −0.663133
\(65\) 3.34544 − 1.34463i 0.414951 − 0.166781i
\(66\) −0.801001 − 0.866149i −0.0985964 − 0.106616i
\(67\) 0.874153 + 0.504693i 0.106795 + 0.0616580i 0.552446 − 0.833549i \(-0.313695\pi\)
−0.445651 + 0.895207i \(0.647028\pi\)
\(68\) 1.60764 − 2.78451i 0.194954 − 0.337671i
\(69\) −4.06697 − 4.39775i −0.489606 − 0.529427i
\(70\) −1.58203 + 0.913387i −0.189089 + 0.109171i
\(71\) 16.2829i 1.93242i 0.257747 + 0.966212i \(0.417020\pi\)
−0.257747 + 0.966212i \(0.582980\pi\)
\(72\) 4.85938 − 7.07688i 0.572684 − 0.834018i
\(73\) 8.99447i 1.05272i 0.850261 + 0.526362i \(0.176444\pi\)
−0.850261 + 0.526362i \(0.823556\pi\)
\(74\) 1.45067 + 2.51264i 0.168637 + 0.292089i
\(75\) −1.68926 + 0.382617i −0.195059 + 0.0441809i
\(76\) −7.67481 − 4.43105i −0.880361 − 0.508276i
\(77\) 0.778350 − 1.34814i 0.0887012 − 0.153635i
\(78\) 5.04601 + 2.38968i 0.571348 + 0.270578i
\(79\) −7.11738 − 12.3277i −0.800768 − 1.38697i −0.919111 − 0.393998i \(-0.871092\pi\)
0.118343 − 0.992973i \(-0.462242\pi\)
\(80\) 0.156924i 0.0175446i
\(81\) 3.23244 + 8.39948i 0.359160 + 0.933276i
\(82\) 5.42274 0.598842
\(83\) 7.52521 − 4.34468i 0.826000 − 0.476891i −0.0264814 − 0.999649i \(-0.508430\pi\)
0.852481 + 0.522758i \(0.175097\pi\)
\(84\) 4.05850 + 1.25925i 0.442819 + 0.137396i
\(85\) −2.31907 − 1.33892i −0.251539 − 0.145226i
\(86\) 8.39713 + 4.84808i 0.905485 + 0.522782i
\(87\) 14.4810 − 3.27994i 1.55252 − 0.351646i
\(88\) −1.09005 − 1.88802i −0.116199 − 0.201263i
\(89\) − 6.84329i − 0.725387i −0.931909 − 0.362693i \(-0.881857\pi\)
0.931909 − 0.362693i \(-0.118143\pi\)
\(90\) −2.21104 − 1.51823i −0.233064 − 0.160035i
\(91\) −1.03847 + 7.29363i −0.108861 + 0.764580i
\(92\) −2.07622 − 3.59611i −0.216461 − 0.374921i
\(93\) 10.6969 − 9.89235i 1.10922 − 1.02579i
\(94\) −0.371484 + 0.643428i −0.0383156 + 0.0663646i
\(95\) −3.69039 + 6.39195i −0.378626 + 0.655800i
\(96\) 7.09926 − 6.56529i 0.724566 − 0.670067i
\(97\) −0.0788540 + 0.0455264i −0.00800641 + 0.00462250i −0.503998 − 0.863705i \(-0.668138\pi\)
0.495991 + 0.868327i \(0.334805\pi\)
\(98\) 2.52563i 0.255127i
\(99\) 2.27861 + 0.178358i 0.229009 + 0.0179256i
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.bt.b.571.35 yes 108
9.7 even 3 inner 585.2.bt.b.376.20 ✓ 108
13.12 even 2 inner 585.2.bt.b.571.20 yes 108
117.25 even 6 inner 585.2.bt.b.376.35 yes 108
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
585.2.bt.b.376.20 ✓ 108 9.7 even 3 inner
585.2.bt.b.376.35 yes 108 117.25 even 6 inner
585.2.bt.b.571.20 yes 108 13.12 even 2 inner
585.2.bt.b.571.35 yes 108 1.1 even 1 trivial