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Properties

Label 567.2.bd.a.17.19
Level $567$
Weight $2$
Character 567.17
Analytic conductor $4.528$
Analytic rank $0$
Dimension $132$
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] = [567,2,Mod(17,567)] 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("567.17"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(567, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([11, 3])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 567 = 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 567.bd (of order \(18\), degree \(6\), not 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: \(4.52751779461\)
Analytic rank: \(0\)
Dimension: \(132\)
Relative dimension: \(22\) over \(\Q(\zeta_{18})\)
Twist minimal: no (minimal twist has level 189)
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label 17.19
Character \(\chi\) \(=\) 567.17
Dual form 567.2.bd.a.467.19

$q$-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)
 
\(f(q)\) \(=\) \(q+(2.06025 - 0.363278i) q^{2} +(2.23328 - 0.812849i) q^{4} +(-0.338534 + 0.123216i) q^{5} +(1.84499 - 1.89632i) q^{7} +(0.682329 - 0.393943i) q^{8} +(-0.652704 + 0.376839i) q^{10} +(2.14028 - 5.88038i) q^{11} +(0.429890 + 1.18111i) q^{13} +(3.11225 - 4.57714i) q^{14} +(-2.37852 + 1.99582i) q^{16} +(0.468609 + 0.811655i) q^{17} +(6.61941 + 3.82172i) q^{19} +(-0.655887 + 0.550354i) q^{20} +(2.27331 - 12.8926i) q^{22} +(-1.68421 - 0.296971i) q^{23} +(-3.73080 + 3.13051i) q^{25} +(1.31476 + 2.27722i) q^{26} +(2.57896 - 5.73471i) q^{28} +(0.731568 - 2.00997i) q^{29} +(1.98478 + 5.45314i) q^{31} +(-5.18821 + 6.18306i) q^{32} +(1.26031 + 1.50198i) q^{34} +(-0.390934 + 0.869301i) q^{35} -3.33513 q^{37} +(15.0260 + 5.46902i) q^{38} +(-0.182451 + 0.217437i) q^{40} +(-8.66669 + 3.15442i) q^{41} +(-0.688669 - 3.90564i) q^{43} -14.8723i q^{44} -3.57777 q^{46} +(-4.74048 - 1.72539i) q^{47} +(-0.192044 - 6.99737i) q^{49} +(-6.54914 + 7.80496i) q^{50} +(1.92013 + 2.28833i) q^{52} +(-3.01037 - 1.73804i) q^{53} +2.25443i q^{55} +(0.511847 - 2.02073i) q^{56} +(0.777037 - 4.40680i) q^{58} +(-4.89583 - 4.10809i) q^{59} +(-3.57768 + 9.82960i) q^{61} +(6.07015 + 10.5138i) q^{62} +(-5.33790 + 9.24552i) q^{64} +(-0.291065 - 0.346878i) q^{65} +(-0.0504771 + 0.286270i) q^{67} +(1.70629 + 1.43175i) q^{68} +(-0.489624 + 1.93300i) q^{70} +(4.18316 + 2.41515i) q^{71} -3.69086i q^{73} +(-6.87121 + 1.21158i) q^{74} +(17.8895 + 3.15440i) q^{76} +(-7.20228 - 14.9079i) q^{77} +(1.79265 + 10.1666i) q^{79} +(0.559294 - 0.968725i) q^{80} +(-16.7096 + 9.64732i) q^{82} +(6.06917 + 2.20900i) q^{83} +(-0.258650 - 0.217033i) q^{85} +(-2.83766 - 7.79642i) q^{86} +(-0.856156 - 4.85550i) q^{88} +(-2.56195 + 4.43743i) q^{89} +(3.03291 + 1.36393i) q^{91} +(-4.00271 + 0.705785i) q^{92} +(-10.3934 - 1.83263i) q^{94} +(-2.71179 - 0.478163i) q^{95} +(7.91567 - 1.39575i) q^{97} +(-2.93765 - 14.3466i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 132 q + 3 q^{2} - 3 q^{4} + 9 q^{5} - 6 q^{7} + 18 q^{8} - 9 q^{10} - 9 q^{11} + 42 q^{14} - 15 q^{16} + 9 q^{17} - 9 q^{19} + 18 q^{20} - 12 q^{22} - 30 q^{23} - 3 q^{25} - 12 q^{28} - 6 q^{29} - 9 q^{31}+ \cdots + 180 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/567\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(407\)
\(\chi(n)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{11}{18}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.06025 − 0.363278i 1.45682 − 0.256876i 0.611545 − 0.791210i \(-0.290548\pi\)
0.845274 + 0.534333i \(0.179437\pi\)
\(3\) 0 0
\(4\) 2.23328 − 0.812849i 1.11664 − 0.406425i
\(5\) −0.338534 + 0.123216i −0.151397 + 0.0551040i −0.416607 − 0.909087i \(-0.636781\pi\)
0.265210 + 0.964191i \(0.414559\pi\)
\(6\) 0 0
\(7\) 1.84499 − 1.89632i 0.697340 − 0.716741i
\(8\) 0.682329 − 0.393943i 0.241240 − 0.139280i
\(9\) 0 0
\(10\) −0.652704 + 0.376839i −0.206403 + 0.119167i
\(11\) 2.14028 − 5.88038i 0.645320 − 1.77300i 0.0109905 − 0.999940i \(-0.496502\pi\)
0.634330 − 0.773063i \(-0.281276\pi\)
\(12\) 0 0
\(13\) 0.429890 + 1.18111i 0.119230 + 0.327582i 0.984923 − 0.172994i \(-0.0553440\pi\)
−0.865693 + 0.500576i \(0.833122\pi\)
\(14\) 3.11225 − 4.57714i 0.831784 − 1.22329i
\(15\) 0 0
\(16\) −2.37852 + 1.99582i −0.594631 + 0.498954i
\(17\) 0.468609 + 0.811655i 0.113654 + 0.196855i 0.917241 − 0.398332i \(-0.130411\pi\)
−0.803587 + 0.595188i \(0.797078\pi\)
\(18\) 0 0
\(19\) 6.61941 + 3.82172i 1.51860 + 0.876762i 0.999760 + 0.0218885i \(0.00696789\pi\)
0.518836 + 0.854874i \(0.326365\pi\)
\(20\) −0.655887 + 0.550354i −0.146661 + 0.123063i
\(21\) 0 0
\(22\) 2.27331 − 12.8926i 0.484672 − 2.74871i
\(23\) −1.68421 − 0.296971i −0.351181 − 0.0619228i −0.00472501 − 0.999989i \(-0.501504\pi\)
−0.346456 + 0.938066i \(0.612615\pi\)
\(24\) 0 0
\(25\) −3.73080 + 3.13051i −0.746160 + 0.626102i
\(26\) 1.31476 + 2.27722i 0.257845 + 0.446600i
\(27\) 0 0
\(28\) 2.57896 − 5.73471i 0.487378 − 1.08376i
\(29\) 0.731568 − 2.00997i 0.135849 − 0.373241i −0.853051 − 0.521828i \(-0.825250\pi\)
0.988899 + 0.148587i \(0.0474725\pi\)
\(30\) 0 0
\(31\) 1.98478 + 5.45314i 0.356477 + 0.979412i 0.980242 + 0.197801i \(0.0633799\pi\)
−0.623765 + 0.781612i \(0.714398\pi\)
\(32\) −5.18821 + 6.18306i −0.917154 + 1.09302i
\(33\) 0 0
\(34\) 1.26031 + 1.50198i 0.216141 + 0.257587i
\(35\) −0.390934 + 0.869301i −0.0660799 + 0.146939i
\(36\) 0 0
\(37\) −3.33513 −0.548292 −0.274146 − 0.961688i \(-0.588395\pi\)
−0.274146 + 0.961688i \(0.588395\pi\)
\(38\) 15.0260 + 5.46902i 2.43754 + 0.887192i
\(39\) 0 0
\(40\) −0.182451 + 0.217437i −0.0288481 + 0.0343798i
\(41\) −8.66669 + 3.15442i −1.35351 + 0.492637i −0.914041 − 0.405621i \(-0.867055\pi\)
−0.439468 + 0.898258i \(0.644833\pi\)
\(42\) 0 0
\(43\) −0.688669 − 3.90564i −0.105021 − 0.595604i −0.991212 − 0.132284i \(-0.957769\pi\)
0.886191 − 0.463320i \(-0.153342\pi\)
\(44\) − 14.8723i − 2.24208i
\(45\) 0 0
\(46\) −3.57777 −0.527514
\(47\) −4.74048 − 1.72539i −0.691470 − 0.251674i −0.0277052 − 0.999616i \(-0.508820\pi\)
−0.663764 + 0.747942i \(0.731042\pi\)
\(48\) 0 0
\(49\) −0.192044 − 6.99737i −0.0274348 − 0.999624i
\(50\) −6.54914 + 7.80496i −0.926189 + 1.10379i
\(51\) 0 0
\(52\) 1.92013 + 2.28833i 0.266275 + 0.317334i
\(53\) −3.01037 − 1.73804i −0.413506 − 0.238738i 0.278789 − 0.960352i \(-0.410067\pi\)
−0.692295 + 0.721614i \(0.743400\pi\)
\(54\) 0 0
\(55\) 2.25443i 0.303987i
\(56\) 0.511847 − 2.02073i 0.0683984 − 0.270032i
\(57\) 0 0
\(58\) 0.777037 − 4.40680i 0.102030 − 0.578641i
\(59\) −4.89583 − 4.10809i −0.637383 − 0.534828i 0.265830 − 0.964020i \(-0.414354\pi\)
−0.903213 + 0.429192i \(0.858798\pi\)
\(60\) 0 0
\(61\) −3.57768 + 9.82960i −0.458075 + 1.25855i 0.468840 + 0.883283i \(0.344672\pi\)
−0.926916 + 0.375269i \(0.877550\pi\)
\(62\) 6.07015 + 10.5138i 0.770910 + 1.33526i
\(63\) 0 0
\(64\) −5.33790 + 9.24552i −0.667238 + 1.15569i
\(65\) −0.291065 − 0.346878i −0.0361022 − 0.0430249i
\(66\) 0 0
\(67\) −0.0504771 + 0.286270i −0.00616676 + 0.0349734i −0.987736 − 0.156134i \(-0.950097\pi\)
0.981569 + 0.191108i \(0.0612079\pi\)
\(68\) 1.70629 + 1.43175i 0.206918 + 0.173625i
\(69\) 0 0
\(70\) −0.489624 + 1.93300i −0.0585213 + 0.231037i
\(71\) 4.18316 + 2.41515i 0.496450 + 0.286625i 0.727246 − 0.686377i \(-0.240800\pi\)
−0.230797 + 0.973002i \(0.574133\pi\)
\(72\) 0 0
\(73\) − 3.69086i − 0.431983i −0.976395 − 0.215992i \(-0.930702\pi\)
0.976395 − 0.215992i \(-0.0692984\pi\)
\(74\) −6.87121 + 1.21158i −0.798762 + 0.140843i
\(75\) 0 0
\(76\) 17.8895 + 3.15440i 2.05207 + 0.361835i
\(77\) −7.20228 − 14.9079i −0.820776 − 1.69891i
\(78\) 0 0
\(79\) 1.79265 + 10.1666i 0.201689 + 1.14384i 0.902565 + 0.430553i \(0.141682\pi\)
−0.700876 + 0.713283i \(0.747207\pi\)
\(80\) 0.559294 − 0.968725i 0.0625309 − 0.108307i
\(81\) 0 0
\(82\) −16.7096 + 9.64732i −1.84527 + 1.06537i
\(83\) 6.06917 + 2.20900i 0.666178 + 0.242469i 0.652901 − 0.757443i \(-0.273552\pi\)
0.0132766 + 0.999912i \(0.495774\pi\)
\(84\) 0 0
\(85\) −0.258650 − 0.217033i −0.0280545 − 0.0235405i
\(86\) −2.83766 − 7.79642i −0.305993 − 0.840709i
\(87\) 0 0
\(88\) −0.856156 − 4.85550i −0.0912665 − 0.517598i
\(89\) −2.56195 + 4.43743i −0.271566 + 0.470367i −0.969263 − 0.246026i \(-0.920875\pi\)
0.697697 + 0.716393i \(0.254208\pi\)
\(90\) 0 0
\(91\) 3.03291 + 1.36393i 0.317935 + 0.142979i
\(92\) −4.00271 + 0.705785i −0.417311 + 0.0735832i
\(93\) 0 0
\(94\) −10.3934 − 1.83263i −1.07199 − 0.189022i
\(95\) −2.71179 − 0.478163i −0.278224 − 0.0490584i
\(96\) 0 0
\(97\) 7.91567 − 1.39575i 0.803714 − 0.141717i 0.243324 − 0.969945i \(-0.421762\pi\)
0.560390 + 0.828229i \(0.310651\pi\)
\(98\) −2.93765 − 14.3466i −0.296747 − 1.44922i
\(99\) 0 0
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 567.2.bd.a.17.19 132
3.2 odd 2 189.2.bd.a.185.4 yes 132
7.5 odd 6 567.2.ba.a.341.4 132
21.5 even 6 189.2.ba.a.131.19 yes 132
27.7 even 9 189.2.ba.a.101.19 ✓ 132
27.20 odd 18 567.2.ba.a.143.4 132
189.47 even 18 inner 567.2.bd.a.467.19 132
189.61 odd 18 189.2.bd.a.47.4 yes 132
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
189.2.ba.a.101.19 ✓ 132 27.7 even 9
189.2.ba.a.131.19 yes 132 21.5 even 6
189.2.bd.a.47.4 yes 132 189.61 odd 18
189.2.bd.a.185.4 yes 132 3.2 odd 2
567.2.ba.a.143.4 132 27.20 odd 18
567.2.ba.a.341.4 132 7.5 odd 6
567.2.bd.a.17.19 132 1.1 even 1 trivial
567.2.bd.a.467.19 132 189.47 even 18 inner