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Properties

Label 567.2.be.a.503.10
Level $567$
Weight $2$
Character 567.503
Analytic conductor $4.528$
Analytic rank $0$
Dimension $132$
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] = [567,2,Mod(62,567)] 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("567.62"); 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([7, 9])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 567 = 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 567.be (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 503.10
Character \(\chi\) \(=\) 567.503
Dual form 567.2.be.a.62.10

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.157445 + 0.432576i) q^{2} +(1.36976 + 1.14936i) q^{4} +(0.725040 - 4.11191i) q^{5} +(2.17886 + 1.50085i) q^{7} +(-1.51017 + 0.871900i) q^{8} +(1.66456 + 0.961033i) q^{10} +(1.36185 - 0.240131i) q^{11} +(-1.43469 - 3.94178i) q^{13} +(-0.992282 + 0.706223i) q^{14} +(0.481603 + 2.73131i) q^{16} +(-1.18976 + 2.06072i) q^{17} +(4.72301 - 2.72683i) q^{19} +(5.71920 - 4.79898i) q^{20} +(-0.110541 + 0.626912i) q^{22} +(1.01769 - 1.21284i) q^{23} +(-11.6836 - 4.25249i) q^{25} +1.93100 q^{26} +(1.25949 + 4.56010i) q^{28} +(-0.495461 + 1.36127i) q^{29} +(-1.18509 + 1.41234i) q^{31} +(-4.69194 - 0.827315i) q^{32} +(-0.704097 - 0.839111i) q^{34} +(7.75112 - 7.87111i) q^{35} +(3.37083 - 5.83846i) q^{37} +(0.435948 + 2.47239i) q^{38} +(2.49023 + 6.84186i) q^{40} +(6.07454 - 2.21095i) q^{41} +(0.819766 + 4.64912i) q^{43} +(2.14140 + 1.23634i) q^{44} +(0.364414 + 0.631183i) q^{46} +(-2.18349 + 1.83217i) q^{47} +(2.49489 + 6.54030i) q^{49} +(3.67905 - 4.38452i) q^{50} +(2.56536 - 7.04826i) q^{52} +11.6368i q^{53} -5.77391i q^{55} +(-4.59906 - 0.366797i) q^{56} +(-0.510844 - 0.428649i) q^{58} +(-1.15353 + 6.54201i) q^{59} +(-1.37654 - 1.64050i) q^{61} +(-0.424356 - 0.735006i) q^{62} +(-1.67684 + 2.90438i) q^{64} +(-17.2484 + 3.04137i) q^{65} +(9.25483 - 3.36848i) q^{67} +(-3.99819 + 1.45522i) q^{68} +(2.18448 + 4.59221i) q^{70} +(9.79394 + 5.65453i) q^{71} +(-7.69650 + 4.44358i) q^{73} +(1.99486 + 2.37738i) q^{74} +(9.60349 + 1.69335i) q^{76} +(3.32769 + 1.52072i) q^{77} +(-2.14212 - 0.779666i) q^{79} +11.5801 q^{80} +2.97580i q^{82} +(-10.9949 - 4.00181i) q^{83} +(7.61088 + 6.38628i) q^{85} +(-2.14017 - 0.377369i) q^{86} +(-1.84726 + 1.55004i) q^{88} +(-4.23831 - 7.34097i) q^{89} +(2.79003 - 10.7419i) q^{91} +(2.78798 - 0.491595i) q^{92} +(-0.448772 - 1.23299i) q^{94} +(-7.78811 - 21.3976i) q^{95} +(-5.77859 + 1.01892i) q^{97} +(-3.22198 + 0.0494945i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 132 q + 12 q^{2} - 12 q^{4} - 6 q^{7} + 18 q^{8} + 18 q^{11} - 3 q^{14} - 24 q^{16} - 12 q^{22} - 12 q^{23} - 12 q^{25} - 12 q^{28} + 48 q^{29} + 6 q^{32} + 36 q^{35} - 6 q^{37} - 12 q^{43} + 18 q^{44}+ \cdots - 126 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.157445 + 0.432576i −0.111330 + 0.305877i −0.982829 − 0.184521i \(-0.940927\pi\)
0.871498 + 0.490399i \(0.163149\pi\)
\(3\) 0 0
\(4\) 1.36976 + 1.14936i 0.684878 + 0.574681i
\(5\) 0.725040 − 4.11191i 0.324248 − 1.83890i −0.190661 − 0.981656i \(-0.561063\pi\)
0.514909 − 0.857245i \(-0.327826\pi\)
\(6\) 0 0
\(7\) 2.17886 + 1.50085i 0.823533 + 0.567268i
\(8\) −1.51017 + 0.871900i −0.533927 + 0.308263i
\(9\) 0 0
\(10\) 1.66456 + 0.961033i 0.526379 + 0.303905i
\(11\) 1.36185 − 0.240131i 0.410614 − 0.0724023i 0.0354742 − 0.999371i \(-0.488706\pi\)
0.375140 + 0.926968i \(0.377595\pi\)
\(12\) 0 0
\(13\) −1.43469 − 3.94178i −0.397912 − 1.09325i −0.963300 − 0.268428i \(-0.913496\pi\)
0.565388 − 0.824825i \(-0.308726\pi\)
\(14\) −0.992282 + 0.706223i −0.265199 + 0.188746i
\(15\) 0 0
\(16\) 0.481603 + 2.73131i 0.120401 + 0.682827i
\(17\) −1.18976 + 2.06072i −0.288559 + 0.499799i −0.973466 − 0.228832i \(-0.926509\pi\)
0.684907 + 0.728630i \(0.259843\pi\)
\(18\) 0 0
\(19\) 4.72301 − 2.72683i 1.08353 − 0.625578i 0.151686 − 0.988429i \(-0.451530\pi\)
0.931847 + 0.362851i \(0.118196\pi\)
\(20\) 5.71920 − 4.79898i 1.27885 − 1.07308i
\(21\) 0 0
\(22\) −0.110541 + 0.626912i −0.0235675 + 0.133658i
\(23\) 1.01769 − 1.21284i 0.212203 − 0.252894i −0.649435 − 0.760417i \(-0.724994\pi\)
0.861638 + 0.507523i \(0.169439\pi\)
\(24\) 0 0
\(25\) −11.6836 − 4.25249i −2.33673 − 0.850499i
\(26\) 1.93100 0.378701
\(27\) 0 0
\(28\) 1.25949 + 4.56010i 0.238021 + 0.861778i
\(29\) −0.495461 + 1.36127i −0.0920048 + 0.252781i −0.977156 − 0.212522i \(-0.931832\pi\)
0.885152 + 0.465303i \(0.154055\pi\)
\(30\) 0 0
\(31\) −1.18509 + 1.41234i −0.212849 + 0.253663i −0.861896 − 0.507085i \(-0.830723\pi\)
0.649048 + 0.760748i \(0.275168\pi\)
\(32\) −4.69194 − 0.827315i −0.829425 − 0.146250i
\(33\) 0 0
\(34\) −0.704097 − 0.839111i −0.120752 − 0.143906i
\(35\) 7.75112 − 7.87111i 1.31018 − 1.33046i
\(36\) 0 0
\(37\) 3.37083 − 5.83846i 0.554162 − 0.959836i −0.443806 − 0.896123i \(-0.646372\pi\)
0.997968 − 0.0637138i \(-0.0202945\pi\)
\(38\) 0.435948 + 2.47239i 0.0707202 + 0.401074i
\(39\) 0 0
\(40\) 2.49023 + 6.84186i 0.393741 + 1.08179i
\(41\) 6.07454 − 2.21095i 0.948684 − 0.345293i 0.179095 − 0.983832i \(-0.442683\pi\)
0.769589 + 0.638539i \(0.220461\pi\)
\(42\) 0 0
\(43\) 0.819766 + 4.64912i 0.125013 + 0.708985i 0.981300 + 0.192484i \(0.0616544\pi\)
−0.856287 + 0.516500i \(0.827234\pi\)
\(44\) 2.14140 + 1.23634i 0.322829 + 0.186385i
\(45\) 0 0
\(46\) 0.364414 + 0.631183i 0.0537299 + 0.0930629i
\(47\) −2.18349 + 1.83217i −0.318495 + 0.267249i −0.787993 − 0.615685i \(-0.788880\pi\)
0.469497 + 0.882934i \(0.344435\pi\)
\(48\) 0 0
\(49\) 2.49489 + 6.54030i 0.356413 + 0.934328i
\(50\) 3.67905 − 4.38452i 0.520297 − 0.620065i
\(51\) 0 0
\(52\) 2.56536 − 7.04826i 0.355751 − 0.977417i
\(53\) 11.6368i 1.59844i 0.601040 + 0.799219i \(0.294753\pi\)
−0.601040 + 0.799219i \(0.705247\pi\)
\(54\) 0 0
\(55\) − 5.77391i − 0.778554i
\(56\) −4.59906 − 0.366797i −0.614575 − 0.0490153i
\(57\) 0 0
\(58\) −0.510844 − 0.428649i −0.0670771 − 0.0562843i
\(59\) −1.15353 + 6.54201i −0.150177 + 0.851698i 0.812886 + 0.582423i \(0.197895\pi\)
−0.963063 + 0.269275i \(0.913216\pi\)
\(60\) 0 0
\(61\) −1.37654 − 1.64050i −0.176248 − 0.210045i 0.670687 − 0.741741i \(-0.265999\pi\)
−0.846935 + 0.531696i \(0.821555\pi\)
\(62\) −0.424356 − 0.735006i −0.0538933 − 0.0933459i
\(63\) 0 0
\(64\) −1.67684 + 2.90438i −0.209606 + 0.363048i
\(65\) −17.2484 + 3.04137i −2.13941 + 0.377235i
\(66\) 0 0
\(67\) 9.25483 − 3.36848i 1.13066 − 0.411525i 0.292126 − 0.956380i \(-0.405637\pi\)
0.838531 + 0.544854i \(0.183415\pi\)
\(68\) −3.99819 + 1.45522i −0.484852 + 0.176472i
\(69\) 0 0
\(70\) 2.18448 + 4.59221i 0.261095 + 0.548874i
\(71\) 9.79394 + 5.65453i 1.16233 + 0.671069i 0.951860 − 0.306532i \(-0.0991686\pi\)
0.210466 + 0.977601i \(0.432502\pi\)
\(72\) 0 0
\(73\) −7.69650 + 4.44358i −0.900807 + 0.520081i −0.877462 − 0.479646i \(-0.840765\pi\)
−0.0233450 + 0.999727i \(0.507432\pi\)
\(74\) 1.99486 + 2.37738i 0.231897 + 0.276364i
\(75\) 0 0
\(76\) 9.60349 + 1.69335i 1.10160 + 0.194241i
\(77\) 3.32769 + 1.52072i 0.379226 + 0.173303i
\(78\) 0 0
\(79\) −2.14212 − 0.779666i −0.241007 − 0.0877193i 0.218693 − 0.975794i \(-0.429821\pi\)
−0.459700 + 0.888074i \(0.652043\pi\)
\(80\) 11.5801 1.29469
\(81\) 0 0
\(82\) 2.97580i 0.328622i
\(83\) −10.9949 − 4.00181i −1.20685 − 0.439256i −0.341237 − 0.939977i \(-0.610846\pi\)
−0.865608 + 0.500722i \(0.833068\pi\)
\(84\) 0 0
\(85\) 7.61088 + 6.38628i 0.825515 + 0.692690i
\(86\) −2.14017 − 0.377369i −0.230780 − 0.0406927i
\(87\) 0 0
\(88\) −1.84726 + 1.55004i −0.196919 + 0.165235i
\(89\) −4.23831 − 7.34097i −0.449260 − 0.778142i 0.549078 − 0.835771i \(-0.314979\pi\)
−0.998338 + 0.0576296i \(0.981646\pi\)
\(90\) 0 0
\(91\) 2.79003 − 10.7419i 0.292475 − 1.12605i
\(92\) 2.78798 − 0.491595i 0.290667 − 0.0512524i
\(93\) 0 0
\(94\) −0.448772 − 1.23299i −0.0462873 − 0.127173i
\(95\) −7.78811 − 21.3976i −0.799043 − 2.19535i
\(96\) 0 0
\(97\) −5.77859 + 1.01892i −0.586727 + 0.103456i −0.459128 − 0.888370i \(-0.651838\pi\)
−0.127599 + 0.991826i \(0.540727\pi\)
\(98\) −3.22198 + 0.0494945i −0.325469 + 0.00499970i
\(99\) 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.be.a.503.10 132
3.2 odd 2 189.2.be.a.104.13 yes 132
7.6 odd 2 inner 567.2.be.a.503.9 132
21.20 even 2 189.2.be.a.104.14 yes 132
27.7 even 9 189.2.be.a.20.14 yes 132
27.20 odd 18 inner 567.2.be.a.62.9 132
189.20 even 18 inner 567.2.be.a.62.10 132
189.34 odd 18 189.2.be.a.20.13 ✓ 132
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
189.2.be.a.20.13 ✓ 132 189.34 odd 18
189.2.be.a.20.14 yes 132 27.7 even 9
189.2.be.a.104.13 yes 132 3.2 odd 2
189.2.be.a.104.14 yes 132 21.20 even 2
567.2.be.a.62.9 132 27.20 odd 18 inner
567.2.be.a.62.10 132 189.20 even 18 inner
567.2.be.a.503.9 132 7.6 odd 2 inner
567.2.be.a.503.10 132 1.1 even 1 trivial