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Properties

Label 567.2.ba.a.341.10
Level $567$
Weight $2$
Character 567.341
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(143,567)] 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("567.143"); 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, 3])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 567 = 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 567.ba (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 341.10
Character \(\chi\) \(=\) 567.341
Dual form 567.2.ba.a.143.10

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.204899 - 0.244189i) q^{2} +(0.329652 - 1.86955i) q^{4} +(2.18935 + 1.83708i) q^{5} +(0.468007 + 2.60403i) q^{7} +(-1.07619 + 0.621337i) q^{8} -0.911030i q^{10} +(2.19030 + 2.61029i) q^{11} +(1.17022 + 3.21516i) q^{13} +(0.539981 - 0.647845i) q^{14} +(-3.19557 - 1.16309i) q^{16} -4.66323 q^{17} +5.52060i q^{19} +(4.15623 - 3.48749i) q^{20} +(0.188615 - 1.06969i) q^{22} +(-0.470293 - 1.29212i) q^{23} +(0.550137 + 3.11998i) q^{25} +(0.545329 - 0.944538i) q^{26} +(5.02264 - 0.0165384i) q^{28} +(2.81504 - 7.73427i) q^{29} +(2.17942 + 0.384290i) q^{31} +(1.22079 + 3.35410i) q^{32} +(0.955489 + 1.13871i) q^{34} +(-3.75918 + 6.56089i) q^{35} +(0.882419 + 1.52840i) q^{37} +(1.34807 - 1.13116i) q^{38} +(-3.49759 - 0.616720i) q^{40} +(4.96713 - 1.80789i) q^{41} +(-1.57831 - 8.95103i) q^{43} +(5.60210 - 3.23437i) q^{44} +(-0.219159 + 0.379594i) q^{46} +(-0.532155 - 3.01800i) q^{47} +(-6.56194 + 2.43741i) q^{49} +(0.649143 - 0.773618i) q^{50} +(6.39667 - 1.12790i) q^{52} +(1.30996 - 0.756305i) q^{53} +9.73859i q^{55} +(-2.12164 - 2.51163i) q^{56} +(-2.46542 + 0.897340i) q^{58} +(6.44995 - 2.34759i) q^{59} +(9.69333 - 1.70919i) q^{61} +(-0.352721 - 0.610930i) q^{62} +(-2.83176 + 4.90475i) q^{64} +(-3.34449 + 9.18891i) q^{65} +(-7.34743 - 6.16523i) q^{67} +(-1.53724 + 8.71812i) q^{68} +(2.37235 - 0.426368i) q^{70} +(9.64722 + 5.56983i) q^{71} +(-5.11937 - 2.95567i) q^{73} +(0.192410 - 0.528643i) q^{74} +(10.3210 + 1.81988i) q^{76} +(-5.77220 + 6.92523i) q^{77} +(8.75642 - 7.34751i) q^{79} +(-4.85952 - 8.41694i) q^{80} +(-1.45922 - 0.842484i) q^{82} +(6.72578 + 2.44798i) q^{83} +(-10.2094 - 8.56673i) q^{85} +(-1.86235 + 2.21946i) q^{86} +(-3.97904 - 1.44825i) q^{88} -12.2586 q^{89} +(-7.82471 + 4.55201i) q^{91} +(-2.57071 + 0.453286i) q^{92} +(-0.627925 + 0.748331i) q^{94} +(-10.1418 + 12.0865i) q^{95} +(2.47531 - 0.436465i) q^{97} +(1.93972 + 1.10293i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 132 q + 3 q^{2} - 3 q^{4} + 9 q^{5} - 6 q^{7} + 18 q^{8} + 9 q^{11} - 3 q^{14} + 3 q^{16} + 18 q^{17} - 18 q^{20} - 12 q^{22} + 6 q^{23} - 3 q^{25} - 12 q^{28} - 6 q^{29} - 9 q^{31} - 3 q^{32} - 18 q^{34}+ \cdots - 27 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(325\) \(407\)
\(\chi(n)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{11}{18}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.204899 − 0.244189i −0.144885 − 0.172668i 0.688721 − 0.725026i \(-0.258172\pi\)
−0.833607 + 0.552359i \(0.813728\pi\)
\(3\) 0 0
\(4\) 0.329652 − 1.86955i 0.164826 − 0.934774i
\(5\) 2.18935 + 1.83708i 0.979106 + 0.821568i 0.983954 − 0.178420i \(-0.0570987\pi\)
−0.00484794 + 0.999988i \(0.501543\pi\)
\(6\) 0 0
\(7\) 0.468007 + 2.60403i 0.176890 + 0.984231i
\(8\) −1.07619 + 0.621337i −0.380489 + 0.219676i
\(9\) 0 0
\(10\) − 0.911030i − 0.288093i
\(11\) 2.19030 + 2.61029i 0.660399 + 0.787033i 0.987443 − 0.157976i \(-0.0504970\pi\)
−0.327044 + 0.945009i \(0.606053\pi\)
\(12\) 0 0
\(13\) 1.17022 + 3.21516i 0.324562 + 0.891726i 0.989462 + 0.144794i \(0.0462518\pi\)
−0.664900 + 0.746932i \(0.731526\pi\)
\(14\) 0.539981 − 0.647845i 0.144316 − 0.173144i
\(15\) 0 0
\(16\) −3.19557 − 1.16309i −0.798893 − 0.290773i
\(17\) −4.66323 −1.13100 −0.565499 − 0.824749i \(-0.691317\pi\)
−0.565499 + 0.824749i \(0.691317\pi\)
\(18\) 0 0
\(19\) 5.52060i 1.26651i 0.773942 + 0.633256i \(0.218282\pi\)
−0.773942 + 0.633256i \(0.781718\pi\)
\(20\) 4.15623 − 3.48749i 0.929362 − 0.779827i
\(21\) 0 0
\(22\) 0.188615 − 1.06969i 0.0402129 − 0.228059i
\(23\) −0.470293 − 1.29212i −0.0980628 − 0.269425i 0.880955 − 0.473200i \(-0.156901\pi\)
−0.979018 + 0.203775i \(0.934679\pi\)
\(24\) 0 0
\(25\) 0.550137 + 3.11998i 0.110027 + 0.623997i
\(26\) 0.545329 − 0.944538i 0.106948 − 0.185239i
\(27\) 0 0
\(28\) 5.02264 − 0.0165384i 0.949189 − 0.00312546i
\(29\) 2.81504 − 7.73427i 0.522741 − 1.43622i −0.344717 − 0.938707i \(-0.612025\pi\)
0.867458 − 0.497511i \(-0.165753\pi\)
\(30\) 0 0
\(31\) 2.17942 + 0.384290i 0.391435 + 0.0690206i 0.365902 − 0.930653i \(-0.380761\pi\)
0.0255331 + 0.999674i \(0.491872\pi\)
\(32\) 1.22079 + 3.35410i 0.215808 + 0.592927i
\(33\) 0 0
\(34\) 0.955489 + 1.13871i 0.163865 + 0.195287i
\(35\) −3.75918 + 6.56089i −0.635418 + 1.10899i
\(36\) 0 0
\(37\) 0.882419 + 1.52840i 0.145069 + 0.251267i 0.929399 − 0.369077i \(-0.120326\pi\)
−0.784330 + 0.620344i \(0.786993\pi\)
\(38\) 1.34807 − 1.13116i 0.218686 − 0.183499i
\(39\) 0 0
\(40\) −3.49759 − 0.616720i −0.553018 − 0.0975120i
\(41\) 4.96713 − 1.80789i 0.775735 − 0.282344i 0.0763420 − 0.997082i \(-0.475676\pi\)
0.699393 + 0.714737i \(0.253454\pi\)
\(42\) 0 0
\(43\) −1.57831 − 8.95103i −0.240690 − 1.36502i −0.830293 − 0.557327i \(-0.811827\pi\)
0.589603 − 0.807693i \(-0.299284\pi\)
\(44\) 5.60210 − 3.23437i 0.844548 − 0.487600i
\(45\) 0 0
\(46\) −0.219159 + 0.379594i −0.0323132 + 0.0559681i
\(47\) −0.532155 − 3.01800i −0.0776228 − 0.440221i −0.998706 − 0.0508551i \(-0.983805\pi\)
0.921083 − 0.389366i \(-0.127306\pi\)
\(48\) 0 0
\(49\) −6.56194 + 2.43741i −0.937420 + 0.348201i
\(50\) 0.649143 − 0.773618i 0.0918027 − 0.109406i
\(51\) 0 0
\(52\) 6.39667 − 1.12790i 0.887058 − 0.156412i
\(53\) 1.30996 − 0.756305i 0.179937 − 0.103886i −0.407326 − 0.913283i \(-0.633539\pi\)
0.587263 + 0.809396i \(0.300205\pi\)
\(54\) 0 0
\(55\) 9.73859i 1.31315i
\(56\) −2.12164 − 2.51163i −0.283516 − 0.335631i
\(57\) 0 0
\(58\) −2.46542 + 0.897340i −0.323726 + 0.117827i
\(59\) 6.44995 − 2.34759i 0.839713 − 0.305630i 0.113874 − 0.993495i \(-0.463674\pi\)
0.725839 + 0.687865i \(0.241452\pi\)
\(60\) 0 0
\(61\) 9.69333 − 1.70919i 1.24110 − 0.218840i 0.485715 − 0.874117i \(-0.338559\pi\)
0.755388 + 0.655277i \(0.227448\pi\)
\(62\) −0.352721 − 0.610930i −0.0447956 − 0.0775882i
\(63\) 0 0
\(64\) −2.83176 + 4.90475i −0.353970 + 0.613094i
\(65\) −3.34449 + 9.18891i −0.414833 + 1.13974i
\(66\) 0 0
\(67\) −7.34743 − 6.16523i −0.897632 − 0.753203i 0.0720942 − 0.997398i \(-0.477032\pi\)
−0.969726 + 0.244195i \(0.921476\pi\)
\(68\) −1.53724 + 8.71812i −0.186418 + 1.05723i
\(69\) 0 0
\(70\) 2.37235 − 0.426368i 0.283550 − 0.0509608i
\(71\) 9.64722 + 5.56983i 1.14491 + 0.661017i 0.947643 − 0.319332i \(-0.103459\pi\)
0.197272 + 0.980349i \(0.436792\pi\)
\(72\) 0 0
\(73\) −5.11937 − 2.95567i −0.599176 − 0.345935i 0.169541 − 0.985523i \(-0.445771\pi\)
−0.768718 + 0.639588i \(0.779105\pi\)
\(74\) 0.192410 − 0.528643i 0.0223673 − 0.0614535i
\(75\) 0 0
\(76\) 10.3210 + 1.81988i 1.18390 + 0.208754i
\(77\) −5.77220 + 6.92523i −0.657804 + 0.789203i
\(78\) 0 0
\(79\) 8.75642 − 7.34751i 0.985175 − 0.826660i 0.000312673 − 1.00000i \(-0.499900\pi\)
0.984862 + 0.173340i \(0.0554560\pi\)
\(80\) −4.85952 − 8.41694i −0.543311 − 0.941042i
\(81\) 0 0
\(82\) −1.45922 − 0.842484i −0.161144 − 0.0930367i
\(83\) 6.72578 + 2.44798i 0.738250 + 0.268701i 0.683653 − 0.729807i \(-0.260390\pi\)
0.0545974 + 0.998508i \(0.482612\pi\)
\(84\) 0 0
\(85\) −10.2094 − 8.56673i −1.10737 − 0.929192i
\(86\) −1.86235 + 2.21946i −0.200822 + 0.239331i
\(87\) 0 0
\(88\) −3.97904 − 1.44825i −0.424167 − 0.154384i
\(89\) −12.2586 −1.29941 −0.649703 − 0.760188i \(-0.725107\pi\)
−0.649703 + 0.760188i \(0.725107\pi\)
\(90\) 0 0
\(91\) −7.82471 + 4.55201i −0.820252 + 0.477181i
\(92\) −2.57071 + 0.453286i −0.268015 + 0.0472583i
\(93\) 0 0
\(94\) −0.627925 + 0.748331i −0.0647655 + 0.0771845i
\(95\) −10.1418 + 12.0865i −1.04053 + 1.24005i
\(96\) 0 0
\(97\) 2.47531 − 0.436465i 0.251330 − 0.0443163i −0.0465636 − 0.998915i \(-0.514827\pi\)
0.297894 + 0.954599i \(0.403716\pi\)
\(98\) 1.93972 + 1.10293i 0.195941 + 0.111413i
\(99\) 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.ba.a.341.10 132
3.2 odd 2 189.2.ba.a.131.13 yes 132
7.3 odd 6 567.2.bd.a.17.13 132
21.17 even 6 189.2.bd.a.185.10 yes 132
27.7 even 9 189.2.bd.a.47.10 yes 132
27.20 odd 18 567.2.bd.a.467.13 132
189.101 even 18 inner 567.2.ba.a.143.10 132
189.115 odd 18 189.2.ba.a.101.13 ✓ 132
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
189.2.ba.a.101.13 ✓ 132 189.115 odd 18
189.2.ba.a.131.13 yes 132 3.2 odd 2
189.2.bd.a.47.10 yes 132 27.7 even 9
189.2.bd.a.185.10 yes 132 21.17 even 6
567.2.ba.a.143.10 132 189.101 even 18 inner
567.2.ba.a.341.10 132 1.1 even 1 trivial
567.2.bd.a.17.13 132 7.3 odd 6
567.2.bd.a.467.13 132 27.20 odd 18