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Properties

Label 567.2.ba.a.341.13
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 341.13
Character \(\chi\) \(=\) 567.341
Dual form 567.2.ba.a.143.13

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.159953 + 0.190625i) q^{2} +(0.336544 - 1.90863i) q^{4} +(-1.62936 - 1.36719i) q^{5} +(2.64099 + 0.158668i) q^{7} +(0.848674 - 0.489982i) q^{8} -0.529283i q^{10} +(0.919336 + 1.09562i) q^{11} +(-1.30709 - 3.59121i) q^{13} +(0.392189 + 0.528818i) q^{14} +(-3.41324 - 1.24232i) q^{16} -0.218194 q^{17} +2.96748i q^{19} +(-3.15782 + 2.64972i) q^{20} +(-0.0618021 + 0.350497i) q^{22} +(-2.64486 - 7.26669i) q^{23} +(-0.0826545 - 0.468757i) q^{25} +(0.475500 - 0.823590i) q^{26} +(1.19165 - 4.98728i) q^{28} +(1.78349 - 4.90009i) q^{29} +(3.81477 + 0.672647i) q^{31} +(-0.979478 - 2.69109i) q^{32} +(-0.0349009 - 0.0415932i) q^{34} +(-4.08618 - 3.86926i) q^{35} +(-3.00625 - 5.20699i) q^{37} +(-0.565677 + 0.474659i) q^{38} +(-2.05269 - 0.361945i) q^{40} +(-10.0627 + 3.66253i) q^{41} +(1.29579 + 7.34878i) q^{43} +(2.40054 - 1.38595i) q^{44} +(0.962159 - 1.66651i) q^{46} +(1.01967 + 5.78285i) q^{47} +(6.94965 + 0.838080i) q^{49} +(0.0761360 - 0.0907353i) q^{50} +(-7.29419 + 1.28616i) q^{52} +(12.1036 - 6.98800i) q^{53} -3.04206i q^{55} +(2.31908 - 1.15938i) q^{56} +(1.21936 - 0.443809i) q^{58} +(1.48515 - 0.540550i) q^{59} +(4.18950 - 0.738722i) q^{61} +(0.481963 + 0.834784i) q^{62} +(-3.27598 + 5.67416i) q^{64} +(-2.78015 + 7.63839i) q^{65} +(11.4276 + 9.58891i) q^{67} +(-0.0734317 + 0.416452i) q^{68} +(0.0839802 - 1.39783i) q^{70} +(4.29452 + 2.47944i) q^{71} +(-2.13740 - 1.23403i) q^{73} +(0.511721 - 1.40594i) q^{74} +(5.66384 + 0.998688i) q^{76} +(2.25412 + 3.03939i) q^{77} +(-2.98202 + 2.50221i) q^{79} +(3.86290 + 6.69073i) q^{80} +(-2.30773 - 1.33237i) q^{82} +(-0.141502 - 0.0515026i) q^{83} +(0.355515 + 0.298313i) q^{85} +(-1.19360 + 1.42247i) q^{86} +(1.31705 + 0.479367i) q^{88} -6.80542 q^{89} +(-2.88221 - 9.69173i) q^{91} +(-14.7595 + 2.60251i) q^{92} +(-0.939256 + 1.11936i) q^{94} +(4.05712 - 4.83509i) q^{95} +(-12.2644 + 2.16255i) q^{97} +(0.951861 + 1.45883i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 132 q + 3 q^{2} - 3 q^{4} + 9 q^{5} - 6 q^{7} + 18 q^{8} + 9 q^{11} - 3 q^{14} + 3 q^{16} + 18 q^{17} - 18 q^{20} - 12 q^{22} + 6 q^{23} - 3 q^{25} - 12 q^{28} - 6 q^{29} - 9 q^{31} - 3 q^{32} - 18 q^{34}+ \cdots - 27 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.159953 + 0.190625i 0.113104 + 0.134792i 0.819626 − 0.572899i \(-0.194181\pi\)
−0.706522 + 0.707691i \(0.749737\pi\)
\(3\) 0 0
\(4\) 0.336544 − 1.90863i 0.168272 − 0.954317i
\(5\) −1.62936 − 1.36719i −0.728670 − 0.611427i 0.201099 − 0.979571i \(-0.435549\pi\)
−0.929769 + 0.368144i \(0.879993\pi\)
\(6\) 0 0
\(7\) 2.64099 + 0.158668i 0.998200 + 0.0599708i
\(8\) 0.848674 − 0.489982i 0.300052 − 0.173235i
\(9\) 0 0
\(10\) − 0.529283i − 0.167374i
\(11\) 0.919336 + 1.09562i 0.277190 + 0.330342i 0.886621 − 0.462497i \(-0.153046\pi\)
−0.609431 + 0.792839i \(0.708602\pi\)
\(12\) 0 0
\(13\) −1.30709 − 3.59121i −0.362522 − 0.996021i −0.978135 − 0.207972i \(-0.933314\pi\)
0.615613 − 0.788049i \(-0.288909\pi\)
\(14\) 0.392189 + 0.528818i 0.104817 + 0.141333i
\(15\) 0 0
\(16\) −3.41324 − 1.24232i −0.853310 − 0.310580i
\(17\) −0.218194 −0.0529198 −0.0264599 − 0.999650i \(-0.508423\pi\)
−0.0264599 + 0.999650i \(0.508423\pi\)
\(18\) 0 0
\(19\) 2.96748i 0.680788i 0.940283 + 0.340394i \(0.110560\pi\)
−0.940283 + 0.340394i \(0.889440\pi\)
\(20\) −3.15782 + 2.64972i −0.706109 + 0.592496i
\(21\) 0 0
\(22\) −0.0618021 + 0.350497i −0.0131762 + 0.0747262i
\(23\) −2.64486 − 7.26669i −0.551491 − 1.51521i −0.831675 − 0.555262i \(-0.812618\pi\)
0.280184 − 0.959946i \(-0.409604\pi\)
\(24\) 0 0
\(25\) −0.0826545 − 0.468757i −0.0165309 − 0.0937514i
\(26\) 0.475500 − 0.823590i 0.0932532 − 0.161519i
\(27\) 0 0
\(28\) 1.19165 − 4.98728i 0.225200 − 0.942508i
\(29\) 1.78349 − 4.90009i 0.331185 − 0.909924i −0.656619 − 0.754223i \(-0.728014\pi\)
0.987804 − 0.155702i \(-0.0497638\pi\)
\(30\) 0 0
\(31\) 3.81477 + 0.672647i 0.685153 + 0.120811i 0.505380 − 0.862897i \(-0.331352\pi\)
0.179774 + 0.983708i \(0.442463\pi\)
\(32\) −0.979478 − 2.69109i −0.173149 − 0.475723i
\(33\) 0 0
\(34\) −0.0349009 − 0.0415932i −0.00598545 − 0.00713318i
\(35\) −4.08618 − 3.86926i −0.690691 − 0.654025i
\(36\) 0 0
\(37\) −3.00625 − 5.20699i −0.494225 − 0.856023i 0.505753 − 0.862679i \(-0.331215\pi\)
−0.999978 + 0.00665534i \(0.997882\pi\)
\(38\) −0.565677 + 0.474659i −0.0917649 + 0.0769999i
\(39\) 0 0
\(40\) −2.05269 − 0.361945i −0.324559 − 0.0572285i
\(41\) −10.0627 + 3.66253i −1.57153 + 0.571990i −0.973341 − 0.229364i \(-0.926335\pi\)
−0.598190 + 0.801354i \(0.704113\pi\)
\(42\) 0 0
\(43\) 1.29579 + 7.34878i 0.197606 + 1.12068i 0.908659 + 0.417540i \(0.137108\pi\)
−0.711053 + 0.703139i \(0.751781\pi\)
\(44\) 2.40054 − 1.38595i 0.361894 − 0.208940i
\(45\) 0 0
\(46\) 0.962159 − 1.66651i 0.141863 − 0.245713i
\(47\) 1.01967 + 5.78285i 0.148735 + 0.843515i 0.964292 + 0.264840i \(0.0853191\pi\)
−0.815558 + 0.578675i \(0.803570\pi\)
\(48\) 0 0
\(49\) 6.94965 + 0.838080i 0.992807 + 0.119726i
\(50\) 0.0761360 − 0.0907353i 0.0107673 − 0.0128319i
\(51\) 0 0
\(52\) −7.29419 + 1.28616i −1.01152 + 0.178359i
\(53\) 12.1036 − 6.98800i 1.66255 − 0.959875i 0.691063 − 0.722795i \(-0.257143\pi\)
0.971490 − 0.237080i \(-0.0761904\pi\)
\(54\) 0 0
\(55\) − 3.04206i − 0.410192i
\(56\) 2.31908 − 1.15938i 0.309901 − 0.154929i
\(57\) 0 0
\(58\) 1.21936 − 0.443809i 0.160109 − 0.0582750i
\(59\) 1.48515 − 0.540550i 0.193350 − 0.0703736i −0.243530 − 0.969893i \(-0.578306\pi\)
0.436880 + 0.899520i \(0.356083\pi\)
\(60\) 0 0
\(61\) 4.18950 − 0.738722i 0.536410 − 0.0945836i 0.101121 − 0.994874i \(-0.467757\pi\)
0.435290 + 0.900291i \(0.356646\pi\)
\(62\) 0.481963 + 0.834784i 0.0612093 + 0.106018i
\(63\) 0 0
\(64\) −3.27598 + 5.67416i −0.409497 + 0.709270i
\(65\) −2.78015 + 7.63839i −0.344835 + 0.947426i
\(66\) 0 0
\(67\) 11.4276 + 9.58891i 1.39611 + 1.17147i 0.962797 + 0.270226i \(0.0870983\pi\)
0.433308 + 0.901246i \(0.357346\pi\)
\(68\) −0.0734317 + 0.416452i −0.00890490 + 0.0505022i
\(69\) 0 0
\(70\) 0.0839802 − 1.39783i 0.0100376 − 0.167073i
\(71\) 4.29452 + 2.47944i 0.509666 + 0.294256i 0.732696 − 0.680556i \(-0.238262\pi\)
−0.223031 + 0.974811i \(0.571595\pi\)
\(72\) 0 0
\(73\) −2.13740 − 1.23403i −0.250164 − 0.144432i 0.369676 − 0.929161i \(-0.379469\pi\)
−0.619839 + 0.784729i \(0.712802\pi\)
\(74\) 0.511721 − 1.40594i 0.0594864 − 0.163438i
\(75\) 0 0
\(76\) 5.66384 + 0.998688i 0.649687 + 0.114557i
\(77\) 2.25412 + 3.03939i 0.256880 + 0.346371i
\(78\) 0 0
\(79\) −2.98202 + 2.50221i −0.335503 + 0.281521i −0.794938 − 0.606691i \(-0.792497\pi\)
0.459435 + 0.888212i \(0.348052\pi\)
\(80\) 3.86290 + 6.69073i 0.431885 + 0.748047i
\(81\) 0 0
\(82\) −2.30773 − 1.33237i −0.254847 − 0.147136i
\(83\) −0.141502 − 0.0515026i −0.0155319 − 0.00565315i 0.334243 − 0.942487i \(-0.391520\pi\)
−0.349774 + 0.936834i \(0.613742\pi\)
\(84\) 0 0
\(85\) 0.355515 + 0.298313i 0.0385610 + 0.0323566i
\(86\) −1.19360 + 1.42247i −0.128709 + 0.153389i
\(87\) 0 0
\(88\) 1.31705 + 0.479367i 0.140398 + 0.0511007i
\(89\) −6.80542 −0.721373 −0.360686 − 0.932687i \(-0.617458\pi\)
−0.360686 + 0.932687i \(0.617458\pi\)
\(90\) 0 0
\(91\) −2.88221 − 9.69173i −0.302137 − 1.01597i
\(92\) −14.7595 + 2.60251i −1.53879 + 0.271330i
\(93\) 0 0
\(94\) −0.939256 + 1.11936i −0.0968769 + 0.115453i
\(95\) 4.05712 − 4.83509i 0.416252 − 0.496069i
\(96\) 0 0
\(97\) −12.2644 + 2.16255i −1.24526 + 0.219573i −0.757169 − 0.653219i \(-0.773418\pi\)
−0.488092 + 0.872792i \(0.662307\pi\)
\(98\) 0.951861 + 1.45883i 0.0961525 + 0.147364i
\(99\) 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.341.13 132
3.2 odd 2 189.2.ba.a.131.10 yes 132
7.3 odd 6 567.2.bd.a.17.10 132
21.17 even 6 189.2.bd.a.185.13 yes 132
27.7 even 9 189.2.bd.a.47.13 yes 132
27.20 odd 18 567.2.bd.a.467.10 132
189.101 even 18 inner 567.2.ba.a.143.13 132
189.115 odd 18 189.2.ba.a.101.10 ✓ 132
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
189.2.ba.a.101.10 ✓ 132 189.115 odd 18
189.2.ba.a.131.10 yes 132 3.2 odd 2
189.2.bd.a.47.13 yes 132 27.7 even 9
189.2.bd.a.185.13 yes 132 21.17 even 6
567.2.ba.a.143.13 132 189.101 even 18 inner
567.2.ba.a.341.13 132 1.1 even 1 trivial
567.2.bd.a.17.10 132 7.3 odd 6
567.2.bd.a.467.10 132 27.20 odd 18