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Properties

Label 162.2.g.a.25.1
Level $162$
Weight $2$
Character 162.25
Analytic conductor $1.294$
Analytic rank $0$
Dimension $72$
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] = [162,2,Mod(7,162)] 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("162.7"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(162, base_ring=CyclotomicField(54)) chi = DirichletCharacter(H, H._module([16])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 162 = 2 \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 162.g (of order \(27\), degree \(18\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [72] 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: \(1.29357651274\)
Analytic rank: \(0\)
Dimension: \(72\)
Relative dimension: \(4\) over \(\Q(\zeta_{27})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{27}]$

Embedding invariants

Embedding label 25.1
Character \(\chi\) \(=\) 162.25
Dual form 162.2.g.a.13.1

$q$-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+(-0.893633 + 0.448799i) q^{2} +(-1.71649 - 0.231626i) q^{3} +(0.597159 - 0.802123i) q^{4} +(-0.267154 + 0.892356i) q^{5} +(1.63787 - 0.563372i) q^{6} +(1.11188 - 2.57764i) q^{7} +(-0.173648 + 0.984808i) q^{8} +(2.89270 + 0.795169i) q^{9} +(-0.161751 - 0.917337i) q^{10} +(4.46341 + 1.05785i) q^{11} +(-1.21081 + 1.23852i) q^{12} +(1.47403 - 0.969483i) q^{13} +(0.163226 + 2.80247i) q^{14} +(0.665260 - 1.46984i) q^{15} +(-0.286803 - 0.957990i) q^{16} +(1.42569 + 0.518907i) q^{17} +(-2.94188 + 0.587652i) q^{18} +(4.12421 - 1.50109i) q^{19} +(0.556246 + 0.747168i) q^{20} +(-2.50559 + 4.16696i) q^{21} +(-4.46341 + 1.05785i) q^{22} +(-2.92223 - 6.77448i) q^{23} +(0.526173 - 1.65019i) q^{24} +(3.45251 + 2.27075i) q^{25} +(-0.882135 + 1.52790i) q^{26} +(-4.78112 - 2.03493i) q^{27} +(-1.40361 - 2.43113i) q^{28} +(-0.418954 + 7.19315i) q^{29} +(0.0651659 + 1.61207i) q^{30} +(-6.24778 - 0.730261i) q^{31} +(0.686242 + 0.727374i) q^{32} +(-7.41639 - 2.84963i) q^{33} +(-1.50693 + 0.176134i) q^{34} +(2.00313 + 1.68082i) q^{35} +(2.36522 - 1.84546i) q^{36} +(-0.516723 + 0.433582i) q^{37} +(-3.01184 + 3.19236i) q^{38} +(-2.75471 + 1.32269i) q^{39} +(-0.832408 - 0.418051i) q^{40} +(-8.46317 - 4.25037i) q^{41} +(0.368950 - 4.84823i) q^{42} +(3.77539 - 4.00168i) q^{43} +(3.51389 - 2.94850i) q^{44} +(-1.48237 + 2.36888i) q^{45} +(5.65178 + 4.74241i) q^{46} +(5.01677 - 0.586377i) q^{47} +(0.270401 + 1.71081i) q^{48} +(-0.604237 - 0.640453i) q^{49} +(-4.10439 - 0.479734i) q^{50} +(-2.32699 - 1.22093i) q^{51} +(0.102583 - 1.76129i) q^{52} +(-1.33883 - 2.31892i) q^{53} +(5.18583 - 0.327285i) q^{54} +(-2.13639 + 3.70034i) q^{55} +(2.34540 + 1.54259i) q^{56} +(-7.42686 + 1.62134i) q^{57} +(-2.85389 - 6.61606i) q^{58} +(4.89941 - 1.16118i) q^{59} +(-0.781729 - 1.41135i) q^{60} +(3.14866 + 4.22938i) q^{61} +(5.91096 - 2.15141i) q^{62} +(5.26600 - 6.57219i) q^{63} +(-0.939693 - 0.342020i) q^{64} +(0.471331 + 1.57436i) q^{65} +(7.90644 - 0.781947i) q^{66} +(0.832311 + 14.2902i) q^{67} +(1.26759 - 0.833706i) q^{68} +(3.44684 + 12.3052i) q^{69} +(-2.54441 - 0.603036i) q^{70} +(-0.0492509 - 0.279316i) q^{71} +(-1.28540 + 2.71067i) q^{72} +(-2.01050 + 11.4021i) q^{73} +(0.267169 - 0.619368i) q^{74} +(-5.40025 - 4.69742i) q^{75} +(1.25875 - 4.20451i) q^{76} +(7.68955 - 10.3289i) q^{77} +(1.86808 - 2.41831i) q^{78} +(3.17920 - 1.59665i) q^{79} +0.931488 q^{80} +(7.73541 + 4.60037i) q^{81} +9.47053 q^{82} +(-9.56164 + 4.80204i) q^{83} +(1.84618 + 4.49813i) q^{84} +(-0.843927 + 1.13359i) q^{85} +(-1.57786 + 5.27043i) q^{86} +(2.38525 - 12.2500i) q^{87} +(-1.81684 + 4.21191i) q^{88} +(1.74720 - 9.90888i) q^{89} +(0.261540 - 2.78220i) q^{90} +(-0.860028 - 4.87746i) q^{91} +(-7.17900 - 1.70145i) q^{92} +(10.5551 + 2.70064i) q^{93} +(-4.21999 + 2.77553i) q^{94} +(0.237708 + 4.08128i) q^{95} +(-1.00945 - 1.40748i) q^{96} +(-4.96298 - 16.5775i) q^{97} +(0.827401 + 0.301149i) q^{98} +(12.0701 + 6.60920i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 72 q + 9 q^{6} + 18 q^{13} - 9 q^{20} - 81 q^{23} + 18 q^{25} - 27 q^{26} - 27 q^{27} + 18 q^{28} - 27 q^{29} - 63 q^{30} - 54 q^{31} + 9 q^{33} - 27 q^{35} - 9 q^{36} + 9 q^{38} - 9 q^{41} + 9 q^{42} + 36 q^{43}+ \cdots + 99 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/162\mathbb{Z}\right)^\times\).

\(n\) \(83\)
\(\chi(n)\) \(e\left(\frac{23}{27}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.893633 + 0.448799i −0.631894 + 0.317349i
\(3\) −1.71649 − 0.231626i −0.991018 − 0.133729i
\(4\) 0.597159 − 0.802123i 0.298579 − 0.401062i
\(5\) −0.267154 + 0.892356i −0.119475 + 0.399074i −0.996716 − 0.0809762i \(-0.974196\pi\)
0.877241 + 0.480050i \(0.159381\pi\)
\(6\) 1.63787 − 0.563372i 0.668657 − 0.229996i
\(7\) 1.11188 − 2.57764i 0.420253 − 0.974255i −0.568223 − 0.822875i \(-0.692369\pi\)
0.988476 − 0.151381i \(-0.0483719\pi\)
\(8\) −0.173648 + 0.984808i −0.0613939 + 0.348182i
\(9\) 2.89270 + 0.795169i 0.964233 + 0.265056i
\(10\) −0.161751 − 0.917337i −0.0511502 − 0.290087i
\(11\) 4.46341 + 1.05785i 1.34577 + 0.318953i 0.839471 − 0.543405i \(-0.182865\pi\)
0.506299 + 0.862358i \(0.331013\pi\)
\(12\) −1.21081 + 1.23852i −0.349531 + 0.357530i
\(13\) 1.47403 − 0.969483i 0.408821 − 0.268886i −0.328398 − 0.944539i \(-0.606509\pi\)
0.737219 + 0.675653i \(0.236138\pi\)
\(14\) 0.163226 + 2.80247i 0.0436239 + 0.748993i
\(15\) 0.665260 − 1.46984i 0.171769 − 0.379512i
\(16\) −0.286803 − 0.957990i −0.0717008 − 0.239497i
\(17\) 1.42569 + 0.518907i 0.345780 + 0.125854i 0.509071 − 0.860724i \(-0.329989\pi\)
−0.163292 + 0.986578i \(0.552211\pi\)
\(18\) −2.94188 + 0.587652i −0.693408 + 0.138511i
\(19\) 4.12421 − 1.50109i 0.946158 − 0.344373i 0.177563 − 0.984109i \(-0.443179\pi\)
0.768595 + 0.639736i \(0.220956\pi\)
\(20\) 0.556246 + 0.747168i 0.124380 + 0.167072i
\(21\) −2.50559 + 4.16696i −0.546764 + 0.909304i
\(22\) −4.46341 + 1.05785i −0.951603 + 0.225534i
\(23\) −2.92223 − 6.77448i −0.609326 − 1.41258i −0.891415 − 0.453189i \(-0.850286\pi\)
0.282088 − 0.959388i \(-0.408973\pi\)
\(24\) 0.526173 − 1.65019i 0.107405 − 0.336845i
\(25\) 3.45251 + 2.27075i 0.690502 + 0.454151i
\(26\) −0.882135 + 1.52790i −0.173001 + 0.299646i
\(27\) −4.78112 − 2.03493i −0.920126 − 0.391622i
\(28\) −1.40361 − 2.43113i −0.265258 − 0.459440i
\(29\) −0.418954 + 7.19315i −0.0777977 + 1.33574i 0.703107 + 0.711085i \(0.251796\pi\)
−0.780904 + 0.624651i \(0.785241\pi\)
\(30\) 0.0651659 + 1.61207i 0.0118976 + 0.294322i
\(31\) −6.24778 − 0.730261i −1.12213 − 0.131159i −0.465271 − 0.885168i \(-0.654043\pi\)
−0.656863 + 0.754010i \(0.728117\pi\)
\(32\) 0.686242 + 0.727374i 0.121312 + 0.128583i
\(33\) −7.41639 − 2.84963i −1.29103 − 0.496057i
\(34\) −1.50693 + 0.176134i −0.258436 + 0.0302068i
\(35\) 2.00313 + 1.68082i 0.338590 + 0.284111i
\(36\) 2.36522 − 1.84546i 0.394204 − 0.307576i
\(37\) −0.516723 + 0.433582i −0.0849487 + 0.0712804i −0.684273 − 0.729226i \(-0.739880\pi\)
0.599324 + 0.800506i \(0.295436\pi\)
\(38\) −3.01184 + 3.19236i −0.488585 + 0.517870i
\(39\) −2.75471 + 1.32269i −0.441107 + 0.211800i
\(40\) −0.832408 − 0.418051i −0.131615 − 0.0660997i
\(41\) −8.46317 − 4.25037i −1.32173 − 0.663796i −0.358763 − 0.933429i \(-0.616801\pi\)
−0.962963 + 0.269633i \(0.913098\pi\)
\(42\) 0.368950 − 4.84823i 0.0569302 − 0.748099i
\(43\) 3.77539 − 4.00168i 0.575742 − 0.610251i −0.372410 − 0.928068i \(-0.621468\pi\)
0.948152 + 0.317817i \(0.102950\pi\)
\(44\) 3.51389 − 2.94850i 0.529739 − 0.444504i
\(45\) −1.48237 + 2.36888i −0.220978 + 0.353132i
\(46\) 5.65178 + 4.74241i 0.833309 + 0.699230i
\(47\) 5.01677 − 0.586377i 0.731772 − 0.0855319i 0.257957 − 0.966156i \(-0.416951\pi\)
0.473815 + 0.880625i \(0.342877\pi\)
\(48\) 0.270401 + 1.71081i 0.0390290 + 0.246935i
\(49\) −0.604237 − 0.640453i −0.0863195 − 0.0914934i
\(50\) −4.10439 − 0.479734i −0.580448 − 0.0678447i
\(51\) −2.32699 − 1.22093i −0.325844 − 0.170964i
\(52\) 0.102583 − 1.76129i 0.0142257 − 0.244246i
\(53\) −1.33883 − 2.31892i −0.183903 − 0.318529i 0.759304 − 0.650737i \(-0.225540\pi\)
−0.943206 + 0.332208i \(0.892206\pi\)
\(54\) 5.18583 − 0.327285i 0.705703 − 0.0445378i
\(55\) −2.13639 + 3.70034i −0.288071 + 0.498954i
\(56\) 2.34540 + 1.54259i 0.313417 + 0.206138i
\(57\) −7.42686 + 1.62134i −0.983712 + 0.214751i
\(58\) −2.85389 − 6.61606i −0.374734 − 0.868732i
\(59\) 4.89941 − 1.16118i 0.637849 − 0.151173i 0.101048 − 0.994882i \(-0.467781\pi\)
0.536802 + 0.843708i \(0.319632\pi\)
\(60\) −0.781729 − 1.41135i −0.100921 − 0.182205i
\(61\) 3.14866 + 4.22938i 0.403144 + 0.541517i 0.956616 − 0.291351i \(-0.0941048\pi\)
−0.553472 + 0.832868i \(0.686697\pi\)
\(62\) 5.91096 − 2.15141i 0.750693 − 0.273230i
\(63\) 5.26600 − 6.57219i 0.663454 − 0.828019i
\(64\) −0.939693 − 0.342020i −0.117462 − 0.0427525i
\(65\) 0.471331 + 1.57436i 0.0584615 + 0.195275i
\(66\) 7.90644 − 0.781947i 0.973216 − 0.0962511i
\(67\) 0.832311 + 14.2902i 0.101683 + 1.74583i 0.535237 + 0.844702i \(0.320222\pi\)
−0.433554 + 0.901127i \(0.642741\pi\)
\(68\) 1.26759 − 0.833706i 0.153718 − 0.101102i
\(69\) 3.44684 + 12.3052i 0.414950 + 1.48137i
\(70\) −2.54441 − 0.603036i −0.304115 − 0.0720766i
\(71\) −0.0492509 − 0.279316i −0.00584501 − 0.0331487i 0.981746 − 0.190198i \(-0.0609130\pi\)
−0.987591 + 0.157049i \(0.949802\pi\)
\(72\) −1.28540 + 2.71067i −0.151486 + 0.319456i
\(73\) −2.01050 + 11.4021i −0.235312 + 1.33452i 0.606645 + 0.794973i \(0.292515\pi\)
−0.841957 + 0.539545i \(0.818596\pi\)
\(74\) 0.267169 − 0.619368i 0.0310578 − 0.0720000i
\(75\) −5.40025 − 4.69742i −0.623567 − 0.542412i
\(76\) 1.25875 − 4.20451i 0.144388 − 0.482290i
\(77\) 7.68955 − 10.3289i 0.876305 − 1.17708i
\(78\) 1.86808 − 2.41831i 0.211519 − 0.273820i
\(79\) 3.17920 − 1.59665i 0.357687 − 0.179637i −0.260876 − 0.965372i \(-0.584011\pi\)
0.618563 + 0.785735i \(0.287715\pi\)
\(80\) 0.931488 0.104144
\(81\) 7.73541 + 4.60037i 0.859490 + 0.511152i
\(82\) 9.47053 1.04584
\(83\) −9.56164 + 4.80204i −1.04953 + 0.527092i −0.888032 − 0.459782i \(-0.847928\pi\)
−0.161494 + 0.986874i \(0.551631\pi\)
\(84\) 1.84618 + 4.49813i 0.201435 + 0.490786i
\(85\) −0.843927 + 1.13359i −0.0915368 + 0.122955i
\(86\) −1.57786 + 5.27043i −0.170145 + 0.568325i
\(87\) 2.38525 − 12.2500i 0.255726 − 1.31333i
\(88\) −1.81684 + 4.21191i −0.193676 + 0.448991i
\(89\) 1.74720 − 9.90888i 0.185203 − 1.05034i −0.740491 − 0.672066i \(-0.765407\pi\)
0.925694 − 0.378273i \(-0.123482\pi\)
\(90\) 0.261540 − 2.78220i 0.0275687 − 0.293269i
\(91\) −0.860028 − 4.87746i −0.0901554 − 0.511297i
\(92\) −7.17900 − 1.70145i −0.748463 − 0.177389i
\(93\) 10.5551 + 2.70064i 1.09452 + 0.280043i
\(94\) −4.21999 + 2.77553i −0.435259 + 0.286274i
\(95\) 0.237708 + 4.08128i 0.0243883 + 0.418731i
\(96\) −1.00945 − 1.40748i −0.103027 − 0.143651i
\(97\) −4.96298 − 16.5775i −0.503914 − 1.68319i −0.709626 − 0.704579i \(-0.751136\pi\)
0.205711 − 0.978613i \(-0.434049\pi\)
\(98\) 0.827401 + 0.301149i 0.0835801 + 0.0304207i
\(99\) 12.0701 + 6.60920i 1.21309 + 0.664250i
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 162.2.g.a.25.1 yes 72
3.2 odd 2 486.2.g.a.73.3 72
81.13 even 27 inner 162.2.g.a.13.1 ✓ 72
81.68 odd 54 486.2.g.a.253.3 72
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
162.2.g.a.13.1 ✓ 72 81.13 even 27 inner
162.2.g.a.25.1 yes 72 1.1 even 1 trivial
486.2.g.a.73.3 72 3.2 odd 2
486.2.g.a.253.3 72 81.68 odd 54