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Properties

Label 324.2.m.a.13.6
Level $324$
Weight $2$
Character 324.13
Analytic conductor $2.587$
Analytic rank $0$
Dimension $162$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [324,2,Mod(13,324)] 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("324.13"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(324, base_ring=CyclotomicField(54)) chi = DirichletCharacter(H, H._module([0, 8])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 324 = 2^{2} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 324.m (of order \(27\), degree \(18\), 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: \(2.58715302549\)
Analytic rank: \(0\)
Dimension: \(162\)
Relative dimension: \(9\) over \(\Q(\zeta_{27})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{27}]$

Embedding invariants

Embedding label 13.6
Character \(\chi\) \(=\) 324.13
Dual form 324.2.m.a.25.6

$q$-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)
 
\(f(q)\) \(=\) \(q+(1.19354 - 1.25517i) q^{3} +(0.462190 + 1.54382i) q^{5} +(1.59904 + 3.70700i) q^{7} +(-0.150919 - 2.99620i) q^{9} +(0.930435 - 0.220517i) q^{11} +(2.77018 + 1.82197i) q^{13} +(2.48941 + 1.26249i) q^{15} +(-2.95953 + 1.07718i) q^{17} +(-4.18090 - 1.52172i) q^{19} +(6.56145 + 2.41738i) q^{21} +(2.07761 - 4.81645i) q^{23} +(2.00767 - 1.32046i) q^{25} +(-3.94088 - 3.38666i) q^{27} +(-0.559405 - 9.60462i) q^{29} +(-4.64944 + 0.543442i) q^{31} +(0.833725 - 1.43105i) q^{33} +(-4.98389 + 4.18198i) q^{35} +(8.17280 + 6.85779i) q^{37} +(5.59321 - 1.30245i) q^{39} +(-0.720394 + 0.361795i) q^{41} +(-0.00600557 - 0.00636553i) q^{43} +(4.55585 - 1.61781i) q^{45} +(-9.82642 - 1.14854i) q^{47} +(-6.38120 + 6.76367i) q^{49} +(-2.18027 + 5.00038i) q^{51} +(-6.20809 + 10.7527i) q^{53} +(0.770477 + 1.33451i) q^{55} +(-6.90010 + 3.43151i) q^{57} +(-4.04484 - 0.958644i) q^{59} +(2.25488 - 3.02884i) q^{61} +(10.8656 - 5.35051i) q^{63} +(-1.53246 + 5.11876i) q^{65} +(0.225665 - 3.87452i) q^{67} +(-3.56576 - 8.35640i) q^{69} +(2.13922 - 12.1321i) q^{71} +(1.58976 + 9.01600i) q^{73} +(0.738824 - 4.09600i) q^{75} +(2.30526 + 3.09650i) q^{77} +(0.699902 + 0.351504i) q^{79} +(-8.95445 + 0.904370i) q^{81} +(-4.09284 - 2.05550i) q^{83} +(-3.03084 - 4.07113i) q^{85} +(-12.7231 - 10.7614i) q^{87} +(-0.189270 - 1.07340i) q^{89} +(-2.32442 + 13.1825i) q^{91} +(-4.86718 + 6.48447i) q^{93} +(0.416900 - 7.15789i) q^{95} +(2.93216 - 9.79410i) q^{97} +(-0.801134 - 2.75449i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 162 q + 27 q^{21} + 27 q^{23} + 27 q^{27} + 27 q^{29} + 27 q^{33} + 27 q^{35} - 18 q^{41} - 54 q^{45} - 54 q^{47} - 63 q^{51} - 54 q^{53} - 54 q^{57} - 63 q^{59} - 54 q^{63} - 90 q^{65} + 27 q^{67} - 90 q^{69}+ \cdots - 90 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/324\mathbb{Z}\right)^\times\).

\(n\) \(163\) \(245\)
\(\chi(n)\) \(1\) \(e\left(\frac{4}{27}\right)\)

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 0
\(3\) 1.19354 − 1.25517i 0.689091 − 0.724675i
\(4\) 0 0
\(5\) 0.462190 + 1.54382i 0.206698 + 0.690419i 0.996979 + 0.0776741i \(0.0247494\pi\)
−0.790281 + 0.612745i \(0.790065\pi\)
\(6\) 0 0
\(7\) 1.59904 + 3.70700i 0.604381 + 1.40111i 0.895774 + 0.444510i \(0.146622\pi\)
−0.291393 + 0.956603i \(0.594119\pi\)
\(8\) 0 0
\(9\) −0.150919 − 2.99620i −0.0503065 − 0.998734i
\(10\) 0 0
\(11\) 0.930435 − 0.220517i 0.280537 − 0.0664884i −0.0879388 − 0.996126i \(-0.528028\pi\)
0.368475 + 0.929637i \(0.379880\pi\)
\(12\) 0 0
\(13\) 2.77018 + 1.82197i 0.768309 + 0.505325i 0.872132 − 0.489271i \(-0.162737\pi\)
−0.103823 + 0.994596i \(0.533107\pi\)
\(14\) 0 0
\(15\) 2.48941 + 1.26249i 0.642763 + 0.325973i
\(16\) 0 0
\(17\) −2.95953 + 1.07718i −0.717792 + 0.261255i −0.674988 − 0.737829i \(-0.735851\pi\)
−0.0428036 + 0.999084i \(0.513629\pi\)
\(18\) 0 0
\(19\) −4.18090 − 1.52172i −0.959164 − 0.349107i −0.185458 − 0.982652i \(-0.559377\pi\)
−0.773706 + 0.633545i \(0.781599\pi\)
\(20\) 0 0
\(21\) 6.56145 + 2.41738i 1.43182 + 0.527515i
\(22\) 0 0
\(23\) 2.07761 − 4.81645i 0.433213 − 1.00430i −0.552137 − 0.833753i \(-0.686187\pi\)
0.985350 − 0.170546i \(-0.0545533\pi\)
\(24\) 0 0
\(25\) 2.00767 − 1.32046i 0.401534 − 0.264093i
\(26\) 0 0
\(27\) −3.94088 − 3.38666i −0.758423 − 0.651763i
\(28\) 0 0
\(29\) −0.559405 − 9.60462i −0.103879 − 1.78353i −0.500685 − 0.865630i \(-0.666918\pi\)
0.396806 − 0.917903i \(-0.370119\pi\)
\(30\) 0 0
\(31\) −4.64944 + 0.543442i −0.835064 + 0.0976050i −0.522869 − 0.852413i \(-0.675138\pi\)
−0.312195 + 0.950018i \(0.601064\pi\)
\(32\) 0 0
\(33\) 0.833725 − 1.43105i 0.145133 − 0.249114i
\(34\) 0 0
\(35\) −4.98389 + 4.18198i −0.842430 + 0.706883i
\(36\) 0 0
\(37\) 8.17280 + 6.85779i 1.34360 + 1.12741i 0.980685 + 0.195591i \(0.0626626\pi\)
0.362914 + 0.931823i \(0.381782\pi\)
\(38\) 0 0
\(39\) 5.59321 − 1.30245i 0.895631 − 0.208559i
\(40\) 0 0
\(41\) −0.720394 + 0.361795i −0.112507 + 0.0565030i −0.504162 − 0.863609i \(-0.668199\pi\)
0.391656 + 0.920112i \(0.371902\pi\)
\(42\) 0 0
\(43\) −0.00600557 − 0.00636553i −0.000915841 − 0.000970735i 0.726916 − 0.686727i \(-0.240953\pi\)
−0.727831 + 0.685756i \(0.759472\pi\)
\(44\) 0 0
\(45\) 4.55585 − 1.61781i 0.679146 − 0.241169i
\(46\) 0 0
\(47\) −9.82642 − 1.14854i −1.43333 − 0.167532i −0.636286 − 0.771453i \(-0.719530\pi\)
−0.797045 + 0.603921i \(0.793604\pi\)
\(48\) 0 0
\(49\) −6.38120 + 6.76367i −0.911600 + 0.966239i
\(50\) 0 0
\(51\) −2.18027 + 5.00038i −0.305299 + 0.700194i
\(52\) 0 0
\(53\) −6.20809 + 10.7527i −0.852746 + 1.47700i 0.0259741 + 0.999663i \(0.491731\pi\)
−0.878720 + 0.477337i \(0.841602\pi\)
\(54\) 0 0
\(55\) 0.770477 + 1.33451i 0.103891 + 0.179945i
\(56\) 0 0
\(57\) −6.90010 + 3.43151i −0.913940 + 0.454515i
\(58\) 0 0
\(59\) −4.04484 − 0.958644i −0.526593 − 0.124805i −0.0412838 − 0.999147i \(-0.513145\pi\)
−0.485309 + 0.874343i \(0.661293\pi\)
\(60\) 0 0
\(61\) 2.25488 − 3.02884i 0.288708 − 0.387803i −0.633980 − 0.773350i \(-0.718580\pi\)
0.922688 + 0.385547i \(0.125987\pi\)
\(62\) 0 0
\(63\) 10.8656 − 5.35051i 1.36893 − 0.674101i
\(64\) 0 0
\(65\) −1.53246 + 5.11876i −0.190078 + 0.634904i
\(66\) 0 0
\(67\) 0.225665 − 3.87452i 0.0275694 − 0.473348i −0.956063 − 0.293161i \(-0.905293\pi\)
0.983632 − 0.180187i \(-0.0576703\pi\)
\(68\) 0 0
\(69\) −3.56576 − 8.35640i −0.429267 − 1.00599i
\(70\) 0 0
\(71\) 2.13922 − 12.1321i 0.253879 − 1.43982i −0.545055 − 0.838400i \(-0.683491\pi\)
0.798934 − 0.601418i \(-0.205397\pi\)
\(72\) 0 0
\(73\) 1.58976 + 9.01600i 0.186068 + 1.05524i 0.924576 + 0.380998i \(0.124420\pi\)
−0.738508 + 0.674245i \(0.764469\pi\)
\(74\) 0 0
\(75\) 0.738824 − 4.09600i 0.0853120 − 0.472965i
\(76\) 0 0
\(77\) 2.30526 + 3.09650i 0.262709 + 0.352879i
\(78\) 0 0
\(79\) 0.699902 + 0.351504i 0.0787451 + 0.0395473i 0.487736 − 0.872991i \(-0.337823\pi\)
−0.408991 + 0.912539i \(0.634119\pi\)
\(80\) 0 0
\(81\) −8.95445 + 0.904370i −0.994939 + 0.100486i
\(82\) 0 0
\(83\) −4.09284 − 2.05550i −0.449248 − 0.225621i 0.209766 − 0.977752i \(-0.432730\pi\)
−0.659014 + 0.752131i \(0.729026\pi\)
\(84\) 0 0
\(85\) −3.03084 − 4.07113i −0.328741 − 0.441576i
\(86\) 0 0
\(87\) −12.7231 − 10.7614i −1.36406 − 1.15374i
\(88\) 0 0
\(89\) −0.189270 − 1.07340i −0.0200626 − 0.113781i 0.973132 − 0.230249i \(-0.0739539\pi\)
−0.993194 + 0.116468i \(0.962843\pi\)
\(90\) 0 0
\(91\) −2.32442 + 13.1825i −0.243666 + 1.38190i
\(92\) 0 0
\(93\) −4.86718 + 6.48447i −0.504704 + 0.672409i
\(94\) 0 0
\(95\) 0.416900 − 7.15789i 0.0427730 − 0.734384i
\(96\) 0 0
\(97\) 2.93216 − 9.79410i 0.297716 − 0.994440i −0.669993 − 0.742368i \(-0.733703\pi\)
0.967708 − 0.252072i \(-0.0811121\pi\)
\(98\) 0 0
\(99\) −0.801134 − 2.75449i −0.0805170 − 0.276837i
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 324.2.m.a.13.6 ✓ 162
3.2 odd 2 972.2.m.a.253.4 162
81.25 even 27 inner 324.2.m.a.25.6 yes 162
81.56 odd 54 972.2.m.a.73.4 162
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
324.2.m.a.13.6 ✓ 162 1.1 even 1 trivial
324.2.m.a.25.6 yes 162 81.25 even 27 inner
972.2.m.a.73.4 162 81.56 odd 54
972.2.m.a.253.4 162 3.2 odd 2