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Properties

Label 324.2.p.a.59.22
Level $324$
Weight $2$
Character 324.59
Analytic conductor $2.587$
Analytic rank $0$
Dimension $936$
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] = [324,2,Mod(11,324)] 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("324.11"); 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([27, 13])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 324 = 2^{2} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 324.p (of order \(54\), 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: \(936\)
Relative dimension: \(52\) over \(\Q(\zeta_{54})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{54}]$

Embedding invariants

Embedding label 59.22
Character \(\chi\) \(=\) 324.59
Dual form 324.2.p.a.11.22

$q$-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+(-0.431752 + 1.34670i) q^{2} +(-1.60792 + 0.643879i) q^{3} +(-1.62718 - 1.16288i) q^{4} +(-0.221174 - 1.89226i) q^{5} +(-0.172884 - 2.44338i) q^{6} +(-0.167545 + 0.333610i) q^{7} +(2.26858 - 1.68924i) q^{8} +(2.17084 - 2.07062i) q^{9} +(2.64380 + 0.519135i) q^{10} +(1.11039 + 2.57416i) q^{11} +(3.36513 + 0.822113i) q^{12} +(2.99565 + 0.709981i) q^{13} +(-0.376933 - 0.369669i) q^{14} +(1.57402 + 2.90021i) q^{15} +(1.29543 + 3.78442i) q^{16} +(-2.21511 - 0.390583i) q^{17} +(1.85123 + 3.81746i) q^{18} +(5.40145 - 0.952421i) q^{19} +(-1.84058 + 3.33625i) q^{20} +(0.0545957 - 0.644298i) q^{21} +(-3.94603 + 0.383951i) q^{22} +(-0.385154 + 0.193431i) q^{23} +(-2.56004 + 4.17686i) q^{24} +(1.33348 - 0.316040i) q^{25} +(-2.24951 + 3.72769i) q^{26} +(-2.15732 + 4.72715i) q^{27} +(0.660574 - 0.348009i) q^{28} +(1.34433 + 0.402465i) q^{29} +(-4.58529 + 0.867555i) q^{30} +(1.43821 - 2.18670i) q^{31} +(-5.65577 + 0.110618i) q^{32} +(-3.44287 - 3.42411i) q^{33} +(1.48237 - 2.81444i) q^{34} +(0.668335 + 0.243254i) q^{35} +(-5.94022 + 0.844843i) q^{36} +(11.0643 - 4.02708i) q^{37} +(-1.04947 + 7.68532i) q^{38} +(-5.27391 + 0.787236i) q^{39} +(-3.69824 - 3.91914i) q^{40} +(-0.432568 - 0.408106i) q^{41} +(0.844102 + 0.351701i) q^{42} +(2.08028 + 1.54871i) q^{43} +(1.18664 - 5.47987i) q^{44} +(-4.39829 - 3.64984i) q^{45} +(-0.0942024 - 0.602199i) q^{46} +(-3.37922 + 2.22255i) q^{47} +(-4.51966 - 5.25097i) q^{48} +(4.09689 + 5.50307i) q^{49} +(-0.150122 + 1.93224i) q^{50} +(3.81321 - 0.798232i) q^{51} +(-4.04883 - 4.63884i) q^{52} +(5.07589 + 2.93057i) q^{53} +(-5.43461 - 4.94621i) q^{54} +(4.62541 - 2.67048i) q^{55} +(0.183458 + 1.03985i) q^{56} +(-8.07188 + 5.00930i) q^{57} +(-1.12241 + 1.63664i) q^{58} +(4.97300 - 11.5287i) q^{59} +(0.811375 - 6.54955i) q^{60} +(-0.301642 - 5.17899i) q^{61} +(2.32387 + 2.88095i) q^{62} +(0.327064 + 1.07114i) q^{63} +(2.29292 - 7.66437i) q^{64} +(0.680913 - 5.82558i) q^{65} +(6.09770 - 3.15813i) q^{66} +(-1.69797 + 0.508339i) q^{67} +(3.15018 + 3.21145i) q^{68} +(0.494752 - 0.559015i) q^{69} +(-0.616144 + 0.795018i) q^{70} +(-8.93715 - 7.49916i) q^{71} +(1.42696 - 8.36444i) q^{72} +(-10.8049 + 9.06636i) q^{73} +(0.646213 + 16.6390i) q^{74} +(-1.94064 + 1.36677i) q^{75} +(-9.89668 - 4.73146i) q^{76} +(-1.04481 - 0.0608530i) q^{77} +(1.21685 - 7.44225i) q^{78} +(10.7707 - 10.1616i) q^{79} +(6.87461 - 3.28831i) q^{80} +(0.425098 - 8.98996i) q^{81} +(0.736357 - 0.406336i) q^{82} +(4.02464 + 4.26587i) q^{83} +(-0.838077 + 0.984901i) q^{84} +(-0.249162 + 4.27795i) q^{85} +(-2.98380 + 2.13284i) q^{86} +(-2.42072 + 0.218450i) q^{87} +(6.86739 + 3.96399i) q^{88} +(8.91268 + 10.6217i) q^{89} +(6.81419 - 4.34733i) q^{90} +(-0.738762 + 0.880423i) q^{91} +(0.851652 + 0.133139i) q^{92} +(-0.904571 + 4.44208i) q^{93} +(-1.53411 - 5.51037i) q^{94} +(-2.99689 - 10.0103i) q^{95} +(9.02283 - 3.81950i) q^{96} +(-15.9056 - 1.85910i) q^{97} +(-9.17981 + 3.14130i) q^{98} +(7.74058 + 3.28892i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 936 q - 18 q^{2} - 18 q^{4} - 36 q^{5} - 18 q^{6} - 18 q^{8} - 36 q^{9} - 18 q^{10} - 18 q^{12} - 36 q^{13} - 18 q^{14} - 18 q^{16} - 36 q^{17} - 18 q^{18} - 18 q^{20} - 36 q^{21} - 18 q^{22} - 18 q^{24}+ \cdots + 153 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(163\) \(245\)
\(\chi(n)\) \(-1\) \(e\left(\frac{41}{54}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.431752 + 1.34670i −0.305295 + 0.952258i
\(3\) −1.60792 + 0.643879i −0.928335 + 0.371744i
\(4\) −1.62718 − 1.16288i −0.813590 − 0.581439i
\(5\) −0.221174 − 1.89226i −0.0989120 − 0.846246i −0.948060 − 0.318091i \(-0.896958\pi\)
0.849148 − 0.528155i \(-0.177116\pi\)
\(6\) −0.172884 − 2.44338i −0.0705796 − 0.997506i
\(7\) −0.167545 + 0.333610i −0.0633261 + 0.126093i −0.923069 − 0.384635i \(-0.874327\pi\)
0.859743 + 0.510727i \(0.170624\pi\)
\(8\) 2.26858 − 1.68924i 0.802065 − 0.597237i
\(9\) 2.17084 − 2.07062i 0.723614 − 0.690205i
\(10\) 2.64380 + 0.519135i 0.836042 + 0.164165i
\(11\) 1.11039 + 2.57416i 0.334794 + 0.776140i 0.999576 + 0.0291027i \(0.00926497\pi\)
−0.664782 + 0.747037i \(0.731476\pi\)
\(12\) 3.36513 + 0.822113i 0.971431 + 0.237324i
\(13\) 2.99565 + 0.709981i 0.830843 + 0.196913i 0.623964 − 0.781453i \(-0.285521\pi\)
0.206879 + 0.978367i \(0.433669\pi\)
\(14\) −0.376933 − 0.369669i −0.100740 − 0.0987982i
\(15\) 1.57402 + 2.90021i 0.406410 + 0.748831i
\(16\) 1.29543 + 3.78442i 0.323858 + 0.946106i
\(17\) −2.21511 − 0.390583i −0.537242 − 0.0947303i −0.101558 − 0.994830i \(-0.532383\pi\)
−0.435685 + 0.900099i \(0.643494\pi\)
\(18\) 1.85123 + 3.81746i 0.436338 + 0.899783i
\(19\) 5.40145 − 0.952421i 1.23918 − 0.218500i 0.484614 − 0.874728i \(-0.338960\pi\)
0.754563 + 0.656227i \(0.227849\pi\)
\(20\) −1.84058 + 3.33625i −0.411567 + 0.746009i
\(21\) 0.0545957 − 0.644298i 0.0119137 − 0.140597i
\(22\) −3.94603 + 0.383951i −0.841296 + 0.0818587i
\(23\) −0.385154 + 0.193431i −0.0803101 + 0.0403332i −0.488501 − 0.872563i \(-0.662456\pi\)
0.408191 + 0.912897i \(0.366160\pi\)
\(24\) −2.56004 + 4.17686i −0.522566 + 0.852599i
\(25\) 1.33348 − 0.316040i 0.266695 − 0.0632080i
\(26\) −2.24951 + 3.72769i −0.441164 + 0.731060i
\(27\) −2.15732 + 4.72715i −0.415177 + 0.909741i
\(28\) 0.660574 − 0.348009i 0.124837 − 0.0657675i
\(29\) 1.34433 + 0.402465i 0.249635 + 0.0747359i 0.409176 − 0.912455i \(-0.365816\pi\)
−0.159541 + 0.987191i \(0.551001\pi\)
\(30\) −4.58529 + 0.867555i −0.837155 + 0.158393i
\(31\) 1.43821 − 2.18670i 0.258311 − 0.392743i −0.682927 − 0.730486i \(-0.739293\pi\)
0.941238 + 0.337744i \(0.109664\pi\)
\(32\) −5.65577 + 0.110618i −0.999809 + 0.0195547i
\(33\) −3.44287 − 3.42411i −0.599326 − 0.596060i
\(34\) 1.48237 − 2.81444i 0.254225 − 0.482672i
\(35\) 0.668335 + 0.243254i 0.112969 + 0.0411174i
\(36\) −5.94022 + 0.844843i −0.990037 + 0.140807i
\(37\) 11.0643 − 4.02708i 1.81896 − 0.662048i 0.823454 − 0.567383i \(-0.192044\pi\)
0.995509 − 0.0946651i \(-0.0301780\pi\)
\(38\) −1.04947 + 7.68532i −0.170246 + 1.24672i
\(39\) −5.27391 + 0.787236i −0.844502 + 0.126059i
\(40\) −3.69824 − 3.91914i −0.584744 − 0.619670i
\(41\) −0.432568 − 0.408106i −0.0675557 − 0.0637355i 0.651723 − 0.758457i \(-0.274046\pi\)
−0.719279 + 0.694721i \(0.755528\pi\)
\(42\) 0.844102 + 0.351701i 0.130248 + 0.0542686i
\(43\) 2.08028 + 1.54871i 0.317239 + 0.236176i 0.743986 − 0.668195i \(-0.232933\pi\)
−0.426747 + 0.904371i \(0.640340\pi\)
\(44\) 1.18664 − 5.47987i 0.178893 − 0.826122i
\(45\) −4.39829 − 3.64984i −0.655658 − 0.544086i
\(46\) −0.0942024 − 0.602199i −0.0138894 − 0.0887895i
\(47\) −3.37922 + 2.22255i −0.492909 + 0.324192i −0.771512 − 0.636215i \(-0.780499\pi\)
0.278602 + 0.960407i \(0.410129\pi\)
\(48\) −4.51966 − 5.25097i −0.652357 − 0.757912i
\(49\) 4.09689 + 5.50307i 0.585269 + 0.786153i
\(50\) −0.150122 + 1.93224i −0.0212305 + 0.273260i
\(51\) 3.81321 − 0.798232i 0.533956 − 0.111775i
\(52\) −4.04883 − 4.63884i −0.561472 − 0.643291i
\(53\) 5.07589 + 2.93057i 0.697227 + 0.402544i 0.806314 − 0.591488i \(-0.201459\pi\)
−0.109087 + 0.994032i \(0.534793\pi\)
\(54\) −5.43461 − 4.94621i −0.739556 − 0.673095i
\(55\) 4.62541 − 2.67048i 0.623690 − 0.360088i
\(56\) 0.183458 + 1.03985i 0.0245156 + 0.138955i
\(57\) −8.07188 + 5.00930i −1.06915 + 0.663498i
\(58\) −1.12241 + 1.63664i −0.147380 + 0.214901i
\(59\) 4.97300 − 11.5287i 0.647430 − 1.50091i −0.205376 − 0.978683i \(-0.565842\pi\)
0.852806 − 0.522228i \(-0.174899\pi\)
\(60\) 0.811375 − 6.54955i 0.104748 − 0.845544i
\(61\) −0.301642 − 5.17899i −0.0386213 − 0.663102i −0.960954 − 0.276706i \(-0.910757\pi\)
0.922333 − 0.386395i \(-0.126280\pi\)
\(62\) 2.32387 + 2.88095i 0.295131 + 0.365881i
\(63\) 0.327064 + 1.07114i 0.0412062 + 0.134950i
\(64\) 2.29292 − 7.66437i 0.286615 − 0.958046i
\(65\) 0.680913 − 5.82558i 0.0844569 − 0.722575i
\(66\) 6.09770 − 3.15813i 0.750574 − 0.388739i
\(67\) −1.69797 + 0.508339i −0.207440 + 0.0621035i −0.388837 − 0.921306i \(-0.627123\pi\)
0.181397 + 0.983410i \(0.441938\pi\)
\(68\) 3.15018 + 3.21145i 0.382015 + 0.389445i
\(69\) 0.494752 − 0.559015i 0.0595611 − 0.0672975i
\(70\) −0.616144 + 0.795018i −0.0736433 + 0.0950228i
\(71\) −8.93715 − 7.49916i −1.06065 − 0.889987i −0.0664726 − 0.997788i \(-0.521175\pi\)
−0.994172 + 0.107801i \(0.965619\pi\)
\(72\) 1.42696 − 8.36444i 0.168168 − 0.985758i
\(73\) −10.8049 + 9.06636i −1.26461 + 1.06114i −0.269439 + 0.963017i \(0.586838\pi\)
−0.995175 + 0.0981197i \(0.968717\pi\)
\(74\) 0.646213 + 16.6390i 0.0751207 + 1.93424i
\(75\) −1.94064 + 1.36677i −0.224086 + 0.157821i
\(76\) −9.89668 − 4.73146i −1.13523 − 0.542736i
\(77\) −1.04481 − 0.0608530i −0.119067 − 0.00693485i
\(78\) 1.21685 − 7.44225i 0.137782 − 0.842669i
\(79\) 10.7707 − 10.1616i 1.21180 − 1.14327i 0.226214 − 0.974078i \(-0.427365\pi\)
0.985583 − 0.169194i \(-0.0541163\pi\)
\(80\) 6.87461 − 3.28831i 0.768605 − 0.367645i
\(81\) 0.425098 − 8.98996i 0.0472331 − 0.998884i
\(82\) 0.736357 − 0.406336i 0.0813171 − 0.0448723i
\(83\) 4.02464 + 4.26587i 0.441761 + 0.468240i 0.909546 − 0.415603i \(-0.136429\pi\)
−0.467785 + 0.883842i \(0.654948\pi\)
\(84\) −0.838077 + 0.984901i −0.0914417 + 0.107461i
\(85\) −0.249162 + 4.27795i −0.0270255 + 0.464009i
\(86\) −2.98380 + 2.13284i −0.321752 + 0.229990i
\(87\) −2.42072 + 0.218450i −0.259528 + 0.0234203i
\(88\) 6.86739 + 3.96399i 0.732066 + 0.422563i
\(89\) 8.91268 + 10.6217i 0.944743 + 1.12590i 0.991901 + 0.127013i \(0.0405390\pi\)
−0.0471583 + 0.998887i \(0.515017\pi\)
\(90\) 6.81419 − 4.34733i 0.718279 − 0.458249i
\(91\) −0.738762 + 0.880423i −0.0774433 + 0.0922934i
\(92\) 0.851652 + 0.133139i 0.0887908 + 0.0138807i
\(93\) −0.904571 + 4.44208i −0.0937997 + 0.460622i
\(94\) −1.53411 − 5.51037i −0.158231 − 0.568351i
\(95\) −2.99689 − 10.0103i −0.307475 − 1.02704i
\(96\) 9.02283 − 3.81950i 0.920889 − 0.389826i
\(97\) −15.9056 − 1.85910i −1.61497 − 0.188763i −0.739885 − 0.672733i \(-0.765120\pi\)
−0.875084 + 0.483970i \(0.839194\pi\)
\(98\) −9.17981 + 3.14130i −0.927300 + 0.317319i
\(99\) 7.74058 + 3.28892i 0.777957 + 0.330549i
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.p.a.59.22 yes 936
3.2 odd 2 972.2.p.a.287.31 936
4.3 odd 2 inner 324.2.p.a.59.4 yes 936
12.11 even 2 972.2.p.a.287.49 936
81.11 odd 54 inner 324.2.p.a.11.4 ✓ 936
81.70 even 27 972.2.p.a.359.49 936
324.11 even 54 inner 324.2.p.a.11.22 yes 936
324.151 odd 54 972.2.p.a.359.31 936
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
324.2.p.a.11.4 ✓ 936 81.11 odd 54 inner
324.2.p.a.11.22 yes 936 324.11 even 54 inner
324.2.p.a.59.4 yes 936 4.3 odd 2 inner
324.2.p.a.59.22 yes 936 1.1 even 1 trivial
972.2.p.a.287.31 936 3.2 odd 2
972.2.p.a.287.49 936 12.11 even 2
972.2.p.a.359.31 936 324.151 odd 54
972.2.p.a.359.49 936 81.70 even 27