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Properties

Label 324.2.p.a.11.3
Level $324$
Weight $2$
Character 324.11
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 11.3
Character \(\chi\) \(=\) 324.11
Dual form 324.2.p.a.59.3

$q$-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+(-1.41031 - 0.105029i) q^{2} +(-0.330952 - 1.70014i) q^{3} +(1.97794 + 0.296246i) q^{4} +(0.00377236 - 0.0322746i) q^{5} +(0.288181 + 2.43248i) q^{6} +(-0.446224 - 0.888506i) q^{7} +(-2.75839 - 0.625539i) q^{8} +(-2.78094 + 1.12533i) q^{9} +(-0.00870995 + 0.0451209i) q^{10} +(2.16786 - 5.02566i) q^{11} +(-0.150943 - 3.46081i) q^{12} +(1.51950 - 0.360127i) q^{13} +(0.535995 + 1.29993i) q^{14} +(-0.0561197 + 0.00426781i) q^{15} +(3.82448 + 1.17191i) q^{16} +(-5.49614 + 0.969119i) q^{17} +(4.04018 - 1.29498i) q^{18} +(0.388087 + 0.0684303i) q^{19} +(0.0170227 - 0.0627196i) q^{20} +(-1.36290 + 1.05270i) q^{21} +(-3.58519 + 6.86004i) q^{22} +(-7.20224 - 3.61710i) q^{23} +(-0.150609 + 4.89666i) q^{24} +(4.86420 + 1.15284i) q^{25} +(-2.18078 + 0.348299i) q^{26} +(2.83357 + 4.35556i) q^{27} +(-0.619388 - 1.88960i) q^{28} +(-1.60710 + 0.481133i) q^{29} +(0.0795943 - 0.000124733i) q^{30} +(-5.26788 - 8.00942i) q^{31} +(-5.27061 - 2.05444i) q^{32} +(-9.26178 - 2.02241i) q^{33} +(7.85304 - 0.789502i) q^{34} +(-0.0303595 + 0.0110499i) q^{35} +(-5.83390 + 1.40199i) q^{36} +(-6.57185 - 2.39196i) q^{37} +(-0.540136 - 0.137268i) q^{38} +(-1.11515 - 2.46417i) q^{39} +(-0.0305946 + 0.0866660i) q^{40} +(1.69763 - 1.60163i) q^{41} +(2.03268 - 1.34148i) q^{42} +(-4.21206 + 3.13576i) q^{43} +(5.77672 - 9.29823i) q^{44} +(0.0258288 + 0.0939989i) q^{45} +(9.77747 + 5.85767i) q^{46} +(7.16150 + 4.71019i) q^{47} +(0.726696 - 6.88999i) q^{48} +(3.58978 - 4.82192i) q^{49} +(-6.73894 - 2.13673i) q^{50} +(3.46660 + 9.02348i) q^{51} +(3.11215 - 0.262164i) q^{52} +(3.35957 - 1.93965i) q^{53} +(-3.53875 - 6.44028i) q^{54} +(-0.154023 - 0.0889253i) q^{55} +(0.675065 + 2.72997i) q^{56} +(-0.0120974 - 0.682449i) q^{57} +(2.31704 - 0.509755i) q^{58} +(3.05504 + 7.08237i) q^{59} +(-0.112266 - 0.00818379i) q^{60} +(0.0769563 - 1.32129i) q^{61} +(6.58811 + 11.8490i) q^{62} +(2.24078 + 1.96873i) q^{63} +(7.21740 + 3.45096i) q^{64} +(-0.00589087 - 0.0503996i) q^{65} +(12.8495 + 3.82497i) q^{66} +(-2.85205 - 0.853849i) q^{67} +(-11.1581 + 0.288645i) q^{68} +(-3.76597 + 13.4419i) q^{69} +(0.0439768 - 0.0123952i) q^{70} +(4.98620 - 4.18392i) q^{71} +(8.37485 - 1.36450i) q^{72} +(1.85640 + 1.55770i) q^{73} +(9.01711 + 4.06363i) q^{74} +(0.350165 - 8.65134i) q^{75} +(0.747340 + 0.250320i) q^{76} +(-5.43268 + 0.316418i) q^{77} +(1.31389 + 3.59236i) q^{78} +(4.37177 + 4.12455i) q^{79} +(0.0522503 - 0.119012i) q^{80} +(6.46727 - 6.25895i) q^{81} +(-2.56240 + 2.08049i) q^{82} +(7.99493 - 8.47413i) q^{83} +(-3.00760 + 1.67841i) q^{84} +(0.0105445 + 0.181042i) q^{85} +(6.26965 - 3.98000i) q^{86} +(1.34987 + 2.57306i) q^{87} +(-9.12354 + 12.5066i) q^{88} +(4.18508 - 4.98758i) q^{89} +(-0.0265540 - 0.135280i) q^{90} +(-0.998011 - 1.18938i) q^{91} +(-13.1740 - 9.28803i) q^{92} +(-11.8737 + 11.6069i) q^{93} +(-9.60521 - 7.39498i) q^{94} +(0.00367256 - 0.0122672i) q^{95} +(-1.74851 + 9.64068i) q^{96} +(16.7230 - 1.95464i) q^{97} +(-5.56914 + 6.42336i) q^{98} +(-0.373168 + 16.4156i) 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 \(\left(\mathbb{Z}/324\mathbb{Z}\right)^\times\).

\(n\) \(163\) \(245\)
\(\chi(n)\) \(-1\) \(e\left(\frac{13}{54}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.41031 − 0.105029i −0.997238 − 0.0742666i
\(3\) −0.330952 − 1.70014i −0.191075 − 0.981575i
\(4\) 1.97794 + 0.296246i 0.988969 + 0.148123i
\(5\) 0.00377236 − 0.0322746i 0.00168705 − 0.0144336i −0.992369 − 0.123307i \(-0.960650\pi\)
0.994056 + 0.108873i \(0.0347242\pi\)
\(6\) 0.288181 + 2.43248i 0.117649 + 0.993055i
\(7\) −0.446224 − 0.888506i −0.168657 − 0.335824i 0.793387 − 0.608717i \(-0.208316\pi\)
−0.962044 + 0.272894i \(0.912019\pi\)
\(8\) −2.75839 − 0.625539i −0.975237 − 0.221161i
\(9\) −2.78094 + 1.12533i −0.926980 + 0.375110i
\(10\) −0.00870995 + 0.0451209i −0.00275433 + 0.0142685i
\(11\) 2.16786 − 5.02566i 0.653634 − 1.51529i −0.191924 − 0.981410i \(-0.561473\pi\)
0.845558 − 0.533884i \(-0.179268\pi\)
\(12\) −0.150943 − 3.46081i −0.0435735 − 0.999050i
\(13\) 1.51950 − 0.360127i 0.421432 − 0.0998813i −0.0144274 − 0.999896i \(-0.504593\pi\)
0.435860 + 0.900015i \(0.356444\pi\)
\(14\) 0.535995 + 1.29993i 0.143251 + 0.347422i
\(15\) −0.0561197 + 0.00426781i −0.0144900 + 0.00110194i
\(16\) 3.82448 + 1.17191i 0.956119 + 0.292978i
\(17\) −5.49614 + 0.969119i −1.33301 + 0.235046i −0.794342 − 0.607471i \(-0.792184\pi\)
−0.538669 + 0.842517i \(0.681073\pi\)
\(18\) 4.04018 − 1.29498i 0.952279 − 0.305230i
\(19\) 0.388087 + 0.0684303i 0.0890333 + 0.0156990i 0.217987 − 0.975952i \(-0.430051\pi\)
−0.128954 + 0.991651i \(0.541162\pi\)
\(20\) 0.0170227 − 0.0627196i 0.00380639 − 0.0140245i
\(21\) −1.36290 + 1.05270i −0.297410 + 0.229717i
\(22\) −3.58519 + 6.86004i −0.764365 + 1.46257i
\(23\) −7.20224 − 3.61710i −1.50177 − 0.754217i −0.507555 − 0.861620i \(-0.669450\pi\)
−0.994216 + 0.107402i \(0.965747\pi\)
\(24\) −0.150609 + 4.89666i −0.0307429 + 0.999527i
\(25\) 4.86420 + 1.15284i 0.972839 + 0.230567i
\(26\) −2.18078 + 0.348299i −0.427686 + 0.0683071i
\(27\) 2.83357 + 4.35556i 0.545321 + 0.838227i
\(28\) −0.619388 − 1.88960i −0.117053 − 0.357101i
\(29\) −1.60710 + 0.481133i −0.298430 + 0.0893442i −0.432518 − 0.901625i \(-0.642375\pi\)
0.134088 + 0.990969i \(0.457190\pi\)
\(30\) 0.0795943 0.000124733i 0.0145319 2.27730e-5i
\(31\) −5.26788 − 8.00942i −0.946138 − 1.43853i −0.898152 − 0.439686i \(-0.855090\pi\)
−0.0479866 − 0.998848i \(-0.515280\pi\)
\(32\) −5.27061 − 2.05444i −0.931720 − 0.363177i
\(33\) −9.26178 − 2.02241i −1.61227 − 0.352056i
\(34\) 7.85304 − 0.789502i 1.34679 − 0.135398i
\(35\) −0.0303595 + 0.0110499i −0.00513169 + 0.00186778i
\(36\) −5.83390 + 1.40199i −0.972317 + 0.233664i
\(37\) −6.57185 − 2.39196i −1.08041 − 0.393236i −0.260347 − 0.965515i \(-0.583837\pi\)
−0.820059 + 0.572279i \(0.806059\pi\)
\(38\) −0.540136 − 0.137268i −0.0876216 − 0.0222678i
\(39\) −1.11515 − 2.46417i −0.178566 − 0.394583i
\(40\) −0.0305946 + 0.0866660i −0.00483744 + 0.0137031i
\(41\) 1.69763 − 1.60163i 0.265125 − 0.250132i −0.542186 − 0.840258i \(-0.682403\pi\)
0.807311 + 0.590126i \(0.200922\pi\)
\(42\) 2.03268 − 1.34148i 0.313649 − 0.206995i
\(43\) −4.21206 + 3.13576i −0.642333 + 0.478199i −0.868368 − 0.495920i \(-0.834831\pi\)
0.226035 + 0.974119i \(0.427424\pi\)
\(44\) 5.77672 − 9.29823i 0.870874 − 1.40176i
\(45\) 0.0258288 + 0.0939989i 0.00385033 + 0.0140125i
\(46\) 9.77747 + 5.85767i 1.44161 + 0.863666i
\(47\) 7.16150 + 4.71019i 1.04461 + 0.687052i 0.951200 − 0.308576i \(-0.0998523\pi\)
0.0934117 + 0.995628i \(0.470223\pi\)
\(48\) 0.726696 − 6.88999i 0.104890 − 0.994484i
\(49\) 3.58978 − 4.82192i 0.512826 − 0.688845i
\(50\) −6.73894 − 2.13673i −0.953029 − 0.302180i
\(51\) 3.46660 + 9.02348i 0.485421 + 1.26354i
\(52\) 3.11215 − 0.262164i 0.431578 − 0.0363556i
\(53\) 3.35957 − 1.93965i 0.461473 − 0.266431i −0.251191 − 0.967938i \(-0.580822\pi\)
0.712663 + 0.701506i \(0.247489\pi\)
\(54\) −3.53875 − 6.44028i −0.481563 − 0.876411i
\(55\) −0.154023 − 0.0889253i −0.0207685 − 0.0119907i
\(56\) 0.675065 + 2.72997i 0.0902093 + 0.364808i
\(57\) −0.0120974 − 0.682449i −0.00160234 − 0.0903926i
\(58\) 2.31704 − 0.509755i 0.304242 − 0.0669341i
\(59\) 3.05504 + 7.08237i 0.397732 + 0.922046i 0.992956 + 0.118486i \(0.0378041\pi\)
−0.595224 + 0.803560i \(0.702937\pi\)
\(60\) −0.112266 − 0.00818379i −0.0144934 − 0.00105652i
\(61\) 0.0769563 − 1.32129i 0.00985325 − 0.169174i −0.989792 − 0.142523i \(-0.954479\pi\)
0.999645 − 0.0266511i \(-0.00848432\pi\)
\(62\) 6.58811 + 11.8490i 0.836691 + 1.50483i
\(63\) 2.24078 + 1.96873i 0.282312 + 0.248037i
\(64\) 7.21740 + 3.45096i 0.902175 + 0.431370i
\(65\) −0.00589087 − 0.0503996i −0.000730673 − 0.00625130i
\(66\) 12.8495 + 3.82497i 1.58167 + 0.470821i
\(67\) −2.85205 − 0.853849i −0.348434 − 0.104314i 0.107804 − 0.994172i \(-0.465618\pi\)
−0.456238 + 0.889858i \(0.650803\pi\)
\(68\) −11.1581 + 0.288645i −1.35312 + 0.0350033i
\(69\) −3.76597 + 13.4419i −0.453370 + 1.61821i
\(70\) 0.0439768 − 0.0123952i 0.00525623 − 0.00148151i
\(71\) 4.98620 − 4.18392i 0.591754 − 0.496540i −0.297029 − 0.954868i \(-0.595996\pi\)
0.888783 + 0.458328i \(0.151552\pi\)
\(72\) 8.37485 − 1.36450i 0.986986 − 0.160808i
\(73\) 1.85640 + 1.55770i 0.217275 + 0.182315i 0.744928 − 0.667144i \(-0.232484\pi\)
−0.527653 + 0.849460i \(0.676928\pi\)
\(74\) 9.01711 + 4.06363i 1.04822 + 0.472388i
\(75\) 0.350165 − 8.65134i 0.0404335 − 0.998971i
\(76\) 0.747340 + 0.250320i 0.0857258 + 0.0287137i
\(77\) −5.43268 + 0.316418i −0.619111 + 0.0360591i
\(78\) 1.31389 + 3.59236i 0.148769 + 0.406755i
\(79\) 4.37177 + 4.12455i 0.491862 + 0.464048i 0.891758 − 0.452512i \(-0.149472\pi\)
−0.399896 + 0.916561i \(0.630954\pi\)
\(80\) 0.0522503 − 0.119012i 0.00584176 − 0.0133060i
\(81\) 6.46727 − 6.25895i 0.718586 − 0.695438i
\(82\) −2.56240 + 2.08049i −0.282969 + 0.229752i
\(83\) 7.99493 − 8.47413i 0.877557 − 0.930156i −0.120484 − 0.992715i \(-0.538445\pi\)
0.998041 + 0.0625588i \(0.0199261\pi\)
\(84\) −3.00760 + 1.67841i −0.328156 + 0.183130i
\(85\) 0.0105445 + 0.181042i 0.00114371 + 0.0196367i
\(86\) 6.26965 − 3.98000i 0.676073 − 0.429175i
\(87\) 1.34987 + 2.57306i 0.144721 + 0.275861i
\(88\) −9.12354 + 12.5066i −0.972573 + 1.33321i
\(89\) 4.18508 − 4.98758i 0.443617 − 0.528683i −0.497182 − 0.867646i \(-0.665632\pi\)
0.940800 + 0.338964i \(0.110076\pi\)
\(90\) −0.0265540 − 0.135280i −0.00279903 − 0.0142598i
\(91\) −0.998011 − 1.18938i −0.104620 − 0.124681i
\(92\) −13.1740 − 9.28803i −1.37349 − 0.968344i
\(93\) −11.8737 + 11.6069i −1.23125 + 1.20357i
\(94\) −9.60521 − 7.39498i −0.990702 − 0.762734i
\(95\) 0.00367256 − 0.0122672i 0.000376797 − 0.00125859i
\(96\) −1.74851 + 9.64068i −0.178457 + 0.983948i
\(97\) 16.7230 − 1.95464i 1.69796 − 0.198464i 0.788728 − 0.614743i \(-0.210740\pi\)
0.909237 + 0.416279i \(0.136666\pi\)
\(98\) −5.56914 + 6.42336i −0.562568 + 0.648857i
\(99\) −0.373168 + 16.4156i −0.0375048 + 1.64983i
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.11.3 ✓ 936
3.2 odd 2 972.2.p.a.359.50 936
4.3 odd 2 inner 324.2.p.a.11.24 yes 936
12.11 even 2 972.2.p.a.359.29 936
81.22 even 27 972.2.p.a.287.29 936
81.59 odd 54 inner 324.2.p.a.59.24 yes 936
324.59 even 54 inner 324.2.p.a.59.3 yes 936
324.103 odd 54 972.2.p.a.287.50 936
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
324.2.p.a.11.3 ✓ 936 1.1 even 1 trivial
324.2.p.a.11.24 yes 936 4.3 odd 2 inner
324.2.p.a.59.3 yes 936 324.59 even 54 inner
324.2.p.a.59.24 yes 936 81.59 odd 54 inner
972.2.p.a.287.29 936 81.22 even 27
972.2.p.a.287.50 936 324.103 odd 54
972.2.p.a.359.29 936 12.11 even 2
972.2.p.a.359.50 936 3.2 odd 2