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Properties

Label 324.2.h.f.215.7
Level $324$
Weight $2$
Character 324.215
Analytic conductor $2.587$
Analytic rank $0$
Dimension $16$
Inner twists $8$

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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(107,324)] 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("324.107"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(324, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 324 = 2^{2} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 324.h (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16,0,0,-8,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.58715302549\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: 16.0.33418400425706520576.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 8x^{14} + 49x^{12} - 104x^{10} + 160x^{8} - 104x^{6} + 49x^{4} - 8x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 215.7
Root \(-0.367543 - 0.212201i\) of defining polynomial
Character \(\chi\) \(=\) 324.215
Dual form 324.2.h.f.107.7

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.919308 + 1.07465i) q^{2} +(-0.309746 + 1.97587i) q^{4} +(1.67303 + 0.965926i) q^{5} +(3.40559 - 1.96622i) q^{7} +(-2.40812 + 1.48356i) q^{8} +(0.500000 + 2.68591i) q^{10} +(-1.01779 - 1.76287i) q^{11} +(-1.23205 + 2.13397i) q^{13} +(5.24379 + 1.85226i) q^{14} +(-3.80812 - 1.22403i) q^{16} +4.76028i q^{17} -6.81119i q^{19} +(-2.42676 + 3.00650i) q^{20} +(0.958800 - 2.71439i) q^{22} +(-3.79845 + 6.57910i) q^{23} +(-0.633975 - 1.09808i) q^{25} +(-3.42591 + 0.637756i) q^{26} +(2.83013 + 7.33804i) q^{28} +(2.00120 - 1.15539i) q^{29} +(-2.49307 - 1.43937i) q^{31} +(-2.18542 - 5.21766i) q^{32} +(-5.11563 + 4.37616i) q^{34} +7.59689 q^{35} +3.73205 q^{37} +(7.31965 - 6.26158i) q^{38} +(-5.46187 + 0.155986i) q^{40} +(-2.77766 - 1.60368i) q^{41} +(3.40559 - 1.96622i) q^{43} +(3.79845 - 1.46498i) q^{44} +(-10.5622 + 1.96622i) q^{46} +(-2.78066 - 4.81624i) q^{47} +(4.23205 - 7.33013i) q^{49} +(0.597230 - 1.69077i) q^{50} +(-3.83483 - 3.09536i) q^{52} -10.1769i q^{53} -3.93244i q^{55} +(-5.28406 + 9.78731i) q^{56} +(3.08137 + 1.08843i) q^{58} +(2.78066 - 4.81624i) q^{59} +(2.86603 + 4.96410i) q^{61} +(-0.745075 - 4.00240i) q^{62} +(3.59808 - 7.14520i) q^{64} +(-4.12252 + 2.38014i) q^{65} +(3.40559 + 1.96622i) q^{67} +(-9.40569 - 1.47448i) q^{68} +(6.98389 + 8.16400i) q^{70} -3.52573 q^{71} -12.6603 q^{73} +(3.43090 + 4.01065i) q^{74} +(13.4580 + 2.10974i) q^{76} +(-6.93237 - 4.00240i) q^{77} +(-0.912526 + 0.526847i) q^{79} +(-5.18878 - 5.72620i) q^{80} +(-0.830127 - 4.45929i) q^{82} +(2.03558 + 3.52573i) q^{83} +(-4.59808 + 7.96410i) q^{85} +(5.24379 + 1.85226i) q^{86} +(5.06629 + 2.73523i) q^{88} +3.62347i q^{89} +9.68994i q^{91} +(-11.8229 - 9.54308i) q^{92} +(2.61949 - 7.41584i) q^{94} +(6.57910 - 11.3953i) q^{95} +(3.73205 + 6.46410i) q^{97} +(11.7679 - 2.19067i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 8 q^{4} + 8 q^{10} + 8 q^{13} + 4 q^{16} - 12 q^{22} - 24 q^{25} - 24 q^{28} - 4 q^{34} + 32 q^{37} - 16 q^{40} - 72 q^{46} + 40 q^{49} - 16 q^{52} - 16 q^{58} + 32 q^{61} + 16 q^{64} + 36 q^{70}+ \cdots + 32 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(163\) \(245\)
\(\chi(n)\) \(-1\) \(e\left(\frac{5}{6}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.919308 + 1.07465i 0.650049 + 0.759892i
\(3\) 0 0
\(4\) −0.309746 + 1.97587i −0.154873 + 0.987934i
\(5\) 1.67303 + 0.965926i 0.748203 + 0.431975i 0.825044 − 0.565068i \(-0.191150\pi\)
−0.0768413 + 0.997043i \(0.524483\pi\)
\(6\) 0 0
\(7\) 3.40559 − 1.96622i 1.28719 − 0.743162i 0.309041 − 0.951049i \(-0.399992\pi\)
0.978153 + 0.207887i \(0.0666586\pi\)
\(8\) −2.40812 + 1.48356i −0.851399 + 0.524519i
\(9\) 0 0
\(10\) 0.500000 + 2.68591i 0.158114 + 0.849359i
\(11\) −1.01779 − 1.76287i −0.306876 − 0.531524i 0.670802 − 0.741637i \(-0.265950\pi\)
−0.977677 + 0.210113i \(0.932617\pi\)
\(12\) 0 0
\(13\) −1.23205 + 2.13397i −0.341709 + 0.591858i −0.984750 − 0.173974i \(-0.944339\pi\)
0.643041 + 0.765832i \(0.277673\pi\)
\(14\) 5.24379 + 1.85226i 1.40146 + 0.495037i
\(15\) 0 0
\(16\) −3.80812 − 1.22403i −0.952029 − 0.306008i
\(17\) 4.76028i 1.15454i 0.816554 + 0.577269i \(0.195881\pi\)
−0.816554 + 0.577269i \(0.804119\pi\)
\(18\) 0 0
\(19\) − 6.81119i − 1.56259i −0.624159 − 0.781297i \(-0.714558\pi\)
0.624159 − 0.781297i \(-0.285442\pi\)
\(20\) −2.42676 + 3.00650i −0.542639 + 0.672274i
\(21\) 0 0
\(22\) 0.958800 − 2.71439i 0.204417 − 0.578709i
\(23\) −3.79845 + 6.57910i −0.792031 + 1.37184i 0.132676 + 0.991159i \(0.457643\pi\)
−0.924707 + 0.380679i \(0.875690\pi\)
\(24\) 0 0
\(25\) −0.633975 − 1.09808i −0.126795 − 0.219615i
\(26\) −3.42591 + 0.637756i −0.671876 + 0.125074i
\(27\) 0 0
\(28\) 2.83013 + 7.33804i 0.534844 + 1.38676i
\(29\) 2.00120 − 1.15539i 0.371614 − 0.214551i −0.302549 − 0.953134i \(-0.597838\pi\)
0.674163 + 0.738582i \(0.264504\pi\)
\(30\) 0 0
\(31\) −2.49307 − 1.43937i −0.447768 − 0.258519i 0.259119 − 0.965845i \(-0.416568\pi\)
−0.706887 + 0.707326i \(0.749901\pi\)
\(32\) −2.18542 − 5.21766i −0.386332 − 0.922360i
\(33\) 0 0
\(34\) −5.11563 + 4.37616i −0.877324 + 0.750506i
\(35\) 7.59689 1.28411
\(36\) 0 0
\(37\) 3.73205 0.613545 0.306773 − 0.951783i \(-0.400751\pi\)
0.306773 + 0.951783i \(0.400751\pi\)
\(38\) 7.31965 − 6.26158i 1.18740 − 1.01576i
\(39\) 0 0
\(40\) −5.46187 + 0.155986i −0.863598 + 0.0246636i
\(41\) −2.77766 − 1.60368i −0.433797 − 0.250453i 0.267166 − 0.963651i \(-0.413913\pi\)
−0.700963 + 0.713198i \(0.747246\pi\)
\(42\) 0 0
\(43\) 3.40559 − 1.96622i 0.519348 − 0.299846i −0.217320 − 0.976100i \(-0.569731\pi\)
0.736668 + 0.676255i \(0.236398\pi\)
\(44\) 3.79845 − 1.46498i 0.572638 − 0.220854i
\(45\) 0 0
\(46\) −10.5622 + 1.96622i −1.55731 + 0.289903i
\(47\) −2.78066 − 4.81624i −0.405600 − 0.702521i 0.588791 − 0.808286i \(-0.299604\pi\)
−0.994391 + 0.105765i \(0.966271\pi\)
\(48\) 0 0
\(49\) 4.23205 − 7.33013i 0.604579 − 1.04716i
\(50\) 0.597230 − 1.69077i 0.0844610 − 0.239111i
\(51\) 0 0
\(52\) −3.83483 − 3.09536i −0.531795 − 0.429249i
\(53\) − 10.1769i − 1.39790i −0.715168 − 0.698952i \(-0.753650\pi\)
0.715168 − 0.698952i \(-0.246350\pi\)
\(54\) 0 0
\(55\) − 3.93244i − 0.530250i
\(56\) −5.28406 + 9.78731i −0.706113 + 1.30788i
\(57\) 0 0
\(58\) 3.08137 + 1.08843i 0.404603 + 0.142918i
\(59\) 2.78066 − 4.81624i 0.362011 − 0.627021i −0.626281 − 0.779597i \(-0.715424\pi\)
0.988292 + 0.152577i \(0.0487571\pi\)
\(60\) 0 0
\(61\) 2.86603 + 4.96410i 0.366957 + 0.635588i 0.989088 − 0.147325i \(-0.0470663\pi\)
−0.622131 + 0.782913i \(0.713733\pi\)
\(62\) −0.745075 − 4.00240i −0.0946246 − 0.508306i
\(63\) 0 0
\(64\) 3.59808 − 7.14520i 0.449760 − 0.893150i
\(65\) −4.12252 + 2.38014i −0.511336 + 0.295220i
\(66\) 0 0
\(67\) 3.40559 + 1.96622i 0.416060 + 0.240212i 0.693390 − 0.720562i \(-0.256116\pi\)
−0.277330 + 0.960775i \(0.589450\pi\)
\(68\) −9.40569 − 1.47448i −1.14061 − 0.178806i
\(69\) 0 0
\(70\) 6.98389 + 8.16400i 0.834734 + 0.975785i
\(71\) −3.52573 −0.418427 −0.209214 − 0.977870i \(-0.567090\pi\)
−0.209214 + 0.977870i \(0.567090\pi\)
\(72\) 0 0
\(73\) −12.6603 −1.48177 −0.740885 − 0.671632i \(-0.765594\pi\)
−0.740885 + 0.671632i \(0.765594\pi\)
\(74\) 3.43090 + 4.01065i 0.398835 + 0.466228i
\(75\) 0 0
\(76\) 13.4580 + 2.10974i 1.54374 + 0.242003i
\(77\) −6.93237 − 4.00240i −0.790017 − 0.456116i
\(78\) 0 0
\(79\) −0.912526 + 0.526847i −0.102667 + 0.0592750i −0.550454 − 0.834865i \(-0.685546\pi\)
0.447787 + 0.894140i \(0.352212\pi\)
\(80\) −5.18878 − 5.72620i −0.580123 − 0.640209i
\(81\) 0 0
\(82\) −0.830127 − 4.45929i −0.0916722 − 0.492446i
\(83\) 2.03558 + 3.52573i 0.223434 + 0.386999i 0.955849 − 0.293860i \(-0.0949399\pi\)
−0.732414 + 0.680859i \(0.761607\pi\)
\(84\) 0 0
\(85\) −4.59808 + 7.96410i −0.498731 + 0.863828i
\(86\) 5.24379 + 1.85226i 0.565452 + 0.199734i
\(87\) 0 0
\(88\) 5.06629 + 2.73523i 0.540068 + 0.291577i
\(89\) 3.62347i 0.384087i 0.981386 + 0.192043i \(0.0615114\pi\)
−0.981386 + 0.192043i \(0.938489\pi\)
\(90\) 0 0
\(91\) 9.68994i 1.01578i
\(92\) −11.8229 − 9.54308i −1.23262 − 0.994935i
\(93\) 0 0
\(94\) 2.61949 − 7.41584i 0.270180 − 0.764885i
\(95\) 6.57910 − 11.3953i 0.675002 − 1.16914i
\(96\) 0 0
\(97\) 3.73205 + 6.46410i 0.378932 + 0.656330i 0.990907 − 0.134547i \(-0.0429580\pi\)
−0.611975 + 0.790877i \(0.709625\pi\)
\(98\) 11.7679 − 2.19067i 1.18874 − 0.221291i
\(99\) 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 324.2.h.f.215.7 16
3.2 odd 2 inner 324.2.h.f.215.2 16
4.3 odd 2 inner 324.2.h.f.215.3 16
9.2 odd 6 inner 324.2.h.f.107.3 16
9.4 even 3 324.2.b.c.323.2 yes 8
9.5 odd 6 324.2.b.c.323.7 yes 8
9.7 even 3 inner 324.2.h.f.107.6 16
12.11 even 2 inner 324.2.h.f.215.6 16
36.7 odd 6 inner 324.2.h.f.107.2 16
36.11 even 6 inner 324.2.h.f.107.7 16
36.23 even 6 324.2.b.c.323.1 ✓ 8
36.31 odd 6 324.2.b.c.323.8 yes 8
72.5 odd 6 5184.2.c.k.5183.2 8
72.13 even 6 5184.2.c.k.5183.8 8
72.59 even 6 5184.2.c.k.5183.1 8
72.67 odd 6 5184.2.c.k.5183.7 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
324.2.b.c.323.1 ✓ 8 36.23 even 6
324.2.b.c.323.2 yes 8 9.4 even 3
324.2.b.c.323.7 yes 8 9.5 odd 6
324.2.b.c.323.8 yes 8 36.31 odd 6
324.2.h.f.107.2 16 36.7 odd 6 inner
324.2.h.f.107.3 16 9.2 odd 6 inner
324.2.h.f.107.6 16 9.7 even 3 inner
324.2.h.f.107.7 16 36.11 even 6 inner
324.2.h.f.215.2 16 3.2 odd 2 inner
324.2.h.f.215.3 16 4.3 odd 2 inner
324.2.h.f.215.6 16 12.11 even 2 inner
324.2.h.f.215.7 16 1.1 even 1 trivial
5184.2.c.k.5183.1 8 72.59 even 6
5184.2.c.k.5183.2 8 72.5 odd 6
5184.2.c.k.5183.7 8 72.67 odd 6
5184.2.c.k.5183.8 8 72.13 even 6