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Properties

Label 324.2.h.f.215.2
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 215.2
Root \(0.367543 + 0.212201i\) of defining polynomial
Character \(\chi\) \(=\) 324.215
Dual form 324.2.h.f.107.2

$q$-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 0.0768413 − 0.997043i \(-0.475517\pi\)
−0.825044 + 0.565068i \(0.808850\pi\)
\(6\) 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 0.977677 − 0.210113i \(-0.0673831\pi\)
−0.670802 + 0.741637i \(0.734050\pi\)
\(12\) 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 \(-0.804119\pi\)
0.816554 − 0.577269i \(-0.195881\pi\)
\(18\) 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 \(-0.542357\pi\)
0.924707 − 0.380679i \(-0.124310\pi\)
\(24\) 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 −0.674163 − 0.738582i \(-0.735496\pi\)
0.302549 + 0.953134i \(0.402162\pi\)
\(30\) 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 0.700963 − 0.713198i \(-0.252754\pi\)
−0.267166 + 0.963651i \(0.586087\pi\)
\(42\) 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 0.994391 − 0.105765i \(-0.0337291\pi\)
−0.588791 + 0.808286i \(0.700396\pi\)
\(48\) 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 \(0.246350\pi\)
−0.715168 + 0.698952i \(0.753650\pi\)
\(54\) 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 −0.988292 − 0.152577i \(-0.951243\pi\)
0.626281 + 0.779597i \(0.284576\pi\)
\(60\) 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 \(-0.432910\pi\)
0.209214 + 0.977870i \(0.432910\pi\)
\(72\) 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 0.732414 − 0.680859i \(-0.238393\pi\)
−0.955849 + 0.293860i \(0.905060\pi\)
\(84\) 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 \(-0.938489\pi\)
0.981386 − 0.192043i \(-0.0615114\pi\)
\(90\) 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.215.2 16
3.2 odd 2 inner 324.2.h.f.215.7 16
4.3 odd 2 inner 324.2.h.f.215.6 16
9.2 odd 6 inner 324.2.h.f.107.6 16
9.4 even 3 324.2.b.c.323.7 yes 8
9.5 odd 6 324.2.b.c.323.2 yes 8
9.7 even 3 inner 324.2.h.f.107.3 16
12.11 even 2 inner 324.2.h.f.215.3 16
36.7 odd 6 inner 324.2.h.f.107.7 16
36.11 even 6 inner 324.2.h.f.107.2 16
36.23 even 6 324.2.b.c.323.8 yes 8
36.31 odd 6 324.2.b.c.323.1 ✓ 8
72.5 odd 6 5184.2.c.k.5183.8 8
72.13 even 6 5184.2.c.k.5183.2 8
72.59 even 6 5184.2.c.k.5183.7 8
72.67 odd 6 5184.2.c.k.5183.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
324.2.b.c.323.1 ✓ 8 36.31 odd 6
324.2.b.c.323.2 yes 8 9.5 odd 6
324.2.b.c.323.7 yes 8 9.4 even 3
324.2.b.c.323.8 yes 8 36.23 even 6
324.2.h.f.107.2 16 36.11 even 6 inner
324.2.h.f.107.3 16 9.7 even 3 inner
324.2.h.f.107.6 16 9.2 odd 6 inner
324.2.h.f.107.7 16 36.7 odd 6 inner
324.2.h.f.215.2 16 1.1 even 1 trivial
324.2.h.f.215.3 16 12.11 even 2 inner
324.2.h.f.215.6 16 4.3 odd 2 inner
324.2.h.f.215.7 16 3.2 odd 2 inner
5184.2.c.k.5183.1 8 72.67 odd 6
5184.2.c.k.5183.2 8 72.13 even 6
5184.2.c.k.5183.7 8 72.59 even 6
5184.2.c.k.5183.8 8 72.5 odd 6