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Properties

Label 324.2.b.c.323.1
Level $324$
Weight $2$
Character 324.323
Analytic conductor $2.587$
Analytic rank $0$
Dimension $8$
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(323,324)] 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("324.323"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(324, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 324 = 2^{2} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 324.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.58715302549\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.5780865024.3
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{6} + 9x^{4} - 16x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 323.1
Root \(1.39033 + 0.258819i\) of defining polynomial
Character \(\chi\) \(=\) 324.323
Dual form 324.2.b.c.323.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.39033 - 0.258819i) q^{2} +(1.86603 + 0.719687i) q^{4} +1.93185i q^{5} -3.93244i q^{7} +(-2.40812 - 1.48356i) q^{8} +(0.500000 - 2.68591i) q^{10} +2.03558 q^{11} +2.46410 q^{13} +(-1.01779 + 5.46739i) q^{14} +(2.96410 + 2.68591i) q^{16} -4.76028i q^{17} +6.81119i q^{19} +(-1.39033 + 3.60488i) q^{20} +(-2.83013 - 0.526847i) q^{22} +7.59689 q^{23} +1.26795 q^{25} +(-3.42591 - 0.637756i) q^{26} +(2.83013 - 7.33804i) q^{28} -2.31079i q^{29} -2.87875i q^{31} +(-3.42591 - 4.50146i) q^{32} +(-1.23205 + 6.61835i) q^{34} +7.59689 q^{35} +3.73205 q^{37} +(1.76287 - 9.46979i) q^{38} +(2.86603 - 4.65213i) q^{40} -3.20736i q^{41} -3.93244i q^{43} +(3.79845 + 1.46498i) q^{44} +(-10.5622 - 1.96622i) q^{46} +5.56131 q^{47} -8.46410 q^{49} +(-1.76287 - 0.328169i) q^{50} +(4.59808 + 1.77338i) q^{52} +10.1769i q^{53} +3.93244i q^{55} +(-5.83403 + 9.46979i) q^{56} +(-0.598076 + 3.21276i) q^{58} -5.56131 q^{59} -5.73205 q^{61} +(-0.745075 + 4.00240i) q^{62} +(3.59808 + 7.14520i) q^{64} +4.76028i q^{65} +3.93244i q^{67} +(3.42591 - 8.88280i) q^{68} +(-10.5622 - 1.96622i) q^{70} -3.52573 q^{71} -12.6603 q^{73} +(-5.18878 - 0.965926i) q^{74} +(-4.90192 + 12.7099i) q^{76} -8.00481i q^{77} +1.05369i q^{79} +(-5.18878 + 5.72620i) q^{80} +(-0.830127 + 4.45929i) q^{82} -4.07116 q^{83} +9.19615 q^{85} +(-1.01779 + 5.46739i) q^{86} +(-4.90192 - 3.01992i) q^{88} -3.62347i q^{89} -9.68994i q^{91} +(14.1760 + 5.46739i) q^{92} +(-7.73205 - 1.43937i) q^{94} -13.1582 q^{95} -7.46410 q^{97} +(11.7679 + 2.19067i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{4} + 4 q^{10} - 8 q^{13} - 4 q^{16} + 12 q^{22} + 24 q^{25} - 12 q^{28} + 4 q^{34} + 16 q^{37} + 16 q^{40} - 36 q^{46} - 40 q^{49} + 16 q^{52} + 16 q^{58} - 32 q^{61} + 8 q^{64} - 36 q^{70} - 32 q^{73}+ \cdots - 32 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(163\) \(245\)
\(\chi(n)\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.39033 − 0.258819i −0.983111 − 0.183013i
\(3\) 0 0
\(4\) 1.86603 + 0.719687i 0.933013 + 0.359843i
\(5\) 1.93185i 0.863950i 0.901886 + 0.431975i \(0.142183\pi\)
−0.901886 + 0.431975i \(0.857817\pi\)
\(6\) 0 0
\(7\) − 3.93244i − 1.48632i −0.669112 − 0.743162i \(-0.733325\pi\)
0.669112 − 0.743162i \(-0.266675\pi\)
\(8\) −2.40812 − 1.48356i −0.851399 − 0.524519i
\(9\) 0 0
\(10\) 0.500000 − 2.68591i 0.158114 − 0.849359i
\(11\) 2.03558 0.613751 0.306876 − 0.951750i \(-0.400716\pi\)
0.306876 + 0.951750i \(0.400716\pi\)
\(12\) 0 0
\(13\) 2.46410 0.683419 0.341709 − 0.939806i \(-0.388994\pi\)
0.341709 + 0.939806i \(0.388994\pi\)
\(14\) −1.01779 + 5.46739i −0.272016 + 1.46122i
\(15\) 0 0
\(16\) 2.96410 + 2.68591i 0.741025 + 0.671477i
\(17\) − 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 \(0.285442\pi\)
−0.624159 + 0.781297i \(0.714558\pi\)
\(20\) −1.39033 + 3.60488i −0.310887 + 0.806077i
\(21\) 0 0
\(22\) −2.83013 − 0.526847i −0.603385 − 0.112324i
\(23\) 7.59689 1.58406 0.792031 − 0.610481i \(-0.209024\pi\)
0.792031 + 0.610481i \(0.209024\pi\)
\(24\) 0 0
\(25\) 1.26795 0.253590
\(26\) −3.42591 − 0.637756i −0.671876 − 0.125074i
\(27\) 0 0
\(28\) 2.83013 − 7.33804i 0.534844 − 1.38676i
\(29\) − 2.31079i − 0.429103i −0.976713 − 0.214551i \(-0.931171\pi\)
0.976713 − 0.214551i \(-0.0688289\pi\)
\(30\) 0 0
\(31\) − 2.87875i − 0.517038i −0.966006 − 0.258519i \(-0.916765\pi\)
0.966006 − 0.258519i \(-0.0832345\pi\)
\(32\) −3.42591 − 4.50146i −0.605621 − 0.795753i
\(33\) 0 0
\(34\) −1.23205 + 6.61835i −0.211295 + 1.13504i
\(35\) 7.59689 1.28411
\(36\) 0 0
\(37\) 3.73205 0.613545 0.306773 − 0.951783i \(-0.400751\pi\)
0.306773 + 0.951783i \(0.400751\pi\)
\(38\) 1.76287 − 9.46979i 0.285975 − 1.53620i
\(39\) 0 0
\(40\) 2.86603 − 4.65213i 0.453158 − 0.735566i
\(41\) − 3.20736i − 0.500906i −0.968129 − 0.250453i \(-0.919420\pi\)
0.968129 − 0.250453i \(-0.0805796\pi\)
\(42\) 0 0
\(43\) − 3.93244i − 0.599692i −0.953988 − 0.299846i \(-0.903065\pi\)
0.953988 − 0.299846i \(-0.0969353\pi\)
\(44\) 3.79845 + 1.46498i 0.572638 + 0.220854i
\(45\) 0 0
\(46\) −10.5622 − 1.96622i −1.55731 − 0.289903i
\(47\) 5.56131 0.811201 0.405600 − 0.914050i \(-0.367062\pi\)
0.405600 + 0.914050i \(0.367062\pi\)
\(48\) 0 0
\(49\) −8.46410 −1.20916
\(50\) −1.76287 − 0.328169i −0.249307 − 0.0464102i
\(51\) 0 0
\(52\) 4.59808 + 1.77338i 0.637638 + 0.245924i
\(53\) 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
\(56\) −5.83403 + 9.46979i −0.779605 + 1.26545i
\(57\) 0 0
\(58\) −0.598076 + 3.21276i −0.0785313 + 0.421855i
\(59\) −5.56131 −0.724021 −0.362011 − 0.932174i \(-0.617910\pi\)
−0.362011 + 0.932174i \(0.617910\pi\)
\(60\) 0 0
\(61\) −5.73205 −0.733914 −0.366957 − 0.930238i \(-0.619600\pi\)
−0.366957 + 0.930238i \(0.619600\pi\)
\(62\) −0.745075 + 4.00240i −0.0946246 + 0.508306i
\(63\) 0 0
\(64\) 3.59808 + 7.14520i 0.449760 + 0.893150i
\(65\) 4.76028i 0.590440i
\(66\) 0 0
\(67\) 3.93244i 0.480424i 0.970720 + 0.240212i \(0.0772170\pi\)
−0.970720 + 0.240212i \(0.922783\pi\)
\(68\) 3.42591 − 8.88280i 0.415453 − 1.07720i
\(69\) 0 0
\(70\) −10.5622 − 1.96622i −1.26242 − 0.235008i
\(71\) −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 \(0.765594\pi\)
\(74\) −5.18878 − 0.965926i −0.603183 − 0.112287i
\(75\) 0 0
\(76\) −4.90192 + 12.7099i −0.562289 + 1.45792i
\(77\) − 8.00481i − 0.912233i
\(78\) 0 0
\(79\) 1.05369i 0.118550i 0.998242 + 0.0592750i \(0.0188789\pi\)
−0.998242 + 0.0592750i \(0.981121\pi\)
\(80\) −5.18878 + 5.72620i −0.580123 + 0.640209i
\(81\) 0 0
\(82\) −0.830127 + 4.45929i −0.0916722 + 0.492446i
\(83\) −4.07116 −0.446868 −0.223434 − 0.974719i \(-0.571727\pi\)
−0.223434 + 0.974719i \(0.571727\pi\)
\(84\) 0 0
\(85\) 9.19615 0.997463
\(86\) −1.01779 + 5.46739i −0.109751 + 0.589563i
\(87\) 0 0
\(88\) −4.90192 − 3.01992i −0.522547 − 0.321924i
\(89\) − 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
\(92\) 14.1760 + 5.46739i 1.47795 + 0.570014i
\(93\) 0 0
\(94\) −7.73205 − 1.43937i −0.797500 − 0.148460i
\(95\) −13.1582 −1.35000
\(96\) 0 0
\(97\) −7.46410 −0.757865 −0.378932 − 0.925424i \(-0.623709\pi\)
−0.378932 + 0.925424i \(0.623709\pi\)
\(98\) 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.b.c.323.1 ✓ 8
3.2 odd 2 inner 324.2.b.c.323.8 yes 8
4.3 odd 2 inner 324.2.b.c.323.7 yes 8
8.3 odd 2 5184.2.c.k.5183.2 8
8.5 even 2 5184.2.c.k.5183.1 8
9.2 odd 6 324.2.h.f.215.3 16
9.4 even 3 324.2.h.f.107.7 16
9.5 odd 6 324.2.h.f.107.2 16
9.7 even 3 324.2.h.f.215.6 16
12.11 even 2 inner 324.2.b.c.323.2 yes 8
24.5 odd 2 5184.2.c.k.5183.7 8
24.11 even 2 5184.2.c.k.5183.8 8
36.7 odd 6 324.2.h.f.215.2 16
36.11 even 6 324.2.h.f.215.7 16
36.23 even 6 324.2.h.f.107.6 16
36.31 odd 6 324.2.h.f.107.3 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
324.2.b.c.323.1 ✓ 8 1.1 even 1 trivial
324.2.b.c.323.2 yes 8 12.11 even 2 inner
324.2.b.c.323.7 yes 8 4.3 odd 2 inner
324.2.b.c.323.8 yes 8 3.2 odd 2 inner
324.2.h.f.107.2 16 9.5 odd 6
324.2.h.f.107.3 16 36.31 odd 6
324.2.h.f.107.6 16 36.23 even 6
324.2.h.f.107.7 16 9.4 even 3
324.2.h.f.215.2 16 36.7 odd 6
324.2.h.f.215.3 16 9.2 odd 6
324.2.h.f.215.6 16 9.7 even 3
324.2.h.f.215.7 16 36.11 even 6
5184.2.c.k.5183.1 8 8.5 even 2
5184.2.c.k.5183.2 8 8.3 odd 2
5184.2.c.k.5183.7 8 24.5 odd 2
5184.2.c.k.5183.8 8 24.11 even 2