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Properties

Label 324.2.b.c.323.3
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 323.3
Root \(1.03295 + 0.965926i\) of defining polynomial
Character \(\chi\) \(=\) 324.323
Dual form 324.2.b.c.323.4

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.03295 - 0.965926i) q^{2} +(0.133975 + 1.99551i) q^{4} +0.517638i q^{5} +2.92163i q^{7} +(1.78912 - 2.19067i) q^{8} +(0.500000 - 0.534695i) q^{10} -5.64415 q^{11} -4.46410 q^{13} +(2.82207 - 3.01790i) q^{14} +(-3.96410 + 0.534695i) q^{16} +2.31079i q^{17} +5.06040i q^{19} +(-1.03295 + 0.0693504i) q^{20} +(5.83013 + 5.45183i) q^{22} -1.51234 q^{23} +4.73205 q^{25} +(4.61120 + 4.31199i) q^{26} +(-5.83013 + 0.391424i) q^{28} +4.76028i q^{29} -7.98203i q^{31} +(4.61120 + 3.27671i) q^{32} +(2.23205 - 2.38693i) q^{34} -1.51234 q^{35} +0.267949 q^{37} +(4.88798 - 5.22715i) q^{38} +(1.13397 + 0.926118i) q^{40} +8.10634i q^{41} +2.92163i q^{43} +(-0.756172 - 11.2629i) q^{44} +(1.56218 + 1.46081i) q^{46} +4.13180 q^{47} -1.53590 q^{49} +(-4.88798 - 4.57081i) q^{50} +(-0.598076 - 8.90815i) q^{52} +4.52004i q^{53} -2.92163i q^{55} +(6.40032 + 5.22715i) q^{56} +(4.59808 - 4.91713i) q^{58} -4.13180 q^{59} -2.26795 q^{61} +(-7.71005 + 8.24504i) q^{62} +(-1.59808 - 7.83876i) q^{64} -2.31079i q^{65} -2.92163i q^{67} +(-4.61120 + 0.309587i) q^{68} +(1.56218 + 1.46081i) q^{70} -9.77595 q^{71} +4.66025 q^{73} +(-0.276778 - 0.258819i) q^{74} +(-10.0981 + 0.677966i) q^{76} -16.4901i q^{77} -10.9037i q^{79} +(-0.276778 - 2.05197i) q^{80} +(7.83013 - 8.37345i) q^{82} +11.2883 q^{83} -1.19615 q^{85} +(2.82207 - 3.01790i) q^{86} +(-10.0981 + 12.3645i) q^{88} -13.5230i q^{89} -13.0424i q^{91} +(-0.202616 - 3.01790i) q^{92} +(-4.26795 - 3.99102i) q^{94} -2.61946 q^{95} -0.535898 q^{97} +(1.58651 + 1.48356i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{4} + 4 q^{10} - 8 q^{13} - 4 q^{16} + 12 q^{22} + 24 q^{25} - 12 q^{28} + 4 q^{34} + 16 q^{37} + 16 q^{40} - 36 q^{46} - 40 q^{49} + 16 q^{52} + 16 q^{58} - 32 q^{61} + 8 q^{64} - 36 q^{70} - 32 q^{73}+ \cdots - 32 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.03295 − 0.965926i −0.730406 − 0.683013i
\(3\) 0 0
\(4\) 0.133975 + 1.99551i 0.0669873 + 0.997754i
\(5\) 0.517638i 0.231495i 0.993279 + 0.115747i \(0.0369263\pi\)
−0.993279 + 0.115747i \(0.963074\pi\)
\(6\) 0 0
\(7\) 2.92163i 1.10427i 0.833754 + 0.552135i \(0.186187\pi\)
−0.833754 + 0.552135i \(0.813813\pi\)
\(8\) 1.78912 − 2.19067i 0.632551 − 0.774519i
\(9\) 0 0
\(10\) 0.500000 − 0.534695i 0.158114 − 0.169085i
\(11\) −5.64415 −1.70177 −0.850887 − 0.525348i \(-0.823935\pi\)
−0.850887 + 0.525348i \(0.823935\pi\)
\(12\) 0 0
\(13\) −4.46410 −1.23812 −0.619060 − 0.785344i \(-0.712486\pi\)
−0.619060 + 0.785344i \(0.712486\pi\)
\(14\) 2.82207 − 3.01790i 0.754231 − 0.806567i
\(15\) 0 0
\(16\) −3.96410 + 0.534695i −0.991025 + 0.133674i
\(17\) 2.31079i 0.560449i 0.959935 + 0.280224i \(0.0904089\pi\)
−0.959935 + 0.280224i \(0.909591\pi\)
\(18\) 0 0
\(19\) 5.06040i 1.16094i 0.814283 + 0.580468i \(0.197130\pi\)
−0.814283 + 0.580468i \(0.802870\pi\)
\(20\) −1.03295 + 0.0693504i −0.230975 + 0.0155072i
\(21\) 0 0
\(22\) 5.83013 + 5.45183i 1.24299 + 1.16233i
\(23\) −1.51234 −0.315346 −0.157673 − 0.987491i \(-0.550399\pi\)
−0.157673 + 0.987491i \(0.550399\pi\)
\(24\) 0 0
\(25\) 4.73205 0.946410
\(26\) 4.61120 + 4.31199i 0.904330 + 0.845651i
\(27\) 0 0
\(28\) −5.83013 + 0.391424i −1.10179 + 0.0739721i
\(29\) 4.76028i 0.883962i 0.897025 + 0.441981i \(0.145724\pi\)
−0.897025 + 0.441981i \(0.854276\pi\)
\(30\) 0 0
\(31\) − 7.98203i − 1.43362i −0.697271 − 0.716808i \(-0.745603\pi\)
0.697271 − 0.716808i \(-0.254397\pi\)
\(32\) 4.61120 + 3.27671i 0.815152 + 0.579247i
\(33\) 0 0
\(34\) 2.23205 − 2.38693i 0.382794 − 0.409355i
\(35\) −1.51234 −0.255633
\(36\) 0 0
\(37\) 0.267949 0.0440506 0.0220253 − 0.999757i \(-0.492989\pi\)
0.0220253 + 0.999757i \(0.492989\pi\)
\(38\) 4.88798 − 5.22715i 0.792934 − 0.847956i
\(39\) 0 0
\(40\) 1.13397 + 0.926118i 0.179297 + 0.146432i
\(41\) 8.10634i 1.26600i 0.774153 + 0.632999i \(0.218176\pi\)
−0.774153 + 0.632999i \(0.781824\pi\)
\(42\) 0 0
\(43\) 2.92163i 0.445544i 0.974871 + 0.222772i \(0.0715105\pi\)
−0.974871 + 0.222772i \(0.928489\pi\)
\(44\) −0.756172 − 11.2629i −0.113997 − 1.69795i
\(45\) 0 0
\(46\) 1.56218 + 1.46081i 0.230331 + 0.215385i
\(47\) 4.13180 0.602685 0.301343 − 0.953516i \(-0.402565\pi\)
0.301343 + 0.953516i \(0.402565\pi\)
\(48\) 0 0
\(49\) −1.53590 −0.219414
\(50\) −4.88798 − 4.57081i −0.691264 − 0.646410i
\(51\) 0 0
\(52\) −0.598076 − 8.90815i −0.0829382 − 1.23534i
\(53\) 4.52004i 0.620876i 0.950594 + 0.310438i \(0.100476\pi\)
−0.950594 + 0.310438i \(0.899524\pi\)
\(54\) 0 0
\(55\) − 2.92163i − 0.393952i
\(56\) 6.40032 + 5.22715i 0.855279 + 0.698507i
\(57\) 0 0
\(58\) 4.59808 − 4.91713i 0.603757 − 0.645651i
\(59\) −4.13180 −0.537915 −0.268957 − 0.963152i \(-0.586679\pi\)
−0.268957 + 0.963152i \(0.586679\pi\)
\(60\) 0 0
\(61\) −2.26795 −0.290381 −0.145191 − 0.989404i \(-0.546380\pi\)
−0.145191 + 0.989404i \(0.546380\pi\)
\(62\) −7.71005 + 8.24504i −0.979177 + 1.04712i
\(63\) 0 0
\(64\) −1.59808 − 7.83876i −0.199760 − 0.979845i
\(65\) − 2.31079i − 0.286618i
\(66\) 0 0
\(67\) − 2.92163i − 0.356933i −0.983946 − 0.178467i \(-0.942886\pi\)
0.983946 − 0.178467i \(-0.0571137\pi\)
\(68\) −4.61120 + 0.309587i −0.559190 + 0.0375429i
\(69\) 0 0
\(70\) 1.56218 + 1.46081i 0.186716 + 0.174601i
\(71\) −9.77595 −1.16019 −0.580096 − 0.814548i \(-0.696985\pi\)
−0.580096 + 0.814548i \(0.696985\pi\)
\(72\) 0 0
\(73\) 4.66025 0.545441 0.272721 − 0.962093i \(-0.412076\pi\)
0.272721 + 0.962093i \(0.412076\pi\)
\(74\) −0.276778 − 0.258819i −0.0321748 − 0.0300871i
\(75\) 0 0
\(76\) −10.0981 + 0.677966i −1.15833 + 0.0777680i
\(77\) − 16.4901i − 1.87922i
\(78\) 0 0
\(79\) − 10.9037i − 1.22676i −0.789789 − 0.613379i \(-0.789810\pi\)
0.789789 − 0.613379i \(-0.210190\pi\)
\(80\) −0.276778 − 2.05197i −0.0309448 − 0.229417i
\(81\) 0 0
\(82\) 7.83013 − 8.37345i 0.864693 − 0.924693i
\(83\) 11.2883 1.23905 0.619526 − 0.784976i \(-0.287325\pi\)
0.619526 + 0.784976i \(0.287325\pi\)
\(84\) 0 0
\(85\) −1.19615 −0.129741
\(86\) 2.82207 − 3.01790i 0.304312 − 0.325428i
\(87\) 0 0
\(88\) −10.0981 + 12.3645i −1.07646 + 1.31806i
\(89\) − 13.5230i − 1.43343i −0.697366 − 0.716716i \(-0.745645\pi\)
0.697366 − 0.716716i \(-0.254355\pi\)
\(90\) 0 0
\(91\) − 13.0424i − 1.36722i
\(92\) −0.202616 − 3.01790i −0.0211242 − 0.314637i
\(93\) 0 0
\(94\) −4.26795 − 3.99102i −0.440205 − 0.411642i
\(95\) −2.61946 −0.268751
\(96\) 0 0
\(97\) −0.535898 −0.0544122 −0.0272061 − 0.999630i \(-0.508661\pi\)
−0.0272061 + 0.999630i \(0.508661\pi\)
\(98\) 1.58651 + 1.48356i 0.160261 + 0.149863i
\(99\) 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.323.3 ✓ 8
3.2 odd 2 inner 324.2.b.c.323.6 yes 8
4.3 odd 2 inner 324.2.b.c.323.5 yes 8
8.3 odd 2 5184.2.c.k.5183.3 8
8.5 even 2 5184.2.c.k.5183.4 8
9.2 odd 6 324.2.h.f.215.5 16
9.4 even 3 324.2.h.f.107.8 16
9.5 odd 6 324.2.h.f.107.1 16
9.7 even 3 324.2.h.f.215.4 16
12.11 even 2 inner 324.2.b.c.323.4 yes 8
24.5 odd 2 5184.2.c.k.5183.6 8
24.11 even 2 5184.2.c.k.5183.5 8
36.7 odd 6 324.2.h.f.215.1 16
36.11 even 6 324.2.h.f.215.8 16
36.23 even 6 324.2.h.f.107.4 16
36.31 odd 6 324.2.h.f.107.5 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
324.2.b.c.323.3 ✓ 8 1.1 even 1 trivial
324.2.b.c.323.4 yes 8 12.11 even 2 inner
324.2.b.c.323.5 yes 8 4.3 odd 2 inner
324.2.b.c.323.6 yes 8 3.2 odd 2 inner
324.2.h.f.107.1 16 9.5 odd 6
324.2.h.f.107.4 16 36.23 even 6
324.2.h.f.107.5 16 36.31 odd 6
324.2.h.f.107.8 16 9.4 even 3
324.2.h.f.215.1 16 36.7 odd 6
324.2.h.f.215.4 16 9.7 even 3
324.2.h.f.215.5 16 9.2 odd 6
324.2.h.f.215.8 16 36.11 even 6
5184.2.c.k.5183.3 8 8.3 odd 2
5184.2.c.k.5183.4 8 8.5 even 2
5184.2.c.k.5183.5 8 24.11 even 2
5184.2.c.k.5183.6 8 24.5 odd 2