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Properties

Label 324.2.h.f.107.8
Level $324$
Weight $2$
Character 324.107
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 107.8
Root \(1.11871 - 0.645885i\) of defining polynomial
Character \(\chi\) \(=\) 324.107
Dual form 324.2.h.f.215.8

$q$-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)
 
\(f(q)\) \(=\) \(q+(1.35299 - 0.411599i) q^{2} +(1.66117 - 1.11378i) q^{4} +(0.448288 - 0.258819i) q^{5} +(-2.53020 - 1.46081i) q^{7} +(1.78912 - 2.19067i) q^{8} +(0.500000 - 0.534695i) q^{10} +(2.82207 - 4.88798i) q^{11} +(2.23205 + 3.86603i) q^{13} +(-4.02461 - 0.935040i) q^{14} +(1.51899 - 3.70036i) q^{16} +2.31079i q^{17} +5.06040i q^{19} +(0.456416 - 0.929237i) q^{20} +(1.80636 - 7.77495i) q^{22} +(0.756172 + 1.30973i) q^{23} +(-2.36603 + 4.09808i) q^{25} +(4.61120 + 4.31199i) q^{26} +(-5.83013 + 0.391424i) q^{28} +(-4.12252 - 2.38014i) q^{29} +(-6.91264 + 3.99102i) q^{31} +(0.532120 - 5.63177i) q^{32} +(0.951118 + 3.12648i) q^{34} -1.51234 q^{35} +0.267949 q^{37} +(2.08286 + 6.84669i) q^{38} +(0.235055 - 1.44511i) q^{40} +(7.02030 - 4.05317i) q^{41} +(-2.53020 - 1.46081i) q^{43} +(-0.756172 - 11.2629i) q^{44} +(1.56218 + 1.46081i) q^{46} +(-2.06590 + 3.57825i) q^{47} +(0.767949 + 1.33013i) q^{49} +(-1.51445 + 6.51852i) q^{50} +(8.01372 + 3.93613i) q^{52} +4.52004i q^{53} -2.92163i q^{55} +(-7.72700 + 2.92927i) q^{56} +(-6.55740 - 1.52348i) q^{58} +(2.06590 + 3.57825i) q^{59} +(1.13397 - 1.96410i) q^{61} +(-7.71005 + 8.24504i) q^{62} +(-1.59808 - 7.83876i) q^{64} +(2.00120 + 1.15539i) q^{65} +(-2.53020 + 1.46081i) q^{67} +(2.57371 + 3.83862i) q^{68} +(-2.04619 + 0.622479i) q^{70} -9.77595 q^{71} +4.66025 q^{73} +(0.362533 - 0.110288i) q^{74} +(5.63617 + 8.40621i) q^{76} +(-14.2808 + 8.24504i) q^{77} +(9.44284 + 5.45183i) q^{79} +(-0.276778 - 2.05197i) q^{80} +(7.83013 - 8.37345i) q^{82} +(-5.64415 + 9.77595i) q^{83} +(0.598076 + 1.03590i) q^{85} +(-4.02461 - 0.935040i) q^{86} +(-5.65891 - 14.9274i) q^{88} -13.5230i q^{89} -13.0424i q^{91} +(2.71488 + 1.33348i) q^{92} +(-1.32235 + 5.69166i) q^{94} +(1.30973 + 2.26852i) q^{95} +(0.267949 - 0.464102i) q^{97} +(1.58651 + 1.48356i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 8 q^{4} + 8 q^{10} + 8 q^{13} + 4 q^{16} - 12 q^{22} - 24 q^{25} - 24 q^{28} - 4 q^{34} + 32 q^{37} - 16 q^{40} - 72 q^{46} + 40 q^{49} - 16 q^{52} - 16 q^{58} + 32 q^{61} + 16 q^{64} + 36 q^{70}+ \cdots + 32 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.35299 − 0.411599i 0.956710 − 0.291044i
\(3\) 0 0
\(4\) 1.66117 − 1.11378i 0.830587 − 0.556890i
\(5\) 0.448288 − 0.258819i 0.200480 − 0.115747i −0.396399 − 0.918078i \(-0.629740\pi\)
0.596880 + 0.802331i \(0.296407\pi\)
\(6\) 0 0
\(7\) −2.53020 − 1.46081i −0.956327 − 0.552135i −0.0612861 − 0.998120i \(-0.519520\pi\)
−0.895041 + 0.445985i \(0.852854\pi\)
\(8\) 1.78912 − 2.19067i 0.632551 − 0.774519i
\(9\) 0 0
\(10\) 0.500000 − 0.534695i 0.158114 − 0.169085i
\(11\) 2.82207 − 4.88798i 0.850887 − 1.47378i −0.0295213 − 0.999564i \(-0.509398\pi\)
0.880409 − 0.474216i \(-0.157268\pi\)
\(12\) 0 0
\(13\) 2.23205 + 3.86603i 0.619060 + 1.07224i 0.989658 + 0.143449i \(0.0458194\pi\)
−0.370598 + 0.928793i \(0.620847\pi\)
\(14\) −4.02461 − 0.935040i −1.07562 − 0.249900i
\(15\) 0 0
\(16\) 1.51899 − 3.70036i 0.379748 − 0.925090i
\(17\) 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 \(0.802870\pi\)
\(20\) 0.456416 − 0.929237i 0.102058 − 0.207784i
\(21\) 0 0
\(22\) 1.80636 − 7.77495i 0.385117 − 1.65763i
\(23\) 0.756172 + 1.30973i 0.157673 + 0.273097i 0.934029 − 0.357197i \(-0.116268\pi\)
−0.776356 + 0.630294i \(0.782934\pi\)
\(24\) 0 0
\(25\) −2.36603 + 4.09808i −0.473205 + 0.819615i
\(26\) 4.61120 + 4.31199i 0.904330 + 0.845651i
\(27\) 0 0
\(28\) −5.83013 + 0.391424i −1.10179 + 0.0739721i
\(29\) −4.12252 − 2.38014i −0.765533 − 0.441981i 0.0657457 − 0.997836i \(-0.479057\pi\)
−0.831279 + 0.555856i \(0.812391\pi\)
\(30\) 0 0
\(31\) −6.91264 + 3.99102i −1.24155 + 0.716808i −0.969409 − 0.245451i \(-0.921064\pi\)
−0.272138 + 0.962258i \(0.587731\pi\)
\(32\) 0.532120 − 5.63177i 0.0940664 − 0.995566i
\(33\) 0 0
\(34\) 0.951118 + 3.12648i 0.163115 + 0.536187i
\(35\) −1.51234 −0.255633
\(36\) 0 0
\(37\) 0.267949 0.0440506 0.0220253 − 0.999757i \(-0.492989\pi\)
0.0220253 + 0.999757i \(0.492989\pi\)
\(38\) 2.08286 + 6.84669i 0.337884 + 1.11068i
\(39\) 0 0
\(40\) 0.235055 − 1.44511i 0.0371654 − 0.228492i
\(41\) 7.02030 − 4.05317i 1.09639 − 0.632999i 0.161117 − 0.986935i \(-0.448490\pi\)
0.935270 + 0.353936i \(0.115157\pi\)
\(42\) 0 0
\(43\) −2.53020 − 1.46081i −0.385852 − 0.222772i 0.294509 − 0.955649i \(-0.404844\pi\)
−0.680361 + 0.732877i \(0.738177\pi\)
\(44\) −0.756172 − 11.2629i −0.113997 − 1.69795i
\(45\) 0 0
\(46\) 1.56218 + 1.46081i 0.230331 + 0.215385i
\(47\) −2.06590 + 3.57825i −0.301343 + 0.521941i −0.976440 − 0.215788i \(-0.930768\pi\)
0.675098 + 0.737728i \(0.264101\pi\)
\(48\) 0 0
\(49\) 0.767949 + 1.33013i 0.109707 + 0.190018i
\(50\) −1.51445 + 6.51852i −0.214176 + 0.921857i
\(51\) 0 0
\(52\) 8.01372 + 3.93613i 1.11130 + 0.545842i
\(53\) 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
\(56\) −7.72700 + 2.92927i −1.03256 + 0.391440i
\(57\) 0 0
\(58\) −6.55740 − 1.52348i −0.861029 − 0.200043i
\(59\) 2.06590 + 3.57825i 0.268957 + 0.465848i 0.968593 − 0.248652i \(-0.0799876\pi\)
−0.699635 + 0.714500i \(0.746654\pi\)
\(60\) 0 0
\(61\) 1.13397 − 1.96410i 0.145191 − 0.251477i −0.784253 − 0.620441i \(-0.786954\pi\)
0.929444 + 0.368963i \(0.120287\pi\)
\(62\) −7.71005 + 8.24504i −0.979177 + 1.04712i
\(63\) 0 0
\(64\) −1.59808 − 7.83876i −0.199760 − 0.979845i
\(65\) 2.00120 + 1.15539i 0.248219 + 0.143309i
\(66\) 0 0
\(67\) −2.53020 + 1.46081i −0.309113 + 0.178467i −0.646530 − 0.762889i \(-0.723780\pi\)
0.337416 + 0.941356i \(0.390447\pi\)
\(68\) 2.57371 + 3.83862i 0.312108 + 0.465501i
\(69\) 0 0
\(70\) −2.04619 + 0.622479i −0.244566 + 0.0744005i
\(71\) −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 \(0.412076\pi\)
\(74\) 0.362533 − 0.110288i 0.0421436 − 0.0128207i
\(75\) 0 0
\(76\) 5.63617 + 8.40621i 0.646513 + 0.964258i
\(77\) −14.2808 + 8.24504i −1.62745 + 0.939610i
\(78\) 0 0
\(79\) 9.44284 + 5.45183i 1.06240 + 0.613379i 0.926096 − 0.377288i \(-0.123143\pi\)
0.136307 + 0.990667i \(0.456477\pi\)
\(80\) −0.276778 − 2.05197i −0.0309448 − 0.229417i
\(81\) 0 0
\(82\) 7.83013 − 8.37345i 0.864693 − 0.924693i
\(83\) −5.64415 + 9.77595i −0.619526 + 1.07305i 0.370047 + 0.929013i \(0.379342\pi\)
−0.989572 + 0.144037i \(0.953992\pi\)
\(84\) 0 0
\(85\) 0.598076 + 1.03590i 0.0648705 + 0.112359i
\(86\) −4.02461 − 0.935040i −0.433985 − 0.100828i
\(87\) 0 0
\(88\) −5.65891 − 14.9274i −0.603241 − 1.59127i
\(89\) − 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
\(92\) 2.71488 + 1.33348i 0.283046 + 0.139025i
\(93\) 0 0
\(94\) −1.32235 + 5.69166i −0.136390 + 0.587050i
\(95\) 1.30973 + 2.26852i 0.134375 + 0.232745i
\(96\) 0 0
\(97\) 0.267949 − 0.464102i 0.0272061 − 0.0471224i −0.852102 − 0.523376i \(-0.824672\pi\)
0.879308 + 0.476254i \(0.158006\pi\)
\(98\) 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.h.f.107.8 16
3.2 odd 2 inner 324.2.h.f.107.1 16
4.3 odd 2 inner 324.2.h.f.107.5 16
9.2 odd 6 324.2.b.c.323.6 yes 8
9.4 even 3 inner 324.2.h.f.215.4 16
9.5 odd 6 inner 324.2.h.f.215.5 16
9.7 even 3 324.2.b.c.323.3 ✓ 8
12.11 even 2 inner 324.2.h.f.107.4 16
36.7 odd 6 324.2.b.c.323.5 yes 8
36.11 even 6 324.2.b.c.323.4 yes 8
36.23 even 6 inner 324.2.h.f.215.8 16
36.31 odd 6 inner 324.2.h.f.215.1 16
72.11 even 6 5184.2.c.k.5183.5 8
72.29 odd 6 5184.2.c.k.5183.6 8
72.43 odd 6 5184.2.c.k.5183.3 8
72.61 even 6 5184.2.c.k.5183.4 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
324.2.b.c.323.3 ✓ 8 9.7 even 3
324.2.b.c.323.4 yes 8 36.11 even 6
324.2.b.c.323.5 yes 8 36.7 odd 6
324.2.b.c.323.6 yes 8 9.2 odd 6
324.2.h.f.107.1 16 3.2 odd 2 inner
324.2.h.f.107.4 16 12.11 even 2 inner
324.2.h.f.107.5 16 4.3 odd 2 inner
324.2.h.f.107.8 16 1.1 even 1 trivial
324.2.h.f.215.1 16 36.31 odd 6 inner
324.2.h.f.215.4 16 9.4 even 3 inner
324.2.h.f.215.5 16 9.5 odd 6 inner
324.2.h.f.215.8 16 36.23 even 6 inner
5184.2.c.k.5183.3 8 72.43 odd 6
5184.2.c.k.5183.4 8 72.61 even 6
5184.2.c.k.5183.5 8 72.11 even 6
5184.2.c.k.5183.6 8 72.29 odd 6