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Properties

Label 1728.3.o.g.1279.6
Level $1728$
Weight $3$
Character 1728.1279
Analytic conductor $47.085$
Analytic rank $0$
Dimension $16$
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] = [1728,3,Mod(127,1728)] 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("1728.127"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1728, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 0, 4])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 1728 = 2^{6} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1728.o (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16,0,0,0,6,0,0] 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: \(47.0845896815\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 3 x^{15} + 7 x^{14} - 30 x^{13} + 76 x^{12} - 144 x^{11} + 424 x^{10} - 912 x^{9} + \cdots + 65536 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{16}\cdot 3^{4} \)
Twist minimal: no (minimal twist has level 36)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 1279.6
Root \(-0.710719 + 1.86946i\) of defining polynomial
Character \(\chi\) \(=\) 1728.1279
Dual form 1728.3.o.g.127.6

$q$-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)
 
\(f(q)\) \(=\) \(q+(1.35609 - 2.34881i) q^{5} +(10.0431 - 5.79837i) q^{7} +(-8.54822 + 4.93532i) q^{11} +(-0.296185 + 0.513008i) q^{13} +8.87968 q^{17} +14.0989i q^{19} +(18.2754 + 10.5513i) q^{23} +(8.82205 + 15.2802i) q^{25} +(10.1764 + 17.6260i) q^{29} +(14.3357 + 8.27670i) q^{31} -31.4524i q^{35} +40.6557 q^{37} +(-21.2177 + 36.7502i) q^{41} +(32.2385 - 18.6129i) q^{43} +(-1.57134 + 0.907211i) q^{47} +(42.7423 - 74.0318i) q^{49} -21.1005 q^{53} +26.7709i q^{55} +(76.6879 + 44.2758i) q^{59} +(-36.4925 - 63.2069i) q^{61} +(0.803307 + 1.39137i) q^{65} +(38.3110 + 22.1189i) q^{67} -111.798i q^{71} -76.2003 q^{73} +(-57.2337 + 99.1316i) q^{77} +(-8.30434 + 4.79451i) q^{79} +(73.6244 - 42.5070i) q^{83} +(12.0416 - 20.8567i) q^{85} -64.7845 q^{89} +6.86958i q^{91} +(33.1157 + 19.1193i) q^{95} +(-3.59139 - 6.22047i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 6 q^{5} + 46 q^{13} - 12 q^{17} - 30 q^{25} + 42 q^{29} - 56 q^{37} - 84 q^{41} + 58 q^{49} - 72 q^{53} + 34 q^{61} + 30 q^{65} + 116 q^{73} - 330 q^{77} + 140 q^{85} + 384 q^{89} - 148 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1728\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(703\) \(1217\)
\(\chi(n)\) \(1\) \(-1\) \(e\left(\frac{1}{3}\right)\)

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 0
\(3\) 0 0
\(4\) 0 0
\(5\) 1.35609 − 2.34881i 0.271218 − 0.469763i −0.697956 − 0.716140i \(-0.745907\pi\)
0.969174 + 0.246378i \(0.0792403\pi\)
\(6\) 0 0
\(7\) 10.0431 − 5.79837i 1.43473 − 0.828339i 0.437249 − 0.899340i \(-0.355953\pi\)
0.997476 + 0.0710013i \(0.0226195\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −8.54822 + 4.93532i −0.777111 + 0.448665i −0.835406 − 0.549634i \(-0.814767\pi\)
0.0582943 + 0.998299i \(0.481434\pi\)
\(12\) 0 0
\(13\) −0.296185 + 0.513008i −0.0227835 + 0.0394622i −0.877192 − 0.480139i \(-0.840586\pi\)
0.854409 + 0.519601i \(0.173920\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 8.87968 0.522334 0.261167 − 0.965294i \(-0.415893\pi\)
0.261167 + 0.965294i \(0.415893\pi\)
\(18\) 0 0
\(19\) 14.0989i 0.742046i 0.928624 + 0.371023i \(0.120993\pi\)
−0.928624 + 0.371023i \(0.879007\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 18.2754 + 10.5513i 0.794583 + 0.458753i 0.841574 − 0.540142i \(-0.181630\pi\)
−0.0469902 + 0.998895i \(0.514963\pi\)
\(24\) 0 0
\(25\) 8.82205 + 15.2802i 0.352882 + 0.611209i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 10.1764 + 17.6260i 0.350910 + 0.607793i 0.986409 − 0.164308i \(-0.0525391\pi\)
−0.635499 + 0.772101i \(0.719206\pi\)
\(30\) 0 0
\(31\) 14.3357 + 8.27670i 0.462441 + 0.266990i 0.713070 − 0.701093i \(-0.247304\pi\)
−0.250629 + 0.968083i \(0.580638\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) − 31.4524i − 0.898641i
\(36\) 0 0
\(37\) 40.6557 1.09880 0.549401 − 0.835559i \(-0.314856\pi\)
0.549401 + 0.835559i \(0.314856\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −21.2177 + 36.7502i −0.517506 + 0.896346i 0.482288 + 0.876013i \(0.339806\pi\)
−0.999793 + 0.0203330i \(0.993527\pi\)
\(42\) 0 0
\(43\) 32.2385 − 18.6129i 0.749732 − 0.432858i −0.0758649 − 0.997118i \(-0.524172\pi\)
0.825597 + 0.564260i \(0.190838\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −1.57134 + 0.907211i −0.0334327 + 0.0193024i −0.516623 − 0.856213i \(-0.672811\pi\)
0.483191 + 0.875515i \(0.339478\pi\)
\(48\) 0 0
\(49\) 42.7423 − 74.0318i 0.872291 − 1.51085i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −21.1005 −0.398122 −0.199061 − 0.979987i \(-0.563789\pi\)
−0.199061 + 0.979987i \(0.563789\pi\)
\(54\) 0 0
\(55\) 26.7709i 0.486744i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 76.6879 + 44.2758i 1.29980 + 0.750437i 0.980369 − 0.197174i \(-0.0631764\pi\)
0.319427 + 0.947611i \(0.396510\pi\)
\(60\) 0 0
\(61\) −36.4925 − 63.2069i −0.598238 − 1.03618i −0.993081 − 0.117431i \(-0.962534\pi\)
0.394843 − 0.918749i \(-0.370799\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0.803307 + 1.39137i 0.0123586 + 0.0214057i
\(66\) 0 0
\(67\) 38.3110 + 22.1189i 0.571807 + 0.330133i 0.757871 − 0.652405i \(-0.226240\pi\)
−0.186064 + 0.982538i \(0.559573\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) − 111.798i − 1.57462i −0.616557 − 0.787310i \(-0.711473\pi\)
0.616557 − 0.787310i \(-0.288527\pi\)
\(72\) 0 0
\(73\) −76.2003 −1.04384 −0.521920 − 0.852995i \(-0.674784\pi\)
−0.521920 + 0.852995i \(0.674784\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −57.2337 + 99.1316i −0.743294 + 1.28742i
\(78\) 0 0
\(79\) −8.30434 + 4.79451i −0.105118 + 0.0606901i −0.551637 − 0.834084i \(-0.685997\pi\)
0.446519 + 0.894774i \(0.352663\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 73.6244 − 42.5070i 0.887041 − 0.512133i 0.0140672 − 0.999901i \(-0.495522\pi\)
0.872973 + 0.487768i \(0.162189\pi\)
\(84\) 0 0
\(85\) 12.0416 − 20.8567i 0.141666 − 0.245373i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −64.7845 −0.727916 −0.363958 − 0.931415i \(-0.618575\pi\)
−0.363958 + 0.931415i \(0.618575\pi\)
\(90\) 0 0
\(91\) 6.86958i 0.0754898i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 33.1157 + 19.1193i 0.348586 + 0.201256i
\(96\) 0 0
\(97\) −3.59139 − 6.22047i −0.0370246 − 0.0641285i 0.846919 − 0.531721i \(-0.178455\pi\)
−0.883944 + 0.467593i \(0.845121\pi\)
\(98\) 0 0
\(99\) 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 1728.3.o.g.1279.6 16
3.2 odd 2 576.3.o.g.319.6 16
4.3 odd 2 inner 1728.3.o.g.1279.5 16
8.3 odd 2 108.3.f.c.91.3 16
8.5 even 2 108.3.f.c.91.2 16
9.2 odd 6 576.3.o.g.511.3 16
9.7 even 3 inner 1728.3.o.g.127.5 16
12.11 even 2 576.3.o.g.319.3 16
24.5 odd 2 36.3.f.c.31.7 yes 16
24.11 even 2 36.3.f.c.31.6 yes 16
36.7 odd 6 inner 1728.3.o.g.127.6 16
36.11 even 6 576.3.o.g.511.6 16
72.5 odd 6 324.3.d.i.163.2 8
72.11 even 6 36.3.f.c.7.7 yes 16
72.13 even 6 324.3.d.g.163.7 8
72.29 odd 6 36.3.f.c.7.6 ✓ 16
72.43 odd 6 108.3.f.c.19.2 16
72.59 even 6 324.3.d.i.163.1 8
72.61 even 6 108.3.f.c.19.3 16
72.67 odd 6 324.3.d.g.163.8 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
36.3.f.c.7.6 ✓ 16 72.29 odd 6
36.3.f.c.7.7 yes 16 72.11 even 6
36.3.f.c.31.6 yes 16 24.11 even 2
36.3.f.c.31.7 yes 16 24.5 odd 2
108.3.f.c.19.2 16 72.43 odd 6
108.3.f.c.19.3 16 72.61 even 6
108.3.f.c.91.2 16 8.5 even 2
108.3.f.c.91.3 16 8.3 odd 2
324.3.d.g.163.7 8 72.13 even 6
324.3.d.g.163.8 8 72.67 odd 6
324.3.d.i.163.1 8 72.59 even 6
324.3.d.i.163.2 8 72.5 odd 6
576.3.o.g.319.3 16 12.11 even 2
576.3.o.g.319.6 16 3.2 odd 2
576.3.o.g.511.3 16 9.2 odd 6
576.3.o.g.511.6 16 36.11 even 6
1728.3.o.g.127.5 16 9.7 even 3 inner
1728.3.o.g.127.6 16 36.7 odd 6 inner
1728.3.o.g.1279.5 16 4.3 odd 2 inner
1728.3.o.g.1279.6 16 1.1 even 1 trivial