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Properties

Label 1728.3.o.g.1279.1
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 1279.1
Root \(-0.523926 - 1.93016i\) of defining polynomial
Character \(\chi\) \(=\) 1728.1279
Dual form 1728.3.o.g.127.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-4.03104 + 6.98197i) q^{5} +(-3.90254 + 2.25313i) q^{7} +(3.25842 - 1.88125i) q^{11} +(3.52605 - 6.10730i) q^{13} -0.517890 q^{17} -16.4164i q^{19} +(-27.7049 - 15.9954i) q^{23} +(-19.9986 - 34.6387i) q^{25} +(9.48394 + 16.4267i) q^{29} +(-13.1355 - 7.58377i) q^{31} -36.3299i q^{35} -0.592061 q^{37} +(-12.3766 + 21.4369i) q^{41} +(-27.8686 + 16.0900i) q^{43} +(52.4682 - 30.2925i) q^{47} +(-14.3468 + 24.8493i) q^{49} -0.664765 q^{53} +30.3336i q^{55} +(-30.5921 - 17.6623i) q^{59} +(-33.7750 - 58.5000i) q^{61} +(28.4273 + 49.2376i) q^{65} +(74.4692 + 42.9948i) q^{67} -56.4434i q^{71} +131.921 q^{73} +(-8.47743 + 14.6833i) q^{77} +(126.869 - 73.2481i) q^{79} +(87.1029 - 50.2889i) q^{83} +(2.08764 - 3.61589i) q^{85} +25.8362 q^{89} +31.7786i q^{91} +(114.619 + 66.1754i) q^{95} +(-48.2534 - 83.5773i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 6 q^{5} + 46 q^{13} - 12 q^{17} - 30 q^{25} + 42 q^{29} - 56 q^{37} - 84 q^{41} + 58 q^{49} - 72 q^{53} + 34 q^{61} + 30 q^{65} + 116 q^{73} - 330 q^{77} + 140 q^{85} + 384 q^{89} - 148 q^{97}+O(q^{100}) \) 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
\(5\) −4.03104 + 6.98197i −0.806209 + 1.39639i 0.109263 + 0.994013i \(0.465151\pi\)
−0.915472 + 0.402382i \(0.868182\pi\)
\(6\) 0 0
\(7\) −3.90254 + 2.25313i −0.557506 + 0.321876i −0.752144 − 0.658999i \(-0.770980\pi\)
0.194638 + 0.980875i \(0.437647\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 3.25842 − 1.88125i 0.296220 − 0.171023i −0.344523 − 0.938778i \(-0.611959\pi\)
0.640744 + 0.767755i \(0.278626\pi\)
\(12\) 0 0
\(13\) 3.52605 − 6.10730i 0.271235 − 0.469792i −0.697944 − 0.716153i \(-0.745901\pi\)
0.969178 + 0.246361i \(0.0792348\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −0.517890 −0.0304641 −0.0152321 − 0.999884i \(-0.504849\pi\)
−0.0152321 + 0.999884i \(0.504849\pi\)
\(18\) 0 0
\(19\) − 16.4164i − 0.864023i −0.901868 − 0.432012i \(-0.857804\pi\)
0.901868 − 0.432012i \(-0.142196\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −27.7049 − 15.9954i −1.20456 − 0.695454i −0.242996 − 0.970027i \(-0.578130\pi\)
−0.961566 + 0.274573i \(0.911463\pi\)
\(24\) 0 0
\(25\) −19.9986 − 34.6387i −0.799946 − 1.38555i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 9.48394 + 16.4267i 0.327032 + 0.566437i 0.981922 − 0.189288i \(-0.0606180\pi\)
−0.654889 + 0.755725i \(0.727285\pi\)
\(30\) 0 0
\(31\) −13.1355 − 7.58377i −0.423725 − 0.244638i 0.272945 − 0.962030i \(-0.412002\pi\)
−0.696670 + 0.717392i \(0.745336\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) − 36.3299i − 1.03800i
\(36\) 0 0
\(37\) −0.592061 −0.0160017 −0.00800083 − 0.999968i \(-0.502547\pi\)
−0.00800083 + 0.999968i \(0.502547\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −12.3766 + 21.4369i −0.301868 + 0.522850i −0.976559 − 0.215250i \(-0.930943\pi\)
0.674691 + 0.738100i \(0.264277\pi\)
\(42\) 0 0
\(43\) −27.8686 + 16.0900i −0.648107 + 0.374185i −0.787731 − 0.616020i \(-0.788744\pi\)
0.139623 + 0.990205i \(0.455411\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 52.4682 − 30.2925i 1.11634 − 0.644521i 0.175879 − 0.984412i \(-0.443723\pi\)
0.940465 + 0.339890i \(0.110390\pi\)
\(48\) 0 0
\(49\) −14.3468 + 24.8493i −0.292791 + 0.507129i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −0.664765 −0.0125427 −0.00627137 − 0.999980i \(-0.501996\pi\)
−0.00627137 + 0.999980i \(0.501996\pi\)
\(54\) 0 0
\(55\) 30.3336i 0.551521i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −30.5921 − 17.6623i −0.518510 − 0.299362i 0.217815 − 0.975990i \(-0.430107\pi\)
−0.736325 + 0.676628i \(0.763440\pi\)
\(60\) 0 0
\(61\) −33.7750 − 58.5000i −0.553688 − 0.959016i −0.998004 − 0.0631460i \(-0.979887\pi\)
0.444316 − 0.895870i \(-0.353447\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 28.4273 + 49.2376i 0.437343 + 0.757501i
\(66\) 0 0
\(67\) 74.4692 + 42.9948i 1.11148 + 0.641714i 0.939213 − 0.343336i \(-0.111557\pi\)
0.172269 + 0.985050i \(0.444890\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) − 56.4434i − 0.794977i −0.917607 − 0.397489i \(-0.869882\pi\)
0.917607 − 0.397489i \(-0.130118\pi\)
\(72\) 0 0
\(73\) 131.921 1.80713 0.903567 − 0.428447i \(-0.140939\pi\)
0.903567 + 0.428447i \(0.140939\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −8.47743 + 14.6833i −0.110096 + 0.190693i
\(78\) 0 0
\(79\) 126.869 − 73.2481i 1.60594 − 0.927191i 0.615677 − 0.787999i \(-0.288883\pi\)
0.990265 − 0.139192i \(-0.0444505\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 87.1029 − 50.2889i 1.04943 − 0.605890i 0.126942 − 0.991910i \(-0.459484\pi\)
0.922491 + 0.386020i \(0.126150\pi\)
\(84\) 0 0
\(85\) 2.08764 − 3.61589i 0.0245604 − 0.0425399i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 25.8362 0.290295 0.145147 − 0.989410i \(-0.453634\pi\)
0.145147 + 0.989410i \(0.453634\pi\)
\(90\) 0 0
\(91\) 31.7786i 0.349216i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 114.619 + 66.1754i 1.20652 + 0.696583i
\(96\) 0 0
\(97\) −48.2534 − 83.5773i −0.497457 − 0.861621i 0.502538 − 0.864555i \(-0.332400\pi\)
−0.999996 + 0.00293363i \(0.999066\pi\)
\(98\) 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.1279.1 16
3.2 odd 2 576.3.o.g.319.8 16
4.3 odd 2 inner 1728.3.o.g.1279.2 16
8.3 odd 2 108.3.f.c.91.4 16
8.5 even 2 108.3.f.c.91.8 16
9.2 odd 6 576.3.o.g.511.1 16
9.7 even 3 inner 1728.3.o.g.127.2 16
12.11 even 2 576.3.o.g.319.1 16
24.5 odd 2 36.3.f.c.31.1 yes 16
24.11 even 2 36.3.f.c.31.5 yes 16
36.7 odd 6 inner 1728.3.o.g.127.1 16
36.11 even 6 576.3.o.g.511.8 16
72.5 odd 6 324.3.d.i.163.5 8
72.11 even 6 36.3.f.c.7.1 ✓ 16
72.13 even 6 324.3.d.g.163.4 8
72.29 odd 6 36.3.f.c.7.5 yes 16
72.43 odd 6 108.3.f.c.19.8 16
72.59 even 6 324.3.d.i.163.6 8
72.61 even 6 108.3.f.c.19.4 16
72.67 odd 6 324.3.d.g.163.3 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
36.3.f.c.7.1 ✓ 16 72.11 even 6
36.3.f.c.7.5 yes 16 72.29 odd 6
36.3.f.c.31.1 yes 16 24.5 odd 2
36.3.f.c.31.5 yes 16 24.11 even 2
108.3.f.c.19.4 16 72.61 even 6
108.3.f.c.19.8 16 72.43 odd 6
108.3.f.c.91.4 16 8.3 odd 2
108.3.f.c.91.8 16 8.5 even 2
324.3.d.g.163.3 8 72.67 odd 6
324.3.d.g.163.4 8 72.13 even 6
324.3.d.i.163.5 8 72.5 odd 6
324.3.d.i.163.6 8 72.59 even 6
576.3.o.g.319.1 16 12.11 even 2
576.3.o.g.319.8 16 3.2 odd 2
576.3.o.g.511.1 16 9.2 odd 6
576.3.o.g.511.8 16 36.11 even 6
1728.3.o.g.127.1 16 36.7 odd 6 inner
1728.3.o.g.127.2 16 9.7 even 3 inner
1728.3.o.g.1279.1 16 1.1 even 1 trivial
1728.3.o.g.1279.2 16 4.3 odd 2 inner