Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1776,2,Mod(529,1776)] 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("1776.529"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1776, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 0, 5])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1776 = 2^{4} \cdot 3 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1776.bz (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,1,0,-6,0,1,0,-1,0,-12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(14.1814313990\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 444)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 1729.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 1776.1729
Dual form 1776.2.bz.a.529.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.500000 + 0.866025i) q^{3} +(-3.00000 + 1.73205i) q^{5} +(0.500000 + 0.866025i) q^{7} +(-0.500000 + 0.866025i) q^{9} -6.00000 q^{11} +(1.50000 - 0.866025i) q^{13} +(-3.00000 - 1.73205i) q^{15} +(6.00000 + 3.46410i) q^{17} +(-3.00000 + 1.73205i) q^{19} +(-0.500000 + 0.866025i) q^{21} -3.46410i q^{23} +(3.50000 - 6.06218i) q^{25} -1.00000 q^{27} +6.92820i q^{29} +1.73205i q^{31} +(-3.00000 - 5.19615i) q^{33} +(-3.00000 - 1.73205i) q^{35} +(-5.00000 - 3.46410i) q^{37} +(1.50000 + 0.866025i) q^{39} +(-6.00000 - 10.3923i) q^{41} -12.1244i q^{43} -3.46410i q^{45} -12.0000 q^{47} +(3.00000 - 5.19615i) q^{49} +6.92820i q^{51} +(-3.00000 + 5.19615i) q^{53} +(18.0000 - 10.3923i) q^{55} +(-3.00000 - 1.73205i) q^{57} +(3.00000 + 1.73205i) q^{59} +(6.00000 - 3.46410i) q^{61} -1.00000 q^{63} +(-3.00000 + 5.19615i) q^{65} +(-2.50000 - 4.33013i) q^{67} +(3.00000 - 1.73205i) q^{69} +7.00000 q^{73} +7.00000 q^{75} +(-3.00000 - 5.19615i) q^{77} +(-1.50000 + 0.866025i) q^{79} +(-0.500000 - 0.866025i) q^{81} +(6.00000 - 10.3923i) q^{83} -24.0000 q^{85} +(-6.00000 + 3.46410i) q^{87} +(9.00000 + 5.19615i) q^{89} +(1.50000 + 0.866025i) q^{91} +(-1.50000 + 0.866025i) q^{93} +(6.00000 - 10.3923i) q^{95} -1.73205i q^{97} +(3.00000 - 5.19615i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{3} - 6 q^{5} + q^{7} - q^{9} - 12 q^{11} + 3 q^{13} - 6 q^{15} + 12 q^{17} - 6 q^{19} - q^{21} + 7 q^{25} - 2 q^{27} - 6 q^{33} - 6 q^{35} - 10 q^{37} + 3 q^{39} - 12 q^{41} - 24 q^{47} + 6 q^{49}+ \cdots + 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1776\mathbb{Z}\right)^\times\).

\(n\) \(223\) \(593\) \(1297\) \(1333\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{1}{6}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0.500000 + 0.866025i 0.288675 + 0.500000i
\(4\) 0 0
\(5\) −3.00000 + 1.73205i −1.34164 + 0.774597i −0.987048 − 0.160424i \(-0.948714\pi\)
−0.354593 + 0.935021i \(0.615380\pi\)
\(6\) 0 0
\(7\) 0.500000 + 0.866025i 0.188982 + 0.327327i 0.944911 − 0.327327i \(-0.106148\pi\)
−0.755929 + 0.654654i \(0.772814\pi\)
\(8\) 0 0
\(9\) −0.500000 + 0.866025i −0.166667 + 0.288675i
\(10\) 0 0
\(11\) −6.00000 −1.80907 −0.904534 − 0.426401i \(-0.859781\pi\)
−0.904534 + 0.426401i \(0.859781\pi\)
\(12\) 0 0
\(13\) 1.50000 − 0.866025i 0.416025 − 0.240192i −0.277350 − 0.960769i \(-0.589456\pi\)
0.693375 + 0.720577i \(0.256123\pi\)
\(14\) 0 0
\(15\) −3.00000 − 1.73205i −0.774597 − 0.447214i
\(16\) 0 0
\(17\) 6.00000 + 3.46410i 1.45521 + 0.840168i 0.998770 − 0.0495842i \(-0.0157896\pi\)
0.456444 + 0.889752i \(0.349123\pi\)
\(18\) 0 0
\(19\) −3.00000 + 1.73205i −0.688247 + 0.397360i −0.802955 − 0.596040i \(-0.796740\pi\)
0.114708 + 0.993399i \(0.463407\pi\)
\(20\) 0 0
\(21\) −0.500000 + 0.866025i −0.109109 + 0.188982i
\(22\) 0 0
\(23\) − 3.46410i − 0.722315i −0.932505 − 0.361158i \(-0.882382\pi\)
0.932505 − 0.361158i \(-0.117618\pi\)
\(24\) 0 0
\(25\) 3.50000 − 6.06218i 0.700000 − 1.21244i
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 6.92820i 1.28654i 0.765641 + 0.643268i \(0.222422\pi\)
−0.765641 + 0.643268i \(0.777578\pi\)
\(30\) 0 0
\(31\) 1.73205i 0.311086i 0.987829 + 0.155543i \(0.0497126\pi\)
−0.987829 + 0.155543i \(0.950287\pi\)
\(32\) 0 0
\(33\) −3.00000 − 5.19615i −0.522233 − 0.904534i
\(34\) 0 0
\(35\) −3.00000 − 1.73205i −0.507093 − 0.292770i
\(36\) 0 0
\(37\) −5.00000 − 3.46410i −0.821995 − 0.569495i
\(38\) 0 0
\(39\) 1.50000 + 0.866025i 0.240192 + 0.138675i
\(40\) 0 0
\(41\) −6.00000 − 10.3923i −0.937043 − 1.62301i −0.770950 − 0.636895i \(-0.780218\pi\)
−0.166092 − 0.986110i \(-0.553115\pi\)
\(42\) 0 0
\(43\) − 12.1244i − 1.84895i −0.381246 − 0.924473i \(-0.624505\pi\)
0.381246 − 0.924473i \(-0.375495\pi\)
\(44\) 0 0
\(45\) − 3.46410i − 0.516398i
\(46\) 0 0
\(47\) −12.0000 −1.75038 −0.875190 − 0.483779i \(-0.839264\pi\)
−0.875190 + 0.483779i \(0.839264\pi\)
\(48\) 0 0
\(49\) 3.00000 − 5.19615i 0.428571 − 0.742307i
\(50\) 0 0
\(51\) 6.92820i 0.970143i
\(52\) 0 0
\(53\) −3.00000 + 5.19615i −0.412082 + 0.713746i −0.995117 − 0.0987002i \(-0.968532\pi\)
0.583036 + 0.812447i \(0.301865\pi\)
\(54\) 0 0
\(55\) 18.0000 − 10.3923i 2.42712 − 1.40130i
\(56\) 0 0
\(57\) −3.00000 − 1.73205i −0.397360 − 0.229416i
\(58\) 0 0
\(59\) 3.00000 + 1.73205i 0.390567 + 0.225494i 0.682406 − 0.730974i \(-0.260934\pi\)
−0.291839 + 0.956467i \(0.594267\pi\)
\(60\) 0 0
\(61\) 6.00000 − 3.46410i 0.768221 − 0.443533i −0.0640184 − 0.997949i \(-0.520392\pi\)
0.832240 + 0.554416i \(0.187058\pi\)
\(62\) 0 0
\(63\) −1.00000 −0.125988
\(64\) 0 0
\(65\) −3.00000 + 5.19615i −0.372104 + 0.644503i
\(66\) 0 0
\(67\) −2.50000 − 4.33013i −0.305424 − 0.529009i 0.671932 − 0.740613i \(-0.265465\pi\)
−0.977356 + 0.211604i \(0.932131\pi\)
\(68\) 0 0
\(69\) 3.00000 − 1.73205i 0.361158 − 0.208514i
\(70\) 0 0
\(71\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(72\) 0 0
\(73\) 7.00000 0.819288 0.409644 − 0.912245i \(-0.365653\pi\)
0.409644 + 0.912245i \(0.365653\pi\)
\(74\) 0 0
\(75\) 7.00000 0.808290
\(76\) 0 0
\(77\) −3.00000 − 5.19615i −0.341882 − 0.592157i
\(78\) 0 0
\(79\) −1.50000 + 0.866025i −0.168763 + 0.0974355i −0.582003 − 0.813187i \(-0.697731\pi\)
0.413239 + 0.910622i \(0.364397\pi\)
\(80\) 0 0
\(81\) −0.500000 − 0.866025i −0.0555556 − 0.0962250i
\(82\) 0 0
\(83\) 6.00000 − 10.3923i 0.658586 − 1.14070i −0.322396 − 0.946605i \(-0.604488\pi\)
0.980982 − 0.194099i \(-0.0621783\pi\)
\(84\) 0 0
\(85\) −24.0000 −2.60317
\(86\) 0 0
\(87\) −6.00000 + 3.46410i −0.643268 + 0.371391i
\(88\) 0 0
\(89\) 9.00000 + 5.19615i 0.953998 + 0.550791i 0.894321 − 0.447427i \(-0.147659\pi\)
0.0596775 + 0.998218i \(0.480993\pi\)
\(90\) 0 0
\(91\) 1.50000 + 0.866025i 0.157243 + 0.0907841i
\(92\) 0 0
\(93\) −1.50000 + 0.866025i −0.155543 + 0.0898027i
\(94\) 0 0
\(95\) 6.00000 − 10.3923i 0.615587 − 1.06623i
\(96\) 0 0
\(97\) − 1.73205i − 0.175863i −0.996127 − 0.0879316i \(-0.971974\pi\)
0.996127 − 0.0879316i \(-0.0280257\pi\)
\(98\) 0 0
\(99\) 3.00000 − 5.19615i 0.301511 − 0.522233i
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 1776.2.bz.a.1729.1 2
4.3 odd 2 444.2.r.a.397.1 yes 2
12.11 even 2 1332.2.bi.f.397.1 2
37.11 even 6 inner 1776.2.bz.a.529.1 2
148.11 odd 6 444.2.r.a.85.1 ✓ 2
444.11 even 6 1332.2.bi.f.973.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
444.2.r.a.85.1 ✓ 2 148.11 odd 6
444.2.r.a.397.1 yes 2 4.3 odd 2
1332.2.bi.f.397.1 2 12.11 even 2
1332.2.bi.f.973.1 2 444.11 even 6
1776.2.bz.a.529.1 2 37.11 even 6 inner
1776.2.bz.a.1729.1 2 1.1 even 1 trivial