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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1860,2,Mod(1141,1860)] 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("1860.1141"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1860, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 0, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1860 = 2^{2} \cdot 3 \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1860.q (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,0,-6,0,6,0,-2] 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: \(14.8521747760\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 3 x^{11} + 29 x^{10} + 405 x^{8} - 117 x^{7} + 2515 x^{6} + 1476 x^{5} + 7995 x^{4} - 2175 x^{3} + \cdots + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{31}]\)
Coefficient ring index: \( 3^{3} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 1741.6
Root \(-1.63261 - 2.82776i\) of defining polynomial
Character \(\chi\) \(=\) 1860.1741
Dual form 1860.2.q.i.1141.6

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.500000 + 0.866025i) q^{3} +(0.500000 + 0.866025i) q^{5} +(2.41160 - 4.17701i) q^{7} +(-0.500000 - 0.866025i) q^{9} +(0.851830 + 1.47541i) q^{11} +(-2.20030 - 3.81103i) q^{13} -1.00000 q^{15} +(-1.56313 + 2.70742i) q^{17} +(4.14540 - 7.18005i) q^{19} +(2.41160 + 4.17701i) q^{21} -0.861460 q^{23} +(-0.500000 + 0.866025i) q^{25} +1.00000 q^{27} -8.52686 q^{29} +(-2.93075 - 4.73400i) q^{31} -1.70366 q^{33} +4.82320 q^{35} +(2.58227 - 4.47263i) q^{37} +4.40060 q^{39} +(-4.69416 - 8.13052i) q^{41} +(-5.75729 + 9.97191i) q^{43} +(0.500000 - 0.866025i) q^{45} -7.10426 q^{47} +(-8.13162 - 14.0844i) q^{49} +(-1.56313 - 2.70742i) q^{51} +(2.32597 + 4.02869i) q^{53} +(-0.851830 + 1.47541i) q^{55} +(4.14540 + 7.18005i) q^{57} +(1.90059 - 3.29191i) q^{59} +6.16455 q^{61} -4.82320 q^{63} +(2.20030 - 3.81103i) q^{65} +(2.65490 + 4.59842i) q^{67} +(0.430730 - 0.746047i) q^{69} +(5.12489 + 8.87657i) q^{71} +(-3.71666 - 6.43745i) q^{73} +(-0.500000 - 0.866025i) q^{75} +8.21708 q^{77} +(-1.28057 + 2.21801i) q^{79} +(-0.500000 + 0.866025i) q^{81} +(-8.67167 - 15.0198i) q^{83} -3.12625 q^{85} +(4.26343 - 7.38447i) q^{87} +8.72295 q^{89} -21.2250 q^{91} +(5.56513 - 0.171101i) q^{93} +8.29080 q^{95} +6.42159 q^{97} +(0.851830 - 1.47541i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 6 q^{3} + 6 q^{5} - 2 q^{7} - 6 q^{9} - 2 q^{13} - 12 q^{15} - 2 q^{17} + 5 q^{19} - 2 q^{21} + 2 q^{23} - 6 q^{25} + 12 q^{27} - 20 q^{29} + 7 q^{31} - 4 q^{35} + 3 q^{37} + 4 q^{39} - 9 q^{41}+ \cdots - 76 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1860\mathbb{Z}\right)^\times\).

\(n\) \(931\) \(1117\) \(1241\) \(1801\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(e\left(\frac{2}{3}\right)\)

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
\(3\) −0.500000 + 0.866025i −0.288675 + 0.500000i
\(4\) 0 0
\(5\) 0.500000 + 0.866025i 0.223607 + 0.387298i
\(6\) 0 0
\(7\) 2.41160 − 4.17701i 0.911499 − 1.57876i 0.0995507 − 0.995032i \(-0.468259\pi\)
0.811948 − 0.583730i \(-0.198407\pi\)
\(8\) 0 0
\(9\) −0.500000 − 0.866025i −0.166667 − 0.288675i
\(10\) 0 0
\(11\) 0.851830 + 1.47541i 0.256836 + 0.444853i 0.965393 − 0.260801i \(-0.0839866\pi\)
−0.708556 + 0.705654i \(0.750653\pi\)
\(12\) 0 0
\(13\) −2.20030 − 3.81103i −0.610254 − 1.05699i −0.991197 − 0.132392i \(-0.957734\pi\)
0.380944 − 0.924598i \(-0.375599\pi\)
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 0 0
\(17\) −1.56313 + 2.70742i −0.379114 + 0.656645i −0.990934 − 0.134353i \(-0.957105\pi\)
0.611820 + 0.790997i \(0.290438\pi\)
\(18\) 0 0
\(19\) 4.14540 − 7.18005i 0.951020 − 1.64722i 0.207797 − 0.978172i \(-0.433371\pi\)
0.743223 − 0.669043i \(-0.233296\pi\)
\(20\) 0 0
\(21\) 2.41160 + 4.17701i 0.526254 + 0.911499i
\(22\) 0 0
\(23\) −0.861460 −0.179627 −0.0898135 − 0.995959i \(-0.528627\pi\)
−0.0898135 + 0.995959i \(0.528627\pi\)
\(24\) 0 0
\(25\) −0.500000 + 0.866025i −0.100000 + 0.173205i
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −8.52686 −1.58340 −0.791699 − 0.610912i \(-0.790803\pi\)
−0.791699 + 0.610912i \(0.790803\pi\)
\(30\) 0 0
\(31\) −2.93075 − 4.73400i −0.526377 − 0.850251i
\(32\) 0 0
\(33\) −1.70366 −0.296569
\(34\) 0 0
\(35\) 4.82320 0.815269
\(36\) 0 0
\(37\) 2.58227 − 4.47263i 0.424523 − 0.735296i −0.571853 − 0.820356i \(-0.693775\pi\)
0.996376 + 0.0850607i \(0.0271084\pi\)
\(38\) 0 0
\(39\) 4.40060 0.704660
\(40\) 0 0
\(41\) −4.69416 − 8.13052i −0.733104 − 1.26977i −0.955550 − 0.294829i \(-0.904737\pi\)
0.222446 − 0.974945i \(-0.428596\pi\)
\(42\) 0 0
\(43\) −5.75729 + 9.97191i −0.877978 + 1.52070i −0.0244216 + 0.999702i \(0.507774\pi\)
−0.853556 + 0.521001i \(0.825559\pi\)
\(44\) 0 0
\(45\) 0.500000 − 0.866025i 0.0745356 − 0.129099i
\(46\) 0 0
\(47\) −7.10426 −1.03626 −0.518132 − 0.855301i \(-0.673372\pi\)
−0.518132 + 0.855301i \(0.673372\pi\)
\(48\) 0 0
\(49\) −8.13162 − 14.0844i −1.16166 − 2.01205i
\(50\) 0 0
\(51\) −1.56313 − 2.70742i −0.218882 − 0.379114i
\(52\) 0 0
\(53\) 2.32597 + 4.02869i 0.319496 + 0.553384i 0.980383 − 0.197102i \(-0.0631530\pi\)
−0.660887 + 0.750486i \(0.729820\pi\)
\(54\) 0 0
\(55\) −0.851830 + 1.47541i −0.114861 + 0.198945i
\(56\) 0 0
\(57\) 4.14540 + 7.18005i 0.549072 + 0.951020i
\(58\) 0 0
\(59\) 1.90059 − 3.29191i 0.247435 − 0.428571i −0.715378 − 0.698738i \(-0.753746\pi\)
0.962814 + 0.270167i \(0.0870789\pi\)
\(60\) 0 0
\(61\) 6.16455 0.789289 0.394645 − 0.918834i \(-0.370868\pi\)
0.394645 + 0.918834i \(0.370868\pi\)
\(62\) 0 0
\(63\) −4.82320 −0.607666
\(64\) 0 0
\(65\) 2.20030 − 3.81103i 0.272914 − 0.472701i
\(66\) 0 0
\(67\) 2.65490 + 4.59842i 0.324348 + 0.561787i 0.981380 − 0.192075i \(-0.0615218\pi\)
−0.657032 + 0.753862i \(0.728189\pi\)
\(68\) 0 0
\(69\) 0.430730 − 0.746047i 0.0518538 − 0.0898135i
\(70\) 0 0
\(71\) 5.12489 + 8.87657i 0.608212 + 1.05345i 0.991535 + 0.129840i \(0.0414464\pi\)
−0.383323 + 0.923615i \(0.625220\pi\)
\(72\) 0 0
\(73\) −3.71666 − 6.43745i −0.435002 − 0.753446i 0.562293 − 0.826938i \(-0.309919\pi\)
−0.997296 + 0.0734914i \(0.976586\pi\)
\(74\) 0 0
\(75\) −0.500000 − 0.866025i −0.0577350 − 0.100000i
\(76\) 0 0
\(77\) 8.21708 0.936424
\(78\) 0 0
\(79\) −1.28057 + 2.21801i −0.144075 + 0.249545i −0.929028 − 0.370011i \(-0.879354\pi\)
0.784952 + 0.619556i \(0.212687\pi\)
\(80\) 0 0
\(81\) −0.500000 + 0.866025i −0.0555556 + 0.0962250i
\(82\) 0 0
\(83\) −8.67167 − 15.0198i −0.951839 − 1.64863i −0.741441 − 0.671019i \(-0.765857\pi\)
−0.210399 − 0.977616i \(-0.567476\pi\)
\(84\) 0 0
\(85\) −3.12625 −0.339090
\(86\) 0 0
\(87\) 4.26343 − 7.38447i 0.457088 − 0.791699i
\(88\) 0 0
\(89\) 8.72295 0.924631 0.462315 − 0.886716i \(-0.347019\pi\)
0.462315 + 0.886716i \(0.347019\pi\)
\(90\) 0 0
\(91\) −21.2250 −2.22498
\(92\) 0 0
\(93\) 5.56513 − 0.171101i 0.577078 − 0.0177424i
\(94\) 0 0
\(95\) 8.29080 0.850618
\(96\) 0 0
\(97\) 6.42159 0.652014 0.326007 − 0.945367i \(-0.394297\pi\)
0.326007 + 0.945367i \(0.394297\pi\)
\(98\) 0 0
\(99\) 0.851830 − 1.47541i 0.0856121 − 0.148284i
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 1860.2.q.i.1741.6 yes 12
31.25 even 3 inner 1860.2.q.i.1141.6 ✓ 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1860.2.q.i.1141.6 ✓ 12 31.25 even 3 inner
1860.2.q.i.1741.6 yes 12 1.1 even 1 trivial