Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1134,2,Mod(109,1134)] 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("1134.109"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1134, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([2, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1134 = 2 \cdot 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1134.h (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,1,0,-1,0,0,5] 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: \(9.05503558921\)
Analytic rank: \(0\)
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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 378)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 109.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 1134.109
Dual form 1134.2.h.n.541.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.500000 - 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{4} +(2.50000 + 0.866025i) q^{7} -1.00000 q^{8} +6.00000 q^{11} +(-2.50000 + 4.33013i) q^{13} +(2.00000 - 1.73205i) q^{14} +(-0.500000 + 0.866025i) q^{16} +(-3.00000 + 5.19615i) q^{17} +(2.00000 + 3.46410i) q^{19} +(3.00000 - 5.19615i) q^{22} +6.00000 q^{23} -5.00000 q^{25} +(2.50000 + 4.33013i) q^{26} +(-0.500000 - 2.59808i) q^{28} +(-3.00000 - 5.19615i) q^{29} +(0.500000 + 0.866025i) q^{31} +(0.500000 + 0.866025i) q^{32} +(3.00000 + 5.19615i) q^{34} +(0.500000 + 0.866025i) q^{37} +4.00000 q^{38} +(3.00000 - 5.19615i) q^{41} +(0.500000 + 0.866025i) q^{43} +(-3.00000 - 5.19615i) q^{44} +(3.00000 - 5.19615i) q^{46} +(3.00000 - 5.19615i) q^{47} +(5.50000 + 4.33013i) q^{49} +(-2.50000 + 4.33013i) q^{50} +5.00000 q^{52} +(3.00000 - 5.19615i) q^{53} +(-2.50000 - 0.866025i) q^{56} -6.00000 q^{58} +(3.00000 + 5.19615i) q^{59} +(0.500000 - 0.866025i) q^{61} +1.00000 q^{62} +1.00000 q^{64} +(0.500000 + 0.866025i) q^{67} +6.00000 q^{68} +12.0000 q^{71} +(-1.00000 + 1.73205i) q^{73} +1.00000 q^{74} +(2.00000 - 3.46410i) q^{76} +(15.0000 + 5.19615i) q^{77} +(0.500000 - 0.866025i) q^{79} +(-3.00000 - 5.19615i) q^{82} +(-3.00000 - 5.19615i) q^{83} +1.00000 q^{86} -6.00000 q^{88} +(-10.0000 + 8.66025i) q^{91} +(-3.00000 - 5.19615i) q^{92} +(-3.00000 - 5.19615i) q^{94} +(-8.50000 - 14.7224i) q^{97} +(6.50000 - 2.59808i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} - q^{4} + 5 q^{7} - 2 q^{8} + 12 q^{11} - 5 q^{13} + 4 q^{14} - q^{16} - 6 q^{17} + 4 q^{19} + 6 q^{22} + 12 q^{23} - 10 q^{25} + 5 q^{26} - q^{28} - 6 q^{29} + q^{31} + q^{32} + 6 q^{34}+ \cdots + 13 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1134\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(407\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.500000 − 0.866025i 0.353553 − 0.612372i
\(3\) 0 0
\(4\) −0.500000 − 0.866025i −0.250000 − 0.433013i
\(5\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(6\) 0 0
\(7\) 2.50000 + 0.866025i 0.944911 + 0.327327i
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) 0 0
\(11\) 6.00000 1.80907 0.904534 − 0.426401i \(-0.140219\pi\)
0.904534 + 0.426401i \(0.140219\pi\)
\(12\) 0 0
\(13\) −2.50000 + 4.33013i −0.693375 + 1.20096i 0.277350 + 0.960769i \(0.410544\pi\)
−0.970725 + 0.240192i \(0.922790\pi\)
\(14\) 2.00000 − 1.73205i 0.534522 − 0.462910i
\(15\) 0 0
\(16\) −0.500000 + 0.866025i −0.125000 + 0.216506i
\(17\) −3.00000 + 5.19615i −0.727607 + 1.26025i 0.230285 + 0.973123i \(0.426034\pi\)
−0.957892 + 0.287129i \(0.907299\pi\)
\(18\) 0 0
\(19\) 2.00000 + 3.46410i 0.458831 + 0.794719i 0.998899 − 0.0469020i \(-0.0149348\pi\)
−0.540068 + 0.841621i \(0.681602\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 3.00000 − 5.19615i 0.639602 − 1.10782i
\(23\) 6.00000 1.25109 0.625543 − 0.780189i \(-0.284877\pi\)
0.625543 + 0.780189i \(0.284877\pi\)
\(24\) 0 0
\(25\) −5.00000 −1.00000
\(26\) 2.50000 + 4.33013i 0.490290 + 0.849208i
\(27\) 0 0
\(28\) −0.500000 − 2.59808i −0.0944911 − 0.490990i
\(29\) −3.00000 − 5.19615i −0.557086 − 0.964901i −0.997738 − 0.0672232i \(-0.978586\pi\)
0.440652 − 0.897678i \(-0.354747\pi\)
\(30\) 0 0
\(31\) 0.500000 + 0.866025i 0.0898027 + 0.155543i 0.907428 − 0.420208i \(-0.138043\pi\)
−0.817625 + 0.575751i \(0.804710\pi\)
\(32\) 0.500000 + 0.866025i 0.0883883 + 0.153093i
\(33\) 0 0
\(34\) 3.00000 + 5.19615i 0.514496 + 0.891133i
\(35\) 0 0
\(36\) 0 0
\(37\) 0.500000 + 0.866025i 0.0821995 + 0.142374i 0.904194 − 0.427121i \(-0.140472\pi\)
−0.821995 + 0.569495i \(0.807139\pi\)
\(38\) 4.00000 0.648886
\(39\) 0 0
\(40\) 0 0
\(41\) 3.00000 − 5.19615i 0.468521 − 0.811503i −0.530831 − 0.847477i \(-0.678120\pi\)
0.999353 + 0.0359748i \(0.0114536\pi\)
\(42\) 0 0
\(43\) 0.500000 + 0.866025i 0.0762493 + 0.132068i 0.901629 − 0.432511i \(-0.142372\pi\)
−0.825380 + 0.564578i \(0.809039\pi\)
\(44\) −3.00000 − 5.19615i −0.452267 − 0.783349i
\(45\) 0 0
\(46\) 3.00000 − 5.19615i 0.442326 − 0.766131i
\(47\) 3.00000 − 5.19615i 0.437595 − 0.757937i −0.559908 − 0.828554i \(-0.689164\pi\)
0.997503 + 0.0706177i \(0.0224970\pi\)
\(48\) 0 0
\(49\) 5.50000 + 4.33013i 0.785714 + 0.618590i
\(50\) −2.50000 + 4.33013i −0.353553 + 0.612372i
\(51\) 0 0
\(52\) 5.00000 0.693375
\(53\) 3.00000 − 5.19615i 0.412082 − 0.713746i −0.583036 − 0.812447i \(-0.698135\pi\)
0.995117 + 0.0987002i \(0.0314685\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −2.50000 − 0.866025i −0.334077 − 0.115728i
\(57\) 0 0
\(58\) −6.00000 −0.787839
\(59\) 3.00000 + 5.19615i 0.390567 + 0.676481i 0.992524 − 0.122047i \(-0.0389457\pi\)
−0.601958 + 0.798528i \(0.705612\pi\)
\(60\) 0 0
\(61\) 0.500000 − 0.866025i 0.0640184 − 0.110883i −0.832240 − 0.554416i \(-0.812942\pi\)
0.896258 + 0.443533i \(0.146275\pi\)
\(62\) 1.00000 0.127000
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) 0.500000 + 0.866025i 0.0610847 + 0.105802i 0.894951 − 0.446165i \(-0.147211\pi\)
−0.833866 + 0.551967i \(0.813877\pi\)
\(68\) 6.00000 0.727607
\(69\) 0 0
\(70\) 0 0
\(71\) 12.0000 1.42414 0.712069 − 0.702109i \(-0.247758\pi\)
0.712069 + 0.702109i \(0.247758\pi\)
\(72\) 0 0
\(73\) −1.00000 + 1.73205i −0.117041 + 0.202721i −0.918594 − 0.395203i \(-0.870674\pi\)
0.801553 + 0.597924i \(0.204008\pi\)
\(74\) 1.00000 0.116248
\(75\) 0 0
\(76\) 2.00000 − 3.46410i 0.229416 − 0.397360i
\(77\) 15.0000 + 5.19615i 1.70941 + 0.592157i
\(78\) 0 0
\(79\) 0.500000 − 0.866025i 0.0562544 − 0.0974355i −0.836527 − 0.547926i \(-0.815418\pi\)
0.892781 + 0.450490i \(0.148751\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −3.00000 − 5.19615i −0.331295 − 0.573819i
\(83\) −3.00000 − 5.19615i −0.329293 − 0.570352i 0.653079 − 0.757290i \(-0.273477\pi\)
−0.982372 + 0.186938i \(0.940144\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 1.00000 0.107833
\(87\) 0 0
\(88\) −6.00000 −0.639602
\(89\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(90\) 0 0
\(91\) −10.0000 + 8.66025i −1.04828 + 0.907841i
\(92\) −3.00000 − 5.19615i −0.312772 − 0.541736i
\(93\) 0 0
\(94\) −3.00000 − 5.19615i −0.309426 − 0.535942i
\(95\) 0 0
\(96\) 0 0
\(97\) −8.50000 − 14.7224i −0.863044 − 1.49484i −0.868976 − 0.494854i \(-0.835222\pi\)
0.00593185 − 0.999982i \(-0.498112\pi\)
\(98\) 6.50000 − 2.59808i 0.656599 − 0.262445i
\(99\) 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 1134.2.h.n.109.1 2
3.2 odd 2 1134.2.h.d.109.1 2
7.2 even 3 1134.2.e.c.919.1 2
9.2 odd 6 1134.2.e.m.865.1 2
9.4 even 3 378.2.g.e.109.1 yes 2
9.5 odd 6 378.2.g.b.109.1 ✓ 2
9.7 even 3 1134.2.e.c.865.1 2
21.2 odd 6 1134.2.e.m.919.1 2
63.2 odd 6 1134.2.h.d.541.1 2
63.4 even 3 2646.2.a.h.1.1 1
63.16 even 3 inner 1134.2.h.n.541.1 2
63.23 odd 6 378.2.g.b.163.1 yes 2
63.31 odd 6 2646.2.a.g.1.1 1
63.32 odd 6 2646.2.a.x.1.1 1
63.58 even 3 378.2.g.e.163.1 yes 2
63.59 even 6 2646.2.a.w.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
378.2.g.b.109.1 ✓ 2 9.5 odd 6
378.2.g.b.163.1 yes 2 63.23 odd 6
378.2.g.e.109.1 yes 2 9.4 even 3
378.2.g.e.163.1 yes 2 63.58 even 3
1134.2.e.c.865.1 2 9.7 even 3
1134.2.e.c.919.1 2 7.2 even 3
1134.2.e.m.865.1 2 9.2 odd 6
1134.2.e.m.919.1 2 21.2 odd 6
1134.2.h.d.109.1 2 3.2 odd 2
1134.2.h.d.541.1 2 63.2 odd 6
1134.2.h.n.109.1 2 1.1 even 1 trivial
1134.2.h.n.541.1 2 63.16 even 3 inner
2646.2.a.g.1.1 1 63.31 odd 6
2646.2.a.h.1.1 1 63.4 even 3
2646.2.a.w.1.1 1 63.59 even 6
2646.2.a.x.1.1 1 63.32 odd 6