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(379,1134)] 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("1134.379"); 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([4, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1134 = 2 \cdot 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1134.f (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,-2,0,-2,4,0,2,4,0,-8,2,0,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(9.05503558921\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\zeta_{12})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 757.1
Root \(-0.866025 - 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 1134.757
Dual form 1134.2.f.r.379.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.500000 - 0.866025i) q^{2} +(-0.500000 + 0.866025i) q^{4} +(0.133975 - 0.232051i) q^{5} +(0.500000 + 0.866025i) q^{7} +1.00000 q^{8} -0.267949 q^{10} +(3.09808 + 5.36603i) q^{11} +(3.23205 - 5.59808i) q^{13} +(0.500000 - 0.866025i) q^{14} +(-0.500000 - 0.866025i) q^{16} -7.00000 q^{17} +0.732051 q^{19} +(0.133975 + 0.232051i) q^{20} +(3.09808 - 5.36603i) q^{22} +(-2.09808 + 3.63397i) q^{23} +(2.46410 + 4.26795i) q^{25} -6.46410 q^{26} -1.00000 q^{28} +(0.767949 + 1.33013i) q^{29} +(-4.09808 + 7.09808i) q^{31} +(-0.500000 + 0.866025i) q^{32} +(3.50000 + 6.06218i) q^{34} +0.267949 q^{35} +10.6603 q^{37} +(-0.366025 - 0.633975i) q^{38} +(0.133975 - 0.232051i) q^{40} +(1.26795 - 2.19615i) q^{41} +(0.732051 + 1.26795i) q^{43} -6.19615 q^{44} +4.19615 q^{46} +(2.36603 + 4.09808i) q^{47} +(-0.500000 + 0.866025i) q^{49} +(2.46410 - 4.26795i) q^{50} +(3.23205 + 5.59808i) q^{52} +9.46410 q^{53} +1.66025 q^{55} +(0.500000 + 0.866025i) q^{56} +(0.767949 - 1.33013i) q^{58} +(2.09808 - 3.63397i) q^{59} +(-1.96410 - 3.40192i) q^{61} +8.19615 q^{62} +1.00000 q^{64} +(-0.866025 - 1.50000i) q^{65} +(3.36603 - 5.83013i) q^{67} +(3.50000 - 6.06218i) q^{68} +(-0.133975 - 0.232051i) q^{70} -6.53590 q^{71} +8.26795 q^{73} +(-5.33013 - 9.23205i) q^{74} +(-0.366025 + 0.633975i) q^{76} +(-3.09808 + 5.36603i) q^{77} +(4.56218 + 7.90192i) q^{79} -0.267949 q^{80} -2.53590 q^{82} +(8.29423 + 14.3660i) q^{83} +(-0.937822 + 1.62436i) q^{85} +(0.732051 - 1.26795i) q^{86} +(3.09808 + 5.36603i) q^{88} -9.92820 q^{89} +6.46410 q^{91} +(-2.09808 - 3.63397i) q^{92} +(2.36603 - 4.09808i) q^{94} +(0.0980762 - 0.169873i) q^{95} +(-5.46410 - 9.46410i) q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{2} - 2 q^{4} + 4 q^{5} + 2 q^{7} + 4 q^{8} - 8 q^{10} + 2 q^{11} + 6 q^{13} + 2 q^{14} - 2 q^{16} - 28 q^{17} - 4 q^{19} + 4 q^{20} + 2 q^{22} + 2 q^{23} - 4 q^{25} - 12 q^{26} - 4 q^{28} + 10 q^{29}+ \cdots + 4 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.500000 − 0.866025i −0.353553 − 0.612372i
\(3\) 0 0
\(4\) −0.500000 + 0.866025i −0.250000 + 0.433013i
\(5\) 0.133975 − 0.232051i 0.0599153 − 0.103776i −0.834512 − 0.550990i \(-0.814250\pi\)
0.894427 + 0.447214i \(0.147584\pi\)
\(6\) 0 0
\(7\) 0.500000 + 0.866025i 0.188982 + 0.327327i
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) −0.267949 −0.0847330
\(11\) 3.09808 + 5.36603i 0.934105 + 1.61792i 0.776222 + 0.630460i \(0.217134\pi\)
0.157883 + 0.987458i \(0.449533\pi\)
\(12\) 0 0
\(13\) 3.23205 − 5.59808i 0.896410 − 1.55263i 0.0643593 − 0.997927i \(-0.479500\pi\)
0.832050 − 0.554700i \(-0.187167\pi\)
\(14\) 0.500000 − 0.866025i 0.133631 − 0.231455i
\(15\) 0 0
\(16\) −0.500000 − 0.866025i −0.125000 − 0.216506i
\(17\) −7.00000 −1.69775 −0.848875 − 0.528594i \(-0.822719\pi\)
−0.848875 + 0.528594i \(0.822719\pi\)
\(18\) 0 0
\(19\) 0.732051 0.167944 0.0839720 − 0.996468i \(-0.473239\pi\)
0.0839720 + 0.996468i \(0.473239\pi\)
\(20\) 0.133975 + 0.232051i 0.0299576 + 0.0518881i
\(21\) 0 0
\(22\) 3.09808 − 5.36603i 0.660512 − 1.14404i
\(23\) −2.09808 + 3.63397i −0.437479 + 0.757736i −0.997494 − 0.0707462i \(-0.977462\pi\)
0.560015 + 0.828482i \(0.310795\pi\)
\(24\) 0 0
\(25\) 2.46410 + 4.26795i 0.492820 + 0.853590i
\(26\) −6.46410 −1.26771
\(27\) 0 0
\(28\) −1.00000 −0.188982
\(29\) 0.767949 + 1.33013i 0.142605 + 0.246998i 0.928477 − 0.371391i \(-0.121119\pi\)
−0.785872 + 0.618389i \(0.787786\pi\)
\(30\) 0 0
\(31\) −4.09808 + 7.09808i −0.736036 + 1.27485i 0.218231 + 0.975897i \(0.429971\pi\)
−0.954267 + 0.298955i \(0.903362\pi\)
\(32\) −0.500000 + 0.866025i −0.0883883 + 0.153093i
\(33\) 0 0
\(34\) 3.50000 + 6.06218i 0.600245 + 1.03965i
\(35\) 0.267949 0.0452917
\(36\) 0 0
\(37\) 10.6603 1.75253 0.876267 − 0.481825i \(-0.160026\pi\)
0.876267 + 0.481825i \(0.160026\pi\)
\(38\) −0.366025 − 0.633975i −0.0593772 − 0.102844i
\(39\) 0 0
\(40\) 0.133975 − 0.232051i 0.0211832 − 0.0366905i
\(41\) 1.26795 − 2.19615i 0.198020 − 0.342981i −0.749866 − 0.661590i \(-0.769882\pi\)
0.947886 + 0.318608i \(0.103215\pi\)
\(42\) 0 0
\(43\) 0.732051 + 1.26795i 0.111637 + 0.193360i 0.916430 − 0.400194i \(-0.131057\pi\)
−0.804794 + 0.593555i \(0.797724\pi\)
\(44\) −6.19615 −0.934105
\(45\) 0 0
\(46\) 4.19615 0.618689
\(47\) 2.36603 + 4.09808i 0.345120 + 0.597766i 0.985376 − 0.170396i \(-0.0545048\pi\)
−0.640255 + 0.768162i \(0.721171\pi\)
\(48\) 0 0
\(49\) −0.500000 + 0.866025i −0.0714286 + 0.123718i
\(50\) 2.46410 − 4.26795i 0.348477 − 0.603579i
\(51\) 0 0
\(52\) 3.23205 + 5.59808i 0.448205 + 0.776313i
\(53\) 9.46410 1.29999 0.649997 − 0.759937i \(-0.274770\pi\)
0.649997 + 0.759937i \(0.274770\pi\)
\(54\) 0 0
\(55\) 1.66025 0.223869
\(56\) 0.500000 + 0.866025i 0.0668153 + 0.115728i
\(57\) 0 0
\(58\) 0.767949 − 1.33013i 0.100837 − 0.174654i
\(59\) 2.09808 − 3.63397i 0.273146 − 0.473103i −0.696520 − 0.717538i \(-0.745269\pi\)
0.969666 + 0.244435i \(0.0786024\pi\)
\(60\) 0 0
\(61\) −1.96410 − 3.40192i −0.251477 − 0.435572i 0.712455 − 0.701717i \(-0.247583\pi\)
−0.963933 + 0.266146i \(0.914250\pi\)
\(62\) 8.19615 1.04091
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −0.866025 − 1.50000i −0.107417 − 0.186052i
\(66\) 0 0
\(67\) 3.36603 − 5.83013i 0.411225 − 0.712263i −0.583799 − 0.811899i \(-0.698434\pi\)
0.995024 + 0.0996351i \(0.0317676\pi\)
\(68\) 3.50000 − 6.06218i 0.424437 − 0.735147i
\(69\) 0 0
\(70\) −0.133975 − 0.232051i −0.0160130 − 0.0277354i
\(71\) −6.53590 −0.775668 −0.387834 − 0.921729i \(-0.626777\pi\)
−0.387834 + 0.921729i \(0.626777\pi\)
\(72\) 0 0
\(73\) 8.26795 0.967690 0.483845 − 0.875154i \(-0.339240\pi\)
0.483845 + 0.875154i \(0.339240\pi\)
\(74\) −5.33013 − 9.23205i −0.619615 − 1.07320i
\(75\) 0 0
\(76\) −0.366025 + 0.633975i −0.0419860 + 0.0727219i
\(77\) −3.09808 + 5.36603i −0.353059 + 0.611515i
\(78\) 0 0
\(79\) 4.56218 + 7.90192i 0.513285 + 0.889036i 0.999881 + 0.0154089i \(0.00490499\pi\)
−0.486596 + 0.873627i \(0.661762\pi\)
\(80\) −0.267949 −0.0299576
\(81\) 0 0
\(82\) −2.53590 −0.280043
\(83\) 8.29423 + 14.3660i 0.910410 + 1.57688i 0.813486 + 0.581584i \(0.197567\pi\)
0.0969238 + 0.995292i \(0.469100\pi\)
\(84\) 0 0
\(85\) −0.937822 + 1.62436i −0.101721 + 0.176186i
\(86\) 0.732051 − 1.26795i 0.0789391 − 0.136726i
\(87\) 0 0
\(88\) 3.09808 + 5.36603i 0.330256 + 0.572020i
\(89\) −9.92820 −1.05239 −0.526194 − 0.850365i \(-0.676381\pi\)
−0.526194 + 0.850365i \(0.676381\pi\)
\(90\) 0 0
\(91\) 6.46410 0.677622
\(92\) −2.09808 − 3.63397i −0.218740 − 0.378868i
\(93\) 0 0
\(94\) 2.36603 − 4.09808i 0.244037 − 0.422684i
\(95\) 0.0980762 − 0.169873i 0.0100624 − 0.0174286i
\(96\) 0 0
\(97\) −5.46410 − 9.46410i −0.554795 − 0.960934i −0.997919 − 0.0644736i \(-0.979463\pi\)
0.443124 − 0.896460i \(-0.353870\pi\)
\(98\) 1.00000 0.101015
\(99\) 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.f.r.757.1 4
3.2 odd 2 1134.2.f.s.757.2 4
9.2 odd 6 1134.2.f.s.379.2 4
9.4 even 3 1134.2.a.m.1.2 yes 2
9.5 odd 6 1134.2.a.l.1.1 ✓ 2
9.7 even 3 inner 1134.2.f.r.379.1 4
36.23 even 6 9072.2.a.bp.1.1 2
36.31 odd 6 9072.2.a.y.1.2 2
63.13 odd 6 7938.2.a.bt.1.1 2
63.41 even 6 7938.2.a.bg.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1134.2.a.l.1.1 ✓ 2 9.5 odd 6
1134.2.a.m.1.2 yes 2 9.4 even 3
1134.2.f.r.379.1 4 9.7 even 3 inner
1134.2.f.r.757.1 4 1.1 even 1 trivial
1134.2.f.s.379.2 4 9.2 odd 6
1134.2.f.s.757.2 4 3.2 odd 2
7938.2.a.bg.1.2 2 63.41 even 6
7938.2.a.bt.1.1 2 63.13 odd 6
9072.2.a.y.1.2 2 36.31 odd 6
9072.2.a.bp.1.1 2 36.23 even 6