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 379.2
Root \(0.866025 + 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 1134.379
Dual form 1134.2.f.s.757.2

$q$-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 \(\left(\mathbb{Z}/1134\mathbb{Z}\right)^\times\).

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.500000 − 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 \(-0.185750\pi\)
−0.894427 + 0.447214i \(0.852416\pi\)
\(6\) 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 −0.157883 + 0.987458i \(0.550467\pi\)
−0.776222 + 0.630460i \(0.782866\pi\)
\(12\) 0 0
\(13\) 3.23205 + 5.59808i 0.896410 + 1.55263i 0.832050 + 0.554700i \(0.187167\pi\)
0.0643593 + 0.997927i \(0.479500\pi\)
\(14\) −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 \(-0.177281\pi\)
0.848875 + 0.528594i \(0.177281\pi\)
\(18\) 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 \(-0.0225381\pi\)
−0.560015 + 0.828482i \(0.689205\pi\)
\(24\) 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 \(-0.878881\pi\)
0.785872 + 0.618389i \(0.212214\pi\)
\(30\) 0 0
\(31\) −4.09808 − 7.09808i −0.736036 − 1.27485i −0.954267 − 0.298955i \(-0.903362\pi\)
0.218231 − 0.975897i \(-0.429971\pi\)
\(32\) 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 \(-0.230118\pi\)
−0.947886 + 0.318608i \(0.896785\pi\)
\(42\) 0 0
\(43\) 0.732051 − 1.26795i 0.111637 − 0.193360i −0.804794 − 0.593555i \(-0.797724\pi\)
0.916430 + 0.400194i \(0.131057\pi\)
\(44\) 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 \(-0.945495\pi\)
0.640255 + 0.768162i \(0.278829\pi\)
\(48\) 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 \(-0.725230\pi\)
−0.649997 + 0.759937i \(0.725230\pi\)
\(54\) 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 \(-0.254731\pi\)
−0.969666 + 0.244435i \(0.921398\pi\)
\(60\) 0 0
\(61\) −1.96410 + 3.40192i −0.251477 + 0.435572i −0.963933 − 0.266146i \(-0.914250\pi\)
0.712455 + 0.701717i \(0.247583\pi\)
\(62\) −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 0.995024 − 0.0996351i \(-0.0317676\pi\)
−0.583799 + 0.811899i \(0.698434\pi\)
\(68\) −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 \(-0.373223\pi\)
0.387834 + 0.921729i \(0.373223\pi\)
\(72\) 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 −0.486596 − 0.873627i \(-0.661762\pi\)
0.999881 − 0.0154089i \(-0.00490499\pi\)
\(80\) 0.267949 0.0299576
\(81\) 0 0
\(82\) −2.53590 −0.280043
\(83\) −8.29423 + 14.3660i −0.910410 + 1.57688i −0.0969238 + 0.995292i \(0.530900\pi\)
−0.813486 + 0.581584i \(0.802433\pi\)
\(84\) 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 \(-0.323619\pi\)
0.526194 + 0.850365i \(0.323619\pi\)
\(90\) 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 0.443124 + 0.896460i \(0.353870\pi\)
−0.997919 + 0.0644736i \(0.979463\pi\)
\(98\) −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.s.379.2 4
3.2 odd 2 1134.2.f.r.379.1 4
9.2 odd 6 1134.2.a.m.1.2 yes 2
9.4 even 3 inner 1134.2.f.s.757.2 4
9.5 odd 6 1134.2.f.r.757.1 4
9.7 even 3 1134.2.a.l.1.1 ✓ 2
36.7 odd 6 9072.2.a.bp.1.1 2
36.11 even 6 9072.2.a.y.1.2 2
63.20 even 6 7938.2.a.bt.1.1 2
63.34 odd 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 9.7 even 3
1134.2.a.m.1.2 yes 2 9.2 odd 6
1134.2.f.r.379.1 4 3.2 odd 2
1134.2.f.r.757.1 4 9.5 odd 6
1134.2.f.s.379.2 4 1.1 even 1 trivial
1134.2.f.s.757.2 4 9.4 even 3 inner
7938.2.a.bg.1.2 2 63.34 odd 6
7938.2.a.bt.1.1 2 63.20 even 6
9072.2.a.y.1.2 2 36.11 even 6
9072.2.a.bp.1.1 2 36.7 odd 6