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(163,1134)] 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("1134.163"); 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([0, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1134 = 2 \cdot 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1134.g (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,-1,0,-1,0,0,-1] 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: \(1\)
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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 163.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 1134.163
Dual form 1134.2.g.b.487.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.500000 + 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{4} +(-0.500000 + 2.59808i) q^{7} +1.00000 q^{8} -4.00000 q^{13} +(-2.00000 - 1.73205i) q^{14} +(-0.500000 + 0.866025i) q^{16} +(-3.00000 - 5.19615i) q^{17} +(-1.00000 + 1.73205i) q^{19} +(1.50000 - 2.59808i) q^{23} +(2.50000 + 4.33013i) q^{25} +(2.00000 - 3.46410i) q^{26} +(2.50000 - 0.866025i) q^{28} -6.00000 q^{29} +(-2.50000 - 4.33013i) q^{31} +(-0.500000 - 0.866025i) q^{32} +6.00000 q^{34} +(-4.00000 + 6.92820i) q^{37} +(-1.00000 - 1.73205i) q^{38} +3.00000 q^{41} +2.00000 q^{43} +(1.50000 + 2.59808i) q^{46} +(1.50000 - 2.59808i) q^{47} +(-6.50000 - 2.59808i) q^{49} -5.00000 q^{50} +(2.00000 + 3.46410i) q^{52} +(-3.00000 - 5.19615i) q^{53} +(-0.500000 + 2.59808i) q^{56} +(3.00000 - 5.19615i) q^{58} +(-6.00000 - 10.3923i) q^{59} +(-4.00000 + 6.92820i) q^{61} +5.00000 q^{62} +1.00000 q^{64} +(-4.00000 - 6.92820i) q^{67} +(-3.00000 + 5.19615i) q^{68} -15.0000 q^{71} +(-5.50000 - 9.52628i) q^{73} +(-4.00000 - 6.92820i) q^{74} +2.00000 q^{76} +(0.500000 - 0.866025i) q^{79} +(-1.50000 + 2.59808i) q^{82} +(-1.00000 + 1.73205i) q^{86} +(-4.50000 + 7.79423i) q^{89} +(2.00000 - 10.3923i) q^{91} -3.00000 q^{92} +(1.50000 + 2.59808i) q^{94} +2.00000 q^{97} +(5.50000 - 4.33013i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} - q^{4} - q^{7} + 2 q^{8} - 8 q^{13} - 4 q^{14} - q^{16} - 6 q^{17} - 2 q^{19} + 3 q^{23} + 5 q^{25} + 4 q^{26} + 5 q^{28} - 12 q^{29} - 5 q^{31} - q^{32} + 12 q^{34} - 8 q^{37} - 2 q^{38}+ \cdots + 11 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.500000 + 0.866025i −0.353553 + 0.612372i
\(3\) 0 0
\(4\) −0.500000 − 0.866025i −0.250000 − 0.433013i
\(5\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(6\) 0 0
\(7\) −0.500000 + 2.59808i −0.188982 + 0.981981i
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) 0 0
\(11\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(12\) 0 0
\(13\) −4.00000 −1.10940 −0.554700 − 0.832050i \(-0.687167\pi\)
−0.554700 + 0.832050i \(0.687167\pi\)
\(14\) −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 −0.957892 − 0.287129i \(-0.907299\pi\)
0.230285 − 0.973123i \(-0.426034\pi\)
\(18\) 0 0
\(19\) −1.00000 + 1.73205i −0.229416 + 0.397360i −0.957635 − 0.287984i \(-0.907015\pi\)
0.728219 + 0.685344i \(0.240348\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 1.50000 − 2.59808i 0.312772 − 0.541736i −0.666190 − 0.745782i \(-0.732076\pi\)
0.978961 + 0.204046i \(0.0654092\pi\)
\(24\) 0 0
\(25\) 2.50000 + 4.33013i 0.500000 + 0.866025i
\(26\) 2.00000 − 3.46410i 0.392232 − 0.679366i
\(27\) 0 0
\(28\) 2.50000 − 0.866025i 0.472456 − 0.163663i
\(29\) −6.00000 −1.11417 −0.557086 − 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 0 0
\(31\) −2.50000 − 4.33013i −0.449013 − 0.777714i 0.549309 − 0.835619i \(-0.314891\pi\)
−0.998322 + 0.0579057i \(0.981558\pi\)
\(32\) −0.500000 − 0.866025i −0.0883883 − 0.153093i
\(33\) 0 0
\(34\) 6.00000 1.02899
\(35\) 0 0
\(36\) 0 0
\(37\) −4.00000 + 6.92820i −0.657596 + 1.13899i 0.323640 + 0.946180i \(0.395093\pi\)
−0.981236 + 0.192809i \(0.938240\pi\)
\(38\) −1.00000 − 1.73205i −0.162221 − 0.280976i
\(39\) 0 0
\(40\) 0 0
\(41\) 3.00000 0.468521 0.234261 − 0.972174i \(-0.424733\pi\)
0.234261 + 0.972174i \(0.424733\pi\)
\(42\) 0 0
\(43\) 2.00000 0.304997 0.152499 − 0.988304i \(-0.451268\pi\)
0.152499 + 0.988304i \(0.451268\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 1.50000 + 2.59808i 0.221163 + 0.383065i
\(47\) 1.50000 − 2.59808i 0.218797 − 0.378968i −0.735643 − 0.677369i \(-0.763120\pi\)
0.954441 + 0.298401i \(0.0964533\pi\)
\(48\) 0 0
\(49\) −6.50000 − 2.59808i −0.928571 − 0.371154i
\(50\) −5.00000 −0.707107
\(51\) 0 0
\(52\) 2.00000 + 3.46410i 0.277350 + 0.480384i
\(53\) −3.00000 − 5.19615i −0.412082 − 0.713746i 0.583036 − 0.812447i \(-0.301865\pi\)
−0.995117 + 0.0987002i \(0.968532\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −0.500000 + 2.59808i −0.0668153 + 0.347183i
\(57\) 0 0
\(58\) 3.00000 − 5.19615i 0.393919 − 0.682288i
\(59\) −6.00000 − 10.3923i −0.781133 − 1.35296i −0.931282 − 0.364299i \(-0.881308\pi\)
0.150148 − 0.988663i \(-0.452025\pi\)
\(60\) 0 0
\(61\) −4.00000 + 6.92820i −0.512148 + 0.887066i 0.487753 + 0.872982i \(0.337817\pi\)
−0.999901 + 0.0140840i \(0.995517\pi\)
\(62\) 5.00000 0.635001
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) −4.00000 − 6.92820i −0.488678 − 0.846415i 0.511237 − 0.859440i \(-0.329187\pi\)
−0.999915 + 0.0130248i \(0.995854\pi\)
\(68\) −3.00000 + 5.19615i −0.363803 + 0.630126i
\(69\) 0 0
\(70\) 0 0
\(71\) −15.0000 −1.78017 −0.890086 − 0.455792i \(-0.849356\pi\)
−0.890086 + 0.455792i \(0.849356\pi\)
\(72\) 0 0
\(73\) −5.50000 − 9.52628i −0.643726 − 1.11497i −0.984594 − 0.174855i \(-0.944054\pi\)
0.340868 − 0.940111i \(-0.389279\pi\)
\(74\) −4.00000 − 6.92820i −0.464991 − 0.805387i
\(75\) 0 0
\(76\) 2.00000 0.229416
\(77\) 0 0
\(78\) 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
\(82\) −1.50000 + 2.59808i −0.165647 + 0.286910i
\(83\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −1.00000 + 1.73205i −0.107833 + 0.186772i
\(87\) 0 0
\(88\) 0 0
\(89\) −4.50000 + 7.79423i −0.476999 + 0.826187i −0.999653 − 0.0263586i \(-0.991609\pi\)
0.522654 + 0.852545i \(0.324942\pi\)
\(90\) 0 0
\(91\) 2.00000 − 10.3923i 0.209657 − 1.08941i
\(92\) −3.00000 −0.312772
\(93\) 0 0
\(94\) 1.50000 + 2.59808i 0.154713 + 0.267971i
\(95\) 0 0
\(96\) 0 0
\(97\) 2.00000 0.203069 0.101535 − 0.994832i \(-0.467625\pi\)
0.101535 + 0.994832i \(0.467625\pi\)
\(98\) 5.50000 − 4.33013i 0.555584 − 0.437409i
\(99\) 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.g.b.163.1 ✓ 2
3.2 odd 2 1134.2.g.g.163.1 yes 2
7.2 even 3 7938.2.a.y.1.1 1
7.4 even 3 inner 1134.2.g.b.487.1 yes 2
7.5 odd 6 7938.2.a.z.1.1 1
9.2 odd 6 1134.2.e.d.919.1 2
9.4 even 3 1134.2.h.c.541.1 2
9.5 odd 6 1134.2.h.m.541.1 2
9.7 even 3 1134.2.e.n.919.1 2
21.2 odd 6 7938.2.a.g.1.1 1
21.5 even 6 7938.2.a.h.1.1 1
21.11 odd 6 1134.2.g.g.487.1 yes 2
63.4 even 3 1134.2.e.n.865.1 2
63.11 odd 6 1134.2.h.m.109.1 2
63.25 even 3 1134.2.h.c.109.1 2
63.32 odd 6 1134.2.e.d.865.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1134.2.e.d.865.1 2 63.32 odd 6
1134.2.e.d.919.1 2 9.2 odd 6
1134.2.e.n.865.1 2 63.4 even 3
1134.2.e.n.919.1 2 9.7 even 3
1134.2.g.b.163.1 ✓ 2 1.1 even 1 trivial
1134.2.g.b.487.1 yes 2 7.4 even 3 inner
1134.2.g.g.163.1 yes 2 3.2 odd 2
1134.2.g.g.487.1 yes 2 21.11 odd 6
1134.2.h.c.109.1 2 63.25 even 3
1134.2.h.c.541.1 2 9.4 even 3
1134.2.h.m.109.1 2 63.11 odd 6
1134.2.h.m.541.1 2 9.5 odd 6
7938.2.a.g.1.1 1 21.2 odd 6
7938.2.a.h.1.1 1 21.5 even 6
7938.2.a.y.1.1 1 7.2 even 3
7938.2.a.z.1.1 1 7.5 odd 6