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,-4] 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 541.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 1134.541
Dual form 1134.2.h.c.109.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.500000 - 0.866025i) q^{2} +(-0.500000 + 0.866025i) q^{4} +(-2.00000 - 1.73205i) q^{7} +1.00000 q^{8} +(2.00000 + 3.46410i) q^{13} +(-0.500000 + 2.59808i) q^{14} +(-0.500000 - 0.866025i) q^{16} +(-3.00000 - 5.19615i) q^{17} +(-1.00000 + 1.73205i) q^{19} -3.00000 q^{23} -5.00000 q^{25} +(2.00000 - 3.46410i) q^{26} +(2.50000 - 0.866025i) q^{28} +(3.00000 - 5.19615i) q^{29} +(-2.50000 + 4.33013i) q^{31} +(-0.500000 + 0.866025i) q^{32} +(-3.00000 + 5.19615i) q^{34} +(-4.00000 + 6.92820i) q^{37} +2.00000 q^{38} +(-1.50000 - 2.59808i) q^{41} +(-1.00000 + 1.73205i) q^{43} +(1.50000 + 2.59808i) q^{46} +(1.50000 + 2.59808i) q^{47} +(1.00000 + 6.92820i) q^{49} +(2.50000 + 4.33013i) q^{50} -4.00000 q^{52} +(-3.00000 - 5.19615i) q^{53} +(-2.00000 - 1.73205i) q^{56} -6.00000 q^{58} +(-6.00000 + 10.3923i) q^{59} +(-4.00000 - 6.92820i) q^{61} +5.00000 q^{62} +1.00000 q^{64} +(-4.00000 + 6.92820i) q^{67} +6.00000 q^{68} -15.0000 q^{71} +(-5.50000 - 9.52628i) q^{73} +8.00000 q^{74} +(-1.00000 - 1.73205i) q^{76} +(0.500000 + 0.866025i) q^{79} +(-1.50000 + 2.59808i) q^{82} +2.00000 q^{86} +(-4.50000 + 7.79423i) q^{89} +(2.00000 - 10.3923i) q^{91} +(1.50000 - 2.59808i) q^{92} +(1.50000 - 2.59808i) q^{94} +(-1.00000 + 1.73205i) q^{97} +(5.50000 - 4.33013i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} - q^{4} - 4 q^{7} + 2 q^{8} + 4 q^{13} - q^{14} - q^{16} - 6 q^{17} - 2 q^{19} - 6 q^{23} - 10 q^{25} + 4 q^{26} + 5 q^{28} + 6 q^{29} - 5 q^{31} - q^{32} - 6 q^{34} - 8 q^{37} + 4 q^{38}+ \cdots + 11 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(6\) 0 0
\(7\) −2.00000 − 1.73205i −0.755929 − 0.654654i
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) 0 0
\(11\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) 2.00000 + 3.46410i 0.554700 + 0.960769i 0.997927 + 0.0643593i \(0.0205004\pi\)
−0.443227 + 0.896410i \(0.646166\pi\)
\(14\) −0.500000 + 2.59808i −0.133631 + 0.694365i
\(15\) 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
\(23\) −3.00000 −0.625543 −0.312772 − 0.949828i \(-0.601257\pi\)
−0.312772 + 0.949828i \(0.601257\pi\)
\(24\) 0 0
\(25\) −5.00000 −1.00000
\(26\) 2.00000 − 3.46410i 0.392232 − 0.679366i
\(27\) 0 0
\(28\) 2.50000 − 0.866025i 0.472456 − 0.163663i
\(29\) 3.00000 − 5.19615i 0.557086 − 0.964901i −0.440652 − 0.897678i \(-0.645253\pi\)
0.997738 − 0.0672232i \(-0.0214140\pi\)
\(30\) 0 0
\(31\) −2.50000 + 4.33013i −0.449013 + 0.777714i −0.998322 − 0.0579057i \(-0.981558\pi\)
0.549309 + 0.835619i \(0.314891\pi\)
\(32\) −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
\(37\) −4.00000 + 6.92820i −0.657596 + 1.13899i 0.323640 + 0.946180i \(0.395093\pi\)
−0.981236 + 0.192809i \(0.938240\pi\)
\(38\) 2.00000 0.324443
\(39\) 0 0
\(40\) 0 0
\(41\) −1.50000 − 2.59808i −0.234261 − 0.405751i 0.724797 − 0.688963i \(-0.241934\pi\)
−0.959058 + 0.283211i \(0.908600\pi\)
\(42\) 0 0
\(43\) −1.00000 + 1.73205i −0.152499 + 0.264135i −0.932145 − 0.362084i \(-0.882065\pi\)
0.779647 + 0.626219i \(0.215399\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 1.50000 + 2.59808i 0.221163 + 0.383065i
\(47\) 1.50000 + 2.59808i 0.218797 + 0.378968i 0.954441 − 0.298401i \(-0.0964533\pi\)
−0.735643 + 0.677369i \(0.763120\pi\)
\(48\) 0 0
\(49\) 1.00000 + 6.92820i 0.142857 + 0.989743i
\(50\) 2.50000 + 4.33013i 0.353553 + 0.612372i
\(51\) 0 0
\(52\) −4.00000 −0.554700
\(53\) −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
\(56\) −2.00000 − 1.73205i −0.267261 − 0.231455i
\(57\) 0 0
\(58\) −6.00000 −0.787839
\(59\) −6.00000 + 10.3923i −0.781133 + 1.35296i 0.150148 + 0.988663i \(0.452025\pi\)
−0.931282 + 0.364299i \(0.881308\pi\)
\(60\) 0 0
\(61\) −4.00000 − 6.92820i −0.512148 − 0.887066i −0.999901 − 0.0140840i \(-0.995517\pi\)
0.487753 − 0.872982i \(-0.337817\pi\)
\(62\) 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 −0.999915 − 0.0130248i \(-0.995854\pi\)
0.511237 + 0.859440i \(0.329187\pi\)
\(68\) 6.00000 0.727607
\(69\) 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 \(-0.389279\pi\)
\(74\) 8.00000 0.929981
\(75\) 0 0
\(76\) −1.00000 − 1.73205i −0.114708 − 0.198680i
\(77\) 0 0
\(78\) 0 0
\(79\) 0.500000 + 0.866025i 0.0562544 + 0.0974355i 0.892781 − 0.450490i \(-0.148751\pi\)
−0.836527 + 0.547926i \(0.815418\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −1.50000 + 2.59808i −0.165647 + 0.286910i
\(83\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 2.00000 0.215666
\(87\) 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
\(92\) 1.50000 − 2.59808i 0.156386 − 0.270868i
\(93\) 0 0
\(94\) 1.50000 − 2.59808i 0.154713 − 0.267971i
\(95\) 0 0
\(96\) 0 0
\(97\) −1.00000 + 1.73205i −0.101535 + 0.175863i −0.912317 − 0.409484i \(-0.865709\pi\)
0.810782 + 0.585348i \(0.199042\pi\)
\(98\) 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.h.c.541.1 2
3.2 odd 2 1134.2.h.m.541.1 2
7.4 even 3 1134.2.e.n.865.1 2
9.2 odd 6 1134.2.g.g.163.1 yes 2
9.4 even 3 1134.2.e.n.919.1 2
9.5 odd 6 1134.2.e.d.919.1 2
9.7 even 3 1134.2.g.b.163.1 ✓ 2
21.11 odd 6 1134.2.e.d.865.1 2
63.2 odd 6 7938.2.a.g.1.1 1
63.4 even 3 inner 1134.2.h.c.109.1 2
63.11 odd 6 1134.2.g.g.487.1 yes 2
63.16 even 3 7938.2.a.y.1.1 1
63.25 even 3 1134.2.g.b.487.1 yes 2
63.32 odd 6 1134.2.h.m.109.1 2
63.47 even 6 7938.2.a.h.1.1 1
63.61 odd 6 7938.2.a.z.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1134.2.e.d.865.1 2 21.11 odd 6
1134.2.e.d.919.1 2 9.5 odd 6
1134.2.e.n.865.1 2 7.4 even 3
1134.2.e.n.919.1 2 9.4 even 3
1134.2.g.b.163.1 ✓ 2 9.7 even 3
1134.2.g.b.487.1 yes 2 63.25 even 3
1134.2.g.g.163.1 yes 2 9.2 odd 6
1134.2.g.g.487.1 yes 2 63.11 odd 6
1134.2.h.c.109.1 2 63.4 even 3 inner
1134.2.h.c.541.1 2 1.1 even 1 trivial
1134.2.h.m.109.1 2 63.32 odd 6
1134.2.h.m.541.1 2 3.2 odd 2
7938.2.a.g.1.1 1 63.2 odd 6
7938.2.a.h.1.1 1 63.47 even 6
7938.2.a.y.1.1 1 63.16 even 3
7938.2.a.z.1.1 1 63.61 odd 6