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,-6,0,-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: \(0\)
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 42)
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.a.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)
 
\(f(q)\) \(=\) \(q+(-0.500000 - 0.866025i) q^{2} +(-0.500000 + 0.866025i) q^{4} -3.00000 q^{5} +(-2.00000 + 1.73205i) q^{7} +1.00000 q^{8} +(1.50000 + 2.59808i) q^{10} -3.00000 q^{11} +(2.00000 + 3.46410i) q^{13} +(2.50000 + 0.866025i) q^{14} +(-0.500000 - 0.866025i) q^{16} +(2.00000 - 3.46410i) q^{19} +(1.50000 - 2.59808i) q^{20} +(1.50000 + 2.59808i) q^{22} +4.00000 q^{25} +(2.00000 - 3.46410i) q^{26} +(-0.500000 - 2.59808i) q^{28} +(4.50000 - 7.79423i) q^{29} +(0.500000 - 0.866025i) q^{31} +(-0.500000 + 0.866025i) q^{32} +(6.00000 - 5.19615i) q^{35} +(-4.00000 + 6.92820i) q^{37} -4.00000 q^{38} -3.00000 q^{40} +(5.00000 - 8.66025i) q^{43} +(1.50000 - 2.59808i) q^{44} +(-3.00000 - 5.19615i) q^{47} +(1.00000 - 6.92820i) q^{49} +(-2.00000 - 3.46410i) q^{50} -4.00000 q^{52} +(-1.50000 - 2.59808i) q^{53} +9.00000 q^{55} +(-2.00000 + 1.73205i) q^{56} -9.00000 q^{58} +(1.50000 - 2.59808i) q^{59} +(5.00000 + 8.66025i) q^{61} -1.00000 q^{62} +1.00000 q^{64} +(-6.00000 - 10.3923i) q^{65} +(5.00000 - 8.66025i) q^{67} +(-7.50000 - 2.59808i) q^{70} +6.00000 q^{71} +(-1.00000 - 1.73205i) q^{73} +8.00000 q^{74} +(2.00000 + 3.46410i) q^{76} +(6.00000 - 5.19615i) q^{77} +(0.500000 + 0.866025i) q^{79} +(1.50000 + 2.59808i) q^{80} +(-4.50000 + 7.79423i) q^{83} -10.0000 q^{86} -3.00000 q^{88} +(3.00000 - 5.19615i) q^{89} +(-10.0000 - 3.46410i) q^{91} +(-3.00000 + 5.19615i) q^{94} +(-6.00000 + 10.3923i) q^{95} +(0.500000 - 0.866025i) q^{97} +(-6.50000 + 2.59808i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} - q^{4} - 6 q^{5} - 4 q^{7} + 2 q^{8} + 3 q^{10} - 6 q^{11} + 4 q^{13} + 5 q^{14} - q^{16} + 4 q^{19} + 3 q^{20} + 3 q^{22} + 8 q^{25} + 4 q^{26} - q^{28} + 9 q^{29} + q^{31} - q^{32}+ \cdots - 13 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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
\(5\) −3.00000 −1.34164 −0.670820 − 0.741620i \(-0.734058\pi\)
−0.670820 + 0.741620i \(0.734058\pi\)
\(6\) 0 0
\(7\) −2.00000 + 1.73205i −0.755929 + 0.654654i
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) 1.50000 + 2.59808i 0.474342 + 0.821584i
\(11\) −3.00000 −0.904534 −0.452267 − 0.891883i \(-0.649385\pi\)
−0.452267 + 0.891883i \(0.649385\pi\)
\(12\) 0 0
\(13\) 2.00000 + 3.46410i 0.554700 + 0.960769i 0.997927 + 0.0643593i \(0.0205004\pi\)
−0.443227 + 0.896410i \(0.646166\pi\)
\(14\) 2.50000 + 0.866025i 0.668153 + 0.231455i
\(15\) 0 0
\(16\) −0.500000 − 0.866025i −0.125000 − 0.216506i
\(17\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(18\) 0 0
\(19\) 2.00000 − 3.46410i 0.458831 − 0.794719i −0.540068 − 0.841621i \(-0.681602\pi\)
0.998899 + 0.0469020i \(0.0149348\pi\)
\(20\) 1.50000 − 2.59808i 0.335410 − 0.580948i
\(21\) 0 0
\(22\) 1.50000 + 2.59808i 0.319801 + 0.553912i
\(23\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) 4.00000 0.800000
\(26\) 2.00000 − 3.46410i 0.392232 − 0.679366i
\(27\) 0 0
\(28\) −0.500000 − 2.59808i −0.0944911 − 0.490990i
\(29\) 4.50000 − 7.79423i 0.835629 − 1.44735i −0.0578882 − 0.998323i \(-0.518437\pi\)
0.893517 − 0.449029i \(-0.148230\pi\)
\(30\) 0 0
\(31\) 0.500000 − 0.866025i 0.0898027 − 0.155543i −0.817625 − 0.575751i \(-0.804710\pi\)
0.907428 + 0.420208i \(0.138043\pi\)
\(32\) −0.500000 + 0.866025i −0.0883883 + 0.153093i
\(33\) 0 0
\(34\) 0 0
\(35\) 6.00000 − 5.19615i 1.01419 − 0.878310i
\(36\) 0 0
\(37\) −4.00000 + 6.92820i −0.657596 + 1.13899i 0.323640 + 0.946180i \(0.395093\pi\)
−0.981236 + 0.192809i \(0.938240\pi\)
\(38\) −4.00000 −0.648886
\(39\) 0 0
\(40\) −3.00000 −0.474342
\(41\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(42\) 0 0
\(43\) 5.00000 − 8.66025i 0.762493 − 1.32068i −0.179069 − 0.983836i \(-0.557309\pi\)
0.941562 − 0.336840i \(-0.109358\pi\)
\(44\) 1.50000 − 2.59808i 0.226134 − 0.391675i
\(45\) 0 0
\(46\) 0 0
\(47\) −3.00000 − 5.19615i −0.437595 − 0.757937i 0.559908 − 0.828554i \(-0.310836\pi\)
−0.997503 + 0.0706177i \(0.977503\pi\)
\(48\) 0 0
\(49\) 1.00000 − 6.92820i 0.142857 − 0.989743i
\(50\) −2.00000 − 3.46410i −0.282843 − 0.489898i
\(51\) 0 0
\(52\) −4.00000 −0.554700
\(53\) −1.50000 − 2.59808i −0.206041 − 0.356873i 0.744423 − 0.667708i \(-0.232725\pi\)
−0.950464 + 0.310835i \(0.899391\pi\)
\(54\) 0 0
\(55\) 9.00000 1.21356
\(56\) −2.00000 + 1.73205i −0.267261 + 0.231455i
\(57\) 0 0
\(58\) −9.00000 −1.18176
\(59\) 1.50000 − 2.59808i 0.195283 − 0.338241i −0.751710 − 0.659494i \(-0.770771\pi\)
0.946993 + 0.321253i \(0.104104\pi\)
\(60\) 0 0
\(61\) 5.00000 + 8.66025i 0.640184 + 1.10883i 0.985391 + 0.170305i \(0.0544754\pi\)
−0.345207 + 0.938527i \(0.612191\pi\)
\(62\) −1.00000 −0.127000
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −6.00000 − 10.3923i −0.744208 − 1.28901i
\(66\) 0 0
\(67\) 5.00000 − 8.66025i 0.610847 − 1.05802i −0.380251 − 0.924883i \(-0.624162\pi\)
0.991098 − 0.133135i \(-0.0425044\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) −7.50000 − 2.59808i −0.896421 − 0.310530i
\(71\) 6.00000 0.712069 0.356034 − 0.934473i \(-0.384129\pi\)
0.356034 + 0.934473i \(0.384129\pi\)
\(72\) 0 0
\(73\) −1.00000 − 1.73205i −0.117041 − 0.202721i 0.801553 − 0.597924i \(-0.204008\pi\)
−0.918594 + 0.395203i \(0.870674\pi\)
\(74\) 8.00000 0.929981
\(75\) 0 0
\(76\) 2.00000 + 3.46410i 0.229416 + 0.397360i
\(77\) 6.00000 − 5.19615i 0.683763 − 0.592157i
\(78\) 0 0
\(79\) 0.500000 + 0.866025i 0.0562544 + 0.0974355i 0.892781 − 0.450490i \(-0.148751\pi\)
−0.836527 + 0.547926i \(0.815418\pi\)
\(80\) 1.50000 + 2.59808i 0.167705 + 0.290474i
\(81\) 0 0
\(82\) 0 0
\(83\) −4.50000 + 7.79423i −0.493939 + 0.855528i −0.999976 − 0.00698436i \(-0.997777\pi\)
0.506036 + 0.862512i \(0.331110\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −10.0000 −1.07833
\(87\) 0 0
\(88\) −3.00000 −0.319801
\(89\) 3.00000 − 5.19615i 0.317999 − 0.550791i −0.662071 − 0.749441i \(-0.730322\pi\)
0.980071 + 0.198650i \(0.0636557\pi\)
\(90\) 0 0
\(91\) −10.0000 − 3.46410i −1.04828 − 0.363137i
\(92\) 0 0
\(93\) 0 0
\(94\) −3.00000 + 5.19615i −0.309426 + 0.535942i
\(95\) −6.00000 + 10.3923i −0.615587 + 1.06623i
\(96\) 0 0
\(97\) 0.500000 − 0.866025i 0.0507673 − 0.0879316i −0.839525 − 0.543321i \(-0.817167\pi\)
0.890292 + 0.455389i \(0.150500\pi\)
\(98\) −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.a.541.1 2
3.2 odd 2 1134.2.h.p.541.1 2
7.4 even 3 1134.2.e.p.865.1 2
9.2 odd 6 42.2.e.b.37.1 yes 2
9.4 even 3 1134.2.e.p.919.1 2
9.5 odd 6 1134.2.e.a.919.1 2
9.7 even 3 126.2.g.b.37.1 2
21.11 odd 6 1134.2.e.a.865.1 2
36.7 odd 6 1008.2.s.n.289.1 2
36.11 even 6 336.2.q.d.289.1 2
45.2 even 12 1050.2.o.b.499.2 4
45.29 odd 6 1050.2.i.e.751.1 2
45.38 even 12 1050.2.o.b.499.1 4
63.2 odd 6 294.2.a.d.1.1 1
63.4 even 3 inner 1134.2.h.a.109.1 2
63.11 odd 6 42.2.e.b.25.1 ✓ 2
63.16 even 3 882.2.a.g.1.1 1
63.20 even 6 294.2.e.f.79.1 2
63.25 even 3 126.2.g.b.109.1 2
63.32 odd 6 1134.2.h.p.109.1 2
63.34 odd 6 882.2.g.b.667.1 2
63.38 even 6 294.2.e.f.67.1 2
63.47 even 6 294.2.a.a.1.1 1
63.52 odd 6 882.2.g.b.361.1 2
63.61 odd 6 882.2.a.k.1.1 1
72.11 even 6 1344.2.q.j.961.1 2
72.29 odd 6 1344.2.q.v.961.1 2
252.11 even 6 336.2.q.d.193.1 2
252.47 odd 6 2352.2.a.n.1.1 1
252.79 odd 6 7056.2.a.g.1.1 1
252.83 odd 6 2352.2.q.m.961.1 2
252.151 odd 6 1008.2.s.n.865.1 2
252.187 even 6 7056.2.a.bz.1.1 1
252.191 even 6 2352.2.a.m.1.1 1
252.227 odd 6 2352.2.q.m.1537.1 2
315.74 odd 6 1050.2.i.e.151.1 2
315.137 even 12 1050.2.o.b.949.1 4
315.254 odd 6 7350.2.a.ce.1.1 1
315.263 even 12 1050.2.o.b.949.2 4
315.299 even 6 7350.2.a.cw.1.1 1
504.11 even 6 1344.2.q.j.193.1 2
504.173 even 6 9408.2.a.db.1.1 1
504.299 odd 6 9408.2.a.bm.1.1 1
504.317 odd 6 9408.2.a.d.1.1 1
504.389 odd 6 1344.2.q.v.193.1 2
504.443 even 6 9408.2.a.bu.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
42.2.e.b.25.1 ✓ 2 63.11 odd 6
42.2.e.b.37.1 yes 2 9.2 odd 6
126.2.g.b.37.1 2 9.7 even 3
126.2.g.b.109.1 2 63.25 even 3
294.2.a.a.1.1 1 63.47 even 6
294.2.a.d.1.1 1 63.2 odd 6
294.2.e.f.67.1 2 63.38 even 6
294.2.e.f.79.1 2 63.20 even 6
336.2.q.d.193.1 2 252.11 even 6
336.2.q.d.289.1 2 36.11 even 6
882.2.a.g.1.1 1 63.16 even 3
882.2.a.k.1.1 1 63.61 odd 6
882.2.g.b.361.1 2 63.52 odd 6
882.2.g.b.667.1 2 63.34 odd 6
1008.2.s.n.289.1 2 36.7 odd 6
1008.2.s.n.865.1 2 252.151 odd 6
1050.2.i.e.151.1 2 315.74 odd 6
1050.2.i.e.751.1 2 45.29 odd 6
1050.2.o.b.499.1 4 45.38 even 12
1050.2.o.b.499.2 4 45.2 even 12
1050.2.o.b.949.1 4 315.137 even 12
1050.2.o.b.949.2 4 315.263 even 12
1134.2.e.a.865.1 2 21.11 odd 6
1134.2.e.a.919.1 2 9.5 odd 6
1134.2.e.p.865.1 2 7.4 even 3
1134.2.e.p.919.1 2 9.4 even 3
1134.2.h.a.109.1 2 63.4 even 3 inner
1134.2.h.a.541.1 2 1.1 even 1 trivial
1134.2.h.p.109.1 2 63.32 odd 6
1134.2.h.p.541.1 2 3.2 odd 2
1344.2.q.j.193.1 2 504.11 even 6
1344.2.q.j.961.1 2 72.11 even 6
1344.2.q.v.193.1 2 504.389 odd 6
1344.2.q.v.961.1 2 72.29 odd 6
2352.2.a.m.1.1 1 252.191 even 6
2352.2.a.n.1.1 1 252.47 odd 6
2352.2.q.m.961.1 2 252.83 odd 6
2352.2.q.m.1537.1 2 252.227 odd 6
7056.2.a.g.1.1 1 252.79 odd 6
7056.2.a.bz.1.1 1 252.187 even 6
7350.2.a.ce.1.1 1 315.254 odd 6
7350.2.a.cw.1.1 1 315.299 even 6
9408.2.a.d.1.1 1 504.317 odd 6
9408.2.a.bm.1.1 1 504.299 odd 6
9408.2.a.bu.1.1 1 504.443 even 6
9408.2.a.db.1.1 1 504.173 even 6