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 = [2,1,0,-1,-2,0,1,-2,0,-4,-4,0,-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: \(2\)
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 757.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 1134.757
Dual form 1134.2.f.j.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)
 
\(f(q)\) \(=\) \(q+(0.500000 + 0.866025i) q^{2} +(-0.500000 + 0.866025i) q^{4} +(-1.00000 + 1.73205i) q^{5} +(0.500000 + 0.866025i) q^{7} -1.00000 q^{8} -2.00000 q^{10} +(-2.00000 - 3.46410i) q^{11} +(-3.00000 + 5.19615i) q^{13} +(-0.500000 + 0.866025i) q^{14} +(-0.500000 - 0.866025i) q^{16} -2.00000 q^{17} -4.00000 q^{19} +(-1.00000 - 1.73205i) q^{20} +(2.00000 - 3.46410i) q^{22} +(4.00000 - 6.92820i) q^{23} +(0.500000 + 0.866025i) q^{25} -6.00000 q^{26} -1.00000 q^{28} +(-1.00000 - 1.73205i) q^{29} +(0.500000 - 0.866025i) q^{32} +(-1.00000 - 1.73205i) q^{34} -2.00000 q^{35} -10.0000 q^{37} +(-2.00000 - 3.46410i) q^{38} +(1.00000 - 1.73205i) q^{40} +(-3.00000 + 5.19615i) q^{41} +(2.00000 + 3.46410i) q^{43} +4.00000 q^{44} +8.00000 q^{46} +(-0.500000 + 0.866025i) q^{49} +(-0.500000 + 0.866025i) q^{50} +(-3.00000 - 5.19615i) q^{52} -6.00000 q^{53} +8.00000 q^{55} +(-0.500000 - 0.866025i) q^{56} +(1.00000 - 1.73205i) q^{58} +(2.00000 - 3.46410i) q^{59} +(-3.00000 - 5.19615i) q^{61} +1.00000 q^{64} +(-6.00000 - 10.3923i) q^{65} +(-2.00000 + 3.46410i) q^{67} +(1.00000 - 1.73205i) q^{68} +(-1.00000 - 1.73205i) q^{70} -8.00000 q^{71} +10.0000 q^{73} +(-5.00000 - 8.66025i) q^{74} +(2.00000 - 3.46410i) q^{76} +(2.00000 - 3.46410i) q^{77} +2.00000 q^{80} -6.00000 q^{82} +(-2.00000 - 3.46410i) q^{83} +(2.00000 - 3.46410i) q^{85} +(-2.00000 + 3.46410i) q^{86} +(2.00000 + 3.46410i) q^{88} +6.00000 q^{89} -6.00000 q^{91} +(4.00000 + 6.92820i) q^{92} +(4.00000 - 6.92820i) q^{95} +(7.00000 + 12.1244i) q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} - q^{4} - 2 q^{5} + q^{7} - 2 q^{8} - 4 q^{10} - 4 q^{11} - 6 q^{13} - q^{14} - q^{16} - 4 q^{17} - 8 q^{19} - 2 q^{20} + 4 q^{22} + 8 q^{23} + q^{25} - 12 q^{26} - 2 q^{28} - 2 q^{29}+ \cdots - 2 q^{98}+O(q^{100}) \) 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
\(5\) −1.00000 + 1.73205i −0.447214 + 0.774597i −0.998203 − 0.0599153i \(-0.980917\pi\)
0.550990 + 0.834512i \(0.314250\pi\)
\(6\) 0 0
\(7\) 0.500000 + 0.866025i 0.188982 + 0.327327i
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) −2.00000 −0.632456
\(11\) −2.00000 − 3.46410i −0.603023 − 1.04447i −0.992361 − 0.123371i \(-0.960630\pi\)
0.389338 − 0.921095i \(-0.372704\pi\)
\(12\) 0 0
\(13\) −3.00000 + 5.19615i −0.832050 + 1.44115i 0.0643593 + 0.997927i \(0.479500\pi\)
−0.896410 + 0.443227i \(0.853834\pi\)
\(14\) −0.500000 + 0.866025i −0.133631 + 0.231455i
\(15\) 0 0
\(16\) −0.500000 − 0.866025i −0.125000 − 0.216506i
\(17\) −2.00000 −0.485071 −0.242536 − 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) 0 0
\(19\) −4.00000 −0.917663 −0.458831 − 0.888523i \(-0.651732\pi\)
−0.458831 + 0.888523i \(0.651732\pi\)
\(20\) −1.00000 − 1.73205i −0.223607 − 0.387298i
\(21\) 0 0
\(22\) 2.00000 − 3.46410i 0.426401 − 0.738549i
\(23\) 4.00000 − 6.92820i 0.834058 − 1.44463i −0.0607377 − 0.998154i \(-0.519345\pi\)
0.894795 − 0.446476i \(-0.147321\pi\)
\(24\) 0 0
\(25\) 0.500000 + 0.866025i 0.100000 + 0.173205i
\(26\) −6.00000 −1.17670
\(27\) 0 0
\(28\) −1.00000 −0.188982
\(29\) −1.00000 − 1.73205i −0.185695 − 0.321634i 0.758115 − 0.652121i \(-0.226120\pi\)
−0.943811 + 0.330487i \(0.892787\pi\)
\(30\) 0 0
\(31\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(32\) 0.500000 − 0.866025i 0.0883883 − 0.153093i
\(33\) 0 0
\(34\) −1.00000 − 1.73205i −0.171499 − 0.297044i
\(35\) −2.00000 −0.338062
\(36\) 0 0
\(37\) −10.0000 −1.64399 −0.821995 − 0.569495i \(-0.807139\pi\)
−0.821995 + 0.569495i \(0.807139\pi\)
\(38\) −2.00000 − 3.46410i −0.324443 − 0.561951i
\(39\) 0 0
\(40\) 1.00000 − 1.73205i 0.158114 − 0.273861i
\(41\) −3.00000 + 5.19615i −0.468521 + 0.811503i −0.999353 − 0.0359748i \(-0.988546\pi\)
0.530831 + 0.847477i \(0.321880\pi\)
\(42\) 0 0
\(43\) 2.00000 + 3.46410i 0.304997 + 0.528271i 0.977261 − 0.212041i \(-0.0680112\pi\)
−0.672264 + 0.740312i \(0.734678\pi\)
\(44\) 4.00000 0.603023
\(45\) 0 0
\(46\) 8.00000 1.17954
\(47\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(48\) 0 0
\(49\) −0.500000 + 0.866025i −0.0714286 + 0.123718i
\(50\) −0.500000 + 0.866025i −0.0707107 + 0.122474i
\(51\) 0 0
\(52\) −3.00000 − 5.19615i −0.416025 − 0.720577i
\(53\) −6.00000 −0.824163 −0.412082 − 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 0 0
\(55\) 8.00000 1.07872
\(56\) −0.500000 − 0.866025i −0.0668153 − 0.115728i
\(57\) 0 0
\(58\) 1.00000 − 1.73205i 0.131306 − 0.227429i
\(59\) 2.00000 − 3.46410i 0.260378 − 0.450988i −0.705965 − 0.708247i \(-0.749486\pi\)
0.966342 + 0.257260i \(0.0828195\pi\)
\(60\) 0 0
\(61\) −3.00000 − 5.19615i −0.384111 − 0.665299i 0.607535 − 0.794293i \(-0.292159\pi\)
−0.991645 + 0.128994i \(0.958825\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −6.00000 − 10.3923i −0.744208 − 1.28901i
\(66\) 0 0
\(67\) −2.00000 + 3.46410i −0.244339 + 0.423207i −0.961946 − 0.273241i \(-0.911904\pi\)
0.717607 + 0.696449i \(0.245238\pi\)
\(68\) 1.00000 − 1.73205i 0.121268 − 0.210042i
\(69\) 0 0
\(70\) −1.00000 − 1.73205i −0.119523 − 0.207020i
\(71\) −8.00000 −0.949425 −0.474713 − 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) 0 0
\(73\) 10.0000 1.17041 0.585206 − 0.810885i \(-0.301014\pi\)
0.585206 + 0.810885i \(0.301014\pi\)
\(74\) −5.00000 − 8.66025i −0.581238 − 1.00673i
\(75\) 0 0
\(76\) 2.00000 − 3.46410i 0.229416 − 0.397360i
\(77\) 2.00000 − 3.46410i 0.227921 − 0.394771i
\(78\) 0 0
\(79\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(80\) 2.00000 0.223607
\(81\) 0 0
\(82\) −6.00000 −0.662589
\(83\) −2.00000 − 3.46410i −0.219529 − 0.380235i 0.735135 − 0.677920i \(-0.237119\pi\)
−0.954664 + 0.297686i \(0.903785\pi\)
\(84\) 0 0
\(85\) 2.00000 − 3.46410i 0.216930 − 0.375735i
\(86\) −2.00000 + 3.46410i −0.215666 + 0.373544i
\(87\) 0 0
\(88\) 2.00000 + 3.46410i 0.213201 + 0.369274i
\(89\) 6.00000 0.635999 0.317999 − 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) −6.00000 −0.628971
\(92\) 4.00000 + 6.92820i 0.417029 + 0.722315i
\(93\) 0 0
\(94\) 0 0
\(95\) 4.00000 − 6.92820i 0.410391 − 0.710819i
\(96\) 0 0
\(97\) 7.00000 + 12.1244i 0.710742 + 1.23104i 0.964579 + 0.263795i \(0.0849741\pi\)
−0.253837 + 0.967247i \(0.581693\pi\)
\(98\) −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.j.757.1 2
3.2 odd 2 1134.2.f.g.757.1 2
9.2 odd 6 1134.2.f.g.379.1 2
9.4 even 3 126.2.a.a.1.1 1
9.5 odd 6 42.2.a.a.1.1 ✓ 1
9.7 even 3 inner 1134.2.f.j.379.1 2
36.23 even 6 336.2.a.d.1.1 1
36.31 odd 6 1008.2.a.j.1.1 1
45.4 even 6 3150.2.a.bo.1.1 1
45.13 odd 12 3150.2.g.r.2899.2 2
45.14 odd 6 1050.2.a.i.1.1 1
45.22 odd 12 3150.2.g.r.2899.1 2
45.23 even 12 1050.2.g.a.799.1 2
45.32 even 12 1050.2.g.a.799.2 2
63.4 even 3 882.2.g.h.667.1 2
63.5 even 6 294.2.e.a.67.1 2
63.13 odd 6 882.2.a.b.1.1 1
63.23 odd 6 294.2.e.c.67.1 2
63.31 odd 6 882.2.g.j.667.1 2
63.32 odd 6 294.2.e.c.79.1 2
63.40 odd 6 882.2.g.j.361.1 2
63.41 even 6 294.2.a.g.1.1 1
63.58 even 3 882.2.g.h.361.1 2
63.59 even 6 294.2.e.a.79.1 2
72.5 odd 6 1344.2.a.q.1.1 1
72.13 even 6 4032.2.a.e.1.1 1
72.59 even 6 1344.2.a.i.1.1 1
72.67 odd 6 4032.2.a.m.1.1 1
99.32 even 6 5082.2.a.d.1.1 1
117.77 odd 6 7098.2.a.f.1.1 1
144.5 odd 12 5376.2.c.bc.2689.2 2
144.59 even 12 5376.2.c.e.2689.1 2
144.77 odd 12 5376.2.c.bc.2689.1 2
144.131 even 12 5376.2.c.e.2689.2 2
180.59 even 6 8400.2.a.k.1.1 1
252.23 even 6 2352.2.q.i.1537.1 2
252.59 odd 6 2352.2.q.n.961.1 2
252.95 even 6 2352.2.q.i.961.1 2
252.131 odd 6 2352.2.q.n.1537.1 2
252.139 even 6 7056.2.a.k.1.1 1
252.167 odd 6 2352.2.a.l.1.1 1
315.104 even 6 7350.2.a.f.1.1 1
504.293 even 6 9408.2.a.n.1.1 1
504.419 odd 6 9408.2.a.bw.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
42.2.a.a.1.1 ✓ 1 9.5 odd 6
126.2.a.a.1.1 1 9.4 even 3
294.2.a.g.1.1 1 63.41 even 6
294.2.e.a.67.1 2 63.5 even 6
294.2.e.a.79.1 2 63.59 even 6
294.2.e.c.67.1 2 63.23 odd 6
294.2.e.c.79.1 2 63.32 odd 6
336.2.a.d.1.1 1 36.23 even 6
882.2.a.b.1.1 1 63.13 odd 6
882.2.g.h.361.1 2 63.58 even 3
882.2.g.h.667.1 2 63.4 even 3
882.2.g.j.361.1 2 63.40 odd 6
882.2.g.j.667.1 2 63.31 odd 6
1008.2.a.j.1.1 1 36.31 odd 6
1050.2.a.i.1.1 1 45.14 odd 6
1050.2.g.a.799.1 2 45.23 even 12
1050.2.g.a.799.2 2 45.32 even 12
1134.2.f.g.379.1 2 9.2 odd 6
1134.2.f.g.757.1 2 3.2 odd 2
1134.2.f.j.379.1 2 9.7 even 3 inner
1134.2.f.j.757.1 2 1.1 even 1 trivial
1344.2.a.i.1.1 1 72.59 even 6
1344.2.a.q.1.1 1 72.5 odd 6
2352.2.a.l.1.1 1 252.167 odd 6
2352.2.q.i.961.1 2 252.95 even 6
2352.2.q.i.1537.1 2 252.23 even 6
2352.2.q.n.961.1 2 252.59 odd 6
2352.2.q.n.1537.1 2 252.131 odd 6
3150.2.a.bo.1.1 1 45.4 even 6
3150.2.g.r.2899.1 2 45.22 odd 12
3150.2.g.r.2899.2 2 45.13 odd 12
4032.2.a.e.1.1 1 72.13 even 6
4032.2.a.m.1.1 1 72.67 odd 6
5082.2.a.d.1.1 1 99.32 even 6
5376.2.c.e.2689.1 2 144.59 even 12
5376.2.c.e.2689.2 2 144.131 even 12
5376.2.c.bc.2689.1 2 144.77 odd 12
5376.2.c.bc.2689.2 2 144.5 odd 12
7056.2.a.k.1.1 1 252.139 even 6
7098.2.a.f.1.1 1 117.77 odd 6
7350.2.a.f.1.1 1 315.104 even 6
8400.2.a.k.1.1 1 180.59 even 6
9408.2.a.n.1.1 1 504.293 even 6
9408.2.a.bw.1.1 1 504.419 odd 6