Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1050,2,Mod(1,1050)] 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("1050.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1050, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1050.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,-1,1,1,0,-1,1,-1,1,0,-4,1,-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: yes
Analytic conductor: \(8.38429221223\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 42)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1050.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -4.00000 q^{11} +1.00000 q^{12} -6.00000 q^{13} -1.00000 q^{14} +1.00000 q^{16} -2.00000 q^{17} -1.00000 q^{18} -4.00000 q^{19} +1.00000 q^{21} +4.00000 q^{22} -8.00000 q^{23} -1.00000 q^{24} +6.00000 q^{26} +1.00000 q^{27} +1.00000 q^{28} -2.00000 q^{29} -1.00000 q^{32} -4.00000 q^{33} +2.00000 q^{34} +1.00000 q^{36} +10.0000 q^{37} +4.00000 q^{38} -6.00000 q^{39} -6.00000 q^{41} -1.00000 q^{42} +4.00000 q^{43} -4.00000 q^{44} +8.00000 q^{46} +1.00000 q^{48} +1.00000 q^{49} -2.00000 q^{51} -6.00000 q^{52} -6.00000 q^{53} -1.00000 q^{54} -1.00000 q^{56} -4.00000 q^{57} +2.00000 q^{58} +4.00000 q^{59} +6.00000 q^{61} +1.00000 q^{63} +1.00000 q^{64} +4.00000 q^{66} -4.00000 q^{67} -2.00000 q^{68} -8.00000 q^{69} +8.00000 q^{71} -1.00000 q^{72} -10.0000 q^{73} -10.0000 q^{74} -4.00000 q^{76} -4.00000 q^{77} +6.00000 q^{78} +1.00000 q^{81} +6.00000 q^{82} +4.00000 q^{83} +1.00000 q^{84} -4.00000 q^{86} -2.00000 q^{87} +4.00000 q^{88} -6.00000 q^{89} -6.00000 q^{91} -8.00000 q^{92} -1.00000 q^{96} +14.0000 q^{97} -1.00000 q^{98} -4.00000 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) −1.00000 −0.408248
\(7\) 1.00000 0.377964
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −4.00000 −1.20605 −0.603023 − 0.797724i \(-0.706037\pi\)
−0.603023 + 0.797724i \(0.706037\pi\)
\(12\) 1.00000 0.288675
\(13\) −6.00000 −1.66410 −0.832050 − 0.554700i \(-0.812833\pi\)
−0.832050 + 0.554700i \(0.812833\pi\)
\(14\) −1.00000 −0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −2.00000 −0.485071 −0.242536 − 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) −1.00000 −0.235702
\(19\) −4.00000 −0.917663 −0.458831 − 0.888523i \(-0.651732\pi\)
−0.458831 + 0.888523i \(0.651732\pi\)
\(20\) 0 0
\(21\) 1.00000 0.218218
\(22\) 4.00000 0.852803
\(23\) −8.00000 −1.66812 −0.834058 − 0.551677i \(-0.813988\pi\)
−0.834058 + 0.551677i \(0.813988\pi\)
\(24\) −1.00000 −0.204124
\(25\) 0 0
\(26\) 6.00000 1.17670
\(27\) 1.00000 0.192450
\(28\) 1.00000 0.188982
\(29\) −2.00000 −0.371391 −0.185695 − 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) 0 0
\(31\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) −1.00000 −0.176777
\(33\) −4.00000 −0.696311
\(34\) 2.00000 0.342997
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 10.0000 1.64399 0.821995 − 0.569495i \(-0.192861\pi\)
0.821995 + 0.569495i \(0.192861\pi\)
\(38\) 4.00000 0.648886
\(39\) −6.00000 −0.960769
\(40\) 0 0
\(41\) −6.00000 −0.937043 −0.468521 − 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) −1.00000 −0.154303
\(43\) 4.00000 0.609994 0.304997 − 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) −4.00000 −0.603023
\(45\) 0 0
\(46\) 8.00000 1.17954
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 1.00000 0.144338
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −2.00000 −0.280056
\(52\) −6.00000 −0.832050
\(53\) −6.00000 −0.824163 −0.412082 − 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) −1.00000 −0.133631
\(57\) −4.00000 −0.529813
\(58\) 2.00000 0.262613
\(59\) 4.00000 0.520756 0.260378 − 0.965507i \(-0.416153\pi\)
0.260378 + 0.965507i \(0.416153\pi\)
\(60\) 0 0
\(61\) 6.00000 0.768221 0.384111 − 0.923287i \(-0.374508\pi\)
0.384111 + 0.923287i \(0.374508\pi\)
\(62\) 0 0
\(63\) 1.00000 0.125988
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 4.00000 0.492366
\(67\) −4.00000 −0.488678 −0.244339 − 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) −2.00000 −0.242536
\(69\) −8.00000 −0.963087
\(70\) 0 0
\(71\) 8.00000 0.949425 0.474713 − 0.880141i \(-0.342552\pi\)
0.474713 + 0.880141i \(0.342552\pi\)
\(72\) −1.00000 −0.117851
\(73\) −10.0000 −1.17041 −0.585206 − 0.810885i \(-0.698986\pi\)
−0.585206 + 0.810885i \(0.698986\pi\)
\(74\) −10.0000 −1.16248
\(75\) 0 0
\(76\) −4.00000 −0.458831
\(77\) −4.00000 −0.455842
\(78\) 6.00000 0.679366
\(79\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 6.00000 0.662589
\(83\) 4.00000 0.439057 0.219529 − 0.975606i \(-0.429548\pi\)
0.219529 + 0.975606i \(0.429548\pi\)
\(84\) 1.00000 0.109109
\(85\) 0 0
\(86\) −4.00000 −0.431331
\(87\) −2.00000 −0.214423
\(88\) 4.00000 0.426401
\(89\) −6.00000 −0.635999 −0.317999 − 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 0 0
\(91\) −6.00000 −0.628971
\(92\) −8.00000 −0.834058
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) −1.00000 −0.102062
\(97\) 14.0000 1.42148 0.710742 − 0.703452i \(-0.248359\pi\)
0.710742 + 0.703452i \(0.248359\pi\)
\(98\) −1.00000 −0.101015
\(99\) −4.00000 −0.402015
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 1050.2.a.i.1.1 1
3.2 odd 2 3150.2.a.bo.1.1 1
4.3 odd 2 8400.2.a.k.1.1 1
5.2 odd 4 1050.2.g.a.799.1 2
5.3 odd 4 1050.2.g.a.799.2 2
5.4 even 2 42.2.a.a.1.1 ✓ 1
7.6 odd 2 7350.2.a.f.1.1 1
15.2 even 4 3150.2.g.r.2899.2 2
15.8 even 4 3150.2.g.r.2899.1 2
15.14 odd 2 126.2.a.a.1.1 1
20.19 odd 2 336.2.a.d.1.1 1
35.4 even 6 294.2.e.c.79.1 2
35.9 even 6 294.2.e.c.67.1 2
35.19 odd 6 294.2.e.a.67.1 2
35.24 odd 6 294.2.e.a.79.1 2
35.34 odd 2 294.2.a.g.1.1 1
40.19 odd 2 1344.2.a.i.1.1 1
40.29 even 2 1344.2.a.q.1.1 1
45.4 even 6 1134.2.f.g.379.1 2
45.14 odd 6 1134.2.f.j.379.1 2
45.29 odd 6 1134.2.f.j.757.1 2
45.34 even 6 1134.2.f.g.757.1 2
55.54 odd 2 5082.2.a.d.1.1 1
60.59 even 2 1008.2.a.j.1.1 1
65.64 even 2 7098.2.a.f.1.1 1
80.19 odd 4 5376.2.c.e.2689.2 2
80.29 even 4 5376.2.c.bc.2689.1 2
80.59 odd 4 5376.2.c.e.2689.1 2
80.69 even 4 5376.2.c.bc.2689.2 2
105.44 odd 6 882.2.g.h.361.1 2
105.59 even 6 882.2.g.j.667.1 2
105.74 odd 6 882.2.g.h.667.1 2
105.89 even 6 882.2.g.j.361.1 2
105.104 even 2 882.2.a.b.1.1 1
120.29 odd 2 4032.2.a.e.1.1 1
120.59 even 2 4032.2.a.m.1.1 1
140.19 even 6 2352.2.q.n.1537.1 2
140.39 odd 6 2352.2.q.i.961.1 2
140.59 even 6 2352.2.q.n.961.1 2
140.79 odd 6 2352.2.q.i.1537.1 2
140.139 even 2 2352.2.a.l.1.1 1
280.69 odd 2 9408.2.a.n.1.1 1
280.139 even 2 9408.2.a.bw.1.1 1
420.419 odd 2 7056.2.a.k.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
42.2.a.a.1.1 ✓ 1 5.4 even 2
126.2.a.a.1.1 1 15.14 odd 2
294.2.a.g.1.1 1 35.34 odd 2
294.2.e.a.67.1 2 35.19 odd 6
294.2.e.a.79.1 2 35.24 odd 6
294.2.e.c.67.1 2 35.9 even 6
294.2.e.c.79.1 2 35.4 even 6
336.2.a.d.1.1 1 20.19 odd 2
882.2.a.b.1.1 1 105.104 even 2
882.2.g.h.361.1 2 105.44 odd 6
882.2.g.h.667.1 2 105.74 odd 6
882.2.g.j.361.1 2 105.89 even 6
882.2.g.j.667.1 2 105.59 even 6
1008.2.a.j.1.1 1 60.59 even 2
1050.2.a.i.1.1 1 1.1 even 1 trivial
1050.2.g.a.799.1 2 5.2 odd 4
1050.2.g.a.799.2 2 5.3 odd 4
1134.2.f.g.379.1 2 45.4 even 6
1134.2.f.g.757.1 2 45.34 even 6
1134.2.f.j.379.1 2 45.14 odd 6
1134.2.f.j.757.1 2 45.29 odd 6
1344.2.a.i.1.1 1 40.19 odd 2
1344.2.a.q.1.1 1 40.29 even 2
2352.2.a.l.1.1 1 140.139 even 2
2352.2.q.i.961.1 2 140.39 odd 6
2352.2.q.i.1537.1 2 140.79 odd 6
2352.2.q.n.961.1 2 140.59 even 6
2352.2.q.n.1537.1 2 140.19 even 6
3150.2.a.bo.1.1 1 3.2 odd 2
3150.2.g.r.2899.1 2 15.8 even 4
3150.2.g.r.2899.2 2 15.2 even 4
4032.2.a.e.1.1 1 120.29 odd 2
4032.2.a.m.1.1 1 120.59 even 2
5082.2.a.d.1.1 1 55.54 odd 2
5376.2.c.e.2689.1 2 80.59 odd 4
5376.2.c.e.2689.2 2 80.19 odd 4
5376.2.c.bc.2689.1 2 80.29 even 4
5376.2.c.bc.2689.2 2 80.69 even 4
7056.2.a.k.1.1 1 420.419 odd 2
7098.2.a.f.1.1 1 65.64 even 2
7350.2.a.f.1.1 1 7.6 odd 2
8400.2.a.k.1.1 1 4.3 odd 2
9408.2.a.n.1.1 1 280.69 odd 2
9408.2.a.bw.1.1 1 280.139 even 2