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Properties

Label 1050.2.o.l.949.1
Level $1050$
Weight $2$
Character 1050.949
Analytic conductor $8.384$
Analytic rank $0$
Dimension $4$
Inner twists $4$

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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(499,1050)] 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("1050.499"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1050, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 3, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1050.o (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,2,0,4,0,0,2,0,8,0,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(14)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(8.38429221223\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 949.1
Root \(-0.866025 - 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 1050.949
Dual form 1050.2.o.l.499.1

$q$-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+(-0.866025 + 0.500000i) q^{2} +(-0.866025 - 0.500000i) q^{3} +(0.500000 - 0.866025i) q^{4} +1.00000 q^{6} +(0.866025 - 2.50000i) q^{7} +1.00000i q^{8} +(0.500000 + 0.866025i) q^{9} +(2.00000 - 3.46410i) q^{11} +(-0.866025 + 0.500000i) q^{12} -4.00000i q^{13} +(0.500000 + 2.59808i) q^{14} +(-0.500000 - 0.866025i) q^{16} +(-2.59808 - 1.50000i) q^{17} +(-0.866025 - 0.500000i) q^{18} +(3.00000 + 5.19615i) q^{19} +(-2.00000 + 1.73205i) q^{21} +4.00000i q^{22} +(-6.06218 + 3.50000i) q^{23} +(0.500000 - 0.866025i) q^{24} +(2.00000 + 3.46410i) q^{26} -1.00000i q^{27} +(-1.73205 - 2.00000i) q^{28} -4.00000 q^{29} +(2.50000 - 4.33013i) q^{31} +(0.866025 + 0.500000i) q^{32} +(-3.46410 + 2.00000i) q^{33} +3.00000 q^{34} +1.00000 q^{36} +(1.73205 - 1.00000i) q^{37} +(-5.19615 - 3.00000i) q^{38} +(-2.00000 + 3.46410i) q^{39} +7.00000 q^{41} +(0.866025 - 2.50000i) q^{42} -2.00000i q^{43} +(-2.00000 - 3.46410i) q^{44} +(3.50000 - 6.06218i) q^{46} +(0.866025 - 0.500000i) q^{47} +1.00000i q^{48} +(-5.50000 - 4.33013i) q^{49} +(1.50000 + 2.59808i) q^{51} +(-3.46410 - 2.00000i) q^{52} +(-1.73205 - 1.00000i) q^{53} +(0.500000 + 0.866025i) q^{54} +(2.50000 + 0.866025i) q^{56} -6.00000i q^{57} +(3.46410 - 2.00000i) q^{58} +(-7.00000 + 12.1244i) q^{59} +(-6.00000 - 10.3923i) q^{61} +5.00000i q^{62} +(2.59808 - 0.500000i) q^{63} -1.00000 q^{64} +(2.00000 - 3.46410i) q^{66} +(-10.3923 - 6.00000i) q^{67} +(-2.59808 + 1.50000i) q^{68} +7.00000 q^{69} -9.00000 q^{71} +(-0.866025 + 0.500000i) q^{72} +(5.19615 + 3.00000i) q^{73} +(-1.00000 + 1.73205i) q^{74} +6.00000 q^{76} +(-6.92820 - 8.00000i) q^{77} -4.00000i q^{78} +(-8.50000 - 14.7224i) q^{79} +(-0.500000 + 0.866025i) q^{81} +(-6.06218 + 3.50000i) q^{82} +4.00000i q^{83} +(0.500000 + 2.59808i) q^{84} +(1.00000 + 1.73205i) q^{86} +(3.46410 + 2.00000i) q^{87} +(3.46410 + 2.00000i) q^{88} +(-3.50000 - 6.06218i) q^{89} +(-10.0000 - 3.46410i) q^{91} +7.00000i q^{92} +(-4.33013 + 2.50000i) q^{93} +(-0.500000 + 0.866025i) q^{94} +(-0.500000 - 0.866025i) q^{96} -7.00000i q^{97} +(6.92820 + 1.00000i) q^{98} +4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{4} + 4 q^{6} + 2 q^{9} + 8 q^{11} + 2 q^{14} - 2 q^{16} + 12 q^{19} - 8 q^{21} + 2 q^{24} + 8 q^{26} - 16 q^{29} + 10 q^{31} + 12 q^{34} + 4 q^{36} - 8 q^{39} + 28 q^{41} - 8 q^{44} + 14 q^{46}+ \cdots + 16 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1050\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(451\) \(701\)
\(\chi(n)\) \(-1\) \(e\left(\frac{2}{3}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.866025 + 0.500000i −0.612372 + 0.353553i
\(3\) −0.866025 − 0.500000i −0.500000 − 0.288675i
\(4\) 0.500000 − 0.866025i 0.250000 − 0.433013i
\(5\) 0 0
\(6\) 1.00000 0.408248
\(7\) 0.866025 − 2.50000i 0.327327 − 0.944911i
\(8\) 1.00000i 0.353553i
\(9\) 0.500000 + 0.866025i 0.166667 + 0.288675i
\(10\) 0 0
\(11\) 2.00000 − 3.46410i 0.603023 − 1.04447i −0.389338 − 0.921095i \(-0.627296\pi\)
0.992361 − 0.123371i \(-0.0393705\pi\)
\(12\) −0.866025 + 0.500000i −0.250000 + 0.144338i
\(13\) − 4.00000i − 1.10940i −0.832050 − 0.554700i \(-0.812833\pi\)
0.832050 − 0.554700i \(-0.187167\pi\)
\(14\) 0.500000 + 2.59808i 0.133631 + 0.694365i
\(15\) 0 0
\(16\) −0.500000 − 0.866025i −0.125000 − 0.216506i
\(17\) −2.59808 − 1.50000i −0.630126 − 0.363803i 0.150675 − 0.988583i \(-0.451855\pi\)
−0.780801 + 0.624780i \(0.785189\pi\)
\(18\) −0.866025 − 0.500000i −0.204124 − 0.117851i
\(19\) 3.00000 + 5.19615i 0.688247 + 1.19208i 0.972404 + 0.233301i \(0.0749529\pi\)
−0.284157 + 0.958778i \(0.591714\pi\)
\(20\) 0 0
\(21\) −2.00000 + 1.73205i −0.436436 + 0.377964i
\(22\) 4.00000i 0.852803i
\(23\) −6.06218 + 3.50000i −1.26405 + 0.729800i −0.973856 − 0.227167i \(-0.927054\pi\)
−0.290196 + 0.956967i \(0.593720\pi\)
\(24\) 0.500000 − 0.866025i 0.102062 − 0.176777i
\(25\) 0 0
\(26\) 2.00000 + 3.46410i 0.392232 + 0.679366i
\(27\) − 1.00000i − 0.192450i
\(28\) −1.73205 − 2.00000i −0.327327 − 0.377964i
\(29\) −4.00000 −0.742781 −0.371391 − 0.928477i \(-0.621119\pi\)
−0.371391 + 0.928477i \(0.621119\pi\)
\(30\) 0 0
\(31\) 2.50000 − 4.33013i 0.449013 − 0.777714i −0.549309 − 0.835619i \(-0.685109\pi\)
0.998322 + 0.0579057i \(0.0184423\pi\)
\(32\) 0.866025 + 0.500000i 0.153093 + 0.0883883i
\(33\) −3.46410 + 2.00000i −0.603023 + 0.348155i
\(34\) 3.00000 0.514496
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 1.73205 − 1.00000i 0.284747 − 0.164399i −0.350823 − 0.936442i \(-0.614098\pi\)
0.635571 + 0.772043i \(0.280765\pi\)
\(38\) −5.19615 − 3.00000i −0.842927 − 0.486664i
\(39\) −2.00000 + 3.46410i −0.320256 + 0.554700i
\(40\) 0 0
\(41\) 7.00000 1.09322 0.546608 − 0.837389i \(-0.315919\pi\)
0.546608 + 0.837389i \(0.315919\pi\)
\(42\) 0.866025 − 2.50000i 0.133631 − 0.385758i
\(43\) − 2.00000i − 0.304997i −0.988304 − 0.152499i \(-0.951268\pi\)
0.988304 − 0.152499i \(-0.0487319\pi\)
\(44\) −2.00000 − 3.46410i −0.301511 − 0.522233i
\(45\) 0 0
\(46\) 3.50000 − 6.06218i 0.516047 − 0.893819i
\(47\) 0.866025 − 0.500000i 0.126323 − 0.0729325i −0.435507 − 0.900185i \(-0.643431\pi\)
0.561830 + 0.827253i \(0.310098\pi\)
\(48\) 1.00000i 0.144338i
\(49\) −5.50000 − 4.33013i −0.785714 − 0.618590i
\(50\) 0 0
\(51\) 1.50000 + 2.59808i 0.210042 + 0.363803i
\(52\) −3.46410 − 2.00000i −0.480384 − 0.277350i
\(53\) −1.73205 − 1.00000i −0.237915 − 0.137361i 0.376303 − 0.926497i \(-0.377195\pi\)
−0.614218 + 0.789136i \(0.710529\pi\)
\(54\) 0.500000 + 0.866025i 0.0680414 + 0.117851i
\(55\) 0 0
\(56\) 2.50000 + 0.866025i 0.334077 + 0.115728i
\(57\) − 6.00000i − 0.794719i
\(58\) 3.46410 − 2.00000i 0.454859 − 0.262613i
\(59\) −7.00000 + 12.1244i −0.911322 + 1.57846i −0.0991242 + 0.995075i \(0.531604\pi\)
−0.812198 + 0.583382i \(0.801729\pi\)
\(60\) 0 0
\(61\) −6.00000 − 10.3923i −0.768221 − 1.33060i −0.938527 − 0.345207i \(-0.887809\pi\)
0.170305 − 0.985391i \(-0.445525\pi\)
\(62\) 5.00000i 0.635001i
\(63\) 2.59808 − 0.500000i 0.327327 − 0.0629941i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 2.00000 − 3.46410i 0.246183 − 0.426401i
\(67\) −10.3923 − 6.00000i −1.26962 − 0.733017i −0.294706 − 0.955588i \(-0.595222\pi\)
−0.974916 + 0.222571i \(0.928555\pi\)
\(68\) −2.59808 + 1.50000i −0.315063 + 0.181902i
\(69\) 7.00000 0.842701
\(70\) 0 0
\(71\) −9.00000 −1.06810 −0.534052 − 0.845452i \(-0.679331\pi\)
−0.534052 + 0.845452i \(0.679331\pi\)
\(72\) −0.866025 + 0.500000i −0.102062 + 0.0589256i
\(73\) 5.19615 + 3.00000i 0.608164 + 0.351123i 0.772246 − 0.635323i \(-0.219133\pi\)
−0.164083 + 0.986447i \(0.552466\pi\)
\(74\) −1.00000 + 1.73205i −0.116248 + 0.201347i
\(75\) 0 0
\(76\) 6.00000 0.688247
\(77\) −6.92820 − 8.00000i −0.789542 − 0.911685i
\(78\) − 4.00000i − 0.452911i
\(79\) −8.50000 − 14.7224i −0.956325 − 1.65640i −0.731307 − 0.682048i \(-0.761089\pi\)
−0.225018 − 0.974355i \(-0.572244\pi\)
\(80\) 0 0
\(81\) −0.500000 + 0.866025i −0.0555556 + 0.0962250i
\(82\) −6.06218 + 3.50000i −0.669456 + 0.386510i
\(83\) 4.00000i 0.439057i 0.975606 + 0.219529i \(0.0704519\pi\)
−0.975606 + 0.219529i \(0.929548\pi\)
\(84\) 0.500000 + 2.59808i 0.0545545 + 0.283473i
\(85\) 0 0
\(86\) 1.00000 + 1.73205i 0.107833 + 0.186772i
\(87\) 3.46410 + 2.00000i 0.371391 + 0.214423i
\(88\) 3.46410 + 2.00000i 0.369274 + 0.213201i
\(89\) −3.50000 − 6.06218i −0.370999 − 0.642590i 0.618720 − 0.785611i \(-0.287651\pi\)
−0.989720 + 0.143022i \(0.954318\pi\)
\(90\) 0 0
\(91\) −10.0000 − 3.46410i −1.04828 − 0.363137i
\(92\) 7.00000i 0.729800i
\(93\) −4.33013 + 2.50000i −0.449013 + 0.259238i
\(94\) −0.500000 + 0.866025i −0.0515711 + 0.0893237i
\(95\) 0 0
\(96\) −0.500000 − 0.866025i −0.0510310 − 0.0883883i
\(97\) − 7.00000i − 0.710742i −0.934725 − 0.355371i \(-0.884354\pi\)
0.934725 − 0.355371i \(-0.115646\pi\)
\(98\) 6.92820 + 1.00000i 0.699854 + 0.101015i
\(99\) 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.o.l.949.1 4
5.2 odd 4 1050.2.i.d.151.1 ✓ 2
5.3 odd 4 1050.2.i.r.151.1 yes 2
5.4 even 2 inner 1050.2.o.l.949.2 4
7.2 even 3 inner 1050.2.o.l.499.2 4
35.2 odd 12 1050.2.i.d.751.1 yes 2
35.3 even 12 7350.2.a.v.1.1 1
35.9 even 6 inner 1050.2.o.l.499.1 4
35.17 even 12 7350.2.a.bs.1.1 1
35.18 odd 12 7350.2.a.e.1.1 1
35.23 odd 12 1050.2.i.r.751.1 yes 2
35.32 odd 12 7350.2.a.ci.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1050.2.i.d.151.1 ✓ 2 5.2 odd 4
1050.2.i.d.751.1 yes 2 35.2 odd 12
1050.2.i.r.151.1 yes 2 5.3 odd 4
1050.2.i.r.751.1 yes 2 35.23 odd 12
1050.2.o.l.499.1 4 35.9 even 6 inner
1050.2.o.l.499.2 4 7.2 even 3 inner
1050.2.o.l.949.1 4 1.1 even 1 trivial
1050.2.o.l.949.2 4 5.4 even 2 inner
7350.2.a.e.1.1 1 35.18 odd 12
7350.2.a.v.1.1 1 35.3 even 12
7350.2.a.bs.1.1 1 35.17 even 12
7350.2.a.ci.1.1 1 35.32 odd 12