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Properties

Label 1050.2.i.r.151.1
Level $1050$
Weight $2$
Character 1050.151
Analytic conductor $8.384$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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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(151,1050)] 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("1050.151"); 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, 0, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1050.i (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,1,1,-1,0,2,-5,-2,-1,0,4,1,8] 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: \(8.38429221223\)
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: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 151.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 1050.151
Dual form 1050.2.i.r.751.1

$q$-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^{3} +(-0.500000 + 0.866025i) q^{4} +1.00000 q^{6} +(-2.50000 - 0.866025i) q^{7} -1.00000 q^{8} +(-0.500000 - 0.866025i) q^{9} +(2.00000 - 3.46410i) q^{11} +(0.500000 + 0.866025i) q^{12} +4.00000 q^{13} +(-0.500000 - 2.59808i) q^{14} +(-0.500000 - 0.866025i) q^{16} +(-1.50000 + 2.59808i) q^{17} +(0.500000 - 0.866025i) q^{18} +(-3.00000 - 5.19615i) q^{19} +(-2.00000 + 1.73205i) q^{21} +4.00000 q^{22} +(-3.50000 - 6.06218i) q^{23} +(-0.500000 + 0.866025i) q^{24} +(2.00000 + 3.46410i) q^{26} -1.00000 q^{27} +(2.00000 - 1.73205i) q^{28} +4.00000 q^{29} +(2.50000 - 4.33013i) q^{31} +(0.500000 - 0.866025i) q^{32} +(-2.00000 - 3.46410i) q^{33} -3.00000 q^{34} +1.00000 q^{36} +(-1.00000 - 1.73205i) q^{37} +(3.00000 - 5.19615i) q^{38} +(2.00000 - 3.46410i) q^{39} +7.00000 q^{41} +(-2.50000 - 0.866025i) q^{42} +2.00000 q^{43} +(2.00000 + 3.46410i) q^{44} +(3.50000 - 6.06218i) q^{46} +(-0.500000 - 0.866025i) q^{47} -1.00000 q^{48} +(5.50000 + 4.33013i) q^{49} +(1.50000 + 2.59808i) q^{51} +(-2.00000 + 3.46410i) q^{52} +(1.00000 - 1.73205i) q^{53} +(-0.500000 - 0.866025i) q^{54} +(2.50000 + 0.866025i) q^{56} -6.00000 q^{57} +(2.00000 + 3.46410i) q^{58} +(7.00000 - 12.1244i) q^{59} +(-6.00000 - 10.3923i) q^{61} +5.00000 q^{62} +(0.500000 + 2.59808i) q^{63} +1.00000 q^{64} +(2.00000 - 3.46410i) q^{66} +(-6.00000 + 10.3923i) q^{67} +(-1.50000 - 2.59808i) q^{68} -7.00000 q^{69} -9.00000 q^{71} +(0.500000 + 0.866025i) q^{72} +(-3.00000 + 5.19615i) q^{73} +(1.00000 - 1.73205i) q^{74} +6.00000 q^{76} +(-8.00000 + 6.92820i) q^{77} +4.00000 q^{78} +(8.50000 + 14.7224i) q^{79} +(-0.500000 + 0.866025i) q^{81} +(3.50000 + 6.06218i) q^{82} -4.00000 q^{83} +(-0.500000 - 2.59808i) q^{84} +(1.00000 + 1.73205i) q^{86} +(2.00000 - 3.46410i) q^{87} +(-2.00000 + 3.46410i) q^{88} +(3.50000 + 6.06218i) q^{89} +(-10.0000 - 3.46410i) q^{91} +7.00000 q^{92} +(-2.50000 - 4.33013i) q^{93} +(0.500000 - 0.866025i) q^{94} +(-0.500000 - 0.866025i) q^{96} -7.00000 q^{97} +(-1.00000 + 6.92820i) q^{98} -4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} + q^{3} - q^{4} + 2 q^{6} - 5 q^{7} - 2 q^{8} - q^{9} + 4 q^{11} + q^{12} + 8 q^{13} - q^{14} - q^{16} - 3 q^{17} + q^{18} - 6 q^{19} - 4 q^{21} + 8 q^{22} - 7 q^{23} - q^{24} + 4 q^{26}+ \cdots - 8 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.500000 + 0.866025i 0.353553 + 0.612372i
\(3\) 0.500000 − 0.866025i 0.288675 − 0.500000i
\(4\) −0.500000 + 0.866025i −0.250000 + 0.433013i
\(5\) 0 0
\(6\) 1.00000 0.408248
\(7\) −2.50000 − 0.866025i −0.944911 − 0.327327i
\(8\) −1.00000 −0.353553
\(9\) −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 \(-0.0393705\pi\)
\(12\) 0.500000 + 0.866025i 0.144338 + 0.250000i
\(13\) 4.00000 1.10940 0.554700 − 0.832050i \(-0.312833\pi\)
0.554700 + 0.832050i \(0.312833\pi\)
\(14\) −0.500000 − 2.59808i −0.133631 − 0.694365i
\(15\) 0 0
\(16\) −0.500000 − 0.866025i −0.125000 − 0.216506i
\(17\) −1.50000 + 2.59808i −0.363803 + 0.630126i −0.988583 − 0.150675i \(-0.951855\pi\)
0.624780 + 0.780801i \(0.285189\pi\)
\(18\) 0.500000 − 0.866025i 0.117851 − 0.204124i
\(19\) −3.00000 − 5.19615i −0.688247 − 1.19208i −0.972404 − 0.233301i \(-0.925047\pi\)
0.284157 − 0.958778i \(-0.408286\pi\)
\(20\) 0 0
\(21\) −2.00000 + 1.73205i −0.436436 + 0.377964i
\(22\) 4.00000 0.852803
\(23\) −3.50000 − 6.06218i −0.729800 − 1.26405i −0.956967 − 0.290196i \(-0.906280\pi\)
0.227167 − 0.973856i \(-0.427054\pi\)
\(24\) −0.500000 + 0.866025i −0.102062 + 0.176777i
\(25\) 0 0
\(26\) 2.00000 + 3.46410i 0.392232 + 0.679366i
\(27\) −1.00000 −0.192450
\(28\) 2.00000 − 1.73205i 0.377964 − 0.327327i
\(29\) 4.00000 0.742781 0.371391 − 0.928477i \(-0.378881\pi\)
0.371391 + 0.928477i \(0.378881\pi\)
\(30\) 0 0
\(31\) 2.50000 − 4.33013i 0.449013 − 0.777714i −0.549309 − 0.835619i \(-0.685109\pi\)
0.998322 + 0.0579057i \(0.0184423\pi\)
\(32\) 0.500000 − 0.866025i 0.0883883 − 0.153093i
\(33\) −2.00000 − 3.46410i −0.348155 − 0.603023i
\(34\) −3.00000 −0.514496
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −1.00000 − 1.73205i −0.164399 − 0.284747i 0.772043 − 0.635571i \(-0.219235\pi\)
−0.936442 + 0.350823i \(0.885902\pi\)
\(38\) 3.00000 − 5.19615i 0.486664 − 0.842927i
\(39\) 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 \(0.315919\pi\)
\(42\) −2.50000 − 0.866025i −0.385758 − 0.133631i
\(43\) 2.00000 0.304997 0.152499 − 0.988304i \(-0.451268\pi\)
0.152499 + 0.988304i \(0.451268\pi\)
\(44\) 2.00000 + 3.46410i 0.301511 + 0.522233i
\(45\) 0 0
\(46\) 3.50000 − 6.06218i 0.516047 − 0.893819i
\(47\) −0.500000 − 0.866025i −0.0729325 − 0.126323i 0.827253 − 0.561830i \(-0.189902\pi\)
−0.900185 + 0.435507i \(0.856569\pi\)
\(48\) −1.00000 −0.144338
\(49\) 5.50000 + 4.33013i 0.785714 + 0.618590i
\(50\) 0 0
\(51\) 1.50000 + 2.59808i 0.210042 + 0.363803i
\(52\) −2.00000 + 3.46410i −0.277350 + 0.480384i
\(53\) 1.00000 − 1.73205i 0.137361 − 0.237915i −0.789136 − 0.614218i \(-0.789471\pi\)
0.926497 + 0.376303i \(0.122805\pi\)
\(54\) −0.500000 − 0.866025i −0.0680414 − 0.117851i
\(55\) 0 0
\(56\) 2.50000 + 0.866025i 0.334077 + 0.115728i
\(57\) −6.00000 −0.794719
\(58\) 2.00000 + 3.46410i 0.262613 + 0.454859i
\(59\) 7.00000 − 12.1244i 0.911322 − 1.57846i 0.0991242 − 0.995075i \(-0.468396\pi\)
0.812198 − 0.583382i \(-0.198271\pi\)
\(60\) 0 0
\(61\) −6.00000 − 10.3923i −0.768221 − 1.33060i −0.938527 − 0.345207i \(-0.887809\pi\)
0.170305 − 0.985391i \(-0.445525\pi\)
\(62\) 5.00000 0.635001
\(63\) 0.500000 + 2.59808i 0.0629941 + 0.327327i
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 2.00000 − 3.46410i 0.246183 − 0.426401i
\(67\) −6.00000 + 10.3923i −0.733017 + 1.26962i 0.222571 + 0.974916i \(0.428555\pi\)
−0.955588 + 0.294706i \(0.904778\pi\)
\(68\) −1.50000 − 2.59808i −0.181902 − 0.315063i
\(69\) −7.00000 −0.842701
\(70\) 0 0
\(71\) −9.00000 −1.06810 −0.534052 − 0.845452i \(-0.679331\pi\)
−0.534052 + 0.845452i \(0.679331\pi\)
\(72\) 0.500000 + 0.866025i 0.0589256 + 0.102062i
\(73\) −3.00000 + 5.19615i −0.351123 + 0.608164i −0.986447 − 0.164083i \(-0.947534\pi\)
0.635323 + 0.772246i \(0.280867\pi\)
\(74\) 1.00000 − 1.73205i 0.116248 − 0.201347i
\(75\) 0 0
\(76\) 6.00000 0.688247
\(77\) −8.00000 + 6.92820i −0.911685 + 0.789542i
\(78\) 4.00000 0.452911
\(79\) 8.50000 + 14.7224i 0.956325 + 1.65640i 0.731307 + 0.682048i \(0.238911\pi\)
0.225018 + 0.974355i \(0.427756\pi\)
\(80\) 0 0
\(81\) −0.500000 + 0.866025i −0.0555556 + 0.0962250i
\(82\) 3.50000 + 6.06218i 0.386510 + 0.669456i
\(83\) −4.00000 −0.439057 −0.219529 − 0.975606i \(-0.570452\pi\)
−0.219529 + 0.975606i \(0.570452\pi\)
\(84\) −0.500000 − 2.59808i −0.0545545 − 0.283473i
\(85\) 0 0
\(86\) 1.00000 + 1.73205i 0.107833 + 0.186772i
\(87\) 2.00000 − 3.46410i 0.214423 − 0.371391i
\(88\) −2.00000 + 3.46410i −0.213201 + 0.369274i
\(89\) 3.50000 + 6.06218i 0.370999 + 0.642590i 0.989720 − 0.143022i \(-0.0456819\pi\)
−0.618720 + 0.785611i \(0.712349\pi\)
\(90\) 0 0
\(91\) −10.0000 − 3.46410i −1.04828 − 0.363137i
\(92\) 7.00000 0.729800
\(93\) −2.50000 − 4.33013i −0.259238 − 0.449013i
\(94\) 0.500000 − 0.866025i 0.0515711 − 0.0893237i
\(95\) 0 0
\(96\) −0.500000 − 0.866025i −0.0510310 − 0.0883883i
\(97\) −7.00000 −0.710742 −0.355371 − 0.934725i \(-0.615646\pi\)
−0.355371 + 0.934725i \(0.615646\pi\)
\(98\) −1.00000 + 6.92820i −0.101015 + 0.699854i
\(99\) −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.i.r.151.1 yes 2
5.2 odd 4 1050.2.o.l.949.1 4
5.3 odd 4 1050.2.o.l.949.2 4
5.4 even 2 1050.2.i.d.151.1 ✓ 2
7.2 even 3 inner 1050.2.i.r.751.1 yes 2
7.3 odd 6 7350.2.a.v.1.1 1
7.4 even 3 7350.2.a.e.1.1 1
35.2 odd 12 1050.2.o.l.499.2 4
35.4 even 6 7350.2.a.ci.1.1 1
35.9 even 6 1050.2.i.d.751.1 yes 2
35.23 odd 12 1050.2.o.l.499.1 4
35.24 odd 6 7350.2.a.bs.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1050.2.i.d.151.1 ✓ 2 5.4 even 2
1050.2.i.d.751.1 yes 2 35.9 even 6
1050.2.i.r.151.1 yes 2 1.1 even 1 trivial
1050.2.i.r.751.1 yes 2 7.2 even 3 inner
1050.2.o.l.499.1 4 35.23 odd 12
1050.2.o.l.499.2 4 35.2 odd 12
1050.2.o.l.949.1 4 5.2 odd 4
1050.2.o.l.949.2 4 5.3 odd 4
7350.2.a.e.1.1 1 7.4 even 3
7350.2.a.v.1.1 1 7.3 odd 6
7350.2.a.bs.1.1 1 35.24 odd 6
7350.2.a.ci.1.1 1 35.4 even 6