Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1470,2,Mod(361,1470)] 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("1470.361"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1470, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1470 = 2 \cdot 3 \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1470.i (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,1,-1,-1,1,-2,0,-2,-1,-1,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(11.7380090971\)
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 30)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 961.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 1470.961
Dual form 1470.2.i.o.361.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.500000 - 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{3} +(-0.500000 - 0.866025i) q^{4} +(0.500000 - 0.866025i) q^{5} -1.00000 q^{6} -1.00000 q^{8} +(-0.500000 + 0.866025i) q^{9} +(-0.500000 - 0.866025i) q^{10} +(-0.500000 + 0.866025i) q^{12} +2.00000 q^{13} -1.00000 q^{15} +(-0.500000 + 0.866025i) q^{16} +(-3.00000 - 5.19615i) q^{17} +(0.500000 + 0.866025i) q^{18} +(2.00000 - 3.46410i) q^{19} -1.00000 q^{20} +(0.500000 + 0.866025i) q^{24} +(-0.500000 - 0.866025i) q^{25} +(1.00000 - 1.73205i) q^{26} +1.00000 q^{27} -6.00000 q^{29} +(-0.500000 + 0.866025i) q^{30} +(-4.00000 - 6.92820i) q^{31} +(0.500000 + 0.866025i) q^{32} -6.00000 q^{34} +1.00000 q^{36} +(-1.00000 + 1.73205i) q^{37} +(-2.00000 - 3.46410i) q^{38} +(-1.00000 - 1.73205i) q^{39} +(-0.500000 + 0.866025i) q^{40} -6.00000 q^{41} -4.00000 q^{43} +(0.500000 + 0.866025i) q^{45} +1.00000 q^{48} -1.00000 q^{50} +(-3.00000 + 5.19615i) q^{51} +(-1.00000 - 1.73205i) q^{52} +(3.00000 + 5.19615i) q^{53} +(0.500000 - 0.866025i) q^{54} -4.00000 q^{57} +(-3.00000 + 5.19615i) q^{58} +(0.500000 + 0.866025i) q^{60} +(5.00000 - 8.66025i) q^{61} -8.00000 q^{62} +1.00000 q^{64} +(1.00000 - 1.73205i) q^{65} +(2.00000 + 3.46410i) q^{67} +(-3.00000 + 5.19615i) q^{68} +(0.500000 - 0.866025i) q^{72} +(-1.00000 - 1.73205i) q^{73} +(1.00000 + 1.73205i) q^{74} +(-0.500000 + 0.866025i) q^{75} -4.00000 q^{76} -2.00000 q^{78} +(-4.00000 + 6.92820i) q^{79} +(0.500000 + 0.866025i) q^{80} +(-0.500000 - 0.866025i) q^{81} +(-3.00000 + 5.19615i) q^{82} +12.0000 q^{83} -6.00000 q^{85} +(-2.00000 + 3.46410i) q^{86} +(3.00000 + 5.19615i) q^{87} +(-9.00000 + 15.5885i) q^{89} +1.00000 q^{90} +(-4.00000 + 6.92820i) q^{93} +(-2.00000 - 3.46410i) q^{95} +(0.500000 - 0.866025i) q^{96} +2.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} - q^{3} - q^{4} + q^{5} - 2 q^{6} - 2 q^{8} - q^{9} - q^{10} - q^{12} + 4 q^{13} - 2 q^{15} - q^{16} - 6 q^{17} + q^{18} + 4 q^{19} - 2 q^{20} + q^{24} - q^{25} + 2 q^{26} + 2 q^{27}+ \cdots + 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1470\mathbb{Z}\right)^\times\).

\(n\) \(491\) \(1081\) \(1177\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{3}\right)\) \(1\)

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
\(5\) 0.500000 − 0.866025i 0.223607 − 0.387298i
\(6\) −1.00000 −0.408248
\(7\) 0 0
\(8\) −1.00000 −0.353553
\(9\) −0.500000 + 0.866025i −0.166667 + 0.288675i
\(10\) −0.500000 − 0.866025i −0.158114 − 0.273861i
\(11\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(12\) −0.500000 + 0.866025i −0.144338 + 0.250000i
\(13\) 2.00000 0.554700 0.277350 − 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) −0.500000 + 0.866025i −0.125000 + 0.216506i
\(17\) −3.00000 − 5.19615i −0.727607 − 1.26025i −0.957892 − 0.287129i \(-0.907299\pi\)
0.230285 − 0.973123i \(-0.426034\pi\)
\(18\) 0.500000 + 0.866025i 0.117851 + 0.204124i
\(19\) 2.00000 − 3.46410i 0.458831 − 0.794719i −0.540068 − 0.841621i \(-0.681602\pi\)
0.998899 + 0.0469020i \(0.0149348\pi\)
\(20\) −1.00000 −0.223607
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(24\) 0.500000 + 0.866025i 0.102062 + 0.176777i
\(25\) −0.500000 − 0.866025i −0.100000 − 0.173205i
\(26\) 1.00000 − 1.73205i 0.196116 − 0.339683i
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −6.00000 −1.11417 −0.557086 − 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) −0.500000 + 0.866025i −0.0912871 + 0.158114i
\(31\) −4.00000 − 6.92820i −0.718421 − 1.24434i −0.961625 − 0.274367i \(-0.911532\pi\)
0.243204 − 0.969975i \(-0.421802\pi\)
\(32\) 0.500000 + 0.866025i 0.0883883 + 0.153093i
\(33\) 0 0
\(34\) −6.00000 −1.02899
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −1.00000 + 1.73205i −0.164399 + 0.284747i −0.936442 − 0.350823i \(-0.885902\pi\)
0.772043 + 0.635571i \(0.219235\pi\)
\(38\) −2.00000 − 3.46410i −0.324443 − 0.561951i
\(39\) −1.00000 − 1.73205i −0.160128 − 0.277350i
\(40\) −0.500000 + 0.866025i −0.0790569 + 0.136931i
\(41\) −6.00000 −0.937043 −0.468521 − 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 0 0
\(43\) −4.00000 −0.609994 −0.304997 − 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 0 0
\(45\) 0.500000 + 0.866025i 0.0745356 + 0.129099i
\(46\) 0 0
\(47\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(48\) 1.00000 0.144338
\(49\) 0 0
\(50\) −1.00000 −0.141421
\(51\) −3.00000 + 5.19615i −0.420084 + 0.727607i
\(52\) −1.00000 − 1.73205i −0.138675 − 0.240192i
\(53\) 3.00000 + 5.19615i 0.412082 + 0.713746i 0.995117 − 0.0987002i \(-0.0314685\pi\)
−0.583036 + 0.812447i \(0.698135\pi\)
\(54\) 0.500000 − 0.866025i 0.0680414 − 0.117851i
\(55\) 0 0
\(56\) 0 0
\(57\) −4.00000 −0.529813
\(58\) −3.00000 + 5.19615i −0.393919 + 0.682288i
\(59\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(60\) 0.500000 + 0.866025i 0.0645497 + 0.111803i
\(61\) 5.00000 − 8.66025i 0.640184 − 1.10883i −0.345207 − 0.938527i \(-0.612191\pi\)
0.985391 − 0.170305i \(-0.0544754\pi\)
\(62\) −8.00000 −1.01600
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 1.00000 − 1.73205i 0.124035 − 0.214834i
\(66\) 0 0
\(67\) 2.00000 + 3.46410i 0.244339 + 0.423207i 0.961946 − 0.273241i \(-0.0880957\pi\)
−0.717607 + 0.696449i \(0.754762\pi\)
\(68\) −3.00000 + 5.19615i −0.363803 + 0.630126i
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0.500000 − 0.866025i 0.0589256 − 0.102062i
\(73\) −1.00000 − 1.73205i −0.117041 − 0.202721i 0.801553 − 0.597924i \(-0.204008\pi\)
−0.918594 + 0.395203i \(0.870674\pi\)
\(74\) 1.00000 + 1.73205i 0.116248 + 0.201347i
\(75\) −0.500000 + 0.866025i −0.0577350 + 0.100000i
\(76\) −4.00000 −0.458831
\(77\) 0 0
\(78\) −2.00000 −0.226455
\(79\) −4.00000 + 6.92820i −0.450035 + 0.779484i −0.998388 − 0.0567635i \(-0.981922\pi\)
0.548352 + 0.836247i \(0.315255\pi\)
\(80\) 0.500000 + 0.866025i 0.0559017 + 0.0968246i
\(81\) −0.500000 − 0.866025i −0.0555556 − 0.0962250i
\(82\) −3.00000 + 5.19615i −0.331295 + 0.573819i
\(83\) 12.0000 1.31717 0.658586 − 0.752506i \(-0.271155\pi\)
0.658586 + 0.752506i \(0.271155\pi\)
\(84\) 0 0
\(85\) −6.00000 −0.650791
\(86\) −2.00000 + 3.46410i −0.215666 + 0.373544i
\(87\) 3.00000 + 5.19615i 0.321634 + 0.557086i
\(88\) 0 0
\(89\) −9.00000 + 15.5885i −0.953998 + 1.65237i −0.217354 + 0.976093i \(0.569742\pi\)
−0.736644 + 0.676280i \(0.763591\pi\)
\(90\) 1.00000 0.105409
\(91\) 0 0
\(92\) 0 0
\(93\) −4.00000 + 6.92820i −0.414781 + 0.718421i
\(94\) 0 0
\(95\) −2.00000 − 3.46410i −0.205196 − 0.355409i
\(96\) 0.500000 − 0.866025i 0.0510310 − 0.0883883i
\(97\) 2.00000 0.203069 0.101535 − 0.994832i \(-0.467625\pi\)
0.101535 + 0.994832i \(0.467625\pi\)
\(98\) 0 0
\(99\) 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 1470.2.i.o.961.1 2
7.2 even 3 30.2.a.a.1.1 ✓ 1
7.3 odd 6 1470.2.i.q.361.1 2
7.4 even 3 inner 1470.2.i.o.361.1 2
7.5 odd 6 1470.2.a.d.1.1 1
7.6 odd 2 1470.2.i.q.961.1 2
21.2 odd 6 90.2.a.c.1.1 1
21.5 even 6 4410.2.a.z.1.1 1
28.23 odd 6 240.2.a.b.1.1 1
35.2 odd 12 150.2.c.a.49.1 2
35.9 even 6 150.2.a.b.1.1 1
35.19 odd 6 7350.2.a.ct.1.1 1
35.23 odd 12 150.2.c.a.49.2 2
56.37 even 6 960.2.a.e.1.1 1
56.51 odd 6 960.2.a.p.1.1 1
63.2 odd 6 810.2.e.b.271.1 2
63.16 even 3 810.2.e.l.271.1 2
63.23 odd 6 810.2.e.b.541.1 2
63.58 even 3 810.2.e.l.541.1 2
77.65 odd 6 3630.2.a.w.1.1 1
84.23 even 6 720.2.a.j.1.1 1
91.44 odd 12 5070.2.b.k.1351.2 2
91.51 even 6 5070.2.a.w.1.1 1
91.86 odd 12 5070.2.b.k.1351.1 2
105.2 even 12 450.2.c.b.199.2 2
105.23 even 12 450.2.c.b.199.1 2
105.44 odd 6 450.2.a.d.1.1 1
112.37 even 12 3840.2.k.y.1921.1 2
112.51 odd 12 3840.2.k.f.1921.1 2
112.93 even 12 3840.2.k.y.1921.2 2
112.107 odd 12 3840.2.k.f.1921.2 2
119.16 even 6 8670.2.a.g.1.1 1
140.23 even 12 1200.2.f.e.49.1 2
140.79 odd 6 1200.2.a.k.1.1 1
140.107 even 12 1200.2.f.e.49.2 2
168.107 even 6 2880.2.a.q.1.1 1
168.149 odd 6 2880.2.a.a.1.1 1
280.37 odd 12 4800.2.f.p.3649.2 2
280.93 odd 12 4800.2.f.p.3649.1 2
280.107 even 12 4800.2.f.w.3649.1 2
280.149 even 6 4800.2.a.cq.1.1 1
280.163 even 12 4800.2.f.w.3649.2 2
280.219 odd 6 4800.2.a.d.1.1 1
420.23 odd 12 3600.2.f.i.2449.1 2
420.107 odd 12 3600.2.f.i.2449.2 2
420.359 even 6 3600.2.a.f.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
30.2.a.a.1.1 ✓ 1 7.2 even 3
90.2.a.c.1.1 1 21.2 odd 6
150.2.a.b.1.1 1 35.9 even 6
150.2.c.a.49.1 2 35.2 odd 12
150.2.c.a.49.2 2 35.23 odd 12
240.2.a.b.1.1 1 28.23 odd 6
450.2.a.d.1.1 1 105.44 odd 6
450.2.c.b.199.1 2 105.23 even 12
450.2.c.b.199.2 2 105.2 even 12
720.2.a.j.1.1 1 84.23 even 6
810.2.e.b.271.1 2 63.2 odd 6
810.2.e.b.541.1 2 63.23 odd 6
810.2.e.l.271.1 2 63.16 even 3
810.2.e.l.541.1 2 63.58 even 3
960.2.a.e.1.1 1 56.37 even 6
960.2.a.p.1.1 1 56.51 odd 6
1200.2.a.k.1.1 1 140.79 odd 6
1200.2.f.e.49.1 2 140.23 even 12
1200.2.f.e.49.2 2 140.107 even 12
1470.2.a.d.1.1 1 7.5 odd 6
1470.2.i.o.361.1 2 7.4 even 3 inner
1470.2.i.o.961.1 2 1.1 even 1 trivial
1470.2.i.q.361.1 2 7.3 odd 6
1470.2.i.q.961.1 2 7.6 odd 2
2880.2.a.a.1.1 1 168.149 odd 6
2880.2.a.q.1.1 1 168.107 even 6
3600.2.a.f.1.1 1 420.359 even 6
3600.2.f.i.2449.1 2 420.23 odd 12
3600.2.f.i.2449.2 2 420.107 odd 12
3630.2.a.w.1.1 1 77.65 odd 6
3840.2.k.f.1921.1 2 112.51 odd 12
3840.2.k.f.1921.2 2 112.107 odd 12
3840.2.k.y.1921.1 2 112.37 even 12
3840.2.k.y.1921.2 2 112.93 even 12
4410.2.a.z.1.1 1 21.5 even 6
4800.2.a.d.1.1 1 280.219 odd 6
4800.2.a.cq.1.1 1 280.149 even 6
4800.2.f.p.3649.1 2 280.93 odd 12
4800.2.f.p.3649.2 2 280.37 odd 12
4800.2.f.w.3649.1 2 280.107 even 12
4800.2.f.w.3649.2 2 280.163 even 12
5070.2.a.w.1.1 1 91.51 even 6
5070.2.b.k.1351.1 2 91.86 odd 12
5070.2.b.k.1351.2 2 91.44 odd 12
7350.2.a.ct.1.1 1 35.19 odd 6
8670.2.a.g.1.1 1 119.16 even 6