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,-2] 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: yes
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.g.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)
 
\(f(q)\) \(=\) \(q+(-0.500000 + 0.866025i) q^{2} +(0.500000 + 0.866025i) q^{3} +(-0.500000 - 0.866025i) q^{4} +(-0.500000 + 0.866025i) q^{5} -1.00000 q^{6} +1.00000 q^{8} +(-0.500000 + 0.866025i) q^{9} +(-0.500000 - 0.866025i) q^{10} +(-1.00000 - 1.73205i) q^{11} +(0.500000 - 0.866025i) q^{12} +2.00000 q^{13} -1.00000 q^{15} +(-0.500000 + 0.866025i) q^{16} +(2.00000 + 3.46410i) q^{17} +(-0.500000 - 0.866025i) q^{18} +1.00000 q^{20} +2.00000 q^{22} +(-4.00000 + 6.92820i) q^{23} +(0.500000 + 0.866025i) q^{24} +(-0.500000 - 0.866025i) q^{25} +(-1.00000 + 1.73205i) q^{26} -1.00000 q^{27} +(0.500000 - 0.866025i) q^{30} +(1.00000 + 1.73205i) q^{31} +(-0.500000 - 0.866025i) q^{32} +(1.00000 - 1.73205i) q^{33} -4.00000 q^{34} +1.00000 q^{36} +(-4.00000 + 6.92820i) q^{37} +(1.00000 + 1.73205i) q^{39} +(-0.500000 + 0.866025i) q^{40} -2.00000 q^{41} -2.00000 q^{43} +(-1.00000 + 1.73205i) q^{44} +(-0.500000 - 0.866025i) q^{45} +(-4.00000 - 6.92820i) q^{46} +(-5.00000 + 8.66025i) q^{47} -1.00000 q^{48} +1.00000 q^{50} +(-2.00000 + 3.46410i) q^{51} +(-1.00000 - 1.73205i) q^{52} +(1.00000 + 1.73205i) q^{53} +(0.500000 - 0.866025i) q^{54} +2.00000 q^{55} +(-2.00000 - 3.46410i) q^{59} +(0.500000 + 0.866025i) q^{60} +(5.00000 - 8.66025i) q^{61} -2.00000 q^{62} +1.00000 q^{64} +(-1.00000 + 1.73205i) q^{65} +(1.00000 + 1.73205i) q^{66} +(-1.00000 - 1.73205i) q^{67} +(2.00000 - 3.46410i) q^{68} -8.00000 q^{69} -12.0000 q^{71} +(-0.500000 + 0.866025i) q^{72} +(-5.00000 - 8.66025i) q^{73} +(-4.00000 - 6.92820i) q^{74} +(0.500000 - 0.866025i) q^{75} -2.00000 q^{78} +(-8.00000 + 13.8564i) q^{79} +(-0.500000 - 0.866025i) q^{80} +(-0.500000 - 0.866025i) q^{81} +(1.00000 - 1.73205i) q^{82} +16.0000 q^{83} -4.00000 q^{85} +(1.00000 - 1.73205i) q^{86} +(-1.00000 - 1.73205i) q^{88} +(-7.00000 + 12.1244i) q^{89} +1.00000 q^{90} +8.00000 q^{92} +(-1.00000 + 1.73205i) q^{93} +(-5.00000 - 8.66025i) q^{94} +(0.500000 - 0.866025i) q^{96} +6.00000 q^{97} +2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} + q^{3} - q^{4} - q^{5} - 2 q^{6} + 2 q^{8} - q^{9} - q^{10} - 2 q^{11} + q^{12} + 4 q^{13} - 2 q^{15} - q^{16} + 4 q^{17} - q^{18} + 2 q^{20} + 4 q^{22} - 8 q^{23} + q^{24} - q^{25}+ \cdots + 4 q^{99}+O(q^{100}) \) 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
\(11\) −1.00000 − 1.73205i −0.301511 − 0.522233i 0.674967 − 0.737848i \(-0.264158\pi\)
−0.976478 + 0.215615i \(0.930824\pi\)
\(12\) 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
\(17\) 2.00000 + 3.46410i 0.485071 + 0.840168i 0.999853 − 0.0171533i \(-0.00546033\pi\)
−0.514782 + 0.857321i \(0.672127\pi\)
\(18\) −0.500000 − 0.866025i −0.117851 − 0.204124i
\(19\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(20\) 1.00000 0.223607
\(21\) 0 0
\(22\) 2.00000 0.426401
\(23\) −4.00000 + 6.92820i −0.834058 + 1.44463i 0.0607377 + 0.998154i \(0.480655\pi\)
−0.894795 + 0.446476i \(0.852679\pi\)
\(24\) 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
\(29\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0.500000 − 0.866025i 0.0912871 − 0.158114i
\(31\) 1.00000 + 1.73205i 0.179605 + 0.311086i 0.941745 − 0.336327i \(-0.109185\pi\)
−0.762140 + 0.647412i \(0.775851\pi\)
\(32\) −0.500000 − 0.866025i −0.0883883 − 0.153093i
\(33\) 1.00000 − 1.73205i 0.174078 − 0.301511i
\(34\) −4.00000 −0.685994
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −4.00000 + 6.92820i −0.657596 + 1.13899i 0.323640 + 0.946180i \(0.395093\pi\)
−0.981236 + 0.192809i \(0.938240\pi\)
\(38\) 0 0
\(39\) 1.00000 + 1.73205i 0.160128 + 0.277350i
\(40\) −0.500000 + 0.866025i −0.0790569 + 0.136931i
\(41\) −2.00000 −0.312348 −0.156174 − 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) 0 0
\(43\) −2.00000 −0.304997 −0.152499 − 0.988304i \(-0.548732\pi\)
−0.152499 + 0.988304i \(0.548732\pi\)
\(44\) −1.00000 + 1.73205i −0.150756 + 0.261116i
\(45\) −0.500000 − 0.866025i −0.0745356 − 0.129099i
\(46\) −4.00000 − 6.92820i −0.589768 − 1.02151i
\(47\) −5.00000 + 8.66025i −0.729325 + 1.26323i 0.227844 + 0.973698i \(0.426832\pi\)
−0.957169 + 0.289530i \(0.906501\pi\)
\(48\) −1.00000 −0.144338
\(49\) 0 0
\(50\) 1.00000 0.141421
\(51\) −2.00000 + 3.46410i −0.280056 + 0.485071i
\(52\) −1.00000 − 1.73205i −0.138675 − 0.240192i
\(53\) 1.00000 + 1.73205i 0.137361 + 0.237915i 0.926497 − 0.376303i \(-0.122805\pi\)
−0.789136 + 0.614218i \(0.789471\pi\)
\(54\) 0.500000 − 0.866025i 0.0680414 − 0.117851i
\(55\) 2.00000 0.269680
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −2.00000 − 3.46410i −0.260378 − 0.450988i 0.705965 − 0.708247i \(-0.250514\pi\)
−0.966342 + 0.257260i \(0.917180\pi\)
\(60\) 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 \(-0.0544754\pi\)
\(62\) −2.00000 −0.254000
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −1.00000 + 1.73205i −0.124035 + 0.214834i
\(66\) 1.00000 + 1.73205i 0.123091 + 0.213201i
\(67\) −1.00000 − 1.73205i −0.122169 − 0.211604i 0.798454 − 0.602056i \(-0.205652\pi\)
−0.920623 + 0.390453i \(0.872318\pi\)
\(68\) 2.00000 − 3.46410i 0.242536 − 0.420084i
\(69\) −8.00000 −0.963087
\(70\) 0 0
\(71\) −12.0000 −1.42414 −0.712069 − 0.702109i \(-0.752242\pi\)
−0.712069 + 0.702109i \(0.752242\pi\)
\(72\) −0.500000 + 0.866025i −0.0589256 + 0.102062i
\(73\) −5.00000 − 8.66025i −0.585206 − 1.01361i −0.994850 − 0.101361i \(-0.967680\pi\)
0.409644 − 0.912245i \(-0.365653\pi\)
\(74\) −4.00000 − 6.92820i −0.464991 − 0.805387i
\(75\) 0.500000 − 0.866025i 0.0577350 − 0.100000i
\(76\) 0 0
\(77\) 0 0
\(78\) −2.00000 −0.226455
\(79\) −8.00000 + 13.8564i −0.900070 + 1.55897i −0.0726692 + 0.997356i \(0.523152\pi\)
−0.827401 + 0.561611i \(0.810182\pi\)
\(80\) −0.500000 − 0.866025i −0.0559017 − 0.0968246i
\(81\) −0.500000 − 0.866025i −0.0555556 − 0.0962250i
\(82\) 1.00000 − 1.73205i 0.110432 − 0.191273i
\(83\) 16.0000 1.75623 0.878114 − 0.478451i \(-0.158802\pi\)
0.878114 + 0.478451i \(0.158802\pi\)
\(84\) 0 0
\(85\) −4.00000 −0.433861
\(86\) 1.00000 − 1.73205i 0.107833 − 0.186772i
\(87\) 0 0
\(88\) −1.00000 − 1.73205i −0.106600 − 0.184637i
\(89\) −7.00000 + 12.1244i −0.741999 + 1.28518i 0.209585 + 0.977790i \(0.432789\pi\)
−0.951584 + 0.307389i \(0.900545\pi\)
\(90\) 1.00000 0.105409
\(91\) 0 0
\(92\) 8.00000 0.834058
\(93\) −1.00000 + 1.73205i −0.103695 + 0.179605i
\(94\) −5.00000 − 8.66025i −0.515711 − 0.893237i
\(95\) 0 0
\(96\) 0.500000 − 0.866025i 0.0510310 − 0.0883883i
\(97\) 6.00000 0.609208 0.304604 − 0.952479i \(-0.401476\pi\)
0.304604 + 0.952479i \(0.401476\pi\)
\(98\) 0 0
\(99\) 2.00000 0.201008
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.g.961.1 2
7.2 even 3 1470.2.a.n.1.1 ✓ 1
7.3 odd 6 1470.2.i.c.361.1 2
7.4 even 3 inner 1470.2.i.g.361.1 2
7.5 odd 6 1470.2.a.p.1.1 yes 1
7.6 odd 2 1470.2.i.c.961.1 2
21.2 odd 6 4410.2.a.e.1.1 1
21.5 even 6 4410.2.a.n.1.1 1
35.9 even 6 7350.2.a.bh.1.1 1
35.19 odd 6 7350.2.a.o.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1470.2.a.n.1.1 ✓ 1 7.2 even 3
1470.2.a.p.1.1 yes 1 7.5 odd 6
1470.2.i.c.361.1 2 7.3 odd 6
1470.2.i.c.961.1 2 7.6 odd 2
1470.2.i.g.361.1 2 7.4 even 3 inner
1470.2.i.g.961.1 2 1.1 even 1 trivial
4410.2.a.e.1.1 1 21.2 odd 6
4410.2.a.n.1.1 1 21.5 even 6
7350.2.a.o.1.1 1 35.19 odd 6
7350.2.a.bh.1.1 1 35.9 even 6