Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2070,2,Mod(1,2070)] 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("2070.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2070, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 2070 = 2 \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2070.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,-2,0,2,-2,0,1,-2,0,2,-1,0,-3] 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: \(16.5290332184\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 230)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.61803\) of defining polynomial
Character \(\chi\) \(=\) 2070.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +1.61803 q^{7} -1.00000 q^{8} +1.00000 q^{10} -3.85410 q^{11} +4.09017 q^{13} -1.61803 q^{14} +1.00000 q^{16} +5.09017 q^{17} -4.85410 q^{19} -1.00000 q^{20} +3.85410 q^{22} -1.00000 q^{23} +1.00000 q^{25} -4.09017 q^{26} +1.61803 q^{28} +4.76393 q^{29} -2.09017 q^{31} -1.00000 q^{32} -5.09017 q^{34} -1.61803 q^{35} -2.47214 q^{37} +4.85410 q^{38} +1.00000 q^{40} +12.3262 q^{41} -3.85410 q^{44} +1.00000 q^{46} -9.70820 q^{47} -4.38197 q^{49} -1.00000 q^{50} +4.09017 q^{52} +8.47214 q^{53} +3.85410 q^{55} -1.61803 q^{56} -4.76393 q^{58} +11.7082 q^{59} +6.32624 q^{61} +2.09017 q^{62} +1.00000 q^{64} -4.09017 q^{65} +5.52786 q^{67} +5.09017 q^{68} +1.61803 q^{70} -7.09017 q^{71} -1.23607 q^{73} +2.47214 q^{74} -4.85410 q^{76} -6.23607 q^{77} +10.4721 q^{79} -1.00000 q^{80} -12.3262 q^{82} -10.9443 q^{83} -5.09017 q^{85} +3.85410 q^{88} +1.52786 q^{89} +6.61803 q^{91} -1.00000 q^{92} +9.70820 q^{94} +4.85410 q^{95} +14.6180 q^{97} +4.38197 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} + q^{7} - 2 q^{8} + 2 q^{10} - q^{11} - 3 q^{13} - q^{14} + 2 q^{16} - q^{17} - 3 q^{19} - 2 q^{20} + q^{22} - 2 q^{23} + 2 q^{25} + 3 q^{26} + q^{28} + 14 q^{29}+ \cdots + 11 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 1.61803 0.611559 0.305780 − 0.952102i \(-0.401083\pi\)
0.305780 + 0.952102i \(0.401083\pi\)
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) 1.00000 0.316228
\(11\) −3.85410 −1.16206 −0.581028 − 0.813884i \(-0.697349\pi\)
−0.581028 + 0.813884i \(0.697349\pi\)
\(12\) 0 0
\(13\) 4.09017 1.13441 0.567205 − 0.823577i \(-0.308025\pi\)
0.567205 + 0.823577i \(0.308025\pi\)
\(14\) −1.61803 −0.432438
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 5.09017 1.23455 0.617274 − 0.786748i \(-0.288237\pi\)
0.617274 + 0.786748i \(0.288237\pi\)
\(18\) 0 0
\(19\) −4.85410 −1.11361 −0.556804 − 0.830644i \(-0.687972\pi\)
−0.556804 + 0.830644i \(0.687972\pi\)
\(20\) −1.00000 −0.223607
\(21\) 0 0
\(22\) 3.85410 0.821697
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) −4.09017 −0.802148
\(27\) 0 0
\(28\) 1.61803 0.305780
\(29\) 4.76393 0.884640 0.442320 − 0.896857i \(-0.354156\pi\)
0.442320 + 0.896857i \(0.354156\pi\)
\(30\) 0 0
\(31\) −2.09017 −0.375406 −0.187703 − 0.982226i \(-0.560104\pi\)
−0.187703 + 0.982226i \(0.560104\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) −5.09017 −0.872957
\(35\) −1.61803 −0.273498
\(36\) 0 0
\(37\) −2.47214 −0.406417 −0.203208 − 0.979136i \(-0.565137\pi\)
−0.203208 + 0.979136i \(0.565137\pi\)
\(38\) 4.85410 0.787439
\(39\) 0 0
\(40\) 1.00000 0.158114
\(41\) 12.3262 1.92503 0.962517 − 0.271220i \(-0.0874270\pi\)
0.962517 + 0.271220i \(0.0874270\pi\)
\(42\) 0 0
\(43\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) −3.85410 −0.581028
\(45\) 0 0
\(46\) 1.00000 0.147442
\(47\) −9.70820 −1.41609 −0.708044 − 0.706169i \(-0.750422\pi\)
−0.708044 + 0.706169i \(0.750422\pi\)
\(48\) 0 0
\(49\) −4.38197 −0.625995
\(50\) −1.00000 −0.141421
\(51\) 0 0
\(52\) 4.09017 0.567205
\(53\) 8.47214 1.16374 0.581869 − 0.813283i \(-0.302322\pi\)
0.581869 + 0.813283i \(0.302322\pi\)
\(54\) 0 0
\(55\) 3.85410 0.519687
\(56\) −1.61803 −0.216219
\(57\) 0 0
\(58\) −4.76393 −0.625535
\(59\) 11.7082 1.52428 0.762139 − 0.647413i \(-0.224149\pi\)
0.762139 + 0.647413i \(0.224149\pi\)
\(60\) 0 0
\(61\) 6.32624 0.809992 0.404996 − 0.914319i \(-0.367273\pi\)
0.404996 + 0.914319i \(0.367273\pi\)
\(62\) 2.09017 0.265452
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −4.09017 −0.507323
\(66\) 0 0
\(67\) 5.52786 0.675336 0.337668 − 0.941265i \(-0.390362\pi\)
0.337668 + 0.941265i \(0.390362\pi\)
\(68\) 5.09017 0.617274
\(69\) 0 0
\(70\) 1.61803 0.193392
\(71\) −7.09017 −0.841448 −0.420724 − 0.907189i \(-0.638224\pi\)
−0.420724 + 0.907189i \(0.638224\pi\)
\(72\) 0 0
\(73\) −1.23607 −0.144671 −0.0723354 − 0.997380i \(-0.523045\pi\)
−0.0723354 + 0.997380i \(0.523045\pi\)
\(74\) 2.47214 0.287380
\(75\) 0 0
\(76\) −4.85410 −0.556804
\(77\) −6.23607 −0.710666
\(78\) 0 0
\(79\) 10.4721 1.17821 0.589104 − 0.808057i \(-0.299481\pi\)
0.589104 + 0.808057i \(0.299481\pi\)
\(80\) −1.00000 −0.111803
\(81\) 0 0
\(82\) −12.3262 −1.36121
\(83\) −10.9443 −1.20129 −0.600645 − 0.799516i \(-0.705089\pi\)
−0.600645 + 0.799516i \(0.705089\pi\)
\(84\) 0 0
\(85\) −5.09017 −0.552106
\(86\) 0 0
\(87\) 0 0
\(88\) 3.85410 0.410849
\(89\) 1.52786 0.161953 0.0809766 − 0.996716i \(-0.474196\pi\)
0.0809766 + 0.996716i \(0.474196\pi\)
\(90\) 0 0
\(91\) 6.61803 0.693758
\(92\) −1.00000 −0.104257
\(93\) 0 0
\(94\) 9.70820 1.00132
\(95\) 4.85410 0.498020
\(96\) 0 0
\(97\) 14.6180 1.48424 0.742118 − 0.670269i \(-0.233821\pi\)
0.742118 + 0.670269i \(0.233821\pi\)
\(98\) 4.38197 0.442645
\(99\) 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 2070.2.a.u.1.2 2
3.2 odd 2 230.2.a.c.1.1 ✓ 2
12.11 even 2 1840.2.a.l.1.2 2
15.2 even 4 1150.2.b.i.599.4 4
15.8 even 4 1150.2.b.i.599.1 4
15.14 odd 2 1150.2.a.j.1.2 2
24.5 odd 2 7360.2.a.bh.1.2 2
24.11 even 2 7360.2.a.bn.1.1 2
60.59 even 2 9200.2.a.bu.1.1 2
69.68 even 2 5290.2.a.o.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.2.a.c.1.1 ✓ 2 3.2 odd 2
1150.2.a.j.1.2 2 15.14 odd 2
1150.2.b.i.599.1 4 15.8 even 4
1150.2.b.i.599.4 4 15.2 even 4
1840.2.a.l.1.2 2 12.11 even 2
2070.2.a.u.1.2 2 1.1 even 1 trivial
5290.2.a.o.1.1 2 69.68 even 2
7360.2.a.bh.1.2 2 24.5 odd 2
7360.2.a.bn.1.1 2 24.11 even 2
9200.2.a.bu.1.1 2 60.59 even 2