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 1.1
Root \(-0.618034\) 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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -0.618034 q^{7} -1.00000 q^{8} +1.00000 q^{10} +2.85410 q^{11} -7.09017 q^{13} +0.618034 q^{14} +1.00000 q^{16} -6.09017 q^{17} +1.85410 q^{19} -1.00000 q^{20} -2.85410 q^{22} -1.00000 q^{23} +1.00000 q^{25} +7.09017 q^{26} -0.618034 q^{28} +9.23607 q^{29} +9.09017 q^{31} -1.00000 q^{32} +6.09017 q^{34} +0.618034 q^{35} +6.47214 q^{37} -1.85410 q^{38} +1.00000 q^{40} -3.32624 q^{41} +2.85410 q^{44} +1.00000 q^{46} +3.70820 q^{47} -6.61803 q^{49} -1.00000 q^{50} -7.09017 q^{52} -0.472136 q^{53} -2.85410 q^{55} +0.618034 q^{56} -9.23607 q^{58} -1.70820 q^{59} -9.32624 q^{61} -9.09017 q^{62} +1.00000 q^{64} +7.09017 q^{65} +14.4721 q^{67} -6.09017 q^{68} -0.618034 q^{70} +4.09017 q^{71} +3.23607 q^{73} -6.47214 q^{74} +1.85410 q^{76} -1.76393 q^{77} +1.52786 q^{79} -1.00000 q^{80} +3.32624 q^{82} +6.94427 q^{83} +6.09017 q^{85} -2.85410 q^{88} +10.4721 q^{89} +4.38197 q^{91} -1.00000 q^{92} -3.70820 q^{94} -1.85410 q^{95} +12.3820 q^{97} +6.61803 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} + q^{7} - 2 q^{8} + 2 q^{10} - q^{11} - 3 q^{13} - q^{14} + 2 q^{16} - q^{17} - 3 q^{19} - 2 q^{20} + q^{22} - 2 q^{23} + 2 q^{25} + 3 q^{26} + q^{28} + 14 q^{29}+ \cdots + 11 q^{98}+O(q^{100}) \) 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
\(7\) −0.618034 −0.233595 −0.116797 − 0.993156i \(-0.537263\pi\)
−0.116797 + 0.993156i \(0.537263\pi\)
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) 1.00000 0.316228
\(11\) 2.85410 0.860544 0.430272 − 0.902699i \(-0.358418\pi\)
0.430272 + 0.902699i \(0.358418\pi\)
\(12\) 0 0
\(13\) −7.09017 −1.96646 −0.983230 − 0.182372i \(-0.941623\pi\)
−0.983230 + 0.182372i \(0.941623\pi\)
\(14\) 0.618034 0.165177
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −6.09017 −1.47708 −0.738542 − 0.674208i \(-0.764485\pi\)
−0.738542 + 0.674208i \(0.764485\pi\)
\(18\) 0 0
\(19\) 1.85410 0.425360 0.212680 − 0.977122i \(-0.431781\pi\)
0.212680 + 0.977122i \(0.431781\pi\)
\(20\) −1.00000 −0.223607
\(21\) 0 0
\(22\) −2.85410 −0.608497
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 7.09017 1.39050
\(27\) 0 0
\(28\) −0.618034 −0.116797
\(29\) 9.23607 1.71509 0.857547 − 0.514405i \(-0.171987\pi\)
0.857547 + 0.514405i \(0.171987\pi\)
\(30\) 0 0
\(31\) 9.09017 1.63264 0.816321 − 0.577598i \(-0.196010\pi\)
0.816321 + 0.577598i \(0.196010\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) 6.09017 1.04446
\(35\) 0.618034 0.104467
\(36\) 0 0
\(37\) 6.47214 1.06401 0.532006 − 0.846740i \(-0.321438\pi\)
0.532006 + 0.846740i \(0.321438\pi\)
\(38\) −1.85410 −0.300775
\(39\) 0 0
\(40\) 1.00000 0.158114
\(41\) −3.32624 −0.519471 −0.259736 − 0.965680i \(-0.583635\pi\)
−0.259736 + 0.965680i \(0.583635\pi\)
\(42\) 0 0
\(43\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) 2.85410 0.430272
\(45\) 0 0
\(46\) 1.00000 0.147442
\(47\) 3.70820 0.540897 0.270449 − 0.962734i \(-0.412828\pi\)
0.270449 + 0.962734i \(0.412828\pi\)
\(48\) 0 0
\(49\) −6.61803 −0.945433
\(50\) −1.00000 −0.141421
\(51\) 0 0
\(52\) −7.09017 −0.983230
\(53\) −0.472136 −0.0648529 −0.0324264 − 0.999474i \(-0.510323\pi\)
−0.0324264 + 0.999474i \(0.510323\pi\)
\(54\) 0 0
\(55\) −2.85410 −0.384847
\(56\) 0.618034 0.0825883
\(57\) 0 0
\(58\) −9.23607 −1.21276
\(59\) −1.70820 −0.222389 −0.111195 − 0.993799i \(-0.535468\pi\)
−0.111195 + 0.993799i \(0.535468\pi\)
\(60\) 0 0
\(61\) −9.32624 −1.19410 −0.597051 − 0.802203i \(-0.703661\pi\)
−0.597051 + 0.802203i \(0.703661\pi\)
\(62\) −9.09017 −1.15445
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 7.09017 0.879427
\(66\) 0 0
\(67\) 14.4721 1.76805 0.884026 − 0.467437i \(-0.154823\pi\)
0.884026 + 0.467437i \(0.154823\pi\)
\(68\) −6.09017 −0.738542
\(69\) 0 0
\(70\) −0.618034 −0.0738692
\(71\) 4.09017 0.485414 0.242707 − 0.970100i \(-0.421965\pi\)
0.242707 + 0.970100i \(0.421965\pi\)
\(72\) 0 0
\(73\) 3.23607 0.378753 0.189377 − 0.981905i \(-0.439353\pi\)
0.189377 + 0.981905i \(0.439353\pi\)
\(74\) −6.47214 −0.752371
\(75\) 0 0
\(76\) 1.85410 0.212680
\(77\) −1.76393 −0.201019
\(78\) 0 0
\(79\) 1.52786 0.171898 0.0859491 − 0.996300i \(-0.472608\pi\)
0.0859491 + 0.996300i \(0.472608\pi\)
\(80\) −1.00000 −0.111803
\(81\) 0 0
\(82\) 3.32624 0.367322
\(83\) 6.94427 0.762233 0.381116 − 0.924527i \(-0.375540\pi\)
0.381116 + 0.924527i \(0.375540\pi\)
\(84\) 0 0
\(85\) 6.09017 0.660572
\(86\) 0 0
\(87\) 0 0
\(88\) −2.85410 −0.304248
\(89\) 10.4721 1.11004 0.555022 − 0.831836i \(-0.312710\pi\)
0.555022 + 0.831836i \(0.312710\pi\)
\(90\) 0 0
\(91\) 4.38197 0.459355
\(92\) −1.00000 −0.104257
\(93\) 0 0
\(94\) −3.70820 −0.382472
\(95\) −1.85410 −0.190227
\(96\) 0 0
\(97\) 12.3820 1.25720 0.628599 − 0.777730i \(-0.283629\pi\)
0.628599 + 0.777730i \(0.283629\pi\)
\(98\) 6.61803 0.668522
\(99\) 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.1.1 2
3.2 odd 2 230.2.a.c.1.2 ✓ 2
12.11 even 2 1840.2.a.l.1.1 2
15.2 even 4 1150.2.b.i.599.3 4
15.8 even 4 1150.2.b.i.599.2 4
15.14 odd 2 1150.2.a.j.1.1 2
24.5 odd 2 7360.2.a.bh.1.1 2
24.11 even 2 7360.2.a.bn.1.2 2
60.59 even 2 9200.2.a.bu.1.2 2
69.68 even 2 5290.2.a.o.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.2.a.c.1.2 ✓ 2 3.2 odd 2
1150.2.a.j.1.1 2 15.14 odd 2
1150.2.b.i.599.2 4 15.8 even 4
1150.2.b.i.599.3 4 15.2 even 4
1840.2.a.l.1.1 2 12.11 even 2
2070.2.a.u.1.1 2 1.1 even 1 trivial
5290.2.a.o.1.2 2 69.68 even 2
7360.2.a.bh.1.1 2 24.5 odd 2
7360.2.a.bn.1.2 2 24.11 even 2
9200.2.a.bu.1.2 2 60.59 even 2