Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [3381,2,Mod(1,3381)] 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("3381.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3381, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 3381 = 3 \cdot 7^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3381.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,-3,-2,3,5,3,0,-6,2,-10,-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: yes
Analytic conductor: \(26.9974209234\)
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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 483)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(1.61803\) of defining polynomial
Character \(\chi\) \(=\) 3381.1

$q$-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)
 
\(f(q)\) \(=\) \(q-2.61803 q^{2} -1.00000 q^{3} +4.85410 q^{4} +3.61803 q^{5} +2.61803 q^{6} -7.47214 q^{8} +1.00000 q^{9} -9.47214 q^{10} +3.47214 q^{11} -4.85410 q^{12} +4.61803 q^{13} -3.61803 q^{15} +9.85410 q^{16} +1.00000 q^{17} -2.61803 q^{18} +3.00000 q^{19} +17.5623 q^{20} -9.09017 q^{22} -1.00000 q^{23} +7.47214 q^{24} +8.09017 q^{25} -12.0902 q^{26} -1.00000 q^{27} +1.47214 q^{29} +9.47214 q^{30} +8.23607 q^{31} -10.8541 q^{32} -3.47214 q^{33} -2.61803 q^{34} +4.85410 q^{36} -7.47214 q^{37} -7.85410 q^{38} -4.61803 q^{39} -27.0344 q^{40} -8.70820 q^{41} -3.09017 q^{43} +16.8541 q^{44} +3.61803 q^{45} +2.61803 q^{46} +11.7082 q^{47} -9.85410 q^{48} -21.1803 q^{50} -1.00000 q^{51} +22.4164 q^{52} -5.61803 q^{53} +2.61803 q^{54} +12.5623 q^{55} -3.00000 q^{57} -3.85410 q^{58} +12.8541 q^{59} -17.5623 q^{60} +10.7984 q^{61} -21.5623 q^{62} +8.70820 q^{64} +16.7082 q^{65} +9.09017 q^{66} +10.8541 q^{67} +4.85410 q^{68} +1.00000 q^{69} -15.0902 q^{71} -7.47214 q^{72} +10.7082 q^{73} +19.5623 q^{74} -8.09017 q^{75} +14.5623 q^{76} +12.0902 q^{78} +9.47214 q^{79} +35.6525 q^{80} +1.00000 q^{81} +22.7984 q^{82} +3.00000 q^{83} +3.61803 q^{85} +8.09017 q^{86} -1.47214 q^{87} -25.9443 q^{88} -9.85410 q^{89} -9.47214 q^{90} -4.85410 q^{92} -8.23607 q^{93} -30.6525 q^{94} +10.8541 q^{95} +10.8541 q^{96} -5.00000 q^{97} +3.47214 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 3 q^{2} - 2 q^{3} + 3 q^{4} + 5 q^{5} + 3 q^{6} - 6 q^{8} + 2 q^{9} - 10 q^{10} - 2 q^{11} - 3 q^{12} + 7 q^{13} - 5 q^{15} + 13 q^{16} + 2 q^{17} - 3 q^{18} + 6 q^{19} + 15 q^{20} - 7 q^{22} - 2 q^{23}+ \cdots - 2 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.61803 −1.85123 −0.925615 − 0.378467i \(-0.876451\pi\)
−0.925615 + 0.378467i \(0.876451\pi\)
\(3\) −1.00000 −0.577350
\(4\) 4.85410 2.42705
\(5\) 3.61803 1.61803 0.809017 − 0.587785i \(-0.200000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(6\) 2.61803 1.06881
\(7\) 0 0
\(8\) −7.47214 −2.64180
\(9\) 1.00000 0.333333
\(10\) −9.47214 −2.99535
\(11\) 3.47214 1.04689 0.523444 − 0.852060i \(-0.324647\pi\)
0.523444 + 0.852060i \(0.324647\pi\)
\(12\) −4.85410 −1.40126
\(13\) 4.61803 1.28081 0.640406 − 0.768036i \(-0.278766\pi\)
0.640406 + 0.768036i \(0.278766\pi\)
\(14\) 0 0
\(15\) −3.61803 −0.934172
\(16\) 9.85410 2.46353
\(17\) 1.00000 0.242536 0.121268 − 0.992620i \(-0.461304\pi\)
0.121268 + 0.992620i \(0.461304\pi\)
\(18\) −2.61803 −0.617077
\(19\) 3.00000 0.688247 0.344124 − 0.938924i \(-0.388176\pi\)
0.344124 + 0.938924i \(0.388176\pi\)
\(20\) 17.5623 3.92705
\(21\) 0 0
\(22\) −9.09017 −1.93803
\(23\) −1.00000 −0.208514
\(24\) 7.47214 1.52524
\(25\) 8.09017 1.61803
\(26\) −12.0902 −2.37108
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 1.47214 0.273369 0.136684 − 0.990615i \(-0.456355\pi\)
0.136684 + 0.990615i \(0.456355\pi\)
\(30\) 9.47214 1.72937
\(31\) 8.23607 1.47924 0.739621 − 0.673024i \(-0.235005\pi\)
0.739621 + 0.673024i \(0.235005\pi\)
\(32\) −10.8541 −1.91875
\(33\) −3.47214 −0.604421
\(34\) −2.61803 −0.448989
\(35\) 0 0
\(36\) 4.85410 0.809017
\(37\) −7.47214 −1.22841 −0.614206 − 0.789146i \(-0.710524\pi\)
−0.614206 + 0.789146i \(0.710524\pi\)
\(38\) −7.85410 −1.27410
\(39\) −4.61803 −0.739477
\(40\) −27.0344 −4.27452
\(41\) −8.70820 −1.35999 −0.679996 − 0.733215i \(-0.738019\pi\)
−0.679996 + 0.733215i \(0.738019\pi\)
\(42\) 0 0
\(43\) −3.09017 −0.471246 −0.235623 − 0.971844i \(-0.575713\pi\)
−0.235623 + 0.971844i \(0.575713\pi\)
\(44\) 16.8541 2.54085
\(45\) 3.61803 0.539345
\(46\) 2.61803 0.386008
\(47\) 11.7082 1.70782 0.853909 − 0.520423i \(-0.174226\pi\)
0.853909 + 0.520423i \(0.174226\pi\)
\(48\) −9.85410 −1.42232
\(49\) 0 0
\(50\) −21.1803 −2.99535
\(51\) −1.00000 −0.140028
\(52\) 22.4164 3.10860
\(53\) −5.61803 −0.771696 −0.385848 − 0.922562i \(-0.626091\pi\)
−0.385848 + 0.922562i \(0.626091\pi\)
\(54\) 2.61803 0.356269
\(55\) 12.5623 1.69390
\(56\) 0 0
\(57\) −3.00000 −0.397360
\(58\) −3.85410 −0.506068
\(59\) 12.8541 1.67346 0.836731 − 0.547614i \(-0.184464\pi\)
0.836731 + 0.547614i \(0.184464\pi\)
\(60\) −17.5623 −2.26728
\(61\) 10.7984 1.38259 0.691295 − 0.722573i \(-0.257040\pi\)
0.691295 + 0.722573i \(0.257040\pi\)
\(62\) −21.5623 −2.73842
\(63\) 0 0
\(64\) 8.70820 1.08853
\(65\) 16.7082 2.07240
\(66\) 9.09017 1.11892
\(67\) 10.8541 1.32604 0.663020 − 0.748602i \(-0.269275\pi\)
0.663020 + 0.748602i \(0.269275\pi\)
\(68\) 4.85410 0.588646
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) −15.0902 −1.79087 −0.895437 − 0.445189i \(-0.853137\pi\)
−0.895437 + 0.445189i \(0.853137\pi\)
\(72\) −7.47214 −0.880600
\(73\) 10.7082 1.25330 0.626650 − 0.779301i \(-0.284425\pi\)
0.626650 + 0.779301i \(0.284425\pi\)
\(74\) 19.5623 2.27407
\(75\) −8.09017 −0.934172
\(76\) 14.5623 1.67041
\(77\) 0 0
\(78\) 12.0902 1.36894
\(79\) 9.47214 1.06570 0.532849 − 0.846210i \(-0.321121\pi\)
0.532849 + 0.846210i \(0.321121\pi\)
\(80\) 35.6525 3.98607
\(81\) 1.00000 0.111111
\(82\) 22.7984 2.51766
\(83\) 3.00000 0.329293 0.164646 − 0.986353i \(-0.447352\pi\)
0.164646 + 0.986353i \(0.447352\pi\)
\(84\) 0 0
\(85\) 3.61803 0.392431
\(86\) 8.09017 0.872385
\(87\) −1.47214 −0.157830
\(88\) −25.9443 −2.76567
\(89\) −9.85410 −1.04453 −0.522266 − 0.852782i \(-0.674913\pi\)
−0.522266 + 0.852782i \(0.674913\pi\)
\(90\) −9.47214 −0.998451
\(91\) 0 0
\(92\) −4.85410 −0.506075
\(93\) −8.23607 −0.854040
\(94\) −30.6525 −3.16156
\(95\) 10.8541 1.11361
\(96\) 10.8541 1.10779
\(97\) −5.00000 −0.507673 −0.253837 − 0.967247i \(-0.581693\pi\)
−0.253837 + 0.967247i \(0.581693\pi\)
\(98\) 0 0
\(99\) 3.47214 0.348963
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 3381.2.a.n.1.1 2
7.6 odd 2 483.2.a.c.1.1 ✓ 2
21.20 even 2 1449.2.a.k.1.2 2
28.27 even 2 7728.2.a.v.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
483.2.a.c.1.1 ✓ 2 7.6 odd 2
1449.2.a.k.1.2 2 21.20 even 2
3381.2.a.n.1.1 2 1.1 even 1 trivial
7728.2.a.v.1.1 2 28.27 even 2