Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [637,2,Mod(1,637)] 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("637.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(637, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 637 = 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 637.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,-3,0,3,0,5,0,-6,4,-5,-6,-15,-2] 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: \(5.08647060876\)
Analytic rank: \(1\)
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 91)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(1.61803\) of defining polynomial
Character \(\chi\) \(=\) 637.1

$q$-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)
 
\(f(q)\) \(=\) \(q-2.61803 q^{2} -2.23607 q^{3} +4.85410 q^{4} +2.23607 q^{5} +5.85410 q^{6} -7.47214 q^{8} +2.00000 q^{9} -5.85410 q^{10} -3.00000 q^{11} -10.8541 q^{12} -1.00000 q^{13} -5.00000 q^{15} +9.85410 q^{16} +1.47214 q^{17} -5.23607 q^{18} +3.00000 q^{19} +10.8541 q^{20} +7.85410 q^{22} -8.23607 q^{23} +16.7082 q^{24} +2.61803 q^{26} +2.23607 q^{27} +4.47214 q^{29} +13.0902 q^{30} +5.00000 q^{31} -10.8541 q^{32} +6.70820 q^{33} -3.85410 q^{34} +9.70820 q^{36} +4.70820 q^{37} -7.85410 q^{38} +2.23607 q^{39} -16.7082 q^{40} -4.47214 q^{41} -8.00000 q^{43} -14.5623 q^{44} +4.47214 q^{45} +21.5623 q^{46} -7.47214 q^{47} -22.0344 q^{48} -3.29180 q^{51} -4.85410 q^{52} -7.47214 q^{53} -5.85410 q^{54} -6.70820 q^{55} -6.70820 q^{57} -11.7082 q^{58} -1.47214 q^{59} -24.2705 q^{60} +3.00000 q^{61} -13.0902 q^{62} +8.70820 q^{64} -2.23607 q^{65} -17.5623 q^{66} -3.00000 q^{67} +7.14590 q^{68} +18.4164 q^{69} -8.94427 q^{71} -14.9443 q^{72} -2.70820 q^{73} -12.3262 q^{74} +14.5623 q^{76} -5.85410 q^{78} -2.70820 q^{79} +22.0344 q^{80} -11.0000 q^{81} +11.7082 q^{82} +3.29180 q^{85} +20.9443 q^{86} -10.0000 q^{87} +22.4164 q^{88} +2.23607 q^{89} -11.7082 q^{90} -39.9787 q^{92} -11.1803 q^{93} +19.5623 q^{94} +6.70820 q^{95} +24.2705 q^{96} +9.41641 q^{97} -6.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 3 q^{2} + 3 q^{4} + 5 q^{6} - 6 q^{8} + 4 q^{9} - 5 q^{10} - 6 q^{11} - 15 q^{12} - 2 q^{13} - 10 q^{15} + 13 q^{16} - 6 q^{17} - 6 q^{18} + 6 q^{19} + 15 q^{20} + 9 q^{22} - 12 q^{23} + 20 q^{24}+ \cdots - 12 q^{99}+O(q^{100}) \) 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 \(0.876451\pi\)
\(3\) −2.23607 −1.29099 −0.645497 − 0.763763i \(-0.723350\pi\)
−0.645497 + 0.763763i \(0.723350\pi\)
\(4\) 4.85410 2.42705
\(5\) 2.23607 1.00000 0.500000 − 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(6\) 5.85410 2.38993
\(7\) 0 0
\(8\) −7.47214 −2.64180
\(9\) 2.00000 0.666667
\(10\) −5.85410 −1.85123
\(11\) −3.00000 −0.904534 −0.452267 − 0.891883i \(-0.649385\pi\)
−0.452267 + 0.891883i \(0.649385\pi\)
\(12\) −10.8541 −3.13331
\(13\) −1.00000 −0.277350
\(14\) 0 0
\(15\) −5.00000 −1.29099
\(16\) 9.85410 2.46353
\(17\) 1.47214 0.357045 0.178523 − 0.983936i \(-0.442868\pi\)
0.178523 + 0.983936i \(0.442868\pi\)
\(18\) −5.23607 −1.23415
\(19\) 3.00000 0.688247 0.344124 − 0.938924i \(-0.388176\pi\)
0.344124 + 0.938924i \(0.388176\pi\)
\(20\) 10.8541 2.42705
\(21\) 0 0
\(22\) 7.85410 1.67450
\(23\) −8.23607 −1.71734 −0.858669 − 0.512530i \(-0.828708\pi\)
−0.858669 + 0.512530i \(0.828708\pi\)
\(24\) 16.7082 3.41055
\(25\) 0 0
\(26\) 2.61803 0.513439
\(27\) 2.23607 0.430331
\(28\) 0 0
\(29\) 4.47214 0.830455 0.415227 − 0.909718i \(-0.363702\pi\)
0.415227 + 0.909718i \(0.363702\pi\)
\(30\) 13.0902 2.38993
\(31\) 5.00000 0.898027 0.449013 − 0.893525i \(-0.351776\pi\)
0.449013 + 0.893525i \(0.351776\pi\)
\(32\) −10.8541 −1.91875
\(33\) 6.70820 1.16775
\(34\) −3.85410 −0.660973
\(35\) 0 0
\(36\) 9.70820 1.61803
\(37\) 4.70820 0.774024 0.387012 − 0.922075i \(-0.373507\pi\)
0.387012 + 0.922075i \(0.373507\pi\)
\(38\) −7.85410 −1.27410
\(39\) 2.23607 0.358057
\(40\) −16.7082 −2.64180
\(41\) −4.47214 −0.698430 −0.349215 − 0.937043i \(-0.613552\pi\)
−0.349215 + 0.937043i \(0.613552\pi\)
\(42\) 0 0
\(43\) −8.00000 −1.21999 −0.609994 − 0.792406i \(-0.708828\pi\)
−0.609994 + 0.792406i \(0.708828\pi\)
\(44\) −14.5623 −2.19535
\(45\) 4.47214 0.666667
\(46\) 21.5623 3.17919
\(47\) −7.47214 −1.08992 −0.544962 − 0.838461i \(-0.683456\pi\)
−0.544962 + 0.838461i \(0.683456\pi\)
\(48\) −22.0344 −3.18040
\(49\) 0 0
\(50\) 0 0
\(51\) −3.29180 −0.460944
\(52\) −4.85410 −0.673143
\(53\) −7.47214 −1.02638 −0.513188 − 0.858276i \(-0.671536\pi\)
−0.513188 + 0.858276i \(0.671536\pi\)
\(54\) −5.85410 −0.796642
\(55\) −6.70820 −0.904534
\(56\) 0 0
\(57\) −6.70820 −0.888523
\(58\) −11.7082 −1.53736
\(59\) −1.47214 −0.191656 −0.0958279 − 0.995398i \(-0.530550\pi\)
−0.0958279 + 0.995398i \(0.530550\pi\)
\(60\) −24.2705 −3.13331
\(61\) 3.00000 0.384111 0.192055 − 0.981384i \(-0.438485\pi\)
0.192055 + 0.981384i \(0.438485\pi\)
\(62\) −13.0902 −1.66245
\(63\) 0 0
\(64\) 8.70820 1.08853
\(65\) −2.23607 −0.277350
\(66\) −17.5623 −2.16177
\(67\) −3.00000 −0.366508 −0.183254 − 0.983066i \(-0.558663\pi\)
−0.183254 + 0.983066i \(0.558663\pi\)
\(68\) 7.14590 0.866567
\(69\) 18.4164 2.21707
\(70\) 0 0
\(71\) −8.94427 −1.06149 −0.530745 − 0.847532i \(-0.678088\pi\)
−0.530745 + 0.847532i \(0.678088\pi\)
\(72\) −14.9443 −1.76120
\(73\) −2.70820 −0.316971 −0.158486 − 0.987361i \(-0.550661\pi\)
−0.158486 + 0.987361i \(0.550661\pi\)
\(74\) −12.3262 −1.43290
\(75\) 0 0
\(76\) 14.5623 1.67041
\(77\) 0 0
\(78\) −5.85410 −0.662847
\(79\) −2.70820 −0.304697 −0.152348 − 0.988327i \(-0.548684\pi\)
−0.152348 + 0.988327i \(0.548684\pi\)
\(80\) 22.0344 2.46353
\(81\) −11.0000 −1.22222
\(82\) 11.7082 1.29295
\(83\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 3.29180 0.357045
\(86\) 20.9443 2.25848
\(87\) −10.0000 −1.07211
\(88\) 22.4164 2.38960
\(89\) 2.23607 0.237023 0.118511 − 0.992953i \(-0.462188\pi\)
0.118511 + 0.992953i \(0.462188\pi\)
\(90\) −11.7082 −1.23415
\(91\) 0 0
\(92\) −39.9787 −4.16807
\(93\) −11.1803 −1.15935
\(94\) 19.5623 2.01770
\(95\) 6.70820 0.688247
\(96\) 24.2705 2.47710
\(97\) 9.41641 0.956091 0.478046 − 0.878335i \(-0.341345\pi\)
0.478046 + 0.878335i \(0.341345\pi\)
\(98\) 0 0
\(99\) −6.00000 −0.603023
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 637.2.a.f.1.1 2
3.2 odd 2 5733.2.a.v.1.2 2
7.2 even 3 91.2.e.b.53.2 ✓ 4
7.3 odd 6 637.2.e.h.79.2 4
7.4 even 3 91.2.e.b.79.2 yes 4
7.5 odd 6 637.2.e.h.508.2 4
7.6 odd 2 637.2.a.e.1.1 2
13.12 even 2 8281.2.a.z.1.2 2
21.2 odd 6 819.2.j.c.235.1 4
21.11 odd 6 819.2.j.c.352.1 4
21.20 even 2 5733.2.a.w.1.2 2
28.11 odd 6 1456.2.r.j.625.1 4
28.23 odd 6 1456.2.r.j.417.1 4
91.25 even 6 1183.2.e.d.170.1 4
91.51 even 6 1183.2.e.d.508.1 4
91.90 odd 2 8281.2.a.ba.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
91.2.e.b.53.2 ✓ 4 7.2 even 3
91.2.e.b.79.2 yes 4 7.4 even 3
637.2.a.e.1.1 2 7.6 odd 2
637.2.a.f.1.1 2 1.1 even 1 trivial
637.2.e.h.79.2 4 7.3 odd 6
637.2.e.h.508.2 4 7.5 odd 6
819.2.j.c.235.1 4 21.2 odd 6
819.2.j.c.352.1 4 21.11 odd 6
1183.2.e.d.170.1 4 91.25 even 6
1183.2.e.d.508.1 4 91.51 even 6
1456.2.r.j.417.1 4 28.23 odd 6
1456.2.r.j.625.1 4 28.11 odd 6
5733.2.a.v.1.2 2 3.2 odd 2
5733.2.a.w.1.2 2 21.20 even 2
8281.2.a.z.1.2 2 13.12 even 2
8281.2.a.ba.1.2 2 91.90 odd 2