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 = [5,4,0,8,-2,5,0,9,3,5,11,-5,-5] 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: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.746052.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - x^{4} - 7x^{3} + 8x + 2 \) 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 1.5
Root \(-1.72525\) 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)
 
\(f(q)\) \(=\) \(q+2.72525 q^{2} +1.34642 q^{3} +5.42699 q^{4} -2.18716 q^{5} +3.66932 q^{6} +9.33940 q^{8} -1.18716 q^{9} -5.96057 q^{10} -1.04815 q^{11} +7.30699 q^{12} -1.00000 q^{13} -2.94483 q^{15} +14.5982 q^{16} +5.29125 q^{17} -3.23532 q^{18} -0.756906 q^{19} -11.8697 q^{20} -2.85648 q^{22} +0.653584 q^{23} +12.5747 q^{24} -0.216314 q^{25} -2.72525 q^{26} -5.63766 q^{27} -3.10408 q^{29} -8.02541 q^{30} -1.02791 q^{31} +21.1050 q^{32} -1.41125 q^{33} +14.4200 q^{34} -6.44273 q^{36} -10.8932 q^{37} -2.06276 q^{38} -1.34642 q^{39} -20.4268 q^{40} -7.32040 q^{41} +0.887771 q^{43} -5.68833 q^{44} +2.59652 q^{45} +1.78118 q^{46} -2.33751 q^{47} +19.6553 q^{48} -0.589510 q^{50} +7.12422 q^{51} -5.42699 q^{52} +4.88814 q^{53} -15.3640 q^{54} +2.29249 q^{55} -1.01911 q^{57} -8.45941 q^{58} +1.04815 q^{59} -15.9816 q^{60} +12.4998 q^{61} -2.80132 q^{62} +28.3200 q^{64} +2.18716 q^{65} -3.84602 q^{66} +4.47889 q^{67} +28.7155 q^{68} +0.879996 q^{69} -6.60274 q^{71} -11.0874 q^{72} +8.28347 q^{73} -29.6868 q^{74} -0.291249 q^{75} -4.10772 q^{76} -3.66932 q^{78} +2.14014 q^{79} -31.9287 q^{80} -4.02915 q^{81} -19.9499 q^{82} +6.66558 q^{83} -11.5728 q^{85} +2.41940 q^{86} -4.17939 q^{87} -9.78914 q^{88} +5.76777 q^{89} +7.07617 q^{90} +3.54699 q^{92} -1.38400 q^{93} -6.37030 q^{94} +1.65548 q^{95} +28.4162 q^{96} +2.88777 q^{97} +1.24433 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 4 q^{2} + 8 q^{4} - 2 q^{5} + 5 q^{6} + 9 q^{8} + 3 q^{9} + 5 q^{10} + 11 q^{11} - 5 q^{12} - 5 q^{13} + 10 q^{16} + 5 q^{17} + 9 q^{18} - 9 q^{19} - q^{20} + 8 q^{22} + 10 q^{23} + 9 q^{25} - 4 q^{26}+ \cdots + 11 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.72525 1.92704 0.963521 − 0.267631i \(-0.0862408\pi\)
0.963521 + 0.267631i \(0.0862408\pi\)
\(3\) 1.34642 0.777354 0.388677 − 0.921374i \(-0.372932\pi\)
0.388677 + 0.921374i \(0.372932\pi\)
\(4\) 5.42699 2.71349
\(5\) −2.18716 −0.978129 −0.489065 − 0.872247i \(-0.662662\pi\)
−0.489065 + 0.872247i \(0.662662\pi\)
\(6\) 3.66932 1.49799
\(7\) 0 0
\(8\) 9.33940 3.30198
\(9\) −1.18716 −0.395721
\(10\) −5.96057 −1.88490
\(11\) −1.04815 −0.316031 −0.158015 − 0.987437i \(-0.550510\pi\)
−0.158015 + 0.987437i \(0.550510\pi\)
\(12\) 7.30699 2.10934
\(13\) −1.00000 −0.277350
\(14\) 0 0
\(15\) −2.94483 −0.760352
\(16\) 14.5982 3.64956
\(17\) 5.29125 1.28332 0.641658 − 0.766991i \(-0.278247\pi\)
0.641658 + 0.766991i \(0.278247\pi\)
\(18\) −3.23532 −0.762572
\(19\) −0.756906 −0.173646 −0.0868231 − 0.996224i \(-0.527671\pi\)
−0.0868231 + 0.996224i \(0.527671\pi\)
\(20\) −11.8697 −2.65415
\(21\) 0 0
\(22\) −2.85648 −0.609005
\(23\) 0.653584 0.136282 0.0681408 − 0.997676i \(-0.478293\pi\)
0.0681408 + 0.997676i \(0.478293\pi\)
\(24\) 12.5747 2.56680
\(25\) −0.216314 −0.0432628
\(26\) −2.72525 −0.534466
\(27\) −5.63766 −1.08497
\(28\) 0 0
\(29\) −3.10408 −0.576414 −0.288207 − 0.957568i \(-0.593059\pi\)
−0.288207 + 0.957568i \(0.593059\pi\)
\(30\) −8.02541 −1.46523
\(31\) −1.02791 −0.184618 −0.0923092 − 0.995730i \(-0.529425\pi\)
−0.0923092 + 0.995730i \(0.529425\pi\)
\(32\) 21.1050 3.73088
\(33\) −1.41125 −0.245668
\(34\) 14.4200 2.47301
\(35\) 0 0
\(36\) −6.44273 −1.07379
\(37\) −10.8932 −1.79084 −0.895418 − 0.445227i \(-0.853123\pi\)
−0.895418 + 0.445227i \(0.853123\pi\)
\(38\) −2.06276 −0.334624
\(39\) −1.34642 −0.215599
\(40\) −20.4268 −3.22976
\(41\) −7.32040 −1.14325 −0.571627 − 0.820514i \(-0.693688\pi\)
−0.571627 + 0.820514i \(0.693688\pi\)
\(42\) 0 0
\(43\) 0.887771 0.135384 0.0676919 − 0.997706i \(-0.478437\pi\)
0.0676919 + 0.997706i \(0.478437\pi\)
\(44\) −5.68833 −0.857547
\(45\) 2.59652 0.387067
\(46\) 1.78118 0.262621
\(47\) −2.33751 −0.340961 −0.170480 − 0.985361i \(-0.554532\pi\)
−0.170480 + 0.985361i \(0.554532\pi\)
\(48\) 19.6553 2.83700
\(49\) 0 0
\(50\) −0.589510 −0.0833692
\(51\) 7.12422 0.997591
\(52\) −5.42699 −0.752588
\(53\) 4.88814 0.671438 0.335719 − 0.941962i \(-0.391021\pi\)
0.335719 + 0.941962i \(0.391021\pi\)
\(54\) −15.3640 −2.09078
\(55\) 2.29249 0.309119
\(56\) 0 0
\(57\) −1.01911 −0.134985
\(58\) −8.45941 −1.11077
\(59\) 1.04815 0.136458 0.0682291 − 0.997670i \(-0.478265\pi\)
0.0682291 + 0.997670i \(0.478265\pi\)
\(60\) −15.9816 −2.06321
\(61\) 12.4998 1.60043 0.800217 − 0.599711i \(-0.204718\pi\)
0.800217 + 0.599711i \(0.204718\pi\)
\(62\) −2.80132 −0.355768
\(63\) 0 0
\(64\) 28.3200 3.54000
\(65\) 2.18716 0.271284
\(66\) −3.84602 −0.473412
\(67\) 4.47889 0.547183 0.273592 − 0.961846i \(-0.411788\pi\)
0.273592 + 0.961846i \(0.411788\pi\)
\(68\) 28.7155 3.48227
\(69\) 0.879996 0.105939
\(70\) 0 0
\(71\) −6.60274 −0.783601 −0.391801 − 0.920050i \(-0.628148\pi\)
−0.391801 + 0.920050i \(0.628148\pi\)
\(72\) −11.0874 −1.30666
\(73\) 8.28347 0.969507 0.484754 − 0.874651i \(-0.338909\pi\)
0.484754 + 0.874651i \(0.338909\pi\)
\(74\) −29.6868 −3.45102
\(75\) −0.291249 −0.0336305
\(76\) −4.10772 −0.471188
\(77\) 0 0
\(78\) −3.66932 −0.415469
\(79\) 2.14014 0.240785 0.120392 − 0.992726i \(-0.461585\pi\)
0.120392 + 0.992726i \(0.461585\pi\)
\(80\) −31.9287 −3.56974
\(81\) −4.02915 −0.447683
\(82\) −19.9499 −2.20310
\(83\) 6.66558 0.731642 0.365821 − 0.930685i \(-0.380788\pi\)
0.365821 + 0.930685i \(0.380788\pi\)
\(84\) 0 0
\(85\) −11.5728 −1.25525
\(86\) 2.41940 0.260890
\(87\) −4.17939 −0.448078
\(88\) −9.78914 −1.04353
\(89\) 5.76777 0.611382 0.305691 − 0.952131i \(-0.401113\pi\)
0.305691 + 0.952131i \(0.401113\pi\)
\(90\) 7.07617 0.745894
\(91\) 0 0
\(92\) 3.54699 0.369800
\(93\) −1.38400 −0.143514
\(94\) −6.37030 −0.657046
\(95\) 1.65548 0.169848
\(96\) 28.4162 2.90021
\(97\) 2.88777 0.293209 0.146604 − 0.989195i \(-0.453166\pi\)
0.146604 + 0.989195i \(0.453166\pi\)
\(98\) 0 0
\(99\) 1.24433 0.125060
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.k.1.5 5
3.2 odd 2 5733.2.a.bm.1.1 5
7.2 even 3 637.2.e.m.508.1 10
7.3 odd 6 91.2.e.c.79.1 yes 10
7.4 even 3 637.2.e.m.79.1 10
7.5 odd 6 91.2.e.c.53.1 ✓ 10
7.6 odd 2 637.2.a.l.1.5 5
13.12 even 2 8281.2.a.bx.1.1 5
21.5 even 6 819.2.j.h.235.5 10
21.17 even 6 819.2.j.h.352.5 10
21.20 even 2 5733.2.a.bl.1.1 5
28.3 even 6 1456.2.r.p.625.2 10
28.19 even 6 1456.2.r.p.417.2 10
91.12 odd 6 1183.2.e.f.508.5 10
91.38 odd 6 1183.2.e.f.170.5 10
91.90 odd 2 8281.2.a.bw.1.1 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
91.2.e.c.53.1 ✓ 10 7.5 odd 6
91.2.e.c.79.1 yes 10 7.3 odd 6
637.2.a.k.1.5 5 1.1 even 1 trivial
637.2.a.l.1.5 5 7.6 odd 2
637.2.e.m.79.1 10 7.4 even 3
637.2.e.m.508.1 10 7.2 even 3
819.2.j.h.235.5 10 21.5 even 6
819.2.j.h.352.5 10 21.17 even 6
1183.2.e.f.170.5 10 91.38 odd 6
1183.2.e.f.508.5 10 91.12 odd 6
1456.2.r.p.417.2 10 28.19 even 6
1456.2.r.p.625.2 10 28.3 even 6
5733.2.a.bl.1.1 5 21.20 even 2
5733.2.a.bm.1.1 5 3.2 odd 2
8281.2.a.bw.1.1 5 91.90 odd 2
8281.2.a.bx.1.1 5 13.12 even 2