Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [5929,2,Mod(1,5929)] 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("5929.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5929, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 5929 = 7^{2} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5929.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [10,-1,3,9,2,9,0,3,9,7,0,-9,1,0,-1,15,19,-17] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(18)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(47.3433033584\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - x^{9} - 14x^{8} + 14x^{7} + 61x^{6} - 57x^{5} - 84x^{4} + 63x^{3} + 20x^{2} - 15x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 77)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.9
Root \(-2.23710\) of defining polynomial
Character \(\chi\) \(=\) 5929.1

$q$-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)
 
\(f(q)\) \(=\) \(q+2.23710 q^{2} -1.33493 q^{3} +3.00462 q^{4} +0.866333 q^{5} -2.98638 q^{6} +2.24744 q^{8} -1.21795 q^{9} +1.93808 q^{10} -4.01098 q^{12} +3.38113 q^{13} -1.15650 q^{15} -0.981487 q^{16} +2.19568 q^{17} -2.72468 q^{18} +3.58857 q^{19} +2.60300 q^{20} +1.66967 q^{23} -3.00019 q^{24} -4.24947 q^{25} +7.56393 q^{26} +5.63069 q^{27} +1.53233 q^{29} -2.58720 q^{30} +8.87690 q^{31} -6.69057 q^{32} +4.91197 q^{34} -3.65948 q^{36} +3.08531 q^{37} +8.02800 q^{38} -4.51359 q^{39} +1.94703 q^{40} -0.657599 q^{41} -9.76359 q^{43} -1.05515 q^{45} +3.73523 q^{46} -13.1955 q^{47} +1.31022 q^{48} -9.50649 q^{50} -2.93109 q^{51} +10.1590 q^{52} +6.56376 q^{53} +12.5964 q^{54} -4.79051 q^{57} +3.42799 q^{58} +14.5692 q^{59} -3.47484 q^{60} -5.79638 q^{61} +19.8585 q^{62} -13.0045 q^{64} +2.92919 q^{65} +5.28376 q^{67} +6.59720 q^{68} -2.22891 q^{69} +7.83420 q^{71} -2.73727 q^{72} +5.91734 q^{73} +6.90214 q^{74} +5.67276 q^{75} +10.7823 q^{76} -10.0974 q^{78} +11.7544 q^{79} -0.850295 q^{80} -3.86275 q^{81} -1.47112 q^{82} +0.563588 q^{83} +1.90219 q^{85} -21.8421 q^{86} -2.04557 q^{87} -3.52203 q^{89} -2.36048 q^{90} +5.01674 q^{92} -11.8501 q^{93} -29.5197 q^{94} +3.10890 q^{95} +8.93148 q^{96} +13.9139 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - q^{2} + 3 q^{3} + 9 q^{4} + 2 q^{5} + 9 q^{6} + 3 q^{8} + 9 q^{9} + 7 q^{10} - 9 q^{12} + q^{13} - q^{15} + 15 q^{16} + 19 q^{17} - 17 q^{18} + 28 q^{19} + 5 q^{20} + 7 q^{23} + 19 q^{24} - 8 q^{25}+ \cdots - 4 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.23710 1.58187 0.790935 − 0.611900i \(-0.209595\pi\)
0.790935 + 0.611900i \(0.209595\pi\)
\(3\) −1.33493 −0.770725 −0.385362 − 0.922765i \(-0.625924\pi\)
−0.385362 + 0.922765i \(0.625924\pi\)
\(4\) 3.00462 1.50231
\(5\) 0.866333 0.387436 0.193718 − 0.981057i \(-0.437945\pi\)
0.193718 + 0.981057i \(0.437945\pi\)
\(6\) −2.98638 −1.21919
\(7\) 0 0
\(8\) 2.24744 0.794591
\(9\) −1.21795 −0.405983
\(10\) 1.93808 0.612873
\(11\) 0 0
\(12\) −4.01098 −1.15787
\(13\) 3.38113 0.937757 0.468878 − 0.883263i \(-0.344658\pi\)
0.468878 + 0.883263i \(0.344658\pi\)
\(14\) 0 0
\(15\) −1.15650 −0.298607
\(16\) −0.981487 −0.245372
\(17\) 2.19568 0.532531 0.266266 − 0.963900i \(-0.414210\pi\)
0.266266 + 0.963900i \(0.414210\pi\)
\(18\) −2.72468 −0.642213
\(19\) 3.58857 0.823275 0.411637 − 0.911348i \(-0.364957\pi\)
0.411637 + 0.911348i \(0.364957\pi\)
\(20\) 2.60300 0.582050
\(21\) 0 0
\(22\) 0 0
\(23\) 1.66967 0.348151 0.174076 − 0.984732i \(-0.444306\pi\)
0.174076 + 0.984732i \(0.444306\pi\)
\(24\) −3.00019 −0.612411
\(25\) −4.24947 −0.849893
\(26\) 7.56393 1.48341
\(27\) 5.63069 1.08363
\(28\) 0 0
\(29\) 1.53233 0.284547 0.142274 − 0.989827i \(-0.454559\pi\)
0.142274 + 0.989827i \(0.454559\pi\)
\(30\) −2.58720 −0.472357
\(31\) 8.87690 1.59434 0.797169 − 0.603757i \(-0.206330\pi\)
0.797169 + 0.603757i \(0.206330\pi\)
\(32\) −6.69057 −1.18274
\(33\) 0 0
\(34\) 4.91197 0.842395
\(35\) 0 0
\(36\) −3.65948 −0.609913
\(37\) 3.08531 0.507221 0.253611 − 0.967306i \(-0.418382\pi\)
0.253611 + 0.967306i \(0.418382\pi\)
\(38\) 8.02800 1.30231
\(39\) −4.51359 −0.722752
\(40\) 1.94703 0.307853
\(41\) −0.657599 −0.102700 −0.0513498 − 0.998681i \(-0.516352\pi\)
−0.0513498 + 0.998681i \(0.516352\pi\)
\(42\) 0 0
\(43\) −9.76359 −1.48893 −0.744467 − 0.667660i \(-0.767296\pi\)
−0.744467 + 0.667660i \(0.767296\pi\)
\(44\) 0 0
\(45\) −1.05515 −0.157293
\(46\) 3.73523 0.550730
\(47\) −13.1955 −1.92476 −0.962382 − 0.271701i \(-0.912414\pi\)
−0.962382 + 0.271701i \(0.912414\pi\)
\(48\) 1.31022 0.189114
\(49\) 0 0
\(50\) −9.50649 −1.34442
\(51\) −2.93109 −0.410435
\(52\) 10.1590 1.40880
\(53\) 6.56376 0.901602 0.450801 − 0.892625i \(-0.351138\pi\)
0.450801 + 0.892625i \(0.351138\pi\)
\(54\) 12.5964 1.71416
\(55\) 0 0
\(56\) 0 0
\(57\) −4.79051 −0.634518
\(58\) 3.42799 0.450117
\(59\) 14.5692 1.89675 0.948373 − 0.317156i \(-0.102728\pi\)
0.948373 + 0.317156i \(0.102728\pi\)
\(60\) −3.47484 −0.448600
\(61\) −5.79638 −0.742150 −0.371075 − 0.928603i \(-0.621011\pi\)
−0.371075 + 0.928603i \(0.621011\pi\)
\(62\) 19.8585 2.52203
\(63\) 0 0
\(64\) −13.0045 −1.62556
\(65\) 2.92919 0.363321
\(66\) 0 0
\(67\) 5.28376 0.645514 0.322757 − 0.946482i \(-0.395390\pi\)
0.322757 + 0.946482i \(0.395390\pi\)
\(68\) 6.59720 0.800028
\(69\) −2.22891 −0.268329
\(70\) 0 0
\(71\) 7.83420 0.929749 0.464874 − 0.885377i \(-0.346099\pi\)
0.464874 + 0.885377i \(0.346099\pi\)
\(72\) −2.73727 −0.322591
\(73\) 5.91734 0.692572 0.346286 − 0.938129i \(-0.387443\pi\)
0.346286 + 0.938129i \(0.387443\pi\)
\(74\) 6.90214 0.802358
\(75\) 5.67276 0.655034
\(76\) 10.7823 1.23682
\(77\) 0 0
\(78\) −10.0974 −1.14330
\(79\) 11.7544 1.32247 0.661237 − 0.750177i \(-0.270032\pi\)
0.661237 + 0.750177i \(0.270032\pi\)
\(80\) −0.850295 −0.0950658
\(81\) −3.86275 −0.429194
\(82\) −1.47112 −0.162458
\(83\) 0.563588 0.0618619 0.0309309 − 0.999522i \(-0.490153\pi\)
0.0309309 + 0.999522i \(0.490153\pi\)
\(84\) 0 0
\(85\) 1.90219 0.206322
\(86\) −21.8421 −2.35530
\(87\) −2.04557 −0.219308
\(88\) 0 0
\(89\) −3.52203 −0.373334 −0.186667 − 0.982423i \(-0.559769\pi\)
−0.186667 + 0.982423i \(0.559769\pi\)
\(90\) −2.36048 −0.248816
\(91\) 0 0
\(92\) 5.01674 0.523031
\(93\) −11.8501 −1.22880
\(94\) −29.5197 −3.04472
\(95\) 3.10890 0.318966
\(96\) 8.93148 0.911565
\(97\) 13.9139 1.41274 0.706370 − 0.707842i \(-0.250331\pi\)
0.706370 + 0.707842i \(0.250331\pi\)
\(98\) 0 0
\(99\) 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 5929.2.a.bx.1.9 10
7.3 odd 6 847.2.e.i.485.2 20
7.5 odd 6 847.2.e.i.606.2 20
7.6 odd 2 5929.2.a.bw.1.9 10
11.5 even 5 539.2.f.g.344.1 20
11.9 even 5 539.2.f.g.246.1 20
11.10 odd 2 5929.2.a.bz.1.2 10
77.3 odd 30 847.2.n.i.9.5 40
77.5 odd 30 77.2.m.b.25.5 yes 40
77.9 even 15 539.2.q.h.312.1 40
77.10 even 6 847.2.e.h.485.9 20
77.16 even 15 539.2.q.h.410.5 40
77.17 even 30 847.2.n.j.366.5 40
77.19 even 30 847.2.n.h.130.5 40
77.20 odd 10 539.2.f.h.246.1 20
77.24 even 30 847.2.n.j.807.1 40
77.26 odd 30 847.2.n.i.753.5 40
77.27 odd 10 539.2.f.h.344.1 20
77.31 odd 30 77.2.m.b.37.5 yes 40
77.38 odd 30 77.2.m.b.58.1 yes 40
77.40 even 30 847.2.n.h.753.1 40
77.47 odd 30 847.2.n.i.130.1 40
77.52 even 30 847.2.n.h.9.1 40
77.53 even 15 539.2.q.h.422.5 40
77.54 even 6 847.2.e.h.606.9 20
77.59 odd 30 847.2.n.i.632.1 40
77.60 even 15 539.2.q.h.520.1 40
77.61 even 30 847.2.n.j.487.1 40
77.68 even 30 847.2.n.j.81.5 40
77.73 even 30 847.2.n.h.632.5 40
77.75 odd 30 77.2.m.b.4.1 ✓ 40
77.76 even 2 5929.2.a.by.1.2 10
231.5 even 30 693.2.by.b.487.1 40
231.38 even 30 693.2.by.b.289.5 40
231.152 even 30 693.2.by.b.235.5 40
231.185 even 30 693.2.by.b.37.1 40
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
77.2.m.b.4.1 ✓ 40 77.75 odd 30
77.2.m.b.25.5 yes 40 77.5 odd 30
77.2.m.b.37.5 yes 40 77.31 odd 30
77.2.m.b.58.1 yes 40 77.38 odd 30
539.2.f.g.246.1 20 11.9 even 5
539.2.f.g.344.1 20 11.5 even 5
539.2.f.h.246.1 20 77.20 odd 10
539.2.f.h.344.1 20 77.27 odd 10
539.2.q.h.312.1 40 77.9 even 15
539.2.q.h.410.5 40 77.16 even 15
539.2.q.h.422.5 40 77.53 even 15
539.2.q.h.520.1 40 77.60 even 15
693.2.by.b.37.1 40 231.185 even 30
693.2.by.b.235.5 40 231.152 even 30
693.2.by.b.289.5 40 231.38 even 30
693.2.by.b.487.1 40 231.5 even 30
847.2.e.h.485.9 20 77.10 even 6
847.2.e.h.606.9 20 77.54 even 6
847.2.e.i.485.2 20 7.3 odd 6
847.2.e.i.606.2 20 7.5 odd 6
847.2.n.h.9.1 40 77.52 even 30
847.2.n.h.130.5 40 77.19 even 30
847.2.n.h.632.5 40 77.73 even 30
847.2.n.h.753.1 40 77.40 even 30
847.2.n.i.9.5 40 77.3 odd 30
847.2.n.i.130.1 40 77.47 odd 30
847.2.n.i.632.1 40 77.59 odd 30
847.2.n.i.753.5 40 77.26 odd 30
847.2.n.j.81.5 40 77.68 even 30
847.2.n.j.366.5 40 77.17 even 30
847.2.n.j.487.1 40 77.61 even 30
847.2.n.j.807.1 40 77.24 even 30
5929.2.a.bw.1.9 10 7.6 odd 2
5929.2.a.bx.1.9 10 1.1 even 1 trivial
5929.2.a.by.1.2 10 77.76 even 2
5929.2.a.bz.1.2 10 11.10 odd 2