Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [9702,2,Mod(1,9702)] 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("9702.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9702, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 9702 = 2 \cdot 3^{2} \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9702.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,-2,0,2,4,0,0,-2,0,-4,-2,0,-2,0,0,2,4,0,-4,4,0,2,4,0,2,2,0, 0,6,0,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(31)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(77.4708600410\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 154)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.41421\) of defining polynomial
Character \(\chi\) \(=\) 9702.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} +3.41421 q^{5} -1.00000 q^{8} -3.41421 q^{10} -1.00000 q^{11} +1.82843 q^{13} +1.00000 q^{16} +7.65685 q^{17} -3.41421 q^{19} +3.41421 q^{20} +1.00000 q^{22} -2.24264 q^{23} +6.65685 q^{25} -1.82843 q^{26} +8.65685 q^{29} -4.00000 q^{31} -1.00000 q^{32} -7.65685 q^{34} -6.58579 q^{37} +3.41421 q^{38} -3.41421 q^{40} +2.58579 q^{41} +5.65685 q^{43} -1.00000 q^{44} +2.24264 q^{46} -6.48528 q^{47} -6.65685 q^{50} +1.82843 q^{52} +11.8995 q^{53} -3.41421 q^{55} -8.65685 q^{58} +8.41421 q^{59} +6.17157 q^{61} +4.00000 q^{62} +1.00000 q^{64} +6.24264 q^{65} +11.2426 q^{67} +7.65685 q^{68} -3.07107 q^{71} -6.58579 q^{73} +6.58579 q^{74} -3.41421 q^{76} -4.75736 q^{79} +3.41421 q^{80} -2.58579 q^{82} -16.1421 q^{83} +26.1421 q^{85} -5.65685 q^{86} +1.00000 q^{88} +4.48528 q^{89} -2.24264 q^{92} +6.48528 q^{94} -11.6569 q^{95} +1.82843 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{4} + 4 q^{5} - 2 q^{8} - 4 q^{10} - 2 q^{11} - 2 q^{13} + 2 q^{16} + 4 q^{17} - 4 q^{19} + 4 q^{20} + 2 q^{22} + 4 q^{23} + 2 q^{25} + 2 q^{26} + 6 q^{29} - 8 q^{31} - 2 q^{32} - 4 q^{34}+ \cdots - 2 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) 3.41421 1.52688 0.763441 − 0.645877i \(-0.223508\pi\)
0.763441 + 0.645877i \(0.223508\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) −3.41421 −1.07967
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) 1.82843 0.507114 0.253557 − 0.967320i \(-0.418399\pi\)
0.253557 + 0.967320i \(0.418399\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 7.65685 1.85706 0.928530 − 0.371257i \(-0.121073\pi\)
0.928530 + 0.371257i \(0.121073\pi\)
\(18\) 0 0
\(19\) −3.41421 −0.783274 −0.391637 − 0.920120i \(-0.628091\pi\)
−0.391637 + 0.920120i \(0.628091\pi\)
\(20\) 3.41421 0.763441
\(21\) 0 0
\(22\) 1.00000 0.213201
\(23\) −2.24264 −0.467623 −0.233811 − 0.972282i \(-0.575120\pi\)
−0.233811 + 0.972282i \(0.575120\pi\)
\(24\) 0 0
\(25\) 6.65685 1.33137
\(26\) −1.82843 −0.358584
\(27\) 0 0
\(28\) 0 0
\(29\) 8.65685 1.60754 0.803769 − 0.594942i \(-0.202825\pi\)
0.803769 + 0.594942i \(0.202825\pi\)
\(30\) 0 0
\(31\) −4.00000 −0.718421 −0.359211 − 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) −7.65685 −1.31314
\(35\) 0 0
\(36\) 0 0
\(37\) −6.58579 −1.08270 −0.541348 − 0.840798i \(-0.682086\pi\)
−0.541348 + 0.840798i \(0.682086\pi\)
\(38\) 3.41421 0.553859
\(39\) 0 0
\(40\) −3.41421 −0.539835
\(41\) 2.58579 0.403832 0.201916 − 0.979403i \(-0.435283\pi\)
0.201916 + 0.979403i \(0.435283\pi\)
\(42\) 0 0
\(43\) 5.65685 0.862662 0.431331 − 0.902194i \(-0.358044\pi\)
0.431331 + 0.902194i \(0.358044\pi\)
\(44\) −1.00000 −0.150756
\(45\) 0 0
\(46\) 2.24264 0.330659
\(47\) −6.48528 −0.945976 −0.472988 − 0.881069i \(-0.656825\pi\)
−0.472988 + 0.881069i \(0.656825\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) −6.65685 −0.941421
\(51\) 0 0
\(52\) 1.82843 0.253557
\(53\) 11.8995 1.63452 0.817261 − 0.576268i \(-0.195492\pi\)
0.817261 + 0.576268i \(0.195492\pi\)
\(54\) 0 0
\(55\) −3.41421 −0.460372
\(56\) 0 0
\(57\) 0 0
\(58\) −8.65685 −1.13670
\(59\) 8.41421 1.09544 0.547719 − 0.836663i \(-0.315496\pi\)
0.547719 + 0.836663i \(0.315496\pi\)
\(60\) 0 0
\(61\) 6.17157 0.790189 0.395094 − 0.918640i \(-0.370712\pi\)
0.395094 + 0.918640i \(0.370712\pi\)
\(62\) 4.00000 0.508001
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 6.24264 0.774304
\(66\) 0 0
\(67\) 11.2426 1.37351 0.686754 − 0.726890i \(-0.259035\pi\)
0.686754 + 0.726890i \(0.259035\pi\)
\(68\) 7.65685 0.928530
\(69\) 0 0
\(70\) 0 0
\(71\) −3.07107 −0.364469 −0.182234 − 0.983255i \(-0.558333\pi\)
−0.182234 + 0.983255i \(0.558333\pi\)
\(72\) 0 0
\(73\) −6.58579 −0.770808 −0.385404 − 0.922748i \(-0.625938\pi\)
−0.385404 + 0.922748i \(0.625938\pi\)
\(74\) 6.58579 0.765582
\(75\) 0 0
\(76\) −3.41421 −0.391637
\(77\) 0 0
\(78\) 0 0
\(79\) −4.75736 −0.535245 −0.267622 − 0.963524i \(-0.586238\pi\)
−0.267622 + 0.963524i \(0.586238\pi\)
\(80\) 3.41421 0.381721
\(81\) 0 0
\(82\) −2.58579 −0.285552
\(83\) −16.1421 −1.77183 −0.885915 − 0.463848i \(-0.846468\pi\)
−0.885915 + 0.463848i \(0.846468\pi\)
\(84\) 0 0
\(85\) 26.1421 2.83551
\(86\) −5.65685 −0.609994
\(87\) 0 0
\(88\) 1.00000 0.106600
\(89\) 4.48528 0.475439 0.237719 − 0.971334i \(-0.423600\pi\)
0.237719 + 0.971334i \(0.423600\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −2.24264 −0.233811
\(93\) 0 0
\(94\) 6.48528 0.668906
\(95\) −11.6569 −1.19597
\(96\) 0 0
\(97\) 1.82843 0.185649 0.0928243 − 0.995683i \(-0.470411\pi\)
0.0928243 + 0.995683i \(0.470411\pi\)
\(98\) 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 9702.2.a.cx.1.2 2
3.2 odd 2 1078.2.a.t.1.2 2
7.2 even 3 1386.2.k.t.991.1 4
7.4 even 3 1386.2.k.t.793.1 4
7.6 odd 2 9702.2.a.ch.1.1 2
12.11 even 2 8624.2.a.cc.1.1 2
21.2 odd 6 154.2.e.e.67.1 yes 4
21.5 even 6 1078.2.e.m.67.2 4
21.11 odd 6 154.2.e.e.23.1 ✓ 4
21.17 even 6 1078.2.e.m.177.2 4
21.20 even 2 1078.2.a.x.1.1 2
84.11 even 6 1232.2.q.f.177.2 4
84.23 even 6 1232.2.q.f.529.2 4
84.83 odd 2 8624.2.a.bh.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
154.2.e.e.23.1 ✓ 4 21.11 odd 6
154.2.e.e.67.1 yes 4 21.2 odd 6
1078.2.a.t.1.2 2 3.2 odd 2
1078.2.a.x.1.1 2 21.20 even 2
1078.2.e.m.67.2 4 21.5 even 6
1078.2.e.m.177.2 4 21.17 even 6
1232.2.q.f.177.2 4 84.11 even 6
1232.2.q.f.529.2 4 84.23 even 6
1386.2.k.t.793.1 4 7.4 even 3
1386.2.k.t.991.1 4 7.2 even 3
8624.2.a.bh.1.2 2 84.83 odd 2
8624.2.a.cc.1.1 2 12.11 even 2
9702.2.a.ch.1.1 2 7.6 odd 2
9702.2.a.cx.1.2 2 1.1 even 1 trivial