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,-2,0,2,0,0,0,-2,0,0,2,0,0,0,0,2,0,0,0,0,0,-2,-12,0,-10,0,0, 0,8,0,0] 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: \(1\)
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: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1078)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{8} +1.00000 q^{11} -4.24264 q^{13} +1.00000 q^{16} -2.82843 q^{17} +4.24264 q^{19} -1.00000 q^{22} -6.00000 q^{23} -5.00000 q^{25} +4.24264 q^{26} +4.00000 q^{29} +7.07107 q^{31} -1.00000 q^{32} +2.82843 q^{34} +2.00000 q^{37} -4.24264 q^{38} +2.82843 q^{41} +10.0000 q^{43} +1.00000 q^{44} +6.00000 q^{46} -12.7279 q^{47} +5.00000 q^{50} -4.24264 q^{52} -2.00000 q^{53} -4.00000 q^{58} +11.3137 q^{59} +9.89949 q^{61} -7.07107 q^{62} +1.00000 q^{64} +8.00000 q^{67} -2.82843 q^{68} -16.0000 q^{71} +8.48528 q^{73} -2.00000 q^{74} +4.24264 q^{76} -8.00000 q^{79} -2.82843 q^{82} -12.7279 q^{83} -10.0000 q^{86} -1.00000 q^{88} +7.07107 q^{89} -6.00000 q^{92} +12.7279 q^{94} +7.07107 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{8} + 2 q^{11} + 2 q^{16} - 2 q^{22} - 12 q^{23} - 10 q^{25} + 8 q^{29} - 2 q^{32} + 4 q^{37} + 20 q^{43} + 2 q^{44} + 12 q^{46} + 10 q^{50} - 4 q^{53} - 8 q^{58} + 2 q^{64}+ \cdots - 12 q^{92}+O(q^{100}) \) 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
\(5\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) −4.24264 −1.17670 −0.588348 − 0.808608i \(-0.700222\pi\)
−0.588348 + 0.808608i \(0.700222\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −2.82843 −0.685994 −0.342997 − 0.939336i \(-0.611442\pi\)
−0.342997 + 0.939336i \(0.611442\pi\)
\(18\) 0 0
\(19\) 4.24264 0.973329 0.486664 − 0.873589i \(-0.338214\pi\)
0.486664 + 0.873589i \(0.338214\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −1.00000 −0.213201
\(23\) −6.00000 −1.25109 −0.625543 − 0.780189i \(-0.715123\pi\)
−0.625543 + 0.780189i \(0.715123\pi\)
\(24\) 0 0
\(25\) −5.00000 −1.00000
\(26\) 4.24264 0.832050
\(27\) 0 0
\(28\) 0 0
\(29\) 4.00000 0.742781 0.371391 − 0.928477i \(-0.378881\pi\)
0.371391 + 0.928477i \(0.378881\pi\)
\(30\) 0 0
\(31\) 7.07107 1.27000 0.635001 − 0.772512i \(-0.281000\pi\)
0.635001 + 0.772512i \(0.281000\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) 2.82843 0.485071
\(35\) 0 0
\(36\) 0 0
\(37\) 2.00000 0.328798 0.164399 − 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) −4.24264 −0.688247
\(39\) 0 0
\(40\) 0 0
\(41\) 2.82843 0.441726 0.220863 − 0.975305i \(-0.429113\pi\)
0.220863 + 0.975305i \(0.429113\pi\)
\(42\) 0 0
\(43\) 10.0000 1.52499 0.762493 − 0.646997i \(-0.223975\pi\)
0.762493 + 0.646997i \(0.223975\pi\)
\(44\) 1.00000 0.150756
\(45\) 0 0
\(46\) 6.00000 0.884652
\(47\) −12.7279 −1.85656 −0.928279 − 0.371884i \(-0.878712\pi\)
−0.928279 + 0.371884i \(0.878712\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 5.00000 0.707107
\(51\) 0 0
\(52\) −4.24264 −0.588348
\(53\) −2.00000 −0.274721 −0.137361 − 0.990521i \(-0.543862\pi\)
−0.137361 + 0.990521i \(0.543862\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) −4.00000 −0.525226
\(59\) 11.3137 1.47292 0.736460 − 0.676481i \(-0.236496\pi\)
0.736460 + 0.676481i \(0.236496\pi\)
\(60\) 0 0
\(61\) 9.89949 1.26750 0.633750 − 0.773538i \(-0.281515\pi\)
0.633750 + 0.773538i \(0.281515\pi\)
\(62\) −7.07107 −0.898027
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) 8.00000 0.977356 0.488678 − 0.872464i \(-0.337479\pi\)
0.488678 + 0.872464i \(0.337479\pi\)
\(68\) −2.82843 −0.342997
\(69\) 0 0
\(70\) 0 0
\(71\) −16.0000 −1.89885 −0.949425 − 0.313993i \(-0.898333\pi\)
−0.949425 + 0.313993i \(0.898333\pi\)
\(72\) 0 0
\(73\) 8.48528 0.993127 0.496564 − 0.868000i \(-0.334595\pi\)
0.496564 + 0.868000i \(0.334595\pi\)
\(74\) −2.00000 −0.232495
\(75\) 0 0
\(76\) 4.24264 0.486664
\(77\) 0 0
\(78\) 0 0
\(79\) −8.00000 −0.900070 −0.450035 − 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −2.82843 −0.312348
\(83\) −12.7279 −1.39707 −0.698535 − 0.715575i \(-0.746165\pi\)
−0.698535 + 0.715575i \(0.746165\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −10.0000 −1.07833
\(87\) 0 0
\(88\) −1.00000 −0.106600
\(89\) 7.07107 0.749532 0.374766 − 0.927119i \(-0.377723\pi\)
0.374766 + 0.927119i \(0.377723\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −6.00000 −0.625543
\(93\) 0 0
\(94\) 12.7279 1.31278
\(95\) 0 0
\(96\) 0 0
\(97\) 7.07107 0.717958 0.358979 − 0.933346i \(-0.383125\pi\)
0.358979 + 0.933346i \(0.383125\pi\)
\(98\) 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.co.1.1 2
3.2 odd 2 1078.2.a.v.1.1 ✓ 2
7.6 odd 2 inner 9702.2.a.co.1.2 2
12.11 even 2 8624.2.a.bz.1.2 2
21.2 odd 6 1078.2.e.o.67.2 4
21.5 even 6 1078.2.e.o.67.1 4
21.11 odd 6 1078.2.e.o.177.2 4
21.17 even 6 1078.2.e.o.177.1 4
21.20 even 2 1078.2.a.v.1.2 yes 2
84.83 odd 2 8624.2.a.bz.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1078.2.a.v.1.1 ✓ 2 3.2 odd 2
1078.2.a.v.1.2 yes 2 21.20 even 2
1078.2.e.o.67.1 4 21.5 even 6
1078.2.e.o.67.2 4 21.2 odd 6
1078.2.e.o.177.1 4 21.17 even 6
1078.2.e.o.177.2 4 21.11 odd 6
8624.2.a.bz.1.1 2 84.83 odd 2
8624.2.a.bz.1.2 2 12.11 even 2
9702.2.a.co.1.1 2 1.1 even 1 trivial
9702.2.a.co.1.2 2 7.6 odd 2 inner