Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [700,2,Mod(1,700)] 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("700.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(700, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 700 = 2^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 700.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,0,0,0,0,0,1,0,-3,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(5.58952814149\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 140)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 700.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{7} -3.00000 q^{9} +4.00000 q^{13} +4.00000 q^{17} +4.00000 q^{19} +8.00000 q^{23} +2.00000 q^{29} -8.00000 q^{31} -8.00000 q^{37} +6.00000 q^{41} +8.00000 q^{43} +8.00000 q^{47} +1.00000 q^{49} -4.00000 q^{59} -6.00000 q^{61} -3.00000 q^{63} +8.00000 q^{67} +12.0000 q^{71} -4.00000 q^{73} -4.00000 q^{79} +9.00000 q^{81} -10.0000 q^{89} +4.00000 q^{91} -12.0000 q^{97} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) −3.00000 −1.00000
\(10\) 0 0
\(11\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) 4.00000 1.10940 0.554700 − 0.832050i \(-0.312833\pi\)
0.554700 + 0.832050i \(0.312833\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 4.00000 0.970143 0.485071 − 0.874475i \(-0.338794\pi\)
0.485071 + 0.874475i \(0.338794\pi\)
\(18\) 0 0
\(19\) 4.00000 0.917663 0.458831 − 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 8.00000 1.66812 0.834058 − 0.551677i \(-0.186012\pi\)
0.834058 + 0.551677i \(0.186012\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 2.00000 0.371391 0.185695 − 0.982607i \(-0.440546\pi\)
0.185695 + 0.982607i \(0.440546\pi\)
\(30\) 0 0
\(31\) −8.00000 −1.43684 −0.718421 − 0.695608i \(-0.755135\pi\)
−0.718421 + 0.695608i \(0.755135\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −8.00000 −1.31519 −0.657596 − 0.753371i \(-0.728427\pi\)
−0.657596 + 0.753371i \(0.728427\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 6.00000 0.937043 0.468521 − 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 0 0
\(43\) 8.00000 1.21999 0.609994 − 0.792406i \(-0.291172\pi\)
0.609994 + 0.792406i \(0.291172\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 8.00000 1.16692 0.583460 − 0.812142i \(-0.301699\pi\)
0.583460 + 0.812142i \(0.301699\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −4.00000 −0.520756 −0.260378 − 0.965507i \(-0.583847\pi\)
−0.260378 + 0.965507i \(0.583847\pi\)
\(60\) 0 0
\(61\) −6.00000 −0.768221 −0.384111 − 0.923287i \(-0.625492\pi\)
−0.384111 + 0.923287i \(0.625492\pi\)
\(62\) 0 0
\(63\) −3.00000 −0.377964
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 8.00000 0.977356 0.488678 − 0.872464i \(-0.337479\pi\)
0.488678 + 0.872464i \(0.337479\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 12.0000 1.42414 0.712069 − 0.702109i \(-0.247758\pi\)
0.712069 + 0.702109i \(0.247758\pi\)
\(72\) 0 0
\(73\) −4.00000 −0.468165 −0.234082 − 0.972217i \(-0.575209\pi\)
−0.234082 + 0.972217i \(0.575209\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −4.00000 −0.450035 −0.225018 − 0.974355i \(-0.572244\pi\)
−0.225018 + 0.974355i \(0.572244\pi\)
\(80\) 0 0
\(81\) 9.00000 1.00000
\(82\) 0 0
\(83\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −10.0000 −1.06000 −0.529999 − 0.847998i \(-0.677808\pi\)
−0.529999 + 0.847998i \(0.677808\pi\)
\(90\) 0 0
\(91\) 4.00000 0.419314
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −12.0000 −1.21842 −0.609208 − 0.793011i \(-0.708512\pi\)
−0.609208 + 0.793011i \(0.708512\pi\)
\(98\) 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 700.2.a.h.1.1 1
3.2 odd 2 6300.2.a.y.1.1 1
4.3 odd 2 2800.2.a.o.1.1 1
5.2 odd 4 140.2.e.b.29.2 yes 2
5.3 odd 4 140.2.e.b.29.1 ✓ 2
5.4 even 2 700.2.a.f.1.1 1
7.6 odd 2 4900.2.a.l.1.1 1
15.2 even 4 1260.2.k.b.1009.1 2
15.8 even 4 1260.2.k.b.1009.2 2
15.14 odd 2 6300.2.a.g.1.1 1
20.3 even 4 560.2.g.c.449.1 2
20.7 even 4 560.2.g.c.449.2 2
20.19 odd 2 2800.2.a.s.1.1 1
35.2 odd 12 980.2.q.d.949.2 4
35.3 even 12 980.2.q.e.569.1 4
35.12 even 12 980.2.q.e.949.1 4
35.13 even 4 980.2.e.a.589.2 2
35.17 even 12 980.2.q.e.569.2 4
35.18 odd 12 980.2.q.d.569.2 4
35.23 odd 12 980.2.q.d.949.1 4
35.27 even 4 980.2.e.a.589.1 2
35.32 odd 12 980.2.q.d.569.1 4
35.33 even 12 980.2.q.e.949.2 4
35.34 odd 2 4900.2.a.m.1.1 1
40.3 even 4 2240.2.g.c.449.2 2
40.13 odd 4 2240.2.g.d.449.2 2
40.27 even 4 2240.2.g.c.449.1 2
40.37 odd 4 2240.2.g.d.449.1 2
60.23 odd 4 5040.2.t.g.1009.2 2
60.47 odd 4 5040.2.t.g.1009.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
140.2.e.b.29.1 ✓ 2 5.3 odd 4
140.2.e.b.29.2 yes 2 5.2 odd 4
560.2.g.c.449.1 2 20.3 even 4
560.2.g.c.449.2 2 20.7 even 4
700.2.a.f.1.1 1 5.4 even 2
700.2.a.h.1.1 1 1.1 even 1 trivial
980.2.e.a.589.1 2 35.27 even 4
980.2.e.a.589.2 2 35.13 even 4
980.2.q.d.569.1 4 35.32 odd 12
980.2.q.d.569.2 4 35.18 odd 12
980.2.q.d.949.1 4 35.23 odd 12
980.2.q.d.949.2 4 35.2 odd 12
980.2.q.e.569.1 4 35.3 even 12
980.2.q.e.569.2 4 35.17 even 12
980.2.q.e.949.1 4 35.12 even 12
980.2.q.e.949.2 4 35.33 even 12
1260.2.k.b.1009.1 2 15.2 even 4
1260.2.k.b.1009.2 2 15.8 even 4
2240.2.g.c.449.1 2 40.27 even 4
2240.2.g.c.449.2 2 40.3 even 4
2240.2.g.d.449.1 2 40.37 odd 4
2240.2.g.d.449.2 2 40.13 odd 4
2800.2.a.o.1.1 1 4.3 odd 2
2800.2.a.s.1.1 1 20.19 odd 2
4900.2.a.l.1.1 1 7.6 odd 2
4900.2.a.m.1.1 1 35.34 odd 2
5040.2.t.g.1009.1 2 60.47 odd 4
5040.2.t.g.1009.2 2 60.23 odd 4
6300.2.a.g.1.1 1 15.14 odd 2
6300.2.a.y.1.1 1 3.2 odd 2