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Properties

Label 700.2.a.b.1.1
Level $700$
Weight $2$
Character 700.1
Self dual yes
Analytic conductor $5.590$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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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,-3,0,0,0,1,0,6,0,-5] 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: \(1\)
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)
 
\(f(q)\) \(=\) \(q-3.00000 q^{3} +1.00000 q^{7} +6.00000 q^{9} -5.00000 q^{11} +3.00000 q^{13} +1.00000 q^{17} +6.00000 q^{19} -3.00000 q^{21} -6.00000 q^{23} -9.00000 q^{27} -9.00000 q^{29} -4.00000 q^{31} +15.0000 q^{33} -2.00000 q^{37} -9.00000 q^{39} -4.00000 q^{41} -10.0000 q^{43} +1.00000 q^{47} +1.00000 q^{49} -3.00000 q^{51} -4.00000 q^{53} -18.0000 q^{57} -8.00000 q^{59} -8.00000 q^{61} +6.00000 q^{63} -12.0000 q^{67} +18.0000 q^{69} +8.00000 q^{71} -2.00000 q^{73} -5.00000 q^{77} +13.0000 q^{79} +9.00000 q^{81} +4.00000 q^{83} +27.0000 q^{87} +4.00000 q^{89} +3.00000 q^{91} +12.0000 q^{93} +13.0000 q^{97} -30.0000 q^{99} +O(q^{100})\)

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
\(3\) −3.00000 −1.73205 −0.866025 − 0.500000i \(-0.833333\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) 6.00000 2.00000
\(10\) 0 0
\(11\) −5.00000 −1.50756 −0.753778 − 0.657129i \(-0.771771\pi\)
−0.753778 + 0.657129i \(0.771771\pi\)
\(12\) 0 0
\(13\) 3.00000 0.832050 0.416025 − 0.909353i \(-0.363423\pi\)
0.416025 + 0.909353i \(0.363423\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 1.00000 0.242536 0.121268 − 0.992620i \(-0.461304\pi\)
0.121268 + 0.992620i \(0.461304\pi\)
\(18\) 0 0
\(19\) 6.00000 1.37649 0.688247 − 0.725476i \(-0.258380\pi\)
0.688247 + 0.725476i \(0.258380\pi\)
\(20\) 0 0
\(21\) −3.00000 −0.654654
\(22\) 0 0
\(23\) −6.00000 −1.25109 −0.625543 − 0.780189i \(-0.715123\pi\)
−0.625543 + 0.780189i \(0.715123\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −9.00000 −1.73205
\(28\) 0 0
\(29\) −9.00000 −1.67126 −0.835629 − 0.549294i \(-0.814897\pi\)
−0.835629 + 0.549294i \(0.814897\pi\)
\(30\) 0 0
\(31\) −4.00000 −0.718421 −0.359211 − 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) 0 0
\(33\) 15.0000 2.61116
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −2.00000 −0.328798 −0.164399 − 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) 0 0
\(39\) −9.00000 −1.44115
\(40\) 0 0
\(41\) −4.00000 −0.624695 −0.312348 − 0.949968i \(-0.601115\pi\)
−0.312348 + 0.949968i \(0.601115\pi\)
\(42\) 0 0
\(43\) −10.0000 −1.52499 −0.762493 − 0.646997i \(-0.776025\pi\)
−0.762493 + 0.646997i \(0.776025\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 1.00000 0.145865 0.0729325 − 0.997337i \(-0.476764\pi\)
0.0729325 + 0.997337i \(0.476764\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −3.00000 −0.420084
\(52\) 0 0
\(53\) −4.00000 −0.549442 −0.274721 − 0.961524i \(-0.588586\pi\)
−0.274721 + 0.961524i \(0.588586\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −18.0000 −2.38416
\(58\) 0 0
\(59\) −8.00000 −1.04151 −0.520756 − 0.853706i \(-0.674350\pi\)
−0.520756 + 0.853706i \(0.674350\pi\)
\(60\) 0 0
\(61\) −8.00000 −1.02430 −0.512148 − 0.858898i \(-0.671150\pi\)
−0.512148 + 0.858898i \(0.671150\pi\)
\(62\) 0 0
\(63\) 6.00000 0.755929
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −12.0000 −1.46603 −0.733017 − 0.680211i \(-0.761888\pi\)
−0.733017 + 0.680211i \(0.761888\pi\)
\(68\) 0 0
\(69\) 18.0000 2.16695
\(70\) 0 0
\(71\) 8.00000 0.949425 0.474713 − 0.880141i \(-0.342552\pi\)
0.474713 + 0.880141i \(0.342552\pi\)
\(72\) 0 0
\(73\) −2.00000 −0.234082 −0.117041 − 0.993127i \(-0.537341\pi\)
−0.117041 + 0.993127i \(0.537341\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −5.00000 −0.569803
\(78\) 0 0
\(79\) 13.0000 1.46261 0.731307 − 0.682048i \(-0.238911\pi\)
0.731307 + 0.682048i \(0.238911\pi\)
\(80\) 0 0
\(81\) 9.00000 1.00000
\(82\) 0 0
\(83\) 4.00000 0.439057 0.219529 − 0.975606i \(-0.429548\pi\)
0.219529 + 0.975606i \(0.429548\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 27.0000 2.89470
\(88\) 0 0
\(89\) 4.00000 0.423999 0.212000 − 0.977270i \(-0.432002\pi\)
0.212000 + 0.977270i \(0.432002\pi\)
\(90\) 0 0
\(91\) 3.00000 0.314485
\(92\) 0 0
\(93\) 12.0000 1.24434
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 13.0000 1.31995 0.659975 − 0.751288i \(-0.270567\pi\)
0.659975 + 0.751288i \(0.270567\pi\)
\(98\) 0 0
\(99\) −30.0000 −3.01511
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.b.1.1 1
3.2 odd 2 6300.2.a.bf.1.1 1
4.3 odd 2 2800.2.a.be.1.1 1
5.2 odd 4 700.2.e.a.449.2 2
5.3 odd 4 700.2.e.a.449.1 2
5.4 even 2 140.2.a.b.1.1 ✓ 1
7.6 odd 2 4900.2.a.u.1.1 1
15.2 even 4 6300.2.k.p.6049.2 2
15.8 even 4 6300.2.k.p.6049.1 2
15.14 odd 2 1260.2.a.h.1.1 1
20.3 even 4 2800.2.g.c.449.2 2
20.7 even 4 2800.2.g.c.449.1 2
20.19 odd 2 560.2.a.a.1.1 1
35.4 even 6 980.2.i.b.961.1 2
35.9 even 6 980.2.i.b.361.1 2
35.13 even 4 4900.2.e.a.2549.2 2
35.19 odd 6 980.2.i.j.361.1 2
35.24 odd 6 980.2.i.j.961.1 2
35.27 even 4 4900.2.e.a.2549.1 2
35.34 odd 2 980.2.a.b.1.1 1
40.19 odd 2 2240.2.a.bb.1.1 1
40.29 even 2 2240.2.a.c.1.1 1
60.59 even 2 5040.2.a.bd.1.1 1
105.104 even 2 8820.2.a.n.1.1 1
140.139 even 2 3920.2.a.bl.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
140.2.a.b.1.1 ✓ 1 5.4 even 2
560.2.a.a.1.1 1 20.19 odd 2
700.2.a.b.1.1 1 1.1 even 1 trivial
700.2.e.a.449.1 2 5.3 odd 4
700.2.e.a.449.2 2 5.2 odd 4
980.2.a.b.1.1 1 35.34 odd 2
980.2.i.b.361.1 2 35.9 even 6
980.2.i.b.961.1 2 35.4 even 6
980.2.i.j.361.1 2 35.19 odd 6
980.2.i.j.961.1 2 35.24 odd 6
1260.2.a.h.1.1 1 15.14 odd 2
2240.2.a.c.1.1 1 40.29 even 2
2240.2.a.bb.1.1 1 40.19 odd 2
2800.2.a.be.1.1 1 4.3 odd 2
2800.2.g.c.449.1 2 20.7 even 4
2800.2.g.c.449.2 2 20.3 even 4
3920.2.a.bl.1.1 1 140.139 even 2
4900.2.a.u.1.1 1 7.6 odd 2
4900.2.e.a.2549.1 2 35.27 even 4
4900.2.e.a.2549.2 2 35.13 even 4
5040.2.a.bd.1.1 1 60.59 even 2
6300.2.a.bf.1.1 1 3.2 odd 2
6300.2.k.p.6049.1 2 15.8 even 4
6300.2.k.p.6049.2 2 15.2 even 4
8820.2.a.n.1.1 1 105.104 even 2