↑

Properties

Label 1700.2.a.f.1.3
Level $1700$
Weight $2$
Character 1700.1
Self dual yes
Analytic conductor $13.575$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1700,2,Mod(1,1700)] 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("1700.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1700, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1700 = 2^{2} \cdot 5^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1700.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(13.5745683436\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.11344.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 4x^{2} + 4x + 3 \) 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 340)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(1.28734\) of defining polynomial
Character \(\chi\) \(=\) 1700.1

$q$-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)
 
\(f(q)\) \(=\) \(q+0.287336 q^{3} +4.67316 q^{7} -2.91744 q^{9} -3.33039 q^{11} -5.04306 q^{13} -1.00000 q^{17} -5.83488 q^{19} +1.34277 q^{21} -7.54754 q^{23} -1.70029 q^{27} +5.83488 q^{29} -2.45601 q^{31} -0.956942 q^{33} -9.83488 q^{37} -1.44905 q^{39} +2.63707 q^{41} -7.04306 q^{43} -0.217146 q^{47} +14.8384 q^{49} -0.287336 q^{51} +4.52041 q^{53} -1.67657 q^{57} +2.43429 q^{59} -1.55991 q^{61} -13.6337 q^{63} +3.61773 q^{67} -2.16868 q^{69} +13.4165 q^{71} +8.30682 q^{73} -15.5635 q^{77} -1.37886 q^{79} +8.26376 q^{81} +11.9008 q^{83} +1.67657 q^{87} -4.27954 q^{89} -23.5670 q^{91} -0.705701 q^{93} -14.9120 q^{97} +9.71622 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{3} - 10 q^{13} - 4 q^{17} - 10 q^{23} - 2 q^{27} + 2 q^{31} - 14 q^{33} - 16 q^{37} + 2 q^{39} - 2 q^{41} - 18 q^{43} + 6 q^{47} + 4 q^{49} + 2 q^{51} - 16 q^{53} - 16 q^{57} - 4 q^{59} + 6 q^{61}+ \cdots + 10 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0.287336 0.165893 0.0829467 − 0.996554i \(-0.473567\pi\)
0.0829467 + 0.996554i \(0.473567\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 4.67316 1.76629 0.883144 − 0.469101i \(-0.155422\pi\)
0.883144 + 0.469101i \(0.155422\pi\)
\(8\) 0 0
\(9\) −2.91744 −0.972479
\(10\) 0 0
\(11\) −3.33039 −1.00415 −0.502076 − 0.864824i \(-0.667430\pi\)
−0.502076 + 0.864824i \(0.667430\pi\)
\(12\) 0 0
\(13\) −5.04306 −1.39869 −0.699346 − 0.714783i \(-0.746525\pi\)
−0.699346 + 0.714783i \(0.746525\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −1.00000 −0.242536
\(18\) 0 0
\(19\) −5.83488 −1.33861 −0.669306 − 0.742987i \(-0.733409\pi\)
−0.669306 + 0.742987i \(0.733409\pi\)
\(20\) 0 0
\(21\) 1.34277 0.293016
\(22\) 0 0
\(23\) −7.54754 −1.57377 −0.786885 − 0.617099i \(-0.788308\pi\)
−0.786885 + 0.617099i \(0.788308\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −1.70029 −0.327221
\(28\) 0 0
\(29\) 5.83488 1.08351 0.541755 − 0.840537i \(-0.317760\pi\)
0.541755 + 0.840537i \(0.317760\pi\)
\(30\) 0 0
\(31\) −2.45601 −0.441113 −0.220557 − 0.975374i \(-0.570787\pi\)
−0.220557 + 0.975374i \(0.570787\pi\)
\(32\) 0 0
\(33\) −0.956942 −0.166582
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −9.83488 −1.61684 −0.808422 − 0.588604i \(-0.799678\pi\)
−0.808422 + 0.588604i \(0.799678\pi\)
\(38\) 0 0
\(39\) −1.44905 −0.232034
\(40\) 0 0
\(41\) 2.63707 0.411840 0.205920 − 0.978569i \(-0.433981\pi\)
0.205920 + 0.978569i \(0.433981\pi\)
\(42\) 0 0
\(43\) −7.04306 −1.07406 −0.537028 − 0.843564i \(-0.680453\pi\)
−0.537028 + 0.843564i \(0.680453\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −0.217146 −0.0316740 −0.0158370 − 0.999875i \(-0.505041\pi\)
−0.0158370 + 0.999875i \(0.505041\pi\)
\(48\) 0 0
\(49\) 14.8384 2.11978
\(50\) 0 0
\(51\) −0.287336 −0.0402351
\(52\) 0 0
\(53\) 4.52041 0.620926 0.310463 − 0.950585i \(-0.399516\pi\)
0.310463 + 0.950585i \(0.399516\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −1.67657 −0.222067
\(58\) 0 0
\(59\) 2.43429 0.316918 0.158459 − 0.987366i \(-0.449347\pi\)
0.158459 + 0.987366i \(0.449347\pi\)
\(60\) 0 0
\(61\) −1.55991 −0.199726 −0.0998632 − 0.995001i \(-0.531841\pi\)
−0.0998632 + 0.995001i \(0.531841\pi\)
\(62\) 0 0
\(63\) −13.6337 −1.71768
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 3.61773 0.441976 0.220988 − 0.975277i \(-0.429072\pi\)
0.220988 + 0.975277i \(0.429072\pi\)
\(68\) 0 0
\(69\) −2.16868 −0.261078
\(70\) 0 0
\(71\) 13.4165 1.59225 0.796123 − 0.605134i \(-0.206881\pi\)
0.796123 + 0.605134i \(0.206881\pi\)
\(72\) 0 0
\(73\) 8.30682 0.972240 0.486120 − 0.873892i \(-0.338412\pi\)
0.486120 + 0.873892i \(0.338412\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −15.5635 −1.77362
\(78\) 0 0
\(79\) −1.37886 −0.155134 −0.0775670 − 0.996987i \(-0.524715\pi\)
−0.0775670 + 0.996987i \(0.524715\pi\)
\(80\) 0 0
\(81\) 8.26376 0.918196
\(82\) 0 0
\(83\) 11.9008 1.30629 0.653143 − 0.757235i \(-0.273450\pi\)
0.653143 + 0.757235i \(0.273450\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 1.67657 0.179747
\(88\) 0 0
\(89\) −4.27954 −0.453630 −0.226815 − 0.973938i \(-0.572831\pi\)
−0.226815 + 0.973938i \(0.572831\pi\)
\(90\) 0 0
\(91\) −23.5670 −2.47050
\(92\) 0 0
\(93\) −0.705701 −0.0731778
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −14.9120 −1.51409 −0.757044 − 0.653364i \(-0.773357\pi\)
−0.757044 + 0.653364i \(0.773357\pi\)
\(98\) 0 0
\(99\) 9.71622 0.976517
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 1700.2.a.f.1.3 4
4.3 odd 2 6800.2.a.bv.1.2 4
5.2 odd 4 340.2.e.a.69.4 ✓ 8
5.3 odd 4 340.2.e.a.69.5 yes 8
5.4 even 2 1700.2.a.g.1.2 4
15.2 even 4 3060.2.g.f.2449.8 8
15.8 even 4 3060.2.g.f.2449.7 8
20.3 even 4 1360.2.e.e.1089.4 8
20.7 even 4 1360.2.e.e.1089.5 8
20.19 odd 2 6800.2.a.bu.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
340.2.e.a.69.4 ✓ 8 5.2 odd 4
340.2.e.a.69.5 yes 8 5.3 odd 4
1360.2.e.e.1089.4 8 20.3 even 4
1360.2.e.e.1089.5 8 20.7 even 4
1700.2.a.f.1.3 4 1.1 even 1 trivial
1700.2.a.g.1.2 4 5.4 even 2
3060.2.g.f.2449.7 8 15.8 even 4
3060.2.g.f.2449.8 8 15.2 even 4
6800.2.a.bu.1.3 4 20.19 odd 2
6800.2.a.bv.1.2 4 4.3 odd 2