Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [3800,2,Mod(1,3800)] 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("3800.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3800, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 3800 = 2^{3} \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3800.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,2,0,0,0,2,0,6,0,3,0,-1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(30.3431527681\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} - 10x^{4} + 16x^{3} + 15x^{2} - 14x - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-0.185519\) of defining polynomial
Character \(\chi\) \(=\) 3800.1

$q$-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)
 
\(f(q)\) \(=\) \(q-0.185519 q^{3} -4.45651 q^{7} -2.96558 q^{9} +2.64623 q^{11} +1.30142 q^{13} -3.51716 q^{17} -1.00000 q^{19} +0.826767 q^{21} -6.52882 q^{23} +1.10673 q^{27} -5.20946 q^{29} +10.8219 q^{31} -0.490925 q^{33} +2.04607 q^{37} -0.241439 q^{39} -3.80044 q^{41} +4.77089 q^{43} +1.49093 q^{47} +12.8605 q^{49} +0.652501 q^{51} +0.225583 q^{53} +0.185519 q^{57} -2.86056 q^{59} -6.31449 q^{61} +13.2161 q^{63} -13.1831 q^{67} +1.21122 q^{69} +12.3310 q^{71} -5.42276 q^{73} -11.7929 q^{77} +14.9688 q^{79} +8.69143 q^{81} +3.84470 q^{83} +0.966455 q^{87} +1.67666 q^{89} -5.79981 q^{91} -2.00768 q^{93} +9.48523 q^{97} -7.84760 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 2 q^{3} + 2 q^{7} + 6 q^{9} + 3 q^{11} - q^{13} - 14 q^{17} - 6 q^{19} + 15 q^{21} + 12 q^{23} + 8 q^{27} + 9 q^{29} + 5 q^{31} + 2 q^{33} - 8 q^{37} + 12 q^{39} + 3 q^{41} + 15 q^{43} + 4 q^{47}+ \cdots - 6 q^{99}+O(q^{100}) \) 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
\(3\) −0.185519 −0.107109 −0.0535547 − 0.998565i \(-0.517055\pi\)
−0.0535547 + 0.998565i \(0.517055\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −4.45651 −1.68440 −0.842201 − 0.539164i \(-0.818740\pi\)
−0.842201 + 0.539164i \(0.818740\pi\)
\(8\) 0 0
\(9\) −2.96558 −0.988528
\(10\) 0 0
\(11\) 2.64623 0.797867 0.398934 − 0.916980i \(-0.369380\pi\)
0.398934 + 0.916980i \(0.369380\pi\)
\(12\) 0 0
\(13\) 1.30142 0.360950 0.180475 − 0.983580i \(-0.442236\pi\)
0.180475 + 0.983580i \(0.442236\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −3.51716 −0.853037 −0.426519 − 0.904479i \(-0.640260\pi\)
−0.426519 + 0.904479i \(0.640260\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) 0.826767 0.180415
\(22\) 0 0
\(23\) −6.52882 −1.36135 −0.680677 − 0.732584i \(-0.738314\pi\)
−0.680677 + 0.732584i \(0.738314\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 1.10673 0.212990
\(28\) 0 0
\(29\) −5.20946 −0.967373 −0.483686 − 0.875241i \(-0.660702\pi\)
−0.483686 + 0.875241i \(0.660702\pi\)
\(30\) 0 0
\(31\) 10.8219 1.94368 0.971839 − 0.235646i \(-0.0757207\pi\)
0.971839 + 0.235646i \(0.0757207\pi\)
\(32\) 0 0
\(33\) −0.490925 −0.0854591
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 2.04607 0.336373 0.168186 − 0.985755i \(-0.446209\pi\)
0.168186 + 0.985755i \(0.446209\pi\)
\(38\) 0 0
\(39\) −0.241439 −0.0386612
\(40\) 0 0
\(41\) −3.80044 −0.593529 −0.296764 − 0.954951i \(-0.595908\pi\)
−0.296764 + 0.954951i \(0.595908\pi\)
\(42\) 0 0
\(43\) 4.77089 0.727554 0.363777 − 0.931486i \(-0.381487\pi\)
0.363777 + 0.931486i \(0.381487\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 1.49093 0.217474 0.108737 − 0.994071i \(-0.465319\pi\)
0.108737 + 0.994071i \(0.465319\pi\)
\(48\) 0 0
\(49\) 12.8605 1.83721
\(50\) 0 0
\(51\) 0.652501 0.0913684
\(52\) 0 0
\(53\) 0.225583 0.0309862 0.0154931 − 0.999880i \(-0.495068\pi\)
0.0154931 + 0.999880i \(0.495068\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0.185519 0.0245726
\(58\) 0 0
\(59\) −2.86056 −0.372413 −0.186206 − 0.982511i \(-0.559619\pi\)
−0.186206 + 0.982511i \(0.559619\pi\)
\(60\) 0 0
\(61\) −6.31449 −0.808488 −0.404244 − 0.914651i \(-0.632465\pi\)
−0.404244 + 0.914651i \(0.632465\pi\)
\(62\) 0 0
\(63\) 13.2161 1.66508
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −13.1831 −1.61058 −0.805288 − 0.592884i \(-0.797989\pi\)
−0.805288 + 0.592884i \(0.797989\pi\)
\(68\) 0 0
\(69\) 1.21122 0.145814
\(70\) 0 0
\(71\) 12.3310 1.46342 0.731711 − 0.681615i \(-0.238722\pi\)
0.731711 + 0.681615i \(0.238722\pi\)
\(72\) 0 0
\(73\) −5.42276 −0.634686 −0.317343 − 0.948311i \(-0.602791\pi\)
−0.317343 + 0.948311i \(0.602791\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −11.7929 −1.34393
\(78\) 0 0
\(79\) 14.9688 1.68413 0.842063 − 0.539379i \(-0.181341\pi\)
0.842063 + 0.539379i \(0.181341\pi\)
\(80\) 0 0
\(81\) 8.69143 0.965714
\(82\) 0 0
\(83\) 3.84470 0.422011 0.211005 − 0.977485i \(-0.432326\pi\)
0.211005 + 0.977485i \(0.432326\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0.966455 0.103615
\(88\) 0 0
\(89\) 1.67666 0.177726 0.0888629 − 0.996044i \(-0.471677\pi\)
0.0888629 + 0.996044i \(0.471677\pi\)
\(90\) 0 0
\(91\) −5.79981 −0.607985
\(92\) 0 0
\(93\) −2.00768 −0.208186
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 9.48523 0.963079 0.481539 − 0.876424i \(-0.340078\pi\)
0.481539 + 0.876424i \(0.340078\pi\)
\(98\) 0 0
\(99\) −7.84760 −0.788714
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 3800.2.a.bc.1.3 yes 6
4.3 odd 2 7600.2.a.ch.1.4 6
5.2 odd 4 3800.2.d.q.3649.7 12
5.3 odd 4 3800.2.d.q.3649.6 12
5.4 even 2 3800.2.a.ba.1.4 ✓ 6
20.19 odd 2 7600.2.a.cl.1.3 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3800.2.a.ba.1.4 ✓ 6 5.4 even 2
3800.2.a.bc.1.3 yes 6 1.1 even 1 trivial
3800.2.d.q.3649.6 12 5.3 odd 4
3800.2.d.q.3649.7 12 5.2 odd 4
7600.2.a.ch.1.4 6 4.3 odd 2
7600.2.a.cl.1.3 6 20.19 odd 2