Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [9200,2,Mod(1,9200)] 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("9200.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9200, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 9200 = 2^{4} \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9200.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [5,0,0,0,0,0,4,0,3,0,0] 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: \(73.4623698596\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.521397.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 9x^{3} - 3x^{2} + 18x + 12 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 4600)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.36629\) of defining polynomial
Character \(\chi\) \(=\) 9200.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1.36629 q^{3} +3.28093 q^{7} -1.13327 q^{9} -3.49709 q^{11} +3.41420 q^{13} -7.46023 q^{17} -3.49709 q^{19} -4.48269 q^{21} -1.00000 q^{23} +5.64722 q^{27} -3.46268 q^{29} -2.01105 q^{31} +4.77803 q^{33} +0.511497 q^{37} -4.66477 q^{39} -7.07954 q^{41} +2.76452 q^{43} +0.889198 q^{47} +3.76452 q^{49} +10.1928 q^{51} -14.2383 q^{53} +4.77803 q^{57} +4.71325 q^{59} +13.4769 q^{61} -3.71817 q^{63} +2.30025 q^{67} +1.36629 q^{69} +10.6214 q^{71} +3.70765 q^{73} -11.4737 q^{77} +7.97978 q^{79} -4.31592 q^{81} -9.42336 q^{83} +4.73101 q^{87} +0.801390 q^{89} +11.2018 q^{91} +2.74766 q^{93} +11.7722 q^{97} +3.96313 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 4 q^{7} + 3 q^{9} - 4 q^{13} - 6 q^{17} - 5 q^{23} + 9 q^{27} + 12 q^{29} + 18 q^{31} - 6 q^{33} - 10 q^{37} - 9 q^{39} - 6 q^{41} + 10 q^{43} + 22 q^{47} + 15 q^{49} + 6 q^{51} - 10 q^{53} - 6 q^{57}+ \cdots + 6 q^{99}+O(q^{100}) \) 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
\(3\) −1.36629 −0.788825 −0.394413 − 0.918933i \(-0.629052\pi\)
−0.394413 + 0.918933i \(0.629052\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 3.28093 1.24008 0.620038 − 0.784572i \(-0.287117\pi\)
0.620038 + 0.784572i \(0.287117\pi\)
\(8\) 0 0
\(9\) −1.13327 −0.377755
\(10\) 0 0
\(11\) −3.49709 −1.05441 −0.527207 − 0.849737i \(-0.676761\pi\)
−0.527207 + 0.849737i \(0.676761\pi\)
\(12\) 0 0
\(13\) 3.41420 0.946928 0.473464 − 0.880813i \(-0.343003\pi\)
0.473464 + 0.880813i \(0.343003\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −7.46023 −1.80937 −0.904685 − 0.426080i \(-0.859894\pi\)
−0.904685 + 0.426080i \(0.859894\pi\)
\(18\) 0 0
\(19\) −3.49709 −0.802288 −0.401144 − 0.916015i \(-0.631387\pi\)
−0.401144 + 0.916015i \(0.631387\pi\)
\(20\) 0 0
\(21\) −4.48269 −0.978203
\(22\) 0 0
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 5.64722 1.08681
\(28\) 0 0
\(29\) −3.46268 −0.643004 −0.321502 − 0.946909i \(-0.604188\pi\)
−0.321502 + 0.946909i \(0.604188\pi\)
\(30\) 0 0
\(31\) −2.01105 −0.361195 −0.180597 − 0.983557i \(-0.557803\pi\)
−0.180597 + 0.983557i \(0.557803\pi\)
\(32\) 0 0
\(33\) 4.77803 0.831748
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 0.511497 0.0840896 0.0420448 − 0.999116i \(-0.486613\pi\)
0.0420448 + 0.999116i \(0.486613\pi\)
\(38\) 0 0
\(39\) −4.66477 −0.746961
\(40\) 0 0
\(41\) −7.07954 −1.10564 −0.552819 − 0.833301i \(-0.686448\pi\)
−0.552819 + 0.833301i \(0.686448\pi\)
\(42\) 0 0
\(43\) 2.76452 0.421586 0.210793 − 0.977531i \(-0.432395\pi\)
0.210793 + 0.977531i \(0.432395\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0.889198 0.129703 0.0648514 − 0.997895i \(-0.479343\pi\)
0.0648514 + 0.997895i \(0.479343\pi\)
\(48\) 0 0
\(49\) 3.76452 0.537789
\(50\) 0 0
\(51\) 10.1928 1.42728
\(52\) 0 0
\(53\) −14.2383 −1.95577 −0.977887 − 0.209132i \(-0.932936\pi\)
−0.977887 + 0.209132i \(0.932936\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 4.77803 0.632865
\(58\) 0 0
\(59\) 4.71325 0.613613 0.306807 − 0.951772i \(-0.400739\pi\)
0.306807 + 0.951772i \(0.400739\pi\)
\(60\) 0 0
\(61\) 13.4769 1.72554 0.862769 − 0.505599i \(-0.168728\pi\)
0.862769 + 0.505599i \(0.168728\pi\)
\(62\) 0 0
\(63\) −3.71817 −0.468445
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 2.30025 0.281020 0.140510 − 0.990079i \(-0.455126\pi\)
0.140510 + 0.990079i \(0.455126\pi\)
\(68\) 0 0
\(69\) 1.36629 0.164481
\(70\) 0 0
\(71\) 10.6214 1.26053 0.630264 − 0.776381i \(-0.282947\pi\)
0.630264 + 0.776381i \(0.282947\pi\)
\(72\) 0 0
\(73\) 3.70765 0.433948 0.216974 − 0.976177i \(-0.430381\pi\)
0.216974 + 0.976177i \(0.430381\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −11.4737 −1.30755
\(78\) 0 0
\(79\) 7.97978 0.897796 0.448898 − 0.893583i \(-0.351817\pi\)
0.448898 + 0.893583i \(0.351817\pi\)
\(80\) 0 0
\(81\) −4.31592 −0.479546
\(82\) 0 0
\(83\) −9.42336 −1.03435 −0.517174 − 0.855880i \(-0.673016\pi\)
−0.517174 + 0.855880i \(0.673016\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 4.73101 0.507218
\(88\) 0 0
\(89\) 0.801390 0.0849471 0.0424736 − 0.999098i \(-0.486476\pi\)
0.0424736 + 0.999098i \(0.486476\pi\)
\(90\) 0 0
\(91\) 11.2018 1.17426
\(92\) 0 0
\(93\) 2.74766 0.284919
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 11.7722 1.19529 0.597644 − 0.801762i \(-0.296104\pi\)
0.597644 + 0.801762i \(0.296104\pi\)
\(98\) 0 0
\(99\) 3.96313 0.398310
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 9200.2.a.cv.1.2 5
4.3 odd 2 4600.2.a.bd.1.4 ✓ 5
5.4 even 2 9200.2.a.ct.1.4 5
20.3 even 4 4600.2.e.w.4049.7 10
20.7 even 4 4600.2.e.w.4049.4 10
20.19 odd 2 4600.2.a.bf.1.2 yes 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4600.2.a.bd.1.4 ✓ 5 4.3 odd 2
4600.2.a.bf.1.2 yes 5 20.19 odd 2
4600.2.e.w.4049.4 10 20.7 even 4
4600.2.e.w.4049.7 10 20.3 even 4
9200.2.a.ct.1.4 5 5.4 even 2
9200.2.a.cv.1.2 5 1.1 even 1 trivial