Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [3040,2,Mod(1,3040)] 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("3040.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3040, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 3040 = 2^{5} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3040.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,1,0,4,0,5,0,1,0,6] 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: \(24.2745222145\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.17428.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 6x^{2} + 4x + 6 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(1.52616\) of defining polynomial
Character \(\chi\) \(=\) 3040.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.52616 q^{3} +1.00000 q^{5} +0.329157 q^{7} -0.670843 q^{9} -0.879127 q^{11} -1.35297 q^{13} +1.52616 q^{15} +5.60228 q^{17} +1.00000 q^{19} +0.502345 q^{21} +8.77078 q^{23} +1.00000 q^{25} -5.60228 q^{27} -2.54997 q^{29} -0.394001 q^{31} -1.34169 q^{33} +0.329157 q^{35} +9.34926 q^{37} -2.06484 q^{39} -3.05232 q^{41} +4.22081 q^{43} -0.670843 q^{45} +1.51487 q^{47} -6.89166 q^{49} +8.54997 q^{51} +0.669597 q^{53} -0.879127 q^{55} +1.52616 q^{57} +0.155969 q^{59} +9.95650 q^{61} -0.220813 q^{63} -1.35297 q^{65} +11.3095 q^{67} +13.3856 q^{69} +3.29406 q^{71} +0.445339 q^{73} +1.52616 q^{75} -0.289371 q^{77} +14.4416 q^{79} -6.53744 q^{81} -0.220813 q^{83} +5.60228 q^{85} -3.89166 q^{87} +4.04762 q^{89} -0.445339 q^{91} -0.601308 q^{93} +1.00000 q^{95} +5.19700 q^{97} +0.589756 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + q^{3} + 4 q^{5} + 5 q^{7} + q^{9} + 6 q^{11} - q^{13} + q^{15} - q^{17} + 4 q^{19} + 5 q^{21} + 5 q^{23} + 4 q^{25} + q^{27} + 3 q^{29} + 16 q^{31} + 2 q^{33} + 5 q^{35} - 8 q^{37} + 13 q^{39}+ \cdots - 10 q^{99}+O(q^{100}) \) 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
\(3\) 1.52616 0.881128 0.440564 − 0.897721i \(-0.354779\pi\)
0.440564 + 0.897721i \(0.354779\pi\)
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 0.329157 0.124410 0.0622048 − 0.998063i \(-0.480187\pi\)
0.0622048 + 0.998063i \(0.480187\pi\)
\(8\) 0 0
\(9\) −0.670843 −0.223614
\(10\) 0 0
\(11\) −0.879127 −0.265067 −0.132533 − 0.991179i \(-0.542311\pi\)
−0.132533 + 0.991179i \(0.542311\pi\)
\(12\) 0 0
\(13\) −1.35297 −0.375246 −0.187623 − 0.982241i \(-0.560078\pi\)
−0.187623 + 0.982241i \(0.560078\pi\)
\(14\) 0 0
\(15\) 1.52616 0.394052
\(16\) 0 0
\(17\) 5.60228 1.35875 0.679377 − 0.733790i \(-0.262250\pi\)
0.679377 + 0.733790i \(0.262250\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 0.502345 0.109621
\(22\) 0 0
\(23\) 8.77078 1.82883 0.914417 − 0.404773i \(-0.132649\pi\)
0.914417 + 0.404773i \(0.132649\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −5.60228 −1.07816
\(28\) 0 0
\(29\) −2.54997 −0.473517 −0.236759 − 0.971568i \(-0.576085\pi\)
−0.236759 + 0.971568i \(0.576085\pi\)
\(30\) 0 0
\(31\) −0.394001 −0.0707647 −0.0353823 − 0.999374i \(-0.511265\pi\)
−0.0353823 + 0.999374i \(0.511265\pi\)
\(32\) 0 0
\(33\) −1.34169 −0.233558
\(34\) 0 0
\(35\) 0.329157 0.0556377
\(36\) 0 0
\(37\) 9.34926 1.53701 0.768504 − 0.639845i \(-0.221001\pi\)
0.768504 + 0.639845i \(0.221001\pi\)
\(38\) 0 0
\(39\) −2.06484 −0.330640
\(40\) 0 0
\(41\) −3.05232 −0.476692 −0.238346 − 0.971180i \(-0.576605\pi\)
−0.238346 + 0.971180i \(0.576605\pi\)
\(42\) 0 0
\(43\) 4.22081 0.643668 0.321834 − 0.946796i \(-0.395701\pi\)
0.321834 + 0.946796i \(0.395701\pi\)
\(44\) 0 0
\(45\) −0.670843 −0.100003
\(46\) 0 0
\(47\) 1.51487 0.220967 0.110484 − 0.993878i \(-0.464760\pi\)
0.110484 + 0.993878i \(0.464760\pi\)
\(48\) 0 0
\(49\) −6.89166 −0.984522
\(50\) 0 0
\(51\) 8.54997 1.19724
\(52\) 0 0
\(53\) 0.669597 0.0919763 0.0459881 − 0.998942i \(-0.485356\pi\)
0.0459881 + 0.998942i \(0.485356\pi\)
\(54\) 0 0
\(55\) −0.879127 −0.118541
\(56\) 0 0
\(57\) 1.52616 0.202145
\(58\) 0 0
\(59\) 0.155969 0.0203054 0.0101527 − 0.999948i \(-0.496768\pi\)
0.0101527 + 0.999948i \(0.496768\pi\)
\(60\) 0 0
\(61\) 9.95650 1.27480 0.637400 − 0.770533i \(-0.280010\pi\)
0.637400 + 0.770533i \(0.280010\pi\)
\(62\) 0 0
\(63\) −0.220813 −0.0278198
\(64\) 0 0
\(65\) −1.35297 −0.167815
\(66\) 0 0
\(67\) 11.3095 1.38167 0.690836 − 0.723012i \(-0.257243\pi\)
0.690836 + 0.723012i \(0.257243\pi\)
\(68\) 0 0
\(69\) 13.3856 1.61144
\(70\) 0 0
\(71\) 3.29406 0.390933 0.195467 − 0.980710i \(-0.437378\pi\)
0.195467 + 0.980710i \(0.437378\pi\)
\(72\) 0 0
\(73\) 0.445339 0.0521230 0.0260615 − 0.999660i \(-0.491703\pi\)
0.0260615 + 0.999660i \(0.491703\pi\)
\(74\) 0 0
\(75\) 1.52616 0.176226
\(76\) 0 0
\(77\) −0.289371 −0.0329769
\(78\) 0 0
\(79\) 14.4416 1.62481 0.812405 − 0.583094i \(-0.198158\pi\)
0.812405 + 0.583094i \(0.198158\pi\)
\(80\) 0 0
\(81\) −6.53744 −0.726382
\(82\) 0 0
\(83\) −0.220813 −0.0242373 −0.0121187 − 0.999927i \(-0.503858\pi\)
−0.0121187 + 0.999927i \(0.503858\pi\)
\(84\) 0 0
\(85\) 5.60228 0.607653
\(86\) 0 0
\(87\) −3.89166 −0.417229
\(88\) 0 0
\(89\) 4.04762 0.429047 0.214524 − 0.976719i \(-0.431180\pi\)
0.214524 + 0.976719i \(0.431180\pi\)
\(90\) 0 0
\(91\) −0.445339 −0.0466842
\(92\) 0 0
\(93\) −0.601308 −0.0623527
\(94\) 0 0
\(95\) 1.00000 0.102598
\(96\) 0 0
\(97\) 5.19700 0.527675 0.263838 − 0.964567i \(-0.415012\pi\)
0.263838 + 0.964567i \(0.415012\pi\)
\(98\) 0 0
\(99\) 0.589756 0.0592727
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 3040.2.a.t.1.3 yes 4
4.3 odd 2 3040.2.a.r.1.2 ✓ 4
8.3 odd 2 6080.2.a.cf.1.3 4
8.5 even 2 6080.2.a.cd.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3040.2.a.r.1.2 ✓ 4 4.3 odd 2
3040.2.a.t.1.3 yes 4 1.1 even 1 trivial
6080.2.a.cd.1.2 4 8.5 even 2
6080.2.a.cf.1.3 4 8.3 odd 2