Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [4840,2,Mod(1,4840)] 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("4840.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4840, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 4840 = 2^{3} \cdot 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4840.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,-2,0,4,0,-7,0,-4,0,0,0,3] 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: \(38.6475945783\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{22 +2 \sqrt{5}})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 3x^{2} + x + 1 \) 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 440)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.09529\) of defining polynomial
Character \(\chi\) \(=\) 4840.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1.29496 q^{3} +1.00000 q^{5} -0.227777 q^{7} -1.32307 q^{9} -3.34478 q^{13} -1.29496 q^{15} +6.04981 q^{17} -2.22137 q^{19} +0.294963 q^{21} -3.47726 q^{23} +1.00000 q^{25} +5.59822 q^{27} +0.0307867 q^{29} +2.21625 q^{31} -0.227777 q^{35} +8.74728 q^{37} +4.33136 q^{39} +10.2404 q^{41} -3.74411 q^{43} -1.32307 q^{45} -5.31399 q^{47} -6.94812 q^{49} -7.83428 q^{51} -10.2028 q^{53} +2.87660 q^{57} +8.34399 q^{59} -2.24553 q^{61} +0.301365 q^{63} -3.34478 q^{65} +3.47330 q^{67} +4.50292 q^{69} -15.0637 q^{71} -3.66389 q^{73} -1.29496 q^{75} -11.6212 q^{79} -3.28027 q^{81} +1.52274 q^{83} +6.04981 q^{85} -0.0398677 q^{87} -10.0058 q^{89} +0.761864 q^{91} -2.86996 q^{93} -2.22137 q^{95} -8.33570 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{3} + 4 q^{5} - 7 q^{7} - 4 q^{9} + 3 q^{13} - 2 q^{15} + 11 q^{17} + 2 q^{19} - 2 q^{21} - 11 q^{23} + 4 q^{25} + q^{27} + 4 q^{29} - 17 q^{31} - 7 q^{35} - 3 q^{37} + q^{39} + 13 q^{41} - 7 q^{43}+ \cdots + 2 q^{97}+O(q^{100}) \) 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
\(3\) −1.29496 −0.747647 −0.373824 − 0.927500i \(-0.621953\pi\)
−0.373824 + 0.927500i \(0.621953\pi\)
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −0.227777 −0.0860917 −0.0430458 − 0.999073i \(-0.513706\pi\)
−0.0430458 + 0.999073i \(0.513706\pi\)
\(8\) 0 0
\(9\) −1.32307 −0.441024
\(10\) 0 0
\(11\) 0 0
\(12\) 0 0
\(13\) −3.34478 −0.927674 −0.463837 − 0.885921i \(-0.653528\pi\)
−0.463837 + 0.885921i \(0.653528\pi\)
\(14\) 0 0
\(15\) −1.29496 −0.334358
\(16\) 0 0
\(17\) 6.04981 1.46730 0.733648 − 0.679530i \(-0.237816\pi\)
0.733648 + 0.679530i \(0.237816\pi\)
\(18\) 0 0
\(19\) −2.22137 −0.509618 −0.254809 − 0.966991i \(-0.582013\pi\)
−0.254809 + 0.966991i \(0.582013\pi\)
\(20\) 0 0
\(21\) 0.294963 0.0643662
\(22\) 0 0
\(23\) −3.47726 −0.725059 −0.362529 − 0.931972i \(-0.618087\pi\)
−0.362529 + 0.931972i \(0.618087\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 5.59822 1.07738
\(28\) 0 0
\(29\) 0.0307867 0.00571695 0.00285848 − 0.999996i \(-0.499090\pi\)
0.00285848 + 0.999996i \(0.499090\pi\)
\(30\) 0 0
\(31\) 2.21625 0.398050 0.199025 − 0.979994i \(-0.436222\pi\)
0.199025 + 0.979994i \(0.436222\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −0.227777 −0.0385014
\(36\) 0 0
\(37\) 8.74728 1.43804 0.719022 − 0.694987i \(-0.244590\pi\)
0.719022 + 0.694987i \(0.244590\pi\)
\(38\) 0 0
\(39\) 4.33136 0.693573
\(40\) 0 0
\(41\) 10.2404 1.59928 0.799641 − 0.600478i \(-0.205023\pi\)
0.799641 + 0.600478i \(0.205023\pi\)
\(42\) 0 0
\(43\) −3.74411 −0.570972 −0.285486 − 0.958383i \(-0.592155\pi\)
−0.285486 + 0.958383i \(0.592155\pi\)
\(44\) 0 0
\(45\) −1.32307 −0.197232
\(46\) 0 0
\(47\) −5.31399 −0.775125 −0.387563 − 0.921843i \(-0.626683\pi\)
−0.387563 + 0.921843i \(0.626683\pi\)
\(48\) 0 0
\(49\) −6.94812 −0.992588
\(50\) 0 0
\(51\) −7.83428 −1.09702
\(52\) 0 0
\(53\) −10.2028 −1.40147 −0.700734 − 0.713423i \(-0.747144\pi\)
−0.700734 + 0.713423i \(0.747144\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 2.87660 0.381015
\(58\) 0 0
\(59\) 8.34399 1.08629 0.543147 − 0.839637i \(-0.317233\pi\)
0.543147 + 0.839637i \(0.317233\pi\)
\(60\) 0 0
\(61\) −2.24553 −0.287510 −0.143755 − 0.989613i \(-0.545918\pi\)
−0.143755 + 0.989613i \(0.545918\pi\)
\(62\) 0 0
\(63\) 0.301365 0.0379685
\(64\) 0 0
\(65\) −3.34478 −0.414869
\(66\) 0 0
\(67\) 3.47330 0.424332 0.212166 − 0.977234i \(-0.431948\pi\)
0.212166 + 0.977234i \(0.431948\pi\)
\(68\) 0 0
\(69\) 4.50292 0.542088
\(70\) 0 0
\(71\) −15.0637 −1.78773 −0.893867 − 0.448332i \(-0.852018\pi\)
−0.893867 + 0.448332i \(0.852018\pi\)
\(72\) 0 0
\(73\) −3.66389 −0.428826 −0.214413 − 0.976743i \(-0.568784\pi\)
−0.214413 + 0.976743i \(0.568784\pi\)
\(74\) 0 0
\(75\) −1.29496 −0.149529
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −11.6212 −1.30749 −0.653744 − 0.756716i \(-0.726803\pi\)
−0.653744 + 0.756716i \(0.726803\pi\)
\(80\) 0 0
\(81\) −3.28027 −0.364474
\(82\) 0 0
\(83\) 1.52274 0.167142 0.0835712 − 0.996502i \(-0.473367\pi\)
0.0835712 + 0.996502i \(0.473367\pi\)
\(84\) 0 0
\(85\) 6.04981 0.656194
\(86\) 0 0
\(87\) −0.0398677 −0.00427426
\(88\) 0 0
\(89\) −10.0058 −1.06062 −0.530309 − 0.847805i \(-0.677924\pi\)
−0.530309 + 0.847805i \(0.677924\pi\)
\(90\) 0 0
\(91\) 0.761864 0.0798650
\(92\) 0 0
\(93\) −2.86996 −0.297601
\(94\) 0 0
\(95\) −2.22137 −0.227908
\(96\) 0 0
\(97\) −8.33570 −0.846362 −0.423181 − 0.906045i \(-0.639087\pi\)
−0.423181 + 0.906045i \(0.639087\pi\)
\(98\) 0 0
\(99\) 0 0
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 4840.2.a.y.1.2 4
4.3 odd 2 9680.2.a.cu.1.3 4
11.5 even 5 440.2.y.a.201.1 yes 8
11.9 even 5 440.2.y.a.81.1 ✓ 8
11.10 odd 2 4840.2.a.z.1.2 4
44.27 odd 10 880.2.bo.d.641.2 8
44.31 odd 10 880.2.bo.d.81.2 8
44.43 even 2 9680.2.a.ct.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
440.2.y.a.81.1 ✓ 8 11.9 even 5
440.2.y.a.201.1 yes 8 11.5 even 5
880.2.bo.d.81.2 8 44.31 odd 10
880.2.bo.d.641.2 8 44.27 odd 10
4840.2.a.y.1.2 4 1.1 even 1 trivial
4840.2.a.z.1.2 4 11.10 odd 2
9680.2.a.ct.1.3 4 44.43 even 2
9680.2.a.cu.1.3 4 4.3 odd 2