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 = [6,0,-3,0,-6,0,7,0,5,0,0,0,-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: \(38.6475945783\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: 6.6.25903625.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 3x^{5} - 7x^{4} + 17x^{3} + 16x^{2} - 20x - 5 \) 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 1.6
Root \(-1.71280\) 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)
 
\(f(q)\) \(=\) \(q+1.71280 q^{3} -1.00000 q^{5} +3.70505 q^{7} -0.0663151 q^{9} -1.17750 q^{13} -1.71280 q^{15} -5.40027 q^{17} -3.84752 q^{19} +6.34602 q^{21} -8.68237 q^{23} +1.00000 q^{25} -5.25199 q^{27} -6.87597 q^{29} +8.75541 q^{31} -3.70505 q^{35} +2.25468 q^{37} -2.01682 q^{39} +1.59733 q^{41} +4.11979 q^{43} +0.0663151 q^{45} -12.6800 q^{47} +6.72743 q^{49} -9.24959 q^{51} -12.3795 q^{53} -6.59004 q^{57} -0.393999 q^{59} +1.80085 q^{61} -0.245701 q^{63} +1.17750 q^{65} -14.3809 q^{67} -14.8712 q^{69} -6.85527 q^{71} +10.0551 q^{73} +1.71280 q^{75} -1.11383 q^{79} -8.79666 q^{81} +0.0173212 q^{83} +5.40027 q^{85} -11.7772 q^{87} -8.49434 q^{89} -4.36270 q^{91} +14.9963 q^{93} +3.84752 q^{95} +10.5155 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 3 q^{3} - 6 q^{5} + 7 q^{7} + 5 q^{9} - q^{13} + 3 q^{15} - 6 q^{17} + 7 q^{19} - 14 q^{21} - 9 q^{23} + 6 q^{25} - 21 q^{27} - 10 q^{29} + q^{31} - 7 q^{35} + 3 q^{37} + q^{39} + 6 q^{41} - 18 q^{43}+ \cdots - 33 q^{97}+O(q^{100}) \) 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
\(3\) 1.71280 0.988886 0.494443 − 0.869210i \(-0.335372\pi\)
0.494443 + 0.869210i \(0.335372\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 3.70505 1.40038 0.700189 − 0.713957i \(-0.253099\pi\)
0.700189 + 0.713957i \(0.253099\pi\)
\(8\) 0 0
\(9\) −0.0663151 −0.0221050
\(10\) 0 0
\(11\) 0 0
\(12\) 0 0
\(13\) −1.17750 −0.326580 −0.163290 − 0.986578i \(-0.552211\pi\)
−0.163290 + 0.986578i \(0.552211\pi\)
\(14\) 0 0
\(15\) −1.71280 −0.442243
\(16\) 0 0
\(17\) −5.40027 −1.30976 −0.654879 − 0.755734i \(-0.727280\pi\)
−0.654879 + 0.755734i \(0.727280\pi\)
\(18\) 0 0
\(19\) −3.84752 −0.882682 −0.441341 − 0.897339i \(-0.645497\pi\)
−0.441341 + 0.897339i \(0.645497\pi\)
\(20\) 0 0
\(21\) 6.34602 1.38481
\(22\) 0 0
\(23\) −8.68237 −1.81040 −0.905200 − 0.424986i \(-0.860279\pi\)
−0.905200 + 0.424986i \(0.860279\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −5.25199 −1.01075
\(28\) 0 0
\(29\) −6.87597 −1.27684 −0.638418 − 0.769690i \(-0.720411\pi\)
−0.638418 + 0.769690i \(0.720411\pi\)
\(30\) 0 0
\(31\) 8.75541 1.57252 0.786259 − 0.617898i \(-0.212015\pi\)
0.786259 + 0.617898i \(0.212015\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −3.70505 −0.626268
\(36\) 0 0
\(37\) 2.25468 0.370667 0.185334 − 0.982676i \(-0.440663\pi\)
0.185334 + 0.982676i \(0.440663\pi\)
\(38\) 0 0
\(39\) −2.01682 −0.322950
\(40\) 0 0
\(41\) 1.59733 0.249461 0.124730 − 0.992191i \(-0.460193\pi\)
0.124730 + 0.992191i \(0.460193\pi\)
\(42\) 0 0
\(43\) 4.11979 0.628262 0.314131 − 0.949380i \(-0.398287\pi\)
0.314131 + 0.949380i \(0.398287\pi\)
\(44\) 0 0
\(45\) 0.0663151 0.00988567
\(46\) 0 0
\(47\) −12.6800 −1.84956 −0.924782 − 0.380497i \(-0.875753\pi\)
−0.924782 + 0.380497i \(0.875753\pi\)
\(48\) 0 0
\(49\) 6.72743 0.961061
\(50\) 0 0
\(51\) −9.24959 −1.29520
\(52\) 0 0
\(53\) −12.3795 −1.70046 −0.850229 − 0.526413i \(-0.823536\pi\)
−0.850229 + 0.526413i \(0.823536\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −6.59004 −0.872872
\(58\) 0 0
\(59\) −0.393999 −0.0512944 −0.0256472 − 0.999671i \(-0.508165\pi\)
−0.0256472 + 0.999671i \(0.508165\pi\)
\(60\) 0 0
\(61\) 1.80085 0.230575 0.115288 − 0.993332i \(-0.463221\pi\)
0.115288 + 0.993332i \(0.463221\pi\)
\(62\) 0 0
\(63\) −0.245701 −0.0309554
\(64\) 0 0
\(65\) 1.17750 0.146051
\(66\) 0 0
\(67\) −14.3809 −1.75691 −0.878455 − 0.477826i \(-0.841425\pi\)
−0.878455 + 0.477826i \(0.841425\pi\)
\(68\) 0 0
\(69\) −14.8712 −1.79028
\(70\) 0 0
\(71\) −6.85527 −0.813571 −0.406785 − 0.913524i \(-0.633350\pi\)
−0.406785 + 0.913524i \(0.633350\pi\)
\(72\) 0 0
\(73\) 10.0551 1.17686 0.588430 − 0.808548i \(-0.299746\pi\)
0.588430 + 0.808548i \(0.299746\pi\)
\(74\) 0 0
\(75\) 1.71280 0.197777
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −1.11383 −0.125316 −0.0626580 − 0.998035i \(-0.519958\pi\)
−0.0626580 + 0.998035i \(0.519958\pi\)
\(80\) 0 0
\(81\) −8.79666 −0.977406
\(82\) 0 0
\(83\) 0.0173212 0.00190125 0.000950626 − 1.00000i \(-0.499697\pi\)
0.000950626 1.00000i \(0.499697\pi\)
\(84\) 0 0
\(85\) 5.40027 0.585742
\(86\) 0 0
\(87\) −11.7772 −1.26265
\(88\) 0 0
\(89\) −8.49434 −0.900399 −0.450199 − 0.892928i \(-0.648647\pi\)
−0.450199 + 0.892928i \(0.648647\pi\)
\(90\) 0 0
\(91\) −4.36270 −0.457335
\(92\) 0 0
\(93\) 14.9963 1.55504
\(94\) 0 0
\(95\) 3.84752 0.394747
\(96\) 0 0
\(97\) 10.5155 1.06768 0.533842 − 0.845584i \(-0.320748\pi\)
0.533842 + 0.845584i \(0.320748\pi\)
\(98\) 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.bb.1.6 6
4.3 odd 2 9680.2.a.dc.1.1 6
11.5 even 5 440.2.y.c.201.3 yes 12
11.9 even 5 440.2.y.c.81.3 ✓ 12
11.10 odd 2 4840.2.a.ba.1.6 6
44.27 odd 10 880.2.bo.i.641.1 12
44.31 odd 10 880.2.bo.i.81.1 12
44.43 even 2 9680.2.a.dd.1.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
440.2.y.c.81.3 ✓ 12 11.9 even 5
440.2.y.c.201.3 yes 12 11.5 even 5
880.2.bo.i.81.1 12 44.31 odd 10
880.2.bo.i.641.1 12 44.27 odd 10
4840.2.a.ba.1.6 6 11.10 odd 2
4840.2.a.bb.1.6 6 1.1 even 1 trivial
9680.2.a.dc.1.1 6 4.3 odd 2
9680.2.a.dd.1.1 6 44.43 even 2