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 1.4
Root \(-0.220878\) 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)
 
\(f(q)\) \(=\) \(q+0.220878 q^{3} -1.00000 q^{5} -2.08772 q^{7} -2.95121 q^{9} +1.87868 q^{13} -0.220878 q^{15} +5.28462 q^{17} -2.92388 q^{19} -0.461133 q^{21} +5.21521 q^{23} +1.00000 q^{25} -1.31449 q^{27} -3.81157 q^{29} +6.68385 q^{31} +2.08772 q^{35} -3.45754 q^{37} +0.414959 q^{39} +5.80287 q^{41} +0.884108 q^{43} +2.95121 q^{45} +7.69696 q^{47} -2.64141 q^{49} +1.16726 q^{51} -9.50720 q^{53} -0.645820 q^{57} -7.17454 q^{59} -7.88806 q^{61} +6.16132 q^{63} -1.87868 q^{65} -1.32232 q^{67} +1.15193 q^{69} -10.2325 q^{71} +8.17103 q^{73} +0.220878 q^{75} +5.23101 q^{79} +8.56330 q^{81} -9.63433 q^{83} -5.28462 q^{85} -0.841894 q^{87} -17.0929 q^{89} -3.92216 q^{91} +1.47632 q^{93} +2.92388 q^{95} -15.3296 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 3 q^{3} - 6 q^{5} + 7 q^{7} + 5 q^{9} - q^{13} + 3 q^{15} - 6 q^{17} + 7 q^{19} - 14 q^{21} - 9 q^{23} + 6 q^{25} - 21 q^{27} - 10 q^{29} + q^{31} - 7 q^{35} + 3 q^{37} + q^{39} + 6 q^{41} - 18 q^{43}+ \cdots - 33 q^{97}+O(q^{100}) \) 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
\(3\) 0.220878 0.127524 0.0637621 − 0.997965i \(-0.479690\pi\)
0.0637621 + 0.997965i \(0.479690\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −2.08772 −0.789085 −0.394543 − 0.918878i \(-0.629097\pi\)
−0.394543 + 0.918878i \(0.629097\pi\)
\(8\) 0 0
\(9\) −2.95121 −0.983738
\(10\) 0 0
\(11\) 0 0
\(12\) 0 0
\(13\) 1.87868 0.521052 0.260526 − 0.965467i \(-0.416104\pi\)
0.260526 + 0.965467i \(0.416104\pi\)
\(14\) 0 0
\(15\) −0.220878 −0.0570305
\(16\) 0 0
\(17\) 5.28462 1.28171 0.640854 − 0.767663i \(-0.278580\pi\)
0.640854 + 0.767663i \(0.278580\pi\)
\(18\) 0 0
\(19\) −2.92388 −0.670783 −0.335391 − 0.942079i \(-0.608869\pi\)
−0.335391 + 0.942079i \(0.608869\pi\)
\(20\) 0 0
\(21\) −0.461133 −0.100627
\(22\) 0 0
\(23\) 5.21521 1.08745 0.543724 − 0.839264i \(-0.317014\pi\)
0.543724 + 0.839264i \(0.317014\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −1.31449 −0.252974
\(28\) 0 0
\(29\) −3.81157 −0.707792 −0.353896 − 0.935285i \(-0.615143\pi\)
−0.353896 + 0.935285i \(0.615143\pi\)
\(30\) 0 0
\(31\) 6.68385 1.20045 0.600227 − 0.799829i \(-0.295077\pi\)
0.600227 + 0.799829i \(0.295077\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 2.08772 0.352890
\(36\) 0 0
\(37\) −3.45754 −0.568416 −0.284208 − 0.958763i \(-0.591731\pi\)
−0.284208 + 0.958763i \(0.591731\pi\)
\(38\) 0 0
\(39\) 0.414959 0.0664467
\(40\) 0 0
\(41\) 5.80287 0.906256 0.453128 − 0.891446i \(-0.350308\pi\)
0.453128 + 0.891446i \(0.350308\pi\)
\(42\) 0 0
\(43\) 0.884108 0.134825 0.0674126 − 0.997725i \(-0.478526\pi\)
0.0674126 + 0.997725i \(0.478526\pi\)
\(44\) 0 0
\(45\) 2.95121 0.439941
\(46\) 0 0
\(47\) 7.69696 1.12272 0.561359 − 0.827573i \(-0.310279\pi\)
0.561359 + 0.827573i \(0.310279\pi\)
\(48\) 0 0
\(49\) −2.64141 −0.377345
\(50\) 0 0
\(51\) 1.16726 0.163449
\(52\) 0 0
\(53\) −9.50720 −1.30591 −0.652957 − 0.757395i \(-0.726472\pi\)
−0.652957 + 0.757395i \(0.726472\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −0.645820 −0.0855410
\(58\) 0 0
\(59\) −7.17454 −0.934045 −0.467023 − 0.884245i \(-0.654673\pi\)
−0.467023 + 0.884245i \(0.654673\pi\)
\(60\) 0 0
\(61\) −7.88806 −1.00996 −0.504981 − 0.863130i \(-0.668501\pi\)
−0.504981 + 0.863130i \(0.668501\pi\)
\(62\) 0 0
\(63\) 6.16132 0.776253
\(64\) 0 0
\(65\) −1.87868 −0.233022
\(66\) 0 0
\(67\) −1.32232 −0.161547 −0.0807733 − 0.996732i \(-0.525739\pi\)
−0.0807733 + 0.996732i \(0.525739\pi\)
\(68\) 0 0
\(69\) 1.15193 0.138676
\(70\) 0 0
\(71\) −10.2325 −1.21437 −0.607186 − 0.794560i \(-0.707702\pi\)
−0.607186 + 0.794560i \(0.707702\pi\)
\(72\) 0 0
\(73\) 8.17103 0.956346 0.478173 − 0.878266i \(-0.341299\pi\)
0.478173 + 0.878266i \(0.341299\pi\)
\(74\) 0 0
\(75\) 0.220878 0.0255048
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 5.23101 0.588534 0.294267 − 0.955723i \(-0.404924\pi\)
0.294267 + 0.955723i \(0.404924\pi\)
\(80\) 0 0
\(81\) 8.56330 0.951477
\(82\) 0 0
\(83\) −9.63433 −1.05750 −0.528752 − 0.848776i \(-0.677340\pi\)
−0.528752 + 0.848776i \(0.677340\pi\)
\(84\) 0 0
\(85\) −5.28462 −0.573197
\(86\) 0 0
\(87\) −0.841894 −0.0902605
\(88\) 0 0
\(89\) −17.0929 −1.81185 −0.905923 − 0.423443i \(-0.860821\pi\)
−0.905923 + 0.423443i \(0.860821\pi\)
\(90\) 0 0
\(91\) −3.92216 −0.411154
\(92\) 0 0
\(93\) 1.47632 0.153087
\(94\) 0 0
\(95\) 2.92388 0.299983
\(96\) 0 0
\(97\) −15.3296 −1.55648 −0.778242 − 0.627964i \(-0.783888\pi\)
−0.778242 + 0.627964i \(0.783888\pi\)
\(98\) 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.1.4 6
4.3 odd 2 9680.2.a.dc.1.3 6
11.3 even 5 440.2.y.c.361.2 ✓ 12
11.4 even 5 440.2.y.c.401.2 yes 12
11.10 odd 2 4840.2.a.ba.1.4 6
44.3 odd 10 880.2.bo.i.801.2 12
44.15 odd 10 880.2.bo.i.401.2 12
44.43 even 2 9680.2.a.dd.1.3 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
440.2.y.c.361.2 ✓ 12 11.3 even 5
440.2.y.c.401.2 yes 12 11.4 even 5
880.2.bo.i.401.2 12 44.15 odd 10
880.2.bo.i.801.2 12 44.3 odd 10
4840.2.a.ba.1.4 6 11.10 odd 2
4840.2.a.bb.1.4 6 1.1 even 1 trivial
9680.2.a.dc.1.3 6 4.3 odd 2
9680.2.a.dd.1.3 6 44.43 even 2