Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [6080,2,Mod(1,6080)] 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("6080.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6080, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 6080 = 2^{6} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6080.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: \(48.5490444289\)
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: no (minimal twist has level 3040)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-2.10710\) of defining polynomial
Character \(\chi\) \(=\) 6080.1

$q$-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)
 
\(f(q)\) \(=\) \(q+2.10710 q^{3} -1.00000 q^{5} +2.43986 q^{7} +1.43986 q^{9} +1.36667 q^{11} +5.47377 q^{13} -2.10710 q^{15} -3.28738 q^{17} -1.00000 q^{19} +5.14101 q^{21} +3.41378 q^{23} +1.00000 q^{25} -3.28738 q^{27} +0.926817 q^{29} +11.0939 q^{31} +2.87971 q^{33} -2.43986 q^{35} +6.90751 q^{37} +11.5338 q^{39} +4.21419 q^{41} -0.486962 q^{43} -1.43986 q^{45} -10.4606 q^{47} -1.04711 q^{49} -6.92682 q^{51} -13.2332 q^{53} -1.36667 q^{55} -2.10710 q^{57} -10.0207 q^{59} +9.48665 q^{61} +3.51304 q^{63} -5.47377 q^{65} +4.01288 q^{67} +7.19316 q^{69} -4.94754 q^{71} +13.3552 q^{73} +2.10710 q^{75} +3.33448 q^{77} +6.97392 q^{79} -11.2464 q^{81} -3.51304 q^{83} +3.28738 q^{85} +1.95289 q^{87} +8.06783 q^{89} +13.3552 q^{91} +23.3759 q^{93} +1.00000 q^{95} -0.546952 q^{97} +1.96781 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - q^{3} - 4 q^{5} + 5 q^{7} + q^{9} - 6 q^{11} + q^{13} + q^{15} - q^{17} - 4 q^{19} - 5 q^{21} + 5 q^{23} + 4 q^{25} - q^{27} - 3 q^{29} + 16 q^{31} + 2 q^{33} - 5 q^{35} + 8 q^{37} + 13 q^{39}+ \cdots + 10 q^{99}+O(q^{100}) \) 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
\(3\) 2.10710 1.21653 0.608266 − 0.793733i \(-0.291865\pi\)
0.608266 + 0.793733i \(0.291865\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 2.43986 0.922179 0.461089 − 0.887354i \(-0.347459\pi\)
0.461089 + 0.887354i \(0.347459\pi\)
\(8\) 0 0
\(9\) 1.43986 0.479952
\(10\) 0 0
\(11\) 1.36667 0.412067 0.206034 − 0.978545i \(-0.433944\pi\)
0.206034 + 0.978545i \(0.433944\pi\)
\(12\) 0 0
\(13\) 5.47377 1.51815 0.759075 − 0.651003i \(-0.225651\pi\)
0.759075 + 0.651003i \(0.225651\pi\)
\(14\) 0 0
\(15\) −2.10710 −0.544050
\(16\) 0 0
\(17\) −3.28738 −0.797306 −0.398653 − 0.917102i \(-0.630522\pi\)
−0.398653 + 0.917102i \(0.630522\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) 5.14101 1.12186
\(22\) 0 0
\(23\) 3.41378 0.711822 0.355911 − 0.934520i \(-0.384171\pi\)
0.355911 + 0.934520i \(0.384171\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −3.28738 −0.632656
\(28\) 0 0
\(29\) 0.926817 0.172106 0.0860528 − 0.996291i \(-0.472575\pi\)
0.0860528 + 0.996291i \(0.472575\pi\)
\(30\) 0 0
\(31\) 11.0939 1.99252 0.996262 − 0.0863841i \(-0.0275312\pi\)
0.996262 + 0.0863841i \(0.0275312\pi\)
\(32\) 0 0
\(33\) 2.87971 0.501293
\(34\) 0 0
\(35\) −2.43986 −0.412411
\(36\) 0 0
\(37\) 6.90751 1.13559 0.567794 − 0.823171i \(-0.307797\pi\)
0.567794 + 0.823171i \(0.307797\pi\)
\(38\) 0 0
\(39\) 11.5338 1.84688
\(40\) 0 0
\(41\) 4.21419 0.658146 0.329073 − 0.944304i \(-0.393264\pi\)
0.329073 + 0.944304i \(0.393264\pi\)
\(42\) 0 0
\(43\) −0.486962 −0.0742610 −0.0371305 − 0.999310i \(-0.511822\pi\)
−0.0371305 + 0.999310i \(0.511822\pi\)
\(44\) 0 0
\(45\) −1.43986 −0.214641
\(46\) 0 0
\(47\) −10.4606 −1.52583 −0.762916 − 0.646498i \(-0.776233\pi\)
−0.762916 + 0.646498i \(0.776233\pi\)
\(48\) 0 0
\(49\) −1.04711 −0.149587
\(50\) 0 0
\(51\) −6.92682 −0.969948
\(52\) 0 0
\(53\) −13.2332 −1.81772 −0.908859 − 0.417103i \(-0.863045\pi\)
−0.908859 + 0.417103i \(0.863045\pi\)
\(54\) 0 0
\(55\) −1.36667 −0.184282
\(56\) 0 0
\(57\) −2.10710 −0.279092
\(58\) 0 0
\(59\) −10.0207 −1.30459 −0.652293 − 0.757967i \(-0.726193\pi\)
−0.652293 + 0.757967i \(0.726193\pi\)
\(60\) 0 0
\(61\) 9.48665 1.21464 0.607321 − 0.794457i \(-0.292244\pi\)
0.607321 + 0.794457i \(0.292244\pi\)
\(62\) 0 0
\(63\) 3.51304 0.442601
\(64\) 0 0
\(65\) −5.47377 −0.678937
\(66\) 0 0
\(67\) 4.01288 0.490252 0.245126 − 0.969491i \(-0.421171\pi\)
0.245126 + 0.969491i \(0.421171\pi\)
\(68\) 0 0
\(69\) 7.19316 0.865955
\(70\) 0 0
\(71\) −4.94754 −0.587165 −0.293582 − 0.955934i \(-0.594847\pi\)
−0.293582 + 0.955934i \(0.594847\pi\)
\(72\) 0 0
\(73\) 13.3552 1.56311 0.781554 − 0.623837i \(-0.214427\pi\)
0.781554 + 0.623837i \(0.214427\pi\)
\(74\) 0 0
\(75\) 2.10710 0.243307
\(76\) 0 0
\(77\) 3.33448 0.380000
\(78\) 0 0
\(79\) 6.97392 0.784628 0.392314 − 0.919831i \(-0.371675\pi\)
0.392314 + 0.919831i \(0.371675\pi\)
\(80\) 0 0
\(81\) −11.2464 −1.24960
\(82\) 0 0
\(83\) −3.51304 −0.385606 −0.192803 − 0.981237i \(-0.561758\pi\)
−0.192803 + 0.981237i \(0.561758\pi\)
\(84\) 0 0
\(85\) 3.28738 0.356566
\(86\) 0 0
\(87\) 1.95289 0.209372
\(88\) 0 0
\(89\) 8.06783 0.855188 0.427594 − 0.903971i \(-0.359361\pi\)
0.427594 + 0.903971i \(0.359361\pi\)
\(90\) 0 0
\(91\) 13.3552 1.40001
\(92\) 0 0
\(93\) 23.3759 2.42397
\(94\) 0 0
\(95\) 1.00000 0.102598
\(96\) 0 0
\(97\) −0.546952 −0.0555345 −0.0277673 − 0.999614i \(-0.508840\pi\)
−0.0277673 + 0.999614i \(0.508840\pi\)
\(98\) 0 0
\(99\) 1.96781 0.197772
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 6080.2.a.cd.1.4 4
4.3 odd 2 6080.2.a.cf.1.1 4
8.3 odd 2 3040.2.a.r.1.4 ✓ 4
8.5 even 2 3040.2.a.t.1.1 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3040.2.a.r.1.4 ✓ 4 8.3 odd 2
3040.2.a.t.1.1 yes 4 8.5 even 2
6080.2.a.cd.1.4 4 1.1 even 1 trivial
6080.2.a.cf.1.1 4 4.3 odd 2