Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [3380,2,Mod(1,3380)] 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("3380.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3380, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 3380 = 2^{2} \cdot 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3380.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,0,0,0,3,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(26.9894358832\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.756.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 6x - 2 \) 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 260)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-2.26180\) of defining polynomial
Character \(\chi\) \(=\) 3380.1

$q$-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)
 
\(f(q)\) \(=\) \(q-2.26180 q^{3} +1.00000 q^{5} +1.11575 q^{7} +2.11575 q^{9} +5.37755 q^{11} -2.26180 q^{15} -7.90116 q^{19} -2.52360 q^{21} +6.49330 q^{23} +1.00000 q^{25} +2.00000 q^{27} +3.63935 q^{29} +3.14605 q^{31} -12.1630 q^{33} +1.11575 q^{35} +7.40786 q^{37} +4.75510 q^{41} +4.78541 q^{43} +2.11575 q^{45} -6.16296 q^{47} -5.75510 q^{49} +0.292106 q^{53} +5.37755 q^{55} +17.8709 q^{57} -11.3776 q^{59} -5.34725 q^{61} +2.36065 q^{63} -11.8709 q^{67} -14.6866 q^{69} -3.43816 q^{71} +4.59214 q^{73} -2.26180 q^{75} +6.00000 q^{77} -10.8157 q^{79} -10.8709 q^{81} +7.40786 q^{83} -8.23150 q^{87} +14.8157 q^{89} -7.11575 q^{93} -7.90116 q^{95} +3.76850 q^{97} +11.3776 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{5} + 3 q^{9} + 6 q^{11} + 6 q^{21} + 6 q^{23} + 3 q^{25} + 6 q^{27} - 6 q^{29} + 6 q^{31} - 6 q^{33} + 12 q^{37} - 6 q^{41} - 6 q^{43} + 3 q^{45} + 12 q^{47} + 3 q^{49} - 6 q^{53} + 6 q^{55}+ \cdots + 24 q^{99}+O(q^{100}) \) 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
\(3\) −2.26180 −1.30585 −0.652926 − 0.757422i \(-0.726459\pi\)
−0.652926 + 0.757422i \(0.726459\pi\)
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 1.11575 0.421714 0.210857 − 0.977517i \(-0.432375\pi\)
0.210857 + 0.977517i \(0.432375\pi\)
\(8\) 0 0
\(9\) 2.11575 0.705250
\(10\) 0 0
\(11\) 5.37755 1.62139 0.810696 − 0.585467i \(-0.199089\pi\)
0.810696 + 0.585467i \(0.199089\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) 0 0
\(15\) −2.26180 −0.583995
\(16\) 0 0
\(17\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) −7.90116 −1.81265 −0.906325 − 0.422582i \(-0.861124\pi\)
−0.906325 + 0.422582i \(0.861124\pi\)
\(20\) 0 0
\(21\) −2.52360 −0.550696
\(22\) 0 0
\(23\) 6.49330 1.35395 0.676973 − 0.736007i \(-0.263291\pi\)
0.676973 + 0.736007i \(0.263291\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 2.00000 0.384900
\(28\) 0 0
\(29\) 3.63935 0.675811 0.337906 − 0.941180i \(-0.390282\pi\)
0.337906 + 0.941180i \(0.390282\pi\)
\(30\) 0 0
\(31\) 3.14605 0.565048 0.282524 − 0.959260i \(-0.408828\pi\)
0.282524 + 0.959260i \(0.408828\pi\)
\(32\) 0 0
\(33\) −12.1630 −2.11730
\(34\) 0 0
\(35\) 1.11575 0.188596
\(36\) 0 0
\(37\) 7.40786 1.21784 0.608922 − 0.793230i \(-0.291602\pi\)
0.608922 + 0.793230i \(0.291602\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 4.75510 0.742622 0.371311 − 0.928508i \(-0.378908\pi\)
0.371311 + 0.928508i \(0.378908\pi\)
\(42\) 0 0
\(43\) 4.78541 0.729768 0.364884 − 0.931053i \(-0.381109\pi\)
0.364884 + 0.931053i \(0.381109\pi\)
\(44\) 0 0
\(45\) 2.11575 0.315397
\(46\) 0 0
\(47\) −6.16296 −0.898960 −0.449480 − 0.893290i \(-0.648391\pi\)
−0.449480 + 0.893290i \(0.648391\pi\)
\(48\) 0 0
\(49\) −5.75510 −0.822158
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0.292106 0.0401238 0.0200619 − 0.999799i \(-0.493614\pi\)
0.0200619 + 0.999799i \(0.493614\pi\)
\(54\) 0 0
\(55\) 5.37755 0.725109
\(56\) 0 0
\(57\) 17.8709 2.36705
\(58\) 0 0
\(59\) −11.3776 −1.48123 −0.740616 − 0.671929i \(-0.765466\pi\)
−0.740616 + 0.671929i \(0.765466\pi\)
\(60\) 0 0
\(61\) −5.34725 −0.684645 −0.342322 − 0.939583i \(-0.611214\pi\)
−0.342322 + 0.939583i \(0.611214\pi\)
\(62\) 0 0
\(63\) 2.36065 0.297413
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −11.8709 −1.45026 −0.725128 − 0.688614i \(-0.758219\pi\)
−0.725128 + 0.688614i \(0.758219\pi\)
\(68\) 0 0
\(69\) −14.6866 −1.76805
\(70\) 0 0
\(71\) −3.43816 −0.408034 −0.204017 − 0.978967i \(-0.565400\pi\)
−0.204017 + 0.978967i \(0.565400\pi\)
\(72\) 0 0
\(73\) 4.59214 0.537470 0.268735 − 0.963214i \(-0.413394\pi\)
0.268735 + 0.963214i \(0.413394\pi\)
\(74\) 0 0
\(75\) −2.26180 −0.261170
\(76\) 0 0
\(77\) 6.00000 0.683763
\(78\) 0 0
\(79\) −10.8157 −1.21686 −0.608431 − 0.793607i \(-0.708201\pi\)
−0.608431 + 0.793607i \(0.708201\pi\)
\(80\) 0 0
\(81\) −10.8709 −1.20787
\(82\) 0 0
\(83\) 7.40786 0.813118 0.406559 − 0.913625i \(-0.366729\pi\)
0.406559 + 0.913625i \(0.366729\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −8.23150 −0.882509
\(88\) 0 0
\(89\) 14.8157 1.57046 0.785231 − 0.619203i \(-0.212544\pi\)
0.785231 + 0.619203i \(0.212544\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −7.11575 −0.737869
\(94\) 0 0
\(95\) −7.90116 −0.810642
\(96\) 0 0
\(97\) 3.76850 0.382633 0.191317 − 0.981528i \(-0.438724\pi\)
0.191317 + 0.981528i \(0.438724\pi\)
\(98\) 0 0
\(99\) 11.3776 1.14349
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 3380.2.a.n.1.1 3
13.5 odd 4 260.2.f.a.181.1 ✓ 6
13.8 odd 4 260.2.f.a.181.2 yes 6
13.12 even 2 3380.2.a.m.1.1 3
39.5 even 4 2340.2.c.d.181.5 6
39.8 even 4 2340.2.c.d.181.2 6
52.31 even 4 1040.2.k.c.961.5 6
52.47 even 4 1040.2.k.c.961.6 6
65.8 even 4 1300.2.d.c.649.2 6
65.18 even 4 1300.2.d.d.649.2 6
65.34 odd 4 1300.2.f.e.701.6 6
65.44 odd 4 1300.2.f.e.701.5 6
65.47 even 4 1300.2.d.d.649.5 6
65.57 even 4 1300.2.d.c.649.5 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
260.2.f.a.181.1 ✓ 6 13.5 odd 4
260.2.f.a.181.2 yes 6 13.8 odd 4
1040.2.k.c.961.5 6 52.31 even 4
1040.2.k.c.961.6 6 52.47 even 4
1300.2.d.c.649.2 6 65.8 even 4
1300.2.d.c.649.5 6 65.57 even 4
1300.2.d.d.649.2 6 65.18 even 4
1300.2.d.d.649.5 6 65.47 even 4
1300.2.f.e.701.5 6 65.44 odd 4
1300.2.f.e.701.6 6 65.34 odd 4
2340.2.c.d.181.2 6 39.8 even 4
2340.2.c.d.181.5 6 39.5 even 4
3380.2.a.m.1.1 3 13.12 even 2
3380.2.a.n.1.1 3 1.1 even 1 trivial