Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [55,2,Mod(32,55)] 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("55.32"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(55, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([1, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 55 = 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 55.e (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.439177211117\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(i, \sqrt{11})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 5x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{4}]$

Embedding invariants

Embedding label 43.2
Root \(1.65831 - 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 55.43
Dual form 55.2.e.b.32.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(1.15831 + 1.15831i) q^{3} -2.00000i q^{4} +(-1.65831 + 1.50000i) q^{5} -0.316625i q^{9} -3.31662 q^{11} +(2.31662 - 2.31662i) q^{12} +(-3.65831 - 0.183375i) q^{15} -4.00000 q^{16} +(3.00000 + 3.31662i) q^{20} +(6.15831 + 6.15831i) q^{23} +(0.500000 - 4.97494i) q^{25} +(3.84169 - 3.84169i) q^{27} +9.94987 q^{31} +(-3.84169 - 3.84169i) q^{33} -0.633250 q^{36} +(-8.47494 + 8.47494i) q^{37} +6.63325i q^{44} +(0.474937 + 0.525063i) q^{45} +(-2.68338 + 2.68338i) q^{47} +(-4.63325 - 4.63325i) q^{48} -7.00000i q^{49} +(-9.63325 - 9.63325i) q^{53} +(5.50000 - 4.97494i) q^{55} -3.31662i q^{59} +(-0.366750 + 7.31662i) q^{60} +8.00000i q^{64} +(1.52506 - 1.52506i) q^{67} +14.2665i q^{69} -3.00000 q^{71} +(6.34169 - 5.18338i) q^{75} +(6.63325 - 6.00000i) q^{80} +7.94987 q^{81} -9.00000i q^{89} +(12.3166 - 12.3166i) q^{92} +(11.5251 + 11.5251i) q^{93} +(-13.4749 + 13.4749i) q^{97} +1.05013i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{3} - 4 q^{12} - 8 q^{15} - 16 q^{16} + 12 q^{20} + 18 q^{23} + 2 q^{25} + 22 q^{27} - 22 q^{33} + 24 q^{36} - 14 q^{37} - 18 q^{45} - 24 q^{47} + 8 q^{48} - 12 q^{53} + 22 q^{55} - 28 q^{60}+ \cdots - 34 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/55\mathbb{Z}\right)^\times\).

\(n\) \(12\) \(46\)
\(\chi(n)\) \(e\left(\frac{3}{4}\right)\) \(-1\)

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 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(3\) 1.15831 + 1.15831i 0.668752 + 0.668752i 0.957427 − 0.288675i \(-0.0932147\pi\)
−0.288675 + 0.957427i \(0.593215\pi\)
\(4\) − 2.00000i − 1.00000i
\(5\) −1.65831 + 1.50000i −0.741620 + 0.670820i
\(6\) 0 0
\(7\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(8\) 0 0
\(9\) − 0.316625i − 0.105542i
\(10\) 0 0
\(11\) −3.31662 −1.00000
\(12\) 2.31662 − 2.31662i 0.668752 − 0.668752i
\(13\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(14\) 0 0
\(15\) −3.65831 − 0.183375i −0.944572 − 0.0473473i
\(16\) −4.00000 −1.00000
\(17\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(18\) 0 0
\(19\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 3.00000 + 3.31662i 0.670820 + 0.741620i
\(21\) 0 0
\(22\) 0 0
\(23\) 6.15831 + 6.15831i 1.28410 + 1.28410i 0.938315 + 0.345782i \(0.112386\pi\)
0.345782 + 0.938315i \(0.387614\pi\)
\(24\) 0 0
\(25\) 0.500000 − 4.97494i 0.100000 − 0.994987i
\(26\) 0 0
\(27\) 3.84169 − 3.84169i 0.739333 − 0.739333i
\(28\) 0 0
\(29\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) 9.94987 1.78705 0.893525 − 0.449013i \(-0.148224\pi\)
0.893525 + 0.449013i \(0.148224\pi\)
\(32\) 0 0
\(33\) −3.84169 − 3.84169i −0.668752 − 0.668752i
\(34\) 0 0
\(35\) 0 0
\(36\) −0.633250 −0.105542
\(37\) −8.47494 + 8.47494i −1.39327 + 1.39327i −0.575396 + 0.817875i \(0.695152\pi\)
−0.817875 + 0.575396i \(0.804848\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(44\) 6.63325i 1.00000i
\(45\) 0.474937 + 0.525063i 0.0707995 + 0.0782717i
\(46\) 0 0
\(47\) −2.68338 + 2.68338i −0.391411 + 0.391411i −0.875190 − 0.483779i \(-0.839264\pi\)
0.483779 + 0.875190i \(0.339264\pi\)
\(48\) −4.63325 − 4.63325i −0.668752 − 0.668752i
\(49\) − 7.00000i − 1.00000i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −9.63325 − 9.63325i −1.32323 − 1.32323i −0.911147 − 0.412082i \(-0.864802\pi\)
−0.412082 − 0.911147i \(-0.635198\pi\)
\(54\) 0 0
\(55\) 5.50000 − 4.97494i 0.741620 − 0.670820i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) − 3.31662i − 0.431788i −0.976417 − 0.215894i \(-0.930733\pi\)
0.976417 − 0.215894i \(-0.0692665\pi\)
\(60\) −0.366750 + 7.31662i −0.0473473 + 0.944572i
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 8.00000i 1.00000i
\(65\) 0 0
\(66\) 0 0
\(67\) 1.52506 − 1.52506i 0.186316 − 0.186316i −0.607785 − 0.794101i \(-0.707942\pi\)
0.794101 + 0.607785i \(0.207942\pi\)
\(68\) 0 0
\(69\) 14.2665i 1.71748i
\(70\) 0 0
\(71\) −3.00000 −0.356034 −0.178017 − 0.984027i \(-0.556968\pi\)
−0.178017 + 0.984027i \(0.556968\pi\)
\(72\) 0 0
\(73\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(74\) 0 0
\(75\) 6.34169 − 5.18338i 0.732275 − 0.598525i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 6.63325 − 6.00000i 0.741620 − 0.670820i
\(81\) 7.94987 0.883319
\(82\) 0 0
\(83\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) − 9.00000i − 0.953998i −0.878904 − 0.476999i \(-0.841725\pi\)
0.878904 − 0.476999i \(-0.158275\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 12.3166 − 12.3166i 1.28410 − 1.28410i
\(93\) 11.5251 + 11.5251i 1.19509 + 1.19509i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −13.4749 + 13.4749i −1.36817 + 1.36817i −0.505128 + 0.863044i \(0.668555\pi\)
−0.863044 + 0.505128i \(0.831445\pi\)
\(98\) 0 0
\(99\) 1.05013i 0.105542i
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 55.2.e.b.43.2 yes 4
3.2 odd 2 495.2.k.a.208.2 4
4.3 odd 2 880.2.bd.d.593.1 4
5.2 odd 4 inner 55.2.e.b.32.2 ✓ 4
5.3 odd 4 275.2.e.a.32.1 4
5.4 even 2 275.2.e.a.43.1 4
11.2 odd 10 605.2.m.a.403.2 16
11.3 even 5 605.2.m.a.233.1 16
11.4 even 5 605.2.m.a.578.1 16
11.5 even 5 605.2.m.a.118.1 16
11.6 odd 10 605.2.m.a.118.1 16
11.7 odd 10 605.2.m.a.578.1 16
11.8 odd 10 605.2.m.a.233.1 16
11.9 even 5 605.2.m.a.403.2 16
11.10 odd 2 CM 55.2.e.b.43.2 yes 4
15.2 even 4 495.2.k.a.307.2 4
20.7 even 4 880.2.bd.d.417.1 4
33.32 even 2 495.2.k.a.208.2 4
44.43 even 2 880.2.bd.d.593.1 4
55.2 even 20 605.2.m.a.282.1 16
55.7 even 20 605.2.m.a.457.1 16
55.17 even 20 605.2.m.a.602.2 16
55.27 odd 20 605.2.m.a.602.2 16
55.32 even 4 inner 55.2.e.b.32.2 ✓ 4
55.37 odd 20 605.2.m.a.457.1 16
55.42 odd 20 605.2.m.a.282.1 16
55.43 even 4 275.2.e.a.32.1 4
55.47 odd 20 605.2.m.a.112.1 16
55.52 even 20 605.2.m.a.112.1 16
55.54 odd 2 275.2.e.a.43.1 4
165.32 odd 4 495.2.k.a.307.2 4
220.87 odd 4 880.2.bd.d.417.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
55.2.e.b.32.2 ✓ 4 5.2 odd 4 inner
55.2.e.b.32.2 ✓ 4 55.32 even 4 inner
55.2.e.b.43.2 yes 4 1.1 even 1 trivial
55.2.e.b.43.2 yes 4 11.10 odd 2 CM
275.2.e.a.32.1 4 5.3 odd 4
275.2.e.a.32.1 4 55.43 even 4
275.2.e.a.43.1 4 5.4 even 2
275.2.e.a.43.1 4 55.54 odd 2
495.2.k.a.208.2 4 3.2 odd 2
495.2.k.a.208.2 4 33.32 even 2
495.2.k.a.307.2 4 15.2 even 4
495.2.k.a.307.2 4 165.32 odd 4
605.2.m.a.112.1 16 55.47 odd 20
605.2.m.a.112.1 16 55.52 even 20
605.2.m.a.118.1 16 11.5 even 5
605.2.m.a.118.1 16 11.6 odd 10
605.2.m.a.233.1 16 11.3 even 5
605.2.m.a.233.1 16 11.8 odd 10
605.2.m.a.282.1 16 55.2 even 20
605.2.m.a.282.1 16 55.42 odd 20
605.2.m.a.403.2 16 11.2 odd 10
605.2.m.a.403.2 16 11.9 even 5
605.2.m.a.457.1 16 55.7 even 20
605.2.m.a.457.1 16 55.37 odd 20
605.2.m.a.578.1 16 11.4 even 5
605.2.m.a.578.1 16 11.7 odd 10
605.2.m.a.602.2 16 55.17 even 20
605.2.m.a.602.2 16 55.27 odd 20
880.2.bd.d.417.1 4 20.7 even 4
880.2.bd.d.417.1 4 220.87 odd 4
880.2.bd.d.593.1 4 4.3 odd 2
880.2.bd.d.593.1 4 44.43 even 2