Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1104,2,Mod(1,1104)] 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("1104.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1104, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1104 = 2^{4} \cdot 3 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1104.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,2,0,-2,0,-2] 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: \(8.81548438315\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 69)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(1.61803\) of defining polynomial
Character \(\chi\) \(=\) 1104.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{3} -3.23607 q^{5} +1.23607 q^{7} +1.00000 q^{9} -4.00000 q^{11} +4.47214 q^{13} -3.23607 q^{15} -7.23607 q^{17} -2.76393 q^{19} +1.23607 q^{21} -1.00000 q^{23} +5.47214 q^{25} +1.00000 q^{27} -4.47214 q^{29} -2.47214 q^{31} -4.00000 q^{33} -4.00000 q^{35} -4.47214 q^{37} +4.47214 q^{39} +6.94427 q^{41} -7.70820 q^{43} -3.23607 q^{45} +4.00000 q^{47} -5.47214 q^{49} -7.23607 q^{51} -0.763932 q^{53} +12.9443 q^{55} -2.76393 q^{57} -12.9443 q^{59} -4.47214 q^{61} +1.23607 q^{63} -14.4721 q^{65} -5.23607 q^{67} -1.00000 q^{69} +8.00000 q^{71} -10.9443 q^{73} +5.47214 q^{75} -4.94427 q^{77} +3.70820 q^{79} +1.00000 q^{81} -4.00000 q^{83} +23.4164 q^{85} -4.47214 q^{87} +3.23607 q^{89} +5.52786 q^{91} -2.47214 q^{93} +8.94427 q^{95} -0.472136 q^{97} -4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} - 2 q^{5} - 2 q^{7} + 2 q^{9} - 8 q^{11} - 2 q^{15} - 10 q^{17} - 10 q^{19} - 2 q^{21} - 2 q^{23} + 2 q^{25} + 2 q^{27} + 4 q^{31} - 8 q^{33} - 8 q^{35} - 4 q^{41} - 2 q^{43} - 2 q^{45} + 8 q^{47}+ \cdots - 8 q^{99}+O(q^{100}) \) 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
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) −3.23607 −1.44721 −0.723607 − 0.690212i \(-0.757517\pi\)
−0.723607 + 0.690212i \(0.757517\pi\)
\(6\) 0 0
\(7\) 1.23607 0.467190 0.233595 − 0.972334i \(-0.424951\pi\)
0.233595 + 0.972334i \(0.424951\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −4.00000 −1.20605 −0.603023 − 0.797724i \(-0.706037\pi\)
−0.603023 + 0.797724i \(0.706037\pi\)
\(12\) 0 0
\(13\) 4.47214 1.24035 0.620174 − 0.784465i \(-0.287062\pi\)
0.620174 + 0.784465i \(0.287062\pi\)
\(14\) 0 0
\(15\) −3.23607 −0.835549
\(16\) 0 0
\(17\) −7.23607 −1.75500 −0.877502 − 0.479573i \(-0.840792\pi\)
−0.877502 + 0.479573i \(0.840792\pi\)
\(18\) 0 0
\(19\) −2.76393 −0.634089 −0.317045 − 0.948411i \(-0.602691\pi\)
−0.317045 + 0.948411i \(0.602691\pi\)
\(20\) 0 0
\(21\) 1.23607 0.269732
\(22\) 0 0
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) 5.47214 1.09443
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −4.47214 −0.830455 −0.415227 − 0.909718i \(-0.636298\pi\)
−0.415227 + 0.909718i \(0.636298\pi\)
\(30\) 0 0
\(31\) −2.47214 −0.444009 −0.222004 − 0.975046i \(-0.571260\pi\)
−0.222004 + 0.975046i \(0.571260\pi\)
\(32\) 0 0
\(33\) −4.00000 −0.696311
\(34\) 0 0
\(35\) −4.00000 −0.676123
\(36\) 0 0
\(37\) −4.47214 −0.735215 −0.367607 − 0.929981i \(-0.619823\pi\)
−0.367607 + 0.929981i \(0.619823\pi\)
\(38\) 0 0
\(39\) 4.47214 0.716115
\(40\) 0 0
\(41\) 6.94427 1.08451 0.542257 − 0.840213i \(-0.317570\pi\)
0.542257 + 0.840213i \(0.317570\pi\)
\(42\) 0 0
\(43\) −7.70820 −1.17549 −0.587745 − 0.809046i \(-0.699984\pi\)
−0.587745 + 0.809046i \(0.699984\pi\)
\(44\) 0 0
\(45\) −3.23607 −0.482405
\(46\) 0 0
\(47\) 4.00000 0.583460 0.291730 − 0.956501i \(-0.405769\pi\)
0.291730 + 0.956501i \(0.405769\pi\)
\(48\) 0 0
\(49\) −5.47214 −0.781734
\(50\) 0 0
\(51\) −7.23607 −1.01325
\(52\) 0 0
\(53\) −0.763932 −0.104934 −0.0524671 − 0.998623i \(-0.516708\pi\)
−0.0524671 + 0.998623i \(0.516708\pi\)
\(54\) 0 0
\(55\) 12.9443 1.74541
\(56\) 0 0
\(57\) −2.76393 −0.366092
\(58\) 0 0
\(59\) −12.9443 −1.68520 −0.842600 − 0.538539i \(-0.818976\pi\)
−0.842600 + 0.538539i \(0.818976\pi\)
\(60\) 0 0
\(61\) −4.47214 −0.572598 −0.286299 − 0.958140i \(-0.592425\pi\)
−0.286299 + 0.958140i \(0.592425\pi\)
\(62\) 0 0
\(63\) 1.23607 0.155730
\(64\) 0 0
\(65\) −14.4721 −1.79505
\(66\) 0 0
\(67\) −5.23607 −0.639688 −0.319844 − 0.947470i \(-0.603630\pi\)
−0.319844 + 0.947470i \(0.603630\pi\)
\(68\) 0 0
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) 8.00000 0.949425 0.474713 − 0.880141i \(-0.342552\pi\)
0.474713 + 0.880141i \(0.342552\pi\)
\(72\) 0 0
\(73\) −10.9443 −1.28093 −0.640465 − 0.767987i \(-0.721258\pi\)
−0.640465 + 0.767987i \(0.721258\pi\)
\(74\) 0 0
\(75\) 5.47214 0.631868
\(76\) 0 0
\(77\) −4.94427 −0.563452
\(78\) 0 0
\(79\) 3.70820 0.417206 0.208603 − 0.978000i \(-0.433108\pi\)
0.208603 + 0.978000i \(0.433108\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −4.00000 −0.439057 −0.219529 − 0.975606i \(-0.570452\pi\)
−0.219529 + 0.975606i \(0.570452\pi\)
\(84\) 0 0
\(85\) 23.4164 2.53987
\(86\) 0 0
\(87\) −4.47214 −0.479463
\(88\) 0 0
\(89\) 3.23607 0.343023 0.171511 − 0.985182i \(-0.445135\pi\)
0.171511 + 0.985182i \(0.445135\pi\)
\(90\) 0 0
\(91\) 5.52786 0.579478
\(92\) 0 0
\(93\) −2.47214 −0.256349
\(94\) 0 0
\(95\) 8.94427 0.917663
\(96\) 0 0
\(97\) −0.472136 −0.0479381 −0.0239691 − 0.999713i \(-0.507630\pi\)
−0.0239691 + 0.999713i \(0.507630\pi\)
\(98\) 0 0
\(99\) −4.00000 −0.402015
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 1104.2.a.m.1.1 2
3.2 odd 2 3312.2.a.bb.1.2 2
4.3 odd 2 69.2.a.b.1.2 ✓ 2
8.3 odd 2 4416.2.a.bm.1.2 2
8.5 even 2 4416.2.a.bg.1.2 2
12.11 even 2 207.2.a.c.1.1 2
20.3 even 4 1725.2.b.o.1174.1 4
20.7 even 4 1725.2.b.o.1174.4 4
20.19 odd 2 1725.2.a.ba.1.1 2
28.27 even 2 3381.2.a.t.1.2 2
44.43 even 2 8349.2.a.i.1.1 2
60.59 even 2 5175.2.a.bk.1.2 2
92.91 even 2 1587.2.a.i.1.2 2
276.275 odd 2 4761.2.a.v.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
69.2.a.b.1.2 ✓ 2 4.3 odd 2
207.2.a.c.1.1 2 12.11 even 2
1104.2.a.m.1.1 2 1.1 even 1 trivial
1587.2.a.i.1.2 2 92.91 even 2
1725.2.a.ba.1.1 2 20.19 odd 2
1725.2.b.o.1174.1 4 20.3 even 4
1725.2.b.o.1174.4 4 20.7 even 4
3312.2.a.bb.1.2 2 3.2 odd 2
3381.2.a.t.1.2 2 28.27 even 2
4416.2.a.bg.1.2 2 8.5 even 2
4416.2.a.bm.1.2 2 8.3 odd 2
4761.2.a.v.1.1 2 276.275 odd 2
5175.2.a.bk.1.2 2 60.59 even 2
8349.2.a.i.1.1 2 44.43 even 2