Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1456,2,Mod(113,1456)] 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("1456.113"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1456, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 0, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1456 = 2^{4} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1456.s (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,1,0,-14] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(11.6262185343\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: 8.0.59066497296.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} + 7x^{6} + 38x^{4} - 16x^{3} + 15x^{2} + 3x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 91)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 113.3
Root \(1.37054 + 2.37385i\) of defining polynomial
Character \(\chi\) \(=\) 1456.113
Dual form 1456.2.s.q.1121.3

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.682410 - 1.18197i) q^{3} +0.741082 q^{5} +(-0.500000 - 0.866025i) q^{7} +(0.568634 + 0.984903i) q^{9} +(-0.682410 + 1.18197i) q^{11} +(0.301907 + 3.59289i) q^{13} +(0.505722 - 0.875935i) q^{15} +(2.07436 + 3.59289i) q^{17} +(3.63303 + 6.29259i) q^{19} -1.36482 q^{21} +(-1.16673 + 2.02083i) q^{23} -4.45080 q^{25} +5.64662 q^{27} +(0.203815 - 0.353017i) q^{29} +2.77245 q^{31} +(0.931366 + 1.61317i) q^{33} +(-0.370541 - 0.641796i) q^{35} +(3.05295 - 5.28787i) q^{37} +(4.45271 + 2.09498i) q^{39} +(-0.627306 + 1.08653i) q^{41} +(-0.870541 - 1.50782i) q^{43} +(0.421404 + 0.729894i) q^{45} +5.85843 q^{47} +(-0.500000 + 0.866025i) q^{49} +5.66224 q^{51} +4.56778 q^{53} +(-0.505722 + 0.875935i) q^{55} +9.91685 q^{57} +(-5.49213 - 9.51264i) q^{59} +(-3.26249 - 5.65079i) q^{61} +(0.568634 - 0.984903i) q^{63} +(0.223738 + 2.66263i) q^{65} +(-6.87983 + 11.9162i) q^{67} +(1.59237 + 2.75807i) q^{69} +(-2.40763 - 4.17014i) q^{71} +6.06987 q^{73} +(-3.03727 + 5.26070i) q^{75} +1.36482 q^{77} +9.12582 q^{79} +(2.14741 - 3.71942i) q^{81} -11.7368 q^{83} +(1.53727 + 2.66263i) q^{85} +(-0.278170 - 0.481805i) q^{87} +(0.880503 - 1.52508i) q^{89} +(2.96058 - 2.05790i) q^{91} +(1.89195 - 3.27695i) q^{93} +(2.69237 + 4.66332i) q^{95} +(-4.76691 - 8.25652i) q^{97} -1.55217 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + q^{3} - 14 q^{5} - 4 q^{7} - 7 q^{9} - q^{11} + 4 q^{13} + 3 q^{15} + 4 q^{17} + q^{19} - 2 q^{21} - 2 q^{23} + 10 q^{25} + 52 q^{27} - q^{29} + 8 q^{31} + 19 q^{33} + 7 q^{35} + 10 q^{37} - 20 q^{39}+ \cdots - 46 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1456\mathbb{Z}\right)^\times\).

\(n\) \(561\) \(911\) \(1093\) \(1249\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\) \(1\) \(1\) \(1\)

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
\(3\) 0.682410 − 1.18197i 0.393989 − 0.682410i −0.598982 − 0.800762i \(-0.704428\pi\)
0.992972 + 0.118353i \(0.0377613\pi\)
\(4\) 0 0
\(5\) 0.741082 0.331422 0.165711 − 0.986174i \(-0.447008\pi\)
0.165711 + 0.986174i \(0.447008\pi\)
\(6\) 0 0
\(7\) −0.500000 − 0.866025i −0.188982 − 0.327327i
\(8\) 0 0
\(9\) 0.568634 + 0.984903i 0.189545 + 0.328301i
\(10\) 0 0
\(11\) −0.682410 + 1.18197i −0.205754 + 0.356377i −0.950373 − 0.311113i \(-0.899298\pi\)
0.744619 + 0.667490i \(0.232631\pi\)
\(12\) 0 0
\(13\) 0.301907 + 3.59289i 0.0837339 + 0.996488i
\(14\) 0 0
\(15\) 0.505722 − 0.875935i 0.130577 − 0.226166i
\(16\) 0 0
\(17\) 2.07436 + 3.59289i 0.503105 + 0.871404i 0.999994 + 0.00358919i \(0.00114248\pi\)
−0.496888 + 0.867814i \(0.665524\pi\)
\(18\) 0 0
\(19\) 3.63303 + 6.29259i 0.833474 + 1.44362i 0.895267 + 0.445530i \(0.146985\pi\)
−0.0617933 + 0.998089i \(0.519682\pi\)
\(20\) 0 0
\(21\) −1.36482 −0.297828
\(22\) 0 0
\(23\) −1.16673 + 2.02083i −0.243279 + 0.421372i −0.961646 − 0.274292i \(-0.911556\pi\)
0.718367 + 0.695664i \(0.244890\pi\)
\(24\) 0 0
\(25\) −4.45080 −0.890159
\(26\) 0 0
\(27\) 5.64662 1.08669
\(28\) 0 0
\(29\) 0.203815 − 0.353017i 0.0378474 − 0.0655536i −0.846481 − 0.532419i \(-0.821283\pi\)
0.884329 + 0.466865i \(0.154617\pi\)
\(30\) 0 0
\(31\) 2.77245 0.497946 0.248973 − 0.968510i \(-0.419907\pi\)
0.248973 + 0.968510i \(0.419907\pi\)
\(32\) 0 0
\(33\) 0.931366 + 1.61317i 0.162130 + 0.280817i
\(34\) 0 0
\(35\) −0.370541 − 0.641796i −0.0626329 − 0.108483i
\(36\) 0 0
\(37\) 3.05295 − 5.28787i 0.501902 − 0.869320i −0.498096 − 0.867122i \(-0.665967\pi\)
0.999998 − 0.00219764i \(-0.000699531\pi\)
\(38\) 0 0
\(39\) 4.45271 + 2.09498i 0.713003 + 0.335465i
\(40\) 0 0
\(41\) −0.627306 + 1.08653i −0.0979688 + 0.169687i −0.910844 − 0.412751i \(-0.864568\pi\)
0.812875 + 0.582438i \(0.197901\pi\)
\(42\) 0 0
\(43\) −0.870541 − 1.50782i −0.132756 − 0.229941i 0.791982 − 0.610545i \(-0.209049\pi\)
−0.924738 + 0.380604i \(0.875716\pi\)
\(44\) 0 0
\(45\) 0.421404 + 0.729894i 0.0628193 + 0.108806i
\(46\) 0 0
\(47\) 5.85843 0.854539 0.427270 − 0.904124i \(-0.359475\pi\)
0.427270 + 0.904124i \(0.359475\pi\)
\(48\) 0 0
\(49\) −0.500000 + 0.866025i −0.0714286 + 0.123718i
\(50\) 0 0
\(51\) 5.66224 0.792872
\(52\) 0 0
\(53\) 4.56778 0.627433 0.313717 − 0.949517i \(-0.398426\pi\)
0.313717 + 0.949517i \(0.398426\pi\)
\(54\) 0 0
\(55\) −0.505722 + 0.875935i −0.0681915 + 0.118111i
\(56\) 0 0
\(57\) 9.91685 1.31352
\(58\) 0 0
\(59\) −5.49213 − 9.51264i −0.715014 − 1.23844i −0.962954 − 0.269665i \(-0.913087\pi\)
0.247940 − 0.968775i \(-0.420246\pi\)
\(60\) 0 0
\(61\) −3.26249 − 5.65079i −0.417719 − 0.723510i 0.577991 − 0.816043i \(-0.303837\pi\)
−0.995710 + 0.0925333i \(0.970504\pi\)
\(62\) 0 0
\(63\) 0.568634 − 0.984903i 0.0716411 − 0.124086i
\(64\) 0 0
\(65\) 0.223738 + 2.66263i 0.0277513 + 0.330258i
\(66\) 0 0
\(67\) −6.87983 + 11.9162i −0.840505 + 1.45580i 0.0489630 + 0.998801i \(0.484408\pi\)
−0.889468 + 0.456997i \(0.848925\pi\)
\(68\) 0 0
\(69\) 1.59237 + 2.75807i 0.191699 + 0.332032i
\(70\) 0 0
\(71\) −2.40763 − 4.17014i −0.285733 − 0.494904i 0.687054 − 0.726607i \(-0.258904\pi\)
−0.972787 + 0.231703i \(0.925570\pi\)
\(72\) 0 0
\(73\) 6.06987 0.710425 0.355212 − 0.934786i \(-0.384409\pi\)
0.355212 + 0.934786i \(0.384409\pi\)
\(74\) 0 0
\(75\) −3.03727 + 5.26070i −0.350713 + 0.607453i
\(76\) 0 0
\(77\) 1.36482 0.155536
\(78\) 0 0
\(79\) 9.12582 1.02674 0.513368 − 0.858169i \(-0.328398\pi\)
0.513368 + 0.858169i \(0.328398\pi\)
\(80\) 0 0
\(81\) 2.14741 − 3.71942i 0.238601 − 0.413269i
\(82\) 0 0
\(83\) −11.7368 −1.28828 −0.644139 − 0.764908i \(-0.722784\pi\)
−0.644139 + 0.764908i \(0.722784\pi\)
\(84\) 0 0
\(85\) 1.53727 + 2.66263i 0.166740 + 0.288802i
\(86\) 0 0
\(87\) −0.278170 − 0.481805i −0.0298230 − 0.0516549i
\(88\) 0 0
\(89\) 0.880503 − 1.52508i 0.0933331 − 0.161658i −0.815579 − 0.578646i \(-0.803581\pi\)
0.908912 + 0.416989i \(0.136915\pi\)
\(90\) 0 0
\(91\) 2.96058 − 2.05790i 0.310353 − 0.215727i
\(92\) 0 0
\(93\) 1.89195 − 3.27695i 0.196186 − 0.339803i
\(94\) 0 0
\(95\) 2.69237 + 4.66332i 0.276231 + 0.478447i
\(96\) 0 0
\(97\) −4.76691 − 8.25652i −0.484006 − 0.838323i 0.515825 − 0.856694i \(-0.327485\pi\)
−0.999831 + 0.0183708i \(0.994152\pi\)
\(98\) 0 0
\(99\) −1.55217 −0.155998
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 1456.2.s.q.113.3 8
4.3 odd 2 91.2.f.c.22.4 ✓ 8
12.11 even 2 819.2.o.h.568.1 8
13.3 even 3 inner 1456.2.s.q.1121.3 8
28.3 even 6 637.2.g.j.373.4 8
28.11 odd 6 637.2.g.k.373.4 8
28.19 even 6 637.2.h.i.165.1 8
28.23 odd 6 637.2.h.h.165.1 8
28.27 even 2 637.2.f.i.295.4 8
52.3 odd 6 91.2.f.c.29.4 yes 8
52.7 even 12 1183.2.c.g.337.1 8
52.19 even 12 1183.2.c.g.337.8 8
52.35 odd 6 1183.2.a.k.1.1 4
52.43 odd 6 1183.2.a.l.1.4 4
156.107 even 6 819.2.o.h.757.1 8
364.3 even 6 637.2.h.i.471.1 8
364.55 even 6 637.2.f.i.393.4 8
364.107 odd 6 637.2.g.k.263.4 8
364.139 even 6 8281.2.a.bp.1.1 4
364.159 even 6 637.2.g.j.263.4 8
364.251 even 6 8281.2.a.bt.1.4 4
364.263 odd 6 637.2.h.h.471.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
91.2.f.c.22.4 ✓ 8 4.3 odd 2
91.2.f.c.29.4 yes 8 52.3 odd 6
637.2.f.i.295.4 8 28.27 even 2
637.2.f.i.393.4 8 364.55 even 6
637.2.g.j.263.4 8 364.159 even 6
637.2.g.j.373.4 8 28.3 even 6
637.2.g.k.263.4 8 364.107 odd 6
637.2.g.k.373.4 8 28.11 odd 6
637.2.h.h.165.1 8 28.23 odd 6
637.2.h.h.471.1 8 364.263 odd 6
637.2.h.i.165.1 8 28.19 even 6
637.2.h.i.471.1 8 364.3 even 6
819.2.o.h.568.1 8 12.11 even 2
819.2.o.h.757.1 8 156.107 even 6
1183.2.a.k.1.1 4 52.35 odd 6
1183.2.a.l.1.4 4 52.43 odd 6
1183.2.c.g.337.1 8 52.7 even 12
1183.2.c.g.337.8 8 52.19 even 12
1456.2.s.q.113.3 8 1.1 even 1 trivial
1456.2.s.q.1121.3 8 13.3 even 3 inner
8281.2.a.bp.1.1 4 364.139 even 6
8281.2.a.bt.1.4 4 364.251 even 6