Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [637,2,Mod(263,637)] 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("637.263"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(637, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([4, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 637 = 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 637.g (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,1,-2,-5,-7] 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: \(5.08647060876\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 91)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 373.4
Root \(1.37054 + 2.37385i\) of defining polynomial
Character \(\chi\) \(=\) 637.373
Dual form 637.2.g.j.263.4

$q$-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+(1.37054 + 2.37385i) q^{2} -1.36482 q^{3} +(-2.75677 + 4.77486i) q^{4} +(0.370541 - 0.641796i) q^{5} +(-1.87054 - 3.23987i) q^{6} -9.63087 q^{8} -1.13727 q^{9} +2.03137 q^{10} -1.36482 q^{11} +(3.76249 - 6.51682i) q^{12} +(-0.301907 - 3.59289i) q^{13} +(-0.505722 + 0.875935i) q^{15} +(-7.68598 - 13.3125i) q^{16} +(-2.07436 + 3.59289i) q^{17} +(-1.55867 - 2.69970i) q^{18} -7.26606 q^{19} +(2.04299 + 3.53856i) q^{20} +(-1.87054 - 3.23987i) q^{22} +(1.16673 + 2.02083i) q^{23} +13.1444 q^{24} +(2.22540 + 3.85450i) q^{25} +(8.11519 - 5.64088i) q^{26} +5.64662 q^{27} +(0.203815 - 0.353017i) q^{29} -2.77245 q^{30} +(-1.38622 - 2.40101i) q^{31} +(11.4370 - 19.8095i) q^{32} +1.86273 q^{33} -11.3720 q^{34} +(3.13518 - 5.43029i) q^{36} +(3.05295 + 5.28787i) q^{37} +(-9.95843 - 17.2485i) q^{38} +(0.412049 + 4.90364i) q^{39} +(-3.56863 + 6.18106i) q^{40} +(0.627306 - 1.08653i) q^{41} +(0.870541 + 1.50782i) q^{43} +(3.76249 - 6.51682i) q^{44} +(-0.421404 + 0.729894i) q^{45} +(-3.19809 + 5.53926i) q^{46} +(-2.92921 + 5.07355i) q^{47} +(10.4900 + 18.1692i) q^{48} +(-6.10000 + 10.5655i) q^{50} +(2.83112 - 4.90364i) q^{51} +(17.9878 + 8.46319i) q^{52} +(-2.28389 - 3.95582i) q^{53} +(7.73893 + 13.4042i) q^{54} +(-0.505722 + 0.875935i) q^{55} +9.91685 q^{57} +1.11734 q^{58} +(-5.49213 + 9.51264i) q^{59} +(-2.78831 - 4.82950i) q^{60} -6.52497 q^{61} +(3.79975 - 6.58137i) q^{62} +31.9557 q^{64} +(-2.41777 - 1.13755i) q^{65} +(2.55295 + 4.42184i) q^{66} -13.7597 q^{67} +(-11.4370 - 19.8095i) q^{68} +(-1.59237 - 2.75807i) q^{69} +(2.40763 + 4.17014i) q^{71} +10.9529 q^{72} +(3.03494 + 5.25666i) q^{73} +(-8.36839 + 14.4945i) q^{74} +(-3.03727 - 5.26070i) q^{75} +(20.0308 - 34.6944i) q^{76} +(-11.0758 + 7.69879i) q^{78} +(4.56291 - 7.90320i) q^{79} -11.3919 q^{80} -4.29482 q^{81} +3.43900 q^{82} -11.7368 q^{83} +(1.53727 + 2.66263i) q^{85} +(-2.38622 + 4.13306i) q^{86} +(-0.278170 + 0.481805i) q^{87} +13.1444 q^{88} +(-0.880503 - 1.52508i) q^{89} -2.31021 q^{90} -12.8656 q^{92} +(1.89195 + 3.27695i) q^{93} -16.0584 q^{94} +(-2.69237 + 4.66332i) q^{95} +(-15.6095 + 27.0364i) q^{96} +(4.76691 + 8.25652i) q^{97} +1.55217 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + q^{2} - 2 q^{3} - 5 q^{4} - 7 q^{5} - 5 q^{6} - 12 q^{8} + 14 q^{9} + 22 q^{10} - 2 q^{11} + 12 q^{12} - 4 q^{13} - 3 q^{15} - 19 q^{16} - 4 q^{17} + 3 q^{18} - 2 q^{19} - 2 q^{20} - 5 q^{22} + 2 q^{23}+ \cdots + 46 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/637\mathbb{Z}\right)^\times\).

\(n\) \(197\) \(248\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.37054 + 2.37385i 0.969119 + 1.67856i 0.698116 + 0.715984i \(0.254022\pi\)
0.271003 + 0.962579i \(0.412645\pi\)
\(3\) −1.36482 −0.787979 −0.393989 − 0.919115i \(-0.628905\pi\)
−0.393989 + 0.919115i \(0.628905\pi\)
\(4\) −2.75677 + 4.77486i −1.37838 + 2.38743i
\(5\) 0.370541 − 0.641796i 0.165711 − 0.287020i −0.771197 − 0.636597i \(-0.780341\pi\)
0.936908 + 0.349577i \(0.113675\pi\)
\(6\) −1.87054 − 3.23987i −0.763645 − 1.32267i
\(7\) 0 0
\(8\) −9.63087 −3.40503
\(9\) −1.13727 −0.379089
\(10\) 2.03137 0.642374
\(11\) −1.36482 −0.411509 −0.205754 − 0.978604i \(-0.565965\pi\)
−0.205754 + 0.978604i \(0.565965\pi\)
\(12\) 3.76249 − 6.51682i 1.08614 − 1.88124i
\(13\) −0.301907 − 3.59289i −0.0837339 − 0.996488i
\(14\) 0 0
\(15\) −0.505722 + 0.875935i −0.130577 + 0.226166i
\(16\) −7.68598 − 13.3125i −1.92149 − 3.32813i
\(17\) −2.07436 + 3.59289i −0.503105 + 0.871404i 0.496888 + 0.867814i \(0.334476\pi\)
−0.999994 + 0.00358919i \(0.998858\pi\)
\(18\) −1.55867 − 2.69970i −0.367383 − 0.636325i
\(19\) −7.26606 −1.66695 −0.833474 − 0.552559i \(-0.813651\pi\)
−0.833474 + 0.552559i \(0.813651\pi\)
\(20\) 2.04299 + 3.53856i 0.456826 + 0.791246i
\(21\) 0 0
\(22\) −1.87054 − 3.23987i −0.398801 − 0.690743i
\(23\) 1.16673 + 2.02083i 0.243279 + 0.421372i 0.961646 − 0.274292i \(-0.0884435\pi\)
−0.718367 + 0.695664i \(0.755110\pi\)
\(24\) 13.1444 2.68309
\(25\) 2.22540 + 3.85450i 0.445080 + 0.770901i
\(26\) 8.11519 − 5.64088i 1.59152 − 1.10627i
\(27\) 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 \(0.154617\pi\)
\(30\) −2.77245 −0.506178
\(31\) −1.38622 − 2.40101i −0.248973 − 0.431234i 0.714268 − 0.699872i \(-0.246760\pi\)
−0.963241 + 0.268638i \(0.913426\pi\)
\(32\) 11.4370 − 19.8095i 2.02180 − 3.50186i
\(33\) 1.86273 0.324260
\(34\) −11.3720 −1.95027
\(35\) 0 0
\(36\) 3.13518 − 5.43029i 0.522530 − 0.905049i
\(37\) 3.05295 + 5.28787i 0.501902 + 0.869320i 0.999998 + 0.00219764i \(0.000699531\pi\)
−0.498096 + 0.867122i \(0.665967\pi\)
\(38\) −9.95843 − 17.2485i −1.61547 − 2.79808i
\(39\) 0.412049 + 4.90364i 0.0659806 + 0.785212i
\(40\) −3.56863 + 6.18106i −0.564251 + 0.977311i
\(41\) 0.627306 − 1.08653i 0.0979688 − 0.169687i −0.812875 − 0.582438i \(-0.802099\pi\)
0.910844 + 0.412751i \(0.135432\pi\)
\(42\) 0 0
\(43\) 0.870541 + 1.50782i 0.132756 + 0.229941i 0.924738 − 0.380604i \(-0.124284\pi\)
−0.791982 + 0.610545i \(0.790951\pi\)
\(44\) 3.76249 − 6.51682i 0.567216 − 0.982447i
\(45\) −0.421404 + 0.729894i −0.0628193 + 0.108806i
\(46\) −3.19809 + 5.53926i −0.471533 + 0.816719i
\(47\) −2.92921 + 5.07355i −0.427270 + 0.740053i −0.996629 − 0.0820357i \(-0.973858\pi\)
0.569360 + 0.822089i \(0.307191\pi\)
\(48\) 10.4900 + 18.1692i 1.51410 + 2.62249i
\(49\) 0 0
\(50\) −6.10000 + 10.5655i −0.862670 + 1.49419i
\(51\) 2.83112 − 4.90364i 0.396436 − 0.686648i
\(52\) 17.9878 + 8.46319i 2.49446 + 1.17363i
\(53\) −2.28389 − 3.95582i −0.313717 − 0.543373i 0.665447 − 0.746445i \(-0.268241\pi\)
−0.979164 + 0.203072i \(0.934908\pi\)
\(54\) 7.73893 + 13.4042i 1.05313 + 1.82408i
\(55\) −0.505722 + 0.875935i −0.0681915 + 0.118111i
\(56\) 0 0
\(57\) 9.91685 1.31352
\(58\) 1.11734 0.146715
\(59\) −5.49213 + 9.51264i −0.715014 + 1.23844i 0.247940 + 0.968775i \(0.420246\pi\)
−0.962954 + 0.269665i \(0.913087\pi\)
\(60\) −2.78831 − 4.82950i −0.359969 − 0.623485i
\(61\) −6.52497 −0.835437 −0.417719 − 0.908576i \(-0.637170\pi\)
−0.417719 + 0.908576i \(0.637170\pi\)
\(62\) 3.79975 − 6.58137i 0.482569 − 0.835834i
\(63\) 0 0
\(64\) 31.9557 3.99446
\(65\) −2.41777 − 1.13755i −0.299887 − 0.141096i
\(66\) 2.55295 + 4.42184i 0.314247 + 0.544291i
\(67\) −13.7597 −1.68101 −0.840505 − 0.541804i \(-0.817742\pi\)
−0.840505 + 0.541804i \(0.817742\pi\)
\(68\) −11.4370 − 19.8095i −1.38694 − 2.40226i
\(69\) −1.59237 − 2.75807i −0.191699 − 0.332032i
\(70\) 0 0
\(71\) 2.40763 + 4.17014i 0.285733 + 0.494904i 0.972787 − 0.231703i \(-0.0744296\pi\)
−0.687054 + 0.726607i \(0.741096\pi\)
\(72\) 10.9529 1.29081
\(73\) 3.03494 + 5.25666i 0.355212 + 0.615246i 0.987154 − 0.159770i \(-0.0510752\pi\)
−0.631942 + 0.775016i \(0.717742\pi\)
\(74\) −8.36839 + 14.4945i −0.972805 + 1.68495i
\(75\) −3.03727 − 5.26070i −0.350713 − 0.607453i
\(76\) 20.0308 − 34.6944i 2.29769 − 3.97972i
\(77\) 0 0
\(78\) −11.0758 + 7.69879i −1.25408 + 0.871716i
\(79\) 4.56291 − 7.90320i 0.513368 − 0.889179i −0.486512 − 0.873674i \(-0.661731\pi\)
0.999880 − 0.0155052i \(-0.00493564\pi\)
\(80\) −11.3919 −1.27365
\(81\) −4.29482 −0.477202
\(82\) 3.43900 0.379774
\(83\) −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
\(86\) −2.38622 + 4.13306i −0.257313 + 0.445679i
\(87\) −0.278170 + 0.481805i −0.0298230 + 0.0516549i
\(88\) 13.1444 1.40120
\(89\) −0.880503 − 1.52508i −0.0933331 − 0.161658i 0.815579 − 0.578646i \(-0.196419\pi\)
−0.908912 + 0.416989i \(0.863085\pi\)
\(90\) −2.31021 −0.243517
\(91\) 0 0
\(92\) −12.8656 −1.34133
\(93\) 1.89195 + 3.27695i 0.196186 + 0.339803i
\(94\) −16.0584 −1.65630
\(95\) −2.69237 + 4.66332i −0.276231 + 0.478447i
\(96\) −15.6095 + 27.0364i −1.59313 + 2.75939i
\(97\) 4.76691 + 8.25652i 0.484006 + 0.838323i 0.999831 − 0.0183708i \(-0.00584795\pi\)
−0.515825 + 0.856694i \(0.672515\pi\)
\(98\) 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 637.2.g.j.373.4 8
7.2 even 3 637.2.f.i.295.4 8
7.3 odd 6 637.2.h.h.165.1 8
7.4 even 3 637.2.h.i.165.1 8
7.5 odd 6 91.2.f.c.22.4 ✓ 8
7.6 odd 2 637.2.g.k.373.4 8
13.3 even 3 637.2.h.i.471.1 8
21.5 even 6 819.2.o.h.568.1 8
28.19 even 6 1456.2.s.q.113.3 8
91.3 odd 6 637.2.g.k.263.4 8
91.9 even 3 8281.2.a.bp.1.1 4
91.16 even 3 637.2.f.i.393.4 8
91.19 even 12 1183.2.c.g.337.8 8
91.30 even 6 8281.2.a.bt.1.4 4
91.33 even 12 1183.2.c.g.337.1 8
91.55 odd 6 637.2.h.h.471.1 8
91.61 odd 6 1183.2.a.k.1.1 4
91.68 odd 6 91.2.f.c.29.4 yes 8
91.81 even 3 inner 637.2.g.j.263.4 8
91.82 odd 6 1183.2.a.l.1.4 4
273.68 even 6 819.2.o.h.757.1 8
364.159 even 6 1456.2.s.q.1121.3 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
91.2.f.c.22.4 ✓ 8 7.5 odd 6
91.2.f.c.29.4 yes 8 91.68 odd 6
637.2.f.i.295.4 8 7.2 even 3
637.2.f.i.393.4 8 91.16 even 3
637.2.g.j.263.4 8 91.81 even 3 inner
637.2.g.j.373.4 8 1.1 even 1 trivial
637.2.g.k.263.4 8 91.3 odd 6
637.2.g.k.373.4 8 7.6 odd 2
637.2.h.h.165.1 8 7.3 odd 6
637.2.h.h.471.1 8 91.55 odd 6
637.2.h.i.165.1 8 7.4 even 3
637.2.h.i.471.1 8 13.3 even 3
819.2.o.h.568.1 8 21.5 even 6
819.2.o.h.757.1 8 273.68 even 6
1183.2.a.k.1.1 4 91.61 odd 6
1183.2.a.l.1.4 4 91.82 odd 6
1183.2.c.g.337.1 8 91.33 even 12
1183.2.c.g.337.8 8 91.19 even 12
1456.2.s.q.113.3 8 28.19 even 6
1456.2.s.q.1121.3 8 364.159 even 6
8281.2.a.bp.1.1 4 91.9 even 3
8281.2.a.bt.1.4 4 91.30 even 6