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(165,637)] 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("637.165"); 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, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 637 = 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 637.h (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,-2,1,10,-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: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 91)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 165.1
Root \(1.37054 + 2.37385i\) of defining polynomial
Character \(\chi\) \(=\) 637.165
Dual form 637.2.h.i.471.1

$q$-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-2.74108 q^{2} +(0.682410 + 1.18197i) q^{3} +5.51353 q^{4} +(0.370541 + 0.641796i) q^{5} +(-1.87054 - 3.23987i) q^{6} -9.63087 q^{8} +(0.568634 - 0.984903i) q^{9} +(-1.01568 - 1.75921i) q^{10} +(0.682410 + 1.18197i) q^{11} +(3.76249 + 6.51682i) q^{12} +(-0.301907 - 3.59289i) q^{13} +(-0.505722 + 0.875935i) q^{15} +15.3720 q^{16} +4.14871 q^{17} +(-1.55867 + 2.69970i) q^{18} +(3.63303 - 6.29259i) q^{19} +(2.04299 + 3.53856i) q^{20} +(-1.87054 - 3.23987i) q^{22} -2.33345 q^{23} +(-6.57220 - 11.3834i) q^{24} +(2.22540 - 3.85450i) q^{25} +(0.827552 + 9.84840i) q^{26} +5.64662 q^{27} +(0.203815 - 0.353017i) q^{29} +(1.38622 - 2.40101i) q^{30} +(-1.38622 + 2.40101i) q^{31} -22.8740 q^{32} +(-0.931366 + 1.61317i) q^{33} -11.3720 q^{34} +(3.13518 - 5.43029i) q^{36} -6.10590 q^{37} +(-9.95843 + 17.2485i) q^{38} +(4.04066 - 2.80867i) 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} +0.842809 q^{45} +6.39619 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} +(-1.66457 - 19.8095i) q^{52} +(-2.28389 + 3.95582i) q^{53} -15.4779 q^{54} +(-0.505722 + 0.875935i) q^{55} +9.91685 q^{57} +(-0.558672 + 0.967649i) q^{58} +10.9843 q^{59} +(-2.78831 + 4.82950i) q^{60} +(3.26249 - 5.65079i) q^{61} +(3.79975 - 6.58137i) q^{62} +31.9557 q^{64} +(2.19403 - 1.52508i) q^{65} +(2.55295 - 4.42184i) q^{66} +(6.87983 + 11.9162i) q^{67} +22.8740 q^{68} +(-1.59237 - 2.75807i) q^{69} +(2.40763 + 4.17014i) q^{71} +(-5.47644 + 9.48548i) q^{72} +(3.03494 - 5.25666i) q^{73} +16.7368 q^{74} +6.07453 q^{75} +(20.0308 - 34.6944i) q^{76} +(-11.0758 + 7.69879i) q^{78} +(4.56291 + 7.90320i) q^{79} +(5.69594 + 9.86566i) q^{80} +(2.14741 + 3.71942i) q^{81} +(-1.71950 + 2.97826i) q^{82} -11.7368 q^{83} +(1.53727 + 2.66263i) q^{85} +(-2.38622 - 4.13306i) q^{86} +0.556340 q^{87} +(-6.57220 - 11.3834i) q^{88} +1.76101 q^{89} -2.31021 q^{90} -12.8656 q^{92} -3.78389 q^{93} +(8.02921 + 13.9070i) q^{94} +5.38474 q^{95} +(-15.6095 - 27.0364i) q^{96} +(4.76691 + 8.25652i) q^{97} +1.55217 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 2 q^{2} + q^{3} + 10 q^{4} - 7 q^{5} - 5 q^{6} - 12 q^{8} - 7 q^{9} - 11 q^{10} + q^{11} + 12 q^{12} - 4 q^{13} - 3 q^{15} + 38 q^{16} + 8 q^{17} + 3 q^{18} + q^{19} - 2 q^{20} - 5 q^{22} - 4 q^{23}+ \cdots + 46 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.74108 −1.93824 −0.969119 − 0.246594i \(-0.920689\pi\)
−0.969119 + 0.246594i \(0.920689\pi\)
\(3\) 0.682410 + 1.18197i 0.393989 + 0.682410i 0.992972 − 0.118353i \(-0.0377613\pi\)
−0.598982 + 0.800762i \(0.704428\pi\)
\(4\) 5.51353 2.75677
\(5\) 0.370541 + 0.641796i 0.165711 + 0.287020i 0.936908 − 0.349577i \(-0.113675\pi\)
−0.771197 + 0.636597i \(0.780341\pi\)
\(6\) −1.87054 − 3.23987i −0.763645 − 1.32267i
\(7\) 0 0
\(8\) −9.63087 −3.40503
\(9\) 0.568634 − 0.984903i 0.189545 − 0.328301i
\(10\) −1.01568 − 1.75921i −0.321187 − 0.556313i
\(11\) 0.682410 + 1.18197i 0.205754 + 0.356377i 0.950373 − 0.311113i \(-0.100702\pi\)
−0.744619 + 0.667490i \(0.767369\pi\)
\(12\) 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
\(16\) 15.3720 3.84299
\(17\) 4.14871 1.00621 0.503105 − 0.864225i \(-0.332191\pi\)
0.503105 + 0.864225i \(0.332191\pi\)
\(18\) −1.55867 + 2.69970i −0.367383 + 0.636325i
\(19\) 3.63303 − 6.29259i 0.833474 − 1.44362i −0.0617933 − 0.998089i \(-0.519682\pi\)
0.895267 − 0.445530i \(-0.146985\pi\)
\(20\) 2.04299 + 3.53856i 0.456826 + 0.791246i
\(21\) 0 0
\(22\) −1.87054 − 3.23987i −0.398801 − 0.690743i
\(23\) −2.33345 −0.486559 −0.243279 − 0.969956i \(-0.578223\pi\)
−0.243279 + 0.969956i \(0.578223\pi\)
\(24\) −6.57220 − 11.3834i −1.34155 − 2.32362i
\(25\) 2.22540 − 3.85450i 0.445080 − 0.770901i
\(26\) 0.827552 + 9.84840i 0.162296 + 1.93143i
\(27\) 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 \(0.154617\pi\)
\(30\) 1.38622 − 2.40101i 0.253089 − 0.438363i
\(31\) −1.38622 + 2.40101i −0.248973 + 0.431234i −0.963241 − 0.268638i \(-0.913426\pi\)
0.714268 + 0.699872i \(0.246760\pi\)
\(32\) −22.8740 −4.04360
\(33\) −0.931366 + 1.61317i −0.162130 + 0.280817i
\(34\) −11.3720 −1.95027
\(35\) 0 0
\(36\) 3.13518 − 5.43029i 0.522530 − 0.905049i
\(37\) −6.10590 −1.00380 −0.501902 − 0.864924i \(-0.667366\pi\)
−0.501902 + 0.864924i \(0.667366\pi\)
\(38\) −9.95843 + 17.2485i −1.61547 + 2.79808i
\(39\) 4.04066 − 2.80867i 0.647023 − 0.449747i
\(40\) −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
\(45\) 0.842809 0.125639
\(46\) 6.39619 0.943066
\(47\) −2.92921 − 5.07355i −0.427270 − 0.740053i 0.569360 − 0.822089i \(-0.307191\pi\)
−0.996629 + 0.0820357i \(0.973858\pi\)
\(48\) 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
\(52\) −1.66457 − 19.8095i −0.230835 − 2.74708i
\(53\) −2.28389 + 3.95582i −0.313717 + 0.543373i −0.979164 − 0.203072i \(-0.934908\pi\)
0.665447 + 0.746445i \(0.268241\pi\)
\(54\) −15.4779 −2.10627
\(55\) −0.505722 + 0.875935i −0.0681915 + 0.118111i
\(56\) 0 0
\(57\) 9.91685 1.31352
\(58\) −0.558672 + 0.967649i −0.0733573 + 0.127059i
\(59\) 10.9843 1.43003 0.715014 − 0.699110i \(-0.246420\pi\)
0.715014 + 0.699110i \(0.246420\pi\)
\(60\) −2.78831 + 4.82950i −0.359969 + 0.623485i
\(61\) 3.26249 − 5.65079i 0.417719 − 0.723510i −0.577991 − 0.816043i \(-0.696163\pi\)
0.995710 + 0.0925333i \(0.0294965\pi\)
\(62\) 3.79975 − 6.58137i 0.482569 − 0.835834i
\(63\) 0 0
\(64\) 31.9557 3.99446
\(65\) 2.19403 − 1.52508i 0.272136 − 0.189162i
\(66\) 2.55295 − 4.42184i 0.314247 − 0.544291i
\(67\) 6.87983 + 11.9162i 0.840505 + 1.45580i 0.889468 + 0.456997i \(0.151075\pi\)
−0.0489630 + 0.998801i \(0.515592\pi\)
\(68\) 22.8740 2.77389
\(69\) −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 \(0.741096\pi\)
\(72\) −5.47644 + 9.48548i −0.645405 + 1.11787i
\(73\) 3.03494 − 5.25666i 0.355212 − 0.615246i −0.631942 − 0.775016i \(-0.717742\pi\)
0.987154 + 0.159770i \(0.0510752\pi\)
\(74\) 16.7368 1.94561
\(75\) 6.07453 0.701427
\(76\) 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 0.999880 + 0.0155052i \(0.00493564\pi\)
−0.486512 + 0.873674i \(0.661731\pi\)
\(80\) 5.69594 + 9.86566i 0.636825 + 1.10301i
\(81\) 2.14741 + 3.71942i 0.238601 + 0.413269i
\(82\) −1.71950 + 2.97826i −0.189887 + 0.328894i
\(83\) −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
\(87\) 0.556340 0.0596459
\(88\) −6.57220 − 11.3834i −0.700599 − 1.21347i
\(89\) 1.76101 0.186666 0.0933331 − 0.995635i \(-0.470248\pi\)
0.0933331 + 0.995635i \(0.470248\pi\)
\(90\) −2.31021 −0.243517
\(91\) 0 0
\(92\) −12.8656 −1.34133
\(93\) −3.78389 −0.392371
\(94\) 8.02921 + 13.9070i 0.828150 + 1.43440i
\(95\) 5.38474 0.552463
\(96\) −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.h.i.165.1 8
7.2 even 3 637.2.g.j.373.4 8
7.3 odd 6 91.2.f.c.22.4 ✓ 8
7.4 even 3 637.2.f.i.295.4 8
7.5 odd 6 637.2.g.k.373.4 8
7.6 odd 2 637.2.h.h.165.1 8
13.3 even 3 637.2.g.j.263.4 8
21.17 even 6 819.2.o.h.568.1 8
28.3 even 6 1456.2.s.q.113.3 8
91.3 odd 6 91.2.f.c.29.4 yes 8
91.4 even 6 8281.2.a.bt.1.4 4
91.16 even 3 inner 637.2.h.i.471.1 8
91.17 odd 6 1183.2.a.l.1.4 4
91.45 even 12 1183.2.c.g.337.8 8
91.55 odd 6 637.2.g.k.263.4 8
91.59 even 12 1183.2.c.g.337.1 8
91.68 odd 6 637.2.h.h.471.1 8
91.74 even 3 8281.2.a.bp.1.1 4
91.81 even 3 637.2.f.i.393.4 8
91.87 odd 6 1183.2.a.k.1.1 4
273.185 even 6 819.2.o.h.757.1 8
364.3 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 7.3 odd 6
91.2.f.c.29.4 yes 8 91.3 odd 6
637.2.f.i.295.4 8 7.4 even 3
637.2.f.i.393.4 8 91.81 even 3
637.2.g.j.263.4 8 13.3 even 3
637.2.g.j.373.4 8 7.2 even 3
637.2.g.k.263.4 8 91.55 odd 6
637.2.g.k.373.4 8 7.5 odd 6
637.2.h.h.165.1 8 7.6 odd 2
637.2.h.h.471.1 8 91.68 odd 6
637.2.h.i.165.1 8 1.1 even 1 trivial
637.2.h.i.471.1 8 91.16 even 3 inner
819.2.o.h.568.1 8 21.17 even 6
819.2.o.h.757.1 8 273.185 even 6
1183.2.a.k.1.1 4 91.87 odd 6
1183.2.a.l.1.4 4 91.17 odd 6
1183.2.c.g.337.1 8 91.59 even 12
1183.2.c.g.337.8 8 91.45 even 12
1456.2.s.q.113.3 8 28.3 even 6
1456.2.s.q.1121.3 8 364.3 even 6
8281.2.a.bp.1.1 4 91.74 even 3
8281.2.a.bt.1.4 4 91.4 even 6