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.h.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 0.598982 − 0.800762i \(-0.295572\pi\)
−0.992972 + 0.118353i \(0.962239\pi\)
\(4\) 5.51353 2.75677
\(5\) −0.370541 − 0.641796i −0.165711 − 0.287020i 0.771197 − 0.636597i \(-0.219659\pi\)
−0.936908 + 0.349577i \(0.886325\pi\)
\(6\) 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 \(-0.667809\pi\)
−0.503105 + 0.864225i \(0.667809\pi\)
\(18\) −1.55867 + 2.69970i −0.367383 + 0.636325i
\(19\) −3.63303 + 6.29259i −0.833474 + 1.44362i 0.0617933 + 0.998089i \(0.480318\pi\)
−0.895267 + 0.445530i \(0.853015\pi\)
\(20\) −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 −0.714268 − 0.699872i \(-0.753240\pi\)
0.963241 + 0.268638i \(0.0865736\pi\)
\(32\) −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 −0.910844 − 0.412751i \(-0.864568\pi\)
0.812875 + 0.582438i \(0.197901\pi\)
\(42\) 0 0
\(43\) 0.870541 + 1.50782i 0.132756 + 0.229941i 0.924738 − 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 0.996629 − 0.0820357i \(-0.0261422\pi\)
−0.569360 + 0.822089i \(0.692809\pi\)
\(48\) −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 \(-0.753580\pi\)
−0.715014 + 0.699110i \(0.753580\pi\)
\(60\) −2.78831 + 4.82950i −0.359969 + 0.623485i
\(61\) −3.26249 + 5.65079i −0.417719 + 0.723510i −0.995710 − 0.0925333i \(-0.970504\pi\)
0.577991 + 0.816043i \(0.303837\pi\)
\(62\) −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 −0.987154 − 0.159770i \(-0.948925\pi\)
0.631942 + 0.775016i \(0.282258\pi\)
\(74\) 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 \(-0.277216\pi\)
0.644139 + 0.764908i \(0.277216\pi\)
\(84\) 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 \(-0.529752\pi\)
−0.0933331 + 0.995635i \(0.529752\pi\)
\(90\) 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 0.515825 − 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 637.2.h.h.165.1 8
7.2 even 3 637.2.g.k.373.4 8
7.3 odd 6 637.2.f.i.295.4 8
7.4 even 3 91.2.f.c.22.4 ✓ 8
7.5 odd 6 637.2.g.j.373.4 8
7.6 odd 2 637.2.h.i.165.1 8
13.3 even 3 637.2.g.k.263.4 8
21.11 odd 6 819.2.o.h.568.1 8
28.11 odd 6 1456.2.s.q.113.3 8
91.3 odd 6 637.2.f.i.393.4 8
91.4 even 6 1183.2.a.l.1.4 4
91.16 even 3 inner 637.2.h.h.471.1 8
91.17 odd 6 8281.2.a.bt.1.4 4
91.32 odd 12 1183.2.c.g.337.8 8
91.46 odd 12 1183.2.c.g.337.1 8
91.55 odd 6 637.2.g.j.263.4 8
91.68 odd 6 637.2.h.i.471.1 8
91.74 even 3 1183.2.a.k.1.1 4
91.81 even 3 91.2.f.c.29.4 yes 8
91.87 odd 6 8281.2.a.bp.1.1 4
273.263 odd 6 819.2.o.h.757.1 8
364.263 odd 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 7.4 even 3
91.2.f.c.29.4 yes 8 91.81 even 3
637.2.f.i.295.4 8 7.3 odd 6
637.2.f.i.393.4 8 91.3 odd 6
637.2.g.j.263.4 8 91.55 odd 6
637.2.g.j.373.4 8 7.5 odd 6
637.2.g.k.263.4 8 13.3 even 3
637.2.g.k.373.4 8 7.2 even 3
637.2.h.h.165.1 8 1.1 even 1 trivial
637.2.h.h.471.1 8 91.16 even 3 inner
637.2.h.i.165.1 8 7.6 odd 2
637.2.h.i.471.1 8 91.68 odd 6
819.2.o.h.568.1 8 21.11 odd 6
819.2.o.h.757.1 8 273.263 odd 6
1183.2.a.k.1.1 4 91.74 even 3
1183.2.a.l.1.4 4 91.4 even 6
1183.2.c.g.337.1 8 91.46 odd 12
1183.2.c.g.337.8 8 91.32 odd 12
1456.2.s.q.113.3 8 28.11 odd 6
1456.2.s.q.1121.3 8 364.263 odd 6
8281.2.a.bp.1.1 4 91.87 odd 6
8281.2.a.bt.1.4 4 91.17 odd 6