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(79,637)] 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("637.79"); 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([2, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 637 = 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 637.e (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,3,0,-3,0] 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: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{5})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + 2x^{2} + x + 1 \) 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 508.2
Root \(0.809017 + 1.40126i\) of defining polynomial
Character \(\chi\) \(=\) 637.508
Dual form 637.2.e.h.79.2

$q$-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+(1.30902 + 2.26728i) q^{2} +(-1.11803 + 1.93649i) q^{3} +(-2.42705 + 4.20378i) q^{4} +(1.11803 + 1.93649i) q^{5} -5.85410 q^{6} -7.47214 q^{8} +(-1.00000 - 1.73205i) q^{9} +(-2.92705 + 5.06980i) q^{10} +(1.50000 - 2.59808i) q^{11} +(-5.42705 - 9.39993i) q^{12} +1.00000 q^{13} -5.00000 q^{15} +(-4.92705 - 8.53390i) q^{16} +(0.736068 - 1.27491i) q^{17} +(2.61803 - 4.53457i) q^{18} +(1.50000 + 2.59808i) q^{19} -10.8541 q^{20} +7.85410 q^{22} +(4.11803 + 7.13264i) q^{23} +(8.35410 - 14.4697i) q^{24} +(1.30902 + 2.26728i) q^{26} -2.23607 q^{27} +4.47214 q^{29} +(-6.54508 - 11.3364i) q^{30} +(2.50000 - 4.33013i) q^{31} +(5.42705 - 9.39993i) q^{32} +(3.35410 + 5.80948i) q^{33} +3.85410 q^{34} +9.70820 q^{36} +(-2.35410 - 4.07742i) q^{37} +(-3.92705 + 6.80185i) q^{38} +(-1.11803 + 1.93649i) q^{39} +(-8.35410 - 14.4697i) q^{40} +4.47214 q^{41} -8.00000 q^{43} +(7.28115 + 12.6113i) q^{44} +(2.23607 - 3.87298i) q^{45} +(-10.7812 + 18.6735i) q^{46} +(-3.73607 - 6.47106i) q^{47} +22.0344 q^{48} +(1.64590 + 2.85078i) q^{51} +(-2.42705 + 4.20378i) q^{52} +(3.73607 - 6.47106i) q^{53} +(-2.92705 - 5.06980i) q^{54} +6.70820 q^{55} -6.70820 q^{57} +(5.85410 + 10.1396i) q^{58} +(-0.736068 + 1.27491i) q^{59} +(12.1353 - 21.0189i) q^{60} +(1.50000 + 2.59808i) q^{61} +13.0902 q^{62} +8.70820 q^{64} +(1.11803 + 1.93649i) q^{65} +(-8.78115 + 15.2094i) q^{66} +(1.50000 - 2.59808i) q^{67} +(3.57295 + 6.18853i) q^{68} -18.4164 q^{69} -8.94427 q^{71} +(7.47214 + 12.9421i) q^{72} +(-1.35410 + 2.34537i) q^{73} +(6.16312 - 10.6748i) q^{74} -14.5623 q^{76} -5.85410 q^{78} +(1.35410 + 2.34537i) q^{79} +(11.0172 - 19.0824i) q^{80} +(5.50000 - 9.52628i) q^{81} +(5.85410 + 10.1396i) q^{82} +3.29180 q^{85} +(-10.4721 - 18.1383i) q^{86} +(-5.00000 + 8.66025i) q^{87} +(-11.2082 + 19.4132i) q^{88} +(1.11803 + 1.93649i) q^{89} +11.7082 q^{90} -39.9787 q^{92} +(5.59017 + 9.68246i) q^{93} +(9.78115 - 16.9415i) q^{94} +(-3.35410 + 5.80948i) q^{95} +(12.1353 + 21.0189i) q^{96} -9.41641 q^{97} -6.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 3 q^{2} - 3 q^{4} - 10 q^{6} - 12 q^{8} - 4 q^{9} - 5 q^{10} + 6 q^{11} - 15 q^{12} + 4 q^{13} - 20 q^{15} - 13 q^{16} - 6 q^{17} + 6 q^{18} + 6 q^{19} - 30 q^{20} + 18 q^{22} + 12 q^{23} + 20 q^{24}+ \cdots - 24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.30902 + 2.26728i 0.925615 + 1.60321i 0.790569 + 0.612372i \(0.209785\pi\)
0.135045 + 0.990839i \(0.456882\pi\)
\(3\) −1.11803 + 1.93649i −0.645497 + 1.11803i 0.338689 + 0.940898i \(0.390016\pi\)
−0.984186 + 0.177136i \(0.943317\pi\)
\(4\) −2.42705 + 4.20378i −1.21353 + 2.10189i
\(5\) 1.11803 + 1.93649i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(6\) −5.85410 −2.38993
\(7\) 0 0
\(8\) −7.47214 −2.64180
\(9\) −1.00000 − 1.73205i −0.333333 − 0.577350i
\(10\) −2.92705 + 5.06980i −0.925615 + 1.60321i
\(11\) 1.50000 − 2.59808i 0.452267 − 0.783349i −0.546259 − 0.837616i \(-0.683949\pi\)
0.998526 + 0.0542666i \(0.0172821\pi\)
\(12\) −5.42705 − 9.39993i −1.56665 − 2.71353i
\(13\) 1.00000 0.277350
\(14\) 0 0
\(15\) −5.00000 −1.29099
\(16\) −4.92705 − 8.53390i −1.23176 − 2.13348i
\(17\) 0.736068 − 1.27491i 0.178523 − 0.309210i −0.762852 − 0.646573i \(-0.776202\pi\)
0.941375 + 0.337363i \(0.109535\pi\)
\(18\) 2.61803 − 4.53457i 0.617077 − 1.06881i
\(19\) 1.50000 + 2.59808i 0.344124 + 0.596040i 0.985194 − 0.171442i \(-0.0548427\pi\)
−0.641071 + 0.767482i \(0.721509\pi\)
\(20\) −10.8541 −2.42705
\(21\) 0 0
\(22\) 7.85410 1.67450
\(23\) 4.11803 + 7.13264i 0.858669 + 1.48726i 0.873199 + 0.487365i \(0.162042\pi\)
−0.0145291 + 0.999894i \(0.504625\pi\)
\(24\) 8.35410 − 14.4697i 1.70527 − 2.95362i
\(25\) 0 0
\(26\) 1.30902 + 2.26728i 0.256719 + 0.444651i
\(27\) −2.23607 −0.430331
\(28\) 0 0
\(29\) 4.47214 0.830455 0.415227 − 0.909718i \(-0.363702\pi\)
0.415227 + 0.909718i \(0.363702\pi\)
\(30\) −6.54508 − 11.3364i −1.19496 − 2.06974i
\(31\) 2.50000 − 4.33013i 0.449013 − 0.777714i −0.549309 − 0.835619i \(-0.685109\pi\)
0.998322 + 0.0579057i \(0.0184423\pi\)
\(32\) 5.42705 − 9.39993i 0.959376 − 1.66169i
\(33\) 3.35410 + 5.80948i 0.583874 + 1.01130i
\(34\) 3.85410 0.660973
\(35\) 0 0
\(36\) 9.70820 1.61803
\(37\) −2.35410 − 4.07742i −0.387012 − 0.670324i 0.605034 − 0.796200i \(-0.293159\pi\)
−0.992046 + 0.125875i \(0.959826\pi\)
\(38\) −3.92705 + 6.80185i −0.637052 + 1.10341i
\(39\) −1.11803 + 1.93649i −0.179029 + 0.310087i
\(40\) −8.35410 − 14.4697i −1.32090 − 2.28787i
\(41\) 4.47214 0.698430 0.349215 − 0.937043i \(-0.386448\pi\)
0.349215 + 0.937043i \(0.386448\pi\)
\(42\) 0 0
\(43\) −8.00000 −1.21999 −0.609994 − 0.792406i \(-0.708828\pi\)
−0.609994 + 0.792406i \(0.708828\pi\)
\(44\) 7.28115 + 12.6113i 1.09768 + 1.90123i
\(45\) 2.23607 − 3.87298i 0.333333 − 0.577350i
\(46\) −10.7812 + 18.6735i −1.58959 + 2.75326i
\(47\) −3.73607 − 6.47106i −0.544962 − 0.943901i −0.998609 − 0.0527200i \(-0.983211\pi\)
0.453648 − 0.891181i \(-0.350122\pi\)
\(48\) 22.0344 3.18040
\(49\) 0 0
\(50\) 0 0
\(51\) 1.64590 + 2.85078i 0.230472 + 0.399189i
\(52\) −2.42705 + 4.20378i −0.336571 + 0.582959i
\(53\) 3.73607 − 6.47106i 0.513188 − 0.888868i −0.486695 − 0.873572i \(-0.661798\pi\)
0.999883 − 0.0152962i \(-0.00486912\pi\)
\(54\) −2.92705 − 5.06980i −0.398321 − 0.689913i
\(55\) 6.70820 0.904534
\(56\) 0 0
\(57\) −6.70820 −0.888523
\(58\) 5.85410 + 10.1396i 0.768681 + 1.33139i
\(59\) −0.736068 + 1.27491i −0.0958279 + 0.165979i −0.909954 − 0.414710i \(-0.863883\pi\)
0.814126 + 0.580688i \(0.197217\pi\)
\(60\) 12.1353 − 21.0189i 1.56665 − 2.71353i
\(61\) 1.50000 + 2.59808i 0.192055 + 0.332650i 0.945931 − 0.324367i \(-0.105151\pi\)
−0.753876 + 0.657017i \(0.771818\pi\)
\(62\) 13.0902 1.66245
\(63\) 0 0
\(64\) 8.70820 1.08853
\(65\) 1.11803 + 1.93649i 0.138675 + 0.240192i
\(66\) −8.78115 + 15.2094i −1.08089 + 1.87215i
\(67\) 1.50000 − 2.59808i 0.183254 − 0.317406i −0.759733 − 0.650236i \(-0.774670\pi\)
0.942987 + 0.332830i \(0.108004\pi\)
\(68\) 3.57295 + 6.18853i 0.433284 + 0.750469i
\(69\) −18.4164 −2.21707
\(70\) 0 0
\(71\) −8.94427 −1.06149 −0.530745 − 0.847532i \(-0.678088\pi\)
−0.530745 + 0.847532i \(0.678088\pi\)
\(72\) 7.47214 + 12.9421i 0.880600 + 1.52524i
\(73\) −1.35410 + 2.34537i −0.158486 + 0.274505i −0.934323 − 0.356428i \(-0.883994\pi\)
0.775837 + 0.630933i \(0.217328\pi\)
\(74\) 6.16312 − 10.6748i 0.716448 − 1.24092i
\(75\) 0 0
\(76\) −14.5623 −1.67041
\(77\) 0 0
\(78\) −5.85410 −0.662847
\(79\) 1.35410 + 2.34537i 0.152348 + 0.263875i 0.932090 − 0.362226i \(-0.117983\pi\)
−0.779742 + 0.626101i \(0.784650\pi\)
\(80\) 11.0172 − 19.0824i 1.23176 − 2.13348i
\(81\) 5.50000 − 9.52628i 0.611111 − 1.05848i
\(82\) 5.85410 + 10.1396i 0.646477 + 1.11973i
\(83\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 3.29180 0.357045
\(86\) −10.4721 − 18.1383i −1.12924 − 1.95590i
\(87\) −5.00000 + 8.66025i −0.536056 + 0.928477i
\(88\) −11.2082 + 19.4132i −1.19480 + 2.06945i
\(89\) 1.11803 + 1.93649i 0.118511 + 0.205268i 0.919178 − 0.393842i \(-0.128854\pi\)
−0.800667 + 0.599110i \(0.795521\pi\)
\(90\) 11.7082 1.23415
\(91\) 0 0
\(92\) −39.9787 −4.16807
\(93\) 5.59017 + 9.68246i 0.579674 + 1.00402i
\(94\) 9.78115 − 16.9415i 1.00885 − 1.74738i
\(95\) −3.35410 + 5.80948i −0.344124 + 0.596040i
\(96\) 12.1353 + 21.0189i 1.23855 + 2.14523i
\(97\) −9.41641 −0.956091 −0.478046 − 0.878335i \(-0.658655\pi\)
−0.478046 + 0.878335i \(0.658655\pi\)
\(98\) 0 0
\(99\) −6.00000 −0.603023
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.e.h.508.2 4
7.2 even 3 inner 637.2.e.h.79.2 4
7.3 odd 6 637.2.a.f.1.1 2
7.4 even 3 637.2.a.e.1.1 2
7.5 odd 6 91.2.e.b.79.2 yes 4
7.6 odd 2 91.2.e.b.53.2 ✓ 4
21.5 even 6 819.2.j.c.352.1 4
21.11 odd 6 5733.2.a.w.1.2 2
21.17 even 6 5733.2.a.v.1.2 2
21.20 even 2 819.2.j.c.235.1 4
28.19 even 6 1456.2.r.j.625.1 4
28.27 even 2 1456.2.r.j.417.1 4
91.12 odd 6 1183.2.e.d.170.1 4
91.25 even 6 8281.2.a.ba.1.2 2
91.38 odd 6 8281.2.a.z.1.2 2
91.90 odd 2 1183.2.e.d.508.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
91.2.e.b.53.2 ✓ 4 7.6 odd 2
91.2.e.b.79.2 yes 4 7.5 odd 6
637.2.a.e.1.1 2 7.4 even 3
637.2.a.f.1.1 2 7.3 odd 6
637.2.e.h.79.2 4 7.2 even 3 inner
637.2.e.h.508.2 4 1.1 even 1 trivial
819.2.j.c.235.1 4 21.20 even 2
819.2.j.c.352.1 4 21.5 even 6
1183.2.e.d.170.1 4 91.12 odd 6
1183.2.e.d.508.1 4 91.90 odd 2
1456.2.r.j.417.1 4 28.27 even 2
1456.2.r.j.625.1 4 28.19 even 6
5733.2.a.v.1.2 2 21.17 even 6
5733.2.a.w.1.2 2 21.11 odd 6
8281.2.a.z.1.2 2 91.38 odd 6
8281.2.a.ba.1.2 2 91.25 even 6