Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1058,6,Mod(1,1058)] 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("1058.1"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1058, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 1058 = 2 \cdot 23^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 1058.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,8,-22,32,68] 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: yes
Analytic conductor: \(169.686081362\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 46)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.41421\) of defining polynomial
Character \(\chi\) \(=\) 1058.1

$q$-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)
 
\(f(q)\) \(=\) \(q+4.00000 q^{2} -2.51472 q^{3} +16.0000 q^{4} +104.711 q^{5} -10.0589 q^{6} +107.917 q^{7} +64.0000 q^{8} -236.676 q^{9} +418.843 q^{10} -692.621 q^{11} -40.2355 q^{12} -763.627 q^{13} +431.667 q^{14} -263.318 q^{15} +256.000 q^{16} +74.8368 q^{17} -946.705 q^{18} +1139.00 q^{19} +1675.37 q^{20} -271.380 q^{21} -2770.49 q^{22} -160.942 q^{24} +7839.33 q^{25} -3054.51 q^{26} +1206.25 q^{27} +1726.67 q^{28} -8085.75 q^{29} -1053.27 q^{30} +563.492 q^{31} +1024.00 q^{32} +1741.75 q^{33} +299.347 q^{34} +11300.0 q^{35} -3786.82 q^{36} -12929.8 q^{37} +4556.00 q^{38} +1920.31 q^{39} +6701.48 q^{40} -5721.35 q^{41} -1085.52 q^{42} -9577.05 q^{43} -11081.9 q^{44} -24782.5 q^{45} -11596.7 q^{47} -643.768 q^{48} -5160.98 q^{49} +31357.3 q^{50} -188.193 q^{51} -12218.0 q^{52} +27158.6 q^{53} +4825.00 q^{54} -72524.9 q^{55} +6906.67 q^{56} -2864.26 q^{57} -32343.0 q^{58} -21848.5 q^{59} -4213.09 q^{60} +10791.6 q^{61} +2253.97 q^{62} -25541.3 q^{63} +4096.00 q^{64} -79959.9 q^{65} +6966.99 q^{66} +34478.7 q^{67} +1197.39 q^{68} +45200.1 q^{70} -82367.3 q^{71} -15147.3 q^{72} +26446.9 q^{73} -51719.1 q^{74} -19713.7 q^{75} +18224.0 q^{76} -74745.5 q^{77} +7681.23 q^{78} -25567.1 q^{79} +26805.9 q^{80} +54478.9 q^{81} -22885.4 q^{82} -41697.7 q^{83} -4342.08 q^{84} +7836.21 q^{85} -38308.2 q^{86} +20333.4 q^{87} -44327.8 q^{88} +35157.2 q^{89} -99130.1 q^{90} -82408.2 q^{91} -1417.02 q^{93} -46386.6 q^{94} +119265. q^{95} -2575.07 q^{96} -39576.9 q^{97} -20643.9 q^{98} +163927. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 8 q^{2} - 22 q^{3} + 32 q^{4} + 68 q^{5} - 88 q^{6} + 312 q^{7} + 128 q^{8} - 100 q^{9} + 272 q^{10} + 12 q^{11} - 352 q^{12} - 1482 q^{13} + 1248 q^{14} + 452 q^{15} + 512 q^{16} - 1768 q^{17} - 400 q^{18}+ \cdots + 260232 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 4.00000 0.707107
\(3\) −2.51472 −0.161319 −0.0806596 − 0.996742i \(-0.525703\pi\)
−0.0806596 + 0.996742i \(0.525703\pi\)
\(4\) 16.0000 0.500000
\(5\) 104.711 1.87312 0.936561 − 0.350505i \(-0.113990\pi\)
0.936561 + 0.350505i \(0.113990\pi\)
\(6\) −10.0589 −0.114070
\(7\) 107.917 0.832422 0.416211 − 0.909268i \(-0.363358\pi\)
0.416211 + 0.909268i \(0.363358\pi\)
\(8\) 64.0000 0.353553
\(9\) −236.676 −0.973976
\(10\) 418.843 1.32450
\(11\) −692.621 −1.72589 −0.862947 − 0.505294i \(-0.831384\pi\)
−0.862947 + 0.505294i \(0.831384\pi\)
\(12\) −40.2355 −0.0806596
\(13\) −763.627 −1.25321 −0.626604 − 0.779338i \(-0.715556\pi\)
−0.626604 + 0.779338i \(0.715556\pi\)
\(14\) 431.667 0.588611
\(15\) −263.318 −0.302171
\(16\) 256.000 0.250000
\(17\) 74.8368 0.0628048 0.0314024 − 0.999507i \(-0.490003\pi\)
0.0314024 + 0.999507i \(0.490003\pi\)
\(18\) −946.705 −0.688705
\(19\) 1139.00 0.723835 0.361917 − 0.932210i \(-0.382122\pi\)
0.361917 + 0.932210i \(0.382122\pi\)
\(20\) 1675.37 0.936561
\(21\) −271.380 −0.134286
\(22\) −2770.49 −1.22039
\(23\) 0 0
\(24\) −160.942 −0.0570350
\(25\) 7839.33 2.50858
\(26\) −3054.51 −0.886152
\(27\) 1206.25 0.318440
\(28\) 1726.67 0.416211
\(29\) −8085.75 −1.78536 −0.892679 − 0.450694i \(-0.851177\pi\)
−0.892679 + 0.450694i \(0.851177\pi\)
\(30\) −1053.27 −0.213667
\(31\) 563.492 0.105313 0.0526567 − 0.998613i \(-0.483231\pi\)
0.0526567 + 0.998613i \(0.483231\pi\)
\(32\) 1024.00 0.176777
\(33\) 1741.75 0.278420
\(34\) 299.347 0.0444097
\(35\) 11300.0 1.55923
\(36\) −3786.82 −0.486988
\(37\) −12929.8 −1.55270 −0.776348 − 0.630304i \(-0.782930\pi\)
−0.776348 + 0.630304i \(0.782930\pi\)
\(38\) 4556.00 0.511828
\(39\) 1920.31 0.202167
\(40\) 6701.48 0.662248
\(41\) −5721.35 −0.531544 −0.265772 − 0.964036i \(-0.585627\pi\)
−0.265772 + 0.964036i \(0.585627\pi\)
\(42\) −1085.52 −0.0949544
\(43\) −9577.05 −0.789879 −0.394939 − 0.918707i \(-0.629234\pi\)
−0.394939 + 0.918707i \(0.629234\pi\)
\(44\) −11081.9 −0.862947
\(45\) −24782.5 −1.82438
\(46\) 0 0
\(47\) −11596.7 −0.765752 −0.382876 − 0.923800i \(-0.625066\pi\)
−0.382876 + 0.923800i \(0.625066\pi\)
\(48\) −643.768 −0.0403298
\(49\) −5160.98 −0.307073
\(50\) 31357.3 1.77384
\(51\) −188.193 −0.0101316
\(52\) −12218.0 −0.626604
\(53\) 27158.6 1.32806 0.664030 − 0.747706i \(-0.268845\pi\)
0.664030 + 0.747706i \(0.268845\pi\)
\(54\) 4825.00 0.225171
\(55\) −72524.9 −3.23281
\(56\) 6906.67 0.294306
\(57\) −2864.26 −0.116768
\(58\) −32343.0 −1.26244
\(59\) −21848.5 −0.817130 −0.408565 − 0.912729i \(-0.633971\pi\)
−0.408565 + 0.912729i \(0.633971\pi\)
\(60\) −4213.09 −0.151085
\(61\) 10791.6 0.371331 0.185665 − 0.982613i \(-0.440556\pi\)
0.185665 + 0.982613i \(0.440556\pi\)
\(62\) 2253.97 0.0744678
\(63\) −25541.3 −0.810759
\(64\) 4096.00 0.125000
\(65\) −79959.9 −2.34741
\(66\) 6966.99 0.196873
\(67\) 34478.7 0.938349 0.469175 − 0.883105i \(-0.344552\pi\)
0.469175 + 0.883105i \(0.344552\pi\)
\(68\) 1197.39 0.0314024
\(69\) 0 0
\(70\) 45200.1 1.10254
\(71\) −82367.3 −1.93914 −0.969570 − 0.244816i \(-0.921272\pi\)
−0.969570 + 0.244816i \(0.921272\pi\)
\(72\) −15147.3 −0.344353
\(73\) 26446.9 0.580854 0.290427 − 0.956897i \(-0.406203\pi\)
0.290427 + 0.956897i \(0.406203\pi\)
\(74\) −51719.1 −1.09792
\(75\) −19713.7 −0.404683
\(76\) 18224.0 0.361917
\(77\) −74745.5 −1.43667
\(78\) 7681.23 0.142953
\(79\) −25567.1 −0.460908 −0.230454 − 0.973083i \(-0.574021\pi\)
−0.230454 + 0.973083i \(0.574021\pi\)
\(80\) 26805.9 0.468280
\(81\) 54478.9 0.922606
\(82\) −22885.4 −0.375858
\(83\) −41697.7 −0.664380 −0.332190 − 0.943213i \(-0.607787\pi\)
−0.332190 + 0.943213i \(0.607787\pi\)
\(84\) −4342.08 −0.0671429
\(85\) 7836.21 0.117641
\(86\) −38308.2 −0.558529
\(87\) 20333.4 0.288013
\(88\) −44327.8 −0.610196
\(89\) 35157.2 0.470477 0.235239 − 0.971938i \(-0.424413\pi\)
0.235239 + 0.971938i \(0.424413\pi\)
\(90\) −99130.1 −1.29003
\(91\) −82408.2 −1.04320
\(92\) 0 0
\(93\) −1417.02 −0.0169891
\(94\) −46386.6 −0.541469
\(95\) 119265. 1.35583
\(96\) −2575.07 −0.0285175
\(97\) −39576.9 −0.427084 −0.213542 − 0.976934i \(-0.568500\pi\)
−0.213542 + 0.976934i \(0.568500\pi\)
\(98\) −20643.9 −0.217133
\(99\) 163927. 1.68098
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 1058.6.a.d.1.2 2
23.22 odd 2 46.6.a.b.1.2 ✓ 2
69.68 even 2 414.6.a.f.1.2 2
92.91 even 2 368.6.a.b.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
46.6.a.b.1.2 ✓ 2 23.22 odd 2
368.6.a.b.1.1 2 92.91 even 2
414.6.a.f.1.2 2 69.68 even 2
1058.6.a.d.1.2 2 1.1 even 1 trivial