Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [98,8,Mod(1,98)] 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("98.1"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(98, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 8, names="a")
 
Level: \( N \) \(=\) \( 98 = 2 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 98.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,16,56,128,238,448,0,1024,1972] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(30.6137324974\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2389}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 597 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 14)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(24.9387\) of defining polynomial
Character \(\chi\) \(=\) 98.1

$q$-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)
 
\(f(q)\) \(=\) \(q+8.00000 q^{2} -20.8774 q^{3} +64.0000 q^{4} +216.755 q^{5} -167.019 q^{6} +512.000 q^{8} -1751.13 q^{9} +1734.04 q^{10} +2581.86 q^{11} -1336.15 q^{12} +8921.97 q^{13} -4525.28 q^{15} +4096.00 q^{16} +11112.9 q^{17} -14009.1 q^{18} +8646.79 q^{19} +13872.3 q^{20} +20654.9 q^{22} -66057.2 q^{23} -10689.2 q^{24} -31142.4 q^{25} +71375.8 q^{26} +82218.0 q^{27} +128836. q^{29} -36202.2 q^{30} +204095. q^{31} +32768.0 q^{32} -53902.5 q^{33} +88903.0 q^{34} -112073. q^{36} +491711. q^{37} +69174.3 q^{38} -186268. q^{39} +110978. q^{40} +623293. q^{41} +422919. q^{43} +165239. q^{44} -379567. q^{45} -528457. q^{46} +1.20543e6 q^{47} -85513.8 q^{48} -249139. q^{50} -232008. q^{51} +571006. q^{52} -1.27297e6 q^{53} +657744. q^{54} +559630. q^{55} -180523. q^{57} +1.03068e6 q^{58} -1.68086e6 q^{59} -289618. q^{60} -2.12255e6 q^{61} +1.63276e6 q^{62} +262144. q^{64} +1.93388e6 q^{65} -431220. q^{66} -3.34933e6 q^{67} +711224. q^{68} +1.37910e6 q^{69} +2.49257e6 q^{71} -896581. q^{72} +3.27353e6 q^{73} +3.93369e6 q^{74} +650171. q^{75} +553395. q^{76} -1.49014e6 q^{78} -4.19839e6 q^{79} +887828. q^{80} +2.11323e6 q^{81} +4.98634e6 q^{82} -3.35527e6 q^{83} +2.40877e6 q^{85} +3.38335e6 q^{86} -2.68975e6 q^{87} +1.32191e6 q^{88} -2.39959e6 q^{89} -3.03653e6 q^{90} -4.22766e6 q^{92} -4.26098e6 q^{93} +9.64343e6 q^{94} +1.87423e6 q^{95} -684111. q^{96} -4.51506e6 q^{97} -4.52118e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 16 q^{2} + 56 q^{3} + 128 q^{4} + 238 q^{5} + 448 q^{6} + 1024 q^{8} + 1972 q^{9} + 1904 q^{10} + 5848 q^{11} + 3584 q^{12} - 1316 q^{13} - 2892 q^{15} + 8192 q^{16} + 47642 q^{17} + 15776 q^{18}+ \cdots + 7639104 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 8.00000 0.707107
\(3\) −20.8774 −0.446429 −0.223214 − 0.974769i \(-0.571655\pi\)
−0.223214 + 0.974769i \(0.571655\pi\)
\(4\) 64.0000 0.500000
\(5\) 216.755 0.775486 0.387743 − 0.921768i \(-0.373255\pi\)
0.387743 + 0.921768i \(0.373255\pi\)
\(6\) −167.019 −0.315673
\(7\) 0 0
\(8\) 512.000 0.353553
\(9\) −1751.13 −0.800702
\(10\) 1734.04 0.548351
\(11\) 2581.86 0.584868 0.292434 − 0.956286i \(-0.405535\pi\)
0.292434 + 0.956286i \(0.405535\pi\)
\(12\) −1336.15 −0.223214
\(13\) 8921.97 1.12631 0.563156 − 0.826350i \(-0.309587\pi\)
0.563156 + 0.826350i \(0.309587\pi\)
\(14\) 0 0
\(15\) −4525.28 −0.346199
\(16\) 4096.00 0.250000
\(17\) 11112.9 0.548599 0.274300 − 0.961644i \(-0.411554\pi\)
0.274300 + 0.961644i \(0.411554\pi\)
\(18\) −14009.1 −0.566181
\(19\) 8646.79 0.289213 0.144606 − 0.989489i \(-0.453808\pi\)
0.144606 + 0.989489i \(0.453808\pi\)
\(20\) 13872.3 0.387743
\(21\) 0 0
\(22\) 20654.9 0.413564
\(23\) −66057.2 −1.13207 −0.566034 − 0.824382i \(-0.691523\pi\)
−0.566034 + 0.824382i \(0.691523\pi\)
\(24\) −10689.2 −0.157836
\(25\) −31142.4 −0.398622
\(26\) 71375.8 0.796423
\(27\) 82218.0 0.803885
\(28\) 0 0
\(29\) 128836. 0.980941 0.490470 − 0.871458i \(-0.336825\pi\)
0.490470 + 0.871458i \(0.336825\pi\)
\(30\) −36202.2 −0.244800
\(31\) 204095. 1.23046 0.615229 − 0.788348i \(-0.289064\pi\)
0.615229 + 0.788348i \(0.289064\pi\)
\(32\) 32768.0 0.176777
\(33\) −53902.5 −0.261102
\(34\) 88903.0 0.387918
\(35\) 0 0
\(36\) −112073. −0.400351
\(37\) 491711. 1.59589 0.797947 − 0.602728i \(-0.205920\pi\)
0.797947 + 0.602728i \(0.205920\pi\)
\(38\) 69174.3 0.204504
\(39\) −186268. −0.502818
\(40\) 110978. 0.274176
\(41\) 623293. 1.41237 0.706185 − 0.708027i \(-0.250415\pi\)
0.706185 + 0.708027i \(0.250415\pi\)
\(42\) 0 0
\(43\) 422919. 0.811181 0.405591 − 0.914055i \(-0.367066\pi\)
0.405591 + 0.914055i \(0.367066\pi\)
\(44\) 165239. 0.292434
\(45\) −379567. −0.620932
\(46\) −528457. −0.800493
\(47\) 1.20543e6 1.69355 0.846777 − 0.531948i \(-0.178540\pi\)
0.846777 + 0.531948i \(0.178540\pi\)
\(48\) −85513.8 −0.111607
\(49\) 0 0
\(50\) −249139. −0.281868
\(51\) −232008. −0.244910
\(52\) 571006. 0.563156
\(53\) −1.27297e6 −1.17449 −0.587247 − 0.809408i \(-0.699788\pi\)
−0.587247 + 0.809408i \(0.699788\pi\)
\(54\) 657744. 0.568432
\(55\) 559630. 0.453557
\(56\) 0 0
\(57\) −180523. −0.129113
\(58\) 1.03068e6 0.693630
\(59\) −1.68086e6 −1.06549 −0.532745 − 0.846276i \(-0.678840\pi\)
−0.532745 + 0.846276i \(0.678840\pi\)
\(60\) −289618. −0.173099
\(61\) −2.12255e6 −1.19730 −0.598651 − 0.801010i \(-0.704296\pi\)
−0.598651 + 0.801010i \(0.704296\pi\)
\(62\) 1.63276e6 0.870065
\(63\) 0 0
\(64\) 262144. 0.125000
\(65\) 1.93388e6 0.873439
\(66\) −431220. −0.184627
\(67\) −3.34933e6 −1.36049 −0.680245 − 0.732984i \(-0.738127\pi\)
−0.680245 + 0.732984i \(0.738127\pi\)
\(68\) 711224. 0.274300
\(69\) 1.37910e6 0.505387
\(70\) 0 0
\(71\) 2.49257e6 0.826502 0.413251 − 0.910617i \(-0.364393\pi\)
0.413251 + 0.910617i \(0.364393\pi\)
\(72\) −896581. −0.283091
\(73\) 3.27353e6 0.984886 0.492443 − 0.870345i \(-0.336104\pi\)
0.492443 + 0.870345i \(0.336104\pi\)
\(74\) 3.93369e6 1.12847
\(75\) 650171. 0.177956
\(76\) 553395. 0.144606
\(77\) 0 0
\(78\) −1.49014e6 −0.355546
\(79\) −4.19839e6 −0.958050 −0.479025 − 0.877801i \(-0.659010\pi\)
−0.479025 + 0.877801i \(0.659010\pi\)
\(80\) 887828. 0.193871
\(81\) 2.11323e6 0.441824
\(82\) 4.98634e6 0.998697
\(83\) −3.35527e6 −0.644102 −0.322051 − 0.946722i \(-0.604372\pi\)
−0.322051 + 0.946722i \(0.604372\pi\)
\(84\) 0 0
\(85\) 2.40877e6 0.425431
\(86\) 3.38335e6 0.573592
\(87\) −2.68975e6 −0.437920
\(88\) 1.32191e6 0.206782
\(89\) −2.39959e6 −0.360804 −0.180402 − 0.983593i \(-0.557740\pi\)
−0.180402 + 0.983593i \(0.557740\pi\)
\(90\) −3.03653e6 −0.439066
\(91\) 0 0
\(92\) −4.22766e6 −0.566034
\(93\) −4.26098e6 −0.549312
\(94\) 9.64343e6 1.19752
\(95\) 1.87423e6 0.224280
\(96\) −684111. −0.0789182
\(97\) −4.51506e6 −0.502299 −0.251150 − 0.967948i \(-0.580809\pi\)
−0.251150 + 0.967948i \(0.580809\pi\)
\(98\) 0 0
\(99\) −4.52118e6 −0.468305
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 98.8.a.k.1.1 2
7.2 even 3 98.8.c.h.67.2 4
7.3 odd 6 14.8.c.a.9.1 ✓ 4
7.4 even 3 98.8.c.h.79.2 4
7.5 odd 6 14.8.c.a.11.1 yes 4
7.6 odd 2 98.8.a.h.1.2 2
21.5 even 6 126.8.g.e.109.1 4
21.17 even 6 126.8.g.e.37.1 4
28.3 even 6 112.8.i.a.65.2 4
28.19 even 6 112.8.i.a.81.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
14.8.c.a.9.1 ✓ 4 7.3 odd 6
14.8.c.a.11.1 yes 4 7.5 odd 6
98.8.a.h.1.2 2 7.6 odd 2
98.8.a.k.1.1 2 1.1 even 1 trivial
98.8.c.h.67.2 4 7.2 even 3
98.8.c.h.79.2 4 7.4 even 3
112.8.i.a.65.2 4 28.3 even 6
112.8.i.a.81.2 4 28.19 even 6
126.8.g.e.37.1 4 21.17 even 6
126.8.g.e.109.1 4 21.5 even 6