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 1.2
Root \(-23.9387\) 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)
 
\(f(q)\) \(=\) \(q+8.00000 q^{2} +76.8774 q^{3} +64.0000 q^{4} +21.2452 q^{5} +615.019 q^{6} +512.000 q^{8} +3723.13 q^{9} +169.962 q^{10} +3266.14 q^{11} +4920.15 q^{12} -10238.0 q^{13} +1633.28 q^{15} +4096.00 q^{16} +36529.1 q^{17} +29785.1 q^{18} +32401.2 q^{19} +1359.69 q^{20} +26129.1 q^{22} +16741.2 q^{23} +39361.2 q^{24} -77673.6 q^{25} -81903.8 q^{26} +118094. q^{27} +43984.4 q^{29} +13066.2 q^{30} -133843. q^{31} +32768.0 q^{32} +251092. q^{33} +292233. q^{34} +238281. q^{36} -579877. q^{37} +259210. q^{38} -787068. q^{39} +10877.5 q^{40} +532967. q^{41} -365375. q^{43} +209033. q^{44} +79098.7 q^{45} +133929. q^{46} +206863. q^{47} +314890. q^{48} -621389. q^{50} +2.80826e6 q^{51} -655230. q^{52} -1.08821e6 q^{53} +944752. q^{54} +69389.9 q^{55} +2.49092e6 q^{57} +351875. q^{58} -161652. q^{59} +104530. q^{60} +844308. q^{61} -1.07075e6 q^{62} +262144. q^{64} -217508. q^{65} +2.00874e6 q^{66} +1.22785e6 q^{67} +2.33786e6 q^{68} +1.28702e6 q^{69} +1.10759e6 q^{71} +1.90624e6 q^{72} -1.63885e6 q^{73} -4.63902e6 q^{74} -5.97135e6 q^{75} +2.07368e6 q^{76} -6.29655e6 q^{78} -4.99421e6 q^{79} +87020.4 q^{80} +936265. q^{81} +4.26374e6 q^{82} +3.38355e6 q^{83} +776069. q^{85} -2.92300e6 q^{86} +3.38141e6 q^{87} +1.67226e6 q^{88} -6.53696e6 q^{89} +632790. q^{90} +1.07143e6 q^{92} -1.02895e7 q^{93} +1.65491e6 q^{94} +688370. q^{95} +2.51912e6 q^{96} -1.35914e6 q^{97} +1.21603e7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 16 q^{2} + 56 q^{3} + 128 q^{4} + 238 q^{5} + 448 q^{6} + 1024 q^{8} + 1972 q^{9} + 1904 q^{10} + 5848 q^{11} + 3584 q^{12} - 1316 q^{13} - 2892 q^{15} + 8192 q^{16} + 47642 q^{17} + 15776 q^{18}+ \cdots + 7639104 q^{99}+O(q^{100}) \) 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
\(3\) 76.8774 1.64390 0.821948 − 0.569563i \(-0.192887\pi\)
0.821948 + 0.569563i \(0.192887\pi\)
\(4\) 64.0000 0.500000
\(5\) 21.2452 0.0760092 0.0380046 − 0.999278i \(-0.487900\pi\)
0.0380046 + 0.999278i \(0.487900\pi\)
\(6\) 615.019 1.16241
\(7\) 0 0
\(8\) 512.000 0.353553
\(9\) 3723.13 1.70239
\(10\) 169.962 0.0537466
\(11\) 3266.14 0.739879 0.369939 − 0.929056i \(-0.379378\pi\)
0.369939 + 0.929056i \(0.379378\pi\)
\(12\) 4920.15 0.821948
\(13\) −10238.0 −1.29245 −0.646223 − 0.763149i \(-0.723652\pi\)
−0.646223 + 0.763149i \(0.723652\pi\)
\(14\) 0 0
\(15\) 1633.28 0.124951
\(16\) 4096.00 0.250000
\(17\) 36529.1 1.80330 0.901650 − 0.432466i \(-0.142357\pi\)
0.901650 + 0.432466i \(0.142357\pi\)
\(18\) 29785.1 1.20377
\(19\) 32401.2 1.08374 0.541868 − 0.840464i \(-0.317717\pi\)
0.541868 + 0.840464i \(0.317717\pi\)
\(20\) 1359.69 0.0380046
\(21\) 0 0
\(22\) 26129.1 0.523173
\(23\) 16741.2 0.286905 0.143452 − 0.989657i \(-0.454180\pi\)
0.143452 + 0.989657i \(0.454180\pi\)
\(24\) 39361.2 0.581205
\(25\) −77673.6 −0.994223
\(26\) −81903.8 −0.913897
\(27\) 118094. 1.15466
\(28\) 0 0
\(29\) 43984.4 0.334893 0.167446 − 0.985881i \(-0.446448\pi\)
0.167446 + 0.985881i \(0.446448\pi\)
\(30\) 13066.2 0.0883538
\(31\) −133843. −0.806920 −0.403460 − 0.914997i \(-0.632192\pi\)
−0.403460 + 0.914997i \(0.632192\pi\)
\(32\) 32768.0 0.176777
\(33\) 251092. 1.21628
\(34\) 292233. 1.27513
\(35\) 0 0
\(36\) 238281. 0.851197
\(37\) −579877. −1.88204 −0.941022 − 0.338345i \(-0.890133\pi\)
−0.941022 + 0.338345i \(0.890133\pi\)
\(38\) 259210. 0.766317
\(39\) −787068. −2.12465
\(40\) 10877.5 0.0268733
\(41\) 532967. 1.20769 0.603847 − 0.797100i \(-0.293634\pi\)
0.603847 + 0.797100i \(0.293634\pi\)
\(42\) 0 0
\(43\) −365375. −0.700809 −0.350404 − 0.936598i \(-0.613956\pi\)
−0.350404 + 0.936598i \(0.613956\pi\)
\(44\) 209033. 0.369939
\(45\) 79098.7 0.129397
\(46\) 133929. 0.202872
\(47\) 206863. 0.290630 0.145315 − 0.989385i \(-0.453580\pi\)
0.145315 + 0.989385i \(0.453580\pi\)
\(48\) 314890. 0.410974
\(49\) 0 0
\(50\) −621389. −0.703022
\(51\) 2.80826e6 2.96444
\(52\) −655230. −0.646223
\(53\) −1.08821e6 −1.00403 −0.502015 − 0.864859i \(-0.667408\pi\)
−0.502015 + 0.864859i \(0.667408\pi\)
\(54\) 944752. 0.816469
\(55\) 69389.9 0.0562376
\(56\) 0 0
\(57\) 2.49092e6 1.78155
\(58\) 351875. 0.236805
\(59\) −161652. −0.102471 −0.0512354 − 0.998687i \(-0.516316\pi\)
−0.0512354 + 0.998687i \(0.516316\pi\)
\(60\) 104530. 0.0624756
\(61\) 844308. 0.476263 0.238131 − 0.971233i \(-0.423465\pi\)
0.238131 + 0.971233i \(0.423465\pi\)
\(62\) −1.07075e6 −0.570578
\(63\) 0 0
\(64\) 262144. 0.125000
\(65\) −217508. −0.0982377
\(66\) 2.00874e6 0.860043
\(67\) 1.22785e6 0.498749 0.249375 − 0.968407i \(-0.419775\pi\)
0.249375 + 0.968407i \(0.419775\pi\)
\(68\) 2.33786e6 0.901650
\(69\) 1.28702e6 0.471642
\(70\) 0 0
\(71\) 1.10759e6 0.367259 0.183630 − 0.982995i \(-0.441215\pi\)
0.183630 + 0.982995i \(0.441215\pi\)
\(72\) 1.90624e6 0.601887
\(73\) −1.63885e6 −0.493069 −0.246535 − 0.969134i \(-0.579292\pi\)
−0.246535 + 0.969134i \(0.579292\pi\)
\(74\) −4.63902e6 −1.33081
\(75\) −5.97135e6 −1.63440
\(76\) 2.07368e6 0.541868
\(77\) 0 0
\(78\) −6.29655e6 −1.50235
\(79\) −4.99421e6 −1.13965 −0.569826 − 0.821765i \(-0.692989\pi\)
−0.569826 + 0.821765i \(0.692989\pi\)
\(80\) 87020.4 0.0190023
\(81\) 936265. 0.195750
\(82\) 4.26374e6 0.853969
\(83\) 3.38355e6 0.649531 0.324765 − 0.945795i \(-0.394715\pi\)
0.324765 + 0.945795i \(0.394715\pi\)
\(84\) 0 0
\(85\) 776069. 0.137067
\(86\) −2.92300e6 −0.495547
\(87\) 3.38141e6 0.550529
\(88\) 1.67226e6 0.261587
\(89\) −6.53696e6 −0.982904 −0.491452 − 0.870905i \(-0.663534\pi\)
−0.491452 + 0.870905i \(0.663534\pi\)
\(90\) 632790. 0.0914978
\(91\) 0 0
\(92\) 1.07143e6 0.143452
\(93\) −1.02895e7 −1.32649
\(94\) 1.65491e6 0.205507
\(95\) 688370. 0.0823739
\(96\) 2.51912e6 0.290602
\(97\) −1.35914e6 −0.151204 −0.0756022 − 0.997138i \(-0.524088\pi\)
−0.0756022 + 0.997138i \(0.524088\pi\)
\(98\) 0 0
\(99\) 1.21603e7 1.25956
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.1.2 2
7.2 even 3 98.8.c.h.67.1 4
7.3 odd 6 14.8.c.a.9.2 ✓ 4
7.4 even 3 98.8.c.h.79.1 4
7.5 odd 6 14.8.c.a.11.2 yes 4
7.6 odd 2 98.8.a.h.1.1 2
21.5 even 6 126.8.g.e.109.2 4
21.17 even 6 126.8.g.e.37.2 4
28.3 even 6 112.8.i.a.65.1 4
28.19 even 6 112.8.i.a.81.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
14.8.c.a.9.2 ✓ 4 7.3 odd 6
14.8.c.a.11.2 yes 4 7.5 odd 6
98.8.a.h.1.1 2 7.6 odd 2
98.8.a.k.1.2 2 1.1 even 1 trivial
98.8.c.h.67.1 4 7.2 even 3
98.8.c.h.79.1 4 7.4 even 3
112.8.i.a.65.1 4 28.3 even 6
112.8.i.a.81.1 4 28.19 even 6
126.8.g.e.37.2 4 21.17 even 6
126.8.g.e.109.2 4 21.5 even 6