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,-14,-448,0,-1024,-908] 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: \(\Q(\sqrt{949}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 237 \) 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 \(-14.9029\) 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)
 
\(f(q)\) \(=\) \(q-8.00000 q^{2} +58.8058 q^{3} +64.0000 q^{4} +424.282 q^{5} -470.447 q^{6} -512.000 q^{8} +1271.13 q^{9} -3394.25 q^{10} +7888.87 q^{11} +3763.57 q^{12} +6717.46 q^{13} +24950.2 q^{15} +4096.00 q^{16} -7075.01 q^{17} -10169.0 q^{18} -26249.2 q^{19} +27154.0 q^{20} -63110.9 q^{22} +12070.5 q^{23} -30108.6 q^{24} +101890. q^{25} -53739.7 q^{26} -53858.7 q^{27} +3061.25 q^{29} -199602. q^{30} -118525. q^{31} -32768.0 q^{32} +463912. q^{33} +56600.1 q^{34} +81352.1 q^{36} -459752. q^{37} +209993. q^{38} +395026. q^{39} -217232. q^{40} +316857. q^{41} -31624.2 q^{43} +504888. q^{44} +539316. q^{45} -96564.3 q^{46} +806553. q^{47} +240869. q^{48} -815120. q^{50} -416052. q^{51} +429917. q^{52} +479328. q^{53} +430869. q^{54} +3.34710e6 q^{55} -1.54360e6 q^{57} -24490.0 q^{58} +674863. q^{59} +1.59682e6 q^{60} -566329. q^{61} +948201. q^{62} +262144. q^{64} +2.85009e6 q^{65} -3.71129e6 q^{66} +1.18470e6 q^{67} -452801. q^{68} +709818. q^{69} +4.61736e6 q^{71} -650817. q^{72} +3.04800e6 q^{73} +3.67801e6 q^{74} +5.99173e6 q^{75} -1.67995e6 q^{76} -3.16021e6 q^{78} -6.90160e6 q^{79} +1.73786e6 q^{80} -5.94716e6 q^{81} -2.53485e6 q^{82} -9.01927e6 q^{83} -3.00180e6 q^{85} +252994. q^{86} +180020. q^{87} -4.03910e6 q^{88} -7.01478e6 q^{89} -4.31453e6 q^{90} +772515. q^{92} -6.96997e6 q^{93} -6.45243e6 q^{94} -1.11370e7 q^{95} -1.92695e6 q^{96} +8.60029e6 q^{97} +1.00278e7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 16 q^{2} + 56 q^{3} + 128 q^{4} - 14 q^{5} - 448 q^{6} - 1024 q^{8} - 908 q^{9} + 112 q^{10} + 2408 q^{11} + 3584 q^{12} + 10724 q^{13} + 26180 q^{15} + 8192 q^{16} - 35098 q^{17} + 7264 q^{18} - 2408 q^{19}+ \cdots + 21971264 q^{99}+O(q^{100}) \) 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
\(3\) 58.8058 1.25747 0.628733 − 0.777621i \(-0.283574\pi\)
0.628733 + 0.777621i \(0.283574\pi\)
\(4\) 64.0000 0.500000
\(5\) 424.282 1.51796 0.758978 − 0.651116i \(-0.225699\pi\)
0.758978 + 0.651116i \(0.225699\pi\)
\(6\) −470.447 −0.889162
\(7\) 0 0
\(8\) −512.000 −0.353553
\(9\) 1271.13 0.581220
\(10\) −3394.25 −1.07336
\(11\) 7888.87 1.78706 0.893532 − 0.448999i \(-0.148219\pi\)
0.893532 + 0.448999i \(0.148219\pi\)
\(12\) 3763.57 0.628733
\(13\) 6717.46 0.848014 0.424007 − 0.905659i \(-0.360623\pi\)
0.424007 + 0.905659i \(0.360623\pi\)
\(14\) 0 0
\(15\) 24950.2 1.90878
\(16\) 4096.00 0.250000
\(17\) −7075.01 −0.349266 −0.174633 − 0.984634i \(-0.555874\pi\)
−0.174633 + 0.984634i \(0.555874\pi\)
\(18\) −10169.0 −0.410984
\(19\) −26249.2 −0.877966 −0.438983 − 0.898495i \(-0.644661\pi\)
−0.438983 + 0.898495i \(0.644661\pi\)
\(20\) 27154.0 0.758978
\(21\) 0 0
\(22\) −63110.9 −1.26365
\(23\) 12070.5 0.206861 0.103431 − 0.994637i \(-0.467018\pi\)
0.103431 + 0.994637i \(0.467018\pi\)
\(24\) −30108.6 −0.444581
\(25\) 101890. 1.30419
\(26\) −53739.7 −0.599637
\(27\) −53858.7 −0.526602
\(28\) 0 0
\(29\) 3061.25 0.0233081 0.0116540 − 0.999932i \(-0.496290\pi\)
0.0116540 + 0.999932i \(0.496290\pi\)
\(30\) −199602. −1.34971
\(31\) −118525. −0.714570 −0.357285 − 0.933995i \(-0.616297\pi\)
−0.357285 + 0.933995i \(0.616297\pi\)
\(32\) −32768.0 −0.176777
\(33\) 463912. 2.24717
\(34\) 56600.1 0.246968
\(35\) 0 0
\(36\) 81352.1 0.290610
\(37\) −459752. −1.49217 −0.746083 − 0.665853i \(-0.768068\pi\)
−0.746083 + 0.665853i \(0.768068\pi\)
\(38\) 209993. 0.620816
\(39\) 395026. 1.06635
\(40\) −217232. −0.536679
\(41\) 316857. 0.717992 0.358996 − 0.933339i \(-0.383119\pi\)
0.358996 + 0.933339i \(0.383119\pi\)
\(42\) 0 0
\(43\) −31624.2 −0.0606569 −0.0303284 − 0.999540i \(-0.509655\pi\)
−0.0303284 + 0.999540i \(0.509655\pi\)
\(44\) 504888. 0.893532
\(45\) 539316. 0.882266
\(46\) −96564.3 −0.146273
\(47\) 806553. 1.13316 0.566579 − 0.824007i \(-0.308267\pi\)
0.566579 + 0.824007i \(0.308267\pi\)
\(48\) 240869. 0.314366
\(49\) 0 0
\(50\) −815120. −0.922204
\(51\) −416052. −0.439190
\(52\) 429917. 0.424007
\(53\) 479328. 0.442250 0.221125 − 0.975246i \(-0.429027\pi\)
0.221125 + 0.975246i \(0.429027\pi\)
\(54\) 430869. 0.372364
\(55\) 3.34710e6 2.71269
\(56\) 0 0
\(57\) −1.54360e6 −1.10401
\(58\) −24490.0 −0.0164813
\(59\) 674863. 0.427793 0.213896 − 0.976856i \(-0.431385\pi\)
0.213896 + 0.976856i \(0.431385\pi\)
\(60\) 1.59682e6 0.954389
\(61\) −566329. −0.319459 −0.159729 − 0.987161i \(-0.551062\pi\)
−0.159729 + 0.987161i \(0.551062\pi\)
\(62\) 948201. 0.505277
\(63\) 0 0
\(64\) 262144. 0.125000
\(65\) 2.85009e6 1.28725
\(66\) −3.71129e6 −1.58899
\(67\) 1.18470e6 0.481222 0.240611 − 0.970622i \(-0.422652\pi\)
0.240611 + 0.970622i \(0.422652\pi\)
\(68\) −452801. −0.174633
\(69\) 709818. 0.260121
\(70\) 0 0
\(71\) 4.61736e6 1.53105 0.765524 − 0.643407i \(-0.222480\pi\)
0.765524 + 0.643407i \(0.222480\pi\)
\(72\) −650817. −0.205492
\(73\) 3.04800e6 0.917034 0.458517 − 0.888686i \(-0.348381\pi\)
0.458517 + 0.888686i \(0.348381\pi\)
\(74\) 3.67801e6 1.05512
\(75\) 5.99173e6 1.63998
\(76\) −1.67995e6 −0.438983
\(77\) 0 0
\(78\) −3.16021e6 −0.754022
\(79\) −6.90160e6 −1.57491 −0.787454 − 0.616374i \(-0.788601\pi\)
−0.787454 + 0.616374i \(0.788601\pi\)
\(80\) 1.73786e6 0.379489
\(81\) −5.94716e6 −1.24340
\(82\) −2.53485e6 −0.507697
\(83\) −9.01927e6 −1.73140 −0.865701 − 0.500562i \(-0.833127\pi\)
−0.865701 + 0.500562i \(0.833127\pi\)
\(84\) 0 0
\(85\) −3.00180e6 −0.530170
\(86\) 252994. 0.0428909
\(87\) 180020. 0.0293091
\(88\) −4.03910e6 −0.631823
\(89\) −7.01478e6 −1.05475 −0.527374 − 0.849633i \(-0.676824\pi\)
−0.527374 + 0.849633i \(0.676824\pi\)
\(90\) −4.31453e6 −0.623856
\(91\) 0 0
\(92\) 772515. 0.103431
\(93\) −6.96997e6 −0.898547
\(94\) −6.45243e6 −0.801264
\(95\) −1.11370e7 −1.33271
\(96\) −1.92695e6 −0.222291
\(97\) 8.60029e6 0.956779 0.478390 − 0.878148i \(-0.341221\pi\)
0.478390 + 0.878148i \(0.341221\pi\)
\(98\) 0 0
\(99\) 1.00278e7 1.03868
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.f.1.2 2
7.2 even 3 14.8.c.b.11.1 yes 4
7.3 odd 6 98.8.c.m.79.2 4
7.4 even 3 14.8.c.b.9.1 ✓ 4
7.5 odd 6 98.8.c.m.67.2 4
7.6 odd 2 98.8.a.d.1.1 2
21.2 odd 6 126.8.g.d.109.2 4
21.11 odd 6 126.8.g.d.37.2 4
28.11 odd 6 112.8.i.b.65.2 4
28.23 odd 6 112.8.i.b.81.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
14.8.c.b.9.1 ✓ 4 7.4 even 3
14.8.c.b.11.1 yes 4 7.2 even 3
98.8.a.d.1.1 2 7.6 odd 2
98.8.a.f.1.2 2 1.1 even 1 trivial
98.8.c.m.67.2 4 7.5 odd 6
98.8.c.m.79.2 4 7.3 odd 6
112.8.i.b.65.2 4 28.11 odd 6
112.8.i.b.81.2 4 28.23 odd 6
126.8.g.d.37.2 4 21.11 odd 6
126.8.g.d.109.2 4 21.2 odd 6