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 1.1
Root \(15.9029\) 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)
 
\(f(q)\) \(=\) \(q-8.00000 q^{2} -2.80584 q^{3} +64.0000 q^{4} -438.282 q^{5} +22.4467 q^{6} -512.000 q^{8} -2179.13 q^{9} +3506.25 q^{10} -5480.87 q^{11} -179.574 q^{12} +4006.54 q^{13} +1229.75 q^{15} +4096.00 q^{16} -28023.0 q^{17} +17433.0 q^{18} +23841.2 q^{19} -28050.0 q^{20} +43846.9 q^{22} -73754.5 q^{23} +1436.59 q^{24} +113966. q^{25} -32052.3 q^{26} +12250.7 q^{27} -98721.3 q^{29} -9838.00 q^{30} -47486.9 q^{31} -32768.0 q^{32} +15378.5 q^{33} +224184. q^{34} -139464. q^{36} -100062. q^{37} -190729. q^{38} -11241.7 q^{39} +224400. q^{40} +489123. q^{41} +299600. q^{43} -350776. q^{44} +955072. q^{45} +590036. q^{46} +962739. q^{47} -11492.7 q^{48} -911728. q^{50} +78628.1 q^{51} +256419. q^{52} +1.83787e6 q^{53} -98005.4 q^{54} +2.40216e6 q^{55} -66894.5 q^{57} +789770. q^{58} -14510.6 q^{59} +78704.0 q^{60} +2.02937e6 q^{61} +379895. q^{62} +262144. q^{64} -1.75599e6 q^{65} -123028. q^{66} -2.96898e6 q^{67} -1.79347e6 q^{68} +206944. q^{69} -4.34296e6 q^{71} +1.11571e6 q^{72} +1.50106e6 q^{73} +800500. q^{74} -319771. q^{75} +1.52583e6 q^{76} +89933.9 q^{78} -1.77236e6 q^{79} -1.79520e6 q^{80} +4.73138e6 q^{81} -3.91299e6 q^{82} -1.57509e6 q^{83} +1.22820e7 q^{85} -2.39680e6 q^{86} +276996. q^{87} +2.80620e6 q^{88} +8.79453e6 q^{89} -7.64057e6 q^{90} -4.72029e6 q^{92} +133241. q^{93} -7.70191e6 q^{94} -1.04491e7 q^{95} +91941.9 q^{96} -1.03493e7 q^{97} +1.19435e7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 16 q^{2} + 56 q^{3} + 128 q^{4} - 14 q^{5} - 448 q^{6} - 1024 q^{8} - 908 q^{9} + 112 q^{10} + 2408 q^{11} + 3584 q^{12} + 10724 q^{13} + 26180 q^{15} + 8192 q^{16} - 35098 q^{17} + 7264 q^{18} - 2408 q^{19}+ \cdots + 21971264 q^{99}+O(q^{100}) \) 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
\(3\) −2.80584 −0.0599983 −0.0299992 − 0.999550i \(-0.509550\pi\)
−0.0299992 + 0.999550i \(0.509550\pi\)
\(4\) 64.0000 0.500000
\(5\) −438.282 −1.56804 −0.784022 − 0.620733i \(-0.786835\pi\)
−0.784022 + 0.620733i \(0.786835\pi\)
\(6\) 22.4467 0.0424252
\(7\) 0 0
\(8\) −512.000 −0.353553
\(9\) −2179.13 −0.996400
\(10\) 3506.25 1.10878
\(11\) −5480.87 −1.24158 −0.620790 − 0.783977i \(-0.713188\pi\)
−0.620790 + 0.783977i \(0.713188\pi\)
\(12\) −179.574 −0.0299992
\(13\) 4006.54 0.505787 0.252894 − 0.967494i \(-0.418618\pi\)
0.252894 + 0.967494i \(0.418618\pi\)
\(14\) 0 0
\(15\) 1229.75 0.0940800
\(16\) 4096.00 0.250000
\(17\) −28023.0 −1.38339 −0.691693 − 0.722192i \(-0.743135\pi\)
−0.691693 + 0.722192i \(0.743135\pi\)
\(18\) 17433.0 0.704561
\(19\) 23841.2 0.797425 0.398712 − 0.917076i \(-0.369457\pi\)
0.398712 + 0.917076i \(0.369457\pi\)
\(20\) −28050.0 −0.784022
\(21\) 0 0
\(22\) 43846.9 0.877930
\(23\) −73754.5 −1.26398 −0.631992 − 0.774975i \(-0.717762\pi\)
−0.631992 + 0.774975i \(0.717762\pi\)
\(24\) 1436.59 0.0212126
\(25\) 113966. 1.45876
\(26\) −32052.3 −0.357646
\(27\) 12250.7 0.119781
\(28\) 0 0
\(29\) −98721.3 −0.751653 −0.375827 − 0.926690i \(-0.622641\pi\)
−0.375827 + 0.926690i \(0.622641\pi\)
\(30\) −9838.00 −0.0665246
\(31\) −47486.9 −0.286291 −0.143145 − 0.989702i \(-0.545722\pi\)
−0.143145 + 0.989702i \(0.545722\pi\)
\(32\) −32768.0 −0.176777
\(33\) 15378.5 0.0744928
\(34\) 224184. 0.978201
\(35\) 0 0
\(36\) −139464. −0.498200
\(37\) −100062. −0.324762 −0.162381 − 0.986728i \(-0.551917\pi\)
−0.162381 + 0.986728i \(0.551917\pi\)
\(38\) −190729. −0.563864
\(39\) −11241.7 −0.0303464
\(40\) 224400. 0.554388
\(41\) 489123. 1.10834 0.554172 − 0.832402i \(-0.313035\pi\)
0.554172 + 0.832402i \(0.313035\pi\)
\(42\) 0 0
\(43\) 299600. 0.574649 0.287324 − 0.957833i \(-0.407234\pi\)
0.287324 + 0.957833i \(0.407234\pi\)
\(44\) −350776. −0.620790
\(45\) 955072. 1.56240
\(46\) 590036. 0.893771
\(47\) 962739. 1.35259 0.676295 − 0.736631i \(-0.263585\pi\)
0.676295 + 0.736631i \(0.263585\pi\)
\(48\) −11492.7 −0.0149996
\(49\) 0 0
\(50\) −911728. −1.03150
\(51\) 78628.1 0.0830008
\(52\) 256419. 0.252894
\(53\) 1.83787e6 1.69570 0.847849 − 0.530239i \(-0.177898\pi\)
0.847849 + 0.530239i \(0.177898\pi\)
\(54\) −98005.4 −0.0846977
\(55\) 2.40216e6 1.94685
\(56\) 0 0
\(57\) −66894.5 −0.0478441
\(58\) 789770. 0.531499
\(59\) −14510.6 −0.00919820 −0.00459910 − 0.999989i \(-0.501464\pi\)
−0.00459910 + 0.999989i \(0.501464\pi\)
\(60\) 78704.0 0.0470400
\(61\) 2.02937e6 1.14474 0.572370 − 0.819995i \(-0.306024\pi\)
0.572370 + 0.819995i \(0.306024\pi\)
\(62\) 379895. 0.202438
\(63\) 0 0
\(64\) 262144. 0.125000
\(65\) −1.75599e6 −0.793097
\(66\) −123028. −0.0526743
\(67\) −2.96898e6 −1.20599 −0.602997 − 0.797743i \(-0.706027\pi\)
−0.602997 + 0.797743i \(0.706027\pi\)
\(68\) −1.79347e6 −0.691693
\(69\) 206944. 0.0758369
\(70\) 0 0
\(71\) −4.34296e6 −1.44006 −0.720031 − 0.693942i \(-0.755872\pi\)
−0.720031 + 0.693942i \(0.755872\pi\)
\(72\) 1.11571e6 0.352281
\(73\) 1.50106e6 0.451614 0.225807 − 0.974172i \(-0.427498\pi\)
0.225807 + 0.974172i \(0.427498\pi\)
\(74\) 800500. 0.229641
\(75\) −319771. −0.0875234
\(76\) 1.52583e6 0.398712
\(77\) 0 0
\(78\) 89933.9 0.0214581
\(79\) −1.77236e6 −0.404444 −0.202222 − 0.979340i \(-0.564816\pi\)
−0.202222 + 0.979340i \(0.564816\pi\)
\(80\) −1.79520e6 −0.392011
\(81\) 4.73138e6 0.989214
\(82\) −3.91299e6 −0.783718
\(83\) −1.57509e6 −0.302366 −0.151183 − 0.988506i \(-0.548308\pi\)
−0.151183 + 0.988506i \(0.548308\pi\)
\(84\) 0 0
\(85\) 1.22820e7 2.16921
\(86\) −2.39680e6 −0.406338
\(87\) 276996. 0.0450979
\(88\) 2.80620e6 0.438965
\(89\) 8.79453e6 1.32235 0.661177 − 0.750230i \(-0.270057\pi\)
0.661177 + 0.750230i \(0.270057\pi\)
\(90\) −7.64057e6 −1.10478
\(91\) 0 0
\(92\) −4.72029e6 −0.631992
\(93\) 133241. 0.0171770
\(94\) −7.70191e6 −0.956425
\(95\) −1.04491e7 −1.25040
\(96\) 91941.9 0.0106063
\(97\) −1.03493e7 −1.15135 −0.575676 − 0.817678i \(-0.695261\pi\)
−0.575676 + 0.817678i \(0.695261\pi\)
\(98\) 0 0
\(99\) 1.19435e7 1.23711
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.1.1 2
7.2 even 3 14.8.c.b.11.2 yes 4
7.3 odd 6 98.8.c.m.79.1 4
7.4 even 3 14.8.c.b.9.2 ✓ 4
7.5 odd 6 98.8.c.m.67.1 4
7.6 odd 2 98.8.a.d.1.2 2
21.2 odd 6 126.8.g.d.109.1 4
21.11 odd 6 126.8.g.d.37.1 4
28.11 odd 6 112.8.i.b.65.1 4
28.23 odd 6 112.8.i.b.81.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
14.8.c.b.9.2 ✓ 4 7.4 even 3
14.8.c.b.11.2 yes 4 7.2 even 3
98.8.a.d.1.2 2 7.6 odd 2
98.8.a.f.1.1 2 1.1 even 1 trivial
98.8.c.m.67.1 4 7.5 odd 6
98.8.c.m.79.1 4 7.3 odd 6
112.8.i.b.65.1 4 28.11 odd 6
112.8.i.b.81.1 4 28.23 odd 6
126.8.g.d.37.1 4 21.11 odd 6
126.8.g.d.109.1 4 21.2 odd 6