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Properties

Label 98.14.a.d.1.1
Level $98$
Weight $14$
Character 98.1
Self dual yes
Analytic conductor $105.086$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [98,14,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, 14); 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, 14, names="a")
 
Level: \( N \) \(=\) \( 98 = 2 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 14 \)
Character orbit: \([\chi]\) \(=\) 98.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,64,1026] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(105.086310373\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 14)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
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)
 
\(f(q)\) \(=\) \(q+64.0000 q^{2} +1026.00 q^{3} +4096.00 q^{4} -4320.00 q^{5} +65664.0 q^{6} +262144. q^{8} -541647. q^{9} -276480. q^{10} -8.78731e6 q^{11} +4.20250e6 q^{12} +2.04209e7 q^{13} -4.43232e6 q^{15} +1.67772e7 q^{16} -1.71946e6 q^{17} -3.46654e7 q^{18} +1.09703e8 q^{19} -1.76947e7 q^{20} -5.62388e8 q^{22} -6.46760e8 q^{23} +2.68960e8 q^{24} -1.20204e9 q^{25} +1.30694e9 q^{26} -2.19151e9 q^{27} +7.28867e8 q^{29} -2.83668e8 q^{30} -1.02805e9 q^{31} +1.07374e9 q^{32} -9.01578e9 q^{33} -1.10046e8 q^{34} -2.21859e9 q^{36} +1.42294e10 q^{37} +7.02099e9 q^{38} +2.09519e10 q^{39} -1.13246e9 q^{40} -4.45445e10 q^{41} -5.46898e10 q^{43} -3.59928e10 q^{44} +2.33992e9 q^{45} -4.13927e10 q^{46} -4.78683e10 q^{47} +1.72134e10 q^{48} -7.69306e10 q^{50} -1.76417e9 q^{51} +8.36441e10 q^{52} -1.69987e11 q^{53} -1.40256e11 q^{54} +3.79612e10 q^{55} +1.12555e11 q^{57} +4.66475e10 q^{58} +3.00766e11 q^{59} -1.81548e10 q^{60} -3.69996e11 q^{61} -6.57951e10 q^{62} +6.87195e10 q^{64} -8.82184e10 q^{65} -5.77010e11 q^{66} -7.87011e11 q^{67} -7.04292e9 q^{68} -6.63576e11 q^{69} +5.59441e11 q^{71} -1.41990e11 q^{72} -1.21138e11 q^{73} +9.10681e11 q^{74} -1.23329e12 q^{75} +4.49343e11 q^{76} +1.34092e12 q^{78} +2.90427e11 q^{79} -7.24776e10 q^{80} -1.38492e12 q^{81} -2.85085e12 q^{82} +3.96511e12 q^{83} +7.42808e9 q^{85} -3.50015e12 q^{86} +7.47818e11 q^{87} -2.30354e12 q^{88} +6.02592e12 q^{89} +1.49755e11 q^{90} -2.64913e12 q^{92} -1.05478e12 q^{93} -3.06357e12 q^{94} -4.73917e11 q^{95} +1.10166e12 q^{96} -1.13028e13 q^{97} +4.75962e12 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 64.0000 0.707107
\(3\) 1026.00 0.812567 0.406284 − 0.913747i \(-0.366825\pi\)
0.406284 + 0.913747i \(0.366825\pi\)
\(4\) 4096.00 0.500000
\(5\) −4320.00 −0.123646 −0.0618228 − 0.998087i \(-0.519691\pi\)
−0.0618228 + 0.998087i \(0.519691\pi\)
\(6\) 65664.0 0.574572
\(7\) 0 0
\(8\) 262144. 0.353553
\(9\) −541647. −0.339735
\(10\) −276480. −0.0874307
\(11\) −8.78731e6 −1.49556 −0.747780 − 0.663947i \(-0.768880\pi\)
−0.747780 + 0.663947i \(0.768880\pi\)
\(12\) 4.20250e6 0.406284
\(13\) 2.04209e7 1.17339 0.586697 − 0.809807i \(-0.300428\pi\)
0.586697 + 0.809807i \(0.300428\pi\)
\(14\) 0 0
\(15\) −4.43232e6 −0.100470
\(16\) 1.67772e7 0.250000
\(17\) −1.71946e6 −0.0172772 −0.00863862 − 0.999963i \(-0.502750\pi\)
−0.00863862 + 0.999963i \(0.502750\pi\)
\(18\) −3.46654e7 −0.240229
\(19\) 1.09703e8 0.534958 0.267479 − 0.963564i \(-0.413809\pi\)
0.267479 + 0.963564i \(0.413809\pi\)
\(20\) −1.76947e7 −0.0618228
\(21\) 0 0
\(22\) −5.62388e8 −1.05752
\(23\) −6.46760e8 −0.910987 −0.455494 − 0.890239i \(-0.650537\pi\)
−0.455494 + 0.890239i \(0.650537\pi\)
\(24\) 2.68960e8 0.287286
\(25\) −1.20204e9 −0.984712
\(26\) 1.30694e9 0.829715
\(27\) −2.19151e9 −1.08862
\(28\) 0 0
\(29\) 7.28867e8 0.227542 0.113771 − 0.993507i \(-0.463707\pi\)
0.113771 + 0.993507i \(0.463707\pi\)
\(30\) −2.83668e8 −0.0710433
\(31\) −1.02805e9 −0.208048 −0.104024 − 0.994575i \(-0.533172\pi\)
−0.104024 + 0.994575i \(0.533172\pi\)
\(32\) 1.07374e9 0.176777
\(33\) −9.01578e9 −1.21524
\(34\) −1.10046e8 −0.0122169
\(35\) 0 0
\(36\) −2.21859e9 −0.169867
\(37\) 1.42294e10 0.911749 0.455874 − 0.890044i \(-0.349327\pi\)
0.455874 + 0.890044i \(0.349327\pi\)
\(38\) 7.02099e9 0.378273
\(39\) 2.09519e10 0.953461
\(40\) −1.13246e9 −0.0437153
\(41\) −4.45445e10 −1.46453 −0.732266 − 0.681019i \(-0.761537\pi\)
−0.732266 + 0.681019i \(0.761537\pi\)
\(42\) 0 0
\(43\) −5.46898e10 −1.31935 −0.659677 − 0.751549i \(-0.729307\pi\)
−0.659677 + 0.751549i \(0.729307\pi\)
\(44\) −3.59928e10 −0.747780
\(45\) 2.33992e9 0.0420067
\(46\) −4.13927e10 −0.644165
\(47\) −4.78683e10 −0.647757 −0.323879 − 0.946099i \(-0.604987\pi\)
−0.323879 + 0.946099i \(0.604987\pi\)
\(48\) 1.72134e10 0.203142
\(49\) 0 0
\(50\) −7.69306e10 −0.696296
\(51\) −1.76417e9 −0.0140389
\(52\) 8.36441e10 0.586697
\(53\) −1.69987e11 −1.05347 −0.526735 − 0.850029i \(-0.676584\pi\)
−0.526735 + 0.850029i \(0.676584\pi\)
\(54\) −1.40256e11 −0.769774
\(55\) 3.79612e10 0.184919
\(56\) 0 0
\(57\) 1.12555e11 0.434689
\(58\) 4.66475e10 0.160896
\(59\) 3.00766e11 0.928304 0.464152 − 0.885756i \(-0.346359\pi\)
0.464152 + 0.885756i \(0.346359\pi\)
\(60\) −1.81548e10 −0.0502352
\(61\) −3.69996e11 −0.919504 −0.459752 − 0.888047i \(-0.652062\pi\)
−0.459752 + 0.888047i \(0.652062\pi\)
\(62\) −6.57951e10 −0.147112
\(63\) 0 0
\(64\) 6.87195e10 0.125000
\(65\) −8.82184e10 −0.145085
\(66\) −5.77010e11 −0.859306
\(67\) −7.87011e11 −1.06291 −0.531453 − 0.847088i \(-0.678354\pi\)
−0.531453 + 0.847088i \(0.678354\pi\)
\(68\) −7.04292e9 −0.00863862
\(69\) −6.63576e11 −0.740238
\(70\) 0 0
\(71\) 5.59441e11 0.518293 0.259147 − 0.965838i \(-0.416559\pi\)
0.259147 + 0.965838i \(0.416559\pi\)
\(72\) −1.41990e11 −0.120114
\(73\) −1.21138e11 −0.0936872 −0.0468436 − 0.998902i \(-0.514916\pi\)
−0.0468436 + 0.998902i \(0.514916\pi\)
\(74\) 9.10681e11 0.644704
\(75\) −1.23329e12 −0.800144
\(76\) 4.49343e11 0.267479
\(77\) 0 0
\(78\) 1.34092e12 0.674199
\(79\) 2.90427e11 0.134419 0.0672095 − 0.997739i \(-0.478590\pi\)
0.0672095 + 0.997739i \(0.478590\pi\)
\(80\) −7.24776e10 −0.0309114
\(81\) −1.38492e12 −0.544845
\(82\) −2.85085e12 −1.03558
\(83\) 3.96511e12 1.33121 0.665606 − 0.746303i \(-0.268173\pi\)
0.665606 + 0.746303i \(0.268173\pi\)
\(84\) 0 0
\(85\) 7.42808e9 0.00213626
\(86\) −3.50015e12 −0.932925
\(87\) 7.47818e11 0.184893
\(88\) −2.30354e12 −0.528760
\(89\) 6.02592e12 1.28525 0.642626 − 0.766180i \(-0.277845\pi\)
0.642626 + 0.766180i \(0.277845\pi\)
\(90\) 1.49755e11 0.0297032
\(91\) 0 0
\(92\) −2.64913e12 −0.455494
\(93\) −1.05478e12 −0.169053
\(94\) −3.06357e12 −0.458033
\(95\) −4.73917e11 −0.0661452
\(96\) 1.10166e12 0.143643
\(97\) −1.13028e13 −1.37775 −0.688875 − 0.724880i \(-0.741895\pi\)
−0.688875 + 0.724880i \(0.741895\pi\)
\(98\) 0 0
\(99\) 4.75962e12 0.508093
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 98.14.a.d.1.1 1
7.2 even 3 98.14.c.b.67.1 2
7.3 odd 6 98.14.c.c.79.1 2
7.4 even 3 98.14.c.b.79.1 2
7.5 odd 6 98.14.c.c.67.1 2
7.6 odd 2 14.14.a.b.1.1 ✓ 1
21.20 even 2 126.14.a.a.1.1 1
28.27 even 2 112.14.a.b.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
14.14.a.b.1.1 ✓ 1 7.6 odd 2
98.14.a.d.1.1 1 1.1 even 1 trivial
98.14.c.b.67.1 2 7.2 even 3
98.14.c.b.79.1 2 7.4 even 3
98.14.c.c.67.1 2 7.5 odd 6
98.14.c.c.79.1 2 7.3 odd 6
112.14.a.b.1.1 1 28.27 even 2
126.14.a.a.1.1 1 21.20 even 2