Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [98,12,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, 12); 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, 12, names="a")
 
Level: \( N \) \(=\) \( 98 = 2 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 98.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,128,-266,4096,-7504,-8512,0,131072,123520] 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: \(75.2976316948\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 101802x^{2} - 3237299x + 1820791991 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{6}\cdot 3\cdot 7^{2} \)
Twist minimal: no (minimal twist has level 14)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-200.794\) 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)
 
\(f(q)\) \(=\) \(q+32.0000 q^{2} -468.588 q^{3} +1024.00 q^{4} +6652.75 q^{5} -14994.8 q^{6} +32768.0 q^{8} +42428.1 q^{9} +212888. q^{10} +121814. q^{11} -479835. q^{12} -1.81677e6 q^{13} -3.11740e6 q^{15} +1.04858e6 q^{16} +5.40225e6 q^{17} +1.35770e6 q^{18} -1.02088e7 q^{19} +6.81241e6 q^{20} +3.89805e6 q^{22} +3.76264e7 q^{23} -1.53547e7 q^{24} -4.56908e6 q^{25} -5.81367e7 q^{26} +6.31277e7 q^{27} -1.87020e7 q^{29} -9.97568e7 q^{30} -2.45594e8 q^{31} +3.35544e7 q^{32} -5.70807e7 q^{33} +1.72872e8 q^{34} +4.34464e7 q^{36} +6.51823e8 q^{37} -3.26683e8 q^{38} +8.51318e8 q^{39} +2.17997e8 q^{40} -6.38428e8 q^{41} +8.54732e8 q^{43} +1.24738e8 q^{44} +2.82263e8 q^{45} +1.20405e9 q^{46} -8.47320e8 q^{47} -4.91351e8 q^{48} -1.46210e8 q^{50} -2.53143e9 q^{51} -1.86037e9 q^{52} -4.22534e9 q^{53} +2.02009e9 q^{54} +8.10399e8 q^{55} +4.78374e9 q^{57} -5.98464e8 q^{58} -2.20451e9 q^{59} -3.19222e9 q^{60} -9.69989e9 q^{61} -7.85899e9 q^{62} +1.07374e9 q^{64} -1.20865e10 q^{65} -1.82658e9 q^{66} -9.72892e9 q^{67} +5.53190e9 q^{68} -1.76313e10 q^{69} -2.83613e10 q^{71} +1.39028e9 q^{72} -3.66835e9 q^{73} +2.08583e10 q^{74} +2.14102e9 q^{75} -1.04539e10 q^{76} +2.72422e10 q^{78} +1.26954e10 q^{79} +6.97591e9 q^{80} -3.70969e10 q^{81} -2.04297e10 q^{82} -4.03358e10 q^{83} +3.59398e10 q^{85} +2.73514e10 q^{86} +8.76353e9 q^{87} +3.99161e9 q^{88} -3.28737e10 q^{89} +9.03243e9 q^{90} +3.85295e10 q^{92} +1.15082e11 q^{93} -2.71143e10 q^{94} -6.79168e10 q^{95} -1.57232e10 q^{96} +1.02279e11 q^{97} +5.16834e9 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 128 q^{2} - 266 q^{3} + 4096 q^{4} - 7504 q^{5} - 8512 q^{6} + 131072 q^{8} + 123520 q^{9} - 240128 q^{10} - 213026 q^{11} - 272384 q^{12} - 1304856 q^{13} + 1137750 q^{15} + 4194304 q^{16} - 8854244 q^{17}+ \cdots + 196707902868 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 32.0000 0.707107
\(3\) −468.588 −1.11333 −0.556666 − 0.830736i \(-0.687920\pi\)
−0.556666 + 0.830736i \(0.687920\pi\)
\(4\) 1024.00 0.500000
\(5\) 6652.75 0.952064 0.476032 − 0.879428i \(-0.342075\pi\)
0.476032 + 0.879428i \(0.342075\pi\)
\(6\) −14994.8 −0.787245
\(7\) 0 0
\(8\) 32768.0 0.353553
\(9\) 42428.1 0.239508
\(10\) 212888. 0.673211
\(11\) 121814. 0.228054 0.114027 − 0.993478i \(-0.463625\pi\)
0.114027 + 0.993478i \(0.463625\pi\)
\(12\) −479835. −0.556666
\(13\) −1.81677e6 −1.35710 −0.678550 − 0.734554i \(-0.737391\pi\)
−0.678550 + 0.734554i \(0.737391\pi\)
\(14\) 0 0
\(15\) −3.11740e6 −1.05996
\(16\) 1.04858e6 0.250000
\(17\) 5.40225e6 0.922796 0.461398 − 0.887193i \(-0.347348\pi\)
0.461398 + 0.887193i \(0.347348\pi\)
\(18\) 1.35770e6 0.169358
\(19\) −1.02088e7 −0.945870 −0.472935 − 0.881097i \(-0.656806\pi\)
−0.472935 + 0.881097i \(0.656806\pi\)
\(20\) 6.81241e6 0.476032
\(21\) 0 0
\(22\) 3.89805e6 0.161259
\(23\) 3.76264e7 1.21896 0.609481 − 0.792801i \(-0.291378\pi\)
0.609481 + 0.792801i \(0.291378\pi\)
\(24\) −1.53547e7 −0.393622
\(25\) −4.56908e6 −0.0935747
\(26\) −5.81367e7 −0.959615
\(27\) 6.31277e7 0.846680
\(28\) 0 0
\(29\) −1.87020e7 −0.169316 −0.0846581 − 0.996410i \(-0.526980\pi\)
−0.0846581 + 0.996410i \(0.526980\pi\)
\(30\) −9.97568e7 −0.749507
\(31\) −2.45594e8 −1.54073 −0.770367 − 0.637601i \(-0.779927\pi\)
−0.770367 + 0.637601i \(0.779927\pi\)
\(32\) 3.35544e7 0.176777
\(33\) −5.70807e7 −0.253900
\(34\) 1.72872e8 0.652515
\(35\) 0 0
\(36\) 4.34464e7 0.119754
\(37\) 6.51823e8 1.54533 0.772663 − 0.634817i \(-0.218924\pi\)
0.772663 + 0.634817i \(0.218924\pi\)
\(38\) −3.26683e8 −0.668831
\(39\) 8.51318e8 1.51090
\(40\) 2.17997e8 0.336605
\(41\) −6.38428e8 −0.860598 −0.430299 − 0.902686i \(-0.641592\pi\)
−0.430299 + 0.902686i \(0.641592\pi\)
\(42\) 0 0
\(43\) 8.54732e8 0.886652 0.443326 − 0.896360i \(-0.353798\pi\)
0.443326 + 0.896360i \(0.353798\pi\)
\(44\) 1.24738e8 0.114027
\(45\) 2.82263e8 0.228027
\(46\) 1.20405e9 0.861936
\(47\) −8.47320e8 −0.538902 −0.269451 − 0.963014i \(-0.586842\pi\)
−0.269451 + 0.963014i \(0.586842\pi\)
\(48\) −4.91351e8 −0.278333
\(49\) 0 0
\(50\) −1.46210e8 −0.0661673
\(51\) −2.53143e9 −1.02738
\(52\) −1.86037e9 −0.678550
\(53\) −4.22534e9 −1.38785 −0.693927 − 0.720045i \(-0.744121\pi\)
−0.693927 + 0.720045i \(0.744121\pi\)
\(54\) 2.02009e9 0.598693
\(55\) 8.10399e8 0.217122
\(56\) 0 0
\(57\) 4.78374e9 1.05307
\(58\) −5.98464e8 −0.119725
\(59\) −2.20451e9 −0.401444 −0.200722 − 0.979648i \(-0.564329\pi\)
−0.200722 + 0.979648i \(0.564329\pi\)
\(60\) −3.19222e9 −0.529981
\(61\) −9.69989e9 −1.47046 −0.735229 − 0.677819i \(-0.762925\pi\)
−0.735229 + 0.677819i \(0.762925\pi\)
\(62\) −7.85899e9 −1.08946
\(63\) 0 0
\(64\) 1.07374e9 0.125000
\(65\) −1.20865e10 −1.29205
\(66\) −1.82658e9 −0.179534
\(67\) −9.72892e9 −0.880345 −0.440173 − 0.897913i \(-0.645083\pi\)
−0.440173 + 0.897913i \(0.645083\pi\)
\(68\) 5.53190e9 0.461398
\(69\) −1.76313e10 −1.35711
\(70\) 0 0
\(71\) −2.83613e10 −1.86554 −0.932770 − 0.360472i \(-0.882616\pi\)
−0.932770 + 0.360472i \(0.882616\pi\)
\(72\) 1.39028e9 0.0846788
\(73\) −3.66835e9 −0.207107 −0.103553 − 0.994624i \(-0.533021\pi\)
−0.103553 + 0.994624i \(0.533021\pi\)
\(74\) 2.08583e10 1.09271
\(75\) 2.14102e9 0.104180
\(76\) −1.04539e10 −0.472935
\(77\) 0 0
\(78\) 2.72422e10 1.06837
\(79\) 1.26954e10 0.464192 0.232096 − 0.972693i \(-0.425442\pi\)
0.232096 + 0.972693i \(0.425442\pi\)
\(80\) 6.97591e9 0.238016
\(81\) −3.70969e10 −1.18214
\(82\) −2.04297e10 −0.608535
\(83\) −4.03358e10 −1.12399 −0.561994 − 0.827141i \(-0.689966\pi\)
−0.561994 + 0.827141i \(0.689966\pi\)
\(84\) 0 0
\(85\) 3.59398e10 0.878561
\(86\) 2.73514e10 0.626958
\(87\) 8.76353e9 0.188505
\(88\) 3.99161e9 0.0806293
\(89\) −3.28737e10 −0.624027 −0.312013 − 0.950078i \(-0.601003\pi\)
−0.312013 + 0.950078i \(0.601003\pi\)
\(90\) 9.03243e9 0.161239
\(91\) 0 0
\(92\) 3.85295e10 0.609481
\(93\) 1.15082e11 1.71535
\(94\) −2.71143e10 −0.381061
\(95\) −6.79168e10 −0.900529
\(96\) −1.57232e10 −0.196811
\(97\) 1.02279e11 1.20932 0.604661 − 0.796483i \(-0.293308\pi\)
0.604661 + 0.796483i \(0.293308\pi\)
\(98\) 0 0
\(99\) 5.16834e9 0.0546208
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 98.12.a.j.1.2 4
7.2 even 3 14.12.c.a.11.3 yes 8
7.3 odd 6 98.12.c.l.79.2 8
7.4 even 3 14.12.c.a.9.3 ✓ 8
7.5 odd 6 98.12.c.l.67.2 8
7.6 odd 2 98.12.a.l.1.3 4
21.2 odd 6 126.12.g.e.109.4 8
21.11 odd 6 126.12.g.e.37.4 8
28.11 odd 6 112.12.i.a.65.2 8
28.23 odd 6 112.12.i.a.81.2 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
14.12.c.a.9.3 ✓ 8 7.4 even 3
14.12.c.a.11.3 yes 8 7.2 even 3
98.12.a.j.1.2 4 1.1 even 1 trivial
98.12.a.l.1.3 4 7.6 odd 2
98.12.c.l.67.2 8 7.5 odd 6
98.12.c.l.79.2 8 7.3 odd 6
112.12.i.a.65.2 8 28.11 odd 6
112.12.i.a.81.2 8 28.23 odd 6
126.12.g.e.37.4 8 21.11 odd 6
126.12.g.e.109.4 8 21.2 odd 6