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 1.4
Root \(305.238\) 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)
 
\(f(q)\) \(=\) \(q+32.0000 q^{2} +543.477 q^{3} +1024.00 q^{4} -7334.23 q^{5} +17391.3 q^{6} +32768.0 q^{8} +118220. q^{9} -234695. q^{10} +48231.8 q^{11} +556520. q^{12} -1.82289e6 q^{13} -3.98598e6 q^{15} +1.04858e6 q^{16} -6.71587e6 q^{17} +3.78304e6 q^{18} +1.79132e7 q^{19} -7.51025e6 q^{20} +1.54342e6 q^{22} -2.08758e7 q^{23} +1.78086e7 q^{24} +4.96277e6 q^{25} -5.83325e7 q^{26} -3.20255e7 q^{27} +2.38808e7 q^{29} -1.27551e8 q^{30} -1.02997e8 q^{31} +3.35544e7 q^{32} +2.62128e7 q^{33} -2.14908e8 q^{34} +1.21057e8 q^{36} -5.21333e8 q^{37} +5.73222e8 q^{38} -9.90699e8 q^{39} -2.40328e8 q^{40} -1.25858e9 q^{41} -1.03048e9 q^{43} +4.93893e7 q^{44} -8.67052e8 q^{45} -6.68024e8 q^{46} +1.59948e9 q^{47} +5.69877e8 q^{48} +1.58809e8 q^{50} -3.64992e9 q^{51} -1.86664e9 q^{52} +4.58472e9 q^{53} -1.02482e9 q^{54} -3.53743e8 q^{55} +9.73540e9 q^{57} +7.64185e8 q^{58} -5.76472e9 q^{59} -4.08165e9 q^{60} -5.83074e9 q^{61} -3.29590e9 q^{62} +1.07374e9 q^{64} +1.33695e10 q^{65} +8.38811e8 q^{66} +1.55836e10 q^{67} -6.87705e9 q^{68} -1.13455e10 q^{69} -7.49669e8 q^{71} +3.87383e9 q^{72} -1.10210e10 q^{73} -1.66827e10 q^{74} +2.69715e9 q^{75} +1.83431e10 q^{76} -3.17024e10 q^{78} +3.57140e10 q^{79} -7.69050e9 q^{80} -3.83474e10 q^{81} -4.02746e10 q^{82} -3.77864e10 q^{83} +4.92557e10 q^{85} -3.29752e10 q^{86} +1.29787e10 q^{87} +1.58046e9 q^{88} -1.77725e10 q^{89} -2.77457e10 q^{90} -2.13768e10 q^{92} -5.59763e10 q^{93} +5.11834e10 q^{94} -1.31379e11 q^{95} +1.82361e10 q^{96} -8.79572e10 q^{97} +5.70196e9 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 128 q^{2} - 266 q^{3} + 4096 q^{4} - 7504 q^{5} - 8512 q^{6} + 131072 q^{8} + 123520 q^{9} - 240128 q^{10} - 213026 q^{11} - 272384 q^{12} - 1304856 q^{13} + 1137750 q^{15} + 4194304 q^{16} - 8854244 q^{17}+ \cdots + 196707902868 q^{99}+O(q^{100}) \) 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
\(3\) 543.477 1.29126 0.645631 − 0.763650i \(-0.276595\pi\)
0.645631 + 0.763650i \(0.276595\pi\)
\(4\) 1024.00 0.500000
\(5\) −7334.23 −1.04959 −0.524795 − 0.851229i \(-0.675858\pi\)
−0.524795 + 0.851229i \(0.675858\pi\)
\(6\) 17391.3 0.913059
\(7\) 0 0
\(8\) 32768.0 0.353553
\(9\) 118220. 0.667355
\(10\) −234695. −0.742172
\(11\) 48231.8 0.0902970 0.0451485 − 0.998980i \(-0.485624\pi\)
0.0451485 + 0.998980i \(0.485624\pi\)
\(12\) 556520. 0.645631
\(13\) −1.82289e6 −1.36167 −0.680836 − 0.732436i \(-0.738383\pi\)
−0.680836 + 0.732436i \(0.738383\pi\)
\(14\) 0 0
\(15\) −3.98598e6 −1.35529
\(16\) 1.04858e6 0.250000
\(17\) −6.71587e6 −1.14718 −0.573592 − 0.819141i \(-0.694450\pi\)
−0.573592 + 0.819141i \(0.694450\pi\)
\(18\) 3.78304e6 0.471891
\(19\) 1.79132e7 1.65969 0.829847 − 0.557991i \(-0.188428\pi\)
0.829847 + 0.557991i \(0.188428\pi\)
\(20\) −7.51025e6 −0.524795
\(21\) 0 0
\(22\) 1.54342e6 0.0638496
\(23\) −2.08758e7 −0.676300 −0.338150 − 0.941092i \(-0.609801\pi\)
−0.338150 + 0.941092i \(0.609801\pi\)
\(24\) 1.78086e7 0.456530
\(25\) 4.96277e6 0.101638
\(26\) −5.83325e7 −0.962847
\(27\) −3.20255e7 −0.429531
\(28\) 0 0
\(29\) 2.38808e7 0.216202 0.108101 − 0.994140i \(-0.465523\pi\)
0.108101 + 0.994140i \(0.465523\pi\)
\(30\) −1.27551e8 −0.958337
\(31\) −1.02997e8 −0.646151 −0.323076 − 0.946373i \(-0.604717\pi\)
−0.323076 + 0.946373i \(0.604717\pi\)
\(32\) 3.35544e7 0.176777
\(33\) 2.62128e7 0.116597
\(34\) −2.14908e8 −0.811182
\(35\) 0 0
\(36\) 1.21057e8 0.333678
\(37\) −5.21333e8 −1.23596 −0.617982 − 0.786192i \(-0.712050\pi\)
−0.617982 + 0.786192i \(0.712050\pi\)
\(38\) 5.73222e8 1.17358
\(39\) −9.90699e8 −1.75827
\(40\) −2.40328e8 −0.371086
\(41\) −1.25858e9 −1.69656 −0.848281 − 0.529546i \(-0.822362\pi\)
−0.848281 + 0.529546i \(0.822362\pi\)
\(42\) 0 0
\(43\) −1.03048e9 −1.06896 −0.534480 − 0.845181i \(-0.679492\pi\)
−0.534480 + 0.845181i \(0.679492\pi\)
\(44\) 4.93893e7 0.0451485
\(45\) −8.67052e8 −0.700449
\(46\) −6.68024e8 −0.478216
\(47\) 1.59948e9 1.01728 0.508640 − 0.860979i \(-0.330148\pi\)
0.508640 + 0.860979i \(0.330148\pi\)
\(48\) 5.69877e8 0.322815
\(49\) 0 0
\(50\) 1.58809e8 0.0718686
\(51\) −3.64992e9 −1.48131
\(52\) −1.86664e9 −0.680836
\(53\) 4.58472e9 1.50590 0.752949 − 0.658078i \(-0.228630\pi\)
0.752949 + 0.658078i \(0.228630\pi\)
\(54\) −1.02482e9 −0.303725
\(55\) −3.53743e8 −0.0947748
\(56\) 0 0
\(57\) 9.73540e9 2.14310
\(58\) 7.64185e8 0.152878
\(59\) −5.76472e9 −1.04977 −0.524883 − 0.851175i \(-0.675891\pi\)
−0.524883 + 0.851175i \(0.675891\pi\)
\(60\) −4.08165e9 −0.677647
\(61\) −5.83074e9 −0.883913 −0.441957 − 0.897036i \(-0.645715\pi\)
−0.441957 + 0.897036i \(0.645715\pi\)
\(62\) −3.29590e9 −0.456898
\(63\) 0 0
\(64\) 1.07374e9 0.125000
\(65\) 1.33695e10 1.42920
\(66\) 8.38811e8 0.0824466
\(67\) 1.55836e10 1.41012 0.705059 − 0.709149i \(-0.250920\pi\)
0.705059 + 0.709149i \(0.250920\pi\)
\(68\) −6.87705e9 −0.573592
\(69\) −1.13455e10 −0.873279
\(70\) 0 0
\(71\) −7.49669e8 −0.0493116 −0.0246558 − 0.999696i \(-0.507849\pi\)
−0.0246558 + 0.999696i \(0.507849\pi\)
\(72\) 3.87383e9 0.235946
\(73\) −1.10210e10 −0.622220 −0.311110 − 0.950374i \(-0.600701\pi\)
−0.311110 + 0.950374i \(0.600701\pi\)
\(74\) −1.66827e10 −0.873959
\(75\) 2.69715e9 0.131241
\(76\) 1.83431e10 0.829847
\(77\) 0 0
\(78\) −3.17024e10 −1.24329
\(79\) 3.57140e10 1.30584 0.652918 − 0.757428i \(-0.273544\pi\)
0.652918 + 0.757428i \(0.273544\pi\)
\(80\) −7.69050e9 −0.262397
\(81\) −3.83474e10 −1.22199
\(82\) −4.02746e10 −1.19965
\(83\) −3.77864e10 −1.05295 −0.526473 − 0.850192i \(-0.676486\pi\)
−0.526473 + 0.850192i \(0.676486\pi\)
\(84\) 0 0
\(85\) 4.92557e10 1.20407
\(86\) −3.29752e10 −0.755868
\(87\) 1.29787e10 0.279173
\(88\) 1.58046e9 0.0319248
\(89\) −1.77725e10 −0.337368 −0.168684 − 0.985670i \(-0.553952\pi\)
−0.168684 + 0.985670i \(0.553952\pi\)
\(90\) −2.77457e10 −0.495292
\(91\) 0 0
\(92\) −2.13768e10 −0.338150
\(93\) −5.59763e10 −0.834350
\(94\) 5.11834e10 0.719326
\(95\) −1.31379e11 −1.74200
\(96\) 1.82361e10 0.228265
\(97\) −8.79572e10 −1.03998 −0.519992 − 0.854171i \(-0.674065\pi\)
−0.519992 + 0.854171i \(0.674065\pi\)
\(98\) 0 0
\(99\) 5.70196e9 0.0602602
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.1.4 4
7.2 even 3 14.12.c.a.11.1 yes 8
7.3 odd 6 98.12.c.l.79.4 8
7.4 even 3 14.12.c.a.9.1 ✓ 8
7.5 odd 6 98.12.c.l.67.4 8
7.6 odd 2 98.12.a.l.1.1 4
21.2 odd 6 126.12.g.e.109.2 8
21.11 odd 6 126.12.g.e.37.2 8
28.11 odd 6 112.12.i.a.65.4 8
28.23 odd 6 112.12.i.a.81.4 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
14.12.c.a.9.1 ✓ 8 7.4 even 3
14.12.c.a.11.1 yes 8 7.2 even 3
98.12.a.j.1.4 4 1.1 even 1 trivial
98.12.a.l.1.1 4 7.6 odd 2
98.12.c.l.67.4 8 7.5 odd 6
98.12.c.l.79.4 8 7.3 odd 6
112.12.i.a.65.4 8 28.11 odd 6
112.12.i.a.81.4 8 28.23 odd 6
126.12.g.e.37.2 8 21.11 odd 6
126.12.g.e.109.2 8 21.2 odd 6