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(67,98)] 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("98.67"); S:= CuspForms(chi, 12); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(98, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([4])) N = Newforms(chi, 12, names="a")
 
Level: \( N \) \(=\) \( 98 = 2 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 98.c (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,-128,-266,-4096,-7504,17024,0,262144,-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: no
Analytic conductor: \(75.2976316948\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} + 101803 x^{6} + 6576400 x^{5} + 8539617914 x^{4} + 333205096780 x^{3} + \cdots + 33\!\cdots\!81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{12}\cdot 3^{2}\cdot 7^{4} \)
Twist minimal: no (minimal twist has level 14)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 79.4
Root \(152.619 + 264.344i\) of defining polynomial
Character \(\chi\) \(=\) 98.79
Dual form 98.12.c.l.67.4

$q$-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+(-16.0000 + 27.7128i) q^{2} +(271.738 + 470.665i) q^{3} +(-512.000 - 886.810i) q^{4} +(-3667.11 + 6351.63i) q^{5} -17391.3 q^{6} +32768.0 q^{8} +(-59110.0 + 102381. i) q^{9} +(-117348. - 203252. i) q^{10} +(-24115.9 - 41769.9i) q^{11} +(278260. - 481961. i) q^{12} +1.82289e6 q^{13} -3.98598e6 q^{15} +(-524288. + 908093. i) q^{16} +(-3.35793e6 - 5.81611e6i) q^{17} +(-1.89152e6 - 3.27621e6i) q^{18} +(8.95659e6 - 1.55133e7i) q^{19} +7.51025e6 q^{20} +1.54342e6 q^{22} +(1.04379e7 - 1.80789e7i) q^{23} +(8.90432e6 + 1.54227e7i) q^{24} +(-2.48139e6 - 4.29789e6i) q^{25} +(-2.91663e7 + 5.05174e7i) q^{26} +3.20255e7 q^{27} +2.38808e7 q^{29} +(6.37757e7 - 1.10463e8i) q^{30} +(-5.14984e7 - 8.91978e7i) q^{31} +(-1.67772e7 - 2.90590e7i) q^{32} +(1.31064e7 - 2.27010e7i) q^{33} +2.14908e8 q^{34} +1.21057e8 q^{36} +(2.60667e8 - 4.51488e8i) q^{37} +(2.86611e8 + 4.96425e8i) q^{38} +(4.95349e8 + 8.57970e8i) q^{39} +(-1.20164e8 + 2.08130e8i) q^{40} +1.25858e9 q^{41} -1.03048e9 q^{43} +(-2.46947e7 + 4.27724e7i) q^{44} +(-4.33526e8 - 7.50889e8i) q^{45} +(3.34012e8 + 5.78526e8i) q^{46} +(7.99740e8 - 1.38519e9i) q^{47} -5.69877e8 q^{48} +1.58809e8 q^{50} +(1.82496e9 - 3.16092e9i) q^{51} +(-9.33320e8 - 1.61656e9i) q^{52} +(-2.29236e9 - 3.97049e9i) q^{53} +(-5.12408e8 + 8.87516e8i) q^{54} +3.53743e8 q^{55} +9.73540e9 q^{57} +(-3.82093e8 + 6.61804e8i) q^{58} +(-2.88236e9 - 4.99240e9i) q^{59} +(2.04082e9 + 3.53481e9i) q^{60} +(-2.91537e9 + 5.04957e9i) q^{61} +3.29590e9 q^{62} +1.07374e9 q^{64} +(-6.68475e9 + 1.15783e10i) q^{65} +(4.19406e8 + 7.26432e8i) q^{66} +(-7.79178e9 - 1.34958e10i) q^{67} +(-3.43852e9 + 5.95570e9i) q^{68} +1.13455e10 q^{69} -7.49669e8 q^{71} +(-1.93692e9 + 3.35484e9i) q^{72} +(-5.51048e9 - 9.54443e9i) q^{73} +(8.34133e9 + 1.44476e10i) q^{74} +(1.34858e9 - 2.33580e9i) q^{75} -1.83431e10 q^{76} -3.17024e10 q^{78} +(-1.78570e10 + 3.09292e10i) q^{79} +(-3.84525e9 - 6.66016e9i) q^{80} +(1.91737e10 + 3.32098e10i) q^{81} +(-2.01373e10 + 3.48788e10i) q^{82} +3.77864e10 q^{83} +4.92557e10 q^{85} +(1.64876e10 - 2.85574e10i) q^{86} +(6.48933e9 + 1.12398e10i) q^{87} +(-7.90229e8 - 1.36872e9i) q^{88} +(-8.88627e9 + 1.53915e10i) q^{89} +2.77457e10 q^{90} -2.13768e10 q^{92} +(2.79882e10 - 4.84769e10i) q^{93} +(2.55917e10 + 4.43261e10i) q^{94} +(6.56897e10 + 1.13778e11i) q^{95} +(9.11803e9 - 1.57929e10i) q^{96} +8.79572e10 q^{97} +5.70196e9 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 128 q^{2} - 266 q^{3} - 4096 q^{4} - 7504 q^{5} + 17024 q^{6} + 262144 q^{8} - 123520 q^{9} - 240128 q^{10} + 213026 q^{11} - 272384 q^{12} + 2609712 q^{13} + 2275500 q^{15} - 4194304 q^{16} - 8854244 q^{17}+ \cdots + 393415805736 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/98\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(e\left(\frac{1}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −16.0000 + 27.7128i −0.353553 + 0.612372i
\(3\) 271.738 + 470.665i 0.645631 + 1.11826i 0.984156 + 0.177307i \(0.0567387\pi\)
−0.338525 + 0.940957i \(0.609928\pi\)
\(4\) −512.000 − 886.810i −0.250000 − 0.433013i
\(5\) −3667.11 + 6351.63i −0.524795 + 0.908971i 0.474789 + 0.880100i \(0.342525\pi\)
−0.999583 + 0.0288710i \(0.990809\pi\)
\(6\) −17391.3 −0.913059
\(7\) 0 0
\(8\) 32768.0 0.353553
\(9\) −59110.0 + 102381.i −0.333678 + 0.577946i
\(10\) −117348. − 203252.i −0.371086 − 0.642740i
\(11\) −24115.9 − 41769.9i −0.0451485 − 0.0781995i 0.842568 − 0.538590i \(-0.181043\pi\)
−0.887717 + 0.460390i \(0.847709\pi\)
\(12\) 278260. − 481961.i 0.322815 − 0.559132i
\(13\) 1.82289e6 1.36167 0.680836 − 0.732436i \(-0.261617\pi\)
0.680836 + 0.732436i \(0.261617\pi\)
\(14\) 0 0
\(15\) −3.98598e6 −1.35529
\(16\) −524288. + 908093.i −0.125000 + 0.216506i
\(17\) −3.35793e6 − 5.81611e6i −0.573592 − 0.993490i −0.996193 − 0.0871746i \(-0.972216\pi\)
0.422601 − 0.906316i \(-0.361117\pi\)
\(18\) −1.89152e6 − 3.27621e6i −0.235946 − 0.408670i
\(19\) 8.95659e6 − 1.55133e7i 0.829847 − 1.43734i −0.0683108 − 0.997664i \(-0.521761\pi\)
0.898158 − 0.439673i \(-0.144906\pi\)
\(20\) 7.51025e6 0.524795
\(21\) 0 0
\(22\) 1.54342e6 0.0638496
\(23\) 1.04379e7 − 1.80789e7i 0.338150 − 0.585693i −0.645935 − 0.763393i \(-0.723532\pi\)
0.984085 + 0.177700i \(0.0568656\pi\)
\(24\) 8.90432e6 + 1.54227e7i 0.228265 + 0.395366i
\(25\) −2.48139e6 − 4.29789e6i −0.0508188 − 0.0880207i
\(26\) −2.91663e7 + 5.05174e7i −0.481423 + 0.833850i
\(27\) 3.20255e7 0.429531
\(28\) 0 0
\(29\) 2.38808e7 0.216202 0.108101 − 0.994140i \(-0.465523\pi\)
0.108101 + 0.994140i \(0.465523\pi\)
\(30\) 6.37757e7 − 1.10463e8i 0.479169 − 0.829944i
\(31\) −5.14984e7 − 8.91978e7i −0.323076 − 0.559583i 0.658045 − 0.752978i \(-0.271383\pi\)
−0.981121 + 0.193395i \(0.938050\pi\)
\(32\) −1.67772e7 − 2.90590e7i −0.0883883 − 0.153093i
\(33\) 1.31064e7 − 2.27010e7i 0.0582985 − 0.100976i
\(34\) 2.14908e8 0.811182
\(35\) 0 0
\(36\) 1.21057e8 0.333678
\(37\) 2.60667e8 − 4.51488e8i 0.617982 − 1.07038i −0.371871 − 0.928284i \(-0.621284\pi\)
0.989853 − 0.142092i \(-0.0453830\pi\)
\(38\) 2.86611e8 + 4.96425e8i 0.586790 + 1.01635i
\(39\) 4.95349e8 + 8.57970e8i 0.879136 + 1.52271i
\(40\) −1.20164e8 + 2.08130e8i −0.185543 + 0.321370i
\(41\) 1.25858e9 1.69656 0.848281 − 0.529546i \(-0.177638\pi\)
0.848281 + 0.529546i \(0.177638\pi\)
\(42\) 0 0
\(43\) −1.03048e9 −1.06896 −0.534480 − 0.845181i \(-0.679492\pi\)
−0.534480 + 0.845181i \(0.679492\pi\)
\(44\) −2.46947e7 + 4.27724e7i −0.0225743 + 0.0390998i
\(45\) −4.33526e8 − 7.50889e8i −0.350224 − 0.606606i
\(46\) 3.34012e8 + 5.78526e8i 0.239108 + 0.414147i
\(47\) 7.99740e8 − 1.38519e9i 0.508640 − 0.880991i −0.491310 − 0.870985i \(-0.663482\pi\)
0.999950 − 0.0100058i \(-0.00318499\pi\)
\(48\) −5.69877e8 −0.322815
\(49\) 0 0
\(50\) 1.58809e8 0.0718686
\(51\) 1.82496e9 − 3.16092e9i 0.740657 − 1.28286i
\(52\) −9.33320e8 − 1.61656e9i −0.340418 − 0.589621i
\(53\) −2.29236e9 − 3.97049e9i −0.752949 − 1.30415i −0.946387 − 0.323034i \(-0.895297\pi\)
0.193438 − 0.981113i \(-0.438036\pi\)
\(54\) −5.12408e8 + 8.87516e8i −0.151862 + 0.263033i
\(55\) 3.53743e8 0.0947748
\(56\) 0 0
\(57\) 9.73540e9 2.14310
\(58\) −3.82093e8 + 6.61804e8i −0.0764390 + 0.132396i
\(59\) −2.88236e9 − 4.99240e9i −0.524883 − 0.909124i −0.999580 − 0.0289746i \(-0.990776\pi\)
0.474697 − 0.880149i \(-0.342558\pi\)
\(60\) 2.04082e9 + 3.53481e9i 0.338823 + 0.586859i
\(61\) −2.91537e9 + 5.04957e9i −0.441957 + 0.765491i −0.997835 − 0.0657725i \(-0.979049\pi\)
0.555878 + 0.831264i \(0.312382\pi\)
\(62\) 3.29590e9 0.456898
\(63\) 0 0
\(64\) 1.07374e9 0.125000
\(65\) −6.68475e9 + 1.15783e10i −0.714598 + 1.23772i
\(66\) 4.19406e8 + 7.26432e8i 0.0412233 + 0.0714008i
\(67\) −7.79178e9 − 1.34958e10i −0.705059 − 1.22120i −0.966670 − 0.256025i \(-0.917587\pi\)
0.261611 − 0.965173i \(-0.415746\pi\)
\(68\) −3.43852e9 + 5.95570e9i −0.286796 + 0.496745i
\(69\) 1.13455e10 0.873279
\(70\) 0 0
\(71\) −7.49669e8 −0.0493116 −0.0246558 − 0.999696i \(-0.507849\pi\)
−0.0246558 + 0.999696i \(0.507849\pi\)
\(72\) −1.93692e9 + 3.35484e9i −0.117973 + 0.204335i
\(73\) −5.51048e9 − 9.54443e9i −0.311110 − 0.538858i 0.667493 − 0.744616i \(-0.267367\pi\)
−0.978603 + 0.205758i \(0.934034\pi\)
\(74\) 8.34133e9 + 1.44476e10i 0.436979 + 0.756870i
\(75\) 1.34858e9 − 2.33580e9i 0.0656203 − 0.113658i
\(76\) −1.83431e10 −0.829847
\(77\) 0 0
\(78\) −3.17024e10 −1.24329
\(79\) −1.78570e10 + 3.09292e10i −0.652918 + 1.13089i 0.329493 + 0.944158i \(0.393122\pi\)
−0.982411 + 0.186730i \(0.940211\pi\)
\(80\) −3.84525e9 − 6.66016e9i −0.131199 − 0.227243i
\(81\) 1.91737e10 + 3.32098e10i 0.610996 + 1.05828i
\(82\) −2.01373e10 + 3.48788e10i −0.599825 + 1.03893i
\(83\) 3.77864e10 1.05295 0.526473 − 0.850192i \(-0.323514\pi\)
0.526473 + 0.850192i \(0.323514\pi\)
\(84\) 0 0
\(85\) 4.92557e10 1.20407
\(86\) 1.64876e10 − 2.85574e10i 0.377934 − 0.654601i
\(87\) 6.48933e9 + 1.12398e10i 0.139587 + 0.241771i
\(88\) −7.90229e8 − 1.36872e9i −0.0159624 − 0.0276477i
\(89\) −8.88627e9 + 1.53915e10i −0.168684 + 0.292170i −0.937957 − 0.346750i \(-0.887285\pi\)
0.769273 + 0.638920i \(0.220618\pi\)
\(90\) 2.77457e10 0.495292
\(91\) 0 0
\(92\) −2.13768e10 −0.338150
\(93\) 2.79882e10 − 4.84769e10i 0.417175 − 0.722568i
\(94\) 2.55917e10 + 4.43261e10i 0.359663 + 0.622955i
\(95\) 6.56897e10 + 1.13778e11i 0.870999 + 1.50861i
\(96\) 9.11803e9 − 1.57929e10i 0.114132 − 0.197683i
\(97\) 8.79572e10 1.03998 0.519992 − 0.854171i \(-0.325935\pi\)
0.519992 + 0.854171i \(0.325935\pi\)
\(98\) 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.c.l.79.4 8
7.2 even 3 98.12.a.l.1.1 4
7.3 odd 6 14.12.c.a.11.1 yes 8
7.4 even 3 inner 98.12.c.l.67.4 8
7.5 odd 6 98.12.a.j.1.4 4
7.6 odd 2 14.12.c.a.9.1 ✓ 8
21.17 even 6 126.12.g.e.109.2 8
21.20 even 2 126.12.g.e.37.2 8
28.3 even 6 112.12.i.a.81.4 8
28.27 even 2 112.12.i.a.65.4 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
14.12.c.a.9.1 ✓ 8 7.6 odd 2
14.12.c.a.11.1 yes 8 7.3 odd 6
98.12.a.j.1.4 4 7.5 odd 6
98.12.a.l.1.1 4 7.2 even 3
98.12.c.l.67.4 8 7.4 even 3 inner
98.12.c.l.79.4 8 1.1 even 1 trivial
112.12.i.a.65.4 8 28.27 even 2
112.12.i.a.81.4 8 28.3 even 6
126.12.g.e.37.2 8 21.20 even 2
126.12.g.e.109.2 8 21.17 even 6