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 79.2
Root \(-100.397 - 173.893i\) of defining polynomial
Character \(\chi\) \(=\) 98.79
Dual form 98.12.c.l.67.2

$q$-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} +(-234.294 - 405.809i) q^{3} +(-512.000 - 886.810i) q^{4} +(3326.37 - 5761.45i) q^{5} +14994.8 q^{6} +32768.0 q^{8} +(-21214.1 + 36743.8i) q^{9} +(106444. + 184366. i) q^{10} +(-60907.1 - 105494. i) q^{11} +(-239917. + 415549. i) q^{12} +1.81677e6 q^{13} -3.11740e6 q^{15} +(-524288. + 908093. i) q^{16} +(2.70113e6 + 4.67849e6i) q^{17} +(-678850. - 1.17580e6i) q^{18} +(-5.10442e6 + 8.84111e6i) q^{19} -6.81241e6 q^{20} +3.89805e6 q^{22} +(-1.88132e7 + 3.25854e7i) q^{23} +(-7.67735e6 - 1.32976e7i) q^{24} +(2.28454e6 + 3.95694e6i) q^{25} +(-2.90684e7 + 5.03479e7i) q^{26} -6.31277e7 q^{27} -1.87020e7 q^{29} +(4.98784e7 - 8.63919e7i) q^{30} +(-1.22797e8 - 2.12690e8i) q^{31} +(-1.67772e7 - 2.90590e7i) q^{32} +(-2.85404e7 + 4.94333e7i) q^{33} -1.72872e8 q^{34} +4.34464e7 q^{36} +(-3.25911e8 + 5.64495e8i) q^{37} +(-1.63341e8 - 2.82916e8i) q^{38} +(-4.25659e8 - 7.37263e8i) q^{39} +(1.08999e8 - 1.88791e8i) q^{40} +6.38428e8 q^{41} +8.54732e8 q^{43} +(-6.23688e7 + 1.08026e8i) q^{44} +(1.41132e8 + 2.44447e8i) q^{45} +(-6.02023e8 - 1.04273e9i) q^{46} +(-4.23660e8 + 7.33801e8i) q^{47} +4.91351e8 q^{48} -1.46210e8 q^{50} +(1.26572e9 - 2.19228e9i) q^{51} +(-9.30187e8 - 1.61113e9i) q^{52} +(2.11267e9 + 3.65925e9i) q^{53} +(1.01004e9 - 1.74945e9i) q^{54} -8.10399e8 q^{55} +4.78374e9 q^{57} +(2.99232e8 - 5.18285e8i) q^{58} +(-1.10225e9 - 1.90916e9i) q^{59} +(1.59611e9 + 2.76454e9i) q^{60} +(-4.84994e9 + 8.40035e9i) q^{61} +7.85899e9 q^{62} +1.07374e9 q^{64} +(6.04326e9 - 1.04672e10i) q^{65} +(-9.13291e8 - 1.58187e9i) q^{66} +(4.86446e9 + 8.42549e9i) q^{67} +(2.76595e9 - 4.79077e9i) q^{68} +1.76313e10 q^{69} -2.83613e10 q^{71} +(-6.95142e8 + 1.20402e9i) q^{72} +(-1.83417e9 - 3.17688e9i) q^{73} +(-1.04292e10 - 1.80638e10i) q^{74} +(1.07051e9 - 1.85417e9i) q^{75} +1.04539e10 q^{76} +2.72422e10 q^{78} +(-6.34771e9 + 1.09946e10i) q^{79} +(3.48796e9 + 6.04132e9i) q^{80} +(1.85485e10 + 3.21269e10i) q^{81} +(-1.02148e10 + 1.76926e10i) q^{82} +4.03358e10 q^{83} +3.59398e10 q^{85} +(-1.36757e10 + 2.36870e10i) q^{86} +(4.38177e9 + 7.58944e9i) q^{87} +(-1.99580e9 - 3.45683e9i) q^{88} +(-1.64368e10 + 2.84694e10i) q^{89} -9.03243e9 q^{90} +3.85295e10 q^{92} +(-5.75412e10 + 9.96642e10i) q^{93} +(-1.35571e10 - 2.34816e10i) q^{94} +(3.39584e10 + 5.88177e10i) q^{95} +(-7.86161e9 + 1.36167e10i) q^{96} -1.02279e11 q^{97} +5.16834e9 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
\(3\) −234.294 − 405.809i −0.556666 − 0.964174i −0.997772 − 0.0667186i \(-0.978747\pi\)
0.441106 − 0.897455i \(-0.354586\pi\)
\(4\) −512.000 − 886.810i −0.250000 − 0.433013i
\(5\) 3326.37 − 5761.45i 0.476032 − 0.824511i −0.523591 − 0.851970i \(-0.675408\pi\)
0.999623 + 0.0274583i \(0.00874136\pi\)
\(6\) 14994.8 0.787245
\(7\) 0 0
\(8\) 32768.0 0.353553
\(9\) −21214.1 + 36743.8i −0.119754 + 0.207420i
\(10\) 106444. + 184366.i 0.336605 + 0.583018i
\(11\) −60907.1 − 105494.i −0.114027 − 0.197501i 0.803363 − 0.595489i \(-0.203042\pi\)
−0.917390 + 0.397988i \(0.869708\pi\)
\(12\) −239917. + 415549.i −0.278333 + 0.482087i
\(13\) 1.81677e6 1.35710 0.678550 − 0.734554i \(-0.262609\pi\)
0.678550 + 0.734554i \(0.262609\pi\)
\(14\) 0 0
\(15\) −3.11740e6 −1.05996
\(16\) −524288. + 908093.i −0.125000 + 0.216506i
\(17\) 2.70113e6 + 4.67849e6i 0.461398 + 0.799165i 0.999031 − 0.0440143i \(-0.0140147\pi\)
−0.537633 + 0.843179i \(0.680681\pi\)
\(18\) −678850. − 1.17580e6i −0.0846788 − 0.146668i
\(19\) −5.10442e6 + 8.84111e6i −0.472935 + 0.819148i −0.999520 − 0.0309748i \(-0.990139\pi\)
0.526585 + 0.850122i \(0.323472\pi\)
\(20\) −6.81241e6 −0.476032
\(21\) 0 0
\(22\) 3.89805e6 0.161259
\(23\) −1.88132e7 + 3.25854e7i −0.609481 + 1.05565i 0.381845 + 0.924226i \(0.375289\pi\)
−0.991326 + 0.131425i \(0.958045\pi\)
\(24\) −7.67735e6 − 1.32976e7i −0.196811 − 0.340887i
\(25\) 2.28454e6 + 3.95694e6i 0.0467873 + 0.0810381i
\(26\) −2.90684e7 + 5.03479e7i −0.479807 + 0.831051i
\(27\) −6.31277e7 −0.846680
\(28\) 0 0
\(29\) −1.87020e7 −0.169316 −0.0846581 − 0.996410i \(-0.526980\pi\)
−0.0846581 + 0.996410i \(0.526980\pi\)
\(30\) 4.98784e7 − 8.63919e7i 0.374753 − 0.649092i
\(31\) −1.22797e8 − 2.12690e8i −0.770367 − 1.33431i −0.937362 − 0.348357i \(-0.886740\pi\)
0.166995 − 0.985958i \(-0.446594\pi\)
\(32\) −1.67772e7 − 2.90590e7i −0.0883883 − 0.153093i
\(33\) −2.85404e7 + 4.94333e7i −0.126950 + 0.219884i
\(34\) −1.72872e8 −0.652515
\(35\) 0 0
\(36\) 4.34464e7 0.119754
\(37\) −3.25911e8 + 5.64495e8i −0.772663 + 1.33829i 0.163436 + 0.986554i \(0.447742\pi\)
−0.936099 + 0.351737i \(0.885591\pi\)
\(38\) −1.63341e8 − 2.82916e8i −0.334416 − 0.579225i
\(39\) −4.25659e8 − 7.37263e8i −0.755452 − 1.30848i
\(40\) 1.08999e8 − 1.88791e8i 0.168303 − 0.291509i
\(41\) 6.38428e8 0.860598 0.430299 − 0.902686i \(-0.358408\pi\)
0.430299 + 0.902686i \(0.358408\pi\)
\(42\) 0 0
\(43\) 8.54732e8 0.886652 0.443326 − 0.896360i \(-0.353798\pi\)
0.443326 + 0.896360i \(0.353798\pi\)
\(44\) −6.23688e7 + 1.08026e8i −0.0570135 + 0.0987503i
\(45\) 1.41132e8 + 2.44447e8i 0.114013 + 0.197477i
\(46\) −6.02023e8 − 1.04273e9i −0.430968 − 0.746458i
\(47\) −4.23660e8 + 7.33801e8i −0.269451 + 0.466702i −0.968720 − 0.248156i \(-0.920175\pi\)
0.699269 + 0.714858i \(0.253509\pi\)
\(48\) 4.91351e8 0.278333
\(49\) 0 0
\(50\) −1.46210e8 −0.0661673
\(51\) 1.26572e9 − 2.19228e9i 0.513689 − 0.889736i
\(52\) −9.30187e8 − 1.61113e9i −0.339275 − 0.587642i
\(53\) 2.11267e9 + 3.65925e9i 0.693927 + 1.20192i 0.970541 + 0.240936i \(0.0774544\pi\)
−0.276614 + 0.960981i \(0.589212\pi\)
\(54\) 1.01004e9 − 1.74945e9i 0.299347 − 0.518484i
\(55\) −8.10399e8 −0.217122
\(56\) 0 0
\(57\) 4.78374e9 1.05307
\(58\) 2.99232e8 − 5.18285e8i 0.0598623 − 0.103685i
\(59\) −1.10225e9 − 1.90916e9i −0.200722 − 0.347661i 0.748039 − 0.663655i \(-0.230995\pi\)
−0.948761 + 0.315994i \(0.897662\pi\)
\(60\) 1.59611e9 + 2.76454e9i 0.264991 + 0.458977i
\(61\) −4.84994e9 + 8.40035e9i −0.735229 + 1.27345i 0.219394 + 0.975636i \(0.429592\pi\)
−0.954623 + 0.297817i \(0.903741\pi\)
\(62\) 7.85899e9 1.08946
\(63\) 0 0
\(64\) 1.07374e9 0.125000
\(65\) 6.04326e9 − 1.04672e10i 0.646023 − 1.11894i
\(66\) −9.13291e8 − 1.58187e9i −0.0897672 − 0.155481i
\(67\) 4.86446e9 + 8.42549e9i 0.440173 + 0.762401i 0.997702 − 0.0677559i \(-0.0215839\pi\)
−0.557529 + 0.830157i \(0.688251\pi\)
\(68\) 2.76595e9 − 4.79077e9i 0.230699 − 0.399582i
\(69\) 1.76313e10 1.35711
\(70\) 0 0
\(71\) −2.83613e10 −1.86554 −0.932770 − 0.360472i \(-0.882616\pi\)
−0.932770 + 0.360472i \(0.882616\pi\)
\(72\) −6.95142e8 + 1.20402e9i −0.0423394 + 0.0733340i
\(73\) −1.83417e9 − 3.17688e9i −0.103553 − 0.179360i 0.809593 − 0.586992i \(-0.199688\pi\)
−0.913146 + 0.407632i \(0.866355\pi\)
\(74\) −1.04292e10 − 1.80638e10i −0.546355 − 0.946315i
\(75\) 1.07051e9 − 1.85417e9i 0.0520898 − 0.0902223i
\(76\) 1.04539e10 0.472935
\(77\) 0 0
\(78\) 2.72422e10 1.06837
\(79\) −6.34771e9 + 1.09946e10i −0.232096 + 0.402002i −0.958425 − 0.285345i \(-0.907892\pi\)
0.726329 + 0.687348i \(0.241225\pi\)
\(80\) 3.48796e9 + 6.04132e9i 0.119008 + 0.206128i
\(81\) 1.85485e10 + 3.21269e10i 0.591072 + 1.02377i
\(82\) −1.02148e10 + 1.76926e10i −0.304267 + 0.527007i
\(83\) 4.03358e10 1.12399 0.561994 − 0.827141i \(-0.310034\pi\)
0.561994 + 0.827141i \(0.310034\pi\)
\(84\) 0 0
\(85\) 3.59398e10 0.878561
\(86\) −1.36757e10 + 2.36870e10i −0.313479 + 0.542961i
\(87\) 4.38177e9 + 7.58944e9i 0.0942526 + 0.163250i
\(88\) −1.99580e9 − 3.45683e9i −0.0403147 − 0.0698270i
\(89\) −1.64368e10 + 2.84694e10i −0.312013 + 0.540423i −0.978798 − 0.204827i \(-0.934337\pi\)
0.666785 + 0.745250i \(0.267670\pi\)
\(90\) −9.03243e9 −0.161239
\(91\) 0 0
\(92\) 3.85295e10 0.609481
\(93\) −5.75412e10 + 9.96642e10i −0.857674 + 1.48554i
\(94\) −1.35571e10 − 2.34816e10i −0.190530 − 0.330008i
\(95\) 3.39584e10 + 5.88177e10i 0.450264 + 0.779881i
\(96\) −7.86161e9 + 1.36167e10i −0.0984056 + 0.170443i
\(97\) −1.02279e11 −1.20932 −0.604661 − 0.796483i \(-0.706692\pi\)
−0.604661 + 0.796483i \(0.706692\pi\)
\(98\) 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.c.l.79.2 8
7.2 even 3 98.12.a.l.1.3 4
7.3 odd 6 14.12.c.a.11.3 yes 8
7.4 even 3 inner 98.12.c.l.67.2 8
7.5 odd 6 98.12.a.j.1.2 4
7.6 odd 2 14.12.c.a.9.3 ✓ 8
21.17 even 6 126.12.g.e.109.4 8
21.20 even 2 126.12.g.e.37.4 8
28.3 even 6 112.12.i.a.81.2 8
28.27 even 2 112.12.i.a.65.2 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
14.12.c.a.9.3 ✓ 8 7.6 odd 2
14.12.c.a.11.3 yes 8 7.3 odd 6
98.12.a.j.1.2 4 7.5 odd 6
98.12.a.l.1.3 4 7.2 even 3
98.12.c.l.67.2 8 7.4 even 3 inner
98.12.c.l.79.2 8 1.1 even 1 trivial
112.12.i.a.65.2 8 28.27 even 2
112.12.i.a.81.2 8 28.3 even 6
126.12.g.e.37.4 8 21.20 even 2
126.12.g.e.109.4 8 21.17 even 6