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 67.1
Root \(-115.837 + 200.636i\) of defining polynomial
Character \(\chi\) \(=\) 98.67
Dual form 98.12.c.l.79.1

$q$-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+(-16.0000 - 27.7128i) q^{2} +(-265.175 + 459.296i) q^{3} +(-512.000 + 886.810i) q^{4} +(-6622.39 - 11470.3i) q^{5} +16971.2 q^{6} +32768.0 q^{8} +(-52061.7 - 90173.6i) q^{9} +(-211917. + 367050. i) q^{10} +(-268422. + 464921. i) q^{11} +(-271539. - 470319. i) q^{12} -1.10710e6 q^{13} +7.02436e6 q^{15} +(-524288. - 908093. i) q^{16} +(-458258. + 793726. i) q^{17} +(-1.66598e6 + 2.88555e6i) q^{18} +(1.86283e6 + 3.22652e6i) q^{19} +1.35627e7 q^{20} +1.71790e7 q^{22} +(-1.12882e7 - 1.95517e7i) q^{23} +(-8.68924e6 + 1.50502e7i) q^{24} +(-6.32981e7 + 1.09636e8i) q^{25} +(1.77135e7 + 3.06807e7i) q^{26} -3.87280e7 q^{27} -1.08153e8 q^{29} +(-1.12390e8 - 1.94665e8i) q^{30} +(-1.25691e8 + 2.17703e8i) q^{31} +(-1.67772e7 + 2.90590e7i) q^{32} +(-1.42357e8 - 2.46570e8i) q^{33} +2.93285e7 q^{34} +1.06622e8 q^{36} +(2.01780e8 + 3.49493e8i) q^{37} +(5.96106e7 - 1.03249e8i) q^{38} +(2.93574e8 - 5.08485e8i) q^{39} +(-2.17003e8 - 3.75859e8i) q^{40} -1.26266e9 q^{41} -1.92415e8 q^{43} +(-2.74864e8 - 4.76079e8i) q^{44} +(-6.89546e8 + 1.19433e9i) q^{45} +(-3.61223e8 + 6.25656e8i) q^{46} +(-5.04901e8 - 8.74515e8i) q^{47} +5.56112e8 q^{48} +4.05108e9 q^{50} +(-2.43037e8 - 4.20952e8i) q^{51} +(5.66833e8 - 9.81783e8i) q^{52} +(-2.26218e8 + 3.91821e8i) q^{53} +(6.19648e8 + 1.07326e9i) q^{54} +7.11039e9 q^{55} -1.97590e9 q^{57} +(1.73045e9 + 2.99723e9i) q^{58} +(-1.81367e9 + 3.14138e9i) q^{59} +(-3.59647e9 + 6.22928e9i) q^{60} +(4.93848e9 + 8.55370e9i) q^{61} +8.04420e9 q^{62} +1.07374e9 q^{64} +(7.33162e9 + 1.26987e10i) q^{65} +(-4.55544e9 + 7.89025e9i) q^{66} +(9.55058e9 - 1.65421e10i) q^{67} +(-4.69256e8 - 8.12776e8i) q^{68} +1.19734e10 q^{69} +5.71008e9 q^{71} +(-1.70596e9 - 2.95481e9i) q^{72} +(4.81805e9 - 8.34511e9i) q^{73} +(6.45695e9 - 1.11838e10i) q^{74} +(-3.35701e10 - 5.81451e10i) q^{75} -3.81508e9 q^{76} -1.87887e10 q^{78} +(-7.83278e9 - 1.35668e10i) q^{79} +(-6.94408e9 + 1.20275e10i) q^{80} +(1.94923e10 - 3.37616e10i) q^{81} +(2.02025e10 + 3.49918e10i) q^{82} -1.10230e10 q^{83} +1.21391e10 q^{85} +(3.07864e9 + 5.33236e9i) q^{86} +(2.86795e10 - 4.96743e10i) q^{87} +(-8.79565e9 + 1.52345e10i) q^{88} +(-2.43330e10 - 4.21460e10i) q^{89} +4.41310e10 q^{90} +2.31182e10 q^{92} +(-6.66600e10 - 1.15458e11i) q^{93} +(-1.61568e10 + 2.79845e10i) q^{94} +(2.46728e10 - 4.27345e10i) q^{95} +(-8.89779e9 - 1.54114e10i) q^{96} +1.33855e11 q^{97} +5.58981e10 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 \(\left(\mathbb{Z}/98\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\)

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
\(3\) −265.175 + 459.296i −0.630036 + 1.09125i 0.357508 + 0.933910i \(0.383626\pi\)
−0.987544 + 0.157344i \(0.949707\pi\)
\(4\) −512.000 + 886.810i −0.250000 + 0.433013i
\(5\) −6622.39 − 11470.3i −0.947720 − 1.64150i −0.750212 − 0.661197i \(-0.770049\pi\)
−0.197507 − 0.980301i \(-0.563285\pi\)
\(6\) 16971.2 0.891005
\(7\) 0 0
\(8\) 32768.0 0.353553
\(9\) −52061.7 − 90173.6i −0.293890 − 0.509032i
\(10\) −211917. + 367050.i −0.670139 + 1.16071i
\(11\) −268422. + 464921.i −0.502526 + 0.870400i 0.497470 + 0.867481i \(0.334263\pi\)
−0.999996 + 0.00291915i \(0.999071\pi\)
\(12\) −271539. − 470319.i −0.315018 − 0.545627i
\(13\) −1.10710e6 −0.826983 −0.413492 − 0.910508i \(-0.635691\pi\)
−0.413492 + 0.910508i \(0.635691\pi\)
\(14\) 0 0
\(15\) 7.02436e6 2.38839
\(16\) −524288. − 908093.i −0.125000 − 0.216506i
\(17\) −458258. + 793726.i −0.0782782 + 0.135582i −0.902507 − 0.430675i \(-0.858276\pi\)
0.824229 + 0.566257i \(0.191609\pi\)
\(18\) −1.66598e6 + 2.88555e6i −0.207812 + 0.359940i
\(19\) 1.86283e6 + 3.22652e6i 0.172595 + 0.298944i 0.939326 − 0.343025i \(-0.111451\pi\)
−0.766731 + 0.641968i \(0.778118\pi\)
\(20\) 1.35627e7 0.947720
\(21\) 0 0
\(22\) 1.71790e7 0.710679
\(23\) −1.12882e7 − 1.95517e7i −0.365697 − 0.633406i 0.623190 − 0.782070i \(-0.285836\pi\)
−0.988888 + 0.148664i \(0.952503\pi\)
\(24\) −8.68924e6 + 1.50502e7i −0.222751 + 0.385816i
\(25\) −6.32981e7 + 1.09636e8i −1.29635 + 2.24534i
\(26\) 1.77135e7 + 3.06807e7i 0.292383 + 0.506422i
\(27\) −3.87280e7 −0.519427
\(28\) 0 0
\(29\) −1.08153e8 −0.979153 −0.489576 − 0.871960i \(-0.662849\pi\)
−0.489576 + 0.871960i \(0.662849\pi\)
\(30\) −1.12390e8 − 1.94665e8i −0.844423 − 1.46258i
\(31\) −1.25691e8 + 2.17703e8i −0.788522 + 1.36576i 0.138350 + 0.990383i \(0.455820\pi\)
−0.926872 + 0.375377i \(0.877513\pi\)
\(32\) −1.67772e7 + 2.90590e7i −0.0883883 + 0.153093i
\(33\) −1.42357e8 − 2.46570e8i −0.633219 − 1.09677i
\(34\) 2.93285e7 0.110702
\(35\) 0 0
\(36\) 1.06622e8 0.293890
\(37\) 2.01780e8 + 3.49493e8i 0.478375 + 0.828569i 0.999693 − 0.0247933i \(-0.00789275\pi\)
−0.521318 + 0.853363i \(0.674559\pi\)
\(38\) 5.96106e7 − 1.03249e8i 0.122043 − 0.211385i
\(39\) 2.93574e8 − 5.08485e8i 0.521029 − 0.902449i
\(40\) −2.17003e8 − 3.75859e8i −0.335069 − 0.580357i
\(41\) −1.26266e9 −1.70206 −0.851030 − 0.525118i \(-0.824021\pi\)
−0.851030 + 0.525118i \(0.824021\pi\)
\(42\) 0 0
\(43\) −1.92415e8 −0.199601 −0.0998004 − 0.995007i \(-0.531820\pi\)
−0.0998004 + 0.995007i \(0.531820\pi\)
\(44\) −2.74864e8 − 4.76079e8i −0.251263 − 0.435200i
\(45\) −6.89546e8 + 1.19433e9i −0.557051 + 0.964840i
\(46\) −3.61223e8 + 6.25656e8i −0.258587 + 0.447886i
\(47\) −5.04901e8 − 8.74515e8i −0.321121 − 0.556197i 0.659599 − 0.751618i \(-0.270726\pi\)
−0.980720 + 0.195420i \(0.937393\pi\)
\(48\) 5.56112e8 0.315018
\(49\) 0 0
\(50\) 4.05108e9 1.83331
\(51\) −2.43037e8 − 4.20952e8i −0.0986362 − 0.170843i
\(52\) 5.66833e8 − 9.81783e8i 0.206746 − 0.358094i
\(53\) −2.26218e8 + 3.91821e8i −0.0743036 + 0.128698i −0.900783 − 0.434269i \(-0.857007\pi\)
0.826480 + 0.562967i \(0.190340\pi\)
\(54\) 6.19648e8 + 1.07326e9i 0.183645 + 0.318083i
\(55\) 7.11039e9 1.90501
\(56\) 0 0
\(57\) −1.97590e9 −0.434964
\(58\) 1.73045e9 + 2.99723e9i 0.346183 + 0.599606i
\(59\) −1.81367e9 + 3.14138e9i −0.330273 + 0.572050i −0.982565 − 0.185918i \(-0.940474\pi\)
0.652292 + 0.757968i \(0.273808\pi\)
\(60\) −3.59647e9 + 6.22928e9i −0.597097 + 1.03420i
\(61\) 4.93848e9 + 8.55370e9i 0.748651 + 1.29670i 0.948469 + 0.316869i \(0.102631\pi\)
−0.199818 + 0.979833i \(0.564035\pi\)
\(62\) 8.04420e9 1.11514
\(63\) 0 0
\(64\) 1.07374e9 0.125000
\(65\) 7.33162e9 + 1.26987e10i 0.783748 + 1.35749i
\(66\) −4.55544e9 + 7.89025e9i −0.447753 + 0.775531i
\(67\) 9.55058e9 − 1.65421e10i 0.864208 − 1.49685i −0.00362356 − 0.999993i \(-0.501153\pi\)
0.867831 − 0.496859i \(-0.165513\pi\)
\(68\) −4.69256e8 − 8.12776e8i −0.0391391 − 0.0677909i
\(69\) 1.19734e10 0.921609
\(70\) 0 0
\(71\) 5.71008e9 0.375596 0.187798 − 0.982208i \(-0.439865\pi\)
0.187798 + 0.982208i \(0.439865\pi\)
\(72\) −1.70596e9 − 2.95481e9i −0.103906 − 0.179970i
\(73\) 4.81805e9 − 8.34511e9i 0.272017 − 0.471147i −0.697361 − 0.716720i \(-0.745643\pi\)
0.969378 + 0.245573i \(0.0789761\pi\)
\(74\) 6.45695e9 − 1.11838e10i 0.338262 − 0.585887i
\(75\) −3.35701e10 − 5.81451e10i −1.63349 − 2.82928i
\(76\) −3.81508e9 −0.172595
\(77\) 0 0
\(78\) −1.87887e10 −0.736846
\(79\) −7.83278e9 − 1.35668e10i −0.286396 − 0.496053i 0.686551 − 0.727082i \(-0.259124\pi\)
−0.972947 + 0.231029i \(0.925791\pi\)
\(80\) −6.94408e9 + 1.20275e10i −0.236930 + 0.410375i
\(81\) 1.94923e10 − 3.37616e10i 0.621147 − 1.07586i
\(82\) 2.02025e10 + 3.49918e10i 0.601769 + 1.04229i
\(83\) −1.10230e10 −0.307164 −0.153582 − 0.988136i \(-0.549081\pi\)
−0.153582 + 0.988136i \(0.549081\pi\)
\(84\) 0 0
\(85\) 1.21391e10 0.296743
\(86\) 3.07864e9 + 5.33236e9i 0.0705695 + 0.122230i
\(87\) 2.86795e10 − 4.96743e10i 0.616901 − 1.06850i
\(88\) −8.79565e9 + 1.52345e10i −0.177670 + 0.307733i
\(89\) −2.43330e10 − 4.21460e10i −0.461903 − 0.800039i 0.537153 − 0.843485i \(-0.319500\pi\)
−0.999056 + 0.0434461i \(0.986166\pi\)
\(90\) 4.41310e10 0.787789
\(91\) 0 0
\(92\) 2.31182e10 0.365697
\(93\) −6.66600e10 − 1.15458e11i −0.993594 − 1.72096i
\(94\) −1.61568e10 + 2.79845e10i −0.227067 + 0.393291i
\(95\) 2.46728e10 − 4.27345e10i 0.327144 − 0.566629i
\(96\) −8.89779e9 − 1.54114e10i −0.111376 − 0.192908i
\(97\) 1.33855e11 1.58267 0.791337 − 0.611381i \(-0.209386\pi\)
0.791337 + 0.611381i \(0.209386\pi\)
\(98\) 0 0
\(99\) 5.58981e10 0.590749
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.67.1 8
7.2 even 3 inner 98.12.c.l.79.1 8
7.3 odd 6 98.12.a.j.1.1 4
7.4 even 3 98.12.a.l.1.4 4
7.5 odd 6 14.12.c.a.9.4 ✓ 8
7.6 odd 2 14.12.c.a.11.4 yes 8
21.5 even 6 126.12.g.e.37.1 8
21.20 even 2 126.12.g.e.109.1 8
28.19 even 6 112.12.i.a.65.1 8
28.27 even 2 112.12.i.a.81.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
14.12.c.a.9.4 ✓ 8 7.5 odd 6
14.12.c.a.11.4 yes 8 7.6 odd 2
98.12.a.j.1.1 4 7.3 odd 6
98.12.a.l.1.4 4 7.4 even 3
98.12.c.l.67.1 8 1.1 even 1 trivial
98.12.c.l.79.1 8 7.2 even 3 inner
112.12.i.a.65.1 8 28.19 even 6
112.12.i.a.81.1 8 28.27 even 2
126.12.g.e.37.1 8 21.5 even 6
126.12.g.e.109.1 8 21.20 even 2