Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [98,6,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, 6); 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, 6, names="a")
 
Level: \( N \) \(=\) \( 98 = 2 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 98.c (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,4,10,-16,84,80,0,-128,143] 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: \(15.7176143417\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 14)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 79.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 98.79
Dual form 98.6.c.d.67.1

$q$-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+(2.00000 - 3.46410i) q^{2} +(5.00000 + 8.66025i) q^{3} +(-8.00000 - 13.8564i) q^{4} +(42.0000 - 72.7461i) q^{5} +40.0000 q^{6} -64.0000 q^{8} +(71.5000 - 123.842i) q^{9} +(-168.000 - 290.985i) q^{10} +(168.000 + 290.985i) q^{11} +(80.0000 - 138.564i) q^{12} -584.000 q^{13} +840.000 q^{15} +(-128.000 + 221.703i) q^{16} +(-729.000 - 1262.67i) q^{17} +(-286.000 - 495.367i) q^{18} +(235.000 - 407.032i) q^{19} -1344.00 q^{20} +1344.00 q^{22} +(2100.00 - 3637.31i) q^{23} +(-320.000 - 554.256i) q^{24} +(-1965.50 - 3404.35i) q^{25} +(-1168.00 + 2023.04i) q^{26} +3860.00 q^{27} +4866.00 q^{29} +(1680.00 - 2909.85i) q^{30} +(-3686.00 - 6384.34i) q^{31} +(512.000 + 886.810i) q^{32} +(-1680.00 + 2909.85i) q^{33} -5832.00 q^{34} -2288.00 q^{36} +(-7165.00 + 12410.1i) q^{37} +(-940.000 - 1628.13i) q^{38} +(-2920.00 - 5057.59i) q^{39} +(-2688.00 + 4655.75i) q^{40} -6222.00 q^{41} +3704.00 q^{43} +(2688.00 - 4655.75i) q^{44} +(-6006.00 - 10402.7i) q^{45} +(-8400.00 - 14549.2i) q^{46} +(-906.000 + 1569.24i) q^{47} -2560.00 q^{48} -15724.0 q^{50} +(7290.00 - 12626.7i) q^{51} +(4672.00 + 8092.14i) q^{52} +(18621.0 + 32252.5i) q^{53} +(7720.00 - 13371.4i) q^{54} +28224.0 q^{55} +4700.00 q^{57} +(9732.00 - 16856.3i) q^{58} +(17151.0 + 29706.4i) q^{59} +(-6720.00 - 11639.4i) q^{60} +(12238.0 - 21196.8i) q^{61} -29488.0 q^{62} +4096.00 q^{64} +(-24528.0 + 42483.7i) q^{65} +(6720.00 + 11639.4i) q^{66} +(8726.00 + 15113.9i) q^{67} +(-11664.0 + 20202.6i) q^{68} +42000.0 q^{69} +28224.0 q^{71} +(-4576.00 + 7925.86i) q^{72} +(1801.00 + 3119.42i) q^{73} +(28660.0 + 49640.6i) q^{74} +(19655.0 - 34043.5i) q^{75} -7520.00 q^{76} -23360.0 q^{78} +(-21436.0 + 37128.2i) q^{79} +(10752.0 + 18623.0i) q^{80} +(1925.50 + 3335.06i) q^{81} +(-12444.0 + 21553.6i) q^{82} +35202.0 q^{83} -122472. q^{85} +(7408.00 - 12831.0i) q^{86} +(24330.0 + 42140.8i) q^{87} +(-10752.0 - 18623.0i) q^{88} +(13365.0 - 23148.9i) q^{89} -48048.0 q^{90} -67200.0 q^{92} +(36860.0 - 63843.4i) q^{93} +(3624.00 + 6276.95i) q^{94} +(-19740.0 - 34190.7i) q^{95} +(-5120.00 + 8868.10i) q^{96} +16978.0 q^{97} +48048.0 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{2} + 10 q^{3} - 16 q^{4} + 84 q^{5} + 80 q^{6} - 128 q^{8} + 143 q^{9} - 336 q^{10} + 336 q^{11} + 160 q^{12} - 1168 q^{13} + 1680 q^{15} - 256 q^{16} - 1458 q^{17} - 572 q^{18} + 470 q^{19}+ \cdots + 96096 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.00000 − 3.46410i 0.353553 − 0.612372i
\(3\) 5.00000 + 8.66025i 0.320750 + 0.555556i 0.980643 − 0.195804i \(-0.0627317\pi\)
−0.659893 + 0.751360i \(0.729398\pi\)
\(4\) −8.00000 − 13.8564i −0.250000 − 0.433013i
\(5\) 42.0000 − 72.7461i 0.751319 − 1.30132i −0.195865 − 0.980631i \(-0.562751\pi\)
0.947184 − 0.320692i \(-0.103915\pi\)
\(6\) 40.0000 0.453609
\(7\) 0 0
\(8\) −64.0000 −0.353553
\(9\) 71.5000 − 123.842i 0.294239 − 0.509636i
\(10\) −168.000 − 290.985i −0.531263 − 0.920174i
\(11\) 168.000 + 290.985i 0.418627 + 0.725084i 0.995802 − 0.0915372i \(-0.0291780\pi\)
−0.577174 + 0.816621i \(0.695845\pi\)
\(12\) 80.0000 − 138.564i 0.160375 − 0.277778i
\(13\) −584.000 −0.958417 −0.479208 − 0.877701i \(-0.659076\pi\)
−0.479208 + 0.877701i \(0.659076\pi\)
\(14\) 0 0
\(15\) 840.000 0.963943
\(16\) −128.000 + 221.703i −0.125000 + 0.216506i
\(17\) −729.000 − 1262.67i −0.611794 − 1.05966i −0.990938 − 0.134321i \(-0.957115\pi\)
0.379144 − 0.925338i \(-0.376219\pi\)
\(18\) −286.000 − 495.367i −0.208058 − 0.360367i
\(19\) 235.000 − 407.032i 0.149343 − 0.258669i −0.781642 − 0.623727i \(-0.785618\pi\)
0.930985 + 0.365058i \(0.118951\pi\)
\(20\) −1344.00 −0.751319
\(21\) 0 0
\(22\) 1344.00 0.592028
\(23\) 2100.00 − 3637.31i 0.827751 − 1.43371i −0.0720476 − 0.997401i \(-0.522953\pi\)
0.899799 − 0.436306i \(-0.143713\pi\)
\(24\) −320.000 − 554.256i −0.113402 − 0.196419i
\(25\) −1965.50 − 3404.35i −0.628960 − 1.08939i
\(26\) −1168.00 + 2023.04i −0.338852 + 0.586908i
\(27\) 3860.00 1.01901
\(28\) 0 0
\(29\) 4866.00 1.07443 0.537214 − 0.843446i \(-0.319477\pi\)
0.537214 + 0.843446i \(0.319477\pi\)
\(30\) 1680.00 − 2909.85i 0.340805 − 0.590292i
\(31\) −3686.00 − 6384.34i −0.688892 − 1.19320i −0.972197 − 0.234166i \(-0.924764\pi\)
0.283305 − 0.959030i \(-0.408569\pi\)
\(32\) 512.000 + 886.810i 0.0883883 + 0.153093i
\(33\) −1680.00 + 2909.85i −0.268550 + 0.465141i
\(34\) −5832.00 −0.865207
\(35\) 0 0
\(36\) −2288.00 −0.294239
\(37\) −7165.00 + 12410.1i −0.860423 + 1.49030i 0.0110986 + 0.999938i \(0.496467\pi\)
−0.871521 + 0.490358i \(0.836866\pi\)
\(38\) −940.000 − 1628.13i −0.105601 − 0.182907i
\(39\) −2920.00 − 5057.59i −0.307412 − 0.532454i
\(40\) −2688.00 + 4655.75i −0.265631 + 0.460087i
\(41\) −6222.00 −0.578057 −0.289028 − 0.957321i \(-0.593332\pi\)
−0.289028 + 0.957321i \(0.593332\pi\)
\(42\) 0 0
\(43\) 3704.00 0.305492 0.152746 − 0.988265i \(-0.451188\pi\)
0.152746 + 0.988265i \(0.451188\pi\)
\(44\) 2688.00 − 4655.75i 0.209314 − 0.362542i
\(45\) −6006.00 − 10402.7i −0.442134 − 0.765799i
\(46\) −8400.00 − 14549.2i −0.585308 − 1.01378i
\(47\) −906.000 + 1569.24i −0.0598251 + 0.103620i −0.894387 − 0.447294i \(-0.852388\pi\)
0.834562 + 0.550915i \(0.185721\pi\)
\(48\) −2560.00 −0.160375
\(49\) 0 0
\(50\) −15724.0 −0.889484
\(51\) 7290.00 − 12626.7i 0.392466 − 0.679771i
\(52\) 4672.00 + 8092.14i 0.239604 + 0.415007i
\(53\) 18621.0 + 32252.5i 0.910570 + 1.57715i 0.813261 + 0.581899i \(0.197690\pi\)
0.0973091 + 0.995254i \(0.468976\pi\)
\(54\) 7720.00 − 13371.4i 0.360274 − 0.624013i
\(55\) 28224.0 1.25809
\(56\) 0 0
\(57\) 4700.00 0.191607
\(58\) 9732.00 − 16856.3i 0.379867 − 0.657950i
\(59\) 17151.0 + 29706.4i 0.641445 + 1.11102i 0.985110 + 0.171923i \(0.0549981\pi\)
−0.343665 + 0.939092i \(0.611669\pi\)
\(60\) −6720.00 − 11639.4i −0.240986 − 0.417399i
\(61\) 12238.0 − 21196.8i 0.421101 − 0.729368i −0.574947 − 0.818191i \(-0.694977\pi\)
0.996047 + 0.0888232i \(0.0283106\pi\)
\(62\) −29488.0 −0.974240
\(63\) 0 0
\(64\) 4096.00 0.125000
\(65\) −24528.0 + 42483.7i −0.720077 + 1.24721i
\(66\) 6720.00 + 11639.4i 0.189893 + 0.328905i
\(67\) 8726.00 + 15113.9i 0.237481 + 0.411329i 0.959991 − 0.280032i \(-0.0903450\pi\)
−0.722510 + 0.691360i \(0.757012\pi\)
\(68\) −11664.0 + 20202.6i −0.305897 + 0.529829i
\(69\) 42000.0 1.06201
\(70\) 0 0
\(71\) 28224.0 0.664466 0.332233 − 0.943197i \(-0.392198\pi\)
0.332233 + 0.943197i \(0.392198\pi\)
\(72\) −4576.00 + 7925.86i −0.104029 + 0.180184i
\(73\) 1801.00 + 3119.42i 0.0395555 + 0.0685121i 0.885125 − 0.465353i \(-0.154072\pi\)
−0.845570 + 0.533865i \(0.820739\pi\)
\(74\) 28660.0 + 49640.6i 0.608411 + 1.05380i
\(75\) 19655.0 − 34043.5i 0.403478 − 0.698844i
\(76\) −7520.00 −0.149343
\(77\) 0 0
\(78\) −23360.0 −0.434747
\(79\) −21436.0 + 37128.2i −0.386435 + 0.669324i −0.991967 − 0.126496i \(-0.959627\pi\)
0.605532 + 0.795821i \(0.292960\pi\)
\(80\) 10752.0 + 18623.0i 0.187830 + 0.325331i
\(81\) 1925.50 + 3335.06i 0.0326085 + 0.0564796i
\(82\) −12444.0 + 21553.6i −0.204374 + 0.353986i
\(83\) 35202.0 0.560883 0.280441 − 0.959871i \(-0.409519\pi\)
0.280441 + 0.959871i \(0.409519\pi\)
\(84\) 0 0
\(85\) −122472. −1.83861
\(86\) 7408.00 − 12831.0i 0.108008 − 0.187075i
\(87\) 24330.0 + 42140.8i 0.344623 + 0.596904i
\(88\) −10752.0 − 18623.0i −0.148007 − 0.256356i
\(89\) 13365.0 − 23148.9i 0.178852 − 0.309781i −0.762636 − 0.646828i \(-0.776095\pi\)
0.941488 + 0.337048i \(0.109428\pi\)
\(90\) −48048.0 −0.625272
\(91\) 0 0
\(92\) −67200.0 −0.827751
\(93\) 36860.0 − 63843.4i 0.441924 − 0.765435i
\(94\) 3624.00 + 6276.95i 0.0423027 + 0.0732705i
\(95\) −19740.0 − 34190.7i −0.224408 − 0.388686i
\(96\) −5120.00 + 8868.10i −0.0567012 + 0.0982093i
\(97\) 16978.0 0.183213 0.0916067 − 0.995795i \(-0.470800\pi\)
0.0916067 + 0.995795i \(0.470800\pi\)
\(98\) 0 0
\(99\) 48048.0 0.492705
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 98.6.c.d.79.1 2
7.2 even 3 98.6.a.a.1.1 1
7.3 odd 6 98.6.c.c.67.1 2
7.4 even 3 inner 98.6.c.d.67.1 2
7.5 odd 6 14.6.a.a.1.1 ✓ 1
7.6 odd 2 98.6.c.c.79.1 2
21.2 odd 6 882.6.a.x.1.1 1
21.5 even 6 126.6.a.f.1.1 1
28.19 even 6 112.6.a.c.1.1 1
28.23 odd 6 784.6.a.i.1.1 1
35.12 even 12 350.6.c.d.99.1 2
35.19 odd 6 350.6.a.i.1.1 1
35.33 even 12 350.6.c.d.99.2 2
56.5 odd 6 448.6.a.e.1.1 1
56.19 even 6 448.6.a.l.1.1 1
84.47 odd 6 1008.6.a.b.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
14.6.a.a.1.1 ✓ 1 7.5 odd 6
98.6.a.a.1.1 1 7.2 even 3
98.6.c.c.67.1 2 7.3 odd 6
98.6.c.c.79.1 2 7.6 odd 2
98.6.c.d.67.1 2 7.4 even 3 inner
98.6.c.d.79.1 2 1.1 even 1 trivial
112.6.a.c.1.1 1 28.19 even 6
126.6.a.f.1.1 1 21.5 even 6
350.6.a.i.1.1 1 35.19 odd 6
350.6.c.d.99.1 2 35.12 even 12
350.6.c.d.99.2 2 35.33 even 12
448.6.a.e.1.1 1 56.5 odd 6
448.6.a.l.1.1 1 56.19 even 6
784.6.a.i.1.1 1 28.23 odd 6
882.6.a.x.1.1 1 21.2 odd 6
1008.6.a.b.1.1 1 84.47 odd 6